0021-3640/00/7205- $20.00 © 2000 MAIK “Nauka/Interperiodica”0245

JETP Letters, Vol. 72, No. 5, 2000, pp. 245–248. From Pis’ma v Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, Vol. 72, No. 5, 2000, pp. 361–365.Original English Text Copyright © 2000 by Filinov, Bonitz, Fortov.

Coulomb systems continue to attract the interest ofresearchers in many fields, including plasmas, astro-physics and solids; see [1, 2] for an overview. The mostinteresting phenomena, such as metallic hydrogen,plasma phase transition, bound states, etc., occur in sit-uations where the plasma is both strongly coupled andstrongly degenerate. However, in this region, the ther-modynamic properties of the plasma are only poorlyknown. The need for the simultaneous account ofstrong Coulomb and quantum effects makes a theoreti-cal treatment very difficult. Among the most promisingtheoretical approaches to these systems are path-inte-gral quantum Monte Carlo (PIMC) techniques; see,e.g., [3, 4].

In this letter, we demonstrate that for many currentproblems in dense warm plasmas (

k

B

T

> 0.1 Ry), directPIMC simulations can, in fact, be carried out withacceptable efficiency. We report results for the internalenergy, equation of state, and pair distribution functionsof partially ionized hydrogen in a wide range of couplingand degeneracy parameters,

Γ

= (4

π

n

e

/3)

1/3

e

2

/4

π

e

0

k

B

T

and

χ

=

n

e

(

λ

e

is the electron thermal wave length

= 2

π

"

2

β

/

m

e

). Furthermore, our calculations predictordering of protons, as well as pairing of electrons, athigh density.

As is well known, the thermodynamic properties ofa quantum system are fully determined by the partitionfunction

Z

. For a binary mixture of

N

e

electrons and

N

i

1

This article was submitted by the authors in English.

λ e3

λ e2

protons,

Z

is conveniently written as

(1)

Here,

q

≡

{

q

1

,

q

2

, …, } comprises the coordinates of

the protons and

σ

= {

σ

1

, …, } and

r

≡

{

r

1

, …, }are the electron spins and coordinates, respectively. Thedensity matrix

ρ

in Eq. (1) is represented in the stan-dard way by a path integral [5]:

(2)

where

∆β ≡ β

/(

n

+ 1) and = 2

π

"

2

∆β

/

m

e

. Further,

r

(

n

+ 1)

≡

r

and

σ

' =

σ

; i.e., the particles are representedby fermionic loops with the coordinates (beads) [

r

]

≡

[

r

,

r

(1)

, …,

r

(

n

)

,

r

]. The electron spin gives rise to the spinpart of the density matrix

6

, whereas exchange effects

are accounted for by the permutation operator andthe sum over the permutations with parity

κ

P

. Follow-ing [3, 6, 7], we use a modified representation (3) of thehigh-temperature density matrices on the r.h.s. ofEq. (2), which is suitable for efficient direct fermionicPIMC simulations of plasmas. With the error of order

e

~ (

β

Ry

)

2

χ

/(

n

+ 1) vanishing with a growing number ofbeads, we obtain the approximation

Z Ne Ni V β, , ,( ) Q Ne Ni β, ,( )/Ne!Ni!,=

Q Ne Ni β, ,( ) q rρ q r σ; β, ,( ).dd

V

∫σ∑=

qNi

σNerNe

ρ q r σ; β, ,( )1

λ i3Niλ∆

3Ne------------------- 1±( )

κP r 1( )…d r n( )d

V

∫P

∑=

× ρ q r r 1( ); ∆β, ,( )…ρ q r n( ) P̂r n 1+( ); ∆β, ,( )6 σ P̂σ',( ),

λ∆2

P̂

ρ q r σ; β, ,( )σ∑ 1

λ i3Niλ∆

3Ne------------------- ρs q r[ ] β, ,( ),

s 0=

Ne

∑=

High-Density Phenomena in Hydrogen Plasma1

V. S. Filinov*, M. Bonitz**, and V. E. Fortov**** Mercator guest professor at Rostock University

** Fachbereich Physik, Universität Rostock, D-18051 Rostock, Germany*** High Energy Density Research Center, Joint Institute for High Temperatures, Russian Academy of Sciences,

Izhorskaya ul. 13/19, Moscow, 127412 RussiaReceived June 27, 2000; in final form, August 1, 2000

A novel path-integral representation of the many-particle density operator is presented which makes direct Fer-mionic path-integral Monte Carlo simulations feasible over a wide range of parameters. The method is appliedto compute the pressure, energy, and pair distribution functions of a hydrogen plasma in the region of strongcoupling and strong degeneracy. Our numerical results allow one to analyze the atom and molecule formationand breakup and predict, at high density, proton ordering and pairing of electrons. © 2000 MAIK “Nauka/Inter-periodica”.

PACS numbers: 52.25.Kn; 52.65.Pp

246

JETP LETTERS Vol. 72 No. 5 2000

FILINOV et al.

(3)

where Ui, , and denote the sum of the binaryinteraction Kelbg potentials Φab [8] between protons,electrons at vertex l, and electrons (vertex l) and pro-tons, respectively.

In Eq. (3), ≡ exp[–π ] arises from thekinetic energy density matrix of the electron with indexp, and we introduced dimensionless distances betweenneighboring vertices on the loop, ξ(1), …, ξ(n). Thus,explicitly, [r] ≡ [r; r + λ∆ξ(1); r + λ∆(ξ(1) + ξ(2)); …]. Theexchange matrix is given by

(4)

As a result of the spin summation, the matrix carries asubscript s denoting the number of electrons having thesame spin projection.

As an example, we present the equation of stateβp = ∂lnQ/∂V = [α/3V∂lnQ/∂α]α = 1:

ρs q r[ ] β, ,( )CNe

s

2Ne

-------- βU q r[ ] β, ,( )–{ }exp=

× φppl det ψab

n 1,s,

p 1=

Ne

∏l 1=

n

∏

U q r[ ] β, ,( ) Ui q( )Ul

e r[ ] β,( ) Ulei q r[ ] β, ,( )+

n 1+-------------------------------------------------------------,

l 0=

n

∑+=

Ule Ul

ei

φppl ξ p

l( ) 2

ψabn 1,

sπλ∆

2----- ra rb–( ) ya

n+2

–

exp

s

,≡

yan λ∆ ξa

k( ).k 1=

n

∑=

βpVNe Ni+------------------ 1

3Q( ) 1–

Ne Ni+------------------ qd rd ξρs q r[ ] β, ,( )d∫

s 0=

Ne

∑+=

(5)

Here, α is a length scaling; α = L/L0, ⟨…|…⟩ denotesthe scalar product; and qpt, rpt, and xpt are the differencesof two coordinate vectors: qpt ≡ qp – qt, rpt ≡ rp – rt, xpt ≡rp – qt, = rpt + , ≡ xpt + , = ,

and ≡ – . Other thermodynamic quantitieshave an analogous form.

We demonstrate our numerical scheme for a two-component electron–proton plasma. In the simulations,we used Ne = Np = 50. To test the MC procedure, wefirst consider a mixture of ideal electrons and protonsfor which the thermodynamic quantities are knownanalytically, e.g., [9]. Figure 1 shows our numericalresults for the pressure, together with the theoreticalcurve. The agreement, up to the degeneracy parameterχ as large as 10, is evident and improves with increas-ing particle number. This clearly proves that ourmethod correctly samples the fermionic permutations.Note that for fast generation of a MC sequence ofN-particle configurations it is necessary to efficientlycompute the acceptance probability of new configura-tions, which is proportional to the absolute value of theratio of the exchange determinants of two subsequentconfigurations, while the sign of the determinants isincluded in the weight function of each configuration.

Let us now turn to the case of interacting electronsand protons. We performed a series of calculations inwhich the classical coupling parameter Γ was kept con-stant while the degeneracy parameter χ was varied, cf.Fig. 2. One can see that, for weak coupling (Γ = 0.4)and small degeneracy parameters χ < 0.5, there is goodagreement with analytical theories and quantum MCsimulations without exchange [10]. However, asexpected, with increasing χ and Γ, the deviations growrapidly. Figure 2 also contains a comparison withrestricted path-integral results of Militzer et al. [11](the large triangles) corresponding to values of the cou-pling parameter in the range of 0.17, … , 0.672. Evi-dently, the agreement, in particular, of the energies, isvery good. The deviations in the pressure are appar-ently related to the fixed-node approximation used in[11] and need further investigation.

× βe2

qpt

---------p t<

Ni

∑ rpt∂∆βΦee

∂ rpt

-------------------p t<

Ne

∑ xpt∂∆βΦie

∂ xpt

------------------t 1=

Ne

∑p 1=

Ni

∑––

– Aptl ∂∆βΦee

∂ rptl

-------------------p t<

Ne

∑ Bptl ∂∆βΦie

∂ xptl

------------------t 1=

Ne

∑p 1=

Ni

∑+l 1=

n

∑

+α

det ψabn 1,

s

----------------------∂det ψab

n 1,s

∂α-------------------------

,

Aptl rpt

l rpt⟨ ⟩

rptl

--------------------, Bptl xpt

l xpt⟨ ⟩

xptl

---------------------.= =

rptl ypt

l xptl yp

l ypl λ∆ ξ p

k( )k 1=n∑

yptl yp

l ytl

Fig. 1. Energy E of an ideal plasma of degenerate electronsand classical protons in excess of the classical energy. PIMCsimulation results with varied particle number are shown:N = 32 (triangles), N = 50 (squares), and N = 90 (circles) arecompared to the exact analytical result (dashed line).

JETP LETTERS Vol. 72 No. 5 2000

HIGH-DENSITY PHENOMENA 247

Fig. 2. (a) Pressure P and (b) energy E of the nonideal plasma as functions of the quantum parameter χ. Curves correspond to dif-ferent values of the coupling parameter Γ given in the inset of the right figure. Large circle denotes quantum MC simulations withoutexchange (QMCNE), and large asterisk denotes the weak coupling model of Riemann et al. (RSDWK), using data from [10]. Thelarge triangles are recent restricted PIMC results of Militzer et al. [11], which are compared to our results (large squares) for threevalues of Γ, from top to bottom: Γ = 0.169, 0.338, and 0.672.

Fig. 3. Electron–electron (gee, full line), ion–ion (gii, dashed line) and electron–ion (gei, dash–dotted line) pair distribution functionsfor dense hydrogen. The line styles in (b) and (c) are the same as in (a). Notice the varying scalings of the curves. The values for thecoupling, degeneracy, and Brueckner parameters are (a) Γ = 2.9, χ = 1.46, rs = 5.44; (b) Γ = 1.16, χ = 0.37, rs = 5.44; (c) Γ = 19.8,χ = 1848, rs = 0.318; and (d) Γ = 53.8, χ = 37.000, rs = 0.117.

A very interesting result is that the energy curves inFig. 2 become almost parallel as the degeneracyincreases. In contrast, for Γ > 0.6, reduction of χ leadsto a rapid decrease in the energy, which is due to the

formation of atoms and molecules, as will be shownbelow.

The main advantage of the presented method is thatit allows one to investigate dense plasmas in a variety

n = 5 × 1025 cm–3

248

JETP LETTERS Vol. 72 No. 5 2000

FILINOV et al.

of physical situations which are very difficult todescribe reliably by other approaches. This includespartial ionization and dissociation, Mott effect, andionic ordering at high densities. To investigate thesephenomena, we show, in Fig. 3, the pair distributionfunctions for the four most interesting physical situa-tions. Figure 3a clearly shows the existence of hydro-gen molecules (cf. the peaks of the proton–proton andelectron–electron pair distribution functions at a sepa-ration of about 1.4aB) and atoms. We note that the peakof the electron–proton function (multiplied by r2) inour calculations appears at r = 1aB if no molecules arepresent (e.g., at lower density). But for the situation ofFig. 3a, the presence of molecules leads to a shift of thepeak to larger distances. Figure 3b shows that, withincreasing temperature, atoms and molecules break up,which is clearly seen by the drastic lowering of thementioned peaks in the pair distribution functions.

Let us now consider the case of higher densities, butkeep the temperature constant. Here, our calculationspredict interesting physical phenomena. In Figs. 3b–3d,one clearly sees increased ordering of protons from apartially ionized plasma (3b, Γ ≈ 1.2), to liquidlike(3c, Γ ≈ 20) and solidlike (3d, Γ ≈ 54) behavior, cf. theproton–proton pair distribution functions.

Notice further a qualitative change in the electron–electron function upon a density increase from Fig. 3bto 3d: gee in Fig. 3b is typical of partially ionized plas-mas, whereas in Fig. 3c a strong peak at small distancesis observed. A further increase in the density leads to analmost uniform electron distribution in Fig. 3d. For bet-ter understanding of the electron behavior, we alsoincluded in Fig. 3c the functions r2gee and r2gii. Theshoulders in these curves indicate that the most proba-ble interelectronic distance is almost two times smallerthan the average distance between two protons. Thereason for the behavior in Fig. 3c is pairing of electronswith opposite spin projections. An analysis of the elec-tronic bead distribution allows us to conclude that the“extension” of the electrons is of the order of the inte-rion distance and that there is partial overlap of individ-ual electrons. Under these conditions, pairing of (partof) the electrons minimizes the total energy of the sys-tem. This effect vanishes with increasing density due tothe growing wave function overlap.

This work was supported by the Deutsche Fors-chungsgemeinschaft (Mercator-Programm) and theNIC Jülich. We acknowledge stimulating discussionswith W. Ebeling, D. Kremp, W.D. Kraeft, andM. Schlanges and thank B. Militzer for providing thedata of [11].

REFERENCES1. H. E. DeWitt et al., W. Ebeling et al., and D. Klakow

et al., in Proceedings of the International Conference onStrongly Coupled Plasmas, Ed. by W. D. Kraeft andM. Schlanges (World Scientific, Singapore, 1996).

2. J. Shumway and D. M. Ceperley, in Proceedings of theInternational Conference on Strongly Coupled CoulombSystems, Saint-Malo, France, 1999; J. Phys. IV 10 (5),3 (2000); J. P. Hansen, D. Goulding, and R. van Roij, inProceedings of the International Conference on StronglyCoupled Coulomb Systems, Saint-Malo, France, 1999;J. Phys. IV 10 (5), 27 (2000); M. Rosenberg, in Proceed-ings of the International Conference on Strongly Cou-pled Coulomb Systems, Saint-Malo, France, 1999;J. Phys. IV 10 (5), 73 (2000).

3. V. M. Zamalin, G. E. Norman, and V. S. Filinov, TheMonte Carlo Method in Statistical Thermodynamics(Nauka, Moscow, 1977).

4. D. Ceperley, The Monte Carlo and Molecular Dynamicsof Condensed Matter Systems, Ed. by K. Binder andG. Cicotti (SIF, Bologna, 1996).

5. R. P. Feynman and A. R. Hibbs, Quantum Mechanicsand Path Integrals (McGraw-Hill, New York, 1965).

6. V. S. Filinov, High Temp. 13, 1065 (1975); 14, 225(1976).

7. B. V. Zelener, G. E. Norman, and V. S. Filinov, HighTemp. 13, 650 (1975).

8. G. Kelbg, Ann. Phys. (Leipzig) 12, 219 (1963); 13, 354(1963); 14, 394 (1964); W. Ebeling, H. J. Hoffmann, andG. Kelbg, Beitr. Plasmaphys. 7, 233 (1967) and refer-ences therein.

9. W. D. Kraeft, D. Kremp, W. Ebeling, and G. Röpke,Quantum Statistics of Charged Particle Systems (Akad-emie, Berlin, 1986; Mir, Moscow, 1988).

10. J. Riemann, M. Schlanges, H. E. DeWitt, and W. D. Kraeft,in Proceedings of the International Conference onStrongly Coupled Plasmas, Ed. by W. D. Kraeft and M.Schlanges (World Scientific, Singapore, 1996), p. 82.

11. B. Militzer and D. Ceperley, submitted to Phys. Rev.Lett.