Spin quantum number
In chemistry and quantum mechanics, the spin quantum number is a quantum number that describes the intrinsic angular momentum of an electron or other particle. It has the same value for all particles of the same type, such as = 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 . The value of is the component of spin angular momentum, in units of the reduced Planck constant, parallel to a given direction. It can take values ranging from + to − in integer increments. For an electron, can be either or .
Nomenclature
The phrase spin quantum number refers to quantized spin angular momentum.The symbol is used for the spin quantum number, and is described as the spin magnetic quantum number or as the -component of spin.
Both the total spin and the z-component of spin are quantized, leading to two quantum numbers spin and spin magnet quantum numbers. The spin quantum number has only one value for every elementary particle. Some introductory chemistry textbooks describe as the spin quantum number, and is not mentioned since its value is a fixed property of the electron; some even use the variable in place of.
The two spin quantum numbers and are the spin angular momentum analogs of the two orbital angular momentum quantum numbers and.
Spin quantum numbers apply also to systems of coupled spins, such as atoms that may contain more than one electron. Capitalized symbols are used: for the total electronic spin, and or for the -axis component. A pair of electrons in a spin singlet state has = 0, and a pair in the triplet state has = 1, with = −1, 0, or +1. Nuclear-spin quantum numbers are conventionally written for spin, and or for the -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. The factor two suggests a fundamental two-fold symmetry for all stable atoms." This 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
The theory of spatial quantization of the spin moment of the momentum of electrons of atoms situated in the magnetic field needed to be proved experimentally. In 1922 Otto Stern and Walter Gerlach observed it in the experiment they conducted.Silver atoms were evaporated using an electric furnace in a vacuum. Using thin slits, the atoms were guided into a flat beam and the beam sent through an in-homogeneous magnetic field before colliding with a metallic plate. The laws of classical physics predict that the collection of condensed silver atoms on the plate should form a thin solid line in the same shape as the original beam. However, the in-homogeneous magnetic field caused the beam to split in two separate directions, creating two lines on the metallic plate.
The phenomenon can be explained with the spatial quantization of the spin moment of momentum. In atoms the electrons are paired such that one spins upward and one downward, neutralizing the effect of their spin on the action of the atom as a whole. But in the valence shell of silver atoms, there is a single electron whose spin remains unbalanced.
The unbalanced spin creates spin magnetic moment, making the electron act like a very small magnet. As the atoms pass through the in-homogeneous magnetic field, the force moment in the magnetic field influences the electron's dipole until its position matches the direction of the stronger field. The atom would then be pulled toward or away from the stronger magnetic field a specific amount, depending on the value of the valence electron's spin. When the spin of the electron is the atom moves away from the stronger field, and when the spin is the atom moves toward it. Thus the beam of silver atoms is split while traveling through the in-homogeneous magnetic field, according to the spin of each atom's valence electron.
In 1927, Thomas Erwin Phipps and conducted a similar experiment, using atoms of hydrogen with similar results. Later scientists conducted experiments using other atoms that have only one electron in their valence shell:. Every time there were two lines formed on the metallic plate.
The atomic nucleus also may have spin, but protons and neutrons are much heavier than electrons, and the magnetic dipole moment is inversely proportional to the mass. So the nuclear magnetic dipole momentum is much smaller than that of the whole atom. This small magnetic dipole was later measured by Stern, Otto Frisch and Immanuel Estermann.
Energy levels from the Dirac equation
In 1928, Paul Dirac developed a relativistic wave equation, now termed the Dirac equation, which predicted the spin magnetic moment correctly, and at the same time treated the electron as a point-like particle. Solving the Dirac equation for the energy levels of an electron in the hydrogen atom, all four quantum numbers including occurred naturally and agreed well with experiment.Electron spin
A spin- particle is characterized by an angular momentum quantum number for spin =. In solutions of the Schrödinger-Pauli equation, angular momentum is quantized according to this number, so that magnitude of the spin angular momentum isThe 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 :
whose solution has only two possible -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:
where is the electron charge, is the electron mass, and is the electron spin g-factor, which is approximately 2.0023.
Its z-axis projection is given by the spin magnetic quantum number according to:
where 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. 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.
Nuclear spin
also have spins. The nuclear spin is a fixed property of each nucleus and may be either an integer or a half-integer. The component of nuclear spin parallel to the -axis can have values, −1,..., −. For example, a N nucleus has = 1, so that there are 3 possible orientations relative to the -axis, corresponding to states = +1, 0 and −1.The spins of different nuclei are interpreted using the nuclear shell model. Even-even nuclei with even numbers of both protons and neutrons, such as C and O, have spin zero. Odd mass number nuclei have half-integer spins, such as for Li, for C and for O, usually corresponding to the angular momentum of the last nucleon added. Odd-odd nuclei with odd numbers of both protons and neutrons have integer spins, such as 3 for B, and 1 for N. Values of nuclear spin for a given isotope are found in the lists of isotopes for each element.