O.72 with the plasma parameter r=100), an iterative process to solve the QHNC ... Liquid metallic hydrogen (LMH) is the simplest system among various liquid .... Here, we have used the following fact that, if a proton is fixed in the LMH, the proton ... Equations (2·34) and (2·35) constitute a closed set of integral equations to ...
940 Progress of Theoretical Physics, Vol. 72, No.5, November 1984
Pair Correlation Functions in Liquid Metallic Hydrogen
J unzo CHIHARA Department of Physics, Japan Atomic Energy Research Institute Tokai-mura, Ibaraki 319~11 (Received March 27, 1984)
§ 1.
Introduction
Liquid metallic hydrogen (LMH) is the simplest system among various liquid metals in the sense that the ion-ion and ion-electron interactions are purely coulombic because of the absence of core electrons. However, for a long time it has been difficult to treat the LMH as a proton-electron mixture, since we need a unified theory which can treat both classical and quantum liquids on the same footing, when the temperature of the LMH is sufficiently high compared with the Fermi temperature of protons. Even when hydrogens are completely ionized at high temperatures, the evaluation of the proton-electron radial distribution function (RDF) on the basis of the classical statistical mechanics leads to its divergence at zero separation owing to the Boltzmann factor, exp(/:Je 2 r), as is well known (see, e.g., Ref. 1) and references therein). In this situation, it is important that the framework of th~ Kohn-Sham-Mermin2 ) can treat a quantum liquid at arbitrary temperature including a classical liquid as its classical limit. Previously we have derived the quantal hypernetted chain (QHNC) equation3 ) which determined the density-response function for a quantum liquid at zero and finite temperatures by using the Kohn-Sham-Mermin formalism: This QHNC equation is reduced to the usual HNC equation for a classical liquid in the classical limit. Therefore, the QHNC equation extended to a binary mixture can be used to calculate the RDF's, gpp between protons and gep between proton and electron, in the LMH, since the density response functions X/P(i = p,e) concerning protons, classical particles, become identical with the structure factors S iP( Q ). Already we have applied this QHNC equation to hydrogen plasmas and calculated the RDF'sand energy levels of an electron bound around a proton. In the present paper, we solve the QHNC equation for the LMH to obtain gpp and gep in the temperature- and density-region where electrons constitute a degenerate quantum liquid and protons form a classical liquid.
Downloaded from http://ptp.oxfordjournals.org/ by guest on January 7, 2016
The quantal hypemetted chain (QHNC) equation, derived from the Kohn-Sham-Mermin theory, is solved for the pair correlation functions in a liquid metallic hydrogen (LMH) modelled as a proton-electron mixture in the range where protons constitute a c1assicalliquid and electrons are in a degenerate state. In the sufficiently high temperature and high density region, hydrogen atoms are found to form a metallic liquid where electrons have no bound state around a proton, and with decreasing temperature or density (for example, near r.=O.72 with the plasma parameter r=100), an iterative process to solve the QHNC equation becomes unstable and diverges owing to the rapid growth in an attractive part of the effective electron-electron interaction_ In addition, it is shown that in the LMH there exists a first-order phase transition which is considered to cause a significant change in the electronic structure: At r.=0.3 and r =20, for example, the QHNC equation has two sets of solutions belonging to two branches; the high temperature phqse and the low temperature phase in which the distribution of electrons reflects more strongly the local order of protons.
Pair Correlation Functions in Liquid Metallic Hydrogen
941
§ 2_
Formalism
In this section we outline the derivation of the QHNC equation for a mixture from the Kohn-Sham Mermin formulation on the basis of a thermodynamic argument and present several formulae necessary to deal with the LMH (for details, see Ref.· 5) ). Let us consider an m-component quantum mixture under the influence of external potentials U i (r}{i=1,2"",m), which couple to the corresponding particles of species i. The density distribution ni{ r) of species i is determined by the functional derivative of the thermodynamic potential Q of this system with respect to li(r)~f-li- Ui(r) at fixed temperature T and volume V (for example, Ref. 6) ), 8Q 8 Ii () r
I
T,V
= - ni(rl{uJ).
(2-1)
Here, f-li is the chemical potential of species i, and {UA} denotes an abbreviation for U1, U2,'" Um. The natural variables of Q are T, V and {/i(r)}: Then, the Legendre transfOmi of Q by the use of (2-1) introduces the intrinsic Helmholtz free energy (2-2) whose natural variables are T, V and {ni(r )}. Thus we can describe this system in terms of {ni( r)} as independent variables, and the external potentials Ui( r) are, alternatively, derived from (2 -2) (2-3)
Now, we define effective external potentials Uieff(r) so as to fulfill the conditions (2-4)
where (2-5)
Downloaded from http://ptp.oxfordjournals.org/ by guest on January 7, 2016
Recently, Dharma-wardana and Perrot (DWP)4) applied the Kohn-Sham-Mermin formalism to the LMH and calculated gpp and gep. In their treatment the HNC equation to gpp was introduced without showing its derivation, while our formalism involves the HNC equation for gpp in a natural way in treating protons and electrons on an equal footing within the Kohn-Sham-Mermin framework. Moreover, their equation is shown to be derived from the QHNC equation by introducing further three approximations in addition to the HNC approximation. Therefore, the meaning of their approximations and the limitation of their treatment can be elucidated in the light of the QHNC equation. In the first part of § 2, by the use of the Kohn-Sham-Mermin formulation some exact relations for the LMH are derived under the condition that protons behave as classical particles. Next, several integral equations for gpp and gep are shown to be derived from these exact relations on the basis of the ·HNC approximation. Numerical procedure to solve these integral equations and their results are displayed in § 3. The last section is devoted to discussion and conclusion.
J
942
Chihara (2'6)
with fJ.io being the chemical potential of noninteracting system of species i. In other words, U/ ff is defined so that the density distribution niO( rl U/ff ) of noninteracting system should become equal to the real density distribution ni(rl{U.l}). Since the intrinsic Helmholtz free energy
