Editing Spin quantum number (section)

==Electron spin==
{{Main|Spin (physics)}}
<!-- This section is linked from [[Caesium]] -->
A spin-{{sfrac| 1 |2}} particle is characterized by an [[angular momentum quantum number]] for spin {{mvar|''s''}} = {{sfrac| 1 |2}}. In solutions of the [[Pauli equation|Schrödinger-Pauli equation]], angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is

<math display="block"> \| \bold{S} \| = \hbar\sqrt{s(s+1)} = \tfrac{\sqrt{3}}{2}\ \hbar ~.</math>

The hydrogen spectrum [[fine structure]] is observed as a doublet corresponding to two possibilities for the ''z''-component of the angular momentum, where for any given direction {{mvar|z}}:
<math display="block"> s_z = \pm \tfrac{1}{2}\hbar ~.</math>

whose solution has only two possible {{mvar|z}}-components for the electron.  In the electron, the two different spin orientations are sometimes called "spin-up" or "spin-down".

The spin property of an electron would give rise to [[magnetic moment]], which was a requisite for the fourth quantum number. 

The magnetic moment vector of an electron spin is given by:

<math display="block">\ \boldsymbol{\mu}_\text{s} = -\frac{e}{\ 2m\ }\ g_\text{s}\ \bold{S}\ </math>

where <math>-e</math> is the [[elementary charge|electron charge]], <math>m</math> is the [[electron mass]], and <math>g_\text{s}</math> is the [[g-factor (physics)#Electron_spin_g-factor|electron spin g-factor]], which is approximately 2.0023.
Its ''z''-axis projection is given by the spin magnetic quantum number <math>m_\text{s}</math> according to:

<math display="block">\mu_z = -m_\text{s}\ g_\text{s}\ \mu_\mathsf{B} = \pm \tfrac{1}{2}\ g_\text{s}\ \mu_\mathsf{B}\ </math>

where <math>\ \mu_\mathsf{B}\ </math> is the [[Bohr magneton]].

When atoms have even numbers of electrons the spin of each electron in each orbital has opposing orientation to that of its immediate neighbor(s). However, many atoms have an odd number of electrons or an arrangement of electrons in which there is an unequal number of "spin-up" and "spin-down" orientations. These atoms or electrons are said to have unpaired spins that are detected in [[electron spin resonance]].