Neutron and proton diffusion in heavy-ion collisions

Volume 104B, number 1 PHYSICS LETTERS 13 August 1981

NEUTRON AND PROTON DIFFUSION IN HEAVY-ION COLLISIONS

A.C. MERCHANT and W. NORENBERG Gesellsehaft fiir Schwerionenforschung [GS1), D-61 O0 Darmstadt, Fed. Rep. Germany

Received 15 April 1981

A statistical theory for the exchange of protons and neutrons in heavy-ion collisions is formulated and applied to recent data. No systematic deviations are observed which could be evidence for (additional) quantal effects in the N/Z equilibration.

Many aspects of dissipative heavy-ion collisions, like mass transfer, angular momentum dissipation and total kinetic energy loss, have been successfully under- stood in terms of statistical processes and well described by transport equations [1,2]. However, the question of whether the equilibration of the mass-to-charge ratio or neutron-to-proton (N/Z) ratio in such reactions can be explained in the same way is not so unequivocal- ly answered. In the study of this N/Z equilibration the conditional charge widths where either the mass or the neutron number is kept constant, have been measured. However, whereas several authors [3-5] have reported experimental charge widths consistent with a dependence on temperature and potential as expect- ed in a statistical model, other authors [6,7] have quoted results which saturate after a rapid rise with increasing kinetic energy loss and hence, are independent of temperature. Although this saturation has been regarded as evidence for a collective mode of charge equilibration (isovector giant vibration) [6,8], it may equally well be interpreted in terms of a model treating the exchange of individual nucleons in a statistical man- ner [9]. Within the semiphenomenological diffusion model [ 10,11 ] generalized to the independent transfer of protons and neutrons separately, we have systematic- ally analysed recent experimental data [5,7,12,13 ] on mass and charge transfer. The results strengthen the case for a statistical explanation of the N/Z equilibration. In this letter we report the main points of the generalized diffusion model and illustrate the results of the analyses with a few typical examples.

We describe the exchange of neutrons and protons

between the two interacting nuclei by the Fokker- Planck equation

Of_O(vNf ) O(vzf )+DN N O2f +Dzz 02f 0t 0N 1 oe 1 ~ ~ (1)

for the distribution function f ( N t , Z 1 ; t) giving the probability of finding N 1 neutrons and Z 1 protons in a specified fragment 1 at time t. The quantities VN, u Z and DNN , DZZ denote the drift and diffusion coefficients for neutrons and protons, respectively. The diffusion coefficients are obtained by generalizing the formulation of Ayik et al. [10] for the mass diffusion coefficient to treat the case where a distinction is drawn between neutrons and protons. For the touching con- figuration the proton transfer is considerably hindered by the repulsive Coulomb interaction. This Coulomb hindrance is the major reason for the smallness of the ratio Dzz/DNN ranging from 1/3 for light nuclei to 1/5 for heavier systems. As compared to this the effects due to the neutron skin are negligible. The mixed diffusion coefficient DNZ vanishes on account of the statistical independence of individual neutron and proton transfers. The drift coefficients are approximately related by the Einstein relations

(2) O N = --(DNN/T ) OU/ON 1 , o Z = - ( D z z / T ) g3U/OZ l

to the diffusion coefficients, the temperature T and the driving potential U(N1, Z1) which is obtained as the sum of the liquid-drop energies of the separated spherical nuclei and the interaction energy (Coulomb plus proximity) at the point of touching. Details of the theory will be described elsewhere [14].

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We solve the Fokker-Planck equation (1) analytical- ly by approximating the driving potential U(N1, Z1) together with the rotational energy by a two-dimen- sional oscillator potential in N1, Z 1 . Along the valley the strength is rather weak (b ~ 0.1 MeV/amu2), whereas it is strong (a ~ 1 MeV/amu 2) perpendicular to the valley. In this case the drift coefficients v N and u Z are represented by algebraic expressions which are linear in N 1 and Z1, and the distribution function therefore has a gaussian form which is completely defined by its first and second moments, i.e. the mean values (NI>, (Z1) , the variances o2N, O2z and the covariance O2Z . These quantities are determined from a set of five coupled first-order differential equations which is derived from the Fokker-Planck equation. The transport coefficients ON, u Z ,DNN andDzz vary with time, the time dependence determined by the (clas- sical) trajectory. We use a phenomenological model [1,2] where this time dependence is replaced by an equi- valent l dependence. This model also relates the nuclear interaction time tint(/) to the total kinetic energy loss (TKEL). We thus obtain the mean values (NI>, <Z 1>

2 O2Z as well as the covariance and the variances aNN ,

a2z as functions of TKE L. We are now in a position to compare quantities like the charge width a2Z, the mass width O2A = O2N + 02Z + 202Z, the neutron-proton correlation coefficient P = O2Z/(aNNoZZ), the charge width at constant mass asymmetry O2z IA1 = (O2NO2 Z -- o47)/02A, and the mass width at constant charge asymmetry O~A IZ1 = ONN -- (O~Z/o~Z) as functions of total kinetic energy loss (TKEL) with the available experimental data for any heavy-ion system.

We have done this for all available data, and now dis- cuss a representative number of examples.

In the upper part of fig. 1 we examine the evolution of the fragment distributions in the N - Z plane with increasing kinetic energy loss for the system 144 Sm + 144Sm (1000 MeV). This is done by setting the ex- ponent of the distribution function equal to some ar- bitrary constant (chosen to b e - 1 / 2 for convenience), which corresponds to plotting contours in the N - Z plane along which f(N1, Z1, t) has fallen to exp( -1 /2) of its value at the mean ((N1), (Z1)). For the sym- metric system illustrated, the mean is of course sta- tionary at (ZI> = 62, (N1) = 82. Elliptic distributions are obtained with major axes which are almost parallel to the neutron axis for small kinetic energy loss, rotate into the direction of the valley defined by the symmetry

~ 64

6 2

6O

N z

o_ j

z~1.0 LL.

8 g o.s

144Sm +144Sm (1000 NIeV)

M e V

78 810 812 81~ 816 " NEUTRON NUMBER, N

l

I0 i i i ~

5 100 150 200 TKEL (MeV)

Fig. 1. The variations of the elliptic neut ron-pro ton distribution in the N - Z plane and the growth of the correlation coefficient p with increasing kinetic energy loss for 144 Sm + 144Sm (1000 MeV).

energy as the loss rises to about 100 MeV, and grow further along that valley direction for greater losses still. The angle, a, made by the major axis of the ellipse with the N-axis is related to the widths by

tan 2a= 2 2 __ 02Z) 20NZ/(ONN , (3)

and by considering the small and large time limits of these widths it can be demonstrated that a -+ 0 ° (parallel to the neutron axis) at small times and kinetic energy losses and increases to 0 (the angle made by the confining valley with the N-axis) at large times and kinetic energy losses. Also shown in fig. 1 is the predicted neutron-proton correlation function

_ 2 p - oNZ/ONNOZZ compared with the experimental data of Hildenbrand et al. [ 12]. The agreement between the two is highly satisfactory, with p rising from 0 initially, towards 1, and reaching 0.9 at about 120 MeV TKE L. The rotation of the elliptic distributions into the confining valley and the growth of p from 0 to 1 go hand in hand and are manifestations of the same phenomenon, namely the hindrance of proton exchange in the early stages of the reaction and the

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confinement of the valley. In our model the neutron diffusion coefficient DNN is greater than the proton coefficient DZZ (essentially because of the Coulomb repulsion inhibiting proton exchange) so that at the beginning of the reaction neutrons are preferentially exchanged, (O2N is then by far the largest width) thus the ellipses have their major axes nearly parallel to the neutron axis and p is very small. As time goes on and more energy is dissipated the protons are able to dif- fuse despite this repulsion and the balance is redressed. For kinetic energy losses greater than around 120 MeV the further changes in direction of the distribution and in the correlation coefficient are very small indeed.

In fig. 2 we compare the predictions of our model with experimental measurements of the mass width at constant charge asymmetry for the system 56Fe (464 MeV) + 165Ho [7] and 129Xe (775 MeV) + 56Fe [13]. The upper curve is the prediction for 56 Fe + 165 Ho, and the experimental points are averaged over A 1 and have error bars of typically +0.2. The lower curve is our prediction for 129Xe + 56Fe, and the experimental crosses are for the isotope Z 1 = 55 and have error bars of +20%. Both sets of data have been corrected for light particle emission. Therefore they may be regarded as reconstructions of the primary distributions and may be fairly compared with our predictions. Similar comparisons between predicted and measured values of a 2 - Iz have been made for the systems 56Fe(464 Me~) +1238U, 56Fe (464 MeV) + 56Fe [7] and

136Xe (800 MeV) + 56Fe [13]. In all cases the agreement obtained is excellent. We note that for large TKE L the values of O2A IZ1 in our model is determined by the temperature and the driving potential, and the long time limit of our expression for O2A IZ 1 is

lira O2AIZt = T/2 _ T t~oo a sin20 +b cos20 2C" (4)

We now turn our attention to the charge width at constant mass asymmetry, a2z IAI" At high energies our model predicts that this quantity too should be determined by the temperature and driving potential, according to

lim ~r2zz IA, = T/2 (5) t ~ a(sin 0 + COS 0) 2 + b(cos 0 - sin 0) 2.

We are thus unable to obtain the saturation of this quantity observed by Breuer et al. [7]. However, it has been shown by Schr6der et al. [9] that specific features of the dynamics of relative motion can pro- duce this saturation in a purely statistical model. Never theless, we see in fig. 3 that our model predictions of a2z IA1 for the system 132Xe(900 MeV) + 197Au are in good agreement with the measurements of Kratz et al. [5] (which also have been corrected for light particle emission). Our model suggests that the dependence of o2 Z IA a on A 1 is very slight so only one curve is presented for comparison with three of the isobars measured. We have made similar calcula-

6#2A(Z,constont)

.o f /

~' ~°~'~ '~Xo×~/~ ' ' / ~ × ~ o 56Fe + 165 HO (/+6/+ MeV)

/ × 129Xe+ 56Fe (775 MeV)

' ' 5'0 ' i~o . . . .

TCEL (MeV)

Fig. 2. Theoretical curves for a2AAIZ1 for S6Fe(464 MeV) + 165 Ho and 129Xe (775 MeV) + 56 Fe as functions of kinetic energy loss compared with experiment.

N2 UZZ ( A, [ensfont] 08

07

06

05

0~

03

0 2 / I * A-197 A=200

01 ]L A:1%

5 100 150 TKEL(MeV)

Fig. 3. Theoretical prediction of a~Z IA 1 for la2Xe(900 MeV) + 197Au as a function of kinetic energy loss compared with the experimental results for the isobars A 1 = 194,197 and 200.

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tions of a2Z IA 1 which agree well with the data obtained for the systems 129Xe +56Fe and 136Xe

+ 56 Fe already mentioned. In addition to the comparisons between the predic-

2 tions of p, O2ZZ I z l , aZZ IA 1' as functions of kinetic energy loss with experiment shown or mentioned here, we have also calculated O~A and Cr2Z as functions of energy loss for the same systems. The experimental data for these latter quantities give points within two standard deviations of the theoretical predictions for all cases. We therefore conclude that with a purely statistical t reatment, without invoking quantum effects, we are able to satisfactorily explain a large body of experimental data on the mass and charge width of several different heavy-ion systems. Overall the deviations between theory and experiment are not systematic and the only blemish is the behaviour of O2ZZ [A1 as a function of energy loss, since it is not completely clear from experiments whether this quant i ty in gener- al saturates or continues to increase with increasing energy loss. In any case, both types of behaviour can be explained within statistical theories of varying degrees of sophistication, and we see no necessity to call on quantum effect to explain any of the observations.

RefeFences

[1] H.A. Weidenmtiller, Progr. Part. Nucl. Phys. 3 (1980) 49; W. N6renberg and H.A. Weidenmtiller, Introduction to the theory of heavy-ion collisions, Lecture Notes in Physics, Vol. 51, 2nd ed. (Springer, Heidelberg, 1980).

[2] A. Gobbi and W. N6renberg, in: Heavy-ion collisions, Vol. 2, ed. R. Bock (North-Holland, Amsterdam, 1980) p. 127.

[3] J. Barrette et al., Nucl Phys. A299 (1978) 147. [4] T.H. Chiang et al., Phys. Rev. C20 (1979) 1408. [5] J.V. Kratz et al., Nucl. Phys. A357 (1981) 437;

J. Poitou et al., Phys. Lett. 88B (1979) 69. [6] M. Berlanger et al., Z. Phys. A291 (1979) 133. [7] H. Breuer et al., Proc. Intern. Workshop on Gross prop-

erties of nuclei and nuclear excitations (Hirschegg, Austria, January 1981).

[8] H. Breuer et al., Phys. Rev. Lett 43 (1979) 191; L.G. Moretto, J. Sventek and G. Mantzouranis, Phys. Rev. Lett. 42 (1979) 563; E.S. Hernandez, W.D. Myers, J. Randrup and B, Remaud, Nucl. Phys. A361 (1981) 483; L.G. Moretto, C.R. Albiston and G. Mantzouranis, Phys. Rev. Lett. 44 (1980) 924; A.C. Mignerey et al., Phys. Rev. Lett. 45 (1980) 508.

[9] W.U. Schr6der, J.R. Huizenga and J. Randrup, Phys. Lett. 98B (1981) 355.

[10] S. Ayik, G. Wolschin and W. N6renberg, Z. Phys. A286 (1978) 271.

[11] C. Riedel, G. Wolschin and W. N6renberg, Z. Phys. A290 (1979) 47.

[ 12 ] K.D. Hildenbrand et al., private communication. [13] D. Schfill et al., private communication. [14] A.C. Merchant and W. N6renberg, to be published.

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Neutron and proton diffusion in heavy-ion collisions