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Adiabatic equation of state and ionization equilibrium of dense plasma
ELSEVIER
Physica A 241 (1997) 719-728
Adiabatic equation of state and ionization equilibrium of dense plasma D i e t e r B e u l e a, W e r n e r E b e l i n g a,., A n d r e a s F r r s t e r b, 1 a Institut fiir Physik, Humboldt-Universitgit zu Berlin, Invalidenstrafle 110, D-lOll5 Berlin, Germany b Department of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan Received 2 December 1996
Abstract
We consider the adiabatic equation of state for partially ionized non-ideal plasma. Plasma isentropes are calculated with the Pad6 technique in the chemical picture. The interplay of ionization/dissociation equilibrium and non-ideality is investigated.
Keywords: Adiabatic change of state; Equation of state; Ionization equilibrium; Dense plasma
O. Introduction
Adiabatic processes are thermodynamic processes which are typically characterised by a relatively fast change of state so that the system undergoing the change does not exchange heat with its surroundings. Reversible adiabatic processes are also isentropic, i.e., they take place with no change in entropy. Adiabatic processes play an important role in plasma physics and astrophysics as well as in technology. The spectra of examples includes sound waves, expansion of plasma or gas into vacuum, ballistic-compressor and shock-wave experiments and molecular-beam technology [1]. As a consequence of hydrodynamic stability, the radial structure of the stars as well as of fluid planets like Jupiter or Saturn follows an adiabat [2]. Finally, we mention the three adiabatic coefficients used in astrophysics as well as their applications in planetary and helioseismology [3]. For many applications of an adiabatic equation of state, the ideal-gas model is insufficient. First, one has to care for the ionization and dissociation equilibrium which determines the number of free electrons, ions of different charges, atoms, and molecules in * Corresponding author. Fax: 20937638; e-maih [email protected]. I Permanent address: Institut fiir Physik, Humboldt-Universit~it zu Berlin, InvalidenstraBe 110, D-10115 Berlin, Germany. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 1 68-4
720
D. Beule et al./Physica A 241 (1997) 71~728
partially ionized plasma. Second, at high densities and/or low temperatures the Coulomb interaction between the charged particles yield negative contributions to, e.g., the pressure and the entropy and modifies in this way the equation of state. Additionally, the relatively light electrons may degenerate and quantum exchange and correlation effects appear. Such plasma are usually called non-ideal or strongly coupled. Third, there is a strong interplay between the Coulomb coupling and quantum effects on one hand and the ionization equilibrium on the other. The more the ionization is developed and charged particles dominate the plasma composition, the stronger are the Coulomb effects in the plasma. On the other hand, the Coulomb interaction shifts the ionization equilibrium to higher ionization states; this effect is also known as pressure ionization. There exist a lot of models for the ionization equilibrium and the equation of states of dense plasma. Usually, they are expressed in dependence on the particle numbers, the volume, and on the temperature. We would like to construct an equation of state which is given in dependence of the entropy instead of the temperature and study the behaviour of the ionization equilibrium and of the equation of state along the adiabates. We will first consider the influence of ionization on the ideal plasma isentropes, then introduce Coulomb forces in the A / 8 approximation that still allows some analytical treatment. Within this approximation we investigate the interference of the direct change of isentropes due to interaction and the indirect change through the change of ionization equilibrium. Finally, we will give an adiabatic equation of state on the basis of a Pad6 approximation for the free energy.
1. Chemical picture We consider hydrogen plasma in the chemical picture [4]. The elementary constituents are free electrons e, bare nuclei i, atoms a, and molecules m. Ne, Ni, Nu, and Arm denote the particle numbers and ne, hi, na, and nm denote the particle densities of these constituents. We do not consider H - and H + since their concentration are found to be negligible (<10 -3) even at low temperatures, T < |04 K [5]. For a given total density of nuclei n = ni + na + 2 nm, the detailed composition can be described by the fraction of atoms s0 na/n and the fraction of molecules tim = 2nm/n contained in the plasma. These quantities are related to the degree of ionization o~ = Ili/n via the relation c~ = 1 - ~0 tim. Any plasma model can be specified by the dependence of a thermodynamic potential on the detailed composition. For given temperature, volume, and particle densities the free energy F(T, V, ne, hi, n~, n,,) is the appropriate thermodynamic potential. Taking electro-neutrality (he ni) into account one particle density can be eliminated and the free energy can be expressed in terms of the degree of ionization ~ and the fraction of molecules tim, i.e., F(T, V, n, ct, fin). Calculation of the equation of state (EOS) involves a minimisation procedure with respect to the detailed composition under the constraints of electro-neutrality and given =
-
=
D. Beule et al./Physica A 241 (1997) 719-728
721
total density n. This constrained minimisation can be expressed by: 6F 6~m
-
~F
0,
-- =0. 6~
(l)
Other thermodynamic properties can be obtained by the usual thermodynamic relations.
2. Plasma isentropes and EOS
2. I. Ideal plasma isentropes First, we consider an ideal model plasma containing only ions, non-degenerate electrons, and atoms with one bound state. The reduced entropy s = S/NkB is given by
Sid(~)~_ ~(l - ~ ) - ~ln (°~FIA3~ -o~ln (°~nA3 ~ \ 9e /I \ 9i /I (2) where ge = 2, gi 2, and ga 4 denote the degeneracy of electrons, ions, and atoms, respectively. A,, denotes the thermal de Broglie wavelength of the different particles species v -- e, i, a, =
Av- ~
=
h .
(3)
Obviously the entropy (2) as well as pressure p = (1 + ~)nkBT depend upon the ionization equilibrium. Ionization equilibrium is determined by the Saha equation 1-~
°~2 -- nASa(T),
(4)
Within this model, the atomic partition function is given by a(T) = exp ( + ) ,
(5)
where I > 0 denotes the ionization energy of the atoms. The adiabatic EOS p(s, T) can be obtained from p(n, T) and s(n, T) by solving s(n, T) with respect to T and inserting the result into p(n, T). One obtains:
p(s,T) = nkB(1 + a)[exp(5/2(1 + ~ ) ~ s ) ] -2/3('+~) L I-I,.[n,,a~/gv] n'/" J '
(6)
where ,4v =(h2/2~mvks) 1/2. In the following section we consider the contribution of the Coulomb interaction.
D. Beule et al./Physica A 241 (1997) 719-728
722
2.2. Adiabatic E O S in the A/8 model
In the A/8 approximation [4,10,11], quantum effects are expressed by just one characteristic length, the thermal de Broglie wavelength A of a particle with reduced mass m = memi/(me-k mi). It leads to a Debye-Hfickel-like formula where the effective diameter of the charged particles is given by A/8. This approximation is valid if the plasma is non-degenerate, i.e., neA~ "~ 1. The A/8 theory starts with an approximate expression for the combined chemical potential [4] ~=~li-~--]Ae=~lid--~O
1 -- - - f f / £ ' q ' - ' ' '
,
(7)
where x denotes the inverse Debye radius, 1 r
2~n~e 2 -
r
.
(8)
In the limit T ~> 1/kB, expression (7) is asymptotically exact up to the order O(n) in the density. Now we construct the simplest Pad6 approximation consistent with (7) in the form e2
K
(9)
# = #ia - 4~ce---o1 + ~A/8 "
From thermodynamic consistency follows that all other thermodynamic functions also have a Debye-Hiickel-like structure. The free energy in this model is given by F = Fia - k e T V ~ - ~ z
(10)
,
with r being z(x) = ~-5 ln(l + x ) - x +
.
(11)
This leads to an reduced entropy K3
s = sia - 2~-~n(Z(x) + 2z'(x)).
(12)
The equilibrium composition can be found by minimisation of (10) with respect to the degree of ionization. This leads to a non-ideal Saha equation 1 - ~ _ nA~ a ( T ) h(~, T ) 0~2
(13)
i.e., due to non-ideality the ionization equilibrium is shifted by a factor to the ideal mass-action constant. This factor h(~,T)=exp\
ksT
J
(14)
D. Beule et al./Physica A 241 (1997) 719-728
723
depends exponentially on the interaction contribution to the combined chemical potential given in (9). It can be interpreted as a lowering of the ionization energy I by AI h(a, T) = exp ( A I/kB T).
( 15 )
In general, non-ideality leads to an increased degree of ionization ct = aid + aint, that cannot be calculated analytically. For high temperatures, the degree of ionization a approaches unity and the ideal case is recovered. At low temperatures, when all charged particles are bound as atoms the interaction contribution of the A/8 model disappears and an ideal atomic-gas model remains. As compared to the ideal case, the pressure P
nkB T
--
Pid + Pint
nk8 T
-- 1 +
~3/2/£3 aid Av ain t -- - - ~ ) ( X ) ,
~ b ( x ) = ~3 [ l + X - x - -'2 1 n ( l + x )
247tn
1
(16)
(17)
is altered in two ways. First, by the change of the ionization equilibrium aint which lead to an increase in pressure and, second, through a non-ideality contribution that generally decreases the pressure. For the entropy one may write K3 S = Sid(aid -~- aint) -- 2 - ~ n ( T ( x )
~- 2qs/(x)) ,
(18)
where sid(cQ is given by (2). In the region where the A/8 approximation is valid, i.e., neA 3 ~ 1 and F < 1, the entropy is increased due to the larger degree of ionization and reduced by the interaction term.
2.3. Padk approximation #t the chemical picture The A/8 approximation is asymptotically correct in the first order with respect to the dimensionless quantum parameter xA. Even at medium densities it yields at least a qualitatively correct picture. A more appropriate description of dense plasma which includes all first-order quantum corrections requires to go beyond the A/8 model. Pad6 approximations of the interaction contributions of thermodynamic functions [ 12] within the chemical picture (PACH) are more complicated than the A/8 model but are still given by analytical formulae. Therefore, the PACH approach (as well as the A/8 model) allows fast access to different thermodynamic properties. An advantage of the PACH model is its validity in wide ranges of density and temperature. The free energy of the plasma in the PACH model is represented by
F = F~a + G + F~ + F,e + Fh~.
(19)
D. Beuleet al./PhysicaA 241 (1997)719-728
724
~a denotes the ideal contribution of the ions, atoms, molecules, and electrons as derived from Boltzmann and Fermi-Dirac statistics, respectively. The excited states of the atoms are accounted for by a Planck-Larkin partition sum o'(T) = Z
m2 exp
-
(20)
m
k~Tm2
"
We also include the additional degrees of freedom of the molecules into the ideal part of the free energy. Our study is mainly devoted to the plasma region where molecules contribute only relatively small corrections. For this reason we shall use only simple approximations for the molecular contributions. For a more advanced treatment of the molecules we refer, e.g., to Saumon and Chabrier [13]. The rotational partition sum of the molecules gives rise to a term approximated by
Frot=gmkBT(ln zr T
Tr 3T
1 90
1~)2 )
,
T~ ----8 5 K ,
(2l)
if T~ ~ T [8]. The contribution of the vibrational degrees of freedom of the molecules is given by [8]: Fvib = N m k B T ( l n ( 1
- exp(-T~,/T))),
Tv = 6210K.
(22)
The interaction contribution F~ and Fi account for the electronic and ionic subsystem, while Fie describes the electron-ion screening in terms of a Pad6 approximation [7]. The Pad6 approximations used here were discussed in detail in [12]. In the lowest order with respect to the parameter KA, they are consistent with the A / 8 approximation and may be considered as a generalization of the later. It has been shown [15] that the Pad6 approximations are in good agreement with recent quantum Monte-Carlo calculations [16]. Finally, Fh,. denotes the hard-sphere contribution, that account for the short-range repulsion due to bound shell electrons.
3. Results and discussion
Within the Pad6 approximation in the chemical picture one has to minimize the free energy with respect to the abundances of the various free and composite particles (2) in order to describe thermodynamic equilibrium for given temperature and density. The dissociation equilibrium of the molecules is coupled to the ionization equilibrium via the relation n i --}- na + 2nm = n. Therefore, the degree of ionization ~ and the density of molecules have to be determined simultaneously. We achieve the minimisation of the free energy by a simulated annealing procedure [14] that was originally developed for multi-component systems. After performing this procedure on a dense grid in the density-temperature plane, one can calculate different thermodynamic properties numerically. First, we calculated the isentropes s = const, for hydrogen plasma. They are plotted in Fig. 1 together with lines of equal ionization. In general, the difference between
D. Beule et al. IPhysica A 241 (1997) 719-728
,"
725
7
1'E 1021 ,
c-
/,
,,
//+ '
102°
,..,. , ,, ,
,/,/ /
, /,
,6;,1 " /
/)"//
.
.
,// /,
/ ,1 /,, ,,; 10
~9
'
'
z
'
~
/
. . . . .
.
.
.... s=21 (PACH) -- - s=21 (ideal) . s=24 (PACH) s=21 (ideal)
...... o~=0.9
,
-
~=0.1
'
10000
100000 T [K]
Fig. 1. Plasma isentropes and degree of ionization in non-ideal plasma. The plasma isentropes as obtained from PACH and in ideal plasma for three differentvalues of reduced entropys = 18, 21, 24. The short dashed lines limit the area where ionization raises from the weakly ionized case ~ =0.1 to the nearly completely ionized case ~ = 0.9.
1.0
0.8 ///
," e"
//
0.6
0.4
0.2
6' / / ,/~/'// (
i
s=21 (PACH) S=21 iideal)"
-d
,
0"~000
-e
,
a
,
l . . ./ . . ~ s=24 / (PACH)
. . . .
t
100000
T [K] Fig. 2. The degree of ionization along plasma isentropes s = 21,24 over temperature for ideal (thin lines) and non-ideal plasma (thick lines).
ideal and non-ideal isentropes increases as density increases and temperature decreases but the interaction effects vanish when the ions and electrons are bound into atoms and molecules. Therefore, the ideal and non-ideal isentropes that nearly coincide at high temperatures merge again after they have crossed the area of partial ionization. Due to the decrease in particle number the adiabates become steeper as atoms and molecule formation takes place for decreasing temperature. The degree of ionization along two of the adiabates is shown in Fig. 2. Lower values of reduced entropy correspond to higher densities or lower temperatures and therefore to larger non-ideality effects and thus to greater differences in the degree of ionization.
D. Beule et al. IPhysica A 241 (1997) 719-728
726
I0 s •
///
fp
104 10 3 .-"
13..
,9//
~//i"'"'" .-'" J ,,/"/'/;'; / /? J J ,/~ j t / / //f / / / /t
" ~ 10 2
10 ~ 10 o /
,,,t /
/'
,
/"
-.... -.....
. . . . . . .
10000
-
S='18(PACH) S=18(ideal) s=21 (PACH) s=21 (ideal) s=24 (PACH) s=24(ideal) fixed ionization
i
,
,
L
100000 T [K]
Fig. 3. Adiabatic equation of state for reduced entropy s--18, 21,24 over temperature. We compare the pressure of an ideal plasma (thin lines) with the pressure obtained from PACH (thick lines). For comparison we also plot an adiabtic EOS for a fixed degree of ionization as obtained from Eq. (23) using Vad = 0.4 (short dashed line).
There are points where the non-ideality contributions to entropy (cf. Fig. 1) cancel each other and the ideal and non-ideal isentropes cross each other. This points are found in the region of highly ionized plasma. Beyond this point the degree of ionization does not differ much for the ideal and the non-ideal case and positive contribution to the entropy due to increased degree of ionization are overcompensated by the direct interaction contributions. Once the isentropes are known, it is easy to give the adiabatic (isentropic) equation of state for different values of entropy, cf. Fig. 3. For high and low temperatures, the ideal and non-ideal pressure almost coincide for a broad range of entropy. For intermediate temperatures non-ideality has significant influence on the adiabatic EOS. The deviation of ideal and non-ideal behaviour seen in Fig. 3 for these temperatures corresponds to deviations in the degree of ionization due to non-ideality (cf. Fig. 2). As for entropy, there exist points in the area of highly ionized plasma where non-ideality contributions cancel each other. In order to demonstrate the effect that ionization has on plasma isentropes, we also plot the adiabatic EOS of an ideal system where no ionization processes take place. For such systems the adiabatic EOS can be obtained from integrating (dlnT~
(23)
voa -- \ d-W. p ) s .
For Boltzmann particles with fixed degree of ionization one finds Vad = 0.4 [2]. Therefore, the adiabatic EOS for such a system is given by a set of parallel straight lines in a log(p) over log(T) plot. One of these lines is shown in Fig. 3. For high temperatures (i.e. a high degree of ionization), the PACH isentropes fullfil Vaa ~ 0.4. But in the area where the formation of bound states takes place the adiabatic EOS deviates significantly from this value. For decreasing temperature, the pressure is
727
D. Beule et al./Physica A 241 (1997) 719-728 . . . . . . . .
i
. . . . . . . .
i
. . . . . . . .
i
'
'
1.00
1
~- 0.90
Q.
0.80
'/
....
.....
s=21
iO
~
.....
ie
.....
"
i&
'
n [cm -3]
Fig. 4. Comparisonof the pressure along ideal and non-ideal adiabates (s = 18,21,24) over density. We plot the quotient of the pressure as obtained by PACH PPACH(S)and from an ideal model plasma Pid(S) for different values of reduced entropy s = 18, 21,24.
decreasing much faster than in a system with fixed degree of ionization. For s = 18 one can even see a second steep decrease in pressure at low temperature where molecule formation sets in. The differences between ideal and non-ideal adiabatic EOS can be seen more clearly in Fig. 4. We plot the quotient of the non-ideal PeAcH(s) and ideal pressure pia(s) along different adiabates over the density. It has to be emphasized that the pressures compared in Fig. 4 belong to different temperatures because ideal and non-ideal adiabates take different paths through the density-temperature plane. For lower values of entropy the difference in pressure along ideal and non-ideal adiabates increase and the minimum of the quotient is shifted towards higher densities. The interaction contributions to the entropy are smaller than other thermodynamic properties like free energy or pressure. This cancellation can also be found in the limiting laws. In contrast to other thermodynamic properties, even in the case of strong coupling, entropy does not deviate much from the ideal behaviour. Nevertheless, the entropy is influenced significantly by the number of particles, i.e., the degree of ionization. The degree of ionization is very sensitive to non-ideality effects. Keeping in mind that Fig. 1 and Fig. 3 are plotted using a logarithmic scale, one realizes that ideal and non-ideal isentropes take quite different paths through the density-temperature plane.
4. Conclusions Bound-state formation significantly alters the adiabatic EOS. In order to give an appropriate description of the adiabatic equation of state one has to give a good description of the ionization equilibrium first. We achieved this by using the Pad6
728
D. Beule et al./Physica A 241 (1997) 719-728
approximation in the chemical picture. As demonstrated within the A / 8 model there is an interference o f the direct change o f isentropes due to interaction contributions and the indirect change through the change o f ionization equilibrium. Deviation o f ideal and non-ideal adiabatic EOS is mainly caused by different degrees o f ionization along ideal and non-ideal plasma isentropes.
Acknowledgements We would like to thank Tino Fuchs for his contributions on the adiabatic equation o f state o f the electron gas. One o f us (A.F.) would like to express his sincere gratitude to Prof. Masuo Suzuki and his group at the University o f Tokyo for their warm hospitality during a research stay in 1996.
References [1] V.E. Fortov, 1.T. Yakubov, Physics of Nonideal Plasmas (in Russian), Inst. Chem. Phys. Chemogolovka, 1984, Hemisphere, New York (English Transl.) 1990. [2l R. Kippenhahn, A. Weigert, Stellar Structure and Evolution, Springer, Berlin, 1994. [3] J. Christenssen-Dalsgaard, W. Dfippen, Astron. Astrophys. Rev. 4 (1992) 267. [4] W. Ebeling, W.D. Kraeft, D. Kremp, Theory of Bound States and lonisation Equilibrium in Plasmas and Solids, Akademie-Verlag, Berlin, 1976; Extended Russ. translation, Mir, Moscow, 1979. [5] G. Chabrier, in: IAU Colloquium 147, The Equation of State in Astrophysics, G. Chabrier, E. Schatzman (Eds.), Cambridge University Press, Cambridge, 1994, p. 287. [6] P. Debye, E. Hfickel, Phys. Z. 24 (1923) 185. [7] W. Ebeling, Contrib. Plasma Phys. 30 (1990) 553. [8] R. Fowler, E.A. Guggenheim, Statistical Thermodynamics, Cambridge University Press, Cambridge, 1952. [9] R. Zimmermann, Many-Particle Theory of Highly Excited Semi-Conductors, Teubner, Leipzig, 1988. [10] G.E. Norman, A.N. Starostin, Teplofiz. Vys. Temp. 8 (1970) 413. [11] G.P. Bartsch, W. Ebeling, Beitr. Plasmaphysik 11 (1971) 393, 15 (1975) 25. [12] W. Ebeling, A. F6rster, V.E. Fortov, V.K. Gryaznov, A.Ya. Polishchuk, Thermophysical Properties of Hot Dense Plasmas, Teubner, Stuttgart, 1991. [13] D. Saumon, G. Chabrier, Phys. Rev. A 44 (1991) 5122. [14] S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Science 220 (1983) 671. [15] W. Stolzmann, T. Blrcker, Phys. Lett. A 221 99 (1996); Astron. Astrophys. 314 (1996) 1024. [16] C. Pierleoni, D.M. Ceperley, B. Bemu, W.R. Margo, Phys. Rev. Lett. 73 (1994) 2145.
Physica A 241 (1997) 719-728
Adiabatic equation of state and ionization equilibrium of dense plasma D i e t e r B e u l e a, W e r n e r E b e l i n g a,., A n d r e a s F r r s t e r b, 1 a Institut fiir Physik, Humboldt-Universitgit zu Berlin, Invalidenstrafle 110, D-lOll5 Berlin, Germany b Department of Physics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan Received 2 December 1996
Abstract
We consider the adiabatic equation of state for partially ionized non-ideal plasma. Plasma isentropes are calculated with the Pad6 technique in the chemical picture. The interplay of ionization/dissociation equilibrium and non-ideality is investigated.
Keywords: Adiabatic change of state; Equation of state; Ionization equilibrium; Dense plasma
O. Introduction
Adiabatic processes are thermodynamic processes which are typically characterised by a relatively fast change of state so that the system undergoing the change does not exchange heat with its surroundings. Reversible adiabatic processes are also isentropic, i.e., they take place with no change in entropy. Adiabatic processes play an important role in plasma physics and astrophysics as well as in technology. The spectra of examples includes sound waves, expansion of plasma or gas into vacuum, ballistic-compressor and shock-wave experiments and molecular-beam technology [1]. As a consequence of hydrodynamic stability, the radial structure of the stars as well as of fluid planets like Jupiter or Saturn follows an adiabat [2]. Finally, we mention the three adiabatic coefficients used in astrophysics as well as their applications in planetary and helioseismology [3]. For many applications of an adiabatic equation of state, the ideal-gas model is insufficient. First, one has to care for the ionization and dissociation equilibrium which determines the number of free electrons, ions of different charges, atoms, and molecules in * Corresponding author. Fax: 20937638; e-maih [email protected]. I Permanent address: Institut fiir Physik, Humboldt-Universit~it zu Berlin, InvalidenstraBe 110, D-10115 Berlin, Germany. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PH S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 1 68-4
720
D. Beule et al./Physica A 241 (1997) 71~728
partially ionized plasma. Second, at high densities and/or low temperatures the Coulomb interaction between the charged particles yield negative contributions to, e.g., the pressure and the entropy and modifies in this way the equation of state. Additionally, the relatively light electrons may degenerate and quantum exchange and correlation effects appear. Such plasma are usually called non-ideal or strongly coupled. Third, there is a strong interplay between the Coulomb coupling and quantum effects on one hand and the ionization equilibrium on the other. The more the ionization is developed and charged particles dominate the plasma composition, the stronger are the Coulomb effects in the plasma. On the other hand, the Coulomb interaction shifts the ionization equilibrium to higher ionization states; this effect is also known as pressure ionization. There exist a lot of models for the ionization equilibrium and the equation of states of dense plasma. Usually, they are expressed in dependence on the particle numbers, the volume, and on the temperature. We would like to construct an equation of state which is given in dependence of the entropy instead of the temperature and study the behaviour of the ionization equilibrium and of the equation of state along the adiabates. We will first consider the influence of ionization on the ideal plasma isentropes, then introduce Coulomb forces in the A / 8 approximation that still allows some analytical treatment. Within this approximation we investigate the interference of the direct change of isentropes due to interaction and the indirect change through the change of ionization equilibrium. Finally, we will give an adiabatic equation of state on the basis of a Pad6 approximation for the free energy.
1. Chemical picture We consider hydrogen plasma in the chemical picture [4]. The elementary constituents are free electrons e, bare nuclei i, atoms a, and molecules m. Ne, Ni, Nu, and Arm denote the particle numbers and ne, hi, na, and nm denote the particle densities of these constituents. We do not consider H - and H + since their concentration are found to be negligible (<10 -3) even at low temperatures, T < |04 K [5]. For a given total density of nuclei n = ni + na + 2 nm, the detailed composition can be described by the fraction of atoms s0 na/n and the fraction of molecules tim = 2nm/n contained in the plasma. These quantities are related to the degree of ionization o~ = Ili/n via the relation c~ = 1 - ~0 tim. Any plasma model can be specified by the dependence of a thermodynamic potential on the detailed composition. For given temperature, volume, and particle densities the free energy F(T, V, ne, hi, n~, n,,) is the appropriate thermodynamic potential. Taking electro-neutrality (he ni) into account one particle density can be eliminated and the free energy can be expressed in terms of the degree of ionization ~ and the fraction of molecules tim, i.e., F(T, V, n, ct, fin). Calculation of the equation of state (EOS) involves a minimisation procedure with respect to the detailed composition under the constraints of electro-neutrality and given =
-
=
D. Beule et al./Physica A 241 (1997) 719-728
721
total density n. This constrained minimisation can be expressed by: 6F 6~m
-
~F
0,
-- =0. 6~
(l)
Other thermodynamic properties can be obtained by the usual thermodynamic relations.
2. Plasma isentropes and EOS
2. I. Ideal plasma isentropes First, we consider an ideal model plasma containing only ions, non-degenerate electrons, and atoms with one bound state. The reduced entropy s = S/NkB is given by
Sid(~)~_ ~(l - ~ ) - ~ln (°~FIA3~ -o~ln (°~nA3 ~ \ 9e /I \ 9i /I (2) where ge = 2, gi 2, and ga 4 denote the degeneracy of electrons, ions, and atoms, respectively. A,, denotes the thermal de Broglie wavelength of the different particles species v -- e, i, a, =
Av- ~
=
h .
(3)
Obviously the entropy (2) as well as pressure p = (1 + ~)nkBT depend upon the ionization equilibrium. Ionization equilibrium is determined by the Saha equation 1-~
°~2 -- nASa(T),
(4)
Within this model, the atomic partition function is given by a(T) = exp ( + ) ,
(5)
where I > 0 denotes the ionization energy of the atoms. The adiabatic EOS p(s, T) can be obtained from p(n, T) and s(n, T) by solving s(n, T) with respect to T and inserting the result into p(n, T). One obtains:
p(s,T) = nkB(1 + a)[exp(5/2(1 + ~ ) ~ s ) ] -2/3('+~) L I-I,.[n,,a~/gv] n'/" J '
(6)
where ,4v =(h2/2~mvks) 1/2. In the following section we consider the contribution of the Coulomb interaction.
D. Beule et al./Physica A 241 (1997) 719-728
722
2.2. Adiabatic E O S in the A/8 model
In the A/8 approximation [4,10,11], quantum effects are expressed by just one characteristic length, the thermal de Broglie wavelength A of a particle with reduced mass m = memi/(me-k mi). It leads to a Debye-Hfickel-like formula where the effective diameter of the charged particles is given by A/8. This approximation is valid if the plasma is non-degenerate, i.e., neA~ "~ 1. The A/8 theory starts with an approximate expression for the combined chemical potential [4] ~=~li-~--]Ae=~lid--~O
1 -- - - f f / £ ' q ' - ' ' '
,
(7)
where x denotes the inverse Debye radius, 1 r
2~n~e 2 -
r
.
(8)
In the limit T ~> 1/kB, expression (7) is asymptotically exact up to the order O(n) in the density. Now we construct the simplest Pad6 approximation consistent with (7) in the form e2
K
(9)
# = #ia - 4~ce---o1 + ~A/8 "
From thermodynamic consistency follows that all other thermodynamic functions also have a Debye-Hiickel-like structure. The free energy in this model is given by F = Fia - k e T V ~ - ~ z
(10)
,
with r being z(x) = ~-5 ln(l + x ) - x +
.
(11)
This leads to an reduced entropy K3
s = sia - 2~-~n(Z(x) + 2z'(x)).
(12)
The equilibrium composition can be found by minimisation of (10) with respect to the degree of ionization. This leads to a non-ideal Saha equation 1 - ~ _ nA~ a ( T ) h(~, T ) 0~2
(13)
i.e., due to non-ideality the ionization equilibrium is shifted by a factor to the ideal mass-action constant. This factor h(~,T)=exp\
ksT
J
(14)
D. Beule et al./Physica A 241 (1997) 719-728
723
depends exponentially on the interaction contribution to the combined chemical potential given in (9). It can be interpreted as a lowering of the ionization energy I by AI h(a, T) = exp ( A I/kB T).
( 15 )
In general, non-ideality leads to an increased degree of ionization ct = aid + aint, that cannot be calculated analytically. For high temperatures, the degree of ionization a approaches unity and the ideal case is recovered. At low temperatures, when all charged particles are bound as atoms the interaction contribution of the A/8 model disappears and an ideal atomic-gas model remains. As compared to the ideal case, the pressure P
nkB T
--
Pid + Pint
nk8 T
-- 1 +
~3/2/£3 aid Av ain t -- - - ~ ) ( X ) ,
~ b ( x ) = ~3 [ l + X - x - -'2 1 n ( l + x )
247tn
1
(16)
(17)
is altered in two ways. First, by the change of the ionization equilibrium aint which lead to an increase in pressure and, second, through a non-ideality contribution that generally decreases the pressure. For the entropy one may write K3 S = Sid(aid -~- aint) -- 2 - ~ n ( T ( x )
~- 2qs/(x)) ,
(18)
where sid(cQ is given by (2). In the region where the A/8 approximation is valid, i.e., neA 3 ~ 1 and F < 1, the entropy is increased due to the larger degree of ionization and reduced by the interaction term.
2.3. Padk approximation #t the chemical picture The A/8 approximation is asymptotically correct in the first order with respect to the dimensionless quantum parameter xA. Even at medium densities it yields at least a qualitatively correct picture. A more appropriate description of dense plasma which includes all first-order quantum corrections requires to go beyond the A/8 model. Pad6 approximations of the interaction contributions of thermodynamic functions [ 12] within the chemical picture (PACH) are more complicated than the A/8 model but are still given by analytical formulae. Therefore, the PACH approach (as well as the A/8 model) allows fast access to different thermodynamic properties. An advantage of the PACH model is its validity in wide ranges of density and temperature. The free energy of the plasma in the PACH model is represented by
F = F~a + G + F~ + F,e + Fh~.
(19)
D. Beuleet al./PhysicaA 241 (1997)719-728
724
~a denotes the ideal contribution of the ions, atoms, molecules, and electrons as derived from Boltzmann and Fermi-Dirac statistics, respectively. The excited states of the atoms are accounted for by a Planck-Larkin partition sum o'(T) = Z
m2 exp
-
(20)
m
k~Tm2
"
We also include the additional degrees of freedom of the molecules into the ideal part of the free energy. Our study is mainly devoted to the plasma region where molecules contribute only relatively small corrections. For this reason we shall use only simple approximations for the molecular contributions. For a more advanced treatment of the molecules we refer, e.g., to Saumon and Chabrier [13]. The rotational partition sum of the molecules gives rise to a term approximated by
Frot=gmkBT(ln zr T
Tr 3T
1 90
1~)2 )
,
T~ ----8 5 K ,
(2l)
if T~ ~ T [8]. The contribution of the vibrational degrees of freedom of the molecules is given by [8]: Fvib = N m k B T ( l n ( 1
- exp(-T~,/T))),
Tv = 6210K.
(22)
The interaction contribution F~ and Fi account for the electronic and ionic subsystem, while Fie describes the electron-ion screening in terms of a Pad6 approximation [7]. The Pad6 approximations used here were discussed in detail in [12]. In the lowest order with respect to the parameter KA, they are consistent with the A / 8 approximation and may be considered as a generalization of the later. It has been shown [15] that the Pad6 approximations are in good agreement with recent quantum Monte-Carlo calculations [16]. Finally, Fh,. denotes the hard-sphere contribution, that account for the short-range repulsion due to bound shell electrons.
3. Results and discussion
Within the Pad6 approximation in the chemical picture one has to minimize the free energy with respect to the abundances of the various free and composite particles (2) in order to describe thermodynamic equilibrium for given temperature and density. The dissociation equilibrium of the molecules is coupled to the ionization equilibrium via the relation n i --}- na + 2nm = n. Therefore, the degree of ionization ~ and the density of molecules have to be determined simultaneously. We achieve the minimisation of the free energy by a simulated annealing procedure [14] that was originally developed for multi-component systems. After performing this procedure on a dense grid in the density-temperature plane, one can calculate different thermodynamic properties numerically. First, we calculated the isentropes s = const, for hydrogen plasma. They are plotted in Fig. 1 together with lines of equal ionization. In general, the difference between
D. Beule et al. IPhysica A 241 (1997) 719-728
,"
725
7
1'E 1021 ,
c-
/,
,,
//+ '
102°
,..,. , ,, ,
,/,/ /
, /,
,6;,1 " /
/)"//
.
.
,// /,
/ ,1 /,, ,,; 10
~9
'
'
z
'
~
/
. . . . .
.
.
.... s=21 (PACH) -- - s=21 (ideal) . s=24 (PACH) s=21 (ideal)
...... o~=0.9
,
-
~=0.1
'
10000
100000 T [K]
Fig. 1. Plasma isentropes and degree of ionization in non-ideal plasma. The plasma isentropes as obtained from PACH and in ideal plasma for three differentvalues of reduced entropys = 18, 21, 24. The short dashed lines limit the area where ionization raises from the weakly ionized case ~ =0.1 to the nearly completely ionized case ~ = 0.9.
1.0
0.8 ///
," e"
//
0.6
0.4
0.2
6' / / ,/~/'// (
i
s=21 (PACH) S=21 iideal)"
-d
,
0"~000
-e
,
a
,
l . . ./ . . ~ s=24 / (PACH)
. . . .
t
100000
T [K] Fig. 2. The degree of ionization along plasma isentropes s = 21,24 over temperature for ideal (thin lines) and non-ideal plasma (thick lines).
ideal and non-ideal isentropes increases as density increases and temperature decreases but the interaction effects vanish when the ions and electrons are bound into atoms and molecules. Therefore, the ideal and non-ideal isentropes that nearly coincide at high temperatures merge again after they have crossed the area of partial ionization. Due to the decrease in particle number the adiabates become steeper as atoms and molecule formation takes place for decreasing temperature. The degree of ionization along two of the adiabates is shown in Fig. 2. Lower values of reduced entropy correspond to higher densities or lower temperatures and therefore to larger non-ideality effects and thus to greater differences in the degree of ionization.
D. Beule et al. IPhysica A 241 (1997) 719-728
726
I0 s •
///
fp
104 10 3 .-"
13..
,9//
~//i"'"'" .-'" J ,,/"/'/;'; / /? J J ,/~ j t / / //f / / / /t
" ~ 10 2
10 ~ 10 o /
,,,t /
/'
,
/"
-.... -.....
. . . . . . .
10000
-
S='18(PACH) S=18(ideal) s=21 (PACH) s=21 (ideal) s=24 (PACH) s=24(ideal) fixed ionization
i
,
,
L
100000 T [K]
Fig. 3. Adiabatic equation of state for reduced entropy s--18, 21,24 over temperature. We compare the pressure of an ideal plasma (thin lines) with the pressure obtained from PACH (thick lines). For comparison we also plot an adiabtic EOS for a fixed degree of ionization as obtained from Eq. (23) using Vad = 0.4 (short dashed line).
There are points where the non-ideality contributions to entropy (cf. Fig. 1) cancel each other and the ideal and non-ideal isentropes cross each other. This points are found in the region of highly ionized plasma. Beyond this point the degree of ionization does not differ much for the ideal and the non-ideal case and positive contribution to the entropy due to increased degree of ionization are overcompensated by the direct interaction contributions. Once the isentropes are known, it is easy to give the adiabatic (isentropic) equation of state for different values of entropy, cf. Fig. 3. For high and low temperatures, the ideal and non-ideal pressure almost coincide for a broad range of entropy. For intermediate temperatures non-ideality has significant influence on the adiabatic EOS. The deviation of ideal and non-ideal behaviour seen in Fig. 3 for these temperatures corresponds to deviations in the degree of ionization due to non-ideality (cf. Fig. 2). As for entropy, there exist points in the area of highly ionized plasma where non-ideality contributions cancel each other. In order to demonstrate the effect that ionization has on plasma isentropes, we also plot the adiabatic EOS of an ideal system where no ionization processes take place. For such systems the adiabatic EOS can be obtained from integrating (dlnT~
(23)
voa -- \ d-W. p ) s .
For Boltzmann particles with fixed degree of ionization one finds Vad = 0.4 [2]. Therefore, the adiabatic EOS for such a system is given by a set of parallel straight lines in a log(p) over log(T) plot. One of these lines is shown in Fig. 3. For high temperatures (i.e. a high degree of ionization), the PACH isentropes fullfil Vaa ~ 0.4. But in the area where the formation of bound states takes place the adiabatic EOS deviates significantly from this value. For decreasing temperature, the pressure is
727
D. Beule et al./Physica A 241 (1997) 719-728 . . . . . . . .
i
. . . . . . . .
i
. . . . . . . .
i
'
'
1.00
1
~- 0.90
Q.
0.80
'/
....
.....
s=21
iO
~
.....
ie
.....
"
i&
'
n [cm -3]
Fig. 4. Comparisonof the pressure along ideal and non-ideal adiabates (s = 18,21,24) over density. We plot the quotient of the pressure as obtained by PACH PPACH(S)and from an ideal model plasma Pid(S) for different values of reduced entropy s = 18, 21,24.
decreasing much faster than in a system with fixed degree of ionization. For s = 18 one can even see a second steep decrease in pressure at low temperature where molecule formation sets in. The differences between ideal and non-ideal adiabatic EOS can be seen more clearly in Fig. 4. We plot the quotient of the non-ideal PeAcH(s) and ideal pressure pia(s) along different adiabates over the density. It has to be emphasized that the pressures compared in Fig. 4 belong to different temperatures because ideal and non-ideal adiabates take different paths through the density-temperature plane. For lower values of entropy the difference in pressure along ideal and non-ideal adiabates increase and the minimum of the quotient is shifted towards higher densities. The interaction contributions to the entropy are smaller than other thermodynamic properties like free energy or pressure. This cancellation can also be found in the limiting laws. In contrast to other thermodynamic properties, even in the case of strong coupling, entropy does not deviate much from the ideal behaviour. Nevertheless, the entropy is influenced significantly by the number of particles, i.e., the degree of ionization. The degree of ionization is very sensitive to non-ideality effects. Keeping in mind that Fig. 1 and Fig. 3 are plotted using a logarithmic scale, one realizes that ideal and non-ideal isentropes take quite different paths through the density-temperature plane.
4. Conclusions Bound-state formation significantly alters the adiabatic EOS. In order to give an appropriate description of the adiabatic equation of state one has to give a good description of the ionization equilibrium first. We achieved this by using the Pad6
728
D. Beule et al./Physica A 241 (1997) 719-728
approximation in the chemical picture. As demonstrated within the A / 8 model there is an interference o f the direct change o f isentropes due to interaction contributions and the indirect change through the change o f ionization equilibrium. Deviation o f ideal and non-ideal adiabatic EOS is mainly caused by different degrees o f ionization along ideal and non-ideal plasma isentropes.
Acknowledgements We would like to thank Tino Fuchs for his contributions on the adiabatic equation o f state o f the electron gas. One o f us (A.F.) would like to express his sincere gratitude to Prof. Masuo Suzuki and his group at the University o f Tokyo for their warm hospitality during a research stay in 1996.
References [1] V.E. Fortov, 1.T. Yakubov, Physics of Nonideal Plasmas (in Russian), Inst. Chem. Phys. Chemogolovka, 1984, Hemisphere, New York (English Transl.) 1990. [2l R. Kippenhahn, A. Weigert, Stellar Structure and Evolution, Springer, Berlin, 1994. [3] J. Christenssen-Dalsgaard, W. Dfippen, Astron. Astrophys. Rev. 4 (1992) 267. [4] W. Ebeling, W.D. Kraeft, D. Kremp, Theory of Bound States and lonisation Equilibrium in Plasmas and Solids, Akademie-Verlag, Berlin, 1976; Extended Russ. translation, Mir, Moscow, 1979. [5] G. Chabrier, in: IAU Colloquium 147, The Equation of State in Astrophysics, G. Chabrier, E. Schatzman (Eds.), Cambridge University Press, Cambridge, 1994, p. 287. [6] P. Debye, E. Hfickel, Phys. Z. 24 (1923) 185. [7] W. Ebeling, Contrib. Plasma Phys. 30 (1990) 553. [8] R. Fowler, E.A. Guggenheim, Statistical Thermodynamics, Cambridge University Press, Cambridge, 1952. [9] R. Zimmermann, Many-Particle Theory of Highly Excited Semi-Conductors, Teubner, Leipzig, 1988. [10] G.E. Norman, A.N. Starostin, Teplofiz. Vys. Temp. 8 (1970) 413. [11] G.P. Bartsch, W. Ebeling, Beitr. Plasmaphysik 11 (1971) 393, 15 (1975) 25. [12] W. Ebeling, A. F6rster, V.E. Fortov, V.K. Gryaznov, A.Ya. Polishchuk, Thermophysical Properties of Hot Dense Plasmas, Teubner, Stuttgart, 1991. [13] D. Saumon, G. Chabrier, Phys. Rev. A 44 (1991) 5122. [14] S. Kirkpatrick, C.D. Gelatt, M.P. Vecchi, Science 220 (1983) 671. [15] W. Stolzmann, T. Blrcker, Phys. Lett. A 221 99 (1996); Astron. Astrophys. 314 (1996) 1024. [16] C. Pierleoni, D.M. Ceperley, B. Bemu, W.R. Margo, Phys. Rev. Lett. 73 (1994) 2145.