Atomic Physics Methods in Modern Research: Selection of Papers Dedicated to Gisbert Zu Putlitz on the Occasion of His 65th Birthday

Lecture Notes in Physics Editorial Board H. Araki, Kyoto, Japan R. Beig, Vienna, Austria J. Ehlers, Potsdam, Germany U...

Author: Klaus Jungmann | Joachim Kowalski | Irene Reinhard | Frank Traeger

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Lecture Notes in Physics Editorial Board

H. Araki, Kyoto, Japan R. Beig, Vienna, Austria J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K. Hepp, Ziirich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Kippenhahn, G6ttingen, Germany H. A. Weidenmiiller, Heidelberg, Germany J. Wess, Mtinchen, Germany J. Zittartz, K61n, Germany Managing Editor

W. Beiglb6ck Assisted by Mrs. Sabine Lehr c/o Springer-Verlag, Physics Editorial Department II Tiergartenstrasse 17, D-69121 Heidelberg, Germany

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K. Jungmann J. Kowalski I. Reinhard F. Tfiiger (Eds.)

Atomic Physics Methods in Modern Research Selection of Papers Dedicated to Gisbert zu Putlitz on the Occasion of his 65th Birthday

~

Springer

Editors Klaus Peter Jungmann Joachim Kowalski Irene Reinhard Physikalisches Institut, Universit~it Heidelberg Philosophenweg 12 D-6912o Heidelberg, Germany Frank Tr~ger Fachbereich Physik, Universit~t Kassel Heinrich-Plett-Strasse 4o D-34132 Kassel, Germany

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Atomic physics methods in modern research : selection of papers dedicated to Gisbert zu Putlitz on his 65th birthday / K. E Jungmann ... (ed.). - Berlin ; Heidelberg ; New York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Pads ; Santa Clara ; Singapore ; Tokyo : Springer, 1997 (Lecture notes in physics ; 499) ISBN 3-540-63716-8

ISSN oo75-845o ISBN 3-540-63716-8 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved,whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Vertag.Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1997 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement,that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready by the authors/editors Cover design: design&production GmbH, Heidelberg SPIN: 10643850 55/3144-543210 - Printed on acid-free paper

Preface

Many of the significant advances in the course of the development of atomic physics were associated with newly invented scientific methods and experimental tools. Today these techniques are successfully employed in a wide spread variety of highly active areas in modern research, which extend from investigations of fundamental interactions in physics to experiments related to applied issues and technical aspects. With increasing importance they are found in areas outside of classical atomic physics in fields such as nuclear and particle physics, physics of condensed matter and surfaces, physical chemistry, chemistry, medicine and environmental research. The spectrum of methods includes among others optical and microwave spectroscopy, molecular beams, spin resonance, spin echo, particle trapping and tunneling microscopy. Laser spectroscopy is one example of a widely used technique: The fundamental process of light interacting with single atomic particles can be investigated especially profitably. Laser spectroscopy is essential in many high precision experiments for determining most accurate values of fundamental constants and for deriving conclusions on basic interactions which are complementary to results obtained in high energy physics. Optical properties of molecules, small clusters and bulk solid state material can be investigated both for revealing elementary processes and for studying, for example, new concepts of optical data storage. Processes in combustion devices can be characterized. Remote sensing of environmental pollution can be carried out with high sensitivity. Laser optical pumping of noble gases nowadays yields novel opportunities for nuclear magnetic resonance imaging in medical diagnostics. In February 1996 an international symposium on Atomic Physics Methods in Modern Research was held in Heidelberg on the occasion of the 65th birthday of Professor Gisbert zu Putlitz. In his scientific work atomic physics with its great diversity of facets has played an essential role with numerous significant contributions being highly esteemed by the community. The nature of the event inspired the authors of this volume, which is dedicated to Gisbert zu Putlitz. It comprises invited lectures and articles selected to give an overview of the manifold of developments in this area.

VI The editors would like to thank all authors for their articles and W. BeiglbSck for publishing this volume. The assistance of T. Katzenmaier, C. Kr~imer, E. Nowak and M. Zinser of the Physikalisches Institut der Universit~it Heidelberg in preparing this volume is gratefully acknowledged. Financial support for bringing the authors together was provided by the Stiftung Universit~it Heidelberg and the companies ABB, BASF, Fibro, Friatec, Lambda Physics, Spectra Physics and B. Struck. We are grateful to all of them. Heidelberg, August 1997 K. Jungmann J. Kowalski I. Reinhard F. Tr/iger

Contents

T w o - P h o t o n M e t h o d for M e t r o l o g y in H y d r o g e n B. Cagnac . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

H i g h Precision A t o m i c S p e c t r o s c o p y of M u o n i u m a n d Simple M u o n i c A t o m s V. W. Hughes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

M u o n i u m A t o m as a P r o b e of Physics Beyond the Standard Model L. Willmann and K. Jungmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

C a n A t o m s T r a p p e d in Solid He Be Used to Search for P h y s i c s B e y o n d t h e S t a n d a r d M o d e l ? A. Weis, S. Kanorsky, S. Land and T.W. H~insch . . . . . . . . . . . . . . . . . . . . .

57

g-Factors of S u b a t o m i c Particles B.L. Roberts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

L a s e r S p e c t r o s c o p y of M e t a s t a b l e Antiprotonic Helium Atomcules T. Yamazaki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Polarized, C o m p r e s s e d SHe-Gas a n d Its A p p l i c a t i o n s E. Often . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105

M e d i c a l N M R Sensing w i t h Laser Polarized 3He a n d 129Xe W. Happer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

Test of Special R e l a t i v i t y in a H e a v y Ion Storage R i n g G. Huber, R. Grieser, P. Merz, V. Sebastian, P. Seelig, M. Grieser, P. Grimm, T. Kiihl, D. Schwalm and D. Habs . . . . . . . . . . . . . . . . . . . . . . .

131

R e s o n a n c e F l u o r e s c e n c e of a Single Ion J.T. Hb'ffges, H.W. Baldauf, T. Eichler, S.R. Helm#led and H. Walther . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

R e s o n a n c e R a m a n Studies of t h e R e l a x a t i o n of P h o t o e x c i t e d Molecules in Solution on t h e Picosecond Timescale W.T. Toner, P. Matousek, A . W . Parker and M. Towrie . . . . . . . . . . . . .

151

F o u r - Q u a n t u m R F - R e s o n a n c e in t h e G r o u n d S t a t e of an Alkaline A t o m E.B. Alexandrov and A.S. Pazgalev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t59

VIII

Hard Highly Directional X - R a d i a t i o n E m i t t e d B y a C h a r g e d P a r t i c l e M o v i n g in a C a r b o n N a n o t u b e V. V. Klimov and V.S. Letokhov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

167

Quasiclassical A p p r o x i m a t i o n in t h e T h e o r y of S c a t t e r i n g of Polarized A t o m s E.L Dashevskaya and E.E. Nikitin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

185

I o n B e a m I n e r t i a l Fusion R. Bock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

211

Spin-Echo E x p e r i m e n t s w i t h N e u t r o n s and with Atomic Beams G. Schraidt and D. Dubbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

231

A N e w G e n e r a t i o n of Light Sources for Applications in S p e c t r o s c o p y M. Inguscio, F.S. Cataliotti, C.- Fort, F.S. Pavone and M. Prevedelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

245

R e m o t e Sensing of t h e E n v i r o n m e n t Using Laser R a d a r Techniques M. Andersson, E. Edner, J. Johansson, S. Svanberg, E. WaUinder and P. Weibring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257

A p p l i e d Laser S p e c t r o c o p y in C o m b u s t i o n Devices V. Sick and J. Wolfrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271

T h e Surface of Liquid H e l i u m - an U n u s u a l S u b s t r a t e for U n u s u a l C o u l o m b S y s t e m s P. Leiderer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

283

A s p e c t s of Laser-Assisted Scanning T u n n e l i n g M i c r o s c o p y of T h i n Organic Layers S. GrafstrSm, J. Kowalski and R. Neumann . . . . . . . . . . . . . . . . . . . . . . . . .

295

O p t i c a l S p e c t r o s c o p y of M e t a l Clusters M. VoUmer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311

N e w C o n c e p t s for I n f o r m a t i o n Storage B a s e d on Color C e n t r e s A. Winnacker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335

E x c i t o n s a n d R a d i a t i o n D a m a g e in Alkali Halides K. Schwartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

351

P o l a r i z a t i o n of Negative M u o n s I m p l a n t e d in t h e Fullerene C60: Speculations A b o u t a Null R e s u l t A. Schenck, F.N. Gyax, A. Amato, M. Pinkpank, A. Lappas and K. Prassides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

367

P o s i t r o n i u m in C o n d e n s e d M a t t e r Studies w i t h Spin-Polarized P o s i t r o n s J. Major, A. Seeger, J. Ehmann and T. Gessmann . . . . . . . . . . . . . . . . . . .

381

IX Light-Induced Liberation of Atoms and Molecules f r o m Solid S u r f a c e s F. T r @ e r

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

On the Shoulders of Giants E a r l y H i s t o r y of H y p e r f i n e S t r u c t u r e Spectroscopy. For Gisbert zu Putlitz P. Br/x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

423

439

Two-Photon Method for Metrology in Hydrogen Bernard Cagnac Laboratoire Kastler-Brossel, Ecole Normale Sup~rieure et Universitfi Pierre et Marie Curie, 75252 Paris Cedex 05, France

1

Introduction

The history of two-photon transitions ( E 2 - E 1 = 2hw) starts with the beginning of quantum mechanics, during this fascinating period around nineteen thirty when all the modern physics was born. The calculation of the twophoton processes was one of the first applications of the time dependent perturbation theory. It was pub!ished in Annalen der Physik in 1931 as the thesis of Maria GSppert-Mayer "Uber Elementarakte mit zwei Quantenspriingen" [1] at the University of GSttingen in Germany. At the end of the paper she thanks professor Born and Weisskopf. Owing to the relatively small probability of such processes, the experimental realization requires a high intensity of the electromagnetic wave. This is the reason why the first experimental observations where done in the radiofrequency range [2,3]. In the optical range, the ruby laser opened in 1961 the era of multiphoton excitation with the experiment of Kaiser and Garret [4] between braod bands in a crystal. The first precise experiment between narrow atomic levels was done by Abella in Cs vapour using thermal tuning of the ruby laser [5]. From 1968 on, the realization of the really tunable dye laser permitted easier experiments [6] and was followed by an explosion of the number of experiments in atomic physics.

2

Two-Photon Method

Figure 1 represents the energy diagram for the process of two-photon absorption, showing the energy defect hAwr = (Er - El) - h~ of the intermediate relay level Er. The square of this energy defect (z~wr) 2 appears in the denominator of the transition probability, calculated in the perturbation theory to 2nd order. (Strictly speaking, it is necessary to carry out the summation over all other levels, but often only one Er has an important contribution). One must compensate the big term (Awr) 2 (the energy defect) in the denominator by a high intensity of the light beam. For that, you can enclose the atoms between two mirrors forming a Fabry Perot cavity. Then the atoms are exposed to two light beams travelling in opposite directions: forward and

Bernard Cagnac

ENERGX

ATOI v -

FRAM :)

E:-'I, BACh(WARD Fig. 1. Energy diagram of a two-photon transition

backward. If you calculate in the rest frame of one particular atom, and if you take into account the Doppler shift of the light frequency due to the velocity v of this atom (the component on the common axis of the two light beams) the apparent frequencies of the two light beams are symmetrically shifted (w + kv), where k is the wave vector, inverse to the wavelength. If the atom absorbs one photon of each oppositely travelling wave, the total amount of the absorbed energy is 2hw independent of the velocity. And the two-photon transition will be Doppler-free. This analysis was done in 1970 by Professor Chebotaev and coworkers in Novosibirsk [7]. In our laboratory, we came later on the same problems [8], but we were interested to study the real feasibility of the two-photon method: the detailed and numerical calculation of the probability showed that the experiment was possible with the low power of dye lasers with very high spectral purity. We analysed the selection rules. We were also able to show that it was possible to overcome the problem of light shifts. We benefited by the vicinity of Professor Cohen-Tannoudji, who had first understood the origin of the light shifts [9], and had the good formulas for their evaluation [10]. The first experiments applying these ideas were carried out almost simultaneously in three groups: in Paris [11], in Harvard [12] and in Stanford [13] in 1974. A review of early experiments can be found in [14]. The proposal for the 1S-2S transition in hydrogen, we gave at this moment [8], was made also independently by Baklanov and Chebotaev [15]. Everybody knows the success obtained later with that experiment by the team of Professor Hgnsch in Garehing [16]. The difficulty was to produce enough light power at the corresponding wavelength of 243nm (two times the L y m a n - a wavelength of 121.5 nm; cf. energy diagram of Fig. 2a showing recent experiments). The 243 nm radiation was produced by frequency doubling pulsed light at 486 nm in a lithium formate crystal.

Two-Photon Method for Metrology in Hydrogen ZERO OF ENERGY [ionizedhydrogen}

I'

~9c

~|

Sw2

Z~ .... M

3/2

D3/2 D5/2 ~ 12=3

~/2

j

i

P312

S C A L E xl I f ULTRAV'IOLK'T

SCALE

I ~3.10~s hcR

x4

~,mb shift

n=2

n=l

-~ [orbit.} 0 ~

(a)

~"'x' ~'~'~.~ .~.,.~='~' ~.,.- O~ (total.]

i12

1

1

I/2

312

2

3/2

5/2

(c)

Fig. 2. Energy diagram of atomic hydrogen showing the transitions recently studied in the indicated laboratories (corresponding wavelengths are indicated at the bottom. a) full diagram from the ground level n = 1 up to ionisation (zero of energy). b) energy scale multiplied by 4 showing the studied transitions from the metastable 2S level, c) fine structure of levels n = 2 and n = 3 depending on the two quantum numbers orbital l and total j.

But the rigorous control of the light frequency during pulses raised m a n y problems. Improvements of such pulsed experiments were obtained via pulseamplified continuous wave (cw) lasers and filtering of the laser pulses. Nevertheless, it is impossible to avoid frequency shifts during the amplification [17]. A detailed study of this frequency chirping effect [18] shows clearly that high precision experiments must be performed with continuous wave light beams.

4

3

Bernard Cagnac

L o n g i t u d i n a l or C o l l i n e a r I r r a d i a t i o n of an A t o m i c Beam

In Paris, the choice was to work in a more conventional range of wavelength, and that permitted to observe in 1985 the first two-photon transition produced with a cw light b e a m in an atomic beam of hydrogen [19], developing the collinear geometry, which has been adopted now in all two-photon precision experiments: the atomic b e a m is irradiated longitudinally, along its own direction, thus producing the m a x i m u m first order Doppler shift, for each of the two counter-propagating laser beams. It does not matter, as the first order Doppler shift is perfectly cancelled. Surely, this collinear geometry increases the very small residual second order Doppler shift; but it increases also largely the interaction time between atoms and light and reduces the transit time broadening below 10 kHz. T h e more conventional wavelengths were found by using the metastabte 2S level of H as the departure of the transitions (instead of arrival; cf. the magnified energy diagram in Fig. 2b). Figure 3 shows the heart of the experiments in Paris for transitions starting from the metastable 2S level of hydrogen: molecular hydrogen is dissociated in a water cooled radiofrequency discharge and the atoms effuse through a nozzle into a first vacuum chamber. A small fraction of them (10 -7) is excited to the metastable 2S state via electron b o m b a r d m e n t . Some of these atoms, deviated through an angle of about 20 ° as a result of the collisions, pass through an aperture into a second vacuum chamber. Another aperture

I Re~ord~ag I~ \

FeedbacA¢

Towards the pumps

Fig. 3. Experimental set up for two-photon transitions from the metastable 2S level, with the longitudinal or colfinear irradiation of the atomic beam.

Two-Photon Method for Metrology in Hydrogen

5

about 50 cm downstream is followed by a detection region in which an electric field is applied to quench the metastable atoms by mixing the 2S and 2P states of neighbouring energy (cf. Fig. 2c) as in Lamb's experiment: the Lyman-~ intensity emitted from the 2P state, recorded with photomultiplier tubes, is proportional to the number of the atoms in the metastable 2S state. The excitation of the resonance is signalled by the depletion of the population of the 2S state as most of the excited atoms fall down in energy to the ground level directly or through the 2P level, avoiding the metastabte 2S state. The apertures help to define the atomic trajectories, a knowledge of which is essential to the interpretation of the atomic spectra. A continuous wave tunable laser propagates along the central axis of the atomic beam. There are, in fact, two intense laser beams as the entire beam apparatus is located inside a high finesse optical cavity. Resonant coupling of the laser light into a TEM00 mode of this cavity ensures both complete overlap of forward and backward beams and enhancement of the optical power available to excite the atoms, both of which are necessary in order to achieve a decent two-photon excitation rate. In the first experiments [19] a dye laser was used, though now a titanium::sapphire laser has replaced it yielding higher power and better frequency stability. In this way, the energy spectral density in the interaction region is equivalent to powers of up to 100 W in a bandwidth of a few kHz. For experiments starting from the 1S ground level, the electron bombardment does not exist; and the hydrogen atoms prepared in the lateral discharge are guided through a Teflon tube [20] to a nozzle, which is a narrow channel centred on the axis of the laser beam. In the case of the transition to the 2S level, like e.g. in Garching, these metastable atoms are also detected at the end of the beam by electrical quenching [21].

4

Light

Shifts

and

Two-Photon

Line

Shapes

A detailed understanding of the line shape is crucial to the success of these experiments. In fact, the most important problem concerning the lineshape comes from inhomogeneous light shifts. When an atom crosses the laser beam, it experiences a varying light intensity and hence a varying light shift. In some instances, this can produce asymmetrical line shapes. However, one can show that if the two-photon transition is far below saturation (two-photon transition probability p(2) ~ < Fe) the light shifts may be much smaller than the natural width F~ [8]. In such cases it is sufficient to work with different optical powers and to extrapolate to zero power. Close to saturation and in precise metrology, one is forced to calculate the light shifts exactly. This was done for example in the experiment on atomic hydrogen performed in Paris on the 2S-nS/nD transitions. The excitation geometry, taken into account in the computer calculation, is indicated in Fig. 4 (the atomic beam is larger than the laser beam). The lineshapes are catcu-

6

Bernard Cagnac o n e particular atomic trajectory

lStdial:)h.

2mldiaph. ,

Fig. 4. Detailed geometry of atomic trajectories through the Gaussian laser beam, taken into account in the computer calculation.

L

lated by summing the contributions of all possible atomic trajectories crossing the laser beams, taking into account the Gaussian distribution of the light power. Figure 5 shows two examples of such lineshapes for the 2S1/2-10S1/2 and the 2S1/2-10D~/2 transitions. As explained previously, these signals appear in absorption. Both signals have asymmetrical lineshapes which are well modelled by calculations. After adjustment of the theoretical curve with the experimental points, the computer is able to calculate the precise true (i.e. unshifted) position of the line [22]. As an illustration we compare in Fig. 6 for each value of the light power: - this true position of the transition (laser frequency) given by the computer calculation after correction of the light shifts - the "brute" half maximum centre of the experimental line The extrapolation of "brute" centres is not exactly linear, whereas the calculated positions are well independent of the light power, which confirms the validity of the theoretical model.

n(2S) n(2S)

(a)

(b) )

! atomic frequency

i

i

i

I

atomicf~uency

Fig. 5. Experimental recordings of two-photon transitions 2S-10S and 2S 10D showing the red-shift for one and the blue-shift for the other in agreement with theory (points are experimental; full lines are theoretical).


7

H a l f - m a x i m u m center of the line 1

- = =:=:-:-::-

....... ! ....... ....... T/i

....... i .......

position

o

Z ~3 +~z.--->1 + 2

(b)

Fig. 10. Principle of two-photon standard in rubidium: a) energy diagram ; b) experimental recording: UV fluorescence at 420 nm versus the laser frequency (points: experimental - full line: fitted Voigt profile).

The precision of the experiment in Paris is limited by the precision of the iodine standard (the width of the Doppler-free line of iodine is rather large, around 5 MHz).

7

D e v e l o p m e n t of a N e w Optical Standard

The problem is to find a new optical standard with a smaller linewidth than iodine, and a wavelength better fitted to the ones used in the hydrogen experiments. The two-photon method has given such a standard with the rubidium atom. Figure 10a shows the energy diagram of the lower levels of the rubidium a t o m with the two-photon transitions. Numerous two-photon transitions have been observed in rubidium, at the beginning of the two-photon method, reaching excited levels with n = 11 to 32 [47] and then up to n = 124 [48]. The lower excited level could not be attained with the dye lasers at that time. Now the Ti::Sapphire laser or the laser diodes working in the near infra-red permit to observe the lower levels. The 5S-5D transition at )~ = 7 7 8 n m is particularly intense because the laser frequency is close to the resonances with 5P1/2 (795 nm) or better 5P3/2 (780 nm). T h a t permits to produce the two-photon transitions with low intensity, without focusing the laser b e a m and to reduce the light shifts to a negligible level. The detection is very easy by collecting the fluorescence light at A = 420 nm in the near UV, emitted in the cascade from the 5P level. Figure 10b shows a particular line chosen inside the hyperfine multiplet [49]; the experimental points are fitted with a Voigt profile. This typical recording is obtained with 100 m W light power each way in an area rt w 2 = 10 m m 2. The width of 500 kHz is ten times smaller than the iodine lines and the signal-to-noise ratio is of the order of 400-500. We have verified

Two-Photon Method for Metrology in Hydrogen ECL

13

~-~merphic plum

Fig. 11.

~eJding

to lock-in ampllfler

Experimental set-up of the two-photon standard (ECL: Extended Cavity laser diode with grating for wavelength selection).

that the position of the line is quasi-independent of the Rb vapour pressure (from 10 -4 Torr to 2.10 -5 Torr by varying the cold point of the Rb cell from 90 °C to 50 °C) and to the change of the cell. The only significant shift is the light shift: in this case, it is much smaller than the line width; and, working with increasing light power up to 600 mW, we have verified that it can be linearly extrapolated. The residual light shift in the case of 100 m W (case of Fig. 10b) is 2 kHz only and can be stabilised with the light intensity. The second step was to build an autonomous standard [50] following the scheme of Fig. 11: The laser diode is mounted in an extended cavity with a grating in order to assure a single mode oscillation. The shape of the light beam is corrected by an amorphic prism before irradiating the Rb cell through 2 or 3 Faraday isolators preventing a spurious feedback to perturb the laser diode. A lateral lens with big aperture collects the UV photons emitted in the return cascade from the 5P level. The two-photon resonance could be observed by simply reflecting the laser beam back onto itself with a mirror without the usual Fabry Perot cavity; but the cavity is better to assure a perfect coincidence of the incoming and return beams. The stabilization is obtained with two servo loops: the fast one reacts on the laser diode current while the slow one controls the piezoelectric transducer (PZT) supporting the grating (Fig. 11). In a set of preliminary measurements the frequency stability has been controlled; the square root of the relative Allan variance is: a(2, r ) / u = 3.1013/V/~

up to

r = 1000sec .

(1)

It is ten times better than a He-Ne laser stabitised on iodine. A preliminary calibration has been done [49], relative to the iodine standard, using the same frequency chain as for the hydrogen experiment [44]. The precision of some kHz is limited by the iodine standard. A new calibration is in development in the Laboratoire Primaire des Temps et Fr6quences (L.P.T.F.) in the Observatory of Paris, using the fact that the rubidium frequency is close to the 13th harmonic of the CO2 laser stabilised on OsO4.

]4

Bernard Cagnac

An optical fiber (less than 3 km) between the Observatory and our university permits to exchange the lightwaves of our Rb standards and to control their stability; it will permit to transfer the calibration without displacement of any standard. The calibration of the hydrogen transition 2S-8S/8D will be obtained by the triple mixing on a Schottky diode of the Ti::Sapphire light wave used for hydrogen, the rubidium light wave, and a 40GHz klystron adjusted to bridge the gap. Surely, that will permit to surpass the precision of 10 -11 . Up to 10 -12 ? - We will see. Anyway, except for the particular transition 1S-2S, taking into account the natural line widths of the hydrogen levels, it seems hardly likely that the experimental precision can be pushed largely beyond 10 -1~. Surely, the highly excited Rydberg levels (particularly "circular" states with l = n - t) are long living; but they are also very sensitive to residual parasitic fields and not adapted to metrology. 8

Confrontation

with

Improvements

of the

Theory

Admitting that the experimental precision can be pushed in the future to the 10 -12 level, these measurements can be interpreted only if the theory reaches the same level of precision, which is not yet the case. Where has the theory arrived now? All calculations start from the formula obtained by Dirac from his relativistic equation [51] modified with the reduced mass/~ = m/(1 + re~M) in place of the true mass m of the electron (M is the proton mass). This modified formula has no exact theoretical justification, as the classical centre of mass has no sense in relativity; nevertheless, it permits to take into account to first order the problem of the nuclear motion. Therefore the binding energy E,~j of the level of quantum numbers n and j is given at zero order by the formula: EnJ :

#c2Z2~ 2

I,.gC2

1

V with

l j(

= J + 2 -

J+

- (Za)2

~

,zo,2

2j + 1

(3)

Ze is the charge of the nucleus (e the elementary charge). (~ = e2/4aotic ,~ 1/137 is the fine structure constant, which gives the order of magnitude of the ratio v/c of the electronic velocity v to the light velocity c. In fact the relativistic calculation and the nuclear motion corrections are intimately intricated and one has to perform an expansion simultaneously in powers of two independent variables m / M and Z a . T h a t explains the


15

five lines in the top of table 2; the well known terms of first order in m/M or in ( Z a ) 2 do not appear in this table, as they are taken into account inside the Dirac formula. This table is intended for non specialists; and for simplification, when we give the order of each term in the expansion, we choose as term of zero order, i.e. as unity, the binding energy E,j (Eq. (2)), which is measured in experiments. (The specialists choose as zero order term the mass energy mc2; i.e. each term must be multiplied by (Za) 2 if you compare with the theoretical formulas, indicated by their reference numbers in some theoretical review papers [52-55].) The radiative corrections, coming from the quantum electrodynamics, necessitate an other expansion in power of c~, which occupies the central part of the table 2. Among these corrections one distinguishes the self-energy (S.E.), the vacuum polarization (V.P.) and the effect of the anomalous magnetic moment, which affects the spin orbit constants in the P and D levels. The most i m p o r t a n t terms correspond to the self-energy in the S states (l = 0) in which the electron has a significant probability density in the nucleus; that raises the degeneracy in l, characteristic for the Dirac formula and produces the so called L a m b shift between the $1/.~ and P1/2 levels (see Fig. 2c). It is yet necessary to calculate the crossed corrections, i.e. radiative corrections to the recoil terms, or recoil corrections to the L a m b shift. The term of lowest order in (m/M)o~(Zo~)2 does not appear in table 2, because it is taken into account by multiplying all radiative corrections with the factor (p/rn) 3 which represents the modification of the wave function at the origin. The terms of higher orders are small but not negligible, and their calculations is not yet quite clear [63,64]. Finally, one must add the correction due to the finite size of the nucleus, which is particularly important for the S levels (l = 0; important electron probability density in the nucleus) and depends on a badly known p a r a m e t e r the r.m.s, proton radius x/(r2). All these calculations use the values of the two parameters rn/M and a. But, since the last adjustment of the fundamental constants [35], the precision has been strongly increased on these two parameters: the electron to proton mass ratio m/M in Penning trap experiments [65], the fine structure constant a in measurement of (g-2) [66] and progresses in its calculation [67]. The present uncertainty on these parameters does not raise problems on the calculation of the energy levels before the 10 -12 level of precision. The limitation comes from the high order necessary in the expansion. The number of terms (or Feynman diagrams) which appear in the expansion increase strongly with the order of the expansion; when you come to the fifth order it can approach a hundred. The calculation is not easy and all terms to the fifth order gave rise in the past years to controversies; the most of which seem now to be settled, owing to very recent advances since two years. These most recent papers are referenced in table 2. The problem arises from the fact that the coefficients found inside some of these terms can be as big as 100, and then these terms are equivalent to

16

Bernard Cagnac

Physical effec~ involved

[S2l 'Order in d~e

~paeston

On;/m"o f

ma~

a~,! -7.=1

n:duced rna~

1

1

E ~ ....

Review- P a l m {531 {5,,1 Sa~tei~ y~ 1990

M~ra" 1995

~admda 1995

(39) (40)

, (19)

(41)

(34)

w,et,k

(2-.4)

(2-7) ~(2-3)

Rela~t~"

(55]

pb.oeon

and

,.,o ~wo

Recoil

(24)

a~xerams

(2-9)

photon

(2-73

(43)

OC.gnms

o~e

loop Radiam,e

~ (Za)~

I0-~

-a(Za)'

to+

~(z~)"

1o-9

(2-10) t2-11)

(C-t)

(2-5) ),a,b.¢

(44)

(36)

(62) • (45) r (50)

S.F_ {21) (23)

$.E.

~' V,P.

[,t61 [4'71 v.P.

(2n

[.561

[5"71

{ss]

[5[[ [S3I oml.~.-tlo~

-a(Za) ~ fc

two Q.E.D.

[59]

to-n

2

(2-1'2)

( ~ ) (Za) z

10-9

(~)~(Za) J

10-|0

(29)

(2-13, (2-153 r (61)

loop diagrams

01)

[61]

(32) (33)

[62]

(37) ~ (38)

{~l [6,t}

(40)

[~]

: Crossed oor~c~ons

m a Za)* (~)~'(

(2-8)

I0"'

(ReL only)

radiative and mvoil m 2~. g l x :

Fimte s g e of nucleus

(2-30)

(2-11)

(64)

(40) I

Table 2. Summary of the power expansion ia the calculation of energy levels.

the terms of preceding order. This was the case in particular for the term in ( a / l r ) 2 ( Z a ) 3 (two loop diagram in the radiative correction) which, after exact calculation, was found ten times bigger than expected, and modified consequently the interpretation of the 1S L a m b shift (see section 5). It remains yet a controversy on the crossed correction of order (re~M)(a/Tr)(Zoo)3 and most of the sixth order terms are unknown. As a consequence, one can now hardly await the 10 -11 precision from the theory.


17

Another limitation comes from the large uncertainty on the proton radius, ( V / ~ , which appears in the nuclear finite size correction (last line of table 2). In fact, the finite size correction and the Q.E.D. corrections are intricately mixed in the exact determination of S levels; and the advances in the Q.E.D. correction will permit to deduce a value of the proton radius. If both theoreticians and experimentalists attain in the future the level of precision of 10 -12 on the totM energy (or a few 10 -7 on the 1S Lamb shift) the proton radius could be determined with a precision better than 1%. On the other hand, the determination of the Rydberg constant can be in some respects independent of the proton radius (and partially of the Q.E.D. corrections) if one utilises some particular combinations of transitions, using the fact that the finite size correction scales as 1/n 3 with successive n levels with the same orbital quantum number I (that is the same for a big part of the Q.E.D. corrections). As it was noticed in [62], for example, the combination (Lls - 8L2s) of the 1S Lamb shift and the 2S Lamb shift is small and quite independent of the proton radius; and the present limit of the theory could nevertheless permit a determination of the Rydberg constant at the 10 -12 level, if the experimentalists continue their advances.

9

Conclusion

At the present time, experimentalists and theoreticians have attained the same level of precision of a few 10 -11 in the determination of the energy levels of hydrogen. One proof is given by the coincidence at this level of precision of the values obtained for the Rydberg constant R ~ from different transitions. Taking into account the big progresses which have been accomplished in the past few years, in experiments and in theory it is possible to hope for further advances in the next years up to a level of 10 -12 with its consequence for the knowledge of the proton radius. The comparisons with other hydrogenic systems (ionic He +, muonium, positronium, ...) surely will help to improve the knowledge of the Q.E.D. corrections and participate to this advancement; these problems are explained in other papers. Nevertheless, it is not sure that it will be possible to pass beyond the experimental limit of 10 -12. Certainly, the exceptional precision of the particular transition 1S-2S will be used in the future for an optical frequency standard with very slow atoms from an hydrogen trap [68]. But the theoretical interpretation, depending on two parameters, R ~ and X / ~ , requires two independent measurements with the same level of precision (supposing that c~ and m / M will follow the same advance...). Anyway, the Rydberg constant R ~ , determined with 10 -12 precision, will remain a corner stone of the adjustments of fundamental constants, far in advance compared to other constants. R ~ is a combination of three fundamental constants m, h and e; two of them h and e are linked by the fine

18

Bernard Cagnac

structure constant a with a precision, better than 10-8; but a third relation at the same level of precision is missing; and m, h and e are known individually with a precision hardly better than 10 -6. Will some advance in the Josephson effect (depending on h/e) be able to bridge this gap? Let me conclude with a personal feeling of perplexity when looking at table 2, which summarizes the calculations: is it not surprising that any correspondence with the reality is obtained after so numerous pages with complicated integrals in the scientific journals and after so many hours of abstract computer work? Nevertheless it works! All right, we believe that physics explains the world and provides us with an understanding of reality i.e. reduces apparent complexity to simplicity. But in what sense is it possible to speak of explanation and understanding when we compare these long and tedious calculations with the spontaneous functioning of any simple electron in all water molecules of the ocean, or in all atoms of the intergalactic clouds?

Note added in proof." The experiments with the new Rb standard and the optical fiber (described in section 7) obtained very recently a new result at a level of 10 -11 for the Rydberg constant R ~ = 109737.3156859 (10) [69].

References [1] Ghppert-Mayer M., Ann.Phys., Lpz 9, 273 (1931) [2] Hughes V.W. et Grabner L., Phys. Rev. 79, 314 and 819 (1950) [3] Brossel J., Cagnac B. et Kastler A., C.R.Acad.Sci.Paris 237, 984 (1953), and J.Physique 15, 6 (1954); Kusch P., Phys.Rev.93, 1022 (1954) and 101, 1022

(1956) [4] Kaiser W., Garrett C.G.B., Phys.Rev.Lett. 7, 229 (1961) [5] Abella I.D., Phys.Rev.Lett. 9,453 (1962) [6] Bonch-Brnevich A.M., Khodovoi V.A., Khronov V.V., JETP Lett. 14, 333 (1971); Agostini P., Ben Soussan P., Boulassier J.C., Opt. Comm. 5,293 (1972) [7] Vasilenko L.S., Chebotaev V.P. and Shish£v A.V., JETP Lett. 12, 113 (1970) [8] Cagnac B, Grynberg G. and Biraben F., J.Physique 34, 845 (1973) [9] Cohen-Tannoudji C., Ann.Phys. 7, 423 et 469 (1962) Alexandrov E.B., BonchBruevich A.M., Kostin N.N., Khodovoi V.A., JETP Lett. 3, 53 (1966) [10] Cohen-Tarmoudji C., Dupont-Roc J., Phys. Rev.A 5, 968 (1972) [11] Biraben F., Cagnac B. et Grynberg G., Phys.Rev.Lett. 32, 643 (1974) [12] Levenson M.D., Blembergen N., Phys.Rev.Lett. 32, 645 (1974) [13] H£nsch T.W., Harvey K., Meisel G. and Schawlow A.L., Optics Comm. 11, 50 (1974) [14] Grynberg G., Cagnac B. and Biraben F., in: "Coherent Non Linear Optics" (Springer) Topics in Current Physics, vol. 21, p 111 (1980) [15] Baklanov E.V. and Chebotaev V.P., Opt.Comm. 12, 312 (1974) [16] H£nsch T.W., Lee S.A., Wallenstein R. and Wieman C., Phys. Rev.Lett. 34, 307 (1975); Lee S.A., Wallenstein R., H/insch T.W., Phys.Rev.Lett. 35, 1262 (1975); Wieman C., H£nsch T.W., Phys.Rev.A. 22, 192 (1980) [17] Tr~hin F., Biraben F., Cagnac B. and Grynberg G., Opt.Comm. 31, 76 (1979)

Two-Photon Method for Metrology in Hydrogen [18] [19] [20] [21]

19

Danzmann K., Fee M.S. and Chu S., Phys.Rev. A 39, 6072-3 (1989) Biraben F. and Julien L., Opt.Comm. 53,319 (1985) Walraven J.T.M. and Silvera I.F., Rev. of Sci.Instr. 53, 1167 (1982) Zimmermann C., Kallenbach R. and H~izlsch T.W., Phys.Rev.Lett. 65, 571-4

(1990) [22] Oarreau J.C., Allegrini M., Julien L. and Biraben F.J., Physique 51, 2263, 2275 and 2293 (1990) [23] Schmidt-Kaler F., Leibfried D., Seel S., Zimmermann C., K6nig W., Weitz M. and HKnsch T.W., Phys.Rev. A 51, 2789 (1995) [24] Lamb W.E. Jr. and Retherford R.C., Phys.Rev. 72, 241-3 (1947) [25] Lundeen S.R. and Pipkin F.M., Metrologia 22, 9 (1986); Hagley E.W. and Pipkin F.M., Phys.Rev.Lett. 72, 1172 (1994) [26] Wieman C.E. and H~insch T.W., Phys.Rev. A 22, 192-205 (1980) [27] Thompson C.D., Woodman G.H., Foot C.J., Hannaford P., Stacey D.N. and Woodgate G.K., J.Phys.B.: At.Mol.Opt.Phys. 25, L1-4 (1992) [28] Weitz M., Schmidt-Kaler F. and H£nsch T.W., Phys.Rev.Lett. 68, 1120-3 (1992); Weitz M., Huber A., Schmidt-Kaler F., Leibfried D. and H~insch T.W., Phys.Rev.Lett. 72, 328 (1994) [29] Berkeland D.J., Hinds E.A. and Boshier M.G., Phys.Rev.Lett. 75, 2470 (1995) [30] Bourzeix S., de Beauvoir B., Nez F., Plimmer M.D., de Tomasi F., Julien L., Biraben F. and Stacey D.N., Phys.Rev.Lett. 76, 384 (1996) [31] Hand L.N., Miller D.G. and Wilson R., Rev.Mod.Phys. 35, 335 (1963) [32] Simon G.G., Schmitt C.H., Borkowski F., Walther V.H., Nucl.Phys. A 333,

381 (1980) [33] Sick I., Phys. Lett. B 116, 212 (1982) [34] Barr J.R.M., Girkin J.M., Ferguson A.I., Barwood G.P., Gill P., Rowley W.R.C. and Thompson R.C., Opt. Comm. 54, 217 (1985) [35] Cohen E.R. and Taylor B.N., Rev.Mod.Phys. 59, 1121 (1987) [36] Biraben F., Garreau J.C. and Julien L., Europhys.Lett. 2, 925 (1986) [37] Zhao P., Lichten W., Layer H.P. and Berquist J.C., Phys.Rev.A 34, 5138

(1986) [38] Zhao P., Lichten W., Layer H.P. and Berquist J.C., Phys.Rev.Lett. 58, 1293 (1987) [39] Beausoleil R.G., McIntyre D.H., Foot C.J., Hildum E.A., Couillaud B. and H£nsch T.W., Phys.Rev./k 35, 4878 (1987) [40] Boshier M.G., Baird P.E.G., Foot C.J., Hinds E.A., Plimmer M.D., Stacey D.N., Swan J.B., Tate D.A., Warrington D.M. and Woodgate G.K., Nature 330, 463-5 (1987) [41] Biraben F., Garreau J.C., Julien L. and Allegrini M., Phys.Rev.Lett. 62,621 (1989) [42] Andreae T., KSnig W., Wynands W., Leibfried D., Schmid-Kaler F., Zimmermann C., Meschede D and H£nsch T.W., Phys.Rev.Lett. 69, 1923-6 (1992) [43] Nez F., Plimmer M.D., Bourzeix S., Julien L., Biraben F., Felder R., Acef O., Zondy J., Laurent P., Clairon A., Abed M., Millerioux Y. and Juncar P., Phys.Rev.Lett. 69, 2326-9 (1992) [44] Nez F., Plimmer M.D., Bourzeix S., Julien L., Biraben F., Felder R., Millerioux Y. and De Natale P., Europhys.Lett. 24, 635 (1993)

20

Bernard Cagnac

[45] Jennings D.A., Pollock C.R., Petersen F.R., Drullinger R.E., Evenson K.M., Wells J.S., Hall J.L. and Layer H.P., Opt.Lett. 8, 136 (1983) [46] Acef O., Zondy J.J., Abed M., Rovera D.G., G6rard A.H., Clairon A., Laurent P., Mill6rioux Y. and Juncar P., Opt. Comm. 97, 29 (1993) [47] Kato Y. and Stoicheff B.P., JOSA 66, 490 (1976) [48] Stoicheff B.P. and Weinberger E., Can.J.Phys. 57, 2143 (1979) [49] Nez F., Biraben F., Felder R. and Millerioux Y., Opt.Comm. 102,432 (1993) [50] Millerioux Y., Touhari D., Hilico L., Clairon A., Felder R., Biraben F. and de Beauvoir B., Opt.Comm. 108, 91 (1994) [51] Dirac P.A.M., Proc.Roy.Soc. A 117, 610 (1928) [52] Erickson G.W., J.Phys.Chem. Ref.Data 6, 831 (1977) [53] Sapirstein J.R. and Yennie D.R., in "Quantum Electrodynamics", edited by T. Kinoshita (World Scientific, Singapore, 1990) [54] Mohr P.J., "Fundamental Physics" in Atomic, Molecular and Optical Physics Reference Book, Drake (ed.) (American Institute of Physics, 1996) [55] Pachucki K., Leibfried D., Weitz L., Huber A., Kgnig W. and H£nsch T.W., Lecture at NATO Advanced Study Institute (Edirne, Turkey) - September 1994 [56] Khriplovich I.B., Milstein A.I. and Yelkhovsky A.S., Physics Scripta T 46, 252 (1993); Fell R.N., Khriplovich I.B., Milstein A.I. and Yelkhovsky A.S., Phys.Lett. A 181, 173 (1993) [57] Pachucki K. and Grotch H., Phys. Rev. A 51, 1854 (1995) [58] Pachucki K., Ann.Phys. (N.Y.) 226, 1 (1993) [59] Molar P.J., Phys.Rev. A 46, 4421 (1992) [60] Eides M.I. and Grotch H., Phys.Lett. B 301,127 and B 308,389 (1993); Eides M.I., Karshenboim S.G. and Shelyuto V.A., Phys.Lett. B 312, 358 (1993); Eides M.I. and Shelyuto V.A., JETP Lett. 61, 478 (1995) [61] Pachucki K., Phys.Rev.Lett. 72, 3154 (1994) [62] Karshenboim S., JETP Lett. 79, 230 (1994) [63] Bhatt G. and Grotch H., Phys.Lett. A 58, 471 (1987) and Arm.Phys. (N.Y.) 178, 1 (1987) [64] Pachucki K., Phys.Rev. A 52, 1079 (1995) [65] Van Dyck R.S., Farnham D.L. and Schwinberg P.B., I.E.E.E. Trans.Instr.Meas. 44, 546 (1995) [66] Van Dyck R.S. Jr., Schwinberg P.B. and Dehmelt H.G., Phys.Rev.Lett. 59, 26 (1987) [67] Kinoshita T. and Lindquist W.B., Phys.Rev. D 42, 636 (1990); Kinoshita T., Phys.Rev.Lett. 75, 4728 (1995) [68] Cesar C.L., Fried D.G., Killian T.C., Polcyn A.D., Sandberg J.C., Doyle J.M., Yu I.A., Greytak T.J. and Kleppner D., Communication to Fifth Symposium on Frequency Standard and Metrology - Woods Hole, October 1995 [69] de Beauvoir B., Nez F., Julien L., Cagnac B., Biraben F., Touahri D., Hilico L., Acef O., Clairon A. and Zondy J.J., Phys.Rev.Lett. 78, 440 (1997); details can be found in the thesis of B6atrice de Beauvoir (Paris, unpublished).

High Precision Atomic Spectroscopy of Muonium and Simple Muonic Atoms Vernon W. Hughes Yale University, Physics Department, J.W. Gibbs, New Haven, CT 06520 USA

Over a period of about 30 years Gisbert zu Putlitz and his colleagues have studied - rapidly and often one by one - some 1015 m u o n i u m atoms, which is equivalent to the number of hydrogen atoms in a bottle of H2 gas with a volume of 0.01 m m 3 at a pressure of 1 atm.

1

Introduction

It's a great pleasure to be able to celebrate Gisbert's 65th birthday with this S y m p o s i u m on Atomic Physics Methods in Modern Research at this great university and in this beautiful city. I first met Gisbert here in Heidelberg in about 1960 when we held the annual Brookhaven Molecular Beams Conference organized by Bill Cohen of BNL which met first here at Heidelberg with Hans K o p f e r m a n n as our host and then at Bonn with Wolfgang Paul as our host. At Heidelberg there were two very competent young associates of Professor Kopfermann who handled the slide projector and related matters. One was Gisbert and the other was Ernst Otten. My next significant meeting with Gisbert was 5 or 6 years later when he visited Yale and we discussed the possibility of his coming to Yale to do research in atomic physics. T h a t Heidelberg meeting and Gisbert's later joining the Yale research faculty was surely one of the most fortunate occasions in my life. It was the beginning of a close scientific collaboration which continues after 30 years. The central theme of our scientific collaboration, which appealed greatly to both of us, was muonium, the #+e- atom, and other related topics in m u o n physics. During Gisbert's memorable stay at Yale research on m u o n i u m was very active for us at the Columbia University Nevis Synchrocyclotron Laboratory where muonium had been discovered in 1960 (Hughes et al., 1960; Hughes et al., 1970). We were involved principally in measurement of the hfs of m u o n i u m in its ground state by microwave magnetic resonance spectroscopy (Thompson et al., 1969; Crane et al., 1971)(Fig. 1). A quite different activity to which Gisbert contributed i m p o r t a n t l y was the organization at Yale of an International Conference on Atomic Physics

22

Vernon W. Hughes

Fig. 1. Experimental setup at Nevis in 1969 showing the Ar gas target and Navy barbettes for shielding.

(ICAP) held at NYU in New York City (ed. by Hughes et al., 1969). Fig. 2 shows Gisbert and L. Wilets at the Conference. Even during the Conference it was not clear that a second ICAP would occur. During a memorable boat trip around Manhattan Kim Woodgate and Pat Sandars agreed to host ICAP 2 at Oxford in 1970. Four years later in 1974 Gisbert hosted ICAP 4 in Heidelberg (ed by zu Putlitz et al., 1975). ICAP has become the principal international conference covering basic atomic physics broadly. The 15th ICAP has just been held in Amsterdam where celebration of the centennial year for the discovery of the Zeeman effect was part of the ICAP program. After the Nevis Synchrocyclotron Laboratory was closed down in the early 1970's and following a short period of muonium research at SREL in Williamsburg, Gisbert and I pursued our muonium experiments at the new meson factories, LAMPF and SIN, where eventually intensity increases of 103 to 104 compared to Nevis were achieved. Fig. 3 shows a three-dimensional electromagnetic coil system - built personally by expert machinist Gisbert zu Putlitz - and used in the study of muonium formation in gases, which was the first published physics research from LAMPF (Stambaugh et al., 1974). The topic I shall talk about is broader than, but includes, muonium. It is high precision atomic spectroscopy of muonium and simple muonic atoms. Several of us: Gisbert and Klaus Jungmann from Heidelberg and Malcolm Boshier and I from Yale have been writing a little article on this topic (Boshier et al., 1996a). My talk includes much of the material we have developed together, but is updated and also includes additional material.

Muonium and Simple Muonic Atoms

F i g . 2. Photograph taken at ICAP in 1968.

Fig. 3. Three-dimensional electromagnetic coil system at LAMPF.

23

24

2 2.1

Vernon W. Hughes

Precision Tests of Q E D Introduction

The principal scientific goal of high precision atomic spectroscopy is to test and study quantum field theory or, more specifically, quantum electrodynamics, the unified electroweak theory and quantum chromodynamics - all now encompassed within the modern standard theory of particle physics. In addition to testing fundamental theory, precision atomic spectroscopy also determines values of fundamental constants including particle masses and magnetic moments, R ~ , ~ and others. Real particles and atoms involve simultaneously the electromagnetic, weak and strong interactions and this often limits the sensitivity of the experimental tests of the theory. Thus for quantum electrodynamics effects of strong interactions or of hadronic structure are at present limiting importantly the QED tests in the simplest one- and two-electron atoms of H and He. With the purely leptonic atoms muonium and positronium the hadronic structure effects are avoided entirely. With simple muonic atoms the hadronic effects can be measured, and hence together with electronic atoms more sensitive tests of QED can be made. As examples we consider briefly the two classic and most important low energy tests of QED - the electron anomalous magnetic moment or g-2 value and the Lamb shift in hydrogen. 2.2

Electron A n o m a l o u s Magnetic M o m e n t and t h e F i n e S t r u c t u r e C o n s t a n t oL

The anomalous g-value a~ = (9 - 2)/2 has been measured to a precision of 3.4ppb (Van Dyck, Jr., 1990) and the QED radiative corrections (Kinoshita, 1990) have been calculated to an even higher precision as a result of a recent improved evaluation of the 6th and 8th order corrections (Kinoshsita, 1996). (This recent evaluation changed the earlier value (Kinoshita, 1995) for radiative corrections by about 50 ppb which is well outside the expected uncertainty.) In order to compare theory and experiment for ae a precise value for the fine structure constant a is needed. Figure 4 shows a plot of the most accurate determinations of a (Kinoshita and Lepage, 1990). In addition to the value recommended by CODATA in the "1986 Adjustment of Fundamental Constants" (Cohen and Taylor, 1987), these include muonium hyperfine structure (Hughes and zu Putlitz, 1990; Mariam et al., 1982), the neutron de Broglie wavelength (Kriiger et al., 1995), the ac Josephson effect (Williams, et al., 1989) (in combination with the gyromagnetic ratio of the proton in water), the quantized Hall effect (Cage et al., 1989), and finally the value obtained from a~ by equating the experimental value to the theoretical expression (Van Dyck, Jr., 1990; Kinoshita, 1996). Determination of a from the neutron de Broglie wavelength is an attractive new method. It involves


25

measuring the ratio of Planck's constant to the mass of the neutron (h/mn) from its velocity and de Broglie wavelength and then obtaining cr using the Rydberg constant R ~ and mass ratios and has no theoretical ambiguities. For comparing a~(theor) with a~(expt) the quantized Hall effect now provides the best value with a precision of about 24 ppb. As shown in Fig. 4 agreement of the various ~ values is not too satisfactory. Such a plot is very useful to test our understanding of the different approaches.

CODATA

muonium

his

~

o

I

~

o

Inutlnm

o

i

~

, t

t ac,l & ~'p'

quantum

Hall

t

o

t

a~

-0.40

I

I

I

I

I

-0.30

-0.20

-0.10

0.00

0.10

( a "1 - 1 3 7 . 0 3 6 0 )

0.20

x 104

F i g . 4. R e c e n t d e t e r m i n a t i o n s of t h e fine s t r u c t u r e c o n s t a n t c~.

For a test of QED, rather than rely upon a determination of a involving condensed matter theory it would be preferable to determine a by a method involving the simplest theoretical assumptions such as that from h/mn, or at least from a simple atom which can be treated within QED. The approach involving hiM in which M is the mass of an atom is attractive if adequate precision can be obtained (Weiss et ah, 1993; Martinos e t a h , 1994). For hydrogen the hfs interval Av in its ground state is known experimentally (Ramsey, 1990) with a precision of better than 1 part in 1012 and At, is proportional to cz2. However, the theoretical value for At, has a contribution from proton structure and polarizability of about 30 ppm and an associated uncertainty of about 1 ppm (Sapirstein and Yennie, 1990). Results of a recent evaluation of the proton size and polarizability contribution to At, are shown in Fig. 5 (Unrau, 1996). For muonium ( p + c - ) Au is known experimentally to 36 ppb (Hughes and

26

Vernon W. Hughes Av~ = 1 420 405 751.766 7(9) Hz (1 part in 10~2) AV~

= AVF(I + ~ + ~ ) AVF = 16aZcR I% 3 " #~o ~=

~+

(uncertainty _" 50 ppb)

~ghero~rmr~

(uncem~inty1/2 of the proton, which has been determined from elastic e p scattering, (Simon et al., 1980; Hand, Miller and Wilson, 1963) results in an uncertainty of 10 ppm in the theoretical value of the Lamb shift. Precise measurements (Weitz et al., 1994; Berkeland et al., 1995; Boirieux et al., 1996) by laser spectroscopy of the 1S-2S transition in H and D determine the Lamb shift in the 1S state and are similarly limited by uncertainty in proton structure. Figure 6 indicates that the Lamb shift measurements can be considered to determine < rp2 > 1 / 2

"small"proton

"big"proton I Lnndeen& Pipkin (2S-2PRF) Hag/ey& Pipkin (2S-2PRF)

~---------_ _ ~ - - - r - - - - - ~ I I ;

I

I I I

Weirs

et

al

(IS Laser)

~------ I-------~

B o ~

et

al

(1S Laser)

I I

"l'nis work

..--.A

m

-30

-20

- 10

I I I

0

(IS Laser)

10

20

30

Deviation from measured Lamb shift (ppm) Fig. 6. Indicating the determination of proton size, < rp2 >1/2 from H spectroscopy. "small" proton < r v2 >x/~ ___0.805(11)fm, "big" proton < rv2 >a/2 = 0.862(12)fm.

28

VernonW. Hughes

There are two approaches to improving the test of QED from the Lamb shift which involve muonium and muonic hydrogen. For muonium this same 1S-2S transition has been observed by laser spectroscopy (Chu et al., 1988; Maas et al., 1994) and the 1S Lamb shift has been determined to 0.8%. It should be possible in the future to achieve very high precision in this measurement where hadronic structure is not involved. The M(2S1/2 - 2P1/2) Lamb shift interval has been measured to about 1% by microwave spectroscopy (Oram et al., 1984; Badertscher, 1985; Hughes and zu Putlitz, 1990; Woodle et al., 1990) and this accuracy could also be substantially improved. Secondly, the Lamb shift in p-p (22S1/2 - 22p1/2) might be studied by laser spectroscopy. (It has been attempted, but so far unsuccessfully (von Arb et al., 1986)). The theoretical level shifts have been evaluated (DiGiacomo, 1969; Pachucki, 1996). In this atom the proton structure effect is very large and an accurate value of < 7~ >1/2 could be measured and then used to improve the theoretical value of the Lamb shift in hydrogen. 3

Current

Muonium

Experiment

at LAMPF

As mentioned in Section 2.1, an experiment is in progress at the Los Alamos Meson Physics Facility (LAMPF) to determine with high precision the hyperfine structure interval Au (to ,-~ 10 ppb) and the muon to proton magnetic moment ratio Pu/#p (to --~ 60ppb) in the ground state of muonium (#+e-). These precision goals correspond to increases in precision for Au and for Pu/Pp by about a factor of 5 compared to present knowledge. The general method of the experiment (Hughes, 1966; Hughes and zu Putlitz, 1990) is microwave magnetic resonance spectroscopy as applied to muonium. It relies on parity nonconservation in the 7r+ --+ #+uu decay to produce polarized p+ and in the p+ -+ e+iuue decay to indicate the spin direction of/~+. The most recent LAMPF experiment provides the present experimental values (Mariam, 1982):

Aue~p = 4 463 302.88(16) kHz (36ppb); Pu/Pp = 3.183 346 1(11) (360ppb) The theoretical expression for Au can be written as: Au(theory) = Av(binding, rad) + Au(recoil) + Au(rad - recoil) + Au(weak) Except for the small term Av(weak) coming from the weak interaction, and a small known contribution arising from hadronic contributions to the photon propagator, this expression arises solely from the electromagnetic interaction of two point-like leptons of different masses in their bound state. The present theoretical value is (Kinoshita and Nio, 1994, 1996; Kinoshita, 1996):

Auth = 4 463 302.38(1.34)(0.04)(0.17)kHz (0.3 ppm) The principal error of 1.34 kHz arises from the uncertainty of Pt,/Pv. The second uncertainty arises from that in c~ based on the electron g-2 experiment,


29

and the third is the estimate of the theoretical error in the latest QED calculation (Karshenboim, 1996). Weak neutral current effects associated with Z exchange in the e - p interaction contribute -0.065kHz or 15 ppb and are included in Auth. The experimental and theoretical values for Au agree well:

z~Vth -- Z~Vexp = --0.50(1.4)kHz ; AUth -- A ~ , ~ p = - - ( 0 . 1 1 ± 0 . 3 ) p p m . Allexp

The Breit-Rabi energy level diagram for ground state muonium is shown in Fig. 7. The history of the various determinations of Au and of IzF,/pp is shown in Fig. 8.

H(kG)

0

2 3 4 5 6 7 8 9 101112131415161718192021.22232425 I I I I I I I I I 111111111111111.J

98 5

+,

(F,Mp)

-3 -

(o,o) ~ - - - . . . . . . ~ t / 2 , - i / 2 )

-5 -6 -7

-8-9 H=

(-I/2,1/2)

0

v31

3

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8 X

9

10

11

12

13

14

15

~ -~ a -~ Ip.J +~t~gj J. H-~-

I~ p, ~Ip.H -~ Stng

AV_VI2+V3

2~g;H

16

4 4" AV [-X "4- ( l + X 2 ) trz]

alh = Av = AW/h=4463 M l l z --Y

F=J+

--)

ip. 1

WF=I±I MF 2 2'

= - -AW

4

V. ,

1

- ILBg~tMFH + - ~ A w ~ ] I + 2 M F x

+ X1

Fig. 7. Breit-Rabi energy level diagram for muorfium in its 12S1/2 ground state in a magnetic field.

The new experiment improves on the earlier one (Mariam, 1982) in several ways. First, the p+ b e a m intensity is now larger by a factor of 3 and is 1.107#+/s, with a duty factory of 6 to 9% and has greater purity, achieved principally by use of an E x B separator to reduce the e + background in the beam. Second, the magnetic field from a commercial Magnetic Resonance

30

Vernon W. Hughes

Av {HICaOO-19TO t ~ * t ' ) [*.eppm) 1O0

I

BERK[L[Y*IgTZIFSR. HZO)

I

I

I

YJLE" HEIOELBERG-1977 ( ~ ' , ' 1 [I.4 ppm|

•

SIN - t 9 7 8 ( ~ W . l r z l (O.Sppm)

•

SIN-1981 l ~ n . e . t l (O.5$Ppml

I

._= YALE-NEIO[LS[RG-Ig@2IF.,-) (0.36ppm| LzJ

'oI ; ~ ' ' , ; [ ~ I p p I I0 ll- 3 1 e 3 3 4 0 |

0

22 ppb

2.2 ppbf 196~

1966

)970

197q

1978

1980

1984

1990

Fig. 8. History of muonium Av and

p,,/pp

measurements.

Imaging superconducting magnet system operating in persistent mode provides a field with a homogeneity of better than 1 p p m over the active region of the microwave cavity and with a stability of 0.01 p p m to 0.1 p p m / h r . Third, an electrostatic chopper in the muon beam line provides a muon beam with an on-period of 4 ps and an off-period of 10#s, which allowed the observation of a resonance line from muonium atoms which had lived longer than the 2.2#s mean lifetime v~ of inuons (Boshier, et al., 1995). Such resonance lines can be narrower than the natural linewidth determined by 7~, and indeed narrower than a line obtained by the conventional method by factors of up to 3. The experimental setup is shown in Fig. 9. The longitudinally polarized #+ b e a m of about 26 MeV/c is stopped in a microwave cavity contained in a pressure vessel filled with krypton at a pressure between 0.5 and 1.5 atm. The entire apparatus is in a region of a strong magnetic field of 1.7T. Muonium is formed in an electron capture reaction between #+ and a Kr atom. The e + from p+ decay is detected in a scintillator telescope. Figure 10 shows the MRI solenoid and indicates the characteristics of this precision magnet which is operated in persistent mode. Resonance lines for transitions /'12 and /'34 are observed by sweeping the magnetic field with fixed microwave frequency.

Muonium and Simple Muonic A t o m s

~//////~//////////.Z/~///////~ $hlmCoils

Shielding

ModulationCoil'1"""'"~*a~Pressm'eV~I,~.

]IIL

/ 3 rail MylwW'~ndow~At-Collimllo¢

!,

PosiU-onT,lcscope.,.~Bij~mmr~

~

!L t.

p*

'-T-'

F i g . 9. E x p e r i m e n t a l setup in 1994/95.

• • • • • •

Manufacturer: Operating Field: Field StabiliLy: Length: Clear Bore 9 : Homogeneity: Modulation Coil:

Oxford Mag'aet Technology (OMT) 1.7 Tesla 10 ppb/hr 2.26 m 1.05 m

shimmed to I ppm over 20 cm dsv to scan field over +_50 G

F i g . 10. L A M P F superconducting solenoid 2 T MRI magnet.

31

32

•

Vernon W. Hughes

Idea: Observationofdecayiogs ~ s

actimesT> C.~nven~on~ method

CO~veNr,o.AL ~.ev.oo

I

..,o0 LPt.f~fi..l'3~J'~l~l~]J

I

] ¢

u,CROW,VES e* ¢,aTc e" CO,.C,OeNCE ~

"oLo" . u o . , u . . E T ~ O 0

e ] ''2 ~Vm = (natural)= Z = 145 kHz Old muonium 8Vm (y, Ib~T) I and I~ 2 are observed to terminate before the K~ x ray is observed. If one calculates the fine-structure splitting for a hydrogen-like atom of nuclear charge + Z e , and includes a Pauli moment term, the energy splitting is given by [2(1] • (ZoO 4 m (14)

A E n ' t = ( 2 +gl) ~n3

e(e+l)

where n is the principal quantum number and ~ is the orbital angular momentum. This calculation is done in the "Pauli approximation" and is accurate up to (and including) terms of relative order (f~/c) 2, where ~ is the expectation value of the electron velocity. 2 The total angular m o m e n t u m is J = L + S, and the total angular mo1 For (g0/2 + gl) < 0 the state with mentum quantum number is j = ~ + ~. highest total angular momentum j lies highest in energy. The difference in weighting between the two terms in Eq. 14, (go + 291), arises from the famous factor of two from Thomas precession [21], which subtracts off an amount from the fine-structure splitting which is exactly half that expected from the Dirac moment term. This fine structure splitting should be compared with the Bohr energy m(Z~)2 n2

En -=- -

(15)

For g = n - 1, n >_ 2, the splitting relative to the total energy is AE,,,_I

(Za) 2 _~

En

(16)

2n2(u - 1)

2 The mean velocity O/c is approximately Zc~/n.

82

B. Lee Roberts

(9,8)

/ /

fi2, i0

TM

(-}

t

Fig. 2. A portion of the atomic cascade. The energy separations are not to scale. In the blow up, the fine-structure states are labelled + for j = g + ½, - for j = g - !2 ~ and the Dirac and Pauli moments are both assumed to be negative.

where the Pauli m o m e n t contribution has been neglected. For the 2p state of h y d r o g e n this ratio is 6.6 × 10 -6. T h e basic features of the a t o m i c cascade are given in Fig. 2. T h e population of the circular (those with ~ = n - 1) energy states tends to become enhanced over the statistical 2 l + 1 weighting as the cascade proceeds towards lower q u a n t u m states. T h e circular transitions, i.e. those between circular states, tend to d o m i n a t e the experimental spectrum. This can be understood by considering the three E1 transitions shown in Fig. 2 which are labelled c~, /~ and 7, where a is the circular transition. T h e radiative transition probability depends on ~3 times the dipole m a t r i x element. T h u s the A n = --2 transition labelled fl is preferred over the "noncircular" A n = - 1 transition labelled 7A s s u m i n g a statistical p o p u l a t i o n of the fine-structure states, the spin flip transition b can be shown to be weak for transitions between two adjacent levels of large principal q u a n t u m n u m b e r [22]. T h e intensity ratio for the three c o m p o n e n t s of a circular transition (i.e. from (n, ~) = (n -4- 1, n) -+ (n, n - 1)) is given by a : b : c = 2n 2 + n -

1 : 1 : 2n 2 - n -

1

(17)

where n is the principal q u a n t u m n u m b e r of the lower state. For n = 10 this ratio is 209 : 1 : 189, where we assume t h a t the state of highest j lies highest. If the a n o m a l o u s m o m e n t were the opposite sign to the Dirac m o m e n t , these two c o m p o n e n t s could reverse in energy (see Eq. 14). Experimentally one

g-Factors of Subatomic Particles

83

would observe two x-ray lines separated in energy by an amount equal to the difference in fine-structure splitting of the upper and lower states.

2.2

Measuring g-Factors with Exotic Atoms

The first measurements of the hyperon magnetic moments were carried out at the Brookhaven AGS and the CERN PS in the late 60s and early 70s. Since the hyperon energies were low, as was the average polarization, these experiments were quite difficult, and did not yield precise measurements. Furthermore, the magnetic moment of the S - , which has a decay asymmetry parameter (cr = -0.068 + 0.008), see Eq. 11, was impossible to measure using these low energy beams. The development of constituent quark models motivated by QCD [2329] permitted theorists to make definite predictions for the moments of the baryons in the SU(3) flavor octet, usually with the proton moment as input to the calculation. Thus in the mid 1970s, the measurement of hyperon magnetic moments became an important testing ground for these models of the quark structure of the baryons. Motivated by the desire to measure the Z - moment, a group working at the Brookhaven AGS developed a new technique for measuring the magnetic moments of the antiproton (p) and ,U-. Since the traditional precession technique did not work for either of these particles (the p does not decay) the exotic atom technique was invented. From Eq. 14 we can see that the x-ray lines observed from # - , p and Z'- will exhibit fine-structure splitting and one can measure the magnetic moment by measuring this splitting. In practice, this technique is useful only for particles with a large anomalous moment, which excludes a precision measurement of the muon moment by this technique. Since the fine-structure splitting goes as ( Z a ) 4 n -5, one wishes to look at the lowest transition not affected by the strong interaction, and in the highestZ target possible. One must choose targets with no static quadrupole moment, i.e. nuclear spin 0 or ½, to eliminate hyperfine structure which would interfere with a measurement of the fine-structure. Even with high-Z nuclei, the finestructure splitting is almost never experimentally resolved, so one has to be very careful to measure the detector resolution and lineshape independently under the same conditions as the data are collected. A deformed nucleus such as 238U, which does not have a static quadrupole moment, can still cause dynamic E2 effects [30]. While the total centroid for the/3 energy levels is shifted, the individual fine-structure components are shifted together to within 8 eV, which was negligible and could be included as a small correction in the analysis. The experimental resolution is such that except for the/5(n = 11 --+ n = 10) transition in uranium, which is shown below, the two components are not well resolved. On the other hand, since the shape and intensities of the two components are known, the chi-square of a fit with no splitting is larger than

84

B. Lee Roberts

I DiracOnly 12.5

(10,9,+)

(,o,s,+)

(

8 '°.°t-

N .~ 5 . 0 -

All Corrections

~ 2.5 0.0

Pauli Moment

-

(Q)

(b)

(c)

Fig. 3. Relative positions of the energy levels of the Pb ,U-(n = t0,~ = 9) and (n = 10, g = 8) states. The states are labelled by (n,e,:J=) where

1 (a) shows the pure Dirac levels, the -4- gives the total j = £ 4- ~. (b) shows the inclusion of all corrections except the anomalous moment, and (c)

shows the fully corrected levels assuming that p ( E - ) = -1.11PN

the m i n i m u m value, which occurs at some finite splitting, by as much as 100 for combined d a t a sets. For small subsets it is larger by at least 10 units as will be shown below. To determine the intensities of the non-circular transitions relative to the circular ones, it is necessary to carry out a simulation of the exotic a t o m cascade [31]. Since the An = - 2 transitions are experimentally observed, their observed intensities relative to the An = - 1 transitions can be reproduced by the cascade calculation by adjusting the initial population and the strong interaction parameters. In addition to the Pauli m o m e n t term, other corrections such as vacuum polarization [32], nuclear polarization [33] and electron screening [34], were included by integrating the Dirac equation [35]. In Fig. 3 we show the effect of these corrections to the Dirac energy levels for S - Pb. Note that the degeneracy of states with the same total angular m o m e n t u m is lifted by the inclusion of these corrections. It has been pointed out that the g-factor of a bound particle is decreased relative to its free value [36]. This effect has been demonstrated [37] to be considerable for a p - bound in the l s level of a high-Z muonic atom. In the experiments discussed below, the lowest principal q u a n t u m state considered was the n = 10 state, where the effect was estimated to be smaller than 0.1% and was neglected. The first magnetic m o m e n t measured with this technique was the p moment measurement carried out at the Brookhaven AGS in the s u m m e r of 1972


85

300

o~ I-- 2 0 0 o o

I00

0~

I

366

I

I

:570

I

I

374

ENERGY (keV)

Fig.4. The/5(n = 11 ~ n --- 10) transition in depleted U. The two fine-structure components can clearly be seen. The non-circular contribution to this transition is negligible.

[38]. In Fig. 4 the #(n = 11 -+ n = 10) transition is shown for a depleted uranium target. For this transition the non-circular intensity is negligible. The solid curve through the d a t a is a free fit of two Gaussians plus background to the data. The measured fine-structure splitting is consistent with a magnetic m o m e n t of p(p) = -2.79, the value expected from the C P T theorem which requires particles and antiparticles to have magnetic m o m e n t s with the same magnitude but opposite in sign. The intensity ratio obtained from the fit is consistent with the expected ratio of 209 : 189. The combined # m o m e n t obtained from the two early experiments [38, 39] was p(#) = (--2.795+0.018)#N. The current value of the 15 magnetic m o m e n t is dominated by an experiment [40] which utilized the unique capabilities of the L E A R facility at CERN to measure the fine-structure splitting in the /5(11 --+ 10) transition in Pb. T h a t experiment obtained the result (-2.800 + 0.0090)pg. While the value of the # m o m e n t is firmly predicted by the C P T theorem, the ~ - m o m e n t was scarcely predicted at all when this technique was developed. Since the Z - lifetime is too short to produce a b e a m and bring it to rest, it was necessary to use a stopped K - beam to produce hyperons through the reaction K - + N -+ ,U- + rr, where N is a nucleon in the target nucleus. Although weak ~ - x rays had been observed in the spectra from low-Z kaonic atoms, Z - x rays had never been seen in high-Z atoms, which would be necessary to measure the magnetic m o m e n t . The first observation of ~ - x rays from high-Z atoms was reported at Brookhaven in 1973 [41] where a limit was placed on the magnetic m o m e n t . With more data, it was possible to measure a value for the magnetic m o m e n t [42], which

86

B. Lee Roberts

7°°0 F

,

6°°° I-

t

=+

soool-f

+

T

f

4oo01.- l I

=

~

|r,

® t

°u .... r, lljL. 1000 -

T+°+ i + ++-

h

~~,~

0~

100

"+

~

t

_~

t

£

I

I

200

300

400

,500

600

ENERGY ( keY} Fig. 5. The x-ray spectrum from kaons stopping in a laminar W target.

was p(~U-) = ( - 1 . 4 8 + 0.37)#N. A similar result was obtained in a second experiment by a Columbia-Yale collaboration [43]. A follow-up experiment at Brookhaven (E723) used a laminar target with sheets of Pb or W placed in a liquid hydrogen bath. A K - beam was stopped in this target, and the reaction K-

+ p -4 ~-

+ ~r+

B . R . = 46%

(18)

was used to produce Z - in the target. The monoenergetic 7r+ was stopped in a range telescope and the 7r+ -4 p+ -4 e + decay chain was observed to tag 2 - production. The x-ray spectrum in coincidence with stopped K - is shown in Fig. 5, and the tagged spectrum is shown in Fig. 6. The Z - (11 -4 10) transition in W is shown in Fig. 7. The tagging improves the 2 - x-ray signalto-noise by a factor of 20 to 30 over the first experiments. To combine all the data, the maps of ~2 vs. fine-structure splitting for the two targets and three detectors were added together. The resulting map showed a one standard deviation preference for the negative sign. The final result [44] was p ( ~ - ) = (-1.105 ± 0.029 + 0.010)#N, where the first error is statistical and the second systematic. Of course, the biggest difference between the first and second generation experiments is signal-to-noise, whose importance cannot be underestimated in such a difficult measurement. With the discovery of polarization in the hyperon beam at Fermilab [45], it became possible to perform precision magnetic moment measurements there, and eventually the S - moment was measured by the precession technique in the same time frame as E723. The current world average [5] is # ( S - ) = (-1.160-t-0.025)pN. The difficulty of this particular measurement is reflected in the fact that the world average includes a scaling up of the combined error


87

200

IZ :::)

0

loo

T

50

,

r

1

-

g

~, t

t

I

i

100

200

300

400

riO(

700

ENERGY(keY) F i g . 6. The tagged spectrum from kaons stopping in a laminar W target.

by a scale factor of 1.7, since the three most precise experiments (E723 and the two most recent Fermilab experiments) do not agree well with each other. All of the hyperon magnetic moments have now been measured well [5]. The models have been further refined. While a specific calculation may come closer to predicting one of the magnetic moments better than the other calculations, the proper test is to look at how one model predicts the whole set. With that criterion, the agreement is only at the 10% level.

3

A N e w E x p e r i m e n t to M e a s u r e t h e M u o n g - F a c t o r

For the muon and electron, the strong interaction contribution to the magnetic moment is very small. The QED calculations of their anomalous moments represent a calculationat tour de force. While the g-factor of the electron has provided a testing ground for QED, the anomalous magnetic moment of the muon has provided an even richer source of information. The muon's mass of 105.7 MeV is quite large compared to the electron's mass of 0.511 MeV, and heavier virtual particles can contribute in a measurable way to its anomalous moment. The relative contribution from heavier particles to the muon anomaly scales as (m~/m~) ~, and in a series of three elegant experiments [46-48] virtual muons and quarks (pions) have been shown to contribute at measurable levels. The current experimental accuracy is not sufficient to observe the W and Z ° gauge boson contributions. At present there is good agreement between experiment and theory [49], and there is no indication of any substructure to the muon.

88

B. Lee Roberts

:1 5ot 160.

°:

150. 140

10

130

-z

.t5

-i

0

-o.s

6.s

i

3O4

1.5

3O6

~{£'1 (Nuclear Magnetons}

(a)

(b)

Fig. 7. (a) A X2 map from a fit to a tagged sample of the W - S - ( 1 1 --+ 10) transition. There are 116 degrees of freedom. (b) The best fit to the observed S transition, corresponding to the X2 minimum. Both the circular and noncircular fine-structure doublets are shown separately. This figure contains ,-- 7% of the total data taken.

Unlike the short lived hyperons, the muon's long lifetime permits a b e a m to be stored in a storage ring, and the spin rotation frequency can be measured. While the decay #+ --4 e+~,ue is a three-body decay, the highest energy positrons go along the electron spin direction, and the parity violation in the weak decay permits one to determine the direction of the spin vector. A charged particle moving in a uniform magnetic field will execute cyclotron motion with the orbital cyclotron frequency we =

eB

(19)

m7

The spin precession frequency in a magnetic field is given by ~

=

geB 2m

eB + (1 - ~ / ) - m"/

,

(20)

with the Larmor and T h o m a s precession terms explicitly separated. Thus the spin vector of a charged particle moving in a uniform magnetic field will precess, relative to the m o m e n t u m vector, with a frequency wa, which is given by the difference between the orbital cyclotron frequency wc and the spin precession frequency we. ~za :

~v~ - w~ :

e

--aB

(21)

m

is directly proportional to the anomalous m o m e n t and independent of the particle's m o m e n t u m . For particles with a Pauli m o m e n t of the same sign


89

as the Dirac moment, the spin vector will lead the m o m e n t u m vector. Experimentally one measures the precession frequency of a particle's spin in a known magnetic field, and determines the magnetic moment. In a real experiment, the field B in Eq. 21 is the average field seen by the ensemble of muons in the storage ring. Vertical focussing must be provided to keep the muon beam stored, which can be accomplished with magnetic multipoles, or with an electrostatic quadrupole field. However, if magnetic multipoles are used, it is difficult to know the average B field to the accuracy needed for a precision measurement of a~. In a region in which both magnetic and electric fields are present, the relativistic formula for the precession is given by [50]

~oa-

dOR _ e dt ~n

[ (') a,B-

a,

72-

1

/3xE

1

'

(22)

where OR = (s,/3) is the angle between the muon spin direction in its rest frame and the muon velocity direction in the laboratory frame. The other quantities refer to the laboratory frame. If the muon beam has the "magic" value of 3' = 29.3, the electric field does not change the relative orientation between the spin and momentum vectors. Thus the precession of the spin relative to the momentum is determined entirely by the magnetic field, and one can use electrostatic quadrupoles for vertical focussing. The magnetic field can be a pure dipole field and can be determined very accurately. This technique was used in the third CERN experiment [48] and will be used in the new experiment at Brookhaven. The famous CERN experiments measured # = 1.001 165 9230 (85) ( e h / 2 r n u ) , a precision of -t-7.3 parts per million (ppm) for a~. While this result tested QED to a high level, and showed for the first time the contribution of virtual hadrons to the magnetic moment of a lepton, this sensitivity was not sufficient to observe the contribution of the W and Z ° gauge bosons. A new experiment [51], is being constructed to measure ( 9 - 2 ) of the muon to better than =k0.35 ppm in order to measure the electroweak contribution to ( 9 - 2). The first data collection run will take place in 1997. This experiment has been described in some detail elsewhere [52, 53], and only a few comments will be made here. The goal of the experiment is to verify the standard model prediction, and to search for physics beyond the standard model. The theoretical value of a u consists of contributions from QED [49], virtual hadrons [54, 55], and virtual electroweak gauge bosons [56-58]. Taking the value of (~ from the electron (9 - 2) experiment [59], the total QED contribution is a QED = 116 584 706(2) × 10 -11. The QED part is calculated to a precision of a few parts per billion, and the agreement between the calculated and measured (g - 2) values for the electron gives us great confidence ill the QED calculation. The hadronic correction cannot be calculated from first principles but can be calculated using dispersion theory and data from e+e - --4 hadrons.

90

B. Lee Roberts

The total hadronic contribution is tL# _Had . ~ 6882(154) x 10 -11. The two recent evaluations [54, 55] agree that the current error on this contribution is ~ 5=1.3 ppm. The latter authors estimate that this error can be reduced to (,-~ 5=59 x 10 -11) or ,.o +0.5 p p m after CMD2, the new Novosibirsk experiment, has finished the analysis of their data. To further reduce this theoretical error, additional e+e - d a t a from x/~ = 1.4 GeV (the m a x i m u m energy for CMD2) to above the J / ¢ threshold will be needed, or perhaps theoretical calculations in this energy region can further reduce this error [60]. The theoretical limit seems to be set by the contribution from hadronic "light by light" scattering [61], aHad(lbl) = --52 (18) × 10 -11, which has an uncertainty of 5=0.15 ppm. This contribution cannot be estimated from data, but might be improved by a lattice QCD calculation. While the single W and Z loop calculations have been available for some time [56], recent higher order calculations which include both fermionic [57] and bosonic [58] two-loop contributions, obtain a higher order contribution which turns out to be surprisingly large. The first order weak contribution of 195(4) x 10 - i t is reduced to 151(4) x 10 -11 (1.3 p p m of au) when the second order terms are included. Since the ability to calculate loop diagrams is intimately tied to the renormalizeability of the theory, this measurement will provide an important test of the renormalization prescription. = 116 591 739 (154) ×10 -11 The total theoretical prediction is a Th~orv u where the theoretical error of 5=1.3 p p m is dominated by the uncertainty on the hadronic vacuum polarization. Since (g - 2)u is very sensitive to W and p substructure, as well as supers y m m e t r y in the mass region below 130 GeV or any SUSY model with large tan/3, a result which agrees with the standard model will place significant new limits on physics beyond the standard model [52, 62]. The experimental goal is nominally stated as a precision of +0.35 ppm, a factor of twenty better than the C E R N experiment. With the option of direct muon injection into the ring, which requires a full aperture kicker to store the muon beam, one m a y be able to do much better. If the estimates of the incident muon flux and capture efficiency into the storage ring for direct muon injection are correct, then the final result for a t could reach a statistical sensitivity of 5=0.12 p p m at the end of the d a t a collection. This statistical error of 5=0.12 p p m happens to be equal to the current estimate of the experimental systematic errors, and the collaboration is studying ways of reducing systematics further. While this statistical error would represent a ten standard deviation measurement of the electroweak contribution, the uncertainty in the current evaluation of the hadronic contribution is much larger, and will need to be improved to realize the full potential of the final experimental result.


4

91

Conclusions

The anomalous g-factors of the muon and electron have played an i m p o r t a n t role in the development of QED, and in our understanding of the nature of these leptons. The current precision of the electron anomaly makes it one of the most precisely measured quantities in nature, and one could argue that QED is so well understood that this measurement can be used to provide the best measurement of the fine-structure constant. The muon g-factor has measurable contributions from virtual hadrons, and soon we hope to measure the W and Z ° gauge boson contributions. This new measurement also opens a window for the discovery of new physics beyond the standard model. For a full interpretation of this experiment, additional data, and calculations, will be needed to permit a more precise evaluation of the hadronic vacuum polarization contribution to a , . Baryon magnetic m o m e n t s is another subject altogether. Because of the difficulties of calculating low energy phenomena in QCD, it is necessary to resort to QCD inspired phenomenological models, which describe the situation to about 10%, decades away from the spectacular agreement between theory and experiment for the leptons. Nevertheless, when the constituent quark models were being developed in the 70s, hyperon magnetic m o m e n t s played an important role in testing their validity. Perhaps some day, computing power will increase to the point where lattice calculations of the static baryon properties can reach an accuracy which would permit a meaningful comparison with experiment. Acknowledgement. I have learned much about exotic atoms from the collaborators listed in the references, especially M. Eckhause, J. Miller, and It. Welsh, as well as from C.3. Batty, M. Blecher and M. Leon who are not listed. The early exotic atom work described here was done in friendly and beneficial competition with C.S. Wu and her collaborators. I am grateful to K. Johnson and S. Glashow for interesting discussions on magnetic moments and baryon structure. T. Kinoshita and W. Mareiano have been invaluable resources on the theory of the muon g-factor. I wish to thank D.W. Hertzog and Y.K. Semertzidis for their helpful comments on this manuscript.

References [1] O. Stern, Z. Phys. 7 (1921) 18. [2] W. Gerlach and O. Stern, Z. Phys. 8 (1922) 110, Z. Phys. 9 (1922) 349, Z. Phys. 9 (1922) 353 and Ann. Phys. 74 (1924) 45. [3] T.E. Phipps and J.B. Taylor, Phys. Rev. 29 (1927) 309. [4] P.A.M. Dirac, Proc. Roy. Soc. 117 (1928) 610, and 118 (1928) 341. [5] Particle Data Group, Review of Particle Properties, Phys. Rev. Dh0 (1994) 1173. [6] R. Frisch and O. Stern, Z. Phys. 85 (1933) 4, and I. Estermann and O_ Stern, Z. Phys. 85 (1933) 17.

92

B. Lee Roberts

[7] Luis W. Alvarez and F. Bloch, Phy. Rev. 57 (1940) 111. [8] J.J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, Reading Massachusetts (1967), p. 110. [9] See M. Gell-Mann, C.I.T. Report CTSL-20 (1961) unpublished, and Y. Ne'eman, Nucl. Phys. 26 (1961) 222, which are reprinted in the classic collection of articles The Eightfold Way, M. Gell-Mann and Y. Ne'eman ed., Benjamin, New York, 1964. [113] S. Col.eman and S.L. Glashow, Phys. Rev. Lett. 6 (1961) 423. [11] E.D. Bloom, D.H. Coward, H. DeStaebler, J. Drees, G. Miller, L.W. Mo, R.E. Taylor, M. Breidenbach, J.I. Friedman, G.C. Hartmann and H.W. Kendall Phys. Rev. Lett. 23 (1969) 930, and M. Breidenbach, J.I. Friedman, H.W. Kendall, E.D. Bloom, D.H. Coward, H. DeStaebler, J. Drees, G. Miller, L.W. Mo, R.E. Taylor, Phys. Rev. Lett. 23 (1969) 935. [12] P. Kush and H.M. Foley, Phys. Rev. 74 (1948) 250. [13] J. Schwinger, Phys. Rev. 73 (1948) 416, and Phys. Rev. 76 (1949) 790. [14] A review of these calculations is given in T. Kinoshita, Quantum Electrodynamics (Directions in High Energy Physics, Vol. 7), T. Kinoshita ed., World Scientific 1990, p 218. These calculations are constantly being extended. [15] R.S. Van Dyck, Jr., P. Schwinberg, and H. Dehmelt, Phys. Rev. Lett. 59 (1987) 26 and in Quantum Electrodynamics, T. Kinoshita ed., World Scientific, 1990, p. 322. [16] E. Fermi, E. Teller and V. Weisskopf, Phys. Rev. 71 (1947) 314, and E. Fermi and E. Teller, Phys. Rev. 72 (1947) 399. [17] M. Conversi, E. Pancini and O. Piccioni, Phys. Rev. 71 (1947) 209. [18] M. Camac, A.D. McGuire, J.B. Platt and H.J. Schulte, Phys. Rev. 88 (1952) 209. [19] V.L. Fitch and J. Rainwater, Phys. Rev. 92 (1953) 789. [20] H.A. Bethe and E. Salpeter, Quantum Mechanics of One- and Two- Electron Atoms, (Springer-Verlag, 1957). The discussion of magnetic moments and freestructure splitting begins in section 10. [21] L.H. Thomas, Nature 107 (1926) 514. [22] See Bethe and Salpeter, op. cit. p. 273. [23] A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [24] T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [25] N. Isgur and G. Karl, Phys. Rev. D21 (1980) 3175. [26] G.E. Brown, M. Rho and V. Vento, Phys. Lett. 84B (1979) 383. [27] S. Theberge and A.W. Thomas, Nucl. Phys. A393 (1983) 252, and references therin. [28] J. Franklin, Phy. Rev. D30 (1984) 1542. [29] Z. Dziembowski and L. Mankiewicz, Phys. Rev. Lett. 55 (1985) 1839. [30] M.Y. Chen, Y. Asano, S.C. Cheng, G. Dugan, E. Hu, L. Lidofsky, W. Patton, C.S. Wu, V. Hughes and D. Lu, Nucl. Phys. A254 (1975) 413. [31] The most modern code, which includes the strong absorption of the orbiting hadrons is by M. Leon and R. Seki, Phys. Rev. Lett., 32 (1974) 132. The code for muonic atoms written by J. H/iffner, Z. Phys. 195 (1966) 365, was widely used in the 1960s. The basic atomic physics is given in Y. Eisenberg and D. Kessler, Nuovo Cim. 19 (1961) 1195. They discuss strong absorption


93

of orbiting hadrons in Phys. Rev. 123 (1961) 1472 and Phys. Rev. 130 (1963) 2352. The most recent review of cascade calculations is by C.J. Batty and R.E. Welsh, Nucl. Phys. A589 (1995) 601. [32] J. Blomqvist, Nucl. Phys. B48 (1972) 95. [33] T.E.O. Ericson and J. H/finer, Phys. Lett. 40B (1972) 459. [34] P. Vogel, At. Data Nuc]. Data Tables 14 (1974) 599 and P. Vogel eta]. Phys. Lett. B70 (1977) 39 and references therein. [35] E. Borie, Phys. Rev. A28 (1983) 555 and references therein. We are grateful to E. Borie for a copy of her code. [36] H. Margenau, Phys. Rev. 57 (1940) 383, and K.W. Ford, V.W. Hughes and J.G. Wills, Phys. Rev. 129 (1963) 194. [37] T. Yamazaki, S. Nagamiya, O. Hashimoto, K. Nagamine, K. Nakal, K. Sugimoto and K.M. Crowe, Phys. Lett. 53B (1974) 117. [38] J.D. Fox, P.D. Barnes, R.A. Eisenstein, W.C. Lain, J. Miner, R.B. Sutton, D.A. Jenkins, R.J. Powers, M. Eckhause, J.R. Kane, B.L. Roberts, M.E. Vislay, R.E. Welsh, and A.R. Kunselman, Phys. Rev. Lett. 29 (1972) 193. The final analysis of these data is reported in B.L. Roberts, Phys. Rev. D17 (1978) 358. [39] E. Hu, Y. Asano, M.Y. Chen, S.C. Cheng, G. Dugan, L. Lidofsky, W. Patton, C.S. Wu, V. Hughes and D. Lu, Nucl. Phys. A254 (1975) 403. [40] A. Kreissl, A.D. Hancock, H. Koch, Th. KShler, H. Poth, U. Raich, D. Rohmann, A. Wolf, L. Tauscher, A. Nilsson, M. Suffert, M. Chardalas, S. Dedoussis, H. Daniel, T. yon Egidy, F.J Hartmann, W. Kanert, H. Plendi, G. Schmidt and J.J. Reidy, Z. Phys. C37 (1988) 557. [41] J.D. Fox, W.C. Lain, P.D. Barnes, R.A. Eisenstein, J. Miller, R.B. Sutton, D.A. Jenkins, M. Eckhause, J.R. Kane, B.L. Roberts, R.E. Welsh, A.R. Kunselman, Phys. Rev. Lett. 31 (1973) 1084. [42] B.L. Roberts, C.R. Cox, M. Eckhause, J.R. Kane, R.E. Welsh, D.A. Jenkins, W.C. Lam, P.D. Barnes, R.A. Eisenstein, J. Miller, R.B. Sutton, A.R. Kunselman, R.J. Powers and J.D. Fox, Phys. Rev. Lett. 32 (1974) 1265, and Phys. Rev. D12 (1975) 1232. [43] G. Dugan, Y. Asano, M.Y. Chen, S.C. Cheng, E. Hu, L. Lidofsky, W. Patton, C.S. Wu, V. Hughes and D. Lu, Nucl. Phys. A254 (1975) 396. [44] D.W. Hertzog, M. Eckhause, K.L. Giovanetti, J.R. Kane, W.C. Phillips, W.F. Vulcan, R.E. Welsh, R.J. Whyley, R.G. Winter, G.W. Dodson, J.P. Miller, F. O'Brien, B.L. Roberts, D.R. Tieger, R.J. Powers, N.J. Colella, R.B. Sutton, and A.R. Kunselman, Phys. Rev. Lett. 51 (1983) 1131, and Phys. Rev. D37 (1988) 1142. [45] G. Bunce, R. Handier, R. March, P. Martin, L. Pondrom, M. Sheaf[, K. Heller, O. Overseth, P. Skubic, T. Devlin, B. Edelman, R. Edwards, J. Norem, L. Schachinger and P. Yamin, Phys. Rev. Lett. 36 (1976) 1113. [46] G. Charpak, F.J.M. Farley, R.L. Garwin, T. Muller, J.C. Sens and A. Zichichi, Nuovo Cim. 37 (1965) 1241. [47] J. Bailey, W. Bartl, B. von Bochmann, R.C.A. Brown, F.J.M. Farley, M. Giesch, H. JSstlein, S. van der Meer, E. Picasso and R.W. Williams, Nuovo Cim. 9A (1972) 369. [48] J. Bailey, K. Borer, F. Combley, H. Drumm, C. Eck, F.J.M. Farley, J.H. Field, W. Flegel, P.M. Hattersley, F. Krienen, F. Lange, G. Leb~e, E. McMillan,

94

[49] [50]

[51]

[52] [53]

[54] [55] [56]

[57] [58] [59] [60] [61] [62]

B. Lee Roberts G. Petrucci, E.Picasso, O. Runolfsson, W. yon Riiden, R.W. Williams and S. Wojcicki, Nucl. Phys. B150 (1979) 1. T. Kinoshita and W.J. Marciano in Quantum Eleetrodynamics (Directions in High Energy Physics, Vol. 7), T. Kinoshita ed., World Scientific 1990, p. 419. V. Bargmann, L. Michel and V.L. Telegdi, Phys. Rev. Lett. 2 (1959) 435, are generally given credit for this formula. As noted by J.D. Jackson in ClassicM Electrodynamics, (John Wiley & Sons, New York, 1975), p. 556, Thomas published an equivalent equation in 1927. Brookhaven National Laboratory AGS E821: D.H. Brown, R.M. Carey, E. Efstathiadis, E.S. Hazen, F. Krienen, J.P. Miller, O. Rind, B.L. Roberts*, L.R. Sulak, W.A. Worstell, J. Benante, H.N. Brown, G. Bunce §, J. Cullen, G.T. Danby, C. Gardner, J. Geller, L. Jia, H. Hseuh, J.W. Jackson, R. Larsen, Y.Y. Lee, R.E. Meier, W. Meng, W.M. Morse*, C. Pai, I. Polk, S. Rankowitz, J. Sandberg, Y.K. Semertzidis, R. Shutt, L. Snydstrup, A. Soukas, A. Stillman, T. Tallerico, F. Toldo, K. Woodle, T. Kinoshita, Y. Orlov D. Winn, A. Grossmann, K. Jungmann, G. zu Putlitz, P.T. Debevec, W. Deninger, D.W. Hertzog, S. Sedykh, D. Urner M.A. Green, U. Haeberlen, P. Cushman, S. Giron, J. Kindem, D. Maxam, D. Miller, C. Timmermans, D. Zimmerman, L.M. Barkov, D.N. Grigorev, B.I. Khazin, E.A. Kuraev, Yu.M. Shatunov, E. Solodov, K. Nagamine, K. Endo, H. Hirabayashi, S. Ichii, S. Kurokawa, Y. Mizumachi, T. Sato, A. Yamamoto, K. Ishida, S.K. Dhawan, F.J.M. Farley, M. GrossePerdekamp, V.W. Hughes*, D. Kawall, R. Prigl, and S.I. Redin (*Spokesmen, §Project Manager). B.L. Roberts, Z. Phys. C56 (1992) S101. V.W. Hughes, et al., Proc. 10th International Symposium on High Energy Spin Physics, Nagoya, 9-14 November 1992, T. Hasegawa, N. Horikawa, A. Masaike, S. Sawada ed., p. 717. S. Eidelman and F. Jegerlehner, Zeit. Phys., C:67 (1995) 585. W.A. Worstell and D.H. Brown, (1996), Phys. Rev. D54 (1996) 3237, and Muon (g-2) technical note # 220 (1995). W.A. Bardeen, R. Gastmans and B Lautrup, Nucl. Phys. B46 (1972) 319; R. Jackiw and S. Weinberg, Phys. Rev. D5 (1972) 157; I. Bars and M. Yoshimura, Phys. Rev. D6 (1972) 374. A. Czarnecki, B. Krause and W.J. Marciano, Phys. Rev. D52 (1995) R2619. A. Czarnecki, B. Krause and W.J. Marciano, Phys. Rev. Lett. 76 (1996) 3267. T. Kinoshita, Phys. Rev. Lett. 75 (1995) 4728. W. Marciano, private communication. M.Hayakawa, T. Kinoshita and A.I. Sanda, Phys. Rev. Lett. 75 (1995) 790 and Phys. Rev. D54 (1996) 3137. See the review by Kinoshita and Marciano, oF). cit. for a more complete discussion.

Laser Spectroscopy of Metastable Antiprotonic Helium Atomcules Toshimitsu Yamazaki CERN, CH-1211 Geneva, Switzerland and Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo, 188 Japan

1

Discovery

of Long-lived

Antiprotons

in Helium

The lifetime of an exotic atom/molecule sets a natural time window which constrains the precision of spectroscopy through the natural width. The high precision studies of muons, muonic atoms and muonium have been facilitated by their long enough lifetime of 2.2 ps. Before 1991 such a favourable situation had not been conceived for antiprotonic atoms. We knew well that even the positron and positronium are short-lived in matter and that the antiproton is destined to annihilate in matter in a few picoseconds via the strong interaction. A surprising situation emerged in 1991, when the University of Tokyo group discovered at KEK that about 3 % of antiprotons stopped in liquid helium survive for microseconds [1], 106 times longer than usually believed. This experiment had been triggered by a preceding one in which they encountered accidentally the longevity of K - mesons in liquid helium [2] while they were searching for £7 hypernuclei [3]. The longevity of negative mesons in liquid helium was suggested by Condo [4] based on their anomalously large free decay components. Condo proposed the formation of metastable exotic atoms X - e - H e 2+ with large principal and orbital quantum numbers (n, l), and this peculiar atom was studied theoretically by Russell [5]. Although Russell calculated the lifetime of antiprotonic helium atoms to be in the microsecond region, it was hardly believed that such metastable atoms are stable in high density medium like liquid helium. So, this extremely interesting subject had remained untouched nearly for two decades till the discovery in 199I. Immediately, a new experimental group (called PS205) was formed to study this phenomena comprehensively by using the Low Energy Antiproton Ring (LEAR) of CERN, which provides a superb beam for this purpose. This monoenergetic low energy antiproton beam made experiments even with gas targets very efficient and productive. In the first period (1991-93) they measured delayed annihilation time spectra (DATS) precisely in various phases of helium and also studied the effect of admixture of other atoms and molecules [6, 7, 8, 9, 10]. Although the information obtained from systematic studies of DATS indicated that the longevity is due to the formation of metastable atoms ~e- He 2+ (=~He +), it is rather "macroscopic" and "integral", and no

96

Toshimitsu Yamazaki

direct proof of the proposed atom was obtained. We needed a new "microscopic" and "differential" method with which individual states and transitions of the metastable atoms can be identified. In 1993 a new breakthrough emerged. The PS205 group succeeded in the first laser resonance experiment on the antiprotonic helium atom [11] based on the proposal by Morita et al. [12]. This success opened a wide field and as of 1995 seven resonance transitions have been found [13, 14, 15, 16, 17]. Parallel to these experiments, theoretical treatments of this atom advanced [18, 19, 20]. In this way the initial curiosity-oriented research of the PS205 group is turning toward high-precision frontier of fundamental physics.

2

Antiprotonic

Helium

Atomcules

The metastability of the antiprotonic helium atom is well understood by now. In the metastable ~He + atom the electron stays in the ls orbit while the antiproton with large-(n,l) undergoes a nearly classical orbital motion (see Fig. 1). The slowly-moving p polarizes the electron in the opposite direction, which helps retard the electric dipole transition. This effect was interpreted as being similar to the nuclear core polarization effects in terms of ls-np configuration mixing [18]. Another approach is to regard the ~ and He 2+ as the two centers of a molecule [19]. The large-(n,l) states of ~He + can be metastable because the Auger transitions to ionized states are suppressed due to the large ionization potential. As shown in Fig. 1, the whole levels are well divided into the radiation-dominated metastable zone and the Augerdominated short-lived zone. Each metastable state is expected to proceed as (n,/) --+ (n - 1 , / - 1) with a radiative lifetime of about 1.5 #s. The typical level spacing (transition energy) is around 2 eV, which is in the range of visible light. The metastable states are hardly destroyed via Stark mixing in collisions with surrounding helium atoms because the degeneracy in I is removed (about 0.3 eV energy difference between (n,l) and (n,l - 1)). Since this atom is neutral and has one electron, characterized as a kind of hydrogenic atom, it does neither stick to nor penetrate into surrounding He atoms. Since this atom possesses a dual character as an atom and a molecule in itself, it is often called "atomcule". Each state (n, l) is interpreted as a member of vibrational states with a vibrational quantum number v. A unique correspondence between (n,l) and (v, J) holds: J = l and v = n - l - 1. It is interesting that this behavior and correspondence result as the n and l become large in a two-body atom. Whether such large-/states can be formed in the p capture by a He atom or not is essential. Yamazaki and Ohtsuki [18] considered the non-statistical distribution of angular momenta brought in by the ~ in a naive way to understand the delayed fraction of 3 %. Korenman [21] made a more realistic calculation. It seems to be interesting that the maximum angular m o m e n t u m

Laser Spectroscopy of Metastable Antiprotonic HeLium Atomcules

97

H e+ A t o m c u l e

:p

E n e r g y (a.u.) He+

-2.0

Ionized ~He++ .... Io = 0.90 a.u. (24.6 eV)

1

He°

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

- - - :": - ~ ~

::: ::: - - - _

........ :.'.'.":-': - - - ~ . . . .

---==!

._. ; - ; : : : ::: ~ _ ~

_

~ "--" ~ " _

_

~

39

~

~ , 0

~;

42

=~/~-38

...::: .... - _ . H , -3.0

outra, P-H+o 30 Fig. 1. (Upper) The p and e - orbital distributions in a typical pile + atomcule state. (Lower) Level scheme of the ~He + atomcule. The bold, broken and dotted Lines represent metastable, short-lived and ionized states, respectively. From [18].

b r o u g h t in the capture process roughly corresponds to the m a x i m u m I ~ 37 when the ~ is b o u n d m o s t likely to a state with n ,~ ( M * / m ¢ ) 1/2 "~ 38. T h e appearance of metastability in antiprotonic a t o m s is a c c o m m o d a t e d by the above independent reasons jointly. There is no possibility in other elements. So, the antiprotonic helium a t o m c u l e is a miraculous existence of an antiparticle coupled to n o r m a l particles. Thus, the a t o m c u l e serves as a unique interface between m a t t e r and a n t i m a t t e r domains.

98

Toshimitsu Yamazaki 1993 v=3 ( 3 9 , 3 5 ) - > ( 3 8 , 3 4 ) r

•

•

i

.

.

.

1994 v=2 (37,34)->(36,33)

.

i

0.010

. . . .

i

•

•

0.010

o cl

o.ooi

0.001 I 1

. . . . Time

v

2.5

I . . . . 2 (,us)

o

1.5

v.

1.0

1 Time

597.259(2)3

2.0

"~

F/2 sidebands begin to appear. For a saturated atom, the form of the spectrum shows three well-separated Lorentzian peaks. The central peak has width F and the sidebands which are each displaced from the central peak by the Rabi frequency are broadened to 3F/2. The ratio of the height of the central peak to the sidebands is 3:1. This spectrum was first calculated by Mollow (1969). For other relevant papers see the review of Cresser et al. (1982). The experimental study of the problem requires, as mentioned above, a Doppler-free observation. In order to measure the frequency distribution, the fluorescent light has to be investigated by means of a high resolution spectrometer. The first experiments of this type were performed by Schuda et al. (1974) and later by Walther et al. (1975), nartig et al. (1976) and Ezekiel et al. (1977). In all these experiments, the excitation was performed by singlemode dye laser radiation, with the scattered radiation from a well collimated atomic beam observed and analyzed by Fabry-Perot interferometers. Experiments to investigate the elastic part of the resonance fluorescence giving a resolution better than the natural linewidth have been performed by Gibbs et al. (1976) and Cresset et al (1982). The first experiments which investigated antibunching in resonance fluorescence were also performed by means of laser-excited collimated atomic beams. The initial results obtained by Kimble, Dagenais, and Mandel (1977) showed that the second-order correlation function g(2)(t) had a positive slope

Resonance Fluorescence of a Single Ion

143

which is characteristic of photon antibunching. However, g(2)(0) was larger than g(2)(t) for t ~ ~ due to number fluctuations in the atomic beam and to the finite interaction time of the atoms (Jakeman et al. 1977; Kimble et al. 1978). Further refinement of the analysis of the experiment was provided by Dagenais and Mandel (1978). Rateike et al. used a longer interaction time for an experiment in which they measured the photon correlation at very low laser intensities (see Cresset et al. 1982 for a review). Later, photon antibunching was measured using a single trapped ion in an experiment which avoids the disadvantages of atom number statistics and finite interaction time between atom and laser field (Diedrich and Walther 1987). As pointed out in many papers photon antibunching is a purely quantum phenomenon (see e.g. Cresser et al. 1982 and Walls 1979). The fluorescence of a single ion displays the additional nonclassical property that the variance of the photon number is smaller than its mean value (i.e. it is sub-Poissonian). This is because the single ion can emit only a single photon and has to be re-excited before it can emit the next one which leads to photon emissions at almost equal time intervals. The sub-Poissonian statistics of the fluorescence of a single ion has been measured in a previous experiment (Diedrich and Walther 1987 and also Short and Mandel 1983 for comparison). The trap used for the present experiment was a modified Paul-trap, called an endcap-trap (Schrama et al. 1993) (see Fig. 1) which produces strong confinement of the trapped ion. Therefore, the number of sidebands, caused by the oscillatory motion of the laser cooled ion in the pseudopotential of the trap, is reduced. The trap consists of two solid copper-beryllium cylinders (diameter 0.5 mm) arranged co-linearly with a separation of 0.56 mm. These correspond to the cap electrodes of a traditional Paul trap, whereas the ring electrode is replaced by two hollow cylinders, one of which is concentric with each of the cylindrical endcaps. Their inner and outer diameters are 1 and 2 mm respectively and they are electrically isolated from the cap electrodes. The fractional anharmonicity of this trap configuration, determined by the deviation of the real potential from the ideal quadrupole field is below 0.1% (see Schrama et al. 1993). The trap is driven at a frequency of 24 MHz with typical secular frequencies in the xy-plane of approximately 4 MHz. This required a radio-frequency voltage with an amplitude on the order of 300 V to be applied between the cylinders and the endcaps, and with AC-grounding of the outer electrodes provided through a capacitor. The measurements were performed using the 32S1/2- 32P3/2 transition of the 24Mg+-ion at a wavelength of 280 nm. The natural width of this transition is 42.7 MHz. The exciting laser light was produced by frequency doubling the light from a rhodamine 110 dye laser. The laser was tuned slightly below resonance in order to Doppler-cool the secular motion of the ion. All the men-

144

J . T . HSffges, H.W. Baldauf, T. Eichler, S.R. Helmfrid and H. Walther

% U1

Uu

U2

UD

? U 0 + U cos Fig. 1. Electrode configuration of the endcap trap. The open structure offers a large detection solid angle and good access for laser beams testing the micromotion of the ion. Micromotion is minimized by applying dc voltages: U1, U2, Uu, Up.

surements of the fluorescent radiation described in this paper were performed with this slight detuning. For the experiment described here, it is important to have the trapped ion at rest as far as possible to minimize the light lost into motional sidebands. There are two reasons which may cause motion of the ion: the first one is the periodic oscillation of the ion within the harmonic pseudopotential of the trap and the second one is micromotion which is present when the ion is not positioned exactly at the saddle point of the trap potential. Such a displacement may be caused by a contact potential resulting, for example, by a coating of the electrodes by Mg produced when the atoms are evaporated during the loading procedure of the trap. Another reason may be asymmetries due to slight misalignments of the trap electrodes. Reduction of the residual micromotion can be achieved by adjusting the position of the ion with DC-electric fields generated by additional electrodes. For the present


145

experiment they were arranged at an angle of 120 ° in a plane perpendicular to the symmetry axis of the trap electrodes. By applying auxiliary voltages (U1 and U2) to these electrodes and Utr and UD to the outer trap electrodes (Fig. 1), the ion's position can be adjusted to settle at the saddle point of the trap potential. The micromotion of the ion can be monitored using the periodic Doppler shift at the driving frequency of the trap which results in a periodic intensity modulation in the fluorescence intensity. This modulation can be measured by means of a transient recorder, triggered by the AC-voltage applied to the trap. There are three laser beams (lasers 1-3 in Fig. 2) passing through the trap in three different spatial directions which allow measurement of the three components of the micromotion separately. By adjusting the compensation voltages Ua, U2, Utr and UD the amplitude of the micromotion could be reduced to a value smaller than )~/8 in all spatial directions. The amount of secular motion of the ion resulting from its finite kinetic energy cannot be tested by this method since the secular motion is not phase coupled to the trap voltage. However, the intensity modulation owing to this motion can be seen in a periodic modulation of the photon correlation signal. For all measurements presented here, this amplitude was on the order of )~/8. This corresponds to a temperature of the ion of 1 mK determined from the kinetic energy. This means that the vibrational sidebands of the trapped ion are populated up to n = 7 which results in less than 50 % of the fluorescence energy being lost into the vibrational sidebands. The heterodyne measurement is performed as follows. The dye laser excites the trapped ion with frequency WL while the fluorescence is observed in a direction of about 54 ° to the exciting laser beam (see Fig. 2). However, both the observation direction and the laser beam are in a plane perpendicular to the symmetry axis of the trap. Before reaching the ion, a fraction of this laser radiation is removed with a beamsplitter and then frequency shifted (by 137 MHz with an acousto-optic modulator (AOM)) to serve as the local oscillator. The local oscillator and fluorescence radiations are then overlapped and simultaneously focussed onto the photodiode where the initial frequency mixing occurs. The frequency difference signal is amplified by a narrow band amplifier and then further mixed down to 1 kHz so that it could be analyzed by means of a fast Fourier analyzer (FFT). The intermediate frequency for this mixing of the signal was derived from the same frequency-stable synthesizer which was used to drive the accousto-optic modulator producing the sideband of the laser radiation so that any synthesizer fluctuations are cancelled out. An example of a heterodyne signal is displayed in Fig. 3, where Aw is the frequency difference between the heterodyne signal and the driving frequency

146


Photo-~l diode,, ................I0 Detector~ ~ 137.001MHz /i/ ~ / / Local t : I "~ Oscill. o

o L- 137 MHz ,//Laser 1

| kk _/ U2 k\ Laser 3 " ~

Fig. 2. Scheme of heterodyne detection. The trap is omitted in the figure with only two of the compensation electrodes shown. Laser 3 is directed at an angle of 22 ° with respect to the drawing plane and Laser 2.

of the AOM. Frequency fluctuations of the laser b e a m cancel out and do not influence the linewidth because at low intensity the fluorescence radiation always follows the frequency of the exciting laser while the local oscillator is derived directly from the same laser beam. The residual linewidth results mainly from fluctuations in the optical p a t h length of the local oscillator or of the fluorescent beam. Both beams pass through regular air and it was observed that a forced motion of the air increased the frequency width of the heterodyne signal. The frequency resolution of the F T T was 3.75 Hz for the particular measurement. The heterodyne measurements were performed at a

~2[2 of 0.7, where A is the laser detuning. saturation p a r a m e t e r s = a2+(r~/4) In this region, the elastic part of the fluorescent spectrum has a m a x i m u m (Cohen-Tannoudji et al. 1992). The signal to noise ratio observed in the experiment is shot noise limited. The signal in Fig. 3 corresponds to a rate of the scattered photons of about 104s -1 which is an upper limit since photons were lost from detection due to scattering into sidebands caused by the secular motion of the ion. In order to reduce this loss as much as possible, a small angle between the directions of observation and excitation was used. Investigation of photon correlations employed the ordinary Hanbury-Brown and Twiss setup with two photomultipliers and a b e a m splitter. The setup


I

I

I

I

147

I

I0

S

::> ..,

6 Hz

6

D

4

-200

- 100

0

100

200

Ao ( Hz ) Fig. 3. Heterodyne spectrum of a single trapped 24Mg+-ion for s = 0.7, A _ --2.5F, /2 ~= 3.9F. Integration time: 267 ms.

was essentially the same as described by Diedrich and Walther (1987). The pulses from the photomultipliers (RCA C31034-A02) were amplified and discriminated by a constant fraction discriminator (EG&G model 584). The time delay t between the photomultiplier signals was converted by a time-toamplitude converter into a voltage amplitude proportional to the time delay. A delay line of 100 ns in the stop channel allowed for the measurement of g(2)(t) for both positive and negative t in order to check the symmetry of the measured signal. The output of the time-to-amplitude converter was accumulated by a multichannel analyzer in pulse height analyzing mode. Two typical measurements, each with weak excitation intensities but with different detunings, are shown in Fig. 4. The generalized Rabi-flopping frequency I2~ = ~ +/22 for the respective measurements are given in the figure caption. For small time delays (< 20 ns) the nonclassical antibunching effect is observed, superposed with Rabi oscillations which are damped out with a time constant corresponding to the lifetime of the excited state. Our experimental results are reasonably well reproduced by the theory for weak excitation (Loudon 1980): = 1 +

xp(-2rt) - 2cos(,at)

xp(-rt).

148


(a)

I

$ 0 ~at)

(b)

2

-30

-20

-I0

0

I0

20

30

40

T i m e ( ns ) F i g . 4.

Antibunching signals of a single 24Mg+-ion. (a) s = 0.7, A = --1.2F, ~ ' = 1.9/'. Integration time: 95 min. (b) s ----0.4, A _-- --0.5F, g2' = 0.8F. Integration time: 220 rain. The sofid line is a theoretical fit, see text for details.

A measurement of g(2) (t) with an averaging time of hours and time delays up to 500 ns resulted in no visible micromotion effects when the compensation voltages U1, U2, Uu and UD were correctly adjusted. Micromotion results in a periodic modulation of the photon correlation at the driving frequency of the t r a p (compare Diedrich and Walther 1987). The stray-light counting rate was so low that there was no need to correct the measurement shown in Fig. 4 (b) for accidental counts. There was actually not a single count in the t = 0 channel within the integration time of 220 min. In conclusion, we have presented the first high-resolution heterodyne measurement of the elastic peak in resonance fluorescence. At identical experimental parameters we have also measured antibunching in the photon correlation of the scattered field, Together, both measurements show that, in the limit of weak excitation, the fluorescence light differs from the excitation radiation in the second-order correlation but not in the first order correla-


149

tion. However, the elastic component of resonance fluorescence combines an extremely narrow frequency spectrum with antibunched photon statistics, which means that the fluorescence radiation is not second-order coherent as expected from a classical point of view. This apparent contradiction can be explained easily by taking into account the quantum nature of light, since first-order coherence does not imply second-order coherence for quantized fields (Loudon 1980). The heterodyne and the photon correlation measurement are complementary since they emphasize either the classical wave properties or the quantum properties of resonance fluorescence, respectively. In a recent treatment of a quantized trapped particle (Glauber 1992) it was shown that a trapped ion in the vibrational ground state of the trap will also show the influence of the micromotion since the wavefunction distribution of the ion is pulsating at the trap frequency. This means that a trapped particle completely at rest will also scatter light into the micromotion sidebands. Investigation of the heterodyne spectrum at the sidebands may give the chance to confirm these findings. It is clear that such an experiment will not be easy since other methods are needed to verify that the ion is actually at rest at the saddle point of the potential .

Acknowledgements We would like to thank Roy Glauber for many discussions in connection with his quantum treatment of a trapped particle. We also thank Girish S. Agarwal for many discussions. We dedicate this paper to Gisbert zu Putlitz on the occasion of his 65th birthday. A major part of his scientific work was pursued using the resonance fluorescence of atoms. We hope that the described new experiments will find his interest

References Cresser J.D., Hgger J., Leuchs G., Rateike F.M., Walther H. (1982): Resonance Fluorescence of Atoms in Strong Monochromatic Laser Fields. Dissipative Systems in Quantum Optics, edited by Bonifacio R. and Lugiato L. (Springer Verlag) Topics in Current Physics 27, 21-59 Cohen-Tannoudji C., Dupont-Roc J., Grynberg G. (1992): Atom-Photon Interactions (J. Wiley & Sons, Inc.) Diedrich F., Walther H. (1987): Non-classical Radiation of a Single Stored Ion. Phys. Rev. Lett. 58, 203-206 Gibbs H.M. and Venkatesan T.N.C. (1976): Direct Observation of Fluorescence Narrower than the Natural Linewidth. Opt. Comm. 17, 87-94 Glauber R. (1992): Proceedings of the International School of Physics "Enrico Fermi", Course CXVIII Laser Manipulation of Atoms and Ions, edited by Arimondo E., Phillips W.D., Strumia F. (North Holland) 643

150


Hartig W., Rasmussen W., Schieder R., Walther H. (1976): Study of the Frequency Distribution of the Fluorescent Light Induced by Monochromatic Excitation. Z. Physik A278, 205-210 Heitler W. (1954): The Quantum Theory of Radiation, (Oxford University Press, Third Edition) 196-204 Jakeman E., Pike E. R., Pusey P.N., and Vaugham J.M. (1977): The Effect of Atomic Number Fluctuations on Photon Antibunching in Resonance Fluorescence. J. Phys. A 10, L257-L259 Kimble H. J., Dagenais M., and Mandel L. (1977): Photon Antibunching in Resonance Fluorescence. Phys. Rev. Lett. 39, 691-695 Kimble H. J., Dagenais M., and Mandel L. (1978): Multiatom and Transit-Time Effects in Photon Correlation Measurements in Resonance Fluorescence. Phys. Rev. A 18, 201; Dagenais M., Mandel L. (1978): Investigation of Two-Atom Correlations in Photon Emissions from a Single Atom. Phys. Rev. A 18, 22172218 Loudon R. (1980): Non-Classical Effects in the Statistical Properties of Light. Rep. Progr. Phys. 43, 913-949 MoUow B.R. (1969): Power Spectrum of Light Scattered by Two-Level Systems. Phys. Rev. 188, 1969-1975 Schrama C. A., Peik E., Smith W.W., and Walther H. (1993): Novel Miniature Ion Traps. Opt. Comm. 101, 32-36 Schuda F., Stroud C., Jr., Hercher M. (1974): Observation of the Resonant Stark Effect at Optical Frequencies. J. Phys. BT, L198-L202 Short R. and Mandel L. (1983): Observation of Sub-Poissonian Photon Statistics. Phys. Rev. Lett. 51, 384-387, and in Coherence and Quantum Optics V, edited by Mandel L. and Wolf E. (Plenum, New York) 671 Walls D.F. (1979): Evidence for the Quantum Nature of Light. Nature 280,451-454 Walther H. (1975): Atomic Fluorescence Induced by Monochromatic Excitation. Laser Spectroscopy, Proceedings of the 2nd Conference, Meg~ve, France, ed. by Haroche S., Reborg-Peyronla J.C., H£nsch T.W., Harris S.E., Lecture Notes in Physics (Springer) 43, 358-369 Wu F. Y., Grove R.E., Ezekiel S. (1977): Investigation of the Spectrum of Resonance Fluorescence Induced by a Monochromatic Field. Phys. Rev. Lett. :15, 1426-1429; Grove R.E., Wu F. Y., Ezekiel S. (1977): Measurement of the Spectrum of Resonance Fluorescence froma Two-Level Atom in an Intense Monochromatic Field. Phys. Rev. A 15, 227-233

R e s o n a n c e R a m a n S t u d i e s of the R e l a x a t i o n of P h o t o e x c i t e d M o l e c u l e s in S o l u t i o n on t h e Picosecond Timescale W.T. Toner ~, P. Matousek ~, A.W. Parker 2 and M. Towrie 2 1 Clarendon Laboratory, Parks Road, Oxford, OX1 3PU, UK 2 Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire O X l l 0QX, UK

1

Introduction

Following the photoexcitation of a molecule in solution, any excess vibrational energy is redistributed a m o n g its vibrational modes and transferred to the solvent as heat, and the excited state molecule assumes its new equilibrium geometry and chemical relationship with its solvent neighbours. These four related processes take place on overlapping timescales extending from a hundred femtoseconds to a few picoseconds. Stilbene has been a particular object of study because it is one of the simplest molecules to undergo a fifth non-radiative process on a similar timescale, relaxation to the ground state via a twisted configuration which gives nearly equal yields of the trans and cis isomers from either excited state. The lifetimes of the excited state trans and cis isomers are --~ 70 ps and ~ 1 ps, respectively, at room temperature. It is natural to study this complex evolution of vibrational populations and structure through vibrational spectroscopy. Several groups have used the time resolved resonance R a m a n scattering technique, since measurements of band frequency, width and intensity are a rich source of structural and dynamical information, particularly in relation to the bonds involved in the dipole coupling to the higher electronic state which is in resonance. But as this paper illustrates, a rather complete set of d a t a is required before one can make full use of the information without ambiguity.

2

Method

The time resolved resonance R a m a n spectrum of a photoexcited molecule is measured using a probe laser pulse tuned to resonance with a higher electronic state, incident at various delay times after photoexcitation by a first (pump) laser pulse. Until recently, dye lasers have been used to generate the tunable probe beams and the p u m p b e a m has been supplied by a harmonic of the laser used to excite the dye, or by a harmonic of the dye laser itself. The molecules studied have therefore been restricted to those whose ground and excited state transitions can both be matched in this way. High average

152

W.T. 'loner, P. Matousek, A.W. Parker and M. Towrie

laser power is required to give the statistical precision needed to extract the Raman scattering signal from fluorescence and other backgrounds which are frequently very strong in comparison. But it is necessary to avoid fluences resulting in saturation and for measurements on the picosecond timescale, peak intensities which could give rise to non-linear artefacts must also be avoided. In practice, a compromise is made between high repetition rate and high peak power. The laser system used for the work described in the first part of this paper had a repetition rate of five kHz and produced tunable probe beams in the wavelength range from 550 to 700nm with up to five microjoules per five picosecond pulse [1]. Pump beams were obtained by frequency doubling with typically 10 to 20 % efficiency. The beams were focused on the surface of a jet of the sample solution flowing from a nozzle and photons scattered through 90 ° were collected and analysed in a high resolution spectrometer equipped with a cooled CCD detector. The strong enhancement of the Raman scattering signal resulting from working close to resonance also varies strongly with probe wavelength. Measurements of the probe wavelength dependence of the scattering, called Raman excitation profiles, are therefore necessary before reliable detailed interpretations of band intensities can be made. A lack of independence of pump and probe tuning prevents the measurement of these profiles under constant photoexcitation conditions. A new laser system based on the amplification of portions of a single white light continuum source in two independently tunable Optical Parametric Amplifiers (OPAs) has now been developed [2] to overcome this limitation. It also gives very broad spectral coverage and a time resolution extending down to ~-, 150 fs. A spectral filter of variable resolution produces probe beam pulses whose width is set by the transform limit of the spectral resolution chosen for any particular experiment. A high repetition rate (40kHz) avoids the fluence and peak power problems discussed above, at some cost to the statistical precision for a given data acquisition time. This system was used for the measurements on quaterphenyl described at the end of the paper.

3

Stokes Spectrum of $1 Trans-Stilbene

Measurements of the resonance Raman spectrum made with picosecond time resolution show that several bands in the Stokes spectrum of S1 trans-stilbene (SITS) shift significantly to the blue and decrease in width in the 25 picoseconds immediately following photoexcitation [3,4]. An example is shown in Fig. 1. The magnitudes of these mode-specific and solvent-dependent changes increase monotonically with the energy of the exciting photon [4,5] and timeindependent changes of a similar kind can be induced by changes in the temperature of the solvent bath, although the ratios of the shifts to the width changes are quite different in the static and dynamic cases [4]. Similar (but

Resonance Raman Studies of the Relaxation of Photoexcited Molecules

153

qD

1600

1400

1200

wavenumber/cm" 1 Fig. 1. Stokes resonance Raman spectra of $1 trans-stilbene at delays of 10ps (solid line) and 80ps (dashed line) after photoexcitation at 285 nm. The probe beam wavelength was 570 nm. From Ref. [4].

much smaller) mode-specific temperature dependencies are seen in the ground state resonance Raman spectrum [6]. In the dynamic case, the changes were taken to represent the cooling of a molecule in approximate internal equilibrium following an initial intramotecular vibrational relaxation cascade too fast for observation. They were related to changes in the absorption spectrum with time [7]. The dynamic and static (temperature induced) changes were attributed by us to the same basic cause, which may be called a temperature dependent solvent shift. Some authors have proposed a connection with the isomerisation process [8]. We will not discuss here the various models of such shifts, but rather concentrate on relating the empirical Raman observations to each other. Further work showed that very similar dynamic shifts and linewidth narrowing effects in the Stokes spectra were exhibited by many molecules, for example, dimethoxystilbene and quinquiphenyl [9]; biphenylyl-phenyl-oxadiazol, bisbiphenylyl-butylbiphenyl-oxadiazol, and biphenylyl-butylbiphenyl-oxadiazol [10]. Quaterphenyl and tetra-t-butyl-p-sexiphenyl showed another kind of behaviour [10], with the disappearance of some bands at late times suggesting that the early time spectra might include components from a short lived isomer. In all these molecules, the excited states are thought to be subject to twists of conjugated chains of C-C and C = C bonds, as in the case of Stilbene.

4

Anti-Stokes Spectrum of $1 Trans-Stilbene

The first measurements of the anti-Stokes spectrum of photoexcited transstilbene [11] up to the band arising from the C = C olefinic stretching mode at 1600 cm -1, shown in Fig. 2, appeared to contradict the idea of approximate internal equilibrium on the picosecond timescale: the intensities of several

154

W.T. Toner, P. Matousek, A.W. Parker and M. Towrie

0 ps

Stokes

300

500

700

900

1100

Wavenumber/cm

1300

1500

1700

-1

Fig. 2. Anti-Stokes resonance Raman spectra of $1 trans-stilbene at delays of (a) 0ps and (b) 50ps after photoexcitation taken with 8ps time resolution. Stokes spectra at 20ps delay are shown in (c) for comparison. Photoexcitation was at 305 nm and the probe beam wavelength was 610 rim. From Ref. [11].

bands decrease markedly with time over c. 10 ps, relative to their neighbours (in the case of the c. 1600cm -1 band, by a factor of at'least 20). Others have reported anti-Stokes spectra showing similar changes [12]. These d a t a present a difficult problem of interpretation. Although the establishment of equilibrium populations in stilbene in solution has been shown [13] to take much longer than the 50 to 500fs commonly supposed, taking the early/late ratios of the band intensities as a direct measure of early time population relative to Boltzmann in this case would imply an unreasonably large difference from equilibrium persisting to the few picosecond timescale. The spectra are also remarkable in a second respect: the ratios of the anti-Stokes to the Stokes intensities of all the bands in the 1100 to 1600 cm -1 region are very much higher than would be expected on the basis of a Boltzm a n n population distribution (in the case of the c. 1530cm -1 band, by a factor of ,-~ 150), even at times as late as 50ps, when time-dependent spectral changes have ceased and thermal equilibrium must be fully established. Since the initial state for anti-Stokes scattering is vibrationally excited, resonance with the v = 0 level in the higher electronic state occurs at a lower probe frequency than for Stokes scattering, but it would require a very narrow resonance to produce such a large change through the ratio of the Stokes and anti-Stokes resonance denominators, even if it was assumed that only one term was dominant. It was not possible to make a quantitative interpretation of these results in the absence of R a m a n excitation profile data.


155

5 Time-Resolved Resonance Raman Excitation Profile of Quaterphenyl The new laser system described above makes it possible to vary the probe b e a m wavelength independently of the p u m p and we present here a preliminary analysis of spectra which contain the first information to be reported [14] on the R a m a n excitation profile of an excited state molecule. Quaterphenyl was chosen for the first trials of the new system in view of its very large cross-section. The dye was dissolved in dioxane at millimolar concentration and the p u m p b e a m had a wavelength of 277 nm. T h e resolution of the probe b e a m spectral filter was set at 20 cm -1, giving a 700fs wide probe pulse. Resonance R a m a n spectra were observed to rise from zero to full intensity between -1 and +1 ps p u m p - p r o b e delay. Stokes spectra at delays of 2 and 50 ps following photoexcitation are shown in Fig. 3 for seven probe wavelengths between 593 and 647nm. There are marked changes in intensity for very small changes in probe wavelength. Figure 4 shows the probe wavelength dependence of the intensities of the two bands having the largest changes. The excitation profile of the c. 1515cm -z band can be fit to a resonance centered at a probe b e a m frequency of ,-- 16250cm - I having a width ,-, 500cm -z (FWHM). We have remarked above on the need for such a sharp resonance in stilbene to account for the anti-Stokes spectrum. In the case

2 ps

50 ps

647nm 635nm 620nm 610nm 605nm 602nm 593nm

t 1200

I 1600

I

~

r 1200 Raman shift, crn "t

I

I 1600

Fig, 3. Stokes resonance Raman spectra of $1 Quaterphenyl for various probe beam wavelengths at 2 and 50ps after photoexcitation at 2 7 7 n m .

156

W.T. Toner, P. Matousek, A.W. Parker and M. Towrie (b) 1616 band

(a) 1616 band at 2 ps 120 I

4~4~/ ~ '

40 t 0 15000

~ 16000

i

120

~

40

f 17000

/

0 15000

at 60 ps

\

: 16000

Probefrequency,cm "~


(c) 1686 band at 2 pa

(d) 1585 band at 50 ps

i"

17000

ll

i° l 0

150OO

[

16000


I

17000

0

i

16000

i

17000

Probe ~ ' ~ l ~ n c y , c m "

Fig. 4. Band intensities versus probe beam frequency for the data of Fig. 3. The values shown for the c. 1585 cm -1 band are sums of components which were not clearly resolved.

of the c. 1585cm -1 band, the profile has the appearance of an interference of the same resonance with a non-resonant background. This requires the involvement of a second electronic state which is distant. Interference of this kind has been observed in ground state resonance R a m a n scattering [15]. Changes of intensity with time delay can also be seen in Fig. 3, particularly for probe wavelengths of 610 and 620 nm ( 16390 and 16130 c m - 1). An increase in the intensity of the c. 1515 band at late time for a probe wavelength of 6 1 0 n m can also be seen in a spectrum of better quality in reference [10]. The excitation profiles in Fig.4 suggest that this m a y be due to a small but significant shift to the blue of the frequency of the resonance, and not to any change in intrinsic strength. This would imply coupling between the electronic and vibrational degrees of freedom during the relaxation. Work is in progress to confirm and extend these preliminary observations. Measurements of the t e m p e r a t u r e dependence of the ground state absorption spectrum of quaterphenyl were made which show that the separation of the broad and structureless absorption bands centered at c. 295 and c. 2 1 0 n m increases by several percent for a cooling of 500. The c. 210 nm band is likely to be due to one of the higher states responsible for the shape of the excitation profile so that this result suggests a connection between the static t e m p e r a t u r e dependence and the cooling of the photoexcited molecule which parallels that previously observed for the vibrational frequencies of stilbene [4,6].


6

157

Conclusions

The experiments described above show the great sensitivity of time-resolved resonance Raman spectroscopy to the changes taking place during the relaxation of photoexcited molecules in solution. During the time that excess energy is transferred to the solvent, there are changes in vibrational frequencies and bandwidths, in level populations and also, if the quaterphenyl results are confirmed, in electronic energies. These dynamic effects all have parallels in the changes which take place on slow timescales with variations in solvent temperature, but the static temperature dependencies and dynamic changes also differ. The sharpness of the quaterphenyl excitation profiles and the possibility that the resonance parameters may change with time confirms the need for measurements of this kind of the anti-Stokes spectrum of stilbene to determine the degree to which non-equilibrium populations contribute to the observed changes in band intensities.

7

Acknowledgements

The vibrational spectroscopy of molecules which have at least 25 atoms too many and are in intimate contact with a solvent, which is the subject of this paper, is far from the main field of this proceedings. But techniques draw together scientists from different fields, and the authors have found it very stimulating and enjoyable to work with Gisbert zu Putlitz and his team on the laser spectroscopy of an atom not included in the periodic table, whose ground state lifetime is 2.2 microseconds. We wish him good health and continued success in research and in the building of bridges between cultures, both national and scientific. We thank our collaborators in the early stages of this work, R.E. Hester, D.L. Faria and J.N. Moore for their participation and many useful discussions, and M. Scully for making the ground state absorbance measurements. This work was carried out at the Central Laser Facility, Rutherford Appleton Laboratory with support from the EPSRC.

References [1] P. Matousek, R.E. Hester, J.N. Moore, A.W. Parker, D. Phillips, W.T. Toner, M Towrie, I.C.E. Turcu and S. Umapathy, Meas. Sci. Technol. 4 (1993) 1090 [2] P. Matousek, A.W. Parker, P.F. Taday, W.T. Toner and M. Towrie, Opt. Comm. 127 (1996) 307 [3] W.L. Weaver, L.A. Houston, K. Iwata and T.L. Gustafson, J. Phys. Chem. 96 (1992) 8956; K. Iwata and H. Hamaguchi, Chem. Phys. Letters 196 (1992) 462 [4] R.E. Hester, P. Matousek, J.N. Moore, A.W. Parker, W.T. Toner and M. Towrie, Chem. Phys. Letters 208 (1993) 471

158

W.T. Toner, P. Matousek, A.W. Parker and M. Towrie

[5] 3.N. Moore, P. Matousek, A.W. Parker, W.T. Toner, M. Towrie and R.E. Hester in Time - Resolved Vibrational Spectroscopy VI (1994), Springer Proceedings in Physics 74, p.89 [6] W.T. Toner, R.E. Hester, P. Matousek, J.N. Moore, A.W. Parker and M. Towrie, ibid, p.115 [7] B.I. Greene, R.M. Hochstrasser and R.B. Weisman, Chem. Phys. Letters 62 (1979) 427; F.E. Doany, B.I. Greene and R.M. Hochstrasser, Chem. Phys. Letters 75 (1980) 206. [8] H. Hamaguchi, Proc. XVII Intl. Conf. on Photochemistry, paper IN8 (1995), to be published. [9] R.M. Butler, M.A. Lynn and T.L. Gustafson, J. Phys. Chem. 97 (1993) 2609; D.L. Morris, Jr. and T.L. Gustafson, Appl. Phys. B 59 (1994) 389; J. Phys. Chem. 98 (1994) 6275. [10] M. Towrie, P. Matousek, A.W. Parker, W.T. Toner and R.E. Hester, Spectrochimica Acta A 51 (1995) 2491 [11] A.W. Parker, P. Matousek, W.T. Toner, M. Towrie, D.L.A. de Faria, R.E. Hester and J.N. Moore, Proc. XIV Int. Conf. on Raman Spectroscopy (1994), Extra booklet, p.E-9; P. Matousek, A.W. Parker, W.T. Toner, M. Towrie, D.L.A de Faria, R.E. Hester and J.N. Moore, Chem. Phys. Letters 237 (1995) 373 [12] J. Qian, S.l. Schultz and J.M. Jean, Chem. Phys. Letters 233 (1995) 9. [13] R.J. Sension, A.Z. Sarka and R.M. Hochstrasser, J. Chem. Phys. 97 (1992) 5239. [14] A.W. Parker, P. Matousek, P.F. Taday, M. Towrie, W.T. Toner and R.H. Bisby, Proc. New Developments in Ultrafast Time-resolved Vibrational Spectroscopy, Tokyo (1995). [15] G.E. Galica, B.R. Johnson, J.L. Kinsey and M.O. Hale, J. Phys. Chem. 95

(1991) 7994

Four-Quantum RF-Resonance S t a t e of an A l k a l i n e A t o m

in t h e G r o u n d

E.B. Alexandrov and A.S. Pazgalev Vavilov's State Optical Institute, St. Petersburg, Russia

1

Introduction

The topic of this paper goes back to the times of the golden age of Optical Pumping in 1950-1960. In those days a lot of refined studies of radiofrequency (RF) spectroscopy of atoms were performed. In particular, multiplequantum transitions were observed and interpreted by the team of A. Kastler when they were studying RF-spectra of optically pumped sodium atoms [1] and by P. Kusch who applied an atomic-beam technique to potassium atoms [2]. These and many other investigations were summarized in an important paper [3] of J. Winter (see also the review of A. Bonch-Bruevich and V. Khodovoi [4]). Later on the interest in multiple-quantum processes shifted from the RF-domain to the optical range where double-quantum transitions in counter-propagating light beams became very popular, because of their ability to suppress Doppler broadening of spectral lines [5]. In the RF-domain, multiple-quantum resonances did not find any further application in research or metrology in spite of their obvious advantage: an n-quantum transition is n times narrower than the single-quantum one (in the limit of low driving field power). But this advantage is heavily depreciated by a rather strong dependence of the resonance frequency on the driving field of the power. Indeed, from the most general point of view the use of the multiple-quantum processes seems to be inexpiable because of the necessity to perturb the system by the much stronger driving field. Nevertheless, it looks like we have found a particular case of RF-induced multiple-quantum transitions in the ground state of an alkaline atom which could be of interest for low magnetic field metrology. We regard a four-quantum resonance m R = 2 ~ m F = --2 in an atom with nuclear moment I = 3/2 (see inlet in Fig.l). Two features of this transition attract attention: (i) the energy of the unperturbed transition is strictly linear with respect to the magnetic field strength H unlike any other transition and (ii) the frequency of this transition seems to be almost unaffected by the driving field H1 again unlike all other multiple-quantum transitions. The latter feature follows from estimates based on a perturbation approach [4]: under the influence of the field H1 both states m R = ±2 are expected

160

E.B. Alexandrov and A.S. Pazgalev r~

1.9

+2 +1 0 -1 -2

F=2

Signal (a.u.) 1.8 1.7

I

/

1.(~

4F=1~.~Hz (='K) -1

0

+I

1.,=

1.4 1.3 1.2

I

,

I

.

I

,

I

,

I

349000 349500 350000 350500 351000

i

Frequency

(Hz) Fig. 1. RF-spectra of the ground state double-resonance signal of 39K at four different values of the driving field Ha strength (in units ofg = fjHl: 1)g = 1; 2)g = 20; 3) g -- 80; 4)g = 190. Inlet: a level diagram displaying the ground state magnetic splitting of an alkaline atom with nuclear moment I = 3/2.

to shift almost equally. This prediction needs to be confirmed by accurate calculations which is the main aim of the following consideration. In the past multiple-quantum processes were theoretically analyzed mainly in the framework of a perturbation approach - as higher terms of a power expansion of the transition probability. Apart from Bloch's well known exact solution of the resonance problem in two-level systems only a three-level system interacting with either a single- or a two-frequency field has been thoroughly analyzed in m a n y publications (see for instance [6-9] and references therein). In principle, using the approximation of rotating waves the exact stationary solution can be obtained analytically for any k-level system. It is noteworthy, however, that for k > 4 it is a highly troublesome task with extremely cumbersome and practically uninterpretable results. But nowadays there exist powerful computers and advanced software which make it possible to get easily numerical or even analytical solutions for a many-level problem. They are valid for any power of the driving field. Below we present the result of a computer simulation of 4 - q u a n t u m RF resonances in optically p u m p e d alkaline atoms.

Four-Quantum RF-Resonance in the Ground State of an Alkaline Atom

2

161

Formulation of t h e P r o b l e m

The frequencies fro,m-1 of the RF-transitions between adjacent magnetic sublevels m and m - 1 of the ground 2S1/2 state obey the well-known Breit-Rabi formula. They are presented below as a power expansion over the magnetic field strength H:

I2,1= L,o = Io,-I = AI,-2 z

a+H - 3bH 2 + cH 3a+H - bH 2

-

cH

3 +

a+H + bH 2 - cH 3-

. .. ...

...

a+H + 3bH 2 + cH 3 + ...

(1)

This results in set of 4 lines within the state F = 2 and of two lines for F -- 1: fl,o = a_H

f0,--1 ---- a _ H

+ bH 2 - cH 3 -

...

- bH 2 + cH 3 + ...

(2)

The coefficients a+ are very close to -t=7G H z / T : a+ = fi ± L ~ ~fY

(3)

where fi = 9 i p B / h , f j = ( g j - 9 i ) # B / ( 4 h ) , 9i and g j are nuclear and electron g-factors, II8 and h are the Bohr magneton and the Plank constant, b

=///Ahfs

c -= 6 b f j / A h f ~

(4)

with Ahfs being the hyperfine splitting of the ground state. The spectrum of the RF-induced transition, at sufficiently low power of the driving field H1 comprises the 6 above lines if the RF-field induced width f j H 1 of the transitions remains much smaller than the frequency difference between the adjacent lines. The same is true for the intrinsic width F i j of the line. Under such conditions the RF-field interacts with the multiple-level system like with an ensemble of independent two-level systems. But as the field grows, the lines are getting broader, their resonance positions shift, and additional two-photons peaks appear with resonance frequencies fJ2) ~ ( f j + l - f j - 1 ) / 2 , where f j is the frequency of level j. Upon further increase of H1 similar tripleand four-quantum transitions appear within the sublevels Of the F = 2 state. It should be mentioned that four-quantum transitions have never been observed experimentally so far, probably, because it could not be distinguished from the background of the overlapping broadened neighbouring lines. To make it detectable, it is necessary to achieve a great sharpness of the spectrum, i.e., a large ratio of the line splittings to the line widths. In the subsequent calculation we will assume the intrinsic line width to be extremely small - 1 Hz - consistent with our recent experimental results [10]. Let us

162

E.B. Alexandrov and A.S. Pazgalev

consider the optically pumped atomic vapour in a so called Mz- configuration, which means that the vapour is being pumped by a circular polarized resonant light beam along the magnetic field H and the magnetic resonance is observed as a change of the absorption ~ of the pumping light:

(5)

t~ ~- E p i w i

where Pi and wi are the population and the absorption probability of the sublevels i. The problem is to calculate the population distribution under the action of optical pumping and a driving RF-field. We search for a steadystate solution of the density matrix equation pjk with phenomenological terms to describe the optical pumping and relaxation processes. The equation for off-diagonal terms reads as follows:

ihOpsk/Ot =

[Ho + V ( t ) , P]jk -

ih(rp)jk

(6)

where the Hamiltonian H0 with the diagonal matrix describes the atom in a constant magnetic field, the operator V(t) takes into account the effect of the RF-field H1 and the matrix Fp describes the relaxation of the coherence Pjk due to thermal processes and optical excitation. We assume, for simplicity, that in the absence of excitation by light all elements pjk relax with the same rate F. Pumping light shortens the life time of atoms and thus additionally broadens the level k by Fk. The coherences pjk relax with the rates Fjk =

r + (rj + r~)/2. Equation (6) neglects the coherence transfer in the course of the optical pumping which corresponds to the buffer gas optical pumping. Dealing with the populations pj - pjj it is necessary to add to Eq. (6) terms describing the optical pumping. In the absence of pumping all populations relax to the same value p = 1/8. With optical pumping the atom at the level j is excited to the states # and after spontaneous decay goes to the sublevel k of the ground state with the probability Bkj = bkjIp which is proportional to the pumping intensity Ip. Finally, the equations for the populations pjj become: i

Opjj/Ot = - ~ [ H 0 + V(t), p]jj - pjj(F + Fj) + F/8 + E pkkBjk

Q

(7)

The probabilities bkj and wi are listed in the paper of Franzen and Emslie [11]. In the rotating wave approximation we have V = V exp(iwt), where w is the driving field frequency. Seeking for the steady-state solution, we will find the coherences pjk(t) in the form pjk(t) = pjk e x p [ i ~ ( j - k)t]. We will also keep in mind that the RF-field does not produce microwave coherences between sublevels belonging to different hyperfine states. It has also non-zero matrix elements Y)k for j - k = +1 only. Presenting pjk as pjk = xjk+ivjk, we can reduce the equations (6) and (7) to a set of 34 linear algebraic equations for 8 populations (steady-state conditions imply Opjj/Ot = 0) and 26 values for Xjk and Yjk.


Signal

163

1.8

(a.u.) 1.7 1.6 1.5 1.4 34996O

*" 350'000

'

350040'

'

Frequency

(Hz) Fig. 2. The four-quantum resonance contour at g = 190.

3

Results

The realistic case of 39K in the average earth magnetic field of about 50 ttT has been simulated 1. T h e pumping light intensity has been chosen such that the main (strongest) single-photon resonance f2,1 became 1.5 times broader compared to its "dark" width F. This corresponds (more or less) to the condition of getting m a x i m u m resolution for the single-photon R F - s p e c t r u m [12]. The set of 34 equations has been solved numerically and a n u m b e r of graphs to(w) has been plotted for different values of H1. Figure 1 shows a family of graphs n(w) for fjH1 from 1 to 190 in units o f f which was assumed to be 1 Hz. The strength of H1 was selected in each case to maximize the sharpness of each n - q u a n t u m resonance in the region from n = 1 to n = 4. Only very weak traces of transitions within the sublevels of the state F = 1 can be found in two of four spectra. Figure 2 presents in more detail the R F - s p e c t r u m in the vicinity of a four-photon resonance. The rather flat background of the resonance is related to the deeply saturated two-photon resonance mF = +1 mR : - 1 at ahnost the same frequency. Figure 3 displays the m a x i m a l steepness S,n (H1) = max[dS/dw], where S is the signal strength. The signal steepness characterizes the accuracy of the resonance peak location, playing the leading role in the evaluation of the resonance line for a real application. Figure 3 compares the steepness of the main ordinary resonance with that of the four-quantum transition. The steepness of the last one is about 7 times higher. Finally, Fig. 4 shows the resonance shift induced by the driving field for the single- and four-quantum lines. For each resonance the field H1 was chosen in the vicinity of its optimal value. One can see t h a t field induced shifts are of the same order being in fact negligible.

i a+/a_ = (7.004666/--7.008639) GHz/T; b = 106.327 GHz/T~; c = 9681 G H z / T 3

E.B. Alexandrov and A.S. Pa~.galev

164

a

a 0.030

-o.o35

dS/dg

Shift

(a.u.) 0 025

( H z ) -oo3o ~0.025

0.020

-.0,020 0015 -0.015 0.010 -0.010 0.005

-0.005

0.000

i 1

, 2

, 3

A 4

O00O

1

2

3

4 g(Hz)

g(Hz) b

..o.o12

Shift

d S l d g 020

(Hz)

(a.u.)

-O.OLO

OA5 -0.0(]8 O.lO -0.004

o o5

~].002 0.00

'

150

200

250

300 g(Hz)

Fig. 3. The maximal steepness of the single-quantum (a) and four-quantum (b) resonances (arbitrary units) vs the driving field strength.

4

15o

200

250

300

g(Hz)

Fig. 4. The frequency shifts for the single-quantum (a) and four-quantum (b) resonances (in Hertz) induced by the driving field in units of fj Hj.

Conclusion

The results of a detailed computer simulation support our initial supposition about attractive features of the four-quantum resonance F = 21 m E = --2 ~ ',, m R = +2 : its quality (frequency discriminating ability) surpasses that of the ordinary resonances and its position is not affected even by a fairly strong driving field. It appears that the four-quantum resonance can compete with the slngle-photon transition in magnetometric application. The strictly linear dependence of these resonance frequencies on the magnetic field strength should also be pointed out. With the optimal intensity of the driving field this resonance can be easily distinguished from adjacent broadened resonances.


5

165

Acknowledgements

This work has been supported by International Science Foundation. E. Alexandrov is grateful also to the Alexander yon Humboldt-Stiftung. Many thanks to Prof. Dr. G. zuPutlitz with whose group the author cooperated in 1994 when the main idea of this work appeared.

References [1] [2] [3] [4] [5]

J. Brossel, B. Cagnac et A. Kastler, C.R. Acad. Sci., Paris, 237, 984 (1954). P. Kusch, Phys.Rev. 93, 1022 (1954). J.M. Winter, Ann. Phys. 4, 745 (1959). A.M. Bonch-Bruevich and V.A. Khodovoi, Sov. Phys. Uspekhi 8, 1 (1965). See for example V.P. Chebotaev, and V.S. Letokhov, Nonlinear laser spectroscopy, Springer-Verlag, Berlin, Heidelberg, New York (1977). [6] V.S. Butylkin, A.E. Kaplan, Yu.G. Khronopulo, and E.I. Yakubovich, Resonance interaction of matter with light, Moscow, Publishing house "Nauka"

(1977). [7] [8] [9] [10]

Th. H~nsch, and P. Toschek, Z. Physik 236, 213 (1970). C. Feuillade, and P.R. Berman, Phys. Rev. A29, 1236 (1984). M. Schubert, I. Siemers, and R. Blatt, Phys. Rev. A39, 5098 (1989). E.B. Alexandrov, V.A. Bonch-Bruevich, and N.N. Yakobson, Sov. J. Opt. Teehnol. 60, 754 (1993). [11] W. Franzen and A.G. Emslie, Phys. Rev. 108, 1453 (1957). [12] E.B. Alexandrov, A.K. Vershovskii, and N.N. Yakobson, Soy. Phys.-Tech. Phys. aa, 654 (1988)

Hard Highly Directional X - R a d i a t i o n E m i t t e d by a Charged Particle Moving in a Carbon Nanotube V.V. Klimov 2 and V.S. Letokhov 1 1 Institute of Spectroscopy, Russian Academy of Sciences, 142092 Troitsk, Moscow Region, Russia, and Optical Sciences Center, University of Arizona, Tuscon, USA 2 p. N. Lebedev Physical Institute, Russian Academy of Sciences, 53 Leninsky Prospect, 117294 Moscow, Russia

1

Introduction

C o n s i d e r a b l e advances have been made in recent years in the synthesis of the so-called carbon nanotubes being important objects for use in nanotechnologies [1-6]. These wonderful objects, a mere nanometer in diameter, may have quite a macroscopic length of up to a few centimeters. Being hollow, nanotubes seem to be natural candidates for transporting both neutral and charged particles in various nanodevices. On the other hand, charged particles propagating in nanotubes interact with their walls and thus can generate a coherent electromagnetic radiation which can be of special interest. The radiation due to charged particles channeling in crystals has now been well studied [7-10]. Such a radiation has a wide range of properties making it of practical use. The investigation of charged particles channeling in nanotubes was started recently [11, 12]. The aim of the present paper is to continue the investigation of the electromagnetic effects taking place in the course of propagation of positively charged particles in single-layer nanotubes. In our view, the most important advantages of nanotubes are due to their large diameter (compared with that of channels in ordinary crystals). The equipotential lines for a particle channeling in a nanotube and in a diamond channel are shown in Fig. 1. In this figure one can see the high azimuthal symmetry of the nanotube potential. This is due to the so called Hermann theorem [13]. Within this potential positrons can oscillate with large amplitudes. The typical trajectory of a channeling positron is shown in Fig. 2. Electrons can channel near the cylindrical nanotube surface. Such channeling has no analogues in usual crystal channeling. The typical trajectory of a channeling electron is shown in Fig. 3. Besides, in a system consisting of parallel nanotubes [1], the ratio between the cross-sectional area of the walls and the total cross-sectional area is small, so that the system can effectively capture sufficiently wide beams and hence emit radiation effectively.

168

V.V. Klimov and V.S. Letokhov

-6 -6

-4

-2

0

2

4

Fig. 1. The equipotential lines for a nanotube of diameter 11/k and for a diamond channel (in the central box).

6

x[£]

The plan of this paper is as follows. In Sect. 2 we will consider the structure of the nanotube and find the expression for the potential acting on a charged particle within the tube. In Sect. 3, the classical picture of motion of a positively charged particle near the nanotube axis will be examined along with the characteristics of the resultant radiation. In Sect. 4 we will consider the motion and radiation of a single electron within classical as well as quantum approaches. In the Conclusion, the results obtained will be summed up and avenues of further investigations outlined.

2

The

Structure

Acting

of the

on a Charged

Nanotube Particle

and Inside

the

Potential

it

The specific electromagnetic effects in the nanotube are entirely due to its structure, and so let us consider this structure in more detail. Best developed today are methods of synthesizing single-layer nanotubes from carbon atoms [1-4]. A nanotube in this case can be visualized as a hexagonal graphite layer rolled into a tube with a diameter of the order of i nm. Generally speaking, one can imagine a number of ways of rolling a hexagonal graphite layer into a tube [5, 6]. Here, for simplicity we consider the case when an elementary cell of four atoms (see Fig. 4) is arranged on the surface of the nanotube so that its side with a period of ax/~ is oriented in the direction of the perimeter of the tube (strictly across the tube axis) and that with a period of 3a along the tube axis. The radius R of the single-layer tube thus obtained can easily be found from the elementary relation associated with the conservation of the carbon

Hard X-Radiation Emitted by a Charged ParticLe

xl

o > j~, j. Due to the conservation of total angular momentum, the quantum numbers g and ~ vary within a relatively narrow interval centered at gr while the latter changes from zero to infinity. The cross sections ~r~,,~(j', j; 1¢) are not necessarily real-valued quantities; in the case j' = j and

r' = r, c~~',~(j,j) , are negative, and they correspond to the total loss of ~ due to transfer of this component to other states. We now perform the quasiclassical analysis of Eq.(3) which will reveal the significance of the g, ~ interference. To this end we turn to the total angular momentum representation of the scattering matrix with R-helicity elements S]~,;jo~. The important property of the quasiclassical S-matrix is that each element of it can be represented as a linear superposition of terms which show quite specific behavior with respect to dependence on the total angular

194

E.I. Dashevskaya and E.E. Nikitin

momentum J: each term can be represented as a product of a rapidly varying exponential and slowly varying preexponential factor. Explicitly, -

i ~

exp

(6)

-7

Now, expressing the S matrix in the J, m, t, n representation through the S matrix in the j,w, J representation, and substituting this into Eq. (4), one arrives at a double sum over total angular momenta, say J and J. This double sum stems from the interference of initial states for different relative angular momenta l and t . Since £ and ~ are close to each other, so are J and ~ . Therefore, when calculating the quantities like (S...)(S...) J 7 • one can account for the different values of J and J only in the exponent of each term in the sum (6) in the first order with respect to A J : J and J , and assume that J and J and which enter into the preexponentials are the same. The final result of the analysis [11,12] yields the cross section in Eq.(5) as the quasiclassical counterpart:

~rr,,~(j 8

"I

-

^

, j ; k ) -- 2~r

/0

,5

-I

,j;£,b)bdb •

(7)

where the transition probability under the integral over the impact parameters is P~'r (J', J; k, b) =

z

.

.

.

(8)

x

.

Eqs.(7) and (8) contain no more large angular momenta which are now incorporated into the classical quantity, the impact parameter b. The objects S... (b) that enter into these equations are related to the partial contributions to the quasiclassical scattering matrix (6) as

Sj,m,;jm(b)~

= Z

Rm',o~'J' (qj,~,j~(b))'Y

• sj,~, "~ j~(b) exp[2i~],~,,j~(b)]J~:m

(9)

~jOJ I

j'

where /~,~, ~,(fl) are the elements of the Wigner matrix for rotation by an

j,

-,

angle fl around the y axis, R,~,,,~, (8) = DJ,~,,~, (0, fl, 0) . Note that quantities ~

Sj,m,;jm (b) are not derived from the scattering matrix as a whole, but rather are synthesized from the partial contributions to the respective elements of the quasiclassical scattering matrix and elements of the rotation matrix. The impact parameter b in the exponents and preexponential factors in the r.h.s. of Eq.(9) replaces the total angular m o m e n t u m J in Eq.(6) via the quasiclassical relation J + 1/2 =- kb , and the deflection angle ~lj~'~,,jw (b) that enters

Quasiclassical Approximation in the Theory of Scattering

195

into the element of the rotation matrix is defined by the usual quasiclassical relation: 2 O[~,~,j~ (b)] (10) b rlJ"~';J"( ) = k cOb A peculiar character of Eq.(9) is that the rotation angle r/ is different for different exit states which is a signature of the multiple-trajectory description of the collision event. If the common trajectory approximation is adopted, the deflection angle is assumed to be independent on the exit state, and Eq.(9) would imply a simple rotation of the quasiclassical S-matrix: through the deflection angle ~ : ~

Sj'm,;jm(b)

j~

~f~R.v,,~,(~] ) .r

_

(11)

03 1

Finally, in the impact parameter approximation when r/is taken to be zero, Sj,,~,jm (b) becomes simply the quasiclassical S matrix expressed via the impact parameter:

~,~,jm(b) = s/,~,,~ I

(12) J=kb--l[2 w~tn

The probability and the cross section for polarization transfer in isotropic collisions can be expressed by formulae which are obtained from Eq.(8) by averaging over all projections s. After this averaging, the dependence of the cross sections and transition probabilities on k disappears, and the quantization axis for electronic angular momenta may be any axis fixed in space. The appropriate expressions are:

~r'(j',j) = 2~r

#

W(j',j;b)bdb

(13)

where

P'(j',j;b)

(-1)J'-~'

-=

"+~

j' j '

j j

rrtl~ms~S,~s

{~,m,;~m(b) S*~,~,;~(b) - 6~,;j 6~,~6,,,,,,,

}

(14)

We now incorporate the locking approximation into the J-helicity scattering matrix. In order to see most clearly the effect of locking, we assume that the molecular scattering matrix is diagonal in ~ and j (no transition occur in the molecular region), and that locking is sudden. Then the S-matrix will be diagonal in j, and it can be represented in the form:

JJ

exp [2i~ (J)]

(15)

196


where ~2 is now identical with % This equation identifies the coefficients in Eq.(6) as

g2

= ~j,jRJ,,~[a(J)]RJ,sl[a(J)]

(16)

Yet another common approximation, used in the quasiclassical calculation of the integral cross section is the so-called random phase approximation. It is based on the observation that the interference terms which are present in Eq.(8) and which contain differences in the phase shifts in bilinear combinations of the scattering matrix for different paths ~ and ~ ' do not provide a noticeable contribution to the integral cross section since they virtually vanish when integrated over impact parameters. The random-phase approximation, (P~,,,.(j', j; ~:, b)), to the transition probability (8) reads: (P:,,~ (j', j; k, b)) = -(fj,,j(f~,,~+

=

~

RJ

r s~

s,"'[rl)'~',J~(b)](-1)J'-5'-J+~

j ' j'

J -J

(17) r'

s'

-~s'

. sja%, j=(b) sj,=,,j~(b) s~. This expression does not contain rapidly oscillating terms, but includes contributions that vary slowly with the deflection angle. Coupled with approximation (14), it describes the probability of the polarization transfer in terms of deflection angles for different trajectories that originate from a single trajectory. The latter is characterized by impact parameter b, collision velocity u and by the elements of the locking matrix. In the sudden locking approximation, the only parameter of the locking matrix is the locking angle a. Some examples of application of the above formulae will be discussed in Sect. 6. 5

Differential

Cross

Sections

for Polarization

Transfer

Application of quasiclassical methods to calculations of differential cross sections for polarization transfer is not as straightforward as calculations of integral cross sections. There are two basic approaches: the eikonal approximation which is valid for the scattering through small angles ((9 > ~d)- Since under quasiclassical conditions [~d is extremely small, two approaches possess a common region of applicability, Od _ 27r) in a SF frame X Y Z . When one considers scattering in the plane, the scattering to the right and to the left corresponds to the same value of 6}, but to two different values of ~v (say ~ = zr/2 and = 37r/2 ); alternatively, one can speak about scattering to the right and to the left by the same angle 6}. Of course, in general, both right and left atoms contribute to the scattering to the right; and both contribute to the scattering to the left. The scattering to the right and to the left are separated by the region of small (diffraction) angles (a cross-hatched region in the upper part of Fig. 3) in which the quasiclassical description is not valid. As an example, Fig. 3 shows the scattering to the right, and the two contributions to the scattering amplitude originating from the right and left atoms. The scattering amplitudes to the right and to the left from the initial state [j, n) to a final state ]j', n') in the NC frame, li~,right j , , , , ; j , , (6}) and i~left , j , , , , ; j , , (t:}] ~v I, can be expressed via the positive and negative J-helicity (JH) scattering amplitudes, fJ+,~';L~(6}) and f~,~,;j,~(6}). Here, n, n' are projections of j,j', onto the Z axis (NC frame), while u, u' are the projections o f j , j ~, onto the J vector (JH) frame. The relation between quantum numbers n and u for the right and left atoms is illustrated in Fig. 3 [12,39]. The appropriate formulae for JH and NC scattering amplitudes read:

f)+ ~,;j,~(O) = - i exp(-iTr/4)

f0 c° e x p ( - i O J )

Sj,,.,;j,,~(J)v/-ffdJ

198


ba

bI

Fig. 3. Quasictassical description of scattering of an atom A(j = 1) on a spherically symmetric atom B positioned in the origin of the natural collision frame XYZ. Shown are two trajectories deflected by the same scattering angle O; trajectory 1 corresponds to repulsive scattering (right atoms are scattered to the right), trajectory 2 to attractive scattering (left atoms are scattered to the right). Total angular momenta J1 and J1 corresponding to these trajectories are shown by vectors originating from the points of closest approach for the respective trajectories. The cones at the initial points of trajectories, each of height n and slant height j, illustrate schematically the polarization state ]j, n) of incoming atoms. The hatched regions in the X Y plane correspond to small impact parameters and small scattering angles where the quasiclassical description is not valid.

--i exp(irr/4) f~,:,,;j,,(O) = -~2-~.kCs~nO

f0 °°

(18)

exp(iOJ) Sj, u,;j,u(J)v/JdJ

and

t~__exp[in,(O F ~,n~;j,n j r ~'-~ i! g h t

~r)]-

[fj+':,,;j,u(O)l ~-~,,-F

(19)

1

Fj~O~t t,n';j,n t~'~ t" ] = exp [-in'(O + ~)][ (-1)J'+U[',.';J, .(°)1J2--:,

" (0 + " (--l'J'+Jf-" "~',"';~,"

"=" )1.,_.,

]

Eq.(18) shows that JH scattering amplitudes have the same s y m m e t r y properties as the JH scattering matrix: it is s y m m e t r i c with respect to q u a n t u m numbers of the initial and final state, and it vanishes for transition between different reflection states. The former s y m m e t r y property disappears in NC scattering amplitudes, Eq.(19), since the s y m m e t r y of the scattering event in a space-fixed frame is lower that the s y m m e t r y of this event in a body-fixed


199

frame. The meaning of the JH positive and negative scattering amplitudes can be seen by referring to elastic scattering and by evaluating the integrals in the stationary phase approximation. One finds then, that f + (O) describes the scattering of atoms under the action of the repulsive forces (right atoms are scattered to the right, and left atoms to the left), while f - ( 0 ) describes the scattering under the action of the attractive forces (right atoms are scattered to the left and vice verse). left /~'~ The manifold of scattering amplitudes Fj~,igh,~j,~(O), ;, and F j,,,v;j,,~t':") , can be used for calculation of differential cross sections which describe the collisional transfer of polarization moments of an atom. If the polarization state of the atom in NC frame is specified by the set of irreducible spherical.state moments ~ s , the cross section of the polarization transfer , ~ s -4 ~,~,1~21~ = JiJ~qAma~2p52s, 3 192, n = -t-2, Qjr,s,;jrs(O), is constructed as a ClebschGordan contraction of NC scattering amplitudes. Now, if one goes back to the X ' Y ' Z t frame, and specifies the polarization state of the atom by the set of polarization moments Pjrp in this frame, the appropriate polarization transfer cross sections qjr,~;j~ (0, ~) can be obtained from the rotation transformation of the matrix of the cross sections Oi~,~,;j~(O) : = exp

- p)]

•

zX;

(20)

where A;a = R;~(rr/2) [40]. The polarization-transfer cross sections qjr,p,;j~p(O; T) are not axially-symmetric and are complex-number quantities; their very simple dependence on azimuthal angle is due to the fact that these cross sections relate irreducible spherical components of the polarization tensor. Had one considered a scattering of an atom whose polarization state is a linear superposition of irreducible spherical components, one should have dealt with objects that are linear combination of qjr'p';jrp(O, ~). The ~-dependence of the latter can be quite complicated. Two examples of the cross sections which are expressed in a simple way via NC scattering amplitudes are the total population-transfer cross section for the scattering of unpolarized atoms, qtot;unpol(O, j -4 jr) , and the right-left difference cross section qaify;h~u~ (0, j, n -4 j') for the scattering of helicopteroriented atoms:

qtot,,,npodO,j -4 j') = (2j + 1) - I E = (2j + 1)

qdiff;helic(O,j, It -+ j') = ~ n t

b~right i'O~ 2 ~J ,v,j,~, ,

(21)

teJ, :

it2)~ ~wright J n';Jn ~v ll 2 -- E

•it;,ieft .,;in(o)?

(22)

n t

The next step in the quasiclassical approximation consists in recovering the JH scattering matrix from the classical attributes of adiabatic motion

200


across the molecular potential curves and from the parameters of the matrices of non-adiabatic transitions. We exemplify this by way of scattering of an atom in an isolated state j = 1 (j = j~ -- 1) for a repulsive interaction in both molecular states (see Fig. 1). The two deflection functions, q0 (b) and 7/1(b), can be used to recover the phase-shifts for elastic scattering from the differential relation (10), 60(J) and 61(J). Other quantities which will enter into the JH scattering matrix are the locking angle L(J) and the slipping probability s(J). The ultimate result of rather simple calculations according to the JH counterpart of Eq.(13) yields the JH scattering matrix which decomposes into two blocks, one with u, u ~-- 4-1 for positive reflection symmetry and the other with u, u ~-- 0 for negative reflection symmetry:

S~I~I =

(1/2)[exp(2i6o 4- 2ia)(1 4- 2s) + exp(2i~l J: 2ia)(1 ~ 2s)]

(23)

So0 = exp(2i61) where all the quantities are functions of b or J. According to Eq.(18), each element of the JH scattering matrix generate two (positive and negative) JH scattering amplitudes. Each amplitude will contain two terms that correspond to the scattering at ~2 -- 0 and ~2 -- 1 molecular potentials, Uo(R) and U1 (R). Finally, if the quasiclassical approximation is pushed further, one solves the integrals over J in the stationary phase approximation; this will express the scattering amplitudes via classical scattering cross sections, appropriate phases, locking angles and two slipping probabilities for different branches of a particular deflection function. All these attributes are generated in the same way as one calculates the quasiclassical elastic scattering cross section from the single deflection function. The only difference being that besides the classical elastic cross section here one has to consider the locking and slipping events, and also the interference between waves scattered elasticity by different potentials. Some examples of application of this approach will be discussed in Sect. 6. For a simple repulsive scattering depicted in Fig. 1 (in this case there is only one branch for each deflection function) all negative JH scattering amplitudes vanish, and four positive JH scattering amplitudes f+(6}), f + (6}), f -+1 - 1 ( 6} ) , f+-1(6}) = f + n ( O ) are expressed as a linear combination of two amplitudes for elastic scattering on $2 -- 0 and ~2 = 1 molecular potentials, £(6}) and f1(6}), with the coefficients depending on locking angles and slipping probabilities, co(6}), a1(6}), and so(O), sl (6}). The elastic scattering amplitudes f a (6}) are expressed via elastic scattering cross sections qa(6}) and phases ¢s~(O) as fa(6}) = ~

exp (i4~a(6}))

(24)

and the meaning of two locking angles, an(6}) , is illustrated in the lower part of Fig. 1. In the approximation of the sudden locking

1[

f+1+1(0) = -~ fo(O)exp(4-2iao(6})) + fl(6})exp(zt:2ial(6}))

]

,

(25)


20]

1

f+1~:i(6)) = f+1+1(0) = ~ [f0(69) - fi(/9)] , f~(O) = f1(8) . A simple relation of the scattering amplitudes in Eq.(25) to the scattering matrix in Eq.(23) is evident.

6

Case Studies of the Recoupling of Electronic Angular Momentum in Collisions

In this section, we discuss three types of simple collision events which illustrate the ability of the quasiclassical description to elucidate the intimate dynamical features of the recoupling of the electronic angular momentum in a collision. Neither of these events can be described within the common trajectory approximation.

Creation of polarization in the beam scattering of unpolarized atoms. When atoms collide under the axially-symmetric conditions, their electronic shell acquires polarization [15], in particular alignment. The creation of polarization from an unpolarized state in a transition j -4 j~ with j~ > j can be easily understood from the simple consideration that the number of magnetic sublevels in the final state are larger than in the initial (unpolarized) state. Therefore, it is quite improbable that the initiM statistical distribution over Zeeman states will result, after a collision, in a statistical distribution over the Zeeman final states. The case of the same j state is different: the question whether the same initially unpolarized state can be collisionally polarized or not, requires a more thorough study. It was proven that a cross section for the creation of the polarization calculated in the impact parameter approximation is zero [41]. Since the impact parameter treatment of relaxation of the state moments via Eq.(13) and (14) with Eq.(12) yielded nonzero cross sections and was believed to provide an accurate quasiclassical limit of the appropriate quantum cross section, it was argued that the above proof is general so that the j-conserving collisions can not create polarization. However, this conclusion is not correct: the vanishing cross section for the creation of polarization in the impact parameter approximation is an artifact related to the superfluous symmetry of the collision event introduced by this approximation. In order to see limitations of the impact parameter treatment we consider a collision as in Eq.(1) taking j = y . Adopting the sudden locking approximation, we substitute sjn,~,j~(b) from Eq. (16) into Eq. (17) and yield for the probability of transfer of the state polarization moment from rank

202


zero (r = 0) to rank r' for the axially-symmetric component (s = s' = 0):

(P°,,o(j, j; 1¢, b)) = E{Pr,(cos[2a(b) +

Tin(b)] ) - £ , , 0 } ( - 1 ) j - n

[,a9 -~2,

$2

(26) where P~,(...) is the Legendre polynomial of order r'; r' can assume values from zero '(in this case the probability corresponds to the loss of population in the initial state) , to 2j. The meaning of dynamic parameters in the r.h.s. of Eq.(26), the locking angle a(b) and the deflection angles r/0 (b) and 7/+1(b) is explained in the upper part of Fig. 1 for the case when both molecular potentials, Uo(R) and U±I (R) are repulsive, and the former is stronger than the latter (therefore) r/0(b) > q+l(b)). For r' = 0, Eq.(26) yields zero probability since the first factor under the sum in the r.h.s, of Eq.(26) vanishes. This simply corresponds to the fact that the total population of the state is not affected by the collision. For odd values of r' (r' = 1 corresponds to the creation of orientation), Eq.(26) also yields zero probability because of the cancellation of contributions to the sum from positive and negative a') (a symmetry property of the Clebsch-Gordan coefficients). This corresponds to the fact that the final polarization component ( r' odd) and the initial component (r = 0) possesses opposite symmetry with respect to a reflection in a plane through the symmetry axis. For even values of r' (r' = 2 corresponds to the creation of alignment), Eq.(26) yields, in general, a nonzero probability. However, if the first factor under the sum in the r.h.s, of Eq.(24) does not depend on a'2 ( the common trajectory approach, the impact parameter approximation in particular) it becomes zero because of the orthogonality relation for the Clebsch-Gordan coefficients. It is thus clear that the vanishing probability of creation of polarization is related to the superfluous symmetry of the problem in this approximation: a common trajectory description of a collision introduces an additional symmetry into the collision picture that actually does not exist when one incoming trajectory generates two trajectories, running on different potentials The integral cross section for the creation of polarization are related to the asymmetry in the Zeeman cross sections for transitions m --+ m' and m' --+ m. For instance, the cross section for creating alignment in a state with j = 1, ~7°,o(l, l; k), determines the difference between two Zeeman cross sections: o'°,0(1, 1;1¢) =

2 [Oh,o(l,l;k:)- O'o,i(1,i;]~)]

(27)

Eq.(27) provides an example when Zeeman cross sections for transitions m --+ m' and m' -+ m are different. Of course, the relation ~5,,.~ (j, j) :~ crS,.~, (j, j) does not contradict the detailed balance principle which requires that the cross sections for a direct, m --+ m ' , and the reverse, - m ' -+ - m , transitions


203

be equal, c~r,,,n (j, j) = trim,_ m, (j, j) , However, for isotropic collisions, the Zeeman cross sections for transitions m ~ m' and m' --+ m are the same. The importance of the multiple-trajectory description for the collisional creation of polarization poses the question to what extent this effect will show up in the values of relaxation cross sections for isotropic collisions. Consider a cross section a r ( j , j ) , and represent it as a sum of a cross section calculated in the impact parameter approximation, o'r'ip(j,j), and the correction Aar(j, j). This correction is associated, via Eq.(13), with the respective correction to the probability of relaxation Ap"(j,j; b). The expression for A P ~(j, j; b) reads [42]:

AP"(j,j;b)-

~

{R~0 (qjo(b)

+ 2c~(b))- R~oo(2o~(b))}

Jl

(28) -1- (2r q- 1~-----)

- - ~ 2S2

{ -2t2,20

(rlj.q(b) -t-

-- Rr--2a,2~

The correction to the transition probability is due to the fact, that the angles of rotation of the molecular axis in the region where the electronic angular momentum is locked to this axis, r r - Oja(b)- 2a(b), are different from the angle of rotation when the system moves along a rectilinear trajectory, 7r- 2~(b). For scattering off a hard core with small impact parameters, when the molecular axis virtually does not rotate at all, the relaxation probability is very small. In this case, the correction Ap~(j, j; b) is almost opposite to P~,iP(j,j;b). We also see from Eq.(26) that the correction Apr(j,j;b) does not vanish when one passes from multiple trajectory description (different deflection angles qjo(b)) to the common trajectory description (a single deflection angle r/different from zero). Of course, Apr(j,j;b) vanishes for rectilinear trajectories, 77= 0.

Interference pattern of the differential scattering of unpolarized atoms. Measurements of the differential cross sections for scattering of excited sodium atoms on Ne at the collision energy 0.15 eV reveals an oscillating structure [33] which can not be explained by interference of waves scattered by the attractive and repulsive portions of the same potential [43]. The closecoupling calculations with realistic potential functions reproduce the structure and suggest that the reason for it is the interference of the waves scattered by two different molecular potentials that arise from the P state of Na [36]. Since the interference requires participation of coherent states one can ask the question how these states can arise from incoherent superposition of atomic magnetic substates. The quasiclassical study of this question reveals that the oscillatory pattern that is due to the interference of the waves scattered by different potentials strongly depends on the locking angles, and that the coherence is created in the locking event.

204


For Na*-Ne collisions, the approximation of a fixed electronic spin is appropriate [44], and the total differential cross section for the scattering in a single fine structure state, 2 p j , j = 1/2 or 3/2, turns out to be equal to the total differential cross section for the scattering of a spinless atom in a P state: qtot,~,,~pot(O;2 Pj) = qtot;,,npot(O; P) (29) The quasiclassieal expression for qtot,,,npot(O; P) can be analyzed in terms of contributions from the scattering at ~ and H potentials with proper account for interference and locking. First we separate the slow- and rapidly-varying contributions to the cross section

qtot;u.vo,(O, P) -~- qtstow rapid .[172, . . . 1"1 ot,unpol ..-. 1(72,p) + qtot,unpol

(30)

The explicit expressions for slow and rapid contributions depends on the number of classical trajectories that are deviated by the same scattering angle O. For Na*-Ne scattering, there are, in general four trajectories; one repulsive trajectory for the ~ potential (Z,rep) one repulsive trajectory for the / / trajectory (H,rep), and two attractive trajectories for the rainbow scattering at the rr potential. For the scattering angles close to the primary rainbow, two attractive trajectories are very close to each other, and can be collapsed in a single trajectory (//,rain). Accordingly, we have three amplitudes for elastic scattering, f.~,rep(O), and f//,rep(O) and fFI,rain(O). slow t ~ p) is just the contribution from The slow-varying portion qtot,unpoltV, elastic scattering by a repulsive E' potential, qt/,rep(O) , from elastic scattering by the repulsive part of a / / p o t e n t i a l , qH,rep(O) , and from elastic scattering by the rainbow part of a / / p o t e n t i a l , qH,rain(O) :

qtot,unpol[O , s. . .l . o p) = (1/3)[q2,rep(O) + 2qr,rep(O) -t- 2q~r,rain(O)]

(31)

We also note that, if the scattering to the right is considered, the first and second term in the r.h.s, of Eq.(31) come from the right atoms, while the third term comes from the left atoms. The rapidly-varying portion is due to interference between waves scattered by the repulsive and rainbow branches of the H potential and between waves scattered by the Z repulsive potential and the rainbow branches of the H potential:

qrapid tot;unpol (O, P) = (2/3)\/qII, repqH,rain Cos( ArN)Fl,rep;Fl,rain) " [2 - sin2(an,~p + an,~in)

(32)

--(2/3),~/qH,repqH,rain Cos(A4:ibH,rep;H,rain) sin2(Otll,rep "4- O'H,rain)] The significance of the locking can be seen from this expression: If the adiabatic approximation were assumed to be valid at all internuclear distances, one would set here all the locking angles equal to zero (the locking of the electronic angular momentum to the collision axis occurs already at infinitely

Quasiclassica] Approximation in the Theory of Scattering

205

r,~pia " " P) in Eq.(32) becomes a familiar large distances). In this case qtot;~,,~pot[~; high-frequency contribution to the elastic scattering by the H potential, and the sum in Eq.(30) a weighted sum of cross sections for independent elastic scattering on the ~ and / / potentials. This is the so-called elastic approximation to the scattering cross section of unpolarized atoms; it is this approximation which failed to explain experimental findings [43]. On the other hand, with proper account of the locking phenomenon, Eq.(30) together with Eqs.(31) and (32) reproduce both the quantum-mechanical strong coupling calculations and the experimental results [44]. Due to the simplicity of the quasiclassical expressions, it helped to correct an earlier inconsistency of the semiclassical description [37] and to reveal the role of spin-decoupling in the collision event. We note also that the amplitude of the oscillations which are due to interference of waves scattered at different potentials serves as a measure of the difference of the locking angles for scattering at these potentials. Right-left asymmetry in the scattering of helicopter-polarized atoms. Differential scattering of unpolarized atoms to the right and to the left by the same angle 69 is characterized by equal cross sections. This evident property of scattering can be related to the mirror symmetry of the initial state with respect to the plane that separates right and left atoms. If atoms are polarized, and the initial state is not symmetrical with respect of reflection in this plane, and one would expect the right-left asymmetry of the scattering. Normally, the reflection of the polarization state of atoms in this plane will change the distribution of the electron density of atoms with respect to the incident velocity vector. As a result, left and right atoms with the same values of the impact parameters will populate different molecular states to a different extent. Now, since different molecular states cause different deflection of trajectories, it is not at all surprising that the scattering of polarized atoms will exhibit a certain right-left asymmetry. One can say, that the rightleft scattering asymmetry can be traced back to the different distribution of electron density for right and left atoms with respect to the molecular axis at the moment of locking. However, there exists one type of polarization for which the mirror reflection does not change the distribution of the electron density. This is the so-called polarization when the atoms are oriented perpendicular to the collision plane. For this type of polarization, the right and left atoms possess exactly the same distribution of the electronic density with respect to the molecular axis. Therefore, the scattering asymmetry should be due to the interaction which is not invariant under the reflection in the mirror plane. As the only difference between the right and left atoms at the moment of the locking is the relative orientation of the electronic angular momentum j to the total angular momentum J (their J-helicity quantum numbers are just opposite) the interaction in question is the Coriolis interaction. Since

206


the Coriolis interaction is very week it would be interesting to see how much asymmetry can be induced by this interaction. Experimentally, the right-left scattering asymmetry has been observed in the scattering of excited -polarized alkali atom on rare gases [34] and for helicopter-polarized Ne(2p53s, 3 P2) atoms in collisions with Ne [35] and Ar [36]. The latter system provide a very interesting case for testing the locking approximation. This system differs in many respects from the case of scattering excited alkali atoms. First, all the potentials of the Ne*(2p53s, 3 P2)-Ar pair are governed mainly by the exchange interaction of electrons of the valence closed shell of Ar with the diffuse 3s orbital of the excited electron of Ne* [45] and therefore are believed to be repulsive. Second, the lifting of degeneracy in the 3 P 2 state of Ne* occurs as a result of core-core interaction, and therefore locking takes place when the trajectory already deviates from the rectilinear path. Third, for low collision energies, there is no appreciable fine structure transition, and therefore one can not speak about decoupling of spin and electronic angular m o m e n t u m . A lengthy analysis [46] yields an approximate expression for the cross section difference A(O) of helicopter-polarized atoms scattered to the right and to the left:

A(O) = --[3q(0)/4] sin4[a0(O) -- al(O)] sin 2A¢0~(O)

(33)

= A,~a~(O) sin 2z5~o1(O) where q is the cross section for scattering of unpolarized atoms, and locking angles a0(O) and a l ( O ) correspond to the scattering off the molecular potentials and that arise from the state j = 2. From Eq.(33) we arrive at an expression for the ratio R(O) of the envelope A,~a,(O) to the cross section q: R(O) -

Ama (O) = 3 s i n 4 [ a 0 ( O ) -q(O) 4

(34)

The difference in the locking angles Aa01 = a 0 ( O ) - a l (6~) depends on the details of the interaction potentials which we do not consider here. However, it is clear from the meaning of the locking angle, that over a range of the scattering angles which are well outside of the backward scattering, Acr01(O) increases starting from zero. Then R(O) should pass a maximum at a certain scattering angle O = ~,,, reaching a value of 0.75; at this scattering angle, the difference in the locking angles equals 7r/8. We turn now to Fig.4. At small angles (say, below Otab = 5°), the asymmetry cross section is much smaller than the total cross section. We interpret this as an indication about the threshold behavior of the asymmetry cross section: for small A a , it increases proportionally to Aa. As seen from Fig. 4, the ratio R increases with increasing O, reaches a maximum of 0.6 at O = 11 °, and then decreases down to 0.3 at O = 21 ° . We identify this maximum with


207

6 I

*4 A

q d •I

0

"*t '

o! Fig. 4. Experimental data on the total differential scattering cross section q, and the envelope of the cross section difference Arnax for scattering of unpolarized and helicopter-polarized Ne*(2p53s, 3 P2, n = 4-2) atoms on Ar at collision energy of 64 meV (after [36]).

6rnax

f

I

I

I

0

I 10 °

I

2

I

a

i

20"

Olab

the m a x i m u m of function R(O) from Eq.(34) and ascribe the difference between the theoretical prediction 0.75 and the experimental value 0.6 to the contribution of other terms neglected in approximate Eq.(34). Finally, we note that the value of Ac~01 at the m a x i m u m , A(~01(Om) = r / 8 , is quite reasonable when one takes into account that the range of variation of individual locking angles is 0 - ~r/2. We can thus say that the behavior of the O-dependent cross section difference for the system Ne*-Ar demonstrates a transition from slippage to locking, and a rather large scattering asymmetry of left- and right helicopter-polarized a t o m s in this case is ultimately due to the fact that the initially prepared atomic states correspond to the largest-possible J-helicity states, ~ = 2 and v = - 2 .

7

Conclusion

In this paper, we have shown that a quasiclassical analysis of the scattering of atoms indeed provides a simple insight into collision dynamics. This is related to the fact that under slow quasiclassical conditions the total inter-

208


Table 1. Different levels of quasiclassical approximation in the description of scattering of polarized atoms T y p e of the process

Lowest level of quasielassical description

Isotropic collisions in the bulk Transfer of polarization moments. Integral cross sections (rr (j', j).

Impact parameter (common trajectory) approximation with sudden locking.

Anisotropic collisions in beams. Transformation of polarization moments. Integral corss sections o'~.,,.(j', j; k ).

Multiple trajectory description; sudden locking. Common trajectory description; non-sudden locking.

Multiple trajectory description; Anisotropic collisions in beams. Transformation of polarization moments. non-sudden locking Differential cross sections q,-, ~;r~.(j', j; k, ~:).

action occurring in a collision can be split into different contributions, which can be analyzed separately. One of these, which is important for the polarization phenomena, is the recoupling of the electronic angular m o m e n t u m from a space-fixed to the body-fixed quantization axis, accompanied by a partial breakdown of the L S coupling in free atoms. This type of reeoupling is most conveniently described in terms of transient Hund coupling cases. The approach along this line was first formulated within a semiclassical picture [3], and later generalized for quantum mechanical formulation [47,48]. An analysis of recoupling of angular momenta allows to identify the interactions which are responsible for specific effects in the scattering of atoms in degenerate electronic states. For instance, in the scattering of unpolarized atoms, the interference structure in the differential scattering is strongly affected by the long-range Coriolis interaction, and the creation of alignment is critically dependent on the difference of the deflection angles for trajectories, corresponding to the same value of the impact parameter. Different levels of the quasiclassical approximation in the description of scattering of polarized atoms are summarized in Table 1. We indeed see that for some processes a rather crude quasiclassical approximation suffices to provide a reasonable result. On the other hand, we believe that the sophisticated quasiclassical multiple-trajectory description of a collision that includes locking and slipping is able to reproduce all the essential features of the polarization transformation in collisions provided the scattering process occurs in the range of classical angles. In conclusion, quasiclassical analysis of scattering, complemented with additional simplifications related to small values of the ratio j / J , provides a very useful tool in the interpretation of both accurate numerical results and


209

experimental findings. This is in line with growing interest in semiclassical methods in Postmodern Quantum Mechanics. It is a pleasure to dedicate this paper to Prof. Gisbert zu Putlitz on his 65th birthday. Our work on the theory of scattering of polarized atoms has been closely connected with the activity of zu Putlitz' group in Physikalishes Institut, Universit/it Heidelberg, and we benefited much from the seminars and discussions there. Finally, we will never forget friendly ties with Gisbert during last, sometimes uneasy, twenty five years.

8

Acknowledgment

We acknowledge very constructive discussions with F. Masnou-Seeuws, J. Baudon and F. Perales. This work was supported by the Technion V.P.R. Fund - Promotion of Sponsored Research, and by the Giladi program.

References [1] L.Landau, and E.Lifshitz, Quantum Mechanics (Oxford, Pergamon Press, 1977). [2] E.E.Nikitin, and S.Ya.Umanskii, Theory of Slow Atomic Collisions (BerlinHeidelberg, Springer, 1984). [3] E.E.Nikitin, in Atomic Physics 4. Edited by G. zu Putlitz, E.W. Weber and A. Winnacker (N.Y., Plenum, 1975), p.529. [4] I.V.Hertel, H.Schmidt, A.BSktring, and E.Meyer, Rep.Prog.Phys., 48, 375.(1985). [5] E.E.B.Campbell, H.Schmidt, and I.V.Hertel, Adv.Chem.Phys., 75, 37 (1988). [6] J.Baudon, R.Diiren, and J.Robert, Adv.At.Mol.Opt.Phys., 30, 141 (1993). [7] O.Carnal, and J.Mlynek, Phys.Rev.Lett., 66, 2689 (1991). [8] P.W.Keith, C.R.Ekortom, Q.A.Gurchette, and D.E.Pritchard, Phys.Rev.Lett., 66, 2693 (1991). [9] J.Robert, Ch.Miniatura, S.Le Boiteux, J.Reinhardt, V.Bocvarski, and J.Baudon, Eur.Phys.Lett., 16 , 29 (1991). [10] E.I.Dashevskaya, and N.A.Mokhova, Optika i Spektr. 33, 817 (1972). [11] E.E.Nikitin, Khimicheskaya Fizika, 3, 1219 (1984). [12] E.E.Nikitin, Khimicheskaya Fizika, 5, 15 (1986). [13] E.I.Dashevskaya, and E.E.Nildtin, Optika i Spektr., 68, 1006 (1990). [14] E.I.Dashevskaya, and E.E.Nikitin, J.Chem.Soc. Faraday Trans., 89, 1567

(1993). [15] E.B.Alexandrov, M.P.Chaika, and G.I.Khvostenko, Interference of Atomic States (Berlin-Heidelberg, Springer, 1993). [16] L.Waldmarm, Z.Naturforsch. B, 13, 609 (1958). [17] R.F.Snider, J.Chem.Phys 32, 1051 (1960). [18] A.Omont, J.Phys. 26, 26 (1965). [191 M.I.D'yakonov, and V.I.Perel., ZhETF, 48, 405 (1965). [20] A.Berengolts, E.].Dashevskaya, and E.E.Nikitin, J.Phys., B26, 3847 (1993).

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[21] P.Hansen, L.Kocbach, A.Dubois, and S.E.Nielsen, Phys.Rev.Lett. 64, 2491

(1990). [22] [23] [24] [25] [26]

C.T.Rettner, and R.N.Zare, J.Chem.Phys., 75, 3636 (1982). B.PouiUy, and M.H.Alexander, Chem Phys., 145, 191 (1990). J.Grosser, Z.Phys.D 3, 39 (1986). L.J.Kovalenko, S.R.Leone, J.B.Delos, J.Chem.Phys., 91, 6942 (1989). A.Berengolts, E.I.Dashevskaya, E.E.Nikitin, and J.Troe, Chem.Phys. 195, 271 (1995). [27] E.E.Nikitin , and R.N.Zare, Molec.Phys., 82, 85 (1994) [28] E.I.Dashevskaya, E.E.Nikitin, and S.Ya.Umanskii, Khimicheskaya Fizika 3, 627 (1984). [29] E.E.Nikitin, Optika i Spektr. 58, 964 (1985). [30] V.N.Rebane, and T.K.Rebane, Optika i Spektr., 33, 219 (1972). [31] A.G.Petrashen', V.N.Rebane, and T.K.Rebane, Optika i Spektr., 35, 408 (1973). [32] E.I.Dashevskaya, and N.A.Mokhova, Chem.Phys.Lett., 20, 454 (1973). [33] G.Caxter, D.Pritchard, M.Kaplan, and T.Ducas, Phys.Rev.Lett., 35, 1144 (1975). [34] R.Diiren, and E.Hesselbrink, J.Phys.Chem., 91, 5455 (1987). [35] J.Baudon, F.PeraJes, Ch.Miniatura, ].Robert, G.Vassilev, J.Reinhardt and H.Haberland H , Chem.Phys. 145, 153 (1990). [36] F.Perales, Effets de Polarisation dans des Collisions aux Energies Thermiques Impliquant des Atomes Metastables de Neon, Thesis, Universit$ de Paris XI, 1990. 37. F.Masnou-Seeuws, M.Phihppe, E.Roueff, and A.Spielfield, J.Phys.B, 12, 4065 (1979). [37] E.I.Dashevskaya, R.Diiren, and E.E.Nikitin, Chem.Phys. 149, 341 (1991). [38] E.I.Dashevskaya, F.Masnou-Seeuws, and E.E.Nikitin, J.Phys.B, 29, 395 (1996). [39] D.A.Varshalovich, A.N.Moskalev, and V.K.Khersonskii, Quantum Theory of Angular Momentum (Singapore, World Scientific), 1988. [40] A.G.Petrashen', V.N.Rebane, and T.K.Rebane, Zhurn. Eksp.Teor.Fiz., 67, 147 (1984) [41] E.l.Dashevskaya, and E.E.Nikitin, Optika i Spektr., 62, 742 (1987). [42] C.Bottcher, J.Phys.B, 9, 3099 (1976). [43] E.I.Dashevskaya, F.Masnou-Seeuws, and E.E.Nikitin, J.Phys.B, 29, 415 (1996). [44] H.Kukal, D.Hennecart, and F.Masnou-Seeuws, Chem.Phys., 145, 163 (1990). [45] E.I.Dashevskaya, E.E.Nikitin, J.Baudon, and F.Perales, (to be published). [46] V.Aquilanti, and G.Grossi, J.Chem.Phys 73, 1165 (1980). [47] V.Aquilanti, S.Cavalli, and G.Grossi, Z.Phys.D, 36, 215 (1996).

Ion B e a m Inertial Fusion R. Bock Gesellschaft fiir Schwerionenforschung, D-64291 Darmstadt, Germany

1

Introduction

The development of thermonuclear fusion to a future energy source is one of the outstanding objectives of present research, its realization one of the great challenges for our scientific community. Consequently, in the present situation where crucial problems of the confinement concepts are not solved yet, all possible options still have to be taken into consideration. The research programs of the United States and of Japan, for example, cover both alternatives of fusion energy research, magnetic and inertial confinement, whereas the European effort is concentrated exclusively to magnetic confinement. Magnetic confinement fusion (MCF), having been studied with an enormous world-wide effort for nearly four decades, is obviously the more advanced concept as compared to inertial confinement (ICF) with respect to its technological and conceptual achievements. With respect to power generation, however, each of the two concepts have their specific advantages and will exhibit their specific problems. It is this long-term perspective, the potential for an economical and environmentally attractive power generation, which requires a continued investigation of both concepts and an evaluation of the two approaches on a comparable basis. For this reason inertial confinement fusion deserves more systematic investigation. Three different driver options have been studied during the last two decades for inertial confinement: Laser beams, light ion and heavy ion beams. The specific advantage of ion beams is based on two outstanding features: (a) the coupling of ion beams to the target is well understood and exhibits a 'classical' behavior, the deposition of the ion energy on the target is nearly 100% and is not impaired by the plasma or any other medium, such as magnetic fields, surrounding the target. (b) the efficiency of heavy ion and light ion drivers is high (around 25%) and the repetition rate for heavy ion accelerators is excellent. Both properties are the key issues for energy generation, which - in the case of the heavy ion accelerator- fulfill the requirements of a power plant already now. The high pulse intensities and the short pulse lengths requested for ignition have not been achieved yet with ion beams. This is a matter of technical development and needs further research. For the heavy ion accelerator [1,2], extrapolations from the operation of existing large facilities indicate that the

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R. Bock

specifications for a reactor driver can be reached. Moreover, a large experienced accelerator community gives confidence in a competent handling of the necessary development programs. For light ion beams [3] the development of pulsed-power technology has made considerable progress during the last decade and the achievements concerning specific deposition power are promising. Some key issues, however, in particular the high repetition rate of light-ion diodes, are not solved yet. According to our present understanding, the various driver options for inertial confinement fusion can be characterized as follows: Based on the enormous effort for the development of laser technology, ignition will, most probably, first be demonstrated with the powerful single-shot Nd-glass laser facilities now under construction, the National Ignition Facility (NIF) in the US and the Mega-Joule Facility in France. They will, however, not meet the requirements for a reactor driver with respect to repetition rate and efficiency. Whether the KrF gas laser will meet these conditions is doubtful. Light ion beam facilities have achieved high deposition power and will provide significant results on beam-target interaction. Both facilities, laser and light ion beams are necessary and indispensable for the fundamental investigations on target performance for both, directly and indirectly driven targets, independent of their reactor-driver capability. The heavy ion accelerator, however, with its excellent repetition rate and high efficiency offers the superior prospects for a reactor driver.

2

The Physics of the Target

Energy generation by inertial confinement fusion is based on the following concept [4,5]: The deuterium-tritium (DT) fuel enclosed in a small spherical shell of some millimeter radius, the 'pellet' or 'target', is compressed isentropically by ablation of the shell and heated up to ignition conditions. The energy necessary for this procedure is supplied by short and adequately shaped pulses of intense laser or particle beams. As in magnetic confinement (MCF), the ignition temperature of about 100 million degrees has to be reached and the Lawson Criterion has to be fulfilled in order to achieve substantial burn. Different from MCF, the confinement time which can be obtained by the inertia of the imploding matter is extremely short, in the sub-nanosecond time scale, so the fuel density at ignition needs to be correspondingly higher by many orders of magnitude. The Lawson Criterion for a shock compressed sphere can be expressed by the relation Pi " Ri > 3 g / c m 2, where Pi and R i designate fuel density and fuel radius at ignition. Consequently, the fuel mass in the pellet and its density to be reached at ignition are inversely related: The smaller the mass of the fuel, the higher is the compression needed. For a pellet explosion to be handled in a reactor vessel, the fuel mass is limited for mechanical reasons to an order of some milligrams, and the corresponding fuel density to be achieved at ignition is about 103 . The beam intensity

Ion Beam Inertial Fusion a) Direct Drive

2]3

b) Indirect Drive

heavy ion

hohlraum radiation heavy ion

'

t

Fig. 1. Principle of inertial confinement. (a) Direct drive: A hollow sphere of 5-10 mm diameter, filled with cryogenic DT fuel, is heated by heavy ion beams and evaporates material by ablation. By the evaporation a high radial pressure is produced by which the fuel is compressed and finally heated up to ignition temperature by shock waves. (b) Indirect drive: The kinetic energy of the heavy ion beam is converted into electromagnetic radiation which is confined by the outer eIlipsoidal high-Z casing (hohlraum radiation) which compresses the inner shell.

required to about 6 MJ in less than as bismuth,

obtain such a compression by a direct heating of the pellet is and the implosion dynamics requires this pulse to be delivered 10 ns (direct drive, see Fig. la). For a beam of heavy ions, such this corresponds to 3 - 1018 ions per pulse.

In order to achieve an isotropic compression and to avoid dynamical instabilities, a high degree of azimuthal symmetry of the incident beam intensity is necessary. In case that the symmetry requirements can not be reached by direct drive with a reasonable number of beams, another concept, indirect drive, has been proposed and is at present widely investigated by computer simulations. In this case the pellet is enclosed in a larger casing of high-Z material with two converters on opposite sides (Fig. lb). The kinetic energy of the heavy ion pulse is deposited in the converters which radiate, according to Stefan-Boltzmann's law, soft x-rays into the casing. By the arrangement of converters and shields inside the casing, this radiation which drives the implosion can be made very isotropic. At a specific deposition power of 1016 W / g the conversion efficiency of the kinetic ion energy into electromagnetic radiation is predicted to be about 90%, and the hohlraum radiation inside the casing is reaching a temperature of 300 eV, sufficient to drive the implosion. Research in Europe on key issues of target design and target dynamics [4-7], such as Rayleigh-Taylor instabilities, radiation symmetrization, conversion of beam energy into radiation, radiation confinement and beam-target interaction, has been carried out by theory groups at Frascati [4], Frankfurt [6], Garching [5] and Madrid [7] with significant results. Recently, Russian groups from Moscow and Arzamas [8] with a long-lasting experience in

214

R. Bock

joined these activities. Experimentally, some work on beam-target interaction at a low power level is going on at the Gesellschaft fiir Schwerionenforschung (GSI), Darmstadt, with heavy ion beams [9] and at the Forschungszentrum (FZ) Karlsruhe with light ions beams [3]. A group of the Max-Planck Institut fiir Quantenoptik (MPQ) in Garching is participating in experiments at the high-power laser facilities in Japan [10] and in France [11]. Japan, with the 30kJ laser facility G E K K O XII in Osaka, has strong activities in this field, and the record of fuel compression, 600 times normal fluid density of hydrogen, has been achieved at this facility some years ago [12]. The main activities in the field of target physics, however, are located outside Europe, partially in classified areas. Most of these - previously classified - results achieved in the US at the high-power laser facilities and with classified computer programs are now becoming published, after the new declassification policy, decided in 1993 by the US government, is effective. Target physics evolved to an interesting world-wide activity with new ideas and concepts for both fusion and basic research. A new ignition concept was suggested recently, the fast ignitor [13], and is presently pursued with great enthusiasm [14]. Different from the usual concept in which the ignition spark is created by shock waves in the very center of the fuel, fast ignition is achieved by a picosecond high power laser pulse (> 1018 W / c m 2) which produces a relativistic electron beam strongly focused in forward direction

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Ion Beam Inertial Fusion

215

and drilling a hole through the pre-compressed fuel. Compression of the target is attained by the heavy ion pulse, ignition by the laser pulse. The most important quantity of the DT-filled target is its gain, the ratio of the produced fusion energy divided by the energy of the beam pulse. According to simulations with hydrodynamic codes [15], ignition and breakeven of an indirectly driven pellet is predicted at a beam pulse energy of about 1-2 MJ. A gain of 100, the working regime for a reactor, needs a pulse energy of about 6 M J, as exhibited in Fig. 2. Existing driver facilities are still far below this energy. With the next generation of laser facilities now under construction, e.g. the National Ignition Facility (NIF), a Nd-glass laser in Livermore with 192 beams and designed to reach 2 M J/pulse, it is the goal to demonstrate ignition. Apart from its key role for energy generation, the target is a fascinating object of basic research. The physics of matter at extreme pressures, densities and temperatures can be studied - in regimes not accessible otherwise - thus opening an exciting perspective for future research on dense and non-ideal plasmas.

3

Reactor

and

Systems

Studies

The ICF power plant has some intriguing features. The clear separation between the reactor chamber and the reactor driver and the absence of large installations for high magnetic fields as in Tokamaks facilitate greatly the design of the reactor chamber, its operation and maintenance. The realization of a liquid protective wall for the first structural wall of the reactor chamber is one of the great advantages of an ICF reactor concept. This protective wall consists of FLiBe or LilTPbs3, an eutectic alloy with extremely low vapor pressure, which allows a high repetition rate of about 5 Hz for a reactor chamber. Systems studies carried out in the early stage of our research program in 1980/81 by KfK Karlsruhe, GSI Darmstadt, MPQ Garching, Giessen University and the University of Wisconsin resulted in a conceptual design study for a power reactor, HIBALL [16], driven by an if-linear accelerator with storage rings. The main goal of this study was to demonstrate the feasibility of such a concept and, in particular, to show whether a heavy ion accelerator can meet the technical and economical requirements of a power reactor. It was based on 20 beams of Bi +, an ion energy of 10 GeV (50 MeV/nucleon) and a pulse energy of 5 MJ. An improved design, HIBALL II, published in 1985 [17], is a concept consisting of 4 reactor chambers operated by the same driver accelerator. With this design a power of 3.6 GW~ was achieved at a reasonable cost level for the produced electricity. In subsequent investigations carried out in the US and in Japan several heavy ion driven reactor concepts have been studied for the two existing types of heavy ion driver accelerators, the induction linac and the rf-linac. They are beyond the scope of this paper. Two new studies, however, carried

R. Bock

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out under the auspices of the US Department of Energy and published in 1992, should be mentioned: OSIRIS [18], and Prometheus H [19], two 1 GWe power plant designs with an induction linac driver. The reactor cavity design of OSIRIS (Fig. 3), for example, has some attractive technological, environmental and safety features. For the first wall and the support structures of the chamber low activation carbon fabric material is used. The reduction of the radioactive inventory after shutdown by orders of magnitude as compared to fission reactors (Fig. 4), but also much better than magnetic fusion designs, is one of the greatest advantages of ICF reactors. The relative cost of the naain components of the OSIRIS plant (i.e. reactor chamber 32%, driver 37%, conventional equipment 22%) are similar to those of HIBALL II. Based on the estimated total cost of 3.1 G$ the resulting cost of electricity is about the same as for other ICF plants and slightly better than for the Tokamak designs. HIBALL fits favorably into these numbers (Fig. 5). For a light-ion driven reactor concept systems studies were carried out in a collaboration between FZ Karlsruhe and the University of Wisconsin. The LIBRA design and improved modifications LIBRA-LiTE and LIBRA-SP [20] have the advantage of a simple modular design, of low cost and of small size reactor units (1 GWe or below). The three design concepts differ mainly by the beam injection concept, which is one of the serious problem areas of a light ion reactor: Channel transport for LIBRA, ballistic focusing for LIBRALiTE and self pinched transport for the SP version. Other problems to be

Ion Beam Inertial Fusion 1011

Fig. 4. Radioactive inventory (30 years operation) as a function of time after shutdown for an ICF reactor (Cascade, LLNL). Comparison with a fission reactor shows that the radioactivity for the ICF reactor (including target materiM and tritium) is smaller by more than a factor of 100.

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solved are the repetition rate of light ion diodes and the pulse shaping of beams. Attractive are the safety features and, compared to heavy ions, the low cost of the driver. In conclusion, existing conceptual design studies of ion inertial fusion power plants exhibit a promising perspective. The separation between driver and reactor chamber allows an optimization of both m a j o r components of an ICF power plant, resulting in the realization of advanced technical concepts with specific advanced low-activation materials and with an easy maintenance. In present design studies, m a n y features remain preliminary, because they can not be investigated on an adequate level of funding. In future research programs this has to be an area of increased activities. 0

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4 4.1

T h e H e a v y I o n A c c e l e r a t o r as I C F D r i v e r General Remarks

In a worldwide frame, two different types of heavy ion driver concepts are investigated. Whereas in the US the induction accelerator [21] is pursued, both a linear and a re-circulator mode [22], research in Europe is concentrating on the combination of the rf linear accelerator with storage rings [1, 23]. Both activities are complementary. At the present stage of modest expenses, research on both concepts should be continued until a clear advantage of one or the other will become evident. The induction linac is a single-pass accelerator with parallel beams through all the induction modules (64 beams conceived for the driver). At the frontend electrostatic focusing, for the rest of the accelerator magnetic focusing is considered (Fig. 6). Research on the induction linac is carried out at the Lawrence Berkeley National Laboratory (LBNL) and the Lawrence Livermore National Laboratory (LLNL), USA. The European approach is based on the tradition of more than 40 years of accelerator research and development for nuclear and particle physics. A large accelerator community is involved in the research and in operations of such facilities. High-current acceleration is one of the main directions of ongoing accelerator research, and the heavy ion driver is one of potential applications. Modern particle accelerator facilities consist of a combination of different accelerating and beam handling modules: linear accelerators, synchrotrons and storage rings, combined with sophisticated diagnostics for beam manipulations and beam control. In principle, most of the components of a driver accelerator already exist, and a wealth of advanced driver technology, such as ultra-high vacuum technology, superconductivity, fast kickers, beam control etc. has been developed for existing machines. The operations of such facilities have achieved a high standard for long-term stability and reliability, and are based on the experience of a large community of physicists and


219

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technicians. The technical specification of a fusion driver accelerator is partially beyond the achievements reached so far, because highest beam quality is required. High currents and ultra-short beam pulses with excellent phase space density are necessary in order to satisfy the needs for the ignition of the pellet. Progress in the development of specific components has been made during the last years, but much more effort is necessary to reach these goals. 4.2

The rf Linac/Storage Ring Concept

During the last decade various concepts of reactor drivers were investigated. The basic structure is about the same (Fig. 7): Acceleration is achieved by a linear accelerator consisting of various types of if-structures, such as rfquadrupole (RFQ), WiderSe, Interdigital-H (IH) and Alvarez structures, delivering a continuous beam of singly charged Bi + ions of between 100 and 400 mA with an energy of 10 GeV (50 MeV/nucleon). The linac beam is injected into a system of storage rings, between 10 to 20 turns each, in order to achieve the necessary current multiplication and the formation of the required bunch structure. Because of space charge limits at low ion velocities, the front-end of the linac consists of 16 parallel channels which are combined by successive funneling with frequency doubling at each step, into a single beam. The rest of the linae, its major part up to a length of about 5 kin, consist of Alvarez structures. New if-structures, such as the RFQ and IH structures were developed in recent years for high-current acceleration, and satisfy the requirement of the front-end part of a driver accelerator. Ion sources for singly charged very heavy ions with the required specification as to current and emittance have been developed at GSI and elsewhere.

220

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The current multiplication of 105 , necessary from the source to the final bunch at the target will be achieved by different techniques: (1) funneling at the front-end, (2) multi-turn injection into the storage rings, (3) bunch merging and (4) final bunching. During acceleration and all beam manipulations, constraints on emittance growth and momentum spread determine the crucial design parameters for the driver facility [24] (Fig. 8). In addition, since the beams in the storage rings are space charge dominated, there is a limitation of storage time in order to avoid instabilities, particularly the longitudinal micro-wave instability. Theoretically, these processes and their influence on beam dynamics and beam quality are well recognized and they are subject of many investigations by computer simulations. Specific experimental investigations on space charge dominated beams are in progress at existing accelerators [25], in particular, at the GSI two-ring facility consisting of a heavy ion synchrotron (SIS) and a storage and cooler ring (ESR), (Fig. 9). According to our present knowledge on beam dynamics and accelerator technology the driver concepts so far envisaged meet the required specification of 6 MJ on the target, for both direct and indirect drive. In case of indirect drive, however, more stringent conditions are requested for focusing and bunching in order to reach the specific deposition power of 1016 W / g in the converter. If further improvement of beam quality should be necessary, non-Liouvillean techniques have to be applied, such as laser cooling in the storage rings [27], telescoping of successive buncher, and the change of the charge state at injection into the storage rings by photoionisation with laser techniques [28].


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High Energy Exp. Areo

K e y Issues and P r e s e n t A c c e l e r a t o r R e s e a r c h

The present driver concept is based on the experience with existing accelerators and on computer simulation of beam dynamics at high intensity and high phase space density. Many problems have been studied during the last decade, some of them have made substantial progress or are considered to be solved. Among the remaining problems a number of key issues need further investigations, such as: high-current performance of linac structures emittanee growth by funneling emittance growth by multi-turn injection - instabilities in storage rings with space charge dominated beams, in particular the longitudinal microwave instability fast bunching and resonance crossing in storage rings fast kickers - beam losses, in particular at injection and extraction - final focusing and repulsive forces between beamlets near the target

-

-

-

-

-

Some of them are under investigation at various existing accelerators, mainly at GSI, at CERN, at Frankfurt University and at Rutherford Appleton Laboratory. An experimental program for the systematic investigation of some of these issues is in progress at GSI. These investigations will be intensified

222

R. Bock

in 1999 after completion of the high-current injector facility now under construction, which will increase the intensity of the heaviest ions by a factor of hundred [26]. Some preliminary results of ongoing experiments on driver relevant issues are quite remarkable and shall be briefly mentioned: - Experiments at the ESR in Darmstadt as well as at the T S R storage ring at the Max Planck Institut (MPI) in Heidelberg and at the Low Energy Antiproton Ring (LEAR) of CERN have shown that beams in storage rings remain stable up to a factor of 10 beyond presently assumed stability limits (Keil-Schnell limit). Bunched beams, now under investigation, are stabilized by the tails of their Landau distribution. - For the fast crossing of an integer resonance in a ring due to the increase of space charge it was demonstrated at the CERN proton synchrotron that only a small increase of emittance occurs. Experiments at GSI have reached 5-101° Ne 5+ ions equal to 50 Joule. Extrapolation to Bi + give evidence that design parameters for a driver accelerator can be reached. - Other experimental activities with high-intensity beams are progressing at GSI in the field of plasma physics [9], in particular measurements of the stopping power and of charge-exchange cross sections for heavy ions in plasma, showing a considerable increase as compared to the cold gas. The most specific feature for heavy ion beams will be the possibility of generating dense plasmas by heating an extended volume of dense matter in a well defined geometry. The investigation of these plasmas, their expansion, cooling and decay, transport properties, measurement of opacities and equation of state, predicted phase transitions to metallic states (metallic hydrogen), are interesting research objectives, with respect to inertial fusion as well as to astrophysical applications. 4.4

A S t u d y G r o u p 'Heavy Ion Ignition Facility'

a) A c t i v i t i e s a n d P a r t i c i p a t i o n . Results in heavy ion inertial fusion research during the last decade, in particular the progress in accelerator technology, have greatly increased our confidence, that the heavy ion accelerator is the superior choice among the driver candidates for a power reactor. In the early Nineties the European Inertial Fusion Community had a series of meetings and workshops in which these achievements and the future strategy and prospects were discussed. It was realized that - after more than a decade of exploratory research in Europe - it is now timely to establish a coherent European program. In a final workshop at CERN in the middle of 1993 with participants from several European countries, the concept of a dedicated facility aiming at ignition was defined as the next logical step. In a proposal submitted to the European Union [29] a Study Group was proposed to elaborate a preliminary design of such a facility. It was submitted to the European Union fusion authorities by ENEA Frascati, DENIM

Ion Beam Inertial Fusion 200

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Madrid, FZ Karlsruhe and GSI Darmstadt. Other research institutes, in particular CERN, Rutherford Appleton Laboratory, MPQ Garching and several university institutes from Germany and France participate with their special knowledge in one of the related research areas in this program. b) Scientific Goals. The Study Group started working in March 1995 and had a series of workshops, mainly on driver issues. The study is concerned with the critical issues of the design of the heavy ion driver, of targets and the means of their production, and of the required reaction chamber with the goal to develop a coherent set of parameters. According to present knowledge ignition would require a pulse energy of about 2 MJ [15] (Fig. 10), delivered on a target of a few mm diameter within about 6 ns, the specific deposition power being of the order of 104 TW/g. For the suggested driver, a 6 - 10 GeV linac and a number of storage rings, scenarios and sets of parameters have been discussed in recent workshops. During the last year scenarios for the injector linac, injection into the storage rings, the final bunch compression and bunch synchronization and the final transport to the reaction chamber were developed. Bunch telescoping and laser cooling in the longitudinal phase space have been proposed as nonLiouvillean techniques for the improvement of beam quality. In the field of target physics, the implosion symmetry and hydrodynamic instabilities represent the key issues for igniting targets. Indirect drive which is accepted as the most appropriate approach to heavy ion inertial fusion, relies on radiation symmetrization inside the target casing to ensure spherical implosion of the fusion pellet. Extensive numerical studies are being carried out [4-6] in order to determine the parameters necessary for the accelerator design. New target designs for an ignition driver have been proposed by the Russian group [8]. It is our understanding that the proposed study shall result in a feasibility report for the next logical step on the route to a driver facility. Whether the goal to achieve ignition can be realized in one step or whether an intermediate step is necessary and useful, depends of the result of the study, especially as

224

R. Bock

the accelerator has a larger potential for the investigation of a number of ICF related issues. The facility to be built should be an optimum choice for addressing the various aspects of an ICF development plan. Obviously, it is the great advantage of the heavy ion approach to fusion that the accelerator technology of an ignition driver is identical to that envisaged for a final power reactor. Heavy ion beams, therefore, offer a direct route towards fusion energy production. 5 5.1

Inertial

Confinement

with

Light

Ion Beams

General Remarks

Since the stopping power for protons is smaller by about a factor of 103 as compared to the very heavy ions discussed before, their energy must be smaller by the same factor in order to meet the target requirements. Consequently, the reduced kinetic energy of the ions has to be compensated by an increase of beam current by the same factor. Therefore, light ion beam currents need to be as high as mega-amperes in order to reach the pulse energy of 6 MJ necessary for ignition. Beam intensities of this order of magnitude can not be handled with the accelerator technology described before. Concept and technology of the light ion driver is completely different from heavy ion accelerator techniques. Light ion beam devices for high currents consist of two main components: (1) The energy is provided by a pulsed-power device, an electrical capacitor array (Marx generator) with a pulse forming line, delivering a short electrical pulse to (2) a diode in which the proton or lithium beam of very high intensity is produced and focused (Fig. 11). This beam pulse is transported to and injected into the reactor chamber to the target. Recent research is concentrating on improvements of diode performance, in particular to ions heavier than protons and at higher voltage. As mentioned before, ongoing research is done using single-shot devices at the Sandia Laboratories in the USA, e.g. with the powerful Particle Beam Fusion Accelerator PBFA II [30], a multi-beam facility, and in Europe with the Light Ion Facility KALIF at the Karlsruhe Research Center [3]. Upgrades to more powerful installations are in progress at both laboratories. The most attractive features of this technology with respect to the inertial fusion application are their high e~ciency and low specific cost. The problems are located mainly in the diode performance, its beam quality and its repetition rate capability, and in the beam transport to the target. Repetitive operation of pulsed-power generators is supposed to be achievable with available technology. 5.2

Present Research: Achievements and Problem Areas

Without going into any technical details, some ideas about achievements and directions of present research shall be given.

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Considerable progress has been made in the understanding of diode physics, both by sophisticated diagnostics and by the application of 3-D simulation codes. The beam divergence is one of the key issues which need further investigations. The achieved value of 17 mrad needs to be reduced to about 10 for an ignition facility and to 5 for a reactor driver. The main sources of beam divergence were identified to be caused by inhomogeneities of the anode plasma and by instabilities of the free electron sheath at the virtual cathode. Two-stage diodes now under consideration may improve existing deficiencies. Great progress is also achieved with the diode voltage and the power density in the target. With a lithium current of 1 MA as high as 10 MV have been reached at Sandia. With these performance parameters a specific power deposition of 1000 T W / g has been achieved in a target, resulting in a

226

R. Bock

plasma temperature of 65 eV, to be compared to 300 eV needed for ignition. With voltage generators of the 1 TW class, such as KALIF in Karlsruhe, 200 T W / g have been reached, an order of magnitude which is already of interest for target investigations. One of the problem areas is the final beam transport. Several transport schemes are considered for a reactor concept. Most promising are ballistic transport combined with solenoidal focusing and self-pinched transport. In the first scheme a background gas provides charge and current neutralization, a disadvantage being the location of the solenoidal lens rather close to the target. No transport device is required for the self-pinched transport scheme. Both schemes need further theoretical and experimental investigation, in particular the self-pinched transport, where little experimental work has been done so far. In conclusion, the light ion approach has the advantage, that already now considerable power densities in targets have been achieved with facilities in operation. They open significant opportunities for investigations in target physics, related to ICF as well as to basic research (Fig. 12). The field of pulsed-power technology is well advanced. The area of diode performance needs a continued research effort, with respect to the required intense lowdivergence beams and the repetition rate capability. Beam transport in the target chamber is another key issue which needs increased investigation.

6

Concluding

Remarks

and Outlook

The investigation of many issues of ion beam fusion, both with light and heavy ion beams is well in progress at several European laboratories. Funding in Europe, however, is by far not sufficient for a balanced program which addresses the key issues of ICF adequately. Light ion activities at the Karlsruhe Research Center have achieved remarkable results with respect to both a conceptual design study for a light ion beam reactor facility and experimental investigations on the application of pulsed-power technique to inertial fusion and to problems of materials research. Collaborations with US laboratories have achieved design studies on light ion beam reactor concepts which have some attractive features, but obviously some essential problems of such concepts are still far from being solved. A collaboration with Sandia Laboratories opens access to larger pulsed-power facilities dedicated to inertial confinement research. In the field of h e a v y ion b e a m s several European accelerator facilities open opportunities for the investigation of problems relevant to heavy ion inertial fusion. In particular, GSI with its intensity upgrading program will after its completion in 1999 - enable an interesting research program on specific driver related issues of beam handling and beam dynamics as well as on problems of plasma physics (Fig. 12). -


227

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On the route to the physics of dense plasmas, heavy ion beams are excellent tools for the generation of volume-heated plasmas at solid state density and above, and they provide a new technique for the investigation of properties of m a t t e r under extreme conditions (and are in some way complementary to those with high-power lasers): beam target interaction, processes in nonideal plasmas, the equation-of-state and hydrodynamics of dense plasmas, probably the discovery of phase transitions. After the injector upgrade at GSI, plasma temperatures of up to 10 eV are expected. A further reasonable step to higher energy density in the plasma with a new facility would be an increase of t e m p e r a t u r e by a factor of 10. At about 100 eV a new regime of phenomena would become accessible: radiation physics - conversion and transport phenomena, opacities - with its relevance to astrophysics and to

ICE" targets. On the route to energy, the European Study Group has started a new effort for a systematic approach to a driver facility and to the relevant problems of targets and systems. The concept of an ignition facility or, alternatively, a staged procedure to achieve this goal will be worked out. The heavy ion accelerator presently investigated by the Study Group is - specific to European accelerator expertise - based on an experienced scientific c o m m u n i t y and - based on a well developed European collaboration. Depending on the results of the study a coherent European research program could be established with a reasonable effort which would allow to place ICF adequately into the context of an international fusion energy strategy.

228

R. Bock

References [1] R.Bock, Status and Perspectives of Heavy Ion Inertial Fusion, Proceedings of the Internat. School of Physics 'Enrico Fermi', Varenna 1990, Course CXVI, p.425-447, North Holland, Amsterdam 1992; C.Rubbia, Proc. of the IAEA Techn.Comm.Meeting on Drivers ]or Inertial Confinement Fusion, (D.Banner and S.Nakai, Eds.), Osaka 1991, p.23; Nucl.Physics A553 (1992) 375-395; Nuovo Cim. 106A (1993) 1429-44; I.Hofmann in Advances of Accelerator Physics and Technology (H.Schopper,Ed.) World Scient.Publ.Co., Singapore 1995, p.348-61; R.Bock, Proc. of the EPS Conference on Large Facilities in Physics, Lausanne 1994 (M.Jacob and H.Schopper, Eds.) World Scient. Publ.Co., Singapore 1995, p.348-61 [2] R.O.Bangerter and RM.Bock in Energy from Inertial Fusion, IAEA, Wien 1995 p.111-135 [3] H.Bluhm and G.Kessler in Physics o] Intense Light Ion Beams, Annual Report 1995 FZKA 5840 (H.Bluhm, Ed.) [4] S.Atzeni, Proc.of the Internat.Symposium on Heavy Ion Inertial Fusion, Frascati 1993, (S.Atzeni and R.A.Ricci,Eds.) Nuovo Cim. 106A (1993) 1429-1995; S.Atzeni in Physics with Multiply Charged Ions, (D.Liesen,Ed.) Plenum Press, New York 1995, p.319-356; Fus.Eng.Design 32/33 (1996) 61-71 [5] J.Meyer-ter-Vehn, The Physics of Inertial Fusion, Proc. of the Internat. School of Physics 'Enrico Fermi', Vareima 1990, Course CXVI, North Holland, Amsterdam 1992, p.395-423; M.Murakami and J.Meyer-terVehn, Nucl.Fusion 31(1991)1315 and 1333; J.Meyer-ter-Vehn, J.Ramirez and R.Ramis, Proc. of the Symp. on Heavy Ion Inertial Fusion, Princeton 1995, Fus.Eng.Design 32/33 (1996) 585; A.M.Oparin, S.I.Anisimov and J.Meyerter-Vehn, Nucl.Fusion 36 (1996) 443-452 [6] K.J.Lutz, J.A.Maruhn, R.C.Arnold, Nuc]. Fusion 32 (1992) 1609; K.H.Kang, K.J.Lutz, N.A.Tahir and J.A.Maruhn, Nucl.Fusion 33 (1993) 17 [7] J.M.Martinez-Val et al., Nuovo Cim.106A (1993) 1873; G.Velarde et al., Particle Accel.37 (1992) 537-542 [8] Yu.A.Romanov, Nuovo Cim 106A (1993) 1913; M.Basko, Nucl.Fusion 33 (1993) 615 and Phys.Plasmas 3 (1996) 4148; Yu.A.Romanov and V.V.Vatulin, Fus.Eng.Design 32/33 (1996) 87-91; V.V.Vatulin et at., Fus.Eng.Design 32/33 (1996) 603 and 609; M.M.Basko, M.D.Churazov and D.G.Koshkarev, Fus.Eng.Design 32/33 (1996) 73-85 [9] D.H.H.Hoffmann et al., Phys.Rev.Letters 65 (1990) 2007; 66 (1991) 1705; 69 (1992) 3623; 74 (1995) 1550; M.Stetter et al., Fus.Eng.Design 32/33 (1996) 503; M.Dornik et al., Fus.Eng.Design 32/33 (1996) 511; R.Bock (Ed.), Annual Reports High Energy Density in Matter produced by Heavy Ion beams 1985-95, GSI Reports, in particular GSI-95-06 and GS1-96-02 [10] R.Sigel et at., Phys.Rev.Letters 65 (1990) 587; T.Loewer et al. Phys.Rev.Letters 72 (1994) 3186 [11] D.Batani, M.Koenig, T.Loewer, A.Benuzzi and S.Bossi, Europhysics News 27 (1996) 210 [12] C.Yamanaka, Proc. of the IAEA Techn.Comm.Meeting on Drivers ]or Inertial Confinement Fusion (D.Banner and S.Nakai,Eds.), Osaka 1992, p.1


229

[13] M.Tabak et al., Phys.Plasmas 1 (1994) 1626-34; A.Caruso, Proc.of the IAEA Tech.Comm.Meeting on Drivers for Inertial Confinement Fusion, (J.Coutant, Ed.), Limeil-Valenton 1995, p.325-39 [14] A.Pukhov and J.Meyer-ter-Vehn, Phys.Rev.Letters 76 (1996) 3975 [15] J.D.Lindl, Nuovo Cim. 106A (1993) 1467-34 [16] B.Badger et al., HIBALL, A Conceptual Heavy Ion Beam Driven Fusion Reactor Study, KfK 3202/VWFDM-450 (1981); D.Boehne et al., Nucl.Eng.Design 72 (1982) 195 [17] HIBALL II, An Improved Conceptual Heavy Ion Beam Driven Fusion Reactor Study, KfK-3840/FPA 84-4/VWFDM-625 (1985) [18] W.R.Meier et al., OSIRIS and SOMBRERO Inertial Confinement Fusion Power Plant Designs, WJSA-92-01 and DOE/ER[54100-1 (2 Vols.), 1992 [19] L.M.Waganer et al., Prometheus L and Prometheus H Inertial Fusion Energy Reactor Design Studies, DOE/ER-54101 and MDC 92E0008 (3 Vols), 1992 [20] G.L.Kulcinski et al., Proceedings of the IAEA Tech.Comm.Meeting on Drivers for Inertial Confinement Fusion, (J.Coutant,Ed.), Paris 1994, p.49-56 [21] R.O.Bangerter, Nuovo Cim. 106A (1993) 1445; R.O.Bangerter, Fus.Eng.Design 32/33 (1996) 27-32 [22] A.Friedman et al., Fus.Eng.Design 32/33 (1996) 235-46; J.J.Barnard et al., Fus.Eng.Design 32/33 (1996) 247-58 [23] I.Hofmarm, Proc.of the IAEA Tech.Comm.Meeting on Drivers for Inertial Confinement Fusion Osaka 1992 [24] I.Hofmann, Nuovo Cim.1O6A (1993) 1457; Fus.Eng.Design 32/33 (1996) 33; Proc. 5th Europ.Part.Accel.Conf. (EPAC'96),Sitges 1996, p.255; I.Hofmann et al. Proc. of the 15th Internat.Conf on Plasma Physics and Controlled Fusion (Seville 1994), IAEA Vienna 1995, Vo.2, p.709-14 [25] U.Oeftiger and I.Hofmann, Fus.Eng.Design 32/33 (1996) 365-70; U.Oeftiger, I.Hofmann and P.Moritz, Proc.of the 5th European Particle Accel.Conf, (EPAC'96) p.1099 [26] B.Franzke, Proc.of the 3rd European Particle Accel.Conf. (EPAC'92) Berlin 1992 p.367-71; K.Blasche und B.Franzke, Proc.of the 4th European Particle Accel.Conf. (EPAC'94) London 1994; N.Angert, Status and Development of the GSI Accelerator Facility Proc.of the 5th European Particle Accel.Conf. (EPAC'96), Sitges 1996, p.125 [27] J.S.Hangst et al. Phys.Rev.Letters 74 (1995) 4432; D.Habs and R.Grimm, Annual Rev.Nucl.Part.Sci. 45 (1995) 391 [28] C.Rubbia, Nuclear Inst. Meth. A273 (1989) 253-265; C.Rubbia, Proc.of the 3rd European Particle Accel.Conf. (EPAC'92) Berlin 1992 p.35 [29] A Study Group 'Heavy Ion Ignition Facility' GSI-Report 95-03 (1995) Proposal to the Commission of the European Union 1994 (G.Plass, Project Manager) [30] A.B.Filuk et aJ., Proc. of the IAEA Tech.Comm.Meeting on Drivers for Inertial Confinement Fusion (J.Coutant, Ed.), Limeil-Valenton 1995, p.233

Spin-echo E x p e r i m e n t s with N e u t r o n s and with Atomic Beams Christian Schmidt and Dirk Dubbers Physikalisches Institut der Universit~it Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany

1

Introduction

Spin-echo spectroscopy was invented in the fifties [1] as a new tool in nuclear magnetic resonance (NMR). Over the years, NMR spin-echo has steadily evolved towards the powerful multiple pulse, multiple frequency, and magnetic gradient NMR-techniques now in standard use in physics and chemistry to probe the various couplings of the resonating nuclei to their atomic neighbours in real space. "In-flight" spin-echo was developed as a high-resolution method in polarized neutron scattering during the seventies [2]. It turned out to be a powerful tool for the investigation of the entangled molecular movements within complex materials. Although conceptually similar to NMR spin-echo, the information content of neutron spin-echo (NSE) is quite different. In-flight spin-echo, too, has much evolved in recent years. The technical aspects of this evolution is well covered by the existing literature. Still, the field is not easily accessible, due to its mathematical intricacies. The present survey tries to fill this gap by discussing the various ramifications of in-flight spin-echo, using only a small number of physical arguments. Finally, an astonishingly simple quantitative description of the spin-echo technique as a magnetic birefringence phenomenon will be presented. In many scattering experiments in physics, the energy and momentum transfer onto the sample is measured via the change in energy and momentum of the scattered beam. In inelastic neutron scattering, the energy state selection of the beam before and after scattering is usually done by Bragg reflection in three-axes spectrometers, or by time-of-flight measurements in pulsed neutron machines. The momentum state selection is accomplished by the choice of the scattering angle. All these scattering methods have in common that, due to the necessity of state selection, high resolution and high intensities are mutually exclusive. What is learned from a conventional neutron scattering experiment? The most important information on the system under study is given by the correlation function G(r, t), which gives the probability that a particle is found at position r at time t when there was a particle at position r = 0 at time t = 0. This correlation function covers all cases of interest, from the structure of ordered crystals or disordered liquids to the study of diffusion and of phonon and other

232

Christian Schmidt and Dirk Dubbers

excitations. (The description via G(r, t) however, is not complete when higher order correlations which depend on the position of more than two particles come into play.) The overall result of neutron scattering theory is that, for a given energy transfer hw and momentum transfer hq onto the sample, the intensity S of the scattered neutrons is the Fourier transform of the correlation function S(q,w) =

2

space-time Fourier transform of G(r, t).

(1)

Classical Neutron Spin-Echo (NSE)

In-beam spin-echo has the specific feature that it allows measurements with very high resolution but with no penalty in intensity. In its simplest possible configuration (which is only rarely realized), spin-echo works as following: Neutrons, for example, in a spin-polarized beam are scattered from a sample into a given direction and are detected after spin analysis. Along the incoming beam, a magnetic field of typically up to B0 = 50 m T is applied, over a typical beam length of 2 m. In this field, transversally polarized cold neutrons of average velocity 500 ms -1 make about 6000 revolutions with Larmor precession frequency WL. As the neutrons travel with various velocities, their transverse polarization is lost after a few mm of flight within the magnetic field. Along the outgoing beam, a magnetic field of equal size, but pointing into a direction opposite to the direction of the first field, is applied. In this second field, the neutron spins will turn back by full - 6 0 0 0 revolutions; that is, at the exit of this second field the neutron polarization is fully recovered, giving what is called a spin-echo signal. The full spin-echo signal, however, is only recovered if the neutron velocity after scattering is the same as before scattering. If, after scattering, the velocity decreases by as little as 10 -5, then the polarization of the outgoing beam is off by an angle of ¢ ~ 20 °, which is easily detectable in a polarization analyser. This explains the extreme sensitivity of spin-echo spectrometers to changes in neutron energy as small as a few neV. In comparison, time-of-flight or three-axes neutron scattering instruments have a resolution of only 0.1 to 1 meV. As all neutrons contribute to the spin-echo signal, independent of their velocity, one does not have to pay a penalty in intensity due to excessive state selection. What is the connection between this spin-echo signal and the observables of condensed matter? During scattering on the sample, the neutrons create or absorb an internal excitation of energy hco, and therefore leave the sample with a higher or a lower energy ("up-scattering" or "down-scattering"). The relation between the physical quantity ~ and the response of the apparatus then is simply: ¢ = wT.

(2)

Here, r is an instrumental quantity of dimension time, called the spin-echo time, given by Ernagn w -- E tTOF C( Bo, (3)

Spin-echo Experiments with Neutrons and with Atomic Beams

233

where tTOF is the time of flight of the neutron through the first arm of the 1 instrument (in our example: ~ 4 ms). Here Emagn _- ghWL = 60neV.B0(Tesla) is the neutron magnetic energy in the magnetic field B0, and E is the initial kinetic energy of the neutron, in our example about 1 meV. Hence, for cold neutrons, a typical spin-echo time is ~- = 20 ns. Relation (3) is derived for instance in [3]. An alternative and physically more appealing derivation is given in section 5 of the present article. In order that the relation ¢ = w~- be applicable, the width of the velocity distribution should not exceed about 20 %. In the following, the connection between the spin-echo signal and the physics of the sample under study will be given only for the simplest case of quasi-elastic neutron scattering, where the energy transfer hoJ is small compared to the quasiparticle excitation energies of the sample. In quasi-elastic scattering there is an equal number of up-scattered and of down-scattered neutrons, and therefore no overall shift of the spin-echo signal is expected. Instead, the size of the signal will decrease. When this decrease is measured as a function of spin-echo time 7-, (i.e. of magnetic field B0), and of scattering angle 0, then one obtains [3]: Spin-echo signal = transversal polarization P(~, B0) = "intermediate scattering function" I(q, T) = spatial Fourier transform of G(r, T) = frequency Fourier transform of

S(q, w).

(4)

The time dependence of the correlation function G of the system under study can thus directly be measured by simply varying the size of the magnetic field along the neutron beam. In contrast to the usual scattering methods, neutron spin-echo does not require a Fourier transform to obtain this time dependence. Instead, the spin-echo instrument performs the Fourier transform itself. Further, the method does not require excessive state-selection. In contrast to NMR spin-echo, in-beam spinecho provides no local probe in real space, but operates in momentum space of the reciprocal lattice. Therefore, by measuring the time dependence of the correlation function for various momentum transfers q, one is able to study the time response of the system separately on different scales of length. An example of a NSE measurement [4] is given in figure 2. The measurement shows the time dependence of the correlation function of a polyethylen-propylen copolymer. According to the reptation model of de Gennes, each chain moves in a "tube" defined by the positions of the neighbouring chains. At large momentum transfer (the lower curves in both plots), one observes the movement on a small scale, which is a fast lateral movement limited by the tube walls. At small momentum transfer (the upper curves of both plots) one sees the movement on a larger scale, which is a slow reptation movement along the tubes. Further, at higher temperatures one finds a larger effective diameter of the tubes. The solid lines in the plot are derived from a model calculation. A real NSE spectrometer is somewhat more complicated than described above. Usually, in both arms of the spectrometer, the magnetic fields point into the same direction in order to

234


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avoid a zero field region on the neutron trajectory in between. Then, to obtain the neutron spin-echo signal, the same tricks have to be applied as in N M R spin-echo. Before the neutron enters the large B0 field of the first arm of the spectrometer, its spin is turned by ~ in a small static magnetic field B1, which is applied at right angles to B0 over a short distance of the beam. Near the sample, another B1 field over twice the length (or twice as strong) as the first BI field "refocusses" the spins via a ~r-flip, much as in N M R spin-echo. At the end of the flight path, another ~ -flip is applied to the neutrons before they enter the analyser. Over the years, the NSE method has seen many sophisticated extensions [3]. Spin-echo methods have been developed for truly inelastic, incoherent, paramagnetic, ferromagnetic, and antiferromagnetic neutron scattering.

3

Zero-Field Neutron Spin-Echo (NRSE)

Several years ago an interesting proposal appeared [5] which claimed t h a t in a spin-echo experiment no B0 fields at all are necessary along the neutron beam, if the various static ~ - and ~r-flip coils are replaced by N M R flip coils. In [5], this claim was proved mathematically, and the proof will not be repeated here. Instead, two different explanations will be given which provide a better physical


235

understanding of the phenomenon. The first explanation is for the reader familiar with NMR, the second explanation is intended for the non-specialist. The principle of this neutron resonance or "zero-field" spin-echo (NRSE) can be derived directly from conventional NSE by going into the frame, attached to the neutron, which rotates at Larmor frequency CJL about the strong B0 field. In this rotating frame, the B0 field is simply transformed away and reappears, with opposite sign, at the position of the static ~- and 7r- flip coils. In the new frame the fields B1 in the flip coils wilt rotate with --COL,that is, the static flip coils have transformed into ordinary NMR flip coils. The spin-echo signal does not change under this transformation. Hence, NRSE is equivalent to NSE. This is the shortes possible proof, but it does not really enhance the basic understanding of the zerofield spin-echo trick. Therefore we give another, more simplistic explanation. NSE, basically, is a time-of-flight method. In everyday life, there are two methods to measure time of flight. One method is to watch one's wrist watch while flying from one place to another, and to read the time difference. This is the method of conventional NSE, where the neutron measures time via its personal Larmor precession clock. Another method is to look at the stationary airport clock at take-off, to remember this time of departure, and to look again at the airport clock upon arrival. One trusts that both clocks are in phase, and calculates the time difference. This is the method of NRSE: After the ~-flip about the momentary direction of the rotating B1 field, the neutron spin displays the time of departure on a dial which lies in the plane of B1. During its flight through the zero-field region, the neutron remembers this time of departure via the fixed direction of its spin. The B1 field of the second flip-coil rotates in phase with the field of the first flip-coil. Via the 7r-flip about the momentary direction of B1 in the second coil, the neutron records the time of arrival and calculates the time difference. Therefore, the neutron spin indicates the time of flight through the first spectrometer arm, in the same way as it does in conventional NSE. In the second arm of the spectrometer the polarization is then refocussed in the usual way. In this context, a word on the celebrated Ramsey method is in order. While, in conventional spin-echo, timing is done by the moving particle's clock, in resonance spin-echo timing is done via the stationary radio-frequency clock. In the Ramsey method, both kinds of time taking are combined, and the moving particle's private clock is compared with the stationary clock of the radio-frequency generator. In an atomic clock, this comparison is done to stabilize the radiofrequency generator with the atomic signal. When applied to basic physics, the comparison is done to search for small anomalies in the particle's clock, as is done for instance in the search for an electric dipole moment of the neutron. For NRSE, too, time has given birth to several extensions of the original method, see [6], and references therein. For instance, it was shown that when not three but four NMR flip coils are used, two 7r-flip coils in the first arm and two in the second arm, then the resolution of the instrument can be doubled for the same size of B0 in the NMR coils. When one uses not four single but four pairs of NMR coils and applies what is called the boot-strap trick [7], resolution

236


can be doubled again. Still, a legitimate question is: If NRSE, in principle, is equivalent to NSE, why take the pain and invest in this new method? There are several distinct advantages to NRSE. Firstly, no large volume B0 fields are necessary. Instead, one only needs simple mu-metal tubes to shield the zero-field region. (On the other hand, it must be reminded that the small NMR coils used in NRSE required a considerable amount of development work.) Another advantage of NRSE is that the system is very insensitive to external perturbations coming, for instance, from moving steel constructions like cranes in an experimental hall. Further, in NRSE, with no Bo field along the main neutron path, there are no problems due to field inhomogeneities, which, in NSE, require fine tuning of numerous correction coils. The geometrical path length variations due to beam divergence, however, remain the same. Conventional NSE also is plagued by the fringe fields of the large coils, which must extend out to the sample position. When the angle between the ingoing and outgoing arms of the spectrometers is changed, the resulting fringe fields change as well and must be newly corrected. The main advantage of zero field spin-echo, however, is that the use of multidetectors for spin-echo work is easily possible. If there is no magnetic field then there is no preferred direction, and the spin-echo trick can be applied to all outgoing neutrons simultaneously, independent of their scattering angle. With conventional spin-echo this is not easily possible, as the field in the second arm usually singles out one scattering direction. While there exist several schemes to introduce large solid angle detection also in NSE, none has been successful yet. Another point in favour of NRSE may appear rather technical, but has important implications. In truly inelastic scattering, spin-echo can be used to measure the lifetimes of the excitations under study. However, a dispersion curve ca(q) usually has a non-zero slope, and a mediocre q-resolution will spoil the very high resolution in energy ha;. It has been shown [3] that when the direction of the B0 field is tilted away from the beam axis by a certain angle, then the dispersion curve can be crossed under right angles, and ca resolution decouples from q resolution. In NSE, to tilt the large volume B0 field is rather akward. In NRSE, to tilt the small NMR coils, on the other hand, is no problem at all. Tilted fields in NRSE can also profitably be used to do small angle elastic scattering [8]. Again, this method combines very high resolution, this time in momentum transfer q, with high overall intensities, as all incoming angles of a divergent beam can be used. This means that the spin-echo trick can be applied also to the momentum variables (instead of the energy variables), and then gives a direct measure of the spatial correlation function G(r). The first prototype of a zero-field neutron spin-echo instrument was developed in our group [9]. In this work we were able to draw on our experience with polarized neutrons in various static, rf, and zero-magnetic field configurations, used in experiments on Berry phases, dressed neutrons, neutron-antineutron oscillations and others, see also the review [10]. At present, two larger NRSE instruments have been developed and are being installed. One instrument was developed in Garehing [11] and is installed at the Orph6e reactor in Paris. The


237

other instrument was developed in Heidelberg and Garching [12] and is temporarily being installed at ILL, Grenoble. The two instruments were built for different purposes, each with its specific technical problems. In the GarchingParis installation, the main emphasis is to build a high-resolution instrument with tiltable coils, but only applicable to monodetector use. In the HeidelbergILL installation, the main emphasis is on a multideteetor system with a large angle of acceptance for each individual coil. As a first application, simple diffusion problems will be measured with our instrument, but later on the main interest is the study of phase transitions, see also next section.

4

Spin-Echo with Atomic Beams

In neutron spin-echo the bulk of a crystal is investigated as a whole, and the method is hardly applicable to surface problems. For surfaces, on the other hand, there exist many sophisticated methods to determine structures (tunnel microscopy, electron diffraction, and others), and also some methods for time dependent studies (mainly laser spectroscopy). What is really wanted, however, is a high resolution method for the combined study of time and space dependent processes on surfaces, in order to obtain the full correlation function G(r, t). It turns out that helium atomic beams can be for surface studies what neutrons are for bulk studies. Like neutrons, slow helium atoms have de Broglie wavelengths of atomic size, and, at the same time, kinetic energies comparable to the typical excitation energies of condensed matter. In fact, both helium diffraction and helium inelastic scattering from surfaces are well established fields now. Inelastic helium scattering [13] uses a chopped supersonic helium beam for time-of-flight measurements. Its energy resolution is limited to roughly 0.1 meV. If one wants to do spin-echo with a helium atomic beam, then one must use 3He, which has a nuclear spin one half. 3He spin-echo would permit an energy resolution orders of magnitude better than the energy resolution obtained with time-offlight methods. Recently, such a 3He spin-echo machine was developed in our group [14]. In this instrument, shown in figure 4, the simplest version of spin-echo is used, i.e. two magnetic fields of opposite sign are applied. A simple trick was used to prove the neV resolution capabilities: The whole apparatus was ramped by an angle of six degrees towards the horizontal, so that the 3He beam had to fly upwards in the earth's gravitational field. Hence, in the second coil, the 3He atoms had lost a kinetic energy of 33 neV, as compared to their energy in the first coil. This energy loss could easily be resolved with the apparatus, see figure 4. One main problem of a 3He spin-echo instrument, as compared to a neutron spinecho instrument, is the requirement that no obstacles like the ~ - or zr-flip coils, customary in neutron spin-echo, are allowed in the atomic beam. This problem was solved by installing zero magnetic field regions between the various regions of high magnetic field. In this way, one can change the direction of the magnetic fields from one place to the other non-adiabatically with respect to the spin

238

Christian Schmidt and Dirk Dubbers scattenng chamber

i

-.

:

~-±

----9

vallable scattering angle

B1

0- 90= spin-echo coil 1

\

\9 ~ / ]

B2

Stem-Gerlach-

3 He supersonm

quadrupole

beam source

polarizer

at T-1.3 K

spin-echo coil 2

Stern-Gedachhexpole analyzer 2350 mm

e-- bombardment detector

Fig. 2. The Heidelberg 3He atomic beam spin-echo apparatus for the study of slow motions on surfaces.

direction of 3He. Another problem was to polarize and to analyze the 3He beam efficiently. To this end, the 3He beam was cooled to a temperature near 1 Kelvin and polarized in a conventional magnetic quadrupole field. An alternative would be to use the transportable high pressure 3He source developed by Hell and Otten, see also the contribution by E.W. Otten in this volume. The new atomic beam spin-echo instrument is now being adapted for surface studies. The instrument is so sensitive that even the residual magnetic fields of antimagnetic inox steel are inadmissibly high. Therefore, titanium was used for the scattering chamber and its accessories. In spin-echo, in general, high energy resolution really means sensitivity to slow motion. In 3He spin-echo, the movement for instance of large molecules or of their aggregates on a surface, or of biological membranes and of their adsorbates can be studied, cases which are of interest to the organic chemist or the biologist. For the physicist, universality in second order phase transitions is a field of high general interest. Within a given universality class, critical behaviour seems to be the same for all systems. For a given symmetry of the order parameter (scalar, vector, etc.), one universality class is distinguished from the other only by the spatial dimension of the system under study. Therefore, also for two spatial dimensions the dynamics of structural phase transitions should be tested as well as possible. But the experimentalist faces the problem that universality holds only asymptotically close to the critical temperature, where the system's movements are very slow ("critical slowing down"). Therefore, to test universality one must be able to study very slow motions. Hence, one main aim of atomic beam spin-echo will be the study of the dynamics of second order phase transitions in two dimensions.

Spin-echo E x p e r i m e n t s w i t h N e u t r o n s a n d w i t h A t o m i c B e a m s

-40

~---7

1.0

-30

-20

-10

0

10

20

30

239

40

Polarization Product

Max.

0.90 _* o.o,

0.6 0.4 i.•

cO .DN

!i I.,.

0.(1 0.2 ~ ~ / i i i -0.2

O

-0.4 ~-0.6 r-0.8 --

- - ! . . . . 4.... J. . . . ......-: . . . .

-1.0

-40

-30

-20

-' -10

........ x . . . . . . . .

:

0

.

10

.

.

.

.

.

.

.

.

.

.

.

20

30

40

20

30

40

0.4 -[

J'Bldl = 9.36 mr.m

.

,~

0.2 I

tO =

o.o-1

N (~

• ~ ~T 1'

~" -O.2 -! I -0.4

t!,

L -i i

-40

-30

-20

-10

0

"10

Magnetic Field Integral Detuning [~T-m]

F i g . 3. aHe a t o m i c b e a m spin-echo curves t a k e n w i t h t h e a p p a r a t u r s s h o w n in figure 2, for a s t r a i g h t b e a m , r a m p e d by an angle of six degrees. T h e lower curve is s h i f t e d to t h e right due to t h e 33 neV energy loss of t h e 3He a t o m s in t h e e a r t h ' s g r a v i t a t i o n a l field.

240


The first physics problem to be studied with the new 3He spin-echo apparatus is the dynamics of a "lattice gas", consisting of flat circular molecules moving on a smooth surface, like a puck on the ice. Further variants of this atomic beam spin-echo technique are possible and are under study. 5

Spin-Echo

In-Beam

as a Birefringence

Phenomenon

As has been pointed out earlier [15, 16], Larmor precession in flight can be regarded as a birefringence phenomenon. In the following, we want to show that, based on this picture, the spin-echo time (3) can be understood as the time difference accumulated by two states that show different group velocities in a magnetic potential on the way to the investigated probe. Because these partial states are thus scattered at different times, the spin-echo signal is naturally found to measure correlations on this time scale [17]. When a beam of spin ~1 particles of mass m and gyromagnetic ratio 7, travelling along the axis z, traverses a region with a stationary magnetic field B(z), then, in the laboratory frame, the Hamiltonian is p2 1 -B(z). H = ~mm - 2 7h a

(5)

When B(z) is sufficiently smooth so that particle reflections on the magnetic potential can be neglected then the WKB approximation gives the two spin states

¢±:exp(-hEt)

exp(h~oZp±(()d~) ,

(6)

where E=p~/2m is the total energy, P0 is the particle's momentum outside the field region, and p±(() = (p2o =kmh'ylB(>F gives " ~,(co)

I -

2 COY

~

CO

,

~, (co) =

60

2

- - 7P r

(3)

CO

For this case the resonance position and shape can be easily determined by inserting (3) into (2). In the vicinity of the resonance the Iineshape is then described by a Lorentzian. In other metals a substantial amount of interband transitions from lower lying bands into the conduction band or from the conduction band into higher unoccupied levels is possible, which alters the simple form of (3) [16]. For the alkali metals the interbaad threshold given by excitations of conduction band electrons to higher levels lies at about 0.64 EFermi, however, their contributions to the dielectric function are small. In contrast, the relevant interband transitions

Quasi-static case:

Z>>2R

E(t=to)

Homogeneouspolarization: dipole excitation

General

case:k ~ 2R

E(t=t0)

Phase shifts in the particles: multipole excitation

Fig, 2. The interaction of light with dusters can be described in a simple way in the quasi static regime (~. >> 2R). In the general case phase shifts of the electromagnetic wave in the particles complicate the optical response.

316

for the noble metals are due to excitation of d-band electrons into the conduction band. They give a positive contribution de 1 to e l, Consequently the dipole resonance frequency given by c I = - 2 em is shifted to lower frequencies, in the case of some noble metals beyond the low frequency interband transition edge. For other metals these effects complicate the optical spectra appreciably and in fact, only a few materials like the alkali and the noble metals as well as aluminum exhibit sharp resonances. With (I, 2) the optical response of metal spheres to incident electromagnetic waves can be calculated. As input parameters the Mie theory uses the phenomenologically introduced dielectric function e (c0) for the dusters. It should be stressed that the Mie theory gives no insight whatsoever to the microscopic excitation mechanisms in the particle material. These are exclusively contained in the applied e((o). Any strong deviations from free electron behavior make it essentially impossible or at least very difficult to derive e(to) from a microscopic theory. Fortunately a huge amount of experimental optical material functions for bulk solids which incorporate all electronic effects is available (e.g. [17]). Figure 3 shows an example of the absorption cross section of single sized sodium dusters as a function of wavelength for fixed radii between 20 um and 100 urn. Input parameters were size independent dielectric functions e (o)) (from [18]). For small duster sizes, i.e. within the quasistatic regime, only a single resonance, the dipolar plasmon polariton is visible. With increasing size, higher order modes come into play. Simultaneously, a red shift of the dipolar mode is observed due to retardation effects (for more details, see e.g. [1, 19]). In comparison, Fig. 4 shows size dependent dielectric functions and the resulting optical absorption spectra within the quasistatic regime. Obviously, the most dramatic change occurs for the damping, since e2(~0) increases for decreasing size. This so called limited mean free path effect accounts for the fact that the duster

t

~"

i~

.

n=2O,,m

31- " ~ V y \ / its, ~, /', " ~o I- L J'-" , ".

FJ/ I!i,:

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

R. ~ R-~OOm

,. kx.,,, 400

500

600

700

WavetenOt~ (nml

Fig. 3. Absorption cross section as function of wavelength for monodisperse free Na dusters in vacuum with mean sizes R = 20, 40, 60, 80, and 100 urn.

317 boundary will impose an additional scattering, i.e. damping mechanism for the electrons if the mean free path of electrons in the cluster exceeds the cluster size. Concerning the optical properties extrinsic (Fig. 3) and intrinsic (Fig. 4) size effects can now be readily attributed to different parts of the theoretical description: For clusters larger than about 10 nm diameter, the optical material functions e(c0) are size independent, having the values of bulk material. The change of the spectra with size is dominated by retardation effects of the electric field across the dimension of the particle which can cause huge shifts and broadening of the resonances (extrinsic size effects). For smaller clusters the optical material functions do no longer have the values of bulk material, but vary as a function of particle size. This is an intrinsic cluster size effect, as the material properties give rise to a change of the optical response. Still, the Mie theory result (Eq. 2) can be a good description, if a proper dielectric function is used. In summary; the position of the dipolar surface-plasmon resonances of small spherical metal clusters is defined by the condition e 1(~0) = - 2 e m which translates into el(to)=- 2 for spheres in vacuum. Larger particles suffer a peak shift due to phase retardation of the electromagnetic waves and the influence of higher multipoles. For clusters in matrices the dielectric surrounding leads to additional shifts usually towards the red - with regard to clusters in beams. The Mie theory gives a constant width for the dipole resonance, yet, additional damping effects show up, described for metallic clusters by the limited mean free path effect. For larger clusters, damping due to retardation causes broadening with increasing size.

,

-2 I

ElO.)

,

,

,

,

,

,

=2R

2.5 2R=25 2.0

-4

2R=3.1nm 1.5

-6

1.0 0.5

-8

I 380

I 400

I 420

I 440 wavelengthInto]

bulk I l_ I I 380 400 420 440 wavelength[rim]

360

400

440

wavelength

480 [nm]

Fig. 4. Size dependence of dielectric functions e 1(~.)and e2(L) for silver clusters computed from bulk optical constants by including the limited mean free path effect (after [20]). The resulting Mie absorption spectra clearly illustrate the broadening for decreasing size.

318

Extension of Mie Theory: Other cluster shapes. Mie's theory was developed in 1908. Only a few extensions of the theory were performed latex They can be classified as dealing with other geometrical particle shapes, core-shell particles, shape dependent substrate effects, diffuse electron density boundaries, non local optical effects, and duster-cluster interactions in duster matter (for more details see [1]). At this point only nonspherical cluster shape effects on the spectra will be briefly discussed. For shapes differing from spheres, dectrodynamic calculations are much more tedious and closed-form expressions for the cross sections are available mostly in special cases like the quasi-static approximation. For this case ellipsoids, cylinders, cubes and other geometries have been treated [1]. For larger sizes, numerical results were obtained. All of the general new features are already present within the quasi-static approximation. As an example, Fig. 5 depicts the result for ellipsoids with three different axes (after 14]).

::3

e6

f,

" AI-Sphere ( 2R < 10 am ) =t

¢0

0 tO

E~

2

A

Ai

0u~ JD

d-electrons --+ electronic excitations of the lattice (1) The energy loss from the charged particle takes place within a time scale of ~ 10-17s (primary atom ionization) to ,-~ 10-14s (thermalization of delectrons) which is much shorter than the relaxation time of lattice excitations and the non-elastic defect creation time. The energy transfer from heavy ions to the lattice is described by the TRIM code [50], and for the lateral energy distribution the models of Katz and Waligorski are used [51,52]. The magnitude of linear energy loss (dE/dx) of various charged particles (electrons, protons, heavy ions) in solids varies on a large scale from 10 -6 e V / ~ (electrons of ~ 1MeV) to 103 e V / ~ (heavy ions of 10 MeV/u) [53]. In such large scale of dE/dx (over 9 orders of magnitude!), the radiation damage creation mechanism can be different at various excitation levels of dE/dx [5,49,50]. Nevertheless, for alkali halides, as demonstrated by Perez et al. [53] and Balanzat et al. [54], the defect creation (at T ~ 15K and at room temperature) and the exciton luminescence (at T ~ 15K) are similar both by irradiation with X-rays and heavy ions. This means that the exciton induced luminescence and defect creation dominate within an extremely large scale of excitation energy density, and the elementary defect creation mechanism by the decay of self-trapped excitons in alkali halides is independent of the type of radiation. Nevertheless, at high magnitude of dE/dx, collective electronic excitations (i.e., superposition of elementary excitations) can modify the exciton defect creation mechanism I34,35]. Various excited electronic states, induced by either charged particles or optical excitation in the fundamental absorption band, rapidly relax to the lowest electronic excitations of the lattice with the energy close to the band gap Eg (self-trapped excitons (e°~),free electrons and holes (e and h)). Only these lowest electronic excitations of the lattice can produce Frenkel defects [5,6,33,36]. The relaxation of high electronic excitations of the lattice is a complicated process where one high energy electronic excitation (E~t > Eg) is converted into several low electronic excitations (Eex ,~ Eg). Such multiplication of electronic excitations is accompanied with a strong electron-phonon interaction, and only one third of the initial energy of E~I(E > Eg) is transformed into excitons and electrons and holes while two thirds are transformed into atomic vibrations [36]:

E~t(E > Eg) --+E~: + E(e) + E(h)

(2)

where E ~ is the energy of the self-trapped exciton, and E(e) and E(h) are the energy of the created electron and hole, respectively. Thus the effective mass

Excitons and Radiation Damage in Alkali Halides

357

of holes in Mkali halides is larger than that of the electrons, holes determine the range of the secondary interaction in the lattice. The energy of the created electrons and holes, usually, exceeds the equilibrium kinetic energy of the charge carriers in the conduction and valence band, i.e., hot electrons and hot holes are produced. In alkali halides, the hot holes either relax to a selftrapped state (within a time of ~ 1- 10ps with a mean free path o f / ~ 10 nm, and a diffusion length of L ~ 100 rim) or can be captured by lattice defects [6,33]. The mean free path and diffusion length of electrons is larger than that e h for holes clue to their effective mass relation (rnef f < meff). The electrons can be trapped by lattice defects or can recombine with a self-trapped hole and create a self-trapped exciton e °. The migration and interaction of excitons, electrons, and holes in solids lead to a remarkable increase (up to 1000/~ and more) of the initial lateral range, determined by the energy transfer from the fast particles. The multiplication of electronic excitations in alkali halides was studied in detail by Lushchik et al. using synchrotron radiation [6,33-35]. The presence of various elementary electronic excitations was observed: primary excitons, secondary low energy excitons, hot electrons and holes, double excitons (2e°), double electron-hole pairs (2(e + h)), etc. [34,35]. The efficiency of Frenkel defect creation by various superpositions of low energy electronic states is different. In KBr the defect creation efficiency is the highest by a superposition of a self-trapped exciton and an electron-hole pair ( e Os, e + h) [34]. The Frenkel defect creation by self-trapped excitons is a relaxation of the lattice in which the electron and the hole are converted into spatially separated defects. Such relaxation can be described as a non-elastic collision, and this is possible if the interaction of the electron and the hole with the surrounding lattice is stronger than their intrinsic interaction in the exciton state (Fig. 1 and Fig. 3). The dominating primary Frenkel defects in alkali halide crystals are the F- and H-centers (Fig. 3). During the further interaction with the lattice most of the F- and H-centers are annihilated, and only a small part of them are separated to stable color centers. A peculiarity of alkali halides is the different chemical binding of the host lattice (ionic) and of various hole centers (covalent). The covalent binding of hole centers is determined by the electronic structure of halogen atoms (ns2pb). Various hole centers are combinations of X~, and the halogen products of radiolysis are covalent molecules X2 [5,6,40,68]. The main radiation damage creation in alkali halides takes place in the anion sub-lattice: the electron centers (F-centers and their aggregates) are created on the anion vacancies; hole centers are created by replaced anions (Fig. 3 and Fig. 4). Nevertheless, cation vacancies are also produced from exeitons states [6,33]. The creation of elementary Frenkel defects occurs from molecular exciton states having a higher probability of electron and hole center separation than that of the atomic self-trapped exciton (Fig. l b and Fig.3) [5,6,33]. Two

358

K. Schwartz

)

(a)

(b)

Fig. 3. The structure of an atomic exciton (a) and of a Frenkel pair (b) with separated F- and H-centers in alkali halides [5]

different Frenkel pairs can be produced: e sO - + v a + I

or

e Os ~ F + H

(3)

where e~° is the self-trapped exeiton, and (F, H) and (Va,I) are the induced Frenkel pairs (Fig. 4). A defect creation similar to (3) can also take place by the recombination of an electron (e) with a self-trapped hole (hs). A conversion of exeitons into electron-hole pairs with a following self-trapping and vice versa is possible (e ° ~ e + h). Thus, Frenkel defects can be created ° e + h) . by the reaction (3) from both electronic excitations ( e ~, As mentioned above, by the decay of excitons various Frenkel pairs can be created (equation (3)). The relation of the concentration of (va + I) and (F + H) pairs in various alkali halides is different [5,6]. Nevertheless, the Frenkel pair (Va + I) corresponds to a positively charged anion vacancy Va and negative interstitial ion X~t , and the probability to capture an electron or a hole by the charged v~ and I centers is high. At higher temperature these defects are transformed into other more stable lattice defects [5]. In alkali halides the H-centers are stable at T _ l l 0 K are mobile which leads to an increase of their interaction radius with the surrounding lattice. Vg-centers can create self-trapped excitons by means of recombination with electrons (VK + e ---+e°). At higher temperature Vg-eenters disappear into more stable hole centers [33,68]. The F-centers produced at low temperatures (T = 4 K) are stable up to higher temperatures (for LiF up to T > 500 K) [68]. At high irradiation doses and higher temperatures, F-center aggregates (F~-centers with n < 4) and colloid centers (macroscopic aggregates of F-centers) are created [57,58].

359

Excitons and Radiation Damage in Alkali Halides F

FAM~

o~

-\+

• ,'-;=, ÷

-

~f.

•

,-,

-

+

:/:

-

+ L:..,

+

-

X +

,e,,'~ +

-

-

+

-

I+

-

+

.

[]

+

-

t+

~

+

-

,+

1~

+

-E:/: ,V~ H

=

+ ~']

+

r'--'l

•

I--'~

* ' - L -' - J ,2e, +

r-~

L-_M

L-.~

--/f÷

--

F'

V~eF

+ ,H, ' ~

-

+

-

+

_

+

--

U e H-V a U a~,

Isotropic

+

--

. , - , "~U- -"J ~ - " --

÷

--

÷

"t"

--

4-

--

MeVoe vova 2e~.__.~., M Anisotropic Electron Centers

[]

.1

I

I --

4=

"t"

--

+

+

nV~e--=nF

4-

-,-

-

+

--

-I-

--

--

÷

--

-t-

--

.,-

Vx:X~.lX-X_ ] H=X2"IX-]

Aggregote Hole

centers

Fig. 4. Various types of defects and color centers in alkali halides: va, vc - anion and cation vacancies; F, F', FA, M (F2) - electron color centers; H, VK - hole centers; nF - aggregates of F-center which correspond to the initial step of colloid center formation [16].

The kinetics of elementary Frenkel pair creation in alkali halides was studied by two photon pulsed laser spectroscopy in a large temperature range (from 4 K up to the melting point) [32]. A relaxation time (r) for stable (F-H) pair creation of r --- 10ps was found. This demonstrates that the Frenkel defect creation time v is much longer than the lattice vibration period rvib~ "~ 10 -13 s. The efficiency of primary Frenkel defect creation (7/) has a complicate dependence on time and temperature. After a short time interval (At ~ 50ps) following the excitation with the laser pulse, the primary efficiency of defect pair creation (r/0) is increasing with the temperature (e.g., in KC1 r/o ~ 1 at T = 880K, and r/0 ~ 0.16 at T ~ 4K). These primary Frenkel defects, however, are rapidly annealed especially at high temperatures, reducing the efficiency 7/of stable Frenkel pairs at optimal conditions to only 5 - 6 %. The dependence of the efficiency of Frenkel defect creation on temperature by optical excitation [32] determines also the efficiency of radiation damage creation at various temperatures. From a general point of view radiation damage creation in alkali halides and in other insulating crystals depends on the irradiation dose, dose rate (i.e., excitation energy density), and temperature. In alkali halides at room temperature the stable Frenkel defects are F- and Vs-centers (V3-centers have a micro structure of v c h X ~ ) [34]. At high irradiation doses and at temperatures above room temperature, complex electron and hole centers are produced (microscopic and macroscopic clusters of F-, X~-- centers, vacancy clusters, covalent halogen molecules X2, etc.) [55,57,58]. At low temperature the defect creation and the exciton luminescence in various alkali halides have a different dependence on the temperature which determines also the efficiency of the Frenkel defect creation at various temperatures [5, 6, 14]. The defect creation efficiency can be described by the

360

K. Schwartz

energy to create one Frenkel pair (AEF). Two different classes of alkali halide crystals exist [12]. For the crystals of the first class (NaCl, NaBr, KI, etc.) at low temperature, AEF is approximately several hundred thousand electron volt and for crystals of the second class (LiF, KC1, KBr, NaF, etc.) A~EF is only several thousand electron volt [6]. At higher irradiation temperatures, the efficiency of defect creation increases, and at room temperature in both classes of alkali halide crystals, the energy to create one Frenkel pair AEF is about 1000 eV. In crystals of the first class, the temperature dependence of the exciton luminescence is anti-correlated with the efficiency of Frenkel pair creation. The magnitude of A E F strongly depends on the excitation energy density (dE/dx). By irradiation with heavy particles (protons, a-particles, heavy ions), AEF increases as the value of dE/dx increases [53, 54, 59, 60]. Such increase of AEF is determined by two processes: (1) recombination (annihilation) of the primary Frenkel defects; (2) aggregation of F-centers to macroscopic metal colloids (see §4). The defect creation by relaxation ofexciton states in dielectric materials is possible if excitons (or electrons and holes) are self-trapped and if the energy of this state Eel: exceeds the energy Ed,e required for defect creation [33,66]. The defect creation energy by the decay of electronic states Ed,e for various dielectrics is about 10 eV which is smaller than the energy of defect creation by elastic atomic collisions (Ed ~ 25 eV) [15, 33, 39, 68]. The defect creation in various solids by electronic excitations of the lattice was analyzed in detail by Itoh [5,26-31], Lushchik [6,33], etc. It was found that dielectrics without self-trapped states have a small probability for Frenkel defect creation, and therefore these materials are resistant under irradiation (Table 2). If the energy of self-trapped excitons in dielectrics is smaller than the defect creation energy (Eej: < Ed,e), Frenkel defect creation from single excitons is not possible (A1203, SiO2, etc., Table 2). Nevertheless, in these materials defects can be created by a superposition of single electron excitations. Itoh has analyzed the exciton damage creation processes in SiO2 and found that the self-trapped excitons in SiO2 correspond to a configuration where the strong intrinsic interaction of the electron and hole prevent their separation and defect creation (Fig. lc) [26,48]. Radiation damage creation by the decay of double excitons was observed in various dielectrics [26,27]. In alkali halides double excitons can produce

Table 2. Self-trapped excitons and radiation damage efficiency of dielectrics [5,6] Type Examples

Self-trapping

1

MgO, ZnO

no

2 3

c-SiO2,A1203, Y203 yes, E ~ < Ea,~ LiF, NaC1 and other alkali halides yes, Eem > Ed,~ AgCI, AgBr etc. CaF2 etc.

Sensitivity to irradiation low low high


361

halogen molecules (2e ° -+ X2) in the bulk or at the surface [27, 29-31, 63-65]. Nevertheless, the efficiency of Frenkel defect creation by the relaxation of two electronic excitations is smaller than by the decay of single self-trapped excitons.

4

Radiation Damage and Heavy Ion Track Formation in Ionic Crystals

Heavy ion induced radiation effects opened a new sphere of radiation damage creation processes in solids [46-49]. The excitation density dE/dx of heavy ions with a specific energy of about 10 MeV/u is several orders of magnitude higher than by conventional irradiation with 7-rays or electrons (of 1 MeV). For such ions the energy transfer to the target electrons is determined by electronic losses [50-52]. Usually only at the end of the ion path (where the velocity of the ion (Vio,~) is below the Bohr velocity (vo) of the electrons in the target atoms) defects are created via elastic collisions (nuclear loss). Thus, the main radiation induced phenomena in solids by heavy ion irradiation occur under an extremely high electronic excitation level. Under heavy ion irradiation, the collective defect creation processes play an important role and these collective excitations are similar in metals and dielectric materials where the defect creation via single electronic excitations is impossible (Table 2) [46-48]. In alkali halides irradiated with heavy ions, however, the simple exciton mechanism of radiation damage creation was demonstrated by Perez et al. [53] and Balanzat et al. [54]. It was shown that the defect creation energy for LiF, NaC1, and KBr crystals under heavy ion irradiation at room temperature and at T ~-, 15K was close to that observed by X-ray excitation. Also the yield of the exciton luminescence in the range of 15 - 200K was the same as that for X-ray excitation. Perez at. al. demonstrated that the main electronic centers of the ion track in LiF crystals irradiated with Ne, Ar, Kr, and Xe ions at room temperature are F- and F2-centers while the presence of macroscopic F-center aggregates was not detected [53]. Nevertheless, aggregates of F-centers and Li colloids were observed in LiF when irradiated with high doses at room or higher temperatures (thermal neutrons, ion implantation, etc.) [57-59,64-66]. Young observed uranium fission tracks in LiF crystals by chemical etching [55]. Gilman and Johnson showed that only macroscopic aggregates of F-center (various Li colloids) can be chemically etched in irradiated LiF crystals [58]. Therefore, we initiated experiments intended to understand the track damage morphology in heavy ion irradiated LiF crystals using optical spectroscopy, small angle X-ray scattering (SAXS), and chemical track etching [59-60]. In LiF crystals irradiated with various heavy ions (U, Au, Pb, Bi, Xe, Se, and Zn) chemical track etching was observed if the value of dE/dx exceeds a critical magnitude of (dE/dx)e,crit ~ 1.2keV/•. SAXS studies demonstrated that the heavy ion

362

K. Schwartz

induced track etching is correlated with a cylindrical damage region with a radius of 10 - 20 A. The radius of the cylindrical track damage region is increasing with dE/dx > (dE/dx)~,cr,t) [60]. Such macroscopic cylindrical damage region is determined by large F-center aggregates (Li-colloids). The radius of the observed track damage region is in good agreement with the lateral track radius for 50 % energy loss (rt~t ~ 10 £) [70]. The heavy ion induced defect structure around the ion path is complicated. Secondary electronic and atomic migration processes in the lattice leads to an extension of the cylindrical radiation damage region around the ion path. Such processes are the relaxation of excitons, electrons, and holes. In alkali halides holes with a larger effective mass than that of electrons, determine the extension of the primary electronic energy loss lateral radius up to rt~t > 1000 ~ [69]. In the extended lateral track damage region single point defects (electron and hole color centers) are dominating, whereas in the central track core region (rtat ~ 20A) defect aggregates and local phase transitions prevail. The aggregates (colloid centers) in the central track core determinate also the chemical etching of the ion track [59,60]. Such large defect aggregates can be created only at an extremely high excitation level. In LiF crystals irradiated with Zn, Se, Xe, Au, Pb, and U the energy to create one Frenkel pair A E F correlates with the magnitude of dE/dx of the ion [60,61]. The increase of AEF at higher excitation energy density leads to the recombination (annihilation) of primary Frenkel pairs which is in good agreement with various experiments [48,54,68]. The increase of AEF can be explained by a higher recombination efficiency of primary Frenkel pairs at high dE/dx, as well as by a more efficient aggregation process from single F-centers to metallic colloids. Such process was observed in LiF crystals irradiated with Se and Xe heavy ions where the initial value of dE/dx at the external surface of LiF was below the critical threshold for chemical etching (dE/dx)~,~it. Under these conditions etchable tracks were produced only in LiF crystals irradiated with Se and Xe ions through a polycarbonate filter which leads to an increase of the magnitude of dE/dz above the threshold value (dE/dx)~,¢~,t [59,60]. The observed chemical etching effect correlates with a decrease of the concentration of F-centers for several times (and a corresponding increase of AEF). These results demonstrate that the formation of colloidal centers in the track core influences the concentration of single F-centers around the ion path (i.e., in the lateral region with a radius rt~t >> 20 A). Nevertheless, the colloid centers and single point defects are created in various track regions by different mechanisms. F-center aggregates and metal colloids in LiF can be created only under definite conditions: (1) the concentration of F-centers (NF) must exceed a critical value (NF > NF,crit,); (2) the hole centers must be spatially separated from the electron centers to prevent their annihilation; (3) thermal or radiation enhanced diffusion of single color centers can realize the aggregation


363

(1) and separation (2) process [33, 59, 60]. Under heavy ion irradiation the formation of F-center aggregates (colloids) at room temperature can be explained as an effect of diffusion of F-centers or by the generation of extremely high local concentration of F-centers (see Fig. 4). Nevertheless, thermal diffusion of F-centers in LiF crystals occurs only at T > 500K, whereas the hole centers can migrate at much more lower temperatures. For both models the hole centers must be spatially separated. Probably, the primary H-centers at the critical" excitation density rapidly migrate to the lateral track region and, therefore, the aggregation of single primary F-centers to large aggregates is possible. The observed critical value of (dE/dx)e,crit for etchable damage creation in LiF crystals demonstrates the important role of the excitation density and a high local irradiation dose in the ion track. Nevertheless, it is difficult to distinguish the role of the dose and the excitation density for the aggregate (colloid) center creation [60]. To understand the elementary mechanism of colloid center formation in heavy ion tracks in LiF crystals, additional experiments at low temperatures are necessary.

5

Conclusion

In alkali halides the excitons are responsible for any energy conversion processes by optical excitation in the fundamental absorption band or by irradiation with charged particles or X-rays with various excitation energy densities. Exciton processes determinate also both luminescence and defect creation under heavy ion irradiation with an extremely high excitation energy density of 103eV/•. The heavy ion track in LiF crystals consist of lithium atom aggregates in the central part (with a radius of rtat < 20 ~) and single color centers in the extended lateral track region. Acknowledgement I am very thankful to Prof. Ch.B. Lushchik (Tartu) and to O. Geit3 (GSI, Darmstadt) for many fruitful discussions and remarks to this review.

References [1] H. Wimmel. Quantum Physics ~ Observed Reality. A critical interpretation of quantum mechanics World Scientific, Singapore, 1992 [2] J.T. Cushing. Quantum Mechanics. Historical contingency and the Copenhagen hegemony University Chicago Press, Chicago, 1994 [3] Ed. W. Neuser, K. Neuser-von Oettingen. Quantenphilosophie Spektrum, Berlin, 1995 [4] J. Frenkel. On the transformation of light into heat in solids I Phys. Rev. 37 (1931) 17 - 44; On the transformation of light into heat in solids//Phys. Rev. 37 (1931) 1276 - 1294 [5] N. Itoh and K. Tanimura. Formation of interstitial-vacancypairs by electronic excitations in pure ionic crystals J. Phys. Chem. Solids 51 (1990) 717 - 735

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[6] Ch.B. Lushchik. Creation of Frenkel Pairs by Excitons in Alkali Halides In: Physics of Radiation Effects in Crystals. Ed. R. A. Johnson, A. N. Orlov. Elsevier Science Publ., Amsterdam, 1986, 473- 525 [7] R.E. Peierls. Zur Theorie der Absorptionsspektren fester KSrper Arm. Physik 13 (1932) 905- 952 [8] H. Dessauer. Uber einige Wirkungen yon Strahlen IV. Zs. Phys. 20 (1923) 288 - 298 [9] W.C. R&ntgen. Uber die Elektrizitfftsleitung in einigen Kristallen und iiber den Einflufl einer Bestrahlung daraufAnn. Phys. 64 (1921) 1 - 195 [10] G. H. Wannier. The structure of electronic excitation levels in insulating crystals Phys. Rev. 52 (1937) 191 - 199 [11] R.S. Knox. Theory of Excitons. Solid State Physics Vol.5, Ed. F. Seitz, D. Turnbull, Academic Press, New York, 1963 [12] H. Rabin and C.C. Klick. Formation ofF-centers at low and room temperature Phys. Rev. 117 (1960) 1005 - 1010 [13] A.L. Shluger and A.M. Stoneham. Small polarons in real crystals: concepts and problems. J. Phys. Condens. Matter 5 (1993) 3049 - 3086 [14] R.T. Williams and K.S. Song. The self-trapped exciton J. Phys. Chem. Solids 51 (1990) 679 - 716 [15] M. Klinger, Ch. Lushchik, T.V. Mashovets, G.A. Kholodar, M.K. Sheikman, M. Elango Defect formation in solids by decay of electronic excitations Sov. Phys. Usp. 28 (1985) 994 - 101 [16] K. Schwartz. The Physics of Optical Recording Springer Verlag, Berlin - Heidelberg, 1993 [17] R. Hilsch and R.W. Pohl. Uber die ersten ultravioletten Eigenfrequenzen einiger einfaeher Kristalle Z. Physik 48 (1928) 384 - 396 [18] A. Smakula. Uber die Verfiirbung der Alkalihalogenidkristalle dutch ultraviolettes Licht Zs. f. Physik 63 (1930) 762 - 770 [19] A. S. Davydov. Theory of Molecular Excitons Plenum Press, New York, 1971 [20] M.N. Kabler and D.A. Paterson. Evidence for a triplet state of the self-trapped exciton in alkali halide crystals Phys. Rev. Lett. 19 (1967) 652 [21] R.A. Kink, G.G. Liidja, Ch.B. Lushchik, and T.A. Soovik. Izv. SSSR, ser. fiz. 31 (1967) 1982 [22] T.G. Castner and W. K/inzig. The electronic structure of V-centers J. Phys. Chem. Solids 3 (1957) 178 - 195 [23] Ch.B. Lushchik, G.G. Liidja, and M.A. Elango. Fiz. Tverd. Tela 6 (1964) 2256 (Sov. Phys. Solid State 6 (1965) 1789) [24] H.N. Hersh. Proposed excitonie mechanism of eolour center formation in alkali halides Phys. Rev. 148 (1966) 928 - 932 [25] D. Pooley. Defect creation mechanism by exeitons Proc. Phys. Soc. 87 (1966) 245 [26] N. Itoh. Self-trapped exciton model of heavy-ion track registration Proc. Intern. Conf. on Radiation Damage, Italy, 1995 - in print [27] N. Itoh and T. Nakayama. Electronic excitation mechanism of sputtering and track formation by energetic ions in the electronic sputtering regime Nucl. Instr. Meth. B 13 (1986) 550- 555 [28] N. Itoh, K. Tanimura, A.M. Stoneham, and A.H. Harker. The initial production of defects in alkali halides: F and H centre production by non-radiative


365

decay of self-trapped exciton J. Phys. C: Solid State Physics 10 (1977) 4197 4209 [29] N. Itoh, K. Tanimura Radiation effects in ionic solids Rad. Eft. 98 (1986) 269 - 287 [30] N. ltoh. Sputtering and dynamic interstitial motion in alkali halides Nucl. Instr. Meth. 132 (1976) 201 - 211 [31] N. ltoh and T. Nakayama. Mechanism of neutral particle emission from electron-hole plasma near solid surface Physics Lett. 92 A (1982) 471 - 484 [32] R.T. Williams, J.N. Bradford, and W.L. Faust. Short-pulse optical studies of exciton relaxation and F-center formation in NaCl, KCI, and NaBr Phys. Rev. B 18 (1978) 7038- 7057 [33] Ch.B. Lushchik, A.Ch. Lushchik. Decay of Electronic Excitations with Defect Formation in Solids Nauka, Moscow, 1989 - in Russian [34] A. Lushchik, I. Kudrjavtseva, Ch. Lushchik, and E. Vasil'chenko. Creation of stable Frenkel defects by vacuum uv radiation in KBr crystals under conditions of multiplication of electronic excitations Phys. Rev. B 52 (1995) 10069- 10072 [35] A. Lushchik, E. Feldbach, R. Kink, and Ch. Lushchik. Secondary excitons in alkali halides Phys. Rev. B 53 (1996) 5379 - 5387 [36] R.C. Alig, S. Bloom. Electron-hole pair creation energies in semiconductors Phys. Rev. Lett. 35 (1975) 1522 - 1525 [37] A. Sumi. Phase diagram of an exeiton in phonon field J. Phys. Soc. Japan 43 (1977) 1286- 1294 [38] F. Seitz. The motion of charged particles through solid matter Disc. Faraday Soc. 5 (1949) 271 - 289 [39] F. Seitz, J.S. Koehler. Displacements of atoms during irradiation Sol. Stat. Phys. 2, ed. F. Seitz and D. Turnbull, N.Y., Acad. Press, London, 1956 [40] F. Seitz. Color centers in alkali halides II. Rev. Mod. Phys. 26 (1954) 1 - 102 [41] H. Oldenburg. Thermoluminescence of fluorites Phil. Trans. Roy. Soc. London 3 (1705) 345 [42] T.J. Pearsall. On the effects of electricity upon minerals which are phosphorescent by heat J. Royal Inst. 1 (1830) 77, 267 [43] K. Przibram. Irradiation colors and luminescence Pergamon Press, London, 1956 [44] K. J. Teegarden. Luminescence of potassium iodide Phys. Rev. 105 (1957) 1222 - 1227 [45] R.S. Knox, and K.J. Teegarden. In: Physics of Color Centers ed. W. B. Fowler, Academic Press, New York, 1968, p. 1 [46] A. Dunlop, D. Lesser, P., H.. Effects induced by high electronic excitations in pure metals: a detailed study in iron Nucl. Instr. Meth. B 90 (1994) 330 - 341 [47] A. Meftah, F. Brisard, J.M. Constantini, E. Dooryhee, M. Hage-Ali, M. Hervieu, J.P. Stoquert, F. Studer, and M. Toulemonde. Track formation in Si02 quartz and the thermal-spike mechanism Phys. Rev. B 49 (1994) 12457 12463 [48] A M. Stoneham. Radiation effects in insulatorsNucl. Instr. Meth. A 91 (1994) 1 - 11 [49] E. Balanzat. Heavy ion induced effects in materials Rad. Eft. 126 (1993) 97 101 -

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[50] J.F. Ziegler, J.P. Biersack, U. Littmark. TRIM 89. The stopping and ranges of ions in solids Pergamon Press, New York, 1985 [51] M.P.R. Waligorski, R.N. Harem, R. Katz. The radial distribution of dose around the path of a heavy ion in liquid water Nucl. Tracks Rad. Meas. 11 (1986) 309- 319 [52] R. Katz, K.S. Loh, L. Daling, and G.-R. Huang. An analytic representation of the radial distribution of dose from energetic heavy ions in water, Si, LiF, NaI, and Si02 Rad. Eft. 114 (1990) 15 - 20 [53] A. Perez, E. Balanzat, J. Dural. Experimental study ofpoint-defect creation in high-energy heavy-ion tracks Phys. Rev. B 41 (1990) 3943 - 3950 [54] E. Balanzat, S. Bouffard, A. Cassimi, E. Dorothyee, L. Protin, J.P. Grandin, J.L. Doualan, and J. Margerie. Defect creation in alkali halides under dense electronic excitations: Experimental results on NaCl and KBr Nucl. Instr. Meth. B 91 (1994) 134- 139 [55] D.A. Young. Etching of radiation damage in lithium fluoride Nature 183 (1958) 375- 378 [56] C.3. Delbecq, P. Pringsheim. Absorption bands in irradiated LiF J.Chem. Phys. 21 (1953) 794 - 800 [57] K.K. Schwartz, A.J. Vitol, A.V. Podins. Radiation effects in pile-irradiated LiF crystals Phys. Status Solidi 18 (i966) 897 - 909 [58] J.J. Gilman and W.G. Johnson. Dislocations, point-defect clusters, and cavities in neutron irradiated LiF crystals J. Appl. Phys. 29 (1958) 877 - 888 [59] K. Schwartz. Electronic excitations and defect creation in LiF crystals Nucl. Instr. Meth. B 107(t996) 128 - 132. [61] K. Schwartz, C. Trautmann. Heavy ion induced radiation damage in LiF crystals GSI Nacrichten GS[ 09-95 (1995) 13- i5 [60] K. Schwartz, C. Trautmarm, and T. Steckenreiter. Ion tracks in LiF crystals GSI Jahresbericht 1995 [6i] D. Albrecht, P. Armbruster, and R. Spohr. Investigation of heavy ion produced defect structure by small angle scattering Appl. Phys. A 37 (1985) 37 - 44 [62] A.T. Davidson, J.D. Comins, T.E. Derry, and F.S. Khumalo. The production of defects and colloids in litthium fluoride crystals by implantation with rare gas ions !Ra& Eft. 98 (1986) 305 - 312 [63] N. Seifert, S. Vijayalakshmi, Q. Yan, A. Barnes, R. Albridge, H. Ye, N. Tolk, and W. Husinsky. Optical absorption spectroscopy of defects in halides Rad. Eft. 128 (1994) 15 - 26 [64] A.E. Hughes. Metal colloids in ionic crystals Adv. Physics 28 (1979) 717 - 828 [65] J.R.W. Weerkamp, J.C. Groote, J. Seinen, and H.W. den Hartog. Radiation damage in NaC1. I - I V P h y s . Rev. B 50 (1994) 9781 - 9801 [66] F. Agullo-Lopez, C.R.A. Callow, P.D. Townsend. Point Defects in Materials Academic Press, London, 1988 [67] G.A. Wagner, P. Van den Haute. Fission-track dating Ferdinand Enke Verlag, Stuttgart, 1992 [68] M. Elango. Elementary inelastic radiation-induced processes American Institute of Physics, New York, 1991 [69] M. Elango. Hot holes in irradiated ionic solids Rad. Eft. 128 (1994) 1 - 13 [70] M. Kr/imer and G. Kraft. Calculations of heavy ion track structure Radiation Environment Biophys. 33 (1994) 91 - 109

Polarization of Negative Muons Implanted in the Fullerene C60: Speculations about a Null Result A. Schenck 1, F.N. Gygax 1, A. Amato 1, M. Pinkpank 1, A. Lappas 2, K. Prassides 2 Institute for Particle Physics of ETH Z/irich, CH-5232 Villigen PSI, Switzerland School of Chemistry and Molecular Sciences, University of Sussex, Falmer, Brighton BN1 9Q J, United Kingdom

1

Introduction

Since the discovery of parity violation in the rr - p - e decay chain it is well known that negative muons (p-) implanted in graphite will preserve about 1/6 of their initial polarization when cascading down to the lowest Bohr orbital (12S1/2) after capture by a C-atom [1]. The loss of polarization during the cascade is due to spin-orbit coupling in a given muonic level E(ng, j = 1 + s) where n is the main quantum number and g the angular momentum. Spin-orbit coupling may be understood to produce a very strong field at the p - causing its spin to turn. Depolarization will happen if the lifetime 1--1 of this state is long enough to allow t h e / J - spin to rotate by a measurable degree, i.e. /7 (w0r)min- Tab. 1 teaches that the radical state [C60p-]' could in principle have been observed directly and consequently possibility (i) should apply. In this case the non-observability of any residual polarization implies a lower limit on u as given by Eq. (5). A similar reasoning also applies to the state pB ° 2p 2P1/2. The derived limits are all physically reasonable and it is not possible on the basis of the available information to determine the correct hyperfine states. However, some additional clue is provided by the results on K3C60. It is reasonable to assume that the hyperfine state formed in this system is not any different from the one in C60. Restoration of the full residual polarization can be achieved by a very much increased electronic fluctuation rate or by a shortened lifetime of the paramagnetic state. Fig. 7 (pertaining to the T F measurements at Be×t -- 0.052 T (see Fig. 4)) shows that in order to push P± to near unity for unchanged lifetime v the fluctuation rate u would have to increase by more than 3 orders of magnitude, which would again push u into an unphysical regime for the two first hyperfine states. On the other hand, if one allows also the lifetime v to become shorter P± will quickly increase. The presence of free charge carriers in K3C60 may indeed shorten 7- as well as increase u. Exchange scattering of the free charge carrier spins off the unpaired local spin will certainly enhance the electronic (Korringa) relaxation.

Polarization of Negative Muons Implanted in the Fullerene C60

P,

,

3~4(5

,.o

~

/ . / . . -_.';--/7"- ~"

~

o.8: , 0.6

_

-4,-J

/

~2/q/o

0.4-

"

,,

,

/ //

o~= 1o

0 . 4 ~-" '73/2

:--~--:~

0.4: ~

-1

/

,' I ,' /~o~= l~ /i ,~-o;~-lo ,/0 /1 /-2- 2 /-1 110 / ./ --/;'/2 / l ,, /

/ 3-3

/

l

-0.5

0

0.5

1

1.5

2

i

1.01 0.80.6

0 :5

, oL__~

o81.--~----;,-/-/--/-/_-" /

~- / 00, / 1 "1 .,,'-1

0'

'

,/

,

~'~---?--7

012.5 -1 -0.5

-4 ." /

J

,- ,' ,' n > _ 2) must be refilled so quickly as to be ineffective. The neutral state pB ° m a y not be stable either since it m a y be energetically favourable to promote the 2p electron into a delocalized state. This would leave us with the diamagnetic singly charged ion pB 1+ 1S0. In C60 in contrast these considerations cannot apply in view of the results. We suggest that the pseudo-boron ion (or atom?) is displaced t o the centre of the Cs0 cage due to the recoil transmitted to the pseudo-boron in the muonic 2 p - ls transition (E.~ = 75 keV) during the cascade providing a recoil energy of 0.25 eV. The remaining disturbed Cs9 m a y rearrange to form the more stable Css. The recoil energy m a y be compared with the binding

378

A. Schenck et al.

energy of 0.6 eV of a C-atom in the C60 molecule [16]. If a C-atom is replaced by a boron atom the binding energy is expected to be smaller since B has only 3 valence electrons so that one double bond is broken [17]. It thus seems that there may be just enough recoil energy to allow the pseudo-boron to leave the 659B net-work. If finally placed inside the cage the pseudo-boron is well shielded from the outside world, enabling it to survive for a certain time relatively undisturbed in whatever state it initially was in. In particular refilling of empty states of the ionic configurations may be sufficiently slowed down. On the other hand in graphite and diamond, where no protective cages are available, the formed pseudo-boron suffers a different fate leading quickly to the mentioned diamagnetic state. This leaves us with the results on K3660. It is reasonable to assume that the final (ionic) pseudo-boron state in this compound is the same as in C60. The presence of free charge carriers is likely to increase u via spin exchange processes (as discussed above) and at the same time to kill the ionic, paramagnetic states by electron capture with the effect that a finite residual polarization may become observable. The fact that the residual polarization in K3C60 depends on temperature is an indication that this conjecture may be right, since temperature may have an effect in particular on v (see, e.g., the result on the electronic fluctuation rate in endohedral muonium in Rb3C60 [13]). Alternatively one may conjecture that the (ionic) pseudo-boron is situated outside of the cage. This situation, however, may have more in common with the situation in graphite or diamond where it is exposed "unshielded" to its neighbours and would thus be expected to transfer very quickly into a diamagnetic state. Unfortunately the available information is too limited to allow to decide which state may actually be formed. What would the next steps have to be in order to unravel the various possibilities? The curves in Fig. 5, 6, 7 teach that one should try to change 7- and u in a systematic way. The few results on K3C60 suggest that detailed studies of temperature dependencies could provide the missing clues. It would be in particular interesting to study K3C60 below the superconducting transition temperature of 19 K, below which temperature the number of unpaired conduction electrons should be reduced. Other metallic and insulating C60 compounds should be tried as well. The measurement of LF-decoupling curves over an extended field range appears also promising provided that x can be made larger than 1. This excludes the state/~B 4+ ls 2S1/2. For instance extending the field range to 5 T would increase the available x to 3.5 for pB ~+ 2s 2S1/2 and to 36 for pB ° 2p 2P1/2. Inspecting in Fig. 5 the plots for w0v ~ 103, one finds appreciable repolarization if U/Wo < 10 -1. The mentioned tentative results above 1 T are very encouraging in this respect and will motivate further measurements. In summary the observed zero residual polarization of # - - i m p l a n t e d into C60 both in T F and LF-measurements has been discussed on the basis of a model which comprises the formation of a paramagnetic state with a strong

Polarization of Negative Muons Implanted in the Fullerene Cs0

379

hyperfine coupling and rapid spin fluctuations of the unpaired electron spin. It is concluded that the paramagnetic state is of a transient nature. This state is susceptible to the presence of free charge carriers as in K3C60. The drastic and unique difference between the p - residual polarization in graphite and diamond on one side and in C60 on the other side seems to point to a rather well shielded paramagnetic state in the latter compound which suggests that the formed pseudo-boron atom is located at the center of a rearranged Css-cage possibly in the ionic configurations [(C60/J-)ls]4+(2S1/2) or

References [1] R.L. Garwin, L.M. Ledermann and M. Weinrich, Phys. Rev. 106 (1957) 1415 [2] I.M. Shmushkevich, Nucl. Phys. 11 (1959) 419 R.A. Mann, M.E. Rose Phys. Rev. 121 (1961) 293 [3] V.S. Evseev, in Muon Physics III, Chemistry and Solids ed. by V.W. Hughes and C.S. Wu (Academic Press, New York, 1975) p. 235 V.S. Evseev, T.N. Mamedov, V.S. Roganov, Negative Muons in Matter (Energoatomistat, Moscow, 1985) (in russian) [4] T. Yamazaki et al., Phys. Rev. Lett. 42 (1979) 1241 [5] Th. Stammler et al., phys. stat. sol. (a) 137 (1993) 381 [6] R. Abela et al., Nucl. Phys. A395 (1983) 413 [7] M. Koch et al., Hyperfine Interact. 65 (1990) 1039 [8] A. Schenck et al., Hyperfine Interact. 86 (1994) 831 [9] E.J. Ansaldo et al., Z. Physik B 86 (1992) 317 [10] V.G. Nosov and I.V. Yakovleva, Soy. Phys.-JETP 16 (1963) 1236 [11] G. Wessel, Phys. Rev. 92 (1953) 1581 [12] P. Kusch and H. Taub, Phys. Rev. 75 (1949) 1477 [13] R.F. Kiefl et al., Phys. Rev. 70 (1993) 3487 [14] I.G. Ivanter and V.P. Smilga, Sov. Phys-JETP 27 (1968) 301, ibid 28 (1969) 796 [15] R.F. Kiefl et al., Phys. Rev. Lett. 69 (1992) 2005 [16] D. Bakowies, W. Thiel, J. Am. Chem. Soc. 113 (1991) 3704 [17] see e.g.W. Andreoni et al., Chem. Phys. Lett. 190 (1992) 159

Positronium in Condensed Matter with Spin-Polarized Positrons

Studied

Jgnos Major 1,2, Alfred Seeger 1,2, JSrg Ehmann 1, and Thomas Gessmann 1'2 1 Max-Planck-Institut ffir Metallforschung, Institut ffxr Physik, HeisenbergstraBe 1, D-70569 Stuttgart, Germany 2 Unlversitgt Stuttgart, Institut fftr Theoretische und Angewandte Physik, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

1 Introduction

and general

background

The positron, the positively charged antiparticle of the electron, had been seen but not recognized in cloud-chamber observations of the cosmic radiation for several years when in 1932 Anderson (1933) identified certain cloudchamber traces as being due to a positively charged particle with a mass of the same order of magnitude as the electron mass. He proposed the name positron for the "new" particle. Its identification with the antiparticle of the electron as predicted by Dirac's relativistic theory of the electron (Dirac 1930) became certain when Klemperer (1934) detected the pairwise emission of the two 511 keV photons resulting from the annihilation of a positron (e +) and an eIectron (e-) according to

e+ + e- ~ 27

(1.1)

By measuring the angular correlation of the annihilation radiation (ACAR) of positrons from the fl+-deeay of 64Cu (obtained by bombarding natural Cu with deuterons) Beringer and Montgomery (1942) confirmed the theoretical prediction, based on the conservation of energy and momentum, that the two 7-quanta in (1.1) should be emitted in opposite directions. For a full account of the discovery and identification of the positron the reader is referred to the book of Hanson (1963). As early as 1934 St. Mohorovi~i~ (1934) suggested that a positron and an electron may form a bound entity, (e+e -), that is analogous to the hydrogen atom but unstable because of the overlap of the e+ and e- wavefunctions and hence of a finite probability of e+e - annihilation. In a discussion of optical and other properties of this "atom", Ruark (1945) proposed the name posiZronium. The "chemical" symbol Ps for positronium appears to have been introduced by McGervey and DeBenedetti (1959; see also McGervey 1905). Independently of Ruark, Wheder (1946) worked out binding energies and annihilation rates for what he called "polyelectrons", viz. entities consisting of small numbers of e+ and e-. Making use of Dirac's (1930) expression for

382

the cross-section of the 27-annihilation of e + and e - , he obtained for the 27annihilation rate of the "bi-eleetron" (e+e - ) in the 1S0 state with principal q u a n t u m number n

lr2 = c ( oe2 2

I

'

(1.2)

where c denotes the speed of Hght in vacuum, me and e mass and electrical charge of the electron, Po := 1 47r. 10- 7 N. A - 2 the permeability of the vacuum, and ~P(0) the value of the e + wavefunction at the e - location in the ground state of Ps. With h : Planck's constant, we may write fo~ the e + density at the e - position ~ . 2 ~.omee2C2 ~ 3

\

(1.3)

/

As will be discussed later, in (1.3) t¢ is a parameter that takes into account the modification of the wavefunction of positronium in matter. The solution of the SchrSdinger equation for Ps in vacuum corresponds to i¢ : 1. As is the case for hydrogen atoms, the hyperfine interaction (in the present case between the e + and e - magnetic moments) splits the IS ground state into a 13S1 (triplet) state, with e + and e- spins parallel, and into a 11So (singlet) state with antiparallel spins of the two particles. In analogy to ortho- and parahydrogen, Ps atoms in the 13S1 state are referred to as orthopositronium (o-Ps), those in the 1 1S 0 state as parapositronium (p-Ps). In the lowest non-vanishing order of perturbation theory ( : fourth-order terms in powers of Sommerfeld's fine-structure constant a : : I~oce2/2h 1/137) the energy of o - P s is larger than that of p - P s by 14

2

2

the so-called hyperfine splitting 2. In (1.4) /z~ = 9 . 2 8 4 . 1 0 -24 J T -1 is the magnetic moment of the electron. In addition to the interaction between the magnetic moments familiar from the theory of the hyperfine splitting of hydrogen and muonium, Eq. (1.4) contains a term of comparable magnitude due to the spin dependence of the virtual e+e - annihilation and pair production. 1 := and ----:mean that the symbol on the side of the colon is defined by the quantity on the side of the equality sign. 2 W e prefer the notation AEo_p for the hyperfine splitting of the 1S state of Ps to the conventional notation AE hf because it is unambiguous. Generally speaking, for Ps the distinction between the fine structure and the hyperfine structure does not play the same r61e as for "ordinary" atoms, since the magnetic moment of the Ps "nucleus" (the e +) differs only in sign from that of the e-. In the present paper, P, D, etc. states of Ps need not be considered since owing to the very small overlap of the e + and e- wavefunctions in these states the annihilation rate is much smaller than the rate of optical transitions to lower-lying states.

Positronium in Condensed Matter If we insert for Pe the Bohr magneton PB := eh/2me,h := m a y be written in terms of the fine-structure constant as AEo_p =

7r,,a4mec2

383

h/2,r,Eq. (1.4) (1.5)

For the corrections of (1.51 due to higher-order terms in a and the comparison with the experimental vacuum value

AEo_p = 0.8412 meV

(1.6)

the reader is referred to a recent summary by Yennie (1992). The agreement between the predictions of quantum electrodynamics (QED / and the (more accurate / experimental value is within the estimated uncertainty of terms that have not yet been calculated. The energy difference (1.6) between the o-Ps and p--Ps is approximately equal to the energy equivalent of 10 K. Since i¢ is not expected to be appreciably larger than unity, in the absence of a magnetic field the ratio of o-Ps and p-Ps formed at not too low temperatures will be given by the statistical weights of the two states; if the electron spin polarization is zero, the ratio is thus equal to three. (This is likely to be true even below 10 K, since Ps formation in condensed matter occurs on such a short time scale that the spins do not yet "feel" the temperature of the host material. This question deserves more detailed attention, however. For the case of applied magnetic fields see Sect. 2.) Although the overlap of the wavefunctions ofe + and c- in the 1381 and in the 11S0 states is the same, it has been known since the early work on Ps that the annihilation rates for o-Ps and p-Ps are very different (Wheeler 1946). Wolfenstein and Ravenhall (1952) showed that this is a consequence of the facts (i) that the interaction mediating the annihilation of e+ and e-, the electromagnetic interaction, is invariant under the so-called charge-conjugation operation, which replaces all particles in a system by their antiparticles while preserving their spins and momenta, and (ii) that the Hamiltonian of selfconjugated systems, i.e. systems that consist of equal numbers of particles and their antiparticles plus an arbitrary number of self-conjugated particles (e.g., photons), does not change under charge conjugation; hence in such systems eigenstates of the Hamiltonian are also eigenstates of the charge-conjugation operator C. The eigenvalues of C are C : (-1) s+z, where S denotes the total spin and I the orbital momentum in units of h. Hence the C eigenvalue of the 1So states is (-110+0 = 1; that of 3S1 is (-1) 1+° = -1. Charge conjugation implies the change of sign of the dectrical charges and the magnetic moments of the particles (and in the case of fermions also of their inner parity) and therefore the reversal of the directions of the electrical and magnetic fields. From this it may be deduced that the C eigenvalue of ~ photons is (-11". It follows from the conservation of C in self-conjugated systems that a Ps "atom" in a 1S0 state can annihilate only into an e v e n number of photons

384

and that in a 3S1 state it can annihilate only into an odd number of photons. Since momentum conservation requires the participation of at least two photons (the probability of one-photon annihilation, with balance of energy and momentum maintained by surrounding matter, is extremely small at the low kinetic energies of e + in Ps; see Heitler 1954) this means that the minimum number of photons resulting from the annihilation of Ps in 3S1 states is three. Prom the standard perturbation treatment of electromagnetic processes it follows immediately that the ratio of the 3-photon annihilation rate to the 2-photon annihilation rate should be of the order of magnitude of Sommerfeld's fine-structure constant a. The detailed calculations of Ore and Powell (1949) gave for the 3~f-annihilation rate of the Ps 3S1 triplet states with principal quantum number r~ the result 2~" 2 91m°c2. ~-~ 3-/'37 : 9-~(~" -h rt3

(1.7/

or, expressed in terms of 1/"27 (Eq. 1.2), 4a" 2 9) 1 r ~ v = 1 .1F2 7 3/'3.r = ~--~(z" • 111---4

(1.8)

Eq. (1.2) predicts for the mean lifetime (= the inverse of annihilation rate) of parapositronium in vacuum 7"p_Ps = 124.5 ps

(1.9)

This time is too short for a high-precision determination. Direct measurements of Tp-ps are therefore not suited for a critical test of QED by comparing them with the QED calculations carried to higher orders in a (Berko and Pendleton 1980). An indirect value for :~p-Ps has been obtained by A1Ramaclhan and Gidley (1994) making use of the admixture of the 11So state of Ps to the 13S1(m = 0) state (cf. Sect. 2). The ratio of the 47-decay rate of the 11So state of Ps to the 27decay rate has recently been determined experimentally by two different groups as aF4.y/1F2.y = [1.48 ± 0.13(statistical) + O.12(systematic)J 10-6 'I

(Adaehi, Chiba, Hirose, Nagayama et al. 1994) a n d 1F47/1/'t27 :

[1.50 ±

O.OT(statistical) ± 0.09(systematic)] • 10 -6 (yon Busch, Thirolf, Ender, Habs et al. 1994). Both results agree with each other and with the QED predictions by various authors (see Adachi et al. 1994). 1F47/IF27 is too small to have an influence on the lifetime of p-Ps. At present, the best value for the mean lifetime of parapositronium in vacuum is rp-ps =: 1/~p_p, = 125.164 ps

(1.10)

obtained from detailed QED calculations (Khriplovich and Yelkhovsky 1990 and references therein) which include the a21na term as highest-order term.

Positronium in Condensed Matter

385

The intrinsic orthopositronium lifetime is in a convenient range for direct precision measurements. However, here one encounters the difficulty that the positrons in triplet Ps m a y undergo 27-annihilation with electrons of opposite spin which they "pick up" from their environment. This so-called pick-off annihilation of Ps m a y reduce the "true" lifetime of o-Ps by orders of magnitude compared to its "intrinsic" value as determined by the 37annihilation referred to above. The dependence of the pick-off annihilation and hence of the o-Ps lifetime on the electron density of the environment allowed M. Deutsch (1951a, b) to demonstrate the formation of o-Ps in gases. Making use of the fact that the jS+-decay of 22Na is accompanied by the emission of a 1.27 M e V ~,-quantum virtually simultaneously with that of a positron and determining from the delays between the recording of a "prompt" 1.27 M e V photon and one of the annihilation photons the individual lifetimes of the positrons (now called positron age, following the usage of MacKenzie and M c K e e 1976; see Sect. 4), Dcutsch (1951a) showed that the admixture of a small amount of nitric oxide to pure N2 gas resulted in a significant decrease of the number of annihilation photons belonging to positron ages that exceed about 100 ns. Measurements of the energy distribution of the annihilation photons with a Nal scintillation detector revealed that the addition of N O increased the number of quanta in the 511 kcV photon peak but decreased the occurrence of photons of lower energies. Both experimental results wcrc interpreted as a reduction of 37-annihilation events due to a rapid conversion of orthopositronium into parapositronium induced by the exchange of electrons between Ps atoms and N O molecules (cf. Sect. 3.2). In further work on e+ annihilation in gaseous CCI2F2 (dichlordifluoromethane = freon) Deutsch (1951b) observed that at pressures above about 4.104 Pa all annihilation events belonging to positron ages that exceeded 80 ns were due to 37-decays. By extrapolating the pressure dependence of the long e+ ages in freon above 4.104 Pa to zero pressure he determined the "intrinsic" o-Ps lifetime and found it to be in agreement with the calculated 37-annihilation rate of o-Ps, thus establishing the existence of Ps "atoms". Deutsch's extrapolation method is stillbeing used to determine the intrinsic lifetime of o--Ps in vacuum. According to an estimate of Labellc, Lepage, and Magnea (1994) of the not yet completely calculated a 2 term in the correction factor to (1.7), quantum electrodynamics gives for the lifetime of o-Ps in vacuum ro-Ps =: I/,~o-es = 142.038 ns

(1.11)

This is to be compared with the most recent experimental values, viz. those of Nico, Gidley, and Rich (1990) and Nico, Gidley, Skalsey, and Zitzewitz (1992) (~'o-Ps = (141.880 + 0.032) ns), and of Asai, Orito, and Shinohara (1995) (~'o-Ps = (142.15 + 0.03 + 0.05) ns). In the latter value the first error limit arises from the statistics of the lifetime spectrum and the time calibration, the second one from the statistics of the photon-energy measurements and

386 systematic errors. The error limits of the two experimental values appear to be incompatible with each other. It may therefore be too early to form a final judgement on whether quantum electrodynamics predicts the intrinsic o-Ps lifetime correctly or not, clearly a question of fundamental importance. In the abovementioned investigations of Beringer and Montgomery (1942) and Deutsch (1951a, b) as well in most later work on positron annihilation, irrespective of whether on fundamental or applied problems, the positrons were obtained from the fl+-decay of neutron-deficient radioactive nuclides 3 that were produced in accelerators or nuclear reactors. The prediction of Lee and Yang (1956) that processes such as fl-decay that are mediated by the Weak Interaction violate invariance under space reflection (= conservation of parity) was verified first by the experiments of Wu, Ambler, Hayward, Hoppes et al. (1957) on the fl--deeay of 6°Co and subsequently by those of Garwin, Lederman, and Weinrich (1957) and of Friedman and Telegdi (1957) on the decay chain lr+ --,/~+ -~ e ÷. The non-conservation of parity has the consequence that positrons and electrons emitted in fl-decay as well as muons (/~±) obtained from the decay of pi-mesons (~r+) are spin-polarized. In dealing quantitatively with spin-polarized positrons or electrons, it is useful to introduce two pseudoscalar quantities, viz. the helicity 7g and the pseudoscalar spin polarization 7~. In the literature they arc sometimes insufficiently distinguished. The helicity of s p i n - l / 2 particles is defined as 7~ := 2 ( s - p / p )

,

(1.12)

where s is the (axial) spin vector, measured in units of h, and p the (polar) momentum vector. (...) denotes ensemble averages. The pseudoscalar spin polarization (in the following abbreviated to "polarization") of particles with spin s is defined as := ( s - i ' ° ) / s

,

(1.13)

where t0 is a unit vector in a fixed space direction. If we choose this as the z-direction of a Cartesian coordinate system, IP is the z-component of the spin-polarization vector P . The modulus P = IPI satisfies P > 7~. Whereas the pseudoscalar spin polarization 9v is determined by the ensemble average of the projection of the spin onto a given direction of space, the helicity 7~ is a measure of the ensemble average of the projection of the spin onto the momentum direction of the spin carriers. We see that 7~ and 9v can coincide only for a collimated beam of particles whose polarization vector P is parallel or antiparallel to the direction of the beam. (Such a beam is often referred to as longilltdinally polarized.) For non-collimated beams we have 9v < 7~. Whereas spin-carrying particles at rest may possess a non-vanishing polarization, helicity can only be assigned to moving particles. As a consequence of a For the conditions a nuclide must satisfy to be an e+ emitter see, e.g., MayerKuckuk (1994).


387

the A-V Universal Fermi Interaction, the helicity of e+ emitted with velocity v from a/~+-active source is x :

(1.14)

For positrons with a kinetic energy of 1 MeV Eq. (1.14) gives us 7~ = 0.94. The polarization of a non-collimated positron beam obtained from a/~+ source may be calculated from (1.14) by integrating over the velocity distribution of the beam. For a practical example see Gessmann, Harmat, Major, and Seeger (1997a). If we wish to exploit the polarization of positrons in condensed-matter studies, we must be able to determine it at the time of annihilation of the positrons. This is by no means straightforward, since the 27-annihilation process, being mediated by the electromagnetic interaction, is invariant under space reflection and hence does not provide information on the e+ spin direction. In this respect positrons are radically different from positive muons (/~+). Since the decay of/z + into a positron, a neutrino and an antineutrino is mediated by the Weak Interaction, the emission probability of the e +, which serves as indicator of the decay and thus plays a r61e similar to that of the 511 keV photons in e+e - annihilation, is asymmetric with respect to the muon spin direction. The various/~+SR techniques ( = m u o n spin rotation, relaxation, resonance) are based on the information on the p+ spins at the time of the/~+ decay contained in the direction distribution of the emitted e + (see, e.g., Chappert 1984, Smilga and Belousov 1994). "Positron polarimeters" can nevertheless be designed by employing a magnetic field B that is strong enough for its direction to establish a quantization axis for the e + spins. If we then identify the B direction with 1'° (Eq. 1.13) and denote by N+ and N_ the numbers of e + with spins parallel or antiparallel to this direction, the polarization of an e + ensemble is given by _ N+ - N_ N++N_

(1.15)

The two main possibilities for determining experimentally the right-hand side of (1.15) are: (i) Modification of the positvonium states by the external magnetic field (Page and Heinberg 1957, Bisi, Fiorentini, Gatti, and Zappa 1962, Dick, Feuvrais, Madansky, and Telegdi 1963; for further references see Berko and Pendleton 1980 and Consolati 1996). (it) Modification of the momentum distribution of the electrons involved in the e+e - annihilation by reversing the magnetization of the sample (Hanna and Preston 1957, Akahane and Berko 1982, Seeger, Major, and Banhart 1987, Banhart, Major, and Seeger 1989). This has been used to study the relaxation of the e + polarization due to the interaction of the e + magnetic moments with spatially varying internal magnetic fields in ferromagnets and the trapping of e + in paramagnetic eentres in additivdy coloured KCI (Lauff, Major, Seeger, Stoll et al. 1993, Deckers, Ehmann, Greif, Keuser et al. 1995).

388

In analogy to the acronym p+ SR for " m u o n spin relaxation", this technique has been dubbed e+SR (= positron spin relaxation) (Seeger et al. 1987). Although both possibilities have been known for a long time, they have only rarely been employed in condensed-matter studies in spite of the fact that their/~+SR counterparts have provided us with very valuable information. The reason for this is clearly that, in contrast to p+SR, investigations making use of the e + spin polarization can be performed only on samples which may serve as polarimeters. The classes of materials that may be investigated are thus severely restricted, particularly in the case of magnetically ordered materials (see Seeger et al. 1987). This is in marked contrast to p+SR, for which the sample requirements are much less restrictive. Nevertheless, since both approaches to measuring e + polarization mentioned above can provide us with information that is hard to obtain otherwise, it is certainly worthwhile to develop them further. The present contribution is one of a series of three papers (see also Gessmann et al. 1997a, Gessmann, Harmat, Major, and Seeger 1997b) that review recent progress in the study of positronium in condensed matter with spinpolarized posilrons. It will be seen that this field is an excellent example for the transfer of atomic-physics methods to other areas of physical research. On the other hand, it is hoped that some of the new results to be reported will have an impact on the atomic-physics research on positronium. Concentrating on the theoretical aspects, the present paper discusses in Sect. 2 the properties, in particular the annihilation characteristics, of positronium in a magnetic field. They are well known for 1S-Ps in vacuum; the present discussion emphasizes the assumptions that have to be made when the results obtained for Ps in vacuum are to be applied to Ps in matter. A new (in our opinion improved) nomenclature for the energy eigenstates of 1S-Ps in a magnetic field will be introduced. Sect. 3 uses the density matrix approach for calculating the evolution of the Ps states when polarized e ÷ form positronlum in the presence of a magnetic field. The treatment includes, in addition to the self-annihilation of Ps, the so-called pick-off annihilation and the spin-exchange processes. Again, some of the topics treated may already be found in the literature. Instead of putting together results obtained by different authors, including ourselves, and trying to harmonize the various notations, we have preferred to give a unified treatment which is believed to be in several respects more straightforward and more transparent than earlier accounts. In Sect. 4 we survey the experimental aspects of the field, including a short account of the recently improved Stuttgart set-up for studying the relaxation of e + spin polarization (e+SR) in matter described in greater detail elsewhere (Gessmann et al. 1997a). Finally, Sect. 5 treats the data analysis and illustrates it by an experimental example obtained with the Stuttgart set-up. The Laplace-transformation treatment on which Sect. 3 is based will be seen to be particularly well adapted to the analysis of the e+SR experi-


389

ments on Ps. For further details and results the reader is referred to one of the two accompanying publications mentioned above (Gessmann et al. 1997b).

2 P o s i t r o n i u m in a m a g n e t i c field The following discussion of the influence of a static magnetic field B on the ground state properties of positronium and on positronium formation in condensed matter is based on two fundamental assumptions: (a) As in the absence of external fields (sec Appendix A), the total Ps wavefunction m a y be separated into a spatial part and into a spin part. The spatial part is not affected by the magnetic field; the infiuencc of the surrounding matter is taken into account by the parameter ~ (Eq. 1.3), which in general differs from its vacuum value unity. (b) The spin Hamiltonian differs from that of vacuum Ps in zero magnetic field only by the Zeeman term -/~P8 • B, where/~p~ is the positronium magnetic moment. This assumption implies that, apart from the Zeeman term, the hypetfine interaction term remains isotropie even in the spin-one states. This is not trivial, since in general spin-one particles do possess an electric quadrupole moment, which may interact with electric-field gradients at the Ps site. The corresponding interaction term may be absorbed in the spin Hamiltonian and gives then rise to an anisotropic hyperfine interaction. However, in self-conjugated systems such as isolated Ps this term is strictly zero (Baryshevsky and Kuten 1977). Therefore, a detectable anisotropic hyperfine interaction for Ps in matter can only arise if, due to a strong interaction with the electronic system of the host, the eJfeclive masses of the positrons and electrons forming positronium are sui~ciently different (Baryshevsky 1984, Bondarev and Kuten 1994). This might be the case if, e.g., Ps atoms play a similar r61e as hydrogen atoms in hydrogen bridges. One approach to the problem of a possible anisotropic hyperfine-interaction term in the spin Hamiltonian is to assume that the hypetfine interaction is isotropic (i.e., that it differs from that of Ps in vacuum only by the normalization factor i¢), to work out the consequences of this assumption, and to compare the i¢ value required to fit the experimental data with the i¢ value that may be obtained from precise measurements of the lifetime of para-Ps (cf. Eq. 1.2). A discrepancy between the "lifetime i¢" and the "hyperfine s¢" will indicate that the assumption of an isotropie hyperfine interaction was inadequate, since there can be little doubt that assumption (a) is valid for the magnetic fields used in laboratory experiments (see Appendix A). An alternative approach to the anisotropy problem is to study Ps in single crystals at different crystallographic orientations of the magnetic field. In the absence of a magnetic field the eigenstates of the spin Hamiltonian are also eigenstates of the S 2 (square of the total spin) and Sz (projection of the total spin onto the axis of quantization) operators. They are the singiet state 1So (total spin zero)

390

Io, o> =

1

and the triplet states 3S 1 (total spin

11, o) =

1

(I T£>-IH>)

(2.1)

one) (I

[ I , I > = I TT> , II,-I) = l

÷ I (2.2)

On the left-hand sides of (2.1) and (2.2) the first number gives the total spin, the second one the so-called magnetic quantum number m. In the symbols on the right-hand side the first one of the up-or-down arrows refers to the e + spin, the second one to the e- spin. As discussed in Sect. 1, the hyperfine interaction between the e + and the e- spins leads to the "hyperflne splitting" AEo_p between the 11S0 and the 13S1 state. If the hyperfine interaction is isotropic, as will be supposed for the remainder of the paper (cf. assumption (b) above), in zero magnetic field all three states (2.2) are degenerate. Neither the 1S0 nor the 3S1 states of positronium carry a magnetic moment, as may be seen in the following. In the singlet state there is no preferred spin direction, hence the magnetic moment must be zero. In the triplet states the magnetic moments of the two particles are opposite, hence there is no total magnetic moment either 4. The vanishing of the magnetic moment of Ps in the absence of an external field means that there cannot be a linear Zeeman effect. The following arguments (Wolfenstein and Ravenhall 1952) show that the 3S 1 states with m : +1 are, in fact, not split at all by a magnetic field B . A rotation by 180 ° around an axis perpendicular to B will interchange the levels m : +1 and reverse the sign of B . By subsequently applying the charge conjugation operation, which leaves the Ps Hamiltonian invariant, the magnetic field can be brought back to its original direction. Hence the two energy levels must be equal. Furthermore, since the sum of the two energies is not affected by the magnetic field, the m : +1 levels must be independent of B. An alternative way to derive the preceding result is the following. If we identify the direction of the magnetic field with the quantization axis, the spin Hamiltonian retains its rotational symmetry around this axis even in the presence of a magnetic field. Therefore, the projection of the total spin remains a good quantum number irrespective of the magnetic field strength. Since the magnetic moments of e + and e- are equal and opposite, the total magnetic moments of the states m : +1 are exactly zero even in large magnetic fields; hence the energies of these two states are not changed by a magnetic field of any strength. 4 In the case of Ps the conventional designation of the eigenvaines of Sz as "magnetic" quantum number is thus somewhat misleading.

Posltronlum in Condensed Matter

391

An analogous reasoning is clearly not applicable to the m : 0 states (2.1) and (2.2), since in the presence of a magnetic field the lotal spin is not a good quantum number and since the "magnetic" quantum number m, which is preserved, is the same for both states. In a finite magnetic field the eigenstates of the Hamiltonian may acquire non-vanishing magnetic moments. The Hamiltonian of a system of two spins 1/2 interacting isotropically with each other in an external magnetic field is known as the Breit-Rabi Hamiltonian (Breit and Rabi 1931). The "Breit-Rabi diagram" (energy eigenvalues E vs. magnetic field, apparently established for the first time by Darwin 1928) of Ps may be obtained from the general case (different gyromagnetic ratios of the two spins) by specialization. Suitable dimensionless variables are E/AEo_p and z ::

4geB/AEo-p

(2.3)

Or

:= Arcsinh z

(2.4)

The use of~ instead o f z has several advantages. It results in more transparent mathematical expressions. At small fields ~ = z + O(~ 3) holds, so that the difference between linear and quadratic Zeeman effects is maintained in E-vs.plots. In large fields, however, ~ grows only logarithmically with increasing B; hence in ~ plots a wider B range can be covered than in z plots. Fig. 2.1 shows the Breit-Rabi diagram for positronium. The zero on the energy scale has been chosen in such way that the sum of the four energy values (which is independent of B) is zero. As discussed above, the energies of the states m = -4-1 are independent of B; with the normalization just mentioned they are

(2.5)

E(fft = ±1) = ZIEo-p/4 The other two energy eigenvalues, given by /

E(fft = 0 ) - : --(AEo_p/4) [1 =t=2~¢/1+z 2) : - - ( A E o _ p / 4 ) ( 1 + 2Cosh~)

show a quadratic Zeeman effect.

,

(2.6)

392

s[T] 0 2

1 ,

2 ,

4

8

16

32 + 1.5

"meikt

+ 1.0 ,13Sl

?

~

odho-Ps (m=.t 1)

+0.5 0.0

LU

t-,--t

>

E

-0.5 -

1.0

-1.5 -2 0.0

I

0.5

1.0

1.5

2.0

2.5

3.0

Fig. 2.1. Energy eigenvalues vs. magnetic field (Breit-Rabi diagram) for 1Spositronium. The energy E is measured in units of the energy difference between o-Ps and p-Ps; the zero of the energy scale is such that the magnetic-field independent sum of the four energy eigenvalues is zero. The right-hand scale corresponds to ~ = 1. For the definition of the dimensionless variable ~ see (2.3) and (2.4), for the nomenclature of the positronium energy eigenstates in finite magnetic fields see Appendix B. The eigenvectors of the m = + 1 states are not changed by the application of the magnetic field; they are given by (2.2). The eigenvector that for B --* 0 reduces to the singlet state (2.1) is [0, O) + 2 - 1 / 2

--

I1, O)

= [Cosh(U2). to, o> + SinhIU2). I', 0)[ = 2-1/2 [exp(~/2) • I t,~) - exp(-~/2) • I ,LT)] v / - S - ~

,

(2.7)

that reducing to the m = 0 triplet state is

-2-1/211-(l+z2)-i/2jr|11/210,0)+

2-1/~[1

+ (1 +z2)-1/2]

1/2 11,o)

= [ - Sinh(~/2). I0,0)+ Cosh(~/2). 11,0)] S ~

= 2-1/2 [exp(-U2). I T~>+ exp(~/2). I~T>[S ~ L

J

(2.8)


393

The left-hand sides of (2.7) and (2.8) give the decomposition of the eigenvectors into the triplet and singlet contributions, whereas the exp(~/2) terms on the right-hand sides correspond to the dominant terms in the high-field regime. Owing to our restriction to magnetic fields that do not affect the spatial part of the Ps wavefunetion (cf. Appendix A), we may argue intuitively that the annihilation rates of the states with m = +1 are equal to that of the triplet states and that the annihilation rates of the states (2.7) and (2.8) are determined by the relative weights of the ringlet and triplet states in them and by the annihilation rates of these states. If for simplicity we restrict ourselves to the principal quantum number n = 1, this reasoning gives us for the annihilation rate of the state (2.7)

)~pp-ps = [Cosh2 (~/2))~p- ps + Sinh2 (~/2)Xo-Ps] Sech~ 1

1

= ~(~p-P8 + ~o-Ps) + ~(~p-P~ - ~o_p~)Sech~

(2.9)

and for that of the state (2.8)

~m-Ps---- [Sinh2(~/2))~p-p~ + Cosh2(~/2)~o-Ps] Seeh~ 1 15 = ~(J~p-Ps -}- ~o-Ps) -- ~( p-Ps -- )~o_Ps)Sech~

(2.10)

The lifetime of the state (2.7) becomes -1

pPs

psi1 Seth,Sinh,,J2(1 Ps]opsj m

"Fp--ps 1 - Sech~ Sinh2(~/2)

(2.11)

'

that of the state (2.8) becomes

Tm-Ps----ro-Ps

[

[1 + Seeh~Sinh2(~/2)(v°-p----k -Tp-Ps

1

+ Sech~ Sinh2(~/2)] -1

To---Ps

Tp_ Ps

J

-1 1)]

(2.12)

(The notations )~pp-v~, )~m-e~, rpp-V~, and rm-ps will be justified below.)

394

B[T] 0

1

2

4

8

16

32

10 4

(a)

~ o r t h o - P s (m =.+ 1)

10 -e

10 3

10 .7

~

{ 102

101

s" (m = O)

10 8 10 .9

"plesiopara-Ps" (m 0 )=~

~

10 o 11S 0 0.0

~

10-1o

para-Ps

J

L

I

I

I

I

0.5

1.0

1.5

2.0

2.5

3.0

B [mT] 0 104

200

400

600

I

I

I

~//

13Sl

(b) ,/ortho-Ps (m = +- 1)

103

10 -8 10 .7

102

"meikto-Ps" (m = 0)

101

10-8 10-9

"plesiopara-Ps" (m = 0)

100

101 o X11S 0

0.00

I

I

I

0.05

0.10

0.15

0.20

X Fig. 2.2. The lifetimes of positronium energy eigenstates (left-hand scale: norrealized by the pazapositronium lifetime rp_p~; right-hand scale: unnormalised for = 1) for two different ranges of the magnetic field. (Top scales: magnetic field for = 1; bottom scales: reduced magnetic fields - for definitions of z and ~ see (2.3) and (2.4)) a) wide B range, b) small magnetic fields (z ~ ~). For the states (2.7) and (2.8) Dick et al. (1963) introduced the n a m e s pseudo-singlet (pseudo 1So) or pseudo-triplet (pseudo 3S1), respectively. T h e y are widely used in the literature (see, e.g., Berko a n d Pencl]eton 1980, Du-


305

pasquier 1981). As argued in Appendix B, we do not consider this to be a good nomenclature. From a practical viewpoint the most important property of these states is the dependence of their lifetimes on the magnetic field. It is determined by the decomposition of the states into the slowly annihilating triplet state I1, 0) and the rapidly annihilating singlet state t0, 0) as exhibited on the left-hand sides of (2.7) and (2.8). For very large magnetic fields the lifetimes of both states approach the value 2rp-Ps. (We neglect here and in the following terms of the order Tp_Ps/To_Ps ~ ]0-3.) However, this limiting case is far outside the regime of terrestrial experiments. At small or moderate magnetic fields the lifetimes of these two states exhibit very different dependences on the magnetic field strength, as is illustrated in Fig. 2.2a and Fig. 2.2b. We may easily deduce that a magnetic field which increases rpp-Ps compared to its zero-field value rp-ps (eq. 2.11) by /3Tp_ps (/3 > 0) leads to a decrease of rm-ps compared to its zero-field value ro-ps (eq. 2.12) by ~To--Ps~p--Ps/~o--Ps ~ 103~To-Ps. To illustrate this result, consider B = 1 T. In the case t¢ = 1 this corresponds to z = 0.276, ~ = 0.273. According to (2.11), rpp-ps is about 1.02rp_Ps, i.e., only by a few picoseconds longer than the zero-field value rp_ps, whereas rm_ps is smaller by a factor of 20 than the zero-field value ro-Ps. We see that the "mixed" character of the m = 0 states (from the point of view of the annihilation modes) makes itself strongly felt in the lifetime of that state that for B = 0 reduces to one of the triplet states, whereas for not too large B the lifetime of the other m = 0 state is almost that of parapositronium. In keeping with the Greek origin of the prefixes ortho and para (op~6s = straight, ~rc~p& = alongside) we propose to call the long-lived m = 0 state meiktopositvonium (m-Ps) and the short-lived one plesioparapositvouium (pp--Ps) 5. As discussed above, an external magnetic field does not affect the 13S1 (m = ±1) states, hence their "intrinsic" lifetime is that of orthopositronium. It is therefore appropriate to refer to them as o-Ps even in strong magnetic fields. Perhaps the most obvious consequence of the "mixing" of the m = 0 states in a static magnetic field is the reduction of the ratio of 37-annihilations (from the triplet states) to that of the 2~,-annihilations, which for isolated Ps in zero magnetic field is 3. This effect is known as the "magnetic quenching of o--Ps". As early as 1951 it was employed by Deutsch and Dulit (1951) in measurements on Ps in freon gas (CC12F2), for the first experimental determination of AEo_p. An early theoretical treatment of the magnetic quenching of o-Ps, including resonance effects due to alternating magnetic fields, was given by Halpern (1956). The fact that the intrinsic lifetime Tm_Ps is significantly shorter than l"o-ps already in rather small magnetic fields has been used by AI-Ramadhan 5 For the etymological justification of the proposed nomenclature the reader is referred to Appendix B.

396

and Gidley (1994) to determine 7"p_Ps experimentally. They determined the m - P s lifetime in a gas mixture of N2 and isobutane in a magnetic field of 0.4 T. The obtained value for the mean lifetime of rp_ps [cf. Eq. (2.12)], rp-ps = (125.142 ± 0.026) ps

,

(2.13)

is in good agreement with the theoretical value (1.10). In condensed matter, the lifetime of o-Ps is always limited and that of m - P s often limited not by the intrinsic annihilation processes discussed so far but by the so-called pick-off annihilation (cf. Sect. 1) or by spin-exchange processes (Ferrell 1958). If this is the case, it may be difficult to distinguish experimentally between o-Ps and m-Ps. A common name for them might then be useful. Since they have in common that they survive at positron ages at which pp-Ps has virtually disappeared, we propose to designate them operationally as leipopositronium (1-Ps).

3 Density m a t r i x description of the spin states of positronium The temporal evolution of the occupation numbers of the four substates 10, 0), I1, 0), I1, 1), and I1,-1) (cf. Eqs. (2.1) and (2.2)) of the positronium ground state (1S) may be described by a 4×4 spin density matrix p(t) obeying the LiouviUe equation of motion

ihdP( ) : np(t) - p ( 0 S + dt

(3.1)

In (3.1) H = Hhf --//Ps" B - ih~/2

(3.2)

is the "magnetic Ps Hamiltonian", where Hhf describes the hyperflne interaction, - / ~ p s . B is the Zeeman term, and H + the adjoint of H. The annihilation term - i h ~ / 2 , to be discussed below, causes the Hamiltonian (3.2) to be nonHermitian. In Subsection 3.1 we shall take into account, in addition to the selfannihilation of Ps discussed in Sects. 1 and 2, the effect of pick-off annihilations. The influence of spin-exchange processes (Ferrell 1958, Mills 1975) on the elements of the spin-density matrix is discussed in Subsection 3.2. It is shown that as long as the spin polarization of the Ps-forming electrons may be neglected, we may derive first-order rate equations governing the dependence of the diagonal elements of the e+ spin-density matrix and hence of the populations of the four Ps substates on the positron age t. Subsection 3.3 solves these equations, making full use of the Laplace-transformation technique. As will be shown in Sect. 5, from these results the quantity required for the comparison with experiment may be easily obtained.

Positroninm in Condensed Matter

397

3.1 P i c k - o f f p r o c e s s e s If in addition to self-annihilation pick-off annihilation processes are taken into account and if a static magnetic field B is applied in the z-direction, the matrix representation of the Ps magnetic Hamiltonian in the basis {J0, 0), ]1, 0), Jl, 1), }1,-1)} reads 1 /-

H=~

0 0 0

~ 0 0

+ ;po -1~

0 0 ~.,' 0 0 ~

2~eB

0 0 0

-|0

0 0 0

o

o

o

0

lo_p~ + Apo

0

0

0

Xo-Ps + J~po

) "

(3.3)

In (3.3) we have introduced hw := AEo_p (see Eq. (1.4)) and the pick-off rate Apo (= rate of 27 annihilation of the e + in Ps with a "foreign" e- of opposite spin). If all annihilation terms are neglected, the diagonalization of (3.3) gives us the energy eigenstates and the energies in a magnetic field as specified in (2.5) - (2.8). The annihilation rates (2.9) and (2.10) of pp-Ps and m - P s are obtained if in (3.3) the pick-off term Apo is neglected and the remaining annihilation terms are treated as small perturbations (this is justified since Ap-ps/~ -~ 5 . 1 0 - 3 ) . The initial condition p(O) required for the solution of (3.1) may be derived as follows, e + and e- are particles with spin 1/2, hence their spin density matrices are given by (Blum 1981) Pe-,e+ -----~

Pio'i

1+

,

(3.4)

where Pi is t h e / - t h component of the spin polarization vector, 1 is the unity matrix, and ~ri are the Pauli matrices. If we choose the direction ore + emission from the/3+-active source as z-axis ( P = (0, 0, 7~)), the spin density matrix of the positrons is given by 1(1+~ Pc+ = ~ 0

0 ) 1 - 7'

(3.5)

In this subsection we assume that the e- involved in the Ps formation are not spin-polarized; hence their density matrix is 0e-=2

0

1

In the basis {I tT), I TJ~), I ~T), I ~J.) } the initial spin density matrix is given by the direct product Pc+ ® Pc-. From this the initial condition in the

398

basis { I0, 0), ll, 0), ll, 1), ll,-1) } is obtained by an orthogonal transformation, giving us 1

p~oj:~,,

1

0

0

0

o l+p

0 o 1-7~

(3.7)

The Liouville equation (3.1) with the Hamiltonian (3.3) leads to a set of coupled first-order differential equations for the components p0(t). Treating the annihilation term A in the Hamiltonian as a small perturbation, the firstorder perturbation-theory solution for the initial condition (3.7) gives us the

followingnon-vanishingcomponents of p(t): P11(t) : - ~ ~

Tanh2~ cos (w t Cosl~)7~ exp(-A t)

+ ¼Sech~ Cosh2 (~/2) (1 + 7~Tanh~) exp(-All t) + ¼Sech~ Sinh 2 (~/2) (1 - 7~Tanl~) exp(-A22 t) ,

(3.8a)

p2~(t) : -~~ T a n h 2 ~

cos(w t Cosh~)7~ exp(-A t) + ¼Sech~ Cosh ~ (~/2) (1 - 7~Tanh~) exp(-X~2 t) + ¼Sech~ Sinh 2 (~/2) (1 + P Tanh~) exp(-A11 t)

,

(3.8b)

p12 (1~) : ¼ [Sech~ cos(w t Cosh~) + i sin(w t Cosh~)] 7~ Sech~ exp(-A t)

1Tanh~ [(1 - 7~Tanh~) exp(-~22 t) m

(1 + "PTanh~) exp(--~ll t)]

,

(3.8C)

P21(t) = ¼ [Seche cos(,~, Coshe) - i sin(,~ t Cosh~)] ~Seche e~p(- A t)

+ 1Tanh~ [(1 + P ~ n l ~ ) exp(-All t)

(3.8d)

- (1 - ~'Wanh~) e x p ( - X ~ t)] p~3(t) = ¼(1 + ~') exp(-~3~ 0 p44(t) = ¼(1 - ~') e x p ( - ~ 0

(3.8e) (3.8f)

, ,

where ~11 :~--- Xpp-Ps -~ "~po ,

X22 := Xm-p, + Apo , X33 := Xo-ps + Xpo ,

(3.9a) (3.9b) (3.9c)

and 1X

1X

(3.10)

Positronlum in Condensed Matter

399

For later use we note that - - in contrast to All and ,~2 - - A33 and A, and therefore also ~11 +~22, are independent of the magnetic field. The parameter characterizing the magnetic field has been defined in (2.3) and (2.4). The diagonal elements of p(t) are the occupation numbers of the basis states { [0, 0), [1, 0), [1, 1), [1,-1) } and hence the quantities determining the annihilation characteristics of Ps. For instance, the fraction of Ps annihilating at a time t by 27 self-annihilation is given by ~p-PsPll. From (3.8a - b) it can be seen that the decay of the occupation numbers of the states 10, 0) and I1, 0) is described by two exponentials with the annihilation rates All or A~2. This is a consequence of the fact that, as discussed in Sect. 2, in a magnetic field pp-Ps and m - P s and not the states I0' 0) and I1, 0) are eigenstates of the Hamiltonian. The "magnetic" elgenstates (2.7) and (2.8) will in the following be denoted as Iepp_ps) and Iem-ps). In the basis { [¢pp_p~), Iem_ps), ll, 1), [1,-1) } the HamUtonlan (3.2) is diagonal. For the diagonal elements of the spin-density matrix the Liouville equation (3.1) reduces to the rate equations ape(t) _ dt

]kii pB(t)

(3.11)

where ~ii stands for the annihilation rate of the i-th basis state (~33 = ~44). The general solution of (3.11) reads

p (t) : p (o) exp(- , t)

(3.12)

The initial values p~(0) may be deduced from (3.7) by means of the orthogonal transformation p~(O) : Mp(O)M t , (3.13) where

M

=

v/se-a~ Cosh(~/2) -v/'~--~Sinh(~/2) 0 0

v / ~ - - ~ Sinh(~/2) V/-S--d~-~Cosh(~/2) 0 0

0 0 1 0

0 0 0 1

) (3.14)

transforms the states 10,0) and I1,0) into the Ps energy eigenstates in a magnetic field, ]@pp_p~) and ]~m-Ps), and where the superscript t denotes the transposed matrix. We thus obtain 1 pB(O) --

1 + ~ rranl~ "P Sech~

~P Sech~ 1 - "P Tanh~

o

o

1+

o

0

0

0

1-~

The only non-vanishing

pB12(*)

0 0

non-diagonal elements

0 0

\

)

are given by

= p~; (t) = plB2(o) e x p ( - i w f Cosh~) e x p ( - A t)

400

where the superscript * denotes the complex conjugate. They are oscillating with the same circular frequency w Cosh~ (_~ 1.3 • 1012 s -1) as the oscillatory terms in (3.8a - d). Since these oscillatory terms cannot be observed experimentally, they may be neglected. The great advantage of the use of the basis { [~pp-Vs), I~m-vs), I1, 1), [1,-1) } is that, in contrast to the nondiagonal elements in the basis { 10,0), I1,0), I1, 1), I1,-1)}, the non-diagonal elements pB2 and pB1 consist of an oscillating term only. Therefore only the diagonal 'elements of the spin density matrix have to be taken into account in the further treatment. This leads to a great simplification of the Liouvil]e equation if spin-exchange processes are to be included (see Subsect. 3.2). Then the spin density matrix in the basis { [0, 0), 11, 0), 11, 1), [1,-1)} may be calculated from (3.12) and (3.16) making use of the transformation

p(t) = M t p B ( t ) M

(3.17)

If the non-diagonal elements (3.16) are disregarded, we o b t a i n - except for the unobservable oscillatory terms - - the same results as in (3.8a - f). The preceding discussion shows that from a practical standpoint the set of tale equations for the diagonal elements of the density matrix in the basis of the eigenstates of Ps in a magnetic field used by Stritzke (1991) give the correct result for the occupation numbers of the four Ps states [0, 0/, 11, 0/, [1, 1), and [1,-1), i.e., the quantities determining the annihilation characteristics (cf. Sect. 2). The subject will be taken up again in Subsect. 3.2, where we shall derive a more general solution that includes spin-exchange processes. 3.2

Spln-exchange processes

The situation becomes more complicated if Ps spin-exchange processes are to be included. A necessary condition for spin-exchange processes to occur is that the system in which Ps is formed contains unpaired electrons, since otherwise the exchange of the electrons in Ps with electrons of opposite spin in the matrix is forbidden by Pauli's exclusion principle. Spin-exchange processes were first observed by Deutsch (1951a) through the decrease of the probability of 37 annihilations due to the conversion of o-Ps into p-Ps in N2 gas to which paramagnetic NO molecules had been added (cf. Sect. 1). The first correct theoretical treatment of spin-exchange processes is due to Ferre]] (1958). Spin-exchange processes (also known as "spinconversion") may play an important r61e in Ps-forming organic liquids containing paramagnetic additions. Well studied examples are methanol and benzene containing 4-hydroxy-2,2,6,6-tetra methylpiperidine-l-oxyl (HTEMPO) (Billard, Abb6, and Duplgtre 1991, Schneider, Seeger, Siegle, Stoll et al. 1993, Major, Schneider, Seeger, Siegle et al. 1995, Castellaz, Major, Schneider, Seeger et al. 1996). The diffusion-reaction problems arising in this context have been treated by Wiirschum and Seeger (1995). We confine ourselves to paramagnetic materials. The spin polarization of the unpaired e- in a magnetic field B is then given by


(#eB'~

Pc- = Tanh k, kB T ]

401 (3.18)

'

where kB is Boltzmann's constant and T the absolute temperature. The probabilities that the spin of the unpaired e- is paralld (T) or antiparallel (,[) to the magnetic field are (1 + ;De-)/2 and (1 - :Pc-)/2, respectively. The spin-density matrix of the unpaired electrons, Pue-, is given by (3.5) with replaced by :Pc-. Following Ferrell (1958), let us picture the individual spin-exchange processes as "collisions" between Ps atoms and unpaired electrons with spin either up (T) or down (1). Since there are four possible Ps states, we have to distinguish eight different eases. As an example, consider collisions of electrons with spin up with Ps in the state I1, 0). The collisions change the state of the system "unpaired electron + Ps" from

I T) I~, o) = I T) ~ (I T 1) + I ~ T)) before the collisions to aex

aex

(gd -- T )

aex

I t) [1, 0) + ~

aex

aex

[ ]) [1, 1) + ~-- I T) I0' 0)

(3.19)

after the collisions, where ad is the "direct" and aex the "exchange" amplitude of the collision. The minus sign of the last term on the left-hand side of (3.19) is due to the fact that this term describes the exchange of two electrons with parallel spins (hence with a symmetrical spin wavefunction) and that this must result in an antisymmetrical spatial part of the wavefunction. The remaining seven possible cases may be handled in analogy to (3.19). Spin-exchange processes can formally be described by a "spin-exchange operator" U which transforms the states before a collision into those after the collision. E.g., U] T)I1, 0) corresponds to (3.19). The spin-density matrix after the collision, ~=on, is then related to that before the collision, ~, by the transformation

~o. = u ~ u+

,

(3.20)

where the superscript + denotes the adjoint operator. The superscript ^ indicates that in (3.20) the spin density matrices are taken in the basis { 1¢1) = 1'I')1o, o),1¢5)= I I)lo, o), I¢a)= I T)ll, O),..., 1¢8) = I I)l 1,-1)}. Since before a collision the spin of the unpaired electron and the Ps spin state are not coupled, ~ is the direct product of the Ps spin-density matrix p and the spin-density matrix Pue- of the unpaired electrons. The matrix elements of ~:on are given by

(0~1~:o. lea = (0~1u ~ u+ lea = ~(¢dUCk)(¢kl.~lCz)(U¢,l¢./) kl

,

(3.21)

402 where the completeness relationship ~-~k [ ~ ) ( ~ k l : 1 has been used. The right-hand side of (3.21) can be calculated from the known effects of U on the wavefunctions I~i) and on the matrix elements of ~. Because in the experiments the spins of the unpaired electrons are not controlled, we have to sum over their different orientations in order to obtain a relationship between the elements of the Ps spin-density matrix before and after the collision. This gives US pcol111: [O,d __ ~a.ex[12 Pll ÷ (~adl* aex -- lI¢/'ex[2) "Pe- PI2 1 ad aex * -- lla-exl 2) ~Oe- P21 ÷ ( ~"

÷ 41 [aex[2 [P22 + (1--'Pc-) P33 + ( l + ' P c - ) P 4 4 ] ,

(3.22a)

p~oo,, : I-d - ~-oxll ~ p ~ + (~adl .ox. - ¼1~.1 ~) ~'e- Pl~ 1 *

+ ( ~ d .ox - ¼1~oxl~) ~'o- p~l +¼1a~x12[p11 + ( 1 - ' P ~ - ) p 3 3 + (1-t-'P~-)p44] coll 1 2 PI2 : lad -- 5aex] P12 +

1

]aex] 2

[

,

(3.225)

(1 -- " ~ e - ) P33

L

(1 -'f- ~e-)

P44 -I- P21] 1 * + (5-d,~o~ - ¼1,~o~l~) ~'o- pll

+ ( ~1 " a " c*x - ¼laexl ~) ~ ' ~ - p ~ ,

(3.22c)

pcoll 1 2 [ 21 :--- lad - ~a0,t ,021 + ¼ la0xl ~ (I

- -

~ 0 - ) .3~

(1 ÷ ' P c - ) P44 ÷ P12] 1

*

+ (~=d =0~- ¼1-~xl ~) ~'~:- pll 1 * ~ox - 11.o~1~) ~,o- p ~ , + (~-~

(3.22d)

p~'~ -- ½ [I.d--.0~l ~ (1 + T~e- ) + lad[ 2 ( 1 - ~ - )

]p33

+ ¼ laexl 2 (1 ÷ ~e-) [Pll ÷ P22 ÷ P21 ÷ P21] , 044 : ~ l a d - a ~ l ~ ( 1 - ' P e - )

÷ ladl 2 ( 1 + 7 ~ e - )

÷ ¼ [aex[2 (1 -- Pc-) [Pll ÷ P22

-

-

P21 -- P21]

(3.22e)

P44

(3.22f)

For e+ polarized in the z-direction, the only non-vanishing non-diagonal elements of the density matrix are P12 and P2I (see (3.7)). Since spin-exchange processes with unpaired e- that are either unpolarized or polarized in zdirection cannot lead to a net polarization in the z- or y-direction, P12-c°nand p~]U will be the only non-vanishing non-diagonal elements of the density matrix after the collision. The conservation of the total number of Ps "atoms" during spin-exchange processes leads to the following relationship between the traces:

Positronium in Condensed Matter Tr(p) := E

p" : Tr(pC°'z)

403 (3.23)

ii

Together with the postulate of random phases, i.e., the fact that the terms a~ aex and ad aex vanish when we average over many spin-exchange collisions, inserting (3.22) into (3.23) gives us ladl 2 + laexl 2 = 1

(3.24)

If laal 2 and the mixed products a~ a~x and ad aex are eliminated from (3.22) by making use of (3.24) and the random-phase postulate we find that the quantities Pij _con -Pij, i.e. the average change of Pij effected by one "collision", is proportional to la~xl2. This suggests that we may obtain the rate of change of Pij by exchange processes, to be denoted by [d Pij/dt]ex, by multiplying pcjol]- pij with the "collision frequency" Ucon. The product UcoUlaexl 2 may be identified with the rate constant kex of spin-exchange processes. This gives us finally

[dpH/dt]ex : hex [ - - ~3P l l -~- ~022 1 -~- 1 (1 - "Pe-) 033 _~.~1 (1 + '~e-) 044 -- 41"Pe- (012 "4- 021)] 3

1

+ 1 (1 + ~ e - )

1 (1 --

,

(3.2 a)

,

(3.25b)

7~,_ ) 033

044 - ~P¢- (012 + 021)

]

[dp12/dl[]e x : kex -- ~'0123 "4- ~'0211 "4- ~'1 (1 - Pe-) 033

,

1 (l_{_~e_) 044 _ i ~ e _ (011 -4-022)

4

[dml/dt]ox=

41 (1 +

[d 033/d $]ex =/¢ex

-f- ~

[dp44/dt]ex

]

(3.25c)

021 + ¼Pl + 1 ( 1 - ~De-) 033

~e-) 044

-- ~ ~lDe- (011 @ 022)

(3.25d)

_ ~1 (1 - ~ e - ) 033

011

022

012

021

l%x

The spin-exchange terms (3.25) may be inserted into the Liouville equation "by hand". For temperatures above 100 K and magnetic fields not exceeding 1 T the spin polarization of the unpaired electrons, P~-, is less than

404 0.01. In many applications we are thus allowed to assume that the two spin directions of the unpaired electrons have equal probability and to set ~eequal to zero. The following calculations make use of this simplification. The fact, demonstrated at the end of Subsect. 3.1, that without spinexchange processes the solution of the Liouville equation can be easily obtained when we choose the energy eigenstates of Ps in a magnetic field as basis functions suggests that it will be advantageous to express the additional spin-exchange terms in terms of the elements of the spin-density matrix in the basis { ICpp_ps), ICm_Ps), I1, 1), I1,--1)}. Transformation of (3.25) to this basis by means of the orthogonal transformation M (see (3.13) and (3.14))" gives US

[dPlB1/dt]ex : _ 1 kex [(2 + Sech2~) plB1 -- p~2 Sech2~ -

(P~2 + P~I) Tanh~ Sech~

- (1 + T a n h O p ~ - (1 - T a n h ~ ) p ~ ]

,

(3.26a)

,

(3.26b)

[dpB2/dt]ex = -¼ k~x [(2 + Sech2~) pB2 - P~I Sech2~ + (P~2 + P~I) Tanh~ Sech~ - (1 - Tanh~)p~3 - (1 + Tanh~)pB4]

[dpBl2/dt]~: _1/%~

[(4 - Sech2~)P~2 - (PRB1 -- P~2)Tanl~ Sech~

-- pB 1Sech2~ - (p3B - pB4) Sech~]

,

(3.26c)

[dp~l/d,]o~ : -¼ k~ [ ( 4 - Sech~0p~l - (p~ - p~2) Tanh~ Sech~ - Pls2Sech2~ -(P~3 - PL)Sech~J

[dpB3/d t] ex ~-----41 k~x [2P3B

(3.26d)

,

- (1 + Tanh~) PlB1

- ( 1 - Tanh~)pB2 - (pB2 + pBi)Sech~]

,

(3.26e)

[dp4B4/d/]~x = - I k e x [ 2 a S - ( 1 - Tanh~)plB1

- (1 + Tanh~) pB2 + (pB2 + pB) Seeh~]

(3.26f)

Eqs. (3.26) show that spin-exchange processes couple the diagonal elements to the non-diagonal elements PP2 and p2B1.However, since in the absence of spinexchange processes these non-diagonal elements oscillate with the circular frequency ~Cosh~ (see (3.16)), we may assume that on time scales large compared to (w Cosh~) -1 _~ 8 . 1 0 -13 s, there is no net effect of transitions between diagonal and non-diagonal elements. Neglecting these transitions then gives us the following rate equations for the diagonal elements of the spin-density matrix:

Positroniurn in Condensed Matter

405

~1---[~11 + lkox(2+ Sech~0].r, + ~ k~x p~ S~ch~ + (1 + T ~ ) +

p~3 ~ + (1 - Tanh~) p~, ,(3.27~)

+

+ ¼k~ [plB1Sech2~+ (1 - Tanh~) p3B3+ (1 + Tanh~) p4B4] ,(3.27b) 1 ~--- - -

P33

1

+ ~kex [(1 -I- Tanl~) PlB1+ ( 1 - - Tanh~) pB2] ,

(3.27c)

+ ¼kex [ ( 1 - 2"~.nh~)p~1 + (1 +Tanh~)P~2] .

(3.27d)

3.3 G e n e r a l s o l u t i o n o f t h e r a t e e q u a t i o n s

Our further strategy is as follows. We shall transform the system (3.27) into a system of 4 linear equations for the Laplace transforms OO

£B(s) := f pB(t) exp(-st)dt

(3.28)

0

These equations may easily be solved for £~i(s). The orthogonal transformation (3.17) then gives us the Laplace transform of the diagonal elements of the density matrix in the basis { 10, 0), [1, 0), 11, 1/, 11, -1 / }, the appropriate basis for determining the Ps fraction annihilating by 2~/self-annihilation and thus responsible for the "narrow components" of the 511 keV photon line (cf. Sect. 5). In the e+SR set-up referred to in Sect. 1 and briefly described in Sect. 4 it is this component that is used to measure the e+ spin relaxation. The set-up integrates over all positron ages. This has the consequence that we do not have to invert the Laplace transforms in order to compare experiment and theory but that it suffices to consider the quantities £~(0). Using the rdationship

f b~(t) exp(-st)dt =

S ~Bii($)

- p~(O) ,

(3.29)

0

we obtain by the procedure just outlined £1B1

+ PTanh~) a2

t

+kex (s+X33+kox) S e c h ~ ,

J

(3.30~)

406

1:~(,) - al - -"~]GeXa2{ [2 (s+)t33)(s+ All)+h~x (2s+ A33+All)] (1-7~Tanh~) +kex (s+X33+ke~) Sech2~,

(3.30b)

J

+ 2~(~ + ~11)(~+ ~ ) T~ ( ~ - ~1~) ko~(~ + k~) Tan~](~ + ~') %

(3.3Oe)

+ kex(al + k~x) (. + a + ko~)Sech~ / with the abbreviations a 1 := 2s + 2~33 -4- kex ,

(3.31a)

a2 := [a1(25-~- 2~tll 3I- kex)- ke2x] [a1(25-4- 2~t22 + kex)- ke2x]

+ kex(al + kex)[al(2s + 2A + kex)- ke2x]Sech2~

(3.31b)

The Laplace transforms of the diagonal dements of the density matrix in the basis { 10,0), I1,0), I1, 1), I1,-1)}, £ii(s;~), are obtained from (3.30) by the orthogonal transformation (3.17). This results in al + kex [(a, + kox)$ + k~, X33] (1 + ~'Tanh~ Se~h~) a2 t

+ ~5 (a, + k~x) Seeh2~ + al Seeh~ [~11Sinh2(~/2) (1 - 7) Tanh~) + )t22Cosh2(~/2) (1 + :P Tanl~) ] /

'

(3.32a)

£22(s;~)- al + kex ~ [(al + /%~)s + k~x:~Z3]( 1 - 3VTanl~ Sech~) a2 (

+ ~ (al + k~,) SeCh~ + a, SeCh~ [~1, Cosh~(~/2) (~ - ~ Tan~)

+ )t~.~.Sinh2(~/2) (1 + 79 Tanh~) ] / •33($; ~) = £B3(8 ] ~) , £44(s; ~) = £B4(S;~)

'

(3.32b) (3.32C) (3.32d)

The time evolution of the diagonal elements of the spin-density matrix in the basis { 10, 0), [1, 0), I1, 1), I1,-1) } is obtained by Laplace-inverting (3.32).


407

To do this one has to find the zeros of the fourth-order polynomial a2 : as(s). In general these four zeros are distinct. This leads to a representation of the population numbers pii(t) of the four Ps states in terms of four exponentially decaying functions. The quantities £ii(O; 4) required for the interpretation of age-integrating experiments [cf. Eqs. (5.1)] may be easily obtained from (3.32) and (3.30). 3.4 E x p l i c i t s o l u t i o n s for s p e c i a l cases Although, as repeatedly emphasized, for the comparison of the theory developed in this paper with age-integrating experiments Eqs. (3.32) need not to be transformed to the time domain, on occasions it may nevertheless be desirable to have explicit expressions for the age dependence of the populations of the various positronium states. As remarked at the end of Sect. 3.3, this requires finding the roots of an algebraic equation of fourth order. In general this is best done numerically. However, there are special eases in which the equation factorizes into algebraic equations of degree two or one, so that simple explicit solutions may be obtained. These cases are as follows: (1) Very large magnetic fields (Sech2~ = 0) (2) No magnetic fields (Tanh2~ = 0) (3) Negligible spin-exchange processes (kex : 0) From these special solutions, approximate solutions in the neighbourhood of the limiting cases may be obtained by perturbation theory. (1) Limit of very large magnetic fields (Seth2~ = 0) In the limit of high magnetic fields (tteB >> AEo_p) terms proportional to Sech2~ = (1 + z2) -1 may be replaced by zero. Then the fourth-order polynomial a2(s), whose zeros give us the decomposition of £ii(s; 4) into partial fractions "(cf. Eqs. (3.31) and Eqs. (3.32)) and hence the representation of £1i(g) as a sum of exponential functions of t, factorizes into two identical second-order polynomials. Their zeros follow from al(2s+2A+kex)=

ke~x ,

(3.33)

where the high-field relationship 2~11 = 2X22 = 2A was used (see (2.9), (2.10), and (3.10)). With (3.31a), Eq. (3.33) becomes (2s + kex) 2 + 2(A + )t33)(25 -~- kex ) -4- 4 A ,)t33

-- ke2x :

0

(3.34)

If we denote by rj (j = 1, 2, 3, 4) the negatives of the roots of a2(s) = 0, i.e. the quantities satisfying a2(-rj) = 0 we obtain from (3.34)

,

(3.35)

408 2rl

2r~ = A + ~33 + kox ± , / ( A - A33)~ + k~x

2r 3 = 2r 4

(3.36)

J Eqs. (3.32) simplify to $ -t- ~t33 + kex

Cl,(S; oo) = £~2(s; oo) = 4(s + rl)(s + r3) C3~(s; oo) }

s + A + kex

'

(3.37a)

(1 +

(3.37b)

C44($; OO) -- = 4(8--'~- ff)"~ T-r3)

Laplace-inversion of (3.37) gives us

o11(0 : p~(0 : ~ + (1 \

-

1 + ~/~-_ ~

¥~x

)

A - ~33 - kex

V/~-~--~333-~ ~ ~¢e2x) e x p ( - - r 3 t ) )

o~p(-r~0 ,

.4 - A33 + k~× + (1 + V/~-~-~333~ T ~e2x) exp(-r3 t) /

(3.38a)

(3.38b)

k

The fact that in the present hmit p11(t) is 7~-independent means that for z >> 1 we cannot obtain information on the e + spin relaxation from observing the 27 self-annihilation. (2) Limit of vanishing magnetic fields (Tanh2~ = O) For arbitrary ~, a2(s) = 0 may be written as tt~

[4(s+ A)(s+ A+/%x)-(Ap-p~- Ao_ p~)2] +4alkex(S+ A33)(s+ A+/%x) =

We see immediately that if ( = 0, (3.39) is satisfied by is solved by -,~ = -(~

a~

= o; hence (3.35)

+ ko~/2)

(3.40)

The remaining zeros of a2(s) have to obey the third-order equation 0,1{($ -~- a ) ( $ --I- A + ~ex) - (A - ~33) 2 }

+ k~x(S + A33)(s + a +/%x) -- 0 or

(3.41a)


(O.1 + ~ex){($ + A)($ + A + kex) - (A - A33 + k e x / 2 ) ( A - A33)} : 0

409

(3.41b)

Eq. (3.41b) is satisfied by (al + kex) : 0; hence

--T3 -- --(J~33 "}- kex)

(3.42)

is a further solution of (3.35). It follows from (3.41b) that the remaining two roots of a2(s) = 0, - r l and - r 2 , have to obey the quadratic equation (s + A) 2 + kex(S + A) - (A - A33)(A - A33 + kex/2) = 0

(3.43)

We thus obtain 2rl / 2 11/2 2 r 2 , = 2A + kex ± [(Ap-p, - Ao-P, + kex/2) 2 + 3k J 4 1

. (3.44)

Since - r 3 is a zero of the denominators of (3.30) and (3.32), for ~ = 0 the Laplace transforms simplify to s + A2~ + kex £11($; 0) = 4(s + rl)($ -~- T2)

'

£22(8; O) =

,

$ "+ All -}" kex 4(s + P1)($ -}- r2)

£33($; O) "~ _-/+(s) £44($; 0) J 8($ + 7'1)($ + r2)($ --~ 1'4)

(3.45a)

(3.45b) (3.45c)

with

f+($)----- [2($+ ~11)($+)~22) -4- kex(2$+ A + ~ 3 3 ) ] ( 1 + ~ ) + kex($ + A + hex)

(3.46)

Performing the inverse Laplace transformation on (3.45) gives us for ke× # 0 1 {1 - (Ap-ps - Ao-ps) + ~ex Pll(t)---~ ~ A } exp(-rlt)

1 (~p-Ps -- ~to-P,) + kex + ~{1 + A } exp(-r2 t),

(3.47a)

1 (J~p-Ps -- Jlo-Ps) -- kex p22(t) : ~{1 + A } e x p ( - r l t) + ~{11 _ (Ap-Ps

-- ~o-PS)A-

kex } exp(-r2 t),

(3.47b)

p33(t) } f+(-ra) e x p ( - r ] t) p44(t), : 4A(Ap_p-~ - ~ o - P s + A) _

f+ ( - r 2 ) exp(-r2 t) 4A(Ap_ps - Ao-Ps - A)

+

f+ ( - r 4 ) 2 [(

p-ps -

o-Ps) 2 - A2]

exp(-r4 t)

(3.47c)

410

with 2 A := [ ( ~ p - P s - Ao-ps q- kex/2) 2 q- 3kex/4 j11/2

(3.48)

In contrast to (3.45), Eqs. (3.47) are not in a convenient form for obtaining the limiting case ]%x : 0. This case is included in the remarks of Subsect. 3.4(3).

(s) Sph-ex h nge processes neg"g ble ( ox : 0) If the spin-exchange processes may be neglected (]%x : 0) the fourth-order equation cz2(s) : 0 separates into four linear equations with three distinct roots - r l , -~'2, -T3 : - e 4 . They are given by gq. (3.9) with ri : J~id ( i : 1 , 2, 3). As it should, the result for the density matrix p(t) agrees with (3.8) if there the terms oscillating with the angular frequency wCosh~ are neglected.

4 Experimental techniques The experimental techniques available for studying e+SR in condensed matter are modifications of the techniques estabfished for e + annihilation studies that disregard the e + spin polarization or use unpolarized e +. The information that may be derived from the e+e - annihilation characteristics perrains either to the positron annihilation rates (i.e., to the overlap of the e + and e - wavefunctions) or to the momentum distribution of the annihilating e+e - pairs. The three "classical" techniques are (see, e.g., Seeger and Banhart 1990) (i) e + lifetime spectroscopy, (ii) angular correlation of the annihilation radiation (ACAR), (iii) Doppler broadening of the 511 keV annihilation line (AE-r). The techniques (ii) and (iii) have in common that they both measure the momentum distribution but differ in the momentum components that they investigate (transverse to the direction of observation in the case of ACAR, along this direction in the measurements of the Doppler shift AE~) and in their experimental characteristics and hence in the information they can provide. ACAR has the advantage of a much higher resolution compared with AE~ measurements, but at the price of a lower data accumulation rate. In AE~ measurements the energy of only one of the annihilation photons has to be recorded. This has the advantage not only of much faster data accumulation compared with ACAR but also of offering the possibility to use the second photon from 27 annihilations for a correlated second AE~ measurement or for determining the e + age of the e + whose annihilation led to the observed Doppler shift. The "positron age" of an individual positron is the time interval between the "zero" of the lifetime measurement and the annihilation event; it is approximately equal to the time the positron has spent in the sample or, in the case of Ps formation, the time interval between Ps formation and annihilation.


411

Lifetime spectroscopy gives us the distribution function of the positron ages. For the "death signal" one of the annihilation photons has to be used. The "birth signal" m a y either be obtained by the "classical n p r o m p t - p h o t o n technique referred to in Sect. 1 (usually based on a 22Na positron source) or by recording the passage of the e + through a scintillator detector located between the e + source and the sample. The latter method, known a s / 3 + 7 technique in contradistinction to the "/'y technique deriving the birth signals from p r o m p t photons, has the advantage of a higher count rate for a given source strength. However, full use of its potential for excellent time resolution can only be made if positrons of relativistic energies are available (for details see Schneider et al. 1993). Since these are also required for achieving a high e + spin polarization, the/3+'), technique is clearly the method of choice for e+SR lifetime experiments. The techniques (i) and (iii) m a y be combined in the so-called A M O C ( = a g e - m o m e n t u m correlation) technique ~. As indicated above, it uses the annihilation photon not required for the AE.~ measurements to measure simultaneously the age of the annihilating positrons and thus to establish the correlation between one of the m o m e n t u m components of the annihilating e+e - pairs and the positron age 7. The e + spin polarimeters based on the modification of the positronium states by a magnetic field B (i.e., possibility (i) of Sect. 1) all make use of the fact t h a t the population of the m = 0 substates of 1S positronium depends on whether the spin polarization is parallel or antiparallel to the magnetic field. Positrons with spins opposite to B are preferentially captured in the 13S1(m = 0) substate at the expense of the 11S0 substate, and vice versa 8 Since e + in Ps "atoms" that without magnetic field would have annihilated 6 In principle an analogous combination of (i) and (ii) is also possible; in practice, however, simultaneous high-resolution measurements of lifetime and ACAR face considerable technical difficulties. 7 Professor Innes MacKenzie of Guelph University informed us in a letter dated August 21, 1996, that when the experimental discovery of positronium in gases by M. Deutsch became known (cf. Sect. 1), Professor W. Opechowski, a theoretical physicist at the University of British Columbia, remarked that one should try to measure time and momentum spectra simultaneously, and that this remark was the initiation for the attempts of Innes MacKenzie and Barry McKee to establish age-momentum correlations when large Ge(Li) detectors became available. The last (unpublished) age-momentum correlation measurements at Guelph University based on the classical ~f~f age measurements, using large CsF scintillators, produced 100 coincidences/s at a time resolution of 290 ps and an energy resolution of 1.3 keV. This is to be compared with about 1.2-10 a coincidences/s, a time resolution of 230 ps, and an energy resolution of 1.3 keV that can be obtained with a 100 mCi 2~Na source at the fl+-y AMOC set-up of the Max-Planck-Institut f ~ Metallforschung in Stuttgart. s This has been used by A1-Ramadhan and Gidley (1994) to increase the measuring effect in their indirect determination of rp-p~ (cf. Sect. 2.).

412

with the characteristics of o-Ps annihilate in the presence of a magnetic field with the characteristics of p-Ps ("magnetic quenching of o-Ps", cf. Sect. 2), any technique that distinguishes p--Ps annihilation from o-Ps annihilation m a y be used as an indicator of the e+ polarization. (Since the Zeeman energy is very small compared with the Ps binding energy, we m a y assume that the formation of positronium "atoms" is not affected by laboratory magnetic fields; cf. Appendix A.) Measuring the ratio between 37- and 2v-annihilation events and its dependence on B might appear to be the simplest way to determine the e + polarization. However, the field strengths required to shift the occupation ratio of the two m = 0 substates of 1S-Ps appreciably are so large that virtually all annihilations in these substates occur by 2"r-processes (cf. Sect. 2) whereas the fraction of 3"y-annihilations from the ]m] = 1 substates is unaffected not only by the strength but also by the direction of the magnetic field. When Page and Heinberg (1957) demonstrated for the first time, by ACAR measurements on various mixtures of argon and propane (C3Hs) gas, that the e + from/3 + decay (in their case of 22Na) had the expected right-handed helicity, they made use of the following effects. Under the experimental conditions chosen the Ps atoms "thermalize" so slowly that their momenta at the average age at which p p - P s annihilates (rpp_p~) cause the corresponding ACAR distribution to be rather wide. At ages of the order of magnitude of rm_ps(_ ~ 3- 10 -9 s in typical magnetic fields), however, the Ps thermalization is virtually complete, so that the 27-annihilations of m - P s give rise to a very narrow ACAR distribution. By measuring the dependence of the 1800 coincidence rate on the direction of the magnetic field the e + polarization could thus be determined. The method of Page and Heinberg is not applicable to Ps in condensed matter because of the much shorter thermalization times. Here the principal possibilities to distinguish p--Ps annihilations from o-Ps annihilations are the following. (1) The great majority of Ps "atoms" annihilating in condensed matter have thermal kinetic energies. Their self-annihilation gives rise to a much narrower momentum distribution than the pick-off annihilation or the annihilation of unbound e + , in which the wide momentum distribution of the host electrons involved in the annihilation characteristic for e- in condensed matter dominates the momentum distribution of the annihilating pairs. Since virtually all p - P s undergo self-annihilation whereas self-annihilation is negligible for o-Ps, the narrow component of the momentum distribution as seen by ACAR measurements is a quantitative indicator of p - P s annihilation. E.g., Greenberger, Mills, Thompson, and Berko (1970) and Herlach and Heinrich (1972) used ACAR to demonstrate the magnetic quenching of orthopositronium in a-quartz and KC1. (2) By applying a large enough magnetic field, the lifetime of m - P s may be lowered below that associated with pick-off annihilation. (In order to reduce


413

Tm_ps t o 2 flS, we reqnire B : 1.8 T i f f : 1, and B : 1.3 T i f f ; : 0.7.) With the help of the expressions given in Sect. 2, measurements of Tm_ps permit the determination of the ratio of o-Ps to p-Ps annihilations. In discussing the pioneering work of Page and Heinberg (1957), Lundby (1960) pointed out that the modification of the lifetime spectrum due to the magnetic quenching of o-Ps may be used to investigate the e + spin polarization. This possibility was realized by Bisi et al. (1962) as wen as by Dick et al. (1963) and later applied extensively to the study of Ps in a wide range of substances by the group at the Politecnico Milano (for a review see Consolati 1996). The two techniques for distinguishing p-Ps from o-Ps referred to above have in common that they both rely on coincidences. Since because of this obtaining good statistics takes a long time, it is dimcnlt to detect by these methods, let alone study quantitatively, the formation of small fractions of Ps. In the present context, lifetime spectroscopy has the additional drawback that the unavoidable magnetic fields interfere with the operation of the photomnltiplier tubes in the photon detectors. This requires the use of fairly long light guides, which reduce the achievable time resolution. As a result, it is not only difficult to determine the lifetime rpp-Ps with an accuracy that allows a reliable determination of the electron-density parameter ~ (of. . Eq. (2.11)) but also often impossible to separate Tp_ps from the lifetime of those e + that do not form Ps. The presence of spin-conversion processes (cf. Sect. 3) and chemical reactions such as complex formation or oxidation (Billard et al. 1991, Castellas et al. 1996) may complicate the identification of ~'m-Ps further. As a non-coincidence technique, Doppler-broadening measurements can be performed with better statistics than ACAR or lifetime measurements. The chances to detect small fractions of Ps-forming e+ are therefore much better. An additional advantage of the Doppler-broadening technique is that, without impairing the lineshape determination, the annihilation photons not required for this measurement may be used for AMOC or, at least, ageselected measurements. The z~ET-based e+SR-measurement set-up built in Stuttgart some years ago (Seeger et al. 1987) and used to study e+ spin relaxation in c~-iron (Banhart 1988, Banhart et al. 1989) as well as positronium formation (Major, Seeger, and Stritzke 1992) has recently been modernized (Gessmann et al. 1997a). It utilizes spin-polarized positrons from a radioactive 6SGe/6SGa source (maximum kinetic energy of the emitted e + 1.88 MeV, average kinetic energy 0.81 MeV) and measures the broadening of the 511 keV annihilation photon line in external magnetic fields of different magnitudes up to 2.6 T that are applied parallel or antiparallel with respect to the e + spin polarization. Reversal of the magnetic field is equivalent to a change of sign of 7~. An improved data-taking system allows the quantitative detection of small fractions of Ps-forming positrons. A practical example will be discussed in Sect. 5.

414

5 Data

analysis

and

results

In condensed matter studies by all three methods (i - iii) mentioned at the beginning of Sect. 4, the most important quantity is the ratio of the number of counts due to 27 self-annihilations to that due to annihilations by pickoff (in the following called yields Ip-ps and Ipo, respectively). According to (3.28) we have oo ~p-Ps(~) = Z~Ps ~p-Ps / P 1 1 ( ~ , ~ ) d ~ o --'~ Nps "~p-Ps ~11(0; ~) ,

(5.1a)

flO

Ipo(~) = ~¢p. ~po / [pl~(~, ~) + p22(~, ~) + p~3(~, ~) + p44(~, ~)] at 0 4 = "NP' ~P° [ E L:ii(O;~)] i-----1

(5.1b)

Since for typical pick-off rates in condensed matter (2 • 10 s - 10 9 s -1) the 3"y-annihilation may be neglected, Nps --- Ip-Ps(~) -}- Zpo(~)

(5.2)

holds. From the Doppler-broadened 511 keV lines as measured at the two opposite fidd directions (indicated by -4-~) we may derive the quantity

p* :

Nps

(5.3)

where K is a dimensionless proportionality factor that depends on the experimental set-up and on the data-handling procedure. Eq. (5.2) holds independently of the sign of the applied magnetic field; hence it follows that

Zpo(-~) - Ipo(+~) = zp_p.(+~) - Zp_p.(-~)

(5.4)

Making use of (5.4) and (5.1), P* may be written as p*

= • ~p_p. [~11(0; +~) - ~1~(0;-~)]

(5.s)

Inserting Eq. (3.32a) into (5.5) and using (2.4) gives us finally p* - 8 K ' ~ ) i p _ P s ) t o ( ~ ° + k e x ) 2a~ C0 ~- C2Z2

with

(5.6)


415

c0=2(~o-~-kex)(2~o'~-kex) [(4~o + kex)~p + 3~o~ex],

(5.7&)

c~ = [(2~o + k~,)(~, + ~o) + 2ko,~o] ~

(5.7b)

0.06

0 1 2 i/...,.......

¢b.

EL

4

B[T] 8

16

32

I

I

(a)

~Z,po=10B1 S-1

'L

0.04

nv: um

0.02 I ~ , ~ ~lpo=l'O7.s-1

....

0.00 0.0

0.06

0.5

0 1 2

1.0

4

1.5

2.0

B[T] 8

2.5

16

32

(b)

Zpo=108 s"1 0.04

3.0

I kex=101° s-1

¢b.

PS in vacuum/

EL 0.02

~[/~.~. " ' / " f " ..... ~ . . . '/./.k.x=lO 8 s"1, Zpo = 0~"~ r.,~4,

0.00 0.0

0.5

..

,

,kex=l01° s~ Zpo=0~"'--.. 1.0

1.5

2.0

2.5

3.0

Fig. 5.1. The dependence of the quantity P * / K 7 ~ on the magnetic field as calculated from (5.6) for ~ = 1. The influence of the pick-off annihilation rate Apo is illustrated for kex = 0 in Fig. 5.la, that of kex for different pick-off rates in Fig. 5.lb.

416

In (5.7) ~p : = ~p-Ps -~- ~po and ]o : = ~o-Ps -~- )(po = )133 are the total annihilation rates of p-Ps and o-Ps, respectively. The field parameter z has been defined in (2.3); Note that through ZlEo_p it depends on ~. P* as given by (5.6) was also derived earlier by Stritzke (1991). The functional form of its dependence on the applied magnetic field agrees with the result of Mills (1975). Fig. 5.1 shows the field dependence of (5.6) as calculated for different values of :~po and kex. (If, in addition, reactions between Ps and "free" positrons are taken into consideration, co, c2, and the factor ~o()io + ]%x)2 in (5.6) have to be modified; for explicit results see Gessmann et al. 1997b.) ~p is usually much larger than either )Lpo and k~x. Then (5.6) and (5.7) may be simplified to

CO = 2(~o "f- /¢ex)(2,~o + ]¢ex)(4~o "}- ]¢ex))Ip-P. 2 C2 : (2)(o "~- kex) 2 '~p-Ps ,

,

(5.8a) (5.8b)

and p. =

8K:PAo ()io + kex) 2 z 2~o + kex 2(go + kex)(4:~o + ]¢ex) + (2)io -~- kex)~p-Ps 22

.

(5.9)

If the spin-exchange rate kex is much larger than the pick-off rate, we have the further simplification p . = 8K7~

~poZ 2kex + ~ p - P s ~ 2

(5.10)

Eq. (5.6) shows that P* is proportional to the initial positron polarization 7~ with a proportionality factor that depends on the annihilation rates, the spin-exchange rate, and the applied magnetic field. Apart from a proportionality factor, P* may be interpreted as the residual polarization of the positrons in Ps at the moment of their annihilation. This is a relative measure of the positron spin relaxation, provided we know the dependence of K and 7~ on the magnetic field (cf. Gessmann et al. 1997b). The maximum of P*/K7 ~ as a function of the magnetic field occurs at z = ~max = (C0/C2) 1/2. If Eqs. (5.8) are applicable, we have

[2(,~o + ~ex)(4,~o -}- kex)] 1/2 Zmax---- L (-2~o ~ ~e--~p----P: J

(5.11)

The maximum is then given by

(P*/KT~)max =

( k2 o +

kox 3/2 [2(4~o + kex))~p-ps] 1/2

(5.12)

We see that as long as the condition )tpo

Atomic Physics Methods in Modern Research: Selection of Papers Dedicated to Gisbert Zu Putlitz on the Occasion of His 65th Birthday

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