Editing Quantum number (section)

==Atomic nuclei==
In [[Atomic nucleus|nuclei]], the entire assembly of [[proton]]s and [[neutron]]s ([[nucleon]]s) has a resultant [[angular momentum]] due to the angular momenta of each nucleon, usually denoted {{mvar|I}}. If the total angular momentum of a neutron is {{math|1=''j''<sub>n</sub> = ''{{ell}}'' + ''s''}} and for a proton is {{math|1=''j''<sub>p</sub> = ''{{ell}}'' + ''s''}} (where {{mvar|s}} for protons and neutrons happens to be {{sfrac|1|2}} again (''see note'')), then the '''nuclear angular momentum quantum numbers''' {{mvar|I}} are given by:

<math display=block>I = |j_n - j_p|, |j_n - j_p| + 1, |j_n - j_p| + 2, \cdots, (j_n + j_p) - 2, (j_n + j_p) - 1, (j_n + j_p)</math>

''Note: ''The orbital angular momenta of the nuclear (and atomic) states are all integer multiples of ħ while the intrinsic angular momentum of the neutron and  proton are half-integer multiples.  It should be immediately apparent that the combination of the intrinsic spins of the nucleons with their orbital motion will always give half-integer values for the total spin, {{mvar|I}}, of any odd-A nucleus and integer values for any even-A nucleus.

Parity with the number {{mvar|I}} is used to label nuclear angular momentum states, examples for some isotopes of [[hydrogen]] (H), [[carbon]] (C), and [[sodium]] (Na) are;<ref name="Krane 1988">{{cite book|title=Introductory Nuclear Physics |first=K. S. |last=Krane |date=1988 |publisher=John Wiley & Sons |isbn=978-0-471-80553-3}}{{page needed|date=February 2019}}</ref>

:{|
| style="text-align:right;" | {{nuclide|link=yes|Hydrogen|1}} || {{mvar|I}} = ({{sfrac|1|2}})<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|9}} || {{mvar|I}} = ({{sfrac|3|2}})<sup>−</sup> || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|20}} || {{mvar|I}} = 2<sup>+</sup>
|-
| style="text-align:right;" | {{nuclide|link=yes|Hydrogen|2}} || {{mvar|I}} = 1<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|10}} || {{mvar|I}} = 0<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|21}} || {{mvar|I}} = ({{sfrac|3|2}})<sup>+</sup>
|-
| style="text-align:right;" | {{nuclide|link=yes|Hydrogen|3}} || {{mvar|I}} = ({{sfrac|1|2}})<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|11}} || {{mvar|I}} = ({{sfrac|3|2}})<sup>−</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|22}} || {{mvar|I}} = 3<sup>+</sup>
|-
| || || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|12}} || {{mvar|I}} = 0<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|23}} || {{mvar|I}} = ({{sfrac|3|2}})<sup>+</sup>
|-
| || || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|13}} || {{mvar|I}} = ({{sfrac|1|2}})<sup>−</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|24}} || {{mvar|I}} = 4<sup>+</sup>
|-
| || || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|14}} || {{mvar|I}} = 0<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|25}} || {{mvar|I}} = ({{sfrac|5|2}})<sup>+</sup>
|-
| || || &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Carbon|15}} || {{mvar|I}} = ({{sfrac|1|2}})<sup>+</sup>|| &nbsp; || style="text-align:right;" | {{nuclide|link=yes|Sodium|26}} || {{mvar|I}} = 3<sup>+</sup>
|-
|}

The reason for the unusual fluctuations in {{mvar|I}}, even by differences of just one nucleon, are due to the odd and even numbers of protons and neutrons – pairs of nucleons have a total angular momentum of zero (just like electrons in orbitals), leaving an odd or even number of unpaired nucleons. The property of nuclear spin is an important factor for the operation of [[NMR]] spectroscopy in [[organic chemistry]],<ref name="Atkins 1977" /> and [[MRI]] in [[nuclear medicine]],<ref name="Krane 1988" /> due to the [[nuclear magnetic moment]] interacting with an external [[magnetic field]].