DIRAC-BRUECKNER APPROACH TO NUCLEAR MATTER IN NON-EQUILIBRIUM

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DIRAC-BRUECKNER APPROACH TO NUCLEAR MATTER IN NON-EQUILIBRIUM

R. Malfliet, B. ter Haar, W. Botermans

To cite this version:

R. Malfliet, B. ter Haar, W. Botermans. DIRAC-BRUECKNER APPROACH TO NUCLEAR MAT- TER IN NON-EQUILIBRIUM. Journal de Physique Colloques, 1987, 48 (C2), pp.C2-287-C2-294.

�10.1051/jphyscol:1987244�. �jpa-00226513�

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JOURNAL DE PHYSIQUE

Colloque C2, supplbment au n o 6, Tome 48, juin 1987

DIRAC-BRUECKNER APPROACH TO NUCLEAR MATTER I N NON-EQUILIBRIUM

R. MALFLIET, B. TER HAAR and W. BOTERMANS

Kernfysisch Versneller Instituut, Zernikelaan 25, NL-9747 A24 Groningen, The Netherlands

Nous prdsentons un traitement unifid de la matiere nucldaire en Lquilibre et non-6quilibre bas6 sur la methode Dirac-Brueckner. L'dquation cindtique quantique qui en rdsulte contient de f a ~ o n self-consistente un champ moyen et un terme de collision. Nous discutons les diffsrences avec les resultats de l'equation VUU.

Abstract

We present a unified treatment of equilibrium and non-equilibrium phenoma in nuclear matter based on the Dirac-Brueckner approach. The resulting quantum kinetic equation contains in a selfconsistent manner a mean field and a collision term. Differences with VUU are discussed.

I. Introduction

Equilibrium properties of nuclear matter such as the equation of state and non-equilibrium phenomena i.e. nucleus-nucleus collisions are considered using the same unified approach. The Dirac-Brueckner formalism with a one- boson-exchange nucleon-nucleon interaction describes .the equilibrium properties of nuclear matter. Based on the same ingredient, the Dirac- Brueckner G-matrix, one can construct a Boltzmann-type kinetic equation which is more general as the well-known Uehling-Uhlenbeck equation. This equation contains a mean field and a collision term which are determined in a selfconsistent way. In the following we give a short account of this approach.

11. Concepts of the Dirac-Brueckner approach

In the self-consistent relativistic Dirac-Brueckner approach [ 1 , 2 , 7 ] a nucleon in the nuclear medium may be viewed as a bare nucleon that is

"dressed" in consequence of its effective two-body interaction with the other nucleons in the medium. So far as the one- and two-particle properties of the nuclear matter go, these can be described by a coupled set of three covariant non-linear integral equations. The self-energy C(k) of the physical nucleon appears in the Dyson equation which relates the bare and physical ("dressed") nucleon propagators:

In the Brueckner formalism this self-energy is given by a summation over all two-body interactions, which leads'to the equation:

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987244

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C2-288 JOURNAL DE PHYSIQUE

The f i r s t term i s u s u a l l y r e f e r r e d t o a s t h e d i r e c t ( o r H a r t r e e ) c o n t r i b u t i o n t o t h e s e l f - e n e r g y and t h e second a s t h e exchange ( o r Fock) c o n t r i b u t i o n . The n o n - r e l a t i v i s t i c f i r s t - o r d e r Brueckner t h e o r y i s o f t e n r e f e r r e d t o a s t h e Brueckner-Hartree-Fock approach. I n g e n e r a l t h e e f f e c t i v e t-matrix r i s a s o l u t i o n of t h e medium dependent Bethe-Salpeter e q u a t i o n which can be w r i t t e n a s :

The d i f f e r e n c e w i t h t h e vacuum c a s e i s t h a t t h e nucleon s t a t e s a r e now d r e s s e d p a r t i c l e s t a t e s . We n o t e t h a t t h e f u l l four-dimensional Bethe-Salpeter

e q u a t i o n i s t e d i o u s t o s o l v e , and i s t h e r e f o r e u s u a l l y reduced t o a c o v a r i a n t three-dimensional form, t h e s o - c a l l e d q u a s i - p o t e n t i a l e q u a t i o n . To c o n s t r u c t t h e OBE NN-interaction, we opted f o r t h e Thompson-reduced form of t h e Bethe- S a l p e t e r e q u a t i o n and f u r t h e r m o r e we r e p l a c e d t h e f u l l nucleon propagator by i t s positive-energy c o n t r i b u t i o n . We s h a l l adopt t h e same procedure f o r n u c l e a r m a t t e r . Therefore i n eq. (2.4) t h e f u l l two-body k e r n e l K i s r e p l a c e d by a q u a s i - p o t e n t i a l U and t h e d r e s s e d nucleon p r o p a g a t o r s G by an e f f e c t i v e two-body propaRator g, t o y i e l d :

where t h e a s t e r i s k s (*) denote t h e f a c t t h a t t h e nucleon s t a t e s a r e d r e s s e d , a s w i l l be explained i n t h e f o l l o w i n g .

The formal s o l u t i o n of e q . (2.1) i s

For i n f i n i t e n u c l e a r m a t t e r , t h e Lorentz s t r u c t u r e of t h e s e l f - e n e r g y assumes t h e g e n e r a l form:

where t h e t e n s o r term h a s dropped o u t because of d i f f e r e n t symmetry r e l a t i o n s . T h i s s t r u c t u r e s u g g e s t s t h e f o l l o w i n g d e f i n i t i o n s :

This e n a b l e s u s t o e x p r e s s eq. (2.6) a s G(k) = (K* - m*)-l w i t h K = y v k

.

^The

o n - s h e l l behaviour of t h e e f f e c t i v e nucleon i s t h e n g i v e n by a Dirac e q 8 a t i o n :

which y i e l d s a positive-energy s o l u t i o n :

* * ¹

E + m + + *

u * = ..* x,

2m * *

w i t h t h e energy on-shell: E +m

from which we deduce t h e s i n g l e - p a r t i c l e energy:

* ⁰ ⁰

E(k) = E - c ( k ) = { ~ ~ ( l + ~ ~ ( i ) ) ~ + (% + ~ ~ ( k ) ) * ) ~ - ⁱ(k) (2.11)

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In general m* is momentum dependent. However it appears that the momentum dependence of C is weak and EV iq much smaller in magnitude than either .XS and EO.

Our OBE interaction contains IT, w , p , E , n and 6-exchange and the pgram te s of the interaction are given in ref. [7]. t E A monopole formfactor A / ( A +4 ) is added to the vertices. Solving the Thompson equation, the OBE interaction (including isobar degrees of freedom) gives a good description of the free NN phase shifts, cross sections and polarization data. With respect to the saturation properties of nuclear matter, the DB approach turns out to be very successful, which has already been pointed out by Shakin et al. and by Machleidt and Brockmann [1,2]. Our calculation [2,7] gives a binding energy of EB = -14 MeV at a saturation density of p = .16 This is closer to the empirical values than conventional non-reqativistic Brueckner calculations.

The corresponding compression modulus K is 250 MeV.

Within the DB model that we described here, single-particle self-energies above and below the Fermi surface can be calculated in exactly the same way.

The dressed nucleon propagators that enter in the calculation are constructed

S 0 v

by using the constants E , ^{C and 2}, which are obtained at the Fermi surface.

The major difference of the self-energy calculation for particles below or above the Fermi surface, is that below the Fermi level C(k) is a real, above the Fermi sea it becomes a complex quantity. The single-particle self-energies above the Fermi sea can be related to the nucleon-nucleus optical model and we obtained good agreement with known data [7,9].

111. Quantum kinetic equation based on the Dirac-Brueckner formalism The Uehling-Uhlenbeck equation [3,4] contains only minimal quantum mechanics through the Pauli blocking of phase space in the collision term. A consistent non-relativistic derivation of this equation, exposing all the necessary approximations and truncations, has been given by R. Snider [81. The starting point is the following operator-form eq. for the one-particle density matrix operator p l:

where the brackets denote commutators, K1 is the one-particle kinetic energy operator and V12 is the two-particle interaction. The equation (3.1) is the first equation of the quantum BBGKY-hierarchy of kinetic equations where low- order density matrices ( P ~ ) are 1in)ced to higher-order ones (p12, the two- particle density matrix operator). A similar equation expresses p12 in terms of the three-particle density ma~rijt p123. Invoking the following assumptions:

a) dilute limit, i.e. three-particle correlations are negligible (only two- particle correlations are included)

b) the Boltzmann molecular chaos assumption: in between collisions (which correlate particles) particles are uncorrelated.

Snider obtained the following solution for pL2 expressed in one-particle density matrices and the interaction V12:

Here hil2 is the Mbller-operator, familiar from two-body scattering theory:

where gl2 is the unperturbed 2-particle propagator including the Pauli blocking operator Q and E is an eigenvalue of K1+K2 corresponding to a scattering state. ~ t s o Q12 can be linked td the so-called t-matrix T12 as follows:

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and clearly T12 obeys a Lippmann-Schwinger equation:

Putting all the equations for h2 together and introducing the result in eq.

(3.1) one obtains the followingl$inetic eq. for p 1 (where we have taken T12 to be anti-symmetric in interchange (1,2)):

From this result we observe that if we neglect the term2between brackets

I...} on the left-hand side (which is of the order of p and thus can be left

out in the dilute limit) we obtain an equation very simllar to the Uehling- Uhlenbeck eq. 131. The right-hand side of (3.7) corresponds to the collision term (with loss- and gain term) and restricting it to on-energy-shell we recover the U-U collision term including the Pauli blocking term since now

and the on-energy-shell t-matrices T +

12 T12= I T12 1 reduce to the familiar (differential) cross section if one neglects Q 2 in (3.4). A more explicit discussion can be found in the work of Snider 181. However it should be clear now that the U-U eq. invokes additional assumptions in order to reduce (3.7) to it. Among them are the neglection of the mean field term U = Tr Re T p

1 12 2

which contains a Hartree and Fock term due to anti-symmetry and furthermore an on-the-energy-shell restriction in the collision term. Also the Pauli blocking of intermediate states in the T12-eq. (3.6) through the Q1 -operator in g12 is left out. Pauli blocking appears only in (3.7) through (3.8) i.e. as a

restriction on the available phase space.

The most significant omission is the neglection of the mean field term

ul. Putting it back as in eq. (3.7) might solve this problem but the resultant mean potential U has not all the appropriate properties for the nuclear matter situation which we like to deal with. The most appealing generalisation 1 would be the Brueckner G-matrix which is a natural extension of the bare t- matrix in nuclear matter taking into account self-consistently specific medium corrections. As we indicated in the previous section the relativistic Dirac- Brueckner approach is ideally suited for our purposes here since it is able to describe nuclear saturation and optical potentials in a very satisfactory manner while starting from a good description of nucleon-nucleon scatteting observables. So the question arises what the form of the kinetic equation is which in equilibrium (i.e. collision term + 0) corresponds to the Dirac-

Brueckner approach for nuclear matter and also has the form of a ~oltzkann equation. The answer 161, lies in the incorporation of certain (Brueckner- like) three-particle correlations into p12 or equivalently into B The

1 2 - resulting modifications as compared to the scheme outline above are the following: The eqs. (3.3) - (3.5) are replaced by:

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where

.., .

and our t-matrix eq. (3.6) becomes a G-matrix equation:

Using these expressions into (3.1) and (3.2) one obtains a kinetic equation like (3.7) bubBnow in terms of (eq. 3.12) instead of T12 and g12 is replaced by g (eq. (3.10)). Agaln restricting the collision term on-the- energy-shell twhere the energy now includes the mean field energy, see i.e.

(2.13)) one obtains a Roltzmann-like eq. but now including mean field terms

ui :

The resulting so-called Brueckner-Boltzmann eq. has the form:

It incorporates the simple U-U limit and furthermora one can see a correlation between the effective transition probabilities r r and the mean fields U1, U2 since r12 depends on U1, through eq. (3.12) at% t3.10). In the next section we shall discuss implications of this cross-correlation between mean field and collision probability. In the Vlasov-Uehling-Uhlenbeck (VUU) approach these are neglected since one takes for U1 a phenomenological mean field while in the collision term free nucleon-nucleon cross sections are used.

We remark that while all the equations used up to now are obtained in a non-relativistic framework one can implement the Dirac-Brueckner r-matrices since they have the same basic appearence as a non-relativistic Brueckner G- matrix calculation (see i.e. [lo]). However the latter does not reproduce correct saturation (unless one introduces an ad-hoc three-body interaction which acts as a fitting parameter) and is not appropriate at higher energies

(i.e. above pion-production thresholds). The most simple relatavistic calculation then amounts to the use of our Dirac-Brueckner results (section 11) into equation (3.13) which should be adapted for relativistic kinematics [ll].

IV. Effective interaction in a nuclear medium

The nuclear mean field can be obtained from the Dirac-Brueckner

expression for the self-energy X(k) (eq. 2.7)). The relativistic "definition"

of a mean potential energy U(,p) can be given as:

where we substracted the kinetic energy of a free particle from the single particle energy E(k,p) (eq. (2.11)).

From the 'L , Co results the single-particle energy can be calculated in the adiabatic lfmit (nucleon with momentum k in nuclear matter at rest) and the resulting mean field U(k,p), eq. (4.1) constructed which contains all medium effects (including also effects of the effective nucleon mass). In ref.

[7] we have shown that the resulting U(k,p) at p v O are similar to the ones obtained through a non-relativistic Brueckner calculation [lo]. However for p a O they differ substantially since the non-relativistic one does not

reproduce the correct saturation behaviour. It is one of the great virtues of the Dirac-Brueckner approach where saturation is correctly reproduced. In fig.

1 we display U(k,p) in the adiabatic limit for three different values of the nucleon momentum k (0.46, 0.75, 1.5 GeV/c) as a function of p/pO where po is

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C2-292 JOURNAL DE PHYSIQUE

the nuclear saturation density. The dotted curve represents a static "Skyrme"

parametrization:

which, through fitting A and B, corresponds to a single-particle energy density E (p ) = sF(p ) + ^{U (p}⁾^where^E (p) is the Fermi energy and which gives correct saturation prope?Pies (E (p ) = F -16 MeV at p = p 0) and has a compression modulus of K = 380 MeV. This particular potential has been used extensively by the Frankfurt group [5] in VUIJ-calculations. It is clear from our comarison in fig. 1 that the momentum dependence which is absent in (4.8) (since it is averaged over a Fermi momentum distribution) is very important. In fact at k =

1.5 GeV/c and p = po we have as much repulsion as the "SkyrmeW static mean field at p = 2p0. This observation was made already some time ago [I21 and its implications have been substantiated in recent calculations using a modified VUU [13]. Consequently a soft equation of state including a momentum dependent mean field can be mocked up by a hard equation of state without momentum dependence. The correct description however, as we showed in section 111, will always contain a momentum dependent mean field and therefore the evidence for hard equations of state seems to be fortuitous.

When nucleons collide in a nuclear medium their cross sections are quenched relative to the free ones since the Pauli principle blocks the occupied states in the nuclear medium. This simple phase-space argument is the basic ingredient in the original Uehling-Uhlenbeck equation and as such it is also used in its applications to nuclear physics (VUU etc.). As we have shown in sect. 111 the nuclear medium influences the collision process in a more profound way through modifications of the bare nucleon-nucleon T-matrix into a Dirac-Brueckner G-matrix. The latter produces the medium effects in the effective cross sections since these are obtained directly from the G-matrices.

*"I ,153

I I I

320 - -

240 -

-

₄ _-

-

a ^z5_-_E _{3 0 -} _,A*--... ^P% ^-

=---- _-m-

30

1 2 3 4 20

P/Po

10

Fig. 1. The mean potential energy U(k,p)

- -

p=2p.

- ,---a--- ^1--

- ^//r'

for different densities p and single par- o 02

&-"---

⁰⁴ ⁰⁶ ⁰⁸

ticle momenta p (0.46, 0.75, 1.5 GeV/c). p (GeV/c) The dashed line is the "stiff" Skyrme

mean potential energy. Fig. 2. Effective total cross sections for different densities. The full lines are the Dirac-Brueckner results including Pauli blocking, the broken curves show the same quantity but now without any medium correction except Pauli blocking.

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Fig. 3. Ratio of effective total cross sections and bare nucleon-nucleon total cross sections. The full curve using our Dirac-Brueckner effective cross sections, the dotted curves based on the Pauli-blocked bare nucleon-nucleon cross sections. The difference between the curves illustrates the effect of genuine medium corrections not included in simple Pauli blocking.

The modifications are two-fold. First of all the Pauli blocking not only affects the final states (phase space) but even so all intermediate states in the scattering equations (see (3.12)). Secondly there is a dispersive effect because of the appearence of the mean field in these same intermediate states i.e. the nucleons propagate "effectively" and are thus dressed. This particular feature introduces a cross-correlation link between the nuclear mean field and the nucleon collisions in the kinetic equation. Since only an explicit solution of this eq. will show the particular effects resulting from this correlation we can only discuss here the behaviour of effective nucleon-nucleon cross sections calculated from the Dirac-Brueckner G-matrices r and integrated (c.q. averaged) over all possible final states.

In absolute magnitude the medium corrected cross sections are smaller than the bare ones. This effect can be seen more clearly in fig. 2 where we show the total effective cross sections as a function of laboratory nucleon momentum and different densities. These are obtained by averaging the

effective differential cross sections over all possible (Pauli-blocked) final states in nuclear matter. The full curves show the complete Dirac-Brueckner calculations, the broken curves correspond to the same calculation but now we have omitted all medium effects (nuclear mean field, Pauli blocking on intermediate states) in the differential cross sections except Pauli blocking on the final states. The latter corresponds thus to the input in the U-U equation. It is obvious that the differences are large. Except for low densities the effective nucleon-nucleon cross sections do not exceed 10 mb.

This again is an interesting observation since small cross sections imply less randomization and thus it will help to preserve collective mean field effects in the asymptotic limit where one performs the experimental observations. It should be noted however that our calculated bare cross-sections are somewhat too low at higher momenta [ 7 ] . In fig.3 we have plotted the ratio R of the total effective cross sections with the bare calculated total nucleon-nucleon cross section. Again the full lines are the complete Dirac-Brueckner results while the dotted ones correspond to the Pauli-quenched bare cross sections.

Both are displayed as functions of laboratory nucleon momentum for different densities p and temperatures T. The latter is an extension for T Z 0 of our Dirac-Brueckner approach and it is described in more detail in ref. 171 where also the influence of temperature T on zS, E 0 and zV is studied, and found to be small. Effective cross sections for vion production and absorption are discussed elsewhere 1141 where also a more detailed account of section IV can be found.

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