TY - JOUR

T1 - Phase transition in a model quantum system

T2 - Quantum corrections to the location of the critical point

AU - Burke, T.

AU - Lebowitz, J. L.

AU - Lieb, E.

PY - 1966

Y1 - 1966

N2 - It was proved by Lebowitz, Penrose, and Lieb that the pressure of a system with an interparticle potential v(r) of the form v(r)=q(r)+γνÏ(γr), where ν is the dimensionality of the space considered and γ-1 is the range of Ï, is given in the limit γ-1→â ž by applying the Maxwell equal-area construction to the generalized van der Waals pressure pÌ(T,Â ρ)=p0(T,Â ρ;Â h)+12αρ2. Here T is the temperature, ρ the particle density, h Planck's constant, p0(T,Â ρ;Â h) the pressure of a system with interparticle potential q(r) and α=â γνÏ(γr)drÏ(y)dy. This system will have a first-order phase transition for α<0. In this paper we apply the above results to obtain the properties of a one-dimensional quantum system for which q(r) is a hard-core potential of "diameter" a. For this system p0(T,Â ρ;Â h) is known exactly and we can thus find how the properties of the phase transition depend on the quantum parameter λ=(h2mkT*)12d*, the ratio of the de Broglie wavelength to the interparticle separation evaluated at the classical critical point, i.e., at the critical point found in the limit h→0. We compute in particular the fractional changes in the critical temperature Tc(λ)Tc*, critical density ρc(λ)ρc*, and the critical ratio (pcρckTc)(pc*ρc*kTc*). We find that Tc(λ)Tc*∼ρc(λ)ρc* decreases rapidly from unity as λ increases, approaching zero as λ→â ž. The critical ratio on the other hand varies only slightly with λ, decreasing from one to about 0.98 as λ varies from 0 to â ž. When we compare this behavior of this model system with that observed experimentally in fluids expected to obey the law of corresponding states, the resemblance is very striking. Taking xenon as our "classical system" to fix the values of Tc*, ρc*, pc* and comparing its critical parameters (in reduced units) with those of argon, neon, and the isotopes of hydrogen and helium we find again Tc(λ)Tc*∼ρc(λ)ρc* decreasing to less than half as we go from Xe to He3, the corresponding values of λ increasing from 0.04 to 1.90. On the other hand, the critical ratio varies by only about 5% (increasing from Xe to He3).

AB - It was proved by Lebowitz, Penrose, and Lieb that the pressure of a system with an interparticle potential v(r) of the form v(r)=q(r)+γνÏ(γr), where ν is the dimensionality of the space considered and γ-1 is the range of Ï, is given in the limit γ-1→â ž by applying the Maxwell equal-area construction to the generalized van der Waals pressure pÌ(T,Â ρ)=p0(T,Â ρ;Â h)+12αρ2. Here T is the temperature, ρ the particle density, h Planck's constant, p0(T,Â ρ;Â h) the pressure of a system with interparticle potential q(r) and α=â γνÏ(γr)drÏ(y)dy. This system will have a first-order phase transition for α<0. In this paper we apply the above results to obtain the properties of a one-dimensional quantum system for which q(r) is a hard-core potential of "diameter" a. For this system p0(T,Â ρ;Â h) is known exactly and we can thus find how the properties of the phase transition depend on the quantum parameter λ=(h2mkT*)12d*, the ratio of the de Broglie wavelength to the interparticle separation evaluated at the classical critical point, i.e., at the critical point found in the limit h→0. We compute in particular the fractional changes in the critical temperature Tc(λ)Tc*, critical density ρc(λ)ρc*, and the critical ratio (pcρckTc)(pc*ρc*kTc*). We find that Tc(λ)Tc*∼ρc(λ)ρc* decreases rapidly from unity as λ increases, approaching zero as λ→â ž. The critical ratio on the other hand varies only slightly with λ, decreasing from one to about 0.98 as λ varies from 0 to â ž. When we compare this behavior of this model system with that observed experimentally in fluids expected to obey the law of corresponding states, the resemblance is very striking. Taking xenon as our "classical system" to fix the values of Tc*, ρc*, pc* and comparing its critical parameters (in reduced units) with those of argon, neon, and the isotopes of hydrogen and helium we find again Tc(λ)Tc*∼ρc(λ)ρc* decreasing to less than half as we go from Xe to He3, the corresponding values of λ increasing from 0.04 to 1.90. On the other hand, the critical ratio varies by only about 5% (increasing from Xe to He3).

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U2 - 10.1103/PhysRev.149.118

DO - 10.1103/PhysRev.149.118

M3 - Article

AN - SCOPUS:0041143112

SN - 0031-899X

VL - 149

SP - 118

EP - 122

JO - Physical Review

JF - Physical Review

IS - 1

ER -