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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