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Reciprocal lattice vectors

  1. Sep 20, 2007 #1

    malawi_glenn

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    1. The problem statement, all variables and given/known data

    "What does a reciprocal lattice vector represent in the real lattice?"


    3. The attempt at a solution

    The answer to that one is that the reciprocal lattice represent all possible k-values for the incoming radiation to be contained in the real lattice. Hence a reciprocal vector represent one of these possible k-values.

    Am I right?
     
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  3. Sep 20, 2007 #2

    Astronuc

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    Here are some discussion of the reciprocal lattice and vectors.

    http://www.matter.org.uk/diffraction/geometry/lattice_vectors.htm

    http://en.wikipedia.org/wiki/Reciprocal_lattice

    http://www.chembio.uoguelph.ca/educmat/chm729/recip/6reci.htm

    http://ocw.mit.edu/NR/rdonlyres/Ear...60155B-B411-4346-ADF3-C89439D43852/0/lec8.pdf

    See pages 6,7 in the last one.

     
  4. Sep 20, 2007 #3
    If you're familiar with vector spaces and their duals, let [tex]\mathbf{e}_\mu[/tex] be the basis vectors for V, and [tex]\mathbf{\theta}^\mu[/tex] be the (co-)basis for V*, such that [tex]\mathbf{e}_\mu \mathbf{\theta}^\nu = \delta_\mu^\nu.[/tex] Let there be a metric, [tex]g_{\mu\nu}.[/tex] The reciprocal vectors are basis covectors [tex]\mathbf{\theta}^\mu[/tex] turned into vectors by the inverse metric.
     
  5. Sep 20, 2007 #4

    malawi_glenn

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    Thanx, I already checked them out. And also I have two good books.

    The things is that the last one, p 6,7 has the Crystallographical definition of reciprocal space; Hence the connection between planes, distance of planes and points in reciprocal space. But the definition in my course is the physics one: i.e you have multiplied with 2pi.

    So I wonder if the points in reciprocal space represent allowed k-values for waves that can be Bragg-difracted in the real-lattice. And if the reciprocal lattice vector represent a certain k-value.
     
  6. Sep 20, 2007 #5

    malawi_glenn

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    Thanx , but not what I looked for ;)
     
  7. Sep 20, 2007 #6
    I know it sounds a bit esoteric, but it's really just geometry. I certainly didn't understand reciprocal lattice vectors until I made that connection -- but maybe I'm just a mathmo trying to pretend to be a physicist ;-)
     
  8. Sep 20, 2007 #7

    malawi_glenn

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    The geometrical connetion I am fine with, but I am after the physical intepretation of the reciprocal lattice vector.

    I know that the answer to this: "So I wonder if the points in reciprocal space represent allowed k-values for waves that can be Bragg-difracted in the real-lattice. And if the reciprocal lattice vector represent a certain k-value."

    Is that the allowed k-vales are contained in the 1BZ, so I now wonder what points in reciporcal lattice space and reciprocal lattice vectors represents. If there are any physical inteprenation of those, or just geometrical.
     
  9. Sep 20, 2007 #8
    If we know that reciprocal vectors are just covectors turned into vectors, then they can represent anything from the covector space. As you've noted, the wave-vector is one such thing, as is momentum (same thing as wave-vector really).
     
  10. Sep 20, 2007 #9

    Kurdt

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    Think about the change in wavevector of radiation incident upon a lattice.
     
  11. Sep 20, 2007 #10

    malawi_glenn

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    You are talking about Bragg condition right?
     
  12. Sep 20, 2007 #11

    Kurdt

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    I am indeed. Now to help you a little more, consider a 1D lattice with an incident wavevector k and a reflected wavevector k'. Take the difference of the wavevectors and see if they look familiar.
     
  13. Sep 20, 2007 #12

    malawi_glenn

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    I know that delta k is equal to reciprocal lattice vector.


    I could write an essay about reciprocal space, diffraction and so on, but I still dont know what to answer to the original question.. =(
     
  14. Sep 20, 2007 #13

    Kurdt

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    If delta k is equal to the reciprocal lattice vector then I would think you would have your answer. The reciprocal lattice of the periodic system is equal to the change in wavevector of the radiation. Or in other words it represents the change in wavevector of the radiation.
     
  15. Sep 20, 2007 #14

    malawi_glenn

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    okay, I go with that answer =) thanx
     
  16. Sep 20, 2007 #15

    Kurdt

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    It is a bit of a vague question but to me I couldn't imagine what else it would be. If its for a homework assignment you could always approach the tutor and ask if thats what they are looking for before the deadline and I'm sure they will be helpful.
     
  17. Sep 20, 2007 #16

    malawi_glenn

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    yes, we have many of those questions. Hate this course=P

    Will ask the guy tomorrow. Thanx
     
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