Foundations of Physics 3B
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Foundations of Physics 3B - Leaderboard
Foundations of Physics 3B - Details
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| 🇬🇧 | 🇬🇧 |
| Binomial probability eqn | P_N (k)=N!/k!(N−k)! p^k (1−p)^(N−k) |
| Macrostate definition | Specification of the state of a system based only on bulk properties containing minimal information to describe the system Hides a lot of the underlying physics, as usually, we have averaged away much of the detail |
| Microstate definition | Complete specification of the state of the system, which is consistent with the theory Things usually change very quickly, switching between microstates The amount of information needed to describe the microstates of a system is huge |
| 3 Types of Ensembles (macrostates with fixed bulk properties) | A. Microcanonical ensemble - macrostate where N, U and V are unchanging (not very easy = number of particles, energy, and volume unchanging) b. Canonical ensemble - macrostate where N, T and V are unchanging (not quite MTV as it sounds least posh and expensive) c. Grand Canonical ensemble - macrostate where µ, T and V are unchanging (µTV as it sounds most posh and expensive - a boujee MTV) - in this µ is the chemical potential or Fermi level |
| Distributions with total energy U and N particles key things to remember | Number of microstates in a distribution option Di: - Ω_i=N!∏1/(n_j !) (To find the total number of microstates sum the Ω_i values) You can then get the probability of a particle being in D_i using Ω_i/Ω You can also find the average number of particles in a state ni using ⟨n_i ⟩=(∑n_(D_i ) Ω_i)/Ω, where n_(D_i ) is the ni in distribution Di |
| Entropy from microstates | S = k_b ln(Ω) |
| 1. Partition function | Sum of exp(-beta*E) , where beta is 1/(KT). epsilon is energy |
| 10. Stirling approximation | In N!= NInN-N |
| Partition function overall dimensions | Dimensionless |
| Z for a particle system with degeneracy | Z = sum of the product of degeneracy and normal exponential term |
| Characteristic Rotational and Vibrational Temperatures | Rotational - k_b T = hbar^2/(2*I) Vibrational - k_b T = hbar*omega |
| Approximating Partition Functions at high and low T justification | High T - sum turns to integral because high energy values are occupied (so sum becomes infinitessimal) Low T - only need to consider first two terms of series as at low T, only low energy levels will be occupied |
| Internal Energy vs Free energy (equations) | U = Nkd/db ln(Z) F = -NkT ln(Z) |
| In metals, magnetism is due to | Delocalised valence electrons |
| Magnetic susceptibility for Pauli paramagnet | Independent of temperature |
| Why are Fe, Co, Ni ferromagnetic? | Large density of states at Fermi level due to narrow energy 3-d band |
| For ferromagnets, spontaneous splitting of spin up and spin down bands can take place due to the internal molecular field Bmf. What are the two competing energy contributions? | (i) increase in kinetic energy and (ii) decreases in potential |
| (Ferromagnetism) decrease in potential energy must be larger than any increase in kinetic energy when spins split, leading to...... | Stoner criterion U(Ef) >1. This Predicts that ferromagnetism is favoured by large Coulomb inter-action and/or high density of states at Fermi level (this is likely in transition metals). |
| J relation for more/less than half-filled shell | J = L+S for more J = |L-S| for less |
| Hund's rules | (i) First maximise S by filling the positive ms side, (ii) maximise L by filling from positive biggest l downwards (iii) selecting J that gives the smallest spin orbit interaction energy (this is J = |L - S| if the band is less than half-full and J = L + S for a band more than half-full). |
| Order of shells in increasing l value | S p d f g h i |
| Energy of magnetic dipole in B field | E = -µ_s . B |
| Μ for an electron | - γ l where l is the angular momentum and γ is the gyromagnetic ratio (=e/2m) |
| Μ for spin/J angular momentum | Μ_(j/s) = -g_(j/s)γ * j/s In this equation it is not j divided by s it is j or s |
| For a nearly free electron solid what can you assume? | Close to zero, electrons are free because far from the bragg condition (k+G)^2 = k^2 and because of 2nd order pertubation theory. This means E = (hbar k)^2/2m* and v_g = 1/hbar dE/dk |
| Relation between r^2 and <r^2> in 2D | R^2 = 2/3 <r^2> |
| Bohr magneton | ΜB = γ hbar |
| Diamagnetic susceptibility equation | X = M/H Used to classify weakly magnetic materials (paramagnets positive, diamagnets negative) |
| Magnetisation | Net dipole moment per unit volume Equal to change in µ * electrons/atom *atoms/volume |
| H for weakly magnetic solids | B = µ0 (H+M) = µ0 H |
| Free electron theory limitations | Predicts spherical Fermi surface, when really there is distortion at boundaries from Bragg diffraction off nuclei (interference between incident and scattered) |
| Origin of magnetocrystalline anistropy: | - crystal field effects arise due to anisotropic shape of electron orbitals - results in orbital quenching where net ang mom L = 0 - orbital quenching modifies SO interaction so Hamiltonian now dependent on crystal direction |
| GL theory: why does free energy only depends on even powers? | - reversing direction of magnetisation vector must leave free energy unchanged - only even powers do this |
| Explain, using classical electromagnetic theory, the origin of diamagnetism. | Electrical charges shield an interior of a body from an applied B field. When the flux through an electric circuit changings, an induced current is set up to oppose the flux change. This current persists as long as the field is present. The B field of the induced current is opposite to applied field. |
| Valence Band definition | Highest Occupied Band |
| Conduction Band Definition | Lowest Unoccupied Band |
| Band Gap Definition | Minimum energy separation between the two bands the energy gap between the valence band maximum and conduction band minimum |
| Direct vs indirect bandgap | In a direct band gap semiconductor the valence band maximum has the same wavevector as the conduction band minimum, for an indirect band gap it does not |
| Electronic transitions for a direct bandgap semiconductor | Only energy (no momentum needs to be supplied) is needed to promote electron, and so this can easily be supplied by photons of the appropriate energy (at least band gap energy). This means light absorption/emission are both strong for direct bandgaps |
| Electronic transitions for an indirect bandgap semiconductor | An electronic transition here involves a change in electron momentum. To conserve momentum during light absorption phonons from the crystal must therefore be either created or destroyed. This means light absorption in indirect bandgap semiconductors is weaker. |
| Alloying: What is it, why is it done and what is the equation of alloying? | Alloying is the process of mixing two different components to form a solid solution. It is done to tune the band gap to a desired energy. Band gap equation is Eg,AB = xEg,A + (1 − x)Eg,B − bx(1 − x), where x is the fraction of A in the alloy, and b is the bowing parameter. |
| Vegard’s law | Lattice parameter in an alloy follows this equation: a_AB = xa_a + (1-x)a_b |
| Why do we need to be conscious of lattice parameter for devices. | Devices feature a THIN layer of semiconductor on top of a substrate, for cheapness. If there is a large difference in lattice parameter between substrate and film then significant strain can build up in the latter, leading to poorer quality devices. |
| Effective mass | As an electron moves through a crystal it will undergo Coulomb scattering with other electrons and atomic nuclei. In free space, a force applied on an electron can be related to it's acceleration using Newton's 2nd Law, however due to coulomb scattering, the applied force is not the only force. The effective mass is the mass required for the electron to have an acceleration a with applied force F. It is calculated using h_bar^2/(d^2E/dk^2) |
| Relating Charge density to E field to electric potential (pn junctions) | ∇.E = p(x)/εrε0 -∇Φ = E |
| Charge density for pn junctions | -eNa on p-side (ionised acceptors on the p-side are negatively charged) eNd on n-side |
| Assumptions when finding potential for pn junctions | ?(wn) = ?_(bi) (Built in potential) ?(-wp) = 0 |
| Clausius-Mossotti Relation | Nα/3ε0 = (εr-1)/(εr+2) |
| Relationship between polarisation and local E field | P = NαE_(local) |
| Electric Susceptibilty | X = P/ε0 E |
| Difference between type 1 and type 2 superconductors | Type 1 - 1 critical B field so superconducting and normal states can't coexist Type 2 - lower and upper critical B field, meaning there's a mixed/vortex phase region where states can coexist |
| Bias current density | J = J0 (exp(-eV/kT)-1) |
| Importance of Tc in Polariation (from condensate energy) | Above Tc, equation for Ps does not give real solutions |