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Variational treatment of rearrangement collisions
Nuclear Physics 64 (1965) 548--566; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher
VARIATIONAL TREATMENT OF REARRANGEMENT COLLISIONS C. J O A C H A I N Universitd Libre de Bruxelles t Received 16 July 1964 Abstract: Variational principles based on the formal theory of scattering and generalizing Schwinger's method for direct collisions are proposed for the treatment of binary rearrangement coUisions. These principles are established both for free and coupled waves, and are compared with the Born development. Application is made to three-body problems, the case of nuclear pick-up and stripping reactions being examined in detail on a specific example.
1. Introduction The variational study of rearrangement collisions has hitherto been approached from tWO different points of view. The first one, initially proposed by Kohn 1) and developed recently by Delves 2) and Spruch and Rosenberg a) generalizes the variational methods for determining the phase shifts. These methods are based on the differential form of the variational principles. They lead to stationary expressions for the elements of the S matrix, the reaction matrix or the derivative matrix. The second point of view, which we adopt here, consists in a generalization of the Schwinger variational principle for the transition amplitude 4). This approach, based on the integral form of the variational principles, has been illustrated successively by Borowitz and Friedman s), Newstein 6) and Lippmann 7). We give in this work new variational principles s, 9), based on the formal theory of scattering and apply them to specific examples. In sect. 2, we establish these variational principles both for free and coupled waves. Sect. 3 is devoted to the application to three-body problems, the case of nuclear pick-up and stripping reactions being examined in detail. Our numerical results are collected in sect. 4. 2. Variational Principles for Rearrangement Collisions 2.1. BASIC F O R M U L A S
Let us consider a rearrangement collision of the kind A+B ~ C+D, t Charg~ de recherches du Fends National de la Recherche Scientifique. 548
(2.1)
REARRANGEMENT. COLLISIONS
549
where there is no creation or destruction of particles. For the initial system (A+B), we have n = H I + Vi, (2.2) where H i is the unperturbed Hamiltonian, particles A and B and where
Vi
is the interaction potential between
Hi ~i = E~i,
(2.3)
#j being the free wave corresponding to the Hamiltonian H i and the energy E. In the same manner n = H f + Vf,
(2.4)
H, ff'f = E # , .
(2.5)
where Denoting by T~f the transition matrix element, we have Tie
:
<~flVflW}+)> = <~P~-)lVil~i>,
(2.6)
where ~ + ) and ~fc-) denote the solutions of the Lippmann-Schwinger equations ~"/(i ,+f ) ~-" ~ i , f m~-~f'J~("I') J i , f TI vi, f ~if/(+) ti, f ,
(2.7)
where G~±) and G(±) are the Green operators associated respectively with Hi and Hr. These equations have the well known formal solutions ~YJ(±)
( l "]- G ( ± ) V i f){lJi: f ,
(2.8)
where the Green operator Gt+) corresponding to H is given by G (±)
G~f)+ Gt+)E , i , f i , f G(±)•
(2.9)
We now introduce the Meller wave operators O(±) i,f
=
l+G(±)Vi, f,
(2.10)
such that, from (2.8),
III(±)i,f = ~'~)~i,f" We
(2.11)
shall also need the transition operator T i f _- Vf+ VfG(+)Vi,
(2.12)
which gives Yif =
<~flTifl~i>.
(2.13)
From (2.10) we have *
Vf[O~+)-1] = [Ol-)-1]tvi t The dagger denotes
Hermitian conjugation.
(2.14)
550
C. ,IOACI.IAIN
while, from (2.9),
1,
[ i _ ~~(+)w/ Ic~(±) 1 , f • i, fA~ai, f
(2.15)
from which one can deduce the relations 7) (2.16)
I] =
[f2~- ) - 1]'[1 - Vi O}+)l = VeG}+), a~-)'[Vf - v" r f ~(+)rzlr°(+) ,~,f r fdLa~i vi
--
II
(2.17)
~
a~-)' l/f O~+)Vi
(2.18)
=
~ f G} + ) Vi ~e~}+ ) .
(2.19)
" |/''i
Let us note also that the transition operator defined by (2.12) may be written, with the aid of (2.10),
T i f = ~f~'~[+)
(2.20)
or also, by making use of (2.14), T if = ~ ( f - ) * ~ + ( V f - Vi). 2.2. V A R I A T I O N A L
(2.21)
PRINCIPLES
Let us consider the expression JR,] = a~-)'VfGtf+)Vi+ l/fQ~+)-fa~-)'[~-~G~+)~][O{+)-l].
(2.22)
Using (2.18) and (2.20) we see that its exact value is given by [R,] = T i'.
(2.23)
Moreover, this expression is stationary for independent variations of the Moller wave operators around their correct values. Indeed, 6[R,] = 6a~-)t~{G~+)l/i- E1 -- G~+)l/f][a~ + ) - 1]} + { 1 - a ~ - ) ' [ 1 - l/fG~+)]}~6a~ +) (2.24) and from (2.16) and (2.15) 6 [R 1] = O.
(2.25)
Thus the expression (2.22) provides a variational principle for the operator T if. As it makes use only of the Green operator G~+), we shall call it the "post" form. A corresponding "prior" variational principle, using the Green operator Gi +), is given by JR2]
(+)
(+)
(-)t
vl
• i..l*'q
•
(2.26)
REARRANGEMENT COLLISIONS
551
Indeed, using (2.21) and (2.19), [R2] = T ie
(2.27)
~[R2] = O.
(2.28)
and, from (2.17) and (2.15),
From (2.22) and (2.26) it is possible to construct a symmetrical variational principle [R3] -- ½{[R1]+ [R2]},
(2.29)
whose connexion with Lippmann's 7) symmetrical stationary expressions can be easily established. The bilinear form of the variational principles is obtained by taking the matrix elements of the stationary expressions between two "free" states ~l and Re of equal energy. Thus, making use of (2.15) and (2.16), we have
[Tid = < ~'~->1v~{1 + 6~+>(~- ~)} I~i> + <~1 v,I ~'~+>> - (,e~-)l v f - vf 6~+~1 ~ + ) )
(2.30)
for the post form, and [T~f] = (~rI{I+(Ve-Vi)G~+>}V~IT~+>)+(~e~-)IVd~i) - < ~ - > I E - ~ G~+>~I~'~+>> (2.31) for the prior form. Lastly, the fractional form of the variational principles is obtained by substituting kv~+) -~ A ~ +),
~ - ) ~ BT~ -)
(2.32)
in (2.30) and (2.31), and varying the amplitudes A and B such that [Tit] be stationary. Starting from (2.30), we obtain in this way for the post form [Tif] = (~f[ Vf[~ + ) > ( ~-)1Vf{1 + G~+)(Vt- Vf)}[~i)
(2.33)
<~e~->lvf - T., r f u~<+>v,~<+>\ f rfl.t i / while (2.31) leads to the prior form
I-T~f] -- < ~ ' $ - > l E l ~ & < ~ d { l + ( V r Vi)G~+)}Vil~ + ) )
(2.34)
2.3. MICRO-REVERSIBILITY The symmetrical aspect of the bilinear and fractional variational principles in the initial and final quantities is directly connected to the micro-reversibility principle. Indeed, let us recall that if K is the time reversing operator lo) and if H is such that KHK + = H,
(2.35)
552
c. JOACHAIN.
one has for the transition matrix element the micro-reversibility relation ll) TKf, K i
= Tif.
(2.36)
Now, it is very desirable to preserve this relation when one writes variational expressions for Tif. This is precisely the case for the proposed pairs of variational principles (2.30)-(2.31) and (2.33)-(2.34). Indeed, as 11) ~Ki ~-~ K ~ i ,
~,(-~)~,
=
r~±),
~)Kf ---- K ~ f ,
~,(~)
=
(2.37)
r ~ ~)
and with the aid of antilinear operator algebra, it can be shown 9) that
[Txf, r,]post = [T~f]pr,o,,
ETxf, Ki]prior
-~-
ET, dpo,,.
(2.38)
thus showing that the proposed variational principles have the desired microreversibility properties. 2.4. DISCUSSION
The pair of fractional variational principles (2.33)-(2.34) we have obtained satisfies all the conditions of a generalization of Schwinger's variational method. Indeed: 1) These variational principles are independent of the amplitude of the trial functions, and of the values of these functions where the interaction potentials vanish. 2) They give directly the stationary value of the transition amplitude Tie. 3) They use the functions ~I +) and ~ - ) , solutions of the Lippmann-Schwinger equations (2.7). 4) They satisfy the micro-reversibility principle because of their form entirely symmetrical in the initial and final quantities. 5) Finally, as expected, they give the fractional form of Schwinger's variational principle in the particular case of direct collisions for which Vi = Vf = V. If in the fractional expressions (2.33)-(2.34) we replace in first approximation the unknown wave functions ~ + ) and We ~-) by the Born free waves #i and 4~f, we obtain the following approximate expressions (which we denote by the subscript B):
[Tlf]Bpost = Tt(fB)
[1+
[Tif]B prior = Ti(B) [I -~"
7
'
7 <~d v~- E a~ +)vd~i>J '
(2.39)
(2.40)
where (2.41) is the transition matrix element in first Born approximation.
REARRANGEMEbrr, COLLISIONS
553
It is interesting to compare the formulas (2.39) and (2.40) with the Born development. If, rin the right,hand side; the second term o f the bracket - whose numerator is o f the second order in the interaction potentials - is neglected, one recovers the first Born approximation. As an offset, if this term is maintained but the second order part o f the denominator is neglected, one finds the second order Born approximation. Together with their many formal advantages, the variational principles proposed here have unfortunately also a major inconvenience, which is the evaluation o f the second order matrix elements appearing in the stationary expressions. This evaluation is often complicated and is in fact the actual limitation of the method. 2.5. EXTENSION TO COUPLED WAVES Let us assume that the interaction potentials Vi and Vf may be written as Ui, fdl-~i,f •
(2.42)
H (i1, )f ffi H i , f.~- Ui, f
(2.43)
~'i,f =
Let us define the new Hamiltonians such that
H(l)v(±) i,f~a'i,f
=
EX(+) i,f,
(2.44)
where X[ ±) and Xft±) are the supposed known coupled * waves having the required asymptotic behaviour. We shall also need the Green operators G~l)(±) and G~1)(±), associated respectively with Hiix) and H(f1), and the Moiler operators t2 (1)c±) i,f
~
1 + G(±)Wi r.
(2.45)
Now, let us assume that the potentials Ui and Uf are such that Ui ~ Ui(ra),
Uf -- Uf(rp),
(2.46)
where the vectors r~ and rp connect respectively the centres of mass o f the systems A, B and C, D. Then, it is well known 12) that for a rearrangement collision, the transition matrix element may be written Tif
(2.47)
= <~['l~-)l~ilX~+)> =.
Now, by means o f a method parallel to the one already developed for the free waves, we can construct variational principles with coupled waves. We obtain for the fractional form 9) (-) (+) (-) r'rflpo,t < X f L : i j= IWfl~ ><~f IWf{l+G~')(+)(W~-Wf)}lXi +)> (2.48)
<'/'~-)lWf- Wf ,.,f":-(1)(+)w' ~(+)\,,f, :i /
[T~d,,,io, = <'e~-)lWflX~+)>
'
~)6~1)(+))~I'I'~+)>
(2.49)
I.,V t2.(1)( + ) [,12" I IV( + ) k vr i ~'mi rril~t i /
t We recall the distinction between the coupled waves X~(±) and X~(±) and distorted waves
which correspond to a particular choice of the potentials Ut and Ui (see for example ref. ,3)).
554
c. JOACttAIN
If the unknown wave functions T~+) and T~-) are replaced in these expressions by the coupled waves X~+) and Xf(-), we obtain the approximate expressions (which we denote by the subscript C) I
[Tif]Cpost= T[fcwB) I +
/ Y ( -)11.~ f~.(1)(+) I,I/" I Y ( + ) \
\~f
l"fvf
/XX (f - ) IV,,f - -
/
"l
-I
"il"Xi W,,f ~v f( 1 ) ( + ) w ' lr v fl.eXY (i+ ) \ / / - I
(2.50) '
IWfGi- - ~ - ~W~lXi > ~ , [Tif]Cp,io, --- T}cwe) i 1 + / y (
I rri -- rri
vi
rr il"x i
(2.51)
/-J
where
T~(rCWB) => =
(2.52)
is the coupled wave Born approximation transition amplitude. Comparison of formulas (2.50) and (2.51) with the Born development of coupled waves is analogous to the one made for the free waves in sect. 2.4. An interesting particular case occurs when the potentials Ul and Uf, already subjected to the conditions (2.46) satisfy also the supplementary conditions of the distorted wave method 12). Then, the coupled waves X~+) and Xrc+) become distorted waves and the approximate formulas (2.50) and (2.51), limited to the first order terms in the potentials Wi and Wf, give a variational justification of the distorted wave Born approximation (DWBA).
3. Application to Three-Body Problems 3.1. G E N E R A L
FORMULAS
Let us consider a system of three particles of masses ml, m2 and m3. For such a system, there exist three types of binary rearrangement collisions: pick-up, stripping and knock-out (or exchange) reactions. We have, for the unperturbed initial and final Hamiltonians, h2 Hi =
-- --
2Mi
zlr,"]- h i ( r 1 ) ,
(3.1)
+h'(r2)"
(3.2)
h2
n, =
2M,
In these formulas, Mi and M e are the initial and final reduced masses, r, and r# are the "relative" vectors fixing the relative position of the centres of mass before and after the collision, r 1 and r2 are the two vectors concerning the initial and final bound systems, and h i and he are the inner Hamiltonians of the initial and final bound systems, such that Rialto.i = 8m, i~/m,i,
hflPn, f = 8n.fl//n,f .
(3.3)
REARRANGEMENT COLLISIONS
555
We assume that the initial and final bound systems are respectively in the states m o and no and we simplify the notations by putting @ i ( r l ) ------@too, i ( r x ) , ~i ~ 8mo. i ,
~/f(r2) -- @.o,f(r2)
(3.4)
~f = e.o, e"
(3.5)
The total energy available in the centre of mass system may be written or
E = e lq- h2k~
E = el+ h2k2--,
2M i
(3.6)
2Mr
where kl and kf are respectively the initial and final propagation vectors. Finally, we have for the initial and final free waves
3.2. C O N S T R U C T I O N
• i(rl, r ) = d~t'"Oi(rl),
(3.7)
• f(rz, r#) = ei~f'"Of(rz).
(3.8)
OF THE GREEN FUNCTIONS
The Green functions ff~+) and ff~+) corresponding respectively to the Green operators G~+) and G~+) in the position representation are given by the formulas ~ + ) ( r l , r~; r l' , r'~) =
- - - M- i
2zch2
~m~lra"(rl)@m'i(rl)' i
*
eikm, d r = - r ' ~ l
[r~--r~[
,
eikm
'
(3.9)
fir# - r ' # l
~f<+)(r2,
,#,. r2, ' r~) =
Mr 2~-~
, , . ~"' f(r2)d/"'f(r2)
[ r # - r~[
(3.10)
In these formulas, the symbol ]l means a summation over discrete states and an integration over the continuum states. Furthermore, the quantities km, i and k~,f are defined by the relations h2k 2 E = •m,i-3 t- m , i (3.11) 2Mi 2 2
= e,,e+ --h kn, f
(3.12)
2Mf with kl - kmo, i,
kf =- kno, f.
(3.13)
The construction of formulas (3.9) and (3.10) can be derived from the structure of the Hamiltonians H i and Hf, it is easy to verify that (E - - Hi) ~ (E--
+
)(rl, r~ ; r~, r~) ---- tS(rI -- r~)t~(r~-- r'.),
(3.14)
r l , r;) = O(r z -- r'2)6(r#-- r~),
(3.15)
Hf)~+)(r2,
r#;
556
c. JoAcru, rN
as expected. Let us note that the Green functions (3.9) and (3.10) introduced into the matrix elements appearing in the calculation of the transition amplitude, take into account the possibility of virtual transitions accomplished by one or the other bound system. In this manner, all the intermediate states of the initial or final system contribute to the transition amplitude through the Green functions (3.9) and (3.10). This is an entirely new process compared to the two-body problem, and it is generally denoted as the polarization effect, by analogy with a similar phenomenon occurring in the study of bound state problems ,4, ~5). 3.3. V A R I A T I O N A L F O R M U L A S
With the aid of the Green functions (3.9) and (3.10) and of the variational formulas (2.39) and (2.40) established for the choice of trial functions ~'/}+) ~--" ~i,
(3.16)
~/~-) = ~ f ,
one obtains for the three-body reactions considered the formulas V
L =
[1÷
11
q
(3.17)
T~g5--I2-I F1
(3.18)
where the second order expressions I,, 12, F, and Fz are explicitly given by
x Vi(r~, r~)@i(r~, r'.),
,2 = f dr~ydr~ f dr~f ' *
(3.19)
r~)~(r,, r2)~+'(,x, r.;r,', r') x E(ri, ri)ei(ri, tO, (3.20)
w, = f dr, f dr; f d,2f drY2¢~(rz, ra)Vf(rl, rz)ff[+)(rz, r,;
r~, r~)
x v,(4, r[)~i(rl, 6), (3.21)
=f dr,f drlf
f de[¢?(r2,ra)Vf(rx,r2)fg +)(r2,r,;r[,r'~) x Vf(rl, rl)¢i(rl,
r'),
(3.22)
and where the free waves @i and @f are given by (3.7) and (3.8). The interaction potentials Vi and Vr depend naturally on the reaction considered. As to the relative vectors r, and ra (or r" and r~) they depend on the vectors r t and r2 (or r~ and r[) according to the reaction considered. For example, in a pick-up reaction where
REARRANGEMENT COLLISIONS
557
the particle 1 is incident on the bound system 2-3 and where the bound system 1-2 is formed in the final state, we have r~ = ~ r l - - r 2 ,
(3.23)
rp = r l - - f l r 2,
where m3
= - - , m2 + m3
ml
fl -
.
(3.24)
m l + m2
Finally, let us remark that the Second Born approximation corresponding to the reactions considered is given by Y(? 2) = T(? ) + I t
(3.25)
Y(fB2) -- Ti(B) --1-F 1
(3.26)
in the prior form, and in the post form. 3.4. NUCLEAR PICK-UP AND STRIPPING REACTIONS
According to the micro-reversibility principle (2.36), the transition amplitude for a stripping reaction is the same as the one relative to the corresponding pick-up reaction. We may thus treat indifferently one of these two reactions. As an example, we study the following model o f (p, d) pick-up reaction. We consider a proton incident on a nucleus composed of an inert and infinitely heavy core and an optical neutron. The interaction of the proton with the core is neglected throughout the reaction (stripping hypothesis). We neglect also the Coulomb effect, so that our treatment works also for the (n, d) pick-up reaction. The interaction potentials between the incident proton and the neutron, and between the neutron and the core are chosen as Vi = - A f ( r 2 ) ,
Vf = - - B 6 ( r l ) .
(3.27)
We assume also that the optical neutron is initially bound in the fundamental state o f an isotropic harmonic oscillator. The initial nucleus is thus described by the Gaussian wave function ~/i(rl)-- Nie -~1"12
(3.28)
with
The parameters Yi and B are chosen so that the mean value of r 2 and the binding energy ei of the optical neutron are correct. The final deuteron is described in its fundamental state by the Gaussian function ~f(r2)----- Nfe rrr22,
(3.30)
558
c. ~OACHAIN
where
This function is easier to manipulate in the subsequent calculations than the more precise Hulth6n function. The parameters ~'f and A are obtained by imposing that the quantity Ro2 = (~bfl(½r2)2l~bf)
(3.32)
and the binding energy gf = - - 2 . 2 3 MeV be correct. The deuteron having only one bound state, it is clear that the post formulas (3.18) and (3.26) are simpler than the prior ones. All the intermediate states of the deuteron are continuum states that we describe by plane waves ~/'.,f(r2) = (2n)-~e 'h'''',
(3.33)
corresponding to intermediate energies h2 8n, f(k') = - - k '2,
(3.34)
M where M is the nucleon mass. Integration over continuum states is thus performed by integrating over all the k' space. Before using the formulas of subsect. 3.3 where Mt = M,
M f = 2M,
= 1, ~ = ½,
(3.35) (3.36)
we must fix the total energy E of the reaction. We have chosen E = 0, an energy for which the outgoing deuteron is emitted with a kinetic energy of 2.23 MeV equal to its inner binding energy. This value of E has two advantages: first, to be in a sufficiently low-energy region, where the variational method earl significantly improve the Born approximation; secondly, to simplify the calculations by giving in the expres: sion of the Green function (3.10) k,,f(k')
(3.37)
= 2ik',
as it is seen with the aid of formulas (3.12), (3.34) and (3.35). We now proceed to the calculation of the transition amplitude in first Born approximation. As the two inner wave functions (3.28) and (3.30) are approximate ones, there is a post-prior discrepancy. The prior form is given by
~ , , t o , = <~flVil~i>
--
(
-ANiNf ~ /~e-Kt2/4rt, \~i /
(3.38)
REARRANGEIdENT COLLISIONS
559
where K 1 is the magnitude of the vector K, =
ki-kt,
(3.39)
while, for the post form, Ti(B) r post = <~e[VfIOi> = -
' BNI Ne (TC~'e-X~2/4,, \~f/
(3.40)
where K2 is the magnitude of the vector
(3.41)
K2 = ½ k t - k,.
It is also possible to eliminate the post-prior discrepancy by using the Schrbdinger equations [h2 [- -2~ A,l + Vf(rl)] d/,(rl) = e,~/,(,,), (3.42)
[-~A,2+Vi(r2,1d/f(ra)=~fOe(r2),
(3.43)
which are verified approximately by the inner wave functions (3.28) and (3.30) for the choice (3.27) of F i and Ff. Thus -Adi(r2)e-"'" ~ [-~ 3,t +eli e -r'22,
(3.44)
-Bdi(rl)e-'"12 ~' [ 2 ~ - A " + ~ ' I e-'m2
(3.45)
and, substituting respectively in the formulas (3.38) and (3.40), we obtain 2q(ifn ) = -NiNfn3(~i~e) -~ ~
K~-,r]
e-rl~/'"e-r221"",
T~n) = --NiN,~a(,,,f)-' ~2h-hMK~-~,l e-'"/4"e-~:~2/" ,
(3.46) (3.47)
which are equivalent, as is easily shown by using the relations (3.6). Table 1 illustrates the results obtained by these three calculation methods of the first Born transition amplitude. As a comparison, we have also calculated T/~fn) in the third method for two other choices of wave functions: the Hulth6n wave function
[tn(n+llo~]'e-"~-e -''=
~ke(r2) = L 2 ~ J
g
,
(3.48)
560
c. JOACICMN TABLE 1
Nuclear pick-up and stripping reactions at total energy E = 0 Born approximations for the transition amplitude (in units 10 -a erg • fln 8) Schr6dinger equation 0
(degrees)
Prior f o r m
Post f o r m Gaussian
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
--3.39 --3.34 --3.17 -- 2.92 --2.61 -- 2.28 --1.94 -- 1.63 -- 1.35 -- 1.11 --0.914 --0.757 --0.635 --0.541 --0.472 --0.422 --0.388 --0.369 --0.363
--3.26 --3.19 --3.00 -- 2.72 --2.38 -- 2.01 "1,66 -- 1.34 -- 1.07 -- 0.849 --0.672 --0.536 --0.434 --0.358 --0.303 --0.265 --0.240 --0.226 --0.221
--4.50 --4.43 --4.23 -- 3.90 --3.44 -- 2.89 --2.32 -- 1.79 -- 1.33 ,-- 0.972 --0.701 --0.506 --0.370 --0.278 --0.216 --0.176 --0.151 --0.137 --0.133
Hulth6n (n = oo)
Hulth6n (n = 7)
--3.03 --2.98 --2.83 --2.61 --2.33 --2.03 --1.73
--3.54 -- 3.47 --3.29 --3.01 --2.66 --2.29 -- i.93
-- 1.45
-- 1.59
--1.20 --0.991 --0.816 --0.676 --0.567 --0.483 --0.421 --0.376 --0.347 --0.330 --0.324
-- 1.30 -- 1.05
--0.855 --0.698 --0.577 --0.486 --0.419 --0.371 --0.340 --0.322 --0.317
where n = 7 and ~ = 0.231 fm " t , and the function t e - ~r2 ~ , /'2
Ipf(r2) =
(3.49)
corresponding to a zero-range neutron-proton interaction potential As a contact potential is clearly a better approximation for the neutron-proton interaction Vt than for the neutron-core interaction Vf, we shall adopt in the following calculations the prior form for Tiree) with the Choice (3.30) o f Gaussian wave function describing the fundamental state of the deuteron. The differential cross section in the first Born approximation is given by Mz tr(al) = 2~n 2 h
kf (Ti¢~))2'
(3.50)
ki
while, for the corresponding stripping reaction, we have M 2 ki (Titfa))2. a¢"1) = 2n2h 4 kt
(3.51)
The numerical values o f these cross sections are given respectively in tables 4 and 5.
REARRANGEMENT COLLISIONS
561
Let us now calculate the second order terms occurring in the post formulas (3.18) and (3.26). Starting from (3.21) and (3.22) and taking into account the relations (3.10), (3.23), (3.35) and (3.36), we have
Fl =
MABNiNf fdri fdr'l fdr2 fdr;e-'h"("-½"=)e -''2z 7~h2
d
d
2~
x 3(rl)
g
d
' e'k"'4''-*''-'',+½"'21 6(r2) e't'" (,',--',)e-,,,','
*
~/" f(r2)~/n' f(r2) Irl - - ½ r 2 -- r1-3u ½1M2I
(3.52)
n
and 7~h 2
X ~(r,)
(3.53)
~/,,f(r2)~/n* f(r2) [rl --½r2 --r~ --~½r21 n
Let us decompose the summation :$, into its discrete part (d) limited to the fundamental state of the deuteron and its continuum part (c) containing the contribution of the continuum states of the deuteron. Thus F1 = --tl~'d() T'tl-L 1~'(¢),
F2 = --21~( d) -,-L-2~'(°)•
(3.54)
We evaluate directly the "discrete" terms F~d) and F2¢d) in configuration space. Details are given in ref. 9). Putting tZt --
8~i~f , 7i -{- 8yf
K 3 -- ½(kf-ki),
1 K4 _ - (Syfki-l-Tikf), 8~)f "~-~i
(3.55)
2 i K5 = ~ki--I--6kf,
(3.56)
we obtain Re Foxd) =
MABNi(Nf) 3 1 x
--
[(kf+ K,)tFl(1; 3; - ( k t +
K4
K,)21~ ')
- (ke- K,)~F~(1 ; 3; - ( k f - K,)2/4~')], Im FOx a) = --
x2MABNt(Nf)3 a'~'h 2
(3.57)
( 1 ~ e_K32/(Syt+~, D \27f + ¼~i/ X - -1
K4
[e-
(kf-- K4)214=" - - e -
(kf + K4)2/4"= ' ] ,
(3.58)
562
C. JOACHAIN
3MB2Ni(Nf) 3 ( ~ ~e-K22/12~. yf h2 \~f]
Re Ft2d) -
x 1 [(½kf+ Ks)IF1(1; 3; _(½kf + Ks)2/~yf) Ks - (½kf- Ks)~Fx(1; k; - (½kf-- K5)2] 3~Tf)], Im F¢2d) =
-
2~2MB2N,(Nf) 3 ( 1
~(3
(3.59)
~e_r~/~2~ ~
× - -1 [ e - (~kf - Ks)2/~Yf - - e - ( -~-kf+ KS)2/~,f].
(3.60)
Ks Evaluation of the terms F~°) and F~c), which are real, implies new difficulties due to the integration over the "intermediate" k' propagation vectors. After some manipulations, we reduced this calculation to the evaluation of a one-dimensional integral, which was performed by the IBM 1620 machine of the University of Brussels t The numerical values of the different second order terms are collected in table 2. Comparing with the first order terms of table 1, we see that the two types of terms are of the same order of magnitude. With the help of the values of F~d), F~~),Fc2a), and F~° we can obtain the transition amplitudes (3.18) - denoted by TiVf ar - and (3.26). They are given in table 3. From these values it is easy to obtain the differential scattering cross sections ava, and ata2) collected in table 4 for the pick-up reaction and in table 5 for the corresponding stripping reaction. 3.5. DISCUSSION OF THE RESULTS
Table 3 collects our results corresponding to the transition amplitude while tables 4 and 5 concern respectively the differential cross sections for pick-up and stripping. We have compared the first Born approximation (prior form), the second Born approximation and the variational method. The numbers in the brackets figuring in tables 4 and 5 correspond to the same quantities but without the contribution of the continuum parts F~°) and F2to). This is to emphasize the specific influence of the discrete and continuum terms on the cross sections considered. Examination of these tables indicates that the introduction of second order terms considerably modifies the results of the first Born approximation. The transition amplitude gets an imaginary part while the differential cross sections are either strongly increased (second Born approximation) or reduced (variational method). t Strictly speaking, the term F,(¢) cannot be evaluated directly because the particular choice (3.27) of V~ leads to divergent integrals. To remedy this situation, we made the supplementary hypothesis
F~o _ F~° Re Ft2d} Re Ft~d)
REARRANGEMENT COLLISIONS
563
TABLE 2 N u c l e a r p i c k - u p a n d s t r i p p i n g r e a c t i o n s a t t o t a l e n e r g y E ---- 0. S e c o n d o r d e r t e r m s o c c u r r i n g i n t h e calculation of the transition amplitude
F1 0 (degrees)
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
F~ (10 -s e r g " f m 8)
R e F~ d)
I m F~ d)
- - 1.80 --1.80 --1.77 --1.73 - - 1.68 - - 1.62 --1.56 --1.48 - - 1.41 --1.33 - - 1.26 - - 1.19 - - 1.12 - - 1.07 - - 1.02 --0.983 --0.956 --0.939 --0.934
--2.79 --2.78 --2.77 --2.75 --2.73 --2.70 --2.66 --2.63 --2.59 --2.54 --2.50 --2.47 --2.43 --2.40 --2.37 --2.35 --2.33 --2.32 --2.32
F~ c)
R e F2(d)
I m F2(d)
--0.944 --0.942 --0.937 --0.929 --0.919 --0.906 --0.891 --0.876 --0.860 --0.843 --0.827 --0.812 --0.798 --0.786 --0.775 --0.767 --0.761 --0.757 --0.756
--3.37 --3.35 --3.30 --3.21 -- 3.10 --2.97 --2.83 --2.68 --2.52 --2.37 --2.22 --2.09 - - 1.97 -- 1.87 -- 1.78 - - 1.71 - - 1.66 -- 1.64 - - 1.63
--2.35 --2.35 --2.34 --2.32 --2.30 --2.27 --2.23 --2.20 --2.16 --2.12 --2.08 --2.04 --2.01 - - 1.98 - - 1.95 -- 1.93 -- 1.91 -- 1.90 - - 1.90
F(2°) - - 1.76 --1.76 --1.74 --1.72 - - 1.69 - - 1.66 --1.62 --1.58 - - 1.54 --1.50 --1.47 - - 1.43 - - 1.40 - - 1.38 - - 1.35 -- 1.34 - - 1.33 - - 1.32 - - 1.32
TABLE 3 N u c l e a r p i c k - u p a n d s t r i p p i n g r e a c t i o n s a t t o t a l e n e r g y E = 0. T r a n s i t i o n a m p l i t u d e T(? )
T(fB2)
0 (degrees)
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
rvarif
(10 -a e r g " f m a)
--3.39 - - 3.34 --3.17 --2.92 --2.61 --2.28 - - 1.94 - - 1.63 - - 1.35 - - 1.10 --0.914 --0.757 --0.635 --0.541 --0.472 --0.422 --0.388 --0.369 --0.363
R e Ti(~2)
I m Ti(fB2)
--6.14 - - 6,07 --5.88 --5.59 --5.22 --4.80 --4.39 - - 3.98 -- 3.61 -- 3.28 - - 3.00 - - 2,76 --2.55 --2.39 --2.27 --2.17 --2.10 --2.07 --2.05
--2.79 - - 2.79 --2.77 --2.75 --2.73 --2.70 --2.66 -- 2.63 - - 2.59 - - 2.54 -- 2.50 -- 2.47 --2.43 --2.40 --2.37 --2.35 --2.33 --2.32 --2.32
R e Tvarif d: 1.I0 + 1.05 +0.915 +0.714 +0.495 +0.294 +0.137 + 0.0301 --0.0338 - - 0.0663 --0.0789 - - 0.0802 --0.0764 --0.0705 --0.0647 --0.0597 --0.0560 --0.0538 --0.0530
I m T~fa ' --0.650 --0.583 --0.411 --0.196 --0.00439 -]-0.133 +0.210 + 0.240 +0.239 + 0.222 +0.199 +0.175 +0.154 +0.136 +0.122 +0.111 +0.104 +0.0994 +0.0980
564
C. JOACHAIN
TABLE 4 Nuclear pick-up at total energy E = 0. Differential cross section
(degrees) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
(10 -~ b/sr) 105 102 92.0 78.0 62.4 47.4 34.4 24.2 16.6 11.3 7.64 5.25 3.68 2.68 2.03 1.63 1.38 1.25 1.21
416 408 387 355 317 277 241 208 181 158 139 125 114 105 98.3 93.5 90.1 88.3 87.6
(318) (312) (294) (268) (237) (206) (177) (151) (130) (114) (100) (90.1) (82.2) (76.2) (71.7) (68.4) (66.2) (64.9) (64.5)
15.0 13.3 9.20 5.01 2.24 0.952 O.576 0.534 0.533 0.492 0.420 0.340 0.271 0.215 O.174 0.145 O.127 0.117 O. 114
(67.3) (62.1) (48.7) (32.3) (18.2) (8.77) (3.74) (1.51) (0.651) (0.347) (0.233) (0.178) (0.142) (0.116) (0.0962) (0.0821) (0.0728) (0.0675) (0.0657)
TABLE 5 Nuclear stripping at total energy E = 0. Differential cross section 0
O(al)
~(a2)
(degrees) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
t~var (10 -~ b/sr)
165 160 144 122 98.0 74.4 54.1 38.0 26.1 1"7.7 12.0 8.23 5.78 4.20 3.19 2.55 2.16 1.96 1.89
653 641 607 557 497 436 378 327 283 248 219 196 178 165 154 147 142 139 138
(499) (489) (461) (420) (372) (323) (277) (238) (205) (178) (158) (141) (129) (120) (112) (107) (104) (102) (101)
23.5 20.8 14.4 7.88 3.51 1.49 0.904 0.839 0.837 0.773 0.659 0.534 0.425 0.337 0.273 0.228 0.199 0.183 0.178
(105) (97.5) (76.4) (50.7) (28.5) (13.8) (5.87) (2.37) (1.02)
(0.544) (0.366) (0.279) (0.223) (0.182) (0.151) (0.129)
(o.114) (0.106) (0.103)
REARRANGEMENT COLLISIONS
565
It is clear that no definite conclusions can be drawn from this exploratory calculation. However, the three following remarks can be made. At the energy E = 0 considered, the second order terms in the interaction potentials are of the same order of magnitude as the first order terms. This result indicates that it is difficult to have confidence in semi-qualitative arguments justifying the first Born approximation at low energies for nuclear direct reactions. Let us note also the importance o f continuum terms in the calculations of the second order terms FI and F2 and their marked influence on the cross sections. Results furnished by the second Born approximation are clearly too high and must be discarded. This situation is similar to the one observed in the variational study o f nuclear potentials 16), The variational method proposed here gives smaller differential cross sections than those of the first Born approximation, the ratio o f the maxima being 7. This result is encouraging, because it is in concordance with the conclusions of the distorted wave method t. The angular distribution has the same satisfactory form as in the first Born approximation with a reduction at small angles analogous to the one observed in the distorted wave method 17). The comparison at large angles is useless because we made the stripping hypothesis. Finally, an important point concerns the behaviour of the different methods at high energies. To this end, we have analysed the first order terms together with the discrete second order terms o f the transition amplitude versus the total energy E. Our results indicate that the second order terms considered remain steadily important even for k I ---, ~ . This last conclusion must be compared with similar results obtained by Drisko is) and Bates and Mac Carroll 19) in the study o f atomic rearrangement collisions. It is evident that such results furnish a supplementary support for a detailed examination of the validity of Born's first approximation at high energies for rearrangement collisions 20, 21), both in atomic and nuclear physics. This work constitutes a part of a thesis which has obtained the Louis Empain Prize 1963 for Physical Sciences. The author wishes to express his gratitude to Professor M. Demeur for his continued advice and guidance. Thanks are also due to Professor L. Rosenfeld for helpful comments. * For example, Horowitz and Messiah 17), using the distorted wave method, obtained for several stripping reactions at l = 0 reduction factors going from 2.42 to 5.81.
References 1) 2) 3) 4) 5) 6)
W. Kohn, Phys. Rev. 74 (1948) 1763 L. M. Delves, Nuclear Physics 29 (1962) 326 L. Spruch and L. Rosenberg, Phys. Rev. 125 (1962) 1407 B. A. Lippmann and J. Schwinger, Phys. Rev. 79 (1950) 469 S. Borowitz and B. Friedman, Phys. Rev. 89 (1953) 441 M. C. Newstein, Technical Report no 4. Project DIC 6915 MIT
566 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
c.
JOACHA|N
B. A. Lippmann, Phys. Rev. 102 (1956) 264 C. Joachain, Bull. Soc. Belg. Phys. (1962) s6rie III, no 2, 133 C. Joachain, Th~se de doctorat, Universit6 Libre de Bruxelles (1963) A. M. L. Messiah, Mdcanique qnantique (Dunod, Paris, 1960) Ch. 15 ibid., Ch. 19, p. 745 E. Gerjuoy, Ann. of Phys. 5 (1958) 58 R. H. Bassel and E. Gerjuoy, Phys. Rev. 117 (1960) 749 M. Demeur and C. Joachain, Nuclear Physics 17 (1960) 329 C. Joachain, Nuclear Physics 7,5 (1961) 317 C. Joachain, Nuclear Physics 64 (1965) 529 J. Horowitz and A. M. L. Messiah, J. Phys. Rad. 14 (1953) 695 R. M. Drisko, Thesis, Carnegie Institute of Technology (1955) D. R. Bates, Collisions between atomic systems, Atomic and molecular processes, edited by D. R. Bates (Academic Press, N.Y. - London 1962) p. 596 20) R. Aaron, R. D. Amado and B. W. Le~, Phys. Rev. 125 (1961) 319 21) E. Gerjuoy, Revs. Mod. Phys. 33 (1961) 544
VARIATIONAL TREATMENT OF REARRANGEMENT COLLISIONS C. J O A C H A I N Universitd Libre de Bruxelles t Received 16 July 1964 Abstract: Variational principles based on the formal theory of scattering and generalizing Schwinger's method for direct collisions are proposed for the treatment of binary rearrangement coUisions. These principles are established both for free and coupled waves, and are compared with the Born development. Application is made to three-body problems, the case of nuclear pick-up and stripping reactions being examined in detail on a specific example.
1. Introduction The variational study of rearrangement collisions has hitherto been approached from tWO different points of view. The first one, initially proposed by Kohn 1) and developed recently by Delves 2) and Spruch and Rosenberg a) generalizes the variational methods for determining the phase shifts. These methods are based on the differential form of the variational principles. They lead to stationary expressions for the elements of the S matrix, the reaction matrix or the derivative matrix. The second point of view, which we adopt here, consists in a generalization of the Schwinger variational principle for the transition amplitude 4). This approach, based on the integral form of the variational principles, has been illustrated successively by Borowitz and Friedman s), Newstein 6) and Lippmann 7). We give in this work new variational principles s, 9), based on the formal theory of scattering and apply them to specific examples. In sect. 2, we establish these variational principles both for free and coupled waves. Sect. 3 is devoted to the application to three-body problems, the case of nuclear pick-up and stripping reactions being examined in detail. Our numerical results are collected in sect. 4. 2. Variational Principles for Rearrangement Collisions 2.1. BASIC F O R M U L A S
Let us consider a rearrangement collision of the kind A+B ~ C+D, t Charg~ de recherches du Fends National de la Recherche Scientifique. 548
(2.1)
REARRANGEMENT. COLLISIONS
549
where there is no creation or destruction of particles. For the initial system (A+B), we have n = H I + Vi, (2.2) where H i is the unperturbed Hamiltonian, particles A and B and where
Vi
is the interaction potential between
Hi ~i = E~i,
(2.3)
#j being the free wave corresponding to the Hamiltonian H i and the energy E. In the same manner n = H f + Vf,
(2.4)
H, ff'f = E # , .
(2.5)
where Denoting by T~f the transition matrix element, we have Tie
:
<~flVflW}+)> = <~P~-)lVil~i>,
(2.6)
where ~ + ) and ~fc-) denote the solutions of the Lippmann-Schwinger equations ~"/(i ,+f ) ~-" ~ i , f m~-~f'J~("I') J i , f TI vi, f ~if/(+) ti, f ,
(2.7)
where G~±) and G(±) are the Green operators associated respectively with Hi and Hr. These equations have the well known formal solutions ~YJ(±)
( l "]- G ( ± ) V i f){lJi: f ,
(2.8)
where the Green operator Gt+) corresponding to H is given by G (±)
G~f)+ Gt+)E , i , f i , f G(±)•
(2.9)
We now introduce the Meller wave operators O(±) i,f
=
l+G(±)Vi, f,
(2.10)
such that, from (2.8),
III(±)i,f = ~'~)~i,f" We
(2.11)
shall also need the transition operator T i f _- Vf+ VfG(+)Vi,
(2.12)
which gives Yif =
<~flTifl~i>.
(2.13)
From (2.10) we have *
Vf[O~+)-1] = [Ol-)-1]tvi t The dagger denotes
Hermitian conjugation.
(2.14)
550
C. ,IOACI.IAIN
while, from (2.9),
1,
[ i _ ~~(+)w/ Ic~(±) 1 , f • i, fA~ai, f
(2.15)
from which one can deduce the relations 7) (2.16)
I] =
[f2~- ) - 1]'[1 - Vi O}+)l = VeG}+), a~-)'[Vf - v" r f ~(+)rzlr°(+) ,~,f r fdLa~i vi
--
II
(2.17)
~
a~-)' l/f O~+)Vi
(2.18)
=
~ f G} + ) Vi ~e~}+ ) .
(2.19)
" |/''i
Let us note also that the transition operator defined by (2.12) may be written, with the aid of (2.10),
T i f = ~f~'~[+)
(2.20)
or also, by making use of (2.14), T if = ~ ( f - ) * ~ + ( V f - Vi). 2.2. V A R I A T I O N A L
(2.21)
PRINCIPLES
Let us consider the expression JR,] = a~-)'VfGtf+)Vi+ l/fQ~+)-fa~-)'[~-~G~+)~][O{+)-l].
(2.22)
Using (2.18) and (2.20) we see that its exact value is given by [R,] = T i'.
(2.23)
Moreover, this expression is stationary for independent variations of the Moller wave operators around their correct values. Indeed, 6[R,] = 6a~-)t~{G~+)l/i- E1 -- G~+)l/f][a~ + ) - 1]} + { 1 - a ~ - ) ' [ 1 - l/fG~+)]}~6a~ +) (2.24) and from (2.16) and (2.15) 6 [R 1] = O.
(2.25)
Thus the expression (2.22) provides a variational principle for the operator T if. As it makes use only of the Green operator G~+), we shall call it the "post" form. A corresponding "prior" variational principle, using the Green operator Gi +), is given by JR2]
(+)
(+)
(-)t
vl
• i..l*'q
•
(2.26)
REARRANGEMENT COLLISIONS
551
Indeed, using (2.21) and (2.19), [R2] = T ie
(2.27)
~[R2] = O.
(2.28)
and, from (2.17) and (2.15),
From (2.22) and (2.26) it is possible to construct a symmetrical variational principle [R3] -- ½{[R1]+ [R2]},
(2.29)
whose connexion with Lippmann's 7) symmetrical stationary expressions can be easily established. The bilinear form of the variational principles is obtained by taking the matrix elements of the stationary expressions between two "free" states ~l and Re of equal energy. Thus, making use of (2.15) and (2.16), we have
[Tid = < ~'~->1v~{1 + 6~+>(~- ~)} I~i> + <~1 v,I ~'~+>> - (,e~-)l v f - vf 6~+~1 ~ + ) )
(2.30)
for the post form, and [T~f] = (~rI{I+(Ve-Vi)G~+>}V~IT~+>)+(~e~-)IVd~i) - < ~ - > I E - ~ G~+>~I~'~+>> (2.31) for the prior form. Lastly, the fractional form of the variational principles is obtained by substituting kv~+) -~ A ~ +),
~ - ) ~ BT~ -)
(2.32)
in (2.30) and (2.31), and varying the amplitudes A and B such that [Tit] be stationary. Starting from (2.30), we obtain in this way for the post form [Tif] = (~f[ Vf[~ + ) > ( ~-)1Vf{1 + G~+)(Vt- Vf)}[~i)
(2.33)
<~e~->lvf - T., r f u~<+>v,~<+>\ f rfl.t i / while (2.31) leads to the prior form
I-T~f] -- < ~ ' $ - > l E l ~ & < ~ d { l + ( V r Vi)G~+)}Vil~ + ) )
(2.34)
2.3. MICRO-REVERSIBILITY The symmetrical aspect of the bilinear and fractional variational principles in the initial and final quantities is directly connected to the micro-reversibility principle. Indeed, let us recall that if K is the time reversing operator lo) and if H is such that KHK + = H,
(2.35)
552
c. JOACHAIN.
one has for the transition matrix element the micro-reversibility relation ll) TKf, K i
= Tif.
(2.36)
Now, it is very desirable to preserve this relation when one writes variational expressions for Tif. This is precisely the case for the proposed pairs of variational principles (2.30)-(2.31) and (2.33)-(2.34). Indeed, as 11) ~Ki ~-~ K ~ i ,
~,(-~)~,
=
r~±),
~)Kf ---- K ~ f ,
~,(~)
=
(2.37)
r ~ ~)
and with the aid of antilinear operator algebra, it can be shown 9) that
[Txf, r,]post = [T~f]pr,o,,
ETxf, Ki]prior
-~-
ET, dpo,,.
(2.38)
thus showing that the proposed variational principles have the desired microreversibility properties. 2.4. DISCUSSION
The pair of fractional variational principles (2.33)-(2.34) we have obtained satisfies all the conditions of a generalization of Schwinger's variational method. Indeed: 1) These variational principles are independent of the amplitude of the trial functions, and of the values of these functions where the interaction potentials vanish. 2) They give directly the stationary value of the transition amplitude Tie. 3) They use the functions ~I +) and ~ - ) , solutions of the Lippmann-Schwinger equations (2.7). 4) They satisfy the micro-reversibility principle because of their form entirely symmetrical in the initial and final quantities. 5) Finally, as expected, they give the fractional form of Schwinger's variational principle in the particular case of direct collisions for which Vi = Vf = V. If in the fractional expressions (2.33)-(2.34) we replace in first approximation the unknown wave functions ~ + ) and We ~-) by the Born free waves #i and 4~f, we obtain the following approximate expressions (which we denote by the subscript B):
[Tlf]Bpost = Tt(fB)
[1+
[Tif]B prior = Ti(B) [I -~"
'
(2.39)
(2.40)
where (2.41) is the transition matrix element in first Born approximation.
REARRANGEMEbrr, COLLISIONS
553
It is interesting to compare the formulas (2.39) and (2.40) with the Born development. If, rin the right,hand side; the second term o f the bracket - whose numerator is o f the second order in the interaction potentials - is neglected, one recovers the first Born approximation. As an offset, if this term is maintained but the second order part o f the denominator is neglected, one finds the second order Born approximation. Together with their many formal advantages, the variational principles proposed here have unfortunately also a major inconvenience, which is the evaluation o f the second order matrix elements appearing in the stationary expressions. This evaluation is often complicated and is in fact the actual limitation of the method. 2.5. EXTENSION TO COUPLED WAVES Let us assume that the interaction potentials Vi and Vf may be written as Ui, fdl-~i,f •
(2.42)
H (i1, )f ffi H i , f.~- Ui, f
(2.43)
~'i,f =
Let us define the new Hamiltonians such that
H(l)v(±) i,f~a'i,f
=
EX(+) i,f,
(2.44)
where X[ ±) and Xft±) are the supposed known coupled * waves having the required asymptotic behaviour. We shall also need the Green operators G~l)(±) and G~1)(±), associated respectively with Hiix) and H(f1), and the Moiler operators t2 (1)c±) i,f
~
1 + G(±)Wi r.
(2.45)
Now, let us assume that the potentials Ui and Uf are such that Ui ~ Ui(ra),
Uf -- Uf(rp),
(2.46)
where the vectors r~ and rp connect respectively the centres of mass o f the systems A, B and C, D. Then, it is well known 12) that for a rearrangement collision, the transition matrix element may be written Tif
(2.47)
= <~['l~-)l~ilX~+)> =
Now, by means o f a method parallel to the one already developed for the free waves, we can construct variational principles with coupled waves. We obtain for the fractional form 9) (-) (+) (-) r'rflpo,t < X f L : i j= IWfl~ ><~f IWf{l+G~')(+)(W~-Wf)}lXi +)> (2.48)
<'/'~-)lWf- Wf ,.,f":-(1)(+)w' ~(+)\,,f, :i /
[T~d,,,io, = <'e~-)lWflX~+)>
'
~)6~1)(+))~I'I'~+)>
(2.49)
I.,V t2.(1)( + ) [,12" I IV( + ) k vr i ~'mi rril~t i /
t We recall the distinction between the coupled waves X~(±) and X~(±) and distorted waves
which correspond to a particular choice of the potentials Ut and Ui (see for example ref. ,3)).
554
c. JOACttAIN
If the unknown wave functions T~+) and T~-) are replaced in these expressions by the coupled waves X~+) and Xf(-), we obtain the approximate expressions (which we denote by the subscript C) I
[Tif]Cpost= T[fcwB) I +
/ Y ( -)11.~ f~.(1)(+) I,I/" I Y ( + ) \
\~f
l"fvf
/XX (f - ) IV,,f - -
/
"l
-I
"il"Xi W,,f ~v f( 1 ) ( + ) w ' lr v fl.eXY (i+ ) \ / / - I
(2.50) '
IWfGi- - ~ - ~W~lXi > ~ , [Tif]Cp,io, --- T}cwe) i 1 + / y (
I rri -- rri
vi
rr il"x i
(2.51)
/-J
where
T~(rCWB) =
(2.52)
is the coupled wave Born approximation transition amplitude. Comparison of formulas (2.50) and (2.51) with the Born development of coupled waves is analogous to the one made for the free waves in sect. 2.4. An interesting particular case occurs when the potentials Ul and Uf, already subjected to the conditions (2.46) satisfy also the supplementary conditions of the distorted wave method 12). Then, the coupled waves X~+) and Xrc+) become distorted waves and the approximate formulas (2.50) and (2.51), limited to the first order terms in the potentials Wi and Wf, give a variational justification of the distorted wave Born approximation (DWBA).
3. Application to Three-Body Problems 3.1. G E N E R A L
FORMULAS
Let us consider a system of three particles of masses ml, m2 and m3. For such a system, there exist three types of binary rearrangement collisions: pick-up, stripping and knock-out (or exchange) reactions. We have, for the unperturbed initial and final Hamiltonians, h2 Hi =
-- --
2Mi
zlr,"]- h i ( r 1 ) ,
(3.1)
+h'(r2)"
(3.2)
h2
n, =
2M,
In these formulas, Mi and M e are the initial and final reduced masses, r, and r# are the "relative" vectors fixing the relative position of the centres of mass before and after the collision, r 1 and r2 are the two vectors concerning the initial and final bound systems, and h i and he are the inner Hamiltonians of the initial and final bound systems, such that Rialto.i = 8m, i~/m,i,
hflPn, f = 8n.fl//n,f .
(3.3)
REARRANGEMENT COLLISIONS
555
We assume that the initial and final bound systems are respectively in the states m o and no and we simplify the notations by putting @ i ( r l ) ------@too, i ( r x ) , ~i ~ 8mo. i ,
~/f(r2) -- @.o,f(r2)
(3.4)
~f = e.o, e"
(3.5)
The total energy available in the centre of mass system may be written or
E = e lq- h2k~
E = el+ h2k2--,
2M i
(3.6)
2Mr
where kl and kf are respectively the initial and final propagation vectors. Finally, we have for the initial and final free waves
3.2. C O N S T R U C T I O N
• i(rl, r ) = d~t'"Oi(rl),
(3.7)
• f(rz, r#) = ei~f'"Of(rz).
(3.8)
OF THE GREEN FUNCTIONS
The Green functions ff~+) and ff~+) corresponding respectively to the Green operators G~+) and G~+) in the position representation are given by the formulas ~ + ) ( r l , r~; r l' , r'~) =
- - - M- i
2zch2
~m~lra"(rl)@m'i(rl)' i
*
eikm, d r = - r ' ~ l
[r~--r~[
,
eikm
'
(3.9)
fir# - r ' # l
~f<+)(r2,
,#,. r2, ' r~) =
Mr 2~-~
, , . ~"' f(r2)d/"'f(r2)
[ r # - r~[
(3.10)
In these formulas, the symbol ]l means a summation over discrete states and an integration over the continuum states. Furthermore, the quantities km, i and k~,f are defined by the relations h2k 2 E = •m,i-3 t- m , i (3.11) 2Mi 2 2
= e,,e+ --h kn, f
(3.12)
2Mf with kl - kmo, i,
kf =- kno, f.
(3.13)
The construction of formulas (3.9) and (3.10) can be derived from the structure of the Hamiltonians H i and Hf, it is easy to verify that (E - - Hi) ~ (E--
+
)(rl, r~ ; r~, r~) ---- tS(rI -- r~)t~(r~-- r'.),
(3.14)
r l , r;) = O(r z -- r'2)6(r#-- r~),
(3.15)
Hf)~+)(r2,
r#;
556
c. JoAcru, rN
as expected. Let us note that the Green functions (3.9) and (3.10) introduced into the matrix elements appearing in the calculation of the transition amplitude, take into account the possibility of virtual transitions accomplished by one or the other bound system. In this manner, all the intermediate states of the initial or final system contribute to the transition amplitude through the Green functions (3.9) and (3.10). This is an entirely new process compared to the two-body problem, and it is generally denoted as the polarization effect, by analogy with a similar phenomenon occurring in the study of bound state problems ,4, ~5). 3.3. V A R I A T I O N A L F O R M U L A S
With the aid of the Green functions (3.9) and (3.10) and of the variational formulas (2.39) and (2.40) established for the choice of trial functions ~'/}+) ~--" ~i,
(3.16)
~/~-) = ~ f ,
one obtains for the three-body reactions considered the formulas V
L =
[1÷
11
q
(3.17)
T~g5--I2-I F1
(3.18)
where the second order expressions I,, 12, F, and Fz are explicitly given by
x Vi(r~, r~)@i(r~, r'.),
,2 = f dr~ydr~ f dr~f ' *
(3.19)
r~)~(r,, r2)~+'(,x, r.;r,', r') x E(ri, ri)ei(ri, tO, (3.20)
w, = f dr, f dr; f d,2f drY2¢~(rz, ra)Vf(rl, rz)ff[+)(rz, r,;
r~, r~)
x v,(4, r[)~i(rl, 6), (3.21)
=f dr,f drlf
f de[¢?(r2,ra)Vf(rx,r2)fg +)(r2,r,;r[,r'~) x Vf(rl, rl)¢i(rl,
r'),
(3.22)
and where the free waves @i and @f are given by (3.7) and (3.8). The interaction potentials Vi and Vr depend naturally on the reaction considered. As to the relative vectors r, and ra (or r" and r~) they depend on the vectors r t and r2 (or r~ and r[) according to the reaction considered. For example, in a pick-up reaction where
REARRANGEMENT COLLISIONS
557
the particle 1 is incident on the bound system 2-3 and where the bound system 1-2 is formed in the final state, we have r~ = ~ r l - - r 2 ,
(3.23)
rp = r l - - f l r 2,
where m3
= - - , m2 + m3
ml
fl -
.
(3.24)
m l + m2
Finally, let us remark that the Second Born approximation corresponding to the reactions considered is given by Y(? 2) = T(? ) + I t
(3.25)
Y(fB2) -- Ti(B) --1-F 1
(3.26)
in the prior form, and in the post form. 3.4. NUCLEAR PICK-UP AND STRIPPING REACTIONS
According to the micro-reversibility principle (2.36), the transition amplitude for a stripping reaction is the same as the one relative to the corresponding pick-up reaction. We may thus treat indifferently one of these two reactions. As an example, we study the following model o f (p, d) pick-up reaction. We consider a proton incident on a nucleus composed of an inert and infinitely heavy core and an optical neutron. The interaction of the proton with the core is neglected throughout the reaction (stripping hypothesis). We neglect also the Coulomb effect, so that our treatment works also for the (n, d) pick-up reaction. The interaction potentials between the incident proton and the neutron, and between the neutron and the core are chosen as Vi = - A f ( r 2 ) ,
Vf = - - B 6 ( r l ) .
(3.27)
We assume also that the optical neutron is initially bound in the fundamental state o f an isotropic harmonic oscillator. The initial nucleus is thus described by the Gaussian wave function ~/i(rl)-- Nie -~1"12
(3.28)
with
The parameters Yi and B are chosen so that the mean value of r 2 and the binding energy ei of the optical neutron are correct. The final deuteron is described in its fundamental state by the Gaussian function ~f(r2)----- Nfe rrr22,
(3.30)
558
c. ~OACHAIN
where
This function is easier to manipulate in the subsequent calculations than the more precise Hulth6n function. The parameters ~'f and A are obtained by imposing that the quantity Ro2 = (~bfl(½r2)2l~bf)
(3.32)
and the binding energy gf = - - 2 . 2 3 MeV be correct. The deuteron having only one bound state, it is clear that the post formulas (3.18) and (3.26) are simpler than the prior ones. All the intermediate states of the deuteron are continuum states that we describe by plane waves ~/'.,f(r2) = (2n)-~e 'h'''',
(3.33)
corresponding to intermediate energies h2 8n, f(k') = - - k '2,
(3.34)
M where M is the nucleon mass. Integration over continuum states is thus performed by integrating over all the k' space. Before using the formulas of subsect. 3.3 where Mt = M,
M f = 2M,
= 1, ~ = ½,
(3.35) (3.36)
we must fix the total energy E of the reaction. We have chosen E = 0, an energy for which the outgoing deuteron is emitted with a kinetic energy of 2.23 MeV equal to its inner binding energy. This value of E has two advantages: first, to be in a sufficiently low-energy region, where the variational method earl significantly improve the Born approximation; secondly, to simplify the calculations by giving in the expres: sion of the Green function (3.10) k,,f(k')
(3.37)
= 2ik',
as it is seen with the aid of formulas (3.12), (3.34) and (3.35). We now proceed to the calculation of the transition amplitude in first Born approximation. As the two inner wave functions (3.28) and (3.30) are approximate ones, there is a post-prior discrepancy. The prior form is given by
~ , , t o , = <~flVil~i>
--
(
-ANiNf ~ /~e-Kt2/4rt, \~i /
(3.38)
REARRANGEIdENT COLLISIONS
559
where K 1 is the magnitude of the vector K, =
ki-kt,
(3.39)
while, for the post form, Ti(B) r post = <~e[VfIOi> = -
' BNI Ne (TC~'e-X~2/4,, \~f/
(3.40)
where K2 is the magnitude of the vector
(3.41)
K2 = ½ k t - k,.
It is also possible to eliminate the post-prior discrepancy by using the Schrbdinger equations [h2 [- -2~ A,l + Vf(rl)] d/,(rl) = e,~/,(,,), (3.42)
[-~A,2+Vi(r2,1d/f(ra)=~fOe(r2),
(3.43)
which are verified approximately by the inner wave functions (3.28) and (3.30) for the choice (3.27) of F i and Ff. Thus -Adi(r2)e-"'" ~ [-~ 3,t +eli e -r'22,
(3.44)
-Bdi(rl)e-'"12 ~' [ 2 ~ - A " + ~ ' I e-'m2
(3.45)
and, substituting respectively in the formulas (3.38) and (3.40), we obtain 2q(ifn ) = -NiNfn3(~i~e) -~ ~
K~-,r]
e-rl~/'"e-r221"",
T~n) = --NiN,~a(,,,f)-' ~2h-hMK~-~,l e-'"/4"e-~:~2/" ,
(3.46) (3.47)
which are equivalent, as is easily shown by using the relations (3.6). Table 1 illustrates the results obtained by these three calculation methods of the first Born transition amplitude. As a comparison, we have also calculated T/~fn) in the third method for two other choices of wave functions: the Hulth6n wave function
[tn(n+llo~]'e-"~-e -''=
~ke(r2) = L 2 ~ J
g
,
(3.48)
560
c. JOACICMN TABLE 1
Nuclear pick-up and stripping reactions at total energy E = 0 Born approximations for the transition amplitude (in units 10 -a erg • fln 8) Schr6dinger equation 0
(degrees)
Prior f o r m
Post f o r m Gaussian
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
--3.39 --3.34 --3.17 -- 2.92 --2.61 -- 2.28 --1.94 -- 1.63 -- 1.35 -- 1.11 --0.914 --0.757 --0.635 --0.541 --0.472 --0.422 --0.388 --0.369 --0.363
--3.26 --3.19 --3.00 -- 2.72 --2.38 -- 2.01 "1,66 -- 1.34 -- 1.07 -- 0.849 --0.672 --0.536 --0.434 --0.358 --0.303 --0.265 --0.240 --0.226 --0.221
--4.50 --4.43 --4.23 -- 3.90 --3.44 -- 2.89 --2.32 -- 1.79 -- 1.33 ,-- 0.972 --0.701 --0.506 --0.370 --0.278 --0.216 --0.176 --0.151 --0.137 --0.133
Hulth6n (n = oo)
Hulth6n (n = 7)
--3.03 --2.98 --2.83 --2.61 --2.33 --2.03 --1.73
--3.54 -- 3.47 --3.29 --3.01 --2.66 --2.29 -- i.93
-- 1.45
-- 1.59
--1.20 --0.991 --0.816 --0.676 --0.567 --0.483 --0.421 --0.376 --0.347 --0.330 --0.324
-- 1.30 -- 1.05
--0.855 --0.698 --0.577 --0.486 --0.419 --0.371 --0.340 --0.322 --0.317
where n = 7 and ~ = 0.231 fm " t , and the function t e - ~r2 ~ , /'2
Ipf(r2) =
(3.49)
corresponding to a zero-range neutron-proton interaction potential As a contact potential is clearly a better approximation for the neutron-proton interaction Vt than for the neutron-core interaction Vf, we shall adopt in the following calculations the prior form for Tiree) with the Choice (3.30) o f Gaussian wave function describing the fundamental state of the deuteron. The differential cross section in the first Born approximation is given by Mz tr(al) = 2~n 2 h
kf (Ti¢~))2'
(3.50)
ki
while, for the corresponding stripping reaction, we have M 2 ki (Titfa))2. a¢"1) = 2n2h 4 kt
(3.51)
The numerical values o f these cross sections are given respectively in tables 4 and 5.
REARRANGEMENT COLLISIONS
561
Let us now calculate the second order terms occurring in the post formulas (3.18) and (3.26). Starting from (3.21) and (3.22) and taking into account the relations (3.10), (3.23), (3.35) and (3.36), we have
Fl =
MABNiNf fdri fdr'l fdr2 fdr;e-'h"("-½"=)e -''2z 7~h2
d
d
2~
x 3(rl)
g
d
' e'k"'4''-*''-'',+½"'21 6(r2) e't'" (,',--',)e-,,,','
*
~/" f(r2)~/n' f(r2) Irl - - ½ r 2 -- r1-3u ½1M2I
(3.52)
n
and 7~h 2
X ~(r,)
(3.53)
~/,,f(r2)~/n* f(r2) [rl --½r2 --r~ --~½r21 n
Let us decompose the summation :$, into its discrete part (d) limited to the fundamental state of the deuteron and its continuum part (c) containing the contribution of the continuum states of the deuteron. Thus F1 = --tl~'d() T'tl-L 1~'(¢),
F2 = --21~( d) -,-L-2~'(°)•
(3.54)
We evaluate directly the "discrete" terms F~d) and F2¢d) in configuration space. Details are given in ref. 9). Putting tZt --
8~i~f , 7i -{- 8yf
K 3 -- ½(kf-ki),
1 K4 _ - (Syfki-l-Tikf), 8~)f "~-~i
(3.55)
2 i K5 = ~ki--I--6kf,
(3.56)
we obtain Re Foxd) =
MABNi(Nf) 3 1 x
--
[(kf+ K,)tFl(1; 3; - ( k t +
K4
K,)21~ ')
- (ke- K,)~F~(1 ; 3; - ( k f - K,)2/4~')], Im FOx a) = --
x2MABNt(Nf)3 a'~'h 2
(3.57)
( 1 ~ e_K32/(Syt+~, D \27f + ¼~i/ X - -1
K4
[e-
(kf-- K4)214=" - - e -
(kf + K4)2/4"= ' ] ,
(3.58)
562
C. JOACHAIN
3MB2Ni(Nf) 3 ( ~ ~e-K22/12~. yf h2 \~f]
Re Ft2d) -
x 1 [(½kf+ Ks)IF1(1; 3; _(½kf + Ks)2/~yf) Ks - (½kf- Ks)~Fx(1; k; - (½kf-- K5)2] 3~Tf)], Im F¢2d) =
-
2~2MB2N,(Nf) 3 ( 1
~(3
(3.59)
~e_r~/~2~ ~
× - -1 [ e - (~kf - Ks)2/~Yf - - e - ( -~-kf+ KS)2/~,f].
(3.60)
Ks Evaluation of the terms F~°) and F~c), which are real, implies new difficulties due to the integration over the "intermediate" k' propagation vectors. After some manipulations, we reduced this calculation to the evaluation of a one-dimensional integral, which was performed by the IBM 1620 machine of the University of Brussels t The numerical values of the different second order terms are collected in table 2. Comparing with the first order terms of table 1, we see that the two types of terms are of the same order of magnitude. With the help of the values of F~d), F~~),Fc2a), and F~° we can obtain the transition amplitudes (3.18) - denoted by TiVf ar - and (3.26). They are given in table 3. From these values it is easy to obtain the differential scattering cross sections ava, and ata2) collected in table 4 for the pick-up reaction and in table 5 for the corresponding stripping reaction. 3.5. DISCUSSION OF THE RESULTS
Table 3 collects our results corresponding to the transition amplitude while tables 4 and 5 concern respectively the differential cross sections for pick-up and stripping. We have compared the first Born approximation (prior form), the second Born approximation and the variational method. The numbers in the brackets figuring in tables 4 and 5 correspond to the same quantities but without the contribution of the continuum parts F~°) and F2to). This is to emphasize the specific influence of the discrete and continuum terms on the cross sections considered. Examination of these tables indicates that the introduction of second order terms considerably modifies the results of the first Born approximation. The transition amplitude gets an imaginary part while the differential cross sections are either strongly increased (second Born approximation) or reduced (variational method). t Strictly speaking, the term F,(¢) cannot be evaluated directly because the particular choice (3.27) of V~ leads to divergent integrals. To remedy this situation, we made the supplementary hypothesis
F~o _ F~° Re Ft2d} Re Ft~d)
REARRANGEMENT COLLISIONS
563
TABLE 2 N u c l e a r p i c k - u p a n d s t r i p p i n g r e a c t i o n s a t t o t a l e n e r g y E ---- 0. S e c o n d o r d e r t e r m s o c c u r r i n g i n t h e calculation of the transition amplitude
F1 0 (degrees)
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
F~ (10 -s e r g " f m 8)
R e F~ d)
I m F~ d)
- - 1.80 --1.80 --1.77 --1.73 - - 1.68 - - 1.62 --1.56 --1.48 - - 1.41 --1.33 - - 1.26 - - 1.19 - - 1.12 - - 1.07 - - 1.02 --0.983 --0.956 --0.939 --0.934
--2.79 --2.78 --2.77 --2.75 --2.73 --2.70 --2.66 --2.63 --2.59 --2.54 --2.50 --2.47 --2.43 --2.40 --2.37 --2.35 --2.33 --2.32 --2.32
F~ c)
R e F2(d)
I m F2(d)
--0.944 --0.942 --0.937 --0.929 --0.919 --0.906 --0.891 --0.876 --0.860 --0.843 --0.827 --0.812 --0.798 --0.786 --0.775 --0.767 --0.761 --0.757 --0.756
--3.37 --3.35 --3.30 --3.21 -- 3.10 --2.97 --2.83 --2.68 --2.52 --2.37 --2.22 --2.09 - - 1.97 -- 1.87 -- 1.78 - - 1.71 - - 1.66 -- 1.64 - - 1.63
--2.35 --2.35 --2.34 --2.32 --2.30 --2.27 --2.23 --2.20 --2.16 --2.12 --2.08 --2.04 --2.01 - - 1.98 - - 1.95 -- 1.93 -- 1.91 -- 1.90 - - 1.90
F(2°) - - 1.76 --1.76 --1.74 --1.72 - - 1.69 - - 1.66 --1.62 --1.58 - - 1.54 --1.50 --1.47 - - 1.43 - - 1.40 - - 1.38 - - 1.35 -- 1.34 - - 1.33 - - 1.32 - - 1.32
TABLE 3 N u c l e a r p i c k - u p a n d s t r i p p i n g r e a c t i o n s a t t o t a l e n e r g y E = 0. T r a n s i t i o n a m p l i t u d e T(? )
T(fB2)
0 (degrees)
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
rvarif
(10 -a e r g " f m a)
--3.39 - - 3.34 --3.17 --2.92 --2.61 --2.28 - - 1.94 - - 1.63 - - 1.35 - - 1.10 --0.914 --0.757 --0.635 --0.541 --0.472 --0.422 --0.388 --0.369 --0.363
R e Ti(~2)
I m Ti(fB2)
--6.14 - - 6,07 --5.88 --5.59 --5.22 --4.80 --4.39 - - 3.98 -- 3.61 -- 3.28 - - 3.00 - - 2,76 --2.55 --2.39 --2.27 --2.17 --2.10 --2.07 --2.05
--2.79 - - 2.79 --2.77 --2.75 --2.73 --2.70 --2.66 -- 2.63 - - 2.59 - - 2.54 -- 2.50 -- 2.47 --2.43 --2.40 --2.37 --2.35 --2.33 --2.32 --2.32
R e Tvarif d: 1.I0 + 1.05 +0.915 +0.714 +0.495 +0.294 +0.137 + 0.0301 --0.0338 - - 0.0663 --0.0789 - - 0.0802 --0.0764 --0.0705 --0.0647 --0.0597 --0.0560 --0.0538 --0.0530
I m T~fa ' --0.650 --0.583 --0.411 --0.196 --0.00439 -]-0.133 +0.210 + 0.240 +0.239 + 0.222 +0.199 +0.175 +0.154 +0.136 +0.122 +0.111 +0.104 +0.0994 +0.0980
564
C. JOACHAIN
TABLE 4 Nuclear pick-up at total energy E = 0. Differential cross section
(degrees) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
(10 -~ b/sr) 105 102 92.0 78.0 62.4 47.4 34.4 24.2 16.6 11.3 7.64 5.25 3.68 2.68 2.03 1.63 1.38 1.25 1.21
416 408 387 355 317 277 241 208 181 158 139 125 114 105 98.3 93.5 90.1 88.3 87.6
(318) (312) (294) (268) (237) (206) (177) (151) (130) (114) (100) (90.1) (82.2) (76.2) (71.7) (68.4) (66.2) (64.9) (64.5)
15.0 13.3 9.20 5.01 2.24 0.952 O.576 0.534 0.533 0.492 0.420 0.340 0.271 0.215 O.174 0.145 O.127 0.117 O. 114
(67.3) (62.1) (48.7) (32.3) (18.2) (8.77) (3.74) (1.51) (0.651) (0.347) (0.233) (0.178) (0.142) (0.116) (0.0962) (0.0821) (0.0728) (0.0675) (0.0657)
TABLE 5 Nuclear stripping at total energy E = 0. Differential cross section 0
O(al)
~(a2)
(degrees) 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180
t~var (10 -~ b/sr)
165 160 144 122 98.0 74.4 54.1 38.0 26.1 1"7.7 12.0 8.23 5.78 4.20 3.19 2.55 2.16 1.96 1.89
653 641 607 557 497 436 378 327 283 248 219 196 178 165 154 147 142 139 138
(499) (489) (461) (420) (372) (323) (277) (238) (205) (178) (158) (141) (129) (120) (112) (107) (104) (102) (101)
23.5 20.8 14.4 7.88 3.51 1.49 0.904 0.839 0.837 0.773 0.659 0.534 0.425 0.337 0.273 0.228 0.199 0.183 0.178
(105) (97.5) (76.4) (50.7) (28.5) (13.8) (5.87) (2.37) (1.02)
(0.544) (0.366) (0.279) (0.223) (0.182) (0.151) (0.129)
(o.114) (0.106) (0.103)
REARRANGEMENT COLLISIONS
565
It is clear that no definite conclusions can be drawn from this exploratory calculation. However, the three following remarks can be made. At the energy E = 0 considered, the second order terms in the interaction potentials are of the same order of magnitude as the first order terms. This result indicates that it is difficult to have confidence in semi-qualitative arguments justifying the first Born approximation at low energies for nuclear direct reactions. Let us note also the importance o f continuum terms in the calculations of the second order terms FI and F2 and their marked influence on the cross sections. Results furnished by the second Born approximation are clearly too high and must be discarded. This situation is similar to the one observed in the variational study o f nuclear potentials 16), The variational method proposed here gives smaller differential cross sections than those of the first Born approximation, the ratio o f the maxima being 7. This result is encouraging, because it is in concordance with the conclusions of the distorted wave method t. The angular distribution has the same satisfactory form as in the first Born approximation with a reduction at small angles analogous to the one observed in the distorted wave method 17). The comparison at large angles is useless because we made the stripping hypothesis. Finally, an important point concerns the behaviour of the different methods at high energies. To this end, we have analysed the first order terms together with the discrete second order terms o f the transition amplitude versus the total energy E. Our results indicate that the second order terms considered remain steadily important even for k I ---, ~ . This last conclusion must be compared with similar results obtained by Drisko is) and Bates and Mac Carroll 19) in the study o f atomic rearrangement collisions. It is evident that such results furnish a supplementary support for a detailed examination of the validity of Born's first approximation at high energies for rearrangement collisions 20, 21), both in atomic and nuclear physics. This work constitutes a part of a thesis which has obtained the Louis Empain Prize 1963 for Physical Sciences. The author wishes to express his gratitude to Professor M. Demeur for his continued advice and guidance. Thanks are also due to Professor L. Rosenfeld for helpful comments. * For example, Horowitz and Messiah 17), using the distorted wave method, obtained for several stripping reactions at l = 0 reduction factors going from 2.42 to 5.81.
References 1) 2) 3) 4) 5) 6)
W. Kohn, Phys. Rev. 74 (1948) 1763 L. M. Delves, Nuclear Physics 29 (1962) 326 L. Spruch and L. Rosenberg, Phys. Rev. 125 (1962) 1407 B. A. Lippmann and J. Schwinger, Phys. Rev. 79 (1950) 469 S. Borowitz and B. Friedman, Phys. Rev. 89 (1953) 441 M. C. Newstein, Technical Report no 4. Project DIC 6915 MIT
566 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
c.
JOACHA|N
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