Nuclear drip line

The nuclear drip line is the boundary beyond which atomic nuclei are unbound with respect to the emission of a proton or neutron.
An arbitrary combination of protons and neutrons does not necessarily yield a stable nucleus. One can think of moving up or to the right across the table of nuclides by adding a proton or a neutron, respectively, to a given nucleus. However, adding nucleons one at a time to a given nucleus will eventually lead to a newly formed nucleus that immediately decays by emitting a proton. Colloquially speaking, the nucleon has leaked or dripped out of the nucleus, hence giving rise to the term drip line.
Drip lines are defined for protons and neutrons at the extreme of the proton-to-neutron ratio; at p:n ratios at or beyond the drip lines, no bound nuclei can exist. While the location of the proton drip line is well known for many elements, the location of the neutron drip line is only known for elements up to neon.

General description

Nuclear stability is limited to those combinations of protons and neutrons described by the chart of the nuclides, also called the valley of stability. The boundaries of this valley are the neutron drip line on the neutron-rich side, and the proton drip line on the proton-rich side. These limits exist because of particle decay, whereby an exothermic nuclear transition can occur by the emission of one or more nucleons. As such, the drip line may be defined as the boundary beyond which proton or neutron separation energy becomes negative, favoring the emission of a particle from a newly formed unbound system.

Allowed transitions

When considering whether a specific nuclear transmutation, a reaction or a decay, is energetically allowed, one only needs to sum the masses of the initial nucleus/nuclei and subtract from that value the sum of the masses of the product particles. If the result, or Q-value, is positive, then the transmutation is allowed, or exothermic because it releases energy, and if the Q-value is a negative quantity, then it is endothermic as at least that much energy must be added to the system before the transmutation may proceed. For example, to determine if ¹²C, the most common isotope of carbon, can undergo proton emission to ¹¹B, one finds that about 16 MeV must be added to the system for this process to be allowed. While Q-values can be used to describe any nuclear transmutation, for particle decay, the particle separation energy quantity S, is also used, and it is equivalent to the negative of the Q-value. In other words, the proton separation energy S_p indicates how much energy must be added to a given nucleus to remove a single proton. Thus, the particle drip lines defined the boundaries where the particle separation energy is less than or equal to zero, for which the spontaneous emission of that particle is energetically allowed.
Although the location of the drip lines is well defined as the boundary beyond which particle separation energy becomes negative, the definition of what constitutes a nucleus or an unbound resonance is unclear. Some known nuclei of light elements beyond the drip lines decay with lifetimes on the order of 10⁻²² seconds; this is sometimes defined to be a limit of nuclear existence because several fundamental nuclear processes occur on this timescale. For more massive nuclei, particle emission half-lives may be significantly longer due to a stronger Coulomb barrier and enable other transitions such as alpha and beta decay to instead occur. This renders unambiguous determination of the drip lines difficult, as nuclei with lifetimes long enough to be observed exist far longer than the timescale of particle emission and are most probably bound. Consequently, particle-unbound nuclei are difficult to observe directly, and are instead identified through their decay energy.

Nuclear structure origin of the drip lines

The energy of a nucleon in a nucleus is its rest mass energy minus a binding energy. In addition to this, there is an energy due to degeneracy: for instance, a nucleon with energy E₁ will be forced to a higher energy E₂ if all the lower energy states are filled. This is because nucleons are fermions and obey Fermi–Dirac statistics. The work done in putting this nucleon to a higher energy level results in a pressure, which is the degeneracy pressure. When the effective binding energy, or Fermi energy, reaches zero, adding a nucleon of the same isospin to the nucleus is not possible, as the new nucleon would have a negative effective binding energy — i.e. it is more energetically favourable for the nucleon to be created outside the nucleus. This defines the particle drip point for that species.

One- and two-particle drip lines

In many cases, nuclides along the drip lines are not contiguous, but rather are separated by so-called one-particle and two-particle drip lines. This is a consequence of even and odd nucleon numbers affecting binding energy, as nuclides with even numbers of nucleons generally have a higher binding energy, and hence greater stability, than adjacent odd nuclei. These energy differences result in the one-particle drip line in an odd-Z or odd-N nuclide, for which prompt proton or neutron emission is energetically favorable in that nuclide and all other odd nuclides further outside the drip line. However, the next even nuclide outside the one-particle drip line may still be particle stable if its two-particle separation energy is non-negative. This is possible because the two-particle separation energy is always greater than the one-particle separation energy, and a transition to a less stable odd nuclide is energetically forbidden. The two-particle drip line is thus defined where the two-particle separation energy becomes negative, and denotes the outermost boundary for particle stability of a species.
The one- and two-neutron drip lines have been experimentally determined up to neon, though unbound odd-N isotopes are known or deduced through non-observance for every element up to magnesium. For example, the last bound odd-N fluorine isotope is ²⁶F, though the last bound even-N isotope is ³¹F.

Nuclei near the drip lines are uncommon on Earth

Of the three types of naturally occurring radioactivities, only alpha decay is a type of decay resulting from the nuclear strong force. The other proton and neutron decays occurred much earlier in the life of the atomic species and before the earth was formed. Thus, alpha decay can be considered either a form of particle decay or, less frequently, as a special case of nuclear fission. The timescale for the nuclear strong force is much faster than that of the nuclear weak force or the electromagnetic force, so the lifetime of nuclei past the drip lines are typically on the order of nanoseconds or less. For alpha decay, the timescale can be much longer than for proton or neutron emission owing to the high Coulomb barrier seen by an alpha-cluster in a nucleus. As a consequence, there are no naturally occurring nuclei on Earth that undergo proton or neutron emission; however, such nuclei can be created, for example, in the laboratory with accelerators or naturally in stars. The Facility for Rare Isotope Beams at Michigan State University came online in mid-2022 and has created many novel radioisotopes, each of which is extracted in a beam and used for study. FRIB runs a beam of relatively stable isotopes through a medium, which disrupts the nuclei and creates numerous novel nuclei, which are then extracted.

Nucleosynthesis

Explosive astrophysical environments often have very large fluxes of high-energy nucleons that can be captured on seed nuclei. In these environments, radiative proton or neutron capture will occur much faster than beta decays, and as astrophysical environments with both large neutron fluxes and high-energy protons are unknown at present, the reaction flow will proceed away from beta-stability towards or up to either the neutron or proton drip lines, respectively. However, once a nucleus reaches a drip line, as we have seen, no more nucleons of that species can be added to the particular nucleus, and the nucleus must first undergo a beta decay before further nucleon captures can occur.

Photodisintegration

While the drip lines impose the ultimate boundaries for nucleosynthesis, in high-energy environments the burning pathway may be limited before the drip lines are reached by photodisintegration, where a high-energy gamma ray knocks a nucleon out of a nucleus. The same nucleus is subject both to a flux of nucleons and photons, so an equilibrium between neutron capture and photodisintegration is reached for nuclides with a sufficiently low neutron separation energy, particularly those near waiting points.
As the photon bath will typically be described by a Planckian distribution, higher energy photons will be less abundant, and so photodisintegration will not become significant until the nucleon separation energy begins to approach zero towards the drip lines, where photodisintegration may be induced by lower energy gamma rays. At kelvin, the photon distribution is energetic enough to knock nucleons out of any nuclei that have particle separation energies less than 3 MeV, but to know which nuclei exist in what abundances one must also consider the competing radiative captures.
As neutron captures can proceed in any energy regime, neutron photodisintegration is unimportant except at higher energies. However, as proton captures are inhibited by the Coulomb barrier, the cross sections for those charged-particle reactions at lower energies are greatly suppressed, and in the higher energy regimes where proton captures have a large probability to occur, there is often a competition between the proton capture and the photodisintegration that occurs in explosive hydrogen burning; but because the proton drip line is relatively much closer to the valley of beta-stability than is the neutron drip line, nucleosynthesis in some environments may proceed as far as either nucleon drip line.