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