Some questions about reciprocal lattice vectors

[patric44]

Homework Statement

i had an assignment with couple questions related to reciprocal lattice vectors :
1- create the reciprocal lattice vectors for both square and rectangular lattice in 2D ?
(i don't understand this question very much is it asking for the derivation for b1,b2,b3 or what !)

2- prove that the reciprocal of the reciprocal vector gives the real vector or the real lattice ?
3- construct 1st and 2nd brillion zones ?

Relevant Equations

R = n1a1+n2a2+n3a3

hi guys
our solid state physics professor introduced to us this new concept of reciprocal lattice , and its vectors in k space ( i am still an undergrad)
i find these concepts some how hard to visualize , i mean i don't really understand the k vector of the wave it elf and what it represents, not to mention of constructing a space for it " the k-space " , what that reciprocal space tell me about the real lattice and why do i ever need it ?
if some some one has some notes or a free book that could help me in that i will really appreciate it

he also give us some assignments :

1- create the reciprocal lattice vectors for both square and rectangular lattice in 2D ?
(i don't understand this question very much , is it asking for the derivation for b1,b2,b3 or what !)
where do i start this derivation from , could i just represent the real lattice with some matrix maybe and do some linear transformation to it (stretch it or rotate it ...) to turn it into a reciprocal lattice vectors .
- can i take the forier transform for δ(x-a1),δ(x-a2),δ(x-a3) to transfer it in another domain , but that just gave me e^iaω/√2π ?

2- prove that the reciprocal of the reciprocal vector gives the real vector or the real lattice ? could i take the forir tansform for the forier transform!
to bring me back to the real space .

3- construct 1st and 2nd brillion zones ? is that asking for the 2d drawing for it ?

i would also appreciate if someone has a program to visualize crystal structure and the reciprocal lattice , brillion zones ...
thanks

 

[jambaugh]

The question is straightforward once you understand what a reciprocal lattice is. That, I believe, is what is unclear to you. If you look at the "Related Threads for: (this thread)" you will find many posts on this topic.

 

[jambaugh]

I apologize if my first post was a bit dismissive. Here (toward the bottom) is a good summary of the linear algebra and geometry of the reciprocal lattice.
Reciprocal Space at dictionary.iurc.org

 

[jambaugh]

I have some time now to answer in more detail.

For a Bravais lattice all of the periodicities must be an integer multiple of a given (typically oblique) set of basis vectors defining minimal lattice translations.

Consider the one dimensional case of a set of points spaced $a$ apart. Any wave propagating along this 1-dimensional space must*[see footnote] have a wave-length that is an integer fraction of this spacing and thus a spatial frequency (cycles per unit of length) that is an integer multiple of $\frac{1}{a}$. This frequency (wave number) lattice is in a distinct space with reciprocal distance units of measure. In this space the set of valid frequencies are thus on a 1-dim lattice with period $\frac{1}{a}$.

In two dimensions you have two basis vectors $\mathbf{a}_1,\mathbf{a}_2$ and for waves to propagate they must* "fit" between the lines spanned by either of these basis vectors and any integer combination thereof. We can define these using a reciprocal basis, $\mathbf{b}_1,\mathbf{b}_2$ residing again in a dual, reciprocal, or frequency space. In this space again the measuring units are reciprocal lengths (cycles per unit distance).

The reciprocal basis vector $\mathbf{b}_1$ is chosen to be orthogonal to $\mathbf{a}_2$ so that it corresponds to a wave moving orthogonal to that direction and must have a component in the $\mathbf{a}_1$ direction of $1/\lVert \mathbf{a}_1\rVert$ so that its wave-length component will fit within the spacing in that direction. In short:
$$\mathbf{b}_1\bullet \mathbf{a}_1 = 1, \mathbf{b}_1\bullet \mathbf{a}_2 = 0$$
Waves with integer multiples ($n$) of this vector frequency as a component will likewise fit an integer number of wavelength components in that directions spacing since it has component in that direction of $\lVert \mathbf{a}_1\rVert/n$.

Similarly you define $\mathbf{b}_2$ such that:
$$\mathbf{b}_2\bullet \mathbf{a}_1 = 0, \mathbf{b}_2\bullet \mathbf{a}_2 = 1$$
Any integer linear combination of these reciprocal basis vectors will correspond to a vector spatial frequency (a.k.a. wave vector) with components of wavelength in each of the original lattices basis directions of the spacing over the corresponding integer. It is "in synch" and in essence a "vector harmonic" of each primitive cell (sort of) of the Bravais lattice.

In three (and theoretically higher) dimensions one constructs the reciprocal (dual) basis to be such that each reciprocal basis vector dots to 1 with the corresponding lattice basis vector and is orthogonal to all the others.

Finally the Brillouin zone is the primitive cell of the reciprocal lattice. It is the locus of all points closer to a central lattice point (its center) than any other lattice point.

*Footnote: I say "must have" here but that is not "must have to propagate" but rather more the opposite, that is what it "must have" in order to maximally interact with the lattice, e.g. to be diffracted. In the 1-dimensional case this is being diffracted backward i.e. reflected. In the general setting I believe one needs to get into the time evolution and shape of the periodic field to describe how strongly it may diffract in a given direction but the crucial point is that diffraction corresponds to exchanges of momentum/wave number only corresponding to those sites on the reciprocal lattice. Basically the (infinite) crystal and (infinite duration) wave do not interact locally in space but rather locally in reciprocal space.

To appreciate this you really do need to delve deeply into the multivariable Fourier transforms including how spreading out the domain of interaction around each lattice point limits the effects of reciprocal lattice points farther from the origin and dually how limiting the extent of the lattice (finite instead of extending to infinity in all directions) has a corresponding broadening effect for the points in the reciprocal lattice. In these realistic cases the interactions are not quite local in reciprocal space nor quite non-local in position space.