Spin quantum number

The spin quantum number helps to explain the magnetic properties of atoms and molecules. A spinning electron behaves like a micromagnet with a definite magnetic moment. If an atomic or molecular orbital contains two electrons, then their magnetic moments oppose and cancel each other.

If all orbitals are doubly occupied by electrons, the net magnetic moment is zero and the substance behaves as diamagnetic; it is repelled by the external magnetic field. If some orbitals are half filled (singly occupied), the substance has a net magnetic moment and is paramagnetic; it is attracted by the external magnetic field.

Early attempts to explain the behavior of electrons in atoms focused on solving the Schrödinger wave equation for the hydrogen atom, the simplest possible case, with a single electron bound to the atomic nucleus. This was successful in explaining many features of atomic spectra.

The solutions required each possible state of the electron to be described by three "quantum numbers". These were identified as, respectively, the electron "shell" number n, the "orbital" number ℓ, and the "orbital angular momentum" number m. Angular momentum is a so-called "classical" concept measuring the momentum of a mass in circular motion about a point. The shell numbers start at 1 and increase indefinitely. Each shell of number n contains n² orbitals. Each orbital is characterized by its number ℓ, where ℓ takes integer values from 0 to n − 1, and its angular momentum number m, where m takes integer values from +ℓ to −ℓ. By means of a variety of approximations and extensions, physicists were able to extend their work on hydrogen to more complex atoms containing many electrons.

Atomic spectra measure radiation absorbed or emitted by electrons "jumping" from one "state" to another, where a state is represented by values of n, ℓ, and m. The so-called "transition rule" limits what "jumps" are possible. In general, a jump or "transition" is allowed only if all three numbers change in the process. This is because a transition will be able to cause the emission or absorption of electromagnetic radiation only if it involves a change in the electromagnetic dipole of the atom.

However, it was recognized in the early years of quantum mechanics that atomic spectra measured in an external magnetic field (see Zeeman effect) cannot be predicted with just n, ℓ, and m.

In January 1925, when Ralph Kronig was still a Columbia University Ph.D. student, he first proposed electron spin after hearing Wolfgang Pauli in Tübingen. Werner Heisenberg and Pauli immediately hated the idea: They had just ruled out all imaginable actions from quantum mechanics. Now Kronig was proposing to set the electron rotating in space. Pauli especially ridiculed the idea of spin, saying that "it is indeed very clever but of course has nothing to do with reality". Faced with such criticism, Kronig decided not to publish his theory and the idea of electron spin had to wait for others to take the credit.[8] Ralph Kronig had come up with the idea of electron spin several months before George Uhlenbeck and Samuel Goudsmit, but most textbooks credit these two Dutch physicists with the discovery.

Pauli subsequently proposed (also in 1925) a new quantum degree of freedom (or quantum number) with two possible values, in order to resolve inconsistencies between observed molecular spectra and the developing theory of quantum mechanics.

Shortly thereafter Uhlenbeck and Goudsmit identified Pauli's new degree of freedom as electron spin.

A spin- 1 /2 particle is characterized by an angular momentum quantum number for spin s =  1 /2. In solutions of the Schrödinger-Pauli equation, angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is

$${\displaystyle \|{\mathbf {S}}\|=\hbar {\sqrt {s(s+1)}}={\tfrac {\sqrt {3}}{2}}\ \hbar ~.}$$

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 z:

$${\displaystyle s_{z}=\pm {\tfrac {1}{2}}\hbar ~.}$$

whose solution has only two possible 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:

$${\displaystyle \ {\boldsymbol {\mu }}_{\text{s}}=-{\frac {e}{\ 2m\ }}\ g_{\text{s}}\ {\mathbf {S}}\ }$$

where ${\displaystyle -e}$ is the electron charge, ${\displaystyle m}$ is the electron mass, and ${\displaystyle g_{\text{s}}}$ is the electron spin g-factor, which is approximately 2.0023. Its z-axis projection is given by the spin magnetic quantum number ${\displaystyle m_{\text{s}}}$ according to:

$${\displaystyle \mu _{z}=-m_{\text{s}}\ g_{\text{s}}\ \mu _{\mathsf {B}}=\pm {\tfrac {1}{2}}\ g_{\text{s}}\ \mu _{\mathsf {B}}\ }$$

where ${\displaystyle \ \mu _{\mathsf {B}}\ }$ 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.

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.

For a solution of either the nonrelativistic Pauli equation or the relativistic Dirac equation, the quantized angular momentum (see angular momentum quantum number) can be written as:

$${\displaystyle \Vert \mathbf {s} \Vert ={\sqrt {s\,(s+1)\,}}\,\hbar }$$

where

• ${\displaystyle \mathbf {s} }$ is the quantized spin vector or spinor
• ${\displaystyle \Vert \mathbf {s} \Vert }$ is the norm of the spin vector
• s is the spin quantum number associated with the spin angular momentum
• ${\displaystyle \hbar }$ is the reduced Planck constant.

Given an arbitrary direction z (usually determined by an external magnetic field) the spin z-projection is given by

${\displaystyle s_{z}=m_{s}\,\hbar }$

where m_s is the secondary spin quantum number, ranging from −s to +s in steps of one. This generates 2 s + 1 different values of m_s.

The allowed values for s are non-negative integers or half-integers. Fermions have half-integer values, including the electron, proton and neutron which all have s = ++ 1 /2 . Bosons such as the photon and all mesons) have integer spin values.

The algebraic theory of spin is a carbon copy of the angular momentum in quantum mechanics theory. First of all, spin satisfies the fundamental commutation relation:

$${\displaystyle \ [S_{i},S_{j}]=i\ \hbar \ \epsilon _{ijk}\ S_{k}\ ,}$$

$${\displaystyle \ \left[S_{i},S^{2}\right]=0\ }$$

where ${\displaystyle \ \epsilon _{ijk}\ }$ is the (antisymmetric) Levi-Civita symbol. This means that it is impossible to know two coordinates of the spin at the same time because of the restriction of the uncertainty principle.

Next, the eigenvectors of ${\displaystyle \ S^{2}\ }$ and ${\displaystyle \ S_{z}\ }$ satisfy:

$${\displaystyle \ S^{2}\ |s,m_{s}\rangle ={\hbar }^{2}\ s(s+1)\ |s,m_{s}\rangle \ }$$

$${\displaystyle \ S_{z}\ |s,m_{s}\rangle =\hbar \ m_{s}\ |s,m_{s}\rangle \ }$$

$${\displaystyle \ S_{\pm }\ |s,m_{s}\rangle =\hbar \ {\sqrt {s(s+1)-m_{s}(m_{s}\pm 1)\ }}\;|s,m_{s}\pm 1\rangle \ }$$

where ${\displaystyle \ S_{\pm }=S_{x}\pm iS_{y}\ }$ are the ladder (or "raising" and "lowering") operators.

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 s occurred naturally and agreed well with experiment.

For some atoms the spins of several unpaired electrons (s₁, s₂, ...) are coupled to form a total spin quantum number S.[9][10] This occurs especially in light atoms (or in molecules formed only of light atoms) when spin–orbit coupling is weak compared to the coupling between spins or the coupling between orbital angular momenta, a situation known as L S coupling because L and S are constants of motion. Here L is the total orbital angular momentum quantum number.[10]

For atoms with a well-defined S, the multiplicity of a state is defined as 2S + 1. This is equal to the number of different possible values of the total (orbital plus spin) angular momentum J for a given (L, S) combination, provided that S ≤ L (the typical case). For example, if S = 1, there are three states which form a triplet. The eigenvalues of S_z for these three states are +1ħ, 0, and −1ħ.[9] The term symbol of an atomic state indicates its values of L, S, and J.

As examples, the ground states of both the oxygen atom and the dioxygen molecule have two unpaired electrons and are therefore triplet states. The atomic state is described by the term symbol ³P, and the molecular state by the term symbol ³Σ⁻
_g.

Atomic nuclei also have spins. The nuclear spin I is a fixed property of each nucleus and may be either an integer or a half-integer. The component m_I of nuclear spin parallel to the z–axis can have (2I + 1) values I, I–1, ..., –I. For example, a ¹⁴N nucleus has I = 1, so that there are 3 possible orientations relative to the z–axis, corresponding to states m_I = +1, 0 and −1.[11]

The spins I of different nuclei are interpreted using the nuclear shell model. Even-even nuclei with even numbers of both protons and neutrons, such as ¹²C and ¹⁶O, have spin zero. Odd mass number nuclei have half-integer spins, such as 3/ 2  for ⁷Li,  1 /2 for ¹³C and 5/ 2  for ¹⁷O, 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 ¹⁰B, and 1 for ¹⁴N.[12] Values of nuclear spin for a given isotope are found in the lists of isotopes for each element. (See isotopes of oxygen, isotopes of aluminium, etc. etc.)

• Total angular momentum quantum number
• Rotational spectroscopy
• Basic quantum mechanics

• Weiss, Michael (2001). "Full treatment of spin – including origins, evolution of spin theory, and details of the spin equations". Department of Mathematics. UC Riverside.