INTERMEDIATE RESONANCES OF NEUTRONS SCATTERED BY DEFORMED NUCLEI

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1 IC/66/109 INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS INTERMEDIATE RESONANCES OF NEUTRONS SCATTERED BY DEFORMED NUCLEI G. PISENT AND F. ZARDI 1966 PIAZZA OBERDAN TRIESTE

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3 Ic/66/lQS INTERNATIONAL ATOMIC ENERGY AGENCY INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS INTERMEDIATE RESONANCES OP HEUTRONS m DEFORMED NUCLEI * G. PISENT** and P. ZAKDI** ; TRIESTE December 1966 To he submitted to Phyeica Letters. Work cp,rr? ed out under contract EURATOll/cmN ** Permanent address: Istituto di Pisica dell'universita di Padova. Lab. dell'acceleratore Van do Graaff, Univ. di Padova.

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5 INTERMEDIATE RESONANCES OF NEUTRONS SCATTERED BY DEFORMED NUCLEI (*) t by G. Pisent and F. Zardi Istituto di Fisica dell'universita di Padova, Lab. dell 1 Acceleratore Van de Graaff, Univ. di Padova 1, As is well known, the cross section of nucleons scattered by "collective" nuclei is expected to show a middle-range modulation due to virtual excitation of the internal degrees of freedom of the target. The first suggestion on the effect has been given by Bohr and Mottelson in the year 1953 The development of the idea led to the generalized optical model and to the coupled channels calcula- ( 2 ) tion techniques. More recently, the nuclear reaction unified theories ( 3 ), owe. and the Feshbach theory m particular, allowedj^to stress the general problem of the easily excitable nuclear structures, and intermediate resonances. It comes naturally now to reformulate the problem of excitation of collective states, in the framework of this more general approach. We consider here the problem of the scattering of neutrons from permanently deformed even-even nuclei. The collective properties of the target are described in the framework of the phenomenological theory developed by Bohr and Mottelson. The schematization introduced in writing down the wave function and the hamiltonian are those familiar in the coupled channels techniques, but the physical (*) Work carried out under Contract EURATOM/CNEN - t-

6 - 2 - interpretation and mathematical manipulation of equatibns are those suggested by the Feshbach doorway states theory. As is well known, the fundamental feature of the Feshbach method lies in separation between open and closed channelssubspaces. The discrete eigenstates of the closed channel system are then correlated with resonances in the scattering channels. At this stage, one is naturally led to introduce into the theory a suitable physical model taken from spectroscopy. It will be seen that the natural choice in our case is application to the Nilsson model. 2. Let us write the total Hamiltonian as follows: (1) H(r", )=T(r)+H t ( )+V(r,, 's), where T is the kinetic energy of relative motion, H. is the target Hamiltonian, and the interaction potential V depends on projectile position r, as well as on target orientation. A spin-orbit interaction is also introduced The target is assumed to be an axially symmetric deformed nucleus. For quadrupole shape, the interaction potential V can be developed as follows (2) V(r i i,t-s)=v o Cr,t-s)-V 1 (r)bro 1 ~V (-)V(?)Y^(O (t) + Given a certain frame of reference x, we denote by x the total coordinate of the point, and by x and x the radial part and angular part respectively. (*) See for example B. Buck quoted in Ref. (2).

7 - 3 - The total wave function f, defined by equation (3) (H-E)S-=O can be written as: J* j * 3 where I is the target spin, and the symbol < ) means vector addition. The target wave function is (I=even). By insertion of eq. (4a) into eq. (3), and projection on each state <* 4 the following system of coupled'differential equations is obtained: CH 00 -E)u 0 +H olul +... H QN u N =O,, H lo u o+ (H U -E)u H ln u N =O (6) (H NN- E)U N =O where the matrix H is defined as (7) H, =(T + e )«-3-

8 and E^ is the target eigenvalue. In equations (6),(7) the index \ (=0,1,...N) stands for the set Ij, and X=0 means entrance channel. The number of coupled equations depends on how many target eigenstates are considered to be effective. The matrix elements <*, V $ > A ' ' y are purely geometrical, and can be derived by standard calculations. 3. We solve now the system (6), and make the basic assumption that the. discrete eigenstates of the closed channel are the collective doorway states (4) as defined by Feshbach. After some lengthy manipulation, a Breit- Wigner formula is obtained for intermediate resonances. For the sake of simplicity and without losing much in generality, we can assume that only the elastic channel is open. One obtains then : (8) exp(2i6)=exp(2ie) ir + 1, 2 - E-($+A )+i(rt + r +,a a a a In equation (8) 6 is the elastic channel phase shift, and 0 is the potential phase shift derived from the scattering equation (9) (T+U-E)u 0 =O where U is an average phenomenological potential. The resonance energy c ^s the eigenstate of the closed channels system. The width r+ and the shift factor A of eq.(8) a a are calculated from the wave functions relative to

9 - 5 - the potential scattering equation and to the bound state equations. In particular, r + is proportional to the probability of the doorway state's growing up and subsequently decaying into the scattering state u 0. The additional width r* takes into account the transition probability of the doorway state to more complicate systems. Xet us consider now the closed channels problem. Since the natural frame of reference for describing the behaviour of a particle bdund in a deformed core is a core-fixed reference, it is obvious at this stage a coordinate transformation. Taking into account equation (5), the inelastic part of the wave function T can be rewritten as where (11) Let us write now the wave function (10) in the fixedto-body frame of reference p. Remembering that one obtains: (13a) 7 JM (13b) -5- "T

10 - 6 - Equation (13a) has the familiar form which is encountered in describing the behaviour of a nucleon bound in a deformed core. The form (13a) is very general, and not influenced by the choice of the number of excitable target levels. As a consequence of the rotation of the frame of reference, the total Hamiltonian (1) assumes the conventional form (m) H=H 0 (p)+h.+rpc " rot where (15) H 0 (p) The term HQ(P) is a single particle Hamiltonian, in interaction with a deformed nucleus. Since we deal with a bound state problem, it comes naturallvto employ the classical approximation introduced by Nilsson. Furthermore, by introduction of the radial wave functions of the three-dimensional harmonic oscillator, one may express X through the Nilsson basic vector NdAE>. In this way, the original differential system becomes a numerical matrix, whose diagonalization is straightforward.

11 ACKNOWLEDGEMENTS We are pleased to thank Professor A. Salam for the hospitality extended to us at the International Centre for Theoretical Physics. Thanks are due to Professor C. Villi for illuminating discussions. The valuable collaboration of Mr A. Pascolini is also gratefully acknowledged. Finally, we thank Professor R.H. Lemmer for some useful comments, and for having informed us that a similar work has been also undertaken at M.I.T.. REFERENCES 1. A. Bohr, B.R. Mottelson, Dan. Mat. Fys. Medd. 22, Nr 16 (1953); 2. S. Yoshida, Proc. Phys. Soc. (London) A69,, 668 (1956); D.M. Chase, L. Wilets, A.R. Edmonds» Phys, Rev. 110, 1080 (1958); B. Buck, Phys. Rev. _130, 712 (1963); T. Tamura, Rev. Mod. Phys 3_2, 679 (1965); G. Pisent, A.M. Saruis, "Virtual Excitation etc.", in press on Nuclear Physics; 3. H. Feshbach, Ann. Phys _19_, 2 87 (1962); C. Bloch, XXXVI Course of Varenna School, in press; U. H. Feshbach, Proc. Antwerp Conference, p (North^Holland Publishing Co., Amsterdam, 1966); 5. S.G. Nilsson, Dan. Mat. Fys. Medd. 2$_ t Nr 16 (1955) -7-

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