145 psi = 1MPa = 106 N / m2 1 GPa = 109 N / m2
1 J = 1 kg m2 / s2 = 1 kPa L = 1 Pa m3
E = hc / λ λ = h / mv = h/p
ρ = nA / VCNA
n = number of atoms associated with each unit cell
VC = volume of unit cell A = atomic weight
NA = Avagadro’s number (6.023 x 1023 atoms / mol)
NV = N e(- Qv / kT)
NV = number of vacancies N = the total of atomic sites
Qv = the energy required for the formation of a vacancy T = temperature (in Kelvins)
k = Boltzmann’s constant = 1.38*10-23 J/atom.K = 8.62*10-5 eV/atom.K
N = NAp / Atomic mass of element p = density N = atoms / m3
weight percent (wt%) C1 = m1 / m1 + m2 *100%
atom percent (at%) C1’ = nm1 / nm1 + nm2 *100% n = m / mm
ASTM… N = 2n-1 N = the average # of grains per square inch at 100x n = Grain #
Speed diffusion occurs (rate of mass transfer) diffusion flux (J), defined as the mass (or,
equivalently, the number of atoms) M diffusing through and perpendicular to a unit cross-
sectional area of solid per unit of time. J = M / At A = area J = kg / m2.s
Frick’s first law J = –D (∆C/∆x)
Constant of proportionality D (diffusion coefficient, m2/s)
The negative sign in the expression indicates that the direction of diffusion is down the
concentration gradient, from a high concentration to a low concentration.
Driving force, Concentration gradient = ∆C / ∆x
D = Doe(-Qd / RT)
Do = a temperature-independent pre-exponential R = the gas constant
Qd = the activation energy for diffusion T = temperature (Kelvins)
Elastic Modulus F / Ao = E (DL /Lo)
F = Force Ao = Cross sectional area E = Elastic Modulus
Lo = undeformed length DL = excess deformed length
Bragg’s law says that constructive interference will occur if the extra path is a multiple of the
wavelength. Formula - nλ=2dsinθ
spacing between planes = d = a / √(h2 + k2 + l2) a = relationship of r = cubic side
Thermodynamics of Vacancies (not covered in text)
Some vacancies lower “free energy” ∆G = ∆H – T∆S Total energy
∆Gtot =V∆Gsol + 4π r2γs= 4/3 πr3∆Gsol + 4π r2γs (for graph surface area increases, surface
energy increases, so volume decreases, volumetric energy decreases)
∆G positive causes nothing, ∆G negative driving force, ∆G = equilibrium
A crystal only has one perfect configuration. Configurational entropy is zero. Vacancies (on
the other hand) give rise to many configurations. Add n vacancies to a lattice with N atoms. So
as n increases S increases G decreases
Create vacancies until G is at a minimum. To find minimum r take derivative.
Xv = n* / N = e(-Qv/RT)
Xsol = e(-Qsol/kT)
q = -k (∆T/∆x) ∆x = xoa∆T q = -k / axo q = heat flux
gradient line is a defined line that can be placed on a graph.
∆Q = wρCP∆T
heat absorbed = wall thickness * density *
heat capacity * temperature difference
w = √(2at) wall width = √ (2 * thermal diffusivity * diffusion time)
a ≡ k / ρCP a ≡ thermal conductivity / ( density * specific heat )
∆Q = (∆T√[2t])√( kρCP)
V = IR P = VI R = ρL/A ρ = 1 / σ ρ = V*A / I*l J = σε
Doping: Add rich Al layers on surface, heat it and then you have a doped semiconductor
regions. Moblity is a measure of a charge carrier’s ability to move through a material.
Since conductivity depends on a materials ability to conduct charge, a higher dopant mobility
also means a higher conductivity value.
Atomic hard sphere model: spheres representing nearest-neighbor atoms touch one another.
Lattice = three-dimensional array of points coinciding with atom positions (or sphere centers).
Coordination number = # of nearest-neighbor or touching atoms
Atomic packing factor (APF) = volume of atoms in a unit cell / total unit cell volume
Simple-cubic Crystal (SCC) # of atoms = 1, Not closed packed
Face-centered cubic (FCC) γ a = 2R√2 CN = 12 V= 16R3√2 ABCABCABC
# of atoms = 8*1/8 (corners) + 6*1/2 (faces) = 4 Closed packed, APF= 0.74 (max)
Body-centered cubic (BCC) α a = 4R / √3 CN = 8
# of atoms = 2 Not close packed, APF = 0.68
Hexagonal close-packed (HCP) a = 2R√2 CN = 12 ABABABAB
Area of hexagon = (3sin60)a2 c/a ratio is 1.633 (can deviate) c = height
# of atoms = 6 Closed packed, APF= 0.74
Polymorphism: Metal/non-metal with more than one crystal structure.
Allotropy: polymorphism found in elemental solids
Single crystal repeated arrangement of atoms without interruption
Point (x,y,z) = h l k, ī is negative, Line is [h l k], Plane (h l k)
Line: start at beginning of line, find vector coords, subtract, multiply by common factor
Plane: find intercepts in x,y,z, find reciprocals, and multiply to get integers
Crystalline = material where atoms are situated in a repetitive three-dimensional pattern, each
atom is bonded to its nearest-neighbor atoms.
Polycrystalline = Crystalline solid composed of collection of small crystals (grains)
Grain boundary = atomic mismatch with the region where two grains meet, occur at angles
Cooled Quickly Smaller Grain Boundary Stronger substance
Noncrystalline/ amorphous = solids which lack systematic/regular arrangement of atoms
Imperfections/ Crystalline defect (tend to change the properties of the substance) = a lattice
irregularity. I.e. point defects (one or two atomic positions), linear (one-dimensional),
Vacancy (point defect) = normally occupied lattice site whom atom is missing. Impossible to
create solid without vacancy. Increases entropy (randomness) of crystal, decreasing free energy
Substitution = replaces atom with different atom
Self-interstitial = pushed into small void space that is not normally occupied, atomic diameter
of interstitial impurity must be substantially smaller than that of the host atoms (less than 10%)
Alloys = are composed of two or more substances
Solvent = major concentration, host atoms Solute = minor concentration
Several features of the solute and solvent atoms that determine the degree to which the former
dissolves in the latter, as follows:
1) Atomic size factor: difference in atomic radii between two atoms is less than ~15%.
2) Crystal structure: crystal structures for metals of both atom types must be the same.
3) Electronegativity: greater difference in EN will form an intermetallic compound instead of
substitutional solid solution.
4) Valances: a metal will have more of a tendency to dissolve another metal of higher valency
than one of lower valency.
Edge dislocation is a linear defect that centers around the line that is defined along the end of
the extra half-plane of atoms. This is sometimes termed the dislocation line.
Screw dislocations are formed by a shear stress that is applied to produce a distortion where
one region is shifted one atomic distance compared to another portion.
Dislocations are normally mixed and there magnitude is measured by Burgers vector.
Diffusion / Material transport / Atomic motion = transfer of mass within a solid
Interdiffusion / impurity diffusion. = Process whereby atoms of one metal diffuse into another
Self-diffusion = Diffusion within pure metals, all atoms exchanging positions are same type
Vacancy diffusion= normal atom to vacancy, limited by number of vacancies in substance
CES Material Index
S = CEWt3 / 12L3 find material index using minimum weight
Our free variable would be w.
m = ρV = ρwLt w = m / ρLt now sub w into original equation
S = CEmt3 / ρLt12L3 = CEmt2 / 12ρL4 we want to minimize m
m = 12SρL4 / CEt2 but using material properties
m = Sρ / E but S was fixed therefore
m = ρ / E minimize and that is final answer.
Crystalline Amorphous Mixed
Metals Usually (steel, brass) Rarely (metallic glass) Never
Ceramics Often (alumina) Often (soda glass) Often (silicon nitride)
Polymers Never (“crystalline” polymers
always partly amorphous) Usually (polyethylene) Sometimes (nylon)
More dislocations lowers entropy (S), Dislocation is always thermodynamically unstable.
Vacancies increase entropy (∆S randomness).
Diffusion is faster for… Diffusion is slower for…
Open crystal structures
Lowering melting T materials
Materials with secondary bonding
Smaller diffusing atoms
higher melting T materials
materials with covalent bonding
larger diffusing atoms
Engineering Stress σ = F / A0
F = Force applied perpendicular to the specimen cross section (N)
A0 = the original cross-sectional area before any load is applied (m2)
σ = Engineering Stress (1MPa = 106 N / m2)
Engineering Strain ε = (li – l0) / l0 = ∆l / l0
l0 = original length before any load is applied (m) li = instantaneous length
ε = Engineering strain (unitless or m/m) (can also be expressed as a percentage)
Shear and Torsional Tests: τ = F / A0
F = Force or load imposed parallel to the upper and lower faces (N)
A0 = Area of faces (m2) τ = Shear stress
Stress-Strain Relationship σ = E ε
E = Constant of Proportionality / Modulus of elasticity / Young’s Modulus (GPa or psi) E
typically ranges from 45GPa to 407GPa
If there is a constant modulus then it is called elastic deformation. The greater the modulus the
stiffer the material, or the smaller the elastic strain that results from the application of a given
stress. Elastic deformation is nonpermanent. E is proportional to dF/dr
τ = Gγ G = shear modulus, slope of linear elastic region
Poisson’s ratio (v) is the ratio of lateral and axial strains. v = - εx / εz = - εy / εz
E = 2G(1+v)
Anelasticity is known as the time that a material takes to return to its original length after a
stress has been applied to it.
Plastic deformation occurs when strains are over about 0.005, where permanent, non-
recoverable deformation occurs.
The stress corresponding to the intersection of this line and the stress-strain curve as it bends
over in the plastic region is defined as the yield strength σy (MPa or psi).
The elastic-plastic transition is very well defined and occurs abruptly in what is termed a yield
After yielding, the stress necessary to continue plastic deformation in metals increases to a
maximum, and then decreases to the eventual fracture. The tensile strength TS (MPa) is the
stress at the maximum on the engineering stress-strain curve.
Resilience is the capacity of a material to absorb energy when it is deformed elastically and
then, upon unloading, to have this energy recovered.
Ur = ∫σdε = ½ σyεy = σ2y / 2E (J/m3)
Toughness is a mechanical term that is used in several contexts; loosely speaking, it is a
measure of the ability of a material to absorb energy up to fracture energy up to fracture.
ρ = Resistivity (Ω . m) σ = electrical conductivity (Ω . m)-1 p = the number of holes per m3
J = current density EF = Fermi energy Eg = Band Gap Vd = drift velocity
µe = constant of proportionality, electron mobility (m2 / V.s) n = electron concentration or free
electrons / m3 (m-3) C = Heat capacity Q= Energy A = temperature-independent constant ( Ω.
m) l = length άl = linear coefficient of thermal expansion (W / m.k) E = modulus of elasticity q
= heat flux k = thermal conductivity
Metal conductivities on the order of 107 (Ω . m)-1
Insulators low conductivities, ranging between 10-10 and 10-20 (Ω . m)-1
Semiconductors intermediate conductivities, generally from 10-6 to 104 (Ω . m)-1
The energy corresponding to the highest filled state at 0 K is called the Fermi Energy, Ef.
When electrons attain this energy they jump the conduction band.
Two final band structures are similar; one band (the valance band) that is completely filled with
electrons is separated from an empty conduction band, and an energy band gap lies between
them. Insulator if band gap is greater than 2eV.
Only electrons with energies greater than the Fermi energy may be acted on and accelerated in the presence of an electric field. These
are the electrons that participate in the conduction process, which are termed free electrons. Another charged electronic entity called
a hole is found in semiconductors and insulators. Holes have less energy than Ef and also participate in electric conduction.
vd = µeε σ = n|℮|µe
Matthiessen’s rule ρtotal = ρt + ρi + ρd t = thermal i = impurity d = deformation
ρ = ρ0 + aT ρi = A*ci (1 – ci) ρi = ράVά + ρβVβ
Intrinsic semiconductors are those in which the electrical behaviour is based on the electronic
structure inherent to the pure metal. When the electrical characteristics are dictated by impurity
atoms, the semiconductor is said to be extrinsic. σ = n|e|μe + p|e|µh σ = e(nμe + ρμh)