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LAURENS JANSEN
Institute for Molecular Physics, University of Maryland, College Park, Md.
Molecular Theory of Dielectric Properties
DIELEOTRK
theory starts from the axwell equation for the divergence of field E. This equation may be written as
n=O
v
denotes the differential operator; N@)is the electric multipole moment of order n of the atom, and [n] indicates that the product of the two nth rank tensors, vnand ( p N ( " ) } ,is contracted rz times. The number of atoms per unit of volume is p , and (pN(")f represents the density of multipole moment of order n. For n = 0 we have the true charge density { p t } ; n = 1 gives the dipole polarization, P, n = 2 the quadrupole density, Q, etc. V.E =
(pt]
V . ( P - '/pV.Q+. . .)
(2)
By introducing the electric displacement vector D =E P - l/pV*Q+ . . . Equation 2 may be written in the familiar Maxwellian form V.D = (ptl I t is customary to terminate the series for D after the dipole term; this may be done if,the macroscopic quantities vary only slowly in space. If the dielectric is linear-i.e., if P = XE, D = EE,with E = 1 X ; x and E are called electric susceptibility and dielectric constant, respectively.
+
+
.
Clausius-Mossotti Equation Still another field is needed, the local field, acting at the center of a specific atom. Again we have a microscopic local field, from which a macroscopic local field can be formed by statistical averaging. Lorentz showed that under certain definite conditions the average local field is equal to E P/3. He derived his formula for a lattice of dipoles arranged with cubic symmetry. The atoms must all have the same constant dipole vector. Another critical assumption lies in the statement that the dipole moment induced in an atom is equal to the polarizability of the atom times the local field-
+
Le., p~ = CYE/Z.
expression gives the correct qualitative behavior, but it is clearly incorrect in the region of very low densities. I t represents approximately only one half of the observed value for compressed argon (7). The deviations due to I1 have recently received considerable attention in the literature. The reason for these deviations is that the polariz, ability is not a dynamical quantity; therefore, it ceases to be a physical concept when the molecules are coupled. W/V) (Pi1 = ( ~ / v ) f f m % I p = {PPI I t occurs in the theory as a variable, (4) and therefore it is of no use for finding Now P = (E- 1) E, and, according to the solution (4-6). The result of the = E P/3, so that: Lorentz, {E,($) quantum mechanical analysis is :
of CY with the polarizability is not at all obvious. The polarizability is defined only in V ~ C U O , and not for an atom coupled with other atoms. The Clausius-Mossotti equation is valid if the Lorentz formula for the local field holds, and if a in Equation 3 may be identified with the polarizability. For a homogeneous medium with N atoms in a volume V, the dipole polarization is given by:
+
-
€ - I Nor €+2v=- 3
(Clausius-Mossottiequation)
(5)
Deviations from ClausiusMossotti Equation
Experimental results show that the Clausius-Mossotti equation is a good approximation if the medium is homogeneous and if the molecules do not possess permanent electric dipole moments. There are small but significant deviations, which are particularly interesting for compressed non(di)polar gases. The left-hand member of Equation 5 first increases with increasing density, reaches a maximum at approximately 200 Amagat units of density, and then decreases, falling below N a / 3 . There are two possible reasons for these deviations: (I) The average local field is not equal to Lorentz' value, E P/3; (11) the quantity a in Equation 4 does not represent the polarizability of an atom. The theory pertaining to f is essentially due to Yvon and Kirkwood (2). The average local field in the gas is equal to the Lorentz field if the molecules carry during their motions always the same dipole vector. An iteration process may now be developed by releasing the condition of constant dipole vector for successive molecules. I n the first order only the dipole of the selected molecule is allowed to fluctuate; in the second order there is one more fluctuating dipole, etc. The result is a virial expansion for the left-hand member of Equation 5 :
+
Literature Cited (1) Boer, J. de, Maesen, F. van der, Seldam, C. A. ten, Physica 19, 265 (1953). (2) Brown, W. F., "Handbuch der Physik,"
vol. XVII, Springer Verlag, Berlin,
1956. (3) Gaizauskas, V., Welsh, H. L., Proc. Conf. on Optical and Acoustical Properties of Compressed Fluids, Paris, July 1957. (4) Jansen, L., Mazur, P., Physica 21, 193. 208 (1955). (5) Jansen, L., 'Solem, A. D., Phys. Rev. 104,1291 (1956). ( 6 ) Mazur, P., Mandel, M., Physica 22, 2 9 8 , 2 9 9 (1956).
RECEIVED for review November 30, 1957
(3)
Here CY is the polarizability of the atom and E$; is the macroscopic local field at the center of atom i for a specific configuration of atoms. The identification
In this expression a. represents the polarizability of the atom, a = 010 f na! is the parameter which so far had been identified with ao,and S gives the effect of translational (and rotational) fluctuations. The expression for may again be given as a series of powers of the density; its evaluation is complicated and requires the knowledge of molecular wave functions. In a few simple'cases it was found that A
