'..-;

..

( : -:>/ = "i WL±y

ma STATES iPARTMENT OF OMMERCE

UBUCATION /wN

IW/

NBS TECHNICAL NOTE 713

Electromagnetic

Multipole Transitions

in the Recouping Picture, or

Electron Scattering

Without Curls

U.S.

WRTMENT

OF

[OMMER^-

Natic QC~

Bun IOO

Standa-^ 5

" 7

^

No. It

(§17-
NATIONAL BUREAU OF STANDARDS

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Ml BUREAU OF STANDARDS

»MAR 2 1972 UNITED STATES DEPARTMENT OF COMMERCE

(\#-V 0CC-Q*f Maurice H. Stans, Secretary

1,,$ NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

No ni3

(4 7^

NBS TECHNICAL NOTE 713

ISSUED FEBRUARY 1972

Nat. Bur. Stand. (U.S.), Tech. Note 713, 20 pages (Feb. 1972)

CODEN: NBTNAE

Electromagnetic Multipole Transitions

in the Recoupling Picture,

or. Electron Scattering Without Curls

J. S. O'Connell

Linac Radiation Division

Institute for Basic Standards

National Bureau of Standards

Washington, D.C. 20234

4** 0F %

""feZ* ^

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C, 20402.

(Order by SD Catalog No. C 13.46:713). Price 30 cents.
Contents

Page

I. Introduction ....... ...... .......... 1

II. Multipole Expansion of the Fundamental Operators ..... 2

III. Single Particle Reduced Matrix Elements ......... 9

A. Coulomb Operator ................... 9

B. Moment Operator. ................... 11

C. Current Operator a .. ....... ......... 12

IV. An Example 13

V. Conclusions ....................... 16

VI. References ........................ 17

II
Electromagnetic Multipole Transitions in the Recoupling Picture

or Electron Scattering Without Curls

J. S. O'Connell

The formation of the multipole operators from

the fundamental charge, current, and moment operators of electron- nucleus scattering is carried

through as an angular momentum recoupling problem

rather than the usual vector algebra derivation.

This point of view allows the electric and magnetic

transition operators to be obtained in a simple and

intuitive manner. The single particle reduced

matrix elements of the charge, current, and moment

operators are then calculated in the j-j coupling

scheme using a flow chart technique developed by

Danos.

Key Words: Electric transitions; electromagnetic

operators; electron scattering; magnetic transitions; multipoles; recoupling.

I. Introduction

The theory of electromagnetic multipole transitions is usually developed in the language of vector algebra: gradients, divergences, and

curlso This approach evolved from the historic connection between 19th

century mathematics and the study of the classical electric and magnetic

fields. An alternative approach to the multipole transitions in quantum

systems is to regard them as essentially angular momentum recouplings.

The quantized rotation theory of Wigner, Racah, and Fano is then brought

to bear on not just the electromagnetic operators, but upon the whole

matrix element, i.e., the spatial integral over operator and initial

and final states of the absorbing or emitting quantum system. This

type of analysis is most easily performed using a diagramatic method
recently introduced by Danos [ l]

In the next section the electron scattering cross section for a

discrete nuclear transition is developed in terms of the single particle

operator. The reduced matrix elements of the charge, current, and moment

operators are calculated in Section III in the j-j coupling scheme. In

12 Section IV a simple example of an Ml transition in C is presented.

Section V summarizes the advantages of the recoupling approach to multipole transitions.

IIo Multipole Expansion of the Fundamental Operators

Electrons interact with the nucleus through the charge, current, and

— — moment operators on each nucleon (e, ep/M, i|J.c X q/2M)e ^*

, where e=l

for a proton and for a neutron, ji is the nucleon magnetic moment

(u = 2.79, U - -l»9l), P is the moment operator, and o the Pauli spin

p ' 'n 7 operator acting on the nucleon at position r. The three momentum transferred to the nucleus is q. For a nuclear transition from an initial

state of angular momentum and parity J. to a final state of J,, the

electron scattering cross section for one photon exchange neglecting

the mass of the electron is given by

da £

dQ "o 2J.+1

f ' i

q,*

X \=±

=* UjJMJ £e.eiq - r j|j.M.)| 2+ (~3- + tax?Q/2)

q ' f f ' . j

J, ii' \2

-*3 ' }

S |<Jf

M

f

I Sfe e

1

^ iA + |i fi.-i. C jjj

"J, >, ±1 j

~ J J-J

1 Figures in brackets indicate the literature references at the end of

this paper.
where

G

o =

e cos.6/2 1 L2e.sin2 e/2j v

T

Hj,

1 +

x 2£. sin2 0/2.-1

We present this formula without proof, but give the following explanation of the origin of some of the terms. The a is the familiar

Mott cross section for the scattering from a point charge by an electron

of energy £,. times a recoil factor associated with the density of final

nuclear states. The (2J.+1) L/ is the sum over final and average

1 M_M.

f l over initial nuclear orientations (with the axis of quantization taken

in the q direction) since they are unobserved. The z_) sums the single

J particle operators over all the nucleons in the nucleus. The first term

in the j

r is the square of the longitudinal form factor, the second an

angular function times the square of the transverse. The longitudinal

component is composed of the Coulomb operator and the longitudinal component of the current operator which, since they are related by current

conservation, can be written as the square of the Coulomb matrix element

2 P n

4

times 11 - ~J = —^ where the four vector a is related to the three

q

2 <f

momentum q and the electron energy loss qQ = &±

- £

f

by q~ = q - qQ

.

The transverse term uses only the \ = ±1 components of the current

and moment operators decomposed along the q direction. The angular

factor in front of the transverse term comes from the one photon exchange analysis.

Because we are interested in the case of a discrete transition with

definite spin and parity change (and eventually definite isospin change),

we are forced to make a multipole decomposition of the fundamental

operators into terms of definite total angular moment J where.

3
'if J. -J,. 1 < J < J.+J_ if and furthermore to combine terms of the current and

moment operators which have the same parity. For the longitudinal

operator this is easily accomplished by the usual expansion formula for

a plane wave into spherical Bessel functions and spherical harmonics

e

iC*- r = Utt)* E (i)

L

L J L(qr) Y^Qj

i

where L = (2L+1) 2 and the z-axis has been taken in the direction of q.

In this paper we will work with the contrastandard element Y

L

which is related to the ordinary spherical harmonic by

Y

[L] = (-i)L

Y Y

M

- { X)

LM

*

Thus the Coulomb matrix element becomes

<J

f

M

£

| e^'r

| J.M^

= U")* J

(-)

J

S <J

f

M

f

| j j \

[J] \JM

±

) •

Using the Wigner-Eckart theorem

= (ATT) 2 E (-) f f J(f M

x;

J f 1

* <J

£

II Jj ^

[J]

II V •

The current and moment operators are more complicated. They involve the coupling of an angular momentum L (from the plane wave) with

one unit of angular momentum (from the vectors p or a X q) to form the

total angular momentum J.
e ±q.V \ (4-TT) 8 y, f ^h

M " M t

(l) J L LO PX

M l

l j L J L

X P\

= l4nl? i e l j i L L J

^i ri[L]

x otOlJ]

In the last step the product of the two contrastandard elements was

replaced by its expansion in terms of elements coupled to a definite

angular momentum J.

Similarly, for the moment operator

.*<•' (c x ^ = (*.)* q. R L S (£ ^>L

[>M x „W

J]

where we have "used (a X q), = V2 cr ^ X q^ ^ |- ^ = M<\

If we hold J fixed, L takes on three values: J+l , J, J-l. The

parity of each current term is tt - (-) but each moment term is

tt = (-) since a is an axial vector. Therefore, to form a current plus

moment operator of definite parity we combine the L = J±l current terms

with the L=^J moment term to form an operator of parity tt = (-) . This

combination is called an electric operator & because the classical

J+l electric multipole field has this parity. The tt = (-) or magnetic

combination ^i- -1 is formed by the sum of the L = J±l moment terms with

the L=J current term.
- [J] = iAMlll ( <^y j /j+i l jv |- y

[j+i] _[i]l[J]

^(VlD^p-13 *^3

+

MS J J X ( J l J

^i [Vj]

x a

[l]

! [j] * 1

»l [J] _ i Utt) 8

f ua

\ M {'¥(3Sr-Sx(«i.0j«['c *

l] ««[lY

]

+ f (j-ijjift1 }.?,: rY

[j-i]

x J!]tj ] -l L

r \

SS (^ij.P.pi'^i o X-X/ j j L k

J

Evaluating the 3~ j coefficients we obtain

[J] = 1(

.)J-i L4ng I (*$ A rY[J+

V" M

J = 1 <- )U *- if

M

1 -i^ JJ+i

i yL

+ f (^ [*m x .[op]

}

"A
n

[j]

1( j M ^ I 2 W J J+1 L \

: -^^[rf^.JW

'2J+1 N

x a

L

7v

2 J

J J

,, i^./1 TM}%

When the electric and magnetic operators are sandwiched between

nuclear states

VA[J] +\[J] KM

i> = (-> Jrt IX m

1 )

<j

f

ll£[J]+^J]

IK> • f- <-<

Therefore, the electron scattering cross section in terms of the multipole operators can be written as

da. G

dQ 2J

2- E {i|<Jf ||Ej yWiij.)! 2 (})

l MM., J q j

^ J/

2 / J J J V

M* M/

f l

+

-1L.

2q

»[ J ] j. JJ

+ t^e/2) S ] I ±i

|Of

|| Si 6$ + ^Jj|| J±>l

V

_ M

, , ^

i2 / J, J J.

,f ^ .A

2,
The completeness relation for 3" j symbols allows us to perform the

M.M. sums

f 1 s. LmT = «-2

M-.M. f

/v i

f ' 1

The sum over \= ±1 simply gives a factor of 2 since the \ dependence

TT TT was removed by the previous step. For a given J

f and J. either the

electric or the magnetic operator will connect the nuclear states for

each allowed value of J since the parity requirement must be satisfied.

We come to the final expression for the cross section in terms of the

reduced matrix elements of the Coulomb, electric, and magnetic

operators.

iS 2_y, j_JJ-

| /TTT, |y!4 v

[J]|| T

TT N ]2

dQ 27^ Ui^ii?^ J K>r

l J q j

+ 5rtt$ + tan2' 2q

e/2

) [

| (j

f

"|| E

g(

Wp^y | 2

+ 1 <j

£

"|| S ^ W

||j .">

| 2

]}

We are now required to compute the reduced matrix elements of the

operators

Jj(qr) I

L

, J L (qr)|^YL X p

L J L J

, J L (qr)jrL J X a

L J j

L J

This type of analysis is most easily performed using the diagramatic

recoupling technique of Danos [l].
III. Single Particle Reduced Matrix Elements

The single particle states will be taken in a spherical basis in

which the orbital angular momentum JL (in the Y

L J representation) is

coupled to the spin 1/2 to a total j.

,[J]

* [j]

> = R„(rfl^ x x

^

^m ' £

v

with the Hermit ian adjoint given by [2]

m

-m

A convenient representation for the spin functions and their adjoint

is

kJ«> = Q , kJ«> -

(?)

<Xi [ *]

| =(10) , <x x [|]| = (0 1)

2 s

The reduced matrix element of an operator L J is related to what

Danos calls a projection integral

(t [j']|| [J]

|| t [j]> = ( _)J ,+J-J [^J^loWl^J]]

where the tilde means transpose which affects only the spin part of

our wave functions.

A. Coulomb Operator

The projection integral of the Coulomb operator is evaluated with

the aid of the recoupling diagram shown in Fig. 1 with the result
^J'] hjYW

u[j]j = [^i[Y[^] x x

C «][r]| j/

[J]

| ^

[Y[^]

x x[«-,[jjj

* 2 J r^ i ji

J J *£ j Ui j J LP o oj

#] |x[l]

j

Y[^]|y[j]| Y[^ ^^J^.!^)^

The recoupling boxes are related to the usual 9~ j symbol by

a b c d e f

-g h i

a b c ., = c f g h -j d e f r g h i

where c = (2c+l) 2

. The projection integral for three contrastandard

spherical harmonics is given by Danos as

Y[^]| Y[^]| Y

[^3]

J

(4tt)*

h k %

o o)

The spin overlap is

~C«

| XL«- = + h = + 2

1

The final result is [3]

(i)

r - J ^

{ p'\^Y\-3\p^) = (.)J+J^

J j* j I

s I f^'i'i-'L /'4'J A>

(4tt) j J6 U ,

: u

) < R^ljjCqr)!^)

10
which may be further simplified by use of the identity

lj i jj Vo o oy

when £' + J + £ = an even integer.

As a special case the electric dipole reduced matrix element is

U)

i'-l-i^V],| r

[l]|| + [J]> = (-)^3'3(„|^i) <R,,|r|R,>

B. Moment Operator

Fig. 2 shows how the spin and orbital functions are recoupled to

give

u

[j']

1Ijt ivu x a[i]f

J]

||t

cj]

(-)

j'-j+j >4 J'~

"4 4

j"

L 1 J ^4 j

^4 j J

|o o_

[^« |o

[i]

|x[Hj|-y[^] |y

[l]

|y

[i]j

X <R.,|j T (qr)|R.> V 1J L

The spin projection integral is (-) 4 1 = + i 6

s therefore the reduced

moment matrix element is [2]

U«- l- 1 - D,Jj'],h

rY

[L]

y ri]-f

j]

lllk [j]

(i) it P L

I-J x a

L llt UJ >

111 2 2

L j'j J11

, vjt« 6

g J L J'JX' j / l 1 1 \ ^'L A/R li (nr)lR > (-) —j {

2

f 2 i;

Vo oJ

<R

jt« |j L (qrj 'V
C. Current Operator

Evaluation of the current operator requires one trick shown in

Fig. 3. A complete set of intermediate angular states is introduced

through the unit operator

Ek [yW(q) X lM(Q.)f°] = 6(0 - Q»)

in order to separate the p

L

-

1 and Y^^ operators. The triangularity

condition between Z, 1, and k in the final result limits the sum to two

terms k = I ± 1.

%y\\ jii^pt'jfi^]]

J J

" * 2 J r *i j ^1 j'

U'i j'J _0 _ ¥«i. ix[ «-

£ k

k

k k 0]

1 1 k I 1

L L] Jfe" X- J 1 £ j£ U'O £'J

kill

_£' j6 jj

Y

[^']| Y

[L]| Y[k]ir

R ^[k],.

p

[i] |Y[^j Y

L ~J R

The projection integral of the momentum operator is given by Danos

as _ v[^l]

lp

[l] |Y[^]-| = .. (m)^|__i (-)

(*- i

M-lf l+l \ Vdr " r

JL(M-l)

1 1 A'

Y[^l]

lp

[l]

IY

[^]-l _. ^I_ +

M-l\ . t

s £/ I

'

, V,5r

3- +

^r V

j6(X-i; 1 1 4

s

0.^

12
We will use the second form to more easily compare our results

with the standard treatment [4J»

(i) U

, -l-i-j&) /1 [j

, ]„. rV[ L ] [i]-l^n,[j] \T I j T

l Y X p

L J

| p

(-)

jfc'+j-l J L J' j i' A J'^j 1 !

U#

rjx 1 j 1 /jI'l wvw i a"

1 / My,, i- /jL.

!_U A'jfrUj VO A 0/ V2i+lA 4

1 lJ L\o^ r/ ' j&

+ U jt'juJ Vo o o A. o o o) \2M) {r

ji> I j l^ + "VAVJ

IV. An Example

We illustrate this formalism with an example to show how the sum

over particles Z_) works and to show the effect of considering a transij

12 tion between states of definite isospin T. C has a ground state

TT -4- TT *4- with J = and T=0 and an excited state at 15.1 MeV with J = 1 and

T=l. The only electromagnetic transition operator that can connect

these two states is one with J=l, tt =.+ which is the magnetic dipole

operator i/t » 12

If the C ground state is taken as the configuration

(lsi)^ ( lp3_) and the 1 excited state as(lsi) (lp3.) ( Ipi) -> then the

2 2 2 2 2 transition can be pictured as simply one nucleon in the p- shell recoupling its orbital and spin from j = 1 + § to j '= 1 - ^. Both neutrons

and protons in the p- shell can make this transition. Because of the way

13
single particle states are normalized (total state functions normalized

to 1 and single particle wave function also normalized to l) we have the

simple relation for closed shell nuclei

<Y [j

f

]

|| £ 0(1) || Y

[J

i

] > i

= E <Y [j,]

|| 0^|| Y

[j]

> Jj'

i.e., the sum over single particle operators between the total state

wave functions is the same as the sum of possible single particle transitions. In our example j = § and j

1 = \ so that only one reduced matrix

element is involved < [vl \ x^ 1 || ^^|| T 1

^x X^~|

2 >•

Electromagnetic transitions can, in general, change the isospin quantum

number of a nuclear state by or 1, an isoscalar or isovector transition. The fundamental interaction operators can be divided into their

isoscalar and isovector components by the replacement of

e.= §( 1 + T. z

) where T |p) - +|p) in the Coulomb and current

operators and LL.= h(\l + T.Z

LL ) where Li = Li + LL = .88 and

|i - \1 - ]1 = J+.70 in the moment operator. In the present example only

the isovector parts of the operators will contribute to the transition.

The single particle operators are therefore multiplied by the factor

|t and LI. is replaced by the |i for all particles. The isospin part

of the single particle matrix element is always of the form

Em, .A i o|| t. z

| o o > = i-^EM, .Ai\\ t. |lo > j (j.J j

' 32 j [1) " J

"

1 1_ M / ill ? l|fe ^ = %

u

1 i_ M / ill 7 lli > = ~.

2 1 n \ s|| T ||S f 1 3

2 2

s
where the M/.\ are the space- spin part of the matrix element. Thus,

the additional specification of the isospin quantum numbers of the

initial and final states reduces the cross section by a factor of 2.

Explicit evaluation of the 3~ 5 6- > and 9~j coefficients and the use of

harmonic oscillator radial wave functions gives (with Tj = qb, b is the

oscillator parameter)

<ipil|j (qr) r^W1 ^

11

,-A

•if A

1 P 3> = -i"^i(l- TTf) ^

(3)

2 b

(4-tt) s

< 1 Pi||Js(q^) W* a" [1]

|1P3>

2 = +i Tf

•TfA

3(6) s

(4tt)

^Pi||jl(qr)

2 rY

[i] XD[i]f

1]

| !lPi> ^^ifi ifA

'J (4")*

i(ipij rf

1]

n ip

f

>i

2 = £ i

[1 - m2 - ^f ^bV2

Taking the isospin factor into account

do

dQ

2q

This equation neglects the momentum dependence of the nucleon form

factor and a center-of-mass correction for the oscillator model. When

compared to experiment this expression is found to overestimate the data

by a factor of 4-« This is thought to be due to the naive assumption that

12 the C ground state is a filled lp3. subshell. More realistic wave

2

15
functions put some particles into the lpi subshell. The magnitude (but

a not the shape) of the form factor of the -* 1 transition is very

sensitive to this effect.

V. Conclusions

We have tried to make two points in this paper. One that the

formation of the electric and magnetic multipole operators is more

transparent when expressed as an angular momentum coupling rather than

the classical method using the curl of the vector spherical harmonic

and other obscure vector algebra tricks. The second point was to show

the relative ease with which one can calculate complex reduced matrix

elements using the flowchart technique of Danos. An important advantage

of this method is that no phases appear in the evaluation of the recoupling boxes. The diagrams also often make selection rules that are

operating in a specific problem more apparent. The recoupling diagrams

serve the same function in angular momentum calculations that Feynman

diagrams do in quantum electrodynamics.

The author thanks Dr. Don Lehman for many useful discussions about

the recoupling technique and the treatment of spin § in particular.

16
VI. References

[l] M. Danos, Ann. Physics 6j, 319 (1971).

[2] The phase convention has been changed from that of Eq. (2)

of [l] on the advice of M. Danos.

[3] The phase factor in front of the reduced matrix element comes

from changing from spherical harmonics to contrastandard

elements. This phase will disappear when absolute magnitudes

of the matrix elements are taken in the cross section expression.

The single particle reduced matrix elements are written this way

in order to simplify the comparison with the standard [4] results,

[4] T. de Forest, Jr. and J. D. Walecka, Adv. in Physics lj>, 1 (1966),

The electric and magnetic operators given in Eqs. (4.85 and 4*86)

in this reference are related to those derived here by

jIj] =i( .)j-i|-ia£aai-]i T

ji

i4J ]=i(-)J- l [^tU]*T^

17
yMxxWI 1

' 1 ,,,

i'

1 2 J

J J

X

1 2

J

X i j

2 ^ i

<Rr I Jj I **>

Fig. 1. Coulomb Projection Integral

18
^;[YW«.?IM]M|fkY

^

.M]W|[ YW IlXl&]D] Rj

V 1 2

J

L 1 J X

12

J

I

12 J

I

12 J

<Rf |i L |Rj)

Fig. 2. Moment Projection Integral

19
[R/ -[Y U1 xX ai

]

1

''1 l[i L Y

[Ll xP[ll F|[YMxX^][i] Rj

"j j'

i fe j

I' Jfe j'

Fig, 3« Current Projection Integral

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1. PUBLICATION OR REPORT NO.

NBS TN-713

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Electromagnetic Multipole Transitions in the Recoupling

Picture, or, Electron Scattering without Curls.

5. Publication Date February 1972

6. Performing Organization Code

7. AUTHOR(S) 8. Performing Organization

J. S. O'Connell

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/ Work Unit No. 2410231

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as No. 9«

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) The formation of the multipole operators from the fundamental charge,

current, and moment operators of electron- nucleus scattering is carried through

as an angular momentum recoupling problem rather than the usual vector algebra

derivation. This point of view allows the electric and magnetic transition operators to be obtained in a simple and intuitive manner. The single particle reduced

matrix elements of the charge, current, and moment operators are then calculated

in the j-j coupling scheme using a flow chart technique developed by Danos.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Electric transitions; electromagnetic operators; electron scattering;

magnetic transitions; multipoles; recoupling.

18. AVAILABILITY STATEMENT

|"X~] UNLIMITED.

I I FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE

TO NTIS.

19. SECURITY CLASS

(THIS REPORT)

UNCLASSIFIED

20. SECURITY CLASS

(THIS PAGE)

UNCLASSIFIED

21. NO. OF PAGES

20

22. Pi

$.30

U S C O MM=D-C- -6 6ZA4- P 7

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