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Unformatted text preview: Chapter 4 Non-ideal gases In this chapter we consider gases of short-range interaction. The “short-range” is defined by that the integration of the potential between two particles over the space converges, i.e., the potential decays faster than r − 3 for r being the distance between the two particles. With the gravitational force neglected, the interaction between neutral particles decays at least as fast as r − 6 . So the short-range interaction condition is satisfied. The case of electrons which carry charges is rather complicated. If the Coulomb potential is screened, the interaction is still short-ranged. But there are cases where the long-range interaction is important, especially for electrons in low-dimensional space and at low density, such as in the case of Wigner lattices formed by 2-dimensional electron gases at low density. 4.1 Cluster expansion for non-ideal classical gases We first consider a classical gas in grand canonical ensemble. For simplicity, we assume that there is only two-body interaction and the interaction has translational symmetry. The potential between two particles has the form u i , j = u ( r i − r j ) , (4.1) which is short-ranged, i.e., lim | r i − r j |→∞ | r i − r j | 3 u i , j = . (4.2) The partition function is Ξ = ∞ ∑ N = 1 N ! ζ N 1 h 3 N ∫ d p 1 d p 2 ··· d p N ∫ d r 1 d r 2 ··· d r N e − β ( p 2 1 + p 2 2 + ··· + p 2 N ) / (2 m ) − β ∑ i < j ≤ N u i , j , (4.3) where ζ ≡ e βµ is the fugacity. Following the calculation for ideal gases, the integration over momentum can be done to obtain Ξ = ∞ ∑ N = z N N ! ∫ d r 1 d r 2 ··· d r N ∏ i < j ≤ N e − β u i , j , (4.4) where z ≡ λ − 3 T exp( βµ ) with λ T ≡ ~ √ 2 πβ/ m being the thermal de Brogile wavelength. 47 48 CHAPTER 4. NON-IDEAL GASES The integration over the real space coordinates is non-trivial as the interaction potential couples all of them. To proceed, we have to expand the integrations in order of some small quantity. A standard way is the cluster expansion which is done with respect to the density of the gas. Noticing that e − β u i , j → 1 quickly as the distance between the two particles increases, we single out the contribution due to the interaction by defining e − β u i , j ≡ 1 + f i , j . (4.5) Thus we can expand ∏ i < j ≤ N e − β u i , j = 1 + ∑ f i , j + ∑ ( i , j ) , ( k , l ) f i , j f k , l + ··· . (4.6) For example, e − β ( u 1 , 2 + u 2 , 3 + u 3 , 1 ) = ( 1 + f 1 , 2 ) ( 1 + f 2 , 3 ) ( 1 + f 3 , 1 ) = 1 + f 1 , 2 + f 2 , 3 + f 3 , 1 + f 1 , 2 f 2 , 3 + f 2 , 3 f 3 , 1 + f 3 , 1 f 1 , 2 + f 1 , 2 f 2 , 3 f 3 , 1 . The point of such expansion is that we can integrate over the variables separately if they are not “connected”. For example, in the integration of f 1 , 2 , the variables r 1 and r 2 can be separated from the other ones. The “connection” is defined as: 1. Two coordinates i and j are connected if the term contains...
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This note was uploaded on 10/14/2010 for the course PHYS 2051 taught by Professor Drkwong during the Spring '10 term at CUHK.
- Spring '10