Crystalline Solids Structure

• Materials Science and Engineering is the title of the presentation, by Engr. Airra Mhae G. Ilagan for the first semester of A.Y. 2023-2024.
• The presentation covers topics within the structure of crystalline solids.

Fundamental Concepts

• Crystalline materials have atoms situated in a repeating, periodic array over large atomic distances.
• Non-crystalline, or amorphous, materials do not crystallize, and long-range atomic order is absent.

Crystals Structure

• Properties of crystalline solids depends on the crystal structure.
• Atoms, ions, or molecules are spatially arranged in a specific manner for cystals.
• An atomic hard-sphere model represents atoms as spheres that touch nearest neighbors.
• A lattice is a three-dimensional array of points coinciding with atom positions or sphere centers.
• Unit cells are small groups of atoms that form a repetitive pattern.
• Unit cells are parallelepipeds or prisms with three sets of parallel faces.
• A basic structural unit or building block of the crystal structure is called a unit cell.
• Crystal structure is defined by unit cell geometry and atom positions.

Metallic Crystal Structures

• Atomic bonding in this group of materials is metallic.
• Metallic crystal structure is non-directional in nature.
• There are minimal restrictions as to the number and position of nearest-neighbor atoms.
• Metallic crystal structures have relatively large numbers of nearest neighbors and dense atomic packings.

Common Metal Crystal Structures: Face-Centered Cubic (FCC)

• Has a cubic unit cell with atoms at each corner and in the center of each cube face.

Common Metal Crystal Structures: Body-Centered Cubic (BCC)

• Has a cubic unit cell with atoms at all eight corners and a single atom at the cube's center.

Common Metal Crystal Structures: Hexagonal Close-Packed (HCP)

• Has a unit cell where the top and bottom faces consist of six atoms forming regular hexagons around a single atom in the center.

Face-Centered Cubic (FCC) Crystal Structure

• Some familiar metals with this structure are copper, aluminum, silver, and gold.
• Spheres or ion cores touch across a face diagonal.
• Cube edge length "a" and atomic radius "R" are related, where a = 2R√2.
• Has a hard-sphere unit cell representation, a reduced sphere unit cell, and an aggregate of many atoms.
• The number of atoms per unit cell for face-centered cubic crystal structures = 4.
• The coordination number, for metals is the same number of nearest-neighbor or touching atoms.
• The coordination number for the face-centered cubic crystal structure is 12.

Atomic Packing Factor (APF)

• Indicates how efficiently atoms are packed in a crystal structure.
• Equals the volume of atoms in a unit cell divided by the total unit cell volume.
• The atomic packing factor for the face-centered cubic crystal structure is 0.74.
• 0.74 APF is the maximum packing possible for spheres of the same diameter.

Body Centered Cubic Crystal Structure

• Has a cubic unit cell with atoms at all eight corners and a single atom at the cube center

Body Centered Cubic (BCC) Crystal Structure.

• Center and corner atoms touch one another along cube diagonals.
• Unit cell length "a" and atomic radius "R" are related by a = 4R / √3.
• Consists of a hard-sphere unit cell representation, a reduced sphere unit cell, and an aggregate of many atoms.
• The number of atoms per unit cell is 2.
• The coordination number for the BCC crystal structure is 8.
• It has a center atom with eight nearest neighbors at its corners.
• The coordination number is less for BCC than for FCC.
• The atomic packing factor is lower for BCC at 0.68, versus 0.74 for FCC.

Simple Cubic Crystal Structure

• Has a unit cell consisting of atoms situated only at the corners of a cube
• None of the metallic elements have this crystal structure because of its relatively low APF
• Polonium is the only simple-cubic element and is considered a metalloid.
• Consists of a hard-sphere unit cell representation and a reduced sphere unit cell.

Hexagonal Close-Packed Crystal Structure

• Not all metals have unit cells with cubic symmetry, and the final common metallic crystal structure has a unit cell that is hexagonal
• The top and bottom faces of the unit cell consist of six atoms that form regular hexagons and surround a single atom in the center.
• Consists of a reduced-sphere unit cell.
• The number of atoms per unit cell is 6.

Volume Determination: Example Problem 2.1

• The volume of an FCC unit cell can be calculated in terms of the atomic radius R.
• In an FCC unit cell, atoms touch across a face-diagonal with a length of 4R.
• Unit cell volume is a³, where a is the cell edge length.
• Given a = 2R√2 the FCC unit cell volume Vc = a³ = (2R√2)³ = 16R³√2.

Density Computations

• Knowing the crystal structure of a metallic solid, theoretical density "p" can be computed using.
• p = nA / VcNa, where:
  • n = number of atoms associated with each unit cell
  • A = atomic weight
  • Vc = volume of the unit cell
  • Na = Avogadro's number (6.022 x 1023 atoms/mol)

Atomic Packing Factor (APF) Computation: Example Problem 2.2

• The atomic packing factor (APF) is defined as the volume of atoms in a unit cell divided by the total unit cell volume.
• APF = volume of atoms in a unit cell / Total unit cell volume = Vs/Vc
• If the volume for a sphere the total FCC atom volume is Vs = (4) 4/3 πR3 = 16/3 πR³
• The total unit cell volume is Vc = 16R³√2
• Therefore, the atomic packing factor becomes APF = Vs/Vc = (16/3 πR3) / (16R3√2) = 0.74.

Copper Density Computation

• Copper has an atomic radius of 0.128 nm, an FCC crystal structure, and an atomic weight of 63.5 g/mol.
• To compute theoretical density, the number of atoms per unit cell, n, is 4 for FCC.
• The unit cell volume VC for FCC is 16R³√2, where R is 0.128 nm.
• By substituting these values into the density equation yields:
  • p = nA / Vc Na = (4 atoms/unit cell)(63.5 g/mol) / [16√2(1.28 x10⁻⁸ cm)³/unit cell](6.022 x 10²³ atoms/mol) = 8.89 g/cm³ The density of copper is 8.94 g/cm³,

Polymorphism

• When metals and nonmetals have more than one crystal structure is called Polymorphism.
• Example: Calcium carbonate in the form of Calcite or Argonite.

Allotropy

• When more than one crystal structure is found in elemental solids it's called Allotropy.
• Example: Carbon in the form of Diamond.

Crystal Systems

• There are seven different possible combinations of axial lengths (a, b, and c) and interaxial angles (α, β, and γ).
• Each unique combination represents a distinct crystal system.
• The seven crystal systems are cubic, tetragonal, hexagonal, triclinic, orthorhombic, rhombohedral, and monoclinic.
• The cubic system (a = b = c, α = β = γ = 90°) has the greatest symmetry.
• The triclinic system (a ≠ b ≠ c, α ≠ β ≠ γ) displays the least symmetry.
• Crystal structures are described in terms of unit cells and are normally more complex than those for FCC, BCC, and HCP.

Crystallography

• Crystallography is the science of measuring crystal structure.
• It utilizes X-ray diffraction as a technique.
• Incoming X-rays diffract from crystal planes.
• The measurement of the critical angle allows computation of planar spacing.
• When dealing with crystalline materials, it is necessary to specify a particular point within a unit cell.
• Crystallographic direction is needed or some crystallographic plane of atoms.

Point Coordinates

• Needed to define the lattice position within a unit cell.
• Point coordinate indices: q, r, and s.
• Indices are fractional multiples of a, b, and c unit cell edge lengths.
  • q = lattice position referenced to the x axis
  • r = lattice position referenced to the y axis
  • s = lattice position referenced to the z axis

Crystallographic Directions

• A line directed between two points, or a vector
• A right-handed x-y-z coordinate system is constructed, with its origin at a unit cell corner.
• Coordinates of two points on the direction vector are determined (tail point 1, head point 2).
• Tail point coordinates are subtracted from head point components (x2 - x1, y2 - y1, z2 - z1).
• These coordinate differences are then normalized in terms of the lattice parameters (a, b, c).
• Normalize the three numbers yields: (x₂-x₁) / a, (y₂ - y₁) / a, (z₂-z₁) / a
• Reduce u, v, and w to integers if needed.
• Use both positive and negative coordinates, with negative indices represented by a bar.

Crystallographic Planes

• Crystallographic planes are specified by three Miller indices (MI).
• If the plane passes through the origin, create a parallel plane or new origin.
• Determine the length of the planar intercept for each axis.
• The reciprocals of these numbers are taken.
• A plane that parallels an axis has an infinite intercept and a zero index.
• Reduce the three numbers to the smallest integers by multiplication or division.
• Enclose the integer indices in parentheses: (hkl).
• An intercept on the negative side of the origin is indicated by a bar over the index.
• Reversing all indices specifies another plane parallel to, on the opposite side, and equidistant from the origin.

Planar (Miller) Indices Example

• To determine Miller indices when the plane passes through the selected origin O, a new origin is chosen at the corner of an adjacent unit cell.
• Each axis now has the following: B = -b and C = c/2 The value can be found using: - h = n(na/A) = 0, - k =n(nb/B) = -1 and - l = n(nc/C) = 2 with the final value (012).

Atomic Arrangements

• The atomic arrangement for a crystallographic plane depends on the crystal structure.
• A "family" of planes contains all planes that are crystallographically equivalent.
• Example: {100} family contains only the (100), (100), (010), and (010) planes because (001) and (001) planes are not crystallographically equivalent.

Hexagonal Crystals

• For crystals having hexagonal symmetry, it is desirable that equivalent planes have the same indices.
• Achieved using the Miller-Bravais system and the four-index (hkil) scheme.
• The use of hkil makes it more clearly identifies the orientation of a plane in a hexagonal crystal.
• Index i is determined through i = -(h + k).
• Three h, k, and l indices are identical for both indexing systems.
• The indices are determined by taking normalized reciprocals of axial intercepts.

Linear and Planar Densities

• Linear density (LD) is the number of atoms per unit length along a specific crystallographic direction:
  • LD = number of atoms centered on direction vector / length of direction vector.
• Units of linear density are reciprocal of length (e.g., nm-1, m-1).
• Planar density (PD) is the number of atoms per unit area centered on a particular crystallographic plane:
  • PD = number of atoms centered on a plane / area of plane.
• Units for planar density are reciprocal area (e.g., nm-2, m-2).

Closed-Packed Crystal Structures

• Centers of all atoms in one close-packed plane are labeled A.
• Associated with this plane are two sets of equivalent triangular depressions formed by three adjacent atoms.
• Triangular vertex-pointing-up are B positions, and the remaining depressions are C positions.
• A second close-packed plane can be positioned with centers over either B or C sites.
• If the B positions are arbitrarily chosen, the stacking sequence is termed AB.

Single Crystals

• A crystalline solid has a perfect periodic arrangement of atoms.
• Arrangement extends throughout the specimen without interruption.
• All unit cells interlock in the same way and have the same orientation.
• Single crystals exist in nature and can be produced artificially.

Polycrystalline Materials

• Most crystalline solids are composed of many small crystals or grains.
• These materials are termed polycrystalline.
• Solidification of a polycrystalline specimen involves small crystals or nuclei forming at different positions.
• Small grains grow by adding atoms from the surrounding liquid.
• Crystallographic orientation varies, and atomic mismatch occurs at grain boundaries.