P R O G R E S S I N LOW TEMPERATURE PHYSICS 111
CONTEh-TS OF VOLU-ME I
c.
J . GORTER,
The two fluid model for supe...
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P R O G R E S S I N LOW TEMPERATURE PHYSICS 111
CONTEh-TS OF VOLU-ME I
c.
J . GORTER,
The two fluid model for superconductors and helium I1 (16 pages)
IIUTV.iL
F R I C T I O X IN THE ~ K I F 0 R M I . Y I
Again a distinction may be made between the cases 2 ! w and 9 w . In the former case the coupling between the two fluids will be small, so that to a good approximation the overall normal fluid motion will not be perturbed by the vortex waves; the first term in eq. (24) then produces a damping of the waves as exp(- ccz), where K rn B,p,k/4e, while the second term gives rise to a fractional change in the effective value of 11 equal to B;,n,/-Le.Both these effects are small. In the latter case the situation is generally more complicated, but it can be shown that in the practical situation described below the mutual friction can have only a negligible effect. Experiments with which this theory of vortex waves can be conipared have been carried out by Hall23.The essential part of the apparatus is a small aluminium can, containing a pile of discs of the type used in the Androniltashvili experiment, which is suspended by a torsion fibre from a head that may be rotated at any constant angular velocity. The experiment consists in filling the can with helium, rotating the torsion head until the whole system is rotating with it, and then measuring the period of small oscillations of the can superimposed on the uniform rotation. Observations were made with various disc separations between 0.B and 7 mm, with periods of oscillat’ion between 3 and 25 sec, and with periods of rotation between 0.1 and 1.1 rad sec-I.
.I$,
(30)
where E is the energy of the “excitation” and fi is the component of its momentum parallel to u, (both referred to coordinates moving with the liquid). Thus a sufficient condition for the existence of ideal superfluid flow a t velocities less than a critical value u, is that there should be no possible “excitation” with E l f < ZI,. I n his original theory, Landau believed that the only excitations possible were the phonons and rotons; for these the minimum value of E/P is about 6 x lo4 cm sec-1, which is of course much larger than the observed critical velocities. However, we now know that quantized vortices can exist in helium, and these provide other types of localised “excitation”. Thus we might consider excitations in the form of vortex rings of radius 6. It can be shown3’ that such rings have kinetic energy and momentum given by
and
(provided the ring is not too close to a solid boundary), so that they will have associated with them a critical velocity Rrftretzccs
p . c56
CH. I,
$ Sj
VORTEX LISES I N LIQUID HELIUM I1
39
In a channel of width d , the radius b will have a maximum value of order d, so that the minimum critical velocity will be given roughly by
And it is clear that a similar result is likely to apply to the generation of vortex line of more general shape. This minimum critical velocity agrees roughly with the experimentally observed critical velocities in both its order of magnitude and its dependence on channel width, and it therefore seems likely that the observed breakdown of ideal superfluid flow is indeed due to the creation of quantized vortices. I n practice these vortices will presumably be farmed in a rather irregular way, and we guess that the final result will probably be a tangled mass of line, or, in other words, a kind of turbulence (as will be seen in the next section, a mechanism exists for stretching any small length of line into a concentrated tangled mass). This is an attractive idea, partly because, as we have seen, it accords with the observed characteristics of the critical velocity, and partly because, as we shall see in the next section, it leads to an explanation of many of the observed properties of the dissipative forces that build up under supercritical conditions. However, there are still difficulties, as is clear as soon as we examine the detailed mechanism for the initial production of the vortex line. Let us consider more carefully the example used already: the creation of a vortex ring of radius b ; similar considerations probably apply t o other configurations of vortex line, including those in which a line is partly attached to a solid boundary (the situation is a little more complicated if the ends of the line are actually s t w k to the boundary, but it can be shown that similar conclusions can still be drawn). It is implicit in the discussion so far that this creation process must involve a direct quantum-mechanical transition, induced by interaction with the wall, between a state of uniform flow and a state of flow that includes the ring. Now we know from experiment 36 that critical velocities can be as low as 3 x 10-2 cm sec-1; and we must therefore assume cm can be created. apparently that rings with radii as large as 3 x But the creation of such a large ring would involve enormous numbers of atoms at considerable distances from the wall, and it is very difficult Refcumces p . 56
40
w.
F. V I N E N
[CH. I,
$6
to believe that such a process could take place with any appreciable probability, even with a rough wall. Indeed it seems alniost as unlikely as the process of slowing the liquid down as a whole. I t should be emphasized that the ring cannot grow gradually from a sinall size to a large size; for the creation of a small ring involves a considerable increase in energy of the liquid, not a decrease. Thus, in a sense, the difficulty is that the creation of vortex line is opposed by a large potential barrier. The size of the largest ring (or similar configuration of line) that is at all likely to be created by a quantum mechanical transition through a perturbation applied at the wall is not easy to estimate with certaint y ; but a rough consideration of the transition probabilities involved suggests that the size cannot be very much larger than an interatomic spacing3R.Thus it is probably safe to assume that it does not exceed 1 0 P cm, which corresponds to a critical velocity (34) of about 7.5 x l o 2 cm SCC-1. In practice all experiments on critical velocities have been carried out at a finite temperature, and we must therefore enquire whether the presence of normal fluid helps in the creation of vortex line. The presence of phonons and rotons as such certainly does not; for (to use again the example of a vortex ring) it can be shown easily that a small ring will move rapidly (under its own velocity field) relative to the normal fluid in such a direction that the mutual friction force acting on it will, through the Magnus effect, causc it to contract. Howevcr, there exists thc possibility that large lengths of vortex line arc occasionally produced by the interaction of phonons and rotons; i.e. that lengths of vortex may occasionally be present in thermal equilibrium (these lengths would act as the basis for the growth of more line). This possibility was mentioned in $ 2 , and, as indicated there, no calculations on it have yet been carried out. Although, as we shall see in the next section, therc is some slight evidence that this effect might be important, it does on the whole seem unlikely that it really is important. In spite of these difficulties, there is considerable evidence, as will be shown particularly in the next section, that the breakdown of idtxl supcdluid flow does lead in fact to the creation of vortex line, and some detailed mechanism for the creation of this linc must therefore bc found. There seem to be four possible ways out of the difficulty, ancl wc consider these in turn. (a) The first and no st obvious possibility is that apparently unI21.fcreitr.cs p . hG
CH. I,
5 61
VORTEX L I N E S I N LIQUID HELIUM I1
41
disturbed helium does always contain lengths of vortex line, but not in thermal equilibrium, and that these lengths can grow as soon as the superfluid velocity exceeds a certain critical value (growth mechanisms are discussed in 3 7). As was explained in 5 4, observations made with the vibrating wire apparatus have shown that such lengths of line can be present, and it was suggested that they are held in metastable equilibrium by having their ends tied to protuberances in the apparatus. (b) The second possibility is that a protuberance can itself cause the creation of a length of vortex line. Suppose superfluid is flowing past a protuberance with a sharp edge. Near this sharp edge the superfluid velocity will be very large, so that it might be possible to satisfy the condition (30) for a ring (or other similar configuration) of vortex line that is small enough to be created by direct quantum mechanical transition (or thermal excitation). Thus a small length of line might be created close to the edge of the protuberance and then be pulled away from the protuberance by the main flow. Detailed examination of this idea38for various idealised shapes of protuberance suggests that the process is probably possible, but only at rather large velocities. Thus, if the protuberance has the form of a knife edge of height H placed perpendicular to the flow, and if the smallest ring or other shape of line that can be created by direct transition has linear dimension 6, then the critical velocity is probably of order (?i/m)(2/H8)+In 6/a,; and if, say, H = cm and 6 = cm this equals about 30 cm sec-l. (c) Thirdly there exists the possibility of creating vortex sheet 4, ', which can subsequently break up into vortex lines (classical vortex sheets tend to break up in this way, owing to instability against any undulatory motion, and presumably the same tendency will exist in helium, although it will then be opposed to some extent by the surface tension in the sheet). The idea here is that a region of stationary fluid is formed, separated from the moving fluid by the vortex sheet. The critical velocity for a region with linear dimension of order d is given by
where o is the surface tension of the sheet (S 2 ) . For a low critical velocity the region must therefore be very large, and thus the same difficulties arise as in the case of vortex line formation. However, there exists again the possibility that a protuberance can help, and indeed it RCJ~ICII.CES p . 56
42
U‘.F. T’INEN
[CH.I ,
96
turns out that the critical velocity for creation of vortex sheet in this way is probably about the same as that for vortex line. Thus for the knife edge protuberance of height H the critical velocity is probably of order ( 2 u / H p , ) +(independent of the size of sheet formed); and if H = cm this is again equal to about 30 cm sec-l (it might, however, be smaller if, as is quite possible, the estimate of G given in 9 2 is too high). (d) Finally the possibility ought to be mentioned that, if the conditions near the centre of a vortex line differ appreciably from those in the simplest picture, then the energy of a very small ring may be appreciably less than that given by eq. (31) ; and vortex line formation would then be correspondingly easier by any of the processes that we have been considering *. Nothing is known about the fourth possibility, and we shall therefore for the present ignore it. Wenote that all the others are based directly on the existence of protuberances in the flow channel, and this is satisfactory since, as we have seen, there is experimental evidence that protuberances might be important. (Unfortunately, however, most of the evidence for this comes from certain heat flow experim e n t ~ ~in ~ which, $ ~ as ~ we . ~shall ~ see in 9 7, the critical velocity may not have been of the ideal type; thus the possibility exists that the effectsobserved are quite unrelated to the mechanisms being considered here.) Process (a) seems to be the most likely one a t low velocities; process (b) or process (c) at the higher velocities. It should perhaps be emphasised that if ideal superflow does break down by one of these mechanisms the critical velocity will depend on different and more complicated considerations than those used t o obtain eq. (34) ; and, as will be seen in the next section, the same is likely to apply to nonideal critical velocities. Thus the fact that eq. (34) does sometimes appear to be obeyed in practice may after all be to extent fortuitous. Furthermore, the critical velocity will depend on the precise experimental conditions, and this may provide an explanation of the fact that the measurements of critical velocities by different authors in channels of the same width have sometimes yielded widely differing results.
* In the special case of t h e film, still another mechanism exists: the formation of quantized surface wavcs or “ripplons”48. References
p . 56
CH. I,
$ 71
VORTEX LINES IS LIQUID HELIUM I1
43
7. Superfluid Turbulence In this section we shall describe some of the characteristics of the dissipative processes that are found to follow the breakdown of ideal superfluid flow, and show that they are consistent with the idea that these processes are associated with the presence in the superfluid of a tangled mass of vortex line, i.e. a kind of turbulence. I n the preceding section we confined our attention to flow in a straight channel; in this section we must confine our attention still further, and we shall in fact consider only the case of a channel that is fairly wide ( >10-2 cm) in which the flow is a counterflow of the two fluids due to a heat current. This case has the virtue that it is probably one of the most simple, and also that it is the one that has been studied most extensively; and it will certainly serve to illustrate the general principles. We begin with a survey of the experimental observations and the qualitative interpretation of them in terms of superfluid turbulence ; and then we indicate briefly the extent to which it is possible to build a quantitative theory. It turns out that this theory can be developed to a large extent without knowledge of the processes discussed in the preceding section. The reason for this is that to a considerable extent these processes appear merely to nucleate the turbulence ; once a small length of line has been produced, turbulence can be built up and maintained simply by the stretching of existing line. Thus, in the basic eq. (38), only the second term on the right hand side depends on these processes, and this term is in fact unimportant in many circumstances. The type of apparatus used for heat flow experiments is shown in Fig. 10. A known power is generated in the heater H; the thermometers T I and T, record the temperature gradient along the channel C ; and the manometer M records the pressure gradient. The channel may be of either circular or rectangular cross section, and it may be constructed of either metal or glass. If it is desired to study the propagation of second sound in the heat current (see below), the channel may be made into a second sound resonator; for example40, it may be made with a rectangular cross section so that a resonance may be established across the heat current (Fig. 10a), the second sound being generated by a wire heater h running along the length of one side of the channel and detected with phosphor bronze resistance thermometer wires (t) running along the opposite side of the channel. Consider first the observations that have been made on the regime References
p . 56
44 W. I;. V I X E N [CH. I, 9 7 obtaining in a heat current in the steady state. Measurements of thc temperature gradient required t o maintain the heat current 33,30, 40 and of the attenuation of second sound in the heat current40 show that the most important force opposing the flow is an essentially linear isotropic mutual friction Fsn = ('8
- vn)
(36)
per unit volume, where in the steady statc thc factor G' takes the form ~
G = Aesen(l v s - v n I - 210)'. (37) A is of order 50 cm sec g-l, increases with increasing temperature, but is approximately independent of channel width and of the nature of
' I Fig. 10. Type of apparatus used for heat flow experiments in lirlium I1 (schematic only). V: vacuum space. (a) Cross-section of channel C (enlarged) whcn used as second suund resonator.
the channel surface (except in the narrower channels) ; (vs - v,) is the instantaneous relative velocity between the two fluids; (vs - v,,) is the mean steady rclativc velocity duc to the steady heat current (these two velocities will not be equal when second sound is superimposed on the heat current); and TI" is a smaIl quantity (less than the velocity corresponding to the critical heat current), which varies sonicHhat with tcmperature and tends to decrease with increasing channel width. Qualitatively, this mutual friction can obviously be 1icfcvr~nr.e.~ p . 56
CH. I,
71
VORTEX LINES I N LIQUID HELIUM I1
45
accounted for in terms of vortex lines in a turbulent superfluid, the amount of vortex line being determined, as one might expect, by the steady heat current independently of any second sound, and the force will be of the same fundamental type as that observed in uniformly rotating helium (§ 3 . 2 ) . The observed dependence of G on ( v s - v,) shows that the length of vortex line per unit volume of superfluid must increase approximately as the square of the steady heat current; the fact that A is approximately independent of both the channel width and the nature of the channel surface suggests that the turbulence is approximately homogeneous and is maintained by the steady relative motion of the two fluids and not by motion relative to the walls; and the isotropy of the mutual friction suggests that the turbulence is also isotropic. At one time it was believed that the mutual friction is the only force acting in a steady heat current, apart from forces due to the normal fluid viscosity, but recent pressure gradient measurem e n t ~have ~ ~ shown - ~ ~that this is probably not true. If it were true, then, as is easily shown42,the pressure gradient along the flow would be equal simply to the value required to maintain ordinary Poiseuille flow of the normal fluid through the channel * ; but it is found that the pressure gradient is appreciably greater than this value. (Analogous effects have been observed in other experiments ; see, e.g., refs. 43, 44.45. ) The existence of this excess pressure gradient shows that there must be some mechanism for transferring momentum within the superfluid. Such a mechanism does indeed exist in a turbulent fluid, owing simply to the fact that there is a continuous transfer of actual matter across the main flow. The turbulent flow behaves to some extent like a laminar flow, but with an extra effective viscosity, which is usually termed an “eddy viscosity”. Values of the eddy viscosity calculated from the observed excess pressure gradient vary somewhat with the experimental conditions and with the assumptions made in the calculation (particularly with the assumed boundary condition for the superfluid at the channel walls), but are usually of order 10-100 ,UP. It should be noted of course that since the turbulence is homogeneous the concept of a well-defined eddy viscosity is perfectly valid. As already indicated, the mutual friction (37) does not hold in small heat currents, and it has been found that the results then depend on * I t is assumcd here that the normal fluid does not become turbulent, and this seems a reasonable assumption, except perhaps for very wide channels or very large heat currents. References p . 66
-16
W. F. V I N E N
[CH. I, $
7
the channel width and perhaps also on the nature of the channel surface. Two sets of detailed experiments have been carried out, and, since the conditions in the two were rather different, it is convenient to discuss them separately. In the first, the “Oxford” experim e n t 33* ~ 34, ~ 44, ~ rather ~ narrow channels (: 1 0 8 1.59 :.: 1 0 - ~ 2.51 >< I O - ~ 3.98 x 10-3 6.30 >' 101.00 x 10 2 1.5!) Y 1 0 - 3 ?..>I T , C,
= C,(T)
-
where C, represent the Debye Function, evaluated at each temperature with the appropriate value of density and velocity of sound. It was found empirically that the best simple fit to the data both above and below the A-point could be obtained by inserting, as a factor in Eq. (l),a simple exponential term after first subtracting out a term representing the theoretical Debye .specific heat. The R ~ f i . w n r r sp . 111
88
12. J . HY(:KISGH.I31 AND If’. $1. F.IIRCrlNK
[CH. 111,
s9
constants in the exponential term that gave the best fit to thc data \\rere found to be 7.10 “ K bclow T , and 3.70 “I< above the ],-point. When these had been dctermjned, the other constants shown in the equation above all followed directly from Eq. (1) setting ‘1‘ = T , in the exponential term. Thus all of thc constants except the “energy gap” in the expoi~entialtcrrn were obtained from the data talien within dcgree of the 31-point. I t is interesting that the presence oi the logarithmic term leads to a rclat ively simple emyiricd expression over the wholc temperatiire rnngc removing t h e difficulty found in tlie past of tinding sucli a n expression. We do not necessarily imply any particular theoretical significance to the actual expression, but it will be discussed again in section 7 . I n summary the specific heat data near the A-point can be representctl by n logarithmic singularity. With this term, which can be deterrnincd from the data within 10 -2 degrec ot the 31-point, it is possible by multiplying by a simple exponcritial term with a teInperatui-e independen t. gap, differcnt on the two sides of the ii line, to fit the data over the entire range of tcmpcrature from the lowest temperature up to 3.8 degrees, due allowance being made for the l k b y c pl-ionon specific heat. Although this does not necessarily reprcscnt the most uscful tlieoretical equation, it points up thc need for the logarithmic term in explaining the data. I t is interesting to note that the constant difference, A , between the two straight lines representing the specific heat above and below the A-point (Fig. 3) is equal to -tR. In this section we have discussed the specific heat under the saturated vapour pressnre, C,. A more interesting quantity for the theory is the specific heat at constant pressure, C,. This is related to C, by,
B being the coefficient of
expansion and (6P/ bT)v,p,v. the slopc of the vapour pressure curve. Below 2.5 “K tlie difference between C, and C, is less than 1 percent so, within experimental error, the two yuantitics can be used interchangeably when discussing the behavior of C , near the A-point, as is done in the discussion which follow\.
Hefewiices
p. 111
CH. 111,
9
31
THE N A T ~ R E01; TIE
,&TRANSITION
3. Thermodynamics of ;”.-Transitions 3.1. )~.-TRAKSITIOXS I n this section we derive rigorously the thermodynamic consequences of A-transitions characterized by the absence of a latent heat, but at which the spec.ificheat at constant pressure becomes infinite. I€ any such transition occurs for a range of pressures, the transition points T,(P) form the %-line,which will in general have a finite slope, ( ZT)?,, on the pressure temperature diagram. Since the integral of the specific heat must be finite, C,, must approach infinity less rapidly than I T - 7’, I--l. The observed properties of liquid helium are consistent with those of a transition of this type, in which C,, becomes infinite like log I1’ - T , i . At such a lL-transition,not only does the specific heat C , become infinite, but so also must the thermal expansion coeficient and the isothermal compressibility. There exist relationships between these quantities and also between C,, ( ?P/ Z).and ( aV/ aP)., which remain finite, if the slope o€ the Mine is finite, as we assume hereafter. These relations, which we derive in the next sections, are useful for comparing different measured quantities and play a role analogous to the Clapeyran and Ehrenfest rclations. 3 . 2 . PROPERTIES AT THE
TIMSSITIOKS
At constant pressure, the entropy and volume are continuous functions of temperature, but achieve an infinite slope at T,. Now from the Maxwell relations we have
iP aT ar (d2, ( d(P 7 d S (w)T (dV (dS iiT
27.
=
61’
= -
‘
(3)
At the transition, (aT[as), and factors on the left in each expression vanish, the other factors remaining finite. This is easily seen by using the relations t
a?’
(4) CP
and 7 \l:e
iisc lierc, and frtiqiicntly in what follows, the identity
where W , S, I’, 2 are functions of state, with two indepcndent variables.
References
fi.
111
Xo\r if C, is infinite along the 1,-line, ( a T /aP),=,
( a T /ZP),, and since the second term on the right in (4) vanishes on the line, ( aT/ aP)s is finite and, by (3), (aT/aV\, must vanish. Then (5) shows that (iT/CP)vreaches the same finite value ( 87'1 %P),as does ( FT/ a$').. I n the same way we see that
as
as
=
(Av 0, W C(A,,; p =
aV
1)
c?P
-
thus at the A-transition the specific heat at constant volume reaches the value
For the velocity of sound we require
aV
2V
FT
as
(ds (%JCP(P), (dc; =
--
Thus at the A-point the velocity of sound, c , is given by
The relations (5)-(7) have been used to calculate the values for liquid helium given in Table 2 of section 4. Each of thcse quantities which remain finite a t the A-transition, achieve the finite value at a sharp cusp, varying near thc transition like Cly1. This variation is examined in the next section. 3 . 3 . PROPEIUIES SEAR
THE
TRWSJTION
In order to study the variation of thermodynamic properties near the A-transition it is convenient to introduce a new variable, the "neighborhood temperature", t, a function of state defined by f = t(T,P ) = 2' - T,(P),
is)
whcre T,(P) is the A-temperature for the preswre P . For convenience we will write 1';. for ( ?I1/aT),, the slope of the A-line, and a function of Refureiirrs p . 111
CH. 111,
$ 31
THE XATURE OF THE
A-TRINSITION
91
pressure. Thus the line t = 0 is the I-line, the line t = t o (constant) js a line parallel, in the P - T plane, to the I-line, but displaced parallel to the temperature axis by an amount to. Thus, by definition,
?I'
aP
2P
( F )=,( F )=,
= -
(T) T'
(9)
14-e also note the following properties of thc new variable:
A simple relationship connecting C , and the expansion coefficient can be obtained from the equation
This equation is exact and we note that if C,/T is plotted as a function of ( aV/ 6 T ) , the resulting curve approaches asymptotically the straight line of slope Pi and intercept (as/Z),t . The displacement from the asymptote is (as/aT), - (as/PT},. It is interesting to note that exactly the same straight line is the asymptote for a corresponding plot of the specific heat at constant volume. Since
8.S
=
(T), - p;
(g)t (%) VI
a curve giving CJ7' as a function of ( 2 P /3T),,, in units of -( aP/ ZV),, will approach a straight line of slopc Pi and intercept (as/aT), as did the plot of C,/T. Unlike the latter, however, thc curve for C, ends at the finite value given by Eq. (6). Another simple relation connects the expansion coefficient and the
t Zn approuniate relatzun of this type has bccn given by Pippaic117. Thl5 is discussed briefl) in Sec. 3.5 below. Referetares
p.
111
'32
31. J. FJYCI 0 (more probably 1 > 1) cooperative effects arise leading to a phase transition in the temperature range 0.05-0.1 OK. Immediately above the transition temperature, the specific heat should be proportional to the temperature with C m 2 C,, where C, is the ideal Fermi gas specific heat23. At the transition temperature T,, a discontinuity in the specific heat is predicted. The actual AC will depend upon the relative contributions from the various 1 and m modes. For example, for 1 = 2, m = 0 , AC = 0.71 C , where C, is the “normal” fluid specific heat at T,. Below the transition temperature, the particle pairs will be correlated for arbitrary directions in the medium. Since the correlation range is only about 400 A, the unperturbed liquid should break up into randomly oriented cells of roughly this dimension, in each of which a correlation axis will exist. In order to observe certain correlation dependent properties, it will be necessary to establish macroscopic polarization axes in some fashion, e.g. by viscous interaction of the flowing fluid with the walls. The properties then measured will be angle dependent. Below the transition temperature, an enhanced fluidity rather than perfect superfluidity should be observed in liquid 3He. It is well known that the latter phenomenon requires the existence of an energy gap, whereas the formation of a correlated phase involves only a reduction in level densityg4. References p . 150
118
E. R. GRILLY AND E. F. HAMMEL
[CH.
IV,
2
2. Theories of Solid 3He 2 . 1. QUALITATIVE
Pomeranchuk*, assuming negligible exchange effects from nuclear spin in the solid, concluded that the spin alignment temperature should be about lo-’ OK. Although the correct value proved to be about 0.3 OK (see Sect. 3.3a), Pomeranchuck’s discussion disclosed the important anomaly in 3He at T < w 1 “K that the solid entropy should be greater than the liquid entropy, which is observed as the negative thermal effect of melting and the minimum in the melting P-T curve (see Sect. 3.3a). Primakoff z5 predicted that nuclear spin alignment in the solid should decrease with incresing pressure. The corresponding entropy increase, at sufficiently low temperatures, led to the prediction that solid 3He will have a negative expansion coefficient (see Sect. 3 .3a). Goldstein 15, 16, 1 7 extended .his theory of the partial spin properties from the liquid (Sect. 1.3) to the solid, with the result that parallel behavior would be exhibited among certain thermal and PVT properties. 2 . 2 . THEORY OF BEKNARDES AND PRIMAKOFF
A quantitative analysis of the properties of solid 3He was made by Bernardes and Primakoff26, who began with a gas-phase LennardJones “12-6” potential modified at small interatomic distances. I n contrast with Porneranchuk, they concluded that exchange effects represent the predominant mechanism for spin alignment in the solid. Their calculations for P M 30 atm and T w 1 “K led to the following conclusions: a) The cohesive energy per atom is about R X 2.5 “ K ; b) the root-mean-square deviation of an atom from its lattice site is about 0.36 times the nearest neighbor distance; (c) the nuclear magnetic susceptibility X follows the Curie-Weiss law X = c/(T - 0) with a Weiss constant e of antiferromagnetic sign 0 w - 0.1 OK; d) the decrease of - 19 w T , with increasing pressure corresponds with a possible transition to ferromagnetic behavior at fi w 150 atm, which could be connected with an observed crystallographic transition (see Sect. 3.3b); e) at T,, the specific heat and susceptibility exhibit singularities (cusp-like or otherwise well-defined maxima) associated with the alignment of the nuclear spins; f ) the thermal expansion coefficient becomes negative below about 0.6 OK; and g) the melting Refersnrrs
p.
150
CH. IV,
9
31
LIQUID AND SOLID
3He
119
curve is characterized by a minimum at T w 0.37 OK and a maximum at T M 0.08 OK. The most striking predictions of this theory appear to be: 1) The singularities in specific heat and susceptibility; and (2) the maximum in the melting curve. The apparent absence of 1) in the liquid is ascribed ultimately to the difference in character between the associated quasi-particles (phonons and magnons or spin waves in solid; individual atoms with m* # m in liquid).
3. Pressure-Volume-Temperature Relations A rather large amount of work has gone into PVT studies of condensed 3He, beginning in 1949 with the vapor pressure and the density of saturated liquid27. The reasons for this great effort lie in 1) the inherent importance of determining the behavior of a second quantum liquid, whose properties were expected to be significantly different from thcse of 4He; 2) the technical need of knowing how to handle the substance in the course of many experiments; 3) the rapid development in the entire field of 3He studies, which naturally brought on simultaneous duplicative investigations. The total effort now covers the tempxature range 0.3 to 3.2 OK between vapor pressures and melting pressures and up to 30 OK along the melting curve. The measurements involved from 0.02 to 12 cm3 of liquid, used a variety of techniques, and usually attempted to obtain high accuracy. At present, therefore, the PVT data on liquid 3He are comparable in extent and quality to those on the much more available and “older” 4He, which in turn has received greater attention than most liquids. The studies on solid 3He have been limited for the most part to the region of the melting curve. 3 . 1 . AT VAPORPRESSURES
The vapor pressure of liquid 3He was measured originally by Sydoriak, Grilly and HammelZ7, then more accurately over the range 1.O-3.3 OK by Abraham, Osborne, and Weinstock28. Measurements were extended down to 0.45 OK by Sydoriak and R0berts2~,who cooled the sample in a liquid 3He bath and determined temperatures from the susceptibilities of two different paramagnetic salts. Sydoriak and Roberts also recomputed the data of Abraham et al. to derivea single equation accurate over the entire range of 0.45 to 3.327 O K (the critical point) and fairly reliable down to 0.28 OK. This equation is References
p.
150
120
E. R. GRILLY AND E. F. HAMMEL
[CH. I\’,
$3
+
In P(mmHg) = 2.3214 In T - 2.53853/T 4.8153 - 0.20644 T 0.08640 T2 - 0,00919 T3,
+
where T = T , is based on the “55E” scale of Clement30. The vapor pressure as T -+0 OK can be calculated from another equation of Sydoriak and Roberts provided that the spin entropy integral can be evaluated. The first serious attempt at high accuracy in saturated liquid density was made over 1.3-3.2 OK by Kerr3I, who tried to limit the error to 0.2 yo.P t ~ k h followed a ~ ~ with a technique that, unfortunately, allowed a possible error of 1 %, but her results agreed with Kerr’s within 0.2 yo up to 2.2” K, then jumped to 0.6% greater in molar volumes. Peshkov’s measurement^^^, through refractive index observation, yielded changes in density, particularly with pressure, more accurately than absolute values. Sherman and E d e s k ~ t yundertook ~~ an ambitious program to determine all of the PVT surface between the vaporization and melting curves from 0.96 to 3.32 OK with great accuracy. Their estimated possible error was less than 0.1% in molar volume, but their results are consistently higher than all the others, at both vapor and melting pressures (the latter comparison being made with the Grilly and Mills35 data)?. A t the time of this writing, Taylor and Kerr36 are remeasuring the molar volumes of the saturated liquid, particularly to determine the behavior of the expansion coefficient below 1 OK. It seems highly desirable to present here a consistent and “bestvalue” summary of PVT data. For molar volumes of saturated liquid, there appear to be three attempts to obtain high accuracy. In the region of overlap, 1.2-1.6 OK, the volumes of Taylor and Kerr are higher than those of K e n by 0.2Sy0and lower than those of Sherman and Edeskuty by 0.28%. It seems adequate, therefore, simply to average the data from these three sources. The resulting numbers are shown in Table 1. For the thermal expansion coefficient ct = V-l( aV/ aT),, the values in Table 1 were chosen as follows: for 0.3 to 1.2 OK, those of Taylor and Kerr; for 1.2 to 3.0 OK, those of Sherman and Edeskuty. The dividing temperature of 1.2 OK was selected because here both sets of data yield the same value and above it the conversion of the V-T slopes of Taylor and Kerr to isobaric derivatives becomes too uncertain. The low-temperature anomalous behaviour of ci is illustrated in Fig. 1, i.e., the values become negative at sufficiently low temperatures. t For explanation, References p . 150
see footnote on p. 122.
CH. IV,
9 31
LIQUID AND SOLID
3He
121
TABLE1 P V T relations of liquid 3He at vapor pressures
TE (OK)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.7 2.8 2.9 3.0 3.1 3.2
P (mmHg at 0 ' C)
(cw3mole-I)
0.00001 0.001 50 0.024 05 0.141 8 0.4985 1.291 2.744 5.092 8.564 19.765 38.03 64.91 101.93 150.55 212.28 288.60 381.02 433.73 491.00 553.01 619.92 691.88 769.04
36.785 36.746 36.722 36.713 36.719 36.741 36.777 36.830 36.899 37.089 37.354 37.705 38.18 38.83 39.67 40.75 42.15 43.03 44.05 45.30 46.90 49.00 52.30
V
loa a
102 ,B
(deg-')
(ah-')
- 8.46
3.68 3.69 3.70 3.73 3.77 3.82 3.89 3.96 4.18 4.49 4.94 5.61 6.58 8.03 10.36 14.43 17.9 23.4 33.6 58.8
-4.29 -0.13 4.12 8.50 13.1 17.9 23.1 34.0 48.7 66.5 89.4 118.3 155.4 207.4 287.1 344.0 439.0 593.0 932.0
The values of compressibility = -V-l( aV/ aP), of saturated liquid 3He come from these sources: the conventional PVT measurements over 1.0-3.0 OK of Sherman and Edeskuty; the index of refraction observations over 1.6-3.1 OK of Peshkov; and the velocity of sound measurements of Laquer, Sydoriak, and Roberts3' over 0.343.14 "K and of Atkins and Flicker3* over 1.2-3.2 OK. The last method gives adiabatic compressibility, which must be multiplied by C,/C,; therefore the uncertainty becomes excessive above 2 OK. The measurements were taken within ?n atmosphere of the vapor pressure except those of Sherman and Edeskuty, which were extended to the melting pressure. All the results are in remarkable agreement, i.e., within 1yo, except Peshkov's, which are lower by as much as 10% at T < 2.7 OK and still more at T > 2.7 O K . The results of the others for saturated liquid are combined in Table 1.
References
p . 150
1'1'1
E. R. GRILL\'
AND E. F. HAMMEL
Temperature
[CH. IV.
$3
(OK)
Fig. 1. Thermal expansion coefficient versus temperature for liquid 3He a t various pressures [from Lee and Fairbank*O]. The dotted curve represents additional results of Taylor and Kerra6 a t vapor pressures.
3 . 2 . AT INTERMEDIATE PRESSURES
Although the PVT area bounded by the vaporization curve, the melting curve, T = 1.0 OK, and T = 3.3 OK was measured by Sherman and E d e ~ k u t y ~there ~ , exists the possibility that their molar volumes are too high. Near vapor pressures, this excess might amount to 0.28% below 1.6 O K and somewhat more above 1.6 OK. Near melting pressures, their volumes are greater than those of Grilly and Mil1s3j by 0.3% in the region of 2.0-2.8 O K and by greater amounts above and below this region. Presenting their extensive array of data is only possible in tabIes such as theirs 7 . Below 1 OK, the major interest has been in the behavior of the thermal expansion coefficient a. From A T / d P measurements of adiabatic expansions covering pressures 1.7 to 22 ?tm, Brewer and Daunt39 derived a values over the range 0.15 to 0.6"K, all of whichwere negative. Up to 1.15 OK, they also obtained T where a = 0. Thus they showed that : a was negative below a temperature which monotonically t .4 recalibration of the cell volume shows that the molar volumes of Sherman and Edeskuty should be lowered by 0.30%. The corrected molar volumes are within 0.08% of those a t vapor pressures by Taylor and Kerr and of those a t melting pressures by Griily and Mills up t o 2.8 "K. Therefore, in Table 1 the values of V at T > 1.6 "K are now derived solely from the results of Sherman and Edeskuty. References
p . I50
CH. IV,
5
31
LIQUID AND SOLID
3He
123
increased with pressure; and a had, at 0.2 O K , minimal values which increased with pressure. More direct values of u, through dielectric constant measurements, were obtained by Lee and Fairbank40 between 0.2 and 29.5 atm over a 0.15 to 1.2 OK range. The behavior was similar to that seen by Brewer and Daunt. In Fig. 1 the data of Lee and Fairbank are compared with those along the vapor pressure curve derived from data of Taylor and From all the present results, it appears that CI ( P = 0) could approach T = 0 approximately as cc = -0.1 T , which was computed by de Boereo, by Goldsteinl6P1*, and by Brueckner and Atkins12. 3.3. AT MELTINGPRESSURES a) Low Region The melting curve of 3He was measured by Weinstock, Abraham, and Osborne41from 1.5 down to 0.16 OK by using the blocked capillary method. They found the melting pressure leveling off to 29.3 atm below 0.4 “K, which effect is to be expected when their technique is used below the temperature of a pressure minimum. Lee, Fairbank, and 40 used a dielectric constant measurement to distinguish between liquid and solid. They concluded there is a minimuminmelting pressure, Pmin, at 29.1 atm and 0.32 OK. The first to report a detailed study of the minimum were Baum, Brewer, Daunt and Edwards43, who measured pressures with a strain gauge cemented to the cell containing sealed-off 3He. The gauge was sensitive to f 0.02 atm and had been calibrated to f 0.1 atm when the vessel contained liquid 3He. The cooling and thermometry were accomplished through paramagnetic salts. The measurements showed Pmin to occur at 29.3 f 0.1 atm and 0.32 OK. Sydoriak, Mills, and Grilly4*used a system in which pressure on the solid could always be measured as the sum of a spring pressure plus liquid 3He pressure. Before and after 3He measurements, their bourdon gauges were calibrated, in sitw, to i 0.02 atm. Cooling and thermometry involved separate compartments of liquid 3He. The melting pressures were within 0.1 atm of those of Sherman and E d e ~ k u t yin~ the ~ short region of overlap between 1.0 and 1.2 OK. Sydoriak et al. found the pressure minimum at 0.330 & 0.005 OK and 28.91 f 0.02 atm, and their points below 0.5 OK were consistent to & 0.02 atm with the empirical equation (where T = T , of ref.29) P(atm) = 28.91 Referenres
p . 150
+ 32.2 (T - 0.330)2.
124
E. R . GRILLY AND E. F. HAMMEL
[CH. IV,
3
They also observed at 0.308 "K the heating effect of melting connected with dP/dT < 0. The results of the various investigations are given in smoothed form in Table 2. The question of a minimum in the melting curve of 3He arose in 1950 when Pomeranchuks suggested its possibility (see Sect. 2 . 1 ) . Fairbank and Walters45 were first to observe the reversal in the heat of melting, at T M 0.4 O K , which corresponds to a negative dP/dT. TABLE2 3He melting pressures (atm) below 1.2 "K
T
(OK)
0.12 0.16 0.20 0.25 0.30 0.33 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20
Ref." 29.3 29.3 29.3 29.3
30.1 31.5 33.2 35.2 37.4 39.9 42.6 45.7
Ref..4s 30.8 30.2 29.8 29.5 29.3 29.3 29.45 30.15 31.4 33.1
Hef.44
Ref.4o
28.94 28.91 29.07 29.80 31.02 32.02 34.50 36.79 39.30 42.08 45.10
29.1 29.1 29.1 30.0 31.5 33.2 35.6
Their nuclear magnetic susceptibility measurements showed that alignment in the solid occurs at about 0.3 OK in contrast with Pomeranchuk's estimate of lo-' OK. From this behavior, Bernardes and Primakoff 26 concluded that solid 3He is a nuclear antiferromagnetic, in the paramagnetic region its Weiss constant O( m - T,) being about - 0.1 OK. The corresponding entropy value led to a predicted minimum in the melting curve at 0.37 OK where P, = 29.1 atm. Below the minimum, their calculated pressures agreed with the measurements of Baum et a1.43,while those computed from T , = 10V "K were much higher. Furthermore, they predicted a maximum in the melting curve at 0.08 OK and 31.7 atm, which did not arise from Pomeranchuk's theory. Such region has not yet been investigated experimentally. To obtain volume measurements below 1 OK and in the vicinity of the melting curve minimum, Sydoriak, Mills and grill^*^ used the apparatus already described for melting curve measurements. The Refcremes p . 150
$ 31
CH. IV,
3He
LIQUID AND SOLID
125
TABLE3 Molar volume of melting liquid (V,) and volume change of melting ( A V,) for SHe below 1.2 "K ~
T("K) Vl(cm3/mole) d V , (cm3/mole)
0.5 25.84 1.19
0.33 25.99 1.20
0.6 25.65 1.18
0.8 25.17 1.14
1.0 24.63 1.10
1.2 24.07 1.06
molar volume of liquid Vl and volume change on melting AVm, with estimated possible errors of 0.1 yo and 1yo,respectively, are presented in Table 3. They joined smoothly the results of Grilly and Mills35 above 1.2 OK. The expansion coefficient is shown in Fig. 2. The anomalous negative values of m1 found a t lower pressures are seen to persist up to the melting pressure. From the relation
and the observation that dVl/dPm was approximately constant as P + Pmin, it appeared likely that ml + 0 as T -+Tmin.Furthermore at Tminthe term in parentheses of eq. (1)seemed to be zero, which would make dorl/dT = 0 , This behavior of ccl does not readily fit in with that expected at low pressure, i.e., the vanishing of al at T = 0 with finite negative slope, and therefore deserves more study. In contrast, the
-
5
I
m
d o T 2
-5
U
-10 0.2
0.01
0.2
1.0
0.6
1
I
0.6
I
I
I .o
1.4
I
I 1.4
Tmt deg K Fig. 2. Thermal expansion coefficient (upper figure) and compressibility coefficient (lower figure) of 3He a t melting pressures [from Sydoriak, Mills and grill^^^]. A Grilly and Mills35; Sherman and Edeskuty34; @ calculated from a , using eq. (1). Broken curves represent gS calculated from eq. (2) - - - : assuming pS = p , ; - - - - - - : assuming ps = 0.99 p,.
References
p . 150
126
E. R. GRILLY A N D E. F. HAMMEL
normal behavior of the liquid compressibility
[CH. IV,
53
is also shown in Fig. 2 ,
as measured and as calculated from eq. (1).
The thermal expansion of the solid was calculated from g,
1 dV, dV, 1 - __ -- + +-
=
V, d T
V, dT
dPm
( B E
- 81) dT
(2)
with the reasonable assumption that 8, 2 /I,. As shown in Fig. 2, it was concluded that g becomes negative at a T, of 1.0 to 1.1 OK, in qualitative agreement with the theories of Goldstein1a.17 and of Bernardes and Primakoff 25,26. b) High Region The melting curve in the region 1.2-31 OK and 50-3400 atm was measured by Mills and grill^*^, using the blocked capillary method under a procedure that insured obtaining equilibrium values within very narrow limits. Estimated possible errors in meltin: pressure were 0.02 and 0.2 atm below and above 240 atm, respectively; temperatures were significant within O.O0lo up to 5", within 0.1' between 5' and 14", and within 0.01' above 14". The results are reproduced by three equations :
+ 15.5053 T2 - 1.35019 T 3 for 1.2 < T < 3.148 = 3.748 + 29.5713 T + 3.95049 T 2 for 3.148 < T < 4.4
P(atm) = 26.379
= 24.35
-
0.62615 T
+ 19.4362 T1.517'38
tor 1.9 < T
OK; O K ;
< 31 OK.
Sherman and E d e ~ k u t obtained y~~ similar results and give the equation P(atm) = 24.559 16.639 T 2 - 2.0659 T3 0.11212 T4 for 1.07 < T < 3.1 O K . Several volume relations near the melting curve were examined by Grilly and Mills35 over the range of 1.3-31 'K and 50-3500 atm. Measured directly were: the molar volume of fluid Vf to 0.1 yo;the volume change on melting d V m to 0.5%; the thermal expansion and compressibility of fluid, and Bf, respectively, to 5%. The A V , measurements led to the unexpected conclusion that there exist two forms of solid 3He. The d V of transition was found, indirectly, to be about 10% of AV,. The P-T curve of transition was determined from the sudden change in compressibility accompanying the phase change. The phase diagram is shown in Fig. 3, and some properties of the transition are listed in Table 4. Subsequently, X-ray diffraction
+
References
p . 150
+
CH. IV,
3 31
LIQUID A N D SOLID
3He
127
T (deg K )
Fig. 3. Phase diagram for condensed 3He [from Grilly and MillsS5].
studies by Schuch, Grilly, and Mills4' showed that the cr-solid, existing at lower pressures, has the body-centered-cubic structure and the ,&solid has the hexagonal-close-packed structure. Furthermore, the lengths of unit cell axes yielded a density equal to 0.154 f 0.004 g/cm3 a t 1.9 "K and 96.8 atm and density equal to 0.172 f 0.004 g/cm3 at 3.3 OK and 177 atm, which are in good agreement with the values derived by extrapolation from the direct measurement^^^ along the melting curve. TABLE4 Properties of the
T (OK)
1.8 2.2 3.6 3.0 3.148
References
p . 150
+ 01 transition in in solid 8He AV
(aW
dPjdT (atmideg)
(cm3/mole)
AS (cal/deg/mole)
107.9 113.2 120.8 131.3 135.9
10.7 16.0 22.4 39.9 33.0
0.068 0.094 0.116 0.123 0.125
0.017 0.035 0.061 0.087 0.098
P
128
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
33
The results on Vf and AVm(cm3/mol)are reasonably represented as functions of melting pressure (atm) by:
V , = -3.248 + 50.841 ( P + 1.04)-0.1a1532for 50 < P < 3440; AV, = 1.55910 - 0.39023 log,, (P - 29.033) for 50 < P < 135.92; A V , = 1.506 15 - 0.30825 log,, ( P - 41.212) for 135.92 < P < 3440. Since AV,(P > 136 atm) was observed to decrease with pressure in a regular fashion, it was interesting to examine the behavior of the
Fig. 4. The thermal expansion coefficient, q ,and the compressibility coefficient, of fluid SHe along the melting curve [from Grilly and Millsss].
pi,
corresponding entropy change AS,, which could be computed from AV, and dP/dT through the Clapeyron equation. A formula for A S , as a iunction of P gave a maximum in A S , at 4080 atm, which is only slightly higher than the experimental range, and indicated that A S , = 0 at 77 x lo3 atm (T = 235 OK). Therefore, while a critical point in a melting curve has never been seen, the requirements of one, A S , = dV, = 0 , could possibly be met in 3He, both in principle and technically. References
p . 150
CH. IV,
3 41
LIQUID AND SOLID
3He
129
The compressibility of fluid seems to behave normally aU along the melting curve, i.e., it decreases regularly with increasing pressure and temperature and never changes sign (see Figs. 2b and 4). Previous discussions brought out the anomalous negative values of thermal expansion below 1.2 OK (see Fig. 2 4 . At higher temperatures, ccI first rises to a maximum at 3.1 OK and 140 atm and thereafter falls in a regular way, as shown in Figs. 4 and 5.
I 0
2000
3000
P,(kg C m " )
Fig. 5. The thermal expansion coefficient of fluid SHeand *He along the melting curve [from Grilly and MillsSS].
4. Thermal Properties 4.1. SPECIFIC HEAT
The history of specific heat measurements on liquid We has reflected the interest in trying to answer the questions: 1) Is there a lambda or other type anomaly in the specific heat-temperature curve ? 2) How does the specific heat extrapolate to 0 OK ? In seeking answers to these questions, investigators successively lowered the temperature limit of measurements. The early measurements of de Vries and Daunt 48 from 0.57 to 2.3 OK were improved and extended by Roberts and Sydoriak'g to 0.37 OK, by Abraham, Osborne and Weinstock50 to 0.23 OK, and by References
p.
150
130
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
54
Brewer, Sreedhar, Kramers and DauntK1to 0.085 OK. The first three series obtained Csst,a specific heat for a change of state while the liquid remains saturated (and thus involves three variables : P,V,T) while the last measured C,, where P was constant at 6 to 14 ern Hg. Qualitative, as well as quantitative, differences can occur between the various specific heats, as shown by Goldstein16 in Fig. 6.
TVK)
Fig. 6. Various molar heat capacities of liquid SHea t vapor pressures [from G o l d ~ t e i n ~ ~ ] The dotted curve represents additional results of Brewer et al."'.
In the region of 0.5 to 1.7 OK, the data of Roberts and S y d ~ r i a k ~ ~ were assigned probable errors of 1.5 to 2.0% and fit the empirical formula C,,, = 0.577 0.388 T 0.0613 T3 cal mol-1 deg-l
+
+
with a mean deviation of 1.0%. Below 1 OK, the merging of C,,, and
C, permits a direct comparison in Fig. 7. These results, combined with the early warm-up observations up to 3.21 "K by Sydonak and Hamme16*, permit us to conclude there is no btype transition in liquid 3He down to 0.085 OK. Furthermore, no maximum of any kind, except possibly in C , at 2.5 "K, appears. Below 0.7 OK, the behavior of C is interesting in that the very small variation down to 0.2 OK rapidly changes, so that the extrapolation of C at 0.085 OK to C = 0 at 0 OK by Brewer et aL61appeared reasonable and consistent with the linearity predicted by the theories of G ~ l d s t e i nl6, ~ ~Landau29 . 4, and Brueckner and GammeP. References p . 150
131
LIQUID AND SOLID 3He
CH. IV, 10
I
09 08
2 07 go6 r05 J
204 0 03
a? 0 0
01
a2
04
03
06
05
07
TEMPERATURE (K)
Fig. 7. C , for liquid *He versus temperature [from Brewer, Daunt, and Sreedhara']. rn De Vries and Daunt48. Roberts and Sydoriak". x Abraham, Osborne and Weinstock &O.
+
TABLE5 Specific heat of liquid SHe in cal mole-' deg-l as a function of P (atmospheres) and T (OK)
P T
V.P.
< 0.1 0.10 0.12 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75
References p . 150
4.00 T 0.400 0.472 0.555 0.640 0.684 0.714 0.737 0.757 0.777 0.793 0.807 0.823 0.845 0.867 0.890
15
29
0.510 0.565 0.609 0.630 0.640 0.648 0.654 0.662 0.671 0.683 0.693
0.522 0.571 0.602 0.612 0.617 0.622 0.627 0.637 0.647 0.659 0.673
5 -
~
0.494 0.560 0.626 0.662 0.685 0.698 0.706 0.717 0.732 0.748 0.764
. -
132
E. R . GRILLY AND E. I;. HAMMEL
[CH. IV,
34
Brewer, Daunt and S r e e d h a ~measured -~~ C , at pressures up to the melting pressure between 0.12 and 0.6 OK. The results, partially reproduced in Table 5, show that ( aC,/ aP),is negative above T w 0.16 "K and positive below this temperature. Near 0 OK, a positive value was predicted by Brueckner and Gammels, Hammel et u Z . ~ ~ , and Goldstein16,whereas a negative value arises from an ideal Fermi-gas model. The lowest temperature, 0.12 "K, was not sufficient to allow reliable extrapolation of C, to 0 "K (but see Sec. 4 . 2 on the entropy). 4 . 2 . ENTROPY
The early m e a s ~ r e m e n t s ~50~ ~of specific heat were not at low enough temperatures to allow extrapolation to 0 "K so as to yield absolute entropies directly. Brewer et uLsl linearly extrapolated their specific heat results (equivalent to Csat) from 0.085 OK to 0 OK thereby obtaining Saatto i 0.03 cal d e g l mole-l. The possible error inherent in this procedure is emphasized by the p r e d i ~ t i o n 2 ~of9 ~a~specific heat anomaly below 0.1 OK. Such an anomaly would influence the limiting slope of C and S, and it might change the values of S above the anomaly temperature. However, the data of Roberts and S y d ~ r i a k ~ ~ , who obtained their absolute entropy values from the thermodynamic vapor pressure equation1, agree with those of Brewer et ~ 1 . Another ~ ~ . way of deriving S,,, is through combination of calculated vapor entropy and measured vaporization heat AH,,, which was done by Abraham, Osborne and W e i n ~ t o c kTheir ~ ~ . measured 499
AH,
= 10.39 & 0.02 cal
mole-1 at TL5S=1.5 "K,
from which Seat(1.5OK) = 2.614 f 0.03 cal d e g l mole-l.
Combining this value with their specific heat measurements 5 0 , which are quite consistent with those of other investigators (see Fig. 7), one finds that their entropy values down to 0.23 OK are higher than those of Brewer et al. by 0.10 f 0.06 cal d e g l mole-l. As Table 6 is based on the Ssatvalues of Brewer et al., one should understand from the above discussion that there is a slight uncertainty in the reference zero of the data presented. At higher pressures, extrapolation of C, to 0 OK was more uncertain. Therefore, C, was used only to derive AS,, which was combined with entropy of compression (S, - Sgat).The latter was computed originally References p . 150
CH. IV,
9: 41
LIQUID A N D SOLID
3He
133
by Brewer and Daunt 39 from their thermal expansion results to yield values of S , as a function of pressure and temperature up to 22 atm and 1 OK, respectively. However, their values were slightly altered, using the more direct expansion coefficients of Lee and Fairbank40, to those given in Table 6. TABLE 6
Entropy of liquid SHe in cal mole-' deg-' as a function of P (atmospheres) and T ("K) P T
V.P.
5
10
15
22
4.00 T 0.476 0.594 0.766 0.914 1.042 1.153 1.253 1.344 1.426 1.503 1.573 1.703 1.822 1.933 2.036
4.44 T 0.516 0.635 0.807 0.951 1.073 1.180 1.273 1.357 1.432 1.603 1.568 1.684 1.792 1.890 1.982
4.77 T 0.547 0.666 0.836 0.977 1.097 1.200 1.290 1.370 1.443 1.511 1.574 1.686 1.790 1.881 1.966
5.12 T 0.581 0.701 0.871 1.010 1.125 1.225 1.312 1.389 1.460 1.524 1.585 1.696 1.792 1.881 1.962
5.55 T 0.619 0.740 0.910 1.046 1.161 1.258 1.342 1.417 1.487 1.549 1.607 1.712 I. 805 1.887 1.964
______
T+O 0.12 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.70 0.80 0.90 1.00
Examination of these -Jta led Brewer and Daunt to t,,e conclusion that as T + 0, SPIT= y = Cp/T,where y is a constant. This observation, then, lends support to the Fermi-liquid model theories of Landau23 4 (see calculations by Khalatnikov and Abrikosov5! ') and of Brueckner and Gammel9 as well as to the nuclear spin theory of Goldstein13r15.From the theory of Brueckner and Gammel, y is expected to be 3.78 cal mole-1 d e g 2 at Psatand to increase with pressure. Using the relation of G01dstein~~. l5 between nuclear magnetic susceptibility and spin entropy, Brewer and Daunt obtained y values of 4, 5, and 6 at 0, 11.2, and 27.6 atm, respectively, which are close to their observed values for total entropy. In Table 6, one can see that the normal variation of entropy with pressure is reversed at low temperatures, starting at T M 1 OK for the higher pressures, and becoming completely reversed at T < 0.6 OK. This behavior is also consistent with the predictions of Goldstein15 and of Brueckner and Gamnielg. References p . 150
13.4
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
$5
5. Transport Properties of Liquid and Solid 3He 5.1. THERMAL CONDUCTIVITY AND
LIQUID Several measurements have been made of both the thermal conductivity and viscosity of liquid 3He with the result that the data now extend from approximately 0.26 OK to the vicinity of the boiling point, as shown in Figs. 8 and 9. Although some slight discrepancies exist between the experimental values from different laboratories, the VISCOSITY OF
0.3
0
A,0,
TPK) Fig. 8. The thermal conductivity of liquid SHe. Lee and Fairbanks*, p = 3 a t m ; Challis and Wilks6'.
+
temperature dependence over the indicated temperature range of each of these quantities now appears to be well established. In the thermal conductivity measurements of Lee and F a i ~ - b a n k ~ ~ , anomalous x values were observed at first below the density maximum for high heat fluxes. This was attributed to a contribution from convective heat transfer in the liquid sample ; consequently below this temperature (0.48 OK at 8 atm) and for the higher heat currents, the direction of the heat flux was inverted with respect to the gravitational field. A t high temperatures although the heat transport through the walls of the containing tube was larger than the heat flow through the sample, accurate corrections were made. At low temperatures, where the corrections were smaller, they were slightly less well known due to the perturbing effect of the Kapitza boundary resistance 62 t . The t A t the VIIth International Conference on Low Temperature Physics, University of Toronto, 29 August-3 September, 1960 (see the Programme, p. 22), J. Jeener and References p . 150
LIQUID AND SOLID 3He
CH. IV, 5 51
135
thermal conductivity values of Challis and WilksS7 have not been corrected for the thermal boundary resistance. These authors estimate
T W )
Fig. 9. The viscosity of liquid aHe from Peshkov and Zinov’eva68. Osborne and Abraham68; Taylor and Dashao;0, x Zinov’evas’.
+
A Weinstock,
F’
>
11#
2
. . ..
I
1.
Fig. 10. The ratio of the thermal conductivity t o the product of viscosity and specific heat a t constant volume as a function of temperature for liquid *He and liquid 4He. The kinetic theory value of this ratio, 2.5, is shown b y the horizontal line. The upper curves are calculated using the viscosity data of Zinov’evaal; the lower using the viscosity values of Taylor and DashGo.See footnote p. 134.
that the application of this correction would increase their x values of the order of 10% and hence bring them into closer agreement with G. Seidel showed that the boundary resistance corrections would raise the conductivities of Lee and Fairbank as T falls below 0.5 “K (by 30% at 0.25 OK). Accordingly, x
increases as T decreases. Riferefices p . 150
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E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
5
those of Lee and Fairbank in the range of overlap. Since tabulations and discussion of the viscosity values can be found in the original articles and also in the review article by Peshkov and Z i n ~ v ’ e v a ~ ~ , these data will not be further reviewed here. Attempts to correlate the properties of liquids usually derive from the assumed similarity of this phase either to a disordered solid or a highly compressed gas. The high zero point energy of liquid 3He and its associated expanded structure suggest that the gas model should be the more applicable. A t “high’’ temperatures, this assumption appears to be justified by the fit of x , 7, and C, to the kinetic equation x = 5 / 1 2 C,q as shown in Fig. 10 taken from the work of Lee and F a i r b a ~ ~ k ~ ~ . The apparent failure of this equation below 1 OK suggests that some new process may be contributing in this temperature region to the transport of energy or momentum. Finally it is of interest to note that below 1 OK the marked change in temperature dependence of the viscosity, which has been interpreted as the beginning of a transition to the expected T-2 dependence of 7 for a Fermi liquid’, is not reproduced by the thermal conductivity (see footnote pag. 134). 5.2. HEATTRANSPORT IN SOLID3He
The heat conductivity in solid 3He has been measured by E. J. Walker and H. A. Fairbank63. For a dielectric solid at low temperatures the thermal conductivity should be given by the expression x = ATne-elbT’,
and the results (shown in Fig. 11) demonstrate that this relationship is obeyed by solid 3He. The changes in slope for the lower density curves have been tentatively identified with the change in the sign of the expansion coefficient of the solid reported by Sydoriak, Mills and Grilly44. The discontinuity in the curve BB’ is attributed to the a+ phase change which occurs at this density as the temperature is reduced (see Fig. 3). 5 . 3 . SELF-DIFFUSION COEFFICIENTFOR LIQUID3He
The coefficient of self-diffusion has been measured in liquid 3He, using spin echo techniques, by Garwin and Reich6*and by Hart and Wheatley‘j5t.The former authors determined the pressure as we1 as the t These measurements have recently been extended to 0.03 “K by Anderson, Hart, and Wheatleye2. The magnetic susceptibility was simultaneously determined. References 9. 150
CH. IV,
4 51
LIQUID AND SOLID
I
I
I
I
3He
I
1
137
I
VT PK-~)
Fig. 11. Thermal conductivity of solid SHe from E. J . Walker and H. A . Fairbankss.
temperature dependence of D.Within the experimental error the coefficient of self-diffusion for pure liquid 3He is given as a function both of T and e, by the empirical equation
D
=
5.9 In
0.16 (T) exp (T/2.8)
applicable between about 1.5 and 4 OK for pressures from 2.4 to 67 atm. A t about 0.55 OK the diffusion coefficient in the saturated liquid passes through a minimum and increases rapidly below 0.2 OK as shown in Fig. 12, taken from the work of Hart and Wheatley. Above approximately 0.6 O K , it is apparent from eq. (3) that the References p. 150
138
E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
36
diffusion process in liquid 3He is neither thermally activated [requiring an exp (-T,/T) type temperature dependence] nor gas-like (for which D = IAV cc TIJa). Garwin and Reich suggest that the observed dependence is explicable qualitatively by considering the diffusion of 3He to be a quantum mechanical tunneling through potential barriers. The increase in D at low temperatures is, according to Hart and Wheatley, probably caused by a decrease in the probability of atomic scattering processes, (equivalent to an increase in the excitation mean free path predicted by Pomeranchuk* and Landau2).
*-
VERAGED DATA, 5-6 CM-HG ATURATED VAPOR PRESSUR GARWIN AND REICH. 238ATM 2
5
Fig. 12. Logarithm of the self-diffusion coefficient of liquid W e vs the logarithm of the temperature. The data of Garwin and Reich are taken from ref.u4.
5 . 4 . SELF-DIFFUSION COEFFICIENTFOR SOLID3He
The preliminary data available is discussed in Sect. 6 . 2 . 6. Nuclear Spin Relaxation in Condensed 3He 6 . 1 . LIQUID3He
The spin-lattice or longitudinal relaxation time T , of a spin system is defined as the time necessary for all but l/e of the spins, following an instantaneous change of state, to reach thermal equilibrium with the other degrees of freedom of the medium containing the nuclei in question. T , is therefore a measure of the coupling or interaction between the nuclear spin system and the “lattice”. According to the References p , 150
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IV,
9 61
LIQUID AND SOLID
3He
139
theory by Bloembergen, Purcell and Pound66,the spin relaxation of a given nucleus in a pure liquid is caused by the Fourier components, at the Larmor frequency, of the fluctuating magnetic fields generated at a given nucleus by the thermal or Brownian motions of adjacent nuclei. The associated relaxation time is given by
where y is the gyromagnetic ratio, b the average interspin distance, z, is the "correlation time" of the motion (z, is a measure of the time interval during which molecular orientation persists and the local field at the nucleus is approximately constant) and w o is the precession frequency in the field Ho(wo= yH,). For mot, 1, a condition fulfilled in all of the spin relaxation experiments carried out in liquid 3He to date (t,M 10-12 sec, w o rn lo7), eq. (4)can be simplified to yield Ti1 = 0.9 '/41i2b-'tc.
References p . 150
146
E. R . GRILLY A N D E. F. HAMMEL
[CH. I V ,
fi
7
7 . 3 . ZERO SOUND
A t some temperature below 0.3 OK, the quantum statistical properties of liquid 3He should also manifest themselves in its sound transmission behavior. Landau3 has suggested that ordinary compressional waves of sound will continue to propagate in the liquid provided the wave length is long compared with the mean free path of the quasi-particles, i.e. ~t 1 where t is the liquid relaxation time. In this region the classical attenuation (eq. (8)) should continue to be obeyed. For a Fermi liquid z cc T-2 however, so that for any given frequency there will be some temperature below which the above inequality will no longer be fulfilled; the wave length of the sound will approach that of the mean free path of the quasi-particles, and the sound wave will be strongly attenuated. At higher frequencies or lower temperatures, for which at 1, Landau predicts the existence of a new type of sound termed zero sound. Since the wave length of zero sound is very much less than the mean free path of the quasiparticles, collisions between the quasi particles are neither essential for its propagation nor capable of establishing local thermodynamic equilibrium in the path of the sound wave. Zero sound is thus a nonequilibrium type of wave propagation. It is characterized analytically by a periodic deformation of the Fermi surface (ie., a time variation in the distribution function). An example would consist of an extension of the Fermi surface at maximum amplitude in the direction of the wave motion and a lesser flattening of the surface in the opposite direction. Half a cycle later the deformation is reversed. The velocity of zero sound in liquid 3Hein the limit T -+0 is estimated to be slightly larger than that of first sound, namely 192 m/sec. Although in principlc, zero sound modes which differ from one another in their angular dependence of both velocity and amplitude are possible in a Fermi liquid, Landau considers it improbable that such modes can be propagated in liquid 3He. Experimentally, zero sound in liquid 3He should be equivalent to an ordinary compression-rarefaction wave in the medium and should be demonstrable by suitable ultrasonic techniques. The attenuation of zero sound will be proportional to T2 (h., to the number of collisions of the quasi-particles, which in turn result in absorption of the sound quanta), and independent of the frequency provided the energy qf the sound quanta is small in comparison with that of the quasi-particles, i.e., tiw kT. In addition both these latter
7.4. SOUND PROPAGATION IN LIQUID3He BELOW THE “PHASE TRANSITION”
According to pair correlation theories, at the transition temperature the attenuation will increase strongly. For temperatures below T , and low enough so that the number of quasi-particles is small, it has been predictedss that ordinary sound will again be propagated in the “superfluid” with a velocity
where p , is the momentum at the Fermi surface and p is the mass of a 3He atom. In the correlated phase the attenuation will be small (similar to *He) and will decrease with decreasing temperature due to the decreasing density of excitations. 8. Summary Since the writing of the article on 3He for this series in 1955,not only has much more experimental work appeared, but also theoretical descriptions of the liquid and solid have become much more sophisticated. In 1955, although the difficulties inherent in the simple ideal FermiDirac description of 3Hewere beginning to be recognized, no alternative References p . 150
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E. R. GRILLY AND E. F. HAMMEL
[CH. IV,
08
treatment had yet appeared. Subsequently, several attempts to introduce the effects of interactions between the “particles” were made, with the result that, at the present time, our understanding of liquid and solid 3He has progressed considerably. Formidable mathematical difficulties still stand in the way of a quantitative theoretical description of 3He, however, and consequently assumptions, approximations, and experimental data have been required to derive theoretical predictions of new 3He phenomena. The degree to which the theoretical conclusions depend upon these approximations and assumptions is as yet not well established and for those cases in which experimental data is available to compare with theory, the correspondence, although sometimes impressive, is more often only fair. But it is probably naive at the present time to expect any theory to provide a complete and quantitative description of 3He. Hence if the different theories arc viewed by experimentalists as alternative approaches to an exceedingly difficult problem, and if the comparison of theoretical predictions with experiment is used by the theoreticians to draw conclusions concerning the validity of the various approaches employed, 3Hewill continue to be a rich and rewarding raw material for both experimental investigation and its complementary theoretical interpretation for some time to come. In summary, it appears that : a) The incipient linear temperature dependence of the specific heat as T -+ 0 provides a satisfactory agreement between quasi-particle theory and experiment, at least at low pressures. The prediction of a sharp maximum or discontinuity in specific heat at 0.03 < T < 0.08 by the pair correlation theories still lacks an experimental check. Even if a transition to a correlated phase is subsequently demonstrated, the quasi-particle description may still be valid in the temperature range T , < T T,. Above 0.2 OK, the specific heat lacks a basic explanation in much the same sense as in all other theories of the liquid state. Further specific heat work, both experimental and theoretical, is warranted on the compressed liquid and on the solid, including for the latter an investigation of the predicted singularity in specific heat at about 0.1 OK. b) In general, the experimental PVT relations of the liquid are fairly well established. The locus of the minimum in thermal expansion with respect to temperature and pressure requires further definition, however. For both liquid and solid along the melting curve, it seems that an anomaly in thermal expansion ( a = 0 ) might occur at about
Fg,0
q ( k , a)e(k)
+ P < P,,C
a
[l -
where 1 - q( k , u) may be regarded as the hole occupation number for k < k,. The excited particles above and the holes below the Fermi surface are to be regarded as the elementary quasi-particle excitations. In a normal metal, e ( k ) --f 0 as k -+ k,, so that only an infinitesimal energy is required to excite an electron at the Fermi surface. As we shall see, the various excited configurations of a superconductor can be described in terms of occupation numbers in k-space as in a normal metal. The excitation spectrum differs in that a finite energy, the energy gap, is required t o excite a particle from the superconducting ground state. Phonons, the quanta of the lattice vibrations, have a Debye specof the order of eV. These fretrum with average energies, &oph, quencies are changed very little by the transition to superconductivity. Plasmons, quanta of plasma oscillations of the electron gas, with energies of rn 10 - 20 eV, are not normally excited and play no direct role in superconductivity. Introduction of the plasma modes is important, however, for an adequate treatment of screening of the electrons, as will be discussed in more detail in 5 13. When a particle is excited out of the Fermi sea, there is a hole left behind. The excited particle and the hole will in general not be bound together in space and they may be regarded as independent excitations and treated in an equivalent manner. Quasi-particle excitations are thus created in pairs from the ground state. 5 . 3 SCREENIXC AND BACKFLOW
A quasi-particle is not a “bare” particle moving independently of all others, but should be regarded as a particle moving in the electron fluid. In the language of field theory, it is “clothed” by interactions with phonons, plasmons and other particles. Surrounding each electron is a “screening hole” with a net deficit of electronic charge which is just equal t o that of the electron in question. A local depletion of the other electrons resulting from Coulomb repulsion leaves a positive uncomKcjerenccs
p . 282
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187
pensated ionic charge which balances that of the electron in the center of the hole. The situation is illustrated in Fig. 1, which also shows schematically the motion of the electron from A to B. Since the screening hole moves with the electron, there is no net transfer of charge. When the electron in question is transferred from A to B there must be a compensating “backflow” in the surrounding electron fluid. The concept of backflow was introduced by Feynman and Cohen55in discussing the nature of the rotons in HeII, which also may be regarded as quasi-particles
A
B
Fig. 1. Screening hole suricxnding each electron. When an electron moves from A to B there is a compensating “backflow” in the surrounding electron fluid.
moving with an associated backflow. At large distances the backflow leads to a current distribution which is dipolar in form. The velocity potential a t large distances is given by: (5.2)
The strength of the dipole, ,u, is proportional to the velocity of the particle. Backflow is a collective motion which may be described in terms of collective variables. In the formalism of Bohm and Pines, in which the long range part of the Coulomb interaction is described in terms of plasmon variables, the backflow at large distances may be viewed as a cloud of virtual plasmons which move with the electron. The problem of backflow for electrons in metals has been discussed by Pines and one of the authorsg. They have pointed out that it is essential to include backflow in order that the quasi-particle excitations satisfy the equation of continuity, a problem closely related to the gauge invariance of the theory. In this connection, there is a marked difference in the description of current flow in longitudinal and transverse waves. In Fig. 2 , we have shown schematically the elementary Rrjerrrtces
p . 2S2
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J. BARDEEN AND J . R . SCHRIEFFER
[CH. VI,
$5
dipoles associated with quasi-particle motion for the two types of waves. There is an exact analogy with longitudinal and transverse waves of magnetization ; the velocity-field corresponds to the H-field from magnetic dipoles and total current density from both particle motion and backflow to B = H 4nM. For long wave lengths, the longitudinal currents are described almost entirely in terms of the collective motion, and thus in terms of plasmon variables. The quasiparticles with their screening holes do not contribute appreciabIy (corresponding to B = 0). The opposite occurs for transverse waves, the backflow from different parts of the wave cancels out (corresponding to H = 0 ) , so that the current is just that which would be obtained from particle motion alone with backflow neglected. The above considerations show that in calculating the response to longitudinal waves, it is necessary to consider collective excitations
+
- - c c c
-+
- - c c c
4-
- - t c c c
- 4
4 - c - c
- 4
Longitudinal waves, B='O
-
c
-
4
c
-
c
-
c
-
4
c
c
c
CCtCCCC
C
C
C
F
I
-
C
C
Transverse waves, H=0
Fig. 2. Longitudinal and transverse waves of magnetization. The velocity-field of the hackflow from the quasi-particles corresponds to thc H-field from the magnetic dipoles and thc total current density from both particle motion and associated backflow to B = H + 4xM. For long wave length longitudinal waves B = 0, while for trans\wse waves H = 0.
explicitly. On the other hand, the response to transverse waves is almost entirely by quasi-particles, and their contribution to the current can be calculated in the usual way, neglecting backflow. This is the essential reason for the London choice of gauge, div A = 0 , which implies transverse waves. In order to make calculations in a general gauge, it is necessary to introduce collective as well as quasi-particle variables, In 5 13 we shall discuss recent gauge invariant calculations of the Meissner effect. Results are essentially the same as those of earlier calculations made in the gauge div A = 0 and which considered explicitly only the quasi-particle excitations. Backflow may also be neglected in considering steady currents flowing in a wire or other conductor. It should be pointed out that a plasmon excitation is simply a coherent superposition of electron-hole excitations which has the form of a density fluctuation of the electron gas. Since the plasmons Rcfucnces
p. 282
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RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
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constitute good elementary excitations for long wavelengths, it is clear that one would be over describing the system by including all possible quasi-particle excitations as well as plasmon excitations. I n many problems, no difficulties arise because of the fact that only the special combination of quasi-particle excitations corresponding to a coherent density fluctuation must be suppressed. Actually, it is possible to carry out a consistent calculation in which no extra variables are introduced by use of generalizations of the Hartree self-consistent field method appropriate to normal and superconducting states (see 5 13). I n addition to the plasmon cloud surrounding each electron, there is a displacement of ions in the vicinity which follows the motion of the electron and which may be described in terms of a cloud of virtual phonons. Further, in describing the phonons, it is important to take into account the screening of the fields of the ions by the electrons. Plasmons, while primarily an oscillation of the electron gas, also involve some ionic motion. 5 . 3 . INTERACTIOKS BETWEEN ELEMENTARY EXCITATIOXS
We shall next discuss the interactions of the quasi-particles with the Fermi sea and with each other. Those important for superconductivity are the electron-phonon interaction and the screened electron-electron interaction. An excited electron can decay or be scattered by emitting or absorbing a phonon, by exciting another electron out of the sea (creating two new quasi-particles) or by interaction with another quasi-particle. The lifetime, z, of quasi-particle excitations is reasonably long a t moderate temperatures because of the restrictions on scattering introduced by the Pauli principle. This accounts for the success of the Bloch individual particle model. A particle with an excitation energy E can knock another particle out of the Fermi sea only if the energy of the latter is within E of the Fermi surface, E,. The energies of both particles after scattering must also be within E of E,. The effect of these restrictions on the available phase space is to increase the free path for electron-scattering by a factor of the order of ( E , J E ) ~This , gives for E , NN lOeV and E M 0.OleV (corresponding to T M 100" K) a free path of the order of (lo6 x 10V) cm, or 10W cm. For values of E of this order or smaller, the free path is restricted by the electron-phonon rather than by electron-electron scattering. Heavy elements with a low Debye temperature have a large electron-phonon interaction, so Refmnces
p . 282
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J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
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that the electron may be readily scattered by excitation of a phonon, with a relatively small mean free path. It is believed that this may account for the anomalous superconducting properties of Pb and Hg, as will be discussed in more detail in 5 9. There is a large number of low-lying excited normal state configurations which correspond to exciting electrons to small energies above the Fermi sea. These may be described, as in the Bloch model, by giving the occupied quasi-particle states in k-space. A typical configuration is shown in Fig. 3. To complete the description, one would have to give the occupation of phonon and plasnion states. It is yresumed that the correlation energies in the ground state are adequately taken into account and that all that prevent the configuration from t
Fig. 3. A typical excited configuration :ii the normal state. Quasi-particle excitations are specified as occupied states aliove and holes below the Fermi surface.
being an exact eigenstate of the Hamiltonian are interactions between the elementary excitations. Thus the configuration is not to be regarded as given by a Bloch determinantal wave function. This type of phenomenological description of a Fermi gas with interactions has been generalized by LandauSs in his theory of the Fermi liquid to include the dependence of the energy of the quasiparticle on the distribution of the particles in k-space in a manner similar to the Hartree-Fock method. He has given a justification for it from basic theory by use of Green's function methods. Long range Coulomb interactions in an electron gas introduce complications not considered explicitly by Landau. This problem has received a great deal of attention from theorists, and considerable progress has been made, but there is as yet no really satisfactory quantitative treatment for the normal range of electron densities. Rq'en~nces p . 2852
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6. Electron-Phonon Interactions Frohlich's suggestion 37 that superconductivity arises from the electron-phonon interaction pointed the way toward the development of a successful theory. His derivation57"of an attractive interaction between electrons from exchange of virtual phonons later was extended by Bardeen and Pines 57b to take Coulomb interactions into account. The interaction may be described in a qualitative way as A particle near the Fermi surface in a state k , emits a virtual phonon of wave vector q and is scattered to a state kl = ki - q. While the electron does not have enough energy to emit an actual phonon, it may do so momentarily because of the uncertainty relation AEAt M fi. A second electron in k , absorbs the phonon and is scattered to kl = k , + q. The effect is to scatter electrons originally in states k,, k , to k;, k; with conservation of wave vector: k,
+ k , = k; + k;.
(6.1)
This corresponds to a pair interaction between the particles, and it is attractive if the energy difference between the electron states involved is less than the energy of the virtual phonon, noph.The criterion for superconductivity is essentially that this attractive interaction dominates the repulsive screened Coulomb interaction. The physical origin of the phonon interaction arises simply from the fact that an electron making a transition from state k , to k , - q gives rise to a charge density fluctuation, de;, of wave vector q and frequency &o(k,, k , - q ) = ~ ( k , ) - &(kl - 9). As a consequence of the electron-phonon interaction, de: can excite a phonon. This phonon will exhibit an ionic (and an associated electronic) charge density fluctuation, de;, which will be out of phase with the initiating electronic charge fluctuation Se; if o(kl, k, - q) is greater than the natural frequency, oqof the phonon. If the reverse is true, de; will be in phase with de;. This process describes the dynamic screening of the electric field set up by the virtual electron transition k , -+ k , - q. I t follows that the strength with which the second electron recoils, ( k , -+ k , q ) , depends on the effectiveness of this screening. If w(k,, k, - q) < w q , over-screening occurs, crudely speaking, by the positive ionic charge fluctuations building up to a value which more than compensates the Coulomb field set up by dp; and the second particle is attracted to rather than repelled from the first. For ~ ( k , k, , 4) > w q ,anti-screening occurs since dp: and are out of
+
+
Ktfrreitces
p . 282
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J. BARDEEN AND J. R. SCHPIEFFER
[CH.
VI, $ 6
phase. As one might expect, the screened interaction due to virtual phonon exchange is proportional to 1
characteristic of the response of a driven harmonic oscillator. It should be mentioned that the phonon interaction and the screened Coulomb interaction also contribute t o the self-energy of the quasiparticles. One such contribution would correspond to the electron in k, emitting a virtual phonon and going to k;, then reabsorbing the phonon and returning to the initial state. It is presumed that all such self-energy corrections are included in the description of the normal state configurations and that these corrections are essentially unaltered in the superconducting state. All that need appear in an effective Hamiltonian for the electrons and phonons are the true interaction terms. The matrix element for the phonon interaction is:
where M , is the matrix element of the electron-phonon interaction and &GO,the energy of the phonon involved. The interaction is attractive (negative) if the energy difference between the electron states is less than &w,. The criterion for superconductivity is that this attractive interaction dominate the screened Coulomb interaction, which may be written 4ne2 VCOUl
=
I k; - k , 12 + k:
’
where k, is a screening constant. This condition may be written symbolically in the form
where the average is taken over an interaction region near the Fermi surface where I E; - E~ 1 < &wc. Here w , is an average phonon frequency, perhaps half the Debye frequency. This criterion has been studied by Pines and in more detail by Morrel References p. ?S?
CH. VI,
9
71
RECEKT DEVELOPMENTS I N SUPERCONDUCTIVITY
193
on the basis of a simplified model5*. They find that the elements most favored are those with a large number of valence electrons per atom, and within this limitation, a low electron density, in agreement with Matthias' empirical rules. The criterion very roughly separates superconductors from non-superconductors. The largest contribution to Vp,, comes from the Umklapp region where ki - k, lies outside of the first Brillouin zone. If the states k are described in the expanded zone scheme, this applies to most of the possible virtual transitions. In the Umklapp region, I k; - k , I may be relatively large (reducing the Coulomb contribution) while the reduced wave vector q = k , - k; K and thus fico, is small. Here K is a lattice vector of the reciprocal lattice space. The role of the Coulomb interactions in counteracting superconductivity will be discussed further in § 9.
+
7. Elementary Excitations in Superconductors The most striking difference between the excitations in normal and superconducting states is the existence of an energy gap for quasiparticle excitations in the latter. Quasi-particles in superconductors may be designated by a wave-vector k and spin u in one-to-one correspondence with those of normal metals. The energy, E,, may be written in the form
where E, is the Bloch energy in the normal state relative to the Fermi energy and A , is an energy gap parameter which is obtained from the theory as a solution of an integral equation. The excitations correspond roughly to particles above and holes below the Fermi surface, although in a superconductor there is no discontinuity in the nature of the excitation as the Fermi surface is crossed. If q(K, 0) = 1, 0 gives the occupation number of the excitation, the total excitation energy above that of the ground state, W,,, is
we,, w,,= k,2 q(k, +%> -
a
(7.2)
in exact analogy with the normal state. The value of A , depends on the distribution of excitations, and so varies with the temperature. Having a maximum at T = 0" K, the energy gap gradually decreases with increasing temperature and vanishes at the transition. In general, the energy gap may be anisotropic and depend on the direction of k as well as on the energy. There is increasing experimental Hefercnccs
p . ?82
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J . BARDEEN AND J. R. SCHRIEFFER
[CH.VI, 5 7
evidence for some anisotropic effects which are, of course, peculiar to the particular metal. Since such effects are not large and are not essential to an understanding of superconductivity, we shall for the most part ignore them and assume that A , is a function only of the energy E~ of the Bloch state involved. Theory indicates that A , should be appreciable over the range for which the attractive phonon interaction is significant, that is within the order of an average phonon energy, fiw,, of the Fermi surface. A typical plot is shown in Fig. 4. For the simplified interaction used by BCS, A is a constant up to a cut-off, no,,and zero thereafter. One is usually interested in excitation energies of no more than a few k,7,.
c
Fig. 4. Variation of the energy gap parameter A(&) near the Ferini surface. In the case of weak coupling, ?iw, % A .
In what is called the weak coupling limit, and which applies to most superconductors, K,T, noc.In this case, one may take d = const over the interesting range without appreciable error. The weak coupling limit does not apply to metals with very low Debye temperatures, such as lead and mercury. Some of the large amount of experimental evidence for an energy gap will be discussed in more detail in later sections. As mentioned in the introduction, the most direct evidence comes from experiments on absorption of electromagnetic waves, either in the microwave or far infrared part of the spectrum. Other evidence comes from experiments on specific heats, absorption of ultrasonic waves, nuclear spin relaxation times, and thermal conductivity. All of these depend on the presence of excited electrons. Experiments done at very low temperatures indicate that the number of excitations drops exponentially, as exp(- bT,/T), which is suggestive of an energy gap. Most of this evidence has been accumulated since 1953, when Brown, Zemansky and Boorsebg showed that the electronic specific heat of vanadium follows an exponential law and Goodmanso found an exponential drop in the thermal conductivity of tin which he interpreted in terms of an energy gap. In Table 2 we have listed empirical values of the gap for several metals.
liw,, and (9.4) becomes. References p . 282
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Here the density of states has been repIaced by N(O),its value at the Fermi surface, clc = 0. In the weak coupling limit,
A
=
2Kw exp (-
l/N(O)V).
(9 * 6)
Practically, the weak coupling limit may be used without appreciable error for N ( 0 )V < 0.5. The energy reduces in this limit to :
w - w = - W ( O ) dm2,
(9 * 7 )
where il(0) is the energy gap parameter at the Fermi surface. Since the interaction, Vkv,is essentially independent of isotopic mass, the isotope effect follows65because the phonon energy, Kw,, which determines the cut-off varies as M-Y2. The solution is such as to make the best use of the available phase space to get a maximum number of pair interactions. The contribution of a pair state k to the energy of condensation is (for k > K,)
The first term represents the Bloch energy of both particles of the pair state k and the second the interaction energy from matrix elements leading to transitions into or out of the pair state k. The maximum contribution comes from states at the Fermi surface, where W , = - A . 9 .2 . COULOMBINTERACTIONS AND LIFE-TIMEEFFECTS
An open question is the role of matrix elements Vllck, of the screened Coulomb interaction, which extend to high energies, of the order of E,, above the Fermi sea. Bogoliubov4 has suggested that if one does not cut off the Coulomb interaction at no,,but allows the pairing to extend t o energies of the order of E,, one can get a superconducting state of lower energy. The result is similar to that one would obtain with a cut-off at Kw,, but with the Coulomb terms reduced by a factor of the order of log (EF/KwC), typically about 5. If this calculation were valid, there would be two serious difficulties: (1) The exponent tc of the isotope effect would be expected to depart significantly from 0.5, contrary to experiment. (2) The effect of the Coulomb interactions would be reduced so much that nearly all metals would be expected to be superconducting. Although one can make only rough estimates, References p . 282
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J . BARDEEN A N D J . R. S C H R I E F F E R
[CH. VI,
99
a factor of five is difficult to reconcile with Pines’ and Morrel’s calculation~~~. One of the authors has suggested that the cut-off, fiw,, may be determined by the life-time of the quasi-particle excitations. When I E, I is large, the excitation may decay so rapidly that it is not well defined. Mathematically, this may be described as an imaginary part of the energy, which, for I E, I large, may become greater than the real part. An estimate given in 9 13, Fig. 33, shows that this occurs for decay from phonon scattering near the Debye energy, and thus may give the desired cut-off. However, there appears to be an energy region beyond this, extending to perhaps 10 6wc, where the excitations are again well defined. At still higher energies, the life-time for decay from scattering by exciting a particle from the sea becomes short. There is thus an uncertainty as to just what determines the cut-off. A reasonable value for the cut-off is obtained if one requires a life-time long enough for a particle to go a coherence distance, but as yet there is no good mathematical justification for this. The essential difficulty is that the superconducting transition energy is only a tiny fraction of the Coulomb correlation energy, and also of the electron-phonon selfenergy. Life-time effects are likely to play an important role for lead and mercury, for which the cut-off is probably not much larger than the energy gap. In these elements, the electron-phonon interaction is particularly strong. 9 . 3 . EXCITATION SPECTRUM
The energy of a “single” quasi-particle excitation may be determined in the following way. If one of the pair, say k T , is occupied and its partner, - kj,, unoccupied in all configurations, the state k is not available for transitions of pairs of equal and opposite momentum because of the Pauli exclusion principle. This subtracts an energy W , from the ground state, giving an increase in energy of E , - E,. The energy of the particle in k t is E ~ so , that the net increase is E,. This is, of course, the reason for the notation we have used; E , is just the quasi-particle energy in the superconducting phase. The minimum value of E , is O(O), the energy gap parameter at the Fermi surface. A pair excitation in k is described by the function (S. €9, which is orthogonal to the ground state. This is the anti-bonding combination which adds an energy E, + E , in place of W,. The energy relative to Kcfwmrcs
p. 282
5 91
CH. VI,
205
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
the ground state is therefore 2E,, just that of two single excitations. In calculating the energy with the reduced Hamiltonian, one need not distinguish between pairs of “single” excitations and true “pair” excitations. Just as in a normal metal, one can describe an excited configuration by giving an occupation number, q ( k , u), equal to unity if there is an excitation and zero otherwise. Occupancy of both k f and - k 4 implies an excited pair wave function. The total excitation energy is We,,
-
W , = 2 q ( k , u)E,. k0
(9.9)
One can see from the above considerations why there is an energy gap in a superconductor but not in normal metals. The breaking of a pair, say by transferring an electron from k t to another state (k d k ) t , infinitesimally close to k, gives single excitations in - k 4 and ( k A k ) 1 . This eliminates two pair states, k and k dk , from virtual transitions with a corresponding increase in energy of 2 I W , 1. In a normal metal, interaction energy arises from the possibility of making virtual transitions to states above the Fermi sea and unoccupied states below. A transfer of a particle from k t to (k d k) f means that ( k + d k) ? is no longer available for such transitions, but k t now becomes available. Since there is no preference for ( k f , - k .f ) transitions in the normal state, the difference in energy becomes vanishingly small as d k approaches zero. I n the above discussion we have considered only the coherent contributions to the energy from scattering of pairs of opposite spin and momentum. The effect of other terms in the interaction Hamiltonian have been estimated and have little effect except in bringing about the collective excitations. As the temperature is raised above 1’ = 0” K, the number of excitations increases and the pairing energy and energy gap decrease. Since the quasi-particle states in k-space may be occupied independently, the entropy is given by the usual expression for particles obeying Fermi-Dirac statistics,
+
+
+
+
TS=
---/-lX {f(k,o)Inf(k,a) -[1
P, 0
-f(k,u)]ln[1 --f(k,o)]}, (9.10)
where ‘I/ = k,?’ and f ( k , a) is the average occupancy of states in the neighborhood of ( k , a). The energy gap parameter is now determined in such a way that the free energy liefcrcmc\
p.
‘?A’?
206
[CH. V I , 5 9
J . BARDEEN A X D J . R . SCHRIEFFER
F
=
W
-
TS
=
is a minimum. By minimizing F with respect to h,, a quasi-particle representation is determined which best represents states typically excited a t temperature T. This leads to an integral equation of the form (9.12)
The maximum value of T for which there exists a non-vanishing solution for A is the critical temperature, T,. As pointed out by Cooper66, the form of the integral equation is such that if there is an energy gap over part of the Fermi surface, there will be one everywhere, except perhaps a t isolated points or lines. To see this, suppose the contrary is true, and that A, is zero everywhere except in a region R of k-space. Then for a point k not in R, it would be required that A,. Ek’ d k = - v k k 3 -tanh -= 0. (9.13) t’ 2E,, 2k,T But if V,,, does not vanish for all k’, there is no reason why the sum over k’ should vanish, except perhaps accidentally a t isolated points. For a general interaction, A , can take on positive as well as negative values. The energy gap, however, is 2 I A , I. I n Fig. 6 , the energy gap A
.
\ - - - - -1 - .0 tI
I
o.2
\: \ -1
i
big 6 ’fhr \ariation of t h e energy gap parameter 4 ( T ) vitli tcniperature as piedictid
by theory. h’ejereweA
p . 28%
CH. VI,
5 101
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
207
is plotted as a function of temperature for the simplified model with V , , = - V for I .sic 1 1,c,,
can be expressed
(+)[3K,(PA)+ K,(BA)l 2
8.5 exp (-
\ 26 exp (-
1.44 T J T ) for 2.5 < T,/T < 6 1.62 T J T ) for 7 < T J T < 11,
( 10. 3)
while at extremely low temperatures the coefficient in the exponent tends to 1.76. The specific heat measurements of Corak et al. on vanadium and Corak and Satterthwaite 6 R on tin exhibit an exponential behavior for T J T > 1.3: ~e' 8 -
YT,
a exp [- bT,/T],
(10.4)
where a = 9.17 and b = 1.50, in good agreement with the predicted curve in this region. The measurements on vanadium and tin have been extended to T,/8 by Go0dman6~,and he finds the vanadium data agree with (10.4) down to the lowest temperature measured, while the tin data show an upward curvature for T J T > 4 on a log c,,/yT, vs. T J T plot. More recent experiments by Chou, White and Johnston70 on niobium can be fitted by (10.4) with the parameters used for vanadium, although their data also fall above this law for lower temperatures. Goodman 69, Zavaritskii'l and Phillips'2 have independently carried out measurements on the specific heat of aluminum. Zavaritskii's and Phillips' data closely follow an exponential law down t o T,/T = 6, with a value of b in (10.4) somewhat smaller than that given by the theory, while Goodman's data definitely show an upward curvature beginning at T J T = 4, as may be seen in Fig. 7. Another example of upward curvature is afforded by measurements on zinc by Phillips, although similar measurements by Zavaritskii fall accurately on an exponential curve. It has been suggested that this upward curvature results from 1) a low density of states located in the energy Krferenccs
p . 28?
CH.
VI, 9 101
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
209
gap, e.g., collective states, or 2) an anisotropic energy gap. Calculations indicate that the density of collective states is too low to account for the effect. The second possibility appears to be not unreasonable since Morse, Olsen and G a ~ e n d a 'have ~ found evidence for an anisotropic energy gap in tin, for which the specific heat curve shows upward curvature for T J T > 4.
Fig. 7. Comparison between the theoretical electronic specific heat with empirical data on several weak coupling superconductors.
Measurements of the critical magnetic field as a function of temperature, when combined with the thermodynamic relation (10.5) 1Ceferencc.s p . 282
210
J. BARDEEN AND J. R. SCHRIEFFER
[CH.VI,
3
10
provide an accurate method of determining c,, at low temperature. The data of Maxwell and Lutes 75 on thallium, indium and tin along with the recent curves for lead by Mapother and c o ~ o r k e r s ~ ~are **b plotted in Fig. 8 as deviations from Tuyn’s 1 - ( T / T J alaw, which agrees with the Gorter-Casimir model. Negative deviations for tin, vanadium, etc., reflect the exponential drop of the electronic specific
Fig. 8. The deviation of the critical magnetic field from the Tuyn’s law H,/H, I - (T/T,)zwhich agrees with the Gorter-Casimir two-fluid model.
=-
heat. The positive deviations for mercury and lead suggest that superconductors with smaller ratios of TJ6, must be considered separately; as we will see later, it is likely that these “bad actors” must be treated by an intermediate or strong coupling theory while the weak coupling approximation appears to hold rather well for the remainder of the superconducting materials. The electronic specific heat derived from the magnetic measurements are plotted in Fig. 9. Thus we conclude from the bulk of the thermal and magnetic data that the exponential variation of c,, for temperatures well below T , is a general property of superconductors and is a consequence of an energy gap in the spectrum of elementary excitations. The smaller References p . 282
CH.
VI, 9 101
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
211
rate of decrease below T J T = 4 may be due to either a low density of states with a smaller energy gap or perhaps more likely an anisotropic energy gap.
Fig. 9. The electronic specific heat in the superconducting state for elements requiring an intermediate to strong coupling theory compared with the prediction of the weak coupling theory.
The jump in specific heat corresponding to the second order phase transition at T,, as given by the current theory3, is (10.6)
while the Gorter-Casimir model gives 2.00 and the Koppe mode177 gives 1.71 for this ratio. Empirical values of the ratio c,,/yT, are given in Table 3. References p . 282
212
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
9 11
TABLE I11
Ratio of the electronic specific heat in the superconducting
and normal states at Tc Element .
Pb I-lg Nb
Sn A1
Ta V Zn TI Theory
__
~ t % ( ~ C ) / Y ~ C
-~
3.65 3.18 3.07 2.60 2.60 2.58 2.57 2.26 2.16 2.44
11. Transition Probabilities and Coherence Effects 1 1 . 1 . THEORY
For many applications of the theory, one needs to calculate matrix elements between excited states of the system of an interaction expressed by asingle particle operator of the form (11.1)
in which Bku,pu'is the matrix element for scattering from ko to k'o' and cto and cka are creation and destruction operators for quasiparticle. excitations in the normal state. In the Bloch approximation, individual particle wave functions, v k a ( x ) , may be defined for a selfconsistent field which is not changed very much by small excitations of the system. In this case, (11.2)
in which x may be defined to include the spin variable. More generally, the matrix element is defined for many-particle normal state functions which include correlation effects and which differ by transfer of a quasi-particle from ko to k'a'. In other words, ko is occupied, k ' d unoccupied in the initial state and k'a' occupied, kcr unoccupied in the final state. Occupation of other quasi-particle states is the same in the initial and in the final state. The matrix element Bka,k'o, may depend weakly on the configuration of the other particles, but such dependence usually is negligible for small excitations of the system. Matrix eleReferences p . 282
CH.
VI, 9 111
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
213
ments of H , between excited states of the superconducting phase may then be calculated in a straightforward manner from the corresponding matrix elements for the normal state. A striking difference between superconductors and normal metals arises from coherence effects associated with the paired wave funct i o n ~ I~n. the normal state scattering from ko to k'a' is entirely independent of scattering from -k', -0' to - k , -a, as well as of all other transitions. The probability of the former is proportional to 1 BkU, k'u, (2, the latter to 1 B.-F,-,,,,-k, 12. Because of the nature of the paired wave functions of a superconductor, these two contributions are coherent, and one must add the matrix elements before squaring. To see that this is true, consider the matrix element of a spin-independent interaction between two excited states of a superconductor. Suppose that in the initial state, k t is singly occupied and the pair k' is either in its ground state or has a pair excitation (see 3 9). In either case, the initial state is a linear combination of normal statelike configurations in each of which k ? is occupied and -k J. unoccupied. As illustrated in Fig. 10, in some configurations (a) the pair k' zi (k' 1' , - k ' $ ) is unoccupied, in others (b) the pair is occupied. We have supposed that in the final state, the pair k is either in its ground state or is excited, and that k ' t is singly occupied. In configurations (c) of Fig. 10 the pair k is unoccupied; in (d) it is occupied. --d
initial
k t
(0)
(bl
-kl
final stole
siate kit
-_o_,-P q --?-*-,m
k l
-kil 0
(c)
(d)
0
-kl 0
kit
-kit
0
o o
Fig. 10. Configurations which enter when a quasi-particle makes a transition from a singly occupied state in k f to one in k' f .
There will be nonvanishing matrix elements of H , between the two configurations (a) and (c), corresponding to scattering from k t to k' t , if the occupancy of all other unspecified states is the same. But there is also a nonvanishing matrix element between the same initial and final states corresponding to scattering of a particle in - k ' $ in configuration (b) to - k $ in (d), again with the same occupancy of all other unspecified states. Since the total number of particles in configurations (a) and (c) must be the same as in (b) and (d), there are two more particles in the unspecified states of the former configurations. References p . 282
214
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
5 11
When the total number of particles is large, this difference has a negligible effect on the weight with which different normal state configurations enter into the sum representing a superconducting state wave function. Depending on the nature of the interaction, the two contributions (a) -+ (c) and (b) -+ (d) may add constructively or destructively. In general, B has the same magnitude for k t k‘f as for -k’J to - k 4, because the wave vector differences are the same, but they may differ in sign. The two cases are: Bk’a’,ku
=
-k
B-k.
--o,
-k’, -d
(case I)
(case 11). B-k, -a, -F, --ol where Ooo. = + 1 for 0 = u1 and Oaul = - 1 for u = - IS’. The first applies to an ordinary potential interaction, such as is involved in calculating the absorption of longitudinal ultrasonic waves, the second to the electromagnetic interaction and to the hyperfine interaction involved in nuclear magnetic resonance relaxation times. The coherence factors may be calculated most simply by use of the quasi-particle operators introduced by Bogoliubov4 and by Valatin6. Both “single” and “pair” excitations of a superconductor may be defined through the operators : &,‘,ko
= - O,,
yf+
Y-fJ
Ilkc,*, ukc-z$
VkC-kS
-k
vkCkt>
(11.3)
(11.4)
,--
where uk = V1 - h, and v, = f i kas in 5 9. A single excitation in ( k , a) is defined by y&Yoand an excited pair in k by yf+y-&!P,,. These correspond to the “single” and “pair” excitations discussed in 5 9. The y operators obey the usual Fermi-Dirac commutation relations. In the normal state ( A = 0 ) , yz+ creates a particle in k.f if k is above the Fermi surface and a hole in - k 4 if k is below. The superconducting ground state may be defined as the vacuum for quasi-particle excitations : (11.5)
With these definitions, \yo must be regarded as an admixture of states with different total numbers of particles, peaked about an average number, n. This simplifies the mathematical formalism, and creates no difficulties for systems with large numbers of particles. This may be seen as follows, Matrix elements of a particle (not quasiKefereitccs p. 282
9 111
CH. VI,
215
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
particle) conserving operator, H,, between states !Pamay be calculated by decomposing !Painto components, !Pan.,each with a fixed number of particles :
!Pa = ;I;An,!Pa,,,
(11.6)
n’
where (Yaw,, !Pan,)= 1. Since H , is particle conserving, we have (YP, H I !Pa) =
X A*,Aw,,(!Pp,,Hl!Pant.) =
n’, n”
(11.7)
As in a grand canonical ensemble, the weights [ A , l2 are sharply peaked about the average n. Since the n-particle matrix element is slowly varying with n, the total matrix element of H , between the !Pas is, to order l/n, equal to the corresponding matrix element between states with n particles. In terms of the y operators, we have for the sum of the two coherent contributions to the matrix element (omitting the common factor Bk‘o’$ ka)
C$&k0
& 6ua!c?k-&-k,-a,
= (f’d,,u, ;f
vg*vk) (y$,yYh
+ (“i% .kVx’)
($dYrk-o
8ua,
*
Y-k-oY--k‘--o’)
(11.8)
f
y--k’--o’yko)*
The first term on the right corresponds to the scattering of quasiparticles and the second to the creation or destruction of two quasiparticles. For example, the matrix element for scattering a quasiparticle in the superconducting state from ko to k ‘ d is &oi‘,ko(t+~l, ? vgnk).The upper signs correspond to case I and the lower to case 11. In a normal metal, creation of a pair of excitations corresponds to exciting an electron from below to above the Fermi surface, with formation of a hole below and an excited particle above. The transition probability from a state of energy E to one of energy E no induced by interaction with a field of angular frequency o is proportional to the square of the matrix element and to the density of final states, N ( E no).To get the net rate of absorption of energy one must take the difference between direct absorption and induced emission and sum over initial states. The ratio CC,/OI, of absorption in the superconducting to that in the normal phase can be simply ex-
+
+
Referdnces
+. 282
216
J . BARDEEN A N D J. R . SCHRIEFFER
[CH. VI,
3
11
pressed if it is assumed that the normal matrix elements, Bk'o,,ko,are independent of the energy difference (although they may depend on the angle) between initial and final states. This should be an excellent approximation because the energy differences involved are generally very small compared with the Fermi energy. The result can be expressed most simply if we abandon our usual convention and define E to have the same sign as E , positive above and negative below the Fermi surface. We then have3
where f ( E ) is the usual Fermi distribution function and N , ( E ) , the density of states in energy in the superconductor, is
We may assume that N, is independent of energy and equal to N ( 0 ) . Note that N , becomes infinite at the Fermi surface, E = 0 or E = f d. The ratio of absorption in superconducting to normal state is then
[E(E L-J tc u,
nw
+ nw) '"1
[ f ( E )-.f(E
[(E' - d'){(E +
+ nw)l dE.
(11.11) Kw)' - ~ l ~ } ] ~ ' ~
The upper sign corresponds to case I, which gives destructive interference for 6w < 2d, the lower to case 11. The difference is particularly marked for very low frequencies, &w A . For case I (ultrasonic attenuation) cc,/cc, drops below T , with an infinite slope. In the limit H!W + 0,
2d(T). The gap decreases with increasing temperature; a knee in the absorption curve occurs at the temperature for which 2d(T) w f i w .
A t very low temperatures, where few particles are thermally excited, the absorption is very small until the frequency exceeds the gap frequency, w g = 2d(O)/jL,corresponding to T = 0" K . Fig. 13 indicates
0.8
0.21
L
Oo
6 --L-
- - - -- I2
18
24
30
36
42
Bw/kT,
Fig. 13. Absorption beyond gap for case I1 a t JI' = 0" I< cxpressed as ratio of conductivity in superconducting and normal phases. Experimental points from early measurements of Glover and Tinkliam lo based on transmission through thin films.
how the predicted absorption increases rapidly to that of the normal metal as w is increased beyond og,as observed in transmission of electromagnetic radiation through thin films lo. We shall next review some of the experimental data bearing on the coherence effects. 11.2. ACOUSTICATTENUATION
A major source of the attenuation of ultrasonic waves in metals at very low temperatures is the interaction with the conduction electrons. We shall first discuss longitudinal waves. From the earliest measurements of the attenuation in superconductors by Bommel and M a ~ k i n n i n it ~ ~was , clear that the rapid drop in attenuation as T h'tfeferencea
p . 282
CH. VI,
5
111
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
219
drops below T , reflects a diminishing number of “normal” electrons. However, the observed drop was so abrupt that it was difficult to reconcile with other estimates of the decrease in the normal component of a two-fluid model; for example, the Gorter-Casimir theory24 predicts p,/@ cc (T/T,)4. In the present theory, the destructive interference for an interaction which follows case I, nullifies the effect of the large density of states near the gap edge, leaving just the Fermi factor (11.12). The rapid drop reflects the sharp increase of the gap below T,. The simple theory leading to (11.12) applies when ql 1, where q is the wave vector of the longitudinal wave and I is the mean free path for impurity scattering. One can then regard the interaction with the electrons as corresponding to emission and absorption of phonons. The opposite limiting case, qi 1, has been considered by Kre~in’~; TsunetoBOhas made a general calculation valid for all 1. The temperature dependence of the attenuation is not very much different than for I -+ 00. Some careful measurements to test the theory have been made by Morse and Bohm12. They have found that (11.12) agrees fairly well with their measurements taken on polycrystalline indium and a very pure single crystal of tin, as indicated in Fig. 11. The temperature dependent energy gap of tin determined by combining (11.12) with the empirical results taken at 33.5 and 54 MHz is shown in Fig. 14. The direction of propagation is along the (001) axis. The best empirical value for the energy gap at T = 0” K is 3.54 k,T,, surprisingly close to the value 3.52 k,T, predicted for all metals by the simplified theory (i.e., constant matrix elements with a cut-off at I E I = no,).More recent mea~urernents~~.of attenuation of waves propagated in different directions have shown appreciable crystalline anisotropy in d(O), so that the agreement is partly fortuitous. Thus, a treatment including anisotropy of the normal state parameters must be carried out before a detailed comparison with experiment can be made. The primary deviation between theory and experiment occurs just below T,, where the experimental results indicate the gap opens more rapidly with decreasing temperature than predicted by theory. The relative attenuation coefficient obtained by Morse and Olsen74 from a very pure tin sample, under conditions ql 1 is shown in Fig. 15 for three crystallographic orientations. For T,/T > 1.5, the curves are well approximated by straight lines from which the values of 2A(O) given in Table 4 were obtained. Because of the requirements of
>
References p . 282
e 4-
0
r 1
-
1
Tin 335 MHz Tin 54 MHz
BCS theory
051 i
t
I
Ob
02
I
-
I-.---
04
06
-L 08
10
T/ T,
Fig. 14. Comparison of the temperature variation of d, as determined by Morse and Bohmr2 from the attenuation of longitudinal acoustic waves in tin with the prediction of the current theory.
[
I
I
2
3
TJT Fig. 15. The crystalline anisotropy of the relative longitudinal acoustic attenuation coefficient obtained by Norse and Olsen from measurements on a very pure tin sample
ql 9 1.
CH. \'I,
9 113
R E C E N T DEVELOPMENTS I N SUPERCONDUCTIVITY
221
TABLE IV
Crystalline anisotropy of d for tin deduced by Morse et af. from attenuation of longitudinal sound waves
2 4 (0) kBT, ~~
parallel to [001] parallel to [I101 perpendicular to [001] and 18" from [loo]
3.2 f 0.1 4.3 0.2 3.5 0.1
*+
energy and momentum conservation in the absorption process, the projection of the quasi-particle group velocity on the direction of sound propagation must equal the speed of sound, s. The majority of the quasi-particles have velocities of the order of vF. Since vF s, only those particles with wave vectors lying in a disc perpendicular to the direction of the wave contribute to the absorption. Experiments on oriented single crystals inherently measure the energy gap averaged over such a disc. Some experimental datas2 on the attenuation of shear waves in
>
-
-
1
g
4t
-
, 1
Discontinuity
0
---
05
,
1.0
-'
I
1
I
I
I
1.5
2.0
2.5
30
3.5
I I
4.0
7 (OK) Fig. 16. The relative acoustlc attenuation coefficient for transverse waves in tin as mrasured by Bohm and Morse. Referencds p . 282
222
J . BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
11
polycrystalline tin with ql> 1, plotted in Fig. 16, show an even more abrupt drop in attenuation at T , followed by a more gradual decrease which appears to follow again the law (11.12). The very sharp drop, which is almost a discontinuity, is most likely due to the strong screening of the transverse fields by the supercurrents, since the Meissner effect garantees that the magnetic field generated by the transverse currents associated with the shear wave will be screened in a distance of the order of the A m 5 x cm. Morse suggests that the more gradual drop of the attenuation below T , is due to the shear strain leading to a change in energy of the electron and therefore to an attenuation. The effect may in part be due to relaxation effects of the type suggested by Kittelss for the normal state. As Morse has pointed out, the shear waves can in principle give more detailed information about the anisotropy of the energy gap since a given transverse polarization will favor certain groups of quasi-particles in the disc perpendicular to q. It is important to develop a better understanding of the attenuation of shear waves in the superconducting state. 11.3. NUCLEAR SPINRELAXATION
An example of coherence effects following the constructive interference of case I1 is given by the relaxation of nuclear spins by the quasi-particles. Simultaneous with the development of the current theory, Hebel and Slichter13, using an ingenious method, were able to measure the zero-field nuclear spin relaxation rate in superconducting aluminum from 0.94” K to 4.2” K. The more recent data of Redfield and Anderson14,are presented in Fig. 17. The relaxation rate exhibits an increase by a factor of two just below T, and a subsequent decrease at lower temperatures. Since the dominant relaxation mechanism is provided by exchange of energy with the conduction electrons, this increased relaxation rate would be impossible to explain on the basis of the conventional two-fluid model because the density of “normal” electrons drops sharply below T,. As Hebel and Slichter have shown, the current theory is in good agreement with their results. The actual energy transfer in the relaxation process, and thus the corresponding no,is extremely small. To get agreement with experiment, Hebel and Slichter assumed that the quasi-particle levels are not perfectly sharp, but are broadened by m 0.01 kBT,, or eV. This avoids the singularity which would otherwise occur in the evaluation of the integral (1 1.ll).The source of this level broadening is uncertain. References p . 282
CH. VI,
§ 111
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
223
Anderson and Redfield l4 have recently extended the measurements on aluminum down to T,/T = 6, and their results, shown in Fig. 17, are in good agreement with the predictions of Hebel and Slichter. The two curves in this figure marked “corrected” represent different choices of level widths, where the energy gap was chosen to be the value obtained experimentally by Biondi and Garfunkelll, 24 = 3.25 k,T,, a value 7.5% smaller than that predicted by the theory, 24 = 3.52 k,T,. Hammonds4 has recently observed an increase of relaxation rate below T , by a factor of about 1.7 in Ga. I t appears that the increased 5.0
T
T I Normal state
t
0
Anderson and Redfield Recent data
i
Fig. 17. The nuclear relaxation time T I of superconducting aluminum as measured by Anderson and Redfield (open circles) and by Redfield (solid dots). The theoretical curves are based on the current theory with the density of states near the gap edge smeared by folding the density of states function (11.7) with a square function of width 2d and height (2d)-I, where A / d = r. The dotted and solid curves were calculated with 2d(O)/kBTc = 3.52 and 3.25 respectively, the latter being the value found by Biondi and Garfunkel from microwave measurements.
nuclear relaxation rate below T , is a general feature of superconductors, although the magnitude of the increase depends upon details of the material. It is important to realize that the observed increase in the nuclear spin relaxation rate and the sharp drop in the acoustic attenuation coefficient as the temperature is lowered from T,, imposes contradictory requirements on the conventional two-fluid model. It is one of References p . 282
224
J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
$ 12
the major successes of the recent theory that the temperature variations of these independent effects follow in a completely natural way from the general formulation of the ground state and the excited states of the system, and give strong experimental verification of the pairing concept. Although the experimentally observed coherence effects in themselves do not contain enough information to infer that only k t , -k J. interactions are important in contrasting the superconducting and normal states, this pairing is consistent with the empirical facts. Other possible pairings will be discussed in 3 13 on collective excitations. It is possible that systems with strong interparticle interactions in odd angular momentum states or strong spin dependent forces tending to line up the quasi-particle spins would be better described by the parallel spin pairing, So far, no evidence has been found for this case, and we will assume the antiparallel spin pairing to be that which leads to the ground state for the systems under discussion.
12. Electromagnetic Properties 12.1. THEORY
A theory of the electromagnetic properties of superconductors requires an expression for the current density for fields which vary arbitrarily in space and in time. The total field acting on the system, that is the sum of the applied field and that due to currents in the metal, is determined with the aid of Maxwell’s equations in a self-consistent manner. In their basic paper, Cooper, and the authorsab obtained an expression for the current due to weak quasi-static fields by treating the electromagnetic interaction in perturbation theory and including only the particle-like excitations of the system. The theory was later extended to treat fields of arbitrary frequency by Mattis and one of the authorss5 and independently by Abrikosov, Gorkov and Khalatnikovld’ a6. In this section we shall give the results without derivation and compare theory and experiment for several phenomena. As we shall see, in general there is excellent agreement between theory and experiment over a wide range of temperatures and frequencies. Before describing the results, we shall make some general remarks on the methods used and also indicate how the Meissner cffect with a non-local relation between current and vector potential is related to the energy gap model. The derivation of the Meissner effect of Cooper and the authors3b Kefermccs p. 282
CH.VI, 9 121
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
225
has been criticized because it is not strictly gauge invariant. There are two reasons for this lack of invariance: (1) The effective interaction with a cut-off for 1 E 1 >&m, is a nonlocal momentum dependent interaction, so that the expression for the current contributed by the quasi-particles should be modified from the usual one. However, estimated errors introduced if the usual expression is used are only of order ( A / & o , ) ~in the weak coupling limit, and so are negligible. To see this, one can start from a strictly gauge invariant theory in which the electron-phonon interaction has not been replaced by an effective interaction between electrons. Such a calculation of the Meissner effect has been given by Rickayzens7, who finds that the correction terms are indeed small if the energies and velocities of the quasi-particles are suitably renormalized to include self-energy corrections. Physically, non-local effects are unimportant because the size of the pair wave functions, of which t ow cm is a good measure, is large compared with the range of the non-local interaction, which is typically of order 10V cm. (2) The second reason is more serious and limits the applicability of the original treatment to transverse electromagnetic waves described in a transverse gauge. In the perturbation expansion, only quasiparticle excitations have been included, and the usual expression for the current contributed by a quasi-particle has been used. In the plane wave approximation, the contribution to the transverse current of a quasi-particle described by a “single” excitation in k is v = ?ik/m*, where m* is the effective mass of the electrons in the normal state. It should be noted that v is not the same as the group velocity of the excitation, 7)
1 aE ?ak i
1 aE ae ?i a& ak
=--=---=V-=V-
aE
a&
E
E ‘
(12.1)
Note that vg vanishes for excitations at the Fermi surface, E = 0. As for rotons in He 11, one may picture the quasi-particle as a vortex ring. The ring as a whole, with the accompanying backflow, moves with a velocity vg, but the expectation value of the velocity of the flow through the center of the ring is v. As discussed in 5 5, the backflow cancels out for transverse waves, so that one gets the correct current by summing v rather than v, over the various excitations. To carry out the calculation in a manifestly gauge invariant way one requires a formalism which is general enough to include backflow and Refertiices
p . 282
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J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
$ 12
collective excitations t. This problem which has been discussed by several authorsg0, most completely by Rickayzen t t, will be treated in 5 13, in which evidence for direct absorption by the collective modes will be reviewed. The theory is developed by considering a metal of infinite extent and calculating the response to a transverse electromagnetic field of arbitrary wave vector q and frequency w , described by a vector potential A = A, exp i(q - r
+ wt),
(12.2)
with div A = 0. The field may be due in part to internal sources and in part to currents induced in the metal by the field. A formulation of this type, first used by Klein to discuss the diamagnetic properties of metals and extended by Lindhardgl to determine the complex dielectric constant of normal metals for transverse and longitudinal fields, has been employed in most of the recent discussions of the electromagnetic properties of superconductors. Let Q0 be the many-particle wave function for the ground state of energy W,, or, at a finite temperature, a quantum state with a quasiparticle distribution appropriate to the temperature, T , and let @, with energy W,, ( j = 1, 2, 3 . . .) represent the spectrum of excited states in the absence of the field. Treating the electromagnetic interaction as a perturbation, one may expand
+ ,Xal(t) exp (-iW,t/fi)Ql,
P ! = exp (- iWot/?i)@,
(12.3)
j=O
where to first order in the field a&) =
( j I HI10) exp i(w - is)t W, - q w - is)
w,
(12.4)
Here s is a small positive constant which indicates that the field was The general structure of the equations for the current density for applied fields of arbitrary wave length and frequency has been discussed by S. Nakajima (see ref.88) and by 0. V. Konstantinov and V. I. Perel’ (see ref.80). They show how the conductivity is related to the current-current correlation function, with use of Kubo’s formalism, and also discuss the sum rules. In an infinite medium, there is a &function singularity in the longitudinal conductivity, corresponding to infinite conductivity, only in the long wave length limit. This limit is discussed in 5 14 in connection with the two-fluid model. t t Rickayzen (ref.8) discussed both the Meissner effect and the complex dielectric constant for longitudinal fields. References p . 282
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121
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RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
turned on in the remote past; the limit s -+ 0 is taken in the final result. An expression of the form (4.2) is used to calculate the current density from the perturbed wave function. As noted in 3 4,there are two contributions, a paramagnetic current, jp,associated with the gradient operator and a diamagnetic or “gauge” current, jD, proportional to the vector potential, A. The latter, jD,depends only on the electron density and is the same for the normal and superconducting states. The difference comes from the expression for jp,in particular from the terms involving a, for which W , - W , is of the order of the energy gap, E,, or less. In a superconductor, aside from collective modes, there are no terms for which W , - W , < E,, and the contribution is greatly reduced below that of the normal metal for W , - W , < M 2E,. For W , - W , > M 2E,, the difference between normal and superconducting states is small. Since the total current, j , = j,, jD, induced by a static magnetic field in a normal metal is extremely small (corresponding to the weak Landau diamagnetism), we have jD m - in,.The net superconducting current is then j , = j,, jD M M I,, - j,?, which is roughly the negative of the contribution of the series for I,,, in the normal state for W , - W , < M 2E,. Prior to the development of the microscopic theory, one of the authors4, used an argument of this sort to show that an energy gap model would most likely lead to a non-local theory of the Meissner effect similar to that suggested by Pippard, and this has been borne out by subsequent developments. The Pippard limit applies if the dominant terms in the expansion for jnphave energy denominators larger than the gap, the London theory if the denominators are less than the gap. The matrix elements for a wave vector q correspond to exciting a particle from state k below the q above. The energy difference, W , - W,, is Fermi surface to k of order fiqv,, where v 0 is the velocity a t the Fermi surface. The dominant q in penetration phenomena are of the order of the reciprocal of the penetration depth, or about 2 x 105 cm-1. With ZI, w 108 cmisec, this gives W , - W , M eV, which is an order of magnitude larger than the energy gap, indicating that the Pippard theory applies. The London equations would apply for q < lo4 cm-l. As indicated by Ferrell and coworkerss2, these arguments can be made more precise by use of Kramers-Kronig relations. For w # 0 , one may express the current in terms of a complex frequency and wave number dependent conductivity :
+
+
+
References p . 282
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J . BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
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(12.5)
u(q, 0)= u,(q, a) - io,(q, o),
where u1 gives the energy loss and u2 the reactive component. These are related by a Kramers-Kronig relation: (12.6)
where P indicates the principle part. The relation between may be written in the form C
j(q, W)
=
-
-K(q,w ) A ( q , o), 4n
i
and A
(12.7)
where (12.8)
This expression may be applied to static magnetic fields by setting o = 0. To have a Meissner effect, K(q, 0 ) must remain finite ( >0 ) as q --f 0. In a London superconductor, for which j , is assumed to vanish, K(q) = l/j12L, where A: = (mc2/4nne2) is the square of the London penetration depth. The matrix elements which enter the sum for K(q) are the same as those which determine absorption of energy at a finite frequency, which can occur for W , - W , M nw. As pointed out by FerrellgZa, one can determine Kfq) from cr,(q, o)if the latter is known for all o. This is closely related to the information that can be obtained about u2(q,w) for small w from ul by use of the Kramers-Kronig relation. An outline of the arguments follows : At frequencies above w, the maximum frequency at which absorption can take place, the response will be that of a system of free electrons, for which u1 = 0 and ne2 u2=-.
(12.9)
mw
When combined with (12.61, this gives the sum rule
me2
~ ~ ( W) 4 ,dm = -, 2m
(12.10)
which must hold regardless of the detailed structure of the system. In a superconductor, part of the contribution to the integral comes References p . 282
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R E C E N T DEVELOPhlENTS I N SUPERCONDUCTIVITY
229
from a d-function at w = 0. If S is the strength of the &function, one may write (12.10) in the form (12.11)
Tinkham and Ferrel192bhave applied this relation to an energy gap model by assuming that ols(q,w) = 0 for w < wg, and olS = oln for w > wg, where &wg M 2Eg. For a free electron model
=o
for w
> voq.
No absorption can take place if w > voq because the velocity of the wave is then greater than the velocity of the electron and one cannot conserve energy and momentum. If wg voq (Pippard limit), the sum rule gives
(12.17) Comparising these two expressions, we find an expression for toin terms of the energy gap : (12.18) References p . 282
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J. BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
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The microscopic theory gives an expression similar to (12.18), with Ko, evaluated explicitly as 7z24(0)/2,which is a little over twice the gap. The &function at w = 0 corresponds to acceleration of the whole group of electrons to give a net current flow. This can occur in a metal but not in an insulator or semiconductor with a gap. The sum rule for the latter is satisfied by absorption at finite frequencies. Andersonea has shown explicitly why a long range order is required for superconductivity. The general expression for the current density j(r, t ) resulting from a field defined by a vector potential in the transverse gauge, div A = 0, may be writtenB5in a form similar to that suggested by Pippard: j(r, t) = I; UJ
e2N(O)v,e*mt J- R ( R * Am(r’))I(w, R, T)e-R’l dr’ 2n%c R4
(12.19)
where R = r - r‘. The kernel I ( w , R,T ) is a rather complicated integral over energies which, except for limiting cases, must be evaluated by numerical methods. The derivation was based on the simplified model of 9 9, with constant matrix elements for the effective electronelectron interaction, but should apply more generally for isotropic Fermi surfaces if the energy gap parameter, d ( T ) ,does not vary much with energy over a range extending within a few k,T of the Fermi surface. This is essentially the weak coupling approximation. One may then regard d ( T ) as a parameter to be determined from experiment. Elastic scattering by impurities described by a mean free path, I, introduces the factor exp (- R/Z) in the integrand. That this factor, suggested by Pippard in his phenomenological theory, occurs in the superconducting case in the same way that it does for normal metals was shown in ref.s5. These authors used, as a basis for the manyparticle superconducting wave functions and the perturbation expansion, wave functions for the individual electrons appropriate to the impure metal with scattering centers present. If yn is one such function, another of the same energy is the complex conjugate, y,*. and in general these can be taken to be orthogonal to one another. The paired states for the ground state configurations of a superconductor are taken to be (y,+,yz+); that is if one of these is occupied in a given configuration, the other is also. It can be shown (9 16) that the pairing interaction energy of an impure metal is not much less than that for the pairing (kt - k 4 ) in a pure metal, even though the mean Referencis
p . 282
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231
free path is much less than the coherence distance. To evaluate the perturbation expansion for the current density, one needs averages of expressionslike (y,*(r)y,(r’)) over random distribution of impurities and over states of the same energy. The averages required for the superconducting case are the same as those required for the evaluation of the normal conductivity and in both cases lead to the factor exp(- R/Z). Derivations by the use of methods of quantum field theory by Edwardsg3and by Abrikosov and Gorkovg4lead to nearly equivalent results. I t is often convenient to use the Fourier transform of (12.19) which gives the relation between the Fourier components of j and A as in (12.7) with ei q Rue-RI1
(1 - d ) I ( w ,R, T ) du dR. (12.20)
Here we have inserted the expression for the London parameter a t T=O: A(O)-l = +e2N(0)vg. (12.21) Several limiting cases of (12.19) are of interest : (1) If the energy gap goes to zero, or more generally, if the frequency is sufficiently high so that Am A , we have
>
I(w,R, 7’)-+ - ninw exp(- iRo/vo),
(12.22)
and the expression for j(r, t) reduces to that of Chambers for normal metals as given by (3.5). The coefficient in front of the integral, N(0)vo,may be evaluated empirically from the surface impedance of the pure metal in the normal state in the extreme anomalous limit (skin depth much less than the mean free path). (2) The limit w --f 0, or Aw A , corresponds to the quasi-static case evaluated by B.C.S. An expression very similar to Pippard’s equation (3.6) is obtained by introducing a function J(R, T ) through
1.5, corresponding to temperatures near T,, but there is some bending below the line with a higher slope for y < 1.5. Prior to the development of the microscopic theory, Lewiss* predicted such a bending on the basis of a Refereitcis p . 282
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243
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY TABLEVII
Y
CLPEL
(erg deg-acm-3) I I1 I11 IV
Sn A1
Pb Cd
1100' 1370k 171@ 5611
17.P 45.58 17-26* 4.lc 12.Od
Do
x 10-8
1061,(0)
(crnlsec)
(cm)
0.65 1.32 0.50 0.29 0.85
3.55 1.57 3.7 11.1 3.8
240 kTe
T,("I 0 , the surface impedance Z is defined by: (12.41)
J
i y ( x I0 ) dx 0
It can be simply expressed in terms of the complex K(q, co) which relates the Fourier components of j and A (12.7). For random scattering,
>
The ratio of the surface impedance, Z,,, in the Pippard limit, to 1, to that in the normal state in the extreme anomalous limit, Z,, can be expressed in terms of the complex conductivity ratios (see Table 5) : (12.43) References p . 282
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J . BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
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Millerg5has evaluated the integral (12.42) for A1 and for Sn and finds that the detailed calculations give large corrections to the surface resistance ratio as calculated in the Pippard limit even for Al, for which E,, m 30 A. Corrections to the reactance ratio are smaller. Biondi and Garfunkel measured energy absorbed from the microwave field by a calorimetric method, and thus determined directly only the resistive part, R, of 2. Their measurements covered such a wide frequency range (from 20 mm to 3 mm wavelength) that they were able to use Kramers-Kronig relations to determine the reactive part, X . The latter may be expressed in terms of an effective penetration depth, d,, by 6,=-.
ex
(12 * 44)
4mo
Figs. 26 and 27 give a comparison of the experimental results with Miller’s calculations. Contrary to the usual practice, the experimental values are plotted as smooth curves and the theoretical values as discrete points, since the former were more complete and more accurate than the latter. The only modification of the simplified model was to take a slightly smaller value for the gap, 3.37 instead of 3.52 k,T,.
ea 0.2 3
ENERGY (in units of kTc)
’
Fig. 26. Frequency dependence of the surface resistance of aluminum as measurcd by Biondi and Garfunkel (smooth curves). Plottcd points are from calculations of Milier. Rejdrmces
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The knees of the surface resistance vs. photon energy plots occur where the photon energy becomes greater than the gap at the corresponding temperature. The increase in absorption beyond the knee comes from excitation of carriers across the gap (creating two quasiparticles). Biondi and Garfunkel have estimated the gap and its temperature variation empirically from the positions of the knee at different temperatures. There have been a great many measurements of the surface impedance of tin, most recently by Kaplan et a1.Io2. They, as well as
9
/I
2
'0
I
2
3
4
5
6
7
8
hv/kT, Fig 27. Frequency dependence of the surface reactance of aluminum, expressed as a penetration depth, from Biondi and Garfunkel (smooth curves). Plotted points are from calculations of Miller.
earlier workers, have found that the experimental results are sensitive to surface imperfections and strain. Pippard has pointed out that over a considerable range (7'from w 0.4 T , to M 0.8 T,, no from M 0.01 k,T, to w 1.0 ksTc), the surface resistance ratio may he expressed as a product of a temperature and a frequency factor,
(12.45) Ikferences
p . 282
250
J. BARDEEN AKD J . R . SCHRIEPFER 10
I
I
I 1 l l l l
'
I
I
,
I I I I ,
I
[CH. VI,
9
12
I 1 1 1 1 I
I
u(kMHz) Fig. -78. Frequency factor A(v)entering the expression for the surface resistance (12.45).
where y ( t ) as a function of reduced temperature is
Pl(t)= t 4 ( 1
(12.46)
- t y (1 - t4)-2.
This empirical result has since been confirmed by other workers. Miller's calculations indicate that these relations are valid within about 10 percent for the same range of t and o.Miller used the simplified model without modification, and thus used the theoretical gap
'P
6
11.2
0.2
T/Tc
Fig. 29. Ilhaikin's measured values of the surface impedance of cadmium compared with theoretical predictions of Abrikosov et al. based on the Pippard limit of the theory. Refcrciicrs p . 262
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of 3.5 k,T,. Fig. 28 gives a comparison of the theory with various experimental determinations of the frequency factor A ( Y ) , The agreement is excellent, considering that no adjustable parameters are involved. The lower curve, based on the Pippard limit, (12.43) is considerably below both the experimental points and the results of the detailed calculations. Khaikin1o3 has measured the surface impedance of cadmium, which has a transition temperature of 0.56" K. The frequency used corre-
a
Vanadium
5.10
0
Tantalum Niobium
4. 9.
o
FREQUENCY (ern-') Fig. 30. Frequency dependence of the electromagnetic power absorbed by bulk superconducting samples in the far infared measured by Richards and Tinkham. Here, Ps and P, are the powers absorbed i n the superconducting and normal states respectively.
sponds to an energy of 0.9 K,T,. Fig. 29 gives a comparison of the observed values with theoretical calculations of Abrikosov et ~ 1 . 8 6based on the Pippard limit (12.43). The agreement again is excellent. We give finally in Fig. 30 some measurements of Richards and Tinkham on the surface impedance in the far infrared region. What is measured is the absorbtion of energy versus frequency of the radiation beyond the gap frequency. Values they have found for the gap in References
p . 282
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13
various materials are given in Table 11. They depart considerably from 3.5 k,T,, varying from 2.8 k,T, for niobium to 4.6 k,T, for mercury. The general trend is inverse to the Debye temperatures. They found some structure in the absorption curves for lead and mercury, indicating possible absorption by collective modes with energies in the gap or anisotropic effects. In general, the observed absorption curves approach the normal state value more rapidly than indicated by the theory. We conclude that on the whole the microscopic theory of the electromagnetic properties is in remarkably good agreement with experiment, particularly if empirical values are used for the energy gap and its temperature dependence. What discrepancies there are can probably be accounted for by the complex band structure of actual metals as compared with the theoretical isotropic model. Only limited progress has been made in understanding the interesting experiments of Spiewak104on the magnetic field dependence of surface impedancelo5.
13. Collective Excitations Thus far, our discussion has been concerned with applications of the theory to problems emphasizing the quasi-particle aspects of the excitation spectrum, for which an independent quasi-particle approximation is valid. Basic to a gauge invariant description of the Meissner effect and a complete account of the system’s response to space-time varying external fields, are the collective excitationss0. These excitations have energies split off from the continuous spectrum as a result of residual interactions not accounted for in the single quasi-particle approximation. As in the normal state, the collective modes may be viewed as coherent superpositions of quasi-particle configurations. In the superconducting state, the plasmon modes continue to exist and are essentially identical to those occurring in the normal state. Due to their high energies (w15 eV), real plasmon excitations do not enter the low frequency phenomena we have been discussing (no< 10WeV). Virtual excitation of plasmons is, however, essential in obtaining a gauge invariant form of the kernel K relating the current density and vector potential. In the Coulomb gauge, div A = 0 , neither real nor virtual plasmon excitations enter due to the longitudinal character of the plasmon current density. For this reason, calculations are often simplified by choosing div A = 0 , as in 5 12. A new feature of the superconducting state is the possibility of excitation-like modes occurring with energies lying in the energy gap, Rejcrcwces
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as pointed out by Anderson6 and Bogoliubov4. The normal state precludes their existence since the finite density of single particle states near the Fermi surface would lead to rapid decay of the collective modes. In the superconducting state the exciton modes are likely to have rather long lifetimes due to the absence of single particle states within the gap. As opposed to plasmon excitations, the excitons may enter both real and virtual processes and physically observable effects are associated with these modes. Real transitions involving exciton creation are reflected in a resonance of the absorptive part of the wave vector and frequency dependent kernel, K(q, w) for q t , < 1 and 6w < 2 4 , while virtual processes have a small but finite effect on the real part of K for all frequencies of interest. Since the nature of the exciton spectrum is closely related to the angular dependence of the residual two-body interaction V (k , k') , the BCS parameter N ( 0 )V does not suffice to determine the types of excitons which occur in a particular metal. For an L state exciton (corresponding to p, d.. . . . .excitons) to exist lo6,it is required that V , be negative, where V , is the L wave part of the interaction V ( k , k'). A plasmon corresponds to an S state exciton whose energy is greatly increased by the long range Coulomb interaction. An approach sufficiently general to treat both indivdual quasiparticle excitations and the collective modes in the high density limit is given by the generalized time-dependent self-consistent field (SCF) or random phase approximation introduced by Anderson 6, and independently by Bogoliubov, Shirkov and Tolmachev4". This method is a generalization of an approach introduced by Bohm and Pineslo8for a description of the normal state. The most complete discussion of the method has been given by Rickayzena, who used it to derive a fully gauge invariant form of the kernel K(q, w) for the superconducting state. Except for small terms due to the excitons, his result for the gauge div A = 0 is identical to that given in the basic paper of Cooper and the authors3. To illustrate the time-dependent SCF approximation, we begin with a discussion of the elementary excitation spectrum of the normal state, valid in the high electron density limit Y, < 1 = 4n/3 (r,aO)3, a, Bohr radius). In this limit, the Coulomb interaction energy is small compared to the kinetic energy of the electrons. Thus, a perturbation treatment starting from the Fermi sea would be a good approximation were it not for the singular nature of the Coulomb interReferences p . 28,"
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J. BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
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action for long wavelengths, that is V(q) = 4ne2/q2--f 30 as q -+ 0. It would appear that the effect on a given electron of the long range part of V can be represented by an average self-consistent potential arising from the coherent motion of the distant electrons, plus a small fluctuating potential associated with their residual random motion. That the fluctuating potential may be neglected for many purposes in the density limit was discussed in detail in the pioneering work of Bohm and Pines. To derive the spectrum of elementary excitations within the SCF approximation it is convenient to study the motion of electron-hole pairs with total momentum $4. Due to the Coulomb forces, an electron and a hole in st:.ies k + q and k respectively may recombine and excite an elec troll into the state k' q leaving a hole behind in state k'. The riew pair may again recombine transferring the excitation to k" $- 7 2' k.'. This ,rcccss is illustrated in Fig. 31a. There are other S ~ Jc d h ! exchange processes possible in which the pair in k + q and k
+
Fig. 31. Hole-electron scattering processes included in the random phase approximation.
scatters directly to k' + q and k' without recombination and creation taking place, as illustrated in Fig. 31b. Since the matrix element for the processes shown in (a) involve the matrix element 4ne2/q2while those in (b) involve 4ne2/ I k - k' 12, the polarization processes (a) are on the average far more important in the limit q -+ 0. Restricting ourselves to direct processes, which as Gell-Mann and Brueckner lo9 have shown is sufficient for Y, < 1, it is clear that certain linear combinations of the electron-hole states will be eigenstates of the Hamiltonian. The plasmon mode which splits off from the continuous spectrum is given by a superposition of the pair configurations in which all states enter with the same sign and with approximately equal weight. The situation is somewhat analogous to the coherent superposition of many particle configurations used to form the ground References p . 287
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state of the superconductor. The remaining linear combinations of electron-hole pairs correspond to scattering states in which the electron and hole are not strongly correlated. The formal procedure of obtaining the elementary excitation spectrum is carried out by finding those linear combinations, pa(q) *, of electron-hole pair operators @ku(q)
(13.1)
= c&quckuJ
which create an eigenstate of the approximate Hamiltonian Hpz(q)*yO = ('lza(q)
+ wO)pa(q)*yoJ
(13.2)
where H includes only those interaction terms leading to polarization processes. Here &Q,(q) is the excitation energy of the ath linear combination of the @ k , ( q ) . The equation (13.2) will be satisfied if the operators pa(q) * obey
LHJ
pz(4)*1
= nQa(q)pa(q)
*'
(13.3)
If @ k , ( q ) is decomposed into an average value &)(q), taken with respect to a self-consistent state, plus a fluctuation &(p) about this average value, the processes shown in Fig. 31(a) are taken into account if only first order terms in @(l) are kept in the full commutator [ H , @ k , ( P ) ] . This is just the self-consistent field approximation as a straightforward calculation showsllO. In the high density limit, it is sufficient to linearize the equations about the average values &,)(9) appropriate to the Fermi sea. In this case only the zero wavenumber component @ k , ( o ) = nku has a nonvanishing average value : (13.4)
I t is straightforward to show that the elementary excitation energies are given by the dispersion relation (13.5)
The solutions of (13.5) are plotted in Fig. 32. Both individual-particle like states describing electron-hole pairs in scattering states, and collective plasmon states appear. While the energies of the holeparticle pairs within the SCF approximation are unaltered from their values, & k + q - f k , in the absence of interactions, the wavefunctions References
p . 282
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J. BARDEEN AND J. R. SCHRIEFFER
[CH. VI,
13
are strongly modified. Each particle (electron or hole) is surrounded by a local depletion of the same type of particle (electron or hole, respectively) and the physical picture of backflow discussed in 4 5 follows in a natural manner. In a more complete treatment, &k+q - &k is shifted by a self-energy term which is complex, the imaginary part corresponding to the finite
%F
d
Fig. 32. Elementary excitation spectrum in the normal state.
lifetimes of the excitationsll'. The imaginary part of the single partic e self-energy, calcuIated by a Green's function approach, is plotted as a function of the real part of the excitation energy in Fig. 33. For energies less than the maximum phonon energy &omax,real phonon emission plays the dominant role, while for E > 10 &wmax,hole-electron pair production takes over and the single particle levels become very broad. The effect of finite lifetime on the energy gap equation is discussed in 5 9. In the superconducting state the pairing effects lead to non-vanishing average values for operators of the form bk
2
C-kJCkf
xk
+ bf) (13.6)
'k
=
bkyO),
as well as for %ko. As Anderson6 and Bogoliubov4 have independently pointed out, an improved description of the elementary excitations in the superconducting state may be given by including residual interactions neglected in the original discussion of Cooper and the authors. They discuss a generalized self-consistent field approximation in which both k k and x k are introduced. The analysis is most simply carried References p. 282
CH. VI,
$ 131
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
253
outs by working with the quasi-particle operators discussed in 3 11. One again seeks those linear combinations p$(g), of the operators ~ : + ~ + y *y-k-qlykt, ~ ~ , and y:+qrryko which create elementary excitations of the system. The normal modes of the linearized equations are somewhat morc complicated to determine in the superconducting state than in normal metals. As pointed out by Rickayzene, the electromagnetic response kernel K and the dielectric constant mey be determined without explicit knowledge of the normal mode operators. His results give a fully gauge invariant description of the Meissner effect with the kernel
production
'
u 0
1
2
3
4
5
6
7
8
9
RebpfiuC
1;ig. 33. The cnergy dependence of the imaginary part of the excitation energy in the normal statc. The cxcitations arc poorly dcfined near the Debye energy and above ten times the Debye encrgy for typical electron densities.
K for zero frequency being identical to that derived by Cooper and the authors if only the s wave part of the two body potential is kept, as in
s 9.
It is explicitly seen that the longitudinal collective excitations contribute a polarization current which, when added to the quasiparticle polarization current, just cancels the diamagnetic gauge current, leading to the gauge invariant form for K . If the s and d wave parts of the potential are chosen to have equal magnetudes, corrections to K are of order and may be safely neglected. , is significantly different from its The dielectric constant ~ ( qu)), \.due in the normal metal only if fico m &v0q < A . Rickayzen's treatR c f r t m c r ~p . ?,(?
258
J . BARDEEN AND J. R. SCHRIEFFER
[CH.VI,
0 13
ment also shows the expression for the acoustic absorption coefficient calculated in 5 11 within the single quasi-particle approximation to be valid to order ( o / ~ ~Mq ) ~and (&v,q/A)aQ 1. Tsunetog6 extended Rickayzen’s analysis to treat the surface impedance at finite frequency. Choosing only the s and d wave parts of the potential to be non-zero, he finds a precursor absorption to exist for frequencies below that of the energy gap, ad/&.His results,’when applied to lead and mercury, predict an absorption due to exciton states in the gap which is an order of magnitude smaller than that observed by Ginsberg, Richards and TinkhamlO in these materials. Since lifetime effects are likely to be important in these strong coupling superconductors, it is desirable to extend the theory to the strong coupling regime before drawing conclusions regarding the role played by collective modes in these experiments. The elementary excitation spectrum of the superconducting state is shown schematically in Fig. 34. As mentioned above, the plasmon
2kF
q-
Fig. 34. Elementary excitation spectrum in the superconducting state.
mode is almost identical to that occurring in the normal state. The broad spectrum of quasi-particle pairs in scattering states is bounded from below by the energy gap 24, The exciton states, having energies lying within the energy gap, may be pictured as a pair of quasi-particles bound together in r e d space moving with center of mass momentum 69. The exciton wavefunction is of the formlosa (13.8)
where
describes the relative motion of the pair having an extent
References p. 282
CH. VI,
w
$ 131
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
259
t oand S,, is the spin function. In the limit q -+ 0, (13.9)
and one has the usual picture of p, d, . . . . state excitons as in an insulator. For larger q, states of different L mix although the mixing is small for qt0 1, while M is a good quantum number for all q if the potential has no crystalline anistropy. Since the quasi-particles are Fermions, the wavefunction must be antisymmetric on interchange of rl, o1 and r2, c2.Thus, a state with L even must have a singlet spin function, while odd L is associated with a triplet spin function. The spectrum of exciton states isstrongly dependent upon the angular dependence of the residual two body potential V (k, k‘). If the potential is decomposed into spherical harmonics
g, coupling constant g, is given by
g,
= - N(O)VL
> 0. The (13.12)
and go = - N(0)Vo = N(O)V, is the coupling constant introduced in $ 9 . The exciton energy for q = 0 is plotted as a function of g, in Fig. 35. For g, >go, the excitation energy is imaginary and the system is unstable when described by the ground state based on the s state pairing discussed in 9 8. For example, if g, is the largest coupling constant, the ground state will be formed from pair functions p, having p like symmetry (see (8.10))
- ‘2)
‘PIM(‘l
= P)l(lrl - ‘2 1)Y1M(e12J v12)
(13.13)
and the triplet spin pairing considered by FisherlOsbwill be appropriate. If the d wave potential is dominant, singlet d functions are appropriate for the ground state, as considered by Anderson et for 3He. The exciton energy is plotted as a function of center of mass momentum in Fig. 36 for several values of gL. The long range Coulomb potential plays no role for M # 0 and these excitons may be thought of as h’cfmences p . 282
2 ti0
J . BARDEEX AND J . R . SCHRIEFFER
0.1 -
Oo
i i i 4 4B
+ b 9 Ib1'1
1'2b
[CH. V I ,
9 13
*
1;ig. 35. Energy of the exciton states within the energy gap for zero center of mass momentum, as. a function of the coupling constant.
0 01
0 2
03 0 4
05
06
07
08
09
10
d o
Pig. 36. Euciton cncrgy as a function of the center of mass momentum hq for t h r magnetic quantum numbrr M f 0.
transverse collective excitations. The M = 0 states may be split from the M # 0 states, with the s state exciton being identified as the plasmon mode if the ground state is described by s state pairing. For non-factorizable potentials, that is V(1 k 1, 1 k' I) can not be expressed as W (I k i)W(1 k' I), more than one bound state for given h'efevencea
p . 28?
CH. TI,
8 131
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
261
L and M may exist, corresponding to states with different principle quantum number, 12, in the hydrogen atom. We turn now to a brief discussion of the electronic spin susceptibility in superconductors. In the normal state the spin susceptibility is given by X, = 2 , 4 N M ( 0 )where , ,u%is the Bohr magneton and N,(0) is an effective density of states at the Fermi surface, differing from N ( 0 ) entering the electronic specific heat by terms arising from the Coulomb exchange energy. In a nuclear resonance experiment the Knight shift K , defined to be the fractional difference in the resonant frequency between a nucleus in a free ion and that same nucleus in a metal, is directly proportional to X,. Since X, is proportional to N,(O), nuclear resonance experiments lead to information about the density of states at the Fermi surface. Reif l7 has measured the Knight shift K , in superconducting mercury colloids consisting of particles mostly less than 500 A in diameter (< A), and finds K , to drop rapidly for T < T,, reaching a saturation value at T m +Tcof about $K,,. Recent data of Androes and Knight l6 taken on thin superconducting platelets of tin ( M 40 A x 140 A) are plotted along with Reif's data in Fig. 37 and show ;I tendency to
0.8 '.OI
x
O
1
0.2
I
O
o
0
:
O/
i
/
,L.-0
r/r, I ig. 37. Temperature dependence of thc rcduccd Knight shift measured b v Kcif on inercury colloids and by 4ndrocs and Knight on tin platelets.
5aturate at K J K , w 0.73 if SnC1, is used as the reference salt. As Yoshida1*8has shown, the microscopic theory leads to a susceptibility X, which vanishes at T + 0" K for a uniform magnetic field, apparent1~Keferenccs
p . 282
262
.
J BARDEEN AND J . R. SCHRIEFFER
[CH.
VI, 5 13
in contradiction with experiment. This result follows because the minimum energy 24 required for creation of two quasi-particles from the ground state is larger than the Zeeman energy p B H ogained by the excitations. Heine and Pippardlla have suggested an alternative form for the matrix elements which enter the theory such that a finite Knight shift is obtained. Thus far it has not proved possible to construct wavefunctions which lead to these matrix elements. Ferrelllls and Anderson1l4 have suggested that as a result of the spin-orbit interaction, a finite value of X , in small specimens near T = 0" K would be obtained because the single particle wavefunctions in the normal state are not eigenstates of the spin. In this case, the magnetic interaction, - pB
S, Ho G H , *
will have non-vanishing matrix elements between the ground state and excited states so that a perturbation calculation of X, starting from the ground state defined in the absence of the magnetic field might be appropriate. A number of authors115have pointed out that even in the absence of spin-orbit effects, a non-zero value of the wave vector dependent susceptibility appropriate to space varying fields is obtained. As a result, a positive Knight shift may be seen within the penetration region of a bulk sample, while a region of reversed spins necessary to satisfy X,(q = 0) = 0 would extend to a distance M to beneath the surface and would not be effective in the observed resonance spectrum. There is no empirical evidence in favor of this picture at present. A wave number dependent susceptibility cannot account for the experimental results on tin mentioned above. Since any state derived by a perturbation series from the singlet ground state not involving spin-orbit effects must have vanishing total spin magnetization, the theory would predict a broadening of the resonance line with little or no Knight shift. The experiments on the contrary exhibit a shift which is at least as large as the line width. As suggested by one of the authors116*, a finite value of X , at T = 0' K applicable to uniform fields might be obtained if the ground state in the presence of the magnetic field H , is formed by a pairing different from that appropriate to the case H , = 0. In analogy to the modified pairing ( k + q t , - k q 4 ) introduced to describe current carrying states, one might begin with the magnetized state appropriate
+
References 9. 282
CH.
VI, 3 141
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
263
to the normal metal, pairing states on the up spin Fermi surface with those on the Fermi surface for down spin. These single particle states are not related by time reversal; however, this condition is not required in the presence of a magnetic field. Thus far no calculations have been carried out with the modified pairing. It would be interesting to investigate the role played by spin-orbit effects by measuring K , for a light metal such as aluminum where the spin-orbit effects are expected to be less effective than in tin and mercury. 14. Two-Fluid Model and Persistent Currents 14.1. TWO-FLUID MODEL
The two-fluid model of He I1 has been extremely successful in predicting and interpreting many of its remarable superfluid properties, such as second sound, heat flow by convection, and various thermomechanical effects. There have been speculations as to whether corresponding effects might be observed in superconductors. In an earlier volume of this series, Gorterll' has given a review and comparison of two fluid models for superconductors and liquid helium. The superfluid component is the part with frictionless flow; it corresponds to flow in the ground state and carries no entropy. The normal component is the part of the flow associated with thermal excitations; it is subject to the usual friction. While the equations for He I1 are formally those of two interpenetrating, non-interacting fluids, Landau118showed that they can be interpreted in terms of the properties of the ground state and the spectrum of elementary excitations of the fluid. Landau's arguments can be formally extended to give a corresponding two-fluid model for superconductors lI9, but for a number of reasons it is much less useful than for He 11. Before outlining the derivation, we shall point out why many of the two-fluid flow phenomena characteristic of He I1 would be difficult to observe in a superconductor. Following the discussion of the theory, we consider applications to persistent currents, critical currents in thin films and the Ginzburg-Landau theory. Some of the complicating factors are: (1) In a superconductor, current flow produces a magnetic field and this field has a strong influence on the flow. For example, when current flows in a superconducting rod, it is confined to a region within a penetration depth of the surface. One can regard References
+. 282
264
J . R A R D E E N A N D J . R. S C H R I E F F E H
LCH. VI,
0 14
thc currcnt as producing a magnetic field and the magnetic field in turn producing Meissner ciirrents which prevent the field from penetrating. The sum of these Meissner currents gives the net current flowing in thc wirc. I t is only whcn thr dimcnsion are of the order of the penetration depth or less that the current density is reasonably uniform. In practicc, this is most easily achieved in thin films. To avoid these complications, w ( s shall in thr following discussion omit effccts of the magnctic field, and suppose that on,' can have a uniform current flow. The electrons still show snperfliiid behavior, wit11 persistcnt currents possible. (2) The excited electrons (normal component) are scattered by and relax to thc lattice. I t is thus difficult to have a normal component of flow in the absence of an electric field; this is one reason why second sound would not be easy t o observe in a superconductor. For simplicity, we shall a t first neglcct relaxation effects, but will consider the consequences later. (3) Uncertainty relations give a much larger minimum size for the excitations in. a superconductor than in He 11, which is mainly a consequence of the large difference in mass. To be reasonably well defined, a quasi-particle excitation in a superconductor cm). I t is only when should bc larger in extent than the coherence distance ( m changcs in motion occur slowly over distances of this order that the local relations of the two-fluid modcl can be used. This puts, for example, limitations on the minimum wavelength a t which one might hope to observe second sound. (4) As indicated above, thermal conduction in He I1 can take place by a counterflow of normal and super components, thc hcat flowing with thc normal component. A similar effcct can occur in a superconductor, but the magnitude is very small compared with the usual electronic thermal conductionlPO.The elementary excitations created when a superconductor is heated correspond to electrons above and holes below the Fermi surface. They are created in nearly equal numbers and tend to flow in the same direction. There is therefore very little net electrical current associated with the hcat flow. In a normal metal this current is related to the thermoelectric effect and is what makes the difference between thermal conductivity with j = 0 and with E = 0, which is known to be very small. The order of magnitude of the effect in a superconductor is correspondingly small.
For the formal derivation of the equations of the two-fluid model, we disregard magnetic fields and electron-lattice relaxation processes and suppose that changes in motion occur slowly over a coherence distance. We also consider for simplicity a free-electron type model for the normal metal. First consider acceleration by an electric field, $. The common momentum wzv, = i(p, - p,) of the ground state pairs (p, 1. , - pz 4) increases according to Newton's cquation m3, = - eb. ,4t the absolute zero there are no thermal excitations and the entire distribution of electrons is displaced in k-space. Because of the energy gap, it is energetically unfavorable to scatter electrons from one side of the Fermi distribution to the opposite until a critical velocity is reached, above which superconductivity is destroyed. If v 0 is the Fermi velocity, this occurs when
+
~ w ( v , v , ) ~- & w ( v~ v , ) ~> E , = 2 4 , Referetiers
p . 282
(14.1)
CH. VI,
4 141
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
265
or when u8 > O/(mu,,). At higher temperatures, thermal excitations can decrease the current, so that only a fraction of the electrons appear to be freely accelerated by the field. London's second equation is (14.2)
where A ( T ) is the temperature dependent London parameter. One may write j , = - ee,v,/m, where e, is the density of the superfluid component. At 7' = 0, e, = Q = nm. In general, (14.3)
This function, defined by ( 1 2 . 2 6 ) , is plotted in Fig. 18. London]" has shown how one may construct from the ground state wave function, !Po(rl, r 2 . . . . . rn), a function Y
=
exp [i
v(r,)]Yo
(14.4)
1
for which u, is a slowly varying function of position. In a local region about r , this corresponds to a displacement of the distribution in k-space by Bk = grad cp(r), and a corresponding flow velocity
v,(r)
= 6m-l
grad
pl(r).
(14.5)
This expression implies potential flow, curl vs = 0. As pointed out by London la, current flowing around a superconducting ring can be described in this way. In this case, y may change by a multiple of 2n in going around the ring. An identical expression for en is obtained by following Landau's derivation of the two-fluid model as extended by Dingle121for Fermi systems. Thermal excitations may give rise to a net current relative to the ground state. First consider the ground state at rest (v, = 0 ) and suppose that there is a net momentum (mass flow) Jn =
Pf(P)
(14.6)
from excitations with a distribution function f(p). The latter may be determined so as to make the free energy F a minimum subject to a given J, by introducing a velocity v as a Lagrange multiplier (14.7)
266
J . BARDEEN AND J . R. SCHRIEFFER
[CH. VI,
9 14
This leads to 1
= 1
+ exp ([E(p)- v - pJ/rZ,T)
(14.8) *
When v is small, J, is proportional to v and the coefficient is defined as the normal density, en:
This expression is in agreement with that derived from (14.3). When there is flow in the ground state, the normal velocity is defined by v, = v, v and the total mass flow is given by
+
J
= ev,
+
env =
egvs
+
envn-
(14.10)
From (14.4) and the kinetic energy associated with v,, the total increase in free energy is found to be (14.11)
One can verify, as DinglelZ1has done, that the entropy flows with the normal component. One must be careful to distinguish between mass flow and flow of the number of excitations, N,,, = X,f(p). The latter move with the normal component so that the flux is given by v,N,,,, which in general differs from v,en/m. Superfluid flow with a velocity v, can be initiated by an electric field in the form of a pulse. All of the electrons are accelerated by the field with an increase in the common velocity of the pairs to v,. Scattering of thermal excitations tends to reduce the current so as to make the free energy a minimum, but such scattering does not change v,. According to (14. l l ) ,the best one can do is to make v, = 0, leaving a net flow J , = e,v,. This is the part which is determined directly from the London equation (14.2). Only a force which acts on all or a large part of the electrons can change v,. In the absence of such a force, the current persists indefinitely. Although second sound would be difficult to observe in a superconductor, it is of interest to estimate the velocity, c2. Formally one can use the same expression'22 as for He I1 (14.12) References p . 282
CH.
VI, 3 141
RECENT DEVELOPMENTS I N SUPERCONDUCTIVITY
267
where S is the entropy and C, the electronic specific heat per unit mass. For an order of magnitude estimate, we may use the Gorter-Casimir model for which @,/en = 1 - t4, and S and C, are proportional to T3 in the superconducting phase and S is equal to the value in the normal phase, yT,,at the transition temperature : (14.13)
If the free electron value is used for y , we find
for tin. This velocity is of the same order as for He 11. To observe second sound of frequency o,one should have ot > 1, (where t is the electron-lattice relaxation time) and to insure that the wavelength is greater than the coherence distance, c2 >WE,. To satisfy both requires that z > to/c2, or, for tin, z > 10V sec, which would be very difficult to realize in practice, since it corresponds to a m.f.p. of about 1 cm. 14.2. CRITICALCURRENTSIN THIN FILMS
The critical current in a bulk specimen is determined by the critical field. As the current is increased, the field at the surface of the specimen increases until it reaches the critical value, H,, at which the specimen reverts to normal or goes into an intermediate state. This is not true for flow in films so thin that the magnetic field can penetrate throughout. Effects of the magnetic field can be minimized by use of a “compensated” geometry. N. I. Ginzburg and A. I. S h a l n i k ~ v lhave ~~ measured the critical current in films of tin deposited on the outside of a cylinder. There are then no edges where abnormally large fields can occur progressively destroying superconductivity. These authors were particularly interested in the critical current near T,, where they find it varies as (T,- T)3/2,as predicted both by the GinzburgLandau phenomenological theory and by the microscopic theory. Earlier studies, in which fewer precautions were taken to eliminate extraneous effects, gave a variation as (T, - T)lj2or (T,- T)2/3. If the London theory applied, one could write for the increase in free energy for a current density, is:
AF References
p . 282
= +e,v,” = &l(T)if.
(14.14)
268
J . BARDEEN A N D J . K. SCHRIEFFER
[CH. V I ,
§ 14
This assumes that A ( T ) is independent of j,, which may not be valid for very large currents. One expects the critical current to be that for which 5 F becomes equal to the energy difference between normal and superconducting phases, H;/Sar, or when (14.15) Near T,, both H , and A(T)plvary as ( T , - T ) ,so that j,, is expected to vary as ( T , - T)3I2.A critical current about 25% smaller is obtained if one takes into account the change in the distribution of quasiparticles and in the energy gap with increase in current, as shown bjRogers 124. The Pippard rather than the London limit applies to very thin films in which the m.f.p., I, is greatly reduced by scattering from the surface. One may express the acceleration of current in terms of the normal conductivity, cr,, and the kernel of the Pippard integral, J ( R ,T ) ,for the limit R + 0 , as follows: (14.16) so that the increase in free energy is
AF
=
$(A(T)t,/J(O, T)Z)j:.
(14.17)
The critical current density is reduced by scattering by a factor of about ( l / E o ) i; the temperature dependence is not affected very much. The critical current is decreased somewhat when changes in the gap with current are taken into account 124. The predicted magnitudes are of the same order as those found experimentally. 14.3. GINZBURG-LANDAU THEORY OF BOUNDARY EX’EHGIES
Some years ago, Ginzburg and Landau3j extended the London phenomenological theory to allow for a space-variation of the cffective concentration of superconducting electrons, n4. This made it possible to treat a number of problems, perhaps the most important of which is the boundary between normal and superconducting regions in the intermediate state. In this case no varies from zero in thc norma1 side to its equilibrium value appropriate to the given temperature in the superconducting side of the boundary, as illustrated in Fig. 38. At the same time, the magnetic field drops from the critical value, H,, IZLfcrcncps
9 . 282
CH. VI,
9:
141
RECENT DEVELOPMEKTS I N SUPERCONDUCTIVITY
269
in the normal side to zero in the superconducting side. The long range of coherence of the superconducting wave functions prevents ns from dropping abruptly at the boundary12j. Ginzburg and Landau assumed that n , ( r ) is proportional to the square of an effective wave function Y s ( r ) . The free energy density F(Y,, T ) depends on Y, (or n J , and the equilibrium value for constant Ysis that which makes F a minimum. They assumed further that in a magnetic field defined by a vector potential, A ( r ) , there is an extra term in the energy proportional to I - ih grad Us ( e A / c ) Y 8 12. To
+
Fig. 38. Variation of the magnetic field and the effective concentration of superelectrons across in normal-supcrconducting phase boundary.
determine the boundary energy, an&,one finds by a variational procedure the functions A ( r ) and Y s ( r which ) make the total integrated free energy a minimum. The parameters of the theory are determined completely from H,(T) and the penetration depth, 1(T).Fairly good agreement is found with values of a,, deduced from experiment, both in absolute value and temperature dependence. However, the theory suffers from the defect that it is based on the London theory rather than on the non-local theory now known to be valid. It is only for temperatures very close to T , that the non-local theory reduces to the London limit. Gor’liov1Z6has extended the microscopic theory so as to allow for a space-variation of the pairing. In the weak coupling approximation valid for most superconductors, the energy gap parameter, O ( r ) , may be regarded as a function of position. Gor’kov formulated the problem in terms of “thermal” Green’s functions, in which temperature is regarded as an imaginary time. While it is not hard to write down the differential equations for the Green’s functions, they are very difficult References p . 282
270
J . BARDEEN AND J . R . SCHRIEFFER
[CH.VI, $ 15
to solve except for limiting cases. Gor’kov carried through the calculation only for temperatures near T , where one expects the London limit to be valid, and found equations almost identical with those proposed by Ginzburg and Landau. The effective wave function, Fs(r) is found to be proportional to A ( r ) . The only difference is that the charge e is replaced by 2e, evidently representing the charge of a pair. Ginzburg127 has pointed out that this change improves agreement with experiment. In his generalized method of compensation, Bogoliubovu has given a different formulation which is also sufficiently general to allow for a space variation of pairing. A pair wave function, p(rl, r 2 ) , need not depend only on the difference, rl - r2, but may depend on rl and r , separately. This approach has not as yet been used to discuss boundary energies. 15. Thermal Conductivity 15.1. LATTICECOMPONENT
The thermal conductivity of superconductors is generally difficult to interpret theoretically because several mechanisms may be effective simultaneously. In the superconducting state, as in the normal state, there are two contributions to the heat current, one due to the conduction electrons and the other due to phononslZ8.The thermal conductivity, x , is given by the sum of the electronic and lattice thermal conductivities, x = x,
+ xg.
(15. 1)
In each case, there are several scattering mechanisms which limit the heat flux. In the normal state one has
B , -1 -- uT2 + Xen T
(15.2)
(15.3)
where the first and second terms in the expression represent the scattering of the electrons by phonons and by static imperfections respectively while the corresponding terms in K& represent the scattering of phonons by electrons and by the boundaries of the specimen. The same scattering mechanisms are effective in the superconducting HefErencm p . 282
CH. VI,
9 151
RECENT DEVELOPMENTS IN SUPERCONDUCTIVITY
271
state ; however, their temperature dependences are distinctly different from those in the normal state. For extremely low temperatures ( 0. This can be done in a formal way by substituting s(y - r(t) - r) into Eq. (1) where s(W) = 1 for W > 0 and s(W) = 0 for W < 0. Now the determination of LIE remains. Let t , be the instant corresponding to the time t of revolution in the orbit (i.e. at the time t, - t the electron was at the centre of the orbit). Then, with a probability given by the free path time z, the electron can obtain energy from t' to t i.e. during the time t - t ' ; t changes from t to - 00. During the time from t' until t' dt' the electron gains the energy ev(t')E(t') dt' where the field strength E should be taken at the point y - r(t) r(t') at a time t, - (t - t') (see Fig. 5 ) ;
+
E(t') exp (id,) = E(y - r(t)
hence References
p . 330
+
+ r(t'))exp [iwt, - iw(t - t')]
9 13
CH. VII,
ELECTRON RESONANCES IN METALS t
ds =
dt’ exp
[
-
(t - ‘‘I t
x exp [id,
297
] ev(t’) E(y - r(t) + r(t‘))
- iw(t
- t’)]
(2)
for simplicity t is considered to be constant. I
y
=
Fig. 5. The electron trajectory in
O
p,
=
const. plane.
Substituting Eq. (2) into Eq. (l),taking the boundary condition into account as described before and changing the limits of integration in Eq. (2) from - 00 and t to t - T o and t, we get a formula for the current density :
I=
---S,dcj[l 2e2 eH c
A3
x
ST’
an,
v ( t ) exp
0
(-
- e ~ p 2nio ( - T - ~ iwt
t
s(y - r(t) - r) dt
(3)
This formula was obtained in by immediate solution of the kinetic equation for the somewhat more general case t = t(p) (although the generalization of Eq. (3) to this case presents no difficulties-see also29). The kinetic equation
- en’(an,/ a&)exp (id,) in the variables E , t, p,, y taking for n = no(&) into account the possibility of introducing the relaxation time has the form : Refereizces p . 330
298
M. YA. AZBEL' AND I. M. LIFSHITZ
v,,a
(i m + - +ay- + -
a l t
at
[CH. VII,
1
n'=v.E. )
The boundary condition for diffuse reflection corresponds to n' I y - 0 , f'O - 0 72' I y - m , vff < o - 0. We return to Eq. (3). As is shown in the beginning of this section, in the formula for ill parallel to the metal surface, which is the only case of interest to us, E,, can be set equal to zero and v * E = v,Ep where the repetition of the greek subscript means summation over x and z. From formula (3) both the existence of cyclotron resonance at frequency 9 = w and frequencies Q = i w , gw, . . ., and the existence of the difference in resonance depth for a quadratic and a non-quadratic dispersion law can be seen. For a quadratic dispersion law Q does not depend on p,, as is shown before, and therefore j, cc 1 - exp (-2nim/Q - 2n/Qt), hence for z -+ 00 the current density at resonance (for w = SZ, 2Q, . . .) tends to infinity in proportion to t / T o . For a non-quadratic dispersion law it can be easily seen that the coincidence of w with one of the frequencies SZ which are different from the extreme value, leads to no characteristic property. When w equals or is a multiple of one of the extreme values of Q (w = qQo, q = 1, 2, . . .) resonance appears, however, in contrast to the case of a quadratic dispersion law, j cc dt/TOi.e. the height of the resonance is considerably smaller. We shall now proceed at once to calculate the surface impedance tensor Zap,given by the relation: ic2 Ea(O) = zu,~,= ( ~ ) z " ~ E ; (; o ) z a p
=z
Rap
t ix,,
(4)
where I, is the total current in the metal
I,
= 0
i A Y ) dY -
For the calculation of Z,, it is necessary to solve the Maxwell equations for El, together with Eq. (3). I t is convenient to extend the field E, to the region y < 0, outside the metal, in such a way that Ea is an even function of y : Ea(- y ) = E,(y). Since the fermi surface is centro-symmetric, a field that is symmetric Kcferences p . 330
CH. V I I ,
§ 11
ELECTROX RESONANCES I N METALS
299
in y is a solution of the equations if we replace s(y - r(t) - r) in Eq. (3) by the equal function for y > 0 ; s(1y - r(t) 1 - r ) . We shall simplify formula (3). Firstly, the largest parameter with tend to infinity in the inner integrals. the dimension of a time---can Secondly, we notice that the deviation of the factor s( I y - r(t) 1 - r ) from unity (which takes into account the “non-effectiveness’’ of the electrons colliding with the surface) cannot essentially affect the results, and leads only to a numerical factor before the impedance, of the order of magnitude of unity. This was proved in starting immediately from the equations; it was shown that this constant factor was almost unity (see also 44). The physical reasonfofor the smal leffect of the boundary condition under anomalous skin effect condition^^^,^^^^^,^ is the fact that at any rate the essential part is played by the “skimming” electrons in
’,
Fig.6. Electron trajectories near the metal surface.
A,,
>
16e2
=-
3h3
9 ((3) f," [ ( 2nio 1 - e x p ----
i=l
B1K
Q = pi
+
1"
52
Qt
e-+x
The tensor aa8 rr 1 is of a rather complicated form7, which is of little interest to us; the reason for the maintenance of the non-resonant
JI 4 Fig.7. Closed fermi surface. Refevemes p. 330
302
M. YA. AZBEL’ AND I . M . LIFSHITZ
[CH.
VII,
9
1
quantity a,b becomes clear in § 1 . 3 ; the variables are the same as in Eq. (6) and the integration extends over the angles corresponding to the “girdle” u, = 0 on the fermi surface (see Fig. 7) ; ql,y 2 . . . rppl are points were D has a given extreme value with respect to vzriation of y ; when taking the cube root of Eqs. (8) and (7) the root corresponding to K, > 0 should be chosen (such a root always exists). IMPEDAXCE 1.3. ANALYSISOF THE SURFACE
An analysis of the surface impedance near resonance is most conveniently made for either dispersion law separately. In experiments it is often the derivatives of R and X (e.g. dRldH, dlnX/dH) instead of R and X that are measured and thus we shall also investigate the derivatives of the impedance, according to &. a) Quadratic disfiersion law. In this case the entire resonant curve can be constructed for any SL! (for values close to the resonance values as well as for those far from them) ; the shape of the curves R(H)/R(O), X ( H ) / X ( O )and X/(RdT) for w z = 1, 10, 50 is depicted in Fig. 8. The small maxima of R and X for w m (q 4)Q are not related to resonance and for Qt -+ 03 the value of the impedance at these points tends to a constant value different from zero. It is important to note that for finite t the depth of the resonance minimum and the frequency shift of the minimum relative to o/q are much different for R and X .
+
The frequency shifts for R and X have different causes. The shift for X is simply related to the fact that a small increase of the magnetic field, which hardly changes the resonance conditions, leads to an advantageous increase in the number of revolutions made by the electrons between collisions. To understand the frequency shift for R we consider how the variation with depth of the phase of the electric field manifests itself. A change in phase destroys the resonance synchronism, diminishes the energy acquired by the electron and thus impairs the resonance. As formula (7)and the graph of X / ( R d 3 )as a function of the magnetic field show, even a small variation of the magnetic field leads to X R,
>
References p . 330
CH. VII,
5
11
ELECTRON RESONANCES IN METALS
303
i.e. the phase of the attenuated field is alsmost unchanged over a depth 6. This proves to be favourable in spite of the fact that after about 1 cu - qQEa I t revolutions the electron appears near the surface when the phase of the field has been changed considerably. We proceed to analyze dRldH and dX/dH. At first sight it may look
XrH) R(H)33
4
3
2
1
Fig. 8. Theoretical resonance curves for WT = 1, 10 and 60 for a quadratic dispersion law. a) R(H)/R(O)vs m/Q;b) X ( H ) /X ( O )vs o/Q; C) X ( H ) / R ( H ~) % v w/Q. S
as if at resonance, where R and X are minimal, their derivatives are equal to zero. However, in fact, resonance does not correspond to zero values of dR/dH and dX/dH but to their maximum values. This is caused by the fact that for cot = 00 the functions R(H) and X(H) do not have a minimum at resonance, but a smallest value (equal to zero) which References p. 330
304
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
5
1
corresponds to a kink in these functions. Thus for w t = 00 we find from formula (7) that in the region where H < HEa, R(H) cc (HE8- H ) t
and dR/dH+O
when approaching resonance from this region, and in the region where H >HgS, R(H) cc (H - HPR,)aand dR/dH --f 00 when approaching resonance from this side (see Fig, 9). The functions X ( H ) and dX/dH behave in an analogous way. Therefore (dZ/dH),,, = Z(0) (q2wz)t/Hie,and for t -+ 00 (dZ/dH),,, -+ 00 (and not to zero) and (w - qQres)/wm (wt)-l; HLa is the magnetic
--H ffres
Fig. 9. Behaviour of R and dRldH as a function of H near resonance for a quadratic
dispersion law, and for a non-quadratic dispersion law for m&.
field corresponding to resonance on the principal harmonic (q = 1). The relative heights of the maxima of dR/dH and dX/dH are significantly larger than the reciprocal value of the relative depth of the minimum for X(H), and the resonance frequency shift is the same as for X and much smaller than for R. b) Non-quadratic dispersion law. In formula (8) for a non-quadratic dispersion law Q(y) enters instead of Q(P,) considered until now, Since the function p,(q) obviously has extreme values only at the points of support of the surface (see Fig. 7) and dQ/dq = (dQ/dp,) (dp,/dy), Q(q) must have extreme values first at the place where Q(p,)]Ealso I\'efcrciiccs
p. 850
13
CH. VII,
ELECTRON RESONANCES I N METALS
305
has them, and secondly at the extreme values of p,(v) i.e. at the elliptical points of support of the surface (which we discussed without proof in § 1.1) (at the hyperbolic points of support m* = 00, 52 = 0 and resonance is impossible). One may think that, since at the elliptical points of support I = 0 , the basic condition for cyclotron resonance, r/6, 1, is not satisfied. However, this condition ( 1 / 6 ~ l), in fact, places no severe restriction on the frequency, as near the point of support I M ( c / e H ) A g , / d R (Fig. 7 ) , and dg, w (ox)-*, and the condition r M ~,(wt)-*7 6, is needed. In fact if we take into account that z 2 rep2 (fi/kO) ( k O / f i ~ ) ~ ; I rn v / w , 6, m c/wo m &c/E,, and EOv/c rn ke, we can easily see that 1/6, 5 d w y Thus, resonance occurs on the central section (where Q(p,) always has an extremum), a t frequencies corresponding to the elliptical points of support (where, as can be shown7, m* = ( v d K ) - l , v and K are the velocity and the Gaussian curvature in the point of support) and at the non-central values of 52 which are extreme with respect to variation of p,. One should notice also that resonance on the central cross-section and on the points of support is definitely different from resonance at the extreme values of $2. In the first case, because of the central symmetry of the fermi-surface, the resonant term in A,, is proportional to the tensor n,(vl)np(yl) one of the principal values of which is equal to zero and the other equals unity. Consequently, only one of the principal values of the impedance has a resonant character. Since at resonance R and X have a minimum, instead of a maximum, for arbitrary polarization of the incident wave on the metal, the impedance will be determined in principle by the large non-resonant principal term, and the resonant part will represent only a small increase in the impedance. A substantial resonance will occur only in the case where the incident wave is polarized along the velocity v o at the point E = e0, 8 = in, v = y1 [for a point of support this direction coincides with the magnetic field direction (Fig. 7 ) ] . The current density corresponding to the electric field perpendicular to vo has a non-resonant character because in this case the electric field in the skin is almost perpendicular to the velocity and correspondingly performs almost no work. The derivative of the impedance with respect to the magnetic field is, in principle, determined by functions strongly depending on H and 4b960
>
>
RcJereiicts
p . 330
306
M. YA. AZBEL' AND I. M. LIFSHITZ
[CH. VII,
5
1
has therefore a resonant character for all directions of E with the exception of the direction perpendicular to yo. However, in agreement with what has been said before, the range of angles in which a strong anisotropy must be observed for R and X as well as for dR/dH and dX/dH, is very large. Thus, for absorption P = R,,(Ep)2 Rpp(EF)2, the range of angles is given by the quantity (R,,,/R(O))*,which is usually not too much different from unity (for the value of Rre8/R(0) see below). I n the case of resonance on the non-central cross-sections, there are at least two centrally symmetric cross-sections on which resonance occurs, and two points (C and D) which contribute appreciably to the current density. The velocities at the points C and D are, in general, not parallel, and for any direction of the electric field the current density (and thus also both principal values of the impedance) has a resonant character. Mathematically this is related to the fact that, in general, none of the principal values of the tensor n,(q~,)n~(p,)n a ( ~ , ) n p ( yis2 ) equal to zero. We shall further analyze only the resonance values of 2, (see formula (8)).The formulae for the resonance values of the impedance have a different form depending on whether the given section has a minimum or a maximum for SZ = eH/m*c. In the formulae for X , this affects the numerical constants only; in both cases XF/X,(O) rn ( q2/wt) *; 1 - qSZ&/o w (or)-'and the tensor aaPdoes not enter. The formulae for R,, however, differ qualitatively for m& and mzaX. Thisdifference canbe explained by the fact that the case of minimum effecis analogous to the case of a quadratic tive mass mzin = (I/gz) ( as/ dispersion law in the respect that a small change in H near resonance leads t o X R and to a significant increase of the resonance depth for R. In the case of maximum m* it is impossible to attain X > R for a small shift of H [all this can be proved starting from Eq. (S)]. Only for the sake of definiteness shall we consider electrons: as/a s > 0 and not holes: as/a& < 0. [All arguments are the same for holes and the result contains I as/8s I ; the case of equal numbers of holes and electrons does not lead to some special property which is reasonable since the Hall-field E, does not enter the formulae at all.] For maximum m*, R, = X , and (w - qSZ&)/w M (wt)-l, the tensor agP,just as in the case of Xu, does not enter. For minimum m*, the resonance depth of R, and the shift of the
+
+
>
References p . 330
CH. VII,
5 11
ELECTRON RESONANCES I N METALS
307
resonance frequency substantially depend on the non-resonant term as in the case of quadratic dispersion, in the given case, on aaD(since for aaP= 0 Rp = 0 would be attained), which for this reason was retained in Eq. (8). For aaP rn 1,
The curves of dR,/dH are also substantially different for the case of maximum or minimum m* 34a. For m& the curves are analogous to those for a quadratic dispersion law (Fig. 9). For m&x close to resonance on the low field side ( H = H,,, - 0) dR,/dH becomes -m and on 0 ) + 00 and the curves are of the the high field side ( H = H,,, form shown in Fig. 10. In either case (mzin and m Z X ) (dRa/dH),a, rn HylR(O)q*((ot)tand the frequency shift (w - q Q m a x ) / W w (wz)-l; H, is the magnetic field
+
Fig. 10. Behaviour of R and dR/dH as a function of H near resonance for a nonquadratic dispersion law, for m;,,.
corresponding to resonance on the principal harmonic (q = 1). It is essential that the relative resonance height for dR,/dH be significantly larger than the reciprocal of the relative resonance depth both for R, and X,, which makes measurements of dRJdH more attractive than those of R, or X,. As is clear from the above, the behaviour a t resonance of R, and Rejeretzces p . 330
308
M.
YA.AZBEL’
AND
I.
M. LIFSHITZ
[CH. VII,
5
1
dRJdH is very sensitive to the properties of the fermi surface (resonances at the points of support, at the central cross-section and a t the non-central sections for minimum and maximum m* are substantially different). Thus cyclotron resonance enables us not only to determine at once the effective mass of the electrons but also to obtain important additional information about the nature of the fermi surface, being an effective tool (together with other methods-De Haas-Van Alphen effect, Shubnikov-De Haas effect, quantum oscillations in high frequency fields, galvano magnetic phenomena, the study of the surface impedance under anomalous skin effect conditions) for constructing the shape of the fermi surface (see also the next section). I . 4. FURTHER DEVELOPMENT OF THE THEORY OF CYCLOTRONRESONANCE
The theory of cyclotron resonance displayed in 9 1 . 2 and 8 1 . 3 is in its essence a pure classical theory in which only the equilibrium function corresponding to a degenerate fermi gas has been introduced. Such a theory is, of course, completely sufficient for the study of the basic effect, since the distance &Q between the levels in a magnetic field is considerably smaller than the fermi boundary energy E~ (for the fundamental groups &ais about E ~ only , for H of the order lo8-lo9 Oe) and the quantum corrections for the classical formulae are quite small, usually even much smaller than the next unwritten anomalous terms. The latter have a relative magnitude of the order (s,/r)+.Nevertheless, the consideration of quantum effects is very significant. I n the first place the quantum effects have an oscillatory character with respect to variation of the magnetic field, with a period of oscillation in terms of the reciprocal magnetic field A ( H - l ) of the order ~ T C , U / E ~( p is the Bohr magneton) which is considerably smaller than the “periods” of cyclotron resonance (i.e. the distance between the harmonics) which are of the order e/mcw. This suggests the possibility of separating the quantum oscillations from the classical phenomenon. The knowledge of the periods of quantum oscillations can basically simplify the problem of the establishment of the form of the fermi surface. The quantum theory of the surface impedance for an arbitrary direction of the magnetic field was established for a general dispersion law E = ~ ( pand ) collision integral in 36 and, independently, somewhat Refeferciaces p. 330
CH. VII,
11
ELECTRON RESONANCES I N METALS
309
later for the special dispersion law E = p2/2m, constant relaxation time z and the magnetic field parallel to the metal surface in 3'. It has been shown in 36 that the periodicity of the oscillations caused by the magnetic field is the same in the h.f. case as in the static effects of De Haas-Van Alphen and Shubnikov-De Haas: O(H-l) = eh/cS, where S, is the extreme cross-sectional area of the fermi surface in a magnetic field parallel to the metal surface and the area of the central cross-section in an oblique field. This makes the experimental analysis of the impedance in a magnetic field very useful for establishing the shape of the fermi surface, and gives in one sample a very precise determination of Sex, from the from the resonant periods1of the quantum oscillations, and of ( as/ frequencies. [The amplitude of the quantum oscillations also includes ( as/a&),,,, but it is very sensitive to mosaic structure, impurities, deformations, etc. and makes the result at the least inaccurate.] The knowledge of Sex,and ( as/a&),,, makes it possible, at least for a convex fermi surface, to establish its form as well as the electron velocity on it (according to the method of Lifshitz-Pog~relov~~), which is one of the fundamental problems of the electron theory of metals. The fact is that because of the fermi statistics, as was indicated above, the form of the boundary fermi surface and the velocity of the electrons on it, in any case fully determine the dynamic characteristics of the conduction electron. In an oblique field the amplitude of the quantum oscillations has a resonance character, whereby the magnitude of this quantum resonant increment is considerably larger than the classical resonant increment. We consider this problem in somewhat more detail. Chambers39 noticed that cyclotron resonance can occur in an oblique magnetic field, for example, on the central cross-section where the electrons do not travel inside the metal (since the average value of their velocity on the central cross-section is equal to zero). However, the relative number of electrons which do not leave the skin layer after the nth revolution is of the order o~d,,/m 1. Their relative contribution to the impedance is of the order d,/r and it is always small (as long as the anomalous skin effect is present i.e. up to frequencies w 2 10%-1, corresponding to a magnetic field H 5 lo6 Oe)', 44. The difference between the central and non-central cross-sections involves only the order of tilting of H , which suppresses the resonance: in the first case it is v'OZ times larger (see also 62).
plc, because of this oz is, in general, impossible - see 5 1.3.1 In order to understand the origin of the gradually attenuated field “splashes” inside the metal, which change the monotonic decrease of the field, we consider in general how a field and a current penetrate into the interior of a metal when a magnetic field is parallel to its surf ace. Therefore, we return first to the motion of an electron in one of its orbits (orbit 1 in Fig. la), which passes through the ordinary skin near the surface, where in any case the electric field is not small (the constant magnetic field is perpendicular to the plane of the figure). In a layer of the order 6 the electrons acquire a directed velocity over an arc of length d/rs(r 6) and produce a current I of density j w IjS. As the electrons move down along the orbit the velocity parallel to the metal surface changes (along which only a current flows), and correspondingly the current changes by a factor cos p, and, secondly, the electrons will spread out through the bulk of the metal finding themselves in a layer of the order of V% sinp instead of 6 (for
>
p?
dV).
Referelices
p . 330
314
M. YA. AZBEL’ AND I. M. LIFSHITZ
[CH. VII,
92
Thus, the current density produced by electrons of a given orbit proves to be of the order (I/d%) cot ‘p, i.e. decreases sharply with the depth and for ‘p m 1 is a factor d/rls smaller than in the layer 6. At a depth y > Y the current density changes sign, remaining small in absolute value compared to I/S until the angle ‘p approaches the value ‘po of the lowest point of the orbit, so that 1 y o - ‘p I 2/6/r. At a depth d the current density increases again sharply and in this case differs only in sign from the current density at the surface. Apparently such a picture is valid for all trajectories of given radius passing through the narrow skin layer parallel to the surface, Therefore,
Fig. 19. Electron orbits passing through the “ordinary” skin.
if all electrons moved along orbits of the same radius (i.e. had the same value of the velocity in the xy-plane), then the current density produced by the electrons skimming along the metal surface in the layer 6, would produce “splashes” of current, and associated with them an electric field at a depth y = d. Such splashes would in turn lead to acceleration of new electrons, skimming along a layer at a depth d , causing the picture to reproduce itself at depths 2d, 3d, etc. The result would be the solution of the problem concerning the selfconsistent system of currents and fields. However, the picture drawn in Fig. 11 is physically clear. The phenomenon changes essentially if orbits of different radii (Grbits 2 and 3), which correspond to different sections of the ferrni surface, are considered (we remember that, for example, for free electrons Y = plc/eH = ( c l e w d 2 m s - $:, and I varies from 0 to (c/eH)1/2mE).The spread in radii leads to the fact that at any depth in the layer of the order 6 only a small part of the electrons collects (of the order 6 / ~ )and , the field, carried through into the metal will of course decrease rapidly, especially in the following “loops”. In the case References p. 330
CH. VII,
$ !
21
ELECTRON RESONANCES I N METALS
315
where the orbital cyclotron frequency f2 does not depend on the section (i.e. in the case of an ellipsoidal fermi-surface), this spread in the radii cannot be eliminated. However, if the frequency depends on the section of the fermisurface, i.e. on ps, then cyclotron resonance can be used to eliminate the large spread in radii, as only electrons near the extreme cyclotron frequencies take part, with a spread in p , of the order M $,dot (since w - SZ w 1/t),p , being of the order of the limiting fermi momentum). The spread in radii given by a certain w t is totally different for each of the following cases : a) for a central cross-section, where, as it is clear from a symmetry consideration, both f2 and d have an extreme value at the same time; b) for points of support, where, as it is shown in $ 1 . 3 d oc dAT; c) for other non-central cross-sections. For points of support Ad M d w d,/dz so that (see 1.3) Ad m d ? 6, always and splashes of current cannot occur. At other non-central cross-sections Ad m d / l / o z and for splashes of field strength and current density to occur it is necessary that Ad M do i.e. w t m ( ~ / 6 , (a ) ~more stringent inequality, as discussed in section 1, is, generally speaking, impossible). For a central section where d’(0) = 0, the spread Ad = +d”(O)$; M M d(wt)-l, and the structure shown in Fig. 11 occurs for c o t T ~ / 6 , , w ? v/d/ldo, which for I M 10-1 cm, 6 , M cm, ZI fw lo8 cm/sec corresponds to a wave length of the order of 1 cm and H M lo4 Oe i.e. values which have already been attained in experiments on cyclotron resonances. It can be easily understood that the negation of this inequality ( w t < ~/6,) leads to a decrease of the magnitude of the consecutive splashes in a geometric progression, proportional to the degree of diffusion of the current at the bottom 6/Ad M ( 6 / r ) o t ,and the splashes are rapidly attenuated. A more precise estimate shows that the splashes for m t < ~ / 6 , , y = ad, are of the order E(0) (u)-+ ( S , / Y ) ~ / (~w t ) s a / l z ;(do = d(O)), that the relative increase of the field near y = do is of the order (wt)’, and that the relative contribution to the impedance in from the splashes is of the order ( S O / ~ ) t ( w zandlfrom )* the following “impeculiar” term of the expansion of the order (dO/~)+(wt)-A. Thus, under certain conditions a characteristic chain of layers of field splashes inside the metal is obtained. References p . 330
316
M. YA. AZBEL’ AND I . M . LIFSHITZ
[CH. VII,
92
It is apparent that the arguments given are not a proof of the existence of such a structure. I n the simplest case of resonance on the central section and when r/do w t (r/80)2,the occurrence of such a structure can be proved by solving equation (5) by the Fourier method and determining E,(y) in the neighbourhood of the points y = ad, ( a is an integer) and in the intervals between them. The basic formal difference from the solution in the “ordinary” case of w t r/do, considered in 7, consists of the fact that the largest parameter is not the parameter of anomaly r/6, but the resonance parameter at.Since the analysis is rather complicated, we refer the reader to lo, 35 for details, where an investigation is given for arbitrary w t and the mathematical reason for such a field structure is explained, namely the proximity of an infinitely degenerate eigen value of the Maxwell equations. We give the results of this consideration only for the central section and w z ? r/So. 1. For y = do(a M-’(’), C’ M , where M m ( ~ / d ~ ) z ( w and ~ ) i is determined by the properties of the ferrni surface a t the point
<
(v/c)(2no/z)*5 1013s-l and the magnetic field H = fiw/2p0 > lo6 Oe. This means that at v/w
References
p . 330
CH. VII,
$ 31
323
ELECTRON RESONANCES I N METALS
x is the magnetic susceptibility; 2 H , is the amplitude of the alternating magnetic field. 2. For thick samples the line has a central structure of width equal to the natural width 1/T, with wings extending over a band l/Tg(Se,,/S)z if 6 < d,,,. 3. The central structure is always markedly more intense than the wings so that under normal experimental conditions the apparent width of the line is of the order Ti1. 4. The characteristic effect of electron diffusion is not t o broaden the line but to make a radical change in its shape. 5. For thick samples with narrow natural line width: D 6,,, d, the absorption line P per unit surface area has an asymmetric form and is given by the formula:
> >
P
rn - ( 4 2 / ~ ) - 1 0 2 H ~ ~ T g ( 6 2sign / 6 e ,a, ), [ d l
+ a: - l]*(l+ a:)-*.
> >
6. For thick samples with broad natural line width D 6 a, the absorption line per unit surface area is described by the formula:
P Fw :w2H;XTg6(1
- al)/(l
+ @?).
7. For thick samples the intensity of the line in the centre is reduced Be,, the integrated intensity of the line comes mainly from the diffuse wings and not from the centre. The theory is in good agreement with the experiments of Feher and Kip 46, who observed paramagnetic resonance in lithium, sodium, beryllium between 4"K and 296" K and in potassium at 4 O K . [The first electron paramagnetic resonance in metals was observed by Griswold, Kip and Kitte153;from the later experiments it should be mentionedsP, that resonance has been observed in solutions of Na, Li, K, Cs, Rb, Ca and in ammonia.] The agreement between experiment and theory becomes obvious from Fig. 15 and from a comparison of the theoretical curves of Fig. 16 and the experimental curves of Fig. 17. All figures are taken fromd6; T , in the figures corresponds to T , of the text :P is the absorbed power and x the magnetic susceptibility of the electron gas. From a comparison of the experiments with theory it is possible to determine two basic parameters characterizing the paramagnetism of the electron gas: the spin-relaxation time together with its dependence on temperature (which in the case of ignorable impurities a factor rn 6/6,,, by the diffusion effect. When 6
References
p . 330
Refevences p. 330
CH. VII,
3 31
327
ELECTRON RESONANCES I N METALS
operator of the total spin of the system and leads to a partial equilibrium with matrix f^eq depending on the energy only, and for a fixed value of the spin moment: dp = Tr
Tr
f d p ; Tr
s
s
Pq6 dp = Tr fa dp.
For sufficiently low temperatures and in sufficiently weak fields (PH
<w ,
The second type of collisions leads to the establishment of a total equilibrium function both for the momenta and for the spins
The form of the kinetic equation enables us to determine in the equilibrium functions the parts giving the current and the magnetic moment, putting f^ = fir^ + f . 6 (3 is the unit matrix). The solution of the equation forfis very complicated and, as in the case of the work of Dyson, we formulate the results only. 1. The total magnetic moment equals M = ~ ( 5 b) where, choosing the z-axis along the d.c. magnetic field direction as we have consistently done b,(5) = Re [40).u$(5)1 (1 Re[~(0).u$(O)ll-l; iHob G b,
+ ib,
+
= u(C) (1
+ Re[~(O)u$(0)]}-~;y cc [zGI.
(10)
The idirection coincides with the inward normal to the surface of the metal; the subscript -“0” in u$ means that u* is taken at resonance, o = SZ,; the bar, as in 1 ,means the average over the fermi surface. The function u([)near to resonance equals, apart from a numerical factor of the order 1 (for 6 d,,,):
30 y) and 176Yb(4 d). The lsoTbdecay scheme is much more complicated than is shown in the figure. I n particular a gamma radiation of 1.18 MeV between the 1.26 and 0.087 MeV levels and a radiation from the 1.26 MeV level to the ground state of lsoDy are not shown. The lssmHo decay scheme as proposed by Grace and modified by Boskma (see ref.Io7b),is somewhat simplified.
crystal, W-(e) was measured under an angle of 0 = 55" with respect to the crystal c-axis, giving an asymmetry effect of - 0.360 & 0.025 upon reversal of the polarizing field. From this it is calculated that dexP(O = 0) = ( - 0.70 f 0.05)vlc for fl = 0.65 and for beta-particle energies between 0.2 and 0.8 MeV. Essentially these results give A - = - v/c as might have been expected for an allowed decay between states I , = 3 and I' = 2. However, according to the value of logft = 8.8 for the predominant branch in the high energy end of the beta spectrum, i.e. for the 0.87 MeV radiation, this is a first forbidden decay. P ~ s t m ashowed ~ ~ from the theoretical formulae of M ~ r i t a ~ ~ that A & for first forbidden decays with allowed spectral shape and References p . 391
364
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
94
with A I = 0 or & 1, is in good approximation equal to the A , value for allowed decay. The conclusion then is that laoTbhas spin 3. The measurements under an angle of 55” with the c-axis were performed in order to observe a P,(cos 0) term in the directional distribution of beta particles emitted from aligned nuclei. No such effect was found within the experimental uncertainty of f 1%.
14Pr dexp M 0.08 was found for the first forbidden negaton decay of 143Prnuclei, polarized in Ce-Mg-nitrate. Since no gamma rays are emitted, fi can only be roughly estimated; assuming ~ ( l ~ ~ wP r ) ,u(l4lPr)= 4pN and I , = -2, a considerably larger dexp should have been expected. The measurements showed that +{W(t ) W ( J.)}was temperature dependent, unlike all other cases discussed above; when the high temperature value is again normalizedto 1 , the low temperature value may be designated by W(O = O,fl = 0 ) , and the data give W (0 = 0 , f i = 0) = 0.028 f 0.002. Measurements of the beta particle intensity in directions perpendicular to the nuclear orientation axis showed correspondingly W(O = in,fl = 0) = - 0.009 & 0.002. This experiment therefore, has shown for the first time the presence of a P,(cos 0) term in the directional distribution of beta particles emitted from oriented nuclei 1 4 6 ~ .This effect will be called “anisotropy” because of the similarity to the anisotropy of directional distributions of alpha and gamma emission from aligned nuclei and also in order t o distinguish it from the mirror asymmetry related to parity nonconservation. The term B , in the theoretical expression for the directional distribution W(6,f i = 0) = 1 B,(v/c) f,P,(cos 0) depends on the various first forbidden nuclear matrix elements and, in general, on the energy. B , would become independent of unknown ratios of nuclear matrix elements only in the case of unique first forbidden transitions ( d l = 2 , cf. I, eq. (1.11)).
+
+
+
+
TIMEREVERSAL INVARIANCE. A measurement of the j3-y-directional correlation for polarized 62Mnnuclei has been reportedg0, in which a search was made for terms which change sign under time reversal. Such a term is A k ) ( j . k ) , having a coefficient (19), as discussed in 3 2. The positons (momentum p) were observed in a direction perpendi-
l*(g
Referelaces p . 391
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ORIENTATION O F ATOMIC NUCLEI I1
365
cular to the nuclear polarization vector, j, in coincidence with gamma rays (momentum k ) emitted perpendicular to p and under angles of 1-45' and -45" with respect to j. The difference between the coincidences of the -45" gamma counter and the coincidences of the +45" gamma counter, normalized to 0 at high temperatures, measures the term of interest; this difference was shown to be (0.012 & 0.022) (v/c)fi (apart from a trivial factor). The approximately zero result may be due to time reversal invariance but may also be due t o lack of sufficient interference between Fermi and Gamow Teller matrix elements. From the upper limit for the effect and from the measured ,&asymmetry Ambler et aLgodraw the conclusion that the phase 8 is restricted by 140" < 8 < 250", where 8 is given by
(8 = 0" or 8 = 180" if time reversal invariance holds). However, this conclusion is based on a value of 0.05 < 1 X 1 = I C,M,/CAMG, I < 0.1. It was discussed under 52Mnthat this seems not yet established beyond doubt so that further experimental evidence concerning time reversal invariance seems desirable. CONCLUSION a) Large asymmetry effects have been observed both in allowed and in forbidden beta transitions; b) the presence of Fermi-Gamow Teller interference has not been definitely established in beta asymmetry experiments with polarized nuclei ; c) spins of beta decaying nuclei could be determined in a few cases e.g. 52Mn,160Tb;the number of possible assignments could be reduced in some other cases; and d) a P,(cos 8) term was found in the directional distribution of a first forbidden decay.
5. Experimental Results on Gamma Radiation The experiments described in this section did not lead to discoveries of such fundamental importance as the beta asymmetry experiments discussed in the preceding section. On the other hand, a considerable amount of experimental data have been gathered in the field of gamma spectroscopy. It is the purpose of our discussion to give a short catalogue of the work done since I was written. A theoretical introducReferences
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95
tion to the subject may be found, e.g. in I ; the experimental methods are well known and in the following we will only mention a few hitherto unused techniques. Practically all experiments were performed by measuring the gamma ray intensity in directions parallel and perpendicular to the axis of nuclear orientation, which intensities will be denoted by W(0) and W(+) respectively if normalized to 1 at high temperatures. In most cases we will quote the magnitude of the anisotropy, E
-{W(b)
- W(O)l/W(:n)
as a, albeit rather arbitrary, figure of merit, thereby complying to the usage of publishing E instead of W(0)and W(&c)separately. The results will be listed according to increasing 2 and A and not in chronological order. Since this' section is complementary to I, recent results will be given relatively more attention.
5zmMn 52Mn in the 21 min isomeric state (Fig. 6) was aligned by Bauer and Deutsch94 at M.I.T. growing 8 h 5aFeinto magnetically diluted nickel fluosilicate crystals, cooled to low temperatures. After Pf-decay of 62Fe the alignment of the 21 min state was measured by means of the gamma ray anisotropy of the 1.43 MeV radiation of 52Cr,populated by P+-decay of 52mMn. Comparison of the gamma anisotropy of the 1.43 MeV radiation with that of the 0.84 MeV E2 radiation from S4Mnin'the same crystal, gives for the nuclear gyromagnetic ratio = 0.36 i 0.08 g(52mMn)
for a pure Fermi p-transition and 0.52 Teller transition.
0.08 for a pure Gamow
52Mn From gamma ray anisotropy measurements of 52Mn(5.7 d) in CeMg-nitrate the nuclear magnetic moment was founde5to be =2 . 8 ~ ~ . ,~4(~~Mn)
Dynamic polarization methods (9 8) have since132 given the more reliable value ~ ( ~ 2 M= n )3.00 f 0.15 pN.This shows that in sufficiently large polarizing fields ( > 500 Oe) the degree of nuclear orientation of Mn nuclei in Ce-Mg-nitrate can, to a reasonable approximation, be References p . 391
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ORTENTATION O F ATOMIC NUCLEI I1
367
described by the paramagnetic resonance data of Trenam on Mn in Bi-Mg-nitrate146f. Bauerg4points to the interesting fact that the nuclear g-values for 52mMnand 52Mnare nearly equal, whereas the spins are widely different, I, = 2 and I, = 6 respectively. 54Mn
Bauerg4oriented 54Mn(310 d) both in Ce-Mg-nitrate and in (10%Ni, 90%Zn) SiF,, 6H20. From the comparison between W(&z)and W(0) for the 0.84 MeV E2 radiation it was concluded that no change in orientation occurs in the preceding 0-decay, and therefore 54Mnhas probably spin 3, in agreement with Oxford results1 and the resonance data of Kedzie et ~ 1 . l ~Whereas ~. Bauer finds p (54Mn)= 2.55 f 0.21 pN, Kedzie et ~ 1 . report ~ ~ 2 p("Mn) = 3.29 f 0.06 pN. From the measurement of the circular polarization of the 0.84 MeV gamma ray, p was found to be positiveg4. 56Mn
Dagley et ~ 1 . 9 6in Oxford aligned S6Mn (2.6 h) in a fluosilicate single crystal, containing Zn, Ni and Mn in the proportions 90 : 10 : 1. This crystal was irradiated by about 5 x 1014 thermal neutrons/cm2, producing 100 pc 66Mnat the start of the experiment. The temperatureentropy relation for this crystal was determined from nuclear alignment experiments with %Coin a crystal of the same composition. The known nuclear magnetic moment and decay characteristics of 58C0 (Fig. 6), in conjunction with paramagnetic resonance data on stable Co, make such a determination feasible. From the known p(56Mn)and paramagnetic resonance data one can calculate the theoretical gamma ray anisotropy as a function of T , e.g. for the 0.845 MeV radiation (Fig. 10) with known multipolarity and spin change. It was found that the degree of nuclear alignment obtained was about 50% of the expected value, which discrepancy is attributed to radiation damage, since annealing could produce an increase of alignment up to 75% of the calculated value. Comparison of the 0.845 MeV anisotropy with the 1.81 and 2.13 MeV radiations gave values for the mixing ratio d(E2/M1) of + 0.19 f 0.02 and - 0.28 f0.02 respectively. Bauer et dg7 have grown 2.6 h 56Mnin Ce-Mg-nitrate and measured directional distributions of various gamma rays. The largest anisotropies, E , obtained with a polarizing field of 450 Oe, were for the 2.13 MeV, 1.81 MeV and the 0.845 MeV radiations E = + 0.40, E = + 0.18 References
p . 391
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[CH. VIII.
95
and E = + 0.27 respectively. By comparison of the last two anisotropies with E = 0.40 for the €22 radiation of 0.845 MeV, the mixing ratios of the 1.81 MeV and 2.13 MeV radiations were found to be d(EZ/Ml) = 0.11 f 0.06 and d(EZ/Ml) = -0.27 0.03 respectively. In a separate experiment 52Mn was added to the source and comparison of the anisotropies of the 52Mn and 56Mn radiations gave g(52Mn)/g(56Mn) = 0.47 f 0.05 or ~ u ( ~ ~ M = n3.35 ) f 0.25 ,uN.Measurements of the circular polarization of the 0.845 MeV radiation showed ,u to be positive. Again, as in 52MnexperimentsQ5,a substantially lower anisotropy was found if the polarizing magnetic field was removed.
+
5 5 ~ 0
Bauer et aLg8polarized 55C0 (18 h) nuclei (Fig. 10) in Ce-Mg-nitrate and measured the angular distribution of the 1.41 and 0.935 MeV gamma rays, simultaneously with the 0.80 MeV E2-radiation from 5 8 C nuclei, ~ which were grown into the same crystals. The largest anisotropies obtained were ~ ( 1 . 4 1MeV) =
+ 0.24 and ~(0.935MeV) = + 0.16 +
compared to ~ ( 0 . 8 0MeV) = 0.13, if a polarizing field of 450 Oe was applied. Using crystals with 55C0 alone, the linear polarization of the 1.41 and 0.935 MeV radiations also were measured simultaneously with the anisotropy. Simultaneous measurements of P , and E may, a t least in principle, lead to a determination of the mixing ratio of a gamma radiation even if the temperature or nuclear magnetic moment is unknown. This originates from the fact that interference terms between dipole and quadrupole radiation contribute very differently to the anisotropy E and to the degree of linear polarization P,, whereas for pure radiations for instance 1 P,(B = in) [ = [ E However, the experimentally measured quantity is not P,, but P,Q instead, where Q is the quality of the analyzer for the determination of the linear polarization; more precisely, Q is the ratio of the linear polarization dependent part of the Compton scattering differential cross section to the differential cross section for unpolarized radiation. Bauer et al. do not evaluate Q but follow a more complicated analysis, leading to the conclusion that the 1.41 MeV radiation is a pure quadrupole radiation and that the spin sequence --f 8 3 C which had not been ruled out by previous experiments, is definitely impossible.
I.
+
References p . 391
CH. VIII,
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ORIENTATION OF ATOMIC NUCLEI 11
369
5 6 ~ 0
The results, reported in I, were extended by Diddens et aLg9 to measurements of the anisotropy and linear polarization of gamma rays of 56C0 nuclei, which were polarized in Ce-Mg-nitrate. The mixing ratios? S’(E2/Ml) were determined to have the following values: 0.15, S’(2.02 MeV) M + 10 S’(1.75MeV) M - 0.03, S’(l.81 MeV) M and S‘(2.13 MeV) M - 0.3. It was further found that the 0.845 MeV, 1.24 MeV, 2.61 MeV and 3.25 MeV radiations are pure E2 transitions, whereas the 1.75 MeV radiation is predominantly M1. The largest values of E and P,, obtained at 1/T = 250°K-1 and for the 0.845 MeV radiation, were: E = 0.32 f 0.01 and P,(&z)= - 0.32 f 0.015. The decay scheme and the assignments of spin and parity given by Diddens et aZ. agree with other investigations, except that there is doubt about the 3+ assignment of the 3.84 MeV level.
+
+
~*CO
Precise measurements of the anisotropy of the 0.81 MeV gamma radiation (Fig. 6) emitted from aligned 5 8 C nuclei, ~ were performed in Oxford100 in order to obtain an accurate value of the ratio I MF/MGTl, of Fermi to Gamow Teller matrix elements in the allowed (A1 = 0) 8+-decay of 58C~. Earlier experiments of Griffing and Wheatleyl had been made in biaxial Co-Tutton salt with the intention of measuring , u ( ~ ~ C Ousing ), the gamma ray anisotropy of s°Co as a thermometer. These experiments also yielded a value I C,MF/C,MG, l2 = 0.12 f 0.04, which was, however, incompatible with the later observed apparent absence of interference effects in the 8-asymmetry of polarized 5 8 C ~ nuclei. Dagley et ~ 1 .chose l ~ the uniaxial Ni-fluosilicate as a cooling salt in which 5 8 C was ~ grown. Only one single crystal was used in order t o avoid inhomogeneous temperatures. After a few small corrections were applied, I C,MF/CAMGT = - 0.003 f 0.005 was found, which agrees with the 8-asymmetry experiments. Later experiments of Wheatley et aZ.ll revealed that, although the gamma anisotropy of 6oCoin Co-Rb-tutton salt does fit the theoretical expectations nicely, a combination of %Co and 6oCoactivities does not give agreement with theoretical curves to the accuracy previously
t 6’ = 1/6,where 6 is defined according to I, ref.e1; elsewhere i n this chapter the convention of e.g. Blin-Stoyle and Gracelo is followed, where 6 has the opposite sign of (and is also the inverse of) the 6 of I, ref.*l. References p . 391
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W. J. HUISKAMP AND H . A . TOLHOEK
[CH. VIII,
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stated for I M,/M,, 12, systematic errors being as large as 0.1 - 0.2 and varying with temperature.
'UPr Grace et aZ.lo2oriented 142Pr(19 h) nuclei in Ce-Mg-nitrate, both in zero external field and under influence of polarizing fields of 300 and 600 Oe along the trigonal c-axis. At T = 0.004 OK in a 300 Oe polarizing field, a gamma anisotropy E = 0.14 was obtained, which is appre-
-kt2+
,
1.57
2.15
9f0b
b+.0.091 2.4 nr +M1+E2 6--Q2*02
*LO '46:Prn66
Fig. 12. Decay schemes of
14*Pr and 147Nd.
ciably larger than the gamma anisotropy of nuclei, aligned in zero field by the anisotropic h.f.s. coupling (gL = B = 0 ) : E = + 0.06 at T = 0.004 "K. The positive sign of E is only compatible with spin 2 for the 1.57 MeV state (Fig. 12). No additional anisotropy increase was found when using a 600 Oe magnetic field. From these results it was concluded that the nuclear magnetic moment p = 0.11 pN, assuming that no angular momentum is carried off in the b-transition 2- --f 2+. If the electron antineutrino field contained one unit of angular momentum, the interpretation of the results would have been ,u = 0.17 pN. Similar results were obtained by Daniels et aZ.lo3; although the highest values of E were somewhat lower (both in a 320 Oe polarizing field and in zero field E w 0.04) and correspondingly the temperatures somewhat higher, practically the same values for p were calculated. Daniels uses only the data at relatively high temperatures and calculates p from the experimentally observed relation E = a/T2,where a = 18.7 x lO-' OK2. References p . 391
CH. VIIL,
9 51
371
ORIENTATION OF ATOMIC NUCLEI II
The experimentally observedlo2-l o 3 dependence of E on the magnitude of the polarizing magnetic field shows that crystalline electric field effects are not negligible, but on the other hand, are a factor 10 smaller than expected on basis of paramagnetic resonance data on stable 141Pr. '47Nd
Most of the results were quoted in I, but paramagnetic resonance have since shown the spin of la7Nd (11 d) to be instead of p, and p = 0.56 pN (Fig. 12). Bishop et ~ 1 . 1 0 5 smeasured the anisotropies, E , of the 0.53 MeV and 0.090 MeV gamma radiations as well as the degree of linear polarization in the 8 = +n direction, Pl(+n).At T = 0.04 OK in Nd-ethylsulfate they find E = 0.20, P,(+n)= - 0.136 for the 0.53 MeV radiation and E = - 0.093, P1(+n)= - 0.055 for the 0.090 MeV radiation; a calculation of p from the 0.53 MeV data and 0.090 MeV data separately gives p = 0.44 & 0.06 pN and p = 0.27 & 0.02 pN respectively. From these results it may be concluded that the nuclear orientation is appreciably disturbed during the 2.4 x s lifetime of the 0.090 MeV level; the discrepancy between p = 0.56 ,uNand the p-value from the 0.53 MeV data may possibly be attributed to other causes than disturbance of nuclear orientation.
+
+
"Prn 149Pm(45d)was aligned by Grace et aZ.1°5bboth in the ethylsulfate and in the double nitrate. Preliminary data could be interpreted by assuming the Pmw ground state to be a non-Kramers doublet in the ethylsulfate and a singlet, influenced by a nearby doublet, in the double nitrate146d.The gamma radiation is probably nearly pure M1. 1WTb Johnson et a1.1O6 aligned 160Tb-nuclei (73 d) in Nd-ethyl sulfate (Fig. 11). The angular distributions at low temperatures could be expressed as W(O)= 1 A,P,(cos O), where A , > 0 for the 0.875 MeV radiation, which is mainly a quadrupole transition, A , < 0 for the 0.30, 1.18 and 1.27 MeV radiations, which are probably dipole radiations with Oyo,1% and 16% quadrupole admixture. Recent meas~rementsinLeiden~~7*gave, at temperature T=0.017 OK, ~ ( 0 . 3 MeV) 0 = 0.40, ~ ( 0 . 8 8MeV) = - 0.14, ~ ( 0 . 9 MeV) 6 = + 0.28,
+
+
References p . 391
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W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
+
35
= 0.25 and ~ ( 1 . 2 7MeV) = +0.17. From these anisotro~ ( 1 . 1 MeV) 8 piesit isconcluded that the 0.30 MeV, 1.18 MeVand 1.27 MeV radiations are dipole radiations with O%, 1% and 10% quadrupole admixturerespectively, assuming the 0.96 MeV radiation (whichis a superposition of a 0.964 MeV 2+ + O+ transition and a 0.960 MeV 3+ --f 2+ transition) to be pure quadrupole. Measurements of the linear polarization of the 0.88 MeV 0.96 MeV and of the 1.18 MeV 1.27 MeV radiations yielded P, = - 0.1 and P, = 0.3 respectively. The P , data agree approximately with the F data, but a more significant result is that the 1.18 MeV and 1.27 MeV gamma rays have a predominantly El M2 character and affirm the negative parity for the 1.262 and 1.359MeVlevels. The temperature dependence of the 0.96 MeV gamma ray intensity does not fit the calculations based on paramagnetic resonance data on Tb3f in Y-ethylsulfateI""". An increase of 50% in ~ ( 0 . 9 MeV) 6 was observed when a polarizing field of 200 Oe was applied in the direction of the large g-value (17.7) of Tb3+. The conclusion is that the nuclear orientation data are internally consistent when adopting the spin and parity assignments of Fig. 11, which assignments are also corroborated by other nuclear spectroscopic data.
+
+
+
166mH0
Measurements of the anisotropy and linear polarization of the 0.817 MeV and 0.706 MeV gamma radiations from aligned 166mH~ ( >30 y) nuclei were performed by Postma et ~ 2 1 . l ~ ' ~A. single crystal of Ndethylsulfate was used at temperatures down to 0.025 OK, but the anisotropies were found to be nearly independent of temperature for 0.025' < T* < 0.050 OK and the degree of nuclear alignment must have been nearly maximal (f2/fZmax > 0.95) in that temperature range. From the results it follows that the 0.817 MeV and 0.706 MeV radiations (Fig. 11) are nearly pure E2 and E l radiations respectively, as can be seen from the comparison of ~ ( 0 . 8 1 7 = ) - 0.44 and P,(&c) = + 0.36 resp. of ~ ( 0 . 7 0 6 = ) - 0.56 andP,(&z) = - 0.62. Consequently the 0.973 MeV and 1.79 MeV levels should be characterized as 5- and 6respectively, reducing the possibilities for the spin of 1 6 6 m H to~6 or 7 (see 3 4). The nuclear magnetic moment was estimated as 3.5 f 0 . 5 , ~ ~ . References p . 391
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9 61
373
ORIENTATION O F ATOMIC NUCLEI 11
175Yb The anisotropies E of the 0.282 MeV and 0.396 MeV radiations of 175Yb (4.2 d) were measured by Grace et uZ.lOs to be 0.08 and - 0.03 respectively at the lowest temperatures obtained in ytterbium ethylsulfate (0.014 OK). From these results, in combination with angular correlation data, it is concluded that the 0.396 MeV level has spin and negative parity and that the 0.396 MeV radiation has a mixing ratio 6(E2/M1) = 0.10 0.03 l(Fig. 11). The nuclear moment was then estimated to be p = 0.15 & 0.04 pN, but it remains possible that the 3.4 x s lifetime of the 0.396 MeV state could be long enough so as to reduce the anisotropies; this might increase the value of p. Daniels et ~ 1 . 1 0 3did not find anisotropies of the gamma radiation if the 175Ybnuclei were incorporated in Ce-Mg-nitrate. Apart from the possibility that the Yb-ion did not enter the crystal lattice in a rare earth position, they suggest another interesting possibility. If the 0.396 MeV state belongs to an electronic configuration 01 the Yb-ion and has 3.4 x s lifetime, then the nuclear orientation in Ce-Mg-nitrate may be strongly disturbed. This is due to the fact that g, for Yb* in Ce-Mg-nitrate may be estimated to be roughly g, 3, whereas g, = 0 in the ethylsulfate.
+
+
+
-
*39Np A small alignment of 239Npnuclei was achieved in uranyl rubidium nitrate in zero external field at liquid helium t e m p e r a t ~ r e s lA ~ ~gamma . anisotropy of about 0.005 at 1.5 OK, compared to the normalization point at 7 OK, was observed. This shows that I > 4. 6. Methods of Nuclear Orientation In I a considerable part of the discussion was devoted to the methods of nuclear orientation; here we want to confine ourselves to those methods in which substantial progress was made since 1956 and in which nuclear orientation was observed by means of radiations from radioactive decay. These results are practically confined to alignment in ferromagnetic and antiferromagnetic substances and to dynamic orientation methods which are discussed mainly from an experimmtal point of view in 9 7 and 3 8 respectively. The other methods will be briefly mentioned here in the same order of succession as in I. References p 391
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W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
$7
As to the external field polarization or brute force method no new results have been published14". The Oxford group reported logs nuclear demagnetization experiments which are of considerable interest for the study of solid state physics and we will leave this subject open to a separate discussion*lo. The orientation of nuclei by magnetic h.f.s. alignment and polarization has been extended to many new nuclei as discussed in $ 5, but surprisingly few new paramagnetic salts have been introduced for this purpose. The latter stateof affairsisrelated to thelackof recentmagnetic data below 1 OK; new paramagnetic resonance data may be found inl03b, 1090. Little is to be reported146aon electric h.f.s. alignment; although the method is in principle applicable t o a large number of nuclei, nuclear electric quadrupole moments or the inhomogeneous electric fields are in most cases too small. Dynamic orientation methods based on paramagnetic resonance saturation have been developed considerably and have so many ramifications in other fields such as maser researchlog*that a separate review article would be appropriate. In 5 7 only some specific achievements are discussed in detail. As to the optical method of polarization the reader is referred t o a recent review article, where the steady and considerable progress in this field is reportedloge. A special case of nuclear polarization is found in the various experiments with polarized neutrons, which we mention, although this is outside the realm of low temperature physics. We want to call attention to two interesting subjects: a) The study of beta decay from polarized neutronsz6,which has set an upper limit to time reversal non-invariant terms in eq. (9). b) The polarization of *Li nuclei by means of irradiation of 'Li by polarized neutrons, which proved to have interesting solid state aspects lo9*. 7. Nuclear Orientation in Ferromagnetic and Antiferromagnetic Substances Whereas the orientation of radioactive nuclei at low temperatures was originally considered to be primarily of interest for studying nuclear properties, it was soon realized that for some nuclei the decay characteristics are known to a much higher precision than the process References p. 391
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ORIENTATION OF ATOMIC NUCLEI I1
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of nuclear orientation itself. Conversely, therefore, the measured gamma ray anisotropy may provide valuable information regarding the solid state environment of the decaying nucleus. We will discuss some specific examples.
FERROMAGNETIC METALS A. Co-metal. As reported in I, Kurti et al.lll*112aligned 6oConuclei in hexagonal Co-metal single crystals, cooled by heat contact with a Cr-alum-glycerol slurry. Ce-Mg-nitrate powder, having a low specific heat, was used as a thermometer by connecting it thermally with the Co-crystal and measuringitssusceptibility. At 1 / T M 30 OK-l thegamma ray anisotropy was E = 0.07; E could be expressed by 6 = a/T2, where a = 0.79 x lP4OK2, leading to an effective field at the position of the Co-nuclei of 193 & 20 kOe. This may be compared to specific heat measurements by Heer et aZ.ll3 and Arp et al.ll& which gave 183 kOe and 200 kOe resp. These results, however, disagree with measurements of Khutsishvilil, who gave E = a / T 2with u = 5 x OK2. In this experiment the nuclei were polarized by a large external magnetic field, applied in the direction of the hexagonal crystal axis, which is also the direction of easy and spontaneous magnetization and of nuclear alignment. Similar experiments in zero field by Daniels OK2, but when the metal crystal was et ~ 1 . yielded ~ ~ 5 a=3 x heated a few minutes to red heat and the experiments repeated, a O K 2 was found. Since Co changes from the value a = 1.5 x hexagonal to the cubic phase above 417 "C, the heat treatment may increase the proportion of cubic regions in the Co-crystal. The degree of nuclear alignment may in the cubic phase be appreciably smaller than in the hexagonal phase, which could explain the various data. However, recent lZ1nuclear resonance measurements of 59C0 in cubic Co metal have yielded a value of 217.5 kOe (extrapolated to 0 OK), which is equal within experimental error to the value of 219 kOe for hexagonal Co, obtained in recent specific heat measurements by Arp et aZ.114b. The large U-values obtained by Khutsishvili and Daniels remain unexplained and, if not due to lack in accuracy in the temperature determination, present an unsolved problem. In this connection calculations of the effective field by Marshall 116should be mentioned as well as recent m e a s u r e m e n t ~ ~of~ 'the effective field in iron metal, which proved to be opposite in direction to what was expected. References
p. 391
376
$7 Still more recently, many new results have been published, which are related to this subject146kJ7m. B. Samoilov et uZ.l18 in Moscow activated a (50% Co, 50% Fe) permendur alloy in a pile. The specimen, containing 3-4 pc 60C0, was soldered to copper, which was then cooled to low temperatures by heat contact with pressed Cr-alum. A small permanent magnet was mounted in the cryostat, such that the alloy was placed between the poles in a field of 1000 Oe. The gamma ray anisotropy was found to be E = 1.2 x 10-4/T2which results in an effective field of 250 kOef14b. C. A 0.1 mm thick disk of a gold-iron alloy, containing 0.3% Au, was irradiated by thermal neutrons, producing 4 pc 198Au. Samoilov et aZ.l19 cooled this sample to a temperature of about 0.015 "K in the same way as mentioned under B. From the magnitude of E and the known value of the nuclear magnetic moment p = 0.5 0.04 pN, the value of the effective magnetic field at the Au-nucleus was calculated to be 600 kOe. I). Samoilov et ~ 1 . also l ~ obtained ~ ~ nuclear polarization of lzzSb and 114mInnuclei, inserted as impurities in Sb-Fe-alloy (0.6% Sb) and In-Fe-alloy (0.5yo In) respectively. Magnetization of the sample was obtained by an electromagnet in the helium dewar, capable of producing a 2000 Oe field. The 0.56 MeV quadrupole radiation (2+ + O+) of lz2Sbshowed an anisotropy E = 0.025 at T = 0.03 OK, from which an effective field of about 190 kOe was derived. Similarly, the anisotropyof the 0.192 MeVgammarays of 114mInwas E = 0.08at T = 0.04 O K , leading to an effective field of 150 kOe if the nuclear magnetic moment is 4.7 pN. So far, anisotropies of gamma radiations of such high multipole order (E4) had not been observed. More recent results1Z0b gave lower limits to the effective fields in Fe at the position of Sb, In and Au nuclei of 280 kOe, 250 kOe and 1000 kOe respectively. The discrepancies with the former results were due to incorrect temperature estimates ; this again illustrates the need for accurate thermometers in indirect cooling experiments below 0.1 OK. E. Kogan et aZ.120C in Leningrad polarized 46Scnuclei solved in iron metal. The concentration of Sc was smaller than 0.6% which was found to be somewhat below the maximum solubility of Sc in Fe metal. Part of the experimental arrangement is shown in Fig. 13. The iron sample is magnetized by the residual field of a superconducting Nb-cylinder (900 Oe), which field was concentrated at the position of the sample by armco iron wedges up to a value of about 1700 Oe. The References p . 391
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[CH. VIII,
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VIII,
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ORIENTATION OF ATOMIC NUCLEI I1
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sample was externally cooled to T w 0.04 OK and a gamma anisotropy E = 0.01 was obtained. This anisotropy is insufficient to make a beta gamma correlation experiment feasible. A j3-y-directional correlation experiment for polarized 4 % nuclei ~ might give very interesting results because of the reported presence of considerable interference between Fermi and Gamow-Teller matrix elements in the 46Scbeta decay and consequently, the experiment might serve as a test for the validity of time reversal invariance in beta decay4915 0 , 51. With the same apparatus but 60Coinstead of 46Sc,a gamma aniso-
CONNECTED WITH MIXTURE OF Cr-ALUM AND FQOPANOL
Fig. 13. Central part of apparatus of Kogan et a2.lz0C for polarizing iron samples which contain diamagnetic impurities. The polarizing field is the residual field of a superconducting Nb-cylinder, concentrated a t the position of the sample by armco iron. The Cr-alum serves as a guard against heat conduction t o the sample; the sample is cooled by a Cr-alum-propanol mixture and the heat contact is provided by a metal rod.
tropy E = 0.2 was obtained, from which an internal field of 300 to 400 kOe at the position of the 6oConucleus was derived (Co concentration of 0.02%) 114b. The results quoted under A-E are of considerable interest for the theory of ferromagnetism in metals particularly as to the question whether the effective field predominantly originates from the influence of conduction electrons, 3d electrons, s electrons from inner shells, or from the interplay between these various electrons. See als0146g. *, 1. ANTIFERROMAGNETIC SALTS It was suggested by Daunt122& and Gorter122bthat it might be possible to align nuclear spins in antiferromagnetic single crystals. Proton References p . 391
VIII, 3 7 resonance experiments by Poulis et aZ.12shad shown that at low temperatures the magnetization of either sublattice remains constant during times ( l W 4 s) which are much longer than the proton precession time M lo-' s). Classically speaking, therefore, the magnetizations of the two sublattices do not interchange frequently enough so as to give a zero average field at the proton position during a single precession. Since the nuclear precession frequency caused by h.f.s. interaction inside a paramagnetic ion is in many cases one or two orders of magnitude higher, the above conclusion would therefore also apply to these nuclei. Antiferromagnetic alignment was first realized by Daniels et aZ.124912s for 64Mnnuclei in MnCl,, 4H,O and MnBr,, 4H,O, which have Nee1 temperatures, T,, of 1.6 OK and 2.2 "K respectively. Cooling of such salts to temperatures below 0.1 "K, as required t o obtain sufficient nuclear alignment, cannot be realized by direct adiabatic demagnetization starting from 1 OK.At 1 "K these salts still have a considerable specific heat and indirect cooling requires a large amount of heat transfer to a coolant paramagnetic salt. Daniels et al. used K-Cr-alum and obtained temperatures of 0.1 "K, in the MnC1, after about half an hour and in MnBr, after several hours. The gamma anisotropy increased with the above time constants to maximal values of respectively E = 0.07 and E = 0.06. Possibly the spin-lattice relaxation time is the dominant factor in the long time needed for cooling. The results also showed that the axis of nuclear alignment coincides with the axis of preference for the electronic angular momenta. No anisotropies of g°Co radiations were found in these salts at 0.1 "K. Miedema et ~ 1 . ~ 2aligned 6 both s°Co and UMn in the Co-NH,-tutton as well as in the Mn-NH,-tutton salt, These salts have transition temperatures of 0.084 "K and 0.14 OK respectively. When starting demagnetizations from 1 OK, final temperatures somewhat below T , can be reached; to obtain temperatures lower than +TN,however, indirect cooling again is needed. Miedema et aZ.laeused supercooled solutions of Paramagnetic salts in propyl alcohol as cooling agents ; in order to increase the heat capacity potassium chrome alum crystals were embedded in the solution as well. A large number of copper wires were assembled in the solution, providing a large contact area with the coolant. The radioactive crystal was glued onto a copper plate, soldered to the wires. 378
References p . 391
W.
J . HUISKAMP AND H. A. TOLHOEK
[CH.
CH, V I I I ,
3 71
379
ORIENTATION O F ATOMIC NUCLEI I1
The gamma anisotropies measured in the paramagnetic region are consistent with nuclear alignment along the two tetragonal axes of the tutton salts, as found earlier1 in the diluted salts. This is indicated in Fig. 14 by 8, m 0 for both 54Mn and 6oCoin Co-NH4-tutton salt above T m 0.08 OK. At lower temperatures there is a change in counting rates, which is interpreted as a rotation of the nuclear spins of 8, w 5" and 8, w 7" for 6OCo and 54Mn respectively towards the bisector, K,, of the tetragonal axes. Susceptibility measurements have shown 127 that below T , the susceptibility strongly decreases along the K,-axis, but increases to 1c
C
eNl -.
I )
0.06
T
j"
OK
I 0.12
Fig. 14. Preferred direction of 64Mn and e°Co nuclear spins in Co-NH,-tutton salt single crystals as a function of temperature. ON represents the angle (in degrees) between the preferred direction of nuclear alignment and the direction of the tetragonal axes, ON being zero in the paramagnetic region. 0: 64Mn A : E°Co.
high values in the K3 direction (K, 1K,). This behaviour is fairly well explained by assuming antiparallel alignment along staggering axes, somewhere between the two tetragonal axes and their bisector, K,. Whereas below T , apparently both electronic and nuclear spins tend towards the K,-axis, quantitative agreement of the nuclear alignment data with a molecular field model could not be obtained. Daniels et ~ 1 . 1 2 5find almost no change in the results for 6oCo in CoNH,-Tutton salt when decreasing T beyond TN. In CoC1, they find an axis of nuclear alignment, which differs from the axis of preference for the electronic angular momenta; here the anisotropies of 54Mnand 60Coare very different: E = 0.18 and E = 0.015 respectively, at T = 0.055 "K ( T , = 3°K). In MnSiF,, 6H,O, for which T , = 0.1 OK, Daniels et aE.125 report References p . 391
380
W.
J . HUISKAMP AND H . A. TOLHOEK
[CH. VIII,
58
large gamma ray anisotropies for both 54Mn ( E = 0.20) and s°Co (8 = 0.19) at about 0.05 O K , which cannot be accounted for by paramagnetic resonance data. Reviewing the progress made since the situation was briefly discussed in I, it may be stated that the feasibility of nuclear alignment in antiferromagnetic salts has been clearly demonstrated. This was also shown to be the case for Co-NH,-tutton salt, for which the previously reported1 negative result was possibly due to the small magnitude of the anisotropy E (which is of the order of 0.01). On the other hand the mechanism of nuclear alignment in the antiferromagnetic state remains a t present largely unknown and probably is fairly complicated and subject to considerable variation with the individual salts. The study of the problem is complicated by the circumstance that strongly antiferromagnetic crystals are difficult to cool as a whole and their nuclear spin system in particular, whereas in feebly paramagnetic substances, for example crystalline fields complicate the situation. It is therefore not possible to conclude definitely whether the antiferromagnetic interaction enhances or, oppositely, decreases the degree of nuclear alignment. Similarly, it is not clear whether the nuclei align along the direction of magnetization of the sublattices or whether they prefer other directions. Probably the differences between the various salts are too great to allow such general statements. 8. Dynamic Methods of Nuclear Orientation It was shown by Jeffries128* 129 that saturation of transitions characterized by A (S, I,) = 0 in paramagnetic resonance spectra can produce appreciably larger nuclear polarization than by saturation of the ordinary A S , = & 1, dI, = 0 transitions. Many of the problems involved had already previously been discussed by Abragani 130; here we follow the discussion of Jeffries131. Consider the system of a paramagnetic ion and a nuclear magnetic moment in an external magnetic field, H,, described by the spinHamiltonian :
+
1,) =
where .!$ &+
=
E-
=
Ell
I s,,I , > + E+ I s, + 1, I , - 1 > +
+ + E! E:
+
=
1,
E,,
E-
[ S,
-
1, I ,
+1>
(30)
w1
+ s, + 1) (1+ 1,)( I I , + 1)B/2g,,BH, (31) 4 s + S,) (S - s, + 1) (1- 1,) ( I + I , + 1)B/2g,,BH,. (32) - d ( S - S,) (S
-
Next it may be shown that introduction of a radiofrequency (r.f.) field H , along the z-axis, considered as a time dependent perturbation 8 1 . 1 . = g1,BHrzSz cos mt induces transitions between the states y(S,, I,) and
y(S,
+ 1 , I , - 1) or y ( S , - 1, I, + 1) g:lB2H;z
since for instance
< Y(S2 + 1, I , - 1) I s, I Y(SP 13
w = ( S - S,) ( S s, 1) ( I 1,) (1- I , 1)B2H&/H2,. This result may be compared to the ordinary transitions
+ +
+
+
>2
(33)
Y(S,, 1,)+ d S 2 i 1, I,), which are induced by r.f. components perpendicular to H,, i.e. Z1.f.
W&B2
-=E Y(S,
+ S-Hrz)
= Sg#(S+Hrx
+ 1>I,)I s+1 d S , , 1,) =
(S
- S,)
>2
(S
(34)
=
+ s, +I) Hk34H;gfl.
(35)
The ratio of transition probabilities of the two types of transitions is, apart from a factor containing I and I,,
which may be orders of magnitude smaller than 1 and consequently, the A(S, I,) = 0 transitions are designated as forbidden transitions. It should be noted that in a microwave cavity the ratio H,,/H,,, averaged over the sample, is neither zero nor infinite. Similar results are obtained if the nuclear spin and electron spin do not belong to the same paramagnetic ion; such cases arise in semiconductors, where a scalar type of coupling AS * / may exist. Also the coupling between S and I may further be anisotropic h.f.s. coupling,
+
References p . 391
382
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
5
8
magnetic dipole-dipole coupling or interaction of the nuclear quadrupole moment. In the latter cases one may have other forbidden transitions like A S , = 0, 4 1, dl, = i 2 etc., which simultaneously provide relaxation mechanisms and may make the attainment of the steady state saturation very complicated. Whereas in the Overhauser method of nuclear p ~ l a r i z a t i o none l ~ ~saturates the A S , = f 1,A I , = (4 -2
1 -1 0 a a a Q T I - ,
_
I
I
I
I
I
L
L
1
-
-
I I
L
-
-
1
-2
-1
,
I
I
-
-
1
1
0
1
2
a I
I
I
L
2
I
I
1
I
I I
,
I I
I
, I
I
--I-IIII
1
I N I T I A L STATE
INITIAL STATE
I
I
I
I
I
I
L-2.-
'
--L
9
1
1 1 IMMEDIATELYAFTER APPLICATION OF RF FIELD
I
1
&
%
FINAL EClUILIBRIUM
A
1
IMMEDIATELY AFTER APPLICATION OF RF FIELD
' STATE
SATURATION OF FORBIDDEN TWNSITIONS
FINAL EOUILBRIUM STATE
B
OVERHAUSER EFFECT
Fig. 15. Diagrams showing the production of nuclear alignment by saturation of forbidden transitions (A) and by the Overhauser effect (B). The drawn arrows represent I,) = 0 relaxation the action of the microwave field, the dotted arrows indicate A ( S , transitions and the dotted lines are the AS, = - 1 relaxations. I n A relaxation by AS, = - 1 only is assumed, in B by A S , = - 1 and d(S, + I,) = 0 only. Since the h.f.s. energy is neglected in the figure, the energy difference between the upper (S, = 4)and lower (S, = - 4) levels is gll/?Hs.Further, a = exp (-gll/?H,/kT) and therefore, as in most dynamic methods, the degree of nuclear polarization does not, i n first approximation, depend on the h.f.s. energy splittings.
+
+
transitions and expects the d(S, + I,) = 0 relaxation to produce nuclear polarization, saturation of forbidden d(S, I,) = 0 transitions gives nuclear polarization directly. A schematic comparison between the two methods is given in Fig. 15 for S = 4 and I = 2; in A the I, = 0 level is filled at the expense of the I , = 1level by the action of the microwave field and subsequently the degree of nuclear polarization is increased by the establishment of thermal equilibrium by A S , = - 1 relaxation; in B there is no nuclear polarization immedi-
+
References 9.391
CH. VIII, §
81
ORIENTATION OF ATOMIC NUCLEI I1
383
ately after application of the microwave field. Other relaxation phenomena, e.g. 01, f 1 may appreciably reduce the nuclear polarization in both A and B. A more detailed analysis shows that such a reduction occurs in A if d I , = & 1competes with A S , = -I, in B if d1,= i.1 competes with d(S, 13 = 0, respectively. Since AS, = - 1 relaxation prevails in many substances, the Overhauser effect may be more difficult to achieve than the saturation of forbidden transitions. Further it can be shown that in cases where the nuclear polarization vanishes (fi = 0) by relaxation mechanisms, nevertheless nuclear alignment (f2)may be appreciable if tc (Fig. 15) is considerably smaller than 1, as is usually the case for HZ w lo4 Oe and T m 1 OK. It is to be noted that the sign of the nuclear polarization obtained by saturation of forbidden lines is opposite to that in the Overhauser method, which makes it experimentally possible to distinguish between the two phenomena, e.g. in cases where the forbidden lines are not resolved from the ordinary lines or if there is only one broad line. 5
+
RESULTS A. Abraham et ~ 1 . lin~Berkeley ~ produced a 10% polarization of 6oCo nuclei in deuterated La-Mg-nitrate with the isotopic abundance ratio Mg : 59C0: W o = lo4 : 60 : 1 and with a 2 mc 6oCoactivity. The 9400 MHz microwave field of approximately 0.1 Oe was applied along the external field of roughly 1500 Oe in the direction of the c-axis of La-Mg-nitrate. Two counters were located closely to the source and under angles 0 and in with [respect to the magnetic field; the difference between the two integrated photomultiplier outputs was directly recorded as a function of the magnetic field, which was swept over the59960 Co++ resonance. The result may be seen in Fig. 16. The magnitude of the observed anisotropy, E m 0.01, was 40% of what could be expected if neither d(S, + I,) = f 2 nor d1, = f 1 relaxation phenomena were active. B. Similar results132 were obtained for 52Mn and 54Mn,for which both I and p could be measured: for 52MnI = 6, p = (3.00 i 0.15) pN and for UMn I = 3, p = (3.29 & 0.06) pN. The number of overlapping paramagnetic resonance lines is very large for Mn, which has S = Q and in particular for 52Mn with I = 6. Therefore, it was impossible here simply to count the lines in the gamma anisotropy as a function of the magnetic field and a more indirect approach was followed. References 9. 391
w.
384
J , HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
38
Fig. 16. Dynamic polarization of SOCo in La-Mg-nitrate. The upper curve represents the paramagnetic resonance spectrum of 6@Co;the calculated peak positions, fitted to a certain extent to the experimental data, are indicated by vertical lines. From the W o resonance peaks the positions of allowed and forbidden resonance peaks of B°Co were calculated and are given in the lower figure by drawn and dashed lines respectively. Also shown is the observed gamma ray anisotropy, E ; the expected values of the anisotropy for saturation of allowed and forbidden lines are indicated by the height of the vertical lines.
Fig. 17. Decay schemes of 7OAs and laaSb.Some weak beta and gamma decay branches have been omitted from the figure. References
p . 391
CH. VIII,
3 81
ORIENTATION OF ATOMIC NUCLEI 11
385
C. Pipkin et ~ 1 . lS5 ~ used ~ 9 the saturation of forbidden transitions for obtaining nuclear polarization in 27 h 76As, incorporated as a donor impurity in Si crystals. The source was obtained by doping a few times l0ls As atoms in about 1 cm3 Si and subsequently exposing the crystal to a flux of l O l B neutrons/cm2 s. The sample was mounted inside a microwave cavity, immersed in liquid helium at 1.25 OK; simultaneous with the saturation of forbidden transitions by sweeping a magnetic field at about 8500 Oe, the 0.56 MeV quadrupole radiation was measured in a direction perpendicular to the magnetic field. To simplify the discussion, we assume that the first forbidden /I-decay preceding the most intense 0.56 MeV gamma radiation does not carry off angular momentum and is characterized by 2-(/?-) 2+, J = 0. Then the /?-decay does not change the populations of the h.f.s. levels and as a result the gamma ray intensity in the perpendicular direction is solely determined by the I , = 0 h.f.s. level population, since the 01,= 1 (I,= f 1 3 I, = 0) and A I , = f 2 ( I , = = rf 2 -+ I, = 0 ) gamma transitions have equal intensities in the &n direction. The A I , = 0 ( I , = 0 -+ I , = 0) gamma transition has zero intensity in the &n direction. The experimentally observed change in counting rate was about 2% at approximately 8225 and 8260 Oe (Fig. 18a, b), which must have been due to (S, = - +, I , = 0)-+ (S, = + Q, I , = - 1) and (-4, 1) + (++,0) transitions. This is more clearly demonstrated in Fig. 19; the sudden increase in counting rate when the microwave field is turned on corresponds to the saturation of forbidden transitions (Fig. 15A) and not to the Overhauser effect (Fig. 15B). Sweeping through the high field resonance resulted in a gamma ray intensity decrease, and therefore was caused by filling the I , = 0 level by the transition (-&, 1) 3 (+&, 0). From the fact that this transition is the high field resonance, one easily finds that the (-4, 1) level has a higher energy than the (-+, 0) level, and consequently S and I are preferably parallel, which leads to a negative sign for the h.f.s. constant A and a negative sign for the nuclear magnetic moment. The accuracy for A is determined by the precision with which one can measure the differencein magnetic fields in Fig. 18a. Double resonance experiments were performed, in which r.f. power was applied to the sample along with microwave power. A special cavity consisting partly of ,thinly silvered lucite was required for admitting the r.f. fields, generated by a variable r.f. oscillator. One of the References p . 391
386
W. J. HUISKAMP AND H. A. TOLHOEK
T
8300
'?
Q
8260
8220
[CH. VIII,
38
a80
MKNETIC FIELD IN OERSTEDS
v
5903
t
C 56 52 50 46 LOW FREOUENCY IN MHz
43
Fig. 18. Gamma ray intensity in a direction perpendicular to the magnetic field versus the magnitude of the magnetic field, showing the production of nuclear alignment of "As when the field is swept through forbidden resonances in the electronic paramagnetic resonance spectrum. The field sweep is from high t o low field in the upper curve (a), and conversely in the middle curve (b). During the time between the low field and the high field resonance, the 1, = 0 level is relatively densely populated in (a)while it is scarcely populated in (b). The lower curve (c) shows a t the left the saturation of a forbidden electronic resonance (-4, I)+($, O),thereby filling the 1, = 0 level, which is thereafter emptied through AT, = 1 and A I , = - 1 nuclear resonances consecutively.
experimental procedures is the following (Fig. 18c). After saturation of a forbidden transition d(S, + I,) = 0 at the appropriate value of the magnetic field, the r.f. power is swept in the region of 50 MHz over a A I z = 3 1 transition and at resonance the gamma anisotropy is destroyed (Fig. 18c). It is seen that there are, in fact, two resonances: (-4,0) 3 (-3, 1) and (-+, 0) + (-Q, -1). In more refined experiments the resonance frequencies could be determined with an accuracy of about 0.1 MHz giving A(76As)= - 93.66 f 0.06 MHz. If the preceding B-transition, which has an allowed shape, does carry away angular momentum, J , the analysis is more complicated. ConReferences #. 391
CH. VIII,
5 81
387
ORIENTATION O F ATOMIC NUCLEI I1
versely, however, the results can give information about J . Double resonance measurements in conjunction with measurements of the gamma ray intensity i n the direction of the magnetic field showed that the ,$-decay is described by a mixture of 50% J = 0, 20% J = 1 and 30% J = 2 matrix elements. The nuclear orientation was shown to persist for 2-3 hours, which agrees with relaxation time measurements in which the A (S , I,) = 0 transitions were found to have a relaxation time longer than 75 min, even though the AS, = f 1 transitions have a relaxation time of 4 minutes.
+
640
1
I
I
I
I
t
TIME IN MINUTES
Fig. 19. Gamma ray intensity of 7'IAs with the magnitude of the magnetic field fixed at a forbidden electronic resonance. After some counts without microwave field, a prompt change in counting rate is observed when the microwave field is switched on; the slower increase thereafter is due to AS, = & 1 relaxation, corresponding t o the attainment of the final equilibrium state in Fig. 15A.
Double resonance experiments on 75As, combined with the known nuclear magnetic moment of 75As, provided the value , u ( ~ ~ A = s) - - 0.903 f 0.005 pN, the error being mainly due to the unknown magnitude of the h.f.s. anomaly. D. Similar e ~ p e r i m e n t s were l ~ ~ performed with lz2Sbin Si; the important difference was that A ( S , I,) = 0 transitions had a much shorter relaxation time for Sb, one of the possible reasons being the higher concentration of l22Sb compared with 76As.The relevant part of the decay scheme is basically the same as in 76As:2- (8) 2+ (E 2) O+ where the p-transition has allowed shape though a non-isotropic betagamma directional correlation was observed. The lzzSb-experimentprovided the interesting situation that nuclear
+
References
p . 391
388
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
98
polarization could be obtained both by saturation of forbidden transitions as well as by the Overhauser effect i.e. saturating a A S , = 1, dI, = 0 transition and waiting for the d(S, I,) = 0 relaxation t o occur. The observed counting rate as a function of time was found to be different in the two cases: in the Overhauser effect a constant counting
+
+
36501
3490
I
[
I
8.45
MAGNETIC
FIELD
I
8.47
I
I
1
8.49
I
8.51
I
I
8.53
IN KILOOERSTED
Fig. 20. Saturation of two forbidden lines in the electronic resonance spectrum of Sb impurities in Si and corresponding changes in counting rate of lp2Sbgamma radiation. During the time between the two resonances the degree of nuclear alignment is practically reduced to zero due to relaxation phenomena.
rate is approached with a time constant of about 10 min. whereas the counting rate changes instantaneously when saturating a forbidden line. Fig. 20 shows the gamma ray intensity as a function of the magnetic field which is swept over the forbidden resonances. It is seen that the nuclear polarization decreases strongly between two resonances, which may be due to a combination of relaxation phenomena. Since no more resonances were found, it could be concluded that the nuclear spin of lzzSb equals 2. By double resonance again the h.f.s. splitting constant, A , was determined as A(lZ2Sb)= - 132.59 f 0.10 MHz from which it follows, with the aid of known A and p for lzlSb, that p('"Sb) = - 1.904 f 0.020 pN. E. Dynamic polarization of non-radioactive 29Si nuclei was ob, 139 at Saclay by saturation of the electained by Abraham et U Z . ' ~ ~ 138, tron resonance in P-doped Si, which was somewhat too broad for showing a 29Si h.f. structure. Microwave power was used for saturation and r.f. fields for measuring the intensity of the 2 5 resonance; the Si-polarization was shown t o be enhanced by a factor of about 100 (fl M 0.001) after saturation of the electron resonance and the sign of the polarization indicated that forbidden transitions A S , = 1, A I , = & 1 were responsible for the 2% polarization and not the Overhauser effect. Refer em es p 931
CH. VIII,
9 81
ORIENTATION OF ATOMIC NUCLEI I1
389
POLARIZED PROTON SAMPLES An important extension of this method was indicated by Borghini and Abragam 140, who obtained polarization of protons by saturation of electronic resonance in the free radical DPPH (diphenyl picryl hydrazyl). Here the electronic spin is coupled to a large number ( m lo2) of proton spins presumably by a dipolar coupling although scalar coupling may simultaneously be present. Whereas in the latter case the forbidden transitions are d(S, + I,) = 0, in the dipolar case more forbidden lines arise like d(S, I,) = f 2 , so that for nuclei with spin 4 (proton, 29Si)more than one forbidden line exists. The sample consisted of 10 mm3 proton rich polystyrene which contained 10% DPPH. 60 mW of 36000 MHz microwave power was applied to the sample at a magnetic field strength of 12000 Oe, whereas the proton resonance was observed with a r.f. spectrometer. It was found that the proton resonance peak became 50 times more intense when saturating the free electron resonance (fl m 0.02). Similar results were obtained by Uebersfeld et aZ.141and by Abraham et ~ 1 . l ~ ~ . It is tempting to speculate on the various applications in nuclear physics, which may be devised with the use of polarized protons in samples which are rich in protons and when degrees of polarization fl of the order of 20% could be reached. One could expect as well many applications in low energy nuclear physics as in high energy physics. A technical difficulty of applications to reactions or scattering with charged particles of energies of a few MeV is that the polarized sample should be a very thin layer. At energies of a few hundred MeV the situation is easier as the samples may be thicker. The beam intensity with charged particles must of course be rather limited because the dissipated heat should not warm up the sample too much. The experiments, which may become feasible in this way, concern polarization effects in proton-proton, proton-neutron, proton-nucleus or protonelectron collisions at lower and higher energies. J e f f r i e ~ let~ ~at. used hydrated paramagnetic crystals like tutton salts and La-Mg-nitrate to obtain proton polarization; at 1.7 OK an enhancement factor of about 20 was observed in the proton resonance intensity compared to non-saturation of forbidden lines AS, = 1, d I z = f 1. This is an order of magnitude less than predicted. More recently144 a proton polarization fl = 0.19 was reported.
+
References Q . 391
390
W. J. HUISKAMP AND H. A. TOLHOEK
[CH. VIII,
99
9. Concluding Remarks
We may conclude with a number of remarks concerning the present position of nuclear orientation at low temperatures (stressing the new points which have arisen since I, tj 4): (1) The discovery of non-conservation of parity (and non-invariance for charge conjugation) in @-radioactivity has been of fundamental importance. As to the fundamental aspects of @-interaction further experiments of importance could be tests of the invariance for time reversal (cf. 5 2). (2) The preceding discovery provides a new tool, useful for nuclear spectroscopy : the asymmetries of ,&rays from polarized nuclei can provide information such as the change of nuclear spin in p-decay and relative magnitude of nuclear matrix elements in /I-decay. (3) Thc “brute force” method of nuclear polarization (which could be applied to all nuclei with a not too small magnetic moment) might become more fcasible because of the technical developments concerning indirect cooling and strong magnetic fields. Further the asymmetries of @-raydistributions should enable one to detect degrees of polarization fi of a few percent only. (4) It has been shown that large internal fields exist at the position of nuclei of diamagnetic atoms when incorporated in iron metal; this has provided a possibility of polarizing Sb, In, Au and Sc nuclei. It may be expected that this method of nuclear polarization in ferromagnets can be extended to more nuclei; progress in this respect is largely determined by problems of metallurgy and radiochemistry. For small concentrations of diamagnetic elements, when other methods for probing the internal magnetic fields like specific heat measurements or nuclear resonance do not seem promising, polarization of radioactive nuclei may be particularly useful. (5) Nuclear alignment was found in some antiferromagnetic single crystals but the relation between the degree and the preferred direction of nuclear alignment on the one hand and the magnetic properties of the crystals below the Nee1 temperature on the other hand, remains largely unsolved. (6) Dynamic methods have produced polarization of some radioactive nuclei to such a degree that the anisotropies of gamma radiations could be measured. The combination of magnetic resonance measurements, both a t microwave and radio frequencies, with gamma References Q. 391
CH. VIII]
ORIENTATION OF ATOMIC NUCLEI I1
391
anisotropy measurements have provided accurate values for h.f.s. splittings and nuclear magnetic moments respectively. (7) The development of certain dynamic methods of orientation may provide samples with, e.g. protons (or deuterons) with a substantial degree of polarization (e.g. fi = 0.10 to 0.20) in a near future. Such samples could be very useful as targets for nuclear reactions (or scattering processes) as well in low energy nuclear physics as in high energy physics (cf. 5 8). REFERENCES
I. THEORY M. J. Steenland and H. A. Tolhoek, Progr. in Low Temp. Phys., Vol. 2, ed. by C. J. Gorter (North-Holl. Publ. Co., Amsterdam, 1957) p. 292. A. R. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, 1957). M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957). U. Fano and G. Rscah, Irreducible tensorial sets (Acad. Press, New York, 1959). L. Roscnfeld, Lectures on Oriented Nuclei, Nordita, Copenhagen (1959) (mimeographed) R. J. Blin-Stoyle, M. A. Grace and H. Halban, Progr. Nucl. Phys. 3, 63 (1953). G. R. Khutsishvilt, Orientation of nuclei, Usp. Fiz. Nauk 53, 381 (1954) (in Russian). R. J. Blin-Stoyle, M. A. Grace and H. Halban, Beta- and Gamma-Ray Spectroscopy, ed. by I n3 = n, maser action will occur at the frequency
If n ,
< n3 = n, maser action will occur at the frequency v3, = h-l(E3 - E J .
Only if n1 = n, = n, no maser action will occur. The steady state population n2 in the presence of the pump power depends of course on the details of the relaxation mechanism between the various levels. This will be discussed more fully in the next section. The general nature of the argument should be emphasized. No assumptions have been made about the frequency separations or the nature of the transitions. It is only required that a strong perturbation is applied which only connects a pair of non-adjacent states. There may be more than one intermediate level. Often the transitions at v I 2and vZ3will be sufficiently separated in frequency from ~ 1 so 3 that the perturbation by the pump has a very small oscillatory character at the other transitions. Selective atomic collisions in a gaseous discharge may also fulfill the pumping function. In the case of optical pumping the frequencies Y,, and vI3may be very close. The separation References p. 427
400
N. BLOEMBERGEN
[CH, IX,
92
of the transitions can sometimes be achieved by the sense of polarization. The polarization or angular momentum pump was first proposed by Kastler 11, although not in connection with masers. The Overhauser effect12 is another example of the pump method. Bassov and Prokhorov13 proposed a pumping scheme for atoms in a beam, while the present author l4 suggested the three level solid state maser. This device which utilizes a paramagnetic material at liquid helium temperature serves as a very low noise microwave amplifier. It will be discussed in more detail in the remainder of this paper. The spontaneous emission which was mentioned at the beginning of this chapter should not be forgotten. I t means that some electromagnetic power w ill be emitted even in the absence of the incident signal. It can be shown that this corresponds precisely to thermal noise of the amplifiers. Its smallness is the very “raison d’etre” of the solid state maser. It will be discussed in more detail in section 5 . In a final section optical pumping in solids at low temperature and some other devices are mentioned which are closely related to masers. A device based on stimulated emission of radiation is defined as maser. Whether the device operates in the microwave region or in another part of the electromagnetic spectrum is less important. Nomenclature such as laser, iraser etc. to designate operation in the visible or infrared region will not be adopted. The term molecular amplifier or generator is also frequently used, especially in the non-english literature, but is less specific. There is a good recent review’ and an introduction in book form on masers 16. They give rather complete references to the literature until early 1959. The most recent contributions may be found in a conference report 16. Many references to recent work also appear at the end of this paper, although no effort has been made to achieve completeness.
2. Paramagnetic Resonance in Maser Materials The splitting of the spectroscopic ground state of paramagnetic ions in crystals can be described by a spin hamiltonian. Excellent reviews of the wealth of theoretical and experimental data have appeared ”. I t should be kept in mind that new data on spin hamiltonians are continually added, especially of ions as impurities in other host lattices and of splittings in the millimeter wave region. Consider the simple spin hamiltonian 2 = gs g * H, 0s; E(S: - s;) (4)
+
Refeyences p. 427
+
CH. IX,
5
21
SOLID STATE MASERS
401
which is adequate if the effective spin S < 2 and nuclear hyperfine interactions are absent. The first term represents the Zeeman energy, the second and third term the crystalline field splitting of the spin levels, E = 0 if a symmetry axis is present, D = 0 in cubic symmetry. The coordinate system has been chosen in such a fashion to diagonalize the crystal field tensor at the position of the ion. Often there are several crystallographically non-equivalent ions in the unit cell. For higher spin values fourth order and sixth order polynomials in the spin components should be added. Nuclear spin interactions present only an undesirable complication from the point of view of maser operation. At the microwave frequencies of interest the nuclear spin is effectively decoupled from the electron spin. Since simultaneous changes in electron and nuclear spin quantization are forbidden in first order, the effect of a nuclear spin I is to increase the number of nonequivalent ions by a factor 21 + 1. Diagonalization of the spin hamiltonian gives 2.5 + 1 energy levels. Transitions between a pair of these energy levels can be induced by a time varying electromagnetic field. A microwave field H,, exp (2nivt) near the resonant frequency k l ( E j - Ei)will induce transitions with a probability per unit time
where g(v - Y,) is a normalized shape function centered around the resonant frequency v $ ~ ,
1,
g ( ~ ~i,) dv = 1.
I t takes into account the distribution of initial and final states by variation of local fields. In very dilute magnetic materials the local variations in the crystalline field splitting parameters and magnetic field variations due to nuclear spin arrangements are most important in determining g(v). In more concentrated materials one has to add the dipole and exchange interactions between neighboring magnetic spins observed at microwave frequencies. If H,, is taken in a principal ( x - ) direction of the g-tensor and can be considered as a small perturbation, the population of the levels per unit volume niand n, will be given by the equilibrium Maxwell-Boltzmann distribution (3). An imaginary part of the susceptibility can be defined by equating the power absorbed to +wx”H$ Combination of eqs. ( 2 ) and (5) then yields Kefcrences
p . 427
402
[CH. IX,8
N. B L O E M B E R G E N W t , ) =
@-lP2g:
I (i I s* I i) l2 gw(4 (% - 4.
2
(6)
The corresponding real part of the susceptibility may be found from the Kramers-Kronig relations. This treatment can be generalized to other modes of polarization of Hrf. The energy levels have been determined for paramagnetic ions in a large number of substances by paramagnetic resonance. As an example the four spin levels of the Cr+++ion in ruby are shown as a function of
FIELD PARAMETER, GI0
r? , J
209
I
1 00 s g
~~~
2
2w
-:
-10
-20
-4
w
g
3
-30
-6
-40
-15 l
-8 0
-50 2 4 6 8 1 0 APPLIED MAGNETIC FIELD,/#. IN kOI
Pig. 3. The energy levels of the Cr+++ion in ruby with the magnetic field parallel t o the trigonal axis.
-20
0
0
l
z
F
2
F
!
J
3
MAGNETIC FIELD I N
KOe
Fig. 4. The energy levels of the Cr+++ion in ruby with the magnetic field perpendicular to the trigonal axis.
magnetic field in the direction parallel and perpendicular to the trigonal axis of the crystal (ruby is A120, with Cr as an impurity). The spin hamiltonian is described1* by eq. (4) with g,, = 1.9840 f 0.0006, gL = 1.9867 f 0.0006, D = - 0.3831 f 0.0002 cm-1. Tables of the matrix elements of the spin operator for arbitrary values of the external magnetic field have been published'. If the field is parallel to the trigonal axis, m, is a good quantum number and transitions with I d m I > 1 are strictly forbidden. This is the case for the straight lines in Fig. 3. Mixing of m,- states occurs for curved characteristics shown in Fig. 4. Ruby with a Cr : A1 atom ratio between 2 : lo4 and 1 : lo2 is an References p. 427
CH. IX,$
21
SOLID STATE MASERS
403
excellent maser material. Large single crystals can be grown and are commercially available. It can be cut and polished with ease. It is chemically and physically stable. It has negligible dielectric losses and a good thermal conductivity. The spin-lattice relaxation time is sufficiently long so that a rather small pump power can produce saturation. The full width of the resonance between points of half maximum intensity is of the order of dv = 6 x lo7 Hz. This width is partly due to the magnetic interactions with A1 nuclei and partly to variations in the crystalline field tensor. The line width will vary with resonant frequency and the pair of energy levels involved. For each transition the line width in gauss is related to Av by dv = ( av/ 8H)dH. If k T is large compared to the overall splitting of the spin quartet, the spin populations in thermal equilibrium with the lattice at temperature T are approximately given by n, - nf = $nohvu/kT,where no is the total number of Cr+++ions per cc. For a relative concentration Cr : A1 = 1 : lo3 corresponding to no = 4.7 x 1019, one finds from eq. (6) X " m 0.01 for vu = 6 kMHz at T = 2" K, if the spin matrix element is put equal to unity. In practice this element will vary widely with geometry. If the Zeeman energy becomes large compared to the crystalline field energy, transitions between non-adjacent states will have a very small matrix element, because they correspond to "forbidden" transitions with I Am, 2 2. The characteristics are then nearly straight. Adequate pumping between non-adj acent levels becomes difficult. For maser operation in the millimeter wave region one needs therefore salts with larger crystalline field splittings. The Cr+++ ion in emeraldlg may be suitable with h-lD = - 26 & 2 kMHz. The Fe+++ ion in Al,03 is another example. This ion has a spin S = 512. Its microwave spectrum has been measured by Prokhorov19, Bogle20 and others2I. In a cubic field with a trigonal component there are three Kramers doublets in zero magnetic field. The intervals between these three doublets are 12.07 kMHz and 19.13 kMHz. This permits the operation of zero field masers which have been discussed by Bogle22. The Fe+++ion in rutile (TiO,) has three doublets separated by 43 kMHz and 81 kMHz. If the magnetic field is applied parallel to the trigonal axis, one finds that the m, = -+- 4 transition is narrower than the others. This indicates that there is a distribution of crystalline field splittings. The frequency of the m8 = 4 --f - 3 transition is independent of the crystal field in first approximation.
I
References p. 427
404
N. BLOEMBERGEN
[CH. IX,
52
The random distribution of magnetic ions, if nothing else, will produce strains and vacancies in the lattice which leads to line broadening. At higher concentrations the magnetic interactions between ions become important. If the magnetic concentration is neither too high nor too low, additional resonance lines due to neighboring ion pairs have been observedzs.z4, The Ni* ion often has excessively broad lines which are hard to saturate. The breadth is undoubtedly due to variations in crystalline field splittings. The Cu++with S = has only two spin levels and an undesirable hyperfine structure. The Mn++has rather small crystalline fields splittings, also complicated by a large hyperfine interaction. The Ti+++is chemically not stable and has a very short relaxation time. The most promising materials for masers in the microwave region therefore contain Cr+++, Fe+++,V++ or Gd+++ions, preferably as impurities in simple oxide structures. These are physically and chemically stable (gems ), and all ions are often in equivalent lattice positions. I t should be mentioned, however, that the first experimental maser utilized dilute gadolinium ethylsulphate 25. Potassium cobalticyanide with Cr as the active magnetic materials has also been used successfully. Recent observations of the microwave spectrum have revealed that this material exhibits polytypism. There are several slightly different unit cells and consequently there are a large number of nonequivalent positions of the Cr atom in a single piece of materialz0. So far an elementary perturbation treatment has been adopted to describe the electromagnetic transitions between spin states. This procedure is not quite satisfactory to describe the operation of a pumped maser, because the pump field necessarily constitutes a large perturbation. The Kramers-Kronig relations are also no longer valid, since the magnetization is a non-linear function of the field under conditions of saturation. The rigorous approach is to start with the equation of motion for the density matrix @
= - i6-1(&@
- @A?)
where the hamiltonian now also contains the time dependent perturbation. The components of spin vector 5 are Tr(S@). The time dependent terms give the microwave susceptibility. Clogston27 has given a detailed algebraic analysis of a three level spin system with an applied microwave field at the pump and at the References p. 427
CH. IX,$
31
SOLID STATE MASERS
405
signal frequency. I t turns out that the diagonal components of the density matrix, corresponding to the populations in the various spin states, are given correctly by the relaxation rate equations from perturbation theory, if the applied pump field is small compared to the line width, H,, < A H . This is the case of interest for solid state masers. The result is not surprising. The phases of the off-diagonal elements are scrambled by the distribution of resonant frequencies g(v). Similarly, perturbation theory would continue to give valid results even in the case H,, A H , if H,, were given a random frequency modulation. Then Hrppresents a hot “black-body’’ radiation field over the width of the resonance, which gives rise to the ordinary saturation phenomenon. Interesting coherence effects, which might give rise to sideband resonances or structure in the microwave susceptibility and which are undesirable in maser applications, can thus be avoided. The following procedure is therefore justifiable. Calculate the populations of the spin levels from the rate equations which utilize transition probabilities per unit time derived from perturbation theory. Then determine X” from eq. (6). The determination of the real part X’ at high power levels is somewhat more involvedz8.Usually the interest is in weak signals at the maser frequency. Then the Kramers-Kronig relations are obeyed near this frequency. Transitions can also be induced by time varying crystal fields rather than Zeeman terms. Ultrasonically induced transitions are well known in nuclear spin systems. Mattuck and Strandberg29have given a discussion of them for electron spins. A periodic variation of D in the Hamiltonian (1) will induce quadrupole transitions. Ultrasonic saturation has recently been achieved at microwave frequencies30. The advent of microwave ultrasonics makes an acoustically pumped maser possible.
>
3. Paramagnetic Relaxation Transitions between the spin levels are not only induced by externally applied electromagnetic or acoustic fields. The modulation of the g-tensor and crystalline fields by the thermal motion of the lattice also causes transitions. They constitute a contact between the spins and lattice, and in the absence of external driving fields are responsible for the establishment of a Maxwell-Boltzmann distribution over the spin levels. The probability per unit time for these spin-lattice transitions will be described by quantities wij,which have the magnitude of References
p. 427
406
N. BLOEMBERGEN
[CH. IX,
53
inverse spin-lattice relaxation times. They satisfy the detailed balancing condition w f I = wg exp (--hv,/kT).
(7)
This relation also follows from the properties of the matrix element of a harmonic oscillator. A lattice quantum or phonon is absorbed when the spin energy increases and emitted when the spin energy decreases. Note that Ep,/(Gp, 1) = exp (--hv/kT), where Z,, is the average excitation quantum number of a lattice oscillator at frequency v and temperature T . Besides the emission or absorption of a single lattice quantum, Raman processes in which two lattice quanta take part may also occur. At liquid helium temperatures, however, the theory of Van Vleck31 predicts that single phonon processes should dominate and the spin-lattice relaxation time should be inversely proportional to T . This feature has been verified experimentally for the C W ion in a number of compounds, but other ions often have a more complicated temperature dependence. The dependence on the frequency of the transition v, which should be w, a v$ in the simplest case has been verified once. The order of magnitude of the transition probabilities wtl at 4’ K is in the range lot2 to sec-’ for Cr++t, F e w and Gd* ions, It varies of course with the matrix element for the particular transition, the nature of the chemical compound etc. Another type of relaxation process takes place entirely within the spin system. No energy is transferred to or from the lattice or a radiation field. A trivial case is the flip-flop between two pairs of equidistant spin levels. I t is possible, however, that processes in the spin system change the population differences between non-equidistant levels, i.e. affect the intensity of well-resolved resonances. The term “crossrelaxation” has been introduced 32 to describe this phenomenon. Higher order processes in which three, four or more spins take part simultaneously are quite important. For example, two downward transitions at frequency vu may be accompanied by one upward transition between a pair of levels with twice the spacing, Y , ~ = = 277,. The transition probability for a cross-relaxation process in which one spin makes the transition from level E, -+E,, a second one from level E , + E l , a third one from level Em --f E , will be denoted by w ~ , ~ ~ , , , ,It, . ,is. not necessary that the energy of the unperturbed levels of the spin hamiltonian be exactly conserved, but a small balance may be absorbed by the dipole-dipole or
+
ipZic
References $. 427
CH. IX,
9 31
407
SOLID STATE MASERS
exchange interaction between electronic spins and/or nuclear spins. These interactions which are not included in the unperturbed spin hamiltonian also determine the line shape functions g(v). For a two spin process the following approximate relationships holds m
wtj.kL
g(v - v k l ) g ( v - y$5> dv.
(T2)-1
(8)
0
The spin-spin phase memory time I', is defined as gmsx(v).The simple physical interpretation of eq. (8) is that the cross-relaxation time is inversely proportional to the overlap of the two resonances. Higher order spin processes are by no means negligible. Although the .characteristic time for a four-spin-flip w;: kz,,,, o p will be much longer than T,, it may still be much shorter than TI. Even higher order processes should sometimes be considered between well-resolved resonances. The population the i t h spin level is thus governed by the following rate equation
Higher order cross-relaxation terms should be added. Higher order spin-lattice terms, which also exist in principle, are usually not important. The equations can be linearized in the n, in the high temperature approximation, hv, k T . The steady state solution of (9), obtained by putting the left hand side equal to zero, describes the balance between the pump action of externally applied field(s), spinlattice and cross-relaxation mechanisms. An important question is whether the lattice vibrations really constitute a thermal bath for the spins. The specific heat of the lattice oscillators in the same frequency range as the spin resonances is very small. It may not be justified to assume that the wu are constants satisfying the relationship (7), where T is the temperature of the helium bath. A set of rate equations for the number of phonons in each lattice oscillator may be juxtaposed to eq. (9)>
nr,an exponential increase in the excitation of the lattice oscillators would result. They can never attain a negative temperature, because their energy levels Krfercnces p . 427
CH. IX,
5
31
409
SOLID STATE MASERS
have no upper bound. A limitation is of course set by the rate at which the spins can supply energy to the acoustic resonator and the latter loses energy through acoustical coupling with the environment or damping mechanisms. With careful control of this acoustic coupling an amplifier or coherent generator for phonons would result. I n practice, however, the phonons serve as a reservoir for the spins rather than vice versa. The hot spins can build up the energy density in the electromagnetic field rather than in the phonon field. The coherent excitation of lattice oscillators should be accomplished by use of the piezo-electric effect rather than the spins ,O. Conversely, such acoustical microwaves can then be used to study the interactions with spins and other phonons and determine the relative magnitudes of the quantities w~ and rph. The solution of three equations of the type (9) for a three level spin system (S = 1) becomes
hN w21v21 - w32v32 (11) 3kT w32 w21 w32 in the absence of cross-relaxation, in the limit of very large pump power at the highest frequency v3, and in the high temperature limit, with an arbitrary signal at the frequency v , ~ . Therefore, eq. (11) is valid only for W,, --f 00, W,, = 0, wtf,k L = 0, hv,, kT. Fig. 5 shows how the effective susceptibility at the pump frequency approaches zero with increasing pump power. This is the usual saturation of a spin resonance. If appreciable phonon heating occurred, the saturation curve would not have the simple theoretical form39. At the maser frequency the susceptibility, measured with small signal power W32 m 0, approaches a negative limiting value for W13 00. The curves were taken for Cr+++ions which have four levels, but if cross-relaxation is avoided the results are very similar to those for a three level system. If the middle level in a three level system comes nearly midway between levels E l and E,, cross-relaxation mechanism w,,,32 becomes dominant. If the middle level moves close to either the upper level or lower level cross relaxations w , , , , ~ or wS1, 12 become important. In either case there will be overlap between closely spaced resonances and the steady state condition becomes n, = n2 = n3 for W13 -+00 and w12,32 wi,. With sufficiently fast cross-relaxation, no maser action is possible in this case. The pump power heats up the spin system as a n,
-
n2 = n,
- n2 = -
+ +
References
9. 427
410
[CH.IX,5 3
N. BLOEMBERGEN
whole. The cross-relaxation processes are responsible for the establishment of thermodynamic equilibrium within the spin system, as postulated by Casimir and du Press. This situation is incompatible with a steady state maser operation. At higher concentrations of magnetic ions higher order spin process gain rapidly in importance. In concentrated magnetic salts the Casimir-du Pre hypothesis is always well satisfied. Only dilute materials can be used in continuous wave masers, at least in the conventional microwave band. It is conceivable that higher magnetic concentrations are permissible with larger spacings
001
aio POWER
10
10 AT
VZ4
IN
100
MILLIWATTS
Fig. 5. The imaginary part of the susceptibility for two transitions between the four levels of Cr in K/(0.996 Co, 0.006 Cr) (CN), as a function of pump power at the frequency va4. The crystal was kept a t 2.6" K. The magnetic field H , = 1176 oersteds made a n angle of 10"with the a-axis in the ac-plane. The drawn curves have the theoretical form given by eq. (9) in the absence of phonon heating.
between the spin levels for ions which have resonances in the millimeter wave region of the spectrum. Various effects of cross-relaxationbetween the four levels of Cr+++ions have been noted by a large number of authors. The solution of the rate equations becomes algebraically involved. A large variety of situations can occur in which one or more cross-relaxation terms are important. S h a p i r ~Maiman40 ~~, and BOlger4lhave shown how cross-relaxationmay inhibit maser action in a certain specimen at low temperature, but not at a higher temperature. What counts is the relative magnitude of the spin-lattice and cross-relaxation time. At high temperatures the former are shorter and cross-relaxation effects become relatively less imporReferences p . 427
CH. IX,
3 31
SOLID STATE MASERS
41 1
tant. As a corollary higher magnetic concentrations may be used in maser materials at higher temperature. Mims42 has noted that sharp variations in the steady state populations occur, whenever the orientation and magnitude of the magnetic field is such that one transition frequency is close or equal to a multiple of another. Several other authors have made similar observation^^^^ 41 and have called the phenomenon harmonic cross coupling. Clearly it is an important special case of the multiple spin transitions discussed in the general framework of cross-relaxation. Harmonic cross-relaxation effects have been reportedls up to the eleventh harmonic! The collaboration of at least twelve spins is of course a highly concentration dependent process. If the concentration is chosen too high, however, the required resolution is not obtainable and everything is washed out in the thermodynamic spin pool of Casimir and du Pre. The observation of these very high order spin processes also requires a long spinlattice relaxation time, T , > 106T2.Cross relaxation effects can also account for certain anomalies in spin-spin relaxation reviewed by G ~ r t e r Additional ~~. dispersive regions in the susceptibility should be expected32near frequencies of the magnitude of each of the wv, The discussion to this point may have given the impression that crossrelaxation is usually detrimental to maser operation. This is not at all true. Cross relaxation is responsible for homogeneous saturation of the pump transition and homogeneous inversion of the population at the maser frequency32.Since only a small fraction of the total number 01 ions have a configuration of surrounding nuclear spins and crystalline field to be at resonance with the pump frequency, one might think that only a small fraction of ions would get pumped and become emissive. In that case there would either be only inversion in a very narrow frequency range, or more likely no maser action at all since there is no one to one correspondence between the pump and signal frequency distributions. Fortunately cross-relaxation effects saturate transitions on both sides of the pump frequency as well, and maser action is obtained over the full width of the signal frequency resonance. It should be noted that homogeneous cross-saturation can only occur if spins in nearby unit cells have different resonant frequencies. At extremely low concentrations of magnetic ions a “hole” can be burned. The same is also true if the line is broadened by large scale inhomogeneities in the external magnetic field or crystalline configurations. Appropriate maser action would still be obtainable, even in the Referenus p . 427
412
N. BLOEMBERGEN
[CH. IX,
3
presence of large scale inhomogeneities, if the pump is frequencymodulated to “cover” the entire resonance. Cross-relaxation between a. pair of levels of a Gd+++ion and an equidistant pair in a Ce+++ion was utilized in the first solid state maser25to obtain a faster relaxation rate for that particular pair of Gd+++levels through coupling with Ce+++which has a much shorter spin-lattice relaxation time. Maser operation in ruby is sometimes also dependent on a fast cross-relaxation process43. Cross-relaxation makes it possible to have a maser frequency higher
€3
1
Pump
Signal
vlj= 9595 MHz
v,J‘10590 M M
€2
6
Big. ti. The energy levels of Cr+++in ruby in a magnetic field of 1675 oersteds perpeudicular to the trigonal axis. Cross-relaxation between the transitions vI4 and vaSsaturatrs the former, n, = n,, when pump power is applied at vps. Maser action results at a higher frequency vlgthan the pump frequency4j.
than the pump frequency. This has been emphasized by Minis and and has been demonstrated experimentally by Geu~ic*~, Higa and A r a m ~Consider ~~. the particular arrangement of energy levels of the C++ ion in ruby shown in Fig. 6. Pump power is applied at the frequency ~ 2 3 ,which is equal to one-half the splitting ~ 1 4 .Harmonic cross-relaxation w , ~ 3, 2 , 1 4 is responsible for saturation of the v I Q transition also, n , = n4. Maser action was observed at Y,, > v12, implying that n3 > n,. Thercis no violation of conservation of energy or other thermodynamic considerations. For the emission of one quantum kv,,, the pump has to supply at least two quanta 2hv3,. The cross coupling converts these into one quantum kv,, > k ~ 1 3 .The difference 2hv,, - hv,, is taken up by the lattice through spin-lattice relaxation. This type of mechanism would be of particular importance in extending the application of masers into the millimeter wave region. This conversion of quanta absorbed by the spin system into larger Kefcrnzccs
p . 427
CH. IX,
5 31
SOLID STATE MASERS
413
quanta -larger level spacings may also be provided by exchange interactions between neighboring spins -is perhaps also helpful in permitting a larger fraction of lattice oscillators to carry away absorbed power. The rate equations (9) clearly also include a description of the case in which more than one transition is pumped. Fig. 7 shows the energy levels of Cr+++in ruby when the magnetic field makes an angle 6 = arc cos (3-Y2)with the trigonal axis. Such push-pull pumping4' has been used to enlarge the population difference at the maser frequency Y,,.
bl
,I
,
--
The rate equations also describe saturation of maser amplification if the signal power becomes too large. The occurrence of W,, in the denominator of eq. (11)is an example of this effect. They also govern the return to equilibrium in a pulsed maser after inversion of the level population. They do not include the effect of a larger coherent transverse time-dependent component of magnetization. The off-diagonal components of the spin density matrix should then be taken into account as was already mentioned before. In this case the energy of the spin system may be radiated away very rapidly. This radiation damping effect 48 is, however, negligible, if the spinspin phase memory time T , is very short compared to the radiation time,
T,
< (~ZM'@O)-~.
(12)
Here M is the transverse magnetization, 7 the filling factor and Qo the References p . 427
414
[CH. IX,3 4
N. BLOEMBERGEN
quality factor of the electric circuit. In pulsed masers care should be taken that this condition is satisfied. Otherwise the material cannot be used as an amplifying medium, since its stored energy is radiated away in a short burst. The combination of the solution of the relaxation rate equations (9) with the equations (2) and ( 5 ) give the (negative) absorbed power of the maser material. They provide the basis for a discussion of the circuit aspects of this type of solid state maser. 4. Maser Circuits
Consider a cavity with a resonance mode at the pump and signal frequency, corresponding to two spin resonances of the paramagnetic RECEIVER
I HELIUM LEWAfi
PARAMAGNETIC
CRYSTAL
Fig. 8. Schematic coupling diagram of a reflection cavity maser.
salt inside the cavity. Fig. 8 shows a reflection type cavity. If the incoming signal stimulates the emission of more power in the salt than the absorption losses in the walls of the cavity, more power will leave the cavity than is incident on it. The action of the paramagnetic salt may be described in terms of a magnetic quality factor QM which is equal to 252 times the electromagnetic energy stored in the cavity divided by the power absorbed per cycle at the paramagnetic resonance. If the whole cavity is filled with the paramagnetic material and if the mode of polarization and the value of X ” for this mode were uniform over the cavity, then the simple relation Q, = (4nX”)-lwould exist. Actually the direction of H,, with respect to the crystal axes will vary over the cavity and also the population difference n, - n, will be a References p. 427
CH. IX, 9 41
415
SOLID STATE MASERS
function of position, because the pump field is not uniform. Therefore the expression for the absorbed power has to be integrated numerically over the volume of the cavity in practical cases. If the pumping field has nodes in the magnetic material, there will always be absorbing regions which are not pumped. If the energy density at the pump frequency is chosen high enough, saturation can still be achieved in nearly the whole specimen with the exception of narrow regions around the pump nodes. It can be shown that spin diffusion is not sufficiently rapid to give saturation in these regions. The suggestion49that hot phonons would produce saturation at the nodes has been disproved by BOlger4l.Maser action is possible because stimulated emission from
\ dM ‘
AVO
Fig. 9. Equivalent circuit of a reflection cavity maser, when both cavity and paramagnetic spins are tuned t o resonance. Reactive parts have to be added off resonance. The equivalent noise current generators associated with the antenna conductance Q,-l at temperature TO.the cavity conductance Qo-l a t temperature T o and the magnetic spin conductance QM-1 a t spin temperature TM are indicated, together with incident and outgoing power.
the rest of the crystal dominates the absorption at the signal frequency near the pump nodes. The reflection cavity maser acts as a negative resistance amplifier with an equivalent circuit as shown in Fig. 9. The reactive parts have been omitted in this diagram. They are, of course, important in determining the band width of the amplifier. The unloaded Qo of the cavity describes all losses in the cavity, other than the spin resonance, such as eddy current and dielectric losses. The latter are usually negligible at liquid helium temperature. The external load coupling is described by Q,. A stable amplifier results for
Q, < 0 and Qol
+ Q,* > I Q,
I-l
> &l.
An oscillator results for
Q, < 0 and Q;’
+ Qi