1p1/2 0d5/2. 2s1/2-0.2 Constant Bound Wave Harmonic Oscillator Bound Wave Woods-Saxon Bound Wave Radius [fm]

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1 Development of the Multistep Compound Process Calculation Code Toshihiko KWNO Energy Conversion Engineering, Kyushu University 6- Kasuga-kouen, Kasuga 86, Japan program \cmc" has been developed to calculate the multistep compound (MSC) process by Feshbach-Kerman-Koonin. radial overlap integral in the transition matrix element is calculated microscopically, and comparisons are made for neutron induced 93 Nb reactions. Strengths of the two-body interaction V are estimated from the total MSC cross sections.. Introduction The quantum-mechanical theory of the pre-equilibrium nuclear reaction by Feshbach, Kerman, and Koonin[] (FKK) has a rather simple and feasible formulation, and it has been applied to analyses of medium and high energy nuclear reactions. The theory distinguishes two types of the pre-equilibrium emission the multistep direct (MSD) and the multistep compound (MSC). To calculate the MSC process, the original FKK assumes constant wave functions within a nucleus because it has a great advantage to evaluate a transition matrix element easily. Milan university group[] adopted more realistic wave functions for a bound and an unbound states, and they have developed a MSC code GMME[3] which calculates the transition matrix elements microscopically. However open questions still exist. The calculated MSC cross section depends on an assumption of the single-particle bound states, normalization of the unbound wave functions, and a limited number of partial waves[]. program \cmc" was designed to show the dierence between the calculations with the constant wave assumption and without it. It calculates the transition matrix element microscopically too. The bound state wave function is calculated with a harmonic oscillator or a Woods-Saxon potential, and quantum numbers of the single particle states are determined according to the shell model. The unbound state wave function is a distorted wave by a spherical optical potential.. The Overlap Integral The MSC energy spectrum is given by[] d du = X k (J + ) h, Ji hd J i J X X N= j h, "j NJ N, (U)i Y h, # MJ i h, NJ i h, M= MJ i ; () where N is the class of the pre-equilibrium states, j is the angular momentum of the emitted particle, h, J i=hd J i is the entrance strength for producing bound p-h states of spin J, h, "j NJ (U)i is the escape width, h, # MJ i is the damping width, and h, NJi is the total width. The

2 escape and the damping widths are factorized by X and Y functions, h, NJ i = X NJ Y N (E), where the Y function contains possible phase space for the transition, and the X function contains the possible angular momentum coupling and a radial overlap integral I(j ;j ;j 3 ;j) between initial and nal states of interaction. The overlap integral with a zero-range interaction is dened as I(j ;j ;j 3 ;j)= Z 3 r3 V u j (r)u j (r)u j3 (r)u j (r) dr r ; () where u j (r) and u j (r) are the single particle radial wave functions for the initial states, u j3 (r) and u j (r) for the nal states, r the radius parameter taken to be. fm, V the strength of residual interaction. When a Yukawa type residual interaction is taken into account[5], the overlap integral is given by Z Z I(j ;j ;j 3 ;j)=v u j (r)u j (r)g L (r;r )u j3 (r )u j (r ) dr dr r r ; (3) where g L is calculated from the modied Bessel functions[6], ( (rr ),= K L+ g L (r;r )= (rr ),= I L+ (r)i L+ (r ) (r r ) (r)k L+ (r ) (r <r ) ; () where, is the range of interaction. ccording to the assumption made by the FKK, the radial wave functions for the bound and the unbound states are constant within the nuclear volume, so that r 3 u B (r) = r; (r <R) (5) R3 and p u j (r) = ktj () 3= r; (6) h where R = r =3, is the reduced mass, k is the wave number of the emitted particle, and T j is the transmission coecient. The unbound wave function carries the single particle state density of free particles inside the nuclear volume V =R 3 =3, Z R ju j (r)j dr = () 3 V k h T j c (E c )T j : (7) To calculate the overlap integral with realistic wave functions, the unbound wave function is replaced by a distorted wave[7] normalized in unit energy, j (r) = p k () 3= h i k n H j (r), S j H j (r) o exp i`; (8) where H j (r) =G j (r)+if j (r) is the outgoing-wave Coulomb function, S j the scattering matrix element, and ` the Coulomb phase shift. The bound wave function is calculated with a harmonic oscillator or a Woods-Saxon potential. The quantum numbers and the binding energies of the bound states are determined according to the spherical Nilsson model. These wave functions for ` =,, and 3, are shown in Fig.. They are the radial wave functions of s =,p =,d 5=, and f 7= states in the Woods-Saxon potential of V = 5 MeV, V so = 7 MeV, r =: fm, and a =:7 fm, and the harmonic oscillator with h! =,=3. Usually there are several congurations for a possible transition for a given angular momentum transfer. For example, both I(s = ;p = ;d 5= ;f 7= )

3 .3. p/ d5/. u(r)/r [fm -3/ ] -. f7/ s/ -. Constant Bound Wave Harmonic Oscillator Bound Wave Woods-Saxon Bound Wave Radius [fm] Fig.: Comparison of the shell model wave functions and the constant wave and I(s = ;p = ;d 3= ;f 7= ) are possible for the case above. These overlap integrals are averaged to give an appropriately averaged matrix element. The overlap integrals for this conguration are, I B =V =:6 for the constant wave, 3:38,5 for the harmonic oscillator, and :,5 for the Woods-Saxon potential. These values strongly depend on a choice of the interacting particles and holes, but the averaged values over various congurations are almost independent. The averaged overlap integrals are, I B =V =5:8,3 for the harmonic oscillator, and 7:7,3 for the Woods-Saxon. The constant wave function approximation generally overestimates the overlap integrals of not only a bound/bound conguration but also a bound/unbound conguration. This overestimate cancels in the ratio of the widths h, "j NJ (U)i and h, # MJ i to the total width h, NJi, and one obtains simple estimates for the ratio regardless of details of the interaction[8]. The approximation has an advantage for the calculation of a composite system decay rate because it contains the ratio only, however it is invalid if one calculates the entrance strength microscopically because it is proportional to the width of the p-h doorway state. 3. The Entrance Strength Function From Eq.(), the emission and the damping probabilities can be calculated regardless to the two-body residual interaction V, since V cancels in the ratio of the emission and damping widths to the total width. The entrance strength still holds V and one can estimate the strength of V if the entrance strength is calculated microscopically[9], h, Ji X j j 3 Q hd J i =()!(; ;E) (Q + )(j 3 +)F (Q)R (j 3 ) I (j ;j ;j 3 ;j); (9) Qj 3 3

4 Q K j j K j j 3 S Fig.: The angular momentum coupling scheme for the entrance channel. The incident particle j is captured in the single particle orbit j, creating the particle-hole pair j and j 3 where the angular momentum coupling scheme is dened in Fig.,!(; ;E) is the p-h state density at the excitation energy E, and F (Q) is the angular momentum density: X ; () F (Q) = X j j (j + )(j +)R (j )R (j ) j where R(j) is the spin distribution of the state. Chadwick and Young[] found that the entrance strength can be evaluated by the optical model transmission coecients corrected by a factor R MSC =! B (; ;E)=!(; ;E), which is the fraction of ux into the bound p-h state. The entrance strength becomes h, Ji hd J i = RMSC T J : () Figure 3 shows the calculated strengths for MeV neutron-induced 93 Nb reactions (multiplied by (J +)=k to give an initial p-h state formation cross section). The distorted wave and the transmission coecient are calculated with the Walter-Guss' global optical potential[]. The single-particle state density parameter g is taken as g = =3 MeV,, and the pairing energy correction =. The solid line is calculated according to the coupling scheme in Fig., and the total reaction cross section is normalized to the value given by Eq.(). t the microscopic calculation, the particle and the hole states which obey angular momentum and energy conservation are included. It restricts the possible nal states, and results in small cross sections for large J. The initial p-h state formation cross section is proportional to V when Eq.(9) is employed, and it is possible to estimate V roughly if one compares the cross sections given by Eq.(9) and those by Eq.(). Figure shows the p-h state formation cross sections for neutron-induced 93 Nb reactions as functions of the incident energy. The solid line is calculated from Eq.(9) with V =: MeV, and the dotted line is Eq.(). The value of V was chosen to give the same cross section at MeV, and it is larger than the value of 5 MeV obtained by Bonetti, et al.[]. j Q

5 If one assumes the constant wave for the entrance strength calculation[], it yields large overlap integrals as indicated in the previous section, and results in small V. Only 5 kev of V gives the same strength at MeV. The p-h formation cross section with the range of fm is shown in Fig. by the dashed line. The eect of inclusion of the nite-range correction is very large, but it can be compensated if one adjusts V appropriately. The calculated cross sections are about 3% of the zero-range results, and 9 MeV of V gives almost the same cross sections.. Conclusion multistep compound process calculation program \cmc" has been developed to calculate an overlap integral microscopically. n entrance strength of the initial MSC process was calculated for 93 Nb+n( MeV), and it gave a rough estimation for V of about. MeV for a zero-range interaction, and 9 MeV for a Yukawa interaction with the range of fm. References [] H. Feshbach,. Kerman, and S. Koonin, nn. Phys., 5, 9 (98). [] R. Bonetti, L. Colli-Milazzo, and M. Melanotte, Nuovo Cimento 3, 33 (98). [3] R. Bonetti, and M. B. Chadwick, \ computer code to calculate multistep compound reaction cross sections according to the theory of Feshbach, Kerman and Koonin," unpublished, Oxford (99). [] R. Bonetti, L. Colli-Milazzo, and M. Melanotte, Phys. Rev., C7, 3 (983). [5] R. Bonetti and L. Colombo, Phys. Rev., C8, 98 (983). [6] M. B. Johnson, L. W. Owen, and G. R. Satchler, Phys. Rev.,, (966). [7] G. R. Satchler, Nucl. Phys., 55, (96). [8] E. V. Lee and J. J. Grin, Phys. Rev., C5, 73 (97). [9] R. Bonetti, M. B. Chadwick, P. E. Hodgson, B. V. Carlson, and M. S. Hussein, Phys. Rep.,, 7 (99). [] M. B. Chadwick and P. G.Young, Phys. Rev., C7, 55 (993). [] R. L. Walter and P. P. Guss, Proc. Int. Conf. Nuclear Data for Basic and pplied Science, Santa Fe, p.79 (985). [] G. rbanas, M. B. Chadwick, F. S. Dietrich, and. K. Kerman, Phys. Rev., C5, R78 (995). 5

6 Partial MSC Cross Section [mb] 5 5 zero range, V =. MeV Transmission J Fig.3: Comparison of the partial cross sections calculated from Eq.(9) and Eq.() MSC Cross Section [mb] zero range, V =. MeV µ= fm -, V =. MeV Transmission E n [MeV] Fig.: Comparison of the total MSC cross sections calculated from Eq.(9) and Eq.() 6