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Phys 460 Lecture 3
(
pdf version - 6 slides/page
)
Wednesday, August 30, 2006
Lecturer: Richard Martin
No new Homework
Reading:
Kittel, Chapt. 2
Diffraction from Crystals and the Reciprocal Lattice
Outline
From last times:
A Crystal is a Periodic Array of (Groups of) Atoms
Crystal Structure = Lattice + Basis
How to study crystal structure?
Need Penetrating radiation, wavelength of order atomic size
X-rays, neutrons, fast electrons
Bragg formulas for diffraction - equivalent to a grating
Viewed as coherent scattering from planes in a crystal
Experimental diffraction using powders and single crystals
Periodic Functions and Fourier Analysis
Any periodic function f(r) can be expanded in Fourier components (harmonics)
Only non-zero Fourier components are vectors G of the reciprocal lattice
Reciprocal lattice
defined by primitive vectors b
i
defined by b
i
dot a
j
= 2 pi delta
ij
Reciprocal lattice vectors G are integer multiples of the b's
Examples of reciprocal lattice
1D, 2D
Orthorhombic, hexagonal
Reciprocal of fcc is bcc (and vice versa)
Diffraction, Fourier Analysis, and the reciprocal lattice
Diffraction is the scattering of waves
Waves form exp(ik
in
r - i omega t) and exp(ik
out
r - i omega t)
Since the scattering is periodic, this leads to the condition that Delta k = G, where Delta k = k
in
- k
out
and G is a reciprocal lattice vector
Elastic scattering (assumed when we set omega
in
= omega
out
)
Magnitude of k
in
equal magnitude of k
out
Leads to 2 k
in
dot G = G
2
Equivalent to Bragg Scattering
Email question/comments/corrections to
rmartin@uiuc.edu
.