Bragg Diffraction angle definitions

[Cruikshank]

I find problems on Bragg diffraction frustrating. I can't tell how the angles are defined, nor the "planes" in the crystal--they look arbitrary. Why can't I just draw a slash through the crystal at any angle I want and get diffraction off the angles I hit with the slash? Is it just that the intensity drops as the spacing of atoms increases? And how can I get more than one "order" of diffraction from the same crystal and wavelength? Isn't the reflection angle completely determined by the "Bragg angle" that defines a slice of crystal? It seems as if given a crystal, one could say, "these are the only possible angles for diffraction. If you use the right wavelength, you can see it. Otherwise, no diffraction anywhere."

To repeat, how are the angles defined? Wikipedia is no help, I already know all of what they say in the entries. I tutor physics, and teach single slit, double slit, diffraction gratings and so forth every year. But Bragg diffraction defeats me. Suggestions?

 

[nickjer]

You can't just draw a slash through a crystal at any angle and define it as a plane. There are defined planes in the crystal written as Miller indices. The simplest being the (100) plane for a cubic crystal as seen on the wikipedia page for Bragg diffraction. Please look up Miller index on wikipedia for more info about these planes. There are different symmetries for different crystal structures leading to different planes.

The reflection angle is equal to the angle of incidence. This is simple physics you learned from mirrors. But certain angles will have constructive interference seen as peaks, which is what that equation describes on the wikipedia page.

How do you get more than one "order" for a single wavelength and crystal?
EDIT: Changed this answer here...
If you alter the angle of the incident ray of light, the path length difference between the reflected light from the surface and the reflected light from the first plane of atoms will change. The total path length difference is defined as $2d\sin(\theta)$. So for first order constructive interference you will need to find the angle of incidence that causes the path length difference to match the length of the wavelength of light. And for 2nd order constructive interference you will need to increase the angle of incidence until two wavelengths can exist in the new path length difference. You would continue to do this until the angle of incidence reaches 90 degrees, meaning you can't see any more higher orders.

 

[Cruikshank]

Thank you. I don't know the Miller indices, I haven't studied crystallography. I'm not actually trying to get that deep into it. I just want definitions of the angles. The book I am using, Bransden and Joachain, for example, defines alpha, theta, and theta(Bragg), but does not make it clear how they are related. When a beam of light is approaching a crystal, which angle is the angle of incidence? When the beam leaves, is that "simple reflection as in mirrors" or is it diffraction? I know how diffraction gratings work, you hit them with light at normal incidence, and you get constructive interference at a series of angles (which I know how to calculate and visualize) and that is NOT angle of incidence = angle of reflection.

I get that there are certain planes at certain angles (and what are THOSE called? Alpha? Something else?) which intersect a lot of atoms in the crystal: when the tangent of the angle between the plane and the surface of the crystal equals 1, or 1/2, or 1/3, or 1/n. Which one or ones are producing the Bragg diffraction? If I choose an incident angle, which planes get involved in the diffraction (reflection?) And given the incident angle AND the choice of a plane, how is the exit angle defined? With reference to what perpendicular?

I believe I could answer the questions in the textbook perfectly well if I could just find out exactly what they want to know and how they are defining their terms. I can derive the geometry, and did (I think) for the case of normal incidence, but then the next problem stated that the incoming beam was "incident at the Bragg angle" and I have absolutely no idea what THAT is supposed to mean.

An angle requires two intersecting rays or lines for definition. I need to know which lines are being used to define the angles, before I can hope to understand the problems. And yes, I tutor students in thin film interference too, all the time, and I understand it well. But I can't solve a puzzle with half the pieces missing.

Does anyone know a textbook that gives a decent explanation of this with clear definitions that does not require me to become a crystallographer just to get three chapter one quantum mechanics problems done? (Sorry, I'm getting frustrated. I'm smart, not psychic.)

 

[nickjer]

You have a crystal and you can shine a beam of light on it at any angle of incidence you want. When the beam leaves it will reflect just like a mirror off of the plane of atoms. Note of warning, this isn't really what happens. This is just a simplified model used to explain Bragg's law. In reality the atoms will radiate and have constructive interference following Bragg's law (but I won't discuss this).

So you want to understand where the diffraction comes from. Well experimentalists will usually take the crystal and crush it up into a powder. That way there will be tiny pieces of crystals all oriented in different directions so that all the atomic planes will be parallel to the surface. This is much more efficient than re-orientating the crystal to expose the different planes. Now they are all exposed simultaneously in the many tiny crystals.

Now you shine a beam of x-rays with a single wavelength at this powder. You will then get a diffraction pattern with peaks of light at different angles, similar to a diffraction grating. The location of the peaks determine the crystal structure. This method is called x-ray powder diffraction.

 

[Cruikshank]

I think I have figured it out. I would appreciate it if someone could tell me whether or not this is correct:

Bragg diffraction is like a diffraction grating in that a single beam can reflect in multiple directions. It is unlike a diffraction grating in that these multiple directions are NOT coming off of a single surface, but rather off of different planes through the crystal tilted at different angles.

Bragg diffraction is unlike a diffraction grating in that angle of incidence DOES always equal angle of reflection; however, those angles must be measured from the normal to the *plane which is currently reflecting* rather than the surface of the crystal.

Bragg diffraction is like thin film interference in that adjacent parallel planes create reflections that interfere. It is unlike thin film interference in that there are many different planes tilted at angles relative to each other.

"Bragg angle" is a confusing redundancy. It's just an angle, and not even always referring to the same one.

One beam hitting a crystal reflects at a few angles with specific intensities that are greater if the density of atoms in the reflecting plane is higher; hence the planes which connect nearby atoms, repeating the pattern over short distances, will have brighter reflections than ones at random angles that miss a lot of atoms. So in theory there is some reflection in nearly every direction off of some plane or other in the crystal; it just has a low intensity except when the Bragg condition is satisfied.

If I have interpreted that correctly, the problems I was looking at become easy.