Nuclear resonance in solid and liquid metals: A comparison of electronic structures

ANNALS

OF

Nuclear

PHYSICS:

8,

173-193

Resonance

(1959)

in Solid and ison of Electronic

W. D. Department

KNIGHT

of Physics,

AND

University

Liquid Metals: Structures A. G.

A Compar-

BERGER*

of California,

Berkeley,

Calijornia

AND

V. Royal

Society

Mend

Laboratory,

HEINE

Tlniversity

of Can&ridge,

Cambridge,

England

The electronic structures of ten solid and liquid metals are investigakd experimentally and theoretically, and the results of recent nuclear resonance measurements are correlated with already published data concerning magnetic susceptibility, electrical conductivity and atomic arrangement. It is concluded that the electronic structure of each of the metals investigated, with the notable exceptions of Bi and Ga, does not, change appreciably at the melting point. Moreover, in Ga and Bi there is a correlation between a change in the electronic structure and structural rearrangements of neighboring atoms at short range; the other metals considered show neither sort of change. A principal conclusion is that a liquid metal possesses a band structure which is very like that of its solid, provided the short-range structure is not altered during me1 ting. Included are hitherto unpublished values for the nuclear magnetic resonance line shifts in liquid Al, Sn, In, and Bi, and the nuclear spin relaxation times in soli’d and liquid Ga. The nmr shifts in solid and liquid Al are equal within experimental error. The same is true for Sn. This is in agreement with published results on the alkali metals and Hg, where the corresponding changes are known to be small, suggesting that the electronic structures of the respective solids and liquids are the same. The large quadrupole coupling in Ga, In, and Bi prevents observation of the nmr in the solid state. However, the value of the nmr shift in liquid In is consistent. with values of the hyperfine coupling constant and the electronic specific heat at low temperatures, a situation which is true for most of the other metals. Ga and Bi are the exceptions for which the nmr shift in liquids is much larger than the hyperfine and low-temperature data would lead one to expect. This is taken as evidence for a reduced density of states at the Fermi surface in f,olid Ga and Bi. Confirmation is provided by the fact that the nuclear spin relaxation time of Ga increases sharply when the metal freezes. The measure* Now

at Boeing

Airplane

Company,

Seattle, 173

Washington.

17-l

KNIGHT,

BERGER

Ah-D

HEISE

ment of relaxation in solid Ga is significant because the nuclear quadrupole resonance is directly observed and the relaxation process is of the same type, so that a change in its magnitude may also be related to a density of states. I. INTRODUCTION

This investigation was originally stimulated by the question: does the result of a nuclear magnetic resonance (nmr) measurement on liquid bismuth give any direct information of value in constructing a theory of the electronic structure of solid bismuth? The abnormally low value of the electronic contribution to the specific heat (1) compared to the sizable value of the nmr shift suggests that the latter measurement is not in fact directly related to the properties of solid bismuth. A second question then presents itself: is bismuth peculiar in this respect, and should most other metals exhibit consistent values for the above quantities? With these questions in mind it seems desirable to examine the nmr of several metals both below and above their melting points. Prior to the present work, no metal had shown any marked change in nmr shift at the melting point. It was, therefore, decided to investigate some other metals, and to compare the results with other known properties that are related to the electronic structure. In nuclear resonance, the magnitude of the nmr shift (5’) AH/H gives directly a quantity xpl (see Section 2 below). Here xp is the paramagnetic spin susceptibility, which is an approximate measure of the density of states N(E,) at the Fermi level EF . The factor l is the average proportion of atomic s-like character in the electron wave functions at the Fermi level. Thus the value of x& is directly related to the electronic structure of the liquid or solid metal. However, since xpl is just a single number, it is difficult to deduce details of the electronic structure from it. In spite of this limitation, it is a very useful index for any changes in the electronic structure of the metal on melting. In particular, we can use it in choosing among different models of the electronic structure of the liquid metal. Two extreme models immediately suggest themselves. On the one hand we might suppose that the disorder in the liquid destroys all directional effects so that the electrons effectively form a free electron gas. On the other hand one could argue that the electronic structure reflects the short range bonds between the atoms and their local arrangements, so that the electronic structure of the liquid should be the same as that of a solid which has the same local (shortrange) structure. We now ask what each of the models predicts for the change in the NMR shift on melting, and how sensitive a test of the two models this provides. In most metals the short range structure does not change very much on melting. For such metals the second model clearly predicts zero change in AH/H, whereas it predicts a considerable change for metals which do change their shortrange order radically on melting. Turning now to the first model, xz, in the solid

ELECTRON

STRUCTURE

OF

LIQUID

METALS

175

is quite sensitive to zone boundary effects since N( EF) may differ by 10 ye-50 yO from the free electron value. The first model therefore predicts a change in AH/H on melting of about this magnitude, which would be considerable, and easily ob;servable. More specifically, xP is sensitive to zone boundary effects since N(JP,) involves l/grad&, which becomes infinite at the zone faces. For instance, in aluminum it is believed that about one-third of the observed N(EF) comes from such Brillouin zone effects (3). Furthermore, on the zone faces the electron states are purely s-like or p-like in character, so that [ also is sensitive to zone bloundary effects. We conclude therefore that the hTMR, shift should be a useful indicator of the electronic structure of liquid as well as solid metals, and in particular that it should provide a sensitive test of the two alternative models proposed for the electronic structure of liquid metals. Unfortunately, since many of the interesting metallic elements possess large nuclear quadrupole moments, the nmr is not observable in the polycrystalline solids, for, if the solid crystal structure does not exhibit a cubic symmetry about the nucleus, the electric quadrupole interaction can be larger than the Zeeman energy of the nuclear magnetic moment, and the nmr line will be much too broad to observe. Nevertheless, the nuclear quadrupole resonance (nqr) of some of these same metals may be observable (4) in the absence of an external magnetic field; the nuclear spin-lattice relaxation time is measurable; and since the relaxation time and the line shift are measures of the same interaction, we may in a sense use them interchangeably for the purpose of investigating the electronic structure. It was decided to attempt a series of such measurements on Ga: above the melting point (nmr), one can measure both the line shift and the relaxation time; below it (nqr) only the relaxation time can be found experimentally. In the following sections, we first review the relationship between the nmr line shift; and the electronic structure. Second, we discuss the effects of quadrupole coupling on the observability of resonances. This is followed by a description of the apparatus. We then survey the available data for the line shift, relaxation time, magnetic susceptibility, electronic specific heat, electrical conductivity, atomic arrangements of the solid and liquid metals, and various transport propties. Correlations among the various properties are found to exist. It is concluded that, for all the metals investigated except bismuth and gallium, there is little or no change on melting, either in the electronic structure or in the short-range atomic structure. The latter two metals appear to show a considerable change in both electronic and in short-range atomic structure, such that they become more metallic in the liquid state. The experimental evidence gives one the strong impres&ort that a liquid metal must have an electronic band structure (5) which is very similar to that of a solid and which is determined essentially by the shortrange atomic structure, as in the second model proposed a.bove. From a theoretical

176

ISIGHT,

BERGER

AND

HEIXE

point of view such a conclusion may be somewhat surprising, since, in the usual presentation of the electronic band structure of solids, the Brillouin zone and all associated effects enter the theory through the long-range crystal structure, which is of course destroyed on melting. In the final section we give therefore a theoretical discussion of the electronic structure of a liquid metal which related it to the short-range order and to the observed long mean free path for scattering. II.

THE

NMR

LINE

SHIFT

AND

RELAXATION

TIME

Although we shall in a later section consider several known properties of metals, a major concern is with the question of how crucial the nmr experiments can be in providing unique information about the metallic structure. Therefore we shall review briefly the theory of the nuclear resonance in metals. The relative enhancement of the local magnetic field at the metallic nucleus compared to the field which would exist in a nonmetallic solid is AH/H

= (%/3)x&P,,

(1)

where PF = ( 1#F(o)

I”>Av

(2)

is the average probability density at the nucleus for electrons at the Fermi surface, and xP is the paramagnetic susceptibility (6) of the electrons per unit mass. M is the mass of one atom. Since the resonance frequency is directly proportional to the magnetic field at the nucleus, the enhancement AH produces a corresponding proportional shift in the observed resonance frequency when the externally applied field is unchanged. The relative enhancement, will be called the nmr line shift, and numerical values will be quoted in percent. It is useful to point out that the condition AH = 0 refers to a condition in a metal such that all electron spins are paired. Lacking a better standard, one assumes that this condition is equivalent to that existing in a nonmetallic salt of the element. At any rate, t’he product xpMPF represents the density of electron spin magnetic moment per unit applied field at the nucleus, and Eq. (l), referring to a metal, is analagous to the Fermi (7) interaction which produces hyperfine structure in the spectrum of a free atom. If we write out the Goudsmit (8) or Fermi-&g& (9) formula and substitute directly for the quantity P, , Eq. (1) is conveniently written AH/H where gI = PI/I and E = P,Il’, atom, and 4s) the appropriate generally be less than unity when duction band contains appreciable

= a(s)xp

f MPgm,

(3)

, PA being the probability density for the free hyperfine coupling constant. The ratio E will the wave function of the electrons in the conadmixtures of p or higher states. We shall not

ELECTRON

STRUCTURE

OF

LIQUID

METALS

be concerned in our discussion with the relatively states. The susceptibility may be w&ten approximately states at the Fermi surface N(EF),

small effects of the non-s in terms of the density

xp = /.dN(Er). This quantity gas yT, where

is directly

or (4)

related to the electronic

specific heat of the electron

y = (7&“/3)N(E,). Therefore, the right As was mentioned T1 measured by nqr the nmr is itself not

177

(5)

side of Eq. (3) may be evaluated in terms of y. in the introduction, the nuclear spin-lattice relaxation time will serve to give a good estimate of AH/H or N(EF) when directly observable, for l/T1 0~ [PFN( EF)]‘, and (10)

2 (6)

(AH1/Hj2’

Equation (6) is correct only in the approximations of free electrons (11) and small p-character of the wave functions, but we believe it to be sufficiently accurate for our purposes. It is also assumed that (i) the dominant relaxation process involves the interaction between the nuclear magnetic moment and the conduction electrons and that (ii) lattice motions may be neglected. In following sections we will show that Eqs. (3) to (6) are consistent with measured values of t’he quantities for all liquid metals considered, and similarly for all the solids except Ga and Bi. III.

THE

IWCLEAR

QUADRUPOLE

COUPLING

The strength of the interactions we consider may be measured in terms of resonance frequencies. Usually for the metals discussed in t’his paper the nmr frequencies are of the order of 10 Mc/sec ( 12). By comparison, quadrupole interactions are often appreciable. For example, Bloembergen and Rowland (1s) estimate that in some copper alloys the presence of a solute atom among the nearest neighbors of a nucleus may produce a quadrupole coupling of the order of 1 Mc,/sec. This is large enough to eliminate the contribution of the particular copper nucleus from an observable resonance line. In beryllium (14) the nmr line is split into three components with a separation of the order of 10 kc. The quadrupole coupling frequencies are 22 Mc/sec for Ga6’ (4) and 45 Me/see for InIl* (25). For the last two, we do not 01 serve the nmr in polycrystalline solids. A. single crystal sample in the form of oriented thin films would have to be used.

178

KNIGHT,

BERGER AND HEINE

One may expect to observe the nmr in a liquid metal, however, since the quadrupole coupling may be partly averaged out by the thermal motions. The residual width is given by l/T* , where (16) l/Tz

= (&J&,

.

(f3a)

For our estimate we must take the quadrupole interaction (6~) for the solid, because we do not know the instantaneous value in the liquid. The appropriate value for 6w in Eq. (6a) will be twice the angular resonance frequency of the nqr in the solid for a nuclear spin 35. For Ga6’, 6~ = 27r22 X lo6 cps. For a correlation time (TV) of 1OP set, derived from diffusion coefficients (171, the residual Tz turns out to be about 6 X 10e6 sec. Since the observed value (18) of Tz is w1o-3 set, we must conclude either that the instantaneous quadrupole interaction in liquid Ga is smaller than the static value in the solid, or that the correlation time, rc , is considerably shorter than 10-l’ sec. The former appears to be the more likely possibility. The principal mechanism producing the nmr line width in liquid Ga is the spin-lattice relaxation time. The line width of the nqr may be a combination of the effects of magnetic dipolar coupling and of inhomogeneous strains in the material. There is no evidence for motional narrowing in the solid near the melting point of Ga (19). IV.

A. RESONANCE

EXPERIMENTAL

TECHNIQUE

APPARATUS

An electronically stabilized electromagnet was operated at various fields between 7 and 10 kilogauss. The marginal oscillator and associated phase-sensitive detector were of a conventional design (,%I). The magnetic field was modulated sinusoidally at a frequency of 100 cps in the usual way, which results in signals possessingthe shape of the differentiated absorption line. The oscillator coil was made of silver wire wound on a fused quartz tube mounted in a cylindrical brass can about three inches long and one inch in diameter. This can was heated by a bifilar winding around its outside and carrying direct current. A platinum versus platinum-rhodium thermocouple junction was permanently mounted inside the can near the sample. The entire assembly was suspended on stainless steel tubes within a fused quartz dewar. A temperature of 500°C was attained with 17 watts input to the heater, the maximum operating temperature of about 700°C being set by the melting point of the hard solder at the points where the heater can was supported. B. EXPERIMENTAL

PROCEDURE

Resonance frequencies were measured by comparison with marker signals from a crystal-calibrated frequency meter, which was standardized against radio station WWV. The nmr shift was measured as a fractional deviation from a calcu-

ELECTRON

STRUCTURE

OF LIQUID

METALS

179

lated resonance frequency for a non-metallic sample. The latter (“salt”) frequency was computed, relative to that from Na23in NaCl, from known frequency ratios (21). Corrections for small time variations in the magnetic field were made by observing the nmr for Naz3 in an external probe in the magnet adjacent to the dewar. The temperature difference between the position of the monitoring thermocouple in the heater can and the sample position was measured at each heater current setting by means of a second thermocouple actually embedded in the NaCl sample mentioned above. The largest source of error was an instability in the mechanical construction of the rf coil assembly which resulted in small erratic frequency changes in the oscillator. In spite of this difficulty, however, the line shift in Al was determined with a precision of 100 cps at 650°C. The accuracy may best be judged by comparing this 100 cps to the line width, which was about 200 cps at this temperature, and. to the total nmr shift, which was about 15 kc. C. SAMPLES The samples consisted of metal powders (99.9-percent purity or better) obtained in the 325-mesh size from A. D. Mackay. The In and Bi samples were mixed with mineral oil in 12-mm Pyrex glasstubes. These and the other samples were sealed off at forepump vacuum. The mineral oil acted as an insulator between adjacent particles, and very little deterioration of the materials appeared to occur at the moderate temperatures required. The dry tin was heated for 8 hours in air at 300°C before sealing its test tube. Since the melting point of Al is rather high (660°C) another means of maintaining small insulated pieces was required, A satisfactory arrangement was a mixture of 1 volume Al powder with 2 volumes silica dust, evacuated and sealed in a fused quartz tube. This scheme was helpful, although evidence was noted of the Al attacking the silica glass after prolonged heating. V. SURVEY

OF DATA

AND

CONCLUSIONS

We consider, in Table I, the following properties of several metals: the spin susceptibility of the conduction electrons, the total or bulk susceptibility, the nmr line shift, the spin-lattice relaxation time, the electrical conductivity, and the spati.al arrangement of atoms. For each of the properties, we wish to compare the values for the respective liquids and solids. It may be interesting further to compare these values with those predicted by free electron theory. A. SPIN SUSCEPTIBILITY In rows 1 and 2 we compare the free electron values with those indicated by measurements of the electronic part of the specific heat at low temperatures. The ratio of a pair of such values is given in row 3. Evaluation in this way usually

TABLE PROPERTIES

-

OF I,IQUID

I AND

SOLID

METALS

Li’ 1. xp (free)&

1.5 3.4 2.3 (f3.4) -0.15 0.050 0.0261 0.0261 55 84 84 1.28

2.XP(-/)

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

m’/m x, x1 - X8 AH/H (calc)b @H/f01 @H/H), TI (-1~)~ (Ti)t (Ti)s r./q (talc)

13. rajq

1.68

14. n, 15. lzll

-

10 8-14

hP

Ga6*

0.66 0.82 1.2 (+0.70)
0.32

0.24

+0.21
+0.20
1.61

10 8-14

1.66 -

0.071 D 0.46 0.15 0.69 2.1 1.5 -0.17 (fO.48) -0.05 -0.13 10 0.23 2.45 0.164 2.45 0.164 0.05t 1.43 2.1 2.0 2.23 2.46 1.82

3.35

11 12

10 12

0.20 , 0.15 0.12 0.21 0.6 1.4 +0.04 f0.34 0.27 1.5 0.44 0.80 0.74 0.20 0.7 2.4 4.5 2.00 d 1.67 e 2.12 i 0.46 11 I0 7 I .2

Sn”9

Bi209

0.16 0.20 1.2 -0.04 -0.06 1.8 0.73 0.75 0.065

0.12 0.0044 0.037 -0.1 +1.1 0.37 1.40 0.096

0.11 3.06

5.0

2.11

0.47

10 10

8 6

-

’ Susceptibilities are in cgs m&88 units X 10-e. b AH/H ia in percent, measured values at the melting c Ti is in milliseconds at the melting point. d Average value. e Value for the c-axis. REFERENCES

point.

FOR TABLE

I

EzectmLic specijic heat (y) Li, Na: L. M. Roberts, Pmt. Phys. Sot. B70,738 (1957); Hg: E. Maxwell and 0. S. Lutes, Phtm. Rar. 95, 333 (1954); Al: D. H. Howling, E. Mendoza, and J. E. Zimmerman, Proc. Roy Sot. A229, 86 (1955), N. E. Phillips, Phys. Rev. 114. 676 (1959); Ga: G. Seidel and P. H. Keesom, Phys. Rev. 112,1083 (1958). In: J. R. Clement and E. H. Quinnell, Phya. Reu. 92, 258 (1953); Sn: W. S. Corak and C. B. Satterthwaite, Phrs. Rar. 102, 662 (1956); Bi: I. N. Kolinkina and P. G. Strekov, Soviet Union JETP 34(7), 616 (1958). Total susceptiMZity (x., xi) Li: E. W. Pugh and J. E. Goldman, Phys. Rex. 99,1633, 1641 (1955); C. Starr and A. R. Keufmann, Phys. Rex. 59, 476 (1941), S. R. Rao and K. Savithri, Proc. Indian Acad. Sci. AN, 207 (1942); Na: R. Bowers, Phys. Reu. 100, 1141 (1955), B. Biihm and W. Klemm, Z. anorg. u. &em. Chem. 243, 69 (1939); K. Venkateawarlu and S. Sriramsn, Z. Natwforsch. 13a, 451 (1958); W. Sucksmith, Phil. Msg. Ser. 7,2,21 (1926); Rb: See Bijhm and Klemm, and Sucksmith above (Na); K. Venkateswarlu and S. Sriraman, J. Sci. Ind. Research India Bl4, 611 (1955); Cs: C. T. Lane, Phil. Msg. Ser. 7, 8,354 (1929), Bahm and Kbmm 888 above (Na); Hg: W. Klemm and B. Hawchulz, Z. Blektrochem. 45, 346 (1939), S. R. Rao and Aravamuthscari, Proc. Indian Acad. Sci. 9A, 181 (1939), L. F. Bates and C. J. W. Baker, Proc. Phys. Sot. 50, 409 (1938), Y. Shimizu, Science Repts. TBhoku Imp. Univ. Ser. 1, 25, 921 (1937), E. Vogt, Ann. Phgsik Ser. 5,21,791 (1935); Al: P. Weiss and W. Klemm, Z. anorg. u. allgem. Chem. 245,288 (1940), 888 Shimizu above (Hg): Ga: A. Marchand, Cmpt. recd. 241,468 (1955); In: J. Verhaege, G. Vandermeersche, and G. Le. Compte, Phya. Rev. 80, 758 (1950); Sn: 888 Shimizu above (Hg), H. Endo, Science Repts. Tbhoku Imp. Univ. 16, 201 (1927); RI: Y. Shimiau, Science Repts. TBhoku Imp. Univ. 21, 826 (1932); J. C. McLennan and E. Cohen, Trans. Roy. Sm. Canada (3) 23, Section III, 159 (1929); 888 Endo above (Sn). Line

shift

(AH/H)

Li: B. R. McGarvey and H. S. Gutowsky, J. Chem. Phys. 21, 2114 (1953); 20,1472 (1952); Na: present work, 888 also reference for Li above; Rb, Cs: See reference for Li above; Hg: F. Reif, Phys. Rev. 106,208 (1957); L. Sarles and II. Loeliger (unpublished): Al: present work; Ga: W. D. Knight, thesis, Duke University, 1950; In, Bi: present work; Sn: present work, 888 also McGtlrvey and Gutowsky (Li) and Bloembergen and Rowhnd, Ref. 1~ in text. Relazation

time

(T,)

Ll, Na, Rb: D. F. Holcomb and R. E. Norberg, Phys. Rev. S&l074 (1955); Al, Sn: J. J. Spokes, thesis, sity of Illinois, 1957; Ga: L. Sarles, 1958 (unpublished). EZectrica2 conductivity (.T8 ) q) Ga: R. W. Powell, Proc. Roy. Sm. A209,525 (1951); for the other metals data were taken from the article Gerritsen, in “Encyclopedia. of Physics,” S. Fliigge, ed., p. 178, Springer, Berlin, 1956. Atomic

Univer-

of A. N.

arrangement

Li: C. Gamertsfelder, J. Chem. Phrs. 9, 459 (1941); Na: F. H. Trimble and N. S. Gingrich, Phys. Rev. 53, 278 (1938); Hg: H. Hendus, Z. Natwforsch. 3A. 416 (1948): Al: 888 Gamertsfelderabove (Li); Ga, In, Sn, Bi: H. Hendus. Z. Naturforsch. 2a, 505 (1947). An extensive review of this subject has been published by I. V. Radchenko, Uspekhl Fiz. Nauk. 61, 249 (1957).

180

ELECTRON

STRUCTURE

OF

LIQUID

METALS

181

gives valu.es for m*/m larger than unity, and irrespective of the interpretation in terms of effective mass, a large ratio indicates a density of states at the Fermi surface in excess of the free electron value. Conversely, an effective mass ratio less than unity implies a relative deficiency in the density of states. The latter situation seems to hold for Ga and Bi. We continue to examine the spin contribution to the susceptibility in rows 4 and 5, where the measured total susceptibilities are compared. Row 4 gives the total susceptibility in the liquid. A clearly referenced tabulation is difficult to make, since available and reliable values may exist for one or two of the quantities, XL , ;cs or XI - xs , but not for all of them. In row 4, parenthesis around a value indicates that we have combined values of xs with xl - xs to obtain xz when the direct measurement is lacking or is of doubtful accuracy. Row 5 gives the change xl - xs on melting. We shall regard a change as being significant if its magnitude is comparable to the magnitude of the spin part of the susceptibility. According to this criterion, we see that only Ga and Bi exhibit large changes. (The susceptibilities given for solid Ga and Bi are averages for the values along three directions in the crystal. The fact of the anisotropies in these metals does not alter the conclusions about the gross behavior.) The susceptibility of each of the two liquids has become less diamagnetic, which is consistent with the assumption that the spin contribution is increased appreciably in these metals, though, of course, we must allow the possibility that other diamagnetic contributions are reduced. B.

I,INE

SHIFT

In row 6 we calculate the line shift AH/H from Eq. (3)) using corrected hyperfine coupling constants, and spin susceptibilities from row 2; ,$ is temporarily assumed to be unity. We follow here a procedure outlined in Ref. 9, where, for other than monovalent elements, the measured value of a(s) in a suitable z&,2 state of a free ion is reduced in order to correspond more nearly to the situation in a metal where each atomic cell is neutral. The reduction is performed empirically according to the dependence (9) of PA on spectroscopic term values. Although the procedure is of limited accuracy, it at least provides a consistent means of comparison among several elements as in Table I. Rows 7 and 8 give the measured values for AH/H in the liquid and solid states. We note first that in all cases where the nmr is observable in the solid, the numbers in rows 7 and 8 are in close agreement. The small differences for Ka, Rb, Cs, and Sn appear to be real. However we regard the values for Hg and Al to be the same within the respective experimental errors (0.05, 0.001). We note further that these values are in fair agreement with those in row 6; differences may usually be ascribed to the .$factor, which is less than unity except for Cs. Ga, In, and Bi are given for the liquid only. For In, the observed value is consistent with Eq. (3), if we take

182

KNIGHT,

BERGER AND HEINE

the spin susceptibility from row 2; a reasonable value of 0.5 is indicated for .$. Similarly, the f value for Al appears to be about 0.7. However following the same procedure, we calculate AH/H for Ga to be 0.26 % which is to be compared with the much larger observed value of 0.45 %, if a [ factor is responsible for this difference, we must assign a value for it which is inconsistent with the value of about one half for the other trivalent elements Al and In above. The difficulty of reconciling the behavior of Bi by adjusting the 5 factor is even greater. Alternatively we prefer to regard the results of the above estimate of AH/H for Ga and Bi as arising from the use of the wrong xP in Eq. (3). That is to say, AH/H is actually observed only in the liquid, while xP in row 2 of Table I is derived from measurement,sin the solid, which may not have the same electronic structure. Larger values of xP would seemto describe better the electronic structure of liquid Ga and Bi. C. RELAXATION

TIME

R,ow 9 gives values for the spin-lattice relaxation time at the melting point, calculated from Eq. (6) ; the value of AH/H for the calculation is taken from row 7. We seethat there is reasonable agreement between the measured values of T1 (rows 10 and 11) and the calculated ones, except for Ga. The value for Sn (0.11) has been obtained by extrapolation from the value measured at 77°K. The measurement in liquid Ga follows the prediction quite well, but the measurement in the solid agrees with neither the predicted value nor with the value measured in the liquid. Furthermore, the disagreement is consistent with the occurrence of a reduction of the density of states in the solid, which would tend to increase the relaxation time. It will be interesting to make similar comparisons for In and Bi when the nqr data are available. D. ELECTRICAL

CONDUCTIVITY

Bi and Ga have usually been cited along with Sb as examples of metals for which an increase in conductivity accompanies a volume contraction on melting. This is in contrast with the behavior of most other metals (see for example rows 12 and 13 of Table I), which suffer a loss in conductivity on melting. Unfortunately, since the conductivity of solid Ga is now known to be highly anisotropic, the casefor this metal is not so clear. At 20°C the resistivities along three principal axes are approximately 8, 17, and 55 &l cm, which, properly averaged, imply a value of 15 polocm for the polycrystalline solid. By comparison, the resistivity in the liquid state at nearly the same temperature is about 25 & cm. We make, therefore, three remarks about the change in conductivity for Ga at the melting point: (1) using the value for the polycrystalline solid, we find us/u2 - s%s, which puts Ga in the class with the “normal” metals. (2) However, we note that the theoretical ratio is relatively larger (4.5) than it is for the

ELECTRON

STRUCTURE

OF LIQUID

METALS

183

latter me-tals. (3) We may say also that with respect to one crystal direction (c axis) the ratio u,/ul - 2%~ is such as to classGa along with Bi. (The anisotropy of Bi is not large, and does not affect the argument.) Returning momentarily to the quest,ion of the theoretical ratio (22) us/u1 , which depends on the relations among latent heat, lattice vibrations, and conductivity, we note the general agreement of experimental and predicted values for all but Ga and Bi, where the discrepancies are consdierable any way we look at it. It seemsfair, therefore, to conclude that processesare occurring in these two metals which are not included in Mott’s, theory, and it is likely that there are in liquid Ga and Bi more electrons free to ta,ke part in conduction, corresponding to a restoration (in the liquid) of the deficiency of density of states manifest in the solid. The Hall Effect and other Transport properties depend on the electronic band structure, and one should in principle be able to relate the appropriate experimental results to the problem at hand. Unfortunately the available data are few. Perhaps the most helpful result is that of Buckel (23)) who finds that amorphous films of Ga and Bi exhibit relatively small Hall effects, characteristic of high electron concentrations as in good conductors. The significant thing is that the amorphous films gave x-ray diffraction pictures which resembled the patterns for liquids. It would be valuable to have measurements on the Hall effect in a wide range of temperature for these metals. Solid lithium is anomalous among all the monovalent metals in being the only one with a positive pressure coefficient of resistance. The anomaly is thought to be due to the very distorted Fermi surface (24). The distortion persists above the melting point, suggesting that the form of the electronic band structure is not destroyed on melting. As regards the optical constants, different measurements are at present not in accord with one another. Schulz (25) finds that the optical constants of liquid mercury and liquid gallium fit the Drude-Lorentz free-electron theory very well. However Hodgson’s measurements (26) on merucry do not, and neither do those of Schulz on the liquid Hg, In alloys. From a theoretical point of view it is also quite uncertain what the relationship is between the optical constants and the electronic structure applicable to steady phenomena such as nuclear resonance and de conductivity. (See Section VI.) It is therefore impossible on both experimental and theoretical grounds to evaluate the significance of these data at present. E. ATOMIC

ARRANGEMENT

Our final concern is with the problem of the spatial arrangement of atoms in the solid and liquid states. In describing the ordering of near neighbor atoms it is convenient to refer to coordination number, radius, and type of crystal. One must recognize, however, that tabulation of these quantities without accompanying qualifications can easily be misleading. In particular, it is clear that the co-

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ordination number, which is supposed to give the time and space average of the number of nearest neighbors, is not sufficient to prescribe the actual structural arrangement of neighbors at any instant. Furthermore, no precise way seems yet to have been devised to arrive at a value of the coordination number, in the usual sense, from the x-ray diffraction data. (Data from neutron and electron diffraction experiments do not help to clarify the situation,) Although it might be useful to reproduce here some examples of the individual radial density plots provided by the x-ray diffraction experiments, we have found that a normalized curve represents very well the similar features and moreover serves as a basis for describing the significant differences among the metals. Figure 1 is such a normalized curve, where the first peak is assigned unit height (representing for many metals an average about of 10 atoms per unit radius) and occurs at unit distance from the origin (representing clusters of near neighbors usually at about 3 A). The cutoff on the side nearer the origin is usually at r = 35, and the first minimum usually occurs close to r = 1.25 if small secondary maxima are neglected. We believe that the short range order in the liquid is determined by the atoms associated with the first major peak, and we prefer to use this number instead of quoting coordination numbers and distances. In rows 14 and 15 of Table I, we compare the area under this peak (nl) with the sum of nearest (nn.) and next nearest neighbors (n.n.n.) for each metal in the solid state (n,). Considering the elements in order, it is first evident that the alkalies must contain more than 8 local neighbors in the liquid, since the distance of the 6 n.n.n. 2

n

I I FIG.

1. Typical

curve representing

I

r

I 2

relative density of neighbor

atoms in a liquid metal

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is about r = 1.1. The peak in the liquid pattern therefore is not sharp. Published values for the coordination numbers of Li and Na vary from 8 to 10, the variation probably arising from the arbitrary way in which one is forced to interpret the curves. In an effort to decide on a reasonable value for inclusion in our table, we have integrated several of the published radial density curves independently. In particular, for Li and Tu’a, it was assumed that the extended contour of the first peak falls to the r-axis at a point equally far to the right of the next nearest neighbor (n.n.n.) position as the cutoff is to the left of the nearest neighbor (n.n.) position. On this basis, liquid Li and Na appear consistently to have about 10 local neighbors, verifying the prior suspicion that neither 8 nor 14 would turn out to be the right number because of the ambiguous situation of the n.n.n. position relative to the first minimum of the radial density curve. The numerical comparison is thus difficult, but we are, nevertheless, unable to conclude that the local sltructures of Li and Na undergo any great changes at the melting point. The case for Hg is similar in that the 4 n.n.n. are situated near the position of the first minimum, and one might question their inclusion under the peak. The results, however, indicate no large changes on melting. Solid Al possesses the close-packed f.c.c. structure, with 12 n.n. ; the next 6 atoms are significantly farther away (T = 1.4) from the first maximum and will not be counted as local neighbors. The number 11 for the liquid is certainly consistent with the hypothesis that no great structural change occurs at the melting point. Crystalline Ga shows an arrangement not unlike that of solid iodine, in which a single nearest neighbor is to be found significantly closer than the three neighboring pairs. One regards the number of local neighbors as being either one or seven, according to the desire to emphasize the similarity to a diatomic molecular crystal. In either case we find that the area under the first peak in Fig. 1 is significantly greater than the number of local neighbors in the solid. Numerically, the change from one or seven to eleven is quite large. Furthermore, the change is unambiguously toward larger numbers. For In and Sn, although there may be trouble in interpreting in detail what the relative populations of the first and succeeding coordination spheres might be in the liquid, at least it is certain that the total numbers under the peak, as we describe things, remains substantially the same during the melting process. Finally we see that Bi behaves more like Ga, and if the change on first consideration does not appear to be remarkable, it is to be noted that the three n.n. in solid Bi are inhabitants of the same layer in the crystal structure; these three may therefore be viewed like the single n.n in. Ga. For both metals we notice a correlation between the change of atomic arrangement toward closer packing and the increase of density on melting, and for both we emphasize the anisotropy in their properties, in which the crystals appear to be of less metallic character in

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certain directions. Hilsch (27) has noticed particularly that amorphous films of both Ga and Bi, deposited at low temperatures, resemble the respective liquids and not the solids in their diffraction patterns, density, and transport properties. We enter in rows 14 and 15 of Table I the approximate numbers of local neighbors near the melting point. One will be cautious about inquiring precisely into the exact numerical changesbecauseof the difficulty of determining the numbers. However, Ga and Bi are set apart from the rest of the metals, for the relative changes are quite large, and nl > ns . The selection of metals above is not exhaustive and, of others that might be added to the list, it seemsappropriate to point out that of the nontransition elements we should expect Pb, Tl, and probably Cd to belong to the group which shows no change at the melting point. On the other hand Sb would probably fit into the other group. F. ALLOYS There are a few data on liquid alloys which are relevant to the general theme of the electronic structure of liquid metals. In the solid phase, the existence of the y-structure at a definite electron per atom ratio is explained by the electronic zone structure of this crystal structure (28). The persistence of something like the y-phase in the liquid would therefore be evidence for Brillouin zone structure of some sort. There is some experimental support for this: Matuyama (29) finds a maximum in the resistivity p and a minimum in dp/dT in liquid CuZn, CuSn, CuCd at concentrations which correlate with the y-phase in the solid; Bernal (SO) mentions a maximum in the diamagnetism. G. CONCLUSIONS The preceeding discussion of the data is summarized in Table II by a simple statement for each metal as to whether there is a considerable change on melting in the electronic properties (Row 1) and in the local atomic arrangement (Row 2). The correlation between the two lines is good: a change on melting in TABLE

1. Change props 2. Change

in

electronic

in structure

3. I (a.u.) 4. Ak,/ksz*

in per cent

II

Li

Na

Rb

Cs

Hg

Al

Ga

In

Sn

Bi

No

No

No

No

No

No

Yes

No

No

Yes

No

Yes

No

No

Yes

65 2.0

Prob. No 20 6

25 2

20 4

20 4

20 5

10 15

Prob. Prob. No No 22 155 3.3 0.6

95 1.2

* Ak, = l/l, ksz = mean radius of Brillouin

zone

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187

the electronic properties accompanies a change in the short range structure, and conversely there appears to be virtually no change in the electronic properties if there is, no major rearrangement of the short range structure. This suggests that a liquid metal possesses some kind of electronic band structure, which is very similar t; that of a solid (including zone boundary effects), and which is determined essentially by the short-range atomic structure. The alloy data, as far as they go, also give some support for the conclusion. Lacking the nmr data, the evidence for the existence of nearly identical band structures in liquid and solid metals near the melting point is suggestive but it is not conclusive: additional correlations between the nmr results and other properties tenId to strengthen the argument; further strength is provided when it is appreciated that the observed small change in AH/H arises from either a correspondingly small alteration of the total electronic structure or from a cancellation of several effects. But AH/H is so sensitive to changes in either xP or PF that it is hard to see how nearly perfect cancellation could occur in such widely different metals such as Na, Al, and Sn for example. On a slightly different point, one might argue that the existence of a spherical Fermi surface would imply an insensitivity of the electronic structure to the catastrophe of melting. However, of all the metals we examine here, the Fermi surface in Na is the most nearly spherical, and yet in fact Na shows a relatively large change of AH/H, which is still only two percent or so of the total value. The other metals certainly are not believed to possess spherical Fermi surfaces. In concluding this section, we propose that the nmr results may be used not only in correlations with other data, but independently to make definite statements ab’out otherwise unknown features of the structure of liquid metals. Two examples follow. First, in an x-ray study of a liquid metal, one can measure the approximate number of nearest neighbors which in the case of many metals is between ten and twelve. However the x-ray measurement cannot determine the geometrical arrangement of those nearest neighbors. Frank (31) has pointed out that there are three such possible arrangements: the two usual ones found in the face-centered cubic and hexagonal close-packed structures, and a third one having dodecahedral symmetry with five-fold axes which is not found in crystals. He has even suggested that the last one might be realized in most liquid metals having nearly twelve nearest neighbors. Now our conclusion about the electronic structure of liquid metals depends on the actual short-range structure and not just on t’he number of nearest neighbors. In aluminum for instance the zero change in AH/H indicates that the face-centered cubic short-range structure is not actually changed by melting, which tends to rule out the dodecahedral arrangement for the liquid in this particular case. Second, most metals increase their volume by a few per cent on melting, which might prompt one to suggest naively that PF and hence t should be reduced because of the increase in the total

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volume over which all the wave functions are normalized. However, as is well known from the x-ray evidence, the volume expansion on melting is not a uniform dilation to which the above remarks would apply, but is more like the introduction of vacancies or rather partial vacancies. At any one instant, the distance between the atoms is close to the correct nearest neighbor distance, but there are gaps or “partial vacancies” left in the middle of irregular clusters (52). [In Ga the melting process is anisotropic (3S).] The absence of positive ionic charge in the vacancies is screened by a reduction in the conduction electron density near them. Thus the charge density 1fi 1’ and hence PF and ,$are not reduced over the regions of the actual atoms and we expect little change in AH/H due to the volume change on melting. In actual fact the simple volume effect appears to be quite small (2 or 3 % at most, see Table I). Considering the specific case of aluminum where the volume change is largest (6.6 %) , the small observed change in AH/H is good experimental evidence for the electronic charge density being bound tightly to the ion cores and moving with them, as envisaged say in a rigid spheres model. VI.

THEORETICAL

DISCUSSION

the last section we concluded on experimental grounds that the electronic band structure of a liquid metal is very similar to that of a solid, and in particular that it exhibits features which correspond to the zone structure in a solid, (e.g., density of states considerably higher or lower than free electron value, and associated band gaps). At first sight this conclusion may be somewhat surprising because in the usual theory of solids it is the long-range periodicity which allows one to introduce the wave vector k to label states, and which therefore leads to the definition of Brillouin zones. This long range order is of course destroyed on melt,ing, so why not the zone structure with it? Recently various authors have calculated the distribution of energy levels in simple models of a disordered structure which indicate in a general way the existence of definite bands (34). We shall give what we believe to be a correct though qualitative physical picture of the band structure of a liquid metal, which accounts immediately for the existence of a zone structure. We start by defining what we mean by the electronic band structure of a liquid metal. A liquid metal does not consist of a uniform electron gas with some ion cores moving about in it at random. Rather the x-ray measurements and other evidence show that the atoms remain as actual atomic units with some vacancies or partial vacancies interspersed between them. Each atom has a number of nearest neighbors and next nearest neighbors which are at any instant arranged round the atom in a definite geometrical structure, subject to the distorting influence of the vacancies. The distortion gets so large after a few orders of near neighbors that the positions of the atoms bear no relation to that of the central In

ELECTRON

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189

atom. We can however idealize the regular structure near the central atom and then imagine it continued regularly outwards to form a fictitious crystal. In this way we can define at the central atom R, a set of what we shall call local crystal axes, which are determined at any instant by the positions of the nearest and next nearest neighbors round R, . Since the motion of the electrons is much faster than that of the ion cores, we shall consider the latter to be stationary in some typical configuration such as discussed above. The local axes allow us not only to define the band structure of the liquid metal, but also to relate it to that of the solid having the same short-range structure. We treat the electrons as independent particles in orbitals 1,5i, j = 1 to N, which are eigenfunctions of the stationary disordered structure. These have a precisely defined energy spectrum which is filled up to some exact Fermi level. The orbitals can be ex:panded near any atom R, in terms of the complete set of Bloch functions tik(r) of the imaginary solid structure with crystal axes defined by the local crystal axes R, ; #j(r)

=

F

dk>

#R(r).

This expansion is of course only meaningful over the region of the central atom R, and its nearest neighbors, but since R, is a representative atom it is sufficient to discuss the band structure near this atom. Summing over all occupied orbitals, we obtain the density matrix.

(8) which gives the probability of a given Bloch state & being occupied. We shall regard p(k) as determining the band structure of the liquid metal which we have therefore defined in an operational fashion. In the solid p(k) is unity below the Fermi surface and zero above it. In the liquid it will drop more gradually, and equal values of p(k) will trace out surfaces in k-space. Thus the Fermi surface is not sharp as it is in a solid, but we can arbitrarily take the contour p(k) = g as the Fermi surface for the liquid. We shall not want to associate energies with our p(k) contours since we already have the energy E(k) of &(r) in the solid. We note that the band structure surfaces of the liquid p(k)

= const.

may, but need not in general coincide with the constant E(k)

= const.

(9) energy surfaces, (10)

We ncnv consider the shape of the p(k) contours (9) and the amount of broadening of p(k) to be expected at the Fermi surface. We first ask whether it is

190

KNIGHT,

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really best to use in (7) the Bloch functions &(r) when k lies on a zone face, say at the center of the zone face where k is half a reciprocal lattice vector K, . For such k, $n in the solid has either s-like (Iclg) or p-like (tip) symmetry. In the nomenclature of orthogonalized plane waves (OPW), #s is not a simple running wave of the form (OPW, but a standing

k) = exp(i ksr)

-

(orthogonalizing

terms)

wave of the form cos k*r or sin kar, i.e., (OPW,

k) f

(OPW,

-k

= k - K,).

This splits the energy and produces a gap of the order of 1 ev between tig and tip , because k is such that the electron is in phase in an s-like or p-like state at each atom. Now in the liquid the electron with wave vector k also has the same phase at the central atom and all the nearest neighbors, so that it can lower its energy just as in a solid by going into a purely s-like or p-like one, whichever has the lower energy. It is reasonable to expect therefore that we obtain a better representation of the #i by using in (7) the standing waves g8 or $, according to the energy and band we are considering, rather than by using running waves of the form (OPW, k) . We next relate the form of (7) to the electronic mean free path. Because of the large mass of the atoms relative to the electronic mass, the scattering by the disorder and the vacancies is to a first approximation elastic, so that each eigenfunction tii (7) will certainly contain all $k’s with k lying on a constant energy surface (10). Moreover the scattering when averaged over the whole liquid is isotropic so that all #k’s with the same energy will appear in (7) with the same amplitude. For a liquid metal we do therefore have that the surfaces defined by (9) and (10) are the same. The variation of 1 ai 1 normal to the constant energy surfaces is given as follows. If 7 is the relaxation time and 1 the electronic mean free path, we have AE N fi/,

or

Ak, -

I/l.

(11)

Here AE is the uncertainty in the energy of a wave packet with definite k, and Ale, the width in k-space over which 1aj(k) 1is large. Equation (11) follows from the uncertainty principle: alternatively Mott (56) has pointed out that 1 may be interpreted as the distance over which a wave packet loses its phase coherence so that its spread in k-space must be -l/Z. Thus p(k) drops from unity to zero over a range of Ic of order l/Z. Near the melting point 1is surprisingly large, namely of the order of 10 interatomic spacings which is several times the range of order. Typical values of I are shown in Table II (line 3)) and we note that the uncertainty AL is only a few per cent of the size of the Brillouin zone (line 4). We conclude that the Fermi surface is fairly sharply defined, and has the same shape

ELECTRON

STRUCTURE

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METALS

191

as in the solid with the same short-range structure. Likewise we expect the density of states N(E,) and other properties to be affected very little by melting. The preceeding picture gives a physical basis for the experimental evidence of Section V for the existence of zone boundary effects in the electronic structure of a liquid metal. At the same time, the picture focuses attention on two questions which are at the crux of the matter. Firstly, why in the expansion (7) should one use the s-like or p-like standing waves +k of the solid rather than the running waves when k lies on a zone boundary? Secondly, why is the mean free path I so long in a liquid metal in view of the short range of order? In the hope of answering these questions in a fundamental way, we have tried to do a band structure calculation for a liquid metal using orthogonalized plane waves (OPW). Although the atternpt was unsuccessful due to the difficulty of treating the near-degeneracy between the OPW’s, some semiquantitative points of interest did emerge which we shall describe briefly. As basis functions we used the simple plane waves exp(i ker), orthogonalized to the ion core of each atom wherever the atom happens to ‘be. These OPW’s had an additional modulating factor to represent the decrease in electronic charge density about the vacancies in the liquid. The matrix element Hw’ of the Hamiltonian H with respect to the OPW’s was then calculated, where H = -V* + cm V(r - R,) and V(r - R,) is a spherical potential centered about the atomic nucleus R, . Inspection showed that Ha’ could be divided very roughly into three parts. (i) A term almost identical with that found in a band structure calculation of a solid. It is largest when k - k’ x K, (a reciprocal lattice vector), and its systematic phase factor tends to couple (OPW, k) and (OPW, k’) at each atom into the standing wave form as in a solid, although the details of this are not clear. (ii) A term due to the disordered arrangement of the atoms, proportional to the variation of the band gap over the zone surface. (iii) A term from the modulation of the charge density round the vacancies. The latter two terms each give rise to scattering of the electrons and a broadening of the drop of p(k) at the Fermi level. Both these effects have been estimated for the case of aluminum using perturbation theory and the band structure results of Heine (36). The terms of type (ii) and (iii) above were found to be of comparable magnitude, and we obtained 1 x 15 a.u. in comparison with the experimental value of above 25 a.u. The broadening at the Fermi surface came out at about 36 ev, in comparison with a total band width of 15 ev and banld gaps of order 2 ev in the solid. Thus the results, as far as they go, are entirely in accordance with our picture of the band structure of a liquid metal. Finally it should be emphasized that the present picture and definition of p(k) is based on the nature of the wave functions near one atom R, . It is therefore only useful for discussing such physical properties of the liquid metal as are

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determined by the local form of the total wave function, e.g., the nmr shift. In particular it is not clear whether the model should be in any way relevant to the optical constants, unlike the solid (25, WS). In an optical experiment the wave length is long compared with both the mean free path and the range of order, so that the light cannot probe the local form of the wave function. By its nature, the light interacts only with some grosser space-averaged property of the wave function. Furthermore the light does not interact with each electron in a particular $i individually (except in an absorptive transition), but produces a coherent motion of the electron gas as a whole. Thus our model cannot be used to discuss the optical constants of liquid metals. However the density of states and electron velocity depend directly on the local band structure, and so does the velocity operator u = (6l2im)

(#* grad 9 - $ grad I/J*),

so that we would expect our picture to apply to dc transport

properties.

ACKNOWLEDGMENTS

We are indebted to Dr. L. Sarles for performing the measurements of the relaxation in Ga. The work has benefited from support by the Alfred P. Sloan Foundation U. S. Office of Naval Research. RECEIVED:

April

and

times the

15, 1959 REFERENCES

1. I. N. KOLINKINA 6. W. D. KNIGHT,

AND‘ P. G. STREKOV, Soviet Union JETP 34(7), 616 (1958). Solid State Physics 2, 93 (1956). 8. V. HEINE, Proc. Roy. Sot. A240,340 (1957). 4. W. D. KNIGHT, R. R. HEWITT, AND M. POMERANTZ, Phys. Rev. 104, 271 (1956). 5. J. D. BERNAL, Trans. Faraday Sot. 33, 27 (1937); B. R. T. FROST, in “Progress of Metal Physics,” Bruce Chalmers, ed., Vol. 5, Chapter Pergamon, London, 1954; R. LANDAUER AND J. C. HELLAND, J. Chem. Phys. 22, 1655 (1954); W. A. HARRISON, Phys. Rev. 110, 14 (1958); T. SUGAWARA, Science Repts. TBhoku Imp. Univ. 38, 238 (1954). 6. W. PAULI, 2. Physik 41, 81 (1927). 7. E. FERMI, 2. Physik 80, 320 (1930). 8. S. GOUDSMIT, Phys. Rev. 43, 636 (1933). 9. E. FERMI AND E. SEGR& 2. Physik 82, 729 (1933). 10. J. KORRINGA, Physica 16, 601 (1954). 11. D. PINES, Solid State Physics 1, 367 (1955). 12. G. PARE, Solid State Physics 2, 1 (1956). 1s. N. BLOEMBERGEN AND T. J. ROWLAND, Phys. Rev. 97, 1679 (1955). 14. W. D. KNIGHT, Phys. Rev. 92, 539 (1953). 16. W. D. KNIGHT AND R. R. HEWITT, Phys. Rev. Lett. 3, 18 (1959). 16. D. PINES AND C. P. SLICHTER, Phys. Rev. 100, 1014 (1955). 1’7. J. PETIT AND N. H. NACHTRIEB, J. Chem. Phys. 24, 1027 (1956).

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METALS

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18. L. R. SARLES (unpublished). 19. M. POMERANTZ, thesis, University of California, Berkeley, 1958. 20. R. V. POUND AND W. D. KNIGHT, Rev. Sci. Instr. 21, 219 (1950); R. V. POUND, Progr. Nuclear Phys. 2.21 (1953). 21. H. E. WALCHLI, ed., “A Table of Nuclear Moment Data,” Oak Ridge Nat.ional Laboratory, June 1, 1953; Suppl. II, Feb. 1, 1955. 22. N. F. MIOTT, Proc. Roy. Sot. A146.465 (1934). 23. W. BUIZKEL, in “Low Temperature Physics and Chemistry,” J. R. Dillinger, ed., p. 326. Univ. of Wisconsin Press, 1958. See also YAHIA and MARCUS, Phys. Rev. 113,137 (1959). 24. M. H. COHEN AND V. HEINE, Phil. Mag. Suppl. 7,395 (1958). 25. L. G. SCHULZ, Advances in Physics 6, 102 (1956). $6. J. N. HODGSON, private communication, 1958. 27. R. HII,SCH, in “Low Temperature Physics and Chemistry,” J. R. Dillinger, ed., p. 276. Univ. of Wisconsin Press, Madison, 1958; private communication. 88. N. F. MOTT AND H. JONES, “The Theory of the Properties of Metals and Alloys,” p. 172. Clarendon Press, Oxford, 1936. 29. Y. MATUYAMA, Science Repts. TBhohx Imp. Univ. 16, 447 (1927). 30. J. D. BERNAL, Trans. Faraday Sot. 33, 38 (1937). 31. F. C. FRANK, Proc. Roy. Sot. A216, 43 (1952). 32. B. R. T. FROST, in “Progress in Metal Physics,” B. Chalmers, ed. Vol. 5, Chapter 3, Fig. 8b. Pergamon Press London, 1954. 33. J. SACLI AND F. SEBBA, Trans. Faraday Sot. 60, 226 (1954). 34. A. I. GUBANOV, Soviet Union JETP 1, 364 (1955) ; R. EISENSCHITZ AND P. DEAN, PTOC. Phys. Sot. A70, 713 (1957); see also Ref. 5. 35. N. F. MOTT, private communication. 36. V. Heine, thesis, University of Cambridge, 1956; Proc. Roy. Sot. A249, 360 (1957).