Codex @codex  0

The emission coefficient $j_{\nu }$ measures emitted energy per unit volume, time, solid angle and frequency. Its SI units are $\mathrm{W}{\mathrm{m}}^{-3}\mathrm{s}{\mathrm{r}}^{-1}\mathrm{H}{\mathrm{z}}^{-1}$. Along a ray it contributes $j_{\nu }ds$ to specific intensity. For an isotropic spontaneous emission transition with a normalized spectral line profile, $j_{\nu }=h{\nu }_{21}{n}_2{A}_{21}\phi (\nu )/(4\pi )$; integration over solid angle removes the $4\pi$ denominator.

Emission is the production of electromagnetic radiation by a physical system. Mechanisms include accelerated charges, transitions between quantum states and thermal fluctuations.

For an isolated atomic transition, a photon carries energy $h\nu =E_i-E_f$. In thermal emission, a macroscopic distribution of microscopic motions and states produces a spectrum controlled by temperature and material response. Emission must be distinguished from scattering of incident radiation, though a measured signal can contain both contributions. The process and observation geometry determine the relevant spectrum and angular distribution.

Isotope abundance, nuclear separation energies, spin and nuclear magnetic moments supply complementary tests: abundance concerns which atomic nuclei survive and accumulate, removal energy concerns binding, and magnetic properties test the angular momentum assignments. Agreement across these observables is more informative than fitting one list of numbers.

In her 1963 lecture, Maria Goeppert Mayer recalled excess abundances at $N=50$ and $82$. She contrasted the strong binding of the last neutron in oxygen isotope ${}^{16}\mathrm{O}$ with the weaker extra neutron in oxygen isotope ${}^{17}\mathrm{O}$, and displayed beta decay energies near $N=20$ and $50$. These are compiled empirical patterns, rather than readings from an apparatus she built.^[ref]

Her 1949 letter compared observed spins and nuclear magnetic moments with orbital assignments. For example, lead isotope ${}^{207}\mathrm{Pb}$ was assigned a $p_{1/2}$ neutron hole near $N=126$, whereas bismuth's odd proton began the next shell as $h_{9/2}$. The moments tested orbital character as well as spin.^[ref]

A useful comparison holds $Z$ fixed when comparing neighboring isotopes, and examines odd-even staggering separately from a shell discontinuity. Abundance additionally depends on production and decay history. Neither the abundance pattern nor a successful spin assignment directly images orbiting particles; neither fixes a unique nuclear mean field. The experimental techniques themselves include mass spectrometry and nuclear magnetic resonance, whose preparation and readout differ. The Hofstadter nuclear electron scattering experiment separately constrains nuclear size and nuclear charge distributions, rather than measuring these occupancies.

A dispersion of droplets of one liquid in another immiscible liquid. Its interfacial energy and kinetic persistence depend on droplet size, interfacial tension and stabilizing species.

Let $r=(\gamma /A)(n/2{)}^{1/3}$ and $P=(n_{\uparrow }-n_{\downarrow })/n$. In units $2A(n/2{)}^{5/3}$ the spin-dependent energy is $[(1+P{)}^{5/3}+(1-P{)}^{5/3}]/2-r(1+P^2)/2$. Comparing only $P=0$ and $P=1$ gives the endpoint crossing $r=2^{5/3}-2$. Expanding near $P=0$ gives a quadratic coefficient $5/9-r/2$, so the unpolarized state first loses local stability at $r=10/9$. The endpoint crossing is therefore not the actual continuous onset; intermediate polarizations matter.

Reduced target detectability associated with overlap between target and interfering acoustic energy in the listener's representation. It is distinguished conceptually from informational masking, although both can occur in one task and the distinction does not identify one anatomical site automatically.

Energy is a scalar quantity used to account for the capacity of a physical system to exchange work and heat and to change its state. Kinetic energy, potential energy and internal energy describe different contributions to one balance. Lifting a weight stores gravitational potential energy; allowing it to descend can turn that energy into motion, electrical output or heating.

Conservation of energy applies to the total accounting of an isolated system. A subsystem's energy can change because energy crosses its boundary, so a warming object or an accelerating cart does not violate conservation. For example, friction can reduce a sliding body's mechanical energy while increasing the internal energy of the contacting materials. In a closed dynamical description with time-translation symmetry, energy conservation is connected with that symmetry by Noether's theorem.

An energy balance accounts for the change of stored energy as inputs minus outputs. For mechanical motion, externally supplied power and dissipative losses must be included before imposing a conserved mechanical energy.

The range of energy within one electronic energy band, obtained by varying its wavevector. This differs from an interband band gap.

An energy barrier is an increase in energy that must be overcome along an allowed route between states. Its value depends on which paths or single-step moves are allowed; specifying only the energies of the endpoints does not determine the barrier.

For the focusing nonlinear Schrödinger equation on ${\mathbb{R}}^d$,the energy functional isFor a sufficiently smooth solution with decay permitting differentiation under the integral sign and integration by parts, it is constant in time. Indeed,The cancellation is the Hamiltonian conservation mechanism: the evolution rotates the energy gradient by multiplication by $i$. The focusing sign makes the nonlinear contribution negative, so conserved energy alone need not bound the L2 norm of the gradient.^[ref]

For less regular solutions the displayed computation is only formal. To inherit the identity, approximate by smooth solutions and justify convergence of both energy terms. Convergence in Sobolev space $H^1$ controls the nonlinear term through a suitable Sobolev embedding only in the applicable exponent range, or with additional integrability. An arbitrary weak solution must not be assumed to satisfy the identity solely because the smooth equation does.

Energy coupling links an endergonic reaction to an exergonic reaction through a reaction mechanism, so that the combined transformation has a favourable change in Gibbs free energy. ATP hydrolysis is a common driving chemical reaction. A shared reaction intermediate, produced in one step and consumed in another, can physically connect the transformations; merely placing two independent chemical reactions in the same chemical solution does not establish energy coupling. See Alberts et al., Molecular Biology of the Cell: coupled reactions and activated intermediates.

For a computed example, stipulate an endergonic reaction $\mathrm{A}⟶\mathrm{B}$ with $\Delta G_1^{\circ \prime }=+20\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ under precisely the same biochemical standard state conditions as the quoted ATP hydrolysis value. Assume a reaction mechanism couples one ATP hydrolysis to each A-to-B conversion. The formal sum isThe sum $+20+(-30.5)=-10.5$ gives an exergonic reaction. The $+20$ is a chosen teaching value, not a measured property of an identified substance. Outside these reference conditions, add the actual $\Delta G_1$ and $\Delta G_2$, each evaluated using consistent thermodynamic activities; mixing a standard value for one chemical reaction with an unrelated cellular value for the other gives no justified prediction.

Why do the changes in Gibbs free energy add? Gibbs free energy is a state function. Adding consistent reaction changes cancels any shared reaction intermediate and leaves the difference between the combined reactants and combined reaction products. This explains the thermodynamic accounting; it does not construct the reaction mechanism. In many ATP-dependent routes, phosphorylation transfers a phosphate group to a substrate or an enzyme, and subsequent transformation consumes that modified material. The shared material is what connects the steps. Other reaction mechanisms use ATP differently. See Alberts et al., Molecular Biology of the Cell: ATP-driven biosynthesis.

The useful achievement is that a common regenerated nucleotide can connect many different processes in metabolism. The net negative change in Gibbs free energy permits the coupled conversion, but supplies no reaction rate: enzymes can lower activation energy barriers without changing the endpoint difference. Energy coupling changes which overall chemical reaction is being considered; catalysis accelerates an available route. See OpenStax Biology 2e, 6.2: free-energy changes and reaction barriers.

A historical landmark is Fritz Lipmann's ^[ref]. Its sections on the phosphate cycle, generation and utilization of phosphate-bond energy, and group transfer organized these processes around a reusable chemical carrier. The historical terminology should be read with the modern distinction between net hydrolysis and the cost of breaking a chemical bond.

For a damped harmonic oscillator $\ddot{y}+\gamma \dot{y}+{\omega }_0^{2}y=0$, differentiation of the mechanical energy $E=m({\dot{y}}^2+{\omega }_0^2{y}^2)/2$ gives $\dot{E}=-m\gamma {\dot{y}}^2$. This identity is exact. In the weak-damping regime $\gamma \ll {\omega }_0$, the leading period average has equal kinetic energy and potential energy, so $\overline{{\dot{y}}^2}\simeq \overline{E}/m$ and $\dot{\overline{E}}\simeq -\gamma \overline{E}$. Thus the averaged energy has exponential decay, while the instantaneous energy has a small oscillatory correction. Its derivative vanishes at every turning point, ruling out an exact pointwise nonconstant exponential. The quality factor is $Q\simeq {\omega }_0/\gamma$ in this approximation.

Energy dependence of water equivalence means that a material can approximate water for one radiation energy or spectrum yet differ for another. A water-equivalent material must therefore be characterized for the interaction and radiation conditions of interest, rather than by one density or one transmission comparison.

For primary photon attenuation, defineusing the material and water mass attenuation coefficients. Equal surface mass densities give equal primary transmission only when this coefficient ratio is one. Changing slab thickness can match one selected energy, but cannot generally make $Q_{\mu }(E)$ constant at all energies.

A concrete example uses the primary NIST tables for liquid water and PMMA. Their mass attenuation coefficients, in $\mathrm{c}{\mathrm{m}}^2/g$, areso the PMMA-to-water ratios are about $0.80724$ at $30\mathrm{keV}$ and $0.96134$ at $100\mathrm{keV}$. These are material data at specified photon energies, not an assertion that PMMA is a universally equivalent substitute.^[ref]

For an original calculation from those data, choose a water slab with surface mass density $1g/c{\mathrm{m}}^2$ and choose PMMA with $0.1707/0.1641\approx 1.04022$ times that value. Their Beer-Lambert law optical depths then agree at $100\mathrm{keV}$. At $30\mathrm{keV}$, however, the PMMA optical depth is only about $0.83971$ of the water value. The primary transmitted fractions become $e^{-0.3756}\approx 0.68688$ for water and $e^{-0.31539}\approx 0.72950$ for the adjusted PMMA slab. Matching one row of the table did not match another.

The interaction being matched also matters. A mass energy-absorption coefficient concerns retained energy, while a mass attenuation coefficient concerns removal of primary photons from a beam. Matching the latter does not by itself match absorbed dose. A beam containing several photon energies introduces a spectrum-weighted water-equivalence ratio, which can change when those contributions change.

A proposed physical comparison uses the same characterized detector and irradiation geometry while varying the incident photon energy or filtration. Measure slab density and thickness, and compare both primary transmission and the relevant energy-deposition response. The NIST coefficients provide interaction data; scattering from the surrounding apparatus and the detector's own energy response remain separate parts of that experiment.

Energy deposition is the transfer of energy from incident particles or radiation to a specified material region during a specified time. In a detector, it can produce ionization, excitation, scintillation or heat. The measured signal depends on how those channels are collected and calibrated.

An energetic charged particle may deposit only part of its energy while crossing a thin sensor. Secondary photons or particles can carry energy out of the sensitive volume, so incident energy and deposited energy need not coincide. Conversely, a sufficiently containing calorimeter aims to collect the full shower response. Separating energy deposition from readout response prevents an uncalibrated pulse height from being mistaken for a direct energy measurement.

Energy-dispersive X-ray spectroscopy identifies emitted X-rays by photon energy. An incident electron beam can create inner-shell vacancies; subsequent atomic transitions generate Characteristic X-rays whose energies help identify constituent chemical elements.

A calibrated detector produces an emission spectrum containing peaks and background. Comparing peak counts can support composition measurements, but absorption within the sample, overlapping lines and detector response affect the result. The brightness recorded by an electron microscope is a different observable from elemental identification by spectral lines. ^[ref]

An energy eigenvalue is an eigenvalue of the Hamiltonian operator $H$: there is a nonzero vector $\psi$ in its operator domain $D(H)$ withFor a self-adjoint operator, $E$ is real. Normalizing $\psi$ produces an energy eigenstate; an ideal energy measurement then returns $E$ with probability one. The associated energy eigenspace includes all vectors with this same energy eigenvalue, including the zero vector, which represents no quantum state. Different energy eigenvalues have orthogonal energy eigenspaces. ^[ref].

The energy eigenvalues form the point spectrum, which can be only part of the spectrum. The term discrete spectrum more specifically means isolated eigenvalues of finite multiplicity. The usual fixed-nucleus hydrogen atom model has discrete negative bound state energy eigenvalues together with a continuum at nonnegative energy. ^[ref]. A quantum system need not have a ladder of discrete energy eigenvalues: a whole-line free particle has continuous spectrum. A plane wave used for a free particle is a generalized eigenfunction of the Hamiltonian operator, rather than a normalized vector in the physical Hilbert space. ^[ref].

For the infinite square well $0<x<L$, the Dirichlet boundary conditions giveSubstitution into $H{\varphi }_n=E_n{\varphi }_n$ and wavefunction normalization verify these statements directly. Here $E_1$ is the ground-state energy, even though its index is one; the labeling convention does not determine which quantum state is lowest. Equal populations of ${\varphi }_1$ and ${\varphi }_2$ have energy expectation value $(E_1+E_2)/2=5E_1/2$, but an energy measurement returns $E_1$ or $4E_1$, each with probability $1/2$. An average of allowed energy eigenvalues need not itself be an energy eigenvalue.

For a computed state $z_n$ and an exact state $z(t_n)$, an energy error is $H(z_n)-H(z(t_n))$ for a specified energy function $H$. If the exact dynamics conserve $H$, the reference is simply $H(z(0))$. This is a scalar diagnostic; it does not by itself bound position or phase errors.

For the unit-frequency harmonic oscillator $q^{\prime }=p$, $p^{\prime }=-q$, the exact energy is $H=(q^2+p^2)/2$. The Forward Euler method givesThus a fixed positive step produces systematic energy growth even though the exact oscillator has constant energy. A bounded oscillatory energy error has a different long-time meaning from such cumulative drift. Comparing methods also requires the same step size, reference trajectory and observation interval.

An energy filter selects electrons according to energy. In cryo-electron microscopy, rejecting energy-loss electrons can reduce background, but filtering cannot reverse chemical damage or remove every source of scattering noise.

Energy flow tracks energy entering an ecosystem, passing through organisms and leaving its boundary. To see its quantitative effect, assume net production at one trophic level is $P_1=1000\mathrm{kJ}{\mathrm{m}}^{-2}\mathrm{da}{\mathrm{y}}^{-1}$ and assume a transfer efficiency $\epsilon =0.10$ at each of two successive steps. Here $P_i$ measures the energy incorporated into new living material per unit area per day, after that level's own cellular respiration; it is not a count of organisms or a standing stock. The joule supplies the energy unit, with $1\mathrm{kJ}=1000\mathrm{J}$.

Only $10/1000=0.01$, or one percent, of the initial production reaches level 3 in this toy food chain. The first step leaves $1000-100=900$ and the second $100-10=90$ outside the next level's new production. These differences can include uneaten material, wastes, and energy released as heat during cellular respiration; their separate amounts have not been assigned. Detritus can still carry energy to decomposers. The 10 percent efficiency is an assumption, not a universal law: transfer efficiencies vary with the organisms, food and conditions. The production ratio $P_{i+1}/P_i$ also differs from the fraction of an individual meal converted into growth. See Clark, Douglas and Choi, Biology 2e, 46.2: production and transfer efficiency.

The shrinking budget explains why supporting another trophic level requires enough production below it. It does not prove an exact maximum length for every food chain, particularly when a food web imports resources or has branching diets. Lindeman's 1942 paper grouped feeding relationships into trophic levels, compared transfers and cellular respiration, and emphasized the ecosystem as a unit connecting living and nonliving components. That viewpoint made different organisms comparable through a common energy budget. See Lindeman, The Trophic-Dynamic Aspect of Ecology (1942).

Material recycling and energy flow describe different bookkeeping. Carbon, nitrogen and phosphorus can be reused through chemical reactions and transfers between organisms and their surroundings. Energy transferred as heat is conserved, but is no longer the same store of chemical energy available to a consumer; photosynthetic primary producers need continuing light input. A decomposer helps recycle material and also releases heat through metabolism. Thus loss from one feeding budget means transfer into other forms or routes, never destruction of energy. See Clark, Douglas and Choi, Biology 2e, 46.3: material cycles. Conservation of energy applies to the whole accounting.