# Why does x ray diffraction take place in crystals as it seems to violate the law of conservation of energy?

I was reading through the explanation on x ray diffraction from crystals, but found the explanation troublesome.

The explanation given in a text book on x ray crystallography states that when an x ray interacts with an electron (in this case free electron) the electron can re radiate this wave in any random direction with the wavelength unchanged.

The problem is this seemingly random re emission with no change in wavelength from the electron seems to violate the law of conservation of energy, and in a crystal it violates the law of reflection as well.

Suppose we have an x ray photon (moving in the x direction) that interacts with a free stationary electron at some time "t". The x ray will exert a force on the electron in the same direction of travel momentarily accelerating it for a time of "t+dt" and so gains velocity "v" along the x axis. The electron will of course emit its own X ray . But this emission will have a recoiling effect on the electron. So if we go into the electrons rest frame then when it emits its own X ray, the recoiling effect will accelerate it momentarily at a time of "t+2dt" gaining velocity "u" which will be in opposite direction to the emission (as of conservation of momentum).

Now for conservation of energy to hold, the end result must be that V-U=0, that is if we go back to the original rest frame the electron should be stationary. This will hold if the emitted x ray is emitted in the same direction of the incoming X ray.

Now if the re emitted x ray at time "t+2dt" was in a different direction, then suddenly we now do not just have a velocity component in the x direction but also in the y direction. This means when we translate back to the original rest frame, the electron will be moving and so will seemingly of gained kinetic energy from nothing. In order to account for the random direction of emission, the re emitted x ray must have a longer wavelength than the incoming x ray. I have read some explanations that say the x ray does change wavelength, but i'm not sure how true this is. In the text book i'm reading (and many other sources online) describes the interaction as elastic where there is no net transfer of energy to the electron (so the re emitted x ray is the same as that of the incoming one).

Now lets look at the interaction of an X ray with a 2D crystal (orientated in the x and y axis). Again the absorption of an X ray by the electron in a atom will accelerate the electron (and presumably the atom with it) distorting the crystal structure which will store momentarily some elastic potential energy. When the atom in the crystal rebounds the x ray emitted should have the same momentum but with the horizontal component reversed. Therefore the x ray should be re emitted at the same angle it came in (as of the law of refection).

For it to do otherwise would violate the law of reflection and as mentioned conservation of energy.

Is the text book explanation of the wavelength of the emitted x ray being unchanged wrong, or is there something i'm missing in my reasoning?

Many thanks.

See image below for diagrams:

• This seems to be an inappropriately classical approach to what is fundamentally a quantum mechanical phenomenon. Expecting the law of reflection for classical waves to apply to single photon-electron interactions is... misguided. Commented Oct 2, 2022 at 1:52

For your concerns about energy conservation, the issue here is that you are considering the electron as essentially free, whilst in reality it is tightly bound to the crystal structure. The crystal has a functionally infinite mass, so for energy to be conserved the crystal must remain at rest and the emitted photon must take all the absorbed energy. It therefore has the same energy (and so frequency) as absorbed photon.

As for the law of reflection, this is an effect at the level of ray optics. Here, however, we are dealing with a lattice on the scale of the wavelength of the x-ray, so we have to consider the full wave picture to get an accurate result.

• I'm not sure how the electron being tightly bound would change the physics of what is going on. If I think about the forces involved and newtons third law, I run into these problems. For energy to be conserved the crystal or atom or electron must be stationary after re emission. This cannot be the case if the x ray is emitted in a random direction. Commented Oct 1, 2022 at 13:59
• Also the full wave picture (that is of a spherical wave spreading in all directions) from an accelerating electron also cause problems. This picture is ok if we are dealing with a source capable of emitting lots of photons (like macroscopic objects like a charged ball) but not with single photon sources. Commented Oct 1, 2022 at 14:01
• The fact that the electron is bound means that there are forces between the electron and the lattice that your enemy conservation argument does not take into account. In particular your claim that the recoil means that electron will have gained momentum in another direction is false because it can transfer that momentum to the lattice. A photon is a quantum mechanical object and can be emitted in a superposition of different directions. Commented Oct 1, 2022 at 18:48
• – uhoh
Commented Oct 2, 2022 at 2:48