Crystal Structure and Analysis

1
Crystal Structure and Analysis

• Physics 133/219
• Lecture 1
• Reference
• N. W. Ashcroft N. D. Mermin,
• Solid State Physics
• Saunders College Publishing (1976)
• Chapters 4 - 6

2
Crystal Lattices - The Bravais Lattice

• An array of discrete points that appears exactly
  the same from any point the array is viewed

R
R
a1
a1
a2
a2
R 3 a1 a2
R 3 a1 a2
For 3D R n1a1 n2a2 n2a2
Coordination number - the number of nearest
neighbors to a given point
3
Unit Cells - Primitive Cells

• A volume of space that, when translated through
  all vectors in a Bravais lattice, just fills all
  of space without either overlapping itself or
  leaving voids is called a primitive cell or
  primative unit cell of the lattice.

Weigner-Seitz Primitive Cell Bisect lines between
nearest neighbors, and draw smallest polygon
4
Lattice with a Basis

• A physical crystal can be described by giving its
  underlying Bravais lattice, together with a
  description of the arrangement of atoms,
  molecules, ions, etc., within a particular
  primitive cell. This arrangement within the
  primitive cell is referred to as the Basis

• A Bravais lattice can also be described as a
  lattice with a basis by choosing a non-primitive
  conventional unit cell. This is often done with
  bcc and fcc crystals to emphasize the crystal
  symmetry.

5
The Reciprocal Lattice

• Suppose you have a Bravias lattice, R, and a
  plane wave, eikr. Generally, for a given k, the
  plane wave will not have the periodicity of the
  Bravais lattice, R. However, for a given R,
  there will be a set of wave vectors, K, that will
  yield plane waves with the periodicity of the
  Bravais lattice.

eiKr eiK(r R) eiKr eiKR thus, eiKR
1 or KR 2?
6
The Reciprocal Lattice

• The Reciprocal lattice is also a Bravais lattice

A brief proof
bi 2? aj x ak / (ai aj x ak) then bi aj
2? ?ij. Furthermore, any vector, k, can be
written as k k1b1 k2b2 k3b3. For any vector
in the direct lattice, R, k R 2?(k1n1 k2n2
k2n2 ), so for eik R to be unity for all R, k
R must be 2? times an integer for any ni. Thus
the coefficients ki must also be integers.
Therefore, the set of vectors, K, are themselves
a Bravais lattice.
7
The Reciprocal Lattice - Some other thoughts.

• The reciprocal lattice of the reciprocal lattice?
• Is just the direct lattice!
• Brillouin Zone
• Brillouin zones are primitive cells that arise in
  the theories of electronic levels - Band Theory.
• However, the first Brillouin zone is the
  Wigner-Seitz primitive cell of the reciprocal
  lattice.

8
Lattice Planes and Miller Indices

• A Lattice Plane is any plane containing at least
  three noncollinear Bravais lattice points.
• Generally, a lattice plane is described by giving
  a vector normal to that plane, and there happens
  to be reciprocal lattice vectors normal to any
  lattice plane.
• Choose the the shortest such reciprocal lattice
  vector to arrive at the Miller indices of the
  plane.
• Thus a plane with Miller indices h, k, l, is
  normal to the reciprocal lattice vector hb1 hb2
  hb3.

9
Determination of crystal structure

• In 1913 W. H. and W. L. Bragg found crystalline
  gave characteristic patterns of reflected x-rays.

• Assuming specular reflection off lattice planes.
• The condition for constructive interference
  becomes
• n? 2d sin ?
• This is known as the Bragg condition - KNOW IT!

?
d
d sin ?
10
The Von Laue formulation

• Doesnt assume specular reflection
• No sectioning by planes
• Rather, at each point on the Bravais lattice the
  incident ray is allowed to be scattered in all
  directions

11
Von Laue Formulation
First, consider only two scatterers.
k kn
k kn
?
d
?
The path difference is then d cos ? d cos ?
d (n - n), Constructive interference is, d
(n - n) m?
k 2??? k 2???
12
Von Laue Formulation

• With an array of scatterers one at each point of
  the Bravais lattice
• The last slide must hold for each d that are
  Bravais lattice vectors, so

R (k - k) 2?m or eiR (k - k) 1!!!!
Compare to the definition for the reciprocal
lattice! The Laue condition - constructive
interference will occur provided that the change
in wave vector, K k - k, is a vector in the
reciprocal lattice.
13
Ewald Construction
Condition for constructive interference, K k -
k, is a vector of the reciprocal lattice.
The RECIPROCAL lattice
14
Some measurements

• Laue - white x-rays
• Yields stereoscopic projection of reciprocal
  lattice
• Rotating-Crystal method - monochromatic x-rays
• Fix source and rotate crystal to reveal
  reciprocal lattice
• Powder diffraction - monochromatic x-rays
• Powder sample to reveal all directions of RL