Nuclear-matter approach to the optical potential for 12C-12C scattering

Nuclear Physics A499 (1989) 413-428 North-Holland. Amsterdam

NUCLEAR-MATTER

APPROACH

TO

THE

OPTICAL

POTENTIAL

FOR

‘*C-‘*C SCATTERING* J. DABROWSKI Institute

for Nuclear

Smdiey,

Hoza 69, 00-681

Warsaw,

Poland

H.S. K6HLER

Received

29 December

1988

Abstract: A simple theory of the heavy-ion optical potential S’= density approximation, is applied to “C nuclei. The energy obtained from the properties of nuclear matter. The frivolous for 3, and ‘i‘, reveal strong energy dependence and show “exact” reaction matrix calculations.

r,+ i’L^, , based on the local frozen density needed for calculating I’, is model is used to calculate ‘I”, Results an overall agreement with results of

1. Introduction In the present paper, the simple nuclear-matter (NM) approach to the heavy-ion potential ‘V = 7f,+ i’Y, presented in ref. ‘) (hereafter referred to as (I)) is applied to “C-“C scattering. The approach is based on the local-density approximation: the two colliding nuclei are described locally as two interpenetrating nuclear matters moving against each other. For the local-density and momentum distribution, the frozen-density model is applied. The real part ‘VRis defined as the difference between the energies

of the overlapping

and spatially

separated

nuclei.

The energy

of the

two nuclear matters is determined directly from known saturation properties of NM. The imaginary potential Y, is calculated from the frivolous model, thus Y, is obtained directly from the NN cross section. In our simple procedure, we bypass the difficult task of solving the two NM problem starting from the NN interaction. This task was undertaken and solved within the Brueckner theory (in lowest order with a continuous choice of the s.p. energies) by Faessler and his collaborators I-‘). To compare our results with the “exact” results for Y of the case of two “C nuclei for obtained in the frozen-density The paper is organized as sect. 2, which contains the

Faessler group, we consider in the present paper the which Ohtsuka et al. “) present these “exact” results approximation for a number of collision energies. follows: Our simple NM approach to 7f is outlined in description of our procedure for Y, including the

* Supported in part by the Polish-U.S. part by NSF Grant PHY-X604602.

Sklodowska-Curie

0375-9474/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V

Fund under

grant

No. P-F7F037P

and in

J. Dahrowski,

414

H.S. Ktihler / Nuclear

ma/fer

approach

inhomogeneity corrections (subsect. 2.1), the frivolous model expression for %, (subsect. 2.2), and the specification of the local-density and momentum distribution according to the frozen-density approximation (subsect. 2.3). Some details of our calculational procedure are described in sect. 3. Our results for the “C-“C system are presented and discussed in sect. 4. In sect. 5 we compare our results with the “exact” results of the Faessler group. Final comments and conclusions are given in sect. 6.

2. Theory of the nucleus-nucleus 2.1.

THE

REAL

POTENTIAL

optical potential

‘I-^,

We consider nuclei 1 and 2 (with masses M, and M2) moving in the center-of-mass system (c.m.s.) with relative momentum K,,, (in units of h) and energy E (see fig. 1). We denote by R the relative position vector between the centers-of-mass (cm.) of nuclei 1 and 2. The real potential Y”‘, between the two nuclei is defined by: .~“R(E, K)= where E’i,( i) is the intrinsic energy of the total system,

Fig. 1. The density

~‘,.,.(K,,I,

K)-~“KLI/~P-

girl(l)-

energy of the isolated nucleus and p = M, M2/( M, + M,).

distribution

in two colliding

nuclei.

The cm.

The conservation of the total energy implies momentum K,,, = K,,,(R) is changing with R: hK,,,( R)‘/2p

that

gi,(2) 9

i, g’,.,. is the nuclear

of nucleus

potential

~‘,.,,.(LI RI = 3

between

nuclei

drff,,,.(K,,,,

c.m.

l(2) is 0,(02).

the instantaneous

+ Y”,( E, R) + ‘V<.(R) = h’K,,,(a3)“/2/.~

where V<.(R) is the Coulomb We write gc.,. in the form

(2.1)

= E,

relative

(2.2)

1 and 2.

R; r) ,

(2.3)

J. Dahrowki,

where H _ is the energy the two nuclei, the system NM of total density p and To determine the energy frozen-density model), we

H.S. Kiihler / Nucleur matter approach

density

(in c.m.s.)

415

at r. For a given distance

same density p as the local density of our system of colliding momentum distribution n,,( kN) = 0( k,- k,), where the Fermi (;n’p)’ ‘. The energy density of this normal NM is

nuclei, and with momentum k, =

(2.4)

H,, =.f(p)p, wheref(p) = EYM/A is the energy energy density in the rest frame of (1.c.m.) frame. Hereafter, we shall energy density in this 1.c.m. frame. in sect. 3. For the differences H - H,,, we H-

R between

is approximated locally (at each point r) by a piece of with momentum distribution n(k,). density of NM for given p and n (for which we use the consider normal NM (i.e. NM in its ground state) of the

per nucleon in normal NM. Obviously, H,, is the NM, which we shall call the local center-of-mass denote by k, nucleon momenta, and by H, the The connection between H and H,,,,. is outlined use the approximate

H,,=4/(2n)j

dk,

[n(h)

-

relation

n,,(k.)le,,(k,) .

(2.5)

The single-particle (s.p.) energy depends on the momentum distribution and is here approximated by e,,(kN), i.e. the ground-state removal energy, calculated with distribution n,,. This is a valid approximation when n - n,, is small. Note that n and n,, give the same density. For the s.p. energy e,, in normal NM, we use the effectivemass approximation: c,,(k,) = e(k,)lv+ where F(k,) = h’kh/2m, and nucleon mass. Eqs. (2.4)-(2.6)

(2.6)

C,

V= m*/m is the ratio of the effective lead to the following result for H: H =.fo+(7.--,,)/v,

where 7 and T,, are the kinetic and in normal NM. For the semi-empirical

energy

function

densities

to the real

(2.7) (in the 1.c.m. frame)

in our system

.f‘(~), we use the form

(2.8) where k,,, is the Fermi momentum at the equilibrium density p,), and the coefficients a, are determined by kFor by the volume energy of NM, F,,,,=~‘(P,,), and by the compressibility K,,,,, = kt,,(d’fldki),,. For u we use the form “)

~=~(P)=ll[~+(l/~,,-I)P/P,,l,

(2.9)

where Y()= v(po). Altogether, to calculate H according the NM parameters: k,,,, E,,,, , K,, ,,,, and v,).

to eq. (2.7), we have to know

J. ~abroulsk~, HS. Kiihler / Nuclear matter approach

416

Variations of the density, especially at the surface of the nuclear system where the effects of the finite range of the nuclear forces are important, lead to density gradient corrections to the energy density H. We take them into account as did Brueckner

et al. ‘). Namely,

we add to the energy

density

the gradient

correction

H~=r~+~~. The gradient correction, is

correction

rv to the kinetic

(2.10)

energy density,

known

as the Weizsacker

TV= (~~/72~)(V~)~/~. (Notice the additional factor correction.) For the gradient correction

$ compared

to the original

7~~ to the potential gr =

(2.11)

energy

form of the Weizsacker density,

we use the form

rtvp)’ ,

(2.12)

and treat n as a free parameter, which we fix by requiring that the calculated intrinsic nuclear energies Ei, (plus the Coulomb energies EcouI) agree with the experimental energies. The energies

Z’i,(i) are calculated

gin(i) =

I

dr

i

according

.~(~i(r))~;(~)+~

”

to

fvp,(r))'/pi(r)+ll(Vpi(r)>'

.

For the Coulomb energy of a nucleus J(r’), we use the expression

with 2 protons

and with the r.m.s. radius

Ec,,,=3ZZe2/5R,, which represents of radius

2.2. THE

(2.13)

}

the Coulomb

energy of an equivalent

(2.14f uniform

charge

distribution

R, = J$( r’).

IMAGINARY

POTENTIAL

?;

In

the case of the nucleon-nucleus optical potential, one may express the imaginary part of the potential directly through the total NN cross section CT.The procedure known as the frivolous model may be easily extended to the case of the nucleusnucleus optical potential. The absorption in the nucleus-nucleus scattering, caused by NN collisions in any volume element of the system, is represented by

?“‘,(E,RI = where for the absorptive

potential

dr u,(fL,, R;

density

U, we have:

rf ,

(2.15)

J. Dabrowski,

H.S.

Kiihler / Nuclear

where k = i( k, - k,) is the relative momentum of the two colliding

nucleons.

principle operator which nucleons of the system. Formally, the frivolous

matler approach

and 2K = k, + kz is the total momentum

By Q(K, k) we denote

prevents model

NN

417

scattering

expressions

the angle-averaged into

states

exclusion

occupied

by other

for Y,, eqs. (2.10) and (2.11),

were

derived in (1) from the equation for the effective NN interaction (the reaction matrix) by applying the optical theorem. Alternatively, one obtains this expression by simple intuitive reasoning “).

2.3.

THE

FROZEN-DENSITY

APPROXIMATION

The frozen-density approximation is the simplest way of determining the local density p and momentum distribution n. In this approximation, all degrees of freedom are frozen, except for R. The density of each of the two nuclei does not relax during the collision. The instantaneous velocity of each point of the nucleus l(2) is the same [in the c.m.s. it is equal to -hK,,,/M,( hK,,,/M,)]. The total density of the combined system at Y is equal to the sum of the original densities of nuclei 1 and 2 (see fig. 1): (2.17)

P(r)=p,(r)+k(lr-RI).

For the local momentum distribution in the combined system at r, we use the two-sphere distribution of fig. 2. The case represented in fig. 2a is simple. The local-momentum distribution at Y of nucleons in nucleus 1 is the Fermi sphere (surface F,) centered in d,, with the local Fermi momentum k,,,, = k,,,,(r)

Similarly for nucleons in nucleus in 6, with the Fermi momentum

= (i~‘p,(

r))“’

.

(2.18)

2, we have the Fermi sphere

k FZIj= kF2,J r) = ($dp(lr

Cal Kr > k,,o+ kFZo Fig. 2. The local momentum

- RI))“’

(surface

F2) centered

.

(2.19)

(b) Kr c &lo+ kFz0 distribution

in two colliding

nuclei.

418

J. Dahrowski,

The distance between

K, between

nucleons

6,

in nucleus

H.S. K6hler

and

&

/ Nuclear

is (twice)

1 and nucleons

matter

approuch

the average

in nucleus

relative

2. It is connected

momentum with K,,,

by

in fig. 2b, when K, < kFlo + kFzO, is slightly more complicated. Here the two Fermi spheres overlap; to resolve the problem of a double occupancy within the overlap region, we apply the prescription of ref. “): we increase both Fermi momenta k,,,, and kkl,, by the same amount 8, The case represented

k,, = &lo+ 6 and determine

6 from the condition

k,, = km, + 6 ,

(2.21)

that

P = pI +P~ = [4/(2#1

V,,

(2.22)

where V, is the volume within F = F, + Fz. With the two-sphere momentum distribution, one may obtain an analytical expression for the angle-averaged exclusion principle operator Q(K, k), given in the appendix of (I). [Let us correct two misprints there: -5 in expression (A.22) for R should be replaced by +[, and the whole right-hand side of this expression should be multiplied by 2r.l

3. Calculational

procedure

To calculate the optical potential Y for the scattering of nucleus 2 (projectile) on nucleus 1 (target), we apply the theory of sect. 2 based on the local density approximation (plus inhomogeneity corrections) and the frozen density model. This means that locally (at each point r, see fig. l), the system is approximated by two pieces of NM with the total density p given by eq. (2.17), and with the two-sphere momentum distribution of fig. 2. As in sect. 2.1, we calculate the energy density H in the 1.c.m. frame of the two pieces of NM. Now, the 1.c.m. frame depends on r. Thus, before calculating the energy %, we go over to the rest frame of piece 1 of NM (the piece which belongs to nucleus l), which we call the “laboratory” (“lab”) frame. This frame - in our frozen-density approximation - coincides with the rest frame of nucleus 1 and does not depend on r. [Of course, it depends on R; at R =a, it coincides with the laboratory (lab) frame of nuclei 1 and 2 with nucleus 1 being the target.] In the lab frame, we have

where H..,,,.. = H + jli’/2m)k:ip,

(3.2)

where k,; is the momentum per nucleon in the locai two pieces of NM in the lab system (see eq. (3.1) of (I)). To determine ‘VR, we use eq. (2.1) and the relation 8&,h.. = %,.,, i (hK$/2( where K, = (~~/~)K~~,

is the momentum

Ml + Ml) ,

of nucleus

Sr,( E, R ) = ~‘.I:,h..(K,,, , R ) - ( AM,K,,,/II)‘/~

2 in the lab frame.

(3.3) We get

M7 - ~i”( 1) - %i,( 2) .

(3.4)

For a given c.m. energy E, we know only K,,,(m) = (2pE)“‘/h, whereas expression (3.4) depends on K,,, = K,,,(R), connected with K,,,(a) by eq. (2.2), which in turn contains YR( E, R ). We solve this problem by iteration, which we start by calculating ‘?“‘$I with the help of expression (3.4) with K,,, = K:,,,(o). In the next step, we calculate ‘7’:’ by applying expression (3.4) with K,,, = K$)( R), obtained from eq. (2.2) with ‘k’, = ‘r-g’. This procedure converges after a few steps. To calculate ‘I/‘,, we apply expressions (2.15) and (2.16). The absorptive potential density O, depends on K,.,, = #,,,( R ). This presents no problem, because after having determined VR, we know K,,,(R} from eq. (2.2). The connection projectile nucleon

between E,,,/A, (the kinetic energy of the projectile in the lab system) and K,(a) [in fm’] is

nucleus

per

(3.5) For the u-integration in (3.1) and (2.15), we use cylindrical coordinates with the z-axis along R. We assume that K, is parallel to R, which reduces the r-integration to a twofold integration. Whereas we have analytical expressions for H, expression These integrations, in cylindrical (2.16) for u, involves integrations over k, and k7. coordinates along K,, lead to eve-dimensional integration. All the integrations have been performed by means of the Gauss formula. In the discussion in sect. 5 we shall present results for two infinite nuclear matters of equal density p, = pZ -= ip, moving against each other with relative velocity K,/m, with the two-sphere momentum distribution n of the combined system. The relative momentum K,,, between the two nuclear matters is connected with K, by: K,= 4K,,,/A [see eq. (2.2)], where A is the total number of nucleons in the combined system. In this homogeneous system, the energy 8_,. = H x (volume of the system) is infinite, and we introduce the energy per nucleon ‘X’,,,,,,/A = H/p, and similarly Y/A. Notice that in the combined system, the 1.c.m. frame coincides with the c.m.s. = and H,.,,,. = H. For the intrinsic energies, we have gi,,( 1)/(A/2) = ~i,(2)/(A/2) .f(ip).

420

.I. Dahrowski, H.S. Kiihler / Nuclear matter approach

4. Results for the ‘*C-‘*C system For the total NN cross section (T = $(gnn+ (T,~), we use the parametrization of Metropolis et al. ‘“) when the laboratory energy Elab > 20 MeV. For Elah < 20 MeV, we use the effective-range approximations with the following values (in fm) of the respective singlet (s) and triplet (t) scattering lengths (a) and effective ranges (r): a,,,,= 5.4, and rtnp= 1.73. aann = -16.1, r,,,, =3.2, as”,,= -23.714, r,,,=2.704, As NM parameters we use: kFO= 1.35 fin-’ (p. = 1.66 fin-‘), &“Ol= -15.8 MeV, K con?= 235 MeV, and vg = 0.83. This value of vg has been determined by Johnson, Horen, and Mahaux “) in their analysis of the nucleon-nucleus optical potential with dispersion relation constraint. It differs from the value of v0 = 0.7 used in (I). In the colliding ‘*C nuclei, we use the three-parameter Fermi distributions determined by electron scattering Ir) p,(r) = p>(r) = b(r) = P(O)]1 +

w(rlc)‘ll[l +ev {(r- c)/zIl

,

(4.1)

with c = 2.334 fm, z = 0.5224 fm, w = -0.149 (p(O) = 0.186 fin-‘, J(r’) = 2.441 fm, R, = 3.151 fm). To obtain YJ’“~~~,~(R) we approximate charge distributions in the two “C nuclei by uniform charge distributions with radius R, and calculate Vc.,,,(R) as the Coulomb interaction between these two uniform charge distributions. We use the value of 7 = 22 MeV. fm5, fitted in ref. “) to the binding energies of I60 and 4”Ca. This value of n used in eq. (2.13) leads for “C to pi, = -102.45 MeV. Together with the Coulomb energy &,,, =9.87 MeV [see eq. (2.14)], we get -92.58 MeV for the total intrinsic energy of 12C, which agrees reasonably with the experimental value of -92.16 MeV. Results obtained for 7fn and Y, are shown in figs. 3 and 4. The most striking feature of our results is the strong dependence of both YR and ‘I’“,on K,(a), i.e. on the energy E, and the R-dependence of Y, (short range repulsiontlong range attraction). The dependence of YfR on K, is nonmonotomic. At small values of K,, Y, decreases (algebraically, i.e. becomes more attractive) with increasing K, up to K,= 1.5 fin-‘. If we further increase K,, LYfR starts to increase and eventually becomes partly repulsive. Let us recall that at small values of K,, we increase the radii of the two Fermi spheres in our momentum distribution, eq. (2.21), to avoid double occupancy in the overlap region, which would violate the exclusion principle. This leads to an increase in the kinetic energy, which gives a repulsive contribution to ‘vK. This Pauli blocking effect becomes less important when K, increases, which explains the initial decrease in w^, On the other hand, the potential energy contribution to v, increases (algebraically) with increasing K,, which may be traced back to the short-range NN repulsion. This explains the increase in cl/‘, at larger values to the position of the “window” of a of K,. Notice that KT= 1.5 fin-’ corresponds minimum in Y,/A for two nuclear matters, discussed in (I).

J. Dabrowski,

_.._._._I____-.r._-

H.S.

.. ..---..

KiMer

1 Nuclear

matter approach

_______

421

I

I

3 f,,- 1

i ______Kr

= 2. 02 fm-1 .:.::'.. _...--:. -----__ __..--1 --_ __.-

---

___L-_-_-__L_

Kr = Kr(ool Kr = Kr(R)

1

/

-100 Fig. 3. Results

R lfml

5

0 for the real “C-“C

potential I”R at K,(m) = 1.1, 2.02 and 3 fm-’ without the R dependence of K,.

calculated

with and

The dependence of Y, on the distance R reflects the dependence of YV,JA for two nuclear matters on the density p discussed in (I). Notice that at small (large) values of R, a substantial part of the combined system has a density p > po(p < p,J. This leads to a repulsive (attractive) contribution to y-R at small (large) value of R. The Pauli blocking

effect occurs

only in the r-region,

kFIO(r)+kl.ZO(Y)=(3~~/2)“3[p,(r)”3+Pz(lr-R1)”3]>

where K,.

This region increases with decreasing R. Thus the repulsive contribution to YR due to the Pauli blocking effect is largest at R = 0. The short distance repulsion and long-distance repulsion in YR is further enhanced by the inhomogeneity (surface) correction, eq. (2.10) [which does not depend on the energy]. Let us start from a large distance R, at which the two “C nuclei are spatially separated. When we decrease R, the two nuclei begin to overlap and the surface of the combined system (originally equal twice the surface of “C) decreases. Since (VP)’ # 0 only in the surface region, this (positive) surface correction to the energy of the combined system decreases, and thus the inhomogeneity correction to ‘YR is negative (attractive). When the two nuclei overlap completely (R + 0), the surface of the combined system becomes minimal (equal to the surface of “C). Here, however, p(r) = 2i(“C; r) and (VP)’ =4(Vp)‘. Thus the increase in (VP)’ by a factor 4 compared to (VP)* becomes more important than the decrease in surface

J. Dabrowski,

422 0

I

_

-

-cl----

--

I _

_--

H.S. Kiihler / Nuclear I

---_rz--

-___

matter approach -T-

--

/y=----

^v, IMeUl

-50

-100

__-I

I

I

I

I

I

5

0 Fig. 4. Similar

as fig. 3 but for the imaginary

/

R [fml potential

I‘,

(by a factor 2, compared to the surface of the two separated “C nuclei). Consequently, the inhomogeneity contribution to Y”/^R at small distances R is positive (repulsive). [The above reasoning applies to the inhomogeneity correction rrY-(Vp)I. A similar reasoning applied to the Weizsacker term TV- (Vp)‘/p shows that its contribution to Y, vanishes at R = 0, and is attractive for R # 0.1 Our final results for YR, the solid curves in fig. 3, were obtained with K, = K,(R), eq. (2.2). The broken curves in fig. 3 indicate results obtained with K,= K,(W). Since the depth of Y, as a function of K, is stationary at K,- 1.5 fm-‘, the shift in Y’“Rdue to the dependence of K, on R approximately vanishes at this value of K,. The behavior of “I/^,is simpler: it depends monotonically on both K, and R. Its depth I”lr,l increases fast with increasing K,. The main reason for this is the decreasing role of the exclusion principle at high momenta K,, where a -+ 1. Furthermore, the average value of the relative NN momentum k increases with K,, and thus the factor k in (2.16) contributes also to this increase in lY,l. The absorptive potential ‘Ir, is concentrated at small distances R. Its range is short compared with the range of Y,. It should be stressed that our absorptive potential arises entirely from the two-body mechanism of incoherent NN collisions. However, the damping of the elastic channel in the heavy-ion scattering may arise also from coherent excitations of collective states. These genuine surface effects, disregarded in our NM approach, are expected to increase the range ofthe imaginary potential 14).

J. Dahron~ski, 5.

The “exact”

H.S. K&h

Comparison

results of Ohtsuka

,I Nuclear

matteruppoach

with the “exact”

et a/. ‘) were obtained

42.1

results

with the Reid 13) soft-core

NN interaction. Their procedure applied at K,= 0 leads to the following NM parameters: k,,,= 1.425 fm ’ (po= 0.195 fm ‘), F,,, = -16.0 MeV, Kc= 188 MeV. These parameters together with ZJ(,= 0.83, are used in our calculations presented in this section to make meaningful the comparison between our results and the results of ref. “). For the same reason, we apply for b(r) the density distribution used in ref. ‘). It Furthermore, has the form of eq. (4.1) with c = 2.355 fm (with z and M’unchanged). we apply the same gradient correction as in ref. ‘): HT = n(Cp)’ with n = (h’/Sm) 8.30 fm3 = 43.03 MeV.fm5, i.e. we disregard the Weizsacker correction. First, we compare our results for the “intrinsic” energy E/A of two nuclear matters of equal density p, =p2=Ap moving against each other with the relative velocity KJ m. E/A=

H/p-(h’/r(m)Kf.

(5.1)

Notice that ,!?/A has the simple meaning of the total energy minus the kinetic energy of the relative motion of the two nuclear matters only when the two Fermi spheres do not overlap (fig. 3a). Results for E/A are shown in fig. 5 as functions of p/c,, (with p,,=O.166 fin-‘) for K, = 0, 1, 2 and 3 fm -‘. Broken curves are the “exact” results. Our solid K,. = 0 curve represents ,!?/A = E/A =,f(p), eq. (2.8), with the same position of the minimum (p,, and F,,,,) and the same curvature at the minimum (i.e. same compressibility K,) as the “exact” broken curve. Although the two curves concide around the minimum, there are slight differences between them away from the minimum. At K,.= 1 fm ‘, At the agreement between our results and the “exact” results remains satisfactory. K, = 2, the agreement is worse, and at K,. = 3 fm ‘, it becomes very poor. Our results are based on approximation (2.5). It is valid for small deformations of the Fermi surface (from the spherical shape) and we do not expect it to work when the two Fermi spheres of our momentum distribution do not overlap (the situation in fig. 2a). Now, the two spheres touch each other at p/p,) = 0.1, 0.8 and 2.7 for K,.= I,2 and 3 fm ‘, and we expect approximation (2.5) to be applicable forp/&,>O.l,O.Sand2.7for K,=l,2and3fm ‘. Actually, the overall agreement of our results for l?/A and the “exact” ones is better than we could expect. This is partly due to the fact that for small densities p, the effective mass ZJ==1, and the dependence of l?/A on K,. becomes very weak. We would like to add that our results, expecially at higher densities, are sensitive to the value of the effective mass v. On the other hand, the accuracy of the “exact” results is worse at higher densities where the convergence of the low order Brueckner theory is poor. Results for I’,(R) are presented in fig. 6. They have been obtained with K,.= K,(W) in a similar manner as the “exact” results shown by broken curves. The only source of the differences between the solid and the broken curves are the differences in

J. Dahrowski,

i__-_-_-_______-____L___ 0

---

H.S. KGhler / Nuclear

I

1

matter approach

I

Q’ PO

----

_J-_---__ 2

Fig. 5. The intrinsic energy per nucleon /?:/A in two colliding nuclear matters of the same density as function of the ratio of the total density p to p,, = 0.166 fmm3. Numbers indicate values of K, in fm-‘. Our results are presented by the solid curves and the “exact” results by the broken curves.

E/A, shown in fig. 5. With increasing K,, the differences become larger. In the case of K, = 3 fin-‘, not shown in fig. 6, our VR is attractive everywhere, whereas the “exact” Y, repulsive at R = 0 and much shallower than our 7f,,, at larger distances.

In conclusion, we find a nice agreement of our results for Y, with the exact results for small values of K,. Around K,- 2 fm-‘, the agreement becomes poor, and for larger values of K, (where our approximate approach is not applicable anyway), our results differ substantially from the “exact” ones. Let us now discuss the imaginary potential. We start with the two nuclear matters we considered in discussing the real potential. Results for V,/A are shown in fig. 7 as functions of p/p<, for K,= 1,2 and 3 fm-‘. The “exact” results are shown as broken lines. In the case of K, = 2 fm-‘, the depth of the absorptive potential IY,I/A increases with increasing p for small values of p. When p exceeds a certain minimal value (p = OSP,,), the exclusion principle becomes increasingly important and /‘V”l/A starts to decrease with increasing p. (For K, = 1 fm-’ this minimal value of p is about p,,/lO, and for K, = 3 it is larger than 2.2po.) This behavior of IY,I/A is revealed by both our frivolous model results and the “exact” results. Both results are reasonably close to each other. The largest differences occur around the minimal value of p mentioned above. Most serious is the discrepancy for K,= 1 fin-’ and

__- -100 -

--

_I_---/

,_/’

_, .’ ...** _,” ,,a” .di Kr 112. 02 fm- 1 I

I

I

0 Fig. 6. The ‘“exact” results curves), both obtained

I 5

for the “C-‘%’ real potential with the same NM parameters

I k (broken and without

I

I R[fm

i 1

curves) and our results (solid the R dependence of K,.

p/po50.5,where our absorptive is possible that the frivolous the free cross section, instead

potential is much weaker than the “exact” one. It model or, more precisely, the approximation to use of a medium-corrected cross section breaks down for

small relative momenta K,.This point will be investigated in future work. In fig. 8 we present our results for ‘YVI^,(R) obtained with I<,=6&(m) and the “exact” results which were also obtained with K, = K,(W).The agreement between our results and the “exact” results is good for K,=2.02 and 3 fin-‘, but not quite as good for K, = 1.1 fm-‘, which simply reflects the situation shown in fig. 7.

with two nuclear

matters

6. Conclusions Our work on the optical potential 7” involves the calculation of the real part and of the imaginary part of an energy density. In Brueckner theory, this energy is obtained from a complex reaction matrix (complex effective interaction). We have, however, found it practical to calculate the two parts of the energy (and thus Y*, and V,) by different methods. The results of our calculations show a remarkable agreement with the much more complicated “exact” calculations as shown in sect. 5. As regards the real part of the energy in our comparison with the “exact” results,

I /

i -.----._-

__--_.-._I_..

._.

_.

“. :

II

-10 0

1

Q/h

Fig. 7. Similar as liy. 5 but for

, ~~ ‘/

X.,/A.

we note that three parameters in E/A were adjusted, but this was only for K, = 0. The change in energy as a result of deformation is well-reproduced by the relatively simple procedure based on eq. (2.5) up to K,--2 fm r. It does fail for larger I(, because our approximation does not take into account the change of s.p. energies with deformation.

They depend

only on density

in our approximation,

but not on

the detailed occupation in momentum space. We plan to improve upon this part of our calculation. The improvement will be based on a Pauli operator and propagatordependent calculation without having to do a full-fledged Brueckner calculation. We will return to this in a forthcoming publication. The imaginary part of the potential is in some sense the easiest to calculate. In our approximation it is expressed directly through the NN cross section. It may be expected a priori that this approximation should work less well for nucleus-nucleus than it does for nucleon-nucleus collisions IX), because it implies replacing an effective interaction with the free scattering matrix in the optical theorem (eq. (3.28) of (I)). Medium effects (especially Pauli blocking) are expected to be substantially larger in nucleus-nucleus than in nucleon-nucleus collisions. However, this does not appear to be so. The reason is the a-function in the collision term that selects energy-conserving collisions. There is one important non-trivial effect in the calculation of the imaginary part of ‘Y,. The s.p. energies in the S-function just mentioned should include both kinetic

J. Dahrowaki. I

I

0

427

H.S. Kiihler / Nuclear matter approach

__-

___--~_---_~

1

_

__

_-

-

--

_=__

I

-__/

y-----

-----------___-

-40

“VI [HeUl

_,_.../--~i

---

Kr

q

1.1

__.,,-’

fm-1

,A ;_,,:::::--_~._<:>4;‘.,.,_.._+ _.i/’ ___.‘-Y ,’ -_-f :,:‘+ K*. 1 2, (Q fm_1 ,,,’

/

*/“.

*/

,‘, /’

-80

,.I I ,’ 1,: Kr 1 3 fm- I .,.X’ ,/:j _;./-::,--’ ._-,

I

I

1

I

/

5

R

I

-120

0 Fig.

8. Similar

as fig. 6 but for

-.

tfml

1”,

and potential energy. The kinetic energy part is of course trivial. Many calculations (e.g. apparently most BUU calculations) neglect the potential energy part. It does in principle need to be calculated as a function of occupations in momentum space, as mentioned above. In our calculations, it is included simply by a density-dependent effective mass, and that seems to be sufficient. Probably the most questionable approximation for the collision between jnite nuclei is the frozen density approximation. Time-dependent calculations (e.g. TDHF and extended TDHF or BUU) show a different behavior. Pauli blocking does not cause a “pileup” of density as assumed in the frozen “sliding” of points in phase-space I”). As a final conclusion, we believe that our methods

density

picture,

but rather

used in this publication

a are

quite accurate for calculation of the optical-model potential in nucleus-nucleus collisions. The method is based upon an assumption of known nuclear properties (saturation density, binding energy and compressibility) and the known NN cross sections, without having to go over an NN potential model. This paper was partly written while one of the authors (J.D.) was visiting the Physics Department at the University of Arizona. He expresses his gratitude to Professor P. Carruthers, the Head of the Department, for his kind invitation, and thanks Professor S. Kiihler for his extraordinary hospitality in Tucson.

428

J. Dabrowski,

H.S. Kiihler / Nuclear

matter approach

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15)

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