Lecture :11
Linear and Planer Densities:
Equivalent directions have identical linear densities(LD), and
plans having the same planer density (PD)are also equivalent .
LD = No. of atoms centered on direction vector / length of
direction vector
Note also that , in general , LD is equal to the reciprocal of the
repeat distance(r) ; between adjacent atoms:
LD 
I
r
PD = No.of atoms centered on a plan / area of plan
Single Crystals and Polycrystalline Materials :
Single crystal: atoms are in a repeating or periodic array over
the entire extent of the material
Polycrystalline material: comprised of many small crystals or
grains. The grains have different crystallographic orientation.
There exist atomic mismatch within the regions where grains
meet. These regions are called grain boundaries.
Anisotropy :
Different directions in a crystal have a different packing. For
instance, atoms along the edge of FCC unit cell are more
separated than along the face diagonal. This causes anisotropy
in the properties of crystals, for instance, the deformation
depends on the direction in which a stress is applied.
In some polycrystalline materials, grain orientations are
random, so bulk material properties are isotropic
Some polycrystalline materials have grains with preferred
orientations (texture), so properties are dominated by those
relevant to the texture orientation and the material exhibits
anisotropic properties
Non-Crystalline (Amorphous) Solids :
In amorphous solids, there is no long-range order. But
amorphous does not mean random, in many cases there is some
form of short-range order.
Schematic picture of amorphous SiO2 structure
X-RAY DIFFRACTION: DETERMINATION OF
CRYSTAL STRUCTURES
X-rays are a form of electromagnetic radiation that have high
energies and short wavelengths—wavelengths on the order of
the atomic spacings for solids.
Diffraction occurs when a wave encounters a series of regularly
spaced obstacles that are
1- capable of scattering the wave,
2- Have spacings that are comparable in magnitude to the
wavelength.
3- Furthermore, diffraction is a consequence of specific
phase relationships that are established between two or
more waves that have been scattered by the obstcales .
we refer to a diffracted beam as one composed of a large
number of scattered waves that mutually reinforce one another.
Bragg’s law: A relationship which stipulates the condition
for diffraction by a set of crystallographic planes.
Where:
n: order of reflection(integer values:1, 2, 3, . . . )
λ: x-ray wavelength
d : interatomic spacing
θ: angle of the diffraction.
If Bragg’s law is not satisfied, then the interference will be
nonconstructive in nature so as to yield a very low-intensity
diffracted beam.
The magnitude of the distance between two adjacent and
parallel planes of atoms (i.e., the interplanar spacing dhkl) is a
function of the Miller indices (h, k, and l) as well as the lattice
parameter(s). For example, for crystal structures having cubic
symmetry,
Ex:
For BCC iron, compute:
(a) the interplanar spacing
(b) the diffraction angle for the (220) set of planes.
The lattice parameter for Fe is 0.2866 nm (2.866 A° ).
Also,
assume that monochromatic radiation having a
wavelength of
0.1790 nm (1.790 A° ) is used, and the order of reflection is
1.
SOLUTION:
(a)
d hkl 
a2
h2  k 2  l 2
d hk l 
(b):
2
(0.2866)2
22  22  12
 0.1013nm  1.013 Ao