Transcript THE “MOST IMPORTANT” CRYSTAL STRUCTURES

THE “MOST IMPORTANT”
CRYSTAL STRUCTURES
THE “MOST IMPORTANT”
CRYSTAL STRUCTURES
NOTE!!
Much of the discussion & many figures in what follows
was again constructed from lectures posted on the web
by Prof. Beşire GÖNÜL in Turkey. She has done an
excellent job of covering many details of
crystallography & she illustrates her topics with many
very nice pictures of lattice structures. Her lectures on
this are posted Here:
http://www1.gantep.edu.tr/~bgonul/dersnotlari/.
Her homepage is Here:
http://www1.gantep.edu.tr/~bgonul/.
THE “MOST IMPORTANT”
CRYSTAL STRUCTURES
• Sodium Chloride Structure Na+Cl• Cesium Chloride Structure Cs+Cl• Hexagonal Closed-Packed Structure
• Diamond Structure
• Zinc Blende
3
1 – Sodium Chloride Structure
• Sodium chloride also
crystallizes in a cubic lattice,
but with a different unit cell.
• The sodium chloride
structure consists of equal
numbers of sodium &
chlorine ions placed at
alternate points of a simple
cubic lattice.
• Each ion has six of the other
kind of ions as its nearest
neighbors.
NaCl Structure
• This structure can also be
considered as a face-centered-cubic
Bravais lattice with a basis
consisting of a sodium ion at 0 and
a chlorine ion at the center of the
conventional cell, at position



a / 2( x  y  z )
• LiF, NaBr, KCl, LiI, have
this structure.
• The lattice constants are of the
order of 4-7 Angstroms.
NaCl Structure
• Take the NaCl unit cell & remove all “red” Cl
ions, leaving only the “blue” Na. Comparing
this with the FCC unit cell, it is found to be that
they are identical. So, the Na ions are on a FCC
sublattice.
NaCl Type Crystals
2 - CsCl Structure
2 - CsCl Structure
• Cesium Chloride, CsCl,
crystallizes in a cubic
lattice. The unit cell may be
depicted as shown.
(Cs+ is teal, Cl- is gold)
• Cesium Chloride consists
of equal numbers of Cs and
Cl ions, placed at the points
of a body-centered cubic
lattice so that each ion has
eight of the other kind as its
nearest neighbors.
CsCl Structure
• The translational symmetry of this structure
is that of the simple cubic Bravais lattice, and
is described as a simple cubic lattice with a
basis consisting of a Cs ion at the origin 0
and a Cl ion at the cube center



a / 2( x  y  z )
• CsBr & CsI crystallize in this structure.The
lattice constants are of the order of 4 angstroms.
CsCl Structure
8 cells
CsCl Crystals
The Ancient “Periodic Table”
4 - Diamond Structure
• The Diamond Lattice consists of 2
interpenetrating FCC Lattices.
• There are 8 atoms in the unit cell. Each atom bonds
covalently to 4 others equally spaced about a given atom.
• The Coordination Number = 4.
• The diamond lattice is not a Bravais lattice.
C, Si, Ge & Sn crystallize in the Diamond structure.
Diamond Lattice
Diamond Lattice
The Cubic Unit Cell
5 – Zinc Blende or ZnS Structure
• The Zincblende Structure has equal
numbers of zinc and sulfur ions
distributed on a diamond lattice, so that
Each has 4 of the opposite kind as
nearest-neighbors.
• This structure is an example of a lattice with a
basis, both because of the geometrical position of
the atoms & because two types of atoms occur.
• Some compounds with this structure are:
AgI, GaAs, GaSb, InAs, ....
5 – Zinc Blende or ZnS Structure
Zincblende (ZnS) Lattice
Zincblende Lattice
The Cubic Unit Cell
Diamond & Zincblende Structures
A brief discussion of both of these structures & a comparison.
• These two are technologically important structures
because many common semiconductors have
Diamond or Zincblende Crystal Structures
• They obviously share the same geometry.
• In both structures, the atoms are all tetrahedrally
coordinated. That is, atom has 4 nearest-neighbors.
• In both structures, the basis set consists of 2 atoms.
In both structures, the primitive lattice
 Face Centered Cubic (FCC).
• In both the Diamond & the Zincblende lattice there are 2
atoms per fcc lattice point.
In Diamond: The 2 atoms are the same.
In Zincblende: The 2 atoms are different.
Diamond & Zincblende Lattices
Diamond Lattice
Zincblende Lattice
The Cubic Unit Cell
The Cubic Unit Cell
Other views of
the cubic unit cell
A view of the tetrahedral coordination
& the 2 atom basis
Zincblende &
Diamond Lattices 
Face Centered Cubic
(FCC) lattices with a
2 atom basis
The Wurtzite Structure
• A structure related to the Zincblende Structure is the
Wurtzite Structure
• Many semiconductors also have this lattice structure.
• In this structure there is also
Tetrahedral Coordination
• Each atom has 4 nearest-neighbors. The Basis set is 2 atoms.
• Primitive lattice  hexagonal close packed (hcp).
2 atoms per hcp lattice point. A Unit Cell looks like
The Wurtzite Lattice
View of tetrahedral
coordination &
the 2 atom basis.
Wurtzite Lattice
 Hexagonal Close
Packed (HCP)
Lattice + 2 atom basis
Diamond & Zincblende crystals
• The primitive lattice is FCC. The FCC primitive
lattice is generated by r = n1a1 + n2a2 + n3a3.
• The FCC primitive lattice vectors are:
a1 = (½)a(0,1,0), a2 = (½)a(1,0,1), a3 = (½)a(1,1,0)
NOTE: The ai’s are NOT mutually orthogonal!
Primitive FCC Lattice
cubic unit cell
Diamond:
2 identical atoms per FCC point
Zincblende:
2 different atoms per FCC point
Wurtzite Crystals
• The primitive lattice is HCP. The
HCP primitive lattice is generated by
Primitive Lattice
Points
r = n1a1 + n2a2 + n3a3.
• The hcp primitive lattice vectors are:
a1 = c(0,0,1)
a2 = (½)a[(1,0,0) + (3)½(0,1,0)]
a3 = (½)a[(-1,0,0) + (3)½(0,1,0)]
NOTE!
These are NOT mutually orthogonal!
Wurtzite Crystals
2 atoms per HCP point
Primitive HCP
Lattice: Hexagonal
Unit Cell
ELEMENTS OF SYMMETRY
• Each of the unit cells of the 14 Bravais lattices
has one or more types of symmetry properties,
such as inversion, reflection or rotation,etc.
SYMMETRY
INVERSION
REFLECTION
ROTATION
Typical symmetry properties of a lattice.
Some types of operations that can leave a lattice invariant.
Operation
Element
Inversion
Point
Reflection
Plane
Rotation
Axis
Rotoinversion
Axes
Inversion
• A center of inversion: A point at the center of the molecule.
(x,y,z) --> (-x,-y,-z)
• A center of inversion can only occur in a molecule. It is
not necessary to have an atom in the center (benzene,
ethane). Tetrahedral, triangles, pentagons don't have
centers of inversion symmetry. All Bravais lattices are
inversion symmetric.
Mo(CO)6
Rotational Invariance &
Invariance on Reflection Through a Plane
Invariance on Reflection
through a plane
Rotational Invariance
about more than one axis
• A plane in a cell such that, when a mirror reflection
in this plane is performed, the cell remains invariant.
Examples
• A triclinic lattice has no reflection plane.
• A monoclinic lattice has one plane midway
between and parallel to the bases, and so forth.
Rotational Symmetry
• There are always a finite number of
rotational symmetries for a lattice.
• A single molecule can have any degree
of rotational symmetry, but an infinite
periodic lattice – can not.
Rotational Symmetries
90º
120°
180°
• This is an axis such that, if the cell is rotated around
it through some angles, the cell remains invariant.
• The axis is called n-fold if the angle of rotation is 2π/n.
Axes of Rotation
Axes of Rotation
5-Fold Symmetry
This type of symmetry is not allowed because it
can not be combined with translational periodicity!
Examples
• A Triclinic Lattice has no axis of rotation.
• A Monoclinic Lattice has a 2-fold axis
(θ = [2π/2] = π) normal to the base.
90°
Examples