a note on the phase diagram of the fermion one-component plasma

Volume 103A, number 5 PHYSICS LETTERS 9 July 1984

A NOTE ON THE PHASE DIAGRAM OF THE FERMION ONE-COMPONENT PLASMA

Hlkaru KAWAMURA Department of Phystcs, College of General Education, Osaka Untverstty, Toyonaka 560, Japan

and

Kunlyoshl EBINA Department of Material Phystcs, Faculty of Engtneermg Science, Osaka Untverstty, Toyonaka 560, Japan

Received 27 March 1984

The phase diagram of the three-dimensional fermlon one-component plasma is constructed by makmg use of the known classical and low-temperature limits The melting curve has a melting density maxunum point m the quantum regmn

The one-component plasma (OCP) is a system con- slstlng of charged particles in a uniform neutralizing background This system is the simplest model of a charged many-body system and has been the object of many active researches It also provides a reasonable model of the dense ionized matter typical of degenerate stellar matter Much information is now available for the OCP both in the classical and in the ground state limits owing to the recent development of extensive Monte Carlo simulations [ 1 -4 ] However, relatively little is known concerning the finite-temperature prop- ertles between the classical and the ground state limits

Let us consider a system of N spin 1/2 fermlons of charge Ze and mass m in a volume V immersed in a uniform and rigid background of opposite charge In the classical limit, the OCP is characterized by a single parameter I" = (Ze)2 /r-kB T , where ~-= ( 3 V/ 47rN) 1/3 IS the mean mterpartIcle distance and T as the temperature Recent Monte Carlo calculations by Slattery et al [2] shows that the classical OCP exhibits a so l i d - liquid transition at I ~ ~ 178 In the ground state limit, the OCP is characterized by the parameter r s = ?/ao, where a 0 = h2/rnZ2e2 IS the Bohr radius associated with the Fermi particle Ceperley and Alder investi- gated the ground state propert ies of the fermion OCP on the basis of the Green function Monte Carlo method [4] and found a sohd-ferromagnet ic-hquid transition at r s ~ 100 and a ferromagnet ic-hquid-paramagnet ic-

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hquld transition at r s ~ 75 Mochkovltch and Hansen drew a phase diagram of

the OCP by interpolating between the known classical and ground state limits with the help of a generahzed Llndemann criterion [5] (The data used by these authors for the classical and ground state hmlts are the ones given in refs [1] and [3] ) The obtained sohd -hqu ld phase boundary exhibits a closed loop In the dens i ty - tempera ture plane On the other hand, one o f the present authors studied the lattice gas ver- sion of the fermlon OCP [6] and found that the melting curve has an additional structure in the quantum region due to its spin freedom it shows a melting-density maximum It IS the purpose of the present letter to examine the sohd-hquId as well as the ferromag- ne t lc -paramagnet lc phase boundaries in the quantum region by making use of the known ground state and low-temperature properties In this procedure, we take account of the spin freedom of fermxons which was missed in the previous approaches based on the Lmdemann criterion [1,5] Then we draw a d e n s i t y - temperature phase diagram of the fermlon OCP by smoothly interpolating between the phase boundary obtained in the quantum region and the known phase boundary m the classical limit The result is shown in fig 2

We first derive a modified Clauslus-Clapeyron relation for the OCP In the usual case, this relation

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IS given by dPc/dT c = ASIAV, where P is the pressure, and AS and 2i V are the entropy and volume differences, respectively In the case of the OCP, however, this relation does not hold in its original form and must be modified It is because the densities of the two coexisting phases must be equal in the OCP due to the existence of uniform and rigid background (Note that coexistence of the two phases with different densities violate the charge neutrality condition microscopical- ly ) As a result, the "thermal pressures" [7] of the two coexisting phases, which are defined as the volume derivative of the free energy under the condition of charge neutrality, are not necessarily equal (Another example of the discontinuous pressure was given by Milton and Fisher in an exactly soluble one-dimensional model [8] ) By equating the Helmholtz free energies of the two coexisting phases along the phase boundary, the modified form of the Clauslus- Clapeyron relation can be obtained as

dgc/dT c = - AS/AP (1)

We rewrite this relation by using the reduced quantities as

d(l/rs)c _ z2~s (2)

dtc 47rr4Zkp '

where t, s and p are the dimensionless quantities de- fin e d by t = k B T/Ry, s = S/(Nk B) and p = el (Ry/a03 ) with Ry = m(Ze)4/(2h 2) If the low-temperature expressions of the entropy and the pressure are available in each phase, one can draw a phase boundary by integrating the Clauslus-Clapeyron relation (2) with the known ground state data as an initial condition (Hereafter, we shall refer to the thermalpressure simply as pressure )

Next we write down the low-temperature expressions o f the entropy and the pressure for each phase In the quantum region, the entropy of the solid phase has the following form

Ss(rs, t) ,~ In 2 + O(t3) , (3)

where In 2 is the contribution o f the spin freedom The second term is the contribution o f the lattice vibration winch will be neglected in the following as a small quantity This expression is valid as far as the temperature is in the range k B Tex "~ k B T '~ ~Wp, where kBTex is the exchange energy associated with the Wigner sohd and COp is the plasma frequency If

we estimate kBTex by using the expression derwed by Carr [9], it is of order 10 -9 Ry at r s = 100 The energy of the plasma frequency IS estimated ~s fiWp ~ 3 X 10 -3 Ry at r s = 100 In this temperature range the pressure is almost temperature independent,

Ps(rs, t) ~ Ps(rs, 0) (4)

For the liquid phase, we consider only the two states, i e , unpolarlzed (paramagnetlc) hquld and completely polarized (ferromagnetic) hquld When the temperature is lower than the Fermi temperature T F (kBT F is about equal to kBT F ~ 4 X 10 -4 Ry at r s = 100), the leading term of the entropy is proportional to the temperature and can be written as

Sp ~ Otp (m*/m)t, O~p = 17r 2 (4/97r)2/3 ' (5)

1 2 Sf ~ af(m*/m)t , c~f = ~ (2/9rr) 2/3 , (6)

for the paramagnetlc and the ferromagnetic phases, respectively, where m* is the effective mass which appears as a parameter In the Fermi liquid theory In writing (6), we have assumed that the main contribution to the low-temperature specific heat of the completely polarized hquld comes from the single particle excitation without spin-flip The contribution of the spin wave is neglected because it gives a term of order t 3/2 The contribution of the plasmon excitation is also neglected because the plasmon has a large gap in three dimensions Within these approximations the pressures of the paramagnetlc and the ferromagnetic liquids are given by

pp(rs, t) ~ pp(r s, 0) + ap(m*/m)r3t 2 , (7)

pf(r s, t) ~ pf(r s, 0) + i f ( m * / m ) r 3 t 2 (8)

These expressions for the hquld phases involve one undetermined parameter m*/m In the non-interact- ing limit, this value goes to unity As far as r s remains small, this quantity can be reliably estimated by using the RPA [10] which shows that m*/m first decreases when r s is increased from zero but it begins to increase when r s exceeds some critical value o f order unity Some authors pointed out, however, that if the short- range correlation effect IS taken into account the value of m*/m is seriously suppressed in the metallic density region as compared with the RPA result [11] At present, It seems very difficult to give a reliable estimation of m*/m from a microscopic calculation in

274


such a low-density region as r s ~ 100 Therefore, in the present letter, we leave m * / m as an adjustable parameter and later determine it rather empmcal ly m such a way that the obtained phase boundary In the quantum region is joined smoothly to the classical phase boundary which has been accurately determined by Monte Carlo simulation

For the expression of the ground state energy of the solid phase, we use the result of the anharmonlc crystal expansion [3,8,12] which has been given by

Es/RY ~ - 1 79185 r s 1 + 2 65724 r s 3/2

- 0 7 3 r s 2 (9)

The ground state energies of the liquid phases were evaluated by Ceperley and Alder [4] on the basis of the Green functzon Monte Carlo method In the present calculation, we use the fitting formula proposed by Vosko et al [13] If these expressions are used for the ground state energies, we find that the system exhibits a sohd-ferromagnet lc-hquld transition at r s 93 and a ferromagnetlc- l lquid-paramagnetic-hquld transition at r s ~ 80 The ground state pressure of each phase can be obtained from the volume derivative of the ground state energy The r s dependence o f the ground state pressure is shown in flg 1 The pressure is negative In this temperature range and the obtained

4grip

-1 35

-140

-I 4E

parQ [ ~ l q ~ferro hq sol

r s

6b 7b 8b 9b 1do 1,0 Fig 1 The r s dependence of the ground state pressure

pressure curve shows discontinuity at each transition point With these ground state data as initial conditions, the sohd -hqu ld as well as the fer romagnet ic-paramagnetic phase boundaries can be drawn by numeri- cally Integrating the Clauslus-Clapeyron relation (2)

At this stage, we present some physical arguments about the expected phenomena In the quantum region. the entropy of the solid is larger than that of the liquid when the temperature is lower than some critical temperature T1, whereas above T 1 it becomes smaller than that of the liquid This observation together with the Clauslus-Clapeyron relation Indicates that the melting curve of the OCP has a meltIng-dens]ty maximum at T = T 1 (Note that the pressure of the solid is always larger than that of the hquzd in this temperature range ) In fact, as has been pointed out In ref [6] , this melting-density maxtmum phenomenon has the same origin as the well-known melting-density m t m m u m phenomenon observed in helium 3 [14] The melting-density minimum phenomenon in 3He appears as a melting- density maximum phenomenon in the OCP because the relative location of the solid and the liquid phases IS reversed in the phase diagram When the temperature is further raised (or the density is further lowered), the pressure difference between the solid and the liquid changes its sign at a certain temperature T 2 This IS caused by the second term in the r h s of eq (7) which gives a positive contr ibution to the liquid-phase pressure It means that the melting curve shows a melting- temperature maximum at T = T 2 and the phase boundary has a loop structure When the density or the temperature is further lowered, the phase boundary finally enters the region where our approximations are no longer valid

As was mentioned previously, we complete the phase diagram by matching the phase boundary obtained m the quantum region with the known class]cal phase boundary which is a straight line in the 1/r s - t

plane The best estimation o f m * / m determined m such a way is m*/m ~ 1 35 We have used this same value both for the polarized and the unpolarlzed phases and the r s dependence o f m * / m is neglected in the temperature range of interest Our final result is sum- marized in the phase diagram fig 2 The melting-density maxLmum and the melting-temperature maximum points are located at t ~ 5 9 × 10 - 5 , r s ~ 81 and t 10 3 × 1 0 - 5 , r s ~ 95, respectively Our estimation of the maximum temperature of the stable solid is almost

275


15

lO0/rs

para hq

, 0

so[

0 5 / / ~

para l ~q

f

5 10 T(lO-SRy)

Fig 2 The phase diagram of the three-dimensional fermlon one-component plasma The abscassa is the temperature m units of Rydberg The straight hne is the purely classmal phase boundary The dash-dotted hne represents the hne given by the relation T = T F

the same as the estlmanon by Mochkovlch and Hansen

[5] The maximum density of the stable solid is about

50% larger than the melting density at zero temperature

and gives a considerable effect on the phase diagram

This phenomenon has been missed in the previous

phase diagrams of the fermlon OCP

Finally it is worth pointing out that a qualitatively

similar phase diagram including the melting-density

maximum phenomenon is expected also in the two-

dimensional OCP, except that the polarized state is

unstable at any finite temperature in two dimensions

and the paramagnetlc-ferromagnetlc phase transmon

takes place only at zero temperature

References

[1] E L Pollock and J P Hansen, Phys Rev A8 (1973) 3110

[2] W L Slattery, G D Doolen and H E DeWltt, Phys Rev A21 (1980) 2087,A26 (1982) 2255

[3] D Ceperley, Phys Rev B18 (1978) 3126 [4] D M Ceperley and B J Alder, Phys Rev Lett 45 (1980)

566 [5] R Mochkowtchand J P Hansen, Phys Lett 73A (1979)

35 [6] H Kawamura, Prog Theor Phys 66 (1981)421 [7] Ph Choquard, P Favre and Ch Gruber, J Stat Phys

23 (1980) 405 [8] G W Milton and M E Fisher, J Stat Phys 32 (1983)

413 [9] WJ Carr, Phys Rev 122 (1961) 1437

[10] M Gell-Mann, Phys Rev 106 (1957) 369, D E DuBms, Ann Phys (NY) 7 (1959) 174, 8 (1959) 29

[11] AW Overhauser, Phys Rev B3 (1971)1888, H Suehlro, K Awa and H Yasuhara, Sohd State Commun 47 (1983) 641

[12] R A Coldwell-Horsfall and A A Maradudm, J Math Phys 1 (1960) 395, W J Carr, R A Coldwell-Horsfall and A E Fern, Phys Rev 124 (1961) 747

[13] SH Vosko, L Wxlkand M Nusatr, Can J Phys 58 (1980) 1200

[14] S B Tncky, W P Ktrk and E D Adams, Rev Mod Phys 44 (1972) 668

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a note on the phase diagram of the fermion one-component plasma

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