DWBA calculations for the reaction 12C(p, d)11C at 155 MeV

I

2.G

I

NuclearPhysics A93 (1967) 145--163; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

DWBA CALCULATIONS

FOR T H E REACTION 12C(p, d ) l l c AT 155 MeV I. S. T O W N E R t

Department of Physics, Battersea College of Technology, London S.W. 11 and Nuclear Physics Laboratory, Oxford Received 15 August 1966 Abstract: Angular distributions for the 12C(p, d)11C reaction to the ground state and three excited

states o f 11C at a proton energy o f 155 MeV have been analysed using a conventional D W B A model. Finite-range effects are treated using the local-energy approximation and a large correction to the zero-range approximation is found for the optical potentials used. The angular distributions are relatively insensitive to the parameters o f the neutron wave function. Experimental spectroscopic factors are in reasonable agreement with the theoretical predictions o f intermediate coupling calculations. Possible mechanisms for transitions to " f o r b i d d e n " states are considered.

1. Introduction

Early work on deuteron pick-up reactions a) at high energies has been analysed using a plane wave theory by Chew and Goldberger 2). Their analysis showed that the cross section depends on the momentum distribution of the initial wave function of the picked-up particle. That is

V(q) = f

dr 2.

(1)

By choosing

F(q) = a/[n(a 2 + qZ)]Z,

(2)

a fit to the experimental data was obtained with ~2/2M = 18 MeV. Brueckner et al. 3) showed that such a momentum distribution could not be obtained from a singleparticle model, and this was interpreted as evidence for a strongly correlated ground state wave function. However this conclusion was based on early experimental data and on plane wave calculations. Further deuteron pick-up experiments at proton energies of 96 MeV (ref. 4)) and 145 MeV (ref. 5)) have been analysed 6-8) using a zero-range DWBA theory and a fit to experiment was obtained using a single-particle wave function for ~n(r). However the results depended quite sensitively on the treatment of the distortion of the wave functions of the incoming and outgoing particles. In the DWBA this distortion * Present address: Nuclear Physics Laboratory, Oxford. 145

146

i . s . TOWNER

is described in terms of the optical model potentials that reproduce the appropriate elastic scattering data. Recently, further results on elastic proton scattering have become available as) together with theories 2a,23) for treating high-energy elastic deuteron scattering, so that it may well be possible to re-examine the contention of Brueckner et al. 3) that high-energy pick-up reactions yield useful information on the initial wave function of the picked-up particle. To investigate this further, we use the DWBA to analyse some recent high-energy data of Radvanyi et al. a o) on a2C(p, d)11C at a proton energy of 155 MeV. In sect. 2, the effects which uncertainties in the optical-model parameters have on the predicted angular distributions are examined. Various approximations for the finite range of the interaction are considered and the possibility of obtaining further information on the neutron wave function is investigated. In sect. 3, the absolute magnitudes of the cross sections for various final states in aaC are determined from the DWBA and compared with the predictions of certain nuclear models.

2. Comparison with experiment The formalism for the DWBA is to be found in many review articles al, 12). We begin by listing the basic sets of parameters chosen for these calculations, while in the ensuing sections the effects that variations and uncertainties in these parameters have on the DWBA predictions are considered. The optical potentials used are of the conventional Saxon-Woods type U(r) = - 1%(1+ exp x ) - i _ i(Wo - 4WDd/dx')(1 + exp x ' ) - i + l/c(+),

(3)

where x = ( r - r o A + ) / a and x' = ( r - r ' o A + ) / a '. The Coulomb potential Vc(r) is approximated by the potential due to a uniformly charged sphere of radius rc A~. A spin-orbit term was included in the optical potential used for the fits to the elastic scattering data but has been put to zero for the following (p, d) calculations. Over the limited angular range 0 ° to 40 ° being considered here subsidiary calculations showed that this approximation made little difference to the shape of the angular distribution but affected the magnitude by about 20 ~ . In table 1 are listed the basic set of parameters used. The proton potential is that of Rolland et aL a s), while the deuteron potential was obtained by fitting the data of Baldwin et al. 22). This is discussed below. The bound state neutron wave function TABLE 1 Basic set of parameters in units of MeV and fm used in the following D W B A calculations

Proton Deuteron Neutron

V0

r0

a

Wo

18.7 100 59

1.13 0.8 1.31

0.72 0.8 0.65

10.42

WD

10

r0'

a'

re

1.18 1.5

0.85 0.5

1.33 1.4

2

30

DWBA

147

was taken to be an eigenfunction of the SchrOdinger equation for a Saxon-Woods potential

V(r) = - Vo(1 + exp x ) - i ..~2(h/ZMc)2(l%/r)d/dr(1 + exp x)- ~l" a

(4)

with x = (r-R)/a and R = r o ( A - I ) ~. The well-depth Vo is adjusted to give the required asymptotic behaviour, while shell-model calculations 36) suggest that r o -- 1.31 fm, a -- 0.65 fm and 2 = 30 are suitable parameters for carbon. 2.1. FINITE-RANGE EFFECTS In the DWBA, the interaction is taken to be that between a free proton and neutron Vpn. This interaction is not known but as it occurs as a product with ~d, the internal

wave function for the deuteron ground state the zero-range approximation is normally used. That is Vpn~d(rpn) ~ D(rpn ) "~ Dob(rpn ). (5) I f further it is assumed that Vpn is the potential which binds the deuteron, then using a Hulth6n form for ~a gives 12) Do2 = 8~ea2(1/~+ 1//3)3,

(6)

where ~ and/3 are the Hulth6n parameters and ed the deuteron binding energy. If the plane wave Born approximation is used no assumption need be made about the interaction Vpn. The transition amplitude is proportional to the Fourier transform of the range function D(rp.), and may be evaluated ~2) using the Hulth6n form for ~'d

G(K) =- f drpne'S"P"D(rp,)

(7)

= 13o/32/(/32 + K2), where K 2 = k 2 +¼k2-kpkd cos 0, kp and kd being the proton and deuteron centreof-mass momenta and 0 the scattering angle. Thus finite-range effects appear as a departure of the factor G(K) from the constant value D o which it assumes in the zerorange approximation. Recently much effort has been directed towards improving the accuracy of zerorange calculations without increasing either computer running time or programming difficulty. Two procedures have been suggested, the effective-mass approximation a4) and the local-energy approximation 15). These are equivalent to first order, and this first-order term has been tested against the full finite-range calculations by Dickens et al. 16). Considerable improvement on the zero-range calculation was obtained. Fig. 1 shows three calculations for the 12C(p, d)alC ground state transition, using the parameters of table 1. Curve X corresponds to the usual zero-range DWBA. Curve Y is the zero-range cross section multiplied by the angle-dependent correction eq. (7). At a proton energy of 155 MeV, k p = 2.5 fm -1, k d - - 3.2 fm -1 and hence the zero-range cross section at 0 ° is reduced by 0.5 (assuming fl = 6.2~ and ~ = 0.23

148

I, S. TOWNER

fm-1). However in the angular range 0 ° to 40 ° considered here, the shape of the angular distribution is altered only slightly. Curve Z corresponds to the use of the first-order term of the local-energy approximation (LEA). Considerable improvement to the fit is obtained, whilst the absolute magnitude of the cross section is virtually unaltered from the zero-range value. c (p,d)

\'r.. d._~e d~

.......... '~"-~

C g.s,

Ep = 154.5

"~.,

M~V

Q=-,6..~v

~'~,,

t = I

mb/sr '-.

x

"..

X

ZERO RANGE

Z

LEA

~xx

03

II 20

10

I 30

0 cm Fig. 1. A n g u l a r distributions for the g r o u n d state transition s h o w i n g the effect t h a t various finite-range corrections have o n the zero-range calculation. T h e p a r a m e t e r s used are listed in table 1. T h e experimental values are t a k e n f r o m ref. 10). E a c h curve h a s the s a m e normalization constant.

The success of the LEA in reproducing the experimental angular distribution was unexpected. Previous calculations 14,16) using this approximation at low energies showed that the effect on the angular distribution was small. However the size of the correction depends on the optical potentials used. The LEA introduces a function

A(r) A(r) = 1 - [ V d ( r ) - Vp(a ~)-

V.(~)- Bd]~2/(~2B~),

(8)

149

ow~a

where a is 1 - 1/A, A is the mass number of the target nucleus, B d the modulus ef the deuteron binding energy, Up(ar) and Ud(r) the complex optical potentials for the proton and deuteron and Vn(r) the neutron bound state potential. The correction vanishes if Ud(r ) = Up(ar)+ Vn(r)+B d for all r. For the potentials listed in table 1, no such cancellations occur, and since the potentials have different radial shapes, A(r) may well differ significantly from unity. These circumstances have been noticed before for light nuclei 17) indicating that the first-order correction may not be sufficient, and that the approximation should be extended to higher orders. The first-order term of the LEA has been used consistently throughout the rest of the work reported here. 2.2. P R O T O N O P T I C A L P O T E N T I A L

High-energy elastic proton scattering on carbon at 150 MeV (ref. 18)) and 180 MeV (ref. 19)) has been reported. The latter data have been analysed by several authors 19-21), and the derived optical model potentials are listed in table 2. Fig. 2 shows the corresponding (p, d) cross sections for these potentials and each gives art TABLE 2 Proton optical potentials in units o f MeV and frn Elab A B C D E

152 183 183 183 183

Vo 18.7 16.0 19.4 25.7 27.3

ro

a

1.13 1.0 0,902 0.827 0.938

0.72 0.5 0.452 0.413 0.571

Wo

WD

10.42 I0 15.6 19.1 10.1

r o'

a'

re

1,18 1,34 1,186 0,656 1.28

0.85 0.5 0.556 0.669 0,715

1.33 1.3 1.33 1.33 0.938

Ref.

18) 19) 2o) 20) 21)

equally acceptable fit to the data. Each curve is normalised to experiment at 0 ° and the ratios of the experimental cross section aexp to the calculated aowBA are listed in fig. 2. The mean of these values is 3.2__0.3. This shows that it is possible to find several proton potentials which fit the elastic and (p, d) angular distributions and yet give absolute magnitudes for the (p, d) cross section that vary by I0 % or more. 2.3. D E U T E R O N O P T I C A L P O T E N T I A L

The only elastic deuteron scattering data on carbon at high energies are those of Baldwin et al. 22) at 156 MeV for 11 angles between 0 ° and 40 °. The absolute error is of the order of 20 %. No optical potential fitting these data has been reported. However there have been several calculations 21,22) assuming the interaction between the deuteron and the nucleus to be a sum of neutron-nucleus and proton-nucleus, optical potentials. At low energies the nucleon-nucleus optical potential has a typical depth of about 50 MeV, thus a first-order approximation to the depth of the deuteron optical potential would be about 100 MeV. There is some evidence 9), that the most acceptable deuteron potential for reproducing stripping cross sections is 100 MeV

150

i.s. TOWNER

deep. At higher energies however, the depth o f the nucleon-nucleus optical potential decreases, therefore at the deuteron energy o f 156 MeV considered here, one would expect a somewhat shallower potential. A reasonable approach is to estimate the deuteron potential using the first-order approximation and then to vary one or two parameters to improve the fit to the somewhat limited data. I

I

t2C (p,d) nC g.s. "Ep

d~ d~

:

154,6

MeV

(}, =-[6.49 MeV

mblsr

L

A B

=

I

°'eXpt/dOWBA 3.7 .....

C D

..........

E

......

2.9 3,1

\

3.3 3.0

03

[

I

I

i0

20

30

0 cm

Fig. 2. Angular distributions for the ground state transition using proton potentials A-E from table 2. Deuteron and neutron parameters are those in table l. Each curve was normalised to experiment at 0° and the normalization constant is given in the figure. Watanabe 24) gives the following expression for a first-order approximation for a deuteron optical potential:

Ua(R)

= f dr ~ (r){ Up(}] R + rl) + U.(½J R - r])},

where R = ½(rp+r.), r =

rp-rn, I~d(r)

(9)

is the internal deuteron wave function and

DWBA

151

Up and Un the proton and neutron optical potentials. If it is assumed that each nucleon in the deuteron has half the deuteron energy, then the potential parameters for Up and Un should be taken from elastic nucleon scattering from carbon at 78 MeV. The closest approximation to this that could be found in the literature was for neutron elastic scattering 25) at 96 MeV and for proton elastic scattering 27) at 90 MeV. The optical-model fit to the former was performed by Hodgson 26) using a Saxon-Woods potential, the real well-depth being 22.8 MeV while the latter data have been analysed by Glassgold and Kellogg 2s) using a Hill-Ford potential giving a well depth of 40 MeV. With these potentials, eq. (9) was used to derive a deuteron optical potential using a Hulth6n form for the internal deuteron wave function. The resulting potential was approximated by a Saxon-Woods form with the following parameters: Vo = 53.4 MeV, ro = 1.2 fm, a = 0.81 fm, Wo = 12.7 MeV, r o = 0.96 fm and a' = 0.79 fm. This potential was then improved in the following manner: (i) Well depths Vo and Wo were varied systematically using a search procedure z9), so as to fit the limited elastic deuteron scattering data 22). TABLE 3 Deuteron optical potentials obtained as outlined in the text

F G H J

Vo

ro

a

14/o

54.1 77.1 108.9 I00

1.2 1.0 0.8 0.8

0.81 0.81 0.81 0.8

36.6 38.6 36.9

WD

10

r o'

a'

re

0.96 0.96 0.96 1.5

0.79 0.79 0.79 0.5

1.4 1.4 1.4 1.4

Units o f MeV and fm are used.

(ii) The radius r o was set at 1.0 and 0.8 and stage (i) repeated to find potentials having Vor ~ ambiguity. (iii) The volume imaginary potential was replaced by a surface imaginary potential. Table 3 lists four potentials which give reasonable fits to the available elastic scattering angular distributions. No spin-orbit term was included. It is interesting to note that the approximate potential gives a very good prediction for the real depth but considerably underestimates the strength of the imaginary potential. Fig. 3 shows the corresponding lZC(p, d)11C g.s. angular distributions for these deuteron potentials. There is a considerable variation in the shape of the distribution with the surface potential giving the better fit to the experimental data. It is clear however that until further data are available on deuteron elastic scattering at high energies, detailed tests of the DWBA theory are not possible. 2.4. N E U T R O N W A V E F U N C T I O N

In the usual DWBA formalism for a reaction involving the transfer of a single

152

1. S. TOWNER

particle, the transition amplitude contains as a factor the overlap of the initial and final nuclear wave functions

(JcMc, TcMTclJAMA, TAMrA)

~C(~)~A(~, x ) d ~ ~

= ~ (JcMcjmIJAMA)(,TcMr~½ZITAMrA)J(j)Ijm,½z).

(10)

jm I

I

I

do' d~ mb/sr

laC (p,d) "C

g,s.

Ep = 154.8

MeV

Q = - 16.49 MeV L = I

SURFACE

\ "~'~.A

VOLUME

T %x

• ..

%. \,

0.1 F

.....

54.1

re 1,2

G

..........

77.1

I,O

5.6

H

.....

108.9

0.8

3.7

I00

0.8

3.7

Vo

J

°'¢xpt/ OWBA " - , ~

I

3.I

I

I0

20

".. \ .\.\ [-""., \~ ..... "\,\ •

"~

i 0

".

3

t

9 era

Fig. 3. Angular distributions for the ground state transition using deuteron potentials F - J from table 3. In the figure the well depth and radius parameters for the real part of the deuteron potential are given, together with the normalisation constants. S U R F A C E and V O L U M E refer to the form of the imaginary part of the deuteron potential. The proton and neutron parameters are those in table 1.

DXVBA

153

Here ~ represents all the coordinates of the residual nucleus C, x the spatial, spin and isospin coordinates of the transferred particle relative to the nucleus C, JA and T A the spin and isospin quantum numbers with projections M a and Mr^ for the target nucleus, while the corresponding quantum numbers for the residual nucleus are denoted with the subscript C. The quantity J ( j ) is an expansion coefficient, essentially a fractional parentage coefficient, and [jm, ½z) is an angular momentum eigenfunction defined by the above expansion. It is convenient to normalize this function (fin, ½zljm, ½z) = 1, (11) such that any arbitrary normalization constant which may arise through the above definition is contained in the coefficient J ( j ) . Clearly to obtain the precise form of Ijm, ½~) requires the functions ~bc and 0A to be known completely. This is a nuclear structure problem which has recently received much attention 30-33). A formal solution gives a series of coupled equations for the function ]fin, ½~) whose asymptotic behaviour is determined entirely by the separation energy for the reaction. The usual DWBA assumption is that Ijm, lz) may be replaced by a single-particle wave function with the required asymptotic behaviour

[jm, ½z) = ~ (IRsa]jm}[iZYt~(~)][uij(r)/r]lsa)1½z) 2a

= Z (1,~sGIjm)O.(r)lso)1½z),

(12)

he,

where the radial function ulj(r ) is determined according to some prescription. We write r as the spatial coordinate in x. The functions ]sa) and ]½z) are spinors in spin and isospin space, respectively. The spectroscopic factor is defined as ~3) ~SP = A X ]J(J)l 2" J

(13)

From the expansion, eq. (10), the value of the spectroscopic factor is seen to depend upon the choice of radial function ulj(r), yet this dependence is usually neglected. The main justification for this 30) is that in the extreme single-particle model the overlap integral, eq. (10), is proportional to a single-particle wave function, however in a more realistic model this simple relationship is destroyed. The usual DWBA prescription for the radial function utj(r) is that it should be an eigenfunction of the single-particle Schr/Sdinger equation for a Saxon-Woods potential eq. (4). The well depth Vo is adjusted to give the required asymptotic behaviour for utj(r ) while the other parameters have typical values; r o --- 1.25 fm, a = 0.65 fm and 2 = 30. The reason for this choice is that Perey 34) has shown that over a wide range of nuclei 27 < A < 197 and for a wide range of energies 9 < E < 22 MeV, this form factor gives good optical-model fits for elastic proton scattering. The use of the same form factor to calculate negative energy bound state neutron wave functions is rarely

154

i . s . TOWNER

commented upon. Some authors 9) take the form factor of the proton optical potential actually used in the DWBA calculation for the neutron wave function. This prescription can be dangerous, for example proton potential D (table 2) with r o = 0.827 fm, a = 0.413 fm required Vo = 109 MeV to fit the separation energy for the 12C(p, d) 11C g.s. reaction, and the resulting angular distribution gave no fit to experiment at all.

~.

Ep = 154.6 MeV (~ = -16.49 MeV L = I

~,~,., *.,..,,

-

>

<>

~

'."',

\

o"expt/ dDWlaA

"..~ ""'-.~\ 10

'..":~', -...~x,

~ ,

I

•

a=0.Bfm -I ........

3.B

a = 0.65 frrr I

3.7

"

-

a:O.5 frn-. . . .

-

3.6

.

I

- -

....... ro-,0* ro 0.1

....

p .......

1.31 frn

r o = 16 f ~

0:1

~' ...........

3.9

i I

1

I0

ZO

-

p:l

30

gem Fig. 4. Angular distributions for the ground state transition showing the effect of varying the neutron parameters: above, varying the diffuseness a; below, varying the radius r0. The p r o t o n and deuteron parameters are those in table 1.

Perhaps the best prescription is to look for independent shell-model calculations involving bound state wave functions. Such calculations are often concerned with explaining electromagnetic properties of nuclei, and necessarily involve proton wave functions. Recent examples for light nuclei have been on Coulomb energies 35),

DWBA

155

electron scattering 36) and also on beta decay matrix elements 37). In the case of electron scattering, a charge density distribution is constructed from proton wave functions, the parameters of which are chosen so that angular distributions for elastic electron scattering and binding energies from (p, 2p) experiments 38) a r e fitted simultaneously. For carbon, this gives 36) the parameters r o = 1.31 fm, a = 0.65 fm and 2 = 30, which are not very different from the average Perey values. Fig. 4 attempts to gauge the sensitivity of these parameters in high-energy pick-up experiments. Varying the diffuseness by 20 ~ causes only a 3 ~o increase in absolute cross section and negligible shape variation over the experimental angular range.

0.8

u (r)

0.4

..'s

"°

-

r fm

7

/ :..:.:f"

-20J.,"

v (r) -

/..."

s.':."" ......-

-40~'"

-60-

.........

....

SAXON

- WOODS

..........

OSCILLATOR

_. ,7..:..-:=:-":':':" 'e'""

Fig. 5. T h e solid curve s h o w s the contribution to the l~C(p, d)11C g.s. cross section f r o m a small distance A r ~ 0.6 f m as a function o f the radial distance r. Also plotted is the b o u n d state n e u t r o n wave function u(r) a n d the n e u t r o n potential V(r) for b o t h S a x o n - W o o d s a n d oscillator forms.

Increasing the radius by 20 ~ gives a 7 ~ increase in absolute cross section and some shape variation. These effects are small and in any case considerably less than the variation caused through the uncertainty of the deuteron optical potential. Values of r 0 = 1.31 fm, a = 0.65 fm and 2 = 30 have been used in all other Saxon-Woods calculations reported herein. It might be argued that oscillator wave functions have had considerable success in shell-model calculations, and as such may well be appropriate here. However they have the incorrect asymptotic behaviour, gaussian instead of exponential. Nevertheless fig. 6 shows that the oscillator gives as good a fit as the Saxon-Woods wave

156

I. S, TOWNER

function with about the same absolute cross section, which indicates that the nuclear interior is important for these high-energy reactions. This statement is supported by fig. 5 in which is plotted the contribution (expressed in arbitrary units) to the 12C(p, d)11C cross section at 0 ° as a function of the radial distance r. The surprising feature is the strong contribution from the nuclear interior between 1 and 2 fm, in addition to the usual surface peak between 3 and 4.5 fro. Also plotted is the bound state wave function u,j(r) and the neutron potential V(r) used in these calculations. The difference between the oscillator and the Saxon-Woods form is small in the important regions, and the differing asymptotic behaviour only appears at distances greater than 4.5 fm. It is obvious that even at the high energies concerned here, the DWBA is rather insensitive to details in the neutron wave function. The angular distribution is dominated by the treatment of the distortion of the incoming and outgoing particles, that is by the choise of optical model parameters. However by making a "physically reasonable" choice of parameters a fit to the experimental data can be obtained using a single-particle neutron wave function, in contrast to the early ideas of Brueckner

et aI. 3). 3. Spectroscopic factors On the assumption that the final state is reached through the transfer of a single /-value, the cross section for pick-up may be written dG

- Sea(Q, l, 0),

(14),

dO where a(Q, l, O) is the reaction term depending essentially on the Q-value for the reaction, the orbital angular momentum of the transferred neutron l and the scattering angle 0. With this assumption, the spectroscopic factor Se appears as a multiplicative constant in the cross section and as such is difficult to determine accurately. In the high-energy 12C(p, d ) l l C experiment of Radvanyi et al. lo), four excited states have been identified in the deuteron energy spectrum. These correspond to exciting the 0.00, 2.00, 4.81 and 6.90 MeV levels in 11C. The energy resolution is 1.2 MeV. Figs. 6 and 7 give the DWBA fits to the angular distributions for these levels. For the first three, an assignment of l -- 1 can be made unambiguously, but for the 6.90 MeV level a unique assignment is not possible. This is most likely due to the admixture of other final states, probably the 6.35 and 6.49 MeV levels. Fig. 8 gives the energy level diagram for J tC. In the right-hand column are the levels which have been experimentally observed up to an excitation of 8 MeV. In the left hand column are the theoretically predicted levels from the intermediate coupling calculations of Boyarkina 4o). The spins and parities of the low-lying levels are wellknown from such experiments 41) as l°B(d, ny)11C. An assignment to the higher levels has been given by Roush et al. 42) from an experiment 9Be(3He, n?)11C

DWBA

] 57

5+ suggesting that the 6.35 MeV level has a spin and parity ~+, 2 the 6.90 MeV level ~and the 7.50 MeV level either ~+ 2 or ½+. These are all positive-parity states. Table 4 gives the spectroscopic factors for the first three populated levels. The theoretical values have been calculated using shell-model expressions given by Macfarlane and French 13). The amplitudes for the intermediate coupling calcula-

" ~ d t~ rnb/sr

Ep - 154.6 MeV "q-, 1~\

-.

I.O

""C"I.

L._

-

o.oo

M,v

"\\

o.,-

t SAXON WOODS ....

0.01

1.0

""

OSCILLATOR

I

10

to.o '-

",,

E)c~

I

i

20

30

Fig. 6. The l ~ ! angular distributions for transitions to the ground state and two excited states of 11C. Both Saxon-Woods and oscillator forms for the neutron wave function are shown, the oscillator length parameter ag) being 1.62 fm. The Saxon-Woods parameters used are listed in table 1. Experimental data are f r o m ref. 10).

158

L S. TOWNER

tions have been taken from tables prepared by B oyarkina 4 o). The experimental values are obtained by normalizing the D W B A calculation. Both Saxon-Woods and oscillator forms for the neutron wave function have been considered. However as was pointed out in sect. 2, the uncertainties of the DWBA theory (e.g. the exact choice of suitable optical model parameters, the neglect of spin-orbit coupling etc.) all contribute to the error in the absolute cross section. Therefore the spectroscopic factor is not expected to be determined to within 30 ~ and may well be worse. Nevertheless, since many of these errors are common to each excited state, the accuracy of the I

i

laC (p,d) tlC 6.90 MeV Ep=154.6 MeV

dec dcx mblsr

i

~:.~7~'~-~. •.

N,

, s,~...." '.

~'W, ..

]"

~ "~ *~ ~"~ \

,,

I

0.1

%

',. ".. ".

\ \ '%

L=O

'\

"-. :

L=I

\ '\ \

t=Z

00,[£2

",.. L=3

20

30

B cm

Fig. 7. Angular distributions for the transition to the 6.90 MeV level of xlC. The/-values 0, 1, 2 and 3 are considered. The parameters used are listed in table 1. Experimental data are from ref. 10). relative spectroscopic factor may well be better than 30 ~ . Consequently the spectroscopic factors have been normalised to give the sum rule 13) Z s ° = 8,

(15)

JcTc

where Jc, Tc are the spin and isospin quantum numbers for the final nuclear states. Table 4 shows that reasonable agreement between experiment and the predictions of intermediate coupling calculations has been obtained.

DWBA

15 9

We now consider the spectroscopic problem of the possible excitation of levels which are forbidden according to the selection rule ½+dA+dc

where JA and Jc are angular momentum. only states with Jc = is allowed. First, consider the

> l >

(16)

IIJA-Jcl-½l,

the initial and final nuclear spins and I the transferred orbital Assuming the ground state of 12C is pure 0 +, then with l = 1 ½- and ~2 - should be populated while with l --- 0, only Jc -- ~'÷ 4.32 MeV (~2-) level in 11C. That this level should be forbidden

69

312

5.7

712

5.2

512

7.50

(1/2~3/Z+)

6.90

( 51z ÷

6.49

7/2 -

635

(3/2 + )

4.81

3/2 -

432

5/2 -

1.9

I/2

2.00

I/2 -

0.0

3/2

0.00

3/? -

IHTERMEDIATE COUPLING

EXPERI MENTAL

Fig. 8. Energy level d i a g r a m for 11C for states below 8 MeV. All states have isospin T = ½. T h e intermediate coupling calculations are t h o s e o f B o y a r k i n a 40).

in the (p, d) reaction is supported by the experiment of Radvanyi in which the excitation of this level appears to be only a few per cent of the 4.81 (~r-) level. However more recent experiments at 50 MeV (ref. 43)) and 30 MeV (ref. 4,)) violate the selection rule by obtaining appreciable excitation of this level. At these energies, the 4.32 MeV level is clearly resolved from the 4.81 MeV level. Second, levels in the region of 6.90 MeV excitation have been populated, yet from the known level scheme (fig. 8) all states in this vicinity violate the selection rule. (A possible exception is an l = 0 transition to the 7.50 MeV level.) To investigate these problems we consider the following two possibilities.

160

I.s. TOWNER

(i) Suppose that there is some inelastic process present, whereby the target nucleus is first excited to the 2 + state at 4.435 MeV, and then a neutron is "picked-up" from this excited state. The possible spin values Jc for the residual nucleus assuming it still to be an l = 1 transition are now ½-, -~-, ~- or ½-. Detailed analyses for treating such inelastic processes have been given by Penny and Satchler 4s). For 185 MeV protons, the experiments 12C(p, p')12C* of Tyr6n et al. 46) show that for scattering angles between 20 ° and 40 ° of interest here, the inelastic cross section for exciting the first 2 + level is indeed comparable with the elastic cross section. Thus it seems reasonable to expect such an inelastic process to contribute to the measured pick-up cross section. (ii) An alternative postulate is to assume that the mechanism is still direct, but that a more complicated description for the ground state of 12C should be used. Goswami and Pal 47) have calculated the collective electric dipole, quadrupole and octupole states of 12C assuming the presence of two hole, two particle (2H-2P) pairs in the TABLE4 Theoretical and experimental spectroscopic factors for the 12C(p,d)11C reaction Theoretical spectroscopic factor

Final state in 11C 0.00 MeV (~-) 2.00 MeV (½-) 4.81 MeV (~)

jj

LS

IC

DWBA fit to experiment SW HO

8

5½ 2a~

5.5 1.4 1.1

6.2 1.1 0.7

6.0 1.2 0.8

Each column has been normalised so as to sum to 8. Theoretical values have been calculated using shell-model expressions 13) for jj, LS and intermediate coupling (IC) schemes. Both Saxon-Woods (SW) and harmonic oscillator (HO) forms for the neutron wave function have been used in the DWBA calculation. ground state. An unperturbed hole-particle state is of the type I h - l p JM; T M r ) , where h and p denote the angular momenta of the hole and particle, respectively, and J and T the total angular momentum and isospin quantum numbers with projections M and M r . Let A*Sr~(h, p) denote the creation operator which operating on the closed shell ground state creates this state. If only the 2H-2P terms in the ground state are considered, and further assume that the 2H-2P states of J = 0, T = 0 are each formed by angular momentum coupling of two hole-particle pairs of equal J and T with opposite projection quantum numbers, then the ground state of 12C can be written as ITo) = N { I 0 ) +

~

(--)J-M+r-M~[2(2J+I)(2T+I)]-÷

hph' p" JMTMT

rST

~fJT r p +,, p)A_M_M~,(h,p)lO)).

~tJT "~'hp, h' p' ZaMMTklL~

Here

10) denotes the closed shell state.

The factors ( - - ) J - M ( 2 J + I ) - ~

(17) and

DWBA

161

(-- )T-MT(2T-{-1) -~t come from the Clebsch-Gordan coefficients, the c~Tp,h,p, are admixture coefficients and N the normalization constant N = [1+ •

(cSvr,h,f)2] -~.

(18)

hph' p' JT

Now if it is assumed that the ground state of 11C corresponds to a p~ hole in the closed shell configuration 10), and the excited states all correspond to removing a particle p from the 2H-2P states in the correlated ground state I~o), then the spectroscopic factor is given by 50 N 2 ~ ,\"'~hp, F Jr h ' p ' ]~2• (19) hh' b" JT

These have been calculated by Goswami and Pal 47) and are given in table 5. It can be seen that qualitatively this prescription allows all possible final states in 11C to be obtained through the transfer of a single nucleon with a unique /-value. Quantitatively, however, the prediction is that the 5- and az- states at 4.32 and 4.81 MeV are TABLE 5 Predictions o f Goswami and Pal 47) for the spectroscopic factor assuming ground state correlations in 1~C wave function Particle state in 1~C g.s. lp~. lf~_ 2p~ ld k lf~_ ld~

Jn for 11C ½~23~+ 7 ~+

/-value o f transition 1 3 1 2 3 2

Spectroscopic factor 0.201 0.013 0.007 0.081 0.101 0.101

weakly excited, whereas the three levels at 6.32 MeV (~+), 6.49 MeV (~-) and 6.90 MeV (I +) will be comparatively strongly excited. This is not the experimental situation. 4. Conclusions

We have shown that conventional DWBA calculations are reasonably successful at high energies for the reaction l ZC(p, d)l 1C. The use of the local-energy approximation is shown to give a large correction to the zero-range calculations for the optical potentials used. However a detailed test of the DWBA is not possible at the present time, since there are insufficient data to allow these optical potentials to be accurately determined. Nevertheless it is quite apparent that the DWBA using a single-particle wave function for the picked-up particle is capable of reproducing the experimental angular distributions and the theoretical absolute cross sections for the lowest excited states in alC. There is some evidence among the higher states that second-order

162

LS. TOWNER

processes are participating in the reaction. T w o such processes are considered. First is the possibility o f coupling to other inelastic channels, and second is the possibility c f including admixtures o f higher states in the shell m o d e l description o f the 12C g r o u n d state. B o t h these processes can qualitatively explain the observed final states in ~1C, but the predictions o f the latter seem to be quantitatively incorrect. T h e a u t h o r is indebted to Drs. D. F. Jackson, R. C. J o h n s o n and P. E. H o d g s o n for m a n y helpful discussions, to Dr. P. R a d v a n y i for c o m m u n i c a t i n g his experimental results in advance o f publication, to the Atlas C o m p u t e r L a b o r a t o r y , D i d co t , Berks, for the c o m p u t i n g facilities p r o v i d e d and to the S.R.C. for financial support.

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DWBA

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