STATISTICAL ASPECTS OF HEAVY-ION REACTIONS

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STATISTICAL ASPECTS OF HEAVY-ION REACTIONS

J. Bondorf, F. Dickmann, D. Gross, P. Siemens

To cite this version:

J. Bondorf, F. Dickmann, D. Gross, P. Siemens. STATISTICAL ASPECTS OF HEAVY-ION REAC- TIONS. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-145-C6-149. �10.1051/jphyscol:1971623�.

�jpa-00214841�

JOURNAL DE PHYSIQUE Colloque C6, suppl6ment a u no 11-12, Tome 32, Novembre-De'cembre 1971, page C6-145

STAT1 STICAL ASPECTS OF HEAVY-ION REACTIONS

J. P. BONDORF, F. D I C K M A N N (*), D. H. E. GROSS (**) a n d P. J. SIEMENS The Niels Bohr Institute, University of Copenhagen, Denmark

Rbume. - Les reactions du transfert de nucleons entre ions lourds, au-dessus de la barriere coulombienne, presentent des caracteristiques que I'on ne peut expliquer ni en termes de processus en une Ctape, ni en supposant la formation d'un noyau compose. C'est pourquoi on introduit le concept d'equilibre statistique partiel. Les resultats experimentaux connus sont compatibles avec la thkorie, dans le cas de transferts de neutrons. Les taux des reactions de stripping de protons sont plus faibles que prevu par la theorie. Cela peut se comprendre comme un effet de la barriere coulombienne entre les ions.

Abstract. - Nucleon transfer reactions between heavy ions above the Coulomb barrier exhibit features that can neither be understood in terms of one step processes nor by assuming the for- mation of a compound nucleus. Therefore the intermediate concept of a partial statistical equili- brium is introduced. The available experimental data are consistent with the theory for neutron transfer processes. Proton stripping reactions are enhanced less strongly than predicted theoreti- cally. This can be understood as an effect of the Coulomb barrier between the ions.

1. Introduction. - In heavy-ion reactions above the Coulomb barrier, e. g. 1 6 0 on 232Th a t an energy of about 120 MeV, the following features are observ- ed [I].

(a) U p t o nine particles are transferred.

(6) The observed Q values differ appreciably from the ground state Q values. The excitation energy Ex ranges from 5 to 60 MeV and depends on how many particles are transferred.

(c) The angular distributions show a maximum a t the grazing angle (400 to 600). At the distance of closest approach of the Coulomb trajectories, the nuclear surfaces are about two fermis apart. Thus only the outer parts of the wave functions overlap.

The collision time is approximately 5 x s (during this time a nucleon can traverse the nucleus three times).

(6) The summed cross section a,,, for transfer reactions a t the grazing angle exceeds 200 mb. This corresponds t o an annulus with a radius of 13 fm and a thickness of more than 0.25 fm.

(e) The logarithm of the cross section depends almost linearly on the ground state Q value Q,, (Fig. I), despite the fact that there usually arc practically n o ground-state transitions (cf. (b) above). It is important t o note that the linear systematics apply even when the nucleus produced is weakly bound- for example "e (neutron threshhold En = 1.7 MeV),

I I Be (En = 0.5 MeV), 14B (En < 1 MeV), and 15B (En < 1 MeV).

L I

- 5 - < a - 5 - 1 0 5 - 3 0 - 3 s - 1 0

a,, i ? c V )

FIG. 1.

Differential crosssections(da/dQ)~o~~ for production of Bc, B, C and N isotopes in the 232Th +

reaction as func-

tions of Q,,, taken from ref. [I].

Point (c) clearly indicates that we are dealing with a direct and fast reaction. O n the other hand, the stan- dard DWBA analysis of such processes normally explains a simple dependence of the cross section on the actual Q value Q = Q,, - Ex, which is not observed here. Further we have found in ref. [3] that the actual Q values of these reactions are always such that there is a maximum matching of the in- and outgoing dis- torted waves, so that from the DWBA point of view the kinematical part of the cross section should be the same for all different final channels. Moreover, points (6) and (e) indicate a statistical reaction, o r at

(*) On leave from Kernforschungszcntrurn. Karlsruhe. least a higher order process. This is also suggested by

( * * ) On lcave from Hahn-Meitner Institut. Berlin. the fact that the weakly bound nuclei mentioned in ( e )

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971623

C6-146 J. P. BONDORF, F. DICKMANN, D. H. E. GROSS AND P. J. SIEMENS

are produced in their ground states, so that the high excitation energy is concentrated almost entirely in the heavy reaction product. This would be expected if there is a thermal equilibrium, because the heavy nucleus, having a large level density, cools the light one.

We reconcile the direct and statistical aspects of the reaction mechanism by introducing the concept of a partial statistical equilibrium in which only certain degrees of freedom participate. A possible model for such a mechanism is sketched in figure 2, where statistical equilibrium is assumed for nucleons moving along the line of centers, while (as argued in (c)

above) the system does not lose the memory of its motion in the direction of the relative velocity, which is orthogonal to the line of centers at the point of closest approach.

FIG. 2.

Heavy ions at the distance of closest approach. The relative size of the ions corresponds to thc case of 232Th and

' 6 0 . The lines drawn illustrate the diffuscness of the surfaces.

The relative motion is indicated by the curve r(f).

We want t o find out which features of the experi- mental cross sections can be understood using this simple concept.

2. Partial statistical equilibrium. - In a statistical reaction thc cross section is given by the formula [4]

Here a, is an overall normalization constant, which does not depend on the properties of the final states.

The function p,(E) is the level density of the final states at the energy E. We are here discussing a sta- tistical equilibrium with respect to energy and par- ticle exchange. This equilibrium is established when the two ions are in contact (Fig. 2). Since the relative velocity of the ions in this configuration is small compared to the velocity of nucleons at the fermi surface, we consider for the moment a static situation.

Nucleons moving along the z-axis (line of centers) can go most easily from one nucleus to the other.

Thus nucleons in single-particle states with magnetic quantum number I, = 0 reach statistical equilibrium first. The transfer of nucleons with larger I, # 0 is hindered by the cylindrical centrifugal barrier in the single-particle wave equation, which holds them

away from the z axis where transfer most easily takes place. In order to illustrate the effect of the restriction to states with 1, = 0 on the single-particle level density, we note that of the single particle states between magic numbers 82 and 184, only about one out of eight has magnetic quantum number 1, = 0.

For lighter nucIei this fraction is of course larger. In the following discussion we will use the symbol /I

to denote the ratio of all single-particle degrees of freedom, to those that participate in the statistical equilibrium, where we have just estimated /I w 8.

The total (many-body) level density of a nucleus with A nucleons can be represented by the Bethe formula (2)

J7( exp 2 JaU

d U ) ⁼ 12 - ~ V V

provided that the excitation energy U is not too small.

It should be noted that the quantity U to be inserted in eq. 2 is the true excitation energy corrected for pairing effects (2). The level density parameter a is proportional to the single-particle level density.

Because the statistical equilibrium does not involve all the degrees of freedom, the single-particle level density, and thus also the parameter a, must be only about l//j of the empirical values given in ref. [ 2 ] . For low excitation energies, a so-called

constant- temperature

formula is used,

p(U) ⁼

exp - To

The parameter To is adjusted empirically (2).

In order to obtain the level density of the system of two ions, we should fold the two individual level densities. It turns out, however, that the temperature T = JUG is approximately 2 MeV. This value is so low that we must use the formula (3) for the level density of the light projectile, e. g. ^{1 6 0} (it may not be at all valid to apply analytic level density formulae for very light nuclei). An extrapolation of the parameters To given in ref. [2] to light nuclei shows that To should be considerably larger than 2 MeV. This means that the light nucleus is not excited at all, which is also found in high-resolution experiments.

The level density of the total system may therefore be approximated by the level density in eq. 2 for the heavier ion, e. g. 13'Th.

The excitation energy E* is determined by the actual and ground-state Q values,

EQ ⁼ Q,, - Q . (4)

The Q value can be expressed in terms of quantities referring to classical orbits in the touching region.

We split Q into three components, following ref. [3]

STATISTICAL ASPECTS OF HEAVY-ION REACTIONS C6- 147

Q, is the change of Coulomb interaction energy due to charge transfer, Q, is the classical change in energy of tangential motion due to mass transfer, and Q,, represents other excitation processes, it is the same for all exit channels. Only the total Q value can be measu- red experimentally, but we have been able to deter- mine its three components separately by applying the method of ref. [3].

The picture of the reaction mechanism outlined in the introduction suggests that the relative motion in the tangential direction is not coupled strongly to the degrees of freedom which are taken to be in statistical equilibrium. Therefore we assume that the part - Q, of the excitation energy due to the tangential motion does not contribute to the statistically distributed energy U of eq. 2. The excitation energy to be inserted into the level density formulae is therefore reduced to

U = E* - P(Z) - P(N) - (- Q,)

= Q,, - Q, - Q,, - P<z> - P(N) (6) where P(Z) and P(N) are pairing corrections discussed in ref. [2]. Lacking detailed understanding of Q,,, we do not exclude it from U (in the present examples Q,, is about 8 MeV). Of the contributions to U in eq. 6, only Q,, depends strongly on the number of neutrons transferred ; thus eq. 6 may be taken as an explanation for the experimental observation that the relative yields of different isotopes of the same ele- ment is a function of Q,,, cf. point (e) in the introduc- tion.

We have compared the cross sections calculated from formula (1) above with all available (1) experi- mental data on proton stripping reactions above the Coulomb barrier, in which sufficiently many final channels have been measured to permit the analysis of ref. [3]. The comparison of theory and experiment is presented in figures 3 to 5. An overall normalization

d6 + ~ 6 ~ '

p = a ,

0 theory wlth g ~ v m O 2 lheoty w~lhoul given 0

-

t . ' ' ^j

FIG. 3, 4, 5. - Comparison of the experimental cross sections with theoretical predictions from eq. (10). The distance of closest approach was chosen to be R = 1.4 (Ail3 + Ail3). The reduc- tion factor /3 of the level density parameter a is given in the figure. As also indicated in the figure the experimental reaction Q-value is not known in some cases. Therefore we have inter-

polated it from Q-value systematics (ref. 131).

has been used for each line of isotopes. The norma- lization constants are discussed in section 3 below.

Even though the experimental cross sections vary

by four orders of magnitude, the deviation between

the theoretical and experimental values is only a factor

1.6 on the average. The largest discrepancies are

observed for reactions which have a very small

cross section. In these cases the calculated excitation

energy U was so small that we had to use the level

C6-148 J. P. BONDORF, F. DICKMANN, D. H. E. GROSS A N D P. J. SIEMENS density formula (3). The choice of values for parameters

entering this formula is very uncertain. We have chosen the parameters so that the constant tempera- ture formula smoothly joins the Bethe formula at an excitation U = 2 MeV. The values for the level density parameter a were taken from ref. 121, neglecting shell corrections and with the reduction factors j? given in the captions to figures 3 to 5. The agreement between theory and experiment is practically unafiected if the reduction factors /I are varied by ^_f 20 %.

Within the statistical model we can calculate the typical number of excitons

Consequently the populated states are not of a very complicated nature. The typical number of particle- hole excitations is similar to the typical number of particles transferred, and since particle transfer occurs with high probability, it does not seem unrea- soilable t o suppose that a statistical distribution of excitation is attainable under the conditions of the reaction.

3. Proton transfer reactions. - The change of the Coulomb interaction energy due to proton transfer between two ions is given by

Z , Z 2 e 2 ( Z , + AZ) ( Z , - AZ) e 2

Thus the system gains excitation energy for proton transfer from the light to the heavy nucleus i. e.

proton stripping reactions. The gain of excitation energy per transferred proton is

where R is the average distance between the charge centres. The term AQ, is the relative shift of the proton Fermi surfaces at closest approach (Fig. 6).

The statistical theory predicts that the cross section depends on the excitation energy U. One would thus expect that reactions leading to neighbouring elements should differ in their cross sections by a

FIG. 6. -The effect of the Coulomb interaction on the proton Fermi levels is shown. The upper part of the figure sketches the ions at infinite distance, the lower part at the closest approach.

enhancement of proton stripping reactions (Fig. 1) ; however, it is not as large as obtained theoretically.

It can be inferred from figure 7 that the theoretical cross sections must be multiplied with factors P = exp(- 1.7) and P ⁼ exp(- 2.3) per transferred proton. Thus the experimental cross sections can be described by a formula

fJ = no. PAZ.j?(u) , ₍₁₀₎

where no is an overall normalization constant.

FIG. 7. -The ratio of the lcvel density of the final channels over the experimental cross section is plotted versus the number AZ of transferred protons. The slope of the curves is thc negative logarithm of the reduction factor P explained in chapter 3.

We find in the case of the 197Au target that In (P) = - 2.3 and in the case of the 232Th target that In (P) - ^{- 1.7.}

factor exp(AQ,/T) if they have the same ground state Q-values. The experiments clearly show an In order to reconcile theory and experiment, we must either (I) change the term AQ, which enters the excitation energy or (2) modify the statistical model itself.

MODIFICATION I. - We have obtained the energy shift AQ, due to proton transfer (see eqs. 8 and 9) by using spherical charge distributions for the ions.

The distance R between the centres of charge at the point of closest approach was supposed to be

One could think of reducing the Coulomb interaction

between the ions without increasing the distance

between their surfaces too much by assuming prolate

deformations. In order to reduce the Coulomb interac-

tion sufficiently, the ratio of the major to the minor

axis of the ions should in this case be two. This is an

STATISTICAL ASPECTS O F HEAVY-ION R E A C T I O N S C6-149

unreasonably large deformation. Nuclear ground state deformations are known to be much snlaller and it is impossible to reach the required deformations dynamically during the interaction time of about

5 x s.

MODIFICATION 2. - We now turn to the possibility of modifying the statistical model for the proton transfer reactions. Apparently the Coulomb barrier between the two ions (Fig. 6) is large enough to pre- vent proton transfer processes fro111 coming into sta- tistical equilibrium. The probability that a proton penetrates the barrier if it hits it once is given by

in the simplest approximation where d is the thickness of the barrier, k is the wave number of the incoming proton that hits the barrier of the height U,, the parameter I is defined by 1. = J(u - E)/E, E being the energy of the particle. Reasonable values for these parameters in the reactions which we have analyzed are

d ⁼ 2 fm, U , = 70 MeV, E ⁼ 45-50 MeV.

Thus the penetrability turns out to be D ⁼ 3-6 "/,.

A proton hits the barrier about three times during the time of collision, i. e. the total penetrability P is of the order of 10-20 %. We have mentioned above that a factor of this size has to be inserted into eq. 10 in order to explain the experiments.

4. Conclusion. - We have shown that transfer cross sections observed in heavy ion reactions above the Coulomb barrier can be understood by assuming the system of colliding ions to be in statistical equili- brium with respect to a subset of the degrees of free- dom. The proton transfer cross sections can be explain- ed by inclusion of a penetration factor. In other words the proton transfer seems not to be in the statistical equilibrium, contrary to the neutron trans- fer.

Further microscopic investigations should be done in order to study the conditions for the partial equili- brium.

Acknowledgment. - We would like to acknow- ledge fruitful discussions with B. R. Mottelson and A. Winther.

References

[ l ] AKTUKH (A. G . ) , AVDEICHIKOV (V. V.), GRIDNEV (G. F.), VOLKOV (V. V.), GRIDNEV ( G . F.), ZORIN (G. N.) and MIKHEEV (V. L.), VOLKOV (V. V.) and WILC- CHELNOKOV (L. P.), NucI. Phys., 1969, A 126, 1.

ZYNSKI (J.), Nucl. Phys., 1971, A 1 6 8 , 321. [2] GILBERT (A.) and CAMERON (A. G. W.), Catradian ARTUKH (A. G.), AVDEICHIKOV (V. V.), ERO (J.), Journal of Physics, 1965, 43, 1446.

GRIDNEV(G. F.), MIKHEEV(V. L.),VOLKOV (V. V.) ^[3] SIEMENS (P. J.), BONDORF (J. P.), GROSS (D. H. E.) and WILCZYNSKI (J.), NUCI. Phys., 1971, A 160, and D~CKMANN (F.), Phys. Letters, 1971, 36B, 24.

51 1 and Phys. Letters, 1970, 33B, 407. [4] ERICSON (T.), Advances in Phys., 1960, 9 , 425.