Equations (NOT IN DATA BOOK) Flashcards by Ollie Cooke

equation for a dipole moment

μ = qr
(vector)

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give the two equations for polarisation (dipole moments and charge)

P = n μ
P = Q/A
(vector)

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give the 2 equations for resultant charge density (displacement fields)

D = εo E + P
(E = electric field, P = polarisation)

D = ε E = k εo P

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give the equation for the change in voltage due to stress on a piezoelectric material

ΔV = dσL /ε = dσL / k*εo

d = piezoelectric constant

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give the equation for the change in voltage due to a change in temperature for a pyroelectric material

ΔV = ΔQL / kεoA = pΔT L / k εo

p = pyroelectric constant

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give the Goldschmidt tolerance factor, what should it equal

t = (rA + ro) / (sqrt(2)*(rB + ro))

it should equal 1

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give the equation for the number of diffusing ions in an electric field (Boltzmann stats)

n = no exp(-qv/kt)

k = Boltzmann constant
q = charge on ion
V = voltage
T = temp

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give the equation derived from the number of diffusing ions in an electric fields for the conc. gradient

dn/dx = -qn/KT dv/dx = qn/KT E

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give the thin lens equation for optical microscopes

1/f = 1/u + 1/v

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give the equation for maxima spacing from a double slit

x = lλ/d

l = distance to slits from screen
λ = wavelength
d = slit spacing

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give the general equation for the phase difference from scattering from the nth atom in a structure (in terms of its coords)

ϕn = 2π (h xn + k yn + l zn)

(xn,yn,zn) are the coords of the nth atom

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give the equation for the multiplicity of a plane (hkl)

M(hkl) = (3!)(2^A) / (n!)

A = number of non-zero indices
n = number of repeated indices

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give the equation for the intensity of a peak intensity for a plane (hkl)

I(hkl) α M(hkl) |F(hkl)|^2 g(θ)

where g(θ) increases with θ

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give the equation for the distance between two points representing planes in reciprocal space in terms of the real spacing of the planes

|d(hkl)*| = 1/|d(hkl)|

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give the equation for determining the planar spacing from a diffraction pattern from a TEM

d(hkl) = λL/R

λ = wavelength of electrons
L = distance from specimen to detector
R = distance on detector from origin to spot

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give the equation for internal energy (differential in three forms, one at constant volume)

dU = δq + δw
dU = CdT - pdV

at const, vol
dU = C(v)dT

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give the equation for absolute internal energy at a given temperature

U = Uo + ∫CvdT (0 to T1) + Σli (on i)

where li is the ith latent heat of transition

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give the non differential AND differential forms of enthalpy, H

give the equation for enthalpy at a constant pressure

H = U +PV
dH = δq + Vdp

dH = δq = C(p) dT
at a const. pressure

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give the equation for absolute enthalpy at a given temperature

H = Ho + ∫CpdT (0 to T1) + Σli (on i)

where li is the ith latent heat of transition

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give the non-differential form for G

G = H - TS

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give the starting equation for ΔHmix

ΔHmix = Hs - Hmm

Hs = enthalpy of the solution
Hmm = enthalpy of the mechanical mixture

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give the equation for the enthalpy of a solution of one mole of atoms

Hs = (NA*Z / 2) ( XA^2 EAA + XB^2 EBB + 2 XA XB EAB)

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give the equation for the enthalpy of a mechanical mixture

Hmm = (NA*Z / 2) (XA EAA + XB EBB)

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using the equations for Hmm and Hs give the final equation for ΔHmix

ΔHmix = Hs - Hmm = (NA*Z /2) XA XB (2EAB - EAA - EBB) = XA XB ψ

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give the defining equation for entropy

S = k ln(w)

what are the equations for the number of distinguishable ways to arrange NA particles in a mechanical mixture and a solution

Ωmm = 1 (A atoms on A sites, B atoms on B) Ωs = NA! / ((XANA)! ((1-XA)NA)!)

give the expression for ΔSmix and give the final equation for ΔSmix, roughly explain how to get from one to the other

ΔSmix = kln(Ωs) - kln(Ωmm) = kln(Ωs) once the Stirling approximation has been used this gives ΔSmix = -R(XA ln(XA) + XB ln(XB))

by combining the two expressions for ΔSmix and ΔHmix, give the overall equation for ΔGmix

ΔGmix = ΔHmix - TΔSmix = XA XB ψ + RT(XA ln(XA) + XB ln(XB))

give Ficks second law, when is it used

∂C/∂t = D ∂^2C/ ∂x^2 where D = diffusivity C = conc.

give the equation for the driving force/change in free energy for the nucleation of a new phase

ΔG = ΔS(Te - T) where Te is the equil. temp

give the expression for the work of nucleation (assuming the nucleus that forms is a sphere) and the work of nucleation if we include a strain term

Wn = (4/3)πr^3 ΔGv + 4π r^2 γ excluding strain Wn = (4/3)πr^3 (ΔGv + U) + 4π r^2 γ where U is the strain energy per unit vol.

from the equations for work done in nucleation what are the equations for critical nucleus radius and critical work done

r* = -2γ / ΔGv Wn* = (16π/3) (γ^3 / (ΔGv)^2)

when considering homogeneous nucleation, what are the two terms that we must consider, give the equations that each of these are given by and hence the equation for nucleation frequency, I

1) population of critical nuclei prop. to exp(-(Wn*/ kT)) 2) rate at which particles can add to make nuclei post-critical prop. to exp(-(Q/kT)) overall I = Cn exp( - ((Wn*+Q)/kT) )

give the relationship between critical work done for heterogenous nucleation and critical work done for homogenous nucleation

Wn*hetero = Wn*homo((2+cosθ)(1-cosθ)^2) /4)

give the equation for the growth rate of a new phase

- the growth rate of a new phase is given by ν = Cg exp(-Q/KT)(1 - exp(VaΔGv / KT)) Cg = const. Q = activation energy K = Boltz. const. T = temp Va = vol of atom ΔGv = driving force per unit vol.

give the equation (relating to TTT diagrams) for the critical rate of cooling to NOT form the phase in question for the TTT diagram

critical rate of T = ΔT(nose)/t(nose) t(nose) = time at nose of 0% curve

give the equation for engineering stress

σ(eng) = F/Ao Ao = init. cross-sectional area

give the equation for engineering strain

ε(eng) = li/lo - 1 li = final length lo = init. length

give the equation for shear stress

τ = F/Ao

give the equation for shear strain

γ = ΔYo/xo = tanφ i.e. distance the side length has shifted

give the equation/definition for poisson's ratio

for normal stress on z εx = εy = -v εz v = poisson's ratio

give the equation linking E and G using poisson's ratio

E = 2G(1+v)

give the equation for how to approximate E using U (potential energy curve) and interatomic spacing, ro

E = (1/ro) (d^2U/dr^2)|r=ro

using the block slab model, give the expression for axial modulus of a composite, Ec, in terms of the moduli of the matrix, Em and fibres, Ef

Ec = EfVf + Em(1-Vf)

using the block slab model, give the expression for transverse modulus of a composite, Ec, in terms of the moduli of the matrix, Em and fibres, Ef

Ec = (EfEm) / (EmVf + Ef(1-Vf))

give the equation for expansion due to temperature using the coefficient of thermal expansion, α

εt = α ΔT

give the equation for how we define radius when considering bending of beams

R = l / θ

give the equation for the deflection at any point on a rectangular beam and hence the max deflection

y = (Fx^2/6EI)(3L-x) δ = FL^3 / 3EI

give the expression for the Peierls-Nabarro stress

τ = 3G exp(-2πw/ b) w = dislocation width G = shear modulus b = |b|

give the expression for the glide 'force' (per unit length) of a dislocation

F = τb b = burgers !vector!

give two expressions for the work done in moving a dislocation

W = FLd d = dislocation distance F = glide force (per unit length) L = length of line of dislocation W = τbLd

give the equation for the critical resolved shear stress

τ(crit) = σy cos(φ) cos(λ) φ = angle of slip plane normal to tensile axis λ = angle of slip direction to tensile axis

what is the Schmid factor

cos(φ) cos(λ)

what is the equation we should consider when thinking about how the tensile axis rotated towards the slip direction on maintained slip

TA = TA(orig.) + n [slip direction]

when two dislocations combine, what is the new burgers vector

b3 = b1+b2

what is franks rule, when is dislocation combination favourable

(b3)^2 < (b1)^2 + (b2)^2

give the equation for dislocation density

ρ = 1/L^2 L = avr. spacing of dislocations on an area

Give the Hall-Petch relationship regarding grain size and yield stress

σy = σo + k/sqrt(d) σy = yield stress σo = Peierls-Nabarro stress k = const. d = avr. grain diameter

give the relationship for the increase in shear stress due to precipitate cutting

Δτ prop. to sqrt(r)

Give the griffith criterion

G = π(σo^2)(c) / E G >= Gc = 2γ

Give the griffith criterion for ductile materials

G >= Gc = 2(γ + γp)

give the equation for the stress intensity paramater

K = σo sqrt(pi*c)

give the equation for viscous liquid flow

τ = -η dγ/dt

Give the equation for the critical undercooling for ice formation on an Ice nucleating agents

ΔT(crit) = -2γ/ ΔSv R

give the Clausius-Clapeyron eq.

dp/dt = ΔS/ΔV

give the inequality which explains why surface melting generally occurs

γsv > γls + γlv

give the equation for strain rate due to creep (generally)

dε/dt = Aσ^n

give the equation for strain rate due to diffusion (coble) creep

dε/dt = (B'σ / d^2) exp(-Q/RT)

give the equation for strain rate due to dislocation creep

dε/dt = (A' σ^n) exp(-Q/RT)

give the general equations for the efficiency of the carnot and brayton cycles for jet engines

Carnot Efficiency = 1 - Tc/Th Brayton Efficiency = 1- T2/T1

give the eq.'s for the max and average transferable energy in radiation damage and define the coeffecient

Max. transferable energy = ζ En Av. transferable energy = ζ En /2 ζ = 4A / (1+A)^2 A = atomic mass number