Reciprocal lattice vectors

  • #1
malawi_glenn
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Homework Statement



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


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?
 

Answers and Replies

  • #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 [Broken]

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

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

http://ocw.mit.edu/NR/rdonlyres/Earth--Atmospheric--and-Planetary-Sciences/12-108Fall-2004/7160155B-B411-4346-ADF3-C89439D43852/0/lec8.pdf [Broken]

See pages 6,7 in the last one.

Reciprocal vectors are defined to be perpendicular to two of the three lattice
vectors and with length equal to 1/length of the third vector.

Linear combinations formed from these reciprocal vectors and the
Miller indices are vectors that are in the same direction as the poles to
the corresponding planes. The vector length of this vector is the
reciprocal of the plane spacing.
 
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  • #3
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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.
 
  • #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.
 
  • #5
malawi_glenn
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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.

Thanx , but not what I looked for ;)
 
  • #6
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Thanx , but not what I looked for ;)

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 ;-)
 
  • #7
malawi_glenn
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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 ;-)

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.
 
  • #8
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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.

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).
 
  • #9
Kurdt
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Homework Statement



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

Think about the change in wavevector of radiation incident upon a lattice.
 
  • #10
malawi_glenn
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Think about the change in wavevector of incident radiation upon a lattice.

You are talking about Bragg condition right?
 
  • #11
Kurdt
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You are talking about Bragg condition right?

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.
 
  • #12
malawi_glenn
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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.

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.. =(
 
  • #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.
 
  • #14
malawi_glenn
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okay, I go with that answer =) thanx
 
  • #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.
 
  • #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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