Even and odd atomic nuclei

In nuclear physics, properties of a nucleus depend on evenness or oddness of its atomic number (proton number) Z, neutron number N and, consequently, of their sum, the mass number A. Most importantly, oddness of both Z and N tends to lower the nuclear binding energy, making odd nuclei generally less stable. This effect is not only experimentally observed, but is included in the semi-empirical mass formula and explained by some other nuclear models, such as the nuclear shell model. This difference of nuclear binding energy between neighbouring nuclei, especially of odd-A isobars, has important consequences for beta decay.

The nuclear spin is zero for even-Z, even-N nuclei, integer for all even-A nuclei, and odd half-integer for all odd-A nuclei.

Even vs. odd mass number (A).

	Even	Odd	Total
Stable	150	101	251
Long-lived	26	9	35
All primordial	176	110	286

The neutron–proton ratio is not the only factor affecting nuclear stability. Adding neutrons to isotopes can vary their nuclear spins and nuclear shapes, causing differences in neutron capture cross sections and gamma spectroscopy and nuclear magnetic resonance properties. If too many or too few neutrons are present with regard to the nuclear binding energy optimum, the nucleus becomes unstable and subject to certain types of nuclear decay. Unstable nuclides with a nonoptimal number of neutrons or protons decay by beta decay (including positron decay), electron capture, or other means, such as spontaneous fission and cluster decay.

Even mass number

Even-mass-number nuclides, which comprise 150/251 = ~60% of all stable nuclides, are bosons, i.e., they have integer spin. 145 of the 150 are even-proton, even-neutron (EE) nuclides, which necessarily have spin 0 because of pairing. The remainder of the stable bosonic nuclides are five odd-proton, odd-neutron stable nuclides (²
₁H
, ⁶
₃Li
, ¹⁰
₅B
, ¹⁴
₇N
and ^180m
₇₃Ta
), all having a non-zero integer spin.

Pairing effects

Even/odd Z, N (Hydrogen-1 included as OE)

p,n	EE	OO	EO	OE	Total
Stable	145	5	53	48	251
Long-lived	22	4	4	5	35
All primordial	167	9	57	53	286

Beta decay of an even–even nucleus produces an odd–odd nucleus, and vice versa. An even number of protons or of neutrons are more stable (higher binding energy) because of pairing effects, so even–even nuclei are much more stable than odd–odd. One effect is that there are few stable odd–odd nuclides, but another effect is to prevent beta decay of many even–even nuclei into another even–even nucleus of the same mass number but lower energy, because decay proceeding one step at a time would have to pass through an odd–odd nucleus of higher energy. Double beta decay directly from even–even to even–even skipping over an odd–odd nuclide is only occasionally possible, and even then with a half-life greater than a billion times the age of the universe. For example, the double beta emitter ¹¹⁶
Cd
has a half-life of 2.9×10¹⁹ years. This makes for a larger number of stable even–even nuclides, with some mass numbers having two stable nuclides, and some elements (atomic numbers) having as many as seven.

For example, the extreme stability of helium-4 due to a double pairing of two protons and two neutrons prevents any nuclides containing five or eight nucleons from existing for long enough to serve as platforms for the buildup of heavier elements via nuclear fusion in Big Bang nucleosynthesis; only in stars is there enough time for this (see triple-alpha process). This is also the reason why ⁸
₄Be
decays so quickly into two alpha particles, making beryllium the only even-numbered element that is monoisotopic.

Even proton, even neutron

There are 145 stable even–even nuclides, forming ~58% of the 251 stable nuclides. There are also 22 primordial long-lived even–even nuclides. As a result, many of the 41 even-numbered elements from 2 to 82 have many primordial isotopes. Half of these even-numbered elements have six or more stable isotopes. The lightest stable even-even isotope is ⁴
₂He
and the heaviest is ²⁰⁸
₈₂Pb
. These are also the lightest and heaviest known doubly magic nuclides.^{[1] 208}
₈₂Pb
is the final decay product of ²³²
₉₀Th
,^[2] a primordial radionuclide with an even proton and neutron number. ²³⁸
₉₂U
is another notable primordial radionuclide with a half life of 4.468 billion years,^[3] and produces almost half of all radioactive heat within the Earth.^[4]

All even–even nuclides have spin 0 in their ground state, due to the Pauli exclusion principle (See Pairing Effects for more details).

Odd proton, odd neutron

Only five stable nuclides contain both an odd number of protons and an odd number of neutrons. The first four "odd–odd" nuclides occur in low mass nuclides, for which changing a proton to a neutron or vice versa would lead to a very lopsided proton–neutron ratio (²
₁H
, ⁶
₃Li
, ¹⁰
₅B
, and ¹⁴
₇N
; spins 1, 1, 3, 1). All four of these isotopes have the same number of protons and neutrons, and they all have an odd number for their nuclear spin. The only other observationally "stable" odd–odd nuclide is ^180m
₇₃Ta
(spin 9), the only primordial nuclear isomer, which has not yet been observed to decay despite experimental attempts.^[5] Also, four long-lived radioactive odd–odd nuclides (⁴⁰
₁₉K
– the most common radioisotope in the human body,^{[6][7] 50}
₂₃V
,¹³⁸
₅₇La
,¹⁷⁶
₇₁Lu
with spins 4, 6, 5, 7, respectively) occur naturally. As in the case of ^180m
₇₃Ta
decay of high spin nuclides by beta decay (including electron capture), gamma decay, or internal conversion is greatly inhibited if the only decay possible between isobar nuclides (or in the case of ^180m
₇₃Ta
between nuclear isomers of the same nuclide) involves high multiples of a change in spin of 1 unit, the "preferred" change of spin that is associated with rapid decay. This high-spin inhibition of decay is the cause of the five heavy stable or long-lived odd-proton, odd-neutron nuclides discussed above. For an example of this effect where the spin effect is subtracted, tantalum-180, the odd–odd low-spin (theoretical) decay product of primordial tantalum-180m, itself has a half life of only about eleven hours.^[8]

Many odd–odd radionuclides (like tantalum-180) with comparatively short half lives are known. Almost invariably, these decay by positive or negative beta decay, in order to produce stable even–even isotopes which have paired protons and paired neutrons. In some odd–odd radionuclides where the ratio of protons to neutrons is neither excessively large nor excessively small (i.e., falling too far from the ratio of maximal stability), this decay can proceed in either direction, turning a proton into a neutron, or vice versa. An example is ⁶⁴
₂₉Cu
, which can decay either by positron emission to ⁶⁴
₂₈Ni
, or by electron emission to ⁶⁴
₃₀Zn
.

Of the nine primordial odd–odd nuclides (five stable and four radioactive with long half lives), only ¹⁴
₇N
is the most common isotope of a common element. This is the case because proton capture on ¹⁴
₇N
is the rate-limiting step of the CNO-I cycle. The nuclides ⁶
₃Li
and ¹⁰
₅B
are minority isotopes of elements that are themselves rare compared to other light elements, while the other six isotopes make up only a tiny percentage of the natural abundance of their elements. For example, ^180m
₇₃Ta
is thought to be the rarest of the 251 stable nuclides.

None of the primordial (i.e., stable or nearly stable) odd–odd nuclides have spin 0 in the ground state. This is because the single unpaired neutron and unpaired proton have a larger nuclear force attraction to each other if their spins are aligned (producing a total spin of at least 1 unit), instead of anti-aligned. See deuterium for the simplest case of this nuclear behavior.

Odd mass number

For a given odd mass number, there is exactly one beta-stable nuclide. There is not a difference in binding energy between even–odd and odd–even comparable to that between even–even and odd–odd, leaving other nuclides of the same mass number (isobars) free to beta decay toward the lowest-mass nuclide. For mass numbers of 147, 151, and 209+, the beta-stable isobar of that mass number has been observed to undergo alpha decay. (In theory, mass number 143 to 155, 160 to 162, and 165+ can also alpha decay.) This gives a total of 101 stable nuclides with odd mass numbers. There are another nine radioactive primordial nuclides (which by definition all have relatively long half lives, greater than 80 million years) with odd mass numbers.

Odd-mass-number nuclides are fermions, i.e., have half-integer spin. Generally speaking, since odd-mass-number nuclides always have an even number of either neutrons or protons, the even-numbered particles usually form part of a "core" in the nucleus with a spin of zero. The unpaired nucleon with the odd number (whether proton or neutron) is then responsible for the nuclear spin, which is the sum of the orbital angular momentum and spin angular momentum of the remaining nucleon.

The odd-mass number stable nuclides are divided (roughly evenly) into odd-proton–even-neutron, and odd-neutron–even-proton nuclides, which are more thoroughly discussed below.

Odd proton, even neutron

These 48 stable nuclides, stabilized by their even numbers of paired neutrons, form most of the stable isotopes of the odd-numbered elements; the very few odd–odd nuclides comprise the others. There are 41 odd-numbered elements with Z = 1 through 81, of which 30 (including hydrogen, since zero is an even number) have one stable odd-even isotope, the elements technetium (
₄₃Tc
) and promethium (
₆₁Pm
) have no stable isotopes, and nine elements: chlorine (
₁₇Cl
), potassium (
₁₉K
), copper (
₂₉Cu
), gallium (
₃₁Ga
), bromine (
₃₅Br
), silver (
₄₇Ag
), antimony (
₅₁Sb
), iridium (
₇₇Ir
), and thallium (
₈₁Tl
), have two odd–even stable isotopes each. This makes a total of 30×1 + 9×2 = 48 stable odd–even isotopes. The lightest example of this type of nuclide is ¹
₁H
(protium) as zero is an even number while the heaviest example is ²⁰⁵
₈₁Tl
. There are also five primordial long-lived radioactive odd–even isotopes, ⁸⁷
₃₇Rb
,^{[9] 115}
₄₉In
,^{[10][11] 187}
₇₅Re
,^{[12] 151}
₆₃Eu
,^[13][14] and ²⁰⁹
₈₃Bi
.^[15][16] The last two were only recently found to undergo alpha decay, with half-lives greater than 10¹⁸ years.

Even proton, odd neutron

Even–odd long-lived

	Decay	Half-life
^113 48Cd	beta	7.7×10^15 a
^147 62Sm	alpha	1.06×10^11 a
^235 92U	alpha	7.04×10^8 a

These 53 stable nuclides have an even number of protons and an odd number of neutrons. By definition, they are all isotopes of even-Z elements, where they are a minority in comparison to the even–even isotopes which are about 3 times as numerous. Among the 41 even-Z elements that have a stable nuclide, only two elements (argon and cerium) have no even–odd stable nuclides. One element (tin) has three. There are 24 elements that have one even–odd nuclide and 13 that have two even–odd nuclides. The lightest example of this type of nuclide is ³
₂He
and the heaviest is ²⁰⁷
₈₂Pb
.

Of 34 primordial radionuclides there exist three even–odd nuclides (see table at right), including the fissile ²³⁵
₉₂U
. Because of their odd neutron numbers, the even–odd nuclides tend to have large neutron capture cross sections, due to the energy that results from neutron-pairing effects.

These stable even-proton odd-neutron nuclides tend to be uncommon by abundance in nature, generally because in order to form and contribute to the primordial abundance, they must have escaped capturing neutrons to form yet other stable even–even isotopes, during both the s-process and r-process of neutron capture, during nucleosynthesis in stars. For this reason, only ¹⁹⁵
₇₈Pt
and ⁹
₄Be
are the most naturally abundant isotopes of their element, the former only by a small margin, and the latter only because the expected beryllium-8 has lower binding energy than two alpha particles and therefore immediately alpha decays.

Odd neutron number

Neutron number parity

N	Even	Odd
Stable	193	58
Long-lived	27	8
All primordial	220	66

Actinides with odd neutron numbers are generally fissile (with thermal neutrons), while those with even neutron numbers are generally not, though they are fissionable with fast neutrons. Only ⁹
₄Be
, ¹⁴
₇N
, and ¹⁹⁵
₇₈Pt
have an odd neutron number and are the most naturally abundant isotope of their element.

References

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