Alpha particle scattering in the rigid projectile approximation

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Alpha particle scattering in the rigid projectile approximation

J A Y A T I G H O S H and V S V A R M A

Department of Physics and Astrophysics, University of Delhi, Delhi 110 007 MS received 8 December 1978

Abstract. We study elastic c-particle scattering off p,¢-particle and taC targets at 17.9 GeV/c incident momentum in the rigid projectile approximation of the Glauber model. Differential and total cross-sections are computed and compared with the data. Reasonable agreement with the observed differential cross-sections is found for small momentum transfers but short-range dynamical correlations in the target will probably have to be taken into account to get better agreement at larger momen- tum transfers, particularly in the case of a-12C scattering.

Kcywords. Alpha particle scattering; Glauber model; rigid projectile approximation.

1. Introduction

Over the years, t h e multiple scattering formalism developed by G l a u b e r (1959) has provided a basis f o r calculations which have described quite successfully particle- nucleus collisions at high energies. The theory has been extended (Czyz and M a x i m o n 1969; F r a n c o a n d V a r m a 1975; Alexander and Rinat 1976; V a r m a and F r a n c o 1977) to cover nucleus-nucleus collisions as more experimental d a t a with composite particle projectiles have b e c o m e available. However, as the projectile becomes m o r e complex, the full G l a u b e r series for such scattering processes b e c o m e progressively m o r e difficult to evaluate, even if one chooses simple forms for nuclear densities and spin- averaged two-particle amplitudes. Various approximations to the full G l a u b e r series have therefore been considered in the literature. O n e approach is to introduce an expansion in t e r m s o f the optical phase shift function. This expansion has been shown to converge fairly rapidly for light and m e d i u m nuclei ( C h a u m e a u x et al 1976;

Franco 1974; F r a n c o and V a r m a 1977). A n o t h e r approach, which is simpler and does not entail assumptions regarding convergence based on an examination o f the first few terms o f a series, is the so-called rigid projectile a p p r o x i m a t i o n (Alkhazov et a l 1977; V a r m a 1978). B o t h these techniques have been applied to the analysis o f the recent measurements performed at Saclay with 1.37 G e V a-particle projectiles (Chau- meaux et a l 1976; Alkhazov et al 1977) and have been successful in explaining these scattering processes in terms o f simple forms o f nuclear densities with spin averaged nucleon-nucleon scattering amplitudes as input.

In the rigid projectile approximation, one treats the projectile as an elementary object a n d assumes that it stays in its ground state t h r o u g h o u t the scattering process.

T h e effect o f the polarisation o f the projectile is thus ignored during scattering. This leads to a tractable f o r m o f the Glauber series which in addition to. its simplicity commends itself because it does not assume, as the optical limit expressions o f the 427



Jayati Ghosh and V S Varma

Glauber series do, that the target also remains in its ground state during the whole of the scattering process. Recent calculations by Varma (1978) and by Viollier and Turtschui (1978) have shown that the rigid projectile approximation is fairly accurate at small momentum transfers and gives, for a-particle scattering, results almost iden- tical to the full Glauber series and may therefore be relied upon as a simple effective approximation for analysing a-particle interactions with light and medium nuclei at high energies.

The main advantage of studying interactions involving a-particles is that their spin and isospin are zero. One therefore has to deal with o n l y t w o elementary a-nucleon amplitudes in contrast to the situation involving protons where in principle as many as 10 elementary nucleon-nucleon amplitudes may contribute. In addition, the a-particle may be considered as the lightest of the heavy ions, so that the effect of the composite nature of the a-particle on the scattering process may also be studied, Recently, experiments on the elastic scattering of 17.9 GeV/c a-particles off a variety o f targets have been performed at Dubna (Ableev

et al

1977) and in the present work we have tried to examine whether or not these can be understood in the framework of the rigid projectile approximation o f the Glauber theory, particularly as this approximation has been applied with considerable success to the lower energy Saclay data.

In § 2 of this paper we set out the expressions for the Glauber series for a-nucleus scattering in the rigid projectile approximation. In § 3 we obtain simple spin- independent parametrisations of c~-p and a - a elastic scattering .amp/~itudes by X"

minimisation. The purpose is to obtain expressions for a-13C scattering not only in terms of two-particle scattering data but also when the a-p and a - a interactions aretreated as elementary. In § 4 we compare our theoretical predictions with the data and finally summarise our results in § 5.

2. Alpha-nucleus scattering-rigid projectile approximation

The rigid projectile approximation in the framework o f the Glauber theory assumes that the projectile nucleus stays in its ground state during the whole of the multiple scattering process. In particular, one assumes that the projectile undergoes scattering off the target nucleus without being decomposed into its constituent nucleons. With this assumption one first constructs the scattering amplitude fol the projectile nucleus and a target nucleon using nucleon-nucleon scattering amplitude as input. This is then used to obtain the expression for the nucleus-nucleus scattering amplitude.

The amplitude for the elastic scattering of a-particles off a nucleus A is given in the Glauber theory by

F~x (L~)

= 2--'~ KA (A)

ik f

d 3 b exp

(i A "b) -~xrt°tal



where l~a-t°tal (b)is the total profile function for a-nucleus scattering, K a ( A ) i s the centre of mass correction for the nucleus A, A is the momentum transfer, k is,the magnitude of the incident momentum and b is the impact parameter.


Assuming the additivity of phase shifts the total profile function for-scattering between the a-particle and the nucleus A can be written as

i c r str°ng (b), (2)

-~avt°tal(b) = l - - e x p [ Xa.4 (b)] + exp [i xcA (b)] -<,4

where xCa is the coulomb phase shift function. Equation (1) can now be written as


F,,,~ ( A )

= K ~ ( A ) F ~ a ( A ) + ~ K A ( / k )

exp (i A-b) •

exp [i c Xaa (b)] -,,A pstrong (b) d z b. (3)

If we assume the coulomb phase shift to originate from the nucleus as a whole and treat both the projectile and target nucleus as point charges, then


~ A ( A ) = - A ~ exp [--2i {r/In


- - arg r (1+i~)}], (4) where

71 =Z,~ Z A e2/hv, v

being the relative velocity between the projectile and the target. In the rigid projectile approximation we have


r . ~ (b) =

1 - [ I - F . N (b)] ~,

-- '


e x p - ( - - i q . b ) Sa (q) f a x (q) d ' q (5) with

FaN (b) 2rrikN

where k N is one fourth the momentum of the incident a-particle and

f,~N (q)

is the

~tmplitude for scattering between the a-particle and a nucleon in the nucleus A.

Assuming the a-particle form factor to be purely Gaussian, viz.

S~ (q) = exp

(--R,~ 2 q2/4)

and the high energy spin-independent pararnetrisation of the nucleon-nucleon ampli- tude to be

fpN (q) -- kN cr (i+p)






is given by


fan (q) -~ -- ikN2 Ka(q)


(4) r-:

L 2*rQ J j (7)


430 Jayati Ghosh and V S Varma where Q = 2 ~ + R a a.

Using (5), (7) and also assuming a gaussian form factor for the nucleus A, viz.

S a (q) = exp (--RAt ql/4), equation (3) takes the following form:


F:~ (A) = KA (A) F ° aa ( A ) - - i k K A ( A )

A ./

j = l k ~ 0




[ k ~ ] i ~ l

X (,'<)<~: X (')c~ c~,_,<,,.,=. ,,.,,.,.

l = O m = 0

exp (-- All4 xsum ) 1F1 (--i~7, 1 ; A~/4


c., = (~)~ cr-" <'-"'1';2,,.<~ .,

j = l, 2, 3,,4

xj~,,,, = j / P ~ + k

( 1 / P r l / P O

+ t O / P , - - I / P d + m (l,/P4-1/P,)

and P j = R a a - - R a i / 4 + Q/j.


Equation (8) is the general expression of the amplitude for the scattering of g-particles off a nucleus A in the rigid pF0jcctile approximation.

3. Parametrisation o f g - A amplitudes

One can parametrise the a-A strong interaction amplitudes in a high energy spin- independent form as a sum of gaussians:



(q) -- (i+Paa! kaa 4~r c'aa ~ Yaaj exp (--flaas qt/2) j = l




y j = 1.


I f now we write the total a-A phase shift as X total = X c + X str°ng

aA aA aA


and treat both t h e a-particle and the nucleus A as point charges, then the elastic scattering amplitude for a-A scattering is given by

Faa ( A ) == l~a a ( A ) q- kaa aaa (i-~'PaA)

r ( l + i n ) 4=


× ~ Ya,4J (2[3aAJ k~,l)'~l exp (-- A'#aAJ2) 1F1 (--i~, 1 ; A=#a,o/2)

(lO) j = l

To obtain a parametrisation of the a--A strong interaction amplitude, the constants occurring in(9)ean be taken to be those which give the minimum value of X~ wben the differential cross-section implied by (10) is compared with the experimental data.

Strong interaction a-p and a - a amplitude parametrised in this manner can be used to obtain the elastic scattering amplitudes for a - a and a-lzC, the calculation being done both for an independent particle as well as the a-particle model for the nC nucleus.

4. Results

4.1. a-p scattering

We first compute the elastic differential cross-section in the Glauber theory using proton-proton data at 4.2 GeV/c (Jenni

et al

1977) as input (this is close to the 4.5

10 3


1 0 2


10 °


F i l l t ~ 1. a-p elastic m o m e n t u m tramf¢r,

( ~ - - T w o particle data as input

Best fit obtained b y

2 . . . .


I I I I ,1

4 8 12 16 20

Ill x 1C)-2lGeV~'c2l differential cross-section at


17.9 G©V/c as a function of


432 Jayati Ghosh and V S Varma

GeV/c momentum per nucleon of the present measurements). We have assumed that the energy is sufficiently large to take proton-neutron parameters equal to the proton- proton values: ppp---0.39,/3pp=7.5 (GeV/c) -2 and %p=42.2 rob. The expression for the =-p scattering amplitude with coulomb effects taken into account in the fashion described above is obtained b y putting A - - l , in (8). t The resulting fit to the data is exceedingly good as is evident from figure 1. The total cross-section predicted is 142 mb to be compared with the experimental value of =t°t----147± 1 rob. ¢zp

We would also like to determine the constants occurring in the spin-independent parametrisation of the a-p scattering amplitude given by (9) (with N = I ) , to enable their subsequent use in computing the a - a and a-l=C cross-sections at this energy.

We take % p = 1 4 7 rob,/3 p=32-5 (GeV/c) -2 as given by Ableev et at (1977). We treat pap as a free parameter and find that the sum of the mean squared deviations of the differential cross-section predicted by (10) from the experimental data is minimum for Pap =--0"09: The best fit corresponding to this value is also displayed in figure l, and is practically indistinguishable from the fit with pp input, upto values of the square momentum transfer J t I,~ 0" 15 (GeV/c) =.




The expression for the elastic scattering amplitude with two-particle data as input in the rigid projectile approximation is given by (8) with A = 4 , R a = R a . The

10 4

10 3





10 ° 0

---- a-p data as input

~ o o two particle data as input

\\ % °o_


t I i /

I I ! i I I 1

4 8 12 16 2 0

Ill x lO-2(GeV2/c2l

Figure 2. a-= elastic differential cross-section at 17.9 GvV/c as a

m o m e n t u m transfer.

*R~ for gaussian form factor of a-particle is 1.366 fm (Bass¢l and Wj!ldn 1968).

function o f


differential cross-section obtained by using the two-particle data of Jenni

et al

(1977) as input is displayed in figure 2. The agreement is reasonably good in the forward direction and again for 0.14 (GeV/c) 2

~ l t l ~ 0 " 2

(GeV/c) s. The theoretical curve shows a distinct minimum near [ t[ = 0.09 (CreV/c) t whereas the experimental data show only a gradual flattening in this region. The value of the total cross-section predicted is 390 mb which does not agree well with the experimentally measured value of a~ t = 4 5 0 + 2 0 mb.

We now use the spin-independent parametrisation of the a-p scattering amplitude obtained in § 4.1 to obtain an expression for the a-a scattering amplitude. Thc expres- sion for a-a scattering is then given by (7) with N replaced by a and the values o f the parameters a, fl, p occurring in this equation are equal to the best fit values for a-p scattering listed in § 4. I. The corresponding differential cross-section is also displayed in figure 2 and the fit is worse than that obtained in the case of two-particle input, the minimum being much deeper. The total cross-section now predicted is however 400 rob.

Finally we determine the constants occurring in the spin-independent parametrisa- tion of the ¢-a scattering amplitude given by (9). The use of N = 1 (i.e. a single gaus- sian) in (9) gives a poor agreement with the experimental data the best fit, displayed in figure 3 being good only upto I t I ~ 0.06 (GeV/c) z. Using the values a~a =450 mb and flaa --- 72"2 (GeV/c) -= given by Ableev

et al

(1977), we find that p¢,=--0.45 gives the minimum mean squared deviation from the data. A reasonably good fit for values o f I t I extending to 0"2 (GeV/c) ~ requires the use of at least a sum o f four



10 4

10 3

10 2

10 1

10 °

- - - - single goussion a sum of 4 gaussions

- - ' t

I I I ~ I I

0 4 8 12 16 2 0

Itw x lO-e(GeV2/c2)

Figure 3. Differ~atial ¢rms-section for clasti¢ ,,-a scattering, showing the best fit CUl'VC$,


434 Jayati Ghosh and V S Varma

gaussians in (9). I n this case we k e e p ~a~ fixed at 450 m b a n d t r e a t paa,/3aaj and Yaaj as free p a r a m e t e r s . T h e best fit is o b t a i n e d for values o f these p a r a m e t e r s listed in table 1 a n d this fit is also shown in figure 3.

4.3. a-x~C scattering

W e first c o m p u t e the a-12C elastic differential cross-section w i t h two-particle input in t h e rigid projectile a p p r o x i m a t i o n and with gaussian f o r m factors* for b o t h the a

1 0 4


10 3


--~. 1 0 2


"10 b


-- --simple harmonic oscillator form factor of 12C simple gaussian form factor for 12C nucleus


0000000 0~

/ " ' \ Q 0


° L ,

,o I I t t

0 4 8 12 16 2 0

Itl x 10-2( GeV2/c 2)

Figure 4. a-lzC elastic differential cross-section at 17-9 GeV/c as a function of momentum transfer for two particle data as input.

Table 1. The best fit parameters for a-a scattering amplitude (sum of four gaussians)

, , - , , , , , i

Paa /~'~J aaaj -- %aj

(C, eV/c)-' ~ a a ~ j


77.5 --3"27 0.399

5.5 --0"31 0"038

4"5 --0.20 0-025

71 "3 --4"41 0"538

*Re for gaussian form factor of " C is 1.96 fm and for the simple harmonic oscillator form factor is 1.59 fro, both corresponding to a rms radius of 2.41 fm (Varma and Franco 1977).


and 1=C nucleus. We also take into account the coulomb scattering using (8) with A=12 and use appropriate two-particle values for p,/3 and o. The results are plotted in figure 4. As in the ease of a-a scattering, we find that the fit is reasonable both for small momentum transfers and in the region beyond the first minimum, but where- as the calculation predicts the existence of a well-defined minimum at

I tl


(GeV/e) ~ the data only show a gradual flattening around this value. The value o f the total cross-section predicted is 853 mb to be compared with the experimentally measur- ed value ot~t=877-q-10 rob.

One can attempt to study the effect of the structure o f the x2C nucleus by using a more realistic form factor t given by

Sc (q) = [] --Re = q=/9] exp (--R= = q=/4).

The corresponding expression for the ~-1~C scattering amplitude including coulomb interaction can be worked out as in the case of the gaussian form factor, and yields a differential cross-section which is also displayed in figure 4. The dip is now shallower but the fit both in the forward direction and in the region beyond the first minimum is not as good as before. The total cross-section now predicted is 811 rob.

We next calculate the a-l~C differential cross-section with a-p input with a gaussian form factor for the lsC nucleus and coulomb interactions taken into account. This fit is shown in figure 5 and is worse than in the case o f the two-particle input. The

,i,0 `4

10 3

~ 10 2

~ S . input.

~ . - - - - simp)e harmonic oscillator form factor of 12C - - . - - simple goussion form

~" f a c t o r of 12C . _

w t I • . ),er-

~ = a - p a r ice model 0T ~.

ilii °°°°%°,

o o

,o' - f,,.,'\

i( \


f l I I j

0 4 8 12 16 2 0

I l l x 10 -2 (GeV2/c 2)

Differential croea-scctioa for elastic a-~=C scatterin 8 with =-p data used as


436 Jayati Ghosh and V S Varma

m i n i m u m is s h a r p e r and the agreement with the experimental data away f r o m the m i n i m u m is poorer. We have repeated these calculations for the realistic xzC form factor and also f o r the a-particle model o f the ltC nucleus (for details please see G h o s h and V a r m a 1978). O f the three cases that we have studied using ~-p input', the a-particle m o d e l seems to give the best fit, but this is in any case not nearly as good as t h e f i t w i t h two-particle input. The predicted total cross-sections in these throe c a s e s a r e 850 rob, 828 m b and 832 m b when the 1~C nucleus is described by an alpha particle model, an independent particle m o d e l and a simple harmonic oscillator model respectively.

Finally we use the parametrisation o f the a-a a m p l i t u d e given in § 4.2 as input to calculate the o~-lzC scattering amplitude within an a l p h a particle model o f the 1~C nucleus (for details see Ghosh and V a r m a 1978). In figure 6 we show the a-lzC differential cross-section calculated for both cases when the ~-a amplitude is parame- trised either as a single ganssian or as a sum o f f o u r gaussians. It can be seen f r o m the plots that the interference m i n i m u m is quite shallow when a single gaussian parametrisation is used, but the fits in the extreme f o r w a r d and higher m o m e n t u m transfer regions are not as good as in the case o f two-particle input. The sum o f f o u r gaussians also leads to poor fits. The total cross-section values predicted are 909 m b for the case o f a single gaussian and 930 m b for the case o f a sum o f f o u r gaussians.

1 0 4

.~. 10 3





10 ~

- - . - sum of 4 gaussions

single goussian pararnetrisalion

i el

0 0

00000 0

- F \ ++.+.


/ " ' - . "°

t / ",.\


I I I , I

4 8 12 1 6

Itl x 10 -2 (GeV2/c 2) , o °

o 2o

Figure 6 . Differential cross-section for elastic a-l'C scattering with a-a data used as input.


5. Conclusions

In the preceding sections we have studied scattering o f a-particles off p r o t o n , a--particle a n d x2C t a r g e t s using a rigid projectile a p p r o x i m a t i o n to t h e G l a u b e r m o d e l o f high energy scattering. W e find that although the ~-p d a t a o f Ableev et al (1977) are well described b y o u r calculations, the rigid projectile a p p r o x i m a t i o n is not as successful in describing the a - ~ a n d the a-l~C d a t a at the present energy as it w a s in the case o f the lower energy Saelay d a t a ( V a r m a 1978; Viollier a n d Turtschi 1978).

T h e rigid projectile a p p r o x i m a t i o n predicts deep interference m i n i m a in these cases at b o t h energies. H o w e v e r , whereas at the lower e n e r g y the experimental d a t a indicate the existence o f such m i n i m a , the higher energy d a t a show only a gradual flattening. O n e is t h e r e f o r e forced to conclude that the rigid projectile a p p r o x i - m a t i o n leads to useful simplification only in the case o f scattering off light nuclei at small m o m e n t u m transfers and at relatively lower energies. I t ceases to be as satisfactory at relatively higher energies possibly on a c c o u n t o f the higher p r o b a b i l i t y o f dissociation o f the projectile inside the target. O u r calculations seem to suggest t h a t the detailed structure o f n C m a y play an i m p o r t a n t role in determining the 1~C differential cross-section even for low m o m e n t u m transfers a n d to get better agree- m e n t in the case o f the heavier nuclei at higher energies. I t seems inevitable t h a t one will h a v e to t a k e into a c c o u n t short-range dynamical correlations in the target.

W e are currently investigating this possibility.


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