Multiple scattering with rearrangement

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Multiple scattering with rearrangement

A. Malecki

To cite this version:

A. Malecki. Multiple scattering with rearrangement. Journal de Physique, 1978, 39 (10), pp.1049-

1053. �10.1051/jphys:0197800390100104900�. �jpa-00208843�

MULTIPLE SCATTERING WITH REARRANGEMENT A. MA0141ECKI (*)

Département ^de Physique ^Nucléaire

CEN Saclay, ^{BP 2, 91190 Gif sur} Yvette, ^France (Reçu ^{le 31 mars} 1978, accepté ^le 19 juin 1978)

Résumé.

La diffusion noyau-noyau ^est étudiée dans le cas où il _y a réarrangement ^{des nucléons}

entre la cible et le projectile. ^Des amplitudes de transition approchées ^sont calculées dans le cadre des formalismes de Watson et de Glauber. Une application ^{formelle au cas} de la diffusion p ⁺ 4He

est proposée.

Abstract.

Nucleus-nucleus scattering ^with rearrangement of nucleons between the projectile

and target nuclei is discussed. The approximated Watson and Glauber formulae for these colli- sions are derived. The example ^{of the} antisymmetrization ^term in elastic proton-nucleus scattering

is considered.

Classification Physics ^Abstracts

25.10 - 25.90

The theory ^of multiple scattering ^{was formulated} by ^{Watson in the} early ^fifties [1]. ^{A few} years later Glauber developed ^the model of shadow scattering [2].

At present the interconnections between the Watson

theory and Glauber approximation ^are well under- stood [3], and the Glauber model is being ^success- fully applied ^for high-energy hadron-nucleus scatter-

ing [4]. Recently ^various extensions of the model which take into ^account the non-eikonal effects have been elaborated [5].

The aim of this work is to develop ^a generalization

of the Watson and Glauber approaches ^{for the}

collisions with exchange of nucleons between the

projectile ^and target ^{nuclei. In} particular, ^these exchanges might ^be quite important in elastic proton- nucleus scattering ^{at backward} angles [6].

Let us consider a rearrangement process with the nuclei A and B in the entrance channel, ^{and the} nuclei C and D in the exit channel ; A, B, C, ^{D will} also denote the atomic numbers of respective ^nuclei.

We consider a transfer of K nucleons from B to C, thus C = A + K, D = B - K. The total Hamil- tonian _may be written ^{as :}

H = Hi + Vi = Hf + Vf (1)

where H;, Hf ^are the free Hamiltonians in the two

channels, ^each being composed ^{of the two nuclear}

Hamiltonians and of the channel kinetic _energy.

For the initial channel we write the interaction

as :

where _r; is the initial channel relative coordinate,

and {rjA, ^{rkB } is the set of A + B - 2 intrinsic}

nucleon coordinates, ^{referred to} the centre-of-masses of the nuclei A and B, respectively.

Similarly, for the final interaction we write :

where rf is the final channel relative coordinate,

and ( ric, rmD } denotes the set of intrinsic nucleon

coordinates, ^{referred to the c.m. of nuclei C and D.}

The two sets of the relative and intrinsic coordi- nates are connected to each other. We have the relations :

where (*) ^{Permanent address :} Instytut Fizyki Jadrowej, ^Krakow

31 342, ^Poland.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197800390100104900

1050

denotes the c.m. of the transferred system, ^measured with respect ^{to the c.m.} of the initial target ^{B. It is}

a characteristic feature of the rearrangement ^colli- sions (K =F 0) that the relative coordinate of a given

channel is coupled ^to the intrinsic coordinates in another channel.

The inverse transformations read :

where

denotes the c.m. of the transferred system, ^measured with respect ^{to the c.m. of} the nucleus C.

Consider now the transition amplitude ^at energy E.

In the case of rearrangement collisions one may define

an operator which induces transitions from the channel i to the channel f either as :

The two operators ^{are different} but, when inserted

«

between the free eigenstates ^{of the} channes 1 ) and 1 f >, ^both give ^{the same} transition amplitude.

Also the first terms of the Born expansions ⁱⁿ equa- tions (4) ^and (4’) ^{have this} property :

which is referred to as microreversibility [7].

Because of equation (5) ^{one is allowed to redefine}

the transition operator Tf; by replacing ^{the first term}

in the expansion (4) ^with V;. Notice that in each term of the Born expansion ^{there is} only ^one Vf.

One _may, therefore, hope ^that by replacing ^{all the} Vf’s ⁱⁿ equation (4) ^with Vi’s the transition amplitude

would not be greatly distorted. Thus we propose the following redefinition of equation (4) :

and analogously ^for equation (4’) :

Equation (6), ^{which is} formally ^{the same as the}

wave equation ^of scattering ^{in the entrance} channel,

will yield ^an approximation ^{to the true transition} amplitude. ^The validity ^{of this} approximation might

be checked, a posteriori, by ^a comparison ^{with the} analogous approximation which would result from

equation (6’). ^In general, ^one may expect ^that equa- tion (6) ^or (6’) ^are good approximations for almost- elastic collisions, ^{i.e. when the number of} transferred nucleons K is small with respect ^to the atomic numbers of nuclei involved in the collision. In that case the two equations ^should give nearly ^{the same result}

for the transition amplitude. On the other hand,

it is obvious that _any large discrepancy ^between

the prior-interaction (eq. (6)) ^{and the} post-interaction

(eq. (6’)) results would put both in doubt.

It can easily be checked that equations (6) ^and (2)

are equivalent ^{to :}

where

the summations running ^{over the} pairs of nucleons from the nuclei A and B. From here a series from of the transition operator may be obtained :

which is formally ^{the same as the} Watson series for

scattering ^of the nuclei A and B [8].

Analogously ^one could obtain from (6’) ^and (2’)

the Watson series for scattering of the nuclei C and D :

where

At moderately high energies ^one may neglect

the nuclear Hamiltonians and the nuclear part ^of the total _energy in the propagators (E - Hi) - 1,

(E - Hf)-1. Consistently, ^the transition operators for the scattering ^{of two} bound nucleons tjk(i), tjk(f) ^can

be well approximated by ^{the free} nucleon-nucleon transition operators. Nearly ^all applications ^{of the} multiple scattering theory adopt ^this impulse approxi-

mation. In the impulse approximation ^{the two series}

of equations (8) ^and (8’) ^should give nearly ^{the same}

transition amplitude of the reaction A + B -> C + D

if AB ~ CD. Since CD

AB - K(A - D ) ^{such a}

situation takes place ^{if K is} small, ^{and if A ~} D ; in the latter case (e.g. exchange ^elastic scattering)

K may be large. ^Thus hopefully, ^it may ^{turn out} that our approximate transition operators (eqs. (4),

and (4’)) may be valid for a wide class of rearrangement

collisions.

Equations (8) ^and (8’) give merely ^{a formal} repre- sentation of the transition amplitude; ^{because of}

their complexity and the infinite number of terms,

they ^{have no} practical ^{use as} they ^{are. In usual scatter-} ing, the Glauber approximation [2], involving ^a

considerable simplification ^of the Watson theory [3],

leads to a tractable description ^of multiple scattering.

Our purpose here is ^to explore what emerges when the Glauber formalism is applied ^to rearrangement

collisions.

"

In order to obtain a Glauber description ^of multiple scattering the free Green’s function of the ^entrance channel is approximated ^{as the eikonal} propagator, independent of nuclear Hamiltonians [3] :

with the z-axis chosen along ^{the vector :}

and JC

Piz

A/C Pfz, ^{P;, Pf being the c.m. momenta} of the nuclei A and C, and v; the initial relative velocity

of the nuclei A and B.

The application ^of equations (6), (9) ^and (3) ^{lead8 to the} following expression for the Glauber transition

amplitude (fully on-shell) :

where _{çoi, çof} denote the intrinsic wave functions of the two nuclei in the initial and final channel, respectively.

It is our choice of the z-axis which has assured that in equation (10) ^{there are no} phase-factors dependent ^on

the longitudinal coordinate.

This expression may be further simplified by neglecting ^the z;-dependence of the final channel wave func- tion _({Jf. Under the assumption ^which may be referred to as the zero longitudinal range approximation :

one obtains from (10) ^the Glauber-type ^{formula :}

with

representing ^the nucleon-nucleon profile.

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It is instructive to write âlso the analogous expression ^{for the} transition amplitude ^which may be obtained

by eikonalizing ^{the Green’s} function of the exit channel. Using equations (6’) ^and (3’) ^one obtains then :

with

the z-axis being ^chosen along ^{the vector}

Equations (12) ^and (12’) constitute the main result of this _{paper. In} comparison with the standard Glauber formula (without rearrangement) ^{there are two diffe-}

rences : the occurrence of the phase-factor describing

the motion of the transferred cluster of nucleons and the dependence (through ^{eqs. (3) and} (3’)) ^{of the}

final (initial) ^{channel wave function on} the initial

(final) impact parameter ^variable.

Equations (12) ^and (12’), although each is invariant under time reversal _pi

_{pf, Pf}

Pi will, ⁱⁿ general, give different results. As has already ^been discussed, ^a negligible discrepancy between the two formulae would, a posteriori, justify ^the validity

of our assumptions : equation (6) ^{and the} impulse approximation. An overall test of our eikonal model would be provided by ^{a careful} comparison ^with

the experimental ^data.

For illustration we shall consider the elastic p-4He scattering ^{with account of} antisymmetrization ^between

the projectile ^and target protons. ^{Since the two} protons ⁱⁿ 4He have opposite spins ^{one has} effectively,

under assumptions ^{of a} spin independent ^N-N interaction, only ^one possibility ^of exchange ^{of the} incoming proton ^{with the} target proton. ^{One is} therefore allowed to treat the ’He nucleus as the

(p ⁺ t) system ; ^{the wave} function of the proton- triton motion in ’He will be denoted /(rpJ. ^The amplitude of elastic scattering may thus be written

as

p’, p being the final and initial c.m. momenta of the proton.

In our two-body approximation the direct scattering

term is :

where

and y,,, yp, ^{are the} proton-proton ^and proton-triton profiles, respectively.

The exchange ^{term which} corresponds ^{to a rear-}

rangment (viz. the triton transfer) may be written, following ^the recipe ^of equation (12), ^{as :}

where

and

the z-axis being along ^{the vector}

The formulae (14) ^and (15) ^{are to be used at A}

^4.

It would be interesting ^to study the connection between equation (15) ^{and the} corresponding ^for-

mula given in reference [6]. ^{In our} approach ^the

basic interaction responsible ^for the triton pick-up by the incident proton ^{as well as the} absorption

effects are treated in the same manner being ^both expressed ^{in terms of the} elementary t-matrices.

In reference [6] the basic interaction is assumed to be that between the proton and triton as bound in the ’He _target while the absorption is described in a semi-phenomenolQgical way.

Acknowledgments.

^The hospitality ^{of Drs. R.} Joly

and R. Bergère ^{extended to} the author at the Centre

d’Etudes Nucléaires de Saclay ^is gratefully ^acknow-

ledged.

References

[1] ^{WATSON, K. M..} Phys. ^{Rev. 89} (1953) ^575.

[2] GIAUBER. R. J., Phys. ^Rev. 100 (1955) 242.

[3] EISENBERG, J. M., Ann. Phys. ⁷¹ (1972) ^542.

[4] CZYZ, W., Adv. Nucl. Phys. ⁴ (1971) ^61.

[5] WALLACE, ^S. J., Phys. ^{Rev. C12} (1975) 179 ;

DYMARZ, ^{R. and} MA0141ECKI, A., Phys. ^{Lett. 66B} (1977) ^413.

[6] LESNIAK, ^{H. et} al., Nucl. Phys. ^{A 267} (1976) ^503.

[7] MESSIAH, A., Mécanique Quantique (Dunod, Paris) ^1964.

[8] ^{WILSON, J.} W., Phys. ^{Lett. 52B} (1974) ^149.