A microscopic theory of α-scattering by odd-mass nuclei

Nuclear Physics 81 (1966) 349--352; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

A MICROSCOPIC THEORY O F a-SCATTERING BY ODD-MASS NUCLEI R. CEULENEER t

Physique Nucldaire Thdorique, Universitd Libre de Bruxelles Received 28 July 1965 Abstract: The formal expression of the DWBA scattering amplitude for inelastic g-scattering by odd-mass nuclei whose states are described through a quasi-particle model is presented. This description is compared with the diffraction model.

1. Introduction

In Blair's theory of inelastic s-scattering by odd-mass nuclei, the excited states of the target nucleus result from the coupling of the odd nucleon to the even core 1). In the weak coupling approximation this model leads to the following rules: (i) To a definite excitation of order Jc for the even core and a definite configuration Ji for the odd nucleon (J, < Jc) corresponds the excitation of a 2J, + 1 multiplet for the odd target nucleus. (ii) The differential cross sections ( d a / d f 2 ) ( J , --, Jr; Jc) corresponding to the excitation of these states, are related to the differential cross section for an excitation of the core by do" (Ji "~ St'; Jc) -

2f

2,2c

d~ (0 ~ dc),

2 -- 2 J + 1.

(1)

The available experimental data are in agreement with these theoretical predictions when the even core is a double closed shell nucleus such as 2°Spb (ref. 2)). When the even core is a single closed shell nucleus such as 62Ni or 64Ni, a more elaborated model is necessary in order to explain, among other things, the excitation of states which cannot be included in a 2J, + 1 multiplet 3). The theoretical transition amplitude for the inelastic s-scattering by odd-mass nuclei whose states are assumed to be linear combinations of three quasi-particle states is presented. In this description the exclusion principle is correctly taken into account for all the nucleons of the odd target nucleus. In Blairs' theory, on the contrary, the odd nucleon is necessarily distinguished from those of the even core. The transition amplitude presented here contains two terms Tt and T2. Only T1 leads, under the assumptions specified in sect. 3 to the above-mentioned rules; the t R6p6titeur ft. l'Ecole Royale Militaire. 349

350

R.

CEULENEER

2J i + 1 rule for instance, is obtained from T~ whatever be the strength of the particlecore coupling. The term /'2, on the other hand, vanishes identically when the odd nucleon is distinguished from those of the even core. This term measures in fact, to what extent the odd nucleon participates to the total excitation of the odd target nucleus and is therefore more important, in principle, when the core is a single closed shell nucleus than when it is a double closed shell one. 2. The Inelastic Transition Amplitude In a quasi-particle model, the ground state ¢'~) of an odd-mass nucleus is described by the one quasi-particle state corresponding to the lowest energy: ¢~0) = [Jj Mi> = dj,u, + lO>.

(2)

-

We use here the same notations as those used in the preceding paper 4). The excited states are described by a linear combination of three quasi-particle states:

O~ ~ = IJtMf> =

E

Jc + d ~ ] s~, d+]~[O>. C~ob~[[do,

(3)

a, b, c, J c

The notation [ ]~t indicates the coupling to the angular momentum J with projection M. The DWBA transition amplitude can be written 4): 47r M ~Tjr = ( 4 + 2 a ) - E ( - ) L E i"-t-Ll½(IL-MMII'O) kik f L,M

t,r

× (lL00lr0) y.

0).

(4)

i

The radial integrals fitly are defined by

fl~,((,) =

:o oz,(kf r)ogL(r, ~,)Xr(ki r)dr.

(5)

The functions X~ are the usual radial distorted waves and the function 09L(r, ~) results from the Slater expansion of the target-projectile interaction. Using expressions (2) and (3) for the states ~ ) and ¢~t~), respectively, the matrix element M = E <@~'((')Ifl~"(¢')YLM((')IO~)((,)

(6)

i

is given by M = T

(7)

+T2,

with a, b,Jc

x

Jo 1°

× [poVb+(-)

0

c Jc

Jc

Jc

#bVo]Ckob,(alflt.rlb),

(8)

et-SCATTERING

BY

ODD-MASS

7.2 = ~ (4~)_,~(_)¢+M+Sc[~°~cj°)¢]c~fL]¢(~ .... s~ x

j~ ½ J°

[jf

Ji

Jc

351

NUCLEI

Ic L ) ( L M 0

[#cVa't"(--)]AaVc][~°ic--(--)

J i Jr) Mi Mf ¢iac]

x (alfl~4c);

(8')

( ) and { } are 3j and 6j symbols, respectively, p° and v. are the coefficients of the quasi-particle transformation. The radial integrals (alfl~,[c) are given by (alfl~,lb) =

Io

. z ~ 2 de, R°(g)fl,.v(¢)R°(¢)¢

(9)

where the functions R.(() are the radial single-particle waves of a j-j coupling shell model. 3. Comparison with Blairs' Theory

Let d~ +) and d~ ±) be the operators which correspond to the nucleons of the even core and d~ +) those corresponding to the odd nucleon. When this nucleon is distinguished from those of the core, the operators d~ +) commute with the operators d~±) and d~±). In this ease the term T 2 which results from the commutators [d~±), d~ ±) ]_ and [d~±), d~ ±) ]_ vanishes identically. In the weak coupling approximation and for a definite excitation arc of the core, the state (3) can be written: V'°b'-'°b , d~]~,lO>

(10)

a,b

where the state ~/ic ---- E ¢~:B~SC] 6) = E (1 +~5°,n)-~q~g E (JaJbm. mt,[JcMc)d:mJ+m,,[ ~)) °, b

°, b

(11)

tna tab

describes the core excitation 4). Comparing (3) and (11) one finds: ¢°bc = - - ( l + 6 ° b )

--

dc ~q~°b

(12)

and 1"1 becomes: Tx = (JcSi M f - M, M,[JfMr)6M. ur-M, Mc( 0 ~ Jc),

(13)

where Mc(O --, arc) is the matrix element (6) for the excitation of a one-boson state for the core with angular momentum arc. Using (4), (7) and (11)-(13) in da_( dO eq. (I) is easily obtained.

/~ ~

)21 kf Z lylf]2, Ji ki Mt,~,

(14)

352

R. CEULENEER

It must be pointed out that in order to obtain this equation one has also to make the approximation/~A ~ /a,~÷ I, kl ~ kr and to assume that the distorted waves corresponding to the elastic ~-scattering by odd nuclei are practically the same as those corresponding to the elastic or-scattering by the neighbouring even nuclei. On the other hand, the Wigner symbol (JcJil~f--lai!aiJJflaf) in T 1 provides the 2Ji + 1 rule whatever be the coefficients tPo,b, i, that is, whatever be the strength of the particle-core coupling. The term T2, on the contrary, does not contain a factor restricting the possible values of Jr. The 2 J i + 1 rule is therefore only obtained if the odd nucleon is distinguished from those of the even core. We wish to thank Professor M. Demeur for many helpful discussions.

References 1) 2) 3) 4)

J. S. Blair, Phys. Rev. 115 (1959) 928 J. Alster, Congr~s International de Physique Nucl~aire Paris 2-8 juillet 1964, vol. II, p. 450 G. Bruge etal., Phys. Lett. 7 (1963) 203 R. Ceuleneer, Nuclear Physics 81 (1966) 339