Center-of-mass corrections to overlap factors for transfer reactions (III)

2.G: 6.B [

Nuclear Physics A330 (1979) 91 - 108; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

CENTER-OF-MASS CORRECTIONS TO OVERLAP FACTORS FOR TRANSFER REACTIONS (III)t W. T. PINKSTON and P. J. IANO

Vanderbilt University, Nashville, Tennessee 37235

Received 12 January 1979 (Revised 21 June 1979) Abstract: A correction is derived for the unphysical c.m. motion of the bound state wave functions used in DWBA calculations of reaction cross sections. The method is an improved version of a recipe reported earlier. It is based upon the observation that intrinsic wave functions, projected from shell-model wave functions by the Vincent procedure, tend to be unphysically larger in the nuclear surface and exterior regions. The new procedure leads to enhancements in cross sections, but they are smaller than those resulting from the earlier recipe. Calculations of two-nucleon overlaps relevant to (t. p) reactions on 160, ~°Ca, 9°Zr and 2°spb result in enhancements in distorted wave cross sections by factors of roughly 2.1, 1.6, 1.3 and 1.2 respectively. The previous cross section enhancement estimate of 4.8 for t60(t, p)lSo is reduced to 2.1. A particularly simple, approximate version of the method consists of replacing the c.m. coordinate ,~ with [(A + 1)/(A+ 2)].# in the two-nucleon overlap function. The underlying assumptions and approximations are investigated by a detailed study of the analogous but better understood problem of single-nucleon transfer.

I. I n t r o d u c t i o n

T h e goal of theories of t w o - n u c l e o n transfer reactions is the accurate prediction of cross sections which are free of a r b i t r a r y n o r m a l i z a t i o n s . T o achieve this goal requires the generalization of the distorted wave m e t h o d to include m u l t i - s t e p processes, finite r a n g e calculations, realistic forces a n d refinements of b o u n d state wave functions. (A striking feature of the data, the c h a r a c t e r i s t i c / - s i g n a t u r e of a n g u l a r distributions, for reactions such as (t, p), tends to be insensitive to reaction m e c h a n i s m a n d n u c l e a r structure a s s u m p t i o n s . Because of this, it has been possible to extract a great deal of spectroscopic i n f o r m a t i o n from data, using rather crude D W B A models. However, this p h e n o m e n o l o g y d e p e n d s u p o n the success of the microscopic a p p r o a c h for its u l t i m a t e justification.) T h e c o n c e r n of the present paper is a p r o b l e m which has received very little a t t e n t i o n , the c.m. corrections to the b o u n d state wave f u n c t i o n s e m p l o y e d in t w o - n u c l e o n transfer calculations. In p r e v i o u s papers 1.2) (referred to hereafter as I and II respectively) it was p o i n t e d out that c.m. corrections could be sizeable in nuclei as light as oxygen; in II a simple * Work supported in part by the US National Science Foundation and the Vanderbilt University Research Council. 91

92

W.T. PINKSTON AND P. J. IANO

recipe was developed for making c.m. corrections. It led to an increase of about 5 in the magnitude of the cross section for 160(t, p)180 (g.s.). This factor of 5 is surprisingly large. Other refinements, such as the use of better bound state wave functions, also result in large increases in absolute cross sections. There is no evidence that a factor as large as 5, in addition to the enhancements due to improved bound state wave functions 3,4), is needed to bring predicted cross sections into agreement with experiment; thus further study of c.m. corrections is in order to determine if the recipe of II overestimates them. The present paper reports on further studies. It is shown that a very similar but m o r e accurate correction, based upon the methods and results of II, yields an enhancement of about 2, rather than 5. The present recipe is based on the discovery that shellmodel wave functions, when corrected for the unphysical c.m. motion, do not have the correct asymptotic behavior in the nuclear surface and exterior region. Sect. 2 gives the background and introduces the notation used in the paper. The new correction is derived in sect. 3 and is applied in sect. 4 to two-nucleon stripping on 160, 4OCa and 9°Zr. The treatment in sects. 3 and 4 is brief and focused on the recipe and its consequences. A more detailed discussion of the underlying physics, the justification of the method and the nature of the approximations, is contained in sect. 5.

2. Background D W B A amplitudes depend on overlap integrals of initial and final bound states. Starting from the shell model, the following program, in principle, should be followed. First, a translation invariant internal function, q~, should be projected from the shellmodel wave function, ~b.

~A(r~" r~ . . . .

rA) --' ~A(~I, ~2

.....

~-1)-

(1)

The r i are laboratory coordinates and the ~ some set of appropriate internal coordinates. The arrow implies some projection procedure. The next step is to form the overlap,

F(~I' ~2) = f d ~ 3 ' '

• d~a+ 1 ~ ( ~ 3 . . . . . ~A+ 1)(~a+ 2(~1 . . . . . CA+ 1)"

(2)

The coordinates ~1 and ~2 locate the transferred nucleons relative to nucleus A. Possible choices are r, p and r 1 - R a , r 2 - R A in fig. 1. In practice the procedure outlined above is not followed. Instead, a shell-model overlap is first calculated: F .... (rl' r2) = f d r 3 " "

drA + 2¢*(r3

.....

rA + 2)~/A + 2(rl

.....

rA + 2)"

(3)

The shell-model overlap is then treated as if it were an intrinsic overlap; the coordinates rl, r 2 in eq. (3) are assumed to locate the nucleons relative to A. In essence,

c.m. C O R R E C T I O N S

III

93

t

Fig. 1. N o t a t i o n for c o o r d i n a t e vectors.

the standard procedure in DWBA calculations is to assume,

F ~ F.... (r l-RA,r 2-RA) ,

(4)

which should be a very good approximation for large ,4. The approach in I and II is based on the analysis of c.m. projection by Vincent 5). Clearly, it is possible, in principle, to expand any shell model function, ~, in terms of a complete set (X) of c.m. functions,

i=O

Vincent has shown how to determine the set of X~ ), ~ ) , for which the convergence in eq. (5) is most rapid, so that ~b can be thought of as being a product function, at least to a good approximation,

~A

~

y(o)~(o) ~

(6)

A~A"

Eq. (6) is the fundamental assumption of the present paper. Although ~ cannot be written as a product, we assume that the neglected terms in eq. (5) are unimportant. If this assumption is made in eq. (3), then it can be written, F . . . . (rl'r2)

=

f dRAXA(RA)XA+2( A ~A RA+

A +12

(r 1+r2))F(~,,~2).

(7)

The zero super scripts have been dropped to simplify the notation. In II arguments were given for approximating the X functions by ls harmonic oscillator functions, e.g. X A(RA) =

Uls(•A; R A ) ,

(8)

with vA = Av. (The notation of the present paper is the same as in II.) If the X's in eq. (7) are ls oscillator functions, then eq. (7) can be inverted. That is, given a shell model overlap, an intrinsic overlap can be obtained from it directly, without resort

94

w.T. P1NKSTON AND P. J. 1ANO

to the projections o n i~t A and ~A+2 implied by eq. (1). The inversion is very simple; it leads to a recipe: Expand F .... in terms of oscillator functions of r, R (see fig. 1 for a definition of coordinates) F ....

=

~ u,l(* ; r)u,~(V • R)u,t;, ,,

v' = ½v, v" = 2v.

(9)

The ct's are known. (The usual procedure, eq. (4), would consist of replacing R by p in eq. (9).) The intrinsic overlap can be expanded in terms of oscillator functions of p, r, with unknown coefficient fl, F = ~ U,l(v';r)u~,~(~;p)fl,lu~ ,

(lO) = 2Av/(A+2).

The fl- and a-coefficients are related by a generalized Talmi-Moshinsky bracket (discussed in the appendix of IlL A ]~'~, ~,l~ = fl,t~,, [_A +-2 3

(ii)

where N is the total energy quantum number. This recipe was applied in II to the reaction, ~60(t, p)~80(g.s.), and resulted in the large enhancement of 5, relative to the value obtained using eq. (4).

3. Improved recipe In order to search for defects in the recipe of II, we adopt a slightly different approach, based on the exact equation 8) for the overlap function, F,

[Ktl2 -~-V(~*)]F'-~-W ' : f A+2 w =

EF',

(12)

d¢3 . . . . . dCA+,4,~,* Y, (V,+V,2)~;,+2.

3

The operator, K'~2, is the kinetic energy of nucleons 1 and 2 relative to A, and E is ,the two-nucleon separation energy. The primes on the functions ~b, F, require an explanation. The functions q7 in eq. (12) are exact solutions of the many-body Schr6dinger equation, and F' is their overlap. The primes distinguish them from many-body wave functions, q~, of the previous section, which are derived from shellmodel wave functions by the projection, eq. (1). Eq. (12) bears a resemblance to the two-particle shell model. In the case of an Anucleus which is a closed shell, the interaction term W contains the interactions of the valence nucleons, 1 and 2, with the closed shell. One might attempt to obtain approximate solutions to eq. (12) by a shell-model ansatz, i.e. replace W' by a sum of one-body interactions operating on F', e.g. W' ~ [U'(r t -RA)+

U'(r2-RA)]F'.

c.m. C O R R E C T I O N S II1

95

This is the phenomenological procedure followed in dealing with one-nucleon overlaps. The result would be a three-body Schr6dinger equation. We feel such an approach would be dangerous and should be avoided. As discussed in I, the problem of insuring that approximate solutions of eq. (12) correspond to normalized, antisymmetric internal wave functions is quite formidable. [A discussion of the difficulties inherent in performing nuclear structure calculations with internal coordinates is given by Avilles 6). Austern and Wackman 7) treated 6Li, in the context discussed above, as a three-body system with Pauli principle constraints. Recently, Philpott 9) has developed methods for nuclear structure calculations with harmonic oscillator internal functions and has successfully applied these methods to some simple systems.] By contrast, the shell-model formulation is simple, at least in principle. The equation analogous to eq. (12), for the case in which A is magic, is 2M

(V~ " + - V 2 ) + g ( l r l ) +

g(r2)+

V(r)

1

F . . . . = EF . . . . •

(13)

Solutions of eq. (13) are constrained only to be normalized and orthogonal to all single-particle states which constitute the closed shell core, A. Very accurate approximate solutions to eq. (13) have been obtained by the approach known as the"extended basis shell model" 3.4). The approach of the present paper is as follows: We regard eq. (7), with the approximation, eq. (8), as a transformation which carries any function of internal coordinates, D(p, v), into a function, Dr(R, r), of laboratory coordinates, i.e.

DTIg, r) - T{D(p, r)}, ( 1 4)

= fdgAu,s(VA ; RA)Ul s(YA + Z ; RA + 2)D(P, r), =

'

, •

- - •R 2

where A+2

P- A+I

/ d'

(15)

and the N is are the normalization constants of the 1s oscillator c.m. wave functions. In what follows all the terms in eq. (12) will be transformed by T, the resulting equation being an equation in the laboratory coordinates. The purpose of the exercise is to determine whether, after suitable approximations, the shell-model equation, eq. (13),'is obtained. The transformation T has no effect on functions or operators which depend only on the coordinate r. Therefore, if one writes the kinetic energy operator as, -h E

K'12 =

M V'2

h2(A +2)

4M~

V2,

96

W.T. PINKSTON AND P. J. IANO

the transform of eq. (12) can be written as, -h

1

,

hZ(A + 2)

M V ; + Vir)JF r -

4MJ[

TiV~F , 2 , ,~+ T IW' I = EF'r.

(16)

Next we make a phenomenological ansatz, consistent with the standard shell model treatment of the interaction terms in W', viz. T{ W' 1 -. [U(r,)+ U(r2)]F'T.

(17)

Thus, the remaining task is to evaluate the transform of the term involving the pLaplacian. According to eq. (14), the integral of interest is,

f dp G(R, p)V,F 2 ,(p, r) = f dp[VZG(R, p)]F'(p, v) =

\A+2/(A+ I~2VZfdpG(R,p)F,(p,r) '

(J8)

which follows from Green's theorem and the functional form of G. From eq. (18) one easily shows that

T{V,]F'}

\,4+2/

exPkA+l}VR

exP\A+lJ

7].

(19)

Eq. (19) is next substituted into eq. (16). The A-dependent factors simplify as follows: A+2/

A

=

AZ+2A

=

If a new function, F, is defined as

(vR2~=F'~/q(R),

F = F~ exp \ A + ~ /

then combining eqs. (16), (17) and (19) results in

M

r

4MV~+U(rl)+U(r2)+V(r)

F = EF.

(20)

Fq. (20) is identical to eq. (13); hence we have F = F .... ¢ F). These mathematical results can be stated in words as follows: |f the inverse transformation T - t is applied to a shell-model overlap, the resulting internal overlap is not expected to be a good approximation to F'. However, if the shell-model overlap is multiplied by an appropriate Gauss factor, F' is obtained, i.e.

q(R)F .... ~ T{F'}.

(21)

The Gauss function in eq. (21) has a rather long range, which means that the recipe previously suggested is accurate in the nuclear interior but fails at large R. Since

c.m. C O R R E C T I O N S

111

97

transfer reactions take place in the surface and exterior, the older recipe is expected to give too large a correction factor. The older recipe is exact for the harmonic oscillator shell model. This is precisely why it is flawed in the case of more reasonable shell models. If one carries further the differentiation indicated in eq. (19), one obtains,

T{VoFr~

V2F'r+

+ (A~I)~ R2 F r +

In the harmonic oscillator shell model, the transformation indicated in eq. (17) can be carried out exactly and results in additional terms, not in eq. (17), which exactly cancel all but the first term in eq. (19'). In addition there is a term which makes the A-dependent factors cancel exactly, with no corrections of order A- 2. The exactness of the older recipe for oscillator overlaps depends very much on the "confining", r 2 behavior of the potential. The reasonableness of eq. (17), without any extra terms, for non-confining potentials was demonstrated by numerical calculations in II. The new recipe can be written as, F . . . . (R, p) = l~' ls(YA)J~' ls(~'a+

z)fdpO(R,

p)F(p, r).

(21')

The procedure for inversion of this equation is as follows: (a) Multiply F .... by q(R); (b) expand the results of (a) in oscillator functions (of appropriately chosen v, see II) as in eq. (9); (c) use eq. (11) to obtain the fl-expansion coefficients of eq. (10). The implementation of the recipe is straightforward. However, simpler, approximate methods would be desirable. The function G in eq. (21') is of short range. In the exterior region, F is a slowly varying function of p; therefore, for values of R in this region G can be approximated roughly by a delta function, with the result, F~'m(R'r) ~ LA(A+I)ZJ

(A+,)

\A+I

F(p, r) ~ F .... A + 2 O, r

'

' (22)

(large p).

For all except the lightest targets, the following is a very good approximation: A+I ~/A A+2 ~

A +2"

The approximation, eq. (22), can be regarded as a reduced mass correction of the kind discussed briefly in I. Further support for this approximation comes from the fact that F .... ( ~ p , v) is an exact solution ofeq. (12) in the asymptotic region. (This does not imply that it is the same as F in the asymptotic region. They obey different boundary conditions near the nuclear surface.) The accuracy of this approximation is discussed in the next section.

98

W . T . P I N K S T O N A N D P. J. IANO

4. Applications The improved recipe of the previous section has been applied to the reaction 160(t, p)180(g.s.) using the ordinary shell model approach, the c.m. correction recipe of II and the improved method. It was shown in II that changes in the overlap tended to have large effects on cross-section magnitudes but relatively small effects on angular distributions. Furthermore, the magnitude of the cross section resulting from the use of a given overlap is roughly proportional to the square of the zero-range form factor obtained from the overlap, evaluated at a point just outside the nuclear surface. The zero-range form factor was obtained by folding a Gauss function, exp ( - [3r2), With fl = 0.0864 fm- 2, into the singlet-S part of the two particle overlap. The resulting from factor was expanded in terms of oscillator S-functions and the c.m. corrections applied to this expansion. For simplicity a Woods-Saxon (ld~) 2 J = 0 wave function was chosen to represent the shell-model overlap. The half-separation energy method was followed, i.e., the depth of the Woods-Saxon well was varied so that the singleparticle energy is half the two-particle binding in 1sO. The bound state well parameters are listed in table 1. TABLE 1 Bound-state potential well parameters

i sO 42Ca 42Ca 9°Zr 2°SPb

no

Binding

Vo (MeV)

r o (fm)

a o (fm)

~'

Id 5 2 I f7,2 2p3~2 2d5,2 2992

6.15 9.92 9.92 7.94 4.56

54.30 55.14 60.43 51.04 47.19

1.25 1.25 1.25 1.25 1.25

0.65 0.65 0.65 0.65 0.65

18.94 18.90 18.90 18.90 20.00

The notation is the same as that of ref. 3). The r 0 and a o values are the same for the central and spinorbit wells.

The form factors are plotted in fig. 2. The shell-model form factor and the c.m. corrected form factors corresponding to the recipe of II and the improved recipe are labeled "s.m.", "old" and "new" respectively. All are shown as functions of the variable p. This makes sense for the shell-model form factor because of eq. (4). The differences in fig. 2 are striking. The improved c.m. correction results in a form factor significantly large in the surface,and exterior than the shell-model form factor but significantly smaller than the form factor resulting from the earlier recipe. It falls more rapidly with a slope very similar b u t slightly less than that of the, shell-model form factor. Since the shape of the corrected form factor is almost exactly the same as that of the shell-model form factor in the exterior, there should be very little effect on angular distributions, and the correction can be considered a numerical multiplier, which is approximately energy independent. The DWBA cross sections are

c.m. C O R R E C T I O N S i

DZ ¢Y rY

i

~

~

111

99 i

OLD NEW

i

SM "\

I- -

N 0 kl-

i

0

i

2

I

P(fm)

I

;

Fig. 2. Zero-range two-particle form factors for the Ida, 2 J = 0 state in '80.

proportional to the squares of the form factors, evaluated at about 5 fm. The ratio of these squares are old: new: s.m. = 4.8: 2.1: 1.0. Thus, the enhancement is reduced from 4.8 to 2.1. In order to get an idea of the A-dependence of the c.m. corrections three other cases were studied. These were the ground states of 42Ca, 92Zr, and 21°pb, treated as (If_;)2, (2dl) 2, and (2g~)2, respectively. The results are very similar to those shown in fig. 2. The c.m. corrections decrease with A as one would expect. In the case of 92Zr F is greater than F .... in the exterior region just outside the surface by 8-12 %; in 21°pb the amplitude enhancement is 7-8 %. In the case of 42Ca, the enhancement in F .... is approximately 20 %. The A-dependence is illustrated in fig. 3, in which the form factor "tails" are plotted for 180 and 42Ca. The approximate recipe, eq. (22), was also tested. In the case of 180, eq. (22) is accurate to about 5 % over the range of p-values of our calculations. The agreement is better if the A-dependent multiplier in eq. (22) is neglected. The recipe tends to worsen with increasing p and overestimates F for large p-values. It is a very poor approximation in the nuclear interior. Its use should be quite reliable for reactions which take place just outside the surface. Its use would be suspect in (a) light-ion reactions in which the interior plays a significant role, (b) low-energy, heavy-ion reactions, such as oxygen on lead, in which transfer takes place at distances greater than those of our calculations. Distorted wave calculations of (t, p) cross sections were performed with all the form factors generated. The computer program D W U C K lo) was used. The aim of

100

W . T . PINKSTON A N D P. J. IANO

'3

'

~5

1~I~8

'7

q2(fm) Fig. 3. Form-factor tails for ~sO and '*2Ca. The upper scale is for 18 0 ; the lower scale is for 42Ca.

these calculations was to get precise values of cross section enhancements and to confirm the conjecture that the c.m. corrections affect cross section magnitudes but not angular distributions. Confirmation is appropriate, since the (t, p) reaction is TABLE 2 The c.m. enhancements of (t, p) distorted wave cross sections (0 + --, 0 +)

Target

Final state

E t (MeV)

Oscillator v(fm 2)

10 160

(1d5,2) 2

Enhancement ") 2.21

0.367 20

2.06

10

1.61

(1 f7/2) 2 20 42Ca

1.59 0.27

10

1.59

20

1.55

(2P3/2) 2

9°Zr

(2ds/z) 2

20

2°Spb

(2g9,2) 2

20

a) Ratio of total cross sections.

0.20

1.29

0.183

1.19

0.156

1.17

c.m, CORRECTIONS 111

101

not entirely peripheral. The optical model parameters were taken from the formulas of Becchetti and Greenlees ~~' 12). Some of the results are given in table 2. The c.m. enhancement given there is the ratio of the total cross section calculated with the corrected form factor to that calculated with the shell model form factor. For ~60 and 4°Ca, calculations were performed for triton lab energies of 10 MeV and 20 MeV, to determine if there is a significant energy dependence. The results in table 2 clearly indicate that the energy dependence is slight, as suggested by the form factor shape. For 4°Ca calculations were made for (lf~) 2 and (2p~) 2 form factors to check for state dependence; the state dependence seems slight. The enhancements in table 2 can be fitted fairly well by the formula, 1.0+ 8.5/A °72. In fig. 4 angular distributions are plotted for 160 and 4°Ca reactions, to show that the c.m. corrections have very little effect on shapes. The cross sections based on shell model and c.m. corrected form factors are labeled s.m. and CM respectively. In the case of 9°Zr and 2°Spb, if the corrected and uncorrected cross sections are normalized so that they agree at 0°; then at all the secondary maxima, theydiffer by less than 1 '~,i. The 2~°Pb calculations should not be taken too seriously, although the 2°spb(t, p) results in table 2 are probably reasonable order-of-magnitude estimates• One of the assumptions underlying our method is that the wave functions of the occupied singleparticle states are approximated very well by oscillator functions. For light nuclei this is a very good approximation, if the oscillator 'parameter v is properly chosen.

i

160(t,p)

\kll

\\

o, CM

,Oca ( CM

E t = 10 M.V

b

~cM 20

40

60

80

100

120

Fig. 4. Distorted wave calculations of (t, p) cross sections, showing the c.m. enhancement.

102

W. T. P I N K S T O N A N D P. J. I A N O

For a nucleus as heavy as Pb, this assumption breaks down, and it is not clear what choice should be made for v. The results of two calculations for 2°SPb(t, p) are listed in table 2, for v-values of 0.1842 fro-2 and 0.156 fm-2. The former value has been used in shell-model calculations 13) in the lead region. The latter smaller value is extrapolated from the v-values of O, Ca and Zr by assuming an A i dependence. There seems to be a weak v-dependence. 5. Further discussion

In this section the validity of the assumptions underlying the development of sect. 3 and the approximations employed are discussed in somewhat more detail. Underlying everything in l, II and the present paper is eq. (6); eq. (7) follows from eq. (6). Shell-model wave functions do not in general separate in a product as in eq. (6) but we treat them as if they did and argue that it is a good approximation to do so. Because of the great difficulty in carrying out in practice the program outlined by Vincent, "approximate separability" must remain a conjecture. Studies 1~) of oversimplified models of the extreme case of A = 2 and the 4He indicate a high degree of separability; however one must be careful. Strictly speaking, what should be implied by an equation such as eq. (6) is that the error in the function on the right hand side does not exceed some acceptable value, no matter what ralues are chosen .[or the independent rariables. It is unlikely that this is the case. What we assume is that eq. (6) is adequate Jor our purposes. This assumption is guided by Vincent's insights but is essentially intuitive. We suspect that one of the reasons why the shell model works so well (even when oscillator functions are not used) is that eq. (6) is adequate for most purposes. The practical implementation of the c.m. correction, eq. (7), depends on a further assumption, eq. (8), namely, that the c.m. functions, X, can be approximated by oscillator 1s functions. At sufficiently large distances this assumption must fail badly, because of the unphysically rapid decay of the Gauss functions. We need to determine whether or not this unphysical behavior poses any serious problems for our model. Could the spurious Gaussian asymptotics vitiate the conclusions of sect. 3? We also need to know over what range the 1s oscillator functions are good approximations. This latter issue has been studied, in one case, in II. The probability distribution of the c.m. coordinate R was calculated for a Slater determinant of Woods-Saxon eigenfunctions corresponding to the ground state of 160, and was compared to /o over three orders of magnitude [UI s(DA;RA)I2. The curves agreed to better than 10 o/ of change. At the value of R (0.58 fm) for which u equals e - 1 of its value at R A = O, the agreement was better than 1 Yd,. The reason for this excellent agreement is that the occupied single-particle states of a light, closed-shell nucleus are very tightly bound and well represented by single oscillator functions. Because of this, eqs. (6) and (8) should both be good approximations. As one goes away from such a closed shell, the valuence orbitals are more weakly bound; this could make eq. (8) less

c.m.

CORRECTIONS

111

103

accurate. However, the c.m. motion is a collective property of all the nucleons, and there are far more nucleons in tightly bound orbitals in 180 than there are in weakly bound valence orbitals. This problem will be addressed more quantitatively in later paragraphs. In order to illuminate some of these issues, we adopt the approach of II and consider one-nucleon transfer instead of two-nucleon transfer. The reason for this is that one-nucleon transfer is understood much better, so much better that it can be used as a test of our ideas. However, it is important that what follows not be misconstrued. We are not developing a theory of c.m. corrections for one-nucleon transfers. As pointed out in I and II, there are no c.m. corrections for one-nucleon transfer except for a multiplicative factor of ((A+ 1)/A) ~N. What is done in the following paragraphs is to determine whether the transformation, analogous to eq. (7), carries one-particle overlaps (which are rather well understood) into one-particle shellmodel functions (which are also rather well understood). In the process of this study, some of the questions posed earlier are answered rather conclusively. The exact equation is) for one-body overlaps, analogous to eq. (12) is,

h2(A + 1) Vx2 j "' + 2MA

f

d~2,

d~A~bA,

....

A2 Lil#)A+ 1

~

'

'

=

Eft.

(23)

The coordinate x is given by, x = r~- R A. The phenomenologicai model used in DWBA calculations consists of simplifying eq. (23) through the ansatz,

f

d¢2 . . . . . dCAq~A*Z "bil~)A ' +1 2

~

U'f' •

This results in an equation which is mathematically similar to but not the same as the single-particle Schr6dinger equation of the shell model. The transformation, analogous to eq. (7), is A

1

fs.m.(rl)= fdRAXA(RA)XA+I I A ~ I R A + A ~

r

1)I(rI--RA) •

(24)

The asymptotic behavior of eq. (24) is illuminating. (One of the nice features of onenucleon overlaps, not shared by two-nucleon overlaps, is that their asymptotic behavior is very simple.) The function X A is of short range; therefore the volume integral is dominated by values ofR A in the neighborhood of the maximum, R °, of the function, R 2 X A . F o r very large values, r 1 ~ AR °, we can write

(r,)

f~m'(rO~ f(rOXa+l A ~ I

x(constant).

(25)

104

W . T . P I N K S T O N A N D P. J. I A N O

At large distances, J", the solution of eq. (23), behaves like .[' ~

e

-- kr

x (spin-angle function), r

k

X/

h2

'

M* = A M / ( A + I ) .

In the shell model, one makes a similar assumption about j~ ..... except that the nucleon mass M is used instead of M*. Let us assume that f and J~.m. in eq. (25) have these properties. We must also make the assumption about X A + I . Suppose it behaves at large R A + 1 as exp ( - 2 R A + 1). Then we would have, )~ A+I

-k

....

-k~--

k

)~½k'.

2A '

This )~-value is much too small to be realistic; it implies that the c.m. function falls to zero at large R a + I much more slowly than the most weakly bound single-particle state. However, we are quite certain that the X functions fall to zero much more rapidly than the single-particle states. We must conclude, for reasonable assumptions about the c.m. functions, that the one-particle intrinsic overlap projected from a shell-model wave function is too large in the surface and exterior. In the above discussion it was pointed out that the integral in eq. (24) is dominated by a small region about the point, R °. For a 1s oscillator function, R ° is exactly the "e-folding" distance, (/~0A)2 = 2/vA. For this value, we have argued that X 2 is approximated to better than 1 "~ by (u~) 2. Thus the Gaussian tail o f X A ~ uls is not a source of error. If R A is set equal to R ° in Xa + 1, then the argument of this function is less than or about equal to 1.0 fm for r I values of 17 fm or less. If the properties O f X A + i are similar to those of X A, then for R A + 1 ~ 1.0 fro, the error should be less than 10 '~i. Thus for r a values appropriate to light-ion reactions and light heavy-ion reactions, we conclude that the unphysical Gaussian fall-off of the c.m. functions does not have serious consequences. A study of the recipe for c.m. corrections to one-nucleon overlaps, analogous to that derived in sect. 3 for two-nucleon overlaps is instructive although of little practical significance. All the steps in the derivation are the same. The transformation of the kinetic energy term is, -hZ(A+l)

2MA

-h 2 (2A+l)

T{V~/}-

2

2M 4A(A+I)fl(rOVI(Tj}/#),2

~

( ~/(rl) = exp [4A + 2 ) ' The one-nucleon recipe is: (a) Multiply Js.m. by ~/(r0; (b) expand the result in oscillator functions, with an appropriately chosen v, to obtain s-coefficients; (c) multiply the

c.m. CORRECTIONS Ill I

I

I

105 I

\

0

Sty1

o

'

x(fm)

'

'

12

Fig. 5. Single-particle radial overlap functions for the lds. 2 state in ~70. the s-coefficients by ((A + 1)/A) iN to obtain the/~ coefficients (which multiply oscillator functions parameterized by v2 = vA/(A + 1) rather than v). The application of steps (b) and (c) discussed in some detail in subsect. 3.2 of II. This recipe has been applied to a ld~ Woods-Saxon eigenfunction, representing fs.m. for ~vO. The results are plotted in fig. 5. As in fig. 2 we use a rather obvious nomenclature, s.m., old, new. The shell-model overlap and that obtained using the improved c.m. correction agree very well with the appropriate Hankel functions out to 12 fro, where our oscillator basis (of order 20) begins to fail. The intrinsic form factor resulting from the old recipe is much too large in the surface and exterior and, as expected, falls too slowly with increasing r 1. The final issue to be clarified concerns the use of eq. (8) for cases other than closed shells. We have approached this in the following way. Let the ground state of 170 be written as an antisymmetrized product of wave functions of the ~60 core, ~,..... and the valence state, qJval ~/A+I = {~J . . . . ~//val}a = { ¢ . . . . Uls(YA'~ RA)~va,}."

(26)

It is assumed in eq. (26) that the wave function of the closed-shell core can be written as the product of an internal function and an oscillator ls function of c.m. motion. The bracket with suffix, a, implies antisymmetrization. We have applied Vincent's procedure 5) to eq. (26); i.e. we have carried out the procedures implied in eqs. (5) and (6) to the extent of determining the function, v(o) -~xA + 1- This function was then used in eq. (24). The resulting overlap f was almost identical to that obtained when a ls

106

W.T. PINKSTON AND P. J. IANO

oscillator function was used for XA+ ~. The technical details of this calculation are discussed in the appendix.

Appendix IMPROVED c.m, WAVE FUNCTION FOR 1:O

The occupied single-particle states in 160 are tightly bound, by 12 MeV or more. As a result the states are represented rather well by oscillator functions, and it is a good approximation to assume that the c.m. behavior is that of a 1 s oscillator function, eq. (8). The neutron separation energy of 170 is only 4 MeV, and the valence orbitals are not so well approximated by oscillator functions. In this appendix we describe a method of improving the 17O c.m. wave function to take into account the effect of this loosely bound valence nucleon. The shell-model wave functions is ~A+I of eq. (26). Let I]]vaI be expanded in an oscillator basis, ~/val = E n

Cnblnlm(V;

rl)'

(A.1)

Because the closed-shell core is LS magic, it is possible to suppress spin indices without error. The brackets in eq. (26) imply multiplication by an antisymmetrizer normalized as follows: |

A+I

A - x//A-+-I ( 1 - ~ Pli).

(A.2)

i=2

If the expansion, eq. (A.1), is substituted into eq. (26), and a Talmi-Moshinski transformation 2, 16) used, ~PA+~ can be expressed in terms'of the variables, RA+ 1, X

¢A+1 = ~ C . ~
(A.3)

NLn'I'

The square bracket in eq. (A.3) implies angular momentum coupling. The symbol symbolizes the internal coordinates ~2, ~3. . . . . ~A+ 1' The angular bracket is a TalmiMoshinsky bracket, and q~,'r is an internal function, given by

O,,'r,,,'(~, x) = { dp.... (~)u,,,r,,,,(v 2 ;x)},.

(A.4)

Clearly eq. (A.3) does not separate into c.m. and internal parts. It is of the more general form, eq. (5). We apply the criterion of Vincent 5) and search for the separable form which is the best approximation t o ~/A + 1, in the least squares sense. We require that the c.m. be in an L = 0 state and expand the c.m. wave function in an oscillator basis. The quantity to be minimized is,

f d~dRa+ ldxl~'a+ 1 - Z 7KUxs(VA-1 ; RA+ l) ~ F,,4~.,,,,.(¢, K

n'

X)[ 2,

c.m. CORRECTIONS 111

107

the outcome of minimisation being the amplitudes, 7, F. The resulting equations are, K

K

E VnKFnon n'

(A.4)

= 7K E FnD,,, 2 n

in which the V,,K are given by

V,,K = ~ c,(KSn'l, I[lsnl, 1).

(A.5)

n

The transformation brackets in eq. (A.6) are particularly simple because there is an angular momentum zero state on each side.

(KSn'l, lilsnl, I)

l(2K-1)!(2n'+21-1)!!(2n

-2)!!/ \A+I/

\A+I/

The Dn, are the normalization integrals of the internal functions, $. These have been derived by Philpott 9). For our simple case, 1

D n= 1 - ~ ( I - N ( A + I ) ) ,

A = 16,

in which N is the energy quantum number, N = / + 2 ( n - 1 ) . Eqs. (A.4) can be converted in to a pair of eigenvalue equations for 7, F. For example,

~?~(Z VnKVn~'Dn)~'K' = 7K2, K' n

(A.6) n

K

Eq. (A.6) has been solved for the 7K, which in turn were used to construct an improved c.m. wave function, x(O) A + I = E ]SK~IKS(~)A+ 1 ; R A + I )" K

This function was used in eq. (22). In place of step (c) of sect. 5 one solves the system

c~, = ~ ?rfln_ K( K S n - KI, l[lsnl, l), K

~1 = 71fl1( lsn/, lllsnl, l) = ;qfll

EX~ij

~2 = 71fl2( l s n + II, lllsn+ II, l)+72fll(2snl, Illsn+ II, l).

108

W. T. PINKSTON AND P. J. IANO

The results of these calculations are essentially negative. X A + 1 differs negligibly from Uls(VA+I; RA+ 1)" The coefficient, 7~, of this term was 0.9995. As a result, the change in the intrinsic function projected from a ld~ W o o d s - S a x o n was negligible. W e feel these results lend a measure of support to the use of eq. (8) for l 8O as well as for ~~'O.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16)

W. T. Pinkston, Nucl. Phys. A269 (1976) 281 W. T. Pinkston, Nucl. Phys. A291 (1977) 342 R. H. lbarra, M. Vallieres and D. H. Feng, Nucl. Phys. A241 (1975) 386 J. Bang and F. A. Gareev, Nucl. Phys. A232 (1974) 45 C. M. Vincent, Phys, Rev. C8 (1973) 929 J. B. Aviles, Ann. of Phys. 42 (1967)403 P. H. Wackman and N. Austern, Nucl. Phys. 30 (1962) 529 R. Jaffe and W. Gerace, Nucl. Phys. AI25 (1969) 1 R. J. Philpott, Nucl. Phys. A289 (1977) 109 P. D. Kunz, private communication F. D. Becchetti, Jr. and G W. Greenlees, Phys. Rev. 182 (1969) 1190 F. D. Becchetti, Jr. and G. W. Greenlees, in Polarization phenomena in nuclear reactions, ed. H. H. Barschell and W. H. Haeberli (University of Wisconsin Press, Madison, 1971) p. 682 W. W. True, C. W. Ma and W. T. Pinkston, Phys. Rev. C3 (1971) 2421 Wen Jui, Ph.D. dissertation, Vanderbilt University, 1978 (unpublished) W. T. Pinkston and G. R. Satchler, Nucl. Phys. All9 (1968) 241 L. Trlifaj, Phys. Rev. C5 (1972) 1534