Editing Spin quantum number

{{Short description|Quantum number parameterizing spin and angular momentum}}
{{Use American English|date=January 2019}}

{{Technical|date=February 2024|talk=Too technical}}

In [[physics]] and chemistry, the '''spin quantum number''' is a [[quantum number]] (designated '''{{mvar|s}}''') that describes the intrinsic [[angular momentum]] (or spin angular momentum, or simply [[Spin (physics)|''spin'']]) of an [[electron]] or other [[Elementary particle|particle]]. It has the same [[Value (mathematics)|value]] for all particles of the same type, such as {{mvar|s}} = {{sfrac|1|2}} for all electrons. It is an integer for all [[boson]]s, such as [[Photon|photons]], and a [[Half-integer|half-odd-integer]] for all [[fermions]], such as electrons and [[Proton|protons]].

The component of the spin along a specified [[Coordinate system|axis]] is given by the '''spin magnetic quantum number''', conventionally written '''{{mvar|m}}<sub>{{mvar|s}}</sub>'''.<ref name="Pauling 1960">{{cite book | last=Pauling | first=Linus | title=The nature of the chemical bond and the structure of molecules and crystals: an introduction to modern structural chemistry | publication-place=Ithaca, N.Y. | publisher=Cornell University Press | date=1960 | isbn=0-8014-0333-2 | oclc=545520 | page=}}</ref><ref>{{cite web|title=ISO 80000-10:2019|url=https://www.iso.org/standard/64980.html|publisher=[[International Organization for Standardization]]|access-date=2019-09-15}}</ref> The value of {{mvar|m}}<sub>{{mvar|s}}</sub> is the component of spin angular momentum, in units of the [[reduced Planck constant]] {{mvar|ħ}}, parallel to a given direction (conventionally labelled the {{mvar|z}}–axis). It can take values ranging from +{{mvar|s}} to −{{mvar|s}} in integer increments. For an electron, {{mvar|m}}<sub>{{mvar|s}}</sub> can be either {{sfrac|+|1|2}} or {{sfrac|−|1|2}} .

== Nomenclature ==
[[File:Quantum projection of S onto z for spin half particles.svg|thumb|Quantum projection of S onto z for spin half particles]]
The phrase ''spin quantum number'' refers to quantized [[spin (physics)|spin angular momentum]]. 
The symbol {{mvar|s}} is used for the spin quantum number, and {{mvar|m{{sub|s}}}} is described as the spin magnetic quantum number<ref>{{cite book | last1=Atkins | first1=Peter |last2=de&nbsp;Paula | first2=Julio | year=2006 | title=Atkins' Physical Chemistry |edition=8th | publisher=W.H. Freeman | isbn=0-7167-8759-8 |page=308}}</ref> or as the {{mvar|z}}-component of spin {{mvar|s{{sub|z}}}}.<ref>{{cite book | last1=Banwell | first1=Colin N. | last2=McCash | first2=Elaine M. | year=1994 | title=Fundamentals of Molecular Spectroscopy | publisher=McGraw-Hill | isbn=0-07-707976-0 | page=135}}</ref>

Both the total spin and the z-component of spin are quantized, leading to two quantum numbers spin and spin magnet quantum numbers.<ref name="Peterson 1989">{{cite journal | last1=Perrino | first1=Charles T. | last2=Peterson | first2=Donald L. | title=Another quantum number? | journal=J. Chem. Educ. | volume=66 | issue=8 | year=1989 | issn=0021-9584 | doi=10.1021/ed066p623 | page=623| bibcode=1989JChEd..66..623P }}</ref> The (total) spin quantum number has only one value for every elementary particle. Some introductory chemistry textbooks describe {{mvar|m{{sub|s}}}} as the ''spin quantum number'',<ref>{{cite book |last1=Petrucci |first1=Ralph H. |last2=Harwood |first2=William S. |last3=Herring |first3=F. Geoffrey |year=2002 |title=General Chemistry |edition=8th |publisher=Prentice Hall |isbn=0-13-014329-4 |page=333}}</ref><ref>{{cite book |last1=Whitten |first1=Kenneth W. |last2=Galley |first2=Kenneth D. |last3=Davis |first3=Raymond E. |year=1992 |title=General Chemistry |edition=4th |publisher=Saunders College Publishing |isbn=0-03-072373-6 |page=196}}</ref> and {{mvar|s}} is not mentioned since its value {{sfrac|1|2}} is a fixed property of the electron; some even use the variable {{mvar|s}} in place of {{mvar|m{{sub|s}}}}.<ref name="Peterson 1989" /> 

The two spin quantum numbers <math>s</math> and <math>m_s</math> are the spin angular momentum analogs of the two [[azimuthal quantum number |orbital angular momentum quantum number]]s <math>l</math> and <math>m_l</math>.<ref>Karplus, Martin, and Porter, Richard Needham. Atoms and Molecules. United States, W.A. Benjamin, 1970.</ref>{{rp|152}}

Spin quantum numbers apply also to systems of coupled spins, such as atoms that may contain more than one electron. Capitalized symbols are used: {{mvar|S}} for the total electronic spin, and {{mvar|m}}<sub>{{mvar|S}}</sub> or {{mvar|M}}<sub>{{mvar|S}}</sub> for the {{mvar|z}}-axis component. A pair of electrons in a spin [[singlet state]] has {{mvar|S}} = 0, and a pair in the [[triplet state]] has {{mvar|S}} = 1, with {{mvar|m}}<sub>{{mvar|S}}</sub> = −1, 0, or +1. Nuclear-spin quantum numbers are conventionally written {{mvar|I}} for spin, and {{mvar|m}}<sub>{{mvar|I}}</sub> or {{mvar|M}}<sub>{{mvar|I}}</sub> for the {{mvar|z}}-axis component.

The name "spin" comes from a geometrical [[Spin (geometry)|spinning]] of the electron about an axis, as proposed by [[George Uhlenbeck|Uhlenbeck]] and [[Samuel Goudsmit|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.<ref>{{cite news |last=Halpern |first=Paul |date=2017-11-21 |title=Spin: The quantum property that should have been impossible |series=Starts with a bang |magazine=[[Forbes]] |url=https://www.forbes.com/sites/startswithabang/2017/11/21/spin-the-quantum-property-that-nature-shouldnt-possess/ |access-date=2018-03-10|archive-url=https://web.archive.org/web/20180310202714/https://www.forbes.com/sites/startswithabang/2017/11/21/spin-the-quantum-property-that-nature-shouldnt-possess/ |archive-date=2018-03-10}}</ref> It was therefore replaced by a more abstract quantum-mechanical description.

== History ==
{{Also | Spin (physics)#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 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.<ref>Manjit Kumar, Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality, 2008.</ref> [[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 <math>N = 2(1 + 2^2 + 2^2 + 3^2 + 3^2 + 4^2)</math>. The factor two suggests a fundamental two-fold symmetry for all stable atoms."<ref>{{Cite journal |last=Langmuir |first=Irving |date=1919 |title=The arrangement of electrons in atoms and molecules |url=https://linkinghub.elsevier.com/retrieve/pii/S0016003219910970 |journal=Journal of the Franklin Institute |language=en |volume=187 |issue=3 |pages=359–362 |doi=10.1016/S0016-0032(19)91097-0|url-access=subscription }}</ref> This <math>2n^2</math> 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.<ref>{{Cite journal |last=Giulini |first=Domenico |date=September 2008 |title=Electron spin or "classically non-describable two-valuedness" |url=https://linkinghub.elsevier.com/retrieve/pii/S1355219808000269 |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |language=en |volume=39 |issue=3 |pages=557–578 |doi=10.1016/j.shpsb.2008.03.005|arxiv=0710.3128 |bibcode=2008SHPMP..39..557G |hdl=11858/00-001M-0000-0013-13C8-1 |s2cid=15867039 }}</ref>{{rp|563}} 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.<ref>Wolfgang Pauli. [https://www.nobelprize.org/uploads/2018/06/pauli-lecture.pdf Exclusion principle and quantum mechanics] Nobel Lecture delivered on December 13th 1946 for the 1945 Nobel Prize in Physics.</ref> This fourth quantum number became the spin [[magnetic quantum number]].

==Electron spin==
{{Main|Spin (physics)}}
<!-- This section is linked from [[Caesium]] -->
A spin-{{sfrac| 1 |2}} particle is characterized by an [[angular momentum quantum number]] for spin {{mvar|''s''}} = {{sfrac| 1 |2}}. In solutions of the [[Pauli equation|Schrödinger-Pauli equation]], angular momentum is quantized according to this number, so that magnitude of the spin angular momentum is

<math display="block"> \| \bold{S} \| = \hbar\sqrt{s(s+1)} = \tfrac{\sqrt{3}}{2}\ \hbar ~.</math>

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 {{mvar|z}}:
<math display="block"> s_z = \pm \tfrac{1}{2}\hbar ~.</math>

whose solution has only two possible {{mvar|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:

<math display="block">\ \boldsymbol{\mu}_\text{s} = -\frac{e}{\ 2m\ }\ g_\text{s}\ \bold{S}\ </math>

where <math>-e</math> is the [[elementary charge|electron charge]], <math>m</math> is the [[electron mass]], and <math>g_\text{s}</math> is the [[g-factor (physics)#Electron_spin_g-factor|electron spin g-factor]], which is approximately 2.0023.
Its ''z''-axis projection is given by the spin magnetic quantum number <math>m_\text{s}</math> according to:

<math display="block">\mu_z = -m_\text{s}\ g_\text{s}\ \mu_\mathsf{B} = \pm \tfrac{1}{2}\ g_\text{s}\ \mu_\mathsf{B}\ </math>

where <math>\ \mu_\mathsf{B}\ </math> 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]].

==Nuclear spin==
[[Atomic nucleus|Atomic nuclei]] also have spins. The nuclear spin {{mvar|I}} is a fixed property of each nucleus and may be either an integer or a half-integer. The component {{mvar|m}}<sub>{{mvar|I}}</sub> of nuclear spin parallel to the {{mvar|z}}-axis can have (2{{mvar|I}} + 1) values {{mvar|I}}, {{mvar|I}}−1, ..., −{{mvar|I}}. For example, a [[isotopes of nitrogen|{{sup|14}}N]] nucleus has {{mvar|I}} = 1, so that there are 3 possible orientations relative to the {{mvar|z}}-axis, corresponding to states {{mvar|m}}<sub>{{mvar|I}}</sub> = +1, 0 and −1.<ref>{{cite book |last1=Atkins |first1=Peter |last2=de&nbsp;Paula |first2=Julio |year=2006 |title=Atkins' Physical Chemistry |edition=8th |publisher=W.H. Freeman |isbn=0-7167-8759-8 |page=515}}</ref>

The spins {{mvar|I}} of different nuclei are interpreted using the [[Nuclear shell model#Other properties of nuclei|nuclear shell model]]. [[Even and odd atomic nuclei|Even-even nuclei]] with even numbers of both protons and neutrons, such as [[isotopes of carbon|{{sup|12}}C]] and [[isotopes of oxygen|{{sup|16}}O]], have spin zero. Odd mass number nuclei have half-integer spins, such as {{sfrac|3| 2 }} for [[isotopes of lithium|{{sup|7}}Li]], {{sfrac| 1 |2}} for [[isotopes of carbon|{{sup|13}}C]] and {{sfrac|5| 2 }} for [[isotopes of oxygen|{{sup|17}}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 [[isotopes of boron|{{sup|10}}B]], and 1 for [[isotopes of nitrogen|{{sup|14}}N]].<ref>{{cite book |last1=Cottingham |first1=W.N. |last2=Greenwood |first2=D.A. |year=1986 |title=An introduction to nuclear physics |publisher=Cambridge University Press |isbn=0-521-31960-9 |pages=36, 57}}</ref> 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.)

== 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 ===
{{main | 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 in science|1922]] (two years before the theoretical description of the spin was created) [[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 {{sfrac|+| 1 |2}} the atom moves away from the stronger field, and when the spin is {{sfrac|−| 1 |2}} 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 in science|1927]] Phipps and Taylor 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: ([[copper]], [[gold]], [[sodium]], [[potassium]]). 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 (about 1836 times), 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, Frisch and Easterman.

===Electron paramagnetic resonance===
For atoms or molecules with an unpaired electron, transitions in a magnetic field can also be observed in which only the spin quantum number changes, without change in the electron orbital or the other quantum numbers. This is the method of [[electron paramagnetic resonance]] (EPR) or electron spin resonance (ESR), used to study [[Radical (chemistry)|free radicals]]. Since only the magnetic interaction of the spin changes, the energy change is much smaller than for transitions between orbitals, and the spectra are observed in the [[microwave]] region.

== Relation to spin vectors ==
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:
<math display="block"> \Vert \mathbf{s} \Vert = \sqrt{s \, (s+1)\,} \, \hbar</math>
where
* <math>\mathbf{s}</math> is the quantized [[spin vector]] or spinor
* <math>\Vert \mathbf{s}\Vert</math> is the [[norm (mathematics)|norm]] of the spin vector
* {{mvar|s}} is the spin quantum number associated with the spin angular momentum
* <math>\hbar</math> is the [[reduced Planck constant]].

Given an arbitrary direction {{mvar|z}} (usually determined by an external magnetic field) the spin {{mvar|z}}-projection is given by
:<math>s_z = m_s \, \hbar</math>

where {{mvar|m{{sub|s}}}} is the '''magnetic spin quantum number''', ranging from −{{mvar|s}} to +{{mvar|s}} in steps of one. This generates {{math| 2 {{mvar|s}} + 1 }} different values of {{mvar|m{{sub|s}}}}.

The allowed values for {{mvar|s}} are non-negative [[integer]]s or [[half-integer]]s. [[Fermion]]s have half-integer values, including the [[electron]], [[proton]] and [[neutron]] which all have {{nobr| {{mvar|s}} {{=}} {{sfrac|+| 1 |2}} .}} [[Boson]]s such as the [[photon]] and all [[meson]]s) have integer spin values.

== Algebra ==
The algebraic theory of spin is a carbon copy of the [[Angular momentum#Angular momentum in quantum mechanics|angular momentum in quantum mechanics]] theory.<ref>[[David J. Griffiths]], ''[[Introduction to Quantum Mechanics (book)]]'', Oregon, Reed College, 2018, 166 p. {{ISBN|9781107189638}}.</ref>
First of all, spin satisfies the fundamental [[Canonical commutation relation|commutation relation]]:
<math display="block">\ [S_i, S_j ] = i\ \hbar\ \epsilon_{ijk}\ S_k\ ,</math>
<math display="block">\ \left[S_i, S^2 \right] = 0\ </math>
where <math>\ \epsilon_{ijk}\ </math> 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 [[Eigenstate|eigenvectors]] of <math>\ S^2\ </math> and <math>\ S_z\ </math> satisfy:
<math display="block">\ S^2\ | s, m_s \rangle= {\hbar}^2\ s(s+1)\ | s, m_s \rangle\ </math>
<math display="block">\ S_z\ | s, m_s \rangle = \hbar\ m_s\ | s, m_s \rangle\ </math>
<math display="block">\ S_\pm\ | s, m_s \rangle = \hbar\ \sqrt{s(s+1) - m_s(m_s \pm 1)\ }\; | s, m_s \pm 1 \rangle\ </math>
where <math>\ S_\pm = S_x \pm i S_y\ </math> are the [[ladder operator|ladder]] (or "raising" and "lowering") operators.

== 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 level]]s of an electron in the hydrogen atom, all four quantum numbers including {{mvar|s}} occurred naturally and agreed well with experiment.

== Total spin of an atom or molecule ==
For some [[Atom|atoms]] the [[Electron magnetic moment|spins]] of several [[Unpaired electron|unpaired electrons]] ({{mvar|s}}{{sub|1}}, {{mvar|s}}{{sub|2}}, ...) are coupled to form a ''total spin'' [[quantum number]] {{mvar|S}}.<ref name=Merzbacher>{{cite book |author-link=Eugen Merzbacher |author=Merzbacher, E. |year=1998 |title=Quantum Mechanics |edition=3rd |publisher=John Wiley |pages=430–431 |isbn=0-471-88702-1}}</ref><ref name=Atkins>{{cite book |author1-link=Peter Atkins |author1=Atkins, P. |author2=de&nbsp;Paula, J. |year=2006 |title=Physical Chemistry |edition=8th |publisher=W.H. Freeman |page=352 |isbn=0-7167-8759-8}}</ref> This occurs especially in light atoms (or in [[Molecule|molecules]] formed only of light atoms) when [[spin–orbit coupling]] is weak compared to the coupling between spins or the coupling between orbital [[Angular momentum|angular momenta]], a situation known as [[Angular momentum coupling#LS coupling|{{math|L S}} coupling]] because {{mvar|L}} and {{mvar|S}} are [[Constant of motion|constants of motion]]. Here {{mvar|L}} is the total ''orbital'' angular momentum quantum number.<ref name=Atkins/>

For atoms with a well-defined {{mvar|S}}, the [[Multiplicity (chemistry)|multiplicity]] of a state is defined as {{nobr| 2{{mvar|S}} + 1}}. This is equal to the number of different possible values of the total (orbital plus spin) angular momentum {{mvar|J}} for a given ({{mvar|L}}, {{mvar|S}}) combination, provided that {{mvar|S}} ≤ {{mvar|L}} (the typical case). For example, if {{mvar|S}} = 1, there are three states which form a [[triplet state|triplet]]. The [[Eigenvalues and eigenvectors|eigenvalues]] of {{mvar|S{{sub|z}}}} for these three states are {{math|+1ħ, 0,}} and {{math|−1ħ}}.<ref name=Merzbacher/> The [[term symbol]] of an atomic state indicates its values of {{mvar|L}}, {{mvar|S}}, and {{mvar|J}}.

As examples, the ground states of both the [[Oxygen|oxygen atom]] and the [[Triplet oxygen|dioxygen molecule]] have two unpaired electrons and are therefore triplet states. The atomic state is described by the term symbol {{sup|3}}P, and the molecular state by the term symbol {{sup|3}}Σ{{su|b=g|p=−}} where the superscript "3" indicates the multiplicity.

== See also ==
* [[Total angular momentum quantum number]]
* [[Rotational spectroscopy]]
* [[Basic quantum mechanics]]

== References ==
{{reflist|25em}}

== External links ==
* {{cite web |first1=Michael |last1=Weiss |year=2001 |title=Full treatment of spin – including origins, evolution of spin theory, and details of the spin equations |website=Department of Mathematics |publisher=[[University of California, Riverside|UC Riverside]] |url=https://math.ucr.edu/home/baez/spin/spin.html}}

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