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Concept# Spin quantum number

Summary

In physics, 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 ms. The value of ms 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, ms can be either + 1/2 or − 1/2 .
The phrase spin quantum number was originally used to describe the fourth of a set of quantum numbers (the principal quantum number n, the azimuthal quantum number l, the magnetic quantum number m, and the spin magnetic quantum number ms), which completely describe the quantum state of an electron in an atom.
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, sometimes using the variable s in place of m_s. Some authors discourage this usage as it causes confusion. At a more advanced level where quantum mechanical operators or coupled spins are introduced, s is referred to as the spin quantum number, and m_s is described as the spin magnetic quantum number or as the z-component of spin s_z.
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 mS or MS 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 mS = −1, 0, or +1. Nuclear-spin quantum numbers are conventionally written I for spin, and mI or MI for the z-axis component.

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