Itional Bureau of Standards

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SEP 1 2 1963

^ecknlcal ^tote

PB 161584

83

ON THE SCATTERING OF

r RAYS BY NUCLEI

U. S. DEPARTMENT OF COMMERCE

NATiONAL BUREAU OF STANDARDS
. . . jf standards

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NATIONAL BUREAU OF STANDARDS

83

NOVEMBER 1960

ON THE SCATTERING OF

y RAYS BY NUCLEI

U. Fano

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Preface

This report consists^ in essence^ of a number of exercises and of

discussions of specific topics^ which were developed in connection

with the experimental program of E. G. Fuller and E. V. Hayward.

TABLE OF CONTENTS

Page

1. Introduction I

2. General Formulas 2

3. The Geometrical Factor 7

4. Polarizabili ty Analysis in Terms of Intermediate

States 13

5. Polarizability Analysis in the Bohr-Mo ttelson Model . . 16

6. Core Polarizability in the Presence of a "Spectator"

Particle 21

7. Appendix 23

II
On the Scattering of y Rays by Nuclei

U. Fano

The theory of scattering by electric dipole interaction . is developed by tensorial techniques^ which permit an early

separation of geometric and dynamic factors. The geometric re- lationships are formulated in terms of variables that represent

arbitrary partial polarization of the incident and scattered

y rays . The relevant dynamic properties of a nucleus are represented by a scalar^ a vector^ and a quadrupole polarizability. These polarizabilities correspond respectively to the

values 0^ I, and 2 of the quantum number j which indicates the

angular momentum transfer in the scattering process. The

analysis of scattering according to angular momentum transfer

is compared to the ordinary theory of angular distributions.

The nuclear polarizability is discussed from the standpoint of

different models. The magnitudes of the three polarizabilities

can be determined by experiments with unpolarized nuclei but

with some degree of circular polarization of the v rays; linear

polarization contributes no additional information. Nuclear

polarization is required to determine the phases of the polarizabilities .

1. INTRODUCTION

The theory of elastic or inelastic p^ -ray scattering by nuclei has

been treated by many authors. It is equivalent^ in essence^ to the

theory of the scattering of light by molecules^ which has been presented

2 in a comprehensive article by Placzek. The geometrical aspects of this

and of related phenomena can be treated in a compact form by means of

techniques of tensorial algebra that have been described in recent

1 This process is a special case of the general theory of angular correlation. Particular aspects of the {'T ,^^ process for deformed nuclei

have been treated by A. M. Baldin, Zh. Exp . Theor. Phys . ^Zj- 202 (1959),

transl. JETP 10, 142 (I960), Zo Marie and P. Mobius, Nuclear Phys., \^,

135 (1959), and E„ G„ Fuller and E. V. Hayward, Phys . Rev„ Letters, \,

1507 (1958).

2G. Placzek, Marx Handb , der Radiologie, Vol. 6, part 2, p. 305 (1934).
works, -^ which will be referred to as FR59 and F60, respectively., An application of these techniques to various aspects of ^-ray nuclear

scattering is presented in the following sections.

The treatment will be limited to the electric dipole interaction,

which gives the main contribution to the scattering and is most prominent

in the "giant resonance" range of photon energies (10-20 Mev). The

treatment will include the effects of ^ -ray polarization and can be

easily extended to take into account nuclear polarization or alignment.

2. GENERAL FORMULAS

Let us define:

r = radius vector from the center of mass of a nucleus to the center

of its charge,

I A, A = unit polarization vectors of the incident and scattered radiation,

= j H - E , - "KcoJ = reciprocal of resonance denominator pertaining to intermediate nuclear state following

photon absorption, with H = Hamiltonian

operator of nucleus, E. = ground state energy

of nucleus, ^ co = energy of incident radiation photon , =H-E. +^<^J = reciprocal of resonance denominator applicable when emission of the "scattered"

photon energy fico precedes the absorption

of the incident photon; co =oofor elastic

scattering.

_

U. Fano and G. Racah, Irreducible Tensorial Sets, Academic Press, (1959)

U. Fano, J. Mathem. Physics, I, 417 (1960).

2
The scattering operator may then be expressed^ in the electric

dipole approximation^ as

S = r-A 0_ r'A + r-A 0+ r-A (1)

According to the conventions of FR59, p. 22-23^ the standard or

contrastandard sets of vector or tensor components will be represented

by letters with shill across in typing or in German type in print with

an appropriate superior index. In this notation, (1) becomes

The parts of this operator that pertain to the radiation can be

separated from those pertaining to the nucleus by the recoupling procedure given at the end of p. 47 of FR. This yields

S = l,^^ [yWx (0..(-l)J 0,)l^f'\jC''> x/l>] <J>

(2)

The index j indicates the angular momentum transferred in the interaction (it was called j on p. 114 of FR59). The first factor on the

right side of (2) may be called the 2J-pole polarizability operator of

the nucleus, and indicated by

_^W= ~^Hx (0. + (-l)J 0+) J^t^^jt^l (2')

Since j = 0, 1, 2, we distinguish a scalar, a vector, and a quadrupole

polarizability; the vector polarizability vanishes in the static case,

since 0_ - 0^ = for w = co = 0, and it may often be disregarded as will

be seen in Sec » 4 and 5.

Following Chap. 18 of FR59 we consider here reduced sets of matrix

elements of the operator S, between the initial and final nuclear states

whose angular momentum quantum numbers I- and I^ may be equal or different. Because the form (2) of S coincides with (18.6) of FR59, its

3
reduced sets of matrix elerpents are given in accordance with (18.7)

[a, I SI I.)]'J^ . (2j . 1)-^/^ (IjUpWllI^) [i'(l> xl(l)]"\ (3)

by^

where the second factor on the right is a reduced matrix element

(Eq. 14.4 of FR59) of the Z-'-pole polarizability . This reduced matrix

element vanishes whenever "triangular" selection rules, such as

|I. - j|.<^I^ < I. + j are violated; in particular its vanishes for

I. = unless I^ = j

.

If the scatterer nucleus is unpolarized, the desired scattering

cross section cT is proportional to the squared modulus of the matrix

element of S between an initial and a final state, summed (or averaged)

over the magnetic quantum numbers of these states. Application of

(18.10) and (18.19) of FR59 and of (3) above yields

-^ = C Z I (I. mf I S I I. m. )|

- C Z. [d^ |S| i.)]^J^*[(i£ I Si i.)]^J^ =

= C Z. (2j + l)"^|(i^ II P^^^ III.) I ^

4 4

where C = ^

_coco; .^. c^ 21 + 1

^ ^

As a result of the expansion of the interaction in Eq. (2), all the

geometrical elements of the y-ray angular distribution and polarization

4 -1/2

The factor corresponding to (2j + 1) was omitted by mistake in (18<,7)

and (18.9) as printed in FR59.
are embodied in the last factor of (4)^ namely, in the hermitian products of \^' ^^^ ^ £^^'^\ by itself, for j = 0, 1, 2. The nuclear

properties, represented by the 2-'-pole polarizabilities, enter as weight

factors in the summation of the hermitian products of different degrees.

If the nuclear orientation were observed before and/or after the

scattering, the calculation would proceed in clQ.se analogy to the derivation of (4). The invariant product of polarization vectors in (4)

would be replaced by an invariant product of the polarization vectors

and of the 2 -pole tensors that describe the nuclear polarization. This

new product contains a factor

(K)

with j' ^ i. Accordingly the Z in (4) would be replaced by a Z

and would include . "rectangular" polarizability factors (I^ || P L-' J || I . )*

X (I. II p'-"'

'11 I.), besides the quadratic factor | (I^ ll P i"" -'(| I. )( which

appears in (4). This result is noteworthy: Whereas the scalar, vector

and tensor polarizabilities give separate contributions the cross section

(4), so that their magnitudes can be determined by experiments with unpolarized nuclei as shown in the next section, the scattering by polarized

nuclei depends on interference effects among the polarizabilities so

that its measurement can provide the relative phases of the different

polarizabilities .

The analysis of the scattering process carried out in this section

differs from the analysis that underlies the familiar theories of angular correlation in nuclear physics. We have emphasized the angular

momentum transferred from the radiation to the nucleus in the complete

scattering process, and expressed the relevant nuclear properties in
terms of Z-'-pole polarizabilities . This analysis is represented by the

angular momentum diagram in figure la, where L. and L represent the

angular momenta of the incident and scattered radiation^ respectively.

(In the electric dipole approximation we have, of course, L. = L = 1.)

1. s The triangle with sides (L . , ^ , j) in figure la corresponds to the

geometrical factor in eq. (4), the triangle (I., If, j) to the nuclear

polarizability factor. Figure lb represents, instead, the analysis of

angular momentum relationships that is performed in the usual theory of

angular correlations. This analysis emphasizes the angular momentum I

of the intermediate nuclear state and deals with the absorption and reemission as separate stages of the scattering process; the separate

stages correspond, respectively, to the triangles (I., L., I) and

(I, L , If)' The treatment of FR59 develops primarily the analysis corresponding to Figure lb which leads to the angular correlation formula

(19.11), and deals only briefly with the analysis corresponding to

Figure la which leads to equations (19.12, (19.13). The equivalence

between the two' types of formulas is established by a geometrical

Figure la Figure lb
identity due to Biedenharn (FR59_, p. 114 and Appendix I)^ and is illustrated in the Appendix of this paper. Note that the analysis according

to angular momentum transfer is particularly convenient in the treatment

of scattering which may result from the alternative sequences absorptionemission and emission-absorption, represented by the two terms on the

right of eq. (1). (Diagrams representing the emission-absorption sequence differ from those of Figure 1 by the interchange of L. and Lj_ .)

The interference between the contributions of the two sequences is

readily represented by the expression of the 2-'-pole polarizabili ty in

eq. (2).

3. THE GEOMETRICAL FACTOR

The product of polarization vectors in (4) is expressed in ordinary

(non-standard) vector notation by

U'^^^=<A^^^P>"U'<'-'xi<^)](j).

(1/3) (A .A^

H(1/2)|a' X a1^ = (1/2)

. 2 ^-(A -A)

.(1/2) l+(l/3) (A =A)

for j - 0,

for j = 1,

for j = 2 (5)

This formula is quite suitable for direct evaluation in two extreme cases,

namely: (1) A and A have definite orientations, i.e., the incident

Vrays have a complete linear polarization and the scattered radiation

is observed with an ideal analyzer for linear polarization (this case is

unrealistic), or (2) the incident y rays are unpolarized and no polarization analysis is made after scattering (this occurs most frequently).

In case 2) one averages over two perpendicular orientations of A and sums

over two perpendicular orientations of A , thereby obtaining the following values of (5)

7
(L/6) (1 + cos^G), for j = 0,

(1/4) (2 + sin^e), for j - L,

(1/12) (13 + cos^e), for j = 2, (6)

where is the angle between the directions of incidence and of scattering.

A partially polarized incident beam may be represented by the

incoherent superposition of two components with orthogonal polarization

ctors A-, and A„ and relative intensities p, and p„ = 1 - ve p, . (The

degree of polarization is^ then^ p - P^.) Polarization analysis of the

scattered radiation may be similarly represented by a maximum response

efficiency e for detection of y rays with a certain polarization A, and

by a minimum efficiency e for y rays with the orthogonal polarization

A . (The degree of selectivity of the analyzer is, then, (e-,-e„)/

(e + e ).) In this general case one must evaluate the expression

22. 1 P. e L,S=1 L S ^ A. 4ri (j) (6')

which reduces to (6) when p = p - \ll and e = e == 1 . Notice that

A 'A in (5) must be replaced by | A -A | when the basic polarizations are

I circular or elliptical, in which case the vectors A. and/or A are compi ex.

Alternatively, the distribution of scattered > rays can be analyzed,

with regard to both direction and polarization, by the general procedure

of harmonic analysis given in Chapter 19 of FR59 . This procedure

represents the kind and degree of polarization of the incident or scattered y rays by means of variables, such as the Stokes parameters, which

8
are always observable whereas the polarization vectors \ and A are

observable only for fully polarized radiation. These variables are, in

essence, mean values of products of polarization vector components,

2. p.

1 1

/. (l>x /(DJ^^^and Z e [^'(O.c /(Dlck)^ where the degree, k.

of the product ranges from to 2. The geometrical factor (5) is expressed in terms of these products by a recoupling transformation. With

reference to FR59, p. 29, p. 37 and eq. (11.10) one finds that

Th. product f/^'\/^^']^'^> .[/(l^x/^l^lC^) in (7) U — (7)

- v-- — ->

\--/

_ ^ ^"^x ^

''"^ ^'"^ in (7) is a harmonic

J [.-^ -^ J function of degree k of the Euler angles between frames of reference

attached to the polarizer of the incident radiation and to the analyzer

of the scattered radiation (FR59, Chap. 6). Accordingly the summation

over k in (7) represents a harmonic expansion of the angular and polarization distribution of the scattered radiation. This harmonic expansion

is obtained automatically in the theories that analyze the distribution

according to the coupling scheme of Figure lb, as emphasized in Chap. 19

of FR59 . When the analysis develops according to the coupling scheme

of Figure 1 a^ as was done in this paper, the harmonic expansion results

from a recoupling transformation and takes the form of (19.13) of FR59,

of which the right hand side of (7) is a special case. Note that the W

coefficient in (7) is the same as one encounters when calculating the

k fl 2 -pole interaction of two p-orbitals (with C=l) coupled with a resultant quantum number j

.
We still have the task of obtaining an explicit expression for the

scalar product on the right hand side of (7) properly averaged over the

polarization intensities and detector efficiencies p.isand e . (This

task has not been carried out in the examples of FR59^ Chap. 19^ where

the final results were expressed in terms of density and detector

matrices and of functions.) In this connection^ it is not always convenient to characterize the polarizations by the vectors A and A

because these vectors are, in general^ complex and their real and imaginary parts merely represent a pair of conjugated half -diameters of the

polarization ellipse. Notice also that the polarization vector A which

appears in cross section formulas is not properly a characteristic of

the scattered radiation but that it characterizes the analyzer detector

whose probability of response is being calculated; similarly A may be

regarded as a property of the polarizer that has prepared the incident

beam. We shall express the cross section in terms of the following parameters :

a) the degrees of linear and circular polarization of the incident

rays, Pg = | A ^ • AJ (p^-p^) and P^ = | A^x A* | (p^-p^);

b) the corresponding selectivities of the analyzer for linear and

circular polarization of the scattered >' rays, Q - |A -A I (e -e )/(e-,+e„)

and Q^= |a^ X a; '1 (^^"^2^ ^ ^^''^2^ ^

c) the total response efficiency ^= 6^+62^ which constitutes only

a normalization constant (simple summation over two alternative polarizations yields ^- 2),

10
d) the angle between the directions of incidence and scattering^

e) the angleT^rfrotn the plane of incident linear polarization,

identified by the polarizer orientation, to the plane of scattering,

f) the angle/'from the plane of scattering to the plane of linear

polarization selected by the analyzer.

We write, then, in terms of these variables

2

1,S=1 "x S _-l -~i J /(^>x/(^> (k)

= 4 -Rk(p , p , Q^, (i^,f, Q,y>)- (8)

The functions R, can be worked out in the real-standard representation

of F60, and are in fact obtained by setting I =E =1 in (26) and (27) of

F60. They are

R = 1/3

o R = (1/2) P Q COS0

1 ^ c^c

R„ = (1/12)(3 cos^e - 1) - (1/4)(P. cos2y-+ Q.cos2(6) sin + (9)

2 + (1/4) PyjQ^ cos2j^ cos2to (1+cos 0) - R? Q^ sin2^ sin26:) cos0.

Expressing the coefficient on the right side of (7) as a matrix

1/3 1/3 1/3

M., = (-1)-

Jk

:+k ,/iiA = W Ilk

1/3 1/6 -1/6

1/3 -1/6 1/3C

we can finally write the geometrical factor (6'), as a function of the

(10)

observable variables of polarization and direction, in the forr

2

2. , p.e i.s=l 1 s -«i

(J)' C'^ii'' -"L (j)

= (2j + l) Z, M., R , ^ -^ k jk k

(11)

This formula reduces to (6) when T^ , P ,Q^, and Q vanish and ^=2.

11
Substitution into (4) of the geometrical factor thus evaluated

yields the cross section for detection of the scattered radiation^ by

an analyzer of given characteristics, in the form

dn

= i c R (12)

Several points may be emphasized regarding this formula:

1) The zero-degree coefficient c„ is 3 times the average value

of the cross section over all the orientation variables. This was expected since the L represents an expansion into harmonic functions of

k

the orientations.

2) When all polarization effects are absent or are averaged out,

the experimental dependence of the cross section on cos^G determines

the value of the coefficient c . 2 3) Once c is thus determined, the dependence of the cross section

on all the linear polarization variables is completely determined through

the expression (9) of R . Therefore: a) the observation of linear

polarization effects contributes no independent information on the

scattering process, and b) conversely, the properties of polarization

considered in this section enable one to utilize the scattering process

as a quantitative polarizer or analyzer tool once c has been determined.

4) Experimental determination of c requires the presence of

circular polarization in the incident beam and the measurement of circular

polarization of the scattered radiation.

12
5) Experimental determination of all three coefficients c c o' 1^ and c (or equivalent information) is necessary and sufficient to determine the magnitudes of the three 2-J-pole polarizabi li ties , by inversion

of (12), which yields

idjip^'^iii,)!^^^ Z, (m"S,^ c,

, jk k (13)

where

-1 i+k -A^j

(M ').j^ - (-l)-J (2j + l)(2k-M) w(^^^j^

1/3 1 5/3

1/^ ^^--s/ajf

5/3 -5/2 5/6

6) All three coefficients c are equal when the vector and quadrupole polarizabilities vanish.

4. POLARIZABILITY ANALYSIS IN TERMS OF INTERMEDIATE STATES

We shall now consider the connection b^^cween the 2J-pole polarizabilities (I ||P "||I.)^ which deteVmine the V-ray scattering^ and other

nuclear parameters. As is well known^ dispersion relations tie the

electric polarizability of a system to its absorption of radiation by

electric dipole interaction. The usual dispersion relations pertain to

scalar polarizability; but in our problem we deal also with a vector

and a quadrupole polarizability. We should^ therefore, derive a more

general dispersion relation.

Since ^L^-l^ i' x (0_ + (-l)-^ 0_,) ?''-' '•"'•', its reduced matrix elements

(l^ II P '- III . ) can be expressed in terms of the reduced matrix elements of

the dipole operator r"- •'pertaining to transitions between the states

I. or I and an intermediate state. The intermediate state can be

1 f classified by its angular momentum quantum number 1=1. -rl, I., or I.-I^

and by an additional quantum number n which may be a continuous variable.

13
The energy of this state will be indicated by E ^ . The intermediate

' nl

state is an eigenstate of the operators 0_ and 0^ which become now

algebraic functions of E . With these conventions^ (15.15) of FR59

nl

yields the following expression of the polarizabilities

(IJIP^^^III,) = (14)

f

(-1)

I.+I +j 1 r-i ^ (2j+l)7 Z^j (If II r^^Jllnl)

(-1)

. nl 1 nl 1

X (nI||rW||Ii) "(^I'ij

This is a Kramers-Heisenberg dispersion formula, generalized

through the W factor to represent vector and quadrupole polarizabilities

besides the scalar one. (In the scalar case, j vanishes and W takes a

simple form.) This formula provides little qualitative information

unless it is discussed from the standpoint of a model that provides some

assumption regarding the matrix elements and the energy levels E y. The

coefficients W can be taken from tables .

Some simplification of (14) is attained for elastic scattering,

when I^ = I., CO =60. The reduced matrix element can then be expressed

in terms of the oscillator strength of the transition I.->-nI (averaged

over magnetic quantum numbers^

5a

b)

A- Simon, J. H. Vandersluis, and L. C. Biedenharn, Tables of the

Racah coefficients, Oak Ridge Nat. Lab. Report 1679 Special (1954).

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, The 3j and

6j Coefficients . Technology press, Cambridge, Mass. (1959). See

particularly "eqs . (2.22-25) for our application.

14
m Fil I. +1-1 rn 2 (I^llrL^Jllnl) (nlllr'-'-^III.) = (-1)

' - -'--L J.'

I +1-1

= (-1) ^ 3(21+1)

^ 2M (E

I (nljlr^ ^lll.)l =

(15)

where M Indicates the proton mass. When I^ = I., the W coefficient in

(14) is the same as one finds in the expression of the 2j-pole interaction energy between systems with quantum numbers I. and 1 coupled

with resultant quantum number I. Ref. 5a^ p. XIII-XIV, also provides

a simplified form of this W coefficient which can be cast in the form

1 1 I

I +1+1 '-'''

,,, F.(I.I) , [3(21.+!)] '/^ J ^ (16)

where

VVi)=<

1 . I(I+l)-I.(I.+l)-2

[81, (I. +1)]

for j =

for j = 1

(17)

6[l(I+l)-I.(I.+l)-2] [l(I+l)-I.(I.+l)-l] -16I.(I.+1)

l60[l.(I.+l)(2l.-l)(2l.+3)]

Substitution of (15) and (16) into (14) yields

(I. II p[^l|ll.) -

1 1

TTT

for j = 2 .

(18)

. 1/2

= (-l)J [3(2l.+l)(2j+l)J Zj F.(I.,I) n 2M(E ^-E.

)

nl 1J<^nl.l,>

(-1)-

nl 1 nl 1

15
Whereas the usual Kramers-Heisenberg dispersion formula provides

a single equation^ which relates the scalar polarizability to a function

of the distribution of oscillator strengths^ we have now two additional

equations^ which relate the vector and quadrupole polarizabilities to

two different functions of the oscillator strength distribution. Experimental determination of the three polarizabilities, as functions

of oa, yields then additional information on the distribution of oscillator strength. More specifically, it is known that the KramersHeisenberg formula can be inverted, at least in principle, to determine

the spectral distribution of oscillator strength, Z / f ^ ^ \ . Since *= ' I \ nl,l^/

the contributions of three values of I, namely I. and I.±l, are pooled

in this spectral distribution, one may inquire whether the availability

of data on the three polarizabilities, with j=0,l,2, would enable us to

sort out the contributions of the transitions to states with different I.

This would be possible if the expression in the braces of (18) were a

function of I but not of j, since (18) would then constitute a system

of three linear equations in the three unknown values of the braces

for 1=1., I. ±1. However, the braces include a factor (-1) so that

(18) includes in fact six separate functions of I, namely

ifi /2m(E -r-E.±^^) for three values of I and two alternative

..^,^^ ' ' nl L signs in the last factor.

5. POLARIZABILITY ANALYSIS IN THE BOHR-MOTTELSON MODEL

It is a characteristic property of the states of a rotational band

that the reduced matrix elements of a tensorial operator between different pairs of states of the band are proportional to a single matrix

16
element^ evaluated in the "intrinsic" coordinate system. The proportionality coefficients are determined by the following geometric

considerations

.

In the intrinsic coordinate system consider the ground state of

the nucleus, call K. its (non-negative) angular momentum quantum number

about the z axis of this system, and indicate this state by |K.). To

this state corresponds in the laboratory system a band of rotational

states indicated by |k I m ), which are represented in terms of IK.) and

L 1 i '1

(I.) , of wave functions of the symmetrical top D ^ K.m. {f , 0, 4o). Similar-

[k]

"" ^ ly a tensorial set of operators t , such as the set of components of

q

the 2 -pole polarizability, is represented in the laboratory system by

Z , t D ^ ^

{'\lr , Q,p), where t indicates an operator of the

q q' q'

q

set in the intrinsic system. It follows that a matrix element

(Kfl^m^ I / iK.I.m.) pertaining to the laboratory system is a linear

combination of matrix elements (K^I'X i|K.) of the intrinsic system,

whose coefficients are integrals over D functions expressed in terms

of Wigner coefficients. (Notice, however, that this procedure is

meaningful only insofar as the operator t'- is independent of the

q

energyBy of the final state Kim. In our problem this condition holds

f f f only in the approximation where ca> in 0, may be taken equal to co, i.e.

when ii (<w - <» ) = E^ - E.'^O, This assumption will be made in the

following .

)

A. Bohr and B. R. Mottelson, Dan. Mat. Fys . Medd., 21_, No. 16 (1953),

p. 109.

17
We shall be interested here in matrix elements between states of

the same rotational band^

, H

in which case K^ = K. and (K^lt ,|K.)

vanishes unless q = or q = 2K. . The matrix elements with q = 2K.

arise insofar as the tensorial operator can reverse the sign of the spin

in the intrinsic system. Considering that | q |-^k, that k < 2 in our

problem, that K. = 1 does not occur in the lowest band of deformed

nuclei, and that K. = 1/2 occurs only in a few instances, we shall

disregard henceforth the contributions from q = 2K^^ 0.

Under this restrictive condition, all matrix elements

(K. I^m j t

w iK.I.m.) are proportional to the single matrix element

M

"^ ^ ^ ^

(K. It qI K. ) of the intrinsic system. Here we need only consider

[k]

the reduced matrix elements (K.I^||T ||K.I,), since the dependence of

the matrix elements on the magnetic quantum numbers (m.,m^) is given

by the Wigner-Eckart theorem (FR59, Eq. (I4.4))which has been taken

into account implicitly in Sect. 2. These matrix elements are given by

(K-I^IItL -JllK.I.) = Ol.+l (I.K.kO I I.klfK.) (K-lT^ "^^ | K.). (19)

In the case of the polarizability matrix of (3) and (4), this equation

reads

(K.I^ll pl^^^HK.I.) = (20)

= V2l7Tl (I^K. jO 1 I. jIjK. ) (K.l t qIk.)

A sum rule follows from this result, namely that the total cross

section for elastic plus Raman scattering within a rotational band

depends only on the polarizability in the intrinsic coordinate system:

18
Zj KK.I^llpL^ilK.I.)) =

= (2I.+1)|(K. I jsl^-^J^lK.)!

1 (21)

Analysis in the intrinsic system . The analysis of polarizability

in terms of dipole transitions to intermediate states^ carried out in

general form in Sect. 4, can also be carried out for the Bohr-Mottelson

model in the intrinsic system. Here there is no further reason to

utilize standard notation and we can express the polarizability operators

in ordinary coordinates. However we keep the complex combinations of

coordinates x±iy because they lead to transitions from the ground state

K. only to intermediate states (n,K) with K = | K, ±1 | , respectively.

Translation . from standard notation to ordinary coordinates yields

LX (0. + (-1) Jo ) ^W]D]

2\/X

2 VT*

2\/T

(x-iy) (0_+0^) (x+iy)4(x+iy) (0_+0+) (x-iy)+2z (0_+0^)z

- (x-iy)(0_-0^) (x+iy) + (x+iy) (0.-0+) (x-iy)

(x-iy){0_+0+)(x+iy) + (x+iy)(0.+0^)(x-iy)-4z(0_+0+)z

(22)

, for j = 0,

for j = 1,

,for j = 2,

Indicating the intermediate states by nK we have^ then,

19
[0] 1 ^n,Ki_ + l "^o

.2 _ ^2.2

(n,K^-fl I x+iy 1 K^)l +

L (E,^K,-M - ^o^ - ^ -'

'njK-ll "^0

(E . 11-E )2-^2co2

(n, K.-l| x-iy|K.)| +

EnKo" %o

o

2 .f»2^2

2

I (nK.|zl K.)|

. til. 1 (K U Ik ) = - ^z 1 1 i/^ n

>r2 2

^^n,K.l E,)2-^2^2

|(n,K.+l| x+iy I K.)| -

2 2

-fi CO ^

2 2 9 (E , ,, - E^) - ^ co^

|(njK.-lll x-iy!K.)r

[2]

VT -^

^n,K.+l

'^ n^K.+l o

- E

- l(n,K.+l|x+iy|Ki)| +

(En, IK.-lj-Eo)^ - -^^^^^

E_^^. - E

^V "'^^^ '"''"'

I (nK^I z 1K.)1 (23)

20
Notice that the matrix element of ? (vector polarizability) vanishes

unless the excitation spectra to states with K = ) K^ ± 1 1 differ. No

indication of such a difference has yet been detected for giant resonance transitions, and none could occur for the even-even nuclei in

which K. = and 1 K. - ll = K. + 1. Similarly the matrix element of

1 L ' 1 [2l ^ (quadrupole polarizability) vanishes unless there is a difference

between transverse and longitudinal excitations. A probable cause of

similarity between the spectra with K = Ik. ± 1 1 is indicated in the

next Section.

6. CORE POLARIZABILITY IN THE PRESENCE OF A "SPECTATOR" PARTICLE

Situations occur in which a nucleus may be regarded as consisting

of a single particle and of a more symmetrical core which are weakly

coupled. In this event the nuclear polarizability should be primarily

determined by the large core and should exhibit the core's symmetry.

209 Examples of this kind may be the Bi nucleus with a single proton outo3

208 side the spherical "double-magic" Pb core, and any odd-A deformed

82 ' nucleus in which the elongated spinless residue of mass A-1 is presumably unaffected by any reversal of the direction of rotation of the odd

particle. It will be shown here how^ when such conditions prevail, the

polarizability of the whole nucleus reduces to the polarizability of the

core and exhibits the core's symmetry.

In a Bi --type nucleus, the ground state may be represented by

83

the coupling scheme symbol (0,1.; I-), where is the angular momentum

of the core, and I. indicates both the angular momentum of the odd proton and the total angular momentum of the whole nucleus. Similarly a

21
state obtained by electric dipole excitation of the core, with spin

I =

'"i"-'-^ ^-f °^ I.+l, may be represented by (nl,I.;I). The reduced

matrix element of the dipole operator r between these two nuclear states

can be expressed in terms of the corresponding matrix element for the

core alone by the projection formula (15.7) of FR59.

r 1

21. ril [- 1 ^/2

(nl,I.;I||rL^J|10,I.;I.) = (-1) ^ (nl|| r

>- J|| 0) [(21 . +1) (2I+1)J

\0 1 I.

1

I-I.+l 1/2 [i] = (-1) 1 (21+1) >/_l_ (nl|l r"-

-^ 1)0), (24)

where the dependence on I is reduced to the weight and phase factor

I-I^+l 1/2

(-1) (21+1) If this result is entered in the polarizability

formula (14), with I - I , and if the weak coupling approximation is

made, E ^ = E , independently of I, the sum over I can be carried out

^ nl nl '

analytically and the polarizability is seen to vanish for j ^ 0. That

is, the spherically symmetric core contributes no vector or tensor

polarizability, as was to be expected. (To verify this result, one must

take into account the conjugation property of the reduced matrix element

(24), namely Eq. (14,8) of FR59, and a special case of the orthonormality

r 1^/2 ii+i+i_

Eq. (11.15) of FR59, namely [3 (21 . +1)J S^(2I+1)(-1) W(I • I . j /ll I) =

-JO

In an odd-A deformed nucleus, the ground state with spin K. may be

represented, in the intrinsic coordinate system, by the coupling scheme

22
symbol (O^K^^K.)^ in which indicates the spin of the elongated residue

of mass A-1 and the first index K. represents the angular momentum of

the odd particle. A state obtained by transverse electric dipole

excitation of the spinless residue may be represented by (nl, K. ;K. ±1)

.

Insofar as the odd particle is weakly coupled to the residue, the excitation energy and the dipole matrix element of the transition will be

independent of whether the transition induces an angular momentum

parallel or opposite to that of the odd particle, i.e. whether their

resultant is K.+l or K.-l. In this event the vector (j=l) polarizability

will vanish, according to (23), as it would do for an even-A spinless

nucleus.

APPENDIX

VECTOR DIAGRAM OF THE BIEDENHARN RECOUPLING IDENTITY

The connection between the ordinary angular correlation formula and

the formulas obtained by emphasizing the angular momentum transferred

in the scattering process is illustrated by Figure 2. In this figure

is reproduced the same pair

of vector triangles, (I. ,L.,I)

and (I,L .1^) as was shown

in Figure lb. Consider now

that the directions of I,L , and L , are not identified in

the scattering process; indeed

even the magnitude of I is not

identified, and those of L. and L

Figure 2

23
would vary if the limitation to dipole transitions were relaxed. The

interference effects arising from alternative vectors I^L-l, and L are

represented by adding to the diagram a second pair of triangles^

(I.,L.' I') and (I' L' I^). The possible magnitudes of the vector

k = L.-Ll - L'-L', namely 0,1, or 2, correspond to the degrees of the

various terms in the harmonic analysis of the distribution of scattered

radiation (Chap. 19 of FR59). The 9 vectors listed thus far form a

triangular bipyramid consisting of two tetrahedra with the common base

(I^I'^k). The edges of these two tetrahedra correspond, respectively,

to the indices of the two W coefficients that appear in a scattering

formula constructed on the pattern of Eq. (19.11) of FR59

.

The angular momentum transfer j is represented by the diagonal

that joins the opposite vertices of the bipyramid. The vector triangles

(L.j.Ij;) and (L.,j,L ) are the same as are shown in Figure la. Notice

that these two triangles, together with (Ll,j,L')^ subdivide the bipyramid into three tetrahedra with the common edge j. The edges

of the tetrahedron (L.,L',L ,L',i,k) correspond to the indices of the

- 1 — i

*- s — s -— ~-^ W coefficient in (7), those of (I.,L.,I ,L ^I,j) to the indices of

the W in (14), those of the third tetrahedron to the indices of the

additional W that appears when the square of (14) is substituted into

(4).

Altogether, the edges of the five tetrahedra that can be seen in

Figure 2 correspond to the indices of the five W coefficients of the

Biedernharn identity in (19.12) of FR59 . The identity of a product of

24
two W and of a sum of products of three W relates to the fact that

the bipyramid can be subdivided alternatively into two or into three

tetrahedra.

25

I

U.S. DEPARTMENT OF COMMERCE

Frederick H. Mueller, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

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