Types of Bonds
Primary 
Ionic 
Electron transfer. 
Nondirectional 
Covalent 
Electron sharing. 
Directional, strong, brittle, high melting temps, less conductive. 
Metallic 
"Sea of electrons." 
Ductile, conductive. 
Secondary 
(van der Waals) 

atom 1 
atom 2 
London 
induced dipole 
induced dipole 
Debye 
permanent dipole 
neutral (nonpolarized) 
Keesom 
permanent dipole 
permanent dipole 
Lattice
A collection of points arranged on a periodic pattern so that the surroundings of each lattice point are identical.
Atomic Radii and Crystal Structures for 16 Metals
Metal 
Crystal Structure 
Atomic Radius (nm) 
Aluminum 
FCC 
0.1431 
Cadmium 
HCP 
0.1490 
Chromium 
BCC 
0.1249 
Cobalt 
HCP 
0.1253 
Copper 
FCC 
0.1278 
Gold 
FCC 
0.1442 
Iron 
BCC 
0.1241 
Lead 
FCC 
0.1750 
Molybdenum 
BCC 
0.1363 
Nickel 
FCC 
0.1246 
Platinum 
FCC 
0.1387 
Silver 
FCC 
0.1445 
Tantalum 
BCC 
0.1430 
Titanium 
HCP 
0.1445 
Tungsten 
BCC 
0.1371 
Zinc 
HCP 
0.1332 
E of Various Crystallographic Directions

Young's Modulus (GPA) 
Metal 
[100] 
[110] 
[111] 
Aluminum 
63.7 
72.6 
76.1 
Copper 
66.7 
130.3 
191.1 
Iron 
125.0 
210.5 
272.7 
Tungsten 
384.6 
384.6 
384.6 
Plane From Miller Index
1. Look at the plane, determine where the plane intersects the various axes (X, Y, and Z intercepts). 
2. Verify that the origin does not intersect with the plane. 
3. Take reciprocals. 
4. Clear fractions (if needed). 
5. Any negative numbers? Use the overbar. 
6. Enclose final result in parentheses. 
Miller Index From Direction
1. Using a righthanded coordinate system, find the coordinates of TWO points that lie in the direction. 
2. Subtract the “tail” point from the “head” point. 
3. Clear fractions and/or reduce the results from step 2 to the lowest integers. 
4. Enclose the numbers with brackets [ ]. If a negative number is present, use a bar over the number. 


Equalibrium Distance
Equilibrium separation occurs when the total energy is minimized and also when the net force is zero.
Coefficient of Thermal Expansion
→CTE = α →∆L/L₀=α(T₂+T₁) →Metal CTE is higher than ceramic CTE. {{nl}→Heat = elongation →Cooling = contraction 
Sources
[1] W.D. Callister, Fundamentals of Materials Science and Engineering, 5^{th} ed. New York, NY, USA: John Wiley & Sons, 2001, pp. 33 
Common Crystal Structures

FCC 
BCC 
HCP 
C.N. 
12 
8 
12 
total atoms/unit cell 
4 
2 
6 
APF 
0.74 
0.68 
0.74 
a 
2R√2 
4R/√3 
td 
Atomic Packing Factor
V_s = (4/3)(pi)(R^{3}) V_c = a^{3}
Theoretical Density of a Metallic Solid
Direction From Miller Index
1. Look at the index values and if any are larger than one, take out a factor such that none of the integer values are greater than one. 
2. For example, we have a Miller index of [2 1 1], we can take out a factor of 2 2* [1 1/2 1/2]. 
3. Essentially we have created a new point to plot with coordinates of x = 1, y = 1/2 and z = 1/2. 
4. Set up your unit cube and it is fine to place the origin at the lower left hand corner as shown below. 
5. Plot the new point (1, 1/2, 1/2) and connect the ‘dots'. 
Terms
Single crystal 
Anisotropic 
Polycrystal 
Grain boundaries 
Isotropic 
Polymorphism 


Stiffness
Stiffness of a material is related to the slope of the F vs atomic separation curve (r). The stiffness (dF/da) of the material is known as the Young’s modulus, E.
Dimensions and Angles
Lattice parameters are the dimensions a,b and c of the unit cell. Interaxial angles are α, β, and γ.
(top to bottom)BCC, FCC, & HCP
Miller Index From Plane
1. Draw the unit cube and right handed coordinate system & be sure that the plane does not pass thru the point you’ve selected to be the origin. 
2. Take reciprocals 
3. Plot the plane. 
4. DO NOT CLEAR FRACTIONS. 
Linear and Planar Density
Linear Density Number of atoms per unit cell whose centers are aligned in a specific direction.

LD= #atoms/unit length

Planar Density Number of atoms per unit area that are centered on the area of the plane.

PD= #atoms centered on plane/area of plane

