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 REACTION 54Fe(16O, 12C)58Ni. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-233-C6-235.
<10.1051/jphyscol:1971652>. <jpa-00214870>
HAL Id: jpa-00214870
https://hal.archives-ouvertes.fr/jpa-00214870
Submitted on 1 Jan 1971
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
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 Q de la section efficace diffkrentielle de la r6action ( 1 6 0 , 1 2 C ) a kt6
evaluke dans le cadre de l'approximation de Born en ondes deformkes. La valeur optimale de Q
pour la reaction ( ' 6 0 , 1 2 C ) sur le 54Fe a 52 MeV est ainsi trouv6e proche de - 6 MeV. La section
efficace dCcroit d'au moins un ordre de grandeur quand le Q dkvie de i 5 MeV de la valeur
optimale.
Abstract. - DWBA calculations have been performed for the dependence of the differential
cross sections of ( 1 6 0 , 1 2 C ) reactions on the Q-value. It is found that the optimum Q-value for
the ( 1 6 0 , 1 2 C ) reaction on 54Fe at 52 MeV incident energy is approximately - 6 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 f 5 MeV.
DWBA calculations of one nucleon transfer reactions 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 at Saclay [3].
The calculations were performed for the 54Fe(160,
12C)58Nireaction 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, E, = 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
at positive Q-values and increase those at negative Q-values by factors 1.5 to appr. 3 (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 sections. Similar calculations involving angular momentum 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
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 differential cross section (four angles are given in Fig. 2)
is rather striking. The inclusion of recoil terms is
expected to increase the drop-off at positive Q-values
and shift the maximum of the curve to more negative
E, = BMaV (constmi):
Q-values. The reaction has a n optimum Q-value of
7 s bound state w . M furtmnx
- 6. MeV (or more). The cross section drops for
Q = - 1.6 MeV to Q = + 2.5 MeV (these are the
Q-values for 40 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 - 3.0 MeV, as well as
O.I
-10
-5
0
+5
+I0
in (160, 14C) reactions, the dynamics of which are very
Q value [MeV]
similar the four-nucleon transfer reaction (see for
FIG. 1. - Q-value dependence of the differential cross section example the comparison of (160, I2C) and (160, 14C)
in the reaction 54Fe(l60, 12C)58Ni for constant bound state in reactions in [2b]). A comparison with the spectra
the final channel, at three angles : 550, 650 and 75O, and 52 MeV
shown in [2] shows a close similarity to the predicted
incident energy.
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 at the nuclear surface.
The strong Q-value dependence of the presently
discussed reaction is present almost at 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 ( q = 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).
Q - value [MGV]
It is interesting to note that for the inverse reaction,
the
four-nucleon pick-up reaction ("C, 160) (or posFIG.2. - Q-value dependence of the differential cross section
sible other choices of the incident particle) will have an
in the reaction 54Fe(16O, 12C)ssNi at four angles (CM) : 40°,
optimum Q-value of approximation + 6.0 MeV in the
55O, 650 and 750 at 52 MeV incident energy.
presently discussed nuclei. In as much as pick-up
ieaction; will have in the most favoured cases, (2-va1u;s
if for each Q-value the correct binding energy is of appr. 0. MeV and more negative values, one can
inserted for the bound state (EB = 8.0 MeV for conclude that alpha-particle pick-up reactions on
Q = 0.MeV) (*). Due to the restriction of the reaction heavier targets must have extremely small cross
to the nuclear surface only the tail of the wave func- sections. Indeed in experiments performed by Volkov
tion enters into the strength of the reaction. A very et al. [5] it was found that four-nucleon pick-up
reactions induced by 12C on Gold and ~ h o r i u khave
cross sections smaller than a few pb.
(*) EB = 2 MeV for unbound states and Q < - 6 MeV.
&-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 (North1 188.
Holland, 1970) and Lectures at ICTP, Trieste
b) LEMAIRE
(M. C.), Lectures at ICTP, Trieste 1971, and
1971.
contributions to 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 GOLDindebted to 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.