Safari, Reza

IACS Institutional Repository

\,ulum J. Phys. 79 (1). 89-92 (2005) o te< U P >A local potential with analytical distorted wave approximation model for elasticscattering of ii o^ns from a nucleus^e;'.u SafariDepartment o f N ucl^ r Physics, Facully o f  Physics,Tabriz University, Tabriz,. IranEmail f*&afari<^tabnzu ac irR eceived 29 April 2004, accepted 20 Sepletni>cr 2004lAh.stract . The ordinary equivalent local form o f Kisslinger Rev 98 761 (I955)J type optical potential together with an analyticalIdistorted wave apptoximation (ADWA) is explained to discuss the elastic scattering of pions with intermediate eneigics from a nucleus. C'alculalion o ff 20HianguKir distribution o f  the scattering o f  negative pion on g., Pb using Born approximation gives an acceptable fit tc» the data The possible values for Ithc potential parameters and ADWA parameters are obtained by comparison of data with the theoretical formula|K eyw ords ■ Nuclear scattering, angular distribution, analytical distorted wave approximation, optical potential.|PA (\S  Nos. 24.IO.Eq, 24 10 Ht, 25 80 DjWe have d iscussed p rev iously  the p ion-nucleus clastic scattering for low energy region | l l .  The study of such interaction with incident energies around the range of the first pion nucleon resonance A (1232) has been a interesting subject for many theoretical and experimental nuclear physicists over the past decade. Theoretical studies arc mostly based upon phenomenological local potential models. The reason for the importance o f this approach is that both s and /> -wave components are present in pion-nucleus scattering. Here, we explain the pion-nucleus elastic scattering for the energy range where 1=0,1 partial waves are in resonance by an equivalent local form [1,2J of a Kissliger optical potential (KOP) [3] using the method of Born approximation (BA). But pion-nucleon two bcxly interaction becomes stronger at intermediate energies for larger distances from the center o f the nucleus which causes pions not to penetrate deeper in to the nucleus. Therefore, the elastic scattering cross section would have a considerable increase which is not consistent with the values obtained from KOP. This needs to increase the strength o f the potential. Regarding the spin and isospin degrees of freedom it is possible to relate the scattering amplitude to the complex phase shifts.  ^he real part o f the phase shifts is measured by an observer vhich give the elastic component of the scattering. On the other land, the imaginary part is responsible for inelastic scattering.The imaginary part indicates that this potential behaves like a source or a well for incident flux. These all indicate that the KOP must be altered from the original form for medium energies. For these reasons, wc use the distorted wave impulse approximation method (DWAM) |5] in our calculation and so, the approximated plane waves are used instead of the usual distorted wave functions. T herefore, a sort o f  analytic  d istorted  wave approxim ation [6-8J should be applied which has been successfully used in calculation of other particles from nuclei. A brief review of the theoretical calculation of optical potential (OP) is de.scribed in Section 2. In Section 3, we present elastic scattering formalism using the ADWA model. Section 4 being devoted to the discussion o f results and conclusions.The actual KOP for pion scattering from a nucleus 11 ] is:2 (0(^(/■) + V a ( r ) V ) . ( 1 )The local transformed form of this potential is calculated from a SchrCdinger like equation which is obtained from a Klein Gordon equation by using the Krell-Ericson transformation [4 J :f//. =q  k}<x f  V~a , (V a)~ aV .^2(0 l - a  4 ( l~ a )^  ^ \ - a (2)©2CX15IACS9 0 Rczci Sa fariThe .V and p-w ave coefficients o f  the potential which are denoted by a ( r )  and p{r) respectively, take the following equations :q{r) -  P^(h^pir) -  e„h^Ap{r))-^ A c fr ) ,a{r) ^ 1 ' + cx^{r) ,J + - g a , ( r )(3)(4)wherea ,  (/-) -  4 na ^{r)  = 4 n(CoPnp (r)-e„CfP(r )Ap{ r )) Aandp(.r) = p „ ( r )  + p ,,(/-) , A p ( r ) =  p „ { r ) - p ^ , ( r ) ,  Pnp = 4 p „ ( /- )p ,,( r ) .(5)(6)(7)(8) (9)Here, e„ takes + /-  sign relative to the + /-  charge state ( f^ pion and P pir)  and p„{r) are proton and neutron density distributions of target nucleus respectively. P, and are kinematics constants which depend on pion energy. The value of the potential parameters presentedin Table 1. These parameters are energy-dependent and their values are obtained by fitting the theoretical angular distribution to the experimental data values for different energy values. In eq.(2), we have ignored the second-order .y-wave contributions and take ^ = 1. The second order Coulom b interaction also hasbeen omitted from these calculations. Here, we have used tw, parameter Fermi distribution for proton and neutron:Po.1 + exp r ~P, =Here, / stands for proton and neutron, Po =0.0815 (fm) -  6.4 fm , c  ^= 6.54 fm, a^  ^= a^  ^= 0.545 fm. The RMS i aclu calculated by eq.( 10) is 5.545 fm. To explain the necessary sireng of the local potential and also U) include the spin and isosp degrees o f freedom, one can argue that KOP should ha\c deformed shape asK/,y. = -^ ^ ^ (r /o < /(r)  + « |V a V ) ,2 ( 0whereby, a^ are / dependent parameters. It is straight forwaid show that these parameters are related to the phase shifts i different spin-isospin states. They also vary with energy ar scattering angle. To explain these features, we replace the incidii wave function calculated from the OP with a distorted pla.- waves:H 'f{r)  = N  exp {i{a  + iP )k .r )  . (i:Here, a  and ft take energy dependent values and A' is functions of /*and scattering angle which may take the fo llo u  in form:These parameters a , . and should also be obtainefrom a fit o f the differential cross section data to the theoreiic. formula resulted from the scattering amplitude:Table 1. txx:al Potential parameter values calculated in this work with g — I Note • In each case , the left number is the real and the right number is the imaginary part.r„(M cV ) 1>Q (fm ) //, (fm ) < y (fnP) (fm ') f  0 r ,  (fm*-)162 0 .1 7 9 ;0 .1  17 -0 .1 2 ;0 .0 0 4 0 .0 3 9 .0 .5 5 4 0 .2 7 6 :1 .6 3 0 0 .0 4 3 .6 2 0 2 .2 2 7 :0 .4 7 1180 - 0 .0 8 5 ;0 .0 4 5  -0,1 2 4 ;0  0 0 7 0 .1 4 6 :0  639 0 0 8 2 :0  159 0 .0 3 0 :2 .1 3 8 0 .1 7 2 :4 .5 1 4291 -0 .0 4 3 :0  135 -0.1 1 8 ,0 .0 2 0 -0 .1 0 1 :0 .6 5 1 -0 .0 5 6 :0 .1 2 1 0 .7 2 8 :0 .4 6 9 -0 .5 9 9 .1 0 1 8 3Table 2, ADW A and kinematic parameter values calculated in this work.7)r(McV) a  P  A , k (fm  *) /:^ /fm ‘‘) P162 1 .3 5  -0 .0 2  4 0 08  0 .7 0 .6  1 .353 1 .8 2 6 8 1 .3 2 4 7  1 .1 6 2 3180 1.33  -0 .0 3  4 .5 0 .0 8  0 .0 8 0 .6  1 .4 5 4 1 .9 3 5 5 1 .3 4 4 0  1 ,1 7 2 02 9 \ 1.2  -0 .1 8 5  6 .2 0 .0 8  0 .7 0.6  2 .0 6 0 2 .5 0 1 4 1 .4 6 3 2  1 .2 3 1 6A local potential with analytical distorted wave approximation model etc f i e U ) = -  Je x p  {~ ik , . r) ( r ).91(14)Real and imaginary parts o f the K isslingcr optical potential tor three energies are presented in Figure 1. The strength o f the repulsive real part has a substantial decrease with increasing energy and the attractive im aginary part also becomes weaker when energy is increased. An attractive minimum in the rc^l part IS gradually produced with increasing energy. We have calculated the angular d istribution  for each energy by using e<|. (14) numerically by taking 400 points o f  integration. The restjtits are compared with data in Figure 2. These results show that although there is difference between the theoretical estimates anid data, but altogether, using the local form o f KOP with the ADWA provides a quite successful theory o f  elastic scattering of negatively charged pions from  “^?Pb target. This can be extended to the other target nuclei. Theoretical results are sensitive to the param eters of the optical potential which are obtained by com paring  the th eo re tica l and experim ental differential cross section. In these case, the optimum values for150.... 100 S 1S  5 0 i  0£a -50 -100: -150\2 4 6 10r(fm)(a)the parameters o f the analytic function o f the ADWA are also calculated We think that both o f the cc and P param eters need to show a smooth variations with the pion energy. A clear fact is obvu)us that the incident and em erging wave functions would provide an overlap function that resem bles the overlap of the two actual distorted  waves in the spatial region o f  integration. In doing so, it is clear that, these wave functions should provide a qualitatively good theoretical background for the study o f inelastic scattering o f pions from nuclei.Figure 1, The real and imaginary parts o f  equivalent local potential, U j, for n~ + at (a): 162 M eV , (b):180 M eV , (c):291 MeV energies,"fhe continuous curves arc the real part and the square shape curves are the imaginary part o f  the potential.theta fsc] (deg)F igure 2. Elastic .scattering differential cross section  o f  negative pton, continuous curves, from ^JfPh target using eq.(14) at (a) ; 162 MeV, (b) 180 Mev , (c) :29l MeV energies. Data points are from references [9], [ 10] and [ \ \ ]  respectively.Referencesf I ] R Safari Int. J, Compu. am i Numeric. Analy.sis an d  Appl. 4  N o. I 71 (2 0 0 3 )(2 ] M B  Johnson and G R Satchlcr Ann. Phy.s. 248  134 (1996)92 Reza Safari(3] L S Kisslmger PIm Rev. 98 761 (1995)[41 S H Lee and F Tabakin NucI PIm. A226 253 (1974) 15j F l)i Marzio and K Amos Au.st J. Pkys. 36 13 (1983) [6] T Janus and I E McCarthy PIm Rev CIO 1041 (1974) (7| K Amos and F l)i Marzio PIm Rev. C29 1914 (1984) [8 1 M Krcll and T E 0  Ericson Nurl. PIm B ll 521 (1969)C Olmcr, D F Geesamann, B Zeidtnan, S Chakrawarti, T S H Lee, R L Boudrie, R H Siemssen. J F Amann, C L Morris, H A Thiessen, G R Burleson, M Devereux, R E vSegel and L W Swenson PIm Rev. C21 254 (1980)[10] N Hintz et al, extracted from Nucl.Phy.\ A540 533 (1992)[11] D F Geesamann. C Olmcr, B 2Jeidman, R L Boudrie, G S Blanpicd M J Devereux, G R Burle.son, R E Scgel, L W Swenson and H A Thiessen Pliyv Rev. C23 2635 (1981)

A Local Potential with Analytical Distorted Wave Approximation Model for Elastic Scattering of Pions from a Nucleus

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A Local Potential with Analytical Distorted Wave Approximation Model for Elastic Scattering of Pions from a Nucleus

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