I Questions About Bragg's Law of X-Ray Diffraction

[tech99]

In the first case, Huygens principle explains the diffraction where propagation direction of the diffracted light isn't random. Every point on the slit acts as a wave source causing a wave to propagate in all directions from that point. In the second case, accelerated electrons scatter the waves in random directions. Waves interfere to give a pattern.

Regarding the comparison between a radiation pattern derived using Huyghen's Principle and one derived using electric currents, the two methods should give the same result. For instance, with a microwave dish, we can find the pattern by assuming an array of Huyghen's Sources spread over the aperture, or by using an array formed by the currents flowing in the metal surface. Whenever we radiate, receive, reflect, refract or diffract an EM wave we require charges, usually electrons, in order to do it. I am not aware of any case of a beam of EM radiation changing its shape in mid flight without the presence of an electric charge.

 

[vanhees71]

Exact vector diffraction theory is very complicated. It was Sommerfeld's habilitation thesis to solve it for the half-space, but Kirchhoff's scalar approximate theory is pretty good, and for (1+3) spacetime dimensions you indeed get Huygens's principle.

 

[Charles Link]

It may be worth mentioning to the OP @Dario56 that mathematically it might get a little complicated to determine where the interference peaks occur at from a 3-D structure which is the crystal. With the Bragg formulation the mathematics gets simplified by observing that the crystal consists of parallel planes that are evenly spaced. Even though all of the atoms participate in the scattering, the atoms in the same plane will always have constructive interference if the angle of incidence equals the angle of reflection. This gives zero path distance difference between the (scattered) rays from all of the atoms in the plane. (If you were to look in any other direction, you assume mostly destructive interference occurs). We can then look at the scattering as (specular) reflections off of parallel thin sheets that are equally spaced. Constructive interference occurs when the path distance difference between adjacent layers $\Delta=2d \sin{\theta} =m \lambda$, where $\theta$ is the elevation angle measured from the plane, and the incident angle is assumed to be the same as the reflected angle. (Notice also that two sheets apart have path distance difference $2 m \lambda$, and 3 sheets apart has $3 m \lambda$, etc. so they all constructively interfere). See also post 9=I'm repeating it here, with a little more detail.

The Bragg formulation employs the above logic to simplify the scattering problem.

 

[vanhees71]

Well, the important point is that you can from the X-ray diffraction pattern infer, how the atoms are ordered in a given macroscopic body or the atoms in (large) molecules. E.g., it was crucial that Watson and Crick had the X-ray spectrocopical results by Franklin to realize that DNA has a helical structure.

 

[hutchphd]

videos
Short analysis of Franklin photo51.
Slightly longer exlanation, also good.

And Watson-Crick and mostly the Wilkins Nature papers
Very sweet

 

[Dario56]

The randomness is in finding a photon, not whether some part of the wavefront is scattered in some direction.

Yes, when I said that electrons scatter x-rays in random directions, this actually refers to the individual photons which is a quantum mechanical perspective of the diffraction.

I think you're running into trouble by not treating the wave as an actual wave, but as some sort of hybrid wave-ray monster. Or wave-x-ray monster. At first glance it makes some level of sense to say that rays are randomly scattered and to say that each ray is a photon, as this fits in with how we detect x-rays and how many other phenomena of light work. But rays are not photons and rays are not scattered randomly. Rays are nothing more than imaginary lines we draw on diagrams to help us see where the wavefront is going. They are just lines that are drawn perpendicular to the wavefront and point in the direction of travel.

I think you are correct that my problem is that I look at the XRD from both classical and quantum perspective which leads to inconsistencies and things not adding up (me not treating wave as the wave, but as a ray-wave hybrid or maybe better, photon-wave hybrid). When explaining phenomena in physics, one needs to pick one of the perspectives and stick to it. In that sense, I can't say that EM waves are scattered in random directions since that refers to photons and quantum view of the diffraction. I started with the classical view (X-rays are modelled as the EM waves exhibiting wave properties like diffraction and interference) and need to be consistent with that.

My problem is still the connection between the Huygens principle view of the diffraction with the Thompson scattering view as these views are quite different. XRD textbook I borrowed (XRD by Warren), explains XRD with the Thompson scattering and electron oscillations.

It seems to me that there is a discrepancy in using both of these models to explain x-ray diffraction since Huygens principle doesn't really take into account that electric field in the EM waves affect motion of the electrons while the opposite is the case with the Thompson scattering which doesn't take into account the wave phenomena like interference and diffraction.

I'm not sure if this is correct, but I think that the solution to this discrepancy is that classical view of the diffraction simply isn't correct. It is simpler to use and interpret than quantum and can give accurate predictions, but since it isn't fundamentally correct, things may not be completely consistent within the classical view. Example of such an inconsistency is what I wrote in the last paragraph. If everything here wants to be explained correctly and consistently, quantum mechanics must be used. What is your view of my thinking?

 

[Charles Link]

I think you are missing a couple of the simpler concepts involving interference phenomena, and you would do well to stick with classical concepts. There is little to be gained by trying to go to more sophisticated explanations. I think you need some practice with several of the interference phenomena, including the diffraction grating, and also the Fabry-Perot effect.
I do recommend you read through this "link" that I previously posted: https://www.physicsforums.com/threa...ic-waves-in-destructive-interference.1015365/

The Bragg explanation for finding an angle of incidence for which you get a scattered ray of high intensity is something that you should be able to follow very readily. The complete subject of scattering by crystals gets much more mathematically complex, but the "inconsistencies" that you are presently finding seem to be key pieces that are all part of classically explained interference theory that you need to learn in detail.

=and please see my post 33. I tried to explain in depth the Bragg condition for the peak scattered intensity, for which the textbooks often seem to omit the fine detail.

 

[hutchphd]

With respect for your efforts, I ask you a simple question. What exactly are you trying to understand?

I think the basic thing you are wanting to know is elastic scattering of waves from a periodic lattice. This can cover a host of physics Including light scattering, or electrons. or low energy helium atoms from periodic crystal surfaces (I know an excellent but now-dated PhD thesis about this!!)

You complicate the physics by worrying about the details of vector EM field and solid state physics and photons and coherence. Formulate a specific specific question please.

 

[Dario56]

With respect for your efforts, I ask you a simple question. What exactly are you trying to understand?

I think the basic thing you are wanting to know is elastic scattering of waves from a periodic lattice. This can cover a host of physics Including light scattering, or electrons. or low energy helium atoms from periodic crystal surfaces (I know an excellent but now-dated PhD thesis about this!!)

You complicate the physics by worrying about the details of vector EM field and solid state physics and photons and coherence. Formulate a specific specific question please.

I'm trying to learn XRD. After reading about it from different sources, I found some things which I didn't understand and which didn't add up. After some discussion, it came out to be:

1) Mixing classical and quantum views doesn't work (which I did)

2) It seems to me that there is a discrepancy in using both previously mentioned models together to explain x-ray diffraction. You can use them separately, but not together. Huygens principle view doesn't really take into account that electric field in the EM waves affect motion of the electrons. It doesn't care about the electrons, it just looks for slits (spaces between atoms). However, if electric fields affect the electron motion in the classical theory than classical theory is problematic in the context of this model. Why? Because this model accurately describes observations, but it doesn't take into account that electrons radiate when affected by EM waves (which classical physics predicts). Therefore, my conclusion is that models made from classical theory are practical and can predict results accurately, but aren't completely consistent if you take all laws of classical physics into account. This isn't particularly surprising since classical physics isn't fundamentally correct, which can especially be seen when we get to the light and small stuff like electrons and atoms.

This is what I concluded and was wondering what do you guys think about my conclusions?

 

[Dario56]

I think you are missing a couple of the simpler concepts involving interference phenomena, and you would do well to stick with classical concepts. There is little to be gained by trying to go to more sophisticated explanations. I think you need some practice with several of the interference phenomena, including the diffraction grating, and also the Fabry-Perot effect.
I do recommend you read through this "link" that I previously posted: https://www.physicsforums.com/threa...ic-waves-in-destructive-interference.1015365/

The Bragg explanation for finding an angle of incidence for which you get a scattered ray of high intensity is something that you should be able to follow very readily. The complete subject of scattering by crystals gets much more mathematically complex, but the "inconsistencies" that you are presently finding seem to be key pieces that are all part of classically explained interference theory that you need to learn in detail.

=and please see my post 33. I tried to explain in depth the Bragg condition for the peak scattered intensity, for which the textbooks often seem to omit the fine detail.

I think you somewhat missed the point of my questions. My questions aren't really about the wave optics, it was about connecting and bringing peace with different perspectives which explain XRD. I got confused as I mixed quantum and different classical views of the subject since I saw all of these in different sources. I think I do understand wave optics well enough to understand Bragg's law since I get what textbooks are saying and I'm familiar with the Young's experiment, diffraction on single slit and diffraction grating. I understand conditions of constructive interference and the derivations of intensity vs angle equations in these.

Not to leave an incorrect impression, thanks for all the answers you provided. I think this discussion helped me a lot.

 

[Drakkith]

2) It seems to me that there is a discrepancy in using both previously mentioned models together to explain x-ray diffraction. You can use them separately, but not together.

On the contrary. You MUST use them together, as they model different things. Huygens principle (more generally Huygens-Fresnel principle) is a model about wave propagation in general. Bragg's law is about the specific condition of an EM wave passing through a crystal structure. Furthermore, you're missing a bit that I'll elaborate on below.

However, if electric fields affect the electron motion in the classical theory than classical theory is problematic in the context of this model. Why? Because this model accurately describes observations, but it doesn't take into account that electrons radiate when affected by EM waves (which classical physics predicts).

I think what you're missing is that Huygen's principle and Bragg's law are both simplified models which are built off of underlying physical laws, and you're not using these laws to their full extent when analyzing the situation. Hugyen's principle says nothing about wave-particle interactions, and Braggs law says nothing about general wave propagation. And neither of them say anything about how electric charges generate or interact with EM waves. That's what the fundamental electromagnetic laws, like Maxwell's equations, are for.

These models (and others) can be derived from fundamental EM laws, but they are not the laws themselves.

Therefore, my conclusion is that models made from classical theory are practical and can predict results accurately, but aren't completely consistent if you take all laws of classical physics into account.

I'd argue that they aren't consistent UNLESS you take all laws of classical physics into account (ignoring quantum effects of course).

 

[hutchphd]

Now I am confused. As usual my confusion is semantic and definitional. lFirst my favorite Feynman about naming things:

In my world the difference between Laue and Bragg diffraction is operational.

1. For Laue diffraction one shoots a (not necessarilly monochromatic) beam of xrays at a crystal and looks at everything that comes out
2. For Bragg diffraction one looks at the specular reflection of a beam of (usually) monochromatic xrays as a function of angle

My definitions may be idiosynchratic (?are they?) but the interactions describing the process are the same physics: scattering of waves from a periodic lattice. So one needs to understand that physics and the rest depends upon the detailed questions one wishes to answer.
Words you may wish to know: Miller indices, reciprocal lattice, direct lattice

 

[vanhees71]

Indeed, and even if you take a QFT point of view and talk of photons, that's the right picture. Photons are no point particles in any sense. They don't even admit the definition of a position observable!

 

[Dario56]

On the contrary. You MUST use them together, as they model different things. Huygens principle (more generally Huygens-Fresnel principle) is a model about wave propagation in general. Bragg's law is about the specific condition of an EM wave passing through a crystal structure. Furthermore, you're missing a bit that I'll elaborate on below.

I think what you're missing is that Huygen's principle and Bragg's law are both simplified models which are built off of underlying physical laws, and you're not using these laws to their full extent when analyzing the situation. Hugyen's principle says nothing about wave-particle interactions, and Braggs law says nothing about general wave propagation. And neither of them say anything about how electric charges generate or interact with EM waves. That's what the fundamental electromagnetic laws, like Maxwell's equations, are for.

These models (and others) can be derived from fundamental EM laws, but they are not the laws themselves.I'd argue that they aren't consistent UNLESS you take all laws of classical physics into account (ignoring quantum effects of course).

My point is actually that you don't need to use both Huygens principle and Thompson scattering together. I don't know about you, but when I studied how intensity of EM waves depend on the angle in Young's experiment, single slit diffraction or diffraction grating, never there was mentioned anything about Thompson scattering. No one modelled that EM waves affect the electrons in the grating material causing them to radiate.

If we take Young's experiment as an easiest example, you would calculate resultant electric field at the point of observation from two slits (as a sum), use some trigonometry to express it as a product of the amplitude and cosine of the phase and express the average value of the Poynting vector at the point from the amplitude of the resultant electric field. Now you have the equation which expresses the intensity as a function of the angle of the rays (for Fraunhofer, parallel ray condition, where distance to the screen is very big compared to the slit separation distance).

Derivation is similar for the single slit diffraction and diffraction grating, but relationship between angle and the intensity turns out to be more complex.

Basically, in these derivations, where do you see anyone mention Thompson scattering? In these derivations, we use laws of classical physics and classically, radiation is the EM wave which should affect charged particles and cause them to radiate. Since electrons are charged particles and part of the material of the grating, why isn't that taken into account?

That is a discrepancy I was taking about, to which I think that the solution is that classical physics is not a correct theory. In some cases, it leads to correct results and is convenient to use, but results may not be completely consistent with all of the laws of classical physics. Example of that is what I wrote in the previous paragraph.

 

[Drakkith]

No one modelled that EM waves affect the electrons in the grating material causing them to radiate.

Of course not. The model you're working with is only concerned about the macroscopic behavior of an EM wave that encounters an obstacle. The details at the subatomic level aren't considered because it's taken as a starting condition that part of the EM wave is being blocked or absorbed at the slit boundaries somehow. Exactly how that works is left to another model.

Basically, in these derivations, where do you see anyone mention Thompson scattering? In these derivations, we use laws of classical physics and classically, radiation is the EM wave which should affect charged particles and cause them to radiate. Since electrons are charged particles and part of the material of the grating, why isn't that taken into account?

Why would it be? It turns out that the details simply don't matter for Young's experiment. You don't need to know how charges absorb and radiate EM radiation if you're doing a simple double-slit experiment. You only need to know that, however it works, the result is part of your wave being blocked, which then leads to the diffraction and interference effects seen in the experiment.

That is a discrepancy I was taking about, to which I think that the solution is that classical physics is not a correct theory.

I don't see any discrepancies in your posts. I see several models that describe different but related things. Huygens principle describes general wave propagation, Bragg's law gives you the angles of highest intensity for EM waves scattering from atomic planes in crystals, and Thompson scattering is concerned with how EM waves scatter off of free charged particles.

You can, of course, derive all of these from more fundamental laws, and Huygens principle may in fact be useful in deriving the latter two (I'm not sure).

 

[vanhees71]

Huygens's principle is just the Green's function of the d'Alembert operator. For slits and gratings you just solve approximately the wave propagation with the openings as sources of the wave equation (Kirchhoff's theory of diffraction) and for the X-ray scattering on crystals the atomic lattice is modelled as a periodic lattice of sources.

 

[Charles Link]

Basically, in these derivations, where do you see anyone mention Thompson scattering? In these derivations, we use laws of classical physics and classically, radiation is the EM wave which should affect charged particles and cause them to radiate. Since electrons are charged particles and part of the material of the grating, why isn't that taken into account?

In more detailed treatments of scattering in crystals, you will see scattering cross sections for the atoms that have an amplitude and phase, and even an amplitude that depends on the direction, rather than just being a delta function at the site of the atom with a spherically symmetric $E$ field that falls off as inverse $r$. The treatment gets very in-depth, but it is important to get a good handle on the fundamentals before taking on this kind of mathematical detail.
See e.g. the book by Squires on Thermal Neutron Scattering. https://www.thriftbooks.com/w/intro...SABEgIfYPD_BwE#idiq=46801417&edition=60567753
This book is one of the better ones for learning some very fine details about scattering, even though it works with neutrons=it treats the incident neutrons as a quantum mechanical wave. You might find it of interest.
Edit: They even account for thermal motion of the scattering sites=the Debye-Waller factor.

 

[tech99]

It turns out that the details simply don't matter for Young's experiment. You don't need to know how charges absorb and radiate EM radiation if you're doing a simple double-slit experiment. You only need to know that, however it works, the result is part of your wave being blocked, which then leads to the diffraction and interference effects seen in the experiment.

If we take the diffraction pattern of a single slit, it appears to me that it will be polarisation dependent. This is because, for a vertical slit, the vertical polarisation will give a maximum on axis, the two edges of the slit re-radiating in-phase. On the other hand, horizontal polarisation should give a zero on axis. This is because for horizontal polarisation the edges are charged oppositely, giving rise to anti-phase radiation. I want to try this tomorrow using microwaves.

 

[hutchphd]

This Bethe guy did a good job: http://www.physics.miami.edu/~curtright/Diffraction/Bethe1944.pdf

 

[Charles Link]

If we take the diffraction pattern of a single slit, it appears to me that it will be polarisation dependent. This is because, for a vertical slit, the vertical polarisation will give a maximum on axis, the two edges of the slit re-radiating in-phase. On the other hand, horizontal polarisation should give a zero on axis. This is because for horizontal polarisation the edges are charged oppositely, giving rise to anti-phase radiation. I want to try this tomorrow using microwaves.

They make a microwave polarizer, which consists of a bunch of thin vertical metal bars, spaced evenly, about 1/4 inch apart. The microwaves that are polarized perpendicular to the bars pass through, while those in the direction of the bars are blocked or canceled by the excitation in the bar.

and I highly recommend the book in post 47 for @Dario56 . I just dug out my copy of the book that I had from my college days. It really treats scattering with the kind of detail that I think @Dario56 is looking for.
See also:
https://en.wikipedia.org/wiki/Scattering_amplitude
and I called it a scattering cross section in post 47, but the proper term is scattering amplitude. The "wiki" "link" describes both of these terms.

@Dario56 See also https://www.physicsforums.com/threads/diffraction-on-periodic-structures.952210/#post-6032867
for a good discussion that is closely related to this one.

 

[Dario56]

You guys provided many new interesting sources to read and expand upon my question. I think that this is a bit too detailed for what I currently need, but who knows. Maybe I'll want to get back to them later.

One new question popped to me. Basically, we take that diffraction centres are different in the XRD and Young's experiment. In the former case, we take that atoms are centres. In the latter, slits are centres which would be equivalent to the interatomic spacing in the context of XRD.

Is it justified to apply Huygens principle to atoms rather than spacing between them? Or in this case, we look at the diffraction from the perspective of Thompson scattering? I think it is the second case, though.

 

[Charles Link]

Interesting question that you are asking. With the XRD the atoms are considered to be the sources that basically send out the $E$ field as a sinusoid in all directions. In general each one scatters just a small portion of the incident amplitude. With the scattering, we are working with an E amplitude and the sinusoids (in a given direction) need to be summed with their phases. Once the sum is computed, the energy/intensity is then computed, (taking $E_{total}^2$), as I think you have got it figured out by now.

One might expect that one could compute the energy scattered in a given direction by each scatterer and sum them all, but that is, perhaps somewhat surprisingly, not how it works. I'm just adding a little extra detail here, even though I think you have it figured out. :)

Edit: Presently you are just computing the intensity pattern with an unknown proportionality factor. In more detailed calculations, the actual power can be computed for the various diffraction patterns, and they can be shown to conserve energy. For an example of the energy conservation, see https://www.physicsforums.com/insights/diffraction-limited-spot-size-perfect-focusing/

 

[Steve4Physics]

Is it justified to apply Huygens principle to atoms rather than spacing between them? Or in this case, we look at the diffraction from the perspective of Thompson scattering? I think it is the second case, though.

It doesn’t matter what the sources of the waves are. They can be atoms, or gaps, or even loud-speakers.

The essential requirements are that:
- the waves radiated by the sources overlap;
- the phase-difference between neighbouring sources is appropriate (e.g. zero for the Young’s double-slit experiment and some constant non-zero value for Bragg diffraction).

 

[sophiecentaur]

Therefore, how can we only consider parallel scattered waves in Bragg's law as non-parallel waves can also create diffraction pattern?

What are you suggesting about "non parallel (incident?) waves"? Diffraction still works - as in the production of holograms but this is not really relevant to Xray diffraction patterns which involve a pretty short coherence length of the incident waves. I have never come across Xray holography - but someone here may know differently.
The basics of Xray diffraction in crystals (Bragg scattering) assume a point source and a plane wave arriving at the crystal. If you could stack layers of optical diffraction gratings you would get a similar effect on the pattern with many gaps in it. Very often, there is not a single, large crystal available to look at and the simple idea of distinct single rays emerging fails. What does remain, though, tends to be cones of scattered rays with particular angles between them. Spacings between atoms can be calculated from the pattern.

 

[sophiecentaur]

Is it justified to apply Huygens principle to atoms rather than spacing between them? Or in this case, we look at the diffraction from the perspective of Thompson scattering? I think it is the second case, though.

I re-read this and I don't understand where you're going with this. Huygens construction is basically the diffraction formula and would be treating a large aperture across the whole crystal and would give you the whole of the pattern of emerging rays. This would produce an intense beam in the straight through direction and also the Bragg maxima when the angles are suitable - i.e. when both i and r are equal, giving an effective reflection. For other angles there is no such 'reflection'.
The basic Bragg formula ignores the spaces between and deals with point sources, which is an interference formula and a lot simpler.
I remember struggling with this in my brief exposure to Crystallography. Life was a lot easier when I started looking at RF transmitting or receiving arrays where there is only an 'incident wave' or an 'output wave'. (the 'other ray' is just in your head)