Linear and Planar Atomic Densities

1
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 Crystallog
raphic 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.
2
Linear Density for BCC

• Calculate the linear density for the following
  directions
• 100
• 110
• 111

3
Planar Density for BCC

• Calculate the planar density for the following
  BCC planes
• (100)
• (110)

4
FCC
Calculate the planar density of the (110) plane
for FCC.
5
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.
6
Several Single Crystals of Fluorite
Single crystals are needed for modern
technologies today. Electronic micro-chips uses
single crystals of silicon and other
semiconductors.
7
Polycrystalline Materials
Composed of a collection of many
small crystals or grains.
8
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.
9
Diffraction

• Diffraction occurs when a wave encounters a
  series of regularly spaced obstacles that
• Are capable of scattering the wave, and
• 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.

10
Constructive and DestructiveInterference
11
Braggs Law
12
Interplanar Spacing
13
Diffraction Angle
14
Polycrystalline a-ironBCC
Sum of the indices h k l even
Problem 3.56
15
Polycrystalline CuFCC
Additional condition for diffraction h, k, and l
must either odd or even. Problem 3.57 and 3.58,
Page 65.