Radiation of Accelerating Positron

A nonrelativistic positron of charge e and velocity v₁ (v₁ << c) impinges head-on on a fixed nucleus of charge Ze (see Figure 1.1). The positron, which is coming from far away (∞), is decelerated until it comes to rest and then is accelerated again in the opposite direction until it reaches a terminal velocity v₂. Taking radiation loss into account (but assuming it is small), find v₂ as a function of v₁ and Z.

Figure 1.1

SOLUTION

In first approximation, we disregard the radiation loss, i.e., we consider the energy to be constant at any given moment:

(1)

From this equation, we may find as a function of r (t) and then calculate

(2)

where ω is the acceleration of the positron. We should check at the end that the energy change due to radiation is small compared to the initial energy. From (1)

(3)

(4)

Substituting (4) into (2) and integrating, we have

(5)

We should integrate (5) from ∞ to r_min and then from r_min to ∞, again only when ∆E << E₀. In our approximation, we can say that the radiation during the deceleration is the same as for the period of acceleration and simply write

(6)

where  Substituting η ≡ r_min/r (6) becomes

(7)

Integrating by parts,

Therefore,

Check our initial assumption:

So