MSE 101 - Lecture 4 - Crystal Structure

8/11/2019 MSE 101 - Lecture 4 - Crystal Structure

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Crystal structure of SolidsCrystal structure of Solids

Crystalline Solid

– a solid that contains a re ular and

repeating atomic or molecular arrangementover lar e atomic distances lon -ran e

order)

–

Non-crystalline Solid

– a so w ou ong-range or er ng a omsor molecules

1

– also termed “amorphous”





DefinitionsDefinitions

n t ce r m t ve e

– the smallest group of atoms possessing thesymme ry o e crys a w c , w en repea ein all directions, will develop the crystal lattice

• lattice constants/parameter : edge lengths, ,

•interaxial angles : angles between axesα

3



Types of unit cells:Types of unit cells:

mp e – a oms are on y pos one a e

corners

2) Body-centered – an additional atom ispositioned at the center of the unit cell

3) Face-centered – atoms are positioned at

the corners, as well at the faces of the unitcell

- –

the corners, as well at two opposite faces of

4

faces)



Crystal systemsCrystal systems

→ a sc eme y w c crys a s ruc ures are

classified according to unit cell geometry;

.e., c ass e accor ng o re a ons ps

between edge lengths and interaxial angles

→ there is a total of seven crystal systems:

cubic, hexagonal, tetragonal, rhombohedral,orthorhombic, monoclinic, triclinic

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1) Cubic1) Cubic

Axial relationships: a = b = c

Interaxial an les: α = = = 90°

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2) Hexagonal2) Hexagonal

x a re a ons ps: a = ≠ c

Interaxial angles: α = β = 90° ; γ = 120°

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3) Tetragonal3) Tetragonal

x a re a ons ps: a = ≠ c

Interaxial angles: α = β = γ = 90°

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4) Rhombohedral/Trigonal4) Rhombohedral/Trigonal

x a re a ons ps: a = = c

Interaxial angles: α = β = γ ≠ 90°

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5) Orthorhombic5) Orthorhombic

x a re a ons ps: a ≠ ≠ c

Interaxial angles: α = β = γ = 90°

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6) Monoclinic6) Monoclinic


Interaxial angles: α = γ = 90° ≠ β

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7) Triclinic7) Triclinic


Interaxial angles: α ≠ β ≠ γ ≠ 90°

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Bravais latticeBravais lattice

→ ma ema ca er va on o e poss enumber of ways of arranging atoms in

space

→ arrived at by combining one of the seven

crystal systems with the basic types of unit

cells

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Bravais latticesBravais lattices

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Bravais latticesBravais lattices

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Principal Metallic StructuresPrincipal Metallic Structures

1. Face-centered cubic (FCC)

– atoms are situated at the corners of the unit cell,

as well as at the centers of each face; each

face atom touches its nearest corner atoms

16(Hard sphere model) (Point model)



FCC structureFCC structure

x. u, ,

Ag, Au

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2. Body-centered cubic (BCC)

– atoms are situated at the corners, as well as at

the (body) center of the cube

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(Hard sphere model) (Point model)






3.Hexagonal close-packed (HCP) – has two basal planes in the form of regular hexagons with

center. In addition, there are three atoms in the form of atriangle midway between the two basal planes

20(Hard sphere model) (Point model)



HCP structureHCP structure

Ex. Cd, Co, Ti, Zn

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Characteristics of a crystal structureCharacteristics of a crystal structure

1. Coordination number

2. Number of atoms per unit cell

→ equivalent number of atoms enclosed

by the unit cell3. Relationship of the cube side, ao and

the atomic radius, r

→ expression relating the lattice constant,

a with the atomic radius r

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Characteristics of a crystal structureCharacteristics of a crystal structure

4. Atomic Packing Factor (APF)

→ ratio of the volume occupied by atoms to the

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Summary of crystal structure characteristicsSummary of crystal structure characteristics

Structure CN Atoms/ Equivalent APF

o

BCC 8 2 0.684 R

2

4 RFCC 12 4 0.74

HCP 12 6 2R 0.74

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Polymorphism and allotropyPolymorphism and allotropy

Polymorphism

→ henomenon wherein solids can ossess

more than one crystal structure

→ polymorphism in elemental solids

x. ure e

- BCC at room temperature

- FCC at 912°C

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Density computationsDensity computations

→ knowledge of the crystal structure permits

computation of theoretical density, ρ :

where,

n = number of atoms er unit cell A = atomic weight

VC = volume of unit cell

N A = Avogadro’s number (6.023x10 atoms/mol)

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Sample problemSample problem

Copper has an atomic radius of 0.128 nm, an

FCC crystal structure, and an atomic weight

of 63.5 g/mol. Compute its theoretical densityand compare the answer with its measured

density.

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MSE 101 - Lecture 4 - Crystal Structure