Spin quantum number

In physics and chemistry, the spin quantum number is a quantum number (designated s) that describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. It has the same value for all particles of the same type, such as s = ⁠1/2⁠ for all electrons. It is an integer for all bosons, such as photons, and a half-odd-integer for all fermions, such as electrons and protons.

The component of the spin along a specified axis is given by the spin magnetic quantum number, conventionally written m_s. The value of m_s is the component of spin angular momentum, in units of the reduced Planck constant ħ, parallel to a given direction (conventionally labelled the z–axis). It can take values ranging from +s to −s in integer increments. For an electron, m_s can be either ⁠++1/2⁠ or ⁠−+1/2⁠ .

The phrase spin quantum number refers to quantized spin angular momentum. The symbol s is used for the spin quantum number, and m_s is described as the spin magnetic quantum number or as the z-component of spin s_z.

Both the total spin and the z-component of spin are quantized, leading to two quantum numbers spin and spin magnet quantum numbers. The (total) spin quantum number has only one value for every elementary particle. Some introductory chemistry textbooks describe m_s as the spin quantum number, and s is not mentioned since its value ⁠1/2⁠ is a fixed property of the electron; some even use the variable s in place of m_s.

The two spin quantum numbers ${\displaystyle s}$ and ${\displaystyle m_{s}}$ are the spin angular momentum analogs of the two orbital angular momentum quantum numbers ${\displaystyle l}$ and ${\displaystyle m_{l}}$.

Spin quantum numbers apply also to systems of coupled spins, such as atoms that may contain more than one electron. Capitalized symbols are used: S for the total electronic spin, and m_S or M_S for the z-axis component. A pair of electrons in a spin singlet state has S = 0, and a pair in the triplet state has S = 1, with m_S = −1, 0, or +1. Nuclear-spin quantum numbers are conventionally written I for spin, and m_I or M_I for the z-axis component.

The name "spin" comes from a geometrical spinning of the electron about an axis, as proposed by Uhlenbeck and Goudsmit. However, this simplistic picture was quickly realized to be physically unrealistic, because it would require the electrons to rotate faster than the speed of light. It was therefore replaced by a more abstract quantum-mechanical description.

During the period between 1916 and 1925, much progress was being made concerning the arrangement of electrons in the periodic table. In order to explain the Zeeman effect in the Bohr atom, Sommerfeld proposed that electrons would be based on three 'quantum numbers', n, k, and m, that described the size of the orbit, the shape of the orbit, and the direction in which the orbit was pointing. Irving Langmuir had explained in his 1919 paper regarding electrons in their shells, "Rydberg has pointed out that these numbers are obtained from the series ${\displaystyle N=2(1+2^{2}+2^{2}+3^{2}+3^{2}+4^{2})}$. The factor two suggests a fundamental two-fold symmetry for all stable atoms." This ${\displaystyle 2n^{2}}$ configuration was adopted by Edmund Stoner, in October 1924 in his paper 'The Distribution of Electrons Among Atomic Levels' published in the Philosophical Magazine.