Deuteron formation in nuclear matter

Nuclear Physics A 668 Ž2000. 137–148 www.elsevier.nlrlocaternpe

Deuteron formation in nuclear matter C. Kuhrts, M. Beyer 1, G. Ropke ¨ FB Physik, UniÕersitat 3, 18051 Rostock, Germany ¨ Rostock, UniÕersitatsplatz ¨ Accepted 21 October 1999

Abstract We investigate deuteron formation in nuclear matter at finite temperatures within a systematic quantum statistical approach. We consider formation through three-body collisions relevant already at rather moderate densities because of the strong correlations. The three-body in-medium reaction rates driven by the break-up cross section are calculated using exact three-body equations ŽAlt–Grassberger–Sandhas type. that have been suitably modified to consistently include the energy shift and the Pauli blocking. Important quantities are the lifetime of deuteron fluctuations and the chemical relaxation time. We find that the respective times differ substantially while using in-medium or isolated cross sections. We expect implications for the description of heavy ion collisions in particular for the formation of light charged particles at low to intermediate energies. q 2000 Elsevier Science B.V. All rights reserved. PACS: 24.10.Cn; 21.65.qf; 21.45.qv

Keywords: Deuteron; Nuclear matter; Few-body equations; Boltzmann equation; Correlations; Quantum statistics

1. Introduction The formation of deuterons is the simplest multichannel reaction process in a heavy ion collision. Within a systematic quantum statistical approach the formation is driven by the Boltzmann collision integral that in turn requires the proper break-up cross sections. Besides electromagnetic disintegrationrformation of the deuteron Ž np g d . that is a generic two-body process, the treatment of three-body collision Ž NNN Nd . is more involved, and requires a solution of the three-body dynamics in medium. Collision rates are a central input into modern microscopic simulations of heavy ion collisions

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1

E-mail address: [email protected] ŽM. Beyer.. Corresponding author: FB Physik, Universitat 3, 18051 Rostock, Germany. ¨ Rostock, Universitatsplatz ¨

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 5 6 1 - 8

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such as the Boltzmann–Uehling–Uhlenbeck ŽBUU. w1–3x or the quantum molecular dynamics approach ŽQMD. w4,5x. For the latest developments and recent applications to multifragmentation in heavy ion collisions see Ref. w6x. Danielewicz and Bertsch w2x started the numerical solution of coupled BUU equations including the deuteron formation. Later this model has been enlarged to include the production of three-particle clusters w7x. So far these approaches are fed with experimental cross sections, e.g. Nd NNN for the deuteron break-up, and the medium effect is taken into account by the Pauli-blocking of final states. However, in principle, as shown earlier, those cross sections are changed substantially, when medium effects are consistently included w8–10x. Besides including these medium dependent cross sections in the respective simulations that is currently on its way, it is an interesting and important question to analyze the change of time scales induced by medium dependent reaction cross sections. The correlated many-particle system is treated within the cluster mean field approach w11–13x that goes beyond the quasiparticle picture and in turn has been applied to a wide variety of many-particle physics problems. This approach is intimately related to the Dyson Equation Approach to Many-Body Green functions w14–18x. Both approaches lead to a systematic decoupling of the equation hierarchy and thus to effective few-body equations. For the two-particle case these are known as Feynman–Galitskii or Bethe– Goldstone equations. Recently, we have derived a corresponding equation for the three-body system for finite temperatures w8,20x based on the formulation of Alt, Grassberger, and Sandhas ŽAGS. w19x. For zero temperatures three-body in-medium equations have been given in Refs. w17,18x for the related 2p–1h problem. The respective one-, two-, and three-body equations are derived in an independent particle basis. That means we do not consider correlations in the surrounding matter that would eventually lead to a self-consistent treatment. However, such a treatment is technically involved and only very recently progress has been achieved towards this direction by including two-body correlations into the one-body spectral function w21x. Since presently our aim is different, i.e. to extract the formation and equilibration time scales, and the densities considered are rather low compared to the ones that require a self-consistent treatment, and since many technical problems still need to be tackled we presently do not consider such an approach for the three-body case. The life time and the relaxation time are determined by linearizing the Boltzmann equation or the corresponding rate equations. Therefore, the respective integrals can be evaluated in thermal and chemical equilibrium. To this end we assume only small fluctuations, i.e. linear response of the system. This will be explained in the following section. The necessary three-body reaction input on the basis of Faddeev type equations will be given in Section 3. The derivation of these equations is based on the Green function approach, e.g. Ref. w22x. In Section 4 we will present numerical results and in the last section we summarize and give our conclusion.

™

2. Reactions The quantity of interest in the quantum statistical approach is the generalized quantum Boltzmann equation for the nucleon f N Ž p,r ., deuteron f d Ž P, R ., etc. Wigner

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distributions. For the time being we assume symmetric nuclear matter. These distributions are driven by a coupled system of Boltzmann equations, see, e.g. Ref. w2x,

E t f N q Ep U P Er f N y Er U P Ep f N s KNin w f N , f d x Ž 1 y f N . y KNout w f N , f d x f N , E t f d q E P U P E R f d y E R U P E P f d s Kdin w f N , f d x Ž 1 q f d . y Kdout w f N , f d x f d ,

Ž 1.

where U denotes the total mean field potential. The coupling between the different species is through the collision integrals K w f N , f d x that are merely shown for the deuteron case in the following two equations relevant in the present context: Kdin Ž P ,t . s d 3 k d 3 k 1 d 3 k 2 <² kP < U < k 1 k 2 :< 2d d ™ d d f d Ž k 1 ,t . f d Ž k 2 ,t . f d Ž k ,t .

H H

q d 3 k d 3 k 1 d 3 k 2 <² kP < U < k 1 k 2 :< 2Nd ™ Nd f N Ž k 1 ,t . f d Ž k 2 ,t . f N Ž k ,t .

H H

q d 3 k d 3 k 1 d 3 k 2 d 3 k 3 <² kP < U0 < k 1 k 2 k 3 :< 2p n N ™ d N

H H

=f N Ž k 1 ,t . f N Ž k 2 ,t . f N Ž k 3 ,t . f N Ž k ,t . q PPP ,

Ž 2.

|

where we used the abbreviations f N s Ž1 y f N . and f d s Ž1 q f d ., the ellipsis denote further possible contributions involving four and more body collisions Že.g. tp dd, hp dd . or processes like np g d, and Kdout Ž P,t . is given by

|

|

Kdout Ž P ,t . s d 3 k d 3 k 1 d 3 k 2 d 3 k 3 <² k 1 k 2 k 3 < U0 < kP :< 2d N ™ p n N

H H

=f N Ž k 1 ,t . f N Ž k 2 ,t . f N Ž k 3 ,t . f N Ž k ,t . q PPP .

Ž 3.

™

The quantity U0 appearing in Eqs. Ž2. and Ž3. is the in-medium break-up transition operator for the Nd NNN reaction and is calculated using the AGS equation as given in the next section. For the isolated three-body problem U0 determines the isolated break-up cross section s bu0 via w19,23–27x

s bu0 Ž E . s

1

1

< Õd y ÕN < 3! 3

3

Hd k

1

d 3 k 2 d 3 k 3 <² kP < U0 < k 1 k 2 k 3 :< 2 2pd Ž EX y E .

= Ž 2p . d Ž3. Ž k 1 q k 2 q k 3 . ,

Ž 4.

where < Õd y ÕN < denotes the relative velocity of the incoming nucleon and deuteron. So far the strategy has been to implement the experimental cross section into Eqs. Ž2. and Ž3.. This has then been solved, for a specific heavy ion collision w2x. Using experimental cross sections respectively isolated cross sections may not be sufficient in particular in the lower energy regime. The cross section itself depends on the medium, e.g. blocking of internal lines or self-energy corrections of the respective three-body Green functions. This has been shown in Refs. w8–10x. While the inclusion of the medium dependent deuteron break-up cross sections in a coupled BUU simulation of a heavy ion collision is a congruous application, it is instructive to study the influence of medium dependent cross sections in a relaxation time approximation, in addition. This

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would enable us to already arrive at conclusions on the possible effect for typical conditions at heavy ion collisions. After linearizing the Boltzmann equation it is possible to define a break-up time for small fluctuations of the deuteron distribution. To do so, we consider only the term given in Eq. Ž3. in the collision integral. For small fluctuations, d f d Ž t . s f d Ž t . y f d0 from the equilibrium distribution f d0 linear response leads to w9,10x

E Et

d f d Ž P ,t . s y

1

t bu Ž P ,n N .

d f d Ž P ,t . ,

Ž 5.

where the ‘‘life time’’ of deuteron fluctuations that depends on the deuteron momentum P and the nuclear density n N has been introduced earlier w9,10x,

ty1 bu Ž P ,n N . s

4

Hdk 3!

N

d 3 k1 d 3 k 2 d 3 k 3

² kP < U0 < k 1 k 2 k 3 : 2 f N0 Ž k 1 . f N0 Ž k 2 . f N0 Ž k 3 .

=f N0 Ž k N . 2pd Ž E y E0 . ,

Ž 6.

which can be related to the break-up cross section using Eq. Ž4.. Chemical equilibration results from break-up and formation reactions, according to the terms that contain the transition operator U0 in Eqs. Ž2. and Ž3.. The change of the total deuteron density is given by the integration of the deuteron momentum in Eq. Ž1.. For a homogeneous system we can neglect the drift terms and only the reaction terms will change the total density, therefore we obtain the following rate equation: d dt

n d Ž t . s ya Ž t . n N Ž t . n d Ž t . q b Ž t . n 3N Ž t . ,

Ž 7.

where the introduced rate coefficient for disintegration a is given by

a Ž t . s d 3 P d 3 k d 3 k 1 d 3 k 2 d 3 k 3 <² k 1 k 2 k 3 < U0 < kP :< 2d N ™ p n N f N Ž k 1 ,t . f N Ž k 2 ,t .

H

H

= f N Ž k 3 ,t .

f N Ž k ,t . f d Ž P ,t . nN Ž t .

nd Ž t .

,

Ž 8.

and an analogous expression holds for the formation rate b . To obtain a time scale for the formation of chemical equilibrium, we consider a near-equilibrium situation. In chemical equilibrium both rate coefficients are constant and related by detailed balance Žtime reversal invariance. resulting in

bs

a n0d

Ž n0N .

2

.

Ž 9.

Here we denote the equilibrium distribution functions by n 0N and n 0d . Assuming now small density fluctuations, viz. n N s n0N q d n N and n d s n 0d y d n N r2, and after insertion into Eq. Ž1., a chemical relaxation time may be defined by d dt

d nd Ž t . s y

1

trel Ž n N .

d nd Ž t . .

Ž 10 .

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Using the definition of the rate coefficient a given in Eq. Ž8. leads to the following expression for the chemical relaxation time: y1 trel Ž n0N . s d 3 k d 3 P f N0 Ž k . f d0 Ž P . < Õd y ÕN < s bu Ž k , P .

H

n 0N q 4 n 0d n 0N n 0d

.

Ž 11 .

The deuteron break-up cross section s bu is defined as in Eq. Ž4. but may now also include medium modifications of the transition operator U0 as will be explained in the next section. The deuteron equilibrium density n 0d is related to the nucleon density via m d s 2 m N for the chemical potentials.

3. Finite temperature three-body equations The basis to derive effective in-medium few-body Green functions for finite temperatures is provided by the cluster mean field approximation. This approximation allows us to truncate the hierarchy of Green function equations in a controlled way. A general introduction may be found in Ref. w11–13x. This approach is related to the Dyson Equation Approach w14–18x extended here to finite temperatures w20,28x. It has been proven useful in the context of strongly coupled systems provided, e.g., in nuclear physics w17,18x. A calculable form is achieved by introducing ladder approximation. We consider generic elementary two-body interactions V2 only. The resulting equations for the decoupled one-, two- and three-body Green functions at finite temperatures Žutilizing the Matsubara technique to treat finite temperatures w22x. will be given in the following. The one-particle Green function reads G 1 Ž z . s R 1Ž0. Ž z . s Ž z y ´ 1 .

y1

.

Ž 12 .

In mean field approximation the single quasiparticle energy ´ 1 is given by

´1 s

k 12 2 m1

q S HF Ž 1 . ,

S HF Ž 1 . s Ý V2 Ž 12,12 . y V2 Ž 12,21 . f 2 ,

Ž 13 .

2

and f 2 s f N0 Ž ´ 2 . the Fermi function2 . This is given by fŽ v. s

1 e

b Ž vy m .

q1

,

Ž 14 .

where b denotes the inverse temperature and m the chemical potential. The equation for the two-particle Green function G 2 Ž z . reads Ž0. G 2 Ž z . s N2 R Ž0. 2 Ž z . q R 2 Ž z . N2 V2 G 2 Ž z . .

™ f for notational simplicity. 2

f N0

Ž 15.

Here and in the rest of the paper we use equilibrium distribution functions only, and therefore we use

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Note that we use a matrix notation for the Fock space indices Ž1,2,3, . . . . for conveŽ z . is then given by nience. The matrix for the two-body resolvent R Ž0. 2 X X R Ž0. 2 Ž 12,1 2 ; z . s

d 11X d 22X

Ž 16 .

z y ´1 y´2

and the Pauli blocking factor N2 by N2 Ž 12,1X 2X . s d 11X d 22X Ž f 1 f 2 y f 1 f 2 . s d 11X d 22X Ž 1 y f 1 y f 2 . .

Ž 17 .

We use the abbreviation f s 1 y f. The respective equation for the three-particle Green function relevant to describe three-body correlations in a medium is given by Ž0. G 3 Ž z . s N3 R Ž0. 3 Ž z . q R 3 Ž z . w N2 V2 x G 3 Ž z . ,

Ž 18.

where we have introduced the notation 3

w N2 V2 x Ž 123,1X 2X 3X . s Ý Ž N2 V2 . Žg . Ž 123,1X 2X 3X . ,

Ž 19 .

gs1

Ž N2 V2 . 3 Ž 123,1X 2X 3X . s Ž 1 y f 1 y f 2 . V2 Ž 12,1X 2X . d 33X . Ž .

Ž 20 .

The last equation is given for g s 3. The channel notation is convenient to treat systems with more than two particles w23–27x. In the three-particle system usually the index of the spectator particle is used to characterize the channel. The Pauli factors N3 and the resolvents R Ž0. 3 read respectively N3 Ž 123,1X 2X 3X . s d 11X d 22X d 33X f 1 f 2 f 3 q f 1 f 2 f 3 ,

ž

X X X R Ž0. 3 Ž 123,1 2 3 ; z . s

d 11X d 22X d 33X z y ´1 y ´2 y´3

/

.

Ž 21 . Ž 22 .

Ž0. Note that w N2 V2 x / w N2 V2 x† and N3 R Ž0. 3 s R 3 N3 . Although we assume a fermionic system the proper symmetrization of the equations is treated separately. If the correlated pair, e.g. X 12X , and the spectator particle, e.g. X 3X , are uncorrelated in Ž z .. We generalize to the the channel Ž3. we may define a channel Green function G Ž3. 3 channel Žg .. The channel Green function is then defined by

G Ž3g . Ž z . s

1 yi b

Ý iG2 Ž vl . G1Ž z y vl . .

Ž 23 .

l

The summation is done over the bosonic Matsubara frequencies vl, l even, vl s plrŽyi b . q 2 m. The equation for the channel Green function is derived in the same way as for the total three-particle Green function given in Eq. Ž18.. The result is Žno summation of g . Ž0. G Ž3g . Ž z . s N3 R Ž0. 3 Ž z . q R 3 Ž z . Ž N2 V2 .

Žg .

G Ž3g . Ž z . .

Ž 24 .

We are now ready to define the proper transition operator Uab , G 3 Ž z . s da b G Ž3a . Ž z . q G 3Ž a . Ž z . Ua b Ž z . G Ž3 b . Ž z . ,

Ž 25.

which in the zero density limit coincides with the usual definition of the transition operator with the correct reduction formula to calculate cross sections w26x. After proper

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three-body algebra we arrive at the following equation for the transition operator in medium, viz. N3Ua b Ž z . s Ž 1 y da b . R Ž0. 3 Ž z.

y1

q

Ž . Ý Ž N2 V2 . g G Ž3g . Ž z . Ugb Ž z . .

Ž 26 .

g/ a

For N3 / 0 the equation may be multiplied by N3y1 from the left. We now connect this equation to the two-body t matrix. To this end we define the following transition channel operator T3Ž g . Ž z .: Ž0. Žg . Žg . Ž0. G Ž3g . Ž z . s N3 R Ž0. 3 Ž z . q R 3 Ž z . N2 T3 Ž z . N3 R 3 Ž z . .

Ž 27.

The definition has been chosen so that it leads to the standard Bethe–Goldstone equation in the two-nucleon subchannel, T3Ž3. Ž z . s T2 Ž z˜. d 33X , where z˜ is the frequency of the two-body subsystem embedded in the three-body system. Inserting Eq. Ž27. into Eq. Ž24. leads to the subchannel Bethe–Goldstone equation, T3Ž g . Ž z . s V2Ž g . q Ž V2 N2 .

Žg .

Žg . R Ž0. 3 Ž z . T3 Ž z . .

Ž 28 .

With the help of this equation it is possible to connect the AGS type equation directly to the two-body sub t matrix, N3Ua b Ž z . s Ž 1 y da b . R Ž0. 3 Ž z.

y1

q

Ý

N2Ž g . T3Ž g . Ž z . R Ž0. 3 Ž z . N3 Ugb Ž z . .

g/ a

Ž 29 . Here we have written the Pauli factors occurring due to the surrounding matter explicitly. Note, that through Eq. Ž28. T3Ž g . Ž z . is also medium dependent. Since N3 ) 0 we may multiply Eq. Ž29. by N3y1 r2 from the left and by N31r2 from the right, and introduce Ugb) s N31r2 Ugb N31r2 . Then ) Uab Ž z . s Ž 1 y da b . R Ž0. 3 Ž z.

y1

q

Ý

) N3y1r2 N2Ž g . T3Ž g . Ž z . N31r2 R Ž0. 3 Ž z . Ugb Ž z . .

g/ a

Ž 30 . We may now introduce the transition channel operator T3) Ž g . s N3y1 r2 N2Ž g . T3Ž g . N31r2 so that the equation looks formally as for the isolated case. The respective channel t matrix equation then is ) Žg . T3) Ž g . s V3) Ž g . q V3) Ž g . R Ž0. . 3 T3

Ž 31 .

The explicit form of the effective potential arising in this equation reads Že.g. in the Ž3. channel. V3) Ž3. Ž 123,1X 2X 3X . s N3y1 r2 Ž 123 . N2Ž3. Ž 123 . V3Ž3. Ž 123,1X 2X 3X . N31r2 Ž 1X 2X 3X . s Ž 1 y f1 y f2 .

1r2

Ž1 y f 3 q g Ž ´ 1 q ´ 2 . .

=d 33X Ž 1 y f 3 q g Ž ´ 1X q ´ 2X . . , Ž 1 y f1 y f2 .

1r2

1r2

y1 r2

V2 Ž 12,1X 2X .

Ž 1 y f 1X y f 2X .

V2 Ž 12,1X 2X . Ž 1 y f 1X y f 2X .

1r2

.

1r2

Ž 32 .

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The last equation holds for f 2 < f only and g Ž v . s we b Ž v y2 m . y 1xy1 . Utilizing this approximation Eq. Ž30. has been solved numerically using a separable potential for the strong nucleon–nucleon interaction w8x.

4. Results Since the Green functions have been evaluated in an independent particle basis the one-, two-, and three-particle Green functions are decoupled in hierarchy, as given in Eqs. Ž12., Ž15., Ž18.. To solve the in-medium problem up to three-particle clusters the one-, two- and three-particle problems are consistently solved. This leads to the single particle self-energy shift equation Ž13., the two-body input including the proper Pauli blocking equation Ž31., and eventually to the three-body scattering state. For technical reasons we have used some reasonable approximations valid at smaller effective densities. The nucleon self-energy has been calculated via Eq. Ž13., but in the three-body code we have approximated the self-energy by effective masses, i.e.

´1 s

k 12 2m

q S HF Ž k 1 . ,

k 12 2 m)

q S HF Ž 0 . .

Ž 33 .

For simplicity and speed of the calculation, we use angle averaged Fermi functions ²² N2 :: and ²² N3 ::, e.g. ²² N3 :: s

1

Ž 4p .

2

Hdcos u

q

dcos up d f q d f p N3 Ž p,q, Pc .m . . ,

Ž 34 .

where the angles are taken with respect to Pc.m . and p, q are the standard Jacobi coordinates w23–27x. We have solved the three-body equation neglecting terms of the order f 2 , using Eq. Ž32. as the driving kernel in the AGS equations, which is sufficient, since the calculation is done below the Mott density of the deuteron. A rank-one Yamaguchi potential w29x has been used in the calculations of the deuteron break-up cross section that includes the coupled 3 S 1 – 3 D 1 and the 1 S 0 channels only. The parameters are given in Table 1. To get an impression of the quality of the calculation the isolated cross section is given in Fig. 1 along with the experimental data on neutron deuteron scattering w30x.

Table 1 Parameters for the separable potential used to solve the three-body AGS equation. The form factors are gk l Ž p . sCk l p l rŽ p 2 q bk2l . l r2 q1. The binding energy of the isolated deuteron is Ed s 2.225 MeV and of the isolated triton Et s 7.937 MeV w31x.

k

C0

C2

wMeV fmy1 x 3

S1 – 3 D11 S0

8.63237 8.87026

b0

b1

wfmy1 x y38.8017

1.24128 1.17328

1.94773

C. Kuhrts et al.r Nuclear Physics A 668 (2000) 137–148

Fig. 1. A comparison of the total, elastic, and break-up cross sections nd experimental data of Ref. w30x.

™ nd, nd ™ nnp

145

with the

Unlike the isolated three-body problem the integral equation depends on the relative momentum of the three-nucleon system with respect to the medium, since the Fermi functions depend explicitly on the center of mass momentum of the three-body system. As a consequence one has to solve the three-body problem at a finite center of mass momentum, Pc.m .s k 1 q k 2 q k 3 , in a medium that may be considered at rest, Pmed s 0. However, technically it is more convenient to let the three-nucleon system rest, Pc.m .s 0, and the surrounding medium move with Pmed s yŽ k 1 q k 2 q k 3 .. This procedure results in the least change of the three-body algebra and is possible since the dependence on Pc.m .y Pmed is only through the Pauli blocking factors thus parametric. For small momenta Pc.m . and small relative energies of the dN-system the influence of the medium is most pronounced. In Fig. 2 we see that the in-medium cross section is significantly enhanced compared to the isolated on. So it is not a justifiable approximation to include medium effects only by the Pauli-blocking factors of the final states Žsee Eq. Ž3... That would effectively decrease the break-up cross section and might lead to wrong conclusions. The dominant effect is the weakening of deuteron binding energy ŽMott-effect.. This provokes also a shift of the threshold to smaller energies with raising

Fig. 2. In-medium break-up cross section at T s10 MeV and Pc.m .s 0. The isolated cross section is shown as a solid line, other lines show different nuclear densities as given in the legend.

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Fig. 3. In-medium break-up cross section at T s10 MeV and ns 0.007 fmy3 for different center-of-mass 0 is shown as solid line. momenta Pc.m .. The isolated cross section s bu

densities. We observe that for higher energies the medium dependence of the cross section becomes much weaker, which a posteriori justifies the use of isolated cross sections Žalong with the impulse approximation. when higher energies are considered w2x. The dependence of the cross section on the center-of-mass momentum Pc.m . is shown in Fig. 3 for a fixed density and temperature. With the definition of a medium dependent cross section according to Eq. Ž4. we can rewrite for the deuteron break-up time Eq. Ž6., < < ty1 bu Ž P . s 4 dk N Õd y ÕN s bu Ž k , P . f Ž k N . .

H

Ž 35 .

To get an impression how much the medium dependence of the cross section shortens the deuteron break-up time we show in Fig. 4 the result of the above Eq. Ž35. with use of the medium cross section s bu) calculated by solving Eq. Ž30. compared to the isolated s bu0 . For small deuteron momenta the break-up time is shorter by a factor of three at n s 0.007 fmy3 and T s 10 MeV.

Fig. 4. Deuteron break-up time at T s10 MeV and nuclear density ns 0.007 fmy3 . Solid line using the medium dependent cross section as given in Fig. 2, and dashed line using the isolated cross section.

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Fig. 5. Relaxation time for small fluctuations of the deuteron density from chemical equilibrium at a temperature of T s10 MeV as a function of the nucleon density. Line coding as in Fig. 4.

We also calculate the chemical relaxation time given in Eq. Ž11.. In Fig. 5 we see again that the medium dependence of the cross section shortens the time scale especially at higher densities.

5. Conclusion and outlook Our results show that medium dependent cross sections in the respective collision integrals lead to shorter reaction time scales. Chemical processes become faster. This also affects the elastic rates that are related to thermal equilibration. The basis of this result is the cluster Hartree–Fock approach that in our approximation includes correlations up to three particles in a consistent way. The equations driving the correlations are rigorous. The respective one-, two- and three-body equations are solved, in particular for the three-particle case FaddeevrAGS type equations that have been given in Ref. w8–10,20x. The AGS approach is particularly appealing since it allows generalizations to n-particle equations in a straightforward way. Results for the three-body bound state in medium has been published elsewhere w31x. As expected from the deuteron case, the triton binding energy changes with increasing density up to the Mott density, where Et s 0. The production rates, distribution of fragments, spectra, etc., of light charged particles in heavy ion collisions at intermediate energies may change because of the much smaller time scales induced through the medium dependence compared to the use of free cross sections Žrespectively experimental cross sections..

Acknowledgements We thank A. Sedrakian for a useful discussion in the early stage of this work. This work has been supported by the Deutsche Forschungsgemeinschaft.

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