Atomic Forces

Atoms when brought near each other experience both repelling and attraction forces – the force of attraction due to the bonding mechanism and the repelling force due to the electron clouds. At some separation distance the force balance will be zero and that will be the resting separation distance for the two atoms.

Integrating each force curve with respect to the separation distance results in a potential energy function. This function has a point of minimum energy which corresponds to the point of zero net force and the rest separation distance for the atoms. The value of the potential energy at this position is the bonding energy and is characteristic of some material properties, such as:

  • Materials having large bonding energies also typically have high melting temperatures and are solids at room temperature
  • Small bonding energies at room temp are typically gases
  • The slope of the E vs r curve at the rest position relates to the value of the Young’s Modulus (the steeper the E vs r curve, the higher the Young’s Modulus)

Atomic Bonding

Three primary types:


Atoms give up valence electrons to nonmetallic atoms in order for one or both to become charged ions. The bonding force then is a Coulomb force. Materials having ionic bonding are typically hard and brittle, and electrically and thermally insulative.


Each atom bonding this way shares at least one electron; the shared electrons then belong to both atoms and provide each with a more stable electron configuration.


Metallic materials have 1, 2, or 3 valence electrons. These valence electrons participate in an overall material electron cloud in which they move from atom to atom. The remaining non-valence electrons of each atom remain at the atom and shield the positively charged nucleus. This unit, termed “ion core”, has a net positive charge equal to the number of valence electrons removed. These ion cores bond to the free roaming electrons.

Metals bonding this way make good conductors of heat and electricity.


van der Walls

This type of bonding results from the presence of atomic or molecular dipoles. The negative end of one dipole attracts to the positive end of another dipole. These types of bonds are transient and fluctuate state with time.

Lattice Systems

All metals, certain ceramics, and some polymers form crystalline structures under normal conditions. These structures can be visualized using the hard sphere model. For metals, each sphere represents an ion core. These structures can be represented by a repetitive pattern, the unit cell. In each unit cell under the hard sphere model, each sphere touches the neighboring spheres.

Characteristics of a crystal structure are:

  • coordination number, which for one atom is the number of other atoms it touches
  • unit cell edge length, a
  • atomic packing factor, equal to the amount of the unit cell occupied by atoms, or \[ \frac{\text{volume of atoms in unit cell}}{\text{volume of unit cell}} \]

Varying the 3 unit cell edge lengths and the 3 angles, there are a total of 7 crystal systems:

1 Cubic

2 Hexagonal

3 Tetragonal

4 Rhombohedral

5 Orthorhombic

6 Monoclinic

7 Triclinic

Two of these systems (cubic and hexagonal) are found most in the common metals, and take the form of face-centered cubic, body-centered cubic, and hexagonal close-packed structures.

Face-Centered Cubic

A cubic system, and has atoms at the corners and centers of each of the 6 faces.

Body-Centered Cubic

A cubic system with atoms at the corners and one at the volumetric center of the cube.

Hexagonal Close-Packed

Consists of layers of close packed atoms staggered and stacked on top of each other.

Point Coordinates/Crystallographic Directions

Used to specify locations and directions within a unit cell.

Point Coordinates

  • X, Y, Z using righthanded csys, with origin at a corner of the cell
  • expressed as “q r s” – with spaces and no commas – for x, y, z
  • values are from 0 to 1, fractions of unit cell length, a

Crystallographic Directions

  • Like points, but taken as a vector
  • specified using the form [ u v w ]
  • pick a convenient length, numbers are reduced/scaled to smallest integer
  • negative indice has bar over number
  • families of equivalent directions are indicated in angle brackets: < >

Crystallographic Planes

  • Defined by Miller indices (hkl)
  • Parllel planes have equivalent Miller indices
  • Vector of coordinates is that which is normal to the plane

Atomic Arrangements

  • The particular arrangement of atoms in a given plane
  • A family of planes are equivalent (e.g. parallel planes), and are represented by curly braces: {hkl}
  • In each atomic arrangement one can infer the following properties:

\[ \text{linear density} = \frac{\text{# of atoms having center on direction vector}}{\text{length of direction vector}} \]
\[ \text{planar density} = \frac{\text{# of atoms having center on plane}}{\text{area of plane}} \]

These characteristics are import with regards to slip/plasticity.


  • A single crystal is a “perfect” segment of unit cells – a grain
  • Most metals are polycrystalline – made up of many crystals or grains
  • Crystals form and meet to create grain boundaries


  1. Callister, William D. and Rethwisch, David G. Materials Science and Engineering: An Introduction. s.l. : John Wiley and Sons, 2009.