lecture+2+mak crystal+structure1
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PHY 3201 FIZIK KEADAAN PEPEJAL
Three-Dimensional
Lattice Types
.
There are seven crystal classes or crystal system, each
of which is a parallelepiped (pa-ra-lel-li-pi-ped). Figure
below shows the scheme whereby these seven types
are defined
3a
1a
2
a
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Three-Dimensional
Lattice Types
System Number of
lattices in
system
Lattice symbols Restrictions on
conventional cell axes and
angles
Triclinic 1 P Monoclinic 2 P, C = = 90
Orthorhombi
c
4 P, C, I, F = = = 90
Tetragonal 2 P, I = = = 90
Cubic 3 Por sc
Ior bccFor fcc
= = = 90
Trigonal 1 R = = < 120, 90Hexagonal 1 P = = 90
= 120
Table 1 The fourteen lattice types in
three dimensions
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The cubic space
lattices
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Characteristics of
cubic lattices
sc bcc fcc
Volume, conventional cell a3 a3 a3
Lattice point per cell 1 2 4
Volume, primitive cell a3 a3 a3
Lattice points per unit
volume1/a3 2/a3 4/a3
Number of nearest
neighbors6 8 12
Nearest-neighbors
distancea 31/2a/2 = 0.866a a/21/2 = 0.707a
Packing fraction 1/6 1/8 3 1/6 2
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Simple cubicThe simple cubic has 1 lattice point per unit cell, with
total area a3
Number of nearest neighbours:6
Conventional= Primitive cell
Note: a= lattice constant
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Wikipedia: Atomic packing fraction is the fraction
of volume in a crystal structure that is occupied by
atoms. It is dimensionless and always less than
unity. For practical purposes, the APF of a crystalstructure is determined by assuming that atoms are
rigidspheres. The radius of the spheres is taken to
be the maximal value such that the atoms do not
overlap.
http://en.wikipedia.org/wiki/Crystal_structurehttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Crystal_structurehttp://en.wikipedia.org/wiki/Crystal_structurehttp://en.wikipedia.org/wiki/Crystal_structure -
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Body-centered Cubic
Figure 9 Body-centered cubic
lattice, showing a primitive cell.
The primitive cell shown is a
rhombohedron of edge 1/23a,and the angle between adjacent
edges is 109o28 . Here, theconventional cubic cell is not
primitive. There are two lattice
points per cubic cell due to the
extra lattice point at the body
centre of the cell.
Figure 10 Primitive translation vectors of
the body-centered cubic lattice; these
vectors connect the lattice point at the
origin to lattice points at the body centers.The primitive cell is obtained on completing
the rhombohedron. In terms of the cube
edge a the primitive translation vectors are
)(2
);(2
);(2
321 zyxa
azyxa
azyxa
a
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Body-centered
Cubic
The body-centered cubic lattice has 2 lattice points per
unit cell
Number of nearest neighbour: 8
Conventional Primitive cell
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bcc
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Three-Dimensional
Lattice Types
Figure 11 The rhombohedral
primitive cell of the face-
centered cubic crystal. The
primitive translation vectors
connect the lattice point at theorigin with lattice points at the
face centers. As drawn, the
primitive vectors are:
).(2
);(2
);(2
321 xza
azya
ayxa
a
The face-centred (fcc) in which there is a
lattice point at the centre of each cube
face. Again, the fcc conventional cubic
cell is not primitive. There are four
lattice points per cubic cell. The
primitive translation vectors for the fccare shown in fig. 7.
From these figures, you can see that the
primitive vectors/cells of both the bcc
and fcc are much too complicated. It is
much easier to classify them as a form of
cubic structure from which symmetry
operations can be visualised more
easily.
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The face-centered cubic lattice has 4 lattice points per
unit cell
Number of nearest neighbour: 12
Conventional Primitive cell
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Three-Dimensional
Lattice Types
Figure 12 Relation of the
primitive cell in the
hexagonal system (heavy
lines) to a prism of
hexagonal symmetry. Here
a = bc. Three suchrhombic prisms can be put
together to form a rightprism of hexagon, hence
the term hexagonal system
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