Welcome to TiddlyWiki created by Jeremy Ruston, Copyright © 2007 UnaMesa Association
*Stephan's Law:
**Black body distribution, <<tex 'B_{\nu}(T)'>>;
**Spectral radiance, <<tex 'L_{\nu}'>>: The power per unit area, per unit solid angle, per unit frequency;
**Irradiance, <<tex 'I_{\nu}'>>: Radiance over on hemisphere;
**Stephan's law: from integrating irradiance over wavelengths, gives an //energy flux// power per unit area emitted by a black body:
><<tex '\displaystyle{\int_{\Theta = 0}^{\pi}} \displaystyle{\int_{\phi = 0}^{\pi / 2}} \displaystyle{\int_{\lambda = 0}^{\infty}} B_{\lambda} (T) d \lambda = F_{BB} = \sigma T^4'>>
*The maximum of <<tex 'B_{\lambda}(T)'>> follows ''Wien's law'':
><<tex '\lambda_{MAX} = \dfrac{hc}{4.96 k T}'>>
*X-ray diffraction seen as x-rays reflecting off lattice planes:
[img[Bragg Diffraction|images/1211305450.png]]
*Extra path difference travelled by subsequent rays: <<tex '2d \sin \theta'>>
*Constructive interference if <<tex '2d \sin \theta = n \lambda'>>, or more conveniently:
><<tex '\lambda = 2 d_{hkl} \sin \theta'>>
*Here higher orders (<<tex 'n > 1'>>) are accommodated for by using multiples of miller indices <<tex 'hkl'>>.
*e.g. <<tex 'n=2'>> reflection from <<tex '(110)'>> planes is equivalent to <<tex 'n=1'>> reflection from <<tex '(220)'>> planes (the <<tex '(220)'>> planes have half the spacing).
*''Systematic Absence'' (missing orders):
**Due to use of non-primitive unit cell.
**E.g:
***reflection off <<tex '(100)'>> planes of a BCC unit cell will be destructively interfered by planes of the central points mid way between the <<tex '(100)'>> planes.
***reflection off <<tex '(110)'>> planes of a FCC unit cell will be destructively interfered by planes of the face points mid way between the <<tex '(110)'>> planes.
*The composition of dry (molar mass <<tex '\sim 29 \textrm{g mol}^{-1}'>>) is as follows
**78% Nitrogen (<<tex 'N_2'>>);
**21% Oxygen (<<tex 'O_2'>>);
**0.5% Water Vapour (<<tex 'H_2 0'>>);
**< 0.1% Other
*Main diatomic constituents don't absorb strongly in visible or infra red (something about dipole moments);
*Tri atomic molecules absorb IR strongly but water vapour also weakly in the visible;
*At atmospheric pressures and temperatures, ideal gas behaviours are observed (due to lack of phase changes).
*Vorticity is the curl of the velocity vector field:
><<tex '\boldsymbol{\omega}(\textbf{r}, t) = \nabla \times \textbf{u}(\textbf{r}, t)'>>
*Measure of local 'spin' or 'rotation'.
*Binding energy:
><<tex 'Q(Z,N) = [ZM_p + NM_n - M(Z,N)]c^2 '>>
*Energy release:
**<<tex '4 \textrm{H} \rightarrow {}^4\textrm{He}, \epsilon = 0.007'>>
**<<tex '56 \textrm{H} \rightarrow {}^{56}\textrm{He}, \epsilon = 0.0084'>>
**Iron is the most stable configuration, but most of the energy is released from <<tex '4 \textrm{H} \rightarrow {}^4\textrm{He}'>>
*Newtonian gravitational force and potential energy given by:
><<tex 'F = -G \dfrac{M m}{r^2}'>>
>
><<tex 'U = -G \dfrac{M m}{r}'>>
*Notes:
**1. Force within a spherically symmetric cavity due to gravity is zero;
**2. The force due to a spherically symmetric concentration of matter is equivalent to that of a point source at the origin with the total mass.
*Smooth distribution, density <<tex '\rho'>>;
*Choose arbitrary sphere radius <<tex 'r'>> within it, with mass <<tex 'M = \frac{4}{3} \rho \pi r^3'>>. Due to note 1, can ignore everything outside;
><<tex 'V = - \frac{4}{3} \pi G m r^2'>>
>
><<tex 'K = \frac{1}{2} m \dot{r}^2'>>
*<<tex 'E_{\textrm{tot}} = K + V'>> so:
><<tex '\dot{r}^2 - \dfrac{8 \pi G}{3} \rho r^2 = \dfrac{2 E_{\textrm{tot}}}{m}'>>
*Using <<tex 'r = a x, \dot{r} = \dot{a} x'>>:
><<tex '\left( \dfrac{\dot{a}}{a} \right)^2 - \dfrac{8 \pi G}{3} \rho = \dfrac{2 E_{\textrm{tot}}}{mx^2} \dfrac{1}{a^2}'>>
*Defining <<tex '-kc^2 \equiv 2 E_{\textrm{tot}} / (m x^2)'>>:
><<tex 'H = \left( \dfrac{\dot{a}}{a} \right)^2 = \dfrac{8 \pi G}{3} \rho - \dfrac{k c^2}{a^2}'>>
*This is the Friedmann equation, or ~Friedmann-Robertson-Walker equation (FRW).
* ''Fermi Energy'': The energy of the state occupied by the highest energy electron in the metal at absolute zero, due to the Pauli Exclusion Principle.
* ''Fermi Temperature'': The temperature relating to the Fermi energy, <<tex 'T_F = E_F / k_B'>>.
* ''Fermi Surface'': The surface of constant energy in k-space that exists at the Fermi energy. Using <<tex 'E = \frac{\hbar^2 k^2}{2m}'>>, we have<<tex '|k_F| = \sqrt{2 m E_F} / \hbar'>> where <<tex '|k_F|'>> is the radius of the Fermi sphere.
* The Fermi surface takes up the volume of <<tex '1/2'>> of 1 Brillouin zone.
*Most stars are in binary or multiple systems;
*Orbital period can range from 11mins to <<tex '10^6'>> years;
*Binary systems have stars far apart which don't interact much;
*Close binaries (~ less than 10 year period) can transfer mass altering their structures;
*Large range of eccentricities of orbits.
*Homogeneous broadening affects all atoms in a sample equally, and all atoms will interact with a radiation beam of specific frequency with the same strength. Examples listed below.
*Inhomogeneous broadening affects different atoms in a sample by different amounts, e.g. Doppler broadening - as not all atoms have the same velocity (velocity distribution given by ~Maxwell-Boltzmann), not all atoms will be Doppler broadened by the same amount.
*The radiative lifetime of an excited atom decays exponentially, according to:
><<tex 'N_k(t) = N_k(0) e^{-t/\tau_\kappa^{rad}'>>
where
><<tex '\gamma = \dfrac{1}{\tau_\kappa^{rad}} = \displaystyle\sum_{E_i < E_k} A_{ki}'>>
>
>(the sum of all the Einstein A coefficients for levels below the current level - i.e. spontaneous emission).
*''Lifetime (or Natural) Broadening'':
**As energy is related to time by the uncertainty principle, <<tex '\Delta E \Delta t \simeq \hbar'>>, a range of energies will be produced due to the finite lifetime of the atom.
**Alternatively, as the electron in the atom is modelled as a damped oscillator, and the frequency distribution is the FT of this oscillation, as the oscillation is damped and not purely harmonic, its FT will give a range of frequencies.
**The FWHM of the line distribution, which is given by a Lorentzian distribution, is:
><<tex '\Delta \omega_N = \gamma'>>
*''Pressure Broadening'':
**This model assumes, that collisions with other atoms in the gas causes a radiating electron to stop radiating;
**This effectively further shortens the radiative natural lifetime of the decay, and broadens the frequency spectrum further.
**The characteristic timescale will be the mean time between collisions, <<tex '\tau_c'>> and we get the expression:
><<tex '\Delta \omega_p = \gamma + \dfrac{2}{\tau_c}'>>
*''Thermal or Phonon Broadening'':
**The energy levels of an atom in a crystal lattice depends on the positions of the neighbouring atoms. Thermal fluctuations change this distance, and cause a rich energy level structure that gives a broadened appearence.
*(Aside: Phonon excitation does not result in emission or absorption of a photon, so population inversion can be helped by exciting atoms to higher states through specific frequency optic phonon absorption - this can be useful in building a laser system.)
*The sky is dark:
**Assume all stars have a luminosity L, and brightness <<tex 'B = L / (4 \pi r^2)'>>;
**Number density of stars <<tex 'n = N/V'>>;
**Gives <<tex 'dN = 4 \pi r^2 n dr'>> for number of stars in a shell;
**Brightness <<tex 'dB = BdN'>>:
><<tex 'B_{\textrm{tot}} = \int_0^{\infty} \dfrac{L}{4 \pi r^2} 4 \pi r^2 n dr = \infty'>>
**So there must be a cutoff: galaxies are moving away from us.
*Can calculate distance to a star, and relative speed by various methods e.g doppler shift of spectral lines
*Hubble measured distances to different galaxies and their velocities, obtained ''Hubble's Law'':
><<tex 'v = H_0 r'>>
*<<tex 'H_0 = 67 \textrm{km s}^{-1} \textrm{Mpc}^{-1}}'>> is Hubble's constant.
*Hubbles law can also be written in terms of coordinates:
**Can write //physical coordinates// <<tex '\textbf{r}(t) = a(t)\textbf{x}'>>, where <<tex '\textbf{x}'>> are //conformal// (fixed in time) coordinates.
**Now:
><<tex '\textbf{v} = \dfrac{d \textbf{r}}{dt} = \dfrac{d (a \textbf{x})}{dt} = \dot{a} \textbf{x} = \dfrac{\dot{a}}{a} \textbf{r} \equiv H \textbf{r}'>>
*Classically, the heat capacity is <<tex 'C = 3 N k_B'>>, and this is independent of temperature. However, this is not actually the case.
*Summing over all modes, where 's' is the branch index, i.e the different optic/acoustic branches etc.
><<tex '\langle E \rangle = \displaystyle\sum_{\mathbf{k}, \mathbf{s}} (n_s (k) + \frac{1}{2}) \hbar \omega_s(k)'>>
*For a simplification, can replace the sum with an integral with the density of states, and <<tex 'n_s(k)'>> is replaced by the Planck distribution:
><<tex '\Rightarrow C = \dfrac{\partial}{\partial T} \displaystyle\int_0^\infty \dfrac{\hbar \omega}{e^{\beta \hbar \omega} - 1} g(\omega) d \omega'>>
*The density of states:
**<<tex 'g(\omega) d \omega'>> = The number of modes in an interval <<tex '\omega \rightarrow \omega + d \omega'>> or similarly, <<tex 'k \rightarrow k + dk'>>.
**<<tex 'g(\omega) d \omega \rightarrow \dfrac{V}{(2 \pi)^3} 4 \pi k^2 dk'>> in 3D.
*For p degenerate brances, <<tex 'g(\omega) d \omega = p g(k) dk'>>
*Can measure ''redshift'' by:
**Doppler shift of spectral lines;
*Can measure distance by:
**Parallax, <<tex 'D = 1 / \alpha'>>;
**Luminosity and brightness comparison: <<tex 'B = L / (4 \pi D^2)'>>, Luminosity from:
***Colour relates to temperature (black body spectra);
***Temperature relates to Luminosity via radius;
***Spectroscopic parallax (up to 10kpc):
****Radius can be inferred from broadness of spectral lines - broader lines, smaller radius.
***Cephid variables (up to 30Mpc):
****Very luminous (<<tex 'L \sim 100 \rightarrow 1000 L_{\odot}'>>), can be seen v far away;
****Luminosity intrinsic to period of brightness oscillations;
****Period of only a few days, easily measurable.
***~Tully-Fisher relation (up to 100Mpc):
****21cm line of Hydrogen doppler shifted by rotating galaxy;
****Rotation of galaxy linked to mass of galaxy;
****More massive galaxy - more stars - more luminous.
****Therefore line broadening relates to luminosity.
***Supernovae (up to 1000Mpc):
****<<tex '10^9 L_{\odot}'>>;
****Luminosity can be known as above, distance from brightness (inverse square law);
****Type <<tex I_a>> - Luminosity related to time it takes to fade after explosion, formed from white dwarf collapse.
*Electron travelling through a solid, <<tex '\psi = e^{ixk}'>>, is Bragg scattered by the crystal potential, by a wave vector <<tex 'G'>>, where <<tex '|G| = 2 \pi / a'>>. This shifts the wavefunction by 1 BZ in whichever direction.
*The crystal potential is periodic, so can be Fourier transformed. The potential wells can also be quite complicated mathematically, <<tex 'V(x) = \sum _G V_G e^{iGx}'>>, so we look at a single Fourier component, which gives a good approximation: <<tex 'V(x) = V_G (e^{iGx} + e^{-iGx})'>>.
*Next we use the Schrodinger equation,
><<tex 'H \psi = - \dfrac{\hbar^2}{2m} \dfrac{\partial^2 \psi}{\partial x^2} + V(x) \psi = E \psi'>>
*Inserting <<tex '\psi = e^{ikx}'>> reveals it is no longer a solution due to the "cross terms" from <<tex 'V(x)'>>.
*Instead we can solve with <<tex "\psi = A_k e^{ikx} + B_{k'} e^{ik'x}">>, where <<tex "k' = k -G">>, i.e. a superposition of the before and after states:
><<tex "H \psi = \dfrac{\hbar^2 k^2}{2m} A e^{ikx} + \dfrac{\hbar^2 k'^2}{2m} B e^{ik'x} + V_G[A e^{i(k \pm G)} + B e^{i(k' \pm G)}]">>
>
><<tex "H \psi = E ( A e^{ikx} + B e^{ik'x} )">>
*Multiply by <<tex "e^{-ikx}">> and integrate from 0 to L, to obtain <<tex 'EA = \frac{\hbar^2 k^2}{2m} A + V_G B'>>.
*Similarly by <<tex "e^{-ik'x}">> to obtain <<tex 'E B = \frac{\hbar^2 (k - G)^2}{2m} B + V_G A'>>
*Solve at the Brillouin boundary, <<tex "|k| = |k'| = \pi / a">>:
><<tex "\begin{array} {|cc|} \dfrac{\hbar^2 (\frac{\pi}{a})^2}{2m} - E & V_G \\ V_G & \dfrac{\hbar^2 (\frac{\pi}{a})^2}{2m} - E \end{array} = 0">>
>
>Eigenvalues: <<tex "\lambda_{\pm} = \dfrac{\hbar^2 (\frac{\pi}{a})^2}{2m} \pm V_G">>
>
>Eigenstates: <<tex "\psi = e^{i(\pi / a)x} \pm e^{-i(\pi / a) x}">>
*This is a standing wave, <<tex "\cos \theta">> and <<tex "\sin \theta">> solutions, <<tex "\Delta E = 2 |V_G|">> between them.
*<<tex "\cos">> has nodes in potential's peaks, and sits in the dips, therefore energy is lowered. <<tex "\sin">> is the opposite: 'fights' with the potential's peaks and its energy must therefore be raised.
*To cover all Fourier frequencies, we can write <<tex "\psi (x) = e^{ikx} u_K (x)">>, where <<tex "u_K (x)">> is a periodic "Bloch function", which represents the full crystal structure.
*Number of allowed modes in a frequency range <<tex '\omega \rightarrow \omega + d \omega'>>is proportional to the volume of the cube.
*For a large enough cavity, the number of modes per unit volume becomes independent of the size or shape, and for <<tex 'V \rightarrow \infty'>>, an essentially continuous spectrum of frequencies is produced, given by a ''mode density'':
><<tex 'g(\omega) d \omega = \dfrac{\omega^2}{\pi^2 c^3} d \omega'>>
*Classically, the energy of a radiation mode is given by:
><<tex 'W_{mode} = \displaystyle\int_{vol} \overline{\left( \frac{1}{2} \mathbf{E \cdot D} + \frac{1}{2} \mathbf{B \cdot H} \right)} d \tau'>>
*However, we need a more complete picture accounting for QM. As the modes oscillate harmonically:
><<tex 'W_{mode} = (n + \frac{1}{2}) \hbar \omega, n=1,2,3...'>>
*In this picture the field can be said to be in the nth excited state, or containing n photons. Number of photons per mode is given by:
><<tex '\langle n \rangle = \dfrac{\displaystyle\sum_{n=0}^\infty n e^{- \beta (n + 1/2) \hbar \omega}}{\displaystyle\sum_{n=0}^\infty e^{- \beta (n + 1/2) \hbar \omega}} = \dfrac{1}{e^{\beta \hbar \omega} - 1}'>>
*Energy density per frequency interval <<tex 'd \omega'>> is then given by number of photons per mode, times no of modes per unit vol, times energy of each photon:
><<tex '\begin{array}{rl} \rho (\omega) d \omega &= \langle n_\omega \rangle \times g (\omega) d \omega \times \hbar \omega \\ & \\ &= \dfrac{\hbar \omega^3}{\pi^2 c^3} \dfrac{d \omega}{e^{\beta \hbar \omega} - 1} \end{array}'>>
*The above equation is ''always'' true, so if the energy density is known the number of photons per mode can be found. This is ''Planck's Law''.
*Apply N.II to a 'blob' of fluid that (ideal case) moves along with the flow, dimensions <<tex '\delta x, \delta y, \delta z'>>;
*<<tex '\lambda \ll \delta x, \delta y, \delta z \ll L'>>, the length scale of the flow.
><<tex 'F = m a'>>
>
><<tex '\rho(\delta x, \delta y, \delta z) a = \delta \textbf{F}'>>
*Net pressure in x direction:
[img[Pressure forces|images/1210586546.jpg]]
><<tex '\delta F= (p(x) - p(x + dx))\delta A'>>
>
><<tex '\delta F = -\delta x \dfrac{\partial p}{\partial x} \delta A = \delta V \dfrac{\partial p}{\partial x}'>>
>
>So:
><<tex '\delta \textbf{F} = - (\delta V) \nabla p'>>
*Previously calculated, <<tex '\rho \propto \dfrac{1}{a^4}'>>
*Consider radiation in thermal equilibrium as a black body;
*Energy density:
><<tex '\epsilon(\nu) \textrm{d} \nu = \dfrac{8 \pi \nu^3 \textrm{d} \nu}{c^3} \dfrac{1}{exp(\frac{h \nu}{kT}) - 1}'>>
*Integrating over all energies gives:
><<tex '\rho_{\gamma} = \dfrac{\pi^2}{15} (k T) \left( \dfrac{k T}{\hbar c} \right)^3'>>
*Showing:
><<tex 'T \propto \dfrac{1}{a}'>>
*For this to be the temperature of the universe:
**everything has to interact with photons, (true at high T);
**radiation must dominate over other forms of matter - be careful!
***Energy density of radiation decreases more quickly, but number density does not;
***Far more photons than baryons in the universe (<<tex 'n_B / n_{\gamma} \approx 10^{-10}'>>).
*We write a general rate equation for e.g. the upper laser level as:
><<tex '\dfrac{d N_2}{dt} = R_2 - (N_2 B_{21} - N_1 B_{12}) \displaystyle\int_0^\infty g_H (\omega - \omega_0) \rho (\omega) d \omega + ...'>>
*i.e the total rate is given by integrating the lineshape with the energy density.
*Can rewrite this as (with <<tex '\omega_L'>> as the central frequency of a the incident narrow beam:
><<tex '\begin{array}{rl} \dfrac{d N_2}{dt} &= R_2 - N^* \displaystyle\int_0^\infty \sigma_{21} (\omega - \omega_0) \dfrac{I_T \delta (\omega - \omega_L)}{\hbar \omega} d \omega + ... \\ & \\ &= R_2 - N^* \sigma_{21} (\omega_L - \omega_0) \dfrac{I_T}{\hbar \omega_L} + ...\end{array}'>>
*This can be viewed as the effective number density of inverted atoms, times their cross sectional area, times the beam's photon flux <<tex 'I_T / \hbar \omega'>>.
*Reciprocal Lattice is the Fourier Transform of the lattice.
><<tex '\displaystyle\sum_{\mathbf{l}} \int \delta (\mathbf{r} - \mathbf{l}) e^{i \mathbf{Q} . \mathbf{r}} d^3 \mathbf{r}'>>
>
><<tex '= \displaystyle\sum_{\mathbf{l}} e^{i \mathbf{Q} . \mathbf{l}}'>>
*and for orthogononal axes:
><<tex '= \displaystyle\sum_{n_1} e^{i Q_x a n_1} \cdot \displaystyle\sum_{n_2} e^{i Q_y b n_2} \cdot \displaystyle\sum_{n_3} e^{i Q_z c n_3}'>>
*carrying on:
><<tex '\propto \displaystyle\sum_h \delta(Q_x - h \frac{2 \pi}{a}) \displaystyle\sum_k \delta(Q_y - k \frac{2 \pi}{b}) \displaystyle\sum_l \delta(Q_z - l \frac{2 \pi}{c})'>>
>
><<tex '\propto \displaystyle\sum_{\mathbf{G}_{hkl}}} \delta(\mathbf{Q} - \mathbf{G}_{hkl})'>>
*Here, <<tex '\{\mathbf{G}_{hkl}\}'>> is the set of reciprocal lattice vectors:
><<tex '\mathbf{G}_{hkl} = h \mathbf{a}^* + k \mathbf{b}^* + l \mathbf{c}^*'>>
*General case:
><<tex '\mathbf{a}^* = \dfrac{2 \pi \mathbf{b} \times \mathbf{c}}{\mathbf{a} \cdot ( \mathbf{b} \times \mathbf{c} )}'>>
*Simpler orthogonal case:
><<tex '\mathbf{a}^* = \dfrac{2 \pi}{a} \mathbf{\hat{a}}'>>
*Notes:
**Reciprocal lattice is FT of crystal lattice;
**Lattice of points in ''Q'' space, basis vectors ''a''*, ''b''*, ''c''*, and dimensions of inverse length;
**<<tex '\mathbf{G}_{hkl}'>> is the normal to the plane <<tex '(hkl)'>>;
***(Can show this by taking two lines in the plane, e.g <<tex '(\frac{\mathbf{a}}{h} - \frac{\mathbf{b}}{k})'>> and <<tex '(\frac{\mathbf{a}}{h} - \frac{\mathbf{c}}{l})'>>, and crossing them to get a normal etc.
**Spacing between <<tex '(hkl)'>> planes is:
><<tex 'd_{hkl} = \dfrac{2 \pi}{| \mathbf{G}_{hkl} |}'>>
*//Poiseuille Flow//
[img[Rectilinear Flow|images/1210603405.jpg]]
>''N.S.:''
>
><<tex '\dfrac{D \textbf{u}}{Dt} \equiv \dfrac{\partial \textbf{u}}{\partial t} + ( \textbf{u} \cdot \nabla) \textbf{u} = - \dfrac{1}{\rho} \nabla p + \dfrac{\mu}{\rho} \nabla^2 \textbf{u}'>>
>
>(Neglecting gravity)
*Setup:
**<<tex '\textrm{u} = (u, 0, 0)'>>;
**<<tex '\nabla \cdot \textbf{u} = 0'>> (Incompressibility) - gives <<tex '\partial u / \partial x = 0'>>
**<<tex '\partial u / \partial t = 0'>> (Steady flow);
**<<tex '\partial u / \partial y = 0'>> (y-independent flow);
**so:
><<tex 'u = u(z)'>>
*Plug into ''NS'':
><<tex '\dfrac{\partial u}{\partial t} + u\dfrac{\partial u}{\partial x} + v\dfrac{\partial u}{\partial y} + w\dfrac{\partial u}{\partial z} = - \dfrac{1}{\rho} \dfrac{\partial p}{\partial x} + \dfrac{\mu}{\rho} \nabla^2 u'>>
>
><<tex '\dfrac{\partial v}{\partial t} + u\dfrac{\partial v}{\partial x} + v\dfrac{\partial v}{\partial y} + w\dfrac{\partial v}{\partial z} = - \dfrac{1}{\rho} \dfrac{\partial p}{\partial x} + \dfrac{\mu}{\rho} \nabla^2 v'>>
>
>etc.
*As <<tex '\textrm{u} = (u, 0, 0) \Rightarrow v, w =0'>> and the y, z components of pressure are zero:
><<tex '\dfrac{\partial p}{\partial y} = \dfrac{\partial p}{\partial z} = 0'>>
*Look at x-component, all that's left is:
><<tex ' \dfrac{\partial p}{\partial x} = \mu \nabla^2 u = \mu \dfrac{d^2 u}{dz^2}'>>
>
>//The viscous force balances the pressure gradient.//
*Boundary Conditions:
**No slip: <<tex 'u = 0'>> at the walls, <<tex 'z = \pm h'>>
*Solve the differential equation gives:
><<tex 'u = \dfrac{G}{2 \mu} (h^2 - z^2)'>>
>
>Where <<tex 'G = - \dfrac{dp}{dx}'>>, the pressure gradient.
*This gives a parabolic flow profile:
[img[Parabolic Flow Profile|images/1210605794.jpg]]
*With volume flux:
><<tex '\displaystyle{\int_{-h}^h} u dz = \frac{2}{3} \left( - \dfrac{dp}{dx} \right) \dfrac{h^3}{\mu}'>>
*Gives intuitive result, higher volume flux if:
**Higher pressure gradient;
**Lower viscosity.
*The space-time invariant interval is:
><<tex 'ds^2 = dx^2 = dy^2 + dz^2 - c^2 dt^2'>>
*Three types of interval:
**<<tex 'ds^2 > 0'>>: Space components are bigger than time components.. ''space-like'' interval. It is impossible for a light ray to travel between these two points in spacetime in the given time;
**<<tex 'ds^2 = 0'>>: The path traced by a light ray;
**<<tex 'ds^2 < 0'>>: A ''time-like'' interval, light can travel between the two points.
*We can express <<tex '\textbf{ds}'>> = <<tex '(cdt, dx, dy, dz)'>>, and <<tex 'ds^2 = \textbf{ds}^T g \textbf{ds}'>>;
**For normal Euclidean space:
><<tex 'g = \left(\begin{array}{cccc} -1&0&0&0 \\ 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \end{array}\right)'>>
**and in spherical coordinates:
><<tex 'g = \left(\begin{array}{cccc} -1&0&0&0 \\ 0&1&0&0 \\ 0&0&r^2&0 \\ 0&0&0&r^s \sin^2 \theta \end{array}\right)'>>
*For the cosmological principle (homogeneity and isotropy) to apply, g can only have a few forms, giving:
**Flat (Euclidean) space, (<<tex 'k = 0'>>);
**Spherical space, (<<tex 'k > 0'>>);
**Hyperbolic space, (<<tex 'k < 0'>>);
*Using spherical polars, we can write the metric for these spaces, with the parameter k as:
><<tex 'ds^2 = -c^2 dt^2 + a^2(t) \left[\dfrac{d^2r}{1 - kr^2} + r^2(d\theta^2 + \sin^2 \theta d \phi^2)\right]'>>
*Clearly <<tex 'k'>> can be expressed as <<tex '\sim 1/R^2'>>, giving <<tex 'R'>> as the characteristic scale at which the curvature can be detected:
**If <<tex 'R'>> is too large, it will be difficult to detect the geometry of the space.
*This is the same <<tex 'k'>> as in the [[FRM Equations|ii. The Friedmann Equations]].
*Electrons attracted rigidly to nucleus - the "adiabatic approximation";
*Amplitude of vibrations is small:
[img[images/1211379604.png]]
*Here:
**<<tex 'x'>> is the displacement from the origin;
**<<tex 'a'>> is the lattice spacing;
**<<tex 'u'>> is an atom's displacement from it's equilibrium position;
**This gives <<tex 'x = a*n + u'>>.
*For <<tex 'u_2 - u_1 \ll a'>>, we can expand <<tex '\phi'>> about <<tex 'x_2 - x_1 = a'>>...:
><<tex '\phi(x_2 - x_1) = \phi(a) + (u_2 - u_1) \dot{\phi}(a) + \frac{1}{2} (u_2 - u_1)^2 \ddot{\phi}(a) + ...'>>
*<<tex '\dot{\phi}(a) = 0'>> by definition (as <<tex 'a'>> is constant), and neglect terms higher than <<tex '(u_2 - u_1)^2'>> (the harmonic approximation);
*Include forces between nearest neighbour atoms only.
*Apparent magnitudes of two stars (relative between the two stars):
><<tex 'm_1 - m_2 = 2.5 log(f_2 / f_1)'>>
5 magnitudes is <<tex '\therefore'>> a factor of 100 brightness.
*Historically, Vega is the 'zero point' of the magnitude system, gives Sun's magnitude as: <<tex 'm_{\odot} = -26.82'>>
*Absolute magnitude, <<tex 'M'>> is the apparent magnitude if it were at a distance of 10pc.
*Bolometric magnitude - absolute magnitude for luminosity integrated over all wavebands. <<tex 'M_{\odot}^{bol} = 4.72'>>
*Magnitude usually defined over limited wavelength bands due to difficulty of measuring entire spectrum:
>*Ultraviolet, apparent <<tex 'U'>> or <<tex 'm_U'>>, absolute <<tex 'M_U'>>.
>*Blue, apparent <<tex 'B'>> or <<tex 'm_B'>>, absolute <<tex 'M_B'>>.
>*Visual, apparent <<tex 'V'>> or <<tex 'm_V'>>, absolute <<tex 'M_V'>>.
*Colours are relative magnitudes in different wavebands, <<tex 'B - V'>>, <<tex 'U - B'>>.
*Bolometric correction: <<tex 'B.C. = M_{bol} - M_V'>>
*Visual extinction: correction due to absorption of light due to gas en-route: <<tex 'A_V'>>, can be up to 30 magnitudes.
*Distance modulus: <<tex '(m - M)_V = 5 \times \textrm{log} (D / 10 \textrm{pc})'>>
*Summary:
><<tex 'M_V = -2.5 \textrm{log} L / L_{\odot} + 4.72 - \textrm{B.C.} + A_V'>>
*For an ideal gas:
><<tex 'P = N k T = \dfrac{\rho}{\mu m_H} k T'>>
**<<tex '\mu'>> is average particle massin a gas, can use X, Y, Z and number of electrons from each to get a value:
><<tex 'N = (2 X + 3 Y / 4 + Z / 2) \rho / m_H'>>
>
><<tex '(1 / \mu) = 2X + 3Y/4 + Z/2'>>
>
>This is a good approximation apart from in cooler, outer regions.
**Can include radiation pressure term (useful for massive stars):
><<tex 'P = \dfrac{\rho}{\mu m_H} k T + \dfrac{a T^4}{3}'>>
*Degenerate gas:
**Deviation from ideal gas law;
**Number density limited by Pauli exclusion principle;
**In a totally degenerate gas, ALL states up to Fermi energy/momentum are filled:
><<tex 'N_e = \displaystyle{\int_0^{p_0}} \dfrac{2}{h^3} 4 \pi p^2 dp = \dfrac{2}{h^3}\dfrac{4 \pi}{3} p_0^3'>>
**From kinetic theory, pressure <<tex 'P'>> is related to momentum <<tex 'p'>> by:
><<tex 'P_e = \frac{1}{3} \int_0^{\infty} p v(p) n(p) dp'>>
**For a non-relativistic, completely degenerate electron gas:
***<<tex 'v(p) = p / m_e'>> for all <<tex 'p'>>:
><<tex '\begin{array}{ll} P_e & = \dfrac{1}{3} \int_0^{p_0} \dfrac{p^2}{m_e} \dfrac{2}{h^3} 4 \pi p^2 dp \\ & \\ & = \dfrac{8 \pi}{15 m_e h^3} p_0^5 \\ & \\ & = \dfrac{h^2}{20 m_e} (3 / \pi)^{2/3} N_e^{5/3} \end{array}'>>
**For a relativistic, completely degenerate electron gas:
***<<tex 'v(p) \sim c'>>
><<tex '\begin{array}{ll} P_e & = \dfrac{1}{3} \int_0^{p_0} pc \dfrac{2}{h^3} 4 \pi p^2 dp \\ & \\ & = \dfrac{8 \pi c}{3 h^3} \dfrac{p_0^4}{4}\\ & \\ & = \dfrac{hc}{8} (3 / \pi)^{1/3} N_e^{4/3} \end{array}'>>
**For ions we can use the normal expression;
**For intermediate electrons, relativistic momentum expression is used.
*Stars are self-gravitating bodies in dynamic equilibrium - gravity must be balanced by internal pressure.
**Force due to pressure from above: <<tex 'P_{r+\delta r} A'>>
**Force due to pressure from below: <<tex '-P_r A'>>
**Force due to gravity: <<tex '(\rho_r A \delta r) G M_r / r^2'>>
**Equate:
><<tex '(P_{r + \delta r} - P_r).A = -\dfrac{G M_r \rho_r}{r^2} \delta r A'>>
><<tex '\dfrac{\partial P_r}{\partial r} = - \dfrac{G M_r \rho_r}{r^2}'>>
*Mass relation:
><<tex 'dM = 4 \pi r^2 \rho_r dr'>>
*Can estimate central pressure using:
><<tex 'P_r = \dfrac{\rho_r}{\mu m_H} k T_r'>>
combined with integrated equation 1, to get:
><<tex 'k T_c \approx \dfrac{G M_s \mu m_H}{R_s}'>>
*Aside: mean molecular weight <<tex '\mu'>> for hydrogen gas is 1/2, due to half the particles being hydrogen and half of them being electrons.
*Pre-main sequence:
**Dust cloud collapses until deuterium fusion can occur (<<tex 'T_c \sim 10^6 K'>>);
**After burning further contraction until H burning (<<tex 'T_c \sim 10^7 K'>>) and main sequence begins.
(Thanks to Will for the table format):
|!Mass / <<tex 'M_{\odot}'>>|!Temperature|!Energy Generation|!Opacity Laws|!Convection Zones|
| < 0.3 | ''Hydrostatic'': <html><br /><br /></html> <<tex '\dfrac{dP}{dr} = - G \dfrac{M \rho}{r^2}'>> <html><br /><br /></html> Can replace with charactersitic quantities: <html><br /><br /></html> <<tex '\dfrac{P_c}{R_S} = - G \dfrac{M_S \rho_c}{R_S^2}'>> <html><br /><br /></html> ''Ideal gas equation'': <html><br /><br /></html> <<tex 'P(r) = \dfrac{\rho(r)}{\mu m_H} kT(r)'>> <html><br /><br /></html> giving: <html><br /><br /></html> <<tex 'kT_c \approx \dfrac{G M_s \mu m_H}{R_S}'>> <html><br /><br /></html> <<tex '\textbf{Core Temp.} \propto \textbf{Mass}'>> | Lower temp -> ''PP Chains'': <html><br /><br /></html> <<tex '\epsilon_{pp} \propto T^4'>> <html><br /><br /></html> <<tex 't_{nuc} \approx 10^{10} \textrm{years}'>> | ''Kramer's Opacity Law'',<html><br /><br /></html> <<tex '\kappa = \kappa_0 \dfrac{\rho}{T^{3.5}}'>> | Extremely high surface opacities cause ''entire star'' to be convective |
| < 1.2 |~|~|~| ''Radiative core'' but opaque surface requires ''convective envelope'' |
| < 1.3 |~| Higher temp -> ''CNO Cycle'' <html><br /><br /></html> Energy generation: <html><br /><br /></html> <<tex '\epsilon_{cno} \propto T^{20}'>> <html><br /><br /></html> Slowest reaction determines cycle speed, second slowest determines time to equilibrium <html><br /><br /></html> <<tex 't_{eqm} \approx 10^6 \textrm{years}'>> |~|~|
| > 1.3 |~|~| High T ''electron scattering'' gives: <html><br /><br /></html> <<tex '\kappa = \textrm{const}.'>> | Vast energy generation can't be transported by radiation -> ''convective cores''. <html><br /><br /></html> In envelope high T leads to H ionisation and low opacity: ''radiative envelope''. |
*Longest phase of star;
*<<tex '\mu'>> increases as fusion occurs and nuclei get heavier, increasing density to prevent collapse, so T increases;
*As T increases, reaction rate increases (<<tex 'T^4'>>) until all H is used up in core, but burning continues in a shell around the core.
*Core is isothermal because there are no reactions and no luminosity, and <<tex 'L \propto dT/dr'>>
*The Ruby Laser is a good example of a 3 level laser.
**Pumping occurs from a low level to a higher level. The high level has a very short lifetime and atoms drop down to the top of the lasing level rapidly, to attempt to cause population inversion.
**If population inversion is achieved, lasing can occur between the middle and lower levels.
**It is therefore hard to acheive population inversion however, unless the pump rate is very high, as there is no drain from the lower level.
**We need over half of the population to be in the upper two levels for lasing to have any chance of happening.
*Analysis of the 3 level laser - PULSED laser operation:
**For small cavity losses:
><<tex '2 \alpha_0 L_g = \delta_{loss} + T_2'>>
>
><<tex '2 \sigma_{21} N^*_{thresh} L_g = \delta_{loss} + T_2'>>
**Level 3 has a very short lifetime, so <<tex 'N_3 \approx 0'>>
><<tex 'N_T = N_2 + N_1'>>
>
><<tex 'N_{thresh}^* = N_2^{thresh} + N_1^{thresh}'>>
*This gives:
><<tex 'N_2^{thresh} = \dfrac{N_T + N_{thresh}^*}{2}'>>
*However, the threshold population inversion density will be small comared to the total ion density, so:
><<tex 'N_2^{thresh} = \dfrac{N_T}{2}'>>
*To achieve lasing, we need an ''absorbed'' pump energy of:
><<tex 'E_{abs}^{thresh} = \dfrac{N_T}{2} \pi a^2 L_g \dfrac{hc}{\lambda_p}'>>
>
>where <<tex 'a, L'>> are the radius and length of the laser rod, and <<tex 'hc/\lambda'>> is the energy required to raise an ion from the lower to upper level.
*This is the //absorbed// energy however, the real factor is around 60x greater:
**2x for uniform pumping - if too much is absorbed only ions on the surface will be raised, no ions in the center of the rod will be.
**8x as only <<tex '\sim 12%'>> of the flashlamp output will be in the bulb bands;
**2x as not all the light from the bulb will hit the rod;
**1.7x as only 60% of the electrical energy is converted to light.
*An example pump energy for this case is around 360J, for a=5mm, L=20mm.
*For CONTINUOUS laser operation:
**The pumping now has to compete with the upper lasing level's spontaneous decay rate, <<tex 'N_2 / \tau_2'>>.
**The pump power is just the single pulse pump energy divided by this level's lifetime, giving a 100kW pumping power.
**This can be decreased with a smaller rod size but is still difficult.
*Two dimensional flow, with:
**<<tex '\mathbf{u} = [ u(x,y,u), v(x,y,t), 0 ]'>>;
**<<tex '\boldsymbol{\omega} = (0, 0, \omega)'>>.
*If <<tex 'curl \mathbf{u} = 0'>>, the vector <<tex '\mathbf{u}'>> can be expressed as the gradient of a potential:
><<tex '\mathbf{u} = \nabla \phi'>>
*In this case, the incompressibility condition becomes ''Laplace's equation'':
><<tex '\nabla^2 \phi = 0'>>
[img[Deep Water Waves|images/1210760045.png]]
*Assumptions:
**Inviscid, <<tex '\nu = 0'>>;
**Irrotational, <<tex '\boldsymbol{\omega} = 0'>>;
**Incompressible, <<tex '\nabla \cdot \mathbf{u} = 0'>>;
**2D (independent of z);
**<<tex 'p = p_0'>> at the surface of the water (atmospheric pressure).
*Laplace's equation:
><<tex '\dfrac{\partial^2 \phi}{\partial x^2} + \dfrac{\partial^2 \phi}{\partial y^2} = 0'>>
*Bernoulli's equation:
><<tex '\dfrac{\partial \phi}{\partial t} = \frac{1}{2} | \mathbf{u} |^2 + \dfrac{p}{\rho} + gy = 0'>>
*Assume <<tex '\mathbf{u}, \phi, \eta'>> are small, so can first neglect <<tex '\mathbf{u}^2'>>, and at the boundary <<tex 'y = \eta'>>:
><<tex '\dfrac{\partial \phi(x, \eta, t)}{\partial t} + \dfrac{p_0}{\rho} + g \eta = 0'>>
*Can redefine <<tex '\phi'>> up to a constant to get rid of the <<tex 'p_0 / \rho'>> term.
*Also we have been solving at the boundary but can get solution at <<tex 'y=0'>> using an expansion:
><<tex '\dfrac{\partial \phi(x, \eta, t)}{\partial t} = \dfrac{\partial \phi(x, 0, t)}{\partial t} + ...}'>>
*Giving at <<tex 'y=0'>>:
><<tex '\dfrac{\partial \phi}{\partial t} + g \eta = 0'>>
*and also the vertical velocity at <<tex 'y=0'>>:
><<tex 'v = \dfrac{\partial \phi}{\partial y} + \dfrac{\partial \eta}{\partial t}'>>
*Guess a travelling wave solution, and substitute into Laplace's equation above:
><<tex '\phi(x,y,t) = f(y) \sin(kx - \sigma t)'>>
>
><<tex "f'' - k^2 f = 0">>
>
><<tex 'f(y) = Ce^{ky} + De^{-ky}'>>
*Boundary condition <<tex 'f(y=\infty) \not= \infty \Rightarrow D = 0'>>
*Using the expressions which differentiate <<tex '\phi'>> with respect to t and y, we get:
><<tex 'C = A \sigma / k'>>
>
><<tex '\sigma^2 = gk'>>
*The second formula is the dispersion relation.
*''Ionic'' bonding:
**Transfer of electrons from one atom to another, to fill atomic shells. Electrostatic force then attracts the ions.
*''Covalent'' bonding:
**Unpaired electron orbitals hybridize to give a lower energy state. Very directional (e.g diamond, silicon).
*''Metallic'' bonding:
**Outer orbitals overlap many other atoms;
**Electrons become mobile, KE reduced by delocalisation;
**Not directional, and weaker than covalent bonding leading to soft and ductile properties.
*''Hydrogen'' bonding:
**Covalent bond between atoms of different atomic species;
**Partial seperation of charge, very weak;
**Can be seen in water/ice.
*''Van der Waals Forces'':
**Force due to fluctuations in dipole moments;
**One atom's dipole moment can induce a dipole in a neighbouring atom - resulting attractive force between the two dipoles;
**Very weak, only important when no other bonds are present;
**Seen in noble gases.
*Translational symmetry can be represented by a lattice of points that fills space. Each lattice point is in an identical environment.
*''Unit cell'':
**Perfect tessellation: when many identical cells are stacked together they fill all of space.
**''Conventional'' unit cell:
***Chosen to display the full symmetry of the crystal;
***Not always the smallest possible choice - can contain more than one lattice point.
**''Primitive'' unit cell:
***Smallest possible unit cell - only contains one lattice point.
*Unit cells are usually 6-faceted polyhedra, formed from 3 primitive translation vectors,
><<tex '\vec{a}, \vec{b}, \vec{c}'>>.
*The length of these vectors are the ''lattice parameters'';
*Volume of unit cell:
><<tex '\vec{a} \cdot (\vec{b} \times \vec{c})'>>
*Starting with [[NS Equation|6. Navier-Stokes equation]]:
**Neglect gravity;
**Steady state solution <<tex '(\partial \mathbf{u} / \partial t = 0)'>>;
**<<tex '\rho = \textrm{const}'>>;
**<<tex '\nu = \textrm{const}'>>;
><<tex '\mathbf{u} \cdot \nabla \mathbf{u} + \dfrac{1}{\rho} \nabla p = \nu \nabla^2 \mathbf{u}'>>
**Oh and negelect the non-linear term compared to the viscous term due to small Reynolds number.
*We get Stoke's flow, which is when the Reynolds number is small (<<tex 'Re = UL / \nu \ll 1'>>, so can be one of the following:
**<<tex '\nu \gg UL'>> (large viscosity);
**<<tex 'L \ll \nu/U'>> (small lengths);
**<<tex 'U \ll \nu/L'>> (small velocity);
**any combination of the above.
*Modified NS gives:
><<tex '\nabla p = \mu \nabla^2 \mathbf{u}'>>
*Take divergence of this, and use incompressibility:
><<tex '\nabla \cdot \mathfb{u} = 0'>>
*and the commutatability of 'div' and 'del-squared' to get:
><<tex '\nabla^2 p = \mu \nabla^2 (\nabla \cdot \mathbf{u}) = 0'>>
*This is Laplace's equation for <<tex 'p'>>, but can't be used very often as the boundary conditions are hard to come by.
*Can take curl instead to get:
><<tex '\mu \nabla^2 (\nabla \times \mathbf{u}) = \mu \nabla^2 \boldsymbol{\omega} = 0'>>
**This is Laplace's equation for all the components of the vorticity.
**Gives reversible solutions - fish swimming in treacle, flap one way moves it forwards, flap back moves it backwards.
[img[Lattice Vibration|images/1211381660.png]]
*Potential energy of nth atom is:
><<tex 'V_n = \phi(x_{n+1} - x_n) + \phi (x_n - x_{n-1})'>>
>
>(Nearest neighbours)
>
><<tex 'V_n = \frac{1}{2} (u_{n+1} - u_n)^2 \ddot{\phi}}(a) + \frac{1}{2} (u_n - u_{n-1})^2 \ddot{\phi}'>>
>
>(Harmonic approximation)
*Using Newton's 2nd Law:
><<tex '\begin{array}{rl} m \dfrac{ \partial^2 u_n}{\partial t^2} &= - \dfrac{\partial V_n}{\partial u_n} \\ & \\ &= C [(u_{n+1} - u_n) - (u_n - u_{n-1})] \end{array}'>>
>
>Here, <<tex 'C = \ddot{\phi}'>>, the force, which is constant.
*Wave-like trial solution:
><<tex 'u_n = A e^{i(k x_n - \omega t)}'>>, where <<tex 'x_n = na'>>.
*Subs into equation of motion:
><<tex '\begin{array}{lr} -m \omega^2 u_n &= C(e^{ika} + e^{-ika} - 2)u_n \\ & \\ &=C(2(\cos(ka) - 1))u_n \end{array}'>>
*Gives the ''dispersion relation'':
><<tex '\Rightarrow \omega = 2 \sqrt{\dfrac{c}{m}} | \sin(ka/2) |'>>
>[img[Dispersion Relation|images/1211389173.png]]
*Notes:
**In the long wavelength limit, <<tex '\omega \simeq ka \sqrt{c/m} '>>. The speed of sound in the medium <<tex 'v_s \equiv a \sqrt{c/m}'>>, so we can write:
><<tex '\omega = v_s k'>>
**The relation is periodic, so all distinct solutions are contained in the range <<tex '-\pi/a \leq k \leq \pi/a'>> (1st Brillouin Zone).
**Normal modes: Each allowed value of <<tex 'k'>> describes a state of motion in which all atoms vibrate with the same frequency.
*Time, age of the universe. Quick estimate:
><<tex 't_0 = H_0^{-1} = 10 - 20 \textrm{Gyr}'>>
*More generally:
><<tex '\dfrac{\dot{a}}{a} = H '>>
>
><<tex 'dt = \dfrac{da}{aH}'>>
>
><<tex 't_0 = \displaystyle{\int_0^1} \dfrac{da}{aH}'>>
>
>From FRW:
><<tex 'H^2 = H_0^2 \left( \dfrac{8 \pi G \rho_{M0}}{3 H_0^2} \dfrac{1}{a^3} - \dfrac{k c^2}{H_0^2} \dfrac{1}{a^2} + \dfrac{\Lambda}{3 H_0^2} \right)'>>
>
>Or:
><<tex 'H = H_0 \sqrt{\dfrac{\Omega_{M0}}{a^3} + \dfrac{\Omega_{K0}}{a^2} + \Omega_{\Lambda0}}'>>
>
>So (for a flat universe):
><<tex 't_0 = H_0^{-1} \displaystyle{\int_0^1} \dfrac{da}{a \sqrt{\Omega_{M0}/a^3 + \Omega_{\Lambda0}}}'>>
*This shows a larger cosmological constant relates to an older universe. This can be seen from the [[Raychauduri equation|2. Raychauduri Equation]]:
><<tex '\dfrac{\ddot{a}}{a} = - \dfrac{4 \pi G}{3} \rho + \dfrac{\Lambda}{3}'>>
*For two universes with the same expansion rate today, one with a larger cosmological constant relates to larger present accelleartion, so the speed of expansion in the past must have been slower, and it must therefore be older.
*General
**''Visual'' binaries: can see larger of the pair 'wobbling' due to interaction of unseen other;
**''Spectroscopic'' binaries: Can see periodic doppler shift of spectral lines:
***Can see movement of lines of either one star or both;
**''Photometric'' binaries: periodic variation of flux/colour;
**''Eclipsing'' binaries: if the plane of orbit is facing earth, stars can pass in front of each other changing luminosity.
*Close binaries:
**Roche lobe is a surface of constant potential = 0 for both stars (forms a kinda figure of 8);
**''Detached'' binaries:
***Both stars underfill their lobes, and do not spill past the potential = 0 boundary;
***Therefore only gravitational interactions can occur.
**''Semidetached'' binaries:
***One star fills its lobe and can transfer mass to ther other lobe;
**''Contact'' binaries:
***Both stars fill their lobes and transfer mass to each other;
***A common photosphere surrounding both components will form.
*Unit cells can be of a bunch of different types, the Bravais lattices cover all possible systems, divided into 7 groups.
*Locations etc:
**Atomic positions (from origin) can be defined inside a unit cell using fractional coordinates:
><<tex 'xyz = x\vec{a} + y\vec{b} + z\vec{c}'>>
**Direction vectors:
><<tex '[uvw] = u\vec{a} + v\vec{b} + w\vec{c}'>>
**Lattice planes:
><<tex '(hkl) = \dfrac{\vec{a}}{h} + \dfrac{\vec{b}}{k} + \dfrac{\vec{c}}{l}'>>
*Spacing
**Distance between two planes can be found by dotting a lattice vector with a unit normal, e.g between <<tex '(110)'>> planes:
><<tex 'd_{110} = \vec{a} \cdot \vec{\hat{n}}'>>
>
><<tex '\vec{a} = (a, 0, 0)'>>
>
><<tex '\vec{\hat{n}} = \dfrac{1}{\sqrt{a^2 + b^2}} (b, a, 0)'>>
>
><<tex '\vec{a} \cdot \vec{\hat{n}} = \dfrac{ab}{\sqrt{a^2 + b^2}} = \dfrac{1}{\sqrt{\dfrac{1}{a^2} + \dfrac{1}{b^2}}}'>>
**''Generally'':
><<tex 'd_{hkl} = \dfrac{1}{\sqrt{h^2/a^2 + k^2/b^2 + l^2/c^2}}'>>
* In a periodic structure:
><<tex '\psi (x) = \psi (x + L)'>>
><<tex 'e^{ikx} = e^{ik(x + L)}'>>
><<tex '\therefore e^{ikL} = 1'>>
* So in a crystal of size <<tex 'L'>>:
><<tex 'k = 0, \pm \dfrac{2 \pi}{L}, \pm \dfrac{4 \pi}{L}, ...'>>
* Density of states = Number of states / Volume per state, (where the factor of 2 is for polarization states):
><<tex 'g(k)dk = 2 \times \dfrac{4 \pi k^2 dk}{(\frac{2 \pi}{L})^3}'>>
*Can equate the Fermi energy to a number density:
><<tex '\int_{0}^{k_F} g(k)dk = N'>>, gives:
><<tex 'E_F = \dfrac{\hbar^2}{2m} (3 \pi^2 n)^{2/3}'>>
*Occupation per mode is ~Fermi-Dirac (~Bose-Einstein) distrubution:
><<tex 'F(p) = \dfrac{g}{exp(\frac{E - \mu}{k T} \frac{-}{(+)} 1})'>>
*Can calculate:
**Number density: <<tex 'n = \dfrac{1}{h^3} \int F(p) d^3 p'>>
**Energy density: <<tex '\rho = \dfrac{1}{h^3} \int E(p) F(p) d^3 p'>>
*Using these relations we can work out that:
**At high temperature, <<tex 'P = \frac{1}{3} \rho c^2'>>, which is the expression for radiation pressure: at high enough temperature matter behaves like radiation.
**At low temperature <<tex 'T \ll M'>> and the quantities are as expected (e.g <<tex 'P = n k T'>>).
*At <<tex 'a(t) =1 '>> (today), we know only the 3 neutrino types behave relativistically, but at earlier times, more particles would exhibit this behavior.
*Can define "relativistic degrees of freedom", <<tex 'g_*'>> as a way of describing the number of particles that behave relativistically, and therefore contribute to radiation terms. This takes the place of <<tex 'g'>>, the degeneracy in the case for a single type of particle, and gives an expression for all particles.
*Today, <<tex 'g_* = 7.25'>> (see problem set q8).
[img[Circular Pipe|images/1210606167.jpg]]
*As in [[rectilinear flow|1. Rectilinear Flow Between Parallel Plates]] except in cylindrical polars, so:
**Flow independent of <<tex '\theta, t'>>
*End up with differential equation:
><<tex '\dfrac{\mu}{r} \dfrac{d}{dr} \left( r \dfrac{du}{dr} \right) = \dfrac{dp}{dx} = -G'>>
*Boundary conditions:
**<<tex 'u(r=a) = 0'>>
**<<tex 'u(r=0) \not= \infty'>>
*Gives:
><<tex 'u = \dfrac{G}{4 \mu}(a^2 - r^2)'>>
*Assume Sun emits as a black body, can use increasing area of a sphere with distance to find the flux at the ~Sun-Earth distance, <<tex '\sim 1367 \textrm{W m}^{-2}'>>;
*Albedo of the Earth is <<tex '\alpha = 0.3'>>, so 30% of the incident energy is reflected away;
*For global energy balance, the Earth must emit as much as a black body as it recieves, so:
><<tex '\sigma T_e^4 = \dfrac{S_0 (1 - \alpha)}{4}'>>
>
>//where the factor of 4 arises due to the fact the Earth emits radiation from a sphere, but only recieves it on a flux projection disk.//
*Gives <<tex 'T_e \sim 255K'>>, compared to surface temperature of <<tex 'T_s = 288K'>>
*For an expression for <<tex 'T_s'>>, see [[transmittance|4. Transmittance, Absorptance and Optical Depth]] page.
*Growth of a beam of finite spectral width;
*We currently have:
><<tex '\dfrac{\partial I}{\partial z} = N^* \sigma_{21} (\omega - \omega_0) I (\omega, z)'>>
*Integrate both sides over the beam's bandwidth:
><<tex '\displaystyle\int_0^\infty \dfrac{\partial I}{\partial z} d \omega = \displaystyle\int_0^\infty N^* \sigma_{21} (\omega - \omega_0) I (\omega, z) d \omega'>>
>
><<tex '\Rightarrow \dfrac{\partial}{\partial z} \displaystyle\int_0^\infty I(\omega, z) d \omega = \displaystyle\int_0^\infty N^* \sigma_{21} (\omega - \omega_0) I (\omega, z) d \omega'>>
>
><<tex '\Rightarrow \dfrac{\partial I_T}{\partial z} = \displaystyle\int_0^\infty N^* \sigma_{21} (\omega - \omega_0) I (\omega, z) d \omega'>>
*Can again bring the cross section outside the integral for narrow-band radiation:
><<tex '\dfrac{\partial I_T}{\partial z} = N^* \sigma_{21} (\omega_L - \omega_0) I_T'>>
*All atoms will interact with a beam of radiation of frequency <<tex '\omega'>> by the same amount, but the strength of this interaction depends on the deviation of this from the peak frequency, <<tex '\omega_0'>>.
*Einstein's postulates are modified, and become (where <<tex 'g_x(\omega - \delta \omega)'>> is the lineshape for that transition, and looking at the range <<tex '\omega \rightarrow \omega + \delta \omega)'>>:
**Spontaneous emission rate: <<tex 'N_2 A_{21} g_A(\omega - \omega_0) \delta \omega'>>;
**Absorption rate: <<tex 'N_1 B_{12} g_B(\omega - \omega_0) \rho(\omega) \delta \omega'>>;
**Stimulated emission rate: <<tex "N_2 B_{21} g_{B'}(\omega - \omega_0) \rho(\omega) \delta \omega">>
*The lineshapes must be normalised, fulfilling the conditions such as:
><<tex '\displaystyle\int_0^\infty N_2 A_{21} g_A(\omega - \omega_0) d \omega = N_2 A_{21}'>>
*Equating these in a similar manner as before, we achieve the same result relations, only with the appropriate lineshape function added to the end of each expression. But, for this to be the case, each lineshape must be identical for all three processes, and we call the single lineshape <<tex 'g_H(\omega - \omega_0)'>>.
*This lineshape will be the lineshape of the frequency spectrum for each level, for example the Lorentzian frequency distribution for lifetime broadening.
*The ''Particle Horizon'':
**Furthest a light ray can travel from between<<tex 't=0'>> and <<tex 't=t'>>:
><<tex "r_p(t) = a(t)c \displaystyle{\int_0^t} \dfrac{dt'}{a(t')}">>
*The ''Event Horizon'':
**All parts of the universe which can ever be reached (<<tex '\rightarrow \infty'>> in e.g radiation/matter universes)
><<tex "r_e(t) = a(t) c \displaystyle{\int_0^{\infty}} \dfrac{dt'}{a(t')}">>
[img[Light Cones|images/1210513786.png]]
*Vertical force on a small cylinder of air, unit area, thickness <<tex '\delta z'>>, density <<tex '\rho'>>:
**Downward pressure on top: <<tex 'p(z + \delta z)'>>;
**Upward pressure on bottom: <<tex 'p(z)'>>
**Force due to gravity: <<tex 'g \rho \delta z'>>
*Equate:
><<tex 'g \rho \delta z + p (z + \delta z) = p(z)'>>
*Expand and such:
><<tex '\dfrac{dp}{dz} = - g \rho'>>
*Next we use the ideal gas law, <<tex 'pV = RT'>> (R is the specific gas constant for dry air <<tex '=R*/m_a'>>, V is the specific volume):
><<tex '\dfrac{dp}{dz} = - \dfrac{gp}{RT}'>>
*Write <<tex 'T = T(z)'>>:
><<tex "ln(p) - ln(p_0) = -\dfrac{g}{R} \displaystyle{\int_0^z} \dfrac{dz'}{T(z')}">>
*Or if over the range T is roughly constant, <<tex 'T \simeq T_0'>>:
><<tex 'p = p_0 exp\left(- \dfrac{gz}{RT_0} \right) = p_0 e^{-\frac{z}{H}}'>>
*Here H is the pressure scale height.
*Background radiation:
**Thermal isotropic bath of cold radiation, the microwave background;
**Black body spectrum:
><<tex '\rho(\nu) d \nu = \dfrac{8 \pi h}{c^3} \dfrac{ \nu^3 d \nu}{ e^{h \nu / k T_0} - 1} }'>>
**Using <<tex 'T_0 = 2.725 K'>> gives a good fit to measured data.
*Abundances of elements:
**3/4 H, 1/4 He, rest negligible.
**At hotter times, everything was seperate due to high energy, as it cools they remain seperate so mostly H.
*''Cosmological Principle'', universe is:
**Homogeneous: everywhere looks the same. Will always measure the same values for any quantity (tempreature, energy, pressure) wherever we are;
**Isotropic: every direction we look looks the same.
**This is clearly not true, but on large enough scales it is a good approximation.
*Using [[Hubble's law|1. Hubble's Law]], can calculate age of universe:
**<<tex 't_0 = r / v = r / (H_0 r) = H_0^{-1}'>>
*Scattered waves from two electrons are superimposed, observing the different paths from source to detector.
[img[Laue Diffraction|images/1211325816.png]]
*One wave scatters off electron at <<tex 'O'>>, extra path difference <<tex '\vec{OB}'>>;
*One wave scatters off electron at <<tex 'j'>>, extra path difference <<tex '\vec{Aj}'>>;
*Phase difference:
><<tex "\begin{array}{rl} \phi_j &= \dfrac{2 \pi}{\lambda} (\vec{Aj} - \vec{OB}) \\ & \\ &= \mathbf{k} \cdot \mathbf{r}_j - \mathbf{k'} \cdot \mathbf{r}_j \\ & \\ &= (\mathbf{k} - \mathbf{k'})\cdot \mathbf{r}_j \\ & \\ &= \mathbf{Q} \cdot \mathbf{r}_j \end{array}">>
*Clearly the total scattered wave is:
><<tex '\psi \propto \displaystyle\sum_j e^{i \mathbf{Q} \cdot \mathbf{r}_j}'>>
*Now:
><<tex '\begin{array}{rl} \displaystyle\sum_j e^{i \mathbf{Q} \cdot \mathbf{r}_j} &= \displaystyle{\int \sum_j} \delta(\mathbf{r} - \mathbf{r}_j) e^{i \mathbf{Q} \cdot \mathbf{r}} d^3 \mathbf{r} \\ & \\ &= \displaystyle\int n (\mathbf{r}) e^{i \mathbf{Q} \cdot \mathbf{r}} d^3 \mathbf{r} \end{array}'>>
>
><<tex '\Rightarrow \psi \propto \textrm{F.T} \left\{ n(\mathbf{r}) \right\}'>>
*Let <<tex '\mathbf{r}_j = \mathbf{l} + \mathbf{d} + \mathbf{s}_d'>>, where:
**<<tex '\mathbf{l}'>> is the nearest lattice vector;
**<<tex '\mathbf{d}'>> is displacement to nearest lattice point
**<<tex '\mathbf{s}_d'>> is electron's displacement relative to nucleus of atom d.
><<tex '\psi \propto \left( \displaystyle\sum_{\mathbf{l}} e^{i \mathbf{Q} \cdot \mathbf{l}} \right) \left( \displaystyle\sum_{\mathbf{d}} [ \displaystyle\sum_{\mathbf{s}_d} e^{i \mathbf{Q} \cdot \mathbf{s}_d} ] e^{i \mathbf{Q} \cdot \mathbf{d}} \right)'>>
*This can be thought of as the FT of the lattice multiplied by the FT of the electron density in a unit cell. But the lattice FT is just the reciprocal lattice! So:
><<tex '\psi \propto \displaystyle\sum_{\mathbf{G}_{hkl}} \delta (\mathbf{Q} - \mathbf{G}_{hkl} )'>>
*Diffraction amplitude is large from a crystal when:
><<tex '\Delta \mathbf{k} \equiv \mathbf{Q} \equiv \mathbf{G}_{hkl}'>>
*This is the ''Laue Condition'', and is identical to the Bragg condition:
**Let <<tex '\mathbf{Q} = \mathbf{G}_{hkl}'>>
><<tex '\begin{array}{rl} \Rightarrow |\mathbf{Q}| &= |\mathbf{G}_{hkl} | \\ & \\ &= \dfrac{2 \pi}{d_{hkl}} \end{array}'>>
>
><<tex '\Rightarrow 2 |\mathbf{k}| \sin \theta = 2 \pi / d_{hkl}'>>
><<tex '\lambda = 2 d_{hkl} \sin \theta'>>, Bragg's Law.
*For smaller stars (<<tex 'M \sim M_{\odot}'>>):
**Core contracts as H starts to run out;
**Higher density increases temperature and shell burning rate increases;
**Luminosity increases causing the envelope to expand, decreasing <<tex 'T_{\textrm{eff}}'>>.
*Bigger stars (<<tex 'M \sim 5 M_{\odot}'>>):
**Whole star contracts as H runs out;
**GPE is released, increasing <<tex 'L, T_{\textrm{eff}}'>> slightly;
**After enough contraction, shell burning begins causing expansion and drop of <<tex 'T_{\textrm{eff}}'>> again.
*Time spent on the MS:
**Due to [[mass-luminosity relationship|5. Mass Luminosity Relationship]], bigger stars life shorter lives;
**Turn off 'peels away' from MS over time;
**Can calculate the age of a cluster by the position of the turn off - age of star at the turn off is approx. the age of the cluster.
**Using:
><<tex 't_{MS} \approx 10^{10} \left( \dfrac{M_{\odot}}{M}} \right)^{3}'>>
(see mass-luminosity page, and plug in numbers for Sun and divide), the age of a star can be determined from its luminosity:
><<tex 't_{MS} \approx 10^{10} \left( \dfrac{L_{\odot}}{L} \right)^{3/4}'>>
*Use periodic boundary condition, <<tex "\psi (x) = \psi (x + L)">>, //(Crystal of length L)//.
*<<tex "\psi (x) = e^{ikx} u_K(x)">>, and <<tex "u_K(x) = u_K(x + ma)">>, where <<tex "m">> is an integer. ( <<tex "u_K (x)">> is a Bloch function).
*<<tex "L = N_x a">>, where <<tex "N_x">> is the number of unit cells in the x-direction, and //a// is the lattice constant and therefore an integer, so <<tex "u_K (x + L) = u_K (x)">>.
*<<tex "\therefore e^{ikL} = 1">>, and so <<tex "kL = 2 \pi n">>, giving <<tex "\Delta k = 2 \pi / N_x a">>.
*In k-space, 1 BZ has a 1-D dimension of <<tex "2 \pi / a">>.
*Number of states is <<tex "2 \times \dfrac{\frac{2 \pi}{a}}{\Delta k} = 2 N_x">>, or in 3D, <<tex "2 \times N_x^3 = 2N">>, the number of unit cells in the crystal.
*''Collimated beam'':
[img[Collimated|images/1211455283.png]]
**Here, <<tex '\rho(\omega) = \langle n_\omega \rangle g(\omega) \hbar \omega = n(\omega) \hbar \omega'>> is the //spectral// photon density;
**So in time <<tex '\delta t'>>, we have <<tex 'n(\omega) \cdot c \delta t \cdot A'>> photons per unit bandwidth corssing area A;
**This gives a ''spectral intensity'' <<tex 'I(\omega) = c n (\omega) \hbar \omega = c \rho(\omega)'>>
*''Isotropic Radiation'':
**Small area, A, within an isotropic radiation field of <<tex 'n(\omega)'>> photons per unit volume per unit frequency interval.
**Assume photons behave as an ideal gas;
**From kinetic theory, num of photons in interval with <<tex '\delta \omega'>> that strikes a side per unit area per second is <<tex '\frac{1}{4} n(\omega) \delta \omega c'>>;
**This gives intensity:
><<tex 'I(\omega) = \frac{1}{4} n(\omega) \hbar \omega c = \frac{1}{4} \rho (\omega) c'>>
><<tex 'V = \frac{4}{3} \pi (a x)^3, E = (\rho V) c^2'>>
>
><<tex '\dfrac{dE}{dt} = c^2 \dfrac{4 \pi}{3} x^3 \left( \dfrac{d \rho}{dt} a^3 + 3 \dfrac{a}{dt} a^2 \rho \right)'>>
>
><<tex '\dfrac{dV}{dt} = \dfrac{4 \pi}{3} x^3 3 \dot{a} a^2'>>
*Using <<tex 'dE + PdV = TdS'>> and assuming adiabatic, <<tex 'dS = 0'>>:
><<tex 'c^2(\dot{\rho} a^3 + 3a^2 \dot{a} \rho) + 3 P \dot{a} a^2 = 0'>>
*Divide by <<tex 'c^2 a^3'>>, ''conservation equation'':
><<tex '\dot{\rho} + 3 \dfrac{\dot{a}}{a}\left(\rho + \dfrac{P}{c^2} \right) = 0'>>
*Differentiate FRM equation, ''ancilliary equation'':
><<tex '\ddot{a} = \dfrac{8 \pi G}{3} a (\dot{\rho} a^2 + 2 \dot{a} a \rho)'>>
*Subs conservation equation in to get ''Raychauduri equation'':
><<tex '\dfrac{\ddot{a}}{a} = - \dfrac{4 \pi G}{3} \left( \rho + \dfrac{P}{c^2} \right)'>>
*Reaction rate is proportional to:
**Number density of particles (<<tex 'n_1, n_2'>>)
**Frequency of collisions <<tex '\rightarrow'>> relative velocity <<tex 'v'>>:
><<tex 'r_{1+2} = n_1 n_2 <\sigma(v) v>'>>
**Probability of penetrating Coulomb barrier:
><<tex 'P_p(v) \propto e^{- 4 \pi^2 Z_1 Z_2 e^2 / h \nu}'>>
**Reaction cross section <<tex '\sigma'>>:
***Usually requires laboratory data
***Can be calculated for p-p theoretically
**Velocity distribution (Maxwell), A is reduced mass:
><<tex 'D(T,v) \propto \frac{v^2}{T^{3/2}} e^{- m_H A v^2 / 2 k T}'>>
*High temperature leads to more collisions and faster rate (except very high temperatures, usually not concerned with this)
*Rate decreases as Z increases, therefore low temperature favours lower Z nuclei.
*Selection rules for lattice types for reflection to be present:
| !Lattice Type | !Condition on <<tex '(hkl)'>> for reflection |
| P (corners) | no constraint. |
| I (interior, BCC etc.) | <<tex 'h+k+l'>> is even integer |
| F (face, FCC etc. ) | <<tex 'h,k,l'>> all even or all odd |
*Effective temperature, <<tex 'T_{\textrm{eff}}'>>:
><<tex 'L_S = 4 \pi R_S^2 \sigma T_{\textrm{eff}}^4'>>
Usually not measurable as <<tex 'R_S'>> is not measurable.
*Colour temperature: use black body spectrum characteristics and it's peak wavelength to determine temperature, can obtain <<tex 'T_{\textrm{eff}}'>> and <<tex 'R_{ph}'>> (the radiating sphere's radius, or photospheric radius).
>Wien's law:
><<tex '\lambda_{max} = \dfrac{b}{T}'>>
*Luminosity classes are based on width of spectral lines, narrower lines correspond to a lower surface pressure and gravity and therefore a bigger star. White dwarfs have very broad lines.
*Absorption lines are caused by cooler material above the photosphere.
*Emission lines are caused by hotter material above the photosphere.
*Spectral lines can tell us:
**Temperature where lines are produced - spectral type
**Chemical composition
**Pressure / Surface gravity
**Rotation (lines are doppler broadened by rotation)
**Orbital velocities (periodic doppler shifts in spectra).
*Viscous fluid moving past a stationary sphere:
**Radius <<tex 'a'>>;
**Flow velocity <<tex '\rightarrow U \mathbf{k}'>> far from the sphere, with <<tex '\mathbf{k}'>> being the direction of flow.
*Interesting to calculate viscous drag if <<tex 'Re \ll 1'>>, i.e Stoke's flow.
*Boundary conditions:
**<<tex '\mathbf{u} \rightarrow U \mathbf{k}'>> far from the sphere;
**<<tex '\mathbf{u} = 0'>> on the surface of the sphere (no slip).
**<<tex 'p \rightarrow p_{\infty} = \textrm{const}'>> far from the sphere
*Can use Stoke's Law for drag on a sphere to calculate this, and from dimensional analysis:
><<tex 'D = K \mu a U'>>
*Here, K is a constant that depends on the geometry of the object, in the case of a sphere from solving Laplace's equation get <<tex 'K = 6 \pi'>>.
*Characteristic timescale it takes for a star to respond to a dynamic change.
*Can be defined in terms of the time it would take the star to collapse completely if pressure forces were negligible:
><<tex 's = \frac{1}{2} g t^2 = \frac{1}{2} (GM_r / r^2)t^2'>>
and with <<tex 's \approx R_S, r \approx R_S, M_r \approx M_s'>>:
><<tex 't_{\textrm{dyn}} \approx \left(\dfrac{2 R_s^3}{GM_s}\right)^{\frac{1}{2}} \approx \left(\dfrac{3}{2 \pi G \bar{\rho}}\right)^{\frac{1}{2}}'>>
*This gives a value of <<tex '(t_{\textrm{dyn}})_{\odot} \approx 40 \textrm{mins}'>>.
*Stars therefore adjust very quickly to changes in pressure and gravity and maintain stability.
*Start with the expression for the heat capacity.
><<tex 'C = \dfrac{\partial}{\partial T} \displaystyle\int_0^\infty \dfrac{\hbar \omega}{e^{\beta \hbar \omega} - 1} g(\omega) d \omega'>>
*The Einstein Model is one method to calculate the density of states, <<tex 'g(\omega)'>>.
*All N atoms vibrate with the same frequencey, the Einstein frequency, <<tex '\omega_E'>>. This gives:
><<tex 'g(\omega) = 3 N \delta(\omega - \omega_E)'>>
*This leads to a heat capacity:
><<tex 'C = 3 N k_B \left( \dfrac{\Theta_E }{T} \right)^2 \dfrac{ e^{\Theta_E / T} }{(e^{\Theta_E / T} - 1)^2}'>>
>
><<tex '\Theta_E = \dfrac{\hbar \omega_E}{k_B}'>>
[img[Einstein Model|images/1211406862.png]]
*At <<tex 'T < \Theta_E'>>, the functional form is:
><<tex '\dfrac{e^{- \Theta_E / T}}{T^2}'>>
*In this laser, the low lasing level lies quite a long way above the ground state, so the thermal population will be small, even during lasing, due to rapid decay to the ground state.
*This is therefore a good example of a four level laser. It is easy to achieve population inversion in this system.
*As before:
><<tex 'N_{thresh}^* = N_2^{thresh} - N_1^{thres}'>>
*however, as <<tex 'N_1 \approx 0'>>
><<tex 'N_2^{thresh} = N_{thresh}^*'>>
*This value is often around 3 orders of magnitude smaller than that for the Ruby 3-level laser.
*The only problem is overcoming losses in the cavity:
**This was easy before in the 3-level laser, as over half the population needed to be in the upper level, so high energy output could easily be obtained by allowing them to drop.
**Here however, any number of ions in the upper level constitutes population inversion, so small levels of inversion will lose all their energy.
**The threshold energy is still small however, <<tex '150 \rm{mJ}'>>, and a CW operation absorbed power of <<tex '11W'>>, and operation power of <<tex '650 \rm{W}'>>. Easily achievable.
*Practical devices can even be pumped by highly efficient diode lasers:
**electrical efficiency near 50%
**large proportion of the output power is in the correct pump band and not other parts of the spectrum
**uni-directional so easier to couple to laser rod.
*Overall efficiency of Nd:YAG can exceed 1% using a diode pump.
*See [[Radiation|6. Energy Transport - Radiation]], where we have (rearranged):
><<tex '\dfrac{dT}{dr} = - \dfrac{3 \kappa L_r \rho}{16 \pi a c r^2 T^3}'>>
*This describes temperature profile dependence on opacity etc.
*Approximate analytical forms for opacity:
**High temperature (constant): <<tex '\kappa = \kappa_1 = 0.020 \textrm{m}^2 \textrm{kg}^{-1} (1+X)'>>
**Intermediate temperature (inverse): <<tex '\kappa = \kappa_2 \rho T^{-3.5}'>> (Kramer's law)
**Low temperature (power): <<tex '\kappa = \kappa_3 \rho^{1/2} T^4'>>
*<<tex '\kappa_1, \kappa_2, \kappa_3'>> are constants for a given chemical composition.
*From the ever useful incompressibility condition on a standard 2D flow:
><<tex '\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} = 0'>>
so we can write:
><<tex 'u = \dfrac{\partial \psi}{\partial y}'>>
>
><<tex 'v = - \dfrac{\partial \psi}{\partial x}'>>
*Where <<tex '\psi'>> is the streamfunction.
*Streamlines mark out a path a blob would take.
*If the flow is also irrotational, we get Laplace's equation again:
><<tex '\nabla^2 \psi = 0'>>
*Also:
><<tex '\mathbf{u} \cdot \nabla \psi = 0'>>
>
>//The velocity is parallel to lines of constant <<tex '\psi'>>, or perpendicular to lines of constant <<tex '\phi'>>.//
>''N.S.:''
>
><<tex '\dfrac{D \textbf{u}}{Dt} \equiv \dfrac{\partial \textbf{u}}{\partial t} + ( \textbf{u} \cdot \nabla) \textbf{u} = - \dfrac{1}{\rho} \nabla p + \dfrac{\mu}{\rho} \nabla^2 \textbf{u}'>>
*Using compressibility, and the vector identity:
><<tex '(\textbf{u} \cdot \nabla) \textbf{u} = \frac{1}{2} \nabla |\textbf{u}|^2 - \textbf{u} \times (\nabla \times \textbf{u})'>>
*This gives:
><<tex '\dfrac{\partial \textbf{u}}{\partial t} + \nabla (\frac{1}{2} |\textbf{u} |^2 = \dfrac{p}{\rho} + gz) - \textbf{u} \times (\nabla \times \textbf{u}) = \nu \nabla^2 \textbf{u}'>>
*Now using
**Vorticity [[definition|1. Definition and Interpretation]]
**<<tex 'curl(grad) = 0'>>
**<<tex '\nabla \times (\nabla^2 \textbf{u}) = \nabla^2(\nabla \times \textbf{u})'>>
*Gives the ''vorticity equation'':
><<tex '\dfrac{\partial \boldsymbol{\omega}}{\partial t} - \nabla \times (\boldsymbol{u} \times \boldsymbol{\omega}) = \nu \nabla^2 \boldsymbol{\omega}'>>
*or:
><<tex '\dfrac{D \boldsymbol{\omega}}{D t} - (\boldsymbol{\omega} \cdot \nabla)\textbf{u} = \nu \nabla^2 \boldsymbol{\omega}'>>
*Using the dispersion relation:
><<tex '\sigma^2 = gk'>>
**Phase and group velocities:
><<tex 'c_p = \dfrac{\sigma}{k} = \sqrt{\dfrac{g}{k}}'>>
>
><<tex 'c_g = \dfrac{\partial \sigma}{\partial k} = \frac{1}{2} \sqrt{\dfrac{g}{k}} = \frac{1}{2} c_p'>>
**We see that these waves are dispersive as <<tex 'c_p'>> depends on <<tex 'k'>>.
*Assume a bulk fluid velocity component in the x-direction, that only depends on z, <<tex 'u(z)'>>.
*Tangential viscous stress in x, (this defines <<tex '\mu'>>, ''dynamic viscosity''):
><<tex '\tau_{xz} = \mu \dfrac{d u }{d z}'>>
*Net viscous force on blob:
><<tex '\delta F= (\tau(z) - \tau(z + dz))\delta A'>>
>
><<tex '\delta F = -\delta z \dfrac{\partial \tau}{\partial z} \delta A = \delta V \dfrac{\partial \tau}{\partial z}'>>
>
><<tex '\delta F_v = \mu \dfrac{d^2 u}{dz^2} \delta V'>>
*More complicated 3d derivation gives:
><<tex '\delta \textbf{F}_v = \delta V \mu [\nabla^2 \textbf{u} + \frac{1}{3} \nabla(\nabla \cdot \textbf{u})]'>>
*<<tex '\nu = \mu / \rho'>> is the ''kinematic viscosity''
*Can calculate number of photons per mode for two examples. General steps:
**Relate ''total'' intensity of the source to the spectral intensity, <<tex 'I_T = I(\omega) \Delta \omega'>>, where <<tex '\Delta \omega'>> is the bandwidth of the source.
**From spectral intensity can find spectral number density of photons, from [[photons per mode for common sources|2. Photons Per Mode for Common Sources]], then photons per mode is given by <<tex '\langle n_\omega \rangle = n(\omega) / g(\omega)'>>.
*''Example 1'' - Single-isotope mercury lamp:
**Single isotope means broadening is only from Doppler effect;
**Bandwidth of <<tex '\sim 0.2 \rm{nm}'>> at <<tex '253.7 \rm{nm}'>>, with intensity of <<tex '1 \rm{W cm}^{-2}'>>.
*''Example 2'' - Frequency stabilized argon-ion laser:
**Delivers <<tex '5 \rm{W}'>> of power in a beam <<tex '2 \rm{mm}'>> in diameter;
**Linewidth of <<tex '0.002 \rm{nm}'>> at <<tex '488 \rm{nm}'>>.
| ! | ! | !<<tex 'Hg'>> lamp | !<<tex 'Ar^+'>> laser |
| Wavelength (nm) | <<tex '\lambda'>> | 253.7 | 488 |
| Bandwidth (nm) | <<tex '\Delta \lambda'>> | 0.2 | 0.0002 |
| Frequency bandwidth (<<tex '\omega = 2 \pi c / \lambda'>>) | <<tex '\Delta \omega'>> | <<tex '6 \times 10^{12}'>> | <<tex '2 \times 10^{10}'>> |
| Intensity <<tex 'W m^{-2}'>> | <<tex 'I_T'>> | <<tex '10^4'>> | <<tex '1.6 \times 10^6'>> |
| Spectral Intensity | <<tex 'I(\omega) = I_T / \Delta \omega'>> | <<tex '1.7 \times 10^{-9}'>> | <<tex '1.0 \times 10^{-4}'>> |
| Spectral Photon Density (using expression for different sources) | <<tex 'n(\omega)'>> | <<tex '7.3'>> | <<tex '8.3 \times 10^5'>> |
| Spectral Mode Density | <<tex 'g (\omega) = \dfrac{\omega^2}{\pi^2 c^3}'>> | <<tex '2.1 \times 10^5'>> | <<tex '5.6 \times 10^4'>> |
| Photons Per Mode | <<tex '\langle n_\omega \rangle = n(\omega) / g(\omega)'>> | <<tex '3 \times 10^{-5}'>> | <<tex '\textbf{15}'>> |
*Here we see that the photons per mode, <<tex '\langle n_\omega \rangle'>> is much higher for lasers, in fact:
**<<tex '\langle n_\omega \rangle \gg 1'>> for laser sources in general;
**<<tex '\langle n_\omega \rangle \ll 1'>> for non-laser sources.
*This is due to the difference of stimulated and spontaneous emission in the sources.
*This time there are two different atoms, with two different masses.
*Nearest neighbour forces only:
><<tex 'm_1 \dfrac{\partial^2 u_n}{\partial t^2} = C(u_{n+1} - 2 u_n + u_{n-1})'>>
>
><<tex 'm_2 \dfrac{\partial^2 u_{n-1}}{\partial t^2} = C(u_n - 2 u_{n-1} + u_{n-2})'>>
*Trial solutions of the form:
**For <<tex 'm_1'>>:
><<tex 'u_n = Ae^{i (kn(a/2) - \omega t)}'>>
**For <<tex 'm_2'>>:
><<tex 'u_{n-1} = \alpha A e^{i (k [n-1] (a/2) - \omega t)}'>>
*Here <<tex '\alpha'>> is the complex amplitude ratio. Subs. into equations of motion, ''dispersion relation'':
><<tex '\omega^2 = \dfrac{C(m_1 + m_2)}{m_1 m_2} \pm C \left[ \left( \dfrac{m_1 + m_2}{m_1m_2}\right)^2 - \dfrac{4}{m_1 m_2} \sin^2 \left( \dfrac{ka}{2} \right) \right]^{1/2}'>>
[img[Dispersion|images/1211403534.png]]
*Consider behavoir at following points:
**''Point 0'':
***<<tex 'ka \ll 1'>>;
***<<tex '\omega \simeq 0'>>;
***<<tex '\alpha \simeq 1'>>;
***''Sound Waves'': both masses oscillate in phase because <<tex '\alpha \simeq 1'>>.
**''Point A'':
***<<tex 'ka \ll 1'>>;
***<<tex '\omega \simeq \sqrt{\dfrac{2 C (m_1 + m_2)}{m_1 m_2}}'>>;
***<<tex '\alpha = - m_1 / m_2'>>;
***Here, both masses oscillate in anti-phase.
**''Points B & C'':
***<<tex 'ka = \pi'>>;
***At B, <<tex '\omega = \sqrt{\dfrac{2 C}{m_2}}'>> and at C <<tex '\omega = \sqrt{\dfrac{2 c}{m_1}}'>>;
***At B, <<tex '\alpha = \infty'>>, and at C <<tex '\alpha = 0'>>;
***At B, <<tex 'm_2'>> is oscillating and <<tex 'm_1'>> is ar rest, and visa versa for C.
*All modes on the accoustic branch are oscillating in the same direction. Near the origin, they're oscillating within a unit cell.
*On the optic branch, near A, atoms oscillate in antiphase in a unit cell.
*Oxygen and Nitrogen do not absorb heavily in the visible or infrared - diatomic.
*Radiatively active constituents:
**Water vapour (IR)
**Carbon dioxide (IR)
**Methane (IR)
**Nitrous Oxide (IR)
**Ozone (IR and UV)
**~CFCs (IR)
**Aerosols and clouds (Visible and IR)
*Using the 1D model, there are 2N states available. For monovalent metals, electrons will fill up half the states, allowing observed conducting behavior as the electrons are free to move around into the empty states.
*But for divalent metals, all 2N states are taken up - the band is full and there is no space to conduct so insulating behavior should be observed. However, this is not the case for many group 2 metals such as Calcium.
*This problem is rectified in 2D:
>[img[Monovalent and Divalent metals in 2D|images/1209140696.png]]
>The hashed area is the Brillouin zone, the white area depicts the filled Fermi states.
*The 'corners' of the 1st band actually lie at higher anergy than the bottom states of the 2nd band (by a factor of <<tex "\sqrt{2}">>) - and therefore the bands overlap, so the electrons still have space to move.
*As electron-lattice interaction strength increases, the Fermi circle deforms into a square, filling the first BZ completely. Here a band gap forms between the 1st and 2nd bands, the electrons have nowhere to go and the material is an insulator.
*30-50% of stars experience mass transfer from Roche-lobe overflow, but usually during late phases of lifetime;
*Quasi-conservative mass transfer:
**One star loses mass, other experiences accretion.
**Mass loser loses its envelope and forms a 'helium star';
**The accretor is rejuvenated with lighter elements and behaves as if its evolutionary clock has been reset;
**The orbit usually widens.
*Dynamic mass transfer:
**Common envelope between two stars (one is usually a red giant);
**Accretor also fills its lobe;
**Donor engulfs accretor;
**Two cores spiraling inside a common envelope;
**Envelope can be ejected forming a very close binary, or
**Two cores can merge into a single, rapidly rotating star.
*Can be observed baryon fraction is <<tex '\Omega_B \sim 0.04'>>;
*Large galaxies rotate much faster than they should! ''Dark matter'', <<tex '\Omega_{DM} \sim 0.25'>>;
*Total <<tex '\Omega = 1 \pm 0.05'>> (measured) meaning the Universe is flat.
**But we only have 0.3, and we need 1 to get a flat universe:
**''Dark energy'' <<tex '\Omega_{DE} \simeq 0.7'>>
*These estimates, with believed <<tex 'H_0 = 67 \textrm{Km s}^{-1} \textrm{Mpc}^{-1}'>>, give age of the universe of <<tex '15 \textrm{Gyr}'>>.
*At early times, before recombination (at 0.308eV - not 13.6eV due to Saha equation), radiation is in thermal equilibrium, and creates a Planck spectrum.
*After recombination, photons are left to propagate freely:
**Only effect is redshifting due to expansion;
**Spectrum is just shifted, indicating the change in temperature.
*History of a photon's lifetime:
**Early universe, hot temperature all in thermal equilibrium;
**Suddenly at recombination, photon is 'released' from ''surface of last scattering'' - moves on its current path in a straight line, but original spectrum is maintained;
**We see today a 'photograph' of the universe at recombination time, age of the universe of around 380,000 years.
*Can be described in terms of a ''lattice'' - the network of points that fills space, and a ''basis'' - a group of atoms assosciated with each lattice point.
*Mathematically:
**A lattice represented by a set of delta functions:
><<tex '\displaystyle\sum_{\vec{l}} \delta (\vec{r} - \vec{l})'>>
**A basis represented as a function of position:
><<tex 'f(\vec{r})'>>
**A crystal structure represented by their convolution:
><<tex '\displaystyle\sum_{\vec{l}} \delta (\vec{r} - \vec{l}) * f(\vec{r}) '>>
>
><<tex "= \displaystyle\sum_{\vec{l}} \int \delta (\vec{r} - \vec{l} - \vec{r}') * f(\vec{r}) d^3 \vec{r}'">>
>
><<tex '= \displaystyle\sum_{\vec{l}} f(\vec{r} - \vec{l})'>>
*Form factor:
**Consider again <<tex '\displaystyle\sum_{\mathbf{s}_d} ... '>> from the derivation of [[Laue diffraction|2. Laue Diffraction Theory]].
><<tex '\begin{array}{rl} \displaystyle\sum_{\mathbf{s}_d} e^{i \mathbf{Q} \cdot \mathbf{s}_d} &= \int n_d (\mathbf{r}) e^{i \mathbf{Q} \cdot \mathbf{r}} d^3 \mathbf{r} \\ & \\ &= f_d (\mathbf{Q}) \end{array}'>>
**Here, <<tex 'f_d(\mathbf{Q})'>> is called the ''atomic form factor'' of d.
*Structure factor:
**Now consider <<tex '\displaystyle\sum_{\mathbf{d}} ...'>> from the same derivation.
**Assume <<tex '\mathbf{Q} = \mathbf{G}_{hkl}'>> for some <<tex '(hkl)'>>:
><<tex '\begin{array}{rl} \mathbf{Q} \cdot \mathbf{d} &= \mathbf{G}_{hkl} \cdot \mathbf{d} \\ & \\ &= (h \mathbf{a}^* + k \mathbf{b}^* + l \mathbf{c}^*) \cdot (x_d \mathbf{a} + y_d \mathbf{b} + z_d \mathbf{c}) \end{array}'>>
**For an orthogonal lattice:
***<<tex '\mathbf{a}^* \cdot \mathbf{a} = \mathbf{b}^* \cdot \mathbf{b} = \mathbf{c}^* \cdot \mathbf{c} = 2 \pi'>>;
***<<tex '\mathbf{a}^*.\mathbf{b} = 0'>>, etc.
><<tex '\mathbf{Q} \cdot \mathbf{d} = 2 \pi (h x_d + k y_d + l z_d)'>>
**This leads to the ''structure factor'', the scattering amplitude:
><<tex 'S_{hkl} = \displaystyle\sum_{d} f_d (\mathbf{G}_{hkl}) e^{ i 2 \pi (h x_d + k y_d + l z_d)}'>>
**N.B. Remember multiplicity of reflection for amplitude...
*''Intensity'':
><<tex 'I_{hkl} \propto | \psi |^2'>>
>
><<tex 'I_{hkl} \propto | S_{hkl} |^2 \delta(\mathbf{Q} - \mathbf{G}_{hkl})'>>
*Example 1:
**Lattice: Primitive Cubic
**Basis: Cs <<tex '000'>>, Cl <<tex '\frac{1}{2}, \frac{1}{2}, \frac{1}{2}'>>
><<tex 'S_{hkl} = f_{Cs} (\mathbf{G}_{hkl}) + f_{Cl} (\mathbf{G}_{hkl}) e^{i \pi (h + k + l)}'>>
**This gives intensities:
***<<tex '(f_{Cs} + f_{Cl})^2'>> if h+k+l is even;
***<<tex '(f_{Cs} - f_{Cl})^2'>> if h+k+l is odd.
*Example 2:
**Lattice: FCC
**Basis: Cu <<tex '000, \frac{1}{2}\frac{1}{2}0, \frac{1}{2}0\frac{1}{2}, 0\frac{1}{2}\frac{1}{2}'>>
><<tex 'S_{hkl} = f_{Cu} (\mathbf{G}_{hkl})\left[1 + e^{i \pi(h+k)} + e^{i \pi(h+l)} + e^{i \pi(k+l)}\right]'>>
**This gives intensities:
***<<tex '16 f_{Cu}^2'>> if h,k,l are all even or all odd;
***<<tex '0'>> otherwise.
[img[Rate Analysis|images/1211552184.png]]
*Rate equations for the laser levels are (where <<tex 'I'>> is the //total// (not spectral) intensity):
><<tex '\dfrac{d N_2}{dt} = R_2 - N^* \sigma_{21} (\omega_L - \omega_0) \dfrac{I}{\hbar \omega_L} - \dfrac{N_2}{\tau_2}'>>
>
><<tex '\dfrac{d N_1}{dt} = R_1 + N^* \sigma_{21} (\omega_L - \omega_0) \dfrac{I}{\hbar \omega_L} + N_2 A_{21} - \dfrac{N_1}{\tau_1}'>>
*The Einstein A term is the spontaneous emission between the levels, and the two sigma terms are the stimulated emission rates. Absorption is included in other terms.
*Steady state solutions:
><<tex 'N_2 = R_2 \tau_2 - N^* \sigma_{21} \dfrac{I}{\hbar \omega_L} \tau_2'>>
>
><<tex 'N_1 = R_1 \tau_1 + N^* \sigma_{21} \dfrac{I}{\hbar \omega_L} \tau_1 + N_2 A_{21} \tau_1'>>
*Can obtain an expression for <<tex 'N^*'>>:
><<tex 'N^* = \dfrac{R_2 \tau_2 [1 - (g_2 / g_1) A_{21} \tau_1] - (g_2/g_1) R_1 \tau_1}{1 + \sigma_{21} \frac{I}{\hbar \omega_L} [ \tau_2 + (g_2 / g_1) \tau_1 - (g_2 / g_1) A_{21} \tau_1 \tau_2]}'>>
*When <<tex 'I = 0'>>, the denominator is equal to 1, so the numerator is the population inversion from pumping in the abscence of the beam, so we can simplify the expression to:
><<tex 'N^*(I) = \dfrac{N^*(0)}{1 + I/I_s}'>>
*where the ''saturation intensity'' is given by:
><<tex 'I_s = \dfrac{\hbar \omega}{\sigma_{21}} \left(\tau_2 + \dfrac{g_2}{g_1} \tau_1 - \dfrac{g_2}{g_1} A_{21} \tau_1 \tau_2 \right)^{-1} = \dfrac{\hbar \omega}{\sigma_{21} \tau_R}'>>
*where the ''recovery time'' is given by:
><<tex '\tau_R = \tau_2 + \dfrac{g_2}{g_1} \tau_1 [1 - A_{21} \tau_2]'>>
*Qualitatively, the expression for <<tex 'N^*(I)'>> tells us that an intense beam 'burns down' the population inversion. The saturation intensity will give a population inversion of <<tex '1/2'>> of that achieved without the beam present. It is therefore a marker to indicate the intensity to which the beam may be amplified before level populations are significantly affected, and marks the boundary between 'high' and 'low' intensity.
*Various approximations to the saturation intensity can be made:
**For four-level lasers, the lifetime of the lower level is much shorter than for the upper level, so <<tex '\tau_2 \gg \tau_1'>>;
**If the upper level's main decay is to the lower level, then <<tex '\tau_2 \approx A_{21}^{-1}'>>;
**For both of these cases:
><<tex '\tau_R \approx \tau_2'>>
>
><<tex 'I_S \approx \dfrac{\hbar \omega_L}{\sigma_{21} \tau_2}'>>
*The saturated gain coefficient can be found from earlier looks at the [[small signal gain coefficient|4. Small Signal Gain Coefficient]]:
><<tex '\alpha_I (\omega - \omega_0) = N^*(I) \sigma_{21} (\omega - \omega_0)'>>
*Or, written in terms of the small signal gain coefficient, <<tex '\alpha_0 (\omega - \omega_0)'>>:
><<tex '\alpha_I (\omega - \omega_0) = \dfrac{\alpha_0 (\omega- \omega_0)}{1 + I/I_s}'>>
><<tex 'H = \left( \dfrac{\dot{a}}{a} \right)^2 = \dfrac{8 \pi G}{3} \rho - \dfrac{k c^2}{a^2}'>>
*Can be seen that if <<tex '\rho \propto 1/a^3'>> or <<tex '1/a^4'>>, the curvature term dominates at late times.
**For <<tex 'k < 1'>>:
><<tex '\left( \dfrac{\dot{a}}{a} \right)^2 = \dfrac{|k|c^2}{a^2}'>>
>
><<tex 'a \propto t'>>, the scale factor grows at <<tex 'c'>>.
**For <<tex 'k > 0'>>:
***Universe grows until:
><<tex '\dfrac{8 \pi G}{3} \rho = \dfrac{k c^2}{a^2}'>>
***Then it stars to shrink... ''big crunch''.
*Can think of this in terms of a ''critical density'':
**When <<tex 'k = 0'>> we have:
><<tex '\rho = \rho_c \equiv \dfrac{3H^2}{8 \pi G}'>>
>
>This corresponds to a balance between gravitational and kinetic energy.
**If <<tex 'k < 0'>>, then <<tex '\rho < \rho_c'>>:
***Kinetic energy is larger, and the universe keeps expanding
**If <<tex 'k >0'>> then <<tex '\rho > \rho_c'>>:
***GPE is larger, and eventually universe collapses.
*This can be summarised with the [[density parameter|4. The Density Parameter]].
* Using <<tex 'E = k_B T'>>, a change in temperature from absolute zero excites electrons just below <<tex 'E_F'>> to just above <<tex 'E_F'>>, by energy <<tex '~ k_B T'>>.
* Number of electrons which are excited = density of electrons at that energy <<tex '\times'>> the energy:
><<tex 'E_F g(E_F) = k_B T g(E_F)'>>
* <<tex '\therefore'>> Total energy change = energy per electron <<tex '\times'>> number of electrons:
><<tex '\Delta E \approx k_B^2 T^2 g(E_F)'>>
*Classically, using the equipartition theorem, <<tex 'C_v = \frac{3}{2} n k_B'>>, but because everything has been 'shifted along' by <<tex 'T_F'>>, the additional term can be seen as scaling the heat capacity.
*More rigorously:
><<tex 'C_v = \frac{\pi^2}{2} n k_B \frac{T}{T_F}'>>
*PPI Chain (times for the Sun, <<tex 'T = 3 \times 10^7 K'>>):
**<<tex '1) {}^1\textrm{H} + {}^1\textrm{H} \rightarrow {}^2\textrm{D} + \textrm{e}^+ + \nu + 1.44\textrm{MeV}, (\tau \approx 14 \times 10^9 \textrm{years}) '>>
**<<tex '2) {}^2\textrm{D} + {}^1\textrm{H} \rightarrow {}^3\textrm{He} + \gamma + 5.49\textrm{MeV}, (\tau \approx 6 \textrm{s})'>>
**<<tex '3) {}^3\textrm{He} + {}^3\textrm{He} \rightarrow {}^4\textrm{He} + {}^1\textrm{H} + {}^1\textrm{H} + 12.85\textrm{MeV}, (\tau \approx 10^6 \textrm{years})'>>
**First two occur twice, second once for each <<tex '4 \textrm{H} \rightarrow {}^4\textrm{He}'>>
**0.26MeV lost by the neutrino, so total of 26.2MeV is luminosity energy.
**First reaction very slow as it's a weak process, and bottlenecks the chain.
**Second reaction by far the quickest, deuterium burns very quickly.
*If there's enough Helium we can also have:
**PPII chain:
***<<tex '3a) {}^3\textrm{He} + {}^4\textrm{He} \rightarrow {}^7\textrm{Be} + \gamma + 1.59\textrm{MeV}'>>
***<<tex '4a) {}^7\textrm{Be} + \textrm{e}^- \rightarrow {}^7\textrm{Li} + \nu + 0.86\textrm{MeV}'>>
***<<tex '5a) {}^7\textrm{Li} + {}^1\textrm{H} \rightarrow {}^4\textrm{He} + {}^4\textrm{He} + 17.35\textrm{MeV}'>>
**PPIII chain:
***<<tex '4b) {}^7\textrm{Be} + {}^1\textrm{H} \rightarrow {}^8\textrm{B} + \gamma + 0.14\textrm{MeV}'>>
***<<tex '5b) {}^8\textrm{B} \rightarrow {}^8\textrm{Be} + \textrm{e}^+ + \nu'>>
***<<tex '6b) {}^8\textrm{Be} \rightarrow {}^4\textrm{He} + {}^4\textrm{He} + 18.07\textrm{MeV}'>>
*Total energy is the same in each case but the neutrino carries away different energies from each.
*All chains operate simultaneously, the second two chains act as catalysts for the conversion of <<tex '{}^3\textrm{He} + {}^1\textrm{H} \rightarrow {}^4\textrm{He}'>>.
*''Multiplicity'':
**planes with the same spacing contribute to the same diffraction peak, so multiplicity makes some peaks larger than others:
**cubic system (<<tex 'a=b=c'>>):
| !<<tex '(hkl)'>> | !multiplicity |
| <<tex '(\pm 1 0 0), (0 \pm 1 0), (0 0 \pm 1)'>> | 6 |
| <<tex '(\pm 1 \pm 1 0), (0 \pm 1 \pm 1), (\pm 1 0 \pm 1)'>> | 12 |
| <<tex '(\pm 1 \pm 1 \pm 1)'>> | 8 |
**similarly tetragonal system (<<tex 'a=b\not=c'>>):
| !<<tex '(hkl)'>> | !multiplicity |
| <<tex '(100), (010)'>> | 4 |
| <<tex '(001)'>> | 2 |
| <<tex '(110)'>> | 4 |
| <<tex '(101)'>> | 8 |
*Circulation definition:
><<tex '\Gamma_C \equiv \displaystyle{\oint_C} \mathbf{u} \cdot d\mathbf{l}'>>
*After a jaunty derivation, ''Kelvin's Circulation Theorem'' comes out as:
><<tex '\dfrac{d \Gamma_C}{dt} = \nu \displaystyle{\oint_C}(\nabla^2 \mathbf{u}) \cdot d \mathbf{l}'>>
>
>Or for an inviscid (non-viscous) flow:
>
><<tex '\dfrac{d \Gamma_C}{dt} =0'>>
*The circulation is constant for an inviscid, uniform-density flow following a closed moving material circuit.
*Atmosphere cooled from 'top down' by radiation, due to layers of atmosphere emitting infrared both up and down, hard for IR on surface to escape.
*''Lapse rate'' defined as:
><<tex '\Gamma = - \dfrac{dT}{dz}'>>
*Using First Law of Thermodynamics, and <<tex 'c_v = (dU / dT)'>>, and for adiabatic (<<tex 'dQ = 0'>>):
><<tex 'dQ = c_v dT + p dV = 0'>>
*Using ideal gas law:
><<tex pdV + Vdp = RdT = (c_p - c_v)dT'>>
*Combining all these:
><<tex 'c_p dT = V dp = - g dz'>>
*So for ''DRY'' air (as we haven't accounted for phase changes), we defined the //dry// adiabatic lapse rate:
><<tex '\Gamma_d = - \dfrac{dT}{dz} = \dfrac{g}{c_p} \simeq 9.7 \textrm{K km}^{-1}'>>
*Stability:
**If ambient rate <<tex '\Gamma < \Gamma_d'>>, the air cools faster than surroundings, is less dense, and gets restored back down - column is stable.
**If <<tex '\Gamma > \Gamma_d'>>, the parcel just accelerates upwards - but this mixes the atmosphere. If there is a cold parcel somewhere, it is heated by IR until <<tex '\Gamma = \Gamma_d'>>, and this is maintained due to mixing.
*Can get oscillations of lapse rate about <<tex '\Gamma_d'>> at the ''buoyancy frequency''.
*In