On multipole radiation

Physica X, no 4

April 1943

ON MULTIPOLE

RADIATION

b y H. A. K R A M E R S

Zusammenfassunl~ Diese Arbeit enthgtlt die mathenmtische Theorie der Multipolstrahlung, sowohl klassisch wie quantentheoretisch, in einer Form, die etwas v o n d e r fiblichen abweicht und die vie]]eicht in mathematischer und physikalischer Hinsicht einige Vorteile bietet. Anstatt mit elektrischer und magnetischer Multipolstrahlung wird hauptsAchlich mit rechts- und linkszirkulater Multipolstrahlung operiert.

§ 1. Introdu, ction. F r o m a m o n o c h r o m a t i c radiation field M satisfying M a x w e 1 l's e q u a t i o n s in v a c u o a new m o n o c h r o m a t i c solution M ' of these e q u a t i o n s is obtained, in general, b y a p p l y i n g to the original solution an a r b i t r a r y r o t a t i o n r o u n d some fix-point in space. \ ¥ e can ask for a finite set of linearly i n d e p e n d e n t solutions ]ill, M2 . . . . ]i,/, such t h a t the r o t a t e d solutions M~ . . . . . M~ can be linearly expressed in t e r m s of the M ' s in a w a y which is irreducible with respect to the c o m p l e t e g r o u p of rotations. The well-known multipole radiation/ields offer precisely the answer to this question; for each positive value of the integer l there exist two sets of 2l + 1 linearly i n d e p e n d e n t M ' s such t h a t the M ' s of each set, on rotation, t r a n s f o r m i r r e d u c i b l y into each o t h e r (i.e. like W e y l's m o n o m i a l s ~*+" .ql--,,, or like the spherical functions P}" ei'$). The t w o sets are u s u a l l y described as electrical 2Lpole radiation, and magnetical 2Lpole radiation; t h e y b e h a v e differently with respect to the reflection t r a n s formation. T h e r e are still two different w a y s in which the m u l t i p o l e - r a d i a t i o n m a y be conceived of. On the one h a n d we m a y consider either o u t going or ingoing w a v e s (or a r b i t r a r y superpositions of these), leaving the fix-point or origin as a singularity in which M a x w e 1 l's e q u a tions are no longer satisfied, on the o t h e r h a n d we m a y restrict ourselves to fields of free s t a n d i n g w a v e s wl~ich r e m a i n finite a n d nonsingular at the origin. Radiation-fields of the l a t t e r t y p e m a y be - -

26t

- -

H.A. KRAMERS

262

analysed in plane waves and vice-versa; they offer therefore the appropriate basis for a quantization of the M a x w e 11 field in terms of mu!tipole light-quanta and have as such been used b y H e i t 1 e r 1). The more general multipole-fields of the former type have been very useful in radiation problems of classical physics 2) ; b u t they have also proved a natural tool in the semi-classical treatment of certain quantum problems 3). The present paper presents the mathematical theory of 'multipoleradiation, both classical and quantum theoretical, in a form differing somewhat from that in literature and offering perhaps some advantages in mathematical and in physical aspect. The irreducibility of the sets of multipole fields will b e our fundamental point of view; and we wish to lay stress on the particular advantage of introducing rigth-hand and le/t-hand circular multipole radiations in stead of electrical and magnetical ones (of which they are the sums and differences). These advantages are analogoLls to those implied in introducing right- and left-hand circular plane waves in the quantum description of the field 4).

§ 2. Expression o/multipol'e ]ields by means o/derivatives. Introducing the complex vector F b y F ---- E + i l l ,

(1)

Maxwell's equations in vacuo take the form i aF rot F = - - - c at'

d N F = 0.

(2)

We look for monochromatic solutions, which contain the time in the form of an exponential factor: F (±~ = F(o+~ e ~:2~v'

(3)

With the abbreviation k =

(4)

(2) takes the form (rot =V k) F¢0+1 = 0,

div 1~o+1 = 0.

(5)

The general solution of (5) is given b y k 2 F(0+) = rot (rot + k) V

=

(k 2

-~- grad div + k rot) V.

(6)

O N MULTIPOLE RADIATIdN

263

where V is any vector field satisfying (a + k 2) V = 0. The multipole-radiations form a ~omplete set of eigen-functions of (5) and correspond to particular choices of V; the components of ¥ are here linear combinations o] partial derivatives with respect to x, y, z (of the(l--1)u order) o/the/u.ndamental spherically symmetric scalar solution G(r) satisfying (A + k 2) G - -

l

d2

r dr 2 (rG) + k 2 G = O.

For G we may choose either e~ikr

G--

(in- or outgoing waves)

kr

or

G -~

sin kr k~

(free standing waves).

Thus, there is a close analogy with the manner in which static multipole potentials by differentiation can be derived from the fundamental solution l/r. We will now indicate in what way derivatives of G have to be combined, in order to yield the irreducible multipoles fields. Introducing a complex vector a (ax, ay, a,) of zero length

a~ + a~ + a~ - 0, we m a y write

k'-' v = . a ( a . v ) ' - ' G.

(7)

Introducing (7) in (6), we get k TM F~0+1 = {k2 a ( a . V) *-I + V ( a . V)* :_~ k[V, a] (a. V) ~-l} G. (8) Representing the components of a b y means of two spinor components *) u, v: ax-

fay

=

'bt,2,

- - a., - - fay = v2, --a.

=

(9)

'lSV,

*) Definition in aeeordanee, n o t w i t h V a n d e r article in Hand- u. Jahrb. d. chem. Phys. I, § 61.

Waerden,

bnt with the author's

264

H . A . KRAMERS

the field (8) appears as a homogeneous polynomial in u, v of degree 21 which, for the sake of suitable normalization, we will write in the form m ~--I • F(0~1 = K ~ u t+ . . . .~ ~ - ' ~ ~ l +2.m ]~" ( :Ll,m ~1 (10) •

tJI : l

The two sets of 21 + 1 radiation fields f defined b y (10) and (8)can be taken as representing the multipole-radi~ions of degree l. Indeed, they are seen from (10) to transform with respect to-rotations con-tragradient to the quantities u s+'' v ~. . . . . (i+,,,) 2, ..~., they correspond precisely to the finite irreducible representation of the rotation group of degree 2l + 1. Since the conventional sl)l~erical harmonics YI" = e'@ P~" ( c o s

~)

leave the expression X ,$$l+,,, vt-,, (__ I) '" YI"

invariant against rotations, it is seen that our fz,,,'s transform like the spherical harmonics 0~m)--~ ( ~ 1)"' Y~" i.e. like the normalized eigen-functions of the square of the angular momentum of a particle as they are ordinaril3~ used in quantum mechanics. K is a normalization factor depending only on 1 and k and will be fixed later on (cf. formula (28)). As will be seen later on, the electrical and magnetical multipole radiation correspond to F (+1 + F I-I and F I+~ - - F I-~ respectively; the F's themselves represent left- and right-handed circular waves. The expression (8) can be given an extremely simple symbolical form, if we introduce two symbolical spinors ~, "~and ~', "0' b y meang of the relations ~' = (a/ax) - - i ( a / a y ) , on' = - - (a/ax) - - i(a/ay), ~.~' - - - - (a/az) - - i k ,

4~'

=

--

(a/a~)

+

(11)

ik.

Since A+k2=__

+ (

Oz

( ~ ik, ,

0' ~

a + ik

....

~ ' • 40' + ;4 •

O,

ON MULTIPOLE RADIATION

265

we .see that any homogeneous polynomial which is of the same degree in ~, -t~ and in ~', "0' constitutes a differential-operator, which is unambiguously defined if it operates on a function/(x, y, z) which, like our G's, satisfies the equation (A + k 2 ) / = O. Introducing a constant null-vector A x A y A , and corresponding spinor U, V Ax--

iA~, = U 2,

-- Ax--

iAy = V 2,

-- A, = UV,

(12)

it is easily shown that the following identities will hold: ( A . F~0+)) = const. (U'~ - - V~) 2 ( . u ~ - v~)~-I ( . u ~ ' - v~') ~+I G, (13a) ( A . Fo~-)) = const. ( U - ~ ' - - V ~ ' ) 2 (u'~--v~) ~+'

(*~30'--V~t)/-I G. (13b)

These expressions define, indeed, two sets of (2l + 1) vector fields. The coefficients of u I+" vI.... correspond to the different fields, those of U 2, U V , l ~ yield their components along the axes in space. Since the right-hand members of (13) are invariant against rotation, the transformations of the fields are of the required irreducible type. Moreover, the form of these members warrants immediately the equations (5) to be fulfilled. Indeed, if a vector-field b is symbolically represented b y a. spinor p, q (p2 = bx - - ibm, etc.), the following identities are easily verified: ( A . b) = - - ½ ( U q - -

(14a)

Vp) 2,

d i v b = - - ½ ( p - ~ , - - q~) ( p ~ ' - - q~'),

(14b)

(A, rot b --~ kb) = - - ~-i(Uq - - Vp) (U~' - - V~') (p~ - - q~),

(14c)

(A, rot b + kb) ---- - - ½i(Uq - - Vp) (U'o - - V~) (p~' - - q~').

(14d)

In case (13a)we may take (~) = (~) × scalar, in case (13b) on the other hand (~) = (~:) × scalar. In both cases the right member of (14b) vanishes, whereas in the former case (14c) and in the latter case (l 3d) is seen to vanish. Formulae (l l) and (1.3) give, so to .say, the deeper insight in t h e mathematical structure of the multipole fields, whereas (8) gives the less simple but more practical expression in cartesian language. § 3. E x p l i c i t representation o[ multipole [idds. The differentiations

implied in (8) or (13) are easily reduced to differentiations of the function G with respect to r. Introducing the abbreviation R : R = kr, R ( X , Y , Z ) = k r ( x , y , z ) ,

G = e+iR/R or s i n R ] R ,

(15a)

266

,

H. A.

KRAMERS

and the differentiation symbol ~ : d f2 = R d ~ -

d d(½R2~ '

(l.5b)

we find {R2fl~+t + ( 2 / + 1) f~,+f~,-t} G = 0 ,

f~G = const. R -~-~ Zt+~(R), (16)

where Z is a Hankelfunction or a Besselfunction. In literature these functions are usually explicitely introduced in the formulae; it appears, however, simpler to preserve the formulae with the symbol fl, which allows also an immediate survey of the behaviour for small and large R. A straightforward calculation gives the following equivalent expressions for F)o+1" F¢o±) = a T t-~ (f2t-~ + l~') G + RTtf~+~G + [R, aJ r '-t f~tG, (17)

--2(A

•

Fo¢±1)- l + 2l+

1 (Uv--Vu) 2T t-lf~l-lG1

1 ( d d)UTt+t,t+lG= V - - ( 2 l + 1)(l + 1) U~-u + V~-V-v i O + V_~v)T,f~,G; Tr(w-v,,l(v

T :-: (a. R) : .x {u2(X + i Y ) - - 2u,vZ - - v 2 ( X - - i Y ) } , r ' = Y, u'+" v' .... ( - - 1)" H/" (X, Y, Z).

8)

(19) (20)

H r is a harmonic polynomial which is simply connected with the conventional spherical harmonics by HI" = R'YI".

(21)

Together with (9) and (10), the formulae (17) and (18) permit us immediately to write down an explicit expression for every multipole field f ~ l in terms of spherical harmonics. In (18) the direction 8, q~ in space appears exclusively in the form of spherical harmonics; H e i t 1 e r's formulae in his paper cited above are a direct consequence of it. Formula (17) is much simpler than (18), and it gives a direct insight in the structure of the field. In the ,,wave-zone" (R >~ 1) the

ON M U L T I P O L E I L A D I A T I O ~r

267

field decreases as R - t , and is practically perpendicular to R; in fact, the formula ( R . Fot+l) --- T' (f~t-, + lf~' + R 2 f2t+') G : - - (1 + 1) T' f~' G shows that the radial component of Fo decreases as R -2. With G ---- eiR/R, we can write for the field in the wave-zone { + ) F ( ± ) ~___ F(0±) e T 2 ~ i v t =

-----{a - - R ( a . R)/R 2 ~ i[R, a]/R} (T/R) '-I eitRq:~vO/R. (22) From this each of the (2l + 1) wave fields corresponding to the upper (lower) sign is seen to represent an outgoing (ingoing) wave which is left-hand (right-hand) circulary polarized in every direction. With G ~ e--iR/R the ingoing left-hand and the outgoing right-hand waves would have been obtained" (--)F (4-) ~ =

{a --

F(o±)

e:Y 2~rivt =

R ( a . R ) / R 2 :F iIR, a]/R} (T/R) l-I e-i~e±2'~°/R.

Adding ½ t+)Ft+l and 4 t-IFt-) we get the field F~, = {a - - R ( a . R ) / R 2 + i[R, a]/R} (T/R) ~-~ cos (R - - 2~vt)/R, which clearly represents an outgoing electrical 2Lpole radiation in the wave zone ; the magnetical 2t-pole radiation is obtained either by subtracting ½ t+lFt+l and ½ t-IFt-I or (the difference being only a phaseshift b y ~/2) b y multiplying Fa b y i. In the case of F,t the tangential electric force is, on a given sphere, distributed in exactly the same way as" the tangential component of the field of a static electrical 2' pole; the magnetic field vector is obtained by rotating, in every point, the electrical field-vecto÷ by 90 ° around the radius-vector*). The complete expressions for F~z and F,nag,j a r e , for outgoing waves, given by cos (R - - 2r~vt) + F,l = {aT t-I (~)t-i + 1~£) + RTt~Y +1} R + i[R, a] T '-~ ~' sin (R ~ 2r~vt) R F,,~g,, =

(23)

'iFa.

With respect to the reflection-transformation R - + ~ R the fields F.t (F,,,.~.) transform either into Fa (and - - F,.~8,,) or into - - Fa *) Compare the a u t h o r ' s analysis of multipole radiation in the wave zone, in H a n d - u. J a h r b u c h d. chem. Phys. I, p. 413.

268

H.A. KRP~MERS

(and + F,,ag,,), depending On the parity of l. The circular fields, however, transform into each other (F (+1 into + F (-I, F(-I into F (+~ .

§ 4. Quantization in terms o/ multipole wave-quanta. Proceeding now to develop an arbitrary nowhere singular radiation-field in terms of multipole fields of the free wave type, we substitute for G in (17) the real expression sin R / R and it is easily seen (using (9) and (10)), that the relation

fl~, )" = (-- 1)" fl,~_~,,,

(24)

will hold. Next, we will follow the ordinary custom of imagining the radiation enclosed in a large spherical shell (ttiameter L), so that the development will imply summation with respect to a set of discrete eigenvalues of k(kl, k2, k3 . . . . k~ . . . . ). If we wish to preserve our circular-polarized fields, the boundary condition to be imposed at the surface of the shell needs must be a fictitious, mathematical one, since physical reflection of some kind of the waves against this surface would change the right-hand into left-hand waves and reversely. No real difficulty will be introduced thereby, since in actual computations we always use the integrals over k, towards which the sums over X tend, when L tends towards infinity. A suitable boundary condition will be sin R = sin (kL/2) = O, k~ = ~,(2~/L), (~, = 1, 2, 3, . . . . ). (25) Summation over ~ tends, for L--~ oo, to integration over k multiplied. b y L/2rc. In order to get formulae, which can directly be compared with the author's analysis and quantization in terms of plane circular waves *) we write q~

F =

E + ilI =

4 ~ I/In---c~!

L

: (~.,,,,,~ fl,,,+~ -t- ~,,*,,~ f~;,~*).

(26)

l,,n,~

Analogous to the case of plane waves, t h e / ' s form an orthogonal set of eigenfunctions and we require such a normalization, that the *) H a n d - u. J a h r b u c h d. chem. Phys. Bd. I, Formulae (102) and (101), p. 433, correspond to (26) and (27) here; formula (167), p. 461, for the vector potentiM should be compared with formula (31) below.

269

ON MULTIPOLE RADIATIO.~

total field energy takes the form

if

/-f=

(F*. F) d v = ~ (~.*~. + ~ * ) h,,a.

•

(27)

~n~,~

Comparing (25) and (27) we see, that each of the f's must be normalized according to"

F r o m (10) we have therefore f (F* . Fo) d V = K 2 (u*u + v'v) ~ X. The left h a n d member can be calculated b y integrating the absolute square of (17) with G = sin R/R. It is sufficient to consider the wave zone only, and so we get

/ (l o* •

~r,X

ev = . [ 4 R2uR k]

T/R [~ - 2

. 2 { ( a * . a) [

--

0

--1

T / R [ ~ } s i n 2 R / R 2 = (a*

•

a)' 2~(/ - - 1 ) ! (l + 1) k 2 ( 2 / + 1) l ! L.

(1 T / R 1~, which means the average of [ T / R 12, over all directions, is equal to (a*. a) t l!/(2l + l) ! !). The result is, that the normalization constant K, appearing in (10) is given b y • K--

4~2(/-1)!(l+

1) =

4~ 2(l+

2Zka (2l + 1) !!

l)l(l--1)!

(28)

k a ( 2 / + 1) !

In a w a y quite similar to that followed for plane circular waves the quantization of the field can now be performed. The c~'s and [3's are to be p r o m o t e d to q-numbers ; t h e y will satisfy the c o m m u t a t i o n rules (n stands for l, m, X) % %,* _ _ e*. ~.,, =

ot~,,[3,,,* - -

I3,*,, ~,,

= ~,,,,,,

(29)

(all others pairs commute).

and can be represented b y J o r d a n Their time-dependency is given b y ~,, =

(ilk)

(I--[~,,-

~.,, Z--D =

--

2ni,~z,

Klein

matrices•

~.,,(t) -~ ~,,(0) e--2"m'A ' , (30) (similar for [~)

and agrees therefore with the result of the field equations (2), when

9-70

H. A. KRAMERS

applied to (26). As soon as the field is in interaction with electrons or other charged particles, additional terms appear in the Hamiltonian containing the vector-potential A of the magnetic field. Since we can write

F :

rot

,/~ ~ (~. f.I

J L

~* fc-~* ' -V.)'

we find A =

,: ,/

x ~

(~,.* f~+,* --

:,.

--

r~

(3~)

The expression for the field F becomes less simple, if we analyse in terms of electrical and magnetical multipole'-quanta. Indeed, if we denote b y ,(~,,,~ and 8~,,,~ the q-number-coefficients appropriate to this analysis, we have, for each set of numbers l, m, ).:

-c = (~ + ~)/,V2, =

(= --

~)/-V2,

=

(~, +

~)/,V2,

:'~ =

(-r --

s)/~,/2.

The c o m m u t a t i o n rules of the T's and ~'s are the same as those of the c,.'s and ~'s. On the other h a n d the expression for A in terms of ¥, ~, T*, $* is h a r d l y less simple t h a n formula (31), and it must not be forgotten, t h a t in most practical applications the use of electrical and magnetical q u a n t a will be far more n a t u r a l t h a n t h a t of circular quanta. The radiation emitted by a system of moving charges will be determined by the perturbation term - - l/c f (j . A) dV, were j is the current-density in the system and A the divergence-less vectorpotential ; this holds both in classical- and in q u a n t u m - t h e o r y . In the latter case both j and A will be q-numbers, depending on the cnumber variables x, 3', z. In both cases the emission is seen to depend on integrals of the type

f (j (k). f(:~)(*) (k)). dV

(32)

where j(k) is the harlnonical component of the current field which corresponds to the frequency v = ck/2~ (in q u a n t u m - t h e o r y : tilt matrix element corresponding to a transition with ck/2r~ as B o h r frequency), k being the wave number appearing in f.

"271

ON MULTIPOLE RADIATION

As an example, let us consider the normalized f-fields of dipole radiation : fJ~) =

4~

fl$) = k -v~' ~

[\o/

{(o) I

+ Yz (X + iY)Q q: \ - - i ( x + i r ) / s

P--

sin R ( P = (1 +D) - R . . . . .

.

zQ~

(+),} .

1

=--fjt_)~ ' ,

•

'

1

2

/~a sin R + ~ - cos R = 3

R2 +

.....

-2sinR __I + ~_T) sin R - - 3 cosR . .1 . . 1 . 3. Q -= ~ - - - R - = ( - - R a , ~ 15 2~0 R --F S = t ~ sinR --

R1a s i n R + ~ - c o1s R = - - ~ +

1

3-~R 2 + ......

If the emitting system is small compared to the wave lengt h . the development in powers of R is useful: in first approximation,only the first term ] in P counts, and substituting in (32) we have the Well known result, that the (electrical) dipole radiation is determined by the (harmonic time components of the) resultant current. In next approximation the term - - ½ in S will count, giving rise to magnetic dipole radiation.

§ 5. The angular momentum o/ the radiation/ield. The resultant total *) angular momentum of the radiation field is given by

J=

(l/4nc) f [r, [E, HI] dV = (-- i/8nc) f [r, [F*, F]] dV. (34)

Inserting the expression (26) for the field F, we are led to integrals of the type ~) j = f Jr, If*, f']] dV.

(35)

The only integrals of the type (35) which do not vanish are J,~+~, = f Jr, [f},+)*(k), f}.+~(k)]] dV, and similarly with ( - - ) sign, (m - - m ' =

0 or i

*) F.J. B e l i n f a n t e , PhysicaO, 887, 1939.

1).

272

o~ MULT-.IP.OLEIbkDIATION

Their evaluation is easiest performed, b y means of (17), (10). In fact, the integrals f [r, IF*, F~]] dv can, for reasons of invariance, only yield results of the types C[a, a*] (a. a*)P, C'a ( a . a*)P' and C"a* (a. a*)P"), where the C's are constants and the p's are integers. The constants C' C" are, however, found to vanish for L -+ oo (they would correspond to ,,transitions" l ~ l + 1), and the only non vanishing integrals are those of the type ! -> l. The factor [ a . a*] ensures us, that onl~ ,,transitions" m - > m (z-component) and m --> m ± 1 (x and y component) give non vanishing results, and the actual calculation shows too, that only ,,combination" between two right-handed or between to left-handed wave fields give results which differ from zero. The integrations over R give results proportional to L, as was to be expected. This shows, that the angu~Iar momentum resides in the wave-zone, contrary to H e i t 1 e r's statement. The final result is = ~:

-~- it.~y

+

~,,*,,,a) m ,

(36)

= /~ '~' ( C(~,,,q..l ,~ ~l,,,,,~ -'['- ~,~,,, ~' [3;,,,,.i,a) v"(l T m) (l ~ m +1). *

I n its essence, this result coincides of course with that of the analysis in H e i t 1 e r's paper. B y calculating ~. Fk --- Fk ~. (i, k = x, y, z) it can explicitly be shown that our expression for J corresponds, as w a s t o b~ expected, to the operator of an infinitesimal rotation *). In our analysis, this property of J a p p e a r s in a more lucid way than in the analysis in terms of plane-wave quanta. R e c e i v e d F e b r u a r y 6th 1943. REFERENCES 1) "~V. H e i t l c r, On the r a d i a t i o n e m i t t e d b y a inultil)ole and its a n g u l a r m o n m n t u t n . Proc. C a m b r . phil. Soc. "lr~, 112, 1936. 2) For l i t e r a t u r e cf. H. B a t e m a n, E l e c t r i c a l and o p t i c a l w a v e - m o t i o n , § 4, § 17, C a m b r i d g e 1915. 3) H . C . B r i n k 11t a n, Zur Q u a n t e n m e c h n n i k d e r M u l t i p o l s t r a h l u n g , Diss. U t r e c h t 1932). (P. Noordhoff, G r o n i n g e n 1932); A. R u b i n o w i c z und J. B 1 a t o u, E r g e b n . ex. N a t u r w i s s . I1, 176, 1 9 3 2 ; H . M . T a y l o r a n d N . F. M o t t, Proc. roy. Soc..4. 142, 215, 1 9 3 3 ; W . W. H a n s e n , Plays. Rev. 47, 139, 1935; J. B l a t o n , AetaPhysicaPolonica6,256, 1937;S. 31. D a n c o f f a n d P . M o r r i s o n , Plays. Rev. 55, 122, 1939. 4) L. d c B r o g 1 i c, N o u v e l l e s R e c h e r c h e s sur Ill l.unli6re, Actual. Sc. et l n d u s t r . 411, 1 9 3 6 ; H . A . N . r a m e r s , H a n d - u. J a h r b . d. chem. Phys. l , § 8 6 s q q . , 1937. • ) C o m p a r e H . A. l ( r a m e r s ,

H a n d - u . J a h r b . d. chem. P h y s i k 1,448. K 2597