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3.1. Single photon-electron scattering

When a photon is scattered by an electron, the energy and direction of motion of both the photon and the electron are usually altered. The change in properties of the photon is described by the usual Compton scattering formula

Equation 21 (21)

where the electron is assumed to be at rest before the interaction, epsilon and epsilon' are the photon energies before and after the interaction, and phi12 is the angle by which the photon in deflected in the encounter (see Fig. 3).

Figure 3. The scattering geometry, in the frame of rest of the electron before the interaction. An incoming photon, at angle theta relative to the xe axis, is deflected by angle phi12, and emerges after the scattering at angle theta' with almost unchanged energy (equation 21). In the observer's frame, where the electron is moving with velocity beta c along the xe axis, the photon changes energy by an amount depending on beta and the angles theta and theta' (equation 25).

For low-energy photons and mildly relativistic or non-relativistic electrons, epsilon << me c2 and the scattering is almost elastic (epsilon' = epsilon). This limit is appropriate for the scatterings in clusters of galaxies that cause the Sunyaev-Zel'dovich effect, and causes a considerable simplification in the physics. Although the scatterings are usually still referred to as inverse-Compton processes, they might better be described as Thomson scatterings in this limit.

Scatterings of this type will also cause Sunyaev-Zel'dovich effects from the relativistic plasma of radio galaxies. The lobes of radio galaxies emit strong synchrotron radiation, and must contain electrons with Lorentz factors gammae gtapprox 108. In the rest frames of such electrons the microwave background radiation appears to have a peak at photon energies ~ 0.1 me c2, and the assumption of elastic scattering will be inappropriate. Little theoretical work has been done on the spectrum of the scattered radiation in this limit, but see Section 5.

In this thermal scattering limit, the interaction cross-section for a microwave background photon with an electron need not be described using the Klein-Nishina formula,

Equation 22 (22)

but rather the classical Thomson cross-section formula which results in the limit epsilon' -> epsilon. Then if the geometry of the collision process in the electron rest frame is as shown in Fig. 3, the probability of a scattering with angle theta is

Equation 23 (23)

where the electron velocity ve = beta c, and µ = costheta. The probability of a scattering to angle theta' is

Equation 24 (24)

(Chandrasekhar 1950; Wright 1979), and the change of photon direction causes the scattered photon to appear at frequency

Equation 25 (25)

with µ' = costheta'.

It is conventional (Wright 1979; Sunyaev 1980; Rephaeli 1995b) to express the resulting scattering in terms of the logarithmic frequency shift caused by a scattering, s (Sunyaev uses u for a related quantity),

Equation 26 (26)

when the probability that a single scattering of the photon causes a frequency shift s from an electron with speed beta c is

Equation 27 (27)

Using (23-25), this becomes

Equation 28a
Equation 28b (28)

where µ' can be expressed in terms of µ and s as

Equation 29 (29)

(from equations 25 and 26), and the integral is performed only over real angles, so that

Equation 30 (30)

Equation 31 (31)

in (28). The integration can be done easily, and Fig. 4 shows the resulting function for several values of beta. The increasing asymmetry of P (s; beta) as beta increases is caused by relativistic beaming, and the width of the function to zero intensity in s,

Equation 32 (32)

increases because increasing beta causes the frequency shift related to a given photon angular deflection to increase.

Figure 4. The scattering probability function P (s; beta), for beta = 0.01, 0.02, 0.05, 0.10, 0.20, and 0.50. The function becomes increasingly asymmetric and broader as beta increases.

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