The reciprocal lattice vector represents the change in wavevector of radiation incident upon a real lattice, specifically correlating to allowed k-values for waves that can be Bragg-diffracted. Reciprocal vectors are defined as perpendicular to two of the three lattice vectors, with a length inversely proportional to the third vector's length. The first Brillouin zone (1BZ) contains these allowed k-values, establishing a direct relationship between reciprocal space and the physical properties of the lattice. Understanding this concept is crucial for interpreting diffraction patterns and wave interactions in crystallography.
PREREQUISITESStudents and researchers in physics, materials science, and crystallography who seek to deepen their understanding of reciprocal lattice vectors and their applications in wave diffraction and solid-state phenomena.
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.
genneth said:If you're familiar with vector spaces and their duals, let \mathbf{e}_\mu be the basis vectors for V, and \mathbf{\theta}^\mu be the (co-)basis for V*, such that \mathbf{e}_\mu \mathbf{\theta}^\nu = \delta_\mu^\nu. Let there be a metric, g_{\mu\nu}. The reciprocal vectors are basis covectors \mathbf{\theta}^\mu turned into vectors by the inverse metric.
malawi_glenn said:Thanx , but not what I looked for ;)
genneth said: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 ;-)
malawi_glenn said: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.
malawi_glenn said:Homework Statement
"What does a reciprocal lattice vector represent in the real lattice?"
Kurdt said:Think about the change in wavevector of incident radiation upon a lattice.
malawi_glenn said:You are talking about Bragg condition right?
Kurdt said: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.