APPENDIX TWO AN INTEGRAL ARISING IN THE DERIVATION OF THE KIRKWOOD G FACTOR IN THE M.S.M

In document Some topics in statistical physics (Page 135-141)

We w i s h to c a l c u l a t e the i n t e g r a l

Q OO

J r 2 h . ( r ) d r (A2.1)

w h e r e h A (r) is a f u n c t i o n o c c u r r i n g in W e r t h e i m ' s s o l u t i o n 21

of the m e a n s p h e r i c a l m o d e l for h a r d s p h e r e s (diameter a) at

d e n s i t y p, w i t h i m b e d d e d p o i n t d i p o l e s (dipole m o m e n t y) . F r o m e q u a t i o n (41) of W e r t h e i m h A (r) 2K(h (r,2Kp) - h ( r , - K p ) ) (A2.2) (2) s - = 9 S (r,p) - 1 is t he s o l u t i o n of the h a r d w h e r e h g (r, p ) sphe r e PY e q u a t i o n for d e n s i t y p. K is a c o n s t a n t d e t e r m i n e d by W e r t h e i m ' s e q u a t i o n (51), 4tt . ß y 2 3 j - ( - ± —) ( p o d ) ö 3 q ( j ( pö 3 ) K) - q (- ~ ( pu 3) K) (A2.3) w h e r e q (x) (l+2x) ( 1 - x ) 4 (A2.4) C o n s i d e r the i n t e g r a l I. F r o m (A2.2), I = (l7'P.° ) 2K (I (2Kp) - I (-Kp) ) (A2.5) w h e r e I (p)

1

" h s (r , p) dr (A2.6)

I r 2 (g 2 (r / p) -l) dr (A2.7)

62

Theile has shown

{ r 2 (g^r,p)-1)dr

i + — -

4npö

(q P 0 3)-1) (A2.8)

so that

I = i [q 1 (2. pa3K) + 2q 1 (- £ p a 3K) - 3] (A2.9)

Thus, if we define the variable

6 = I (pa3)K

(A2.10)

then our integral is given by

I = i [q 1 (2C) + 2q 1 (-€) - 3] (A2.ll)

where £ satisfies (from (A2.8))

4 7T , 3 P 2 , 3 x

_ ( b ü _ ) ( p ö d )

3

a

3

q(2£) - q(-£) (A2.12)

with the Wertheim restriction

§6.

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