THEORY OF A NUCLEAR JOSEPHSON EFFECT IN REACTIONS BETWEEN HEAVY IONS

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THEORY OF A NUCLEAR JOSEPHSON EFFECT IN REACTIONS BETWEEN HEAVY IONS

K. Dietrich, K. Hara, F. Weller

To cite this version:

K. Dietrich, K. Hara, F. Weller. THEORY OF A NUCLEAR JOSEPHSON EFFECT IN REAC- TIONS BETWEEN HEAVY IONS. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-183-C6-184.

�10.1051/jphyscol:1971635�. �jpa-00214853�

JOURNAL DE PHYSIQUE Co/loque C6, supplkment au no 1 1-1 2, Tome 32, Novembre-De'cembre 1971, page C6-183

THEORY OF A NUCLEAR JOSEPHSON EFFECT IN REACTIONS BETWEEN HEAVY IONS

K. DIETRICH, K. HARA, and F. WELLER Institute for Theoretical Physics, University of Heidelberg

and Kernforschungszentrum Karlsruhe

Rhumb. - Dans le traitement semi-classique de reactions mettant en jeu un transfert multiple de paires, chaque noyau est reprksente par un systeme de deux etats degkneres. Le modele rend compte du transfert preferentiel de paires vers les etats fondamentaux et les etats de vibration de paires des noyaux en interaction.

Abstract. - In the semi-classical treatment of reactions involving multiple pair transfer, each nucleus is represented by a system of two degenerate levels. The model displays enhanced transfer of pairs into the ground states and pairing vibrational states of the interacting nuclei.

1. Introduction. - It has been demonstrated in a simple model that we should expect enhanced mul- tiple transfer of pairs between superconducting nuclei in a heavy ion reaction [I]. In this model the interacting nuclei were described by one degenerate level. We generalize the theory by representing each nucleus by two degenerate levels. This generalized quasi-spin model was applied to the isotopes of Ni, Sn and Pb with reasonable success [2]. This improved version of the quasi-spin model can describe seniority 0 excita- tions, among others pairing vibrations. We describe the model in section 2 and present results in section 3.

2. Theory. - The Hamiltonian of the model is

where Ha is the Hamiltonian of nucleus a(a = 1,2)

and Hi,, represents an effective pairing interaction between the two nuclei

A A 6

Here S, = S,, S,, is the total quasi-spin of nucleus a, while SA,(SB,) is the quasi-spin associated

with level A(B) of nucleus a. The quantities ^6, and e, are defined by

where B~,(E~,) is the energy of the level A(B). The time-dependent strength V ( t ) of the pairing interaction

between the systems is defined in ref. [I]. Because of the presence of the last term in eq. (2), the eigenstates of H, are superpositions of states of different quasi- spin S, with the same s:.

The total wave function I Y(t) > is expanded in terms of the eigenstates of Ho. Using the interaction representation (superscript I), we have

The index v designates the following quantum numbers : v = (S1, n, 0 ; s:, n,) where n, counts the states of given s:. The time-dependent equation for the amplitudes f,(t) is ( h = 1) :

where Ev is the eigenvalue of H,. The enhancement of the transfer is produced by the special form of the matrix-elements < I Hin,(t) I ^@,, > .

3. Results. - We choose, as an example, the scattering of lZ2Sn on ,06Pb because '08Pb is known to display pairing vibrations and the tin isotopes are strongly superconducting nuclei.

The matrix-elements < @, ( Hin,(t) I @,. > are large, if the unperturbed states I ^@, > and I @,, > represent the ground states of the two interacting nuclei or the pairing vibration of 2 0 8 ~ b . This can be seen from the following values of matrix-elements :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971635

C6-184 K. DIETRICH, K . HAKA A N D F. WELLER

< '"Sn(gr), '08Pb(p. v.) l ~ ( ' ~ ~ ~ n ( ~ r ) , behaviour of the cross-section as a function of the 2 0 6 p b ( ~ ~ ) > ⁼ 3.52 scattering energy E (and scattering angle 6 ) which is

< 120Sn(gr), 2 0 8 ~ b ( 2 . ~ x c . ) J M ~ ' ' ~ s ~ ( ~ ~ ) , reminiscent of multiple Coulomb excitation. This 20"b(gr) > ⁼ 0.13. is displayed in figures 1 and 2.

The smallness of the third number implies that the 2nd excitcd seniority 0 state of 208Pb is not of a coherent nature.

This property of the matrix clements leads to mul- tiple transfer of pairs between the ground states of the tin isotopes and the ground states, and, to a smaller extent, the pairing vibrational states of the lead iso- topes. The multiple transfer results in an oscillatory

L I Z L 1 5 L20 LX WO 13s LID LL5 1%

\ &=I0 L MeV

LIZ L15 LM LX LM 135 LLO LL5 450

E q y lMeV1

Rc;. 2. -- As in figure 1 : transition into ground and excited states of ZoxPb. a = transition into ground state of 2osPb ; b --- transition into pairing vibration of 2"XPb ; c - = transition

into 2 n d excited seniority 0 state of 2onPb.

FIG. 1. -Probability of pair transfer for the scattering of

"2Sn on 206Pb as a function of the scattering energy E : transi- We believe that our model gives a qualitatively tion into ground states. ( N I , N2) = configuration of N, pairs correct description of the effects to be expected. On in nucleus a ; SZ = number of available pair states of the dege- the other hand, it is possible that an improved repre-

nerate Icvel. Pairing constant G for Pb is adjusted to give the

correct energy of the pairing excitation ; G for tin from ref. 2 : of the between the two G ~ , , = 0.095 ( M ~ v ) and G ~ , , --. 0.187 (MeV), ~ ( t ) as in ref, 1 (above all ill~lUSi0n of t~nnelling) will lead to an

with G = 0.15 (MeV). appreciable modification of our results.

References

[ I ] DIETRICH (K.), HARA (K.), and WELLER (F.), Phys.

Letters, 1971, 35B, 201.

[2] HARA (K.), Zeits. f. Phys., 1967, 202, 504.