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

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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.  
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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. 
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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
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FCC - 4 atoms per unit cell CN 12  HCP - 6
atoms per unit cell CN 12
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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) 
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Lattice Sites in an Orthogonal Coordinate
System i.e. Simple Cubic
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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 
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Indices of a Family or Form 
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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.  
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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.
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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.
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Miller Bravais indices are h,k,i,lwith i
-(hk). Basal plane indices (0 0 0 1) Prism
planes -- 1 0 0 family  Direction Indices in
HCP Unit Cells -- hkil where hk -i 
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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.
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VOLUME, PLANAR, AND LINEAR DENSITY  Volume
density --   Planar density --   Linear
Atomic density --   
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X-ray Diffraction and Braggs Law n order
of reflection, whole number of reflections?
x-ray wavelengthdhkl spacing between planes
with indices (hkl)? angle between incident
x-ray beam and crystal planes (hkl)
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Figure 3.29
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Figure 3.28
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X-ray Diffraction and Braggs Law n order
of reflection, whole number of reflections?
x-ray wavelengthdhkl spacing between planes
with indices (hkl)? angle between incident
x-ray beam and crystal planes (hkl)
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For cubic crystals
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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)
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POLYMORPHISM OR ALLOTROPY  Existence of more
than one equilibrium crystallographic form for
elements or compounds at different conditions of
temperature and pressure.  
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Example  Iron liquid above 1539
C. d-iron (BCC) between 1394 and 1539
C. ?-iron (FCC) between 912 and 1394
C. a-iron (BCC) between -273 and 912 C.