UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations

3D Bravais Lattices, Lattice Planes and the
Reciprocal Lattice
Shyue Ping Ong
Department of NanoEngineering
University of California, San Diego

Readings
¡Chapter 3 of Structure of Materials
NANO 106 - Crystallography ofMaterials by Shyue Ping Ong - Lecture 2

Brief Recap
¡Concepts of symmetry – translational and
rotational
¡Lattice vectors and parameters
¡Derivation of the five 2D Bravais nets
¡Definition of unit cells

Derivation of 3D Bravais lattices
¡ Following the same general principle as 2D Bravais nets,
we consider possible values for the six lattice parameters
¡ All 3D crystal systems can be derived by adding an
additional vector to existing an 2D crystal system
a = Length of a
b = Length of b
c = Length of c
α = Angle between b and c
β = Angle between a and c
γ = Angle between a and b
a,b,c,α,β,γ{ }

Conventional for Crystallographic Axes
¡ Always right-handed
¡ c lattice vector is usually the direction of highest rotational
symmetry (if unique)
a.(b×c) > 0
a
b
c

Triclinic lattice
¡ No special conditions
¡ Lowest symmetry (no rotational
symmetry at all)
a ≠ b ≠ c
α ≠ β ≠γ
Example: K2Cr2O7
{7.574Å, 7.632Å, 13.661Å, 95.374°,
97.483°, 90.955°}

Monoclinic lattice
¡ What symmetry does the
monoclinic lattice have? (hint:
think of the relationship with a
particular 2D net)
α =γ = 90°
β ≠ 90° Example: Monoclinic sulfur
{10.668Å 10.690Å 10.816Å, 90°, 95.637°, 90°}

Hexagonal lattice
a = b
α = β = 90°
γ =120°
Example: ZnO
{3.25Å, 3.25Å, 5.21Å, 90°, 90°, 120°}

Rhombohedral/Trigonal lattice
a = b = c
α = β =γ ≠ 90°
Example: CaCO3
{6.36Å, 6.36Å, 6.36Å, 46.1°, 46.1°, 46.1°}
Note: All rhombohedral cells
can be represented in a
hexagonal setting

Orthorhombic lattice
α = β =γ = 90°
Example: BaSO4
{8.87Å, 5.54Å, 7.15Å, 90°, 90°, 90°}

Tetragonal lattice
a = b ≠ c
α = β =γ = 90°
Example: SnO2
{3.784Å, 3.784Å 9.514Å, 90°, 90°, 90°}

Cubic lattice
a = b = c
α = β =γ = 90°
Example: CsCl
{4.115Å, 4.115Å, 4.115Å, 90°, 90°, 90°}

File formats and Viewing Software for
Crystals
¡ VESTA (Visualization for Electronic and STructural
Analysis)
¡ Open-source 3D visualization program for structural models,
volumetric data such as electron/nuclear densities, and crystal
morphologies.
¡ Create lattice planes, calculate atomic distances, etc.
¡ Download at http://jp-minerals.org/vesta/en/.
¡ Main supported file formats used in NANO106
¡ Crystallographic information file (CIF) for crystals
¡ XYZ files for molecules (mainly for demonstration of point group
symmetries)

Where to get Crystal Structures
¡Inorganic Crystal Database (ICSD)
¡ http://icsd.fiz-karlsruhe.de
¡The Materials Project
¡ https://www.materialsproject.org/

Demo
NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2

3D crystal systems
¡ The lattices you have seen thus far are all primitive and corresponds
to the seven 3D crystal systems.
¡ The 3D crystal systems can be ranked according to their symmetry.
Distort along [001]
direction.
Distort along [111]
direction to make
all angles ≠ 90.
Make all lengths
unequal butone lattice
vector perpendicular to
other two Make all angles
unequal
Distort along
[100] or [010].
Decreasing
symmetry
Make one angle ≠
90.
Make lengths unequal
and all angles = 90.

The 7 3D Crystal Systems and
Primitive Lattices

Adding new lattice nodes
¡ Remember how we derived the additional centered
rectangular net by adding a node to the primitive
rectangular net?
¡ In 3D, there are five possible locations where an
additional node can be placed:
¡ In the center of the cell at (½, ½, ½) [Body centering, denoted by
the symbol I]
¡ At the center of all faces of the cell at (½, ½, 0), (½, 0, ½), (0, ½, ½)
[Face centering, denoted by the symbol F]
¡ At the center of one of the faces at (½, ½, 0) or (½, 0, ½) or (0, ½,
½) [Base centering, denoted by the symbols A, B or C,
depending on whether the additional node is on the b-c, a-c or
a-b base planes respectively.]

Generating additional 3D lattices
¡ We now apply all five node placements (I, F, A, B, and C) to all seven
primitive 3D lattices (35 possibilities).
¡ Not all node placements are compatible with all primitive 3D lattices,
and some do not result in a new lattice.
Example: Tetragonal lattice
• Analogous to the 2D square net, C-
centering does not result in a new
lattice.
• A or B centering can be redefined into
monoclinic cell.
• The I node placement result in a new
tetragonal lattice.
• F-centering does not result in a new
lattice (problem set exercise)

The 14 3D Bravais Lattices
P: primitive
C: C-centered
I: body-centered
F: face-centered
(upper case for 3D)
a: triclinic (anorthic)
m: monoclinic
o: orthorhombic
t: tetragonal
h: hexagonal
c: cubic

3D unit cells
¡ Infinite number of unit cells for all 3D lattices
¡ Always possible to define primitive unit cells for non-primitive lattices,
though the full symmetry may not be retained.
¡ The Wigner-Seitz cell is similarly defined for 3D.
Conventional cF cell Primitive unit cell Wigner-Seitz cell

Crystallographic
Calculations
NANO 106 - Crystallography of Materials by Shyue Ping Ong - Lecture 2

Readings
¡Chapter 4 of Structure of Materials

Positions and Directions in Lattices
¡Recall that the lattice vectors form a basis for 3D
space, i.e., any point in 3D space can be
represented as a linear combination of the lattice
vectors
¡We then denote the coordinates as (u, v, w)
ua + vb+ wc

Calculating Distances in Lattices
¡Consider the (1, 1, 1) points in two lattices, one
cubic and another triclinic. What is the distance
from the origin (0, 0, 0) to (1, 1, 1)?

Calculating Distances in Lattices
a) Effectively a Cartesian frame. From Pythagoras’ Theorem, we can calculate the
distance as
b) We can convert this to a Cartesian frame by multiplying by the basis vectors,
and then calculating the distance
Though this is a perfectly valid approach, we will introduce an alternative, general
method of calculating distances using what is known as a metric tensor (for now,
just think of this as a 3x3 matrix, we will get into a more formal definition of a tensor
later)
d = a2
+ a2
+ a2
= a 3
d = 1a +1b+1c

Derivation of the Metric Tensor
Blackboard

Using the metric tensor in dot products
¡By an extension of the previous derivation, we
can see that a dot product in crystal coordinates
can be computed as:
x.y = pT
gq

Form of the metric tensor
¡How can we write the metric tensor in terms of
the lattice parameters?
¡What is the special form of the metric tensor for a
hexagonal lattice? What about the orthorhombic
and tetragonal lattices?
Blackboard

Metric Tensors for the 7 Crystal
Systems
gtriclinic =
a2
abcosγ accosβ
abcosγ b2
bccosα
accosβ bccosα c2
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
,gmonoclinic =
a2
0 accosβ
0 b2
0
accosβ 0 c2
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
gorthorhombic =
a2
0 0
0 b2
0
0 0 c2
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
,gtetragonal =
a2
0 0
0 a2
0
0 0 c2
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
,gcubic =
a2
0 0
0 a2
0
0 0 a2
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
ghexagonal =
a2
−
a2
2
0
−
a2
2
a2
0
0 0 c2
⎛
⎝
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
⎟
⎟
⎟
,grhombohedral =
a2
a2
cosα a2
cosα
a2
cosα a2
a2
cosα
a2
cosα a2
cosα a2
⎛
⎝
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟

Example
¡ You have a orthorhombic lattice with lattice parameters {4,
6, 8, 90, 90, 90}.
¡ What is the length of a vector from the origin to (0.5, 0.5, 0.5)?
¡ You have two atoms P and Q at locations (0.2, 0.2, 0.2) and (0.4,
0.5, 0.6) respectively. What is the distance between them? What is
the angle between them and the origin, i.e., the angle POQ?
Blackboard example

UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Andere mochten auch

Andere mochten auch (18)

Ähnlich wie UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations

Ähnlich wie UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations (20)

Mehr von University of California, San Diego

Mehr von University of California, San Diego (16)

Kürzlich hochgeladen

Kürzlich hochgeladen (20)

UCSD NANO106 - 02 - 3D Bravis Lattices and Lattice Computations