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Electron-phonon interactions in cadmium measured by inelastic neutron scattering

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1979 J. Phys. F: Met. Phys. 9 1983

(http://iopscience.iop.org/0305-4608/9/10/009)

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J. Phys. F: Metal Phys., Vol. 9 No. 10, 1979. Printed in Great Britain

Electron-phonon interactions in cadmium measured by inelastic neutron scattering

A A Chernyshovt, V V Pushkarevt, A Yu Rumyantsevt, B Dorneri and R PynnS t I V Kurchatov Institute of Atomic Energy, Moscow, USSR $ Institut Laue-Langevin, Grenoble, France

Received 2 February 1979

Abstract. Precision measurements of the dispersion relations for some low-energy phonons in Cd have been performed at 77 K by inelastic neutron scattering. Phonon group veloci- ties have been extracted from the data with an accuracy of 5-10%. Several abrupt changes of group velocity as a function of wave vector have been identified as Kohn anomalies. The wave vectors and strengths of the anomalies are compared with calculations based on a model electron-ion potential and a spherical Fermi surface. Discrepancies between experiment and theory are qualitatively explained in terms of deformations of the Fermi surface from sphericity. These deformations are consistent with those determined by R W Stark and L M Falicov. Extension of the calculations to include third-order terms in the model potential leads to the prediction of further phonon anomalies, none of which are observed.

1. Introduction

In metals the phonon dispersion curves exhibit a fine structure which reflects features of the electron-ion interaction and the properties of the itinerant electrons. The exist- ence of a sharp, well defined Fermi surface results in abrupt changes of phonon frequencies at wave vectors which are related to extrema1 dimensions of the Fermi surface. Such ‘kinks’, known as Kohn anomalies, have been extensively studied (Woll and Kohn 1962, Taylor 1963, Roth et a1 1966) since they were first predicted twenty years ago (Kohn 1959). The presence of Kohn anomalies in both transition (Powell et a1 1968, Shaw and Muhlestein 1971, Dutton et al 1972) and nontransition (Stedman et al 1967, Brockhouse et al 1961, Vosko et al 1965) metals is now well established.

Brovman and Kagan (1974) in their treatment of nontransition metals have pre- dicted a hierarchy of phonon anomalies related to the indirect interionic interaction mediated by the conduction electrons. In addition to the Kohn anomalies which appear when this interaction is central, Brovman and Kagan find further anomalies which reflect the noncentral character of the interionic forces. Expressed somewhat differently, the Kohn anomalies appear in perturbative calculations of phonon fre- quencies which are correct to second order in the electron-ion pseudopotential. To find the anomalies predicted by Brovman and Kagan a third-order calculation is required. Recently, fine structure which appears to confirm the existence of these third-order (or three-particle) anomalies has been found in the phonon spectra of aluminium (Rumyantsev et al 1978).

0305-4608/79/101983 + 13 $01.00 1983 @ 1979 The Institute of Physics

1984 A A Chernyshoc et a1

In this paper we present measurements of the frequencies of some low lying phonon branches in cadmium. This element shows a clear departure from the ideal cja ratio of HCP materials. Pseudopotential calculations (D Weaire, private communica- tion) incorrectly predict this cia ratio but show an unusually weak dependence of the lattice cohesive energy on cja. In this situation, any effect omitted from the pseudopo- tential calculation can be decisive in determining the c/a ratio actually observed. One such effect is the noncentral interionic forces discussed by Brovman and Kagan (1974). As a corollary of the non-ideal c/a ratio, the phonon modes propagating in the basal plane and polarised along the hexad axis are of low frequency. For these modes there is thus a significant cancellation of electrostatic and bandstructure contributions to the dynamical matrix and, once again, noncentral forces may play an important role.

Measurements of the phonon spectrum of cadmium have already been made at the IRT-M reactor at the Kurchatov Institute (Chernyshov et al 1975). Detailed measurements have also been performed to search for some of the anomalies discussed above. Although some anomalies were observed, the results were neither complete nor of sufficient accuracy for detailed comparison with theory. For this reason, the present investigation of the group velocities of low-frequency modes was undertaken at the Institut Laue-Langevin.

2. Experimental details

Since naturally occurring cadmium absorbs neutrons strongly (cross section - 2650 barn), this experiment has been performed with a monocrystal of the isotope "'Cd which has an absorption cross section of 5 2 barn. The sample, grown by the Bridg- man technique, was in the form of a cylinder of volume - 20 cm3 and mosaic spread (FWHM) - 16'. To avoid deformation of the sample during the experiment it was mounted in an aluminium sleeve of thickness 0.2 mm.

Phonon measurements were made on the three-axis spectrometer IN2 at the Insti- tut Laue-Langevin. The constant-& mode of operation was used with fixed incident neutron energy. For measurements of the low-energy branches both monochromator and analyser pyrolytic graphite (002), and the wavelength of incident neutrons was i = 2.36 A. A pyrolytic graphite filter before the sample served to reduce the second- order contamination in the beam. For high-resolution measurements at small q(q 5 0.1) the incident wavelength was i. = 4.054 A. In this case cooled Be was used as a high-order filter. A Cu(220) monochromator (i. = 1.5 A) and pyrolytic graphite (004) analyser were used for measurements at large energy transfer (w 2 1.2 x 1013 rads- '). As far as possible, the beam collimations were chosen to equalise the contributions of monochromator and analyser to the energy resolution.

Since the aim of this experiment was to measure the phonon group velocity dw/dq, very high relative accuracy of the phonon frequencies was required. A typical phonon of the T3 branch at w = 0.8 x 10l3 rads- ' had an energy width of 0.1 x 10l3 rads- ' , which represents the instrumental resolution. All phonons were fitted to Gaussians modified by the normalisation factor k3cot0, (Dorner 1972), where k is the wave vector of the scattered neutrons and QA the Bragg angle of the analyser. From these fits the phonon frequencies and their standard deviations were deduced. Neither the standard deviations nor the differences between repeated scans exceeded 0.0012 x l o i3 rads- ' for transverse phonons or 0.003 x 1013 rads- ' for longitudinal modes.

Electroiz-phonon interacrioris in cadrnium 1985

The group velocity dw/dq was calculated as AwlAq, where Aw and Aq are the differences of frequency and wave vector between adjacent measured points. Although the absolute accuracy of the measured phonon frequencies is only -0.01 x 10' rads- ' . the relative accuracy is sufficiently good to allow group veloci- ties to be extracted with an error of 5-10°/0. Intervals Aq used in the determination of Ao;Aq were: 0.042 A - ' for the direction T, 0.031 A - ' for the direction X and 0.028 A - ' for the direction A. These numbers should be compared to typical trans- verse and longitudinal momentum resolutions of 0.09 A - ' (mosaic spread included) and 0.04 A - respectively.

3. Experimental results

The measured phonon dispersion curves are given in figure 1 and a list of phonon frequencies is presented in table 1. In figures 2 to 5 the measured dispersion curves and the derived group velocities do/dq are plotted for the low-frequency branches along the high-symmetry directions T, T', C, A and U (see figure 6(d) for the symbols).

For 4-0 our group velocities agree well with those calculated from measured elastic constants (Garland and Silverman 1960) which are represented by filled sym- bols at q = 0 in figures 2 to 4.

Several abrupt changes of group velocity (anomalies) are evident in figures 2 to 5. In addition, the transverse branches T, and Z,, which are polarised in the [OOl] direction, have group velocities which, at small q, decrease rapidly as q in- creases. The latter behaviour is very different from that found in other simple metals

1 0-13

D

3 e - x u C 3 W

U

r T - K T L M --I: r A - A

Phonon wavevec tor q

Figure 1. Dispersion curves of "'Cd at 77 K. Different symbols are used for different branches to distinguish in regions where they come close to each other. Symmetry labels are explained in figure 6 ( d ) .

1986 A A Chernyshoc et a1

like A1 or Cu (Weymouth and Stedman 1970, Nilsson and Rolandson 1974) and may indicate an incipient lattice instability.

4. Discussion

It is known that the static and dynamic properties of the majority of nontransition metals can be well described by (almost) free-electron models. Therefore we shall

Table 1. Measured values of phonon frequencies ( x 10l3 rads- ' ) and their standard devi- ations for five branches measured at 77 K. Values of q are in units of 4 ~ / a for the T-T' direction, units of 4n/\ '5a for the Z direction and units of ZR/C for the A and U directions. For q = 0.3 and 0.35 in the A direction two frequencies are given. These represent the overlap between (low-frequency) measurements made with a PG(002) monochromator and analyser and measurements with a Cu(220) monochromator and a PG(004) analyser. On the X3 branch there is a similar overlap between low-frequency Be filter measurements and PG filter measurements.

0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.1 1 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 030 0.3 1 0.32 0 3 3 0.34 0.35

0.1307 0.1855 0.2315 0.2810 0.3142 0.3431 0,3727 0.3988 0.4209 0.4418 0,4553 0.4706 0.4822 0.4964 0,5053 0.5157 0.5292 0,5373 0.5474 0,5556 0.5630 0.5777 0.5943 0.6114 0,6358 0.6540 06754 0.6950 0,7205 0.7467 0,7689 0,7902 0.8042 0.8 137

04)009

0.0010 OQ009 04009 OGQ09 0.001 1 0~0008 0.001 1 0.00 14 00012 0.00 10 0~0011 OG009 0.00 1 2 0.00 16 0.00 18 0.00 14 0~0018 00014 0.0022 0.00 1 3 0.00 19 0~0020 0.0020 0.0020 0~0020 0.0021 0.00 16 00021 0.002 1 00016 0.0026 0,0026

0.0250 0.0375 0,0500 0,0675 0,0750

0,0875 0.1oOo

0,1125 0.1250 0,1375 0,1500 0,1625 0,1750 0.1875 0.2000 0,2125 0.2250 0,2375 0.2500 0.2625 0.2750 0,2875 0,3000 0,3125 0,3250 0.3375 0.3500 0.3625 0,3750 0,3875 0~4O00 0,4125 0,4250

0.1022 0,1458 0,1849 0.2195 0,2505 0,2596 0.2791 0.3064 0.3137 0.3385 0,3672 0.3893 0.4090 0,4268 0,4426 0.458 1 0.4728 0.4860 0.5010 0,5123 0,5262 0,5381 0.5497 0.5581 0.5711 0.5855 0.6019 0.6198 0.6373 0.6587 0,6798 0.6974 0.7206 0,7422 0,761 7

0.0007 OQC07 0~0006 0.0005 0.001 1 0,0007 0,0007 0.0007 0.0009 0~0007 0~00 12 0.001 1 00008 0.0008 0 0 " OQO 12 0 4 " Oaoll 0~00 12 0.00 10 0.0012 04lO13 0.0009 OQ016 0.0012 0.00 12 0~0010 0~0011 0.001 4 0.00 12 Oft013 0.0013 0.00 13 0.00 14 oao 1 2

0.150 0.200 0.250 0.300

0,350

0.375 0.400 0.425 0.450 0,475 0.500 0,525 0.5 50 0.600 0.650 0.700 0.800 0.900 1.000

0.4254 0.0035 0,5592 0,6780 0.0035 0,7990 0.0035 0,8257 0.0053 0,9060 0.0020 0,9317 0.0043 0.9840 0,0060 1.0394 00050 1.1021 OQ050 1,1540 0,0050 1,2265 0.0040 1,2997 0.0040 1,3660 0,0025 1,4311 0.0035 1,5229 0.0030 1,6290 0.0045 1,7153 0,0045 1.8582 0.0075 1.9541 OG070 1,9942 OQO70

4

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

As, A6 w A u

0.1909 0.0030 0,2721 0.0010 0,3493 0.0007 0,4119 0.0010 0,4774 0.0007 0.5296 0.0010 0,5831 0.0010 0.6233 0.0008 0,6631 0~0010 0.6968 00013

Electron-phonon interactions in cadmium 1987

Table 1. Continued

T, + Ti 2 3 A I , A2

4 0 Aw 4 w f Aw 4 w f Aw

0.36 0,8213 0.37 0.8221 0.38 0.8296 0.39 04564 0.40 0,9053 0.41 0.9574 0.42 0.9945 0.43 1.0256 0.44 1,0489 0.45 1,0676 0.46 1.0860 0.47 1.0990 0.48 1,1068 0.49 1.1115 0.50 1,1160

04)028 0.00 19 0.0028 0.0016 0.0013 04"4 00014 0.00 14 O+M19 00015 0.00 17 00022 0.00 19 OG018 00023

0.4375 0,7810 0.0012 0.60 0,4500 0,7994 OCQ12 0.65 04625 0.8167 0.0015 0.70 0.4750 0,8304 OC4I15 0.75 0.4875 0.8374 0.0015 0.80 0.5000 0,8432 0,0016 0.85

0.90 0.95 1.00

0.7264 0.0010 0.7518 04"2 0.7726 OW12 0.7955 00018 0.8037 0,0017 0,8136 0.0020 08214 0.0022 0.8231 0,0016 0.8234 0~0021

U2 4 w 'I A w

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

0.8722 0.8696 0,8419 0,8020 0,7917 0.8 164 0.8489 08607 0.8696 0.8744 0.8869

04085 0,0088 0q055 0w60 0g1m7 0,0044 0g040 0.0025 0,0026 0w44 0.0035

begin the analysis of the observed fine structure in the Cd dispersion curves on the basis of results obtained for a spherical Fermi surface.

In this case the wave vectors q at which Kohn anomalies occur are determined by the relation

+ l q + G ( h k l ) / = kF (1)

where G ( h k l ) is a reciprocal lattice vector and kF is the radius of the Fermi sphere. In figure 6 the Fermi sphere is drawn superposed on the Brillouin zone scheme

for cadmium; the first three zones are filled or partially filled by electron states. According to equation (1) the positions of expected Kohn anomalies may be found by the following procedure:

(i) choose a point corresponding to G(hk1)/2 in figure 6 (several such points are marked) ;

(ii) proceed from the chosen point in the direction q (normally a symmetry direc- tion) until the Fermi surface is encountered;

(iii) if the distance between the chosen (starting) point and this Fermi surface intersection is 14/21, the Kohn anomaly will be found at wave vector q.

The Kohn anomalies which this simple geometrical model predicts for the various symmetry directions are numbered 1 to 12 in figure 6 and are listed in table 2. It is to be noted that, in the absence of interaction between the Brillouin zone boundaries and

1988 A A Chernyshoc et al

Table 2. Cadmium Kohn anomalies. The columns give respectively the branch label, the value of q (for units see table 1) at which anomalies are calculated to occur for a spherical Fermi surface, the reciprocal lattice vectors required in equation (1) to produce the anomaly, the sense of the anomaly and its measured position. The anomaly numbers (column 2) are the same as those given in figure 6.

Reciprocal No. Calculated lattice Expected

Branch of position of vectors type of Experimental position label anomaly anomaly (4) (equation (1)) anomaly of anomaly

T3 1

2

3

0,037

0.197

0,230

max.

max.

min.

absent

absent

0.210

4

5

0.397

0.390

002 0 07 111 11T

0.407

0,367

max.

min.

TO1 TOT 011 TI1 011 T l T 102 1 02

0.055

0.105

0.3 15

absent

not conclusive

0280

max.

max.

min.

101 011 TO1 011 IT1 TI1 002

Ai.5 9 0.240 absent

0,485

max.

max. 10

A2.6 11

0,490

0,240 101 011 TO1 011 IT1 TI1

absent min.

002 TO2

U2 12 0.240 0.250 max.

the Fermi surface, a Kohn anomaly can only occur for a phonon whose polarisation vector has a component parallel to the correspnnding kF. Thus, for example, vectors G(h, k, 0) cannot contribute in equation (1) to anomalies in the T3, T; and Z3 modes which propagate in the basal plane and are polarised (by symmetry) along [OOl].

Examining the data shown in figures 2-5 we find five anomalies which occur at wave vectors close to those calculated. There are three strong maxima of do/dq (anomalies 4, 10, 12) in the T', A and U directions and two somewhat weaker minima (numbers 3 and 8 in figure 6) in the Z3 and T3 branches. More details of our assign- ments of the anomalies are given in table 2, which lists both the expected positions of the anomalies, the G vectors satisfying equation (1) and the expected senses of the anomalies. The latter are determined according to whether G/2 lies inside (maxi- mum in dw/dq) or outside (minimum in do/dq) the Fermi surface. It is noticeable

Electron-phonon inreractions in cadmium 1989

0 0.1 0.2 03 OL 0.5 r T K T ' M

(I-

Figure 2. (a), dispersion curves (full circles) and group velocity do/dq (open circles) for the T3-T3 branch at 77 K. At q = 0 the group velocity obtained from the elastic constants (Garland and Silverman 1960) is represented by a full circle. The line is a guide to the eye. (b), theoretical predictions of the group velocity for the T,-T; direction. The full line is calculated in perturbation theory including second-order terms in the potential; the broken line including third-order terms in the potential. The numbers refer to the anomalies listed in table 2.

in table 2 that in no case has a measured anomaly been assigned to G (101), and that the reciprocal lattice vectors involved are either (1 ll), (102) or (002). To determine the reasons for these observations quantitative calculations of the strengths of the anomalies are required.

Calculations were carried out using a local, Heine-Abarenkov model potential fitted to the Cd phonon spectrum by Chernyshov et a1 (1975). The bandstructure contribution to the dynamical matrix was calculated in lowest order with the results displayed as solid curves in the lower parts of figures 2 to 5 . From the calculated curves it is evident that the anomalies related to vectors G (101) (numbers 2, 6, 9 and 11) are predicted to be comparable to or stronger than those anomalies which we have observed. The absence of these anomalies in the data is one of the most striking discrepancies between calculation and experiment and indicates the need to include in the calculation distortions of the Fermi surface from sphericity.

For nonspherical Fermi surfaces two types of electron transitions contribute to Kohn anomalies; those for which the wave vectors of the initial and final electron states are related by inversion symmetry and those for which no such symmetry exists. Kagan et al(1979) have shown that the second type of transition has a relatively small effect on phonon frequencies in Cd. Thus, for this discussion, we shall consider only transitions between diametrically opposite electron states. This allows us to use the

1990 A A Chernyshov et ai

I ~ I I ~

( a )

1.75

c

Ll 0.50 I

3

3.25

3

3

r f M

q- Figure 3. (a) and (b), dispersion curve, group velocity and theoretical calculations for the Z, branch. For explanation of symbols see figure 2.

same geometrical construction as above (cf. figure 6), albeit with a modified nonspheri- cal Fermi surface.

A great deal of experimental and theoretical work (for example by Visscher and Falicov (1972), Stark and Falicov (1967) and Gaidukov (1970), and references cited by these authors) has been devoted to the investigation of the Fermi surface in Cd. As shown in figure 6 the lattice potential strongly distorts the Fermi surface from sphericity and the number of conduction electron states is reduced to about one-half of its value for a spherical Fermi surface. The conduction electron states, which for a spherical Fermi surface lie in the third zone beyond the zone boundary which bisects the reciprocal lattice vector [ lol l , are thought to be suppressed in the real system. These electron states comprise several 'butterflies' which would, for a spherical Fermi surface, fill in the unshaded oval regions of figure 6(a). All the unshaded areas in figure 6 correspond to filled electron states bounded by a zone boundary and a corresponding energy gap.

In the present experiment the Kohn anomalies numbered 1, 2, 6, 7, 9 and 11 which correspond to the use of G(101) in equation (1) are absent. This can be under- stood if the Fermi surface shown in figure 6 is the correct one. In this case the absence of the 'butterflies' implies that equation (1) is satisfied with G(101) and q = 0. Anomalies 1, 2, 6, 7, 9 and 11 thus occur at q = 0 rather than at the finite wave

Electron-phonon interactions in cadmium 1991

Figure 4. ( a ) and (b), group velocities and 0 0.5 1 0 theoretical calculations for the A,-A2

and A5-A6 branches. The results are pre- r A A A r sented in an extended zone. For explana-

9- - 9 tion of symbols see figure 2.

vectors obtained with a spherical Fermi surface. The anomalies will be weakened but not necessarily suppressed by the existence of the band gap at the [loll boundary of the second Brillouin zone.

It should be noted that ‘butterflies’ are expected to occur in Zn and in Mg. Unfortunately inelastic neutron scattering data on Zn (Almquist and Stedmann 197 1) and on Mg (Pynn and Squires 1972) are not sufficiently detailed to give decisive information about Kohn anomalies corresponding to the ‘butterflies’.

Our experimental observations indicate a rapid increase of group velocity towards q = 0. This effect, which is more pronounced for the T and E directions (see figures 2 and 3) than for the A direction (figure 4) is substantially greater than would be expected in normally behaved metals. It may indicate an anomaly at q = 0. The strength of this effect depends on the angle between q and the second Brillouin zone boundary. An anomaly at q = 0 would explain the failure of the lattice dynami- cal calculations based on a spherical Fermi surface to describe adequately the elastic moduli (Chernyshov et al 1975).

From figure 6 it is seen that the shape of the Fermi surface near the CO021 direction is not distorted from sphericity. This is confirmed in the present experiment by anomaly number 10, for which the measured and calculated positions agree closely. The strong anomaly number 12 on the U, branch is also in the position expected from a spherical Fermi surface.

A third group of anomalies (numbers 3, 4 and 8) corresponds to a region in which the spherical Fermi surface is thought to be distorted. All three anomalies

1992 A A Chernyshoc et a1

Figure 5. (a) and (b), dispersion curve, group velocity and theoretical calcula- tions for the lowest branch in the U direction. For explanation of symbols see figure 2.

L

lie on the plane + (hk2) . which is the boundary of the second Brillouin zone. The deformation due to the presence of this Brillouin zone boundary is less that that caused by the [ lo l l boundary. The anomalies are all found (crosses in figure 6) to be outside the k , sphere and to require a k F in equation (1) which is 2% larger than the free-electron value. It is possible that more information could be extracted from these anomalies by an analysis of their strengths and shapes. However, such is beyond the scope of this paper.

In figures 2-5 we present a second theoretical curve (broken curve), the calculation for which includes non-pair interionic interactions mediated by the conduction elec- trons (Brovman and Kagan 1974). In addition to the Kohn anomalies found in the second-order calculation (full curve), additional anomalies appear which arise from third-order terms in the perturbation expansion of the electron energy in powers of the model potential. Several such anomalies appear to have a magnitude similar to those of observable Kohn anomalies. However, we are not able to identify unambi- guously any three-particle anomalies in our data. In the T and X directions we observe some small irregularities of dw/dq in the regions of expected three particle anomalies but the data do not permit a positive identification. For the T’ direction the calculated three-particle anomaly is probably too close to the strong observed Kohn anomaly to be separable from the latter. Finally, in the A direction, the pre- dicted, strong three-particle anomaly is certainly absent.

A three-particle anomaly is expected to occur at wave vector q if the vertices of a triangle whose sides are G , + q, G 2 + q and G 3 (Cl, GZ, G 3 are reciprocal

Electron-phonon interactions in cadmium 1993

lattice vectors) lie on the Fermi surface. The detailed shape of the Fermi surface is thus important in determining both the positions and magnitudes of these anom- alies. In our calculations, for example, the absence of the [loll ‘butterflies’ has the effect of reducing the three-particle anomaly in the A, branch. Given that the geo- metrical condition described above is satisfied, the magnitude of the three-particle anomaly depends on the product of the Fourier components of the pseudopotential with wave vectors G, + q, G 2 + q and G,. The strength of the anomaly is thus sensitive to the detailed shape of the pseudopotential rather than to its value at a single wave vector. Relatively small changes of the pseudopotential, which have little overall effect on phonon frequencies, could cause a drastic change in the magni- tudes of three-particle anomalies and explain our failure to observe such anomalies. Finally, it is possible that the use of a local pseudopotential, fitted to phonon frequen- cies, is not adequate for a discussion of three-particle anomalies in cadmium.

Figure 6. The Fermi surface of Cd in the extended zone scheme. (a), projection on the basal (T-X) plane. The Fermi surface for free electrons is shaded while the boundaries of filled bands (boundaries of the second Brillouin zone) are unshaded. (b ) and (c ) are projections on the A-T and A-Z planes respectively. The hatched regions comprise the second Brillouin zone. In parts (a), (b ) and (c ) the numbers identify the anomalies listed in table 2 and discussed in the text. The full circles are the calculated positions of the anomalies while the crosses are the observed positions. (d), first Brillouin zone with sym- metry labels for HCP structures.

1994 A A Chernyshov et a1

1

0

r T K T' M q -

Figure 7. The group velocity of phonons for the Z3 and T,-T; branches measured at 300 K.

In figure 7 some earlier room temperature measurements made at the IRT-M reactor are presented. Comparison with figures 2 and 3 shows that a change of temperature from 77 K to 300 K does not substantially alter the observed anomalies. In fact the differences between the measurements can be explained in terms of the difference in resolution of the spectrometers used for the measurements.

5. Conclusion

We have made accurate measurements of several branches of the phonon spectrum of Cd and have deduced the group velocities of these branches. Several anomalies in the group velocities as a function of wave vector have been identified as Kohn anomalies. Although we have explained, at least qualitatively, some discrepancies between our results and the predictions of simple calculations based on spherical Fermi surfaces, a number of puzzles remain. The observed anomalies are, for example, different in both strength and shape from those calculated. These differences are especially unexpected in the case of anomaly number 10 which involves electron states on an ostensibly spherical part of the Fermi surface, for which our model should work well. Evidently, further effort is necessary if these remaining discrepancies are to be resolved.

Electron-phonon interactions in cadmium 1995

Acknowledgments

The authors express their gratitude to Yu M Kagan and N A Chernoplekov for constant support and valuable discussions, to P Flores for help with the measure- ments and to M N Severov for preparation of the sample. A Chernyshov and A Rumyantsev thank ILL for their hospitality and for the use of their facilities.

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Dorner B 1972 Acta Crystallogr. A28 319-27 Dutton D H, Brockhouse B N and Miller A P 1972 Can. J . Phys. 50 2915-27 Gaidukov Yu N 1970 U s p . F i z Nauk 100 449-66 (1970 Soc. Phys.-Usp. 13 194-203) Garland G W and Silverman J 1960 Phys. Rel;. 119 1218-22 Kagan Yu, Pushkarev V V and Kholas A 1979 Soc. Phys.-JETP in press Kohn W 1959 Phys. Rev. Lett. 2 3 9 3 4 Nilsson G and Rolandson S 1974 Phys. Rec. B9 3278-84 Powell B N, Martel P and Woods A D B 1968 Phys. Ret . 171 727-36 Pynn R and Squires G L 1912 Proc. R . Soc. A326 347-60 Roth L M. Zeiger H J and Kaplan T A 1966 Phps. Reo. 149 519-21 Rumyantsev A Yu. Pushkarev V V, Zemlianov M G. Parshin P P and Chernoplecov N A 1978 Zh.

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(Amsterdam: North-Holland) pp 191-300

Fiz . 68 347-52 (1975 Soc. Phps.-JETP 41 169)

Eksp. Teor. F i z . 15 712-22