Q-VALUE DEPENDENCE IN THE FOUR-NUCLEON TRANSFER REACTION 54Fe(16O, 12C)58Ni

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Q-VALUE DEPENDENCE IN THE FOUR-NUCLEON TRANSFER REACTION 54Fe(16O, 12C)58Ni

W. von Oertzen

To cite this version:

W. von Oertzen. Q-VALUE DEPENDENCE IN THE FOUR-NUCLEON TRANSFER REAC- TION 54Fe(16O, 12C)58Ni. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-233-C6-235.

�10.1051/jphyscol:1971652�. �jpa-00214870�

JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 11-12, Tome 32, Novembre-Dkcembre 1971, page C6-233

Q-VALUE DEPENDENCE

IN THE FOUR-NUCLEON TRAN SFER REACTION 4Fe(1 0 , 'C)' Ni

W. von OERTZEN

Max-Planck-Institut fiir Kernphysik, Heidelberg, W-Germany

Rhumb. -

La dkpendance en

de la section efficace diffkrentielle de la r6action

a kt6 evaluke dans le cadre de l'approximation de Born en ondes deformkes. La valeur optimale de

pour la reaction

sur le 54Fe

52 MeV est ainsi trouv6e proche de

MeV. La section efficace dCcroit d'au moins un ordre de grandeur quand le Q dkvie de

5 MeV de la valeur optimale.

Abstract. - DWBA

calculations have been performed for the dependence of the differential cross sections of

reactions on the Q-value. It is found that the optimum Q-value for

the

reaction on 54Fe at 52 MeV incident energy is approximately

MeV, and that

the cross section drops by more than one order of magnitude if the real Q-value deviates from the optimum value by

5 MeV.

DWBA calculations of one nucleon transfer reac- tions above the Coulomb barrier [ l ] based on the approximations of Buttle and Goldfarb [2] have shown to yield reliable relative spectroscopic factors and shapes of angular distributions. It is the aim of the present note to discuss the application of these methods to the four-nucleon transfer reactions on nuclei with Z

20

30, induced by 160, which have been recently studied a t Saclay [3].

The calculations were performed for the 54Fe(160, 12C)58Ni reaction for which angular distributions of the elastic scattering as well as the transfer channels exist [3 b]. The procedure was equivalent to that applied in one-nucleon transfer reactions. The bound state is approximated by a n equivalent Hankel function, and the final transition amplitude is calculated by an equivalent zero range integral neglecting the recoil terms (Code DWUCK [4]). In the integral the Hankel function multiplied by the normalization constant of the bound state is again replaced by a bound state wave function calculated in a Woods-Saxon well.

In the present case a 7 S (center of mass motion of four nucleons in the 2P - 1F shell) wave function was used with r ,

1.35 fm, a

0.65 fm as parameters for the potential well (its depth ranges from 110 to 130 MeV). The post representation was chosen, because it can be shown [2b], that the influence of recoil terms can be minimized in transfer reactions from a light projectile to a heavier target nucleus. This is achieved by choosing the representation in which the bound state in the integral is taken to be in the heavier nucleus, and the interaction which is responsible for

the transition to the light donor nucleus. The potential parameters given in [3b] for the elastic scattering imply complete absorption of the small partial waves and rather small contributions from internuclear distances smaller R

1.5(16'l3 + 54'13) fm.

The scattering parameters were chosen to be equal in the initial and final channels. Calculations with an energy dependent imaginary potential in the final channel gave principally the same results except for the very extreme Q-value for which the cross sections are very small. The extreme Q-values being cases with bad matching conditions have the larger intrinsic errors due to the optical model parameters.

In figure 1 the dependence of the cross section on the reaction Q-value is shown for a given bound state (7 S,

8.0 MeV). It illustrates the effect of the overlap of the initial and final distorted waves.

At Q-values smaller

10 MeV the overlap becomes very poor and the cross section drops rather fast.

The inclusion of the recoil terms in the post representation will diminish the cross section a t positive Q-values and increase those a t nega- tive Q-values by factors 1.5 to appr.

(see for example ref. 2b). Calculations using other bound state parameters or principal quantum numbers did not change the curves except for the absolute cross sec- tions. Similar calculations involving angular momen- tum transfer 1

2 give exactly the same result. For angular momentum transfer different from zero the discussion of the recoil effects, however, has to be done with more care.

Figure 2 shows the result of the calculations

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971652

Ce234 W. VON OERTZEN

E, = BMaV (constmi):

7 s bound state w . M furtmnx

O.I -10 -5 0 +5 +I0

Q value [MeV]

FIG. 1. - Q-value dependence of the differential cross section in the reaction 54Fe(l60, 12C)58Ni for constant bound state in the final channel, at three angles : 550, 650 and 75O, and 52 MeV

incident energy.

Q - value [MGV]

FIG. 2. - Q-value dependence of the differential cross section in the reaction 54Fe(16O, 12C)ssNi at four angles (CM) : 40°,

55O, 650 and 750 at 52 MeV incident energy.

if for each Q-value the correct binding energy is inserted for the bound state (EB

8.0 MeV for

MeV)

Due to the restriction of the reaction to the nuclear surface only the tail of the wave func- tion enters into the strength of the reaction. A very

strong additional Q-value dependence is therefore obtained (often expressed in terms of a dependence of the normalization constant of the bound state wave function relative to its asymptotic form

on EB), which gives approximately a factor of 2.5 for change of E, by 2. MeV.

The result for the total dependence of the diffe- rential cross section (four angles are given in Fig. 2) is rather striking. The inclusion of recoil terms is expected to increase the drop-off a t positive Q-values and shift the maximum of the curve to more negative Q-values. The reaction has a n optimum Q-value of

-

6. MeV (or more). The cross section drops for Q

- 1.6 MeV to Q

+ 2.5 MeV (these are the Q-values for

Ca and 44Ca, ground state transitions) by factor 10. From these curves it also becomes clear that strong lines in (160, I2C) reactions can be observed for Q-values between

10. and

MeV, as well as in (160, 14C) reactions, the dynamics of which are very similar the four-nucleon transfer reaction (see for example the comparison of (160, I2C) and (160, 14C) reactions in [2b]). A comparison with the spectra shown in [2] shows a close similarity to the predicted strength curve. However, it is clear that for transitions involving larger angular momentum transfer the shape of the curves shown in figure 2 can be different. Also strong lines in spectra can be due to the promotion of particles to higher shells (increased number of nodes in the radial wave function of the bound state) because they yield more strength a t the nuclear surface.

The strong Q-value dependence of the presently discussed reaction is present almost a t all angles (it does not depend therefore on the exact shape of the angular distributions, they are smooth with a maximum around 500 to 600 CM) and is a result of the combined effects of the large Coulomb parameter

15) and the localization of the reaction on the nuclear surface.

A decrease of the incident energy will shift the maximum in the curves of figures 1 and 2 to more positive Q-values because the absolute energy of the outgoing particle will be nearer to the Coulomb barrier in the final channel. From the same argument one can extrapolate that the maximum in the curves of figures 1 and 2 will shift to more n:gative Q-values with increasing incident energy (see for example contributions to this Conference).

It is interesting to note that for the inverse reaction, the four-nucleon pick-up reaction ("C, 160) (or pos- sible other choices of the incident particle) will have a n optimum Q-value of approximation + 6.0 MeV in the presently discussed nuclei. In as much as pick-up ieaction; will have in the most favoured cases, (2-va1u;s of appr. 0. MeV and more negative values, one can conclude that alpha-particle pick-up reactions on heavier targets must have extremely small cross sections. Indeed in experiments performed by Volkov et al. [5] it was found that four-nucleon pick-up reactions induced by 12C on Gold and ~ h o r i u k have

(*) EB = 2 MeV for unbound states and Q <

-

cross sections smaller than a few pb.

&-VALUE DEPENDENCE IN THE FOUR-NUCLEON TRANSFERT

References

[I] VON OERTZEN (W.) et al., Proc. of Conf. Reactions [3] a) FAIVRE (J. C.) et al., Phys. Rev. Letter-s, 1970, 24, induced by Heavy Ions, Heidelberg (North- 1 188.

Holland, 1970) and Lectures at ICTP, Trieste b) LEMAIRE (M. C.), Lectures a t ICTP, Trieste 1971, and

1971. contributions t o this Conference.

121 a) GOLDFARB (L. J. B.), Ioc. cit., ref. 1, and references [4] KUNZ (P. D.), DWBA code Dwuck. The author is there in ; b) BUTTLE (P. J. A.) and GOLD- indebted t o Mr. Wenneis for his help with the

FARB (L.J.B.),Preprint, University of Manchester, code.

1971. [51 VOLKOV (V. V.) el al., Nucl. Phys., 1969, A 126, 1.