Crystallography
Part 1: Crystal lattice and Indices
Part 2: Morphology and Symmetry Basis
Part 3: Crystal Class and Bravais lattice
Part 4: Crystal Point Group
Part 5: Crystal Space Group
Part 6: Reciprocal Space
Reference: Kristallographie/ Crystallography by W. Borchardt-Ott
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Part 1-1: Crystal lattice
1.1 Crystalline Status
1.2 Crystal structure
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Crystal status
Gas:
form- and volume variable
statistic homogeneity
isotropy
Liquid:
form variable, volume invariant
statistic homogeneity
isotropy
Crystal:
form- and volume constant
periodic homogeneity
anisotropy
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Basis conception: Crystal status
Ein Kristall ist ein anisotroper Körper, der eine
dreidimensional periodische Anordnung der Bausteine
besitzt.
A crystal is an anisotropic, homogenous body consisting of a
three-dimensional periodic ordering of atoms, ions or
molecules.
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Anisotropy
Anisotropy - different values of a physical property in different directions
isotropy
anisotropy
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Homogeneity
Statistic homogeneity
Number of lentils/buttons in certain
area tends to be the same as the
areas considered become larger
: gases, liquids
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Periodic homogeneity
On each square, there are
precisely two lentils, periodically
arranged with respect to each
other.
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Crystal structure - Basic conception
Atoms
A
B
C
b
Crystallstrukture
a
Lattice constant:
a
b
c(not shown)
=
Basis
+
Gitter/Lattice
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Crystal structure - Definition
Basis: the arrangement of atoms within a unit cell.
Basis
Lattice translation: reproduce the atoms throughout the entire lattice.
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Number of chemical formula units per unit cell - Z
An important quantity of any structure is Z, the number of chemical formula
units per unit cell.
Cs+
Cs+
I-
I-
Caesium iodide: CsI
Density - Z:
ρ=
Z=1:
one Cs+ ion and
one I- ion
per cell
Simplified by gravity center
m
g ⋅ cm −3
V
Z .M
m=
NA
ρ=
Z ⋅M
g ⋅ cm −3
N A ⋅V
m: the mass of atoms in unit cell
V: the volume of the cell
The mass of one chemical formula is M/NA,
where M is the molar mass
NA is the Avogadro number(6.023×10-23 mol-1.
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Number of atoms in unit cell - N
atoms in corner: 1/8 for cubic unit cell
atoms in face center: 1/2
atoms in body center: 1
N=1/8 × 8
=1 atom
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N=1/8 × 8 + 1
=2 atoms
N=1/8 × 8 + 1/2 × 6
=1 + 3
=4 atoms
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Summary - Lattice structure
1. Crystal status: 3-dimensional periodisch, anisotrop
2. Cristal structure = lattice + Basis
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Part 1-2: Crystal Indice
1. Lattice point
2. Lattice direction
3. Lattice plane - Miller Indice
4. Zone
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Lattice point
Coordinate:
For example:
r
r
v
v
τ = ua + vb + wc
• uvw •
100
110
111
A unit cell is described by six lattice parameters:
Length of lattice translation vectors
IaI=a0
IbI=b0
IcI=c0
interaxial lattice angles
a∧b=γ
a∧c=β
b∧c=α
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τ
c
αb
β
a
γ
Crystallographic axes a, b and c
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Lattice direction - 1
Direction indice:
[uvw]
[100]
[010]
[001]
[111]
For example:
c
α
β
a
γ
[uvw] describe:
b
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A lattice line through the origin and the point uvw.
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Lattice direction - 2
[uvw] describes:
Infinite set of lines which are parallel to it
and have the same lattice parameter
b
The smallest triple of the ratio u:v:w
is used to define the lattice line.
I.e. [240]→[120]
a
[310]
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[110]
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Lattice direction indice - indexing steps
[111]
c
a
1,1/3,0
b
1,-1/3,0
1. Decide the origin according to the sign of coordinate;
2. Find the lattice point;
3. Connect the origin and the point.
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Lattice plane - definition
Miller Indice:
(hkl);
The smallest integral multiples of the
reciprocals of the plane intercepts on the
axes.
c
C
For example:
m=2, 1/m=1/2
n=5, 1/n=1/5
p=2, 1/p=1/2
M
O
a
A
b
B
× smallest integral multiples=10
h=5, k=2, l=5
(525)
Miller indice (hkl) is integral and teilerfremd.
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Draw a plane with known indice
c
C
For example: (525)
h=5, 1/h=1/5
k=2, 1/k=1/2
l=5, 1/L=1/5
× smallest integral multiple=10
O
a
intercept:
m=2, n=5, p=2
the intercepts are:
m‘=2/5
n‘=1
p‘=2/5
0,0,2/5
2/5,0,0
B
A
c
then in one unit cell:
b
Steps:
1. Reciprocal of indice hkl
2. Times multiplie
3. Connect the intercepts
0,1,0
b
a
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Lattice plane - Miller indice characteristics
The Miller indice (hkl) represents not only the position of a single plane, but
also the position of an infinite set of parallel planes.
The plane with higher Miller indice has smaller inter-plane spacing.
(100)
(100)
(110)
(110)
(210)
(210)
b
a
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(310)
(310)
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Zone - Definition
Zone: A set of planes whose intersection lines are all parallel to each
other.
The inter-section line is called Zone axis.
[uvw]
[uvw]
Zone axis
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Zonal equation
Zonal Equation(Zonengleichung):
A crystal plane (hkl) belong to one Zone [uvw]
when:
For example:
hu + kv + lw = 0
Topas
For example:
zone [120]: (21w), (21w)
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Zonal equation - application
A) Two lattice lines [u1v1w1] and [u2v2w2] will describe a lattice plane (hkl):
u1
u2
v1
v2
w1
w2
V1w2-v2w1
u1 v1 w1
u2 v2 w2
w1u2-w2u1
h
k
u1v2-u2v1
[121]
l
[101]
B) Two lattice planes (h1k1l1) and (h2k2l2) intersect in the lattice line [uvw]:
h1
h2
k1l2-k2l1
k1
k2
l1
l2
h1
h2
l1h2-l2u1
u
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v
k1
k2
l1
l2
h1k2-h2k1
w
Try (101) and (112)
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Special case: Hexagonal - 1
Axis system
a1 = a2 ≠ c
α = β = 90°, γ = 120°
=> Miller Indice (hkl)
c
a2
a1
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a2
a1
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Special case: Hexagonal - 2
Axis system
=> Miller-Bravais- Indice
(hkil)
[uvtw]
c
a1 = a2 = a3 ≠ c
a3
a2
a3
a1
Convert: Miller-indice ↔ Miller Bravais-indice
Plane:
h + k + i = 0, ↔ i = -(h + k)
Direction:
[UVW] = [u-t v-t w] = [2u+v u+2v w]
a1
[uvtw] = [(2U-V)/3 (2V-U)/3 (-U-V)/3 W]
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a2
Special case: Hexagonal - indice
[120]:
u=1
v=2
w=0
a2
a1
c
(123) or (1233):
h=1
k=2
l=3
a3
a1
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a2
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Summary - Indice
diskret
symmetry equivalent
Lattice point
• uvw •
intercept
: uvw :
Lattice direction
[uvw]
intercept
<uvw>
Lattice plane
(Miller Indice)
(hkl)
inverse of intercept
{hkl}
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