Influence of the momentum dependence of nuclear interactions on heavy-ion potentials

Volume 206, number 3

INFLUENCE OF THE MOMENTUM ON HEAVY-ION POTENTIALS ~

PHYSICS LETTERS B

26 May 1988

DEPENDENCE OF NUCLEAR INTERACTIONS

Volker K O C H ~, Ulrich MOSEL, Toni REITZ, Christoph J U N G and Koji N I I T A Institut fiir Theoretische Physik, Universitiit Giessen, D-6300 Giessen, Fed. Rep. Germany Received 19 January 1988

We analyse the momentum dependence of nucleus-nucleus interactions and compare the predictions for this dependence obtained from nonrelativistic Skyrme forces, from relativistic mean field models and from phenomenological ans~itzeused in data analyses. In all cases the momentum dependence is determined by the effective mass alone and is therefore the same for relativistic and nonrelativistic models.

Ainsworth et al. [ 1 ] have recently stressed that the equation o f state as extracted from high-energy heavyion d a t a is much stiffer than that for nuclear m a t t e r in equilibrium. This follows from a very strong mom e n t u m d e p e n d e n c e o f the m e a n field. Ainsworth et al. have estimated this m o m e n t u m dependence on the basis o f a relativistic m o d e l involving vector- and scalar exchange. It is the purpose o f this letter to compare the m o m e n t u m dependence obtained in this way with that inherent in nonrelativistic models a n d with phenomenological ans~itze used recently in d a t a analyses by means o f the V U U equations. F o r a nonrelativistic t r e a t m e n t we consider the m o m e n t u m d e p e n d e n c e o f the nucleus-nucleus interaction on the basis o f the phenomenological Skyrme force. W h e n this force is applied to a system o f m o v i n g nuclei one obtains for the energy density in the m e a n field a p p r o x i m a t i o n [ 2 ]

~ k = ½a ( a r - j 2) ,

( 1)

where p is the spatial density, z the kinetic energy density a n d j the m o m e n t u m density, c~ is a positive strength p a r a m e t e r [c~= ~ (3tt + 5t2) in terms o f the Skyrme p a r a m e t e r s ] . The first term in parentheses on the R H S o f eq. ( 1 ) is a consequence o f the (reSupported by BMFT and GSI Darmstadt. t Present address: Department of Physics, SUNY, Stony Brook, NY 11973, USA. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )

pulsive) m o m e n t u m dependence o f the Skyrme force, the second restores Galilei invariance. In terms o f the phase-space distribution f ( x , p ) can be written as

°f

)((Sk= 4(2/t) 6

d3pd3p' f ( x , P ) f ( x , P ' ) ( P - P ' )

2,

(2) where the phase-space density is n o r m a l i z e d according to

if

(2~Z)3 f ( x , p ) d 3 x d 3 p = N , N is the n u m b e r o f particles. Goeritz and Mosel [2], Z i n t [3] a n d Brink and Stancu [4] have more than a decade ago p e r f o r m e d calculations o f ~ with the purpose o f extracting the energy dependence o f heavy-ion potentials. It was found then that these potentials first become more attractive as a function o f b o m b a r d i n g energy because due to the increasing separation o f the two nuclei in m o m e n t u m space the Pauli principle loses its importance. At higher energies, however, the energy dependence o f the potential changes sign (at about 20 M e V / u ) so that at high energies, above about 120 M e V / u , the potentials are purely repulsive [4 ]. Eq. ( 2 ) makes the physical reason for the repulsion obvious: in a high-energy collision the m o m e n t u m spheres o f the two colliding nuclei are well separated so that the relative m o m e n t a between any 395

Volume 206, number 3

PHYSICS LETTERS B

two nucleons in different nuclei can be quite large and the repulsive, momentum-dependent component of the nucleon-nucleon interaction wins over the attractive, static parts. The single particle potential is obtained from (2)

26 May 1988

P Uoso (x, p)

- - p o A 2 (2n) 3

d3p' f(x,p')(p-p')

(7)

2.

The effective mass is given by

as

l / m * - 1/m= ( 1/pF ) dg/dplpF

Usk (x,p) = (2n 3) (all/agO (x,p)

°f

= - (2clA 2)p/po,

d3p'f(x,p')(p-p') 2 .

-- 2(2n)3

(3)

The effective mass due to ~Sk is then

l / m * - l / m = ( 1/pv )dUsk/dplpv=O~# .

(4)

This density-dependent effective mass determines the repulsion due to the first term in eq. ( 1 ) which is also present for a single, static, time-reversal-invariant nucleus for w h i c h ] disappears. For two colliding nuclei, however, ] is in general nonzero, making ~Sk more attractive. It vanishes only in the overlap region because there the currents of the two individual nuclei just cancel each other. Whereas outside the overlap the ]-dependent term is attractive, in the overlap region its disappearance there leads to a strong repulsion. We now show that the forms (2) and ( 3 ) agree in lowest order with the momentum-dependent potential recently employed by Gale, Bertsch and DasGupta [5] in their V U U analysis of high-energy heavy-ion data. These authors work with an energy density

(8)

where we have used that ( p ) = 0 for nuclear matter where m* is defined. We note that as in the case o f the Skyrme force this expression is proportional to p. As an illustration we show in fig. 1 the single-particle potential of a n 1 6 0 projectile nucleon at a bombarding energy o f 100 M e V / u impinging on a 160 target. The Wigner function appearing in eq. (7) has been approximated by that of two boosted nuclei using a semiclassical approximation and empirical densities. The potential shown is that for a nucleon at the Fermi surface moving in beam direction. The spatial distance of the two nuclei is such that the density in the middle between target and projectile is just flat. The potential in fig. 1 shows a striking x dependence: whereas it has the usual depth of about - 5 0 MeV for the projectile nucleon inside the projectile this particle runs virtually against a wall of about 30 MeV height when it traverses into the target nucleus. This is a consequence of the p dependence exhibited in eq. (7); its origin is exactly the same as discussed

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1 f d3pd3p'f(x'P)f(x'P') aUdGBG=Lc (2/r)6 -10 x{l+ [(p-

)/A]2} -' ,

(5) >

-20

where c is a negative parameter, A a cutoff and ( p ) is a local m o m e n t u m average at x. Up to terms in second order of p - ( p ) the momentum-dependent part of agGB6 can be rewritten as c

1

1

~¢~B6-- poA22 (2n)6 × (p_p,)2.

f

d3pd3p' f ( x , P ) f ( x , P ') (6)

~ B G has exactly the same form as ~Sk. It is therefore not surprising that also the single-particle potential in this order looks the same as in the Skyrme case: Thus

396

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-4

-2

0

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(fm) Fig. 1. Single-particle potential of an '60 projectile nucleon colliding at a beam energy of 100 MeV/u with another ~60 nucleus calculated with the model of eq. ( 5 ). The potential shown is that for a projectile nucleon at the Fermi surface moving in beam direction. The spatial distance between the two nuclei is about 4 fm.

Volume 206, number 3

PHYSICS LETTERS B

above for the Skyrme force: whenever a nucleon moves in the current field of the other nucleus, it experiences a strong repulsion. Finally, we write down the corresponding expressions for ~ , U and m* for a relativistic mean field model involving a scalar and a vector meson [6]. Here one obtains in the local density approximation 1 2 ~ e ~ = ~1 c v ( p 2 v - j 2) -~c~p~ + < ~P(y'p+ m) ~ ) ,

(9)

where p~ and Pv are the scalar and vector densities, respectively, and Cv and c~ are positive constants; here j is the expectation value o f the spatial part of the relativistic current density. The energy-dependent part of the Schr6dinger equivalent single particle potential [ 7 ] is

UrEel = ( c v / m ) p , , e - ( c , , / m ) p . j + ( c 2 / 2 m ) j 2 ,

(10)

where e = E - m and E is the total energy of the particle. In order to obtain also here the effective mass, which is determined by the m o m e n t u m dependence o f the potential at the Fermi m o m e n t u m [see eq. ( 8 ) ] , we perform a nonrelativistic expansion. This yields for the momentum-dependent term WrPel = ( C v / 2 m 2 ) p v p 2 - ( C v / m ) p . j

c~ 2 (2n) 1 3 f d3p'f(x,p')(p-p') -- 2m

2,

(11)

26 May 1988

that the factor oe is directly determined by the effective mass in eq. (4). Exactly the same result actually holds also for the relativistic model. With P, =Y(Pl,v "+-P2,v ),

P, =Pl,s "q-P2,s ,

J = Y(P,,vP-P2,v/g),

(14)

where Pi., and p~,s are the Lorentz boosted vector and scalar densities, respectively, of the two nuclei, one obtains from eq. (9) for the interaction energy densities of two nuclei ~Pnt ----~Lcv~ 2 ( 2Pl,vPZ,v ) -- ICv~2fl2( --2pl,vP2,v )

=Cvy2p,,vPZ,v( 1 .q_/~2 ) =Cv72pl,vP2,v( 1 --~81 "/g2 ) •

(15)

Using now the exact, relativistic relations 7 2 = 1 -4-pZ/m 2,

~22~1 "P2 =Pl "P2/m 2 ,

one obtains Jg,Pn, = (Cv/2m 2 )Pl,vP2,v (Pl - P 2 )2

=2(cv/m2)p~,vP2,vP 2

(16)

with p=p~ = -P2. This result had also been obtained in refs. [1,8]. Introducing here again the effective mass form eq. ( 13 ) gives agPm = ( 2 / p ) ( 1 / m * - 1/m)p~p2p 2 .

(17)

so that the effective mass is given by

l/m*-

1 / m = (¢v / m Z ) p v .

(12)

Again the effective mass is directly proportional to the density, as in the other two cases. We consider now the collision of two nuclei at so high energies per nucleon that the energy of the Fermi motion can be neglected. In this case the phase-space density is given by

f ( x , p ) = (2rc)3 [p, (x )~(p--p] ) -k p2 (x )~(P--P2 ) ] • One then obtains for the interaction energy of two equal nuclei in the CM system with the Skyrme force 'Y/~Skt = IOLpl "P2 (el --P2 )2=20ep,P2P 2

= ( 2 / p ) ( 1/m*-- 1 / m ) p t p 2 p 2 ,

(13)

where pt and P2 are the CM m o m e n t a per particle of nuclei 1 and 2, respectively, and p = p j = - P 2 . Note

Note that this result does not involve a nonrelativistic expansion. In the framework of the G B G model the momentum dependence is more complicated. However, for large cutoffs A one obtains from eq. (6) with the same phase-space distribution as above

°¢tC~BG "~ -- ( C/poAZ)PlP2 (Pl --P2 )2 = ( 2 / p ) ( 1 / * - 1/m)p,p2 p2,

(18)

i.e. an energy dependence identical to that obtained with the Skyrme force [eq. (13) ]. The comparison of eqs. ( 13 ) and (17) shows that the m o m e n t u m dependence o f the nucleus-nucleus interaction is exactly the same for the Skyrme force and in the relativistic model. In the G B G model [eq. ( 18 ) ] it has the same form for the lowest-order terms quadratic in p; for higher m o m e n t a quantitative differences arise in this model from the higher-order 397

Volume 206, number 3

PHYSICS LETTERS B

terms but this does not affect the following arguments. The origin o f the repulsion in the nucleus-nucleus potential in the nonrelativistic models is the disappearance o f the current in the overlap zone o f the two nuclei as discussed above. In the relativistic case repulsion is affected by the d i s a p p e a r a n c e o f the space c o m p o n e n t s o f the vector field 09 o f the two nuclei which p r o v i d e in general attraction; the sources o f those c o m p o n e n t s are again the currents. In the overlap zone, however, these c o m p o n e n t s from the two colliding nuclei just cancel each other; this cancellation leads to repulsion. In all cases the m o m e n t u m d e p e n d e n c e o f the nucleus-nucleus potential is completely d e t e r m i n e d by the effective mass that is adjusted to the empirical value in all three models. Note that the energy dependence is quite different in both cases since in one case p2~E and in the other p2=E2+m% most B U U codes, however, use relativistic kinematics anyway so that this difference plays no role. There is, therefore, as far as the m o m e n t u m dependence o f heavy-ion interactions is concerned besides the " n o r m a l " relativistic effects no new physics contained in using relativistic m e a n field models for the description o f heavy-ion interactions. The " a d d i t i o n a l repulsion" o b t a i n e d in ref. [ 9 ] in the sudden limit from the dif-

398

26 May 1988

ferent relativistic t r a n s f o r m a t i o n properties o f the scalar a n d vector c o m p o n e n t s o f the self-energy finds at low energies its correspondence in the pz term o f the Skyrme model. A m o n g the " n o r m a l " effects in a d d i t i o n the retardation o f the mesons fields and excitations o f the Dirac sea which have been neglected in the considerations above m a y have nonnegligible effects but these can be reliably d e t e r m i n e d only in a t i m e - d e p e n d e n t description o f the process. The authors gratefully acknowledge m a n y stimulating discussions with G.E. Brown.

References [ 1] T.A. Ainsworth, E. Baron, G.E. Brown, J. Cooperstein and M. Prakash, Nucl. Phys. A 464 (1987) 740. [2] G.H. Goeritz and U. Mosel, Z. Phys. A 277 (1976) 243. [3] P.G. Zint, Z. Phys. A 281 (1977) 373. [ 4 ] F. Stancu and D.M. Brink, Nucl. Phys. A 243 ( 1975 ) 175. [ 5 ] C. Gale, G. Bertsch and S. DasGupta, Phys. Rev. C 35 ( 1987 ) 1666, [6] D. Walecka, Ann. Phys. (NY) 83 (1974) 491. [7] M. Jaminon, C. Mahaux and P. Rochus, Nucl. Phys. A 365 (1981) 371. [8] G.E. Brown, The equation of state of dense matter, Stony Brook preprint ( 1987 ). [9] B. ter Haar and R. Malfliet, Phys. Lett. B 196 (1987) 414.