Selfconsistent equation of state

Volume 101B, number 5 PHYSICS LETTERS 21 May 1981

SELFCONSISTENT EQUATION OF STATE

R.K. TRIPATHI Institute of Physics, A/105 Saheed Nagar Bhubaneswar, 75100 7 India 1 and Institut ffir Theoretische Physik, Universitiit 7Tibingen, D- 7400 Tiibingen, West Germany

Received 10 December 1980 Revised manuscript received 2 March 1981

We investigate the effects of a selfeonsistent single-particle potential and the Pauli principle on the equation of state for dense systems at low temperatures. It is found that the selfconsistent potential plays an extremely important role for the equation of state near nuclear matter density.

The equation of state is an essential ingredient for the understanding of stellar collapse and supernova explosions [1-10] . Its utility in the area of high- energy heavy-ion collisions is unquestionable [11]. Consequently it is vital that the equation of state must be determined with great accuracy.

Calculations of the equation of state involving the strong nuclear interaction, amongst other things, should accurately account for the strong nature of the force. Most of the existing [7-9] finite-temperature calculations consider only the temperature in- dependent model effective interaction and consequently neglect the scattering to intermediate states (i.e., the Pauli operator is not taken into account). Others [3] do not consider the selfconsistencies of the potential and the degeneracy and are consequently valid only for very low densities. To our knowledge there does not exist any calculation in the literature where these effects have been taken into account in a selfconsistent way in the calculation of the equation of state. And that is the purpose of this note. The basic differences between the present calculation and the previous calculations are the following. Our eq. (9) (discussed later), which is the direct outcome of the linked cluster-like expansion for the grand potential, implies that our results are valid up to densities much higher than the nuclear matter densities. In addition

1 Permanent address.

eq. (7) (also discussed later) implies that we take into account the scattering to the intermediate states selfconsistently (i.e. the temperature dependence is built in selfconsistently) in the interaction. Besides by taking into account the double selfconsistencies [eqs. (9) and (11)] we determine the degeneracy selfconsistently as well.

At zero temperature Brueckner theory is commonly used to account for the short-range repulsion. How- ever, due to the compression the matter in the supernova bounce and also in heavy-ion collision gets heated up. Consequently one should bring in the temperature effects in the formalism as well.

In the f'mite temperature case one starts by calcu- lating the grand thermodynamic potential per unit volume (~2):

~2 = - P = - Tin tr exp [ - T - I ( H - / a n ) ] , (1)

where H, P, T,/~ and n are the hamiltonian, pressure, temperature, chemical potential and number density, respectively. The reason for doing this is that the grand thermodynamic potential can be expressed as a linked cluster expansion analogous to the zero- temperature Brueckner-Goldstone expansion, i.e.

~2 = ~20 + ~21 + ~22 + .... (2)

where ~20, ~21, ~22, ... are the contribution to the thermodynamic potential due to the unperturbed part, one-body part (single-particle potential), and two-

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body part (binary collision) of the hamiltonian. In the present note we discuss the equation of state for neutron matter valid up to very high densities. The single-particle energy is defined as

e(k) = (hk)2/2m + U(k) , (3)

where k is themomentum, m is the neutron mass, and U(k) is the single-particle potential at k (this potential has to be determined selfconsistently). The unperturbed part of the thermodynamic potential is given by

f de f l 0 _ 1 d k k 3 ~-~ n ( k ) , (4 ) 31r2 0

where n(k), the Fermi distribution function, is given by

n(k) = (1 + exp[(e(k) - p)/T] } - 1 . (5)

We emphasize that n(k) depends on the single-particle energy [eq. (3)] and hence on the single-particle potential U(k). In addition it depends on the temperature T(MeV) and the chemical potential p. As emphasized later on in the text, both e(k) and/a have to be determined selfconsistently. The contribution to the thermodynamic potential due to the single- particle potential U(k) is given by

~ 1 - - Of dk k2n(k)(klUlk) . (6)

The contribution from binary collision in degenerate media has been studied by Bloch and Dominicis [12] in the ladder approximation with particle intermediate states. As the temperature increases, however, one should also take into account the non- occupancy of the hole states. This has the advantage of treating both particle and hole lines symmetrically. Consequently a general formula accounting for the contribution of binary collision to the grand potential can be written as follows:

1 2 2 ~2 = - - f f dkl dk2 klk2g(Es' k l ' k2)

47r2

where

g(Es, kl , k2) = tan-1 [7rPEaK(Es) ]/7rpza. (7)

Here PE is the single-particle level density and the K- matrix satisfies the integral equation

K(Es) = V + V[Q/(E s - H0)] K(Es), (8)

where V is the nuclear potential, Q is the Pauli operator and E s is the starting energy.

The selfconsistent single-particle potential can be defined in analogy with the zero-temperature case, namely,

! f dk2 n(k 2) U(kl) = 4n2 0

X g(E s = (lt2/2m)(k 2 + k 2) + U(kl) + U(k2)) • (9)

It is important to emphasize the following con- cerning eq. (9). One notes that the single-particle potential is needed in n(k2) and g in order to calculate the single-particle potential itself. Consequently one must calculate it by iteration. Besides, since in eq. (9) one integrates up to ~ (and not up to kF, which has no meaning in the finite-temperature case), it will be unfair to set the single-particle potential equal to zero after a certain momentum as is commonly done in the zero-temperature case. As we shall see later on it is extremely important to treat the single-particle potential accurately. In addition the symmetric defini- tion of the particles and holes is more satisfactory here. Our single-particle potential which is determined selfconsistently accounts properly for the scattering to intermediate states. This is due to the fact that g [eq. (7)] depends on the self consistent Pauli operator which goes as an input in eq. (9). In addition the temperature dependence of the single- particle potential is insured through n(k) and through g [Q depends on n(k)]. These effects have been neglected in the previous calculations [3,7-9].

Using eqs. (4)-(7) one arrives at the following equation of state,

dk k2n(k)[~kde/dk + l U(k)] . (10) I

P = ~ O

The chemical potential p is determined by the number-density constraint, i.e.

2 f d 3 k n ( k ) (11) n = (2") 3

It should be noted that eqs. (9) and (10) warrant the double self consistency, which must be satisfied with respect to the selfconsistent potential and chemical potential. The method of solution is as follows.

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For a given number density n and temperature T, one makes an initial guess for the single-particle potential. Eq. (11) is then integrated numerically by varying the chemical potential until one gets the given number density. Eq. (9) is then solved, with g calculated from eq. (7) for different approximations of the Pauli operator for the chosen interaction, to get the new guess for the single-particle potential. The pro- cess is repeated until one gets the same value of the single-particle potential and chemical potential as fed in eqs. (9) and (11), thus satisfying both self-consis- tencies. This means that the degeneracy is calculated selfconsistently as well and is not fed in as imput in- formation. It is worth emphasizing that as the method is based on the linked cluster expansion, it is reliable up to very high densities.

In order to test the sensitivity of the equation of state on the Pauli principle we discuss here three different approximations of the equation of state. These approximations can be incorporated in eq. (7).

Approximation A:

g(~s, k l , k2) = K(Es, k l , k2)" (12)

This is similar to the Beth-Uhlenbeck [13] approximation. This obviously becomes a better approximation when the quantity inside tan -1 is small, so that it can be expanded. This means this would be a better approximation when Q is extremely small and g is small as well. This is usually the case for small densities. This is similar to the approximation used in ref. [7].

Approximation B: This is akin to the approximation of Bloch and Dominicis [12] and amounts to taking the ladder summation with the particle intermediate lines. As such, this is in philosophy very similar to the approximation of the zero temperature Brueckner theory. Here we take

a = [1 - n(k i ) ] [1 - n(k2) ] . (13)

Approximation C: Finally, a symmetric treatment of the particle and hole intermediate states leads to the following prescription of the Pauli operator:

a = 1 - n(kl) - n(k2). (14)

It should be noticed, however, that approximations B and C differ only by the second-order effect of the distribution function. Clearly a comparison of approximations A and B would emphasize the tern-

perature dependence of the interaction as well. The main purpose of this note is to demonstrate

the importance of the selfconsistent potential [eq. (9)] on the equation of state [eq. (10)] and to study the effect of different prescriptions of the Pauli principle on the equation of state, thus giving a reliable equation of state which can be used in the supernova explosion and also in heavy ion collisions. In order to look at the sensitivity of these effects we keep the same input reaction matrix in all the cases. It is known [ 14,15 ] that at zero temperature the self consistency on the potential and the Pauli operator do play an important role as well. The interaction chosen [7,16] fits the gross properties of nuclear matter. Any other interaction would not effect the results discussed here qualitatively.

In order to see the effect of the selfconsistent single-particle potential on the equation of state we have done two sets of calculations: (a) here the effect of the single-particle potentials has been neglected completely. This is called the non-selfconsistent calculation and (b) here the single-particle potential has been calculated selfconsistently. This is called the selfconsistent calculation. These two sets of calculations have been done for the above mentioned three prescriptions of the Pauli principle both for the total pressure and for the pressure due to binary collisions.

We display here some of the results for a typical temperature of T = 20 MeV. Fig. la shows the ratio of the total pressures under approximation (a) to the total pressure calculated with the fully selfconsistent prescription given by eq. (9) i.e. prescription (b). The solid line gives the ratio of the total pressure without selfconsistency in the single-particle potential to the total pressure with a fully selfconsistent single- particle potential for prescription A of the Pauli principle. The dashed and the dotted lines give the same ratios for prescription B and C of the Pauli principle. As seen from fig. 1, the inclusion of the selfconsistent single-particle potential does make a tremendous dif- ference near and at nuclear matter density, with the non-selfconsistent calculation overestimating the pressure up to a factor of 1.85, 1.73 and 1.72, respectively, under the different approximations of the Pauli principle. Fig. lb shows the comparison of the selfconsistent calculations amongst themselves i.e. it shows the effect of the different approximations of the Pauli principle. Here the solid line represents the

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Volume 101B, number 5 PHYSICS LETTERS

2,0 (a)

1.0

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1.5

(b)

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N M

-2, 55 -I,55 - 0.55 log p [frn -3]

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-555 ' ' -2,55 , -1.55 - U.55 log p [fni ~]

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Fig. 1. (a) (Top figure) ratios of the total pressures versus density: solid line - (PA n°nself/PA self), dashed line - (PB nonself/PB self), dotted line - (Pc n°nself/Pc self); (b) (lower figure) ratios of the total selfconsistent pressures under different Pauli approximations: solid line - (PA self- PB self), dotted line - (Pc self/PB self). The arrow on the density axis represents the density of nuclear matter. PA self, for example, means total pressure with selfconsistent single- particle potential and approximation A for the Pauli principle.

ratio of the selfconsistent total pressure under approximation A of the Pauli principle to the total selfconsistent pressure with approximation B for the Pauli principle. The dashed line represents the ratio of the selfconsistent pressure with approximation C to the selfconsistent pressure with approximation B for the Pauli principle. It is seen that approximation A can be off as much as by 15%, while approximations B and C give very similar results practically for all the densities.

Fig. 2a demonstrates the same results as for fig. la but now only for binary collisions. It is seen that near nuclear matter density non-selfconsistent calculation underestimates the pressure due to binary collisions up to a factor of 5. Fig. 2b shows the comparison amongst the selfconsistent calculations for different approximations for the Pauli operator for binary collision. It is seen that approximation A gives an overestimate up to a factor of 1.4, while again the approximations B and C are very similar

throughout.

21May 1981

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1.0 . . . . . . . . . . . . . . .

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0-54.55 - 3'.55 -2155 -1,551 { -0,55 Iogp [fro -3]

Fig. 2. Same as fig. 1 except here we compare pressures due to binary collisions only.

It is of interest to see the effect of the single- particle potential on the chemical potential ~ ) , the entropy and the free energy per particle. The entropy has been calculated by the usual formula:

1 2 fd3k s - n (2rr)3

X {n(k)ln n(k) + [1 - n(k)l ln[1 - n ( k ) ] } . (15)

Fig. 3 demonstrates these results. The solid lines are the results under approximation B for the Pauli principle with self consistency in the single-particle potential and the dashed lines are the corresponding results without selfconsistency but with the same approximation B for the Pauli principle. Here again we notice that the selfconsistency in the single-particle potential does have an important effect near and at nuclear matter density.

In fig. 4 we show the temperature and degeneracy dependence of our equation of state. In addition we compare it with the equation of state of ref. [7]. One finds that both equations of state agree completely for all temperatures at lower densities. There are, however, departures at higher densities with our equation of state being more stiff than that of ref. [7]. This is in part due to the fact that our method includes the scattering to intermediate states (i.e. the Pauli opera-

372

Volume 101 B, number 5 PHYSICS LETTERS 21 May 1981

c~

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Fig. 3. Free energy, chemical potential and entropy. Solid line - results obtained with selfeonsistency under approximation B of the Pauli operator. Dashed line - results obtained without selfeonsistency under approximation B of the Pauli operator.

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tor is treated selfconsistently) while this has been neglected in ref. [7]. The same is true for the nuclear equation of state calculated in ref. [8] since the method.followed there is the same as that of ref. [7].

Our calculations do show clearly that selfconsistency in the single-particle potential and the Pauli operator gives very important effects in determining the equation of state and other gross thermal properties of neutron matter around nuclear matter density.

The author acknowledges Professor A. Faessler for the excellent hospitality at the University of Tiibingen where part of this paper was written. He is

extremely grateful to the referee for this extremely valuable comments and suggestions which have im- proved the presentation tremendously.

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log p [fm3l

Fig. 4. Equation of state for neutron matter with prescription B for the Pauli operator at various temperatures. Solid lines show the results of this work and dashed lines are from E1 Eid and Hflf [7]. The numbers at the top of the figure are the self consistent neutron degeneracies for T = 31.62 MeV, T = 10 MeV and T-- 1 MeV, respectively, from the top down- wards at the densities indicated by the arrows.

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[8] M.F. El Eid and W. HiUibrandt, Astron. Astrophys. Suppl. 42 (1980) 215.

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Selfconsistent equation of state