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Crystallography studies crystal structures and their symmetries. A structure determination connects measurements such as crystal scattering patterns to a model of the unit cell and its atomic basis of a crystal.

X-ray crystallographic structure inference makes that connection explicit: compare proposed crystal structures through the positions and wave intensities of their predicted reflections. The main difficulty is that a diffraction pattern encodes wave interference, so it must be interpreted through a model rather than read as a direct picture of the chemical atoms.

A surface reconstruction rearranges chemical atoms at a crystal surface relative to a simple termination of its bulk crystal structure.

By Nicholas Harvey, posted under the byline nickhar on 1 February 2012. The institutional course page links this post as lecture 9 of CPSC 536N: Randomized Algorithms; its scheduled lecture date, 2 February 2012, is distinct from the blog publication date. WordPress.com is the host.

Online text: nickhar.wordpress.com/2012/02/01/lecture-9-polynomial-identity-testing-by-the-schwartz-zippel-lemma/

Cited passages: §1, formal coefficient-based zero and evaluation access; §1.2, Lemma 1 and proof; §1.3, evaluation rule and independent repetition; §1.4, formal polynomials versus induced functions, extension fields and representation limits.

Observational cosmology tests models of the Universe through observations such as galaxy redshifts, distance indicators, element abundances and the cosmic microwave background. Each observable constrains a combination of cosmological parameters through an explicit model.

For example, a galaxy redshift measures a wavelength ratio directly; converting it to a distance requires an expansion model and assumptions about peculiar motion. Combining several kinds of observation can break degeneracies that one dataset cannot resolve. Instrument response, selection effects and foreground signals therefore belong in the inference alongside the cosmological equations.

Physical cosmology connects the universe's history to the measurable behaviour of its contents. The Friedmann equations describe cosmic expansion; atomic physics, radiation pressure and gravity determine what that expanding universe looks like. A particularly revealing example is the route from cosmological recombination through the photon-baryon fluid to the Cosmic microwave background acoustic peaks. The universe supplies the initial conditions and the laboratory; the prediction must still pass through a model of the telescope before it can be compared with observations.

The practical reward is that one sky contains several different tests. The Cosmic microwave background power spectrum carries information about the sound horizon, the cosmological baryon density and cosmic geometry. These are related quantities with different observational signatures. Angular sound-horizon degeneracy explains why measuring a characteristic angle alone cannot count the universe's baryons.

Ruby is red corundum, with chromium ions substituting for a small fraction of the aluminium in $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$. The local crystal field modifies chromium's electronic energy levels.

Selective absorption produces the red appearance, and characteristic emission supports the classic ruby laser. This connects a trace substitution in a crystal with a macroscopically visible optical property.

Sapphire is gem-quality corundum, crystalline $\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3$, conventionally excluding the red variety called ruby. Trace impurities and defect centers can produce several colors.

Synthetic sapphire is also valued as a hard, optically useful substrate and window material. The material's structural identity is aluminium oxide; a color name alone does not specify which impurity or electronic transition is responsible.

For weak drive in this homogeneous model, half maximum occurs at $|\Delta |=1/T_2$. The absorption full width at half maximum is thereforeThe example gives $2.653\mathrm{Hz}$ or $0.06230\mu \mathrm{T}$. At finite drive the model width is multiplied by $\sqrt{1+s}$. Field inhomogeneity and unresolved components can also broaden an actual line, so its width alone does not determine pure dephasing.

The same width appears by an original Fourier transform calculation. For $a=1/T_2$ and a free homogeneous signal demodulated exactly at resonance,This connects exponential memory to a Lorentzian absorptive line. It is a mathematical connection, not evidence that the 1946 teams performed pulsed Fourier transform spectroscopy. Instrumental signal bandwidth is a separate choice: narrowing a receiver band can reject thermal electrical noise, but too narrow a band distorts a changing signal. A sample linewidth and a receiver bandwidth are not interchangeable.

Drive reduces the longitudinal population imbalance when excitation competes with recovery. The dimensionless $s$ in the steady Bloch equations measures that competition.

Condensed matter comprises physical systems such as solids and liquids in which interactions among many constituents determine the observed properties. The term refers to the material system; condensed matter physics studies its structure, dynamics and phases.

A crystal illustrates the change of viewpoint. Its chemical atoms can have collective vibrational modes described as phonons, while its electronic response can involve energy bands rather than isolated-atom energy levels. Collective excitations and phase transitions show why a many-constituent system can display useful organizing principles that are not apparent from one atom alone.

Electron correlation concerns the dependence between electronic degrees of freedom that cannot be captured by treating each electron as moving independently. In a quantum-chemical use of the term, correlation usually means effects beyond the best single-Slater determinant Hartree-Fock method; that method already includes exchange associated with fermionic antisymmetry.

The distinction matters when comparing approximations. Dynamic electron correlation describes additional instantaneous avoidance even when one electronic configuration dominates. Static electron correlation becomes important when several configurations have comparable significance, as during some bond-dissociation processes. A correlation energy is therefore defined relative to a stated reference calculation, and the physical need for multiple configurations is different from merely allowing a single orbital to change shape.

By Jeremy Orloff, credited on the first page. Authored course notes in Complex Variables with Applications, MIT 18.04, Spring 2018. Spring 2018 identifies the course offering; no separate note publication or revision date is verified.

Online text: ocw.mit.edu/courses/18-04-complex-variables-with-applications-spring-2018/330e301bd727c7bdaa679cf44cb75fe3_MIT18_04S18_topic2.pdf

Cited passages: §2.5, printed pp. 5–6, one point at infinity and exterior neighbourhoods; §2.5.2, printed p. 7, equatorial unit-sphere projection and north-pole cap. The first page gives the title, author and heading 2 Analytic functions.

Allocate $n$ distinct objects to $k$ labelled bins of specified nonnegative sizes $n_1,\dots ,n_k$, with ${\sum }_i{n}_i=n$. Order within a bin is ignored. The number of allocations is the multinomial coefficientTo prove this, order all the objects in a permutation and assign the first $n_1$ to the first bin, the next $n_2$ to the second, and so on. Every allocation comes from exactly $n_1!\cdots n_k!$ internal orderings. These disjoint classes exhaust the $n!$ permutations, proving the formula by the multiplication principle. Empty bins are allowed because $0!=1$.

For two bins of sizes $m$ and $n-m$, the count is $\left(\genfrac{}{}{0ex}{}{n}{m}\right)$ and every allocation has multiplicity $m!(n-m)!$ in the permutation sample space. Exchanging two equal-sized labelled bins generally gives a different allocation. In contrast, a set partition uses unlabelled nonempty blocks; forgetting bin labels changes the counting problem and must not be done silently.

A screened Coulomb pair potential models the effective electrostatic interaction of particles in an electrolyte. For identical weakly charged spheres of radius $a$, charge $Q$, and Debye–Hückel screening length ${\kappa }^{-1}$, a common pairwise approximation is $U(R)=Q^2{e}^{-\kappa (R-2a)}/[4\pi \epsilon (1+\kappa a{)}^{2}R]$ for $R\ge 2a$. The near-contact interaction can require more detailed boundary conditions or nonlinear screening. Charge, screening length and temperature all matter to a resulting phase diagram.

DLVO theory combines screened electrostatic repulsion from overlapping electric double layers with attractive Van der Waals forces to model colloid stability. A repulsive barrier can inhibit aggregation even when a deep short-distance attractive minimum exists. Increasing electrolyte concentration reduces the screening length and can lower that barrier. Molecular surface forces and polymer-mediated interactions require additions to this description.

A depletion force is an effective interaction caused by exclusion of smaller particles or macromolecules from regions around larger objects. Overlap of the excluded regions increases the accessible volume for the smaller species. At fixed reservoir osmotic pressure ${\Pi }_p$, the associated pair free energy is $-{\Pi }_p$ times the overlap volume in the ideal dilute-depletant approximation. It is an entropic attraction for two large hard particles. Surface adsorption can introduce different interactions, so adding an arbitrary polymer does not by itself guarantee depletion attraction.

Steric stabilization is resistance to particle aggregation supplied by attached or adsorbed molecular layers. Bringing two coated particles close compresses or overlaps the layers, increasing their free energy through conformational and osmotic effects. It differs from the depletion force of nonadsorbing dissolved chains.

Polymer bridging occurs when one polymer chain attaches to more than one particle, producing an effective attraction or network connection. Surface affinity and incomplete coverage are essential to this mechanism. It should be distinguished from both steric stabilization by densely coated particles and the depletion force produced by nonadsorbing chains.