- Email: [email protected]
Off-energy-shell effects in multiple scattering
ANNALS
OF PHYSICS
81,
591-624 (1973)
Off-Energy-Shell
Effects in Multiple
Scattering*
E. KUIAWSKI+ Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 AND
E. LAMBERT Institut de Physique de l’ilniversitt!,
Neuchritel, Switzerland
Received March 2, 1973
The elastic scattering from a two-body bound system is investigated in order to determine what information about the nuclear structure and the projectile-nucleon interaction may be obtained at intermediate energies. Emphasis is placed on the kinematics and the importance of properly treating the overlapping potential region. The kinematics of the “fixed scatterer” and “impulse” approximations result in differences which are still significant at intermediate energies. As far as scattering information is concerned, the overlapping potential region is the interesting one since it requires explicit knowledge of the interaction. Our results suggest that for nucleon scattering at intermediate energies the corrections due to properly taking into account of&energy-shell effects are likely to be comparable with the contributions due to the short-range twobody correlations. The relevance of the above results to various models currently employed in analyzing high- and intermediate-energy nuclear scattering is also discussed.
1. INTRODUCTION The full multiple scattering equations are very complicated, and with the exception of a few pedagogical examples and the three-body problem it is impractical to solve them exactly. Two general approaches have then been pursued: (i) attempts to approximately solve the equations using simplified propagators [l-3]; and (ii) attempts to evaluate various multiple scattering terms in the Watson-Faddeev multiple scattering series [4-71. In this work we follow the second * This work is supported in part through funds provided by the Atomic Energy Commission under Contract (ll-l)-3069 and the Swiss National Science Foundation. + Present address: Department of Physics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213. Copyright All rights
0 1973 by Academic Press, Inc. of reproduction is any form reserved.
591
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KUJAWSKI
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approach because we think that it lends itself more readily to study which information may be extracted from high- and intermediate-energy elastic scattering. The essential simplification when considering elastic intermediate- and highenergy nuclear scattering results from the applicability of the so-called “fixed scatterer” approximation (FSA) [3, 81, which corresponds to performing closure on the intermediate target eigenstates assuming that they are completely degenerate. The projectile-target scattering amplitude is then obtained by calculating the scattering from instantaneously fixed scatterers using the reduced mass of the total system and averaging over the internal eigenstates. The problem is thereby reduced to that of potential scattering, where, however, we no longer have spherical symmetry. The results obtained from studying scattering off a sum of potentials then shed light on which information can be obtained from elastic high- and intermediate-energy nuclear scattering. For nonoverlapping potentials the complete scattering is explicitly determined in terms of the individual on-energy-shell scattering amplitudes [9-121. In contrast the scattering from overlapping potentials depends on the off-shell behavior of the specific interaction, and it has been suggested that by supplementing the two-body data with many-body scattering data it may be possible to distinguish among phase equivalent potentials [IO]. As a result of averaging the scattering amplitude over the ground state density the potentials due to the individual target nucleons to which the projectile is subjected generally overlap, the amount of overlapping being sensitive to the short-range behavior of the internal target wave function. Elastic scattering then has the unfortunate feature of mixing up information about the two-body interaction with two-body correlations. In this work we study nucleon-deuteron elastic scattering within the FSA for a variety of two-body interactions and wave functions in order to obtain insight into the importance of the contributions arising from properly treating the overlapping potentials and to learn what information may be extracted from the experimental data. We now outline the succeeding sections. In Section 2 the FSA and impulse approximation are discussed and compared. In Section 3 we develop the double scattering term in the multiple scattering series stressing the dependence on the off-energy-shell two-body t-matrices. In Section 4 we present the exact result for scattering by two S-wave separable potentials. In Section 5 we present two approximations whereby the scattering is expressible simply in terms of on-shell information. In one case we assume the potentials do not overlap while in the other we neglect the principal part of the propagator. In Section 6 we introduce the various nucleon-nucleon potentials which we consider in our calculations. We include both local and nonlocal phase equivalent potentials and present their off-energy-shell matrix elements. In Section 7 we present the results of our numerical studies. The conclusions are given in Section 8.
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2. THE KINEMATICS One of the basic problems in any multiple scattering formulation is to relate the matrix elements of the relevant many-body modified operators to the free two-body scattering matrix elements whose on-shell values are determined in elastic scattering experiments. For high-energy elastic scattering two formulations seem relevant for use in a multiple scattering expansion: (i) the “fixed scatter approximation” (FSA) kinematics [3, 81; and (ii) the “on-energy-shell impulse approximation” (OEI) kinematics [13]. The essential difference is that in the FSA the struck nucleon is assumed to recoil with the target as a whole while in the OEI the struck nucleon essentially recoils freely. The significance of the above two approaches is best illustrated by specifically considering the three-body problem with arbitrary masses. Let the indices 1, 2, and 3 refer specifically to the projectile and two target “particles” with masses m, , m2 , m3 , lab momenta kl , k, , k, and lab coordinates R, , R, , R, . In the usual manner we introduce the “canonical” parameters [14]. Let qi be the relative momentum of particles j and k, pi the relative momentum of particle i with respect to the center of mass of j and k, P the total momentum, ri , xi , R the corresponding conjugate coordinates, and pLi , pjk , M the associated masses: p. = hi 2 qi =
P=
+ mk> ki - m4kj + kk) M
mjk, - m,ki +-km,
3
(2.la)
’
iki, i-l m&
ri = Ri -
+
mkRk
mj+mk
’
xi = Rk - Ri,
(2.ld) (2.le)
R
R = 1/M c maRi,
(2.lf)
i-1 Mm* pi
=
pjk
=
m3
’
(2.lh)
mjmk mj
M=
+
M
irni. $4
f
mk
’
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Assuming that the particles interact via two-body forces only, the resulting Schriidinger equation in the total center-of-mass system reads PI2 (-+E a4
2
v12
+
+
~12
+
~13
+
02.9
W,,
1
rl)
=
EW,,
r3,
(2.2a)
Xl>,
(2.2b)
where =
VlZ('l
(dm2)
Xl),
01s
=
~13(b
-
(P22/%>
and E denotes the appropriate nonrelativistic energy which is the sum of the kinetic energy of the projectile and the binding energy of the target. We now proceed to discuss various approximations of interest for high- and intermediate-energy nuclear scattering. The FSA In the FSA the internal target Hamiltonian 412/&‘2,
where Es is the target binding then reads
+
V23h)
is neglected, i.e., we set =
-EB
7
(2.3)
energy. The resulting Schrodinger equation (2.2) (2.4)
where E is the kinetic energy in the total center-of-mass system (E = E + EB). The solution to (2.4) is clearly of the form (2.5) where &,(x1) denotes the initial internal wave function of the target. #(xl ; rl) satisfies the Schrodinger equation specified by (2.4) where x1 is now a parameter. As a result of neglecting the internal target Hamiltonian the problem has been reduced to that of scattering from two potentials located at -(y.Jm,) x1 and This is a nontrivial problem since the resulting total or global +(P23h) Xl * potential is no longer spherically symmetric. The exact solution then involves solving a system of coupled equations coupled through angular momenta. In practice this system of coupled equations must be limited by truncation; unfortunately, for high-energy scattering where the wave length is short compared to the size of the scatterer this involves a rather sizable system of coupled equations. The total or composite elastic scattering amplitude is then obtained by averaging
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the amplitude calculated for various positions internal wave function &(x), j&k’,
k) = --2&4rr
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of the target nucleons over the
1 dx 1 &(x)12 1 dr e-ik’*rV(x, r) @(x; r),
(2.6)
where #+ is the solution to (2.4), and Vx, r) f u12+ o13. It is important to realize that although the relative distance between the two target constituents remains fixed during the scattering, their center-of-mass motion is properly taken into account. On-Energy-Shell Impulse Kinematics
In the Faddeev method the three-body scattering amplitude three amplitudes Ti which satisfy the equations [14]
T is broken up into
T,(E) = h(E) + f,(E) G,(E) 1 G(E), i#i
(2.7)
where G, is the free propagator for the three particles and ki now an operator. The variables are defined by Eqs. (2.1). Zi is a many-body operator because the propagator appearing in it contains the entire kinetic energy; however, it can readily be related to the two-body t operator in its own center-of-mass system. For the sake of definiteness let us consider 2, which involves only u12. Making use of the fact that u12 depends only on the coordinate conjugate to q3 one obtains (P319alP’ I W)
I PSQap) = %P’ - P) %P,’ - P3) x
(2.8a)
where all the quantities are as previously defined. t is the two-body t-matrix in its own center-of-mass. In terms of the appropriate momenta p1 and q1 defiend by (2. I) the above t-matrix element reads (2.8b)
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where a’ = e - (PI’ - pa m2 CL?.3’
WC------ 1 21112
(%P,
+
91~
-
EB
9
To obtain the elastic scattering off the deuteron in the impulse approximation the above matrix element then needs to be averaged over the deuteron wave function. As is well known this involves calculating the two-body t operator off-the-energy shell. At high energies one often neglects the internal momentum and binding energy [13], namely
*PI’ + (2.9)
where 3 = &
(%Pl)“.
We refer to the above approximation as the OEI. The resulting t-matrix element is exactly the one measured in the relevant two-body scattering experiment. Comparison of the FSA and OEI As already discussed the FSA corresponds to neglecting the internal target Hamiltonian or performing closure over the intermediate target eigenstates neglecting their excitation energy. The center-of-mass motion is properly taken into account and the corrections due to neglecting the internal Hamiltonian may systematically be investigated. It, however, has the unfortunate feature that the relevant matrix elements differ from those determined in the corresponding projectile-nucleon scattering due to the different effective mass relevant in the two scatterings. We then see that in the impulse approximation or Faddeev-Watson multiple scattering series the target nucleon in lowest order is effectively treated as recoiling freely. Unless the internal motion is neglected as in the OEI, off-shell effects need to be taken into account. Furthermore, as for elastic scattering the target nucleus recoils as a whole the corrections to the propagator or a proper treatment of the recoil may then need to be explicitly included. For the sake of definiteness and clarity we now write down the single scattering terms relevant for N-D scattering within the FSA and OEI. These matrix elements may be denoted by (cm’ I f (+$)
I cPl>,
(2.10)
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where c = $ in the OEI and c = 1 in the FSA. At this point it is worth emphasizing that the two kinematics coincide at high energies where the Glauber eikonal approximation [l] is valid. This may readily be seen by observing that in the Glauber eikonal approximation the relevant parameter is the velocity and cp,/cpl = pI/pI . However, at intermediate energies the two t-matrices may still differ significantly, and in our work we will compare these results. The usefulness and preferability of a multiple scattering formulation based on the FSA has also been conjectured by Kowalski and Pieper [15] and appears to be supported by their calculations. The multiple scattering series may be expanded in terms of rather general t-matrices [5]; however, since in practice the series will be truncated the usefulness of the approach will depend on the specific t-matrix employed. We wish to point out that the FSA and OEI kinematics would also coincide if the dynamics can be parametrized as f(q) = 2
(1 - ia) e-8Q2/2,
(2.11)
where CT,01,and /3 are energy independent. Such parametrizations are the usual ones employed at high energies [ 1, 71 and are characteristic of the high-energy limit. 3. THE DOUBLE SCATTERING
AMPLITUDE
In this section we develop the Watson-Faddeev multiple scattering series up to double scattering terms with special attention to its off-energy-shell dependence. For the sake of definiteness we consider N-D scattering. As shown in Section 2, in the FSA the problem is then reduced to that of scattering from two potentials. Assuming the potentials to be identical and located at &R/2 the resulting Schriidinger equation then reads (3.1) + u(r - W2N + 4~ + @/WlrCI = 4 where ~1 = Qm, E = k2/2p with k being the momentum in the three-body centerof-mass system, and m is the nucleon mass. The multiple scattering series for the resulting scattering operator is given by L.-(V,2/+4
T = Tl + 7-2-I- T,G,T, f T,G,T, i- ....
(3.2a)
where (R’,
r’ 1 G 1 R r) = 0 >
2p @‘-” 47r ( t - r’ ( 6(R - R’),
= e-iq.a/2
(3.2~) (3.2d)
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t
q = k’ - k, and is the collision operator for scattering from the potential D located at the origin. The composite scattering amplitude for elastic scattering is then obtained by averaging the resulting scattering amplitude over the internal deuteron wave function Q0
p(k’,@oI TIk, @o> Pj- d3Rp(Wk’I TI k),
foo(k’, k) = -CW2
= -CW2
(3.3)
where T is given by (3.2) and p denotes the deuteron density. We now consider the scattering only up to double scattering terms. Keeping our numerical calculations in mind we consider the following idealized situation: (i) Spherically symmetric target density. We assume that the two target particles are in a relative S-state. Note that the deuteron has a D-state component. (ii) Selective scattering in S-states only. Since the potentials are assumed to scatterer S-waves only there will be no admixture of angular momenta due to multiple scatterings. This greatly simplifies the algebra. Although the above assumptions need not be introduced at this point, they greatly simplify the resulting expressions and make the essential physics more transparent. The resulting scattering amplitude is then given by
ho@‘, k> =
-‘NW3
jm R2 dR p(R) r,,(k’, k; R),
(3.4a)
0
where
To@‘,k; R) =
joW2)
I to(e) I k) +
(2y/2n2)
m p dp --m
(P
$2743 Zo(k
R>
=
-
2R
s
eipR
j,(QR) Z,(k, R),
1 to(E)
1 k>2
k2 - p2 + i8 ’
(3.4b) (3.4c)
q = k’ - k, Q = (k + k’)/2 and (p 1 t,(e) 1k) is the right-half off-shell S-wave t-matrix with l = k2/2p. Although the above expressions have been determined consistently within the FSA, their generalization to the OEI is obvious and simply involves using the corresponding two-body t-matrices as specified in Eq. (2.10). The corrections, however, will be different. In the remainder of this section we then address ourselves to the evaluation of IO . This is the term of interest since it contains double scattering and possible off-shell information. We consider u to have a finite range a. Before proceeding it is useful
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to list some relevant properties of the half-off-shell t-matrix elements ( p 1 tl j k)HOS in the complex p-plane [lo]. For an interaction of finite range a, in the complex p-plane (p I tl 1k)HOS is an entire function with the following properties: (i) (ii)
for R > a, liml,l,, e*i*R(p / tl I k)HOS = 0 for Imp >( 0, it has a zero of order 1 at the origin.
Using the above properties of the t-matrix elements the integral 1, for R > 2a can readily be evaluated by contour integration in the complex p-plane. Proceeding as indicated one finds I,,(k, R > 2a) = - 9
(k j t,, ($-)
tL
1 k)2 eikR.
We then see that for nonoverlapping potentials the double scattering term is expressible entirely in terms of the on-shell two-body scattering amplitudes. It is important to realize that the above statement does not mean that there is no off-shell scattering but rather that for nonoverlapping potentials the scattering is independent of the off-shell amplitudes. This is not to be confused with neglecting the principal part of the propagator or assuming that the scatterers are far apart so as to enable one to use the asymptotic form of the scattered waves [16]. Although we have only considered up to double scattering terms, it can in general be shown that for nonoverlapping potentials the full scattering amplitude depends only on the individual on-shell scattering amplitudes [ 111. Assuming finite range interactions with radius a the composite scattering amplitude for our “deuteron problem” may then be broken up into two contributions (3.6a)
ho@‘, k> = .Mk’, W + f&k’, W, where f2 corresponds to the scattering from the nonoverlapping potential which as previously discussed depends only on on-shell information, and f,(k’> k) = --4~(24~
1’” R2 dR p(R) T&‘, 0
k; R),
region
(3.6b)
which is the contribution arising from the overlap of the scatterers. The evaluation of (3.6b) requires a knowledge of the off-shell t-matrix elements, and we will calculate it for a variety of interactions. The information of interest (off-shell effects and two-body correlations) is then contained in (3.6b), and we investigate its importance for elastic intermediate-energy nuclear scattering.
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KUJAWSKI
4. SCATTERING
AND LAMBERT
FROM A SUM OF SEPARABLE POTENTIALS
In this section we examine the scattering from two mobile S-wave separable potentials. Since the scattering can be exactly solved this is a very interesting case for investigating various approximations. Before proceeding a few words are in order concerning the connection between the FSA and the closure approximation assuming degeneracy for the target eigenstates and Galilean invariant separable interactions [17]. In this case, closure [3] no longer leads to a result identifiable with the FSA because the two-body interaction is nonlocal in both the target-particles and projectile coordinates. However, it is consistent to make the FSA provided the target-particles are much heavier and slower than the projectile [18]. We also wish to stress that one may, nevertheless, discuss the Watson-Faddeev multiple scattering series presented in Section 3 using separable t-matrices. For scattering by a sum of separable nonlocal potentials Foldy and Walecka were able to write down the exact result in closed form. A simple derivation for the scattering by two S-wave separable nonlocal potentials using the Faddeev approach is presented in Appendix A. The scattering amplitude for scattering by two S-wave separable potentials
(4.1)
located at -&R/2 is given by (k’ 1 T(E, R) ) k) = ‘:““$$’
[co, (q * +)
+ h(R) cos(Q . R)],
(4.2)
where q = k’ - k, Q = (k + k’)/2, E = k2/2p,
W I t(c) I 6 = Mk’) g(k)/44 x h(R) = d(~) I d”l! eiveRE _ d(e) = 1 - 47rh IOrnp2 dp
(4.3a)
1
(py2p)
+ is g”(P)*
1 E - (p2/q4
+ is g”(P)*
(4.3b) (4.3c)
For nonlocal interactions of finite range a and R > 2u, h(R) can be explicitly evaluated and expressed in terms of on-shell information illustrating that for nonoverlapping potentials there is no off-shell information h(R > 2a) = -(2~)~ p(k’ 1t(k2/2p) ) k) eikR/R.
(4.4)
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Equation (4.2) is exact and clearly exhibits the multiple scattering effects. The numerator is simply the result up to double scattering terms. The denominator represents the sum of all intermediate reflections off the scatterers and gives rise to what is commonly referred to as the “effective or local field corrections” [8]. For separable potentials the exact result will be compared with the scattering up to double scattering terms thereby allowing us to study the convergence of the multiple scattering series.
5. APPROXIMATIONS
RELEVANT
AT HIGH
ENERGIES
The essential simplification when considering elastic intermediate- and highenergy nuclear scattering results from the applicability of the FSA which corresponds to performing closure on the intermediate target eigenstates. As shown in Section 2 the FSA reduces the problem to computing the scattering amplitude from N fixed potentials, where N is the number of target nucleons, and then averaging it over the internal target wave function. In Section 3 the scattering amplitude for the scattering by two local potentials was given up to double scattering terms while in Section 4 the exact result was given for two separable potentials. At high energies the multiple scattering series is supposedly rapidly convergent [6], most of the scattering being taken into account by the first two terms. However, it is interesting whenever feasible to examine its convergence. In addition to the FSA and truncated versions of the Faddeev-Watson multiple scattering series we also examine various approximations which result in the full scattering amplitude being expressed purely in terms of the individual on-shell amplitudes. For nonoverlapping potentials the scattering depends only on the two-body on-shell partial amplitudes. However, in the FSA as a result of averaging the scattering amplitude over the target ground state the potentials due to the individual target nucleons to which the projectile is subjected generally overlap. This greatly complicates the problem since the scattering can no longer be expressed simply in terms of on-shell information and its calculation requires explicit knowledge of the interaction. Two approximations then come to mind whereby the scattering is expressible simply in terms of on-shell information: (i) The nonoverlappingpotential approximation. The scattering amplitude is calculated assuming that the potentials do not overlap. The scattering amplitude up to double scattering terms, presented in Section 3, is approximated by
fiioP(k’, k) = ---4/42~)~ j-m Ra dR p(R) 7NoP@t,k; R), 0
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where
7NoP&',k; R>= jo (4+)
P-Y
CLq jot,, (k I to(4I k>2. (5.1 b)
All the quantities are as defined in Section 3. A similar approximation may be applied to the exact result given in Section 4. Assuming that the potentials do not overlap, Eq. (4.3b) simply reduces to hNoP(R) = -(27+
hg2(k) eikR P yqqR 3
(5.2)
and the multiple scattering amplitude involves using (5.2) in (4.2) for all values of R. As previously discussed in Section 4, the resulting scattering amplitude obtained using (5.2) contains the exact propagator and all orders in the multiple scattering terms but the potentials are assumed not to overlap. Another approximation which leads to a (ii) On-energy-shell scattering. scattering amplitude which is also expressible simply in terms of on-shell information is the one resulting from neglecting the principal part of the propagator or the Green’s function. In the on-energy-shell scattering approximation the exact propagator in the multiple scattering series is approximated by 1 k2 - pa + iS
Making
--t -in&k2
the above substitution los(k, R) = - &
- p2) = - -&
(5.3)
in 1, (3.4~) we readily find (24s
The resulting scattering amplitude Section 3 then reduces to
+‘,
[6(k + p) + 6(k - p)].
(k [ t,(e) 1k)2 ( ezkR ; ‘-‘,,).
(5.4)
up to double scattering terms as given in
k R>= j. (4 -&)
~(279~ ( ePkR yRe-‘““)
j,(QR)
(k I to(E) 1k>2,
(5.5a)
and
f&W, k) = --4p(27~)~ j-0 R2 dR p(R) @(k’, k; R).
(5.5b)
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A similar approximation may be applied to the exact result given in Section IV. In this case h(R) as given by (4.3b) simply reduces to
).
(5.6)
The above result obtained by simply keeping the on-energy-shell part of the propagator and double scattering terms is reminiscent of the result provided by the Glauber multiple scattering model. Such an analogy may however be misleading since the latter result depends on the use of the high-energy eikonal approximation or a linearized propagator, and for local interactions the principal part of the double scattering term is exactly cancelled by all the higher order multiple scattering terms [12, 19,201. In contrast Eq. (5.5) is a phenomenological variant of the Glauber multiple scattering formulation which is often used in data analysis. We close this section by comparing the two approximations presented above. For a fixed position of the scatterers the double scattering term within the above two approximations appear to be quite different $OP(k’,
k; R) z -(2~)~
Tz6(k’, k; R) 2 -(27r)2
p(k
1 t,(e) ) k)2 eikR
eiQR2ii-i*R
p(k 1 t,,(e) 1k>2 ( e’kR ; ‘-“““)(,
), ezQR2i;-IVR
),
(5.7b) where Q = (k + k’)/2. In the forward direction Q = k. It may now readily be seen that as a result of averaging over the target density that at high energies and forward angles the two approximations are very similar. When averaged over the target density only the nonoscillatory terms of the expressions (5.7a) and (5.7b) contribute, and both may be approximated by $0P*8(k’,
k; R) w
-(2~)~
p(k I &,(E) I k>2 (i/2kR).
(5.74
k-rm
As pointed out by Tobocman and Pauli [21] such a cancellation or interference arising from the spatial smearing (or time variation) of the location of the scattering centers partly explains the validity or success of the Glauber multiple scattering model. At larger angles the higher order multiple scattering terms cannot be neglected and the two approximations may differ when applied to the full scattering amplitude. This latter case will be studied numerically using separable potentials.
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6. OFF-SHELL
DEPENDENCE
AND LAMBERT
OF THE MULTIPLE
SCATTERING
TERMS
In this section we study several phase equivalent potentials and their off-energyshell t-matrix elements. These expressions are then employed to calculate the corresponding multiple scattering terms relevant to our numerical studies. As shown in Sections 3 and 4 the evaluation of the second- and higher-order multiple scattering terms for overlapping potentials requires a knowledge of the off-energy-shell t-matrix. There are two possible ways of obtaining off-energy-shell t matrices: (i) direct parametrization or analytic continuation of the scattering amplitude [22]; (ii) determination of a potential and subsequent calculation of the offenergy-shell t-matrix elements. We follow the second approach because of the greater physical insight associated with potentials. We first need to recall a few results of the inverse scattering problem [14]. Given a phase shift of a given angular momentum for all energies and provided there is no bound state there exists a unique local energy-independent potential associated with it. There are several well known procedures for constructing a local potential given a phase shift for all energies. A separable potential of the form
is also uniquely determined [23]. Although these two potentials are phase equivalent they have different off-shell t-matrix elements. For the sake of simplicity we now restrict ourselves to S-wave scattering for a particle with “effective” mass p, where p is at present left unspecified. We study the S-wave scattering from a square well and the corresponding phase equivalent separable potential. It is important to realize that the phase equivalent potential depends on the mass; this point is further discussed below as well as in Section 7. For N-N scattering the observable phase shifts are specified for jZ = (m/2). Consider a local potential
(6.1)
where 0 is the step function. The corresponding S-wave scattering matrix element is given by (6.2a) &(k> = fW/Y(--k), where f(k) = e-4”a(cos 2~ + i(k/ R) sin ~a), (6.2b) : = (k2 - 2@VO)1/2.
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As shown by Gourdin and Martin S-wave scattering is given by
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605
[23] the equivalent separable local potential for
where bW12 = G/ww2
kx..f(k) -f(-k)l.
(6.3b)
In coordinate space the separable potential is of the form (r’ 1 V 1r)s = C(I’) t?(r),
(6.3~)
where d denotes the properly normalized Fourier transform of v. Although the above two potentials (6.1) and (6.3) are phase equivalent they have different off-shell t-matrix elements. The N-N t-matrix may directly be employed in the multiple scattering series using the OEI kinematics. This is to be contrasted with the multiple scattering series within the FSA where the relevant mass (p) is the total reduced mass, and for N-D scattering p = sm. In that event the elementary on-shell t-matrix elements are no longer equal because they refer to a mass different from the one (p = m/2) appearing in the phase equivalent potentials. The calculation of off-energy-shell t-matrix elements is widely discussed in the literature, and we only write down the results. For the local square well potential the half-off-shell S-wave t-matrix element is given by r? sin pa cos Za - p sin f?a cos pa 17sin ka cos r?a - k sin Za cos ka
I k)L >
(6.4a)
where
I to (+)
I k)L
=
&$
k sin ka cos r7a - k sin I?a cos ka 3 r? cos t?a - ik sin Ra
(6.4b)
fT2 = k2 - ~/.LV~.
(6-k)
and For the equivalent separable potential the half-off-shell t-matrix element is given by
k 112 sin pa cos &a - (p/G) sin &a cospa U2 I k)s = (>) ( sin ka cos ria - (k/R) sin r?a cos ka ) x
I kh
3
(6.5a)
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where I? sin ka cos Ea - k sin .?a cos ka - 1) + : cos t?a - ik sin Ra ’
r?e-ika(plp
(6Sb)
and iP=k2-2~Vo,
(6.52)
52 =p”-2pv,.
The phase equivalence of the above two potentials for the mass fi can be readily seen from Eqs. (6.4b) and (6Sb) which lead to
(k I to (+)
I 6, = (k I to
($-)
I k)s
(6.6)
.
Equations (6.1)-(6.6) refer to the individual scattering amplitudes, and we now employ them to study the multiple scattering series, where the relevant mass depends on the kinematics under consideration. With the above expressions we may then proceed within the FSA to evaluate the double scattering term for R < 2a as given in Section 3. For the above interactions the expression Z,(k, R) may be explicitly evaluated by suitable integration in the complex plane. Since the involved integrations are somewhat subtle we outline their calculation in Appendix B. Our results are (i)
Square well
Z,(k, R < 2a)L = w
I k>“L
[r7 sin ka cos r?a - k cos ka sin r7a12 x
$(k
R),
(6.7a) where Io(k, R) = eika sin kR[C cos da - ik sin t?a] x [k cos ka sin Ra - ri sin ka cos Ea] +
k2(k2 - I?“) sin r?R[i?(2a 43
+ &- [k2cos k2 +
2~2
k2
R) - sin 2Ha]
r7R - ri2 cos kR][k2
sin2 Ea + e2 cos2 Ra]
[kz cos RR + (g2 - k2) cos2 r?a];
(6.7b)
MULTIPLE
(ii)
Equivalent
607
SCATTERING
separable potential
x
-
[COS
k(R - a) + i $ sin k(R - a)]
A ____ 4
I -,,
cos q(R - a) sh(A2 - q2)1/2 a q2 - k2 [
, (A2 -qq2)l,2 sin q(R - a) ch(h2 - @)1/2 a11
(6.8)
where h = (2fiV0)1/2 andf(k) is the Jost function given by (6.2b). The integral is performed numerically. We also consider the Yamaguchi potential (6.9a)
(k’ 1 V ) k) = X/(k’2 + f12)(k2 + j12).
The on-shell single scattering t-matrix is given by
I kh = (k2 : 83 D -
(x/2M~2~~2
p/N + ifl)2)1’
(6.9b)
The term I,(k, R) may readily be evaluated, and the scattering amplitude in Section 3 up to double scattering terms is then given by
defined
(p/n2>j,(QR)Zo(k R),
(6.104
Tc2Yk’, k; RI = jo(q(W))
I t(k2/&4
I 4,
+
where Il,(k, R) = --(21~)~ 5
/eikR - cR* [ 1 + +
(P + k2)] 1
I k)2, . (6.1Ob)
In this case we may also give the exact result as presented in Section 4. The exact scattering amplitude is given by ,(‘)(k’
7(kly k’ R, = 1 -{(k
k* R)
) t(k2/2p) ) k), ((27~)~/R) pfeiiR - (1 + (k2 + /@R/2/3)) e-sR]}2 * (6.11)
608
KUJAWSKI
AND LAMBERT
The contributions of the multiple scattering terms as well as the off-energy-shell contributions are well exhibited. These are more fully discussed in the next section.
7. NUMERICAL
STUDIES
We now investigate the various approximations and concepts discussed in the previous sections for a variety of two-body interactions and so-called deuteron wave functions. ’ Three types of interactions are considered for the calculation of the various relevant S-wave f-matrix elements: (i)
Square well interaction, V(r) = V,0(r - a),
(ii) Equivalent separable potential. described in Section 6. (iii)
Yamaguchi
This
(7.la) interaction
is constructed
as
potential, (k’ 1 V 1k) = -A/(/32 + k’2)(j32 + k2).
(7.lb)
The parameters of these potentials are, respectively, V, = -14.5 MeV, A = 0.03245 fm-2,
a = 2.55 fm, j3 = l.l77fm-l,
and they were determined so as to reproduce the 5, nucleon-nucleon length and effective range, a,, = -23.74fm,
scattering
r,, = 2.67 fm.
For the densities we used three of the wave functions given by Franc0 and Glauber [24]. We write them as p2(r> = c2e-6*r(sh y2rh2rY,
(7.2a)
pa(r) = k c,ie-“di’(sh y4r/y4r)2,
(7.2b)
i-1
(7.2~) These densities multiplied
by r2 are plotted in Fig. 1.
MULTIPLE
DEUTERON -
FIG. I.
609
SCATTERING
DENSITIES
1=2
Deuteron densities specified by Eqs. (7.1) multiplied
by P.
In the presentation of the results emphasis will be placed on the following points: (i) The kinematics. The various kinematics are discussed in Section 2. All results are considered for both the FSA and OEI kinematics. (ii) Convergence of the multiple scattering series. As discussed in Section 4, for separable potentials the exact result can be explicitly obtained, in which case we compare it with the result keeping up to double scattering terms. (iii) Sensitivity of the result to various interactions. The scattering series up to double scattering terms will be compared for the square well and equivalent separable potential. (iv) Sensitivity of the results to various densities. (v) Sensitivity of the results to the various approximations discussed in Section 5. (vi) Energy and angular dependence of the results. Since we no longer have any implicit small angle approximation, the differential cross section is calculated for all angles. The results are considered for 100 and 500 MeV nucleons. It should be pointed out that the employed interactions do not provide a realistic description of the scattering of 500 MeV nucleons, but they, nevertheless, may be of relevance in studying the energy dependence of the various approximations. As discussed in Section 2 the OEI and FSA kinematics are only identical in the Glauber eikonal approximation. At intermediate energies the corresponding scattering amplitudes may differ significantly. Furthermore, as discussed in Section 6 the phase equivalent potentials are determined for the N-N system, and, consequently, they are no longer phase equivalent when using the reduced mass of
610
KUJAWSKI
AND
LAMBERT
TABLE I Total cross sections for N-D scattering using the square well interaction at 100 and 500 MeV for various densities and approximations discussed in the text.
dmb>
Single and double scattering Exact
E(MW
P PZ4 P4 PS Pa
500
P4 P6
FSA
NOP OEI
14.2 13.8
FSA
30.5 29.7 32.4
15.1
0.857 0.846 0.872
OEI
14.6 13.9 15.9
1.31 1.29 1.35
34.1 31.4 39.1
0.858 0.846 0.875
TABLE
On-shell
1.32 1.29 1.36
FSA 14.2 13.9 15.1
OEI 32.3 31.2 35.7
0.857 0.846 0.872
1.32 1.29 1.36
II
Total cross sections for the phase equivalent separable interaction using the square well interaction at 100 and 500 MeV for various densities and approximations discussed in the text,
O T N-N
Single and double scattering Exact EWeV)
100
On-shell
P
FSA
OEI
FSA
OEI
FSA
OEI
;:
14.0 13.7 14.5
30.3 29.3 32.0
15.0 14.2 16.3
34.1 31.4 39.1
14.5 14.2 15.5
32.3 31.2 35.7
P6
500
NOP
Pa Pa PS
0.859 0.851 0.870
1.31 1.29 1.34
0.864 0.851 0.880
1.32 1.29 1.36
0.862 0.852 0.877
1.32 1.29 1.36
the N-D system. These points are well illustrated in Tables I and II. Although the NOP (nonoverlapping potentials) and on-shell approximations depend solely on the on-shell individual scattering amplitudes the corresponding total cross sections are identical only within the framework of the OEI kinematics. This effect, however, is not as crucial as the explicit dependence on the kinematics. The scattering appears
MULTIPLE
611
SCATTERING
to be quite sensitive to the kinematics especially at 100 MeV, and as expected the difference decreases with increasing energy. At intermediate energies the kinematics then has important consequences on the multiple scattering. The calculation of the scattering from two separable potentials was discussed in Section 4. The convergence or importance of the higher order multiple scattering terms is illustrated for the Yamaguchi potential and density no. 2 at 100 MeV in Fig. 2. It is interesting to note that the multiple scattering series based on the FSA is well approximated by the single and double scattering terms. In contrast using the OEI kinematics the convergence seems quite poor at large angles, and for angles larger than 120” the result provided by the first two multiple scattering terms differ significantly from the exact one. Similar results apply using the two other densities. At 500 MeV the exact result including all multiple scattering terms hardly differs from the so-called exact double scattering result at all scattering angles. Although
05
0
1.5
10
20
25
‘CM
[fm -‘I
13
, -
POTENTIAL
VAMAGUCHI
E i~B
MeV
= 100
-
ALL -
SCATTERING
SINGLE AND SCATTERING
TERMS
DOUBLE ONLY
I
I
I
30
60
90
I 120
I 150 @c M [deg
180 1
FIG. 2. N-D scattering at 100 MeV using the Yamaguchi interaction and density no. 2. The curves denoted by OEI and FSA are calculated using the corresponding kinematics as explained in the text. This figure illustrates the convergence of the multiple scattering series.
612
KUJAWSKI
AND
LAIdBERT
as the energy increases the agreement between the exact and various approximations to the double scattering term improves, the importance of the off-shell effects remains significant especially at the larger angles. This is illustrated in Figs. 3 and 4 where we compare the exact result up to double scattering terms with those calculated using various approximations leading to on-shell scattering amplitudes at 100 and 500 MeV. In Fig. 5 the exact results for the Yamaguchi interaction at 100 MeV are shown for the three densities given above. As expected the results from densities no. 2 and no. 4 are in reasonable agreement and differ significantly from density no. 6. This is readily understood by examining Fig. 1. By comparing Figs. 3 and 5 it becomes apparent that the differences due to two equally realistic densities are likely to be as small or large as those due to an approximate treatment of the off-shell t-matrix elements. At 500 MeV the effects of the various on-shell approximations have somewhat decreased in importance in comparison to the various densities. This 0 T t
0.5
1.0
10
1.5
2.0
I
2.5
(1Jfm-‘l
EI KINEMATICS
2
VP bC1:
10
-
EXACT
0.1 -
-
NOP APPROX.
-.-
ON-SHELL
APPRO
0.01 DEUTERON
DENSITY : p2
POTENTIAL:
YAMAGUCHI
E L*B = 100 MeV SINGLE
I
0
I 30
AND
DOUBLE I 60
SCATTERING I 90
ONLY I 120
I 150
%.M.rdeg I
FIG. 3. N-D scattering at 100 MeV using the Yamaguchi interaction and density no. 2 calculated up to double scattering terms for various approximations discussed in the text.
Ok-+-
’
6. qC.MIfme’ I
410 ’
01 ” ”
DEUTERON
DENSITY P,
POTENTIAL
YAMAGUCHI
1
ALL SCATTERING TERMS -
EXACT
-
-
NOP APPROX
-.-
0
30
ON - SHELL APPROX.
60
90
120
150
180
CM [deg 1
FIG. 4. Same as Fig. 3 but including all terms at 500 MeV. 0 10
05 I
WTENilAL.
10 I
15 I
20 I
25 I
s,,,Cfm-‘3
YAMAGLICHI
E L*B = 100 Mev ALL SCATTERING 001
TERMS
,
--P, -i-p
p2
6
FIG. 5. N-D scattering at 100 MeV using the Yamaguchi interaction densities. The scattering is calculated to all orders.
for various deuteron
614
KUJAWSKI
AND
LAMBERT
01 POTENTIAL
ALL
0.0001
VAMAGUCHI
SCATTERING
TERMS
-
120
150 h
FIG.
180
[deg 1
6. Same as Fig. 5 but at 500 MeV.
may readily be seen by comparing Figs. 4 and 6. However, as already mentioned our interactions are not realistic at 500 MeV. At the higher energy it may then be possible to supplement elastic electron scattering studies to discriminate among various two-body densities. Because of the off-shell effects and the sensitivity to the kinematics such a determination may still be rather unreliable. Using the phase equivalent interactions described in Section 6 we now systematically study the multiple scattering series up to double scattering terms within the framework of the various approaches given in Section 5: (i) Exact treatment of the double scattering term taking into account the off-shell dependence of the t-matrix elements. (ii) Nonoverlapping potential assumption. The scattering is calculated assuming the potentials do not overlap even when averaged over the target density. The relevant expressions are given by Eqs. (5.1). (iii) On-energy shell approximation. The principal part of the propagator in the multiple scattering terms is neglected. The resulting scattering is then again given in terms of fully on-shell two-body scattering amplitudes and the relevant expressions are given by Eqs. (5.5).
MULTIPLE
615
SCATTERING
The various above approximations for the square well interaction at 100 MeV using the density no. 2 are shown in Fig. 7. The corresponding results for the equivalent separable potential are shown in Fig. 8. It is interesting to observe that the nonoverlapping and on-energy shell approximations lead to very similar results, but they differ significantly from the exact results. The close similarity between the nonoverlapping and on-energy shell approximations is in keeping with the discussion presented in Section 5. To further emphasize the importance of the off-energy-shell dependence of the multiple scattering series, in Fig. 9 we have plotted the exact single plus double scattering terms for the square well and equivalent separable interactions already presented in Figs. 7 and 8. The corresponding results at 500 MeV are shown in Figs. 10 and 11. Although the validity of the various approximate treatments or models improves with increasing energy, a proper treatment of the multiple scattering terms seems essential for an analysis to be of significance. Finally we have also examined the dependence of the scattering on the densities. The results are similar to that for the Yamaguchi interaction. The results for the square well at 100 MeV are shown in Fig. 12, and those for the equivalent separable n,oo
05
10
15
20
25
5 2 4%
:
10
EXACT .-
NOP APPROX ON-SHELL
120
AWROX
150 % ,., [deg 1
160
FIG. 7. N-D scattering at 100 MeV using the square well interaction and density no. 2 for various approximations discussed in the text. The kinematics is as discussed in the text. The calculation is up to double scattering terms.
10
05
2.0 I
15
i?l
-
qcM km-‘1
2.5
I
EXACT
0.1 -
001
-
NOP APPROX
-- OEUTERON
DENSITY : Pz
POTENTIAL 3 EQUIVALENT
SEPARABLE
ELAB = 100 MeV SINGLE
1 0
AN0
DOUBLE
I
I
30
60
SCATTERING
I 90
ONV
I
I
120
150
I 160
@c,a [d-w 1
FIG. 8. Same as Fig. 7 but for the equivalent separable interaction. 10
0.5 I
3 10'r;,
15 I
I OEI
20 I
25 I
4,.
[tm-‘I
KINEMATICS
f ? al,zf 10 L
\
\
FSA
\
KINEMATICS\
\
01
0 01I-
OEUTERON
DENSITY : p2
E iAB E loo
MeV
SJNGLE AN0
DCHJBLE SCATTERING “NL”
*
l\
-
O
SOUARE -
WELL
EQUIVALENT I 30
I 60
I
POTENTIAL
SEPARABLE I 90
POTENTIAL _I 120
I 150 oc M. [de9
1E 1
FIG. 9. Comparison of N-D scattering at 100 MeV for the square well and separable interaction using density no. 2.
-
EXACT
-
-
NOP APPROX ON-SHELL
I 0
I/ 30
1 90
1 60
APPROX
/ 120
I 150
180
8c M [deg
FIG.
;
10. Same as Fig. 7 but at 500 MeV. “L
I
I
1,O 2,O -
0
L,O
SQUARE
-
01
,
-
, WELL
EQUIVALENT
6,O
‘CM:‘“;’
POTENTIAL
I
SEPARABLE POTENTIAL
I
I
I
ALL..
30
60
90
120
j 150 ec M [deg
FIG.
180
1
11. Same as Fig. 8 but at 500 MeV.
oo
05 I
10 ,
15 ,
20 I
25 I
%k41fme’l
‘\ *\ ‘1.\ ‘\ ‘\ + OEI
FSA
KINEMATICS
I
KINEMATICS
-p2
1.5
ho
,
,
so
120
,
i
150 % M. h’eg 1
160
FIG. 12. N-D scattering at 100 MeV using the square well interaction The calculation is up to double scattering terms.
,oo
05
10
15
2.0 I
25 I
1
< POTENTIAL: EQUIVALENT
0,
30
60
so
SEPARABLE
120
160
150 ec. ..L-
FIG.
for various densities.
1
13. Same as Fig. 12 but for the equivalent separable interaction.
MULTIPLE
SCATTERING
619
interaction in Fig. 13. One readily sees that it is rather difficult at intermediate energies to discriminate or distinguish among various approximations, phase equivalent interactions or reasonable densities. As the energy increases the validity of the various high-energy approximations improves, the importance of the off-shell effects decreases, and the sensitivity to the density increases. Nevertheless, the various approximations are such as to question their predictive powers. 8. CONCLUDING
REMARKS
In this work we have studied elastic scattering from a two-body bound system using a variety of simplified two-body interactions. The goal was to obtain insight into the importance of the contributions arising from properly treating the overlapping potentials and various kinematics. Our calculations are of relevance to high- and intermediate-energy nuclear scattering. We now summarize the essential ideas in this work: (i) The kinematics of the FSA and OEI are quite different especially at low energies, but become identical in the high-energy limit. A proper treatment of the kinematics is also expected to be crucial for x-nucleus scattering in the neighborhood of the (3, 3) resonance [25]. (ii) For nonoverlapping potentials the composite scattering is completely determined in terms of on-shell information. This should be kept in mind, and it suggests that in the optical potential formulation [5] the second order term is likely to cancel some of the off-shell dependence of the first order optical potential [26]. (iii) The multiple scattering is such that the off-shell effects of the two-body interaction are strongly coupled with the two-body correlations. This observation plus the sensitivity on the kinematics suggest that elastic scattering is not well suited to obtain information about two-body correlations or the two-body interaction. Such a conclusion is in keeping with the one reached by Feshbach [27] within the framework of the eikonal approximation, Kujawski [12] using onedimensional models, and Adelberg and Saperstein [28] using the optical potential formulation [5]. (iv) At intermediate energies the predictions provided by the nonoverlapping potential and on-energy-shell approximations deviate sufficiently from the exact FSA results to render ambiguous the extraction of the distribution function or two-body interaction in terms of either of these two models. These conclusions about using elastic scattering to obtain information about two-body correlations or the two-body interactions are rather discouraging, but they should be kept in mind when analyzing experiments. On the optimistic side
620
KUJAWSKI
AND LAMBERT
let us state that there may exist an energy or angular region where the off-shell effects are weak, and one can, therefore, be able to determine two-body correlations. More theoretical and experimental work is needed before such studies are fruitful. Finally, let us remember that there are various other approaches [29] better suited than high-energy elastic scattering for investigating nuclear structure and two-body interactions.
APPENDIX
A:
SCATTERING
BY Two SEPARABLE NONLOCAL
INTERACTIONS
For the scattering from two fixed potentials vi (i = 1,2) the Faddeev equations (2.7) reduce to TO
= T,(E) + T,(E), 1,2,
j#i= T,(E) = W) + 4(E) Go@) T,(E), ti(E) = vi + ui(E - (k2/2,u) + 8)-l &(E),
(A.la) (ASb) (A.Ic)
where ka is now an operator. We now consider two real identical separable potentials (k’ I 0 I k) = Mk’)
64.2)
g(k).
Assuming that they are located at &r we obtain
(A.3a)
where q = k’ - k, and
W I t(E) I k) = hd(E) dk’) g(k),
Substituting
(A.3b)
(A.3) into (A. 1b) and using the explicit expression for Tj in terms of
Ti , (A. 1b) reduces to
[
e*“l”g(k) +
I
s dp
Go(E) IP) g"(p)
dp’ el d”‘rd”)
e--
] 1, (A.4)
MULTIPLE
621
SCA'ITERING
where
OF PHYSICS
81,
591-624 (1973)
Off-Energy-Shell
Effects in Multiple
Scattering*
E. KUIAWSKI+ Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 AND
E. LAMBERT Institut de Physique de l’ilniversitt!,
Neuchritel, Switzerland
Received March 2, 1973
The elastic scattering from a two-body bound system is investigated in order to determine what information about the nuclear structure and the projectile-nucleon interaction may be obtained at intermediate energies. Emphasis is placed on the kinematics and the importance of properly treating the overlapping potential region. The kinematics of the “fixed scatterer” and “impulse” approximations result in differences which are still significant at intermediate energies. As far as scattering information is concerned, the overlapping potential region is the interesting one since it requires explicit knowledge of the interaction. Our results suggest that for nucleon scattering at intermediate energies the corrections due to properly taking into account of&energy-shell effects are likely to be comparable with the contributions due to the short-range twobody correlations. The relevance of the above results to various models currently employed in analyzing high- and intermediate-energy nuclear scattering is also discussed.
1. INTRODUCTION The full multiple scattering equations are very complicated, and with the exception of a few pedagogical examples and the three-body problem it is impractical to solve them exactly. Two general approaches have then been pursued: (i) attempts to approximately solve the equations using simplified propagators [l-3]; and (ii) attempts to evaluate various multiple scattering terms in the Watson-Faddeev multiple scattering series [4-71. In this work we follow the second * This work is supported in part through funds provided by the Atomic Energy Commission under Contract (ll-l)-3069 and the Swiss National Science Foundation. + Present address: Department of Physics, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213. Copyright All rights
0 1973 by Academic Press, Inc. of reproduction is any form reserved.
591
592
KUJAWSKI
AND
LAMBERT
approach because we think that it lends itself more readily to study which information may be extracted from high- and intermediate-energy elastic scattering. The essential simplification when considering elastic intermediate- and highenergy nuclear scattering results from the applicability of the so-called “fixed scatterer” approximation (FSA) [3, 81, which corresponds to performing closure on the intermediate target eigenstates assuming that they are completely degenerate. The projectile-target scattering amplitude is then obtained by calculating the scattering from instantaneously fixed scatterers using the reduced mass of the total system and averaging over the internal eigenstates. The problem is thereby reduced to that of potential scattering, where, however, we no longer have spherical symmetry. The results obtained from studying scattering off a sum of potentials then shed light on which information can be obtained from elastic high- and intermediate-energy nuclear scattering. For nonoverlapping potentials the complete scattering is explicitly determined in terms of the individual on-energy-shell scattering amplitudes [9-121. In contrast the scattering from overlapping potentials depends on the off-shell behavior of the specific interaction, and it has been suggested that by supplementing the two-body data with many-body scattering data it may be possible to distinguish among phase equivalent potentials [IO]. As a result of averaging the scattering amplitude over the ground state density the potentials due to the individual target nucleons to which the projectile is subjected generally overlap, the amount of overlapping being sensitive to the short-range behavior of the internal target wave function. Elastic scattering then has the unfortunate feature of mixing up information about the two-body interaction with two-body correlations. In this work we study nucleon-deuteron elastic scattering within the FSA for a variety of two-body interactions and wave functions in order to obtain insight into the importance of the contributions arising from properly treating the overlapping potentials and to learn what information may be extracted from the experimental data. We now outline the succeeding sections. In Section 2 the FSA and impulse approximation are discussed and compared. In Section 3 we develop the double scattering term in the multiple scattering series stressing the dependence on the off-energy-shell two-body t-matrices. In Section 4 we present the exact result for scattering by two S-wave separable potentials. In Section 5 we present two approximations whereby the scattering is expressible simply in terms of on-shell information. In one case we assume the potentials do not overlap while in the other we neglect the principal part of the propagator. In Section 6 we introduce the various nucleon-nucleon potentials which we consider in our calculations. We include both local and nonlocal phase equivalent potentials and present their off-energy-shell matrix elements. In Section 7 we present the results of our numerical studies. The conclusions are given in Section 8.
MULTIPLE
593
SCATTERING
2. THE KINEMATICS One of the basic problems in any multiple scattering formulation is to relate the matrix elements of the relevant many-body modified operators to the free two-body scattering matrix elements whose on-shell values are determined in elastic scattering experiments. For high-energy elastic scattering two formulations seem relevant for use in a multiple scattering expansion: (i) the “fixed scatter approximation” (FSA) kinematics [3, 81; and (ii) the “on-energy-shell impulse approximation” (OEI) kinematics [13]. The essential difference is that in the FSA the struck nucleon is assumed to recoil with the target as a whole while in the OEI the struck nucleon essentially recoils freely. The significance of the above two approaches is best illustrated by specifically considering the three-body problem with arbitrary masses. Let the indices 1, 2, and 3 refer specifically to the projectile and two target “particles” with masses m, , m2 , m3 , lab momenta kl , k, , k, and lab coordinates R, , R, , R, . In the usual manner we introduce the “canonical” parameters [14]. Let qi be the relative momentum of particles j and k, pi the relative momentum of particle i with respect to the center of mass of j and k, P the total momentum, ri , xi , R the corresponding conjugate coordinates, and pLi , pjk , M the associated masses: p. = hi 2 qi =
P=
+ mk> ki - m4kj + kk) M
mjk, - m,ki +-km,
3
(2.la)
’
iki, i-l m&
ri = Ri -
+
mkRk
mj+mk
’
xi = Rk - Ri,
(2.ld) (2.le)
R
R = 1/M c maRi,
(2.lf)
i-1 Mm* pi
=
pjk
=
m3
’
(2.lh)
mjmk mj
M=
+
M
irni. $4
f
mk
’
594
KUJAWSKI
AND
LAMBERT
Assuming that the particles interact via two-body forces only, the resulting Schriidinger equation in the total center-of-mass system reads PI2 (-+E a4
2
v12
+
+
~12
+
~13
+
02.9
W,,
1
rl)
=
EW,,
r3,
(2.2a)
Xl>,
(2.2b)
where =
VlZ('l
(dm2)
Xl),
01s
=
~13(b
-
(P22/%>
and E denotes the appropriate nonrelativistic energy which is the sum of the kinetic energy of the projectile and the binding energy of the target. We now proceed to discuss various approximations of interest for high- and intermediate-energy nuclear scattering. The FSA In the FSA the internal target Hamiltonian 412/&‘2,
where Es is the target binding then reads
+
V23h)
is neglected, i.e., we set =
-EB
7
(2.3)
energy. The resulting Schrodinger equation (2.2) (2.4)
where E is the kinetic energy in the total center-of-mass system (E = E + EB). The solution to (2.4) is clearly of the form (2.5) where &,(x1) denotes the initial internal wave function of the target. #(xl ; rl) satisfies the Schrodinger equation specified by (2.4) where x1 is now a parameter. As a result of neglecting the internal target Hamiltonian the problem has been reduced to that of scattering from two potentials located at -(y.Jm,) x1 and This is a nontrivial problem since the resulting total or global +(P23h) Xl * potential is no longer spherically symmetric. The exact solution then involves solving a system of coupled equations coupled through angular momenta. In practice this system of coupled equations must be limited by truncation; unfortunately, for high-energy scattering where the wave length is short compared to the size of the scatterer this involves a rather sizable system of coupled equations. The total or composite elastic scattering amplitude is then obtained by averaging
MULTIPLE
the amplitude calculated for various positions internal wave function &(x), j&k’,
k) = --2&4rr
595
SCATTERING
of the target nucleons over the
1 dx 1 &(x)12 1 dr e-ik’*rV(x, r) @(x; r),
(2.6)
where #+ is the solution to (2.4), and Vx, r) f u12+ o13. It is important to realize that although the relative distance between the two target constituents remains fixed during the scattering, their center-of-mass motion is properly taken into account. On-Energy-Shell Impulse Kinematics
In the Faddeev method the three-body scattering amplitude three amplitudes Ti which satisfy the equations [14]
T is broken up into
T,(E) = h(E) + f,(E) G,(E) 1 G(E), i#i
(2.7)
where G, is the free propagator for the three particles and ki now an operator. The variables are defined by Eqs. (2.1). Zi is a many-body operator because the propagator appearing in it contains the entire kinetic energy; however, it can readily be related to the two-body t operator in its own center-of-mass system. For the sake of definiteness let us consider 2, which involves only u12. Making use of the fact that u12 depends only on the coordinate conjugate to q3 one obtains (P319alP’ I W)
I PSQap) = %P’ - P) %P,’ - P3) x
(2.8a)
where all the quantities are as previously defined. t is the two-body t-matrix in its own center-of-mass. In terms of the appropriate momenta p1 and q1 defiend by (2. I) the above t-matrix element reads (2.8b)
596
KUJAWSKI
AND
LAMBERT
where a’ = e - (PI’ - pa m2 CL?.3’
WC------ 1 21112
(%P,
+
91~
-
EB
9
To obtain the elastic scattering off the deuteron in the impulse approximation the above matrix element then needs to be averaged over the deuteron wave function. As is well known this involves calculating the two-body t operator off-the-energy shell. At high energies one often neglects the internal momentum and binding energy [13], namely
*PI’ + (2.9)
where 3 = &
(%Pl)“.
We refer to the above approximation as the OEI. The resulting t-matrix element is exactly the one measured in the relevant two-body scattering experiment. Comparison of the FSA and OEI As already discussed the FSA corresponds to neglecting the internal target Hamiltonian or performing closure over the intermediate target eigenstates neglecting their excitation energy. The center-of-mass motion is properly taken into account and the corrections due to neglecting the internal Hamiltonian may systematically be investigated. It, however, has the unfortunate feature that the relevant matrix elements differ from those determined in the corresponding projectile-nucleon scattering due to the different effective mass relevant in the two scatterings. We then see that in the impulse approximation or Faddeev-Watson multiple scattering series the target nucleon in lowest order is effectively treated as recoiling freely. Unless the internal motion is neglected as in the OEI, off-shell effects need to be taken into account. Furthermore, as for elastic scattering the target nucleus recoils as a whole the corrections to the propagator or a proper treatment of the recoil may then need to be explicitly included. For the sake of definiteness and clarity we now write down the single scattering terms relevant for N-D scattering within the FSA and OEI. These matrix elements may be denoted by (cm’ I f (+$)
I cPl>,
(2.10)
MULTIPLE
SCATTERING
597
where c = $ in the OEI and c = 1 in the FSA. At this point it is worth emphasizing that the two kinematics coincide at high energies where the Glauber eikonal approximation [l] is valid. This may readily be seen by observing that in the Glauber eikonal approximation the relevant parameter is the velocity and cp,/cpl = pI/pI . However, at intermediate energies the two t-matrices may still differ significantly, and in our work we will compare these results. The usefulness and preferability of a multiple scattering formulation based on the FSA has also been conjectured by Kowalski and Pieper [15] and appears to be supported by their calculations. The multiple scattering series may be expanded in terms of rather general t-matrices [5]; however, since in practice the series will be truncated the usefulness of the approach will depend on the specific t-matrix employed. We wish to point out that the FSA and OEI kinematics would also coincide if the dynamics can be parametrized as f(q) = 2
(1 - ia) e-8Q2/2,
(2.11)
where CT,01,and /3 are energy independent. Such parametrizations are the usual ones employed at high energies [ 1, 71 and are characteristic of the high-energy limit. 3. THE DOUBLE SCATTERING
AMPLITUDE
In this section we develop the Watson-Faddeev multiple scattering series up to double scattering terms with special attention to its off-energy-shell dependence. For the sake of definiteness we consider N-D scattering. As shown in Section 2, in the FSA the problem is then reduced to that of scattering from two potentials. Assuming the potentials to be identical and located at &R/2 the resulting Schriidinger equation then reads (3.1) + u(r - W2N + 4~ + @/WlrCI = 4 where ~1 = Qm, E = k2/2p with k being the momentum in the three-body centerof-mass system, and m is the nucleon mass. The multiple scattering series for the resulting scattering operator is given by L.-(V,2/+4
T = Tl + 7-2-I- T,G,T, f T,G,T, i- ....
(3.2a)
where (R’,
r’ 1 G 1 R r) = 0 >
2p @‘-” 47r ( t - r’ ( 6(R - R’),
(3.2~) (3.2d)
598
KUJAWSKI
AND
LAMBERT
t
q = k’ - k, and is the collision operator for scattering from the potential D located at the origin. The composite scattering amplitude for elastic scattering is then obtained by averaging the resulting scattering amplitude over the internal deuteron wave function Q0
p(k’,@oI TIk, @o> Pj- d3Rp(Wk’I TI k),
foo(k’, k) = -CW2
= -CW2
(3.3)
where T is given by (3.2) and p denotes the deuteron density. We now consider the scattering only up to double scattering terms. Keeping our numerical calculations in mind we consider the following idealized situation: (i) Spherically symmetric target density. We assume that the two target particles are in a relative S-state. Note that the deuteron has a D-state component. (ii) Selective scattering in S-states only. Since the potentials are assumed to scatterer S-waves only there will be no admixture of angular momenta due to multiple scatterings. This greatly simplifies the algebra. Although the above assumptions need not be introduced at this point, they greatly simplify the resulting expressions and make the essential physics more transparent. The resulting scattering amplitude is then given by
ho@‘, k> =
-‘NW3
jm R2 dR p(R) r,,(k’, k; R),
(3.4a)
0
where
To@‘,k; R) =
joW2)
I to(e) I k) +
(2y/2n2)
m p dp --m
(P
$2743 Zo(k
R>
=
-
2R
s
eipR
j,(QR) Z,(k, R),
1 to(E)
1 k>2
k2 - p2 + i8 ’
(3.4b) (3.4c)
q = k’ - k, Q = (k + k’)/2 and (p 1 t,(e) 1k) is the right-half off-shell S-wave t-matrix with l = k2/2p. Although the above expressions have been determined consistently within the FSA, their generalization to the OEI is obvious and simply involves using the corresponding two-body t-matrices as specified in Eq. (2.10). The corrections, however, will be different. In the remainder of this section we then address ourselves to the evaluation of IO . This is the term of interest since it contains double scattering and possible off-shell information. We consider u to have a finite range a. Before proceeding it is useful
MULTIPLE
599
SCATTERING
to list some relevant properties of the half-off-shell t-matrix elements ( p 1 tl j k)HOS in the complex p-plane [lo]. For an interaction of finite range a, in the complex p-plane (p I tl 1k)HOS is an entire function with the following properties: (i) (ii)
for R > a, liml,l,, e*i*R(p / tl I k)HOS = 0 for Imp >( 0, it has a zero of order 1 at the origin.
Using the above properties of the t-matrix elements the integral 1, for R > 2a can readily be evaluated by contour integration in the complex p-plane. Proceeding as indicated one finds I,,(k, R > 2a) = - 9
(k j t,, ($-)
tL
1 k)2 eikR.
We then see that for nonoverlapping potentials the double scattering term is expressible entirely in terms of the on-shell two-body scattering amplitudes. It is important to realize that the above statement does not mean that there is no off-shell scattering but rather that for nonoverlapping potentials the scattering is independent of the off-shell amplitudes. This is not to be confused with neglecting the principal part of the propagator or assuming that the scatterers are far apart so as to enable one to use the asymptotic form of the scattered waves [16]. Although we have only considered up to double scattering terms, it can in general be shown that for nonoverlapping potentials the full scattering amplitude depends only on the individual on-shell scattering amplitudes [ 111. Assuming finite range interactions with radius a the composite scattering amplitude for our “deuteron problem” may then be broken up into two contributions (3.6a)
ho@‘, k> = .Mk’, W + f&k’, W, where f2 corresponds to the scattering from the nonoverlapping potential which as previously discussed depends only on on-shell information, and f,(k’> k) = --4~(24~
1’” R2 dR p(R) T&‘, 0
k; R),
region
(3.6b)
which is the contribution arising from the overlap of the scatterers. The evaluation of (3.6b) requires a knowledge of the off-shell t-matrix elements, and we will calculate it for a variety of interactions. The information of interest (off-shell effects and two-body correlations) is then contained in (3.6b), and we investigate its importance for elastic intermediate-energy nuclear scattering.
600
KUJAWSKI
4. SCATTERING
AND LAMBERT
FROM A SUM OF SEPARABLE POTENTIALS
In this section we examine the scattering from two mobile S-wave separable potentials. Since the scattering can be exactly solved this is a very interesting case for investigating various approximations. Before proceeding a few words are in order concerning the connection between the FSA and the closure approximation assuming degeneracy for the target eigenstates and Galilean invariant separable interactions [17]. In this case, closure [3] no longer leads to a result identifiable with the FSA because the two-body interaction is nonlocal in both the target-particles and projectile coordinates. However, it is consistent to make the FSA provided the target-particles are much heavier and slower than the projectile [18]. We also wish to stress that one may, nevertheless, discuss the Watson-Faddeev multiple scattering series presented in Section 3 using separable t-matrices. For scattering by a sum of separable nonlocal potentials Foldy and Walecka were able to write down the exact result in closed form. A simple derivation for the scattering by two S-wave separable nonlocal potentials using the Faddeev approach is presented in Appendix A. The scattering amplitude for scattering by two S-wave separable potentials
(4.1)
located at -&R/2 is given by (k’ 1 T(E, R) ) k) = ‘:““$$’
[co, (q * +)
+ h(R) cos(Q . R)],
(4.2)
where q = k’ - k, Q = (k + k’)/2, E = k2/2p,
W I t(c) I 6 = Mk’) g(k)/44 x h(R) = d(~) I d”l! eiveRE _ d(e) = 1 - 47rh IOrnp2 dp
(4.3a)
1
(py2p)
+ is g”(P)*
1 E - (p2/q4
+ is g”(P)*
(4.3b) (4.3c)
For nonlocal interactions of finite range a and R > 2u, h(R) can be explicitly evaluated and expressed in terms of on-shell information illustrating that for nonoverlapping potentials there is no off-shell information h(R > 2a) = -(2~)~ p(k’ 1t(k2/2p) ) k) eikR/R.
(4.4)
MULTIPLE
6OL
SCATTERING
Equation (4.2) is exact and clearly exhibits the multiple scattering effects. The numerator is simply the result up to double scattering terms. The denominator represents the sum of all intermediate reflections off the scatterers and gives rise to what is commonly referred to as the “effective or local field corrections” [8]. For separable potentials the exact result will be compared with the scattering up to double scattering terms thereby allowing us to study the convergence of the multiple scattering series.
5. APPROXIMATIONS
RELEVANT
AT HIGH
ENERGIES
The essential simplification when considering elastic intermediate- and highenergy nuclear scattering results from the applicability of the FSA which corresponds to performing closure on the intermediate target eigenstates. As shown in Section 2 the FSA reduces the problem to computing the scattering amplitude from N fixed potentials, where N is the number of target nucleons, and then averaging it over the internal target wave function. In Section 3 the scattering amplitude for the scattering by two local potentials was given up to double scattering terms while in Section 4 the exact result was given for two separable potentials. At high energies the multiple scattering series is supposedly rapidly convergent [6], most of the scattering being taken into account by the first two terms. However, it is interesting whenever feasible to examine its convergence. In addition to the FSA and truncated versions of the Faddeev-Watson multiple scattering series we also examine various approximations which result in the full scattering amplitude being expressed purely in terms of the individual on-shell amplitudes. For nonoverlapping potentials the scattering depends only on the two-body on-shell partial amplitudes. However, in the FSA as a result of averaging the scattering amplitude over the target ground state the potentials due to the individual target nucleons to which the projectile is subjected generally overlap. This greatly complicates the problem since the scattering can no longer be expressed simply in terms of on-shell information and its calculation requires explicit knowledge of the interaction. Two approximations then come to mind whereby the scattering is expressible simply in terms of on-shell information: (i) The nonoverlappingpotential approximation. The scattering amplitude is calculated assuming that the potentials do not overlap. The scattering amplitude up to double scattering terms, presented in Section 3, is approximated by
fiioP(k’, k) = ---4/42~)~ j-m Ra dR p(R) 7NoP@t,k; R), 0
602
KUJAWSKI
AND
LAMBERT
where
7NoP&',k; R>= jo (4+)
P-Y
CLq jot,, (k I to(4I k>2. (5.1 b)
All the quantities are as defined in Section 3. A similar approximation may be applied to the exact result given in Section 4. Assuming that the potentials do not overlap, Eq. (4.3b) simply reduces to hNoP(R) = -(27+
hg2(k) eikR P yqqR 3
(5.2)
and the multiple scattering amplitude involves using (5.2) in (4.2) for all values of R. As previously discussed in Section 4, the resulting scattering amplitude obtained using (5.2) contains the exact propagator and all orders in the multiple scattering terms but the potentials are assumed not to overlap. Another approximation which leads to a (ii) On-energy-shell scattering. scattering amplitude which is also expressible simply in terms of on-shell information is the one resulting from neglecting the principal part of the propagator or the Green’s function. In the on-energy-shell scattering approximation the exact propagator in the multiple scattering series is approximated by 1 k2 - pa + iS
Making
--t -in&k2
the above substitution los(k, R) = - &
- p2) = - -&
(5.3)
in 1, (3.4~) we readily find (24s
The resulting scattering amplitude Section 3 then reduces to
+‘,
[6(k + p) + 6(k - p)].
(k [ t,(e) 1k)2 ( ezkR ; ‘-‘,,).
(5.4)
up to double scattering terms as given in
k R>= j. (4 -&)
~(279~ ( ePkR yRe-‘““)
j,(QR)
(k I to(E) 1k>2,
(5.5a)
and
f&W, k) = --4p(27~)~ j-0 R2 dR p(R) @(k’, k; R).
(5.5b)
MULTlPLE
603
SCATTERING
A similar approximation may be applied to the exact result given in Section IV. In this case h(R) as given by (4.3b) simply reduces to
).
(5.6)
The above result obtained by simply keeping the on-energy-shell part of the propagator and double scattering terms is reminiscent of the result provided by the Glauber multiple scattering model. Such an analogy may however be misleading since the latter result depends on the use of the high-energy eikonal approximation or a linearized propagator, and for local interactions the principal part of the double scattering term is exactly cancelled by all the higher order multiple scattering terms [12, 19,201. In contrast Eq. (5.5) is a phenomenological variant of the Glauber multiple scattering formulation which is often used in data analysis. We close this section by comparing the two approximations presented above. For a fixed position of the scatterers the double scattering term within the above two approximations appear to be quite different $OP(k’,
k; R) z -(2~)~
Tz6(k’, k; R) 2 -(27r)2
p(k
1 t,(e) ) k)2 eikR
eiQR2ii-i*R
p(k 1 t,,(e) 1k>2 ( e’kR ; ‘-“““)(,
), ezQR2i;-IVR
),
(5.7b) where Q = (k + k’)/2. In the forward direction Q = k. It may now readily be seen that as a result of averaging over the target density that at high energies and forward angles the two approximations are very similar. When averaged over the target density only the nonoscillatory terms of the expressions (5.7a) and (5.7b) contribute, and both may be approximated by $0P*8(k’,
k; R) w
-(2~)~
p(k I &,(E) I k>2 (i/2kR).
(5.74
k-rm
As pointed out by Tobocman and Pauli [21] such a cancellation or interference arising from the spatial smearing (or time variation) of the location of the scattering centers partly explains the validity or success of the Glauber multiple scattering model. At larger angles the higher order multiple scattering terms cannot be neglected and the two approximations may differ when applied to the full scattering amplitude. This latter case will be studied numerically using separable potentials.
604
KUJAWSKI
6. OFF-SHELL
DEPENDENCE
AND LAMBERT
OF THE MULTIPLE
SCATTERING
TERMS
In this section we study several phase equivalent potentials and their off-energyshell t-matrix elements. These expressions are then employed to calculate the corresponding multiple scattering terms relevant to our numerical studies. As shown in Sections 3 and 4 the evaluation of the second- and higher-order multiple scattering terms for overlapping potentials requires a knowledge of the off-energy-shell t-matrix. There are two possible ways of obtaining off-energy-shell t matrices: (i) direct parametrization or analytic continuation of the scattering amplitude [22]; (ii) determination of a potential and subsequent calculation of the offenergy-shell t-matrix elements. We follow the second approach because of the greater physical insight associated with potentials. We first need to recall a few results of the inverse scattering problem [14]. Given a phase shift of a given angular momentum for all energies and provided there is no bound state there exists a unique local energy-independent potential associated with it. There are several well known procedures for constructing a local potential given a phase shift for all energies. A separable potential of the form
is also uniquely determined [23]. Although these two potentials are phase equivalent they have different off-shell t-matrix elements. For the sake of simplicity we now restrict ourselves to S-wave scattering for a particle with “effective” mass p, where p is at present left unspecified. We study the S-wave scattering from a square well and the corresponding phase equivalent separable potential. It is important to realize that the phase equivalent potential depends on the mass; this point is further discussed below as well as in Section 7. For N-N scattering the observable phase shifts are specified for jZ = (m/2). Consider a local potential
(6.1)
where 0 is the step function. The corresponding S-wave scattering matrix element is given by (6.2a) &(k> = fW/Y(--k), where f(k) = e-4”a(cos 2~ + i(k/ R) sin ~a), (6.2b) : = (k2 - 2@VO)1/2.
MULTIPLE
As shown by Gourdin and Martin S-wave scattering is given by
SCATTERING
605
[23] the equivalent separable local potential for
where bW12 = G/ww2
kx..f(k) -f(-k)l.
(6.3b)
In coordinate space the separable potential is of the form (r’ 1 V 1r)s = C(I’) t?(r),
(6.3~)
where d denotes the properly normalized Fourier transform of v. Although the above two potentials (6.1) and (6.3) are phase equivalent they have different off-shell t-matrix elements. The N-N t-matrix may directly be employed in the multiple scattering series using the OEI kinematics. This is to be contrasted with the multiple scattering series within the FSA where the relevant mass (p) is the total reduced mass, and for N-D scattering p = sm. In that event the elementary on-shell t-matrix elements are no longer equal because they refer to a mass different from the one (p = m/2) appearing in the phase equivalent potentials. The calculation of off-energy-shell t-matrix elements is widely discussed in the literature, and we only write down the results. For the local square well potential the half-off-shell S-wave t-matrix element is given by r? sin pa cos Za - p sin f?a cos pa 17sin ka cos r?a - k sin Za cos ka
I k)L >
(6.4a)
where
I to (+)
I k)L
=
&$
k sin ka cos r7a - k sin I?a cos ka 3 r? cos t?a - ik sin Ra
(6.4b)
fT2 = k2 - ~/.LV~.
(6-k)
and For the equivalent separable potential the half-off-shell t-matrix element is given by
k 112 sin pa cos &a - (p/G) sin &a cospa U2 I k)s = (>) ( sin ka cos ria - (k/R) sin r?a cos ka ) x
I kh
3
(6.5a)
606
KUJAWSKI
AND
LAMBERT
where I? sin ka cos Ea - k sin .?a cos ka - 1) + : cos t?a - ik sin Ra ’
r?e-ika(plp
(6Sb)
and iP=k2-2~Vo,
(6.52)
52 =p”-2pv,.
The phase equivalence of the above two potentials for the mass fi can be readily seen from Eqs. (6.4b) and (6Sb) which lead to
(k I to (+)
I 6, = (k I to
($-)
I k)s
(6.6)
.
Equations (6.1)-(6.6) refer to the individual scattering amplitudes, and we now employ them to study the multiple scattering series, where the relevant mass depends on the kinematics under consideration. With the above expressions we may then proceed within the FSA to evaluate the double scattering term for R < 2a as given in Section 3. For the above interactions the expression Z,(k, R) may be explicitly evaluated by suitable integration in the complex plane. Since the involved integrations are somewhat subtle we outline their calculation in Appendix B. Our results are (i)
Square well
Z,(k, R < 2a)L = w
I k>“L
[r7 sin ka cos r?a - k cos ka sin r7a12 x
$(k
R),
(6.7a) where Io(k, R) = eika sin kR[C cos da - ik sin t?a] x [k cos ka sin Ra - ri sin ka cos Ea] +
k2(k2 - I?“) sin r?R[i?(2a 43
+ &- [k2cos k2 +
2~2
k2
R) - sin 2Ha]
r7R - ri2 cos kR][k2
sin2 Ea + e2 cos2 Ra]
[kz cos RR + (g2 - k2) cos2 r?a];
(6.7b)
MULTIPLE
(ii)
Equivalent
607
SCATTERING
separable potential
x
-
[COS
k(R - a) + i $ sin k(R - a)]
A ____ 4
I -,,
cos q(R - a) sh(A2 - q2)1/2 a q2 - k2 [
, (A2 -qq2)l,2 sin q(R - a) ch(h2 - @)1/2 a11
(6.8)
where h = (2fiV0)1/2 andf(k) is the Jost function given by (6.2b). The integral is performed numerically. We also consider the Yamaguchi potential (6.9a)
(k’ 1 V ) k) = X/(k’2 + f12)(k2 + j12).
The on-shell single scattering t-matrix is given by
I kh = (k2 : 83 D -
(x/2M~2~~2
p/N + ifl)2)1’
(6.9b)
The term I,(k, R) may readily be evaluated, and the scattering amplitude in Section 3 up to double scattering terms is then given by
defined
(p/n2>j,(QR)Zo(k R),
(6.104
Tc2Yk’, k; RI = jo(q(W))
I t(k2/&4
I 4,
+
where Il,(k, R) = --(21~)~ 5
/eikR - cR* [ 1 + +
(P + k2)] 1
I k)2, . (6.1Ob)
In this case we may also give the exact result as presented in Section 4. The exact scattering amplitude is given by ,(‘)(k’
7(kly k’ R, = 1 -{(k
k* R)
) t(k2/2p) ) k), ((27~)~/R) pfeiiR - (1 + (k2 + /@R/2/3)) e-sR]}2 * (6.11)
608
KUJAWSKI
AND LAMBERT
The contributions of the multiple scattering terms as well as the off-energy-shell contributions are well exhibited. These are more fully discussed in the next section.
7. NUMERICAL
STUDIES
We now investigate the various approximations and concepts discussed in the previous sections for a variety of two-body interactions and so-called deuteron wave functions. ’ Three types of interactions are considered for the calculation of the various relevant S-wave f-matrix elements: (i)
Square well interaction, V(r) = V,0(r - a),
(ii) Equivalent separable potential. described in Section 6. (iii)
Yamaguchi
This
(7.la) interaction
is constructed
as
potential, (k’ 1 V 1k) = -A/(/32 + k’2)(j32 + k2).
(7.lb)
The parameters of these potentials are, respectively, V, = -14.5 MeV, A = 0.03245 fm-2,
a = 2.55 fm, j3 = l.l77fm-l,
and they were determined so as to reproduce the 5, nucleon-nucleon length and effective range, a,, = -23.74fm,
scattering
r,, = 2.67 fm.
For the densities we used three of the wave functions given by Franc0 and Glauber [24]. We write them as p2(r> = c2e-6*r(sh y2rh2rY,
(7.2a)
pa(r) = k c,ie-“di’(sh y4r/y4r)2,
(7.2b)
i-1
(7.2~) These densities multiplied
by r2 are plotted in Fig. 1.
MULTIPLE
DEUTERON -
FIG. I.
609
SCATTERING
DENSITIES
1=2
Deuteron densities specified by Eqs. (7.1) multiplied
by P.
In the presentation of the results emphasis will be placed on the following points: (i) The kinematics. The various kinematics are discussed in Section 2. All results are considered for both the FSA and OEI kinematics. (ii) Convergence of the multiple scattering series. As discussed in Section 4, for separable potentials the exact result can be explicitly obtained, in which case we compare it with the result keeping up to double scattering terms. (iii) Sensitivity of the result to various interactions. The scattering series up to double scattering terms will be compared for the square well and equivalent separable potential. (iv) Sensitivity of the results to various densities. (v) Sensitivity of the results to the various approximations discussed in Section 5. (vi) Energy and angular dependence of the results. Since we no longer have any implicit small angle approximation, the differential cross section is calculated for all angles. The results are considered for 100 and 500 MeV nucleons. It should be pointed out that the employed interactions do not provide a realistic description of the scattering of 500 MeV nucleons, but they, nevertheless, may be of relevance in studying the energy dependence of the various approximations. As discussed in Section 2 the OEI and FSA kinematics are only identical in the Glauber eikonal approximation. At intermediate energies the corresponding scattering amplitudes may differ significantly. Furthermore, as discussed in Section 6 the phase equivalent potentials are determined for the N-N system, and, consequently, they are no longer phase equivalent when using the reduced mass of
610
KUJAWSKI
AND
LAMBERT
TABLE I Total cross sections for N-D scattering using the square well interaction at 100 and 500 MeV for various densities and approximations discussed in the text.
dmb>
Single and double scattering Exact
E(MW
P PZ4 P4 PS Pa
500
P4 P6
FSA
NOP OEI
14.2 13.8
FSA
30.5 29.7 32.4
15.1
0.857 0.846 0.872
OEI
14.6 13.9 15.9
1.31 1.29 1.35
34.1 31.4 39.1
0.858 0.846 0.875
TABLE
On-shell
1.32 1.29 1.36
FSA 14.2 13.9 15.1
OEI 32.3 31.2 35.7
0.857 0.846 0.872
1.32 1.29 1.36
II
Total cross sections for the phase equivalent separable interaction using the square well interaction at 100 and 500 MeV for various densities and approximations discussed in the text,
O T N-N
Single and double scattering Exact EWeV)
100
On-shell
P
FSA
OEI
FSA
OEI
FSA
OEI
;:
14.0 13.7 14.5
30.3 29.3 32.0
15.0 14.2 16.3
34.1 31.4 39.1
14.5 14.2 15.5
32.3 31.2 35.7
P6
500
NOP
Pa Pa PS
0.859 0.851 0.870
1.31 1.29 1.34
0.864 0.851 0.880
1.32 1.29 1.36
0.862 0.852 0.877
1.32 1.29 1.36
the N-D system. These points are well illustrated in Tables I and II. Although the NOP (nonoverlapping potentials) and on-shell approximations depend solely on the on-shell individual scattering amplitudes the corresponding total cross sections are identical only within the framework of the OEI kinematics. This effect, however, is not as crucial as the explicit dependence on the kinematics. The scattering appears
MULTIPLE
611
SCATTERING
to be quite sensitive to the kinematics especially at 100 MeV, and as expected the difference decreases with increasing energy. At intermediate energies the kinematics then has important consequences on the multiple scattering. The calculation of the scattering from two separable potentials was discussed in Section 4. The convergence or importance of the higher order multiple scattering terms is illustrated for the Yamaguchi potential and density no. 2 at 100 MeV in Fig. 2. It is interesting to note that the multiple scattering series based on the FSA is well approximated by the single and double scattering terms. In contrast using the OEI kinematics the convergence seems quite poor at large angles, and for angles larger than 120” the result provided by the first two multiple scattering terms differ significantly from the exact one. Similar results apply using the two other densities. At 500 MeV the exact result including all multiple scattering terms hardly differs from the so-called exact double scattering result at all scattering angles. Although
05
0
1.5
10
20
25
‘CM
[fm -‘I
13
, -
POTENTIAL
VAMAGUCHI
E i~B
MeV
= 100
-
ALL -
SCATTERING
SINGLE AND SCATTERING
TERMS
DOUBLE ONLY
I
I
I
30
60
90
I 120
I 150 @c M [deg
180 1
FIG. 2. N-D scattering at 100 MeV using the Yamaguchi interaction and density no. 2. The curves denoted by OEI and FSA are calculated using the corresponding kinematics as explained in the text. This figure illustrates the convergence of the multiple scattering series.
612
KUJAWSKI
AND
LAIdBERT
as the energy increases the agreement between the exact and various approximations to the double scattering term improves, the importance of the off-shell effects remains significant especially at the larger angles. This is illustrated in Figs. 3 and 4 where we compare the exact result up to double scattering terms with those calculated using various approximations leading to on-shell scattering amplitudes at 100 and 500 MeV. In Fig. 5 the exact results for the Yamaguchi interaction at 100 MeV are shown for the three densities given above. As expected the results from densities no. 2 and no. 4 are in reasonable agreement and differ significantly from density no. 6. This is readily understood by examining Fig. 1. By comparing Figs. 3 and 5 it becomes apparent that the differences due to two equally realistic densities are likely to be as small or large as those due to an approximate treatment of the off-shell t-matrix elements. At 500 MeV the effects of the various on-shell approximations have somewhat decreased in importance in comparison to the various densities. This 0 T t
0.5
1.0
10
1.5
2.0
I
2.5
(1Jfm-‘l
EI KINEMATICS
2
VP bC1:
10
-
EXACT
0.1 -
-
NOP APPROX.
-.-
ON-SHELL
APPRO
0.01 DEUTERON
DENSITY : p2
POTENTIAL:
YAMAGUCHI
E L*B = 100 MeV SINGLE
I
0
I 30
AND
DOUBLE I 60
SCATTERING I 90
ONLY I 120
I 150
%.M.rdeg I
FIG. 3. N-D scattering at 100 MeV using the Yamaguchi interaction and density no. 2 calculated up to double scattering terms for various approximations discussed in the text.
Ok-+-
’
6. qC.MIfme’ I
410 ’
01 ” ”
DEUTERON
DENSITY P,
POTENTIAL
YAMAGUCHI
1
ALL SCATTERING TERMS -
EXACT
-
-
NOP APPROX
-.-
0
30
ON - SHELL APPROX.
60
90
120
150
180
CM [deg 1
FIG. 4. Same as Fig. 3 but including all terms at 500 MeV. 0 10
05 I
WTENilAL.
10 I
15 I
20 I
25 I
s,,,Cfm-‘3
YAMAGLICHI
E L*B = 100 Mev ALL SCATTERING 001
TERMS
,
--P, -i-p
p2
6
FIG. 5. N-D scattering at 100 MeV using the Yamaguchi interaction densities. The scattering is calculated to all orders.
for various deuteron
614
KUJAWSKI
AND
LAMBERT
01 POTENTIAL
ALL
0.0001
VAMAGUCHI
SCATTERING
TERMS
-
120
150 h
FIG.
180
[deg 1
6. Same as Fig. 5 but at 500 MeV.
may readily be seen by comparing Figs. 4 and 6. However, as already mentioned our interactions are not realistic at 500 MeV. At the higher energy it may then be possible to supplement elastic electron scattering studies to discriminate among various two-body densities. Because of the off-shell effects and the sensitivity to the kinematics such a determination may still be rather unreliable. Using the phase equivalent interactions described in Section 6 we now systematically study the multiple scattering series up to double scattering terms within the framework of the various approaches given in Section 5: (i) Exact treatment of the double scattering term taking into account the off-shell dependence of the t-matrix elements. (ii) Nonoverlapping potential assumption. The scattering is calculated assuming the potentials do not overlap even when averaged over the target density. The relevant expressions are given by Eqs. (5.1). (iii) On-energy shell approximation. The principal part of the propagator in the multiple scattering terms is neglected. The resulting scattering is then again given in terms of fully on-shell two-body scattering amplitudes and the relevant expressions are given by Eqs. (5.5).
MULTIPLE
615
SCATTERING
The various above approximations for the square well interaction at 100 MeV using the density no. 2 are shown in Fig. 7. The corresponding results for the equivalent separable potential are shown in Fig. 8. It is interesting to observe that the nonoverlapping and on-energy shell approximations lead to very similar results, but they differ significantly from the exact results. The close similarity between the nonoverlapping and on-energy shell approximations is in keeping with the discussion presented in Section 5. To further emphasize the importance of the off-energy-shell dependence of the multiple scattering series, in Fig. 9 we have plotted the exact single plus double scattering terms for the square well and equivalent separable interactions already presented in Figs. 7 and 8. The corresponding results at 500 MeV are shown in Figs. 10 and 11. Although the validity of the various approximate treatments or models improves with increasing energy, a proper treatment of the multiple scattering terms seems essential for an analysis to be of significance. Finally we have also examined the dependence of the scattering on the densities. The results are similar to that for the Yamaguchi interaction. The results for the square well at 100 MeV are shown in Fig. 12, and those for the equivalent separable n,oo
05
10
15
20
25
5 2 4%
:
10
EXACT .-
NOP APPROX ON-SHELL
120
AWROX
150 % ,., [deg 1
160
FIG. 7. N-D scattering at 100 MeV using the square well interaction and density no. 2 for various approximations discussed in the text. The kinematics is as discussed in the text. The calculation is up to double scattering terms.
10
05
2.0 I
15
i?l
-
qcM km-‘1
2.5
I
EXACT
0.1 -
001
-
NOP APPROX
-- OEUTERON
DENSITY : Pz
POTENTIAL 3 EQUIVALENT
SEPARABLE
ELAB = 100 MeV SINGLE
1 0
AN0
DOUBLE
I
I
30
60
SCATTERING
I 90
ONV
I
I
120
150
I 160
@c,a [d-w 1
FIG. 8. Same as Fig. 7 but for the equivalent separable interaction. 10
0.5 I
3 10'r;,
15 I
I OEI
20 I
25 I
4,.
[tm-‘I
KINEMATICS
f ? al,zf 10 L
\
\
FSA
\
KINEMATICS\
\
01
0 01I-
OEUTERON
DENSITY : p2
E iAB E loo
MeV
SJNGLE AN0
DCHJBLE SCATTERING “NL”
*
l\
-
O
SOUARE -
WELL
EQUIVALENT I 30
I 60
I
POTENTIAL
SEPARABLE I 90
POTENTIAL _I 120
I 150 oc M. [de9
1E 1
FIG. 9. Comparison of N-D scattering at 100 MeV for the square well and separable interaction using density no. 2.
-
EXACT
-
-
NOP APPROX ON-SHELL
I 0
I/ 30
1 90
1 60
APPROX
/ 120
I 150
180
8c M [deg
FIG.
;
10. Same as Fig. 7 but at 500 MeV. “L
I
I
1,O 2,O -
0
L,O
SQUARE
-
01
,
-
, WELL
EQUIVALENT
6,O
‘CM:‘“;’
POTENTIAL
I
SEPARABLE POTENTIAL
I
I
I
ALL..
30
60
90
120
j 150 ec M [deg
FIG.
180
1
11. Same as Fig. 8 but at 500 MeV.
oo
05 I
10 ,
15 ,
20 I
25 I
%k41fme’l
‘\ *\ ‘1.\ ‘\ ‘\ + OEI
FSA
KINEMATICS
I
KINEMATICS
-p2
1.5
ho
,
,
so
120
,
i
150 % M. h’eg 1
160
FIG. 12. N-D scattering at 100 MeV using the square well interaction The calculation is up to double scattering terms.
,oo
05
10
15
2.0 I
25 I
1
< POTENTIAL: EQUIVALENT
0,
30
60
so
SEPARABLE
120
160
150 ec. ..L-
FIG.
for various densities.
1
13. Same as Fig. 12 but for the equivalent separable interaction.
MULTIPLE
SCATTERING
619
interaction in Fig. 13. One readily sees that it is rather difficult at intermediate energies to discriminate or distinguish among various approximations, phase equivalent interactions or reasonable densities. As the energy increases the validity of the various high-energy approximations improves, the importance of the off-shell effects decreases, and the sensitivity to the density increases. Nevertheless, the various approximations are such as to question their predictive powers. 8. CONCLUDING
REMARKS
In this work we have studied elastic scattering from a two-body bound system using a variety of simplified two-body interactions. The goal was to obtain insight into the importance of the contributions arising from properly treating the overlapping potentials and various kinematics. Our calculations are of relevance to high- and intermediate-energy nuclear scattering. We now summarize the essential ideas in this work: (i) The kinematics of the FSA and OEI are quite different especially at low energies, but become identical in the high-energy limit. A proper treatment of the kinematics is also expected to be crucial for x-nucleus scattering in the neighborhood of the (3, 3) resonance [25]. (ii) For nonoverlapping potentials the composite scattering is completely determined in terms of on-shell information. This should be kept in mind, and it suggests that in the optical potential formulation [5] the second order term is likely to cancel some of the off-shell dependence of the first order optical potential [26]. (iii) The multiple scattering is such that the off-shell effects of the two-body interaction are strongly coupled with the two-body correlations. This observation plus the sensitivity on the kinematics suggest that elastic scattering is not well suited to obtain information about two-body correlations or the two-body interaction. Such a conclusion is in keeping with the one reached by Feshbach [27] within the framework of the eikonal approximation, Kujawski [12] using onedimensional models, and Adelberg and Saperstein [28] using the optical potential formulation [5]. (iv) At intermediate energies the predictions provided by the nonoverlapping potential and on-energy-shell approximations deviate sufficiently from the exact FSA results to render ambiguous the extraction of the distribution function or two-body interaction in terms of either of these two models. These conclusions about using elastic scattering to obtain information about two-body correlations or the two-body interactions are rather discouraging, but they should be kept in mind when analyzing experiments. On the optimistic side
620
KUJAWSKI
AND LAMBERT
let us state that there may exist an energy or angular region where the off-shell effects are weak, and one can, therefore, be able to determine two-body correlations. More theoretical and experimental work is needed before such studies are fruitful. Finally, let us remember that there are various other approaches [29] better suited than high-energy elastic scattering for investigating nuclear structure and two-body interactions.
APPENDIX
A:
SCATTERING
BY Two SEPARABLE NONLOCAL
INTERACTIONS
For the scattering from two fixed potentials vi (i = 1,2) the Faddeev equations (2.7) reduce to TO
= T,(E) + T,(E), 1,2,
j#i= T,(E) = W) + 4(E) Go@) T,(E), ti(E) = vi + ui(E - (k2/2,u) + 8)-l &(E),
(A.la) (ASb) (A.Ic)
where ka is now an operator. We now consider two real identical separable potentials (k’ I 0 I k) = Mk’)
64.2)
g(k).
Assuming that they are located at &r we obtain
(A.3a)
where q = k’ - k, and
W I t(E) I k) = hd(E) dk’) g(k),
Substituting
(A.3b)
(A.3) into (A. 1b) and using the explicit expression for Tj in terms of
Ti , (A. 1b) reduces to
[
e*“l”g(k) +
I
s dp
Go(E) IP) g"(p)
dp’
e--
MULTIPLE
621
SCA'ITERING
where
= (E - (~~12~) + W. The solution to (A.4) is clearly of the form (k’ 1 T,(E) ( k) = (h/d(E)) eiik”‘g(k’) Q&E, k),
where by substituting
(ASa) into (A.4) one finds
Q(E,
k)
=
[email protected](k)
*
+
e*2ik”h(r)
,
1 - F(r) h WI = __ d(E) s 4 ezipW Combining
(A.Sa)
I G,(E) I P> g”(p).
(A.5b) (A.5c)
the above expressions one readily obtains Eq. (4.2).
APPENDIX Substituting
B:
DERIVATION
OF EQS. (6.7b) AND (6.8)
(6.4a) into (3.4c) we obtain
x [I? sin pa cos r7a - p sin ria cos pa12.
(B.1)
Integral (B.l) is conveniently broken up into a sum of three integrals which are computed by closing the path of integration in the upper and lower half-complex p-plane, as is appropriate. Proceeding as indicated we obtain Eq. (6.7b). The substitution of (6.5a) into (3.4c) leads to
x [cos iiiia sin pa - (p/G) sin &a cos pa],
CB.2)
with G = (p2 - 2/zV”)1/2.
We now sketch the evaluation rewritten as
of the integral in (B.2). This integral may now be W)
595/8+-17
-
z2m
(B.3a)
622
KUJAWSKI
AND
LAMBERT
where
40=i~ffike-$+i8 (w)
sin[(ir, + p) a] sin pR,
(B.3b)
sin[(& - p) a] sinpR.
(B.3c)
The Riemann surface of the integrands is formed by making connecting the two branch points of ~5 as shown in Fig. 14.
a single cut
Imp Complex -III
C+i f-i
-A
III III
p-plane
III Re P
x
---I---
FIG. 14. (a) Riemann surface for integrands in Eqs. (B.2). Z denotes the appropriate (p’ - ~FV,)‘/~, A = (2,6V,,)‘/z. (b) Contour %I . (c) Contour V, .
function
Let us now consider Z, for R < 2~. This integral is conveniently broken up into a sum of four integrals, II(R) = - $ Jamk2 _ f
+ i6 (v)
+ exd-W
+ P) a + ~41
-
+
ew-W
P) a -
texp[i[(G + p) a + pRl1 -
exp[W
+ P) a - pRl1
PRIII
= ZF’( R) + Z,(2)(R)+ Z?‘(R) + Z?‘(R).
(B.4)
MULTIPLE
SCATTERING
623
The relevant paths of integration WI and %‘$ are shown in Figs. 14b and 14c, respectively. I:” and Zp’ are computed along @I and Zi2’ and I:“’ along q2 . The technique for Z,(R) is similar. Proceeding as indicated we obtain Eq. (6.8). We wish to point out that our result differs from Eqs. (3.19) and (3.20) given by Beg [lo]. Our result has the proper limit for R = 2a. ACKNOWLEDGMENTS The authors would like to thank H. Feshbach, J. Gillespie, A. Kerman, C. Shakin, and C. Wilkin for useful and enlightening comments and discussions. This work was started while one of us (E. L.) was a guest of the Center for Theoretical Physics at M.I.T.
REFERENCES in “Lectures in Theoretical Physics,” (W. E. Brittin and L. G. Dunham, Eds.), Vol. 1, lnterscience, New York, 1959. R. L. SUGAR AND R. BLANKENBECKER, Phys. Rev. 183 (1969), 1387. E. KUJAWSKI, Ann. Phys. (N.Y.) 74 (1972), 567. M. L. GOLDBERGER AND K. M. WATSON, “Collision Theory,” Wiley, New York, 1964. A. K. KERMAN, H. MCMANUS, AND R. M. THALER, Ann. Phys. (N.Y.) 8 (1959), 551. N. M. QUEEN, Nucl. Phys. 55 (1964), 177. H. FESHBACH AND J. H~~FNER, Ann. Phys. (N.Y.) 56 (1970), 268; H. FESHBACH, A. GAL, AND J. H~~FNER, Ann. Phys. (N.Y.) 66 (1971), 20; E. LAMBERT AND H. FESHBACH, Phys. Left. 38B (1972), 487; Ann. Phys. (N.Y.), 76 (1973), 80. L. L. FOLDY AND J. D. WALECKA, Ann. Phys. (N.Y.) 54 (1969), 447. K. A. BRUECKNER, Phys. Rev. 89 (1953), 834; S. DRELL AND L. VERLET, Phys. Rev. 99 (1955). 849. M. A. BACI B&G, Ann. Phys. (N.Y.) 13 (1961), 110. D. AGASSI AND A. GAL, Ann. Phys. (N.Y.) 75 (1973), 56. E. KUJAWSKI, Phys. Rev. C7 (1973), 18. K. L. KOWAUKI AND D. FELDMAN, Phys. Rev. 130 (1963), 276. R. G. NEWTON. “Scattering Theory of Waves and Particles,” McGraw-Hill, New York, 1966. K. L. KOWALSKI AND S. C. PIEPER, Phys. Rev. C 4 (1971), 74. N. F. MOTT AND H. S. W. MASSEY, “The Theory of Atomic Collisions,” Chapter VIII, Oxford University Press, Oxford, 1971. The work of Foldy and Walecka [8] has recently been extended by G. E. Walker to finite-mass target particles and Galilean invariant interactions; Indiana University preprint. A. K. KERMAN, private communication. D. R. HARRINGTON, Phys. Rev. 184 (1969), 1745. T. A. OSBORN, Ann. Phys. (N.Y.) 58 (1970), 417. W. TOBOCMAN AND M. PAULI, Phys. Rev. D 5 (1970), 417. H. S. PICKER, E. F. REDISH, AND G. J. STEPHENSON, Phys. Rev. C 4 (1971), 287. M. GOURDIN AND A. MARTIN, Nuovo Cimento 6 (1957), 757; 8 (1958), 699.
1. R. G. GLAUBER, 2.
3. 4. 5. 6. 7.
8. 9.
10. 11.
12. 13. 14. 15. 16.
17. 18. 19.
20. 21. 22. 23.
624 24. 25. 26. 27.
KUJAWSKI V. C. F. H,
AND
LAMBERT
FRANCO AND R. J. GLAUBER, Phys. Rev. 142 (1966), 1195. SCHMIT, Nucl. Phys. Al97 (1972), 541. SCHECK AND C. WILKIN, Nucl. Phys. B49 (1972), 541. FESHBACH, in “In Honor of Philip M. Morse,” (H. Feshbach
and K. V. Ingard, Eds.),
p. 190, M.I.T., Cambridge, MA, 1969. 28. M. L. ADELBERG AND A. M. SAPERSTEIN, Phys. Rev. C 7 (1973), 63. 29. A rather complete and excellent account of research in nuclear physics and elementary particle studies at intermediate energy is “High-Energy Physics and Nuclear Structure,” (S. Devons, Eds.), Plenum Press, New York, 1970.