Editing Semi-empirical mass formula (section)

===Pairing term===
[[File:Pairing_term_nuclear_physics.gif|thumb|300x300px|Magnitude of the pairing term in the total binding energy for even–even and odd–odd nuclei, as a function of mass number. Two fits are shown (blue and red line). The pairing term (positive for even–even and negative for odd–odd nuclei) was derived from binding energy data.<ref>{{cite journal | doi=10.1088/1674-1137/36/12/002 | title=The Ame2012 atomic mass evaluation | date=2012 | last1=Audi | first1=G. | last2=Wang | first2=M. | last3=Wapstra | first3=A.H. | last4=Kondev | first4=F.G. | last5=MacCormick | first5=M. | last6=Xu | first6=X. | last7=Pfeiffer | first7=B. | journal=Chinese Physics C | volume=36 | issue=12 | pages=1287–1602 | bibcode=2012ChPhC..36....2A }}</ref>]]
The term <math>\delta(A, Z)</math> is known as the ''pairing term'' (possibly also known as the pairwise interaction). This term captures the effect of [[Spin (physics)|spin]] coupling. It is given by<ref name=":0">{{Cite book |last=Martin |first=B. R. |title=Nuclear and particle physics: an introduction |date=2019 |author2=G. Shaw |isbn=978-1-119-34462-9 |edition=Third |location=Hoboken, NJ |page=62|oclc=1078954632}}</ref>

: <math>\delta(A, Z) = \begin{cases}
 +\delta_0 & \text{for even } Z, N ~(\text{even } A), \\
 0 & \text{for odd } A, \\
 -\delta_0 & \text{for odd } Z, N ~(\text{even } A),
\end{cases}</math>

where <math>\delta_0</math> is found empirically to have a value of about 1000&nbsp;keV, slowly decreasing with mass number&nbsp;''A''. Odd-odd nuclei tend to undergo beta decay to an adjacent even-even nucleus by changing a neutron to a proton or vice versa. The pairs have overlapping wave functions and sit very close together with a bond stronger than any other configuration.<ref name=":0" /> When the pairing term is substituted into the binding energy equation, for even ''Z'', ''N'', the pairing term adds binding energy, and for odd ''Z'', ''N'' the pairing term removes binding energy.

The dependence on mass number is commonly parametrized as
: <math>\delta_0 = a_\text{P} A^{k_\text{P}}.</math>

The value of the exponent ''k''<sub>P</sub> is determined from experimental binding-energy data. In the past its value was often assumed to be&nbsp;−3/4, but modern experimental data indicate that a value of&nbsp;−1/2 is nearer the mark:
: <math>\delta_0 = a_\text{P} A^{-1/2}</math> or <math>\delta_0 = a_\text{P} A^{-3/4}.</math>

Due to the [[Pauli exclusion principle]] the nucleus would have a lower energy if the number of protons with spin up were equal to the number of protons with spin down. This is also true for neutrons. Only if both ''Z'' and ''N'' are even, can both protons and neutrons have equal numbers of spin-up and spin-down particles. This is a similar effect to the asymmetry term.

The factor <math>A^{k_\text{P}}</math> is not easily explained theoretically. The Fermi-ball calculation we have used above, based on the liquid-drop model but neglecting interactions, will give an <math>A^{-1}</math> dependence, as in the asymmetry term. This means that the actual effect for large nuclei will be larger than expected by that model. This should be explained by the interactions between nucleons. For example, in the [[nuclear shell model|shell model]], two protons with the same quantum numbers (other than [[Spin (physics)|spin]]) will have completely overlapping [[wavefunction]]s and will thus have greater [[strong interaction]] between them and stronger binding energy. This makes it energetically favourable (i.e. having lower energy) for protons to form pairs of opposite spin. The same is true for neutrons.