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Magnetic quantum number
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{{short description|Number describing angular momentum along an axis}} {{more citations needed|date=May 2016}} In [[atomic physics]], a '''magnetic quantum number''' is a [[quantum number]] used to distinguish quantum states of an [[electron]] or other particle according to its [[angular momentum]] along a given axis in space. The '''orbital magnetic quantum number''' ({{mvar|m<sub>l</sub>}} or {{mvar|m}}{{efn|{{mvar|m}} is often used when only one kind of magnetic quantum number, such as {{mvar|m<sub>l</sub>}} or {{mvar|m<sub>j</sub>}}, is used in a text.}}) distinguishes the [[Atomic orbital|orbitals]] available within a given [[Electron shell|subshell]] of an atom. It specifies the component of the orbital angular momentum that lies along a given axis, conventionally called the ''z''-axis, so it describes the orientation of the orbital in space. The '''spin magnetic quantum number''' {{mvar|m<sub>s</sub>}} specifies the ''z''-axis component of the [[Spin (physics)|spin angular momentum]] for a particle having [[spin quantum number]] {{mvar|s}}. For an electron, {{mvar|s}} is {{frac|1|2}}, and {{mvar|m<sub>s</sub>}} is either +{{1/2}} or β{{1/2}}, often called "spin-up" and "spin-down", or Ξ± and Ξ².<ref name="NIST 2019">{{cite web |last1=Martin |first1=W. C. |last2=Wiese |first2=W. L. |title=Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas |url=https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas |website=National Institute of Standards and Technology, Physical Measurement Laboratory |publisher=NIST |date=2019 |access-date=17 May 2023}}</ref><ref name="Atkins 1991">{{cite book | last=Atkins | first=Peter William | title=Quanta: A Handbook of Concepts |edition=2nd | publisher=Oxford University Press, USA | date=1991 | isbn=0-19-855572-5 | page=297}}</ref> The term ''magnetic'' in the name refers to the [[magnetic dipole moment]] associated with each type of angular momentum, so states having different magnetic quantum numbers shift in energy in a magnetic field according to the [[Zeeman effect]].<ref name="Atkins 1991" /> The four quantum numbers conventionally used to describe the quantum state of an electron in an atom are the [[principal quantum number]] ''n'', the [[azimuthal quantum number|azimuthal (orbital) quantum number]] <math>\ell</math>, and the magnetic quantum numbers {{mvar|m<sub>l</sub>}} and {{mvar|m<sub>s</sub>}}. Electrons in a given subshell of an atom (such as s, p, d, or f) are defined by values of <math>\ell</math> (0, 1, 2, or 3). The orbital magnetic quantum number takes integer values in the range from <math>-\ell</math> to <math>+\ell</math>, including zero.<ref>{{Cite book |last=Griffiths |first=David J. |title=Introduction to quantum mechanics |date=2005 |publisher=Pearson Prentice Hall |isbn=0-13-111892-7 |edition=2nd |location=Upper Saddle River, NJ |pages=136β137 |oclc=53926857}}</ref> Thus the s, p, d, and f subshells contain 1, 3, 5, and 7 orbitals each. Each of these orbitals can accommodate up to two electrons (with opposite spins), forming the basis of the [[periodic table]]. Other magnetic quantum numbers are similarly defined, such as {{mvar|m<sub>j</sub>}} for the ''z''-axis component the [[Total angular momentum quantum number|total electronic angular momentum]] {{mvar|j}},<ref name="NIST 2019" /> and {{mvar|m<sub>I</sub>}} for the [[Spin quantum number#Nuclear spin|nuclear spin]] {{mvar|I}}.<ref name="Atkins 1991" /> Magnetic quantum numbers are capitalized to indicate totals for a system of particles, such as {{mvar|M<sub>L</sub>}} or {{mvar|m<sub>L</sub>}} for the total ''z''-axis orbital angular momentum of all the electrons in an atom.<ref name="Atkins 1991" />
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