Inverse Compton Scattering

The HERA electron storage ring at Hamburg, Germany contains circulating electrons with an energy of 27 GeV. Photons of wavelength 514 nm from an argon-ion laser are directed so as to collide head-on with the stored electrons. Calculate the maximum scattered photon energy.

SOLUTION

Write the energy and momentum conservation laws in 4-notation, where we use units in which c = 1:

(1)

where p_γ, p_e, and  are the 4-momenta of the photon and electron before and after the collision, respectively. After subtracting  from both sides of (1) and squaring, we obtain

(2)

Here we used  and p^μ_γ p_{γ μ} = 0. We may rewrite (2) in the form

(3)

Introducing the angle θ between the direction of the incident and scattered photons, we have

(4)

From (4)

(5)

The maximum energy  in (5) corresponds to the minimum of the denominator which yields cos θ = -1, θ = π (backscattering). Therefore

(6)

Using the fact that m_e << ε_e (p_e ≈ ε_e) and expanding the denominator, we can rewrite (6) in the form

(7)

We may now substitute values into (7): ε_γ = hc/λ = 2.42 eV, m_e = 0.51 MeV, and ε_e ≈ p_e = 27 GeV, so