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Stage 2 SEM2, Stat Mechanics (Partition function \[z_1=\sum_i e^{…

- - - - \[P=-\frac{\partial F}{\partial V}\]
        
        \[S=-\frac{\partial F}{\partial T}\]
- - - - Bose-Einsten distribution
        \[f_{BE}(\varepsilon)=\frac{1}{e^{(\varepsilon -\mu)/kT}-1}\]
        Valid for non-interacting and indistinguishable bosons
        
        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
      - Fermi energy of a degenerate Fermi gas
        \[\varepsilon_f=\frac{\hbar ^2}{2m}(3\pi^2m)^{2/3}\]
        
        At \(T=0\), \(f_{FD}=1\) for \(\varepsilon\leq\varepsilon_F\) and \(f_{FD}=0\) for \(\varepsilon >\varepsilon_F\)
- - - - 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.
- - - - 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 \(a_n\)
- - - - Picking \(b_0=0\)
        \[a_0=\frac{1}{2l}\int_{-l}^l f(x)\mathrm{d}x\]
        
        NOTE \(-l\) and \(l\) can be replaced with the start
        and end points of a \(f(x)\) not centred on the origin
- - - - 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
        \[\mathbf{\hat{r}}\cdot\mathrm{d}\mathbf{r}=\mathrm{d}r\]
        
        For two parallel vectors
        \[\mathbf{a}\cdot\mathbf{b}=|\mathbf{a}||\mathbf{b}|\]
  - - - Electrostatic potential and the gradient theorem
        \[\int_a^b\pmb{\nabla}\phi\cdot\mathrm{d}\mathbf{r}=\phi(b)-\phi(a)\qquad\phi(\infty)=0\]
  - - - The dot product with a unit vector gives the component in that direction, e.g.
        \[\mathbf{D}\cdot\mathbf{\hat{n}}=D_n\]
  - - - Stoke's theorem (applies for \(\mathbf{B}\), \(\mathbf{E}\), etc)
        \[\oint_C\mathbf{B}\cdot\mathrm{d}\mathbf{l}=\int_A (\pmb{\nabla}\times\mathbf{B})\cdot\mathbf{\hat{n}}\,\mathrm{d}A\]
        
        Gauss' law of magnetism
        \[\pmb{\nabla}\cdot\mathbf{B}=0\]
        for any magnetic field
- - - - Antennae act as a link between waves travelling down a transmission line and TEM waves travelling through free space
  - - - 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
        
        Heaviside condition \(L'G'=R'C'\)
  - - - \(\eta \approx \sqrt{\frac{\mu}{\varepsilon}}\)
- - - - Coaxial cables are larger and require more precise measurements
  - - - 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
- - - - Electron affinity is the energy gain from creating a negative ion by adding an electron
    - - Finding the binding energy
        \[\frac{\mathrm{d}\phi}{\mathrm{d}r}=0\Rightarrow B=(...)\Rightarrow E_B=\phi(r_0)\]
      - \(r_{ij}=r_0p_{ij}\) 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
        \[\phi_i=\sum_{i\neq j}\phi_{ij} \Rightarrow \Phi=\frac{N}{2}\phi_i\]
        
        Generally \(r\rightarrow p_{ij}r\) for finding \(\phi_{ij}\)
        
        When finding the Madelung constant \(A\), remember to double if only one half was considered
  - - - 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
- - - - The speed of sound in a material is the phase velocity \(v_{phase}=\frac{\omega}{k}\) under the long wavelength limit, as \(k\rightarrow 0\)
  - - - Arising from the linearity of \(N\) vs \(k\)
        \[g(\omega)=\frac{\mathrm{d}N}{\mathrm{d}\omega}=\frac{\mathrm{d}N}{\mathrm{d}k}\frac{\mathrm{d}k}{\mathrm{d}\omega}=\frac{\Delta N}{\Delta k}\frac{1}{v_g}=\frac{Na}{2\pi}\frac{1}{v_g}\]
    - - Phase velocity describes the velocity of individual maxima and minima
        \[v_{phase}=\frac{\omega}{k}\]
  - - - 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