Crystal Structure Notes

Unit Learning Outcomes

At the end of this unit, you should be able to:

• Demonstrate understanding of the basic concepts of crystal structure.
• Differentiate metals and non-metals by understanding their characteristics.
• Explain the concept of polymers.
• Show understanding of the concept of engineered nanomaterials.

Key Definitions

• Amorphous: A solid lacking the long-range order characteristic of a crystal.
• Atomic Packing Factor (APF): The ratio of the volume occupied by atoms in a unit cell to the total volume of the unit cell.
• Body-Centered Cubic (BCC): A unit cell with lattice points at the eight corners plus an additional point at the center.
• Bravais Lattice: The fourteen distinguishable ways of arranging points independently in three-dimensional space.
• Coordination Number: The number of equidistant nearest neighbors that an atom has in a given structure.
• Crystal Lattice: The symmetrical three-dimensional arrangement of atoms inside a crystal.
• Crystalline: A solid material with a highly ordered microscopic structure, forming a crystal lattice that extends in all directions.
• Crystallographic Axes: Lines drawn parallel to the lines of intersection of any three faces of the unit cell which do not lie in the same plane.
• Density: Mass per unit volume.
• Face-Centered Cubic (FCC): A unit cell with lattice points at the eight corners plus additional points at the centers of each face.
• Hexagonal Close-Packed (HCP): Layers of spheres are packed so that spheres in alternating layers overlie one another; each sphere is surrounded by 12 others.
• Interfacial Angles: The angles between three crystallographic axes.
• Lattice Parameters: A collective term for primitives (unit cell dimensions) and interfacial angles.
• Primitives: The dimensions of a unit cell.
• Simple Cubic (SC): The simplest repeating unit with lattice points at each corner of the cube; an atom, ion, or molecule can be found in the crystal.
• Space Lattice: An infinite array of points in three dimensions where every point has identical surroundings.
• Unit Cell: The smallest group of atoms of a substance that has the overall symmetry of a crystal, and from which the entire lattice can be built up by repetition in three dimensions.
• Void Space: Vacant or unutilized space in a unit cell, also known as interstitial space.

Basic Concepts of Crystal Structure

Crystal structure is the ordered arrangement of atoms, ions, or molecules in a crystalline material.

• Ordered structures arise from the tendency of constituent particles to form symmetric patterns that repeat along principal directions in 3D space.
• The unit cell is the smallest repeating group of particles that defines the symmetry and structure of the entire crystal lattice.
• The crystal lattice is built by repetitive translation of the unit cell along its principal axes. These repeating patterns are located at the points of the Bravais lattice.
• Lattice constants or lattice parameters are the lengths of the principal axes (edges) of the unit cell and the angles between them.
• The symmetry properties of the crystal are described by space groups; there are 230 possible space groups.
• Crystal structure and symmetry determine physical properties like cleavage, electronic band structure, and optical transparency.
• Understanding crystal structures is crucial in materials science and engineering, and is fundamental to techniques like X-ray diffraction (XRD) and Transmission electron microscopy (TEM).

States of Matter and Solids

Matter exists in solid, liquid, or gas states. According to modern concepts, classification is specified as condensed state (solids and liquids) and gaseous state.

• Solids are characterized by incompressibility, rigidity, and mechanical strength.
• This indicates that the constituent molecules, atoms, or ions are closely packed in a well-ordered arrangement.

Classification of Solids

Solids are classified into:

• Crystalline: Particles are orderly arranged (long-range order).
• Amorphous: Particles are randomly oriented.

Crystal Structure

• A crystal structure is formed when atoms or molecules are uniquely arranged in a crystalline solid or liquid.
• A crystal possesses long-range order and symmetry.
• Periodicity is the main property, resulting from the arrangement of atoms/molecules in lattice points.
• The crystal structure represents the repetition of the unit cell; the shape of the unit cell is consistent for a given crystal but varies from crystal to crystal.

Unit Cell Geometry

The unit cell is defined as the smallest repeating unit having the full symmetry of the crystal structure.

• The geometry is defined by a parallelepiped with six lattice parameters: the lengths of the cell edges ($a$, $b$, $c$) and the angles between them ($\alpha$, $\beta$, $\gamma$).
• Positions of particles inside the unit cell are described by fractional coordinates ($x<em>i$, $y</em>i$, $z_i$) along the cell edges, measured from a reference point.
• Only the coordinates of the smallest asymmetric subset of particles need to be reported; other particles are generated by symmetry operations.
• The collection of symmetry operations is expressed as the space group of the crystal structure.

Primitive vs. Non-Primitive Unit Cells

• A primitive cell is a minimum volume unit cell and has only one lattice point.
• A non-primitive cell contains more than one lattice point.

Bravais Lattices

The number of lattices that can fill two- or three-dimensional space with periodically repeating units without gaps or overlaps is limited.

• Different crystalline solids may crystallize according to the same pattern, even with different lattice metrics, provided the symmetry is the same.
• Lattices which fill space without gaps are called Bravais lattices; there are five in two dimensions and 14 in three dimensions.

Bravais Lattices in Two Dimensions

• Oblique Lattice: The most general and least symmetric.
• Rectangular Lattice: Angle between lattice vectors is 90°; can be primitive or centered.
• Square Lattice: Higher symmetry; unit cell vectors have the same length.
• Centered Rectangular Lattice: Has an extra lattice point (atom) at the center; can also be described as a primitive lattice with lower symmetry.
• Hexagonal Lattice: Lattice vectors are the same length, and the angle is 120°.

Bravais Lattices in Three Dimensions

Based on lattice parameters $a$, $b$, $c$, $\alpha$, $\beta$, and $\gamma$, there are 14 possible lattices in three dimensions: one general (triclinic) and thirteen special. These are divided into seven crystal systems:

• Triclinic
• Monoclinic
• Orthorhombic
• Tetragonal
• Trigonal (Rhombohedral)
• Hexagonal
• Cubic

Atomic Coordination

• Coordination number (ligancy) is the number of atoms, molecules, or ions bonded to a central atom/molecule/ion.
• The surrounding atoms/molecules/ions are called ligands.
• Simple cubic, face-centered cubic (FCC), and body-centered cubic (BCC) have coordination numbers 6, 12, and 8 respectively.

Atomic Packing Factor (APF)

• APF indicates the arrangement of atoms/ions in solids and gives the efficiency of space filling.
• It is the ratio of the volume of atoms in the unit cell to the volume of the unit cell.
• $APF = <br /> Volume of atoms in unit cell <br /> Volume of unit cell$

Examples:

1. Simple Cubic (SC)

• Atoms at the corners of a cube of side 'a', with atoms of radius 'r' such that $a = 2r$.
• Each corner atom is shared by 8 unit cells, contributing 1/8 of an atom per unit cell.
• No. of atoms per unit cell = $(1/8) * 8 = 1$
• Volume of unit cell = $a^3 = (2r)^3 = 8r^3$
• Volume of atoms in unit cell = $(1) * (4/3) <br /> pi r^3 = (4/3) <br /> pi r^3$
• $APF = <br /> frac{(4/3) <br /> pi r^3}{8r^3} = <br /> frac{<br /> pi}{6} <br /> approx 0.52<br /> = 52$%

2. Body-Centered Cubic (BCC)

• Atoms at corners and at the body center of a cube of side 'a', with atoms of radius 'r'.
• Length of body diagonal: $<br /> sqrt{3}a = 4r$
• Atoms per unit cell = $(1/8) * 8 (corner atoms) + 1 (body center) = 2$
• Volume of atoms in unit cell = $2 * (4/3) <br /> pi r^3 = (8/3) <br /> pi r^3$
• Volume of unit cell = $a^3 = (<br /> frac{4r}{<br /> sqrt{3}})^3 = <br /> frac{64r^3}{3<br /> sqrt{3}}$
• $APF = <br /> frac{(8/3) <br /> pi r^3}{<br /> frac{64r^3}{3<br /> sqrt{3}}} = <br /> frac{<br /> sqrt{3} <br /> pi}{8} <br /> approx 0.68<br /> = 68$%

3. Face-Centered Cubic (FCC)

• Atoms at corners and face centers of a cube of length 'a', with atoms of radius 'r'.
• Length of face diagonal: $<br /> sqrt{2}a = 4r$
• Atoms per unit cell = $(1/8) * 8 (corner atoms) + (1/2) * 6 (face atoms) = 4$
• Volume of atoms in unit cell = $4 * (4/3) <br /> pi r^3 = (16/3) <br /> pi r^3$
• Volume of unit cell = $a^3 = (<br /> frac{4r}{<br /> sqrt{2}})^3$
• $APF = <br /> frac{(16/3) <br /> pi r^3}{(<br /> frac{4r}{<br /> sqrt{2}})^3} = <br /> frac{<br /> pi}{3<br /> sqrt{2}} <br /> approx<br /> 0.74<br /> = 74$%

4. Hexagonal Close-Packed (HCP)

• The unit cell is a hexagonal prism containing six atoms.
• Atoms: 3 in the middle layer, 1/2 * 2 at the top and bottom, 1/6 * 6 * 2 at the vertices: 3 + (1/2)×2 + (1/6)×6×2 = 6.
• Each atom touches 12 others.
• If $a=2r$, then $c = 4 <br /> sqrt{\frac{2}{3}} r$
• Volume of the hcp unit cell is $<br /> frac{3<br /> sqrt{3}}{2} a^2c = 24<br /> sqrt{2}r^3$
• $APF = <br /> frac{6 * (4/3) <br /> pi r^3}{<br /> frac{3<br /> sqrt{3}}{2} (2r)^2 (4<br /> sqrt{\frac{2}{3}}r)} = <br /> frac{<br /> pi}{<br /> sqrt{18}} = <br /> frac{<br /> pi}{3<br /> sqrt{2}}</li> </ul> <p>approx 0.74048$%

  Density Calculation

  The density of a solid is that of the unit cell.

  • $\rho = frac{nA}{V<em>c N</em>A}$
    • $\rho$ = density
    • $n$ = no. of atoms
    • $A$ = atomic/formula weight
    • $V_c$ = volume of unit cell
    • $N_A$ = Avogadro's number

  Examples:

  1. Polonium (Po)

  • Crystallizes in a simple cubic pattern with a unit cell length of 3.36 Å ($1 Å = 1x10^{-8} cm$)
  • Given: $n = 1 atom$, $a = 3.36x10^{-8} cm$, $A = 209 g/mol$
  • $\rho = <br /> frac{(1 atom) (209 g/mol)}{(3.36x10^{-8} cm)^3 (6.022x10^{23} atoms/mol)} = 9.15 g/cm^3$

  2. Aluminum (Al)

  • Face-centered cubic unit cell with an atom at each corner and the center of each face.
  • Al-Al distance ($2r$) = 0.2863 nm
  • Mass of aluminum atom = 26.98 amu
  • Given: $n = 4 atoms$, $A = 26.98 g/mol$, $2r = 0.2863 nm = 0.2863x10^{-7} cm$
  • $<br /> (4r)^2 = a^2 + a^2$
  • $16r^2 = 2a^2$
  • $a^2 = 8r^2$
  • $a=2<br /> sqrt{2}r$
  • $\rho = <br /> frac{(4 atoms)(26.98 g/mol)}{(<br /> sqrt{2}(0.2863x10^{-7} cm))^3 (6.022x10^{23} atoms/mol)} = 2.7 g/cm^3$

  Additional Resources

  • Hermann, K. (2017). Crystallography and surface structure: An introduction for surface scientists and nanoscientists. Chapter 2, Bulk Crystals: Three-Dimensional Lattices, pp 7-90.

  Exercises

  1. Molybdenum forms body-centered cubic crystals and at 20 oC the density is 10.3 g/mL. Calculate the distance between the centers of the nearest molybdenum atoms.
  2. An element crystallizes in a body-centered cubic lattice. The edge of the unit cell is 0.286 nm, and the density of the crystal is 7.92 g/cm3. Calculate the atomic weight of the element.
  3. Austenite form of iron has FCC crystal lattice structure, whereas its alpha form has BCC crystal lattice structure. Assuming closest packed arrangement of iron atoms, what will be the ratio of density of Austenite to that of alpha iron?
  4. Sodium crystallizes in the body-centered cubic structure with a=0.424 nm. Calculate the theoretical density of Na in kg/m3.
  5. Find the distance (in nm) between the body-centered atom and one corner atom in Na, given a=0.424 nm.

  Activities

  Let’s Analyze

  • Activity 1. Compare and contrast the 7 crystal system.
  • Activity 2. Compare and contrast the 3 Bravais lattice of the cubic crystal system.

  In a Nutshell

  • Activity 1. Illustrate the: (a) simple cubic, body-centered, and face-centered cubic of the cubic crystal system; and (b) hexagonal crystal system.
  • Activity 2. How is learning crystal structure relevant in the study of engineering?