The proof for a photon cannot transfer all its energy to an isolated electron is a follows
By Conservation of Energy
hν + m0C2 = mC2 ...(1)
Where m0 = Rest mass of an electron
m = Relativistic mass of electron after complete photon absorption
C = Speed of light in vacuum
m = γm0
Here γ2 = 1/ (1 - β2) or γ = 1/ (1 - β2)1/2
and β = V/C
Here V = velocity of electron after complete absorption of photon
Equation (1) is written as
hν = m0C2 (γ – 1) ...(2)
By conservation of momentum
hν/C = mV = γm0 βC
or hν = γm0 βC2 ...(3)
Equating (2) & (3) we obtain
m0C2 (γ – 1) = γm0 βC2
or (γ – 1)= γβ
or 1/ (1 - β2)1/2 - 1 = β/(1 - β2)1/2
Solving which we obtain
β(1-β) = 0
Hence either = 0, and the electron is at rest after the interaction which is not possible as conservation of momentum will not hold good otherwise.
or β = 1 or V = C, i.e. the electron is moving with the speed of light. This is again impossible.
Consequently simultaneous conservation of energy & momentum requires the presence of a third body (namely, the nucleus).
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