Stage 2 SEM2

Differentials

Stat Mechanics

Solid-state

Electro

The first of Maxwell's equations: Gauss's law


∇∇⋅E=ρε0or∮AE⋅dA=qINε0

∫V∇∇⋅EdV=∮AE⋅ˆndA

Section 2

Any analytic function has a unique convergent power series


\(f(x)=\sum_{n=0}^{\infty}a_n(x-x_0)^n\)


Convergent series can be used to solve second order ODEs. They can be added, multiplied, differentiated and integrated :

If both \(p(x)\) and \(q(x)\) in


\(y''+p(x)y'+q(x)y=0\)


are analytic in \(|x-x_0| < R\), then any solution \(y(x)\) is also analytic, with \(y(x)=\sum_{n=0}^{\infty}a_n(x-x_0)^n\)

A function is analytic at \(x=x_0\) if its Taylor series at \(x=x_0\) exists.

A function is singular at \(x=x_0\) if \(f(x_0)\) or any of its derivatives are undetermined

If \(p(x)\) or \(q(x)\) are not analytic, \(x\) describes a singular point


A singular point is described as regular at \(x=x_0\) if \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) are analytic; otherwise, it is described as essential, and is generally unsolveable

A coefficient can be introduced to allow for singular points:


\(y(x)=\color{green}{x^{\lambda}}\sum_{n=0}^{\infty}a_n(x-x_0)^n\) with constant \(\lambda\)

Power series solutions

Differentiate for \(y'\) and \(y''\) and simplify the resulting sums - factorisation, index swapping etc. Pulling out terms can allow terms to start at the same index, e.g. \(\sum_{n=0}c_n\Rightarrow c_0 + \sum_{n=1}c_n\)


With the resulting sum, compare coefficients to determine \(\lambda\) and a recurrence relation for \(a_n\)

Useful stuff

Potential difference
\[\mathbf{E}=-\pmb{\nabla}\phi\]\[\phi =\int_a^b\mathbf{E}\cdot\mathbf{\hat{n}}\mathrm{d}l\]

Closed integrals over an area or volume can be evaluated with
\[\oint_A\mathrm{d}A=A\qquad\oint_V\mathrm{d}V=V\]

Differentials can also be converted, e.g.
\[V=\frac{4}{3}\pi r^3 \Rightarrow \mathrm{d}V=4\pi r^2\mathrm{d}r\]

Vector differentials can dot product with unit vectors


\[\mathbf{\hat{r}}\cdot\mathrm{d}\mathbf{r}=\mathrm{d}r\]

In a conservative vector field (electrostatics), a line integral is independent of path, a closed integral is zero, and the curl is zero:
\[\oint_C\mathbf{E}\cdot\mathrm{d}\mathbf{l}=0\qquad\pmb{\nabla}\times\mathbf{E}=0\]

For two parallel vectors
\[\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\]

Dielectrics are insulators that can be polarised in an electric field


By taking free and bounded charge densities \(\rho_f\) and \(\rho_p\), we obtain a more generalised
\[\pmb{\nabla}\cdot\mathbf{D}=\rho_f\qquad \mathbf{D}=\varepsilon_0\mathbf{E}+\mathbf{P}=\varepsilon_0\varepsilon_r\mathbf{E}\qquad \rho_b=-\pmb{\nabla}\cdot\mathbf{P}\] where \(\mathbf{P}\) is the polarisation

Magnetostatics

Lorentz force in an electromagnetic field
\[\mathbf{F}=q\left(\mathbf{E}+\mathbf{B}\times\mathbf{v}\right)\]

Capacitors


\[E=\frac{\sigma}{\varepsilon}=\frac{V}{d}\qquad C=\frac{Q}{\Delta\phi}=\frac{Q}{V}=\frac{A\varepsilon}{d}\]

To find the boundary conditions: evaluate the following and determine whether \(\mathbf{E}\) and \(\mathbf{D}\) are continuous normally and tangentially
\[\oint_A\mathbf{D}\cdot\mathrm{d}\mathbf{A}=\oint_A\mathbf{D}\cdot\mathbf{\hat{n}}\;\mathrm{d}A=Q_f\]


This should be evaluated with \(h\rightarrow 0\) where \(h\) is the normal height from the contact area

The dot product with a unit vector gives the component in that direction, e.g.
\[\mathbf{D}\cdot\mathbf{\hat{n}}=D_n\]

Electrostatic potential and the gradient theorem


\[\int_a^b\pmb{\nabla}\phi\cdot\mathrm{d}\mathbf{r}=\phi(b)-\phi(a)\qquad\phi(\infty)=0\]

Free charges in a conductor are always located at the surface; a result of Gauss's law

Stoke's theorem (applies for \(\mathbf{B}\), \(\mathbf{E}\), etc)
\[\oint_C\mathbf{B}\cdot\mathrm{d}\mathbf{l}=\int_A (\pmb{\nabla}\times\mathbf{B})\cdot\mathbf{\hat{n}}\,\mathrm{d}A\]

\(\mathbf{D}\) is generally parallel to \(\mathbf{E}\) (?)

Section 5

Functions can be expanded into infinite series under a functional basis other than polynomials. The Fourier series is an expansion into sines and cosines
\[f(x)=\sum_{n=0}^\infty \left(a_n\cos{\frac{\pi nx}{l}}+b_n\sin{\frac{\pi nx}{l}}\right)\]

Using orthogonality relations and multiplying by \(\sin{\frac{\pi mx}{l}}\) and \(\cos{\frac{\pi mx}{l}}\) we obtain the coefficients
\[a_m=\frac{1}{l}\int_{-l}^l f(x)\cos{\frac{\pi mx}{l}}\mathrm{d}x\]\[b_m=\frac{1}{l}\int_{-l}^l f(x)\sin{\frac{\pi mx}{l}}\mathrm{d}x\]

Picking \(b_0=0\)
\[a_0=\frac{1}{2l}\int_{-l}^l f(x)\mathrm{d}x\]

The even extension is symmetric about the y axis; the odd extension is mirrored on the x axis. Expanding to \(-l\leq x\leq l\) produces the odd and even expansions \[\widetilde{f}_e=\sum_{n=0}^\infty a_n\cos{\frac{\pi nx}{l}}\]\[\widetilde{f}_o=\sum_{n=1}^\infty b_n\sin{\frac{\pi nx}{l}}\]Noting the series index of \(\widetilde{f}_o\) starts at \(n=1\)

Gauss' law of magnetism
\[\pmb{\nabla}\cdot\mathbf{B}=0\]
for any magnetic field

NOTE \(-l\) and \(l\) can be replaced with the start
and end points of a \(f(x)\) not centred on the origin

Distributions

Maxwell-Boltzmann distribution
\[n_i=\frac{N}{Z}e^{-\varepsilon_i/kT}\qquad f(\varepsilon)=e^{(\mu-\varepsilon)/kT}\]
Valid for non-interacting and indistinguishable particles

Partition function
\[z_1=\sum_i e^{-\varepsilon_i/kT}\]

For distinguishable particles \(z=z_1^n\): atoms are localised and associated with fixed lattice sites (in a solid)


For indistinguishable particles \(z=\frac{z_1^n}{n!}\): atoms are not localised

\[E=-N\frac{\partial\ln{z_1}}{\partial\beta} \qquad F=-Nk_bT\ln{z_1} \]

\[c_V=\frac{\partial E}{\partial T}\]

\[P=-\frac{\partial F}{\partial V}\]

\[S=-\frac{\partial F}{\partial T}\]

Bonding

Ionic

Ionisation energy is the energy needed to remove one electron from a neutral atom to create a positive ion

The potential between two ions
\[\phi_{ij}=\pm\frac{e^2}{4\pi\varepsilon_0 r_{ij}}+\frac{B}{r_{ij}^n}\delta_{i=j\pm 1}\]

Finding the binding energy
\[\frac{\mathrm{d}\phi}{\mathrm{d}r}=0\Rightarrow B=(...)\Rightarrow E_B=\phi(r_0)\]

Electron affinity is the energy gain from creating a negative ion by adding an electron

Covalent

The atomic orbital is a wavefunction describing electron behaviour in an atom

\[\sum_m H_{nm}c_m=Ec_n\Rightarrow \begin{pmatrix}E_0 & -h \\ -h & E_0\end{pmatrix}\begin{pmatrix}c_1 \\ c_2\end{pmatrix}=E\begin{pmatrix}c_1 \\ c_2\end{pmatrix}\]
The smaller resulting \(E\) describes the bonding orbital; the larger describes the antibonding orbital

Dirac notation for s orbitals
\[\varphi_n = Ae^{\frac{r_n}{\rho}}=\lvert{1}\rangle\]

Find eigenvalue \(E\) and eigenvectors; normalise with \(c_1^2+c_2^2 =0\) and construct the orbital wavefunctions
\[\phi_{bonding}=c_1\lvert{1}\rangle + c_2\lvert{2}\rangle \qquad\phi_{antibonding}=c_1\lvert{1}\rangle - c_2\lvert{2}\rangle\]


For large differences in energy, \(\varphi\) reduces to an ionic bond

\(r_{ij}=r_0p_{ij}\) where \(p_{ij}\) is a function of \(i\) and \(j\); e.g. in a linear chain, \(p_{ij}=|i-j|\)


The sign \(\pm\) can also be written as a function of \(i\) and \(j\)

The total potential
\[\phi_i=\sum_{i\neq j}\phi_{ij} \Rightarrow \Phi=\frac{N}{2}\phi_i\]

When finding the Madelung constant \(A\), remember to double if only one half was considered

VdW

Momentary dipole moment in one atom polarizes a neighbouring atom; resulting dipoles (weakly) attract

Vibrations

\[\beta=\frac{1}{L}\frac{\mathrm{d}L}{\mathrm{d}F}\approx\frac{1}{ka} \\ F=-k\delta x \qquad L=N(a+\delta x)\]

Normal modes describe the collective oscillation of particles moving at the same frequency; they are orthogonal to one another

Standard dispersion relation
\[\omega=2\sqrt{\frac{\kappa}{m}}\left|\sin{\frac{ka}{2}}\right|\]

\(\mathrm{d}N\) is the number of modes that fit in the interval \(\omega\rightarrow\omega+\mathrm{d}\omega\)


Hence the density of states \(g(\omega)=\frac{\mathrm{d}N}{\mathrm{d}\omega}\)

Arising from the linearity of \(N\) vs \(k\)
\[g(\omega)=\frac{\mathrm{d}N}{\mathrm{d}\omega}=\frac{\mathrm{d}N}{\mathrm{d}k}\frac{\mathrm{d}k}{\mathrm{d}\omega}=\frac{\Delta N}{\Delta k}\frac{1}{v_g}=\frac{Na}{2\pi}\frac{1}{v_g}\]

Group velocity describes the velocity of the wavepacket
\[v_g=\frac{\mathrm{d}\omega}{\mathrm{d}k}\]

Phase velocity describes the velocity of individual maxima and minima
\[v_{phase}=\frac{\omega}{k}\]

Infinite geometric series with constant coefficients
\[S_{\infty}=\frac{a}{1-x}\]

Fermi-Dirac distribution
\[f_{FD}(\varepsilon)=\frac{1}{e^{(\varepsilon -\mu)/kT}+1}\]
Valid for electrons and bosons. For bosons, multiply by \(N\) (multiple can occupy one state) (?)

Bose-Einsten distribution
\[f_{BE}(\varepsilon)=\frac{1}{e^{(\varepsilon -\mu)/kT}-1}\]
Valid for non-interacting and indistinguishable bosons

Einsten vs Debye models for \(C_V\)

Factorisation of the partition function; the energy can be split into independent contributions from each degree of freedom
\[\varepsilon_{ijk}=\varepsilon_i^{trans}+\varepsilon_j^{rot}+\varepsilon_k^{vib} \\ \Rightarrow Z=\sum_{i,j,k}e^{-\frac{\varepsilon_i}{kT}}e^{-\frac{\varepsilon_j}{kT}}e^{-\frac{\varepsilon_k}{kT}} = \sum_ie^{-\frac{\varepsilon_i}{kT}}\sum_je^{-\frac{\varepsilon_j}{kT}}\sum_ke^{-\frac{\varepsilon_k}{kT}} \\ \Rightarrow Z=Z_{trans}Z_{rot}Z_{vib} \]

In paramagnetic solids energy is a function of the magnetic moment \(\mu\) with
\[\varepsilon_{parallel}=-\mu B\qquad\varepsilon_{antiparallel}=\mu B\]

The speed of sound in a material is the phase velocity \(v_{phase}=\frac{\omega}{k}\) under the long wavelength limit, as \(k\rightarrow 0\)

3D monatomic chains

For a lattice with displacement vector \(\textbf{u}_{l,m,n}=x_{l,m,n}\textbf{x}+y_{l,m,n}\textbf{y}+z_{l,m,n}\textbf{z}\)
\[\phi_x =\frac{\kappa}{2}(x_{l-1,m,n}-x_{l,m,n})^2 \\ \phi_y =\frac{\kappa}{2}(y_{l,m-1,n}-x_{l,m,n})^2 \\ \phi_z =\frac{\kappa}{2}(z_{l,m,n-1}-x_{l,m,n})^2 \\ \Phi_T = \Phi_{eq}+\sum_{l,m,n}(\phi_x+\phi_y+\phi_z) \]

Equation of motion
\[\mathbf{F}_{l,m,n}=-\pmb{\nabla}\Phi_T =m\ddot{x}_{l,m,n}\]
This may be split componentwise for diatomic chains

For a wavevector \(\mathbf{k}=(k_x,k_y,k_z)\) with \(\mathbf{r}=(la,ma,na)\) we use trial function
\[\mathbf{u}_{l,m,n}=A_xe^{i\omega t-i\mathbf{k}\cdot\mathbf{r}}\mathbf{\hat{x}}+A_ye^{i\omega t-i\mathbf{k}\cdot\mathbf{r}}\mathbf{\hat{y}}+A_ze^{i\omega t-i\mathbf{k}\cdot\mathbf{r}}\mathbf{\hat{z}}\]

Equate \(-m\omega ^2A_xe^{(...)}\) with \(\mathbf{F}_{l,m,n} \) using the trial functions


Construct matrices, identify eigenvalues/eigenvectors and solve for \(\omega\)

\(\mathbf{v}_s=\left.\frac{\omega_x}{k_x}\right|_{k_x\rightarrow 0}\mathbf{\hat{x}}+(...)\)

\[\begin{vmatrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{vmatrix}=a\cdot b\cdot c\]

NOTE for diatomic chains, final \(\omega\) may be left in a complicated root form; no further simplification needed

click to edit

Generally \(r\rightarrow p_{ij}r\) for finding \(\phi_{ij}\)

Any potential can be shown to be quadratic close to its minimum by taking the Taylor expansion:
\(\left.\Phi(r)\right|_{r\rightarrow r_0}=\Phi(r_0)+0+\frac{\kappa_2}{2!}(r-r_0)^2+...\)

Fourier transforms

Similarity theorem
\[\mathcal{F}[f(kx)]=\frac{1}{|k|}\hat{f}(k/a)\]

Shift theorem
\[\mathcal{F}[f(x-a)]=e^{-ika}\hat{f}(k)\]

Modulation theorem
\[\mathcal{F}[e^{iax}f(x)]=\hat{f}(k-a)\]

\[\mathcal{F}[e^{-|x|}]=\sqrt{\frac{2}{\pi}}\frac{1}{1+k^2}\]

Harmonic oscillators

In 3 dimensions we have 3N harmonic oscillators


\(\varepsilon_i =(i+\frac{1}{2})\hbar\omega\) where \(\hbar\omega\) is the energy spacing

Fermi energy of a degenerate Fermi gas
\[\varepsilon_f=\frac{\hbar ^2}{2m}(3\pi^2m)^{2/3}\]

Breaks down at low \(T\) and high density; quantum effects become apparent if
\[n\geq n_Q=\left(\frac{2\pi mkT}{h^2}\right)^{3/2}\]

Einstein model - treat solids as 3N harmonic oscillators at the same frequency \(\omega_E\); a set of collective normal modes


Debye model - treat lattice vibrations as sound waves passing through a solid with \(\omega=v_sk\) (where \(k\) is the wavenumber); correct number of modes given as \(\omega\) is bounded by the lattice

High \(T\): both models predict \(C_V\rightarrow 3NkT\)


Low \(T\): Debye predicts \(C_V\propto T^3\), whereas Einstein predicts exponential


Debye more accurate as \(T\rightarrow 0\) as experiments show \(C_V\propto T^3\)

\(\omega\propto k\) near \(\omega=0\) with Debye model, hence is a better fit for acoustic modes

\(\omega_E\approx\text{const}\) near \(\omega=0\) with Einstein model, hence is a better fit for optical modes

At low \(T\), only the lowest excitations are possible

For low temperature metals, Debye predicts \(C_V\approx AT+BT^3\)


Linear term arises from the free electron gas in a metal.

Bose-Einstein condensation arises when many particles fall into the ground state as \(T\rightarrow 0\)


Two or more fermions, as per the Pauli exclusion principle, cannot occupy the same quantum state


Consequences include a discontinuity in \(C_V\) and superfluidity, where particles in the ground state no longer contribute to viscosity

Define the Fermi temperature by \(kT_F=\varepsilon_F\)


If \(T >> T_F\), Maxwell-Boltzmann or dilute gas models are appropriiate


If \(T << T_F\), Fermi-Dirac model is needed and the gas is said to be degenerate

At \(T=0\), \(f_{FD}=1\) for \(\varepsilon\leq\varepsilon_F\) and \(f_{FD}=0\) for \(\varepsilon >\varepsilon_F\)

1/2NkT energy per degree freedom


3:2:2 degrees trans/rot/bin


Rot 10k, vib 1000k

Waves and T lines

Co-axial cables consist of a central 'go line' (which transmits the signal intended for the load) and an external 'return line' to complete the circuit

Good conductors strongly attenuate space (EM) waves; perfect conductors stop waves entirely


The return line helps to mitigate noise by preventing space waves from reaching the go line

The propagation constant \(\gamma=\alpha+i\beta\) where \(\alpha\) is the attenuation constant and \(\beta\) is the phase constant


If \(\arg{Z}>0\), the potential leads the current (equiv. the current lags the potential)

Tutorial shit

Coaxial and twisted pairs

Coaxial cables more expensive due to materials + installation

Coaxial cables are larger and require more precise measurements

Twisted pair; no shielding from space waves due to lack of an attenuating return line


EMI from the environment causes noise in the cable; poor for frequency-sensitive applications


Some EM leakage into the environment

Return line in coaxial cables attenuates space waves, prevents from reaching the core (go line)

\(F=\frac{mv^2}{r}=qvB\)

Mass spectrometry + Thomson (mass/charge ratios)

Lorentz force in an electromagnetic field
\[\mathbf{F}=q\left(\mathbf{E}+\mathbf{B}\times\mathbf{v}\right)\]

Thomson: electrons accelerated with a pd, translates into kinetic
\[q\phi=\frac{1}{2}m_ev^2\]
Electrons pass into magnetic field to find m/e ratio (using centripetal eq)

In a mass spectrometer: ions pass through the velocity selector (2 charged plates + B field), only ions with \(F_E=F_B\) pass through

Antennae act as a link between waves travelling down a transmission line and TEM waves travelling through free space

Lumped element model

Resistivity \(\rho=\frac{RA}{l}\)


Conductance \(G=\frac{1}{R}=\sigma\frac{A}{l}\)


Generally assumes the dielectric conductance is zero

Impedance \(Z=\sqrt{\frac{L'}{C'}}\)


Impedances must match at boundaries to prevent reflection

Cyclotrons use alternating \(E\) fields and a static \(B\) field to accelerate charged particles


At high energies, relativistic effects cause variation in \(m\)

\(n=\sqrt{\varepsilon_r\mu_r}\)

\(v=\frac{1}{\sqrt{\varepsilon\mu}}\)

Poynting vector (power density)
\[\mathbf{S}=\mathbf{E}\times\mathbf{H}\]

\(\mathbf{P}_{\text{avg}}=\frac{1}{T}\int_0^T\mathbf{S}\;\mathrm{d}t\approx \frac{1}{2}\frac{E^2}{\eta}\mathbf{\hat{k}}\)

\(\eta \approx \sqrt{\frac{\mu}{\varepsilon}}\)

\(\mathbf{J}=\sigma\mathbf{E}\)

Heaviside condition \(L'G'=R'C'\)

Brewster's angle \(\theta_B=\arctan{\frac{n_2}{n_1}}\)