Elastic constants of solid krypton at T = 77 K determined by inelastic neutron scattering

J. Phys. Chem, Solids, 1973, Vol. 34, pp. 255-265.

Pergamon Press.

Printed in Great Britain

ELASTIC CONSTANTS OF SOLID KRYPTON T = 77 K D E T E R M I N E D BY INELASTIC NEUTRON SCATTERING*

AT

H. PETERt(a), J. SKALYO, JR.~t(b), H. GRIMM(b), E. L~ISCHER(a) and P. KORPIUN(a) (a) Physik-Department der Technischen Universit~t Miinchen, 8 Miinchen 2, Arcisstr. 21, Germany (b) Institut fiir Festk~irperforschung, KernforschungsanlageJiilich, 517 Jiilich, Germany (Received 25 April 1972) Abstract--Long-wavelength acoustic phonons have been studied for each of the [~00]T, [~00]L, [ ~ 0 ] L and [~:0] Tl branches in solid Kr at T = 77 K by means of inelastic neutron scattering utilizing "cold neutrons' as they are available in the long-wave length tail of the pile spectrum. The raw data have been corrected for resolution effects taking into account the curvature of the dispersion surface and variation of mode eigenvectors. It has turned out, that this yields appreciable shifts of the raw data. The results of our experiment give c , = 4.25---+0.10, c44 = 2.04-----0-03, c~2 = 2.82±0.12 and a value for B = ( c , + 2c12)/3 = 3.30 -----0.09 × 101° dyne/cm2. Available thermodynamic data for Kr gives a derived value for B *a = 2.58 ±0.06 × 101° dyne/cm2 indicating a large difference between zero sound and first sound in solid Kr at high temperatures. 1. INTRODUCTION

TaE FIRST inelastic neutron scattering experiment on one of the rare gas solids has been done on solid krypton by Daniels et al. [1]. The authors have used a clamped crystal grown at high pressure with the measurements being performed at 79 K at a pressure of about 0.3 kbars. Subsequently information has been obtained on both argon[2-4], neon[5, 6], and helium in both the h.c.p.[33] and b.c.c.[34] phases. To investigate anharmonic effects is one goal in the actual research on these relatively simple solids. Helium of course, with its extremely large zero point motion appears to be a class by itself. One manifestation of anharmonicity is a difference in the velocities of zero and first sound. Such an effect has already been predicted [7] and observed[8] in alkalide crystals and in quartz [9]. Calculations *Supported in part by the Deutsche Forschungsgemeinschaft. tThis work is part of a Ph.D. thesis at the TU Miinchen. ~Guest scientist from Brookhaven National Laboratory, Upton, N.Y. 11973, U.S.A.

on this effect in inert gas solids have been done by Goldman et al.[10] and Niklassen [11]. To establish an experimental basis for these theories, phonons with very small wavevectors have to be measured by inelastic neutron scattering. The high resolution necessary for such an experiment is obtained by using as high scattering angles for the monochromator and analyzer as physically permitted by the spectrometer construction and by utilizing 'cold neutrons' as they are available in the long-wavelength tail of the pile spectrum. In this article we present the results of our measurements which have been done on a free standing krypton crystal at T = 77 K along the directions [ 100] and [ 110] in the f.c.c, lattice. The measurements of Daniels et al.[1] cannot be used to deduce reliable elastic constants. Their data have been analyzed in terms of interatomic force constants tak!ng account of nearest neighbors only and then again considering interactions out to second nearest neighbors. The variation in the elastic constants between the two models is large

255

256

H. P E T E R et al.

(c~ differs by 13 per cent, (C44--C12) cfianges sign). This is an indication of the need to have low q data, the slopes of which will not depend critically on the number of neighbors considered. 2. EXPERIMENTAL DETAILS

A krypton crystal, with a diameter of 32 mm and a length of 60 mm, was grown at a temperature of T = 111 K (melting temperature T = 115.95 K) using a method similar to that of Gs~inger et al. [12]. We used Kr gas of the natural isotopic mixture with an impurity concentration < 0.01 per cent. After the crystal had been cooled down to liquid nitrogen temperature it was transferred into an aluminium sample holder. The sample was placed therein on an aluminium pedestal covered with Cd and fixed in a w a y that prevented it from turning. T h e distance between the crystal and the wall of the holder was approximately 5 mm which facilitated elimination of aluminium powder lines in the neutron experiment by proper masking. Finally the holder was transferred into a bath cryostat filled with liquid nitrogen. T h e neutron measurements were performed with the triple axis spectrometer at the FRJ-2 reactor at Jiilich. T h e krypton sample contained a single crystalline grain whose volume we guessed from neutron photography to be between 6 and 10 cm a. Its growth direction was about 6 ° away from the [ 112] direction and the mosaic spread was 17 min F W H M . T h e cryostat could be tilted on the spectrometer and therefore all measurements were made in a [001] zone. A few additional crystallites in the sample were of no consequence in the present experiment due to their small size and masking of the neutron beam to the size of the main grain. In an inelastic phonon scattering experiment momentum and energy have to be conserved according to the relations KF--K~= Q = q+G,

(I)

E = h2/2m (Kj 2 -- KF2),

(2)

Kl=2~/hl,

(3)

where KF and K~ are the scattered and incident wave vectors of the neutrons respectively, hQ is the momentum transfer and E is the energy transfer of the neutrons, q is the wavevector of an excitation and G is a reciprocal lattice vector of the sample. In the following we shall denote by ,k the wavelength of the neutrons that are selected out of the pile spectrum near its peak by the monochromator (hi = 1-836___0.001 ,~,). In order to obtain high resolution at the triple axis spectrometer we utilized the (004) reflection of pyrolytic graphite (PG) at both the monochromator and analyzer positions. F o r k, the (004) PG nearly gave the instrument limit (70 °) for the scattering angle 20M of the monochromator. According to the Bragg equation h = 2dhkt'sin 0M one obtains in addition to h, neutrons of wavelength 2k from (002) PG, 2/3k from (006) PG etc. We shall show later that the '2h neutrons' are extremely useful in measuring phonons with small q in spite of not being in the peak of the pile spectrum. As neutrons of different wavelength are falling on the sample and as the analyzer is reflecting the scattered neutrons on different sets of planes (hkl) too, more than one excitation may be observed in a single constE or const-Q scan. This is depicted in Fig. 1. Figure la shows the scattering triangle for the measurement of a longitudinal phonon in the [ 100] direction in a [001] zone just as we have performed it. We have plotted the triangles for neutrons of wave-length h and 2k respectively. One can see that when measuring with h near (400) we simultaneously measure at the same scattering angle, Os, near (200) utilizing 2k. It is obvious too that for a fixed scattering angle the length of the q vectors differ by a factor of 2 and the energy transfers differ by a factor of 4 (compare equation (I) to (3)); therefore if the mode energy has a q dependence the two peaks obtained will separate. This can be visualized

SOLID KRYPTON

[o~o]

,/¢%~% ,o 240e

440e

640e

040i 020e

220...=e~

e420

~~,,~' 0

"~'I

2

200

400

e620

,~oo]

600

(a)

Ec

EO 4 qi

(b)

qz

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Fig. l(a). [001] reciprocal lattice plane with scattering triangles for a measurement of long. phonons in the [100] direction using simultaneously neutrons with wavevectors that differ by a factor of 2. l(b). Path of the two different neutron groups in the E - q plane for a const-E

scan. for the case of a const-E scan, starting over the Bragg-peaks and going out in the [100] direction (in a const-E scan, K~ and KF are kept constant and the final point of KF and KF/2 respectively is moved in [ 100] direction [14]). This is shown in Fig. lb in the E--q plane. We see that the dispersion curve is intercepted first at ql and later at q2. The phonon at ql is measured utilizing '2hneutrons' and the phonon at qz is measured utilizing 'h-neutrons'. In order to understand the aforesaid consequences, it is worth noting that the resolution of a triple axis spectrometer at the elastic scattering position is directly proportional to E : c o t Ou when the monochromator and analyzer crystals have the same d-spacing [ 15]. In our experiment E~ of the 'h-neutrons' was 24 meV, the '2h-

257

neutrons' had an energy E l = 6 meV and cot Ou is of course the same for both energies. Such low energies normally can only be utilized at cold neutron sources ! It is therefore worth considering whether one can see phonons generated by '2hneutrons' in the present instance. In the pile spectrum the intensity of the '2h-neutrons' compared to that of the 'k-neutrons' will be smaller by a factor of 10-15. The absorption cross section of the specimen which is proportional to K -a, will increase by a factor of 2([16]O'abs=25"0-----0"8 barn for neutrons at 23 meV). On the other hand, the reflectivity, r(h), of our monochromator and analyzer crystals with the (002) reflection will be almost 100 per cent for neutrons with Et = 6 meV which has to be compared to about 32 per cent for neutrons measured with the (004) reflection at E~ = 24 meV (r(h) is improving with increasing h[ 13]). The coherent double differential cross section (02o-/01~Oe)coh for phonon creation in the harmonic one phonon approximation is dependent on the BoseEinstein population factor, nr(T), and on the energy of the phonon, E, in the following way [13]

[ ~o" ~

~ nE(T) +

\O~'~OE//eoh

!

(4)

E

This factor gets very big for small E at T----77 K and cancels both the smaller intensity of the incoming '2h-neutrons' and the higher absorption in the specimen. All the other wavelengths, that are picked out of the pile spectrum by the monochromator do not give any detectable signal. This can be worked out using equations (1-4). In contrast to the measurements of [ 100]L and [ 100] T phonons the scans for measuring phonons in the [ 110] direction have exhibited one peak only. The reciprocal space available for a neutron scattering experiment depends on K~, KF and the maximum scattering angle permitted by the spectrometer. Using Er = 24 meV ('h-neutrons') we only were able-to

258

H. PETER et al.

by Gaussians and the background by a straight line. Figure 2 shows all the longitudinal phonons we have measured in [ I00] direction utilizing neutrons with E , = 2 4 m e V (',kneutrons') and the 2,k component (EI = 6 meV) of the pile spectrum. Open circles are the measured intensities, the plotted curves are the results of our least squares fits. Note that the scale of the x-axis is correct only in terms of 'h-neutrons'. T h e correct q-values for the phonons at the left side of Fig. 2 one gets by dividing the scale by 2. Figure 2 shows clearly that as E decreases the two phonon peaks in a single scan are running together preventing lower q-values from being reached. This is due to the effect that the difference of the two scattering angles at which the resonances occur, goes to zero approaching the Bragg peaks (compare Figs. l(a) and l(b)). F o r phonon scattering we have a surface in E, q space for which the cross section is finite. F o r small q the dispersion surface has an appreciable curvature, and because of the finite resolution of a triple axis spectrometer an asymmetry will be introduced into an experimental peak. The observed maxi-

measure near (220). The 2h component with EI = 6 m e V then probes excitations near (110) which is the zone boundary X point[20] and hence would not be detected. In order to utilize the better resolution of the '2hneutrons' we increased Et so that we could use the 'cold neutrons' at (220) directly. Their energy El had been 8.5 and 8 meV for measurements in the [ l l 0 ] L and [ l l 0 ] T 1 direction respectively. Consequently in this case the excitations of the 'h-neutrons' (Et = 34 and 32 meV respectively) could no longer be detected for their intensity was down due to the pile spectrum. 3. ANALYSIS OF THE RAW DATA AND RESULTS

Longitudinal and transvers phonons have been measured in the [100] and [ l l 0 ] directions in a [001] zone. Normally the spectrometer has been operated in a const-E mode. Only the low lying transverse phonons in the [ 110] direction had to be measured by constQ scans because the slope of this branch is extremely small. We have analyzed the raw data by least squares fits. T h e phonons have been described I 2X

2ooI

For ;~: E = 2.410 meV o ~ X^

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0"30

0"35

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0-45

Fig. 2. Observed neutron groups measured in the [100] direction by const-E scans. The q scale is given in terms of the h peak; divide by 2 for the 2h peak. The energy for the 2h peak is I/4 that of the h peak.

SOLID

mum intensity will be shifted in comparison to the nominal position of the one phonon resonance. To correct for the shift of the peak we have used an expanded version of a program originally written by Pynn and Werner[17]. The main alteration was the extension to f.c.c, materials and the inclusion of the four collimator resolution function of Cooper and Nathans[15]. This program utilizes the Born-yon K~irmfin force constants to determine the energy, Ej(q), and the j t h mode eigenvectors, Ej(q), over the resolution function of the three-axis spectrometer. The cross section is then calculated according to e_DQ2(Q.Ej) 2 n(Ej) + 1 E~ '

(5)

and folded with the resolution function to calculate the intensity for each spectrometer setting. By this method the line shape of a phonon is calculated point wise and the position of the maximum is determined by plotting the phonon. The raw data have been corrected by the following procedure. The positions of the maxima of the observed phonons we have taken from least squares fits and we will denote them by ~ob~for const-E scans and by Eob~ for const-Q scans respectively. In cases where two phonons have been observed, the fit was to a double gaussian in order to take account of the possible shift in the observed positions due to the running together of the peaks. The corresponding values taken from the folding program we denote be ~eold and Erold respectively. Finally we have to calculate the dispersion curves in the appropriate principal symmetry directions, using the same force constants as in the folding program. From these calculations we deduce ~symmand Esymm respectively for the phonons under investigation. ~ is the reduced wave vector. Now we can define a correction, a, by the relations ~symm - O~ =

~fold

~fold

for const-E scans,

(6)

KRYPTON

259

Es~,mm --

O~ =

Erold

for const-Q scans.

(7)

Efol d

One gets a corrected value of the observed data by applying ~cor = ~obs(l + o0,

(8)

Eco r = E o b s ( I q- ~x).

(9)

The set of ~eor and Ecor is then used to determine a better set of force constants and the whole procedure is repeated until ot does not change. To start the first folding program we have used a set of Born-von K~rmfin force constants, taken from Brachner[18] who has analysed the measurement of Daniels [1]. The Debye-Waller coefficient has been taken from Goldman [ 19]. Altogether the correction procedure was done three times. For the 2 na and 3 ~a folding we have utilized first nearest neighbors force constants derived from our measurement only. The results of our experiment are given in Table 1 and Figs. 3 and 4. In Table 1 E1 denotes the energy of the neutrons, selected out of the pile spectrum and E denotes the energy transfer we have used in the const-E scans (note that only [~0]T 1 was done in a const-Q mode). ~* obs gives the the length of q in reduced units after the correction a* had been applied to the raw data. (FWHM)obs gives the full width half maximum of our observed phonons, taken from the least squares Gaussian fit and weighted variance is a measure how well the assumed function fits to the measured points (weighted variance should be 1). The errors given in Table 1 have been derived as follows: The absolute energy of the neutrons reflected by the monochromator has been determined to 0.1 per cent calibrating the triple axis spectrometer with a-Al~Oz. As we are determining the energy transfer, E, by a relative measurement, this error can be neglected. Rocking of the analyzer in Bragg peaks showed that the analyzer had been

260

H. P E T E R et al.

Table 1. Experimental phonon energies in krypton at E1 meV

Branch

[~00]r

[t~00] ~.

[~/~0]L

[~0]r,

T = 77 K

E meV

Co . reduce~umts

in %

(FWHM)obs reduced units

24 24 24 6 6 6

2.182___0.012 1.903---0.012 1.618_-.0.012 0"546±0.003 0.476 ± 0.003 0.405 ± 0.003

0"381 ___0.005 0.326±0.003 0.273_-_0.003 0.093___0.001 0.082 ± 0.001 0"068 ± 0"001

3"3 4.5 6"3 2.3 2.7 3-5

0'104___0.012 0.093---0"006 0.096---0"006 0.024±0"002 0.022 ± 0.002 0"023 - 0.002

1"12" 1.05" 1.03" 1.12" 1.05" 1.03"

24 24 24 6 6 6

2"410---0"012 1"996---0"012 1"618±0.012 0.603 ± 0.003 0.499 ± 0.003 0.405±0.003

0"286---0"005 0"234±0-004 0.179±0.004 0.072 ± 0.002 0.059 ± 0"002 0.046±0.001

4"0 4-8 5-8 6"5 8-7 11.1

0.084±0.009 0"105---0"008 0.105±0.009 0.019 _ 0.003 0"030 --- 0"004 0.021±0.002

1.19" 1.12" 1"15" I. 19" 1' 12* 1.15"

24 24 24 8"5 8.5

1-617---0-012 1.423 ± 0-012 1-177+__0-012 0"598 ± 0.004 0.491 ± 0"004

0-117___0.003 0-102 ± 0"002 0.085±0.003 0.044 ± 0.001 0-039 ___0.002

11.2 14.8 22.9 17.6 32" 1

0.051 ---0"005 0.052 ± 0.004 0.054±0-003 0.029 ± 0'002 0.026 ± 0-002

1.14 1 "22 0-58 1" 18 1-92

8 8

0.622 ± 0.024 0.581___0.014

0 ~123 0.117

0.36 ± 0-02 m e V 0.35 _ 0 - 0 2 m e V

t .46 0.67

-- 4.3 --4.8

. Weighted variance

El: energy of the neutrons incident on the specimen. E: energy of the phonons. ~*~.: length o f the p h o n o n w a v e v e c t o r in reduced units after the correction a * had been applied. ot~*~ correction o f the raw data d u e to the finite resolution of the triple axis s p e c t r o m e t e r and the curvature o f the dispersion surface. (FWHM),bs: full width half m a x i m u m o f observed phonons. Weighted variance: m e a s u r e h o w well o u r least s q u a r e s fits describe the m e a s u r e m e n t , w.v. should be 1. *Indicates that the 6 to 24 m e V data were fit at the s a m e time, using a double gaussian.

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0

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Fig. 3. Observed longitudinal, L, and transverse, T, dispersion curves for Kr at T = 77 K in [100] direction; the straight lines give their slopes. For the low ~ phonons in the [~00]r branch the uncorrected raw data are not plotted, because the corrections arc too small.

aligned always better than 1 min (on the 2Oa scale). Using A (20A)= 1 min we have

derived the errors in the 3 rd column. These errors should therefore be upper limits for the const-E scans. The lattice constant of Kr measured with observed Bragg peaks has been a0 = 5.744___0-005 ,~, (limited by our Al2Oa calibrate in absolute units) which is in agreement with the result of Losee and Simmons [21] who have found a0 = 5-7404/~, at T ----77 K. Again here the accuracy of q is high as it has been determined relative to the nearest Bragg peak. The raw data have been analyzed by least squares fits assuming a Gaussian shape of the measured phonon. As Cooper and Nathans[15] have pointed out, this is a good approximation only for phonons with long lifetimes and for a planar dispersion surface. The latter is not true for phonons with small wave vectors. The statistics of the measurement however do not permit the

SOLID KRYPTON l

i

i

i

• Corrected

+

l

1

the phonons the error bars are too small to be plotted. Open circles with error bars have been taken from Daniel et al. [1].

raw data

Uncorrected

raw data

}. Danielsetal.{ll >m 2-0 E %,

4. DISCUSSION

[g Co]

+ .,~/

ill

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261

TI ~

0-1

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0-2

Fig. 4. O b s e r v e d longitudinal, L, and low transverse, TI, dispersion curves for Kr at T = 77 K in [110] direction; the straight lines give their slopes. For the p h o n o n s in the [Eg0]r, branch the uncorrected raw data are not plotted, b e c a u s e the corrections are too small.

detailed deviation from Gaussian to be observed; we have however multiplied the standard deviation given by our least squares fit program by 1-5 if the weighted variance is reasonable, otherwise we have used a factor of 3 (in four cases). T h e s e resultant errors are given in column 4 and should be taken as one standard deviation of ~o*bs"The errors given for (FWHM)obs are the standard deviations taken from the least squares fit. Finally we have calculated the whole line shape of one phonon (E = 2.410 meV [~00]L) using our folding program and first nearest neighbors force constants derived from the results, printed in Table 1. T h e folding program has yielded F W H M = 0.092 reduced units which is in reasonable agreement with FWHM=0.084_0.009 reduced units obtained for the Gaussian fit. In Figs. 3 and 4 we have plotted our results. T h e dots give the corrected raw data and the uncorrected raw data taken from the Gaussian fits have been plotted as crosses. F o r most of

T h e only inelastic neutron scattering experiment done on solid Kr in the past is that of Daniels et al.[l]. T h e i r experiment has been performed at T = 7 9 K and at an estimated pressure of 0.3 kbars along all three principal symmetry directions. Their measured lattice constant was 5.725_0-010,~,. In Figs. 3 and 4 we have included those measurements of Daniels et al. that overlap with the present results. N o phonon data from their experiment are available for the low lying transverse branch in [110] direction, denoted by T,; the measurement of this branch is highly desirable for an accurate determination of c12. Our experiment has been performed at about 1 bar and T = 77 K. T o compare the results of both measurements one has to take into account the volume and temperature shift in the phonon energies, E. T h e volume shift can be estimated using the prediction of the quasi-harmonic approximation. E ( T , V) = E ( T , V o ) [ 1 - - y - ~ - ] .

(10)

y = 2.9 is an average Griineisen parameter assumed to be independent[22] of q and polarisation direction, E ( T , Vo) describe Daniels et al.'s phonon energies. One then calculates for the volume shift AE v AV E ( T 1 V o ) = -- Y--V- = --2"3%.

T h e temperature shift of 2 K is expected to give rise to a negligible shift in energy. However there is an additional effect that should be kept in mind. Daniels et al. have not corrected their data for resolution effects as previously described. As all their phonons have been measured in the const-Q mode their energies have to be shifted to smaller

262

H. PETER et al.

values (compare in Table 1 our phonons of the [ ~ 0 ] T 1 branch). Without knowing their spectrometer parameters we are not able to make any quantitative statement on how much their energies have to be lowered. H o w e v e r it is known that their higher flux has permitted a tighter collimation than in the present experiment, thus giving a smaller or* than we obtained. The lack of resolution correction appears to have been compensated by what appear to be overly large error bars; the agreement of both sets of measurements with one another is well within the indicated errors. In order to compare our measurements with ultrasonic experiments[23-25] and measurements of the isothermal compressibility[26, 27] we have to extract elastic constants from the slopes of the dispersion curves. In an harmonic crystal the slope of the dispersion curves for q ~ 0 yield sound velocities which are identical to the velocities deduced from ultrasonic measurements. From these one gets the adiabatic elastic constants (in a harmonic crystal this would be identical to the isothermal elastic constants) directly and the bulk modulus for a cubic crystal is given by B =~(cH+2cl,,). (11) In an anharmonic crystal the velocity of sound depends on the relationship of its frequency 11 to the inverse lifetime F of most of the phonons in the crystal[7]. Ultrasonic measurements are normally done at frequencies much smaller than F. Velocities measured in this way belong to the so called first sound regime whereas phonon frequencies measured through neutron scattering experiments are normally done at 11 -> F and fall in the zero sound regime. Cowley[7] has pointed out that there could be a difference in the velocity of first and zero sound. The elastic constants determined in the first sound region are the adiabatic constants[31]. In the zero sound region, it is also shown[7] that measured elastic constants may, unlike the macroscopic

elastic constants, not have the symmetry of a second-rank tensor in addition to the point group symmetry of the crystal. Numerical calculations on this problem have been done for rare gas solids by Goldman et a/.[10] and Niklasson[l I]. Goldman et al. have reported values for clamped Kr and Xe crystals using the lowest-order selfconsistent phonon scheme to calculate the first sound elastic constants. The procedure has been described in Ref. [29]. T h e y have calculated the velocity of zero sound from the limiting slopes of phonon dispersion curves. Utilizing a L e n n a r d - J o n e s (12-6) nearest neighbor model, they have found that the neutron sound velocity was never more than 4 per cent above the first sound velocity and that the usual symmetry properties seem to hold very well for neutron elastic constants. Niklasson's paper pertains to the transition between the low-frequency hydrodynamic regime and the high-frequency zero sound regime. His calculations are based on the assumption of small anharmonicity and therefore the validity of his theory may be restricted to temperatures roughly below one third of the melting temperature. As our measurements have been done at 0-66 of the melting temperature his numerical results may not be applicable. In Figs. 3 and 4 the straight lines give the slopes of the dispersion curves. T h e straight lines plotted for the [~0011. [~O0]T and [~O]T, branches we have obtained by a linear least squares fit where we only have used the phonons with Et = 6 meV and Et = 8 meV respectively. Note that for both branches in [100] direction these straight lines are even good fits to the phonons that are about four times further out in the reciprocal lattice! T h e two phonons with E~ = 8.5 meV in the [~0]L branch however appear to have a higher uncertainty, in part probably because of the size of a*; we have therefore used all the phonons of this branch to determine the slope by a straight line fit. It is clear that non linear dispersion would possibly influence

SOLID KRYPTON the r e s u l t s w h i c h w e h a v e d e r i v e d in this w a y , a n d this p o i n t will be d i s c u s s e d later. F r o m the s l o p e s w e h a v e o b t a i n e d the f o l l o w i n g s o u n d v e l o c i t i e s v: [~00]r [~0011. [~¢;0]r, [~0]L

v=816+7m/sec, v = 1191+21m/sec, v = 491 -+3 m / s e c , v = 1343 + 18 m / s e c .

U s i n g the r e l a t i o n s f o r pV 2 a n d ci/'a, p b e i n g the d e n s i t y , o n e o b t a i n s [~00]~ [~00]L [~0]T, [~/~0]L

Ov 2 = c44,

o r " = c,,, pV 2= I[2(C,I--Cv,_), p v " = 1/2(C,~ + 2C44+C1,,).

(12)

T h e c~j d e r i v e d f r o m t h e first t h r e e b r a n c h e s a r e g i v e n in T a b l e 2 c o l u m n l a b e l e d b y (a). A s w e h a v e d e r i v e d t h e s l o p e s o f the f o u r dispersion curves independently from one a n o t h e r o n e m a y t e s t the t e n s o r s y m m e t r y o f the e l a s t i c c o n s t a n t s . U s i n g the cij o f c o l u m n (a) in T a b l e 2 o n e o b t a i n s for [1 10]L (OV2)cal. = 5.42-----0"11

10 l° d y n e / c m 2.

263

T h i s has to be c o m p a r e d with t h e v a l u e f r o m the s t r a i g h t line fit (/gV2)measured = 5"31 + 0 " 1 4

10 ~° d y n e / c m 2.

W i t h i n the e r r o r s w e c a n d e t e r m i n e no d e v i a tion f r o m the t e n s o r s y m m e t r y . T h i s is in a g r e e m e n t with C o w l e y [7] a n d G o l d m a n [ l 0] w h o p o i n t o u t t h a t e v e n for a p p r e c i a b l e d i f f e r e n c e s b e t w e e n first a n d z e r o s o u n d e l a s t i c c o n s t a n t s the v i o l a t i o n o f t h e t e n s o r s y m m e t r y s h o u l d be small. In c o l u m n (b) o f Table 2 we have given the results of a Bornv o n K ~ r m ~ n i n t e r a t o m i c f o r c e c o n s t a n t fit t a k i n g into a c c o u n t n e a r e s t n e i g h b o r s ( 1 N N ) only. The limited q range of the measurement e s s e n t i a l l y d e t e r m i n e s an e f f e c t i v e f o r c e c o n s t a n t o n l y : h o w e v e r the co w o u l d be c o r r e c t l y given. N o t e t h a t s u c h a fit i m p o s e s the t e n s o r s y m m e t r y a u t o m a t i c a l l y on t h e e l a s t i c c o n s t a n t s a n d a l s o t a k e s into a c c o u n t a n o n linear d i s p e r s i o n . U t i l i z i n g t h e v o l u m e s h i f t e d d a t a o f D a n i e l s et al. [ 1] t a k e n t h r o u g h o u t t h e z o n e , t o g e t h e r with o u r low q m e a s u r e m e n t s , a I N N fit g i v e s t h e s a m e co as in c o l u m n (b) e x c e p t w i t h s m a l l e r e r r o r s . A t w o n e i g h b o r fit r e s u l t e d in o n l y a d e c r e a s e o f c44

Table 2. Elastic constants for krypton

co in 101°dyne/cmz cN c44 c12

Current experiment T=77K (a) 4.17___0-15 1.96±0-03 2-75±0.15

1NNfit T=77K (b)

Q.H. appro. V = Vexp. Ultrasound (12--67 L.J. (13--6) L.J. T=77K T=80"24K T=81"80K (c) (d) (e)

4.25___0.10 4.03±0-16 2-04--+0"03 1.75±0.15 2.82±0-12 1.86___0.12

3.72 1-55 2-01

3.95 1-63 2'13

S.C. appr. V = Vexp. (12--67 L.J. T=77K (f) 3"82* 1"75" 1-95"

(a) Elastic constants derived from the slopes of the dispersion curves we have measured. (b) Values derived from a Born-van K~.rmfin interatomic force constants fit to our data taking into account nearest neighbors, I N N, only. (c) c~d experimental taken from [25]. (d), (e) Calculations of c~a based on the quasiharmonic approximation and using different Lennard-Jones potentials-calculations have been done for experimental volumes and the temperatures T given in the columns [28]. (f) Calculations of c~a taken from [29], lowest order self consistent phonon theory has been used. *These values have been derived by linearly interpolating between isothermal values at T = 85 K and T = 55 K given by Klein et al. [29] and then applying an adiabatic correction which is given for T = 85 K.

264

H. PETER et al.

to 1.98+0-02.10 l° dyne/cm 2. With regard to the present agreement now obtained for 1NN and 2 N N fits, we have therefore seen the importance of essentially fixing the co by low q measurements. T h e only elastic constants determined by ultrasonic experiments at T = 77 K are reported by Korpiun et al. [25]. We give their results in column (c) of Table 2. Korpiun et al. have used the pulse-echo technique utilizing frequencies of about 107 c/sec. Their measurements therefore clearly belong to the first sound regime, cll and c44 the authors have deduced from the measurement of longitudinal and transverse sound velocities performed in two crystals. T h e y have assumed that these crystals were single crystals and that the direction of their measurement was [100]. T h e y have also measured transverse and longitudinal sound velocities in polycrystalline kyrpton. Using c1~, c44, and Bexp.adwhich is the adiabatically corrected isothermal bulk modulus as measured by Coufal et al. [26] and calculating the polycrystalline sound velocities they found agreement with their measurements. Thus they obtained c12 = ( 3Baa exp.-- c11)/2. Column (d) and (e) of Table 2 give calculations of the adiabatic elastic constants using a quasiharmonic approximation[28] and column ( f ) gives results based on the lowestorder selfconsistant phonon theory (SC) [29]. These calculations give first sound adiabatic elastic constants. Note the calculations are all much lower than our observations, while the ultrasonic data on c1~ and c44 would tend to overlap within two standard deviations. One can however consider the experimental isothermal bulk modulus which is known very well for Kr over a large termperature range (for details see Ref. [26, 27] and [30]). F o r T = 77 K the adiabatic bulk modulus has been determined from the isothermal data as ad = 2"58 --_+0"06 Bexp.

101° dyne/cm 2

T h e connection between B ad and cfid (equa-

tion (I1)) is very well established for the hydrodynamic region. Numerical calculations [10, 11] show that the zero sound elastic constants, ci%, are higher than the first sound c o n s t a n t s , c /ad m linear combination of c?. j " 13 should therefore give a number that can be compared with B ead . We define xo B ° = I/3 (c~, + 2c~.,). Using our results from Table 2 column (b) we get B ° = 3.30__+0-09

10 ~° dyne/cm 2.

One observes that B ° is about 22 per cent aap, and that this difference is higher than Bex much bigger than the errors of both values. Although unknown systematic errors can never be ruled out in experiments we feel that the difference between B ° and B~o. is too big to be explained only in this manner. The case of Kr at T = 77 K contrasts with that of neon at 5 K where recent low q neutron measurements [6] give B ° in agreement to within 1 per cent of the isothermal value obtained by Batchelder et al.[32]. T h e s e latter measurements were X-ray studies of the lattice spacing as function of pressure and temperature. At 5 K the correction to obtain the adiabatic value is negligible. 5. CONCLUSION

Inelastic neutron scattering in solid Kr at T----77 K has given a strong indication that there is an appreciable difference between zero sound and first sound elastic constants. At present only one numerical calculation [ 10] of this difference is available for krypton. H o w e v e r it seems that the model Goldman et al.[10] use predicts too small a velocity difference. As Niklasson[l 1] has pointed out, the numerical differences between the adiabatic and zero sound velocities depend critically on the anharmonic part of the interaction potential. T h e r e f o r e the present measurement may be a critical test for the

SOLI D KRYPTON

different potentials used in actual calculations on rare gases. Further accurate measurements, especially on the temperature dependence of the elastic constants, would give valuable information. T h e most striking difference b e t w e e n first and zero sound occurs in the damping[l 1]. H o w e v e r , for the near future it s e e m s to be extremely improbable that this effect can be measured in both regions. Acknowledgement-One of us (H.P.) would like to acknowledge the kind hospitality and help of the Neutron Scattering Group of Prof. T. Springer during the course of this experiment. Another of us (J.S.,Jr.), while echoing the aforesaid would like to additionally acknowledge many discussions of various aspects of the present investigation and related problems with Dr. W. Press and Dr. F. Hossreid which were held during his one year visit as a guest of the Neutron Scattering Group. We would like to thank Dr. H. Stiller for his direct interest, encouragement and discussions. We would also like to thank Prof. W. Gl~iser, who has made possible our success by the loan of two polyrolytic graphite crystals. We are indebted to Dipl. Phys. H. J. Coutal and Dipl. Phys. H. Meixner for help during the crystal growth and to Dr. W. Press for expeditious help on the neutron scattering. Grateful thanks are due to Dr. R. Pynn for many discussions concerning the details of the folding program. One of us (H.P.) has benefited greatly from very useful discussions on the content of this paper with Prof. W. G6tze, Dr. V. V. Goldman and Dr. J. Kalus. REFERENCES 1. D A N I E L S W. B., S H I R A N E G., F R A C E R B. C., U M E B A Y A S H I H. and L E A K E J. A., Phys. Rev. Lett. 18, 548 (1967.).. 2. E G G E R H., G S A N G E R M., L O S C H E R E. and D O R N E R B., Phys. Lett. 2,8A, 433 (1968); DORNER B. and E G G E R H., Phys. Status Solidi 43b, 611 (1971). 3. B A T C H E L D E R D. N., COLLINS M. F., HAYWOOD B. C. G. and SIDEY G. R., J. Phys. C: SolidSt. Phys. 3, 249 (1970). 4. B A T C H E L D E R D. N., H A Y W O O D B. C. G. and S A U N D E R S O N D. H., J. Phys. C: Solid St. Phys. 4,910(1971). 5. L E A K E J. A., DAN1ELS W. B., SKALYO J. Jr., F R A C E R B. C. and S H I R A N E G., Phys. Reo. 181, 1251 (1969). 6. SKALYO J. Jr., M I N K I E W l C Z V. J., A X E J. D., SH1RANE G. and DANIELS W. B., Phys. Rev. B6, December (1972). 7. COWLEY R. A., Proc. Phys. Soc. Lond. 90, 1127 (1967).

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