Ionic correlations in fused salts

Volume 51A,

number 6

PHYSICS LETTERS

7 April 1975

IONIC CORRELATIONS IN FUSED SALTS* D.K. CHATURVEDI and K.N. PATHAK Department ofPhysics, Panjab University, Chandigarh-160014, India Received 31 January 1975 The expression for the fourth frequency moment of current correlation function in a two-component fluid is de-

rived. This result is used to obtain certain exact sum rules, in the long-wavelength limit, for charge current correlations in fused salts.

Recently [1,2] there has been interest in the study of ionic correlations in fused salts. It has been found from the second moments of appropriate linear combinations of longitudinal and transverse current correlations that these exhibit acoustic and optic phonon like behaviour as in solids. In view of this and other applications of the moments, in this letter we present the general result for the fourth frequency moment of current correlation function in a two-component fluid. From this result, the moments of the appropriate linear combinations of the correlation functions corresponding to the fluctuations in mass and charge current densities in fused salts can be obtained easily. In the long wavelength limit, we obtain certain exact sum rules for charge current correlations, which may be considered as the next higher analogue of the well known sum rules in a rigid-ion solid. We define the space Fourier transform of current densities of particles of type 1 and type 2 as j~(t)=

~_~___

~{p7(t),

exp (—iq -h7(t))};

(a = 1,2)

(1)

,

where p7, i~ and ma are the momentum, the position and the mass of the jth particle of type a. The current-current correlation functions are defined as S~(q,~)(~-(j~(0), J~q(t)})~,;

(a,13= 1,2).

(2)

Where the indices p and v refer to Cartesian components and the notation (---)~ represents the Fourier transform of the statistical average over equilibrium ensemble. The frequency moments of current correlation function are given by the relation

K

w~’~(q)= (—1)” kBTlirn-~ [ddt~~~

/~(t),

f~q(t)]).

(3)

To evaluate these moments we write the Hamiltonian of the two-component liquid in the form

(4)

H=EE~-~—+4~~V~(Ir7—r~I), 1/ a,~3 / cs2ma

where the primed sum excludes the case i = / if a = ]3. The function Va~represents the interionic potential. With the definition of equation of motion for the current fluctuation operator and using eq. (4) and (3) we obtain the general expression for the fourth moment of the current correlation function as JCBT ~ =

+

k~Tn2q26 m~

nq4

3&a~

~

~

{6.6,~,+4’M~~

[6w,—~

~V)2]V 7~(r)

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Volume 51A, number 6

3kBTn2

PHYSICS LETTERS

7 April 1975

]

~

+ mnm~

+___)fdrg~(r)sinq -r(q -V)V~VVV~(r)

~ (_L + -J_)fdrgy~.j(r)(v7M V 7~(r))- (V V ~, V7~(r))

mm~

(~~)f +

~

m~p

~

drg~(r)cosq r(VV~V~(r)) ~

(5)

V~(r))

ffdr dr’ g~77(r,r’)(V-V’)VMV~V7O(r)V7~(r’)

(1 +1) ~ ffdrdr’ g~(r,r’) cosq

r(V-V’)V~V~V~(r)V7~(r’)

~

where g0p(r) andg~~7(r, r’) are respectively two and three particle correlation functions for different species. From this equation, one can easily obtain expressions for the longitudinal and transverse current correlation functions, which for one component case reduce to that of Forster et a]. [3]. For fused salts it is useful to get the moments of the appropriate linear combinations of the correlation functions corresponding to the fluctuations in mass and charge current densities. These correlation functions are defined as s,~Q~M(q,w) = ~-_(/‘~(0)Jg(t) s~M(q,w)

=

+!~(0)/~(t))~

(6)

I (jM+(0)jZ(t))w

(7)

S~f2(q,w)(j~(0)j~q(tp~.

(8)

In eqs. (6—8) m is the mean mass density and p is the reduced mass of an ion pair. The fluctuation in mass and charge current densities are respectively defined as f~(t)= mi/q’(t) + m24(t)

/qQ(t) =j,~(t)—j~(t) .

(9, 10)

Using the above definitions and the result (5) it is easy to obtain the results of the fourth frequency moment of the fluctuations in mass-mass, mass-charge, and charge-charge current densities. From these results, it is seen that in the 2 whereas the moment long wavelength limit thedensity moments of mass-mass mass-charge current densities go as q of charge-charge current tends to a finite and value. In the long wavelength limit, the fourth moment of charge-charge current correlations simplifies considerably a coulomb and short-range part, i.e.

if we divide the potential V~(r) into V~(r)= V~(r)+ V~(r).

(11)

Then using Laplace theorem for coulomb part, we obtain ~

(4)(q) + 2w~4~]

rn 1rn2~(fdrgi2(r)(vvv12(r))2

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Volume 51A, number 6

+~

~‘

[J~

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PHYSICS LETTERS

7 April 1975

2 V~(r)V ~2!L (.1 +_i_) + ~~1”~] ffdr dr’g~~7(r, r’)(V-V’) 7(r’))~

(12)

where primed sum denotes a * j3 * ‘y, and ti1 = 1 and n2 = —1. This result may be considered to be the next higher analogue of the first Szigeti relation in a rigid-ion solid. If we neglect, like Brout, the short range interactions between like ions, eq. (12) further simplifies to 4)(q) + 2w~~4kq)J = m urn [Q( 1m2 ~ ~fdrgi2(r)(vvv~2(r))2 (13) +n ffdrdr’[_~ gi22(r~r’)+_~g2fl(r,r’)] (V~Vl)2V~2(r)V~2(rl)) Another exact result we obtain is 4~(q) w~~~4~(q)] +~ [~~)~Q(2)(q)w~~2)(q)J2) = mi (4 irne2 )2 (14) urn [w~~ which establishes a simple relationship between the second and fourth moments of charge current correlations in the long-wave-length limit. Other applications of the fourth frequency moment will be in the estimations of the conductivity and viscosities of fused salts. In the one component case this has already been done and at present work is in progress to generalize these theories for fused salts as well.

{

Work supported by

‘~‘

—

the U.S.

—

National Science Foundation under the grant No. GF-36470.

References [11 M.C. Abramo, M. Parrinello and M.P. Tosi, J. Nonmetals 2 (1973) 57, 67. [21 M.C. Abramo, M. Parrinello and M.P. Tosi, 1. Phys. Cl (1974) 4201. [31 D. Forster, P.C. Martin and S. Yip, Phys. Rev. 170 (1968) 155.

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