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

3D Bravais Lattices, Lattice Planes and the Reciprocal Lattice

Shyue Ping OngDepartment of NanoEngineeringUniversity of California, San Diego

Readings¡Chapter 3 of Structure of Materials

NANO 106 - Crystallography of Materials 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 ab = Length of bc = Length of cα =Angle between b and cβ =Angle between a and cγ =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 ≠ cα ≠ 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 = bα = β = 90°γ =120°

Example: ZnO{3.25Å, 3.25Å, 5.21Å, 90°, 90°, 120°}

Rhombohedral/Trigonal lattice


a = b = cα = 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 ≠ cα = β = γ = 90°

Example: SnO2{3.784Å, 3.784Å 9.514Å, 90°, 90°, 90°}

Cubic lattice


a = b = cα = 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


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 but one 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: primitiveC: C-centeredI: body-centeredF: face-centered(upper case for 3D)

a: triclinic (anorthic)m: monoclinico: orthorhombict: tetragonalh: hexagonalc: 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


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 = pTgq

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 accosβabcosγ b2 bccosαaccosβ bccosα c2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,gmonoclinic =

a2 0 accosβ0 b2 0

accosβ 0 c2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

gorthorhombic =a2 0 00 b2 00 0 c2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,gtetragonal =

a2 0 00 a2 00 0 c2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟,gcubic =

a2 0 00 a2 00 0 a2

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

ghexagonal =

a2 − a2

20

− a2

2a2 0

0 0 c2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟

,grhombohedral =a2 a2 cosα a2 cosα

a2 cosα a2 a2 cosαa2 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

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