To obtain the differential cross section, we simply average over in .NET framework

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To obtain the differential cross section, we simply average over
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dt/! da = r; 1 + cos {3 dO. 2
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Scattering is maximal both in the forward and backward directions. The total Thomson cross section is the integral of (1-200) over 0.:
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(1-201)
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For an electron this yields 0.66 x 10- 24 cm 2 . We recall that this expression holds for weak fields and small frequencies w me 2jh. When hw ;G me 2 or A ;:; hjme (the quantum mechanical Compton wavelength), the initial frequency is no longer
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Ui=;;
Figure 1-4 Bremsstrahlung.
conserved, we encounter both quantum mechanical and relativistic effects, and the process is then called Compton scattering (see Chap. 5). To a good approximation the cross section is then given by the Klein-Nishina formula [(5-116) and (5-117)J. Let us only quote here the lowest-order correction in hw/me 2 to Thomson scattering:
lT tot =
8n 3 re
2(1 - 2hw + ...) me
(1-202)
We can study along the same lines classical bremsstrahlung, i.e., radiation by a charge suddenly accelerated. Let Ui = pdm and ul = PI/m be the initial and final four-velocities (see Fig. 1-4). Choose the origin of coordinates at the spacetime point of acceleration, which can be thought of as the idealization of a collision process. The space-time trajectory is parametrized as
x(r)
(1-203)
m PI r m
while the current
= - ~fd4k e- ik . x (p~
(2n)4
pi. k
pI. k
p{ )
(1-204)
takes after Fourier transformation the simple expression
] (k)
-ie ( ~- ~ =]*(-k) p"k pI'k 11
We require that as t -+ - 00 the electromagnetic field reduces to the Coulomb field of the incident particle, since current conservation forbids the vanishing of jf1(x) when t -+ - 00. In the Lorentz gauge we write, using the definition (1-173),
QUANTUM FIELD THEORY
AI'(x)
f d4x' Gret(x - x')jIl(x') (1-205)
Gadv(X -
= f d4x' G(-)(X - x')jIl(X') + f d4x'
x')jIl(X')
We have exhibited the structure of AI'(x) as the sum of a radiated field (solution of the homogeneous Maxwell equation) and a Coulomb field attached to the particle. Therefore
A~ad(X) = f
d4x' G(-\x - x')jIl(x')
(1-206)
e-ik. x
_i_ fd4k (2n)3
8(k )i5(P)JI'(k)
while
F~;d(X) = (2~ f d4k e- ik .
8(ko)i5(P)[kI'JV(k) - kvJI'(k)]
(1-207)
The density of radiated energy is given by (1-117):
0 =
For a light-like vector kO vectors 8 a satisfying
FoI'FI'
+ igOOFI'VFl'v
[k [, we define two orthogonal space-like polarization (1-208)
We readily derive that the emitted energy t! at a positive time is 3 3 00 1 fd k "[ ';' [2 t! = f d x 0 (t, x) = (2n)3 2k ko a='T,2 8a J(k) o
(1-209)
Let us anticipate the interpretation of this radiation in terms of light quanta, i.e., photons of energy momentum hk. Henceforth set h = 1. Then the energy emitted in a phase space element d 3 k will read 3 1 d k 2 /8a pi 8a pI 12 (1-210) dt! = (2n)3 "2 e a=~'2 k. pi - k. pI This semiclassical calculation enables us to obtain the number of emitted photons of polarization 8 by dividing by ko the energy of an individual quantum: dt! 2/ 8 pi 8 pI 12 d 3k (1-211) dN = kO = e k. pi - k. pI 2(2n)3kO This result agrees, in fact, with the full quantum mechanical treatment (see Secs. 5-2-4 and 7-2-3). By integrating over k (for small k) we note that the total energy is finite but the total number of photons is not. This is the infrared catastrophe to be discussed in greater detail later.
CLASSICAL THEORY
The angular distribution given by (1-211) is of interest. In the frame where k" == I k 1 (1, 1,0,0), sq = (0,0,1, 0), s~ = (0,0,0,1), taking into account that k ](k) = 0 implies / = /, we see that
(1-212)
Therefore,
This can be rewritten in terms of the particle initial and final velocities, with k = k/l k 1 as 3 2 2 d _ e2 d k 2(1 - Vi' Vf) _ m _ m ] Iff - 2(2n)3Ikf L(1 - k' vi)(1 - k' Vf) Er(I - k' Vi)2 EJ(I - k' vf)2
(1-213)
For the emission of soft quanta (I k 1---> 0) the radiation is strongly peaked in the directions of the initial and final velocities, a typical property of bremsstrahlung. To evaluate the total number ofradiated photons, we integrate (1-211) from a lower value kmin , needed because of the infrared catastrophe, up to some maximum momentum kmax, needed because of the unrealistic sharp angle in the trajectory (1-203). A typical cross section dlTcoll will describe the collision process and the total cross section dlTbrems will include the final emitted photons. To integrate (1-211) we introduce the notation q2 = (pf - Pif '" - 4E2 sin 2 (J12 in the ultrarelativistic limit where Ei ~ Ef = E with (J the scattering angle, while in the nonrelativistic limit 1Vi 1 ~ 1Vf 1 = v. We use Feynman's integral representation:
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