Formula sheet

Gauss's law (electric)

$\mathbf{E}$ electric field, $\rho$ charge density, $\varepsilon_0$ permittivity of free space

Charge makes an electric field.

Gauss's law (magnetic)

$\mathbf{B}$ magnetic field

There are no single magnetic charges.

Faraday's law

$t$ time

A changing magnetic field makes an electric field.

Ampère–Maxwell law

$\mu_0$ permeability of free space, $\mathbf{j}$ current density, $c$ speed of light

Current (and a changing electric field) makes a magnetic field.

Mass scaling

$m$ mass, $L$ length scale

Bone cross-section requirement

$A$ cross-section area needed

Bone area has to grow as fast as mass to carry the load.

Uniform-scaling area

$A$ area, $L$ length

What the area actually does if you stretch every dimension equally.

Molecular volume

$v$ volume per molecule

Area–volume scaling

$A$ area, $v$ volume

Surface tension scaling

$\Delta H_{vap}$ enthalpy of vaporisation, $\sigma$ surface tension, $v$ volume per molecule

Plot $\Delta H_{vap}$ against $\sigma v^{2/3}$ and you get a straight line.

de Broglie wavelength

$\lambda$ de Broglie wavelength, $h$ Planck's constant, $p$ momentum

Compare it with the size of the system to choose classical or quantum.

Planck's constant

$h$ Planck's constant

Momentum

$p$ momentum, $m$ mass, $v$ velocity

de Broglie relation

$p$ momentum, $h$ Planck's constant, $\lambda$ wavelength

Ties how much the electron wave is squeezed to how much energy it carries.

Momentum

$m$ mass, $u$ speed

Electron kinetic energy

$\text{KE}$ kinetic energy

Squeeze the wave (small $\lambda$) and KE grows fast.

Electron potential energy

$Z$ nuclear charge number, $e$ elementary charge, $\epsilon_0$ permittivity, $r$ separation

The slide takes $r \approx \lambda$.

Total energy vs wavelength

$E$ total energy, $\lambda$ matter wavelength

Energy minimum

$\langle r\rangle$ mean radius in units of $a_0$

Generic multipole potential

$V$ potential energy, $r$ separation, $n$ and $m$ equal 1 for a monopole, 2 for a dipole, 4 for a quadrupole

Gives the falloff table: ion-ion $1/r$, ion-dipole $1/r^2$, dipole-dipole $1/r^3$, quadrupole-quadrupole $1/r^7$.

Coulomb potential

$q_1, q_2$ charges, $r$ separation

Dipole–dipole potential

$\mu_1, \mu_2$ dipole moments

Induced dipole–induced dipole

$C_6$ dispersion coefficient, $\alpha$ polarisabilities, $I$ ionisation energies

Rotation operator

$n$ order of the axis

A $C_2$ axis means a half turn, a $C_3$ axis a third of a turn.

Power identities

$E$ identity operation

Doing the full set of turns brings you back to where you started. Two quarter turns make a half turn.

Improper rotation

$S_n$ improper rotation axis of order $n$, $\sigma_h$ horizontal mirror plane, $C_n$ rotation

Turn, then flip in the plane across the axis. The order does not matter.

Low-order improper rotations

$\sigma$ mirror plane, $i$ inversion centre

So every operation is either a proper or an improper rotation.

Closure

$G$ point group, $\diamond$ combining two symmetry operations

Doing one operation then another is itself an operation in the group. Beyond the scope of the course, per the slide.

Identity

$E$ the identity operation

Associativity

$A, B, C$ symmetry operations in the group

Inverse

$A^{-1}$ the operation that undoes $A$

Commutativity (abelian group)

Holds only if the group is abelian

Position in a unit cell

$\mathbf{r}$ position vector, $\mathbf{a},\mathbf{b},\mathbf{c}$ unit cell vectors, $x,y,z$ fractional coordinates

Say where an atom is, as a fraction of each cell edge.

Position anywhere in the crystal

$u,v,w$ integers (which cell), $x,y,z$ fractional coordinates (where in the cell)

Translation of an atom

$x,y,z$ fractional coordinates

An atom here means a copy one cell over, or three cells over.

Cell lengths and angles

$a, b, c$ lengths of the cell vectors, $\alpha, \beta, \gamma$ angles between the cell vectors

Space group count

Point group symmetry combined with translation in the 14 lattices

Only some point group symmetries fit each lattice, which is why the product is not 32 times 14.

Screw axis

$m$ order of the rotation, $n$ translation index

Examples from the slide: $2_1$ (½, 180°), $3_1$ (⅓, 120°), $4_2$ (²⁄₄, 90°), $6_3$ (³⁄₆, 60°).

Glide translations

$\mathbf{a},\mathbf{b},\mathbf{c}$ cell edges; a glide is a reflection plus this translation

fcc vertex atoms

8 vertex atoms, each shared by 8 cells

fcc face atoms

6 face atoms, each shared by 2 cells

ccp atoms per cell

vertex atoms plus face atoms

Four atoms in each fcc cell.

fcc packing efficiency

fraction of space filled by spheres in fcc

Octahedral hole radius ratio

$r_h$ radius of the hole, $r$ radius of the packed spheres

Tetrahedral hole radius ratio (Lecture 5 slide)

$r_h$ radius of the hole, $r$ radius of the packed spheres

Lecture 7 gives 0.225 for this ratio. The two slides disagree.

Octahedral edge holes per cell

12 edge holes, each shared by 4 cells

Octahedral holes per cell

edge holes plus the centre hole

Tetrahedral holes per cell

one per octahedron face, or per cube vertex

Tetrahedral holes per ccp atom

8 holes per cell divided by 4 atoms per cell

Na count in rock salt

Na atoms per cell: 8 corners shared by 8 cells, 6 faces shared by 2 cells

Cl count in rock salt

Cl atoms per cell: 12 edges shared by 4 cells, 1 at the centre

Rock-salt formula

cell contents reduced to the empirical formula

Zinc blende composition

ccp S²⁻ with Zn²⁺ in half the tetrahedral holes

Antifluorite composition

ccp O²⁻ with Li⁺ in all tetrahedral holes

CsCl composition

two interpenetrating simple cubic cells

Rutile composition

distorted hcp O²⁻ with Ti⁴⁺ in half the octahedral holes

Miller plane

$(h\,k\,l)$ integers, $\mathbf{a},\mathbf{b},\mathbf{c}$ cell vectors

The slide ties each index to an axis intercept of $1/h$, $1/k$, $1/l$.

Bragg equation

$n$ integer order of reflection, $\lambda$ X-ray wavelength, $d$ lattice plane spacing, $\theta$ incidence angle

Use it to get $d$ from a measured angle, or the angle from $d$.

Path difference

extra distance travelled by beam 2 compared with beam 1

Constructive interference needs this to equal $n\lambda$.

Charge neutrality, general form

$m$ and $p$ are set by charge neutrality

The total positive charge equals the total negative charge.

Octahedral hole ratio

$r$ radii of the octahedral hole and of the packed spheres

Tetrahedral hole ratio

$r$ radii of the tetrahedral hole and of the packed spheres

Lecture 5 gives 0.255 for the same ratio, which does not match.

Coulomb potential

$q_1,q_2$ charges, $\varepsilon_0$ permittivity, $r$ separation

Lattice reaction

$\Delta H^0_L$ lattice enthalpy

The energy to break the crystal into separate gas-phase ions.

Born–Mayer equation

$N_A$ Avogadro's number, $e$ electronic charge, $z_A, z_B$ ionic charges, $\varepsilon_0$ vacuum permittivity, $d = r_A + r_B$ separation, $d^*$ repulsion constant (0.345 Å), $A$ Madelung constant

For one structure, only $d$, $z_A$ and $z_B$ change.

Madelung series

sum over successive coordination spheres of NaCl

Madelung form

$A$ Madelung constant

First three NaCl sphere terms

six nearest neighbours attract, twelve next neighbours repel, eight further ones attract

Band-gap thermal condition

$E_g$ band gap, $k$ Boltzmann constant, $T$ temperature

For a material to be an intrinsic semiconductor the gap must be small enough that thermal energy, of order $kT$, can lift electrons across it. The slide writes it as a band gap of order $kT$.

Krogmann salt oxidation

$\text{X}$ is Cl or Br

The halogen removes some electrons from the Pt chains. That makes the band partly empty, so the chain conducts.

Electron count in C60

4 valence electrons per carbon atom, times 60 carbon atoms

Use it to count the electrons that fill the molecular orbitals of C60.

BCS critical temperature

$k_B$ Boltzmann constant, $T_c$ critical temperature, $\hbar\omega$ Debye energy, $V$ strength of the attraction between electrons, $N(E_f)$ density of states at the Fermi energy

A larger $N(E_f)$ or a stronger attraction $V$ raises $T_c$.

Debye temperature

$\hbar$ reduced Planck constant, $\omega$ phonon frequency, $k_B$ Boltzmann constant

The slide describes it as the collective (acoustic) vibration of the lattice.

Conductivity and resistivity

$\sigma$ conductivity (S cm$^{-1}$), $\rho$ resistivity

Conductivity is how well a material carries current. Resistivity is how much it resists.

Conductivity from carriers

$n$ carrier concentration, $e$ electronic charge, $\mu$ carrier mobility

More carriers, or carriers that move more easily, mean higher conductivity.

Carrier mobility

$e$ electronic charge, $\tau$ relaxation time between scattering events, $m^*$ effective mass

Longer time between scattering gives a higher mobility.

Drude conductivity

$n$ carrier concentration, $e$ electronic charge, $\tau$ relaxation time, $m^*$ effective mass

Put the mobility into the carrier equation and you get this. Use it to see why a metal loses conductivity when $\tau$ falls on heating.

Semiconductor carrier density

$E_g$ band gap, $k$ Boltzmann constant, $T$ temperature

Heating a semiconductor puts many more carriers across the gap.

Capacitance (definition)

$C$ capacitance, $Q$ charge on either plate, $\Delta V$ potential difference between the plates

Capacitance says how much charge the plates hold per volt.

Parallel-plate capacitance in vacuum

$\varepsilon_0$ permittivity of vacuum, $A$ plate area, $d$ plate separation

Bigger plates or a smaller gap give more capacitance.

Energy stored in a capacitor

$U$ electric potential energy, $C$ capacitance, $\Delta V$ potential difference

Capacitance with a dielectric

$\varepsilon_r$ relative permittivity, $\varepsilon_0$ permittivity of vacuum, $A$ plate area, $d$ plate separation

Put in a dielectric and the capacitance goes up by the factor $\varepsilon_r$.

Polarisation

$P$ polarisation, $\varepsilon_0$ permittivity of vacuum, $\chi_e$ electric susceptibility, $E$ applied electric field

For a paraelectric, polarisation grows in proportion to the field. The slope at the origin is $\varepsilon_0\chi_e$.

Electric susceptibility

$\chi_e$ electric susceptibility, $\varepsilon_r$ relative permittivity

Vacuum has $\varepsilon_r = 1$ so $\chi_e = 0$, meaning no polarisation response.

Magnetisation

$\mathbf{M}$ magnetisation (magnetic moment per unit volume), $\chi$ magnetic susceptibility, $\mathbf{H}$ applied magnetic field

Susceptibility tells you how strongly the material responds to a field.

Diamagnetic susceptibility

$\chi$ magnetic susceptibility, which is negative and does not change with temperature

The sign tells you the material is repelled by a field.

Curie law

$\chi$ susceptibility, $C_{\text{Curie}}$ Curie constant, $T$ temperature

Use it for a paramagnet. Susceptibility falls as temperature rises.

Total spin of high-spin Cr2+

$S$ total spin quantum number, $s = \pm\tfrac{1}{2}$ for each electron; four unpaired electrons

Total spin of low-spin Cr2+

$S$ total spin quantum number, $s = \pm\tfrac{1}{2}$ for each electron; three up and one down

Bohr magneton

$\mu_B$ Bohr magneton, the unit of an electron magnetic moment

Nuclear magneton

$\mu_N$ nuclear magneton, the unit of a nuclear magnetic moment

Curie-Weiss law

$\chi$ susceptibility, $C$ Curie constant, $T$ temperature, $\theta$ Weiss constant

Use it above the ordering temperature of a ferromagnet or antiferromagnet. It reduces to Curie law when $\theta = 0$.

Weiss constant of a ferromagnet

$\theta$ Weiss constant, $T_C$ Curie temperature

Weiss constant of an antiferromagnet

$\theta$ Weiss constant, $T_N$ Néel temperature

Extrapolating a straight line of $\chi^{-1}$ against $T$ back to the axis hits $-T_N$.

Harmonic oscillator levels

$h$ Planck constant, $\nu$ frequency, $n$ quantum number

Harmonic oscillator levels, zero-point energy ignored

$h$ Planck constant, $\nu$ frequency, $n$ quantum number

Evenly spaced rungs. Dropping the constant half does not change the gaps.

Factorial

$n$ an integer

Weight of a configuration

$W$ number of microstates, $N$ total molecules, $n_i$ occupation of state $i$

Count the microstates behind a configuration.

Worked weight

The blue configuration of three quanta on three oscillators.

Stirling approximation

$n$ a large integer

Turns awkward factorials into something you can differentiate.

ln W to maximise

$N$ total molecules, $n_i$ occupation of state $i$

Fixed number of molecules

$n_i$ occupation of state $i$, $N$ total molecules

Fixed total energy

$\varepsilon_i$ energy of level $i$, $E$ total energy

Boltzmann distribution

$p_i$ probability of level $i$, $\beta$ Lagrange multiplier (inverse temperature)

Gives the share of molecules in each level at equilibrium.

Partition function

$\varepsilon_i$ energy of level $i$, $\beta$ as above

Population ratio

$n_i, n_j$ populations of levels $i$ and $j$

Only the energy gap matters.

Definition of temperature

$k$ Boltzmann constant, $T$ temperature

Air in a room, all on one side (very rough)

Chance that more than 51% of the air in a 40 m$^3$ room is on the left

Expectation value

$O$ an observable, $p_i$ probability of state $i$, $O_i$ its value in state $i$

A weighted average: each state counts as often as it occurs.

Average energy per molecule

$\varepsilon$ energy, $\beta = 1/kT$

Total energy of N molecules

$E$ total energy, $N$ number of molecules, $n_i$ occupation of level $i$

For identical, independent molecules.

Boltzmann distribution

$p_i$ probability of level $i$, $n_i$ its occupation, $N$ total molecules

Definition of temperature

$k$ Boltzmann constant = $1.380649 \times 10^{-23}$ J K$^{-1}$, $T$ absolute temperature

beta has units of inverse energy.

Population ratio

$n_i, n_j$ populations of levels $i$ and $j$

Absolute-zero limit

$\varepsilon_0$ ground-state energy

Ground-state occupation at T = 0

$n_0$ ground-state occupation

All molecules in the ground state

At absolute zero

Partition function

$\varepsilon_i$ energy of state $i$, $\beta = 1/kT$

Roughly the number of thermally accessible states per molecule.

Occupation relative to the ground state

$n_i$ occupation of state $i$, $n_0$ ground-state occupation

Oscillator levels

Case where $h\nu = \beta^{-1}$

Oscillator partition function

A geometric series, summed in closed form.

Oscillator probabilities, Case 1

$q = 1.58$

Oscillator probabilities, Case 2 (q = 2.54)

Levels brought closer together, so $q$ is larger

Energy from the partition function

$E$ total energy, $N$ molecules, $k$ Boltzmann constant, $T$ temperature

Bulk quantities are derivatives of q.

Derivative of q

Partition function over levels

$g(\varepsilon_i)$ degeneracy of the level with energy $\varepsilon_i$

Sum over levels, with each level counted g times.

Level population ratio

$g$ degeneracies of the two levels

Characteristic temperature

$\Delta E$ typical energy spacing, $k$ Boltzmann constant

Roughly the lowest temperature at which excited states start to fill.

Population versus characteristic temperature

$n_j, n_i$ populations of an upper and a lower level

Worked values for Theta = 300 K

Ratio of first excited state to ground state

Internal energy (microscopic)

$n_i$ occupation of microstate $i$, $\varepsilon_i$ its energy

Total energy of the occupied states.

First law

$Q$ heat entering the system, $W$ work done on the system

The lecture convention. Work done ON the system is positive.

First law, alternate sign convention

$W$ is the work done by the system

Used by some books and Wikipedia. Not the lecture convention.

Work

$\mathbf{F}$ generalised force, $\mathbf{x}$ generalised displacement

Expansion work

$p_{\text{ext}}$ external pressure, $dV$ change in volume

Total expansion work

$V_i$ initial volume, $V_f$ final volume

Surface expansion work

$\gamma$ surface tension, $\sigma$ surface area

Extension work

$f$ tension, $l$ length

Electrical work

$\phi$ electric potential, $Q$ charge

Energy balance

Isolated-system relation

What the system gains, the surroundings lose.

Microscopic differential of U

First sum is work (levels shift), second sum is heat (populations change)

Second law, version 1

For an isolated system

Dog-flea jump probability

$R$ half the number of fleas, $N_{\text{Rover}}$ fleas on Rover

Chance that the next jump goes from Rover to Rex.

Reverse-jump probability

Equilibrium in the dog-flea model

Direction of change

$W$ number of microstates, at constant $N$ and $E$

Entropy

$k$ Boltzmann constant, $W$ number of microstates

Second law for an isolated system

Weight of a configuration

Stirling form of ln W

Total differential of ln W

With $N$ constant, so $dN = 0$

Equilibrium population

Expansion of the differential

The first term vanishes because $N$ is constant

Reversible heat relation

Thermodynamic entropy

$Q_{\text{rev}}$ heat in a reversible process, $T$ temperature

Entropy change, reversible process

Slide wording. See the aside about the inequality.

State-function equality

Entropy depends only on the state, not the path.

Entropy from q (1)

$U$ internal energy, $q$ partition function

Entropy from q (2)

Entropy from q (3)

The lecture says there is no need to memorise these.

Additivity of entropy

Two independent, isolated systems $x$ and $y$

Total entropy on thermal contact

$x$ and $y$ two systems in thermal contact, isolated from everything else

Heat terms in the entropy sum

Combined form

Direction of heat flow

Heat flows into $x$ when $x$ is the colder system

Heat flows spontaneously from hot to cold.

Third law, perfect crystal

One microstate at $T = 0$ for a perfect crystal of a pure substance

Residual entropy of CO

$N$ molecules, each in one of two orientations

Total entropy with a bath

$S$ system entropy, $S_{\text{bath}}$ bath entropy, $Q$ heat entering the system

Clausius form of the second law

$S$ system entropy, $Q$ heat

Energy balance

Combined inequality

Helmholtz free energy

$A$ Helmholtz free energy, $U$ internal energy, $T$ temperature, $S$ entropy

A is for Arbeit, German for work.

Helmholtz change at constant T

Work bound

At constant T, A is the maximum work you can extract.

Spontaneity at constant T and V

No pV work and no non-pV work

Internal energy differential

No non-expansion work

Constant-volume heat

At constant $V$

Derivative identity

Enthalpy

$H$ enthalpy, $p$ pressure, $V$ volume

Enthalpy differential

Constant-pressure heat

At constant $p$ and no non-pV work

Exothermic and endothermic usually refer to delta H.

Gibbs free energy

$G$ Gibbs free energy

Gibbs bound

Constant T and p

Spontaneity at constant T and p

No non-pV work

Spontaneity summary

Gibbs from the partition function

$N$ molecules, $q$ partition function