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Nuclear Physics A361 (198 1) 307 - 325; @ North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

ON THE ABSORPTIVE POTENTIAL IN HEAVY ION SCAITERING

R. A. BROGLIA, G. POLLAROLO + and A. WINTHER

The Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen Q, Denmark

Received 21 August 1980

Abstract: A preliminary investigation of the nuclear imaginary potential to be used for the analysis of

elastic scattering data of heavy ions is presented. The derivation is carried out in the framework of

the semiclassical description. The resulting potential is angular momentum independent and shows

two components. A long range part due to transfer reactions and a short range part due to nuclear

inelastic scattering. Coulomb excitation has not been taken into account. Simple closed expressions

are derived for the transition amplitudes associated with the transfer and inelastic processes, including

the Q-value dependence which can be used for the analysis of reaction data.

1. Introduction

In the analysis of elastic scattering of nucleons on nuclei one has introduced,

besides the average (real) potential, U, an imaginary potential, W. It accounts for

the depopulation of the elastic channel due to residual interactions. A part of this

absorption is associated with the mean free path of nucleons in nuclear matter. This

part of the absorption, WV, is constant over the nuclear volume and vanishes in the

surface region more or less proportional to the density of the nucleus. The volume

part, WV, has been determined experimentally to have a smooth dependence on the

bombarding energy and is rather well understood ‘).

Besides the interaction of the nucleon with the bulk of the nucleus, depopulation

of the entrance channel will take place through the coupling to specific channels

like the excitation of surface modes and the pick-up of nucleons. Attempts to

calculate a surface absorptive potential,

not been successful. In fact the potentials calculated from the interpretation

in terms of a position dependent mean free path are strongly

Nevertheless it has been possible to describe

adjusting a suitably parametrized function W = WV+ W,, which is independent of

angular momentum. The surface contribution

than the volume contribution for bombarding energies below 40-50 MeV [ref. 3)].

The concept of an imaginary potential has been used also in the analysis of elastic

scattering of heavy ions with similar success 4). It is expected that the surface

W,, which describes these effects have

of W,

non-local functions

by empirically

2).

elastic scattering

to the absorption is usually larger

+ On leave of absence from Istituto di Fish Teorica, Universita di Torino, Torino, Italy.

307

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R. A. Broglia et al. / Absorptive potential

absorption is in this case completely dominant in that all exit channels including

fusion can hardly be reached except via channels excited during the approach of the

nuclear surfaces.

In this note we show how one may understand the empirically found absorptive

potential, when the strict interpretation of WJr) in terms of a mean free path is

discarded. Explicit expressions for Ws are given.

2. Semiclassical treatment

The semiclassical equations of motion for the amplitudes associated with the

different reaction channels /? are 5,

h?,(t) = C (cop, (V,-

Uy)~y)ei(E,-Ey)f’n~y(t),

Y

where Ep = Eb + EB is the sum of the energies of the two nuclei in channel fi, while

E, is similarly defined in channel y. The quantities $, and wB are related to the

product wave functions associated with the channels y and p respectively. The

ion-ion potential U, is the expectation value of the interaction V, in channel y. The

matrix elements are functions of the relative position of the ions and are functions of

time as the ions move on the classical trajectory of relative motion. For each term

the relative motion is chosen as the average between the two channels connected

by the matrix element, Eqs. (1) are to be solved for a given impact parameter with

the initial condition that cs( - 00) = S(/?, a), where a is the entrance channel (cf.

appendix A).

The question we want to address is, whether one can solve the set of equations (1)

in terms of a small number, S, of active channels, including the entrance channel,

introducing at the same time an imaginary potential, i.e.

iAt, = 1 (cob, ( Vy+ iWj”- Uy)lC/ y)ei(EB--Ey)t’ficy(t).

The index S indicates that W would depend on the size of the subspace of states

which are explicitly treated by eqs. (2). In fact W would vanish if S included all

channels. In so far as the imaginary potential

elements independent of the channel label one may solve (2) in terms of the solutions

c’ of the correspondingly truncated set of equations (I), i.e.

W,!“’ has only diagonal matrix

c&t) = c;(t) exp {i ~~~W(r(r))dr],

where r(t) indicates the classical trajectory of relative motion which we assumed

to be similar within the group of channels S. The total probability of finding the

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R. A. Broglia et al. / Absorptive potential

309

system in one of the channels S after the collision is

’

Ps = C Ic~@)l exp E

BES

_

{’ S_IP(r(t))dt) = exp { - f ~~~IP’(r(r))dr} ,

(4)

where we used the unitarity of any truncated set of equations (1).

As the subset S we first consider only the entrance channel a. In order to evaluate

W we view the total depopulation of the entrance channel as due to elementary

transitions like excitation of collective modes and transfers of single nucleons. In

this picture multinucleon transfer is the result of the successive transfer of nucleons.

We shall assume (i) that these elementary transitions are independent of each other,

and (ii) that the probability,

P,, that a specific transition n occurs during the

collision is a small number. In order to evaluate in this model the probability, P,,

of remaining in the entrance channel after the collision, we must envisage that there

are so many possible transitions n that p = &,,

The probability

P, can thus not be estimated by 1 -p,

expression

p, = ~~UYP,) = exp{- CP,>.

> 1 although all p. are small.

but one should use the

In the first expression we used assumption (i), while to obtain the final form we

also used assumption (ii).

Although these two assumptions may seem crude, they may be rather well

fulfilled in actual cases. The first assumption of independence, essentially amounts to

describe the nuclear states as product states of fermion excitations, particle-hole

phonons and pairing phonons. The limitation of this description is associated

with the overcounting of the degrees of freedom as well as with the effects of the

residual interactions, which couple e.g. the transferred nucleons into pairing and

particle-hole phonons. The second assumption is rather well fulfilled except for

the collective surface vibrational or rotational states. To the extent that one can

describe the vibrations as purely harmonic, one can substitute a collective vibrator

by a large number of independent vibrators each of them being excited only weakly.

Formula (5) is actually an exact expression for the probability of staying in the

ground state of a harmonic oscillator. For the excitation of rotational states this

argument does not apply and the expression (5) will lead to a poor approximation for

the imaginary potential associated with the elastic scattering on deformed nuclei.

In order to evaluate the exponent in (5) we should calculate the transition

probabilities

p, for stripping and pick-up of single nucleons as well as for the

excitation of surface and pairing vibrations in target and projectile. We use the

expression

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R. A. Broglia et al. 1 Absorptive potential

where cd_ is the transition amplitude

(7)

in perturbation

Inserting (6) into (5) and comparing to (4) we obtain the following equation which

is to be satisfied by the imaginary potential

s

Since this equation does not provide a prescription to determine the function W(r),

additional constraints have to be imposed.

If one adhers to the concept of a depopulation

(8) for an arbitrary upper limit t. One then finds

theory (cf. appendix B).

m

-m

W(“(r(t))dt = +h C lc,,,J2.

B

(8)

local in time one should enforce

( Wco’(r(0)), = h Re 1 ~~,,(t)c_.#)~

B

where c,+,(t)

leads to a @‘(r(t)) which is different for t = & It 1, although r(t) = r( - t). Com-

promising on the locality in time one may enforce the locality in r by the

prescription

g

is iven by (7) with t as an upper limit instead of co. This prescription

( Wco)(r)),, = 9 Re c (&(t)c,,,(t) + $,J - t)c,+,( - t)),

II

(10)

which also satisfies (8). This prescription leads to a function which depends strongly

on the energy and the angular momentum, and which coincides, for the case of

Coulomb excitation, with the expressions given in ref. 12).

Since the imaginary potential used in the analysis of experimental data is angular

momentum independent, it seems more natural to use this criterion as the sub-

sidiary condition to determine W(r) from (8). This can be done by interpreting (8)

as an identity in the angular momentum.

c,,@‘s in (8) are mainly determined by the distance of closest approach

condition in 1 is largely equivalent to an interpretation

Using a parabolic expansion of the trajectory

appendix B, eq. (lo)), one finds

For nuclear interactions, where the

ro, the

of (8) as an identity in ro.

around the turning point (cf.

(11)

i, being the acceleration

amplitudes c can be parametrized as

at the turning point. As shown in appendix B, the

C a-l?

z Ke-‘Oh.

(12)

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R. A. Broglia et al. 1 Absorptive potential

In this case, the integral equation (11) for W can be solved leading to the result

311

(13)

For the transfer channels, the quantity cc0 is of the order of 1.2 fm, while for nuclear

inelastic scattering it is of the order of 0.6 fm. This implies that the total absorptive

potential has two components; a long range part with the difuseness of N 0.6 fm

due to transfer, and a short range with a difuseness of N 0.3 fm due inelastic pro-

cesses, Some experimental evidence for such two-component

found 13).

When Coulomb excitation is included, the transition amplitudes for inelastic

scattering cannot be parametrized as in (12). In fact the amplitudes would consist of

two terms with opposite sign, leading to a short range potential due to inelastic

scattering, a long range potential due to Coulomb excitation, and an interference

contribution which is in fact a source term.

In the present paper we shall neglect Coulomb excitation and the results are

therefore valid for relatively light ion scattering.

Since Coulomb excitation, at bombarding energies of few MeV per nucleon, leads

only to states of low excitation energy, such reactions are often counted experi-

mentally as elastic events. Because Coulomb

trajectory of relative motion to any appreciable extent, one might, for the analysis

of such “elastic” experiments, use an absorptive potential which consists merely of

the W due to transfer plus a W due to inelastic scattering to high-lying states which

can only be excited by nuclear interaction.

potential have been

excitation does not change the

3. Calculation of the absorptive potential

In this section we evaluate the absorptive potential

utilizing the approximate

appendix B. The contributions to W from nucleon stripping and pick-up are given by

W(r) defined in eq. (13),

amplitudes expressions for the transition given in

1

** 2

0

& (W(r)),,,,,, =

(21, + 1)(2Z* + 1) EJ-

(I( C:Q,“=,l* + I(&?)I;~ = ,),

LJ

(14)

where the summation is to be carried over the quantum numbers associated with

the index 8, that is, Z,M,, ZB and M,. Furthermore, the sum also implies an averaging

over the initial orientation, that is, a sum over A4, and MA.

Utilizing the expressions (B.19), (B.29), (B.44) and (B.47) we can write (14) in

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the form

R. A. Broglia ef al. 1 Absorptive potential

The orbital integrals over the single-particle

and (B.45) for nucleon stripping and nucleon pick-up respectively except that the

fiMJ’ form factors are substituted by (B.30) and (B.48) respectively. In deriving (15)

we have neglected the dependence of dE on In and Zb for a given fixed single-particle

orbital, that is, the variation of the Q-value within the energy interval over which

the single-particle strength is spread. The quantities

probabilities

form-factor are given by (B.28)

V2 are the occupation

that the single-particle

U2 = 1 - V2 is the corresponding

Using the approximate expression (B.41) for the orbital integral we find

orbitals a; are occupied in the nucleus a. The quantity

probability that the state is empty.

VW),,,“,, = 4

a&,&)

~----.-

a,tlja ~wol~2

((2j; + l)V’(u,zA)I/~(u;z~)~f~~dO~~Ns~(u, $I2

+(2j, + W2(Q,) ~2(~,~A)If”nb”~‘NP’(O~ r)12)s&z),

(17)

where we assumed p’ = 0 and define

The function g determines the adiabatic cut-off. The quantities a and b are related

to the quantity q in eq. (B.42). They are given by

a=

where the optimum Q-value is

w

(20)

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R. A. Broglia et al. / Absorptive potential

313

and

(21)

For small collision times the quantity (18) is unity. One can estimate the function g

by utilizing the approximation

and substituting the sum over p by an integral. One thus obtains

gA(a, b) = k

s

oexp [ - Re (a-b cos 8)‘] dtI.

(22)

(23)

This function has been evaluated numerically for real values of a and b and the

result is given in fig. 1.

Q

I.

2. 3.

a(Q)

Fig. 1. The adiabatic cut-off function g,(Q) which describes the ratio of the actual transition probability

to the same quantity in the sudden approximation. It is defined in (18) and is given as a function of the

dimensionless parameters a and b. The parameter a which depends on the Q-value is defined in (19)

while b which depends on the angular momentum transfer I, is defined in (21).

The contributions to W from nuclear inelastic scattering is similarly given by

Cl

P

I)

zPE

A#

2

7

(24)

where the orbital integrals for target and projectile excitation are given by (B.3) and

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R. A. Broglia et al. / Absorptive potential

(B.7) with the form factors (B.4a) and the nuclear part of (B.8) respectively. Inserting

the result (B. 12) we find

W”k) = 1 J-

The adiabatic cut-off function g is given by the same expression as (18) with

Q,,, = Oand with d ff a i useness parameter a instead of a,r. The sum of the form factors

appearing in (25) can be written in terms of the total zero-point fluctuation u of

the two nuclear surfaces defined by

fJ2 = c (21 + 1) (

of all states below the adiabatic cut-off, i.e.

16rr; ,p (If~N’TE’w12

0

+ v-~‘PE’w12h(Q).

(25)

1

I:

2 (R:“‘)2 + $ (Rr’)2

1

>

g#o,),

A

A

(26)

K&9 = J16nFo,h2

c2 (%>‘.

(27)

The total absorption is thus equal to

W(r) = W,ransf(r) + Winei(r).

(28)

In the derivation of this potential we assumed (cf. eq. (12)) that the transition

amplitudes were exponential functions of the distance of closest approach. This

ansatz seems reasonable, because although

also on i,, this dependence appears as a multiplicative factor under the square root.

However the expressions (17) and (27) are singular for i, = 0 which in the classical

description happens when the angular momentum of relative motion is equal to

the grazing value 6. This is however a fictitious problem since the existence of an

imaginary potential means that the classical trajectories should be complex. In

terms of the effective potential for the radial motion Ueffr the turning point de-

termined from

the transition amplitudes depends

E = &(r,) + iVr,),

is complex and the acceleration

(29)

i, = - + d (Uert+iW),,ro

aA a’

is in general non-vanishing. In fact the use of the expression (28) for W(r) requires

that 7, and W should be determined self-consistently from eqs. (28)-(30).

For simple estimates one may use for the modulus of the acceleration

estimate based on a Coulomb trajectory which is

an

2E

Z Z e2

- _A+.-

2E-E

x B maAfO = _

TO r0 ‘B

(31)

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R. A. Broglia et al. / Absorptive potential 315

where EB is the height of the Coulomb barrier and rB the radius of the barrier

(rJr,,, = l.O7(Af + Ai) + 2.72.

(32)

The absorptive potential determined self-consistently from eqs. (28)(30) is local as

far as is independent of 1. It does show however an energy dependence through the

parameter f,. Besides entering as a multiplicative factor, f, enters in the adiabatic

cut-off parameters (19) and (21). For increasing bombarding energies it is expected

that i, increases. The corresponding decrease in W, because of the square root

factor, is more than compensated by the change in the adiabatic cut-off which

implies that more states contribute to the depopulation of the entrance channel.

Preliminary numerical calculations of the imaginary potential on the basis of the

expressions given above are in good agreement

Systematic calculations are in progress and will be presented elsewhere.

with the experimental data.

4. Conclusions

In this paper we have derived explicit expressions for the imaginary potential to

be used in the analysis of elastic scattering data. This was done by imposing the

condition that W should not depend on the angular momentum, and making use

of the assumption that the different channels contributing to W are independent.

Only the effects due to nuclear interaction were included. In the derivation we

used the semiclassical description and the corresponding

transition probabilities. The explicit expressions for these quantities are useful1 for

the discussion of the Q-value dependence of transfer reactions and inelastic scattering

of heavy ions. The resulting W has a long range component due to single-particle

transfer, and a short range part due to inelastic scattering.

expressions for the

Appendix A

SEMICLASSICAL EQUATIONS OF MOTION

The coupled equations (1) are complicated by the fact that the channel wave

functions I//? describing various exit channels in transfer

orthogonal, i.e. the overlap matrix

reactions are non-

a&, = (& JI,)

(A. 1)

is only diagonal at t = f co.

From the time-dependent Schriidinger equation the coupled equations are directly

obtained in the form

(A-2)

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In order to write them in the simple form (1) we introduce the adjoint channel

wave functions oy as

WY = 1 (u- i)&$D,

P

(A.3)

where a-r is the reciprocal matrix to (A.l). For small overlap (grazing collisions)

where

a = l+&,

(A.4)

with E < 1 we may use

a-l = l--E+&‘+....

(A.5)

Defining adjoint amplitudes

ts = 1 ap,cyei(Efl-&)tift

Y

(A@

as the expansion of the state vector on the adjoint channel wave functions (A.3) we

find that they satisfy the equations

ihtD = C ($,, (I$ - UB)$r)ei(Efl-EY)*“cy

= 1 (I)@, (VP- Ug)Oy)ei(Eg--E,)t’hCy.

Y

(A.7)

We used here the post-prior symmetry relation

d

dt

(A.8)

= ih _ (a

By

ei(EB-&)r/fi),

derived in ref. ‘) (cf. ibid, eq. (4.2)).

While the quantities lcB12 or lC,12 at time f cc signify the probability of being in

channel /? they do not satisfy unitarity at t x 0. We may at intermediate

rather use the quantity

J’,(t) = Re {$(t)c,(t)),

times

(A.%

for which it is easy to prove unitarity, i.e.

Appendix B

SEMICLASSICAL FIRST ORDER PERTURBATION THEORY

(A. 10)

In this appendix we quote the results of the semiclassical perturbation

of transfer and inelastic reactions between heavy ions [cf. refs. 5-7)].

theory

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R. A. Broglia et al. / Absorptive potential

317

We consider a collision between projectile a and target A.

spherical nuclei and assume that excitations only take place

projectile. For target excitation the first order amplitude is

We shall consider

in either target or

where ZAMA and Z;MX are the spin quantum numbers of the target states before

and after the excitation, while EA and Ei are the corresponding energies. We may

write (B.l) in the form

cTE = - i 1 (Z,M,L - pIZ;M;)

AP

p

J212+ 1 ac’

(-l)“_” z

with

1 O”

_ m

ZA, = h s

dr exp C~&lflPWN Y&W

(B-3)

where r(t) indicates the time-dependent

respect to the target and hw; is the excitation energy. The form factorfiA)(r) is to a

good approximation ‘) given by

position vector of the projectile with

f?)(r) = fl’A’(r) +fi(A’(r),

with

j-,“‘“‘(r) = -( - 1)”

f d-

4nZ,e

$Rf)

A

% JZTi,

(B.4a)

f,“‘“‘(r) = (- 1)”

(2A +

1)

(rZIId,(EA))lO)r-"-

‘,

(B.4b)

where dw

U,,(r) is the ion-ion potential and R, (‘) the radius of nucleus A. The second term is

the Coulomb excitation part of the interaction, Za indicating the charge number

of the projectile. The reduced matrix element of the electric multipole moment

may be written

is the zero point amplitude, (l,,la,,lO), of the vibration, while

(&M(En)llO) = 3zAF’

J

2 ,/m,

L

where R; is the charge radius of nucleus A, while Z,e is its charge.

For projectile excitation one finds a similar expression

cPE = -i(ZaA4,A-pIZ~M~)

(- 1)’

pITi zw

(B.5)

03.6)

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R. A. Broglia et nl. / Absorptive potential

with

and

IAt, = k s

00

dt exp Ciw~tJ~~)(r(t))Y,,(~(t)),

CO

03.7)

p(r) = (- 1)”

03.8)

The difference in sign between (B.6) and (B.2) is due to the fact that P is oriented

from A to a.

The orbital integrals (B.3) or (B.7) are most easily evaluated in a coordinate system

where the z-axis is perpendicular to the plane of the trajectory along the angular

momentum of relative motion, and the x-axis is pointing towards the projectile at

the point of closest approach. In this case

q&P(t)) = YAp(* n, 0) eip+(‘),

(B-9)

where # is the azimuthal angle.

Since the nuclear part of the form factor has a short range we may evaluate the

orbital integrals by a series expansion around the time (t = 0) of closest approach, i.e.

r(t) = r. +fi,t2,

440 = &t,

(B. 10)

where the acceleration F0 and the angular velocity &, at the closest distance r0 are

positive. Utilizing an exponental form for iJaA, i.e.

ava,

ar * U' ( ro) e- "- ' """,

(B.I 1)

with a x 0.6 fm, we find

I$ = Y,& O)~~(~*) f s ; exp [-d~~t2+iw,t2/a+io,t+i~~iotldt

00

= Y;#(fK 0)

(B. 12)

The quantity m

exponential factor the orbital integral is the product of this time and the form factor

at the distance of closest approach in units of h. The exponential function gives rise

to the adiabatic cut-off at nuclear frequencies w1 which exceed the inverse collision

time. We may estimate the quantity maAPO (Q,

bombarding energy in the c.m. system divided by r0 [cf. (31)]. The adiabatic cut-off

indicates the collision time, zinel, and in the absence of the

being the reduced mass) by the

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R. A. Broglia et al. / Absorptive potential

319

energy is

(B.13)

where ehiev is the laboratory bombarding energy per nucleon measured in MeV and

= (& + R,),,

is the distance of closest approach

h& is similarly given by

(%l

in fm. The quantity

(B.14)

where lg is the grazing angular momentum L, in units of /i.

Nuclear states can thus only be excited if the excitation energy AE = hw satisfies

the inequality

AE+(AL&, 5 ;,

(B.15)

1nel

where (AL), = +$I indicates the angular momentum,

relative motion in the direction of the relative angular momentum L. In a purely

classical description with a continuous energy and angular momentum loss, the

two quantities on the left-hand side of (B.15) would be equal since

which is gained by the

L

-*AL%

L

s

L maA

dt (r x F(t))% z L

f

8. F(t)dt = - AE/q$,,

where F is the effective dissipative force acting on the projectile.

The expression (B.12) has been compared *) to the results of the WKB

approximation for the radial matrix elements in a DWBA treatment of inelastic

scattering. If one uses for r e, t, and 4, the average values of the corresponding

quantities in entrance and exit channel, which are in general complex numbers, one

finds quite accurate agreement even for large Q-values.

The orbital integrals (B.3) for the Coulomb excitation part have been evaluated

numerically and are given e.g. in ref. Q).

Next we consider a single particle stripping reaction, where the residual nucleus

B contains one nucleon more than the target A while the scattered nucleus b has

lost a nucleon.

The first order amplitude for this reaction is

x exp W,, + EB - E. - E&+ r,dOllfi,

(B.16)

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R. A. Broglin et al. ,I Absorptive potential

where we have indicated the nuclear states by their spin quantum numbers. The

phase r,,(t) is given by [cf. ref. ‘) eq. (2.52)]

t

0

r@(t) =

J

W,(t) - ~,@)W,

(B.17)

where Z(f) is the lagrangian of relative motion, i.e.

y&t) = 3maAvz - ~,,C~),

(B.18)

the (average) position and velocity of relative motion being r(t) = $tr,,+r,,)

y(r) = f(t), respectively.

We shall write the amplitude (B.16) in the form [cf. ref. 6, sects. 2 and 31

and

CNS

aA-+bB =

-i 1

(IAMAJ~~IBMB)(IbMbJ’M’~IaM,>(~~JM~J’M’)Znfl,

(B. 19)

JJ’2.

MM’p

with

(B.20)

The dependence of the form factor on the velocity of relative motion through

is associated with the recoil effect, md = ma-m, being the mass of the transferred

nucleon.

The form factors can be expressed in terms of the intrinsic form factors through

the relation

(B.22)

The eulerian angles 9; are defined by the rotation from the laboratory system to

the intrinsic (1, 2, 3) system where the 3-axis is along r and the l-axis is in the

direction of the orbital angular momentum. If we chose to evaluate the amplitude

(B.19) in the coordinate system (A) used above where

we may write

x i J 1 dt exp [i(dEt+ Y&,(t) + hCl~(t))lA}f~~‘lk3(t), k,(t), r(t)).

(B.24)

4,

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R. A. Broglia et al. / Absorptive potential 321

Examples of the intrinsic form factors are shown in ref. ‘=), figs. 6-8. The main part

of the longitudinal recoil effect associated with k, may be extracted as an average

phase factor

(B.25)

In the latter equation we used an approximation

which is valid for not too high bombarding energies [cf. ref. 6, eq. (4.36)]. The

phase 0 is given by (cf. ref. 5), eq. (3.16))

for the transverse recoil effect

6 = k&t) R,- --!k- r

’

h m,+mA

md

mb+mB

z - ~ i(t)(R,m, - RamB),

(B.26)

where we used that r x Ra-t- R,. The transverse momentum k, can be expressed in

terms of the angular momentum L. The product

k2(t)r(t) = -2 ;

ZIA

(B.27)

is thus a constant. Inserting these results in (B.24) we find

x i

s

y dt exp [@Et + ya.(r) + #S(L) + @&r))/h]f$‘(O,

m

r(r)). (B.28)

The radial form factors can be expressed in terms of form factors describing the

transition between the single-particle

a, = (n,l,ji) in A, i.e.

configuration a; = (n;r;j;) in a and

f;i’(O, r) = 1 C*(Z,a, ; Z,)C(Z,a; ; ZJ xf$P’(O, r)&j,, J )S(j;, J ’),

alai

(B.29)

where the C’s are spectroscopic

factors for p’ 1 0 are given by [cf. ref. 6), eqs. (6.18) and (6.9)]

amplitudes and where the single particle form