Computational Physics

References 195

(b) We assume that the incoming rays have intensityI 0. Show that the average total

intensity of waves with wave vectork 1 arriving at the detector is given by

I(k 1 ,r)=I 0

〈N

∑

l,j= 1

eik(rl−rj)

〉

withk=k 1 −k 0.

(c) Show that this expression is equal toI 0 NS(k), whereSis the static structure

factor, defined in terms of the correlation functiongas

S(k)= 1 +n

∫

d^3 rg(r)eikr.

(nis the particle densityN/V.)

7.2 The magnetic susceptibility of the Ising model on anL×Lsquare lattice is defined
byχ=∂m/∂H, wheremis the magnetisation andhthe magnetic field.
(a) Show that the magnetic susceptibility can be written as

χ=

1

L^2 kBT

∑

i,j

(〈sisj〉−〈si〉^2 ).

(b) A scaling exponentηassociated with the magnetic correlation function (see

Eq. (7.48)) is defined by

g(r)∝r^2 −d−η.

Assuming that close to the critical point this form extends to a distanceξ, whereξ

is the correlation length, find the following scaling relation betweenγ,ηandν:

γ=ν( 2 −η).

References

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[4] F. Reiff,Fundamentals of Statistical and Thermal Physics. Kogakusha, McGraw-Hill, 1965.
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