Exciton-Dependent Pre-formation Probability of Composite Particles

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1 Commun. Theor. Phys. (Beijing China) 47 (27) pp c International Academic Publishers Vol. 47 No. 6 June Exciton-Dependent Pre-formation Probability of Composite Particles ZHANG Jing-Shang WANG Ji-Min and DUAN Jun-Feng China Institute of Atomic Energy P.O. Box 275(41) Beijing China (Received August 7 26) Abstract In Iwamoto Harada model the whole phase space is full of fermions. When the momentum distributions of the exciton states are taken into account the pre-formation probability of light composite particles could be improved and the exciton state-dependent pre-formation probability has been proposed. The calculated results indicate that the consideration of the momentum distribution enhances the pre-formation probability of [1 m] configuration and suppresses that of [l > 1 m] configurations seriously. PACS numbers: s Key words: pre-formation probability pre-equilibrium emission 1 Introduction The Iwamoto Harada model is a successful method to give the pre-formation probabilities of light composite particles used in the pre-equilibrium emission rates. [1 4] However the practice calculations indicate that this method overestimates the pre-formation probabilities because the whole momentum space is full of fermions so no restriction exists in the momentum space integration in Iwamoto Harada model. Obviously when an incident particle bombards a target nucleus to form compound nucleus with a certain excitation energy E then the maximum momentum can be determined by p max = 2mE. If this restriction is added in the momentum space integration then the pre-formation probabilities of composite particles could be reduced properly [56] which is called E- dependent pre-formation probability. But this condition only restricts the integration limit in momentum space while it is still full of fermions below p max. In practice there is very rare momentum distribution above the Fermi-surface in an n-exciton state. In order to improve the Iwamoto Harada method the momentum distribution in various exciton states will be taken into account then the exciton-dependent preformation probability of composite particles has been proposed. Once introducing the momentum distribution in Iwamoto Harada model the pre-formation probabilities of [l > 1 m] configurations are suppressed seriously only the [1 m] configuration needs to be considered in low energy reactions. In Sec. 2 the Iwamoto Harada model is introduced briefly. In Sec. 3 the reduced occupation number equation is employed to describe the momentum distributions of each exciton state. In Sec. 4 the description of the exciton-dependent pre-formation probability of light composite particles and the related results are given. A summary is given in the last section. 2 Iwamoto Harada Model In the pre-equilibrium emission process of nuclear reactions the pickup mechanism is introduced in terms of the pre-formation probabilities for light composite particle emissions. In Iwamoto Harada model for the light composite particles such as d 3 He t and α particle the single particles are all in the 1s states of the harmonic oscillation model. The basic idea of the Iwamoto Harada model is that the normalized pre-formation probability of each configuration of light composite particle is determined by the number of the phase square occupied by this configuration. The volume of each phase square is (2π h) 3. The various pre-formation probabilities of light composite particles such as d 3 He t and α particle as well as the unstable cluster 5 He have been obtained. [347] If the mass number of the composite particle is denoted by A b the intrinsic wave functions have the intrinsic degree of freedom of A b 1. For deuteron there is only one intrinsic degree of freedom. In the harmonic oscillation model each intrinsic degree of freedom provides the energy (3/2) hω. The energy conservation relation reads p 2 r m mω2 dr 2 = 3 2 hω d (1) where m is the mass of nucleon. If ε Ab is the observable energy of the emitted composite A b with the energy conservation relation we have ε b = p2 b 2A b m hω b(a b 1) A b ε f B b (2) where the total momentum of A b particle is denoted by p b ε f refers to the Fermi energy and B b stands for the binding energy of A b composite particle in its compound nucleus. The excitation energy is E = ε b + B b. Hence equation (2) is rewritten into the following form: p 2 b 2A b m = E 3 4 hω b(a b 1) + A b ε f. (3) The pre-formation probabilities of the configuration [l m] for light composite particles means that l nucleons are above the Fermi surface and m nucleons are blow the Fermi surface with l + m = A b. For deuteron there are three configurations of [ 2] [1 1] [2 ] and the number of the phase square occupied by [l m] configuration reads F [lm] (ε d ) = C (2π h) 3 [lm] d p r d r. (4)

2 No. 6 Exciton-Dependent Pre-formation Probability of Composite Particles 117 With the normalization condition F [lm] (ε d ) = 1 (5) lm the normalization factor is obtained by C = (4/3) 2. [34] For convenience in calculation the following dimensionless variables are defined as x d = p d p f G = 3 hω 4ε f z = p r Gpf. (6) Carrying out the integration over r in Eq. (4) we get F [lm] (ε d ) = 16 (1 z 2 ) 3/2 z 2 dzd cos β (7) π [lm] where β is the angle between p d and p r as shown in Fig. 1. Fig. 1 The relation sketch of p d p r and p 1 p 2. For d nucleus the momenta of p 1 and p 2 are given by the intrinsic coordination of p r and cos β Let p 2 12 = 1 4 p2 d + p 2 r ± p d p r cos β. (8) x = 1 x d/2 1 4x 2 y = d /4 G G B = p2 f (p2 d /4) p2 r. (9) p d p r Therefore the integration limit of cos β can be divided into three areas [47] (i) 1 cos β 1 (ii) B cos β B (iii) 1 cos β B; B cos β 1. Thus the integration regions of cos β for [2 ] [1 2] [ 2] configurations for different values of x d and z are listed in Table 1. When the normalized pre-formation probabilities F [lm] (ε) of d nucleus are obtained then the pre-formation probabilities F [lm] (ε) of α particle can be easily given by the following formula in which the dimensionless variables are still used 1 1 F lm (x α ) = dxs(x) d cos β R lm (1) 1 where with S(x) = 16 π x2 (1 x 2 ) 3/2 (11) x α = p α p f (12) R 4 = F 2 (x + )F 2 (x ) R 31 = F 2 (x + )F 11 (x ) + F 2 (x )F 11 (x + ) R 22 = F 2 (x + )F 2 (x ) + F 11 (x )F 11 (x + ) + F 2 (x + )F 2 (x ) R 13 = F 11 (x + )F 2 (x ) + F 11 (x )F 2 (x + ) R 4 = F 2 (x + )F 2 (x ) where x 2 x ± = α 4 + 2Gx2 ± 2Gx α x cos β. (13) The pre-formation probabilities of d and α particle of all kinds of configurations calculated by Iwamoto Harada model are shown in Figs. 2 and 3 respectively. Since the pre-equilibrium emissions occur at suitable high excitation energies so the excitation energy region from 1 MeV to 1 MeV is considered in this paper. Fig. 2 The pre-formation probabilities of d. Table 1 The integration regions of cos β. x d z [2 ] [1 1] [ 2] < x d < 2(1 G) < z < 1 (1) 2(1 G) < x d < 2( 1 G) < z < x (1) x < z < 1 (3) (2) < z < x (1) 2( 1 G) < x d < 2 x < z < y (3) (2) y < z < 1 (2) (3) 2 < x d < 2(1 + G) < z < x (1) x < z < 1 (2) (3) 2(1 + G) < x d < z < 1 (1) Fig. 3 The pre-formation probabilities of α.

3 118 ZHANG Jing-Shang WANG Ji-Min and DUAN Jun-Feng Vol. 47 In these calculations the whole momentum space is occupied by fermions as assumed in Iwamoto Harada model. Hence the overestimated pre-formation probabilities of configurations [l > 1 m] are given. Meanwhile figure 2 shows that the pre-formation probability of the configuration [2 ] is larger than that of configuration [1 1] at the excitation energy region over 2 MeV. In particular for the α particle figure 3 shows that the pre-formation probability of the configuration [2 2] is larger than that of configuration [1 3] at whole excitation energy region. However if the momentum distribution is taken into account in the momentum integration of the Iwamoto Harada model the pre-formation probabilities should be changed obviously. Therefore to give the momentum distributions of each n- exciton state becomes the key point. To solve this problem in the next section the reduced occupation number equation is employed with this method the momentum distributions of each n-exciton state could be obtained. 3 Reduced Occupation Number Equation The relaxation process of the single particle occupation number n(ε t) towards statistical equilibrium could be described by the nonlinear reduced occupation number equation [89] which reads n t = V ε [n(1 n + D 2 n ε 2. (14) The diffusion coefficient D accounts for the spread of the occupation number and the drift coefficient V determines their shift behavior. The n(1 n) term explicitly introduces Pauli blocking and causes the equilibrium solution to be the Fermi type n eq (ε) = [1 + exp(β(ε ε f ) 1 (15) where β = V/D with the respective saturation values of the transport coefficients as time goes to infinity. In equilibrium limit the system thus attains a temperature KT = β 1. The constant transport coefficients is used in this paper due to existing analytical solution. The initial occupation number box distribution n (ε) reads n (ε) = [1 Θ(ε ε f + Θ(ε ε 1 )[1 Θ(ε ε 2 (16) where Θ(x) = for x < and Θ(x) = 1 else and ε f is the Fermi energy of the target nucleus. In the case of single nucleon induced reaction as the same in exciton model the constant single particle level density g = A/1 is used where A is the mass number of the target nucleus. So the initial total particle number is N = g n (ε)dε = gε f + 1. (17) The total energy of the fermion system at the initial state is E = g The excitation energy is obtained by εdε = g 2 ε2 f (ε 1 + ε 2 ). (18) E = 1 2 (ε 1 + ε 2 ) ε f (19) when the fermion system reaches the equilibrium state the total particle number can be obtained analytically by N = g n eq (ε)dε = gε f. (2) The closed fermion system conserves the total particle number so from Eqs. (17) and (2) the Fermi energy of the compound system and that of the target nucleus have the relation ε f = ε f + 1 g. (21) The total energy at the equilibrium state is given by E eq = εn eq (ε)dε = g 2 ε2 f + gπ2 6β 2. (22) Based on the energy conservation E = E eq the relation between nuclear temperature and excitation energy is obtained 6E KT = gπ 2. (23) Using the equation g(ε 2 ε 1 ) = 1 for nucleon induced reaction and Eq. (19) the parameters in the initial box distribution can be determined ε 12 = E + ε f 1 2g. (24) Regardless the relaxation procedure only the occupation number distributions of each exciton state are required therefore new time is defined as = Dt. (25) Thus the reduced occupation number equation is rewritten into the form n = β ε [n(1 n + 2 n ε 2. (26) In this case the analytical solution has been obtained [9] n(ε ) = A(ε ) + E(ε ) A(ε ) + B(ε ) + E(ε ) + F (ε ) with ( εf ε + β ) A(ε ) = erf B(ε ) = exp{β(ε ε f )} [ ( ε1 ε β ) ( erf 2 εf ε β erf 2 E(ε ) = exp{ β(ε f ε 1 )} [ ( ε2 ε + β ) ( erf 2 ε1 ε + β erf 2 F (ε ) = exp{β(ε ε f + ε 1 ε 2 )} [ ( ε2 ε β 1 erf 2 (27) where erf(x) is the error function. The time corresponding to the occupation number distribution of the n-exciton state is denoted by (n).

4 No. 6 Exciton-Dependent Pre-formation Probability of Composite Particles 119 As an example A = 56 ε f = 3 MeV and E = 2 MeV corresponding to KT = MeV the relaxation of the occupation number distributions at the exciton number n = and the Fermi distribution are shown in Fig. 4. a little bit weak. The increasing of single particle level density means the occupation number becomes even more rare then the suppression effect becomes stronger. This situation is shown in Fig. 6 with the mass numbers of A = Fig. 4 The occupation number distributions of (n = ) and the equilibrium Fermi distribution with A = 56. Fig. 5 The pre-formation probabilities of d at exciton states n = 3 5 with A = 56. From Fig. 4 one can see that at low exciton states the occupation situation is different from that of the Iwamoto Harada model obviously. The occupation numbers above the Fermi surface are very rare as described by the reduced occupation number equation. Evidently once the momentum distribution is taken into account in the Iwamoto Harada model the suppression effect for the configuration [l m] would be very strong. 4 Exciton-Dependent Pre-formation Probability Based on the reduced occupation number equation the momentum distributions can be obtained for each exciton state. Substituting the momentum distribution into the Eq. (7) the exciton-dependent pre-formation probabilities of light composite particles could be obtained by F [lm] (ε d ) = 16 (1 z 2 ) 3/2 J(n)z 2 dzd cos β. (28) π [lm] The momenta of particles 1 and 2 are determined by Eq. (8) for each of the values of p r and cos β. Then the corresponding energies ε 12 = p 2 12/2m can be given. In this case the momentum distribution function J(n) in Eq. (28) of n-exciton state is given by J(n) = n(ε 1 (n))n(ε 2 (n)). (29) After re-normalization the exciton-dependent preformation probabilities of d nucleus can be obtained. Then substituting the normalized deuteron pre-formation probabilities into Eq. (1) the exciton-dependent preformation probability of α particle can also be obtained. The pre-formation probabilities of deuteron at exciton states n = 3 5 are shown in Fig. 5. Obviously when the occupation number is taken into account the configuration [1 1] is enhanced while the configuration [2 ] is reduced. Since n = 5 exciton state has more particles above the Fermi surface so the suppression effect becomes Fig. 6 The pre-formation probabilities of d at exciton state n = 3 with A = Anyway the comparison of Figs. 5 and 6 with Fig. 2 indicates that once the occupation number is introduced in Iwamoto Harada model the configuration [1 1] becomes the dominant part at low excitation energies. In addition the pre-formation probabilities of α particle have been calculated for exciton states n = 3 5 respectively. The n = 3 exciton state has [ 4] [1 3] [2 2] configurations due to only two particles above the Fermi surface while the n = 5 exciton state has one more configuration of [3 1] due to three particles above the Fermi surface. This situation is quite different from Iwamoto- Harada model. The calculated results are shown in Figs. 7 and Fig. 8 respectively. The results indicate that the pre-formation probabilities of α particle have very different picture from the results in Fig. 3. The configuration of [1 3] enhanced obviously while the other configurations are suppressed seriously at low excitation energy region. The pre-formation

5 111 ZHANG Jing-Shang WANG Ji-Min and DUAN Jun-Feng Vol. 47 probability of configuration [2 2] is smaller than that of configuration [1 3] below 5 MeV. Fig. 7 The pre-formation probabilities of α at exciton state n = 3 with A = 56. Fig. 8 The pre-formation probabilities of α at exciton state n = 5 with A = 56. Figure 8 shows that the configuration [3 1] is always very small in the whole energy region. This result implies that the composite particle emissions in n 5 exciton states with the configuration [3 1] could be ignored in pre-equilibrium processes. 5 Summary The reduced occupation number equation is able to get the momentum distributions of each exciton state. Adding the momentum distributions into the Iwamoto- Harada model the exciton-dependent pre-formation probabilities of light composite particles could be obtained. The including of the momentum distribution suppresses the pre-formation probabilities of light composite particles seriously. In fact the particle number at n exciton state is p = (n + 1)/2 oppositely if all of the momentum space is fully occupied above the Fermi surface then the particle number would be p = ge. Hence in the preformation probabilities of light composite particles with [l m] configuration the suppression factor roughly is estimated by ( n + 1 ) l S [lm] (n). (3) 2gE After re-normalization the pre-formation probabilities of the configurations have been changed obviously in comparison with the Iwamoto Harada model. In a configuration [l m] the more particle number l corresponds to the smaller suppression factor. The calculated results indicated that the [1 m] is the dominant configuration while the configuration [l > 1 m] could be neglected at low energy reactions. In the statistical model UNF code [1] only the configuration of [1 m] is taken into account for the preequilibrium emission process it is reasonable physically. On the other hand for the composite particles like 3 He t even 5 He [7] the similar physical picture should be obtained with the method mentioned above. In general when the momentum distribution is added in the Iwamoto Harada model the pre-formation probabilities of light composite particles could be improved. In middle energy nuclear reactions once the configurations of [l > 1 m] are needed to describe the preequilibrium emissions the correction by the momentum distribution is required to be performed accordingly. References [1] A. Iwamoto and K. Harada Phys. Rev. C 26 (1982) [2] K. Sato A. Iwamoto and K. Harada Phys. Rev. C 28 (1983) [3] J.S. Zhang S.W. Yan and C.L. Wang Z. Phys. A 344 (1993) 251. [4] J.S. Zhang Y.Q. Wen S.N. Wang and X.J. Shi Commun. Theor. Phys. (Beijing China) 1 (1988) 33. [5] J.S. Zhang and S.J. Zhou Chin. J. of Nucl. Phys. 18 (1996) 28. [6] J.S. Zhang Proc. Int. Conf. Nuclear Data for Science and Technology Gatlinburg Tennessee May Vol. 2 and American Nuclear Society (1994) p [7] J.F. Duan L.Y. Yan and J.S. Zhang Commun. Theor. Phys. (Beijing China) 42 (24) 587. [8] G. Wolschin Phys. Rev. Lett. 48 (1982) 14. [9] J.S. Zhang and G.Z. Wolschin Z. Phys. A 311 (1983) 177. [1] J.S. Zhang Nucl. Sci. & Eng. 152 (22) 27.

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