On the theory of nuclear heavy-ion direct transfer reactions

On the theory of nuclear heavy-ion direct transfer reactions

ANNALS OF PHYSICS 117, 323-359 (1979) On the Theory of Nuclear Heavy-Ion Direct Transfer Reactions B. J. B. CROWLEY Department of‘ Theoretic...

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117, 323-359 (1979)

On the Theory

of Nuclear





B. J. B. CROWLEY Department

of‘ Theoretical


1 Keble



OX1 3NP,


Received October 11, 1977 We review the distorted-wave approach to direct transfer reactions and draw attention to some of the shortcomings of current theories. We show that a reformulated form of the distorted-wave Born approximation (DWBA) for transfer can lead to important simplifications of the theory, which are valid for nuclear heavy-ion induced reactions at energies 2 10 MeV/nucleon. In particular, in the semiclassical limit, it leads to a new and simple formula for the transfer t-matrix which includes all the essential physics while offering several important advantages over standard “full-recoil finite-range” DWBA. One such advantage is that the new formula is more transparent in that it is amenable to interpretation and analytical manipulation. At high-energy it is shown to reduce to one earlier deduced using eikonal-DWBA. The conditions for the validity of the new theory are discussed in detail. They are shown to be generally well satisfied for small-mass transfer between heavy-ions at energies at or above those which particularly favour transfer (2 10 MeVinucleon for transfer of valence nucleons). The restriction to small mass is not due to any recoil approximation; in fact, it is only a necessary restriction at certain energies. The theory treats recoil exactly. Consideration of the optimum dynamical conditions for transfer leads to a set of matching conditions. The presence of hitherto neglected absorption, arising from dynamical effects of poor matching, is suggested and qualitatively discussed. Conditions under which such absorption may be neglected are derived. Results of numerical calculations are presented showing that the theory is capable of good agreement with standard full-recoil finite-range DWBA, and that it is capable of giving at least as good an account of experimental data for nucleon-transfer between heavy-ions at energies -10 MeV/nucleon.

1. INTRODUCTION This paper describessomeideasconcerning the theory of nuclear heavy-ion induced transfer reactions at energies well above the Coulomb barrier. These reactions, in which nuclear matter is transferred between the target nucleus and the projectile nucleus during the collision, are currently attracting much interest. They are important as a means of studying selectednuclear states [l], particularly those in which only the valence nucleons are active. The discussionin this paper is mainly given in the context of direct single-particle transfer, and generalisations to include more complicated types of processare only briefly discussed. Much apparent successin the treatment of direct nuclear reactions has been achieved using distorted-wave theories such as the distorted-wave Born approximation (DWBA) [2-l l] and its coupled-channels generalization (CCBA) [12]. However, a standard calculation of a “finite-range full-recoil” DWBA t-matrix element for transfer involves the evaluation of a complicated integral of not lessthan 323 OOO3-4916/79/020323-37$05.00/O All

Copyright Q 1979 rights of reproduction

by Academic Press, Inc. in any form rzscrvcd.






six dimensions. The full calculation can be cumbersome, particularly at high energies, and requires the use of a large computer. Furthermore, as much of the “physics” is hidden in the numerical calculation, DWBA codes used for routine analysis of experimental data may take on the appearance of “black boxes.” This is highlighted by the need to adopt an experimental approach [13] to determine the relationships between the results of a distorted-wave calculation and the input parameters. Little positive information may be yielded when a distorted wave theory fails to reproduce all the features of the experimental data. What is needed, in fact, is more insight into the workings of the theory, and its limitations, as well as a means of reliably assessing the accuracy of quantitative information extracted from experimental data by means of DWBA or CCBA analyses. More intuitive semiclassical approaches [13-161 to direct reactions have been useful as aids to interpretation and understanding, but are considered to lack the quantitative reliability attributed to quanta1 distorted-wave theories. However, some recently developed semiclassical theories [16] have proved to be capable of yielding as acceptable agreement with experiment as do full DWBA calculations, while involving considerably less expense on the computations. In this paper we consider the semiclassical (‘Vi + 0”) [17] and high-energy limits of the DWBA applied to direct transfer reactions. In doing so we exploit a number of simplifying features exhibited by high-energy direct reactions; in particular: (i) that the net impulse attritubable to the reaction is much less than the momentum of relative motion; and (ii) that during much of the time the nuclei are close together the motion does not differ significantly from that undergone during elastic scattering. The latter largely accounts for the adequacy of distorted-wave theories as a means of describing these reactions. During a nuclear reaction the individual nuclei involved are subjected to impulses which not only affect their states of relative motion but also may give rise to internal excitations. In elastic and inelastic scattering the net impulse is determined by the properties of the initial and final channels. Effects of internal excitations leading to other channels, not strongly coupled to either of the channels of interest, can be described in terms of absorption by treating the nuclei as “inert” particles interacting via a complex potential. This description may be extended to the n-body problem by means of a Faddeevtype treatment [ 181 in which the 3-body t-matrix is expressed as a multiple scattering series involving energy-dependent 2-body t-matrices. Each such 2-body t-matrix describes a complete 2-body scattering with the third particle as a spectator. The 2-body subsystem therefore defines an instantaneous inertial frame in which internal excitations leading to loss of flux into other channels are determined solely from the 2-body scattering. The situation is then just as for elastic scattering. Each 2-body t-matrix in a Faddeev-type formalism may therefore be given in terms of the appropriate energy-dependent complex potential. The Faddeev-Watson series contains each 2-body t-matrix to all orders, and this implies (in general) a nonlinear dependence





of net absorption on the absorption occurring in each ‘-body channel. This is to be expected in a model which relates the absorption to the real impulse (for which excitation probability is not proportional to the applied impulse). Formal 3-body theory does not, however, provide a practical approach to direct transfer. Distorted-wave theories describe the “pseudo 3-body problem” in which the 3-body system is (explicitly) coupled only to 2-body channels. Such a theory includes a full description of the elastic scattering in the entrance and exit channels, and this may be given in terms of the appropriate complex potentials. The process of transfer may give rise to additional impulses partly in order to conserve energy, momentum and angular momentum (see Sect. 2.6). If, as is usually assumed, the nuclei are effectively inert, any such impulse is absorbed into the relative motion. In reality. one would expect that a large impulse, (K ‘2, kE ym Fermi momentum of the constituent nucleons in any of the nuclei), would give rise to a significant probability of further excitation, thus leading to an attenuation of flux into the reaction channel. Thus, there is an additional source of absorption, which we shall call woction-induced absorption, which is not described by any distorted-wave theory involving the assumption of inert cores. When the interaction time is long compared with the nuclear response time (i.e., low energies), the total absorption is the sum of the reaction-induced absorption and that produced by elastic scattering. At high energies, when the nuclear response time 2 interaction time, the total absorption is related to the net impulse and would not be expected to be given linearly in terms of the elastic-scattering and reaction-induced components. The latter situation is representative of many high-energy heavy-ion transfer reactions since it is when the velocity of relative motion matches that of internal nucleon motion, that direct transfer is most favoured. When the various impulses are small (K < RF), they are unlikely to lead to excitations, and reaction-induced absorption may be neglected. The “inert cores approximation” is then valid. The (minimum) magnitude of K is a measure of the strength of the forces required to induce the reaction-the smaller the impulse, the easier it is for the reaction to proceed. We therefore expect to find experimentally that states populated by reactions involving small reaction impulses should be selected in preference to those involving large impulses. The necessary conditions for K to be small in the classical problem leads to a set of matching conditions which are derived in this paper. In this paper we confine our attention to situations in which the relative-motion momentum, p, is such that P>

6, .

(Note that in the case of heavy-ions this condition is satisfied when the nucleon velocity at the Fermi surface is comparable in magnitude with the relative motion velocity). Therefore, if K 2 IF, it follows that

which is just assumption (i) above. The main results of this paper are summarized in Section 4.





2.1. Notation We consider a reaction proceeding from an initial channel 01to a final channel /3, in which the rearrangement nature of the reaction is defined by: A+a*B+b where A = b + c and B = a + c. The nuclear cores are denoted by a and b, while c denotes the transferred particle which may be a single nucleon or a cluster of nucleons.

b a FIG.


Coordinate system used for describing transfer reactions (for explanation,


The coordinate system is shown in Fig. 1; R is the relative coordinate of the cores, a and b; and St and s, are the coordinates of the transferred cluster c relative to b and a, respectively. The channel coordinates, ri and rf , are given by: ri = R - CiSi 2

r, = -R - Efsf ,


where ei = mclmA = pilmb ;

Ef = mclmB = prIma ;


and m, , ma and m, are the masses of the cores, a and b, and the transferred cluster, c. The reduced mass of the initial bound-state system A is pi = mCmb/mA ; and the reduced mass of the final bound-state system, B, is pf = m,m,(m, . Reduced masses of the (A + a), (B + b) and (a + b) systems are, respectively, mi
















(3) where m, = mb + m, and mB = m, + m, .


The Hamiltonian




for the system is H = H, + V, = Ha + V, = H, + K,, , etc.,

where states

H,, . HB ,..., are the channel Hamiltonians @, , / Qslt,... . These satisfy H,


whose eigenstates are the channel

@,, ’ = (w, + h”ky3/2m,) 1oy,i = E ! @.,.,,

y E \‘X, p,...;,


where -w,, --wg ,..., are the binding energies of the A emub -- c. B =I a $- c,.... systems; and k, . k, , are the relative motion wave numbers for r’, i Coordinate representations of the channel eigenstates are

2.2. Distorted- Watre Approximatiotls DWRA


elements [2-51 for the reaction N --f /3 are (7a) (7b)

where C,; is the potential acting between u and c in the exit channel; and Vi acts between b and c in the entrance channel. The distorted-waves 1EA+” and @L--) are solutions of the Schrodinger equations, (H,-(H,3 f




I/,,+) ! @A-’ y = E : @a-’ ,


with outgoing-wave and ingoing-wave solutions satisfy the Lippman-Schwinger

= EjEi)

boundary equations,




G:,-’ = l/(E - H./ -- i0) is a channel propagator. These equations take into account the possibility of U, and UO leading to transitions to other channels (y # iy or y # /3). Most choices of U, and rl, restrict the number of such open channels, often to just the single channel, (1 or p. In such cases, equations in which y denotes a closed channel reduce to trivial identities and may be ignored.



The potentials

U, and U, should be chosen so that1

in the case of the prior form; or [email protected]’ I (Vr - vi + u/3,)I @,,> + ,


in the case of the post form; together with higher-order terms which we may represent by means of the single expression, (@j-j / (V, - U6) G’+‘( I’, - U,) / EL+‘)

(where G(+) is the interacting propagator, [E - H .+ iO]-I), are small. The standard method of allegedly achieving this is to choose U, and U, to be optical potentials, U,“““(I’,) and UiPt(r,), which reproduce the e&cts of V, and V, in the elastic channels. Provided that c is bound to one or other core in both channels (wi < 0, wf < 0), potentials, such as L/aoPtor UfPt, which operate only in the space of ri or rf , cause the left-hand matrix elements in (9) to vanish identically. One then has only to argue that the right-hand ones are negligibly small (or include them in the calculation). For the purpose of this paper we shall choose U, and U, in a way which is different from that described above. In this way, we derive not an “approximation to the DWBA” as given by (7) but rather a different form of it. We take both U, and U, to be effective potentials acting only in the space of the relative core coordinate, R = & These potentials may still be chosen so as to reproduce the effects of V, and V, in the elastic channels. Exactly how U,(R) and U,(R) might be constructed so as to achieve this is left until later in the discussion. We now seek solutions of the distorted-wave Schriidinger equations in which the potentials U, and U, have the forms described above. In appendix A we deduce the following approximate solutions valid at high energy when the typical momentum2 associated with internal motions of A = (b + c) or B = (a + c) bound systems can be considered to be small compared with the momentum of relative motion of target and projectile: s)‘(R, si) N #i+)(R) e-ikeu’Reik~“ix,(s,), (‘0) @L-‘(R, s2) N I@‘(R) e-iko6’Re-ik13”~~XB(s~), where $L+)(R) is the physical scattering solution (comprising an incident plane-wave, eikoa*R, plus outgoing spherical waves) of the 2-body Schriidinger equation: [-(+iz/2mo~ OR2 + U,(R) - h%,/2m,l

#,(W = 0;


1 The lirst term in each of the following expressions denotes the amplitude for transitions between a and fi induced by U, or Us. The second terms are DWBA amplitudes depending on the residual interactions, V,, - U, or VCa - Us. z The fact that these conditions involve the momenta rather than the velocities suggests that the approximation is a dynamical one, rather than a kinematical one such as an adiabatic approximation.



and t&)(R) is the scattering solution plus ingoing spherical waves) of [-(++/2,??,)


T;Rp + C’;(R)



an incident

- /~‘k;~jJZm,] &(K)

plane-wave. 0.

P~‘@~‘~, (I Ib)

where Substituting yields:

the distorted-wavefunctions

Tq,>‘v f t&‘*(R)

I/J:-‘(R) e

(IO) into the prior form of the DWBA



This differs from the more usual form of the DWBA [2. 5, 61) which is

which results from using the distorted W


k,, = (m,,/mf) k,, .

k,, = (m,/‘mi) k, .


,x$(s+) xJs,) d3si d3sf



( I3a)

for TBI (e.g., Refs.

waves, = 6ka)(rf) x&)~

o(-) == $&~(rJ Xn(Si), “R which are exact eigenfunctions Ff, + UEPt*(r,). Equation (13a) reduces to

of the distorting

&, ru f z,@‘*(R)





p) d3R,


where F(R, p) = &+R = e-*ip’R

~c x:(s) .1


V,(s) x& - R)



$- R) e+~.~ d3s - R) xa(s) e-lp’s



and P = Eika + crk, = qt - q, , qu = (1 - &)

k,, - (1 - 44 km = &(qi + qf),

9, = k, - (1 - cJ k, , The vectors qi and qf have the property


%=(I-ef)k,-k,. that

k, . rf + k, . ri =


. sf



. si





and they are related to the reaction Q-value by &fi’(qi”IPi

- qf2/pf) = Q G ~?i2(ks2/m, - k,2/mi).


The form of the DWBA t-matrix for transfer as given by (14) and (1.5) will be recognized as that of the familiar Dodd and Greider [I93 expressions when one makes the replacements, &-j(R)

eikoO’R-+ J+‘(R),

#p)(R) e--iko,.R -+ B?‘(R). The argument used by Dodd and Greider is basically a semiclassical one. fn the semiclassical limit the integrand of the DWBA is dominated by the behaviour of the phases which give rise to rapid oscillations in the function over much of the range of integration. Functions B,!Ti(r,) and @;B)(T~)appearing in the usual DWBA expression (13b) are presumed to be slowly varying, when they may be reasonably well approximated by Bj++J(R) and B:;,‘(R) respectively for small l i and Ed. Approximations such as this, which depend upon Ei and Ef being small, are known as recoil approximations. A proper treatment of the Dodd and Greider recoil approximation is given by Hasan [ 161. This makes explicit use of semiclassical wavefunctions and the result involves local momentum variables in place of some of the asymptotic momenta appearing in Dodd and Greider’s expression. The derivation of the approximation given here does not depend directly on the assumption that ei , Ed Q 1. Our approximation is therefore not a recoil approximation. It depends upon assumptions concerning the magnitude of the typical momenta of internal motions compared with that of the relative motion. Constructing the potentials ZJ, and U, so as to reproduce the elastic scattering in the entrance and exit channels leads to an approximation which is indeed very similar to that originally proposed by Dodd and Greider. On respect in which the approximations differ, however, is that the wavefunctions, #L+)(R) and &-j(R) appearing in (14), respectively depend on the wave vectors &,,, and k04, whereas the equivalent functions appearing in the Dodd and Greider expression, viz. $jt,‘(R) and &i(R), depend on k, and k, . At high energies our approximation should be better than Dodd and Greider’s. An important case when assumptions leading to (10) are valid is when m, < m, , mb (typically true for heavy-ion transfer reactions) and internal motion velocities are comparable with velocities of relative motion. Under these conditions transfer is particularly favoured. For the transfer of valence-shell nucleons between heavy-ions the optimum energy is -10 MeV/nucleon [I, 41. 2.3. The Distorting Potentials, U, and U,

It is convenient to describe the potentials UN and U8 in terms of the phase-shifts that they generate in the 2-body scattering systems described by Eqs. (11). These potentials should reproduce the average motion of the cores during the reaction.





When allowance is made for the angular momentum carried by the transferred particle, the generated phase-shifts, @) and Sy), should therefore lead to the same deflections as do the phase-shifts describing elastic scattering in the enetrance and exit channels. If we assume: (a) that, during a process in which direct transfer takes place, most of the angular momentum is associated with the relative motion of the two cores (as is the case when the relative-motion angular momentum is >>ti, as in the semiclassical limit, while the angular momentum of the transferred particle relative to either core is O(A).) (b) that the derivative, a@/%, of the nuclear deflection-function, @ = 2 a 8>N)/2/, with respect to Y is small (as in the semiclassical limit when a@/U is O(h).) then (i) the change in angular momentum associated the cores is small compared with the average value.

with the relative motion


(ii) the dynamics of the cores’ relative motion should be well described by the same nuclear phase-shifts as describe the elastic-scattering in the entrance and exit channels.

Thus the potentials, U, and Us, may be chosen to be optical potentials which generate (in the contexts of Eqs. (11)) the phase-shifts which describe the elasticscattering in the entrance and exit channels. However, with such a choice, one does not recover the exact elastic s-matrices from wavefunctions having the form of (10). This is to be expected, not only because of the approximation made in deriving these wavefunctions. It is our view that it is not the average nuclear potential, as used to describe elastic scattering, which should be used for describing distortion effects in high-energy direct reactions. Rather, one should use an interaction which is more appropriate to those nuclear configurations which (near minimal distance of approach) are (most) likely to result in a transition to the reaction channel. In the case of transfer, favourable configurations are those in which the transferred particle lies between the cores’ centres (when its angular momentum relative to the system c.m. is close to minimum); as opposed to unfavourable configurations when the transferred particle and one core are on opposite sides of the other core (when the transferred particle’s angular momentum relative to c.m. is close to maximum). The long-range Coulomb potential is felt out to large distances of separation and hence for long times. Tn this case the usual average may be appropriate. In practice, the average nuclear interaction in the form of elastic-scattering phaseor a phenomenological optical potential may represent the only information available. The arguments above then enable us to invoke assumption 2.3(a), and hence obtain a first guess at U, and U0 by relating them to the known elastic phase-shifts, as described. In fact, as will be shown, one may not need to determine U, and U, at all, as the transition amplitude Tadcan be expressed in terms of the phase-shifts themselves. Complete potential sets (Upor, Up, V,) and (UtoSt, Ufost, Vi) should be constructed as follows: SOS/1




Prior representation Vf = ZaZce2/sr + V$’ + Vj”), Ujrio’ = UiN’(R) + Z,Z,e’/R, Up’

= UjN)(R) + ZbZBe2/R;

Post representation V.z = ZbZ,e2/s. I + I+‘+ bc post a


VfN) 13

= ULN)(R) + Z,ZAe2/R,


= UcN)(R) + ZbZ,e2/R. 6

In these equations Z, , Zb , Z, , Z,, , and Z, denote nuclear charges and “e2” is the appropriate constant in the expression for the Coulomb potential (with the Z’s expressed in units of the proton charge, e2 has the value (ec2)sr x lo2 = 1.43998 MeV.fm); the superscript (N) denotes the specifically nuclear part of the potential; and vi:) and VJs’ contain any short-range modification to the Coulomb parts of V, and V, ((similar terms appearing in U, and U, can usually be ignored or their effects simulated in Ui”) and Uj”)). The effective nuclear potentials, ULN) and UjN), may initially be constructed to yield, in the presence of the complete Coulomb potentials, Z,Z,e”/R and ZbZBe2/R, respectively, the elastic phase-shifts describing the entrance and exit channel interactions. The potentials, V,C”) and VjN), which are the specifically nuclear parts of Vi and V, , will usually be related to some nuclear model with parameters chosen to reproduce known nuclear properties such as binding energies. (Note that this prescription gives a description, to at least first order, of all Coulomb monopole terms in the various interactions-including V, and Vi ; cf. discussion in Ref. [8]). 2.4. Momentum-Space



(13) in the form of a momentum-space

integral yields

= I ~~:,*






= (2~r)-~/~ J” Vf(s) x@(s) eCiq” d3s

_ (q / Iif j xs\ ==(% - ~4VP.pKcI I x3: -= GJ3- f52c12/2pf) r&?(n); S(q) = j &j*(R)

&+'(R) e-%,-koo+o).R


(20b) (21)

The integrand in (19) separates into a part, (22)

43ab-l) = 63% - 9) 2dat - 919

which depends upon the kinematics and the properties of the bound states; and a part, S(q), which depends only on the relative motion dynamics. The function S(q) bears a resemblance to an elastic s-matrix. Indeed, in the high energy limit, S(q) becomes an eikonal s-matrix [20]. In another paper [21] we consider the semiclassical limit of the integral (21) and hence derive approximations for S(q). We are able to interpret S(q) as the amplitude for an impulse t = (k,, - k,, - q) applied in the relative-motion frame of the cores leading to a transition from the scattering state LX(+)to the scattering state fl(-). The spectrum of this impusle is determined by the form-factor floe . The product, S(q)F,,(q), integrated over all permitted values of q, thus yields the transition amplitude, TBu. 2.5. Approximations

for S(q)

The function S(q) has a particularly simple form when eikonal approximations are used for the distorted-waves. Such approximations are not expected to be valid for heavy-ions at energies below several GeV and are therefore unlikely to yield results of much practical use. However, the calculation, which is described elsewhere 14, 201, did prove worthwhile, as it immediately suggested a more general approximation and illuminated a possible approach to the problem using semiclassical methods. The semiclassical limit of (21) is discussed in an earlier paper [21]. Here we just quote the result which is valid for a peripheral reaction (in which contributions come only from a limited range ~(dL)fi of angular momenta centred about Lh (L > AL > 1)) provided that the energy, E, , is at least of the order of Q0 AL and Peff AL: Serf(q) N +$ S (*

f (28 + 1) e2isePp(cos~(4)).

- y)



where ko A b’eff

D = fik,/m,




= +$I, + q)



Qo cos

kos), 30 -



(= average relative velocity),




(254 PO = PCRI) = (L + 8)lR1, p(R) = ${[k&

Q. = fi’(k$


- 2moU,(R)/A2]1/2 + [k$ - 2moUo(R)/fi2]1/2},


- k&)Pm,

= moCU/w - l/m)E

E = @“(ks2/mf

+ Hlh


+ ka2/mi) = $(Ea + EJ

Q = &?i2(kB2/mf - k,2/m,)

= EB - E,

(= average c.m. energy),


(= reaction Q-value),


Qen = effective Q-value = impulsive change of K.E. of cores = Q, - AU where d U is the potential discontinuity N U,(R,) - U,(R,). (29 P{(z) = Legendre polynomial cos()((q))

= [$$ cos;

of degree J’ and with argument z. e] [ 1 - $&I

+ [ 1 - 2 cos ; e] cos 8.


8 = scattering angle = angle between k, and k, = angle between b, and k,,, . s(z) = very sharply peaked function of z resembling the Dirac a-function which can reasonably be taken as an approximation to s(z) provided that flOll(q) is not a very rapidly varying function of q. 8, = $@~~ + SiB)) = average of the phase-shifts (as defined in Section 2.3) describing elastic scattering in entrance and exit channels. We write

where 8jN) is the arithmetic mean of the nuclear phase shifts, and Sic) is the Coulomb phase-shift which can be taken to be given by g(C) = arg T(l + L + in), e where r = g($,) + q@)) = arithmetic

mean of the Sommerfeld






defined in terms of the Coulomb parts of U, and U, as given in Section 2.3. It is convenient to define an average charge product “i&Z;’ by

so that If there is no charge transfer (Z, = 0) then ZlZ2 = z,z,


otherwise (z~z2)pr’or

= Z,&

+ Z,Z,d/Z?, = Z,(Z,, A-- [zQya, L zQ3)lZ,),



= z,z,

+ Z,Z,d/v,


= Z,(Zh + [zj,/Ctl, + zyj)] Z,).

Other conditions for the replacement of S(q) in (19) by Sefr(q), as given by (23), yielding a valid approximation to T,, are: (i)

that values of q are restricted

to values such that

I t/p0 I2 -+ I&,, - k,, - q)/po I2 is small, so that



and I 1 - (k,/p,)

cos 40 I i t/pa I2 < 7T/L;


(ii) that the potentials, U, and CJ, , give rise to similar scattering at the respective energies, E, and E, . More precisely: cos(~(O,

- 0,)) * I,

where 0, and 0, are the deflections associated with the incoming branches of a classical trajectory with angular momentum --L;

(35) and outgoing

(iii) that the quantities A Uj U, Q/E, QJE,, , 2m,Qeff/h2p,,“, where E,, = h”k,“/2m,, lJ = E0 - h2p02/2m, are all sufficiently small to be treated only in first order; and for their squares, e.g., (Q/E)“, to be much less than Z-/L. (iv) that an approximation of locally rectilinear motion is valid for / R x p,, , N L, R ‘v RI, for which a sufficient condition is iK&M while a necessary condition



<< 1,


I(I/P,,~) ap(R)I%R IR+

<< 1.



Other conditions listed in Ref. [21] are, if not implied by the above, either unnecessary or follow as a consequence of the semiclassical saddle-point approximation.



The S-function in (23) expresses energy conservation for trajectories with angular momentum -AL. The energy conservation condition is @*PM&

- 6x) = @2/%)(~,2

- pi2) + (Qo - Qen)

which, with the aid of (26) may be expressed as

Qefe= (fi*/&)P * (PI - Pi), where p = &(p, + pi). But (pI - pi) is just the impluse, t = kOB- k,, - q, while P 533+p(R)($

+ $) set $0.

Therefore bf - pi) . p = t . p = P(koB - k,,) cos ge - q - p = Qo(~op/fi2ko) COS he - Q - p. Hence, energy conservation for angular momenta -fiL is expressed (approximately) by Qeff = Qo(~olko)~0s $0 - (fi2/moh - p. or A Vefflfiv = q * Polk0 . A further reasonable approximation which we find convenient to make, and which is consistent with (ii) and (iii) above, and the assumption that F&(q) is not a rapidly varying function of q, results from neglecting AV,ff in (23) altogether. In fact, the statement

AVeff = 0 is equivalent to assuming that there is negligible longitudinal (in the direction of po) due to elastic scattering, viz.

(3% momentum


(koB - ko,) COS +e = Pf - Pi,

and obviates the need to make assumptions about AU which is rarely precisely known. This assumption leads to the effective Q-value being given by

Qeff = (PO/k,) Q, cos i$e,


while Serf(q) becomes: Se”(q) Y Z$ S(q * po) f (2L + 1) eBisePc(cosx(q)). 0



The quantities p. , R, , & , and cos x appearing in (41) are in general complex as a result of U, and U, being complex. The phase-shifts, & for 8 = 0, 1, 2, 3,..., which may be determined accurately, describe all the main dynamical and absorptive effects





produced by these potentials. Absorption gives rise to a nonzero (positive) imaginary part of 8, which describesthe attenuation of the partial wave. In heavy-ion reactions. the low partial-waves are believed to be strongly attenuated and so play little part in the reaction. Recent evidence [22] for low partial-waves producing observable effects in heavy-ion reactions may require some modification of this picture. At high energies in particular, these effects are apparent only at large angles (0 2 90’) and when attention is confined to forward anglesthey may be ignored. What importance, if any, should be attached to the complex nature of the other quantities above. An estimate of Im( p,,)/Re( p,,) may be obtained by assuming that the free path near the point of closest approach of a trajectory with angular momentum NL is 2 ;R, . This is a reasonable assumption for all except very badly matched reactions (for which, for other reasons, the theory is likely to break down). Hence, one can deduce that

arg(po) = Wpo)lRe(po) 5 l/L.


L > 1, the imaginary part of p,, (and of R,) may be considered, for many purposes, to be negligible. These simple arguments are borne out by detailed calculations. For example: for the 114 MeV 26Mg(11B,1°B) 27Mg neutron transfer reaction leading to the ground states of l”B and 27Mg, discussedin Section 3.3, L N 45 and arg(p,) N 0.0027. Combining Eqs. (19), (22), and (41) leads to the following approximation for Tsi, : Since


where 4 is the azimuthal angle and the integral is carried out in the plane q pn 7 0:

We propose Eqs. (43)-(45) as a practical meansof calculating the transition amplitude for a heavy-ion transfer reaction satisfying previously mentioned criteria. As well as permitting a much more efficient calculation than does the original DWBA, this approximation is also amenable to analytical manipulation and easier interpretation. The function ,4(q) given by (45) has the form of an elastic-scattering amplitude, and can be treated using short-wavelength semiclassicaltechniques, as described by Ford and Wheeler [30] and by Berry and Mount [17]. By writing cos x, as given by (30) in the form



we deduce the following high-energy approximation eff


to (41):

&q - ii) f (2L + 1) eWJt(l

N E 0


which is valid if

- qZ/2k,3,



I t II

1 - 2 cos ; 0 j < z-/L



is satisfied in addition to (33)-(38). Except for a constant factor, k,/p, , which is close to unity, the approximation (46) is just the formula, originally suggested by the form of the eikonal approximation for S(q), which we have already proposed as a means of treating high-energy direct transfer reactions [4,20]. The precise conditions for its validity are now revealed and are as stated above. The strongest condition is likely to be (47). Use of Eq. (46) leads to Tso, being given by (43), with FBiool given by (44) and A(q) by A(q) N A&q)

= &

f (24 + 1) ezistPe(l - q2/2k0’), 0 e=o


which contains no explicit reference to the distorting potentials. It turns out, however, that the more exact form of A(q), (45), can be calculated with very little extra effort, so that little, if anything, is gained by making the high energy approximation. For this reason, we no longer advocate its use, and we mention it in passing only to draw attention to the connection with earlier work. 2.6. Conservation Laws and Selection Rules Conservation of total energy and of angular-momentum during the interaction are (for spin-independent forces) exactly expressed by the equations Aw + Q = 0,

(49) (50)

4 + I, = I, + 1, ,

where Aw = w0 - w, is the separation energy which is the change in internal energy of the system, & and Z, are the orbital angular-momenta associated with the initial and final internal states, and Ii and 1, are the initial and final orbital angular-momenta associated with the relative motion (conjugate to ri and rf). The various impulses due to the transfer process can be resolved into an impulse t acting in the cores’ c.m. frame, and an impulse K, acting on the transferred particle in the c.m. frame of the complete system (the inertial c.m. frame). Conservation of linear-momentum leads to the impulses, X, and %b, acting on each core in the inertial frame, being given by: %


t -




hz/(m. -



mb)] +



= Kc

t =



, -




%, ,




where j3 = nz,/(m,, $- q,). By conservation of angular-momentum, R x t + rr x H, = 0.


During the process of transfer the particle c is likely to lie on or close to the line ab of the cores’ centres. We therefore take rGto be given, on average, by rc = (y - P)R


FIG. 2. Diagramillustrating impulses acting in a 3-body system during “instantaneous” transfer. The point G is the centre of mass of the [a + b] system. The impulses X, and t act in the intertial c.m. frame and the cores’ cm. frame, respectively. Conservation of linear momentum implies X, + % -t- X, = 0, so that all impulses shown are co-planar. where y = distance(bc)/distance(ba) = (si.:/R = 1 - (,q.,/R, specifying the most probable3 position of c on ab during transfer. A rough estimate of the range of values permitted to y for peripheral reactions in heavy-ion systems is y N (my * l)/(m~” + m;‘3). In practice, the actual value of y representing a particular reaction will be found to be shifted towards the preferred optimum value (v.infra). 3 The number y specifies a position of maximum overlap for distances of approach =R,, i.e. y -value of x (0 Q x < 1) for which --u&I - x)R~bV~((l- x)R~)u,(xR,),where U,(S) and ~(3) are respectively,s x radial partsof X&S) and X&S), has a maximum.




(52) and (53) yields R x (t + (y - p) K,) N 0.

Hence the transverse (perpendicular tudes K eL =



to R) components of X, , xb and w, have magni-

‘%L = 6’ - I>&@


- I-W,

KCL = A/‘$’ - YM, (54)

where (I = 1 A j and A = R x t is the impulsive change in angular-momentum of the cores due to transfer. In Appendix B the following approximate expression for A is derived using classical arguments:

where Q and Ef are given by (2). Treating Ei , Ef , e, and & as small parameters, allows (55) to be simplified to

A = (b’ - ‘Y)[(G+ 4L - [l/U - 1414s- (l/y) 4,


where L = R x p, and p and L are average values of the momentum momentum associated with the cores’ motion during transfer. By defining the basis (E,j, fr) according to i = L/L,

J^= P x L/PL

and angular

k = PIP,


we express A in the form A = hi + vj + &.


Using (56) and taking & and I, to be quantized in the z-direction so that fi . I, = m, )

L * I, = ??Jfl

and z,2 = f,(k, + I),

ZB2= l&,

+ 11,

we deduce the following properties of the components of A:


I x2 I < I x I < I A, I,


P = (Y - mJ%/(l - Y) + WJrl;


A, = (/3- y)[(ci + Ef)L 5 (Z/32 - JJQ2)1’“/~1 - y) + (Cm2 - %2)1’2/yl, h,














(zE2 -




(59c) +

x]l”} VW





where, in the last equation, X is chosen in the range \ X \ :.< 2(,/&l - r) 7 $&,, so as to give the expression in ( 1 brackets its smallest absolute value. It will also be shown that

In (59d) fihax :~ (2dL/L)(~/(y ~- /3))“; dL is the angular-momentum “window.” The classical impulse, t, acting in the cores’ system. may be expressed in terms of the effective Q-value, Qerf, as follows:

where U, = E,, - fi2p02/2m, = li2(ko2 -- p~z)/2m, ,

z-R*&, --p-“/L

(61a) (6lb)

= v.


In the usual model of a peripheral reaction, the interaction is confined by the short range of the nuclear interaction, and by strong absorption at small nuclear separations, to a thin spherical shell of radius -RI and thickness A& . The corresponding angularmomentum window is centred at ( - L N poR, , and has width dL cz p,, AR,. Assuming L ?> AL. we find that z is restricted to the range: 3 5 2RIARI. Consequently, p 2 p. and v2 ( ~“(2d L/L), the latter following from (6 lc). Combining with (58) and (59) yields the range of values permitted to fl”: A,? + p2 2 112 5 X,” 1. p2( 1 -t 2dL,‘L). which, together with (54), yields upper and lower ponents of H,, , x,, . and x, , e.g.,

bounds on the transverse


The magnitudes of w, and t are direct measures of the strength of the nuclear interaction required to induce the transfer reaction. When K, and t have to be large, the reaction is only likely to occur in those partial waves for which there is large nuclear overlap. However, competing effects of other channels, in the form of absorption, suppress the total reaction probability. Small K,. and t. however, enable



the reaction to occur more easily in the higher partial-waves where the absorption is less strong, and this results in an enhancement of the reaction rate. The optimum conditions for transfer are therefore those which permit small values of K~ and 1. The conditions are:



(which follows from (59d)). Conditions (62b) and (62~) together imply a preference for &ff m 0 3 Q, = 0 (velocity matching). In Section 1 we remarked that if any of K, , Kb or K, were to be large so as to be greater than, or of the order of, the Fermi momentum (1.3 fm-l in gas model) of the nucleons in the corresponding nucleus, then there would be a significant probability of that nucleus being excited into other channels. This additional absorption further suppresses the reaction rate when K is large, and is not described in the usual DWBA formalism. Necessary and sufficient conditions for all of K, , Kb , K, to be sufficiently small for reaction-induced absorption to be negligible are: t Q cG)j , Kc < &&)F)j7

j = a, b; j = a, b, c.

These imply (using (60), (54) and (58)) that ((A - &WY

Q wwJ5) k21 (k3/L)2 Q tj2, (P/W < @ - Y12 kj29 CxIL12 < (P - Y)” fj23

where fi = (J&I&


j= j = j = j =

a,b; a,b; a, b, c; a, 6, c;

= a, b or c denoting the nucleus, and A,, = LQ,tfI2(&




The above can be satisfied for all (classicaiiy) allowed values of A and p, given 8, , m, , &, mB and L, if E,2, j = a, b; (B - Y)* (43* - m6*Y2 + (L2 - ma2Y2 * g * I 2L L* [ Y l--Y


j = a, b, c; WV j = a, b. c.




The conditions (64a-c), together with

Qeff c X-4 - UI)

- yj2L4-. [-” 12< &‘. (%+ 9)2@ 211~ I3

.i = a, b,


are sufficient to ensure that reaction-induced absorption is negligible, provided that h, falls within the range of allowed values of h = A . i. For given R, and AR[ , (64a,b) do not involve the reaction energy and thus impose restrictions only on the properties of the nuclear states by, in effect, imposing upper limits on the internal angular momenta. Condition (64~) imposesan upper limit on the energy below which reactioninduced absorption is negligible. The related necessary condition, which, at high energy, when (ei + E,)L > to/(1 - r), f,/y, takes the form:


PO< P/(G + 41V~ + ~U&>Gl(~ - Y) + f&4


which defines an energy above which the reaction is strongly suppressed. The conditions (33)-(35) for the validity of the theory described in Section 2.5 depend upon the quantity / r/p0 j2. Using (60), this may be expressedas

WPO)”= (p/W + Go - wPoz)2+ (ho/L)“,


where ho is defined by (63), and 1 2 ( P~z)~ 5 L AL. This permits expression of the validity conditions (33)-(45) in terms of known parameters of a reaction. To summarize, we have derived classically optimum conditions for transfer in more precise and general forms than previous statements thereof. Our conditions (62b,c) are related to one of the conditions originally derived by Brink4 [23, Eq. (l)}. (See Note added in proof.) We have also shown that the conditions (33)-(35) for the validity of the theory, developed in Section 2.5, are likely to be satisfied in heavy-ion reactions at energiessuch that p. > $ under conditions given by (64)-(65). The latter are also required by standard DWBA.





3. I. Method of Calculation Calculations of transfer-reaction angular distributions have been performed using the computer code, WOMBLE ONE [24, 251. This code exists in two versions: 4To derivethe form of Brink’sconditiononemustredefinethe directionof the angular-momentum quantisationaxisto bethat of the unit vector,i. TheconditionH, . (i x R) = 0 impliesh/(p - y) 5 0 which is Brink’s equation.



version 02 which evaluates the differential cross-section du/dQ as a function of center-of-mass angle using the approximation for the t-matrix, TBa, given in Section 2.5 by Eqs. (43), (44), and (48); and version 03 which evaluates TBoaccording to the more accurate formulas (43)-(45). Bound-state wavefunctions were calculated for a Woods-Saxon + Coulomb potential using the computer code BOUND [24]. (Latest marks of this programme use the symmetrized Woods-Saxon form [26], 1 + cosh(R/u)

f-Sk>= cosh(r/u) + cosh(R/u) ’ instead of the more usual, though less useful, standard form,

f(r) = [l + exp((r - WW.)








eCM (degrees) FIG. 3. Comparison of calculations (solid curve) using the ‘high-energy’ form of the theory described in this work (code: WOMBLE ONE V02) with similar calculations (dashed curve) using standard finite-range full-recoil DWBA (code: LOLA). The reaction is the hypothetical transfer, lTO*(l(LO, 170*)160, at E(lBO) = 200 MeV. Reaction parameters are given in Table I.



TABLE Table of Bound-State

and Scattering Potential




Parameters Used in Numerical

Bound states

Elastic scattering .~~__ Reaction --___ 1,0* + ‘60 2eMg + “B


R, (fm)





(it%) (f2) ________.0.49 20 0.80



(f2) ..-6.8



(,Z) ~--_ 0.53






(tx, ~~ --


a Elastic scattering potentials used were of the form: U(r) = -U,,/ll + exp{(r - R&a,}] -iW,/[l + exp{(r - &.)/ad] + U@‘(r), where the final term is the Coulomb potential. Bound-state potentials used to calculate single-particle wavefunctions were taken to be of the form: vs(r) = - V&l + exp{(r - &)[a~1] + I$$+) where & = reA$&, and V, is determined from the experimental binding energy. V$‘(r) is the Coulomb potential (protons only) which describes the interaction between a point charge and a uniformly charged sphere of radius -RB

CICM (degrees


FIG. 4. 114 MeV zaMg(llB, *lB)ssMg elastic scattering data [ll] and results of optical-model calculation using JWIU3 approximation. Optical-model parameters are given in Table I. i-i



The depth of the potential well is adjusted so as to reproduce experimental binding energies. The elastic nuclear phase-shifts were calculated from a given optical potential fitted to the entrance-channel elastic scattering, using the single-reflection complex JWKB method [4,24,27]. Coulomb phases were calculated exactly from the effective interaction, ZlZze2/r, with ZIZ2 given by (32a). Parameters used in calculations of all reactions described are given in Table I. Unless otherwise indicated, all calculations were carried out using version 3 of the code. 3.2. Comparison with Standard D WBA Figure 3 illustraties a comparison, for a perfectly matched (t = 0) reaction, between a calculation performed with version 2 of the code WOMBLE ONE, and one using

0.11 0






'20 25 ecM (degrees)

5. 114 MeV ssMg + llB induced neutron stripping to ground state of ‘OB. Data from Ref. Solid curve is calculation using the code WOMBLE ONE V03.





using identical parameters, performed with the DWBA code, LOLA 17, 281. Results describe a hypothetical elastic neutron transfer between 160 and the first excited state of 170, and are in very close agreement. Various other calculations (not illustrated) have been performed over a range of reaction energies and values of parameters describing the nuclear interaction. At low energies (250 MeV in c.m.), some discrepancy was apparent in the magnitudes of the cross sections. This is not surprising in view of the assumptions of high energy and/or on-shell behaviour made in deriving the approximation used. Breakdown of the simple single-reflection JWKB approximation is also expected at those energies. Less serious discrepancies were also apparent when (unrealistically) weakly absorbing potentials were used (except U, = UP = 0, when results of all calculations were indistinguishable from exact plane-wave Born approximation). These would seem to be due to contributions from low partial-waves (L ry 1) which are not treated properly.

&,-,(degrees) FIG. 6. 114 MeV 2aMg + ‘IB induced neutron stripping to ground states of l”B and loBe. Data from Ref. [I 11. Solid curves are calculations using the code WOMBLE ONE V03.



3.3. Comparison with Experiment

Figures 5-8 illustrate analyses of experimental data [l 1] for single-nucleon transferreactions induced by 114 MeV llB ions incident on zsMg. Elastic scattering parameters were deduced by an optical-model fit to the entrancechannel elastic scattering data, and are given in Table 1. The elastic scattering data and the angular distribution calculated using the JWKB method are illustrated in Fig. 4. The JWKB calculation yields results that are virtually identical to the exact calculation over the range of angles covered by the data. Seven single-nucleon transfer reactions, including stripping and pick-up of both neutrons and protons, were analysed using WOMBLE ONE V03. The results (Figures 5-8) show excellent agreement with experiment. Whenever the data shows pronounced structure (as in p + s transfers) the fits obtained are consistently better than those using results of LOLA calculations [ll]. Deduced spectroscopic factors are given in Table 11 and are generally consistent with values obtained from light-ion experiments [29]. Spectroscopic factors obtained from the same data using LOLA [l I] are not directly compatible with our results, as the LOLA analyses made use of different optical potentials in calculating the distorted waves5 100 dd dR “b/s,


I,,,,l,,,,l,,,,l,,,,1 o



15 20 25 e,, (degrees)

FIG. 7. 114 MeV zaMg + llB induced proton stripping to ground state of “‘Be; proton stripping to excited state of 27A1. Otherwise as for Fig. 6. b It is claimed [I l] that these differences had no significant effects on the quality of fits either to elastic scattering or (using LOLA) to transfer data, but that they were significant in determining spectroscopic factors.





! g 26Mg(‘1B,‘2B)



O.lC I







20 El,,

25!? (degrees)

FIG. 8. I 14 MeV 26Mg + ‘lB induced single nucleon pick-up to nuclear ground states. Otherwise as for Fig. 6.

The seven reactions have also been analysed using WOMBLE ONE V02 which usesthe less accurate form of the theory [20] in which A(q) is given by Eq. (48). The results (not illustrated) provide equally good fits to the data---almost indistinguishable from those provided by V03. However the yielded spectroscopic factor!, (see Table Ii) are consistently about 30”,,; z IO”,, smaller. but nevertheless still in good agreement with light-ion data.



We have presented here an alternative theory of heavy-ion transfer reactions which combines considerable computational efficiency with much higher accuracy for many cases, than previous approximations to DWBA. The theory. which is applicable to medium- and high-energy transfer, is developed from a general form of the DWBA, and leads to expressions for the transition amplitude, 7;,, , in which the











~s/e h







O+ 41s


I Plh PSI2



0+ ~s/a Jcc&/a



3+ Pa/a








4.60 * 1.4 2.00 * 0.5

0.71 0.12 I

046 . + p

0.35 f 0.1

2.3 - 2.5

2.6 - 3.00 -____

0.42 - 0.65

0.25 - 0.33

0.13 - 0.20

0.58 - 0.60

0.75 - 0.83

Ref. [29]


0.51 f ;;’


.__ S(aaMg&iV)

2.0 & 0.3

4.14 f 0.5

1.4 f 0.3

2.35 h 0.3

0.92 & 0.09 0.74 T;;;

0.56 & 0.03 0.39 f O.&i

0.22 h 0.03 0.17 h 0.02

0.59 I 0.06 0.37 & 0.05

0.81 $ ;E



Spectroscopic factors

0.18 + 0.04

0.46 i 0.1

0.57 f 0.2

Ref. [ll]








-5.016 --5.996

Ref. [31]

S(llB&N)l( (MeV)


a The first two pairs of columns to the right of the reaction identification give the excitation energy (in MeV above nuclear ground state) and spin and parity of residual nucleus and ejectile nucleus, respectively. Columns labelled “Transfer, i -f” indicate the Li quantum numbers of the inital and final states of the transferred nucleon. Q denotes reaction Q-value. Spectroscopic factors are given as for zaMg(llB, x)Y with S(“B f N) denoting the spectroscopic factor y{(llB, X)}, and S(aeMg + N) denoting S{*8Mg(,) n. The last columns contain values of Y{eSMg(,)Y} deduced from experimental data using WOMBLE ONE and taking values of Y{(llB, X)} from Ref. [31]. For comparison, other values of y{*aMg(,) Y} taken from the literature are quoted. These are obtained from light ion experiments 1291 and from results [l l] of a standard DWBA fit to the same data. Confidence limits, inserted on deduced spectroscopic factors in columns 10 and 12, are intended to give an approximate indication of (relative) qualities of theoretical fits to the experimental data. These figures take no account of possible normalisation errors in the experimental data, nor of any systematic errors occurring in numerical calculations (believed to be less than a few percent). * Results of calculation using WOMBLE ONE V03 (see text). c Results of calculation using WOMBLE ONE V02 (see text).


*6Mg(11B, =B)=Mg














0.0 ,,







Residual nucleus

*OMg(llB, l*C)*sNa

*@Mg(ltB, lOBe)*‘Al


TABLE II Particulars of ssMg -t “B Induced Single Nucleon Transfer Reactions’






w 8





main distorting effects are described in terms of the elastic phase-shifts. These expressions take the form of partial-wave sums. Our approximation for TBatakes the form of Eqs. (43)-(4% with F,,(q), x(q) and p. defined by (22), (25), (30), together with preceding equations. The phase-shifts, 6, , e = 0, 1) 2 ,...) are averages of the elastic phase-shifts for the entrance and exit channels, as described in Section 2.5. The validity conditions are expressed by (33)(37), where t is given classically by (67) in terms of the components of the angular impulse, A, which is given by (55)-(58). A further approximation, involving the additional assumption (47), leads to an even simpler formula for Tso,in which (48) replaces (45). This result is recognizable as just that originally deduced by anzatz from the eikonal-distorted-wave expression for T,, [4,20]. However, there is probably little advantage to be gained from making this last approximation, particularly since calculations using the more accurate formula involve very little extra effort. Calculations, some of which are presented here (in Sect. 3), which had already been performed using the original simpler form of the theory before the more accurate formula came to light, nevertheless proved to be capable of giving a good account of experimental data for single nucleon transfer. Some advantages of the new theory are: (i) the form of the final expression for T,, permits more efficient computation than does quanta1 full-recoil finite-range DWBA. (ii) the transition amplitude, TBa, involves elastic phase-shifts which are, in principle, directly measurable-rather than a potential which is not. Whereas the simpler formula (43), (44), (48) depends only upon the phase-shifts, our more general result (43)-(45) contains some potential-dependent effects relating to local kinematics during the transfer process. (Calculations show that results are not very sensitive to assumptions made concerning these). In both formulas, all the main dynamical effects-including those due to absorption-are contained in the phase-shifts. (iii) the theory is amenable to analytical techniques currently widely applied to elastic scattering [17, 27, 301. These are important tools for interpreting features of observed angular distributions. (iv) the theory has been demonstrated to be at least as good as full-recoil finite-range DWBA for describing direct nucleon-transfer between medium mass heavy-ions at -10 MeV/nucleon incident energy. Higher-order processes, involving inelastic excitations in entrance or exit channels (usually treated using CCBA), may be included in the theory by generalizing the function S(q) (see Sect. 2.5) so as to include terms analogous to the inelastic part of the s-matrix. Other conclusions of this paper concern the form of the DWBA for transfer. It is pointed out that the usual full-recoil finite-range formula (in either post or prior forms) has no claim to uniqueness. We show that alternative formulations in which the “three-body” nature of the problem is made more explicit can offer some advantages. These advantages are not only of a formal nature: it is possible to identify



kinematical and dynamical aspects of the various two-body subsystems and thus interpret features of a reaction in terms of classical models. Such a visualization can be useful in understanding the effects of recoil and of kinematical mismatching during the reaction. Fundamentally, the problem is not a three-body one but a many-body one. In a two-body system, one can often treat the effects of other open channels by means of absorption in the form of a complex optical potential. In a three (or more)-body problem, there are difficulties in treating absorption this way owing to additional degrees of freedom affecting the number of ways the particles can interact, in a pairwise manner, in attaining a given final state, This difficulty is highlighted in the “three-body” picture of transfer when one wishes to consider the effects of impulses on the internal nuclear states. This problem may be rendered tractable in a formulation in which one can identify the various two-body interactions, and hence determine the impulses felt by each nucleus. Absorptive effects, hitherto neglected by making the usual “inert-cores” approximation, may then be reinserted.s In Section 6 we have identified those situations in which such additional absorption is unimportant, and they are shown to be consistent with conditions under which the theories here described are applicable. Conservation of total energy, linear momentum and angular momentum in a peripheral reaction model leads to estimates of the classical impulses (54 et seq.) to which the nuclei are subjected. Simple arguments show that the optimum conditions for transfer are just those for which these impulses are minimized. These conditions are expressed by (62) in cases of small mass transfer. (More general conditions are obtainable from the formulas given in Appendix Bthough these may be more conveniently applied after substitution of actual numerical values of ei , Ed, /3 and y).




In this appendix we consider a “three-particle” scattering problem in which a particle, a, is scattered off a bound state, A = b + c, of two particles, b and c. The scattering potential U(R) is assumed to act only between a and b. B One possible way of including the effects of core excitation in giving rise to absorption is to replace the form factor F&(q) in (44) by the modified form factor q&I)

= F&lNl

- C.W(l

- Cds)>/(l - iL(dXl

- IErml”a

where xi(q) is the static form factor [32] for the nucleus i, and is the probability that the impulse q leads to excitation of the system; and where s = 1kOg - koa I. The static form factor is given by an integral over energy of the dynumic form factor (or responsefunction), R(q, w),viz. C(q>=s,” R(q, w)dw For a zero temperature “infinite” fermion system, c(q)



where 8(x) is the step function, Jr,


S(T) d7.






‘%q -



The wavefunction,


!P, satisfies the three-body





where B is the binding-energy (B < 0) of b T c in A, and k is the incident vector. The rest of the notation is as described in Section 2.1. For convenience, we adopt here the slightly revised notation,


(and the results become applicable to b scattering off B = a + c when particle and channel labels are suitably interchanged). The coordinates are illustrated in Fig. 9.

FK. 9. Coordinate

system used in Appendix

A for describing a 3-body scattering problem.

The masses,mi and pd , may be written in terms of the reduced mass, m, , of the cores, a and b, according to: mi = ma/(1 - 4, where

Equation (A.l) then becomes:



-‘8)v~-&($)v~+ 0

=(l-++‘. = I 2m,

Vs) + U(r + es)/ Y (A.41



This equation involves the coordinates r and s. It may be transformed involving just R and s, where

into one

R = r + ES,


by v,= 3 VR2 whereupon


the trial solution, Y = Z/J(R)&Sk’R[x(s) e-irk-#],

where x(s) is the unperturbed equation,

bound-state wavefunction satisfying the Schrodinger

-~~2i2/[email protected]

V,zx + Wx

and where #(R) is a function to be determined, -(fi2/2mo)


xv29 - (fi2i”lmo>B(V$ - &,#I

= --Bx;

yields after some manipulation: - Vx + WI

x4 = (fi2~02/2wJ xh (A.9

where k, = (1 - #k.


In general, Eq. (A.9) cannot be satisfied by a function 4 which is independent of s. However, we can make an approximation which has the effect of replacing Ox, in (A.9), by zero, thus eliminating the dependence of the equation on s. This can be illustrated by substituting,

which reduces (A.9) to an equation for g:

--(~“/[email protected][xV2g+ Wg) * orox- iPx>] -I- Urn) gx = 0. If the incident energy is sufficiently high, then the magnitude of #?Vx will be everywhere, except near the nodes of x, be much less than that of k,x. That is to say: for sufficiently large k. the main momentum components which make up the boundstate will be negligible in comparison with the momentum of relative motion. In such




a situation we can neglect the internal momentum by putting Vx = 0 in (A.9) which then reduces to a two-body equation. The resulting approximation to the solution of (A.6) is thus found to be: (A.1 1)

y(R, s) = $(R) eir6k.Re-irk’sx(s), where z&R) is the scattering solution of the two-body Schrbdinger equation,


-(fi2/2m3 V2$(R) + U(R) +(R) = V%2!2mo) ICI(R),

with the appropriate boundary conditions, and where k, is given by (A.lO). The wavefunction # is a solution of the two-body scattering problem describing scattering of the cores, a and b, by the potential, U(R), at an energy which corresponds to the mean asymptotic relative velocity, u = fik/mi . The wave-vector k, corresponds to this relative velocity in the [a + b] system, viz.: k, = m,v/h = (m,/mJk. Using (A.2,3,5,10), (A.ll)

can be reduced to: Y(R, s) = #(R) e-ikO.Reik’rx(s),

where t = R - ES.








Using the coordinate conventions defined in Section 2.1: R = ri + cisi = -rf

- l fsf

li = ii + EiSi = ---if - Efkf Hence p. = m,li = m$i + c,m& = -m$,

= @dmi) Pi + (Em,l~J

- qm&

pi = (mO/mf) or - (Erm,lpJ

7~ ,

where pi is momentum

conjugate to ri in two-body system of [a + A];

pr is momentum

conjugate to -rf in two-body system of [b + B];

7i is momentum

conjugate to si in two-body system of [b + c];

TVis momentum

conjugate to sf in two-body system of [a + c];

p0 is momentum

conjugate to R in two-body system of [a + b].




The angular momentum

in the a + b system is

L = R x PO = (~o/mi)(R = (mo/mz)(R

x pi) + (wo/~i>(R

x ~3

x PA -

x 4.



We take the instantaneous position of c, during a transfer process, to be on the line joining the centres of a and b, so that si = yR, sf = (y - l)R

(0 < y < 1).

It then follows that Ti = R -r,


= R(l - E~Y),

= R + qsf = R(l - cf(l - 7)).


R X pi = [l/(1 - l iy)] R x pr = [-l/(1


X pi

= [l/(1 -

l n)]

li ,

- Ef(l - r))l rf x p, = [l/(1 - Es(l -


where 8, and 8, are the initial and final angular momenta of relative motion. R X pi = (l/y)(si

X it) = (l/r) 2, )

--R x sf = H/U - r)l(s, x 4 = [l/(1 - r)l z+e ,

(B-3) Also (B-4)

where L’, and fD are angular momenta of c relative to b and a, respectively. Combining (B.3) and (B.4) with (B.2) yields

=- m. mf 1 -


l f(l

1-B - r) ” + 1 - y “.

Since mo/mi == 1 - E$ and moJmf = 1 - Ef(l - /?), where /3 = m,l(m, have that

+ mJ, we

(B.5) = ( l1 --f(l - Ef(l --“)lf+(*)z6. 7)





If (Ii , &) and (I,, &) respectively refer to the initial and final states of a system. then the impulsive change in L is given by

If angular momentum

is conserved, then:


For small ei , Ed, /,, , f13 :

* ‘v CP - Y)M% +

CEf -

Ei -

+ Ef+ l i2y i- Ef2U- y))(& t l,) l/Y)





Ei +






Note added in proof. The above analysis takes no account of the longitudinal (radial) component of the impulse X, . (All components of t are considered through (60)). This impulse component is zero if the change Q,“” in the energy in the [a + b] subsystem satisfies (2;”

EZ -;,

.x, = -ic . (A x IQ/(/3 - r)R2 - -A - M/[&g - y)P],


where and M = m@

x i3 = &l

- 2tW f ds/w)C - ((1 - IWO -



We note that Qtf’ above is the impulsive change in the cores’ energy (as distinct from Q,, (Eq. (26)) which is the change in the average energy). Starting with the relations pi = p. - j3cOmBic, pf = p0 + (1 - &,moi,, where Q = cJp(l - $3) =
= Qzff + (q,im,R2)L * M + (1 - 2/?)(@ * M + A . L)/(2m,(j3 - r)P)

+ 0(eo2),


where EK = $(pt2/mt + piz/mi) (27) and 1 err = &e/m, - pzzjmi). Combining (N.l) and (N.2), and using the approximation (56), yields a second optimum condition for transfer in the form: (moR?L%%,,

- Q,““) + &Rev4

- 2B)lL - i(ci + l ,,)L + A . ~[E,,/(E~+ c,) + B/(/3 -- Y)]




where& =lo!.i~L,X,=i,.F<
ACKNOWLEDGMENTS The author would like to thank Brian Buck for much help and encouragement, particularly during the early stages of this work; Ray Mackintosh and David Brink for illuminating discussions and constructive criticism; members of the Oxford Nuclear Physics heavy ion group for discussions and for making their data available to me; Dr. R. Rook for supplying the original bound-state routine; and the Computing services of the Universities of Oxford and Manchester (UK) for their help and cooperation with the development and running of the programmes. Financial support from the Science Research Council is gratefully acknowledged.


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