The theoretical physics                                                             

                         Assoc. Prof. PHD Mirteymur Mirabutalybov  M.

                         Azerbaijan State Oil Academy, Baku Az1010, Azadlyg 20.

              Analytic Distorted-Wave Approximation for Intermadiate

                             Energy Proton Scattering Calculatons

The increased interest to studies the structure of nuclei by elastic and

inelastic scattering of nucleons is due to the numerous precise experimental data on  a number of nuclei at different energies and large transferred momentum. By getting a simple and accurate analytical expression for the scattering amplitude of the processes, a lot of important information about the structure of nuclei can be drawn from these data. To study the properties of highly excited states of nuclei by inelastic scattering of protons, considering the energy loss we use the scattering amplitude obtained in analytical form by the method of distorted waves in [1].

We write the differential cross section in the form:


                                                 (1)

                                
Resonant excitation is considered here in Breit – Wigner form, where the energy loss is equal to the energy difference between the incident and scattered particles.

The nuclear form factor    obtained in [1] has the following form:
                ,                            (2)

where                      

                                                                     (3)
Function , arising on account the distortions in the incident and outgoing

waves, has the form:

                                    (4) 
         Explicit expressions of potential in the center of the nuclei - , the parameters , and  , depending on the density distribution of nucleons in nuclei, as well as a parameter  (the slope of the diffraction peak), which is a part of the amplitude of the free NN – interaction, are given in [1].

To calculate the integral (2) we choose a coordinate system in which   and designating, the impulse transmitted to the core (considering energy loss of the incident proton) is written:

                   (5)

      Angle of scattering  and deflection angles of the incident () and the scattered particles relatively oõ -axis in three-dimensional coordinate system

 (Fig.1) are related as follows:

                                                             (6)

                                                                                    (7) 

      After integrating over the angles, the form factor is reduced to one-dimensional integral
                                  ,                                                        (8)

where

                             ,                                  (9)

                         ,                   (10)                                                                     

                      ,                                                              (11)

                                                  (12)

                                                   (13)

                                                    (14)

   and

                                                                            (15)

 

 

 

 

 

 

 

 

 

 

 


Fig. 1. Impulses of incident () and scattering ()   particles in three-  

Fig. 1. Impulses of incident () and scattering ()   particles in three-   dimensional coordinate system with the transfer impulse .

 

The final expression of the differential cross sections for inelastic scattering of nucleons on nuclei, we write in the form:

                    ,                                (16)    

where

                                                                                            (17)

  - is the cross-section of nucleon - nucleon scattering in nucleus.

To study the dependence of the cross section on the final energies of the scattered particles, the general expression of the double differential cross section is written:

                        (18)

Using the quantum hydrodynamic model of the nucleus [2], we investigate the properties of highly excited states of nuclei in the energy region of giant resonances.

         In order to consider in the excited nucleus the connection of a giant resonance with the vibrations of the nuclear surface, we combine the low-energy
 collective degrees of freedom with the high-energy one. The interaction between these motions is very strong, so it can significantly affect the structure of the giant resonances [2]. Let’s present the proton density of the excited nucleus as the sum of the equilibrium proton density  and the density of fluctuations, responsible for the giant resonances  extending from the center to the surface, and vibrations of the nucleus surface  as:

                                                                  (19)

         According to the collective model  can be written as:

                                                                      (20)

and the of fluctuations density on the surface of the nucleus we present in the form of an expansion in the collective coordinate   -:

                                                              (21)

Values   are normalization factors which are determined from the normalization conditions:                                                

                                                              (22)

         As shown in [2], the potential velocity is a solution of equation

                                 ,                                                        (23)

which can be represented as an expansion 

                                                              (24)

         Collective oscillations of protons density respectively neutrons lead to a harmonically varying deformation of proton substance near the initial spherical equilibrium shape   ()  

                                                             (25)

Using condition

                  ,                                                     (26)    

we find the relation between the coefficients  of the expansion of the  potential of velocities with  coefficients ,  which define the shape of the density distribution of the nucleons on the surface of the nucleus

                                                                                      (27)

         Nucleons density distribution in the ground state of the nucleus  we choose in the form of the Fermi - functions:

                                                                                       (28)

 To reveal the surface effect, the equilibrium density is expressed in the form of two terms [3]:                

                                                              (29)

 where  - a step function,  with and  under , and -derivative of  - function.

         Low-energy spectra of spherical nuclei are often the  typical spectra of almost harmonic surface vibrations quadruple type [2]. Therefore, in expression (21), limiting by the term  for the transition density, describing the quadruple oscillations of the nuclear surface, we obtain:

                                                                            (30)          

  and for the so-called mass parameter  in general form, we get

                 ,                                             (31)

 At that the excitation energy is:

                                                                                                  (32)                            

Here, the stiffness coefficient is determined by the expression

                      ,                                             (33)

where -   coefficient of surface tension of the core [2].                     

         Thus, after integrating (9) for the form factor responsible for the quadruple oscillations of the nuclear surface, we have

                ,                                                   (34)

   where

                                                           (35)                           

Then the form-factor responsible for the giant resonances takes the form
                                            (36)

         This theory has been applied to the inelastic scattering of protons with incident energy 800 MeV on nucleus. In this case for the nuclear radius and the thickness of the surface layer characterizing the distribution nucleons in the ground state, we used the values derived from the elastic scattering of electrons (,).
         Comparison of the obtained results for the double-section in the energy
dependence of the scattered protons with the experimental [4] and theoretical data obtained by Glauber approximation method [5], at scattering angle, are shown in Fig. 2.

At this scattering angle the energy loss of incident protons is ~ 45 MeV. This means that at this angle of scattering and these energy losses of incident protons the giant dipole and quadruple resonances with energies and widths of excitation  MeV and  MeV;  MeV and  MeV, as well as vibration of the surface of the nucleus with energy -  MeV and width MeV may appear in the nuclei.

 

 

 

 

 

 

 

 

 

 

 

 


          Fig.2. Dependence of the double differential cross section at energy of
incident protons 800 MeV, scattering angle  for  nuclear   on the energy of the scattered protons, solid line-the obtained results, points - experimental data [4], and a bar line –  the theoretical calculations obtained in [5].

        

          The parameter characterizing the mean-square deformation of the excited nucleus is determined by the expression

                             ,                                             (37)

for which at the parameter of rigidity MeV we get .
         As seen from Fig. 2, the calculated theoretical cross section correctly predicts the location of excitation in the nucleus.

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Fig. 3. The dependence of the differential cross section on the scattering

 angle protons in the nucleus at  MeV for the dipole and quadruple giant resonances, as well as for quadruple  vibration of the nuclear surface. Solid line - received results, points experimental data [6], bar-line results calculated in DWBA [6].

 

However, for all values of the energies of the scattered protons double differential cross section is somewhat underestimated. In addition, the value of the dynamic deformation when compared with obtained from the experiment is a little underestimated, too. Apparently, this is due to the fact that we used nuclear model which is still not quite perfect.

         Besides, the calculation of differential cross sections of giant
dipole and quadruple excitations   was held   as well as the cross sections of the quadruple vibrations of the nuclear surface at small scattering angles. The results of calculations in comparison with experimental and theoretical data derived in the distorted-wave Born approximation (DWBA) [6], are shown in Fig. 3.

 As seen from this figure the curves obtained DWBA, and the results of the present work coincide. However, at certain scattering angles corresponding to maxima cross sections, discrepancy appears.

                                               Conclusion
         By the method of distorted waves in the analytical form the expression for the amplitude of inelastic scattering of nucleons by nuclei was obtained. Applying this theory of scattering for studying excited states of nuclei we assumed that the vibration of the nuclear surface is a consequence of the decay of giant multiple resonances arising in the center of the nucleus. This allowed expressing in terms of collective coordinates the fluctuations of density and their frequencies on the base of the collective model of the nucleus, the degrees of freedom the quadruple deformation of the surface of the nucleus, described by collective coordinates.

         By comparison of the calculated double differential cross sections with
the experimental data at scattering angle of protons   in the nucleus, the losses of energy were defined.

         The angular dependence of cross sections of highly excited giant dipole and quadruple excitations, as well as low-energy excitations with quadruple surface vibrations of the nucleus was studied.

         Analyzing the results we revealed that the expression obtained for the
scattering amplitude was very sensitive to the nuclear parameters in the ground state, what’ll allow, investigating the important properties of excited nuclei applying more sophisticated nuclear models.


                                               Literature:
1. Mirabutalybov  M.M. Rus. Ac. Sci. Yad. Fiz. 67, 12, 2178 (2004).

2. Eisenberg I., Greiner W. Models nuclei. M. 275 (1975).

3. Bohr A., Mottelson B., Structure of atomic nucleus. M. 161(1971).

4. Chrien R.E. et.al., Phys. Rev. C.21, 1014 (1980).

5. Esbensen H., Bertsch G.F., Phys. Rev. C 34, 1419 (1986).

6. Adams G.S., Cook D.B. et al. Phys. Rev. C 33, 2054 (1986).

 

                                 Information about authors

1. Mirabutalybov Mirteymur Mirkyazym  (1949).

4. Physics Department, Azerbaijan State Oil Academy.
5. Assoc. Prof. PHD
6.Adres: AZ1007, Az. Rep, Baku, Azadlyg 90, 35.
7. E-mail
    mmmteymur@yahoo.com
8. tel. 440-38-85