Glauber - sitenko screening in elastic and inelastic diffraction of hadrons and light nuclei

GLAUBER - SITENKO SCREENING IN ELASTIC AND INELASTIC

DIFFRACTION OF HADRONS AND LIGHT NUCLEI

^uldashev B.S., 1,2Ismatov E.I., ^azilova Z.F., 2Ajniyazova G.T., 2Kurmanbai M.S., 2Shunkeev K.Sh., 3Tskhai K.V., 2Medeuova A.B.

1Institute o f Nuclear Physics, Tashkent, Uzbekistan; 2State Pedagogical Institute, Aktobe, Kazakhstan;

3State University, Aktobe, Kazakhstan;

The basic reason of usages theory of repeated diffraction scattering for analysis of nuclear - nuclear scattering serves it success in description of processes of hadron - nuclear interactions. At the nuclear-nuclear scattering theoretical analysis we use the same approximations, that at hadron - nucleon scattering [1-15],

In the basic of hadron-nuclear scattering theory in the eikonal approximations there are three physical assumptions: first of all, that during the time of flight of relativistic particle through nucleus the nucleon does not change its position inside the nucleus, which means that we have scattering on the system of fixed scattering centers; secondly, we suggest that the phase shift of impact parameter in the space takes place, thirdly, usage of the completeness of wave functions of target nucleus final states [1-8],

Using previous notations in [1-3, 7, 8, 15], results of scattering cross-section of Ai nucleus on the target nucleus A2 can be present as:

d a \ q , e l

\ d Q J _k=1

f l P.U (r)dP ■ exp [- j [co(b, - s; {5})]+ m (b2- s ; {s} )/, (5 )ttt

(1)

• {;xp [[ co(b, - 5 ; {s}>y* (b2 - s ; {s})7^ (5 ) d s ] -1}

For calculation of cross-sections of quasielastic scattering it is necessary to elaborate approximated procedure calculation. One of probable possibilities consists of simplifications in the decomposition values in series, on so-called multiplicities of quasielastic scattering, mathematically corresponding to the exponent expansion in series in curly brackets of expressions ( 1 )

Section II. Basic problems o f nuclear physics

where d a \d Q j

Z f— r

f — T

p i n ) = j£_ j dbexpiqb p j fa y 3 ^ . Y \ 0)(b - s ; {s})exp[- f a(b - s ; {5})- T(s)d2s ] (4)

2-^ k= \ 1=1

Such procedure is possible, if in the sum (2) the main contribution comes from the low order terms (n=l,2,3,.. As it follows from the analysis of proton-nuclear quasielastic scattering of [7,8], this condition is valid only at small values of transferred momentum. Relative role of the n- multiplicities of quasielastic scattering, corresponding to backing-out from target nucleus of n- nucleons is determined by the factor [12]

s n 1 exp ——^-Bq2,

n \67jB (5)

where a t - total cross section of NN-interactions, B- slope parameter of elastic NN-scattering. From general reasons leads that for the qualitative estimations of the contributions of different terms in the sum (2) one can use relationships (5). After trivial transformations

atm —>oAN, B=BNN^ B 4N, and neglected by the effects of 3 and 4 multiplicities screening the profile function can be presented as a sum of two terms [12,15]

£9, = Y ^û){b-s - Sj) i=l

4

co2 - s - 5,)- co(b - s - s k) i^k

According to (6) an amplitude F(1) can be presented as difference of Fi(1) amplitudes and in the case of ccA- scattering we have

F ,(1) (q, s ) = ---(#) [ db ■ exp iqb cox(b - s ,

{s})-2n J • exp \ û) ( b- s, {.v})7', (,v>/2.v f l Pa (A )dr, i = l (6) and F2(1) (V) F2(])(q,s) = ~ — K(q)\db ■ exp/^^c92(^ - 5,(5})-2 K J • exp[- J"co2(b — s, (s})exp[ - 1ct)(b-s, {.v})/', (s)d2s (8) Let's remind, that in the approximation of rigid projectile nucleus an expression for cross­ sections of all processes of light nucleus Ai scattering on heavy nucleus A2 has the form which coincides with expression for similar values of hadron - nuclear scattering processes but only with difference that hN-scattering amplitude therein should be replaced by AiN- scattering amplitudes. Technically it means that the procedure of composite functions averaging for (0A N (b - s, {5}) operators over Ai coordinates is reduced to their replacement by functions from mean values of these operators, which can be coincideres as profile functions of elastic AiN scattering. In particular, total cross-section of quasielastic scattering in the approximation of hard a- particle is given by expression

where

n(b,b2) = j a M (b, - s ) T ^ s ) d 2s

If, as usual, the density distribution of nucleons in a - particle is chosen to exponential, then the profile function a)aN can be written as linear combinations of Gauss exponents. Partial cross­ sections of n-multiplicities of quasielastic aA - scattering are obtained in [11,12],

Let's consider the process of Ai and A2 nuclei interaction in the result of which Ai nucleus remains in its ground state and A2 nucleus can be arbitrary excited. In this case it is convenient at first to proceed to optical limit on the nuclear to a number of conserved Ai nuclei and to write the profile function of Ai+A2 -> Ai+A2* process in the form [ 1,7,8]

ib) = { ^ / (fi • • Wi (fi • • ) n <0a1 (A,, M ),

k = 1

(A,, M) = [l - exp[- J û)AıN (s, {?})Ta (F - s ) f / 2 (10)

The sum of Ai nucleus scattering cross-sections, accompanying by possible excitations of nucleus A2 with completeness condition for wave functions is given by expression

r d a ^ r p ^ ^ dCl j el -a,el ) a2 J dbxdb2 exp iq(bj - b2}- n k = TA2 i sk ) d s k Ao

• {1 - exp 7 \o) A2n{s,{s})TAx (b - s )r/2s ] - exp \a * a2n{s s ,{s})TAx(bx s ) d 2s

-(11)

- e x p [-\o) A2N{s,{s })TAi (b - s > /2s j - - e x p [ - \ o *a2n ^ ,r ^ Condition for real values of operator co

i b2

A N is equivalent to the neglection of real

coa2n ~ coa2n

part of NN- scattering amplitude in comparison with imaginary part that at high energies is a good approximation. By excluding from the total cross-section (11) the cross-section of elastic scattering, it is possible to obtain the cross-section of Ai nucleus quasielastic scattering on target nucleus A2 in optical limit of atomic numbers for both nuclei in the following form:

d a dQ 2 71, ^dbxdb2 expiq(bx - £2)-Q{b^b2), where Q(bA) = exp— [d 2s ■ f vJ - exp— [d ^{T](b] - s ) + T](b2-s)),T2(s)^ (b, - s);T2(s) + y f e (b2 - s ) ^T2(s) (12)

From (12) one can see that of AiA2 quasi elastic scattering cross-section calculation in optical approximation over atomic numbers of colliding nuclei requires operations of five-time integration that, in general, is a simple problem at the presence of oscillating factor expiqfa - b2) . However on the base of qualitative analysis carried out in [8-14], it is possible to restore cross-sections of quasielastic scatterings by integral characteristics of a , and <q2> types, which are determined in [12] As an example, the cross-section of quasielastic ocA-scattering can be calculated by using formula (12) and reduced values of ^ - ( 0 ) and BaA

dt

parameters and adding to it a cross-section of elastic scattering, calculated in [12], As a result we

2

Section II. Basic problems o f nuclear physics

can obtain the total cross-section. It is possible to compare the calculated results with experimental data [2,8,11], Results of such comparison are shown on the Fig. 1 and Fig.2.

Differential cross sections of a+ 27A1 and a + 64Cu scattering in optical approximation at Pa =17,9 GeV/s in the region of transferred momentum 0 < q < 2 fm'1.

Solid curves I and II describes total (elastic and quazielastic ) cross section. Lower curve correspond to the case, when both contributions from elastic and inelastic scattering are calculated in the framework of optical approximation. Upper curve correspond to the case, when contribution from elastic scattering was calculated in the frame of perturbation theory by shadow effects. Separately (dashed) is shown contribution from quazielastic scattering , calculated in the optical approximation [12,15], For illustration of Coulomb contributions, on the same pictures is presented contribution from elastic scattering , calculated in optical approximation with taking into account coulomb effects (dashed - dotted curve) and without coulomb effects (dashed curve).

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Section II. Basic problems o f nuclear physics