Why Is the Length of the Reciprocal Lattice Vector Equal to 2π/d_hkl?

[Nikitin]

Hi. This is a very simple and stupid question: Why is the length of the reciprocal lattice vector $G_{hkl}$ equal to $2 \pi / d_{hkl}$, where $d_{hkl}$ is the distance between the $(hkl)$ planes. Just like the length of the wave vector $k$ equals $2 \pi / \lambda$

I remember that you get $G_{hkl}$ by fourier-transforming the real lattice in some way or the other, but I fail to see the big picture. Can somebody explain to me a bit? Charles Kittel's book is almost impossible to read for a beginner.

 

[M Quack]

d_hkl is a crutch used in Bragg's law for people that feel uncomfortable in reciprocal space. A real solid state physicist is at home in reciprocal space and tries to stay away from "real" space and the strange people over there. Communication via the interwebs is kinda tolerated, though.

All diffraction physics is essentially taking place in reciprocal space only. Therefore it makes more sense to write Bragg's law as

| G_hkl | = 2k sin(theta) = 4 pi/lambda sin(theta)

The complete diffraction condition is G_hkl = k_in - k_out, where all 3 variables are vectors and k_in and k_out describe the wave vectors is the incident and diffracted beams. You can easily derive above equation from this.

The proper way is to construct the reciprocal lattice by a* = 2 pi b x c /(a . (b x c)) etc. and G = h a* + k b* + l c* as you have seen in Kittel's book.

BTW I very much prefer Ashcroft and Mermin.
.

 

[Nikitin]

Yeah. Right now I hate real space too :(

Anyway so G_hkl is pretty much defined from the bragg condition in reciprocal space? I think I see more now

 

[M Quack]

No, the reciprocal lattice and G_hkl are defined by the crystal structure.

Bragg's difraction and crystallography in general are ways to observe the reciprocal lattice. From these measurements the real space crystal structure can then be obtained - the more complex the crystal structure, the more information you need, i.e. the more Bragg reflections with different hkl you need to measure.

 

[read]

Just a comment. To be precise, the reciprocal lattice, i.e. its basis and any vectors G_hkl in this basis are completely defined by the real lattice basis, and independent on the details of atomic positions (i.e. the crystal structure). I think M Quack first sentence should be modified to "... G_hkl are defined by the crystal metric." Or maybe by Bravias lattice.

 

[M Quack]

I stand corrected.

 

[Nikitin]

By the way, concerning the Bragg condition: Is it derived from

1) Assuming specular reflection from Bragg planes and finding the difference between rays (and hence constructive interference at certain incoming angles).

or

2) Assuming Rayleigh scattering from atoms and finding the paths where we get constructive interference from the spherical waves?

Because condition 2) alone is not equivalent to condition 1). Condition 1 says that incoming and outgoing angles must be the same, but that is not necessary for 2...

And thank you for all the help provided. It is much appreciated.

 

[M Quack]

2) usually assuming Thomson scattering, but I am not completely certain if in the early days a different interpretation was used. After all, it was not clear from the beginning that crystals are periodic lattices of atoms, however evident this seems today.

If you look carefully there is a whole zoo of effects beyond the simple "Bragg"s law" in x-ray diffraction that occur when photon get scattered more than once.

André Authier has complied a very interesting history of the early days of x-ray crystallography.

http://ukcatalogue.oup.com/product/9780199659845.do

 

[Nikitin]

But still, why is assumption (2) by itself not correct? It says here http://en.wikipedia.org/wiki/Bragg's_law that

"In physics, Bragg's law (or "http://en.wikipedia.org/w/index.php?title=%D0%92%D1%83%D0%BB%D1%8C%D1%84_%D0%93%D0%B5%D0%BE%D1%80%D0%B3%D0%B8%D0%B9_%D0%92%D0%B8%D0%BA%D1%82%D0%BE%D1%80%D0%BE%D0%B2%D0%B8%D1%87&action=edit&redlink=1 -Bragg's condition" in postsoviet countries) gives the angles for coherent and incoherent scattering from a crystal lattice. When X-rays are incident on an atom, they make the electronic cloud move as does any electromagnetic wave. The movement of these charges re-radiates waves with the same frequency, blurred slightly due to a variety of effects; this phenomenon is known as Rayleigh scattering (or elastic scattering). The scattered waves can themselves be scattered but this secondary scattering is assumed to be negligible."

So why can't you just calculate the paths of constructive interference for the scattered spherical waves? Why do you need to use the assumption of specular reflection from bragg planes (1) ?

 

[M Quack]

You can and do just that.

The result is that scattering occurs when the scattering vector (momentum transfer) is a reciprocal lattice vector, which can be associated with a plane in the crystal.

So the planes are a result of the calculation, not an a-priori assumption.

 

[Nikitin]

You can and do just that.

The result is that scattering occurs when the scattering vector (momentum transfer) is a reciprocal lattice vector, which can be associated with a plane in the crystal.

So the planes are a result of the calculation, not an a-priori assumption.

But in post #4 here, you say that " setting the detector at the right angle is not the only condition that must be met". https://www.physicsforums.com/threads/solid-state-physics-x-ray-scattering.796525/ In the beginning I calculated $\theta$ using solely assumption 1, but that did not work out for me too well. I had to use the condition of specular reflection too, i.e. incoming angle = reflected angle.

As for the scattering vector: But laue's condition says that the scattering vector must equal a reciprocal lattice vector, and each point in reciprocal space represents a bragg plane. So in both bragg and laue conditions you assume reflection off bragg planes, as far as I can see?

 

[M Quack]

There is no contradiction here. You have to remember that in Rayleigh scattering, the scattered radiation is in phase with the incident electric field's oscillation at the position of the scatterer. Therefore the phase of atoms within a plane depends on the angle between the plane's normal (i.e. reciprocal lattice vector) and the direction of the incident beam.

The scattering vector=momentum transfer vector has both a magnitude and a direction, as does a reciprocal lattice vector. Since we are talking about elastic scattering, the incident and scattered wave vectors both have the same magnitude, therefore they must lie symmetric wrt the reciprocal lattice vector - which is the same as saying you have specular reflection from the plane.

The Laue condition comes directly out of Rayleigh scattering calculations. The interpretation that this means specular reflections from planes in the crystal comes later.