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The Faddeev-Lovelace equations in the eikonal approximation
ANNALS
OF
PHYSICS:
73, 136-155 (1972)
The Faddeev-Lovelace
Equations
in the Eikonal
Approximation
R. K. JANEV* Institute of Physics, Beograd, Yugoslavia
AND
A. SALIN+ Observatoire de Paris, 92 Me&m,
France
ReceivedSeptember 8, 1971
The three-bodyproblemwith two heavy particlesand a light one is considered. It is shownthat the Faddeev-Lovelace equationscanbe simplifiedby usingthe eikonal approximation.Two coupledintegralequationsareobtained,the potentialbetweenthe heavyparticlesbeingaccountedfor with the help of the distortedwavemethod.
INTRODUCTION
The Faddeev approach to the three-body problem has been widely used to study either bound states or collisions. In nuclear and particle physics, the advantage of this approach stems from the fact that the three-body T matrix may be calculated by using information on the two-body t matrix. When the two-body interactions are dominated by a few bound states and resonances, the subsequent simplifications in the formulation of the problem lead to important advances toward the quantitative evaluation of three-body processes. In atomic physics, where the two-body potentials (coulomb potentials) are well known, the situation is rather different. This is certainly the reason why the Faddeev equations have not been so widely used. Nevertheless, the nice formal properties of the Faddeev approach made it a convenient starting point to study suchproblems asthe charge exchangeof hydrogen
atoms with protons at high energies [l, 21. Some work has also been performed on electron hydrogen-atom collisions and the usefulness of separable approximations for the two-body t matrix has been explored [3]. In this case some simplification in * Permanent address: Instituteof NuclearSciences “Boris KidriE,” 11001Beograd,Yugoslavia. t Presentaddress: Laboratoired’Astrophysique, UniversitedeBordeauxI, 33 Talence,France. 136 Copyright AU rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
FADDEEV
EQUATIONS
IN
EIKONAL
FORM
137
the analysis is achieved by using a partial-wave expansion. On the other hand, the partial-wave expansion is not very convenient to study three-body problems involving two heavy particles (for example, two protons and one electron). In the latter case a more successful approach is via the eikonal approximation. This old method [4] has received more attention recently, following the work of Glauber [5]. In this work, Glauber did not give a general derivation of the eikonal approximation for the case of multichannel scattering, but restricted his theory by some further approximations which are referred to as the “Glauber approximation.” Most applications of the eikonal approximation to nuclear physics have been restricted to the Glauber approximation. In atomic physics, the well-known “impact. parameter method” was shown to be connected with the eikonal approximation [6, 71. The fact has also been underlined that the eikonal approximation is suitable for heavy colliding systems. In this work, we wish to incorporate the simplification achieved by the eikonal approximation in the Faddeev-Lovelace equations [8, 91. Some ideas along this line can already be found in the work of Osborn [lo]. But, in fact, the work of Osborn is, in some sense, opposite to our work since its objective is to maintain Glauber hypothesis on many-body reactions (scattering from a composite target sum of scattering from fixed centers) but to drop the eikonal approximation. The consequence of this is that the matrix elements of Osborn’s eikonal operators always contain delta functions. In our work the simplification achieved by these delta functions is accounted for in the equations themselves. This avoids the necessity of repeating part of the derivation of the eikonal approximation for the calculation of every matrix element. The organization of our work is as follows. We consider a three-body problem with the masses of two particles much bigger than the mass of the third. Section 1 gives the definitions and basic equations, Because of the mass ratio between the third particle and the other two, it is not very convenient to “eikonalize” directly the system of three coupled equations of Lovelace [9]. Therefore we first consider the case when the potential between the heavy particles is zero (Sections 2 and 3). In Section 2 we show how coupled equations of a form similar to those of Lovelace can be obtained when we use the eikonal approximation from the very beginning. Section 3 shows that the equations of Section 2 can be obtained directly from the Faddeev-Lovelace equations in the case of the “straight-line eikonal approximation” [1 I]. In Section 4, we consider the case of a nonzero potential between the heavy particles and the introduction of the distorted-wave method, following the work of Dodd and Greider [12]. In conclusion, we consider briefly some possible applications of our theory.
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1. GENERAL CONSIDERATIONS AND DEFINITIONS We consider three particles labeled 1, 2, and 3 of mass m, , m2 , m3 . We further consider the case when the mass of particle 3 is much smaller than the mass of particles 1 and 2. We call p the reduced mass of particle 1 and 2. Therefore (m3/d Q 1. Let us call ri the position vector of particle i with respect to some fixed origin. We shall use the coordinate systems {pi , rjk} where pi = ri - (mjrj + mkrk)/(mj + mJ; rij = ri - rj .
(1)
We also define x = r32, s = r31, R = rlz , r = p3.
69
With those definitions, x (resp. s) is the position vector of particle 3 with respect to particle 2 (resp. 1). r is the position vector of particle 3 with respect to the center of mass of particles 1 and 2. We define the reduced masses: t4
=
m&j
+
mk)l(mi
+
mj
+
mk>,
j&i
=
mimJ(mi
+
md, p = plz .
(3)
We label by i the “arrangement-channel” [13, Section 16.2.11 in which two particles not labeled by i are bound together and by 0, the arrangement channel consisting of three free particles. The relevant “channel hamiltonian” is called H, . Let H be the hamiltonian for the three-particle system. We call Vi the potential between the particles not labeled by i. H = Hi + c
Vk
k#i
We use the Green functions Gi(s) = [s - H&l,
S+‘(s)= [s - HI-l.
(5)
We shall use the Faddeev equations in the form given by Lovelace [9]. First the following transition operators are introduced:
k#j
L#i
P-9
FADDEEV EQUATIONS IN EIKONAL
Coupled equations are easily obtained
FORM
139
between these operators, [9], namely;
P’b) In the case of breakup reactions, we have, for example;
(8) In the case of two heavy particles and a light one, Dodd and Greider [12] have shown that these three coupled equations can be reduced to two coupled equations. That this is so is not astonishing if one considers the very special role played by I’, . As is well known, the total cross sections in our case are independant of F’, (except for elastic scattering). But the differential cross sections do depend on I’, and therefore a consistent treatment of this potential is still necessary. The dependance on V3 in the coupled equations (7) is introduced via the coupling with the transition operator U& or iJ8<. If i and j are different from 3, this is a very awkward situation since the physical process which Ui; describes is one in which the two heavy particles are bound together in the final state. In order to simplify the arguments in Sections 2 and 3, we shall first put V, = 0. But our results do not depend on this assumption. We shall explain in Section 4 what our expressions are when V3 f 0.
2. FADDEEV EQUATIONS IN THE EIKONAL
APPROXIMATION
a) Summary of Multichannel Eikonal Approximation The eikonal approximation has proved to be a very successful method in atomic, nuclear or particle physics. The eikonal approximation for potential scattering has been derived in its more general form by Chen and Watson [14]. Nearly all the applications have been restricted to the “straight-line eikonal approximation” or “impact-parameter method” as derived by Glauber [5]. The extention of the “straight-line eikonal approximation” to multichannel cases has been accomplished by McCarroll and Salin [15, 61 and Wilets and Wallace [7]. These authors have pointed out that the “straight-line eikonal approximation” may be considered as a first-order approximation in an expansion in m,/p. Other authors have considered the case of hyperbolic rather than straight-line trajectories (e.g., [ 161). But in the latter case only the total cross sections could be obtained, whereas in the
140
JANEV
AND
SALIN
straight-line eikonal approximation, expressions are also obtained for the differential cross sections. In the following sections, all the quantities defined in the eikonal approximation will be distinguished by a tilde. In any form of the eikonal approximation, one is led to solve the equation [i(d/dt) - 2P(t)]!Pm+(r, t) = 0,
(9)
where )j&
P,*(r,
t) = 6jf(r,
t).
(10)
t) = 0,
(11)
The wavefunction 6 is a solution of [i(d/dt) - Hi(t)]&yr,
where i is the arrangement channel and 01represents the quantum numbers necessary to define a given state in channel i. The hamiltonian s(t) is the one for particle 3 in the field of particles 1 and 2 fixed. In this section we suppose the potential between the heavy particles to be zero. This is not a problem in the eikonal approximation since this potential can always be accounted for easily. Nevertheless, the discussion of this point is postponed to Section 4. The dependance of X and Hi on t arises from the dependance of R on t. For example, in the straight-line eikonal approximation R = b + vt, where b is the impact parameter and v the relative velocity of the incoming particle and the target in the initial channel. But in this section the form of R(t) will remain unspecified. A set of independant variables must be chosen from now on. We choose this set to be r and t. The 6 functions can then be written explicity in the case of the straight-line eikonal approximation as d;,l = ya(x) exp { -k,t
- ipvr - (i/2m3)p2u2t},
(124
CFf12 = &s) exp {--i+
+ iqvr - (i/2m3)q2u2t},
(12b)
where yor(x) and qa (s) are eigenfunctions for the bound states of particles (1 + 3) and (2 + 3) respectively; E,,~ are the corresponding eigenenergies. P = mlm31(ml
+ nz2); 4 = wdh
+ m2>.
(13)
The exponential factors allow for the fact that particles 1 and 2 are moving with respect to the origin of coordinates (r = 0) with velocity qv and pv, respectively. b) Basic Definitions and Equations We shall now construct formal expressions in the eikonal approximation using the identity of form of (9) with the Schrodinger equation for the time-dependent
FADDEEV
EQUATIONS
IN
EIKONAL
141
FORM
scattering theory. The main difference arises from the signification
of the variable
t in our case. Note also that &’ and Hi depend on t. In this section we do not give
the detailed derivation of expressions that are obtained in the same way as in the time-dependent scattering theory [13]. We first define the Green functions [i(d/dt) - Hi]Gi*(t)
= 8(t),
(14)
[i(d/dt) - ~]~:“(t)
= s(t),
(15)
with the initial conditions G+(t) = g+(t) = 0, t < 0, G-(t)
= P(t)
= 0, t > 0.
It is shown in exactly the same manner as in [13] that Pa*@, t) = Gmi(r, t) + J”+” dt’ G&t -m
- t’) Vj(t’) Pa*(f)
(174
5 &ai(r, t) + I+” dt’ g(t - t’) Vj(t’) &N’,i(t’). --m
(17b)
These equations may be called the Lippmann-Schwinger equations for the eikonal approximation. We note the following properties of the Green functions:
c$:j+(t’ - t) &i(t) = -i!Bmj(t’), t’ > I, CT-(t’- t)@(t)= &j(t’), t’ < t, e-(t- t’)G+(t’- t) = 1, t’ > t, e+cr- t’)G-(t’- t) = 1, t’ < t.
(184 (18b) WC)
WW
Corresponding expressions are obtained for 8. Therefore the Green functions are related to the evolution operators for the state of particle 3 in the timedependent field created by the movement of the heavy particles. In the case of the straight-line eikonal approximation, a Born expansion of Eq. (17) yields immediately the result of Moiseiwitch [17]. c) Definition
of the Transition
Operators
The quantities which have a physical meaning are b,, = F-2 (6J/pm+)
= ,“m- <%@O-/6,,i>.
+
From these quantities, the differential and total cross sections corresponding
(19) to
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JANEV
AND
SALIN
process N ---fB can be calculated. b,, plays a role equivalent to the S matrix. We define a matrix ri’,j whose elements are
with
We first show that
The proof runs as follows:
But
Therefore
Q.E.D. In conclusion
In the case of break-up reactions, expressions (22) to (23) are correct only if a wave-packet is used to describe the free states of particle 3. The same condition is also necessary in the usual formal scattering theory [13, p. 5041. A formal expression for Fij can now be written. Using the Lippmann-Schwinger equation for pm+, we obtain dt’ Gj+(t KB= g ($2 JCrn -cc + Jim dt’ dt” Gj+(t --m
- t’) V,(t’) G,-(t’
-
t)
t’) Vi(t’) ?F+(t’ - t”) Vj(t”) lT*-(t” - t) G-‘(t)).
(24)
FADDEEV
EQUATIONS
IN
EIKONAL
143
FORM
Therefore; Ttj = 2-2 j+* dt’ Gj+(t - t’) Vi(t’) &(t’ -02
- t)
+ j+” dt’ dt” Gj+(t - t’) vi(t’) @+(t’ - t”) V&“) et-(t” --m
- t).
(25)
G,+(t” - t).
(26)
We show in exactly the same manner that Tsi = f’+n~j+m dt’ Cj-(t - t’) vj(t’) C,+(t’ - t) -cc + j+m dt’ dt” @(t -al
- t’) Vi(t’) @+(t’ - t”) I’&“)
It is natural to introduce the transition
operators
Dz(t, t’) = Gj+(t - t’) Vi(t’) + j+m dt” cj+(t - t”) Vi(f) @(t” - t’) Vj(t’), --m D&t’, t) = Vj(t’) Gi+(t’ - t) + j+m dt” V%(t’) g(t’ - t”) I’#‘) --m
(27a)
Gi+(t” - t). (27b)
One has, of course; dt’ ?7;(t, t’) C&-(t’ pij+m --m
t) = ,1I& j+m dt’CJt -00
- t’) QJt’,
t) = i’i’ij,
(28)
but this is not necessarily the case for any value oft. This property is analogous to that of the U operators of Lovelace (6). These last operators are equivalent on the energy shell to the T matrix, but correspond to two different extensions of the T matrix off the energy-shell. d) Faddeev Equations Starting from the expression for the transition operators oij , it is possible to obtain coupled integral equations. The derivation is parallel to that of Lovelace [9]. We consider first the case i, j # 0, (i #j). We use the equation1 g*(t - t’) = cj*(t - t’) + j dt” &(t
- t”) vi(f)
ej+(f
7 t’).
(29)
1 In this section we omit the boundary on the integrals which are always calculated from - to to +a. 595/73/I-10
144
JANEV
Substituting
AND
SALIN
this expression for G in the expression for 0: , one obtains
o;(t, t’) = ej+(t - t’) Vi(t’) + j dt” Gj+(t - t”) F~‘$~:i+(t” - t’) v&‘) + j dt” dt”’ Cj+(t - t”‘) vi(t”‘)
C&P - t”) y#“)
G,+(t” - t’) y3(t’). (30)
If we now introduce I?+&, t’) = ej+(t - t’) Vi(t’) + j dt” Gj+(t - t”) v&“)
@‘+(t” -
t’) vi(t’),
(31)
we obtain u;(t,
t’) = ei+(t - t’) V,(t’) + j- dt” o;(t,
t”) c?,+(t” - t’) vi@‘).
(324
In a similar way; u;(t,
t’) = G,+(t - t’) Vi(t’) + j dt” 7?+;(t, t”) &+(t”
A corresponding
WI
system can also be obtained for Us;, namely,
t?-G(t’, t) = V&t’) Gi+(t’ U;(t’,
- t’) I’&‘).
t) = vj(t’) C#*(t
t) + j dt” Ifi
ci+(t
- t”) o&“,
- t) + j dt” Vj(t - t”) O&t”,
t),
t).
The case of breakup reactions presents no more difficulty example,
(3W W)
and we obtain,
for
Gi,
+ Vj(t’))
+ j dt” O&(t, t”) Gi+(t” - t’) Vj(t’),
(34a)
ait,(t> t’) = G,+(t - t’)(Vi(t’)
+ Vi@‘)) + j dt” o$,(t, t”) C=?,+(t” - t’) V<(t’).
(34b)
The interest of these coupled equations compared to (7) is that now we only operate on functions of the three-dimensional vector r and t, whereas the operator U of (6) operates on fonctions of r and R. Therefore we have included the simplification achieved by the eikonal approximation in the Faddeev-Lovelace equations themselves.
FADDEEV EQUATIONS IN EIKONAL
145
FORM
3. CONNECTION
BETWEEN THE RESULTS OF SECTION 2 AND THE FADDEEV-LOVELACE EQUATIONS
In the last section we showed how coupled equations similar in form to the Faddeev-Lovelace equations could be obtained when the relative motion of the heavy particles is treated in the eikonal approximation. In this section, we show that the Eq. (32) can be obtained directly from the Faddeev-Lovelace equations in the case of the straight-line eikonal approximation. As shown by McCarroll and Salin [6, 151 and Wilets and Wallace [7] the straightline eikonal approximation is a first-order approximation in an expansion in m,/p. Therefore, starting from Eq. (7) we shall develop every quantity to first order in m,/p. The wavefunctions describing free propagation in channels 1 and 2 can be written as tDa,’ = eik+&c), (354 Go2= -%%p&y).
(35b)
k, (resp., -k,) is the momentum of particle 1 (resp., 2) with respect to the center of mass of the atom (2 + 3) (resp., 1 + 3). We define a vector k such that (1/2p2)km2 + E, = (1/2&kB2
+ 4 = (1/2p2)k2 = E.
(36)
ii = ii,;k = pv.
(37)
It is easy to show that
m12m21tml + m2+ m&m1 + m2)l>--2w, k, = k + tW2db12m2/(ml + m2121 -4dk) + W/P).
k,2 = k2{1 + (llp)t
@a)
In the same manner, k, = k + W/41 Using the definitions of ~1 =
Qi
m22m3/h + m2>“l-&W
+ WP).
(38b)
and R, we have
R - h/m2)h/h
p2 = --R i- b%/m2>h/tml
-I- m2)lR- (m&,>r + 0(1/p2)(R+ r>, i- m31R - (m3/m2)r+ 0Wp2D
-t r).
(394 WV
To simplify the following arguments, we shall choose i = 1, j = 2. The proof is similar for any choice of i and j. We also put V, = 0, the case V, # 0 being discussed in Section 4.
146
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AND
SALIN
Equations (7) reduce to m>
= Vl + U,:(s) G2b) v2 3
u,‘,(s)
= VI + u,f,(s) G,(s) J’, .
(40)
We first carry out an expansion to first order in m,/p of the expression G,(E) V,@,l and we calculate this expression in configuration space. We use ewN3 G&%
~2 , ~2),
s, s’>
=
2
C
B
cps(s)
I pz I P2 -
P2’
Therefore, the expression for G,(E) V,@,’ in configuration of the form
~2’
I)
(ps*cs,j
I
*
(41)
space contains integrals
(42)
where ),I means that an integration is carried out over all s’ space. Using (38) and (39), the expression for CD: is transformed into @,(~2
, $1 =
&4
ew
(--ikp2
-
vp, - i 5 vp, - im,vr . )
k cmyyA2j2
(43)
The integral now is I =
exp(ik6
a&
s
\ P2 -
I P2 -
Pzl
P; 1) Fcp2tj
e-ikps
,
I
where F(p;) = (ye
V, exp(-im,vr’)
QJ”,(x’)),* exp (-
&
p’vp; - 2 VP,‘).
The relations between x’, s’, r’, pz’ are, of course, the same as between x, s, r, pa. The quantity p is defined in (13). The integral may now be calculated to first order in rn.Jp by the method of stationary phase [13, p. 5821. Using expression (38b) for kB we obtain G,(E) V2@,1= -i 1 v&s) exp (ikR - & I3 t
x
I
-co
q2u2t-I- iqvr - iq$) 3
dt’ ei’@‘<&s’) V, exp(-im3vr’)
x exp ( -iiE,t’ - &
(p” - q2) u”t’).
~Jx’)>~, (45)
FADDEEV
EQUATIONS
IN
EIKONAL
147
FORM
In this expression, we have used cylindrical coordinates for pz such that p,=b+vt=b+GZ
(46)
and b is a vector perpendicular to v. This expression may be written in a simplified form by using the definitions of Section 2: G,(E) V,@,l = ieikR
+m dt’ Gz(t - t’) Vz~~-(t’ 1 -m
- t) &l(t).
(47)
In going from (45) to (47) we introduce no new approximation since the operator e, is here defined by its action on a complete set of Gf12functions. The same is true for (?I also since expression (47) does not depend on the particular state 01chosen. The quantity that we wish to calculate is a matrix element of a U operator between states @al and QD2. Consider the expression <~~2/U~(EY@2)z,s
= (@~~lU~W@~~)z.s
+ (@Zulu,+,
G,(E)
, (48)
v,l@t>z.s
where the integration over p2 is only carried out over its z component from - cc to z. The matrix elements in (48) are therefore functions of b and z. An expansion of the last term of (48) to first order in m,/p is carried out by using (47): (sPB2/U,f,(E) G,(E) V,/@2)z,s
N v s” dt” @(t”) e-ik’p%J2+2(E) eikp; --m +* x --m dt’ G2f(t” - t’) V2(t’) Cl-(t’ s
t>m,l(tQ (49)
with $‘=j&;
p2” = b + vt”;
k’ = k;
pa = b + vt’.
(50)
This expression can also be written as
G,(E) V&D2> N v ( sB2(t) jT X
dt” &(t
+m dt’ G2+(t” s -co
- t”) e-“k’““U,+,(E) t’) V2(t’) el-(t’
-
ezkp; t) @2(f))r
* (51)
In an analogous manner, the term on the left of Eq. (48) can be written as <@B~U&(E) @a9 NV
+* dt’ C2+(t - t’) e-ik’Pa’U,+,(E) eikPZ’G,-(t’ -co
- t)/@
>r
. (52)
148
JANEV
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The first term on the right of (48) is a Born matrix element and previous works have already given its first-order approximation in an expansion in m$p (Crothers and Holt [18], McCarroll and Salin [15], Vinogradov [19]). <@f32~2@21)Z.r dt’ e-ik’pa’CS,(t - t’) V2(t’) CT-(?’ - t) &l(t)
e+ikpz’ . >
(53)
Therefore if we define the operators z&t, t’) = Gz+(t - t’) e-ik’pGJ,,(E)
eikP,‘,
(54)
it is readily seen that the equations satisfied by these operators are identical to (32). Note that they operate also on 4 functions. Our task is not finished here. We have shown that equations identical to (32) may be obtained from (40) as a first-order approximation in rnJp. It is now necessary to show that the signification of the u operators is the same as for the r7 operators, namely, p+c tij(t) = pi% s dt’ z&t, t’) e,-(t’
- t) = I& .
(55)
For that purpose, we calculate an element of the T matrix:
T,, = <@02U&(E) @a’>,,.,.
(56)
The matrix elements of U& are calculated on the energy shell and the integration now carried out over all pz and s space. Now, (djp2U&(E)
@al)p,s, N s db v+e (6;e”(t)/e-ik’p2t&(t)
eikp2/6jaz(t)>,
is
(57)
which, to first order in mJ7 and for small scattering angles, may be reduced in the manner shown by Glauber [5] (see also [6]) to <@B~U&(E) W>,,.s = s db eiQbv+% (6B2(t) t&(t) &a1(t)),
(58)
where q is the “elastic” momentum-transfer 7 = 2~2~sin (O/2);
7-p = 0,
and 8 is the scattering angle. This relation is the same as the relation connecting T,, and F12 [6]. Therefore ut2 can be identified with O& .
149
FADDEEV EQUATIONS IN EIKONAL FORM
4. FADDEEV EQUATIONS
AND THE DISTORTED-WAVE
METHOD
In Sections 2 and 3 we have supposed that V, - 0. This was only done to achieve a greater simplicity in the exposition of the main ideas. Furthermore, the existence in (7) of a coupling with channel 3 makes very ackward the search for the first-order approximation in mJp of (7) without any transformation. This transformation should take into account the special role played by V, . In general, when V, # 0, Dodd and Greider [12] have shown that the system of Eq. (7) can also be reduced to a system of two coupled equations by using the distorted wave approximation. We first discuss briefly their results and show how the corresponding approximation can be incorporated into the eikonal form of the Faddeev-Lovelace equations. a) Discussionof Dodd and Greider Theory Let us first define the potentials vi = H - Hi
(60)
CL@= 1 + [E - Hi - wa f ic]-lwi ,
(61)
and the distorted wave operators
where wi is a distorting potential introduced for conveniance. Greider and Dodd [20] have shown that, if the following condition is fulfilled: l$
i~(@sj/(wi-)+/@~i) = 0,
(62)
then u;(s) = (co-)’ (Vj - wj+)[l + S’(S)(Vi - Wi)] Wif.
63)
For example, when i f j, a potential that produces only elastic scattering in the final channel will certainly be one. But, obviously, the condition (62) cannot be satisfied for any j and j3 since the only possible choice would be wj = 0. For example, if wi produces only elastic scattering in channel j, then condition (62) will not be satisfied for @& = @jai. More generally, if wj # 0 it is always possible to find at least one j and /3 such that condition (62) is not satisfied. For such values of j, iJ& cannot be written as (63). When taking this fact into account, care must be exercised in the interpretation of some of the expressions of Dodd and Greider. In their work, Dodd and Greider write coupled equations between U$ and some other operator ZJ$ of the form (63). Now, condition (62) may be valid for U$ but not for U& . Therefore no straightforward physical signification can be given to the matrix elements of Uz , though the coupled equations of Dodd and Greider are correct.
1.50
JANEV
AND
SALIN
We first show how this difficulty can be overcome in some special cases. We assume that condition (62) is fulfilled for i # j only. This means that there exists /3 such that
It is then shown that if i = j
where %A. = (Oj-)+ (Vj - Wj’)[l + S’(S)(Vi - Wi)] wi+,
(66)
while, for i #,j,
The expressions for %& are the same as (63). The only difference is that the matrix elements of %$ are not necessarily equal to those of&j on the energy shell. Note also that the first term in (65) does not describe any rearrangement process since it is not present for i # j. Therefore all the coupling with rearrangement channels is included in 9; and therefore we need only look for coupled equations for the @ operators and not the U operators. Of course, we have restricted our choice for Wi but it is obvious that in most applications this is not really a restriction since Wj is nearly always chosen so that this extra condition is satisfied. The form of the ??Zoperators being equivalent to that of the article of Dodd and Greider [12] it is now obvious that all their results are valid for our % operators, namely, 92; = (mj-)+ (vj - wjt) wi+ + %;jG,+(vi %zj = (a~~-)+ (uj - wj+) wi+ + @;Gi+u, ,
- wi) wi+,
(684 Wb)
where G,+ = [E - H - u, + ic]-l
and v, and wj are such that (1) condition (62) is verified for i # j (2) v, and wi are chosen in such a way that no disconnected diagram is obtained from (68). Corresponding
results can also be obtained for @‘ij .
FADDEEV
EQUATIONS
IN
EIKONAL
FORM
151
b) Complete Treatment of the Potential Between the Heavy Particles We now choose wj zzzwi zz2w = v, ; q* = qf
= &,
(69)
and we introduce the operators
(co-)+ WA= a;.
(70)
These W operators operate on the space of x functions
xi+ = &Dai.
(71)
Coupled equations will now be obtained for W$ . As a possible choice for v, , we take v, = vj * (72) This choice is consistent with the conditions on wj and v, . The result is W; = Vi + W;Gj+Vi
,
(734
W; = Vi + W;Gi+Vi
.
WI
These equations are the same as (7) if we put V, = 0. The only difference is that we are now interested in matrix elements of W between x functions. On practical grounds, Eq. (73) may appear no more simple than the starting ones since the influence of V, should be accounted for exactly in the calculation of x. Anyhow we will show that they are a better starting point to study the limit of heavy particles 1 and 2. The expression for Tij is Tij = &(o-(E))+
w + (u-)+ Wij(E)o+.
(74)
c) Eikonal Theory A theory similar to that of Dodd and Greider can be constructed in the eikonal approximation by using the ideas explained in Section 2. We shall briefly sketch the procedure, the detailed arguments being avoided. We define the distortedwave operators as c&*(t) = 1 + s dt’ g’i*(t - t’) w,(t’) Gi-(t’ - t),
(75)
where [
i $ - Hi - wi] ii+(t)
= s(t)
(76)
152
JANEV
AND
SALIN
with the usual boundary conditions. It is then easy to show that if ,‘jz then Yi,j =
(cFB$&-(t))+ @>,i) = 0,
(77)
+m s --m dt (@f3ycz-(t))+ (Uj - Wj) P,+(t)>.
(78)
We can now choose wi such that condition (77) is fulfilled except for i = j. Therefore, ?-& = S&;+(z+
-
wj+) + j &;(t,
t’) 6,+(f)
G,-(t’
- t) dr’,
(79)
where &;(t, t’) = q+(t
- t’) aj-(t’)+ (Uj - wi’)
+ j dt” C&+(t - t”) Gj-(t”)+ (uj - wit) @+(t” - t’)(ui - wi).
(80)
Then coupled equations can be obtained by the usual procedure between @$(t, t’) and an operator &I$(& t’). As an example, we consider again the case wj
=
wi
=
w
=
v,
; ij,*
zz
(pt.
(81)
We can define as in the preceeding paragraph @(t, t’) operators acting on the 2 functions gt) = m&yf). (82) The coupled equations obtained for w(t) are @$(t, t’) = g;(t c(t,
t’) = &+(t -
t’) Vi(f)
+ j dr” c(t,
t”) &+(t”
t’) Vi@‘) + j dt” t@;(t, t”) &+(t”
-
t’) &(t’),
(W
- t’) Vi(C).
(8%)
Now it is obvious that Xi*(t) = exp IF j:, &*(t - t’) = ef(t
Vdf’) dt’l saW,
- 1’) exp (pi jl
V3(t”) dt”).
Therefore the coupled equations satisfied by the operators I@, t’) exp (- jl, V3(P) dt”)
(84) (85)
FADDEEV EQUATIONS IN EIKONAL FORM
153
are identical to (32), that is, the equation satisfied by l? when V, = 0. Also the matrix elements of pj reduce to gij = exp (--i j+w V3(t) dt) j+m dt’ (&(t) -02 --m
o;(t, t’) c&t’)).
m
The influence of V, reduces to merely a phase factor, a result that could be shown directly from (9) (see [21]). This last result can be used to simplify a great deal the calculations of cross sections. The fact that the influence of V, reduces to merely a phase factor in the calculation of & (except for elastic scattering) ensures that to first order in m3/p the total cross sections are independant of V, since they are only functions of / zj 12. For the calculation of differential cross sections, two methods can be used: (1) All the calculations are performed in the eikonal approximation. This allows one to calculate & as a function of b. This information can then be used to calculate Zj . For example, in the case of the straight-line eikonal approximation, the relation between qj and zj is SLij = 1 db exp(iqb) K@).
(87)
(2) It is also possible to calculate & by using (7) with V, = 0. Let us call the result thus obtained tij . Then to first order in m,/p we have qj = &
J” dq exp(-iirlb)
tij($
exp (-i
J$
k’a(I b + vt I) d).
(88)
Therefore, Kj
= -
(2$
J dq’ tij(q’) J db exp (i(q - Q’) b -
i J.*w V3(/ b + vt I) dt). --m
(89)
All the dependance on V, is taken into account by the second integral which, in some cases, may be very easy to calculate.
CONCLUSION
We have shown how the six-dimensional coupled integral equations of Lovelace could be reduced to two coupled four-dimensional integral equations in the eikonal approximation. These coupled equations can be as well written as a single integral equation (see 32) but this is not necessarily a simplification. The use of a
154
JANEV AND SALIN
suitable expansion for the two-body operators will further reduce the problem to two coupled integral equations in a single variable (t). The problem is then amenable to, at least, a numerical treatment. In atomic physics, the interest of the equations thus obtained is that the coupling with rearrangement channels is automatically included in the formulation. This is an interesting property to study, for example, proton-hydrogen collisions in the medium energy range (10-100 keV). If an eigenfunction expansion is used to solve these integral equations, one obtains coupled equations which greatly differ in form from the traditional ones [22,23], the most striking difference being that we obtain coupled integral equations instead of coupled differential equations. The only difficulty that remains to be solved for such applications is the coulomb character of the interactions between charged particles. Some of these applications will be considered with more details in future publications.
ACKNOWLEDGMENTS A. Salin wishes to thank the Institute of Physics and the Institute “Boris KidriE,” Beograd, Yugoslavia, for their warm hospitality while this work was performed.
REFERENCES 1. R. MCCARROLLAND A. SALIN, Proc. Royal Sot. A 300 (1967), 202-209. 2. C. P. CARPENTERAND T. F. TUAN, Phys. Reu. A 2 (1970), 1811-1821. 3. J. C. Y. CHEN AND K. T. CHUNG, Phys. Rev. A 2 (1970), 1449-1457. 4. G. MOLIERE,2. Naturforschung A 2 (1947), 133-40. 5. R. J. GLAUBER,“Lectures in Theoretical Physics,” (W. E. Brittin and L. G. Dunham, Eds.), Vol. 1, p. 315, Interscience, New York, 1959. 6. R. MCCARROLLAND A. SALIN, J. Phys. B 1 (1968), 163-171. 7. L. WILETSAND S. J. WALLACE,Phys. Rev. 169 (1968), 84-91. 8. L. D. FADDEEV,Sou. Phys.-JETP 12 (1961), 1014-1019. 9. C. LOVELACE,“Strong Interactions and High Energy Physics” (R. G. Moorhouse Ed.), Oliver and Boyd, London, 1964. 10. T. A. OSBORN,Ann. Phys. (N.Y.) 58 (1970), 417-453. 11. J. C. Y. CHEN AND K. M. WATSON,Phys. Rev. 188 (1969), 236-256. 12. L. R. DODD AND K. R. GREIDER,Phys. Rev. 146 (1966), 675-686. 13. R. G. NEWTON, “Scattering Theory of Wave and particles,” McGraw-Hill, New York, NY, 1966. 14. J. C. Y. CHENAND K. M. WATSON,Phys. Reu. 174 (1968), 152-164. 15. R. MCCARROLLAND A. SALJN, C. R. Acad. Sci. (Paris) 263 (1966), 329-332. 16. K. ALDER, A. BOHR, T. Huus, B. M~TTEL.~ON AND A. WINTHER, Rev. Mod. Phys. 28 (1956), 432-542. 17. B. L. MOISEIWITCH,Proc. Phys. Sot. 87 (1966), 885-888. 18. D. S. F. CROTHEIUAND A. R. HOLT, Proc. Phys. Sot. 88 (1966), 75-81.
FADDEEV
19. 20. 21. 22. 23.
A. K. A. D. L.
EQUATIONS
IN
EIKONAL
V. VINOGRAWV, Opt. Spectrosc. 22 (1967), 361-362. R. GREIDER AND L. R. DODD, Phys. Rev. 146 (1966), SALIN, J. Phys. B3 (1970), 937-951. R. BATES, Proc. Royal Sot. A 247 (1958), 294-301. WILETS AND D. F. GALL-R, Phys. Rev. 147 (1966),
FORM
671-675.
13-20.
OF
PHYSICS:
73, 136-155 (1972)
The Faddeev-Lovelace
Equations
in the Eikonal
Approximation
R. K. JANEV* Institute of Physics, Beograd, Yugoslavia
AND
A. SALIN+ Observatoire de Paris, 92 Me&m,
France
ReceivedSeptember 8, 1971
The three-bodyproblemwith two heavy particlesand a light one is considered. It is shownthat the Faddeev-Lovelace equationscanbe simplifiedby usingthe eikonal approximation.Two coupledintegralequationsareobtained,the potentialbetweenthe heavyparticlesbeingaccountedfor with the help of the distortedwavemethod.
INTRODUCTION
The Faddeev approach to the three-body problem has been widely used to study either bound states or collisions. In nuclear and particle physics, the advantage of this approach stems from the fact that the three-body T matrix may be calculated by using information on the two-body t matrix. When the two-body interactions are dominated by a few bound states and resonances, the subsequent simplifications in the formulation of the problem lead to important advances toward the quantitative evaluation of three-body processes. In atomic physics, where the two-body potentials (coulomb potentials) are well known, the situation is rather different. This is certainly the reason why the Faddeev equations have not been so widely used. Nevertheless, the nice formal properties of the Faddeev approach made it a convenient starting point to study suchproblems asthe charge exchangeof hydrogen
atoms with protons at high energies [l, 21. Some work has also been performed on electron hydrogen-atom collisions and the usefulness of separable approximations for the two-body t matrix has been explored [3]. In this case some simplification in * Permanent address: Instituteof NuclearSciences “Boris KidriE,” 11001Beograd,Yugoslavia. t Presentaddress: Laboratoired’Astrophysique, UniversitedeBordeauxI, 33 Talence,France. 136 Copyright AU rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
FADDEEV
EQUATIONS
IN
EIKONAL
FORM
137
the analysis is achieved by using a partial-wave expansion. On the other hand, the partial-wave expansion is not very convenient to study three-body problems involving two heavy particles (for example, two protons and one electron). In the latter case a more successful approach is via the eikonal approximation. This old method [4] has received more attention recently, following the work of Glauber [5]. In this work, Glauber did not give a general derivation of the eikonal approximation for the case of multichannel scattering, but restricted his theory by some further approximations which are referred to as the “Glauber approximation.” Most applications of the eikonal approximation to nuclear physics have been restricted to the Glauber approximation. In atomic physics, the well-known “impact. parameter method” was shown to be connected with the eikonal approximation [6, 71. The fact has also been underlined that the eikonal approximation is suitable for heavy colliding systems. In this work, we wish to incorporate the simplification achieved by the eikonal approximation in the Faddeev-Lovelace equations [8, 91. Some ideas along this line can already be found in the work of Osborn [lo]. But, in fact, the work of Osborn is, in some sense, opposite to our work since its objective is to maintain Glauber hypothesis on many-body reactions (scattering from a composite target sum of scattering from fixed centers) but to drop the eikonal approximation. The consequence of this is that the matrix elements of Osborn’s eikonal operators always contain delta functions. In our work the simplification achieved by these delta functions is accounted for in the equations themselves. This avoids the necessity of repeating part of the derivation of the eikonal approximation for the calculation of every matrix element. The organization of our work is as follows. We consider a three-body problem with the masses of two particles much bigger than the mass of the third. Section 1 gives the definitions and basic equations, Because of the mass ratio between the third particle and the other two, it is not very convenient to “eikonalize” directly the system of three coupled equations of Lovelace [9]. Therefore we first consider the case when the potential between the heavy particles is zero (Sections 2 and 3). In Section 2 we show how coupled equations of a form similar to those of Lovelace can be obtained when we use the eikonal approximation from the very beginning. Section 3 shows that the equations of Section 2 can be obtained directly from the Faddeev-Lovelace equations in the case of the “straight-line eikonal approximation” [1 I]. In Section 4, we consider the case of a nonzero potential between the heavy particles and the introduction of the distorted-wave method, following the work of Dodd and Greider [12]. In conclusion, we consider briefly some possible applications of our theory.
138
JANEV
AND
SALIN
1. GENERAL CONSIDERATIONS AND DEFINITIONS We consider three particles labeled 1, 2, and 3 of mass m, , m2 , m3 . We further consider the case when the mass of particle 3 is much smaller than the mass of particles 1 and 2. We call p the reduced mass of particle 1 and 2. Therefore (m3/d Q 1. Let us call ri the position vector of particle i with respect to some fixed origin. We shall use the coordinate systems {pi , rjk} where pi = ri - (mjrj + mkrk)/(mj + mJ; rij = ri - rj .
(1)
We also define x = r32, s = r31, R = rlz , r = p3.
69
With those definitions, x (resp. s) is the position vector of particle 3 with respect to particle 2 (resp. 1). r is the position vector of particle 3 with respect to the center of mass of particles 1 and 2. We define the reduced masses: t4
=
m&j
+
mk)l(mi
+
mj
+
mk>,
j&i
=
mimJ(mi
+
md, p = plz .
(3)
We label by i the “arrangement-channel” [13, Section 16.2.11 in which two particles not labeled by i are bound together and by 0, the arrangement channel consisting of three free particles. The relevant “channel hamiltonian” is called H, . Let H be the hamiltonian for the three-particle system. We call Vi the potential between the particles not labeled by i. H = Hi + c
Vk
k#i
We use the Green functions Gi(s) = [s - H&l,
S+‘(s)= [s - HI-l.
(5)
We shall use the Faddeev equations in the form given by Lovelace [9]. First the following transition operators are introduced:
k#j
L#i
P-9
FADDEEV EQUATIONS IN EIKONAL
Coupled equations are easily obtained
FORM
139
between these operators, [9], namely;
P’b) In the case of breakup reactions, we have, for example;
(8) In the case of two heavy particles and a light one, Dodd and Greider [12] have shown that these three coupled equations can be reduced to two coupled equations. That this is so is not astonishing if one considers the very special role played by I’, . As is well known, the total cross sections in our case are independant of F’, (except for elastic scattering). But the differential cross sections do depend on I’, and therefore a consistent treatment of this potential is still necessary. The dependance on V3 in the coupled equations (7) is introduced via the coupling with the transition operator U& or iJ8<. If i and j are different from 3, this is a very awkward situation since the physical process which Ui; describes is one in which the two heavy particles are bound together in the final state. In order to simplify the arguments in Sections 2 and 3, we shall first put V, = 0. But our results do not depend on this assumption. We shall explain in Section 4 what our expressions are when V3 f 0.
2. FADDEEV EQUATIONS IN THE EIKONAL
APPROXIMATION
a) Summary of Multichannel Eikonal Approximation The eikonal approximation has proved to be a very successful method in atomic, nuclear or particle physics. The eikonal approximation for potential scattering has been derived in its more general form by Chen and Watson [14]. Nearly all the applications have been restricted to the “straight-line eikonal approximation” or “impact-parameter method” as derived by Glauber [5]. The extention of the “straight-line eikonal approximation” to multichannel cases has been accomplished by McCarroll and Salin [15, 61 and Wilets and Wallace [7]. These authors have pointed out that the “straight-line eikonal approximation” may be considered as a first-order approximation in an expansion in m,/p. Other authors have considered the case of hyperbolic rather than straight-line trajectories (e.g., [ 161). But in the latter case only the total cross sections could be obtained, whereas in the
140
JANEV
AND
SALIN
straight-line eikonal approximation, expressions are also obtained for the differential cross sections. In the following sections, all the quantities defined in the eikonal approximation will be distinguished by a tilde. In any form of the eikonal approximation, one is led to solve the equation [i(d/dt) - 2P(t)]!Pm+(r, t) = 0,
(9)
where )j&
P,*(r,
t) = 6jf(r,
t).
(10)
t) = 0,
(11)
The wavefunction 6 is a solution of [i(d/dt) - Hi(t)]&yr,
where i is the arrangement channel and 01represents the quantum numbers necessary to define a given state in channel i. The hamiltonian s(t) is the one for particle 3 in the field of particles 1 and 2 fixed. In this section we suppose the potential between the heavy particles to be zero. This is not a problem in the eikonal approximation since this potential can always be accounted for easily. Nevertheless, the discussion of this point is postponed to Section 4. The dependance of X and Hi on t arises from the dependance of R on t. For example, in the straight-line eikonal approximation R = b + vt, where b is the impact parameter and v the relative velocity of the incoming particle and the target in the initial channel. But in this section the form of R(t) will remain unspecified. A set of independant variables must be chosen from now on. We choose this set to be r and t. The 6 functions can then be written explicity in the case of the straight-line eikonal approximation as d;,l = ya(x) exp { -k,t
- ipvr - (i/2m3)p2u2t},
(124
CFf12 = &s) exp {--i+
+ iqvr - (i/2m3)q2u2t},
(12b)
where yor(x) and qa (s) are eigenfunctions for the bound states of particles (1 + 3) and (2 + 3) respectively; E,,~ are the corresponding eigenenergies. P = mlm31(ml
+ nz2); 4 = wdh
+ m2>.
(13)
The exponential factors allow for the fact that particles 1 and 2 are moving with respect to the origin of coordinates (r = 0) with velocity qv and pv, respectively. b) Basic Definitions and Equations We shall now construct formal expressions in the eikonal approximation using the identity of form of (9) with the Schrodinger equation for the time-dependent
FADDEEV
EQUATIONS
IN
EIKONAL
141
FORM
scattering theory. The main difference arises from the signification
of the variable
t in our case. Note also that &’ and Hi depend on t. In this section we do not give
the detailed derivation of expressions that are obtained in the same way as in the time-dependent scattering theory [13]. We first define the Green functions [i(d/dt) - Hi]Gi*(t)
= 8(t),
(14)
[i(d/dt) - ~]~:“(t)
= s(t),
(15)
with the initial conditions G+(t) = g+(t) = 0, t < 0, G-(t)
= P(t)
= 0, t > 0.
It is shown in exactly the same manner as in [13] that Pa*@, t) = Gmi(r, t) + J”+” dt’ G&t -m
- t’) Vj(t’) Pa*(f)
(174
5 &ai(r, t) + I+” dt’ g(t - t’) Vj(t’) &N’,i(t’). --m
(17b)
These equations may be called the Lippmann-Schwinger equations for the eikonal approximation. We note the following properties of the Green functions:
c$:j+(t’ - t) &i(t) = -i!Bmj(t’), t’ > I, CT-(t’- t)@(t)= &j(t’), t’ < t, e-(t- t’)G+(t’- t) = 1, t’ > t, e+cr- t’)G-(t’- t) = 1, t’ < t.
(184 (18b) WC)
WW
Corresponding expressions are obtained for 8. Therefore the Green functions are related to the evolution operators for the state of particle 3 in the timedependent field created by the movement of the heavy particles. In the case of the straight-line eikonal approximation, a Born expansion of Eq. (17) yields immediately the result of Moiseiwitch [17]. c) Definition
of the Transition
Operators
The quantities which have a physical meaning are b,, = F-2 (6J/pm+)
= ,“m- <%@O-/6,,i>.
+
From these quantities, the differential and total cross sections corresponding
(19) to
142
JANEV
AND
SALIN
process N ---fB can be calculated. b,, plays a role equivalent to the S matrix. We define a matrix ri’,j whose elements are
with
We first show that
The proof runs as follows:
But
Therefore
Q.E.D. In conclusion
In the case of break-up reactions, expressions (22) to (23) are correct only if a wave-packet is used to describe the free states of particle 3. The same condition is also necessary in the usual formal scattering theory [13, p. 5041. A formal expression for Fij can now be written. Using the Lippmann-Schwinger equation for pm+, we obtain dt’ Gj+(t KB= g ($2 JCrn -cc + Jim dt’ dt” Gj+(t --m
- t’) V,(t’) G,-(t’
-
t)
t’) Vi(t’) ?F+(t’ - t”) Vj(t”) lT*-(t” - t) G-‘(t)).
(24)
FADDEEV
EQUATIONS
IN
EIKONAL
143
FORM
Therefore; Ttj = 2-2 j+* dt’ Gj+(t - t’) Vi(t’) &(t’ -02
- t)
+ j+” dt’ dt” Gj+(t - t’) vi(t’) @+(t’ - t”) V&“) et-(t” --m
- t).
(25)
G,+(t” - t).
(26)
We show in exactly the same manner that Tsi = f’+n~j+m dt’ Cj-(t - t’) vj(t’) C,+(t’ - t) -cc + j+m dt’ dt” @(t -al
- t’) Vi(t’) @+(t’ - t”) I’&“)
It is natural to introduce the transition
operators
Dz(t, t’) = Gj+(t - t’) Vi(t’) + j+m dt” cj+(t - t”) Vi(f) @(t” - t’) Vj(t’), --m D&t’, t) = Vj(t’) Gi+(t’ - t) + j+m dt” V%(t’) g(t’ - t”) I’#‘) --m
(27a)
Gi+(t” - t). (27b)
One has, of course; dt’ ?7;(t, t’) C&-(t’ pij+m --m
t) = ,1I& j+m dt’CJt -00
- t’) QJt’,
t) = i’i’ij,
(28)
but this is not necessarily the case for any value oft. This property is analogous to that of the U operators of Lovelace (6). These last operators are equivalent on the energy shell to the T matrix, but correspond to two different extensions of the T matrix off the energy-shell. d) Faddeev Equations Starting from the expression for the transition operators oij , it is possible to obtain coupled integral equations. The derivation is parallel to that of Lovelace [9]. We consider first the case i, j # 0, (i #j). We use the equation1 g*(t - t’) = cj*(t - t’) + j dt” &(t
- t”) vi(f)
ej+(f
7 t’).
(29)
1 In this section we omit the boundary on the integrals which are always calculated from - to to +a. 595/73/I-10
144
JANEV
Substituting
AND
SALIN
this expression for G in the expression for 0: , one obtains
o;(t, t’) = ej+(t - t’) Vi(t’) + j dt” Gj+(t - t”) F~‘$~:i+(t” - t’) v&‘) + j dt” dt”’ Cj+(t - t”‘) vi(t”‘)
C&P - t”) y#“)
G,+(t” - t’) y3(t’). (30)
If we now introduce I?+&, t’) = ej+(t - t’) Vi(t’) + j dt” Gj+(t - t”) v&“)
@‘+(t” -
t’) vi(t’),
(31)
we obtain u;(t,
t’) = ei+(t - t’) V,(t’) + j- dt” o;(t,
t”) c?,+(t” - t’) vi@‘).
(324
In a similar way; u;(t,
t’) = G,+(t - t’) Vi(t’) + j dt” 7?+;(t, t”) &+(t”
A corresponding
WI
system can also be obtained for Us;, namely,
t?-G(t’, t) = V&t’) Gi+(t’ U;(t’,
- t’) I’&‘).
t) = vj(t’) C#*(t
t) + j dt” Ifi
ci+(t
- t”) o&“,
- t) + j dt” Vj(t - t”) O&t”,
t),
t).
The case of breakup reactions presents no more difficulty example,
(3W W)
and we obtain,
for
Gi,
+ Vj(t’))
+ j dt” O&(t, t”) Gi+(t” - t’) Vj(t’),
(34a)
ait,(t> t’) = G,+(t - t’)(Vi(t’)
+ Vi@‘)) + j dt” o$,(t, t”) C=?,+(t” - t’) V<(t’).
(34b)
The interest of these coupled equations compared to (7) is that now we only operate on functions of the three-dimensional vector r and t, whereas the operator U of (6) operates on fonctions of r and R. Therefore we have included the simplification achieved by the eikonal approximation in the Faddeev-Lovelace equations themselves.
FADDEEV EQUATIONS IN EIKONAL
145
FORM
3. CONNECTION
BETWEEN THE RESULTS OF SECTION 2 AND THE FADDEEV-LOVELACE EQUATIONS
In the last section we showed how coupled equations similar in form to the Faddeev-Lovelace equations could be obtained when the relative motion of the heavy particles is treated in the eikonal approximation. In this section, we show that the Eq. (32) can be obtained directly from the Faddeev-Lovelace equations in the case of the straight-line eikonal approximation. As shown by McCarroll and Salin [6, 151 and Wilets and Wallace [7] the straightline eikonal approximation is a first-order approximation in an expansion in m,/p. Therefore, starting from Eq. (7) we shall develop every quantity to first order in m,/p. The wavefunctions describing free propagation in channels 1 and 2 can be written as tDa,’ = eik+&c), (354 Go2= -%%p&y).
(35b)
k, (resp., -k,) is the momentum of particle 1 (resp., 2) with respect to the center of mass of the atom (2 + 3) (resp., 1 + 3). We define a vector k such that (1/2p2)km2 + E, = (1/2&kB2
+ 4 = (1/2p2)k2 = E.
(36)
ii = ii,;k = pv.
(37)
It is easy to show that
m12m21tml + m2+ m&m1 + m2)l>--2w, k, = k + tW2db12m2/(ml + m2121 -4dk) + W/P).
k,2 = k2{1 + (llp)t
@a)
In the same manner, k, = k + W/41 Using the definitions of ~1 =
Qi
m22m3/h + m2>“l-&W
+ WP).
(38b)
and R, we have
R - h/m2)h/h
p2 = --R i- b%/m2>h/tml
-I- m2)lR- (m&,>r + 0(1/p2)(R+ r>, i- m31R - (m3/m2)r+ 0Wp2D
-t r).
(394 WV
To simplify the following arguments, we shall choose i = 1, j = 2. The proof is similar for any choice of i and j. We also put V, = 0, the case V, # 0 being discussed in Section 4.
146
JANEV
AND
SALIN
Equations (7) reduce to m>
= Vl + U,:(s) G2b) v2 3
u,‘,(s)
= VI + u,f,(s) G,(s) J’, .
(40)
We first carry out an expansion to first order in m,/p of the expression G,(E) V,@,l and we calculate this expression in configuration space. We use ewN3 G&%
~2 , ~2),
s, s’>
=
2
C
B
cps(s)
I pz I P2 -
P2’
Therefore, the expression for G,(E) V,@,’ in configuration of the form
~2’
I)
(ps*cs,j
I
*
(41)
space contains integrals
(42)
where ),I means that an integration is carried out over all s’ space. Using (38) and (39), the expression for CD: is transformed into @,(~2
, $1 =
&4
ew
(--ikp2
-
vp, - i 5 vp, - im,vr . )
k cmyyA2j2
(43)
The integral now is I =
exp(ik6
a&
s
\ P2 -
I P2 -
Pzl
P; 1) Fcp2tj
e-ikps
,
I
where F(p;) = (ye
V, exp(-im,vr’)
QJ”,(x’)),* exp (-
&
p’vp; - 2 VP,‘).
The relations between x’, s’, r’, pz’ are, of course, the same as between x, s, r, pa. The quantity p is defined in (13). The integral may now be calculated to first order in rn.Jp by the method of stationary phase [13, p. 5821. Using expression (38b) for kB we obtain G,(E) V2@,1= -i 1 v&s) exp (ikR - & I3 t
x
I
-co
q2u2t-I- iqvr - iq$) 3
dt’ ei’@‘<&s’) V, exp(-im3vr’)
x exp ( -iiE,t’ - &
(p” - q2) u”t’).
~Jx’)>~, (45)
FADDEEV
EQUATIONS
IN
EIKONAL
147
FORM
In this expression, we have used cylindrical coordinates for pz such that p,=b+vt=b+GZ
(46)
and b is a vector perpendicular to v. This expression may be written in a simplified form by using the definitions of Section 2: G,(E) V,@,l = ieikR
+m dt’ Gz(t - t’) Vz~~-(t’ 1 -m
- t) &l(t).
(47)
In going from (45) to (47) we introduce no new approximation since the operator e, is here defined by its action on a complete set of Gf12functions. The same is true for (?I also since expression (47) does not depend on the particular state 01chosen. The quantity that we wish to calculate is a matrix element of a U operator between states @al and QD2. Consider the expression <~~2/U~(EY@2)z,s
= (@~~lU~W@~~)z.s
+ (@Zulu,+,
G,(E)
, (48)
v,l@t>z.s
where the integration over p2 is only carried out over its z component from - cc to z. The matrix elements in (48) are therefore functions of b and z. An expansion of the last term of (48) to first order in m,/p is carried out by using (47): (sPB2/U,f,(E) G,(E) V,/@2)z,s
N v s” dt” @(t”) e-ik’p%J2+2(E) eikp; --m +* x --m dt’ G2f(t” - t’) V2(t’) Cl-(t’ s
t>m,l(tQ (49)
with $‘=j&;
p2” = b + vt”;
k’ = k;
pa = b + vt’.
(50)
This expression can also be written as
G,(E) V&D2> N v ( sB2(t) jT X
dt” &(t
+m dt’ G2+(t” s -co
- t”) e-“k’““U,+,(E) t’) V2(t’) el-(t’
-
ezkp; t) @2(f))r
* (51)
In an analogous manner, the term on the left of Eq. (48) can be written as <@B~U&(E) @a9 NV
+* dt’ C2+(t - t’) e-ik’Pa’U,+,(E) eikPZ’G,-(t’ -co
- t)/@
>r
. (52)
148
JANEV
AND
SALIN
The first term on the right of (48) is a Born matrix element and previous works have already given its first-order approximation in an expansion in m$p (Crothers and Holt [18], McCarroll and Salin [15], Vinogradov [19]). <@f32~2@21)Z.r dt’ e-ik’pa’CS,(t - t’) V2(t’) CT-(?’ - t) &l(t)
e+ikpz’ . >
(53)
Therefore if we define the operators z&t, t’) = Gz+(t - t’) e-ik’pGJ,,(E)
eikP,‘,
(54)
it is readily seen that the equations satisfied by these operators are identical to (32). Note that they operate also on 4 functions. Our task is not finished here. We have shown that equations identical to (32) may be obtained from (40) as a first-order approximation in rnJp. It is now necessary to show that the signification of the u operators is the same as for the r7 operators, namely, p+c tij(t) = pi% s dt’ z&t, t’) e,-(t’
- t) = I& .
(55)
For that purpose, we calculate an element of the T matrix:
T,, = <@02U&(E) @a’>,,.,.
(56)
The matrix elements of U& are calculated on the energy shell and the integration now carried out over all pz and s space. Now, (djp2U&(E)
@al)p,s, N s db v+e (6;e”(t)/e-ik’p2t&(t)
eikp2/6jaz(t)>,
is
(57)
which, to first order in mJ7 and for small scattering angles, may be reduced in the manner shown by Glauber [5] (see also [6]) to <@B~U&(E) W>,,.s = s db eiQbv+% (6B2(t) t&(t) &a1(t)),
(58)
where q is the “elastic” momentum-transfer 7 = 2~2~sin (O/2);
7-p = 0,
and 8 is the scattering angle. This relation is the same as the relation connecting T,, and F12 [6]. Therefore ut2 can be identified with O& .
149
FADDEEV EQUATIONS IN EIKONAL FORM
4. FADDEEV EQUATIONS
AND THE DISTORTED-WAVE
METHOD
In Sections 2 and 3 we have supposed that V, - 0. This was only done to achieve a greater simplicity in the exposition of the main ideas. Furthermore, the existence in (7) of a coupling with channel 3 makes very ackward the search for the first-order approximation in mJp of (7) without any transformation. This transformation should take into account the special role played by V, . In general, when V, # 0, Dodd and Greider [12] have shown that the system of Eq. (7) can also be reduced to a system of two coupled equations by using the distorted wave approximation. We first discuss briefly their results and show how the corresponding approximation can be incorporated into the eikonal form of the Faddeev-Lovelace equations. a) Discussionof Dodd and Greider Theory Let us first define the potentials vi = H - Hi
(60)
CL@= 1 + [E - Hi - wa f ic]-lwi ,
(61)
and the distorted wave operators
where wi is a distorting potential introduced for conveniance. Greider and Dodd [20] have shown that, if the following condition is fulfilled: l$
i~(@sj/(wi-)+/@~i) = 0,
(62)
then u;(s) = (co-)’ (Vj - wj+)[l + S’(S)(Vi - Wi)] Wif.
63)
For example, when i f j, a potential that produces only elastic scattering in the final channel will certainly be one. But, obviously, the condition (62) cannot be satisfied for any j and j3 since the only possible choice would be wj = 0. For example, if wi produces only elastic scattering in channel j, then condition (62) will not be satisfied for @& = @jai. More generally, if wj # 0 it is always possible to find at least one j and /3 such that condition (62) is not satisfied. For such values of j, iJ& cannot be written as (63). When taking this fact into account, care must be exercised in the interpretation of some of the expressions of Dodd and Greider. In their work, Dodd and Greider write coupled equations between U$ and some other operator ZJ$ of the form (63). Now, condition (62) may be valid for U$ but not for U& . Therefore no straightforward physical signification can be given to the matrix elements of Uz , though the coupled equations of Dodd and Greider are correct.
1.50
JANEV
AND
SALIN
We first show how this difficulty can be overcome in some special cases. We assume that condition (62) is fulfilled for i # j only. This means that there exists /3 such that
It is then shown that if i = j
where %A. = (Oj-)+ (Vj - Wj’)[l + S’(S)(Vi - Wi)] wi+,
(66)
while, for i #,j,
The expressions for %& are the same as (63). The only difference is that the matrix elements of %$ are not necessarily equal to those of&j on the energy shell. Note also that the first term in (65) does not describe any rearrangement process since it is not present for i # j. Therefore all the coupling with rearrangement channels is included in 9; and therefore we need only look for coupled equations for the @ operators and not the U operators. Of course, we have restricted our choice for Wi but it is obvious that in most applications this is not really a restriction since Wj is nearly always chosen so that this extra condition is satisfied. The form of the ??Zoperators being equivalent to that of the article of Dodd and Greider [12] it is now obvious that all their results are valid for our % operators, namely, 92; = (mj-)+ (vj - wjt) wi+ + %;jG,+(vi %zj = (a~~-)+ (uj - wj+) wi+ + @;Gi+u, ,
- wi) wi+,
(684 Wb)
where G,+ = [E - H - u, + ic]-l
and v, and wj are such that (1) condition (62) is verified for i # j (2) v, and wi are chosen in such a way that no disconnected diagram is obtained from (68). Corresponding
results can also be obtained for @‘ij .
FADDEEV
EQUATIONS
IN
EIKONAL
FORM
151
b) Complete Treatment of the Potential Between the Heavy Particles We now choose wj zzzwi zz2w = v, ; q* = qf
= &,
(69)
and we introduce the operators
(co-)+ WA= a;.
(70)
These W operators operate on the space of x functions
xi+ = &Dai.
(71)
Coupled equations will now be obtained for W$ . As a possible choice for v, , we take v, = vj * (72) This choice is consistent with the conditions on wj and v, . The result is W; = Vi + W;Gj+Vi
,
(734
W; = Vi + W;Gi+Vi
.
WI
These equations are the same as (7) if we put V, = 0. The only difference is that we are now interested in matrix elements of W between x functions. On practical grounds, Eq. (73) may appear no more simple than the starting ones since the influence of V, should be accounted for exactly in the calculation of x. Anyhow we will show that they are a better starting point to study the limit of heavy particles 1 and 2. The expression for Tij is Tij = &(o-(E))+
w + (u-)+ Wij(E)o+.
(74)
c) Eikonal Theory A theory similar to that of Dodd and Greider can be constructed in the eikonal approximation by using the ideas explained in Section 2. We shall briefly sketch the procedure, the detailed arguments being avoided. We define the distortedwave operators as c&*(t) = 1 + s dt’ g’i*(t - t’) w,(t’) Gi-(t’ - t),
(75)
where [
i $ - Hi - wi] ii+(t)
= s(t)
(76)
152
JANEV
AND
SALIN
with the usual boundary conditions. It is then easy to show that if ,‘jz then Yi,j =
(cFB$&-(t))+ @>,i) = 0,
(77)
+m s --m dt (@f3ycz-(t))+ (Uj - Wj) P,+(t)>.
(78)
We can now choose wi such that condition (77) is fulfilled except for i = j. Therefore, ?-& = S&;+(z+
-
wj+) + j &;(t,
t’) 6,+(f)
G,-(t’
- t) dr’,
(79)
where &;(t, t’) = q+(t
- t’) aj-(t’)+ (Uj - wi’)
+ j dt” C&+(t - t”) Gj-(t”)+ (uj - wit) @+(t” - t’)(ui - wi).
(80)
Then coupled equations can be obtained by the usual procedure between @$(t, t’) and an operator &I$(& t’). As an example, we consider again the case wj
=
wi
=
w
=
v,
; ij,*
zz
(pt.
(81)
We can define as in the preceeding paragraph @(t, t’) operators acting on the 2 functions gt) = m&yf). (82) The coupled equations obtained for w(t) are @$(t, t’) = g;(t c(t,
t’) = &+(t -
t’) Vi(f)
+ j dr” c(t,
t”) &+(t”
t’) Vi@‘) + j dt” t@;(t, t”) &+(t”
-
t’) &(t’),
(W
- t’) Vi(C).
(8%)
Now it is obvious that Xi*(t) = exp IF j:, &*(t - t’) = ef(t
Vdf’) dt’l saW,
- 1’) exp (pi jl
V3(t”) dt”).
Therefore the coupled equations satisfied by the operators I@, t’) exp (- jl, V3(P) dt”)
(84) (85)
FADDEEV EQUATIONS IN EIKONAL FORM
153
are identical to (32), that is, the equation satisfied by l? when V, = 0. Also the matrix elements of pj reduce to gij = exp (--i j+w V3(t) dt) j+m dt’ (&(t) -02 --m
o;(t, t’) c&t’)).
m
The influence of V, reduces to merely a phase factor, a result that could be shown directly from (9) (see [21]). This last result can be used to simplify a great deal the calculations of cross sections. The fact that the influence of V, reduces to merely a phase factor in the calculation of & (except for elastic scattering) ensures that to first order in m3/p the total cross sections are independant of V, since they are only functions of / zj 12. For the calculation of differential cross sections, two methods can be used: (1) All the calculations are performed in the eikonal approximation. This allows one to calculate & as a function of b. This information can then be used to calculate Zj . For example, in the case of the straight-line eikonal approximation, the relation between qj and zj is SLij = 1 db exp(iqb) K@).
(87)
(2) It is also possible to calculate & by using (7) with V, = 0. Let us call the result thus obtained tij . Then to first order in m,/p we have qj = &
J” dq exp(-iirlb)
tij($
exp (-i
J$
k’a(I b + vt I) d).
(88)
Therefore, Kj
= -
(2$
J dq’ tij(q’) J db exp (i(q - Q’) b -
i J.*w V3(/ b + vt I) dt). --m
(89)
All the dependance on V, is taken into account by the second integral which, in some cases, may be very easy to calculate.
CONCLUSION
We have shown how the six-dimensional coupled integral equations of Lovelace could be reduced to two coupled four-dimensional integral equations in the eikonal approximation. These coupled equations can be as well written as a single integral equation (see 32) but this is not necessarily a simplification. The use of a
154
JANEV AND SALIN
suitable expansion for the two-body operators will further reduce the problem to two coupled integral equations in a single variable (t). The problem is then amenable to, at least, a numerical treatment. In atomic physics, the interest of the equations thus obtained is that the coupling with rearrangement channels is automatically included in the formulation. This is an interesting property to study, for example, proton-hydrogen collisions in the medium energy range (10-100 keV). If an eigenfunction expansion is used to solve these integral equations, one obtains coupled equations which greatly differ in form from the traditional ones [22,23], the most striking difference being that we obtain coupled integral equations instead of coupled differential equations. The only difficulty that remains to be solved for such applications is the coulomb character of the interactions between charged particles. Some of these applications will be considered with more details in future publications.
ACKNOWLEDGMENTS A. Salin wishes to thank the Institute of Physics and the Institute “Boris KidriE,” Beograd, Yugoslavia, for their warm hospitality while this work was performed.
REFERENCES 1. R. MCCARROLLAND A. SALIN, Proc. Royal Sot. A 300 (1967), 202-209. 2. C. P. CARPENTERAND T. F. TUAN, Phys. Reu. A 2 (1970), 1811-1821. 3. J. C. Y. CHEN AND K. T. CHUNG, Phys. Rev. A 2 (1970), 1449-1457. 4. G. MOLIERE,2. Naturforschung A 2 (1947), 133-40. 5. R. J. GLAUBER,“Lectures in Theoretical Physics,” (W. E. Brittin and L. G. Dunham, Eds.), Vol. 1, p. 315, Interscience, New York, 1959. 6. R. MCCARROLLAND A. SALIN, J. Phys. B 1 (1968), 163-171. 7. L. WILETSAND S. J. WALLACE,Phys. Rev. 169 (1968), 84-91. 8. L. D. FADDEEV,Sou. Phys.-JETP 12 (1961), 1014-1019. 9. C. LOVELACE,“Strong Interactions and High Energy Physics” (R. G. Moorhouse Ed.), Oliver and Boyd, London, 1964. 10. T. A. OSBORN,Ann. Phys. (N.Y.) 58 (1970), 417-453. 11. J. C. Y. CHEN AND K. M. WATSON,Phys. Rev. 188 (1969), 236-256. 12. L. R. DODD AND K. R. GREIDER,Phys. Rev. 146 (1966), 675-686. 13. R. G. NEWTON, “Scattering Theory of Wave and particles,” McGraw-Hill, New York, NY, 1966. 14. J. C. Y. CHENAND K. M. WATSON,Phys. Reu. 174 (1968), 152-164. 15. R. MCCARROLLAND A. SALJN, C. R. Acad. Sci. (Paris) 263 (1966), 329-332. 16. K. ALDER, A. BOHR, T. Huus, B. M~TTEL.~ON AND A. WINTHER, Rev. Mod. Phys. 28 (1956), 432-542. 17. B. L. MOISEIWITCH,Proc. Phys. Sot. 87 (1966), 885-888. 18. D. S. F. CROTHEIUAND A. R. HOLT, Proc. Phys. Sot. 88 (1966), 75-81.
FADDEEV
19. 20. 21. 22. 23.
A. K. A. D. L.
EQUATIONS
IN
EIKONAL
V. VINOGRAWV, Opt. Spectrosc. 22 (1967), 361-362. R. GREIDER AND L. R. DODD, Phys. Rev. 146 (1966), SALIN, J. Phys. B3 (1970), 937-951. R. BATES, Proc. Royal Sot. A 247 (1958), 294-301. WILETS AND D. F. GALL-R, Phys. Rev. 147 (1966),
FORM
671-675.
13-20.