Three-body approach to direct nuclear reactions involving weakly bound systems

ANNALS OF PHYSICS: 44, 363-425 (1967)

Three-Body Approach to Direct Nuclear Reactions*t

P. E. SHANLEY

Cracker Nuclear Laboratory, University of California, Davis, California

AND

R. AARON

Department of Physics, Northeastern University, Boston, Massachusetts

Recently developed methods of treating the three-body problem are applied to nuclear rearrangement collisions involving a neutron, a proton, and

a heavy nucleus assumed to have no internal structure. The two-body interactions are parameterized in terms of s-wave separable potentials and the

resulting three-body equations are solved exactly on a high-speed computer. Result,s are obtained for a spinless model which neglects the proton-core inter-

action and these are compared with the general behavior of experimental results although no attempt is made to fit a particular nucleus. These exact

results are also used as a test of the distorted-wave Born approximation and of some high energy diffraction models. The theory is then generalized to

include spin, isospin, and a charge independent nucleon-core interaction, para.meterized with the core assumed to be 160. Numerical results are obtained for t,his system which resemble some general features of the experimental

situation but detailed agreement is not obtained. Finally, further generaliza- tions of t,he model are discussed and the possibility of applying three-body

methods to other nuclear systems is considered.

I. INTRODUCTION

Nuclear scattering experiments almost always involve composite particles and in such reactions either the incident or target system is capable of breaking up into two or more subsystems. The many-body and multichannel nature of such collisions has put them beyond the scope of exact theoretical treatment and has necessitated the introduction of ingenious approximation methods such as the optical model, the distorted-wave Born approximation, and the impulse approximation. The two-body problem in potential theory is of course well under- stood and hence the scheme underlying most of these approximate methods is

* Supported in part by the National Aeronautics and Space Administration, National

Science Foundation, and the U. S. Atomic Energy Commission. t Based in part on a dissertation submitted by P. E. Shanley to Northeastern University

in partial .fulfillment of the requirements for the degree of Doctor of Philosophy.

363

364 SHANLEY AND AARON

to treat the many-body system as an effective two-body one, thereby making it amenable to theoretical description. Alt’hough the success of such methods in describing many nuclear reactions is considerable, in the process of reducing the many-body system to an effective two-body problem large portions of the interaction must be treated phenomenologically with free parameters adjusted to fit the experimental data. Hence, the interesting details of the reaction mechanism are difficult to extract because they are buried in these parameters and also because the calculational scheme is often not based on a rigorous mathematical foundation.

Deuteron stripping and pickup are by their nature three-body problems if the core nucleus is assumed to act as a reasonably inert unit. In this paper it will be shown that these reactions can be studied in the context of a soluble three-body model proposed by Amado (I), and used successfully to describe the three-nucleon system (d), (3). The approach involves the use of particularly simple two- particle interactions (separable potentials) which reproduce the basic features of the low-energy two-body systems such as, for example, bound-state wavefunc- tions, scattering lengths, and effective ranges. These potentials reduce the three- body problem to the solution of coupled sets of one dimensional Fredholm integral equations, and allow complicated three-body effects such as bound-state breakup and the coupling between elastic and inelastic channels to be taken into account exactly.

The contents of this paper are arranged as follows: In Sec. II we first consider a simplified problem in which all particles are spinless and the proton-core interaction is omitted. Even in this case the model stripping amplitudes have qualitative resemblance to experimental ones. We have given preliminary results con- cerning this model previously (4) but here we discuss this simplified problem in great detail since the absence of spin and isospin permits a clearer exposition of our theoretical approach. Here, we also display the partial wave phase shifts and inelasticity factors obtained and consider their behavior in the light of recent research on generalized Levinson theorems for multichannel problems. In Sec. III we study the particular reaction d + 160 + p + “O* (0.871-MeV level) including spin, isospin and a proton-core interaction, but neglecting Coulomb effects (“0, and “F* are assumed to have the same binding energy and same single-particle wavefunction). We obtain some agreement with experiment and discuss the possibility of using our model to obtain reduced widths and spectroscopic factors from stripping reactions. In Sec. IV we examine the distorted-wave Born approximation (DWBA) and some high-energy diffraction models by comparing the predictions of these models with our exact theoretical results. Also we question the usual interpretation and derivation of some recent high-energy diffraction formulae and further, suggest and motivate their use in the realm of low- and medium-energy nuclear reactions. In Sec. V we present discussion and

THREE-BODY APPROACH TO NUCLEAR REACTIONS 365

conclusions. Numerical met,hods are discussed in Appendix A and a table of numerical results is given in Appendix B.

II. SPINLESS MODEL

A. THREE-BODY METHODS

A mathematically sound integral formulation of the nonrelativistic three-body problem was first given by Faddeev (6) who succeeded in writing a set of coupled integral equations which describe scattering and reactions as well as bound states in three-body systems, while avoiding the difficulties (6) associated with the well-known Lippmann-Schwinger equation (7). The resulting set of integral equations are linear Fredholm equations and require the off-energy-shell two- body scattering amplitudes in each of the three two-body subsystems as input information.

Although formally attractive, the Faddeev equations involve too many vari- ables to attempt a numerical solution with presently available computers. This is not surprising since the central difficulty of the problem, the multiplicity of coordinates, is still present in this formulation. However, a few restricted three- body problems are amenable to calculation if one is able to reduce the number of coordinates needed to specify intermediate states to one. This reduction is exact if, for example, one introduces into the Faddeev equations a potential between pairs which is separable in momentum space. Alternatively, one may introduce a quasi-particle through which pairs interact. These alternatives are exactly equivalent in the limit of vanishing wavefunction renormalization constant for the quasi-particle introduced (8) and in the present work we choose the latter procedure. The use of separable interactions in the three-body problem was introduced by Mitra (9), (IO). A theoretical justification for the use of these interactions has subsequently been given by Lovelace (11) for the case where the two- body amplitudes are dominated by a few bound states and resonances. Along other lines, a reformulation of the distorted-wave Born approximation in a rigorous three-body formalism has been given by Greider and Dodd (12).

B. FORMULATION OF THE INTEGRAL EQUATIONS

We now derive a set of linear one-dimensional integral equations to describe various deuteron-nucleus reactions. The three-body equations are formulated using the field-theoretic method of Amado (1) rather than the Faddeev formalism. For separable two-body interactions, Rosenberg (IS) has shown that the two approaches are equivalent. The model deals with five spinless particles n, p, d, A, and B, with n, p, and A possessing no internal degrees of freedom. All particles are free to move and nonrelativistic kinematics are employed. The masses of n, p, and cl are chosen to be those of the neutron, proton, and the deuteron, respectively. The particle A may be thought of as some closed-shell nucleus and B


is a composite state of n and A. In this simple theory only the following s-wave interactions are allowed :

n+p+-+d, n+AttB.

(2.1)

Note that no p-A interaction is permitted and that in the two-body sector the n-p and n-A scattering occurs only in s-states since it must proceed through the spinless d and B particles, respectively. The above processes may be described by a momentum-space interaction Hamiltonian of the form (fi = 2m = 1, m = nucleon mass)

H, = Yd g, fd (In+ i> +d+(n + n’)qn(n)*b’)

(2.2) QB+(k + k’)Pa(k)K(k’) + h. c.

where Yd and yB , and fd andfB are the renormalized coupling constants and vertex functions associated with the processes (2.1). @ and \k are annihilation operators for the appropriate particles and A is the mass number of the core nucleus A. Since spin is neglected, all field operators obey the usual canonical commutation relations. Associated with each of the composite particles d and B are wavefunction renormalization constants .??d and 2, which measure the degree of “elementarity” of d and B. It has been shown (13) that when all wavefunction renormalization constants are zero, the Amado model is equivalent to a separable potential theory. We restrict ourselves to this case.

Having described the two-body interactions, we now go on to the three-body sector of our theory where the following reactions occur:

(a> d+A+d+A,

(b) d + A --+ P + B,

Cc> d+A--tp+n+A,

(4 P + B * P + B,

(4 p+B-+d+A,

(f) p+B-+p+n+A.

(2.3)

All of the above processes are driven by the exchange of a neutron between the states d and A, and p and B. The basic collision diagram is the Born approximation for deuteron stripping (d + A + p + B) which is represented schematically in Fig. 1. The amplitude for this second-order process may be obtained using the

THREE-BODY APPROACH TO NUCLEAR REACTIONS

A B

FIG. 1. Schematic representation of Born approximation to stripping amplitude.

Hamiltonian (2.2) and is given by

fd I k/2 - k’ I Ifs (I i&-&k

B&k’, k; E) = */d YS l+A I> E - (k - k’)2 - h+/A - lct2 ’ (2.4)

where k and k’ are the incident d--4 and final p-B center-of-mass momenta, respectively, and E is the center-of-mass energy. The same type of expression is obtained for the stripping amplitude in plane-wave Born approximation when local potentials are used. The advantage and simplification of the separable approach is t,hat the higher-order terms can be summed into an exact integral equation. Typical terms in the formal expansion of the p-B elastic scattering and deuteron stripping amplitudes are shown in Fig. 2. Only two types of processes can occur in our theory; the neutron may exchange between cl and A or p and B, and the composite particles cl or B may virtually disassociate in any internal line. It should be stressed that the formal expansions depicted in Fig. 2 do not in general represent a perturbation series of the exact amplitude, since this series may diverge if the couplings are strong enough to produce three-body bound states. The elastic p-B scattering and stripping amplitudes may be formally summed into exact integral equations shown schematically in Fig. 3. The dark line presents the sum of “bubbles” for d or B particles in intermediate states. A similar set of equations may be formulated that couple d-A elastic scattering to the pickup reaction (p + B -+ d + A). The specific form of the integral equations are

T&c’, k; E) = Bst(k’, k E) + h3 (2.5a)

T,dk’, k; El = & / c&l B,,(kl , k; E)~dA(kl ; E)%t(k’, 4 ; E), (2.5b)


T,,(k’, k; E) = B&k’, k; E)

+ & s d% B,u(k, k; Eha(h ; -Wcdk’, 4 ; El, (2.6a)

Tdfr’, k; El = &, j- d3h B& , k; E)7p&l ; E)T,,(k’, 4 ; E). (2.6b)

Ts, , TUB , T,u , and TdA are the T-matrices for stripping, p-B scattering, pickup,

and d-A scattering, respectively and B,, is the Born term for pickup and is re-

lated to the stripping Born term by B,,(k, k’; E) = B.,(k’, k; E). The above set of integral equations are equivalent to the Faddeev equations with separable potentials. rpB and rdA represent the propagators for the p-B and d-A intermediate states and are given by

SB ~pB(h ; E) = (

E - s k: + fB + iv)

E-2+A 2 ’

(2.7a)

m kl + eI3 + irl

Sci TdA(h, El = (

E - G iit + cd + iq>

E-2fA 7 (2.7b)

2~ h2 + Ed + f$

where cd and tg are the deuteron and nucleus B binding energies (the binding

n = x+...+x+... B B BAB B A BAB A B

(a)

P

+ n +...+y-J%y- A .B A B

+ . . .

(b)

FIG. 2. (a) The sum of graphs for p-B scattering showing typical higher order terms. The box represents the full p-B amplitude. In general there may be an arbitrary number of both ladder rungs and bubbles on internal d or B lines; (b) The sum of graphs for stripping. The circle represents the full stripping amplitude.

THREE-BODY APPROACE TO NUCLEAR REACTIONS 369

A B A B

P P

IL B B

= B A B

(a)

(b)

FIG. 3. (a) Diagrammatic representation of coupled integral equations for stripping and elastic p-B amplitudes. The pickup and elastic d-A amplitudes are coupled analogously; (b) Renormalized propagator for either d or B.

energy of nucleus A is set to zero), and Sd and 8, represent the sum of bubbles as discussed previously. The forms of these functions are

[s,(x)]-’ = 1 - Td”CC s d3n fd?n) - (27r)3 (2d + ed)(x - ed - 2n2) ’

L‘L(x)l-’ = 1 - &I7 s f;(n)

>(

(2.8b) x - EB -!A..&2 *

A > The integral equations may be reduced to one-dimensional equations by partial- wave analysis. That is, writing each amplitude and Born term as

T(k’, k; E) = F (21 + l)!!“(lc’, k; E)P~(cos e),

B(k’, k; E) = F (21 + l)B’(k’, k; E)Pl(cos e), (2.9)

cos e = 8.1

370 SHANLET AND AARON

and integrating over the angles, we obtain the following set of one-dimensional integral equations for each partial wave:

T:&t’, k; E) = B&‘k; E)

+ k2 lm iit cZk~B:t(k~, k; Wpdk; EP’;~(~‘, h; EL (2.10a)

T&(k’, k; E) = & I- k; dh B;,(h, k; E)rcu(h ; E) T:&‘, h ; E), (2.10b) 0

T;,(k’, k; E) = B;,(k’, AC; E)

+ $2 I- kt dh @&I, k; Wa(kl; E)Thdk’, h ; E), (2.11a)

T:a(lc’, k; E) = &I- k: &B&I, k; E)rpdh; E)T;,(k’, h; E). (2.11b) 0

The above discussion completes the prescription for calculating the amplitudes for the processes (2.3) with two particles in the initial and final states. The breakup amplitudes for processes (2.3~) and (2.3f) are not calculated explicitly, but the total breakup cross section may be obtained from Eqs. (2.10) and (2.11) together with the optical theorem. It should be remarked that the breakup amplitude for deuteron-nucleus collisions can be obtained from the off-energy- shell elastic scattering and stripping amplitudes without solving any further integral equations. Breakup distributions have been calculated by Aaron and Amado (5) for the reactions n + d -+ n + n + p.

C. CHOICE OF VERTEX FUNCTION

For both the deuteron and nucleus B vertex functions we choose the Hulthen form

f(k) = l/e2 + P”> (2.12)

where p is some measure of the inverse range of the potential. This choice gives the Hulthen wavefunction in configuration space for the s-wave bound state and this wavefunction can be made to fit the deuteron quite well, but does not lead to a particularly realistic nuclear wavefunction; however, it has the advantage of simplicity. More complicated form factors will be considered in Section III. The stripping Born function then becomes

Bsdk’, k; E) = [E _ (k _ k’)2 _ k:/Ayd-y;“]&,2 _ k’)2 + &] (2.13)

f-&k’-k 2+Ps > 1 -1


The advantage of the Hulthen form for the vertex is that it allows the partial- wave projection of the Born function to be carried out analytically. The partial- wave amplitude is defined as

Bit@‘, k; El = $3 f; d/&dk’, k; EPz(cL) (2.14)

where p = L l L’. Substituting for Bst (k’, k; E) and carrying out the integration we obtain

Cl+ A) Bit@‘, k; E) = -ydyB 4Akk,

-[ () Qz 5

(2.15)

(a - b)(a - c) + (6 1 a) + (c - a)(c - b) ’

where Q1 is the Legendre function of the second kind and we have introduced the notation

u=~[2ktz+1~k2-E-i~],

b = lG2/4 + lcr2 + pi, (2.16)

l+A C=2A 2 k’2 + ii2 + pB2 1 .

For E < 0 (below breakup threshold), the function Bi,( k’, lc; E) is real and Eq. (2.15) is amenable to numerical evaluation. For the case E > 0, however, the term QI(u/Iclc’) may become complex and develop logarithmic singularities that are difficult to treat accurately with straightforward numerical procedures. Hence Eq. (2.15) is used only below the breakup threshold with other procedures, discussed later, introduced for E > 0.

The integrals involving the S-functions, defined in Eqs. (2.8a) and (2.8b) may also be carried out analytically. We first consider #B(z). Define

EB = [(I + A)IAlw2, 6B2 = 1 ag2 - M/(1 + A)lrz: I>

(2.17)

where in the integral equation the variable J: takes on the values E - [( 2 + A) / (1 + A)lkt + EB - For E < 0, Ss(z) is real and we obtain

1 1 (6B + /%z)’ - (aB + PB>’ 1 *

(2.18)


Above breakup threshold, when E > 0, Se(z) is complex and we have

PB2 - 6B2 1 (pB2 + V)2 - (as + &,)2 1 ’ (2.19)

Im [k%(x)]-’ = YB26B A

471x&~ + 6B2)2 1+A *

The corresponding expressions for Sd(z) may be obtained from the above results by setting the mass number A equal to one and replacing all B parameters with their d counterparts.

D. THRESHOLD AND ANALYTIC PROPERTIES OF THE AMPLITUDES

In considering the various thresholds for the amplitudes recall that we are restricting the consideration to exothermic stripping reactions ( ee > Q) and we choose units such that ed = 0.5 (2.225 MeV). To consider the threshold properties it is useful to write the PB elastic scattering amplitude as a single uncoupled integral equation. This may be obtained by substituting Eq. (2.10a) into (2.10b).

~;rs(k’, k; El = f

*s

k ’ & B;,(kl , k; E)& m 1 ( E - ‘g kt + Ed + iv) B:t(k’, kl ; E)

0 Em2+A 2A k12 + ci + irl

+ & lm k: dkl lm k,” dkz, (““)

B:,(h, k; E)Sd ( E - +4k:+tcz+in B:t(kz,kl;E)

>

. Se ( E - ‘&; k; + tg + iv) T;B(k‘, kz; E)

E-2+8 2~ k12 + Ed + iv >(

2+A E - 1 + A k&B + irl

>

The total center-of-mass energy variable is

E,2+Ak2 l+A pB

- cB in the p-B channel,

E=2+A 2 -g-- ku - Ed in the d-A channel.

The inhomogeneous term of the integral equation is the impulse approximation of the model, with higher-order terms appearing as iterations of this amplitude.


Three-body bound states, if any, will occur for E < - Q and will be associated with homogeneous solutions of Eq. (2.20). As the energy E is increased, the amplitude develops branch points first at E = -Q corresponding to the pB elastic scattering threshold due to the B particle propagator, second at E = - cd corresponding t’o the threshold for the pickup reaction p + B -+ d + A due to the deuteron propagator, and finally at the breakup (p + B + p + n + A) threshold. The breakup singularities arise from three sources. Both & and Se develop square-root branch points at’ E = 0 (onset of breakup), as can be seen from Eq. (2.19). The Born function also developes logarithmic singularities’ for E > 0. These singularities occur in all four amplitudes. Since we have restricted ourselves to the case eB > Ed , the p-B elastic threshold is at E = - 6B , while the thresholds for stripping, pickup, and d-A scattering open at E = - cd and the breakup channels open at E = 0.

E. CHOIC:E OF PARAMETERS

We now choose the parameters used in the spinless calculation with Hulthen vertex function. The units employed are such that 6 = 2m = 1 (where m is the nucleon mass) and the binding energy of the deuteron Ed = 0.5. We have chosen the parameters of n-P system to give the binding energy of the deuteron and a scattering length equal to the triplet n-p value of 5.38F. This choice requires

(14)

@d = 6.225ad ,

yd2 = 32rffdbd(ffd + @d13,

C&j = ( &?)1’2.

(2.21)

In the n--A system, the relations between the parameters are

2 (OB + @Bj3,

(2.22)

The parameter Bs is chosen to give a reasonable RMS radius to the B nucleus. It should be recalled that we are taking the wavefunction renormalization constants of d and B equal to zero, so that we are working in the potential theory limit. The bound-state wavefunction of B is the usual Hulthen form

+B(r) = 2aB@B(%3 + pB) epuBr - emBB’

(aB - PB)’ ( 7

r

1 See Appendix A for it further discussion of these singularities.

(2.23)


from which the RMS radius may be calculated:

s OD &MS = r~#B2(r> dr = ~~B~(QB + 0~)~ + ~B’((YB + ,%I3 - 1fh3b3 . (2.24)

0 ~c~z+~/~B~(C~B + PB)'(~B - Pi4

In these preliminary calculations we are making no attempt to fit a particular nucleus, so that we choose parameters for convenience and ease of computation. We have done most of the calculations for two particular B nuclei with the following parameters

(a) eB = 2.0 (8.90 MeV), fiB = 4.0, TRMS = 1.69F, ClB = 1.414,

(b) CB = 1.0 (4.45 MeV), /3B = 4.0, TRMS = 2.14F, aB = 1.0,

where in both cases we have set the mass number of nucleus A to infinity. Note that the resulting B nucleus is quite small, corresponding to very light nuclei. Numerical difficulties prevent us from obtaining larger radii. The fault lies with the choice of a Hulthen wavefunction for the bound state-a large nuclear radius corresponds to 0 1: (Y, and the numerical solutions are unstable unless /3 is quite a bit larger than cr. In the next section we overcome the above difficulty by using a separable square well for our nuclear potential.

F. METHODS OF NUMERICAL SOLUTION

An inspection of the kernels of the integral Equations (2.10) and (2.11) indi- cates that they are sufficiently well behaved to allow Fredholm methods to be used in their solution. Three different techniques have been employed in different energy regions.

(1) Bound States. If the interactions assumed in the two-body subsystems give rise to three-body bound states, these states will show up as zeros of the Fredholm determinant of Eq. (2.10)) for energies E < - ee .

(2) Scattering, E < 0. For these energies the breakup channels are closed and accurate results are possible using standard numerical procedures. For

EB < (2.10)

E < - cd , the pair of Eqs. ( 2.11) are below threshold, and only Eq. need be solved. For - cd < E < 0, all four channels are open and both

(2.10) and (2.11) must be solved. The solution of the integral equations are carried out by approximating the integrals by finite sums and rewriting the equations in matrix form. If there are no bound states in a particular partial wave (I 2 1)) the Neumann series for the integral equation converges and the solution is obtained by iteration. If bound states exist, full matrix inversion is necessary to obtain solutions since the Neumann series diverges, at least at the energies calculated. For each partial wave, the solution of the integral Eqs. (2.10) and (2.11) yields T-matrix elements which are in general a function of k, k’, and E. For scattering calculations with E < 0, the final momentum k’ and the energy E are


assigned .their energy-shell values while the remaining momentum variable k is allowed to assume general off -energy-shell values. The physical amplitude will be the one with the initial momentum also on the energy shell.

(3) Xcattering, E > 0. Above the breakup threshold, the kernel develops logarithmic singularities which are difficult to treat by usual numerical methods. These singularities arise from the Born term in the kernel. As can be seen from Eq. (2.15) for the Born function, the expression 1 a(E)/kk’ 1 may be less than one if E > 0. This leads to logarithmic singularities, and accurate solution of the integral equations is not possible in this case, the solutions under some circum- stances being strongly dependent on the integration mesh employed. To circum- vent this difficulty a technique introduced by Hetherington and Schick (15) in a related problem has been used. The integral equations are solved along appropriate pabhs in the complex momentum plane along which the kernel is extremely smooth. Since there are no singularities between our new path and the old one along the real axis, we may use the integral equation to analytically con- t,inue our solution to physical values of the momentum variable. This procedure is discussed in Appendix A.

G. RESULTS

In the bound-state calculations we have two parameters at our disposal, the binding energy EB of nucleus B and the range parameter PB of the n-A potential. All calculations with reasonable choices of these parameters lead to one s-wave bound state and none in higher partial waves. The bound states are found by searching the homogeneous versions of Eqs. (2.10a) and (2.10b) for zeros of the corresponding Fredholm determinant. Figure 4 shows the type of results obtained for two choices of EB , with PB varying. A zero of the Fredholm determinant D(E) gives the three-body binding energy. The binding energy is seen to be not particularly sensitive to the range parameter PB . Increasing 0s , corresponding to a shorter range potential in configuration space, gives tighter binding in the three-body system. The order of five-MeV additional binding relative to the two- body nucleus B is obtained in the three-body system.

Above scattering threshold the amplitudes become complex and the matrix size doubles, compared to the bound-state problem. We must solve2 four coupled equations corresponding to the real and imaginary parts of the p-B elastic scattering and stripping amplitudes and another four for the d-A elastic and pickup amplitudes. This must be done for a sufficient number of partial waves such that the contribution of higher waves becomes negligible.

From t’he solution of the above integral equations phase shifts, angular distributions, etc. may be computed. The real and imaginary parts of the elastic phase shifts and the inelasticity parameter 7~ are given in terms of the elastic

* Numerical results of these calculations may be found in Appendix B.


c

f, z

0.64 v.+ \s

0.4

0.2

i

-0.6{ }

FIG. 4. S-wave Fredholm determinant as a function of energy below scattering threshold for two choices of l g and various BB (Units: h = 2m = 1, E,J = 0.5).

T-matrix elements by

(2.25) Re 61(k) = f tan-’ (-Iccc/a> I-k TZN) 1 + @P/T) Im T@) ’

Im 6z(k) = -a log{ I+ : [Im Al + g 1 ~z(k) IZ])“‘~ (2.26)

w(k) = exp L-2 Im &(k)l, (2.27)

where k is the center-of-mass momentum in the elastic channel and ,.J is the reduced mass in either the p-B or d-A channel. In our units (2m = 1, m = nucIeon mass) the reduced masses are

ll+A A ~PB=ijz+~, paA = 2 + A *

(2.28)

In the energy range - Ed < E < 0, the amplitudes obey the following unitarity relations :

Im TkB(k’, k’) = -$ s 1 T;B(~‘, k’) 1’

lc” 2A (2.29a) -- 4?r 2+A I T;-&“, k’) I’,


(2.29b).

where k’ and k” are the pB and cl-A center-of-mass momenta, respectively. For E< - ed , only p-B elastic scattering is possible and therefore only the first. term on the right of Eq. (2.29a) contributes. For E > 0 the above relations are not satisfied since there are contributions from the breakup amplitude which are not calculated explicitly, although it is still possible to obtain the total breakup cross section.

The real parts of the elastic phase shifts are shown in Figs. 5 and 6 for p-B and cl-A scattering in the low partial waves for the two cases Q = 2.0 and 1.0. The p-B phase shift is real for - cB < E < - Ed where it is the only channel open and becomes complex at the pickup threshold E = - Ed. The d-A phase shift is always complex since the elastic and stripping thresholds are at E = - ed . The breakup threshold is at E = 0 in both channels. The phase shifts are nor- malized according to what one might naively expect from Levinson’s theorem

-lr 2.0 - 1.0 -

ENERGY

.0.5 0.0 1.0 2.0 L I , I I I

2:o 4.0 6.0 t 1 I I

0

FIG. 5. Real part of the pB and d-A phase shifts versus energy for various 1 and Ed = 2.0, fin = 4.0. The reaction thresholds are indicated (Units: h = 2m = 1, cd = 0.5).

L I.0 -0.5 0.0 1.0 -I 2.0 4.0 6.0 , I I I I

FIG. 6. Real part of the p-B and d-A phase shifts versus energy for various I and es = 1.0, fib = 4.0. The reaction thresholds are indicated (Units h = 2m = 1, ed = 0.5).

SHANLEY AND AARON

- B(p,p)B --- A (d.d)A

c6= 1.0 (4.45 MeV)

(16). There is one s-wave bound state and the difference between the phase shifts at zero and infinite energy seems to be ?r for both p-B and d-A. There are no bound states in higher partial waves so these phase shifts start and end at zero. The only exception to this is the p-wave d-A phase shift which has a difference of ?r between threshold and large energies. This behavior of the p-wave phase shift is discussed further in Sec. III. Another feature of the real part of the d-A phase shifts is that they are negative for II > 1. One would normally associate this behavior with repulsive forces but the connection is not necessarily true in the presence of inelastic channels. The s-wave p-B phase shift has the expected cusp behavior at the pickup threshold, but nothing detectable occurs at the breakup threshold in either channel.

To compare the amount of absorption in each of the elastic channels, the inelasticity parameter qz , previously defined, has been calculated and is shown in Fig. 7. For E < 0, the breakup channels are closed and unitarity requires VP” = qiA. Above breakup threshold, the absorption in the d-A channel generally exceeds that in the p-B channel since the deuteron is more loosely bound and inelastic processes are more likely.


0.8 -

0.8-

0.6-

TPE 11

0.4 - cb)

p+B - p+B

FIG. 7. Inelasticity parameter nl = exp (-2 Im 62) in (a) the d-A and (b) the p-B channel as a function of 1 for various E, sB = 2.0, j3* = 4.0 (Units: h = 2m = 1, ed = 0.5).

A pair of elastic angular distributions are shown in Fig. 8 for a center-of-mass total energy E = -0.1. The elastic angular distributions generally show little structure and are forward peaked.

The total cross sections for all processes initiated from an incoming p-B state are shown in Figs. 9 and 10 for two choices of eB . An interesting feature is the rise in the elastic cross section as the first inelastic (pickup) threshold is approached, the peak becoming more prominent for smaller EB . This type of threshold peak has been seen previously (17) in two-body coupled-channel calculations, and the mechanism has been used by Ball and Frazer (18) in an attempt to explain certain peaks in the pion-nucleon cross section. The mechanism that gives the elastic peak is the opening of a strong inelastic channel with the inelastic cross section


300-

p- 8 Elastic Scattering d-A Elastic Scattering , E,= 8.45 MeV Ed= 1.78 MeV

IOO-

0 I I 0 20 45) 60 80 100 120 140 160 I60 0 20 40 60 80 100 120 I40 160 I

C. M. ANGLE IN DEGREES C. M. ANGLE IN DEGREES

IO

Fro. 8. p-B and d-A elastic angular distributions for E = -0.1. The proton and deuteron kinetic energies are indicated. eg = 2.0, PB = 4.0.

rapidly achieving a sizeable fraction of its unitarity limit. Note that the peak is more pronounced for smaller EB , the pickup cross section increasing for small EB since the neutron is less tightly bound. The effect is dominated by the p-wave amplitude as can be seen in Fig. 11 which shows the partial-wave elastic cross sections. The s-wave cross section falls quickly from its threshold value, exhibiting the expected (19) downward step behavior at the inelastic threshold. The p-wave cross section rises rapidly and exhibits a resonance shape in the vicinity of the pickup threshold. It does not appear possible by varying the parameters of the model to make the phase shift pass through gr and achieve an actual resonance. Such phenomena, an interesting feature of this model, also occur in actual nuclear reactions (19) as a much smaller scale effect. In actual nuclei there is probably a sufficient background inelasticity present such that the opening of one more channel does not cause such a major perturbation on the elastic cross section. Also, Coulomb effects are expected (90) to suppress such threshold phenomena.

The total cross sections versus energy for all processes initiated from an incoming d-A state are shown in Fig. 12. The cross sections are rather smooth compared to the p-B channel.


Several angular distributions for the deuteron stripping process are shown in Figs. 13 and 14 for eB = 1.0 and 2.0, with the total center-of-mass energies E = -0.4, -0.1, 1.0, and 4.0. The corresponding deuteron laboratory kinetic energies are indicated in MeV. Also shown are the angular distributions for the plane-wave Born approximation. The difference between the Born and exact

TOTAL CROSS SECTIONS

-B(P,P)B 1.2 -

E,=2.0 , & =4.0

m %

0.4-

& PICK-UF BREAK-W

-2.0 -1.0 0.0 1.0 2.0 ENERGY

FIQ. 9. Cross sections for all processes initiated by an incident p-B state as a function of energy E for tiB = 2.0 (Units: h = 2m = 1, Ed = 0.5).

WvvP~ TOTAL cRoss SECTIONS Bb,d)A---, Btp,p)B- Ee=l.O = , & 4.0

5i 2.0 -

z 5 1.6 -

5 1.2- F: ; 0.8-

3 & 0.4-

OP -1.0 0.0

PICK-UP , BREAK-UF

2.0 ENERGY

FIG. 10. Cross sections for all processes initiated by an incident p-B state as a function of energy E for cB = 1.0 (Units: h = 2m = 1, ~,j = 0.5).


\ \ B( p,p) B PARTIAL WAVE CROSS SECTIOF

- B(p,d)A Threshold Qg = 1.0 (4.45 MeV)

I

0’2

0 -1.0 -0.5 0.0 0.5 1.0 1.5 2

ENERGY

FIG. 11. Partial-wave elastic p-B cross section versus energy E for Q = 1.0 (Units: 2m = h = 1, 6j = 0.5).

results is a measure of the importance of the effects of distortion and of competing channels. The Born approximation is particularly bad at low energies where distortion effects are very important. As the incident deuteron energy increases, the forward peak of the exact result grows until the angular distribution resembles the characteristic s-wave stripping pattern found experimentally. By comparing the Born and exact answers, it is apparent that the effect of including distortion and competing channels is to sharpen the forward peak and to reduce considerably its magnitude.

We now discuss the significance of the above results. Let us first point out implications about the general structure of stripping amplitudes. It is clear that many of the qualitative features of deuteron stripping are contained in this spinless model-the reason for this can be made plausible if we assume that the basic mechanism responsible for a “real” A(& p)B reaction is the exchange of a neutron to form either a deuteron or nucleus B. Recall that the Born approximation is much too large and violates s-wave unitarity by an order of magnitude. Clearly, the Born term itself is a poor approximation to the true stripping amplitude. What we request is a sum of diagrams involving the basic neutron exchange mechanism which gives amplitudes in all channels satisfying three-body


TOTAL CROSS SECTIONS d-A

SCATTERING AND REACTIONS

Es = 2.0 (El.90 MeV)

0

ENERGY

FIG. 12. Cross sections for all processes initiated by an incident d-A state as a function of energy E for EB = 2.0 (Units: h = 2m = 1, 41 = 0.5). flB = 4.0.

unitarity, Solving the integral equations presented in this section or equivalently solving the three-body Schrijdinger equation with separable potentials gives exactly this unitary sum of ladder diagrams. While in actual reactions the mechanism for stripping is much more complicated than in our model, our results do imply that nucleon exchange is an extremely important ingredient. The present results seem encouraging enough to warrant a generalization of the model to include the effects of spin, isospin, and a direct p-A interaction using a more realistic nuclear potential. This procedure is carried out in the next section.

III. MODEL INCLUDING SPIN AND ISOSPIN

A. CHOICE OF SEPARABLE INTERACTION

In the preceding section we studied a spinless model of deuteron stripping that exhibited many features of experimental stripping results. Here the model is generalized to include spin, isospin, a more realistic nuclear wavefunction, and a direct proton-core interaction. The remaining severe limitation of the model, the assumption that the nucleon-nucleus interaction proceeds through a single s-wave separable interaction may not be removed at present since, even with the


I I Ed = 6.67 MeV I I l B = 4.45 WV

I I I I Born Approx. \ \

I

140

9 CM

Ed = 1.76 MeV

l B 1 4.45 t&V

I I Ed = 20.02 MeV

I eB = 4.45 MeV

- \

I I Born Approx.

I I 0 20 40 60 80 100 120

9 CM

0

FIG. 13. Stripping angular distributions given by our exact theory (-) and by the Born approximation (- - -) for EB = 1.0 (4.45 MeV) and various deuteron kinetic energies Ed ,fl~ = 4.0.

above degree of generality, the available (IBM-7094) computational facilities are severely taxed.

The previously used parameterization of the B-n-A vertex in terms of the Hulthen wavefunction was shown to have drawbacks. In related calculations in nuclear physics such as the DWBA, the Woods-Saxon potential is used almost exclusively to represent the nucleon-nucleus interaction. A separable potential of this type was considered as a possibility in the present calculations, but was rejected because excessive computing time would be required to obtain our amplitudes. The reason is that to solve our integral equations accurately above the deuteron breakup threshold the path of integration must be deformed into the complex momentum plane, thereby requiring the wavefunction for complex values of the momentum. The analytic continuation can be carried out by per-

THREE-BODY APPROACH TO NUCLEAR REACTIONS 3%

forming a Fourier transformation of the configuration-space wavefunction, but tuis procedure is very time consuming. To reduce the computing time, an interaction should be chosen that yields an analytic expression for the momentum- space wavefunction. Therefore, the simple square-well potential was adopted; that is, we chose a separable potential that gives the same momentum-space

200

150

50

C

I

I \ Ed = 6.67 MeV

I

I “0 = 8.90 MeV

I

20 40 60 SO 100 120

Ed = 1.78 MeV

l B = 8.90 MeV

\

\

I Ed = 20.02 MeV

I

I f0

= 8.90 MeV

FIG. 14. Stripping angular distributions given by our exact theory (-) and by the Born approximation (- - -) for eg = 2.0 (8.90 MeV) and various deuteron kinetic energies Ea ;Be = 4.0.


bound-state wavefunction as the local square-well potential. In configuration space we have for square well.

V(r) = -VII, l/L(r) = ys@$ r < r,,

V(r) = 0, l/b(r) = sq, r > r,,

where y and 6 are normalization constants, and (2m = fi2 = 1)

aL3 = MEEI(l + A)P2,

p = [ A(Vo - Q> 1 l’z

1+24 ’ (3.2)

r, = 1.3P3,

where A is the mass number of nucleus A. The corresponding wavefunction in momentum space is

4 ck) _ 4v B

sin (k - b>m _ sin (k + a>~% k [ m - P> 2@ + P)

(3.3) sin pr,

+- k2 + ag2

(ai3 sin kr, + k cos km)].

To fit a particular nucleus, the binding energy Ed and the nuclear radius r, are fixed and Vo or equivalently /3 is obtained by solving the transcendental equation

tan pm = -p/a, , (3.4)

which ensures the continuity of the logarithmic derivative of the wavefunction at the nuclear radius. The normalization constant y is given by

-112

y = 4ir [ ( sin 2@r, r,/2 - ___

4P + sin” fir, ~

2aB >I - (3.5)

To ensure that the nonlocal separable potential has the same bound-state wavefunction as the local square-well potential, the vertex function must have the form

Fs(k) = [(I + A)lAk2 + ‘B] ,J rE

(k) B * (3.6)

Defining the coupling constant rB as

rE = 4?r/[(l + AI/Al, (3.7)


we obtain the vertex function

\ (3.8) + sin pr,(ff, sin kr, + k cos kr,) .

This is the final form of the vertex function used in the calculation. It has the advantage of having a simple analytic form land allows parameterization in terms of a radius parameter r,, and the binding energy EB . Also the number of nodes of

the function may be varied by taking appropriate solution of Eq. (3.4).

B. INCLUSION OF SPIN AND ISOSPIN

In this section we show how to include spin, isospin, and a proton-core nucleus interaction in our model. We retain the assumption of s-wave nucleon-nucleus interaction and assume charge independence. The theory now includes the three processesclt,n+p,B~n+A,andCttp+AsothatthenucleiBandC are isobaric analogs of each other. The deuteron has spin 1 and we assume the core nucleus ,4 to have spin and isospin zero; therefore, their field operators obey canonical commutation relations. All other particles have spin s and their field operators obey anticommutation relations. Our interaction Hamiltonian now becomes

(3.9)

.(.!5 - ?4 1 ‘W4 - ?4 Bm-udk + k’>Nt,-dk’>A;t;o(k)

.Cn,dq + q’>N:,uz(q’)A&dq) + h. c.

The bracket (JM 1 jlmlj2m2) is the Clebsch--Gordon coefficient3 for adding J = j1 + j2 and M = ml + m2 ; the first bracket in each term refers to spin and the second to isospin. The capital letters refer to field operators for the particular particle (D for deuteron, etc.); i.e., Nz, - l/2( k + k’) is the creation operator for a nucleon (neutron) with spin (isospin)-projection quantum numbers m( - M)

3 See for example Rose, Reference (21).

3ss SHANLEY AND AARON

and momentum k + k’. The 1/2 in the d-n-p term is inserted to obtain the proper triplet n-p amplitude.

With the introduction of the C particle there are more reactions to consider in the three-body sector. If we just look at processes originating with an incoming p-B state, we have the four reactions

p+B+p+B,

p+B*n+C,

p+B*d+A, (3.10)

p+B-+n+p+A.

The charge-exchange reaction p + B + n + C did not occur in Dhe preceding spinless model since there was no p-A interaction. Including the breakup channels there are a total of sixteen processes to consider. However, the amplitudes for these are not independent, as we shall see later. To derive the integral equations for the system, consider an initial state which will consist of some pair d, B, or C scattering with the third particle and going to a final state which may or may not contain the same pair. From the spinless analysis we expect the form of the amplitude to be

T = B + BrT, (3.11)

where B and T are the Born and propagator functions, respectively. We shall assume L-S coupling. To specify a state we must, for example, give the overall spin S and isospin I of the system and the projections S, and I, , and also give the spin and isospin of the composite pair 2, T with corresponding projections c and 7, and finally a center-of-mass momentum vector k. The state vector so specified is written

1 X, S, ; I, I, ; I;, a; T, 7; k). (3.12)

Since there are no spin-orbit couplings present, S, S, , I, and I, are conserved in any scattering and these are denoted by n for convenience of notation. Then taking matrix elements of Eq. (3.11) and inserting a complete set of states, we obtain the set of equations

(n; 2, a; T, 7; k 1 T(E) 1 n; Z’, a’; T’, 7’; k’)

= (n; 2, u; T, 7; k 1 B(E) / n; Z’, a’; T’, 7’; k’)(l - 62x/)

+ c (n; 2, u; T, T; k 1 B(E) 1 n; Xv, cr”; T”, 7’; k”)Tx*(lc”) (3.13)

X (n; z”, an; T”, 7”; k” I T(E) I n; 2’, 0’; T’, 7’; k’),

where the inhomogeneous Born term vanishes for elastic scattering since it is not


possible with three nonidentical particles to obtain an elastic process with single- particle exchange. The sum represents a summation over (Z”, a’ ; TN, 7”)) an integration over k” and gives the coupled integral equations we must solve. Since there are three choices for the double-primed quantum numbers, corresponding to d-A, p--B, and n-C, we obtain three sets of three coupled int.egral equations. This degree of complexity may be reduced somewhat by assuming that the nucleon-core interaction is charge independent. We illustrate this simplification by writing the above integral equations in abbreviated form, suppressing all spin and isospin labels and denoting the channels d-A, p-B, and n-C by 1, 2, 3,

Ti, = Bij( 1 - S,j) + 2 BikTkTkj) k=l (3.14)

i, j = 1, 3.

IIence there are a total of nine amplitudes to be considered. The reciprocity of the T-matrix (22) requires that, on the energy shell, the off-diagonal elements obey the relations Tij = Tji . This leaves 6 independent amplitudes, but the imposition of charge independence implies that T12( cZA + pB) = T13( dA + nC) and that Tzs(pB + pB) = T,(nC + nC). Thus there are only 4 independent amplitudes and we have the relations

Tzz = Ts , (3.15)

Tsz 7’2s .

We can take the independent amplitudes to be

Tn(dA -+ dA), Tzz(pB -j PB),

TddA -+ pB), T&B * d’) (3.16)

and all others may he obtained from these using (3.15). The set of coupled integral equations (3.14) simplify considerably by assuming charge independence, since we require es = cc , yB = 7~ , andfB(IC) = fc(k). It then follows that Q = 73 and B12 = RIP and hence the three equations with j = 1 reduce to a pair of equations involving only T11 and T~I . The set of three equations for j = 2 can be reduced to a single equation involving (T32 - Tz2) and a Iair of equations coupling Tlz and ( Tz2 + Ts2). The remaining equations with j = 3 are redundant and need not be considered. Thus we have five integral equations to solve with no more than two amplitudes coupled together at one time.

We now return to the full equations with spin and isospin indices, but employ- ing the simplifications indicated in (3.15). We have written the integral Eqs. (3.13) in a representation in which the total spin and isospin and their projec-


tions are diagonal. However, to carry out the explicit calculation of the Born functions with the interaction Hamiltonian (3.9), a representation in which the spin, isospin, and projections of the individual particles are specified is most convenient. The connection between the representations can be seen by writing the Born term

(S, S, ; I, I, ; 2, a; T, 7; k ( B(E) 1 S, S, ; I, I, ; Z’, a’; T’, 7’; k’)

= c (SS, 1 Za/wz)(IIz 1 T~iiz)(XSz ) 2’0’/.&‘)(11s ) T’/i’&‘) (3.17)

where CL, pZ ; i, i, , and B’, pZ’; i’, i,’ refer to the spin and isospin of the uncorre- lated particle in the initial and final state, respectively, and the sum is over all projection quantum numbers. The matrix elements on the right of Eq. (3.17) are now calculated with the interaction Hamiltonian (3.9) and transformed using (3.17). There are only two independent Born functions, one for stripping (d + A + p + B) and one for the charge exchange process (p + B + n + C), and suppressing spin labels we obtain

&(k’ k. E) = YeYd fd k/2 - k’ I)fs(l M/(1 + Nlk - k I) 7 7

ti E - (k - k’), - h+/A - k’2 (3 Is)

*

Bk@‘, k El = $fdl k’ + k/(1 + A) I)fs(l k + k’/(l + A) I)

.

C

E _ (k + ,'I2 _ k2 _ km 1 -' (3.19)

A >

where as before, A is mass number of the core nucleus. Note that in the static limit (A 4 co ) , the charge-exchange amplitude loses all angular dependence and hence will affect only s-waves. For all but the very lightest nuclei, the angular dependence of this amplitude turns out to be quite weak and only the very low partial waves contribute significantly. In field-theoretic language we would say that a massive particle being exchanged gives rise to very short-range force.

The partial-wave analysis of the Born functions and of the integral equations proceeds as in the spinless case and yields linear one-dimensional integral equations for each partial wave. The definition of the S-functions is that given by Eq. (2.8).

C. RESULTS

The numerical solution of the integral equations is carried out using the procedures discussed in Sec. II. To compare the results of the calculation to experiment we want to consider some fairly light closed-shell nucleus, so that Coulomb effects are minimized, and also there must be a strongly populated s-state in the


residual nucleus following a stripping reaction. For these reasons we have chosen I60 to be the core so that we will be studying specifically the reaction d + 160 --+ p + “O* (0.871-MeV level in “0). The shell model puts the last neutron in 170* in a 2s level with a binding energy of 3.273 MeV. To fit these data with the separable square-well potential Equation (3.8), we give the vertex function one node and choose the depth of Vo = 53.8 MeV.

The corresponding three-body bound states, if any, will be those of lsF. The experimental energy levels of the various relevant nuclei are given in Table I relative to 160. In this region of the periodic table, the shell model is filling the 2s and Id shells. The ground and second-excited state of “0 are Id levels and their effect neglected in the present calculation. The first-excited state is the 2s level that we are describing with the separable potential. The many negative parity levels of 170 have been left out of the table since they are due either to core excitation or to raising of the last neutron to the next p-state. Also the excited levels of 18F are not shown. The three-body calculation gives two s-wave bound states in ‘*F and none in higher partial waves. The binding energies relative to 160 are 4.15 and 4.50 MeV which are much smaller than the experimental values for 16F. That we obtain too little binding could have been anticipated from the shell-model studies of Elliot and Flowers (d3) and more recently of Inoue et al. (24). Our 2s separable potential produces an 18F nucleus with the configuration (s&” and neglects all configuration mixing due to the Id shell which the shell- model studies have shown is very important. It would be interesting to study the effect of including the Id level of “0 into the theory and to compare the resulting 18F energy levels with the shell-model calculations. One would need two addi- dional separable interactions that could be parametrized to fit the dg12 ground state and the dzi2 excited state of 170. From the point of view of computer space, the resulting four-channel problem could be easily handled since for bound states the kernels are real and sufficient storage space is available.

In the scattering domain, the numerical solution of the integral equations is

TABLE I

EXPERIMENTAL ENERGY LEVELS

Nucleus J’, T Binding Energy (MeV)

160 o+, 0 0

170* %‘+, w -0.94 170* Pi+, w 3.27

“0 x+, 35 4.14

‘SF 1+, 0 9.74


carried out using the procedures outlined in Sec. II and Appendix A and gives the amplitudes for the four independent processes of the model,

d+A+d+A,

d+A+p+B,

p+B+p+B, (3.20)

with all other amplitudes (excluding breakup amplitudes) obtainable by invok- ing time-reversal invariance or charge independence. We have constrained the neutron and proton always to be in a relative spin-triplet, isospin-singlet state, so that for the last two processes in (3.20) we only calculate the contribution per- taining to this set of relative quantum numbers. In Table II the thresholds for the various reactions for incident p-B and d-A states are given. The thresholds and entire kinematics for the n + C reactions are identical to those for p + B and we have assumed tg > ed . Less extensive calculations than those presented in Sec. II for the spinless case have been carried out for the present model with spin since considerably more computing time is required. For example, to calculate an angular distribution requiring six partial waves for each of the processes listed in (3.20) requires approximately forty minutes of IBM-7094 time, or about four times that necessary for the corresponding spinless calculation. Most of this additional time is needed to calculate the integrals used in defining the partial wave Born and the X, functions numerically; these integrals could be done analytically for the Hulthen vertex.

Phase shifts angular distributions, etc. can be assembled as outlined in Sec. II. The differential cross sections, after summing partial waves, are obtained by

TABLE II REACTION THRESHOLDS

Reactions Threshold

d+A+d+A d+A+p+B d+A+n+C d+A-+p+n+A

E= -sB -eB

- Ed

0

E= --a -u - Cd

0


averaging over initial and summing over final spins to obtain

da ccl Pf kf 1 -=--- cm c

(2?r)2 ki (28 + 1) (2/J + 1) 0.P. O’CP

(3.21)

x I(% a; PJ Lcz , . T, 7; i, i, ; ki ( T(E) 1 Z’, 0’; /.L’, pi; T’, 7’; i’, iz’; kf)j2,

where the spin notation is that of Eq. (3.7). The T-matrices obtained from the solution of the integral equations (3.13) are not in the above representation, but in one in which the overall spin and isospin are diagonal. The necessary change of representation can be obtained by inverting the equivalent of Eq. (3.1’7) for the T-matrix.

The real parts of the d-A and p-B phase shifts for s and p-waves are shown in Fig. 15. Roth the p-B and d-A phase shifts are complex over the entire energy range since inelastic channels are open even at the elastic threshold. Since there are two three-body bound states, we start the s-wave phase shifts at 2a; however, they approach ?r rather than z,ero at high energy. This behavior is discussed in the next section. Nothing very dramatic occurs in the d-A channel and in the p-B channel there is strong damping at low energies which evidently prevents the

,iP,dl

+ .

P=I -

‘\\

-(d,d) (d,P) C&II)

-lP*Pnl

- -

P=O

-\ -\,a=0

‘. ---

. . --. +=I

\ ‘---

-(d.ml

I I I ) 0.5 1.0 1.5 2

E

Real Elastic Phase Shifts

I====-- --- ,,+A d+A -

- p+B - p+B

---_ ---------

I I I I 2.0 4.0 6.0 6.0 10.0 .O

FIG. 15. Real part of the p-B and d-A phase shifts versus energy E for various 1. The reaction thresholds are indicated (Units: h = 2m = 1, l d = 0.5).


O-20 2.0 4.0 6.0 8.0 10.0 12.0 E

FIG. 16. Inelasticity parameter TLZ = exp (-2 Im 6~) in the d-A and p-B channels as a function of E for various 1 (Units: h = 2m = 1, Q = 0.5).

occurrence of the threshold peaks seen in the spinless model at pickup threshold. This is evident in Fig. 16 which displays the corresponding inelasticity parameters r]l. Both s and p-wave inelasticity parameters become quite small in the p-B system; the minimum in vl for I = 0 is apparently related to the return of the real phase shift back through 2~. There is more absorption present in this model than in the spinless model previously considered. This is evident in Fig. 17 which shows our d-l”0 elastic scattering angular distributions for two energies. The 15.0-MeV curve exhibits diffraction oscillations characteristic of experimental deuteron angular distributions. The positions of the second and third maxima are in qualitative agreement with the recent 12-MeV data of Alty et al. (25) although detailed comparison is not possible since Coulonb effects have not been included in our calculation. Angular distributions for p-B elastic scattering are shown in Fig. 18 and they have less structure than the Cl-A scattering results. Our B nucleus is “O* and of course there is no experimental data for this process, nor is there any for the charge-exchange reactions (p + B * 1~ + C) shown in Fig. 19. There is little resemblance between our results for these two processes and the general trend of experimental results for light nuclei.4 We obtain less diffraction structure than the experiments exhibit presumably because our simple reaction mechanisms do not provide enough absorption to reproduce these features. As previously discussed, the charge-exchange Born term, which dia- grammatically is represented by core exchange between pB and nC, has little angular dependence since it has only significant contributions from s and p waves. The forward peaking present in the (p, n) angular distribution arises, not from this Born term, but from impulse-approximation-type diagrams in which

4 For (p, p) and (p, n) reactions see, for example, References (96) and ($?7), respectively.


d + 016 - d+016

NOTE TWO SCALES

I I I I I I I I I I 0 20 40 60 SO 100 120 140 160 180

8 CM

39.5

FIG. 17. Elastic d-A angular distributions for two energies.

there is double nucleon exchange and their iterations. The amplitude for the (p, p) process is dominated by the same type of diagram and hence the angular distributions for these processes are quite similar, particularly at higher energies.

D. STRIPPING

Stripping angular distributions are calculated and compared with experiment at deuteron energies of 4.41 MeV (5’8) and 15.0 MeV (29) in Figs. 20 and 21. Although agreement is poor at larger angles, fairly close agreement with experiment is obtained in the region of the forward peak. We believe that this agreement is not accidental and shall discuss its significance below. We first should note, however, that the large angle part of the angular distribution represents


I I , I , I I I I

0 20 40 60 00 100 120 140 160 I

9 CM

FIQ. 18. Elastic p-B angular distributions for three energies.

0

collisions involving large momentum transfers which are sensitive to the precise form of the nucleon-nucleus interaction and to the amount of absorption present in the low partial waves. Since we have put into the theory only enough interaction to cause a few simple reactions to occur, we cannot expect to reproduce these high angle details of the stripping angular distributions. The forward peak which we have fit fairly well gets its main contribution from the higher partial waves which are not as sensitive to the details of the interaction. Finally, note that at 15.0 MeV the plane-wave Born approximation itself fits the data well in the forward direction.

We attempt to explain the above phenomena by modifying an argument orig- inally proposed by Amado (SO) to explain the general forward peaked nature of


p+B -n+C

- Exact Results

I I , I I I I I 0 20 40 60 60 100 120 140 160

8 CM

FIG. 19. Charge-exchange angular distribution for two energies. Also shown is the Born approximation.

stripping angular distributions. He suggested that, in the sense of dispersion theory, the peaked distributions were due to the dominance of a nearby single- particle-exchange pole in the momentum-transfer variable (the pole being con- tributed by the Born term)--this idea also offered the possibility of obtaining reduced widths and spectroscopic factors by extrapolation to the pole. It has since been realized that aside from explaining qualitatively the forward peaking, pole dominance alone is not a useful concept in the study of stripping reactions since in general distortion (deviation from the Born approximation) prevents an accurate extrapolation even when one is ostensibly “near the pole” (31). Our claim is that the major part of this distortion comes from the requirements of


three-body unitarity (as discussed in Sec. II.G, the s-wave contribution of the Born term violates unitarity by roughly an order of magnitude) and is independent of the detailed dynamics of the reaction. In other words, we suggest that the forward part of the stripping distribution is given by single-particle exchange

100 -

Exact Theory

Experiment

Born Approx.

I I I

I I :

: : I 1

::

! !

-

0 20 40 60 80 100 120 140 160 180

8 CM

FIG. 20. Deuteron stripping angular distributions at Ed = 4.41 MeV as given by our exact, theory (--), experiment (- - -), and the plane-wave Born approximation(...).


plus three-body unitarity, and as we have explained earlier, solving the three-body Schrodinger equation with separable potentials incorporates just the above in- gredients. Taken seriously, the results of Figs. 20 and 21 would imply that the spectroscopic factor of the 2s state in “0 is the order of unity. We avoid quantitative statements however, since Coulomb distortion should be taken into account.

d+o'6 >p+o'7 0.871 MeV Level Ed = 15.0 MeV

100

\

\

\

\

!

- Exact Theory -- Experiment

---- Born Approx.

1.0 -

0 LO 40 60 80 100

8 CM

120 140 160 180

FIG. 21. Deuteron stripping angular distributions at Ed = 15.0 MeV as given by our exact theory (-), experiment (- - -), and the plane-wave Born approximation (. . .).


Even assuming success, the above calculation must still be taken with some skepticism since we have obtained results only in a particular simple reaction- one in which the spectroscopic factor is probably close to unity and as we have seen, one in which the plane-wave Born approximation itself is fairly successful. At 15.0 MeV, it certainly does as well in the forward direction as our full theory in describing experiment. The real test of our theory would be to obtain spectroscopic factors for a large number of stripping reactions in a variety of nuclei-we are presently attempting such a program. Particularly interesting reactions to study in this respect would be endothermic reactions in which the momentum transfer pole is as close as possible to the physical region (82).

E. LEVINSON'S THEOREM

We now discuss the behavior of the real parts of the phase shifts shown in Figs. 5, 6, and 15. In two-body-potential theory, Levinson’s theorem (18) states that the difference between the elastic phase shift between zero and infinite energy is a multiple of ?r, given by

W) - s(m) = NT, (3.22)

where N is the number of bound states in the particular partial wave. For systems of many coupled two-body channels a generalized Levinson’s theorem has been obtained 18), (88) of the form

F MO) - M tfr >I = n7r, (3.23)

where 6, is the eigenphase shift in the nth channel and N is again the number of bound states. A similar result has been obtained for three-body systems by Wright, (34) where there is the additional problem that the S-matrix has dis- connected parts and thus a continuous spectrum. It is not possible to apply these results depending on the eigenphase shifts in the present problem since we have not calculated the amplitudes for all reactions in the three-body sector (i.e., we do not know the amplitude for elastic scattering involving three free particles in initial and final states) and hence the eigenphases cannot be determined. What we have calculated and what experimentalists usually obtain are the complex phase shifts for bound-state scattering and any discussion of Levinson’s theorem should involve these quantities.

A generalization of Levinson’s theorem valid when inelastic channels are open and involving the real part of the phase shift has recently been given by Hartle and Jones ($5) who find

Rea (threshold) - Re6(m) = ?r(N - N1), (3.24)

where, as usual, N is the number of bound states and Nr is the number of inelastic resonances. These inelastic resonances are connected with the presence of


zeros and the S-matrix on the physical sheet. Bander, Coulter, and Shaw (36) and others (37) have studied a soluble two-channel model and have shown that if the coupling constant in Channel 2, for example, is increased beyond a critical point, a zero of the Channel 1 S-matrix moves through the inelastic cut onto the physical sheet. Since the usual proof of Levinson’s theorem gives the difference in phase at threshold and infinite energy as the number of poles minus the number of zeros of S, such a zero gives a contribution of --a.

In the present calculations including spin both the p-B and d-A s-wave phase shifts change by A between threshold and the maximum energy calculated although there are two s-wave bound states. This is the behavior one would expect if a zero of the S-matrix for these processes migrated through one of the inelastic cuts onto the physical sheet Figure 16 shows that both of these partial waves are quite inelastic, particularly the one in the p-B channel. Since we have the relation ~~12 = St * SI, a small value of ql is consistent with there being a nearby zero of Sl on the physical sheet. In both the calculations, with and without spin, the p-wave d-A phase shift seems to change by ?r between threshold and infinite energy even though there are no p-wave bound states. There seems to be no simple explanation for this behavior in terms of the above generaliza- tions of Levinson’s theorem, but it is probably related to the strong absorption present i.n these partial waves. This phenomenon could be studied further, for example, by searching for zeros of the S-matrix off the physical cut; however’ this is probably best studied in simpler models than the present one. The question of whether such phenomena occur in actual nuclear scattering must await the availability of detailed phase-shift analyses for an answer, although the answer is almost certainly in the affirmative,

In our model the effective potentials in the bound-state scattering channels are highly energy-dependent and nonlocal and it is not surprising that deviations occur from the various forms of Levinson’s theorem discussed above.5 The previous discussion does indicate, however, that in general physical systems the behavior of phase shifts may be quite unpredictable and consequently that the more naive forms of Levinson’s theorem may not be valid in the analysis of such syst#ems.

IV. APPROXIMATION METHODS

A. INTRODUCTION

In the previous two sections we have seen that our three-body model, at least as presently formulated, does not give quantitative fits to experimental data. Further possible improvements are discussed in the final section. In the present

6 For nonlocal interactions, deviations from the usual forms of Levinson’s theorem have been previously noted (38).


discussion, however, we assume that the model is sufficiently realistic to serve as a testing ground for various approximation methods currently used in nuclear as well as elementary particle physics. In particular, we consider the distorted- wave Born approximation (DWBA) used extensively in direct nuclear reactions, and certain diffraction models developed recently to partially unitarize single- particle exchange amplitudes in high-energy physics. The two methods6 have the same starting point, a Born term which is a poor approximation to the full amplitude because it neglects the effects of distortion and of competing channels and for the low partial waves usually violates the limits imposed by unitarity.’ In the DWBA, one retains the interaction to first order, but replaces the entrance and exit-channel plane waves by distorted waves generated by optical potentials which reproduce the elastic scattering. In the high-energy diffraction models the Born amplitude is multiplied by factors depending on elastic S-matrices in the incident and outgoing channels. The major effect of both techniques is a reduction of the contribution from the low partial waves which results in considerable improvement over the simple Born approximation.

We use the spinless model of Sec. II for testing; that is, we take the results of those calculations in all channels as “experimental data” and see to what extent the above approximation methods can reproduce the exact answers. In direct nuclear reactions there seems to be no previous calculations of this sort to test the DWBA due to the lack of a sufficiently realistic soluble model, at least for reactions which are of a three-body natures such as deuteron stripping, where actual rearrangement occurs.

Recently a three-body model similar to the present one has been proposed by Mitra, (10) who also formulates a distorted-wave approximation within the context of his model. Since the present calculations seem to shed some light upon Mitra’s model it will also be discussed in the present section. The high energy diffraction models have been tested recently by Wills, Ellis, and Lichtenberg (43) in a calculation simulating pion-nucleon scattering and reactions by solving, coupled Schrbdinger equations with as many as four channels included, and also by Tobocman and Giltinan (44) who takes over the distorted-wave method of direct nuclear reactions to treat one-meson exchange processes at high energy.

13. DISTORTED-WAVE BORN APPROXIMATION

We first treat the DWBA as an approximation to our known stripping amplitude. Recall that our model of Sec. II deals with the following spinless channels:

6 For a discussion of the DWBA see, for example, Tobocman ($9) and Austern (40). The diffraction models are treated by Durand and Chiu (41) who also give references to other

work. 7 The Born term, of course, intrinsically violates unitarity because it is real. * In a two-body scattering context, a comparison of the DWBA with the more exact

coupled-channels method has been carried out (42) for inelastic scattering involving excitation of collective nuclear states.


d+A+d+A, P+B+p+B,

d+A+p+B, p+B+d+A, (4.1)

d+A+p+n+A, p+B-+p+n+A,

and that when the potentials between pairs are separable in momentum space the results are exact and unitary in all channels. We now want to use these results in an attempt to study and gain insight into the DWBA. This approximation treats the n-p potential to first order but differs from the ordinary Born approximation in that the interaction is “sandwiched” between distorted waves chosen to fit the elastic data at the same energy, and hence includes rescattering corrections to the inelastic amplitude in initial and final states. In practice the distorted waves are generated by optical potentials in the elastic channels and all inelastic processes, including the one in question, are assumed to produce only a damping of the incident and outgoing waves. This damping is described by the imaginary part of the potential. The form of the distorted wave matrix element has been discussed by many authors (45) and is given by

T DWBA = J x;-‘(k,, r,)*(B 1 v,, 1 A) d+YJcd, rd) d3r, d3f-d, (4.2)

where x(lc, r) are distorted waves for the scattering of a pair of particles with relative momentum Ic and separation r. The remaining factor in the integrand is a matrix element of the proton-neutron interactions integrated over all coordinates independent of r, and rd and contains the deuteron and nucleus B wave- functions. The zero-range approximation (V,, = 6(r, - rP)) is used in the evaluation of Eq. (4.2) and Coulomb forces are omitted since there are none in the model. To generate the distorted waves the elastic d-A and p-B angular distributions calculated by numerical solution of the three-body equations are used as input in an automatic search code (46) which varies parameters in an optical potential such as to minimize the quantity x2, defined as

x2 = ; -$ [uth(e;lu- gPt(ei)l’) % 1 exp

(4.3)

where N is the number of angular points & , oth anduexpt are the theoretical and experimental differential cross sections, and Auexpt is the “error” in the experi- m.ental cross section taken to be 0.5%. The optical potential employed is of the Woods-Saxon type

V(r) = - (V+ilV)[1+exp(r-:A1’3)~.

The param.eters V, W, r. , and a are continuously varied until a minimum in x2 is found. The final entrance and exit channel parameters are shown in Table


TABLE III

OPTICAL MODEL PARAMETERS

Proton Paramefers E -% (MeV) m(F) a(F)

V (MeV) W (MeV)

-0.1 8.45 170 13.2 .002 .757

0.5 12.12 225 27.3 - .025 .789

1.0 13.35 333 47.7 - .063 ,802

2.0 17.80 143 27.4 .019 .736

E Ed (MeV)

-0.1 1.78

0.5 4.45

1.0 6.67 2.0 11.12

Deuteron Parameters

V (Mev) W (MeV)

260 88.1

326 45.4

277 51.5 573 87.3

n(F) a(F)

.263 .145

.239 .075

.218 .098

.170 .141

Reaction Cross Section

E

-0.1 0.5 1.0

2.0

Exact WA 6-h)

206

204 191 205

%&b)

87 152

184 202

Optical Model CdA bb) %Bhb)

354 143

201 176

206 172 202 157

III for four energies, with the binding energy of nucleus B set at 8.90 MeV. These solutions are the ones with the lowest x2 for the energies searched and also provide the best fit to the known elastic phase shifts. The parameters of Table III differ considerably from those usually found in the analysis of real nuclei (47). The wells are deeper and the radius parameter r. is quite small; r. being even negative in some cases. Presumably shallower wells with larger radius parameters could have been found giving equally good fits to the data since there is usually a strong correlation between these parameters. Notice also that the diffuseness parameter a is very small in the d-A channel, leading to a very square optical potential. This squareness is apparently necessary to provide the generally negative phase shifts found in this channel. Fig. 22 shows the quality of the fits in each channel for two energies. The agreement is seen to be quite good in both channels. Less success is obtained in reproducing the elastic phase shifts. Figs. 23 and 24 compare the real phase shifts and inelasticity para.meters ql( = exp ( -2 Im 61)) as a function of energy. The fits are superior in the deuteron channel, particularly at higher energies.


250 p- 8 Elastic Scattering

- Optical Model --- Exact Calc.

250

i

d-A Elastic Scattering

- Optical Modei

--- Exact Calc.

0

FIG. 22. Elastic d-A and p-B angular distributions. Comparison of best optical model fits and exact results for two energies (Units: h = 2m = 1, Q = 0.5).

We now turn to the stripping amplitude. This is calculated by evaluating (4.2) using the distorted waves generated by the best, fit parameters of Table III. The code JULIE (48) was used for this purpose. The resulting DWBA angular distributions are compared with the exact results in Figs 25-27 for three energies. The general forward peaking of the exact, result is obtained but neither the magnitude of the cross section nor the angular dependence in the backward hemisphere are accurately reproduced. The DWBA has a second diffraction minimum even at very low energies which does not occur in the model calculations. The closest. agreement occurs at the highest energy where the magnitude and shape of the forward peak and the position of the first minimum are fairly close to the model calculation.


- Optical Model

--- Exact Calc.

\ d -A Channel

0.6-

FIG. 23. Real phase shifts and inelasticity parameters in d-A channel versus energy E. Comparison of exact results and best optical model fits for various I (Units: h = 2m = 1,

B,j = 0.5).

At first glance, the ability of the optical model and distorted-wave theory to reproduce the model stripping results is rather disappointing since it is usually more successful when applied to real nuclear stripping reactions. On the other hand it is somewhat remarkable that the DWBA works as well as it does (gives the major features of the cross section) since all processes--elastic scattering, stripping, and breakup-take place through intermediate stripping. Supposedly a necessary condition for the validity of any distorted-wave treatment of inelas-


lr 1

- Optical Model

___ Exact Calc.

p-8 Channel

407

FIG. 24. Real phase shifts and inelasticity parameters in p-B channel versus energy E. Comparison of exact results and best optical model fits for various 1 (Units: h = 2na = 1,

Bd = 0.5).

tic process is that it be weakly coupled to the elastic channel. In actual deuteron- nucleus systems this condition is satisfied by the existence of many competing inelastic channels including stripping to various levels as well as other deuteron- induced reactions such that stripping to the particular level in question does not dominate the total inelasticity. The existence of many open inelastic channels also aids in the fitting of the deuteron and proton optical potent,ials, since this fitting is usually more successful if these channels are strongly absorbing. In the model problem there are only two inelastic channels present for incident deu-


IOC

IO

t

1; E

.r

c: -0 . b

u

0.1

17 \

\

\

cl+A - p+B

1 Exact Cole.

\ m - m DWBA

‘\ \ \ \ \ \ \ \

I(

I /- I ’ ‘\ I \ / /- \ / I \ -k I I ’ I 1 1 / ! 1 : YI E = -0.1

Eg = 0.90 MeV

Ed = 1.78 MeV

0 20 40 60 120 80 100

8 CM

140 160 0

FIG 25. Comparison of exact deuteron stripping angular distribution with that of the distorted-wave Born approximation for Ed = 1.78 MeV.

terons-stripping to one particular level and deuteron breakup. We are planning further study of the DWBA in the context of our model; for example, we would like to compare optical potential chosen by the search code with our exact optical potential. Also we might examine the validity of certain approximation procedures such as the zero range approximation. The above results, though preliminary, show the types of questions that can be asked and answered in a program of this nature.

It seems appropriate here to discuss a model of stripping reactions proposed


1000

100

I

-

\

\

0

d+A-p+B

- Exact Colt. --- DWBA

E = 0.5

Es = 8.90 MeV

Ed= 4.45 MeV

I I I 1 I I 1 1 20 40 60 80 100 120 140 160 0

9 CM FIG. 26. Comparison of exact deuteron stripping angular distribution with that of the

distotred-wave Born approximation for Ed = 4.45 MeV.

by A. N. Mitra (10). He considers two spinless identical nucleons interacting with a spinless static core and with themselves through s-wave separable potentials. The model is intermediate in complexity between our spinless model neglecting p-A interaction and our model with spin discussed in Sec. III. Mitra works in a Schrijdinger formalism which allows him to examine the structure of his equations and, by neglecting certain terms, to write down a distorted-wave approximation directly within the context of his model. Without numerical computations, but based on his experiences with the three-nucleon system, he


Exact Calc.

--- QWBA

E = 2.0

E B = 8.90 MeV

Ed = II.1 MeV

IO

FIG. 27. Comparison of exact deuteron stripping angular distribution with that of the distorted-wave Born approximation for Ed = 11.1 MeV.

conjectures that the stripping amplitude is an order of magnitude larger than the elastic scattering amplitude and then concludes that his distorted-wave approximation is a valid approximation to the exact stripping amplitude. In the explicit calculations carried out here we find that the elastic amplitude generally exceeds the stripping amplitude, at least for the dominant low partial waves. This result is independent of the specific details of the model being considered such as whether spin, recoil, or a pA interaction are included. It is obvious that in Mitra’s model the relative size of the elastic and stripping amplitudes will be


similar to what we find and the Mitra’s distorted-wave approximation is not valid in general.

C. HIGH-ENERGY DIFFRACTION MODELS

The other types of approximation considered in this discussion are some high energy models recently developed to treat elementary particles reactions. At energies of the order of 1 BeV, many elementary-particle scattering processes develop distinctly forward-peaked angular distributions. There has been some success in fitting these angular distributions with the so-called peripheral model where one assumes that the amplitude for the particular process is dominated by single-particle exchange diagrams, with the lowest mass particle being exchanged. This would give the longest range part of the interaction and the required forward peaking of the angular distribution. The experimental angular distributions are often smaller in magnitude and are more forward peaked than the predictions of such single-particle exchange models. This situation has led to many attempts to introduce inelastic effects into the peripheral model. The most widely used model was introduced by Sopkovich (49) who gives the following prescription for calculating a Born amplitude modified by initial- and final-state absorption

T2($)2 = S( iy’2Bfj( &y, (4.5)

where Si2 and Sj’ are initial- and final-state elastic S-matrices, which are in principle available from experiment, and B:j is unmodified Born amplitude. Durand and Chiu (41) have considered the theoretical justification for this expression and find that it should be valid for small angle scattering at high energy if the range of the inelastic interaction is smaller than that of the elastic interaction. For inelastic interactions of long range compared to the elastic channel, Durand and Chiu obtain

Tk:‘” = $$j[Si’Bfj + BfjSj”]. (4.6)

In both cases it is assumed that the energy is sufficiently high that the associated wavelength is smaller than the range of either the elastic or inelastic interaction. Either expression will reduce the contribution from the low partial waves if there is considerable absorption in the elastic channels. The validity of the above approximations has been shown only for the case of nonrelativistic potential scattering involving many coupled two-body channels; however, it is generally assumed that they might have a much greater range of validity.

We now propose to examine these approximations by using them to calculate the stripping amplitude and then to compare the results with the exact calculations of Sec. II. In direct analogy with the unmodified peripheral model, the stripping Born amplitude is given by a single-particle exchange diagram and is


E = 1.0

Es = 8.90 MeV

Ed = 6.67 MeV

too

t

1, E

.r IO

c

% -0

I

I I I I I t I I I I

0 20 40 60 SO 100 120 140 160 I

6 CM

0

FIG. 28. Comparison of the exact deuteron stripping angular distribution with various approximation methods for Ed = 6.67 MeV. The curve labeled Born is the plane-wave Born

approximation, T(l) is the Sopkovich formula, and 2’ c2) the Chiu-Durand approximation.

forward peaked at high energy. Not only is the width of the forward peak larger than that of the exact result, but also the Born cross section exceeds unitarity limits for the low partial waves. The aim of the present calculations is to see to what extent the two methods of partially unitarizing the Born approximation discussed above can cure these inadequacies of the plane-wave results.

The evaluation of the modified Born expressions are readily carried out using the known elastic phase shifts from Sec. II. Figures 28-30 show resulting strip-


d+A-p+B

E = 2.0

E B = 8.90 MeV

Ed= iI.1 MeV

\ \ \

PWBA

J&y/ / \ \ \ 6

\ fl-\ ‘1 a

6

Yll II \

-\ ‘\ i \ Exact

\ I I

I I I , I I I I I 0 20 40 60 80 100 120 140 160

8 CM

0

FIG. 29. Comparison of exact deuteron stripping angular distribution with various approximation methods for Ed = 11.1 MeV. DWBA means distorted-wave Born approximation. See caption to Fig. 28 for the meaning of other symbols.

ping angular distributions for deuteron laboratory energies of 6.67, 11.1, and 28.9 MeV. Included in the figures are the exact result from solving the three-body equations, the Sopkovich formula (denoted T(l)), the Chiu-Durand approximation (denoted !P2)), and the plane-wave Born approximation. For comparison, Fig. 29 also shows the DWBA result previously discussed. As expected, the uni- tarized cross sections have their magnitudes and widths considerably reduced from the plane-wave results. In general the Sopkovich formula gives better fits


iii a E

.c IO-

c T!

-2

I-

O 20 40 60 00 100 120 140 160 I

d+A-p+B

E = 6.0

Eg=8.90 MeV

Ed = 28.9 MeV

8 CM

FIQ. 30. Comparison of exact deuteron stripping angular distributions with various approximation methods for Ed = 28.9 MeV. See caption to Fig. 28 for meaning of symbols.

to exact results than the Chiu-Durand approximation, at least in the region of the forward peak. At large angles the cross section for Sopkovich’s formula is too high but since both approximations are only valid for small angles this is not too significant. It is a little surprising that the Chiu-Durand approximation gives inferior results since this method is supposedly valid if the range of the inelastic interaction exceeds that in the elastic channel. This is the situation that occurs in our model which can be seen by recalling that the stripping process may proceed through a single neutron exchange whereas in elastic scattering the lowest-


E =2.0

Es = 8.90 MeV

Ed’ 11.1 Me’?

, I I I 0 I 2 3 4

e

FIG. 31. Comparison of exact partial-wave stripping cross sections with those of various approximation methods as a function of 2 for Ed = 11.1 MeV.

order diagram involves two-neutron exchange. Hence the stripping is a longer range interaction. This is also evident from the fact that the stripping amplitude contains significant contributions from high partial waves which are negligible for the corresponding elastic amplitude (see Appendix B). To compare the relative cont’ributions from various I, the partial-wave cross sections are shown in Figs. 31 and 32. At 11.1 MeV, both methods still violate unitarity limits for s-waves although all low partial waves contribute considerably less than the Born term. Figure 33 shows the total stripping cross section versus energy. The energy range shown corresponds to deuteron kinetic energies from 2.2 to 37.8 MeV. At low energies the diffraction formulas show little improvement over the Born approximation since there is little absorption in the elastic channels. At


0 I 2 3 4 5 6

0

FIG. 32. Comparison of exact partial-wave stripping cross sections with those of various approximation methods as a function of I for .Ed = 28.9 MeV.

higher energies the diffraction-model predictions are reduced considerably from the Born result and are quite close, though somewhat smaller than the exact total cross section.

These calculations indicate that the diffraction models, particularly the Sop- kovich method, could be useful as a simple method of calculating small-angle scattering for nondiagonal processes in nuclear physics even at quite low energies.9 We speculate once again that the success of this approximation is related to the nearness of a pole in the momentum transfer variable. The partial uni- tarization provided by the Sopkovich procedure is sufficient to give the remarkable fits to the exact results in the region of the forward peak.

g In this connection see references (4) and (60).


d+A-p+B

Total Cross Section

I I I , I , I

01234567 E

FIG. 33. Comparison of exact total stripping cross section with that of various approximation methods as a function of energy E (Units: h = 2m = 1, ed = 0.5).

V. DISCUSSION AND CONCLUSIONS

In the preceding sections we have given a three-body formulation of certain nuclear rearrangement collisions and have compared the predictions of the model with experiment and with some approximate methods of solution. Since no more than qualitative agreement with experiment was obtained we will discuss in this section the limitations of the model and possible extensions of it that might make it more realistic. The major shortcomings of the model are the simplified two- body nucleon-nucleus interaction that we have used and the neglect of Coulomb forces. For most nuclei, the number of bound and resonant states is sufficiently large that it is not possible to include enough separable interactions to be realistic, and still be able to solve the equations. As we have seen, the use of an abbreviated nucleon-nucleus interaction leads to too little absorption in the elastic


channels and therefore such characteristic features as diffraction oscillat)ions in the elastic angular distributions are not obtained. For t’he case of l60 that we studied in Sec. III, it is clear that more reaction channels must’ be included if improved agreement with experiment is to be expected. This brings us to the question of how many bound or resonant two-body states it is necessary or feasible to include in the two-body nucleon-nucleus interaction. For Dhe nucleon- 160 compound system there are in excess of twenty states up to lo-MeV nucleon excitation and the inclusion of this number of separable terms is out of the question. It may not be necessary, however, to include all two-body bound st’ates and resonances in order to perform a realistic three-body calculation. In a stripping reaction involving l60 for example, only those reactions in which the neutron is placed in the single-particle d&j2 ground state, sl12 bound state, and cl,, resonance of 170 have appreciable stripping cross sections and possibly the inclusion of just the d states will improve the results of our stripping calculation considerably. In our l60 calculation in which only the s 112 interaction was included, we obtained fairly close agreement with experiment in the region of the forward peak but not at larger angles. It is expected that a nucleon-nucleus interaction that includes the d-states will improve the situation at high angles.

In addition to l60, another nucleus which would be of interest to study is 40Ca In both 160 and %a it is expected that the levels formed by adding an additional neutron to these closed shell nuclei would be fairly pure single-particle levels characterized by spectroscopic factors equal to unity. Recent distorted-wave studies (25)) (51) obtain values less than unity and it is not known whether this is a property of the nucleus or a shortcoming of the DWBA. Therefore, it would be valuable to perform calculations for these nuclei by methods ot’her than the DWBA to try to resolve this question of spectroscopic factors.

It does not seem worthwhile at present to consider heavier nuclei than those discussed above since the level density in such nuclei is too great for our separable approach to be applied with confidence. In the realm of lighter nuclei, the most promising system for study is the alpha particle-two nucleon system. As a core nucleus, the alpha particle is particularly attractive because of its large binding energy and also because nucleon-alpha scattering has been extensively studied and is dominated by only three partial waves up to about 10 MeV. Since the 1s shell is closed, the effective ~112 interaction is repulsive. The p-wave amplitude has a strong spin-orbit interaction and the low-energy scattering is dominated by a ~313 resonance and by a broad ~112 state. Including a separable potential to describe each of these interactions plus the triplet neutron-proton potential, results in a four-channel problem in the three-body sector and this degree of complexity could be handled on the larger faster computers now becoming available. In such a calculation of deuteron-alpha scattering one would be including the inelastic processes of stripping to the p3/3 and ~112 levels of 5He


and 5Li with the subsequent decay of these nuclei, and also direct dueteron breakup. Work on this system is now in progress.

ACKNOWLEDGMENTS

We would like to acknowledge the collaboration of R. H. Bassel with the distorted-wave calculations and for the hospitality he extended to one of us (P.E.S.) at the Oak Ridge

National Laboratory where these calculations were carried out. We would also like to ex-

press our appreciation to Professor Y. N. Srivastava, Professor M. T. Vaughn, and Pro- fessor M. S. Weiss for helpful discussions.

Generous grants of computing time from the computation centers at the Goddard Space Flight Center, Massachusetts Institute of Technology, and Oak Ridge National Labora-

tory are gratefully acknowledged.

APPENDIX A. CONTOUR DEFORMATION

A procedure for obtaining accurate numerical solutions of the integral equations arising in Sets. II and III is the subject of this appendix. The method is outlined for the spinless model of Sec. II. In order to calculate the amplitudes for all processes we must solve the four coupled equations (2.10) and (2.11). In this discussion we will consider only the first pair of equations.

T:$(k’, k; E) = B:t(k’, k; E)

khlklB:t(kl,k;E)& E-~k:+~~)TSx(k’,kl;E) (

(Al)

E-2+Ak:+e, l+A

k; dkl B;,(kl , k; E)Sd E - v kt + Q) Tidk’, kl ; E) (A2)

E/.2+A z yjyg- kl + ea

where E is assumed to have a small positive imaginary part. The center-of-mass energy is given by

where k’ is the p-B and k” is the &A center-of-mass momentum. For stripping and p-B scattering at a given E, the final momentum is common to both processes; therefore, we fix E on the final momentum via (A3), and then T-matrices are functions of the off-shell variable k.


To see that a deformation of contour is desirable, we will first consider the singularities of the kernel for real momentum. For E < 0, the only singularities present are the propagator poles which occur when k, = k’ in (Al) and when kl = k” in (A2). The resulting principal-value intergrals can be carried out accurately by standard numerical procedures. For E > 0, other singularities arise. Sd and Se develop square-root branch points at kI = [2A/( 2 + l)E]“’ and kl = ii1 + A)/(2 + ALw2, respectively. We now consider the branch points of the Born function-we have shown in Sec. II that this function is given by

ltd- Qzblkk') Qz Wkk') Lsk(k 7 4 &) = - /̂d -Ye 4Alclc' L(a _ b)@ - c) + (b _ (.&)(b - (?)

Qddkk') (A4)

+ (c _ a)(c _ b) 1 = &m(k, k’; E),

where

a = a [2kr2 + F k* - E] ,

b = k2/4 + k’* + Pi, (A.51

l+A c=zi-

’ k12 + k* + /&* 1 ,

For the special case of either k or 1~’ = 0 we have

B:t(k’, k; E) = ya ys 8zo X E - y k* - 2k’*] [k2/ 4 + kr2 + P:]

2k’2 + k* + /3s2

e-1 (-46)

where Q1(x) in Eq. (A4) is the Legendre function of the second kind which has logarithmic singularities at x = f 1. In the last two terms of (A4) ] b/kk’j and 1 c/kk’ ] are always > 1, but for E > 0, 1 a/kk’ ] may be < 1. In fact, the Born function in the kernel of (Al) is singular along the two curves

[(l + A)/‘A]k* f 2kkl + 2k: - E = 0 (A7)

and complex in the region between them, while the Born function in the kernel of (A2) has similar behavior along the curves

[( 1 + A)/A]k: f 2kkl + 2k2 - E = 0. (A81

For 1 = 0 and k = 0, these logarithmic singularities degenerate into poles at k, = (E/2)l’*in (Al) and at kI = [AE/(l + A)]“” in (A2). In the presence of these singularities, the results depend quite sensitively on the choice of integration


mesh. Thus we adopt the procedure of Hetherington and Schich (15) and deform the path of integration to avoid these singularities. We let k and kI take on the complex values z and z1 and solve Eqs. (Al) and (A2) along a path in the complex plane for which the kernels are particularly smooth. This gives amplitudes of the form T( k’, z; E) for each process. The integral equation itself may then be used to analytically continue back to physical momenta by substituting the above amplitudes on the right of (Al) and (A2), performing the integral to give the on-energy-shell amplitudes T&k’, k’; E) and TSt( k’, k” ; E). These procedures are valid if no singularities are crossed in the process of deforming the contour and if the integrands vanish rapidly enough so there is no contribution along an arc of infinite radius between the original and the deformed contour.

The contour deformation used for the calculations involving the Hulthen vertex is the one introduced by Hetherington and Schick which is a rotation into the fourt,h quadrant by an angle +,

k + z = kt?*, --i* kl + x1 = kle . (A9)

Singularities of the Born functions must be investigated for both arguments complex and also for just one argument complex. For both arguments complex there are no singularities between the original and deformed contours if + < 1r/4. The Born terms with one argument complex arise form the inhomogeneous stripping term Bi,( k’, x: E) in solving the integral equations along the arc in the complex plane and also we must consider the singularities of Bit(xl , k’; E) a,nd Bi,(zI , Ic’: E) when performing the integrals to get back onto the energy shell. Branch points arise in these functions at the momenta

A *l+A

k’ f i fhQ2 f i(Cd/2)“2,

A *l+A

k’ f i/3* , *l+A N l+A

Ak +i-rpBJ

the angle @ must be chosen sufficiently small that none of these branch points are crossed.

For the square-well vertex function used in Sec. III, the contour must be chosen somewhat differently. The vertex function is

(All) + sin @,(aB sin lcr, - k cos kr,)} .

422 HHANLEY AND AARON

TABLE IV

NUMERICAL RESULTS FOR SPINLESS CALCULATION

E = -1.5

1 ReT;, ImT;,

0 8.74 -10.47 1 -3.24 -0.61

2 -0.09 0.00

1 ReT;, ImT;,

E = -0.5 1

ReTd.4 2

ImTd, ReTi, ImT,: Bft

0 3.24 -9.11

1 -5.09 -4.49 2 -0.62 -0.04

3 -0.06 0.00

0 1.60 -7.91 1 -4.12 -5.12

2 -0.93 -0.22

3 -0.11 -0.01

4 -0.01 0.00

1 ReT;, ImTk,

0 -0.23 -6.05

1 -2.74 -4.65

2 -1.11 -0.84

3 -0.22 -0.13

4 -0.04 -0.02

0 -1.22 -3.35

1 -1.21 -3.22 2 -0.52 -1.35

3 -0.15 -0.45 4 -0.04 -0.15 5 -0.01 -0.05

7.92 -0.66

0.00 0.00 0.00 0.00

0.00 0.00

E = -0.1

2.60 -4.71 1.75 -0.83

0.18 -0.10 0.02 -0.01 0.00 0.00

E=l.O 1

ReTd, 2

ImTd.4 -0.05 -2.96

1.32 -1.65 0.40 -0.40

0.10 -0.07 0.02 -0.01

E = 4.0 -0.42 -1.16

0.16 -1.22

0.15 -0.52 0.07 -0.17 0.03 -0.06 0.01 -0.02

-0.88 2.46 -18.83 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.63 1.65 -14.68 -1.39 -0.88 - .390 -0.90 -0.07 -0.96 -0.24 0.00 -0.24 -0.06 0.00 -0.06

1.85 -0.16 -8.82 -1.35 -0.41 -4.00 -1.41 -0.07 -1.65 -0.65 0.00 -0.67

-0.27 0.00 -0.27

0.66 -1.03 -3.76

-0.77 -0.20 -2.38 -1.06 0.00 -1.35 -0.69 0.01 -0.73 -0.39 0.00 -0.39 -0.21 0.00 -0.21

The trigonometric functions have essential singularities at k = i 0~ so that the simple rotation of contour into the fourth quadrant is not possible. A suitable contour is one along a ray in the fourth quadrant for a finite distance joined to a straight line path parallel to the real axis. This is the contour used in the square- well calculations.


TABLE V NUMERICAL RESULTS FOR CALCULATION WITH SPIN

E = 2.673

1 2 ReTd,

1 ImTd, ReTi, ImTj, B1 .3t

0 0.84 1 0.02 2 -0.19

3 -0.10

4 -0.03 5 0.00

6 0.00

1 ReT;,

0 1.51 -2.10 0.38 -1.90 7.00 1 -0.89 -0.84 -1.06 -0.84 0.36 2 0.04 -0.86 0.04 -0.86 0.00

3 0.00 -0.43 0.00 -0.43 0.00 4 0.00 -0.15 0.00 -0.15 0.00 5 0.00 -0.05 0.00 -0.05 0.00

6 0.00 -0.01 0.00 -0.01 0.00

-0.66 -0.26 -0.44 -2.14 -0.10 0.47 0.13 0.54

-0.48 1.03 -0.01 1.26 -0.28 0.86 0.01 0.94

-0.10 0.54 0.00 0.55 -0.03 0.30 0.00 0.30

-0.01 0.17 0.00 0.17

ImTk, ReT;, ImTA, BZ ce

The rotated contour procedure allows considerable savings in computer space since fewer mesh points are required to perform the integrations along the deformed contour. For the solution along the real axis about 35 integration points were used per coupled equation, leading to 140 X 140 matrices. Above breakup there is considerable sensitivity to the choice of mesh points due mainly to the logarithmic singularities discussed above. With the deformed contour approximately 25 points per equation are needed and the resulting solutions are quite stable against mesh variation.

APPENDIX B. NUMERICAL RESULTS

In this appendix we present representative numerical results of the calculations carried out in Sets. II and III. Our units are such that 2m (nucleon mass) = fi2 = 1 and the binding energy of the deuteron Ed = 0.5. The numerical results are given as a function of the three-body center-of-mass energy E.

For the spinless calculation we present results for the B nucleus which is static and has the range parameter ps = 4.0 and binding energy Q = 2.0. The real and imaginary parts of the partial-wave amplitudes for p-B and d-A scattering and stripping and also the Born approximation for stripping are given in Table IV.


Table V contains sample numerical results for the calculation with spin carried out in Sec. III. In this calculation the core is I60 and nucleus B is “O* (0.871- MeV level) so that we choose the depth of the square-well separable potential Vo = 53.8 MeV and the binding ener,q Q = 0.736. The T-matrices are given in the total spin-isospin representation as directly obtained from the solution of the integral equations; the four independent amplitudes are p-B and d-A elastic scattering, stripping, and charge exchange. The Born terms for stripping and charge exchange are also included.

RECEIVED: April 28, 1967

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Three-body approach to direct nuclear reactions involving weakly bound systems

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