# Reciprocal space

1. Aug 5, 2008

### Foyevtsov

Hello all,
I don't understand some things concerning reciprocal space. I know how it appears from quantum mechanics (it comes from gas of free electrons model, then applying Fermi-statistics for it and solve Heisenberg equation and get some kx,y,z which have to be integer. After that we call the space of these k-values as k-space, or reciprocal [Ashkroft, Mermin for instance]).
But I also know that there is another reciprocal space which used for crystallography (i.e. for X-ray). This one comes from Laue equation.
Both of these spaces are linked somehow, nevertheless it came from different time and different (a bit) physics.
Excuse me for the my more general question which cannot be answered with a few words, but maybe someone experienced can understand my lack of knowledge and guide me. Thanks for your attention.

2. Aug 5, 2008

### malawi_glenn

Well the reciprocal lattice in Crystallography is also the momentum representation of the lattice. The reciprocal lattice's basis vectors are orthogonal to all the real lattice vectors, so you can say that the reciprocal lattice is the 'orthogonal' lattice to the space lattice. And why this is important? Well you can first think of plane waves, how do they behave? The wave fronts are orthogonal to the velocity (momentum) vector. And that waves can interfer destructivly and constructivly. Another way to see it is to look at the scattering amplitude in position space, and you'll see from mathematics why the concept of reciprocal space is important.

Try to make your question more explicit, what exactly are you don't understanding? Have you worked out the mathematics?

3. Aug 5, 2008

### Foyevtsov

OK, I'll try.
I am not a mathematician, but physicist. So, probably some mathematical views are not so close to my view.
Why the reciprocal space is so convenient for describing of x-ray diffraction from physicist point of view?
(Let say here http://capsicum.me.utexas.edu/ChE386K/html/laue_recip.htm). They just introduce vector R which would be a vector of reciprocal space, but is it not enough to know just Bragg's law (or Laue) to explain diffraction?
There is must be something reason for introduction this R vector (which might simplify our life)
(Maybe my question is naive, but be patient, thanks)

Last edited by a moderator: Apr 23, 2017
4. Aug 5, 2008

### malawi_glenn

Apperently, reciprocal space is in physics books. Never forget that Math is the language of physics, you can NEVER get away with the math when you are doing physics!

Yes, Braggs law is enought, but when you come to other parts of Solid State physics, like phonons (crystal vibration), electrons in solids, energy bands etc. You'll use the reciprocal space all the time. And things like Brillouin zones will be your 'new tools'.

The thing is that some parts is easier to handle in a certain 'space'. Like when you do signal analysis, you can switch between time and frequence domain by doing a fourier transform. And the same in Quantum mechanics, you can go from position space to momentum space by fourier transforms.

So with the reciprocal space you go from position space to momentum space (k-space), and since you'll also treat phonons and electrons as wave's - the reciprocal space will be extremley useful for you. That's why we use the Reciprocal space in Solid State physics. If you have not encounter these things before, then either take a solid state course or continue reading (and doing all the steps!) in Ashkroft &Mermin

5. Aug 7, 2008

### Foyevtsov

That is exactly what I am doing.
Is the reciprocal space (actually k-space) which comes from the free electron gas model (almost in the begining of Ashkroft&Mermin) the same as reciprocal space from diffraction of x-ray (or newtrons, doesn't matter), namely from Laue equations?
It is a bit confusing for me.
Thanks.

6. Aug 7, 2008

### malawi_glenn

There is a difference between the reciprocal space (momentum space)(Where you'll have the Fermi Sphere and all that stuff) and reciprocal LATTICE space.

In the free electron model, you don't even have a lattice. In the 'Nearly free electron mode', you will take into account the lattice.

http://en.wikipedia.org/wiki/Nearly-free_electron_model

And here, you will start using the reciprocal lattice space as you had in the crystal diffraction.

7. Aug 8, 2008

### Modey3

Foyevtsov,

This is a good question! The reciprocal lattice exists in reciprocal space. Momentum space and reciprocal space only have a 1:1 correspondence for the free-electron model . The 1st Brillouin Zone (BZ) is centered around the gamma-point on the reciprocal lattice (i.e. the {000} reciprocal lattice point). Consider the a simple-cubic reciprocal lattice. The BZ boundary is halfway between the gamma-point and the adjacent reciprocal lattice point.

For any sort of diffraction to occur whether it be x-rays, electron, or neutrons, the Laue Conditions have to be met: ko - k = n G. Which also gives: k$$\cdot$$G(unit-vector) = 1/2 G(magnitude). When the 1st BZ is filled to the BZ-boundary, as in the case of a insulator, the Fermi-Surface will intersect the BZ-boundary. The wave-vectors ending on or near the BZ boundary get diffracted because the electrons interact strongly with the lattice. The possible directions that the electron gets diffracted in are given by the Laue Conditions, which are several possible directions in the BZ. At any rate, the diffracted electron becomes a standing wave (this is easy to prove for the 1-D case!). The overall wavefuction of the electron is given by the superposition (linear combination) of the wavefunctions describing the incident and diffracted waves. This superposition gives many different solutions to the SchrÃ¶dinger's Equation - hence different energy eigenvalues, which implies the formation of energy gaps.

If you want to know more read up on the electron diffraction used in a TEM (transmission election microscope). I hope this helps connect the two separate view points of reciprocal space.

modey3

8. Aug 11, 2008

### mesogen

Hello,

This has me curious about selection rules. I never quite got the physical origins of diffraction selection rules. I always just understood it as "that reflection doesn't meet the condition." Is the physical origin of a forbidden reflection simply a lack of constructive interference for that lattice plane at that particular angle?

9. Aug 11, 2008

### malawi_glenn

Yes, destructive interference. You can also see this if you do the scattering amplitude with a periodic potential -> Leads to Bragg's rule

10. Aug 11, 2008

### Modey3

mesogen,

As a Materials Scientist, I can say that the selection rules have a major physical significance, but interference does take place in "off-Bragg" conditions for a small crystal, which makes the situation more interesting! Generally, the Laue Conditions must be satisfied so that diffraction exists, but to use the Laue Conditions you need a reciprocal lattice. Lets take the FCC real space lattice for example. It turns out that the reciprocal lattice is BCC, but why? The selections rules for FCC are all even or all odd. So what does this give us? {111}, {200}, {222} etc... These are reciprocal space vectors, which give fully constructive interference when considering the amplitude at a point away from the lattice. The way to understand this is to add up the amplitude at a point from all the individual scatterers in the lattice. In the summation, you get a ko - k term.

modey3