Recoil corrections for single-nucleon transfer reactions

I

2*G

I

Nuclear Physics

A223 (1974) 394-408;

@

North-Holland

Publishing

Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

RECOIL CORRECTIONS FOR SINGLE-NUCLEON TRANSFER REACTIONS+ PAUL s. HAUGE It Cyclo.tron Laboratory,

Physics Department,

Michigan

Michigan

State University,

East Lansing,

48823

Received 11 January 1974 Abstract: A formalism is developed for approximately treating the recoil bf the projectile in singlenucleon transfer reactions. The “no-recoil” term is evaluated exactly by rotating the nuclear matrix element to a coordinate system where the lighter projectile lies along the z-axis relative to the target nucleus. Recoil terms are constructed by expanding the “recoil phase factor” in terms of spherical harmonic and Bessel functions, and they are again evaluated exactly. Numerical results are then obtained for some representative light-ion reactions of high incident energies. For these reactions, we find that although the first-order recoil term may be very large, higher order terms are correspondingly less important, and convergence is always easily obtained. However, absolute magnitudes of cross sections at forward angles predicted by the present theory and by the local energy approximation differ by up to 50% for the reactions we consider.

1. Introduction Recoil corrections to single-nucleon transfer reactions have been, considered many times in the past, both by exact ’ - “) and approximate 5*“) means. Unfortunately, the exact finite-range calculations have been found to be quite costly and time consuming, and so there still remains an active interest in finding alternate ways of suitably treating recoil. Such a procedure is developed in the present work. One limitation of the two approximate methods that have so far been considered for recoil 5*“) is that they both impose some rather severe restrictions on the wavefunction of the transferred particle bound in the target nucleus. Dodd and Greider ‘) approximate this wavefunction with harmonic oscillator orbitals, which are realistic only for deeply bound wavefunctions, while Nagarajan 6), on the other hand, considers a spherical Hankel function valid only for weakly boufid particles. In the present work, we formulate a theory valid for wavefunctions of arbitrary numerical shape. Although the resulting form factors are not analytic, we will show that they can be rapidly computed by numerical means and readily inserted into standard zerorange DWBA codes. In sect. 2, we present our method for effectively computing the recoil terms. Then, in sect. 3, we specialize the formalism to light-ion projectiles where the transferred t Supported by the National Science Foundation. tt Present address: Exploration Systems Division, Esso Production 2189, Houston, Texas, 77001. 394

Research Company, P.O. Box

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395

CORRECTIONS

nucleon exhibits an s-wave orbital in the projectile. Quantitative calculations are also performed. Finally, in sect. 4, we discuss the results, and draw some conclusions about the importance of recoil corrections. 2. General formalism We begin by considering a stripping reaction of the form afA

-+ b+(x+A)

E (bfx)fA

= b+B,

(1)

where x represents a nucleon that is transferred from the incident projectile, a, to the final target, B. The notation that we use is similar to Satchler’s ‘), and the coordinates needed for specifying the interaction are shown in fig. 1. One can show in a straightforward rnamer [see for example, refs. “>“)I that the transition amplitude can be f

a

b

i+ T 7

%

X

?; B

A

Fig. 1. Vector diagram showing the coordinates

needed in specifying the present reaction.

decoupled (via two spectroscopic amplitudes and four Clebsch-Gordan coefficients) to a reduced transition amplitude, Bf;t2, involving a six-dimensional integration over two of the five coordinates in fig. 1. The most common way of expressing this quantity is Btj&(kb,

k,) =

s

94x+@,>~~)CII/ZI,I,(Y~>V(~,)~~*~~(~~)I, (2)

d3r, d3rbX-(ks

where E, (I,) and 2, (A,) specify the orbital angular momentum quantum numbers of the bound state wavefunction about the core b(A). We choose to work in the post formalism *), and V(r,) therefore describes the potential between the transferred nucleon and the outgoing projectile b. The functions inside the square brackets of eq. (2) are normally referred to as the “nuclear matrix element”, and they can be recoupled to a “transfer function” of definite angular momentum defined as GE(ra

p5) =A~2Wi&L; -h,h, WCC--_)“‘G, -,,(rl>v(r1W~l.,(r2)l.

(3)

Following Nagarajan’s choice of integration coordinates 6), we express the vectors r, and rb in terms of r and rl, as A rb = -r3

x

-rl,

B

r, = r-t Xrl, a

P. S. HAUGE

396

where A, B, a, and x refer to the masses of the respective particles in fig. 1. We now formally expand the scattered waves about Y, and apply a local WKB approximation to the translation operator of the incident wave. The result is x-(kb,

YJ = e-(xiB)rl ’ “x-

x+(k,,

r,)

=

Me

e+(x/@Q * vx+(k,

,

r)

M

ei(x/drl

* k”‘x+(ka,

$

(5)

where ka(r) and kb(r) are the local momenta ‘) of the projectiles at distance r. Substituting the approximations of eq. (5) into eq. (2), we find that the reduced transition amplitude can be written as J%?,(k, 2 k) = /d’rX-

(k, y i r) x+ (k,, r)ld3r,

ei” ’ Q(r)~~12(r2)V(r1)~111.1(Y1),(64

where Q(r) is the “recoil momentum” of the transferred particle defined as Q(r) = f k,(r)+

$ kb(r).

(6b)

The four functions inside the second integral of eq. (6a) can be appropriately referred to as an “effective transfer function” ‘), and they differ from the normal transfer function of eq. (3) by the “recoil phase factor”, exp(ir, * Q). As will be seen shortly, our present formalism is simplified considerably when Q(Y) is taken to be along the lab z-axis. For the present, we achieve this condition [as is done in ref. “)I by neglecting the recoil due to the target nucleus [i.e., the second term of eq. (6b)], and by considering only stripping reactions, for which k, is automatically defined to be on the z-axis. Pickup reactions can of course be evaluated by considering the equivalent stripping reaction through means of the principle of detailed balance. At the end of this section, we also show how the recoil of the target can be included without too much additional complexity. The explicit form that we take for the recoil momentum is

where VaA(r) is the real part of the optical potential, and K is the reduced mass of the (a+ A) system. In order to test the validity of the local WKB approximation of eq. (7a), we repeated some of our calculations using only a plane-wave approximation, where Q(Y) is constant and everywhere equal to its asymptotic value, Q(r) =

zk,Z.

PI

Upon comparing the results of the two approximations, we found virtually no difference in either the shapes or magnitude of the final cross sections. Thus, the

RECOIL

CORRECTIONS

397

omission of the imaginary part of the optical potential in eq. (7a) should be a very good approximation. The integral over rl in eq. (6a), which we hereafter denote as 121t2(Q, Y), is very difficult to compute, because $&,(rz) is a function of both r and rl . One way of formally treating the problem is to expand this wavefunction as

and solve for g$,(r, rJ by numerical means. As has already been mentioned, both Nagarajan “) and Dodd and Greider “) approximate $&(~z) by relatively simple functions, so that analytic forms for the gifl(y, or) can be obtained. In the present work we evaluate this integral exactly by rotating the “effective transfer function” to a coordinate system where the vector Y (c.f., fig. 1) lies along the z-axis. The exact procedure is given in detail in appendix A. The final result, which includes recoil corrections to all orders, is

x c(ll, E,; OOO)c(lZ,2,; 02, Il)c(Z,

Z2 2;

-A,

A,M)F;$(Q,r)Y;,(P),

(9)

where I arises from the spherical expansion of the recoil phase factor, 2 describes the effective L-transfer of the term under consideration, and the form factor, Fy$(Q, r) is computed by eq. (A.7). The feasibility of evaluating eq. (9) is of course dependent on how rapidly IynZ, (Q, Y) converges with Z, and subsequently (as can be inferred from appendix A) on the value of 1QR I, where Q is the recoil momentum of eq. (7b) and R is some typical value for the distance between the transferred nucleon and the core projectile. In the next section we show that this procedure is, in fact, very practical for reactions induced by certain light ions of high incident energies. The no-recoil term in our present formalism can be computed exactly by substituting unity for the recoil phase factor in eq. (6a) [or equivalently setting _$Q ~1) = a,, o in eq. (9)]. The result is I?;& = ,c,‘-)%(I,

I, L; -A1 1, M)F~z2(Q, r)YtM(P),

(104

where FF”(Q, r) is the no-recoil form factor given by F;““(Q, r) = 872 c ~(2, I, L; v, -v, 0) Y X

~o~~:d~,Cv(r,)s,(r,)lSl~d(cos WX~&,,, &z).

The numerical integrations must be carried r = pz-rl is along the z-axis. Two previous cross section have been developed in the past. and Tobocman 9), also evaluates the no-recoil

w-4

out in a coordinate system where methods of computing the no-recoil One of these, formulated by Sawaguri term exactly. But their expression for

398

P. S. HAUGE

the form factor is entirely different and more complicated than the compact form given in eq. (lob). The other method, developed by Buttle and Goldfarb *, lo), assumes that the transferred particle is weakly bound in the target nucleus, so that $&(Y~) can be effectively replaced by a spherical Hankel function. Both of these no-recoil methods have had some success in explaining heavy-ion transfer reactions at sub-Coulomb energies ‘l). Q uan t a t ive calculations have also been carried out with Nagarajan’s formalism “) for including first order recoil terms, and these corrections were found to be quite important in many cases 12). Finally, we note that for some heavy-ion reactions, the masses of the projectile and target nuclei are of comparable size, so that the neglect of the second term in eq. (6b j is not justified. This problem can be resolved by expanding the scattered waves about Y,,instead of Y [c.f. eq. (5) and fig. 11. The form factor integral can again be evaluated by the manner shown in appendix A. Since all of our quantitativecalculations are for projectiles much lighter than the target, we shall not consider this problem further. 3. Applications to light-ion reactions The form factor integral of eq. (9) is simplified considerably when the transferred particle is bound in an s-wave orbital about the projectile, b, as is the case for all light-ion reactions with a, b 5 4He. For these kinds of projectiles, I, = 0 and I2 (which in this section we denote as L) is equal to the angular momentum of the transfer function in eq. (3). Eq. (9) then reduces to

I,";(Q, P) = ; i’c(ZLc!Z;0MM)FzL2(Q, r)Y&(P),

(114

with the form factors given by [c.f., eq. (A.7)] FzLZ(Q, r) = 471% ; c c(lL2; SQ x

s

v, -v, 0)

r?dr, d(cos e,)[j,(Qr,)T/(r,)Ul,(r,)l~,(P,)~L,

-dud

(lib)

Remember that the numerical integrations must be carried out in a “body” coordinate system where Y = r2-r1 is along the z-axis. The reduced transition amplitude can now be evaluated by combining eqs. (6a) and (1 la, b), and applying a partial wave expansion to the scattered waves. The final result is I$$(&)

k,)

=

X

c iz+“e-Lc(zL.L?; 12 ?.Tk, kb

OMM)

B c c(Z, m,; A zalb

M,

-M,

O)c(Z,

23,;

OOO)i’“-‘b-~

RECOIL

399

CORRECTIONS

Since L, is parallel to the lab z-axis, the two spherical harmonics are proportional to an associated Legendre function, and the last equation can be more conveniently written as

where B is the angle between k, and k,, and j3EM can be found by equating the bracketed terms of the last two equations. TABLE 1

Bound state neutron parameters System A+n

lHe+n 1ZC+n 49Tifn

S~g~e-p~icle contiguration of neutron (n, &j)

Neutron separation energy (MeV)

used in the present calculation Woods-Saxon

parameters

(M:V)

(f&

(f:)

-50.4 -44.8 -55.1 -51.5

1.79a) 1.25 1.25 1.25

0.65 0.65 0.65

I

-20.58 - 4.94 - 1.09 - to.95

OS* On+ Od& Of%

0.65

“) 25.0 25.0 25.0

The assumed potentia1 is of the form

where f(x) = I1 -f-exp(x)l-l, x = (~--R_4~)/e, and A is the mass of the system minus that of the transferred nucleon. In all cases, V. was varied to yield the observed neutron separation energy. “) Varied to predict the experimental rms radius of the transferred neutron wave-function, as explained in the text. “) S-wave orbitals are not affected by spin-orbit splitting. TABLE 2

Optical model parameters Target phts projectile

‘2C_tc(‘) 13C+ 3He “) 5eTi-i-3He d, @Ti+o( “)

used in the present work

Optical - mode1 parameters Lab energy (MeV) 139.0 109.1 ‘) 70.0 81.8

(ZV)

(zj

(2)

---10&l -110.5 -103.6 -127.5

1.22 1.09 1.24 1.25

0.76 0.83 0.75 0.78

(M:V)

(2)

-16.9 -16.7 79.6 -18.9

(2)

(tzl)

1.85 1.80 1.25 1.63

0.41 0.57 0.80 0.53

(fZ) 1.3 1.3 1.3 1.4

The potential is of the form V(r) = VRJC(XII)+i~:fCXI)l-iWLfl(XI)+ vc, wheref(x) = [I Sex&))-‘, x = (r-&4)/u, f’(x) is the total derivative off(x), and A is the mass of the target nucleus. The Coulomb potential, V,, is that of a charged sphere of radius &ii+. As seen from the footnotes, some of the optical model parameters have not yet been pub~shed. “) Optical model parameters taken from ref. l’). “) Optical model parameters taken from ref. 22). “) Energy of final projectile for Op+ transfer of fig. 2. Corresponding energy for the Ode transfer is 104.3 MeV. d, Optical model parameters taken from ref. I*). ‘) Optical model parameters taken from ref. 23).

P. S. HAUGE

400

One might expect recoil effects to be very important for light-ion reactions, simply because the mass ratio x/a in the recoil momentum of eq. (6b) is much larger than it is in a typical heavy-ion reaction. Nevertheless, recoil corrections for light-ion reactions have so far not been considered in detail. The primary reason is that an alternate theory, commonly referred to as the local-energy approximation (LEA) ’ 3, ’ 4), has been developed for s-wave projectiles, and is, at the present time, a very popular method for including some finite-range corrections to zero-range calculations. Therefore, in this section, we shall perform some quantitative recoil calculations, and compare them with the usual LEA cross sections, which are readily calculated from the computer code DWUCK 15). The recoil cross sections were calculated to all significant orders in I by evaluating the relevant form factors via eq. (1 lb) and numerically feeding them into the DWBA code DWUCK ’ “). Since DWUCK could handle at most five L-transfers at a time, we needed to store the /?EMof eq. (13) on permanent file, and read them back into a small final program that computed the final cross section. This procedure did not present too much of a problem, because the normal version of DWUCK already had an option for printing out a quantity proportional to the /3EMvariables. The numerical accuracy of our final code was checked against analytical results that are possible in the plane-wave limit 16). We shall now compare our recoil formalism with two recent reactions induced by highly energetic Light ions, namely 12C(a, 3He)13C at 139 MeV alphas “) and 5oTi(3He a)49Ti at 70 MeV helions I*). The bound-state parameters for the 3He+n system giien in the top row of table 1will be used in all of the recoil calculations, and they have been determined by (i) setting the diffuseness parameter, a, equal to the reasonable value of 0.65 fm, and (ii) fitting the potential depth, V,, and the range, roA*, to the experimental neutron binding energy (Err = -20.58 MeV), and the experimental rms radius of one of the four nucleons relative to the c.m. of the other three. We assume this latter quantity to be 2.17 fm, or $ that of the experimental rms charge radius of the cc-particle ’ “). With these values, we compute the normalization, Do, and range, R, of the interaction to be

Do =

f d3rlJTrl)$drl) rf dr, V(r,)Uo(rl)

R = [(60,)-l

/d3r,

= 359 MeV * fm’,

rf V(r,)$,(r,)]

* = 0.91 fm,

Pa)

W)

respectively, where tio(rl) = Uo(r1)/2/&. Note that our definition for the range is the same as that used in the computer code DWUCK ’ “) and in ref. 20), but is only half that of the definition used in many other works 17y21).The absolute normalization of the theoretical cross sections relative to experiment is, of course, very sensitive to

RECOIL

401

CORRECTIONS

the value of D, , and we shall consider this problem further at the end of this section. All of our calculations will be carried out with nonlocality parameters of p = 0.2 for both the incident and exit channels, and we shall assume a finite range parameter of R = 0.9 fm for our LEA calculations, as suggested by eq. (14b). In fig. 2, we show the experimental and three different theoretical curves for the reaction l*C(a, 3He)13C at 139 MeV alphas “). The bound-state and optical model parameters that we use are similar to those in ref. I’), and they are listed in tables 1 I

1

I

I

I

I

IO’

MeV

I

I

I

ALPHAS

io” 10-l

____ -.-‘- - - - -

10-2

RECOIL NO-RECOIL LOCAL ENERGY

10-3

10-4 L

Ok’; TRANSFER Q = -15.64 MeV E, = 0.0 MeV

‘\ .\ ‘1. \

IO2 IO' IO0 10-l 10-2 1O-3 OD $ TRANSFER Q = -19.49 MeV E, = 3.85 MeV

10-4 10-5 10-6 t 0

I

20

I

40

I

t

I

I

I

I

I

60

80

100

120

140

160

180

e,.,.

(deg)

Fig. 2. Theoretical and experimental cross sections for the (c(, He3), reaction to two different states in 13C. The recoil and LEA curves are normalized to the experimental data, but the no-recoil cross section is plotted relative to the recoil curve.

and 2. The optical model parameters were uniquely determined in both the incident and exit channels by past experiments 17722). Both the recoil and LEA theoretical cross sections of fig. 2 (denoted by the solid and dashed curves respectively) are norqalized to fit the experimental data. Notice that both of these theoretical curves compare equally well with experiment for the Op+ transfer, but that recoil does

P. S. HAUGE

402

1=0+1+2+3

0

5

0

5

DISTANCE

0

BETWEEN

5

0

5

THE TWO CORES-

0 r

5

(fm)

Fig. 3. Cross sections and form factors resulting from the four lowest orders of recoil corrections are displayed for the 0~;. transfer of fig. 2. (a) The four cross sections that arise from including only those form factors with 1 $0 through 1 d 3 respectively. (b) Zero-range and recoil form factors, both of which correspond to a normalization constant of Do = 359 MeV * fm*. Form factors of odd 1 are drawn with opposite phase so that they can be more conveniently displayed with positive amplitude near I = 0.

50T~(3He, aJ4’Ti

-

70 MeV HELIONS KEY:

-

10-5

RECOIL

-.-.-

NO-RECOIL

-

-

- -

LOCAL

ENERGY

-

-

0

20

40

60

60

100

120

140

160

180

8c.m. (W Fig. 4. Experimental and theoretical cross sections for the reaction 50Ti(3He, ar)49Ti at 70 MeV helions. The recoil and LEA cross sections are normalized to the experimental data, but the norecoil cross section is plotted relative to the recoil curve. Note that only the recoil curve predicts the correct slope at angles above &.,. = 75”.

RECOIL CORRECTIONS

403

slightly better than LEA for the Od, transfer. The third curve plotted in each diagram represents the no-recoil term, which is found by setting j,(Qrl) = 6,, e in eq. (11). The curve is not normalized to the experimental data, but is plotted relative to the recoil curve. The large separation between the recoil and no-recoil curves in fig, 2 is therefore physical, and demonstrates how important recoil corrections are to both the shape and absolute magnitude of the no-recoil cross section. We do not have any explanation for the violent oscillations that appear in the no-recoil curve for the 04%transfer. They did not occur in any of our other calculations. In order to make sure that these oscillations were not caused by mathematics inaccuracies, we increased both the matching radius and number of partial waves, and found no change in the computed cross sections. Note that the recoil corrections dampen these fluctuations considerably. A comparison of the recoil and LEA cross sections in fig. 2 reveals that the two curves have similar shapes for 0_. 6 50”, but differ greatly in both shape and magnitude for large angles. This variance is very large for the Ops transfer where the recoil cross section is smaller than that of LEA at 0 = 180” by several orders of magnitude. The reason for this discrepancy is probably because the LEA theory employs an expansion parameter that brea,ks down for large &,+ _ Th:: exact form of the parameter is (k,- (~~B)k~) - rl [refs. 13,14)], where k, and ki, are the relative momenta of the two projectiles. For small angles, k, and k, are nearIy parallel, and the expansion term is of the same order as that used in recoil, namely (x/a)k,, [c.f., eq. (6b)]. For large angIes, however, k, and k, are nearly antiparallel, and the expansion term is very large. Since LEA truncates the expansion after second order, we would expect the theory to fail at large angles. In fig. 3a, we show explicitly how important the various orders of recoil are to the total recoil cross section for the Op+transfer considered in fig. 2. The results displayed here are typical of all our calculations. The I = 0 cross section is not the same as the corresponding no-recoil curve in fig. 2, because of the additional function j,(QrJ in eq. (llb). Their shapes are quite similar, but their ma~itudes differ by almost a factor of two. ~though the I = I terms produce most of the recoil corrections, the I = 2 terms are also important, even in the forward angles where we see that they approximately double the magnitude of the cross section. The 2 = 3 terms contribute noticeably only at very small cross sections. The relevant form factors for this calculation are shown in fig. 3b. For comparison, we also show the zero-range form factor in the extreme left-hand diagram. The analytic expression for this form factor, FZR= D,U,(r), can be readily found by substituting the function Dc,~~(v,)/JG for the range term, V(rI)U,(rl) in eq. (llb). Note that the main effect of recoil is to greatly reduce the amplitude of the zero-range form factor, and to redistribute it over many different values of 3, the effective angular momentum transfer. From our discussion so far, we see that the experimental cross sections must be measured out to relatively Iarge angles in order to differentiate between the recoil and

P. S. HAUGE

404

LEA theories. Such an experiment, namely the pickup reaction 50Ti(3He, a)49Ti at 70 MeV helions, has recently been performed at the Michigan State University Cyclotron ‘*). The experimental curve, as well as the predictions of no-recoil, LEA, and recoil theories, are displayed in fig. 4. The bound state and optical model parameters used in the calculation are listed in tables 1 and 2. As expected, only the recoil curve can correctly describe the experimental curve at backward angles. The 70 MeV 3He optical model parameters, which are the only ones in table 2 to include an imaginary surface term, were taken from ref. l*), while the a-parameters were averaged over optical potentials derived from 80 MeV alphas on Ca and Ni isotopes ““). Spectroscopic

TABLE3 factors for the two ground state transitions

Description of spectroscopic factor [A+(W; Bl

Assumed spectroscopic factor S

[lW+op,. 2’ W] [49Ti+Ofz.z’ 50Ti]

0.6 “) 6 9

considered in the present paper Do needed to produce the assumed spectroscopic factor LEA recoil 246 306

278 236

The first column identifies the spectroscopic factor, and the most reasonable values for these two numbers are listed in column 2. Then we show, in the third column, for both the LEA and recoil theories, the normalization constants that are needed to yield the assumed spectroscopic factors of column 2. “) This number (to one significant figure) is obtained by the two independent shell-model calculations of ref. z4). “) This number (to one significant figure) is an average of two numbers (one experimental and one theoretical) quoted in ref. 25).

In order to test the valid&y,of our theoretical normalization constant in eq. (14a), Do = 359 MeV - fin?“, we must use the following relation between the experimental cross section for stripping, and the corresponding cross section that one computes with DWUCiT do

C-1 dS2

where C is the usual isospin Clebsch-Gordan coefficient, S is the spectroscopic factor, v is the number of identical nucleons available for the reaction [v = 2 for (a, 3He) reactions] 21), and Do is fhe normalization constant defined in eq. (14a). A similar relation holds for pick-up reactions ’ ‘). The results for the present reactions are shown in table 3. The first column specifies the two ground state transitions considered in this section, and in column 2 we give, to one significant figure, the best previously determined value for the spectroscopic factor of these two cases 24’25), as explained in the footnotes of table 3. If we extract the spectroscopic factors by means of eq. (15), using our theoretical value of eq. (14a), we find that our computed

RECOIL CORRECTIONS

40.5

spectroscopic factors, for both the LEA and recoil theories, are a factor of two to three larger than the ones listed in column 2. Therefore, in the third column, we give the normalization constant that is needed to predict the assumed spectroscopic factor of column 2. For comparison, we note that these extracted numbers are close to the value of Dc = 276 MeV * fm”- which was recently obtained from an analysis of the reaction 40Ca(a, 3He)“1Ca at 104 MeV alphas ‘“). We should not be too concerned that our extracted values of Do are roughly 50% smaller than our theoretical value of eq. (14a). Unlike other light-ion reactions, the normalization constant for (CL,3He) or (3He, a) reactions is still not well determined, and past experiments at lower incident energies have published values for (D~~I04~ ranging from 15 to 30 MeV’ * fm3 fiefs. 19Z20Z27)].This corresponds to a Do up to 50% larger than our computed value of 359 MeV * fm*! These reactions at low incident energies, however, usually consider optical model potentials with a much deeper real well depth than the high energy ones listed in table 2 and ref. “). This difference in potentials surely contributes to some of the ambiguity in Do, and it is certainly a problem that requires further study. 4. Conclusion A formalism has been presented for effectively taking into account all orders of recoil introduced by the recoil phase factor of eq. (8). Our procedure for evaluating these recoil terms has been to rotate the nuclear matrix element to a coordinate system where r, the vector connecting the two core nuclei of fig. 1, is parallel to the z-axis. The idea is basically very simple, and it leads to a very compact expression for both the recoil and no-recoil form factors [eqs. (9) and (lOa, b) respectively]. From the quantative calculations of sect. 3, we see that the method appears to be a practical way of treating recoil. The formalism can be generalized to m~ti-nucleon transfers in a straightforward manner, and recoil corrections are presently being performed for the reaction i3C(3He, 6He)ioC at 70 MeV helions. Previous calculations for this reaction, using only the no-recoil theory 28), failed completely in describing the shapes and relative magnitudes of the observed cross sections. There is some question, of course, regarding the validity of replacing the gradient operator of eq. (5) with the local momenta of the projectiles. Nagarajan has recently considered this problem 29), by replacing the recoil phase factor with the quantity (1+ i(~/a)r~ * V), and then formally operating on the distorted wave xf(&, r,). He again uses the Buttle-Goldfarb approximation for the bound state wavefunction in the target nucleus. Baltz and Kahana 30) have extended this procedure by again including the gradient operator, and also evaluating the relevant form factors exactly through the method of Sawaguri and Tobocman “).,In order to see how important such corrections might be in our present work, we repeated the recoil cross section of fig: 4 using only the plane wave approximation of eq. (7b), and found virtually no difference in either the shape or magnitude of the final cross section. So additional

406

P. S. HAUGE

corrections, due to the explicit use of the gradient operator, will most likely have little effect on the results of sect. 3. Finally, we note that there are still noticeable differences between the LEA and recoil theories in the magnitudes of the total cross sections at forward angles. This is evident from the extracted normalization constants in table 3. The LEA does include part of the first order recoil theory exactly by means of Green’s theorem and gradient identities 13*14). Bu t 1‘t a 1so neglects many of the finite-range effects that are included even in a no-recoil calculation. It would be very informative if our recoil calculations, using the parameters of tables 1 and 2, were repeated with an exact finite-range code lT4). 0 ne could then decide definitely which approximation, recoil or LEA, offers the best method for extracting spectroscopic information from highly energetic light-ion reactions. The author is indebted to G. Bertsch for many helpful discussions, and also for suggesting the present method of evaluating the form factors. I am also grateful to R. Doering and A. Galonsky [ref. ‘“)I for allowing me to show one of their unpublished (3He, a) cross sections in fig. 4 and their 3He optical model parameters of table 2. Finally, I wish to thank D. Goldberg and H. H. Chang [refs. “, ““) respectively] for granting me permission to list their unpublished optical model parameters in table 2. Appendix A THE EVALUATION

OF

We begin by expanding exp (ir, . Q) in terms of spherical harmonics so that

where Ul,(r,) is the radial part of the function $rIRZ(rl). If we neglect the recoil of the target nucleus B, and consider only stripping reactions, Q is parallel to the lab z-axis (as explained in the text), and eq. (A.l) can be reduced to p2 ata2

=

4~T

i’ $===d3r, 4r,(Q, r,)Y,,(h)Y,,,,(~,)( -)%2, -&z),

(A.3

where 1 zzz(2Zf l)* and J4rr1(Q, rl) re p resents the three square-braced terms of eq. (A.l). There are two equivalent ways of evaluating eq. (A.2) depending on how one wishes to couple the three angular functions to overall angular momentum 9. The numerical calculations are somewhat simpler if we first couple the two harmonics of

407

RECOIL CORRECTIONS

coordinate rl to a new harmonic Y,,,,(P,)

[ref. “)I. We then have

We now transform the two angular harmonics to a “body-coordinate” system where Yis along the z-axis. Using Rose’s definition 31) for the rotational matrices, we have

where the primed (unprimed) coordinates refer to the rotated (unrotated) coordinates. Since the two angular wavefunctions are now specified from the z-axis, we see from fig. 1 that the only terms in eq. (A.4) which will survive the azimuthal integration are k,+ k2 = 0.Using this fact, along with the coupling relation of the rotation matrices 31), we find

s

d4XY,,dhk,, -&))

=C c(z, z22; a,, -a,, k

-A)

where in the last equation, we have recoupled the angular functions to angular momentum 9, performed the azimuthal integration, and used the fact that 13+I,-/-9 is an even number. Combining eqs. (A.3) and (A.5), we obtain our final result J:$,(ka,

k,)= C i'(-)"'c(ZZ1 2,;000) 2id x C(ZZ1 13; oa,a,)c(z3 z29; -a1a,d)F::‘?(Q,

r)Y_&(P).

(A.6)

The form factors are given by

where the brackets again denote angular coupling. Note that we have suppressed the primes on the integration coordinates for clarity. The function Iyi2 can now be computed as a function of r by evaluating the double integral for various positions of particle b on the z-axis (c.f., fig. 1). All functions are well-behaved, so that the integrations can be rapidly performed.

408

P. S: HAUGE

RefkrenCes 1) N. Austern, R. M. Drisko, E. C. Halbert and G. R. Satchler, Phys. Rev. 133 (1964) B3; R. M. Drisko and G. R. Satchler, Phys. Lett. 9 (1964) 342 2) R. M. DeVries and K. I. Kubo, Phys. Rev. Lett. 30 (1973) 325; R. M. DeVries, Phys. Rev. C8 (1973) 951 3) L. A. Charlton, Phys. Rev. C8 (1973) 146; Phys. Rev. Lett. 31 (1973) 116 4) C. A. McMahan and W. Tobocman, Nucl. Phys. A212 (1973) 465 5) L. R. Dodd and K. R. Greider, Phys. Rev. 180, (1969) 1187; Phys. Rev. Lett. 14 (1965) 959 6) M. A. Nagarajan, Nucl. Phys. Al96 (1972) 34 7) G. R. Satchler, in Symposium on heavy-ion transfer reactions, vol. 1 (Argonne National Laboratory, Argonne, Ill., 1973) p. 145 8) P. J. A. Buttle and L: I. B. Goldfarb, Nucl. Phys. 78 (1966) 409 9) T. Sawaguri and W. Tobocman, 5. Math. Phys. 8 (1967) 2223 10) P. 3. A. Buttle and L. J. B. Goldfarb, Nucl. Phys. All5 (1968) 461; Al76 (1971) 299 11) F. Schmittroth, W. Tobocman and A. A. Golestaneh, Phys. Rev. Cl (1970) 377; U: C: SchlotthauerVoos, H. G. Bohlen, W. Jon Oertzen and R. Bock, Nucl. Phys. Al86 (1972) 385 12) D. G. Kovar, B.G. Harvey, F. D. Becchetti, J. Mahoney, D. L. Hendrie, H. Homeyer, W. von Oertzen and M. A Nagarajan, Phys. Rev. Lett. 30 (1973) 1075 13) P. J. A. Buttle and L. J. B. Goldfarb, Proc. Phys. Sot. 83 (1964) 701 14) F. G. Perry and D. S. Saxon, Phys. Lett. 10 (1964) 107; Gy. Bencze and J. Zimanyi, Phys. Lett. 9 (1964) 246 15) P. D. Kunz, University of Colorado 16) N. K. Glendenning, Ann. Rev. Nucl. Sci. 13 (1963) 191 17) S. M. Smith, G. Tibell, A. A. Cowley, D: A. Goldberg, H. G. Pugh, W. Reichart and N. S. Wall, Nucl. Phys. A287 (1973) 273 18) R. Doering, R. Bortins, A. Galonsky and R. Hinrichs, private communication 19) R. F. Frosch, J. S. McCarthy, R. ,E. Rand and M. R. Yearian, Phys. Rev. 160 (1967) 874 2b) D. H. Youngblood, R. L. Kozub, J. C. Hiebert and R. A. Kenefick, Nucl. Phys. Al43 (1970) 512 21) R. Stock, R. Bock, P. David, H. H. Duhm and T. Tamura, Nucl. Phys. A104 (1967) 136 22) D. A. Goldberg and S. M. Smith, private communication 23) H. H. Chang, N. King and B. Ridley, private communication 24) S. Cohen and D. Kurath, Nucl. Phys. 73 (1965) 1; P. S. Hauge and S. Maripuu, Phys. Rev. C8 (1973) 1609 25) J. D. McCullen, B. F. Bayman and L. Zamick, Phys. Rev. 134 (1964) B515 26) G. Hauser, R. Lohken, G. Nowicki, H. Revel, G. Schatz, G. Schweimer and J. Specht, Nucl. Phys. Al82 (1972) 1 27) R. R. Betts, H. T. Fortune and R. Middleton, Phys. Rev. C8 (1973) 660 28) E. Kashy, W. Benenson, I. D. Proctor, P. Hauge and G. Bertsch, Phys. Rev. C7 (1973) 2251 29) M. A. Nagarajan, Nucl. Phys. A209 (1973) 485 30) A. 3. Baltz and S. Kahana, Bull. Am. Phys. Sot. 18 (1973) 1383; to be published 31) M. E. Rose, Elementary theory of angular momentum, (Wiley, New York, 1957) p. 52, 61