Compton scattering

Today I want to talk about one of the bases of quantum mechanics or one of the demonstrations of the existence of such a mechanism is the Compton effect.

Arthur Compton, an American physicist obsessed with quantum mechanics, was circling the photoelectric effect and the collisions between photons and electrons. And he was obsessed to know how much energy was transferred from a photon or any particle with a certain wavelength and an electron from an atom.

To check the reason and calculate the wavelength of the photon or particle after colliding opto to use very energetic particles (x-rays, of very small length and therefore high energy) against coal due to its quantity of electrons and its capacity of valencia (let’s join four other things).

We have that by electromagnetic theory the energy and momentum of a wave are related by:

E = pc

We also have that the energy of a photon is:

E = hf

Where h is the Planck constant and f is the frequency.

Putting it all together, we have to:

p=\frac{E}{c}=\frac{hf}{c}=\frac{h}{\lambda }

That is the momentum of a photon.

If the shock is like that of the image that heads the post, and we assume that the electron is at rest, we will have that the initial linear moment must be equal to the final linear moment, obviously. And the initial linear momentum is only that of the photon (because the electron is at rest) while the final momentum is the sum of the photon and the electron. That is, vectorially we will have to:

\overrightarrow{p_{initial foton}}=\overrightarrow{p_{final foton}}+\overrightarrow{p_{final electron}}

As pulling with vectors is not plan, we can find the scalar multiplying by itself, an old mathematical trick in exchange for adding the cosine of the angle of departure with respect to the input.

p^{2}_{foton imitial}=p^{2}_{foton final}+p^{2}_{electron final}-2p_{foton final}p_{electron final}cos\Theta

Now, since this is not classical mechanics but relativistic mechanics, surprise, we have an equation that relates energy to its momentum:

E^{2}=p^{2}c^{2}+m^{2}c^{4}

So, if we apply it to the moment (curiously, we have it squared, that stuff) and knowing the wonderful principle of energy conservation, we have to.

p_{foton initial}c+m_{electron}c2^{2}=p_{foton final}c+\sqrt{p^{2}_{electron final}c^{2}+(m_{electron}c^{2})^{2}}

Where there is little to be said that c is the speed of light in the void.

If from all this confusion we eliminate the moments of the electrons of the equations we have in the end that:

\lambda _{final} - \lambda _{inicial}=\triangle \lambda=\frac{h}{m_{electron}c}(1-cos \Theta )

That is the Compton equation that gave the Nobel Prize and gives us the difference of frequency between an input particle and the output particle or, which is also important, know the angle of dispersion of the particles.

Also the Compton effect explains something very important and is that all the particles vibrate (although they vibrate less than their size of atom) explaining the particle / wave duality of everything that we know and, mainly, of the light to being, in that easy time of measuring its wavelength.

It is curious how such a small formula coming from the classic mechanics where we have put a term of quantum mechanics unites both worlds and, above all, explains a very important phenomenon.

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