Physica 125B (1984) 293-296 North-Holland, Amsterdam

PHONON DISPERSION IN DILUTE C r - W ALLOYS

S A D H A N A G A R G , H.C. G U P T A , T.K. B A N S A L and B.B. T R I P A T H I Department of Physics, Indian Institute of Technology, Hauz Khas, New Delhi, India

Received 17 May 1984

An investigation has been carried out to study the clustering of atoms in Cr-W alloys containing different concentrations (0.3, 0.8 and 1.6 at% W), based on a concentration-dependent theory using a modified rigid ion model within the Green function technique. An inference is drawn that there is no clusterings of atoms in such alloys for low concentrations of impurities.

1. Introduction

In the past few years several investigations of the phonon dispersion relation in substitutional disordered alloys have been carried out by neutron inelastic scattering [1-6]. The following features are expected theoretically: systems with sufficiently heavy impurities in a matrix of light atoms have resonance modes corresponding to vibrations with rather low frequencies while systems with light impurities in a matrix of heavy atoms have localized phonon modes, having the frequency above the host phonon band. Localized modes were observed in Cu(AI) [1], Ge(Si) [2] and Rb(K) [3], and resonance modes in Cu(Au) [4], Cr(W) [5], Ni(Pt) [6] etc. The theoretical approaches used to discuss the phonon aspects of the binary alloys utilise molecular dynamics technique [7] and also employ coherent potential approximation [8], which have met with limited success to explain the local and the resonant modes in the binary alloys of the type Al-xBx.

In the present work we have used the Green function technique [9] to explain the phonon dispersion in the case of Cr-W. The tungsten concentration for the alloys are 0.3, 0.8 and 1.6 at% W. The theoretical results are compared with the experimental results of Cunningham [5] et al. in the (k, 0, 0) and (k, k, 0) symmetry directions.

2. Theory

Consider the binary alloy of the type A~-~Bx in which the lattice is occupied by atoms of type A and B in the b.c.c, structure. If the occupancy of the lattice is randomly distributed in such a way that the concentration of A is (1 - x) and that of B is x, then the situation is of a homogeneous binary alloy. In this case, the dynamical matrix, on similar lines as eq. (8) of Kutty [9], will be given by

I ( W 2 M 2 - 05~ )8~a8~ - C~05~(q)l = 0 , (1)

where ct and fl are sublattice indices, 1 and l' are lattice sites, ~ and u denote the atomic type and M and 05 are, respectively, mass and force constant.

0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

294 Sadhana Garg et al. / Phonon dispersion in dilute C r - W alloys

In a typical binary alloy A~-xBx, we have

c ~ - - l - x , C ~ = x ,

MA M ~ - Ms M A = I - - x ' x

also

q5 AA = (1 - x ) P , 6 ~ = x P ,

d91~1 = P , cb ~ ¢ = P ,

here ' P ' is the dynamical matrix of order 3 x 3, the e lements of this dynamical matrix P are the same as of b.c.c, s t ructure and are given as follows:

P~I 8~ (c~ + 2a ' ) (1 - GC2C3) + 4c~2S21 + 4c~3(2S12 + S 2 + S 2 - 2 S 2 S ~ - 2S2~S~)

+ 2a"(2 - cos 27rak2 - cos 27rak3),

PI2 ~-- 8 (Oll _ o~ ' )SIS2C 3 q- 8ol3S1S2C1C2, (2)

where a~, a2, a3 are the central force constants up to third nearest ne ighbour for the binary alloy and a ' , a " are the angular force constant up to second nearest ne ighbour for the alloy and Ci = cos rraki,

S i --- sin rraki, where i = 1, 2, 3. Substi tut ion of these equat ions in eq. (1) gives

I(1 - x)2p - M A W 2 x(1 - x ) P x ( 1 - x ) P x2 P _ M B W 2 = 0 . (3)

In the limit x ~ 0, the roots of eq. (3) give the phonon frequencies which cor respond to the host lattice. The behav iour is r epea ted for the impuri ty a toms when x ~ 1. While in order to explain the local or resonant m o d e behaviour in the case of substi tut ional binary alloys, we propose that the distr ibution of the a toms is non- random, i.e. the (1 - x ) A a tom interacts with the (1 - x ) A a tom as well as with xAB, where A is a non - r andomness pa ramete r varying with the concentra t ion. Similarly the xB a tom will interact with the xB a tom and also with the (1 - x ) a A atom. In this manner we have incorpora ted the effect of clustering of a toms in the case of binary alloys. Hence the dynamical matrix will now be given by

I ( 1 - x ) ( 1 - x + xA )P - M A W 2

x(1 - x)(1 - a ) P x(1 - x)(1 - A)P

x [ x + (1 x ) , X l P - M B W z = 0 . (4)

The solution of the dynamical matrix will provide us the local or resonant mode behaviour depending upon the type of impurity.

3. Results and discussion

The five unknown force constants al, a: , a3, a ' and a" involved in the solution of eq. (2) are de te rmined with the help of two elastic constants and three transverse phonon frequencies at k -- 0.5 in

Sadhana Garg et al. / Phonon dispersion in dilute C r - W alloys 295

the (k, 0, 0), (k, k, 0) and (k, k, k) symmetry directions. To evaluate the force constants of the binary alloys we employ the linear relation

a~loy = (1 - X)acr + Xaw.

The relavent force constants for all the three concentrations are given in table I.

Table I Input data (in units of 10 +3 dyn/cm)

Concentrat ion a i o~ t

Cr 0.3 at% W 64.57 -24 .47 Cr 0.8 at% W 64.93 -24 .52 Cr 1.6 at% W 65.52 -24.61

0(2 eta' ~ 3

50.90 1.91 18.09 51.07 1.81 18.12 51.33 1.64 18.18

Eq. (4) which is in the form of a 6 x 6 matrix is solved for various symmetry directions. The value of non-randomness parameter h for a particular concentration is determined by fitting one of the experimental phonon frequencies for the Cr -W system given by Cunningham et al. [5]. We observe that for A = 0 itself, the theoretical results are in satisfactory agreement with the experimental ones. The present theoretical results (solid curves) and the experimental data (open circles) are shown in fig. 1.

It is obvious from fig. 1 that there is a marked agreement between the theoretical results and the experimental ones, which shows that the phonon frequencies in Cr -W alloys can be well explained by the present model, the value of A = 0 suggests that there is no clustering of atoms in the lattice at such a low concentration. Our earlier study [11] for the Rbl-xKx system suggested that there is a clustering of atoms in the lattice for x = 0.06, 0.18 and 0.29, A having the values 5.1, 2.73 and 2.3, respectively. These h values suggest that the clustering effect goes on decreasing as we increase the concentration [11]. It was apperant in the study of Rbl-xKx that, as x ~ 1, the local mode must evolve into the band mode of the impurity lattice.

n- 7 k -

> - 0

,,=,s C9 uJ

,,=3

I ( k ,k ,O } I

Cr 1.6 o f % W

I k,O,O )

° °

I

' ' f ° °

[

REDUCED WAVE VECTOR (k)

7 I , -

> -

~5 t lJ

0 Lid

3 i i

1 ( k,k,O } I (k.O,O) (k.0,0}

Cr 0.3 at % W Cr 0.8 at %W

/ 00 0000 O 0 0000000 1 ~ ; 4 o 1 2 3 4 0 1 2 3 ~

REDUCED WAVE VECTOR (k)

Fig. 1. Open circles represent the experimental points of Cunn ingham et al. while solid curves represent the present theoretical results for C r - W alloys.

296 Sadhana Garg et al. / Phonon dispersion in dilute C r - W alloys

The earlier theoretical work done by Elliott and Maradudin [10] on the Cr -W system assumes that the force constants between the impurity and its neighbours are similar to those of the host lattice and that the perturbation is caused only by the mass change at the impurity site. Almost all the attempts, made on the basis of the mass defect theory to explain the observed shifts and widths of the phonons due to the presence of impurities in the crystal, were found to be inadequate. The earlier theoretical study could not predict clearly the reasons for the discrepancy between theory and experiment. The present investigation leads to the inference that for such low concentrations, there is no clustering of atoms (A = 0) and the phonon frequency shifts can be satisfactorily explained by considering the force constant variation to be linearly dependent on the concentration,

Acknowledgement

One of the authors (SG) gratefully acknowledges the financial support from C.S.I.R., India.

References

[1] R.M. Nicklow, P.R. Vijayaraghavan, H.C. Smith and M.K. Wilkinson, Phys. Rev. Lett. 20 (1968) 1245. [2] N. Wakabayashi, R.M. Nicklow and H.G. Smith, Phys. Rev. B4 (1971) 2558. [3] W.A. Kamitakahara and J.R.D. Copley, Phys. Rev. B18 (1978) 3772. [4] E.C. Svensson, B.N. Brockhouse and J.M. Rowe, Solid State Commun. 3 (1965) 245. [5] R.M. Cunningham, L.D. Muhleskin, W.M. Shaw and C.W. Tompson, Phys. Rev. B2 (1970) 4864. [6] N. Kunitomi, Y. Tsunoda and Y. Hirai, Solid State Commun. 13 (1973) 495. [7] G. Jacucci, M.L. Klein and R. Taylor, Phys. Rev. B18 (1978) 3782. [8] M. Mostoller and T. Kaplan, Phys. Rev. B19 (1979) 3938. [9] A.P.G. Kutty, Solid State Commun. 14 (1974) 213.

[10] R.J. Elliott and A.A. Maradudin, Inelastic Scattering of Neutrons, Vol. 1 (Intern. Atomic Energy Agency, Vienna, 1965) p. 231.

[11] Sadhana Garg, H.C. Gupta and B.B. Tripathi, Solid State Commun. 51 (1984) 41.