Volume 8, number 2
PHYSICS LETTERS
15 January 1964
INTENSITY EFFECTS IN COMPTON SCATTERING
I. I . GOLDMAN
Physical Institute of the State Committee
on the Use of Atomic Energy, Erevan, USSR
Received 6 November 1963
Does Klein-Nishina's well-known formula describe correctly the Compton-effect at very high densities
of photon available for example in lasers? And how probable is the absorption process of several photons
accompanying the emission of a harder one? These questions arouse special interest when connected to
the recently discussed method for obtaining hard y-quanta by the reflection of laser beam from relativistic electrons 1-3) .
The calculation of these processes given below is correct for any photon density . The dimensionless
relativistically invariant characteristic parameter for a plane wave is
2 2e2I1N
m2 c3
where N is the photon density and l is the wavelength ; the Klein-Nishina formula is correct only if l; << 1 .
We shall consider this problem in the following way . The interaction of an electron with the incident
photons is described by the exact Volkov solution 4) of the Dirac equation for an electron moving through
a electromagnetic plane wave . After that the emission of the photon is treated as a first order perturbation .
Let us write the Volkov solution in the following way
rn+X+Cr T~
'y = \QZ(m-X)+oar/ ve i[pr+S(z-t)-Xt]
(1)
Here all vectors are perpendicular to the direction of the wave propagation, v is a two-component spinor,
and
S(T) =2"J (~r2+m2-A2) dT ; r=P-eA .
(2)
The energy E and the momentum Pz of the electron before and after interaction are expressed by X as
follows
E = (p2+m2+X2)/2X,
To normalise ip we shall consider that A and
in this case we get
p
2irn
L '
Pz = (p2+m2-a2)/2x .
(3)
are periodic on the surface of a cube of volume L3 = 1 ;
[p2 + m2(1+1;2)- x2]/2a =
27T nT
L ,
(4)
where n, nT are integers, t ;2 = e2A2/ m2 and it is assumed that A = 0 . One can prove that the qi form a
complete orthogonal set at any time and the normalising constant to be put in (1) is
c = {2[p2+ m2(1+~2)+ \2]}-Z
(5)
The matrix element of Hint for emission of a photon K, Kz, w
(1pXIHint 10p'X') =e„12ir/w el(X-~I+w-hz)ty+Mv' ,
(6)
v+Mv' =1 i/i+(ea+ezaz) J' e-1(Kr+K2T)drdT ,
(7)
where (e, ez) is the unit vector of polarisation, a, az - are the Dirac matrices and primes indicate the
initial state of electron .
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Volume 8, number 2
PHYSICS LETTERS
15 January 1964
In the formula for the transition probability per unit time
2
dW = 27r I
(1pX
I
Hint 1 0p' A' )
1 2
d
3 d (w-Kz +a )/ d w '
(2a)
w
(8)
the last factor is a little unusual (see the time dependence of matrix element) . The reason is that the
unperturbed problem - an electron in the plane wave field - is nonstationary .
In (7) after the integration over the space coordinates the matrix M expressed by Pauli matrices will
have the form
M = 2cc'b
p+K f
(bo+icibi)
e-i(S-S+KzT)dT
,
(9)
where
+;e X)
bo = e(ncX'
+ e z (m 2 -X'X +rz'7r) ,
bl = e y m(a-X') - ez m (7ry -7r y ) ,
(10)
b2 = -ex m(A-A') + ez m(7r x'-Trx ) ,
b3 = e x(Air y'-a'7r y ) - e y (a7rz -A'7r x) + e z (7rx7ry '-7rz a y ) .
There are four different functions under the integral sign in (7) and accordingly we introduce the following notation
[I, F, G,1~]
e-i(S-S+KzT)[1,
= f
- e A x, - e A
m
m Y
e2 (A2-AA )] dT .
(11)
Let the coordinate system be such that
p' = 0, -px = Kx = w sin 9, Kz = w cos 9
(12)
From (6) we get also the "conservation law" :
A = A' - w(1-cos 9) .
(13)
M = 2cc' b -K (Bo +iBi vi) ,
(14)
Now the integration over T in (9) gives
where
Bo = -e x [a'px I + m(x+A')F] + e y m(A+A')G +ez {[m 2 (1+t2) - Aa']1 + m p xF+ m21} ,
B 1 = ey m(a -XI) I ,
(15)
B2 = -ex m (a - A') I - ez mpx I ,
B3 =ex m(A-A')G +ey [A'px I-m(A-a')F]+ez mpx G .
The summation over the spin directions of the electron after the scattering process and the averaging
over the spin directions in the initial state yield
i
I va+
MvR ' 12 =
3
' Tr M+M = 4 c 2 c' 2 bp ;K ~,
p
a, R
I Bi ~ 2 .
(16)
i=0
It will be convenient to consider the emitted radiation as composed of two linearly polarised components
e x(l) = ez (l) = 0, a (1) = 1 ; ex (2) = cos 9, e y (2) = 0, e, (2) = -sin 9 .
According to (12), (13) and (15) for the first polarisation state we obtain
E IB (1) I
where p = (XI/m) cot -1. 0 .
1 04
2
= m 2 (a'-X)
2 (111 2
+
IpI- F1
I G12 ,
2)
+ m 2 (a'+a) 2
( 17 )
Volume 8, number 2
PHYSICS LETTERS
15 January 1964
If the radiation is emitted in the second polarisation state we get
(
E B (2)
2
= nz2 (a' -A) 2 (II1
2
+ G1 2 ) + nt 2 (a' +A)I p1 - F
2
Taking into account the fact that the incident electromagnetic wave is monochromatic, we see that the
enlargement of the exponent S-S'+wT cos 9 must have the form 27rn if T is changed by the period 27r/cw' :
X
2 [p 2
+ In
2 (1 +
2I ["12 (1 +
2) - a 2] -
2 ) - X12 ]
+ w cos 9 = nw'
(19)
Then the frequency of the emitted photon is according to (12) and (13), given by
2nkw'
X' (l +cos 9) + [2nw' + rn2(1+t2)/ ;'](1- cos 9)
w
(20)
This expression depends on the angle 9 and the positive integer n . If one assumes n = 1 and t << 1
it coincides with the Compton formula (a' = ni for an electron initially at rest) . We see from (20) that w
increases with n, but never surpasses
w max
A,
= 1- cos 9
(21)
The expression for the vector potential of the monochromatic elliptically polarised electromagnetic
field can be written
-e A x =,/2 4m sin a cos w' T ,
(22)
-e A y =,/ ~ in cos a cos (w'
T
+/3) .
Then the Stokes parameters ~i can be expressed as follows
~1 = sin 2a cos $,
t2 = sin 2a sin /3, ~ 3 =cos 2a .
(23)
In this case ~ 2 describes circular polarisation, ~ 1 and ~ 3 the linear polarisation of the wave . Taking into
account this expression for A, the integration of (2) gives
S
(T) =
2
[px+
2
191
(1+
2
)-a
2
n9 sin
tPx
]T/ ta t
a sinw T+
Xw'
(sin2 a +cos 2 a cos2(3)~2m2
sm2w
4aw'
T+
+ cos 2 a sin 2/3
42m2 cos 2w'
4Aw'
(24)
T .
The expression (11) for I becomes
7r
in (s,q,r) =
2
f exp{-i(nx - s sin x + q sin 2x +r cos 2x} dx ,
(25)
-7r
where
2 sin anp~
S -
1+p 2 +~ 2
(sin 2 a+cos 2 a cos 2/3) n~ 2
'
q =
'
2(1+~ 2 +p2 )
cos 2 a sin 2/3 nt 2
~
r -2(1+ 2+p2)
(26)
The other integrals in (11) can be expressed in terms of 137 :
Fn =,/
2 sin a (In-1 +' n+ l)'
Hn = Ik 2 (sin 2 a+cos2 a e 2i
i/i
i(3
cos a (e In- I + e-
G =
/2
I
72+1 ) I
(27)
o)'n-2+`-z~ 2 (sin2
a+cost ae -2i13 )ln+2 •
Here we shall consider circular polarisation, the only case leading to the closed formulas . Then q = r = 0,
s = 2npJ(1+~2+p2) . Then all these integrals can be expressed by Bessel functions
In (s,0,0) = In(s),
In =
1+~ 2 +p2
79i(s),
2p
G 32
=
i~ 1, 1'(s),
Hy1 = 0 .
(28)
Inserting these expression in (17) and (18) we get
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Volume 8, number 2
PHYSICS LETTERS
15 January 1964
1
2 )[1 ,2 +(n2 -1)I 2
EIB (1,2) 2 = Zm 2 t 2 (X2 +x'
n ] - 4m2 xx' 1n 2 .
n
P2
(29)
To find the transition probability it remains only to compute dx/dw in (8)
2x(w sing 0 + x cos 9)
dX
dw
(30)
w 2 sin20 + x 2 + J (1+42 )
Putting together all factors one finds
e2 (m 2p 2 +x ,2 )IBij 2 wdSZ
(31)
drv = 4rrm 2 x' 2 (m 2 (1+~ 2 )+x' 2](1+~ 2 +p 2 ) '
To obtain the effective cross section we must divide by the flux density
2 m 2 w' (1-v')
(32)
4r e
where v' = vZ is the initial electron velocity .
The differential cross section now becomes
2e4(w/nw')gdst {(' +a')[In'2(s) + (n2 - 1)I 2 (s)]
(33)
- 2t-21 n2 (s)},
n
(1-v')[x'g+m2(1+t2)] x x
p2
where x' = m '/(1-v' )/ (1+v' ) .
The analysis of this formula shows, that for t >> 1 the absorption of many photons (n ~ 3 ) is the most
probable, the angle 0 0 of the emitted quanta is determined by the condition cot 20 0 = ( m/x'),/1+t2 .
In other case Q << 1), the principal term in the cross section for multiple absorption is as follows
d0=
uo) ( w, L
2 n+ nO
~ t
d 0n =
mg
- sing 0) cp n d SZ ,
(34)
where
n2n+1
n1 2
n
sin 0 2n-2
(35
s2in
)
and the electron is assumed to be initially at rest .
The exact expression of w will have the form
w -
nw,
1 + [
m
n w'
~2
+ 2 ](1 - cos 0)
(36)
For n = 1 eq . (34) represents the well-known Klein Nishina formula 5 ) . For n = 2 we have cp2 =
2t 2 sin 2 0 and after integration over angles we get total cross section for two photons absorbing and
emitting a harder one
02 =
4rt 2 e4 [(1+a)(2+4a-a2)
yn2
a5
In (1+2 a) +
_ 2(6-12a-a2)]
1
1+2a
3 a4
a
2W'
m
(37)
In conclusion we note that the parameter
10-4 and 02 - 10 - 3 2 cm 2 for the laser beams of maximum
power reported up to now . It is possible to increase t by focusing laser beams by a factor 10 3 .
The author should like to thank V . Arutunian and G . Nagorsky for discussions on this topic .
References
1) F . A . Arutunian and V .A . Tumanian, Physics Letters 4 (1963) 176 .
2) R . H . Milburn, Phys . Rev . Letters 10 (1963) 75 .
3) F . A . Arutunian, I . I . Goldman and V . A . Tumanian, J . Exptl . Theoret . Phys . (USSR) 45 (1963) 312 .
4) D . M . Volkov, Z . Physik 94 (1935) 250 .
5) A . I . Akhiezer and V . B . Berestetskii, Quantum electrodynamics (1959) § 28 .
106