P. S. MAHESH and B. DAYAL: Debye-Waller Factors of Cu and Au 399

phys. stat. sol. 7, 399 (1964)

Department of Physics, Banaras Hindu University

The Debye-Waller Factors of Copper and Gold by de Launay’s Method

BY P. S. ~KAHESH~) and B. DAYAL

The DEBYE-WALLER exponents for copper and gold are calculated from the frequency distributions derived from DE LAUNAY’S electron model. Frequencies for 1000 points of the first BRILLOUIN zone are considered. The computation has been made by BLACHMANN’S sampling technique. The results are compared with available experimental data obtained a t different temperatures. Good agreement between theory and experiment is found for gold, but not for copper.

Die Debye-Waller-Exponenten fur Kupfer und Gold werden aus der Frequenzvertei- lung, die von DE LAUNAY’S Elektronenmodell hergeleitet wurden, berechnet. Es werden Fre- quenzen fur 1000 Punkte der ersten BRILLouINechen Zone verwendet. Die Berechnung wurde mit Hilfe von BLACKMANN’S Probenverfahren durchgefiihrt. Die theoretischen Ergebnisse werden fur beide Metalle mit den experimentellen Daten verglichen. Nur fur Au konnte eine gute ubereinstimmung zwischen Theorie und Experiment erzielt werden, jedoch nicht fur Cu.

1. Introduction It is well known that there is a reduction in the intensity of X-ray diffraction

maxima with the rise of temperature on account of thermal vibrations. This effect was first considered theoretically by DEBYE 111 who showed that this decrease could be expressed by an exponential factor e c Z M . The treatment was reconsidered by WALLER [2] who found that the value of M was twice that given by DEBYE. The expression for 1M has been calculated assuming the vibration spectrum of a solid to be given by the Debye theory and is expressed in terms of a well known function involving the characteristic temperature 0. The experi- mental values of M are determined from the X-ray intensities of the Bragg re- flections and are used to compute the Debye characteristic temperatures. These are compared with values obtained from the specific heats and the elastic con- stants. A review article by HERBSTEIN [3] shows that the agreement between the X-ray values and the others is rather poor. This is not suprising because there is no crystal whose vibration spectrum is given by the Debye theory. It has been shown by BLACKMANN [4] that the actual distribution has a peak in the middle and bears no resemblance to the one given by the Debye theory. The characteristic temperature itself is an average parameter which has no physical meaning but is nevertheless useful for a discussion of the average thermodynamic and related properties. As the averaging over the frequency distribution in the X-ray case is different from that used in the specific heat theory, there is no reason why the 0 obtained from them should be the same. In fact the X-ray values of the characteristic temperature are found to be slightly lower than those obtained

On leave from D.A.V. College, Muzaffarnagar.

400 P. S. M . h n m H and B. DAYAI,

from the thermal data (COCHRAN [ 5 ] ) . If the X-ray intensities are used to obtain the dependence of 0 on temperature, the resulting @ vs. T curve is found to differ considerably from the one given by the specific heat data of HERRSTEIN [S].

Instead of using a hypothetical Debye distribution which is different from reality. i t would certainly be preferable to compute the Debye-Waller factor from an actual frequency distribution. A number of studies of the vibration spectrum have been made on the basis of Born’s lattice dynamics. Even though these are based on approximatc elastic force models (cf. Reviews by PARKINSON 161 and COCHRAN [ 5 ] ) , the frecpcncy distributions givcn by them lie much nearer t o the actual ones than those of DEBYE. Of these, de Launay’s modcl [ 7 ) is one of the best arid has been shown, in this laboratory, t o give a fairly satisfactory inter- prctntion of the specific heat data of solids (cf. 1)AYAL and SHARAN [8. 91, DAYAL antl SINGI~ [lo, 111). An examination of DAYAL and SHARAN’S results [9] on sodium shows that the frequencies calculated on the basis of this theory give a satisfactory agrcemerit with the neutron diffraction data of WOODS rt al. 1121. It was, therefore, thought proper to calculate the Uebye-Wallcr factor of metals on the basis of this theory. The results for copper antl gold are presented in this paper.

2. Numwical Computation

The following expression for the exponent 2 M of the Debye-Waller factor can be easily derived (cf. JAMES 1131) :

cop.j is the angular frequency, n,? the average occupation number of the cp j lattice mode of vibration and WL the mass of the atom. evj is the polarisation vector of the pj j mode, N the total number of atoms and s is a vector perpendicular to the reflecting plane of the crystal with Is1 = 2 sin 0, 8 being the glancing angle of incidence. npj is given by Planck’s theory as

From considerations of cubic symmetry, it can be shown that the factor ( y s . epj)2 may be replaced by its average value outside tlie summation so that

equation (1) becomcs

In the Debye-Waller theory, the summation was replaced by an integration on the basis of tJhe Debye distribution. We have calculated w p j for the 1000 points of the first Rrillouin zone by means of the de Launay’s electron model [7]. The secular determinant for the evaluation of the characteristic frequencies has been quoted by XRIVASTAVA [I41 who has also derived the 48 points of the irreducible section (l/48) of the first Brillouin zone which correspond to a mesh of 1000 points in the uniformly subdivided zone. The numerical values of wpj were taken from the paper of SRIVASTAVA [I41 in the case of copper and from that of DAYAL and XWGH [ 111

Debye-\Taller Factors of Cu and Au by de Launay's hietlod 40 1

for go d. If a frequency m a , corresponding to one of the 48 points (nth point) is

The ex- cquivalent to q frequencies by symmetry, its statist cal weight is --*- P Y 1000 .

pression for 2 ilf then becomes

The computation was made by Blackmann's sampling technique. The whole vibra- tion spectrum was divided into intervals of 0.4 x 10l2 for copper and 0.2 x 10l2 in the case of gold and the midpoint of each frequency step was taken as the represen- tative of any particular interval whose statistical weight was given by the number of frequencies lying in it. In sampling, the central point of the Brillouin zone requires some attention because its frequency is zero. Since occurs in the denominator, this would give an infinite value of M . However, this is the fre- quency of the non-vibrating lattice for which M should obviously be zero. We have. therefore. taken the zero value for the contribution of this point.

3. Discussion If pT and Po are the experimentally measured integrated reflections from the

crystal a t temperatures T and To, respectively, after allowance has been made for the corrections due to absorption and various other geometrical factors (cf. OWEN and WILLIAMS [151), we have

( 5 ) 9" - - e-"diT/e-2 >lo

90

where MT and Mo are the values of M a t 7' and To, respectively. This gives

It can be seen from (4) that both sidcs of (6) arc independent of 3, and sin 0 and are the same for all reflections. The numerical values of - -- (ZLWo - 2 M T ) ,

as calculated from (4), for copper and gold have been plotted at various tempe- ratures in Figs. 1 and 2. Eor this purpose To has been taken as 293 OK. The

have been shown as circles and are taken experimental values of

from the work of OWEN and WILLIAMS 1151. FLINK et al. [16] have also made measurements for copper but they have not given the detailed values. They

(si: H)s 1 2. 2 977

b n 0 ) e 2 9 3

!i

402 P. S. MAHESH and B. DAYAL

‘D Fig. 2. (A)‘ log ?for gold. Ttie circles rcpre-

scnt rxpcriniental data of OWEN and WILLIAMS [I51 eo

have expressed their results in terms of the Debye temperature 0, using the usual

Debye-Waller formula. We have, therefore, calcuIated ~ eT ( A )z from the

corresponding values of 0 and have shown them in Fig. 1 by crosses. In the case of copper, the experimental values are definitely lower than those

calculated theoretically. SRIVASTAVA’S [ 141 results show that the specific heats calculated from de Launay’s theory are equivalent to 0 = 320 O K of the Debye theory a t temperatures higher than 80 OK. The X-ray data are, however, best interpreted in terms of the Debye-Waller theory by 0 = 295 O K . This can be

seen from the lower line in Fig. 1 which gives the values of -- ( 2 Mo - 2 MT) as calculated from the Debye-Waller formula with 0 = 295 OK.

It is usual to explain the above discrepancy on the basis that there is a decrease in the characteristic frequency at high temperatures due to thermal expansion (cf. ZENER and BLINSKY [17]). OWEN and WILLIAMS have tried to estimate this effect from thermal expansion and the Gruneisen constant. There is no doubt that the discrepancy observed in our case can also be due to the same effect. However, no detailed discussion is possible. The Gruneisen constants of various frequencies are usually quite different from each other (see DAYAL [IS]). They also change their values as the temperature rises (see TRIPATHI [19]). At high

temperatures the factor in equation (4) depends on __. The low frequency peaks

in the frequency distribution, therefore, contribute much more to the value of M than the higher frequencies. SRIVASTAVA’S results show that the frequency distribution can be approximated by two peaks. The high frequency peak is. centered a t v = 7.9 x 10l2 s-l and covers a smaller area. The other peak is centered at about 4.1 x s-l and covers a much larger area. It is the latter which must largely govern M and the intensity of X-ray lines. Since the average Grun- eisen constant of the peak and its dependence on temperature is not known, no quantitative assessment of the effect of thermal expansion on the X-ray intensities is possible.

In the case of gold, DE LAUNAY’S model gives a good agreement with the X-ray experimental data. This results seems to be different from that of copper. How- ever, SINGH’S calculations show that the specific heats given by DE LAUNAY’S model do not agree with the experimental c,. The characteristic temperatures given by this theory are about 6 to 10% lower than those given by experimental c,. It is thus apparent that the peaks in the de Launay distribution occur a t smaller

g,,, sin@

( slIilo)z

1 fl&

Debye-Waller Factors of Cu and Au by de Launay's Method 403

values than in the true distribution. It is quite likely that the latter decrease to DE LAUNAY'S values a t higher temperatures due to thermal expansion, thus causing a good agreement to subsist between the X-ray results and DE LAUNAY'S theory.

The authors express their thanks to SHRI P. L. SRIVASTAVA for discussions.

References [l] P. DEBYE, Ann. Phys. (Germany) 43, 49 (1914). [2] I. WALLER, Z. Phys. 17, 398 (1923). [3] F. H. HERBSTEIN, Adv. Phys. 10, 313 (1961). [4] M. BLACKMANN, Proc. Roy. SOC. A 159, 416 (1937). [5 ] W. COCHRAN, Rep. Progr. Phys. 26, 1 (1963). [6] D. H. PARKINSON, Rep. Progr. Phys. 21, 239 (1958). [7] J. DE LAUNAY, Solid State Physics 2. 220, Academic Press Inc., New York 1956. [S] B. DAYAL and B. SHARAN, Proc. Roy. Soc. A 25'3, 361 (1960). [9] B. DAYAL and B. SHARAN, Proc. Roy. Soc. A 262, 136 (1961).

[lo] B. DAYAL and S. P. SINGH, Proc. Phys. Soc. A 76, 777 (1960). [ll] B. DAYAL and S. P. SINGH, Proc. Phys. SOC. A 78, 1495 (1961). [12] A. D. B. WOODS, B. N. BROCKHOUSE, R. H. MARCH, and R. BOWERS, Proc. Phys. SOC.

[I31 R. W. JAMES, The Optical Principles of Diffraction of X-Rays, Bell & Sons, London

[14] P. L. SRIVASTAVA, phys. stat. sol. 2, 713 (1962). [15] E. A. OWEN and R. W. WILLIAMS, Proc. Roy. SOC. A 188, 509 (1947). [16] P. A. FLINN, G. M. MCMANUS, and J. A. RAYNE, Phys. Rev. 123, 809 (1961). [17] C. ZENER and S. BILINSKP, Phys. Rev. 50, 101 (1936). [IS] B. DAYAL, Proc. Indian Acad. Sci. 20 A, 70 (1944). [191 B. B. TRIPATHI, Indian J. pure and appl. Phys. 1, 278 (1963).

79, 440 (1962).

1950.

(Received July 17, 1964)