Why Are Only Maximal-Intensity Waves Observed in Bragg's Law?

[Roo2]

I took a course on protein crystallography last year and there's one thing I couldn't figure out then, and still can't figure out now. My understanding of Bragg's law hinges on the fact that in-phase scattered waves constructively interfere, and the requirement to be in-phase is met only when the extra distance traveled is an integral multiple of the photon's wavelength. With this premise and a diagram of two atoms scattering a photon, it becomes apparent that nλ = 2dsin(θ).

What I don't understand is why only maximal-intensity waves are observed. Unless the two waves are π radians out of phase, there should still be some constructive interference that gives rise to amplitude at the detector plate. Therefore, there should be some intensity at almost any value of θ. The wikipedia page on Bragg's law acknowledges this fact:

It should be taken into account that if only two planes of atoms were diffracting, as shown in the pictures, then the transition from constructive to destructive interference would be gradual as the angle is varied. However, since many atomic planes are interfering in real materials, very sharp peaks surrounded by mostly destructive interference result.

Unfortunately, I don't see a clear and unbroken line of reasoning in the explanation. So many atomic planes means more complex interference... why is that guaranteed to produce mostly destructive interference? For help, I turned to Gale Rhodes' Crystallography made Crystal Clear. He explains it as follows:

For other angles of incidence θ' (where 2d(hkl) sinθ does not equal an integral multiple of λ), waves emerging from successive planes are out of phase, so they interfere destructively, and no beam emerges at that angle. Think of it this way: If X-rays impinge at an angle θ' that does not satisfy the Bragg conditions, then for every reflecting plane p, there will exist, at some depth in the crystal, another parallel plane p' producing a wave precisely 180° out of phase with that from p, and thus precisely cancelling the wave from p.

This is starting to approach an answer but I don't think it's complete. Why must there exist some parallel plane that produces a 180 degree phase shift? Why do such cancelling planes not exist for the set of planes that comply with Bragg's equation?

Thanks for any help!

 

[DrDu]

If there is a phase shift $\phi$ between rays reflected on adjacent planes (distance d) then there will be a phase difference $n\phi$ between planes with distance nd.
Hence if there are many planes you will observe constructive interference between all the reflected waves only if $\phi< n/N$ where N is the total number of planes in the crystal, i.e. D/d, where D is the diameter of the crystal (or better that of domains in the crystal). As D is many orders of magnitude larger than d, you get very sharp reflections.

 

[Roo2]

I might be extremely obtuse but I still don't get it. I drew a picture to visualize what you're saying:

Unfortunately, I don't see the relationship between

If there is a phase shift $\phi$ between rays reflected on adjacent planes (distance d) then there will be a phase difference $n\phi$ between planes with distance nd.

and

Hence if there are many planes you will observe constructive interference between all the reflected waves only if $\phi< n/N$

From my picture, it becomes clear that if d$\phi$ = λ/2, then every neighboring plane should destructively interfere. Likewise, if d$\phi$ = λ/4, then every second plane should destructively interfere. What you seem to be saying is that if there are very many planes in the crystal, it is statistically likely that the phase shift from one of them would be $\phi$, and from another would be $\phi$ + λ/2, or very close to it, resulting in destructive interference and consequent loss of amplitude. However, wouldn't this apply to the planes spaced to give the Bragg angle as well? Isn't it just as likely that there is some plane between them such that its diffracted photons are of phase $\phi$ + λ/2? Why is the diffraction from the Bragg planes "special", and not wiped out from the diffraction pattern?

Sorry again if I'm being stupid; I just don't see it clearly yet.

 

[DrDu]

First I wonder why you are asking this question in the biology forum and not in the general physics forum where you could calculate with much more answers.
Now to your question.
The fields of the reflected waves sum up to the total field whose square is proportional to the intensity.
Now this sum is something like $E=E_0\sum_{j=-N/2}^{N/2} \sin(2jd\sin(\theta)/\lambda)$. The latter sum can be calculated exactly. The point is that it will be nearly zero as long as $2d\sin(\theta)/\lambda$ is not a multiple of 2π because only for this value all $\sin(2jd\sin(\theta)/\lambda)=1$ and no two planes will interfere destructively.

 

[LudusRex]

I can understand your confusion about Bragg's Law and its application in crystallography. Let me try to provide a clear and unbroken line of reasoning to help you understand why only maximal-intensity waves are observed and why there is a transition from constructive to destructive interference as the angle is varied.

Firstly, it is important to understand that Bragg's Law is derived from the fundamental principles of wave interference. When X-rays are scattered by a crystal, they form a diffraction pattern due to the interference of the scattered waves. The intensity of the diffraction pattern is directly related to the amplitude of the scattered waves. Now, let's consider a simple scenario where only two planes of atoms are diffracting, as shown in the pictures you mentioned. In this case, the transition from constructive to destructive interference would indeed be gradual as the angle is varied, as you pointed out. This is because there are only two waves (from the two planes) interfering with each other. As the angle is varied, the waves may be in-phase at one point and out-of-phase at another, resulting in a gradual change in intensity.

However, in real materials, there are many planes of atoms that are diffracting. This means there are many more waves interfering with each other, resulting in a much more complex interference pattern. The waves from different planes will interfere with each other in a way that some will be in-phase and some will be out-of-phase, resulting in constructive and destructive interference. This is similar to what happens when you throw multiple pebbles into a pond at different angles - the ripples from each pebble will interfere with each other, resulting in a complex pattern of waves.

Now, let's consider what happens when the angle of incidence θ' does not satisfy the Bragg conditions. In this case, the waves from different planes will be out-of-phase, resulting in destructive interference. This is because for every reflecting plane p, there will exist another parallel plane p' producing a wave precisely 180° out of phase with that from p, as explained by Gale Rhodes. This is due to the fact that the distance between parallel planes is equal, and the difference in path length between the waves from these planes is always an integral multiple of the wavelength. This results in complete cancellation of the waves and no beam will emerge at that angle.

On the other hand, when the Bragg conditions are met, the waves from different planes will