Solid State Communications Vol.2, PP , Pergamon Press, Inc. Printed in the United States. LATTICE VIBRATIONS OF TUNGSTEN*

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1 Solid State Communications Vol.2, PP , Pergamon Press, Inc. Printed in the United States. LATTICE VIBRATIONS OF TUNGSTEN* S. H. Chen and B. N. Brockhouse Department of Physics, McMaster University, Hamilton, Ontario, Canada. (Received 29 January 1964) The frequency wave number dispersion relations of the lattice vibrations in the symmetric directions of body-centered cubic tungsten at room temperature have been measured by means of inelastic neutron scattering. The measurements were made with the Chalk River triple-axis crystal spectrometer using the constant Q method. ~ The results in the three major directions [001], [110] and [lit] are reported here together with a Born - Von Kárm~ngeneral force model analysis from which is inferred the range and magnitude of the forces between the atoms in this metal. The connection between the results and the unusual elastic behavior of tungsten is discussed. BECAUSE of the relatively high thermal the neutron groups for these phonons were neutron absorption cross-section (19.2 b.) poorly formed. The frequencies for L[crd and low coherent scattering cross-section with ~ = 0. 6, 0. 7, 0.8 are also more than (2. 74 b.) of tungsten, four individual usually uncertain. The following general cylindrical single crystals (about 0. 6 cm in features of the curves should be noted: diameter and 6. 3 cm in length) were used as 1. The two transverse branches in a composite specimen. The crystals were the [001] and [111] directions are dengenerate purchased from the Materials Research Corp. by symmetry. and were cut from a 25 cm length crystal 2. Zone boundary points in both [001] having the [110] direction roughly along the and [111] directions are equivalent and at cylindrical axis. In the experiment the four these points all branches are degenerate by pieces of crystal were mounted vertically in symmetry. an aluminum frame. The alignments of the 3. The longitudinal branch in the [001] crystals in the frame were individually direction has a maximum at about ~ = adjusted. The relative alignment of the This implies a large second neighbour force crystals is believed to have been within 1/4. constant~. The frame was so arranged that either (110) 4. l he two transverse branches in or (001) planes of the crystals could be in the [110] direction are almost degenerate for (horizontal) plane of the spectrometers. The small ~ and they cross at about ~ [110] T 1 branch was measured in the (001) The small dip in the T2 branch near c= ~ plane; the other branches in the (110) plane. implies the existence of very long range forces. The results are shown in Fig. 1 (p. 74) where the frequencies (in units of 1012 c/s) The qualitative features of the measured are plotted against the component of the dispersion curves can be understood however reduced wave vector ~ defined by ~ = in terms of relatively short range general (2~/a)~for example, along the [110] forces. A preliminary analysis of the curves direction ~ = (~0). The frequencies of using ~ Born - Von K~rmangeneral force some modes at special points in reciprocal model was carried out for forces of varying space are given in Table 1 (p. 74). The range, out as far as 8th neighbors. The frequencies for the LftrO ] branch near the results for the 3rd neighbor force model zone boundard are especially uncertain because (seven force constants are plotted as broken * Experimental work carried out at the Chalk River Laboratories of Atomic Energy of Canada Ltd. 73

2 74 LATTICE VIBRATIONS OF TUNGSTEN Vol. 2, No [~oj [oos] Z.~B. I, / ~ ~,1. /._( -~ ~ L ~ 5.0 ês L /.- _., _., ~ H ~ ~4.,i T, 0~5 (~ ~0) (000) (IH)-(00I) (~~) (000) Dispersion curves for the lattice vibrations in the three major symmetric directions in tungsten at room temperature TABLE 1 Frequencies of Phonons at Room Temperature at Some Specific Points in Reciprocal Space Coordinate (in unit of 2iVa) and mode (a = A) Frequency (1012 c/s) (0 0 1) All modes degenerate ± 0. 1 (1/2 1/2 1/2) LandT degenerate 5.50 ±0.05 (1/2 1/2 0)L Maximum point ±0. 15* (1/2 1/2 0)T ± 0.05 (1/2 1/2 0)T2 4.15±0.05 ( )L 6.30 ±0.07 ( )T ±0.03 ( )T ±0.05 *This value may possibly be in serious error because of poorly formed neutron groups.

3 Vol. 2, No. 3 LATTICE VIBRATIONS OF TUNGSTEN 75 TABLE 2 Stretching and Bending Force Constants in Units of 1O 4 dynes/cm 1)3 Stretching force constant Bending force constant s C 1(s) = CB(s) = p~/d i = = ~2= ~ ~2 = n~+.y3 =0.81 B lines in Fig. 1. The values of force constants perpendicular shear elastic waves (for an thus obtained (in units of ~ dynes/cm and in the notation of reference 2) are as follows: direction of propagation) have the same velocity. This fact has the special consequence ~ ± 0 02 that the slopes of [1l0]T1 and [i10]t2 are the 2..JO same in the long wave length limit (~= 0). = 1. 92±0.03 This is clearly seen also in our experimental results. In fact, we have tested the consistency = 4 73±0 05 of our results in the long wave length limit 0.2 with the elastic constants by imposing or not = ±0.03 imposing the condition that the slopes of the dispersion curves must auproach the correct = 0 32 ± 0 01 sound velocities given by the elastic constants; Ct3 the two sets of force constants so obtained B3 = ± agreed within the experimental errors. By = 0 49 ± 0 02 the method of long waves, 5 it can be shown that the elastic constants are related to the It is seen that the qualitative features of the force constants by the relations (up to 3rd curves are reproduced in this model. Taking fl ig r. C into account additional neighbors up to 6th a 11 = 2ci.~ (15 force constants) does not improve the agreement significantly. If atoms up to 8th ac = 2a + 2~ + 4ct + 4~ neighbors are included, the agreement becomes 44 1 ~2 3 3 better, although the hump in the [il0]t2 branch near the zone boundary still cannot be a(c11-c12)/2 = 201-2B12+B2-~-6a3-~-2B3~4y3 reproduced. Thus it must be concluded that the detailed nature of the forces is quite Thus, the condition of isotropy imposes the complicated in tungsten in that, besides the strong short range forces, there exists a weak condition upon these force constants that the combination 2B1-a2+82-2Ct3+2B3+4y3 = 0. but long range force system analogous to, but This condition is approximately satisfied by weaker than, thqj found for niobium by our values of force constants (the combination = Nakagawa etal. A few qualitative conclusions 0.08). For forces extending only to 2nd can be based on the 3rd neighbor force model: neighbor, it can be shown easily the [llo}t1 1. According to the recent measure- and [l10]t2 should be degenerate all the way ment of adiabatic elastic constants of tungsten to the zone boundary; this is not the case and by Featherston and Neighbours, 4 the condition consequently the forces must extend beyond of isotropy, i.e., C44-1/2 (Cli - C12), is 2nd neighbors. satisfied to within 1% in the temperature range 2. In order to discuss the nature of the from 0 Kto 3000K. Physically, C44 and interaction between the atoms let us adopt the 1/2 (C11 - C12) are two shear moduli and their central force model, i.e., assun~that the equality implies that the two mutually interation is binary and that it can be derived

4 76 LATTICE VIBRATIONS OF TUNGSTEN Vol. 2, No. 3 from an effective potential depending only on between an atom and its first neighbor is the distances between atoms. The condition attractive and the forces between it and its of validity of this model is the Cauchy relation second and third neighbors are probably C 12 = C~, which is satisfied to within 22% repulsive, in agreement 8, and with by conclusions Featherston & according constants4, toor theequivalently reported values the Born-Huang of elastic Neighbours4, reached by Isenberg from consideration of the elastic equilibrium condition5 2 that the combination constants B = 0, which is satisfied approximately (experimentally the combination= We have also calculated the frequency ). In such 5 toalinear modelcombinations the force constants of first and can be second related derivatives of the effective potential distribution obtained fromfunction the 8thusing neighbor the 23 best force fit. const Theants frequency distribution curve shows two function ~, where s denotes different pronounced peaks, at 4. 6 x 1012 and at neighbors. If we take linear combinations of 6. 3 x 1012 c/sec. The Debye temperature force constants in such a way that they are 8p (T) calculated from this frequency proportional to ~ and ~ respectively, we get distribution function is 3800K at absolute zero; the result in Table 2, where d 5 is the it gradually decreases through values equilibrium distance to the neighbors s. The Kat T = 20 Kand 318 Kat 8D = 312 K, 50 Kto proportional combinationstof4~ and force constants ~ can be which interpreted6 are an foressentially temperatures constant from value, 100 Kto 300 K. as bond-stretching ( C 1(s~)and bond-bending (CB(s) ) force constants respectively. This research was supported by a grant Assuming the same functional form for the from the National Research Council of Canada. potentials of all neighbors s, it is clear from We are grateful to Dr. A. D. B. Woods, the relative signs of C1 (s) and CB(S) in Table 2 Dr. 0. Dol ling and Mr. E. A. Glaser for that it is quite impossible to construct a cooperation and advice in carrying out the sensible potential function between atoms, whereas do so. However, for sodium, if the it was functional found possible form of 7 the to experiments, his least square to Dr. fit programs Y. Nakagawa and for use his of helpful discussions, and to other members ~ is different for different neighbors, then the of the staff of Atomic Energy of Canada Ltd. signs of C 1(s) and CB(S) suggest that the force for their cooperation. References 1. BROCKHOUSE B. N., Inelastic Scattering of Neutrons by Solids and Liquids p~113 International Atomic Energy Agency, Vienna (1961) 2. WOODS A. D. B., BROCKHOUSE B. N., MARCH R. H. and STEWART A. T., Phys. Rev. 128, 1112 (1962) 3. NAKAGAWA Y. and WOODS A. D. B., paper submitted to Copenhagen Conference on Lattice Dynamics (1963); Phys. Rev. Letters 11, 271 (1963) 4. FEATHERSTON F. H. and NEIGHBOURS J. R., Phys. Rev. 130, 1324 (1963) We are grateful to Dr. Neighbours for sending us results prior to publication. 5. BORN M. and HUANG K., Dynamical Theory of Crystal Lattices. Oxford University Press, London (1954) 6. LEHMAN G.W., WOLFRAM T. and DE WAMES R. E., Phys. Rev. 128, 1593 (1962) 7. WOODS A. D. B., Inelastic Scattering of Neutrons by Solids and Liquids p. 3, Vol 2. International Atomic Energy Agency, Vienna (1963) 8. ISENBERG I., Phys. Rev. 83, 637 (1951).

5 Vol. 2, No. 3 LATTICE VIBRATIONS OF TUNGSTEN 77 On a mesuré les courbes de dispersion des fréquences des oscillations thermiques se propageant suivant les axes de symétrie du tungstè avec la spectroscopie neutronique. Les resultats (analyses selon la théorie Born-Von Karmán) se montrent que les forces interatomiques s étendent au d~lales 8-e voisins. Cependant, les voisins l-r et 2-e les contributions principales aux forces; celles-ci sont, respectifment, attrayantes et repoussantes.