Collective modes in hot asymmetric nuclear matter at variable densities

Nuclear Physics A 665 Ž2000. 13–45

Collective modes in hot asymmetric nuclear matter at variable densities Fabio ´ L. Braghin Nuclear Theory and Elementary Particle Phenomenology Group, Instituto de Fısica, UniÕersidade de Sao ´ ˜ Paulo, C.P. 66.318, CEP 05315-970, Sao ˜ Paulo - SP, Brazil Received 7 April 1999; received in revised form 7 June 1999; accepted 21 June 1999

Abstract A nearly exact expression for the response function of hot asymmetric nuclear matter is derived for Skyrme type effective interactions and the resulting strength distribution is analyzed for the four channels of the particle–hole interaction. Several proton–neutron asymmetries are considered as well as different total densities. In the isovector channel the strength presents a very collective behaviour Žzero sound type. which becomes still more collective with increasing asymmetry. For higher nuclear densities it may be collective or unstable depending on the effective interaction. This is also the behaviour for the spin-isospin channel. The other channels may have the collectivity increased for a highly asymmetric nuclear matter at higher densities. In the spin channel zero sound modes are found for higher enough p-n asymmetries andror densities higher than the saturation density. The static limit of the polarizabilities are considered yielding the symmetry energy coefficients for isovector, spin and spin-isospin channels. The dependence of the polarizabilities on p-n asymmetry is analyzed, in particular in the isovector channel which is of interest, for example, for the supernovae mechanism. q 2000 Elsevier Science B.V. All rights reserved. PACS: 21.30.-x; 21.60.Jz; 21.65.qf; 26.50.qx; 26.6.qc

Keywords: Neutron-proton asymmetry; Variable nuclear matter density; Response function; Zero sound; Symmetry energy

1. Introduction The study of symmetric and asymmetric nuclear matter provides relevant information concerning systems like large nuclei, dynamics of neutron stars and the supernovae mechanism, as well as effects present in highrmedium energy heavy ion collisions. Among many usually employed approaches, the linear response method for non-relativistic symmetric and asymmetric nuclear matter at zero and finite temperatures has been 0375-9474r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 4 7 4 Ž 9 9 . 0 0 3 3 1 - 0

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F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

investigated in Refs. w1–7x. It corresponds to a summation of chain diagrams and it is well-suited for the study of collective modes taking into account the Žin medium. mediumrlong range correlations w8x. In the following some motivations for the present work are discussed. First, in heavyrneutron rich nuclei there is a non-negligible asymmetry in the neutron-proton number and it is interesting to evaluate its influence on the collective motion and consequently on the nuclear dynamics. Thus one can extract information for the effective interaction dependence on the neutron proton asymmetry. In the present work this is indeed done by comparing the results from interaction SGII w9x and SLyb w10x. With this comparison the role of the velocity-dependent and density-dependent terms is studied. As far as collective modes are concerned, pieces of information about giant resonances are expected to be found. In particular, the Isovector Dipole Giant Resonance ŽIVDGR. has been observed in light exotic nuclei w11–13x Žcontaining the so-called ‘soft mode’., and an halo region has been recently predicted for the exotic 122 Zr nucleus w14x which would contain up to 6 neutrons Žin this calculation a pairing interaction with density-dependent effective interaction of zero range was used.. As pointed out in w6x one can hope the IVDGR as well as its soft mode to take place in exotic heavier nuclei which would possess a larger neutron halo. However, the IVDGR Ž a good review may be found in w15x. is not the only collective dipolar mode which takes place in nuclei: scalar w16,17x, spin-isovector w18x and spin w19x excitations have also been observed as well as the double-IVDGR w20–22x. Nuclear matter collectivercoherent modes Žin the four channels. are expected to be, at least, qualitatively related to these resonances in finite nuclei in spite of important surface effects. Besides that, asymmetric nuclear matter and neutron matter have been extensively studied with several astrophysical motivations w7,23–26x. For instance, during the collapse and explosion of massive stars Žsupernovae., neutrino scattering from nuclei produces its trapping for hot dense nuclear matter. The inelastic contribution from particle–hole excitation Žvia Z0 . and consequently from its phonons Ždensity and spin-density fluctuations. are important phenomena which need a consistent quantum treatment. In neutron stars, a strong correlated system, neutrino scattering by density fluctuations influences the neutrino mean free path. Neutrinos also interact with asymmetric nuclear matter Žin the proto-neutron star stage. via the charged weak current, whose effect is thus also relevant for the neutrino mean-free path w7x. In this system one may expect zero sound in the spin-isovector for high n-p asymmetries and in the spin channel for densities r N M 0 2. r 0 w24,25x. This behaviour is shown in Section 4 of the present paper. Notwithstanding, in collapsing supernovae the nuclear matter density is assumed to be r , .915 r 0 , where r 0 is the saturation density and the neutron-proton asymmetry is taken to be a s 2 rn y r 0 s 1r3, i.e. in terms of the asymmetry coefficient used in this paper b s 1, with pressure P s 0. The equilibrium density in these systems are not the saturation density in large nucleir nuclear matter. Thus a study with variable density, smaller and bigger than r 0 , .17 fmy3 , appears to be important. The low density regime is also interesting since it sheds light on the growth of instabilities at finite temperatures, which are responsible for fragmentation w27x. In the context of heavy ion collisions thermally excited nuclear matter is studied in the determination of the amount of energy

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

15

which can be absorbed by nuclear matter with increasing temperature. In this work, however, nucleon-densities are considered to not depend on the temperature. Another important aspect of the response function of nuclear matter is that its static limit is directly related to the energy symmetry coefficient in the corresponding channel. For instance in w3x this was shown for the isovector channel which gives place to the neutron-proton symmetry energy coefficient of Žsymmetric. nuclear matter Ž at .. In the present paper this is worked out in the four channels. If the isovector symmetry energy coefficient of nuclear matter Ž at . increases with temperature Žor density or n-p asymmetry. electron capture in pre-supernovae phase is hindered and this causes an increase in electron fraction at trap which produces a stronger shock wave as long as less energy is lost for dissociating nuclei. With the usually assumed temperature dependence of the symmetry energy the calculated shock wave is nearly half of the observed w28,29x. It is suggested below that the static polarizability of Ža.symmetric nuclear matter may be more suitable than the standard coefficient at at zero and finite temperatures. It provides a natural parametrization for the symmetry energy coefficient dependence on the density, temperature and p-n asymmetry. The spin and spin-isovector symmetry energy coefficients are also analyzed. In Refs. w1–3x there were motivations for the study of the response function from the possibility of getting information about the effective nucleon–nucleon interaction and about the theoretical description of the nuclear isovector dipole giant resonance ŽIVDGR. on the ground state and with increasing excitation energy. Indeed, an investigation of the dependence of the collective modes on the effective interaction ŽSkyrme type. was done. In these two works the response function of the symmetric nuclear matter to small external isovector perturbation using Skyrme interactions was calculated and a zero sound collective mode was found at low temperatures for reasonable values for the effective mass when density-dependent forces were considered. This phonon has the following dispersion relation: v res s c 0 q, where c 0 is the zero sound velocity and q the momentum transfer. The energy of the mode is mainly determined by the effective mass together with a function V0 which is written in terms of the Skyrme interaction parameters for each channel Žand it is directly proportional to the respective Landau parameter w4x.: higher the effective mass Ž and V0 . lower Žhigher. is the mode frequency, being more collective in both cases. This phonon typical from correlated Fermi liquids indicates that many-body correlations are not negligible in nuclear matterrheavy nuclei, since heavier the nucleus more collective the mode. The model worked out in these references was based on the Steinwedel–Jenssen one w30x, considering that the transferred momentum between protons and neutrons is related to the mass number of the analyzed nucleus. In particular, for heavy nuclei one should consider small q. With the increase of temperature the collective mode couples to the particle–hole spectrum making zero sound damped, until its disappearance by the time when the temperature is of the order of some MeV. This critical temperature above which there is no more zero sound depends strongly on the used effective interaction. The charge longitudinal response of nuclei Žincluding scalar and isovector channels. has been investigated for large momentum transfer e.g. w4,8x and a Fermi gas coherent behaviour Žtypical from an independent particle approximation. has been found for light nuclei. An investigation of the other three channels of the response function was done by Hernandez et al. in Ref. w5x. ´

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F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

The linear response for an asymmetric hot nuclear matter over a time-dependent Hartree–Fock approximation was presented in the isovector channel in Ref. w6x where two prescriptions for the density fluctuations were analyzed. The one which leads to a more collective behaviour was chosen corresponding to a sounder physical picture. This effect is observed in the isovector dipole giant resonances in nuclei. In Ref. w7x the linear response has been performed over a Hartree calculation for non-relativistic nuclear matter Žwith Skyrme interactions. and for a relativistic case. As it will be discussed in the present article, the response function in this second work only yields the same result as the former in a specific approximation of the response function worked out in the present article for the symmetric nuclear matter limit. In spite of that, some results are in agreement for the asymmetric cases, depending on the Skyrme interaction and on the p-n asymmetry used as discussed below. However, in the present work a more extensive analysis of the response function is exhibited while in w7x only some cases of astrophysical interest were showed in the frame of their approach. In the language of Landau’s Fermi liquid theory, Reddy et al neglect the l s 1 particle–hole interaction parameters, which are considered here. In the present paper some care is also taken when considering Fermi surface modification due to the increasing of temperature. A self-consistence check is done by verifying the energy-weighted sum rule. Moreover, here the Lindhard functions are explicitly calculated instead of using a dispersion relation to obtain the real parts from the imaginary ones. In the present article a detailed derivation of a nearly exact expression for the response function of asymmetric hot nuclear matter using Skyrme effective interactions is considered, departing from a time-dependent Hartree–Fock approximation. It is employed for a further investigation in the four channels of the effective interaction: scalar, spin, isovector and spin-isovector, for different neutron-proton asymmetry coefficients. Zero sound modes are found in the isovector Žas in symmetric nuclear matter w3x., spin and spin-isospin channels for high enough n-p asymmetry at different nuclear densities depending on the Skyrme interaction. The importance of the effective interaction used is stressed as long as there happens collective modes only for very repulsive forces. Qualitative agreement with the work of Reddy et al. is found for the cases where the system Žin the isovector and spin-isovector channels. shows the tendency to undergo a phase transition for higher densities when the interaction SLyb from w10x is used. Analytical expressions are presented for the Lindhard functions which compose the response function and the results of Reddy et al. w7x are reproduced in particular cases of the present work. An investigation of the response dependence on the density of nucleons is done for r s .5 r 0 and r s 2 r 0 , where r 0 is the saturation density. The static polarizabilities are given for symmetric and asymmetric nuclear matter with increasing temperature. The static polarizabilities are obtained in the limit vrq 0 and its dependence with temperature and neutron-proton asymmetry are studied. Some results are shown and related to other works where the dependence of the symmetry energy coefficient with temperature was calculated using different techniques w26,28,29x. In these three articles the relevance of the p-n energy coefficient for the supernovae mechanism is discussed. In the spin and spin-isospin channels the respective Žspin and spin-isospin. symmetry energy coefficients also correspond to the static polarizabilities and they are calculated considering neutron excess. In the scalar channel the polarizability defines a ‘‘dipolar compressibility ’’ which is directly relate to the usual volume

™

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

17

Fig. 1. Distribution of strength per unit volume for the operator t expŽ iqPr . Žin fmy2 . – isovector channel – as a function of the energy v Žin MeV.. Asymmetry bs 0 Žsolid line. at T s 0 MeV, bs 0.54 for different values of the temperature T s 0 and 6 MeV Ždotted lines., bs 2 Žshort dashed line., bs8 Žlong dashed. and bs 32 Ždotted-dashed.. Using interaction SGII with q s 0.23 fmy1 .

compressibility K` as it will be shown below. Effects of the proton–neutron asymmetry are discussed. The energy-weighted sum rule is verified and well satisfied in all cases but two which have been analyzed. This is discussed in the text.

2. Linear response calculation The time-dependent Hartree–Fock equation, shown below, determines the temporal evolution of the one-body density matrix r . In the presence of an infinitesimal external perturbation, the equation reads, in natural units, i E t r s w W q Ve , r x ,

Ž 1.

where W is the Hartree–Fock energy of protons or neutrons and Ve is the small amplitude Ž e . external field of the form:

ˆ yi qP r eyiŽ vqih .t . Ve s e Oe

Ž 2.

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F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

Fig. 2. The same as Fig. 1 using interaction SLyb.

In the above equation h is an infinitesimal number which corresponds to the adiabatic switching on of the external source. The operator Oˆ may be t 3 Žthird isospin Pauli matrix., s 3 Žthird spin Pauli matrix., t 3 s 3 or a unit matrix, respectively for the isovector, spin, spin-isovector and scalar channels. Each of these perturbations induces small amplitude density fluctuations around the static solution so that Eq. Ž1. can be linearized. In the following, the calculation of the response function for the isovector channel with an asymmetry in the neutron-proton densities is showed, but the procedure is the same for the other channels. The mean energy for Skyrme-type effective interaction w31,32x is expressed in terms of the nucleon density Ž r i ., kinetic energy density Žt i . and momentum density Ž j i ., where i stands for protons or neutrons. The resulting equation for the total fluctuation density in momentum space, dr s rn y r p , is i E t²k < dr
Ž 3.

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

19

Fig. 3. Distribution of strength per unit volume – isovector channel neglecting function of interaction V1 using force SGII. Parameters: bs 0 Žsolid line., 0.54 Ždotted. and 8 Ždashed. at T s 0 MeV.

where e 0X p Žk. s k 2 Ž1 q ac .r2 m )p , and m )p is the proton effective mass. In this expression f i Žk. is the fermionic occupation number and three asymmetry coefficients have been used: as

m )n m )p

y 1,

bs

r0n r0 p

y 1,

cs

drn dr

.

Ž 4.

The coefficient b is related to a frequently used asymmetry coefficient Ž a s 2 r 0 n y r 0 . by bs

2a

Ž1ya .

.

Ž 5.

For the induced difference between proton and neutron density distribution we consider the following time-dependent prescriptions, analogously to the form of Ve : ²r < dr
²r < dt
²r < d j
Ž 6.

Due to the definition of the used densities w31,32x the above prescriptions are required to satisfy the following equalities:

Ž a , b ,g . s H

d3 k

Ž 2p .

3

ž

1,k P Ž k H q . ,

1 q2

/

Ž 2k H q . P q ²k < dr Ž t s 0 .
F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

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Table 1 Right-hand side ŽRHS. of the energy weighted sum rule ŽMeV = fmy3 . compared to the contribution of the particle–hole spectrum of the strength SŽ v . to the integral weighted by the energy of expression Ž23. obtained numerically for T s 0, 3 and 6 MeV for some cases analyzed in the text Žisovector channel.. The missing strength is due to the pole b, r r 0

RHS

m1 ŽT s 0.

m1 ŽT s 3.

m1 ŽT s6.

RHS Ž q s 2U q 0 .

m1 ŽT s 0.

0, 1 0.54, 1 0.54, 1, no V – 1 8, 1 0.54, 0.5 0.54, 2

53 53 45 53 22 141

28 16 14 0.44 6 94

53 53 45 0.44 22 140

53 53 45 0.47 22 141

212 212 – 212 – –

212 212 – 2 – –

From here on the index Ž s,t . for each Žspin,isospin. channel is used. The deviation in the energy density of neutrons Žand protons. can be written as Wn y W0 n s d Wn s 2 Ž V00,1 q V20,1 . drn Ž r,t . q 2V10,1= P drn Ž r,t . = q 2V10,1dtn Ž r,t . q 2 iV10,1 Ž = P d j n q d j n P = . .

Ž 8.

0,1

These functions Vi are related to the parameters of the Skyrme interaction by the expressions of Appendix B. In order to conclude this calculation another asymmetry coefficient is needed, a kinetic energy density dependent coefficient bn ds . Ž 9. b In Ref. w6x two prescriptions were considered for the coefficients c Žfrom expression Ž4.. and d. They were taken either to be equal to 1r2 Žprescription A. or calculated in terms of the saturation densities at zero temperature Žprescription B.. As discussed in that paper, prescription A leads to more reasonable results and will be considered. This means that density fluctuations should be proportional to the respective density of nucleons instead of being proportional to the perturbation amplitude. One considers 1qb 1 cs , ds . Ž 10 . 2r3 2qb 1 q Ž1 qb. By rearranging Eq. Ž3. it is straightforward to show that the matrix elements at t s 0 satisfy: ²k < dr Ž t s 0 .
Ž 11 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

21

Fig. 4. Same as Fig. 3 but for the force SLyb.

For this expression an ansatz for the total nucleon density fluctuation has been considered:

dr Ž t . s dr Ž t s 0 . eyi Ž v qih .t ,

Ž 12 .

X

In expression Ž11. d f i s f i Žk . y f i Žk., and another asymmetry coefficient is used: cXX s gnrgp .

Ž 13 .

Its value was considered to be cXX s c. The resulting response function does not depend sensitively on cXX . Multiplying Eq. Ž11. respectively by 1, k P Žk H q. and Ž2k q q. P q and integrating them over k a set of linear equations for a , b and g is obtained, namely

a s Ž V00,1 q V20,1 . P0 c a q V10,1 P2 c a q V10 ,1 P0 d b q 2g M p)v V10,1 P0 cXX q

ž

e 2

Ž P0n qP0 p .

/

,

Ž 14 .

b s Ž V00,1 q V20,1 . P2 c a q V10,1 P4 c a q V10,1 P2 d b q 2g M p)v V10,1 P2 cXX q

ž

gsy

e 2

Ž P2n qP2 p .

/

,

2 M p)v q 2  1 y 4V10,1 M p) Ž r 0 n cXX q Ž 1 y cXX . r 0 p 4

Ž 15 . a.

Ž 16 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

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Solving the above linear system for a leads to the expression of the retarded response function P R Ž v ,q .. This function is the polarizability, i.e. the ratio of the density fluctuation to the field strength, P R Ž v ,q. s are . It yields the following expression for the polarizability in the particular Ž s,t . channel, P Rs ,t Ž v ,q . 1 2

s

1

Ž P 0n q P 0p . Ž 1yV1s ,t P2 d . q V1s ,t Ž P 2n q P 2p . P0 d 2

2

1yV0s ,tP0 ,c yV1s ,t Ž P2 ,c q P2 , d . qV1s ,tV0s,t Ž P0 ,c P2 , d y P2 ,c P0 , d . q Ž V1s ,t . Ž P2 ,c P2 , d y P4 ,c P0 , d .

.

Ž 17 . we have used M p) s m )p rŽ1 q ac . q 1 y Õ P 2pi v ,q . XX

In the above expression and P2 i ,Õ s Õ P 2ni Ž v ,q . Ž . Ž . Ž 18 . Ž In this expression Õ s c,d,c they act as weights for the proton and neutron Lindhard functions. and i s 0,1,2. The functions P 2i N are referred to as generalized Lindhard functions. They are defined as 4 fi Ž k q q. y fi Ž k. N P 2i N s P 2i N Ž v ,q . s d3 k Ž k P Ž k H q. . , 3 v q ih y e pX Ž k . q e pX Ž k q q . Ž 2p .

H

Ž 19 . Expressions for

P 2i N Ž v ,q .

are shown in Appendix A.

Fig. 5. Distribution of strength per unit volume, isovector channel, using force SGII for higher momentum transfer: q s 2=0.23 fmy1 . Symmetric nuclear matter limit Žsolid line. and bs 0.54 for temperatures T s 0, 2, 4 and 6 MeV Ždotted, dashed, long dashed, dotted-dashed lines respectively..

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

23

In Eq. Ž17. we have used a modified coefficient V00,1 defined by V0s,t s V0s,t q V2s,t y

ž

2 M p)v q

2

/

2V1s,t 1 y 4V1s,t m )p r 0 p

.

Ž 20 .

This modified coefficient arises because the change in the momentum density induced by the external field and it is also due to the asymmetry between protons and neutrons. The complete expressions for all the four channels in terms of Skyrme parameters are shown in Appendix B. Expression Ž17. generalizes that of Ref. w3x to which it reduces when considering a symmetric nuclear matter. In order to reproduce the results of w7x one considers that V1 s 0: the final expression is not the same for the asymmetric case but it is very similar. Numerically it provides nearly the same results in several cases, as shown in Section 4.

3. Some properties of the dynamic and static polarizability The strength distribution per unit volume, SŽ v ,q ., corresponds to real transitions of the system, being thus proportional to the imaginary part of the dynamic polarizability.

Fig. 6. Distribution of strength per unit volume in the isovector channel using force SV for bs 0.54 Žsolid line. and bs8 Ždotted line. at T s 0 MeV.

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F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

It is proportional to the photoabsorption strength distribution S a b s as a direct consequence from the fluctuation–dissipation theorem w33,34x. Both are proportional to the imaginary part of the polarizability: S Ž v ,q . s Ž 1 y eyv r T . S a b s Ž v ,q . s y

1

p

I P Rs,t Ž v ,q . .

Ž 21 .

The denominator of the polarizability determines the existence of a collective mode. When there is a pole in P Rs,t Ž v ,q ., the real part of the denominator yields the energy of the resonance while its imaginary part is directly related to the width. As far as the momentum transfer is concerned with the resonant frequency it is worth to recall that in the model worked out in Refs. w1–3x q is directly related to the radius of a nucleus as long as the nucleus is identified with a ‘‘box’’ inside nuclear matter. For a heavy nucleus the momentum transfer is small, in particular, for the nucleus of lead it was considered that: q s prŽ2 R . s 0.23 fmy1 . In this relation the usual nuclear radii were considered to be R s r 0 A 1 3. This parametrization leads to the usual fit of the IVDG resonance frequency for mediumrlarge nuclei, v s BAy , where B is a constant. 1 3

Fig. 7. Distribution of strength per unit volume, isovector channel, using force SGII for r s r 0 r2. Asymmetry coefficients bs 0 Žsolid line. and bs 0.54 Ždotted..

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

25

It is important to check the results by considering the energy weighted sum rule ŽEWSR.. For the dipolar modes considered here one has to consider the following expression: ²0 < Oˆ s,t , w H ,Oˆ s,t x <0:

m1s,t s

2

.

Ž 22 .

For the asymmetric nuclear matter it yields m1s,t s

`

H0 d v S

s,t ab s

Ž v ,q . v s q 2

ž

r0 p m )p

r

Ž 1 y 2V1s,t m)p . q m0)n Ž1 y 2V1s,t m)n . n

/

.

Ž 23 . It is still interesting to remember that the following dispersion relation hold for the real Ž x X . and imaginary Ž x XX . parts of the retarded response function:

x X s d v X S Ž q, v X . P

H

ž

2 vX

v 2 y vX 2

/

,

x XX s yp Ž S Ž q, v . y S Ž q,y v . . . These dispersion relations were used in Refs. w5,7,33x.

Fig. 8. The same of Fig. 7 using SLyb.

Ž 24 . Ž 25 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

26

As will be shown in Section 4, at low frequencies Ž v < qÕF . the strength distribution of particle–hole spectrum have the general behaviour SŽ v . A v . It was pointed out in Refs.w3,35x that a simplified model can be used without missing the main features of the complete response function given above. It corresponds to neglect the function V1s,t. This roughly corresponds to the idea that the main effect of the spatial non-locality Žvelocity dependence. of the effective interaction is present in the effective mass calculation and almost absent in what concerns the collectivity. The expression of the response function becomes 1

P Rs,t

Ž v ,q . s

2

Ž P 0n q P 0p . 1 yV0s,tP0,c

.

Ž 26 .

This expression is tested in Subsection 4.1. As discussed above, this expression is not equal to that obtained from a linear response built with a Hartree approximation Ždone in Ref. w7x., but it provides similar results in some cases as discussed below. The main difference comes from the approximations which have been done in the present work which lead to the use of the asymmetry coefficients a, c and d. These approximations do not invalidate the results, since the results proved to be consistent and numerically the results of the present paper are very similar to those of Ref. w7x. A Thomas–Fermi

Fig. 9. The same as Fig. 7 for r s 2 r 0 . Asymmetry coefficients: bs 0 Žsolid., bs 0.54 Ždotted. and bs8 Ždashed..

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

27

Fig. 10. The same as Fig. 9 but for the force SLyb.

type approximation also produces similar results for the symmetric nuclear matter case w3x. 3.1. Static polarizabilities

™

In the static limit Ž vrq 0. of the response function one obtains the static polarizabilities for the respective channel at finite temperature and for asymmetric matter for a given density. One obtains r0 P 0,1 Ž v 0,q 0,T ,b . s , 2 At Ž T ,b, r . r0 P 1,0 Ž v 0,q 0,T ,b . s , 2 As Ž T ,b, r . r0 P 1,1 Ž v 0,q 0,T ,b . s , 2 Ast Ž T ,b, r .

P

0,0

™ ™ ™ ™ ™ ™ 3r . Ž v ™ 0,q ™ 0,T ,b . s K Ž T ,b, r . 0

D

Ž 27 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

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Fig. 11. Distribution of strength per unit volume – scalar channel – using SGII. Parameters bs 0 Žsolid line., 0.54 Ždotted. and 8 Ždashed., transferred momentum q s 0.23 fmy1 at the saturation density.

™ ™

At zero temperature in the symmetric nuclear matter limit these static polarizabilities reduce to the symmetry energy coefficients usually defined as at s At ŽT 0,b 0., the p–n symmetry energy coefficient; as s As ŽT 0,b 0., the spin symmetry energy coefficient; ast s Ast ŽT 0,b 0. the spin-isospin symmetry energy coefficient, and K D s A K ŽT 0,b 0., a ‘‘dipolar compressibility’’ of nuclear matter. This last coefficient is related to the usual nuclear matter volume compressibility Ž K` . by the following expression:

™ ™

™ ™ ™ ™

K D s K` q

4 5

TF y 2V1 k F2 r 0 q

3 4

t 3 r 0aq1 ,

Ž 28 .

where k F is the momentum at the Fermi surface. The values for these coefficients will be shown below. The symmetry energy coefficients for the symmetric nuclear matter may then be written in the following form: at s

TF

ast s

q

r0

3 TF 3

2 q

V00,1 q k F2 r 0 V10,1 , as s

r0 2

TF

q

3

V01,1 q k F2 r 0 V11,1 , K D s 3

r0 2

ž

2 3

V01,0 q k F2 r 0 V11,0 ,

TF q r 0 V00,0 q 2 k F2 r 0 V10,0 ,

/

Ž 29 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

29

where TF is the kinetic energy at the Fermi surface and V0s,t and V1s,t are given in Appendix B. In terms of the Fermi liquid parameters Ž F0 , F0X ,G 0 and GX0 . one can write the symmetry energy coefficients as at s

r0 N0

ats s

Ž 1 q F0X . , as s

r0 N0

r0 N0

Ž 1 q GX0 . , K` s

Ž 1 q G0 . ,

9r 0 N0

Ž 1 q F0 . .

Ž 30 .

It is clearly seen from expressions Ž27. that for asymmetric nuclear matter, with different densities at finite temperatures one gets corrections to the usual expressions in the Fermi liquid theory shown above.

4. Results In this section the resulting strength distributions SŽ v ,q . are plotted for several cases with different asymmetry coefficients b Žequivalent to a by expression Ž5.. and at

Fig. 12. The same as Fig. 11 but using SLyb.

30

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

different densities, from 0.5 r 0 up to 2 r 0 in the four channels. In this work the SGII w9x and SLyb w10x are used in all but one of the figures. The former was adjusted considering spin and spin-isospin observables as well as ground state observables like compressibility and, as other Skyrme interactions, reproduces well several ground state properties. The second was fitted to reproduce neutron matter properties obtained from a microscopic calculation w36,37x. There are important differences mainly concerning the non-local terms, i.e. dependent of momentum which are bigger for the latter. These interactions are not supposed to provide good results for higher densities since one expects relativistic effects to be important. A non-realistic Skyrme force, SV, which has no density dependence, is used in one example in order to stress the importance of this term. Finally the static limit is studied. 4.1. Asymmetric polarizabilities in the isoÕector channel In Fig. 1 the strength distributions are shown for cases of different asymmetry coefficients compared to the symmetric case Ž b s 0. considering force SGII. First, for b s 0.54 Žequivalent to a , 0.22, and corresponding to the asymmetry of the nucleus of 208 Pb., at T s 0 MeV and at T s 6 MeV. The collectivity for such a small asymmetry does not change a lot, but the frequency of the mode is nearly 1 MeV higher. With

Fig. 13. The same as Fig. 11 but for a higher density: r s 2 r 0 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

31

increasing temperature the zero sound mode of T s 0 MeV becomes damped by the Landau damping mechanism, which is noticed for T s 6 MeV in the figure. A more detailed analysis of the temperature dependence of the strength was done in w3x. The other curves correspond to higher asymmetry coefficients b s 2,8 and 32. As stated before, the more frequently used asymmetry coefficient a is related to b by b s 2 arŽ2 q a .. The zero sound modes become more collective, increasing the resonant frequency with higher neutron proton asymmetry whereas the particle–hole spectra at smaller energies get suppressed. There is a small bump in the lower part of the spectrum Žparticle–hole excitations. for the case b s 2 and one still smaller for b s 8. In Fig. 2 the same cases are shown for the force SLyb. The modes are less collective and the collectivity is even suppressed at low asymmetry Žcases b s 0 and 0.54.. The frequencies of the resonances are smaller than in the case of force SGII. These differences are due to the higher values of the velocity-dependent terms of the SLy Skyrme interaction w10x. The strength for the simplified model discussed in Section 3 Žwith V1 s 0. is shown in Fig. 3 for b s 0, 0.54 and b s 8 at zero temperature with force SGII. For low b there is a complete agreement with Ref. w7x. Comparing the collectivity by regarding the poles contribution to the total sum rule, shown in Table 1, it is noted that the collectivity diminishes without considering the function V1 which is composed by non-local Žmomentum-dependent. terms. This is the opposite tendency of the results obtained for symmetric nuclear matter. The energy of the mode also is a little smaller – to be

Fig. 14. The same as Fig. 13 but using SLyb.

32

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

compared with Fig. 1. The same for the force SLyb is shown in Fig. 4. In this case, however, the collectivity and the frequency are slightly increased neglecting V1 terms for non-zero asymmetry. The curves for symmetric nuclear matter are in agreement with w3,5,7x, In Ref. w7x phenomenological values for the Fermi liquid parameters were used ŽFig. 22 of that article. for a RPA using a Hartree approximation with q s 0.253 fmy1 . In Fig. 5 the case of a higher transferred momentum Ž q s 2 = 0.23 fmy1 . is exhibited for the cases of b s 0 and 0.54 at zero temperature with force SGII: it is twice the momentum associated with the nucleus of lead as discussed in Section 3 Ž q s 0.23 fmy1 .. This figure should be compared to Fig. 1. This dependence on q is expected to happen as long as the zero sound takes place Ž Eres s c 0 q ., and it happens for small q. It is seen that the energy of the mode is twice the original one and Žconsequently. they become less collective. Moreover, the collective mode contribution to the EWSR is nearly the same Žthe greater the asymmetry the closer the percentual contribution of the modes due to q s 0.23 and q s 0.46 fmy1 .. This can be noted in the last columns of Table 1. The energy of the resonance is twice the original, reminiscent of the basic characteristics of the double-IVDGR mode in nuclei w20–22x. The relevance of the effective mass and of the n-p coefficient b is explicated by fitting the resonant energies by the following expressions. In symmetric nuclear matter the energy of the resonance, which is mainly determined by the effective mass for fixed V0 and q, can be parametrized nearly by E res , Cqrm ) , where C is a constant w3x. In

Fig. 15. Distribution of strength per unit volume – spin channel – using SGII. Parameters bs 0 Žsolid line., 0.54 Ždotted. and 8 Ždashed., transferred momentum q s 0.23 fmy1 at the saturation density.

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

33

asymmetric nuclear matter, for the force SGII, the energy shift due to the asymmetry may be written Žfor 32 ) b ) 0. as

D E res Ž b . , Alog Ž Bb . q Cb,

Ž 31 .

with C s 2 MeV, B s 4 and A , 2.3 MeV for the SGII, while for SLyb one gets A , 2.7 MeV, B , 2.6 and C s 0.8 MeV. In Fig. 6 the response function calculated with the Skyrme force SV, which contains no density-dependent terms, is shown for b s 0.54 and b s 8. For low asymmetry it exhibits no qualitative difference from that of the symmetric nuclear matter: there is no collectivity w3x. Indeed, its effective mass is too low Ž m ) , 0.4 m.. However, for a much bigger asymmetry there is a collective mode, since the interaction become strongly repulsive, even without a density-dependent term. For the very collective cases showed above there is always a critical temperature at which the pole Ževen when it is an imaginary pole. in the response function disappears. It is very interaction-sensitive and some values are given below in the case of the interaction SGII. For b s 0 this disappearance is found to happen at nearly T , 4–5 MeV. The absence of the interaction function V1 makes it change a little to T , 5–6 MeV, indicating more collectivity. For the Skyrme interaction SkM w38x, not analyzed in the rest of the paper, which differs from SGII mainly by the compressibility and symmetry energy coefficient Žthey are bigger than the values obtained with SGII.

Fig. 16. The same as Fig. 15 using interaction SLyb.

34

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

Tc , 7–8 MeV. Increasing the neutron–proton asymmetry makes the critical temperature greater, for instance, considering b s 0.54 and b s 2 we found Tc , 11 MeV and Tc , 15 MeV respectively. For the supernova Žcontracting. matter one usually considers b s 1 and r s 0.915 r 0 . For this case Tc , 8–9 MeV is obtained in the isovector channel and Tc , 6 MeV in the spin-isovector one Žwhich will be analyzed in the following.. It can be shown that in this last channel for these values of b and r there is a collective zero sound which does not take place for symmetric nuclear matter at normal density Žshown in Subsection 4.3.. For the force SLyb there is no zero sound in this channel up to b , 2 at the saturation density. As far as the temperature dependence is concerned, some remarks on the damping mechanism are useful. For the zero sound mode there are two mechanisms of damping: first the collisions between nucleons Žnot considered in this paper. which may be present even at low T, and also the Landau damping which occurs at low frequencies smaller than that of the Fermi surface v ( qÕF . In spite of that, zero sound does not propagate in the full collision regime which takes place at non-zero temperatures but in a Žat least nearly. non-equilibrium system w33x. The transition of the zero sound regime to the first sound regime was analyzed in w39x. These two mechanisms may be relevant for the shape of the strength distribution of the giant resonances usually assumed to be a lorentzian.

Fig. 17. The same as Fig. 15 but for a higher density: r s 2 r 0 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

35

4.2. Varying nuclear density The variation of nuclear matter density is achieved by adjusting the chemical potential present in the Fermi–Dirac occupation number which, by the way, is an important piece of the Lindhard functions. In Fig. 7 Žstill in the isovector channel., the response function for a density half the equilibrium one Ž r s r 0r2. is shown for a low asymmetry coefficient Ž b s 0 and 0.54. with force SGII. The mode is less collective than at the saturation density as can also be seen from Table 1, and its energy is smaller if compared to Fig. 1 Žnormal density case.. In Fig. 8 the same case for the interaction SLyb is shown. In this last force however the mode is a little bit more collective for b s 0.54 than it was for the case with the saturation density with a smaller vres . This effect of reducing Eres for decreasing r is present in exotic finite nuclei, in spite of the differences between the two Skyrme forces analyzed in this paper. A higher density case Ž r s 2 r 0 . is shown in Fig. 9 for the force SGII. The modes become more Žless. collective for big Žsmall. proton–neutron asymmetry than it is for nuclear matter at the saturation density Žcomparing to Fig. 1.. The frequency of the modes are higher than in Fig. 1, mainly for high b. This illustrates the superluminal problem for the Skyrme type interaction, where the sound velocity, and in the present case the zero sound velocity Ž c 0 s Eresrq ., becomes higher than the velocity of sound w40x. The effective interaction Žin the case of large b . is more repulsive than for the case

Fig. 18. The same as Fig. 17 using interaction SLyb.

36

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

where r s r 0 . For the case of the force SLy this effect is the opposite Žseen in Fig. 10.: the effective particle–hole interaction becomes attractive at higher densities, causing an instability. In this last case for b s 8 the energy-weighted sum rule is not satisfied anymore due to the presence of the corresponding phase transition. 4.3. Normal and double density asymmetric polarizabilities in the spin, spin-isospin and scalar channels The case of the scalar channel is exemplified in Figs. 11 and 12 at the saturation density respectively with interactions SGII and SLyb. In both figures the strength functions for b s 0.54 and b s 8 are compared to that of symmetric nuclear matter at T s 0 MeV. There is strong collective effect only for b s 8, when the interaction becomes repulsive enough. In the case of symmetric nuclear matter Ž b s 0. the results are in agreement with Refs. w5,7x. For a proton–neutron asymmetry as big as b s 8 the results may be compared to the neutron matter response function of Ref. w7x. In the present work the energy of the mode is placed at higher energies than Reddy et al. found in neutron matter for the interaction SLyb. First of all, in this last paper a smaller momentum transfer Žnearly half of q used in the present work. pushes the resonant energy down to smaller energies. Besides that, as discussed above, their RPA with a Hartree approximation neglects functions of the velocity-dependent terms, which are much more important Žmainly for the SLyb force. for asymmetric nuclear matter. This

Fig. 19. Distribution of strength per unit volume – spin-isovector channel – using SGII. Parameters bs 0 Žsolid line., 0.54 Ždotted. and 8 Ždashed., transferred momentum q s 0.23 fmy1 at the saturation density.

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

37

last difference may be observed in other cases. The collective mode for the scalar channel at a density r s 2 r 0 does not take place anymore, as shown in Figs. 13 and 14 respectively for forces SGII and SLyb. The corresponding p-h interaction at this density Žfor SGII. is more attractive. The spin channel is analyzed in Figs. 15 and 16 for the saturation density using respectively forces SGII and SLyb. In the first figure the cases for b s 0, 0.54 and b s 8 are exhibited and, as expected, the mode becomes more collective with increasing asymmetry w24,25x. The effective interaction SLyb ŽFig. 16. favors a more repulsive interaction in this channel making the strength be more collective. The double density case is shown in Figs. 17 and 18 and the energy of the collective mode Žwhich becomes still more collective. is pushed away from the particle–hole spectrum. There is a spin zero sound at higher n-p asymmetry b and also at r s 2 r 0 . It is stronger for the interaction SLyb. The same analysis as done above for the differences between the present work and that of Reddy et al. holds. Two cases of intermediate asymmetry coefficients has also been done in w7x Žfor x p s 0.049 – corresponding to b s 18 – at r 0 and for x p s 0.22, corresponding to b s 2.54 at 3 r 0 . and they are in qualitative agreement with the present work. Finally the spin-isospin channel at the saturation density is shown in Figs. 19 and 20 respectively for interactions SGII and SLyb. In both cases the strengths become more collective for higher n-p asymmetry. The force SLyb is more repulsive than SGII at higher asymmetry b but not for low b. Examples for higher density Ž2 r 0 . are shown in

Fig. 20. The same as Fig. 19 using interaction SLyb.

38

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

Figs. 21 and 22. For very low p-n asymmetry the forces tend to become attractive at higher densities. This is seen in Fig. 22 for the b s 0 case using interaction SLyb. The strength presents an unusual shape. However, for bigger asymmetries, b s 8 for instance, the repulsive character of the p-h interaction is still dominant yielding zero sound-modes. The symmetric nuclear matter limit was investigated in Ref. w5x, being in complete agreement with the present article. Neglecting V1 terms nearly leads to the results of Ref. w7x. In short there are collective motions in these three channels only for higher proton–neutron asymmetries or higher densities. For Žvery. small b, these channels exhibit rather a behaviour identified with a non-correlated Fermi gas which seems to be in qualitative agreement with experimental results from the Žspin and scalar. giant dipole resonances. The coherent modes do not have the zero sound relation dispersion but the collective ones do have. Another tendency is that the p-h interaction with increasing densities may become attractive Žin the spin-isospin channel. giving place to a phase transition beyond which one may expect to find pion condensation w41x. 4.4. Static polarizabilities In Table 2 the denominators of the static polarizabilities are shown as well as their values for different asymmetry coefficients of nuclear matter with interaction SGII. The temperature dependence is negligible mainly because, in this work, the density is kept

Fig. 21. The same as Fig. 19 but for a higher density: r s 2 r 0 for bs 0 and bs8.

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

39

Fig. 22. The same as Fig. 21 using interaction SLyb.

constant, independently of the temperature for a fixed asymmetry coefficient b. Also shown in this table are the most probable values for these coefficients after the analysis of w42x with the parametrization of Landau theory for Fermi liquids. An extensive analysis for these symmetry energy coefficients for variable densities, p-n asymmetries at finite temperature will be shown elsewhere w43x. For the ‘‘dipolar compressibility’’ one notes that its value decreases rapidly with increasing n-p asymmetry, an expected behaviour discussed in w10x. It is seen that the

Table 2 Static polarizabilities in the four channels: isovector channel Ž At s r 0 r2 P 0,1 ., spin-isovector Ž Ast s r 0 r2 P 1,1 ., spin Ž As s r 0 r2 P 1,0 . and scalar channel Ždipolar compressibility: A K s r 0 r2 P 0,0 ., for different proton–neutron asymmetry coefficients b in the case of interaction SGII. In the column ‘‘Exp’’ the most probable experimental values after the analyses of Ref. w42x based on Landau Fermi liquid theory are shown. In the last column the variation of the coefficients with the increase of the temperature, for DT s6 MeV when bs 0.54 Channel, r 0

bs 0

bs 0.54

bs 2

Exp.

D A s,t

At ŽMeV. At , s ŽMeV. As ŽMeV. A K ŽMeV.

26.9 23.4 15.7 14.7

34.7 29.5 15.9 11.2

59.5 48.4 55 y2.6

28 39 10

0.05 0.06 0.01 0.2

™ 38 ™ 41 ™15 –

40

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

static polarizabilities associated with the energy symmetry coefficients in each channel depend strongly on the asymmetry coefficient b. One can expect these polarizabilities Ž A s,t . to be more appropriate that the symmetry energy coefficients Ž a s,t s A s,t ŽT s 0,b s 0.. if a finite temperature asymmetric nuclear matter calculation is considered, as, for example, in the electron capture in hot asymmetric supernovae matter w24,25x. For this system, the larger the symmetry energy contribution Ž at or At . the more quenched the neutronization and, consequently, the higher the lepton fraction at trapping. This helps the supernovae explosion causing a stronger shockwave which would hopefully be more compatible with experimental observations. In Ref. w28x an approach from a Fermi gas model was considered Ži.e. considering that at has well-defined potential and kinetic parts.. An increase of nearly 2.5 MeV for the asymmetry coefficient was found with the rise of temperature from 0 to 1 MeV. This would happen because the total effective mass was considered, i.e. the coupling to the vibrational states of the nuclear medium yields a temporal non-locality to the mass, increasing it at low temperatures. In contrast to this result, a Monte Carlo calculation of w29x shows almost no variation for that coefficient. In the present work the temperature dependence is also absent, as seen in the last column of Table 2, because the matter density does not depend on T. Precise experimental fits to test these results as well as the spin and spin-isospin energy symmetry coefficients Ž as , As and ast , Ast . are needed. A more extensive analysis of this subject will be carried out elsewhere w43x.

5. Conclusions An almost exact expression for the response function of hot asymmetric nuclear matter at different densities of nucleons was calculated, extending the work of Ref. w6x and generalizing the case previously studied of symmetric nuclear matter with fixed density w3x. Some results were compared to the work of Ref. w7x which has been independently developed. In the isovector channel the collectivity of the response was analyzed as well as its evolution with increasing excitation energy until the disappearance of the zero sound whose critical temperature depends very strongly on the effective interaction, nucleon density and on the proton–neutron asymmetry coefficient b. At normal density the spin and the scalar channel show no collective behaviour that can happen for very asymmetric nuclear matter and at higher densities. The spin-isospin channel exhibits some coherence which increases a lot with the parameter b. In these last cases the increase of b causes the appearance of a zero sound in the respective channel at the respective channel. At higher densities the isovector Žfor low n-p asymmetry. and the scalar channels have the tendency of getting attractive p-h interactions favoring instabilities typical from phase transitions. This is also the case for the spin-isospin channel for low p-n asymmetries at high densities mainly for the interaction SLyb as expected. The zero sound phonon is a coherent superposition of states that can be obtained by giving extra momentum "q to one of the quasiparticles. This disturbance propagates only at very low temperatures or very high frequency. It propagates in a collisionless regime, and collisions mainly start to take place with increasing energy excitation, destroying collectivity. The presence of such collective modes indicates that nuclear

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

41

matterrlarge nuclei are reasonable correlated systems and these many-body correlations present in the linear response approach should be taken into account in the description of many important observables mainly at zero temperature. It is a behaviour typical for correlated quantum liquids where zero point motion prevents the forming of a solid. The critical temperature beyond which zero sound does not take place anymore Žnor the momentum transfer between nucleons. are expected to be measured. It is important to keep in mind that the Skyrme effective interaction SGII has been fitted at normal density and, in principle, are not suitable for neither much higher densities, when relativistic effects are very important, nor very low densities. On the other hand, the other force used ŽSLyb. was fitted for very asymmetric nuclear matter. Static polarizabilities were also explicitly calculated exhibiting almost no variation with the temperature, what happens mainly because the densities were considered to not depend on T. These static polarizabilities Žwith T, r and b dependence. may be more suitable than the usual zero temperature symmetry energy coefficient for phenomena where asymmetric Žat different densities. nuclear matter is considered. Acknowledgements The author wishes to thank FAPESP, Brazil, for the financial support, and D. Vautherin who introduced the astrophysical subject to him. Appendix A. Expressions of the asymmetric generalized Lindhard functions In this appendix we give the explicit expressions of the real and imaginary parts of the generalized asymmetric Lindhard functions defined in Eqs. Ž19.. For this purpose we need to define the following integrals: 2 Ž f i Ž k H q. y f i Ž k. . 3 Ž2 N . I2i N s d kk . Ž A.1 . 3 v q ih q e pX Ž k H q . y e pX Ž k . Ž 2p .

H

The imaginary parts of the generalized Lindhard functions are given by I P 0i Ž v ,q . s y

I

P 2i

Ž v ,q . s y

m )i M p)

p qb

log

2 m )i 2 M p)

pb 2 q

1 q e b Ž m iyEy 1qe

b

ž

m) i

.

,

M p) b Ž m iyE q ) . mi

M p) m )i

qLi 2 1 q e b Ž m iyEq

M p)

(E

q Ey

log

1 q e b Ž m iyEy 1 q e b Ž m iyEq

M p) m) i

.

/

y Li 2 1 q e b Ž m iyEy

ž

M p) M p) m) i

H

(

.

M p) m) i

I P4i Ž v ,q . s yq 2 m )i Eq I P 2i q 2 I I4i y 2 2 m )i Eq q I I2i , where Li 2 is the Euler dilogarithmic function w44x x log Ž t . Li 2 Ž x . s dt, 1 ty1

(

.

m) i

.

/

,

Ž A.2 . Ž A.3 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

42

and E "s

M p) 2 q2

ž

v"

2

q2 2 M p)

/

.

Ž A.4 .

The expressions of the functions I I2i and I I4i are I I2i Ž v ,q . s y

m )i 2 M p)

p qb 2



M p)

b Eq

m )i

qLi 2 1 q e b Ž m iyEq

ž

I

I4i

Ž v ,q . s y

2 M p) 3 m )i 2

p qb

`

H1 d zz log

2 Eq

log



1 q e b Ž m iyEy 1qe

log

1 q eyb

q2

.

1 q e b Ž m iyEy 1 q e b Ž m iyEq z Eq

1 q eyb

ž

M p) m) i z Eq

yv M

.

b Ž m iyE y

/ y Li ž 1 q e 2

ž

m) i

M p) b Ž m iyE q ) . mi

M p) m) i

M p)

m) i

.

/

0

,

M p) m) i

.

M p) m) i

M p)

.

ym i

m) i

/

)

m) i

M p)

ym

/

0

.

/

,

Ž A.5 .

At zero temperature the real parts of the P 2 N are given by R P 0 Ž T s 0. s

R P 2 Ž T s 0. s

M p) k F

p

2

M p) k F3 2p

2

ž ž

y1 q

kF 2q

f Ž xq . q f Ž xy .

2 2 y3 q xq xyq xq q xy q

kF 2q

2 =f Ž xq . q Ž 1 y xy y 2 xq xy . f Ž xy .

ž

Ž 1 y xq2 y 2 xq xy .

/

,

(

R P4 Ž T s 0 . s 2 R I4 Ž T s 0 . y 2 q 2 M p) Eq R I2 Ž T s 0 . qq 2 M p) Eq R P 0 Ž T s 0 . q

1 3p 2

M p) q 2 k F3 ,

/

where R eI2 Ž T s 0 . s

M p) k F3 4p

ž

2

ž

2 2 y3 y xq xyy xq q xy q

2 q 1 q xy q

4 m )v k F2

/

f Ž xy .

/

,

kF 2q

Ž 1 q xq2 . f Ž xq .

Ž A.6 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

R I4 Ž T s 0 . s y

M p) k F6 2p 2 q

1

žž

6

1 3 q with x "s

ž

1 3

k F2

Ž 1 q xy2 . q

3 2 M p)v xy

2 M p) 2v 2 k F4

/

5q

Ž 1 q xq2 q xq4 . f Ž xq . q 3k 6

3 xy q

k F2

M p)v

1

=f Ž xy . q

q

Ž 1 q xy2 q xy4 . q

43

1 3

q

3 xq q

2 M p)v xy k F2

4 M p) 2v 2 xy k F4

/

q

F

8 q 2 xq k F2

/

,

q

5 xy

3

q

5 xq

3

Ž A.7 .

)

q 2 kF

1 < "M p v, and f Ž x . s Ž1 y x 2 .log < xx y q1 . qk F

For non-zero temperature the expression of R P 2 N Ž N s 0,1,2. is an average of the zero temperature functions calculated for the same values of v and q, but with various values of the Fermi momentum k F distributed with a weight factor which is just the derivative of the Fermi occupation number. Explicitly one has the following formula: R P 0i Ž v ,q,T . s y R P 0 Ž v ,q,T s 0,k F s k . d f i Ž k ,T . ,

H

R P 2i Ž v ,q,T . s y R P 2 Ž v ,q,T s 0,k F s k . d f i Ž k ,T . ,

H

R P4i Ž v ,q,T . s y R P4 Ž v ,q,T s 0,k F s k . d f i Ž k ,T . ,

H

Ž A.8 .

where f i Ž k,T . is the occupation number. For the case of zero temperature we have d f i Ž k . s yd Ž k y k Fi . d k , yielding the above expressions for these functions.

Appendix B. Functions Vis,t In the isovector channel one obtains t0

ž ž

V00,1 s y

2

q2 y 16 V10,1 s

1 16

V20,1 s t 3

ž

x0 q

1 2

/

y

t3 12

ž

x3 q

1 2

/

r 0a

/

Ž 3t1Ž 1 q 2 x 1 . q t 2 Ž 1 q 2 x 2 . . Ž 1 q bc . ,

Ž t 2 Ž 1 q 2 x 2 . y t1Ž 1 q 2 x 1 . . , 1 2

q x 3 r 0ay1 Ž c rn0 y Ž c y 1 . r p 0 . r12,

/

Ž B.1 .

F.L. Braghinr Nuclear Physics A 665 (2000) 13–45

44

where r 0 n , r 0 p and r 0 are the proton, neutron and total saturation densities of asymmetric nuclear matter. For the other channels

ž

V00,0 s 3 V10,0 s 3

t0 4

t1 16

q Ž a q 1. Ž a q 2. q Ž5 q 4 x2 .

ž

V01,0 s y Ž 1 y 2 x 0 . qŽ 1 q 2 x 2 . V11,0 s y Ž 1 y 2 x 1 .

t0 4

t2 32 t1 16

t2 16

t3 16

r 0a q q 2 9

, V20,0 s

y Ž1 y 2 x3 .

//Ž

ž

t3 12 t3 24

t1 32

y Ž5 q 4 x2 .

t2 32

//Ž

1 q bc . ,

Ž x 3 q 0.5 . Ž c rn q Ž c y 1 . r p r 0Ž ay1. . , r 0a y q 2 3 Ž 1 y 2 x 1 .

ž

t1 32

1 q bc . ,

q Ž1 q 2 x2 .

t2 16

,

t3

r ay1 Ž 2 q a . Ž rn c q r p Ž c y 1 . . , 48 0 t0 t3 a t1 t2 V01,1 s y y r 0 q q 2 y3 y Ž 1 q bc . , 4 24 32 32 V21,0 s y

ž

V11,1 s

//

ž

yt1 q t 2 16

, V21,1 s ya

t3 24

r 0Ž ay1. Ž rn c y r p Ž c y 1 . . .

Ž B.2 .

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