mean field and effective cross sections in hot nuclei

Nuclear Physics A529 (1991) 209-233North-Holland

N FIELD AN

Received 20 June 1990(Revised 21 January 1991)

1 . Introduction

H.S . KÖHLER

Physics Department, University ofArizona, Tucson, AZ85721, USA

Abstract : Collisions between heavy nuclei produce nuclear matter of high density and excitation .Brueckner methods are used to calculate the momentum- and temperature-dependent mean fieldfor nucleons propagating through nuclear matter during these collisions. The use of a potentialmodel for the NN interactions is bypassed by calculating the Brueckner reaction matrix dirt-flyfrom the NN phase shifts using a version of Brueckner theory previously published by the author.Arndt phase-shift solutions up to 1600 MeV in energy are used. The binding energy of nuclearmatter is normalized to --15.5 MeV by adjusting one free parameter. This gives a saturation densityof 0.16 fm-3 and the compressibility is 200-250 MeV . The temperature and density dependence ofthe effective interaction is shown. The mean field is complex and the real part relates to "one-body"collisions while the imaginary part is related to the "two-body" collisions i .e . the collision term intransport theory . The error in using free cross-sections when calculating the imaginary part is ingeneral less than 20% but can be at least as large as 75% . The influence ofthe mean field (effectivemass) on the collision term is emphasized.

One primary purpose of heavy-ion (HI) collisions is to explore the properties ofhot nuclear matter, both static and dynamic,. It is of particular interest to analysethese collisions in order to learn about nucleonic degrees of freedom . However,before proceeding to introduce some model of these into a theory of hot nuclei itseems appropriate to investigate the consequence of incorporating important many-body erects in a "nucleons only" theory . This has not yet been done satisfactorily .A first step is to find the "effective" force :éff in hot nuclear matter from the known"free" NN interaction . Most "microscopic" HI calculations have been done withforces fitted to zero-temperature properties, like binding and compressibility, anddepend on local density only, but not on other variables of the medium. Althoughthis might be adequate at low energies, it is well known that the medium dependenceof the effective force in nuclear matter is far more complicated . Not only is theforce momentum dependent, it also depends on the deformation in momentumspace, i .e . on excitation ; it is to a simplest parametrization temperature dependent .The momentum distribution in the interior of a ground-state nucleus is essentially

isotropic and with a sharp Fermi surface as in a Thomas-Fermi approximation . In

0375-9474/91/$03.50 cQ 1991 - Elsevier Science Publishers B.V . (North-Holland)

210

U& KAIer / Mn field

the Mal stage of a Mgh-energy cal « An bOween two such nuclei, the momentum

istributi mn, at some point where the two nuclei overlap in coordinate space, is

strongly deformed. 8t ito a first that oftwo Fermi spheres separated

by thu, relative momentum. of the colliding ions . This is shown in fig. 8 at t =0 taken

from a calculation in which two-body collisions are incorporated by the relaxation-

time method ').In this p^ementation of those results the distribution is averaged over

coordinate space. These calculations also show that the mean field distorts this

averaged distribution only slightly (left-hand contours) during the course of a

collision, while two-body collisions Khermalizes it (right-hand contours).

Ina model of HD collimions where theeffective two-body interaction V ff is assumed

to be local / i . e . independent of the relative NN momentum), the resultant mean

field will also be local and independent of the distribution in momentum space;other than that of the zero moment of this distribution, i .e . the local density . Inmost of the earlier calculations like VUU 2\, BUU3) or lFDH lPRX') and VRX 4\

this assumption (local V n \ is made more for simplicity rather than physical reality .In a calculation of nuclear-matter properties from realistic nuclear forces, it isextremely important to include the momentum dependence, for example, to calculatethe saturation density or the compressibility. In calculations of collisions between

ENTA 84 MeV 160

.160

LO

1~a5 00 -35 GO 3.5 -2u 00 -25 0.0 25

p-Par. P-par .

P-par . P-por.Fig./ . The left -hand figurrvhowsuommou , p!omu[xhrmumcruumd istrü»udon( WiBocryuocdonaveruâcdover coordinate space) as u function of time in units u[ l0-o v, for u Tl]BF calculation i.e . without ucollision term, while thr figure to the right-hand side is with collisions included by thc relaxation timemethod / ).This figure shows that collisions drive thc system from u distribution looking like twoFenni

spheres tuu thcnna\iscd system .

t= 004

WC .

_-& 012 inc 0.04

t= 0.08

bas 002 ina 004t= 1) 12 t= 0 .16

t= 020 t= 0.24

heavy ions made by Stöcker et al. 5 ) using a local

interaction

eff , it was foundnecessary to use quite a large compressibility in order to obtain the experimentallyobserved perpendicular flow.One can, however, argue that the large stiffness in the equation of state is

necessitated by neglecting the momentum dependence, which is already known tobe important for many other effects in nuclei . The argument goes as follows : ydeforming or heating a zero-temperature Fermi distribution (while keeping thedensity constant) the repulsive part of the energy due to the momentum dependenceof the force will increase. We have already seen in fig . 1 that collisions between theions result in such deformation in momentum space. If the energy is assumed tobe independent of the deformation this increased repulsion has to be compensatedfor, by increasing the compressibility, making use of the fact that the density alsoincreases in «hc interaction region .

This situation is exemplified by fig . 2 where the lower curve shows the result ofa theoretical saturation curve. The spheres serve to indicate that the momentumdistributions in this calculation are taken to be zero-temperature Fermi distribu änncSuppose we would like to test this theory by doing HI collisions . As discussed aboveand shown in fig . l , the distribution is then deformed in momentum space. Thermal-ization makes the distribution isotropic but energy is conserved . The exclusionprinciple will force a larger density to require a higher beam energy but this willalso lead to a larger deformation in momentum space. We are, therefore, obligedto consider a calculation of the energy of this system of two Fermi spheres. For aqualitative study let us consider doing this calculation with two different assumptionsabout the effective interaction Veff .

aDcw

H.S. Köhler / Mean field

11

TOTAL ENERGY

N I

0 .0 0 .5 1 .0 1 .5 2 .0

Density plpo

Fig . 2 . The lower curve shows the binding-energy per particle as a function of density calculated forzero-temperature Fermi distributions as indicated by the circles . The two other curves show the bindingenergy per particle for the two sphere problem as indicated by the deformed circles . The middle curve

is for force (1), while the upper is for force (I1) as discussed in the text .

212 '.S. Kôhler /

40 =

%= 49(A

'pan field

UP) -his is a form of interaction used in many microscopic HI calculations .

ost nuclear HF calculations use a form of this type, e.g . Skyrme, Gogny etc.oth of these effective forces are assumed to contain parameters which are fitted

to exactly reproduce the saturation curve; i.e. the lower curve. The results of thecalculations with the two forces (1) and (11) for the two-sphere case, are shown bythe middle curve (curve (1)) and the upper curve (curve (11)) in fig . 2 .Curve (1) lies above the saturation (the lower) curve because the deformation

increases the kinetic energy, while the potential energy is the same as for thesaturation curve, simply because the interaction, depending only on the density, isindependent of the deformation under constant density.

ut if the interaction is momentum dependent (like the force (11)) the potentialenergy will increase with deformation . Curve (11) will, therefore, lie above curve (1) .The implication of this discussion is that the uppermost curve corresponds closest

to representing an "experimental" saturation curve . This is, however, for deformedFermi spheres only . After energy-conserving, thermalizing collisions it is true forheated nuclear matter too. In order to use this "experimental" saturation curve tocalculate the zero-temperature saturation curve, which we want to compare withthe theoretical saturation curve, one needs to know the deformation energy inmomentum-space which depends on the momentum-dependence of the force . Tocomplicate matters the force itself does also depend on the deformation or tem-perature.

Considerations like these have served as an incentive to pursue the microscopicinvestigations presented in this paper.The calculations by Welke et a1. 6 ) show tii~ Importance of the momentum

dependence of the force in VUU calculations . It inay be that the simple argumentsgiven above may have to be modified . It may not be just a question of the energyfunctional. Rather, it appears that the dynamics especially with regards to theperpendicular flow is different in the two separate cases . Increasing the stiffness ofthe equation of state by increasing the density dependence of Veff results in a largermean-field repulsion in the overlap region of the ions . Nucleons hitting this repulsionin a non-central collision will be reflected sideways . If the force, and consequentiythe mean field, is momentum dependent, a different mechanism described as a"coherence" in phase space seems to enter'). A similar effect was observed whendisplaying the Wigner functions for low-energy (head-on) collisions between slabswith a momentum-dependent force of quadratic (Skyrme) form 8) . It was then foundthat a purely density-dependent force tended to break up the distribution in phasespace much more than the momentum-dependent force, although the compressibilitywas the same for both .Whatever the mechanism is, the calculations do indicate that the momentum

dependence is important ') and one of the purposes of this paper is to calculate

this microscopically as a function of density and temperature in the framework ofBrueckner many-body theory .The Brueckner

-matrix is at zero temperature a functional ofthe Pauli operator,resulting in an important density dependence. If the distribution is deformed atconstant density, the Pauli operator will change so that the K-matrix (and Veff ) willdepend on this deformation, leading to a temperature dependence. It is not easy tofully incorporate the erects of deformation in momentum space that have beendiscussed above. To find out to what extent this would be justified some quantitativework is necessary.Ahhough the reaction matrix, K, is formally defined for any distribution of

occupied states, calculations are in practice restricted to some simple cases . esimplest is the zero-temperature Fermi distribution, for which of course manycalculations have been done. Non-zero temperature Brueckner calculations havealso been done 9,1° ) and in relativistic Brueckner theory by

alfliet l' ) . While aheated Fermi sphere would be the appropriate representation for the final stages ofthe collision, the distribution in tire initial stage of an 1 II collision is better representedby two spheres, separated by the relative momentum of the two ions . Calculationsfor this system have also been done as a step towards calculating an optical modelpotential for HI collisions 12,13) .

This investigation is a contribution to the understanding of the effective forcesin hot nuclei, especially with regard to the mean field and the two-body dissipativecollisions . Results of calculations of the mean field potential, both real and imaginaryparts, in hot nuclear matter are shown. A report of similar work, using a version ofBrueckner theory 14) was given earlier ' 5 ) . The recent work by Malfliet et al. 11 "16)

presents a comprehensive Green function formalism for both static and dynamicnuclear systems using the Keldysh methods. The formalism for the static system(such as the expression for the effective force) resembles that of the Bruecknertheory, but there are important differences . The formalism for the dynamic systemcan via quasi-particle approximations be reduced to the more familiar semi-classical transport equations of Vlasov- and Boltzmann-type (with Uehiing-Uhlenbeck/ Nordheim modification) but contain quantum and other corrections .

The formalism of 1Vlalfliet et al. overlaps to some extent with previous works ofKadanoff and Baym 17

), and Danielewicz 18) . One potentially important aspect of

these theories that we plan to incorporate are the non-energy conserving NN

collisions in the medium that were already emphasized by Danielewicz 19) . As a

first step, the formalism will be applied in the quasi-particle approximation andneglecting hole-hole ladders . The formalism is then practically identical to Brueckner

theory .Although the results in this paper are restricted to the problem of one hot Fermi

sphere, calculations have also been made for the two-sphere problem in momentum

space, but we shall reserve those results for a later presentation .+

+ ofou

calculatingt "the~_ _

~_:_reac 1or lÎlatilx was presented in ref. 14)

Our nleth

. For

completeness and because of several misprints in the earlier work it is shown again

H.S. Köhler / Mean feld 21 3

214

in sect . 2 and the connection between the scattering and bound-state problem whichis crucial for our calculation is now shown in more detail .

Sect . 3 shows the results of the binding-energy calculations . Two models and twoapproximations are defined. The definition of single-particle energies and the resultsof the calculations of the real part of the mean field are shown in sect . 4 . The resultsfor the imaginary part of the mean field which is related to the two-body collisionsin the transport equation are shown in sect . 5 . Sect . 6 contains a summary andconclusions .

2 . The effective interaction in nuclei

M& Köhler / Meran faeld

The effective interaction V fF (here also referred to as the reaction matrix K) iscalculated with Brueckner theory') . A major difference from the conventionalapproach is that we shall calculate here the K-matrix directly from the scatteringphase shifts, rather than from an NN potential . The formal relation between thescattering and the reaction matrix is of course simple and shown below. To solvefor the reaction matrix one does, however, need the off-diagonal matrix elementsof the phase-shift "operator", which are not accessible from NN scattering (butcould be calculated via a potential model). To overcome this problem we simplyassume this operator to have separable matrix elements in momentum space . Underthis assumption it is easy to solve the equations for the diagonal elements of Kneeded in a nuclear matter calculation . The assumption of separability may seemad hoc. We think it is justified by the agreement with calculations from NN potentialsshown below, and because of the comparative ease with which calculations can bedone for the more complicated distributions of nucleons in momentum space thatwe also do. Actually, it was initially thought of as a qualitative and rough methodfor checking certain effects . It should not be regarded as a serious many-body theoryfor calculating binding energy, even thoug'a we will see that it does give with easethe "correct" saturation properties, etc . It is more regarded as a method ofcomparingdifferent approximations and formalisms . The method has ;~~r example also beenextended to Malfliet's formalism (already referred to in the Introduction) in thequasi-particle approximation, the results will be published in the near future . Theplan is to also apply it to the more complete formalism, which is a far morecomplicated task but still feasible to execute .

Underl~-ng the assumption of separability is of course also a potential model(although unknown) which does satisfy the requirement of fitting the observed phaseshifts . Our method should not be confused with the method of using a separablepotential . We are in fact not using a potential model (at least not explicitly) andwe consider it an advantage in itself to be able to bypass the need for such a modelin our calculations .The phase shifts themselves will actually be the first approximation to our effective

interaction . Our approach may seem more sensible if we rise that in the low-density,

high-temperature limit, and in the limit of large relative momentum, K asymptoti-cally approaches this first approximation. This is shown in fig . 3 to be discussed later.We now describe some details of deriving the expression for the

-matrix usingphase shifts . We start with the definition of the operator K. It is defined by

Here Q = Qpp indicating that only particle ladders are included, as is customaryin Brueckner theory. The q (not necessary in original Brueckner theory at zerotemperature) is included because our definition of the energy denominator (seesect . 4) will sometimes lead to a pole in the propagator. The traditional procedureis to calculate K, assuming that the interaction v, the "free" NN interaction, isknown. Our procedure is different.We first note that with Q =1 and e = eo in eq . (1), with eo being kinetic energy

one obtains the T-matrix equation :

A reactance matrix Yi^ is defined by taking the principal value . Thus

where P denotes that the principal value is to be taken when there is a pole . Thediagonal elements of the reactance matrix in some particular state is related to thecorresponding phase shift 0 by

The relation between T and Y( is


21 5

K = v+v

Q

K.

(1)e+i~

T=v+ v

1

T.

(2)eo+ ir7

Y1=tan(8(k))/k .

(4)

T=X-i-rr8(eo)XT

(5(e,,) is here delta function of eo), which is consistent with

T=e - " sin (8(k))/k .

We note for future reference that the imaginary part of T is sine 8(k)/k whilethe first iteration of eq. (5) gives tang 8(k)/k.Some caution is necessary to make the connection between the equations for T

and X, which are for continuum scattering states, and the K -matrix equation, whichwill also include bound states . In a low-density limit, one may be tempted to simply

identify K with X by seeting Q == P and e = eo in eq.(1) to get K oc tan 8 (k )/ k. Someearly calculations were in fact made in this approximation 2',22) . One indication that

this is wrong is that this disagrees with the calculation of the second virial

coefficient 2i). It is actually rather trivial to show that the energy shift which results

fro

two particles interacting in a large but finite enclosure is proportional to thease shift itself, rather than the tangent function of the phase shift '4' 2s) .

°The correct limiting procedure in the transition from formal scattering theory todiscrete spectra

theory was derived by Ire

itt`~) and independently by Riesenfeldand

atson `;) (see also refs . 'a,'')) .

the limit of low density, the effectiveinteraction between a pair of particles in a "large but finite enclosure'° can beo tamed exactly . In r-illo in- ig er perturbation theory it is given by (with

`vith e® = E® -

° .

is the energy shift of the unperturbed level E° due to theinteraction, i.e .

is the matrix element of I~° between the unperturbed pair of states .~, is as ususal the unperturbed hamiltonian and Q° projects out the states of the

interacting particles .'® is related to the reactance operator by

e propagator-difference sterns from the fact that Q° relates to a discrete problemwhile

denotes a principal value integration and there is an energy shift, ®, in the® pro agator . This energy shift is proportional to the energy spacing of the discrete

spectra

and will, therefore, g® to zero as the size of the b,~x in ".-greases . It ~s hoWevcierroneous to simply let

~0 in eq. (3 ) to get I{o~~`~~ tan ( ~ )/ k. The correctrocedure shows that')

giving

A dia pole ; hence the S(e° ) term in P~. (9) .

'e then find

.S. FC®Bader /

leaaa ,~edd

e°+®

eo

tan S

S

erence between the bound and scattering states is that the latter always have

I--~+Q~r(

~

-1 8(e)

(10)e+ ir

e~,

~tan ~

S

This equation relates the

-matrix to the phase shifts and 's the main result in thissection .

n approximation to this equation is obtained by putting ~ = 1 and e = e° ineq . ( 5 ) to get


which is usually referred to as the impulse approximation. This implies using freecross sections in the collision term as discussed below.Although eq. (10) is readily solved for K under our assumption of separability

of the matrix Ko, it is practical to first introduce another matrix Kp by

which is related to K by

21 7

Kp = Ko+ KoQP-

P+Q'ff

1

-1

Ble) Kp ,

(11)e

eo

tan S

S

K=Kp(1- i-rrKpQS(e))-' .

(12)

Eq. (11) is readily solvable in momentum space for the diagonal elements

p ifwe assume the Ko matrix to be separable. In fact we find Kp = D(k) Ko, where Dis a function of relative momentum k only and is related to the integral

fxI(k) =

1 Ko(k')j

QP_P+Q~r(

1

_ 1

S(e)

dk' ,

(13)o

e

eo

\tan S

S

where D(k) =1/(I - I(k) ). The reaction matrix K is then obtained from eq. (12)and it is in general complex. A non-zero imaginary part is obtained whenever wehave a pole, that results in energy-conserving transitions . For two nucleons interact-ing below the Fermi surface at T = 0 there is no pole and K = Kp .The phase-shift solutions of Arndt 28) were used . All phases with .T _ 5 and lab

energies up to 1600IVieV were included . The integration in eq. (13) was cut of at10 fm- ', which implies that phases up to g GeV are needed' Tins illustrates the factthat the theory of nuclear matter is actually a high energy, relativistic problem,which is now also widely accepted to be the case . That this high cutoff is requiredis not specific to our method, but necessary in any theory of nuclear matter thatproposes to include short-ranged correlations . For the energies higher than 1600 MeV(up to which energy the Arndt phases are available) a model is needed. If one were

to use an NN potential model, one would specify the nature of the short-rangedrepulsion. Our model is specified by extrapolating all phases to some fixed values

at 9 GeV but always normalising the result so that the binding energy per particleis -15.5 NIeV/A . Toge~her with some alternative approximations this resulted insome models to be specified later . The coupled states ('S'D) require a specialtreatment . For these states the contribution from the coupling is simply added tothe right-hand side of eq. (13) and this procedure was found quite adequate . The-

3tensor coupling between other states (3 P, F2 , etc .) was neglected .The experimental phase shifts, their asymptotic extrapolation and the stipulation

of a separable matrix constitutes our NN interaction model .The K -matrix is calculated with an angle-averaged Pauli operator . (For T> 0 see

ref. 29)] . Only particle ladders are included . Hole-hole ladders should be included

at higher densities and this contribution will be considered in an extension of these

calculations to Malfliet"s Green-function formalism ' ° ) . The K -matrix calculated as

218

described above is density and temperature dependent. This is shown in fig . 3 . Theleft-hand curves show the diagonal real part of the K-matrix elements, summedover all states, referred to as V,,ff , twice, once and a tenth of the normal density, aswell as the free (KO) elements at zero temperature as function of relative momen-tum P. The interaction decreases in strength with increasing density. The matrixelements (except the free elements) depend of course also on centre-of-mass momen-tum. The curves are plotted with centre-of-mass momentum equal to twice therelative momentum. There is an increased attraction of nucleons interacting closeto the Fermi surface . This is especially pronounced at the lowest density . The right-hand pan of fig . 3 shows the temperature dependence at normal density. Shown areQ w T= 0, the upper curve, and T = 50 MeV, the middle curve, compared withthe free interaction, the lowest curve . All curves in both figures approach the free-interaction curve at large momentum . Furthermore, by decreasing the density and/orincreasing the temperature the free curve is approached at all relative momenta .An approximation of eq. (11) is obtained by expanding the denominator in

eq . (12 ) to get

omenturn P fm''

H.S. KiAler / Mean field

K = Kp+ i7TI KP12Q5 (e) .

(14)

This decouples the real and imaginary part . This approximation will be used insome calculations presented below. (Approximation A.1) . One should note thatformally replacing the last Kp in eq. (14) by the free scattering matrix T impliesusing free cross sections in the absorptive optical model or collision term of atransport equation . We shall investigate this approximation also numerically . (Seesect . 5.)

EFFECTIVE INTERACTION

T=O

T--50

P-1-0p,

0.0

1.0

2.0

3,0 0.0

1.0

2.0

3.0Momentum P fm-'

Fig . 3 . The left-hand curves show the effective interaction as a function of relative momentum at zerotemperature ( T = 0) at three densities (twice, once a tenth the normal nuclear-matter density) comparedwith the low-density interaction shown by the lowest curve . The interaction decreases in strength as thedensity inLre-ases. Right-hand curves show the temperature-dependence of the interaction at normal

nuclear matter density . See text for further comments .


219

Calculations are done as a function of density and temperature . The uncorrelatedmedium is described by a Fermi distribution . For T> 0 this requires that the energyspectrum is known . The calculation of the real part of the mean field is discussedin sect . 4 and should be used to define this energy spectrum. Tleis requires aself-consistent calculation, as the calculated mean field (obviously) will depend onthe Fermi distribution . Our T> 0 calculations are, however, at this stage incompleteand we use instead an effective-mass approximation with m* defined by 30)

1

-'m* =

1 ~-

Mo-1

R

where m 0* = 0.83 and R = p/ p0 with p0=0.166 fm-3 .With the energy spectrum determined by this effective mass and for a given

temperature the chemical potential is calculated from the density. Because the energyspectrum is not calculated self-consistently this chemical potential is not the chemicalpotential in the thermodynamic sense, but serves only as a parameter to detertninethe uncorrelated Fermi distribution . These problems are discussed in detail in ref. "o)

and references therein. In the terminology of these references our calculation isnon-conserving . A fully self-consistent and conserving calculation requires, forexample, the second-order rearrangement energy (see sect . 4), which is not includedin the present work but will be included in a forthcoming publication . It is notbelieved that the numerical results presented in this publication are affected by theapproximate Fermi distribution we used . It would be a serious defect, however, ifother thermodynamic functions are calculated as fully discussed in inc referencesmentioned above. It is our purpose to do such calculations in the future . It is infact for this reason, and to implement a complete self~~nsi;te:a Green functionformalism that goes be'and the quasi-particle formalism by including the self-consistent spreading of the spectral function in the nucleon propagators, that thepresent simplified computation of the K -matrix from phase shifts has been invented .

3, . The binding energy

The binding energy is in Brueckner theory obtained from summing the single-particle Fermi-gas kinetic energies and the two-body K-matrix interaction energies .Thus

E-, _

niti +

1,

E

nin jKij .)i=1

i=1 j=1

(15)

where the n i are Fermi-distribution occupation numbers. The Brueckner expansioncalls for on-energy-shell insertions in hole-lines and off-energy-shell insertions inparticle lines . To simplify the calculations we use on-energy-shell K -matrix insertions

220

in both hole and particle lines, i.e . a continuous spectrum. The mean fields, realand imaginary are discussed in more detail in sects. 4 and 5, respectively .

It is of interest to know how much the final result will depend on various possiblevariations of the theory and approximations used in the literature . The calculationswere, therefore, carried out under some different assumptions and approximations .

e binding energy (but not the saturation density or other data) was in each casenormalised to -115 MeV/A. Except for the S-waves, all phases were set to zero at9 GeV. Models M I and M2 were defined as follows:

odel MI .

The normalisation was done by varying the S-wave phase shifts at9 GeV (assumed the same

for 'S and 3S) . The binding energy is shown by the lowercurve in fig . 4, from which we find S,-o=-140°, and this defines model MI .

Wei jW2. This differs from MI in the method of normalisation . The twoS-waves were assumed to have constant phases in the region 1 .6-9 GeV, i .e . -50'and -65' for the triplet and singlet state,, respectively . The normalisation wasachieved by adding an energy contribution F3 = 1 .04 x pl po to simulate higher-order(three-body) contributions . This model may be more realistic than model MI .

In addition to these two models two approximations were also investigated .Approximations Al and A2 were defined as follows:

pproximation Al :

Two approximations are made.(i) The approximate relation eq. (14) is used, rather than the exact eq. (12) .

(ii) The last term (Qv(l1tan8-lQ)Qe)) of eq (10) A also neglected . Thedependence of energy on the 9 GeV S-wave phases is shown by the upper curve infig . 4, from which one finds 5 1=0= -70'. This defines approximation Al.

pproximation A2.

Only the approximation (ii) of Al is invoked . The energyas a function of the S-wave phase shift is now shown by the lower curve in fig . 4

H.S. Köhler / Aitinfield

=L ENE GY

T!

r

I0.0 .50.0 -100.0 -150 .0

Phaseshift Degreesdig . 4 . The binding-energy per particle at normal saturation density (0.166 fm--3 ) is plotted as a functionof the two S-wave phase shifts at 9 GeV, assumed to be the same at this energy . Upper curve is forapproximation Al while the lower is for model M1 and approximation A2 . See text for definition of

models and approximations .

(i.e . the same as for model M 1) and the normalisation to the same binding as beforegive a phase of -140° and this defines approximation A2.

Fig . 5 shows the binding energies as a function of density at zero temperaturefor these four cases . The lower curve shows 1 and A2 that overlap, while theupper curve is from the M2 model. The middle curve shows Al . All curves showsaturation at or slightly below the normal nuclear-matter density of 0.166 fm-3. Thecompressibility of these curves vary between 200-250 MeV. The higher com-pressibility is found for model Ml, which may be the most realistic of the two.Al and A2 diner in approximation (i), i.e . in the imaginary part. At zero tem-

perature only the real elements Kp , contribute to the binding energy . There is,however, still a difference in binding energy between Al and A2 because the meanfield used to calculate K is according to ec- . (12) (i.e . for A2) approximation coupledto the imaginary field . The result is that this field is more attractive for large momentafor A2 than it is for Al . This results in the larger binding at higher densities forA2, even though the 9 GeV phases are more negative, i.e . more repulsive fnr the A2approximation.The approximation A2 diners from model

1 only in the scattering-correctionterm, (Qlr(1 /tan S -115)8(e)) of eq. (10) . This can only indirectly affect the energycalculation, because the neglected term is only present for scattering states . The twobinding-energy curves actually overlap, but we shall find that there is an importantdifference in the imaginary parts of the field at large momenta .

Both the 9 GeV S-wave phase-parameters as well as the three-body contributionin model M2 have to be called reasonable. The reason for normalising the bindingenergy is simply that it is not to be expected a priori that this calculation shouldgive the "true" saturation curve because higher-order diagrams and relativistic effects

H.S. Köhler / Mean field 22 1

Fig. 5 . Binding-energy per particle as a function of density (expressed as ratio to normal nuclear-matterdensity) for model M2 (upper curve), model M1 (lowest curve) and for approximation A1 (middle

curve) . The saturation curve for approximation A2 overlaps with that of MI. See text for definitions ofmodels and approximations .

222

MA Kôhler / "tafield

are neglected. These are of the order of a few MeV [ref. -31 )] and the contributionof three-body forces is not included either -12) (other than added on parametricallyin M .2) . Our calculation shares of course this deficiency with most other calculations .

rueck"er theory in its most generoi brat of a diagrammatic expansion in termsof Goldstone diagrams is in fact believed to be only slowly convergent . This hasespecially been attributed to the medium-ranged correlations induced by the tensorcomponent of the force -l ") . These problems are expected to be especially trouble-some at higher densities . The aim of this work is not to explore these convergence-problems, nor to calculate saturation properties . Just as in other non-relativisticlow-order Brueckner and HNC calculations we do not expect a priori, nor do weobtain in our model saturation at the empirical point. It is, however, remarkable,maybe fortuitous, that our method, after introducing reasonable parameters repro-duces known saturation properties so well ; and that is really all we require of ourmethod because the aim has not been to develop a fundamental many-body theorybut rather a simple approximate scheme containing some of the well-knowningredients of such a theory . The method will serve our purpose of investigatingthe temperature and momentum dependence of the mean field and the effectivecross section in the collision term . Other approximations and improvements suchas the contribution of hole-hole propagators have also been evaluated and a reportwill be forthcoming.Many nuclear matter calculations include states with P; 2 only, while ours include

all states J

5. We find the contribution to the binding-energy per particle fromstates 2 < J

5, to be given by (in units of MeV and fm)

"J

2-5=

30.43 + 0.67k F

4. Real part of the mean field

The binding energy is calculated from E2 (and E3) as discussed above . Thesingle-particle energies in Brueckner theory deserves a rather lengthydiscussion 20,36)

. Here we only point out the important steps. In the calculation ofthe K-matrix defined in eq. (1) we choose to calculate the single-particle energiesei from

ei = ti + 1 nj Re(Kij ) .(16)

Note that the K-matrix in general is complex and that only the real part is includedat this step of the calculation .The K-matrix in eq. (16) is defined as in eq. (1) so that the energies ei are to be

calculated iteratively (self-consistently) . The definition of the energy denominatorin eq . (1) implies that a particular subset of terms (Goldstone diagrams) is includedin the energy calculation . All diagrams with first-order K-matrix insertions in hole

H.S. Köhler / Mean field 223

lines are for example included (cancelled), because they can be put on the energyshell . Insertions in particle lines are not cancelled exactly, because they are off-the-energy shell 20,37,38) One may argue, however, that they are part of the three-bodydiagrams, which should then also be summed for consistency 39 ) and this is quitecomplicated to do 3'

) . One would of course like to define the spectrum of e; in sucha way that these diagrams are summed as well as possible . The "continuous" choiceseems to satisfy this criterion fairly well 4o) .

For the propagation of real particles another set of diagrams are important also .These are the third-order (rearrangement energy) diagrams . These diagrams aretogether with the first-order diagrams (first order in K) discussed above, generatedby defining the single-particle energy E; by 33)

i .e . by a change in energy when a particle is added or removed. Returning to previousdefinitions we find that the differentiation gives the following contributions :

(i) The single-particle kinetic energy t; from the differentiation ofthe total kineticenergy in eq. (12) .

(ii) The first-order contribution included in the definition of e; in eq. (16) . Thiscomes from the differentiation with respect to the n; in the potential energy termof eq. (15) .

(iii) The second-order Brueckner rearrangement energy diagram . This is generatedfrom a change in the Pauli operator ofthe K -matrix as a nucleon is removed or added.

(iv) Two third-order Brueckner rearrangement energy diagrams . These are gener-ated when making a change in the single particle- and hole-energy, respectively, inthe denominator of the K matrix as defined by eq. (1) .

(v) The differentiation of the energy E3 used in defining model M2 in sect. 3results in a contribution V3 to the single-particle energy given by

V3 = 3.12 x ( P/Po) 2 .

(18)

It is to be noted that if we were to include this energy V3 in the single-particle e;

in eq. (16) it cancels when taking the difference between particle and hole energiesbecause it is momentum independent .The third-order diagrams (iv) are conveniently calculated by explicitly doing the

indicated differentiation numerically . They are added to the energy e; to obtain themean field energy E; of a particle propagating through the nucleus. (They couldalso be included in the definition of e; to irciude more diagrams in the bindingenergy but one has then to be careful with double counting 36).) This third-ordercorrection (rearrangement energy) corrects for the fact that nucleons are correlatedand partly excited out of the chosen (model) unperturbed Fermi sea .The second-order contribution is not included in the s.p.e . E; . It would not be

consistent to do so because it actually belongs to a class of diagrams with hole-hole

2-24

0

00

P-1-01P.

P-i

-+

a

o.0 1.0

2.0

3.0 4.o

5.0 &0 1.0

20

3.0

4.0

5.0Momentum P frro-'

Momentum P fm-'

Fig . 6 . The momentum dependence ofthe mean field at the two temperatures indicated in MeV. Left-handcurves are for model M1 at normal density and right-hand curves are for M 1 at twice normal density.

M2 and A2 give practically the same result.

propagators which are not included in the definition of the

-matrix eq. (1) . Itdoes contribute to the removal energy because the removal of a particle creates ahole into which particles can physically scatter . (This type of diagram will bediscussed again in a later publication with hole-hole propagator included in thedefinition of the Pauli operator in the K-matrix .)

e mean field for model

1, at normal and twice the normal density is shownin fig . 6 at T = 0 and 50

eV as a function of particle momentum . The approximationAl gives the result shown in fig . 7 . Model M2, as well as approimation A2, givepractically the same result as M 1 and are not displayed .

FHS. Milder / Xfean,ield

S.P. POTENTIAL

0.0 1 .0

2.0

3.0

4.0

5.0 0.0 1 .0

2.0

3.0

4.0

5.0omentum P fm-'

Momentum P fm'Fig . 7 . Similar to fig . 6 but for approximation Al . Results are seen to be similar to that for M1 at normaldensity except that the approximation Al shows a slightly less temperature-dependence and somewhatless repulsion at large momenta . At the larger density the approximation Al is seen to lead to a ratherdifferent conclusion regarding temperature dependence . This approximation also gives a too large

repulsion at large momenta compared to M 1 . See discussion in text .

Wiringa 34) has calculated single-particle potentials at zero temperature, usingHNC methods and various NN interaction-potential models. Although the methodsof calculation are widely different, both calculations define the s.p.e . by eq. (17) .Wiringa states that he keeps the correlations independent ofthe occupation numbersn; when differentiating which probably implies that second-order rearrangement isnot included, just as in the present calculations . He does, however, not have acomplex mean field, with coupling between the real and imaginary parts. Hiscalculation should, therefore, be compared with our approximation Al . Comparingour results (at T= 0) for this approximation (fig. 7) with Wiringa's, the agreementis overall satisfactory . There is a difference, our curve at twice the normal density(right-hand curves) goes through zero at P = 3 fm-' while his goes through zerocloser to 2 fm- ' . Some ofhis results are also somewhat (10-20 MeV) higher at 5 fm- ' .However, Wiringa "normalised" his curves for different potentials to go throughsome common point, so that a direct comparison is not really relevant .The main difference between the calculations shown in figs . 6 and 7, respectively,

is that our approximation Al (fig. 7) does not include propagations in an absorptivemedium . We see that this effect is not small, especially at twice the normal density.The repulsion at 5 fm' is heäc: suppressed by about 35 MeV. At normal density theeffect is somewhat smaller (about 10 MeV). This latter result agrees with results inref. 41),

In addition to being momentum- dependent, the mean field is also temperaturedependent. This is, of course, at least partly a consequence of the momentumdependence of the two-body interaction Vef .. To investigate this further we comparewith the predictions of the parametrization used by Welke et aL 6) -who use atemperature-independent phenomenological interaction . Fig . 8 shows the mean field

O

OrC

OT

S.P. PO

p-1.0p.

T--50

T--0


225

S.P.

0.0

1.0

2.0

3.0

4.0

5.0

0.0

1.0

2.0

3.0

4.0

50

Momentum P fm'

Momentum P fm'Fig. 8 . The momentum dependence of the mean field predicted by the momentum dependent (but

temperature-independent) interaction used in the VUU calculations of ref. 6 ) . Left-hand curves are at

normal density, the right-hand ones at twice the normal density of nuclear matter. Comparison with

fig . 6 shows the effect of the increase in interaction strength with temperature .

calculated Wth their interaction at normal and twice the normal density and indicatedtemperatures . Comparing 6g. 8 with figs . 6 and 7 we see that the overall agreementis good, but there is a noticeable difference in temperature dependence, especiallyat the higher density . This effect agrees with our fig . 3b which shows that theinteraction in the medium increases with temperature as the Pauli blocking decreases .

and relates to dissipative two-body collisions 43). Results are shown in fig. 9 . Results

from other calculations 29,41,44,4:5) (and experimental data) are only available at T = 0and normal density . A comparison indicates that our calculations show about 3 MeVlarger absorption at P = 2 fm- ' MeV than the others but for P > 4 fm-' the agreementis better . The reason for the discrepancy is not fully understood ; maybe becauseour calculations at T = 0 were, for computational reasons, actually made at T=

eV. Another reason may be that our calculations include all states for J

5while other calculations use much less states, e.g . J -- 2 [ref. 41)] . Our calculationsalso show a shoulder in the curves at about 2 fm- ', which is not so pronounced inother calculations . The reason for this is not known. It may be related to the factthat the S-wave contributions are peaked at this momentum .More important is that these calculations of the imaginary potential at various

temperatures and densities give us a chance to test the customary use of free cross

2

The imaginary part of the atean field is calculated from

.MCWt% R 7

P-1.0p.

No

T=50

H.S. K61der / Mean field

rt of the mea

I --I

0.0 1 .o 2.o 3.o 4.o 5.0 6.0 0.0Momentum P fm"

n, lm (Kjj )

(19)j=1

T=O

I

_F_ I

1.o 10 3D 4.0 5.0 6.0

Momentum P fm- 'Fig . 9. The imaginary (absorptive) part of the optical model W at normal (left-hand side) and twicethe normal density (right-hand side) at zero and 50 MeV temperature . This is for model 1V1 .1 . Model M.2

gives practically identical result .


sections in the collision term of transport equations. To this end we defineoc

227

free by

free,i

E

njcIQX m*(kij) sin`' sl(kij) ,

20)1,j=1

where Q is the Pauli operator and subscript l denotes the states . The cl are statistical(and other) factors while m*(k) is the effective mass. This is equivalent to usingthe free total cross section when calculating Wfree .

For the discussion of the ratio W/ Wfree we note that with eq. (19) expressed interms of Kp one gets

Wi = É

njcl

Qm*IKp,1I

2

2 .(21)J,j=1

1 + ( Qn1

IKp,i I )One expects that W=> Wfree at low density and high temperature, because in thislimit Kp, l goes to tan (SI ) as Q =:> 1 and e==> eo [Of course then also K =* T as definedby eqs . (1) and (2) and Im Tocsine (3) .] One should also expect this lirnit to bereached for large momenta P, because Q and e should then also approach thefree-scattering limits . This is indeed seen in fig. 10 showing the ratio"sigma/sigma(free)" as a function of momentum at a tenth the normal density, with"sigma" and "sigma(free)" referring to in-medium and free cross sections,respectively . (Note that the plot actually shows the ratio W/ Wfree-) It is interestingto observe that at T = 0 the ratio is as large as 1 .75 while at T= 50 MeV it isessentially 1 . At higher density the situation has changed . Fig . 11 shows that modelsM 1 and M2 give a ratio of about 1 .2 instead of the expected 1 .0 for large momentaP at both densities . A further investigation showed that the approach to 1 .0 is quiteslow and is not reached until P -- 9 fm - ' . This can be understood, at least qualita-tively, because with Qm* < 1 (for a P = 4 fm- ' nucleon it is --0.65) and withKp oc tan (S) the ratio is always larger than 1 . (See also discussion below in relation

0

SIGMA/SIG Atree)

00.0 1.0 2.0 3.0 4.0 5.0 6.0

Momentum P fm-'

Fig;. 10. The ratio W/ Wfre, at 0 .1 x normal nuclear matter density and temperatures of zero and 50 MeV.Model M2 is used here . The figure shows that at low density and high temperature the ratio goes to one .

C4

0à -

P W,MomentumP W'

Fib. 11, The left-hand curves show the ratio of W to III,,, calculated with the same mean field, as afunction of the momentum P ofthe nucleon for model M I (middle curve at large moments), approxima-tion A I (upjwr curve at lame momenta) and A2 (lower curve) . Model M2 is essentially identical to Ml.These curt;, are for normal nuclear matter density at zero temperature . Note that both W and Wf,,,eare zzero below the Fermi-momentum . The right-hand curves show the same ratio at double density andT = -SO MeV . The ratio is overall smaller (~l.2) for model M I (and M2) than it is for approximation

All, for which it increases with momentum.

to fig. 13.) Comparing the T=0 and T= 50 MeV results shown in fig. 12, it appears,however, that the convergence to the high-temperature limit is also slow at thisdensity.

e approximations Al and A2, on the other hand, give very wrong limits forlarge R is is easily understood because if Q=:>1 and peen one finds that for Al,the potential W_~ 32' rather than W=:;>sin2('6), which explains the large value of

in this approximation for large P. For A2 one finds W~32/(1+32) < sin2(16)which explains the too small value in this case=

KWer / Afcai4 field

SIGMA/SIGMA(fred

Pnn

omentum P fm&0

lO

2.0

ÏO

4.0 - ÈO

éo

Fig . 12 . The temperature dependence of the ratio W1 Wfre, at normal density is shown for model M2.The two curves are at zero temperature and 50 MeV, respectively.


Cugnon et al. ") have presented results of Brueckner calculations that are innature very similar to ours. The results agree on the whole. W/ W,,,,, is in both caseson the average larger than 1 by about 10-20% for momenta P> 2.5 fm - ' . Formomenta just above the Fermi momentum our results differ qualitatively, but theabsorption is also relatively smaller for these momenta . At the lowest density(0.0166 fm-3), shown in fig. 10 we find a peak in the ratio of 1 .75 at a momentumof =1 fm - ' . The result of Cugnon and coworkers at a comparable density of0.023 fm -3 appears to be qualitatively similar but shows a peak as large as 5. This

ratio is apparently more sensitive to the method of calculation than any of the otherquantities that we have shown. This is also illustrated by the results for approxima-tions Al and A2, which have already been discussed .Many BUU (VUU, etc.) calculations are done without considering the mean field

in the propagator and this corresponds to putting m* =1 in eq. (21) . Fig . 13 showsthe effects of this approximation at normal density and zero temperature . e lowercurve is for m* =1 while the upper is the "exact" . The effect is very similar to thatshown in ref. 46 ) . The approximation gives a ratio that is less than 1 fc.r smallmomenta and slightly above but closer to 1 for large P than the exact. This isexpected from the discussion of this problem above.The relativistic calculations of ref. 47) compares the Pauli-blocked free cross

sections (without effective mass correction) wi`:~ Dirac-Brueckner results getting aratio which is stnâllci tl:ar 1 . This agrees qua1_ita_ti_vely with. our fig_ 113 but the ratio

in ref. 47) seems to be smaller than ours .Previous investigations by Dabrowski and Kbhler' 3,29 ) show that using free cross

sections to calculate the imaginary part of the optical model, even for the case of

two Fermi spheres, gives good agreement with "exact" Brueckner calculations . Butit was pointed out that the real part of the mean field (the effective mass) and of

O

ac

SIGMA/SIG

P-t®P"

T=0

free

O --'i--I0.0 10 20 3.0 ~.0 9.0 6 .0

Momentum F frn"

229

Fig . 13. The upper curve shows the ratio W1 Wfree calculated with the self-consistent mean field while

the lower is the same ratio but neglecting the mean field (m* =1) in the calculation with free cross

sections . Model M 1 is used in the calculation of W.

210

course the Pauli operator has to be the same for this comparison to make sense.liese results are on the whole confirmed by our present results, although we are

able to be a little more precise in our conclusions .e can now conclude that the use of bee cross sections in the calculation of the

imaginary part ofthe nucleon-nucleus optical potential underestimates the collisionssomewhat (cxcepi for nucleons in cold nuclear matter just above the Fermi surfacewhere they are overestimated) . The error is in general less than 20% although wehave seen that it sometimes can be as large as 75% . This assumes that the correcteffective mass is used also in the -free" calculations . The VUU calculations areapparently often made not only with free cross-sections but also with an effectivemass m* =-- 1 . The correction will then mostly be less rather than larger than 1 .0.The point of major interest in this regard is if collisions in nuclei are indeed due

to binary collisions only as assumed in this investigation or if other effects areimportant, such as a density-dependent coupling to the pion field as suggested byrown "8) . The consequences of a nucleons-only model must, however, first be

evaluated and in combination with an analysis of the "in-medium" cross sectionfrom experimental data 49,50), one will hopefully reach some conclusion with regardsto the experimental evidence for these more exotic phenomena.

S Kôhler / Alean field

Summary and conclusions

The many-body theory that has been used for the calculations in this paper isessentially that of Brueckner. The method to calculate the reaction matrix directlyfrom tbe p1hase shifts, rather than from an NN potential fitted to the phase shiftsis somewhat unconventional . It has, however, been demonstrated to be a usefulmethod because of its simplicity, which makes it possible to relatively easy explorevarious effects . The calculations are also relatively easy to extend to include themore complicated contributions to the mean field and the in-medium cross sectionsthat are present in Green function techniques of, for example, refs . 11,47) . Somecuafidence in the procedure is instilled by the agreement with other many-bodycalculations . This agreement verifies the long-time experience that essentially allpotentials that reproduce the phase shifts, result in similar properties of the many-body system . A potential model is defined only implicitly in this work but it doessatisfy the requirement of reproducing, the free-space scattering phases . It wasprompted to include a parameter in the calculations . This is anaingnus to theparameter describing the short-ranged repulsion in a potential model and/or thecoatribution of three-body terms. With a reasonable value of this parameter nuclearmatter is found to saturate at a density of =:--0.16 fns-3 with a binding of -15.5 MeVand compressibility of 200-250 MeV. The apparent agreement with other many-bodycalculations should not be taken too seriously . The conventional potential model(fitted to phase shifts) is one approach to the problem of defining off energy-shellinteractions in the nuclear medium not provided by free NN scattering . The model


described in this paper provides another approach to this problem . The potentialmodel (especially the recent Paris, Bonn, etc.) is of course more fundamental, basedon meson exchanges. The aim of this work is certainly not to introduce anotherbona fide many-body theory of nuclear matter. The model is introduced for ease ofcalculation . It is of course gratifying that the agreement with the serious models isas good as it is and this justifies a continued application of the model.

Results have been obtained not only for zero but also higher temperatures andit was decided to show the T = 50 MeV results . These latter results were not obtainedself-consistently but calculated with a Fermi-distr ution function defined withenergies calculated in the effective-mass approximation as specified at the end ofsect. 2 . There is room for improvement in these T> 0 calculation and this is beingconsidered .The mean field in nuclear matter has been calculated as a function of momentum

and temperature . The momentum dependence ofthe mean field is at zero temperaturewell approximated by deriving it from an effective interaction such as used in ref. 6 ) .Our T>0 results do, however, suggest that the interaction should be ai:nwed tF_'

increase in strength with temperature as shown by the right-hand part of fig . 3 andalso seen by comparing figs . 6 and 7 with fig. 8, the latter being from a temperature-independent interaction .

Based on our nucleon-nucleus optical model calculations the effective, in mediumcross section that should be used to evaluate the collision term in the VUU theoryor to calculate the relaxation-time in TDHFRX is found to be anywhere between1 .2 and 1 .75 time the free cross section, depending on density, temperature andmomentum . In agreement with general argument, the correction decreases withdecreasing density and increasing temperature of the medium, and increasingmomentum of the nucleon . The largest corrections occur at zero temperature for anucleon just above the Fermi surface . If one likes to use are average correction factorit would probably for 100 MeV/A HI collisions be about 1 .2 . It should, however,be noted that this is the correction to choose together with a self-consistent mean-field(effective mass) . If the free propagator (only kinetic energies) is used in the collisionterm as apparently often is done in VUU calculations then the correction factor isless than 1 .0 . It is important to realise that the collision term in fact is dependentof the mean field . It is expressed by the expressions for W in sect . 5, showing thatW is essentially proportional to the effective mass. When evaluating W (or thecollision term) the effective mass may in fact be the source of more uncertainty thanthe effective cross section . The mean ïield that enters in the collision term of thepresent work and in most calculations of this kind that have been done is only thefirst-order K-matrix insertions as defined by eq. (16) . The result may after plannedimprovements in the self-consistency have to be revised (but we think only slightly) .The results presented here are for a thermally equilibrated system while calcula-

tions for the initially un-equilibrated state reached in a HI collision (see fig . 1)would be more relevant because this is when important dynamics takes place . The

231

232

H. S. Köhler / Mean field

method of calculation used here is also relatively easy to do for the system of twoFermi spheres and this has been done. Results will be presented later. The generalconclusions appear to be the same however . The main difference is really only incomplexity ; in the case of two spheres the mean field in momentum space is notisotropic and the results are more complicated to display .

This work was initiated while the author was on sabbatical leave with the nucleartheory group at LBL, Berkeley . He wishes to express his thanks to LBL with specialthanks to Dr. Jorgen Randrup for the hospitality .

This work has evolved from discussions with Professors Janusz Dabrowski andRudi Malfliet . I wish to thank both . I also wish to thank Professor R.A. Arndt forproviding the NN phase shifts . This work was supported in part by the NationalScience Foundation, under grant PHY86-04602; and in part by the Director, Officeof High Energy Research, Office of High Energy Nuclear Physics, Division of HighEnergy Physics, of the U.S. Department of Energy under Contract No. DE-AC03-76SFOO098 .

eferences

I ) H.S . Köhler and B.S. Nilsson, Nucl . Phys. A477 (1988) 318 ;H.S . Köhler, Nucl . Phys . A494 (1989) 281

2) G.F . Bertsch, H . Kruse, and S . Das Gupta, Phys . Rev. C29 (1984) 6733) W. Bauer, Nucl . Phys . A471 (1987) 6044) H .S . K8hler and W. Bauer, Phys. Rev. C40 (1989) 11715) 311 . St6cker and W. Greiner, Phys . Reports 137 (1986) 2796) G.M. Welke, M. Prakash, T.T.S . Kuo, S . Das Gupta and C. Gale, Phys . Rev . C38 (1988) 21017) S . Das Gupta, private communication8) H .S . Köhler, unpublished9) J.R . Buchlcr and S.A. Coon, Astrophys . J . 212 (1977) 807

10) P . GrangE, J . Cugnon and A. Lejeune, Nucl . Phys . A373 (1987) 36511) R . Malfliet, Prog . Part . Nucl . Phys . 2 1 (1988) 20712 ; F . Beck, K.-H . Mueller and H.S . Köhler, Phys. Ref. Lett . 40 (1978) 837 ;

T. Izumoto, S . Krewald and A . Faessler, Nucl . Phys . A357 (1981) 47113) J . Dabrowski and H.S . Kö%i,:r, Nucl . Phys . A489 (1989) 30314) H.S . Köhler, Nucl . Phys . A415 (1984) 3715) H.S. Köhler, in Proc . of NATO Advanced Study Institute on the nuclear equation of state, Penisc ,,)Ia,

Spain (May 1989) to be published16) W. Botermans, Thesis Umversity of Groningen ( 11989)17) L.P. Kadanoff and G. Baym, Quantum statistical mechanics (Benjamin, New York,18) P . Danielewicz, Ann . of Phys . 152 (1984) 23919) P . Danielewicz, Ann . of Phys . 152 (1984) 30520) K.A . Brueckner and J .L . Gammel, Phys . Rev . 109 (1958) 102321) K.A . Brueckner, Phys . Rev. 97 (1955) 135322) Reifman, DeWitt an Newton, Phys . Rev . 101 (1956) 87723) W.B . Riesenfeld and K.M . Watson, Phys . Rev . 104 (1956) 49224) N . Fukuda and R.G . Newton, Phys . Rev . 103 (1956) 155825) Kurt Gottfried, Quantum mechanics (Benjamin, New York, 1966)26) B.S . DeWitt, Phys . Rev . 103 (1956) 1565


233

27) K.A . Brueckner, in The many body problem, ed . C . DeWitt (Wiley, New York, 1959) p.4728) R.A . Arndt, private communication29) J . Dabrowski and W. Piechocki, Acta Phys . Polon . B16 (1985) 109530) B . Friedman and V.R . Pandharipande, Phys . Lett. 11100 (1981) 20531) B . D . Day, Phys. Rev . C31 (1981) 226632) I.E. Lagaris and V.R . Pandharipande, Nucl . Phys . A359 (1981) 34933) K.A . Brueckner, J.L . Gammel and J.T. Kubis, Phys . Rev . 118 (1960) 143834) R.B . Wiringa, Phys . Rev. C38 (1988) 296735) H.S . Köhler, Nucl . Phys. A328 (1979) 1736) H .S . Köhler, Phys . Reports C18 (1975) 21737) Sidney A. Coon and Janusz Dabrowski, Phys . Rev . 140 (1958) B28738) Nucl . Phys. A204 (1973) 6539) H .A . Bethe, Phys . Rev . B138 (1965) 80440) C . Mahaux and R . Sartor, in Nuclear matter and heavy ion collisions, ed . M . Soyeur, H . Fiocard,

B . Tamain and M. Porneuf (Plenum, New York, 1989)41) J.-P. Jeukenne, A . Lejeune and C. Mahaux, Phys . Reports 25 (1976) 8342) H .S . Köhler, Nucl . Phys. 128 (1969) 27343) H.S . Köhler and B.S. Nilsson, Nucl. Phys . A417 (1985) 54144) S . Fantoni, B.L . Friman and V.R . Pandharipande, Phys . Lett . B104 (1981) 8945) F.A. Brieva and J.R. Rook, Nucl . Phys . A291 4(1977) 29946) J . Cugnon, A. Lejeune and P. Grangé, Phys. Rev . C35 (1987) 861 .47) B. ter Haar and R. Malfliet, Phys . Reports 149 (1987) 20748) G.E. Brown, Nucl. Phys . A488 (1988) 68949) J . Jiang, J . Cogar and D. Keane, Proc. Workshop rn nuclear dynamics (1990), ed. J . Randrup

(LBL-28709)50) G.D . Westfall, W. Bauer, D.A . Cebra, S. Howden, J. Karn, Proc . Workshop on nuclear dynamics

(1990), ed . J . Randrup (LBL-28709) . 11 (1960) 65

mean field and effective cross sections in hot nuclei

Documents