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MULTI-NUCLEON TRANSFER REACTIONS
I. Rotter
To cite this version:
I. Rotter. MULTI-NUCLEON TRANSFER REACTIONS. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-113-C6-117. �10.1051/jphyscol:1971614�. �jpa-00214832�
MULTI-NUCLEON TRANSFER REACTIONS
I. ROTTER
Zentralinstitut fiir Kernforschung Rossendorf bei Dresden, GDR
Rbumk. - Les proprietes caracteristiques des reactions de transfert de plusieurs nucleons induites par ions lourds sont discutees. Si I'ion a une structure simple des informations spectro- scopiques sans ambiguite peuvent gtre obtenues sur la structure a plusieurs particules des Ctats nucleaires.
Abstract. - The characteristic properties of multi-nucleon transfer reactions induced by heavy ions are discussed. I f the ion has a simple structure unambiguous spectroscopic information on the many-particle structure of nuclear states can be obtained.
I . Introduction. - Multi-nucleon transfer reac- tions induced by heavy ions may be a useful tool in order to study nucleon correlations in nuclei. However, there is as yet no general theory of these reactions.
Calculations have been performed only for some special cases in which simplifying assumptions could be done.
For the description of heavy ion reactions with transfer of one nucleon some methods have been worked out in the last years. Since the reaction dynamics does not depend sensitively on the type of the interacting particles these methods are supposed to be applicable also to reactions with transfer of nucleon groups.
For this purpose, the nuclear structure part, containing all characteristic features of the interacting particles, must be written in a suitable manner.
In this talk, the characteristic properties of multi- nucleon transfer reactions induced by heavy ions will be discussed. The aim is to write the nuclear structure amplitude in such a manner that distorted wave calculations become possible in analogy to those for one-nucleon transfer reactions.
2. Many-particle properties of nuclei and their des- cription in the framework of the shell model. - Expe- rimental data on the many-particle structure of nuclear states can be obtained, with the refined experimental techniques today, from a study of reactions in which knock-out or transfer of a nucleon group takes place.
As an example, the reaction 12C(p, pa)'Be is mentioned which is studied some years ago [I] leading to qualita- tive but valuable results. In this reaction, the 'Be nucleus is formed in its ground state and in excited ones.
By that time, there are more experimental data which indicate that in a given low-lying nuclear state some different structures coexist with a certain pro-
bability each 121-[4]. As has been shown in several
papers [6]-1101 the probability for the different many- particle structures can be calculated in the frame- work of the shell-model.
As to the basic equation, the wave function @, of a nucleus consisting of A nucleons, which are moving in an oscillator potential, can be expanded into a set of wave functions Q),-, of a nucleus consist- ing of A - k nucleons [ l l ]
Here, the symbol $, stands for the configuration of the remaining k nucleons. The coefficients c i are generally [6]-[lo] expressed by the fractional parentage coefficients. Such a formulation is convenient because tables of fractional parentage coefficients exist for some nuclei A + A - 2, A - 3 and A - 4. But another formulation using the second quantization instead of the concept of fractional parentage coefi- cients is also possible 1131.
According to the basic assumption of the shell model the nucleus is described as a system of identical particles. Consequently, the wave functions @ A - k
and IC/, overlap each other completely what means
that they are extended over the same volume as a,.
Exchange of nucleons between the nucleon groups ..4 - k and k takes place.
The complete overlapping of nucleon groups inside a nucleus is, naturally, an idealization also for light nuclei. For higher excitation energies and for heavier nuclei, effects coming from a partial spatial separation of the nucleon groups must surely be taken into account. Doubtless, the complete overlapping is a disadvantage of the model. The description of many- particle properties in the framework of the shell model has, however, the advantage that one can use the mathematical technique elaborated in the frame of
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971614
C6-114 I. ROTTER the model. One is able to obtain quantitative results relatively easy.
3. Calculation of spectroscopic factors for the sepa- ration of nucleon groups from nuclei. - The spec- troscopic factors are defined as the probability to find a certain many-particle configuration formed by A - k and k nucleons in a state of the A-nucleon system. They are proportional to the overlap integrals
transfer reactions in order to get spectroscopic information. Recently, it has been shown experimen- tally 1161 that in the reaction 12C('4N, 6Li)20Ne even eight nucleons are transferred, with a great probability, in a one-step process. This reaction, therefore, is suitable to obtain information on eight- nucleon substructures in nuclei. It seems to be jus- tified to assume that also in other reactions nucleon groups are transferred as a whole in a one-step process.
Let us consider the multi-nucleon transfer reaction
where @, is the wave function of the final nucleus under the following assumptions on the transfer formed by k nucleons. The wave function of the rela- process :
tive motion of the nuclei A - k and k is designed by
(i) rt is a one-step process. - -
~ N L M .
(ii) It is a direct one.
The spectroscopic factors depend on the model
used for their calculation. In the shell model they (iii) No mutual excitation of the nuclei A and B have the form given in [6]-[lo] takes place before the transfer process.
A I , V + L I Z Then the transfer reaction can be written schema-
s,K/----) A - k { G . K ~ ) ' (3) tically as
with A A-k
Here, the quantity G is the overlap integral of the shell model wave function @iM of the initial nucleus, having A nucleons, with the shell model wave function
@ z M k of the final nucleus. having A - k nucleons, and the-shell model wave function $:M of the k remaining nucleons. This overlap integral is proportional to the fractional parentage coefficient. The quantity KO is the overlap integral of the wave function $fM of the k separated nucleons with the wave function
$rL(Rk) @,(ak) of the nucleus formed by these k nucleons where @&(ak) stands for the internal motion and $,,(Rk) for the motion of the centre of mass.
The overlap integral KO can be expressed by Moshinsky coefficients [14].
According to eq. (3) the spectroscopic factor consists, in the main, of two parts. One part is the overlap integral G containing the nuclear structure information and the other one is the overlap intcgral KO containing the probability for formation of a nucleus by the k separated nucleons. The part decisive for the value of the spectroscopic factor is, in general, the overlap integral G because the dependence of KO on the quantum numbers of the nuclear levels is weaker than that of G.
The two vertices in eq. (7) are
This involves a certain analogy of transfer reactions to decay processes which will be used in the following.
In decay processes, the nucleon group consisting of the k nucleons is observed. It must be, therefore, in its lowest state. In multi-nucleon transfer reactions the group consisting of the k nucleons is not observed.
It can be in its lowest state as well as in higher states.
Consequently, the cross section of the reaction is, in contrast to decay processes, determined by all the overlap integrals
which are not in contradiction with the quantum numbers of the initial and final nuclei.
In analogy to eq. (3), the structure amplitude of multi-nucleon transfer reactions can be written in the form [I71
4. Calculation of the nuclear structure part of A N t KL B + k N ' t %L'
multi-nucleon transfer reactions. - The question ( A - S ) (3-1K(n1, N L , L,) x whether nucleon groups are transferred in a one-step
process is important for the analysis of multi-nucleon x K ( n l , N' L, LL)
with
The K(nl, NL, L,) are the overlap integrals of the wave function of the k separated nucleons with the wave function of the group of the k nucleons, which is transferred from the nuclcus A to the nucleus B and has the internal quantum numbers nl and the relative quantum numbers NL characterizing the centre-of- mass motion. The K(n1, NL, L,) can be expressed by Moshinsky coefficients [17].
In order to get the cross section of the reaction one must multiply the structure amplitudes (9) by the corresponding dynamical amplitudes B : ; ~ and sum coherently over all intermediate states. The dynamical amplitudes containing all the radial expressions depend on the quantum numbers of the centre of mabs of the transferred group. They will be similar, there- fore, to the dynamical amplitudes of one-nucleon transfer reactions.
If the nucleon group is transferred as a whole one can assume that the group has quantum numbers S,, T, and f, (spin, isospin and Young scheme) as well as n and 1 which are not changing in the transfer process. This assumption has been discussed in [17].
With this assumption, the structure amplitude (9) is not separable into two parts one of which depends only on the nuclei A and A - k and the other one on the nuclei B and B + k because the two overlap integrals K(n1, NL, L,) and K(n1, N' L', L;) are connec- ted by the internal quantum numbers n, I of the group of the transferred nucleons. Consequently, two different reactions (A,, A , - k) and (A,, A, - k ) on the same target nucleus B lead, generally, to different excitation spectra of the final nucleus B + k.
Let us consider the differences between the structure amplitudes of one-nucleon transfer reactions and those of multi-nucleon transfer reactions. For one- nucleon transfer processes, it follows from eq. (10) that K(tz1, NL, L,) = K(n1, N' L', LA) = 1. The sums over nl, NL, N' L' do not appear, therefore, in one- nucleon transfer reactions. Moreover, the orbital angular momenta L, and LL have definite values in one-nucleon transfer processes if one considers for simplicity pure configurations (without configuration mixing) of the initial and final nuclei whereas one has to sum over L, and Li in multi-nucleon transfer reactions.
Because of these coherent sums, the theory of multi- nucleon transfer reactions is much more complicated than that of one-nucleon transfer reactions.
The dependence of the integrals K(n1, NL, L,) and K(nl, N' L' L;) on the quantum numbers of a nuclear state is much weaker than the dependence
of the integrals G on the same quantum numbers.
The value of the structure amplitude (9) is, therefore, determined primarily by the fractional parentage coefficients including all intermediate configurations.
This statement is analogous to that which is known when calculating the spectroscopic factors for the emission of nucleon groups.
5. Discussion. - In the last time, a lot of spec- troscopic factors for the emission of nucleon groups have been calculated in the frame of the shell model [4], [7]-[lo], [18]-[20]. Generally, the calculated spectro- scopic factors are in agreement with the experimental data.
In some cases, the shell-model results are different from the assumptions of cluster models. This gives the possibility to proof directly the statements of the shell model on many-particle structures in nuclei.
For example, a large probability for an observation of the configuration 12C,.,. + a in the ground state of 160 has been assumed in every cluster model.
The shell model leads to another result : The pro- bability for the configuration 12C,.,. + a is only 6 % whereas the probability for other configurations like 12C4.4,,, + a is much larger [8], [lo]. This result is not in contradiction with the experimental data [4] although a final quantitative proof could not be done as yet. An analogous situation holds for
1 2 C,.,. + 'Be + a.
Results similar to those for 160 have been obtained also for other a-particle nuclei [4], [7]-[lo], [19].
With increasing number of nucleons in the last unfilled shell the a-widths enlarge with regard to the excited states of the final nucleus as compared with those with regard to the ground state. The theoretical results are in a qualitative agreement with the experi- mental ones obtained in (a, 2 a) reactions [2], in (d, 6 ~ i ) reactions [2,4] and in (3He, 7Be) reactions [19].
Some few other examples of shell-model predictions for spectroscopic factors of nucleon groups will be mentioned [8]-[lo], [18], [20]. The spectroscopic factor for 'OB,.,. 4 6 ~ i , . s . + a is also small in compa- rison with those for 'OB,.,. + 6Li* + a. It is
Experimentally, it has been found a larger value of this ratio [4] although a final quantitative value could not be given as yet. In other cases, the spectro- scopic factors for the processes A,,,. -+ (A - 4),,,, + a
are larger than those for the processes A,,,, 4 (A - 4) + a and A -, (A - 4),,,, + a . This holds for the a-particle emission from 14N and 13C states.
The spectroscopic factors for A,.,. -, (A - 4) + a
can be measured in reactions with knock-out of an a-particle like (p, pa), (a, 2 a) or (d, 6Li) reactions but the spectroscopic factors for A 4 (A - 4),,,. + a follow from transfer reactions like (6Li, d) or ('Li, t).
Comparing the quantitative results obtained in the
C6-116 I. ROTTER
different reactions one must take into account that where, because of the structure of the nuclei A and transfer reactions with heavy ions have surface A - k, the four nucleons are transferred mainly in an character whereas this is not the case for knock-out intermediate state with the configuration of an reactions induced by high-energy particles. Differences a-particle.
of such a type have been found experimentally when As one can see from the considered examples, comparing the results of (P, pt) ~eactions on 6Li transfer reactions induced by ions with a simple with those of (6Li, 3He) reactions [15]. configuration in their ground state can be very Let us consider some examples [17], [21] of transfer suitable for spectroscopic studies. Nevertheless,
reactions complete calculations taking into account the structure
(i) Transfer of two nucleons from the 1 p-shell of properties of the nuclei in a Proper manner have not the nucleus A to the 1 p-shell of the nucleus B + 2. been done as yet. Recently, DWBA calculations have been performed for the 12C('80, 160)14C reac- The intermediate state of the two particles can
tion [121. But in this case, the assumption has been have both the symmetry [21 and the symmetry ['11 made that the two neutrons are exchanged between if the corresponding fractional parentage coefficients the cores A - and without any rearrangement of are different from zero. For the 6Li having there cores which are shells. Clearly, this is a n the wave I421 13S the parentage approximation which can be done in only a few cases.
coefficient [42] + [4] + [I I] is zero. The two particles
can be transferred in ( 6 ~ i , a) reactions only with the With some assumptions on the structure of only symmetry [2]. Consequently, the excitation spectrum two nuclei, for example of A and A - k, the analysis of of the final nucleus B + 2 in the (6Li, a) reaction is the transfer reaction also simplifies. The point is determined approximately by the reduced deuteron that in this case spectroscopic information on the widths of the states of the nucleus B + 2. In ("B, 9Be) nuclei B and B + k can be obtained. Examples for reactions, however, there is no reason why the two such nuclei are lithium ions which have, in a good particles could not be transferred also with the ap~roximation, a Pure configuration : 1 / / 6 ~ i = [42I
1 3
symmetry [I I]. In the experiment, it is impossible to S, and 1 / / 7 L i = [43] 22P3/2.
separate the parts with different intermediate confi- The four n~lcleons transferred in (6Li, d) and gurations. ~t is difficult, therefore, to determine the ( ' ~ i , t) reactions come from the 1 S-shell of the lithium reduced deuteron widths from a (1 IB, 9Be) reaction. ~ U C ~ ~ U S what leads to some simplifications in the transfer amplitude. Neglecting spin-orbit coupling, it (ii) Transfer of three nucleons from the 1 p - shell
of the nucleus A to the 1 p - shell of the nucleus follows from eq. (9) B + 3.
In (7Li, a) reactions the excitation spectrum of the final nucleus B + 3 is determined, to a good approxi- mation, by the reduced triton widths of the states of with
the final nucleus because of the purity of the 7Li B + k N ' + %L;
wave function. In ('IN, 12C) or ("N, "B) reactions. S: = 1 ~rL:;f,
(-B)
xhowever, the three particles can be transferred with a configurations different from the configuration of a
triton or 3He. The excitation spectra of the final K(n = 0 1 = 0, N' Li, Lk) (12) nucleus taken in (7Li, a) and (I5N, 12C) and (14N,
"B) reactions on the same target nucleus will, and
therefore, be different from each other also if the b = Bh'I-i
reaction mechanism is a pure three-particle transfer L; (13)
mechanism.
(iii) Transfer of four nucleons from the nucleus A to the nucleus B + 4. In (6Li, d) and (7Li, t) reactions the transferred particles have the configuration of an a-particle. The (6Li, d) and (7Li, t) reactions are expected, therefore, to be very suitable for a study of the a-particle structures in nuclei. As to other reactions like (lOB, 'Li) the excitation spectrum of the final nucleus is not given by the corresponding reduced a-widths. This has been confirmed in recent expe- riments [5]. In ('OB, 6Li) reactions on 12C, strong transitions have been observed which are forbidden for a-transfer (Of -t 2 - ) whereas only very weak transitions have been found in such reactions on 12C
The sum on the left of eq. (1 1) runs over L, a,, L; a;, nl, NL, N' L' where a,, a; stand for all quantum numbers characterizing the nuclear states without L, and L;, respectively. The factor bLk contains the dynamical amplitudes BNjL' and, generally, depends on L; = L'. The other factor, s;i2, is the amplitude of the spectroscopic factor which also depends on the orbital momentum L;.
When the transferred nucleons do not come from the 1 s-shell but from any other shell of the nucleus A , the amplitude A:? also simplifies if the nucleus A has a pure configuration. Examples are (6Li, a) and (7Li, a) reactions in which the transferred nucleons
come from the Ip-shell of the lithium nuclei. One gets, in analogy t o eq. (1 I), the following expression
where
and
K(rz1, N L , L,) K(n1, N' L', Li)
.,,.
x B ~ L- (I6)
K(O0, Nb Lk, LL)
The coefficients dL; are somewhat more complicated than the corresponding bL;. But they can be calculated in the interesting cases (*) where K(O0, NA Li, Lk) # 0.
Thus, unambiguous spectroscopic information on the many-particle structure of nuclear states can be obtained from heavy-ion reactions with transfer of k nucleons if the nucleus A has a simple structure.
The coefficients g;LL' can be calculated in analogy to those for one-nucleon transfer reactions taking into
account the correspondence of the quantum numbers of the centre of mass of the transferred nucleon group to those of the transferred nucleon.
6. Conclusions. - In the foregoing, the problem has been discussed in what manner the amplitudes of spectroscopic factors for the separation of nucleon groups from nuclei can be calculated theoretically and obtained from experimental data. The shell- model calculations lead to results whose accuracy is higher than that of results following, at present, from an analysis of the experimental data. Thus, it is not necessary, for the present, to take into account corrections on the theoretical values coming from the shell-model approximations like complete overlapping of the clusters. Rather, calculations must be done with good shell-model wave functions for other than 1 p-shell nuclei,
The problem which must be solved is the analysis of the experimental data. Knock-out reactions induced by high-energy particles such as (p, pu) reactions can be handled relatively easily. But an analysis of multi- nucleon transfer reactions, where a lot of experimental data exist, is not performed as yet. Surely, there are many troubles. Every reaction must be calculated individually. If one restricts, however, to reactions induced by ions with a simple nuclear structure the (*) If K(O0, No LI, L k ) are not defined, an equation ana-
logous to eq. (14) can bc written down without introduction analysis of multi-nucleon transfer reactions simplifies of these factors. and can give valuable information on nuclear structure.
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