REFERENCE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC

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1 REFERENCE IC/73/193 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS DIRECT INTERACTION IB (n,a) REACTION FROM DEFORMED TARGETS D. Pal T. JSa.^rajan and M.Y.M. Hassan INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1973 MIR AM ARE-TRIESTE

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3 IC/73/193 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS 4. DIRECT INTERACTION IN (n,a)reaction PROM DEFORMED TARGETS D. Pal *, T. Hagarajan ** and M.Y..M. Hassan *** international Centra for Theoretical Physics, Trieste, Italy. ABSTRACT A direct interaction study of (n,a) reactions from deformed targets 3 has been made "by describing the target as a bound system where a He is attached to a deformed core. Numerical calculations are made for MeV neutrons on Dy target. An agreement with experiment is obtained from a DWBA calculation with deformed Woods-Saxon potential. In view of the large errors in the experimental data,a full coupled channel calculation appears to be hardly worthwhile. MIRAMARE - TRIESTE December To be submitted for publication. * On leave of absence from Saha Institute of Nuclear Physics, Calcutta, Indi a. ** On leave of absence from Department of Nuclear Physics, University of Madras, India. *** On leave of absence from Physics Department, Faculty of Science, Cairo University, Cairo, Egypt.

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5 I. INTRODUCTION The analysis of (n,a) reactions with neutron energies ip to lu MeV and target nuclei with mass number A < 100 is fairly systematic and exhaustive It seems well established that for low-energy neutrons the reaction proceeds mostly via formation of a compound nucleus. Whereas for Ik MeV neutrons the direct interaction and the compound nuclear reaction mechanisms seem compete with each other. There are several (n,a) angular distribution data from light nuclei in this energy range which can he explained as a knock- 2) 3) 3 M out or pick-up of He or as a mixture of "both. But for targets with mass number A > 100 and medium energy neutrons the compound nuclear picture fails completely. Also the smallness of the reaction cross-section in this region is responsible for a lack of adequate experimental information, and consequently a thorough theoretical analysis has not been made. Nevertheless the high Coulomb Vu-rier ( ' -; 20 KeV) for to ct particles suggests that the direct process should be predoni-art in this mass and energy region. is quite clear that a simple Butler's picture will not be adequate here and a DWBA calculation seems worthwhile. The experiments performed by the Polish group with Dy isotopes as targets appear to be quite interesting. showa in Fig.l the sharp fall of the (n,a) cross-section near the maximum energy "a" suggests that the reaction might have proceeded with the excitation of only a few degrees of freedom; in other words, entirely via direct interaction. We intend to interpret the experimental data "by assuming a direct interaction type of reaction mechanism and ascribing a simple model to the target nucleus. In our model, the target nucleus is described as a core consisting of A-3 nucleons with a 3-nucleon cluster ( He) bound to it. The incident neutron interacts with the extra-core cluster and picks it up to form the a particle in the exit channel. Since Dy is known to "be de- 3 formed,we shall have the He bound to a deformed core. A coupled channel calculation with adiabatic approximation should explain the experimental data if the reaction mechanism is a direct reaction. It As In Sec.II we shall formulate the problem in general and in Sec.Ill we shall discuss various approximations. Sec.IV will be devoted to the description of our calculation and result. II. FORMULATION OF TEE PROBLEM Let 7(1,2,..,,A+l) be the scattering wave function satisfying the Schrodinger equation >,. (1) -2-

6 with appropriate boundary condition, where A is the mass number of the target nucleus, H is the Hamiltonian of the system and E is the total energy in the CM system. H is given by A A A T± V + T + ij A+l i > J - V ia + l +V C In the above equation S\ is the kinetic energy operator for i th nucleon, V ij is the tw -^od y nuclear interaction between the nucleons i and j i and v c (target) is the Coulomb interaction in the target nucleus. Let us write the total scattering wave function } * / ] n as X n (A-2,A-l,A,A+l) A n r o> r (3) In Eq_.(3) for convenience of notation we have denoted all the spacial coordinates JL>2_...,A_3 by and the spacial co-ordinates of the four remaining nucleons as r., r o, r_ and r. The wave functions 1 "-c ***j n satisfy the Schrb'dinger equation n where T. + A - 3 i> J Multiplying the left-hand side of Eq.(l) by <$ (C)] and using Eqs.(3) and one obtains - E + T. + i h V., + V i J C {h particles) m (5) Eq.(5) is a system of coupled equations with.an infinite number of coupling terms. In solving this equation one must truncate the infinite summation appearing on the right-hand side by introducing certain reasonable approximations, -3-

7 A coupled channel calculation for determining x consists in treating a few of the states $ (?) to be strongly coupled to the state $ ( ) through Ei n the residual interaction and the rest to be considered as weakly coupled. In the weak coupling limit (DWBA) the matrix elements h are interpreted as the imaginary part of the optical potential due to the target at state "n". weak coupling assumption holds for all states DWBA turns out to he a good approximation when the $ ( ), otherwise one must take recourse to a coupled channel calculation, treating rigorously those channels which are strongly coupled via the residual interaction. Bearing in mind cur simple model for the target nucleus,let us write the wave function describing the target at state E, as A X.U.r r r ) A.1 d >j m E C * { ) n (r r r m ma >o, M?) Ti.(r.rr ) ma m A.*1*"2~»3 (6) The X, satisfy the equation where (T) + ) ' \ + ^ \ V AJ + V c (3 particles) + ) (8) The expansion coefficients c ^ satisfy the completeness condition and C*, C. = 6., ma my Xy (9) C. C -, = 6 ma na mn Using Eqs.(U), (6), (T) and (8), one obtains the equation satisfied by r), as na -k-

8 '"n " B X Ein 3. (10) In principle, Eq.(10) can be solved, for bound 3,-particle cluster vave functions n, (r., r :r ) "by truncating the summation appearing on the right-hand side at a reasonable point. We then expand the total scattering vave function in terms of target states and the single neutron wave function as A comparison between Eqs.(3) and (ll) gives (11) (12) Substituting for obtains L (E, - E) + T(r ) + A n in Eq.(5) from Eq.. (12 ) and using Eq. (11) one finally m Eq.(l3) immediately tells us that the (an) differential cross-section vill be given by a (8) = * * mm k n a n S ]frtti (8), m* = reduced mass, with K where *K (r ) satisfies the equation- A n -5-

9 (E, - E) + T(r } m ^X ^rn'' mean:i - n S outgoing vave in the neutron channel. S ~ appropriate averaging in the initial channel and summing over final channel. reciprocity theorem the (na) cross-section will be given by With the,_2 (15) n k and k being the wave numbers corresponding to the a particle and neutron, respectively. III. VAEIOUS APPROXIMATIONS In Sec.II we have derived the direct interaction reaction crosssection without specifying the nature of the core $ ( ). For a noncollective target in our simple model we can take the core to "be spherical, and the coupling of various core states via the residual interaction will be small and DTOA will be a good approximation. In this approximation Eq..(5) will reduce to,) = o (16) after writing * * *+ 1I1L> *««* ^"^ (IT) X. (p,s,r) is the internal wave function for the a particle satisfying int " ~ *"" the equation T(r) + T(p) + T(s) + V(p,s) I i. *», ««n ot where T(r) T(s) V m is the nucleon mass -6-

10 with p = r - r } L = J (^l +^2 +^3 is the binding energy of the a particle and S u = - E + e. ma m a Similarly, for a non-collective target one may set up the uncoupled equation for the 3-particle cluster wave function n,(r ) as na 3 where [T(R ) + v (R )1 n AKJ L *~3 3 3 j ma 3 ma (Rj, (18) (19) r l,- v, + (p» s _J satisfies the equation T(p) + Tls) + V(p,s) >,s) = e n..{p,s) i AV> j int -w «where e is the binding energy of the 3-*particle cluster which is a J He in our problem. In this approximation the target state X, instead, of being described by the expression (6) will look like X\ = * (0 H i( r -i r o r -D) A Ei ma 12 3 can always obtain the expansion coefficients C, by diagonalizing the Hamiltonian matrix K in the basic states X,, with some suitable assumption about the residual interaction r. 3 One!The DWBA approximation in the neutron channel reduces Eq.(l3) to [(E X -E) + T(r n ) + V n (r n (20) -7-

11 with = 0 (21) The above weak coupling assumptions will not hold if one must deal with collective targets. It is well known that in such cases a coupled channel calculation will "be necessary, since the ground and the low-lying excited states of the target will be strongly coupled via the residual interaction due to / their collective nature. We should point out here that within the framework of our simple model the collective aspect of the target will "be ascribed entirely to the core to which a He will be attached to form the target. Adopting a phenomenological approach we shall describe the collective motions in terms of deformations (permanent or vibrational) of the core and the interaction felt by the projectile or the outgoing particle would be a deformed optical potential V(r,6,(J>). According to this picture, the bound ^He should also see a deformed potential and consequently the optical potentials appearing in Eq_s.(l6), (l8) and (21) should be replaced by the.corresponding deformed potentials in the respective channels. Derivation of coupled equations with deformed potentials is beautifully reviewed by Tamura. We shall refer to Tamura's article for a full description of the situation and write down the coupled equations for oc particle, the bound He and the neutron for rotational target. With a Woods-Saxon radial dependence v(r f e.*) = -C (1+e)' X = ir-meson + V p _ Compton wavelength e = exp[(r-r)/a] e = exp[(r-r)/a] H' = Ro 1 + R = R, \ - 'o * 1/3 \ " 'o A1/3 A is the mass number of the core and G 1 refers to the body fixed system. The non-spherical potential V(r,0,< >) can be written as -8-

12 (22) where v diag 7^'< r) (,)-v 50 (^ 2P C- 2 J E (r-bj, C (23) coupl with -fv + iv) cp exp [r-r 0 [l -JD C ' <i(cos9) L 1 + exp The symbols are self-explanatory and are explained in Ref.6 (22), (23) and (2h), Eqs.(l6), (l8) and (21) become Using expressions n + 1).0-1 'np 11 ' n where P = K r, K is the wave number for the particle p * u D v n n X being the spin wave function of the projectile, while 0 T.. is the wave function of the core in its n state. Eq..(25) must be solved for the three channels with proper asymptotic form of the respective wave functions. -9-

13 In adiabatic approximations one neglects the rotational motion of the well deformed target nucleus compared with the projectile motion and Eq.(25) is 6) reduced to np Jl'j 1 V 1 E (26) primes attached to Y n XII denote a reference to the "body fixed co-ordinates, and m is the projection of the total angular momentum j of a partial wave of J the projectile along the nuclear symmetry axis. The "body fixed co-ordinate system is preferred in writing Eq.(26),to avoid the coupling "between the partial waves with different values of KL., in addition to the coupling between those with different values of j. IV. NUMERICAL CALCULATION AND RESULT In order to obtain an order of magnitude estimate we have calculated ^ -Dy (na),-i Gd- cross-sectionswith MeV neutrons without including the coupling. We have taken a 5-function interaction for 2 t ^. whose 3 strength -is determined by producing the a-particle binding energy. For He and a-particle intrinsic wave functionswe have taken the following form: n int (p»s) = n a exp[-a 2 (2p 2 + s 2 )], p,s,r)= U B exp[-3 2 ( p 2 + 2s r 2 )], where N and H o are normalization constants, and a and fj are related to the He and a-particle binding energies, according to the relations 2 2m-, 2 2m a = -^e 3 and p - -j a. The 6-function interaction gives V(r) x int (p>s,r) = D Q 6(r) with D n = L3p>) -10-

14 The scattering wave function ^ (r ) and X (*O neutron and a channels and the bound He wave function n.(rg) are computed by solving Eq.. (25) with appropriate boundary conditions ignoring the coupling term. The value of the deformation parameter 3 n is taken to be 0.3 from BE2 measurements of,.ga?. The potential parameters 161 for bound He are fixed by producing the binding energy of The optical model parameters for neutron and a particle are chosen by looking into the elastic scattering situation in this mass and energy region. The solid curve in Fig.l is the calculated curve. Our rough calculation tells us that we are on the right track. A better set of experimental data is really necessary for a complete understanding of the situation. The level information on the residual nucleus Gd -is taken from experiment. The DWBA cross-sections for the levels of Gd J are convoluted with Gaussian response for the detector with full width at half maximum equal to 0.5 MeV in 2-2 <" " da dftde c(e i ) where c can be regarded as an average spectroscopic factor for the residual nuclear states. In the present case, a value of d ~ 0.21 brings about a gross agreement with the experiment. Even though the agreement with experiment is fairly good, the deviation at maximum <x energy is perhaps due to the attenuation of the <X -energy spectrum due to the thick target used. ACKNOWLEDGMENTS The authors would like to thank Prof. Abdus Salam, the International Atomic Energy Agency and UNESCO for kind hospitality at the International Centre for Theoretical Physics, Trieste. Special thanks are due to Prof. L. Fonda for reading the manuscript. -11-

15 REFERENCES 1) ' B. Cinclro, Rev. Mod. Phys. 38, 391 (1966). 2) R.A. Al Kital and R.A. Peese, Jr., Phys. Rev. 130_, 1500 (1963). 3) M.L. Chatterjee and B. Sen, Uucl. Phys. 51_, 583 {196*4).. h) M.L. Chatterjee, Uucl. Phys. 65_, 635 (1963). 5) L. Glowacka, M. Jaskola, M. Kozlovski, W. Osakienwicz, J. Turkiewicz and M. Zembo, Report Ho. 1U68/I/PL/A (1973), Institute of Nuclear Research, Warsaw. 6) Taro Tamura, Rev. Mod. Phys. 37, 679 {1965). -12-

16 158 O 150 f C3 100 :. (M The theoretical curve (solid line) is compared "with the experimental data. -13-