Crystal Structure Continued!
• NOTE!! Much of the discussion & many figures in what
follows were 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/.
Crystal Lattices
Bravais Lattices
Non-Bravais Lattices
(BL)
(non-BL)
All atoms are the same kind
All lattice points are equivalent
Atoms are of different kinds.
Some
lattice
points
aren’t
equivalent.
Atoms
are of
different
kinds.
Some
A combination
2 or
more BL
lattice
points areofnot
equivalent.
2 d examples
Lattice Translation Vectors
In General
• Mathematically, a lattice is defined by 3 vectors called
Primitive Lattice Vectors
a1, a2, a3 are 3d vectors which depend on the geometry.
• Once a1, a2, a3 are specified, the
Primitive Lattice Structure
is known.
• The infinite lattice is generated by translating through a
Direct Lattice Vector: T = n1a1 + n2a2 + n3a3
n1,n2,n3 are integers. T generates the lattice points. Each
lattice point corresponds to a set of integers (n1,n2,n3).
2 Dimensional Lattice Translation Vectors
Consider a 2-dimensional lattice (figure). Define the
2 Dimensional Translation Vector: Rn  n1a + n2b
(Sorry for the notation change!!)
a & b are 2 d Primitive Lattice Vectors, n1, n2 are integers.
Point D(n1, n2) = (0,2)
Point F(n1, n2) = (0,-1)
• Once a & b are specified by the
lattice geometry & an origin is
chosen, all symmetrically equivalent
points in the lattice are determined by
the translation vector Rn. That is, the
lattice has translational symmetry.
Note that the choice of Primitive
Lattice vectors is not unique! So,
one could equally well take vectors a
& b' as primitive lattice vectors.
The Basis
(or basis set)
 The set of atoms which, when placed at each
lattice point, generates the Crystal Structure.
Crystal Structure
≡ Primitive Lattice + Basis
Translate the basis through all possible lattice vectors
T = n1a1 + n2a2 + n3a3
to get the Crystal Structure or the
Direct Lattice
• The periodic lattice symmetry is such that
the atomic arrangement looks the same from
an arbitrary vector position r as when viewed
from the point
r' = r + T
(1)
where T is the translation vector for the lattice:
T = n1a1 + n2a2 + n3a3
• Mathematically, the lattice & the vectors a1,a2,a3 are
Primitive
if any 2 points r & r' always satisfy (1) with
a suitable choice of integers n1,n2,n3.
• In 3 dimensions, no 2 of the 3 primitive lattice
vectors a1,a2,a3 can be along the same line. But,
DO NOT think of a1,a2,a3 as a mutually
orthogonal set! Often, they are neither mutually
perpendicular nor all the same length!
• For examples, see Fig. 3a (2 dimensions):
The Primitive Lattice Vectors a1,a2,a3 aren’t
necessarily a mutually orthogonal set!
Often, they are neither mutually
perpendicular nor all the same length!
• For examples, see Fig. 3b (3 dimensions):
Crystal Lattice Types
Bravais Lattice 
An infinite array of discrete points with an
arrangement & orientation that appears exactly the
same, from whichever of the points the array is
viewed. A Bravais Lattice is invariant under a
translation T = n1a1 + n2a2 + n3a3
Nb film
Non-Bravais Lattices
•In a Bravais Lattice, not only the atomic
arrangement but also the orientations must appear
exactly the same from every lattice point.
2 Dimensional Honeycomb Lattice
• The red dots each have a neighbor
to the immediate left. The blue dot
has a neighbor to its right. The red
(& blue) sides are equivalent &
have the same appearance. But, the
red & blue dots are not equivalent. If
the blue side is rotated through 180º
the lattice is invariant.
 The Honeycomb Lattice is
NOT a Bravais Lattice!!
Honeycomb
Lattice
It can be shown that, in 2 Dimensions, there are
Five (5) & ONLY Five Bravais Lattices!
2-Dimensional Unit Cells
Unit Cell  The Smallest Component
of the crystal (group of atoms, ions or molecules),
which, when stacked together with pure translational
repetition, reproduces the whole crystal.
b
S
a
S
S
S
S
S
S
S
S
S
S
S
S
S
S
Unit Cell  The Smallest Component
of the crystal (group of atoms, ions or molecules),
which, when stacked together with pure translational
repetition, reproduces the whole crystal.
Note that the choice of unit cell is not unique!
S
S
S
2-Dimensional Unit Cells –
Artificial Example: “NaCl”
Lattice points are points with
identical environments.
2-Dimensional Unit Cells: “NaCl”
Note that the choice of origin is arbitrary!
the lattice points need not be atoms, but
The unit cell size must always be the same.
2-Dimensional Unit Cells: “NaCl”
These are also unit cells!
It doesn’t matter if the origin is at Na or Cl!
2-Dimensional Unit Cells: “NaCl”
These are also unit cells.
The origin does not have to be on an atom!
2-Dimensional Unit Cells: “NaCl”
These are NOT unit cells!
Empty space is not allowed!
2-Dimensional Unit Cells: “NaCl”
In 2 dimensions, these are unit cells.
In 3 dimensions, they would not be.
2-Dimensional Unit Cells
Why can't the blue triangle be a unit cell?
Example: 2 Dimensional, Periodic Art!
A Painting by
Dutch Artist Maurits Cornelis Escher (1898-1972)
Escher was famous for
his so called “impossible
structures”, such as
Ascending &
Descending, Relativity,..
Can you find the “Unit Cell” in this painting?
3-Dimensional Unit Cells
3-Dimensional Unit Cells
3-Dimensional Unit Cells
3 Common Unit Cells with Cubic Symmetry
Simple Cubic
(SC)
Body Centered
Cubic (BCC)
Face Centered
Cubic (FCC)
Conventional & Primitive Unit Cells
Unıt Cell Types
Primitive
Conventional
(Non-primitive)
A single lattice point per cell
More than one lattice point per cell
The smallest area in 2 dimensions, or
The smallest volume in 3 dimensions
Volume (area) = integer multiple of
that for primitive cell
Simple Cubic (SC)
Conventional Cell =
Primitive cell
Body Centered Cubic (BCC)
Conventional Cell ≠
Primitive cell
Face Centered Cubic (FCC)
Structure
Conventional Unit Cells
• A Conventional Unit Cell just fills space when
translated through a subset of Bravais lattice vectors.
• The conventional unit cell is larger than the primitive
cell, but with the full symmetry of the Bravais lattice.
• The size of the conventional cell is given by the lattice
constant a.
FCC Bravais Lattice
The full cube is the
Conventional Unit
Cell for the FCC
Lattice
Conventional & Primitive Unit Cells
Face Centered Cubic Lattice
Primitive Unit Cell
(Shaded)
Lattice Const.
Primitive Lattice
Vectors
a1 = (½)a(1,1,0)
a2 = (½)a(0,1,1)
a3 = (½)a(1,0,1)
Note that the ai’s are
Conventional Unit
Cell (Full Cube)
NOT Mutually
Orthogonal!
Elements That Form Solids
with the FCC Structure
Body Centered Cubic (BCC) Structure
Conventional & Primitive Unit Cells
Body Centered Cubic Lattice
Primitive Lattice
Primitive Unit Cell
Vectors
a1 = (½)a(1,1,-1)
a2 = (½)a(-1,1,1)
Lattice a3 = (½)a(1,-1,1)
Constant
Conventional Unit
Cell (Full Cube)
Note that the ai’s are
NOT mutually
orthogonal!
Elements That Form Solids
with the BCC Structure
Conventional & Primitive Unit Cells
Cubic Lattices
Simple Cubic (SC)
c
b
Primitive Cell = Conventional Cell
Fractional coordinates of lattice points:
000, 100, 010, 001, 110,101, 011, 111
a
b
c
Body Centered Cubic (BCC)
Primitive Cell  Conventional Cell
a
b
c
Fractional coordinates of the lattice points
in the conventional cell: 000,100, 010,
001, 110,101, 011, 111, ½ ½ ½
a
Primitive Cell = Rombohedron
Conventional & Primitive Unit Cells
Cubic Lattices
Face Centered Cubic (FCC)
Primitive Cell  Conventional Cell
The fractional coordinates of lattice
points in the conventional cell are:
000,100, 010, 001,
110,101, 011, 111,
½ ½ 0, ½ 0 ½, 0 ½ ½,
½ 1 ½, 1 ½ ½ , ½ ½ 1
b
c
a
Simple Hexagonal Bravais Lattice
Conventional & Primitive Unit Cells
Points of the Primitive Cell
Hexagonal Bravais
Lattice
Primitive Cell =
Conventional Cell
c
b
a
Fractional coordinates of lattice
points in conventional cell:
100, 010, 110, 101,
011, 111, 000, 001
Hexagonal Close Packed (HCP) Lattice:
A Simple Hexagonal Bravais Lattice with
a 2 Atom Basis
The HCP lattice is not a
Bravais lattice, because the
orientation of the environment of a
point varies from layer to layer
along the c-axis.
General Unit Cell Discussion
• For any lattice, the unit cell &, thus,
the entire lattice, is UNIQUELY
determined by 6 constants (figure):
a, b, c, α, β and γ
which depend on lattice geometry.
• As we’ll see, we sometimes want to
calculate the number of atoms in a unit cell.
To do this, imagine stacking hard spheres
centered at each lattice point & just
touching each neighboring sphere. Then,
for the cubic lattices, only 1/8 of each
lattice point in a unit cell is assigned to that
cell. In the cubic lattice in the figure,
Each unit cell is associated with
(8)  (1/8) = 1 lattice point.
Primitive Unit Cells & Primitive Lattice Vectors
• In general, a Primitive Unit
Cell is determined by the
parallelepiped formed by the
Primitive Vectors a1 ,a2, &
a3 such that there is no cell
of smaller volume that can
• The Primitive Unit Cell
volume can be found by
• As we’ve discussed, a Primitive vector manipulation:
V = a1(a2  a3)
Unit Cell can be repeated to fill
space by periodic repetition of it • For the cubic unit cell in
through the translation vectors
the figure, V = a3
be used as a building block
for the crystal structure.
T = n1a1 + n2a2 + n3a3.
Primitive Unit Cells
• Note that, by definition, the Primitive Unit Cell
must contain ONLY ONE lattice point.
• There can be different choices for the Primitive
Lattice Vectors, but the Primitive Cell volume must
be independent of that choice.
2 Dimensional
Example! j
P = Primitive Unit Cell
NP = Non-Primitive
Unit Cell