Critical ultrasonic attenuation in a disordered binary alloy

Volume 73A, number 3

PHYSICS LETTERS

17 September 1979

CRITICAL ULTRASONIC ATTENUATION IN A DISORDERED BINARY ALLOY B.K. CHAKRABARTI’ and G.K. ROY Saha Institute of Nuclear Physics’, Calcutta 700 009, India Received 8 June 1979 The expression for ultrasonic attenuation in a binary alloy (annealed disordered) system has been obtained solving the Green function equation of motion and performing configurational averaging. Its typical dependence on the configurational fluctuation correlation is seen to lead to a non-divergent behaviour in the ultrasonic attenuation near the order—disorder transition point, where the correlations diverge. The results compare well with the experimental observations on /3-brass.

Ultrasonic experiments in disordered binary alloys indicate that the critical exponent associated with the attenuation coefficient near the order—disorder transition point (Tc) is zero. The acoustic attenuation in j3-brass exhibits [1] a strictly flat region in the neighbourhood of the critical point and then falls off on either side of the critical region. This behaviour is different from that obtained in other transitions (e.g., ferromagnetic, ferroelectric etc.) where the attenuation is proportional to the correlations which diverge at Tc with finite critical exponents [2—41. To understand this behaviour, an investigation of the phonon modes in such an annealed disordered system has been undertaken. After configurational averaging the attenuation is found to depend on the correlations in a way that can account for such a temperature dependence. The acoustic attenuation is taken to arise from phonon scattering by configurational fluctuations which become critically correlated at Tc. A pseudo-spin representation [5], (1) of the cooperative fluctuations will be used while the lattice hamiltonian will be taken in its harmonic form,

H2=L—+-’-~ 2rni 2~j,~/3 p~u7uf,

(2)

g,o

where rn

=

=

(1

—

c~) rn~+ cimB +c

1(p~ — 4A) + c1(p~—

~r)

+ cjj(~p~+ ~BB

—

2~AB)

(3)

The occupation number c1 = 0 or I according as the ith site is occupied by an A- or B-type atom. The implicit coupling of the phonons with the fluctuations is contained in the relation C1 = + o~. The phonon self-energy, whose imaginary part gives the attenuation, is obtainable from the lattice Green function ((u~(t);uf(0)))

,

(4)

where the thermal averages (denoted by the two overbars) involve traces over both phonon and configuration variables. For an annealed system considered here, the order in which traces are performed is immaterial. Summing over the lattice variables first, the co transform of (4) is obtained from the Green function equation of motion in Address from October 1979: Department of Theoretical Physics, Oxford.

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Volume 73A, number 3

PHYSICS LETTERS

17 September 1979

matrix form (5)

21—K)b=M,

(w

where, for simplicity, we have taken the phonon modes to be degenerate [6] which makes the matrix b diagonal in the cartesian indices a, j3. In terms of the off-diagonal (in site indices if) force constants, the matrix K is given by K

1~

11

=

—

and

m~’ c,O~j ,

[M]~,= m~’~

(6)

.

m~

Defining the lowest-order configurationally averaged Green function D0 by (7)

(~2Ij~)~O=M,

one can write eq. (5) in the form

b = M-l(DOM + D°AD).

(8)

For an algebraic mean type of disorder, the matrix A is given by A,~= K~,

—

14

E ~A

=

—

~Aj

~ +

~ ~,

~

pftAp 1

—

~A~1]

~9)

,

with M (1 CB)MA + CBMB and the configurational fluctuation operator p~= c1 (c1) = (a,). The parameters ~ and are defined for nearest neighbour interactions as ~ = (~B/mB)/(c~A/mA) 1, e = pAB/~pAA 1. We next introduce the following definitions: —

—

—

—

b~q,q‘)=N~

—

p~q)=N~ Ep~e~~Ri.

~J~11e~qi~q’1?/,

(10,11)

Because does not become translationally invariant till the thermal averages respect hamiltonian 0(q,q’) with = D0(q) ~(q to the q’). Ising By taking the (1) are taken, its Fourier transform depends explicitly on q and q’ as D double Fourier transform of eq. (8), we obtain the equation —

D = D0 +

(i7/iIif)

D0p +M1D0A0,

(12)

where we have adopted the matrix definitions (qIbIq’)b(q,q’), and

(qlDO~q’)~b0(q)6(q—q’), (qIAIq’)=A(q,q’)

,

(qlplq’)p(q —q’)

flr~MB_MAl/rnB_l/rnA. With these definitions, one gets A(q,q’)

=

p(q

—

q’) v(q,q’) = p(q

q’) {(~/mA)[~A(0) p~(q’)] + (e/rn~)[~A(q q’)

—

—

—

—

pAA(q)]}

(13)

We now iterate eq. (12) to obtain =

+

(b°(A/i~b°+b°(A/M)ö°(A/M)~°+ ...) +i~D0(p/M)+b0(A/M) b0(p/M) +...),

(14)

which we have to average, term by term, over all possible configurations of the system given by eq. (1). This gives rise to multiple pseudo-spin correlation functions which we approximate by factorisation into all possible pairs of configuration variables p(q). This is most conveniently accomplished by representing D0 and A by a line and a dot, respectively; the process of taking the trace over the configuration variables then corresponds to linking two dots at a time in all possible ways. The first series of eq. (14) defines the phonon self-energy and the contributions from diagrams of the type

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Volume 73A, number 3

PHYSICS LETTERS

17 September 1979

/1

/

I’

I’__

+

~

A

~

~s

/

-1-

~

q

1

q2

q3

q1

q2 q3

q4

q5

can be summed to infinity. The self-energy is then given by

—4--2 E q

H(wq)

NM +

—~-

2M4 N

V(qq1) V(q1q)g(q

q1)bO(~,q1)

1

~

2(c,iq,)DO(wq V(qq1) V(q1q2) V(q2q1)V(q1q)g(q —q1)g(q1 _q2)(bO)

2)

q 1q2

—4~-3M6 q ~

+

—

N

V(qq1) V(q1q2) V(q2q3) V(q3q2) V(q2q1) V(q1q)

1q2q3

2(wq X g(q — q1)g(q1

—

q2)g(q2

—

q3)(fiO)

2(~q 1)(~O)

2)~O(wq3) +...,

(15)

where N(p(q) p(q’)) = g(q) ö(q + q’), the ~ term coming from the translational symmetry of the pseudo-spin hamiltonian (1) and g(q) being the2 +fluctuation correlation (determined by hamiltonian (I)) for near which we q2)—(l —‘i12), K IT function Te I”. Since g(q) becomes inordinately large T~ forcan q -~0, use the form [7]: g(q) (K the major contributions to the q, summations of eq. (15) come from the region where q I
—

—

E

M2V2(qq)g(O)N1 H(~q)=

Iq—kI<~

—

E

1 —M2V(qq)g(0)N’

(16)

.

—

(B~)2(wk)

Iq —k I <~

Making the analytical continuation w to integrals to yield N

I

~ ~.,

~ Iq—kI
w

+

i, putting w = cq with q -~0,the summations of(16) are converted

~ I

____

D (wk)——-——i2w~--c~ln 4ir2c3\

Iq—kI
N~

-~

(b0)2(wk)

=

W~+C~.)/

_~~~_N1 3w2

E

D0(wk) =

Iq—kI<~

4irc3 ~

(-!--

In

47~2c3\2(JJ

~

~

—

—

~

°~

‘~I + i ~

8irc3~’

(17)

where is the volume of the crystal unit cell. In the neighbourhood of the order—disorder point, g(0) ~ I so that the temperature dependence in eq. (16) cancels out and the acoustic attenuation is obtained, using the smallness of w~,as &~

,

a = (2wc)1 Im H(wq) = (4/ircw~)W2 ,

T

Tc,

(18)

in the long wave length limit. Thus the temperature dependence of the acoustic attenuation acquires a flat region near the critical point. On the other hand, far away from T~,g(0)~l,so that, using V(qq) = (~ + ~)w2, one gets ITTcI~O, 208

(19)

Volume 73A, number 3

PHYSICS LETTERS

17 September 1979

which falls off with an exponent ‘y = 2v(l i~/2)as one moves away from the critical point. In view of the decoupling used in (15) and the approximations made in the frequency sums in (16), the most natural choice of the exponents would be the Ornstein—Zernike (OZ) values: v = 0.5, t~= 0,7 = 1, rather than those of the Fisher—Burford [8] form: v = 0.643, i~= 0.056, ~ = 1.25, because in this form gT~(q)is not necessarily maximum at q = 0. Also, in fact, the OZ form fits well with the neutron-scattering data for 13-brass, which is not very sensitive to any subtle deviation from that form [7]. 0)to another (a w2IT— TCI°, The cross-over from one type of behaviour (a w4IT— TCI~,IT— T~I~’ T T~)occurs when the real part of the denominator in (16) becomes ~‘~SO. The width of the flat region then turns out to be of the order of [(~+ )2/2ir2c3M2w~]w4. Ultrasonic attenuation arises here solely from phonon scattering by configurational fluctuations. The pseudo (Ising) spin model (eqs. (1), (2)) of the binary alloy does not include any relaxation mechanism for the fluctuations (strictly “Ising” spins do not have any dynamics). As such, the singular “hydrodynamic” form of the attenuation, resulting from the phonon scattering by the critically relaxing fluctuations, does not appear. The absence of a divergent behaviour in the attenuation and its fall away from T~is quite in accord with Young’s observations on 13-brass, where a linear dependence on frequency was obtained. Apart from approximations made in the frequency integrations in (16), the different frequency dependence shown by our calculations, performed in the low-frequency region, may be due to neglecting the couplings with other relevant variables of the system such as the electronic degrees of freedom etc. —

-‘

‘-

The authors are grateful to Profs. S.K. Sinha and B. Dutta Roy for some valuable discussions. One of the authors (BKC) is grateful to CSIR (India) for financial support. References [1] P.L. Young, Solid State Commun. 9(1971)1211. [21 B. Luthi, T.J. Moran and R.J. Poffina, J. Phys. Chem. Solids 31(1970)1741. [31 K. Kawasaki, Phys. Rev. A3 (1971) 1097. [41 K.K. Murata, Phys. Rev. B13 (1976) 4015. [51R.A. Tahir—Kheli, Phys. Rev. 188 (1969) 1142. [61 Yu. A. lzyumov, Adv. Phys. 14 (1965) 569. 171 J. Als-Nielsen, in: Phase transitions and critical phenomena, eds. C. Domb and M.S. Green (Academic Press, 1976) Vol. 5A, p 87. [8] M.E. Fisher and R.J. Burford, Phys. Rev. 156 (1967) 583.

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