The neutron–proton ratio (N/Z ratio or nuclear ratio) of an atomic nucleus is the

ratio In mathematics, a ratio shows how many times one number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lan ...

of its number of neutrons to its number of protons. Among stable nuclei and naturally occurring nuclei, this ratio generally increases with increasing atomic number. This is because electrical repulsive forces between protons scale with distance differently than

strong nuclear force The strong interaction or strong force is a fundamental interaction that confines quarks into proton, neutron, and other hadron particles. The strong interaction also binds neutrons and protons to create atomic nuclei, where it is called the ...

attractions. In particular, most pairs of protons in large nuclei are not far enough apart, such that electrical repulsion dominates over the strong nuclear force, and thus proton density in stable larger nuclei must be lower than in stable smaller nuclei where more pairs of protons have appreciable short-range nuclear force attractions. For each element with atomic number ''Z'' small enough to occupy only the first three nuclear shells, that is up to that of

calcium Calcium is a chemical element with the symbol Ca and atomic number 20. As an alkaline earth metal, calcium is a reactive metal that forms a dark oxide-nitride layer when exposed to air. Its physical and chemical properties are most similar ...

(''Z'' = 20), there exists a stable isotope with ''N''/''Z'' ratio of one, with the exception of

beryllium Beryllium is a chemical element with the symbol Be and atomic number 4. It is a steel-gray, strong, lightweight and brittle alkaline earth metal. It is a divalent element that occurs naturally only in combination with other elements to form ...

(''N''/''Z'' = 1.25) and every element with odd atomic number between 9 and 19 inclusive (''N'' = ''Z'' + 1). Hydrogen-1 (''N''/''Z'' ratio = 0) and

helium-3 Helium-3 (3He see also helion) is a light, stable isotope of helium with two protons and one neutron (the most common isotope, helium-4, having two protons and two neutrons in contrast). Other than protium (ordinary hydrogen), helium-3 is the ...

(''N''/''Z'' ratio = 0.5) are the only stable isotopes with neutron–proton ratio under one. Uranium-238 has the highest ''N''/''Z'' ratio of any

primordial nuclide In geochemistry, geophysics and nuclear physics, primordial nuclides, also known as primordial isotopes, are nuclides found on Earth that have existed in their current form since before Earth was formed. Primordial nuclides were present in the ...

at 1.587, while lead-208 has the highest ''N''/''Z'' ratio of any known stable isotope at 1.537. Radioactive decay generally proceeds so as to change the ''N''/''Z'' ratio to increase stability. If the ''N''/''Z'' ratio is greater than 1,

alpha decay Alpha decay or α-decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle (helium nucleus) and thereby transforms or 'decays' into a different atomic nucleus, with a mass number that is reduced by four and an ato ...

increases the ''N''/''Z'' ratio, and hence provides a common pathway towards stability for decays involving large nuclei with too few neutrons.

Positron emission Positron emission, beta plus decay, or β+ decay is a subtype of radioactive decay called beta decay, in which a proton inside a radionuclide nucleus is converted into a neutron while releasing a positron and an electron neutrino (). Positron emis ...

and electron capture also increase the ratio, while

beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For e ...

decreases the ratio. Nuclear waste exists mainly because nuclear fuel has a higher stable ''N''/''Z'' ratio than its

fission products Nuclear fission products are the atomic fragments left after a large atomic nucleus undergoes nuclear fission. Typically, a large nucleus like that of uranium fissions by splitting into two smaller nuclei, along with a few neutrons, the release ...

Semi-empirical description

For stable nuclei, the neutron-proton ratio is such that the

binding energy In physics and chemistry, binding energy is the smallest amount of energy required to remove a particle from a system of particles or to disassemble a system of particles into individual parts. In the former meaning the term is predominantly use ...

is at a local minimum or close to a minimum. From the liquid drop model, this bonding energy is approximated by empirical Bethe–Weizsäcker formula :

E_ = a_ A - a_ A^ - a_ \frac - a_ \frac \pm \delta(A,Z).

Given a value of

A

and ignoring the contributions of nucleon spin pairing (i.e. ignoring the

\pm \delta(A,Z)

term), the binding energy is a quadratic expression in

Z

that is minimized when the neutron-proton ratio is

N/Z \approx 1 + \frac A^

References

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Semi-empirical description

See also

References