On the scattering of an electromagnetic wave packet

M a h a u x , C. 1966

P h y s i c a 32 1319-1332

ON THE

SCATTERING OF AN ELECTROMAGNETIC WAVE PACKET by C. MAHAUX*)

Physique Nucl6aire Th6orique, Universit~ de Liege, Liege, Belgique

Synopsis A d o p t i n g A k h i e z e r a n d B e r e s t e t s k i ' s d e f i n i t i o n of t h e s c a t t e r i n g m a t r i x , t h e elastic s c a t t e r i n g of a n e l e c t r o m a g n e t i c w a v e p a c k e t b y a s p h e r i c a l l y s y m m e t r i c c e n t e r is s t u d i e d . V a n K a m p e n ' s m a c r o s c o p i c c a u s a l i t y c o n d i t i o n is easily a d a p t e d t o t h e p r e s e n t f o r m a l i s m . T h e case of a w a v e w i t h n o n v a n i s h i n g a n g u l a r m o m e n t u m z comp o n e n t is discussed. I t is also s h o w n t h a t o n e m u s t associate w a v e s w i t h o p p o s i t e angular momentum components to construct a sharp-front wave packet. A resonance e x p a n s i o n of t h e s c a t t e r i n g m a t r i x is given, w i t h a n a p p r o p r i a t e t h r e s h o l d factor.

1. Introduction. In a celebrated paperl), V a n K a m p e n proposed a definition and studied the properties of the scattering matrix corresponding to the elastic scattering of an electromagnetic wave by a fixed center with spherical symmetry. He succeeded in finding the analytical properties of that scattering matrix on the basis of his causality condition, which m a y be stated as follows: "The scattered wave can be detected only after the amount of time needed by the incident packet to reach the scatterer and by the scattered wave to reach the detector". Although the results obtained by V a n K a m p e n are quite general and of deep physical interest, the formulation presented in his paper is somewhat obscured because of the following two reasons. Firstly, he describes the electromagnetic wave by its associated complex Debye potential2), which is a somewhat cumbersome and unfamiliar procedure; the separation of the wave in multipoles of given parity is also rather difficult. Secondly, he uses a definition of the collision matrix which appears to be somewhat complicated. As a result of this, his scattering matrix is not diagonal in k (the wave number) and ~¢t (the angular momentum z component). In this paper, we shall rather adopt the definitions proposed by A k h i e z e r and B e r e s t e t s k i i 3 ) for the multipoles and for the scattering matrix. The basic quantity is here the vector potential, as proposed by m a n y authors 4). We believe that the treatment presented here is thereby simpler than Van *) Chercheur I.I.S.N. **) Institut de Math6matique, 15, avenue des Tilleuls, Liege, Belgium. --

1319

1320

c. MAHAUX

Kampen's. The case of an incoming wave with angular momentum z compoponent (~') different from zero is also studied. It is shown that it is necessary to combine the (~¢~, ~ ) and ( ~ , --~¢) waves to form a wave packet with a sharp front, ~ being associated to the total angular momentum. This result corresponds to the necessity of combining particles and antiparticles to construct a localized state in field theory 5) and to the fact that, in classical electrodynamics, one must associate left-hand and right-hand circularly polarized plane waves to form a localized electromagnetic signal. We finally introduce a resonance expansion for the scattering matrix, including a "threshold factor". The analogy with the case of the elastic scattering of a particle 6) b y a central potential is emphasized.

2. The multipoles. In this section, we briefly define some quantities which will be extensively used in the following. We follow the procedure of A kh i e z e r and B e r e t e t s k i i 3 ) ; we, however, take the vector potential as the basic quantity (rather than the electric field) and work in a different system of units. We also briefly discuss the relation with other representations of the field 7) s) 9). Let a be the radius of a sphere centered at the origin of the reference frame and enclosing the spherically symmetrical scatterer. We use the solenoidal gauge and the Gauss' system of units. For r > a, the vector potential A(r, t) obeys the following free-field equations div A(r, t) = 0 1

V2A(r, t)

(la)

02

cz ~t~ A(r, t) = O.

(lb)

Let us introduce the three-dimensional Fourier transform A(k, t) of A(r, t) : a (r, t) = f a (k, t) e' k., dk. With A k h i e z e r and B e r e s t e t s k i i 3 ) , vector ](k, t) b y the equations

it(k, 0 = - i

a'(k, t) = -ck

(2)

we can now define a complex

[.f(k, 0 - . t ' ( - k , 0"]

(h), ~ [.f(k.O +.f(-k,

t)*l,

(3a) (3b)

where o~ =

Ikl c =

kc.

(4)

Owing to the normalization factors introduced in eqs. (3), the vector

ON T H E S C A T T E R I N G OF AN E L E C T R O M A G N E T I C W A V E P A C K E T

1321

f(k, t) is a solution of the following equations, equivalent to eqs. (1), \

it~ ~ ](k, t) = EJ(k, t) k .J(k, t) = o,

(sb)

E = t~kc.

(6)

with The quantity f(k, t) m a y be shown 3) to play the part of the photon wave function in momentum space. K u r s u n o g l u 7) and G o o d s) proposed as photon wave function in mom e n t u m space the vector ¢(k, 0 = c(k){E(k, t) + ill(k, 0},

(7)

where c(k) is a suitable normalisation factor. The quantities E(k, t) and H(k, t) are the Fourier transforms of the electric and magnetic fields, respectively. One has, from Maxwell's equations,

ik × n(k, t) = c-l~(k, t).

(8)

Hence, eq. (7) can be given the form

¢(k, t) = c(k) E(k, t) + ~ - n x i(k, t),

(9)

n = k/Ikl.

(lo)

with

From eqs. (3), it is also easy to show that

](k, t) = o(k) ~(k, 0 + ~

(11)

where

M o s e s 9) chooses the wave function ~p(k, t) given by

~(k, 0 = --i¢(k, t).

(12)

Comparison between eqs. (9), (11) and (12) shows that the various wave functions are in fact very similar. These equations also exhibit the reason why we prefer to use f(k, t) rather than •(k, t) or ~p(k, t) to represent the field in momentum space. Indeed, the vectors E(r, t) and H(r, t) have polar and axial character, respectively; consequently, the wave functions ~b(k, t) and q2(k, t) lead to a definition of miltipoles which are not eigenstates of the

1322

c. MAHAUX

reflection operatorS). On the contrary, the vector f(k, t), as confirmed by the equations given below, is an eigenstate of that operator. Let us now introduce the eigenvectors of the operators s 2 and Sz for particles of spin one: (0) Zo=

1 (1)

0 1

,Zl-

~/2

1 (

i 0

,Z-l=

~/2

1 ) -i 0

.

(13)

The vector spherical harmonics are defined by the equations

Yz, z,m(e) --~ ~ (lrns#l~dg> Yzm(e) Z~,

(14)

m

with

e = r/]r I.

(15)

It should be noted that these quantities do not transform according to the D(1) representation of the rotation group. The vectors Yz, z,~(~¢ = --~q~, .... + ~q~) form an irreducible tensor of rank ~¢. One easily checks that the vectors 3) ~g

(e) =

2£¢' q- 1

Y'~'~+l'~(e) +

2~.W q- 1

y(0) (e) = Y~,w,~ (e)

(leb)

verify the following relations: e " v(~) .. ~

(e) -----0

Yg)(,)

(~ = 1, O)

(17a)

. ~ m (--e).

-- (-1)

(17b)

We shall also need the longitudinal vector

Y~-~)(e) = Y~m (e). e --

2~ + 1

The vectors v(a~ l:

Y'~'~e-l'~(e) -0, 1,

2~,~ + 1

(18a) Y~'w+l'~(e)"

(18b)

1) are orthonormalized according to

f Y ~ ( e ) * . Y~:~,(e)* de : bwz,b~m,~z.

(19)

Let us write the vector potential as follows:

A(r, t) = ,~(r, t) e -iEt/* -~ ~ ( r , t)* eiEt/*.

(20)

The electric (upper index 1) and magnetic (upper index 0) multipoles are defined by the expressions3): ~ z . (17 ~(r)

=

V~(~

2~e+

+ 1) [g.~_l(k,) + g.~+l(kr) ] Y ~ ) ( e ) + 1

+ [ 2~q~+l ~ + 1 g~-l(kr)

2 ~ oW + 1 g~+l(kr) ~Y~.g(e) (it

(21a)

ON THE SCATTERING OF AN ELECTROMAGNETIC WAVE PACKET

ag(o) k.~./.¢

= g.~(kr) y(o) tkr~ £f ~ k l,

1323 (21b)

with

(22) the quantity

J.v+ ~(kr) being the familiar Bessel function of half-integer order.

3. The scattering matrix. Let us consider the following wave packet: OO

a(r, t) =fag(r, o~) e -t°~t d o + c.c.,

(23)

0

where the complex vector ag(r, co) is a solution of the equations

div ag(r, o~) = 0

(24)

Wag(r, o~) + k2ag(r, ~) = o.

(25)

Under such conditions, the left-hand side of eq. (23) satisfies eqs. (1). According to the results presented in section 2, the general solution of eqs. (24) and (25) can be written as 1

ag(r, co) = - - Z [x(ff~(k)

~(ff~(kr, e) + y~)~(k)] ~.v~¢(a)(kr, e)]

for real positive values of o~ = outgoing and incoming solutions:

kc. We have introduced the following

i ~÷1 1 ~ / ~ ( ~ e + 1) I 2~t'+ 1

(9~)~(kr , e) +

[ "~

+

1

2£P+ 1

[O~_l(kr ) + O~e+l(kr)] Y(-~)(e)-~(27a)

O£a_ 1(kr)

O~o)_~.~lkr,e) = i"~+10.~(kr) Y~)(e) ~'(1) ,kr e) i.~+ 1 { CoW(LP + 1) [I.~ _ l(kr) + I~e + 1 (kr)] ~e~t~ , = 20Z' + 1

+ 1 [ ~ + 2~+ 1

I.~_l(kr)

L#

2'~¢+ 1

I~71> (e) +

(28b)

O~(kr) and I,(kr) are defined as usual6): Ot(kr) = iW~l)(kr), Idkt) = --i~i2)(kr),

where

(27b)

i.~+l(kr)] y ~ ( e ) } (28a)

~¢~0,~.~,,k.,e) = / ~ + 1 ~ ( k , ) v ~ ( e ) . The functions

(26)

(29)

9ff~l)(kr) and W~)(kr) are the spherical Hankel functions of the first and

1324

c. MAHAUX

second kind, respectively. The following relations hold ~):

(30a)

O~(r, k)* = I~(r, k)

(30b)

O~(r, --k) = (--i)~ Idr, k), and, for large values of kr,

Odkr ) ~-~ exp i(kr -- l~r), I,(kr) ~-~ exp --i(kr -- 1½n).

(31)

From eqs. (27), (28) and (31), we get the following asymptotic behaviours for large value of kr.

(9(~)(kr), e)

Ply°) .(kr, e)

__i.We i ( k r - ~ ) Y~m(e) (1)

(32a)

i "v+l e i ( k r - ~ ) Y(~)~(e)

(32b)

dc(~!cc(kr, e) ~-~ i "~ e -i(kr-'~½~) Y ( ~ ( e )

(33a)

d ¢(°).~rkr, e) ~-~ i ~+1 e -i(kr-~=) Y ~ ( e ) .

(33b)

We shall define the collision matrix for positive values of k by the following equation: _

x(A',b,(k)

y.~,~ (k )

,

(34)

when the ~Lfd¢~ channel is the only one containing an incoming wave. If the scatterer is removed, we have S(~',~) Im = ~ , ~ , ~ £a, lg,,gevlgVvl

~,,

(35)

Because of the invariance under rotation and reflexion, the collision matrix is diagonal in ~ , J / , ,t and independent of J/a) ; hence we may write

S(~', (m ~ , ~ ,~) ,~V~]

:

S(~)(h) ~ . ~ , ~ , ~ , .

(36)

4. Properties o/the outgoing and incoming solutions. From eqs. (27), (28) and (30b), we can easily prove that

( 9 ~ b ( - k r , e) = (--1) ~+~ ~ ¢ ~ ( k r , e)

(37a)

(9~b(-kr,

(37b)

e) = (-- 1)z+a 3#~b(kr, e).

Adopting Condon and Shortley's 10) phase convention for the spherical harmonics, i.e. Ylm(e)* = (--1) m Y1--m(e), (38) we get, from eq. (14), the relation

Y~,,,~(e)* = (--1) z+l-z-~" r.~,z,-.K(e) •

(39)

From eqs. (16), (18) and (39), we obtain the relation

Y~(e)*

= (--])~.+.~¢t'A-I

Y~_~(e)

(40)

ON T H E SCATTERING OF AN ELECTROMAGNETIC WAVE PACKET

1325

which, together with eqs. (27), (28), (30) and (37) leads to the following equations"

~)~(kr, e)* = (-- 1)a+dc+se ,P~)_~(kr, e) J(~) ,(kr,

(41)

= (--1) ~a ~)~k.K(--kr, e)

(42a)

=

(428)

~¢~)_~(--kr, e).

Since

Ytm(e) = (--1)t Ytm(--e), we also have the relations

~(~(kr, e) -~ (--l) a~+~ V~(kr , --e)

(43a)

~¢¢~)-(kr, e)

(43b)

(-- l) ~+~ J ~ ( k r , --e).

In the particular casedt' = O, eqs. (42) become

V o(kr, e)* = (A 0(kr, e)* = J(A

e) 0(-kr, e).

(44a) (44b)

5. Analytical properties o] the scattering matrix. Henceforth, we shall discuss the case Jr' = 0 only. According to eq. (36), this does not restrict the validity of the results which are to be obtained in this section, for the scattering matrix. We shall come back to the casedt' v~ 0 in the next section. For simplicity, we shall drop the index Jr' = 0 in the following equations. Eq. (25) reads, taking eqs. (26) and (44) into account, co

rAy(r, t) = • { f [x~)(k) ~P~)(kr, e) + y~)(k) ~¢~)(kr, e)]. a

o

oo

e-'or d~o +f[xg)(k) * ~)g)(--kr, e) + y~)(k)* ~¢~)(--kr, e)] .e `c°t do. o

(45)

Let us now introduce the following incoming and outgoing amplitudes, which are now defined for positive and negative values of k: for positive values of k

--- xg)(-k)*

for positive values of k

=

(46a)

for negative values of k

(46b)

for negative values of k.

Then, the following relations hold, for real positive or negative k, (47) (47b)

1326

c. MAHAUX

Eq. (45) can now be given the form oo

rA~(r, t) = Y~ fdo)[~(~)(k) (9~)(kr, e) + Y.v-(~)(k)d~(~)(kr, e)] e -u°t. ).

(48)

--oo

The latter equation leads us to extend the definition of the scattering matrix to negative frequencies as follows: 9(~)(k) "

(49)

For positive values of k, the function S~)(k) is of course identical with S(~)(k) defined in eq. (34). From eq. (47), we have,

Hence, the latter relation is a direct consequence of the reality of the field; this was also shown by A. B o h r 5) in the case of scalar field particles of zero mass. We shall write eq. (48) in the form

A v(r, t) : X EAin,£z(r, (a) t,. t)J, t) + a(~) J o u t , Aako,

(51)

2

where +00

rAin,.v(r, t) = / d o

.2~ ) (k) d;(~)(kr, e) e -*°~t,

(52a)

--oo

and +co

rao.t,.~(r, t) ~-- f d o £~)(k) O~)(kr, e) e -lot.

(52b)

--oo

Taking eqs, (32) and (33) into account, we have the following asymptotic behaviours, for large values of kr, +0o O) I", t) "" i ~+ 1-~Y~)(e) f d o Y~e rAi~..~( -(a)(k) I~(kr) e -*~°t

(53a)

--¢o

+00

r a o(a) tr , t) u t , £a ~

~

i_~+ 1-~ Y~)(e) f d o ~ ) ( k ) O~(kr) e-*~t

(53b)

--¢o

Inserting eq. (31) into relations (53), we get co

rain,.~(r, t) ~.~ (__I)£a (i)1-~ Y(~)(e) f Y~e -(a)(k) e-i~(t+ (r/c) d o (a)

(54a)

--oo co

(~) (r , t) rAout,.~

i1+~ Y~0) (e) f x~ ~(z)(k) e -i~°[t-(r/c)] doJ.

(54b)

--oo

Let us now compute the incoming energy in a large sphere of radius

ON THE SCATTERING OF AN ELECTROMAGNETIC WAVE PACKET

1327

R ( k R ~>~ 1). We have E(4) /R t ) =

1

in'Aak '

C

c~ A~),~e(R,t)~.~

~t

1 1)~+1. - ~ C i--A(-co

• r~)(e)..I" o)Y'~ (k

e--i~[t+(R/c)]

do)

(55 )

--co

"'i~,~u(4) = curl A~)

t) ~.~ e × ~i~,.~--,

(55b)

On the sphere, the P o y n t i n g vector reads thus 1

1

S ~ ( R , t) ~ -- 4n---c e ]Y~)(e)l 2. R2 co

•

co

09

e-io~Et+ (R/e)]

)

do)

--co

o) y~

e -i°''Et+(R/o)l do'.

(56)

--co

The total incoming energy is given b y W~4~se = f a t

~ R 2 dD IS(R, t)[.

(57)

--oo

The integral over the time variable contributes a 0-function and we eventually get co

1~/(4) in,.LP rr

k 2 [9(~)(k)12 dk.

=

(58a)

--oo

We find in the same w a y the value of the outgoing energy: oo

uzc4)

k 2 I~<~)(k)Ie dk.

(58b)

--co

The conservation of energy reads thus, using eq. (49), / k 2 [Y.~ -(4)(k) 12 dk --oo

=

f cok2S~)(k)~q~)(k) * Ip~)(k)[ 2 dk.

(59)

--co

The latter relation must be true for any square integrable function p~)(k), hence we have, for real k, the relation

,~((k)*.~(~)(k)----1.

(60)

W e can now discuss the problem of the analytic continuation of the scattering matrix to complex values of k. We shall use Van K a m p e n ' s causality condition 1). Since from now on the argument is close to Van

1328

c. MAHAUX

Kampen's we shall be rather brief. We recall that the sphere of radius a encloses the scatterer. The causality condition can be worded as follows. "If the amplitude A in,.SPk (a) /R , t) of the incoming wave packet vanishes for t smaller than tl, one must have Aout, z(

,t)=0

for any time t such that R--gt

t ~ t l + 2 - -

"

C

Translated in time-space, this principle can be given the following form: " I f (~) (R , t) = 0 Ain,~O

for

ct + R < O,

(61)

one has A~) : o t) : 0 o u t , £ a k ~= ,

for

ct--R<--2a"

(62)

The Pailey-Wiener theorem 11) can now be invoked. Condition (61) implies the existence of a function y~((k) analytic in the upper-half of the complex k plane, and almost everywhere identical to 2~)(k) on the real axis. Condition (62) corresponds to the existence of a function 2~)(k) having similar properties. The function ~)(k) defined in the complex k plane b y the relation

~.~)(k) : --S(~)(k) y(~)(k)

(63)

is therefore analytic in the upper-half plane too. Indeed, the amplitude ~(~)(k) can always be chosen such that it has no zero in the upper-half plane and the function ~q~)(k) is independent of the choice of :~z(*):k~;j hence the function ~(~)(k) defined b y eq. (63) has no pole at a zero of ~)(k). The definition of ~q~)(k) in the lower half-plane must be such that eq. (60) is fulfilled on the real axis. Accordingly, we define ~q~)(k) for I m k ~< 0 b y the relation

(64)

=

According to eqs. (47), the following relation holds in the whole complex k plane: =

(6s)

Hence the poles of the function ~q(~)(k), if any, lie in the lower half-plane only, symmetrically with respect to the imaginary axis. Just like V a n K a m p e n 1), B e c k and N u s s e n z v e i g l 2 ) , and R o s e n f e l d 1 3 ) , we could now show that the poles correspond to radioactive states decaying exponentially in time. 6. The case.,/[ ~ O. In the case Jr' :/: 0, one must take some caution when defining amplitudes for negative values of the wave number, i.e. quantities analogous to ~(~)(k) and 9~)(k). This has to do, as will be shown below, with

ON THE

SCATTERING

OF AN ELECTROMAGNETIC

WAVE

PACKET

1329

the impossibility of constructing a sharp-front wave packet with a single value ofdt'. One has to associate the (~q~,~') and (~qo, _ Jr') waves. Let us consider the wave packet oo

A(r, t) --r~(~)_j~t/r, oJ) e -lot

d o + c.c.

(66)

0

where

-~'~e.K

-~

(k)

(kr, e) +

(k)

(kr, e).

(67a)

According to eqs. (42), the complex conjugate of the righthand side of eq. (67a) reads (-- 1 ) ~ + ~ + a [ x ~ ( k ) * d~)

~(k,, e) + y ~ ( k ) O~)_~,(kr, e)].

(67b)

We now try to apply the same procedure as in section 5, i.e. we change k in --k in the quantity (67b), so as to transform the integral appearing in eq. (66) in an integral ranging from --co to + c o . In doing so, and taking eqs. (37) into account, we get ( - - 1 ) ~ [ x ~ ( - - k ) * O~)

~(kr, e) + y~)~(--k)* d~)_~,(kr, e).

(67c)

In contrast to what happened in section 5, an index --de' now appears in expression (67c); this hinders us from writing the right-hand side of eq. (66) in the form of an integral ranging from - - c o to + co b y simply introducing tilded amplitudes for negative frequencies as in eqs. (46). The solution of this difficulty of course consists in introducing incoming waves of equal amplitudes in the channels oWJt'2 and A°--J/it:

yg~(k) =

ygL~(k).

(r, ~o) reads r,~g~(r, o~) : x ~ ( k ) ~ (~)k r , e) + y ~ ( k ) fg(~(kr, e),

(68)

The corresponding vector "~" --4(~) .W.JI

(69)

where we have introduced the quantities

e) :

e) + O L (kr, e)

W~(kr, e) = M~(kr, e) + M~)__~(kr, e).

(70a) (70b)

We have, from eqs. (42),

rag~(r, o~)* = Y, ( - - 1 ) ~ [ x ~ ( k ) * ~ ( - - k r ,

e) + y ~ ( k ) ~ ( - - k r ,

e)].

(71) Let us define the amplitudes

~(k)

: x~(k) = (--1) ~

for positive values of k

x~(--k)* for negative values of k

y~(k) ~ y~(k) ~-- ( - - 1 ) ~ y ~ ( k )

for positive values of k for negative values of k.

1330

c. MAHAUX

Eq. (25) gives, for the expression of the corresponding wave packet, oo

rA(r, t) = f [ 2 ~ ( k ) ~o~6~(a)(kr, e) + p ~ ( k )

~(a)~se,,(kr, e)] e -*or do.

--oo

Now the Pailey-Wiener theorem can again be applied: the incoming wave packet will have a sharp front provided yze~,(k) "(a) is analytic in the upper-half of the complex k-plane. Eqs. (68) and (70), however, show that multipoles ( ~ , ~ ' ) and (oW, - - ~ ' ) had to be associated to construct such a wave packet.

7. Resonances and threshold/actors. Let DN be a bounded region in the complex k plane enclosing N poles k = k ~ (n : 1 ... N) of the function S~)(k). If we assume all poles to be simple, we m a y write, in D2v, N

~(~)

T~)(k) -= S~)(k) -- 1 = R~eV)(k) + E vsen n=~ k - - b(a) ' '~ n

(72)

where the function R(ff~)(k) is analytic in D2v. The quantity y..COn ~(z) is the residue of the function S(~)(k) at k = ,b(~)~n. The elastic scattering cross section is given bye) 3"I a - - 2k 2 ~X (2~q~ +

1)IS~ )-

1[ 2.

The expansion (72) is, however, not well-adapted to the low-energy behaviour of the cross section (or to the "threshold behaviour", by analogy with the terminology used in nuclear reaction theory). Indeed, the quantity S~)(k) -- 1 goes to zero with k, while this is not true for the one-pole approximation derived from expansion (72). As in the case of particle scatteringl4), it is therefore of interest to write down another expansion for T~)(k) where each pole term would tend toward zero with k like the function T~)(k) itself. Let us consider a fixed value of r ( > a) and sufficiently small values of k, such that kr ~ 1. According to eqs. (21), (27) and (28), we must have the following asymptotic relation:

l

ow + 1

+ I 02_m(kr) --y~(k)

,.W

+ 1

O~+l(kr)~ ~-~

[ 2,LP +1 + 1 L _l(kr)

2 ~ + 1 I'~+l(kr)

'

since the electric and magnetic fields remain finite. Therefore, we have - - - -

- -

- -

(.~q~ + 1)[I.~_l(kr) -- O.~_l(kr)] oo.~[Iq~+l(kr ) -- O L.q+l(kr)J (-L~° + 1) O~,_l(kr ) - - ~ q ~ O ~ + l ( k r ) -

-

(73)

ON T H E S C A T T E R I N G OF AN E L E C T R O M A G N E T I C W A V E P A C K E T

1331

For kr ~ 1, one has Ot(kr) ,'~ I~(kr) oc (kr) -t

(74a)

Ot(kr) -- Iz(kr) oc (kr) t+l.

(74b)

We have, therefore, from eqs. (73) and (74),

T(A)(k) oc k 2 ÷1.

(7s)

for small values of k. In the case of a dipole wave, this leads to the familiar Rayleigh law (oc 2-4) for the cross section. We can now e x p a n d the q u a n t i t y t(~a)(k) = k - 2 £ a - 1

T(~)(k)

(76)

in the same w a y as T(~)(k) in eq. (73). We get, with obvious notations, = k

)(k) + Z k ~ ' ~ b(*) /r n

(77)

- - i~o~n ...a

The corresponding one-level approximation for the cross section reads 0 (~) 2

a~ ) = ½z~(2.Lf+ 1) k 4"v (k - - ~(~)~2 ~ + ~,,~nl

,(~)2, 7..~n

(78)

with k(~)

,~(a)

~. ,(a)

the quantities ~x~ ~(~) a n d zx~ ~'(x) being real. We are indebted to Professor J. H u m b l e t and a careful reading of the manuscript.

for m a n y fruitful discussions

Received 17-12-65

REFERENCES 1) v a n K a m p e n , N. G., Phys. Rev. 90 (1953) 1072. 2) B o u w k a m p , C. J. and C a s i m i r , H. B. G., Physica 20 (1954) 539. 3) A k h i e z e r , A. I. and B e r e s t e t s k i i , V. B., Q u a n t u m Electrodynalnics (Interscience Publ. 1965). 4) See e.g. S i m o n , Phys. Rev. 92 (1953) 1050. M o r i t a , M., S u g i e , A. and Y o s h i d a , S., Progr. theor. Phys. 12 (1954) 713. N e l i p a , N. F., Relation between Photoproduction and Scattering of Mesons (Pergamon Press, 1961). M a h a u x , C., Nuclear Phys. 68 (1965) 481. 5) B o h r , A., Lectures in Theoretical Physics, vol. III (Interscience Publ. Co., 1961). 6) H u m b l e t , J. and R o s e n f e l d , L., Nuclear Phys. 26 (1961) 529. 7) K u r s u n o g l u , B., J. mat. Phys. 2 (1961) 22.

1332 8) 9) 10) 1 I) 12) 13) 14)

ON T H E S C A T T E R I N G OF AN E L E C T R O M A G N E T I C W A V E P A C K E T

G o o d , J r . , R. H., Ann. Physics 1 (1957) 213. M o s e s , H. E., Phys. Rev. 113 (1959) 1670. C o n d o n , E. U. and S h o r t l e y , G. H., Theory of Atomic Spectra (Cambridge U.P., 1953). T i t c h m a r s h , E. C., Introduction to the Theory of Fourier Integrals (Oxford U.P., 1948). B e c k , G. and N u s s e n z v e i g , H., Nuovo Cimento 16 (1960) 416. R o s e n f e l d , L., Nuclear Phys. 70 (1965) 1. H u m b l e r , J., Nuclear Phys. 50 (1964) 1.