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ADVANCES IN ELECTRONICS

VOLUME I V

This Page Intentionally Left Blank

ADVANCES IN ELECTRONICS Edited by L. MARTON National Bureau of Standards, Washington, D . C.

Editorial Board T. E. Allibone W. B. Nottingham E. R. Piore H. B. G. Casimir L. T. DeVore M. Ponte W. G. Dow A. Rose L. P. Smith A. 0. C. Nier

VOLUME IV

1952

ACADEMIC PRESS INC., PUBLISHERS NEW YORK, N. Y.

COPYRIGHT@

1952 BY ACADEMIC PRESS INC.

ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS

ACADEMIC PRESS INC. 111 FIFTHAVENUE NEW YORK,NEW YORK 10003

United Kingdom Edition Published by

ACADEMIC PRESS INC. (LONDON) LTD. BERKELEY SQUARE HOUSE, LONDON W. 1 First Printing, 1952 Second Printing, 1964

PRINTED I N T HE UNITED STATES OF AMERICA

CONTRIBUTORS TO VOLUME IV

J. S . DONAL,JR., Radio Corporation of America, R C A Laboratories Division, Princeton, New Jersey WINFIELDE. FROMM, Airborne Instruments Laboratory, Inc., Mineola, New York

H. S. W. MASSEY,F. R. S., Department of Mathematics, University College, London, England G. A. MORTON, Radio Corporation of America, R C A Laboratories Division, Princeton, New Jersey M. G. PAWLEY, National Bureau of Standards, Corona, California C. V. L. SMITH,Ofice of Naval Research, Washington, D. C.

W. E. TRIEST,International Business Machines Corporation, Poughkeepsie, New York ALDERT VAN DER ZIEL, Department of Electrical Engineering, Institute of Technology, University of Minnesota, Minneapolis, Minnesota


PREFACE

Another volume of “Advances in Electronics” the fourth in the series, is now presented to the scientific community. It is with deep satisfaction that members of the Editorial Board note the growing recognition of these volumes. I n fact the book reviews of the past years have been on the average so favorable that, more than anything else, the obligation to keep up with the various reviewers’ expectations, has set the level of this and future volumes. It is sincerely hoped that the present volume will find as favorable a reception as its predecessors.

L. MARTON Washington, D.C.


CONTENTS CONTRIBUTORS TO VOLUME IV . . . . . . . . . . . . . . . . . . . . . .

v

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

Electron Scattering in Solids

BY H . S. W . MASSEY,F.R.S:, Department of Mathematics. University College, London. England

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 I1. Elastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 3 I11. Inelastic Scattering . . . . . . . . . . . . . . . . . . . . . . . . 17 IV. Multiple Scattering . . . . . . . . . . . . . . . . . . . . . . . . 32 V. Energy Loss of Electrons in Passage through Solids . . . . . . . . . . 48 VI . The Mobility of Electrons in Metals, Alloys and Semi-Conductors . . . 58 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 The Scintillation Counter

.

BYG A . MORTON, Radio Corporation of America, RCA Laboratories Division, Princeton, New Jersey

I . Introduction . . . . . . . . . . . . . . . . . I1. The Photomultiplier . . . . . : . . . . . . . I11. Multiplier Performance . . . . . . . . . . . . . IV . Phosphor Crystals . . . . . . . . . . . . . . . V. Scintillation Counter Applications . . . . . . . References . . . . . . . . . . . . . . . . . .

. . . . . . . . . . 69 . . . . . . . . . . . 71 . . . . . . . . . . 78 88 . . . . . . . . . . . . . . . . . . . . . 94 . . . . . . . . . . 106

Fluctuation Phenomena B Y ALDERTVAN DER ZIEL, Department of Electrical Engineering, Institute of Technology. University of Minnesota, Minneapolis, Minnesota

I . Introduction . . . . . . . . . . . . . I1. Fourier Analysis of Fluctuating Quantities I11. Application t o Various Noise Generators . IV. Noise in Receivers . . . . . . . . . . . References . . . . . . . . . . . . . .

. . . . 110 . . . . 112 . . . . 117 . . . . 147 . . . . 153

Electronic Digital Computers

BY C. V . L. SMITH.Ofice of Naval Research. Washington. D . C .

.

I Introduction . . . . . . . . . . . . I1. Input-Output . . . . . . . . . . . . I11. Internal Storage’. . . . . . . . . . . I V . Arithmetic and Control Organs . . . V. Whirlwind . . . . . . . . . . . . . ix

157 . . . . . . . . . . . . . . . 160 . . . . . . . . . . . . . . . 161 . . . . . . . . . . . . . . . . 171 . . . . . . . . . . . . . . . 174

. . . . . . . . . . . . . . .

X

CONTENTS

.

VI SEAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V I I . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180 185

Modulation of Continuous-Wave Magnetrons BY J . S. DONAL, JR.,Radio Corporation of America, R C A Laboratories Division. Princeton. N e w Jersey

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Frequency Modulation or Control by Spiral Electron Beams . . . . . . I11. Frequency Modulation by Electron Clouds . . . . . . . . . . . . . . IV . Voltage Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . V . Amplitude Modulation Using Absorption by a Spiral Electron Beam . . . VI . Amplitude Modulation by Means of the Electron Coupler . . . . . . . VII . Contro! or hfodulation by Injection Phase Locking . . . . . . . . . . VIII . Amplitude Modulation by Plate Modulation of a Magnetron with Simultaneous Frequency Control . . . . . . . . . . . . . . . . . . . . . IX . The Injection Magnetron as the Possible Means of Producing Amplitude or Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . X . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

188 194 201 207 211 219 225

234 247 253 254

The Magnetic Airborne Detector BY WINFIELDE . FROMM, Airborne Znstruments Laboratory, Inc., Mineola, N e w York

I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Types of Magnetic Anomaly Detectors . . . . . . . . . . I11. Historical Development of the Magnetic Airborne Detector . IV. The Saturable-Core Magnetometer . . . . . . . . . . . V . Magnetic Stabilization and Orientation . . . . . . . . . . VI . Magnetic Airborne Detector System . . . . . . . . . . . VII . The Noise Problem . . . . . . . . . . . . . . . . . . . VIII Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

.

. . . . . 258 . . . . . . 263 . . . . . . 267

. . . . . .

268 . . . . . . 279 . . . . . . 292 . . . . . 295 . . . . . 298 . . . . . 298

Multichannel Radio Telemetering BY M . G . PAWLEY A N D W . E . TRIEST, National Bureau of Standards, Corona, California, and International Business Machines Corporation, Poughkeepsie, N e w York I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 I1. Evolution of Radio Telemetering . . . . . . . . . . . . . . . . . . 302 111. Basic Systems of Radio Telemetering . . . . . . . . . . . . . . . . 304 312 IV . Typical Telemetering Systems . . . . . . . . . . . . . . . . . . . . V . Future Trends in Telemetering . . . . . . . . . . . . . . . . . . . 328 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 Author Index .

. . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . .

331

336

Electron Scattering in Solids H. S. W. MASSEY, F. R. S. Department of Mathematics, University College, London, England CONTENTS Page I. Introduction.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 11. Elastic Scattering.. . . . . . . .......... 1. Elastic Scattering of F a a. Scattering by Free Atoms.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 b. Relativistic Correction ............ ... 8 c. Validity of Born’s App .......................... 9 d. Comparison of Born’s Approximation with Observation. . . . . . . . . . . . . 10 13 . . . . . . . . 13 . . . . . . . . . . . . . . . . . . 15 ........

b. Effect of Atomic Binding Forces..

........

III. Inelastic Scattering. ...... .......... ................. 1. Inelastic Scattering of s-Born’s Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a. Angular Distribution of the Totality of Inelastically Scattered Electrons.. .... .. ...... b. Total Cross Sections for Inelastic and Total Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c. Relativistic Modifications. . . . . . . . .......... 2. Experimental Evidence on Inelastic S 3. Influence of the Solid Binding.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a. Dynamical Polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Study of Low Energy-Loss Collisions with a Solid.. . . . . . . . . . . . . . . . IV. Multiple Scattering. ........... ................. 1. The Boltzmann Equation .......................... 2. The Angular Dist ................... a. Momentum Loss Cross Section and Mean Free P a t h . . . . . . . . . . . . . b. Small-Angle Multiple Scattering. ....................... c. Multiple Scattering Distribution in Terms of Projected Angle of Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d. Mean Values.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e. Allowance for Energy Loss.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f. Experimental Evidence on Multiple Scattering of Fast Electrons in Foils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g . Multiple Scattering of Electrons in Photographic Emulsions. . . . . . . . 3. Space Distribution of Multiply Scattered Electrons-Absorption of Electrons in Plates.. . . . . . . . . . . . . . . ......................... 1

17

18

23 28 28 30 32 33 34 35 36 40 41 41 42 43 43

2

H. S. W . MASSEY

Page 4. The Diffusion Stage.. . . ................... V. Energy Loss of Electrons in hrough Solids. . . . . . . 1. Stopping by Free Atoms ................................. 49 2. Effect of Atomic Interaction in the Solid State.. . . . . . . . . . . . . . . . . . . . 51 3. Attempts to Detect Atomic Interaction Effects.. . . . . . . . . . . . . . . . . . . . . 55 4. The Range of Electrons in M a t t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 V I . The Mobility of Electrons in Metals, Alloys and Semi-Conductors. . . . . . . . 58 1. Scattering by Lattice Vibrations-Resistance of Pure Metals.. . . . . . . . . 61 2. Scattering by Foreign Atoms-Resistance of Alloys.. . . . . . . . . . . . . . . . . . 63 3. The Resistance of Semi-Conductors.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 a. Non-degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 b. Degenerate Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 References.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

I. INTRODUCTION Many phenomena associated with the scattering of electrons by solids are of great importance and application in various branches of physics. Electron diffraction provides a valuable supplementary technique t o the diffraction of x-rays and of neutrons for the exploration of the structure of solid materials. The electron microscope, now proving such a n important tool in many fields of research, depends for image formation on small angle scattering by the specimen. I n recent years the photographic plate has become a most useful medium for the investigation of the properties of high-energy particles including, as a result of the latest developments, electrons. For these studies it is essential t o have reliable information on the rate of energy loss of electrons in the solid material of the plate as well as of the probability of multiple scattering. The determination of the range of electrons in a chosen solid material has long been a n important method for measuring the initial energy of the electrons. Secondary electron emission is another phenomenon which has been put t o use in electron multiplier tubes and in any case has always t o be reckoned with in any apparatus in which electrons impinge on a solid. The electrical resistance of conductors and semi-conductors is another property of great practical importance which is determined by the probability of scattering of the conduction electrons within the material. I n addition t o all these specifically solid state phenomena a great number of fundamental investigations on single scattering of electrons have been carried out, using of necessity solid scatterers. Included among these have been the attempts, now partly successful, t o detect the polarization of electrons by double scattering. It is clearly out of the question in the present review t o attempt even a cursory description of all these aspects of the subject. Instead we shall discuss a selection only. Secondary electron emission has already been the subject of a review in this series,' and we shall omit any further dis-

ELECTRON SCATTERING IN SOLIDS

3

cussion of it here, Another major subject we shall exclude is electron diffraction, as many books on this subject2 already exist. The first section will be devoted to a discussion of single elastic scattering of fast electrons which may be treated, apart from superposed coherent diffraction effects, in much the same way as scattering by single atoms of the material. Single inelastic scattering mill form the subject of the second section. Attention will be directed particularly toward obtaining formulas which are likely to be useful in application t o the electron microscope. The rather meager information available about the probability of energy losses due to excitation of the loosely bound electrons in a solid mill also be reviewed. The theory of multiple scattering and diffusion of electrons in a solid scatterer forms the subject of the third section, together with a brief discussion of experimental evidence. Formulas are obtained which allow for energy loss in passing through the material, but a detailed discussion of the determination of the rate of energy loss is reserved for the next section. This section includes a n elementary account of the way in which the dynamic polarization of the medium influences the energy loss. The final section is concerned with the consideration of the relative importance of the different scattering processes which determine the electrical resistance of metals, alloys, and semi-conductors.

11. ELASTIC SCATTERING

If a fast electron enters a solid the chance that it will undergo a n elastic collision in a small distance 6x mag be calculated to a good approximation by regarding the atoms of the solid as free. This is particularly true of collisions in which the electron is scattered through a large angle. Such collisions involve close approach of the electron and a n atomic nucleus so that the modification of the interaction due to the solid binding is quite negligible. On the other hand the probability of distant collisions in which the electron suffers only a very small deviation may be markedly influenced by the solid binding. Thus in a metal in which the atoms are ionized and the valence electrons more or less free, the charge distribution of the latter electrons is very different from t h a t in the free atom, and this will be reflected in the small angle elastic scattering. We shall begin by a discussion of the elastic scattering by isolated atoms and then consider in what way the results are modified by the solid binding. 1 . Elastic Scattering of Fast Electrons a. Scattering by Free Atoms. The scattering of fast electrons by a n atom may be treated by regarding the atom as a static center of force which exerts a force of potential energy V ( r ) on a n electron at distance r

H. S. W. MASSEY

4

from it. Polarization and other distortion effects which are important when the velocity of the electron is comparable with t h a t of the atomic electrons may be neglected. The potential energy V ( r ) of the atomic field can only be calculated accurately for atomic hydrogen but for other atoms approximate methods exist which are accurate enough for many purposes. For atoms which are not too light the most convenient approximation is obtained by treating the atom as a statistical assembly of electrons obeying the Fermi-Dirac statistics. This method, due t o Thomas3 and Fermi4 gives, for a neutral atom,

where Z is the nuclear charge and the length

p

is given by

p = 0.885aoZ-~,

(2)

where a0 is the radius, 0.53 X lo-* cm, of the first Bohr orbit of hydrogen. The function 4, which represents the effect of the atomic electrons in screening the nuclear charge, has been tabulated by Bush and C a l d ~ e l l . ~ Various analytical approximations for exist but for many purposes i t is sufficient t o write

+

+(r/p) =

e-ar/P

(31

where s is a constant of order unity t o be determined empirically [see (30)]. The actual function falls off much more slowly at large distances but a t such distances the statistical theory considerably over-estimates the field so t h a t (3) probably gives as good a n average representation of the field as may be obtained with any simple formula. For light atoms the statistical model is not very accurate. I n such cases the Hartree-Fock self-consistent field method may be used. This has t o be determined separately for each atom. Results exist in tabular form for a number of atoms and ions.6 We first calculate the scattering in the nonrelativistic approximation. The Schrodinger equation for the motion of an electron of kinetic energy E ( = k2h2/2m)is given by V2+

+ ( k 2 - 2mV/h2)+ = 0.

(4)

The electron incident in the direction of the unit vector no is represented by a plane wave A exp (ikno . r). At a great distance from the scattering atom the scattered electrons will be represented by an outgoing spherical wave Ar-1eikrf(0,4). Remembering that the current density of electrons


5

represented by a wave function J. is given b y i€h 2m

j = - (J.* grad # - J. grad #*),

(5)

we have t ha t the number of electrons incident per square centimeter per second is vAA* and th at the number scattered into the solid angle dw about (B,+) is vAA*If(B,+)I2dw. The ratio of these gives the differential elastic scattering cross section

I(%,+>dw= lf(@)I2dw.

(6)

The total elastic cross section Qo is then given by

Thus, if the atom were actually a spherical obstacle of cross section Q o the number of incident electrons colliding with the sphere per second would be vAA*Qo exactly as in the above formulation. To solve the scattering problem it is therefore necessary to obtain a solution of the equation (4) which, while being a well-behaved function throughout space, has the asymptotic form J.

-

eikno.r

+ r-leikrf(8,4).

(8)

Born’s approximation, satisfactory for the discussion of the scattering of fast electrons by most atoms, may be obtained by treating the scattering potential ‘v as a small perturbation and regarding the scattering probability as small. Writing (4)in the form V2#

+ k2# = 2mVJ./h2.

(9)

we then substitute for J. on the right-hand side of (9), simply the part exp (&no . r) representing the incident wave. This gives VY,

+ k2#

2mVeikno’r/h2,

(10) with the right-hand side a known iunction of r. This equation may be solved by Green’s theorem to give as a well-behaved solution’ =

which has the asymptotic form (8) with

where nlis a unit vector in the direction (%,+)measured with respect to the direction of incidence as polar axis.

6

H. S . W. MASSEY

Since Ino - rill

=

2 sin +6,

we have

sin 0’ dr’ do’ d+’, -

-

V(r’) sin (2kr’ sin @)r’ dr’.

h2k sin $3

(14)

The scattered amplitudef(O,+) may be related to the atom form factor F for scattering of x rays by writing

where p ( r l ) is the density of the atomic electrons. (12) and use of the formulas

Ir - rll

dT1

4R

= -

.

’ lim

K2

LT+O

/om

On substitution in

1 e-ar sin Xrdr = xy

(16)

we find

p(r’)

sin (2 kr’ sin + e ) r f dr’,

is the atomic form factor. In this form the term F ( 0 ) represents the shielding effect of the atomic electrons. If it is omitted f(e,+) takes the well-known form for the Rutherford scattering of electrons by a charge ZE. If the form (1) is substituted for V ( r ) in (14) we have

=

(Zp2/ao)G(krsin + O ) ,

where

It follows then that I(e)

= =

lf(o)121

( Z 2 p 4 / u O 2 ) J ( ksin p ae),

where J(Y)

=

(G(?4)12.

(19)


7

Since p is given by (2) we have

I ( 0 ) = 0.613ZMao2J(0.885kaoZ-~~ sin +O).

(22)

Given the function 4(z) in tabular form G(y) and hence J ( y ) may be obtained by numerical integration. From a table of J ( y ) over a sufficient range Z(0) may thus be determined for all angles, electron velocities and scattering atoms. A table of J ( y ) has been given by Bullard and Massey* but does not cover small angle scattering very effectively. In any case the statistical model is not very accurate for these angles and the simple approximation (3) may be used to give useful results. Substituting (3) for 4 in (20) we find

r(e) = 4m2v4(sin2&e + a2)-2 Z2€4

~

where (Y

=

O.%~Z>’S(~T~~/~V).

Once again the cross section is exhibited in a form which shows the close relation to Rutherford scattering which is obtained if a ---f 0. The mean velocity of atomic electrons is of the order 2?rZ”e2/h so that, provided the incident electrons are fast, a is small and the scattering is as given by the Rutherford formula except for angles e < 2a. It follows from (22) that the total elastic cross section can be written in the form Qo = 1.23sao2ZZ6q (0.885kaoZ-%), (26) where q(u) =

Jo”

~ ( sin u

+e) sin e de.

Thus Q02-nis a function of vZ-35 so that a single numerical tabulation of q(u) will give, in principle, Qo for all electron velocities and all atoms. No such table is available at present although Bullard and Massey* have given a diagram illustrating QOZ-SS as a function of vZ->$.However, if the form (3) is used for 4 a good approximation to Qo is obtained by integrating the expression (24)for I ( 0 ) . This gives 7rZ2€4

Qo

= -

YG7 ay1

1

+

a2)

Z+Sh2 1 1.28m2v2s21 a2 2 3.13~Z++/k~s~,

+

for fast electrons, a being then very small compared with unity.

8

H. S. W. MASSEY

This may be compared with the approximation given by Debye.lo We have, from (26) QO

/6' - 'w /oUo

=

-

1.23ra0~Z3~ J(0.885kaoZ-% sin +e) sin

e do,

J ( u ) u du,

where uo = 0.885kaoZ-4s.

The contribution from the upper limit is so small th a t we may take the limit as 00 without appreciable error. This gives

'r/om

Q 0 =.A J ( u ) u du

which is of the same form as (27) except th a t the numerical factor is different. Numerical evaluation of the integral gives 256

Qo = 7 . 1 4 ~ 7 -

k

This agrees with (27) if we take s

= 0.66 as we shall do henceforward. Other approximate forms for I ( @and ) Qo have been given by different authors. l 1 s S 2 It is probable that for applications t o the solid state in which the charge distribution of the valence electrons is not well known the formulas (24) and (27) or (30) are as accurate as the neglect of modifications due t o the solid justifies, except for the lighter atoms. For these the Fermi-Thomas interaction should be replaced by the appropriate HartreeFock field. Some tables of I(0) calculated from the Hartree field of atoms are given in The Theory of Atomic Collisions,12but these do not cover very small angles of scattering. Marton and Schiff13 have given data from which the formulas (24) and (27) may be corrected for light atoms. The correcting factors are.given in Table I.

TABLE I. Factors to replace 2 9 s in formulas (24) and (27) for elastic scattering cross sections. Atom Li C N 0 F Ne Factor replacing Z56 2.7 11 10.5 9.5 13.5 9.8

b . Relativistic Correction. The only correction to the small angle scattering formula arises from the Lorentz contraction which changes the relation between the momentum k and the energy or velocity of the electron. The formula (24) remains unchanged provided k is given by k = mvy/h,

y =

(1 - v2/c2)-%,

(31)

9


instead of by mv/h and the mass m is replaced by my. Thus (22) is replaced by I ( 0 ) = 0.613Z~ao2J(0.8851ca,Z-~ sin +0), (32) and (24) by Z9e4

I(0) = 243 (sin2 +e 4mvy

+

(33)

d/y2)-2.

The angle a t which the effect of the screening produces departure from the relativistic form of the Rutherford formula is thus reduced by a factor y. The total cross section QOis obtained from (27) by interpreting k as mv/h, not as mvy/h. All these corrections ignore spin-orbit coupling effects. These may be included if Dirac’s equations for the electron ape used in place of the Schrodinger equation (4). It is found’l that they introduce an additional factor 1 - p2 sin2 $8 into the expression (32) for I ( @ ) . This factor is negligible a t small angles and has no important influence on Q,,. c. Validity of Born’s Approximation. l6 The approximation will certainly be valid if the part of the approximate wave function representing a plane wave is always large compared with that corresponding to the scattered wave. Referring to (11) it will beseen that this requires

+

for all r. The right-hand expression in (11) is probably a maximum a t If this is so then the approximation will be a good one if T = 0.

el/ V(T’)exp Iikr’(1 + cos 8’))r‘sin 8‘ dr’ do’ d+’ h2

I

10". The energy range in which effects due to the finite size of the nucleus become apparent was first investigated by Rosez4using Born's approximation. Roughly speaking it is to be expected that even for the heaviest nuclei the effect will not be appreciable except at very large angles of scattering unless the electron energy is greater than 20 mev. Eltonz5 has carried out detailed calculations for the scattering of 20-mev electrons by gold in which exact solutions of the Dirac equations have been obtained, assuming the field of the gold nucleus to be given by


13

(39) and

The first case corresponds to a nucleus in which the charge is uniformly distributed throughout a sphere of radius ro, the second to one in which the charge is distributed uniformly over the surface of the sphere. The radius ro was taken as 8.2 x cm in accordance with the usual estimates of nuclear radii. Figure 2 illustrates the results obtained in terms of the ratio of the scattering by the extended nucleus to that by a point nucleus. The modifications are quite marked and indicate that the scattering of very energetic electrons by nuclei is likely to yield very useful information about the charge distribution within the nucleus. Preliminary results obtained experimentally by Lyman, Hanson, and Scott26for the scattering of 16.5-mev electrons by gold are comparable with those predicted from Elton's calculations. The comparison is given in Table 111. More extensive calculations have since been carried TABLE111. Comparison of observed ratio of scattering by gold to that expected for a point nucleus, with calculated ratios. Angle of scattering 30' 90" 150" Observed ratio 0.87 0.59 0.43 Calculated ratio 0.99 0.52 0.41 Case A 0.97 0.35 0.32 Case B

out by Achesongsfor electrons of energies between 15 and 35 mev scattered by aluminium, copper, tin, and gold, assuming the same two interactions (39) and (40). Parzen*' has carried out similar calculations for the scattering of 100 mev electrons by lead nuclei and a potential energy of the form (39) with ro = 8.09 x cm. At these energies diffraction maxima and minima appear in the angular distribution as may be seen by reference to Fig. 3. 2. Modifications Introduced b y the Solid Binding a. Effectof Perturbation of Valence Electrons. When atoms are bound to form a solid the valence electrons are strongly perturbed, but the

14

H. S. W . MASSEY

inner electrons remain unaffected. The scattering by the free atom is modified only in so far as the valence electrons contribute to it, i.e., a t very small angles. To estimate at what angle a perturbation of the valence electron is likely to influence the scattering we may proceed as follows. If ro is the radius of the outer electron shell in an atom, the atomic field will be markedly modified by the solid binding only a t distances greater than ro. Referring to (14) we see that the scattering at an angle

Angle of scattering

FIG. 3. Calculated cross section for scattering of 100-mev electrons by lead nuclei. Curve I, treating the nucleus as a point charge; curve 11, assuming the cm. nuclear charge distributed uniformly throughout a sphere of radius 8.09 X

0 comes mainly from the scattering field a t a distance r where

2kr sin 46 N 1.

(41)

Hence a modification of the field for r > ro will become apparent for angles t9 < O0 where eo N l / k r o . (42) Since the radius of the outer shell will be of the order ao, 0 0 will be of the order Z-sa where a is given by (25). For the calculation of total cross sections for fast electrons the perturbation of the valence electrons is therefore negligible (see Table IV). This applies even when the atoms are ionized as in a metal. Although no finite cross section exists for an unscreened Coulomb field, the cross section calculated for the neutral atom will give a good approxi-

15

ELECTRON SCATTERING I N SOLIDS

mation as the free electrons in the metal will provide sufficient screening to reduce the contribution from the net positive charge on the ions to negligible proportions. TABLE IV. Values of the three limiting angles 01, ea, and Or and of the total cross section QOfor elastic scattering of fast electrons by solid carbon and gold. Electron energy (ev) 10,000 50,000 100,000

a

00

(radians) C Au

(radians) Cand Au

0.024 0.011 0.007

0.055 0.025 0.018

ev

0.013 0.006 0.004

(radians) C Au 2 X 4 X 2 X

2 X 4 X 2 X

&o

(in 10-18 em*) C AU

9.24 1.82 0.9

287 56.6 29

b. E$ect of Atomic Binding Forces. A second point concerns the effect of the binding of the atoms within the solid lattice on the transfer of energy from a fast electron to a lattice atom. If the atom were free and of mass M , an electron would lose a fraction of order 2(m/M)02of its energy to an atom in undergoing elastic scattering through a small angle 8. It is important in certain considerations regarding the coherence of the incident and scattered electron waves to know the extent to which this is modified by the binding'of the atom within the lattice. We may obtain an estimate as follows of the minimum angle of scattering for which the binding exerts any restrictive effect on the energy transfer. If v is the frequency of vibration of the atom, the energy transfer t c the vibration will be negligible if the time of collision is so long that the Fourier analysis of the perturbing force due to the colliding electron contains no component of frequency v with appreciable amplitude. If the electron is incident in such a direction as to pass the center of the atom at a distance p (the impact parameter) the time of collision is of order p / v where v is the electron velocity. The restrictive effect will be confined to collisions in which p / v > l/v. (43)

Making use again of (41) we see that the restrictive effect will only be apparent for collisions in which

e < e,

=

=

?iv/mv2, 2b/E

(44)

where E is the kinetic energy of the incident electron. Although this argument is essentially a classical one it is clearly valid, for p is so large that the electron is practically undeviated from the straight path and may be treated as a center of force moving classically.

16

H. S. W. MASSEY

A similar argument will be given in Sec. V.1 in which the transfer of energy from the incident to atomic electrons is discussed. It is instructive to compare the magnitudes of the three limiting angles a , 00, and 0.. Typical values are given in Table IV. It will be seen that the effect of the solid binding on the transfer of energy between the incident electron and the struck atom is confined to angles much less than a so that in all but a very small proportion of collisions the fractional energy transfer is of order 2me2/2M. 3. Elastic Reflection of Electrons from Metal Surfaces

As an illustration of the opposite extreme to the conditions we have been assuming, in which the perturbation of the atoms by their neighbors is very ineffective, we shall briefly discuss the elastic reflection of electrons of much lower energy (< 100 ev) by a metal surface. The metal is now treated as a single entity which exerts a force on an electron approaching its surface. This force is a combination of the so-called image force and the potential drop a t the surface. If the metal is taken to occupy the space on the negative side of the plane, 2 = 0, the potential energy acting on an electron incident normally on the metal may be written

v

= -vo = -€‘/4X01 = -$/4x,

x

> 50.

X

< 20, (45)

+

The potential drop Vo is given by 4 p where 4 is the work function of the metal and p is the maximum Fermi energy of an electron in the metal. 4 may be obtained from experiment and p is given by (3no/?r)” h2/8mwhere no is the number of free electrons per cubic centimeter in the metal. Typical values of Vo are 10.2 ev for silver and 5.7 ev for barium. MacCol128 has calculated the coefficient r for elastic reflections of electrons a t the surface of a metal assuming the surface interaction given by (45). For small electron energies r N h2Vo/8e4rn, and for high energies E r

-

hzVo4/8r4mE3.

Observations of the elastic reflection of slow electrons from copper have been made by Gimpel and Richardsonz9 (electron energy range 0.35-0.85 ev), and by Farnsworth30 (electron energy range 5-10 ev) while Bruining31 has measured r for electrons of energy ranging from 2.5 to 25 ev reflected from Ba, Ag, and BaO. A comparison of Bruining’s results with MacColl’s calculations for silver is given in Fig. 4. Although there is qualitative agreement, the observed values are considerably


17

larger than the calculated and the observed rate of decrease of r with increase of electron energy is smaller than the calculated.

5

10 15 Electron energy in ev.

20

FIO.4. Reflection coefficients for electrons incident normally on a metal surface. Curves I and I1 calculated by MacColl*a for V o = 10 and 20 ev respectively. X observed for silver (V,= 10.2 ev) by Bruining.sl There is clearly scope for more extensive investigations, both experimental and theoretical, in this field. 111. INELASTIC SCATTERING

Just as for elastic scattering it is convenient at first to regard the inelastic collisions of electrons in passing through a solid as if they were taking place with isolated free atoms of the solid. This is clearly a very good approximation for collisions in which ionization of the atoms in the solid occurs from inner shells. These are only very slightly perturbed by the solid binding so that a purely atomic theory of the ionization of the K or L shells of heavy atoms is quite accurate. On the other hand we must expect that when the collisions involve energy losses of the order of the excitation energies of the electrons the free atom approximation is no longer so satisfactory. Nevertheless, in attempting a theoretical estimate of the probability of inelastic scattering it is usually only possible to work to this approximation. We shall first discuss the theory of the inelastic scattering of fast electrons by free atoms with special reference to applications to the solid state. This will be followed by a short account of investigations, both experimental and theoretical, which have been carried out for solids specifically. The effect of inelastic collisions in determining the energy loss and hence the range of electrons in a solid will form the subject of Sec. V.

18

H. 8. W. MASSEY

1. Inelastic Scattering of Fast Electrons by Free Atoms-Born’s

Approximation

To discuss the inelastic scattering of electrons of initial energy k2h2/2mby an atom containing Z electrons we write the Schrodinger wave equation for the system, atom colliding electron, in the form

+

N

-V2 -

Ha(rl,

-

. . , rN)

+ Eo + 2m

s=l

+ 7 )*(r,rl, Z€2

.

. . , r,)

=

0.

(46)

Ha(rl, . . . , rN) is the Hapiltonian of the atom and EOthe energy of the atom in its ground state. 2e2/1r - rs( is the interaction energy between the incident electron and the atomic electrons which gives rise to the finite probability of energy transfer between the atom and the incident electron. Let h 4 - 1 , . . . , rN), &(rl, . * . , rN) be the respective wave functions for the ground state and for the nth excited state of the atom so that (Ha - Eo)4o = 0y ( H a - En)A = 0(47) We may expand the function P in terms of these atomic functions in the form \k = 2&(r1, . * * , rN)Fn(f). (48) Substituting in (46) we have

In the absence of the interaction

P

=

$1 e2/lr

-

ral we have

40(r1, . . . rN)eikao.r, )

(50)

where nois a unit vector in the direction of incidence. If the interaction is treated as a small perturbation, we may substitute this expression for P on the right-hand side of (49) to give


19

Multiplying both sides by +,* and integrating over the configuration space of the atomic electrons gives

where

Provided kn2 > 0 the excitation of the nth state is energetically possible. T o obtain the differential cross section for this excitation it is only necessary to obtain a solution of ( 5 2 ) which has the asymptotic form Fn ,.,

T-l

eiknrfn(e,+)-

(54)

This corresponds t o a spherical outgoing electron wave of wave number k,, leaving the atom in the n th state. The outgoing flux of these electrons may be calculated as for the elastic scattering in Sec. II.la, and we find for the differential cross section I,(O,+) for a n inelastic collision in which the nt h state is excited

Proceeding as with equation (10) we have, from ( l l ) ,

and where Kn

=

kno - k,nl,

nlbeing a unit vector in the direction (e,+).

we have

N

Since

H. S. W. MASSEY

20

In terms of the angle of scattering we have ~2

= k2

+ k,2

- 2kk, cos e.

Also, for fast electrons, for which

we have, for all inelastic collisions of appreciable probability, where

Hence, for 0

> aom2/2k2we have K 2N 4k2sin2

while, for B

=

0

K = k - k,, 2 ao,2/2k.

To a close approximation, over the whole angular range

K 2 N 4k2 sin2 40

+ (aon4/4k2) cos 0.

(67)

It is clear then that fn(O,+) falls off very rapidly as 0 increases so that the major contributions to the integrand come from the first nonvanishing terms in the expansion of eixn'r*in power series. Thus

S

where zll = II

- r..

As the product +o&* will be very small for values of r > rg, the radius of the shell concerned in the excitation, (68) is essentially the first term in an expansion in powers of Kro. It can therefore be used provided Kro < 1 or, in terms of 8, provided 8

< l/kro.

(69)

If Eo is the energy of an electron in its initial state then l / r o 21~ 87r2mlEol/h2,

(70)


21

so the expansion will be valid provided

e < ( I E , I / E )=~ e~1, say,

(71) where E is the energy of the incident electron. The cross section for scattering through angles > 0, will be very small and for many purposes, such as the calculation of the total cross section, may be neglected. The expression (68) will be a good approximation for 8 < O1 unless the integral vanishes, which will be so if the transition to the n th state from the ground state is optically forbidden. For optically disallowed transitions the first nonvanishing terms involve the matrix elements (ZZ,)~,. The cross sections for excitation of such collisions are considerably smaller than for optically allowed ones except possibly when the electron energy is close to the excitation threshold. At much higher energies they become less and less important as the energy increases. We shall ignore them henceforward. We have now

where EH is the binding energy of the ground state of hydrogen, E is the kinetic energy of the incident electron and AEn = En - Eo. We see then th at the angular distribution of the electrons scattered after exciting a particular optically allowed transition falls off steadily with increasing angle. As the electron energy increases the steepness of the distribution increases. Thus

(73) and (74)

for 8

> AEn/El

so that, whereas the zero angle limit increases as the energy, the intensity takes on its asymptotic form a t an angle which decreases as the energy increases. The asymptotic form itself varies inversely as the energy. a. Angular Distribution of the Totality of Inelastically Scattered Electrons. The angular distribution of all inelastically scattered electrons will be given by

H. S. W. MASSEY

22

provided the condition e < (IEo(/E)$$is satisfied for all transitions which contribute appreciably t o the sum. For excitation from a particular shell the only transitions of appreciable probability are those for which AE is of the same order as (Eel. If we introduce a suitable mean value A T for AE we may then write

For e > (E,/E)'* where E, is the ionization energy of the most firmly bound electrons, we may return t o (60) and substitute ( 6 5 ) for K t o give

where

H e i ~ e n b e r ghas ~ ~ shown how S may be calculated if the Thomas-Fermi statistical atom model is used. It takes the form

S

=

I - R ( k sin

+el

where R tends to zero as k sin &O increases. high electron energies,

(79)

Thus, in the limit of very

which is the Rutherford formula for scattering of an electron by N free electrons initially a t rest. For the calculation of total cross sections the contributions from 0 > may be neglectredso t h a t the formula (76) is adequate, If, on the other hand, i t is necessary to calculate the mean energy loss due t o inelastic collisions, contributions from larger angles can no longer be neglected as they involve relatively large energy losses. For such purposes a different procedure is desirable, as sketched in Sec. V.l. I n the present section we shall be concerned only with the angular range for which (76) is valid. A convenient method of calculation is as follows. Using the usual summation rules,

el

Z f l J ( W o n 1= 2 [(2,z,)2100.

(81)

23


If, further, we represent the wave function of the ground state of the atom as an antisymmetrized product of one electron wave functions then

where i and j refer to occupied one-electron states of the atom.

Iin(e,4)= ( B E x / E ){sin240

Thus

+ (aE/4E)2cos el-*.

(83)

where

Formulas closely similar to (83) have been given by Marton and Schiffla and by K ~ p p e . The ~ ~ former authors have calculated B for a number of atoms using Slater wave functions for the individual electrons. They have also estimated the appropriate value of AE as a mean of the binding energy of individual electrons weighted in proportion to their contribution to B. Table V gives the values they obtain. K ~ p p eignored ~ ~ the cross terms Izii12 in B which may then be

2

iPi

related to the atomic diamagnetic susceptibility x which is given by

so that

B

=

18mc2x/e2.

a

Observed values of x may then be used to determine €3. was taken as the ionization energy of the atom concerned. Koppe’s estimates of B and are also included in Table V . b. Total Cross Sections for Inelastic Scattering, Inner Shell Ionization, and Total Ionization. The total cross section Qnfor excitation of the nth state of an atom is given by Q,,

=

%

/o” I,(e) sin e de.

(87)

To evaluate this integral the approximate form (72) for I,,(O) may be used. It is not accurate for large angles of scattering, but the contribution of the integral from such angles is so small that this inaccuracy is unimportant. We find then that Qn

= ( 2 E x / E )I( zzJon I log

(4E/AE,)

(88)

If the excited state is one of the continuous spectrum, a similar formula may be used provided the kinetic energy E, of the ejected electron is not

24

H. S . W. MASSEY

Xoccurring in the formula for the TABLE V. Values of the quantities B and A angular distribution of the totality of inelastically scattered electrons. B (in units lo-'* cm*) Atom or ion Marton and Schiff H Li+ C N

0 FNs+ P S

K+ Ca++ cu+ BrRb+ Ag+ Sb

I-

Cs+

29.7 7.6 78.2 62.6 51.5 58.1 29.1 179 156 96.1 73.7 203 268 197 446 492 402 268

Koppe

AE (in ev)

Marton and Schiff

31

366 240

13.5 98 44 63 83 91 180 63 74 140 180 130 130 170 170 180 150 230

Koppe

11.2

11.1 10.4

much greater than Eo. Thus if Q,dK is the cross section for ionization of a n atom in which a n electron with energy

Ex = ~ ' h ~ / 2 m

(89)

is ejected then

Q ~ K (~EH/E)\(XZ~)O./~ log ( 4 E / A E , ) (90) , being the energy of the state in where AEx = E+ - Eo ~ % ~ / 2 r nE+ which the positive ion is left after the ionization has occurred. I n calculating (zs)Ox the wave function for the ejected electron must of course be appropriately normalized. The sum Qin of all cross sections for inelastic scattering may be evaluated by using the expression (83) to give

+

&in = (2BEx/E) log ( 4 E / n E ) , (91) are as defined in (84)and (76). where B and Another cross section, involving an incomplete sum of individual cross sections for excitation and ionization of an atom, is t h a t which refers t o ionization of the inner shell of an atom, important in the generation of x-rays.


25

I n this case i t is best to proceed as follows. Let the inner shell from which ionization occurs be the nl-shell. If there were no outer atomic electrons the total cross section for removal of a n electron from the shell in a collision would be given by &is

where

=

( ~ B ~ E x /log E )( 4 E / D u ) ,

(92)

is the mean excitation energy from the shell and Bis

= Znl(Z2)nl,ni,

(93)

Zz, being the number of electrons in the shell. Owing, however, t o the excluded transitions, Bi, must be modified to take the form

the sum being over all excluded transitions. should also be somewhat higher than it would be in the absence of outer electrons. We therefore obtmain finally &is

=

(2B’Ex/E) log (4EIZEis)

(95)

where B’ is of order Znz(z2)nl,n~. Finally there is the total cross section for all ionization processes, a knowledge of which is very important in the interpretation of many experimental results. This will be obtained by integrating (90) over all possible energies of the ejected electron. It is true th a t (90) is not accurate for those collisions in which the incident electron loses a considerable fraction of its initial energy, but the probability of such collisions is so low that they may be ignored. This cannot be done in cslculating the rate of energy loss as the low probability is partly offset by the large energy loss involved. Betheg has shown that, with these assumptions, the total cross section for ionization Q takes the form, for electrons of velocity v %re4 jl 2mv2

&i =

where I is a mean ionization energy and f l is a quantity of order unity which varies from atom to atom. jl and I cannot be obtained with any accuracy from theory and are best derived from experiments at one electron energy for each substance. Table VI gives some values of cross sections Qi, for different atoms and various electron energies calculated using values of B and I given by Marton and Schiff.

26

H. S. W. MASSEY

TABLEVI. Calculated cross sections for excitation of all inelastic collisions by imuact of fast electrons. Electron energy (kev) 10 50 100

Atom or ion-Cross section in C

0

1.5

0.89 0.22 0.12

0.36 0.20

Na+ 0.43 0.11 0.06

K+

Ca++

Rb+

1.5 0.38 0.21

1.1 0.29 0.16

3.0 0.76 0.42

cm2 Ag+

Sb

Cs+

7.0 1.7 0.96

7.6 1.9 1.0

3.8 1.0 0.55

c. Relativistic ModiJications. As long as we are not concerned with inelastic scattering through large angles, the only important modification introduced by relativity arises from the change in the relation between the wave number k , the kinetic energy and the mass of the incident electron. I n relativity theory the kinetic energy E is related to the wave number IG by E mc2 = ch(k2 l c o 2 ) ~ P 1 (97) where k o = mc/h, k = ymv/h, y = (1 - v2/c2)-ss

+

+

Equation (53a) must therefore be replaced by

(kn2

+ ko2))6 = ( k 2 + ko2))fr - AE,/ch,

(98)

and (63) becomes

k,

k -CYO,~/~~,

where aOn2

(2ym/h2)(En - Eo)

=

I n place of (72b), (83), (99), (95), and (96) we now have respectively

B[sin2 +O

+ (ae/2F)2 cos el-',

(102) (103)

(L) (e B log (2F/@, 137 v) 2

&in

= 2

+ ')'

2

&in =

2('->137

e(e

+ 2) B' log (2F/L\e,,),

27


where e

=

E/mc2, Ae

=

AE/mc2, F

=

ye(e

+ 2 ) / ( e + 1)2,

E being the kinetic energy of the incident electron. 6. Experimental Evidence on Inelastic Scattering by Atoms Direct experimental evidence in support of the above theoretical considerations is still very meager. It would be out of place in this review t o discuss the evidence available from the study of electron scattering in gases, It is sufficient t o say that from this evidence there is little doubt of the general correctness of the theory.34 I n view of the

1-: No

t

I I

I

1

I

1

I

I

I

Energy of incident electron (in units of ionization energy of shell)

FIG.5. Comparison of observed and calculated cross sections for inner shell I. observed cross sections;3611. calculated cross sections.36

ionization.

difficulty of providing quantitative theoretical cross sections for inelastic scattering by heavy atoms, most of the detailed comparison must be done in experiments on excitation of hydrogen or helium. I n other cases the lack of accurate theoretical knowledge of such quantities as the mean excitation energy hE and the quantity B in (83) precludes any detailed test. There is in fact a serious need for experiments devoted t o a determination of these quantities for different atoms.

28

H . S . W. MASSEY

The only processes involving heavy atoms which are reasonably predictable in detail are those in which inner shell ionization occurs, particularly of a K or L shell. As in these cases very few electrons are involved and their wave functions are very nearly hydrogenic, i t is possible t o calculate Iis(O) and &is more accurately than given by the formula (95). This has been done by B ~ r h o pfor~ ~ ionization of the K and L shells of Nil Ag, and Hg. I n Fig. 5 his results are compared with observed cross sections obtained by various authors.36 On the whole the agreement obtained is very satisfactory in view of the fact that the comparison is between calculated and observed absolule values. Also in Burhop's calculations no relativistic modifications are included. When these are allowed for the calculated value is increased relatively a t the higher incident energies [see (lOS)] by a n amount which very substantially reduces the discrepancies apparent at these energies. I n some ways the agreement is even better than would be expected a t the lower energies because Born's approximation would not be expected t o give good results when the energy loss is comparable with the incident energy. 3. Influence of the Solid Binding

So far we have neglected any consideration of the effect of binding of the atoms in a solid on the inelastic scattering. This will take the form principally of a modification of the excitation energies AE and D . I n general such modifications can only be detected by comparing observations of inelastic scattering by the same atoms in the free and bound states, experiments which are very difficult t o carry out. Some measurements have been made, using mainly electrons of medium velocity, which provide evidence on the relative probabilities of different energy losses being suffered by a n electron in scattering from a solid. Before describing the results of this work and the theoretical interpretation in terms of the quantum theory of the solid state we shall consider the effect of the dynamic polarizability of the medium which is, paradoxically, of great importance when the incident electrons are of very high energy. a. Dynamical Polarization. A detailed discussion of this effect will be deferred until Sec. V.2 when i t is considered in relation t o the energy loss of electrons in passing through a solid, but we shall give here the main resu!ts as far as inelastic scattering is concerned. By following a correspondence principle argument as in Sec. V.2 we would expect the polarization t o change the natural frequencies of the electrons bound in the atoms of the solid from AE,/h t o AE,'/h, where

+

(AE,I/h)' = ( A E n / h ) 2

vo being given by vo2 =

ane2/rm,

VO',

(107) (108)


29

where n is the number of electrons per cubic centimeter and a is a constant of order unity. This follows from the classical electron theory of the refractive index or dielectric constant of a solid medium and is correct under nonrelativistic conditions. The modification introduced by (107) is usually not very important except for scattering by the nearly free conduction electrons in a metal. For these electrons the minimum value of AE,, would be effectively zero were it not for (107) which ensures that the effective value cannot be less than hvo. It follows then, as in (SO), t ha t the contribution of the conduction electrons t o the scattering of incident electrons of energy E may be expressed in terms of a differential cross section

nl being the number of conduction electrons per cubic centimeter. (hvo/E))’” is usually so small that the form of the cross section a t smaller angles is of no practical interest. Thus for metallic lithium hvo ‘v 7.5 ev. As discussed in Sec. V.2 a remarkable effect arises when the incident electron has a velocity comparable with that of light. On account of the possibility of interaction through electromagnetic radiation, the influence of the interaction between the electrons of the material becomes relatively much more important. (107) now takes the form where y = (1 - v2,’c*)+, v being the velocity of the incident electron. Thus, in particular, the total cross section (106) for primary ionization becomes, when y >> 1,

so that the cross section now tends to a finite limit instead of increasing logarithmically as y -+ cc , according to (106). So far no very definite evidence in confirmation of this effect has been found though some support for it is provided by the measurements of Hayward37 on the ionization produced by very energetic cosmic ray electrons. The closely similar effect on the stopping power of very fast electrons is discussed in more detail in Sec. V.2. Observations of the variation of grain density along the tracks of fast particles in a photographic emulsion have not been easy to reconcile with the theory, but Messel and R i t s ~ have n ~ ~recently pointed out that t o be effective in producing photographic action a secondary electron must be absorbed in a grain. High-speed electrons will not be absorbed and when this is allowed for the discrepancy is largely removed.

30

H. S . W. MASSEY

b. Study of Low Energy-Loss Collisions with a Solid. Some progress has been made toward a detailed theory of inelastic scattering of electrons of medium speed by metals. This has become.possible because of the experiments carried out in the first instance by R ~ d b e r g . ~ ~ I n these experiments a homogeneous beam of electrons of a few hundred elecbron volts energy was allowed t o impinge a t a n angle of 45 degrees on a solid target T. Secondary electrons emitted from this target were largely collected on a cylindrical shield C enclosing T so t h a t the primary current could be measured as the sum of the currents t o T and C. Those electrons emitted from T in a particular narrow range

I

I

10

I

20

I

30

Energy loss in ev.

FIG.6. Energy distributions of electrons scattered from a copper surface observed by R ~ d b e r g . ~ Numbers $ refer to the init,ial energy of the electrons in electron volts.

of angles about a direction a t right angles to the primary beam passed through a slit in the shield C into a magnetic analyzer. The velocity distribution of these emitted electrons could therefore be determined. Typical curves obtained for copper targets are illustrated in Fig. 6. It will be seen that the shape of the curves is independent of the electron energy, showing that the maxima correspond t o energy losses determined by the structure of the metal. Similar results were obtained for Au and Ag. I n view of the comparatively low penetrating power of the primary electrons, Rudberg investigated the possibility that the effects arise from collisions of electrons with surface atoms or ions and are not determined by the bulk structure of the metal. He found that even if a layer of CaO, Ba, or BaO several atoms thick is present on the surface of a silver


31

target, the chief features of the velocity distribution curve for silver are still present. This indicates that the effects are not determined solely by the purely surface properties of the metal. Slater and Rudberg40 therefore attempted a theoretical description of the results, based on the quantum theory of the solid state. In a solid the allowed energy levels fall in bands between which there are gaps in which no stationary states exist (see Sec. VI). These bands can be regarded as arising from the broadening of atomic energy levels due to the mutual interaction of the lattice ions. In the normal metal the allowed energy levels are filled up so that each electron occupies the lowest level accessible according to Pauli's principle. In copper the least firmly bound electrons occupy levels in a band arising from the perturbation of the 3d and 4s levels of atomic copper. The band is only partially filled, the occupied levels arising from the 3d state of the atom and the vacant levels partly from 4s as well as 3d. Incident primary electrons are considered to cause transitions from the occupied 3d levels of the band to 4s levels and others above. To calculate the relative probability of different energy losses it is necessary to know the density N ( E ) of the electronic levels in the metal as a function of energy E (such that N ( E ) d E is the number of levels with energies between E and E d E ) . It is also necessary to know the wave functions corresponding to each of these levels. Sufficient information was available about both energies and wave functions from applications of quantum theory t o metallic copper to enable Slater and Rudberg40 to make a theoretical estimate of the shape of velocity distribution curve for the inelastically scattered electrons. These results are compared with the observations in Fig. 7. I t will be seen that there is good general agreement as to the position of the first two maxima although for higher energy losses the theory predicts two further maxima which are not observed. Reasons are given by Slater and Rudberg why their calculations, which are only approximate, are likely to be less satisfactory for these losses. On the whole the agreement is encouraging, but no further theoretical work on these lines has been carried out. Further experimental work has been done by Rudberg, 4 1 Turnbull and Farnsworth, 4 2 and Ruthemann. *3 The work carried out by the last author is of special interest in that he studied the energy distribution of faster electrons (2.8 kev energy) which had been scattered through very small angles (up to 4 O ) in passing through thin films (100-500 A thick) of collodion, A1203, Be, Al, and Ag. I n all cases, with the thinnest foils, a sharp maximum was found for a single energy loss at 21.4, 22.3, 19.0, 14.7, and 22.6 ev for the respective materials studied. As the foil thickness was increased, energy losses closely equal to integral multiples of these respective values were found,

+

32

H. 5. W. MASSEY

showing that multiple inelastic collisions were occurring. No evidence for other discrete energy losses was found. Rudberg’s results for electrons of much lower energy (129-248 ev) scattered by gold do show evidence of a maximum in the neighborhood of 24 ev, which may correspond to the loss observed by Ruthemann. No evidence of the maxima found at lower energies by Rudberg is found by Ruthemann, although his resolving power should have been adequate t o find them. This discrepancy may be due to the different conditions of the two experiments, but

20

10 Energy loss ev.

FIG.7. Comparison of observedJ9and calculated40energy distribution of electrons of 180 electron volts incident energy scattered from copper. Calculated.

- - - _ - - -Observed.

it is clearly desirable that more investigations should be carried out on these lines. The results would be of interest not only for the theory of metals but also for checking estimates of inelastic scattering cross sections for use in electron microscopy and in other cases in which the scattering is due to solids rather than gases.

IV. MULTIPLE SCATTERING It is clear that in passing through a densely packed set of atoms as in a solid the chance of an electron undergoing more than one collision in passing through the solid is likely to be high unless the solid is very thin. Thus, using the cross sections of Table IV, it is seen that the chance of an electron of 100,000 ev energy undergoing an electron collision in gold in traversing a small distance 61 is given by p

=

2.9

x

10-17~61


33

where N is the number of atoms per cubic centimeter. In gold N = 5.9 X so that p is already approaching unity when 61 is ‘v lo-’ cm. The theory of the scattering which can arise when the path length in the solid is sufficient for many collisions to occur is, in its most general aspect, very complicated. As the electron proceeds through the solid its mean angle of scattering increases so that the path length traversed in passing through a foil may be considerably greater than that given by the thickness of the solid. Furthermore, inelastic as well as elastic scattering occurs so that the mean energy of the electron also changes on its way through the solid. Since the differential cross section for both elastic and inelastic scattering depends on the energy, the ultimate angular distribution of electrons emerging from the foil will be influenced by the rate of energy loss. For these reasons it is necessary to consider the problem in stages each of which has a useful range of validity. The simplest case to consider is that in which the electrons are quite fast so that even after making a great number of collisions the mean angle of deviation is small. The thickness of the foil or plate through which the electrons pass is supposed to be large enough for many collisions to occur in passing through so that the problem can be treated statistically. On the other hand the foil must not be too thick, for then the mean scattering is so large that the path length of the electrons is much greater than the foil thickness. I t is also usually supposed that the thickness of the foil is small enough for the energy loss to be ignored although this restriction can be raised, at least in principle. We call this case one of multiple scattering through small angles. A second, rather extreme, case arises when the plate is so thick that all trace of the initial direction of motion is lost and the electrons diffuse through the material over a large part of their path. The intermediate cases between single scattering, small angle multiple scattering, and diffusion are very much more difficult to treat. It is usual to refer to the case in which the probability of an electron making more than one collision in passing through the solid is considerable, but not large enough for statistical considerations to apply accurately, as that of plural scattering. Some progress has been made in dealing with this case, but little has been achieved toward developing a theory of the transition stage between multiple scattering through small angles and true diffusion. 1. The Boltzmann Equation The theory of multiple ~ ~ a t t e r i nmay g ~be~ based * ~ ~on~the ~ Boltz~ ~ ~ ~ mann equation which gives the rate of change of the distribution function of the electrons in coordinate and velocity space as they pass through the scattering medium. In deriving this equation it is assumed that only

34

H. S. W. MASSEY

elastic scattering occurs so that the magnitude v of the velocity v of the electrons remains unchanged. Approximate methods of allowing for energy losses will be described later. Let f(r,e,+,t) sin 6 dB d 4 dr be the probability of finding an electron in an element d7 of volume a t the point r moving in the direction (0,4) at the time t. The local time rate of change off arises partly from convection and partly from scattering. The former contributes an amount -V . grad f. We thus have af =

at

- v . grad f

+ n+ - n-,

(112)

where n+ and n- represent respectively the number of electrons entering and leaving the distribution per second by scattering. If I({,v) sin l d{ d x is the differential cross section for elastic scattering into the solid angle sin l d{ dx then

where N is the number of scattering atoms per unit volume and cos 1

=

cos 8 cos e

+ sin 8 sin e cos (a - 4).

(114)

We thus have

where v3 = v and 6s is the path length traversed by the electron in time St. The initial condition to be satisfied by f if we suppose the electrons incident at the origin in the direction e = 0 is that

We shall not attempt to deal with the full equation (115) but consider simplified forms applicable to special problems.

2. The Angular Distribution of Multiple Scattering The angular distribution function is obtained by integrating the function j over all space to give

F(O,4,s) = lff(r,e,4,s)d7.

(117)


35

F(B,+,s) sin 8 dt9 d+ is the probability of finding the electron moving in the direction (e,+) after traversing a total path length s in the scattering medium. It follows from (115) that

I/

as = N

I([,v)[F(e,a,s)- F(B,+,s)] sin 1 d r d x

the term involving grad f vanishing on integration. The boundary condition (116) becomes q i - cos e) F(e,+,o) = 2*

(118)

(119)

As we are concerned with the scattering of a homogeneous beam of electrons incident a t a point of a plane surface of the scattering material in a direction normal to the surface the function F will not depend on 9 and we may write F = ZFi(S,v)Pl(COS e), ( 120) where aFl

-

as

+ K~FZ= 0,

(121)

with Kl

=

[

2 n ~ ~ ( ~ , v )[ l~ l ( c o [>I s sin

Since

qaz

c dl.

+ i)Pl(cos e) = 26(1 - cos el,

(122) (123)

the boundary condition for F Z is that Fi(0)

=

21 -t

+1

4a

so we have, integrating (121)

z=o

a. Momentum Loss Cross Section and Mean Free Path. The expression ~1 plays a n especially important part in the theory of multiple scattering, diffusion, and energy loss. It may be written in terms of the momentum loss or diffusion cross section &Om which is defined as =

2a

/o" I ( C , ~ ) ( I- cos 1) sin

dy,

(126)

and occurs in the theory of many processes such as gaseous diffusion,

36

H. S. W. MASSEY

ion mobility and electron mobility in gases and in solids (see Sec. VI). 1 'NQom can be regarded as the momentum loss free path X so that K 1 = 1/x. For the scattering of fast electrons we have, using (33) for I(l,v)

where p = v/c. b. Small-Angle Multiple Scattering. If single scattering is mainly confined t o small angles and we consider thicknesses d of scattering medium which are not too large so the average value of cos 0 in the multiple scattering distribution is still close t o unity, then we may replace s by the actual thickness d t o give the formula of Goudsmit and Saunder~on~~ m

i=e

where in terms of the total elastic collision cross section QO

=

v =

I(l,v) sin 1 d l , 2aNd h r I ( { , v ) sin 1 d{ 2a

(129)

and I(e,v)

= ( & 0 / 4 r ) Z ( 2 1 + i)pzpz(cose>.

(130)

v is thus the total number of collisions made by the electron in passing through the material. T o obtain Fl(0,s) in a form which exhibits not only the Gaussian distribution of multiple scattering but also the transition t o single scattering as 8 increases50 we use the approximation Pl(cos e) N Jo(ZB)for small e. Substituting in (122) we find then

where

A = 8rZze4/m2v4yz. Now since JO(Xy) =

* sin (Xg cosh

t)dt,

(133)


37

we have

1

=

cosh t

-,A

cosh t dt

KI(X) is here the Bessel function49 whose series expansion is

where

-c + 2 r

F(r)

=

n-1,

n= 1

C being Euler's constant,

=

0.5772

Hence K

Z

= ~

Nd[Qo

*

...

+ (Ah/4a)K1(2la/-~)l

(137)

as the upper limit in (137) may be taken as co without appreciable error. To the same approximation

=

so that

Ay2/8(r2,

(138)

+ (2Za/~)K1(21a/~)l, + (2la/-Y)K1(2Ea/-Y)l,

(139) (140) v being, as before, the total number of collisions made in passing through the foil. Hence on substitution in (125) and transformation of the sum to an integral we have K Z = ~

NdQ&

= 41

F ( 0 ) = 27r Since

1

zJo(z0) exp [ - v { 1

+ (2az/7)R1(2awc/y)1]dz.

v = NQod = NAY2d/8a2,

(141) (142)

38

H. S. W. MASSEY

we have, on changing the variable of integration to

where

y = (NAd/2)&,

(143)

x

(145)

= (NAd/2)-550.

We are interested now in the case where v is large. we have, from (136))

For small values of

IJ/V>+

(146)

where r = ec. The first term of this series is a good representation provided y < v34. For larger values of y the exponent is already so large that the contributions to the integral are negligible. We may therefore write

where

b

=

log v

+ 1 - 2C.

Finally, provided log v is considerably greater than unity, F ( 6 ) may be expanded in powers of b or more conveniently, in powers of B , where b = B - log B.

(149)

If we change to a new variable u = B54y

then F ( 0 ) ‘v

/n

(B”))’r

1 m d

uJo(xB-’*u) exp

where

and

+*

=

e2/e22,

where

e2 = (NAdB/2)55.

(153)


39

The first term in (151) is the Gaussian distribution of the multiple scattering. It has the standard form

F(e)

=

(1/1re2z)e-~1/8~'

where 0 2 the mean square angle of deflection is given on substitution for A and B from (149), (148), (142), and (132) in (151) by* 0 2 = (8*NdZ2e4/m2v4y2)log { 6 0 3 Z ~ ( N d~)~ / m v c )

(154)

if log B is small compared with B. The remaining terms in (151) represent the correction to the Gaussian distribution and the transition t o single scattering. When 0 > Oz these terms become important and the transition to single scattering is obtained by considering their asymptotic form when e >> ez so $ >> 1. By expanding exp ( -+v2) in the integrand of (152) and integrating term by term we find

--

+

+

+

Fi($) (2/#') (8/#6) (36/#') ..., F ~ ( ~ L )(16/+6)(10g - Q) ~ / # 8 ) ( l o g - A+),

w

-

+

w

(155)

and so on. Hence, for large values of 6/62,

F(e)

NAd/2d4

= 4Z2e4Nd/m2v4y204,

the single scattering law. I n order that the scattering through an angle e should be effectively the result of a single deflection we must therefore have

e >> e2.

The transition from the Gaussian to the Rutherford distribution is a rather gradual one as will be seen by reference to Fig. 8. The functions F1 and F z have been tabulated by M ~ l i B r eand , ~ ~Table VII reproduces his results. The expression (151) for the multiple scattering distribution ignoring higher terms in Fs, F,, etc. is a correct representation of the integral (150) to order B-3. The approximation made of ignoring the effect of the higher terms in the expansion (146) is of order v-l = e-B. Errors * In Molibre's analysis a more accurate expression for Z(0) is used. The h a 1 formula again follows from (153), (149), and (148)but with

Y

now given by

In a recent paper Hanson, Lanzl, Lyman, and Scott*' have found that the observed multiple scattering distribution of electrons of 15.7 mev energy after passing through thin gold foils agrees with Molibre's theory within the experimental accuracy of 2-3 % (see (f)).

H. 8. W. MASSEY

40

arising from both sources will lead to an inaccuracy of less than 1 per cent if B > 4.5, i.e., if the average number v of collisions made in passing

Of 02

FIQ.8. Illustrating the transition from the multiple scattering to the single scattering distribution as the angle of scattering increases. Distribution including multiple scattering.

_ _ - - - - -Single scattering distribution.

8 2 is the mean square angle of scattering and the number of collisions per centimeter path is taken as e1O.

through the scatterer is > 20. On substitution for v we can express the condition that the scattering should be largely multiple in the form

c. Multiple Scattering Distribution i n Terms of Projected Angle of Scattering. In practice it is often convenient to measure a scattering distribution not in terms of the angle 8 but of its projection $I on a particular plane. This applies not only to investigations of the trails of fast particles in cloud chambers but also to the tracks of such particles, including electrons, in photographic emulsions. The multiple scattering distribution in terms of 4 may be obtained without difficulty by resolving e into two perpendicular components $I and w and integrating over w . ~ O If f(+)d+ is the chance of observing a projected deflection between $I and 4 d 4 then

+

41


f(+)d4 =

e2-l(

+

( 2 / ~ r 9 e - + ' / ~ + ~B*- W 1 ( 4 / G 42) B-%t(4/& 42)

where

+

+

*

+

*

1 d4

42 = e 2 / d ,

(157) (158)

is the root mean square projected angle of scattering. The functions 01 and Gz are tabulated in Table VII. TABLE VII. Functions occurring in multiple scattering formula.

* 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3 3.2 3.5 4

Fz(J.)

Fl (*)

0.8456 0.700 0.343 -0.073 -0.396 -0.528 -0.477 -0.318 -0.147

O.OO0

+O .080 0.106 0.101 0.082 0.062 0.045 0.033 0.0206 0.0105

2.49 2.07 1.05 -0.003 -0.606 -0.636 -0.305 0.052 0.243 0.238 0.131 0.020 -0.046 -0.064 -0.055 -0.036 -0.019 ,0052 +.001

+

GI(*)

GI(*)

+0.0206

+O. 416 0.299 0,019 -0.229 -0.292 -0.174 +o. 010 0.138 0.146 0.094 +O. 045 -0.049 -0.071 -0.064 -0.043 -0.024 -0.010 +o. 001

-0.0246 -0.1336 -0,2440 -0.2953 -0.2630 -0.1622 -0.0423 +o. 0609

0.1274

0.147 0.142 0.1225 0.100 0.078 0.059 0.045 0.0316 +O. 0194

0.006

d. Mean Values. Using the distribution functions (151) and (157) mean values of various functions of the angles 0 or 4 may be obtained. I n particular the mean values 8, 4 are given by '

=

(e2/T>s

0.982

--0.117 - - B2

e. Allowance for Energy Loss. It is possible to make some allowance for energy loss as follows. The mean energy will be a function of the distance s traversed in the material so that the quantities KI in (121) which depend, through the cross section I(T,v), on the energy, will be functions of s. The function Ft appearing in (120) will now be given by

H. 6. W. MASSEY

42

where E is the mean energy. Knowing the rate of energy loss dE/ds (see Sec. V.l), Fl may be calculated. The subsequent procedure then follows the same lines as in Sec. IV.28b but is necessarily much more complicated. f. Experimental Evidence o n Multiple Scattering of Fast Electrons in Foils. The most comprehensive and precise investigation of the multiple scattering of fast electrons in metal foils is that of Kulchitsky and T,aty~chev.~~* They obtained a narrow and almost parallel beam of 2.25 mev electrons by analyzing the @-raysemitted from a thin-walled glass tube containing radium emanation. The angular divergence of the beam was less than 0.8 degree and the energy spread less than 1 per cent. The beam was allowed to impinge on a metal foil mounted at the center of a scattering chamber. Scattered electrons were observed by means of two counters in a coincident arrangement which ensured that the electrons observed could only have arisen from a small volume of the target and not from any other part of the apparatus. Table VIII gives the results they obtained for the angle at which the multiple scattering distribution falls to half-value. Included in the TABLEVIII. Comparison of Observed and Calculated Half-widths of Multiple Scattering Distributions. Half-width angle (degrees) Scattering material A1

Fe cu Mo Ag

Sn

Ta Au Pb

Y

Obs.

Calc .

60.2 41.4 46.8 36.6 35.1 34.2 28.7 29.4 26.1

9.50 9.60 10.40 10.25 10.20 10.65 9.85 9.90 9.70

9.40 9.60 10.50 10.35 10.30 10.65 10.95 11.35 10.85

table are the values of this angle calculated according to the theory of Goudsmit and Saunderson48 which, for small angles of scattering, is equivalent to (151). The quantity v of (156) which must be greater than 20 in order that the scattering should be multiple over a large part of the angular range is also included in each case. The agreement between the observed and calculated values is very good except for the heavy elements, where the disagreement is well outside experimental error. The reason for the discrepancy in these cases has been traced by Moli61-e~~ as due t o the form assumed for the angle *See footnote on p. 139.


43

scattering distribution. The calculated values in Table VIII are obtained, using a distribution essentially of the form (24). This is accurate enough for the light elements but not'for heavy elements. For these elements MoliGre found that the function I ( 8 ) obtained by use of the true function 4 in (19) may be represented for small angles of scattering, by

relativistic effects being included, the notation being as in Sec. II.la. When this is used in the multiple scattering formula the discrepancies for the heavy elements are greatly reduced. g . Multiple Scattering of Electrons in Photographic Emulsions. It has recently been showns3that multiple scattering may be used in conjunction with measurements of grain density to determine under certain circumstances the momentum and mass of the fast particle producing a track in a photographic emulsion. The multiple scattering is measured by dividing the track into elements of length and measuring the average angular deviation 4 between successive chords. This will be given by (159) with the appropriate substitutions corresponding to the constitution of the emulsion. Ignoring the changes in the logarithmic term, 4 is simply given by k / p v where p is the momentum and v the velocity of the particle, assumed to be singly charged, and k is a constant depending on the material. The grain density along the track may be determined as a function of the velocity by observing it as a function of the rate of energy loss in the plate, use being made of the formula (197) of Sec. V.l for the latter quantity. It is not within the scope of the present review to discuss these techniques as they are concerned with mesons and heavier particles rather than electrons, but the technique of multiple scattering measurement has rece'ntly been applied by CorsonS4to check the theory for multiple scattering of 115-mev electrons in Ilford G-5 plates. The mean angle 6,considering only angles less than four times the mean and obtained between chords of interval length 100 microns, was found to be 0.17 f 0.02'. This compares very well with the value 0.20" calculated from the theory of Snyder and which is equivalent, apart from small details, to that of Moli6resodescribed in Sec. IV.2c. 3. Space Distribution of Multiply Scattered Electrons-Absorption

of

Electrons in Plates

The difficult problem of determining the distribution in space of multiply scattered electrons arising from a narrow incident beam is one which occurs quite frequently in practice. Thus if a homogeneous beam

44

H. 8. W. MASSEY

of electrons is incident normally on a thick plate, it is of interest to determine the flux and energy distribution of the electrons which penetrate the plate and also of those which are “back-diffused” to pass out again through the surface on which the beam is incident. Even when the energy loss is neglected, the problem is a complicated one. has used the Boltzmann equation (115) to calculate the values of such quantities as zn where z is the distance traversed through the scatterer, and also for angular correlation functions such as z cos 8. Although his formulas provide in principle all the information required, they are not very convenient for use in practice, but it has not proved possible so far to obtain space distribution functions without using the Fokker-Planck approximation t o the Boltzmann equation (I18). This is equivalent to assuming that

instead of merely taking PZ(C0S

as has been done in Sec. IV.2. - = -3.

as

r> = JLm,

It gives in place of (115) grad f

+ (1/X)V2f,

where h is the momentum loss free path defined in Sec. IV.2a and V2 is the Laplace operator in the angular ooordinates 8 and 4-

vz

i a ae (sin

= __ sin 8

06)

1 a2 +s e a(p2-

It has the disadvantage of giving a purely Gaussian distribution for the multiple scattering and does not exhibit the transition to single scattering a t large angles. Bethe, Rose, and Smith44have applied the equation (161) to discuss the penetration of electrons through a thick plate in which there is no energy loss. In this problem, in which there is symmetry about the axis of the incident beam if it is incident normally on the plate, the equation (161) for the steady state takes the form

where z is measured normal to the surfaces of the plate and p = cos 8. By solving this equation it is found that the fraction r of the incident beam transmitted through a plate of thickness d is given by T

= 0.862/(0.719

+ d/h).


45

The distribution function a t any point of the plate is given in terms of an expansion in unfamiliar functions which are solutions of the equation

Furthermore, in many applications it is preferable to work with a distribution in terms of projected angles of scattering. Provided the mean angle of scattering is small, a simple distribution function of this kind may be obtained.55*56 Let W(x,y,41,4~,z)dxdyd~ld~z be the probability that, after penetrating a distance z in the scatterer, the coordinates of the particle in a perpendicular plane are (x,y) and it is moving in a direction whose projections in t&e xz and yz planes make angles and 42 with the axis of z. If 41 and 42 are small, the equation (161) for the steady state takes the form

aw aw + =+ 41%

m)'

42-aw au = -xl r w + a2w

since 82 'V

412

+

( ~ ~ 2 .

This equation is separable so that, writing

W

=

Wl(X,41,Z)W2(M,42,z)

we have

If the incident beam passes through the origin in a direction parallel to the z-axis Wl(X,41,0) = %)6(41). (169) The solution of (168) satisfying (169) was found by FermiK6in the form

W1

=

, e x p 2u2

z

Integration over x gives the Gaussian angular distribution with the mean square angle of scattering = (2z/X)" which agrees fairly closely with the more accurate value (153) obtained without the Fokker-Planck equation. Similarly, integration over 41 gives the lateral distribution in the Gaussian form.

The distribution function (170) is often useful for interpretation of the

H. S. W. MASSEY

46

tracks of charged particles in cloud chambers and photographic emulsions as well as the study of the passage of electrons through foils.

4. The Difusion Stage We now consider the stage in which almost all trace of the original direction of motion is lost and we are concerned with the spatial distribution only. If the distribution is nearly isotropic and axially symmetrical we have f(r,e,s) =fdr,s)

where

j,,

= 2s

+ fl(r,s) cos 8,

(172)

l,”f(r,s,s) sin e de,

fl = 21

cos 0 f(r,e,s) sin 0 do.

(173)

f,, is thus the total electron density p and f is the magnitude of the total electron flux density j. It is convenient now t o work in terms of p and j . Substituting in the equation (161) and integrating over all angles we have

Again, multiplying (161) first by cos 0 and then integrating over all angles gives

For true diffusion to prevail p and j must not vary much in a free path so that aj/as is small compared with 2 j / X and may be neglected in (175). We then have, on elimination of j between (174) and (175), the standard diffusion equation

So far we have taken no account of energy loss. To do this approximately we note that X is uniquely determined by the distance traversed in the plate as this determines the mean energy. We define a new variable r so that r(s)

=

&

lu”

X(s)ds.

This may be rewritten in terms of the electron energy in the convenient form


47

E , being the initial electron energy and E the energy after traversing a path s in the material. This form for r is convenient because explicit formulas are available both for X(E)[see (127)] and dE/ds [see (197)l. Substituting for r we find (176) takes the form

There is no difficulty in obtaining solutions of this equation satisfying definite boundary conditions, but the formulation of these conditions in problems of the type we have been considering is not easy. This is because the diffusion stage is only reached after passage from the stage of low-angle multiple scattering through an intermediate transition stage which is very difficult to deal with. Bethe, Rose, and Smith44 overcame this difficulty in an approximate manner by ignoring the transition stage and laying down a reasonable criterion to determine when the behavior could be regarded as changing abruptly from the first stage to that of diffusion. When a homogeneous beam of electrons enters a plate, the mean angle of deviation from the incident direction due to elastic scattering increases very slowly while energy loss occurs due to inelastic collisions. When the mean value of cos e becomes small enough, the conditions are those of diffusion. Bethe, Rose, and Smith44assume that there is an abrupt change to this stage when cos 8 = Remembering that, in the 'notation of Sec. IV.2, = 47rF1, we have, using (125),

e

a.

a

-

cos 8 = exp (- Jkl ds) = exp ( - Jds/X).

(179) The change to diffusion will then occur when the path s1 traversed in the plate is given by l ' d s / h = $7

(180)

or, in terms of the energy, when the mean energy El is given by

The average penetration in the first, nearly straight, path is then given by (E)1 = ds. (182)

lo" cos)

To follow the process further the diffusion equation (178) is used with the boundary condition that the source of the diffusing electrons is a point source located within the plate a t a depth z =

48

H. 8. W. MASSEY

The fraction of electrons emerging with energy between E and E from the back of the plate is then given by

+ dE

where

The fraction of electrons absorbed by the plate, i.e., the fraction which is brought to rest within the plate is then given by 1- rln(E)dE.

Bethe, Rose, and Smith have applied this approximate theory to the absorption in aluminium of electrons produced by y-rays and the agreement with observed datas7 on the half-value thickness as a function of y-ray quantum energy is very satisfactory. There have been many investigations made of the phenomenon of back diffusion of electrons from plates of different materials. Thus BotheS8has recently measured the energy distribution and “back diffusion coefficient” (the fraction of the electrons which diffuse out through the front face of the plate or foil) for electrons of energies in the range 10-700 kev and a number of materials (C, All Cu, Sn, Pb). B ~ t h e ~ ~ has given a rough theoretical interpretation of these- results but they have not yet been discussed from the viewpoint of the theory of Bethe, Rose, and Smith.44 V. ENERGY Loss

OF

ELECTRONS IN PASSAGE THROUGH SOLIDS

In the preceding section we have discussed the passage of electrons through foils or plates in terms of the rate of energy loss dE/ds per unit of path traversed. To apply the formulas obtained it is necessary to know, either from theory or experiment, or from a combination of both, this rate of energy loss as a function of the material concerned and of the energy of the electrons. I n view of its great importance in the physics of high-energy particles very much attention has been devoted to the theory of the processes by means of which energy is given up by a fast charged particle to the medium through which it passes. I n order to bring out clearly the physical principles involved we shall first give a simplified theory which nevertheless exhibits the main features of the phenomenon.

49


I. Stopping by Free

Atoms69-60

Consider a fast electron passing through matter containing n electrons per cubic centimeter. If these electrons can be regarded as free the number p dB of collisions which the incident electron will make per centimeter path in which it is deviated through a n angle between e and e dB is given by the relativistic form of the Rutherford scattering formula (80) 8 m e 4 d0 p d O = --

+

.

I

y2m2v4

83’

where v is the velocity of the incident electron, y = (1 - v 2 / c z ) - ~ +and the angle 6 is supposed small. In suffering a deflection between 0 and 0 de the energy transferred to the struck electron is approximately h v 2 y 2 t 1 2 . Hence the energy loss per centimeter caused by collisions in which e > eminwill be given b y

+

This formula may be applied to the rate of energy loss in a n actual and Omin. To the material if we make appropriate substitution for , O, accuracy with which we are working Omax can be taken as unity, but the choice of em, requires much more careful consideration. The formula (186) breaks down at sufficiently small angles because the electrons of the stopping material are not free but subject to binding forces which limit their response t o a force which varies with the time. Under classical conditions the response will be small if the time of collision is long compared with the period of oscillation 1/v of a n electron about its equilibrium position under the binding forces. I n a collision in which the incident electron would, if undeviated, pass a bound electron at a minimum distance p , the time of collision is given roughly by p / v y where v is the velocity of the incident electron and the factor y arises from the Lorentz contraction. Hence, if a classical treatment is valid, the effect of the binding is to limit the energy loss t o collisions in which P/VY

< 1/5

(189)

i.e., t o those in which the impact parameter p satisfies

P < YV/V. (190) It has been pointed out in Sec II.2a th a t scattering through an angle < e arises from the interaction between colliding particles at

50

H . S. W. MASSEY

distances > h/mv9. If collisions for which p > yv/v are t o be regarded as ineffective in producing energy loss, Omin in (188) must be taken as hlmvp = hv/ymv2,t o give

- d-E-- ds

my2

log (mv2y/hv),

Provided p >> d where d is of the order of the amplitude of vibration of the bound electrons, the classical determination of the limiting impact parameter p will be valid-the incident electron will be only slightly perturbed and can be regarded as a center of force following a classical trajectory. For a complete translation to the quantum description it is only necessary to interpret the frequency of oscillation. Corresponding to each possible process of excitation of a particular electron involving a transition from the ground state (of energy Eo) to a particular excited state (of energy Em)there will be a frequency and a corresponding oscillator strength f m a such that

1

fm"

=

1.

(193)

m

If x. is the fraction of electrons with this set of possible transitions and oscillator strengths then

1

- dE = %?T4 xafmslog (mv2y/hvm). ds mu2 -

(194)

8-m

This may be put into a more convenient form by introducing a mean excitation energy I so that

- - dE _ ds

-

(195)

with log I

=

Zx,fmalog hv,'.

(196)

A more accurate discussion reproduces this result with a small correction so that61*62 dE = -ds

mu2 (log (mv2y/l)

+

(197)

The calculation of I cannot be carried out with any accuracy except for atomic hydrogen but Bloch,62 who treated the problem as one of the perturbation of a sphere of charge with density corresponding to the Fermi-Thomas atom model, showed that I should be given approxi-


51

mately by 13.52 electron volts, where 2 is the atomic number of the material concerned. I n practice, it is convenient t o determine I once and for all for a particular substance by a n observation for electrons or other particles of a convenient energy. The variation of dE/ds with electron energy according t o (197) is illustrated in Fig. 10 for different materials. It exhibits a minimum a t a n energy of the order of a few mev. for each material after which it increases steadily in the ultra-relativistic region. This minimum arises from the presence of the factor y in the argument of the logarithm in (197). The introduction of this factor can be traced t o the decrease of the collision time for a given impact parameter due t o the Lorentz transformation and is therefore a purely kinematic effect of special relativity. 2. E$ect of Atomic Interaction in the Solid State

I n deriving the formula (195) it was implicitly supposed t h a t the binding forces concerned are those which bind electrons t o individual atoms. For the calculation of the rate of energy loss in a gas at normal pressure this is correct, but in a condensed material other possibilities arise which can lead t o substantial modification of the formula (195), particularly for very high electron energies. I n a solid or liquid medium the displacement of a n electron from its equilibrium position sets up a polarization of the surrounding medium which tends t o oppose the initial displacement. If we are concerned with purely nonrelativistic phenomena, the effect may be represented as a modification of the effective electric force acting on the electron. If in the absence of the polarization a field E would act, then when polarization is allowed for i t becomes E 47raP where a is a constant of order unity and P is the polarization. Since P is given b y -ntr where r is the average displacement of each electron, the polarization gives rise t o an additional restoring force equal t o nt2a times the displacement. The effective frequency of the electron oscillations about equilibrium due t o atomic binding forces is therefore changed from urn to v,’ where

+

vmr2

= v2

+

vo2,

with v02

=

ne2a/rm.

Y,’ will be recognized as the absorption frequency of the medium in the classical theory of dispersion. When a = 1, v 0 is the frequency of oscillation of a free electron gas of concentration n. Under most nonrelativistic circumstances v o is very small compared with ‘,v for all m and s and has little influence. Its effect is only important when v,’ is especially small or zero as for the free

52

H. S. W. MASSEY

electrons in a metal. The contribution of these electrons to the stopping power is obtained by taking their effective binding energy as hvo. It was shown by Kramersa3th at this is indeed the limiting factor and not the resistance of the metal as originally suggested by von W e i ~ z a c k e r . ~ ~ The situation is quite different when the velocity of the incident particle is comparable with th at of light. The reaction of the surrounding electrons on a particular electron of a solid when all are disturbed by the passage of a n ultrarelativistic particle can no longer be treated as a quasi-static polarization effect. Most of the interaction arises from the radiation field and is relatively much stronger than under nonrelativistic conditions. As a result the maximum effective impact parameter p,,, due t o the atomic interaction effects does not increase in the same way as does t ha t due to atomic binding. Thus as v 4 c, p,,,, due to atomic binding, -+ yc/vm8whereas p,.,, due to atomic interaction effects, -+ c/vo no factor y appearing. Hence, at sufficiently high incident energies, for which y > I/hvo the limitation arises from the atomic interaction and not the atomic binding forces. These conclusions may be rendered plausible by the following argument due t o A. Bohr.60 Consider a fast electron following the path A B through the medium which is colliding a t time t with an electron at the point P where P N , the perpendicular from P t o A B , equals p . The electron a t P is also under the action of the fields due to the acceleration of the surrounding electrons. The field a t P due t o an electron at a point Q where PQ = r is given by

E

= (E

sin +/c2r)a(t - T / c ) ,

(200)

where a(t - T / C ) is the acceleration of the electron a t the retarded time and 4 is the angle between the acceleration and the direction PQ. The only electrons effective in influencing the motion of the one a t P will be those which, a t the retarded time, were themselves undergoing collision. The volume in which these electrons are located may be determined as follows. We choose P as origin with the x-axis measured in a direction parallel to A B . An electron a t Q for which x 2 = r2 - p 2 (see Fig. 9) will, a t the retarded time, be in a phase of collision earlier than that of P a t the time t by a n amount r given by r = - r- - . x c v Points for which r is a constant will therefore lie on the hyperboloid

When r

=

0 the hyperboloid degenerates to a cone of angle 2 arcsin (I/y)


53

extending backwards from P , which intersects the path of the particle at a point for which z = ypv/c. Electrons which a t the retarded time were in the same stage of collision as is the one a t P a t time t lie on the surface of this cone. Since the collision time is of order p l v y , the electrons which effectively interact with the one a t P a t the time t will lie within the two hyperboloids for which 7 = + p / 2 y v . Electrons within this volume a t points for which x < y v p / c all lie on the same side of the path of the incident electron as P , and their contributions t o the net force on the electron a t P will be additive. Electrons in the remainder of the volume will give a much smaller contribution as they will be

FIG.9

accelerated in all directions. Hence if v 'v c s o y >> 1the electrons eff cctive in producing acceleration of the one .at P are confined within a volume of the same order as that of the cone PRS of height y p and semi-vertical angle arcsin (l/y), i.e., a volume of order p 3 y . If a is the acceleration of the electron a t P a t time t due t o the field of the incident electron, i t follows from (ZOO), since sin 4 N 1, that the surrounding electrons produce a net force on the electron a t P a t time t of order (202) ? 2 ( E 2 / C 2 ) ( P 3 Y )W P Y ) = (E"P2/c2)a, in a sense opposite t o a. This force is negligible compared with the total force ma acting on the electron, only if p

',v is certainly satisfied for all frequencies ',v of importance. The observed rate of energy loss was 1.82 & 0.08 mev/g cm-2. The loss due to radiation was 0.13 mev/g cm-2 leaving 1.69 k 0.08 mev/g cm-2 arising from electron collisions. The theory uncorrected for atomic interaction gives 1.93 mev/g cm-* whereas the corrected value gives 1.72 mev/g cm-2 in much closer agreement . Further less definite evidence favoring the modified formula has been obtained by Hereford71and by Paul and R e i ~ h . ~ ~ Evidence obtained from a study of the specific primary ionization produced by fast electrons has been referred to in 111, 4a.

4. The Range of Electrons in Matter The total path traversed through matter by an electron of initial energy E , before it is brought to rest is given by s = rdE/ldE/dsl.

(205)

Owing to scattering, the path will not be straight and an important distinction must be made between the range of electrons in a solid absorber and the total path traversed before coming to rest, The observed range is usually associated with the distance traversed in a direction normal to the absorbing plate whereas the actual path in the material must be much greater. However, in an electron sensitive emulsion the total path may be measured and is defined as the range of electrons of a given initial energy in the emulsion. This is an important quantity for all quantitative research on electron phenomena using these emulsions. It has been measured by Zajac and Ross13for electrons with energies ranging from 30 to 250 kev in Kodak NT4 photographic emulsions. Their results are given in Table IX and have been compared with the ranges calculated by the same authors, using the formula (197) with a mean excitation energy I = 125 ev. The agreement obtained is remarkably good.

56

H. S. W . MASSEY

TABLE IX. Observed and calculated mean ranges of electrons in Kodak NT4 photographic emulsions. Electron energy (kev)

Number of tracks examined

Mean range in microns with standard error

Percentage standard deviation

Calculated range (microns)

30 40 50 60 80 100 147 200 250

25 55 25 51 25 50 25 26 27

7.0 k 0.3 10.8 ? 0 . 4 15.8 f 0 . 5 21.4 ? 0 . 6 32.7 k 1 . 6 46.7 k 1 . 5 95.4 k 1.2 141 5 6 201 8

23 28 16 20 24 21 6 21 20

6.0 9.9 14.6 20.0 32.8 47.8 92.0 149 21 1

*

-

There have been a great number of experimental investigations of the absorption of electrons in solids. Summaries of our present knowledge of the subject have been given recently by Bleuler and Ziintilh and by Glendennin.Is

Absorber thickness

FIQ. 11. Illustrating the distinction between effective range (R,) and actual range (Ro) of electrons in ~b solid absorber.

Provided the initial energy of the electrons is not too low the shape of the absorption curve for electrons of homogeneous energy is as indicated in Fig. 11. There is a considerable “tail” to the curve which renders practical determination of t,he actual range rather arbitrary. It i s


57

customary to overcome this difficulty by defining an effective range by interpolation of the linear part of the absorption curve as shown in Fig. 11. This is not satisfactory when the initial electron energy is less than 10 kev for then no part of the absorption curve is straight enough for unambiguous determination of an effective range. For this reason it is to be expected that results obtained by different observers for electrons of these energies will show discrepancies due to the arbitrariness of definition. Most attention has been devoted to the range of electrons in aluminum. Bleuler and Zunti74give the following semi-empirical expression for the maximum range Ro in this material for electrons with energies below 3 mev:

where E , is the kinetic energy and mc2 the rest mass energy of the electrons measured in mev. The rate of energy loss as a function of energy derived from this expression agrees with that calculated from the formula (197) with I = 125 ev, provided the electron energy is greater than 1 mev, but for lower energies this agreement becomes less satisfactory. This is to be expected because scattering increases as the energy decreases SO that the actual path traversed becomes substantially greater than the thickness of the material. Fowler, Lauritsen, and L a ~ r i t s e nhave ~~ extended the range energy curve to higher energies by evaluating

L6%)

where El is the maximum energy, 3 mev, considered by Bleuler and Ziinti,74the theoretical expression (197) with I = 125 ev being used for IdE/dsl. This gives for Eo > 3 mev

Ro = &Ei(log 316 (Eo

+ m c z ) ) - 0.39 em,

(207)

where Ei denotes the exponential integral. In the same way, for the effective range R,, Bleuler and Zunti74give

R,

=

+ +

EO me2 0.22Eo E~ 2mc2 cm)

which has been extrapolated by Fowler, Lauritsen, and L a u r i t ~ e n ’to ~ give R - ltoEi(log 316(Eo mc2)1 - 0.44 cm. (209)

+

Hereford and Swarm'? have tested the formula (209) by measuring the effective range in aluminum for electrons with energy between

H. 6. W. MASSEY

58

3 and 12 mev. The comparison of their results with those calculated from (209) is shown in Fig. 12. It will be seen that there is a considerable discrepancy which has been traced by Hereford and Swannll t o the effect of multiple scattering.

Energy in mev.

FIG.12. Effective range of electrons in aluminum as a function of electron energy. X

Observed by Hereford and Swan.??

____ For energies less than 3 mev as given by Bleuler and Z t i r ~ t i . ~ ~ - - - - - - - Extrapolated by Fowler, Lauritsen, and Lauritsen.’e VI. THE MOBILITYOF ELECTRONS IN METALS,ALLOYS,A N D SEMI-CONDUCTORS If a swarm of electrons of mass m and charge E is diffusing through a medium under the action of a uniform electric field F , a steady state will be reached in which the swarm drifts with a steady velocity u in the direction of the field. I n this state the energy gained from the field per second is equal t o that lost per second in collisions with the atoms of the medium. The drift velocity u is proportional t o the field strength and the constant of proportionality is known as the mobility p of the charges under the prescribed conditions. Thus u = Fp.

The mobility is determined by the frequency with which the diffusing particles make collisions with the atoms of the medium. If n is the


59

number of scattering centers per unit volume then I.L =

E/nrniiQm,

(210)

where D is the mean velocity of the swarm and Qn is the so-called momentum loss or diffusion cross section (see Sec. IV.2a) for collisions of the particles with an atom of the medium. Qm is given by

Q~

=

2a

/o”

(1

- cos e ) i ( e ) sin e de,

(211)

where I ( B ) d w is the differential scattering cross section as defined in Sec. II.la. If Qm is a function of the velocity of the electrons, then in (210), DQ, must be replaced b y

Jo

* vf (v)Qm(v)dv,

(212)

where f ( v ) d v is the fraction of electrons with velocity between v and v dv. Many experimental and theoretical studies have been devoted to the determination of the mobilities of electrons and of positive ions in gases, but the mobility of an electron in a solid is also of great importance. The electrical conductivity c of a solid metal, alloy, or semi-conductor is essentially determined not only b y the number iV of electrons able t o move through the solid but also by their mobility p. Thus

+

u = hrcp.

(213)

The only feature which does not arise in the gaseous case is the presence of the periodic field of the solid lattice which modulates the otherwise plane waves of the “free” electrons responsible for the conductivity. Electrons move through a perfect lattice without undergoing scattering. It is only deviations from the perfect lattice which are effective in this respect. I n a pure metal lattice distortion arises from the heat motion of the lattice ions, and it is this which gives rise to the resistance. I n an alloy consisting of a pure metal in which a foreign metal is present t o a small extent an otherwise perfect lattice of the pure metal will be distorted in the neighborhood of the foreign ions. Such regions will also act as scattering centers and give rise t o additional resistance. Again a semi-conductor owes its conductivity t o the presence within an otherwise perfect nonconducting crystal of small amounts of impurity which either releases electrons t o wander freely through the crystal or removes electrons t o leave freely movable positive holes. The conductivity due t o the flow of these electrons or positive holes under the action of a n electric field is limited not only by scattering due t o the

60

11. S .

R. MASSEY

vibrations of the crystal lattice but also by scattering a t the centers where the inipurity is located. The ca1cul:ition of the cross section Q m in these cases can often be c a n i d out by a method similar in principle t o that used in Sec. 11. The only essential difference is that the undisturbed motion is represented by planc wa\-rs modulated by the pcrfect crystal lattice field. Thus, provided thc distortion of the perfect lattice is small enough, the differential cross section n hich appcars in (21 1) is given by

U is thc potential energy due t o the distortion of the perfect lattice. and J.,are the wave functions of the initial and final state of the

#%

electron and arc givcn by

#$ = zik(r)evLnoT, #,

=

uA,(r)eZL’nl r.

(215)

The modulating fuiictioiis u k , u j b ’ have the periodicity of the crystal lattice. The actual mass nz is replaced by an effective mass meffwhich is determined by the crystalline field and the electron energy. The integral in (214) is taken over the whole of the crystal but this is not necessary. If the crystal is divided up into regular polyhedra centered round each ion, a good a p p r o ~ i m a t i o nt o~ ~uk(r) ~ ~ ~may be obtaincd by solving the equation

within a polyhedron, V being the potential energy of an electron in the field of the central ion. The function is t o be a proper function within the polyhedron and must satisfy the condition that its normal derivative shall vanish on the surface of the polyhedron. I n order that such a solution should exist, the energy must have ccrtain proper values. For a particular proper value E,, say, there will be a solution The allowed energy states of an electron in the crystal mill then be given approximately hy

with the corresponding wave functions having the form

#

= eikna.r+s(T),

in each polyhedron, r being measured from the center. assured by the boundary conditions satisfied by the I#J8.

(218)

Continuity is


61

The energy level diagram thus consists of a number of bands characterized by the different values of s and of width, depending on k,.,". In a crystal in its normal state the electrons will fill the lowest accessible states. If the number of electrons is such t h a t the uppermost occupied levels only partially fill a band, the crystal will be a conductor. The electrons which take part in the conduction will be those with energies near the minimum. The energy gained from the field is small but, according t o Pauli's principle, i t must be sufficient t o excite an electron t o a n unoccupied level. This will clearly be possible when there are unoccupied levels in the uppermost partly occupied band, but if the number of electrons is such that the uppermost occupied levels completely fill a band, the pure crystal will be a n insulator. The electrons can be accelerated by the field only if they acquire sufficient energy t o jump t o a state in the nearest unoccupied band, across the gap between the bands. This will not be possible with electric fields of ordinary strength. An insulator will be converted t o a semi-conductor if impurities are present which either donate electrons to the crystal or capture electrons from the crystal. I n the former case, that of a n excess semi-conductor, the donated electrons must fill the lowest vacant levels which will be at the bottom of the lowest unoccupied band. These electrons can be accelerated by the field, giving rise t o conduction. If the impurities capture electrons these will come from the uppermost levels of the completely filled band leaving that band only partially filled so that conduction can occur. 111 this case the material is a defect semi-conductor. To apply the theory sketched above i t is necessary t o replace the polyhedra by spheres of radius ro so that

where the integration is taken over a sphere of radius ro surrounding a scattering center. We now consider the contributions t o the scattering from the different kinds of lattice distribution outlined above. 1 . Scattering by Lattice Vibrations-Resistance

of Pure Metalsaosal

The conductivity of a pure metal is limited by the scattering due t o lattice vibrations. Except at very low temperatures this may be calculated by adding the contributions due t o scattering from separate ions. If we assume that the displacement of a n ion from the center of a polyhedron does not affect the field outside the polyhedron, (219) may be applied with U = R * grad V ,

62

H. S. W . MASSEY

where R is the displacement of the ion from the center. differential cross section is then given by

The mean

as the motion of the ions maybe treated classically. R 2 ,the mean square displacement of the ions of mass M , is given by h 2 T / M k 0 2where T is the absolute temperature, 8 the characteristic temperature of the solid, and li Boltzmann's constant. To complete the calculation it is convenient t o transform the volume integral t o one over the surface of the polyhedron. This may be done by using the fact that 4.eikno.*is a solution of the equation

GoGi* grad

1' = grad (V$&*) - l'Ic.o grad $I* - V$,* grad $o, =

[

grad ( V

- E,

- 2m

$o$l*

h2 +[J.oV2 grad 2m

$I*

h2

- 2m - $0v2$1*] - V*+Ograd $I*].

(222)

The first term in brackets vanishes as satisfies (22lj, and the second may be transformed t o a n integral over the surface of the polyhedron t o give

s

$I*

grad Vfi0 d7

= 2m h2

1

($I*

-$grad

- grad

__ a'1d*)r

dS.

(223)

For metals such as the alkali metals &(ro) N 1 and +s'(ro) = 0 so that

Over the surface of a polyhedron

as & satisfies (221) and, for the alkali metals, is spherically symmetrical. Hence

.c

grad V $ od7

=

4rro2 cos x[V(ro) - Es(ro)l (sin Kro - Kro cos KrO)/K2r02, (225)


where K = 2k sin 40 and Thus

63

x is the angle between no - nl and grad V . (sin Kro - Kro cos K r o ) 2 / K 4 r ~ 4(226) .

It remains to determine the value of k appropriate to the conduction electrons. This may be done as follows. The number of energy levels per unit volume between k and k dk is

+

k2dk/8n3,

(227)

so that, if the total number of electrons occupying levels in the s-band is N , per unit volume N , = km3/31r2, (228) where k, is the value of k for the electron in the highest occupied level. Thus k , = (31r2N,)’. (229) Finally, for the alkali metals, of unit valency, there will be one “free” electron per atom, i.e., one per polyhedron so

N . = 3/4?rr03. This gives kro

=

(97r/4)%= 1.92.

On substitution in (226) we find that Qm

4?rm.rr2Tro4(v - E ) 2 1 - cos 0)f(3.84 sin 40) sin 8 de, h2M~e2 0 = 2.2$merr2Tro4(V - E)2/h2M~82, (232)

=

where

f(z)

=

(sin x - x cos z)~z-~.

(233)

The conductivity obtained using this expression is of the correct order of magnitude and exhibits the observed variation with temperature,80s81and there is little doubt that $he principal source of resistance of pure metals arises from scattering by lattice vibrations. 2. Scattering by Foreign Atoms-Resistance of Alloys

A foreign atom introduced within the lattice will produce a disturbance of the lattice field in its immediate neighborhood. This will lead to additional scattering and resistance. If the replacement of an ion within a polyhedron by a foreign ion changes the potential energy by an amount U ( r ) then, provided this perturbation is small, the formula

64

H. S. W. MASSEY

(219) may be used to calculate the momentum loss cross section due to this effect. hIottg2has applied the Fermi-Thomas statistical method to determine U ( r ) for an alloy consisting of a metal such as copper, silver, or gold, with one valence electron per atom, containing in solid solution a small proportion of a metal with Z 1 valence electrons per atom. He obtains V ( r ) = (Ze2/r)e--r/a, (234) where ) (235) a2 = ( ? r / 3 ~ ,%2/4me2,

+

N Obeing the number of atoms per unit volume. If the perturbation method may be used and the electrons treated as completely free, the calculation of I ( @for this case is exactly similar to that for scattering of electrons by free atoms which has been discussed in Sec. 11.1. I ( @is given by (24) with a = h/2mva,

(236)

and Qm may be calculated as in (127) to give

Thus the increase in resistance due to 1 per cent of foreign metal in solid solution is, according to (237),

where y = h2/4m2v2a2,

(239) v being the velocity of the conduction electrons which is given from (229) by h v = - (3P2NS)5'd. m

This formula agrees with experimental results on the resistance of dilute solid solutions in copper, silver, '*and gold83 provided a is taken as 0.3 A, which is rather smaller than the value given by the statistical model. 3. The Resistance of Semi-Conductors

Since there are necessarily impurities present in a semi-conductor, both the sources of scattering discussed in Secs. VI.l and VI.2 must be present. In the course of a detailed study of the properties of silicon both as an excess semi-conductor (containing a small amount of phosphorus as


65

impurity) and as a defect semi-conductor (containing a small amount of boron as impurity) Pearson and BardeenS4have determined the mobilities of the electrons responsible for the conduction, and we shall conclude with a brief discussion of their results. a. Non-degenerate Case. When the impurity concentration is not too high, the electrons or holes produced form an assembly of such low concentration that it may be treated by classical statistics. I n general the impurities will be only partially ionized so that scattering will arise not only from ionized but also neutral impurities. The contribution from the former collisions to the momentum loss cross section, and hence to the reciprocal of the mobility, has been calculated by Conwell and Weisskopf.S6 They simply consider the scattering of free electrons with a Maxwell distribution of energies corresponding t o a temperature T . Q,,, may be calculated for electrons of velocity v by taking I ( e ) for e > eo as given by the Rutherford scattering formula for a charge e / K , K being the dielectric constant. The limiting angle O0 is so small that the contribution from smaller angles can be neglected. Thus

as the upper limit may be taken as unity without serious error. To determine eo i t is supposed that the interaction of a particular ion and an electron is screened when they are at distances apart > d, 2d being the average distance between neighboring ions. For ordinary temperatures T , the velocity v of the electrons is so low that Born's approximation is no longer valid for the description of the collisions [see (37)]. On the other hand the classical theory gives satisfactory results. According to this theory the field a t a separation d determines the scattering at a n angle eo where eo = 2e2/mv2Kd. Hence Q~ N _ ( 2 r e 4 / ~ 2 m 2 v 4 ) log ( 2 2 / 4 ~ 4 . (242) To complete the calculation vQ, must be averaged over the Maxwell velocity distribution. Conwell and WeisskopfE6obtain finally

where

x = 6KduT/e2, N l being the number of ionized impurities per unit volume. The contribution from the scattering by the neutral centers is difficult to determine as it is likely to depend sensitively on the precise form of the scattering field, and we shall ignore it henceforward.

66

H. S. W. MASSEY

Whereas the resistance due to lattice vibration increases as T , (243) shows that the contribution from ionized impurity scattering varies as T-34 and should be mainly responsible for the resistance a t low temperatures. Pearson and Bardeens4 showed that in the samples they studied the mobility could be represented by an expression of the form l/p =

aT-35

+ bTM,

where a is of the order of magnitude to be expected from (243). b. Degenerate Case. If the electron or hole concentration due to the impurities is so dense that the assembly must be treated as a degenerate one, all the impurities can be assumed to be ionized and the scattering by neutral centers ignored. I n this case the effect of the ionized impurity scatterers may be calculated in the same way aa the increase of resistance in an alloy due to lattice distortion in the neighborhood of the foreign atom. The scattering potential (234) used in the latter case (Sec. VI.2) is replaced by U ( r ) = (E2/Kr)e-7/a,

(244)

where K is the dielectric constant and a is given by (235) with n the number of impurity centers per cubic centimeter. This gives as in Sec. VI 2, if Born’s approximation is applicable, l/p =

(2eam2/3~K2h3)f(y)

where

f(Y) and

=

log (1

+ Y> - Y/(l + Y)

y = (h2/4me2)(3n/7r)36.

(245) (246)

(247)

In this case the contribution to l/p is independent of temperature and (245) should represent the value of l/p at low temperatures where the effect of lattice scattering is negligible, Comparison with observation on five samples showed that (245) gives results of the correct order if f(y) N l.84 If the screening constant a is taken as given by Mott’s statistical treatments2 a smaller value of f(y) would be obtained, but it is hardly to be expected that the rather crude theory using Born’s approximation would give results of greater accuracy. REFERENCES 1. McKay, K. G. Secondary Rlectron Emission. Advances in Electronzcs, I, 66 (1948). 2. Thomson, G. P., and Cochrane, L. Theory and Practice of Electron Diffraction, Macmillan, New York, 1939. 3. Thomas, L. H. Proc. Cambrtdge Phzl. SOC.,23, 542 (1927). 4. Fermi, E. 2.Physik, 47, 73; 49, 550 (1928). 5. Bush, V . , and Caldwell, S. H. Phys. Rev., 38, 1898 (1931).


67

6. Hartree, D. R. Rept. Prog. Phys. XI (1946-7). 7. hlott, N. F., and Massey, H. 8. W. The Theory of Atomic Collisions, Clarendon Press, Oxford, 2nd Edition, p. 114. 1949. 8. Bullard, E. C., and Massey, H. S. W. Proc. Cambridge Phil. Soc., 26,556 (1930). 9. Bethe, H. Ann. Physik, 6,325 (1930). 10. Debye, P. Ergeb. tech. Rontgenkunde, 3, 11 (1933). 11. Borries, B. von. Z. Naturjorsch., 4a, 51 (1949). 12. Mott, N. F., and hlassey, H. S. W. The Theory of Atomic Collisions, Clarendon Press, Oxford, 2nd Edition, p. 188. 1949. 13. Marton, L., and Schiff, L. I. J . Applied Phys., 12, 759 (1941). 14. Mott, N. F., and Massey, H. S. W. The Theory of Atomic Collisions, Clarendon Press, Oxford. 2nd Edition, p. 119. 1949. 15. Schiff, 1,. I. Quantum Mechanics, p. 169. 1949. 16. hIott, N. F., an'd hlassey, H. S. W. The Theory of Atomic Collisions, Clarendon Press, Oxford, 2nd Edition, Ch. IX. 1949. 17. Massey, H. S. W.,and Burhop, E. H. S. Electronic and Ionic Impact Phenomena, Clarcndon Press, Oxford, Ch. 111. 1952. 18. Buerhner, W. W.,van der Graaff, R. J . , and Feshbach, H. Phgs. Reu., 69, 452 (1946); Buechner, \T. W., van der Graaff, R. J., Sperduto, A., Burrill, 5;. A., and Feshbach, H., zbid., 72, 678 (1947). 19. hIott, N. F., Proc. Roc. SOC.(London),A124, 425 (1929); 136, 429 (193%. 20. Bartlett, J. H., and Welton, T. A. Phgs. Rev., 69, 281 (1941). 21. McKinley, W., and Feshbach, H. Phys. Rev., 74, 1759 (1948). 22. Massey, H. S. W., and Mohr, C. B. 0. Proc. Roy. SOC.(London), A177, 341 (1941). 23. Mohr, C. B. 0. Proc. Roy. SOC.(London),A182, 189 (1944). 24. Rose, M. E. Phys. Rev., 73, 279 (1948). 25. Elton, L. R. B. Proc. Phys. Soc. (London),A63, 1115 (1950). 26. Lyman, M., Hanson, A. O., and Scott, hi. B. B d l . Am. Phys. SOC.,26,3 (1950). 27. Parzen, G. Phys. Reti., 80, 355 (1950). 28. MacColl, L. A. Phys. Rev., 66, 699 (1939). 29. Gimpel, I., and Richardson, 0. Proc. Roy. Soc. (London), A182, 17 (1943). 30. Farnsworth, H. E. Phys. Rev., 20, 355 (1922); 26, 41 (1925). 31. Bruining, P. Physicu, 6,913 (1938). 32. Heisenberg, W. Physik Z., 32, 737 (1931). 33. Koppe, H. 2. Physik, 124, 658 (1948). 34. Mott, N. F., and hlassey, €1. S. W. The Theory of Atomic Collisions, Clarendon Press, Oxford, 2nd Edition, Ch. XI. 1949. 35. Burhop, E. H. S. Proc. Cambridge Phil. Soc., 36, 43 (1940). 3G. Clark, J. C. P h p . Reo. 48, 30 (1935), and Webster, D. L., Hansen, W. W., and Duveneck, F. B. tbzd.. 43, 851 (1933), for I< ionization of Ag; Smick, A. E., and Kirkpatrick, P., zbrd., 67, 153 (1945), and Pockman, L. T., Webster, D. L., Kirkpatrick, P., and Hanorth, K. ibid., 71,330 (1047) for K ionization of Ni; Webster, D. L., Pockman, L. T., and Kirkpatrick, P. ibzd., 44, 130 (1933) for L shells of Au. 37. Hayward, E. Phys. Rev., 72, 937 (1947). 38. Messel, H., and Ritson, D. hf. Phil. Mag., 41, 1129 (1950). 39. Rudberg, E. Phys. Reti., 60, 138 (1936). 40. Slater, J. C., and Rudberg, E. Phys. Rev. 60, 150 (1936). 41. Turnbull, J. C., and Farnsworth, H. E. Phys. Rev., 64, 509 (1938). 43. Reichertz, P. P., and Farnsworth, H. E. Phys. Rev.,76, 1903 (1949). 43. Ruthemann, G. Ann. Physik, 6, 113, 135 (1948).

68 44. 45. 46. 4i. 48. 40. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. i2. 73. 74. 75. 76. 77. i8. 79. 80. 81. 82. 83. 84. 85. 86. 87.

H. S. W. MASSEY

Bethe, H., Rose, M. E., and Smith, L. P. Proc. Am, Phil. SOC.,78, 573 (1938). Snyder, H. S., and Scott, W. T. Phys. Rev., 76, 220 (1949). Lewis, H. W. Phys. Rev., 78, 526 (1950). Butler, S. T. Proc. Phys. SOC.(London), A63, 599 (1950). Goutlsmit, S., and Saunderson, J. L. Phys. Rev., 67,24 (1940); 68, 36 (1940). \Yhittakcr, E. T., and Watson, G. N. Modern Analysis, Cambridge Univ. Press, Cambridge, 4th Edition, p. 373. 1927. Moli&re,G. 2. LVuturforsch., 3a, 78 (1948). Iculrhitsky, I,. A, and Latyschev, G. D. Phys. Rev., 61, 254 (1942). MoliBrc, G. 2. .?‘aiurforsch., 2a, 133 (1947). Goldschmidt-Clermont, G., King, D. T., Muirhead, H., and Ritson, R. M. Proc. Phys. SOC.(London), 61, 183 (1948). Corson, D. R. Phys. Rev., 80, 303 (1950). Rossi, B., and Griesen, K. Revs. Modern Phys., 13, 240 (1941). Scott, W.T. Phys. Rev., 76, 220 (1949). Fleischmann, R. 2. Physik 103, 113 (1936); Mitchell, A., and Langer, R. Phys. Rec., 62, 137 (1937). Bot.he, W. Ann Physik, 6, 44 (1949). Williams, E. J. Revs. Modern Phys., 17, 217 (1945). Bohr, -4. Kgl. Danske Vidensk. Sqlskab, 19, 1 (1948). Bethe, H., and Fermi, E. 2.Physik, 77, 296 (1932). Bloch, F. Z. Physik, 81, 363 (1933). Kramers, €1. A. Physicu, 13, 401 (1947). von Weiszacker, C. F. Ann. Physik, 17, 869 (1933). Swann, W.F. G. J. Franklin Inst., 226, 598 (1938). Fermi, E. Phys. Rev., 67, 485 (1940). Halpern, O., and Hall, H. Phys. Rev., 67,459 (1940); 73, 477 (1948). cercnkov, P. Compi. rend. acad. sci. U.R.S.S., 2, 451 (1934). Frank, I., and Tamm, Ig. Compt. rend. acud. sci. U.R.S.S., 14, 109 (1937). Crane, H. R., Olesen, N. L., and Chao, K. T. Phys. Rev., 67, 664 (1940). Hereford, F. 1,. Phys. Rev., 74, 574 (1948). Paul, W., and Reich, H. Z . Physik, 127, 429 (1950). Zajac, B., and Ross, M. Nature, 164, 311 (1949). Bleulrr, l?., and Ziinti, W. Helv. Phys. A d a , 19, 376 (1946). Glendennin, L. E. Nucleonics, 2, 12 (1948). Fowler, W.A,, Lauritsen, C. C., and Lauritsen, T. Revs. Modern Phys., 20, 236 (1948). Hereford, F. L., and Swann, C. P. Phys. Rev., 78, 726 (1950). Mott, S . F., and Jones, H. Properties of Metals and Alloys, Clarendon Press, Oxford, Ch. 11. 1936. Seitz, F. Modern Theory of Solids, Ch. IX. 1940. Mott, K. F., and Jones, H. Properties of Metals and Alloys, p. 252. 1936. Seitz, F. Modern Theory of Solids, Ch. XV. 1940. Mott, N . F. Proc. Cambridge Phil. SOC.,32, 281 (1936). hlott, N. F., and Jones, H. Properties of Metals and Alloys, Clarendon Press, Oxford, p. 289. 1936. Pearson, G. L., and Bardeen, J. Phys. Rev., 76, 865 (1949). Conwell, E., and Weisskopf, V. F. Phys. Rev., 69,258A (1946). Acheson, L. K., Phys. Rev., 82,488 (1951). Hanson, A . D., Land, L. H., Lyman, E. M., and Scott, M. B. Phys. Rev., 84, 634 (1951).

The Scintillation Counter G. A. MORTON Radio Corporation of America, RCA Laboratories Division, Princeton, New Jersey CONTENTS

Page I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 11. The Photomultiplier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Multiplier Performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 1. Photoelectron Collection Efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2. Multiplication Statistics ....... .. .. .. . . 80 3. Gain . . . . . . . . . . . . . . . . . . ................................... 83 4. Spurious Pulse Output.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Resolving Times.. . . . . . . . . . . . . . . . . . . . . . IV. Phosphor Crystals.. . . . . . . . . . . . . . . . . . . . . . . V. Scintillation Counter Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Radiation Detection and Monitoring.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Scintillation Counter Spectrometry. . . . 3. Time Measurements.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

94 94 103 106

I . INTRODUCTION In the few years that the scintillation counter has been in general use it has been developed into one of the most valuable tools in the field of nuclear research for the detection of high-energy radiations. As such, it has been the subject of considerable analysis itself, in addition to having been the means of obtaining the solution of a number of interesting problems of nuclear physics. It is extremely difficult to trace out the history of the scintillation counter in any detail. The device is the logical evolution of one of the earliest forms of nuclear particle detectors, the Spinthariscope. The idea of electronically counting the flashes of light produced in the phosphor of a Spinthariscope by x-rays or nuclear radiation is not new but was not applied until quite recently. The first published account of a scintillation counter appears to be that by H. Kallman' in 1947 describing work that he had carried out in Berlin at that time. This was closely followed by articles by Coltman and Marshall2 and other workers in the field. The essential components of a scintillation counter are a phosphor crystal, a secondary emission multiplier, and some form of presentation 69

70

G . A . MORTON

device: for example, a pulse height selector and pulse counter, arranged as illustrated in Fig. 1. The nuclear radiation to be measured falls on the phosphor crystal. As each nuclear emission strikes the crystal, it gives up some or all of its energy and excites a fluorescent scintillation. The light from this flash causes the release of photoelectrons from the photocathode of the secondary emission multiplier. These initial electrons are multiplied by the cascaded dynodes of the tube and when collected result in a current pulse at the output. The electrical pulses thus formed are analyzed and counted by the circuit which follows the multiplier. The scintillation counter is superior in a number of very important respects to detectors which depend on the ionization of gas for their action. Phosphor crystals can be found which are quite efficient at PHOTOMULTIPLIER

FIG.1. Scintillation counter.

converting the energy of the incident particles into photons of radiation suitable to excite the multiplier. Therefore, nearly all the particles which are absorbed by the crystal produce a countable pulse. Furthermore, since the density of the crystal can be fairly high, a large fraction of the incident radiation, whether it be material particles, such as alpha or beta rays or photons as in the case of gamma rays, is absorbed by the phosphor. As a consequence, the efficiency of the device is quite high, for example, efficiencies of 50 per cent or more for gamma rays can readily be achieved with a scintillation counter. On the other hand, with the gas-ionization type detector, the interaction of the nuclear radiation is either with the walls of the counter or with the relatively low density of gas with which the counter is filled, so that it can only detect a relatively small fraction of the particles passing through it. Therefore, the efficiency of this class of detector is low, being for most types of Geiger counters, for example, only a fraction of a per cent. Also, since the number of light photons produced in a given scintillation is proportional to the energy absorbed by the crystal from the incident particle, the number of electrons released by the photocathode and consequently the size of the current pulse a t the output is a measure of the energy of the incident particle. This means that the scintillation detector may be employed as an energy analyzer or spectrometer without sacrifice of any of its other desirable qualities.

THE SCINTILLATION COUNTER

71

Finally, the scintillation counter has an extremely short resolving time. The multiplier being an electronic device operating at fairly high voltages is extremely fast in its action. Phosphor crystals have been found for which the duration of the flash representing a single scintillation is only a per cent or less of a microsecond. This means that the scintillation counter has a resolving time of only a few billionths of a second. On the other hand, an ionization type of detector, such as the Geiger counter, will not resolve events separated by less than a few microseconds. Therefore, the scintillation counter can be used to determine the chronological relationship between nuclear events where the time intervals are several orders of magnitude smaller than has heretofore been possible. This capability is of extreme importance in experimental nuclear research. Before considering in greater detail the performance of this device or describing some of the results which it has made possible, it would be well to consider the major components making up the scintillation counter in some detail. Two in particular are characteristic of this device, namely, the photomultiplier and the phosphor crystal. Each of these will be treated in some detail in succeeding sections. 11. THE PHOTOMULTIPLIER The photomultiplier3-6 is a device for obtaining a large electrical signal from a small amount of incident visible light radiation. It consists of a photocathode and a series of electrodes known as dynodes which have been so surfaced that electrons striking them produce secondary emission. The electrodes of a multiplier are so shaped that electrons from the photocathode are directed onto the first dynode. The secondary electrons produced at the first dynode are in turn made to focus on the second dynode. This cascading process is repeated for the subsequent dynodes and finally the secondary electrons from the last dynode are collected by an anode. It is evident that with this arrangement, if the secondary emission ratio of each dynode (that is, the number of secondary electrons which is on the average produced by each primary bombarding electron) is u, the initial photocurrent will be amplified by a factor uk where k is the number of dynodes. The factor uk is known as the gain G of the multiplier. Thus, a secondary emission photomultiplier is in fact a photocell and amplifier assembled in a single electron tube. At this point it might be asked why, since in nearly eyery case elaborate circuitry involving a number of vacuum tubes must follow the multiplier in a scintillation counter, an ordinary phototube and thermionic vacuum tube amplifier cannot be used instead of the photo-

72

G . A . MORTON

multiplier. The reason is that only a few photoelectrons are released from a photocathode by the light from a scintillation. This minute photocurrent must be amplified with an amplifier having an extremely short rise time (i.e., second), if it is to be useful as a scintillation counter. However, it is not possible to design an amplifier with the required extremely short rise time which a t the same time has the very high input impedance and high signal-to-noise ratio required to amplify the very small initial current. The photomultiplier, on the other hand, has, as will be discussed in greater detail later in this section, the required rapid rise time and at the same time functions as an almost completely noise-free amplifier. To date, the photomultiplier is the only photosensitive device known which meets the demands placed on this element in a scintillation counter. I

MAGNETIC

FIG.2.

2

3

MULTIPLIER

Magnetically focused multiplier.

A number of different classes of photomultipliers exist which are fundamentally suitable for scintillation counting. These differ primarily in the means for directing electrons from one dynode to the next. The most important classes are the magnetically focused multiplier, the electrostatically focused multiplier, and the unfocused multiplier. The magnetic multiplier depends upon crossed magnetic and electric fields which produce cycloidal-like electron trajectories, for directing the electrons from one stage to the next. Figure 2 illustrates schematically the electrode arrangement for this type of multiplier. The magnetic multiplier is not available commercially and has not been used for scintillation counting. One of the reasons for this is that it is rather critical in adjustment and requires a magnetic field which must be accurately controlled in intensity and orientation, if the multiplier is to operate satisfactorily. It does, however, possess certain advantages which will be discussed later on in this article. The electrostatically focused multiplier is the type most widely used in this country. I n this type of multiplier the dynodes are so shaped as to produce an electrostatic potential distribution between successive dynodes which satisfies two important conditions. The first of these is that electrons from the working surface of one dynode strike the working


73

area of the next. The second is that the electrostatic field at the surface of each dynode is in such a direction as to draw electrons away from it. Other conditions which must be satisfied by the electrode design of the tube are that sharp points and close spacings must be avoided to eliminate cold cathode discharge effects, free paths between widely separated dynodes must be eliminated in order to prevent ion feedback, and practical configurations permitting good tube design must be used. The structure of a typical unfocused multiplier is illustrated in Fig. 3. This is a venetian blind type of multiplier used fairly extensively abroad. The electrons strike the dynode from the left, producing secondary elec-

COLLECTOR

d

SCREEN

FIG.3.

“Venetian blind” multiplier.

trons which are drawn through the structure by the potential difference between the mesh screen and the venetian blind-like dynode. After passing through the screen these electrons strike the next dynode where they produce a second generation of secondary electrons. This process is cascaded for as many stages as is required to obtain the desired total gain. In this multiplier no attempt is made to cause the electrons to follow definite predetermined paths. A number of the multipliers which have been successfully used for scintillation counting and which are commercially available in this country and abroad are listed in Table I.6 The salient features of some of these multipliers warrant further discussion. All the multipliers listed in Table I have photocathodes based on an intermetallic compound Cs,Sb of cesium and antimony. The sensitive material is thought to be an internal photoemitter, that is, photons are absorbed in the bulk material in a depth of a few tens of angstrom units causing the excitation of photoelectrons which escape from the surface. This is in distinction to surface photoemitters, such as cesium-cesium oxide-silver, where the absorption of photons effective in releasing electrons occurs only at surface atoms. Two forms of cesium antimony photocathodes are employed, these

TABLEI. Tuhe Type Photocathode type area (cm2) spectral class peak response (angstroms) longwave cutoff (angstroms) sensitivity ( p a j l ) Gain number of stages volts per stage average gain Capacity coll. to last dynode (jtpf) coll. to total structure (ppf) Voltage overall (maximum) coll. to last dynode Current coll. (maximum, average) (ma) dark current (pa) Dimensions length (in.) diameter (in.) See footnote page 78.

Characteristics of commercial photomultipliers.

RCA 931A

RCA 1P21

RCA 1P22

RCA 1P28

RCA 5819

EM1 4588

EM1 5060

EM1 5311

Internal 1.9

Internal 1.9 s-4 4000 7000 40

Internal 1.9 5-8 4200 8000

Internal 1.9 s-5 3400 7000 15

Tube end 11 S-9 4800 7000 40

Internal 20 Like S-4

Tube end 0.7 Like S-9

Tube end 5 Like s-9

40

20

20

9 100 105

10 90 6 X lo5

9 150 106

11 160 107

11 160 107

s-4 4000 7000 10

3

9 100 108

9 100 2 x 108

9 100 2 - x 106

4 6.5

4 6.5

4 615

4 6.5

5 8

1250 250

1250 250

1250 250

1250 250

1250 150

1500 150

1 .o 0.25

0.1 0.1

1.o 0.25

2.5 -

0.75 0.05

0.03

3"16

3"16

3"16

1A

1A

56 2t

10 2

2

x

0

p g

o

2

0 2

8

1

180

180

1 0.01

1 0.1

S#

8%

2

2

75


being illustrated in Fig. 4, together with their respective spectral responses. For the first, the material is deposited on a solid metal back and electrons are emitted from the side on which the light is incident. The second form is a thin semi-transparent layer of the material on a transparent glass backing. Here the light is incident on the surface through the glass and electrons are emitted from the side opposite t h a t from which the light falls. It will be seen from Fig. 4 t h a t the spectral response has a maximum a t a slightly shorter wavelength for the first form of cesium antimony photocathode than for the second. Further. more, the quantum efficiency of the first form is slightly higher than t h a t

-GLASS

3000

4000 WAVE

SPECTRAL

5OOO LENGTH

6000

A

RESPONSE

FIG.4. Spectral response of CssSb photocathodes.

of the second. The quantum efficiency of the metal-backed materia a t its peak response is about 0.3 per cent per microamp per lumen response. Since sensitivities of 40 microamps per lumen or more can be easily achieved with this surface, quantum efficiencies above 10 per cent can be readily obtained. The semi-transparent cesium antimony photocathode has a somewhat lower quantum efficiency, it being 0.2 per cent per microamp per lumen. Thus, with the same sensitivity t o white light, that is 40 microamps per lumen, its quantum efficiency will be about 8 per cent. Obviously, if the wavelength of the incident light does not correspond t o that of the peak response, the quantum efficiency will be lower. The quantum efficiency for any wavelength can be readily determined with the aid of the spectral response curve given in Fig. 4,together with a n equation which takes account of the variation of energy per photon with wavelength. If yx, is the quantum efficiency a t Xo, the wavelength of the peak of the spectral response, the quantum efficiency ^/x a t wsvelength X will be:

where Rx is the response a t wavelength X as indicated by the response

76

G. A. MORTON

curve. A high quantum efficiency is of considerable importance in scintillation counting, as will become more apparent when the statistics of the situation are considered in a later section. The thermionic emission from the photocathode is a factor which cannot be ignored. If the thermionic emission a t room temperature causes a large number of electrons to enter the multiplier structure, these will contribute to spurious pulses in the output, which will form a background that may be objectionable or even intolerable. The thermionic emission from the cesium antimony type of photocathode at a given temperature varies'somewhat with the specific processing which has been

I

I

10

100

I

1000

1

l0,OOO

VOLTS

FIG.5. Secondary emission ratio of CstSb as a function of voltage.

given the material. Measurements on a number of these surfaces indicate that a fair value is 5000 electrons per square centimeter per second a t a temperature of 30°C. The thermionic emission increases rapidly as the temperature is raised, as would be expected from the Richardson equation. As has already been mentioned, the dynode surfaces are sensitized to have a high secondary emission ratio. The material used in this sensitization is essentially the same as that used on the photocathodes, namely, an intermetallic compound of cesium and antimony. The specific method of depositing and activating the material, however, may differ considerably from that employed for the photocathodes. The secondary emission ratio of the material is a function of the velocity of the impinging electrons. Figure 5 shows the variation of the secondary emission ratio of a typical cesium antimony surface as a function of the potential through which the bombarding primary electron falls.


77

This material is found t o be quite satisfactory as a secondary emitter. While the maximum secondary emission ratio is perhaps 25 per cent lower than the best practical secondary emitter known, namely, the cesium-sensitized cesium oxide-silver surface, i t is more stable and also more easily produced. Not only is the material found t o be fairly insensitive t o temperature changes, but also i t is quite stable under electron bombardment as long as current densities of 50 microamperes per square centimeter are not exceeded. At higher current densities there is a falling off of secondary emission ratio as the bombardment continues. Complete recovery from this fatigue effect can be expected if the secondary emissive surface is allowed t o remain a t room 6. diagram of type 931A multitemperature without bombardment for a pe- plier structure. riod of a few minutes t o several hours. Of the multipliers listed in Table I the RC'A 931A, 1P21, 1P22, and 1P28 are identical in their electrode design. The photocathodes in these tubes are metal electrodes followed by nine-stage multiplier units with the dynodes arranged in a circular pattern in order to obtain maximum space utilization. Figure 6 shows schematically the arrangement of this structure. The area of the photocathode in this tube is rather small, and the electrode is mounted well back inside the tube. This makes i t difficult t o obtain good optical coupling between the phosphor crystal and the photocathode. Other characteristics such as uniformity, stability, gain, and life are well suited for scintillation counting. The 931A has a metalbacked cesium-antimony photocathode of the Type S4. The RCA 1P21 is similar t o the 931A, but is processed t o have high gain and stability, low dark current and maximum uniformity. The same type of photocathode is used in the 1P28, but the tube is sealed in a Corning 9741 glass envelope which is transparent t o ultraviolet radiation out t o 2000 angstroms. The RCA 5819 eliminates the objection of the small, inaccessible photocathode mentioned in connection with the 931A. The photocathode is a l+-in. diameter disk forming the glass end of the tube, followed by a ten-stage multiplier structure. The arrangement of the photocathode and dynodes is illustrated in Fig. 7. The electrode structure is the same as that of the 931A, but the surface which in the former is a photocathode is in the 5819 so processed t h a t it can serve as the first dynode. The large end-on photocathode of this tube makes i t possible

@

78

G . A. MORTON

t o obtain good optical coupling with a phosphor crystal and, therefore, this tube has been found t o give excellent results for scintillation counting. The EM1 4588 and 5060 are both venetian blind type multipliers. The 4588 has a large area internal photocathode followed by a nine-stage structure. Because of its size, fairly efficient optical coupling can be obtained with this multiplier in spite of the fact that the photocathode is a n internal electrode. From the characteristics listed in Table I it will be seen that the gain for a given overall voltage is somewhat less than

J

PHOTOCATHODE

I 1 I

1

ACCELERATING

1 ELECTRODES$

'.

\

FIG 7. Schematic diagram of type 5819 multiplier structure. for a focused multiplier. Furthermore, the inefficiency characteristic of this t$ypeof unfocused multiplier makes the statistics slightly poorer. I n spite of the two unfavorable characteristics mentioned, the tube has given very good performance in scintillation counters. The EM1 5060 employs a n on-the-glass photocathode, but the area of this photocathode is only about one twenty-fifth t h a t of the 4588. The small area is t o minimize the thermionic current, and the on-the-glass cathode permits good optical coupling. This tube has also given excellent results. *

111. MULTIPLIER PERFORMANCE I n considering the performance characteristics of a multiplier which makes it suitable for scintillation counting, one must examine the factors which contribute t o obtaining a current pulse output which is most nearly proportional t o the number of photons striking the photocathode; those which permit the fastest rise time and those minimizing the spurious pulse output. The principal cause of the departure of proportionality between the current pulse from the multiplier and the number of photons striking the photocathode is the poor statistics involved. A scintillation causes the release of some small number 6 of electrons from the photocathode. Since the emission is a random process, there is a probable deviation in 6 proportional t o the square root of 6 itself. Therefore, the

* Added

in press.

Additional photomultipliers have been introduced by both

-

RCA and EMI. Some of these are: RCA 6199, like the 5819 but with 135-inch diameter; EM1 6262 approx. 3 in.2 photocathode area, 14 stage, gain lo8.


79

fractional root mean square deviation cannot be less than 1/43, where 6 is the number of electrons entering the multiplier. Furthermore, the whole multiplication process occurring in the multiplier is statistical in its nature which further increases the deviation. As will become increasingly apparent as the discussion proceeds, among the several ways of decreasing the deviation the most effective way of improving the statistics of the situation is by making 6 as large as possible. 1. Photoelectron Collection Eficiency

The two factors which determine the number of electrons entering the multiplier structure for a given number of incident photons on the photocathode are the quantum efficiency of the photocathode and the collection efficiency of the electron optical system which directs the electrons from the cathode on to the first dynode. The quantum efficiency of the photocathode has been discussed briefly in the preceding section. Its improvement rests on the discovery of new materials for sensitizing the photocathode or of new ways of activating known materials. The efficiency of photoelectron collection in a multiplier is primarily a matter of the electrode design. A multiplier which is faulty in this important aspect cannot be made t o perform well, no matter how it is used. . It is possible, however, b y incorrectly using a well-designed multiplier t o spoil the collection efficiency. The two most common causes of poor collection efficiency, where the inefficiency is due t o the manner of usage rather than of multiplier design, are the presence of a magnetic field in the neighborhood of the photocathode or incorrect voltages used on the multiplier. Occasionally a multiplier will incorporate magnetic materiais in its structure, which may become permanently magnetized causing poor collection efficiency even when a n external source of magnetic interference is absent. Such a multiplier can be readily demagnetized by passing it through a coil carrying alternating current (in doing this care should be taken not t o turn off the current through the coil until the multiplier has been moved some distance away from it). Where a multiplier is being used in an experiment demanding a high correlation between the intensity of light flashes and the current output pulses, as is the case for energy spectrometry, etc., it is well t o measure not only the gain and photosensitivity of the multiplier used, but also its collection efficiency. There are a number of procedures for doing this. One which uses essentially the same equipment required for spectrometric measurements is as follows. A small amount of light is allowed t o fall on the photocathode of a multiplier which has a pulse height discriminator and pulse counter as output circuit. An integrated pulse height curve is made of the pulse output by plotting the number of

80

G . A . MORTON

pulses per second n which exceed a given pulse height p as a function of p . Obviously, when this curve is extrapolated to zero pulse height, the pulse rate n,‘ must equal the number of electrons which enter the multiplier. A curve of this type is illustrated in Fig. 8. The number of electrons COUNTING

I

“P

RATE DO00

2

I

PULSE

HEIGHT

3

(ELECTRON

HEIGHTS)

FIQ.8. Photoelectron pulse height distribution.

per second n, which leave the photocathode can be calculated from the output current ioand the gain G of the multiplier by the relation:

The collection efficiency q p will therefore be:

If a given scintillation yields N photons at the photocathode of a multiplier, the number of electrons 6 which enter the multiplier and are useful in determining the size of the output current pulse will be 6

= YVPN

I t is evident that the quantum efficiency y of the photocathode and the collection efficiency q p are equally important in determining the effectiveness of the multiplier. 2. Multiplication Statistics

It was also mentioned that the process of multiplication contributes somewhat to increasing the mean square deviation of pulse height from their expected value. This is because the whole process of secondary emission is statistical in nature. In other words, if the average gain of the multiplier is given as G, this does not mean that every electron


81

passing through the multiplier leads t o exactly G electrons at the output. G is actually only the expected number of electrons and the probable deviation from this value must be calculated. The secondary emission ratio u of a dynode is in itself a statistical quantity and, as has been pointed out, the multiplier simply cascades the values of u. The exact distribution of secondary electrons from a dynode is not known. The distribution is, however, approximately that given by Poisson's law, namely, if u is the secondary emission ratio, the probability of exactly z electrons leaving the surface when it is bombarded by an electron is given by

Nevertheless, there is reason to believe that the actual distribution in the case of real secondary emitters and in particular of dynodes as used in a mulbiplier departs somewhat from this relationship. If a Poisson distribution is assumed, the root mean square deviation from the expected number of electrons z = u will be

49

= d u

I n striking the next dynode each of these electrons produces its secondary electrons in accordance with the same distribution. This type of cascading problem can be handled by means of a special generating function from the theory of probability. The form of this function F ( s ) is such that z = F'(1) AZ2 = F"(1) 4-F(1) - [F'(1)]'

For the multiplier problem the function takes the form for k stages: F(x)

. - fi-l(fi(fi+l

= fO(fl(f2(. m

*

-

. (f&>>>>)>>

2=0

and for the dynodes f l

- .

fk

j,(z>=

2

= e-cesz Z!

2=L)

Forming the generating function as indicated above leads to F ( % ) = e-6e6e-"eue . . . '

-

repeated k timea

82

G. A. MORTON

By differentiation and substituting x

=

z=

1

6Uk

as was t o be expected and

Since k is a relatively large number and moment reduces t o

-

AZ2 - 1

u

> 1 the

fractional second

a

6u-1

2 2

If photoelectrons are released one a t a time from the photocathode (note: this does not mean 6 = 1 but rather -fo = x) the fractional mean square AZ2 1 deviation of pulse heights will be 7= __. a-1 If, on the other hand, the distribution of secondary electrons for a dynode differs somewhat from a Poisson distribution so that for a secondary emission ratio of u the root mean square deviation is & the fractional mean square deviation for a k stage multiplier becomes

z

and for electrons emitted singly from the photocathode:

A22 2 2

-

E

a-1

The relation for 6 electrons from the photocathode is a n important one and will be used when the problem of employing a scintillation counter as an energy spectrometer is considered. A measurement of the distribution curve of the pulse heights obtained from a multiplier can be used t o evaluate e in the relation €/a - 1. Such a curve is made as described in connection with the determination of the photoelectron collection efficiency. For a typical RCA 5819 this type of measurement gives a value E = 1.54. The photoelectron distribution curve is very valuable in appraising the performance of a multiplier in a scintillation counter system. I n addition t o giving a means of evaluating E and q p it permits a rather direct determination of the pulse height produced on the average by an electron entering the multiplier.


83

3. Gain

The gain of a multiplier used for scintillation counting is not a t all critical as long as it remains constant throughout the measurement. The gain should be high enough so that the output pulse can be readily handled by ordinary vacuum tube circuits following the multiplier. This sets a gain of a few hundred thousand as the practical lower limit for a multiplier used for this purpose. If the gain is too high, that is 1 0 9 or more, the output current may become space-charged limited unless special precautions are taken. Multiplication factors lying anywhere between these two extremes are found to be perfectly satisfactory for scintillation counting. Since it is important to know the gain of the multiplier in order t o determine the. statistics of the situation, a word might be said about methods of making gain measurements. Since the ratio of output current to photoelectron current a t the photocathode is very large, it is usually impossible t o make a gain measurement simply by measuring these two currents. There are, however, several ways of getting around this difficulty. One is t o divide the multiplier into two or more sections. For example, a ten-stage multiplier with a gain of a million can be measured by first measuring the input and output current for the first five stages and then the input and the output current of the last five, the total gain being the product of the two values determined. An alternative method is t o first operate the multiplier a t a reduced voltage so that the overall gain is low. Under these conditions the gain of the multiplier can be measured simply by measuring input and output current. Keeping the multiplier voltage constant, the light level is then reduced until the output current is very small. The input current is then calculated. Next, keeping the incident light on the photocathode constant the voltage of the multiplier is raised to its normal value. The output current is then measured and the gain calculated from the ratio of the measured output current to the calculated input current. If the operating gain of a multiplier is extremely high, it may be necessary to carry out this procedure in three steps instead of two. Since the secondary emission ratio is a function of the potential difference through which the primary bombarding electron falls, the overall gain of a multiplier is extremely sensitive to the overall voltage. This can be readily seen from the following equations: a-kV G = uk ,- ( I G V ) ~ AG AV G,-k-v

84

G . A. MORTON

Referring to Fig. 5, it will be seen that the secondary emission ratio varies only a little less rapidly than linearly with bombarding voltages in the range normally used between dynodes of a multiplier. The gain is therefore almost proportional to the lcth power of the voltage where lc is the number of stages of the multiplier. Actual measurements of the variation of gain with voltage for a nine-stage 1P21 shows it to be proportional to the seventh power of the voltage. This means that where a scintillation counter is to be used under conditions where a quantitative measurement is made of the output current pulses, the overall voltage of the multiplier must be maintained at a very constant value.

4. Spurious Pulse Output When a secondary emission photomultiplier is measured in complete d a r k n e s ~ ,a~certain ?~ number of pulses per second will be observed in the output. Some few of these may be due t o cosmic rays reacting directly with the photocathode or other parts of the multiplier, but by far the largest number of the spurious pulses are internally generated. A number of causes may operate to produce the. dark current pulses. All but one of these causes are, however, nonfundamental in nature. Thermionic emission from the photocathode and to a lesser extent from the dynodes will contribute output pulses which are basic to the photomultiplier. For a given type of photocathode they cannot be eliminated except by lowering the temperature of the multiplier. Other nonfundamental causes of dark current pulses in the output are the ionization of any small amount of residual gas left in the tube and field emission from sharp points or edges of electrodes where high gradients may develop. Electrical leakage over insulating surfaces may contribute to a dark current a t the output, but usually does not produce spurious pulses. With a well-manufactured photomultiplier operated with normal voltages, the nonfundamental spurious pulse sources should be almost completely absent. The dark current pulse outputs of an RCA 931A and an RCA 5819 are shown in Fig. 9. The pulse height distribution curve for the 5819 is almost identical with the distribution curves obtained for photoelectrons from the photocathode. This is to be expected, since the large size of the photocathode makes it the major source of electrons producing the dark current pulses. With the 931A where the dynodes have nearly the same area as the photocathode, the form of the distribution curve is noticeably different. The curves shown are plotted in terms of electron heights entering the multiplier. In this form the distributions are essentially independent of voltage, until the voltage is raised to a point where other effects than thermionic emission appear.

85

THE SCIXTILLATION C O U N T E R

Since the primary cause of the dark pulses is thermionic emission, it is obviously strongly effected by the temperature a t which the tube is operated. By cooling the multiplier t o dry ice or liquid air temperature, the dark pulse rate can be reduced t o a very lorn value. I n the neighborhood of room temperature the number of dark pulses per second of any

0

2

4

PULSE

HEIGHT

6

8 (ELECTRCN

10

HEIGHTS]

FIG.9. Spurious pulse output from types 931A and 5819 photomultipliers.

given pulse height doubles with each ten-degree centigrade rise in temperature. 5 . Resolving Time

It has already been pointed out t h a t one of the important attributes of the scintillation counter is its very short resolving time. The limit to the resolving time is determined in part by multiplier performance and in part by the properties of the phosphor crystal employed. As will be shown somewhat later, i t is possible successfully t o use crystals whose flash durations are less than lop8second and where the rise time of the flash is even shorter. A cooled crystal of trans-stilbene is a n example of a phosphor showing this performance. Such time durations are very small indeed, in fact quite comparable

86

G . A. MORTON

with electron transit times. Therefore, i t is necessary t o examine the operation of the multiplier rather critically t o determine its limitations in this respect. When an electron starts from the photocathode of the multiplier and proceeds t o produce generations of secondary electrons which are eventually collected a s a current at the anode, a certain amount of time is required. This transit time itself does not, however, limit the resolving time of the multiplier. If all the electrons took exactly the same length of time in performing this operation, this time would simply constitute a delay which could be measured and allowed for in the experiment. The electrons in traversing the multiplier, however, do not necessarily take exactly the same length of time. The resultant transit time spread is the thing which ultimately limits the resolving time of a multiplier. A number of processes which occur as part of the operation of a secondary emission multiplier may contribute t o the transit time spread. Among the more important ones, which must be examined in detail, are: ( u ) emission time of secondary electrons, ( b ) initial velocity effects, (c) electron trajectory differences, and (d) space charge effects. Considering these in order: The exact emission time for secondary electrons has not yet been determined; however, some recent measurements seem t o indicate time constants as long as 3 X second. Other experiments have indicated t h a t a t least a certain fraction of the secondary electroiis are emitted in times less than second. Secondary electrons are emitted with a rather large range of initial velocities. A good secondary emitter will emit most of its electrons with initial velocities in the range of from 0 t o 3 v. It is quite obvious that an electron emitted with 3 v velocity in the direction of the next dynode mill reach i t before a n electron starting with zero initial velocity. It can be readily shown t h a t the fractional decrease in transit time of the higher velocity electrons over those emitted with zero initial velocity is approximately given by the relation

"=JZ t

Assuming now t h a t the multiplier is operated with 100 v per stage making V o = 100 v, a n electron leaving with a n initial velocity of 3 v directed toward the next dynode will require 17 per cent less time than one leaving with almost no velocity. Since for the type of structure under consideration, the transit time delay is the order of 3 X lop9 second, this decrease in transit time amounts t o about 5 X 10-lO second. Differences in trajectory lengths of electrons leaving various points on the dynode is perhaps the largest factor contributing t o the spread in


87

transit time. The effect of path length differences has been calculated for a typical linear electrostatic multiplier structure of the type illustrated in Fig. 10. Here the time required to traverse the shortest path is found to be 3 X second and the time difference between the longsecond. Including both the effect of est and shortest path 1.7 X initial velocity and the differences in path lengths, the maximum time spread in one stage is the order of 2 X second. The probable spread is then less than but close to half this value. Since the differences in transit times are more or less random in nature, the total spread for a

FIQ.10. Typical linear multiplier structure.

multiplier having k stages is not k times that for a single stage, but more nearly equal to the square root of k times this value. Thus, the transit time spread for a sixteen-stage multiplier of the type shown will be about 4 X 10-9 second. Experimentally with a sixteen-stage multiplier having a structure corresponding to that illustrated and operated under conditions such that the collector current is space charge limited to 35 milliamps, the maximum charge on a 150 micromicrofarad condenser developed by pulses corresponding to single electrons emitted from the photocathode is found to be about 1; v. This means that the time spread for the pulse from a single electron is approximately 5 x second, which is in good agreement with the value arrived a t above. It should be pointed out that for many experiments it is not necessary t o base the time accuracy upon the arrival of the entire charge which results from a single electron a t the photocathode. Frequently the circuitry following the multiplier can be triggered by as little as 10 per

88

G . A. MORTON

cent of the charge from a single electron. When this type of measurement is possible, it obviously leads to a considerable improvement in time resolution. Indeed, almost one order of magnitude in time resolution improvement can be obtained.

IV. PHOSPHOR CRYSTALS A wide variety of phosphorss-'7 are available for scintillation counters. The selection of the particular phosphor to be used depends upon the type and energy of the radiation investigated and upon the particular type of measurement that is being made with the scintillation counter. For alpha particle measurements,S~9the phosphor is generally in the form of a relatively thin screen of powdered fluorescent material mounted on a glass or plastic plate. This screen is mounted as close to the photocathode of the multiplier as possible in order to obtain good optical coupling. Screen thicknesses of only a few milligrams per square centimeter are optimum for this type of measurement, since the penetrating power of alpha particles is rather small. Silver-activated zinc sulfide has been found to be one of the most efficient materials to use for this type of screen. A well-prepared screen of this material will yield one photon per each one hundred electron volts or less energy of the incident particle. This type of phosphor, however, is rather slow, the duration of the scintillation being several tens of microseconds. For most alpha particle measurements this is not a serious objection. Where coincidences between alpha particles or an alpha particle and other types of radiation are being investigated, a zinc sulfide screen may not have adequate resolving time. Zinc oxide has only about 20 per cent the efficiency of zinc sulfide, but the duration of the scintillation is considerably shorter being about one microsecond. For still shorter resolving time the organic crystals may be used, but these are quite inefficient under alpha particle excitation. For gamma and beta rays the problem is somewhat different. Here the radiation is quite penetrating and it is necessary to use a phosphor crystal having a relatively large volume and high optical transparency, so that light generated inside the crystal can escape without too much attentuation. Materials which have been successfully used for scintillation counting of beta and gamma rays are of two classes, namely, inorganic phosphors and organic phosphors. The behavior of these two classes is quite different and probably the mechanisms of fluorescence are fundamentally unlike. All the organic phosphors which have been found satisfactory so far contain conjugated double bonded carbon atoms in benzene rings. The fluorescence is probably due to intramolecular energy transitions. The


89

fluorescent behavior of the organic phosphors is characterized by a very short duration of luminescence, which duration decreases as the temperature is decreased. The efficiency of the release of photons per unit energy of the incident particle is fairly good, but decreases a s the resolving time of the material decreases. One of the interesting properties exhibited by the organic phosphors is that they will form quite efficient mixed crystal phosphors. For example, naphthalene, which in itself is a rather inefficient phosphor, when mixed with 1 per cent or less of anthracene (a good phosphor) and crystallized forms a solid solution which is crystalline and nearly, if not wholly, as efficient as anthracene itself. So many examples are found of this property of forming a n efficient phosphor when a small amount of one material, usually a good phosphor, is dissolved in a large amount of material which in itself is inefficient, t h a t processes which are fundamental t o the basic mechanism of the production of fluorescence in organic materials must be involved. At present, the mechanism of the transfer of energy from the inert solvent t o the active centers of the solute is not understood. One possibility is t h a t ultraviolet radiation serves as the transfer link. Ultraviolet may be produced having a wavelength slightly longer than that of certain characteristic absorption edges of benzene rings of the solvent. The very electronic configuration that makes the solute a good phosphor may shift these characteristic absorption edges t o slightly longer wavelengths. Therefore, the solute strongly absorbs the ultraviolet produced in the solvent, transforming this energy into visible radiation by the fluorescent mechanism in the solute molecules. It is interesting t o note that this property holds for both liquid and solid solutions. I n the case of the inorganic materials the fluorescence is probably the result of transition involving the energy band structure of the crystal and energy levels associated with impurities or imperfection activator centers in the crystal rather than with transitions in individual molecules. This mechanism rules out the possibility of forming the type of mixed crystal phosphors mentioned in connection with the organic materials. The scintillation behavior of inorganic materials is characterized by a fluorescent flash of longer duration than that for organic materials, and the duration of the flash in general increases with decreasing temperature. The efficiency of some of the inorganic phosphors is considerably higher than that of most organic materials known. Furthermore, since some of the inorganic phosphors have a very high density, the net efficiency per volume for converting gamma rays t o visible light photons is great. Table I1 lists a number of phosphors which have been found useful

90

G. A. MORTON

for scintillation counting. These materials are grouped into inorganic phosphors, organic phosphors, mixed crystal phosphors, and liquid phosphors. The most important characteristics of the materials, namely, density, index of refraction, wavelength of radiation, duration of flash, and efficiency, are included in the table. I n giving the efficiencies no attempt is made to place the efficiency of the material on an absolute basis. Rather, the numbers from 1 to 5 are used to order the material in terms of their effectiveness; 1 being the most efficient and 5 the least. As a guide t o the significance of these numbers, materials designated as 1 produce more than 10,000 visible light photons per mev energy of the incident particle. Those numbered 2 produce approximately 10,000 photons per mev; 3 approximately 5000; 4, 2000; and 5, 1000 photons per mev. TABLE11. Crystal Phosphors. Name

ComSpectromposition Density Index eter

Inorganic 6.06 1.934 3.67 1.7745 7.90 2.2-2.3 4.06 1.955 3 . 1 8 1.434 Organic Anthracene 1 . 2 5 1.595 Trans-stilbene 1 . 1 6 1.622 1.23 Terphenyl Naphthalene 1 . 1 5 1.618 M i z e d Organic Naphthalene (anthracene) 1 . 1 5 1.618 Naphthalene (distyryl) 1 . 1 5 1.618 Diphenylene oxide (anthracene) Liquid Phosphors Terphenyl in xylene 0.8641 1.500 Terphenyl in benzene 0.879 1.501 Anthracene in phenyl ether 1.073 1.583

Calcium tungstate Sodium iodide Cadmium tungstate Lithium iodide Fluorite

Caw04 NaI(T1) CdWOh LiI(T1) CaFz

4300 A 4100 5200

Blue ~4000 4400 4100 -4000 -3500 4400 N4500 -4400

-4000 -4000 -4400

Time

>

3 x 10-7 > 10-6 10-6 10-7 3-4 2-1 -2 -5

x x

x x

10-8 10-8 10-8 10-8

4 x 10-8 1-3 X 10-8 -10-8

(fy;oapt)

Efficiency 1 1-2 1 2 3 2 3-4 2-3 5

3 2 3 >4 4

5

A number of the materials listed have such wide usage that they deserve separate mention. Sodium iodide activated with thalliumll is a very effective inorganic phosphor. Its efficiency as a phosphor is good, and the relatively high atomic number of the iodine makes it an excellent absorber of gamma rays. The flash duration of this material at room temperature is the order of 5 X 10-7 second. The material forms very nice transparent crystals; but the crystals are extremely sensitive to moisture. A rela-


91

tively short exposure t o the atmosphere may completely ruin one of these crystals. The instability of the material toward water vapor together with the rather long flash duration somewhat curtails its usefulness. The other alkali iodides activated with thallium are also useful as phosphor crystals. Cadmium tungstateI2 is an efficient phosphor and has a very high density. It can also be made into clear very stable crystals. Howei7er, the flash duration is quite long, being several microseconds so that it is not suited for work where speed is important. Calcium fluoride13 can be obtained in large clear crystals which are extremeIy stable. The flash duration for this material is the order of lo-' second, but the material is not a very efficient phosphor, which limits its usefulness. The organic crystal phosphor, anthracene,14 is probably the most widely used material in scintillation counters for the detection of beta and gamma rays. Although the efficiency of this material is not as good a s some of the inorganic phosphors, it is nevertheless good enough so that the deficiency in this property is not serious. The important feature of the crystal is that its flash duration is quite short, being 3 t o 4 X lo+ second a t room temperature and 1 t o 2 x 10-8 second when cooled t o liquid air temperature. Because of its great usefulness, i t is worth devoting a few paragraphs t o the description of its properties and method of preparation. Anthracene consists of three benzene rings linked together in a straight line. The rings are so joined as t o permit the formation of conjugated double-bonded carbon, which in turn allows the existence of T electrons, that, in accordance with present theories of organic fluorescence, are responsible for the emission of light. The material forms monoclinic crystals having a specific gravity of 1.25. The melting point of these crystals is about 215"C, a temperature which is high enough for most practical applications in scintillation counter work without being so high as t o make crystal synthesis difficult. The fluorescence is in the form of a relatively narrow band having a peak a t 4140 angstroms. The index of refraction of the crystal is about 1.5948. T o prepare a crystal of anthracene, first dissolve 20 g of the material in 100 cc of pure ethylene glycol. The ethylene glycol and anthracene are co-distilled and then precipitated, the anthracene being filtered from the solution. The ethylene glycol is recycled. After this process has been completed, the anthracene is washed in hot distilled water t o remove most of the ethylene glycol. The material is next dried in a vacuum desiccator. I n order t o remove the last traces of ethylene glycol the material should then be melted under vacuum. Next, the anthracene is

92

G . A . MORTON

distilled into the crucible in which the crystal is t o be grown. A crucible for this amount of material might consist of a pyrex tube with a conical tip ending in a small bulb a t the lower end and an entrance tube through which thc material is distilled into the vessel at the upper end. Eyelets or hooks as required t o support the crucible should be sealed near the top of the container. The furnace in which the anthracene is crystallized consists of a cylindrical pyrex or ceramic tube wound with heater wire in such a way t h a t the temperature in the upper half of the furnace is above the melting point of anthracene and there is a gradient of approximately 60°C per inch throughout the length of the furnace, the lower part being held only slightly above room temperature. The windings should be arranged in such a way that the isothermal lines are as flat and uniform as possible. .4n outsidc cylindrical baffle will increase the efficiency of the furnace and decrease its sensitivity t o external convection currents, etc. The crucible containing the anthracene is suspended in the furnace by a flexible wire which is wrapped around a drum that is turned by a slow speed motor mounted above the furnace. The speed of the motor is adjusted so as to lon-er the crucible a t a rate of about & in. per hour. The anthracene will be melted a s long as the crucible is in the upper part of the furnace, but as it is lowered through the 215°C isothermal surface, crystallization takes place starting a t the tip. A small single crystal (to act as a seed) should first form, and then as the crucible continues t o lower, the crystal should grow from the bottom up until the entire charge has crystallized. Occasionally more than one crystal will start a t the tip of the crucible. I t is then necessary t o raise the crystal back t o the upper high temperature part of the furnace and remelt the anthracene. Usually two or three tries are sufficient to obtain a good, single crystal. If the anthracene has been carefully prepared and purified, it should be possible t o obtain water-clear, single crystals of several cubic inches in volume by following this procedure. Trans-stilbene is another organic material which has been found to be very useful for certain types of investigations using scintillation counting. The efficiency of the material is lower than that of anthracene, but its flash duration is shorter, making i t useful where time relationships are t o be measured. The flash duration of the material a t room temperature is about 10-8 second. When the material is cooled t o liquid nitrogen second. temperature, the flash duration becomes less than 5 X Naphthalene itself is a poor phosphor, a t least in the visible region. However, the material may be useful a s the solvent for mixed crystals. With distyryl as a n activator the mixed crystal has an efficiency slightly

T H E SCINTILLATION COUKTER

93

higher than that of anthracene and there is some indication that a crystal grown of the solution may be somewhat more transparent to its own radiation than is a pure anthracene crystal. The flash duration of this mixed crystal lies between that of anthracene and that of trans-stilbene. Finally, i t is somewhat easier t o obtain large clear crystals of this material since the melting point of the solvent material is lower than the melting point of anthracene. As is evident from Table I1 the liquid phosphors are much less efficient than the crystalline material. However, the liquid material can be used in any size or shape desired which is a very great advantage for certain experiments. I n cosmic ray work, for example, where the energy of the incident particles is high so that the question of efficiency is not very important, this class of phosphor has found i m p x t a n t application. If liquid phosphors can be found having higher efficiencies than those known a t present, it may well be that they will replase entirely the crystal phosphors now in use for gamma-ray detection. Because of the necessity of a container, they are not as well suited for beta-particle detection except where the energies involved are extremely high. The detection and measurement of neutrons with scintillation counters is a much more specialized application. However, i t should be mentioned briefly. Where fast neutrons are involved, the organic materials and especially liquid phosphors are quite successful in converting the neutron energy into visible light scintillations. Here the low atomic number material of the phosphor serves to rapidly slow the neutron down absorbing its energy through collisions. These collisions in turn excite the fluorescent centers and give rise t o the emission of visible light photons. For thermal neutrons the situation is somewhat different in that the neutrons themselves do not have sufficient energy t o excite fluorescence. Therefore, the phosphor must incorporate nuclei which will capture neutrons and give up energy in doing so. Phosphors containing boron (10) and indium have both been successfully used for this purpose. Obviously i t would be possible t o incorporate fissionable material in the phosphor crystal and make use of the energy of fission t o excite the material. However, the latter method is a very extravagant way of accomplishing a rather simple end. A good deal remains t o be done in the field of phosphor materials for scintillation counting and active work in many research centers is in progress towards a better solution of this problem. As this work proceeds, marked improvements in available phosphors can confidently be expected.

94

G . A. MORTON

Ti. SCINTILLATIOX COUNTER APPLICATIONS 1 . Radiation Detection and Monitoring

The first application of the scintillation counter that might be considered is t h a t of using the device for detecting nuclear radiation. This may be for the purpose of making surveys for radioactive contamination, for observing tracer materials, the monitoring of nuclear reactors, or similar tasks. An example of this class of survey device which has shown considerable promise is the scintillation counter for alpha-particle detection. The rather low penetrating power of alpha particles makes them rather difficult t o detect with devices depending upon the ionization of a gas. It has been virtually impossible t o make a Geiger counter with thin enough malls t o be of much use for the detection of alpha particles. Proportional counters and ionization chambers can be equipped with windows covered by very thin organic films t o make them suitable for the detection of alpha particles. However, such devices are quite complicated in their electronics and difficult t o maintain in operation. On the other hand, the alpha-particle scintillation detector simply consists of a thin fluorescent screen close t o the photocathode of a multiplier. Where the device is to be used with ambient light, it is necessary t o coat the phosphor with a light-tight film. Such opaque films can be made thin enough so that the air-equivalent range loss of the alpha particles is only a few millimeters. Since in general alpha particles are quite energetic, the flashes produced a t the phosphor are large and consequently large output pulses are obtained. It is, therefore, not difficult t o arrange the circuits which follow the multiplier in such a way that only a few background counts are observed per hour in the absence of alpha-emitting material, and yet every alpha particle that strikes the screen will be counted. An alpha-particle sensitive scintillation counter which is t o be used as a survey instrument should h a r e as large a fluorescent screen as possible because of the very low tolerance level that has been established t o guard against the health hazard of alpha emitters. There is, however, a relationship between the size of the fluorescent screen that can be used and the size of the photocathode of the multiplier. It can be shown that if A , is the area of the fluorescent screen, and R its reflectivity t o its own radiation, and A , the area of the photocathode, the maximum fraction of the light produced by a scintillation that can be collected by the photocathode is given by the relation:


95

Inasmuch a s the value of R rarely exceeds 0.5 it is evident from this equation that, if a large fraction of the light from the scintillation is to be usefully collected, the fluorescent screen and photocathode should have approximately equaI areas. Consequently, a multiplier with a large photocathode is important for an alpha survey instrument. The scintillation counter is very sensitive a s a device for detecting gamma rays. The relatively high density of crystal phosphors makes them quite efficient at intercepting gamma rays and converting their energy t o visible scintillations. Furthermore, crystals can be made which are very transparent so that large volumes of the material may be used without too severe optical losses. Therefore, large crystals in combination with good multipliers can form gamma-ray detectors which have sensitivities th at are orders of magnitude greater than can be achieved with any device employing gas ionization. When such an arrangement is used as a survey instrument or radiation monitor, obviously the background due to cosmic rays and natural radioactivity will increase by the same factor as the radiation to be detected. However, because of the improved statistics thus achieved, a lower intensity of radiation being sought will give a meaningful signal. For example, a Geiger counter instrument with a n integration time of 1 second might have a natural background reading of ten counts per second. The root mean square deviation of this reading will be the square root of 10. Therefore, it would be difficult to detect a n increase of radiation of less than 30 per cent of background. On the other hand, a scintillation counter using a large crystaI might have a natural background count of 1000 counts per second. The statistical fluctuations a t such a counting rate would give a root mean square deviation of about 3 per cent, therefore a change of radiation intensity of a few per cent of background would give a recognizable signal. Thus, such a device would, for example, give warning of a source of radioactivity a t a considerably greater distance than would an instrument based upon a Geiger counter. Where the energy of the radiation to be detected is small and the ambient operating temperature high, it may be necessary t o use a pair of multipliers receiving light from the same crystal and connected in coincidence in such a way th at only when pulses occur a t the output of both multipliers simultaneously will the pulse be registered on the presentation instrument. The random pulses generated as a result of thermionic emission in the two multipliers will only rarely produce accidental coincidences which are counted, thus the multiplier background through such an arrangement is very greatly reduced. When the scintillation counter is used with a simple rate counting circuit, the meter reading is simply proportional to the number of

96

G.

A . MORTON

gamma-ray photons per second, quite independent of their energy SO that such a device does not read in Roentgen units and does not measure the radiation health hazard. More elaborate output circuits, however, can be used to cause the meter to read Roentgen units. Where the survey device is to be used a t fairly high radiation levels it is not necessary to employ a counting technique a t all. Instead, the photomultiplier simply integrates the light from the phosphor crystal and the multiplier output current is a measure of this light. * By properly selecting the crystal (e.g., a crystal composed of atoms having approximately the same atomic number as those of air) such an arrangement can also be made t o read in Roentgen units directly. The scintillation counter also has promise of being a very useful survey type beta-ray detector. It avoids the problem of a thin vacuum window which is a very serious problem with a Geiger counter. For beta-particle detection an organic crystal similar to that used for gammaray detection may be used, but its thickness can be considerably less because of the lower penetrating power of the beta particle. A l-mev beta particle will penetrate only about cm of anthracene. In order to give maximum discrimination against gamma rays the phosphor should be as thin as will give output pulse heights which can be adequately separated from background noise. Again the phosphor crystal must be covered with an opaque coating to exclude ambient light. This film may be much thicker, however, than for alpha particles. I n addition to their application for survey purposes, such devices may be used in connection with beta-particle detection for radioactive tracer work and for such devices as penetration type thickness gages for thin metals or plastics. It is relatively simple to make a scintillation counter which will detect either fast or slow neutrons. In almost all cases, neutrons are only found in the presence of fairly large amounts of gamma rays, and so far no satisfactory way has been devised for discriminating in favor of neutrons in a strong gamma-ray background. However, in certain special devices such as for neutron spectrometers, the scintillation counter has been used quite effectively as a neutron detector.

+

2. Scintillation Counter Spectrometry

It has already been pointed out that the number of photons from a phosphor crystal is proportional to the amount of energy absorbed from the incident nuclear particle. If these photons are directed onto the photocathode of a multiplier they will produce a number of photoelectrons proportional to their number. Consequently, the size of the output pulse from the multiplier will be related to the energy lost by the nuclear particle. Where the particle loses all its energy in the phosphor


97

or a known fraction thereof, obviously the device can be used as an energy spectrometer. However, since all of the processes are statistical, the system must be examined in some detail in order to determine the significance in terms of an energy spectrum of a given pulse height distribution obtained from the multiplier. In the discussion of the statistics of a multiplier, it was shown that if a series of pulses, each one due to the emission of 6 electrons from the photocathode, were examined a t the output of a multiplier, a distribution in pulse heights or amounts of charge per pulse would be found which had a standard deviation A, delta given by:

This means that if in a scintillation counter spectrometer, a sequence of monoenergetic nuclear particles which give up their entire energy to the phosphor crystal and thus each produced the same number of photons were detected, a distribution in pulse heights would be obtained rather than simply a single value of pulse height for the entire series. The position of the maximum of this distribution would correspond to the energy of the particle. Again, if the nuclear radiation whose spectrum is to be analyzed consists of two groups of monoenergetic particles with fairly widely separated energies, the pulse height spectrum obtained from the scintillation counter would be in the form of two separate approximately Gaussian distributions whose maxima would indicate the two energies of the particles being analyzed. When, however, the energies of the incident particles are so close together that the distributions corresponding to these energies overlap, the problem of analyzing the pulse height distribution obtained into the energy spectrum of the incident particles becomes more difficult. I n the most general case the incident particles will have a continuous energy distribution. To analyze this general case assume that the energy distribution curve of the incident particles is represented by G ( p ) in Fig. 11. Here the abscissa p is the number of photoelectrons released (average) at the multiplier by the particle of energy E , with E , = constant p . When such particles are measured with the scintillation counter they will giv? rise to a pulse height distribution curve given by F ( p ' ) shown in the same figure. A consideration of what is taking place in the system will show that the curve G ( p ) and F ( p ' ) are related by the following integral equation: F(P')

=

lo- G ( P ) f ( P , P

- P'WP

Where f(p, p - p') is the multiplier distribution curve for p electrons

98

G . A. MORTON

from the photocathode. Thus, each point on the pulse height distribution curve of the multiplier output is the result of scanning the curve G ( p ) by the distribution f(p, p - p ’ ) with p’ fixed.

PULSE.

HEIGHT

P

ENERGY

(PULSE

HEIGHT)

FIG.11. Representative energy and pulse height distributions.

The function f(p, p - p ’ ) can be fairly closely approximated by the Gaussian distribution: f(p, p - p’)

1 .\/ 2*kp

= ____

e-(p-p”~/2kp

The problem is now to reconstruct G ( p ) from the measured curve F(p’). It is not possible to obtain an exact solution of this integral equation, but by making the appropriate approximations it can be expanded and integrated giving the wanted distribution in terms of the measured pulse height distribution. The relations thus obtained are the following :

kP G(P) = F ( P ) - 4 F”(P) G’(P)

k

=

F’b) - ;iF”(P)

Higher order approximations can, of course, be derived if necessary. These formulas make it possible to compute the energy spectrum of the


99

radiation being observed from the pulse height distribution curve measured with the scintillation counter spectrometer. Beta-ray energy spectra are perhaps the most easily measured with a scintillation detector. Even here, however, it is necessary to take many precautions in order to avoid spurious effects. The sample should be mounted on a thin film of low-density material in order to avoid back scattering. This sample is placed close to the crystal in such a way as to minimize the fraction of beta particles striking the crystal but not giving up their entire energy to it. For example, the beta emitter is frequently

2000

loo0 PULSL

FIG.12. Pulse height distribution of

HEIGHT

Csl37

(redrawn from Nucleonics, ref. 19).

embedded entirely in the crystal, or it is placed between two crystals which avoid a good many of the problems associated with back scattering, reflection of beta particles from the crystal, etc. The beta particles which enter the crystal give up all their energy, to it producing scintillations whose integrated luminosities, i.e., total number of photons, are closely proportional to the energy of the particle. The light thus produced is received by the multiplier photocathode and gives rise to the pulse a t the output. These pulses are examined by means of pulse height discriminators, and a pulse height distribution is obtained. This distribution can be analyzed as outlined above to obtain the energy spectrum. This has been done for a number of beta emitters with very good results. Figures 12 and 13 indicate results

100

G . A. MORTON

obtained in this way by P. R. Bell for cesium (CsL3’),phosphorus and yttrium (ITg1). Figure 12 shows the beta-ray spectrumlg of Cs13’ together with the internal conversion electron a t 630-kev energy. Figure 13 shows a Kurie plot of phosphorus (F) compared with a similar plot’g for Ygl. It will be observed that the P32curve is a straight line on the Kurie plot (except at low energies where instrumental error introduces some curvature),

.5

I. 5

1.0

2.4

MEV

ENERGY

FIG.13. Kurie plots of

P3*

and

Y91

(redrawn from Nucleonics, ref. 19)

indicating that it is an allowed transition, whereas the Ygl plot cannot be represented by a straight line. Correcting for a degree of “forbidness,” the Ygl spectrum can be reduced to a straight line. A number of other substances have been examined, including such very weak beta-ray sources as Belo and K40. Gamma-ray energy spectrum measurements are less straightforward than those of beta rays becauce of the more complicated mechanism involved in the transfer of energy to the crystal phosphor by the incident radiation. At least three mechanisms of energy transfer may be involved in this process. The first is photoelectric conversion where the gamma ray interacts with one of the firmly bound electrons of an atom of the crystal and gives up essentially all its energy to this electron, thus causing it t o be ejected photoelectrically from the atom. With this type of


101

conversion the electron and, hence, the crystal receives essentially all the energy of the gamma particle. The second type of energy conversion is through so-called Compton collisions. Here the incoming gamma ray interacts with an essentially free electron in the crystal in a two-body collision. The gamma-ray energy will be only partially transferred to the electron with which it is interacting, and the fraction of this energy which the electron receives is a matter of probability. Consequentially, the scintillations will not have a luminosity corresponding t o the full energy of the gamma rays. The maximum energy which can be transferred to the crystal through a Compton collision is given by:

The energy transfer to the crystal forms a continuum from this maximum energy down to zero energy according to distribution laws governing the Compton process.

-PAIR

PROD.

ENERGY

FIG.14.

Cross section for absorption for iodine (redrawn from Nucleonics, ref. 20).

Finally, the gamma ray, if it is of sufficiently high energy may interact with the phosphor by producing an electron-positron pair. Here the amount of energy transferred to the crystal will be equal t o that of the gamma ray, minus the energy required for the pair-formation (approximately 1 mev). Figure 14 gives the theoretical cross-section curves for the three con-

102

G . A. MORTON

version processes in iodine of sodium iodide as computed by Hofstadter and McIntyre.20 Based on these curves, Figs. 15 and 16 show the theoretical pulse distribution expected for 0.51- and 2.04-mev particles

0.51 MEV

I

RAY

\

L.22 10

2

4

6

PULSE

8

HEIGHT

FIG. 15. Calculated pulse height distribution for a 0.51-mev gamma ray in NaI(T1) (redrawn from Nucleonics, ref. 20).

PULSE

HEIGHT

FIG. 16. Calculated pulse height distribution for a 2.04-mev gamma ray in NaI(T1) (redrawn from Nucleonics, ref. 20).

as computed by the above authors. Measurements made on the gamma rays from a number of materials, such as Au19*,Na24,and Ga66,confirmed these results. It will be evident from the above that the interpretation of pulse heights distributions obtained from a complicated gamma-ray spectrum

103


may be difficult indeed. It is fortunate, however, that gamma spectra are generally in the form of the discrete lines rather than a continuous distribution of energy as with beta rays. Therefore, unless the line structure of the spectrum being analyzed is extremely complicated, it is usually possible t o obtain an interpretation from the measured pulse heights distribution curves. An interesting procedure for using the Compton effect in measuring gamma-ray spectra was worked out by Hofstadter.21 It consists of arranging two scintillation counters, A and B, as shown in Fig. 17, in PHOTOMULTIPLIER

/

RECOIL ELECTRON

Fro. 17. Two-crystal gamma-ray spectrometer.

such a way t ha t their outputs are in coincidence, and only if a pulse occurs in counter B will the pulse be recorded from counter A . It can be shown from the Compton interaction equations th at if e is the angle formed at counter A between the incident gamma ray and the Compton scattered gamma ray 'which reached counter B, the following energy relations must be fulfilled:

E

=

' l1 4- [' 4- E,,,.

- E ,, 2

2mc2 hav

e

I")

If the angle 0 is kept constant, the energy given t o the Compton electron in crystal A will be proportional to the energy of the incident gamma ray. Therefore, a pulse height distribution from counter A for pulses which occur only when the scattered gamma ray reaches B will be a measure of the energy spectrum of the source. Figure 18 illustrates the spectrums obtained from ( 2 0 6 0 (1.17- and 1.33-mev gamma ray) as measured in this way. 3. Time Measurements I n addition to the example cited above, the coincidence method in general is a very important adjunct to the scintillation counter technique. It makes possible the reduction of spurious signals due t o multiplier

104

G . A. MORTON

dark current pulses. It provides means of distinguishing between different kinds of nuclear phenomena. It permits taking advantage of the extremely short resolving time capabilities of the device for the determination of chronological relationships between nuclear events, and can aid in many other ways. A large number of circuits have been developed for using scintillation counters in coincidence. Figure 19 illustrates a typical circuit using three

PULSE

HEIGHT

(VOLTS)

d B

FIG.18. Energy spectrum of C o ' O (redrawn from Phys. Rev.,:ef. a ' A

21).

CRYSTAL CHAVACTERISTICS

ZP' d

B

d'r2a

a

2a

+ DYNOOE

COLLECTOR COLLECTOR

DYNODE

FIG.19. Three-crystal diode coincidence circuit.


105

crystal diodes22in such a way that only when pulses appear from the two multipliers simultaneously does a pulse reach the presentation device. The coincidence method, employing a circuit with a delay line in one of its branches has been used in connection with the decay of short-lived isomers. The problem here is to determine the half-life of radioactive daughters of an initially radioactive atom. The arrangement of the apparatus, as used by DeBendetti, McGowan, and is illustrated CRYSTALS

I

AMPLIFIER

I COINCIDENCE

I REGISTER DELAY

FIG.20. Schematic diagram of delayed coincidence system

in Fig. 20. The material being investigated is placed between two scintillation counter chains. One includes a delay element and is arranged to respond to the nuclear particle released by the initial transition. The second chain responds to the radiation released by the daughter atom. When a source having only a single transition is placed in the apparatus, the delayed coincidence measurements give only a very small constant background due to accidental delayed coincidences. This is illustrated in the dotted curve of Fig. 20. On the other hand, when there is a double transition, the second having a half-life of the same order of magnitude as the transfer time of the delay element, the delayed coincidence gives a straight line on a semi-logarithm plot. The slope of the line is a measure of the half-life of the isomer. Some examples of the half-lives found for such isomers are given below: Ga72

8-

Ge72*

Y

___ -+

14 hr

2.9 x 10-7

P-

Y

sec Ge72

Yb177 --+ Lu177* A sec Lu177 1.8 hr 1.3 x 10-7 B- Au'97* Y ---f sec Aulg7 Hg197 2.3 hr 7 x 10-9 I

~

106

G . A . MORTON

I n order t o use the coincidence method to reduce the thermionic background of secondary emission multipliers two multipliers are arranged so they look a t a single phosphor crystal. The scintillation flash excites photoelectrons in both multipliers simultaneously and, consequently, gives a recordable pulse through the coincidence circuit. The thermionic emission, on the other hand, being a random process, produces only a relatively few number of accidental coincidences. By making the resolving time of the system short, that is comparable with the flash duration of the phosphor, the normal thermionic background of a multiplier can be reduced to a matter of a few counts per minute or even a few counts per hour. This is very useful when weak radioactive sources are being studied. I n the above article it has not been possible to cover completely either the operation of a scintillation counter or its many applications in full detail. However, the discussion may serve to indicate the general principles of the scintillation counter, its limitations, and its uses in nuclear physics. With the rapidly widening usage of the device and the continued study of its operation, it can be confidently expected that many improvements will be made in the performance, reliability, and applicability of the scintillation counter. However, even without these improvements the scintillation counter represents one of the most powerful detection means available to the nuclear physicists. REFERENCES 1. Kallmann, H. Natur u. Tech., July, 1947. 2. Coltman, J. W., and Marshall, F. H. Photomultiplier Radiation Detector. Phys. Rev., 72, No. 6, 528 (1947). 3. Zworykin, V. K., Morton, G. A., and Malter, L. The Secondary Emission Multiplier-A New Electronic Device. Proc. Inst. Radio Engrs., 24, No. 3, 351 (1936). 4. Rajchman, J. A., and Snyder, R. L. An Electrostatically Focused Multiplier Phototube. Electronics, 13, 20 (1940). 5. Engstrom, R. W. Multiplier Phototube Characteristics: Application to Low Light Levels. J. Optical SOC.Am., 37, 420 (1947). 6. Morton, G. A. Photomultipliers for Scintillation Counting. R C A Rev., 10, No. 4, 525 (1949). 7. Morton, G. A,, and Mitchell, J. A. Performance of 931-A Type Multiplier in a Scintillation Counter. R C A Rev., 9, No. 4, 632 (1948). 8. Sherr, R. Scintillation Counter for the Detection of Alpha Particles. Rev. Sei. Instruments, 18, 767 (1947). 9. Graves, J. D., and Dyson, J. P. Scintillation Counter for Laboratory Counting of Alpha Particles. Rev. Sn'. Instruments, 20, 560 (1949). 10. Properties of Scintillation Materials. Nucleonics, 6, No. 5, 70 (1950). 11. Hofstadter, R. The Detection of Gamma-Rays with Thallium Activated Sodium Iodide Crystals. Phys. Rev., 7 6 , 796 (1949). 12. Gillette, R. H. Calcium and Cadmium Tungstate as Scintillation Counter Crystals for Gamma-Ray Detection. Rev. Sci. Instruments, 21, 294 (1950).


107

13. MacIntyre, W. J. Decay of Scintillations in Calcium Fluoride Crystals. Phys. Rev., 76, 1439 (1949). 14. Bell, P. R. The Use of Anthracene as a Scintillation Counter. Phys. Rev., 73, 1405 (1948). 15. Feazel, C. E., and Smith, C. D. Production of Large Crystals of Naphthalene and Anthracene. Rev. Sci. Instruments, 19, 817 (1948). 16. Gettings, H. T., et al. Relative Sensitivities of Some Organic Compounds for Scintillation Counting. Phys. Rev., 76, 205 (1949). 17. Hofstadter, R., Liebson, S. H., and Elliott, J. 0. Terphenyl and Dibenzyl Scintillation Counters. Phys. Rev., 78, 81 (1950). .18. Reynolds, G. T. Liquid Scintillation Counters. Nucleonics, 6 , NO.5 , 68 (1950). 19. Jordan, W. H., and Bell, P. R. Scintillation Counters. ATucleonics, 6 , NO.4, 30 (1949). 20. Hofstadter, R., and McIntyre, J. A. Gamma-Ray Spectroscopy with Crystals of NaI(T1). Nucleonics, 7 , No. 3, 32 (1950). 21. Hofstadter, R., and McIntyre, J. A. Measurement of Gamma-Ray Energies with Two Crystals in Coincidence. Phys. Rev., 78, 619 (1950). 22. Morton, 0.A., and Robinson, K. W. A Coincidence Scintillation Counter. Nucleonics, 4, No. 2, 25 (1949). 23. DeBenedetti, et al. Self-Delayed Coincidences with Scintillation Counters. Phys. Rev., 73, 140 (1948).


Fluctuation Phenomena ALDERT VAN DER ZIEL Department of Electrical Engineering. Institute of Technology. University of Minnesota. Minnea.polis, Minnesota CONTENTS

Page 110 1 Correlation . . . . . . . . . . . . . ......................... 110 112 2 . Fourier Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Fourier Analysis of Fluctuating Quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 1. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2 . Methods of Calculation of the Fourie; Spectrum. . . . . . . . . . . . . . . . . . . . . 114 a . From Statistical Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 b . F r o m t h e Time Average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 c . From the Fourier Analysis of a Single Elementary E v e n t . , . . . . . . . . . 115 d . From x ( t ) X ( t + w ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 I11. Application to Various Noise Generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 117 1. Thermal Noise Generators . . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 2 . Shot Noise in Diodes and Triodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 a . Noise in Diodes at Low Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b . Noise in Triodes at Low Frequencies. . . . . . . . . . . . . . . . . . . . . . . . . . 123 3 . Noise in Diodes and Triodes at High Frequenci . . . . . . . . . . . . . . . . . . . . 124 a . Noise in Saturated Diodes at High Frequencies . . . . . . . . . . . . . . . . . b . Exponential Part of the Characteristic . Total Emission Noise ... c . High-Frequency Noise in Space-Charge Limited Diodes . . . . . . . . . . . . 126 129 d Noise in Triodes a t High Frequencies., . . . . . . . . . . . . . . . . . . . . . . . . . . . e . Noise in Electron Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 f. Induced Grid Noise in Triodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g. Discussion of the Results of the Last Two Sections . . . . . . . . . . . . . . . . . 132 4 . Partition Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 133 a . Partition Noise in Pentodes and Hexodes., . . . . . . . . . . . . . . . . . . . . . 134 b . Induced Grid Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Secondary Emission Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6. Noise in Gas Discharge Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7 . Noise in Mixer Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a . Triode, Pentode, and Hexode Mixers . . . . . . . . . . . . . . . . . . b . Diode Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 c. Deflection Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 139 8 . Noise in Photocells and Photomultipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 9 Shot Noise in Semi-conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 10. Flicker Noise in Cathodes and Semi-Conductors . . . . . . . . . . . . . . . . . . . . . 11 Noise in Crystal Diodes and Transistors . . . . . . . . . . . . . . . 109

I . Introduction . . . . . . . . . . . . . .

.

.

. .

.........................

110

ALDERT VAN DER ZIEL

Page IV. Noise in Receivers.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 ..................................... 147 1. Noise Figure . . . . . . . . . . . 2. Noise Figures of Various rcuits.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 a. Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 b. Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 150 c. Noise in Triode and Pentode Circuits.. . . . . . . . . . . . . . . . . . . . . . . . . . . . d. Diode Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 References. .. ................................. . . . 153

I. INTRODUCTION The random motion of the free electrons in a conductor gives rise to a fluctuating voltage across its terminals; this effect is known as thermal noise. The random emission of electrons by the cathode of a vacuum tube gives rise to fluctuations in the tube current; this effect is known as shot noise. The name “spontaneous fluctuations of electricity ” would be the most appropriate one. However, in view of their acoustical effects, these fluctuations are now generally known under the name “noise.” Noise sets a very serious limit to the detection of all kinds of weak signals.1l In the discussion of some noise problems, an extensive mathematical analysis2~12-15-is cannot be avoided. In many cases, however, a simple Fourier analysis is sufficient. It is the aim of this paper to give a simple discussion of some important noise problems. In the measurement of noise, one takes the average value of the noise power over a sufficiently long time. In the calculations, however, we shall take average values over a large number of identical systems subjected to independent fluctuations (ensemble), in order to avoid confusion. Both methods give identical results. An average value will be denoted by . A quantity X ( t ) will be called a fluctuating quantity in this paper if X ( t ) = 0 but Xz(t) # 0. If X(t) # 0, one might better introduce [ X ( t )- X(l)] as the fluctuating quantity. It may in general be assumed that Xz(t> taken over an ensemble is independent of time. For if the elements of the ensemble all started at t = 0 with X ( t ) = XO,then x”(t)= Xo2 at t = to. Independent of the initial value Xo2, the expression Xz(t> will approach a stationary value X 2 for sufficiently large values of t ; X 2 is usually the only item of interest. 1. Correlation In our discussion of noise problems, we have to introduce the concept of correlation between fluctuating quantities.I7 Let x and y be two fluctuating quantities such that Z = Q = 0, then the correlation coefieient c is defined as:

FLUCTUATION PHENOMENA

111

= 3 . = 0 and c = 0; in that case If x and y are independent then the quantities are said to be uncorrelated. If # 0 the coefficient c is a good measure for the dependence of the one quantity upon the other one. If 0 < [cl < 1 then the quantities are said to be partially correlated; if Icl = 1, the quantities are said to be completely correlated. The most important case in noise problems is the case of linear correlation (at any rate as long as linear circuit theory can be applied) :

xy

y where a is a constant and

2

=

ax

+ z,

(1.2)

is uncorrelated with x. Then:

which shows that IcI 5 1. Correlation is very important if two fluctuating quantities x and y have to be added and the mean square value is taken:

c = 0 (no correlation) means that the quantities have to be added quadratically; - (1.4a) (29 y)Z = 2 2 y2,

+

+

]cI = 1 (complete correlation) means that the quantities have to be added linearly; = 2 5 2 dS*.y’i+ y2= (dz& dz)2. (I.4b) In many cases the correlation between the values of a certain fluctuating quantity X ( t ) a t different instants is also of interest.17 We shall often use correlation coefficients c(w) of the type:

+

c(w) = X ( t ) X ( t w)/X2(t); c ( - w ) = c(w), (1.5) c(w) is equal to unity a t w = 0 (normalization) and decreases more or less rapidly for w > 0. The decrease can usually be characterized by a single correlation time T O (for exceptions compare section 111.10) such that c(w) = 0 for w >> T O . In the case of fluctuating currents in tubes, T O is usually equal to the transit time of the electrons, for fluctuating signals across tuned circuits TOis equal to the time constant of the circuit. * * c ( - w ) = c(w) because X(u)X(u’) = X(u’)X(u). Putting u‘ = u + w, we have:

+

X ( u ) X ( u w ) = X(u’)X(u’ - w) = X(u)X(u- w), since an average value, taken over an ensemble, does not depend upon u.

112

ALDERT VAN DER ZIEL

2. Fourier Analysis

The most convenient way of calculating mean square values of fluctuating quantities is the method of Fourier analysis. I n order to show that it is permissible to use this method consider a fluctuating quantity Y ( t ) of a certain system (e.g., the noise output voltage of an amplifier) which is caused by a fluctuating quantity X ( t ) (e.g., the noise of the first input circuit or the noise of the first tube). X ( t ) is developed into a Fourier series for 0 < t < To,the sum X l ( t ) of this series is equal to X ( t ) for 0 < t < T o but differs from X ( t ) outside that interval. Though the values of X ( t ) for t < 0 may have some influence upon the values of Y ( t ) for t > 0, this influence will have died out after a certain time T I , depending upon the time constant of the system. Hence it is allowed t o use the Fourier representation X , ( t ) of X ( t ) for the interval - 00 < t < T o in order to calculate the values of Y ( t ) and of for 7'1 < t < T O . As is independent of time this will give the right value of For that reason i t i s allowed to apply a-c circuit analysis to noise problems in tubes and circuits and to apply the laws of vacuum tube electronics to noise problems in radio tubes.

m.

11. FOURIER ANALYSIS OF FLUCTUATING QUANTITIES

I. Theory16.16 Let X ( t ) be a fluctuating quantity which is continuous except in a To IX(u)Idu finite number of points of the interval 0 < t < T osuch that exists; then it is allowed to develop X ( t ) into a Fourier series for the time interval 0 < t < T o and the sum of this series is equal to X ( t ) for that time interval if X ( t ) is properly defined a t the discontinuities:*

/o

n=-

(D

f

To

(11.2)

The Fourier component x, for the frequency w,, is: Z, = a,eiw*t + a e - i w d

(11.3) * Sometimes discontinuous fluctuating quantities are introduced in the theoretical

discussion of a problem, b u t they are never measured as such; as every physical instrument has a finite rise time, these discontinuities will always be smoothed out. A perfect example is the convection current at a certain point in a radio tube; it is infinite if an electron passes that point and zero at all other instants. However, the fluctuating currents in the outer circuit are perfectly continuous.

113


and its mean square value taken over a n ensemble is:

=

/

4Af

X(u)X(u

0

=

+

W)

cos wnw dw

F(fniAf,

(11.4)

where Af = 1/To and (u'- u) = w.* Equation (11.4) shows that the Fourier spectrum F ( f n ) of X ( t ) is: F(fn>

= 4

lom

x(~>x(U

+ w ) cos wnw dw.

(11.5)

Having calculated the mean square values of the Fourier coefficients of X ( t ) , we can now draw the following conclusions: a. If 7 0 is the correlation time, then the Fourier spectrum F ( f n ) is independent of frequency if wnr0 3r. This means that the initial conY ection current fluctuations will then have disappeared.

131


The small residual amount of noise in the minima was found t o be due t o the interception of a small fraction of the electron beam by the cavity (partition noise). (2) T h e theory gives the right magnitude of the noise. The only discrepancy was that the theoretical curve had t o be shifted somewhat in order to obtain the best agreement with experiment. This small discrepancy is thought t o be due t o the use of (111.35) and (111.36), otherwise these equations seem t o be the right starting point. 10

I

I

I

I

11

w u)

9 12

5In

13

3 14

9

15

m

a

z 16 5 17 z

0

a in

; rJ

19

20 0

2 3 4 DISTANCE F R O M ANODE IN I N C H E S

5

6

FIG.3. Results of Cutler’s experiment on the noise power versus distance along a n electron beam. The distance is measured from the anode of the electron gun producing the beam. (Courtesy of Dr. Cutler.)

f. Induced Grid Noise in Triodes.20#60 Turning back t o triodes we observe that the noise current il in the cathode lead and the noise current iz in the anode lead are completely correlated. But as they have a phase difference, a current (il - i,) has t o flow t o the grid. Using (111.30) and (111.34) and employing a series expansion of the various functions we obtain for moderately high frequencies: (il

- iz) =

io(+jjw~1

+

g j j w ~ 2 ) ;(il

- i2)2

+

= Z2&o2(~1

2 ~ 2 ) ~ .(111.37)

The induced grid noise shouId therefore be proportional t o u 2 in a wide frequency range. This was verified experimentally by Bakker;,O i t cannot be considered as a verification of the above theory, however, as i t follows from very general considerations. Bellzza has expressed (il - iz)2 in terms of the increase AC in input capacity and van der Ziel

132

ALDERT VAN DER ZIEL

has given a n extension of his result with the help of the LlewellynPeterson equations.90a The induced grid noise should be completely correlated with tube noise. Strutt and van der %elsoshowed that under those conditions the induced grid noise might be used in order t o eliminate part, if not all, tube noise by properly detuning the input circuit. Kleen48proved that some noise reduction could be obtained in this way, whereas van der Ziel and Versnela6 showed t h a t a t least part of the induced grid noise was correlated with the tube noise and that a t 7-m wavelength this part had a phase difference of 90 degrees with respect t o the tube noise, as required by (111.37). Recent experiments carried out by the authorgob showed that for 654 and 6AC7 tubes (the latter used as a triode) a major part of the induced grid noise (about two-thirds of it) was not correlated t o tube noise. This is a serious difficulty of the above theory, but i t explains why the noise reducing schemes mentioned previously did not eliminate more tube noise. g. Discussion of the Results of the Last Two Sections. Due t o the occurrence of this uncorrelated part of the induced grid noise it does not seem t o be worth while t o investigate whether (111.37) agrees numerically with experiment as it was derived under the assumption of a complete correlation. The question is now where this uncorrelated part comes from. Most uncorrelated noise can be accounted for by the fact that the electrons can pass the grid wires at various distances.22 The shape of the current pulses flowing t o the grid will then vary a t random, which gjves rise t o an uncorrelated part of the induced grid noise. Part of the uncorrelated noise can be explained by tube inhomogeneit i e s g 7 If il = il’ illf and i2 = i,’ i2“ where i,’ and i,” and also iz‘ and i2” are uncorrelated, but il’and i2’ and also i,” and iz” are completely correlated, then il and iz are no longer completely correlated in general. It is also likely that part of the uncorrelated induced grid noise is caused by the fact that the three assumptions of Sec. 1 1 1 . 3 ~upon which the theory is based are rather crude, especially for tube structures in which the distance between cathode and potential minimum is a n appreciable fraction of the cathode-grid distance, as is true in many modern tubes. I n t h a t case it is understandable that at least part of the induced grid noise might not be correlated t o the tube noise, as one would not expect the theory t o hold well in that instance. On the other hand, this lack of correlation should not affect the beam experiment. For even if the convection noise current close t o the potential minimum were not correlated at all t o the velocity fluctuations, then

+

+

133


(111.35) and (111.36) would still be valid, provided that ur1 i&sufficiently large. For in that case the original convection current would become negligible at the anode. 4, Partition Noise

a. Partition Noise in Pentodes and Hexodes. I n studying the motion of electrons in multi-grid tubes one usually finds the following condition to prevail. Whether an electron arrives a t the anode or at the screen grids is compfetely random, mainly due to the random velocity component parallel to the cathode and the absence of space-charge effects outside the cathode-grid region. Due to the random distribution of the electrons over the various positive electrodes a new type of noise, partition noise, occurs. In the same notation as before we have for a pentode, if 2 is the noise in the cathode lead : ',I2

Af;

-

iC2= 2e(Ia

+ I2)rC2Af,

(111.38)

The interpretation of these equations is as follows. We split the instantaneous cathode current into a constant part and a fluctuating part i,. The part I z / ( I u I,) of the fluctuation will go to the screen grid, and the part Ia/(Ia 12)to the anode. This gives the first term in the equations. The constant current is divided a t random between screen grid and anode, it gives rise to a fluctuating current i, flowing from the screen grid to the anode (partition noise) ; this accounts for the second term in the equations.* The partition noise was treated simultaneously by Schottky74and Bakker18 and independently by North.Kg According to the above theory the noise currents in the screen grid lead and the anode lead are correlated. By feeding back the right part of the noise in the screen grid lead into the input circuit in the proper phase it is possible to get rid of the partition noise. This was first demonstrated by Strutt and van der !Gelsoand verified experimentally by

+ +

* The magnitude of 2 follows from the following theorem. Let certain similar independent events have the probability X to occur in the form A and the probability (1 - A) to occur in the form R, such that the form of an individual event is completely random. Counting n events, n1 are found to be of the form A and n2 of the form B, such that:

n1

= An;

122

=

(1 - X)n; (nl - ii,)'

=

(n, -

?&)2

= X(1

- X)n.

134

ALDERT VAN DER ZIEL

I n a hexode a similar theory can be applied.5g Noise is generated here a t the cathode, the first screen grid and the second screen grid. The ' partition noise generated at the second screen grid flows to the anode, the partition noise generated a t the first screen grid is properly distributed between the second screen grid and the anode and the shot noise generated a t the cathode is distributed between the various positive electrodes a t the proper rate. It is then found that the expression (111.39) for 2 also holds for a hexode, provided that I 2 represents the current flowing to the positive grids. The more positive electrodes are inrerted between cathode and anode, the more current will be intercepted and the larger the noise; for that reason the noise resistance of a hexode is usually much larger than for a pentode. Equation (111.39) can also be applied to those pentagrids in which the anode current is not influenced by the space charge in front of the second control grid. I n some cases, however, the space charge in front of the second control grid limits and controls the anode current; the current fluctuations will thus be partly suppressed by the space charge and 2 may be much smaller than would follow from (111.39). We obtain therefore for the noise resistance R, of a pentode (or hexode), if gm is the transconductance of the anode current and e afactor of 2.5-4 (Sec. 111.2a), according t o (111.39) and (111.13) : (111.40) This formula shows that partition noise is especially important for tubes with a large ratio 1 2 / g , . b. Induced Grid Noise. Due to the fact that a fluctuating current is flowing through the second control grid of a hexode or pentagrid, induced grid noise occurs. Some measurements have been carried out by Bakker,20from which it seems to follow that induced grid noise is much larger for these tubes than for a pentode. There are two reasons for it: (1) the transit times are usually much longer; (2) the tube current a t the second control grid is usually much noisier as it contains a large amount of partition noise. A good theoretical discussion of this kind of induced grid noise is still lacking for most hexode and pentagrid tubes; Bakker's theory is correct for the type of tube he measured. 5 . Secondary Emission N o i ~ e ' ~ . ~ ~

Consider a secondary emission cathode for which on the average each primary electron liberates 6 secondaries. If every primary liberated exactly 6 secondaries then no extra noise would result, but as the number

135


of secondaries per primary is fluctuating, a new type of noise, usually called secondary emission noise arises. Let Bn be the probability that a primary electron liberates n electrons. Then :

the latter equation defines a quantity K which will be used below. Suppose the primary beam is saturated and the primary current is I,. Then the part Ipn= /?,Ip of this primary current will liberate n electrons per primary. I,, shows fluctuations; if all the secondaries are collected a t the anode:

2

=

2

-

(2eIpnAf) n2 = 2eI,~6Af;

(111.42)

n=O

Zieglerg3measured K and 6 for various secondary emitters. K is usually 30 to 50 per cent larger than 6. If every primary had liberated 6 secondaries we would have had:

-

ia2= 2eIPa2Aj, so that the part:

-

',i

(III.42a)

= 2 e I , ( ~- S)SAj

(111.43)

is realIy the noise due to the fluctuations in secondary emission. If the primary current is not saturated we obtain:

-

-

+ 2 = 2e1,6(rP26 +

- 6)Af (111.44) is the primary current fluctuation and rP2 the noise suppression i,' = i p 2 6'-

K

where ?a factor of the primary current.

6. Noise in Gas Discharge Tubes

I n radio tubes containing small traces of gas the noise is found to be several times higher than would be expected according to the preceding theory. The reason for this large increase in noise is understood as follows. In an evacuated triode the noise is partially suppressed by space charge and? = 2eI,r24f where r2,the noise suppression factor, may have a value of 0.10 or less. If some gas is present, so that there is a small probability p that a primary electron will ionize a gas molecule, then this in itself would give a contribution 2eIapAf to 2. I n addition, however, the positive ions moving through the potential minimum with a very low speed (much smaller than of the average velocity of the electrons in the potential minimum) give rise to a large temporary increase in

136

ALDERT VAN DER ZIEL

anode current as they reduce the space charge in the potential minimum (Fig. 1). This increases the original current fluctuation SI due to the ionization by a large factor m, so that the actual contribution to ? becomes 2eI,pm2Af, this can be much larger than the shot noise current 2eI,I’*Af, even for small values of p . For a more exact theory compare Thompson and North’s paper.g2 Gas discharge tubes have been used recently as microwave noise standard^.^' The discharge tubes wbich are rather thin are usually inserted in a wave guide such that their axis makes an angle of about 10 degrees with the axis of the guide, this provides about a perfect match over a rather wide range of temperatures and frequencies. The “noise temperature ” of the gas discharge is approximately equal t o the “electron temperature” of the discharge. Noise ratios around 16 db are common; this property makes these discharge tubes valuable for measuring noise figures (Sec. IV) of 20 db or less. The fact that the noise temperature of the discharge is about equal to the electron temperature T,can be easily understood in the following way. The positive ions move so slowly that their influence can be neglected. The plasma electrons in the discharge can be considered as a free electron gas of temperature T. except for the convection current I flowing in the direction of the axis of the tube. The presence of the gas ions makes itself felt only in the fact that the free electrons suffer collisions. Let the average time between two collisions be r ; during two consecutive collisions the electrons move in a straight path and give rise to a current pulse in the outer circuit as discussed previously. If an electric field is applied to the gas, the electron gas is found to have a conddctivity G ( w ) . The random motion of the plasma electrons constitutes thermal noise for which Nyquist’s theorem holds (noise temperature T,). Superimposed upon this random motion is a forced motion of the electrons in the direction of the axis of the tube. The randomness in the free path length which is traveled between two collisions gives rise to a shot noise current i in the direction of the tube which has to be added quadratically t o the thermal noise. But as this shot noise current flows in a direction making an angle e with the direction of the E vector of the wave guide, the current in the direction of the E vector is i cos 0. The shot noise current can be calculated in a way as shown in Sec. 11.2d. We quote here the result obtained by Parzen and Goldstein‘j2for the available noise power P, of a gas discharge tube:

Pa = kT,Af

+3 cos2 0 (1 + N

2 +

w2T2)

Af

(111.45)


137

where N is the number of free electrons in the plasma and P o the d-c power dissipated in the tube. The second term in (111.45) is usually only a few per cent of the first one especially because in practical applications of gas discharge noise standards e is about 80 degrees.

7. Noise in Mixer Tubes a. Triode, Pentode, and Hexode mixer^.^^,^^*^^^ I n mixer tubes the anode current changes in the rhythm of the local oscillator voltage and may be zero during part of the cycle. I n order to find the i-f noise current in the output of the tube one has to take the average value of the noise over one complete cycle of the local oscillator voltage. Denoting this average by ( )*,, we have for a triode mixer in which local oscillator voltage and signal are both applied to the grid, if gmis the instantaneous transconductance of the tube :

-

(ia2)Av =

e

*

4ikT(gm)~vAf,

(111.46)

as gm is the only quantity which changes during a cycle. For a pentode and hexode mixer, things are a little more complicated. Let I,, Ia,and I2 denote the instantaneous cathode current, the instantaneous anode current, and the instantaneous screen grid current, respectively, and let gm denote the instantaneous transconductance of the cathode current; I , = (Ia I 2 ) of course. We then have for a pentode mixer in which the local oscillator voltage and the signal voltage are both applied to the grid, according to (111.39) and (111.13):

+

@)av

= e

*

4kT(gm)~,X2Af -k 2e(Ic)AvX(l- X)Af,

(111.47)

where = I a / I C . I n this case only I , and gmchange during the cycle but remains constant. In a hexode mixer, however, in which the local oscillator voltage is applied to the second control gird, I , and gm remain practically constant, but X changes in the rhythm of the local oscillator voltage. Hence:

-

(ia2)dV

=

€ ’

+

4kTgm(h2)rdf 2eIC(X(1- X)jAvA.f.

(111.48)

If g,, is the conversion transconductance, one can introduce the noise resistance of the tube according to the definition: (23.4,

= 4kTRnAf * gmc2.

(111.49)

For large oscillator voltages (gm)Av = g,, and gmcis about equal to one-fourth of the maximum transconductance so that the noise resistance of a triode mixer is about four times the noise resistance of a corresponding h-f triode under the most favorable circumstances. For hexodes values for R, of 100,000-200,000 ohms are common. Much larger values

138

ALDERT VAN DER ZIEL

occur for small local oscillator voltages because gmc becomes so small in that case. b. Diode Mixers.s An interesting noise compensation occurs in a diode mixer circuit.39 Let the input circuit have a conductivity Gi and let the output be short-circuited. Let w i be the input frequency, wm the intermediate frequency, and let v h cos be the local oscillator voltage. As the transconductance gm is a periodic function with period Wh, we develop it into a Fourier series: gm

=

90

+ 2gmc cos + uht

*

*

*

(111.50)

1

where g o = ( g m ) A v and gmcis the conversion transconductance. Let a small fluctuation peak in the diode current occur a t t = T . Developing this fluctuation peak into a Fourier series we obtain an h-f component a cos w i ( t - T ) and an i-f component u cos wm(t - T ) . If the input were also short-circuited, the latter would be the total i-f noise current. But if the input circuit has a conductivity Gi, an h-f voltage will be developed across the input and if wm = (oi - W h ) it will give rise to an i-f current: Fc

u cos [ ~ m ( t- T )

+

@h7];

Fc

=

gmc/(Gi

90).

(111.51)

Combining the two i-f currents, the square of the amplitude becomes: U2(1 - 2Fc COS WhT

+ Fez)

(111.52)

instead of a2. But according to (111.13) the instantaneous value of u2 a t a certain instant T is proportional to the instantaneous value of gma t that instant. Using (111.13) and substituting t = T into (111.50), the i-f noise is found by taking the average of (111.52) over a cycle:

whereas it would be equal to e .4kTg&f if the input were short-circuited, Therefore a compensation takes place, which is practically a complete one if gmc = go and Gi v appreciably with surface treatment (sand blasting, presumably due to the changes in surface recombination) or by changing the bias voltage. V>

=

V Z i


145

The single crystals used by Montgomery and Shockley were electron conductors, but some holes with short life time ( = 10W sec) were present which generated the excess noise. The observations fitted best with the idea of lumped sources of holes. * These measurements were carried out a t audio frequencies. Herzog and van der Ziel measured the noise ratio n of such single crystals over a wider range of frequencies and varying currents and found that the experiments could be described by a formula :44a

n

=

1

+ A/f + B/(1 + w 2 ~ 1 2 )+ C/(l + w27z2)

(111.62)

with T~ = 10-6 sec and 7 2 much smaller; A , B, and C were proportional to 12. The first term is due t o thermal noise, the second to the excess noise mentioned above, the third term is ascribed t o the shot effect of the holes causing the excess noise (because the value of T~ corresponds with the life time of the holes) and the fourth term is ascribed to the shot effect of the electrons. 1 1 . Noise i n Crystal Diodes and Transistors

A marked difference exists between the noise in the backward and in the forward direction of a crystal diode. I n the forward direction there is some Flicker effect a t lower frequencies, but a t higher frequencies the noise often remains more or less constant; this constant part might be interpreted as shot noise. In the backward direction the noise decreases with increasing frequency even above 30 megacycles; it might be identified largely as Flicker effect, though some shot noise will be present. A good summary of earlier work is found in Torrey and Whitmer, Crystal rectifier^.^ As a typical example of recent measurements we give some data obtained by Kno1.t Knol measured the ratio I d / I in 1N34 crystals and the ratio T , / T between 1 and 15 megacycles with I as a parameter; T is room temperature, T , the equivalent noise temperature of the differential resistance R = d V / d I , and I d is the equivalent saturated diode current of a crystal carrying a d-c current I ; T , / T = 20RId of course. I n the forward direction Id/I waa practically constant between 1 and 15 megacycles. Values for Ia/I which were much smaller than unity were observed, especially for currents between 1 and 5 ma. I n an ordinary diode one would call this “space charge suppression of shot noise,” *T.R.E. Conference on electron tubes and solid state devices. Ann Arbor, Michigan, June, 1950. t Unpublished. I am indebted to Dr. Knol, N. V. Philips’ Gloeilampen-fabricken, Eindhoven, HoIland, for allowing me to publish these data.

146

ALDERT VAN DER ZIEL

but in this case at least part (if not all) of it is due to a feedback effect caused by the bulk resistance of the germanium.9 I n the backward direction the noise was strongly frequency dependent, and Id/I was always much larger than unity; Id/I varied somewhat slower than f-' at room temperature and somewhat faster than f-' at -70°C. In the forward direction Ia/I decreased with decreasing temperature; in the backward direction Id/I increased with decreasing temperature, especially a t the longer wavelengths. T,/T was found t o be rather high in the backward direction; quite apart from the Flicker effect this would be expected even for pure shot effect, because the differential resistance R is so large in that case. T,/T = 1 for zero current of course; with increasing current, T,/T goes

F

FIG.4. Equivalent network for the noise of a transitor; Z,, Zb, Zc, and Z, are the characteristic impedances of the transitor; V , represents the emitter noise emf, V , the collector noise emf, i, is the emitter noise current.

through a minimum which may be as low as 0.50 and then increases again. A value T,/T = 4 is not surprising in the absence of Flicker effect, for the characteristic in the forward direction is exponential for low voltages and in a diode with an exponential characteristic T J T , was also found to be (Section III.2a). We mentioned Knolls results about the frequency dependence of the Flicker noise. Other measurements seem to indicate a f-' frequency d e ~ e n d e n c ewhereas ,~~ Mooer~ found ~ ~ ~several cases in which the Flicker noise in crystal diodes varied as f-36 at low frequencies and as f-34 a t higher frequencies. This shows that different crystal diodes have widely different noise properties. Noise in transistor^^^ is closely related to noise in crystal diodes, because a transistor can be considered as a crystal diode with two point contacts instead of one. In an N type transistor, the input contact, which is called the emitter ( e ) , is positive with respect to the base ( b ) ; the output contact, which is called the collector ( c ) , is negative with respect to the base. An equivalent circuit of a transistor is represented in


147

Fig. 4. The noise has to be represented by an input noise emf v6 and an output noise emf v c ; both are partly correlated. This correlation is due to the conduction mechanism in a transistor, the current from the emitter largely consists of holes; a large part of this hole current flows to the collector and modulates the current voltage characteristic of the collector. Both C and 2 vary about inversely proportional to the frequency, and 2 is usually much smaller than 2.. It is common to characterize the noisiness of a transistor by mentioning its noise figure a t 1000 cycles. A noise figure of 50 db a t 1000 cycles is not uncommon. This huge amount of noise is mainly due to 2, it increases if the collector voltage is made more negative and it also depends slightly upon the emitter current I,. The fact that 2 is so large is in agreement with the fact that a crystal diode operated in the back direction a t rather large currents (0.5-1 ma) gives rise to a huge amount of Flicker noise a t 1000 cycles. N-P-N junction transistors have a much lower noise figure.

IV. NOISE IN RECEIVERS

I. Noise Figure6,sVLo Just as the noise ratio n characterizes the noisiness of a noise generator, so the noise figure F of a receiver characterizes the noisiness of a ‘ receiver.33,35s4**s0It is defined as follows. Let the actual antenna of the receiver be replaced by a dummy antenna (noise ratio of 1) kept a t room temperature. The noise output power of the receiver is such as if the receiver itself were noiseless and the dummy antenna had a noise ratio F.* If the actual antenna has a noise ratio n,, then the eflective noise figure Feffof the receiver13 is defined in analogy with the noise figure F ; the only difference is that the words “dummy antenna” have to be replaced by “actual antenna.”t Obviously : (IV.1) F.fr = n, (F - 1).

+

Noise figures are often expressed in db. ; the reason is that in comparing two receivers, the ratio of their noise figures is the determining factor. If several amplifier stages are connected in cascade, having noise figures F1, F z , Fa, etc., respectively, then the noise figure F of the whole amplifier can be determined by a method first given by F r i i ~ . ~ ~

* Another definition is: F is the ratio of the available noise output power of the receiver to t h a t part which is due to the thermal noise of the dummy antenna. t An ideal receiver is one for which F = 1. Usually, one tends to design a receiver such that F is a minimum. If n, > ( F - l ) , then F,rf = n,; in that case, very little can be gained by decreasing F . This is the case for good receivers in the 5- to 30megacycle region, as n, is very large in that region.13

148

ALDERT VAN DER ZIEL

I n order to discuss this method, the circuit elements belonging to the antenna, to the first stage, to the second stage, etc., must first be defined properly. The dummy antenna is considered all by itself. The coupling elements which couple the dummy antenna to the first stage are considered to belong to the first stage; the coupling elements which couple the output electrode of the first stage to the input of the second stage are considered to belong t o the second stage, etc. We can now define the noise figure F of the first stage for that particular coupling between antenna and input, which is actually used. One can also define the available power gain 91, for that particular coupling as: 91

=

Available signal power a t output electrode Available signal power at the antenna

The first stage as seen from its output electrode has an internal resistance

R 1and is coupled t o the input of the second stage. The noise figure F z of the second stage is defined for that particular coupling. The definition of gz for that particular coupling is similar to the definition of gl, as is the output resistance Rz of the second stage. We are now able to calculate the noise figure F of a number of stages. Looking a t the output of the first stage, the noise ratio of that output is: n1 = F l g l .

(IV.2)

This means that in analogy to (IV.l), the noise of the first two stages combined is such as if the output of the first stage had a noise ratio n =

n1

+ ( F z - 1) = F l g l + (Fz- 1).

(IV.3)

In analogy to (IV.2), we have for the noise figure of the two combined stages : F = n/gi = Fi (Fz - l)/gi. (IV.4)

+

In an analogous way, we have for more stages in cascade Friis' formula:

This formula holds as long as the available gain can be defined for each of the stages, that is as long as R1, R2,etc., are positive.* We see that F = F1if g1 is sufficiently large. In other cases, the noise of the second stage is also important.

* Formula (IV.4) shows t h a t it is sense t o use an h-f stage of noise factor F 1 in front of a receiver of noise figure F t if F < F f . Obviously this means F1 < Fz;but there is also a condition for the available gain g1 of the stage as F < Fs means: F1

+ ( F z - l)/gl

< F z or g1 > (Fz - 1 ) / ( F 2 - PI).

If g1 < 1 the stage cannot serve any useful purpose.


149

This is especially the case with radar receivers having a crystal mixer as their first stage. The “gain” of the mixer stage is actually a power loss ( g l < l), the noise figure of the receiver is determined by the output noise ratio n of the crystal mixer, the power gain g 1 of the mixer stage and the noise figure Fz of the first i-f stage. Combining (IV.4) and (IV.3), we have therefore for the noise figure F of the radar receiver9 (IV.5) 2. Noise Figures of Various Circuits

a. Measurement. The noise ratio n of a noise source can be measured by comparing its noise output to the output from a known noise source.8s As the noise figure F of a receiver is in fact a noise ratio, F can be measured in a similar way. Saturated diodes are often used as a standard noise may be used as well. ~ o u r c ebut , ~ hot ~ ~ ~ ~ ~or gas discharge One way is t o compare the noise output power of a receiver a dummy antenna of noise ratio 1 to the noise output powtx of receiver dummy antenna of known noise ratio nu (e.g., gas discharge tube). If the measured noise powers are Po and PI, respectively, then according to (IV.1):

+

+

+ F - 1)/F

(nu

=

P l / P o or F

=

(nu - l ) P 1 / ( P 1- PO). (IV.6)

For the best accuracy, nu and F should have the same order of magnitude. If a noise diode is used in parallel with the dummy resistance R , the saturated diode current is changed from zero t o such a value Id that the output noise power of the receiver is doubled. Then:*

F

=

(e/2kT)IdR = 2QIdR.

(IV.7)

I n a wide-band receiver, one has t o distinguish between the “overall” noise figure F’,8 which is the value measured in the above way and the noise figure F for a small region of the amplified frequency band. If one wants t o measure F , one has t o use the wide-band amplifier as a preamplifier for a narrow-band receiver; tuning the latter t o the various frequencies gives F as a function of frequency. The available gain g also depends upon frequency; the relation between F’ and F ( f ) is:

F’

k- g ( f M h=F ( f ) g ( f ) d f . =

(IV.8)

If g ( f ) is constant inside a band of width B and zero outside t h a t band, then F’ corresponds to the average value of F ( f ) taken over the band.

* According

to the definition of F :

2eZ&fR2 = F . 4 k T R A f ; F

=

(e/2kT)ZdR.

150

ALDERT VAN DER ZIEL

F‘ generally depends upon bandwidth whereas F does not. Strictly speaking, the discussion of Sec. IV.l holds for F and not for F‘. If the bandwidth of the input circuit of the first amplifier stage is equal t o or larger than the overall bandwidth of the amplifier, F’ and F will differ very little. I t is often useful t o measure F as a function of the transformed anteima resistance R, in order t o find the most suitable In t h a t case, one may connect the dummy transformed antenna resistance R, and the noise diode right across the input circuit of the amplifier. One usually finds, experimentally :

F

=

A

+ B / R , + CR,,

(IV.9)

which has a minimum value:

+- 1/m

F,,, = A 2 for R, = 2/B/C. (IV.10) If a wide band is required, it is often better to choose a value of R, which is a few times smaller than d / B / C ; as the minimum in (IV.9) is rather flat, this does not increase F very much, and i t results in a flatter frequency response. Another way to niden the bandwidth of the input circuit is to use an input band-pass filter; it does not improve the noise figure, hoivever. 5iib I n all measurements, it is advisable to check the linearity of the amplifier with a noise diode and to make sure that the detecting instrument really measures power. -1crystal diode acts as a quadratic detector as long as the detected current does not escrecl a few microamperes. 0. C a l ~ z i l a f i o n . ~I n~ order to calculate the noise figure of an amplifier stage. one considers the output of the stage to be short-circuited* and determines the contributions of antenna noise, tuned circuit noise, induced grid noise, and tube noise to the mean square value ? of the noise current in the short-circuited output lead. Let those contributions be a?,b2, c 2 , and d2, then according to the definition of F : t F

= (a2

+ b2 + c2 + d’)/a2.

(IV.11)

One usually finds a theoretical formula of the type (IV.9). c. Noise in Triode and Peirtode Circuits. 3 . 6 , 1 0 I n low-noise amplifiers, one often uses a neut,ralized triode or a grounded grid circuit as a first stage. A favored combination is a neutralized triode followed by a * T h i s does not alter the signal-to-noise ratio of the stage, as the signal-noise uol/aye ratio at the o\itput terminals is equal to the signal-noise current ratio in the lead short-circuiting the terminals. This short circuit eliminatcs the feedback and thus simplifies the calciilations. iBecause of the correlated part of induced grid noise, it is riot always allowed to add c arid d quadratically (compare Sec. III.3f).

151


grounded grid.g1a I n the first circuit, the anode impedance is kept lo^ and the anode grid capacity is tuned, both in order to reduce feedback. I n the grounded grid circuit, the cathode is used as the input electrode; i t has a low input impedance ( l / g m ) and a low anode-cathode capacity (especially if some screening is applied) so that i t is perfectly stable too. It is best suited for u-h-f disk-seal triodes. Though the grounded grid triode has a lorn input impedance, both circuits have identical noise figures under identical conditions. *9 This is due t o a feedback effect;s7 the tube noise flows through the input circuit as well as through the output circuit.* If the constants A , B, and C of (IV.9) are calculated for a grounded grid circuit, one finds :

A

=

1

4-2Rn/R1; B

as usually (R,/R1)2

=

(e/ZkT)Id 4-Rn/RI2

=

iveis usually the greatest, for the space-charge cloud acts like a n expanding metallic conductor as the anode voltage is increased. This produces a large increase in the resonant wavelength of the reactance section. From

MODULATION O F CONTINUOUS-WAVE MAGNETRONS

205

Fig. 10, er is negative, and operation is carried out far from the perturbations found near the cyclotron frequency) when the ratio of the operating angular frequency t o the cyclotron angular frequency ( w f / w c ) is less than 0.36. This corresponds to a high magnetic field, well above that which could make the operating frequency equal t o the cyclotron frequency. The increase in Xo shown below 450 v in Fig. 9 is due t o a negative value of er. Welch and Brewer16have confirmed the predictions based upon Fig. 10 by hot-resonance measurements on a magnetron. The radius of the

FIG.10. Dielectric constant of magnetron space charge as a function of w ~ / w c .

space-charge cloud is constant for a constant ratio of anode voltage t o the square of the magnetic field. For a small value of E,/B2, B can be made quite high before oscillation starts, so that w f / w C can be made less than 0.36. As B is reduced, with E,/B2 maintained constant, E, is increasingly far from the values necessary for oscillation, SO that high values of w f / w e can be investigated. The experimental values of Xo varied with B in a manner to be expected from Fig. 10. 3. Experimental Applications

In general, two separate anode structures, within the same envelope and electrically connected) have been used for frequency modulation. One of the most successful forms of the tube, designed for a power output of about 500 watts at a wavelength of 13 cm, is shown in Fig. 11.16 The coaxial cavity was one-half wavelength long and loaded a t its two highvoltage points by vanes protruding from the outer conductor through slots in the center conductor. The cathodes of the two interdigital structures were held a t different dc potentials so that the potential of the modulator

206

J. S. DONAL, JR.

cathode could be varied. This resulted in a variation in the diameter of the space-charge cloud in the modulator section and, therefore, in the capacitance presented by this cloud. Low values of w , / w ~ were used so that the expanding cloud increased the resonant wavelength of the system. In order to obtain a useful change in capacitance it was necessary t o provide for a large increase in cloud diameter before the anode voltage attained a value resulting in oscillation of the modulator section. Welchg

FIG.11. Oscillating and modulating structures enclosed in a single envelope.

has shown that this end can be accomplished by using fewer anode segments in the modulator section without using impractically low ratios of anode to cathode radii. For a tube of the type shown in Fig. 11, the expected changes in the diameter of the modulator space-charge sheath were duplicated by metal cylinders inserted in a model of the tube. The results predicted a change in the operating frequency of about 2.5 per cent. In a tube oscillating a t 13 cm a change in anode voltage of the modulator section from zero to about 500 Y resulted in a frequency change of about 1 per cent.

4. Evaluation This work is still in progress and has not yet reached the point at which experimental measurements of frequency modulation have been


207

obtained in quantity. Several other tube structures are under investigation. The theoretical treatments in the reports already issued have in themselves been a contribution to the art. The method is potentially capable of producing very large frequency changes but, as the workers a t the University of Michigan have pointed out, there may be serious losses in the space charge which could result in associated amplitude modulation. In the experimental results so far obtained, some current has been drawn t o the modulator anodes. It will be necessary to obtain frequency modulation without drawing excessive anode current in order t o limit the required modulating power. IV. VOLTAGE TUNING

It will be remembered from the Introduction that the frequency of oscillation of a normal magnetron is determined by the susceptances of the electron stream, of t%e tank circuit, and of the effective load. The resonant frequency of the tank circuit is usually the determining factor as regards the frequency of operation, so that the latter varies a few per cent a t most from the cold-resonant frequency. Figure 9 illustrated this fact for a representative tube. After oscillation begins, the power output of a conventional tube increases very rapidly, while the frequency increases perhaps 1 per cent of the carrier frequency. It is well known that the pushing, or change of frequency with voltage or current, can be increased markedly by increasing the loading on the magnetron, but even under this condition the total increase in frequency, from the start of oscillation to maximum safe power output, is seldom more than a few per cent. Wilbur and Peters’? have found, however, that under certain special conditions the change in frequency with anode voltage may be as much as a factor of 4 t o 1. The effect was observed in split anode magnetrons and tubes of similar structure. The necessary conditions for “voltage tuning,” as the above effect has been called, are ( a ) extremely heavy loading of the magnetron and ( b ) cathode emission partially or completely temperature limited. With these limitations it would appear that the conditions in the electron stream, rather than resonance in the tank circuit, are the factors determining the frequency of operation. The pushing, in megacycles per volt, is several times that of conventional tubes. More important, however, is the fact that while a relatively small increase in voltage will draw a current resulting in the destruction of a conventional tube, under the conditions of voltage tuning the anode voltage may be increased several fold before dissipation limits are exceeded. The power output and efficiency are comparatively low, compared t o that expected for a magnetron of similar structure operated under normal conditions, particularly

208

J. S. DONAL, JR.

if it is desired t h a t the power output remain relatively constant in order t o avoid amplitude variations when the frequency is varied. 1. Experimental Results

All the results reported by Wilbur and Peters were obtained with magnetrons generally similar t o split anode or interdigital tubes, in which leads extending through the envelope wall permitted very heavy loading. The operating frequencies were in the range below 1000 megacycles. One group of tubes investigated had output powers from 10 to 200 watts. v)

5 V 2 0

W

I

750-

{ 160

650 700

I

$

600-

Z

3

0

1400

O 1600

1

' " 2000 2200 ANODE VOLTAGE

I800

'

~ 2400

~ 2600

0 ~

FIG.12. Voltage tuning of a 4-vane tropotron.

A s an example, the results obtained with a four-vane "tropotron" are shown in Fig. 12. Two different values of external loading are indicated by the subscripts L I and Lz. For the first loading (LI),in particular, the frequency was nearly doubled. The power output was a more sensitive function of loading than was the frequency variation. The power output was rather low in the range in which i t was reasonably independent of voltage. A large number of miniature magnetrons were tested by Wilbur and Peters. These split anode and interdigital tubes were capable of producing up t o 1 or 2 watts of output power when operated a t normal loading and with space-charge ,limited emission. With the heavy loading and temperature-limited emission required for voltage tuning, the power output was only a few milliwatts, but the frequency could be varied by

~

~

MODULATION OF CONTINUOUS-WAVE MAGNETRONS

factors of a t least four t o one. had a low noise factor.

209

The tubes were stable in operation and

2. Discussion of Voltage Tuning

I n spite of the reduced efficiency and power output of the tubes, voltage tuning may come t o be of very considerable importance as a means of obtaining frequency modulation. Certainly little is known a t this time of the mechanism behind the effect,and much further investigation will be required t o see if useful radiated power is obtainable under the required loading conditions. With the support of the Signal Corps, a comprehensive study of voltage tuning has been started at the University of Michigan. ‘ The object of the program is the development of useful voltage-tuned tubes at frequencies of several thousand megacycles. H. W. Welch, Jr., and his co-workers have pointed out12 several unusual properties of voltage tuning in addition to the heavy loading, low efficiency, and temperaturelimited emission noted above. For example, a t the very low Q used it is possible that Q’s of the same order in the transmission line t o the load will result in difficulties due to long line effect. Since the low Q must increase the circuit efficiency, the observed decrease in overall efficiency must be due to a marked decrease in electronic efficiency. The decrease in maximum-current boundary, usually associated with 1ow-Q operation, is not found under the conditions used for voltage tuning. Finally, voltage tuning is essentially the usual pushing enhanced by the heavy loading. The Michigan work has been initiated only recently and has SO far been devoted to a general study of pushing, current boundaries, 1ow-Q operation and space-charge theory of the magnetron. By assuming that the space-charge swarm is a modified constantcurrent generator and by considering the variation of the total conductance and susceptance of the tank circuit and load as a function of the change of frequency with plate voltage, it has already been possible to calculate a pushing characteristic in reasonable agreement with experimental results. The qualitative description of pushing is perhaps of more interest here. Between the voltage a t which the electron angular velocity becomes synchronous with the rotating electromagnetic wave and the voltage producing oscillation, the space-charge spokes expand in constant phase with respect to the wave on the anode. A spoke passes under an anode segment when the voltage on this segment is a maximum, but the current induced in the anode has already passed through a maximum and is zero a t this instant. The spoke thus contributes a loading current and increases the effective system capacitance,

210

J. S. DONAL, JR.

with a reduction in resonant frequency. As oscillation builds up, power must be contributed b y the electron stream. There must be a component of current in phase with the rf field. To accomplish this end, the spacecharge spokes advance in phase, and the induced current advances in phase with respect t o the rf field. This reduces the capacitative effect of the spokes and the resonant wavelength of the system decreases (Fig. 9) t o yield the usual pushing phenomenon. By application, t o the assumed space-charge distribution, of methods for calculating induced currents, i t is expected that this qualitative picture can be placed on a quantitative footing so that pushing under conditions of very heavy loading can be better understood. At the upper current boundary a magnetron either stops oscillating or shifts modes. It is rather strange that the effect does not appear to be unduly serious under the conditions of voltage tuning, although heavy loading reduces the boundary current in a tube operated under normal conditions. Welch has pointed out that the current boundary is affected by both cathode and space-charge limitations, by a space-charge density insufficient t o induce adequate current in the anode structure, b y transittime limitation of current through the spokes, and by inadequate phase focusing a t very high currents. Under normal operating conditions the maximum current is usually, but not always, increased by a n increase in cathode temperature. An explanation must be found for the elevated current boundary observed by the General Electric workers when the Q is low and the emission is limited. A tube has been constructed12 in which the resonant cavity is outside of a glass envelope containing a n interdigital anode structure. This will permit gross changes in the cavity and loading for a study of 1ow-Q operation. Fundamental studies of magnetron space charge, as a n aid t o a n understanding of voltage tuning, are being carried on with a diode magnetron, termed a “trajectron.” With this tube the electron trajectories in the space charge will be studied in a manner similar to that employed by Reverdin and Marton.lgab A beam of electrons will be passed through the diode in a n axial direction. Except for the axial velocity, the electrons in the beam may be expected t o have the same motion as electrons emitted from the cathode. Thus the beam displacement, observed on a fluorescent screen, should duplicate the radial motion of emitted electrons during the time of transit of the beam through the interaction space. S. Evaluation To date, voltage tuning has been restricted t o special types of tubes and t o low frequencies. It has not been demonstrated that power can


211

be delivered t o a useful load without practical difficulties arising from long-line effects. The method is of such potential value, however, that i t merits the above description of the initial phase of the Michigan work on the subject. The problems that must be solved are interesting because they require a study of precisely those magnetron characteristics which are the most complex and, heretofore, the most neglected.

V. AMPLITUDEMODULATION USINGABSORPTION BY A SPIRAL ELECTRON BEAM The method for frequency modulation described in See. I1 used a beam of electrons projected through a resonant cavity which might be a cavity of the magnetron, for example. Equation (10) gave both components of the resulting electronic admittance in terms of e = ( B e / m - W ) T , and Fig. 4 showed that the conductance term, that producing a change in the Q of the resonant circuit, is a maximum a t e = 0 while the reactive effects vanish. A method of amplitude modulation by absorption has been developed which makes use of these properties of the spiral beam.20 Beams placed within a magnetron for purposes of frequency modulation could be used t o produce pure amplitude modulation. This is inadvisable for two reasons, however. First, the adjustment of 0 t o zero means that w = w,, or that the magnetic field applied t o the beams is that for which the carrier frequency is cyclotron frequency. Without a special structure, this same magnetic field is applied t o the magnetron, and it is too low for good efficiency in the case of most tubes. Secondly, absorption of power by the beams results in a higher value of GT in ( l ) , in a higher G,, and, usually, in a higher electronic efficiency. Thus, the power developed in the tube is increased, which tends t o cancel the power reduction in the load and t o reduce the range of amplitude modulation for a given change in beam current. For these reasons the absorption beam is placed in a n external cavity coupled t o the magnetron by means of a circuit described below. i. Attainable Electronic Conductance and Power Absorption Using (10) and Fig. 4 the electronic conductance presented by the beam a t 0 = 0 is L2 I0 G,(O) = - - * 8d2 Vo Since r

=

L/(2e/mVo)s6,the conductance may be written

G,(O)

=

e I. r2 - - . -. md2 4

212

J. S. DONAL, JR.

The form of the envelope of the beam for 0 = 0 is shown in Fig. 13. An instantaneous view would show the electrons lined up along the directrix of a cone or, in the practical case, forming a pencil beam the size of the cathode. Equations (15) and (16) are independent of the rf voltage between the plates, but if this is so high t h a t electrons strike the pole faces, r and G,(O) become functions of Vfi. The power absorbed is a function of Vrt under normal nongrazing conditions, for

P

1 v2 Vd2Ge = - . - . -L2 I 0 watts. 2 16 d2 Vo

=-

If Vrt and V o are both several hundred volts and L/d is about 5, values within the usual ranges employed, a n ampere of beam current will

BEAM AT ANY INSTANT

FIG.13. Envelope of a n absorption beam.

absorb several hundred watts. From (15), which is in practical units, an ampere of beam current a t a beam potential, Vo, of 1000 and L / d = 5 indicates a conductance of 0.003 mho. This is transformed by the loop coupling the external absorption tube t o the transmission line t o a much higher conductance presented t o the line. With the usual coupling loop the corresponding resistance presented t o the line would be a few ohms, so that the modulation possibilities are extremely good. The efficiency of absorption by electrons in a spiral beam can be considerably greater than is the case for a triode, for example. If each


2 13

of the N electrons per second in a beam absorbs a n energy of $mu2where v is the velocity of collection, i t can be shown that the energy absorbed by each electron is P I N and, using (17),

The energy absorbed by each electron in a triode is simply (V,,)e, but the term in brackets in (18) can be made substantially larger than unity. The absorption efficiency of the spiral beam is the higher of the two by this factor.

CATHODE

-T UT

POLE FACES OF L E N G T H " ~ ' FIG. 14. Placement of absorption-beam cathode.

Just as was the case for the frequency change in (11) and (12) of See. 11, L / d or T cannot be decreased indefinitely in (15) or (16), for collection of electrons will occur and G will not be a linear function of 10. I n the rf circuit t o be described, a t least, the fact that r becomes a variable after collection occurs results in a very poor modulation characteristic. The rotational energy absorbed by each electron, proportional t o wc2r2 may be used t o express the total power absorbed, and using (17) the maximum radius attained by the electrons, which determines the power absorbed, is

If the poles are spaced apart by d and the cathode is a strip of width t, rmaris limited t o $(d - t ) . (See Fig. 14.) The optimum design of the absorption tube is obtained by a series of simple approximations. One starts with a n rf voltage determined from circuit considerations, a maximum allowable cathode current density

214

J. S. DONAL, JR.

with an assumed pole spacing, pole length, and cathode size, and with the maximum allowable cathode current. The beam voltage is then calculated from relations such as those of Haeff.21 This gives the conductance or power absorbed, from (15) or (17), but the r,,, of (19) must be less than +(d - t ) . If r,,, can be increased, the length (and transit time) can be increased t o increase G, but if r,,, is too large, the length and the resulting G must be decreased. If the value of G calculated is unsatisfactory, the dimension of the cathode (if it is a strip) parallel t o the faces of the poles may be increased, the type of cathode may be changed or structures may be used in parallel.

FIG.1.5. RF circuit used for absorption modulation.

2. Example of a n RF Circuit for Magnetron Modulation Figure 15 shows a circuit which has been used for absorption modulation of a magnetron. It is basically the circuit used earlier by Parkerzz for absorption modulation with triodes. The power output of the magnetrons used n as found t o be closely proportional t o the conductance ( G L L ) , presented t o the tube, to a power u = 0.33. A 400-ohm shunt load prevented complete unloading of the magnetron and possible poor spectrum. The conductance, G,, presented by the absorption tube is the electronic conductance after transformation by the coupling loop. Keglecting losses and the shunt load, i t is obvious that with the absorption beam off the main transmission line is short circuited, which reduces the power in the load t o zero. The magnetron efficiency is reduced by reducing its loading. Conversely, a very high beam conductance would make GJIrvery lo\\ and would present oiily GL t o the magnetron. This would make the poner in the load a maximum. A straightforward circuit analysis yields the powers of Fig. 16, plotted in terms of their ratios t o the matched-load poncr output of the magnetron. From a krtou-ledge of the transformed beam conductance, as a function of beam current or control-grid voltage, the system performance can be determined.


215

3. Predicted Modulation Characteristics

Assuming a 400-ohm shunt load and a 50-ohm useful load it can be shown from a n analysis of the circuit of Fig. 15 th a t the rf voltage across the line outside the absorption tube would be (200P,,)55 where Po, is the matched load output of the magnetron, if a = 0, i.e., if there were no losses in the absorption tube with the beam cut off.' For a 1-kw magnetron, for example, such voltages would be expected to result in electron

a=& = ABSORPTION GL

TU B E CONDUCTANCE CONDUCTANCE OF U S E F U L L O A D

FIG.16. Variation of RF powers in absorption system.

collection since within the tube they are increased by the coupling coefficient. Theoretical modulation characteristics have been determined20 and are shown in Fig. 17. At low values of coupling coefficient and high values of Po, the undesirable dotted curves would be followed at low beam currents. Fortunately, the small losses in the absorption tube, corresponding t o a value of a (Fig. 16) of about 0.04, reduce the rf voltage and prevent electron collection a t the poles, although such losses reduce the maximum modulation factor to about 0.75.

4. A Developmental Absorption Modulation System I n a laboratory modulation system an 850-Mc 1-kw cw magnetron was amplitude modulated b y the tube and cavity shown in Fig. 18. T o provide easy replacement and experimental control of the cavity resonant frequency and the coupling t o the circuit of Fig. 15, the beam was enclosed in a glass capsule with wall coatings to provide pole faces. The necessary axial magnetic field was provided by a solenoidal magnet,

J. S. DONAL, JR.

216

FIG.17. Theoretical modulation characteristics. ABSORPTION T U B E

COLLECTOR W A L L COATING

A N T CAVITY

L I N E TO MAGNETRON AND A N T E N N A

FIG.18. View of absorption tube in its resonant cavity.


FIG.19. Photograph of developmental absorption modulation system.

217

2 18

J. S. DONAL, JR.

shown encased, in the photograph of the equipment of Fig. 19. The grid-controlled electron beam in the absorption tube was operated a t 600 v and about 150 ma. The maximum value of conductance developed by the beam was about 0.04 mho, corresponding to a value of a. of two in Fig. 16, so th a t the maximum power in the load was approximately 60 per cent of the matched-load power output of the magnetron. Actually, the power in the load was limited t o about 500 watts by failure of the glass walls of the absorption tube under the high rf voltages in the tube a t low beam currents, but this cause of failure could have been eliminated by building the tubes of a material having lower loss, such as a ceramic. With the magnetron used, the system efficiency was somewhat less than 40 per cent a t the peaks of the modulation cycle. The average efficiency during the modulation cycle was thus quite low, a n unfortunate property of all absorption modulation systems, although this low efficiency was partially compensated by the absence of the usual driver stages of a conventional system. The grid was a t all times negative, the grid swing was only about 15 v, and the grid capacitance was only a few micromicrofarads. The power delivered by the modulator t o its plate load was therefore less than 1 watt. The modulation bandwidth of the system was in excess of 10 megacycles. The depth of modulation attainable corresponded t o a modulation factor of 0.75 and there was excellent linearity of the rf voltage across the load as a function of grid voltage. The frequency stability of the system during modulation is of fundamental interest in a comparative study of methods of modulation. From the nature of the rf circuit of Fig. 15 the loading of the magnetron was changed during the modulation cycle, although this change was one of conductance only. As discussed in the Introduction, a change i n G, resulting from such a change in GL" causes a variation of Be and of the frequency. I n the absorption system, however, it was possible to introduce a very slight amount of reactance, by adjustment of the line lengths and magnetic field, so that the point of operation on the Rieke diagram followed a contour of constant frequency very closely. As a result the frequency varied less than +15 kc in 825 megacycles. This change could have been reduced further, and the average frequency could have been controlled, by a feedback ldop such as that described in Sec. VIII. 5. Evaluation The method meets all of the desired performance criteria for a n amplitude modulation system except that of high average efficiency

MODULATIOK O F CONTINUOUS-WAVE MAGSETRONS

219

during the modulation cycle. I n common with all absorption systems, this arrangement has poor efficiency. The small frequency change during the modulation cycle could be corrected by a feedback loop. The linearity, depth of modulation, and bandwidth are satisfactory. The output power required from the modulator is very low.

TI. AMPLITUDE AIODULATION BY J I E A N S COCPLER

O F THE

ELECTRON

The spiral-beam absorption tube of Sec. V varied the load on the oscillator slightly and, as a result, caused a small amount of frequency modulation during the modulation cycle. While the frequency change U H F POWER SOURCE

ELECTRON COUPLER

FIG.20. Basic schematic diagram for use of coupler.

was only 15 kc, this variation would produce a large phase modulation a t low frequencies and would require a further control mechanism, such as a feedback loop, for some applications. The electron ~ o u p l e r , on ~~~*~ the other hand, is a device which provides variation of the power in the useful load with no change in the load presented to the oscillator and, hence, no change in frequency. I n principle, the electron coupler absorbs a constant amount of energy from the oscillator and transfers this energy t o a second circuit in which the energy can be dissipated or sent t o the load without any reaction on the primary circuit. The basic block diagram is shown in Fig. 20. I n practice, a resonant cavity forms the only load on a magnetron, and a constunt-current spiral beam traverses this cavity t o absorb a constant amount of power. A longitudinal magnetic field makes the cyclotron frequency of rotation of electrons in the beam equal t o the resonant frequency of the resonant cavity (and equal t o the oscillator frequency) precisely as in the case of the absorption tube of Sec. V. Instead of dissipating the energy a t a collector plate, however, the beam passes through a n aperture into a second resonant cavity shown in

220

J . S. DONAL, JR.

Fig. 21, tuned to the frequency of the oscillator. The beam excites this second cavity and gives up all or a controllable portion of its energy. The energy given up is transferred t o the load; the energy retained by the beam is dissipated a t a collector plate beyond the output cavity. The modulation is thus by absorption and results in a low average efficiency during the modulation cycle as was the case for the system of Sec. IT. Equations (1.5) and (16) for the determination of the electronic conductance of the absorption beam are applicable t o the coupler. This conductance is transformed by the loop coupling the input cavity of the electron coupler t o the transmission line and, if the line is matched, forms OUTPUT RESONANT CAVITY I N P U T RESONANT CAVITY

COLLECTOR

ELECTRON GUN

FIG.21. Internal structure of electron coupler.

the load presented to the output of the magnetron. Since the input cavity of the coupler is heavily loaded by the electron beam, it is necessary only that the load presented to the magnetron be constant. I t is usually most convenient t o match the coupler t o the line, however, for othernise the magnetron load would be a function of the transmission line length. The design criteria for the beam and input cavity of the electron coupler are the same as those of Sec. II-. I t is particularly important that electrons are not collected a t the pole faces, since such collection reduces the power remaining in the beam and the power available in the output cavity. 1. Behavior of the Beam in the Output Cavity of the Coupler The resonant cavitics used in the coupler were cylindrical, with the pole faces mounted on extensions from the walls as shown in Fig. 21. I n practice the second cavity was rotated 90 degrees with respect t o the

MODULATION O F COSTINUOUS-TVAVE MAGSETRONS

22 1

input cavity t o minimize coupling between the cavities. X s a result, there was no detectable power in the output cavity when the beam 11-as turned off and no evidence that conditions in the output cavity affected the input cavity. The cone-directrix beam in the first cavity becomes the element of a cylinder in the field-free space between the cavities and ~ r o u l dcontinue as an element of a cylinder through the second cavity if there were no rf fields there. If the output cavity were driven 180 degrees out of phase (neglecting the 90-degree physical rotation of the structure) with the rf voltages in the input cavity, the beam would be collapsed. Actually, the rotating electrons of the beam induce currents in the output cavity which, in turn, produce rf voltages which are 180 degrees out of phase with those in the input cavity (again neglecting the 90-degree physical rotation) and the spiral beam collapses and gives up its energy t o the cavity. I n the special case of identical geometries, beam voltages and rf fields in the two cavities, the envelopes of the spiral beams would be identical except for inversion of the second cone. All the energy absorbed in the first cavity would be transferred t o the second. Only the dc energy of the beam, attained before it enters the first cavity, would be dissipated a t the collector. If the transit time in the second cavity were decreased, by increasing the beam voltage, the beam would fail t o collapse and energy would be left t o be dissipated a t the collector. An increase in transit time would cause the beam t o collapse, giving up its energy, but i t would then reabsorb energy again from the fields in the cavity and dissipate this portion a t the collector. It can be shown that a n increase in the loading of the output cavity by the useful load produces the same effect upon the beam as does a decrease in transit time, and vice versa. C u c ~ i a has * ~ solved the equations of motion for the electrons in the second cavity (Region 111 of Fig. 22) for the components of velocity and displacement in the plane perpendicular t o the pole faces. These are as follows: 1 -lel [(E1rl- E 3 ~ 3 ) ] /?&I = W,TO - -2 le' E3r3 = m 2m

where w,ro is the angular velocity a t the entrance to the second cavity, wC is the cyclotron angular frequency and the symbols E and r indicate the rf fields and the transit times in the respective regions. Both velocity and displacement are zero when

222

J. S. DONAL, JR.

If lr is the length of the poles in the input cavity, the distance into the

w

a

El

z 0 W

W

a d

FIG.22.

Envelopes of Electron Trajectories.

pole-face region of the output cavity at which convergence occurs is given by

where the values of V are the beam voltages in the respective cavities. 6. Modulation of the Power in the Load

Several points of beam convergence in the output cavity are shown in Fig. 22. Suppose that the lengths of the poles and the beam voltages


223

are equal in the two cavities It can be shown that the rf fields will be equal, and convergence will take place at the exit of the pole faces in the secondary cavity if the load resistance, transformed into the tube and presented to the beam, is given by

Variation of the beam voltage in the second cavity only, to alter the transit time there, will shift the convergence point as predicted by (22). If the beam voltage is reduced to give-convergence at I , (Fig. 22),.the power given up will be reabsorbed and dissipated at the collector, for the power content of the beam is proportional to the square of its radius of rotation and the exit and entrance radii are equal. Modulation by increase of transit time encounters space-charge limitations and instabilities, however, and an increase of beam voltage is preferable. If the convergence is moved to l d , for example, or to a point twice the poleface length from the entrance, one-quarter of the power originally in the beam remains to be dissipated while three-fourths of the original power has been extracted and transferred to the useful load. Modulation by decrease in transit time is, of course, preferable to modulation of the current by varying the potential of the control grid of the gun, for such modulation would result in a large change in the loading on the oscillator and in a very large change in the system frequency. 3. Construction a.nd Performance of a Typical Coupler A photograph of a typical coupler for use a t 825 megacycles is shown in Fig. 23. The diameter of the structure is about 4 in. Tuning of each cavity is accomplished by varying, through bellows, capacities in shunt with the poles. The tuners and the seals closing off the coaxial lines are in planes 90 degrees apart because of the rotation of the poles mentioned earlier. The grid-controlled electron gun provides about 100 ma of beam current at about 500 v. The collector plate forming the top of the envelope is water cooled to dissipate the dc power of the beam plus rf power not transferred to the load. The rf power output of the experimental tubes was from 50 to 70 per cent of the rf input. The principal cause of the loss was probably dispersion of the beam and loss of outer electrons a t the pole faces of the input cavity. In the 800-900 megacycle range, the peak power in the load was about one-half kilowatt, limited by the oscillator used and the transfer efficiency of the coupler rather than by the power handling capability of the beam. A modulation characteristic is shown in Fig. 24. The available

I

0

~

"

N 0

"

b

"

"

0 m

'

0

m

PERCENT OF MAXIMUM VOLTAGE APPEARING ACROSS OUTPUT LOAD 0

-


225

modulation factor approaches 100 per cent in voltage. The linearity is not particularly good and the modulation voltage is high, although no current need be handled by the modulator, for the electrons are collected at an electrode, held at a constant potential, beyond the cavity. Those determinations of bandwidth made so far indicate merely t h a t i t is in excess of 5 megacycles. There was no measurable change in the system frequency during the modulation cycle. 4. Evaluation The most important advantage of the electron coupler is the elimination of changes of loading on the oscillator, with consequent changes in system frequency, as the power in the load is altered. Methods of modulation which require less modulating voltage and provide better linearity are under investigation. The most obvious disadvantage of the method lies in the fact t h a t it is an absorption system with the characteristic low average efficiency over the modulation cycle. The efficiency is further reduced by the transfer efficiency of the coupler. Since the frequency of the magnetron can be stabilized, against variations caused by changes in temperature or voltage, by the inclusion of frequency control guns and very simple circuits, the electron coupler should be useful in systems in which frequency stability is more important than efficiency. Of course the oscillator t o be modulated need not be a magnetron. I n considering spiral beams for handling power at higher frequencies, it should be remembered that for the same spiral radius the power absorbed by the beam varies as the square of the frequency. VII. CONTROL OR MODULATION BY INJECTION PHASE LOCKING Locking or synchronization of conventional oscillators has been studied by a number of workers. The term locking implies that not only are the frequencies of the locking and controlled oscillators the same but there is also a definite phase relationship between the rf voltages. From what has been said concerning the low inherent stability of self-excited oscillators, the locking of such oscillators t o a stable control oscillator js of obvious importance in the microwave field. E. E. Davidz5rz6J78-b has described a method for the phase control of microwave tubes which utilizes the injection of power via the output circuit of the oscillator t o be controlled. I n addition t o frequency stabilization, this procedure permits phase or frequency modulation or, using two oscillators, amplitude modulation by variation of their relative phases. The general phase-control characteristics of the system are as follows.

226

J.

S.

DONAL, JR.

If the frequency of an uncontrolled oscillator is modulated by an amount independent of modulation frequency, the rf phase modulation is inversely proportional to the modulation frequency. In the case of the phaselocked oscillator, however, the phase modulation is independent of modulation frequency up to perhaps 10 kc. Therefore, the frequency deviation of the locked oscillator is proportional to modulation frequency up to perhaps 10 kc. At modulation rates much above the audio range, the locked phase modulation decreases, as will be seen in greater detail below. For a constant-phase deviation, it is obvious that a t modulation frequencies approaching zero, the locked frequency deviation approaches zero. 1. Phase Locking under Static Conditions David has described the operation of the injection-locking system by reference to the chart giving the frequency and power output of the magnetron or klystron t o be controlled as a function of the magnitude

FIQ.25. Schematic Diagram of Injection Locking System.

and phase of the reflection coefficient. Suppose the oscillator delivers power to a matched load, i.e., the reflection coefficient, p, is zero. If the oscillator frequency, w’, is altered by a change in its dc conditions, it is obvious that a change in loading could correct the frequency change. The injected locking power performs this function automatically. Thus it is found that a strong injected signal of frequency w 1 causes the oscillator to assume the frequency w1 and t o ‘(track” until w 1 - LO’ becomes too great, at which point the lock is broken. The point of operation of the oscillator shifts, in general, to a new power output and to a point on the frequency contour wl. The locking power appears as if reflected from a load and the magnitude and phase of the resulting reflection coefficient uniquely specify the new power and the new frequency, WI. The simplified locking circuit is shown in Fig. 25. It is assumed that the locking power, P,, propagates only toward the controlled oscillator 0, that none of the output of 0 reaches the locking oscillator, OL,and that the frequency, phase and amplitude of the voltage of O L are independent variables. If PI is the total power going to the right in the line from the locked oscillator and lpl is the magnitude of the reflection coeEcieni,

MODULATION O F CONTINUOCS-WAVE MAGNETRONS

PL

227 (24)

= lPl2PI.

From (24) and the fact t h a t Po, the power output of the oscillator as specified on the reflection coefficient chart, must be zero when lpl = 1, we have Po = PI - P,. (25)

a. Graphical Treatment. If the locking power is held constant, as is usually the case, the oscillator must see a fictitious load which changes

-3162 FREQUENCY (MC)

FIG.26. Rieke diagram of velocity modulation tube.

with frequency in such a manner that the reflected power is constant. The contours of constant locking power, or loci of points of operation of the oscillator with the locking power constant, may be calculated from (24) and (25). Figure 26 is the chart of a klystron in the reflection coefficient plane. If a point such as -4 is chosen, a t a Po of 14 mw and with a IpI of 0.6, (24) and ( 2 5 ) yield a value of PL of 7.8 mw. A number of such calculations permit the contours of constant locking power of Fig. 27 t o be constructed. The value of PL calculated as a n example is shown a t point B. It is interesting that point A' of Fig. 26, on the 10-mw contour gives a value of PL of 12.7 mw, or a value of P L larger than Po.

228

J. S. DONAL, JR.

The oscillator must operate a t a point on the contour of known PI,, specifically a t the intersection of this contour with the frequency contour corresponding t o the locking frequency. If there is no such intersection the osciiiator will not be locked a t this frequency. Figure 28 shows the locking contours superimposed on the frequency contours of Fig. 26. The dots a t the points of targency are the breakout points. It is to be noted particularly that less locking power is required for control of the oscillator over a smaller range. David’s experimental contours were in 180.

FIG.2’7. Contours of constant locking power.

escellent agreenient with the theoretical contours, as mere the experimentally determined frequeiicies a t which the lock was broken. Since the locking contours intersect each frequency contour twice there 11-ould appear t o be two possible values of power output for each frequency but it has been shown t h a t the stable points are those only on the portions of the contours which are concave upward in the figures. Since the contours of constant PL can be closely approximated by circles for small P L , a rapid means for computing the synchronized behavior of an oscillator is available. b. A n a l y t i c a l E x p r e s s i o n for the Locking R a n y c . Only the case of operation into a matched load, before the application of the locking


229

signal, is considered here and in ( a ) , above. Pitts and Wickszs have analyzed the mismatched load case. SlaterZ9has shown that for GL = I and B L = 0 (normalized valu&) the complete normalized load admittance is given by y L = 1 - ~[,,[~~[(wI-o)c+BI = 1 - 2[,,[@" (26) The second form holds for the locked condition in which w1 = W , where w is the instantaneous frequency of the oscillator. I n ( 2 6 ) , Ipl is the reflec-

- - - - REFLECTED

OR LOCKING F R E Q U E N C Y (MC)

POWER(MW)

FIG.28. Locking contours superimposed on frequency contours.

tion coefficient of the apparent load and 6 is both the phase of the reflection coefficient and the phase difference between the locked and locking oscillators. If this value of Y Lis substituted in the ordinary expression for the condition for oscillation [see (4)and ( 5 ) of the Introduction] with GL = 1 and BL = 0,

230

J. S. DONAL, JR.

A comparison with the corresponding expressions for a n ordinary matched load shows t h a t the locking signal adds the last term in each expression. These are the load conductance and susceptance introduced by the locking signal. If 1pI is set equal t o zero in (as), t o give the matched load condition, u1 becomes u’, the matched-load angular frequency, or (29) Combining this with (28),

If lpI is assumed t o be constant ( P Lsmall), w 1 - w’ is a maximum when sin 0 = k 1, giving the basic expression relating the static locking range and the system parameters

This locking range is consistent with the tangency condition on the locking range of Fig. 28 for the assumed small values of IpI. The equality of the phase of the reflection coefficient with the phase difference between the two oscillators is now obvious, as well. I n addition, (31) permits (with 24) the calculation of the required locking power for a desired locking range. 2. Frequency Control of Modulated Oscillators The static case, treated above, is of interest for the adjustment of the oscillator frequency or for stabilization of this frequency against, very slow changes such as those caused by variations in temperature of the oscillator. Often, however, it is desired t o stabilize the frequency against power pack hum or, a most important case, against changes in frequency resulting from deliberate voltage changes incident t o plate modulation. An entirely different situation would arise if the oscillator were t o be frequency stabilized against slow frequency changes but phase or frequency modulated in addition. All these cases require a knowledge of the phase deviations of the locked oscillator. From these the frequency deviations can be determined, for +_Af

=

fm(+AO),

(32)

where Af and A0 are the amplitudes of the frequency and phase deviations, respectively. From (30), the phase angle, 0, between the two oscillators is given by


23 1

It is assumed that, as a contour of constant locking power is traversed, PO is constant, a reasonable assumption as may be seen from Fig. 28. IpI is then constant. The phase angle between the oscillators, or the phase angle of the locked oscillator if the phase of the locking oscillator is held constant, is a function only of the free-running frequency, w'. Since it will simplify the discussion if e is varied symmetrically about its zero value, changes in 0 will be assumed to be small so that sin 0 can be replaced by 0. An equal purpose would be served by assuming that W' varies in such a manner as t o produce a linear variation in 8. I n either STABILIZED KLYSTRON

-

PHASE SENSITIVE .DETECTOR OSCILLOSCOPE

MAGNETRON

FIG.29. Practical injection-locking circuit.

case e is sinusoidally modulated by sine-wave modulation of w' with a deviation A d . As long as the amplitude of the phase deviation A0 is constant, Af is proportional t o the modulation rate from (32). a,. Variation of Phase Modulation with Static Phase Diference. The phase modulation for a given Aw' is not constant over the static locking range. David shows that if the phase deviation, AO, is small, a deviation, Aw', of the free running frequency results in a phase deviation

Thus, A 0 is a function of 0 and rises from a minimum at w1 = w', or e = 0, to a very high value near the breakout point, e = -190'. This is of importance when the oscillator is to be stabilized to give a minimum phase or frequency deviation. b. In,jection Locking for Automatic Frequency Control. A practical injection circuit is shown in Fig. 29. A directional coupler with an attenuation of about 10 db was used to direct the major portion of the magnetron power to the useful load. This means, however, that 90 per cent of the locking power was dissipated and only 10 per cent was

232

J. S. DONAL, JR.

directed toward the magnetron, although this is not necessarily a permanent limitation of the system. The stabilization fa.ctor, H , of the control system is defined as the ratio of the unlocked frequency deviation t o the locked frequency deviation. This can be shown t o be

The stabilization factor increases as the square root of the locking power and varies inversely as urn,which follows, of course, from (32) if A0 is independent of urn. For slow changes such as temperature variations the locked frequency deviations are limited only by the stability of the control oscillator, since am.in t h a t case would be very small.

FIG.30. Vector diagrams of out-phase amplitude modulation

As a n example, for a 300-Mc tube with a n external Q of 3000 and with a IpI of 0.1, H would be 8 X lo3 at a 120-cycle hum frequency. H would decrease t o 8, however, a t frn = 120 kc. Equation (35) holds only for H much greater than unity, or, for the example chosen, for fm < < lo6. A t higher frequencies the phase is unable t o change sufficiently rapidly and i t would appear that an increase in H is t o be expected. Since the phase cannot keep up with the variation in w', neither (33) nor (34) hold a t high modulation rates. 3. Out-Phase Modulation

Out-phase modulation is a procedure for obtaining amplitude modulation, familiar in the literature of conventional tubes. The relative rf phase of two oscillators is varied in such a manner that when the voltages from the two oscillators are in phase a t the reference plane, the total power goes t o the useful load, but when the rf voltages are 180 degrees out of phase, all the power is dissipated and the power in the load is zero. Use of injection locking t o accomplish this end was suggested by E. E. David and tested experimentally by W. P. S ~ h n e i d e r . ~ The ~ system vectorial relationships are shown in Fig. 30. The two portions of the figure represent the two halves of the modulation cycle. The vectors


233

W , represent the carrier voltage of the control oscillator, the two portions of the vector indicating that this voltage is inverted in phase as presented t o one of the two controlled klystrons. Elf, and Erf,are the vectors representing the carrier voltages of the controlled klystrons, while el and 0 2 are the rf phase differences between the locking oscillator and the controlled oscillators. I n the unmodulated condition both klystron oscillators are in phase with the locking source. The repeller voltages were then modulated in push-pull t o advance the phase of one oscillator and retard that of the other. Here the inverse sinusoidal variation of e given in (33) is most useful, because the total rf voltage is proportional t o sin 8 and, hence, linear with variations in the free-running frequency w'. It is necessary t o choose two oscillators of equal output and equal linear variations of free-running frequency with repeller voltage. Furthermore, this frequency must not vary with temperature, or the unmodulated phase relationships will be disturbed. With the modulation shown in Fig. 30 only the amplitude modulation sidebands are supplied. If the unmodulated phase difference between the two oscillators were made 90 degrees and each oscillator were modulated +45 degrees, the carrier would be supplied. The modulation characteristic would no longer be inherently linear, however. Schneider's experimental results were excellent, although the particular circuit used required a total locking power greater than the output power of the oscillators. The minimum power in the load was 30 d b below the maximum power and the linearity in the audio frequency range was almost perfect. The carrier frequency was 9200 megacycles. Both David and Schneider considered the bandwith a t some length. Present theory is adequate only for low modulation factors, where sin 0 may be replaced by 8, with the limit on Aw' placed by Schneider a t 0.7 [Iplwo/(QeXt)o]. Under these conditions the rf amplitude is down 3 db a t

Thus for a 3000-Mc tube with = 300 and IpI = 0.1, the response would be down 3 db at f m = 1 megacycle. Under the condition of higher carrier frequency and greater IpI used in Schneider's experimental work the conventional bandwidth was both calculated and measured t o be about 3 megacycles. It is interesting that the expected value of fm for a decrease in response by 3 d b is IpI times the total rf bandwidth expected from the ratio w/&. While the bandwidth has been determined for only small modulation factors, it may well be roughly the same for deeper modulation. Preemphasis should be quite practical for the improvement of bandwidth.

234

J. S. DONAL, J R .

4. Evaluation As a method of frequency control of magnetrons, injection locking offers the advantage t h a t no frequency-control guns are required. However, the locking power may be rather large if a wide locking range or a high stabilization factor is desired. The stabilization factor, H , has not been measured at high modulation r a t p and, simultaneously, high modulation depths, nor can i t be calculated from existing theory. This theory, applicable t o either the frequency control or t o out-phase modulation, predicts a n increase in H at high rates provided the modulation factor is kept low. If phase control by injection is used t o produce out-phase amplitude modulation, the overall efficiency is low; the arrangement is effectively a n absorption system since the input t o the oscillators is constant. This may not be a serious disadvantage for low-power systems. The available modulation factor is extremely high, approaching unity, and the linearity is intrinsically good. Any spurious frequency modulation is probably associated with second-order nonlinearities arising from assymmetries. The bandwidth is as yet little understood when the modulation factor is high, but i t appears t o be limited a t low modulation factors t o a value considerably less than that determined by the loaded Q of the oscillator. Pre-emphasis should be effective for improving the response. This method is in a n early stage of development, and it is highly probable t h a t some of its limitations will be removed. I n particular, i t should be possible t o reduce or t o eliminate the loss of locking power resulting from the use of the directional coupler of Fig. 29. VIII. AMPLITUDE MODULATION BY PLATEMODULATION OF

MAGNETRON WITH

A

SIMULTANEOUS FREQUENCY c O N T R O L 3 1 * 3 2

Amplitude modulation by the spiral-beam absorption tube (Sec. V), by the coupler (Sec. VI), and by out-phase modulation (Sec. VII) are all absorption methods, in that the input remains constant while the power output is varied. The average efficiency during the modulation cycle is low for all these procedures, a consideration which is particularly import a n t in high-power systems. Plate modulation of a magnetron, on the other hand, would yield much higher efficiency, but it has been believed that the characteristics of such a system would be poor with regard t o linearity, bandwidth, attainable depth of modulation and, in particular, frequency stability. I n the last case, the pushing during the modulation cycle would be expected t o give an intolerable amount of mixed frequency modulation. However, the simplicity and high efficiency of a platemodulation system warranted serious consideration of the method,


235

particularly since frequency control by spiral beams (Sec. 11) offered a means for the automatic stabilization of the frequency. In considering the stabilization of a magnetron it is not always obvious whether one should use a phase-control or a frequency-control system. The first yields, ideally, a controlled phase deviation which is independent of modulation rate, while in the second case the frequency deviation should be independent of rate. Since A0 = Af/fm, the phase control system gives a vanishingly small frequency deviation as the modulation rate is reduced to zero and, since the carrier frequency of any system must be extremely stable, the phase control system might be preferred. While there is evidence that frequency modulation can under certain circumstances cause severe mhltipath effects, such frequency modulation is usually of the constant deviation type with very large phase modulation a t low frequencies, again pointing to phase control as the better choice for use with amplitude modulation for television applications. Both phase and frequency control will be considered here, with greater emphasis on the first because of the high carrier stability it affords. While the frequency control may be of interest independently, it will be treated largely as a means of improving the phase control. The modulation characteristics to be expected from plate modulation will be considered first, followed by a description of the methods of stabilization. 1. Performance of a Plate-Modulated Magnetron The tube employed was the developmental 1 kw cw tube described in Sec. 11. The carrier frequency in this work was 825 megacycles. The magnetron was equipped with either nine or eleven internal frequencycontrol beams with a total beam current of about 1 amp at 500 v. The total range of frequency control available with this tube was 6 or 7 megacycles. A schematic diagram of the modulation system is shown in Fig. 31. The magnetron anode is grounded so that the rf lines may be a t ground potential. The modulator output stage is in series with the “plate” supply and serves as a variable impedance for control of the potential on the magnetron cathode. Thus when the drop across the modulator is increased from 200 to 400 v, the magnetron cathode-anode potential difference is decreased by 200 v. The phase-control signal, described below, is applied to the grids of the control guns in parallel with a frequency-compensation signal derived directly from the modulator chain. a. Modulation Factor. The modulation factor is defined as the difference between the minimum and maximum amplitudes of the modulation envelope divided by twice the carrier amplitude. It is

236

J. S. DONAL, JR.

important t h a t this factor be high if intelligence is t o be transmitted without waste of carrier power and production of interference. For example, in television practice the modulation factor is approximately 0.75, corresponding t o a minimum power of about 2 per cent of the peak power. Magnetron mode shifts a t high currents limit the peak power and, with this upper limit fixed, low-current discontinuities in power or frequency determine the available modulation factor. With the cw tubes used, the upper mode shift never occurred below 2.5 kw. The lowMAGNETRON

TO

T O PHASE-CONTROL

MAGNETRON HEATER AND

FIG.31.

Block diagram of magrletron plate modulation.

current discontinuities often occurred a t 60 t o 100 watts output statically but dynamically the power could be reduced smoothly t o 30 t o 40 watts. The available modulation factor was thus 0.75 or more. b. Modulation Linearity. Since the dynamic impedance of a magnetron is low the power output is a linear function of the current t o a first approximation. Because of a n efficiency rising with current and the slight increase in voltage, the rf power output varied, with these tubes, approximately a ,t':i or the rf voltage across the load varied as ia35. Since the current in magnetrons is roughly linear with change in voltage, this is by no means the desired characteristic of rf voltage linear with applied voltage. However, the current in the magnetron, which is the load for the tetrode modulator output stage, varied as the three-halves power of the modulator input, so that the rf voltage was linear with the grid voltage of the modulator. A typical linearity characteristic is shown in Fig. 32. The rf voltage across the load is plotted vertically,


237

and the grid drive is plotted horizontally. The required peak to peak modulating voltage at the magnetron cathode is 200 to 250 v. RF voltage versus anode voltage characteristics of magnetrons usually show deviations from smooth curves due to slight changes in the magnetron impedance. It is most important from a practical standpoint that the modulator used by insensitive to these impedance changes so that deviations from linearity will not occur in the overall characteristic. c. Amplitude Modulation Bmdwidlh. As might be expected, the frequency response of the amplitude modulation is roughly that to be expected from the loaded Q of the tube. The loaded Q of these tubes

FIG.32.

Oscilloscope presentation of modulation characteristic.

was slightly in excess of 100, and at 825 megacycles the response was down the conventional 3 db at a modulation rate somewhat less than 4 megacycles. The response curve deviates somewhat from that of a single RC circuit. d. Cho.racteristics of the Frequency-Control Beams. These characteristics have an important bearing on the operation of the phase of frequency-control systems. It is obvious, for example, that the total frequency-control range of the beams must exceed the pushing during the modulation cycle. The frequency change must be independent of modulation rate over a wide range of rates. Theoretically, the bandwidth should be in excess of 100 megacycles under the conditions employed (Sec. 11). The small amount of loading actually mixed with the frequency modulation, even at the magnetic field for which such loading should disappear, makes experimental determinations of bandwidth difficult and inaccurate, due to the unknown phase relationships between the frequency and amplitude modulations. It may be said that the bandwidth of the beams is certainly in excess of 5 megacycles and may approach the theoretical value.

238

J. S. DONAL, JR.

The frequency change due t o the beams is roughly proportional t o beam current and, therefore, t o the three-halves power of the controlgrid voltage. The megacycles per volt yielded by the beams is therefore a function of the average bias and of the amplitude of the correcting signal a t the grid. e. Pushing. The frequency change during the modulation cycle, to be corrected by the control systems, was about 6 megacycles for a modulation factor of 0.75 and a peak power of 2.5 kw. The frequency change per unit change in current was much less above the mid-current range than below this range. Figure 33 shows a curve of frequency

u $

3

0

U

z

-I

W

3

0 W

a -2

Lr

z

0

= -3 W

z

20 -4 w

>

c d

i

mJ

a

-5

0

2

6

8

P R O P O R T I O N A T E R - F VOLTAGE

FIG.33.

10 ACROSS

12

14

THE LOAD

Magnetron frequency versus RF output.

versus rf voltage across the load. It is obvious that if the pushing is t o be minimized while maintaining the modulation factor constant the tube should be operated at the highest practical peak power. Pushing decreases rapidly as the load is mismatched t o the line in such a direction as t o decrease the magnetron loading. This is t o be expected from the increased stability a t the higher loaded Q. The decrease in pushing in the high-current range is relatively greater than for the low-current range. An interesting observation is t h a t any residual loading by the control beams reduces the Q and increases the pushing. When the correcting signal on the guns is varied as the tube is modulated, the Q and pushing vary slightly during the modulation cycle, an effect which should be taken into account in designing the frequency control system.


239

6. Phase-Control System

The method of phase control was largely the development of D. S. Bond and D. G. Moore.32 A schematic diagram of the circuit is shown in Fig. 34. About 0.1 watt a t 825 megacycles, derived from a crystal controlled oscillator, and about 1 watt from the magnetron are supplied to the input terminals of a phase-comparison system which is a balanced modulator consisting of a diplexer, 3 3 filters, matching sections, and diode rectifiers. A wide-band combination of a dc amplifier and a cathode follower couples from the diodes to the grids of the frequency-control 825 MC

AMPLIFIER

AMPLIFIER (FIG. 31)

FIG.34. Block diagram of phase-control system.

guns. When the magnetron and locking-oscillator frequencies are not synchronous, but the beat note is within the pass band of the system, the beat signal brings the two oscillators into synchronism. After this occurs, there is not only zero difference in frequency between the oscillators but there is also a definite rf phase relationship maintained, for the output of the rectifiers and the signal a t the grids of the control beams is of a polarity and amplitude proportional t o the phase departure from quadrature. I n the static case, for example, when the free-running frequency of the magnetron is equal t o that of the control oscillator, there is no dc signal, other than bias, supplied t o the guns. When the freerunning frequency of the magnetron is altered, the locked magnetron frequency is unaltered but a new equilibrium is set up in which the relative rf phase at the diplexer differs from 90 degrees. The static locking range is exceeded, and breakout occurs, when the rf phase differ-

240

J. S. DONAL, JR.

ence is more than 590" from the quadrature condition. The range of phase locking is thus identical with that, of the injection locking of Sec. VII. The phase-control system is a form of feedback loop. As in feedback loops in general, spontaneous oscillation will occur a t the frequency for which the total phase shift around the circuit is 180 degrees unless the Nyquist criterion3* for stability is met, i.e., unless the loop gain is less than unity at this frequency. Fortunately, a correction network can be used t o so shape the gain and phase that the gain is high a t low modulation frequencies, yet the stability requirement is satisfied. CIRCUIT OF FIG. 3 4 F R O M D I P L E X E R TO FREQUENCY C O N T R O L GUNS I N C L U S I V E

\

FIG.35. Schematic representation of feed-back loop.

We are interested chiefly in the stabilization factor or compression ratio, H , which is the high ratio of the unlocked frequency deviation, due t o the pushing of the magnetron, t o the locked frequency deviation. This must be distinguished from pb, the gain of the feedback loop. Figure 35 shows the loop schematically. E,, is analogous t o AfuL, the unlocked frequency deviation, and Eoutis analogous t o AfL, the locked frequency deviation. From the figure, the output is passed through the loop with a gain of p@, so that

The quantities in (37) and (38) are vectors. The phase-control loop has two very important properties. First, the signal a t the gun grids produces a frequency change, but the signal from the diodes is proportional to a phase change. T h i s integration process results in there being a 90-degree phase shzft in the loop at all modulation rates, in addition to the usu0.1 phase shifts in the circuit components.

24 1


Secondly, it can be shown that, associated with the 90-degree phase shift, the loop g k , NO, is proportional to l/f,,,,aside from the gazn characteristics of the circuit components. a. Performance of Loop with Circuit Components of Infinite Bandwidth. This is an illuminating if somewhat academic case. An assumed gain characteristic is shown in Fig. 36. The total phase shift around the loop is 90 degrees and independent of frequency. In the Nyquist diagram of Fig. 37, each successive pp vector is plotted, as usual, a t an angle, a, from the negative end of the line O A , equal to the total phase shift in the loop. In the present case this phase shift is always 90 degrees.

0

0.2 0.4 0.6 0.6 1.0 1.2 1.4 MODULATION RATE f m ( M E G A C Y C L E S )

1.6

FIG.36. Gain versus modulation frequency for circuit of infinite bandwidth.

The unit vector is OA and the values of H = 1 - pp have been drawn in the figure. The values of compression ratio, H,are plotted as a function of rate, fm, in Fig. 38. The values of rf phase deviation, AO, are of particular interest. The magnetron pushing was assumed to be f l megacycle a t a rate of 1 megacycle and the unlocked phase deviations Since the compreswere calculated from the relation * A 8 = *Af/fm. sion ratio applies to the phase as well as to the frequency, the locked phase deviations were obtained from the relation AOL = AOUL/H. From Fig. 38, H never becomes less than unity in the infinite bandwidth case, so there can never be amplification of the phase or frequency deviations. The locked phase deviation, whether the components hove Jinite or infinite bandwidth, is independent of rate at loco frequencies, since unity is negligible compared to pB and H , itself, is proportional to l/fm as is A&,. From Fig. 37, when ,u/3 = 1, H = 4 2 and AOL is 0.707 of AOuL. AOL is also 0.707 of its value a t very low frequencies, or the locked phase deviation has fallen 3 db.

242

J. S. DONAL, JR.

FIG.37. Nyquist diagram for circuit of infinite bandwidth.

FIG.38. Performance of feed-back loop with circuit of infinite bandwidth.

From See. VII it will be remembered that expression for the compression ratio, H , of (35) holds only when H is large, while the bandwidth of the phase control, (36), holds only for low depths of modulation. With these limitations the analogy t o the present infinite bandwidth case is very interesting. If David’s H is assumed t o be actually p/3 (the


243

H of this section approached p/3 a t low frequencies), one would expect from the preceding paragraph t h a t the phase deviation should be down 3 d b when H = 1. This is the case, for placing H = 1 in (35) gives precisely the condition of (36). b. Performance of a Pmctical Loop. I n a practical loop the circuit components add enough phase shift t o make the total phase shift 180 degrees a t a frequency of a few megacycles. I n Fig. 39 the circuitcomponent phase shift is assumed t o be linear with fm, the case often found in practice. For simplicity, p/3 is again assumed t o be proportional to l/fmsince the individual circuit components are usually quite flat t o a high frequency. The Nyquist diagram is shown in Fig. 40.

FIG.39. Gain and phase curves for circuit of finite bandwidth.

Each value of p/3 is plotted a t its corresponding phase angle, or 90 degrees plus the phase shifts of Fig. 37. The loop cannot oscillate, for po is less than unity when the total phase shift is 180" degrees, but when H = 1 - p/3 becomes less than unity (the tip of the p/3 vector within the unit circle) the phase and frequency deviations are amplified rather than compressed. I n Fig. 41, 1/H becomes much greater than one and this amplification extends over a rather wide range of frequencies about the frequency a t which the phase shift is 180 degrees. The degree of amplification shown is not particularly significant since it is a function merely of the amount by which pP differs from unity a t the singing frequency. It is quite obvious, however, that the bandwidths of the components of the loop should be made as wide as possible. c. Experimental Results. At this writing, the quantitative results have been obtained largely a t modulation rates below 20 kc. The

J. S. DONAL, JR.

244

C I R C U I T COMPONENT P H A S E A N G L E FOR

#,g FOR

fm=(fm)i

fm=(fm)c

H=I

-fib

FOR

(frn)i,~T~. PHASE SHIFT A R O U N D LOOP \

\

I

P\l/ \

A

I

I

FIG.40. Nyquist diagram for circuit of finite bandwidth.

FIG.41. Performance of feed-back loop with circuit of finite bandwidth.

magnetron was modulated with a modulation factor of about 0.75, for which the uncontrolled pushing mas k 2 megacycles. The correction network was used to eliminate singing yet increase the loop gain a t low frequencies. At a modulation rate of 5 kc, H was 1300 or the locked frequency deviation was about 1.5 kc. H was inversely proportional to

MODULATION O F CONTINUOUS-WAVE MAGNETROSS

245

the frequency as expected, so that i t rose t o 6500 a t 1 kc and about 50,000 a t the hum frequency of 120 cps. Since H varied as l/fm,the phase modulation was approximately constant throughout the audio range. From the value of H given above, the phase modulation was only k 18 degrees a t 5 kc, for example. This compares favorably with existing conventional transmitters. The megacycles per volt of the control beams is a factor in the loop gain. For low modulation factors, such as those t o be expected from hum modulation, the guns could be biased t o the steepest portion of their characteristics t o increase the loop gain about a factor of two and raise the compression of hum modulation t o a factor of 100,000. The loop gain a t lorn frequencies could be increased still further by the correction network. I n this method the low-frequency compression ratio is not limited by the locking power as was the case for injection locking (equation 35). 3, Frequency Control Used in Addition to Phase Control

While few experimental results have been obtained a t modulation rates in the megacycle region, amplification as predicted in paragraph 3(b) above has been encountered. I n addition t o broad banding of the loop, two other methods of control, used in parallel with the phasecontrol system, have been investigated. The first of these is direct compensation, whereby a signal derived from the modulator chain (see Fig. 31) is adjusted in phase, shape, and amplitude and applied t o the grids of the guns. This type of correction differs in two respects from the phase control. It yields a , m u c h lower compression ratio at low modulation rates but, contrary t o the results with the phase control system, the frequency correction can be constant with modulation rate and, furthermore, no amplification need be encountered. The improved control a t high frequencies is largely due t o the fact that the correcting signal can occur a t the same time as the signal t o be corrected, i.e., there need be no phase delay as in the phase-control loop. As much as 15 t o 20 d b of compression ratio has been obtained over a frequency range of one octave in the megacycle range of modulation rates, with 10 db over a range of frequency of 20 octaves. If the highgain region corresponded t o the amplification region of the phase-control loop, the amplification could be corrected t o compression. Thus, when the direct compensation and its phase-control loop are used together they act independently, and the total compression ratio is the product of the individual ratios of the two systems. The phase-control system acts as usual, but upon a magnetron with a pushing reduced by the compensation. Although the compression ratios quoted in the last paragraph may be

246

J. S. DONAL, JR.

obtained without difficulty b y use of the direct compensation, the longtime stability might be improved if a frequency discriminator were used in a second feedback loop. This is in a n early developmental state. As with compensation, H may be constant over a wide frequency range, rather than proportional t o l/fm. To correct any amplification produc2d by the phase-control loop, the singing frequency of the frequency-control loop must be very much the higher of the two. There must be a frequency margin wide enough t o permit the loop gain t o be adjusted from less than unity at the singing frequency t o a high value a t the singing frequency of the phase-control loop. This broad banding is made easier, however, by the fact that there is no initial 90-degree phase shift in the frequency-control loop.

4. Evaluation This work is still in progress a t this writing (November, 1950). It offers a very good chance of making magnetrons useful devices for conventional amplitude modulation, as is also the case for the injectionlocking system of Sec. VII. If the spurious frequency modulation can be controlled adequately, advantage can be taken not only of the intrinsically high efficiency of the magnetron but in particular of the fact that plate modulation maintains this high efficiency whereas absoqtion modulation seriously reduces i t during the modulation cycle. The bandwidth and modulation characteristics of the magnetron appear t o be satisfactory for most purposes. At low modulation rates the compression ratio of the phase control loop can be made very high. The values attained can probably be approached by injection locking, a t least by t h a t of the form described in See. VII, only for values of p close t o unity, or values of locking power nearly equal t o the power delivered t o the load, yet the power required from the control oscillator is only 0.1 watt for a 1-kw magnetron. If the unmodulated carrier of an uncontrolled magnetron is heterodyned into the audio range, low frequency f m noise and hum modulation result in a n extremely rough note. The locked magnetron, on the other hand, sounds like a crystal-controlled oscillator. Since H approaches infinity a t zero modulation rate, the carrier frequency stability is equal t o that of the crystal-controlled oscillator. At very high modulation rates, theory predicts amplification by the phase-control loop, and this is confirmed experimentally, although direct compensation or a frequency-control loop could help t o correct the situation. A fundamental improvement would result from an increase in the loop singing frequency. I n the case of injection phase control, on the other hand, the theory predicts a decrease in phase modulation and,

MODULATION OF CONTINUOUS-WAVE MA G X ETK O N S

247

hence, on increase in compression ratio a t high modulation rates. At present, however, the injection theory is valid only for low modulation factors and little is known of the behavior of the compression ratio wit,h frequency under other conditions. IX. THEINJECTION MAGNETRON AS THE POSSIBLE: MEANSOF PRODUCING AMPLITUDE OR FREQUENCY h$ODULATION Some of the disadvantages of the magnetron, when plate modulated for the production of amplitude modulation, were mentioned in Sec. VIII. Thus, with some tubes it is difficult to obtain deep modulation and the inherent nonlinearity of the modulation must be corrected. Compared with grid modulation, both the current and the voltage to be furnished by the modulator are high. Perhaps most important, the frequency change during the modulation cycle must be reduced by a large factor if the system is to be acceptable for the transmission of information. The injection magnetron proposed by A. M. Clogston* yields a good spectrum a t much lower relative powers than is the case for a conventional tube. Although there appears t o be no significant, improvement in the inherent linearity and the modulation voltage is somewhat higher th a n in conventional plate modulation, the current t o be supplied by the modulator is greatly reduced. From preliminary results, the frequency change during the modulation cycle is reduced by a factor of a t least ten compared with the pushing of conventional tubes at the same frequency. Finally, a linear frequency modulation can be produced if desired. 1. The Principles of the Injection Magnetron

I n the conventional tube a sheath of space charge surrounds the cathode and expands in diameter as the anode voltage is increased. The angular velocity of the electrons in the sheath increases as a function of the sheath diameter until, without rf fields, the outer edge of the sheath reaches the anode. Before this happens, currents induced in the anode structure cause a traveling wave t o appear on the anode if the applied magnetic field is in the correct range. These rf fields cause a grouping of the outer electrons of the sheath into spokes due to a phasing process. During this phasing process the power output of the tube jumps rather suddenly from zero to a finite value and thereafter increases rapidly as

* A. M. Clogston, United States patent 2,530,948, November 21, 1950, and The Injection Magnetron, unpublished report of the Bell Telephone Laboratories, October 30, 1947. Preliminary work was begun at the M.I.T. Radiation Laboratory during 1946. Some of this material was presented at thc 1948 I.R.E. Conference on Electron Devices. Permission to discuss this material has been granted by the author.

248

J. S. DONAL, JR.

the anode voltage is raised, as the phase difference between the induced currents and the rf voltage decreases. The latter change results in a rather large change in the frequency of oscillation. The chain of events in the injection magnetron is quite different because the anode voltage is held constant while the anode current is controlled by a separate means. A simplified sectional drawing of the tube is shown in Fig. 42." As used b y Clogston the anode was a conventional sixteen-vane structure for 4000 megacycles, but the usual cathode was replaced by a molybdenum rod. The emitting cathode was coaxial with this rod but separated from it. Surrounding the emitting cathode was a control cylinder with a n internal diameter equal t o t h a t of the anode.

MAGNETRON

FIG.

42. Simplified sectional view of injection magnetron.

The current t o the resonant-anode structure was controlled by the control-anode voltage. The mechanism of control may be explained as follows. In a conventional magnetron, omission of cathode-end shields causes large currents t o flow out the ends of the anode structure t o the lids of the tube a t anode potential. I n the injection magnetron the hat between the structures is omitted and the axial flow of electrons supplies a sheath of electrons around the molybdenum rod. This sheath is expanded by the anode voltage, which is higher than that on the control anode, and electron resonance and oscillation occur. As pointed out by Clogston, this process of axial repulsion by the space charge is t o be expected, for the space-charge density in the sheath surrounding the hot cathode is very large, possibly fifty times t h a t in a cylindrical diode without magnetic field. The calculated voltages corresponding t o the longitudinal motion have been found t o be a substantial fraction of the

* Figures 42 to 45 are adapted from figures in the report of A. M. Clogston, loc. c i t .


249

voltages corresponding t o the rotational motion of the electrons in the space charge.

2. Characteristics with Control-Anode Voltage Constant The mode of operation of the injection tube is best seen from the performance when the oscillating-anode voltage, V A ,is varied as would be done in a conventional tube. With constant magnetic field, and the rod and cathode voltages both zero, the control-anode voltage, Vc, was held a t 500 v. Typical results are shown in Fig. 43. The most notice-

200

600

1000

1400

I800

A N O D E VOLTAGE (VA)

FIG.43. Performance when the anode voltage of a n injection magnetron is varied.

able features are the current and power peaks over a range of V Aof about 200 v. At low V1 most of the electrons contribute t o the control-anode current, I,. I A then increases much as it does in a conventional tube before oscillation starts although in the case of the latter tube, in particular, such currents are not explained by theory. At the outset of oscillation there is (Fig. 43) the rapid increase in current and power t o be expected, followed by a decrease in both. The decrease in power, due the electronic angular velocity becoming too high, is only rarely seen in conventional tubes, but a decrease in efficiency is always seen, due either to this effect or t o the very high rf fields collecting electrons, as the cut-off voltage is approached, before proper interaction occurs. The current never decreases in a conventional tube although its rate of increase may decrease if the cathode does not fail first. The action of the injection system might be considered as analogous t o an emission limitation. The power peak in Fig. 43 occurs between the Hartree voltage ( V H )

250

J . S. DOWAL, JR.

and the cut-off voltage (V,) where it is usually assumed to appear in conventional tubes, although in the latter case oscillation often occurs, as pointed out by Clogston, below the Hartree voltage. 3. Performance as a Function of Control-Anode Voltage

From the viewpoint of amplitude modulation, the most significant results were obtained when the anode voltage was held a t a value near that giving the peaks in Fig. 43 and the control-anode voltage was varied. The result is shown in Fig. 44. As in Fig. 43 changes in the values of the nominally constant voltages change the details but not the general character of the curves. In Fig. 44, the power output rises smoothly

CONTROL ANODE VOLTS (Vc)

FIG 44. Performance of an injection magnetron as a function of control-anode voltage.

from a very low value, as the efficiency increases, and is nearly linear with V,. Above a low power the power output is very roughly proportional t o the current, as in conventional tubes. The saturation of the power curve sets in as the control-anode current increases, evidence that when the sheath diameter in the control system becomes rather large some electrons enter the main anode a t too great a radius for either efficient spoke formation or for return t o the cathode of electrons in such a phase as t o absorb energy from the rf field. It is to be noted particularly that the oscillator frequency is double valued and that its change is very small. This result is a plausible effect of the rod potential remaining constant. Over the same range of power the frequency change in a conventional tube might be expected t o be ten or twenty times as great.


25 1

4. Variation of Rod Potential Although the results so far obtained are very preliminary in nature, it is this form of operation which provides a linear frequency change. With V Aa t 1250 v and V , a t 500 v as in Figs. 44 and 43, respectively, the altering of V Rgave the variations of Fig. 45. The decrease of Po t o zero at negative and high positive values of VR would be expected from the behavior of the rod current, In. When IR is reduced t o below zero no electrons are returned t o the magnetron cathode and rejection of

ROD VOLTAGE (VR)

FIG.45.

Performance of a n injection magnetron as a function of rod potential.

unwanted power-absorbing electrons is faulty. When V Ris SO high that most of the injected electrons are collected directly, the number available for interaction is reduced. The frequency variation, while small compared t o the carrier frequency, is quite linear with V E ,and for some purposes the associated amplitude modulation would not be harmful. The range of frequency variation is probably susceptible of great improvement. 5. Low-Current Behavior of the Injection Magnetron

For most conventional tubes the leakage current t o the anode is 5 t o 10 per cent of the normal operating current a t the point of onset of the oscillation. Oscillation starts discontinuously a t a few per cent of the rated power output (see Sec. VIII), and the spectrum is likely t o be poor a t the very low powers. By contrast, the three or four samples of the injection magnetron tested by Clogston oscillated smoothly, when the control-anode voltage was reduced, down to 2 ma compared with the normal operating current of about 50 ma. The power decreased smoothly

252

J. S. DONAL, JR.

t o perhaps 100 mw out of the normal rating of 25 watts. Most important of all, the spectrum was not impaired at the low powers; on the cont rnry measurements showed that the spectrum improved a t low p o ~ ~ e r sBelow . a few hundred milliwatts the noise figure was comparable t o that of a lclystron. 6 . Evaluation

It should be realized that a t the time this is written the development work on the injection magnetron is incomplete. It is obvious that other types of injection mechanism could be employed. Only a few tubes have been tested and all these were of a similar structure. Adaptation t o other frequencies and powers will doubtless be made, with much more complete and improved results. The low noise levels found when operation is carried out a t very low currents offer several possibilities. The available depth of amplitude modulation is increased. The tube might be useful as a local oscillator. Finally, good low-current operation means t h a t such a tube for operation a t wavelengths of 1 ern or less could have a size large enough for convenient construction yet operate a t cathode current densities available from eyisting cathodes. Although the available frequency modulation may be considered small by some standards and the band of frequency deviation that is free from appreciable amplitude modulation may be considered narrow, both these aspects may well be improved by further development. During amplitude modulation the power output is proportional t o the control voltage, as in most conventional magnetrons. It would, of course, be more desirable t o have the square root of the power output proportional t o the control voltage. The required modulation voltage is higher than for plate modulation with ordinary tubes, but the anode current need not be carried by the modulator. The disadvantages cited in the preceding paragraph could in some cases be minor, however, compared with the advantage of the small frequency changr observed during the modulation cycle. This would simplify the problem of frequency control by control guns (Scc. VII) or injection locking (Sec. VIII). There are as yet no available data on control bandwidth characteristics peculiar t o the injection magnetron or, for that matter, on the bandwidth of the amplitude modulation. However, no inhcrrnt limitations are t o be expected. Pulsed injection magnetrons w o d d require less pulse power. In addition, the starting of oscillation a t lower levels might make locking of the pulsed tube, by injection from a controlled source, a less difficult procedure.

MODULATION O F CONTINUOUS-WAVE M A G N ETRO N S

253

X. CONCLUSIONS Each of the methods described above has been evaluated to some degree a t the ends of the respective sections. Space will not be taken here t o summarize the performance characteristics of the modulation systems. The above evaluations would perhaps bear rereading in the light of the performance criteria stated in the Introduction. It should be re-emphasized that much of this work is not only unpublished but is still under development. Present inadequacies of performance will doubtless be removed by further investigation. Work carried on outside of the United States has not been included in this discussion for the reasonthat the writer could not conveniently, as in the case of other methods, add t o the interpretation of the literature by personal contact with the originators of the work. Specific attention should be directed, however, t o the papers of Gutton and O r t ~ s i . ~ ~ , ~ ~ Spiral electron beams should perhaps be considered the standard method for frequency modulation, for the performance of the beams is nearly ideal except for limited deviation. Reactance-section tuning and voltage tuning are not yet fully developed, but they promise great improvement in the attainable deviation. Of the methods for amplitude modulation, the procedures employing spiral-beam absorption, the electron coupler and out-phase modulation are those most free from mixed frequency modulation. These are all absorption methods, however, with low efficiency averaged over the modulation cycle. Plate modulation has the desired high efficiency and feedback loop or injection locking should effectively eliminate the associated frequency pushing. The injection magnetron, also, would be expected to have high efficiency. The somewhat lower pushing of this tube should yield t o control by the methods suggested for use with plate modulation. Frequency-control procedures, making use of a feedback loop or injection locking, are important in themselves, since many other systems, including those based upon pulse modulation, could be improved by application of these control methods. The large amount of effort being devoted t o the modulation and control of magnetrons is evidence of the possibilities these tubes are considered to have for uses in which high powers and high frequencies are required. ACKNOWLEDGMENTS The following journals have kindly granted permission for the use of the figures listed: Reviews of Modern Physics, Fig. 2 ; Proceedings of the Institute of Radio Engi-

254

J, S . DONAL, JR.

neers, Figs. 3 to 10, inclusive, and 15 to 19, inclusive; RCA Review, Figs. 20, 21, and 22; lhecore is nonsaturated). On this basis a computation can be made with straightforward and elementary procedures. It must be remembered however that this assumption neglects hysteresis

274

WINFIELD E. FItOMM

losses, curvature of the corners of the magnetization curve, and the small but important finite inductance of the magnetometer in saturation. Experience shows t h a t these approximations, with the exception of the latter, can be made with the materials that are usually employed in the 70 r

-20 -30

-

UNBALANCE RESISTOR=480 OHMS

-40 -

-50 -60

-

-70

FIG. 7. Pulse output of unbalanced magnetometer bridge with zero external magnetic field.

magnetometer. The finite inductance present during saturation prevents the very steep discontinuities in current shown, for instance, in Fig. 8. With reference t o Fig. 8 the current is computed in the coil of a magnetometer whose characteristics are as follows: impedance in saturation, 120 j 0 ohms; impedance out of saturation a t 400 cycles, 240 j550 ohms; saturation level, 15 ma; internal impedance of generator, 120 j 0 ohms; maximum emf of the generator, 19.7 v (sinusoidal).

+

+

+

MAGNETIC A I R B O R N E D E T E C T O R

275

L

-90

FIG.8. Current waveforms in single-coil magnetometer with zero external magnetic field.

The curves drawn in Fig. 8 are as follows. Curve A is the current t h a t would flow in the magnetometer coil if it were always in saturation. We call this current the saturation current. Curve B is the current t h a t would flow in the magnetometer coil if its impedance were always that of the unsaturated case. We call this the below-saturation current.

276

WINFIELD E. FROMM

The solid line (curve C) is the net current that mill actually flow under the conditions listed above. Beginning from point P I the current is above saturation and therefore the current of the magnetometer is controlled by curve A down t o an angle of about 170” at which the magnetometer goes out of saturation. The current a t this point Pz cannot change suddenly because of the presence of the inductance and therefore a transient takes place along curve P2-P,. This transient is computed with reference t o curve B because the current tends to approach exponentially the “permanent” current curve represented by it. A t point P a the current has gone through zero and increased negatively to the negative saturation value a t point P,. The current then rises in a steep discontinuity to point Pq corresponding again to curve A, the saturation current. (Were the finite inductance actually present during saturation taken into account, the rise from P 3 to Pq would be exponential.) Curve A is followed until the magnetometer goes out of saturation at point Pg and a transient takes place from point P s t o point P s identical to that described for the transient from P z t o Pa. The previous procedure leads to the determination of the net current (curve C, solid line, Fig. 8) flowing in the coil of the magnetometer. Once this current is determined all problems regarding the application of a magnetometer can be solved. For instance, if one desires to determine the shape of the voltage across the magnetometer coil, one must subtract the voltage drop across the internal impedance of the generator from its generated emf. When this is done a voltage of the type shown in Fig. 9 will be obtained. In order to determine the behavior of a magnetometer in the presence of an external field, one must visualize the difference introduced in the curves of Fig. 8 by a change in the value of the current a t which saturation occurs. For instance, for the magnetometer of the characteristics described above, a negative field (i.e., one aiding a negative current) of 10 gammas requires an increase in the positive saturation level from 15 to about 15.0004 ma and a decrease of the negative saturation level from 15 to 14.9996 ma. An accurate graphical analysis for this case cannot be carried out of course without a large increase in the scale of the diagram, but an understanding-of the nature of the phenomenon can be obtained by direct observation of the current shape of Fig. 8. Assuming such a field is present, the current required for saturation will be increased a t points Pz and Pe and will be decreased a t points P , and Pa. For this reason the sharp increase in current from point Psto point P I will occur slightly later than the time shown in the diagram. The rise in current from point P3 to point P , will occur slightly earlier. For this reason the current shape is no longer 180” antisymmetrical; this means

MAGNETIC AIRBORNE DETECTOR

277

that what occurs around 76" in the positive cycle differs from what occurs 180' later a t 256" in the negative cycle. The tlitfcrence in timing of each rise is very small. For instance, a n exact aiialysis of the case shown in Fig. 8 would show that a field of 10 gammas in this magnetometer would cause a time difference of the order of 3 millimicroscconds. The effect of inductance during saturation can be computed t o a first approximation by assuming that these pulses are applied t o a circuit containing resistance R and inductance L in series. The result is that

r

L Voltage waveform across single-coil inagnctometer with zero extcriid

FIG. 9. magnetic field.

the width of the pulses is controlled by the time constant R / L of the circuit, while their amplitude is a function of the product of the width times the amplitude of the narrow pulses predicted by the simplified theory outlined above. If the balanced circuit of Fig. 5 is employed, the current present a t the detector output will consist of the difference between two curves each sirnilair t o curve C of Fig. 8 but with modified saturation levels because of the presence of an external field. The saturation levels of the two magnetometers are always shifted in opposite directions (one increased, one decreased) since the coils are wound in opposite directions. It is quite evident that a small shift in the value of the saturation current

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WINFIELD E. FROMM

levels will leave the regions of the curve between PI and P Zand between

P , and P 6 unchanged so that during the time corresponding t o the intervals between about 76" and 170" and between 256' and 350" the output a t the detector terminals is zero. At 76" and a t 256" there will be a pulse of about 65-ma amplitude, which according t o this simplified theory would be a few millimicroseconds wide. During the time interval corresponding t o the transients between P z and P B ,and P5 and P,, there will be a current yave shape corresponding to the difference between the transients, and opposite in polarity from the pulse immediately following it. \;Ii hile it is impossible to draw current shapes corresponding to the output of a balanced bridge for fields as low as 10 gammas, the shape of the current can be visualized if one refers t o that computed for much larger outside fields. For instance, for a n outside field of the order of 250,000 gammas the theoretical shape of the current a t the detector output has a form shown in Fig. 6 . Of course the width of the pulses is much larger, and in this case is equal t o 250 microseconds. It is obvious from a n observation of Fig. 6 that only even harmonics are present a t the detector output as was explained in a qualitative fashion earlier. The effect of an unbalance resistor RB across one of the windings of the balanced magnetometer bridge of Fig. 5 is much more difficult t o visualize because a resistance in parallel with one winding has the effect of changing several of the important parameters. I n this case winding L1 behaves as if it were supplied by a generator at a lower emf and lower internal resistance. It is possible by a n analytical approach to determine the actual shape of the current output in the case of small unbalances as those employed in practice; however, the complication of the mathematics is such as t o make it much more desirable t o resort again to a graphical illustration. For instance, in Fig. 7 is given the shape of the current a t the detector output for the case of the magnetometer bridge of Fig. 5 with unbalance resistor R3. The value of the unbalancing resistor is here assumed t o be 480 ohms. This is between 50 and 150 times lower than the values employed in practice. For this reason, while the shape of the current is representative of the effect of unbalancing resistors, the width of the corresponding pulse has been greatly increased. In the case of Fig. 7 it is apparent that since the computation was made assuming that no external field was present, the current wave contains only odd harmonics. The width of the pulse is about 125 microseconds. The effect of a n external field is t o increase the width of the pulses in proportion with the magnitude of the field in the same manner as described for the balanced bridge. The amplitude of the pulses in the case where the saturation impedance is purely resistive changes only slightly with an external field present. I n practical cases, however, the

.


279

amplitude becomes nearly proportional to the field because of the reasons given above. We must remember that in our analysis we have neglected the curvat)ure of the magnetization curve. This as well as the finite inductance of the magnetometer in saturation prevents steps of current as sharp as those shown in Fig. 8. I n practice, where these characteristics may not be neglected, the amplitude as well as the width of the pulses changes both for the balanced (even harmonic) and unbalanced (peak type) magnetometers. The finite inductance of the magnetometer in saturation is probably more important in determining the rise time of the steps than the curvature of the magnetization curve. The balanced output pulses of the unbalanced bridge definitely change therefore in amplitude as the external field changes. The change is in one direction, t h a t is, positive pulses increase and negative pulses decrease, or vice versa, as we would expect with the presence of even harmonics. The unbalanced bridge has an advantage over the balanced in that the signal pulses are above the “noise” level, even with zero external field. Consequently, it is often used as the magnetometer bridge in both detector and servo applications. I n this section we have indicated how the saturable-core elements may be used as magnetometers, either singly or in bridges. Two types of bridges have been described, the balanced and the unbalanced. The unbalanced is characterized by a n electric unbalance purposely inserted by means of a resistive (or sometimes reactive) element, and the output of which consists of pulses symmetrical about the base line in the absence of a n external field. The balanced bridge is characterized by even harmonic output only in the presence of an external magnetic field. The methods by which both of these bridges, particularly the unbalanced one, can be used in magnetic stabilization are described in the next section. V. MAGNETIC STABILIZATION A N D ORIENTATION Each of the five types of magnetometers suitable for airborne use produce outputs that are proportional t o that component of field parallel or perpendicular t o some specific axis of the magnetometer, except for the coil inductor whose output, as we have seen, is proportional t o the change in a component of the field. Because of this and the fact that the earth’s magnetic field is always present as a strong ambient field, a magnetometer must be stabilized accurately if the detector noise level is t o ’ b e kept low enough so that very small signals can be identified. This section will consider some of the aspects of magnetic stabilization, the importance of proper orientation of the magnetometer, methods of

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W I N F I E L D E. FROMM

magnetic stabilization and orientation, and special methods by which the stabilization requirements may be relaxed. 1. Stabilization and Orientation in the Ea,rth’s Magnetic Field Figure 10 shows a simplified diagram of a saturable-core magnetometer stabilized with respect t o the earth’s field. Usually i t is desired t o stabilize the magnetometer parallel t o the earth’s field, and that is the intention in Fig. 10. However, a n initial misalignment angle 4 and a dynamic stabilization error angle 6 can both affect the orientation angle of the magnetometer. These angles are indicated in the figure.

/”

EARTH’S MAGNETIC FIELD

MAGNETOMETER

0=

INITIAL MISALIGNMENT ANGLE OF MAGNETOMETER WITH EARTH FIELD VECTOR

8 = DYNAMIC STABILIZATION ERROR ANGLE

FIG. 10. Diagram showing initial misalignment and dynamic stabilization error angles of a detector magnetometer.

In some geophysical applications it is necessary t o measure continuously the magnitude of the earth’s field. I n such cases the initial misalignment angle 4 causes a direct, static error equal t o H(1 - cos 4). If a stabilization error angle 6 exists, the dynamic error then becomes H[1 - cos (4 6)]. This may be greater or less than the static error but, t o a first approximation, its average value will equal the static error. If we are interested in measuring only the changes in the earth’s field (or, more strictly, the external field H ) , as is the case in submarine detection from aircraft and in some geophysical applications, the dynamic error is the only one of significance. It is, however, a function of both angles 4 and 6. I n this case the noise signal caused by imperfect stabilization of the magnetometer is

+

MAGNETIC A I R B O R N E D E T E C T O R Xn =

H[COS 4 - cos (4

= H[cos 4 (1

where

+ e)] + sin 4 sin el

- cos 0 )

28 1 (2) (31

X, = noise signal H 4

magnitude of earth's magnetic field initial misalignment angle of magnetometer with the earth field vector 0 = dynamic stabilization error angle. From (2) we note first that if e = 0, the noise signal is zero, no matter what the misalignment. Secondly, if 0 is small (as i t is in practice) S,, as given by (3), is a minimum when 4 = 0" and a maximum when 4 = 90". Thus, if 4 = O", S,, = H(1 - cos e), and, if 0 is small, X,, = H ( l - 1 P / 2 ) = H 0 2 / 2 H , where e is in radians. If 4 = go", S,,, = H sin 8, and, if 0 is small, S,,, = H e , where 0 is in radians. The equations for S , a t 0" and 90" are of fundamental importance in magnetic stabilization. From them we obtain the ratio = =

+

A common value for 0 is about 5 minutes, or 0.0015 radian, in which case the ratio S,,,/S,, = 1330. This ratio indicates clearly the tremendous advantage t o be gained in aligning or orienting the detector magnetometer parallel t o the earth magnetic field vector. When the magnetometer is oriented exactly parallel t o the earth's field, the equation for X,, may be used t o give directly the noise signal caused by imperfect stabilization. Figure 11 is a graph of noise sig,ial versus dynamic stabilization error angle for this case. The earth's field has been assumed t o have a typical value of 50,000 gammas. Although (4) has indicated the importance of orienting the magnetometer parallel t o the earth's field, i t does not show the accuracy with which the orientation should be made. From (3), however, we see t h a t if 0 is very small and 4 is small but not zero, S,

=

H

(g +

98)

(5)

Figure 12 is a graph of noise signal versus 4, the angle by which the magnetometer is misaligned from the earth's field, for a dynamic stabilization error angle of 0.0015 radian (5 minutes). The earth's field has again been assumed t o have the typical value of 50,000 gammas. From this graph we see that very precise orientation is required for this type

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W I N F I E L D E . FROMM

e = STABILIZATION

/ 0

2

4

6

/

ERROR, RADIANS

MAGNETOMETER ORIENTED PARALLEL TO EARTH FIELD

I

I

I

I

I

1

8

I0

12

14

16

la

DYNAMIC STABILIZATION ERROR IN MINUTES

FIG.11. Noise versus dynamic stabilization error. 05

r SN= H p * + H 8 0 H: 50,000 GAMMAS

8 = .0015 RADIAN ( 5 MINUTES)

0

I

I

I

I

I

I

I

I

I

MISALIGNMENT ANGLE 0 IN MINUTES

FIG.12.

Noise versus detector magnetometer misalignment angle.

of magnetometer if the noise signal is t o be kept small. For instance, in this example a misalignment of only 2.3 minutes doubles the noise from 0.05 gamma t o 0.1 gamma. It should be realized that the discussion thus far and Figs. 10, 11, and 12 apply only t o stabilization of the magnetometer in one plane, or in one


283

degree of freedom. The detector magnetometer actually must be stabilized in two degrees of freedom. This is normally done by stabilizing a platform mounted in gimbals similar to those used in compass supports. The two perpendicular axes of rotation ordinarily are called the pitch and roll axes. Assuming that the stabilization error is the same in each degree of freedom and that the two errors are maximum a t the same time, the resultant and effective stabilization error angle is 1.4 times the error of each; This angle may then be used with Figs. 11 and 12 to give the resultant noise signal. One further point should be mentioned. It so happens that most of the energy in the noise due to stabilization errors occurs in the same restricted frequency spectrum as the energy of the submarine or geologic signal. The signal-to-noise ratio is therefore little affected by the frequency characteristics of the detector circuits, and the noise levels given by Figs. 11 and 12 are representative of those obtained in practice from stabilization and orientation errors. 2. Methods of -Magnetic Stabilization and Orientation

The saturable-core magnetometer is ideally suited to function as a stabilization control element because of its small size and high sensitivity. Furthermore, it was shown in See. IV that both the peak and even harmonic types of magnetometers supply amplitude and sense information when operated about the zero value of axial field. At this zero value the peak type magnetometer produces balanced output pulses of opposite polarity. If an axial field appears (such as might be caused by the magnetometer rotating out of a plane perpendicular to the earth’s field) the absolute difference of the pulse heights is proportional to the axial field. The polarity of the difference is determined directly by the polarity of the field. In a similar fashion, the even harmonic magnetometer produces an even harmonic output proportional to the axial field. The phase of each even harmonic reverses as the axial field reverses in polarity. The outputs of both magnetometers therefore possess the necessary characteristics of amplitude and sense for servo control. It is highly desirable to position the stabilizer magnetometers in the manner just described, because, as shown above, maximum useful output is obtained for a given stabilization error when the magnetometer is perpendicular ( 4 = 90’) to the earth’s magnetic field. In general, then, the detector magnetometer should be oriented parallel to the earth’s field to minimize noise due to stabilization errors. Conversely, stabilizer magnetometers should be mounted in a plane perpendicular to the earth’s field vector H to provide servo operation with maximum sensitivity.

284


Figure 13 is a schematic diagram of a magnetically stabilized magnetometer head employing the principles just outlined. The stabilized platform Carrying detector magnetometers D-D and stabilizer magnetometers 1-1 and 2-2 is mounted on a shaft driven by rotary drive 1. We may call the axis of this shaft the pitch axis. The pitch axis shaft is carried in bearings mounted in the roll gimbal which is driven by rotary drive 2 . The axes of rotary drives 1 and 2 are frequently called respectively the inner and outer axes. All the magnetometers are connected as pairs in bridge circuits for increased sensitivity. Magnetometers 1-1 control rotary drive 1.

MAGNETIC FIELD ROTARY DRIVE

FIG.13. Schematic diagram of a magnetically stabilized magnetometer head.

Whenever a component of H exists parallel to their lengths, a signal (pulses or even harmonics) is produced. This is directed to a servo amplifier that usually drives an electric motor (though it may control a gyroscopeg) connected t o rotary drive 1. The sense of the resulting rotation is such as to reduce the component of H along the magnetometers t o zero. Magnetometers 2-2 control rotary drive 2 in a similar fashion. Thus the magnetometers 1-1 and 2-2 maintain themselves and the stable platform perpendicular to H despite the movements of the aircraft in flight. The detector magnetometers D-D mounted normal to the stable platform are thus maintained parallel to H . In Fig. 14 is shown a photograph of one type of magnetometer head used extensively for submarine detection during World War 11. The various parts of this head are easily identified. Figure 15 is a photograph


285

of the complete head and shows the driving motors a t the lower end of the head frame. The head and frame must be essentially nonmagnetic t o avoid unwanted magnetic fields a t the detector magnetometer. For this reason the slightly magnetic motors are mounted some distance away from the magnetometers.

FIG.14. Two-axis magnetometer head showing pitch and roll axes.

The possibility of using either peak or even harmonic magnetometers for stabilizer elements has been noted. Both types have been used in practical systems. The even harmonic system suffers the disadvantage of requiring substantial filtering t o accept the desired even harmonic and t o reject unwanted even and all odd harmonics. For instance, if the second harmonic is accepted, both the fundamental and third (as well

286

WINFIELD E. FROMM

as higher) harmonics must be rejected. The amplitudes of both of these harmonics always are much larger than the small second harmonic signal; therefore excellent filters (in this case, sharply tuned bandpass filters) are needed. However, the severe phase shifts introduced by such filters must be carefully considered from the standpoint of servo stability. Ordinarily, these phase shifts complicate considerably the design of high-performance servos. In general, peak magnetometer systems are less complicated, require fewer components, and give satisfactory performance. A combined schematic-block diagram of the electronic portion of a single stabilizer channel with peak magnetometers is shown in Fig. 16. Conventional portions of the circuit are shown in block form. The diagram shows that the output pulses of the magnetometer, occurring at a pulse repetition frequency of f cps, are amplified and passed through a diode charging circuit that includes the R-C network. This network acts both as a pulse stretcher and an antihunt circuit for the servo system. The balanced sawtooth voltages, 180" out of phase, are combined in the mixer circuit. When the sawtooth voltages are equal, the mixer output contains only a sawtooth voltage of frequency 2f. However, a slight unbalance of pulses due to stabilization error results in a fundamental component (f cps) a t the mixer output. The phase of the fundamental component depends upon the direction of the stabilization error. The fundamental component is filtered, amplified, and applied to the control field of a high-performance two-phase FIG. 15. Two-axis servo motor with a low-inertia rotor. The magnetometer head with system is so phased that the rotor turns in a servo motor assembly. direction that reduces the stabilization error. In analyzing the stabilizer servomechanism, two distinct aspects of the problem must be considered. The first is the static condition in which the aircraft is motionless. Here the system is an electromechanical closed loop and it may easily be unstable, in which case the supposedly stable platform will oscillate violently. Antihunt circuits in the system usually provide reasonable stability for this condition. The second


'

287

condition is the dynamic one, in which the aircraft imparts translatory and semi-rotary motion t o the magnetometer frame. Under these dynamic conditions, the platform must not rotate (with respect to the earth's field), except for very small stabilization errors. The forcing or input function is the displacement of the frame. The output function is the equivalent displacement of the motor shaft after a mechanical reduction. The difference of the input and output functions is exactly equal to the small displacement of the stable platform and is the stabilization error. It is important to realize that in the dynamic condition one is primarily interested in dynamic response, or minimum stabilization

I

I I I

I

I

a

FIG.16. Schematic-block diagram of stabilizer channel with peak type magnetometer bridge.

error, and that the inertia of the stable platform is directly coupled to the error shaft of the system. Thus, increasing the inertia of the stable platform and decreasing the friction of the bearings will effectively improve the dynamic response of the system. It is interesting to note the restrictions on the use of the two-axis gimbal system for stabilizing the detector magnetometer. There are actually three possible mountings for this system, and these may be defined with respect to the airplane as follows: (a) outer axis horizontal and in line with the direction of flight; ( b ) outer axis horizontal and perpendicular t o the direction of flight; (c) outer axis vertical and perpendicular t o the direction of flight. Mounting (a,) is that used in the head of Figs. 14 and 15. This mounting is especially useful in regions where the magnetic dip angle is greater than 45". For dip angles below this, and especially for small or zero dip angles near the magnetic equator, operation of the system will

288

WINFIELD E. FROMM

fail in a lateral turn of the aircraft. This is because a unique position of the gimbal system about the outer axis is not defined when the detector magnetometer is in line with the outer axis. This undefined position occurs for instance on a north or south heading a t the magnetic equator. I n such a case, the slightest deviation in heading from north or south immediately defines a unique position of the system about the outer axis, and very high gimbal velocities will be required t o stabilize the platform properly without introducing noise signals due t o stabilization errors. Practical systems are unable t o correct errors this rapidly, SO that "gimbal locking" must be avoided. This is done by operating mounting (a) only a t the higher dip angles. Mounting (b) has limitations similar t o those of (a,) and a n additional one. I n (b) gimbal locking also occurs when the detector magnetometer is in line with the outer axis. This alignment would occur a t the magnetic equator on a n east or west heading. However, this could also occur for the same headings in a region of 45" dip angle if the aircraft banked 45" in a direction such as t o cause the axes of the detector magnetometer and outer gimbal t o coincide. Because bank angles are higher than pitch angles [by which ( a ) is similarly affected] mounting ( b ) is less practical than ( a ) and operation with i t is limited t o dip angles above about 60". Mounting (c) is limited t o equatorial regions where the dip angle is small. With this mounting the complement of the dip angle must always exceed the angle of bank. Assuming a bank angle of 45" and a leeway of 15", the mounting is satisfactory only for regions with a dip angle of less than 30". A further disadvantage of ( c ) is that successive turns of the aircraft in one direction require continuous, unobstructed rotation about the vertical axis of the gimbal system. This bars the use of pigtail electrical connections and requires slip rings. One advantage of (c) is that only a 90" change in mounting converts an equatorial unit t o a polar unit. None of the mountings ( a , b, or c) is satisfactory in itself for all dip angles. From the preceding discussion it is evident that a ('universal" head that will function a t all dip angles can be made by adding a third gimbal t o the two gimbals already described. The three gimbals of such a head are shown in Fig. 17. The entire head is shown in Fig. 18. This head employs a two-axis gimbal system with mounting ( b ) , but with a n added third gimbal axis horizontal and in line with the direction of flight. Neither the first (inner) nor second (outer) is fixed with respect t o the aircraft; both may rotate about the third axis. The function of the third axis is t o defeat the tendency of the detector element t o come in line with the outer axis a t one particular heading. T o

MAGNETIC A I R B O R N E DETECTOR

289

accomplish this, motion of the gimbal system about the third axis is made a function of the movement about the first axis of the detector element from its neutral position (with respect to the outer axis). When motions

FIG.17. Three-axis magnetometer head showing pitch, roll, and third axes.

about the first and third axes are set for approximately one-to-one ratio, these two axes share equally the movement of the aircraft relative to the earth’s magnetic field, and alignment of the detector element and outer axis is impossible. The gimbal system of the universal head therefore functions satisfactorily a t all magnetic latitudes. In reality, the third axis provides a variable combination of mountings ( b ) and (c). The motion about the third axis need only be approxi-

290

WINFIELD E. FROMM

mately as fast as the average rate of aircraft turn or bank. Thus the third axis servo need not be one of high performance. Control for the third axis motion may be achieved in several ways. The head in Figs. 17 and 18 employs pick-up coils mounted on the second axis gimbal. The alternating magnetic field of the driving current in the detector magnetometer coil induces a voltage in the pick-up coils, and this voltage is used as part of the control signal for the third axis servo. Alternative methods of control include the use of low-friction switches or potentiometers mounted on the second axis gimbal, with the variable arm of the switch or potentiometer controlled by motion about the first axis. The control system is ustially complicated by the necessity of introducing for comparison an electrical signal representing the position of the third gimbal with respect to the head frame. 3. Special Methods

a. Three-Component Clnstabilized Tota.1 Field Magnetometer. In a three-component unstabilized total-field magnetometer there are three similar, mutually perpendicular magnetometer elements mounted rigidly in a framework that is unstabiliaed. The outputs of the three magnetometers are squared and added; the resultant is then proportional to the square of the total magnetic field. Such a system has been used to explore the earth's magnetic field at very high altitudes.'O Although the three-component system is simple, operation a t very low noise levels places stringent requirements on its electrical and mechanical FIG. 18. Three-axis characteristics. For instance, an error of 4 minm a g n e t o m e t er head utes in the perpendicularity of one magnetometer w i t h servo m o t o r can result in 0.1 gamma noise with an ambient assembly. field of 50,000 gammas. For the same noise, the allowable departure from perfect squaring for any single magnetometer output is only 0.001%. Requirements on linearity and summing are also very strict. Because of these practical difficulties, the three-component unstabilized magnetometer has been used only where relatively insensitive detection is adequate.

MAGNETIC A I R B O R N E D E T E C T O R

29 1

b. Three-Component Stabilized Total Field Magnetometer. The requirements on alignment, squaring, etc., of the three-component magnetometer are eased considerably if the system is stabilized with respect t o the earth's magnetic field in a manner somewhat similar to that described earlier. Two of the orthogonal elements are used as stabilizer elements and supply signals of such phase and amplitude that continuous control servos maintain the elements perpendicular to the ambient field. The third orthogonal element is thus maintained parallel to the ambient field exactly as in simpler stabilized magnetometers. However, each of the three elements is also a part of the detection system and the squared outputs of each are added t o give a quantity proportional t o H 2 . A magnetometer of this type was developed and used during World War I1 and has been employed since 1945 in geophysical surveying." The thrcc-component stabilized magnetometer has the advantage over the single-detector stabilized magnetometer in that stabilization requirements are eased somewhat if accurate squaring and proper alignment between elements can be achieved. We have seen t h a t the noise of the single-detector stabilized magnetometer is

8, where 0

=

H

(a +

$0)

(5)

dynamic stabilization error angle misaligiimciit angle of magnetometer with the earth field vector. The corresponding expression for the three-component stabilized magnetometer can be shown t o be (assuming perfect squaring, summing, and linearity) S , = H+O (6) =

+ = initial

Here 4 is defined as above with the understanding that it is a t the same time the misalignment angle of the third element with respect to a true perpendicular t o the two stabilizer elements. I n other words, the stabilizer elements are oriented exactly perpendicular to the earth's field. It should be noted a t this point that in the case of the single-detector stabilized magnetometer, can be made zcro or very small by the simple expedient of static electrical adjustments on either or both servomechanism channels (first and second axes). However, this cannot be done in the case of the three-component stabilized magnetometer where the magnitude of is determined solely by the mechanical misalignment of the magnetic axes of the magnetometers. I n practice this is usually at least several minutes of arc and may easily be much more.

+

292

=INFIELD

E. FROMM

Equations (5) and (6) are plotted as curves A and B respectively in Fig. 19. For simplicity, i t is assumed that the misalignment angle 4 is 2 minutes in both cases, but it should be remembered t h a t although this is easily achieved in the single-detector case, it is a very optimistic figure for the three-component case. I t is evident that the three-component system permits greater stabilization error for a given noise, particularly for relatively large noise levels. However, for small noise levels (below 0.1 gamma) there is little practical difference in the servo accuracy required in the tn-o systems. When the added requirements of accurate 07

06

.

t i = 50,000 GAMMAS 0 = MISALIGNMENT ANGLE = 2 MINUTES (ooO6 RADIAN) 8 = STABILIZATION ERROR, RADIANS

05 ul

5

04

0

G

#

0

03

02

STABILIZED SINGLE-DETECTOR

01

0 2

4

6

8

10

12

14

16

18

DYNAMIC STABILIZATION ERROR IN MINUTES

FIG. 19. Xoise versus dynamic stabilization error for stabilized single-detector and three-component niagnctomcters.

squaring, summing, etc., in the three-component magnetometer are considered, it appears that the advantages o f its use in very high-performance magnetic airborne detectors are not outstanding.

TI. ~ I A G X C .IIRI~ORSE TIC DETECTOR Swrm I n this section we shall consider the overall system of a universal magnetic airborne detect or employing a single-detector stabilized magnetometer. This is tlic type used most extensively up t o the end of M’orld War I1 for military uses and is of a kind employed today in geophysical prospecting. The block diagram for the single-detector magnetometer employing

293


three axes of stabilization is shown in Fig. 20. The equipment consists of four principal units: (1) universal head, ( 2 ) drive amplifiers, (3) servo amplifiers, and (4)detector. These will be considered briefly. 1. Universal Head This is the magnetically sensitive portion of the equipment. It is usually mounted on the aircraft in a region of low spurious magnetic fields and gradients. It may also be towed in a nonmagnetic housing behind the aircraft. The head contains the detector and stabilizer magnetometers, a n appropriate gimbal system with three axes, and a PRIMARY POWER-I

-

i I

REGULATED POWER WPPLY

I

PULSE OR EVEN I

AMPLIFIER

I II

STABILIZER

PHASE 2 0 MOTOR

DETECTOR

131 AXIS

MOTOR

PHA5E

wixlsz 0

THIRD A X I S SENSE COIL

MOTOR -

1

I _ _ _ _ _ _ _ _ _!?1IE_R_sLL?kXJ%LEE--_-___-1 L

FIG.20.

Block diagram of universal MAD system.

servo motor for each axis. The motors are mounted a t some distance from the magnetometer t o reduce undesired magnetic effects. The gimbal system of the universal head was described in Sec. V. The system is "universal" in that one mounting is adequate for all magnetic latitudes. 2. Drive Amplifiers This unit includes a stable audio oscillator operating with negligible harmonic distortion a t a compromise frequency satisfactory for both magnetometers and servo motors. The oscillator output is amplified in the three separate amplifier stages to drive respectively the detector magnetometers, the stabilizer magnetometers, and the fixed phases of the three servo motors. The detector magnetometer drive amplifier

294

WINFIELD E. FROMM

must be carefully designed t o achieve the stability.and very low harmonic distortion essential for sensitive detection. 3. Xeroo Amplifiers

The universal MAD has three servo amplifiers. The first (inner) axis and second (outer) axis servo amplifiers are normally driven from peak magnetometers as shown in Fig. 16. These two amplifiers must be carefully designed t o make possible the servo accuracy of a few minutes of arc required for a high performance MAD. This servo accuracy must be maintained for all normal maneuvers of the aircraft. The third axis servo amplifier is normally driven from a pick-up coil sensitive in phase and amplitude t o the detector magnetometer driving flux. This servo amplifier is a relatively insensitive one because the response of the third axis servo system need only be approximately as rapid as the average maneuver rate of the aircraft.

4. Detector A magnetic airborne detector is in reality a magnetic anomaly detector. The MAD indicates the presence of a n anomaly in the earth’s magnetic field. Such a n anomaly (caused by a submarine) is illustrated in Fig. 1. An anomaly may also be caused by variations in the magnetic characteristics of the earth’s surface that cause fairly extensive effects on the main field of the earth. The extent of the anomaly may thus be localized, as in the case of the submarine, or it may cover a wide area, as in the case of geological effects. I n any event, the detector portion of the system must be designed t o detect signals from desired anomalies and reject signals from undesired anomalies. The manner of detecting a magnetic anomaly, as we have sren earlier, is to detect the component of the anomaly field parallel t o the earth’s field, and thus parallel t o the longitudinal axis of the detector magnetometer. The magnitude of this component affects directly the output of the magnetometer bridge. Modulation of the output appears as a difference in the pulse heights of a peak type detector, or as a difference in the even harmonic output of the magnetometer bridge. It should be noted here that in describing MAD systems, the term ‘ I detector magnetometer” is used t o include both single-coil magnetometers and magnetometer bridges, since both types are used in practical systems. Both peak and even harmonic types of detection have been employed. The block diagram of Fig. 20 indicates both. I n either case, the carrier (pulses with pulse repetition rate f , or a n even harmonic off) is amplitudemodulated a t a signal frequency determined by the extent of the anomaly


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and the altitude and speed of the aircraft. The amplitude-modulated carrier is amplified, then demodulated. The modulation or signal frequency may vary from essentially zero to a few cps. Consequently the following d-c or very low frequency amplifier incorporates a bandpass filter designed to accept desired signals and reject undesired ones. I n general, for anomalies of great extent such as are encountered in geophysical surveying, signal frequencies are lowest and d-c amplifiers with low-pass filters are used. For more localized anomalies, bandpass filters are employed. The very low-frequency signal is usually amplified sufficiently to drive a recorder producing a permanent inked trace. The time scale of the recorder is useful in mapping the magnetic contours.

VII. THENOISEPROBLEM The noise or interference th at appears as continual or intermittent fluctuation in the MAD output is important because it defines the smallest detectable signal. There are three classes of background interference: (1) instrumental noise, (2) aircraft noise, and (3) noise from sources external to the aircraft. 1 . Instrumental Noise

Instrumental noise originates within the MAD itself. It may be divided into four types. a. Electrical Noise. As in all high-gain systems, fluctuations in supply voltages or microphonics in vacuum tubes may cause noise. b. Magnetic Noise. Unless transformers, inductors, and vacuum tubes in low-level portions of the circuits are magnetically shielded, movement or tilt of the equipment in the earth’s magnetic field may cause noise. c . Stabilization Noise. This was discussed in detail in Sec. V. The noise t o be expected for a stabilized single-detector system is plotted in Figs. 11 and 12. d. Magnetometer Noise. The noise from the saturable-core magnetometer (or any other detecting device) is basic and represents the absolute minimum noise level for the system. I n a saturable-core magnetometer there is a small noise output due to thermal agitation effectsI2in the coil, but a more serious noise results from the Barkhausen effect.12,13,14 It is well-known that the process of magnetization is a discontinuous one because of the domain structure of ferromagnetic materials. It has also been demonstrated (Zoc. cit.) that a representative magnetization

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curve for these materials can be divided into three parts: (1) the initial portion where reversible domain boundary displacements occur, ( 2 ) the middle portion where irreversible boundary displacements occur discontinuously, and (3) the upper portion where reversible rotation of the domain magnetic moments occurs smoothly. Barkhausen discontinuities occur when the magnetization is changing along the steep, middle portion (2) of the magnetization curve. The discontinuities correspond t o the irregular fluctuations in the motion of the domain boundary (termed the Bloch wall) as the applied magnetic field is changed. The Barkhausen discontinuities are not repeated exactly from cycle t o cycle, so that the amplitude and phase of the harmonics generated by the nonlinearity of the magnetometer are altered from one cycle t o the next. The random variation in amplitude and phase of the harmonics appears as noise in the magnetometer output. Saturable-core magnetometers have been built with a n equivalent noise level of the order of 0.02 gamma for a bandwidth of 1 or 2 cps. 2. Aircraft Noise Aircraft (or carrier) noise is defined as that caused by a relative movement of the MAD head (carrying the detector magnetometer) with respect t o magnetic fields generated by ferromagnetic or conducting parts in the aircraft. The ferromagnetic parts (engines, landing gear struts, etc.) may have permanent or induced moments that produce strong fields a t the detector. Movements of the aircraft may cause the component of the resultant field parallel t o the magnetometer axis t o change considerably, thus causing noise. Conducting parts (wing surfaces, etc.) may support eddy currents caused by movements of the aircraft in the earth’s magnetic field, and the magnetic fields set up in turn by these eddy currents may cause serious noise signals. Aircraft noise in general may be reduced by locating the MAD head appropriately on the aircraft in a region of small spurious magnetic fields and gradients. Such a region may be found a t the wing tip or aft of the tail structure. The noise may be greatly decreased by towing the MAD head in a nonmagnetic streamlined housing well behind or below the aircraft. Both types of installation have been employed. I n aircraft installations where spurious fields are not small, noise signals may be reduced by applying a systematic compensation technique t o cancel all or most of the locally generated fields a t the detector magnetometer. Considerable success has been achieved with this procedure, hut it is not a simple one. Small permanent magnets must be oriented properly t o neutralize fields due t o permanently magnetized parts, strips


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of high permeability material are oriented t o neutralize fields due t o induced moments, and fields due t o eddy currents are neutralized by the magnetic field of a n output coil controlled by the amplified signal of a properly oriented pick-up coil. The compensation procedure must be repeated a t frequent intervals because of changes in the magnetic characteristics of the aircraft structure. 3. Noise f r o m Sources External to the Aircraft The noise is caused by time and space variations in the magnetic field of the earth. a. T i m e Variations in the Earth’s Fzeld. An exhaustive survey over the entire frequency spectrum of time variations in the earth’s field has not been made. The amplitude of such variations will depend t o a considerable extent on the geographical location. For instance, in high latitudes time fluctuations may be of very high amplitude, particularly in the zones of maximum occurrence of aurora, while in the temperate or equatorial zones the fluctuations may be quite small. b. Space Variations zn the Earth’s Field. Space variations may be divided into two types: uniform and nonuniform. Uniform variations are those due t o the normal vertical and north-south horizontal gradients of the earth’s field. Nonuniform variations are those caused by geological discontinuities in the earth’s crust with resultant variations in the earth’s magnetic field. I n geographical prospecting signals caused by such variations may or may not be regarded as noise. The vertical gradient of the earth’s field lies between 0.005 gamma per foot a t the magnetic equator and 0.01 gamma per foot a t the poles. Altitude fluctuations of 10 ft can therefore produce noise signals of about 0.1 gamma. The north-south horizontal gradient is about 0.0015 gamma per foot, and fast maneuvers may produce some noise from this source. The signals due t o variations in geologic structure may be very large. For instance, over shallow water or land where magnetic rocks are very close t o the surface, the magnetic anomalies may have amplitudes of hundreds of gammas. Small signals are usually completely masked in such regions. On the other hand, over deep water or in regions where the magnetic rocks are a t great depth, the geologic anomaly may not be measurable. I n geological surveys, of course, space variations in the earth’s field are of vital importance. By using MAD systems with direct-coupled detectors it is possible t o plot fairly accurate magnetic contour maps of a region. From these maps certain areas can be pinpointed for more detailed ground surveys. I n such work, naturally, signals due t o the space variations are the desired signals, and all other signals are regarded as noise.

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VIII. CONCLUSION I t is evident from our consideration of the characteristics of magnetic fields from ferromagnetic bodies, and the spurious magnetic fields which may result from various noise sources, that the magnetic airborne detector is essentially a low-range device. The actual range of detection depends of course on the magnitude of the magnetic moment causing the anomaly in the earth’s field. The magnetic airborne detector will undoubtedly find increased use in future geophysical prospecting. It is often possible (though not always) t o save considerable time and money in examining vast areas, particularly when covered by water, if aerial magnetic surveys are made rather than by using the older ground methods of prospecting. The usefulness of the magnetic method depends however upon the existence of a recognizable relationship between the magnetic anomalies and the geological structure being sought, and also upon the random space variations in magnetic field (noise) due to superficial rocks and soil over the terrain being surveyed. The extreme sensitivity of the saturable-core magnetometer has been described. Applications will be found for laboratory and industrial instruments using this device in the detection and measurement of extremely small magnetic fields and direct currents. Detection of fields as low as 0.015 gamma with toroidal magnetometers has been reported12 by the British, and several laboratories in this country have achieved similar results. Direct currents of below 7 x 1O-lo amp, corresponding t o about lo-’’ watt, have also been measured using these instruments.

ACKNOWLEDGMENTS During World War I1 many individuals, both in the United States and Great Britain, contributed greatly to the development of the magnetic airborne detector. Their achievements should be recognized despite the fact that most of their wartime reports must remain unpublished. The author is indebted to Dr. J. W. Joyce, formerly of the Bureau of Aeronautics, Department of the Navy, and presently with the State Department, Washington, for his helpful suggestions regarding this article. R. F. Simons, I. Kasindorf, and F. Rockett of Airborne Instruments Laboratory, Inc., have made valuable critical comments. Special thanks are due Dr. E. G. Fubini of the same laboratory for the material of Sec. IV and for his continuous help and encouragement. REFERENCES 1. Chapman, S., and Bartels, J. Geomagnetism; Vol. I. Oxford University Press, London, 1940. 2. Heiland, C. A. Geophysical Exploration. Prentice-Hall, Inc., New York, 1940.


3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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Thomas, H. P. U. S. Patent No. 2,016,977 (1931). Vacquier, V. V. U. S. Patent No. 2,406,870 (1946). Squier, R. T. Electronics, 20, 121 (1947). Wurm, 51. Z. angew. Physik, [2] 6, 210 (1950). Wyckoff, R. D. Geophysics, 13, 182 (1948). Bemrose, J., Heggblom, J. C., Holt, T. C., Richards, T. C., and Watson, R. J. Geophgsics, 16, 102 (1950). Vacquicr, V., Simons, R. F., and Hull, A. W. Rev. Sci. Instruments, 18,483 (1947). Maple, E., Bowen, W. A., and Singer, S. F. J . Geophys. Research, 66, 115 (1950). Felch, E. P., Means, W. J., Slonczewski, T., Parratt, L. G., Rumbaugh, L. H., and Tickner, A. J. Elec. Eng., 66, 680 (1947). Wil!iams, F. C., and Noble, S. W. J . Inst. Elec. Engrs. (London), 97, Pt 11, 445 (1950). Bozorth, R. M. Revs. Modern Phys., 19, 29 (1947). Kittel, C. Revs. Modern Phys., 21, 541 (1949).


Multichannel Radio Telemetering M. G. PAWLEY

AND

W. E. T R I E S T

National Bureau of Standards, Corona, California, and International Business Machines Corporation, Poughkeepsie, New York CONTENTS

Page 301 11. Evolution of Radio Telemetering.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 1. Television Telemetering .... . . . . . . . . . . . . . . . . . 303 ........................ 2. Instrument Telemeterin 3. Oscillogram Telemetering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Basic Systems of Radio Telemetering ........ . . . . . . . . . . . . 304 1. Frequency-Division Telemetering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 2. Time-Division Telemetering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 a. Pulse Position Modulation and Demodulation. . . . . . . . . . . . . . 306 b. Pulse Interval Modulation and Demodulation. . . . . . . . . . . . . . 309 3. Factors Determining Choice of Telemetering System.. . . . . . . . . . . . . . . . . 311 IV. Typical Telemetering Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 1. Early Telemetering Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 2. Example of Frequency-Division Telemetering System. 3. Examples of Time-Division Telemetering System. . . . . . . . . . . . . . . . . . . . . 316 a. Typical Pulse Position-Modulated Telemet,ering System. . . . . . . . . . . . 31 6 b. Typical Pulse Interval-Modulated Telemetering System. . . . . . . . . . . . 320 c. Typical Mechanically Commutated Telemetering System. . . . . . . . . . . 325 V. Future Trends in Telemetering., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I. INTRODUCTION A telemeter is defined b y Websterl as an electrical instrument for measuring a quantity, transmitting the result t o a distant station, and there indicating or recording the quantity measured. This definition was satisfactory in the days when single-channel telemetering was the vogue, but now, with the advent of multichannel telemetering, the definition should be revised to include the measurement of one or more quantities. It is the purpose of this review t o describe briefly some advances in the art of radio telemetering which have been made since the period just prior t o World War 11. The multichannel radio telemetering systems t o be described were developed largely for military applications and hence have been illustrated chiefly for application t o the flight testing of pilot30 1

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less aircraft. It is likely that in the future, there will be more and more applications of telemetering to industry, where a telemeter would provide means for transmitting over a single mire (or radio) link a number of varying measurements for remote indication or recording. E'sually important are the applications of telemetering techniques to remote control, where many separate control signals may be transmitted simultaneously over a single wire (or radio) link. The material in this review is largely descriptive and reference should be made to other publications for theory related t o t e l e m e t e r k ~ g . ~ ~ ~

11. EVOLUTION OF RADIOTELEMETER~NG Prior to World War 11,the principal applications of telemetering were industrial and usually involved single-channel telemetering.* The Radio Sonde15which was developed at the National Bureau of Standards early in 1937 for meteorological applications, was a simple form of multichannel radio telemeter. However, a need for more adequate multichannel radio telemetering grew out of the rapid growth of the airplane industry where, with the increased speed and cost of airplanes, the taking of data during flight tests of models became increasingly important. The target drone, or radio-controlled target aircraft, and a number of guided missiles were developed during World War 11. Since these vehicles carried no pilot, some new means had to be devised t o obtain flight test data. I n the first flight tests, the pilot, with pad and pencil, made notes from meter observations. Later, with the introduction of pilotless vehicles, photo-panels mere installed in which recording cameras photographed a panel of test instruments dwing flight. Photo-panel recording necessitated recovery of the test vehicle, or at least recovery of the record magazine, in order to obtain the record of flight test data. Experience showed, however, that vehicle recovery was the exception rather than the rule. Loss of a model, representing a considerable expenditure of time, effort, and money, meant the loss of expensive recording equipment and often the loss of the record indicating the causes of failure of the tests. Multichannel radio telemetering equipment that was relatively compact, inexpensive, and expendable might well mean the difference between total loss of flight test data and a successful test, since the data obtained during flight tests are essential for future design and development. For the above reasons we find that the greatest advances in the art of multichannel telemetering have been associated with the flight testing of military aircraft, particularly in the United States where the great need has been supported by the appropriation of large sums of money for research and development. It is not possible in this review to give just

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credit to agencies in Europe and elsewhere who contributed to the development of multichannel radio telemetering, although it is recognized that t.he contributions of scientists throughout the world have been vital in this development. Wartime liaison between American and British scientists resulted in fruitful exchange of ideas which contributed to the success of telemetering in both countries. 1 . Television Telemetering

As an outgrowth of the first test flights wherein the pilot, with pad and pencil, made notes from meter observations, and of the later ones when photo-panels were used, the use of television for flight test observations came naturally. Compact narrow-band airborne television equipment was produced in large quantities during World War I1 for purposes other than telemetering. Some of this equipment was adapted for telemetering so that the camera viewed several test meters, modified with special high-contrast indicator dials. In some applications, a bank of oscillograph elements was also viewed by the camera, and this, together with a view of the test meters, was televised to a remote ground station during flights. The television telemetering system proved to be unsatisfactory for a number of reasons. The method was unnecessarily complex because of double conversion of electrical signals to optical signals. An unnecessarily wide bandwidth was imposed by the television link. The equipment was bulky and very expensive, particularly as an expendable item. For these reasons, we find that television telemetering lost favor. 6. Instrument Telemetering

The first radio telemetering systems, developed specifically for flight testing of pilotless aircraft during the early part of World War 11, or just prior to it, were instrument telemetering systems wherein the indications of flight instruments were transmitted by radio to a remote location. This was another logical extension of the test pilot pad and pencil method. In these so-called instrument telemetering systems, existing flight test instruments were modified by such devices as the coupling of lightweight phase-transformer rotors or microtorque potentiometers to the shafts of needle instruments, or by the use of small magnets, differential coils, or photocell pickoffs. Corresponding remote indicating dials were observed visually and notes were made from them during flights. 3. Oscillogram Telemetering

Many of the early telemetering systems were extremely complex, costly, unstable, and not practicable for multichannel operation. The

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accuracy of these telemeters suffered from the loading of the meters, which themselves were not very accurate. The accuracy fell far short of that required for securing research test and evaluation data. I n many cases, the frequency response of the system was inadequate to transmit the desired information. I n the early flight tests, when only a few channels were required and tests were usually of short duration] little attention was paid to the manner in which the data were recorded. As the need arose for the transmission of many data channels during a test, it became apparent that data recording and data analysis could not be adequately accomplished by manual recording methods. There arose a need for automatic recording of channel signals as continuous functions of time, together with reference time signals. With this new requirement, we find that the more recent telemetering systems, developed during the war, used multi-element recording oscillographs, such as had been developed for use in the seismic method of geophysical prospecting. The growing need for accommodating more channels of information a t increased frequency response and for improving accuracy in the transmission of flight test data led to the development of improved telemetering sensory elements or pickups, and to the development of more adequate telemetering systems. A number of systems were built by the Naval Aircraft Experiment Station, the Princeton Laboratories, the National Defense Research Committee, the Naval Research Laboratory] the Applied Physics Laboratory of Johns Hopkins University, several aircraft companies] and by numerous other organizations.

111. BASICSYSTEMS OF RADIOTELEMETERING Two basic systems for multichannel telemetering grew out of this period of development. The first is known as frequency-division or subcarrier telemetering, and the second is known as time-division or commutation telemetering.

I. Frequency-Division Telemetering h frequency-division or subcarrier system of telemetering is defined as one in which a number of information channels to be transmitted are synthesized or multiplexed by combining a number of subcarrier frequency signals, each carrying a particular information channel, into a composite electrical signal. Each measurement, utilizing one channel, modulates a subcarrier oscillator. A number of these subcarriers, differing in frequency] are mixed, and the resulting signal modulates the radio frequency carrier. It is possible, and customary, to choose the sub-


305

carrier frequencies in such a manner as to minimize cross-modulation effects. Subcarrier telemeters differ in the manner in which the subcarriers are modulated by the input signals, which vary in accordance with the information to be transmitted and in the manner in which the radio frequency carrier is modulated by the composite signal. For example, in a typical subcarrier telemeter to be described later, the subcarriers are frequency-modulated by the corresponding channel signals, and the radio frequency carrier is frequency-modulated by the composite signal which is the sum of the frequency-modulated subcarriers. Such a telemetering system is commonly referred to as an FM-FM subcarrier telemetering

FIQ. 1. Block diagram for a typical N-channel frequency-division telemetering system.

system. At the receiving station, the output of a radio receiver passes to band-pass filters which separate the frequency-modulated subcarriers. These channel signals separately pass to conventional FM discriminators and thence to corresponding recording elements in the recording oscillograph. The practical limit to the number of subcarriers which can be transmitted by a single radio frequency carrier a t the present time is between 10 and 14. With more subcarriers than this, cross-modulation effects and noise become serious obstacles, principally because it is not possible to build a transmission system that is perfectly linear in response. With any number of subcarriers, each channel modulation must be limited so that the composite signal does not overmodulate the radio frequency carrier. The restricted number of channels is a disadvantage of the subcarrier system of telemetering. An advantage of the subcarrier

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system, however, is that it can transmit a relatively high frequency continuously in each channel. Also, with certain types of pickups as described later, the telemetering transmitter unit can be made quite simple and compact, particularly if the radiated power is low. Figure 1 shows a block diagram for a typical N-channel frequencydivision telemetering system. A more detailed description of a system of this type is given in a later section of this review. 2. Time-Division Telemetering

A time-division or commutation system of telemetering is defined as one in which the transmission of a number of information channels is divided in time. The channels are rapidly switched or commutated in a fixed sequence. Each channel modulates the carrier fully during that part of the commutation cycle assigned to that particular channel. The time-division or commutation system of telemetering can transmit a much larger number of channels than a subcarrier system before cross-modulation becomes a serious limiting factor. The channel frequency response, however, depends upon the commutation speed and upon the number of channels transmitted, and electronic methods of commutation must be employed if many channels of high frequency response are required. Since pulse techniques are used in the electronically commutated time-division telemetering systems, high peak power may be obtained with relatively compact radio transmitters. In some applications where long ranges are required, this is an advantage over the continuous wave transmitters used in the subcarrier telemeters. A disadvantage of the pulse system in some applications is the relatively wide bandwidth required to adequately pass the short pulses. Figures 2 to 5 show block diagrams for typical N-channel time-division telemetering systems. More detailed descriptions of systems of these types will be given in a later section of this review, only the principles of operation being outlined here. a. Pulse Position Modulation and Demodulation. Figure 2 shows one type of multichannel time-division telemetering transmitter utilizing pulse position modulation, in which the sampling rate is established by a sine wave master oscillator. For an N-channel system, the phase splitter gives N sine wave outputs equally distributed in phase or time, thereby setting the zero-modulation phase for each of the N channels. The phase modulator shown in each channel shifts the phase in accordance with the information from the corresponding pickup or sensory element. The sine wave output from the phase modulator passes to a pulse generator which develops a short pulse locked in phase with the

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signal from the phase modulator. I n a separate path, the master oscillator signal passes t o a frame synchronizing pulse generator which develops a coded signal, usually two or three pulses in close time sequence. This coded signal identifies the initiation of the sequence of channel pulses in the frame. The frame synchronizing signal and the separate channel pulses pass to a mixer, and the composite signal from the mixer modulates the radio transmitter. The signal transmitted consists of a repeating sequence or frame of N-channel pulses, marked by a coded synchronizing signal. Each channel pulse is displaced in time from its zero-modulation

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position in accordance with the signal from the corresponding telemetering pickup. The type of modulation illustrated here is referred to as pulse position modulation, sometimes abbreviated as PPM. Figure 3 shows a block diagram of another N-channel time-division telemetering transmitter utilizing pulse position modulation. The type of signal transmitted is identical with that transmitted by the system illustrated in Fig. 2; However, in this case, the frame sampIing rate is established by a ring counter chain which is triggered by a master pulse generator. The pulses from the generator cause successive tubes in the chain to conduct in sequence and to generate pulses distributed uniformly in time. The pulse from one of the counter chain tubes is coded and is


308

used as a frame synchronizing signal. Each channel pulse produced in the counter chain passes to a corresponding pulse interval modulator which generates a pulse delayed with respect to the pulse from the counter chain, the delay being proportional to the information signal from a corresponding telemetering pickup. The N pulses from the pulse interval modulators, together with the coded frame synchronizing signal, pass to a mixer, and thence to the modulator of the radio transmitter as in the system illustrated in Fig. 2.

I FIG.3. Block diagram for a typical N-channel time-division telemetering transmitter utilizing pulse position modulation and a master pulse generator.

Figure 4 shows a block diagram for a typical W-channel time-division telemetering receiver and recorder for pulse position modulated signals as generated by the telemetering transmitters shown in Figs. 2 and 3. The multichannel sequence of pulses from the radio receiver passes to a channel synchronizing pulse generator which electronically separates the coded frame synchronizing pulse from the incoming pulse train and develops a sequence of channel synchronizing pulses which mark the reference time or phase for the separate channels. Each of these channel synchronizing pulses triggers a variable pulse width generator in which the pulse width is initiated by the channel synchronizing pulse and terminated by the corresponding channel signal pulse which is also fed into the pulse width generator. Thus, for each channel, a pulse is generated whose width varies in accordance with the modulation produced

309


by the corresponding sensory element or pickup in the telemetering transmitter. These variable-width chanpel pulses, recurring a t the sampling rate generated at the transmitter, pass to suitable metering circuits and to corresponding elements in the recording oscillograph. Alternately, the receiving system illustrated in Fig. 4 may be modified to utilize a cathode-ray oscillograph with recording film. This will be illustrated in more detail in the description of a typical pulse positionmodulated telemeter, t o be given later in this review.

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0. Pulse Interval Modulation and Demodulation. Figure 5 shows a block diagram for a typical N-channel time-division telemetering system which utilizes pulse interval modulation. The upper part of the diagram shows the transmitter. The sampling rate is established by a master, free-running multivibrator, the pulse output of which triggers channel 1 monostable multivibrator, whose pulse width is determined by the instantaneous DC voltage from the channel 1 sensory element or pickup. The trailing edge of the pulse from the channel 1 multivibrator triggers the channel 2 multivibrator, which functions in a similar fashion t o that of channel 1. Thus, the channel multivibrators are triggered in sequence and the width of the pulse generated by any channel monostable multivibrator varies in accordance with the instantaneous voltage from the corresponding telemetering pickup. The trailing edges of the synchroniz-

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M. G . PAWLEY AND W . E. TRIEST

ing pulse and of the channel pulses are differentiated t o give short pulses t o mark accurately the channel pulse intervals. These pulses are mixed and pass t o the radio transmitter. Note t h a t the composite signal in this system is different from t h a t generated in the pulse position-modulated telemeter. I n this pulse interval-modulated system, the information t o be transmitted in any channel varies the interval between the corresponding channel pulse and the preceding channel pulse. As any one channel is modulated, all succeeding channels are shifted equally in time by the modulation, but the interval is changed only for the channel being modulated.

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A t the receiving station, illustrated in the lower part of the block diagram of Fig. 5, the pulse sequence from the radio receiver passes to a synchronizing pulse separator where the synchronizing pulse is recognized by virtue of the relatively long time interval between the X t h channel pulse and the synchronizing pulse. The separated synchronizing pulse triggers ON the channel 1 monostable decoder multivihrator, which in turn is triggered OFF by the incoming channel 1 pulse. Similarly, and in time sequence, each channel decoder multivihrator is triggered O N by the trailing edge of the preceding channel decoder pulse, and triggered O F F by the corresponding channel pulse. Each channel monostable decoder multivibrator, therefore, generates a pulse whose width varies with the corresponding channel signal. By means of


31 1

suitable metering circuits in the channel decoder, the sequence of variable width pulses in each channel is converted to a varying DC voltage which is recorded on the multi-element oscillograph. The pulse interval modulated telemeter, requiring only one tube envelope per channel in the transmitter coding unit is simpler than other multichannel pulse telemeters. This is important where space is at a premium and where simplicity is desired. Such a system will be described in more detail later in this review. 3. Factors Determining Choice of Telemetering System

Many telemetering systems have been built combining features of the basic systems described above. Because of added complexity in these hybrid systems, the current tendency is to use one or the other of the basic systems; the subcarrier system when fewer than ten channels are required, and a time-division telemetering system when ten or more channels are required. However, as described later, it is common practice to subcommutate with motor-driven multicontact switches, one or more subcarriers in a frequency-division system. It is thus possible to accommodate many additional channels of information a t the expense of the lowered sampling rate of the mechanical switch and with greatly increased difficulty in reduction of data. This subcommutated frequency-division system is not as complex as some of the hybrid systems which have been built. Factors which determine the choice of a telemetering system in a particular application include required accuracy, desired channel frequency response, information handling ability, size, weight, cost, stability, durability, effectiveness under specified operating conditions, and availability of transmitting, receiving, decoding, and recording apparatus. Availability is an important factor because relatively few telemetering systems can be purchased on the market today. There have been many specialized military needs for telemetering equipment, and there has been considerable duplication of effort, with but little standardization. Only recently has there been a concerted effort toward standardization of telemetering equipment. This effort of the Research and Development Board in the United States is currently in progress. In general the best overall accuracy which can be expected with existing telemetering equipment is approximately 2 per cent. The channel frequency response in subcarrier telemetering systems is of the order of 100 or 200 cps and is limited by the recording galvanometer frequency response and by the particular channel subcarrier frequency involved.2 If higher frequency response is required, a correspondingly

312

M. G . PAWLEY AND W. E. TRIEST

high subcarrier frequency is used, and the recording generally is made with a cathode-ray oscillograph and camera. The channel frequency response in time-division (commut,at.ion) telemetering is of the order of one-fifth the channel sampling rate, although this response may also be further limited by recording galvanometer response. The channel frequency response may be increased by suitable integrating methods to a value approaching one-half the channel sampling rate.2 The choice between a subcarrier type of telemetJer or a commutat.ion type may depend upon a number of factors, perhaps the most important of which is the number of channels desired. Since, at present, ten t,o fourteen channels (not counting subcommutated channels) are the most, that can be used in subcarrier telemetering without excessive cross-talk, a commutation type telemeter might be chosen if it were essential that more channels be provided. I n some applications, the high peak power available in the commutation (pulse type) system is a distinct advantage, particularly where space for telemetering equipment is at, a premium, and long ranges of transmission are involved. Considerably more effort has been expended in the development of subcarrier telemetering systems than in the development of electronic commutation systems. Therefore we find more equipment of the subcarrier type available on the market, and we can look for earlier standardization of this type of equipment. Extensive development of timedivision systems is currently in progress, but it is likely that several years will elapse before these latter syst,ems can be adequately tested and standardized.

IV. TYPICAL TELEMETERING SYSTEMS 1. Early Il'elemetering Systems

The Naval Aircraft Experimental Station Telemetering System, which was one of the first usable systems, used six amplitude-modulated subcarriers which in turn frequency-modulated the radio frequency carrier. This is an example of an AM-FM subcarrier telemetering system. The Vultee Telemetering System,s which was one of the first commutation systems, used .a mechanical commutator to connect the signals from seventy-two channels to an amplitude-modulated transmitter. The information in each channel varied the frequency of an oscillator, and this frequency was transmitted over a radio link. One of the first telemetering sets to use high-speed electxonic commutation with a relatively high channel frequency response was developed by Princeton University under sponsorship of the National Defense Research Committee (NDRC).'

MUT,TICH.4NNE:T, RADIO TELEMETERING

313

A compact four-channel subcarrier type telemeter was also developed (luring World War I1 by Princeton University under NDRC contract. Although these early telemetering systems served a useful purpose, they proved t o be inadequate with increasing flight test requirements. However, they did serve as a pattern for extensive development which has continued ever since. More than fifty telemetering syhtems were in existence at the end of World War 11, and new systems are currently being developed. It is beyond the scope of this review t o tlesc*ribe many of these in detail. Rather, me are concerned in pointing out thc trend in advancement of the art of multichannel radio telrmetering. I n the detailed descriptions t o follow, typical telemetering systems of the two basic types are described, the choice having been made from among systems which are currently being used successfully in flight testing and which, in the authors’ opinion, represent good engineering design. Other telemetering systems are heing used successfully, but in general no new principles are involved in their operation beyond thow dcscaribed in the following typical examples. 2. Example of Frequency-Division Tclpmctcring System During World War 11, Princeton University developed a subcarrier telemetering system utilizing four audio subcarriers frequency-modulated 5 7 . 5 per cent of center frequency by the intelligence t o be transmitted. The Applied Physics Laboratory of the .Johns Hopkins University (APL/ JHU) developed a similar system with different subcarrier frequencies. The designation AN/AKT-5 was applied to a system employing the APL/JHU frequencies and four voltage-controlled multivibrator type subcarrier oscillators. This system was used by several of the military services. When the Telemetering Group a t APL/JI-IU was consolidated, i t was decided that the Princeton subcarrier frequencies would be used in the APLIJHU Telemetering System. As this work progressed, i t became evident that greater intelligence carrying capacity was required. TWO subcarriers were added, making a six-band system. One version of this system, with a fixed scheme of sulwommutation, was designated the AN/DKT-3. Later, the need for a channel with higher frequency response was satisfied by the addition of one more subcarrier, making a scven-band system. The suhcarrier oscillators of the APT,/JHU seven-band system are frequency-modulnt ctl by the intelligence to be transmitted. These subcarriers are mixed a t the proper levels in linear circuits, and the composite signal applied to the input of a frequency-modulated radio fre-

314


quency transmitter. The combined information is transmitted through a radio frequency link t o the ground station. Here it is received and the R F carrier demodulated by conventional F M circuitry. The composite audio subcarrier output of the receiver is applied to the proper number of band-pass filters and audio subcarrier discriminators. The output of each subcarrier discriminator is a voltage representing the original quantity measured. Here i t is passed through a low-pass filter whose cut-off frequency depends upon the frequency response of that particular band. A number of different recording schemes may be applied a t the output of the low-pass filters, depending upon the type of information and data display desired. The band-pass filters provide the subcarrier selection for each individual discriminator. These filters have an attenuation of not more than 3 d b within the rt7.5 per cent subcarrier deviation. The skirts of the response curve are down 20 db a t f l l per cent of the center frequency and 40 db a t +13 per cent and beyond. The audio discriminators have sufficient linearity and stability so that only three frequency points need be used t o determine the calibration curve. The low-pass filters decrease the high frequency noise above the intelligence band. The output impedance of the low-pass filters is 330 ohms. The low impedance of the recording galvanometer elements is increased by series resistance in order to match the output of the filters. This type of impedance matching network has adequate efficiency t o drive the elements to full output, since they are essentially current-operated elements. The sensitivity of the discriminator is such that its output provides a current of & 10 ma for full subcarrier deviation. Recordings are made on standard electromagnetic recording oscillographs with 12-in. photographic paper running a t 6 in. per second. A camera and cathode-ray oscillograph are used t o record the information from the high frequency subcarrier. A ten-channel FM-FM telemetering system, recently developed a t APL/JHU, uses an improved radio receiver and improved audio frequency discriminators in order t o increase the overall telemetering accuracy. Three types of audio frequency subcarrier oscillators are commonly used in the APL/JHU telemetering system. The first is a n 1,C type in which conventional Hartley resistance-stabilized oscillators are used. The frequency variation is accomplished by variation of inductance. The second type of oscillator is used for the measurement of voltage and utilizes a n RC phase shift netmoik t o determine the frequency. The


315

plate resistance of a modulator tube is varied with the applied intelligence voltage, which thereby shifts the oscillator frequency. A third type of oscillator is used t o transmit intelligence having a frequency of several thousand cycles per second. This oscillator is a multivibrator frequencymodulated by the input voltage. The following sensory elements or pickups of the variable-inductance type have been used extensively with the APL/JHU and other subcarrier systems. Accelerometer. An oil-damped accelerometer is made by attaching a mu-metal slug t o a thin, corrugated beryllium copper diaphragm. The mass of the slug moves the diaphragm toward or away from the air gap of an inductance coil, thereby varying its inductance. The resonant frequency of the diaphragm is 50 t o 75 cps. Pressure Gages. Absolute and differential pressure gages use two chambers with a corrugated heryllium copper diaphragm hetween them. A high-mu pad, which varies the air gap of an inductance, is attached to the diaphragm. I n the differential gage, the chamhers are each connected t o a pressure source. I n the absolute pressure gage, one of the chambers is sealed a t fixed pressure. Motion Meter. With the motion meter, the rotation of a control surface through 10" or 15" is mechanically amplified by means of gears to cause a 180" or 360" rotation on a threaded shaft. When the shaft advances, the air gap of an inductance coil is varied. Tachometer. The tachometer consists of a disk which rotates in front of an inductance coil air gap. The disk consists of mu-metal which is mounted eccentrically, thereby varying the inductance of the air gap with each revolution and causing a frequency change of the associated subcarrier oscillator with each revolution. The speed of the shaft can be determined by counting the number of cycles per second of oscillator frequency shift. Gyro Position Indicator. A wedge-shaped piece of mu-metal is attached t o a gyroscope cage so that rotation of the cage causes a thicher or thinner portion of the wedge to be across the air gap of an inductance coil. I n addition t o the above variable-inductance type pickups, microtorque or other potentiometers are usrd frequently t o vary the voltage input t o voltage-controlled oscillators. These potentiometers may be used t,o telemeter angular shaft displacement or linear displacement when properly geared t o the moving part. Thermocouples are used t o measure temperature by allowing the output of the couples t o saturate a reactor, thereby varying its inductance and the frequency of the associated subcarrier oscillator.

316

M . G. PAWLEY AND W. E. TRIEST

3. E.uniples

OJ

l’ime-l>ivision l’elemetering System

u. l‘gpicul Pulse Position-ill oduluted l’elemctering System. ‘l’hc pulse position-modulated (PPM) telemeter t o be described was developed by the Rocket Sonde Research Branch of the Naval Research Laboratory (YRL) shortly after World War 11. It is referred t o by N l t L as the Matrix Telemetering System. The transmitting unit is designated the XN/DKT-2. The system has been used successfully in telemetering

I

I

AMmnNA

FIG.6.

Block diagrarii of A N /l>IiT-2 tt:lcrnctcring transmittcr.

from a number of V-2 rockets launched a t the White Sands Proving Grounds for upper atmosphere research. The matrix telemetering system utilizes pulse position modulation t o convey thirty channels of input information. Data in the form of varying DC voltages is supplied t o the equipment from the various instrument pickups. The data channels are sampled successively and the whole process is repeated at a 312.5-cycle rate. A coded triple pulse identifies the start of each group and establishes a time reference framework, or matrix, of thirty-two intervals of 100 microseconds each. Within thirty of these intervals, single data pulses occur, the position of the pulse within its interval indicating the voltage being measured on the channel. This

MULTICHANNEL RADIO TELEMETERIXG

317

chain of pulses is used to modulate an RF transmitter operating a t a frequency of 1025 megacycles per second with an average power in the pulse of about 4 km. Figure 6 shows a block diagram of the matrix telemeter transmitting unit. This diagram shows the 10-kc oscillator t o be the independent initiating part of the circuit. The sine wave output of this oscillator is fed to a multivibrator which serves to shape it t o derive a series of pulses accurately spaced a t 100-microsecond intervals. These 'pulses are fed through a cathode follower to a bus feeding each of the thirty-two tubes of the electronic commutator. These tubes are arranged as a chain so connected that only one tube can conduct a t a time. As any one of these thyratron tubes fires, it readies the next tube in the chain. Upon the appearance of the next pulse on thc triggering bus, conduction shifts to the next succeeding tube, and so on down through the whole chain of thirty-two tubes, and the process is repeated when the last tube readies the first tube for conduction. As each of the first thirty of the chain tubes fires in turn, it generates a sawtoothed wave form which is added electrically to the input voltage of a channel pick-off thyratron. When the sum of these two voltages reaches a predetermined level (as it will sometime during the 100 microsecond interval), it will cause its corresponding pick-off thyratron to fire, producing a time-modulated output pulse. The greater the input voltage, the sooner the sum of the input voltage and sawtooth will reach the firing voltage. The output pulses from the various channels are collected on a common bus. For synchronization of the time base generators a t the ground station, a coded pulse group consisting of three pulses each separated by 7.9 microseconds is generated in a multivibrator circuit triggered by channel 32. This multivibrator has an LC ringing circuit with a 7.9-microsecond period in the plate of the normally ON tube, with associated limiting and differentiating circuits. This output is combined with the channel video pulses on the common bus. The output of the common bus triggers a blocking oscillator to form a 0.9-microsecond pulse which is used to key ON a coaxial L band VHF cavity by means of a pair of 3E29 modulators. The output of this cavity feeds a 50-ohm line with an average pulse power of 4 kw a t a frequency of about 1025 megacydes per second. For calibration, each input channel is connected to its data input via a single-pole double-throw microswitch operated by a synchronized motor-driven cam unit. For about 2 per cent of the time, data input is removed from the input channel and a calibrating voltage, consisting of 1 volt steps from 0 to + 5 volts, is substituted. This step voltage is repeated on each channel in sequence every 10 seconds during flight. For

318

M . G. PAWLEY AND W. E. TRIEST

any channel requiring continuous input data recording, this calibration can be removed. For ease of disassembly of the unit from the rocket, aircraft type quick-disconnect straps are used for mounting, with a vee-ring pressure seal using similar quick-disconnect toggle snaps to seal the pressurized transmitter case to its top bulkhead. The total weight of the telemeter transmitter set, with batteries suitable for thirty-minute operation, is about 130 lb. Two 28-volt plastic-cased storage batteries are used in conjunction with a vibrator type converter unit to provide the various plate voltage and bias supplies, and four 2-volt plastic-cased storage batteries for the filament supplies. Total power requirements for the system are 15 amperes at 28 volts and 20 amperes at 8 volts. Matrix Ground Station. Each ground station comprises four racks, the monitor rack containing receiving and decoding apparatus as well as a monitoring oscilloscope, and three identical recording racks. The ground station receives the recurrent train of pulses from the remote transmitter in a suitable receiver. The triple pulse is recognized by a discriminator and is used to generate a 10-kc matrix in exact phase with the matrix in the transmitter. From this, signals are obtained at any particular time in the matrix which are used to generate the sweep voltages for the cathode-ray tubes on which the data are presented as intensity modulation. Pulses generated a t the beginning of each matrix interval are also applied to the display and the scopes are photographed on a continuous film camera. There results a graph of the variation of the input voltage with time, with reference lines (which do not vary in position) between each of the data lines. Each ground display has six cathode-ray tubes upon which the thirty data channels can be applied in any combination. Sweeps of appropriate duration and timing in respect to the triple pulse are so arranged that one tube after another is swept in succession until all channels are recorded. Additional information is also supplied on the record by the application of signals to the cathode-ray tube to indicate time from a standard source. Referring to the block diagram for the matrix system ground station, Fig. 7, main signal paths are indicated by arrows. The receiver delivers the video signal on two lines, one to the recording racks, and another into the synchronizing pulse discriminator where the triple pulse is recognized and supplies a pulse which is used to phase and frequency lock the 10-kc matrix oscillator to the airborne oscillator. The output of the oscillator is shaped and fed into the scale-of-32 binary counter. This counter operates in synchronism with the chain counter in the airborne unit. It is reset by the synchronizing pulse, when necessary, so that particular


319

states correspond in both units. By means of resistor networks from different stages of the counter, the state selector delivers a timed series of 100-microsecond pulses to a patchboard on the panel of the unit. Suitable connections are made to the trigger and gate unit to form gates for controlling the sweep circuits in the recorder racks. There are several auxiliary units in the monitor rack. The frequency monitor determines whether the matrix oscillator is operating on the proper harmonic of the 312.5-cycle synchronizing signal and indicates the

r--(

ANTENNA

loe5

RECEIVER

*oNITwI RACK

d7-G I

VIDEO

TRIPLE PULSE

I

MONITOR

D-Li \

I

I COUNTER STATES

FIG.7. Block diagram of matrix system ground station.

proper frequency by means of lights on the face of the unit. The monitor oscilloscope is used for adjustment of the units and for monitoring the signal during operation. It is swept from associated generators which are synchronized from 312.5-cycle and 10-kc pulse sources. Appropriate power supplies are included in the unit, as are switches to control the operation of the recording racks. Each of the three recorder racks contains a double recording unit consisting of tivo type 5 R P l l cathode-ray tubes and a continuous 9x-in. film camera. Appropriately timed gates from the monitor rack allow sweep generators to operate one after another and, at the same time, unblank the cathode-ray tube so that the video pulses from the receiver

320

‘


in the monitor rack can be presented as intensity modulation on the tube. Necessary video amplifiers are included in this unit, as are circuits for mixing the video with timing signals and with the reference pulses generated by the 10-kc matrix oscillator on the ground. There are also controls for positioning and focusing the cathode-ray tubes, as well as necessary power supplies. The image on the face of the cathode-ray tubes is focused on a slit on the front plate of the unit by means of two lenses, placed side by side. A film magazine is placed so that the image from both tubes is recorded on it as the film is drawn continuously past. The ground station operates on 110 volts, 60 cycles AC with a power consumption of about 3000 watts. Voltage regulation of the primary power for these units is obtained from constant voltage transformers. All plate supplies under 600 volts are electronically regulated against load variations. The high voltage supplies required for the cathode-ray tubes are - 2000 volts obtained from a conventional transformer-rectifier circuit, unregulated, and +10,000 volts obtained from an R F power supply operated from an electronically regulated 300-volt power supply. Owing to the complexity of the ground station circuits and the number of tubes involved; two complete and separate ground stations are utilized for every flight. In normal flights, six 9X-inch film records of about 120 ft. in length are produced, corresponding to about 450 seconds of flight. These films give a complete graph of voltage versus time for all thirty channels and are reproduced in the form of black line paper prints. b. Typical Pulse Interval-Modulated Telemeterhg System. The pulse interval-modulated (PIM) telemeter to be described was originally developed in a twenty-three-channel form by the Rocket Sonde Research Branch of the Naval Research Laboratory near the end of World War 11. The transmitter unit was later repackaged in a ten-channel form which carries the designation AN/AKT-1 A. This model of the transmitter unit, which was designed by the Control and Report Links Branch, R.D. 3, of the Naval Research Laboratory, has been successfully used in about sixty-five flight tests. The operation of the AN/AKT-1A telemetering transmitter unit may be explained by reference to Figs. 8 and 9. Referring to Figure 8, the sampling rate of the pulse interval modulator is controlled by the master multivibrator 2-101, which is of the “freerunning ” type, preset to give approximately 400-cycle-per-second square wave pulses a t point (A). The output is coupled to: (a) Channel No. 1 monostable multivibrator (2-102) through a differentiating circuit consisting of C-103 and the grid to ground impedance of V-104 and (b) to the

321


mixer (Z-112) through a differentiating circuit consisting of C-104 and R-106. The positive surge from 2-101 a t point (A) does not affect Channel No. 1 (2-102) because V-104 is already conducting due t o the grid's being returned to a positive potential by R-109, and the surge does not pass t o the mixer because i t is blocked by the crystal diode, CR-101. At the end of a pulse from 2-101, a negative surge is impressed on the grid of V-104 and is also passed through CR-101 t o the mixer (Z-112). The IMELLWNCE INPUT

----------- j I - "?~

+IsoV

""_ ---

INTEUIBENCE

- - - -r c - 2 2

m m

~

-+ell* 5

FIG.8. Schematic diagram of AN/AKT-1A telemetering transmitter.

potential across the common cathode resistors, R-111 and R-112, drops until V-103 begins t o conduct. Conduction of V-103 produces a negative surge a t its plate which is coupled t o the grid of V-104 through capacitor C-105 and drives the grid of V-104 further negative. The channel remains in this temporary or unstable condition until the grid potential of V-104, rising a t a n exponential rate determined by the discharge of c-105 through R-109 (see Fig. 9b), reaches a sufficiently positive value for conduction to begin again in V-104. The cathode potential again rises and a positive surge occurs a t the plate of V-103 which applies a further positive surge t o the grid of V-104, returning thc channel t o its stable condition. During this time, a positive pulse is generated

322


a t point (B), in Fig. 8, which is coupled to (a) Channel No. 2 monostable multivibrator (2-103) through a differentiating circuit consisting of C-107 and the grid to ground impedance of V-106 and (b) to the mixer (2-112) through a differentiating circuit consisting of C-106 and R-113 initiating in Channel No. 2 the same action that occurred in Channel No. 1. At the same time, a negative surge resulting from the negative swing of the positive pulse a t point (B) is passed through the crystal diode CR-102 to the mixer (2-112).

FIQ.9. Waveforms in the AN/AKT-1A telemetering transmitter.

This action continues from channel to channel, the return of one channel to the stable state triggering the next channel into its unstable condition and simultaneously delivering a negative pulse to the mixer. When the action in the last channel is completed, the circuit is quiescent until the next pulse from the master multivibrator 2-101 triggers 2-102 again. The input of each channel, which is the grid of the non-conducting tube when the channel is in the stable condition, is connected to a sensory element or pickup which supplies a variable voltage from 0 to +4 volts. When this positive voltage is applied to the input of a channel such as 2-102, V-103 will conduct more heavily than it would if this voltage were zero when the channel is triggered. A large negative surge is impressed


323

on the grid of V-104 and therefore a longer time is required for it to leak off through R-109. (See Fig. 9b.) This produces a positive pulse a t the plate of V-104 whose width is linearly proportional to the DC voltage applied to the grid of V-103. (See Fig. 9c.) Consequently, time T (see Fig. 9d) is governed by the intelligence voltage applied to Channel No. 1 and varies from a minimum of t l to a maximum of T . In the same manner, the time that Channel No. 2 (2-103) is in its temporary state is governed by the intelligence voltage applied t o the input of Channel No. 2, and so on for the succeeding channels. The intelligence in any particular channel is independent of that in any of the other channels. The crystal diodes, CR-101 through CR-111, allow the negative surges from the master multivibrator and from the channel multivibrators to pass t o the mixer, and in addition, prevent feedback between channels. In this manner, each sampling of ten channels of intelligence which appears at the input of the mixer, V-123 (point D, Fig. 8) is represented by a group of negative pulses, equal t o the number of channels, plus a frame synchronizing pulse. (See Fig. ge.) The number of these sampling groups per second is equal to the pulse frequency of the master multivibrator, approximately 400 cps. The frame time, or interval between successive Synchronizing pulses, therefore remains constant and is equal to the reciprocal of the sampling frequency or approximately 2500 microseconds. (See Fig. 9e.) At point D, the negative pulses decrease the small positive bias maintained by R-114 on the amplifier or mixer, V-123. This grid change causes amplified positive surges or pulses on the plate of V-123 which are passed through the cathode follower, V-124, t o key the blocking oscillator, V-125. (See Fig. 8.) The time relationship between the pulses is the same as it was at the input t o the mixer-cathode follower, 2-112. The pulse group from the output of 2-112 is made up of pulses approximately 15 microseconds long. In order t o keep the average power consumption in the radio frequency oscillator as small as possible, each pulse is reshaped in the blocking oscillator, V-125, to be approximately 2 microseconds long and 180 volts in amplitude. Capacitor C-111 and resistor R-120 in the cathode of V-125 determine the duration of the output pulse. Resistor R-121, a t the output grid of V-125, is a damping resistor and prevents the blocking oscillator from delivering more than one output pulse for each input pulse. The driver tube, V-126, is a cathode follower providing a low impedance driving source directly coupled to the grid of the radio frequency oscillator, V-127. The oscillator is biased by applying a fixed positive voltage t o the cathode of V-127. The oscillator consists of a “lighthouse” high frequency triode, V-127,

324

M . G. PAWLEY AND W. E. TRIES”

xvhich excites an “ L ” band cavity (2-113) shown in Fig. 8. The cavity is capable of being tuned from 500 t o 580 me per second but is normally tuned to approximately 525 mc per second. At this frequency, the transmitter delivers approximately 400 watts peak power a t the cavity. The radio frequency energy is removed from the cavity by the inductive coupling loop on 5-104. This energy is fed to receptacle J-105 on the front panel through a short coaxial cable. The quarter-wave antenna is connected with coxial cable to the receptacle 5-105. The batteries, which include the 2000-volt supply for the oscillator, weigh 21 lb. and occupy a volume of 350 cu. in. They are housed in a cylindrical metal enclosure and are held in the container with shockabsorbent padding. The transmitter unit and power supply are enclosed in a cylindrical metal housing 944 in. in diameter by 2648 in. long, and the assembly weighs 78 lb. The Telemetering Group a t the National Bureau of Standards has recently completed repackaging of the AN/AE(T-l A Telemetering Transmitter. For flexibility, the equipment was broken down into smaller units. The overall weight and space occupied by this repackaged version is approximately one-half its former weight and volume. As a result of experience with the former version, minor but important circuit changes were incorporated in the newer version, which is designated the AN/AKT-lA(XN-2). The ground station receiving equipment used with this pulse interval modulated (PIM) system is the same as originally developed a t the Naval Research Laboratory. Its principle of operation was described earlier in this review in connection with the block diagram for the PIM telemetering system shown in Fig. 5. The pulse interval modulated telemeter described above requires approximately 0 to + 4 volts input to the channels. The sensory elements or pickups must therefore, either directly or indirectly, supply this varying DC voltage t o each channel. For this reason, we find microtorque potentiometer type pickups commonly used, or ordinary potentiometers when the mechanical load on the sensory element is not excessive. A small battery across a number of these potentiometers supplies the excitation for the corresponding channels. Microtorque potentiometer type accelerometers, pressure gages, and vane type instruments are available on the market. Considerable ingenuity is frequently required in adapting the many and varied test measurements involving telemetering to the 0- to 4-volt DC channel voltage input requirement.

MULTICH.INNEL

325

R.IDIO T E L E M E T E R I S G

c. Typical Mechanically Commiitated Telemeteriny S y s t e m . The system* to be described in this section was developed by the General Electric Company to meet requirements of a large number of information channels and rigid limitations on space and weight. Basically, this is a time-division system, and it has the inherent limitation of a lo\\. frequency response due t o the low (mechanical) switching rate. The recorded data appears in a form difficult t o reduce. These disadvantages may, in some applications, be outweighed by the relatively high number of channels and compactness of the system.

w PICKUP

0 3

OMITTED FOR FRAME SYNCHRONIZATION

T

FIQ.10. Block diagram of mechanically commutated telemetering system.

The entire transmitter is packaged in two pressure-tight cylindrical cans, each 4 in. in diameter and of lengths 15 in. and 16 in. The total weight is 23 Ib., and the total space is 0.2 cu. ft. for this equipment. One cylindrical can contains a high voltage dynamotor, which also drives the commutator used for sequencing the channels (to be described below) ; the second cylinder contains the transmitter and other electronic equipment. As shown in Fig. 10, the mechanical commutator sequentially connects voltages from the instrument pickups t o the transmitter unit. The commutator is driven a t 2000 rpm, through reduction gearing t o the dynamotor, which operates a t GO00 rpm, speed-regulated t o 2 per cent over a 20 per cent change in voltage. The commutator has sixty segments, of which alternate contacts are used t o give break-before-make

326

M. G . P A W L E Y AND

W.

E. TRIEST

action. Two contacts are used for synchronization of the ground (receiving) station, leaving twenty-eight useful channels. The transmitter unit receives a series of pulses of amplitude 0 to 5 volts from the commutator, and the premodulator chassis converts these t o variable width pulses, 100 microseconds corresponding to 0 volts and 700 microseconds to 5 volts. This conversion is done with a monostable multivibrator, giving an output pulse width linear with input amplitude. An antibounce circuit uses a capacitor to maintain the input voltage on the premodulator input despite any commutator bounce and discharges this condenser, between successive channels, to - 15 volts. Any pickup can be used which gives a variable DC output of 0 to 5 volts. However, it was found desirable t o include a variable transformer gage, a moving core being used to vary the coupling between primary and secondary windings, in this system. For this purpose, a 10-kc audio oscillator was included in the telemeter and is shown as the pickup exciter in Fig. 10. The output of the oscillator is applied t o two sets of variable transformer windings connected in a bridge so that movement of the core simultaneously increases the coupling in one set and decreases the coupling in the other. The output of the bridge is rectified and filtered to produce the required 0 to 5 volts DC. The variable pulse width output of the premodulator is applied to a 6AK5 reactance tube which frequency-modulates a 6C4 oscillator. A 2326 power amplifier then delivers 7 watts at 150 mc per second with 21 100 kc per second frequency deviation. The ground system, shown in block diagram in Fig. 11, uses a modified communications receiver with a band width of 300 kc per second in the IF amplifiers and discriminator. The output of the receiver is a reproduction of the width-modulated pulses generated in the transmitter. These pulses are passed through a clipper and differentiated t o obtain pulses at the leading edges and at the trailing edges of the channel pulses. The frame synchronizing pulse separator generates a frame synchronizing pulse, using the large interval preceding the synchronizing pulse t o charge a condenser t o a higher level than it can be charged during the intervals between channel pulses. These pulses are used to generate waveforms necessary t o produce three different types of recording on 35-mm film. For all types of film record, a 5 R P l l cathode-ray tube is used; this tube has high sensitivity, good resolution, and an optically flat face. The P I 1 phosphor is especially adaptable t o oscilloscope photography. Film in the 35-mm camera is moved vertically across the cathode-ray tube during the flight with camera shutter continuously open. In the ground station, separate tubes and cameras are used for the three methods of recording which will be described.

327


Figure l l b shows the type of record obtained with a ('lines" presentation. Successive channels appear as parallel bright lines, each of length proportional to the quantity measured by the corresponding channel in the missile. Each frame gives one reading on each channel, and it can be seen that the omission of two contacts in the commutator of the transmitter separates the groups of lines so that each frame is

I

I DIFFEREN-

FN RADIO

TIATOR & CLIPPER

RECEIVER

1 f A J BLOCK

+ONE

J

~-~.{-l INDICATOI1-4J

'IpMERA

DIAGRAM OF GROUND STATION

FRAME4

--

CXRLCTION OF FILM NOTION

f B ) " L l N E " TYPE OF 3 S M M FILM RECORD

FIG.11. Block diagram of ground station for mechanically commutated telemeter and a typical record.

recognized. Small marker pips are put on the channel signals at 50-microsecond intervals to facilitate direct reading of the recording. A film motion of 15 in. per second is used for this recording, to make successive pulses appear as lines. In the "dash" type of presentation, the ground station generates a twenty-eight-step sweep, which successively '(jumps" the cathode-ray beam horizontally for each channel and allows the channel modulation

328

M. G.

P A W L E Y AND W. E. TRIEST

to vary the level of its step. Thus, as the 35-mm film moves vertically across the face of the cathode-ray tube, twenty-eight plots of channel modulation versus time are obtained in the form of dashes, each of width proportional t o channel modulation. The twenty-eight plots are spaced across the width of the 35-mm film. For this presentation, a film speed of about 55 in. per second is used. The accuracy is reduced because each channel is displayed on one twenty-eighth of the film width, but the record is compact and useful for initial examination and inclusion in reports. The third type of presentation is by LLdots.’’Each channel is recorded as a time variation of the quantity measured, successive samples of a quantity taken a t 445-second intervals appearing as adjacent dots which permit easy tracing of the time variation of the channel. It is not feasible t o put twenty-eight channels on one film without confusion; in this system, the twenty-eight channels are divided into four groups, and seven channels are recorded on one film. T o prevent possible confusion with seven channels crossing and recrossing, (‘sequential intensification ” is used. The dots are intensified in a fixed sequence and, in the data analysis, this helps to eliminate ambiguity in separation of the channel traces. A film speed of $6 in. per second is used. This method of presentation is more compact than the “lines” method and has equal accuracy since the full width of the film is used t o record one channel. The principal difficulty with this method, however, is that identification of the channels is sometimes not easy, especially if several channels operate a t about the same level of modulation.

V. FUTURE TRENDS IN TELEMETERING Increasing demands of telemetering user-groups are for more channels and higher frequency response, compactness and ruggedness, and increased overall accuracy, stability, and flexibility. Because of the labor required for data reduction from multichannel telemeter records, the cost may run as high as a dollar per second of flight per channel. Hence, there is a n increasing demand for automatic, or semiautomatic equipment for aiding in the reduction of data. Such equipment should include automatic means for applying instrument calibration corrections and for plotting of corrected data in a form suitable for inclusion in reports. New recording methods should be investigated with a view t o eliminating the time delay required for photographic processing. I n this connection i t seems likely that magnetic recording techniques mill become increasingly important. New measurement problems are continuing t o arise for which new

M U L T I C H A N N E L RADIO T E L E M E T E R I N G

329

sensory elements or pickups must be developed. Existing pickups must be improved t o give better accuracy. It is likely t h a t continued studies in information theory and continued progress in digital computer development will lead to advancements in the a r t of telemetering. I n this connection, pulse code modulation (PCM) methods may become important in telemetering, since they possess relatively high signal-to-noise ratio capabilities and involve binary pulse coding which matches with digital computer techniques and may lead t o better integrated overall telemetering, recording, and data reduction performance. REFERENCES 1. Webstcr's New International Dictionary, 2d ed., G. and C. lferriani, Springfield Mass., 1949. 2. Nichols, M. H., and Raiich, L. L. Rev. Sci. Instruments, 22, 1-29 January, 1951. 3. Leifer, M., and Schreihcr, 11'. F. Advances in Electronics, 3, 306-343 (1951). 4. Borden, P. A., and Thyncll, G. M. Principles and hfethods of Tclcmctering, Reinhold, New York, 1948. 5. Diamond, €I., Hinman, JV. S., Jr., and Dunmore, F. W. J . Aeronaut. Sci., 4, 241-248 (1 937). 6. Giffen, Harvey D. Ae1,onaut. Eng. Rev., 2, 9-21 (1943). 7. Rauch, L. L. Electronics, 20, 241-248 (1947). 8. Neelands, I . J., and Hausz, W. Radio-Electronic Eng., 10, 3-6 (1948).


Author Index Numbers in parentheses are reference numbers. They are included to assist in locating references in which the authors' names are not mentioned in the text. Numbers in italics refer to the page on which the reference is listed in the bibliography a t the end of each article. 131 (20), , 153, indicates t h a t this author's article is reference 20 Example: Bakker, C. J.. . on page 131 and is listed on page 153.

..

A Acheson, L. K., 13, 68

B

...

Brown, G. W., 239 (33), 256 Bruining, P., 16, 67 Buechner, W. W., 10 (18), 12 (181, 67 Bullard, E. C., 7, 67 Burgess, R., 118, 153 Burhop, E. H. S., 10 (17), 28, 67 Burrill, E. A., 10 (18), 12 (18), 67 Bush, R. R., 152 (44), 154, 201 (8), 211 (20), 215 (20), 254, 255 Bush, V., 4, 66 Butler, S. T., 33 (47), 68

Bakker, C. J., 118, 122, 131 (20), 133, 134, 15s Ballantine, S.,124, 153 Bardeen, J., 65, 66, 68 Barnes, R. B., 110 ( l l ) , 163 Bartels, J., 259 (l), 298 L Bartlett, J. H., 11, 67 Batten, H. W., 209 (18), 255 Caldwell, S. H., 4, 66 Begovich, N. A., 126 (23), 153 cerenkov, P., 54, 68 Behrend, W. L., 239 (33), 256 Bell, P. R., 88 (14), 91 (14), 99 (19), Chandrasekhar, S., 110 (12), 153 Chao, K. T., 55, 68 100 (19), 107 Chapman, S., 259 ( l ) , 298 Bell, R. L., 131, 132 (22), 15s Christensen, C. J., 140 (28), 153 Bemrose, J., 268 (8), 299 Clark, J. C., 28 (36), 67 Bernamont, J., 117, 118, 119 (24), 155 Cochrane, L., 3 (2), 66 Bethe, H., 25, 33 (44), 44, 47, 48, 67 Black, J. R., 202 (10, 12), 205 (15, 16), Collins, G. B., 190 (3), 192 (3), 193 (3), 209 (12), 210 (12), 254, 255 254 Coltman, J. W., 69, 106 Bleuler, E., 56, 57, 58, 68 Conwell, E., 65, 68 Blewett, J. P., 204, 254 Corson, D. R., 43, 68 Bloch, F., 50, 68 Crane, H. R., 55, 68 Bogle, H., 140 (25), 143, 144 (25), 153 Cuccia, C. L., 201 (8), 219 (23, 24), 221, Bohr, A., 49 (60), 52, 68 254, 255 Bond, D. S., 234 (32), 239, 256 Cutler, C. C., 130 (28a), 131, 153 Borden, P. A., 302 (4), 329 Borries, B. von, 8 ( l l ) , 67 D Bothe, W., 48, 68 Bowen, W. A., 290 (lo), 299 David, E. E., Jr., 225, 232, 233, 255 Bozorth, R. M., 295 (13), 299 Brewer, J. R., 202 (10, 12), 205 (15, 16), Davydov, B., 117, 140 (29), 154 De Benedetti, 105, 107 209 (12), 210 (12), 254, 255 Debye, P., 8, 67 Brillouin, L., 140 (26), 153 331

.-.

332

AUTHOR INDES

Diamond, H., 302 (5), 329 Dienier, G., 122. 151, 154 Donal. J. S., Jr., 201 (8), 211 (20), 215 (20), 219 (24), 234 (31), 234 (32), 2s4, 25s Dresel. I, A. C., 124 (50), 125 (50), 129 (SO), f54 Dunmore, F. &’., 302 (5), 529 I h r a l l , G. 15.) 116 (32), 124, 125, 164 Duveneck, F. B., 28 (36), 67 Dgsoti, J. P., 88 (9), 106

E I*:lliott, J. O., 88 (17), 107 Illton. L. It. B., 12, 67 Engstrom, R. IV.,71 (5), 106

F Farnsworth, €I. E., 16, 31, 67 Frazel, C. E., 88 (l5), 107 Felch, I

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