Nuclear Physics B59 (1973) 477-492. North-Holland Publishing Company
ABSORPTIVE
CORRECTIONS
TO EXCHANGE
REACTIONS IN POTENTIAL SCATTERING
*
A.N. KAMAL Department of Physics and Theoretical Physics Institu re, University of Alberta, Edmonton, Alberta, Canada
Received 2 April 1973 (Revised 14 May 1973) Abstract: Absorption prescriptions are derived for exchange reactions in potential scattering using the eikonal approximation.
1. Introduction Central to the idea of the absorption models [1,2] of exchange reactions in high energy particle phenomenology is the Sopkovich [3] formula. The only derivations [4, 5] of the Sopkovich formula that the present author knows of are based on potential scattering where the "absorption" or "distortion" of the incident or the outgoing waves occurs through the elastic potentials - these are the diagonal potentials V11 and V22 in a two channel two-body problem with transitions from channel 1 to channel 2 occurring to first order in VI2, the off:diagonal potential. In the present work we have set up a multi-channel two-body model in potential scattering. Such a model has the advantage that it defines rather clearly what an elastic or an inelastic process is and it also has the built-in Hermiticity requirements on the potentials. We have solved such a model in the eikonal [6] approximation and have derived the Sopkovich formula [3] and its analogue in the presence of other competing channels. The formulae we have derived, though not precisely the same as those advocated by the Michigan group [2]~ share some common features with their prescription [2]. Alternative absorption prescriptions are also derived.
2. Derivation of the Sopkovich formula for a two-channel problem Consider two channels (two-body, spinless) with reduced masses m 1 and rn 2 satisfying the coupled Schr6dinger equations (r = (b, z) where b is the impact parameter) * This research was partially supported by the National Research Council of Canada.
478
A.N. Kamal, Potential scattering
(V2 + kl ) ~1 (b'z)=2rnl [VII (b,z) Ol (b,z)+ V12(b,z )
~(b,z)],
[V21 (b, z ) ~ 1 (b,z)+ V 2 2 ( b , z ) ~ 2 ( b , z ) ] ,
(g2+k~)¢2(b,z)=2m2
(2.1) (2.2)
where m i and k i are the reduced masses and particle momenta in the centre-of-mass respectively. ~j are Hermitian potentials. Defining (2.3)
~/(b, z) = e ikiz 4~] (b, z),
one obtains a set of coupled linear equations in ~/(b, z),
a~ 1 (b,z) ml 3z - i k 1 [ V l l ( b , Z ) O l ( b , z ) + V l 2 ( b , z ) O2(b,z)],
(2.4)
a~ 2 (b,z) m2 Oz - ik 2 [ V21 (b, z) ~1 (b, z) + V22 (b, z) ~b2 (b, z)].
(2.5)
To zero order in V12 the solution l\)r 01 (b, z) is (mI z I ~0) ( b ' z ) = exp {ikl-lt f VII (b, Z')dz' J ,
(2.6)
which satisfies the boundary condition (k~O) (b,
co) = 1.
(2.7)
Next, we seek a solution to q~2 (b, z) of the form
(2.8)
O2 (b, z) : ~o) (b, z) ~2 (b, z), with ~b~()~ (b, z) = exp t ii:-[ f
}
V22 (b, z') dz' .
(2.9)
Substituting (2.8) into (2.5) we get 0~2 m 2 i(kl e Oz - ik 2
k2)z
V21 (b, z)
Ol (b,z) - ,
~0) (b, z)
(2.1o)
with a solution m2 / ei(k 1 kz)z, O1 (b,z') dz'. ~2 (b, z) = 1~¢2 _= V21 (b, z') q~o) (b, z')
(2.11)
A.N. Kamal, Potential scattering
479
We now make two approximations; (i) evaluate ~2 (b, z) to first order in V21 which implies that we set 41 (b, z') = q~]0) (b, z')
(2.12)
in eq. (2.1 1), and (ii) ignore the changes in longitudinal momenta in the exponential. One then gets
V21(b'z')dz'exP[ikl
4~1) ( b ' z ) ; / - k 2 7 _
J
Vll (b'~') d~"
x expt,ik 2 } gzz(b,~) dr~ .
(2.13)
The scattering amplitude for the exchange reaction 1 -+ 2 is then obtained as
ik2 r
TI2 (q) = - ~ j e
iq± .
b
d2b ~b(21)(b, co).
(2.14)
4~1) (b, oo) is obtained from eq. (2.13) which says that the incident wave is "distorted" via V 11 up to z' where it converts to channel 2 v i a V21. The outgoing wave is then "distorted" via V22. Note that as Vii are real the term "distortion" is more appropriate than the term "absorption" as the exponents are purely imaginary. We shall see later that in presence of other competing channels (channel 3 and higher channels) the term "absorption" will become more appropriate. In proceeding further one makes the assumption that V21 is short ranged compared to Vii so that the z' integration in V 11 (b, z') can be terminated at z' = 0 and that of V22 (b, z') can be started at z' = 0. This allows one to write m2 ~ qS(1) (b, °°) = T~- f
V21(b,z' )
dz' ~i(Xll(b)+ X22(b))
e
,
(2.15)
z
where Xjj are the eikonal phases [6]: mj
=
Xjj ('b) = - V -f ~ Vj] (b, ~') d~'.
(2.16)
Insertion of eq. (2.15) in (2.14) then yields the Sopkovich formula. Note that if m1 rn 2 k--l- Vll (b, z) = ~ V22 (b, z ) ,
(2.17)
then the short range assumption is not needed to enable us to write eq. (2.15).
A.N. Kamal, Potential scattering
480
3. Next order correction to the Sopkovich formula in a two-channel model
The next order correction to the Sopkovich formula of order (V t 2) 3 is obtained by using 4~2) (b, z), which is 41 (b, z) correct to order (VI2) 2, in eq. (2.11). 4] 2) (b,z) is obtained in turn by inserting 4~1) (b, z) (eq. 2.13) in eq. (2.4) and solving for 41 (b. z), that is,
a41 (b, z) mi az -ik I [Vll(b,z)Ol(b,z)+V12(b,z)4~l)(b,z)].
(3.1)
Eq. (3.1) is solved by setting
41 (b, z) = 4 I°~ (b, z) ~t (b, z),
(3.2)
where ~1 (b, z) satisfies the equation a ~ (b,z) m t 4~ ) (b,z) , 3z ik I Vl2(b'z) 4(lO)(b,z )
(3.3)
with a solution (boundary condition ~1 -+ 1 as z ~ ~1 ( b ' z ) = 1 + - ikl
~
Vl2(b,z')
~), (3.4)
4(01) (b,z')
Substituting for 4~ ) (b, z) from eq. (2.13) one obtains ~1 (b, z) to order (V12) 2. Substituting ~1 (b,z) in eq. (3.2) one obtains 4(I2) (b, z). ~(23) (b, z) is then generated via eq. (2.1) and 4~~ (b, z) : 4~°~ (b, z) ~23~ (b, ~),
(3.s)
giving 4~3) (b, z)
ik~-
{m2
x exp ik 2 /
dz' V21 (b, z')~
(b, z')
1
(3.6)
V22 (b, 9) dr3 ,
with m 1 fi'
. 4(21) (b, 7-) \
(3.7)
A.N. Kamal, Potentialscattering
481
It
Z
Fig. 1. Exchange reaction 1 --* 2 proceeding via an interaction of order (VI2)3. Solid line represents channel 1, dashed line channel 2. and
9¢01) (b, r) = ~
V21 (b, f)
(b, ~-) dfexp ~
Using eq. (3.7) and (3.8) in (3.6) one gets 4)(3) (b, co) = ik~
V21 (b, z')
f
V22 (b, r/) dr/ .
{m2;
exp /~2 , V22 (b, r) dr
_oo
(3.8)
}
Z
mira 2 ~' 3 klk2 d3 g12 (b,/3) f V21 (b, ~') d~"
X exp
m 2 .3
VI 1 (b, ~)
da]
3 × exp
Vii (b, O)
a )t
(3.9)
The first term inside the bracket produces the conventional Sopkovich-like amplitude where the conversion from channel 1 to channel 2 occurs at z' and the incoming wave is distorted via V11 and the outgoing one via 1122. The second term represents a third order conversion, see fig. 1, where channel 1 is distorted up to ~"where conversion to channel 2 occurs, which then propagates being distorted to/3 where channel 1 is regenerated, which proceeds being distorted up to z' where finally channel 2 is produced and continuesbeing distorted thereon. With the pictorial aid of fig. 1 and a little expertise in handling the factors that enter the theory it is not difficult to write down q~2 (b, z) to any order in V12"
A.N. Kamal,Potentialscattering
482
The corrected Sopkovich formula is then
ik 2 feiqi. 'b d2b 4~3)(b, oo),
(3.1 o)
T12 (q) = - 2--~with
ik~-~_
~ dz' V21 (b, z') (1-A
/{"/
J mlm2 i '
A (b, z) - klk2
X
(b,z')) (3.11)
3 d3 V12(b,3) f
d~" V21(b,~")
[ml g" (i~lf Vll(b,7?)d~}exp (m2 ~
d@
(3.12)
A (b, z') represents the correction term and it is in general complex. If mI m2 kW Vii = k 2 V22'
(3.x3)
then
mlm 2 ~' A(b,z')- klk2
3 dfi V12(b, 3) f d~" V21 (b,~'),
(3.14)
which in the two-channel model is real and positive due to Hermiticity of the potential. If eq. (3.13) is assumed to be true then m2 ? Tl2(q)=-y~f e~qz'b d2b X (1-A (b, z)) ei×~ l(b) .
dz V21 (b,z)
(3.15)
Note that in our two-channel model Vii are Hermitian and therefore e& 11(b) is strictly oscillatory rather than absorptive. Through the definition of Xl 1 (b), eq. (2.16), one notices that there is no distortion of waves coming in with large impact parameters as Xll (b) b -~ 0.
A.N. Kamal, Potential scattering
483
4, More than two channels So far we have confined the discussion to only two channels interacting via Hermitian potentials. Let us now assume that there are many other competing channels. To fix ideas let us in particular consider a third channel. The exchange reaction 1 -+ 2 can now proceed via channel 3 as 1 -+ 3 -+ 2 i.e. a process of order V13 V32. Such a transition will give rise to an optical potential and the effect of channel 3 will be describable in terms of this optical potential. If there were N channels and we were interested in the exchange reaction 1 ~ 2 we could in principle eliminate all channels except channels 1 and 2 and describe the effect of the eliminated channels through a non-local non-Hermitian energy-dependent optical potential. In the following we have set up a three-channel model, channel 3 being the extra channel. The model is defined by the following set of linearized Schr~dinger equations (for simplicity the impact parameter label b has been dropped from the arguments), 3¢ 1 32
3¢ 2
m1 -
ik 1
(4.1)
[V21 (z) ¢1 (z) + V22 (z) ¢2 (z) + V23 (z) ¢3 (z)],
(4.2)
[g31 (z) ¢1 (z) + V32 (z) ¢2 (z) + V33 (z) ¢3 (z)] .
(4.3)
~tt2
OZ - ik 2
0¢3
[VII (z) ¢1 (2) + Via (z) ¢2(Z) + V13 (z) ¢3 (z)],
m3
OZ - ik 3
~i are Hermitian potentials. In order to reduce the problem to an effective two-chan. n~l problem we eliminate channel 3 from the above set of equations. This procedure can also be carried out for a higher number of channels. We solve for ¢3 (z) in terms of ¢1 (z) and ¢2 (z) by defining ¢3 (z) = ¢(0) (z) ~3 (z),
(4.4)
where ¢(30) (z) = exp
V33 ( z ) dz'
(4.5)
One then obtains
m3f
~3 (Z) : / ~ 3
[V31(z')¢l(Z')+V32(z')¢2(z')]
[¢~0) (z')] -1 dz'.
(4.6)
484
A.N. Kamal, Potential scattering
Substituting eq. (4.6) in eq. (4.4) and finally 4~3 (z) in eq. (4.1) and (4.2) one gets the effective two-channel model: ~ t
oo, m z - b z - i k - ' 1 [Vll (z)~ 1 ( z ) + f Vll(Z,Z')~f(z')~' l
Z
+ Vl2(Z)~b 2 ( z ) + f
V12(z,z')(b 2(z')dz'] ,
(4.7)
and Oq~2 m 2 z Oz = ?k2 [V22 (z) ~b2 (z)+ f V22 (z, z') ~b2 (z') dz' Z
+ V21 (z) ~bI (z) + f
V21 (z, z') ~bl (z') dz'] ,
(4.8)
where the non-local potentials are m3 VII (z, z') = t~33 V13 (z) ~b~O) (z) V31 (z') [q)~O)(z')] 1,
V 1 2 ( g , z ' ) = i ~ 3 V13(z )
(z) V32(z' )
(z')]
m3 V21 ( z , z ) - T 6 - V23(z) O(30)(z) V31 (z') [q5(O)(z')] 1 ,
(4.9)
(4.1 o)
_
(4.11)
and m3 V22 (z, z ) = ik 3 V23 (z) qS~O) (z) V32 (z') [~b~O) (z')] 1 ,
(4.12)
Though the many channel problem can be reduced to a two-channel one with the eliminated channels producing the non-local optical potential in the eikonal approximation the problem is intractable beyond this stage. However, in low energy nuclear physics an adequate phenomenological description of the experimental data in a limited energy region is secured by using local optical potentials. Ixl the same spirit, if we simplify the problem by assuming that the optical potentials are local then the problem reduces to that discussed already in sects. 2 and 3. The only difference would be that we would be using optical potentials (symbol ~)i/) which are non-
A.N. Kamal, Potential scattering
485
Hermitian. To first order in VI2 the amplitude for the exchange reaction is then (analogues of eqs. (2.13) and (2.14))
ik2 (" iq±.b T12 (q) = - ~ - j e
(4.13)
d2b ~2 (b,~),
m2; X exp
} (4.14)
V22 (b, ~) d~ . Z
The hat on top of 4~2 and Vii distinguishes this case from that discussed in sect. 2. It must be emphasized that strictly V 1| is not the optical potential in channel 1 since channel 2 has not been eliminated. However for a large number of open channels the distinction will be only academic. The imaginary parts of V11 and V22 will be negative definite and therefore one now does indeed get an "absorption" of the waves in channels 1 and 2 in the sense of the decay of the wave amplitude with the penetration inside the potential. For a spherical potential, it is also seen from eq. (4.14) that the absorption for waves with small impact parameters is larger than that for waves with large impact parameters. The amplitude of eq, (4.14) is obtained by eliminating channel 3 and therefore ^ ^ is exact (within^ the eikonal approximation) to all orders in VI3 and V23 but approximate to order V21 in the coupling of channel 1 to channel 2. Such a procedure can be followed for any number of inelastic channels and a similar result exact to all orders in~i j (1)12 excluded) can be derived but one would still be working to first order in V21. Notice in particular that to order V2/the effective parameter X of the Michigan group [2] has not appeared and for purely imaginary V 11 and V22 the absorption cannot lead to a reversal in sign of the amplitude in the impact parameter space. The formula is more like that of the Argonne-Imperial College Group [ 1]. One must however realize that one ought to work to all orders in V12 before one has a complete description of the exchange reactions. The next order correction in ~" ~ 3 V I 2 (of order (VI2) ) can be made exactly as discussed in sect. 3. One gets to order (V12) 3 ,
rn2? ~(23) (b, oo) = t ~ 2
/i
X exp ~_1
ikl - ~
^
,,
V21 (b, z) dz- (1 - A (b, z))
VII (b, ~') d~
l{m ; exp ~
z
}
V22 (b, ~) d~j ,
(4.15)
A.N. Kamal,Potentialscattering
486 with
mlm2 ~ ^ 3 ~k(b,Z)-klk2 d3V12(b,3) f d~" V21 (b'~') (rn I ~ XexPtt~l!
}
[m2 ~
Vll (b, r/) dr/ expl/c~-2! I~'22(b,~)d~
) •
(4.16)
The parameter/k (b, z) does not have to be real. Particularly simple forms for /~ (b, z) can be obtained in two situations: A
^
(i) V 11 = V22 = 0, then
mira2 z
)X(b,z)- klk2f ^
^
(3
d/3 V12(b,3) f
k l k 2 2!
d~ V21 (b,~')
d13V12 (b, 3
(4.17)
A
fo r
VI 2 = V21"
(ii) If
m1 ^ m 2 ,, kl VII = ~ 2 V22'
then again eq. (4.17) can be derived. In the following sects. 5 and 6 we have demonA strated that in the two situations described above one can work to all orders in V12 and get an expression for the exchange reaction amplitude in impact parameter space in a closed form.
5. Model I
An extreme model, I~2, 4=0,
~.i : 0.
The model is defined
via the coupled linearized equations
3 ^ ml ^ ^ q~l (b, z) =/~1 V12 (b, z) q52 (b, g), Uz-z
(5.1)
3,, m2^ ^ 3-z ~b2 (b, z) = ~ V21 (b, z) ~b1 (b, z).
(5.2)
A.N. Kamal, Potential scattering
487 A
The formal solution with the boundary conditions, 4)l (b, _oo) = 1 and }2 (b'-°°) = 0, is r n l j ~ d.z' ~bI (b, z) = 1 + t ~ 1 ~ V12 (b, z') ~b2 (b, z'),
(5.3)
^ ( b , z ' ) aqS1 (b,z') " q~2 (b, z) = / ~m2 2 f_~ dz' V21
(5.4)
These equations can be solved by an iteration and a resummation of the series. One gets ~ (b,z) ~1 (b,z) = cos XI2
(5.5)
~" z) = ikrn2 ^ (b,z')cos~(12(b,z') ~b2,(b, 2 /j_~ dz ' V21
(5.6)
A
A
with VI2 = V21 , and 1
^
(mlm2t-f ~
X12 (b, z) = ~ k ~ "
,^
_~o da V12 (b, z').
(5.7)
~2 (b, z) is proportional to sin X12 (b, z). The scattering amplitudes are then
ikl ~ 2
Tll (q) = - 7-~Jd
*
b eiql'b (cos X12 (b, oo) - 1 ) ,
T 1 2 ( o O = - ~ 2 f d 2 b e iql"b f dz V21(b,z) cos~12(b,z ). _
(5.8)
(5.9)
c~a
A
Comparing eq. (5.9) with eq. (4.15) (setting VII = 0 in the latter equation) one finds ^
A
A (b, z) = 1 - cos ×12 (b, z).
No useful bounds on A (b, z) can be set even in this extreme model. X12 will in general be complex. Only for real X12 (b, z) one finds that 0 ~< A (b, z) ~< 2. In^ this extreme^ model there is no absorption of the waves in channels 1 and 2 via V11 and V22. Nevertheless the interaction V 12 is being taken into account to all orders and the attenuation of the wave occurs because of transitions from channel 1 to channel 2 and vice versa.
A.N. Kamal,Potentialscattering
488 6. Model I1
This model is defined through the assumption mI ^ m9 ^ V l l = k2 V22" k1 In sect. 4 we have already used such an assumption. The simplification brought about was that the exponentials of the kind appearing in eq. (4.16) can~el out. Such a cancellation can be demonstrated to take place to all (odd) orders in V 12" The calculation of ~2 (b, z) is then considerably simplified. ~b2 (b, z) is calculated through an iteration of two equations (see eq. (3.6) and (3.7) of sect. 3): ~{"+ w2 ~) (b,
m2 / A _~ dz~ V21 (b, ,z1) $I ") (b, z I )
z) : ~
X e.xp
j,. V22(b , ~') d~- ,
(6.1) ZI
A
× v,2 (b, z 2)
~(2n-l)(b, z2)i
Sl0-(;,
i
(6.2)
The superscript represents the order of I)t2 to which ~i is being calculated. Under the assumptions m1 A
m2 ^ V22'
(i)~l VII = ~ 2 A
A
(ii) V12 = V21, the iterated series can be summed up rather easily yielding
-~2 (b,z) = ~m2 ,x /~ 7 / ~z d'Zl V21(b, zl)(1- a(b, Zl)) X expl. b- ]
VII (b, ~) d~ ,
(6.3)
A.N. Kamal, Potential scattering
489
where A
1
A
(6.4)
A(b, Zl)=COSXl2(b, Zl).
The exchange reaction amplitude is then obtained as 00
r,2
(q) =
-m 2 fd2b
e iqz'b
f
/~ (b, z) (I - A A (b, z)) eiX ~(b), dz V2!
(6.5)
where
mif
(b) = - ~
V.i (b, z) dz.
(6.6) A
The usual Sopkovich formula is obtained by setting A (b, z) = 0 or expanding cos ~12 (b, z) and keeping only the first term of the expansion. Notice that the formula is not that written down by the Michigan group [2] but the possibility does exist that the amplitude in impact parameter space (62 (b, co)) reverses sign as a result of the absorption. A
7. Alternative absorption prescriptions to first order in V12" We start with a two channel model with the implicit assumption that all other channels have been eliminated generating the optical potentials Viy (i, ] = 1,2). We shall assume that these optical potentials are local so that the computation can be done. Thus the coupled problem is defined via (suppressing the impact parameter label); A
A
A
/"
A
(V 2 + k~) ~1 (z) = 2m I (VI1 (z) ~1 (z) + V12 (z) ~2 (z)),
(7.1)
(V2 + k2) ~2 (z) = 2m 2 (V21 (z) ~1 (z) + V22 (z) ~2 (g))'
(7.2)
Let us then solve these equations using the Green function [7], (r,,.')
i
=-
ikj(z-z') 82 (b-b'),
(7.3)
where j is the channel label. Then the solutions are,
ikxz
2rnli
1 (z) = e A
4k 1 ~"
feikl(z-z ')
[ [711 (z') ~ 1 (z')
t
+ V12(z') ~2(z)] dz',
(7.4)
A.N. Kamal, Potentialscattering
490 /, ~2 (z) -
2m2i f A A ,', A 4k2 e#~2 (z-z') [V21 (z') ffl (z') + V22 (z') if2 (z')] dz'.
(7.5)
If we now define
(7.6)
22 (z) = eikhz 2/(z),
and ignore the differences in the longitudinal momenta k I k 2 in the exponentials, then one gets
2mli ; 21 (z) = 1
~2 (z) -
4k-~
2m2i f 4k 2
A
A
A
[Vll (z') ~1 (z') + V 12 (z') ~b2 (z')] dz',
,', A ^ ^ dz'. [V21 (z') ~b1 (z')+ V22(z)~2(z')]
(7.7)
(7.8)
These equations ((7.7) and (7.8)) can be iterated and resummed, yielding
"
~b2(b,'~) =
X (1
(
2mli' ; 1 + ~_~
^ Vll(b,~')d
2m2i + ~
~.)-1 ~2m2i )j~ " ~
d~) -l . f
V21(b,z) dz
(7.9)
322(b,~)
The exchange reaction amplitude is then obtained as T12 (q) = --~-n
X
--~ Xll (b)
V21(b,z) dz 1--~' ix22(b
(7.10)
The absorption factors are the same as one would expect on the Blankenbecler-Goldberger [8] type of theory. There is one rather subtle point, namely, that we have used the Green function of eq. (7.3) which does not yield the wave function which Blankenbecler and Goldberger [8] write down even though it yields the same on-shell amplitude. The reader is referred to ref. [7] for a detailed discussion of this point. We also discuss this point to some extent in the next section. If we use the Blankenbecler-Goldberger wave functions [8],
2mli z
1+ 4-q-if
) Vll(b,z')da')dz'-2,
(7.11)
A.N. Kamal,Potentialscattering ~ o u t ( b , z ) = (1 2
2rn2i ? A 4k 2
V22 (b, z') dz
)-2
491
(7.12)
,
Z
then the resulting amplitude for the exchange reaction is T12 (q) = -
d2b e iq±'b
J=
dz V21 (b, z) ( ~ u t (b, z))
_oo
× ~i~ (b, z).
(7.13) A
A
For short range forces VI1 and V22 we can set z = 0 in eq. (7.11) and (7.12). For symmetrical and weak potentials with short range assumption,
__ ( i + 2mli f V^ l l ( b , z ' ) d z , 4k I _~
) ( -2
1+ 4 ~ 1 _~ VII (b, z') dz'
),
"
(7.14)
In this approximation then, the absorption formula of eq, (7.13) reduces to that of eq. (7.10),
8. Summary In sects. 2 and 3 we discussed a two channel two-body problem where the potentials were Hermitian. The Sopkovich formula (sect.2) and a corrected form of the Sopkovich formula to order (V12) 3 (sect.3) were derived. In sects. 4, 5 and 6 the problem where there were more than two channels interacting via Hermitian potentials was discussed. All channels but two were then eliminated (in the particular model the third channel was eliminated) and the problem reduced to a coupled twochannel problem with the eliminated channels giving rise to an optical potential. The Sopkovich formula was again derived in the approximation where the optical potentials were local. Models were discussed in sects. 5 and 6 where the interactions Vii and Vii were taken into account to all orders. It was shown that to first order in VI2 the absorption formula has the features of the Argonne - Imperial College prescription, The sign of the unabsorbed Born amplitude in impact parameter s~ace cannot be reversed through absorption. However if one works to all orders in V12 the possibility of the reversal in sign of the exchange reaction amplitude in impact parameter space exists. This is the feature of the Michigan absorption prescription. Our formula does not have the same structure as that of the Michigan group though it shares the same qualitative features. Alternative absorption prescriptions yielding Blankenbecler-Goldberger type of absorption factors were derived in sect. 7. The derivation of the reaction amplitude given in eq. (7.10) is based on the use of the Green function
, i eikj(z-z ') 62 ( b - b ' ) , u0 (r,r)=-
(8.1)
A.N. Kamal, Potential scattering
492
to solve the coupled channel SchriSdinger equation. This Green function allows the propagation o f the particle through the potential with unchanged longitudinal mom e n t u m , i.e. the particle stays on the (eikonal) energy shell [7], so that the wave f u n c t i o n carries the same wave n u m b e r and is altered only in its normalization. It was shown in ref. [7] that his wave function though different from the wave function, eq. (7.11), written down by Blankenbecler-Goldberger [8] does yield the same on-shell scattering amplitude. In contrast the S o p k o v i c h formula with Glanber like (exponential) absorption factors (eq. 2.15) could have been derived through the use o f the Green f u n c t i o n ([7] and Osborn [6])
~
(r,r') -
2k.i ~2 (b b') 0 (z- z') e iki(z z') ,
(8.2)
1 which allows some o f f (eikonal) energy shell scattering through the potential.
References [l] R.C. Arnold, Phys. Rev. 153 (1967) 1523; R.C. Arnold and M.L. Blackmon, Phys. Rev. 176 (1968) 2082; M.L Balckmon and G.R. Goldstein, Phys. Rev. 179 11969) 1480; B.J. Hartley, R.W. Moore and K.J.M. Moriarty, Phys. Rev. 187 (1969) 1921; P.A. Collins, B.J. Hartley, R.W. Moore and K.J.M Moriarty, Nucl. Phys. B20 (1970) 381. [21 F. Henyey, G.L. Kane, J. Pumplin and M.H. Ross, Phys. Rev. 182 (1969) 1579: M.H. Ross, F.S. Henyey and G.L. Kane, Nucl. Phys. B23 (1970) 169; R.L. Kelly, G.L Kane and F.S. Henyey, Phys. Rev. Letters 24 (1970) 1511. [3] N.J. Sopkovich, Nuovo Cimento 26 (1962) 186. [4] K. Gottfried and J.D. Jackson, Nuovo Cimento 34 11964) 735. [5] Loyal Durand and Y.T. Chiu, Phys. Rev. 139 (1965) B646. [6] R.J. Glauber, Lectures in theoretical physics, vol. 1, ed. W.E. Brittin and L.G. Dunham (lnterscience, New York, 1959) p. 315; T.A. Osborn, Ann. of Phys. 58 (1970) 417. [7] A.N. Kamal, Can. J. Phys. 50 (1972) 1862. [8] R. Blankenbecler and M.L. Goldberger, Phys. Rev. 126 (1962) 776.