Linear and Planar Atomic Densities
Linear Density:
Directional equivalency is related to the atomic linear density in the sense that
equivalent directions have identical linear densities.
The direction vector is positioned so as to pass through atom centers.
The fraction of line length intersected by these atoms is equal to the linear
density.
Planar Density:
Crystallographic planes that are equivalent have the same atomic planar density.
The plane of interest is positioned so as to pass through atom centers.
Planar density is the fraction of total crystallographic plane area that is occupied
by atoms.
Linear and planar densities are one- and two-dimensional analogs of the atomic
packing factor.
Linear Density for BCC
Calculate the linear density for the following directions:
a. [100]
b. [110]
c. [111]
Planar Density for BCC
Calculate the planar density for the following BCC planes:
a. (100)
b. (110)
FCC
Calculate the planar density of the (110) plane for FCC.
Crystalline and Non-Crystalline
Materials
Single Crystal:
The periodic and repeated arrangements of atoms is
perfect or extends throughout the entirety of the
specimen without interruption.
All unit cells interlock in the same way and have the
same orientation.
Single crystals exist in nature, but they may also
produced artificially.
They are ordinarily difficult to grow, because the
environment must be carefully controlled.
Several Single Crystals of Fluorite
Single crystals are needed for modern technologies today.
Electronic micro-chips uses single crystals of silicon and other semiconductors.
Polycrystalline Materials
Composed of a collection of many small crystals or grains.
Anisotropy
Physical properties of single crystals of some substances depend on the
crystallographic direction in which measurements are made.
This directionality of properties is termed anisotropy, and it is associated with the
variance of atomic or ionic spacing with crystallographic direction.
Substances in which measured properties are independent of the direction of
measurement are isotropic.
Diffraction
Diffraction occurs when a wave encounters a series of regularly spaced
obstacles that
(1) Are capable of scattering the wave, and
(2) Have spacings that are comparable to the wavelength.
Furthermore, diffraction is a consequence of specific phase relationships
that are established between two or more waves that have been scattered
by the obstacles.
Constructive and Destructive
Interference
Bragg’s Law
2d hkl Sin  n
Interplanar Spacing
d hkl 
a
h2  k 2  l 2
.
Diffraction Angle
Polycrystalline α-iron
BCC
Sum of the indices:
Problem 3.56
h + k + l = even
Polycrystalline Cu
FCC
Additional condition for diffraction: h, k, and l must either odd or even.
Problem 3.57 and 3.58, Page 65.