Lecture on CRYSTALLINE SOLIDS SPACE LATTICE AND UNIT CELLS Space Lattice -- atoms arranged in a pattern that repeats itself in three dimensions. Unit cell -- smallest grouping which can be translated in three dimensions to recreate the space lattice.
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Lecture on CRYSTALLINE SOLIDS
SPACE LATTICE AND UNIT CELLS
Space Lattice -- atoms arranged in a pattern
that repeats itself in three dimensions.
Unit cell -- smallest grouping which can be translated in three dimensions to recreate the
space lattice.
CRYSTAL SYSTEMS AND BRAVAIS LATTICES
Seven crystal systems are each described by
the shape of the unit cell which can be translated to fill space.
Bravais lattices -- fourteen simple and complex lattices within the seven crystal
systems. The complex lattices have atoms centered
either in the center of a "primitive" unit cell or in the center of two/or more of the unit cell
faces.
PRINCIPAL METALLIC CRYSTAL STRUCTURES
We will concentrate on three of the more densely packed crystal structures, BCC - body centered cubic, FCC - face centered
cubic, and HCP - hexagonal close packed.
BCC - 2 atoms per unit cell CN = 8
0.68 = V
V = APF 3
R 4 = a
cell unit
atomsBCC
FCC - 4 atoms per unit cell CN = 12
HCP - 6 atoms per unit cell CN = 12
0.74 = V
V = APF 2
R 4 = a
cell unit
atomsFCCFCC
0.74 = V
V = APF
; 1.633 = a
c ; R 3.266 = c ; R 2 = a
cell unit
atomsHCP
HCP
HCPHCP
BCC FCC HCP
EQUIVALENT SITES (ATOMIC POSITIONS) IN CUBIC UNIT CELLS
Simple Cubic, SC - one per unit cell - corner
atoms only (0,0,0) (1,0,0) (1,1,0) (0,1,0)
(0,0,1) (1,0,1) (1,1,1) (0,1,1) Body Centered Cubic, BCC - two per unit cell - corner atoms as above, plus (1/2, 1/2, 1/2) Face Centered Cubic, FCC - four per unit cell -
corner atoms as above plus (1/2, 1/2, 0) (1/2, 0, 1/2) ( 0, 1/2, 1/2)
(1, 1/2, 1/2) (1/2, 1, 1/2) (1/2, 1/2, 1)
BCC FCC HCP
Lattice Sites in an Orthogonal Coordinate Systemi.e. Simple Cubic
DIRECTIONS IN CUBIC LATTICES 1 . Vector components of the direction are resolved along each of the coordinate axes
and reduced to the smallest integers. 2. All parallel directions have the same
direction indices. 3. Equivalent directions have the same atom
spacing.4. The cosine of the angle between two
directions is given by
) l+ k+ h()l+k+h(
ll+kk+hh =
222222
cos
Indices of a Family or Form
00]1[],1[000],1[0[001],[010],[100], >100<
negativessixtheplus
][],[],[],[],[],[ >< 110101011101011110110
][],[],[],[
][],[],[],[ ><
111111111111
,111111111111111
MILLER INDICES FOR CRYSTALLOGRAPHIC PLANES
Definition: Miller Indices are the
reciprocals of the fractional intercepts (with fractions cleared) which the plane makes with
the crystallographic x,y,z axes of the three nonparallel edges of the cubic unit cell.
! cleared fractions with
c
1,
b
1,
a
1 = Indices Miller
Spacing between planes in a cubic crystal
where dhkl = inter-planar spacing between
planes with Miller indices h,k,and l.
a = lattice constant (edge of the cube)h, k, l = Miller indices of cubic planes
being considered.
l + k + h
a = d
222hkl
CRYSTALLOGRAPHIC PLANES AND DIRECTIONS IN HEXAGONAL UNIT
CELLS
Miller-Bravais indices -- same as Miller indices for cubic crystals except that there are
3 basal plane axes and 1 vertical axis.
Basal plane -- close packed plane similar to the (1 1 1) FCC plane.
contains 3 axes 120o apart.
Miller Bravais indices are h,k,i,l
with i = -(h+k). Basal plane indices (0 0 0 1)
Prism planes -- {1 0 0} family
Direction Indices in HCP Unit Cells -- [hkil] where h+k = -i
COMPARISON OF FCC, HCP, AND BCC CRYSTAL STRUCTURES
Both FCC and HCP structures are close
packed APF = 0.74.
The closed packed planes are the {111} family for FCC and the (0001) plane for HCP.Stacking sequence is ABCABCABC in FCC
and ABABAB in HCP.
BCC is not close packed, APF = 0.68. Most densely packed planes are the {110} family.
VOLUME, PLANAR, AND LINEAR DENSITY
Volume density --
Planar density --
Linear Atomic density --
cell tvolume/uni
cell mass/unit = = metal ofdensity Volume v
plane of area selected
dintersecte centers atom # = =density atomic Planar p
line of length selected
dintersecte diameters atom # = =density atomic Linear l
X-ray Diffraction and Braggs Law
n = order of reflection, whole number of
reflectionsλ = x-ray wavelength
dhkl = spacing between planes with indices
(hkl)θ = angle between incident x-ray beam and
crystal planes (hkl)
sin2 hkld = n
Figure 3.29
Figure 3.28
X-ray Diffraction and Braggs Law
n = order of reflection, whole number of
reflectionsλ = x-ray wavelength
dhkl = spacing between planes with indices
(hkl)θ = angle between incident x-ray beam and
crystal planes (hkl)
sin2 hkld = n
For cubic crystals
l + k + h
a = d
222hkl
sin2154.0222 lkh
a = nm
sin2 hkld = n
1n nmKCu 154.0,
Selection Rules for Observing X-ray Peaks
FCC : (h k l) must all be either odd or even
BCC : sum h + k + l must be even
(Otherwise, an in between plane will cancel
the reflection)
POLYMORPHISM OR ALLOTROPY
Existence of more than one equilibrium crystallographic form for elements or compounds at different conditions of
temperature and pressure.
Example:
Iron liquid above 1539 C.δ-iron (BCC) between 1394 and
1539 C.γ-iron (FCC) between 912 and
1394 C.α-iron (BCC) between -273 and
912 C.
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