POTENTIAL MODEL FOR NUCLEAR SCATTERING OF PROTON BELOW 30 MeV BY TARGET NUCLEI WITH MASS NUMBER BELOW 70

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1 POTENTIAL MODEL FOR NUCLEAR SCATTERING OF PROTON BELOW 30 MeV BY TARGET NUCLEI WITH MASS NUMBER BELOW 70 Budi Santoso ABSTRACT POTENTIAL MODEL FOR NUCLEAR SCATTERING OF PROTON BELOW 30 MeV BY TARGET NUCLEI WITH MASS NUMBER BELOW 70. Optical potential model for nuclear scattering of proton with the energy below 30 MeV by target nuclei with mass number below 70 is presented. The model is a modification of Bechetti-Oreenless (BO) potential in which new parameters are introduced. Choice of the parameters are discussed so as to fit the experimental data. Calculations for the cross sections are done by integrating the radial wave of the SchrOdinger equation. Cowell's method of integration has been used by applying the matching procedure in a chosen radius to obtain complex phase shifts. From the known phase shifts one can calculate the cross sections. The procedure of calculation has been applied to proton scattering by Fe-58, Ni-58, Cu-60, Cu-65, Zn-64, Zn-66 and Zn-68 in on attempt to estimate the cross sections. The cross sections are required to get the optimum yield in the isotope production using proton bombardment. Results of calculations are compared with experimental data and with calculatied data made by using standard potential. From the result of this study, the modified potential has shown reasonable inprovement. INTRODUCTION Recently development of optical models have attracted nuclear scattering physicists because the models can be easily modified, meanwhile the models represent empirical form of the scattering system. Some of the models commonly used are the Wilmore-Hodgson for atomic mass target, A>40, and neutron energy E<IO MeV, the Bechetti-Greenless for 24<A<209, E=l MeV and Maddland for actinides, E<IO MeV, while for proton the known models are the Perey model for 30<A<100, E<20 MeV, the Bechetti-Greenless model for A<40 and 20<E<50. The model have not accounted for some energy range below 30 MeV and atomic taget with mass number below 70. This range of energy and atomic massjarget are important in isotopic production using proton bombardment. A cyclotron facility (Cs-30) is available in Serpong-Jakarta to fabricate short lived isotopes required for medical scanning. ' Methods of calculation have been available such as the SCA TI, ALICE and ABAREX but their validity are limited to a range of calculations only. The range of interest in which the calculation required for estimating cross section in isotope production using cyclotron Cs-30 in Serpong is not completely available. It is therefore necessary to develop the optical model Pusat Pengkajian Teknologi Nuklir - SATAN 47

2 in the range mentioned above. However the model is first empirically formulated to fit the experimental data at some point of values and subsequently tested whether other values can fit well with the model calculations. Numerical integration applied in finding the phase shifts is the Cowell method although the Runge-Kutta method can also be used. THEORETICAL FORMULATION Assuming a single proton projectile scattering by a single target nucleus can represent the scattering system, one can write down the Schrodinger equation for the radial wave function d2 2 ( +1) -2 Uf (r)+(k - 2 dr r -U(r))uf(r)=O (1) where Ue is radial wave function containig phase shifts information, k2 = 2mE/112 in which m is the proton mass, E its energy, 11 = h/21t, h being the Planck constants; I = the quantum number of angular momentum, I = 0,1,2,...,; U(r) is the optical potential representing the interaction. In general, the emperical form U(r) contains Ue(r) the Coulomb potential, Ur(r) the volume potential, User) the surface imaginary potential, Uy the volume imaginary potential and Uso the spin orbit potential; written as U(r) = Uc (r) + U y(r) + i(us(r) + Uy(r)) +.s Uso(r) Ue(r) = azjz2e2 I r for r> Rc = (aziz2e2 12Rc)(3-r2 1Rc2) for r<rc (2) (3) where Rc is the nuclear radius given by Rc = I,2A 113, A being the atom ic mass number, Zle and Z2e are the charges of the projectile and the target respectively and a = 2mE/112 where RJ = rja1/3, rl and a] are the reduced radius and volume diffusivity coefficients, Vr is a volume real potential strength parameter (4) 48

3 R2 = r2a1/3, r2 and a2 are the reduced radius and surface diffusivity coefficients, Ws is as surface imaginary potential strength. (5) R3 = r3aii3, r3 and a3 are the reduced radius and imaginary volume diffusivity. (6) ~ = r4a1/3,r4 and a4 are the reduced radius and spin orbit diffusivity, Cso is a coupling constant, the value of which near 2 fm2. f. s is the angular momentum and spin interaction. f. S = [j(j + 1) - ( + 1) - s(s + 1)] / 2 in which 1 - sl :::;j :::; + s The values of rj and aj (I = 1,2,3,4) mentioned above are taken to be energy dependent as follows rj = rei) + re(i)e aj = a(i) + ae(i)e (7) (8) where rei), a(i), rei) and ae(i) are the parameters to be chosen. Other parameters that are energy dependent are Vn Ws, Wyand Vso. Ifone refers to Vr = Vh Ws = V2, Wy to V3 and Vso to V4 then the energy dependent is taken to be The combination of chosen values of ri, aj and Vi determine the model in which one refers to. The problem now is to determine the values of the parameters. Starting from the standard model Bechetti-Greenless (B-G) one can try to shift one parameter while others are kept constant. In this way one tries to see the sensitivity of the cross section with respect to the change of this parameter. This procedure is applicable to all other parameters. The shift is only limited to 5% of the related original value. There are 24 independent parameters and therefore contains 24! combinations, if one wants to find all combinations. (9) 49

4 In practice this will be time consuming and impracticable. Combinations are therefore only taken from sensitive parameters. METHODS OF INTEGRATION As has been mentioned before, Cowell method will be used in integrating the radial wave function. Writing the second order defferentia] equation as U" + Fu = 0 (J 0) The solution for un+!(un+lis refer to u at rn+1and rn+1is (n+j)h, h being the step length of integration and n is an integer number) is given according to Cowell 2 - J 2 2 Un+1= [l+(h /12)Fn+d [(2-5h Fn/6)un - {(l+h Fn_J)/12}un.d (1 1) Similarly for u'n_], the solution is given by The phase shifts is obtained by comparing the solution with and without nuclear potentia] presents. If Fi and G, are the regular and irregular solution of the Coulomb wave function then the complex phase shifts 8/ j is given in the form (12) T] (j = exp(8 (j) - (u;;f;... - uf;f;') - i(up;g; - u;;gt) (uuf'r' - u;;f'r) - i(ufig; - u:;gt) (13) The values of Uf j, U't j, Fr, F'b G1 and dl are evaluated at a matching radius rm = mh where rm is chosen to be around 1.25fm. The cross section is given by (J 4) 50

5 where 1;. " = 1-I... 1'1,f " T,+ = 1;; for j = i+ l/2 1;- =~; for j = i- l/2 RESULTS AND DISCUSSIONS The standard model of B-G uses the following parameters rm = 1.25, r, = 1.17, r2 = 1.32, r4 = 1.01, al = a(l) = 0.75, a2 = m*, VII = m* + O.4Z/(A ),VI2 = -0.32, V21 = m*, V22 = -0.25, V31 = -2.7, V32 = 0.22, V41 = 6.2.m* = (A-2Z)/A, Z = target atomic number. The values of rj and aj are in femtometer (fm), while V are in MeV. New parameters using the procedure mentioned above are as follows rl = 1.0 I, r3 = 1.34 and al = Table I shows the improvement over B-G standard model for Fe-56, Ni-58, Cu-63. For Cu-65 and Zn-66, there are some exceptions where the new parameters give less accurate than the B-G model at relatively low energy and for Fe-56 at relatively high energy. CONCLUSIONS Modification of the optical model has been obtained by replacing the Bechetti-Greenless parameters using a parameter shifting procedure. Only three parameters have been taken into account in this study because the change of these parameters give significant effect on the cross sections. Those parameters are related to the volume potential parameters, indicating that the volume interaction is more sensitive to the parameters being used. Improvement is quite significant, while the highest improvement in this example is achieved for Ni-58 target by 14.5 MeV proton as projrctile. Further examination is required to obtain a more consistent improvement in the whole range energy of interest. ACKNOWLEGEMENTS Thanks are due to IAEA Data Section who provides with the experimental data. CalCulations have also been verified by Mr. Silakhuddin as partial requirement for his M.Sc. thesis. 51

6 REFERENCES 1. QAIN, S.M., "Medical Radioisotopes and Nuclear Data INDC-195/GZ", IAEA, (1988) 2. GREEN, A.E.S., SAWADA, T., and SAXON, D.S., "Toe Nuclear Independent Particles", Academic Press, (1968) 3. BERSILON, 0., "Optical Model Calculation and Used of The Computer Code SCA T2", ICTP, (1992) 4. NEA DATA BANK, "OECD", Data Bank, France, 14 Nov. (1992) 5. PEREY, F.G., "Optical Model Analysis of Proton Elastic Scattering in the Range of9 to 22 MeV", Phys. Rev. 131, (1963) 52

7 Table I. Results of calculation using new parameters. Target Energy % 0.4% 5.1% 12.4% % 7.4% 13.7% 21.4% 5.7% 8.~ 8.8 New 1.2% 18.4% 4.2% 10.8% 0.5% 8.2% 26% 9.5% 9.8% 11% 13% 16% 1.6% 10% 15% 1.5% 0.7% 5.0% calculation ± B-G ± 7.9% % % % %719 6% % % % % % 4.8% 1.8% % 1.4% Present Relative (mb) (j parameter errors ((jexp - (jcal) / (jexp * Data obtained from IAEA Nuclear Data Section through private communication. 53