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The Theory of Electric and Magnetic Susceptibilities: Unfinished Second Edition

TEXTUAL BACKGROUND

John Van Vleck published his classic monograph “The Theory of Electric and Magnetic

Susceptibilities” in 1932 while on the faculty of the University of Wisconsin. In his later years,

he worked on a second edition which unfortunately was never completed. After his death his

wife Abigail gave his notes for the second edition to Chun Lin. These notes contain additions

and other, mostly minor, changes to Chapters I, II, III, IV, and VI of the first edition. There are

no notes for Chapters VII - XIII. Chapter V, “Susceptibilities in the Old Quantum Theory,

Contrasted with the New” in the original edition was to have been omitted. In its place was an

entirely new chapter entitled “The Local Field and the Dielectric Constants of Condensed

Media” that was inserted after Chapter III. The old Chapter IV, “The Classical Theory of

Magnetic Susceptibilities” would become Chapter V in the new edition.

In view of the long-standing interest in the first edition, it would be worthwhile to present

here Van Vleck’s draft of the new Chapter IV that he had planned for the revised edition. It is

hoped that researchers interested in this area may find it beneficial to have available this part of

Van Vleck’s work which results from his life-long association with this field.

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Text of a new chapter in the unfinished second edition of

J. H. Van Vleck’s The Theory of Electric and Magnetic Susceptibilities

©2011 The University of Wisconsin Foundation

IV

THE LOCAL FIELD AND THE DIELECTRIC CONSTANTS

OF CONDENSED MEDIA

25. Depolarizing (or Demagnetizing) Corrections

In §5 we stressed that the local field eLOC is not the same as the macroscopic field E. The

primary purpose of the present chapter is to derive and examine various formulas which have

been proposed for the local field, as the latter plays a fundamental role in the theory of the

dielectric constants of condensed media such as liquids and solids where the difference between

E and eLOC is very important.

However, before proceeding to the investigation of eLOC, it is advisable to first look into

the relation between E and E0, the field in the absence of the body. It will be assumed that E0

has a constant value through the space to be occupied by the body and that the insertion of the

body does not modify the charge distribution producing E0 (or in the magnetic case the current

distribution generating H0). If the body is of arbitrary shape, E will vary from point to point

within it even though E0 is constant and even though the body is homogeneous and isotropic, as

we suppose throughout the chapter. Only if the sample is cut in the shape of an ellipsoid does

the constancy of E0 imply that of E. The situation simplifies considerably if in addition one of

the principal axes of the ellipsoid is parallel to E0, for then one can show that

E = E0 – NP. (1)

The proportionality factor N does not depend on the dielectric but only on the shape of the

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ellipsoid. It is called the depolarizing factor. In the magnetic case there is a corresponding

relation

H = H0 – NM, (2)

and then N is called the demagnetizing factor. This term one encounters more frequently in the

literature of physics than one does the corresponding depolarizing factor in the electric case. The

reason is that is it relatively easy to measure the drop in the electric potential across the body in

the case it fills all the space across a condenser. On the other hand this can not be

correspondingly done in the magnetic case, and one usually measures the field H0 in the absence

of the body. As the permeability µ is the ratio B/H rather than B/H0 a knowledge of the

demagnetization corrections is essential to the determination of µ from experiment unless µ is

nearly unity.

The proof of the legitimacy of the assumption (1) or (2) and the mathematical theory for

the calculation of N for an arbitrary ellipsoid is a classical problem in electrostatics first solved

by Maxwell. The analytical formulas for N are fairly simple in case the ellipsoid is one of

revolution, and tables of N for variously shaped ellipsoids have been published in the literature

[1]. The values of N appropriate to the three principal axes of the ellipsoid always obey the sum

rule

Nx + Ny + Nz = 4π. (3)

We shall give the values of N only for simple limiting cases, viz.

N = 0 (very prolate ellipsoid, needle or thin plate || E0),

N = 4π (very oblate ellipsoid or thin plate ⊥ E0),

N = 4π/3 (sphere). (4)

We have assumed that the long axis of the prolate ellipsoid and the short axis of the oblate one

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are parallel to E0. If instead the latter is perpendicular to E0, the value of N is 4π. A thin plate

or slab is equivalent to a very oblate ellipsoid except for corrections near the edges. The

appropriateness of the values N = 0 and N = 4π for the cases listed above are obvious

consequences of the continuity of, respectively, the tangential component of E and the normal

component of the induction E + 4πP at the bounding surface of the body. The value N = 4π/3 is

established by a simple electrostatic calculation, which we now give now in fine print.

We will solve the slightly more general problem of a sphere of dielectric constant ε1 and radius a

embedded in an infinite medium of dielectric constant ε2. The present case corresponds to ε1 = ε and ε2 =

1 , but the general results are needed for later work. If the applied field is along the x-axis, the

electrostatic potential function for this problem is

φ = φ2 = −E∞x −[3ε2 /(2ε2+ ε1) − 1]E∞(a3/r

3)x (r ≥ a), (5)

φ = φ1 = −3ε2 /(2ε2+ ε1)E∞x (r ≤ a), (6)

as Laplace’s equation is satisfied, also the boundary condition − ∂φ/∂x = E∞ at r = ∞ and the continuity

equations

φ2 = φ1 ε2∂φ2/∂r = ε1∂φ1/∂r, (7)

at r = a (continuity of the tangential component of the field and normal component of the induction).

That these conditions are met is readily verified when φ is expressed in polar coordinates by setting x =

rcosθ. In the present context, where ε2 = 1 and ε1 = ε, we can identify E∞ with E0, and −∂φ1/∂x with E.

One thus has E = 3E0/(2 + ε). When one remembers that by definition ε − 1 = 4πP/E one sees that

E = E0 − (4πP/3) in agreement with (1) if N = 4π/3.

26. The Lorentz Local Field and its 4π/3 Catastrophe

The classic expression for the local field is that of Lorentz [2], viz.

eLOC = E + (4π/3)P (or hLOC = H + (4π/3)M). (8)

This result is obtained by assuming that the appropriate local field is that at the center of a small

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spherical cavity in the dielectric (or magnetic) body and that the polarization of the body, apart

from the cavity, is “frozen” or in other words is unconscious of the existence of the cavity.

Perhaps the easiest way to derive the result (8) is to note that with the model just

described, the field at the center of the cavity must be the same as the field before the cavity is

created minus the field arising from the polarization of the matter which is “punched out” in

creating the cavity. In other words, one has

eLOC = E − eSP, (9)

where eSP is the field at the center of a uniformly polarized sphere, as the latter is supposedly so

small that the microscopic polarization is substantially uniform over its dimensions. The

magnitude of eSP is most readily computed by regarding the polarization as generated as a

limiting case of two equally but oppositely charged uniform spheres. By an elementary

application of Gauss’ theorem, the field inside a uniform symmetric charge distribution of

density ρ is radially directed and of magnitude −4πρr/3. Hence if the two spheres be centered at

+ ½∆x and − ½∆x, the field at x = 0 is −2(4π/3)½ρ∆x, and in the limit ∆x → 0, ρ∆x → P this

reduces to −(4π/3)P and (9) is equivalent to (8).

We now give an alternative derivation of (8) which is rather illuminating, especially as

regards the relation of the Lorentz field to the depolarization corrections. Also, it may be more

satisfying to readers who may have qualms about the rigor of the calculation in the preceding

paragraph inasmuch as the field due to the dipole has a singularity at the origin. The analysis in

Chap. I, especially its Eq. (23), shows that the field in the presence of a dielectric equals E0 in its

absence plus the field due to the body’s polarization (with of course the understanding that the

presence of the body does not modify charge distribution producing E0, so that E0 is unaffected).

Hence we have

eLOC = E0 − ∫∫∫grad(Pr/r2) dV

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where with the Lorentz model the volume integral includes all the body except the cavity. The

volume integral can be transformed immediately into two surface integrals, so that for say the x-

component

eLOCx = E0x − ∫∫CAVPr(cos(x,n)/cos(r,n))dω − ∫∫EXTPr(cos(x,n)/cos(r,n))dω, (10)

where dω is a small element of solid angle, emanating from the origin, which we take as the

center of the cavity where the local field is being computed. The notation Pr is used for the

component of P in the direction of the radius vector corresponding to the solid angle dω. The

two integrals are, respectively, over the wall of the cavity and the exterior boundary of the body,

the only boundary it would have were it not for the cavity. The outer normal with respect to the

surface of the body is denoted by n. As the cavity is spherical and as the outer normal with

respect to the body points inward along the radius, the factor cos(r,n) in the denominator of the

second right-hand member of (10) is − 1 and so this member is

∫∫Σq = x,y,z Pqcos(q,r)cos(x,r)dω = (4π/3)Px, (11)

or in other words has the same value as the term (4π/3)P in (8).

What about the third term in (10)? One immediately suspects that this yields the

depolarization (or demagnetization) corrections, inasmuch as without the cavity, the calculation

would be a completely macroscopic one. We will verify that this is the case for the simple

geometries treated in §25. We assume that the body is an ellipsoid (or thin plate, needle or slab)

so that the polarization is uniform over the body. Then (10) should reduce to

eLOCx = E0x +(4π/3)Px − NxPx, (12)

where Nx is the depolarizing (or demagnetizing) explained in §25. For the needle or very prolate

ellipsoid or thin plate parallel to the field, which we suppose is in the x-direction, cos(x,n) is zero

or negligible so that the last member of (10) vanishes in agreement with the value N = 0 given in

§25. For the slab or very oblate ellipsoid perpendicular to the field, we can take take Pr =

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Pxcos(r,x), cos(r,n) = ± cos(r,x), cos(x,n) = ± 1, with the two sign choices corresponding to the

two surfaces of the slab. The last member of Eq. (10) is then − Px∫∫dω = − 4πPx, as each surface

contributes 2π to the integral. Again there is agreement with §25. The proof that the

depolarization factor has the value 4π/3 for a sphere is immediate if the origin, i.e. the cavity, is

at the center of the sphere for then the second integral in (10) is identical to the first except for

sign, and its contribution to the local field is thus the negative of (11). In other words, the

Lorentz and depolarization fields cancel. This result is also obvious from the fact that any

spherically symmetric distribution of parallel dipoles, e.g. a spherical shell, exerts zero force at

the origin as long as the distribution does not embrace the origin. The proof that N = 4π/3 even if

the origin is not at the center of the sphere is more subtle, but is readily effected if one pairs

together the contributions of two oppositely directed pencils, 1,2 of each small solid angle dω1 =

dω2 = dω. This paired contribution to the last member of (10) is

Σq = x,y,zPq[cos(q,r1)(i⋅⋅⋅⋅n1)/cos(r1,n1) + cos(q,r2)(i⋅⋅⋅⋅n2)/cos(r2,n2)] (13)

where i and n denote unit vectors, respectively, along the x-axis and the outer normal for the

element of surface corresponding to the pencil in question. The trick is now to express i as the

sum i|| + i⊥⊥⊥⊥ of vectors parallel and perpendicular to r1 (or to r2 which is oppositely directed to r1).

Since cos(r1,n1) = cos(r2,n2), cos(i⊥⊥⊥⊥,n1) = cos(i⊥⊥⊥⊥,n2), and cos(q,r1) = − cos(q,r2), the parts of

(13) involving i⊥⊥⊥⊥ cancel one another. On the other hand, the i|| parts are equal, and since i⋅⋅⋅⋅ni =

cos(x,ri)cos(ri,ni), the denominators of (13) can be cancelled so that the last member of (10)

becomes the negative of (11).

In connection with the latter parts of the present chapter, it is advisable to make a cursory

examination of the state of affairs when the cavity is ellipsoidal. If the coordinate axes coincide

with the principal axes of the ellipsoid, then considerations of symmetry show that there can be

no cross terms, so that

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eLOCq = E0q + aqPq −NqPq , (q = x, y, z) (14)

where

aq = ∫∫CAV[cos(q,r)cos(q,n)/cos(r,n)]dω.

Since cos(r,n) = cos(x,n)cos(x,r) + cos(y,n)cos(y,r) + cos(z,n)cos(z,r), we see immediately that

ax + ay + az = 4π (15)

It should be pointed out that the calculation given in the preceding paragraph assumes

that the polarization is uniform over the ellipsoid, something we have not verified in the general

case, as it would require more detailed analysis. On the other hand, we have established the

legitimacy of this assumption in the special case of the sphere by means of an elementary

calculation of the electrostatic potential given §25, and also alternatively through the

demonstration given in the preceding paragraph that the expression for the depolarizing factor is

independent of where the origin is chosen inside the sphere. This fact shows that at least the

assumption of a uniform polarization yields self-consistent results.

Independence of Lorentz Field of Size of Cavity. The Lorentz expression for the local

field has the convenient property that is independent of the size of the cavity, provided only that

it be small from the macroscopic standpoint. This is in contrast to the Onsager field which we

will treat later on and in which ambiguity creeps in when induced polarization is included. In

particular, the cavity need not have a size equal to the atomic or molecular volume, and

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presumably should be large enough to include a number of molecules in order that the part of the

body outside the cavity be treated not as a discrete structure but as a continuum, as is done in

deriving the expression for the field. The use of the Lorentz expression for the local field is then

warranted if the given molecule in which we are interested and which is supposed to be at

the center of the cavity experiences , at least on the average, negligible fields from the other

molecules inside the cavity. It is not clear that this condition is always fulfilled. This point is

discussed by Lorentz himself, to whose work the reader is referred for a more detailed study.

If the molecules are mutually parallel dipoles of equal magnitude distributed spacially with

cubic symmetry, then the field they exert at the center is indeed zero, as is readily verified by

a simple electrostatic calculation.

The “4π/3 Catastrophe” of the Lorentz Field. In highly polar materials the Lorentz

model of the local field often leads to bizarre results not in accord with experiment for the

following reason. The true susceptibility χ in a condensed medium is not the same as χ0

calculated for the rarefied case in which the molecular interactions are neglected, i.e. the

distinction between E and eLOC is disregarded. Essentially, by the definition of eLOC, we must

take P = χ0eLOC rather than P = χ0E, and so if eLOC has the Lorentz form E + (4π/3)P, we have

χ = χ0/[1 − (4πχ0/3)]. (16)

In Chap. II we found that for a system of molecules carrying a permanent dipole moment

µ, the expression for χ0 had the Langevin-Debye form χ0 = (Nµ2/3kT) + Nα provided the

molecules are free to rotate or constrained only by potentials having a center of symmetry.

As a first approximation in a highly polar material we can usually neglect, at least for qualitative

purposes, the contribution Nα of the induced polarization in comparison to the leading term

arising from the permanent dipole moments. Then (16) takes the form

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χ = N(µ2/3k)(T − TC)

−1, with TC = 4πNµ2

/9k. (17)

According to (17) there is a critical or “Curie” temperature TC at which the susceptibility and

hence ε = 1 + 4πχ becomes infinite. This does not mean that the dielectric constant and

polarization really increase without limit. Instead at temperatures inferior to TC, it is not

allowable to treat the polarization effects as linear in the field strength, as presupposed in

obtaining (16) and (17). Below TC, saturation effects should enter, and one should expect

the electric analog of ferromagnetism. In fact, the mathematical formalism leading to the result

(17) is very similar to that in the well-known Weiss molecular field theory of ferromagnetism,

except that in the latter the coefficient in the local field is much greater than 4π/3. It is true

that “ferroelectric” materials actually exist. However these substances are rather esoteric, and

exhibit abnormal deportment usually only for certain particular directions. In conventional

isotropic media, such as liquids, the anomalous behavior predicted by (17) does not arise. It is

convenient to call the vanishing of the denominator of (16) or (17) a “4π/3 catastrophe”, for the

difficulty comes from the presence of the factor 4π/3 in the Lorentz local field. If the critical

Curie temperatures TC given by (17) were unobtainably or even inordinately low, the difficulty

of the 4π/3 catastrophe would not be serious. Actually, one calculates according to (17) that TC

equals 1200O K and 260

O K for H2O and HCl whereas it is a matter of common knowledge that

water and hydrochloric acid do not behave ferroelectrically, at least at ordinary temperatures.

How is this difficulty to be explained?

One possibility that comes to mind is that below a certain critical temperature free

rotation of the molecules is suppressed because of a hindering potential, and the dielectric

constant is then reduced provided this potential is unilateral, i.e. lacks a center of symmetry. In

§18 we discussed the Debye-Fowler model based on this situation which in principle is capable

of averting a 4π/3 catastrophe. Indeed in many materials, there are critical temperatures TD at

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which there are discontinuities in the dielectric constant and anomalies in the specific heat

which, it is generally agreed, are associated with the onset of hindered rotation when the

temperature is reduced below TD. If TC were less than TD, the 4π/3 catastrophe would be

averted, but actually the reverse is true. For instance, according to Pauling [3], HCl has TD =

100O K and HBr TD = 90

O K, whereas according to (17) the values of TC are 260

O K and 120

O K.

The non-occurrence of the 4π/3catastrophe hence cannot be attributed to hindered rotation, and

about the only alternative is that the Lorentz formula for the local field is not correct.

27. The Onsager Local Field

For over half a century the expression for the local field was regarded as practically

sacrosanct because it was obtained by the great Lorentz, but in 1936 Onsager [4] proposed a

different expression which averts the 4π/3 catastrophe and comes closer to physical reality in

polar media. Actually, as one might expect, the results of Lorentz were correct for the case he

was considering. He was dealing only with induced polarization and so could make his

calculation with the harmonic oscillator model. The error was made by other physicists in

seeking to apply his results to a case which he did not have in mind, namely, that of polar

molecules where the polarization is caused primarily by the orientation of permanent dipoles.

In both the Lorentz and Onsager models the given molecule is regarded as at the center

of a spherical cavity. In the calculation of Lorentz the polarization of the medium is regarded

as “frozen” before the cavity is constructed, and the lines of force are undeflected by the cavity.

On the other hand, in the Onsager model, more physical reality is given to the cavity, the lines

of force take cognizance of its existence, and are bent [5]. A simple electrostatic calculation

shows that the field inside the cavity is connected with that at large distances from it by the

relation

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eC = 3εE/(2ε+1). (18)

In this connection, the cavity is regarded as small from the macroscopic standpoint, so that

the cavity can be considered as deflecting an otherwise uniform field. To prove (18) we have

merely to use the electrostatic calculation given in fine print in §25, taking ε2 = ε, ε1 = 1

and identifying E∞ with E and − dϕ1/dx with eC.

First let us consider the case that there is no induced polarization, so that the dielectric

properties are determined entirely by the orientation of permanent dipoles. Then with the

Onsager model

P = (Nµ2/3kT)eCAV = (Nµ2

/kT)εE/(2ε + 1). (19)

Since P = (ε − 1)E/4π on solving for ε we have the result

ε = ¼ + (3/4)ψ + (3/4)(1 + (2/3)ψ + ψ2)1/2

, (20)

where ψ = 4πNµ2/3kT. With this model there is no catastrophe regardless of how large µ is.

When one includes induced polarization, the story is not so simple. One must not

overlook the fact that a dipole moment in a cavity, be it induced or permanent, creates a field

in the surrounding dielectric, and hence polarizes it. This state of affairs gives rise to an

additional field in the cavity, which, following Onsager, we may appropriately call the reaction

field, while we call (18) the cavity field.

The magnitude of the reaction field is readily determined by an elementary electrostatic

calculation as follows. Let p be the vector dipole moment at the center of the cavity of radius a.

The electrostatic potentials inside and outside the cavity are, respectively

ϕ1 = p⋅⋅⋅⋅r/r3 − a

−3[2(ε − 1)/(2ε + 1)]p⋅⋅⋅⋅r , (21)

ϕ2 = [3/(2ε + 1)] p⋅⋅⋅⋅r/r3. (22)

Inasmuch as ϕ1 and ϕ2 are both solutions of Laplace’s equation, ϕ1 has a singularity at the

origin corresponding to a dipole, ϕ2 vanishes at infinity, and ϕ1 and ϕ2 satisfy proper

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continuity equations (Eq. (7) with ε1 = 1, ε2 = ε) at the surface r = a of the cavity. The reaction

field at the center of the cavity is thus

eR = (8π/3V)[(ε − 1)/(2ε + 1)]p. (23)

where V = 4πa3/3 denotes the volume of the cavity.

The moment of the molecule is the vector sum of its permanent moment µµµµ and its

induced moment, which is characterized by a coefficient of induced polarizability α and which

responds to the total field, i.e. the sum of the cavity and reaction fields. We thus have

p = µµµµ + α(eC + eR). (24)

On substituting (23) in (24) we obtain

p = γµµµµ + γαeC. (25)

where

γ = [1 − (8πα(ε − 1)/3V)/(2ε + 1)]−1

. (26)

It is instructive to examine the physical meaning of Eq. (25). Both terms of (25)

are amplified by a factor γ compared to what they would be if the reaction field were

neglected. Hence one immediately suspects that instead of

P = (Nα + Nµ2/3kT)eC,

which would be the Langevin-Debye formula in a pure cavity field one has instead

P = (Nγα + Nγ2µ2/3kT)eC (eC = 3εE/(2ε + 1)). (27)

To prove this result more rigorously and also to have more physical insight into the results, we

note that from (24) and (25) it follows that

eR + eC = γeC + (γ − 1)α−1µµµµ. (28)

The reaction corrections have the effect of amplifying the applied cavity field eC by a factor

of γ, and the first member of (27) is thus accounted for. Correspondingly the potential energy

associated with the orientation of the permanent dipole moment µ in the cavity field is

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amplified by a factor γ. Since the extra factor γ thus enters in the exponent of the Boltzmann

factor associated with the orientational energy, as well as in (26), we can immediately adapt the

conventional derivation of the Langevin-Debye formula given in Eq. (3) of Chap. II to the

present model simply by replacing µ with γµ, thus yielding the second term of (27). An

alternative derivation of (27) will also be given in §28.

It is particularly to be emphasized that the term (γ − 1)α−1µµµµ of (28), representing a

field parallel to µµµµ, is incapable of giving rise to any torque orienting µµµµ, and so does not figure

in the orientational Boltzmann factor. This was Onsager’s key observation, and is the reason

why the reaction field could be omitted entirely until we included the amplifying effect

associated with the induced polarization. Indeed Eq. (27) reduces to (19) if α = 0, γ = 1.

The reason that the Lorentz field gives a 4π/3 catastrophe for a system of permanent dipoles

is that it has an excessively orienting field because it improperly replaces the instantaneous

value of the reaction field by its average component in the direction of the applied field.

From (18), (26), (27) and the relation 4πP/E = (ε − 1) it follows that

ε − 1 = [12πεN/(2ε + 1)](γα + γ2µ2/3kT). (29)

Because the definition (26) of γ involves V, Eq. (29) shows that now the induced polarization

has been included, and the expression for the dielectric constant is no longer independent of the

volume V of the cavity. The usual procedure, following Onsager, is to take V equal to the

atomic or molecular volume 1/N. The reason for so doing is that then, and only then, Eq. (29)

reduces to the standard Clausius-Mossotti relation (ε − 1)/(ε + 2) = 4πNα/3 in the non-polar case

µ = 0. Therefore we will henceforth take V = 1/N and (26) becomes

γ = (ε∞ + 2)(2ε + 1)/(6ε + 3ε∞) with (ε∞ − 1)/(ε∞ + 2) = 4πNα/3. (30)

The quantity ε∞ has the physical significance of being the dielectric constant at frequencies too

high for molecular realignment; under such circumstances all the polarization is of the induced

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type, and it is legitimate to use the Lorenz-Lorentz relation (Eq. (36) of Chap. I).

It is convenient and illuminating to throw (29) into a form which can be more easily

compared with the formula

(ε − 1)/(ε + 2) = (4πN/3)(α + µ2/3kT). (31)

of the Debye type obtained with the Lorentz field. To do this, we multiply Eq. (29) by 1/γ(ε + 2)

and utilize (30). We thus find that (29) can be written as

(ε − 1)/(ε + 2) = (4πN/3)[α + 3εγ(µ2/kT)((ε + 2)(2ε + 1))

−1]. (32)

Buckley and Maryott [6] have calculated the dipole moments for over fifty different polar

liquids by using the Onsager relation (32) together with the experimental values of ε and ε∞ at a

given T. They then compare the moments thus obtained with the “true” values furnished by one

of the accurate methods for the gaseous phase described in our Chap. III. As they say at the

beginning of their paper, “The Onsager theory has been notably successful in relating electric

dipole moments as determined in the vapor phase with the “static” dielectric constants of polar

liquids. Exceptions are liquids in which association occurs, usually through hydrogen bonding,

as in water and the alcohols, or for which the dipole moment is variable as in ethylene chloride.

For a large number of the more normal liquids, such as ketones, nitriles, and alkyl and aryl

halides, the calculated values of the dipole moments fall with more or less equal frequency above

and below the measured gas values. The Onsager relation thus represents a rather satisfactory

average. However, the agreement between observed and calculated values is far from perfect

and discrepancies of the order of 10 to 20 percent are relatively common”.

We reproduce here the results of Buckley and Maryott’s comparison for eight molecules

selected more or less randomly from their tables. These examples are rather typical and include

some of the cases of best and poorest agreement. We also have included the corresponding

results when the Debye rather than the Onsager relation is used. In the final column we have

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added the results for water, for which both the Debye and Onsager theories are notoriously

unsuccessful.

LIQUID C2H5I C2H5CN C2H5Cl C6H5NO2 CH3NO2 CHCl3 (CH3)2O (C2H5)2O H2O

µONS2/µ0

2 0.74 0.73 0.73 0.96 1.05 1.49 1.16 1.57 2.7

µDEB2/µ0

2 0.5 0.2 0.5 0.2 0.2 1.1 0.8 1.2 0.2

TABLE 1. RATIOS OF SQUARES OF DIPOLE MOMENTS DEDUCED FROM THE ONSAGER OR

DEBYE FORMULA (Eq. (32) or (31)) TO THE ACCURATE VALUE µ0 DEDUCED FROM REFINED

MEASUREMENTS ON GASES. DATA FROM REF. 6.

With a few exceptions, the Onsager formula is much more successful than the Debye one. The

results given in the table are based on the use of Eq. (32) or (31) with values of ε for T in the

neighborhood of 300O K. Buckley and Maryott also usually compute the ratio of µONS

2/µ0

2 for

the same molecule at a number of different temperatures. Since the Onsager model is a

semi-phenomenological one, it is not surprising that the values of the ratio depend on which

temperature is selected. As one might expect, this dependence is usually least in the cases where

µONS2/µ0

2 is closest to unity, i.e. when the theory works best. For instance, in the case of C2H5I,

µONS2/µ0

2 turns out to be 0.62 at 193

O K and 0.77 at 333

O K. In C6H5NO2, this ratio changes

only from 0.97 to 0.91 in going from 283O K to 473

O K.

Mixtures and Dilute Solutions. Any model of a pure polar liquid cannot be more than an

approximation, and, as we already stressed in Chap. III, dilute solutions of polar materials in

17

non-polar solvents should be more amenable to theory. However, the Onsager theory turns out

to be not as great an advance over the Debye one for dilute solutions as for a pure polar

substance. What is actually done in the case of mixtures or solutions is to assume that in a given

field E the contributions of the two constituents to the polarization can be computed additively

from Eq. (29) so that

ε12 −1 = [12πε12L(2ε12+1)−1

/V12](f1γ1α1 + f2γ2α2 + f1γ12µ1

2/3kT + f2γ2

2µ22/3kT). (33)

Here the suffices 1,2 refer to the two components of the mixture, L is Avogadro’s number, V12

is the molar volume, f1, f2 are the mol fractions of the two materials, and ε12 is the dielectric

constant of the mixture. Particular attention should be called to the definition of γi, which is a

generalization of (26) and which is conventionally employed in (32), viz.

γi = [1 − (8π/3)αi(ε12 − 1)Ni(2ε12 + 1)−1

]−1

. (34)

In other words the cavity volume Vi associated with component I is taken as the molecular

volume 1/Ni = V12/Lfi associated with this material. Clearly this is a rather arbitrary procedure,

as the interactions between molecules which are associated with the local field at a given

site are determined at least partly by the distance between molecules, i.e. the sums of the atomic

or molecular radii, rather than just by the size of the molecule where the local field is being

computed. The extrapolation of the Onsager model to mixtures is, in fact, not unique or free

from ambiguity. For instance, instead of (33), we might, by generalizing from (32) take

(ε12 − 1)/(ε12 + 2) = (4πL/3V12)[f1α1 + f2α2 + 3ε12({f1γ1µ12 + f2γ2µ2

2} /kT)((ε12 + 2)(2ε12 + 1))

−1]. (35)

The Debye form analogous to (35) is

(ε12 − 1)/(ε12 + 2) = (4πL/3V12)[f1α1 + f2α2 + (f1µ12 + f2µ2

2) /3kT)] . (36)

A criticism which can be leveled against (33), unlike (35) is that (33) does not acquire the

Clausius-Mossotti form when both components are non-polar, i.e. (33) does not lead to

(35) when µ1 = µ2 = 0. It is known experimentally (Chap. III) and theoretically (§28) that this

18

form is an exceedingly good approximation when both components are non-polar. However,

it probably does not make too much difference whether (33) or (35) is used since (33) and (35)

are equivalent when there is no induced polarization and since for dilute solutions of polar

materials in non-polar solvents the contribution of the polar component arises primarily from its

permanent moment, while practically all the contribution of the induced polarization arises

from the solvent. Equation (35) corresponds to additivity of the two contributions

in a given applied field E0, whereas (33) does in a given macroscopic field E. Since the

distinction between E and E0, the field after and before the insertion of the body, is caused

by interactions internal to the body, it may be that the assumption of additivity has a better

justification for a given E0 than for a given E.

Since the Onsager theory is more arbitrary and ambiguous for mixture than a one-

component system, it is not surprising that the Onsager model does not represent as great an

improvement over the Debye one for dilute solutions of polar materials in non-polar

solvents as it does for pure liquids. It gives somewhat better numerical values of the moment,

but the main advantage is that it reduces the so-called solvent effect, i.e. a spurious

dependence of the value of the dipole moment on the nature of the solvent. This feature is

discussed in some detail by Buckley and Maryott [6], and a typical example taken from their

paper is the following, which is for the dipole moment of C2H5Br as determined alternatively by

the application of (33) and (36) to dilute solutions in various solvents at 20O C

Solvent n-C6H14 C6H12 CCl4 C6H6 CS2

µONS2/µ0

2 0.95 0.93 0.95 0.94 0.90

µDEB2/µ0

2 0.97 0.92 0.94 0.90 0.76

19

TABLE 2. RATIOS OF THE SQUARE OF THE DIPOLE MOMENT OF C2H5Br TO µ02 IN

SOLUTIONS WITH VARIOUS SOLVENTS. THE VALUES ARE DEDUCED FROM THE

ONSAGER OR DEBYE FORMULA ( Eq.(35 or 36)). T = 20O C. DATA FROM REF. 6.

Generalization of the Onsager Model to Non-Spherical Cavities. Various investigators

have developed and studied interesting modifications of the Onsager theory in which the cavity

around the molecule is ellipsoidal rather than spherical [7]. The physical justification for this

procedure is that a given molecule is not spherical and the distance to other molecules is

consequently different in different directions. When µONS2/µ0

2 < 1 with the unmodified spherical

molecule, the ellipsoid is taken as prolate, and when µONS2/µ0

2 > 1 as oblate. This gives

corrections in the right direction, inasmuch as the value of the cavity field is decreased or

increased as compared with the corresponding value for the sphere according as the ellipsoid

is prolate or oblate, and correspondingly the value of the dipole moment which will make the

theoretical formulas yield the observed dielectric constant are respectively increased or

decreased. Buckley and Maryott, with their ellipsoidal model, obtain quite good agreement

between the dipole moment as obtained from their theory as applied to a number of pure polar

liquids and the “true” values furnished by measurements on gases. Usually the discrepancy does

amount to more than a percent or so. It is a little hard to know how much of the success of their

theory to attribute to physical reality, and how much to the fact that, compared to the spherical

model, it contains an extra adjustable parameter, viz., the degree of ellipticity. There are two

pieces of evidence that their model has physical meaning. One is that for the given selected

value of the eccentricity, the value of the dipole moment deduced from experimental data shows

much less variation with the temperature selected than in the case of the spherical model. The

other is that their eccentricities show rough qualitative agreement with what is known about

molecular shape from other considerations (the so-called Fisher-Hirshfelder-Taylor atomic

models). However, Buckley and Mariott conclude that “in view of the rather arbitrary

20

assignment of cavity volume, inadequate allowance of atomic polarizability, neglect of optical

anisotropy, and other factors that could lead to an overly complicated molecular model, the exact

values derived for the eccentricities are not of particular significance and should be regarded

only as useful molecular parameters” [6].

Buckley and Maryott have also applied their theory based on ellipsoidal cavities to dilute

solutions of polar materials in non-polar solvents. Again there is improvement in numerical

results s compared with the conventional spherical Onsager theory, but the improvement is not

as striking as for pure polar liquids.

28. Dipole Interaction Treated by Statistical Mechanics.

All the theory of the local field we have presented so far in this chapter is semi-empirical

- a hybrid of microscopic and macroscopic considerations. The question naturally arises

whether there cannot be a more rigorous and fundamental treatment by statistical mechanics of

the effect of the dipolar interaction which is responsible for the difference between E and eLOC

[8]. As we shall see, this is possible in principle, but, except for harmonic oscillators, the results

can be expressed only in the form of a poorly converging series of which only the leading terms

can be computed. Nevertheless, this calculation, even though the series cannot be carried far, is

illuminating because of the light which it can throw on the comparative degree of validity of the

Onsager formulas and those of the Clausius-Mossotti type based on the Lorentz local field. We

therefore start afresh and treat the system as a sort of glorified molecule whose Hamiltonian

function is

HZ = H0 −ΣiE0piz, (37)

where H0 is independent of the applied field, which we suppose directed along the z-axis, and piz

is the z-component of the electric moment of atom or molecule i. It is particularly to be noted

21

that the calculation is made not in terms of E, the actual field in the body, but rather in terms of

E0, the field which would exist without the presence of the body. The reason for this procedure

is, of course, that the dipolar interaction which is responsible for the distinction between E and

E0 is incorporated in H0, and so E0 is an external parameter. We now assume that the body is cut

in the form of a sphere, a great convenience in our calculation. The polarization is then uniform

over the body, as one knows from the general theory of the depolarization factor (§25). We also

assume that all the molecules of which the body is composed are identical, and similarly situated.

The polarization per unit volume is then N times the polarization of a single molecule, which is

computed by averaging over all the coordinates, weighted with the Boltzmann factor

corresponding to the Hamiltonian function (37). We neglect saturation, and so are interested

only in the part of the polarization which is linear in E0. In consequence, we may make the

approximation

exp[−H/kT] = exp[−H0/kT](1 + ΣiE0piz/kT). (38)

The polarization of the body then arises from the second right-hand term of (38), as it is

unpolarized when E0 = 0 (unless we were dealing with a ferroelectric). We thus have

P = N<pizΣjpjz>AVE0/kT, (39)

where < >AV denotes the statistical average without including the last term of (37), i.e.

the contribution of the applied field, in the Boltzmann factor. Since the body is a sphere, E

and E0 are connected by the relation

E0 = E + (4π/3)P,

as we showed in our discussion of depolarization corrections in §25. Inasmuch as 4πP/E =

ε − 1, it follows from (39) that

(ε − 1)/(ε + 2) = (4πN/3kT) <pizΣjpjz>AV. (40)

Since the susceptibility is the same in all directions because of the spherical symmetry, an

22

equivalent and convenient form of (40) in vectorial notation, one used extensively by

Kirkwood and others [9], is

(ε − 1)/(ε + 2) = (4πN/9kT) <pi⋅⋅⋅⋅Σjpj>AV. (41)

We henceforth will assume, unless otherwise stated, that all the coupling between

different molecules is of dipolar origin, so that

H0 = ΣiH0i + Σj>i pi⋅⋅⋅⋅dij⋅⋅⋅⋅pj, (42)

where H0i is a function only of the coordinates of molecule i, and where dij is a symmetrical

second rank tensor whose components are

(dij)xx = rij−3

[1 − 3cos2(rij,x)], (43)

(dij)xy = rij−3

[ −3 cos(rij,x) cos(rij,y)], (44)

etc., with the understanding that dii = 0. The meaning of the tensor notation is that

pi⋅⋅⋅⋅dij⋅⋅⋅⋅pj = Σq = x,y,zΣr = x,y,zpiq(dij)qrpjr,

Vanishing Effect of Dipolar Coupling in First Order. The dipolar interaction is such a

complicated function that it is not in general feasible to make a calculation with it occurring

in the exponent. Instead we make the expansion

exp[−H0/kT] = exp[−ΣiH0i/kT][1 − Σj>i pi⋅⋅⋅⋅dij⋅⋅⋅⋅pj/kT] (45)

After the dipolar terms are removed from the exponent, the statistical average for the whole body

factors into products of averages which can be taken over each molecule separately and

independently, as there is nothing in the exponent to couple them together. We shall use the

notation < >0, to be distinguished from < >AV, for a statistical average in which the factors

23

involving the coordinates of each atom i are evaluated by using the Boltzmann factor appropriate

to its “private” energy H0i. Since the molecules are alike, we have <pi2>0 = <pj

2>0. Clearly

<pim

>0 vanishes if m is odd, as do expressions such as <pixpiy>0. Thus if we keep only terms

through first order in the dipolar interaction, Eq. (41) becomes

(ε − 1)/(ε + 2) = (4πN/9kT)<pi2>0 − (4πN/27k

2T

2)[<pi

2>0]

2Σj[(dij)xx + (dij)yy + (dij)zz]. (46)

Now if the surroundings of a given molecule have cubic symmetry, one has

Σj(dij)xx = Σj (dij)yy = Σj (dij)zz = 0, (47)

inasmuch as in (43) the mean square of a direction cosine is 1/3 and in (44) the mean product

of two orthogonal ones is zero [10].

Actually in, say, a liquid, the molecules around a given molecule are not arranged with

cubic symmetry, but if on the average there is isotropy, we can also use the relation (47) on the

average, inasmuch as in an isotropic medium the fluctuations from cubic symmetry are ironed

out either by taking the time average for a single molecule or averaging over a number of lattice

sites in a physically small volume element. This type of argument cannot, however, be used in

connection with the higher order terms to be considered later in which the dipolar energy does

not occur linearly.

General validity of the Lorentz Field in First Order. When account is taken of the fact

that the second right hand member of (46) vanishes because of (47), the expression (46)

reduces to the generalized Langevin-Debye formula of Ch. II except that (ε − 1) = 4πχ is

replaced by 3(ε − 1)/(ε + 2). This is what would be obtained from the calculation of Ch. II if

the field polarizing the molecule is taken to be E + (4πP/3) rather than E. We thus see that

to first order in the dipolar coupling, the use of the Lorentz local field model is generally

warranted without the need of specializing the model as long as the material is isotropic,

24

and the resulting formulas have the Clausius-Mossotti type of structure in which 3(ε − 1)/(ε + 2)

enters as the fundamental entity. In other words, it is a better approximation to take

3(ε − 1)/(ε + 2) rather than ε − 1 as proportional to χ0, where χ0 is the susceptibility computed

for an isolated molecule.

Specialization of the Model for Calculations in Higher Order. When we seek to include

the effect of the dipolar interaction beyond the first order, simple and illuminating results are

obtained only if the model is specialized both as regards inter-molecular spacings and the

molecule itself. In the first place we shall assume that the molecules are arranged in a crystal

grating such that at each point where a molecule is located there is cubic symmetry at each

instant of time. We will further assume that the molecule carries a permanent moment and that

the effect of the induced polarization can be represented by an isotropic harmonic oscillator.

This latter assumption is a considerable drawback from the standpoint of physical reality, as

actually the induced polarizability of a molecule is anisotropic, i.e. described by a tensor rather

than a scalar when referred to axes fixed in the molecule. However, our simplified model

probably gives at least a qualitative representation of the effect of induced polarization, since

presumably the isotropic polarization constant in the model can be regarded as a sort of

average of the actual anisotropic ones, as a rough approximation.

With this model, the dipole moment of a given molecule has the form

pi = e'ri + µµµµi with |µµµµi| = µ, (48)

where e' is the effective charge carried by an oscillator whose displacement is specified by a

radius vector ri. (We assume there is only one oscillator per molecule – the extension to the

case that there are several is straightforward, as long as they are isotropic). For reasons

explained in detail in Chap. II, the kinetic energy can for our purposes be omitted entirely

from the Hamiltonian used in the Boltzmann factor and we can thus take

25

H0i = ½ ari2,

The volume element to be used in the computing the statistical average for molecule i is

dωidxidyidzi where dωi is an element of solid angle associated with the orientation of µµµµi. We thus

have

<pim

>0 = C∫…∫pim

exp[−(1/2)ari2/kT] dωidxidyidzi, (49)

where the constant of proportionality C is as usual determined by the normalization condition

that the total probability be unity.

Cole’s Method. Our procedure at this point follows, with minor modifications, one

used in a paper by Cole [11]. His key observation is that the polarization arising from the

harmonic oscillators in the field E0 is unaffected by dipolar interactions, including that with the

permanent dipoles, provided the body is a sphere and the local symmetry is cubic.

The first step in the proof is to note that even though the oscillators have their

amplitudes distributed in accordance with Boltzmann statistics, their polarization is the same

as though it were computed electrostatically. Namely, if r0i denotes the position of equilibrium

in the presence of both the dipolar interaction and the applied field, then if we introduce in place

of the ri new variables ri' = ri − r0i, the exponent in the Boltzmann factor is of the form A + B,

where A is a homogeneous quadratic function of the various ri', in fact the same function as if

there were no dipolar and applied fields, and where B involves E0 and the r0i but is independent

of the ri'. (The coefficient B can contain the oscillator momentum coordinates but we already

saw in Chap. II the inclusion of vibrational kinetic energy does not influence the value of the

induced polarization.) The fact that the r0i are equilibrium positions insures the absence of

linear terms in the ri'. Hence, since the ri' are odd as regards the simultaneous reflection of all

the ri' in their origins, while A is even, the Boltzmann average of the ri' is zero, and so

<Σi ri>AV E0 = Σi r0i.

26

Here we have used the subscript AV E0 in order to indicate clearly that the average is to be

taken in the presence of the applied field E0, unlike the averages < >AV and < >0 so frequently

used in the present section.

Our problem is now to compute the r0i, which are determined by the equilibrium

conditions

∂H/∂xi = ∂H/∂yi = ∂H/∂zi = 0 (i = 1, …, N). (50)

Here we have assumed for convenience, as we do henceforth, that the body is one of unit

volume , so that N denotes the total number of molecules and the range of the index i, as

well

as the number of molecules per unit volume. With our model, the explicit form of (50) is

ar0i + e'Σjdij⋅⋅⋅⋅r0j + e'Σjdij⋅⋅⋅⋅µµµµj −e'E0 = 0 (i = 1, …, N). (51)

Because the µµµµj are not systematically directed, it would be exceedingly difficult to determine the

individual r0i, but when we sum over all the molecules, everything simplifies because of the

relation (47) expressing the cubic symmetry. One thus has the exceedingly simple result for the

induced polarization

PINDUCED = Σie'r0i = αE0 with α = e'2a

−1. (52)

In other words the polarization directly ascribable to the oscillators is just the same as though

there was no dipolar coupling between molecules. It is to be emphasized that this result is

inclusive of all powers of the dipolar interaction and also of the field strength E0. In other

words the induced polarization is a strictly linear function of E0. (It is, however, not necessarily

a linear function of E, as saturation effects connected with the orientation of the permanent

dipoles may make it cease to be linear in E in very strong fields.)

Since we have taken account rigorously of the contribution of the harmonic oscillators, in

virtue of (52), we can now rewrite (41) in the form

(ε − 1)/(ε + 2) = (4πN/3) [α + (1/3kT)<µµµµ i⋅⋅⋅⋅Σjpj>AV], (53)

27

where pj = µµµµ j + e'r0j. Furthermore we can omit here the term e'Σjr0j since the average in (53) is

to be taken at zero field strength where the right hand side of (52) also vanishes. This procedure

is legitimate since (52) is valid regardless of how the permanent dipoles are oriented as long as

the lattice is cubic. Hence (53) can be simplified to read

(ε − 1)/(ε + 2) = (4πN/3) [α + (1/3kT)<µµµµ i⋅⋅⋅⋅Σjµµµµj>AV]. (54)

Effect of Dipolar Interaction in Second Order. From a superficial examination of (54)

one might conclude that the part of the polarization arising from the orientation of the permanent

dipoles is unaffected by the existence of the induced polarization, but this is not really the case,

for the oscillator coordinates are coupled to the permanent dipoles through the dipolar interaction

and so enter into the exponents of the Boltzmann factors involved in taking the < >AV in (54).

This can be seen when we now proceed to take into account the contribution of the permanent

dipoles to the polarization inclusive of second order terms in the dipolar interaction.. The form

(54) makes it comparatively easy to do this. We carry out the expansion of the Boltzmann factor

one step further than in (45). The first order terms, exhibited explicitly in (46), have been shown

to vanish. It is convenient to segregate the term j = i from the rest of the sum in (54) as µµµµi⋅⋅⋅⋅µµµµi is

simply the square of the permanent dipole moment µ. For the remaining part of the sum it is

helpful to utilize the fact that because of the cubic symmetry, the contributions arising from the

x, y, z parts of the scalar product are equal on the average. So through second order terms in the

dipolar interaction, (54) can be written as [12]

(ε − 1)/(ε + 2) = (4πN/3) [α + (µ2/3kT) + (1/2k

3T

3)Σj (j≠i),k,l,m,n< µµµµi⋅µµµµj(pk⋅⋅⋅⋅dkl⋅⋅⋅⋅pl)(pm⋅⋅⋅⋅dmn⋅⋅⋅⋅pn)>0], (55)

where, as before, < >0 denotes a “decoupled average”, i.e. without the dipolar interaction in

the Boltzmann factor. The only terms which give a non-vanishing factor are those involving

three pairs of equal indices corresponding to a system of three molecules at the vertices of a

triangle. Other arrangements, including simple pair interactions (two triply coincident indices)

28

vanish because odd powers of the moments are involved. When we utilize the fact that

<r0i>0 =0, <µiypix>0 = 0, <pix2>0 = <piy

2>0, etc. Eq. (55) becomes

(ε − 1)/(ε + 2) = (4πN/3) [α + (µ2/3kT) + (k

3T

3)−1

<µiz2>0<µjz

2>0<pkz

2>0A], (56)

where

A = Σj (j≠i),k[(dik)zz(dkj)zz + 2(dik)zx(dkj)xz]. (57)

If we add the excluded term j = i to both sides of (57), we can make use of the rigorous

relation (47) for a cubic lattice, and then (57) becomes

A = − Σk[(dik)zz2 + 2(dik)zx

2]. (58)

We now introduce the explicit forms (43) and (44) of the dik. Because the susceptibility is

isotropic, for a cubic material one can average over all directions of the coordinate axes

relative to the cubic axes, or equivalently to all directions of rij. When this is done, (58)

reduces to

A = − 2Σkrik−6

(59)

inasmuch as <cos2(rij,z)>AV = 1/3, <cos

4(rij,z)>AV = 1/5 and <cos

2(rij,z) cos

2(rij,x) >AV = 1/15.

At this stage the explicit form (48) of pkz is substituted in (56) and the averaging < >0 is

performed, which is elementary since <µiz2>0 = (1/3)µ2

, and by equipartition <(1/2)ax'k2>0 =

(1/2)kT. In view of (59) the expression (56) then becomes

(ε − 1)/(ε + 2) = (4πN/3) [α + (1/3)µ2/kT − 2/(27k

3T

3)(µ6

+ 3µ4e'

2a

−1kT) Σkrik

−6]. (60)

It will be noted that the appearance in (60) of terms proportional to µ4e'

2 shows that the

interaction of the induced polarization with the permanent dipoles modifies the latter’s

29

contribution to the polarization, as already stated, even though the latter does not modify the

contribution of the former.

At this point it is illuminating to compare the Debye and Onsager results to the second

order in the dipolar energy with the exact calculation we have made to this order with our

model. To do this, we expand the factor 3εγ/((2ε + 1)(2 + ε)) in (32) in a Taylor series in

(ε − 1) and α, utilizing the definition (26) for γ and dropping terms beyond the second order. To

this degree of approximation one thus finds

3εγ/((2ε + 1)(ε + 2)) = 1/3 + (8πNα/27)(ε − 1) − (2/27)(ε − 1)2. (61)

On the right hand side of (61) one can introduce the lowest order of approximation ε − 1 =

4πN(α + (1/3)µ2/kT) as here (ε − 1) and α enter only in the second order. When this is done, the

Onsager formula (32) becomes identical with (60) if in (60) one takes A = − 2(4πN/3)2 instead of

being defined by (59). Since the Onsager model utilizes a continuum rather than a discrete

lattice structure, one immediately wonders if perchance the two values of A would agree if the

discrete sum in (59) is replaced by an integral extending down to the cavity radius a = (3/4πN)1/3

employed in the Onsager model. The question can be answered in the affirmative inasmuch as

Σ(1/r6) → 4πN 4

ar dr

∞−

∫ = (4πN/3)a−3

= (4πN/3)2

if 4πa3/3 = 1/N. (62)

30

However the continuum hypothesis overestimates the sum in (59). It is larger than the actual

value of the sum by a factor of 2.4 for a face-centered cubic lattice and 2.1 for a simple cubic

one. On the other hand, the Debye formula, Eq. (31) is equivalent to taking A = 0. The results

of the exact calculation are thus about half way between those of the Onsager and Debye

models.

Effect of the Dipolar Interaction in Third Order. We have carried the series development

in powers of the dipolar interaction only through terms of the second order. The extension of the

calculation to include terms of the third order has been made by Rosenberg and Lax [13] for

permanent dipoles and by Cole [14] who included in addition the effect of induced polarization

as portrayed by harmonic oscillators. Cole finds that when the grating sums are evaluated

accurately for a cubic lattice, the third order terms are intermediate between those obtained with

the Lorentz and Onsager models, as one might guess. Cole notes that it is rather surprising that

the dielectric constants for the high temperature solid phases of HBr, HCl and H2S do not fall

between the values calculated with the Debye and Onsager models, especially inasmuch as

these materials have face-centered cubic lattices , and so, unlike dielectric liquids, have the

geometrical arrangement assumed in the series expansion [15]. He also observes that it is

remarkable that the observed values agree within ten percent, usually on the low side, with

31

those predicted by the Onsager theory.

Continuum Approximation in the Statistical Approach. We have seen that the Onsager

model and the statistical approach agree through the second order if a sum is replaced by an

integral. Cole shows that there is agreement in the third order except for one minor term (called

by Cole the “shuttle term”). The question arises whether the two procedures can be made to

agree in all orders if a continuum approximation is made in the treatment by statistical

mechanics. This can indeed be done, as was shown by Kirkwood [9], by Buckingham [16] and

by Cole [14], whose method we follow. The trick is to replace the factor Σjpj in (53) by Pµ,

where Pµ is the macroscopic polarization produced in a spherical dielectric medium by having a

dipole of strength µ at the center of a cavity corresponding to the site i, which we can suppose

located at the center of the sphere. This polarization will of course be co-directional with the

central dipole and is inclusive of the latter’s contribution. (In this connection do not forget that

the average in(53) is to be taken in the absence of an applied field.) Thus (53) is approximated

by

(ε − 1)/(ε + 2) = (4πN/3) [α + (1/3kT)<µµµµ i⋅Pµ>AV]. (63)

The problem is now to compute Pµ. This is a simple exercise in electrostatics, viz. the

determination of the potential Bp by a dipole of strength p at the center of a hollow

32

sphere of inner and outer radii a,b with b >> a. As this potential has the dipolar form, Bp can

be identified with Pµ. An electrostatic calculation, given in an accompanying footnote [17],

shows that

Pµ/µ = B = 9ε/((ε + 2)(2ε + 1)) (64)

At first thought, one is tempted to identify p with µ, but this is not correct, as one must allow

for the fact that the reaction field induces a dipole moment in the central molecule as we

have seen in §27. With (64), Eq. (63) becomes identical with the Onsager formula (32). This

procedure impresses me as a somewhat more rigorous derivation of this formula than that given

in §27. The relation (64) used in the present proof is happily independent of the size of the

cavity as long as a << b. However, choice of a particular cavity size has not really been avoided

for the relation (32) is a consequence of the Onsager theory only if the cavity volume entering in

(26) is taken equal to the molecular volume.

Mixtures and Dilute Solutions. The extension of the statistical mechanical approach to

systems with more than one component loses its simplicity unless it is assumed that there is

local cubic symmetry at the sites of the molecules of both components, as our preceding

calculations in the present section (except for first order terms in the dipolar interaction) have

leaned heavily on the cubic symmetry. When the full cubic symmetry is thus preserved, then

33

(63) is readily generalized into (36) if the continuum approximation is made. However, this

requirement is too restrictive to be of much use except in special situations. Some simplification

can arise when we are dealing with very dilute solutions of polar molecules in non-polar

solvents, and then all interactions between permanent dipoles can be disregarded. However, the

elimination of the interactions between permanent and induced moments then leads to a formula

of the simple structure (36) with µ2 = 0 and f2 nearly unity, only if it is assumed that all the

harmonic oscillators be on a cubic grating and that the permanent dipoles be symmetrically

spaced relative to this grating. These conditions will be met if either (a) the polar molecules

carry no induced moment and be situated at a central interstitial position or (b) that it replaces

one of the non-polar molecules on a regular cubic site and somehow happens to have the same

characteristics as regards polarizability (same effective charge and coefficient of restitution) as

the molecule it replaces. With analogous assumptions the formula (33) based on the Onsager

model reduces to (36) in case (b) but not case (a) [18]. Admittedly these two cases involve

rather artificial assumptions, but nevertheless the results are not devoid of interest. They show

that the dielectric behavior depends on the geometry of how the solute molecules are embedded

in the solvent, and that the Onsager model is a better approximation for some geometries than for

others.

34

The Spherical Model. Returning now to pure polar liquids or solids, we should by all

means mention one rather esoteric-appearing model which lends itself particularly well to

treatment by statistical mechanics. We have seen that even with simple dipoles a rigorous

treatment of the dipolar interaction is possible only by resorting to a series expansion for which

it is feasible to compute only a few terms. The situation changes, however, if one replaces the N

constraints µi2 = µ2

(i = 1,…, N) corresponding to permanent dipoles by a single constraint

Σi µ2 = Nµ2

. An explicit expression can then be obtained for the dielectric constant inclusive of

all orders of dipolar interaction provided certain integrals are evaluated numerically. This is the

so-called spherical model studied by Lax and by Toupin and Lax [19]. It may seem rather

unrealistic physically but yields surprisingly good results. Cole [14] shows that through second

order in the dipolar interaction it yields results identical with those of a calculation based on the

usual, more realistic assumption of a permanent moment for each molecule. Also, he shows that

in third order there is again agreement except for a rather minor (“shuttle”) term. He thus

concludes that the full potentiality of the spherical model has not been fully exploited.

Additional evidence to this effect is furnished by the fact that with this model Toupin and Lax

calculate that for HBr there should be a discontinuity in the slope of the dielectric constant

against temperature at 90O K, and a discontinuity is actually observed at 89

O K. The usual

35

theories are incapable of yielding or predicting such a discontinuity.

Semi-Empirical Formulas. The formulas for the dielectric constant obtained with the

spherical model have a definite theoretical foundation on the basis of the model assumed. The

other extreme is represented by a number of more or less empirical formulas connecting

dielectric constants with dipole moments. They are contrived to yield values of the dipole

moments in better agreement than do the Debye and Onsager models. The formulas are often

useful because of their analytical simplicity, but have their main theoretical justification as

merely convenient interpolations between the results of the Debye and Onsager theories.

However, these various semi-empirical theories furnish little insight into what is really going on.

Instead the more modern approach is to try and explain deviations from the Onsager model along

more mechanistic lines, in which the effect of short range order or correlation is studied

explicitly and more or less quantitatively [20].

29. Short Range Order – Kirkwood’s Formula

Hitherto we have supposed that the coupling between the various molecules is entirely of

a dipolar type, a dominantly long range effect since dipolar coupling energy varies only as the

inverse cube of the distance. Actually there are short range forces, be they electrostatic or

quantum mechanical, which are operative at short distances. One can, however, still employ

formula (53) if one includes the effect of short range forces in computing the average value

36

of µµµµi⋅⋅⋅⋅ Σjpj. In other words one calculates this average by using a Hamiltonian in the Boltzmann

factor which differs from (42) by including short range interactions as well. The ingenious idea

of segregating out from the summation the members involving short range correlations as far as

the central molecule i is concerned is due to Kirkwood [9]. Let us assume for simplicity that the

short range coupling is important only between nearest neighbors. Then we can write

µµµµi⋅ Σjpj = µµµµi⋅(pi + Σnn - ipj + Σrestpj) (65)

or if there are z neighbors

<µµµµi⋅⋅⋅⋅ Σjpj>AV = µ2(1 + z<cosα>) (66)

where <cosα> is the mean cosine of the angle between the dipole at i and one of the neighboring

dipoles. (In passing from (65) to (66) we may appear to have overlooked the induced

polarization which molecule i creates on its neighbors, but this effect will average out if the

neighbors are arranged with cubic symmetry, and anyway, the important applications of the

Kirkwood theory are to substances where the influence of induced polarization is minor.)

We may to a good approximation evaluate the last member of (66) by electrostatics in the same

way as was done in obtaining Eqs. (63) and (64), but now the sum entering in this member is to

be regarded as coming from the molecules exterior to a sphere of radius large enough to include

the given molecule and its neighbors. In other words, a cluster of molecules rather than a single

molecule is embedded in the central cavity, and the Kirkwood model is thus a cluster one. We

must in consequence identify p in (66) , or [17], with γµ(1 + z<cosα>) rather than γµ, as

there is now a correlation between the directions of the dipole moments involved in the central

cluster. We assume that the “feedback factor” expressing the polarization induced back on the

central molecules has the same value as before, i.e. given by (26). How the feedback effect is to

be handled in any cluster model is not free from ambiguity. Kirkwood treated the induced

polarization in a somewhat different fashion, as he assumed that it responded only to the

37

cavity field (Eq. (18)). Our procedure follows Cole’s [11,

14] and is, we believe, somewhat

better, but the difference is unimportant, as the Kirkwood model is used only in highly polar

materials.

The preceding paragraph shows that, with the approximations there used, the

effect of the short range correlations or associations is to replace µ2 by gµ2

in relations such as

(29) or (32), with

g = (1 + z<cosα>) (67)

The quantity g can be regarded as a sort of amplification factor reflecting the correlation

between adjacent dipoles. With the aid of (29) the thus modified Eq. (32) can now be thrown

into the form [14]

ε = ε∞ + [3ε/(2ε + ε∞)](ε∞ + 2)/3)2(4πNµ2

g/3kT). (68)

The most noted application of the Kirkwood theory is to water [9, 21]. He showed that if

one uses the Bernal-Fowler model of ice or water [22], which is confirmed by x-ray evidence,

then one obtains a value of g which when substituted in (68) yields almost the correct value

relation between the dielectric constant and the true dipole moment, something the Onsager

model could not do. We will not attempt to describe the geometry of, or give diagrams of, the

structure proposed by Bernal and Fowler, as it is quite complicated and often described in the

literature. Suffice it for us to say that there is a tetrahedral arrangement, making z = 4, and the

average value of cosα is nearly 1/3 so that g = 7/3. If we set g = 7/3 in (68) and introduce the

experimental values of ε and ε∞ (viz. 78 and 1.77) then for (68) to be satisfied the value of µ

should be 2.08x10−18

e.s.u.. The correct value as determined by more accurate methods, such as

measurements on gases, is 1.88x10−18

. The value 1/3 which we have used for <cosα> is only an

approximate one. It would be correct if the angle ϕ1 between the two arms of the water molecule

were the same as the angle ϕ2 between the two tetrahedral diagonals, for the water molecules

38

could then be fitted simply into the Bernal-Fowler picture. Actually the value of ϕ1 is 105O

and that of ϕ2 is 109O; it is the near coincidence of the two that is basic to the model. The

agreement is all that can be expected in view of the fact that the model involves many

approximations, so many that it seems rather pointless to try and go into such refinements

as correlations with next to nearest neighbors, etc..

The Kirkwood theory has also be successfully applied to a number of other polar liquids

in which there is a high degree of association. In the case of five alcohols, it yields the correct

dipole moment within 5 to 10 percent. In general, it is difficult to calculate or even estimate g

a priori and the best procedure is to regard g as an unknown parameter to be determined from

(68) with the aid of known values of ε, ε∞ and µ [23]. When g deviates appreciably from unity,

i. e. if the unmodified Onsager theory does not work well, then this is an indication that there are

important correlation or short range order effects.

30. Limits of Validity of the Clausius-Mossotti Formula – The Translational Fluctuation

Effect

Hitherto we have probably given too much the impression that the Clausius-Mossotti (or

Lorenz-Lorentz) ratio based on the Lorentz local field is sacrosanct and completely rigorous for

non-polar materials. However, this is not really the case. To be sure, the C-M ratio

(ε − 1)V/(ε + 2) on the whole remains remarkably independent of density in a non-polar material,

changing by only a few percent in going from the gaseous to the liquid or solid state. However,

if measurements are made even in the gaseous phase at high pressures, there is found to be a

small dependence on density and pressure which is the concern of the present section.

From a theoretical standpoint, there are two reasons, which will call I and II, why the

Clausius-Mossotti formula should not be exact.

39

I. All of our previous treatment by statistical mechanics in §28 was on the assumption

that there is cubic symmetry in the force field at each site. Actually, this is a property

which can be achieved only in an ideal solid. In a liquid the migrations of the

molecules make any given molecule instantaneously fail to have a cubic environment,

even though this is so on the average. In a mixture with more than one component,

e.g. solutions, the redistribution of the two or more types of molecules gives rise to

further fluctuations. In a gas there will be fluctuations because of occasional

collisions or near collisions between molecules.

II. Even in a cubic lattice perturbations by intermolecular forces, be they dipolar,

quadrupolar or short range, mix together different quantum states, which in general

have different polarizabilities. So the polarizability, instead of being constant, will be

altered as the material compressed. Only the harmonic oscillator has the agreeable

property that its polarizability is the same in all quantum states, and it is for this

reason that the calculations with the harmonic oscillator model could appropriately be

included in the classical chapters of the present volume.

Effect I was first treated by Kirkwood [24] and by Yvon [25] and is called by Kirkwood

“the translational fluctuation effect”. Effect II was first examined for gases by Jansen and

Mazur [26] and is called the dependence of mean polarizability on density. The two effects are

only approximately additive, as fluctuations in position influence the mixing of different states.

When I and II are taken into account, the Clausius-Mossotti ratio takes the form

(ε − 1)V/(ε + 2) = A0 + A1/V + A2/V2 + … (69)

instead of being independent of volume. If when the volume of the system is changed, the

molecules always remained cubically spaced, then the coefficient An on the right hand side of

(69) would arise from the nth power of the dipolar interaction, since this interaction is

proportional to the inverse cube of the interatomic or intermolecular spacing, and so

inversely proportional to the volume in a cubic lattice. Under such circumstances the

coefficient A1 would vanish, since we saw in §28 that there is no first order correction from the

dipolar interaction. In fact all the coefficients would vanish if the non-polar molecules were

really isotropic oscillators.

40

The situation is changed when we consider a gas, in which the molecules

continually approach or recede from each other. In a regular grating, which serves as a rough

approximation even to a liquid, the only dimensionless parameter available for a series

expansion is α/V, as the polarizability α has the dimensions of a length cubed. In a gas or

liquid whose molecules fluctuate in position, an additional parameter becomes available, viz.

the distance a0 of closest approach. (There must be some cut-off in how close molecules can

get to avoid singularities in the dipolar or other energies of interaction.) In consequence all

powers of the dipolar interaction, beginning with the second, contribute to the coefficient A1

in (69). For instance, the second form of the integral in (62) ceases to be applicable, as the

lower limit of integration is no longer the Onsager cavity radius proportional to V1/3

, and is

instead independent of V; an expression such is (62) is then proportional to 1/V rather than

1/V2. (In this connection remember that the number N of molecules per unit volume is

proportional to 1/V.) The fact that A1 becomes non-vanishing when at least two powers of

the dipolar interaction are included is a general consequence, as shown in §28, of the fact that

in an isotropic material the average symmetry is cubic even though there are instantaneous

deviations or fluctuations. This absence of a first order contribution is the reason that the

Clausius-Mossotti formula is usually such a good approximation in non-polar materials.

This section omits discussion in any detail of the deviations of the Clausius-

Mossotti formula, as the subject is a highly specialized one, and is very difficult

experimentally and theoretically. On the experimental side the measurements must be made

at high pressures and with high precision because of the smallness of the effects; often the

results in different laboratories do not agree very well. The theoretical calculation of

fluctuation Effect I is complicated by the fact that it is too naïve an approximation to assume

that the molecules are randomly distributed except for a cut-off at the closest allowed

41

distance of approach. Because of the potential energy associated with intermolecular forces,

especially those of long range, the Boltzmann factor is different for different relative

positions of the molecules, which are consequently not randomly distributed. It is also not

feasible to calculate Effect II, the dependence of mean polarizability on density, even for a

hydrogenic atom, without making an approximation, viz. the use of a mean energy of

excitation rather than taking account of the fact that actually the energies of excitation to the

various upper quantum states are different. Furthermore the resulting mixing in of the upper

states into the ground state is highly dependent on the nature of the intermolecular forces

responsible for this blending, and it is again an approximation to assume they are entirely

dipolar.

In view of these complexities and the specialized scope of this subject of small deviations

from the C-M formula, we give it only limited attention. For fuller discussion the reader is

referred to W. F. Brown’s review [27]. Brown concludes his discussion on the topic

with the remark ”further progress in reconciling theory and experiment in this field can come

only through great labor at both ends: further refinement of the theory requires difficult

analytical and numerical calculations, and further refinement of the measurements requires a

precision difficult to obtain under high-pressure conditions”.

31. Inadequacy of the Local Field Concept

The analysis in the preceding three sections makes it quite clear that it is too simple a

picture of molecular interactions, dipolar or otherwise, and even with fixed molecular spacings,

to regard the molecule as subject to a local field of definite magnitude. We have seen that any

really accurate expression for the dielectric constant can be obtained in the form of a poorly

converging and intractable series.

42

Particularly striking and succinct quantitative evidence on the inadequacy of the concept

of the local field is furnished by recent experiments on the nuclear magnetic resonance of polar

liquids whose nuclei have quadrupole moments, when taken in conjunction with conventional

measurements of dielectric constants [28]. We shall not attempt to go into the theory involved in

these experiments, as it would take us into the quantum mechanics of magnetic resonance, and

we shall simply state that, inasmuch as the quadrupole moment is known from other resonance

experiments, all the proportionality constants are known in the appropriate expressions for the

resonance pattern, and Hilbers and MacLean are consequently able to evaluate the expression

<3cos2θ − 1>E0 , where θ is the angle between the molecular dipole moment and the applied

field, which is also the direction of the local field. The subscript E0 indicates that the average is

to be taken in the presence of an applied field. Indeed, without such a field, the average of this

expression in any isotropic material is zero. Similarly, an ordinary measurement of the

dielectric’s polarization P determines θ inasmuch as P = Nµ<cosθ>E0. In this connection we

assume that the value of the permanent dipole moment µ is known, and neglect the induced

polarization, as is warranted in the highly polar materials studied by Hilbers and MacLean.

The trick is now to notice, following Buckingham [29], that as long as the spatial distribution

is determined by a local field, so that the Boltzmann factor is exp[ − H0/kT + NµE0cosθ/kT],

then there is a ratio which has an unambiguous, inescapable value

<3cos2θ − 1>E0/<cosθ>E0

2 = 6/5 = 1.2 (70)

To prove (70) we have merely to expand the Boltzmann factor involved in the averaging of

the numerator through terms of order µ2E0

2/2k

2T

2, and then perform the angular integrations,

which are elementary. Similarly, <cosθ>E0 can be evaluated by expanding through terms of first

order in µE0/kT. The numerator and denominator are thus proportional to µ2 E0

2/k

2T

2, and

this quantity consequently cancels out in (70). It is to be emphasized that no explicit formula

43

for eLOC has be assumed. It has only been supposed that eLOC exists as a combined representation

of the effects of the applied field and intermolecular interactions of any sort, so that for a given

E0 there is a definite value of eLOC.

Experimentally, the value of the left hand side of (70) often deviates materially from 1.2.

The values for nitrobenzene and nitromethane are respectively 3.1 and 2.7. It is not surprising

that the observed ratios are greater than the theoretical value 1.2 as any dispersion in the

instantaneous value makes the mean square greater than the square of its mean. So the ratio

appearing on the left hand side of (70) is underestimated by assuming that eLOC has a definite

value. Incidentally, Hilbers and MacLean show that, for some reason, the Onsager local field

is a better approximation for evaluating the numerator than the denominator of (70).

Buckingham shows that the ratio (70) can also be evaluated by using measurements of

the Kerr or the Cotton-Mouton effect. In addition, he also has an ingenious method for

determining it by studying the nuclear magnetic resonance of molecules containing two

nuclear moments even though these nuclei (usually protons) do not have quadrupole moments.

None of these methods appear to have yet achieved sufficient experimental accuracy to be as

reliable as the experiments on nuclear quadrupole resonance.

32. Incipient Saturation Effects

So far in the present chapter we have assumed that the field is so weak that the

polarization can be considered linear in the field strength. Actually in very strong fields

deviations from linearity can be detected even at ordinary temperatures in condensed media, so

that the polarization is given by an expression of the form

44

P = a1E + a3E3 + … (71)

It is not too difficult to calculate the coefficients a1 , a2, … for an ideally rarefied gas [30] and let

us denote by a10, a20, … the values of such coefficients. If such a thing as a definite local field

existed in the condensed phase and is linear in the applied field, then clearly we would have

P = a10eLOC + a30eLOC3 + … (72)

and we would have a1/a10 = x, a3/a30 = x3, … where x = eLOC/E. However, the considerations of

the preceding section show that this is too naïve a way of looking at things. Furthermore the

theory for even the linear response to an applied field, we have seen, is a difficult subject except

for a highly disperse material, and non-linear effects are much more complicated to treat. On the

experimental side the coefficient a3 in (71) is hard to measure. Only if the dielectric constant is

fairly high does it become measurable. Fields of the order of 100,000 volts/cm are required

even in liquids [31]. Early experiments did not agree with each other well. So we give only

brief mention of this subject—in fact giving it less attention than in the first edition of this book,

simply because it has not progressed in a particularly illuminating way in the interim in

comparison with other more striking modern developments. For references and some discussion

the reader is referred to the monographs of Smyth [20], Brown [27] and Böttcher [31]. Suffice it

to say here that use of the Lorentz field in (72) grossly overestimates the coefficient a3 in (71),

45

but this fact can not be considered as an argument for the Debye model of hindered rotation

discussed in Chap. III [32], as we have seen that the Lorentz field gives far too much polarization

in materials of high dielectric constant. The Onsager model appears to yield values of a3 of a

reasonable order of magnitude, but because of the non-linearity does not lend itself to reliable

quantitative calculation.

References

[1] J. A. Osborn, Phys. Rev. 67, 351 (1945); E. C. Stoner, Phil. Mag. 56, 803 (1945). The

quantity N/4π is usually tabulated rather than N itself.

[2] See, for instance, H. A. Lorentz, The Theory of Electrons, 2nd

edition ( Dover Publications,

New York (1952)) Section 117 and note 54. Lorentz first proposed his theory of the local field in

1878.

[3] L. Pauling, Phys. Rev. 36, 430 (1930).

[4] L. Onsager, J. Amer. Chem. Soc. 58, 1486 (1936); our µ is µ0 in his notation.

[5] J. H. Van Vleck, Ann. N. Y. Acad. Sci. 40, 293 (1940).

[6] F. Buckley and A. A. Maryott, J. Res. Natl. Bur. Std. 53, 229 (1954).

[7] J. A. Abbbott and H. C. Bolton, Trans. Faraday Soc. 48, 422 (1952); I. G. Ross and R. A.

Sack, Proc. Phys. Soc. (London) 63, 893 (1956); A. D. Buckingham, Austral. J. Chem. 6, 93

(1953); T. G. Scholte, Physica 15, 437 and 450 (1949); also especially Buckley and Maryott, l.c..

[8] Apparently the first attempt to treat dipolar coupling in dielectrics by means of statistical

mechanics, using this interaction in the partition function instead of employing electromagnetic

theory, was made by the writer (J. Chem. Phys. 5, 320, and especially, 556 (1937)). The

analysis in the present section follows to a considerable extent that in this earlier work, but is

presented in improved and somewhat more condensed form. Certain details are discussed more

fully in the original articles. The criticism of the writer’s calculation made by J. A. Pople (Phil.

Mag. 44, 1276 (1953)) has been shown by B. K. P. Scaife to be unfounded (ibid. 46, 903 (1955)).

46

[9] J. G. Kirkwood, J. Chem. Phys. 7, 911 (1939).

[10] The reader may object that if atom i is not at the center of a sphere, symmetry

considerations do not obviously require these statements to be true. However, dependence of the

sums (47) on where the site is selected would make the macroscopic polarization cease to be

uniform over the sphere contrary to the results of macroscopic electrostatic potential theory.

Furthermore, we have verified in §26 that the field at the center of a spherical cavity in a

uniformly polarized sphere is zero, even though the spheres are not concentric. This is

equivalent to proving that (47) vanishes if one includes in the sum only the molecules exterior to

a cavity sufficiently large that a continuum model may be used or in other words the sum

replaced by an integral. (In §26 we considered explicitly only effects corresponding to the

diagonal part (43) of dij but the extension to the non-diagonal portion (44) is obvious.) The cubic

symmetry requires that the sum over the molecules interior to the cavity vanish, as its center can

be taken at the site i. Since both the interior and exterior contributions vanish, the relation (47) is

established.

[11] R. H. Cole, J. Chem. Phys. 27, 33 (1957).

[12] It may appear that in our series expansion we have overlooked the fact that the

normalization of the proportionality constant in the Boltzmann factor is unaffected by the dipolar

energy. It actually is in second order, but this effect does not need to be taken into account for

present purposes, since <µµµµi⋅µµµµj>0 vanishes in zeroth order unless i = j, and µ2 is a constant whose

average does not involve questions of normalization. It was to permit this simplification that we

separated out the term i = j from the j summation in going from (54) to (55). In the more

involved calculations of the translational fluctuation effect to be briefly mentioned in §30, it is

essential to take into account the modulation of the normalization constant by the dipolar

interaction.

[13] R. Rosenberg and M. Lax, J. Chem. Phys. 21, 424 (1953).

[14] R. H. Cole, J. Chem. Phys. 39, 2602 (1963).

[15] N. L. Brown and R. H. Cole, J. Chem. Phys. 21, 1920 (1953); R. W. Swenson and R. H.

Cole, ibid. 22, 284 (1954); S. Habrilak, R. W. Swenson and R. H. Cole, ibid. 23, 134 (1955).

[16] A. D. Buckingham, Proc. Roy. Soc. 238A, 235 (1956).

[17] Proof of Eq. (64). Because the dielectric no longer extends to ∞, the boundary value

problem is slightly more complex than that treated in §27 in connection with Eqs. (21) and (22).

There are now three regions, viz.

I. 0 ≤ r ≤ a II. a ≤ r ≤ b III. b ≤ r

47

The potential ϕ1' is practically the same as that given in (21) as long as a << b. In II and III the

forms are respectively

ϕ2' = ϕ2 + A(p⋅r) ϕ3' = B(p⋅r)/r3,

with ϕ2 the same as in (22). Satisfaction of the relations

ϕ2' = ϕ3' and ε∂ϕ2'/∂r = ∂ϕ3'/∂r ,

gives immediately the expression (64) for B and

A = 6(ε −1)b−3

/[(2ε + 1)(ε + 2)].

Because of the addition of the A term, the boundary conditions at r = a are not quite satisfied, but

the resulting error is negligible for our purposes as long as a << b.

It is the A not the ϕ2 part of ϕ2' which gives region II its polarization, as grad ϕ2 has zero

mean value when averaged over a spherical shell. The A term, which is a reflection of surface

effects connected with the outer boundary, represents a field of constant magnitude − Ap, and so

generates a moment (ε − 1)Ap(b3/3) in the spherical shell since the latter’s inner radius is

negligible. This moment and that p of the central dipole add to Bp, as they should.

[18] One takes µ1 = 0 and makes the approximation ε12 = 1 + 4πNα1 in the part of (33)

multiplying µ22, as f2 is very small compared to f1. In case (a) γ1 is defined by (34) while α2 = 0,

γ2 = 1. In (b) one takes α1 = α2 = α and γ1 = γ2 = γ, with γ defined by (30). One finds that (33)

yields (36) in case (b) but not in case (a).

[19] M. Lax, J. Chem. Phys. 20, 1351 (1952); R. A. Toupin and M. Lax, J. Chem. Phys. 27, 458

(1957).

[20] For a review and discussion of semi-empirical formulas see C. P. Smyth, Dielectric

Behavior and Structure (McGraw-Hill, New York, 1955) pp 34-51; see also Ref. 5.

[21] G. Oster and J. G. Kirkwood, J. Chem. Phys. 11, 175 (1943).

[22] J. D. Bernal and R. H. Fowler, J. Chem. Phys. 1, 515 (1933).

[23] Ref. 20, p 33.

[24] J. Kirkwood, J. Chem. Phys. 4, 592 (1936).

[25] J. Yvon, C.R. Hebd. Seances Acad. Sci. (France) 202, 35 (1936).

[26] L. Jansen and P. Mazur, Physica 21, 193, 208 (1955).

48

[27] W. F. Brown, Jr., Encyclopedia of Physics, Vol. XVII (Springer, Berlin, 1956).

[28] C. W. Hilbers and C. MacLean, Mol. Phys. 16, 275 (1969).

[29] A. D. Buckingham, Chem. Brit. 1, 54 (1965).

[30] K. F. Niessen, Phys. Rev. 34, 253 (1929).

[31] C. J. F. Böttcher, Theory of Electric Polarization (Elsevier, Amsterdam, 1952) pp. 193-

198.

[32] J. H. Van Vleck, J. Chem. Phys. 5, 556 (1937) p. 562.

49

EDITORIAL NOTE:

The source for this chapter was typed text supplemented with hand written notes and

equations. Some of the text was illegible due to fading; in those cases we either filled in what

appeared to be missing or, when that was not possible, omitted the material altogether. It should

be emphasized that the chapter, as presented, is a draft of a draft; in other words, it, too, is a work

in progress. We welcome comments.

David Huber and Chun Lin

February 2011