This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 4 Nonideal gases In this chapter we consider gases of shortrange interaction. The “shortrange” 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 shortrange interaction condition is satisfied. The case of electrons which carry charges is rather complicated. If the Coulomb potential is screened, the interaction is still shortranged. But there are cases where the longrange interaction is important, especially for electrons in lowdimensional space and at low density, such as in the case of Wigner lattices formed by 2dimensional electron gases at low density. 4.1 Cluster expansion for nonideal classical gases We first consider a classical gas in grand canonical ensemble. For simplicity, we assume that there is only twobody 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 shortranged, 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. NONIDEAL GASES The integration over the real space coordinates is nontrivial 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...
View
Full
Document
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
 DrKwong

Click to edit the document details