On the theory of optical activity. II

Terwiel, R. H. 1964

Physica 30 1027-1037

ON THE THEORY OPTICAL

OF OPTICAL ACTIVITY.

11

ROTATORY POWER OF AN ISOTROPIC TWO-COMPONENT SYSTEM

by R. H. TERWIEL Instituut-Lorentz,

Leiden,

Nederland

synopsis The theory of a previous paper is extended to the case of an isotropic twocomponent system. A generalized Lorentz-Lorenz formula is obtained as well as an expression for the optical rotatory power and, in the case of a solution of an optically active substance in an inactive solvent, an expression containing the non-specific influence of the solvent.

9 1. Introduction.

We shall generalize

the results of a previous paper l) *)

to the case of a medium consisting of a mixture of arbitrary molecules, eventually specializing to the case that has been studied in an extensive way experimentally, namely that of the optical rotatory power of a solution of an optically active substance in an inactive solvent. This will lead to an expression for what Kuhn 2) calls the non-specific influence of the solvent, i.e. the influence which persists in the absence of specific effects such as the formation of chemical compounds or of association products between the various molecules. In 5 2 we derive the integral equations for the statistical averages of the field strengths for a medium consisting of s components. Whereas for the one-component system one obtains 2 coupled equations, the statistical correlation and fluctuation effects provide an additional coupling between the partial contributions from each molecular species, thus yielding a system of 2s intricately coupled equations. In $3 the case of a two-component system is treated along the same lines as the one-component system. This results in a generalized Lorentz-Lorenz formula, an expression for the optical rotatory power and, in the case of an inactive solvent, an expression containing the non-specific influence of the solvent. $ 2. The integral equations for the statistical averages in an s-component system. We consider a mixture of s components. Let the number of molecules *)

Hereafter

referred

to as I.

-

1027 -

’

1028

R. H. TERWIEL

of species m be Nm(m = be X’, 5 N,

1, 2, . . . . S) and let the total

= N. The molecules

number

of species m are labelled

of molecules

(ml),

(m2),

. . .,

m ~.1

(mN,). Let the positions of the centres of the molecules be rnai. The equations for dynamical equilibrium (cf. (I. 99) and (I. 100)) can now be written

as

(1)

(2) Assuming the interaction energy of any pair of their orientations, in the absence of an the system with a configuration distribution coordinates only (cf. I S 5). Let the number

of molecules to be independent external field, we can describe function depending on position density of each species be uni-

form, .\’,8Z C (6(rmi j ~_1

.Y,,, r) f>,, I = C id(r,f

~ r) I>, = pm = Z$A,

i =I (HZ

S 7 z pm ,,1~~ 1

--

1,

2, . . . . s)

.V p ~~ ~~ > V

(3) (4)

where I/ is the total volume of the system. The contribution from the molccules of species m to the average exciting field per unit volume, (ll”,(m, r), and their contribution to the average derivative of the exciting field per unit volume fZ,$)(m, r), are &,(m,

r) = eg
r) j>,,,

= pm En(w

r) i,,,,

-= pm E,,(m,

4,

I).

(5) (6)

Substituting (1) and (2) into (5) and (6), performing the orientational averaging, using the same approximation as in the one-component case, with the notations (cf. (1.64), (I. 65) and (I. 95)): 0 ax8 (m) = go(m) &B I w anfly Cm) = iii

iyn) = I’

a(m)

&Y

g2(m)

&s~

/

J

I

(7)

ON THE

THEORY

and using the abbreviations

g,(m, r) =

Pm

2

+

i=l

OF OPTICAL

ACTIVITY.

1029

II

(I. 117), we have

E:(r) + go(m’)
E

6(rmi-r) f> +

m’i’imi

*v?n +

lx

i=l

rmt, rm,v)

Ern’i*o

d(rmi

z

gz(m’)


X m’i’fmi

gl(m’)


rm’v)

Ern’t’py

<@$(rmt,

rmpr)

Ern*t’P d(rmt

m’i’/mi

-

r) f> +

NPn +

bLf)(m,

C i=l

r) =

pm E&(r)

d(rmt

-

r) I>,

(8)

+

Nlr, +

Z i=

Z 1 m’i’fmi

go(m’)

-

r) f>.

note that formally we could have performed the orientational in (1) and (2), i.e. we could have taken as basic equations

averaging

We

E mia = Ez(rmt)

+

2 go(m’) m’i’imi

A$‘(rnlt,

+

C gl(m’) m’i’ # mi

A$\l)y(rmi,

E mtas =

We now transform (I. 114).

First

EzJrmi)

+

rmrv)

Ern’i’o +

rm,t,) Ern*t’by,

CI go(m’) m’i’fmi

(8) and (9) in exactly

we write

B$(rmt,

(10)

rwi*)

Ern’ti’b.

(11)

the same way as (I. 113) and

and Em’i’bY as Em*g’o +

Em’i’B

(9)

(Emftts

-

EmIt<@)

i.e. we split off the contributions to Ern’i’PY + (Ern’i’By -Em~t~~v), &,(m, r) and &t)(m, r) due to statistical fluctuations around the average values. We call these “fluctuation terms” J,(m, r) and Jii)(m, r) respectively. Using the definition of the pair distribution functions

and

i) for two molecules

of different

species

r') = El t:l
ii) for two molecules

r')f>,

r) d(rmtf

r’)

(12)

of the same species

PmPmg cm3m)(r,r') = 2

$

i=i i’=i i’fi

as well as the properties

r) d(rmftf-


-

of the pair correlation

functions

-

f),

(13)

(cf. (I. 126)), and

K.

1030

H. TERWIEL

using also the abbreviations

&2(r) = c go

(WqJdr’ A$‘(r, r’)

c

+

&s(f% r’) +

1.(r)

111

g2

(m)/dr’ A$)(r,

r’)

Gp(m, r’) $

r’)

&jj~‘(wz,r’),

1’ ( Y )

,,I

1'

cI gl Cm) ./ dr’

t

,,*

l-Cl.1

A$( r,

c”ht)fz(r) = C go (wzjj’dr’ ,,‘ I-(r)

BL$(r,

(14)

r’) Bs(m, r’),

(15)

we find r) = p,Ef(r)

8&2,

+ frill&Z(r) + go(m’)(!,dr’

--I- pm 2

A$)(r,

r’) dp(m’, r’) {g(“‘, ““)(r, r’) -

1I -f

,,I’ +

fm

+

Pm

A$‘(r,

2 gz (vz’)ydr’

m’

c

,,I’

T(T)

r’) (gon, ‘n’)(r, r’) -

(ni:!cTlr' A’,:,‘,,(r, r’) &~~,‘(m’, r’) {g”7’2‘“‘)(I, r’) -

gl

+ J&h

r’) 8p(m’,

l} + I> + (16)

4,

&il,)(m, r) 1 pmEzc(r) + pm ahf”‘(r) + + pnLx go(m’;Err WI ’ + JhB+c where Ja(m,

B$j(r,

(‘7)

rL

r) and ,flf’(m,

r) are given by:

Ja(m,

A-, r) = C

X

/dr’


rm,i,)(E m’i’B - E,,i,a)

.

i= 1 nr’i’ I mi

*vm + 2

c

i= 1 m’i’ # mi

.ldr’

i-l

so@4

.

c

7TL’i’P,,li

d(r,t

-

r) 6(rln,if -

r’) f> -1

6(r,i

-

r) d(r,,i,

r’) j?

gz(m’)


217, + 1c

r’) &;p(m’, r’) {go”* ““)(r, r’) ~ 1) -t

Ia

E,,i,o)

.

(A$j,(rmi, rm,~,)(Em,i,~Y - Em’i’py) S(r,i sm j~~)(rn, r) = 2 2 g&z’)Tdr’. i= 1 tn’i’#mi /hr’

.
-

rrn,i,)(Em’i’o -

E,,i,s)

S(r,t-r)

-

r) b(rnl,i, -

d(r,,it-r’)

f>.

r’)f>,

(‘8)

(19)

ON THE THEORY

OF OPTICAL ACTMTY.

1031

11

The statistical effects in (17) and (18) are contained in the fluctuation terms Ja(m, I) and JE’(m, r) and in the socalled (cf. I 5 5) “correlation terms”, consisting of integrals with the factor {&“I m’)(r, r’) - l} in the integrand. Upon neglecting these statistical correction terms alltogether, (17) and (18) reduce to s&%

r) = p&Lo)(r)

c?~~)(wz,r) = p,ELt)(r)

+ Pm&:(r),

(20)

+ pmdLt)d(r).

(21)

It follows that E,(m, r) = [c?~(wz,r)]/pm and El(m, r) = [d$~)(m, r)]/pm are then independent of m, so that (20) and (2 1) can be written, using (14) and (15)

E:(r)

= E:‘)(r) + Go/Vdr’A$)(r, r’) G(r’) v(r) + GzJVdr’

w(r)

+

A$)(r, r’) EB(r’) + .

(24 -

E,,(r)

= EiE(r) + Go/vdr’Bj$r,

r’) G(r’),

(23)

v(r)

where (; = 0, 1, 2).

2 pmgi(m) = G, m

(24)

Eqs. (22) and (23) are the two coupled integral equations which Hoek 3) obtained by averaging over physically infinitesimal volume elements. However, (14) and (15) as they stand form a set of 2s intricately coupled equations, since the statistical effects provide an additional coupling between the partial contributions from each molecular species to the average fields. Rather than take all these statistical effects into account, we shall use the same approximations as in the one-component case (cf. the discussion after formula (I. 130)). Firstly we drop all first order correlation and fluctuation

terms in (16) so that this relation

8,(m, r) =

pm

+

pm

5

reduces to

E:(r) + pm&z(r)+

ga(m~([TW A$‘(r, r’)

bu(m’, r’)

{g’m’m’)(rJr’) - 1) +

+ .L(w r), where Ja(m, Ja(%

(25)

r) is now given by

r) = =

Z

C

i= 1 rn’i’#rni

*d(rrnt

-

go(m’)l’clr’rm’t’)(Em’vu
Y) 6(Ymsi, - r’) f>.

Ern,t,a)

*

(26)

i032

R. H. TERWIEL

Secondly

we neglect

the influence

of zeroth

order statistical

effects

small partial contribution to &d(r) containing gl(cf. (14)), GFY’(yn, t’) to satisfy Hoek’s equation (21). The other first contribution to &f(r), i.e. the term containing g2, can now same approximation also bc simplified. Since in the absenccl effects

in the

i.e. we take order partial by using the of statistical

we have

x

g+,,Jt) Q&n,

r) = 2:

,,I

111

we write with negligible ning g.2:

-:‘Jgo(m) a&,

r),

(27)

JO

error for the partial

contribution

to f:(r)

contai-

Note the difference with the one-component case: in the many-component system the approximation of neglecting zeroth order statistical effects in the first order contributions to B’,‘(r) implies neglecting the statistical to the averages of the fields, from coupling of the partial contributions, each molecular species, so that this approximation affects both first order contributions to &t(r), whereas in the one-component system, where the statistical coupling between contributions from different species is absent, the approximation affects only the first order contribution to 8$(r) containing gl. We shall now show that Ja(rut, r) can be written as

whet-c the kernel

K(wz, m’; r, r’) is a symmetric

/‘Kafl(wz,

~2’; r, r') dr'

=

In order to prove this we introduce

= E:(r) Let

E,i(r,i =

+

x fio(m’)Jdr’ IIL’

r) be the

r>u,j&,

m’;

a quantity

A$‘(r,

tensor such that PI,

E*(m,

. ...

ps,

T, co).

(30)

r) defined by

r’) t;‘p(wz’, r’) go”, ““I(,,

E,i,as given by (lo), corresponding

r’).

(3’)

to the confi-

ON THE THEORY

guration

(rll,

-&ta(rmi

= r) = Ez(m, r) +

OF OPTICAL

. . . . rl_v’l, . . . . rmt = r, . . . . rsN,).

2 go 7FS’i’#?Pti

ACTIVITY.

1033

II

Then we have:

A$)(r,

rmftit)Em,i,o(rmt = r) -

Furthermore E,ia(r) 1 + __

= Ez(m NVII z C

pm j=l

r) + go@“)

<&$(rm~,

rm”iu)

(Em”~~~-Em~~~d

d(rmj-

r)

f>.

(33)

m”j”fmj

Substituting by iterative

(32) and (33) into (26) one obtains an expression which can use of (32) and (33) be transformed into a series. The expression

becomes proportional to E* and, in view of the definition written as in (29). For K(m, m’; r, r’) one finds, to second

(31) may be order in the

go(m) 7 Kaa(m, m’; r, r’) = CTdr” go(m”) ALt)(r, r”) go(m’) A$)(r”, r’). m” PmPm”{g(?%Ttb”, m’)(r, r”, r’) _ g(W n”“)(r, r”) gCmu3 m’)(rn, r’)} + gdm”) A:;‘( r, r”) go(m’) A$)(r”,

+ ,_hf ’ ~m’mprnn g where

distribution

the

of different

function

cm, m”)(r,

r”) d(r

_

I’),

for three particles,

(34)

e.g. for three particles

species is defined as rn’XnL,,

PmPm’Pm”

g

r’) +

(na, m’, l’lO)(r, I’, r”) =

2

‘i

x

(d(rmz

_

r) d(r,,p

r') .

-

i-1 i’=l i”=l .d(r,-in

-

rn)

f>.

5 3. Comfiutation of the optical rotatory power in a two-component We shall now prove that (25) and (21) have a solution satisfying Al(m,

r) + k%z2 &‘(m, r)

= 0,

div B(m, r) = 0, A&Lt)(m, r) + k%z2&‘~~)(m,r) = 0,

(35) system.

(m =

1, 2, . . . . s)

(36)

(m=

1,2,...,s)

(37)

(m = 1, 2, . . . . s)

(38)

by assuming (36)-(38) to hold, and showing that n can then be chosen so that (25) and (21) become identities. Finally n can be identified with the physical refractive index. With (21), (37) and (38) one shows (cf. (I. 145)): &Y@?(m,

r) =

-

pm 2

go@‘)

?I&’

.,“”

1

rota &(m’, r).

(39)

R.

1034

H. TERWIEL

Choosing for rS(m, r) a solution of (36) and (37) representing plane wave propagated in an arbitrary direction, so that r’) = e ink.(r’-r) a@,

s(m,

r),

(m =

1,2,

a transverse

. . . . s)

(40)

where the vector k is the product of k and the unit vector in the direction of propagation, we can write for (25): c

{&n?n~ +

P&M

go@‘)

+

C

Rmm”prn~

D,-,,

go(nz’))

6&z’,

r)

=

VE’

/Idr’ &~(m,

+ C IN’

pmEO,(r) + p,&:(r)

m’; r, r’) Lprn, Ei(r’)+pmflB$(r’)] (41)

with the definitions - -&,t

P _jdr’

n, A:$+,,

r’) no ei,lk.(r’--r) ,&,(& ,JL”(r, r’) _

l),

naKn~(m, m’ ; r, r’) np eink,(r’-r),

l? mm’ e /dr’

(42) (43)

n being a unit vector orthogonal to k. With (36), (37) and (40) one can show, using (14) and (28), that (41) leads to two equations, one describing the extinction of the field EO, the other (cf. (I. 152)) determining the parameter 12, c {&Un~ + p&km,

go(m’)

+

X &m~pm-Dm-m~ ,n ”

m’

-

IX,{(pm +

z, Rrnrn~prn~) go@‘))G

go(m’)}

ga(m’,

r)

=

$- $;; ,;T1 rotn &(nz’, r). (4)

Finally the interpretation of n as the refractive index of the medium can be established in complete analogy to the one-component case. We shall now investigate the propagation of the light through the medium. Let us write (44) in a more compact form by introducing a matrix notation. Let B&(r) be a “vector” with s components &,(m, r), (m = 1, 2, . ..) S) and let U, V and W be square matrices with elements u,,*

-

&,m,,

V,,*

== (pm&,,

W mm’

=

(pm

+

(45) +

C

R,,-

IJL”

C &m-pm”) na”

P~-~w)

go@‘).

go@‘),

(46) (47)

ON THE

THEORY

OF OPTICAL

ACTIVITY.

1035

II

Then we can write (44) as

4Jt

---

ns -

Gi

1

Wsrot,

8.

(48)

For the sake of simplicity we now specialize to a two-component system (s = 2). In zeroth order (i.e. setting gi(m) = gs(m) = 0, (m = 1,2)), we obtain the generalized Lorentz-Lorenz equation for an isotropic two-component consisting of non-polar molecules with constant polarizabilities : U+V---mW

476 fz; + 2

=O.

(49)

0

Since we have, according

to (47), det W = 0,

(50)

(49) reduces to

92;- 1 ____=_

47~ go(l) det F(i) + ga(2) det F(s)

ni+2

det (U + V)

3

(51)

’

with the definitions 1 ’ F(l)

s

1 go(l)

Wll

1+

v12 ,

v/11

F(2) -=

1

__ go(4

w12

.

I

__

w21

1 +

v21

v22

\ adl)

,

-go(2)

(52)

w22,

Upon neglecting statistical effects, we have V = 0, det F(i) = pi, det ~(2) = = ps, so that (51) reduces to ni -

4Z

1

ni+2

=3

(53)

Go,

where (24) has been used. Let us now turn to the first order theory. As in the one-component case (cf. the discussion after formula (I. 157)) we replace in the right hand side of (48) the factor 4n(ns + 2)/3(ns - 1) by 4n(ni + 2)/3(nz - 1) and then substitute eq. (53). We obtain 4~ n2+2

(

u+v---

3

ns-1

Let the direction we have a,@,

w

>

*CYa=-

of propagation

r) = a,(m)

&

( G2LG1)

W.rota 8.

of the plane waves be the z-direction.

eiknz; b&z,

r) = L?,(m) eiknz; b,(m, r) = 0.

(54)

Then (55)

1036

R. H. TERWIEL

Substitution Denoting

into (54) yields 4 equations

“+“-?.

for G,(l),

B,(l),

g&2)

n2 -t 2 __~--W by A and. ~~4,2 (?%$?.) n2-1 ?a2 - 1 >

3

we find, upon setting

the coefficient

determinant

and g,(2).

WbyB

equal to zero and using

(50) (56) with solutions B,(M) T i&Y,(m) = 0, (m = 1, 2), the (+) sign corresponding to a right (left) circularly polarized wave with refractive index +z,(nl). In terms of the matrices U, V, F(1) and F(2) relation (56) takes the form det (U + V) -

-:-

-1;:

& -;ET

-t {go( 1) det F(l) + go(2) det F(z)} = (32-5)

(go(l) det F(l) + go(2) det F(z)).

Multiplication of (57) by ~2 - 1 and subtraction (with the - sign) form the first yields, using (51) :

of the second

equation

{go( 1) det f(1) + go(2) det F(g)) det (U $- V) -(go( According

to (51) this relation

~~~ 1) det F(1) + go(2) det F(z))

may then be written

(57)

(58)

as

(go(l) det f(l) + go(2) det F(2))

(59)

or, alternatively ‘2

n r-

Substitution

into Fresnel’s

nz I= 4nk(G1 + Gz) SG.’ ’ formula

. 0

(60)

(I. 1) yields

(61) which is the desired generalization of (I. 166). Finally we note that in the case of a solution of an optically active substance in an optically inactive solvent (gl(2) = g2(2) = 0) the non-specific influence of the solvent is contained in the factor

/(no)=

ni - 1 4nGo

’

(64

ON THE

This is an extension

which is correct

THEORY

OF OPTICAL

only in the absence

of statistical

effects,

after formula

(I. 166)).

1 I 12-63

REFERENCES I) Terwiel,

II

1037

of Hoe k’s result. He obtained

from (51) and (55) (cf. the discussion Received

ACTIVITY.

2)

Kuhn,

R. H., and Ma z UT, P., Physica 30 W., 2. phys. Chem. (B) 30 (1935) 356.

3)

Hoe!+

H., Thesis

(Leiden

1939) Physica

(1964) 625.

8 (1941)

209..

as can be seen