Analysis of 20 GeV proton-nuclei scattering

Nuclear Physics Al87 (1972) 153-160; Not to be reproduced

ANALYSIS

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SCATTERING

P. M. SOOD Physics

Department,

Physics Department,

Panjab

University,

Chandigarh,

S. K. TULI Banaras Hindu University,

India

Varanasi,

India

Received 11 May 1971 (Revised 18 January 1972) Abstract: The semi-empirical model for the elastic and quasi-elastic scattering of nucleons by nuclei in the GeV region, proposed by Steenberg, has been applied for the analysis of data for the scattering of z 20 GeV protons from the light nuclei 9Be and 12C and the medium weight nucleus 27A. The agreement between the model and the scattering data has been found to be fairly reasonable. Also, in the case of light nuclei, the best-fit parameter x concerning the ratio of the real and imaginary parts of the proton-nucleus scattering amplitude agrees well with the ratio y between the real and imaginary parts of the p-nucleon forward-scattering amplitude at this energy.

1.Introduction The differential elastic cross sections from 2.5 to 20 mrad and the total elastic and inelastic cross sections of z 20 GeV protons for 6Li, ‘Li, ‘Be, “C, “Al, Cu, Pb and U have been measured by Bellettini et al. ‘). In this experiment, the momenta of the protons before and after scattering were estimated with an accuracy of + 50 MeV/c. The kinematical analysis carried out by these authors shows that for momentum changes within 50 MeV/c, events having a scattering angle less than I mrad in the proton-nucleus c.m. system are mainly due to the situation when the nucleus remains close to its ground state (coherent elastic scattering). The events at larger angles, instead, can also be due to cases in which one of the nucleons inside the nucleus gets enough recoil energy to leave the nucleus (incoherent or quasi-elastic scattering). The experimental results depict that the elastic scattering differential cross sections for light and medium weight nuclei decrease monotonically with slopes different at small and large scattering angles. At small angles, the scattering shows the characteristic central diffraction peak due to coherent scattering. The slopes at large scattering angles are independent of mass number. This phenomenon at large angles can be interpreted as being mainly due to proton-nucleon quasi-elastic scattering inside the nucleus. The heavy nuclei Cu, Pb and U, on the other hand, exhibit diffraction rings which are typical of coherent scattering. Since the coherent scattering increases with mass number, the contribution of the quasi-elastic process is insignificant for these heavy nuclei. Because there are hundreds of partial waves participating in the scattering process 153

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P. M. SOOD AND S. K. TULI

at this high energy, a direct analysis of the data by integrating the wave equation for some complex potential becomes impracticable. Bellettini et al. ‘) have tried to interpret coherent scattering results by a sharp cut-off black-sphere model and incoherent scattering by the known results of p-p scattering and the impulse approximation. More detailed information about the nucleus can only be derived by analysing the data through a model which is based upon a partial-wave analysis. A very satisfactory analysis of whole of the elastic scattering data has been carried out by Frahn and Wiechers 2), and Dar and Varma “) by means of a smooth cut-off generalized strongabsorption model “). This model is basically a five-parameter fit and involves long and laborious computational work. Steenberg ‘) has developed a semi-empirical optical-type model for the elastic and quasi-elastic scattering of nucleons by nuclei in the GeV region. This model is essentially a two-parameter fit and we have applied it for the analysis of elastic scattering data for Be, C and Al in the present work. 2. Outlines of the method The general expression as

for the differential

cross section at any angle can be written

z =[F,(e)+F,(e)]2+F:(e),

where FC is the Coulomb scattering amplitude, FR is the real and F, the imaginary part of the nuclear scattering amplitude. The Coulomb interaction has been calculated with the help of Glauber’s expression “) for a uniform charge distribution with a radius R, = (R,),A’ and screened at a distance w lo-* cm which is > R,: F,(q)

= qexp

[-iB

In (qRc)-0.1160]+i@

4 x[exp(iBIny)-exp((-iB)(~(4-yZ)~1-y~-ln(1+J~)y2)))]ydy,

(2)

B = 2Ze2//3hc.

The value of (R,), for the Coulomb interaction has been chosen to be that calculated by Elton for a uniform charge distribution ‘). The variable q = (Ml) is used in place of 8, since for small angles, the momentum transfer is almost perpendicular to the momentum of the incoming particle and thus q is invariant for the transformation between the lab and c.m. systems. In Steenberg’s model, in order to take into consideration the granular nature of nuclear matter, the nucleus is regarded as a randomly arranged system of A (= mass number) identical grey discs, each having a transparency a and radius 6. The whole system is confined to a nuclear volume of radius R = R,A* and 6 < R,. When the projectile is traversing the nuclear medium, owing to screening of one nucleon by the other nucleons, all the nucleons of the nucleus are not seen by the projectile. The

PROTON-NUCLEUS

155

SCATTERING

shadowing effect, for nuclei with A* > 2, is accounted cient n in the cross sections:

for by introducing

the coeffi-

~tod~)= %(l)A”,

err(1) = 2(1-a)7c62,

(3)

Q&)

= oa(l)A”,

a,(l)

= &(l--a2),

(4)

= cs(l)A”,

o,(l)

= r&(1-a)!

(5)

a,,,,,(A)

In this picture n = 1 if there is no screening and n = 3 if the shadowing is maximum. While deriving these expressions, the nuclear model considered is the sharp cut-off model. Coherent scattering is assumed to take place when the projectile interacts with the nucleus as a whole and the incoherent scattering when the projectile interacts only with one of the nucleons inside the nucleus. Spin effects are believed to be small and are neglected. The expression for the imaginary part of the nuclear cross section is:

(s), = ($)’ [gz(6*)(1-G2)~]i.~.h+ (~)2L~2(4G21c,... ’

=

(f)’

(l+

$3(‘-

;+A&,,)

(6)

’

6*2=s2[1-(y2].

In these expressions, PF (in units of h) is the maximum Fermi momentum of a nucleon in the nucleus; g is the nucleon form factor and G is the nuclear form factor. When the target particles are in motion, the net probability is such that the incident particles are scattered through a larger scattering angle than if they were stationary. This effect of the Fermi motion in the incoherent part is represented by a reduced nucleon radius 6” in the form factor g occurring in the incoherent part of expression (6). The parameter C is more or less a correction factor to take care of the defects of the approximations made during the development of the expressions of this model and it is determined by demanding j do = rr,,,,,. At such high energies, z 20 GeV, it is very difficult to differentiate experimentally between true elastic scatterings and quasi-elastic scatterings. Bellettini et al. ‘) measured the cross sections using CERN sonic spark chambers and did not separate the quasi-elastic events from the true elastic events. Steenberg’s model has the merit that it includes in it the contribution of quasi-elastic processes as well and thus can be applied to the experimental data of Bellettini et al. For the light and medium weight nuclei, where the diffraction rings are not at all prominent, the form factors g2 and G2 can be taken to be of Gaussian form: &3) &a*)

= e-“(46)*, = e-*(4a*)*,

G2 = e-$(M)2.

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Because of the restriction A* > 2 for the relations (3), (4) and (5) to be true and the nature of the form factors chosen, we have applied Steenberg’s model only for the analysis of the 9Be, “C and “Al data in the present work. The real part of the scattering amplitude can be taken to be the square root of expression (6) but multiplied by a factor X, a measure of the contribution of real part of nuclear scattering. Assuming the Coulomb scattering amplitude to be real, Fe(q) can be calculated from expression (2) by taking the absolute value of F,-(q). Finally the expression for the differential scattering cross section becomes:

do(B)=k2

-&fdl+

da

3. Calculations and results The experimentally observed absolute cross sections of Ashmore et al. 8), Coor et al. 9), Atkinson et al. lo), Pantuev et al. 11) and Bellettini et al. ‘) can be fitted with expressions (3), (4) and (5) wh en n has the value 0.67 to 0.72 [refs. 59“)I. The average value of 6 can be calculated by substituting the experimentally observed cross sections ‘) in, say, expressions (3) and (4). We have calculated the value of 6* from expression (7) by taking PF = 1.20 fm-I. Keeping close to these average values of n, 6 and the observed crto,,the values of x and R. are then adopted so as to get the best fits to the experimentally observed differential cross sections. The best-fit parameters for Be, C and Al are tabulated in table 1. The figures in brackets in this table are the ones observed experimentally or calculated from expressions (3) and (4) using the experimental data. The best-fit values for R. found by Bellettini et al. ‘) for 9Be, 12C and 27A1 are 1.67, 1.42 and 1.38 fm respectively. However, much significance cannot be attached to these values as the model used for the analysis does not fulfil all the kinematical details of the problem, and because the finite transparency of the nuclear interior is also neglected. All these and our values of R, are, of course, larger than the value R o x 1.15 fm estimated by Frahn and Wiechers ‘). This is as expected because in this smooth cut-off model, as in the Woods-Saxon form for the radial distribution of the optical-model potential, R( = R,A*) is defined to be the distance at which the nuclear effect falls to half of its central value. In a sharp cut-off model, on TABLE

Best-fit parameters Element

n

6 best fit

‘Be W =A1

0.67 0.67 0.67

1

for the elastic scattering data

1.35 1.30 1.35

RO

o,,,(b) calculated (1.35) (1.35) (1.35)

best fit 0.270 0.335 0.687

observed (0.278 &0.004) (0.335*0.005) (0.687f0.010)

x

(fm) 1.50 1.42 1.42

-0.26 -0.30 -0.12

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157

the other hand, the parameter R refers to the radius of the nuclear matter after which the nuclear effect ceases to exist. In figs. 1, 2 and 3 the experimental data are plotted, while the continuous curves are the ones calculated from Steenberg’s model using the best-fit parameters listed in table 1.

IO3-

s

E < d .

.

b

to*-

t

i

d

:

IO’

I

4.5

2.5

I

6.5 -

I

I

IO.5

8.5 8

I

I

I

I

12.5

I&_

14.5

lb.5

(mrod)

Fig. 1. Differential scattering cross section of 19.3 GeV/c protons by ‘Be in LS. The curve is based upon Steenberg’s model.

to3-

2 3

< d b ,O*: d

.

I ,O'

I

2.5

I

I

I

4.5

I

I

6.5 -

8.5 e

I,,

, 10.5

12.5

I

,

14.5

,

,

16.5

(wad)

Fig. 2. Differential scattering cross section of 21.5 GeV/c protons by 12C in LS. The curve is based upon Steenberg’s model.

158

P. M. SOOD AND S. K. TULI

IO’

I

I

2.5

I

I

4.5

I

I1

I

6.5 -

,

8.5 0

I

IO.5

I

I

12.5

I

I

I

14.5

I

lb.5

(mrod>

Fig. 3. Differential scattering cross section of 19.3 GeV/c protons by *‘Al in LS. The curve is based upon Steenberg’s model.

4. Proton-nucleon and proton-nucleus interactions

The interference between the real part of the nuclear scattering amplitude (= xF,) and the Coulomb scattering amplitude F,(q) has been found to be constructive in the above analysis. This fact reveals that at an energy w 20 GeV, the nucleon-nucleus interaction is repulsive in nature. The proton-proton small-angle elastic scattering at z 20 GeV has been studied by Bellettini et al. 13), who analysed their data on the assumption of a complex spin-independent coherent scattering amplitude F,,(t) (t is the four-momentum transfer) using the relation: F,(t) = 6 %(PP)[~ cxp (3Ci t) + ~(0 exp (+CR91.

(9)

To simplify the procedure, they further assumed that C, = CR = C and that y(t) = y is independent of t. The best-fit parameters at 19.3 GeV/c thus determined are C = lO.O(GeV/c)-’ and y = -0.33+0.03 [ref. ‘“)I. Dispersion-relation predictions at this momentum as calculated by Soding 14) give y = -0.25, while the estimates of Levintov et al. 15) suggest a value between -0.20 and -0.26. Thus the protonnucleon scattering amplitude F”(t) at w 20 GeV also has an appreciable negative real part. From these considerations, one therefore expects that the nucleon-nucleus amplitude may in some way be connected with the nucleon-nucleon amplitude. It would be

PROTON-NITC’LEUS

SCATTERING

159

instructive, for instance, to examine the extent to which the scattering by a complex nucleus can be obtained by a superposition of the nucleon-nucleon scattering amplitudes. In the impulse approximation, the amplitude Fcoh(f) is connected with &F,(t) by the relation: Fcoh(t) = AF,(r)G(tL

(10)

where A is the target mass number and G(t) the nuclear form factor. Choosing a Gaussian form factor G(t) = exp(SBt) with the parameter B = &R2, from expressions (9) and (10) we get

(11) From this expression, the forward-scattering

amplitude FcFEoh(0) is

From expression (8), the forward-scattering

amplitude &‘fOh(0)is (13)

Comparing (12) and (13), we have Y = x.

Because of the various approximations, eq. (11) is valid for small momentum transfers and light nuclei. From table 1, the values of x for Be and C are respectively - 0.26 and -0.30, which are in reasonable agreement with the value y = -0.333-0.03. These various features reveal that Steenberg’s model is a fairly reasonable and simple description of p-nucleus coherent and incoherent scattering at small momentum transfers. In applying this model to heavier targets like Cu, Pb and U, one should perhaps choose the form factors for g and G which are sensitive to diffraction minima. Work is in progress on these lines to interpret the scattering data of these heavier nuclei on the basis of Steenberg’s model.

The authors are indebted to Dr. G. Bellettini for making available all the details of the data analysed in this work. Thanks are also due to the computer staff, especially Mr. Bansal, of the IBM 1620 Computer Centre, Panjab University, for their help in the program of the problem. One of the authors (S.K.T.) thanks the Department of Atomic Energy, Government of India for financial assistance which partially supported this work.

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P. M. SOOD AND S. K. TULI

References 1) G. Bellettini, G. Cocconi, A. N. Diddens, E. Lillethun, G. Matthiac, J. P. Scanlon and A. M. Wetherell, Nucl. Phys. 79 (1966) 609 2) W. E. Frahn and G. Wiechers, Arm. of Phys. 41 (1967) 442 3) A. Dar and S. Varma, Phys. Rev. I&t. 16 (1966) 1003 4) W. E. Frahn and R. H. Venter, Ann. of Phys. 24 (1963) 243; R. H. Venter, Ann. of Phys. 25 (1963) 405. 5) N. R. Steenberg, Nucl. Phys. 32 (1962) 381; 35 (1962) 455. 6) R. J. Glauber, Lectures in theor. phys. 1 (1958) 315 7) L. R. B. Elton, Nuclear sires (Oxford University Press, 1961) p. 31 8) A. Ashmore, G. Cocconi, A. N. Diddens and A. M. Whetherell, Phys. Rev. Lett. 5 (1960) 576 9) T. Coor, A. D. Hill, W. F. Homyak, L. W. Smith and G. Snow, Phys. Rev. 98 (1955) 1369 10) J. H. Atkinson, N. W. Hess, V. Perez-Mendez and R. W. Wallace, Phys. Rev. Lett. 2 (1959) 168 11) V. S. Pantuev and M. N. Khachaturyan. JETP (Sov. Phys.) 15 (1962) 626 12) D. J. Holthuizen, Nuovo Cim. 34 (1964) 1413 13) G. Bellettini, G. Cocconi, A. N. Diddens, E. Lillethun, J. Pahl, J. P. Scanlon, J. Walters, A. M. Whetherell and P. Zanella, Phys. Lett. 14 (1965) 164 14) P. Sbding, Phys. Lett. 8 (1964) 285 15) I. I. Levintov and G. M. Adelson-Velsky, Phys. Lett. 13 (1964) 185 16) H. Bethe, Ann. of Phys. 3 (1958) 190