Back to 460 Home
Calendar
Previous Lecture
Next Lecture
Phys 460 Lecture 5
(
pdf version - 6 slides/page
)
Monday, September 11, 2006
Lecturer: Richard Martin
No new homework today
Reading:
Kittel, Chapt. 2
Diffraction from Crystals and the Reciprocal Lattice -Continued
Outline
From last time:
Study of crystal structure reuires penetrating radiation, wavelength of order atomic size: X-rays, neutrons, fast electrons
Bragg formulas for diffraction - viewed as coherent scattering from planes in a crystal
Periodic Functions and Fourier Analysis - f(r) = sum of Fourier components
Reciprocal lattice
Non-zero Fourier components are f
G
Reciprocal lattice vectors G are integer multiples of the b's, the primitive vectors of the reciprocal lattice
Examples of reciprocal lattice: fcc, bcc, ......
Diffraction, Fourier Analysis, and the reciprocal lattice
Delta k = G, where Delta k = k
in
- k
out
Elastic scattering: |k
in
| = | k
out
|
Leads to 2 k
in
dot G = G
2
Ewald Consruction
Geometric construction of k
in
- k
out
= G
with the elastic condition on magnitudes
Interpretation of formula 2 k
in
dot G = G
2
Equivalent to Bragg condition -proof uses relation of magnitude of G to spacing of planes (homework problem, Kittel, prob. 2-1)
Geometric interpretation: k vector is on perpendicular bisector plane for a G vector
Same as construction of Brillouin Zone
Consequence: No diffraction for k in first Brillouin Zone
(Will be important later in course)
Bragg formula can be written sin (2 theta) = (2 pi/lambda) |G|
Each Lattice has characteristic ratios of sin (2 theta)
Examples of lattices: Cubic, Lower symmetry
Fourier Analysis of the basis
Form factors for each atom
FCC, BCC viewed as simple cubic with a basis
Diamond structure
Non-periodic crystals - quasicrystals - very brief comments
Email question/comments/corrections to
rmartin@uiuc.edu
.