Radioactive decay ($\overline{N}(t)=N_0{e}^{-\lambda t}$)

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus or nuclear state. Its timing can be unpredictable for an individual atomic nucleus while a large population has a reproducible average behavior. A constant nuclear decay constant $\lambda >0$ gives the familiar exponential decay law. The half-life describes how quickly the parent population falls, while radioactive activity describes its rate of transformations.^[ref]

This connection between randomness and a regular population law makes radioactive decay useful for measuring elapsed time and for understanding sources of radiogenic heating. It also distinguishes two questions in an experiment: what the source emits, and what an instrument records. Becquerel's uranium-radiation experiment used a photographic plate; the Curie ionization-current measurement measured an electrical consequence of radiation. Their readouts reveal different aspects of the same investigation.

The probability rate for decay of a nucleus in a simple exponential-decay model.

A succession of radioactive decays in which a daughter can itself be unstable. For a closed unit-branching chain $a\to b\to c$ with stable $c$, the population rate equations are ${\dot{N}}_a=-{\lambda }_a{N}_a$, ${\dot{N}}_b={\lambda }_a{N}_a-{\lambda }_b{N}_b$ and ${\dot{N}}_c={\lambda }_b{N}_b$. They conserve $N_a+N_b+N_c$. Sequential radioactive decay gives the two-member transient solution; coincident nuclear decay constants require the continuous limit containing $t{e}^{-\lambda t}$.

The expected number of alpha decay events per bulk material volume per unit time. Its units are events per cubic metre per second, distinct from absorbed radiation dose or total activity.

Heat deposited by the radioactive decay of radioactive constituents. It persists after a reactor stops sustaining a nuclear fission chain reaction and affects spent-fuel and waste management.

A radionuclide is a species of atomic nucleus with an unstable nuclear state that undergoes radioactive decay. A sample's expected surviving population decreases with its nuclear decay constant. Carbon-14 is the radioisotope used in radiocarbon dating.

Radiometric dating estimates elapsed time from changes in radionuclide populations, using a known nuclear decay constant and assumptions about the initial composition and subsequent closure to matter exchange.

Age determination from uranium decay to lead, commonly in zircon; initial lead, inheritance and later loss or growth must be assessed.

With no initial radiogenic argon-40 and complete daughter retention, the surviving potassium-40 population is $N_K=N_{K,0}{e}^{-\lambda t}$ and the accumulated daughter population is $N_{\mathrm{Ar}}=b(N_{K,0}-N_K)$. Here $\lambda$ is the total nuclear decay constant and $b$ is the argon-producing branching fraction. Their ratio gives the displayed age. The result measures closure of the mineral's isotope system under those assumptions.

Radiocarbon dating uses the loss of carbon-14 relative to stable carbon after carbon exchange stops. It can measure either an isotope ratio by accelerator mass spectrometry or radioactive activity per fixed carbon content. Interpreting an age requires radiocarbon calibration and control of contamination and reservoir differences.

A radiocarbon age estimates elapsed time from a measured carbon-14 ratio $R$, an assumed initial ratio $R_0$ and a specified nuclear decay constant $\lambda$. The displayed exponential decay inversion gives the age on that model's timescale. A conventional radiocarbon age uses a prescribed reference and decay convention; it is not generally a calendar age. Radiocarbon calibration accounts for changes in the atmospheric isotope ratio and maps the measurement to a calendar-age estimate.

Radiocarbon calibration converts conventional radiocarbon ages into calendar-age information using independently dated material and models of changes in reservoir isotope ratios. Conventional ages use the historical $5568$-year half-life, whereas the physical carbon-14 half-life is about $5730$ years.^[ref]

Radioactive decay by emission of an alpha particle, leaving a daughter atomic nucleus with two fewer protons and two fewer neutrons.

The alpha preformation probability is a model-dependent nuclear-structure factor multiplying an interior assault frequency and barrier transmission probability in an alpha-decay rate estimate. It accounts for the availability of an alpha-like cluster at the matching surface. It is not fixed by the exterior Coulomb tunnelling exponent alone.

In a semiclassical escape model, an alpha-particle assault frequency is the interior rate scale multiplying the barrier transmission probability. Confinement alone gives $p\sim \hslash /L$ and hence $\nu \sim \hslash /(m_{\alpha }{L}^2)$ up to geometric factors. Counting encounters with either wall differs by a factor of two from returns to one wall. A stationary bound wavefunction does not itself follow a classical collisional trajectory.

In two-body alpha decay of a parent at rest, momentum conservation makes daughter and alpha momenta equal and opposite. The nonrelativistic kinetic energies therefore satisfy $T_d/T_{\alpha }=m_{\alpha }/M_d$, givingAn excited daughter reduces the available Q value. To leading recoil order, an alpha line separation $\Delta T_{\alpha }$ corresponds to excitation $E_x=(1+m_{\alpha }/M_d)\Delta T_{\alpha }$. Thus ratios of excitation energies inferred from line separations do not need the common recoil factor. These formulas assume kinetic energies small compared with rest masses; an exact relativistic two-body decay calculation can replace them when necessary.

The ground state of an even-even nucleus has $J^P=0^{+}$, and the emitted alpha particle also has $0^{+}$. In an alpha decay to daughter spin $J_Y$, addition of angular momentum permits total zero only when the relative orbital angular momentum is $L=J_Y$. Parity conservation gives $+1={\pi }_Y(-1{)}^L$, hence the daughter has natural parity, $0^{+},1^{-},2^{+},3^{-},\dots$. This rule does not itself restrict $J_Y$ to even integers. An even-$J$ sequence such as $0^{+},2^{+},4^{+}$ requires additional nuclear structure, for example a positive-parity ground-state nuclear rotational band.

The Geiger-Nuttall law expresses the extreme sensitivity of an alpha decay half-life to its Q value. In a restricted range of heavy nuclei, a useful local form is ${\log }_{10}[T_{1/2}/T_{\ast }]\simeq a{Z}_d/\sqrt{Q_{\alpha }}+b$, where $Z_d$ is the daughter atomic number, and the constants depend on the chosen units and nuclear family. It is named for Hans Geiger and John Mitchell Nuttall.

The Coulomb barrier outside the daughter has $V(r)=2Z_d\alpha /r$ in natural units. Its outer turning point is $r_t=2Z_d\alpha /Q_{\alpha }$. The WKB approximation gives a penetration probabilitywhere $\mu$ is the daughter-alpha reduced mass. The leading large-barrier exponent is $2\pi (2Z_d\alpha )\sqrt{\mu /(2Q_{\alpha })}$, producing the approximate $Z_d/\sqrt{Q_{\alpha }}$ dependence. Finite nuclear size modifies the exponent and the slowly varying intercept. The nuclear decay constant also contains an assault frequency and an alpha-cluster formation probability. Extra orbital angular momentum raises the barrier further, and odd-nucleon structure can introduce substantial hindrance. Thus extrapolation from even-even decays gives an order-of-magnitude reference rather than a precision prediction for an odd-odd nucleus.

For ground-to-ground beta-minus decay of bare nuclei, the nuclear mass difference supplies the electron rest energy as well as the kinetic energy shared among products. Neglecting recoil and neutrino mass, $T_{\max }=Q_{{\beta }^{-}}$. For neutral atomic masses the additional final atomic electron cancels the emitted electron rest-mass subtraction, giving $Q\simeq [M_{\mathrm{atom}}(i)-M_{\mathrm{atom}}(f)]c^2$ up to electron-binding corrections. A daughter excitation reduces the endpoint by its excitation energy.

A kurie plot removes known electron phase-space and Coulomb factors from a beta spectrum and takes a square root. For negligible neutrino mass/recoil and constant nuclear shape factor, its ordinate is proportional to $Q-T$, giving a straight line whose kinetic-energy intercept is the beta endpoint. The precise factors depend on whether the measured spectrum is per momentum or per energy. Forbidden transitions generally need an additional shape correction.

If a beta spectrum has nonconstant shape factor $C(T)$, divide by it as well as by the Coulomb and phase-space factors before taking a square root. A leading unique first-forbidden shape contains $p_e^2+p_{\nu }^2$ in natural units, with Coulomb and nuclear corrections. Dividing only by a Coulomb Fermi function does not remove that shape. A straight uncorrected line must therefore be treated as an approximation, not an exact consequence of forbidden-transition selection rules.

For counts $N(p)dp$ per electron momentum interval, the allowed spectrum is proportional to $p^{2}F(p)(Q-T{)}^{2}dp$ when recoil and neutrino mass are neglected. Thus the displayed ordinate is linear in kinetic energy under a constant-shape assumption. Counts per kinetic-energy interval instead have the factor $p{E}_e$ in natural units; the Jacobian $dT/dp=p/E_e$ accounts for the difference. Mixing those two binning conventions changes the inferred ordinate.

A nuclear beta decay connects initial and final quantum states through vector and axial-vector parts of the weak charged current. At leading order in the lepton wavelength expansion, Fermi transitions carry lepton-pair spin $S=0$, while Gamow-Teller transitions carry $S=1$. If the lepton pair carries orbital angular momentum $l$, its total angular momentum is $J=l$ for $S=0$, or $J=|l-1|,\dots ,l+1$ for $S=1$. The nuclear states must obey $|J_i-J_f|\le J\le J_i+J_f$, and their parities satisfy ${\pi }_f/{\pi }_i=(-1{)}^l$ in this leading orbital classification. The least permitted $l$ labels the degree of forbiddenness. Real forbidden transitions involve several weak multipole operators; the elementary spin classification does not assert that only one operator or nuclear matrix element exists.

The product of a dimensionless beta-decay statistical phase-space factor $f$ and the partial half-life $t$ for the transition. The tabulated $\log ft$ normally means ${\log }_{10}[ft/(1\mathrm{s})]$. Factoring out this approximate phase space dependence allows comparisons of nuclear weak matrix elements. Strong superallowed Fermi transitions have small $\log ft$; forbidden beta decays generally have larger values, with nuclear-structure exceptions and overlaps between categories.

A beta decay transition for which the leading $l=0$ lepton orbital term vanishes by the angular momentum or parity selection rules. The least contributing orbital order $l=n\ge 1$ gives the name $n$th forbidden. First-forbidden transitions change parity and can carry total lepton angular momentum $0,1,2$; second-forbidden transitions preserve parity and can carry $1,2,3$, with Fermi-type $S=0$ contributions restricted to $J=l$. Forbidden is suppression by higher powers of lepton momentum times nuclear size, rather than an absolute prohibition.

An $l$th-forbidden beta decay transition with the maximal leading angular-momentum transfer $l+1$. Only the spin-dependent $S=1$ coupling can attain that transfer at this orbital order. Thus a $0^{+}\to 2^{-}$ transition is first-forbidden unique, whereas a same-parity $\Delta J=2$ transition is second-forbidden nonunique. Unique refers to the leading weak multipole structure, not a claim that every correction to the decay vanishes.

Using the orbital order after 28 supplied in the simple shell model, proton counts 32 and 38 fill $2p_{3/2}$ and $1f_{5/2}$, so the odd 39th proton occupies $2p_{1/2}$. With paired valence neutrons this gives a $1/2^{-}$ yttrium-91 ground-state prediction. The daughter has paired 40 protons and one neutron beyond the 50-neutron closure in $2d_{5/2}$, giving $5/2^{+}$. The inferred ground-state beta transition is therefore first-forbidden unique. This is a stated paired-shell model deduction; an endpoint diagram alone does not determine the spin assignment or an evaluated isotope mass.

A first-forbidden unique beta decay has $|\Delta J|=2$ with a parity change. Its leading lepton shape factor contains two additional powers of momentum relative to an allowed beta decay. In the massless, no-Coulomb limit, ${\int }_0^Q{E}^2(Q-E{)}^2[E^2+(Q-E{)}^2]dE=2Q^7/105$. Thus its microscopic reduced nuclear matrix element multiplies a seventh-power phase-space factor, rather than the fifth power in the Sargent rule. A fifth-power comparison across different forbiddenness classes instead extracts convention-dependent effective strengths.

A beta decay transition whose leading lepton orbital term has $l=0$. Fermi transitions allow $\Delta J=0$ and Gamow-Teller transitions allow $\Delta J=0,1$, excluding a Gamow-Teller $0\to 0$ transition. Both preserve nuclear parity. Allowed describes the selection rules; cancellations in the nuclear matrix element can still make an allowed transition slow.

The approximate fifth-power scaling of an allowed beta decay rate with available energy when electron mass and Coulomb distortion are negligible and the nuclear matrix element is held fixed. It follows from ${\int }_0^Q{p}^2(Q-p{)}^{2}dp=Q^5/30$. Different nuclear matrix elements or nonrelativistic electron kinematics spoil an exact universal law.

With a momentum-independent allowed nuclear matrix element, negligible nuclear recoil and negligible neutrino mass, the electron spectrum in natural units is $d\Gamma /dp=CF(Z,E)p^2(E_0-E{)}^2$, where $E=\sqrt{p^2+m_e^2}$. Transforming to an energy spectrum requires the Jacobian determinant $dp/dE=E/p$.

The electron-nucleus Coulomb-distortion factor multiplying an otherwise plane-wave beta decay spectrum. It depends on daughter charge, lepton charge and lepton energy. Its energy dependence means that a pure fifth-power phase-space rule is not generally exact.

An especially strong allowed beta decay, conventionally including pure Fermi transitions between $0^{+}$ isobaric analog states. Those states have closely related spatial and spin structure, so the isospin ladder operator has a large overlap. A $0^{+}\to 0^{+}$ assignment alone fixes the allowed Fermi character, but does not establish analog-state overlap or a large rate; a small comparative half-life supplies additional evidence.

The spin-dependent axial-vector contribution to an allowed beta decay. Its rotational rank is one, permitting $\Delta J=0,1$ with unchanged parity, but excluding $0\to 0$. Here ${\mathbf{\sigma }}_k$ acts on nucleon spin and ${\tau }_k^{\pm }$ on isospin. Higher-orbital spin-dependent contributions are conventionally described as Gamow-Teller-type forbidden transitions.

The spin-independent vector contribution to an allowed beta decay. The isospin operator ${\tau }_k^{\pm }$ changes a neutron into a proton or the reverse; the sign convention depends on the assigned nucleon isospin projection. Its rotational rank is zero, so $J_f=J_i$ and the parity is unchanged. The phrase Fermi-type forbidden contribution extends the $S=0$ classification to higher orbital angular momentum.

The Fermi transition operator is an isospin ladder operator. Acting between normalized members of one isospin multiplet gives the displayed squared matrix element. An edge transition in a $T=3/2$ multiplet has Fermi strength three, whereas the transition between the two $T=1/2$ mirror members has strength one. This fixes the vector contribution; nonzero-spin analogue transitions can also have a Gamow-Teller transition contribution that is not fixed by this counting.

A nuclide is beta-stable if it has no energetically allowed single beta-minus decay, beta-plus decay, or electron capture to a lower-mass neighboring nuclear isobar, with electron thresholds treated consistently. Beta stability need not imply stability against alpha decay, double beta decay, or nuclear fission.

The valley of stability is the region of nuclear masses near the minima with respect to proton-neutron imbalance. Its beta-stable band bends toward neutron excess for heavier nuclei, as nuclear Coulomb energy competes with the cost of neutron-proton asymmetry. The band is distinct from the particle-emission boundaries set by nuclear separation energies.

Double beta decay changes the atomic number of a nucleus by two while preserving its mass number. The ordinary two-neutrino process can connect even-even nuclear isobars when the intermediate odd-odd nucleus is too heavy for a single beta decay. Neutrinoless variants would violate lepton number and require additional physics.

A lepton-number-violating nuclear transformation emitting two electrons without neutrinos. A signal appears near the nuclear Q value in summed electron energy, after recoil and detector response are accounted for. Its interpretation requires the decay mechanism and nuclear matrix elements.

Two charged-current beta vertices can be connected by a virtual Majorana neutrino. With light exchanged masses relative to nuclear momentum scales, the chirality connection makes the amplitude proportional to ${\sum }_i{U}_{ei}^2{m}_i$, not an incoherent mass average. Squaring introduces the effective Majorana mass, nuclear matrix elements and phase space. Other lepton-number-violating operators can also mediate the same final state, so translating a rate into a mass requires the stated mechanism.

A contained neutrinoless event deposits approximately the nuclear Q-value in the two-electron sum, apart from recoil and losses. Ordinary two-neutrino double beta decay has a continuous electron-sum radiation spectrum. Resolution, escape losses and backgrounds broaden or mimic features, so a peak assignment requires calibrated response, sidebands and appropriate event topology. A null peak search is a limit on a decay rate under an efficiency/background model, not proof of Dirac neutrinos.

A search in which isotope-enriched detector material is also the decay source. germanium detectors combine a contained energy deposit with energy resolution, pulse information and material-background controls. Background peaks and resolution tails must be tested before assigning a signal.

The coherent electron-flavor mass combination entering the usual light-neutrino-exchange amplitude. Its phases can cancel contributions, so a null double-beta result does not imply vanishing individual masses.

An electron or Positron emitted in beta decay.

In the reaction displayed here, an electron antineutrino interacts with a proton to produce a Positron and a neutron. It turns an otherwise hard-to-observe incident particle into two products with different detection signatures. A prompt Positron signal followed by neutron capture can identify a candidate much more selectively than a single energy deposit. Inverse beta decay is a weak interaction; the incident electron antineutrino is detected through its reaction products.

The neutron and Positron together have more rest mass than the initial proton, so a finite incident energy is necessary even before recoil is considered. In the proton's rest frame, use four-momentum conservation and a negligible neutrino mass. The initial squared center-of-mass energy is $m_p^2{c}^4+2m_p{c}^2{E}_{\nu }$; the least final squared center-of-mass energy is $(m_n+m_e{)}^2{c}^4$. Equating them gives the threshold above. At threshold the final particles share one center-of-mass velocity; they are not both at rest in the laboratory.

Writing $\Delta =m_n-m_p$, the result separates intoThe positive second term is the laboratory recoil correction. With rounded rest energies $m_p{c}^2=938.272\mathrm{MeV}$, $m_n{c}^2=939.565\mathrm{MeV}$ and $m_e{c}^2=0.511\mathrm{MeV}$, the calculated threshold is about $1.806\mathrm{MeV}$. These rounded inputs follow the recommended particle masses.^[ref]

As a further ideal energy balance, suppose the Positron stops and annihilates with an electron, and all prompt annihilation gamma rays are contained. Its prompt visible energy is $T_{e^{+}}+2m_e{c}^2$. Neglecting neutron recoil gives $E_{\mathrm{vis}}\simeq E_{\nu }-(m_n-m_p)c^2+m_e{c}^2\simeq E_{\nu }-0.782\mathrm{MeV}$. Escaping gamma rays, detector response and recoil change that measured sum. The delayed capture signal is a separate energy deposit, not an extra contribution to the incident electron antineutrino's prompt energy reconstruction.

Clyde Cowan, Frederick Reines and collaborators tested whether a particle produced by a nuclear reactor could interact again after leaving its source. At Savannah River, water targets containing cadmium chloride lay between liquid scintillators. Opposite detectors recorded prompt positron annihilation gamma rays; Neutron moderation preceded delayed neutron capture signals in cadmium. The arrangement used two target layers and three scintillator layers, with the middle scintillator shared.^[ref]

The original functional diagram separates the two signal mechanisms and their timing. It omits dimensions and phototube counts, and is not a reproduction of a historical drawing. The measured objects are detector pulses; identifying their common cause also requires calibration and background tests.

The 1956 paper describes positron-source and neutron-source calibrations, pulse-height gates and photographic pulse records. Diluting the target with heavy water halved the proton density and reactor signal while preserving neutron-detection efficiency. Changing the cadmium concentration altered capture timing; removing it removed the reactor signal. Added shielding tested competing external radiation. These interventions tested the reaction's ingredients, beyond simply observing a reactor-dependent count rate.^[ref]

Why not accept every delayed pulse? A longer window captures more genuine neutron capture signals, but also admits more accidental coincidences. Consider an original model with a capture waiting time following an exponential distribution of mean $\tau >0$. Assume detection after a prompt signal accepts $0<t<W$, with perfect time measurement and no lower cut. Integrating the normalized density givesThis assumes that moderation, competing capture channels, boundaries and electronic recovery can be represented or neglected in a single effective waiting-time model. It is not a fitted historical capture law.

Let independent prompt-like and delayed-like background streams have Poisson process rates $R_p$ and $R_d$. The accidental coincidence pair rate in a directed window of full width $W$ is $R_p{R}_{d}W$. Electronics producing at most one accepted delayed match per prompt instead give $R_p(1-e^{-R_{d}W})$. These definitions coincide to first order when $R_{d}W\ll 1$. Correlated backgrounds, such as one external neutron producing both a prompt recoil and a later capture, require separate controls; an independent-stream estimate cannot remove them.

For chosen $\tau =20\mu \mathrm{s}$ and $W=50\mu \mathrm{s}$, the accepted fraction is $1-e^{-2.5}\simeq 0.918$. With chosen $R_p=40{\mathrm{s}}^{-1}$ and $R_d=30{\mathrm{s}}^{-1}$, the pair rate is $0.0600{\mathrm{s}}^{-1}$, while the at-most-one rate is approximately $0.05996{\mathrm{s}}^{-1}$. These are teaching parameters, not reported Savannah River rates. A measured uncorrelated background can be tested using displaced time windows; a reactor-off measurement addresses a different question and cannot by itself characterize every reactor-correlated background.

If genuine pairs arrive at rate $S_0$ before the timing cut, an exposure $T$ gives $S=S_{0}T(1-e^{-W/\tau })$ and expected accidental pairs $B=R_p{R}_{d}WT$. In a background-dominated large-count approximation, maximizing $S/\sqrt{B}$ reduces to maximizing $f(x)=(1-e^{-x})/\sqrt{x}$ for $x=W/\tau >0$. Differentiation gives the positive stationary solution $e^x=1+2x$, or $x\simeq 1.25643$. The endpoint $x=0$ solves the rearranged equation but is outside the domain of this derivative calculation; $f(x)$ tends to zero at both positive-domain extremes.

Thus the chosen $\tau =20\mu \mathrm{s}$ gives $W\simeq 25.13\mu \mathrm{s}$ for this particular criterion, accepting about $0.715$ of genuine delayed signals. This differs from maximizing signal counts alone, which would favor longer windows. Small samples, appreciable signal fluctuations, unknown background, dead time and multiple-candidate vetoes require a fuller likelihood model. The calculation isolates a tradeoff rather than prescribing an experimental cut.

In the constant-rate mathematical model, an undecayed atomic nucleus has conditional probabilityof decaying during the next positive interval $\Delta t$. The nuclear decay constant has units of inverse time. Its value describes a rate; it does not specify a scheduled death time for one atomic nucleus. The conditional probability is independent of the age $t$, the memorylessness of the exponential distribution.

Take $N_0$ independently decaying atomic nuclei, each with lifetime $T_i$ drawn from an exponential distribution of rate $\lambda$. With no replenishment, the survivor count isEach indicator variable has success probability $p(t)$, so the binomial distribution givesThe smooth population law is the expected value $\overline{N}(t)=\mathbb{E}[N(t)]$, satisfying ${\overline{N}}^{\prime }=-\lambda \overline{N}$. One realization instead loses whole atomic nuclei at discrete times. An observed count above the smooth curve need not contradict the mathematical model.

For an original chosen example, let $N_0=1000$ and half-life $3\mathrm{h}$. Then $\lambda =\log 2/(3\mathrm{h})$. After $9\mathrm{h}$, the survival probability is $1/8$, givingThe standard deviation describes variation between repeated populations governed by these assumptions. It is not an uncertainty interval for a noiseless deterministic curve.

The half-life and the mean lifetime answer different questions:The first is the median of one lifetime, or the time when half the population survives on average. The second is the lifetime's expected value. In this example they are $3\mathrm{h}$ and approximately $4.33\mathrm{h}$.

With $n$ parent atomic nuclei present, the conditional expected number of transformations in a short interval is approximately $\lambda n\Delta t$. The corresponding radioactive activity is $A=\lambda n$. Averaging over the finite-population radioactive decay model gives $\overline{A}(t)=\lambda \overline{N}(t)$. Its unit is the becquerel.^[ref]

For a finite counting interval, depletion matters. Suppose each transformation is recorded independently with detection efficiency $\epsilon$, with at most one count per transformation and no background. The expected value of the count between $t$ and $t+\Delta t$ isOnly when $\lambda \Delta t\ll 1$ does this become approximately $\epsilon \overline{A}(t)\Delta t$. A count rate therefore needs a detector model before it can be interpreted as the source's radioactive activity.

radioactive activity per volume, in $\mathrm{Bq}{\mathrm{m}}^{-3}$. This differs from mass-specific radioactive activity, usually expressed per kilogram.

One becquerel is a radioactive activity of one nuclear transformation per second.

The probability that an event of a specified kind produces a registered count in a specified arrangement. For a radiation detector, geometry, intervening material and the detector response can all affect detection efficiency.

In his 2 March 1896 report, Henri Becquerel described uranium salt crystals acting on a photographic plate through opaque wrapping. A copper cross weakened the exposure behind it. Plates prepared during intermittent sunshine were left in a drawer and developed on 1 March; unexpectedly strong images prompted deliberate dark tests. He placed a plate and crystals inside two opaque boxes in a drawer; one crystal was separated from the emulsion by glass. After five hours, both produced images. The separation tested transmission without direct chemical contact.^[ref]

The result challenged a dependence on immediate illumination. Henri Becquerel still considered long-lasting invisible phosphorescence a possibility, and explicitly withheld a final explanation. The photographic readout established an effect through intervening material; it did not count nuclear transformations or measure a nuclear decay constant.^[ref]

Marie Curie's thesis describes powdered material on the lower plate of a capacitor, biased by a battery. Radiation made the intervening air conduct. The upper plate connected to an electrometer and a loaded piezoelectric crystal of quartz. Adjusting the load supplied a known compensating electric charge while keeping the electrometer at zero; charge divided by elapsed time determined the electric current. Voltage scans identified a collection plateau. Contact potentials affected weak-field readings; excessively strong fields could instead introduce additional ionization. Fixed geometry and plateau operation made comparisons meaningful.^[ref]

The null method turns a weak signal into a balance against a known quantity. In the diagram's chosen constant-current model, $I=10\mathrm{pA}$ for $\Delta t=30\mathrm{s}$ givesSupplying $-300\mathrm{pC}$ over the same interval leaves zero net electric charge at the sensing node. The electrometer detects imbalance; the compensating-charge calibration supplies the numerical measurement.

This electric current measures collected charge produced by ionization. Converting it to becquerels additionally requires the energy deposition and charge-collection response per transformation. The apparatus can compare preparations without already possessing that complete source-to-signal calibration.

Radiogenic heating is thermal energy deposited by radioactive decay. If its specific heating rate is $Q$ in watts per kilogram, the volumetric rate is $\rho Q$ and the heat input per horizontal area to a column of depth $H$ is $\rho QH$. The distinction between specific and volumetric rates matters in dimensional analysis of a convective heat flux law.

In a chain $N_1\to N_2\to N_3$ with distinct decay constants, the Bateman equation expresses each population as a linear combination of the exponential functions $e^{-{\lambda }_{i}t}$. When two consecutive constants coalesce to $\lambda$, the repeated Laplace transform pole produces a term proportional to $t{e}^{-\lambda t}$.

The Bateman equation gives the populations in a sequential radioactive decay chain. For a parent with population $N_1(0)$ and an initially absent daughter, distinct decay constants give $N_2(t)=N_1(0){\lambda }_1(e^{-{\lambda }_{1}t}-e^{-{\lambda }_{2}t})/({\lambda }_2-{\lambda }_1)$; the first population satisfies $N_1^{\prime }=-{\lambda }_1{N}_1$.