P. S. MAHESH: The Debye-Waller Factors of Aluminium, Silver, and Sodium 1051

phys. stat. sol. 7 , 1051 (1964)

Department of Physics, Banurus Hindu Cniwrsity

The Debye-Waller Factors of Aluminium, Silver, and Sodium by de Launay’s Method

BY P. S. MAHESH’)

The Debye-Waller exponents for aluminium, silver and sodium have been calculated from de Launay’s electron gas model. 9 sampling technique has been used, as in a n earlier paper of MAHESH and DAYAL. These i s agreement between theoretical and experimental values up to 600 “K in the case of aluminium and silver and up to 200 “K for sodium.

Die Debye-Waller-Exponenten fur Al, Ag und Na sind mit Hilfe von de Launays Elek- tronengasmodell berechnet worden. Wie in einer vorangegangenen Arbeit von MAHESH und DAYAL wurde auch hier ein Probenverfahren benutzt. Bei A1 und Ag nurde bis zu 600 “K und fur Na bis 200 OK eine gute Ubereinstirnmung zwischen Theorie und Experiment ge- funden.

1. Introduction In an earlier paper, hereinafter referred as paper I,MAHESH ~ ~ ~ D A Y A L [l] had

calculated the Debye-Waller factors of copper and gold from de Launay’s theory. The frequency spectrum was replaced by 3000 discrete frequencies corresponding to 1000 evenly distributed points in the Brillouin zone and a sampling technique was adopted to calculate the Debye-Waller factor. The central point of the zone with zero frequency was neglected because it corresponded to a nonvibrating lattice for which the Debye-Waller exponent must obviously be zero. In the present paper the author has calculated the Debye-Waller factors for aluminium, silver and sodium for which the x-ray data about the intensities of reflection at various temperatures are available. The method of computation is exactly the same as in paper I referred to above.

2. Numerical Computation The following expression for the exponent 2 M of the Debye-Waller factor was

derived in paper I for cubic lattices

where the summation is carried out over the points lying in the 1/48th irreducible part of the Brillouin zone, q is the number of the symmetrically equivalent points corresponding to each of them and wp jn are the circular frequencies for the nth point of the zone for the p j mode.

If eT and eo 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 the absorption and various other factors, we have as in

1 ) On leave from D.B.V. College, Muzaffarnagar.

1052

paper I

P. S. MAHESH

The expression on the right-hand side represeni I the theoretical quantity while the other is the experimental one. These have been plotted in the figures for the different metals against temperatures.

2.1 Aluminium The effect of temperature on the intensity of the x-ray reflections by aluminium

has been studied by BACKHURST [ 2 ] , COLLINS [3], JAMES, BRINDLEY and WOOD [4], OWEN and WILLIAMS [ 5 ] , and CHIPMAN [6], the last two investigations being thorough and extensive.

As pointed out by JAMES [7], the observations of BACKEURST and COLLINS were not very accurate as these were much affected by recrystallisation and annealing due to the heating which would influence the primary and the second- ary extinctions. The first reliable measurements are those of JAMES et al. [4] who made a set of absolute measurements of the integrated reflections from a single crystal of aluminium a t the liquid air and room temperatures (To = 290 OK). These lie on a smooth curve with the high temperature measurements made by later workers.

OWEN and WILLIAMS [5] took an extensive series of observations from room temperature (To = 293 OK) to 903 OK. They made microphotometric measure- ments on lines in the x-ray reflections from powder specimen maintained in vacuum and the temperatures were estimated from lattice parameter changes. The fall in intensity was found to be regular up to 600 "K and beyond this tem- perature the diminution of intensity was too rapid to be accounted for by mere volume changes.

CHIPMAN [6] measured the integrated intensities of the high angle x-ray diffract- ion peaks and corrected them for the thermal diffuse scattering. His results are in good agreement with those of OWEN and WILLIAMS which were obtained long before the importance of this correction was appreciated. It is possible that their method of intensity measurement was such that the thermal diffuse scattering was corrected unintentionally.

The theoretical calculations of Debye-Waller exponent were made with the circular frequencies wPjn taken from the paper of SINGH and TRIPATHI [S]. For the purpose of sampling, the frequencies were divided in the intervals of Av = 0.4 x x 1012 s-1. The values of the ordinate Y have been plotted in Fig. 1. To has been taken to be 293 OK. There is good agreement between the theory and experiment up to 600 OK. At higher temperatures, the decrease in the integrated intensity is greater than expected theoretically.

500 700 goo

Pig. 1. Y for aluminium (To = 293 OK). Experi- mental data are

X CHIPMAN -3.0

The Debye-Waller Factors of Aluminium, Silver, and Sodium 1053

2.2 Silver

Four independent sets of x-ray measurements of silver have been made by ANDRIESSEN [9], BOSCOVITS, ROILOS, THOEDOSSIOU and ALEXOPOULOS [ 101, SPREADBOROUGH and CHRISTIAN [ll] and HAWORTH [12] using slightly different experimental techniques.

ANDRIESSEN [9] worked with silver powder and though his measurements ex- tended from room temperature up to 875 OK, he has only tabulated results for the temperature region 607" to 875 OK. His results do not agree with those of later workers. BOSCOVITS et al. [lo] have pointed out that his results were probably affected by the change of extinction effects and oxidation of the surface of the scattering material.

BOSCOVITS et al. [lo] made careful measurements of the integrated intensities of diffraction lines (111) and (422) from a silver wire a t six different temperatures in the range 81" to 774 OK. The high temperatures were estimated from the angular shift of the line (422) within a possible error of 10". In order to eliminate the errors due to the fluctuations of the primary intensity and of the efficiency of the Geiger counter, the ratio of the intensities of the two lines was measured in a number of runs and an average was taken. The expression (2) for the data of BOSCOVITS et al. is transformed as

where the unprimed and primed angles refer to the lines (111) and (422), respect- ively. To has been taken as 291 OK. The data of BOSCOVITS et al. have been plotted in Fig. 2 in the above form only.

SPREADBOROUGH and CHRISTIAN [ l l ] have carried out only high temperature measurements of the peak intensities of diffracted powder lines for silver. After stabilizing the grain size by a high temperature anneal, the peak intensities were measured for the three highest angle lines.

HAWORTH'S measurements extended from room temperature (To = 286 OK)

to 1100 OK and his work can be considered to be most reliable as he used a diffrac- tometer, measured integrated as well as peak intensities, allowed for the tempera- ture diffuse contribution to the Bragg's peaks and used a specimen repeatedly annealed at 1220 OK. His results are in good agreement with those of SPREAD- BOROUGH and CHRISTIAN [ll] but differ appreciably from those of BOSCOVITS et al. [lo] (at temperatures above 700 OK) and ANDRIESSEN [9].

The theoretical calculations of M for silver have been made with the circular frequencies coPjll taken from the paper of SINGH [13]. For the sampling purpose, the frequencies were divided into steps of dv = 0.3 x 1012 s-l. The values of the ordinate Y have been plotted in Fig. 2 against temperature as before with To =

Fig. 2 . Y for silver ( T o = 291 OK). Experimental data arc

0 B o s c o v ~ ~ s et al. x ANDRIESSES

0 HAWORTH + SPREADBOROUQH and CHRISTIAN

+ . -3.0 t

1054 P. S. MAIIESH

=T 291 O K . The experimental values of for all the workers have been

shown for comparison. In case of Boscovits et al. (,h ~ @)i has been replaced by A2

sin2 8 - sin2 6’ as discussed above.

The theoretical results agree with those of ANDRIESSEN [9] very well throughout the temperature range studied. However, this agreement is not very important in view of the later criticism about the accuracy of this experimental measure- ments. As discussed by HERBSTEIN [14], the data of other workers nearly agree with each other. There is a good agreement with theoretical values up to 600 O K .

The discrepancy gradually increases with the rise of temperature and is in the same direction as in the case of aluminium.

2.3 Sodium

Sodium has been investigated for x-ray intensities by DAWTON [15] for the temperature range 115 to 370 OK. He found strong hysteresis in the reflecting power which vanishes when the crystals are suddenly chilled with liquid air.

The intensity ration %for chilled single crystals have been reported for the

(440), (400), (310), (220), (200) and (110) reflections. Of these the (400), (310) and (220) reflections have been considered to be most reliable. In the Fig. 3, the average value of Y obtained from these three reflections only have been plotted.

For the theoretical calculations, the circular frequencies have been taken from the work of DAYAL and SHARAN [16]. The frequencies have been divided into steps of 0.2 x 10l2 s-l and the values of Y have been plotted against temperature in Fig. 3.

The curve from de Launay’s theory shows an agreement with the experimental points only up to 200 OK and thereafter the divergence between the theoretical and the experimental curves goes on increasing regularly.

TPKI-

QT

0.0

-4.0 1 \\

rig. 3. I’ for sodiuni (T, = 117 O R )

Expcrirrrcntdl data of D A % ~ O V die mdiked as 0 -8.0

3. Discussion The study of Debye-Waller factor from de Launay’s theory in this and the

earlier paper [l] reveals a common featurc. The theory is able to account for the experimental observations up to a certain temperature. Thereafter the experi- mental value of the exponent 2 M increases systematically above its theoretical value. This is not unexpected because no account has been taken of the variation of the vibration frequencies with increase of volume a t higher temperatures. This depends on the Griineisen constant considered as an average paramcter. TRIPATHI’S [ 171 theoretical calculations on potassium iodide show that the average Griineisen parameter at first increases slowly with the increase of volume. At higher volumes,

The Debye-Waller Factors of Aluminium, Silver, and Sodium 1055

however, the increase in its value is very rapid. Our results also appear to show the same feature. As the temperature rises above the reference temperature To, the Griineisen constant a t first increases slowly. The variation in frequencies with the rise of temperature in this region is small and its effect on the value of M is almost completely covered by the experimental errors in the intensities of the integrated reflections. As the temperature rises further, the frequencies change their values considerably resulting in a rapid increase in the value of Debye-Waller exponent 2 M .

The author expresses his deep sense of gratitude to Dr. B. DAYAL for his con-

The author also thanks SRI MAMCHANDRA AGRAWALA of the National Physical

Thanks are also due to the authorities of the D.A.V. College, Muzaffarnagar,

stant interest and guidance.

Laboratory, New Delhi, for helping him in finding out some data on silver.

for encouragement.

References [I] P. S. MAHESH and B. DAYAL, phys. stat. sol. 7,399 (1964). [2] I. BACKHURST, Proc. Roy. Soc. A lo?, 340 (1922). [3] E. H. COLLINS, Phys. Rev. 24, 152 (1926). [4] R. W. JAMES, G. W. BRINDLEY, and R. G. WOOD, Proc. Roy. SOC. A 125,401 (1929). [5] E. A. OWENS and R. W. WILLIAMS, Proc. Roy. Soc. A 168, 509 (1947). [6] D. R. CHIPMAN, J. appl. Phys. 31, 2012 (1960). [7] R. W. JAMES, The Optical Principles of Diffraction of X-rays, Bell & Sons, London

[8] S. P. SINGH and B. B. TRIPATHI, B.H.U. Jn. Sc. Res. XI1 (2 ) , 245 (1961/62). [9] R. ANDRIESSEN, Physica 2, 417 (1935).

1950.

[lo] J. BOSCOVITS, M. ROILOS, A. THEODOSSIOU, and K. ALEXOPOULOS, Acta cryst. Camb.

[ll] J. SPREADBOROUGH and J. W. CHRISTIAN, Proc. Phys. SOC. 74, 609 (1959). [12] C. W. HAWORTH, Phil. Mag. 6, 1229 (1960). [13] S. P. SINGH, phys. stat. sol. 2, 1165 (1962). [I41 F. H. HERBSTEIN, Phil. Mag. 8, 863 (1961). [15] R. H. V. M. DAWTON, Proc. Phys. SOC. 49, 294 (1937). [16] B. DAYAL and B. SHARAN, Proc. Roy. Soc. A 262, 136 (1961). [17] B. B. TRIPATHI, Ind. J. pure and appl. Phys. 1, 8, 278 (1963).

11, 845 (1958).

(Received September 23,1964)