Order 1238142: Condensed matter
tutorthammy
Lect-5-Crystal-Structure2.pdf
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• Bravais lattice (or “Lattice”), unit cell, • primitive cell, conventional cell, basis • Wigner-Seitz (primitive) cell • Common crystal structures
at the end of this lecture you should understand....
Crystal Structure
See Ch4 Ashcroft and Mermin; Ch7 p 112-119
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Some formal definitions
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The crystal lattice: “Lattice” or “Bravais lattice” A Lattice is an infinite set of points, defined by integer sums
of a set of linearly independent primitive lattice vectors
The lattice looks exactly the same from every point
In two dimensions
(m,n integers)
A Lattice is an infinite set of vectors where addition of any two vectors
in the set give a third vector in the set
A Lattice is a set of points where the environment of any given point
Is equivalent to the environment of any other given point.
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The crystal lattice: Bravais lattice (3D)
This reflects the translational symmetry of the lattice
called
“simple cubic”
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A unit cell is a region of space such that when many identical
units are stacked together it tiles (completely fills) all of space and
reconstructs the full structure
Unit Cell, Primitive Cell
A unit cell is a repeated motif which is the elementary building block of the
periodic structure
A primitive unit cell for a periodic crystal is a unit cell containing only a single
lattice point
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The crystal lattice: primitive unit cell
a1
a2
In two- and three-dimensions the choice
of primitive lattice vectors is not unique
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The crystal lattice: primitive unit cell Primitive unit cell: any volume of space which containing one lattice point, when
translated through all the vectors of the Bravais lattice, fills space without overlap
and without leaving voids
a1
a2
We can put these unit cells wherever we want - the primitive unit cell contains only one lattice
point - the non-primitive unit cells contain more than one.
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The crystal lattice: Wigner-Seitz cell
Wigner-Seitz cell: special choice of primitive unit cell: region of points closer to
a given lattice point than to any other.
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Unit cells: conventional, primitive, Wigner-Seitz
Sometimes useful to define a unit cell that is not primitive
so that it is simpler to work with usually with orthogonal axes
– called a conventional unit cell
Wigner-Seitz
unit cell
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Bravais lattice (2D)
• The number of possible Bravais lattices (of fundamentally different symmetry) is limited to 5 (2D) and 14 (3D).
Θ
Ashcroft and Mermin; Ch7 p 112-119
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Body centred cubic
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perspective view plan view
Primitive lattice vectors
defines lattice
Face centred cubic
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perspective view plan view
Primitive lattice vectors
defines lattice
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Primitive and Conventional Cells
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Primitive and conventional cell of
the face centred cubic Bravais Lattice.
The volume of conventional cell is four
times larger than that of the primitive
cell.
Primitive and conventional cell of
the body centred cubic Bravais Lattice.
The volume of conventional cell is two
times larger than that of the primitive
cell.
BCC FCC
Wigner-Seitz cell: bcc fcc
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bcc fcc
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The crystal lattice: Basis
• We could think: all that remains to do is to put atoms
on the lattice points of the
Bravais lattice.
• But: not all crystals can be described by just a Bravais
• BUT: all crystals can be described by the combination
of a Bravais lattice and a
basis.
• The basis is what one “puts on the lattice points”.
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The crystal lattice: one atomic basis
• The basis can also just consist of one atom.
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The crystal lattice: basis
• Or it can be several atoms.
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The crystal lattice: basis
• Or it can be molecules, proteins and pretty much anything else.
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The crystal lattice: one more word about
symmetry
• The other symmetry to consider is point symmetry. The Bravais lattice for these two crystals is identical:
four mirror lines
4-fold rotational axis
inversion
no additional
point symmetry
Example of a Lattice with a Basis
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The Basis The lattice points are at:
centre atom of basis at:
other 4 atoms of basis at:
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Another example: triangular lattice with basis
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Coordinates of grey circles
with respect to reference point (black) Primitive lattice vectors
a1
a2
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• What structure do the solids have? Can we predict it? • Could just put the spheres together in order to fill all space. • A simple cubic structure? fcc? bcc?
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Consider Atomic Packing Factor
• Atomic Packing Factor (APF) is defined as the volume of atoms within the unit cell
divided by the volume of the unit cell.
Crystal Structure 24
1-CUBIC CRYSTAL SYSTEM
Simple Cubic has one lattice point so its primitive cell.
In the unit cell on the left, the atoms at the corners are cut
because only a portion (in this case 1/8) belongs to that cell.
The rest of the atom belongs to neighboring cells.
Coordinatination number of simple cubic is 6.
a- Simple Cubic (SC)
a
b c
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Atomic Packing Factor of SC
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Body Centered Cubic (BCC)
BCC has two lattice points (in the
conventional cell) so BCC is a non-
primitive cell.
BCC has eight nearest neighbors.
Each atom is in contact with its
neighbors only along the body-
diagonal directions.
Many metals (Fe,Li,Na..etc),
including the alkalis and several
transition elements adopt the BCC
structure. a
b c
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0.68 = V
V = APF
3
R 4 = a
cell unit
atoms BCC
2 (0.433a)
Atomic Packing Factor of BCC
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Face Centered Cubic (FCC)
• There are atoms at the corners of the unit cell and at the center of each face.
• Face centered cubic has 4 atoms in its conventional cell. • Many of common metals (Cu,Ni,Pb..etc) crystallize in FCC
structure.
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Crystal Structure 29
4 (0.353a)
0.68 = V
V = APF
3
R 4 = a
cell unit
atoms BCCFCC
0.74
Atomic Packing Factor of FCC
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Hexagonal close-packed: hcp
• The hcp lattice is NOT a Bravais lattice. It can be constructed from a hexagonal Bravais lattice with a basis containing two
atoms.
• the packing efficiency is exactly the same as for the fcc structure (74 % of space occupied).
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Crystal Structure 31
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All the three-dimensional
Bravais lattice types
Ashcroft and Mermin; Ch7 p 112-119
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Some examples
of real crystals with
simple structures
