Nucleon clusters in light nuclei

Nuclear Physics 52 (1964) 3 4 5 - - 3 5 2 ; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

N U C L E O N CLUSTERS IN L I G H T NUCLEI R. I. J I B U T I , T. I. K O P A L E I S H V I L I a n d V. I. M A M A S A K H L I S O V

Tbilisi State University, Tbilisi, USSR Received 1 July 1963 Abstract: T h e (?,, pt) reaction is considered in t e r m s o f the n u c l e o n cluster model. Expressions for the distribution over the relative energies o f o u t g o i n g particles a n d the total cross section o f the reaction are obtained. T h e results are c o m p a r e d with experimental data. A possible m e c h a n i s m o f the transition between cluster configurations in light nuclei is considered.

1. Introduction

The assumptions on the possible production of temporary nucleon clusters in light nuclei and their role in nuclear processes were first stated by Mamas~dchlisov in 1953 1). He also put forward new considerations in favour of the a-particle grouping of nucleons in light nuclei. Nucleon clusters in a nucleus must decrease the effective number of nuclear particles (or the degrees of freedom) if of course it is assume that the internal degrees of freedom of these substructures do not show up in many processes (in this the idea of substructure just lies). Thus the presence of a-particle or other formations should be sought in an analysis of such processes which depend on the number of particles with given mass and charge constituting the nucleus. The density of nuclear levels is one of such quantities essentially depending on the number of particles. It is of interest therefore to compare the theoretical values of the nuclear level densities obtained under the assumption that the number of independent particles in the nucleus equals that of neutrons and protons with the corresponding experimental values for a given nuclear excitation. If the theoretical values of the nuclear level densities prove to be always larger than the experimental values, this could warrant the contention that the number of degrees of freedom of the nuclear system is less than the value following from the proton-neutron model of the nucleus. The corresponding calculations by the model in which the nucleus was considered as a mixture of two Fermi gases were performed by Ma~nasakhlisov in 1955 2). It was shown that to achieve agreement with the experimental values of the nuclear level densities it is necessary to decrease the number of particles by approximately a factor of 3 for heavy nuclei and by a factor of 1.5 to 2 for medium nuclei. Thus, the effective number of particles in the nucleus with respect to the production of energy levels proved to be considerably less than the number of neutrons and protons in the nucleus under study. Hence two conclusions are possible: either in the nucleus there 345

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are complex particles (e.g. ~-particles, tritons, etc.) whose transition into excited states determined the observed picture of energy levels, or there occur simultaneously two, three or four-particle excitations. On the basis of the model of substructures in the form of ~-particles, tritons, He 3, and deuterons, Chilashvili 3) considered some nuclear reactions with T-quanta accompanied by the emission of these substructures and estimated the magnetic moment of a number of nuclei. The model of substructures (clusters) is in principle the next approximation as compared with the shell model taking into account the residual interaction of nucleons in the nucleus by the mixing of only close adjacent configurations, since the cluster model, introducing three and four-particle correlations, takes into account effectively stronger and more distant interactions. It should be noted that the assumption of the division of the nucleus into substructures has an approximate character. If the exchange of nucleons is small, the nucleus divides well into substructures and the 'properties of the nucleus under study reduce mainly to the properties of these substructures and those additional properties which are determined by the interaction of these substructures. It is possible that the properties of substructures inside the nucleus differ in the general case from their properties in the free state. Obviously, strong nucleon-nucleon interactions may lead to the destruction of some and the production of other clusters, the division of the nucleus into groups of nucleons being dependent not only on the nucleus under study but also on its energy state. Since 1958 Wildermuth et al. 4) undertook several investigations in which the energy spectrum of some light nuclei (He s, Li 5, Li 6, Li 7) was studied in terms of the cluster model taking into account nucleon exchange and they reached satisfactory agreement with experiment. From the condition of an energy minimum for the system of nucleons of the nucleus Li 6 divided into two subsystems in the form of a quasi-~-particle and a deuteron Kopaleishvili et al. 5) found the average distance between the latter. The distance proved to exceed the proper size of the stibstructures, which confirms the cluster structure of the nucleus Li 6 in the ground and excited state with 3.50 MeV. Clustering of nucleons in the shell model was investigated by Balashov et aL 6). At present there is ample experimental evidence confirming the existence of nucleon clusters in the ground and excited states in the form of ~-particles, tritons, He 3 and deuterons in Li 6, Li 7, Be 7, Be s, Be 9, C 12 and other nuclei 7). For a further check of the model of nuclear substructures it is of interest to investigate nuclear reactions induced by y-quanta. In particular, the (~, pt) reaction on the C 12 and 016 nuclei may furnish valuable information on the existence in them of 4-nucleon clusters in the form of quasi-~t-particles. This reaction on C 12 was observed experimentally by Maikov s). Since the threshold of this reaction lies above the giant resonance, the reaction

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occurs mainly through the direct interaction of a y-quantum with a nucleon or a nuclear nucleon cluster. Two mechanisms of this process are possible: (a) a y-quantum ejects directly a proton from the s shell after which the residual nucleus in a strongly excited state emits a triton, and (b) the products o f the reaction, a proton and triton, are obtained as a result of the direct fission by the y-quantum of an intranuclear a-particle cluster. The (~,, pt) reaction was studied by Balashov and Fetisov 9) under the assumption that the transition occurs through the first mechanism. The same reaction is considered in this paper in terms of the second mechanism. It can readily be seen that the difference between these mechanisms may show up in the energy spectrum and angular correlation of the decay products. If the calculations are made on the basis of the first mechanisms, out of all possible states of the nucleus B 11 the main contribution to the process under study comes, as is shown by Balashov and Fetisov 9), from the state with an energy of the order of 20 MeV. Then, bearing in mind that the energy of emission of a triton from the B 11 nucleus is of the order of 11 MeV, we can see that the subsequent decay of the B 1 nucleus will yield tritons of energies of 6 or 7 MeV mostly. The tritons produced in the first act of the process proved to possess the same energies since the main maximum of the cross section lies, according to the experiment s), at the y-quantum energy of 43 MeV. If, on the other hand, the second mechanism is used, an outgoing proton and triton must possess approximately equal momenta. Indeed, the energy for the motion of the centre-of-mass of an a-particle in the nucleus is much lower than that for the relative motion of a proton and triton in an a-particle. Therefore an a-particle absorbing a y-quantum may in the first approximation be considered at rest since the momentum of the y-quantum can be neglected. The approximate equality of the momenta of an outgoing proton and triton means that in most cases the proton energy Ep is higher than the triton energy Et. From the Maikov experiments it follows that in 70~ of all cases Ep > Et. This fact shows the predominating role of the second mechanism in the process under study. It should be noted, however, that the experimental data are rather rough since they ere based on insufficient statistics. Therefore our conclusion about the predominating role of the second mechanism may prove not to be quite accurate. In this connection a more elaborate experimental study of the spectra of products of the (7, pt) reaction is o f interest. To elucidate the validity of either mechanism of the reaction it is also important to investigate the angular correlation of the products of the reaction or distribution in the relative energies of a proton and triton. Obviously, if the process occurs in accordance with the first mechanism, the correlation between the products of the reaction will be very weak because of the independence of the processes of the emission of a proton and the subsequent emission of a triton from the excited nucleus B 11. If the second mechanism applies, this correlation will be essential.

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T o show this the Ca2(y, pt) reaction must be considered in more detail in terms of the second mechanism. 2. Investigation of

the cl2(y,

pt) Reaction

Since the reaction occurs at high energies (up to 150 MeV) the retardation has to be taken into account in the Hamiltonian of the interaction of a y-quantum with an e-cluster. Introducing variables convenient for the calculation and assuming that in the process under study the internal state of a triton does not change, we can write for the Hamiltonian of the interaction

H = ieh {eik,o. (~+ i,)[¼(e. VR) + (e- V,)] + e 'k'' (R- •,)[¼(e. VR)-- ~(e. V,)] }, (1) Me where R is the radius vector of the centre-of-mass of a quasi-e-particle with respect to the residual nucleus, r is the radius vector of the proton in the e-particle with respect to the triton centre-of-mass, e is the y-quantum polarization vector, k,o is the y-quantum wave vector, and M is the nucleon mass. As the wave function of the C 1z nucleus we take an expression allowing for the a-particle structure of this nucleus. This wave function was used by Biel when considering the stability of the C 12 nucleus. Since the a-particle binding energy in C 12 is much less than the internal energy of the e-particles the role of the exchange of nucleons between the e-clusters is small. Therefore, for the wave function of C ~z we may take ~i = Ae-I~R~ e-lU'ZqTt ~Be8x~X~,

(2)

where ¢Pt and ~B,~ are the internal wave functions of the triton and the Be s nucleus, and X~ and X~' are the spin and isospin functions of the intracnulear e-particle. The parameters 2 and/~ are taken from the investigation of Biel (2 = 0.0694 fm -2, /~ = 0.3156 fm -2) and .4 is the normalization coefficient. Since we are only interested in the qualitative characteristics of the process we neglect the interaction between the decay products in the final state. As a result we obtain for the distribution in the relative energies of the triton and proton

d_~a= 4n dt

ME r

y2ox/t(1-t)e-(3gl2a)x°2-°'s'°2F(xo, Yo, t),

(3)

where Yo--

2M(E~ Q) E~ Ept , ~2 = t-hell hcx/~ ' Er-Q' -

Q = 27.16MeV.

Here Ept is the relative energy of a proton and triton, E r is the y-quantum energy, and

F(x°'y°'t)=e-(4"l~-°'5)~°2(1 -o [O./#W { 72~x~ [etX°2fl+(e-'°2124+et=°i)f2] 8 [etX°2f3+ (e-~°2/24 - 3eiX°e)f4]} + x~

(4)

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Here we have sinh u) sinh v A

=

cosh

f2 =

(

u

--

-

U

0

,

sinh u ) s i n h ~v -

cosh u

u

(5) f3 = ( cosh v - si 7 v ) s i n h u U

f4 =

(

sinh ~v~ sinhu ~v ] u '

cosh½v

where U --

2x/6x° Y°2 #

x/-~--t,

v

½~/3XoYo~/t.

For the total cross section of the process we have 4

tr = 1.7

Y°e-6"alax°2-°'sY°2A(xoYo) e~

fm 2,

(6)

where A is a slowly varying function of the order of unity and E r is the y-quantum energy in MeV. The theoretical curve representing the total cross section as function of y-quantum energy represented in fig. 1 is in good agreement with the experimental data in the main maximum region.

g3 02 O.J

Jo

~o

50

6o

7o

Fig. 1. Cross section for the CI~(7, pt)Be 8 reaction plotted against F-quantum energy.

Fig. 2 represents the curve of distribution in relative energies given by eq. (3). It is clear that the maximum of this distribution curve lies at a value t close to unity, which indicates that a proton and triton fly out mainly in opposite directions. This fact points out a strong angular correlation between the products of the reaction,

350

n. L JmorI e t aL

whereas in the case of the first mechanism there should be no such correlation and the distribution curve must be symmetric with respect to t = 1/2. If the experimental data confirm the presence of correlation between the products of the reaction this will indicate that the ~-particle adsorption of a 7-quantum occurs in the C 12 nucleus in the reaction. dE "'f.O

a8 0.6 .O.k O.Z i

0.6

,

*

I

0.8

Fig. 2. F u n c t i o n o f distribution in relative energies o f a p r o t o n a n d triton in the Cx~(7, pt)Be s reaction.

3. Transition From One Cluster Structure to Another

It has been indicated above that the cluster structure of the nucleus may change with the excitation energy. For example Phillips and Tombrello lo) showed that the reduced widths of the He 5 nucleus with the structure ( n ÷ ~ ) in the ground state IQ1+412~ 1 whereas with an excitation energy 16.69 MeV the reduced width is of the order of unity already for the ( d + t ) configuration, i.e. IQ2+312~ 1. For the Li 7 nucleus in the ground and first excited states the (t + ~) configuration is mainly realized, whereas the (Li 6 + n ) configuration predominates for 8.47 MeV. A similar situation occurs for the Li 6 nucleus. The (~+ d) configuration is realized in the ground and first excited states up to 5 MeV.. At higher energies the Li 6 nucleus proves to be unstable with respect to the decay into He 3 and H 3. Therefore it ought to be expected that the H e 3 + H 3 configuration predominates in the Li 6 nucleus in this region of energies. A question arises: how does the transition from one cluster structure to another occur? In the above examples the configuration changed in most cases through the transition of one nucleon from one cluster to another. For example in the Li 6 nucleus the (~+ d) configuration transforms into the H e 3 + H 3 configuration as a result of the transition of a neutron from an ~-particle to a deuteron cluster. It is natural therefore to suppose that the binding of the nucleon in the ~-particle cluster weakens'somewhat owing to another cluster nearby. The effect of the second cluster can be taken into account in the first approximation under the assumption that there arises an additional field for the nucleon under study which as a result finds

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itself in the field of two centres. Separately there are no excited levels for neutrons in the potential wells of eithe r an u-particle or a triton, but as these wells approach gradually the nucleon starts moving in the c o m m o n field represented by a stepped potential well with wider walls as is shown in fig. 3. The widening o f the potential well is known to lead to additional levels. Therefore it can be supposed that one configuration transforms into another as a result of the transition of a nucleon in this c o m m o n field to a higher level. The appearance of an additional level is connected with the attainment of a certain intercluster distance. Our calculations show that such a level does not arise for the Li 6 nucleus when the distance between the clusters d < dl + d 2 where dl and d 2 are the widths of the potential wells for a nucleon in an u-particle and triton in free states,

1 Fig. 3. Curve of the potential energy of the interaction of a nucleon moving in the field of an orparticle and triton.

while an additional level does arise when d = dl +d2, the corresponding energy being of the order o f 11 to 12 MeV. The wave function of the neutron in this state has a m a x i m u m at r satisfying the condition dl + d2 > r > d 1, which corresponds to the He 3 + H 3 configuration of the Li 6 nucleus with the excitation energy 11 to 12 MeV. We have determined the depth of the wells V1 and 1/2 proceeding from the neutron binding energy in He 4 and H 3 and from the sizes of these nuclei (d 1 ~ d2 ~ 1.6 fm, V1 = 71 MeV, V2 = 52 MeV). The energy of the additional level of 11 to 12 MeV thus found corresponds to the threshold of the fission of the Li 6 nucleus on He 3 and H 3.

References 1) V. I. Mamasakhlisov, JETP 24 (1953) 190 2) V. I. Mamasakhlisov, Proe. Physical Institute of the Georgian SSR 3 (1955) 31 3) G. A. Chilashvili, Proc. Georgian Academy of Sciences Tbilisi, XVIII (1957) 279; J. Ch. Vashakidze and G. A. Chilashvili, JETP 29 (1955) 157 4) K. Wildermuth and K. Kaunellopoulos, Nuclear Physics 7 (1958) 150, 9 (1959) 449; L. D. Pearlstein, J. G. Tang and K. Wildermuth, Phys. Rev. 120 (1960) 224, 123 (1961) 548; Nuclear Physics 18 (1960) 23

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5) T. I. Kopaleishvili, I. Ch. Vashakidze, V. I. Mamasakhlisov and G. A. Chilashvili, JETP 38 (1960) 1758; Nuclear Physics 23 (1961) 430 6) V. V. Balashov, V. G. Neudachin, Yu. F. Smirnov and N. P. Yudin, JETP 37 (1959) 1385; Yu. F. Smirnov and D. Chlebowska, Nuclear Physics 26 (1961) 306 7) E. Anderson, Proc. 2nd Conf. on Reactions Between Complex Nuclei (New York-London, 1960) p. 67; G. Morrison and M. Huberman, Proc. 2nd Conf. on Reactions Between Complex Nuclei (NewYork-London, 1960) p. 246; T. Cooding and G. Igo, Phys. Rev. Lett. 7 (1961) 28; L. Brown and H. Knowles, Phys. Rev. 125 (1962) 1339 8) V. N. Maikov, Nuclear reactions for low and average energies (Moscow, 1958) p. 414 9) V. V. Balashov and V. N. Fetisov, Nuclear Physics 27 (1961) 337 10) G. C. Phillips and F. A. Tombrello, Nuclear Physics 19 (1960) 555 11) S. Biel, Proc. Phys. Soc. A70 (1957) 806