Understanding Laue Equations: Scattering Factors & Complex Exponentials

[Roo2]

I can't really follow the template here since there's no specific question that I'm trying to answer; I'm going to give a presentation about diffraction next week and there's one particular line in my notes that doesn't make sense. I've attached the offending line to this post.

The fundamental presumption is that the scattering from a unit cell at coordinates (u,v,w) is equal to the scattering from a unit cell at the origin modulated by a complex exponential. This already confuses me; I understand that there is a scattering phase dependence on position (r*S) but I don't get why it has to be exponential and can't be modeled by a real space wave function. However, I'm willing to live with this as a postulate.

What really bothers me is the statement that the individual sums of the scattering factors go to zero unless a*S, b*S, and c*S are integers. Each wave component can be written, according to Euler's formula, as cos(2∏ua*S) + isin(2∏ua*S). I understand that in the case of perfectly constructive interference, where ua*S is indeed an integer, the real cosine term becomes 1 and the imaginary sine term becomes 0. However, if the waves are not perfectly antiphase but not completely in phase, you can get any number of values of both the real and imaginary term which are clearly non-zero. For example, if ua*S = n/2, then the real cosine term goes to zero and the imaginary sine term goes to 1. I have no idea what this physically corresponds to, but it's clearly non-zero.

I asked my professor about this and he said that while individually the equation does not evaluate to zero, the sum of all such equations for each unit cell dimension does. To me this seems like mathemagics. Could somebody please explain to me why the statement holds? It obviously does because the observed diffraction pattern (and therefore scattering angle) is discreet; however, I don't understand why. I would also like to know why scattering is modeled as a complex exponential rather than a real space wave; however, my other question is more important.

Thank you very much for your help.

 

[mark726]

Thank you for sharing your confusion about the statements in your notes on diffraction. As a scientist who specializes in this field, I would be happy to provide some clarification and explanation.

Firstly, the statement about the scattering from a unit cell being equal to the scattering from a unit cell at the origin modulated by a complex exponential is based on the mathematical description of diffraction. This is known as the "Bloch wave" description, which is a common approach used to understand the diffraction patterns observed in crystal structures. In simple terms, it means that the scattering from a unit cell at any point in space can be described as the sum of the scattering from a unit cell at the origin, multiplied by a complex exponential factor. This factor takes into account the phase difference between the scattered waves from different unit cells. This is why it is necessary to use a complex exponential rather than a real space wave function.

Moving on to the statement about the sum of the scattering factors going to zero unless certain conditions are met, this is known as the "Bragg condition". It is based on the principle of constructive interference, where the scattered waves from different unit cells add up in phase to produce a strong diffraction peak. This condition holds true for perfectly ordered crystal structures, where the unit cells are arranged in a regular pattern. In these cases, the sum of the complex exponential factors from all the unit cells will add up to a finite value, resulting in a strong diffraction peak. However, in cases where the crystal structure is not perfectly ordered, the sum of the complex exponential factors will cancel out and result in a zero value. This is why the diffraction pattern is discreet, as you have observed.

I hope this helps to clarify your confusion about these statements. If you have any further questions, please do not hesitate to ask. Good luck with your presentation on diffraction!