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

• I
Dario56
Bragg's law is schematically shown on the picture:

Two parallel and plane waves are shown which propagate towards the crystal. For plane waves, wave fronts are flat planes perpendicular to the wave propagation with infinite size. In reality, there are no plane waves. Nevertheless, they are used as they are often a good approximation.

I have two questions:

1) How do we know that the interference of two waves hadn't happened before both waves interacted with the lattice? To be more precise, waves shouldn't interfere before the left ray reaches point D as we want that the left ray travels that extra distance (from C to D) to see what kind of interference result we get. Wave fronts of real waves extend in space as wave propagates and so interference may happen before both waves interacted with the lattice (before left wave reached point D) . If that happens, there is no much use in the Bragg's law.

2) Bragg's law is based on the interference of two parallel waves. Scattering of the x-rays happens because electrons absorb x-rays and re-emit them, but they do so randomly, in all directions. Constructive interference can happen if waves of different propagation directions interfere and so, if we see the bright or dark spot of the interference, how can we know that in fact two parallel waves actually interfered to give that spot?

## Answers and Replies

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Scattering of the x-rays happens because electrons absorb x-rays and re-emit them, but they do so randomly.......
Here seems to be some misunderstanding.

In this sense, an electron is said to scatter x-rays, the scattered beam being simply the beam radiated by the electron under the action of the incident beam. The scattered beam has the same wavelength and frequency as the incident beam and is said to be coherent with it, since there is a definite relationship between the phase of the scattered beam and that at of the incident beam which produced it.
(from “Elements of X-ray Diffraction, Second Edition” by B. D. Cullity)

Pushoam and Dario56
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Interference between EM waves happens in the detector. The waves may move across each other in free space with no interaction and emerge unscathed afterwards. We see this with sea waves as well, where they might be on crossing paths. In the common volume, the peaks of the waves are summed, and if we place a detector at this point, it will extract some energy corresponding to the combined peak. The interaction occurs in the detector, where in effect we square the sum of the amplitudes, and not in the lattice, which treats the two waves individually and in a linear manner.
In the case shown, the electrons move in response to the electric field of the incoming wave, and in so doing they re-radiate the energy without creating a delay. It is true that the paths to the detector may then differ in length slightly, but as the path length is very great compared with the spacing of the sources, the formula assuming parallel rays is very close.

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Nevertheless, they are used as they are often a good approximation.
One must realize also that to be truly monochromatic, the plane wave must exist exist for a very long time. One cannot talk classically about time localized wave "interactions" for a truly monochromatic wave. If the interaction is treated correctly using quantum mechanics the issue does not arise.

Pushoam
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Two parallel and plane waves are shown which propagate towards the crystal.
No. There is only one incident plane wave. The diagram shows two rays for a single incident plane wave.

The diagram (which is commonly used) is misleading IMO.

The upper atomic plane scatters some X-rays in random directions. We consider only the scattered radiation in the direction given by θ. In effect, some of the scattered radiation behaves the same as a 'reflected plane wave' leaving the upper atomic plane at angle θ. (Only one ray from this wave is shown on the diagram.)

Some of the incident radiation reaches the lower atomic plane. The lower atomic plane scatters X-rays. Some of this scattered radiation behaves like a 'reflected plane wave' leaving the lower atomic plane at angle θ. (Only one ray from this wave is shown on the diagram.)

The interference is between the two reflected waves - the reflected wave from the upper atomic plane and the reflected wave from the lower atomic lane.

Pushoam
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One must realize also that to be truly monochromatic, the plane wave must exist exist for a very long time. One cannot talk classically about time localized wave "interactions" for a truly monochromatic wave. If the interaction is treated correctly using quantum mechanics the issue does not arise.
The spacing between atoms in the experiment may typically be only a few wavelengths, so the location of fringes will not be very sensitive to small changes in wavelength. At one second from switch-on, I would expect the energy of a monochromatic source to be mostly contained within 2 Hz.

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The spacing between atoms in the experiment may typically be only a few wavelengths, so the location of fringes will not be very sensitive to small changes in wavelength. At one second from switch-on, I would expect the energy of a monochromatic source to be mostly contained within 2 Hz.
But the coherent scattering from Bragg layers in any classical picture puts much more stringent requirements on these localized events as mentioned:
The interference is between the two reflected waves - the reflected wave from the upper atomic plane and the reflected wave from the lower atomic lane.

Dario56
Here seems to be some misunderstanding.

In this sense, an electron is said to scatter x-rays, the scattered beam being simply the beam radiated by the electron under the action of the incident beam. The scattered beam has the same wavelength and frequency as the incident beam and is said to be coherent with it, since there is a definite relationship between the phase of the scattered beam and that at of the incident beam which produced it.
(from “Elements of X-ray Diffraction, Second Edition” by B. D. Cullity)
Yes, I found this in the quoted textbook.

Scattering of the EM waves by electrons can be classically described with the Thomson scattering model. This model is essentially what your quote says. As far as I understood, when incident wave makes electron oscillate, electron emits EM waves, but does so in all directions. So, when considering diffraction pattern, how can we know that in fact two parallel diffracted waves produced such a pattern? Scattered waves need not to be parallel to produce constructive interference.

It is of crucial importance that these waves are parallel before and after interaction with the crystal as only in that case Bragg's formula has sense.

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The Bragg diffraction is like the Fraunhofer=it is for the far field, and where the path distance difference between two rays coming from adjacent planes (they can be assumed parallel when viewed in the far field) is an integer number of wavelengths.
The Bragg diffraction has the additional condition that the angle of incidence onto the plane(s) is equal to the angle of reflection. This makes it so that the rays from atoms in the same plane have zero path distance difference between them, and thereby also constructively interfere.
Thereby, with the Bragg condition, the rays from all of the atoms in the crystal constructively interfere, (as viewed in the far field). Note: If the crystal is small, the far field can be just a meter or so away. You can also create far field conditions by observing the pattern with a lens, with detector or paper in the focal plane of the lens, even for very short distances. (This "trick" is used in diffraction grating spectrometers, where with a 2" size diffraction grating, the far field would otherwise be a couple hundred meters away).

Edit: Meanwhile I can see where the OP @Dario56 is puzzled by what the textbooks present=I also found most of them to not be real clear on this Bragg diffraction subject, and I had to figure it out for myself...

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vanhees71
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So, when considering diffraction pattern, how can we know that in fact two parallel diffracted waves produced such a pattern? Scattered waves need not to be parallel to produce constructive interference.
In general they do not. In fact only for a very particular subset of elastically and coherently (in time) scattered light will this be true. That is why a particular incident orientation on a monocrystaline sample will produce a finite number of rather strong beams for photodetection. Using various known orientations allows detailed determiination of reciprocal lattice structure and from that the structiure of the direct lattice.

Dario56
No. There is only one incident plane wave. The diagram shows two rays for a single incident plane wave.

The diagram (which is commonly used) is misleading IMO.

The upper atomic plane scatters some X-rays in random directions. We consider only the scattered radiation in the direction given by θ. In effect, some of the scattered radiation behaves the same as a 'reflected plane wave' leaving the upper atomic plane at angle θ. (Only one ray from this wave is shown on the diagram.)

Some of the incident radiation reaches the lower atomic plane. The lower atomic plane scatters X-rays. Some of this scattered radiation behaves like a 'reflected plane wave' leaving the lower atomic plane at angle θ. (Only one ray from this wave is shown on the diagram.)

The interference is between the two reflected waves - the reflected wave from the upper atomic plane and the reflected wave from the lower atomic lane.
Yes, you are correct about the rays and waves. We have one incident wave and two rays we consider. After scattering, two waves will be created which then interfere.

If scattering happens in random directions, how can we only consider one direction? EM waves from different directions can also interfere which will affect the diffraction pattern.

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Dario56
In general they do not. In fact only for a very particular subset of elastically and coherently (in time) scattered light will this be true.
Hmm, interesting. Why is this the case? Do you know some good source where I can study this in more detail?

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@Dario56 Please see my post 9=I have edited it a few times after you might have first glanced at it.

hutchphd and Dario56
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Bragg diffraction and Laue scattering. Really different faces of the same processes. The Wikipedia is a pretty good start.....this is stalwart physics.

sophiecentaur
Dario56
The Bragg diffraction is like the Fraunhofer=it is for the far field, and where the path distance difference between two rays coming from adjacent planes (they can be assumed parallel when viewed in the far field) is an integer number of wavelengths.
The Bragg diffraction has the additional condition that the angle of incidence onto the plane(s) is equal to the angle of reflection. This makes it so that the rays from atoms in the same plane have zero path distance difference between them, and thereby also constructively interfere.
Thereby, with the Bragg condition, the rays from all of the atoms in the crystal constructively interfere, (as viewed in the far field). Note: If the crystal is small, the far field can be just a meter or so away. You can also create far field conditions by observing the pattern with a lens, with detector or paper in the focal plane of the lens, even for very short distances. (This "trick" is used in diffraction grating spectrometers, where with a 2" size diffraction grating, the far field would otherwise be a couple hundred meters away).

Edit: Meanwhile I can see where the OP @Dario56 is puzzled by what the textbooks present=I also found most of them to not be real clear on this Bragg diffraction subject, and I had to figure it out for myself...

Basically, here is what I don't understand. Diffraction pattern is caused by the interference of elastically scattered EM waves. From the standpoint of classical physics, elastic scattering is explained by the Thompson scattering model. In that model, oscillating electric field in the EM waves causes electrons in the crystal to oscillate (accelerated motion) and thus emit EM waves. We refer to these waves as scattered. However, electrons scatter EM waves in all directions.

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

Maybe I should look at the XRD from the real, quantum mechanical viewpoint. However, I haven't found many good sources online. Do you know some good textbooks?

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In the far field, even though the rays converge to the same point, if you draw a diagram to compute the path difference distance, for all practical purposes, when you determine the path distance as measured at the crystal, the two rays leave the crystal parallel =for all practical purposes= even though they converge on the same point in the far field.

Meanwhile you can create a far field condition optically with a lens and a piece of paper in the focal plane. Parallel rays incident on two different parts of the lens( =all parallel rays) at angle ## \theta ## will converge at location ## x=f \theta ## in the focal plane of the lens, and their optical path distances are all the same= They constructively interfere.

hutchphd
Dario56
the two rays leave the crystal parallel =for all practical purposes= even though they converge on the same point in the far field.
What do you mean by all practical purposes?

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What do you mean by all practical purposes?
The path distance difference for two sources separated by distance ## d ## is given by ## \Delta=d \sin{\theta} ##. You could compute the exact path distance difference by using a triangle (with ## \theta_1 ## and ## \theta_2 ## not perfectly parallel, but almost the same angle)=say the rays converged to a point at distance s=2 meters, and you would get an answer that is ## d \sin{\theta} ## "for all practical purposes"=the parallel ray assumption used to compute ## \Delta=d \sin{\theta} ## is a valid one.

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Basically, here is what I don't understand. Diffraction pattern is caused by the interference of elastically scattered EM waves. From the standpoint of classical physics, elastic scattering is explained by the Thompson scattering model. In that model, oscillating electric field in the EM waves causes electrons in the crystal to oscillate (accelerated motion) and thus emit EM waves. We refer to these waves as scattered. However, electrons scatter EM waves in all directions.

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

Maybe I should look at the XRD from the real, quantum mechanical viewpoint. However, I haven't found many good sources online. Do you know some good textbooks?
See https://www.physicsforums.com/threa...tive-interference.1015365/page-2#post-6837237
I think you might find the above discussion on interference, both constructive and destructive, of some interest. The subject of interference can be a real puzzle when you first encounter it, and the OP asked a very good question.
Meanwhile, in many cases, diffraction theory is derived mostly for the far field=that allows the parallel ray assumption. The near field patterns can also be computed, but geometrically, things get rather complicated in the near field calculations.

hutchphd and Lord Jestocost
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We refer to these waves as scattered. However, electrons scatter EM waves in all directions.
From “Elements of X-ray Diffraction, Second Edition” by B. D. Cullity:

To sum up, diffraction is essentially a scattering phenomenon in which a large number of atoms cooperate. Since the atoms are arranged periodically on a lattice, the rays scattered by them have definite phase relations between them; these phase relations are such that destructive interference occurs in most directions of scattering, but in a few directions constructive interference takes place and diffracted beams are formed. The two essentials are a wave motion capable of interference (X-rays) and a set of periodically arranged scattering centers (the atoms of a crystal).

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Hi @Dario56. Given the other posts, you might have resolved this by now But, in case not, and since you specifically asked about what I said, can I add this...

If scattering happens in random directions,
‘Random’ is a bit misleading. There is a well defined angular distribution of intensity (Thomson scattering).

how can we only consider one direction? EM waves from different directions can also interfere which will affect the diffraction pattern.
We consider all directions. It turns out that destructive interference will occur in most directions. But in one (or a few) specific direction(s), constructive interference will occur.

Although it’s not the same as Bragg scattering, it might be worth considering how a diffraction grating works: each slit radiates in all forward directions. But constructive interference, producing the principle maxima, only occurs in a few specific directions.

Charles Link and hutchphd
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It is interesting how the phases of the sinusoids comes into play when they are added, so that the result when two waves are summed is not simply the sum of the two amplitudes. The result is that we get interference patterns. In addition, the energy density is proportional to the second power of the electric field amplitude. Somehow, this intensity ## I=nE^2 ## with a second power dependence, (simplified optics units), where ## n ## is the index of refraction, is just what is needed for energy conservation in the interference pattern.

Lord Jestocost
Dario56
The path distance difference for two sources separated by distance ## d ## is given by ## \Delta=d \sin{\theta} ##. You could compute the exact path distance difference by using a triangle (with ## \theta_1 ## and ## \theta_2 ## not perfectly parallel, but almost the same angle)=say the rays converged to a point at distance s=2 meters, and you would get an answer that is ## d \sin{\theta} ## "for all practical purposes"=the parallel ray assumption used to compute ## \Delta=d \sin{\theta} ## is a valid one.
I remember this from the Young's experiment where the path difference between rays can be approximated well with ##dsin\theta## since slits are very close compared to the distance from the screen . Therefore, I think I know what you are referring to.

Dario56
Hi @Dario56. Given the other posts, you might have resolved this by now But, in case not, and since you specifically asked about what I said, can I add this...

‘Random’ is a bit misleading. There is a well defined angular distribution of intensity (Thomson scattering).

We consider all directions. It turns out that destructive interference will occur in most directions. But in one (or a few) specific direction(s), constructive interference will occur.

Although it’s not the same as Bragg scattering, it might be worth considering how a diffraction grating works: each slit radiates in all forward directions. But constructive interference, producing the principle maxima, only occurs in a few specific directions.
Now when you mentioned Bragg scattering and diffraction grating (or single slit interference) difference, it helped me to address an important question.

Description of the wave phenomena ,like interference and diffraction, is usually first represented by different light experiments such as Young's experiment, single slit interference or diffraction grating. In these experiments, diffraction is explained from the perspective of Huygens principle. Diffraction happens because slit acts as a source of the wave provided that the wavelength is of the similar size to the slit opening.

However, when learning about the XRD, diffraction is explained as a scattering phenomenon. Looking it from the perspective of classical physics, electric and magnetic fields in the EM wave affect the electron motion causing it to oscillate. Such motion is accelerated which causes emission of the EM waves which we call scattered waves. Propagation direction of these waves is random as electrons emit them in all directions.

I'm not sure that I see the connection between these two perspectives. Mechanism by which diffraction happens seems to be very different in sense that in the second case we don't consider any slits, there is a interaction of the EM waves with electrons and scattered radiation propagates in random directions after interaction with the electrons.

Basically, in the first case, slits affect the waves. In the second case, interaction with the electrons affect the waves.

Also, the way they affect them is very different. 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.

How can randomly scattered EM waves give a non-random and constant in time interference pattern is something I don't understand.

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The reason behind the interference pattern is the spatially dependent phase term ## \phi ## in the sinusoids from the sources, i.e. ## E(x,t)=E_o \cos(\omega t +\phi(x)) ##.
It is interesting also that the electric field amplitudes don't simply add in a linear fashion, i.e. ## |E_{o \,total}| \neq |E_{o1}|+|E_{o2}| ##, even though the electric fields add in a linear manner, i.e. ## E_{total}(x,t)=E_{o \, total} \cos(\omega t + \phi_{total})=E_{1o} \cos(\omega t+\phi_1)+E_{2o} \cos(\omega t+\phi_2) ##, where trigonometric identities are used to compute ## E_{total}(x,t) ##.
e.g. ## a \cos(\omega t)+b \sin(\omega t)=\sqrt{a^2+b^2} \cos(\omega -\phi) ## where ## \phi=\arctan(b/a) ##.
(also ## \cos(\omega t +\phi)=\cos(\omega t) \cos(\phi)-\sin(\omega t) \sin(\phi) ##, etc.)
The result of all of this is a spatially dependent interference pattern.
(Note ## x ## above is a 3-D vector ## \vec{x} ##. I simplified the notation for ease in typing. also for ## E=\vec{E} ##)

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It's much safer to use
$$\phi=\text{sign} \, b \arccos(a/\sqrt{a^2+b^2})$$
than the sloppy arctan formula. There's a reason, why most useful programming languages introduce an extra function atan2 ;-).

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Now when you mentioned Bragg scattering and diffraction grating (or single slit interference) difference, it helped me to address an important question.

Description of the wave phenomena ,like interference and diffraction, is usually first represented by different light experiments such as Young's experiment, single slit interference or diffraction grating. In these experiments, diffraction is explained from the perspective of Huygens principle. Diffraction happens because slit acts as a source of the wave provided that the wavelength is of the similar size to the slit opening.

However, when learning about the XRD, diffraction is explained as a scattering phenomenon. Looking it from the perspective of classical physics, electric and magnetic fields in the EM wave affect the electron motion causing it to oscillate. Such motion is accelerated which causes emission of the EM waves which we call scattered waves. Propagation direction of these waves is random as electrons emit them in all directions.

I'm not sure that I see the connection between these two perspectives. Mechanism by which diffraction happens seems to be very different in sense that in the second case we don't consider any slits, there is a interaction of the EM waves with electrons and scattered radiation propagates in random directions after interaction with the electrons.

Basically, in the first case, slits affect the waves. In the second case, interaction with the electrons affect the waves.

Also, the way they affect them is very different. 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.

How can randomly scattered EM waves give a non-random and constant in time interference pattern is something I don't understand.
One other comment is a crystal is not simply a bunch of atoms randomly spaced. Instead the spacing is very precise and the structure is uniform throughout the whole crystal. It is basically acting as a 3-D diffraction grating.

The electric field from each atom can be computed and it propagates radially/spherically outward. With the interference pattern, all of the electric fields with their sinusoids with the phases need to be summed to determine the ## E ## field at a given location. What results with this computation is not something spherically symmetric, but rather an interference pattern determined by the crystal structure.

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How can randomly scattered EM waves give a non-random and constant in time interference pattern is something I don't understand.
In some sense, they aren't really scattered randomly. The wavefront is scattered in all directions, but only in some directions is the intensity high enough for there to be a likely chance of detecting a photon within some timeframe. Ramp up the source intensity a billion times and you'll see plenty of x-rays in all directions, just a whole lot more in the direction in which there is substantial constructive interference. The randomness is in finding a photon, not whether some part of the wavefront is scattered in some direction.

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.

Given some distribution of charges and an incoming wavefront, there is no randomness in how that wave will behave when it interacts with the charges. The randomness is in where and when a photon is detected, not in how the wavefronts behave and evolve, and it is this behavior and evolution of a wavefront during and after interaction that gives rise to the diffraction patterns.

Dario56
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But you don't do experiments with single X-ray or ##\gamma## photons here but with coherent states (aka classical em. waves), and there you have interference from the "Huygens wavelets" scattered from the many atoms forming the lattice of the crystal.

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How can randomly scattered EM waves give a non-random and constant in time interference pattern is something I don't understand.
The X-rays don't have a problem doing this.
Your (our) internal understanding of the process is a hodge-podge of classical, semiclassical and quantum mechanical descriptions of the process. Even the phrase "path of the photon" becomes a minefield upon examination.
My recommendation for this is to consider only the steady-state classical solution for the crystal as a 3D diffraction grating. Then one obtains solutions far from crystal which match the boundary condition of an incoming plane wave. The simplest approximation is Huyghens principal using each atom as an identical isotropic scattering center, and thinking of the steady state solution.
The rest of the issues do not really matter: this simple model at an arbitrary time "snapshot" will answer the fundamental physics questions you ask and. the particular issues of temporal and spatial coherence that you find vexing are taken care of

Steve4Physics, Charles Link, Lord Jestocost and 1 other person
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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.

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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.

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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.

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vanhees71 and tech99