In chemistry and quantum mechanics, 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 .
Nomenclature
The phrase spin quantum number refers to quantized spin angular momentum. The symbol s is used for the spin quantum number, and ms is described as the spin magnetic quantum number or as the z-component of spin sz. 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 ms 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 ms. The two spin quantum numbers s {\displaystyle s} and m s {\displaystyle m_{s}} are the spin angular momentum analogs of the two orbital angular momentum quantum numbers l {\displaystyle l} and m l {\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 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.
History
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 model of the atom, Arnold 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 N = 2 ( 1 + 2 2 + 2 2 + 3 2 + 3 2 + 4 2 ) {\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 2 n 2 {\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. The qualitative success of the Sommerfeld quantum number scheme failed to explain the Zeeman effect in weak magnetic field strengths, the anomalous Zeeman effect. In December 1924, Wolfgang Pauli showed that the core electron angular momentum was not related to the effect as had previously been assumed. Rather he proposed that only the outer "light" electrons determined the angular momentum and he hypothesized that this required a fourth quantum number with a two-valuedness. This fourth quantum number became the spin magnetic quantum number.
Name The name "spin" comes from a geometrical spinning of the electron about an axis, as proposed by George Uhlenbeck and Samuel 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.
Detection of spin When lines of the hydrogen spectrum are examined at very high resolution, they are found to be closely spaced doublets. This splitting is called fine structure, and was one of the first experimental evidences for electron spin. The direct observation of the electron's intrinsic angular momentum was achieved in the Stern–Gerlach experiment.
Stern–Gerlach experiment
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