Direct interaction theory of inelastic scattering. Part I

ANNALS

OF

PHYSICS:

Direct

2, 471-498

Interaction

(1957)

Theory

of Inelastic

CARL A. LEVINSON Palmer

Physical

Laboratory,

AND

MANOJ

Princeton

University,

Scattering.

Part

I”

K. BANEnJEEt Princeton,

In this paper the direct interaction theory of inelastic on nuclei is developed. The method of distorted waves plane wave approximation for the projectile is not sufficient ysis of these processes. Several interesting selection rules parison of these rules with experiments are presented. In how information about level spins in nuclei can be derived tering experiments.

New

Jersey

scattering of nucleons is required since the for a detailed analare derived and comparticular, it is shown from inelastic scat-

I. INTRODUCTION

The low excited and ground states of nuclei seem, in many cases, to be well described by the independent particle model. Recent calculations in the lp shell (I), in the regions of 016 (2), Ca4’ (3), and Pbzo8(4) have succeededremarkably in predicting the correct ground state magnetic moments and spins and the energy spacings and spins of observed low-lying levels. More recent’ly the elastic scattering of nucleons on nuclei has been successfully described by an optical well model (5, 6) whose parameters are quite consistent with those used in the shell model. The main difference is that there is an imaginary part to the scattering potential which is needed to describe the attenuation of the elastic wave function due to inelastic processes. In view of these two successfulmodels which give the bound and elastic scattering wave functions one is tempted to try to compute the inelastic scattering using these wave functions in a distorted wave calculation. Such calculations have been carried out by Lamarsh and Feshbach (7) and Kajikawa, Sasakawa, and Watari (8). Similar calculations have been successful in dealing with the problem of inelastic scattering of electrons on atoms (9). In these problems the effect of electron distortion is often quite appreciable. In inelastic processes’ (12) such as (p, p’), (n, p), (a, p), and ((Y, cy’) one ob* This work was supported in part by the U. S. Atomic Energy Commission and The Higgins Scientific Trust Fund. t On leave from the Institute of Nuclear Physics, Calcutta, India. 1 These points were discussed in detail by H. McManus who discussed the direct interaction theory developed in collaboration with Austern and Butler (10) at the conference on statistical aspects of the nucleus held at Brookhaven during January, 1955. (11) See also the comments by P. C. Gugelot at this same conference. 471

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LEVINSON

AND

BANERJEE

serves that reactions leading to residual nuclei in low excited states show a behavior quite distinct from that predicted on the basis of the statistical theory of nuclear reactions. The differential cross sections are usually peaked in the forward direction and are not symmetric about 90”. The magnitude of the cross sections is often much larger than that predicted from statistical arguments. On the other hand this behavior is qualitatively consistent with a direct interaction picture in which a compound nucleus is not formed but only a few nucleons take part in the reaction. For example, the mean free path for 17-Mev protons in nuclear matter is about 4 X lo-l3 cm. (6) Hence, one would not expect multiple inelastic collisions to occur when protons of this energy are incident on nuclei. One would expect about one inelastic event per incident proton at the nuclear surface which is hardly sufficient to form a compound nucleus. Of course in the region of a well-defined compound nucleus level these arguments are not valid since the mean free path estimate given in the optical model is only an average and does not reflect the fluctuations that occur about distinct compound levels. Since potential well depths of something like 40-55 Mev (6) are necessary to describe elastic scattering in optical model calculations, it follows that the incident and the. outgoing scattered waves in an inelastic process are not really plane but are distorted by the strong optical well. Particularly in the energy region of an elastic shape resonance the effects of such distortions should be considerable. In this paper the distorted wave formalism will be discussed as it applies to intermediate coupling shell model targets, and projectiles under the influence of optical wells. The structure of the target in the initial and final states will be shown to lead to several valuable selection rules concerning the inelastic differential scattering cross sections. The predictions of these selection rules will be compared with the results of various experiments carried out at Princeton. Detailed formulas are given for calculations involving distorted waves and finite range interactions. The effect of the Pauli principle is taken into account. In a second paper, the theory of (p’, y) angular correlation following an inelastic scattering will be discussed. It will be shown how the amplitudes derived for the inelastic scattering determine the angular correlation. Hence (p’, y) angular correlation experiments serve as an additional check on the inelastic theory presented here. In several respects the angular correlation experiments provide information about nuclear wave functions not readily derivable from the inelastic scattering experiments. In a third paper the results of detailed numerical investigations will be reported. These calculations were carried out on an I.B.M. 704 electronic computor at the Los Alamos Scientific Laboratories. A comparison of these calculations with the extensive experimental work of Sherr and Hornyak (IS) on angular correlation and of Peelle (14) on angular distributions will be given.

THEORY

OF

II. DERIVATION

INELASTIC

473

SCATTERING

OF INELASTIC

CROSS SECTION

In this derivation it is simplest to fix our attention on a particular example. We will discuss any nucleus which can be considered as made up of a magic core and two extra core particles which can couple in various ways to form the low-lying excited states of the target. The inelastic process will consist of a collision with a nucleus in which the extra core particles absorb energy and recouple to form an excited state. We shall use the isotopic spin formalism and hence demand that the scattering wave function be fully antisymmetric, since the neutrons and protons are treated on the same footing. Figure 1 shows our example schematically. Particles (1) and (2) are bound to the core by potentials VI, Vz , and interact with each other through a potential VIZ. Particle (3) interacts with this system by potentials V, , VU, I’23 as shown. We use space, spin, and isotopic spin variables (ri , ui, rJ which we denote collectively by si . Denoting the kinetic energy of the ith particle by Ti we have the Hamiltonian :

H = 7’1 + Tz + Ts + VI + Vz + Vo + VIZ + VIS + Vu .

(1)

We also introduce the Hamiltonian of the target

H, = TI + TP + VI + V, + VIZ

(2)

and its eigenstates $, and eigenvalues E, obey the relations

H&da

,4

= E&da

, sd.

(3)

These states #n(~l , s2) are antisymmetric in sl and st . Denote by *(sl , s:!, sa) the solution of the total scattering problem

(H, + Ta + V, + Vu + V2Ms1 ,

s2

, sa) = E*(sl

, sz , 4.

(4)

*(s, , s2, sa)is antisymmetric in s1, s2, ssand describesthe scattering of a beam

FIG. 1. Schematic formulas.

description

of potential

bonds used in deriving

the distorted

wave

474

LEVINSON

AND

BANERJEE

of particles incident on the target in its ground stat,e corresponding to #i(sisJ with energy E’; . Then E = E+ + (K?/Zm) where Ki2/2m is the energy of the we can expand projectile. Since \k,(sl , sp) is a complete set of functions Wsl , sz , ~3) as WSl ) $2 ) s3) = ~nbz(Sl ,

(5)

S2)4&3).

Equation (5) is simply a definition of the functions &(.~a). In order to find the elastic scattering given by \E(sl , sz , sg) we look at the behavior of the integral

s

#;*(a , s~)*(si

,

SP

, s&a

dsz for large values of the variable

ra . This integral

is simply I&(.$ which should have an asymptotic behavior corresponding to an incident plane wave and spherically scattered waves. Its behavior is given phenomenologically by the optical well model wave function (5) which has been so successful in describing elastic scattering of nucleons by nuclei, In this paper we shall always use the optical model solution for &(ss). Similarly for n # i, but E, < 0 &(s3) describes the inelastically scattered particles which leave the target in a bound state of energy E, . These functions &(.~a) corresponding to E, have only outgoing spherical waves asymptotically and it will be the purpose of the remainder of this section to derive an expression for their asymptotic behavior. In order to investigate c&(.Q) further we go back to the Schriidinger equation (4), multiply this by #J*(s~ , sp), the wave function corresponding to the final target state and integrate over s1 and sp t,o obtain, in view of Eq. (3) and Eq. (5),

where K:/2m = E - E, and IQ is the momentum of the inelastically scattered part(icle. We use units in which h = c = 1 throughout so that wave numbers and momenta are numerically equal. We now introduce the basic two channel approximation. We approximate on the right side of Eq. (6) *l(s1 , s:, , s3) =

(1 -

PI3 -

P23)bfqSl ,

s2h(s3)

+ $f(-% ,

s2h(s3)1,

(7)

where Pii is the particle interchange operator. The first factor simply insures antisymmetrization and physically allows for exchange scattering to take place. The approximation restricts our consideration to only two channels and hence ignores virtual transitions to channels other than initial and final ones. All virtual transitions within these channels, however, are treated exactly. These transitions contribute to the distortion effects. A more detailed treatment of this problem can be given in the framework of the Watson-Francis (15) many body

THEORY

OF

INELASTIC

475

SCATTERING

scattering formalism. In this treatment the effects of other channels can be treated. The main modification of our final Eq. (13) that results from the more detailed treatment is that the potentials VI3 and I’23 must be replaced by an operator which is approximately the scattering amplitude used in the impulse approximation. The net result might therefore be designated as the distorted impulse approximation. Inserting Eq. (7) into Eq. (6), we obtain

Define V,(.ss) by J

$,%l , 4w13 + V*,l[l -

P&h

PI3 -

) S?MS3)

(9

X da dsz = V,(s,)r$,(s,). Vf(s3) is simply the effective potential including exchange effects that a particle sees when interacting with the two extra core particles in the target in a state #!(a1 , sz). If we bring this term to the other side of the equation in Eq. (8) it combines with the term V3r$,(s3) to give + VfL93)14f(S3),

w3

which can be interpreted as the product of the wave function and the effective elastic one-body potential produced by the excited final state. We have then (T3

+

V3 +

V,(sJ - 2)

d&d

= - 1 #r*(a , 11

-

PIS

-

s2)W13

P23lAb,,

+

V231

(10)

sz)4i(sd da da.

We can now solve Eq. (10) for the asymptotic behavior of +f(sa). (16) In order to do this we introduce the solution to the homogeneousequation (T3

+

v3

+

v/k%)

-

K,) = 0,

$)&(s3,

(11)

which has an asymptotic form iKfn

d&

, 6)

-

eiKf”%+ k

g,(e)

(12)

&,(s3 , K,) is an elastic scattering solution and should not be confused with 4(s3), the inelastic wave function. xf(a3 , TV)is the spin isotopic spin wave func-

476

LEVINSON

AND

BANERJEE

tion for the scattered particle and 0 is the angle between Kf and r3 . We now make the reasonable assumption that the potential V, + Vf(sI) is close to the potential for elastic scattering on the ground state of the target nucleus. We assume that a low excited nuclear state presents the same potential to an elastically scattered particle as does the ground state. r$,(ss , K,) is then given by the elastic optical scattering wave function for an energy K:/2p and gr(0) is the elastic amplitude. The inelastic scattering amplitude fine,(e) is then given by:

, sz)[V1;3+ V,,l

- &)+,*(a

K, and Ki are the final and initial momenta of the inelastically scattered particle and or,is its reduced mass with respect to the target. &(sg) is the optical elastic wave function and is also equal to &(s3 , KJ in our notation. 0 is the angle between Kf and K; . Note the minus sign in the momentum argument of 4, which is always required in final distorted states. The initial and final target states tii and $, are taken as the intermediate shell model solutions for the states in question and they are therefore antisymmetrical in si and s2 . In this derivation the effect of the core was represented by the potentials VI , V2 , and V8 . This was only done for convenience. All of the degrees of freedom in the problem can be explicitly taken into account by starting with the Hamiltonian A+1

l-A+1

H = C Ti + i-1

C i>i

Vij

and setting up a representation of the target states &(sl , . . . SA) which is the full shell model wave function. ri is the position of the ith particle with respect to the center of mass of the target. One then expands * as: \k(Sl , . * * and repeats the derivation integral

s $f*(sl , . . .

SA)

(8

vi,

SA+l)

=

~n,h(Sl

,

. . ’

given above. In this derivation

A+,)(~

-

2

pi9

(14)

SA)d&A+I)

one encounters

the

A+l)

#r(a , . **

SA)+,(SA+l)

da . * . dS.4,

which is to be identified with the expression (V, + V,(S,))+,(S~) obtained in the previous derivation. We note that the center of mass of the target is not handled correctly in this approach but that the same error is involved as in conventional shell model calculations. The error approaches zero as the number of particles increases.

THEORY

In general, therefore,

OF

INELASTIC

477

SCATTERING

we have

as the inelastic scattering amplitude. Equation (16) reduces to Eq. (13) in the case where the core particles in the wave function are unexcited and only two “valance” particles are excited. Only the coordinates of the particles which are excited or recoupled in the final state enter essentially into the expression for f,inei,(e). Hence the sum over i can be taken over the extra-core particles only, and the functions #f, J/i can be taken as the antisymmetric wave functions describing the extra-core particles. Equation (13) can be further reduced. The term involving PI3 is exactly the same as the term involving P23 because of the antisymmetry of the target wave functions, so we can write

-K&P*(s~

, sz)(Vu + V,dO

- 2Pza)

(17)

#i(a , SP)&(SQ , Ki) dsl dsz dss . Next we consider the V,,P,, term which is proportional -K,>$~*(sI s 6 (8s ,

, s2)V&(s1

, s&(s2

to

, Ki) dn dszdsa.

(1%

Physically this corresponds to an event where an extra-core target particle, No. 3 in this case, is emitted by the interaction VI3 as the scattered final particle. The incident particle No.2, replaces the emitted target particle. The integral over s2is

s

i‘f*(sl , sz)di(sz , Ka) dsz

(19)

and corresponds to the absorption of the incident particle by the target without any interaction Vii causing the transition. This process is not possible and so term (18) must be zero in a rigorous formulation. Another way of seeing this is to note that the integral (19) must be essentially zero since &(sz , KJ is a continuum wave function corresponding to the same potential which determines the radial dependence of the bound state wave function #f(sl , sp).This potential is in both casesthe optical or Hartreee potential and so the wave functions should be orthogonal. Neglecting, therefore, the V13Pyyterm and making use of the antisymmetry of #f(sl , sp) and #i(si , sg)we obtain our final form

478

LEVINSON

AND

BANERJEE

(20) .Ic’ih , s&(s~, The same arguments fincl(e)

=

-

E, (“I)“26

Ki) dsi dsz dss .

applied to Eq. (16) lead to the result /&(sn,

- K,)+*(-

si ...I

x #i(. * f si . . .)cj;(s,,

$

V~,.(I

(21)

- pi,)

K,) dsl . . . dsA ds, ,

wherex: designates a sum over the extra core particles in the target. The target wave functions depend only on the extra-core particle variables and sn is the variable for the scattered particle. The expression

for the differential

cross section therefore

+(.

. . si

. .

becomes

.)c$i(s,, Ki) ds, . . . ds,

,

where, now, the sum is explicitly over the particles in the outermost shell. The symbol S denotes the summation over the final states and averaging over the initial states, Since $i and $, are completely antisymmetrical wave functions and the operator

is symmetric over the particles of the outermost shell, every term in the sum gives the same contribution to the integral. Hence we can replace the operator by NVi,(l - Pin), where N is the number of extra-core particles. It is advantageous to use the channel spin formalism. Let J and S stand for the spins of the nucleus (in a definite state) and the projectile, respectively, and I for the channel spin. Then I = J + S. Let ~(a, m,, r, mr) denote the spin and isotopic spin wave function of the projectile, where u = x is the spin and r = W is the isotopic spin and m, and mr are the respective components, and F(K, m) denote the spatial part of the projectile wave function. Then +(s, , K) = ~(a, m, , 7, m,) F(K, r,). Similarly, exhibiting explicitly the quantum numbers J, the total spin, T, the isotopic spin, and the MJ and T, their respective components, we may symbolize the nuclear wave function by 1#(JMTTz)) and suppress the other quantum numbers. The vectorially coupled channel wave function is then

THEORY

9(J, T, Tz), x(u,T,~IM)

OF

INELASTIC

[” u8’ll

= &

MJm

#(...

si

M

. .

.)JM.tTTA I ~(a, m,, 7, m)),

[ 1 Ja

where we have used the symbol

(JMam, 1JaIM). The isotopic may now be written as

479

SCATTERING

I

Mm,M

for the Clebsch-Gordan

spins have been left uncoupled.

coefficient

Equation

(22)

(24) where

9 = f#,(J,T,T,),

xbm)IfMf

IJ

F/(-K,

, r,)Vi,(l

- Pi,) (25)

.F(Kir,)

d3r,

~i(JiT,T,)x(uTm,)IiMi::

The symbol [J] stands for 2J + 1. III.

SHELL

MODEL

AND

SELECTION

RULES

We shall make two simplifying assumptions to facilitate the discussion of the shell model aspect of the problem and the nuclear selection rules. The first assumption (A) is the use of a zero range force, Vin(rin) = VoS(rin). Any special effect of finite range of the force shall be mentioned during the discussion wherever pertinent. At the end of this section a brief discussion of the effects of the finite range will be given. The second assumption (14) is the use of plane waves for the scattering wave functions F(K, r,). Fi(Ki F/(-K,,

, r,) = exp (z&.x-,), r,)

= exp (-zK,.r,J.

(26)

Most of the conclusions drawn in this section are not materially altered when these two assumptions are replaced by more rigorous considerations. With a zero range force the space exchange operator Pi, may be replaced by its only possible value unity. Hence the total exchange operator Pin = PinzPinuPinT reduces to Pin”Pi,’ only, where Pi,” is the spin exchange operator and P,,’ is the isotopic spin exchange operator. Now using the plane wave assumption expressed by Eq. (26), the integral in Eq. (25) becomes

480

LEVINSON

AND

= v, c i”[kljk(QPi)(l

BANERJEE

- PinTinT)

c z,“(n*>*z,“(D1),

k

P

where q = Ki - K, = momentum respectively, and

transfer,

Q2,, !& = polar angles of q and rl ,

where YPk(Q) is the normalized spherical harmonic of order Ic as defined by Condon and Shortley. (17) The use of the unnormalized spherical harmonics ZPk(fi) avoids the appearance of ?r and many other unnecessary factors in the problem. In obtaining the form (27) we have used the partial wave expansion of the plane wave. We have thus been able to express the amplitude for the inelastic transition for the zero-range case as the matrix element of an operator between two channel wave functions. The space coordinates of the projectile have been integrated over and we can proceed to analyze the problem in terms of the channel wave function quantum numbers. This circumstance is made possible by the fact that Pin’ can be replaced by unity with a zero range force. The operators which connect the initial and the final channel spin wave functions are of the form jk(qrl)zpk(Ql)

(1 - PiRPinT).

(28)

We note that PC = ?$ + xdl-d, and Pifl7 = 35 + 34~~. 7%, where d and T are the spin and isotopic spin operators. There are four distinct types of operators which affect the nuclear wave function. These are ZPk(Q1), Zpk(Ql)dl , Zpk(Ql)~, and Zrk(f21)dlzl . When ic is nonzero each of these operates on the space-part of the nuclear wave function. The second and the third affect the spin and the isotopic spin variables, respectively. The fourth affects all the three types of variables. The operator ZPk(l - PUP’) is an irreducible tensor of rank k in the channel spin representation. Hence we get the selection rule Ii + If + k = 0. Equation (29) is a symbolic inequalities Ii+I,-k>O;

representation

(29)

of the following

If + k - Ii > 0

and

three triangular

k + Ii - II > 0.

(30)

In the nuclear space the tensors Z,“(&) and the vector dl combine to produce tensors of ranks k, k + 1, and Ic - 1. Hence the selection rules are

Ji + J, + k = 0 or,

Ji + J, + (k f

1) = 0.

(31)

THEORY

OF

INELASTIC

SCATTERING

481

The selection rule on isotopic spin change is

T; - Tf = 0,

fl.

(32)

The selection rules (29), (31), and (32) are not modified if one uses the distorted waves rather than the plane waves. The integral over r, can still be performed and the only real change is the replacement of [k]jk(qr) in (27) by the appropriate expression. With a zero range potential the rank Ic is associated with the function ZP’(&). It is therefore subject to the parity selection rule as will be clear in Eqs. (39) and (44). When the force is given a finite range the simple meaning of Ic is somewhat obscured. Although the final result can be represented effectively in terms of tensors of rank k, it is no more subject to the parity rule. Further discussion of this point will follow after the derivation of the parity rule and also in Sec. 4. In the intermediate coupling shell model the nuclear wave function ] #(JM,TT,)) may be written as c

LWI

@LSldT 1~(Z”[hlLSJ&TTz)).

(33)

It is implied that the nucleus consists of a core and an I” configuration. The wave function has been written in the L-S coupling representation. The quantum numbers L and S stand for the total orbital angular momentum and the total spin, respectively. These vectorially couple to give the total angular momentum J. [X] denotes the symmetry and other quantum numbers. Defining the reduced matrix element of the irreducible tensor T,k as’

(jm j T,k [ j’m’)

=

[ 1 ;, ; j,

(j (1T” 11j’),

and using Eq. (34) and Eq. (27) in Eq. (25) we get

x Z”(n,)(l where we have omitted a detailed description for brevity. A convenient way of evaluating 2 With

this

choice

of phases

and normalization,

- Pis”P~in’) I) JiliTiT,~m,), of the nuclear states [see Eq. (33)] the reduced matrix element is to we have

(I-l(h’)L’S’T’ / Zn(X)LS!I’) is the fractional parentage coefficient as defined by Racah and the particle one state has been (18, 19). The parent state 1 I”-‘(X)L’MI,‘SMs’T’T,‘) vectorially coupled to form the state 1 In-I(x)L’S’T’; Z’, LM.&VsTT~).

482

LEVINSON

decouple the wave function function using the fractional

1 2”[A]LM,SM,TT,)

=

AND

BANEEUEE

of particle one from the rest of the nuclear wave parentage expansion (18)

[h,l,gs,,*, r-‘[m’~‘~’

I mmw) (36)

X ) Z’%‘lL’S’T’;

1; LM&&TT,),

where [A’], L’, S’ and T’ are the quantum numbers describing the parent state of the I*-’ configuration. Then we may recouple the particle one wave function to the spin wave function of the projectile using the Racah coefficients. l(.h j2>Jl2&J)

= Jg U(j,j*Jj,

The orthogonal transformation coefficient Racah coefficient in the following way:

; J&23)

1jl(j2jdJ23J).

U(j&Jj,

(37)

; J12Jz3) is related to the

U(j,j,Jj, ; JuJzd = ([J~21[Jd>““W(j~~J~~; JnJd.

(38)

Since the operator Eq. (28) depends on the variables of the projectile and particle one only, the advantage of the procedure is obvious. The actual evaluation can be best handled with Racah algebra and some relations specially useful in connection with the present problem are listed in Appendix (A). The final expression for the reduced matrix element becomes:

JiTi

= [Xi] [AflcLiSi L/QT; T/

%iSi[Xil

X (I”-‘[X]LST

X

%fs/[X,l

J’Tf rLzT (l”-‘NLST

( Z”[Xi]LiSiTi)U(LLr;

ILi)

[ 1(-)Lf-Li+ri-rf I, 0”I,

cJ U(fS2~JK; J,Ii) U(S.tJfLiK; LiJ)

(j,(qr)) is the expectation of particle one.

value of jK(qr)

I Z”[~,lL,S,T,>

(39)

&,s$T,T&J~

over the radial wave

function

Us

THEORY

OF

INELASTIC

11 SI jl [ 1 12 s2 j2

=

483

SCATTERING

((hsdjl(l2s2)j2J

(40) j (Gl2Mw2bSJ).

LSJ

The symbol defined above is the LS-jj transformation of the Racah and the LS-jj transformation coefficients It follows from Eqs. (24), (25), (35), and (39) and Clebsch-Gordan coefficients that the differential cross

coefficient. Useful tables are available. (20) the orthogonality of the section is (40

where

WI2

=c [kl [;o”o” 12sk rirf 2[Ji]

- I (JfrfTfTmn,

(42)

1 1Zk(Q1)(l - P.CPiZ)

1 1JiIiTiTzmr)

1.

The selection rules for k mentioned before in Eqs. (29) and (31) are present in Eq. (39) in the form of nonvanishing conditions of the second, third, and the fourth U-coefficients. These conditions are given in the Appendix. The first U-coefficient and the Clebsch-Gordan coefficient

lkl 000 [I 1

in Eq. (39) imply the following

selection rules: l+l+k=O,

(43)

1 + 1 + k = even,

(44)

Li + L, + k = 0. One additional

selection rule follows from the two terms within AS = Sf -

Rules

Si = 0, fl.

(45) the curly bracket. (46)

(45) and (46) are of importance if the nuclear states approach the pure Rules (43) and (44) are valid when the excited and the ground states belong to the same configurations. When the transition involves excitation of a nucleon from an orbital 1 to an orbital I’, the necessary change in Eq. (43) and Eq. (44) may be made by replacing one of the Z’s by 1’. The selec-

L-S coupling limit.

484

LEVINSON

AND

BANEBJEE

tion rule (44) is the parity conservation rule. Even if we do not employ an individual partical description of the nucleus parity considerations will impose a restriction on k as strong as Eq. (44) provided we use a zero range force. However, the selection rule (44) depends on the shell model description in an essential way and in the next section we shall see an example where its role can be perceived. We mention again that with a finite range force the parity rule (44) is no longer strictly satisfied. The space exchange term produces an effective tensor whose rank lc is not required to satisfy the condition (44). In Sec. 4 we shall see that the amplitude of this new type of term, in general, will be very small. IV.

EXPERIMENTS

WITH

LIGHT

NUCLEI

Since Eq. (41) was derived under the assumption that there is no distortion of the plane waves and that the target is given by the shell model wave function one would expect that it would be most valid for targets containing few nucleons which nevertheless can be treated in the shell model approximation. The ideal target in this sense is therefore Li6 whose struct,ure has been successfully treated by many authors (21) on the basis of the intermediate shell model. Sherr and Hornyak (22) have studied the angular distribution of 20-Mev protons inelas-

FIG. 2. Theoretical (plane wave and zero range force approximation, arbitrary normalization) and experimental results of Sherr and Hornyak for the inelastic scattering process Li6(p, p’)Li@*(Q = -2.19 Mev) to the first excited state of Li.6 The circles correspond to Ep = 19.4 Mev, the crosses to 14.8 Mev. The dashed theoretical line is for 19.4 Mev with a target shell model parameter b = 3.1 fermis. The abscissa p is the momentum transfer in units of 0.715 X 1Ol3 cm-l.

THEORY

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SCATTERING

485

tically scattered from Li6. They observed the scattering corresponding to the first three excited final target states. Their results are shown in Fig. 2. From the viewpoint of shell theory, Li6 consists of an alpha-particle core with a p” configuration of nucleons about it. This configuration gives rise to states of isotopic spin zero and spins I+, 2+, and 3+ as well as states of isotopic spin one and spins O+, l+, and 2+. Figure 3 shows the experimental (2~) positions of the first few levels and their spin assignments. Table I, defines the L-S coupling admixtures in Li6 and He’ which are possible under the assumption of a pure p2 configuration. Only the states are listed which are of interest in analyzing the scattering experiments. These mixture parameters are normalized so that the sum of their absolute squares is equal to one for each state. We note that the isotopic spin zero states of spin 3+ and 2+ of Li6 are pure 3D states since no other p2 L-S state can be constructed. Concerning the ground state of Li6 there is strong evidence that the 3S1 component is dominant. First the experimental magnetic moment of 0.822 n.m. is to be compared with the theoretical magnetic moment of (0.879a2 + 0.500/3” + 0.316y’) n.m. Next the square of the beta decay matrix element in the super-allowed He6-Li6 transition is given theoretically by 6(2(u - 1~p/d3), whereas the experimental value, obtained by comparing the f2 value of this process with that of the decay of a free neutron and assuming GF2 = GOT2,is 5.65. These results imply that x and (Y are close to unity. Since the spin zero state of Li” at 3.6 Mev is a member of the same isotopic spin (T = 1) multiplet as is the ground state of He6 we can conclude that the value of x deduced for He6 holds for Li6. We next consider the differential cross section for exciting the various levels of Li6. In terms of the mixture parameters for the various states of Li’, Eq. (41) yields the results given in Table II. We note that only the terms 1(j,,(pr)) I2 and 1 (j&r)) 1’ occur. This is a direct consequence of the selection rules (43), (44) for

LP The

FIG. 3. Experimental labels A, B, C, and

positions

D identify

and spin assignments states in Table I.

of some

levels

in

Li6

and

He6.

486

LEVINSON

AND

TABLE L-S

COUPLING ADMIXTURES UNDER THE ASSUMPTION

BANEBJEE

I

IN Li6 AND He6 WHICH ARE POSSIBLE OF A PURE p2 CONFIGURATION* L-S expansion

Level

The

a Charge states

independence

of nuclear forces in Fig. 3.

implies

that

states

C and

D are equivalent.

A, B, C, D are shown

TABLE DIFFERENTIAL STATES

II

CROSS SECTIONS FOR INELASTIC SCATTERING TO THE FIRST Two EXCITED IN Li6 UNDER THE ASSUMPTION OF A ZERO RANGE Two BODY FORCE V&r1 - rz) AND PLANE WAVES FOR THE SCATTERED PARTICLEB Transition

Differential

cross sections

2 (u,le/6) l 3.85d + 0.79p + 0.539 + 0.86~~~ \

(0.388 y + 0.41~~ + 0.25y&* [ (jz(qr)) 12

a See Fig. ters.

3 and

Table

I for

the

significance

of the state

labels

and

admixture

parame-

k, the subscript of j&r). The value k = 4 in the l+-3+ transition is eliminated solely by the selection rule 1 + 1 + k = 0, which is a consequence of the individual particle shell model configuration assumedto describe the L? wave function. In the l+ -+ 3+ transition k = 0 is ruled out by the selection rules on channel and nuclear spins. So the theoretical prediction is that the angular distribution should be purely 1(j,(qr)) 1’. Figure 2 gives a comparison of the theoretical prediction with the experimental data of Sherr and Hornyak obtained with 14% Mev and 19-Mev protons. The expectation value of 1(j,(pr)) 1’ has been calculated with a lp oscillator wave function, and these are

THEORY

(j&r)

OF

INELASTIC

SChTTERING

= ’ -6h’qz exp (-F);($*r))=yexp(-T),

487 (47)

where

The oscillator well parameter b has been adjusted to fit the peak of the experimental angular distribution. The agreement between the experimental distribution and the theoretical prediction of 1 ($(qr)) I2 is only qualitative. It may be noted that the cross section does not change very much when the energy is changed from 14.8 Mev to 19 Mev. This slow variation with energy is an important feature of the direct interaction process. For a fixed momentum transfer the cross section depends on energy only through the factor K,/Ki and hence the variation with energy should be very small. The most notable discrepancy between the experimental distribution and the / (j&r)) I2 distribution is in the forward direction where the theoretical value is vanishingly small whereas the experimental value is quite large. This difficulty can be resolved if we use distorted waves in place of plane waves to describe the initial and final states of the proton. Such calculations will be reported in Paper III. In the 1+ ---f O+ transition, 1 (j&r)) I2 and / (j2(4r)) I2 both appear. But only / (j&r)) I2is of importance as the coefficient of I (j&r)) 1’ is very small because p, y, and y are very small. So the operator responsible for this transition is the term p&(qr)Zk(dl *dn)(10*4

(48)

in Eq. (28). In particular, the coefficient of j&r) is proportional to the GamowTeller operator for the allowed beta decay. So the coefficient of I {j,(qr)) I* is inversely proportional to the ft value of the superallowed beta decay between the ground state of He6, which is the counterpart of the O+ state of Li6 to the ground state of Li6. No accurate measurement of the angular distribution of the inelastically scattered protons in the I+ + Of transition is available. Crude estimates (24) of the angular distribution show a steady decreaseas the angle of scattering increases. This is consistent with the theoretically predicted I (Jo) I2 shape. The remark about the coefficient of I (j&r)) I2 in the l+ -+ O+ transition in Li6 applies to the similar situation in N14.This is so becausewhereas Li6 contains two lp particles, N’* has two holes in the closed lp shell. Theft value of Cl4 -+ N14 beta decay is of the order of 10’. (See ref. 24.) The abnormally large ft value for this superallowed beta decay is customarily explained (25) as a result of an accidental cancellation of the beta decay matrix element. The same argument will

488

LEVINSON

AND

BANERJEE

also predict that the angular distribution in the I+ ---f O+ transition is almost pure 1 (j&r)) 1’ in contrast to the corresponding situation in Li6, where the distribution is almost pure 1 (j&r)) 12.Unfortunately, this angular distribution has not been measured so far with sufficient) accuracy. In view of the remarks made here, accurate measurement of these two angular distributions will indeed be of some interest. The structure of Li’ has been studied by many people (1) on the basis of the intermediate coupling shell model. The level scheme of Li’l along with experimentally (22) known excitation energy, spin and parity are shown in Fig. 4. The magnetic moment of the ground state of Li7 and the beta decay ft value of the mirror transition Be’ - Li7 agree fairly well with the theoretical predictions based on the assignment of a pure L-S coupling 22P3/2wave function belonging to the highest space symmetry to the ground states of these mirror nuclei. More detailed studies by Kurath (1) and Meshkov and Ufford (1) show that the admixtures of other states are indeed small. In fact, Kurath finds that a spin-orbit interaction of relatively low strength, indicated by the ratio a/K = 2, explains the level structure of Li7 moderately well. Both Kurath and Meshkov and Ufford expect the 4.61-Mev level to be 22F7/2. According to Meshkov and Ufford the well-known odd parity level of spin 94 at 7.5 Mev should be represented by 24P6,2, which belongs to a space symmetry lower than that of the ground state.

24

P

512

22

F

512

22F

712

22

31

Pl/2

22P

312

Li’ FIG.

4. Level

structure

in Li7. The

L-S coupling

assignments

are given

beside

the levels.

THEORY

OF

INELASTIC

489

SCATTERING

They expect the 22Fa,2level to have an excitation energy of about 6.5 Mev. A broad even parity level is known to exist at 6.6 Mev. There is no definite evidence to show the existence of an odd parity level near 6.5 Mev. Meshkov and Ufford cite a recent work by Chromochenko and Blinov (26) who found a level of unidentified spin and parity at 6.53 Mev. Assuming this state to be the 22F6/2level and using pure L-S coupling states for simplicity the cross sections for exciting the different states by inelastic scattering of protons on the ground state of Li7 may be easily calculated. Using Eqs. (41) and (42), we find that (22F,,z) : (22Fs,g) : (24F&

= 1: $&: No.

(49)

Maxson and Bennett3 have measured the spectrum of protons inelastically scattered by Li7 at various angles with proton beams of initial energy of 17.5 Mev. Some sample spectra are shown in Figs. 5. A glance at these figures will show that the relative orders of magnitudes of these cross sections are in agreement with the prediction given in Eq. (49). An inelastic proton group corresponding to the excitation of the 7.5-Mev level is hardly seen. A crude estimate of the experimental ratios are given below

and

a(6.6 Mev)/a(4.61

Mev)

= >$’

~(7.5 Mev)/a(4.61

Mev)

< $50

(50)

It may be seen that no other levels are excited by inelastic scattering. The f$level at 0.48 Mev could not be studied as it was difficult to separate the corresponding proton group from the elastic scattering group because of limited energy resolution in the experiment of Maxson and Bennett. This encourages us to believe that Meshkov and Ufford’s assignment are indeed correct and that there is a 4$- level close to 6.5 Mev. Theoretical angular distributions for the “F3/z - “F712 and the 22P3,2- 22F6,2 in each case by the selection rule on are pure I (j2@>) I2 as Ic = 0 is forbidden total orbital angular momentum. The beta decay matrix element between 22P3j2 and 24P5,2is zero as they belong to different space symmetries. Hence k = 0 is forbidden in this transition and the angular distribution is given by 1(j,(pr)) I2 in this case as well. However, the last point is only of academic interest as the cross section is too low to permit measurement of angular distribution. A reliable measurement of the angular distribution is available for the 22P3,2- 22F7,2transition. This has been shown in Fig. 6 along with the theoretical plot obtained by using an oscillator lp wave function. One important aspect of the direct interaction theory is that the nuclear excitation results from the interaction of the projectile with a single nucleon. Hence 3 We are deeply their experimental

indebted to D. R. Maxson data prior to publication.

and E. F. Bennett

for making

available

to us

400. 350.

A. BLob, = 12’

300.

100 50. C’

16

‘.

62

CWANNEL-

600550500450B eL,, = 32” 400w' p 350: E 300. 0 f 5 2508 200150100 -

0 30 “~“‘*‘*‘a’*‘*’ 3Q 30 FIG. 5. Spectrum taken Li7 at Ep = 17.6 Mev.

by Maxson

42

CH%NEL

and

Bennett 490

50

-

54

of protons

56

62

inelastically

scattered

from

THEORY

OF

INELASTIC

491

SCATTERING

9

8

700

, 600

i 500

;-J

1300

200

100

!1

0

20 ,

24

28

32

36 CXANNEL

40

44

48

L 52

-

FIG. 5.-Continued

the probability of exciting two nucleons will be lessthan that of exciting a single nucleon. The second excited state Cl2 at 7.7 Mev is a Of state (23). This level cannot be easily interpreted as a member of the ground configuration of eight lp nucleons. The wave function of this state must contain, to a very large extent, wave functions describing two-particle excitation or collective breathing mode and the weight of the ground configuration should be fairly small. This implies that the cross section for exciting this state should be very small compared to that for exciting the first excited stae which is a member of the ground configuration. Peelle measured (I,$) these cross sections at several energies, the results of which are presented in Table III. The results support our conclusions very well. A similar comment may be made about the low-lying O+ state of many even-evennuclei, e.g., 016, Ne*‘, Mg24, etc. Unfortunately, in most of these cases there are other levels very close to these levels and on account of the limited energy resolution of the experimental setup accurate measurement of the individual cross sections is very difficult. However, it is possiblethat (p, p’y) angular correlation study in such casesmay provide us with a crude estimate of the relative cross sections.

492

LEVINSON

AND

BANEBJEE

26 -

4. I..*. Oo

*....’

(.,.I.*. 0.4

0.;

0.6

0.8

I.I.I.1.. 1.2

1.0 q

1.4

1.6

‘-.-. ... ... .... I.6 2.0

FIG. 6. Theoretical (plane wave and zero range force approximation, arbitrary normalization) and experimental results of Maxson and Bennett for the inelastic scattering process LiT(p, p’)Li’(Q = -4.61 Mev) to the second excited state of Li. Ep = 17.5 Mev. The abscissa p is the momentum transfer in units of 0.715 X lo-l2 cm-l.

TABLE MEASUREMENTS

Energy

OF CROSS 7.7 Mev AND

of proton

16.7 Mev 17.8 Mev 18.9 Mev

SECTIONS THE ti

III FOR LEVEL&

EXCITING THE O+ LEVELS AT 4.45 Mev

v Co+)

(I (z+)

4.7 mb 4.4 mb 3.7 mb

160 mb 145 mb 125 mb

AT

dz*)/do+)

34 33 34

a Denoted as u(O+) and u(2+), respectively. V. DISTORTED

WAVES

AND FINITE

RANGE

INTERACTION

In this section we shall discussthe method of calculation of the inelastic scatdifferential cross section using distorted waves in place of plane waves and a finite range interaction. The interaction is assumedto be of spin independent Serber exchange type. To seethe effects of these modifications we refer back to the expression Eq. (25) for 9. On account of the operator identity tering

ppin(l

+ Pin”)(l

- Pi,“Pin”p,n’)

9 now becomesequal to

= gvin(l

+ PinZ)(l - Pin”pin’),

(51)

THEORY

OF

INELASTIC

493

SCATTERING

While the spin and isotopic spin operators have remained the same the spatial operator is different since Pin2 can no longer be replaced by unity. We still use the recoupling procedure described in Sec. III to evaluate 4. As a result one ultimately deals with integrals involving variables 1 and n only. The rest of the integrations contribute factors involving the recoupling coefficients. Since the effect of the spin and isotopic spin operators are unchanged we consider only the space integrals and a typical integral is of the form J

udrd Ym”‘CW’f(

- kf , m) Win)

(’ +2pi’)

Fi(ki,

r,) Y,,‘(Q)

(53) X d3r d3r,, .

With assumptions of plane waves for the F’s and zero range for V(rin), as used in Sec. III, the integral becomes

(54) where q&= ’ = ik[kl(jk(Qf9)

iI1k11z,=(L!q)*. () 0 ()

All the results of the previous sections may be rewritten us introduce the quantity Ak defined as

in terms

(J$fTfTsrm,

=

11jk(~~>Z”(G> (1 - P,,“Pi,‘)

11JiI&TiTzm,)

(j,(d).

The left-hand

(55)

of FPk*. Let

(56)

side is given in Eq. (39). Then we may write

(57) and expression

(24) for the differential

cross section becomes (58)

494

LEVINSON

AND

BANERJEE

Sk has been defined by Eqs. (42) and (39). We may now compute the integral (53) with the new modifications. The scattering wave function may be expanded in terms of partial waves in the following manner: Fi(K;

, r,) = CLZL[LlfL(Kirn)ZgL(~2,).

(59)

CL denotes the polar angles of rn measured in a coordinate system whose Z-axis is along the direction of Kc. rft(f(a) is the solution of the radial Schrijdinger equation d2 dr2-

L(L

+ 1) r2

V(r) + Ki2

1

rfL(Kir)

= 0.

V(r) represents the net potential of the projectile due to the nucleus. Of course, asymptotically F(Kir) will go over into an incoming plane wave along the Z-axis and an outgoing spherical wave. The final scattering wave function is Ff(--

Kfr,)

=

c

L’M’

(-i)L’[L’lf~,(K~,)Z,~“‘(~2,)Znr~L’(Q,)*.

(61)

Q, and Qj are the polar angles of r, and Kf measured in the same coordinate system. Asymptotically Ff( --K/ , T) will appear as an outgoing plane wave and an incoming spherical wave. fL!(K,r) is really the radial wave function in the effective potential due to the excited state of the nucleus. However, we shall ignore the small difference between the two potentials. The two body interaction potential may be expanded in the following way:

Win> = g [Wxh, G&&YQ1)~,“*w Introducing

(62)

Eqs. (59), (61), and (62) in (53), we obtain (‘3.3)

for the integral

in Eq. (53). Where

the first term within I LL ‘k D=

the curly bra.cket is the direct term

s

ul’(Tl)fL(kirla)fL’(lcrr,)

V&l,

m)f-1

2TtI 2 drl dr,

(65)

THEORY

OF

INELASTIC

495

SCATTERING

and the second term is the exchange term, where

I LL ‘XEC

s u,(r,>u,(r,>f~(Kcl)~~,(~,rl)

So the only change is the replacement

WI,

~,h2~n2 d-1 dr, .

cf.33

of the old F,,;“* by the new Tpk*. We obtain

(67) and the differential

cross section is given by

where T,“* is given by Eq. (64). An important change is the appearance of values of k which do not satisfy the parity rule (44). The parity rule is imposed by the Clebsch-Gordan coefficient

1 kl 000 [ 1

which appeared with Tsk* as may be seen in Eq. (55). In the new TFk*, given in Eq. (64), only the direct term contains this factor and is therefore subject to the parity rule (44). The exchange term, which connects a bound state orbital 1 with some orbital L of the scattering wave function, does not have the factor

1kl1 [000

and is therefore not subject to the parity rule (44). So we may get some terms corresponding to values of k which do not satisfy the parity rule. As an example, let us consider the transition between two states belonging to the p” configuration. Here the rule (43) allows the values 0, 1, and 2 for k. With a zero range interaction, the parity rule (44) would have been strictly valid and would have ruled out k = 1. But now the exchange part gives rise to such a term, particularly when L = L’. However, its amplitude is always very small compared to the k = 0 or 2 term. The reason is that firstly the odd k term is associated with the exchange integral alone and the exchange integrals are always smaller than the direct integrals. Secondly, the two exchange integrals IL,L,L+IE and I,.,,,-,E have the same sign and so do the geometrical factors

for X = L f

1 when k is even. But the geometrical

factors have opposite signs

496

LEVINSON

AND

BANERJEE

when k is odd. Therefore, whereas for even k the X = L + 1 and X = L - 1 terms add up, for odd k we get their difference. So the selection rule (44) is quite strong even with the finite range interaction although not strictly rigorous. DISCUSSION

We have set down the formulas necessary for computing inelastic scattering cross sections in general under the physical assumptions outlined in the Introduction. The structure of the target in initial and final states has been shown to lead to several interesting selection rules which have been tested for various experiments on light nuclei and shown to be valid. The model presented here is really a natural extension of the shell model to include inelastic scattering processes. For bound state problems in the shell model one always takes the single particle levels as given empirically and proceeds to calculate level splittings as due to the effects of interactions between the “valence” particles. In the same spirit one must take the elastic scattering wave functions as given phenomenologically and then calculate inelastic scattering due to interactions between the “valence” particles and the scattered particle. The main deterrent to this approach lies in the extreme complexity of the calculations needed to study distortion and finite force range effects. One must at last appeal to electronic computers. Such calculations have been carried out and the detailed results will be reported in a forthcoming paper. The first requirement in doing a distorted wave calculation is the knowledge of the scattering wave function. The recent success of the optical model leads us to believe that the cloudy crystal ball potential produces a satisfactory asymptotic form of the scattering wave function. However, we require a knowledge of the wave function inside the nucleus as well. The cloudy crystal ball model is not expected to provide this information. It has been argued that once the nucleon penetrates deeply inside the nucleus the chances of multiple collision increase and consequently a compound nucleus may be formed. Once a compound nucleus is formed with excitation energy exceeding 20 Mev there will be such a large number of channels opened for it that the width for its decaying into the particular inelastic channel will be very small. Hence, one should leave the interior of the nucleus out of the integration. However, one may also argue that the imaginary part of the cloudy crystal ball well, when obtained correctly, should represent the effect of any enhanced probability of formation of a compound nuclear state. Then such a well will automatically produce appropriate attenuation of the scattering wave function inside the nucleus and a further “cutoff” in the integral is not called for. ACKNOWLEDGMENTS We are deeply grateful In particular, we sincerely

to Professor appreciate

Rubby Sherr who originally his very close cooperation

suggested this problem. and the very many con-

THEORY

OF

INELASTIC

497

SCATTERING

versations we had with him concerning the experimental as well as the theoretical aspects of this problem. We would like to thank Dr. William Tobocman for the many discussions we had with him concerning the basic ideas in this paper. We deeply appreciate the comments and helpful criticisms of Professor Eugene Wigner. RECEIVED:

July

11, 1957 APPENDIX

(1) The U-coefficient inequalities are violated: a+b+e=O;

U(abcd;ef)

vanishes

c+d+e=O;

if any of the following

a+c+f=O;

triangular

b+d+f=O.

(2) Consider two systems 1 and 2. Initially they have angular momenta j, and j, and the total angular momentum of the system is j. In the final state the individual angular momenta are jr and j,’ and the total angular momentum is j’. The matrix element of the irreducible tensor operator T,‘“(2), which operates only on the second system, between the two combined wave functions is related to the reduced matrix element of the same operator between the two states of system 2 alone in the following way:

(3) Let S be the total spin of (T - Z) particles, each having spin s. Initially S is coupled to the spin s of the rth particle to form the initial total spin S; of the system of r particles. Si is coupled to the total orbital angular momentum L of the system to form the initial total angular momentum Ji. Ji is then coupled to the spin s of an external particle to form the channel spin I. The spin exchange operator Pr,R wi 11exchange the spins of particles r and n. As a result the newly formed group of r particles shall have a total spin 8, . The total angular momentum of the system will now be J. But the channel spin will remain unaltered as p. c is a scalar in the channel spin space. The matrix element of P,,,” between thentwo channel wave functions is

(lL(Ss)S,)

JsI 1)Pm” 11(L(Ss)Si) JisI)

The symbol on the right is defined in Eq. (40).

=

[ 1 IS J< s S Si . J S, L

REFERENCES 1. J. P. ELLIOTT, Proc. Roy. Sot. 218, 345 (1953); D. KURATH, Phys. Rev. 101, 216 (1956). 2. S. MESHKOV AND C. W. UFFORD, Phys. Rev. 101, 734 (1956); J. P. ELLIOTT AND B. H. FLOWERS, Proc. Roy. Sot. A229, 536 (1955); M. G. REDLICH, Phys. Rev. 99, 1427 (1955). 3. C. LEVINSON AND K. W. FORD, Phys. Rev. 166, 13 (1955).

498

LEVINSON

AND

BANERJEE

4. M. H. L. PRYCE, Proc. Phys. Sot. A66, 773 (1952) ; D. E. ALBURGER AND M. H. I,. PRYCE, Phys. Rev. 96, 1482 (1954); M. H. L. PRYCE, Nuclear Phys. 2, 226 (1956); W. W. TRUE, Phys. Rev. 101, 1342 (1956). 5. H. FESHBACH, C. E. PORTER, AND V. F. WEISSKOPF, Phys. Rev. 96, 448 (1954); R. D. WOODS AND D. S. SAXON, PAYS. Rev. 96, 577 (1954). 6. M. A. MELKANOFF, S. A. MOSZKOWSICI, J. NODVIK, AND D. S. SAXON, Phys. Rev. 101, 507 (1956). 7. J. R. LAMARSH AND H. FESHBACH, Phys. Rev. 104, 1633 (1956). 8. R. KAJIKAWA, T. SASAKAWA, AND W. WATARI, Progr. Theoret. Phys. Japan 16, 152 (1956). 9. H. S. W. MASSEY, Revs. Modern Phys. 28, 199 (1956). 10. N. A~JSTERN AND S. T. BUTLER, Phys. Rev. 92, 350 (1953). 11. H. MCMANUS, Brookhaven National Laboratory Report BNL-331 (Cal), 1955. 12. D. C. PEASLEE, “Annual Review of Nuclear Science,” Vol. 5 Annual Reviews, Inc., Stanford, 1955. 13. R. SHERR AND W. F. HORNYAK, Bull. Am. Phys. Sot., [21 1 (4), 197 (1956). 14. R. W. PEELLE, Phys. Rev. 106, 1311 (1956). 15. K. M. WATSON, Phys. Rev. 89, 575 (1953); N. C. FRANCIS AND K. M. WATSON, Phys. Rev. 92, 291 (1953) ; N. C. FRANCIS AND K. M. WATSON, Phys. Rev. 93, 313 (1954). 16. N. F. MOTT AND H. S. W. MASSEY, “The Theory of Atomic Collisions,” 2nd ed. Oxford Univ. Press, London and New York, 1949. See pages 111-115 for a detailed description of this procedure. 17. E. U. CONDON AND G. H. SHORTLEY, “The Theory of Atomic Spectra,” Cambridge Univ. Press, London and New York, 1951. 18. G. RACAH, Phys. Rev. 63, 367 (1943). 19. H. A. JAHN AND H. VAN WIERINGEN, Proc. Roy. Sot. A209, 502 (1951). 20. T. ISHIDZU, II. HORIE, S. YANGAWA, Y. TANABE, AND M. SATO, Annals of The Tokyo Astronomical Observatory University of Tokyo, second series, Vol. 4, Numbers 1, 2, and 3, (1954); W. T. SHARP, J. M. KENNEDY, B. J. SEARS, AND M. G. HOYLE, Tables of Coefficients for Angular Distribution Analysis, CRT 556, Atomic Energy Commission Limited Report No. 97; J. M. KENNEDY AND M. J. CLIFF, Transformation Coefficients Between L-S and j-j Coupling, CRT 609, Energy Commission Limited Report No. 224; L. C. BIEDENHARN, J. M. BLATT, AND M. E. ROSE, Revs. Modern Phys. 24, ‘249 (1952); A. SIMON, 3. H. VANDER SLUIS, AND L. C. BIEDENHARN, Tables of the Racah Coefficients, Oakridge National Laboratory Report ORNL-1679; L. C. BIEDENHARN, Tables of the Racah Coefficients, Oak Ridge National Laboratory Report ORNL-1098. 21. D. R. INGLIS, Revs. Modern Phys. 26, 390 (1953). 22. R. SHERR, W. F. HORNYAK, AND H. YOSHIKI, Bull. Am. Phys. Sot., [211 (4), 231 (1956). 23. F. AJZENBERG AND T. LAURITSEN, Revs. Modern Phys. 27, 77 (1955). 24. R. SHERR AND W. F. HORNYAK (private communication). 25. R. SHERR, J. B. GERHART, H. HORIE, AND W. F. HORNYAK, Phys. Rev. 100, 945 (1956); B. JANCOVICI AND I. TALMI, Phys. Rev. 96, 2.89 (1954). 26. 1,. M. CHROMCHENKO AND V. A. BLINOV, J. Ezptl. Theoret. Phys. (U. S. S. R.) 28, ‘741 (1955).