Relativistic derivation gives the same formula

The Compton scattering formula has been derived above using the classical expressions for kinetic energy and momentum of the electron. If the electron velocity v is significant compared to the speed of light, the corresponding relativistic expressions must be used.

We will meet the methods of relativity later in this book (Appendix 16.2), but as happens when applied to Compton scattering, the relativistic way is simpler, requires no approximation, and gives the same final result. It is given here for completeness:

Electron: m0 = rest mass E = total energy p = momentum but T = hf - hf' as before fi p2c2 = (hf )2 + (hf ')2 - 2h2ff' + 2m0c2 (hf - hf') but p2c2 = (hf )2 + (hf ')2 - 2h2ff' cos Q as before fi 2h2ff' + 2m0c2 (hf - hf') = -2h2ff' cos 0 fi 2m0c2 (hf - hf') = 2h2ff' (1 - cos 0)

Ac c

Ê h ^

fi

l '-1 =

(1 - cos q)

1 m0c V

Compton formula (via relativistic derivation)

Similar the classical formula, with m0 = rest mass instead of m = classical mass

It is interesting to note that the Compton formula calculated relativistically is identical to the formula calculated using classical mechanics. The classical method uses the approximation that the square of AAA be ignored while the relativistic method is exact. The higher the photon energy, the less valid the approximation as AA becomes a greater fraction of A. The distinction between me and m0 becomes meaningful only at very high energy, when the speed of the electron approaches the speed of light.

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