Elastic and inelastic scattering of nucleons and light nuclei at intermediate and high energies with in the frame work of multiple scattering theory

EL A ST IC AND IN E L A ST IC SC A T T E R IN G O F N U CLEO N S AND L IG H T N U C L E I A T IN T E R M E D IA T E AND H IG H E N E R G IE S W IT H IN T H E FR A M E

W O R K O F M U L T IPL E SC A TT ER IN G T H E O R Y

Ye.I.Ismatov1, Sh.Kh.Djuraev2, Sh.P.Esanniazov2, S.B.Strigina1, B.K.Paluanov3 1) Institute o f Nuclear Physics, Uzbekistan Academy o f Science, Uzbekistan

2) Termes State University, Uzbekistan 3) Nukus State Pedagogical University, Uzbekistan

The elastic and inelastic scattering of nucleons and light nuclei on nuclei at intermediate and high energies has been studied for a long time. The scattering at such energies has diffraction character and Glauber-Sitenko diffraction multiple scattering theory (GSDMST) [1-6] is successfully applied to analyze the processes. The main assumptions underlying GSDMST are enumerated and the general structure of amplitudes and cross-sections of scattering is analyzed. The way is shown for calculation of cross-section of elastic and inelastic nucleus-nucleus scattering with using the condition of completeness of wave functions in the final state. In terms of GSDMST the amplitude of elastic nucleus-nucleus scattering may be written as

Fe1'A2(<]) = J^ l d p e

X ¥ A 2 \rU r2 - rAl A

)n

«(p;{sıKs2})ri^3//

Û d \

A\ Aj

co(p;{sıl{s2})

= l -

n FID-

°)jk

(p +

s\j +S2k)\

i=\ k-\

( i)

(2)

Fj and are nucleon radius-vectors of nuclei Ai and A2, Sj and s/- - are their projections on impact parameter planes.

Nucleus 6Li, Study of 6Li structure showed for sure, that the most of experimental data can not be explained within the framework of the plain shell model. Dynamic base of calculations is the Glauber-Sitenko theory of multiply scattering (GSDMST) [1-4], which is used successfully for description of interaction between high-energy particles with nuclei. We will calculate the amplitude of scattering of particles by 6Li nuclei assuming that nucleons are in its mass center. In this case the amplitude of elastic scattering of particles on nuclei 6Li is the same as that of

Fe (( q ) = 2 ^ J d p e 'qpa ( p ) ’ (3)

where the profile function in the case of the target nuclei transition from the ground state | 0> to an excited state I f >

® ( P ) =<

1

- n (

.

0 >

(4)

Elastic scattering. Using methods described in [7,10], the final expression for the differential crass-section of elastic scattering is the following

= lF0 ^ 0 (k> q )2 (5)

l d n ) e(

Inelastic scattering. Expression for amplitude of inelastic scattering for 6Li with excitation to 3+T=0 (2.18 MeV) state is obtained in the same way as that for elastic scattering. The only but substantial difference is, that there is the spherical function Y2M(x5) under the integral in (11). Using it’s concrete form [7,10], it is not difficult to obtain an amplitude of the transition into states M= 0, ±1, ±2.

The angular distributions of protons at elastic and inelastic scattering on nuclei were calculated by the expressions: d a

d n ) 3+

=0

I lF0 0 ^

,

)2

Figures 1 and 2 show that the cluster model provides a good accordance with experimental data [7,14].

Fig.1. Differential cross-section of elastic scattering of the protons at nuclei 6Li on two cluster model (c.m.)

1 - Po(R) = 1 - 8/9pR2; 2 - Po(R) = 1 - 8/9ajR2. Numeral calculations by formula (6). The dashed curve - the shell model limiting (x=y=1). The dots are experimental data. [14] for the energy E=152 MeV (light dots) and E=185 MeV (dark dots).

Fig.2. Differential cross section of elastic scattering of protons with E=185 MeV at nuclei 6Li in two cluster model (c.m.) Numeral calculations by formula (7).

1 - Po(R) = 1 - 8/9 pR2; 2 - Po(R) = 1 - 8/9a1R2. The dashed curve - the shell model limiting (x=y=1). The dots are experimental data.

7Li nucleus. Associative effects in nuclei 7Li are pronounced not so well as in 6Li, the most of the experimental data can not be explained in the framework of the simple shell model. Moreover, the one-parameter shell model does not explain satisfactorily the inelastic form- factor at any value of the oscillator parameter. The most proper model, taking into account the discrepancy between the wave functions of 7Li and that of shell model, is the cluster model. It gives good correlation between the data on elastic and inelastic scattering at the same values of wave function’s parameters.

The differential cross-section of elastic scattering is determined by [7, 20]

' d a dQ = 3 { \ 1 2 = l | F1M ^ 1 M ' (kqJ = Jet M M '=-1 2 \Fn (kq)\2 + |F1-1(k,q)|2 + |F oo(k,q)|2 } (7)

The amplitude of inelastic scattering with excitation of the 7/2 T=1/2 (4.63 MeV) state is calculated in the same way as that of elastic scattering. The only difference is that the function

*

Y2M should be under the integral instead of the spherical function Y l m • Using its concrete form, it is not difficult to obtain all the amplitudes of the transition. Because of symmetry it turns out, that only 6 of 21 transitions 1M ^3M ' should be considered.

The differential cross-section of inelastic scattering has the form [7,20]:

f d a \ 1 1 1 -ITT 1 = 3 I I \FM ^ M ' (kq) l dQ J7 /2 - T= - 3 M = -1M =-3 2 = 3 {lF00(k, q)|2 + 2 |f 1 3 ^ kf + |F11(k, q)|2 + 2 2 +1^1,-1Cq,k ^ + |Fı,-3 (q, k ^ + \Fo2(q, k )| 2 2 (8)

The angular distributions of protons at elastic and inelastic scattering on 7Li nuclei were calculated with the expression (9) and (10). As it is seen at figure 3, the cluster model agrees well with the experimental data [16-18].

Fig.3. Differential cross-sections of elastic and inelastic scattering of protons with E=185 MeV on nuclei 7Li (c.m.) in two cluster model

1 and 2 calculations by formulae (6) according to x-0,4 and 0,5. 3-shell model limiting (x=y=1). The dots are experimental data for the energy E=152 MeV (light dots) [15] and E=185 MeV (dark dots) [16,17]. Nucleus 12C. The cross-sections of nucleon and light nuclei scattering on 12C nuclei is described in the framework of the Glauber-Sitenko theory of diffraction nuclear reactions. The wave function of 12C nuclei is presented as a determinant composed of one-particle functions OJm

2

(r,G) ty , that are eigen functions of square total moment J of a nucleon and it’s projection T2

on Z-axis, and also square isospin t of nucleon and it’s projection on Z-axis.

Taking the wave function from [7,10] as that of ground state of 12C nuclei, the amplitude of elastic scattering of protons and a-particles on 12C nuclei will be in the following form

F o ^ o ( q ) = 2 ^ exp_2n

q < r

dpeiqp

N 0

dr\...dr

_r

a

+

-20a* M

These functions are antisymmetrisized over all arguments. Antisymmetrization of ^ s-function over all coordinates is not necessary, because the operators are not dependent on spin and isospin of particles.

The differential cross-sections of elastic scattering of protons and a-particles on 12C nuclei are calculated with the expression (11) and shown in Fig.4.

Fig.4. Differential cross-section of elastic and inelastic scattering of protons with energy E=1 GeV on nuclei

12C (The solid curve is obtained at < r 2 >1/2 = 2.29 fm). Numerical calculations by formulae (11) and (12).

1 - The deformed nucleus (e = 1,35); 2 - The spherical symmetrical nucleus (e = 1). The dots are the experimental data.

Taking the wave functions 0 0 and 0 2M from [7,10], we obtain the amplitude of inelastic protons with excitation of 2 level of 12C:

f0 0 ^ 2 M 0?) = N 0 N 2M

2a

f ik_ ^ \ 2 n J

exp q2 < r2 >/104

•Jdpe*qp -Jdr\...dry2^'s(r^ . r^)-exp

f \ — — J r 2 a2 j j V J J (10) {Q2M + ..)-f , \ | 12 1+ — - J ( - 0MqM Q -m +... V 40a 4 M 2 - E (1 - Y j) j =1

The differential cross-sections of inelastic scattering of 1 GeV-protons on 12C nuclei are calculated with the expression (28) and shown in Fig.4b. The parameters of nucleus and nucleon-nucleon amplitude are the same as those for the elastic scattering. The cross-section calculated in this way is better than that in [21,22], however it is not in a good agreement with the experimental values.

Fig.5 .Differential cross-section of elastic scattering for the a - 12C reaction at 1,37 GeV. The dashed curve is obtained by formula (11). The solid curve is obtained with phase variation [y2 = 2(GeV/c)-2] from the work [9].

The results on the elastic scattering of a-particles, 12C and 40Ca at the energies of 1.37, 1.44, 1.503, 2.4 GeV, calculated by us with the help of the cluster model within the framework of the GSDMST, were compared to the corresponding experimental data [19] and the theoretical results in [9]. For comparing our approach with the calculations in [9], the figure 5 shows calculated and experimental [19] values of the diffraction cross-section of the elastic scattering of a-particle on 12C nuclei at the energy Ea= 1.37 GeV.

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