p848ps4 - Physics 848: Problem Set 3 Due Tuesday, November...

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Unformatted text preview: Physics 848: Problem Set 3 Due Tuesday, November 8 at 11:59 PM Note: each problem is worth 10 points unless otherwise stated. 1. In class it was shown that the Ornstein-Zernike correlation function Γ( r- r ) was given by Γ( ρ ) = Z 1 a/ 2 + Ck 2 e i k · ρ d 3 k (1) in three dimensions. (Here ρ = r- r .) Show that this integral is given by Γ( ρ ) = K ρ exp(- ρ/ξ ) , (2) where K is a constant and ξ = q 2 C/a is the correlation length. [Thus, if a ∝ ( T- T c ), ξ ∝ ( T- T c )- 1 / 2 .] Hint: first introduce spherical coordinates such that the polar angle θ is the angle between k and ρ . Then carry out the angular integrals. The remaining integral can be arranged so that it can be done as a contour integral involving simple poles. 2. Critical exponents for the Van der Waals equation of state. As given in class, the Van der Waals equation of state may be written P + 3 V 2 ¶ V- 1 3 ¶ = 8 3 T. (3) Here P = p/p c , V = v/v c , and T = t/t c , where p , v , and t are the pressure, specific volume (volume per atom) and temperature, and...
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This note was uploaded on 07/17/2008 for the course PHYS 848 taught by Professor Stroud during the Fall '05 term at Ohio State.

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p848ps4 - Physics 848: Problem Set 3 Due Tuesday, November...

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