Some studies of realistic form factors for (d, p) and (p, d) reactions in the lead region

I

2.G

[ NuclearPhysics A138 (1969) 671-----684;(~) North-HollandPublishino Co., Amsterdam I

Not to

be reproducedby photoprint or microfilmwithout wriUenpermission from the publisher

S O M E S T U D I E S OF R E A L I S T I C F O R M F A C T O R S F O R (d, p) AND (p, d) R E A C T I O N S I N T H E LEAD R E G I O N V. R. W. EDWARDS t

Nuclear Physics Laboratory, Oxford Received 15 August 1969 A formalism is developed for the calculation of the form factors of one-nucleon transfer reactions using a pet turbation treatment which is correct to all orders and techniques originally employed in photonuclear reaction theory to sum the contributions of bound and continuum terms. A simplified version of this formalism is used to analyse the reactions 2°6pb,(d, p)2°VPb, aOgBi(d' p)al OBi and 210pb(p' d)2°gPb and the results are employed to determine the sensitivity ofthe ground state wave function predictions for the target nuclei to the form factor prescription. It is shown that significant corrections are required in certain cases.

Abstract:

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

In the usual form of D W B A theory 1) for a single nucleon transfer reaction A(ct,/~)B all the nuclear structure information can be factored into the overlap integral (~A(¢)I ~e(~, x))¢ where x represents the space and spin coordinates of the transferred nucleon and ~ those of the other nucleons. The quantities ~ ^ and ~B are the internal wave functions for nuclei A and B respectively. The integration is over ~ and the overlap integral is thus a function of x. It is customary to expand it in terms of components of definite total and orbital angular m o m e n t u m j ' and l' ( ~^(¢)1 ~'a(~, x))~ = ~ (J^ M^ j ' M e - M^lJa Ms> R~':^(r) ~ , ( ~ ) .

j,

(1)

r

Here JA, M^, Je, Ms are the total angular m o m e n t u m and its z-projection for the nuclei A and B respectively. The quantity aftj, (~) is an orthonormal spinor spherical harmonic which is a function of the angular and spin coordinates :¢ of the transferred nucleon. The quantum numbers 1' and the z-projection of j ' have not been explicitly attached to this function, but it is understood that the symbol j ' includes them. The dependence of the overlap on the radial coordinate r is represented by the functions /~j,A:^(r), which are called the form factors. The factor of r - 1 in eq. (1) is not included by some authors. Until recently the form factors were always assumed to be proportional to singleparticle wave functions. Attempts 2-12) have been made to calculate them directly from the effective two-nucleon interaction. These computations are complicated by ~' Present address: Wheatstone Laboratory, King's College, Strand, London W.C.2., England. 671

672

V.R.W.

EDWARDS

the need to go beyond the simple approximations of the common form of the independent particle model. Because the overlap integral must have an accurate radial shape with the correct asyrnptotic ff)rm, it is necessary to use trial functions with arbitrary radial dependence. Two suggestions have been made for dealing with this problem. The earliest paper 2) proposed the use of a form of first-order perturbation theory in which a sum was included over the principal quantum numbers of the basis functions. Although this approach illuminated the underlying physics, it was not developed to a stage suitable for practical calculations because the problem of summing contributions from the 'continuum' of unbound states was left untackled and because configuration mixing was ignored. Later authors 3..,) developed a more complex formalism based on sets ofcoupled differential equations which was employed in a number of calculations 5-12). The present paper contains two developments. The first is an attempt presented in the next section to remove the objections mentioned above to the perturbation treatment. The aim here was to obtain a simpler formalism for practical calculations and to clarify the relationship between the perturbation and coupled equations approaches. This attempt is only partially successful since adding the continuum terms destroys, the simplicity of the perturbation formalism of ref. 2) and indeed leads to coupled differential equations identical to those derived by earlier workers 3). However, the relationship between the two treatments is clearly demonstrated. The coupled equations are formally shown to arise from summing all the contributions made to the form factor by both the bound and the unbound states. The second concern of this paper, dealt with in sect. 3, is with the application of the above formalism to stripping experiments in the region of 2°8pb which give information on the wave functions of two nucleons outside the N = 126 Z = 82 doubly magic core. The results are of particular interest because of the ideal conditions that exist in such nuclei for the application of the independent particle shell model. All lengths in this paper are in units of fm and all energies are in MeV. 2. Perturbation formalism

We consider the single nucleon stripping reaction A(ct, fl)B. Nucleus A is assumed to consist of an inert core of closed shells whose coordinates are not explicitly indicated in our wave functions, and n - 1 active nucleons, whose coordinates are written as ~. The spin and space coordinates r of the additional nucleon in B are denoted by x. The internal wave function of B may be written in terms of a parentage expansion

x) =

c(p,

(2)

Py

where the curly brackets denote vector coupling i.e. { ~P(¢)Or(X)}JaMn = ~, (']e M e JmlJB M . ) ~ , Mp(¢)~,(x). ,n Mpm

(3)

FORM FACTORS FOR (d, p) AND (p, d)

673

The parent states ~v~,(¢) are antisymmetric in the coordinates of all the nucleons in A. The spouse states ~ ' ( x ) are eigenfunctions of the single-particle Hamiltonian for the transferred nucleon •Y~(x) = T(r)+ V ( r ) (=2 / a r 2 ) - l I 1 2 - h 2 0 -ar

T(r)

( r 2 O)l

+ V(r),

(4)

where the kinetic energy operator has been expressed in terms of the orbital angular momentum operator 12 for later convenience. The quantity is the optical potential between the transferred nucleon and the inert core and/~ is the reduced mass of this nucleon. Jp, Mp, j, m, JB and MB are respectively the angular momentum and its z-projection for ~ P ( ¢ ) , and ~'~*(¢, x). The subscripts P and ~ denote all the non-magnetic numbers of the parent and spouse states. Where no confusion is likely to arise the magnetic number superscripts on the wave functions are discarded. Only a few of the eigenfunctions o f ~ ( x ) are bound. The sum in eq. (2) is assumed to be over both the bound and the unbound levels. The methods employed by Buck and Hill 13) in their work on photonuclear cross sections can he used to sum the contributions from both types of term. Expressing the coefficients in eq. (2) in terms of the perturbation integral solution 14) we have

V(r)

O'S(x)

~B(¢, x) = Z <{ ~'p(¢')~,(x')},.~,.I V(¢', x')l ~'e(¢', x')>~,x, { ~'p(¢)O,(x)},.~,..

(5)

Here eB, er and ~r are the energy eigenvalues associated with ~e(~, x), ~v(~) and • y(x). This expression is exact providing that the perturbation series converges. In the earlier work 2) first-order theory was used. For this approach ~B and ea in eq. (5) are replaced by their zero order approximations. The dummy variables of integration in the matrix elements have been labelled ~' and The quantity is the sum of the two-particle interactions between the last nucleon and each active nucleon. The first step in summing (5) is to remove the energy denominator from the sum over y by exploiting the fact that {~p(¢)O~(x)};.ue is an eigenfunction o f ~ ( x ) with eigenvalues ey. Hence

x'.

V(~', x')

x)

(

•

,

•

,

x {Pp(¢)'i~,(x)},.u.)V(¢', x')PB(¢', x')d¢' dx'.

(6)

It is possible to evaluate the sum over ~ using closure relationships similar to those in ref. 13) Y 7

-- Z

•

(7)

Mp

On substituting from (7) into (6); taking the overlap with ~A(¢); expanding the overlap integral using (1); multiplying through by e B - e ^ - . ~ ( x ) ; taking the inner product

674

v.R.W.

EDWARDS

of both sides of the resulting equation with (JAMAjMB--M^IJBMe)~(x) and summing over all possible values of MA one obtains a set of differential equations for the form factors identical to that obtained by Pinkston and Satchler in ref. 3) d2R]^'^(r)dr 2

I - - / (1/ + )r.2

+

+ ~Sn

V(r))] R~^S^(r) v(¢'.

l

x). (8)

where I is the orbital angular momentum associated withj. The RHS terms of eq. (8) do not affect the asymptotic behaviour. Thus the present theory correctly predicts that the asymptotic form of R~AS^(r) should be governed by the separation energy e n - e A. In contrast, perturbation theories which include only bound states yield asymptotic forms governed by the binding energy of a singleparticle in the well V(r). Eq. (8) is unsuitable for practical calculations because it contains the unknown wave function 7'B(¢', x). Pinkston and Satchler 3) expressed ~uB(¢', x) in terms of the form factors R~^S^(r) and reduced eq. (8) to a set of coupled equations. In the present work we have employed the more convenient approach of ref. 6), replacing ~PBby the ordinary independent shell-model wave function ~/,~)h¢.and thus converting eq. (8) into an inhomogeneous differential equation. This is equivalent to a first iteration of the coupled equations. Since our calculations are concerned with 21 opb ' 206pb and 2t°Bi it is convenient to specialize the treatment to two active nucleons. For this case we may write ~)h"l(x,, x2) = X

a'~t'{Os.(X')q),b(X2)}SB,VtB"

(9)

s

where the a, are coefficients and a~¢'~ antisymmetrizes the wave function over the coordinates of the last two nucleons. These have been labelled xl and x 2 in place of and x. If this wave function is substituted in eq. (8) it reduces to dZR~^S^(r2)

V- l(l + 1)

+ L

2#

+

1

=

(10)

where

P(r2) = 212 ~ a.[Nls Rsb(r2) Vjj~b(r2 A,o ) + N2., g~.(r2) Uji~o(r2)], A~ Asa em.(r2) =

dxt d.~2 {~I(X,)@'~(~2)}SBMBV(xl, X2){~.,~(Xt)~0'j,b(:~2)}SsM~,

(11) (12)

and NI, = 1 and N2~ = 0 for s,---s, otherwise NI~ = 2- ½and N2~ = 2-* ( - 1)s~ ÷ r~-j~,-j,~. The resultant isospin Tn of the two active particles has not been included in the previous equations in order to save space, but its value is important when the effects of antisymmetrization have to be explicitly calculated.

FORM FACTORS FOR (d, p) AND (p, d)

675

3. Calculations The calculational methods will be described in connection with z°6Pb(d, p)2°TPb as most of our work was concerned with this reaction. The present paper is restricted to transitions to single-hole as distinct from 2 h - l p states of 2°Tpb. The two most extensive studies of such transitions at 10.85 MeV [ref. 15)] and at 15 MeV [ref. 16)] were selected for analysis. Other measurements are described in refs. 17-2o). Figs. io o

o

10-'

o{

5

1~1

~

X 0025

~ 10-41 O

oo

I

~-__ I 60 SCATTERING ANGLE

2

120 DEGREES

180

Fig. i. Angular distributions ~5) and D W B A fits for the three lowest transitions in the reaction 2°6Pb (d, p)2°TPb at 10.85 MeV.

1 and 2 show the experimental angular distributions obtained for the lowest transitions leading respectively to the p~ ground state, the 0.57 MeV ft- state and the 0.89 MeV P~r state of 2°Tpb. At 15 MeV there were also three data points for the 1.63 MeV i~, level. The quantity P(r) was calculated for each transition from eqs. (11) and (12) using a Gaussian Serber two-particle interaction 22)

V(xl , x2) = Vo exp (r,21ro)P,, 2 z

(13)

with Vo = - 3 2 . 5 MeV and r o = 1.85 fm; Psc is the singlet even projection operator. The function ~p~h,, was also taken from ref. 2z) 7~h'U(Xl, X2) = 0.8653(p{)-2+0.3077(f,}) -z +0.3765(p~}) -z-0.1216(i'~) -2. (14)

676

v.R.W.

EDWARDS

The optical potential employed in these calculations and later for the generation of distorted waves in DWBA calculations was of the form

x l+exp(r-ruA~) -2_4Wiexp(r-rwA÷ ) l+exp(_r:Ou

(15)

aW

o

o

o

c •

2

10:

E

~

F

,

~,-

XO01

o

lo. p~O~,(D,p ) p~O: ~5 MeV

___

I0 -a

_--:___

Fig. 2. Angular distributions

I

40 SCATTERING

o

~6)

i

ANGLE

I

80 DEGREES

.t_

___

~20

and DWBA fits for the threc lowest tzansitions in the reaction 2°6Pb(d, p)2°Tpb at 15 MeV.

where the various parameters have their usual significance and took the values listed tg) in table 1. The Coulomb potential V"c is due to a uniformly charged sphere of radius r u,4 ~ and .4 is the mass number of the stripping target nucleus. Eq. (15) has been made sufficiently general to cover all the cases involved. The Coulomb and imaginary terms were not used in the form factor calculations. The spin orbit term was not employed in calculating distorted waves. Eq. (10) was numerically integrated employing a modified D W B A code supplied by Smith 23), the value of the well depth U being adjusted to fit the observed sep-

FORM FACTORS FOR (d, p) AND (p, d)

677

aration energies. Since this is an inhomogeneous equation it is necessary to specify the form factor normalization before attempting to solve it. Two alternative prescriptions were employed. TABL~ I Data used in calculations in 2°~Pb(d, p)2°TPb a) Energy levels of 2°~Pb and Q-values for 2°ePb(d, p)2°TPb [ref. 21)] State 3p½ 2f4 3p~ li~

Energy

Binding energy

Q

0.00 0.57 0.89 1.63

7.39 7.96 8.28 9.02

4.505 3.935 3.615 2.865

b) Optical p a r a m e t e r s 19) Particle

U

r,

a,

proton deuteron neutron

54.23 120.0

1.25 I. 10 1.23

0.65 0.91 0.65

W 17.78 40.0

rw

aw

1.25 1.23

0.47 0.76

U,...

7.0

The first was based on approximating ~ , ( ¢ , x) in eq. ( I ) by t/'~hcn(xl, X2). For the common case in which only one set o f f contributes significantly to the sum on the right-hand side of this equation, the normalization of R~j^~^(r) can be shown to reduce to the usual result #a 2 where as is the coefficient of the configuration (q~ACj) in eq. (9) for ~,~hc, and g = I or 0.5 for equivalent and inequivalent particles respectively. In the second prescription the normalization was adjusted so that eq. (10) yielded the same U as that which fitted the binding energy of one neutron hole ¢ i inside the 2 08pb core. These two prescriptions would not be expected to yield identical normalizations because the first one neglects the higher principal quantum number components and is usually based on harmonic oscillator wave functions. However, as table 2 shows, the methods agree moderately well and this provides a useful check on the formalism and on the programming. Figs. I and 2 show the fits obtained to the lowest three transitions using the normal separation energy prescription. These angular distributions differed by less than 2 from those predicted by the inhomogeneous equation. Slightly larger differences were observed for the i~ transition which has not been plotted on the figures because so little data was available. The absolute magnitudes of the cross sections predicted by the two prescriptions differ considerably for the ft and the i t, transitions as will be seen from the spectroscopic factors listed in table 3. The ratio of the homogeneous to the inhomogeneous

678

v.R.W. EDWARDS

cross sections at 10.85 MeV are respectively 1.32 and 3.42 for these transitions. The results for the ie transition are extremely uncertain because there arc only two d a t a points available. The present calculation is based on the 60 ° point and yields a h o m o g e n e o u s spectroscopic factor ten times greater than theory. The 30 ° point gives a s p e c t r o s c o p i c factor which is a further nine times larger. It seems p r o b a b l e that a weak transition o f this kind would be very sensitive to an a d m i x t u r e o f a low l c o m p o n e n t . TABLE 2

Form factor normalizations for 2°6pb(d, p)2°~Pb Transition prescription l p½ ftPt i9

Normalization p~escription 2

0.75 0.095 0.142 0.0148

1.16 0.085 0.173 0.0105

TABLE 3 Spectroscol.ic factors for 2°6Pb(d, p)2°TPb Transition

p½ ft" p~

10.85 MeV spectroscopic factors 15 MeV spectroscopic factors homogeneous prescription I prescription 2 homogeneous prescription I prescription 2 0.647 0.097 0.065

i~ ") ~--.~jSj(j-]-~) b) 1.079 YjSjO'+½) c) 1.219

0.644 0.128 0.065

0.646 0.126 0.065

1.171 1.663

1.167 1.675

0.658 0.115 0.087 0.022 1.189 1.329

0.667 0.152 0.088 0.072 1.311 1.803

0.669 0.149 0.088 0.074 1.304 1.812

") The i¢ spectroscopic factors are subject to great uncertainty. t,) Sj for the i~a state was taken as 0.0021 from theory. c) Sj for the i~t state was calculated from the experimental 15 MeV (d, p) cross section at 60"

[ref. 2,1)].

The s p e c t r o s c o p i c factors S i should satisfy the sum rule Z i S i ( j + }) = I. It will be seen f r o m the last line o f table 3 that the sum is c o n s i d e r a b l y greater than one, p a r t i c u l a r l y for the i n h o m o g e n e o u s calculations. In the second line from the b o t t o m o f the table we have recalculated the sum a s s u m i n g that the i¥ spectroscopic factor is 0.0021 from t h e o r y a n d the results are c o n s i d e r a b l y closer to unity. The confidence one has in the spectroscopic factors is further strengthened by the close c o r r e s p o n dence that exists between the 10.87 MeV a n d the 15 MeV values. We chose to examine the reaction 2°6pb(d, p)2°Tpb in greatest detail because o f the well-known d i s a g r e e m e n t which exists between the e x p e r i m e n t a l s p e c t r o s c o p i c factors a n d the p r e d i c t i o n s o f the classic shell-model calculation o f True a n d F o r d 22). In c o l u m n six under the h e a d i n g ' ( d , p)', we list the squares o f the coefficients c./of the

Present work

0.28

0.13

0.015

(fl)-2

(p~)-2

(ig) -2

0.015

0.12

0.33

0.53

0.12

0.12

0.25

0.51

0.29

0.087

0.24

0.37

0.12

0.12

0.20

0.54

0.57

0.17

0.20

0.54

inhomo ( d , p ) ~ ) (d,t)~) ( p , p ' ) a )

Other analyses

0.015

0.14

0.095

0.75

0.012

0.13

0.16

0.68

ref. 2z) tel. z~) !

Theory

i) These results have been obtained assuming that c~" -----0.015 from theory for the i,~ transitiol all four transitions or so that ~jc~ 2 ~- 0.97 over the filst three transitions. b) Ref. =4). ~) Ref. as). a) Ref. as).

0.57

homo*) inhomo') homo

(p½)-2

Component

TABLE 4 Comparison of the present results for cj 2 in the "°6Pb ground state wave function

680

V . R . W . EDWARDS

components in the ground state wave function of 2°~'Pb as calculated by Mukherjee and Cohen 24) from the 15 MeV (d, p) data using the relation c] = (j+~)S i. Their results are confirmed by (d, t) and (p, p') measurements as shown in columns seven and eight. The theoretical values in column nine show much less configuration mixing. The recent calculations with realistic forces by Freed and Rhodes 28) predict still smaller admixtures, as shown by column eleven. This disagreement led True 27) to modify his original calculations by introducing an additional ,°2 type force. As column ten of table 4 show~ this improvement slightly reduced the (p½)- 2 component, though by not nearly enough to conform with ref. 2,,). It also correctly predicted that the (f~)-2 component was somewhat larger than the (p,/)- 2 component. Our own results are presented in columns 2-5 of table 4. They have been obtained from table 3 by normalizing the spectroscopic factors so that ~ic] = 0.99 and then averaging over energy and normalization prescription. Because of the uncertainty in the i,~ spectroscopic factors we have presented one set of figures in which the (iv) -2 coefficient is assigned the theoretical value of 0.015 and another set which uses the experimental results. The homogeneous figures agree closely with Mukherjee and Cohen. It will be seen that employing the theoretical i v strength considerably reduces the discrepancy between the theoretical and the experimental (p~z)-2 coefficients, though it somewhat worsens the (f.1.)-2 fit. It was our hope that the remaining discrepancies could be removed by introducing a realistic treatment of the form factor. However, the effect is actually to worsen the disagreement. This trend is particularly marked when the i v spectroscopic factor is taken from experiment because this quantity is much larger for the realistic treatment and considerably reduces the (p½)-2 strength through its effect on the normalization. The fact that th.~ inhomogeneous ( i v ) - 2 spectroscopic factor is 24 times the theoretical value further strengthens the case for rejecting the present i v data. We therefore regard the results in column three obtained with the realistic form factors and the theoretical i,t spectroscopic factor as the most reliable. This calculation yields an if.t)-' strength which is twice the value obtained in ref. 27) and a (p½)-2 component which differs by 30 9/0 from theory. Although the fits to the angular distributions are not perfect, the choice of the point at which theoretical and experimental curves are matched is unlikely to explain the present discrepancies. Thus the (p½)-2 strength is extremely insensitive to any error because of its dominant effect on the normalization. The discrepancy associated with (ft) -z component is far too large to be attributable to this cause even though the 10.85 MeV fit is not completely satisfactory. These discrepancies are larger than the uncertainties in the DWBA calculations. These were estimated for the more accurate 10.57 MeV data. Because this was obtained below the Coulomb barrier, the spectroscopic factors should be insensitive to the choice of cut off and distorted wave optical parameters. The main uncertainties at this energy arise from the finite range effects, from the non-locality of the distorted

FORM FACTORSFOR (d, p) AND (p, d)

681

wave and bound state optical potentials and from the choice of bound state parameters. The tests made on these three sources of error all employed the theoretical (i~) -2 strength for normalization. The finite range corrections and the effect of non-locality on the distorted waves were investigated using the first-order treatment developed in ref. 30) and the local energy approximation 31) respectively. The non-locality range constant was taken as 0.85 fm for protons and 0.54 fm for deuterons. For all the transitions the combined effect of these two corrections was to increase the cross sections by ~ 2 ~ and to make a negligible change in the relative spectroscopic factors. There is some evidence 32) that the local energy approximation over-estimates the corrections for the bound state non-locality. When this effect was included using a range of 0.85 fm it increased the p~, ft and p~ cross sections by ~ 30 ~o and the i,/ by ,~, 60 ~ . The form factor employed in this calculation was obtained from the inhomogeneous equation using the second normalization prescription. The resulting coefficients after normalization are shown in the 14th column of table 4 under the heading 'parameter sensitivity tests I'. These values differ only slightly from those of column three. TABLE 5 Bound state parameters used in the lead region Set and source A 19) B aa) C a'*) D aS) E a2) F a~) G a6)

ru

1.230 1.349 1.190 1.250 1.225 1.200 1.320

au

0.650 0.700 0.750 0.650 0.700 0.650 0.635

U

).

U,.,.

46.5 40.5 50.9

29.1 ") 36.9 ") 20.5 25.0 25.0 25.0 24.0 ")

7.0 7.7 ") 5.4 ")

41.9

5.2

") Approximate equivalent o f ;t or U,.,. calculated from /3",.,. = 0.00517 U2.

A systematic investigation of the effect of varying the bound state parameters would be very time consuming. Instead, we examined the bound state parameter sets used by the principal investigators in this mass region, which are summarized in table 5, and repeated the DWBA calculations using sets B and C of that table, the two which differed most from our own set A. The inhomogeneous equation and the second normalization prescription were employed, and corrections were made for finite range and for non-locality in the distorted waves. Column 15 of table 4 shows the results obtained with set B. A significant improvement in the (p~)-2 fit is obtained, but the disagreement in the (ft)-2 strength remains large. The results obtained with set C are shown in the last column and are almost identical to those obtained with our original set.

682

V . R . W . EDWARDS

If we adopt the most favourable combination of options i.e. the dubious bound state non-locality correction and set B of bound state parameters, then the theoretical and experimental (p½)-2 coefficients, 0.68 and 0.62 respectively, are reconciled, but the (f?)-2 coetficients, 0.16 and 0.24, remain in discordance. The parameters of set B do not give unit spectroscopic factors for the 2°sPb(d, p)2°gPb reaction and are theretore somewhat suspect in the present context. Our view is that the discrepancies are significant and that they may have an explanation in terms of strong collective admixtures in the wave functions of 2°6pb and 2°Tpb. Calculations are in progress to test this hypothesis. The surprising results obtained for the i,~ transition are also worth following up. We hope to obtain a complete angular distribution for this transition by using (z, 3He) instead of the (d, p) reaction. The former favours high momentum transfers. The present results can be compared with the unpublished analysis of Jones 29) of his own 8.0 and 18.7 MeV (d, p) data. It will be seen from columns 12 and 13 of table 4 that his coefficients agree slightly better with theory than our comparable figures in column two. Calculations were also performed for the Zl°Pb(p, d)2°9pb reaction studied at 20.5 MeV by Barnes, Flynn and [go. The author is grateful to them for supplying the results prior to publication. As the cross sections were unnormalized and these calculations were of a preliminary character, the results alone will be presented without further details. In table 6 the ratio of the inhomogeneous to the homogeneous spectroscopic factors has been tabulated for each of the 7 single-particle states of TABLE 6 T h e ratio o f the i n h o m o g e n e o u s to the h o m o g e n e o u s spectroscopic factors for 21°Pb(p, d)2°gpb Transition

Energy

Ratio plescrip, l

g~ i¥ j9 d,!. s½ gtd t.

0.00 0.77 1.41 1.56 2.03 2.47 2.52

0.952 0.764 0.694 0.849 0.798 0.729 0.770

- prescrip. 2 0.940 0.702 0.632 0.847 0.857 0.780

2t°Pb observed in this experiment. It will be seen that the two form factor normalization prescriptions yield almost identical results and that for every case the ratio is less than one. Since the dominant g~ transition is the least altered, the main effect of using the more accurate form factor prescription is to reduce the apparent configuration mixing in the ground state of 21opb ' the reverse of the effect observed for 2 ° 6 p b . The change in the relative spectroscopic factors is most marked between the gl and j~ transitions where it amounts to ~ 30 ~ .

FORM FACTORS FOR (d, p) AND (p, d)

683

The last reaction studied, 2°gBi(d, p)210Bi ' is o f p a r t i c u l a r interest because it gives i n f o r m a t i o n on b o t h the T = 0 and the T = 1 c o m p o n e n t s o f the two t w o - p a r t i c l e interaction. The D W B A analysis o f this reaction was f o u n d to be very insensitive to the choice o f form factor prescription. This was not unexpected because the s e p a r a t i o n a n d b i n d i n g energies are almost equal for all transitions. F o r the strong transitions with form factor n o r m a l i z a t i o n s larger than 0.4 the difference is less than 0.8 MeV a n d for weaker transitions it is less than 0.11 MeV. These figures should be c o m p a r e d with the typical 5 MeV differences in 2 o 6pb(d, p)207 Pb a n d 21 o p b ( p ' d)209pb. A p r e l i m i n a r y a c c o u n t o f some of the w o r k described in this p a p e r has been circulated in r e p o r t form 37). This r e p o r t c o n t a i n e d a n u m b e r o f errors which have been corrected in the present paper. The a u t h o r is grateful to Prof. W. T. P i n k s t o n for bringing these faults to his attention. The d i s a g r e e m e n t m e n t i o n e d in the r e p o r t between o u r own calculations and those o f ref. 6) a p p e a r s to have been due to a mistake in the preprint we received from the a u t h o r s a n d does not a p p l y to their final published results. This research has been s p o n s o r e d in p a r t by the Airforce Office o f Scientific Research O A R t h r o u g h the E u r o p e a n Office o f A e r o s p a c e Research, United States Airforce, under c o n t r a c t F61052.67.C.0104. We also express o u r g r a t i t u d e to S R C for further financial s u p p o r t ; to the Atlas C o m p u t i n g L a b o r a t o r y , Harwell for the use o f their facilities; to Dr. W. R. Smith for supplying the special stripping code used to solve the i n h o m o g e n e o u s f o r m factor e q u a t i o n , a n d to Dr. P. E. H o d g s o n for his m a n y valuable c o m m e n t s on the work.

References 1) G. R. Satchler, Lectures in theoretical physics VIII C (University of Colorado Press, Boulder, 1966) p. 73 2) N. Austern, Phys. Rev. 136 (1964) 1743 3) W. T. Pinkston and G. R. Satchler, Nucl. Phys. 72 (1965) 641 4) T. Berggren, Nucl. Phys. 72 (1965) 337 5) R. l-luby and J. L. Hutton, Phys. Lett. 19 (1966) 660 6) A. Prakash and N. Austern, Ann. of Phys. 51 (1969) 418 7) A. Prakash and N. Austern, Phys. Rev. Lett. 20 (1968) 864; A. Prakash, Ph.D. thesis, University of Pittsburgh (1967) unpublished 8) E. Rost, Phys. Rev. 154 (1967) 994; Phys. Lett. 21 (1966) 87; Report PUC-937-226 (University of Colorado, Boulder, 1966) 9) M. Kawai and K. Yazaki, Prog. Theor. Phys. 37 (1967) 638 10) K. Sugawara, Nucl. Phys. A l l 0 (1968) 305 11) R. J. Philpot and W. T. Pinkston, Nuel. Phys. All9 (1968) 241; W. T. Pinkston, R. J. Philpot and G. R. Satchler, preprint 12) R. Stock and T. Tamura, Phys. Lett. 22 (1966) 304; R. Stock, R. Book, P. David, H. H. Duhm and T. Tamura, Nucl. Phys. AI04 (1967) 136 13) B. Buck and A. D. Hill, Nucl. Phys. A95 (1967) 271 14) P. M. Morse and H. Feshbach, Methods of theoretical physics (McGraw-Hill, 1953) p. 1002 15) M.T. Meellistrem, H. J. Martin, D. W. Miller and M. B. Sampson, Phys. Rev. 111 (1958) 1636 16) B. L. Cohen, R. E. Price and S. Mayo, Nucl. Phys. 20 (1960) 370 17) R. H. Stokes, Phys. Rev. 121 (1961) 613

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18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37)

v.R.W.

EDWARDS

D. W. Miller, H. E. Wagner and W. S. Hall, Phys. Rev. 125 (1962) 2054 W. Darcey, A. F. Jeans and K. N. Jones, Phys. Lett. 25B (1967) 599 J. A. Harvey, Can. J. Phys. 31 (1953) 278: Phys. Rev. 81 (19511 353 M. Dost, W. R. Herring and W. R. Smith, Nucl. Phys. A93 0967) 357 W. W. True and K. W. Ford, Phys. Rev. 109 (19581 1675 W. R. Smith, Atlas Laboratory Program Library Report 7, Atlas Computing Laboratory, Harwell, 1967 P. Mukherjee and B. L. Cohen, Phys. Rev. 127 (19621 1284 R. Tickle and J. Bardwick, Phys. Rev. 166 0968) 1167 P. Richard, 1'4. Stein, C. D. Kavoloski and J. S. Lilley, Phys. Rev. 171 1"19691 1308 W. W. True, Phys. Rev. 168 (1968) 1388 N. Fleed and W. Rhodes, Nucl. Phys. A126 (19691 481 K. N. Jones, Ph. D. thesis, Oxfold University 0969) W. R. Smith, Nucl. Phys. 94 (1967) 550 F. G. Perey and D. S. Saxon, Phys. Lett. 10 (1967) 107 G. Muehllener, A. S. Poltorak and W. C. Parkinson, Phys. Rcv. 159 (1967) 1039 E. Rost, Phys. Lett. 26B (19681 184 S. A. A. Zaidi and S. Darmodj, Phys. Rev. Lett. 19 (19671 1446 G. J. Igo, P. D. Barnes, E. R. Flynn and D. D. Armstrong, Report LA-DC 9147, Los Alamos Scientific Laboratory, University of California L. Cranberg, C. D. Zafiratos, J. S. Lcvin and T. A. Oliphant, Phys. Rev. Lctt. 11 (19631 341 V. R. W. Edwards, Report 31/68, Nuclear Physics Labolatory, Oxford University 1968