Reaction atlas
Glossary
The words the atlas uses, in plain language. Each reaction page links here the first time a term appears.
To put these words to work: exercises.
- Chart of nuclides
A map of all atomic nuclei. Each square is one nucleus: one step up adds a proton (and so makes the next element), one step to the right adds a neutron (a heavier isotope of the same element). Capturing a proton moves a nucleus up, capturing a neutron moves it right, and a β+ decay, in which a proton turns into a neutron, moves it one step down and one to the right. On the atlas map the stable nuclei are filled in grey and the radioactive ones are outlined.
See also: Reaction notation
- Reaction notationtarget(in,out)product
Physicists write a reaction as target(in,out)product. 14N(p,γ)15O reads: a nitrogen‑14 nucleus (the target) captures a proton (p, what comes in) and becomes oxygen‑15 (the product), giving off a gamma ray (γ, what comes out). The small number before the symbol is the mass number, the count of protons and neutrons. Common particles have short names: p for a proton, n for a neutron, d for a deuteron (a proton and a neutron), α for a helium‑4 nucleus, γ for a gamma ray, e+ for a positron and ν for a neutrino.
See also: Chart of nuclides, Q-value
- keV and MeVkilo-electronvolt, mega-electronvolt
Units of energy for single particles. One electronvolt (eV) is the energy an electron gains across 1 volt; a keV is a thousand of them and an MeV a million. In the centre of the Sun the typical thermal energy of a nucleus is about 1.3 keV, while the energy released by one fusion reaction is a few MeV.
- Cross sectionσ; barn (b), picobarn (pb)
The probability that a reaction happens, expressed as an effective target area that each nucleus presents to an incoming one. It is measured in barns: 1 barn is 10−24 cm2, roughly the size of a large nucleus. At stellar energies cross sections are tiny, often picobarns (10−12 barn) or less, which is why so few reactions happen in an experiment and why they are so hard to count.
See also: Astrophysical S-factor, Gamow window
- Differential cross sectiondσ/dΩ
The cross section per unit of solid angle in a given direction: how likely it is that the product of a reaction, or a scattered nucleus, flies off at that angle. Summed over all directions it gives the total cross section. Elastic scattering data, which are often used in R-matrix fits, are differential cross sections, usually divided by the Rutherford value expected from the electric repulsion alone.
See also: Cross section, R-matrix
- Astrophysical S-factorS(E), S(0); keV b
The cross section with its steepest, well-understood parts taken out. Because nuclei must tunnel through their electric repulsion, the cross section falls by many orders of magnitude as the energy drops. Writing σ(E) = S(E) exp(−2πη) / E, that is, dividing out the tunnelling probability (the Gamow factor exp(−2πη)) and a factor 1/E that comes from the quantum-mechanical size of the colliding nuclei, leaves the S-factor, which changes only slowly with energy (except near resonances) and is therefore much easier to extend from measured energies down to stellar ones. S(0) is its value extrapolated to zero energy; it is quoted in keV b or MeV b. For two carbon nuclei a modified S-factor, S*, divides out a further factor that remains in such a heavy system.
See also: Cross section, R-matrix
- Coulomb barrier
The electric repulsion between two positively charged nuclei, which keeps them apart until they come close enough for the strong nuclear force to take over. For a proton and nitrogen‑14 the barrier is about 2–3 MeV high, while in the Sun the typical energy is about a thousand times smaller; fusion still happens, rarely, by quantum tunnelling through the barrier. A neutron has no charge and feels no barrier.
See also: Gamow window, Astrophysical S-factor
- Gamow windowGamow peak, E0
The narrow band of energies at which a reaction happens in a star. Fast nuclei are rare in a hot gas, but slow ones almost never get through the electric repulsion; the product of the two is a peak, at an energy well above the typical thermal one. For 14N + p at the centre of the Sun it lies near 27 keV. Measuring a cross section inside, or close to, the Gamow window is the goal of experiments like LUNA.
See also: keV and MeV, Cross section, Temperatures in stars
- Resonanceresonance strength ωγ
When the energy of the two colliding nuclei matches an excited state of the nucleus they form together, the cross section rises sharply, often by orders of magnitude, over a narrow range of energy. A narrow resonance is characterized by its energy and by its strength (ωγ). A single low-energy resonance can dominate the rate of a reaction in a star, so finding and measuring them is a large part of the work.
See also: R-matrix
- Excited stateenergy level
A nucleus, like an atom, can exist only at certain energies. The lowest is the ground state; the others are excited states, which usually give up their extra energy within a tiny fraction of a second by emitting a γ-ray or a particle. Each nucleus has its own pattern of levels, and the ones just above the energy at which two nuclei can merge decide where a reaction has resonances.
See also: Resonance, Compound nucleus
- Compound nucleus
The nucleus formed for a moment when two nuclei merge: a proton and nitrogen‑14 form oxygen‑15, two carbon nuclei form magnesium‑24. It is born in an excited state and at once gets rid of the extra energy, by emitting a γ-ray (a capture) or by breaking up into other particles. Which way it decays does not depend on how it was formed, only on its energy and its other properties.
See also: Excited state, Branching ratio, Resonance
- Q-value
The energy released by a reaction, in MeV. It is the difference between the masses before and after, converted with E = mc2. A positive Q-value means the reaction gives off energy, which is what heats a star; a negative one means energy must be supplied. The atlas computes it from the measured atomic masses of the 2020 Atomic Mass Evaluation (AME2020).
See also: Reaction notation
- Threshold and endothermic reactionsendothermic, Q < 0
A reaction with a negative Q-value is endothermic: it takes energy instead of releasing it, and it can happen only if the colliding nuclei bring at least that much energy with them, the threshold (|Q| in the centre of mass). In a star only the fastest nuclei are above it, so for such a reaction the atlas shows the range of energies just above the threshold, not the ordinary Gamow window, which would lie partly or wholly below it.
See also: Q-value, Gamow window
- Branching ratiobranch
When a nucleus can decay, or a reaction can end, in more than one way, the branching ratio is the fraction that goes each way. Two carbon nuclei, for example, end up as neon‑20 and an alpha particle or as sodium‑23 and a proton; the split between the two decides how much sodium a star makes.
See also: Compound nucleus
- Reaction rateN_A⟨σv⟩
How many reactions happen per second in a given amount of stellar matter. It combines the cross section at every energy with the distribution of speeds of the nuclei at the temperature of the star. Stellar models use tables of rates against temperature, which is why the atlas says how well each rate is known. What the tables give is N_A⟨σv⟩, the rate per pair of particles, in cm3 mol−1 s−1; the number of reactions per cubic centimetre and per second is r = n1n2⟨σv⟩ / (1 + δ12), with n1 and n2 the number densities of the two nuclei and δ12 = 1 for identical ones, so that each pair is counted once.
See also: Cross section, How well a rate is known
- How well a rate is knownWell known, Improving, Uncertain, Unmeasured
Each reaction page carries one of four labels. Well known: the rate is known well enough for the stellar models that use it. Improving: recent measurements have narrowed it, but some uncertainty remains. Uncertain: the uncertainty is large enough to change the models that depend on it. Unmeasured: there are no direct data at the energies that matter, and the rate rests on extrapolation, indirect methods or theory. The label is the author's summary of the evaluations cited on the page.
- Well knownThe rate is known well enough for the stellar models that use it.
- ImprovingRecent measurements have narrowed the rate; some uncertainty remains.
- UncertainThe rate is uncertain enough to change the models that depend on it.
- UnmeasuredNo direct data at the energies that matter; the rate rests on extrapolation or indirect methods.
See also: Reaction rate
- Temperatures in starsMK, GK, T9
Stellar temperatures are given in millions of kelvin (MK) or billions of kelvin (GK, gigakelvin); T9 is the temperature in GK. The centre of the Sun is at about 15.7 MK, or T9 = 0.0157; helium burns at about 100– 300 MK, and the hottest explosive burning reaches a few GK.
See also: Gamow window
- Solar massM☉
The mass of the Sun, about 2 × 1030 kg, used as the unit for the masses of stars. A 15 M☉ star has fifteen times the mass of the Sun; the mass decides how hot a star's core gets and so which reactions run in it.
- Main sequence
The long, stable phase in which a star burns hydrogen into helium in its core, by the pp chains or the CNO cycle. Stars spend most of their lives there; the Sun has been on the main sequence for about 4.6 billion years. When the hydrogen in the core runs out, the star leaves it and becomes a red giant.
See also: Solar mass
- Half-lifet½
The time after which half of a sample of a radioactive nucleus has decayed. It ranges from fractions of a second to billions of years. The atlas takes half-lives from the NUBASE2020 evaluation; in a star, where the atoms are ionized, a half-life that depends on electrons (electron capture) can be different.
- AGB starasymptotic giant branch
A late stage in the life of stars of roughly 1 to 8 solar masses. The star has a core of carbon and oxygen, burns hydrogen and helium in thin shells around it, and swells into a cool giant that loses much of its mass in a slow wind. AGB stars make about half the nuclei heavier than iron, through the s-process, and enrich the Galaxy with carbon and nitrogen.
See also: s-process, Solar mass
- s-processslow neutron capture
The slow neutron-capture process. Nuclei capture neutrons one at a time, slowly enough that an unstable nucleus usually decays before it captures the next one, so the process climbs along the valley of stable nuclei. It runs mainly in AGB stars and in massive stars, with neutrons from the reactions 13C(α,n)16O and 22Ne(α,n)25Mg, and makes about half the nuclei heavier than iron.
See also: r-process, AGB star, Main and weak s-process, Branch point, s-only and r-only isotopes, Maxwellian-averaged cross section
- r-processrapid neutron capture
The rapid neutron-capture process. In an environment flooded with neutrons, nuclei capture many of them in a fraction of a second, far faster than they can decay, and then decay back towards stability. It makes the other half of the heavy elements, including gold and uranium. Merging neutron stars are one confirmed site, from the kilonova seen with the gravitational-wave event GW170817 in 2017.
See also: s-process
- Main and weak s-processmain and weak components
The two parts of the s-process found when its products are compared with the Solar System. The main component comes from low-mass AGB stars, with neutrons mostly from the 13C pocket, and makes most of the s-process nuclei from strontium to lead. The weak component comes from massive stars, with neutrons from 22Ne at the end of core helium burning and in the carbon-burning shell, and makes mostly nuclei lighter than mass 90.
See also: s-process, 13C pocket, Thermal pulse
- 13C pocket
A thin layer in an AGB star, between the helium shell and the hydrogen envelope, where a few protons mixed into helium-burning ashes turn carbon-12 into carbon-13. Between two thermal pulses the carbon-13 captures helium and releases neutrons through 13C(α,n)16O, at a thermal energy of about 8 keV. Most of the heavy s-process elements are made in it.
See also: Thermal pulse, AGB star, Neutron poison
- Thermal pulsehelium-shell flash
In an AGB star, helium accumulates in a shell under the burning hydrogen until it ignites in a brief, violent flash, which drives convection through the shell. The flash recurs again and again. At its peak, around 270 million kelvin, it briefly starts the second neutron source of the star, 22Ne(α,n)25Mg, with a higher neutron density than the 13C pocket.
See also: 13C pocket, AGB star
- Neutron poison
A nucleus that captures neutrons that would otherwise reach the iron seeds of the s-process. A poison may capture neutrons only rarely, but if it is far more abundant than iron it still takes a large share: nitrogen-14 in the 13C pocket, and oxygen-16 and neon-22 in massive stars.
See also: 13C pocket, s-process
- Neutron density
The number of free neutrons per cubic centimetre in a star. In the s-process it is low: about 107 per cubic centimetre in the 13C pocket of an AGB star, where a nucleus with a typical cross section of a tenth of a barn waits about 250 years between two captures, up to about 1010 in the brief 22Ne bursts of the thermal pulses, and about 1011–1012 in the carbon-burning shell of a massive star. In the r-process it is higher still by many orders of magnitude, above 1020. Branch points of the s-process measure it.
See also: Branch point, s-process, r-process
- Magic numbersclosed neutron shells
Protons and neutrons fill shells inside the nucleus, much as electrons do in an atom. A nucleus with 2, 8, 20, 28, 50, 82 or 126 neutrons has a closed shell and is especially stable and reluctant to capture another neutron. At N = 50, 82 and 126 the s-process slows down, which makes the abundance peaks at strontium, barium and lead.
See also: s-process, Chart of nuclides
- Branch point
A radioactive nucleus on the s-process path that lives long enough to capture a neutron before it decays, but not always. There the path forks, and the share of each branch depends on the neutron density and sometimes on the temperature of the star. The abundances of the nuclei beyond, such as krypton-86 or the samarium isotopes, therefore act as a gauge of neutron density or as a thermometer.
See also: Neutron density, s-process, Branching ratio
- s-only and r-only isotopes
A stable isotope is s-only when a stable isobar of lower Z (a nucleus with the same mass number and fewer protons) shields it from the decays that follow the r-process, while the s-process path passes through it; samarium-148 and lead-204 are examples. An r-only isotope is a neutron-rich one the s-process path skips, such as ytterbium-176. Their abundances separate the two processes.
- Maxwellian-averaged cross sectionMACS
In a star the neutrons have the spread of energies of a hot gas, the Maxwell–Boltzmann distribution. The Maxwellian-averaged cross section is the capture cross section averaged over that distribution at a given temperature, which is what stellar models need. By convention it is quoted at a thermal energy kT = 30 keV, about 350 million kelvin. The measured values are collected in KADoNiS, the Karlsruhe Astrophysical Database of Nucleosynthesis in Stars (Dillmann et al. 2014), and now in its successor ASTRAL, which recalculates them from the raw data of each measurement (Vescovi et al. 2023).
σ(E) is the cross section at neutron energy E (centre of mass), kT the thermal energy. See also: Cross section, Time of flight, Activation
- Time of flight
A way to measure a neutron's energy: neutrons of all energies are made in a short burst, and the time each takes to cover a known distance, tens to hundreds of metres, gives its speed and so its energy. One experiment then measures a cross section at many energies at once. n_TOF at CERN, GELINA in Geel and LANSCE in Los Alamos work this way.
See also: Maxwellian-averaged cross section, Activation
- Activation
A way to count reactions after the fact: a sample is irradiated, and the radioactive nuclei the reaction made are counted afterwards from their decay. For the s-process, neutrons with a spectrum close to that in a star, made by protons on lithium, irradiate the sample beside a gold foil, whose well-known capture serves as the reference.
See also: Maxwellian-averaged cross section, Time of flight
- LUNA and underground laboratoriesLaboratory for Underground Nuclear Astrophysics
At the Earth's surface, cosmic rays leave far more signals in a γ-ray detector than the few reactions of a measurement at stellar energies. A laboratory deep underground is shielded by the rock above it. LUNA, at the Gran Sasso National Laboratories of INFN in Italy, has run accelerators under about 1.4 km of rock since the early 1990s; the rock reduces the cosmic-ray muons by about a factor of a million. Besides LUNA's 400 kV accelerator, the Bellotti Ion Beam Facility (BIBF), inaugurated there in 2023, hosts a 3.5 MV machine for helium and carbon burning. Other deep underground accelerators run in China (JUNA) and the United States (CASPAR); Felsenkeller in Dresden, under 45 m of rock, is a shallow-underground laboratory that adds active muon vetoes.
- At the surface, cosmic rays make a steady rain of muons, which leave signals in any γ-ray detector.
- About 1400 m of rock (3800 m water equivalent) stop all but about one muon in a million.
- LUNA 400 kV: proton and helium beams for hydrogen burning, since the early 2000s.
- The 3.5 MV machine of the Bellotti Ion Beam Facility (BIBF), inaugurated in 2023: proton, helium and carbon beams for helium and carbon burning.
- The beam hits a target; a detector around it counts the γ-rays or particles from the few reactions.
Schematic, not to scale.
See also: Metres water equivalent, Gamow window
- Metres water equivalentm w.e.
A way of comparing the depth of underground laboratories under different rock: the thickness of water that would stop cosmic rays as well as the rock does. The roughly 1.4 km of rock above the Gran Sasso laboratories corresponds to about 3800 m w.e.
See also: LUNA and underground laboratories
- Electron screening
In a laboratory target the nuclei are inside atoms, and their electrons partly hide the nuclear charge from an incoming nucleus. This lowers the electric barrier and makes the measured cross section at the lowest energies larger than that of bare nuclei. In a star the screening comes from the free electrons of the plasma and is different, so laboratory data must be corrected; the enhancement measured in the laboratory is often larger than atomic physics predicts.
See also: Cross section, Trojan Horse method
- R-matrixR-matrix theory, AZURE2
A theoretical framework for fitting nuclear reaction data. Inside a certain radius the nucleus is described by a set of levels, each with an energy and widths; outside it, the particles move under the known electric force. Fitting these parameters to all the data on a nucleus at once gives a model that can be extended to the low energies of stars. AZURE2 is an R-matrix code, and the evaluations on this site use it.
See also: Astrophysical S-factor, Resonance, Asymptotic normalization coefficient
- Asymptotic normalization coefficientANC
When a capture at very low energy goes directly, without passing through a resonance, it happens mostly far outside the nucleus (it is peripheral), where only the tail of the final nucleus's wave function matters. The ANC measures the size of that tail. It can be found from transfer reactions at higher energy, where the cross section is large, and then gives the direct, non-resonant part of the capture at stellar energies without measuring the capture itself; resonances are added separately.
See also: R-matrix
- Spin and parityJπ
How a nuclear level spins, and how it looks in a mirror: the two labels that decide which levels two colliding nuclei can form. Every level carries them, written together as Jπ, for example 3/2−. J is the spin, the level's total angular momentum, counted in units of ħ; the parity π, + or −, says whether its wave function stays the same or changes sign when mirrored through the centre. Both are conserved in a reaction, so they fix which levels can form and with how much orbital angular momentum the two nuclei must come together.
See also: Excited state, Resonance
- Partial and reduced widthΓ, γ2
How readily a level decays through one channel, for example by emitting a proton or a γ-ray. The partial width Γ is an energy (eV, keV): the larger it is, the faster that decay and the broader the resonance; the widths of all channels add up to the level's total width. The reduced width γ2 is the same quantity with the penetrability of the Coulomb and centrifugal barriers divided out, so it measures the nuclear structure alone. An R-matrix fit adjusts these widths, together with the level energies, to the data.
See also: R-matrix, Resonance, Asymptotic normalization coefficient
- Particle–γ coincidence
Recording a particle and a γ-ray only when they arrive together, within a few billionths of a second. A real reaction often gives both, a proton and the γ-ray of the nucleus left behind, while most background gives only one; requiring the two together removes most of the background, at the cost of missing the decays that give no γ-ray.
See also: Cross section
- Normalizationscale factor, systematic uncertainty
Every measured data set carries an overall scale uncertainty, from the beam current, the number of target nuclei or the detector efficiency, which moves all its points up or down together. In a fit such as an R-matrix evaluation each data set may be given a normalization factor, allowed to move only within its quoted scale uncertainty.
See also: R-matrix, χ2 (chi-squared)
- χ2 (chi-squared)χ2/N
A measure of how well a model fits data: for each point, the difference between model and measurement divided by the measurement's uncertainty, squared, and summed over all points. Divided by the number of points, χ2/N close to 1 means the model agrees with the data within their uncertainties; much larger values mean it does not, or that the uncertainties are underestimated.
See also: R-matrix, Normalization
- Trojan Horse methodTHM
An indirect method. The reaction of interest is hidden inside another one at higher energy: the projectile is carried in by a larger nucleus (the Trojan horse, for example a deuteron carrying a proton), which passes the Coulomb barrier easily. Selecting the events in which the rest of the carrier is only a spectator gives the energy dependence of the low-energy reaction, without Coulomb suppression or electron screening. The result is normalized to direct data at higher energy.
See also: Electron screening
- Inverse kinematics
Swapping the roles of beam and target: the heavier nucleus is accelerated and hits a target of hydrogen or helium. The products then fly forward in a narrow cone, where they can be caught and identified, and the heavy nucleus can be a short-lived one that could never be made into a target.
See also: Recoil separator
- Recoil separator
A set of magnets and electric fields that picks out the few product nuclei of a reaction in inverse kinematics from the beam, which is billions of times more intense and moves in almost the same direction. Counting the products directly gives the cross section. DRAGON at TRIUMF, ERNA and St. George are examples.
See also: Inverse kinematics
- Fusion hindrance
A fall of the fusion probability of two heavy nuclei, at energies far below the Coulomb barrier, faster than the usual models predict. If it happens for two carbon nuclei, the S-factor reaches a maximum and then drops towards stellar energies, and carbon burns more slowly than the standard rate says. Whether it does is still debated.
See also: Astrophysical S-factor
- Superburst
A rare, very long X-ray burst from a neutron star that pulls matter from a companion star. Ordinary X-ray bursts, from helium and hydrogen burning near the surface, last seconds to minutes; superbursts last hours and are thought to start when carbon ignites deeper down, in the ashes of earlier bursts.