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Scattering amplitude of diffracted beam by a crystal

  1. Jan 4, 2013 #1
    1. The problem statement, all variables and given/known data
    The scattering amplidtude, F, of a, by a crystal, diffracted beam is defined to be:

    (1) [itex]F=\sum{_\vec{G}}\int n_{\vec{G}}e^{i(\vec{G}-\delta\vec{k})\cdot \vec{r}}\mathrm{d}V[/itex]

    The integral is over all the volumeelements, dV, of the crystal.
    [itex]n_{\vec{G}}[/itex] is the local electron concentration of the crystal in dV, [itex]\vec{G}[/itex] is a reciprocal lattice vector and the sum is over the set of all the reciprocal lattice vectors. [itex]\vec{r}[/itex] is the position vector of dV
    further [itex]-\delta\vec{k}=\vec{k}-\vec{k'}[/itex] where [itex]\vec{k}[/itex] is the wave vector of the incomming beam and [itex]\vec{k'}[/itex] is the wave vector of the outgoing scattered beam.

    The beam could be
    fotons, electrons neutrons etc. It is descibed as a plane wave.

    It is easy to see that [itex]F=n_{\vec{G}}V[/itex] when [itex]\delta \vec{k}=\vec{G}[/itex]

    But how can it be shown that F is negligibly small when [itex]\delta \vec{k}[/itex] differs sginificantly from any [itex]\vec{G}[/itex]?


    2. Relevant equations

    I thought some expression for the summation could be used.

    Maybe

    (2) [itex]\sum{_m=0}^{M-1}x^m=\frac{1-x^{M}}{1-x}[/itex]




    3. The attempt at a solution

    I cant find a away to make (1) fit in to (2). And if i just look at (1) i really am lost.

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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