Innuclear physics, thevalley of stability (also called thebelt of stability,nuclear valley,energy valley, orbeta stability valley) is a characterization of the stability ofnuclides toradioactivity based on their binding energy.^[1] Nuclides are composed ofprotons andneutrons. The shape of the valley refers to the profile of binding energy as a function of the numbers of neutrons and protons, with the lowest part of the valley corresponding to the region of moststable nuclei.^[2] The line of stable nuclides down the center of the valley of stability is known as theline of beta stability. The sides of the valley correspond to increasing instability tobeta decay (β⁻ or β⁺). The decay of a nuclide becomes more energetically favorable the further it is from the line of beta stability. The boundaries of the valley correspond to thenuclear drip lines, where nuclides become so unstable they emitsingle protons orsingle neutrons. Regions of instability within the valley at highatomic number also include radioactive decay byalpha radiation orspontaneous fission. The shape of the valley is roughly an elongatedparaboloid corresponding to the nuclidebinding energies as a function of neutron and atomic numbers.^[1]

The nuclides within the valley of stability encompass the entiretable of nuclides. The chart of those nuclides is also known as a Segrè chart, after the physicistEmilio Segrè.^[3] The Segrè chart may be considered a map of the nuclear valley. The region of proton and neutron combinations outside of the valley of stability is referred to as the sea of instability.^[4][5]

Scientists have long searched for long-lived heavy isotopes outside of the valley of stability,^[6][7][8] hypothesized byGlenn T. Seaborg in the late 1960s.^[9][10] These relatively stable nuclides are expected to have particular configurations of "magic" atomic andneutron numbers, and form a so-calledisland of stability.

All atomic nuclei are composed of protons and neutrons bound together by thenuclear force. There are 286primordial nuclides that occur naturally on earth, each corresponding to a unique number of protons, called theatomic number,Z, and a unique number of neutrons, called theneutron number,N. Themass number,A, of a nuclide is the sum of atomic and neutron numbers,A =Z +N. Not all nuclides are stable, however. According to Byrne,^[3] stable nuclides are defined as those having ahalf-life greater than 10¹⁸ years, and there are many combinations of protons and neutrons that form nuclides that are unstable. A common example of an unstable nuclide iscarbon-14 that decays bybeta decay intonitrogen-14 with ahalf-life of about 5,730 years:

¹⁴
₆C →¹⁴
₇N +e⁻
+ν
_e

In this form of decay, the original element becomes a new chemical element in a process known asnuclear transmutation and a beta particle and an electronantineutrino are emitted. An essential property of this and all nuclide decays is that the total energy of thedecay product is less than that of the original nuclide. The difference between the initial and final nuclide binding energies is carried away by the kinetic energies of the decay products, often the beta particle and its associated neutrino.^[3]

The concept of thevalley of stability is a way of organizing all of the nuclides according tobinding energy as a function of neutron and proton numbers.^[1] Most stable nuclides have roughly equal numbers of protons and neutrons, so the line for whichZ =N forms a rough initial line defining stable nuclides. The greater the number of protons, the more neutrons are required to stabilize a nuclide; nuclides with larger values forZ require an even larger number of neutrons,N >Z, to be stable. The valley of stability is formed by the negative of binding energy, the binding energy being the energy required to break apart the nuclide into its proton and neutron components. The stable nuclides have high binding energy, and these nuclides lie along the bottom of the valley of stability. Nuclides with weaker binding energy have combinations ofN andZ that lie off of the line of stability and further up the sides of the valley of stability. Unstable nuclides can be formed innuclear reactors orsupernovas, for example. Such nuclides often decay in sequences ofreactions calleddecay chains that take the resulting nuclides sequentially down the slopes of the valley of stability. The sequence of decays take nuclides toward greater binding energies, and the nuclides terminating the chain are stable.^[1] The valley of stability provides both a conceptual approach for how to organize the myriad stable and unstable nuclides into a coherent picture and an intuitive way to understand how and why sequences of radioactive decay occur.^[1]

• 
  Chart of nuclides (isotopes) by binding energy, depicting the valley of stability. The diagonal line corresponds to equal numbers of neutrons and protons. Dark blue squares represent nuclides with the greatest binding energy, hence they correspond to the most stable nuclides. The binding energy is greatest along the floor of the valley of stability.
• 
  Chart of nuclides by half-life. Black squares represent nuclides with the longest half-lives, hence they correspond to the most stable nuclides. The most stable, long-lived nuclides lie along the floor of the valley of stability. Nuclides with more than 20 protons must have more neutrons than protons to be stable.
• 
  Chart of nuclides by type of decay. Black squares are stable nuclides. Nuclides with excessive neutrons or protons are unstable to β⁻ (light blue) or β⁺ (green) decay, respectively. At high atomic number, alpha emission (orange) or spontaneous fission (dark blue) become common decay modes.

The protons and neutrons that comprise an atomic nucleus behave almost identically within the nucleus. The approximate symmetry ofisospin treats these particles as identical, but in a different quantum state. This symmetry is only approximate, however, and thenuclear force that binds nucleons together is a complicated function depending on nucleon type, spin state, electric charge, momentum, etc. and with contributions from non-central forces. The nuclear force is not a fundamental force of nature, but a consequence of the residual effects of thestrong force that surround the nucleons. One consequence of these complications is that althoughdeuterium, a bound state of a proton (p) and a neutron (n) is stable, exotic nuclides such asdiproton ordineutron are unbound.^[11] The nuclear force is not sufficiently strong to form either p-p or n-n bound states, or equivalently, the nuclear force does not form apotential well deep enough to bind these identical nucleons.^{[citation needed]}

Stable nuclides require approximately equal numbers of protons and neutrons. The stable nuclidecarbon-12 (¹²C) is composed of six neutrons and six protons, for example. Protons have a positive charge, hence within a nuclide with many protons there are large repulsive forces between protons arising from theCoulomb force. By acting to separate protons from one another, the neutrons within a nuclide play an essential role in stabilizing nuclides. With increasing atomic number, even greater numbers of neutrons are required to obtain stability. The heaviest stable element,lead (Pb), has many more neutrons than protons. The stable nuclide²⁰⁶Pb hasZ = 82 andN = 124, for example. For this reason, the valley of stability does not follow the lineZ = N for A larger than 40 (Z = 20 is the elementcalcium).^[3] Neutron number increases along the line of beta stability at a faster rate than atomic number.

The line of beta stability follows a particular curve ofneutron–proton ratio, corresponding to the most stable nuclides. On one side of the valley of stability, this ratio is small, corresponding to an excess of protons over neutrons in the nuclides. These nuclides tend to be unstable to β⁺ decay or electron capture, since such decay converts a proton to a neutron. The decay serves to move the nuclides toward a more stable neutron-proton ratio. On the other side of the valley of stability, this ratio is large, corresponding to an excess of neutrons over protons in the nuclides. These nuclides tend to be unstable to β⁻ decay, since such decay converts neutrons to protons. On this side of the valley of stability, β⁻ decay also serves to move nuclides toward a more stable neutron-proton ratio.

The mass of an atomic nucleus is given by

${\displaystyle m=Z{m}_p+N{m}_n-\frac{E_B}{c^2}}$

where${\displaystyle m_p}$ and${\displaystyle m_n}$ are the rest mass of a proton and a neutron, respectively, and${\displaystyle E_B}$ is the totalbinding energy of the nucleus. Themass–energy equivalence is used here. The binding energy is subtracted from the sum of the proton and neutron masses because the mass of the nucleus isless than that sum. This property, called themass defect, is necessary for a stable nucleus; within a nucleus, the nuclides are trapped by apotential well. A semi-empirical mass formula states that the binding energy will take the form

${\displaystyle E_B=a_{V}A-a_S{A}^{2/3}-a_C\frac{Z^2}{A^{1/3}}-a_A\frac{(A-2Z{)}^2}{A}\pm \delta (A,Z)}$^[12]

The difference between the mass of a nucleus and the sum of the masses of the neutrons and protons that comprise it is known as themass defect. E_B is often divided by the mass number to obtain binding energy per nucleon for comparisons of binding energies between nuclides. Each of the terms in this formula has a theoretical basis. The coefficients${\displaystyle a_V}$,${\displaystyle a_S}$,${\displaystyle a_C}$,${\displaystyle a_A}$ and a coefficient that appears in the formula for${\displaystyle \delta (A,Z)}$ are determined empirically.

The binding energy expression gives a quantitative estimate for the neutron-proton ratio. The energy is a quadratic expression inZ that is minimized when the neutron-proton ratio is${\displaystyle N/Z\approx 1+\frac{a_C}{2a_A}{A}^{2/3}}$. This equation for the neutron-proton ratio shows that in stable nuclides the number of neutrons is greater than the number of protons by a factor that scales as${\displaystyle A^{2/3}}$.

The figure at right shows the average binding energy per nucleon as a function of atomic mass number along the line of beta stability, that is, along the bottom of the valley of stability. For very small atomic mass number (H, He, Li), binding energy per nucleon is small, and this energy increases rapidly with atomic mass number.Nickel-62 (28 protons, 34 neutrons) has the highest mean binding energy of all nuclides, whileiron-58 (26 protons, 32 neutrons) andiron-56 (26 protons, 30 neutrons) are a close second and third.^[13] These nuclides lie at the very bottom of the valley of stability. From this bottom, the average binding energy per nucleon slowly decreases with increasing atomic mass number. The heavy nuclide²³⁸U is not stable, but is slow to decay with a half-life of 4.5 billion years.^[1] It has relatively small binding energy per nucleon.

For β⁻ decay, nuclear reactions have the generic form

^A
_ZX →^A
_Z+1X′ +e⁻
+ν
_e^[14]

whereA andZ are themass number andatomic number of the decaying nucleus, and X and X′ are the initial and final nuclides, respectively. For β⁺ decay, the generic form is

^A
_ZX →^A
_Z−1X′ +e⁺
+ν
_e^[14]

These reactions correspond to the decay of a neutron to a proton, or the decay of a proton to a neutron, within the nucleus, respectively. These reactions begin on one side or the other of the valley of stability, and the directions of the reactions are to move the initial nuclides down the valley walls towards a region of greater stability, that is, toward greater binding energy.

The figure at right shows the average binding energy per nucleon across the valley of stability for nuclides with mass numberA = 125.^[15] At the bottom of this curve istellurium (₅₂Te), which is stable. Nuclides to the left of₅₂Te are unstable with an excess of neutrons, while those on the right are unstable with an excess of protons. A nuclide on the left therefore undergoes β⁻ decay, which converts a neutron to a proton, hence shifts the nuclide to the right and toward greater stability. A nuclide on the right similarly undergoes β⁺ decay, which shifts the nuclide to the left and toward greater stability.

Heavy nuclides are susceptible to α decay, and these nuclear reactions have the generic form,

^A
_ZX →^A-4
_Z-2X′ +⁴
₂He

As in β decay, the decay product X′ has greater binding energy and it is closer to the middle of the valley of stability. Theα particle carries away two neutrons and two protons, leaving a lighter nuclide. Since heavy nuclides have many more neutrons than protons, α decay increases a nuclide's neutron-proton ratio.

The boundaries of the valley of stability, that is, the upper limits of the valley walls, are the neutron drip line on the neutron-rich side, and the proton drip line on the proton-rich side. The nucleon drip lines are at the extremes of the neutron-proton ratio. At neutron–proton ratios beyond the drip lines, no nuclei can exist. The location of the neutron drip line is not well known for most of the Segrè chart, whereas the proton and alpha drip lines have been measured for a wide range of elements. Drip lines are defined for protons, neutrons, and alpha particles, and these all play important roles in nuclear physics.

The difference in binding energy between neighboring nuclides increases as the sides of the valley of stability are ascended, and correspondingly the nuclide half-lives decrease, as indicated in the figure above. If one were to add nucleons one at a time to a given nuclide, the process will eventually lead to a newly formed nuclide that is so unstable that it promptly decays by emitting a proton (or neutron). Colloquially speaking, the nucleon has 'leaked' or 'dripped' out of the nucleus, hence giving rise to the term "drip line".

Proton emission is not seen in naturally occurring nuclides. Proton emitters can be produced vianuclear reactions, usually utilizinglinear particle accelerators (linac). Although prompt (i.e. not beta-delayed) proton emission was observed from an isomer incobalt-53 as early as 1969, no other proton-emitting states were found until 1981, when the proton radioactive ground states oflutetium-151 andthulium-147 were observed at experiments at theGSI in West Germany.^[16] Research in the field flourished after this breakthrough, and to date more than 25 nuclides have been found to exhibit proton emission. The study of proton emission has aided the understanding of nuclear deformation, masses and structure, and it is an example ofquantum tunneling.

Two examples of nuclides that emit neutrons areberyllium-13 (mean life2.7×10⁻²¹ s) andhelium-5 (7×10⁻²² s). Since only a neutron is lost in this process, the atom does not gain or lose any protons, and so it does not become an atom of a different element. Instead, the atom will become a newisotope of the original element, such asberyllium-13 becomingberyllium-12 after emitting one of its neutrons.^[17]

Innuclear engineering, aprompt neutron is aneutron immediately emitted by anuclear fission event. Prompt neutrons emerge from the fission of an unstablefissionable orfissile heavy nucleus almost instantaneously.Delayed neutron decay can occur within the same context, emitted afterbeta decay of one of thefission products. Delayed neutron decay can occur at times from a few milliseconds to a few minutes.^[18] The U.S.Nuclear Regulatory Commission defines a prompt neutron as a neutron emerging from fission within 10⁻¹⁴ seconds.^[19]

The island of stability is a region outside the valley of stability where it is predicted that a set of heavyisotopes with nearmagic numbers of protons and neutrons will locally reverse the trend of decreasing stability inelements heavier than uranium.The hypothesis for the island of stability is based upon thenuclear shell model, which implies that theatomic nucleus is built up in "shells" in a manner similar to the structure of the much larger electron shells in atoms. In both cases, shells are just groups of quantumenergy levels that are relatively close to each other. Energy levels from quantum states in two different shells will be separated by a relatively large energy gap. So when the number ofneutrons andprotons completely fills theenergy levels of a given shell in the nucleus, thebinding energy per nucleon will reach a local maximum and thus that particular configuration will have a longer lifetime than nearby isotopes that do not possess filled shells.^[20]

A filled shell would have "magic numbers" of neutrons and protons. One possible magic number of neutrons for spherical nuclei is 184, and some possible matching proton numbers are 114, 120 and 126. These configurations imply that the most stable spherical isotopes would beflerovium-298,unbinilium-304 andunbihexium-310. Of particular note is²⁹⁸Fl, which would be "doubly magic" (both itsproton number of 114 andneutron number of 184 are thought to be magic). This doubly magic configuration is the most likely to have a very long half-life. The next lighter doubly magic spherical nucleus islead-208, the heaviest known stable nucleus and most stable heavy metal.

The valley of stability can be helpful in interpreting and understanding properties of nuclear decay processes such asdecay chains andnuclear fission.

Radioactive decay often proceeds via a sequence of steps known as a decay chain. For example,²³⁸U decays to²³⁴Th which decays to^234mPa and so on, eventually reaching²⁰⁶Pb:

${\displaystyle \begin{array}{l}\\ {}_{92}{}^{238}\text{U}\underset{4.5\times {10}^9\text{y}}{\overset{\alpha }{\to }}{}_{90}{}^{234}\text{Th}\underset{24\text{d}}{\overset{{\beta }^{-}}{\to }}{}_{91}{}^{234\text{m}}\text{Pa}\underset{1\text{min}}{\overset{{\beta }^{-}}{\to }}{}_{92}{}^{234}\text{U}\underset{2.4\times {10}^5\text{y}}{\overset{\alpha }{\to }}{}_{90}{}^{230}\text{Th}\underset{7.7\times {10}^4\text{y}}{\overset{\alpha }{\to }}\\ {}_{88}{}^{226}\text{Ra}\underset{1600y}{\overset{\alpha }{\to }}{}_{86}{}^{222}\text{Rn}\underset{3.8\text{d}}{\overset{\alpha }{\to }}{}_{84}{}^{218}\text{Po}\underset{3\text{min}}{\overset{\alpha }{\to }}{}_{82}{}^{214}\text{Pb}\underset{27\text{min}}{\overset{{\beta }^{-}}{\to }}{}_{83}{}^{214}\text{Bi}\underset{20\text{min}}{\overset{{\beta }^{-}}{\to }}\\ {}_{84}{}^{214}\text{Po}\underset{164\mu \text{s}}{\overset{\alpha }{\to }}{}_{82}{}^{210}\text{Pb}\underset{22\text{y}}{\overset{{\beta }^{-}}{\to }}{}_{83}{}^{210}\text{Bi}\underset{5\text{d}}{\overset{{\beta }^{-}}{\to }}{}_{84}{}^{210}\text{Po}\underset{138\text{d}}{\overset{\alpha }{\to }}{}_{82}{}^{206}\text{Pb}\\ \end{array}}$

With each step of this sequence of reactions, energy is released and thedecay products move further down the valley of stability towards the line of beta stability.²⁰⁶Pb is stable and lies on the line of beta stability.

Thefission processes that occur withinnuclear reactors are accompanied by the release of neutrons that sustain thechain reaction. Fission occurs when a heavy nuclide such asuranium-235 absorbs a neutron and breaks into nuclides of lighter elements such asbarium orkrypton, usually with the release of additional neutrons. Like all nuclides with a high atomic number, these uranium nuclei require many neutrons to bolster their stability, so they have a large neutron-proton ratio (N/Z). The nuclei resulting from a fission (fission products) inherit a similarN/Z, but have atomic numbers that are approximately half that of uranium.^[1] Isotopes with the atomic number of the fission products and anN/Z near that of uranium or other fissionable nuclei have too many neutrons to be stable; this neutron excess is why multiple free neutrons but no free protons are usually emitted in the fission process, and it is also why many fission product nuclei undergo a long chain of β⁻ decays, each of which converts a nucleusN/Z to (N − 1)/(Z + 1), whereN andZ are, respectively, the numbers of neutrons and protons contained in the nucleus.

When fission reactions are sustained at a given rate, such as in a liquid-cooled or solid fuel nuclear reactor, the nuclear fuel in the system produces manyantineutrinos for each fission that has occurred. These antineutrinos come from the decay of fission products that, as their nuclei progress down a β⁻ decay chain toward the valley of stability, emit an antineutrino along with each β⁻ particle. In 1956,Reines andCowan exploited the (anticipated) intense flux of antineutrinos from a nuclear reactor in the design ofan experiment to detect and confirm the existence of these elusive particles.^[21]

• The Live Chart of Nuclides - IAEA with filter on decay type
• The Valley of Stability (video) – a virtual "flight" through 3D representation of the nuclide chart, byCEA (France)
• The nuclear landscape: The variety and abundance of nuclei – Chapter 6 of the bookNucleus: A trip into the heart of matter by Mackintosh, Ai-Khalili, Jonson, and Pena describes the valley of stability and its implications (Baltimore, Maryland:The Johns Hopkins University Press), 2001.ISBN 0-801 8-6860-2

• Nuclear physics
• Radioactivity
• Isotopes

• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from October 2019