Unit cell
fundamental repeating structure that is found in all crystalline structures
lattice parameters
a - define a unit cell’s size and shape
unit cell shapes
- cubic
- tetragonal
- orthorhombic
- rhombohedral
- hexagonal
- monoclinic
- triclinic
Bravais lattices
expands number of lattices to 14 because you have to take into account location of atoms
BCC Lattice definition
body centered cubic - atoms at corner of cell and one in the middle - a = 4R/sqrt(3)
bcc lattice characteristics
at room temp is quite open so not in lowest energy state
bcc lattice number of atoms
of interior atoms = 1
# of corner atoms = 8 * 1/8 = 1
total = 2
fcc lattice definition
face centered cubic - atoms at corners and centered on each cube face - a = 2Rsqrt(2)
fcc lattice number of atoms
of face-centered atoms = 6*1/2 = 3
of corner atoms = 8 *1/8 = 1
total = 4
hcp definition
hexagonal close packed - two hexagons with atoms at each corner and one in the faces’ centered stacked on top of each other with three interior atoms - a/c = 1.63333 (a = edge length of hexagon, c = height of prism)
hcp number of atoms
of corner atoms = 12 * 1/6 = 2
# of face atoms = 2 * 1/2 = 1
# of interior atoms = 3
total = 6
theoretical density of a material
rho = nA/V(c)N(a)
n = number of atoms in a unit cell
N(a) = Avogadro’s number
A = atomic weight
V(c) = volume of cell
Miller - Bravais
four axis system for representing hcp - a1, a2, a3, evenly spaced axis in xy plane and z is same - [uvtw] - can convert from [UVT] using formulas
crystal planes rules
- pick origin
- determine intercepts of plane with respect to the axis
- take inverse numbers of intercepts
- make these numbers integers
- these are your (h k l) indices
important notes for crystal plane indices
no intercept - infinity (1/infinity = 0)
-1 = 1 bar (1 with line above it)
crystal directions for cubic crystals
n is usually 1
use starting point and end point
since unit cell, a=b=c=1
APF
atomic packing factor
volume of atoms - use sphere formula (4piR^3/3) and number of atoms
volume of cell - use ‘a’ parameter and formula of cube volume
LD
linear density - number of atoms per unit length whose centers lie on the direction vector - must take into account proportion of atoms in that unit cell
PD
planar density - number of atoms per unit length on a plane
the higher the density, the more likely _______
it is for electrons and atoms to interact with each other - (111) plane is more active than (100) for fcc
fcc pattern
ABCABC
hcp pattern
ABAB
what are the most important planes
highest density planes or close packed planes
x-ray diffraction principle
like skipping rocks, if the speed (energy) and direction (angle) are right then we diffract the incoming x-ray beam
