MIAE 221 (MIDTERM EXAM) 2025 PREP Structure of Crystalline Solids Key Concepts and Atomic Arrangemen

MIAE 221 (MIDTERM EXAM) 2025 PREP Structure of Crystalline Solids: Key Concepts and Atomic Arrangements Concordia University

Structure of Crystalline Solids Atomic arrangement (crystal structure) determines some properties of materials Solid materials: Amorphous (non-crystalline) ⇒ atoms bonded to each other but there is no repeating pattern (short range order) - water - glass Crystals ⇒ atoms bonded together in regular 3D pattern with a long range order (wall paper pattern or brick wall)

metals alloys Atomic hard-shape model ⇒ atoms (ions) are thought of as being solid spheres having well-defined diameters spheres representing nearest-neighbor atoms touch one another

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Lattice ⇒ three-dimensional array of points coinciding with atoms positions

Unit Cell Smallest repetitive volume which contains the complete lattice pattern of a crystal. smallest entity that exhibits the chemical and physical properties of the material defines the crystal structure by virtue of its geometry and the atom p ositions within

−9

size of unit cells are given by lattice parameters ⇒ on the order of 10

m

Important More than a single unit cell may be chosen for a particular crystal structure; however, we generally use the unit cell having the highest level of geometrical symmetry.

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All sides are equal(a ⇒ lattice parameter) All angles are equal (=90∘)

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Atomic Packing Material’s crystal structure * Arrangement of atoms in their crystalline solid material

Dense atomic packing for metals

bonds between metal atoms are non-directional nearest neighbor distances tend to be small in order to lower bond energy high degree of shielding (ion cores) provided by free electron cloud

Energy and Packing Bond Length (Equilibrium Bond Distance) Equilibrium distance where the attractive and repulsive forces balance out, creating a stable bond The energy value at this point is the minimum bonding energy ⇒ it requires additional energy input to break the bond.

Random Packing (non-dense)

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Ordered Packing (dense)

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Important Ordered structures tend to be nearer the minimum in bonding energy and are more stable

Materials Crystalline Atoms arranged in periodic, 3D arrays most metals (> 99%) all ceramics expect for glasses (amorphous) some polymers

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Noncrystalline (Amorphous) Atoms have no periodic arrangement

complex structures most polymers

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Semi-crystalline

Crystalline + Amorphous

some polymers (plastics)

Metallic Crystal Structures Different ways of arranging atoms in crystals Each type of atom has a preferred arrangement depending on Temperature and Pressure (most stable configuration) these patterns are known as space lattices (defined by their unit cells)

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Metals have the simplest crystal structures Densely packed because: only one element present (uniform atomic radii) metallic bonding is not directional lower bond energy electron cloud shields cores from each other

Most metals crystallize into one of the following structures:

1. Body Centered Cubic (BCC) 2. Face Centered Cubic (FCC) 3. Hexagonal / close packed (HCP)

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Coordination Number

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Simple Cubic Structure (SC)

Centers of atoms located at the eight corners of a cube Very rare due to poor packing

Only polonium (Po) has this structure

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Atomic Packing Factor for Simple Cubic APF for a simple cubic structure = 0.52

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Body-Centered Cubic Structure (BCC) Atoms touch each other along cube diagonals (all atoms are identical) atoms located at 8 cube corners with a single atom at cube center

Atomic Packing Factor for BCC

APF for a body-centered cubic structure = 0.68

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Face-Centered Cubic Structure (FCC) Atoms located at 8 cube corners and at the centers of the 6 faces

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Atomic Packing Factor for FCC APF for a face-centered cubic structure = 0.74

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FCC Plane Stacking Sequence

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Hexagonal Close-Packed Structure (HCP) ABAB Stacking Sequence not simple hexagone → has extra plane of atoms in the middle

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Important Because of Hexagonal arrangement, have two lattice parameters

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Polymorphism (Allotropy) Some elements/compounds can exist in more than one crystal form requires change in temperature or pressure

Note Change in crystal structure can often lead in volume change of material

Density for Metals The theoretical density ρ of a material can be calculated knowing its crystal structure:

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Density of Cr

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In general:

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Metals close-packing (metallic bonding) large atomic masses

Ceramics lighter elements Polymers low packing density (amorphous) lighter elements

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Crystal Systems Group crystals depending on shape of unit cell: x, y and z → axes of lattices

α, β and γ → angles lattice constants → sides of the unit cell ⇒ a, b and c

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There are seven possible crystal systems:

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Crystallographic Point Coordinates A point coordinate is a lattice position in a unit cell determined as fractional multiples of a, b and c unit cell edge lengths

Upper Corner Lattice position is

Divide by unit cell edge lengths (a, b, and c) and remove commas

3. Point coordinates for unit cell corner are

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Example

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Crystallographic Directions Line between two points or vector (using 3D coordinate axes → x, y, z)

1. Position vector so that it passes through origin 2. Length of vector projected onto the three axes (x, y and z) is determined in terms of unit cell dimensions (a, b and c)

x2 − x1 y2 − y1 z2 − z1 a b c 3. Multiply or divide by common factor to reduce to lowest common integers

4. Enclose in square brackets with no commas [uvw] - ! negative numbers given by bar over number

[11 2̄], [111], [2 1̄2] The u, v and w indices can be determined using the following equations:

( x −a x ) y −y ) v=n( b z −z ) w=n( c u=n

2

1

2

1

2

1

where n is the factor that may be required to reduce u, v and w to integers

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Common Crystallographic directions Family of directions ⇒ all direction that are crystallographically equivalent (same atomic spacing), indicated by angle brackets

[100], [ 1̄00], [010], [01̄0], [001], [00 1̄]

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Cube internal diagonals

Cube face diagonals

Important

parallel vectors have same indices changing sign of all indices give opposite direction

Hexagonal Crystals

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Conversion from three-index system (a1-a2-z coordinate axes) to the four-index system

[UV W ] → [uvtw]

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Directions in hexagonal crystals:

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Crystal Planes Planes specified by Miller Indices (hkl) used to describe a plane (surface) in a crystal parallel planes are equivalent (same miller indices)

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Algorithm for determining Miller Indices of a plane:

1. If plane passes through selected origin, establish a new origin in another unit cell 2. Read off values of intercepts of plane (designated A, B, C) with x, y, and z axes in terms of a, b, c 3. Take reciprocals of intercepts 4. Normalize reciprocals of intercepts by multiplying by lattice parameters a, b, and c

a b c A B C 5. Reduce to smallest integer values 6. Enclose resulting Miller Indices in parentheses

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Example

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Family of Planes Family of planes use {hkl} They are crystallographically similar (same atomic arrangements)

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Important In cubic system only, directions are perpendicular to planes with same indices

Plane in HCP

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Linear Atomic Density How well packed atoms are in a given direction (atoms/unit length)

Planar Atomic Densities How well packed atoms are in a given plane (atoms/unit area length)

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Linear Density of Atoms (LD)

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Planar Density of Atoms (PD)

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Packing on Planes

FCC and HCP are both close-packed structures ⇒ APF = 0.74

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BCC is not close packed ⇒ APF = 0.68

Types of Crystals Single Crystals A piece of material is made up of one crystal ⇒ all unit cells are aligned up in the same orientation

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Polycrystals Many small crystals (grains) with different orientations joined together most materials/metals Grain Boundary ⇒ regions where grains (crystals) meet

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Anisotropy and Isotropy Many properties depend on direction in crystal in which they are measured.

If property varies with direction - Anisotropic.

If no variation of property with direction - Isotropic

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single crystal ⇒ show variation polycrystalline⇒ randomly oriented and leads to average values in all directions

X- Ray Diffraction Diffraction ⇒ occurs when a wave encounters a series of regularly spaced obstacles that

1. are capable of scattering the wave 2. have spacings that are comparable in magnitude to the wavelength

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X-rays ⇒ form of electromagnetic radiation that have high energies and short wavelengths

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Finding out the Type of Crystal Structure => X-Ray Diffraction

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Crystallographic planes diffract incoming X-rays

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