A contribution to the theory of the dielectric constant of polar liquids

Physica XV, no 5--6

J u l i 1949

A CONTRIBUTION TO THE THEORY OF THE DIELECTRIC CONSTANT OF POLAR LIQUIDS by TH. G. SCHOLTE Laboratorium voor Anorganisehe en Physisch_~ Chemie d~r Rijksuniversiteit te Leiden

Summary S o m e s i m p l i f i c a t i o n s used in O n s a g e r ' s t h e o r y o n t h e d i e l e c t r i c c o n s t a n t of p u r e l i q u i d s are c o n s i d e r e d . A t first, i n s t e a d of a s p h e r i c a l , a n ellipsoidal s h a p e of t h e m o l e c u l e s is a t t e m p t e d a n d t h e c a v i t y field a n d r e a c t i o n field of a dipole are c o m p u t e d i n t h i s case. I t a p p e a r s t h a t t h e r e a c t i o n field of a d i p o l e in a n ellipsoidal c a v i t y is s m a l l e r t h a n t h e r e a c t i o n field i n a s p h e r i c a l c a v i t y of t h e s a m e v o l u m e if t h e l o n g e s t axis is i n t h e d i r e c t i o n of t h e dipole. I n t h e case t h a t t h e s h o r t e s t axis is in t h e d i r e c t i o n of t h e dipole, t h e r e a c t i o n field is g r e a t e r t h a n i n a s p h e r i c a l c a v i t y . Also a d i s c u s s i o n o n t h e r a d i u s of t h e c a v i t y is given. A m o d e l is a s s u m e d in w h i c h for t h e p o l a r i s a b i l i t y effects t h e c o n t i n u o u s e n v i r o n m e n t b e g i n s a t a d i s t a n c ~ e q u a l t o t h e r a d i u s of t h e m o l e c u l e s w h i l e for t h e i n f l u a n c e of t h e d i p o l a r i n t e r a c t i o n t h e c o n t i n u o u s e n v i r o n m e n t b e g i n s a t t w i c e as g r e a t a distance.

§ 1. Introduction. In 0 n s a g e r's theory on the dielectric constant 1) and in B 6 t t c h e r's modification and extension of this theory 2) 3) two conceptions are of great importance : the "reaction field" of a dipole and the " c a v i t y field". By the reaction field of a dipole in a cavity we understand the electric field in the cavity if, without external field, the surroundings are fixed and then the dipole is taken away. No doubt 0 n s a g e r's opinion that the contribution of the reaction field to the internal field must be subtracted from the latter to get the directing field, means a considerable improvement in the theory of the dielectric constant. For B 6 t t c h e r showed 3) t h a t the mere introduction of the reaction field R, whatever its value - - 437 - -

438

TH. G. SCHOLTE

m a y be, leads for a pure dipole liquid to the formula: e--1 4~rN( + t '2 ) -a e + 2 3 . 3kT + ~R in which :

e a /, N

= = = =

(1)

dielectric constant. polarisability of the molecule. dipole m o m e n t of the molecule. n u m b e r of molecules per cm 3.

This formula is identical with the empirical formula of v a n A rk e 1 and S n o e k 4) which accounts roughly for the deviations of De b y e ' s equation in undiluted dipole liquids. However, it is still an open question which formula for the reaction field is the best and which part of the internal field has to be s u b t r a c t e d to get the directing field, because for the calculation of the reaction field 0 n s a g e r used several simplifications. E.g. the dipole molecule was considered to be spherical with a " m a t h e m a t i c a l " dipole in the centre. Its surroundings were t r e a t e d as a continuous dielectric and were assumed to follow the m o v e m e n t s of the dipole without measurable relaxation. Moreover, for the c o m p u t a t i o n of the reaction field and the c a v i t y field O n s a g e r used a spherical c a v i t y with a content equal to the average volume available per molecule, thus 4~rr3 N / 3 = 1 if r is the radius of the cavity. Using all these simplifications, formulae for the reaction field and the c a v i t y field are derived which result in the following formula for a pure dipole liquid : /~2 _ 9kT ( e - n 2) (2e + n 2) 4~rN e(n 2 + 2) 2

(2)

n being the refractive index. Now B 6 t t c h e r showed 5) t h a t the errors m a d e b y the simplifications m e n t i o n e d above cancel each other to such an e x t e n d t h a t the dipole m o m e n t s calculated with the aid of formula (2) from d a t a of undiluted dipole liquids are in satisfactory agreement with those calculated from gases and dilute solutions of dipoles in a non-polar liquid. To obtain these results n had to be t a k e n equal to noo (n for infinitive wave length, extrapolated from the visible and ultra-violet part of the spectrum), so the atomic polarisation h a d to be neglected. Norton WilsonS) showed t h a t if this is not done m u c h

A C O N T R I B U T I O N TO T H E T H E O R Y OF T H E D I E L E C T R I C C O N S T A N T

439

lower values of/~ are obtained. However, for a n o t h e r reason formula (2) has to be corrected in such a w a y t h a t the calculated values of # increase again. For in a series of papers on the refraction of organic liquids and electrolyte solutions B 5 t t c h e r 7) pointed out t h a t the radius r of the c a v i t y should be t a k e n about equal to the average radius of the particle itself. In the present paper a discussion is given of the modifications of O n s a g e r's formulae if not all of his simplifications are used. Successively an ellipsoidal cavity, a non-central dipole and finally the question of the radius of the c a v i t y are considered.

§ 2. Computation o~the cavity field and the reaction field in case o/an ellipsoidal 7barticle. T h e c a v i t y f i e 1 d. We consider an isotropic dielectric having a dielectric constant ~, in which there is a homogeneous electric field E. Placed in this field we imagine an ellipsoid of which the semiprincipal axes are a, b and c and the dielectric constant is %. One of the axes, e.g. the one having a length 2a, is situated in the direction of the external field E. The calculation of the field inside the ellipsoid is a well-known electrostatic problem s). Its solution is a homogeneous electric field in the eUipsoid, parallel to the external field and with field intensity:

Eh, =

'1 E 8t + ( % - - e l ) A 1

(3a)

in which:

abc F

ds

(4a)

At ---- 2 J (s + a2)a.2 (s + b2)~'2 (s + c2)~'°0

If the position of the ellipsoid was such t h a t the axis with a length

2b would be situated in the direction of the external field, the field in the ellipsoid would be:

Eh~ =

el E el + ( e 2 - ~1)A2

(3b)

in which: A2=

ds 2 J (s + a2) ~'~ (s + b2)~/~"(s + c2)''~ 0

(4b)

440

TH. G. SCHOLTE

In case the third axis would be placed in the direction of the external field, the field in the ellipsoid would be: El, 3. --

~ E el + ( e 2 - el) A3

(3c)

in which: oo

abc i" ds A 3 = ~ - , !/ (S -~- a2) 1'2 (S -~ b2) ]~ (s -[- c2) 32 0

(4c)

So with every ellipsoid there are three numbers A, each of t h e m corresponding with one of the axes. These numbers are only dependent on the ratio between the axes of the ellipsoid, not on the absolute magnitudes of the axes. The values of the A's lead directly to the relation : A 1+ A2 + A3 : 1 (5) In case of a spheroid the integrals and consequently the A's as well can be expressed in elementary functions. Two cases are possible : a) Prolate spheroid. a > b = c. Let the ratio a/b be called p. --1 p AI -- p 2 1 + x/~-l~131n (p + x/PU--1) "

(6)

g

b) Oblate spheroid.

a
a/b=p.

1

A1-- l--p2

P

If a = b = c it follows t h a t A 1

Eh --

1

x / l - - p 23c°s p" =

A 2 =

3el E. 2el + e2

(7)

A 3 = ~ and (8)

This is the well-known expression in case of a sphere. In table I examples are given of the values t h a t A 1 and Ehl can have for some of the ratios between the axes. W i t h a n y given position of the ellipsoid with respect to the direction of the external field E the latter can be resolved into three components, viz. in the directions of the three axes. If the angle between E and the first axis is 0 and the angle between the component of E perpendicular to the first axis and the second axis is 9, the three

A CONTRIBUTION

TO T H E T H E O R Y

OF THE DIELECTRIC

CONSTANT

'~41

TABLE I The l a s t t h r e e c o l u m n s i n d i c a t e , for v a r i o u s v a l u e s of 6 and p, the r a t i o of the c a v i t y field in a s p h e r o i d a l c a v i t y and the c a v i t y field in a s p h e r i c a l c a v i t y w i t h the s a m e e x t e r n a l field P

At

0 0.2 0.4 0.6 0.8 0.9

1 0.784 0.583 0.464 0.394 0.362 0.333 0.308 0.276 0.233

1 1.1 1.25 1.5 2.0 4.0 oo

Ratio 6=4

0.174 0.075 0

Espherold/Esphtre 6=6

3.00 1.82 1.33 1.15 1.06 1.03 1 0,97 0.94 0.91 0,86 0.80 0.75

4.33 2.08 1.40 1.18 1.08 1.03 1 0.97 0.94 0.90 0.84 0.77 0.72

6=10 7.00 2.38

1.47 1.20 1.09

1.04 1 0.97 0.93 0.89 0.83 0.75 0.70

components of E are, respectively, E cos 0, E sin 0 cos ~, and E sin 0 sin ~. Each of them causes a homogeneous field in the ellipsoid as illustrated by formula (3). Therefore the total field inside the ellipsoid is also homogeneous, but geherally no longer parallel to the external field. The field intensity in the direction of the external field is:

J

81

cos 2 0 +

B e =/e I + (%-

el)A1

el el -{- (e2 - - e l ) A 2

+

sin 2 0 cos 2 9

sin 2 0 sin 2 ~o E

(9)

e l + ( % - - e l ) A3 In case of a normal average direction spread, i.e. all directions being equally probable, the mean field intensity in the ellipsoid is: 81

=

e1+(e

el

el)

+ el+(*

el

el)

+

1

I S.

If the eccentricity of the ellipsoid is not too great the above differs only very slightly from 3e 1 E/(2e I + %), in other words: from the field intensity inside a sphere with a dielectric constant %. The reaction field. For the computation of the reaction field of a dipole in the centre of an ellipsoid and parallel to one of the principal axes, the following method is adopted.

442

TH.G. SCHOLTE

We imagine an ellipsoid of which the polarisability is evenly distributed all over the volume. Consequently there is the same dielectric constant t h r o u g h o u t the ellipsoid, which will be denoted b y e i (internal diel. const.). The polarisabilities of the ellipsoid in the directions of the three axes are a 1, ch and a 3. T h e y can be defined as the ratio of the dipole caused in the ellipsoid when the latter is introduced into a homogeneous field in a v a c u u m in the given direction and the intensity of this field. B y the average polarisability a is m e a n t the average value of this ratio. As with small fields all possibilities as to the orientation are equally probable it follows t h a t a = (a I + a 2 + %)/3. If the ellipsoid is placed in a v a c u u m with a homogeneous external field E parallel to the first axis, the field intensity in the ellipsoid is : I

Eh, = 1 +

(e i - -

E.

,(I 1)

1) A 1

Then the dipole induced into the ellipsoid is: 4zt

ei - - 1

y., = ~ a b c

4zt

1

E

1 --}- (e i -

(12)

1) A l

Besides : g'v = al E.

(13)

(I 2) and (13) lead to" ei --

al=3{l

1 _ _

+(ei--l)

(14)

abc.

A0

and: abc+ ei =

3(1 - - A 0 a I

(15)

abc - - 3 A lal

Now if the same ellipsoid is in a m e d i u m having a dielectric cons t a n t e in which, again, exists an external field E in the direction of the first axis, the field in the ellipsoid is: Eh =

e + ( s ; - e) A 1

E

(16)

Then the induced dipole is: 4z~

si-

1

e

E

(17)

A CONTRIBUTION TO THE THEORY OF THE DIELECTRIC CONSTANT 4 4 3

Substitution of e; (15) gives: e

1

lz=e+(1--e)

Al al

3alAl(1--Al)(e--1) abc e + (1-- ~) A1

1

E

(18)

This means t h a t the induced dipole m o m e n t is not equal to the field i n t e n s i t y in a v a c u u m c a v i t y as big as the ellipsoid multiplied b y a t, b u t t h a t in addition it has to be multiplied b y the factor:

1 1

At(1 - - A t )

3al

abc

( e - - 1)

e + (1-- e) A1

The reaction field is responsible for this. If a certain dipole m with a polarisability a causes a reaction f i e l d / m parallel to m , the result is t h a t the dipole is increased b y the factor 1/(1 - - / a ) . In the case of the ellipsoid, therefore, the factor of the reaction field is: 3 AI(1 - - A t ) ( e - - 1)

/t = ~ C

e + (1--e) A t

(19)

T h e n the reaction field is"

3 A,(1 - - A t ) ( e - - I) R1 -- abc e + (1 - - e) A l V~

(20)

In case of a sphere (a = b = c = r, A t = 1/3) this expression for R turns into the well-knov/n expression:

1 2 ( e - - 1) R - - r3 2e + ~ P'"

(21)

It follows from this inference t h a t this is the reaction field of a dipole evenly distributed all over the ellipsoid. If the reaction field originates from a dipole induced b y an external field this is the correct value, the polarisability being supposed to be evenly distributed all over the ellipsoid. For the reaction field of a p e r m a n e n t dipole this value is not quite correct. However, as in case of a sphere it makes no difference w h e t h e r the dipole is supposed to be a mathematical dipole in the centre or evenly distributed t h r o u g h o u t the volume, we assume t h a t for ellipsoids not differing too m u c h from the spherical shape this will h a r d l y make a difference either.

444

TH.G.

SCHOLTE

In table II the magnitude of the reaction field f a c t o r / l for some values of the ratio of the axes is given in case of a spheroidal cavity. TABLE II R e a c t i o n field f a c t o r of a dipole i n a s p h e r o i d a l c a v i t y

abc./,

I ~=4 0.4 0.8 1

1.25 2.0

0.583 0.394 0.333 0.276 0.174

0.972 0.763 0.667 0.567 0.372

I

e=6 1.182 0.889 0.769 0.649 0.420

]

e=

I0

1.381 0.999 0.857 0.718 0.460

§ 3. The reaction field o/ a non-central dipole. Another fact we have to take into account is that in many cases the permanent dipole is not in the centre of the particle under consideration, but eccentric. D e k k e r s) has calculated the magnitude of the reaction field for cases in which the dipole is assumed to be "mathematical" and situated in a spherical cavity in such a manner that the centre of the cavity lies in prolongation of the dipole. It appears that this is no longer a homogeneous field. Being interested in the increase of the permanent dipole b y the inducing action of the reaction field and supposing the polarisability of the spherical particle to be evenly distributed over it, it is essential to know the average reaction field all over the volume of the particle. As this average value turns out to be independent of the eccentricity of the dipole, and therefore equal to the reaction field of a dipole having a central position, the eccentricity need not be taken into consideration in the calculation of the factor 1/(1--/a). This holds for every arbitrary direction of the eccentric dipole. In other calculations, e.g. when computing the energy of a dipole in its own reaction field, which energy is of considerable importance for the ~ohesion energy of a dipole liquid, we must know the magnitude of the reaction field in the place of the dipole itself. In such cases the eccentricity of the dipole is of great interest. With a rigid dipole in the direction of the centre, the energy of the dipole in its reaction field is approximately 1.3 times as great as with a central dipole, in case the distance dipole-centre is ¼ of the radius. If that distance is half .the radius, the energy is about three times as great as with a central dipole. For e > 2 this ratio depends only in a small degree on e.

A CONTRIBUTION TO THE THEORY OF THE DIELECTRIC CONSTANT 4 4 5

§ 4. The radius o/the cavity. In § 1 it was already m e n t i o n e d t h a t it is an i m p o r t a n t question which value must be used for the radius r of the c a v i t y in the calculation of the c a v i t y field and the reaction field. O n s a g e r used 4~r 3 N/3 = 1 in which N is the n u m b e r of molecules per cm 3. B 6 t t c h e r, however, showed in his papers on the refractive index t h a t r is a b o u t equal to the average radius a of the particle. On the o t h e r hand, K i r k w o o d 10), who gave a statistical t h e o r y of the dipolar interaction, arrived at a formula of the same t y p e as 0 n s a g e r's b u t in which r is equal to the d i a m e t e r 2a of the particle. Now the difference between K i r k w o o d's and B 6 t t c h e r's results is only a seeming one. F o r the polarisability of a particle is e x t e n d e d over the whole v o l u m e of it, while for 0 n s a g e r's and K i r k w o o d's calculations the dipole is supposed to be in the centre of the particle. T h u s the nearest distance from the centre of a particle where the polarisable surroundings begin is a, but the nearest distance where a n o t h e r dipole can be is 2a. These considerations lead to the conclusion t h a t it would be useful to c o m p u t e the internal field for a case in which for the polarisability effects the continuous dielectric begins at a distance a from the centre of the molecule whereas for the interaction of the dipoles the continuous dielectric begins at a distance 2a from the centre. In o t h e r words: at distance a the h o m o g e n e o u s dielectric begins with a dielectric c o n s t a n t n 2 between a and 2a and from 2a a dielectric c o n s t a n t e. This model enables us to calculate the c a v i t y field and the reaction field electro-statically in the usual way. a) Calculation of the c a v i t y field. We imagine two concentric spheres h a v i n g radii R and R ' ( R < R ' ) . The inner sphere is e m p t y (diel. const. = 1), between the two spheres there is a substance with a dielectric c o n s t a n t e2, the e n v i r o n m e n t h a v i n g a dielectric c o n s t a n t e3. We will use spherical polar coordinates, the origin of the coordinate s y s t e m being the centre of the spheres. In the e n v i r o n m e n t exists an electric field which, b u t for the disturbance caused b y the two spheres, would be homogeneous with the field i n t e n s i t y E in the direction of the axis 0 = 0. The field inside the inner sphere will t h e n be the c a v i t y field. The p o t e n t i a l inside the sphere with radius R is 9t, between the two spheres it is 92 and outside the sphere with radius R' it is 93. The

446

TH. G. S6HOLTE

potential conforms to L a p 1 a c e's equation which is in spherical coordinates: _~) 1 0 ( r2 r 2 ar

-9

1 r 2 sin 0

~ 0 ( sin o F )0

+

-~

1

~#~o-- 0.

r 2 sin 2 0 a~2

(21)

Because of the s y m m e t r y round the axis O = 0 is oF/OF ---- 0, so the last t e r m of the left-hand part of equation (21) can be omitted. Also, the potential m u s t c o m p l y with the following b o u n d a r y conditions : 1. At a great distance the field becomes homogeneous again, the field i n t e n s i t y being E, so ~v3 --> - - E r cos 0 if r --> oo. 2. On the sphere r = R' is ~v2 = ~3.

oF2

oF3

3. On the sphere r = R' is s2 ~ = ~ s ~ . 4. On the sphere r = R is

V)~ = ~2.

5. On the sphere r -- R is oF__!= ~2 oF___22 6. In the origin (r = 0) ~,, m u s t be regular. In view of the first b o u n d a r y condition we t r y :

~o = / ( r ) cos 0.

(22)

Substitution in L a p ] a c e's equation produces: 2/,

/ " - + - -r

2 ---~l

= 0.

(23)

The general solution of this differential equation is: P

/ = -~ + 0 r.

(24)

Therefore :

Br)cos;

25a,

+ o,/cos 0

125b,

~v3 ---- ~ + Gr co~ e

(25c)

A C O N T R I B U T I O N TO T H E T H E O R Y OF T H E D I E L E C T R I C C O N S T A N T

447

The constants A, B, C, D, F and G follow from the b o u n d a r y conditions. A and B are : A = 0

(26) 98283

B =

--

(283 +

1) R3/R '3 E.

82) (282 -[- 1) + 2(83 - - 82) (82 - -

(27)

So it appears t h a t a homogeneous electric field exists in the inner sphere, parallel to the external field E and with the field intensity: 98283

Eh = (283 + 82) (282.+ 1) + 2(83 - - 82) (e2 - - 1)

R 3 / R '3

E.

(28)

I n the case under consideration is:

R

=

a, R'

----

2a, 82

=

n 2, 8 3

Then for the c a v i t y field one has: 368n 2 E h = 1 7 8 n 2 + 7n 4 q- 78 q-

=

8.

(29)

5n 2 E.

Table I I I shows, for some values of 8 and •2 the ratio of the c a v i t y field as calculated above a n d the c a v i t y field c o m p u t e d b y assuming a single spherical cavi~:y in the dielectric with a dielectric constant e. T A B L E III Correction factor cavity field

4 6 10

1.8

2.0

2.5

1.067 1.091 l.ll3

1.069 1.098 1.125

1.063 1.108 1.144

b) Calculation of the reaction field. We t a k e the same model as under a) but in this case w i t h o u t the external field. In the centre of the spheres there is a m a t h e m a t i c a l dipole with a m o m e n t m , directed along the axis ~ = 0 . The potentials are again taken to be ~v1, ~v2 and ~v3 for the three spatial parts. Again 0~v/a~0 = 0 and the potential complies with L a p 1 a c e's equation :

~ Or

~

+ r 2 sin----~ O0 sin 0 ~

The b o u n d a r y conditions are in this case: 1. ~0a - * 0 if r --* oo. 2, 3, 4 and 5 as under a).

= 0.

(30)

448

TH. G. SCHOLTE

m 6. ~v1 ~ ~- cos 0 if R ~ oo, for with a v e r y large R, the ordinary

dipole field will exist. In view of the sixth b o u n d a r y condition we t r y :

=/(r)

cos 0

(31)

This leads to the equations: ~l =

+ B r cos 0

(32a) "

~2=

~+Dr

cos0

(32b)

~v3 =

~-+Gr

cos0

(32c)

The constants A, B, C, D, F and G follow from the b o u n d a r y conditions. A and B are: A -----m

(33)

2(283 + 82) (82 - - 1) R '3 -Jr- 2(83 ~ 82) (82 -{- 2) R 3 m B = - - (283 + 82) (282 + 1) R '3 + 2(83 - - 82) (82 - - 1) R 3 R -~

(34)

The potential in the inner sphere is : m ~vI = -fi cos 0 + B r cos 0

(35)

the second t e r m being the potential caused by the reaction field. So the reaction field is homogeneous and has a field intensity: [m

2(283 -{- 82) (82 - - 1) R '3 -{- 2(83 - - 8 2 ) (82 -Jr- 2) R 3 m = (283 + 82) (282 + 1) R '3 + 2(83-- 82) ( 8 2 - 1) R 3 R 3

(36)

In our case is : R = a, R ' ~- 2a, 82 = n 2, 83 = 8.

The reaction field, therefore, is:

/m

-

-

2m a3

- n2--1+

3(8

-

178 + 7n 2 3(8 n 2) -

2n 2 + 1

n 2)

-

-

(37)

178 + 7n 2

As the value of 3(8 - - n2)/(178 + 7n 2) is only small (3/17 at most) this do not differ much from 2 m ( n 2 - - 1)/aa(2n 2 + 1), i,e. from the formula applying when only the electronic and atomic polarisation

A C O N T R I B U T I O N TO T H E TI~EORY OF T H E D I E L E C T R I C CONSTANT

449

of the environment are taken into consideration. Therefore, the influance of the dipoles of the environment, owing to their being at a greater distance, is ony small. This contribution to the reaction field is even smaller in case we suppose the dipoles of the environment not quite to follow the movements of the considered particle itself. The change in 1/(1 - - / a ) caused by counting in the reaction field in respect of the dipoles of the environment is generally about 30/0 . Table IV shows, for some values of e and n 2 and for a/a 3 -. 0.3, a / a a = 0.4 and a / a 3 = 0.5, the ratio of the factor 1/(l-/a) as calculated above and this factor calculated according to / = = 2(n 2 - 1)/aa(2n 2 + 1). TABLE IV Correction factor in 1/( 1 - - / a ) 1.8

2.0

2.5

0.3

4 6 10

1.014 1.020 1.024

1.012 1.018 1.022

1.008 1.013 1.018

0.4

4 6 10

1.020 1.027 1.033

1.017 1.028 1.031

1.011 1.019 1.025

0.5

4 6 10

1.026 1.037 1.044

1.023 1.033 1.042

1.015 1.026 1.035

I n a subsequent paper in this journal the formulae derived will be applied for the calculation of polarisabilities, molecular radii and dipole moments of some organic molecules. Received January 24th, 1949. REFERENCES 1) 2) 3) 4) 5) 6) 7)

L. O n s a g e r , J. Am. chem. Soc. 58, 1486, 1936. C.J.F. B6ttcher, Physica 9, 937, 945, 1942; Rec. Trav. chim. 6,% 119, 1943. C.J.F. BSt tcher, PhysicaS, 635, 1938. A . E . v a n A r k e l a n d J . L. S n o e k, Phys. Z. 33, 662,1932; 35, 187, 1934. C.J.F. B6ttcher, Physica B, 59, 1939. J. N o r t o n W i l s o n , Chem. Review 2.5, 377, 1939. C.J.F. B~ttcher, Rec. Tray. chim. 62, 325, 503, 1943; 6.5, 14, 19, 39, 50, 91, 1946. 8) E.g.: J. A. S t r a t t o n, Electromagnetic Theory, New York, London, 1941, Ch. III. 9) A . J . D e k k e r , Physica 12, 209, 1946. 10) J . G . K i r k w o o d , J. chem. Phys. 4, 592, 1936. Physica XV

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