The inelastic scattering of deuterons from 12C

Nuclear Physws 1S (1960) 678--682, ~

North-Holland Pubhsh,ng Co, Amsterdam

Not to be reproduced by photoprmt or mmrofflm wxthout written permmsmn from the pubhsher

THE

INELASTIC

SCATTERING W

OF D E U T E R O N S

FROM

12C

M FAIRBAIRN

Mathematws Department, The College o/ Sczence and Technology, Manchester Recexved 23 N o v e m b e r 1959 A b s t r a c t : The a n g u l a r d m t r l b u t l o n of d e u t e r o n s s c a t t e r e d melastically f r o m x2C is determined using a direct interaction t h e o r y whtch a s s u m e s t h e excitation of a single one-particle level in t h e mtermedaate s t a t e Comparison is m a d e w i t h t h e expermaental d a t a for two b o m b a r d i n g energies The a g r e e m e n t is good and the m e t h o d could be extended to o t h e r nuclei to o b t a i n r e f o r m a t i o n a b o u t nuclear s t r u c t u r e

1. Introduction There are three derivations of the differential cross-section for the inelastic scattering of deuterons under the assumption that only one of the constituent nucleons of the deuteron interacts with the bombarded nucleus ( H u b y and Newns 1), Fairbairn 2), Sawicki 3)). Sawicki has claimed that the inelastic scattering from ~Mg considered b y H u b y and Newns and b y the present author should be analysed using his formula which assumes excitation of the collective rotational levels of the bombarded nucleus. Since 24Mg is an even nucleus which shows rotational level structure this claim is justified. The purpose of the present paper is to consider a reaction which is more likely to proceed b y the excitation of single particle states and to analyse the experimental data using the formula derived previously b y the author.

2. The One-Level Formula The formula (ref. 2), eq. (5.4)) for the differential cross-section when deuterons are scattered inelastlcally assumes that the reaction proceeds through intermediate states in which only one oi the nucleons of the deuteron reacts with the bombarded nucleus to form a compound nucleus. Such one-particle excitation of the nucleus implies that in those reactions the structure of the final excited level must differ from that of the ground level only b y the state of one of its constituent nucleons. Both the ground state and the excited state of the bombarded nucleus must be possible "parent states" for the intermediate compound nucleus state formed b y the addition of one nucleon This requirement restricts the number of allowed intermediate states. If the actual nuclear states were shell model states, then for many reactions only one intermediate state would be allowed The actual wave functions for 678

679

T H E I N E L A S T I C S C A T T E R I N G OF D E U T E R O N S FROM 12C

these states contain admixtures of other shell model configurations even when the shell model is a good approximation, but we consider the case in which the shell model wave functions form the major portion of the actual wave func-' tions and in which there is only one possible intermediate state for these shell model states. In the formula cited above * we require that, given (Jo, 0) and (it, t), the product UJe/* YoO~n* (kSn0 ' tO)Us'. U t t~n (kSnt ' t o )

is appreciable for only one (JB, s) with corresponding fixed values of (1, I*, ln, ln*) Under these conditions the integration over kp will give a contribution only at the zero kp -----kps of b u.~oOz.(kp) which corresponds to the state (JB, s). The resulting expression for the differential cross-section is at(0o) =

3z# ( 2 A + 1 ) 2~2Kt k~s (2jo+l)K ° [u/d~ttta

/2L+l~

f!

w(1½

(ksnt' ro)ujd.,oot=.(kSo,

ro)[ 2

- s)w2(zij* io; A L)z (l°* ½lo½;/* z, ps; L)

_~:o) h . ( , X o _ k ~ [ r o )

1½Ko--kpsl

ro (~ro 3 --/tst) ],.(/Kt--kps,ro)

IIK --kp t

vq

'nn ~nn.

(1)

2.

Ctn*,na* ~ 'nn Y~a* 'nn* ( K ° - - k p

s) Y~n ms ( K t - - k v s ) d ~ p s

The above formula is obtained using 1-1 couphng. A similar expression can be found when L-S coupling is used. The factors insade the modulus signs are unaltered, the difference occurs in the Racah W-functions and z-function which show how the various spins and angular momenta are coupled However, these terms only give a numerical factor in the differential cross-section and the angular distribution is determined by the energies involved and by the values of ln*, In and P For a particular intermediate state and given energy of incident deuterons the quantum numbers ln, ln* and the energies are fixed The equations

P

= In+In*,

P = S + L = ~+2+Jo+Jt I I • •

determine the allowed values of P, and thus the angular distribution of the inelastically scattered deuterons can be calculated from eq. (1). ' T h e n o t a t i o n IS t h e s a m e as t h a t used in ref 2) t h e suffix 0 a n d the starred q u a n t u m n u m b e r s refer to t h e g r o u n d s t a t e a n d t h e suffix t to t h e excited s t a t e of the b o m b a r d e d nucleus, t h e suffix s ts associated w i t h t h e p a r t i c u l a r i n t e r m e d i a t e c o m p o u n d state c o n t r i b u t i n g to t h e r e a c t m n

680

W

M

FAIRBAIRN

3. Results

A convenient reaction to analyse is 12C(d, d')12C * with Q-value - 4 . 4 3 MeV. This reaction leaves the 12C nucleus in its excited state at 4.43 MeV which is known to be a 2 + state: the ground state of I2C is a 0 + state. The angular distribution has been determined at two energies of bombardment; at 15.0 MeV b y Haffner 4) and at 19.1 MeV b y Freemantle et al ~) Both energies are in the laboratory system In L - S coupling the ground state of 1~C is 1S[4,4] and the excited state is 1DE4, 43: the only possible intermediate state is ~P[4, 4, IJ which corresponds to the ground state of 1~C or 1aN. The corresponding states in j-j coupling are (pt)S, (pt)V (p½), (pt)S (p½). Using the known properties of these nuclear levels (their energies, spins and parities) the angular distribution can be calculated b y eq. (1). We have ]o ---- O, Jt ----- 2, .Is = ~, 1 ln -----In* = 1, I = {, I* = ~,x" these give L ---- 2 and P = 1 or 2. The value r 0 = 4.47 x 10-13 cm has been chosen as in the corresponding (d, p) reaction. 10 9 8 7 O 5 4 3 2 1

c 10

i° ~B

,C; 150

90

Ii}0

150

(c,r~,~ m ~ r e e s

Fig 1 Experimental and calculated angular dlstrlbutxons

Since the calculation involves the evaluation of a double integral for each value of the scattering angle it was not convenient to perform it b y hand and

THE

INELASTIC

SCATTERING

OF

DEUTERONS

FROM

12C

681

the calculation was performed on the Mercury computer at Manchester University. The experimental and the calculated angular distributions are shown in figure 1. 4. D i s c u s s i o n

The model used to determine the angular distribution of the inelastically scattered deuterons has been the shell model with either L - S or j-j coupling. It has been shown b y Almqvist et al ~) that the intermediate coupling shell model gives a very good representation of the properties of the low lying levels in 12C, and thus the two states which are the initial and the final states in the reaction belong principally to the (ls)4(2p) 8 configuration. Because the coupling is intermediate the ground state of lsc (or 1aN) m a y not be the only level of the compound nucleus which contributes as an intermediate state in this (d, d') reaction, but since the predicted maxima and minima of the angular distribution are in good agreement with the experimental ones it seems that it provides the major portion of the direct interaction cross-section and that any other levels which contribute can be treated as background. At both energies the predicted magnitude for the cross-section is too large b y a factor five. This is in accordance with previous results on (d, p) stripping reactions, and if the Coulomb interaction and the proton-nucleus interaction were taken into account the theoretical cross-section should be reduced. It is unlikely that this would alter appreciably the form of the angular distribution because the Butler stripping amplitude is a very good approximation to the correct one and it is the angular dependence of the stripping amplitude which determines the angular distribution in the inelastic scattering reaction Haffner 4) has shown that the formula obtained b y H u b y and Newns 1) using Born approximation gives a reasonable fit to the experimental data at 15 0 MeV. At 19.1 MeV there is similar agreement. The expression obtained b y Sawicki 3) would replace the zeros of the Born approximation formula b y non-zero minima as eq. (1) does Thus all three formulae predict angular distributions which are in good agreement with the data on the l~C(d, d')12C* reaction. It would be helpful if measurements could be made more nearly in the forward direction as the present data are inconclusive as to whether or not the angular distributions fall off towards zero scattering angle. For the present reaction all the direct interaction theories predict a forward minimum whereas the electric interaction theory predicts a forward maximum (Mullin and Guth ~) ) It is intended to extend the programme which was used to evaluate the double integral in formula (1) so that other (d, d') reactions can be investigated. In this w a y it should be possible to obtain information about the structure of nuclear levels in general and of the lower excited states of the p-shell nuclei in particular.

682

w

M. F A I R B A I R N

References 1) 2) 3) 4) 5) 6)

R Huby and H C Newns, Ptnh Mag 42 (1951) 1442 Vf M Faxrbalrn, Proc Roy Soc A 238 (1957)448 J. Sawlckl, Nuclear Physics 6 (1958) 613 J W Haffner, Phys. Rev 10S (1956) 1398 R G Freemantle, V~ M Gibson and J Rotblat, Phil Mag 45 (1954)1200 E Almqvxst, D A Bromley, A J Ferguson, H E. Gove and A. E. Lxtherland, Phys. Rev. (1959) 1040 7) C G Mulhn and E Guth, Phys Rev. 82 (1951) 141

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