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Atomic spectroscopy studies the emission, absorption and scattering associated with the energy levels of chemical atoms and ions. The positions of spectral lines identify differences of energy, while their strengths and polarization also depend on populations and selection rules.

A dilute-gas discharge can excite chemical atoms; a spectrometer then separates the emitted light by wavelength. Comparing the resulting lines with an independently calibrated reference can identify an element or reveal an isotope shift. Changing a magnetic field tests Zeeman splitting, and improving energy resolution can reveal fine structure. Different experimental changes therefore isolate different pieces of the atomic model.

Energy-loss near-edge structure is the detailed shape near an inner-shell excitation edge in an electron energy loss spectroscopy measurement. Transition probabilities sample accessible unoccupied electronic states, so local bonding and orientation affect the edge shape as well as elemental identity.

Comparing two specimens containing the same element can therefore reveal different electronic environments. Interpretation must account for instrumental energy resolution, the incident beam's energy spread and lifetime broadening of the states involved. ^[ref] A small peak difference alone does not uniquely determine one atomic arrangement.

Energy resolution specifies the characteristic energy separation or broadening that an instrument can distinguish, with a stated width convention and measurement geometry. Resolving a phonon in inelastic scattering requires measuring a small energy transfer as well as momentum transfer. It is distinct from the incident beam's central energy.

A photon of energy $E$ fully absorbed in silicon produces on average $E/w$ electron-hole pairs, with variance $FE/w$ rather than necessarily a Poisson variance. Thus ${\sigma }_E=\sqrt{FwE}$ and Gaussian FWHM is $2\sqrt{2\ln 2}{\sigma }_E$. Representative silicon values $w\simeq 3.65$ eV and $F\simeq 0.12$ give about $156$ eV FWHM at $10$ keV. read noise in split-event pixels, trapping and gain errors broaden that limit; optical pixel size alone does not specify the energy resolution.^[ref]

George Ewan helped introduce germanium detectors into experimental nuclear physics, improving the energy resolution of gamma ray measurements. His university credits him with the first nuclear experiment using such a detector. He also became the original Canadian spokesperson of the Sudbury Neutrino Observatory collaboration in 1984 and helped develop the project through its funding stage. These contributions connect instrument development with the organization of a large experiment capable of testing solar neutrino physics.^[ref]

A physicist whose research included nuclear physics and germanium gamma ray detection.^[ref]

Inelastic X-ray scattering measures energy and momentum transferred from X-rays to matter. With sufficiently high energy resolution, it can map a phonon dispersion relation by measuring inelastic peaks at different momentum transfers. Crystal momentum is determined modulo a reciprocal lattice vector.

A Majorana neutrino is its own antiparticle, with a mass term that can violate lepton number by two units. It is not thereby a state of both readily observable helicities with equal weak rates: charged-current vertices still select chirality. In light-neutrino exchange, the needed helicity connection between two beta vertices supplies a neutrino-mass factor.

Ordinary two-neutrino double beta decay is $(A,Z)\to (A,Z+2)+2e^{-}+2{\bar{\nu }}_e$ and conserves lepton number. It can occur when the direct single-beta route is energetically forbidden or highly suppressed but the double change is allowed. neutrinoless double beta decay, $(A,Z)\to (A,Z+2)+2e^{-}$, violates lepton number by two. For the conventional light-Majorana exchange in neutrinoless double beta decay mechanism,Here $U$ is the lepton mixing matrix, $G_{0\nu }$ the phase-space factor in this convention and ${\mathcal{M}}_{0\nu }$ a nuclear matrix element. Complex phases can cancel contributions to the effective Majorana mass; oscillation mass differences alone do not fix it. Other lepton-number-violating mechanisms can also cause the process, so a rate is not a model-independent measurement of $m_{\beta \beta }$.

The experimental signature is a peak in the summed electron energy at the nuclear Q-value, apart from small recoil and energy losses. Two-neutrino decay instead gives a continuous sum-energy radiation spectrum because neutrinos carry away variable energy. Isotope enrichment, large exposure, good energy resolution and radiopure shielding improve sensitivity; gamma backgrounds, radon, surface events and the two-neutrino tail require calibration and sideband controls. Coincidence or track information can test whether the topology is compatible with two electrons. A later concrete example is GERDA's 2013 germanium-76 source-detector search, which used pulse-shape discrimination and background controls rather than identifying any isolated energy deposit as a decay.^[ref]

TODO: a numerical estimate from the missing MINOS spectrum cannot be supplied. On source page 7 the graph is absent: PDF Form /Im3 invokes undefined image /Im1 with no image resources, and the archived bundle has the same defect. In particular, no measured minimum energy survives. For maximal mixing angle, the first minimum of muon neutrino survival obeysA restored spectrum or fitted data are needed to choose $E_{\min }$. This is a relation for the survival minimum; a minimum in the raw event spectrum also depends on flux, cross section and smearing. For context only, accelerator and atmospheric results put the splitting at a few $10^{-3}{\mathrm{eV}}^2$, versus the solar splitting about $7.5\times 10^{-5}{\mathrm{eV}}^2$, roughly thirty times smaller. This known scale is not a reconstruction of the absent graph. ^[ref]

Finite event statistics, hadronic shower energy resolution, the muon scale, near-to-far flux extrapolation, interaction and nuclear modeling, backgrounds and detector calibration limit the precision; the baseline itself is comparatively well known. In a two-flavour vacuum fit the sign of $\Delta m^2$ is inaccessible. Evidence for dominant ${\nu }_{\mu }\to {\nu }_{\tau }$ includes the zenith-angle-dependent atmospheric deficit in Super-Kamiokande, approximately preserved active neutral current rates, small electron-flavour appearance at this scale, atmospheric tau appearance and direct tau appearance in OPERA. The latter had already reported four candidates and $4.2\sigma$ evidence in 2014; a later five-candidate discovery result is not needed for this June 2015 answer. ^[ref] These tests constrain a dominant sterile conversion as well as conversion into electron neutrinos.

Starting with the electron neutrino state in a, the survival probability amplitude isTaking its squared absolute value givesThis derives the supplied survival probability from the evolution operator rather than from a Hamiltonian entry. It also shows why the two written forms agree. For the stipulated phase and $\theta =22.5^{\circ }$, ${\sin }^{2}2\theta =1/2$, so at $L=79\mathrm{km}$,The phase decreases monotonically from $7.9$ to $0.79$ over the open interval $1<E<10$. Differentiating gives $d{P}_{ee}/dE=[7.9/(2E^2)]\sin (15.8/E)$. The interior maxima occur at $7.9/E=n\pi$ and the interior minima at $7.9/E=(n+1/2)\pi$. Enumerating those phases givesThe minimum just above $1\mathrm{MeV}$ must not be lost by rounding the lower endpoint. Neither excluded endpoint is an additional extremum.

For solar neutrinos propagating in the same vacuum model, the vastly longer baseline makes the oscillations extremely fine in energy, with the same bounds $1/2$ and $1$. The figure uses a deliberately schematic enlargement of the oscillation spacing to show those bounds. A $1\mathrm{MeV}$ detector energy resolution averages very many periods, so $⟨{\cos }^2\phi ⟩=1/2$ andThe detector averages event rates weighted by source spectrum, interaction cross section and response; the displayed constant follows when these vary slowly across each tiny oscillation period. Reactor production physically gives electron antineutrinos. In this two-flavor vacuum approximation their survival probability is identical; the positive matter-potential discussion in d concerns solar neutrinos.

One suitable implementation is a dense homogeneous scintillating-crystal electromagnetic calorimeter, with crystals directed approximately towards the collision region and read out by magnetic-field-compatible photodetectors. An incident electron emits bremsstrahlung photons; these produce electron-Positron pairs, which radiate again. This electromagnetic shower develops over successive radiation lengths until the daughter energies approach the critical energy of an electromagnetic shower, below which ionization dominates. The deposited energy creates scintillation light, and calibrated integrated detector signals estimate the incident energy.

A depth of roughly $25$ radiation lengths contains most of a high-energy electromagnetic shower. Small transverse cells resolve its lateral profile, distinguish nearby deposits and permit position reconstruction. Crystal material, optical transmission, gain and electronics must remain stable under the LHC radiation exposure. Fast shaping, timing, segmentation and radiation-tolerant readout permit operation at high collision rates; collision pileup and particle showers from nearby particles require reconstruction and calibration. A dense crystal array offers good intrinsic energy resolution because it measures almost all the shower deposition rather than only the sampled fraction.

Independent fluctuations add as variances. A useful parametrization of the electromagnetic calorimeter energy resolution isThe convention makes $a,b,c$ dimensionless when $E$ is inserted in GeV. The three terms describe different mechanisms.

The stochastic term comes from fluctuations in the electromagnetic shower, scintillation production and detected photoelectron counts. If the number of detected photoelectrons is $N=kE$, Poisson distribution gives ${\sigma }_N=\sqrt{N}$, hence ${\sigma }_E/E=1/\sqrt{kE}\propto E^{-1/2}$. A sampling electromagnetic calorimeter also has fluctuations in the fraction deposited in its active material, contributing to the same approximate scaling. A homogeneous detector suppresses that sampling contribution.

The noise term represents approximately fixed absolute electronics and dark-current noise for a chosen reconstruction window. Additional collision pileup fluctuations can behave similarly over a specified operating range. A nearly constant ${\sigma }_E$ gives ${\sigma }_E/E\propto E^{-1}$.

The constant term describes residual fractional cell-calibration errors, gain drifts, temperature dependence, nonuniform light collection, dead material, and imperfect correction for longitudinal and transverse leakage. These effects often scale approximately with the incident energy, so they approach a constant fractional error. For electrons, upstream bremsstrahlung and conversion of the radiated photons spread the deposit over multiple cells; material mapping and clustering are essential. Leakage and material losses can also produce non-Gaussian tails, so the three-term parametrization describes a resolution core rather than every feature of the response. Noise dominates at low energy, stochastic fluctuations at intermediate energy, and the constant term eventually limits performance.

A compact silicon-tungsten sampling electromagnetic calorimeter can serve as the small-angle luminosity monitor. Alternate dense tungsten absorber plates with finely segmented silicon sensor planes. Incident electrons and Positrons create electromagnetic showers in the absorber; their charged daughters create electron-hole pairs by ionization in depleted silicon. An electric field collects the charge, and the summed calibrated signal estimates shower energy.

Radial and azimuthal sensor segmentation gives precise shower positions and hence polar angles using the measured distance to the interaction point. Multiple layers measure shower development and improve the centroid estimate. Opposite-side coincidence, beam-time synchronization and an energy requirement reject beam backgrounds. Mechanical survey, alignment and edge-response calibration define the small-angle fiducial boundary: high position precision matters especially because the Bhabha scattering rate changes steeply with angle. The monitor's overall energy resolution need not match the central precision crystal detector; it must accurately reconstruct its own fiducial selection and efficiency.

The relevant resolution condition is on the oscillation phase, not solely on which mixing angle is being measured. Since $\Delta \propto L/E$,At the same medium baseline, the fast ${\theta }_{13}$-dependent ripples have phases about $37.5$ times larger than the slow ${\theta }_{12}$ component. Resolving those ripples therefore requires a much better fractional energy resolution. At 47 km, $|{\Delta }_{23}|\simeq 59.5$; avoiding substantial phase smearing requires a resolution appreciably better than $1/59.5\simeq 1.7\%$.

However, a short-baseline ${\theta }_{13}$ experiment near 1.24 km and a ${\theta }_{12}$ experiment near 46.5 km each operate near their relevant first maximum, $\Delta \simeq \pi /2$. Their intrinsic fractional phase-resolution requirements are then comparable. The smaller ${\theta }_{13}$ disappearance amplitude demands careful relative calibration and background control, but it does not by itself impose a universally finer energy resolution. The proposal is conditional on baseline and analysis, rather than a general requirement.

To determine a crystal structure, scatter probes whose wavelength is comparable to atomic spacings, typically $0.1\mathrm{nm}$. X-ray diffraction, neutron diffraction and electron diffraction probe periodic order through coherent elastic scattering. With incident and outgoing wavevectors ${\mathbf{k}}_i,{\mathbf{k}}_f$, define $\mathbf{Q}={\mathbf{k}}_i-{\mathbf{k}}_f$. Elastic scattering has $|{\mathbf{k}}_i|=|{\mathbf{k}}_f|$ and constructive interference occurs when $\mathbf{Q}=\mathbf{G}$, a reciprocal lattice vector. The equivalent Bragg's law iswhere the detector scattering angle is $2\theta$, not $\theta$. Peak positions reveal plane spacings and the unit cell; systematic absent reflections constrain centring and the atomic basis.

For X-ray diffraction, a laboratory tube or a synchrotron supplies a collimated beam. In a monochromatic single-crystal experiment a goniometer rotates the specimen and an area detector records reflection positions and integrated wave intensities. A powder experiment uses many randomly oriented crystallites; the reflections form cones recorded as rings or peaks versus $2\theta$, permitting indexing and determination of lattice constants without selecting one crystallite. For a cubic cell, for example, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. The white-beam Laue method is especially useful for crystal orientation.

The crystal basis structure factor isafter geometric and thermal corrections, including the Debye–Waller factor. For X-rays, $f_j$ describes scattering from the atomic electron density. Thus positions provide lattice geometry while relative wave intensities constrain atomic positions and species. Wave intensities do not directly give the phases of $F_{\mathbf{G}}$: structure refinement or other constraints must address the phase problem.

Neutron diffraction uses a reactor or spallation source, wavelength selection and a detector bank or time-of-flight detection. Neutrons interact with nuclei through nuclear scattering lengths that vary between elements and isotopes rather than simply with atomic number. This makes them useful for light atoms, isotope contrast and distinguishing some neighbours in the periodic table; their magnetic interaction also probes magnetic order. Electron diffraction in a transmission electron microscope gives strong signals from very small thin crystallites, although strong multiple scattering often complicates quantitative wave intensity analysis. These techniques complement one another.

A phonon dispersion relation requires both momentum and energy transfer, so elastic diffraction alone is insufficient. In coherent inelastic neutron scattering, the measured transfers for phonon creation areHere $\mathbf{q}$ is the phonon wavevector reduced into a Brillouin zone, and $s$ labels an acoustic phonon branch or optical phonon branch. Negative energy transfer corresponds to annihilating an already occupied phonon. For neutrons $E={\hslash }^2{k}^2/(2m_n)$, so wavelength and energy selection are linked. Typical thermal-neutron energies of a few to tens of millielectronvolts ($1\mathrm{meV}=10^{-3}$ electronvolts) are well matched to many lattice vibrations.

A triple-axis spectrometer uses a monochromator crystal to select ${\mathbf{k}}_i$, the sample as the second scattering axis, and an analyser crystal to select ${\mathbf{k}}_f$ before detection. Rotate these elements and the specimen to hold $\mathbf{Q}$ fixed while scanning $E_i-E_f$; inelastic peaks locate ${\omega }_s(\mathbf{q})$. Repeat at different $\mathbf{q}$ along symmetry directions to map the phonon dispersion relation. The dynamic structure factor governs peak weights, including the selection of phonon polarization by the scattering geometry. A time-of-flight spectrometer instead infers outgoing neutron energy from travel time over a known flight path and records many directions simultaneously.

Inelastic X-ray scattering can also resolve phonons when very high energy resolution is available. Optical Raman scattering and Brillouin scattering measure vibrations near the zone centre because photon momentum transfers are small compared with a typical reciprocal lattice spacing; they complement, rather than generally map the entire zone like, momentum-resolved neutron measurements. Ultrasound measurements give the long-wavelength sound speed, the initial slope $d\omega /dq$ of an acoustic phonon branch.

Spectroscopy investigates how a system emits, absorbs or scatters signals as a function of frequency, wavelength or energy. A spectrum can identify transition energies, response strengths and line shapes. In atomic spectroscopy, a photon connects two energy levels through $h\nu =|E_f-E_i|$; in inelastic scattering, both energy and momentum transfers can matter.

A simple optical experiment passes source light through a sample, disperses the transmitted light and compares it with a reference. An absorption line identifies reduced transmission over a spectral interval, but its depth also depends on the sample amount and instrumental response. Wavelength calibration and energy resolution distinguish an intrinsic feature from a shifted or broadened recording. The inference therefore uses a response model as well as a list of line positions.