Physics

The study of atomic nuclei, their structure, reactions and interactions. Nuclear forces between nucleons and nuclear binding energies connect it to the low-energy consequences of Quantum chromodynamics.

An energetically allowed weak transition converts a neutron into a proton, emitting an electron and an electron antineutrino. Free neutron decay is an example; in nuclei, the nuclear binding energy determines whether the transition is possible.

Fusion combines nuclear reactants into heavier compound nuclei or products. A net decrease in rest mass releases energy; heavier fusion reactions can instead require energy. Stellar nuclear fusion uses exothermic networks to power stars, and quantum tunnelling through the Coulomb barrier permits many charged-particle reactions at stellar temperatures.

Repulsion between positively charged nuclei creates this electrostatic energy barrier outside the short-range nuclear interaction. Quantum tunnelling allows reaction at thermal energies below its classical height and contributes the penetration factor in the Gamow peak.

A sufficiently energetic photon can break up a nucleus, consuming energy for an endothermic breakup. In neon burning and silicon burning, such steps combine with captures to form a larger reaction network whose net energy balance can be positive.

An energetically allowed weak nuclear transition converts a proton into a neutron, emitting a positron and electron neutrino. Positron annihilation deposits additional electromagnetic energy, while the neutrino can escape. These transitions are steps of the CNO cycle and the pp III branch; an isolated free proton cannot undergo this decay.

A nucleus can capture an electron when energetically allowed, converting a proton into a neutron and emitting an electron neutrino. Beryllium-7 electron capture participates in one of the proton-proton chain branches. Stellar capture rates depend on ionization, mass density and electron phase space.

The residual strong interaction between nucleons. Its long-distance part involves pion exchange and depends on spin and isospin. It is distinct from directly exchanging gluons between color-singlet nucleons.

The central system of protons and neutrons in an atom. Its mass number $A$ is its baryon number in ordinary nuclear states; $Z$ is the proton number and $N$ the neutron number.

The helium-four atomic nucleus, containing two protons and two neutrons. Its ground-state spin and isospin are zero. A quantized cubic four-Skyrmion supplies a corresponding nuclear state in the soliton description.

The bound atomic nucleus containing one proton and one neutron. Its ground-state spin is one and its isospin is zero. The appropriately quantized toroidal two-Skyrmion models these quantum numbers.

The positive energy needed to separate a bound atomic nucleus into its constituent protons and neutrons. It is the mass defect times $c^2$. In a Skyrmion model of a nucleus, both separated nucleon masses and the multi-soliton state require a consistent quantization and calibration.

A semiconductor has an electronic band gap and a conductivity strongly affected by temperature, impurities and carrier injection. Optical absorption can create mobile electrons and holes, enabling a photodiode.

Mercury cadmium telluride is a semiconductor whose composition tunes its band gap and hence its long-wavelength absorption cutoff. It is widely used as the absorbing material in infrared detector arrays.

Sonar uses sound propagation and reflected signals to detect objects or measure distances. Upward-looking sonar can record sea-ice draft along a moving platform's track; beam width, return selection and processing affect the inferred profile.

An acoustic waveguide confines sound transversely while allowing longitudinal propagation. Rigid walls impose a homogeneous normal-velocity condition, producing Neumann transverse modes. Each transverse eigenvalue gives a longitudinal wave number: modes below their cutoff decay evanescently, while cut-on modes carry outgoing sound. An open semi-infinite guide can be studied with the Wiener-Hopf method.

For an even field between rigid plates at $y=\pm b$, the transverse modes are $\cos (n\pi y/b)$, with longitudinal wave numbers $k_n=\sqrt{k_0^2-(n\pi /b{)}^2}$. With time dependence $e^{i\omega t}$, outgoing modes into $x>0$ use $e^{-i{k}_{n}x}$: choose positive real $k_n$ for cut-on modes and negative imaginary $k_n$ for cut-off modes. A lower-half-plane Fourier inversion extracts these modes by residues. Their decaying components are evanescent waves, not backward-propagating waves. At an exact cutoff one takes the radiation continuation before its limit.

An acoustic analogy rearranges fluid conservation equations into a chosen linear acoustic propagation operator plus effective forcing. The rearrangement can be exact before source approximations are made. A term classified as forcing can also contain propagation effects omitted from the reference operator; this produces propagation-source ambiguity in an acoustic analogy. The Lighthill acoustic analogy is a familiar homogeneous reference-medium example.

The distinction between generation and propagation depends partly on the reference operator chosen in an acoustic analogy. For example, a homogeneous-Laplacian operator with spatially varying reference sound speed omits the first-gradient terms of the stratified acoustic pressure equation. These terms then reappear as effective forcing, even for a freely propagating small disturbance in a static medium. A right-hand-side term is therefore not automatically an independent physical noise source.

Eliminating momentum between the continuity equation and conservation of momentum gives ${\rho }_{tt}={\partial }_i{\partial }_j(\rho u_i{u}_j-{\sigma }_{ij})+{\nabla }^2(p+\chi )$ when the force density is $-\nabla \chi$. For $q=p-(P-\chi )$ and time-independent reference fields,No energy equation is needed for this identity. The arbitrary reference fields should match the stationary medium where the outgoing sound is evaluated.

Small-height Dirichlet scattering expands the acoustic boundary condition about a mean plane for a fixed regular surface profile. The zeroth-order field includes the flat reflection; higher orders solve outgoing Helmholtz equation problems with induced boundary data. Small $|kh|$ controls the incident-field Taylor expansion, but rapidly varying roughness can require additional control of normal derivatives and spectral moments.

Through second order in height, a normally incident acoustic plane wave has the displayed mean reflected amplitude for a Dirichlet boundary. The propagating part of the power spectrum of surface height contributes a positive real correction to the negative flat reflection coefficient, while the evanescent part contributes an imaginary correction. The mean linear rough correction is zero.

The quadratic coherent correction weights the power spectrum of surface height by $\beta (q)$, so equal root mean square heights can give different mean fields. Only an additional long-spatial-scale approximation, with spectrum concentrated at $|q|\ll k$, reduces it to $2k^2{\sigma }^2$. Second-order surface scattering therefore contains information absent from first-order coherent reflection.

For a real zero-mean stationary height process, its power spectrum of surface height is the Fourier transform of the height covariance function. It is even and nonnegative, and ${\sigma }^2=(2\pi {)}^{-1}\int S_h(q)dq$. More general stationary processes have a spectral measure rather than a density. Weighted spectral moments control the existence of boundary derivatives and second-order mean scattering corrections.

For normal incidence on a Dirichlet rough surface, the scattered trace on the moving boundary is exactly $-e^{-ikh(x)}$. Its trace on the mean plane is instead $-1+2ikh+2kh\mathcal{B}h+\cdots$. The two traces are related by a normal Taylor expansion and must not be identified. Their ensemble averages can therefore have different quadratic coefficients without inconsistency.

For normal plane-wave incidence, the second-order Dirichlet boundary condition gives ${\psi }_s^{[2]}=-h{\partial }_z{\psi }_s^{[1]}$ at the mean plane, since ${\partial }_z^2{\psi }^{[0]}$ vanishes there. The Dirichlet-to-Neumann map for a Helmholtz half-space then yields $2kh\mathcal{B}h$. This nonlocal expression is not generally $2k^2{h}^2$; that local limit needs roughness varying slowly compared with the wavelength.

Taylor expansion of the Dirichlet boundary condition gives the first-order boundary trace $-h{\partial }_z{\psi }^{[0]}$ at the mean plane. The outgoing angular spectrum then propagates it. A zero-mean surface gives zero mean for this linear rough correction, while the complete scattered field still contains its nonzero flat reflection.

The outgoing angular spectrum propagates prescribed scalar boundary data into a half-space. For $e^{-i\omega t}$ time dependence, $\beta (q)=\sqrt{k^2-q^2}$ has nonnegative real part on propagating modes and nonnegative imaginary part on evanescent modes. Its trace at the boundary is $g$, and each mode solves the Helmholtz equation. The spectral representation can be interpreted through distributions for nondecaying plane-wave data.

A progressive small-amplitude pressure perturbation in a uniform inviscid fluid depends on one travelling coordinate. Linear momentum balance gives its velocity parallel to propagation and proportional to pressure with impedance ${\rho }_0{c}_0$.

Recover a compactly supported acoustic source $Q$ from measurements of its outgoing field, such as its far-field pattern. The Helmholtz equation forward problem is specified with the Sommerfeld radiation condition. At a single frequency, the source-to-far-field operator at fixed frequency has invisible fixed-frequency nonradiating sources, and its compactness makes recovery of the visible component unstable.

A source is nonradiating at frequency $k$ if its outgoing field has zero far-field pattern. For every $\chi \in C_c^{\infty }(A)$, the source $Q=(\Delta +k^2)\chi$ has outgoing field $\psi =\chi$ and is invisible outside $A$. Integration by parts shows that its Fourier transform vanishes on $|\mathbf{\xi }|=k$, so it belongs to the kernel of a linear map defined by the source-to-far-field operator at fixed frequency. This family proves nonuniqueness independently of the instability of the inverse.

For $(\Delta +k^2)\psi =Q$, the outgoing Green function for the three-dimensional Helmholtz equation yields the far-field patternThe sign corresponds to the Green function satisfying $(\Delta +k^2)G=-\delta$. This Hilbert-Schmidt operator samples the source Fourier transform on the sphere $|\mathbf{\xi }|=k$; for a ball of radius $r_0$, its squared Hilbert-Schmidt norm is $r_0^3/3$.

The adjoint source-to-far-field operator composed with the source-to-far-field operator at fixed frequency has volume kernel $K(\mathbf{r},{\mathbf{r}}^{\prime })=(4\pi {)}^{-1}\mathrm{sinc}(k|\mathbf{r}-{\mathbf{r}}^{\prime }|)$, where $\mathrm{sinc}(s)=\sin (s)/s$ and $\mathrm{sinc}(0)=1$. This follows from ${\int }_{S_1}{e}^{ik\mathbf{\theta }\cdot \mathbf{d}}dS=4\pi \mathrm{sinc}(k|\mathbf{d}|)$. Its smoothing action helps explain the unstable normal equation for a linear inverse problem.

With complex inner products linear in the first argument, the adjoint operator of the source-to-far-field operator at fixed frequency isIt is a normalized Herglotz wave function. Complex conjugation reverses the phase sign, while retaining the real factor $-1/(4\pi )$.

Multiple scattering can enlarge the angular diversity reaching a receiver array. Reversing the paths then makes the array act like a larger aperture, improving the focus in appropriate regimes of wave propagation in a random medium.

For a linear operator $H$ mapping transmitted signals to recorded signals, the positive operator $H^{\ast }H$ describes adjoint back-propagation after recording. In a reciprocal array, physical time reversal acoustics implements this operation up to a final complex conjugation and the chosen signal conventions.

The DORT method uses eigenvectors of a time reversal operator to focus on individual point scatterers. A separate nonzero eigenvalue corresponds to each scatterer in the ideal weak-scattering, well-resolved limit; coherent overlap and multiple scattering can change this correspondence.

A collision can transfer kinetic energy to an atom or ion and excite a bound Electron. Subsequent radiative de-excitation removes that energy from an optically thin gas. The thermal excitation rate is strongly reduced when the transition energy exceeds $k_{B}T$, giving the threshold behavior of atomic line cooling.

Photoionization removes a bound electron when an atom or ion absorbs a sufficiently energetic photon. The threshold is its ionization energy. The excess photon energy becomes kinetic energy of the products; subsequent collisions can heat the gas or cause secondary ionizations. A photoionization rate and a photoionization equilibrium are distinct concepts: equilibrium also requires specified recombination and other reaction rates.

Absorption of an ionizing photon leaves energy in a photoelectron above the binding energy. Thermalization supplies heat to the gas. Photoionization heating competes with the astrophysical cooling function, can establish thermal equilibrium, and can raise the pressure enough to inhibit baryon accretion into shallow dark-matter haloes.

A free Electron can be captured by an ion while a photon carries away binding and kinetic energy. This radiative recombination process contributes free-bound radiation to plasma cooling and balances collisional ionization in a collisional-equilibrium plasma.

In collisional ionization, an energetic collision frees a bound Electron. The process consumes kinetic energy and changes the Electron and ion populations controlling plasma cooling. In a thermal gas its rate competes with recombination and depends strongly on the ionization threshold relative to temperature.

In collisional ionization equilibrium, collision-driven ionization and recombination balance for the populated ion stages, determining their relative abundances at a given temperature. The resulting cooling curve includes the loss of line-emitting bound states as temperature increases. Rapidly cooling gas or external photoionization can invalidate this equilibrium assumption.

A multipole moment is a coefficient in the far-field angular expansion of a localized source. Successive moments describe progressively finer departures from spherical symmetry.

Compactification makes some spatial dimensions compact, so modes along them acquire discrete momenta. At energies below the first nonzero compact-mode mass, a dimensional reduction to the zero modes gives a lower-dimensional effective field theory. The compact volume fixes the gravitational kinetic coefficient in an unwarped product background.

For an unwarped product metric with fixed compact volume $V_n$, integrating the higher-dimensional Einstein-Hilbert action gives $M_{\mathrm{Pl}}^2=M_{\ast }^{n+2}{V}_n$ when the action coefficient is $M_D^{D-2}/2$. The reduced Planck mass convention must be distinguished from the unreduced one, and stabilizing the internal metric is a separate dynamical issue.

A Kaluza-Klein mode is a coefficient in a field expansion in eigenfunctions on compact extra dimensions. On a circle of circumference $L$, momentum is $2\pi \ell /L$, and a higher-dimensional massless field has lower-dimensional squared mass $(2\pi \ell /L{)}^2$. Nonzero modes give exponentially suppressed static-potential corrections at distances much larger than $L$.

On three noncompact spatial dimensions times a circle of circumference $L$, the method of images gives ${\sum }_{j\in \mathbb{Z}}[r^2+j^2{L}^2{]}^{-1}=\pi \coth (\pi r/L)/(Lr)$ at equal compact coordinates. This interpolates from $1/r^2$ at short distances to $\pi /(Lr)$ at long distances. A gravitational circle-radius scalar can change the long-range coefficient if it remains unstabilized.

Dimensional reduction retains fields independent of chosen internal coordinates. In a flat-torus zero-mode reduction, a metric supplies a metric, internal-component vectors and internal metric scalars; a massless p-form gauge field supplies lower-rank fields with internal indices. Fluxes, twists or projections can alter the spectrum and preserved supercharges.

A spacetime diagram plots one spatial coordinate against time. Curves on it represent trajectories, wavefronts, or characteristics, and their slopes encode propagation speeds.

A continuum limit replaces lattice variables that vary slowly over many lattice spacings by smooth fields. Lattice sums become spatial integrals and finite differences become derivatives, while short-distance information survives in effective coefficients and higher-derivative corrections.

Added mass is the effective inertia contributed by fluid accelerated together with an immersed body.

A sphere of volume $V$ translating through unbounded inviscid incompressible fluid has fluid kinetic energyIts added mass is therefore one half of the displaced fluid mass.

A sphere of density ${\rho }_s$ falling in inviscid fluid of density $\rho$ has effective inertia $({\rho }_s+\rho /2)V$ and net weight $({\rho }_s-\rho )Vg$. Hence

The Compton wavelength of a particle of mass $m$ is ${\lambda }_C=h/(mc)$.

For a photon scattered through angle $\theta$ by an initially stationary electron,

For a charged Dirac particle, incoming momenta $p,k$ and outgoing momenta $p^{\prime },k^{\prime }$ obey $p+k=p^{\prime }+k^{\prime }$. The two QED diagrams have internal fermion momenta $p+k$ and $p-k^{\prime }$, giving $s$ and $u$ channels. The amplitude is proportional to $\bar{u}(p^{\prime })[/{ϵ}^{\prime \ast }R(p+k)/ϵ+/ϵR(p-k^{\prime })/{ϵ}^{\prime \ast }]u(p)$, where $R(q)=(/q+m)/(q^2-m^2+i0)$. It is related to particle-antiparticle two-photon annihilation by crossing symmetry, with appropriate changes to external wavefunctions and conjugations.

Double Compton scattering is $e^{-}+\gamma \to e^{-}+\gamma +\gamma$, together with its inverse. The forward reaction has one incoming and two outgoing photons, so it changes their number and can source the divergence of the photon number current. Ordinary Compton scattering has one incoming and one outgoing photon and conserves photon number. Forward and inverse rates can balance without either reaction individually conserving it.

The physical dimension of a quantity records its powers of chosen base dimensions, independently of units. For example, velocity has dimension $L{T}^{-1}$ and force has dimension $ML{T}^{-2}$. Dimensional analysis compares these expressions in physical equations.

Natural units choose units so that selected fundamental constants have numerical value one. In relativistic quantum field theory the standard choice is $\hslash =c=1$: energy, mass and inverse length then have the same units. This is a convention, not a physical change of the constants; other natural-unit systems may also normalize other constants.

For uniform rotation, the Newtonian gravitational potential plus the centrifugal potential generates the non-Coriolis part of the equation of motion in a rotating frame. Stationary particles obey $\nabla {\Phi }_{\mathrm{eff}}=0$. The Coriolis force changes perturbation dynamics, so curvature of this potential alone is not a general rotating-frame stability criterion.

Effective gravity in a uniformly rotating frame is minus the gradient of the sum of the gravitational and centrifugal potentials. A fluid surface in rotational equilibrium is an equipotential surface and is therefore normal to this effective gravity.

After reducing conserved cyclic momenta, an equilibrium at a strict local minimum of the resulting effective potential is stable in the reduced one-degree-of-freedom dynamics.

Motion-dependent induced current can produce a magnetic force opposing motion and dissipating energy resistively.

An ideal superconducting loop supports persistent current that preserves linked magnetic flux when resistance vanishes.

A uniformly charged spherical shell produces zero field inside and the field of a point charge outside.

The Lyman-alpha line is the hydrogen $2p\leftrightarrow 1s$ electric-dipole transition, with photon energy about $10.2\mathrm{eV}$ and vacuum wavelength about $1216$ angstroms. Its emission cools collisionally excited gas, while absorption traces neutral hydrogen along a background-source sightline.

A neutral hydrogen atom absorbs a photon near the Lyman-alpha resonance by excitation from $1s$ to $2p$. Absorption lines in a quasar spectrum can therefore locate hydrogen along the sightline by their redshift, subject to peculiar-velocity and line-profile effects.

For a population with comoving number density $n_c$ and proper interception area ${\sigma }_p$, the absorption incidence is the proper density multiplied by cross-section and proper light-path length $cdz/[H(z)(1+z)]$. The displayed relation assumes one counted absorption system per interception. Completeness, covering fraction, evolution and the distinction between observed and rest-frame wavelength separation must be stated in an inference from a spectrum.

A circular, opaque, geometrically thin disk has projected area $\pi R^2|\cos i|$. For isotropically oriented normals, $|\cos i|$ is uniform on $[0,1]$, so the mean interception area is half the face-on area. A nonunit covering fraction multiplies this cross-section. Finite thickness, orientation-dependent absorption thresholds and incomplete coverage can change it; this is an average over randomly placed disks, not the inclination distribution conditional on an observed interception.

Radiative decay from the hydrogen $2p$ level to $1s$ produces a Lyman-alpha photon. In an optically thin collisionally excited gas, escape of this photon removes the excitation energy deposited by an Electron collision. The excitation threshold is responsible for the rapid temperature sensitivity of this cooling channel.

The Bohr model assumes circular electron orbits with quantized orbital angular momentum $N\hslash$. It reproduces the hydrogen energy levels, but wavefunctions in quantum mechanics describe spatial probability distributions rather than definite trajectories. In particular, spherically symmetric Coulomb bound states have zero orbital angular momentum.

For a hydrogenic bound state, the principal quantum number is the positive integer $n$ that determines the unperturbed energy. Its allowed orbital quantum numbers are $\ell =0,\dots ,n-1$.

The ionization energy is the minimum energy needed to remove an electron from an isolated atom or molecule in a specified state. Ground-state hydrogen has ionization energy $13.6\mathrm{eV}$.

For two particles of masses $m_1,m_2$ interacting through a central Coulomb potential, separation of the centre of mass leaves the relative-coordinate Schrödinger equation with

For reduced mass $\mu$, the Coulomb length scale is

A no-radial-node Coulomb eigenstate has principal quantum number $n=l+1$ and radial factorIts energy is $-e^2/[8\pi {ϵ}_{0}a(l+1{)}^2]$.

For $R(r)\propto r^l{e}^{-r/[a(l+1)]}$, the radial probability density is proportional to $r^{2l+2}{e}^{-2r/[a(l+1)]}$ and

Neglecting spin, bound hydrogen eigenstates are labelled $|n,\ell ,m⟩$, where $\ell =0,\dots ,n-1$ and $m=-\ell ,\dots ,\ell$. Their unperturbed energy depends only on $n$.

Neglecting spin, the hydrogen level with principal quantum number $n$ has degeneracy

For the $z$ component of position,requires $\Delta m=0$ and $\Delta \ell =\pm 1$. The first condition follows from $[L_z,z]=0$; the second follows from the vector-operator angular-momentum relation together with odd parity.

The even, spherically symmetric hydrogen ground state has no permanent electric dipole, so a uniform electric field gives no first-order energy shift. Odd-parity excited states mix in at second order, yielding a negative shift quadratic in the field.

The $n=2$ eigenspace has basis

A perturbation $g{L}^2$ preserves the hydrogen eigenbasis and shifts $|n,\ell ,m⟩$ byIn the first excited eigenspace, the $2s$ state is unshifted and the three $2p$ states receive the common shift $2g{\hslash }^2$.

The operators $X_3$ and $P_3$ commute with $L_3$, so they preserve $m$. Both are odd under parity, so they connect only states of opposite orbital parity. Within the $n=2$ eigenspace, only the pair $|2,0,0⟩$ and $|2,1,0⟩$ can therefore have nonzero matrix elements.