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Bonds and Crystal Structures of Metals Cheat Sheet (DRAFT) by

ME 313 || Materials Science || Mechanical Engineering

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Types of Bonds

Primary
Ionic
Electron transfer.
Non-di­rec­tional
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
(non-polarized)
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
[1, Tab 3.1]

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 right-­handed 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, 5th 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)(R3)
V_c = a3

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