On the absorptive potential in heavy ion scattering
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 Qvalue dependence which can be used for the analysis of reaction data.

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Nuclear Physics A361 (198 1) 307  325; @ NorthHolland Publishing Co., Amsterdam
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ON THE ABSORPTIVE POTENTIAL IN HEAVY ION SCAITERING
R. A. BROGLIA, G. POLLAROLO + and A. WINTHER
The Niels Bohr Institute, University of Copenhagen, DK2100 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 Qvalue 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 pickup 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 4050 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,
nonlocal 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
ionion 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(EBEy)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, particlehole
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
particlehole 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 pickup 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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310
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 al?
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 twocomponent
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 highlying 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 pickup 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
Page 6
312
the form
R. A. Broglia ef al. 1 Absorptive potential
The orbital integrals over the singleparticle
and (B.45) for nucleon stripping and nucleon pickup 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 singleparticle
orbital, that is, the variation of the Qvalue within the energy interval over which
the singleparticle strength is spread. The quantities
probabilities
formfactor are given by (B.28)
V2 are the occupation
that the singleparticle
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 cutoff. The quantities a and b are related
to the quantity q in eq. (B.42). They are given by
a=
where the optimum Qvalue is
w
(20)
Page 7
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 (ab 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 cutoff 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 Qvalue 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
Page 8
314
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 cutoff 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 zeropoint fluctuation u of
the two nuclear surfaces defined by
fJ2 = c (21 + 1) (
of all states below the adiabatic cutoff, 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 nonvanishing. In fact the use of the expression (28) for W(r) requires
that 7, and W should be determined selfconsistently 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+.
2EE
x B maAfO = _
TO r0 ‘B
(31)
Page 9
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 selfconsistently 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
cutoff 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 cutoff 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 Qvalue dependence of transfer reactions and inelastic scattering
of heavy ions. The resulting W has a long range component due to singleparticle
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 timedependent Schriidinger equation the coupled equations are directly
obtained in the form
(A2)
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R. A. Broglia et al. / Absorptive potential
In order to write them in the simple form (1) we introduce the adjoint channel
wave functions oy as
WY = 1 (u i)&$D,
P
(A.3)
where ar is the reciprocal matrix to (A.l). For small overlap (grazing collisions)
where
a = l+&,
(A.4)
with E < 1 we may use
al = lE+&‘+....
(A.5)
Defining adjoint amplitudes
ts = 1 ap,cyei(Efl&)tift
Y
(A@
as the expansion of the state vector on the adjoint channel wave functions (A.3) we
find that they satisfy the equations
ihtD = C ($,, (I$  UB)$r)ei(EflEY)*“cy
= 1 (I)@, (VP Ug)Oy)ei(EgE,)t’hCy.
Y
(A.7)
We used here the postprior symmetry relation
d
dt
(A.8)
= ih _ (a
By
ei(EB&)r/fi),
derived in ref. ‘) (cf. ibid, eq. (4.2)).
While the quantities lcB12 or lC,12 at time f cc signify the probability of being in
channel /? they do not satisfy unitarity at t x 0. We may at intermediate
rather use the quantity
J’,(t) = Re {$(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. 57)].
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
(B3)
where r(t) indicates the timedependent
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
$Rf)
A
% JZTi,
(B.4a)
f,“‘“‘(r) = ( 1)”
(2A +
1)
(rZIId,(EA))lO)r"
‘,
(B.4b)
where dw
U,,(r) is the ionion potential and R, (‘) the radius of nucleus A. The second term is
the Coulomb excitation part of the interaction, Za indicating the charge number
of the projectile. The reduced matrix element of the electric multipole moment
may be written
is the zero point amplitude, (l,,la,,lO), of the vibration, while
(&M(En)llO) = 3zAF’
J
2 ,/m,
L
where R; is the charge radius of nucleus A, while Z,e is its charge.
For projectile excitation one finds a similar expression
cPE = i(ZaA4,ApIZ~M~)
( 1)’
pITi zw
(B.5)
03.6)
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318
R. A. Broglia et nl. / Absorptive potential
with
and
IAt, = k s
00
dt exp Ciw~tJ~~)(r(t))Y,,(~(t)),
CO
03.7)
p(r) = ( 1)”
03.8)
The difference in sign between (B.6) and (B.2) is due to the fact that P is oriented
from A to a.
The orbital integrals (B.3) or (B.7) are most easily evaluated in a coordinate system
where the zaxis is perpendicular to the plane of the trajectory along the angular
momentum of relative motion, and the xaxis is pointing towards the projectile at
the point of closest approach. In this case
q&P(t)) = YAp(* n, 0) eip+(‘),
(B9)
where # is the azimuthal angle.
Since the nuclear part of the form factor has a short range we may evaluate the
orbital integrals by a series expansion around the time (t = 0) of closest approach, i.e.
r(t) = r. +fi,t2,
440 = &t,
(B. 10)
where the acceleration F0 and the angular velocity &, at the closest distance r0 are
positive. Utilizing an exponental form for iJaA, i.e.
ava,
ar * U' ( ro) e " ' """,
(B.I 1)
with a x 0.6 fm, we find
I$ = 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 cutoff 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 cutoff
indicates the collision time, zinel, and in the absence of the
being the reduced mass) by the
Page 13
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), = +$I indicates the angular momentum,
relative motion in the direction of the relative angular momentum L. In a purely
classical description with a continuous energy and angular momentum loss, the
two quantities on the lefthand side of (B.15) would be equal since
which is gained by the
L
*AL%
L
s
L maA
dt (r x F(t))% z L
f
8. F(t)dt =  AE/q$,,
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 Qvalues.
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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320
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) = $tr,,+r,,)
y(r) = f(t), respectively.
We shall write the amplitude (B.16) in the form [cf. ref. 6, sects. 2 and 31
and
CNS
aA+bB =
i 1
(IAMAJ~~IBMB)(IbMbJ’M’~IaM,>(~~JM~J’M’)Znfl,
(B. 19)
JJ’2.
MM’p
with
(B.20)
The dependence of the form factor on the velocity of relative motion through
is associated with the recoil effect, md = mam, being the mass of the transferred
nucleon.
The form factors can be expressed in terms of the intrinsic form factors through
the relation
(B.22)
The eulerian angles 9; are defined by the rotation from the laboratory system to
the intrinsic (1, 2, 3) system where the 3axis is along r and the laxis is in the
direction of the orbital angular momentum. If we chose to evaluate the amplitude
(B.19) in the coordinate system (A) used above where
we may write
x i J 1 dt exp [i(dEt+ Y&,(t) + hCl~(t))lA}f~~‘lk3(t), k,(t), r(t)).
(B.24)
4,
Page 15
R. A. Broglia et al. / Absorptive potential 321
Examples of the intrinsic form factors are shown in ref. ‘=), figs. 68. The main part
of the longitudinal recoil effect associated with k, may be extracted as an average
phase factor
(B.25)
In the latter equation we used an approximation
which is valid for not too high bombarding energies [cf. ref. 6, eq. (4.36)]. The
phase 0 is given by (cf. ref. 5), eq. (3.16))
for the transverse recoil effect
6 = k&t) R, !k r
’
h m,+mA
md
mb+mB
z  ~ i(t)(R,m,  RamB),
(B.26)
where we used that r x Rat R,. The transverse momentum k, can be expressed in
terms of the angular momentum L. The product
k2(t)r(t) = 2 ;
ZIA
(B.27)
is thus a constant. Inserting these results in (B.24) we find
x i
s
y dt exp [@Et + ya.(r) + #S(L) + @&r))/h]f$‘(O,
m
r(r)). (B.28)
The radial form factors can be expressed in terms of form factors describing the
transition between the singleparticle
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