Haque, Sanaul

Haque, Sangita

Rahman, A

Rahman, S N

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Available at: http://www.ictp.trieste.it/˜pub_off IC/2004/36   United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency  THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS    PROTON-NUCLEUS INTERACTION AT LOW AND INTERMEDIATE ENERGIES   S. Nasmin Rahman Industrial Physics Division, B.C.S.I.R., Dhaka, Bangladesh,  Md. Sanaul Haque, Sangita Haque Department of Physics, University of Dhaka, Dhaka, Bangladesh  and  Md.A. Rahman1 Department of Physics, University of Dhaka, Dhaka, Bangladesh and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.     Abstract  The elastic scattering of protons from a number of nuclei between 12C and 208Pb is the subject matter of the present study at the projectile energies 30.3, 66.5 and 1044 MeV within the framework of the generalized diffraction models of Frahn and Venter. The best fit parameter values, the cut-off angular momentum T, the rounding parameter ∆ and the real nuclear phase shift µ are obtained from the elastic scattering analyses of the entrance channel angular distributions. The interaction radius R, diffuseness d and the reaction cross section σr , have been estimated from the best fit parameters. Energy dependence of T, ∆ and σr/πR2 and mass dependence of R are discussed. Finally, the inelastic scattering of protons exciting to the lowest 2+ collective states in 42,44,48Ca, 48Ti and 148,154Sm and 3- collective state in 40Ca are studied to check the validity of the derived elastic scattering parameters. The deformation parameters β2 and β3, so extracted are comparable with the available values in literature.      MIRAMARE – TRIESTE June 2004                                                   1 Regular Associate of the Abdus Salam ICTP.  21. Introduction  The nuclear projectiles n, p, τ, 3He, 4He and the like are strongly absorbed by the target nuclei. The diffraction model or the so-called strong absorption model (coined as SAM) was initiated by Frahn and Venter1) to study the interaction of hadrons and various other nuclear probes with target nuclei as an alternative to the optical model. The simplicity of the approach is that the elastic scattering phenomena become easily describable without any knowledge of the absorption mechanism. The scattering function ηl of the model is expressed as a function of the angular momentum of the projectile, thereby avoiding the concept of potential as used in the optical model approaches. The scattering function ηl can be made complex to account for the non-elastic processes. The model under discussion has been used by some of the present authors for the description of the interaction of pions, kaons, nucleons, deuterons, 3,4He and heavy ions with various target nuclei2). We, in the present communication, study the elastic scattering of protons from several nuclei between 12C and 208Pb at projectile energies 30.3, 66.5 and 1044 MeV 3-5). Angular distributions data for the inelastic scattering of protons leading to the lowest 2+ collective states in 42,44,48Ca, 48Ti and 148,154Sm5) and 3- collective states in 40Ca are then studied using the best fit SAM elastic scattering parameters and the corresponding deformations parameters are determined.    2. The SAM formalism  The strong absorption model of Frahn and Venter1) is used for the calculation of the differential cross section. The scattering function ηl of the model is parameterized through the angular momentum of the projectile and an analytic expression is attained for differential cross section of the elastic scattering making suitable approximations6). The parameters of the model are T, ∆ and µ for the reproduction of the elastic scattering cross section. The first two parameters are related semi-classically to the interaction radius R and surface diffuseness d respectively through the relations:    T  =  ( ) 2/121 KRnKR −  (1)  and   Λ = Kd  − KRn1 ( ) 2/121 −− KRn  (2)  The strong absorption model form of ηl was given by Potgieter and Frahn7) in the generalized scattering amplitudes of the adiabatic distorted waves, a pioneering work of Austern and Blair8) and an analytic expression for the inelastic scattering amplitude is obtained as the first derivative of the elastic scattering amplitude in terms of the parameters, namely T, ∆ and µ, all fixed from the elastic scattering. The fourth parameter, the normalization constant, of theory to experiment, is the only free parameter proportional to the reduced matrix element of the inelastic scattering interaction. The deformation parameter βL of different multiple modes and connected to reduced matrix element, can be determined from the collective excitation of the target nucleus.    3 Fig.1. Elastic Scattering of protons from 12C at Elab = 30.3 MeV.    Fig.2. Elastic Scattering of protons from 16O at Elab = 30.3 MeV.    43. Results and Discussion  3.1 Elastic scattering analyses  The best fit SAM parameters of the elastic scattering are summarized in Table 1.   Table  1.  The  SAM  parameters for the elastic scattering of protons.  SAM  parameters Derived parameters  Nucleus     Ep (Mev)    T     ∆ µ/4∆   R (fm)   r0  (fm)  d (fm)   σr (mb) σr/πR2 12C 30.3       3.75 0.01 0.0 4.60      1.35 .01 600 0.94 16O          30.3 4.60 0.025 0.01 4.27 1.40 0.001 518 0.91 40Ca 30.3 6.30 0.025 0.0 5.87 1.32 0.004 906 0.84 56Fe 30.3 6.70 0.02 0.0 6.33 1.33 0.001 1018 0.81 59Co 30.3 6.90 0.025 0.0 6.52 1.33 0.021 1075 0.81 58Ni 30.3 6.80 0.025 0.01 6.46 1.33 0.021 1045 0.80 60Ni 30.3 6.90 0.03 0.017 6.54 1.33 0.026 1076 0.80 62Cu 30.3 7.1 0.35 0.043 6.73 1.35 0.30 1248 0.88 120Sn 30.3 8.35 0.025 0.01 8.29 1.39 0.021 1546 0.72 208Pb 30.3 10.0 0.25 0.10 10.53 1.52 0.203 2303 0.70 148Sm 66.5 12.5 0.73 0.068 7.76 1.23 0.41 1765 0.93 154Sm 66.5 12.75 0.75 0.017 8.0 1.22 0.42 1837 0.98  Comparisons between the experimental angular distributions and the SAM - predictions in some typical cases are made in Figs.1-5. Measured angular distributions display diffractive oscillations in heavier nuclei which wash out as one moves on to lighter nuclei. It may be mentioned that both the elastic and inelastic scattering experimental data are digitized from various refs.3-5). The  quoted uncertainties range over a  value of (5–7) % in the case of elastic scattering and the uncertainties in the case of inelastic scattering range over a value of  about (15-20)%. A reasonably good description of the elastic scattering is given by the model. The fit is generally very poor in very light nuclei particularly for 12C at low energies. The description to the data improves as the incident energy increases. This is evident in the description of the proton elastic scattering angular distributions from 40,42,44,48Ca and 49Ti a 1044 MeV. It is observed that the agreement between theory and experiment improves gradually as both the target mass and the projectile energy increase. The poor fit of the model to lighter nuclei at lower projectile energies and the perfect matching between theory and experiment for higher target mass number and at higher projectile energies are in conjunction with the fact that the SAM conditions are (2 π T> >1, ½ π (∆/T) << 1) are less satisfied in the former case while they are satisfied in the latter case respectively.    5 Fig.3. Elastic Scattering of protons from 40Ca at Elab = 1044 MeV.   Fig.4. Elastic Scattering of protons from  48Ca   at Elab = 1044 MeV.   Table  2.   The SAM parameters for the elastic scattering of protons at 1044 MeV.  SAM  parameters Derived parameters  Nucleus         T                    ∆               µ/4∆           R             r0            d              σr            σr/πR2 (fm)         (fm)        (fm)         (mb) 40Ca 33 5.25 0.097 4.8 1.08 0.76 1003 1.39 42Ca 34.1 5.40 0.095 4.95 1.11 0.78 1067 1.38 44Ca 34.4 5.40 0.095 4.92 1.10 0.78 1081 1.38 48Ca 35.5 4.80 0.096 5.15 1.14 0.72 1098 1.32 48Ti 35.75 5.20 0.096 5.18 1.12 0.75 1134 1.36   We then determine the quantities like the interaction radius R, the surface diffuseness d and reaction cross section σr. These are tabulated in Tables 1 and 2 respectively. The values of R and d are obtained from eqns. (1) and (2) and σr is determined from the expression1):   σr =  π T2/K2  [ 1 + 2 ( ∆/T ) + π2/3 ( ∆/T )2 – 1/3 ( π/∆ )2 ( ∆/T ) 2] (3)   6The standard nuclear radius parameter r0 (= R/A1/3) from 30.3 MeV data analyses yields a value ≈ 1.37 fm and that from 1044 MeV data analyses gives a value ≈ 1.11 fm. A somewhat smaller value of r0 at much higher energy is consistent with the smaller de Broglie wave length than at the 30.3 MeV data analysis. The least squares method furnishes us with the following relation for the interaction radius:    Fig.5. Elastic Scattering of protons from  154Sm at  Elab = 66.5 MeV.   R = 1.6634 A1/3 + 0.20465     for 30.3  MeV  and R  =  1.6094 A1/3 -  0.6805     for   1044 MeV  The diffuseness d of the nuclear surface assumes a negligible value up to 0.30 fm for the present target mass region 12C to 208Pb at 30.3 MeV projectile energy while d remains practically constant at the value =  0.758 ± 0.17 fm for the mass region A= 40- 48 at the proton energy 1044 MeV. The mass number dependence of reaction cross section σr, as given by the model, was then studied. Reaction cross section σr is dependent both on target mass as well as on the projectile energy. Reaction cross section increases with the increase in target mass number while the projectile energy remains fixed. This fact is substantiated for 30.3 MeV as well as for the 1044 MeV as shown in Tables 1 and 2 of SAM parameters. The quantity 2Rr πσ  is perhaps more meaningful than the quantity σr itself as the former remains fairly constant for various target masses at the same projectile energy. The fairly constant value of the quantity 2Rr πσ  is evident from the present study (Tables 1 and 2). The least squares relations for reaction cross section σr are as follows:  σr = 0.2044 A2/3 + 3.3684      for 30.3  MeV and σr =  0.5448 A2/3 + 25.62      for 1044 MeV  The A2/3 dependence of σr indicates the strong surface absorption situation.       73.2 Inelastic scattering analyses  The inelastic scattering angular distributions of 65 and 1044 MeV4,5) leading to collective states, such as the lowest 2+ and 3- states in several nuclei, have been undertaken using SAM parameters obtained from the description of the corresponding  elastic scattering channels (Tables 1 and 2 ).  Table 3. Quadrupole and octupole deformation parameters from inelastic scattering of protons.   Deformation parameter βL Nucleus Ep (MeV) Ex (MeV) Jπ (a) (b) 42Ca 1044 1.524 2+ 0.205±0.08 0.209b) 44Ca 1044 1.157 2+ 0.216±0.09 0.218b) 48Ca 1044 3.383 2+ 0.0952±0.04 0.0854b) 48Ti 1044 0.983 2+ 0.2284±0.06 0.24b) 148Sm 66.5 0.055 2+ 0.17 0.12c) 154Sm 66.5 0.082 2+ 0.25 0.253c) 40Ca 1044 3.737 3- 0.31042±0.08 0.33-0.36d)   a)  Present Analysis  b)  Summary from electromagnetic measurements Ref.9)      c)  Ref.5)   d)  Range of β values as quoted in Ref.10)    The fits to the inelastic scattering angular distribution data are displayed in Figs. 6 and 7. The deformation parameters βL are obtained from the relation:  δL  = (T/K) βL  where, δL is found from normalizing the theoretical cross sections to experimental ones in the respective nuclei. The deformation parameter values β2 and β3 extracted are summarized in Table 3. The βL values from previous other works are also presented in the Table 3 for comparison.      8 Fig.6. Inelastic Scattering of protons from 154Sm   leading to 2+ collective state in 154Sm.   Fig.7. Inelastic Scattering of protons from 40Ca leading to 3- collective state in 40Ca.   The general characteristics, such as diffractive oscillations in the angular distributions, practically all of them are reproduced by the model throughout the whole angular range. Besides minor discrepancies in the description of the experimental angular distributions of Sm isotopes by the present model, all the other states such as 2+ states in 42,44,48Ca and 48Ti and 3- state in 40Ca are fairly well described by the model in comparison with the quality of fit obtained by models like DOMP / CCBA methods5). The latter approaches are more complicated and have many more degrees of flexibility at their disposal. The deformation parameter values βL for the lowest 2+ and 3- collective states are shown in Table 3 along with previous other works. The values of the present work compare reasonably well with other values studied through DOMP / CCBA analyses and with adopted values 9,10).  4. Conclusion  The three parameters version of the simple generalized diffraction model of Frahn and Venter is fairly successful to account for the elastic and inelastic scattering data from several target nuclei between 12C and 208Pb at different projectile energies such as 33.3, 66.5 and 1044 MeV. The fit quality is generally poor particularly at larger angles, at lower energy and lower target mass number (cf. Figs.1 and 2). The quality of fit to the angular distribution is excellent at high energy i.e. at 1044 MeV. The SAM is thus successful in  9extracting standard nuclear radius r0 to be ≈ 1.37 fm for the mass A = 12-208 at 30.3 MeV projectile energy and this turns out to be 1.10 fm for 40,42,44,48Ca and 48Ti nuclei at 1044 MeV. The so obtained values of r0   are reasonable and consistent. The mass and energy dependence of reaction cross section are discussed. The quantity σr/πR2 remains fairly constant for various target masses at the same projectile energy, as expected. The deformation parameters β2  and β3 are determined for the available 2+ collective states in 42,44,48Ca and 48Ti and for the available 3- collective state in 40Ca respectively. These values are compared with those obtained from the CCBA method and with adopted values. The values of the present analyses are in excellent agreement with the values from other studies. The SAM parameters obtained are thus reliable and credible in reproducing the inelastic scattering data with a minimum number of parameters as against several of the optical model, DOMP, CCBA and other microscopic methods.   Acknowledgements  This work was carried out within the framework of the Associateship Scheme of the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Financial support from Swedish International Development Cooperation Agency is acknowledged.  References  1) W.E. Frahn and R.H. Venter, Ann.Phys. (N.Y.), 24 (1963)243; W.E. Frahn, in Fundamental in Nuclear Theory (International Atomic Energy, Vienna, 1967), p.1.  2) Md.A. Rahman, H.M. Sen Gupta and M. Rahman, Phys. Rev. C41 (1990) 2305; Phys. Rev. C44 (1991)2484; Md.A. Rahman, S.C. Paul, H.M.Sen Gupta and M.Rahman, Nuovo Cim. A105(1992)851; D.R.Sarker, Md.A. Rahman, H.M.Sen Gupta and M.Rahman, Nuovo Cim. A107(1994)511; S.Soheli, E. Hai, Md.A.Rahman, H.M. Sen Gupta and M.Rahman, J. Phys. Soc. (Japan ), 63 (1991)2572.  3) R.W. Ridley and J.F. Turner, Nucl. Phys. A58 (1964)497.  4) G.D. Alkhazov, T. Bauer, R. Beurtey, A. Boudard, G. Bruge, A.Chaumeax, P. Couvert, G. Cvijanovich, H.H. Duhm, J.M. Fontaine, D.Garreta, A.V. Kulikov, D. Legrand, J.C. Lugol, J. Saudinos, J. Thirion and A.A. Vorobyov, Nucl. Phys., A274(1976)443.  5) A. Guterman, D.L. Hendrie, P.H. Debenham, K. Kwiatkowski, A. Nadasen, L.W. Woo and R.M. Ronningen, Phys. Rev. C39(1989)1730.  6) R.H. Venter, Ann. Phys. (N.Y.), 25(1963)405.  7) J.M. Potgieter and W.E. Frahn, Nucl.Phys. 80(1966)434.  8) N. Austern and J.S. Blair, Ann. Phys., (N.Y.), 33(1965)15.  9) S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor (Jr.) and P. H. Stelson, Atomic Data & Nucl. Data Tables, 36(1987)1.  10) R.H. Spear, Atomic Data and Nucl. Data Tables, 42(1989)55. 

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