Solid State Physics - Crystal Structure Notes

Solid Materials

• Crystalline: Exhibits long-range order.
• Amorphous (Non-crystalline): Exhibits short-range order.
• Distinguished by the size of the ordered region.
• The structure of materials determines mechanical, thermal, electrical, and magnetic properties.
• A crystal is a solid composed of atoms or microscopic particles arranged in an orderly, repetitive array.
• Examples of amorphous materials: glass, plastic, and gel.

Elementary Crystallography

• Short-range order vs. Long-range order.
• Crystal Structure = Crystal Lattice + Basis.

Crystal Lattice

• An infinite periodic array of points in space with identical surroundings, referred to as lattice points.
• A 3-D (or 2-D) representation of lattice points in space is called a Space Lattice.
• If all points are identical, it's a Bravais Lattice.

Crystal Structure

• Formed by attaching atoms, groups of atoms, or molecules (the basis or motif) to the lattice points of the crystal lattice.
• $Crystal Structure = Crystal Lattice + Basis$
• Examples:
  • Cu and Na: Basis is a single atom.
  • NaCl, CsCl: Basis is diatomic.
  • $CaF_2$: Basis is triatomic.

Bravais Lattice vs. Non-Bravais Lattice

• Every Bravais lattice is a space lattice, but not every space lattice is a Bravais lattice.
• Bravais Lattice (BL):
  • All atoms are of the same kind.
  • All lattice points are equivalent.
  • Translational and orientational (rotational) symmetry.
• Non-Bravais Lattice (non-BL):
  • Atoms can be of different kinds.
  • Some lattice points are not equivalent.
  • A combination of two or more BL.
  • Only translational symmetry.

Translational Lattice Vectors – 2D/3D

• A space lattice is a set of points where translation from any point in the lattice by a vector T locates an exactly equivalent point.
• In 2-D: $T = n<em>1 a + n</em>2 b$
• In 3-D: $T = n<em>1 a + n</em>2 b + n_3 c$
• Where a, b, and c are lattice vectors, and $n<em>1$, $n</em>2$, and $n_3$ are integers.
• The choice of lattice vectors is not unique.

Unit Cell in 2D

• The smallest component of the crystal (group of atoms, ions, or molecules) that, when stacked together with pure translational repetition, reproduces the whole crystal.
• The choice of unit cell is not unique.

2D Unit Cell Example - (NaCl)

• Lattice points are points with identical environments; the choice of origin is arbitrary.
• Lattice points need not be atoms, but the unit cell size should always be the same.
• Empty space is not allowed in a unit cell.

Unit Cell Parameters

• The unit cell is uniquely determined by six lattice constants: a, b, c (lattice parameters in x, y, and z directions) and α, β, γ (angles between bc, ca, and ab).
• Only 1/8 of each lattice point in a unit cell can be assigned to that cell.
• Each unit cell can be associated with $8 <br /> eq <br /> eq1/8 = 1$ lattice point.

Primitive Unit Cell

• A primitive unit cell must have only one lattice point.
• Different choices for lattice vectors are possible, but the volumes of the primitive cells are all the same.
• P = Primitive Unit Cell
• NP = Non-Primitive Unit Cell

Primitive Unit Cell and vectors

• A primitive unit cell is made of primitive translation vectors $a<em>1$, $a</em>2$, and $a_3$ such that there is no cell of smaller volume that can be used as a building block for crystal structures.
• A primitive unit cell will fill space by repetition of suitable crystal translation vectors, defined by the parallelepiped $a<em>1, a</em>2$, and $a_3$.
• The volume of a primitive unit cell is $V = a<em>1.(a</em>2 <br /> eq a_3)$ (vector products).
• Cubic cell volume = $a^3$

Seven Crystal Systems

• There are only seven different shapes of unit cell which can be stacked together to completely fill all space (in 3 dimensions) without overlapping.
• 3D – 14 Bravais Lattices and the Seven Crystal System.
• Cubic Crystal System (SC, BCC, FCC)
• Hexagonal Crystal System (S)
• Triclinic Crystal System (S)
• Monoclinic Crystal System (S, Base-C)
• Orthorhombic Crystal System (S, Base-C, BC, FC)
• Tetragonal Crystal System (S, BC)
• Trigonal (Rhombohedral) Crystal System (S)

Unit Cell Types

• P = Primitive
• I = Body-Centred
• F = Face-Centred
• C = Side-Centred
• 7 Crystal Classes → 14 Bravais Lattices
• Cubic: $a=b=c$, $α = β = γ = 90°$
• Tetragonal: $a=b<br /> eq c$, $α = β = γ = 90°$
• Orthorhombic: $a<br /> eq b<br /> eq c$, $α = β = γ = 90°$
• Hexagonal: $a=b<br /> eq c$, $α = β = 90°$, $γ = 120°$
• Monoclinic: $a<br /> eq b<br /> eq c$, $α = γ = 90°$, $β <br /> eq 120°$
• Triclinic: $a<br /> eq b<br /> eq c$, $α <br /> eq β <br /> eq γ <br /> eq 90°$
• Trigonal: $a=b=c$, $α = β = γ <br /> eq 90°$

Coordination Number

• Coordination Number (CN): The number of nearest neighbors to a given point in the Bravais lattice.
• Because the Bravais lattice is periodic, all points have the same coordination number.
• Simple cubic: CN = 6
• Body-centered cubic: CN = 8
• Face-centered cubic: CN = 12

Atomic Packing Factor

• Atomic Packing Factor (APF) is defined as the volume of atoms within the unit cell divided by the volume of the unit cell.
• $APF = \frac{Volume of Atoms in Unit Cell}{Volume of Unit Cell}$

Cubic Crystal System

• Simple Cubic (SC):
  • Has one lattice point, so it's a primitive cell.
  • Atoms at the corners are shared with neighboring cells.
  • Coordination number = 6.

Simple Cubic (SC)

• Coordination # = 6 (number of nearest neighbors).
• $APF = \frac{Volume of atoms in unit cell}{Volume of unit cell}$ (assuming hard spheres).
• Volume of atom = $\frac{4}{3}π (0.5a)^3$
• Volume of unit cell = $a^3$
• APF for a simple cubic structure = 0.52

Body Centred Cubic (BCC)

• BCC has two lattice points, so BCC is a non-primitive cell.
• BCC has eight nearest neighbors.
• Atoms are in contact with neighbors along the body-diagonal directions.
• Many metals (Fe, Li, Na, etc.) choose the BCC structure.

Body Centred Cubic (BCC)

• Coordination number = 8.

Cubic Packing - BCC

• $\sqrt{3}a = 4R$
• $a = \frac{4R}{\sqrt{3}}$

Atomic Packing Factor: BCC

• Close-packed directions: length = 4R = $\sqrt{3} a$
• Unit cell contains: 1 + 8 x 1/8 = 2 atoms/unit cell.
• $APF = \frac{2 <br /> eq \frac{4}{3} π(\sqrt{3}/4 a)^3}{a^3}$
• APF for a body-centered cubic structure = $\frac{\sqrt{3}}{8}$ = 0.68

Face Centred Cubic (FCC)

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

Face Centred Cubic (FCC)

• Coordination number = 12.

Atomic Packing Factor: FCC

• Close-packed directions: length = 4R = $\sqrt{2} a$
• Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell.
• $APF = \frac{4 <br /> eq \frac{4}{3} π(2/\sqrt{2}a)^3}{a^3}$
• APF for a face-centered cubic structure = $\frac{π}{3\sqrt{2}}$ = 0.74 (best possible packing of identical spheres).

Packing efficiency

• Describes the total volume occupied by atoms in the unit cell.
• For a given atom radius, the higher the coordination, the higher the packing efficiency:
  • Simple cubic: Coordination # 6, Packing Efficiency 52%
  • Body-centered cubic: Coordination # 8, Packing Efficiency 68%
  • Face-centered cubic: Coordination # 12, Packing Efficiency 74%
  • Hexagonal: Coordination # 12, Packing Efficiency 74%
• Most metallic elements pack in hexagonal closest packing lattice.
• Many ionic compounds are FCC. (NaCl - two interspersed fcc lattices.)

Theoretical Density, ρ

• Density = mass/volume
• mass = number of atoms per unit cell * mass of each atom
• mass of each atom = atomic weight/Avogadro's number
• $ρ = \frac{nA}{V<em>C N</em>A}$
  • n = # atoms/unit cell
  • A = Atomic weight (g/mol)
  • $V_C$ = Volume/unit cell (cm³/unit cell)
  • $N_A$ = Avogadro's number ($6.023 <br /> eq 10^{23}$ atoms/mol)
  • $V_C = a^3$

Theoretical Density, ρ - Example

• Example: Copper has a FCC structure with 4 atoms/unit cell. Its atomic weight and atomic radius R are 63.55 g/mol and 0.128 nm respectively. Find out the density of Copper.
• Solution: $V<em>C = a^3$; For FCC, $a = \frac{4R}{\sqrt{2}}$; $V</em>C = 4.75 <br /> eq 10^{-23} cm^3$
• Result: theoretical $ρ_{Cu} = 8.89 g/cm^3$

Theoretical Density, ρ - Example

• Ex: Cr (BCC)
  • A = 52.00 g/mol
  • R = 0.125 nm
  • n = 2 atoms/unit cell
  • a = $\frac{4R}{\sqrt{3}}$ = 0.2887 nm
• $ρ_{theoretical} = \frac{2 <br /> eq 52.00}{a^3 <br /> eq 6.023 <br /> eq 10^{23}} = 7.18 g/cm^3$
• $ρ_{actual} = 7.19 g/cm^3$

Numerical Examples

• For NaCl: $M_A = 58.5$, $ρ = 2180 kg/m^3$. Calculate the spacing between the nearest neighboring ions. Ans: 0.282 nm
• Zinc has hcp structure. The height of the unit cell is 0.494nm. The nearest neighbor’s distance is 0.27nm. The atomic weight of the Zinc is 65.37. Calculate the volume of the unit cell and density of Zn. Ans: V = $9.35 <br /> eq 10^{-29} m^3$ $ρ = 6968 kg/m^3$
• Calculate the number of atoms /$mm^2$ in SC structure and FCC structure Note: $N_A=6.02 <br /> eq 10^{26}/kmol$ No. of molecules in a unit cell of NaCl=4

Polymorphism

• Two or more distinct crystal structures for the same material (allotropy/polymorphism)
• Iron system
  • α, B-Ti
  • Carbon

Crystal Planes

• Within a crystal lattice, it is possible to identify sets of equally spaced parallel planes, called lattice planes.

Miller Indices

• Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes.
• To determine Miller indices of a plane:
  1. Determine the intercepts of the plane along each of the three crystallographic directions.
  2. Take the reciprocals of the intercepts.
  3. If fractions result, multiply each by the denominator of the smallest fraction.

Miller Indices - Examples

• Example-1: Intercept points (1, ∞, ∞). Reciprocals (1, 0, 0). Miller Indices (100)
• Example-2: Intercept points (1, 1, ∞). Reciprocals (1, 1, 0). Miller Indices (110)
• Example-3: Intercept points (1, 1, 1). Reciprocals (1, 1, 1). Miller Indices (111)
• Example-4: Intercept points ($\frac{1}{2}$, 1, ∞). Reciprocals (2, 1, 0). Miller Indices (210)
• Example-5: Intercept points (2, ∞, 1). Reciprocals ($\frac{1}{2}$, 0, 1). Miller Indices (102)
• Example-6: Intercept points (-1, ∞, $\frac{1}{2}$). Reciprocals (-1, 0, 2). Miller Indices (102)

Miller Indices - Reciprocal numbers

• Plane intercepts axes at $\frac{3}{2}a, \frac{2}{3}b, \frac{2}{3}c$
• Indices of the plane (Miller): (2,3,3)
• Indices of the direction: [2,3,3]

Miller indices and intercepts

• ∞$a<em>1$: ∞ $a</em>2$: 1 $a_3$ = (001)
• ∞$a<em>1$: 1 $a</em>2$ ∞$a_3$ = (010)
• 1$a<em>1$ ∞$a</em>2$ ∞$a_3$ = (100)
• 1$a<em>1$: 1 $a</em>2$: 1 $a_3$ = (111)
• ∞$a<em>1$: ∞ $a</em>2$ 1 $a_3$ = (001)
• 1$a<em>1$∞$a</em>2$ ∞$a_3$ = (100)

Indices of a Family or Form

• When the unit cell has rotational symmetry, several nonparallel planes may be equivalent. These planes are grouped in the same Miller Indices using curly brackets {hkl}.
• {111} ≡ (111),($\bar{1}$11),(1$\bar{1}$1),(11$\bar{1}$),($\bar{1} \bar{1} \bar{1}$),($\bar{1} \bar{1}$1),($\bar{1}$1$\bar{1}$),(1$\bar{1} \bar{1}$)
• {100} ≡ (100),(010),(001),($\bar{1}$00),(0$\bar{1}$0),(00$\bar{1}$)

Three Important Crystal Planes

• (001), (110), (111)
• Parallel planes are equivalent

EXAMPLE: CRYSTAL PLANES

• Construct a (0,-1,1) plane

FCC & BCC CRYSTAL PLANES

• Atomic packing different in the two cases
• Family of planes: all planes that are crystallographically equivalent-that is having the same atomic packing, indicated as {hkl}
• For example, {100} includes (100), (010), (001) planes
• {110} includes (110), (101), (011), etc.

LINEAR & PLANAR DENSITIES

• Linear density (LD) = $\frac{number of atoms centered on a direction vector}{length of direction vector}$
• LD (110) = $\frac{2 atoms}{4R} = \frac{1}{2R}$
• Planar density (PD) = $\frac{number of atoms centered on a plane}{area of plane}$
• PD (110) = $\frac{2 atoms}{[(4R)(2R\sqrt{2})]} = \frac{1}{4R^2\sqrt{2}}$
• LD and PD are important considerations during deformation and