Introduction to crystallography

What shall we study?

• Lattice, Motif, Crystal structures, Bravais lattices
• Plane and directions in crystals
• Packing and voids
• Ordered structures
• Symmetry, point group, space group
• Basics of stereographic projection

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What is crystallography??????

• According to WIKI: Crystallography is the science of the arrangement of atoms in solids.
• Taxonomy: The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.

Brighter atoms are Cu and darker ones are Al.

Σ3 boundaries and a Σ9 boundary in nanocrystalline palladium

Radial distribution function

In statistical mechanics, the radial distribution function, (or pair correlation function) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.

Amorphous vs.crystalline

Diffraction pattern from a thin film of amorphous carbon

Why should we study crystallography?????

• Atomic level: Electronic configuration: Elemental properties e.g. Semiconductors, Magnetic materials
• Bonding: Intrinsic properties e.g. Elastic modulus, Polymers (Graphene), Ceramics
• Crystal structure: Extrinsic properties
• Smart materials (Pizoelectric, ferroelectric)
• Bio-medical: Protein structure
• Anisotropic behaviour of single crystals
• Polycrystalline nature of most of the material

Beauty of crystals

What makes them?

Johannes Kepler hypothesized in “Strena seu de Nive Sexangula” (1611) that the hexagonal symmetry of snowflake crystals was due to a regular packing of spherical water particles.

So what is crystal ?????

Short range order

No order

Long range order

Short range order

What is a Crystalline solid?

• A crystalline solid is a solid material, whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions.
• For example:
• Non-Metallic crystals: Ice, Carbon, Diamond, NaCl, KCl etc…
• Metallic Crystals: Copper, Silver, Aluminium, Tungsten, Magnesium etc…
• An ideal crystal is a periodic array of structural units
• It can be constructed by the infinite repetition of these identical structural units in space.
• Structure can be described in terms of a lattice, with a group of atoms attached to each lattice point.
• The group of atoms is the basis.

Space filling

Lattice

One dimensional lattice

Two dimensional lattice

Two Dimensional Lattices

• There is an unlimited number of possible lattices, since there is no restriction on the lengths of the lattice translation vectors or on the angle between them.

Three dimensional lattice

Basis

• A group of atoms or molecules identical in composition is called the basis.

OR

• A group of atoms which describe the crystal structure.

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 (crystalline solid).

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Primitive Unit Cell

• A primitive cell or primitive unit cell is a volume of space that when translated through all the vectors in a Bravais lattice just fills all of space without either overlapping itself or leaving voids.
• Primitive cell must contain precisely one lattice point

Choice of origin is arbitrary - but unit cell size should always be the same.

This is NOT a unit cell even though they are all the same - empty space is not allowed!

Crystal structure

• Crystal structure can be obtained by attaching atoms, groups of atoms or molecules which are called basis (motif) to the lattice sides of the lattice point.

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Crystal lattice + basis = Crystal structure

Lattice parameters

Crystal systems

Bravais lattice

• Considering
• Maximum Symmetry
• Minimum Size
• There are not more than 4 ways of arranging spheres in any shape of unit cell
1. 8 Corners (P)
2. 8 Corners and 1 body center (I)
3. 8 Corners and 6 face centers (F)
4. 8 corners and 2 centers of opposite faces (A/B/C)

14 Bravais lattices

• In 1850, M. A. Bravais showed that identical points can be arranged spatially to produce 14 types of regular pattern. These 14 space lattices are known as ‘Bravais lattices’.

Why you can’t get a base centered cubic unit cell?

Nitrogen - simple cubic

Face centered cubic

Body centered cubic

Tetragonal (P) �

Body centered Tetragonal (BC)

Orthorhombic (Simple)

Orthorhombic (Base-centred)�

Orthorhombic (BC)

Orthorhombic (FC)

Lanthanum (La) - hexagonal

Ice - hexagonal

Rhombohedral (R) or Trigonal (S)

• Triclinic minerals are the least symmetrical.
• Three axes are all of different lengths, and none of them are perpendicular to each other.
• These minerals are the most difficult to recognize.

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Monoclinic (Simple)

Triclinic (Simple)

Monoclinic (Base Centered) �

14 BRAVAIS Lattices

Coordination Number �

• Bravais lattice points closest to a given point are the nearest neighbours.
• Total number of nearest neighbors = Coordination number.
• Because the Bravais lattice is periodic, all points have the same number of nearest neighbours or coordination number.
• It is a property of the lattice.

Coordination polyhedra

Space filling polyhedra

Archimedean solids

Atomic Packing Factor

Effective number of atoms

�Simple Cubic (SC)�

• Simple Cubic has one lattice point so its primitive cell.
• Only a portion (1/8) belongs to that cell.
• The rest of the atom belongs to neighboring cells.
• Coordinatination number of simple cubic is 6.

Atomic Packing Factor

• APF for a simple cubic structure = 0.52

Coordination Number & Atomic Packing Factor �

Exercise

Assuming atoms are represented by hard spheres, calculate the atomic packing factor for a material having BCC and FCC structures.

Body Centered Cubic (BCC)

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• BCC has two lattice points so BCC is a non-primitive cell
• BCC has eight nearest neighbors. Each atom is in contact with its neighbors only along the body-diagonal directions
• Many metals (Fe, Li, Na.. etc), including the alkalis and several transition elements possess the BCC structure.

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Atomic Packing Factor: BCC

• APF for a body-centered cubic structure = π√3/8 = 0.68

Face Cantered Cubic (FCC)

• There are atoms at the corners of the unit cell and at the center of each face
• Face centered cubic has 4 atoms so its non primitive cell
• Many of common metals (Cu, Ni, Pb etc.) crystallize in FCC structure

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Atomic Packing Factor: FCC

• APF for a body-centered cubic structure = π/(3√2) = 0.74

(best possible packing of identical spheres)

The primitive, body-centred and face-centred cubic unit cells

Close Packed Crystals

• The term close packed crystal implies closest packed crystal (having a packing fraction of 0.74).
• Cubic Close Packed (CCP- commonly called FCC crystal) and Hexagonal Close Packed (HCP) are two common examples of close packed crystals.
• Every atom in these structures has a coordination number of 12.
• The common starting point is a close packed layer of atoms with 6-fold symmetry.
• Identical layers are stacked one on another with a shift.
• The shift is such that the atoms in the above (and below) layers sit on valleys formed by a layer.

Square grid vs. Hexagonal grid

• Square packing covers 78% of the area, while hexagonal packing yields 91% coverage.

• Starting Point → Hexagonal layer.
• Three positions A (the first layer atomic positions), B & C (Valleys)
• The second layer (of hexagonal packing of atoms) can be positioned in valley B (or equivalently in valley C).

Step-1

Step-2

A

AB

• The third layer can be positioned with atoms directly above the A layer (Option-1) or with atoms above the C layer (Option-2).

Step-3

(Option-1)

(Option-2)

C-site vacant

ABC

ABA

ABAB stacking: HCP structure

ABCABC stacking: CCP structure

FCC and HCP: Unit cell & close-packing

• ABCABC… & ABAB… are just but two amongst the infinite possibilities
• At each stage of construction an atomic layer can be added at A, B or C position
• Possibilites include (but not limited to):�ABCAB/ABCAB/ABCAB…� ABCABCAB/ABCABCAB/ABCABCAB…
• Crystals with larger and larger unit cells (e.g. SiC, Alumina)

Atomic Packing Fraction: HCP

Ordered structures (superlattice)

L1₂ ≡ Cu₃Au

L1₀ ≡ CuAu

L2₀ ≡ B₂ ≡ CuZn

DO₃ ≡ Fe₃Al

L2₀ ≡ Cd₃Mg

Order-disorder transformation

• Atomic disorder→order transformation: change of phase
• Change in the crystallographic symmetry of the disordered (high temperature) phase to usually a less symmetric (low temperature), atomically ordered phase.

High Temperature, disordered phase (FCC)

Low Temperature, ordered phase (L1₀)

Order parameter

• Order parameter is normally a quantity which is zero in one phase (usually above the critical point), and non-zero in the other
• It characterises the onset of order at the phase transition
• For a ferromagnetic system undergoing a phase transition, the order parameter is the net magnetization
• For liquid/gas transitions, the order parameter is the difference of the densities
• For crystal structures, it is the change in lattice parameter (determined from super-lattice diffraction peaks in XRD)

Location of a point

Atomic positions

Point coordinates for a BCC

Crystal Directions

• We choose one lattice point on the line as an origin
• Choice of origin is completely arbitrary, since every lattice point is identical.
• Then we choose the lattice vector joining O to any point on the line, say point T. This vector can be written as:

R = la + mb + nc

• The length of the vector projection on each of the three axes is determined; these are measured in terms of the unit cell dimensions a, b, and c.
• These three numbers are multiplied or divided by a common factor to reduce them to the smallest integer values.
• To distinguish a lattice direction from a lattice point, the triple is enclosed in square brackets e.g. [l, m, n]
• [l, m, n] is the smallest integer of the same relative ratios.
• Negative directions can be expressed by shifting the origin
• http://www.materials.ac.uk/elearning/matter/crystallography/indexingdirectionsandplanes/indexing-lattice-directions.html

Examples of crystal directions

X = 1 , Y = 0 , Z = 0 ► [1 0 0]

Which one is direction [112]?

Crystal Directions in Cubic system

There are 3 special directions in a cubic crystal. These directions are perpendicular to the associated planes

Miller Indices for Crystallographic Planes

• William Hallowes Miller in 1839 was able to give each face a unique label of three small integers, the Miller Indices.
• The orientation of a crystal plane is determined by three points in the plane that are not collinear to each other.
• Definition: Miller Indices are the reciprocals of the fractional intercepts (with fractions cleared), which the plane makes with the crystallographic x, y, z axes of the three nonparallel edges of the cubic unit cell.

How to determine Miller Indices

Crystallographic planes – Miller indices

• intercepts = 2, 1, ∞
• invert = 1/2, 1, 0
• smallest whole number set 1, 2, 0

(120)

Interrelation between directions and planes

• Cubic system: (hkl) ⊥ [hkl]
• Tetragonal system: only special planes are ⊥ to the direction with same indices:�[100] ⊥ (100), [010] ⊥ (010), [001] ⊥ (001), [110] ⊥ (110)�([101] not ⊥ (101))
• Orthorhombic system: �[100] ⊥ (100), [010] ⊥ (010), [001] ⊥ (001)
• Hexagonal system: [0001] ⊥ (0001) �(this is for a general c/a ratio; for a Hexagonal crystal with the special c/a ratio = √(3/2) the cubic rule is followed)
• Monoclinic system: [010] ⊥ (010)
• Other than these a general [hkl] is NOT ⊥ (hkl).

Family of directions : Multiplicity

Family of planes : Multiplicity

Family of planes : Multiplicity

Multiplicity factor

* Altered in crystals with lower symmetry (of the same crystal class)

Summary of notations

A family is also referred to as a symmetrical set

Plane spacing

Angle between planes

Planar & linear atomic density

{100}: Face planes

• The densest-packed planes can dictate growth, surface chemistry, and deformation
• Lowest surface energy (facets)
• Planes for dislocations movement (slip) and twining
• Planes on which cracks form (cleavage planes)

End