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Visualization of a reciprocal lattice

  1. Sep 9, 2007 #1
    I have been learning kittel's solid physics,

    But find it hard to have a firm grab of what a reciprocal lattice is like and cant understand it's relationship with e original lattice.. Is there a picture that draws a lattice n its reciprocal lattice into e same picture? so that i can visualize what a reci latt really is like..

    I also find kittel's fourier analysis of a lattice rather hard and disliked it very much..
    Is there a simpler approach?
    I mean its just a bunch of periodic repetitions cells so what's the meaning of introducing such an complex method?
     
  2. jcsd
  3. Sep 9, 2007 #2
    Try the simplest case of 1-dimensional crystal with the lattice constant a. The crystal consists of atoms at points [itex] \ldots, -2a, -a , 0, a, 2a, \ldots [/itex]. Every physical function [itex] f(x) [/itex] in the lattice (e.g., the electron density) must be periodic with the period a. Examples of such periodic functions are "plane waves" [itex] \exp(iknx) [/itex], where n is integrer [itex] n = \ldots, -1, 0, 1, 2, \ldots [/itex], and [itex] k = 2 \pi/a [/itex]. Now, the important Fourier theorem says that any periodic function can be represented as a linear combination of these plane waves

    [tex] f(x) = \sum_{n=- \infty}^{\infty} A_n \exp(iknx) [/tex]

    with coefficients [itex] A_n [/itex]. The reciprocal lattice is just a method to visualize this Fourier series. One can imagine a real line with points [itex] kn [/itex] placed on it equidistantly: [itex] \ldots, -k, 0, k, 2k, \ldots [/itex] and with the coefficient [itex] A_n [/itex] assigned to each point. This set of points is called "reciprocal lattice". The reciprocal lattice cannot be drawn together with the "direct lattice", because numbers [itex] kn [/itex] are not related to positions in space. They can be (loosely) interpreted as values of momentum.

    The reciprocal lattice for 3-dimensional crystals is just an immediate generalization of these ideas to 3D. Any 3D periodic function can be equivalently represented as a set of numerical coefficients [itex] A_{lmn} [/itex] specified at nodes in the abstract "reciprocal lattice".

    Eugene.
     
  4. Sep 9, 2007 #3
    that was a good explanation.

    thanx, meopemuk..
     
  5. Jan 20, 2008 #4

    LydiaAC

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    Gold Member

    Reciprocal lattice is a mathematical concept which is used to translate the very easy rules used for orthogonal vectors into the working of non-orthogonal vectors.
    In an orthogonal set of vectors, any vector is orthogonal to all other vectors. In a non-orthogonal set of vectors, you can look for a vector which is orthogonal to all but one vector. The vector that you find and the only excluded vector are reciprocals to each other. You can repeat the process for each vector and you will find a completely new set of vectors, with which you can build a reciprocal space.
    Orthogonal vector systems meet with
    "vector n" point "vector m" is Kronecker delta nm
    Non-orthogonal vector systems meet instead with
    "vector n" point "reciprocal vector m" is Kronecker delta nm
    It is easy to find the reciprocal vectors for a bidimensional system. Draw two non-orthogonal vectors in a sheet of paper and call them "a" and "b". Draw the vector which is orthogonal to "a" and call it "reciprocal b". Then, draw the vector which is orthogonal to "b" and call it "reciprocal a". You can find the length of reciprocal vectors obliging them to meet with "a" point "reciprocal a" is one and with "b" point "reciprocal "b" is one.
    Units in reciprocal vectors are inverse that those in original vectors. If your original vectors are in meters, reciprocal vectors are inverse of meter or, if they are meant to represent a spacial frequency, cycles/meter. Christalographers prefer frequencies in radians/meter so they do not normalize their reciprocal vectors to one, but to 2 Pi.
    Formulae for reciprocal vectors in Kittel are only for tridimensional systems. They do not hold for bidimensional, four dimensional or more dimensional systems.
    A reciprocal lattice is a set of points which is found using only integer multiples of reciprocal vectors. Reciprocal lattice inhabits reciprocal space (or Fourier space) but it is not the same of it. The first is a set of points and the other is a full n-D space.
    Reciprocal vectors or reciprocal functions are not used only in Chrystallography. In Quantum Mechanics, Kets are reciprocal of Bras. In Electromagnetics, you need a vector set for electric fields and another set for magnetic fields and these two vector sets are reciprocals to each other.
     
  6. Jun 11, 2011 #5
    You can find the length of reciprocal vectors obliging them to meet with "a" point "reciprocal a" is one and with "b" point "reciprocal "b" is one.

    Can you explain this point a little more please?
     
  7. Jun 11, 2011 #6

    LydiaAC

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    Gold Member

    I think I was not able to write a good sentence in English in 2008.

    You have four vectors, "a", "b", "reciprocal a" and "reciprocal b".

    "Reciprocal a" is orthogonal to "b", "reciprocal b" is orthogonal to "a".

    The shadow that makes "reciprocal a" on "a" (or vice versa) must be 1.

    The shadow that makes "reciprocal b" on "b" (or vice versa) must be 1.

    Mathematically

    "Reciprocal a" dot "b" equal to zero
    "Reciprocal b" dot "a" equal to zero

    "Reciprocal a" dot "a" equal to one
    "Reciprocal b" dot "b" equal to one

    With the first two you can find the direction of the reciprocal vectors. With the last two you can find the magnitude of the reciprocal vectors.
     
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