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Spin quantum number

From Wikipedia, the free encyclopedia
(Redirected fromNuclear spin)
Quantum number parameterizing spin and angular momentum

Inchemistry andquantum mechanics, thespin quantum number is aquantum number (designateds) that describes the intrinsicangular momentum (or spin angular momentum, or simplyspin) of anelectron or otherparticle. It has the samevalue for all particles of the same type, such ass =⁠1/2⁠ for all electrons. It is an integer for allbosons, such asphotons, and ahalf-odd-integer for allfermions, such as electrons andprotons.

The component of the spin along a specifiedaxis is given by thespin magnetic quantum number, conventionally writtenms.[1][2] The value ofms is the component of spin angular momentum, in units of thereduced Planck constantħ, parallel to a given direction (conventionally labelled thez–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

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Quantum projection of S onto z for spin half particles

The phrasespin quantum number refers to quantizedspin angular momentum. The symbols is used for the spin quantum number, andms is described as the spin magnetic quantum number[3] or as thez-component of spinsz.[4]

Both the total spin and the z-component of spin are quantized, leading to two quantum numbers spin and spin magnet quantum numbers.[5] The (total) spin quantum number has only one value for every elementary particle. Some introductory chemistry textbooks describems as thespin quantum number,[6][7] ands is not mentioned since its value⁠1/2⁠ is a fixed property of the electron; some even use the variables in place ofms.[5]

The two spin quantum numberss{\displaystyle s} andms{\displaystyle m_{s}} are the spin angular momentum analogs of the twoorbital angular momentum quantum numbersl{\displaystyle l} andml{\displaystyle m_{l}}.[8]: 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:S for the total electronic spin, andmS orMS for thez-axis component. A pair of electrons in a spinsinglet state hasS = 0, and a pair in thetriplet state hasS = 1, withmS = −1, 0, or +1. Nuclear-spin quantum numbers are conventionally writtenI for spin, andmI orMI for thez-axis component.

History

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See also:Spin (physics) § History

During the period between 1916 and 1925, much progress was being made concerning the arrangement of electrons in theperiodic table. In order to explain theZeeman effect in theBohr model of the atom,Arnold Sommerfeld proposed that electrons would be based on three 'quantum numbers',n,k, andm, that described the size of the orbit, the shape of the orbit, and the direction in which the orbit was pointing.[9]Irving Langmuir had explained in his 1919 paper regarding electrons in their shells, "Rydberg has pointed out that these numbers are obtained from the seriesN=2(1+22+22+32+32+42){\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."[10] This2n2{\displaystyle 2n^{2}} configuration was adopted byEdmund Stoner, in October 1924 in his paper 'The Distribution of Electrons Among Atomic Levels' published in thePhilosophical Magazine.

The qualitative success of the Sommerfeld quantum number scheme failed to explain the Zeeman effect in weak magnetic field strengths, theanomalous 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.[11]: 563  Rather he proposed that only the outer "light" electrons determined the angular momentum and hehypothesized that this required a fourth quantum number with a two-valuedness.[12] This fourth quantum number became the spinmagnetic quantum number.

Name

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The name "spin" comes from a geometricalspinning of the electron about an axis, as proposed byGeorge Uhlenbeck andSamuel 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.[13] It was therefore replaced by a more abstract quantum-mechanical description.

Detection of spin

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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 theStern–Gerlach experiment.

Stern–Gerlach experiment

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Main article:Stern–Gerlach experiment

The theory of spatial quantization of the spin moment of the momentum of electrons of atoms situated in themagnetic field needed to be proved experimentally. In1922 (two years before the theoretical description of the spin was created)Otto Stern andWalter 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 createsspin magnetic moment, making the electron act like a very small magnet. As the atoms pass through the in-homogeneous magnetic field, theforce 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++ 1 /2⁠ the atom moves away from the stronger field, and when the spin is⁠−+ 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, Thomas Erwin Phipps andJohn Bellamy Taylor [de] conducted a similar experiment, using atoms ofhydrogen 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.

Theatomic 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,Otto Frisch andImmanuel Estermann.

Energy levels from the Dirac equation

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In 1928,Paul Dirac developed arelativistic wave equation, now termed theDirac equation, which predicted thespin magnetic moment correctly, and at the same time treated the electron as a point-like particle. Solving theDirac equation for theenergy levels of an electron in the hydrogen atom, all four quantum numbers includings occurred naturally and agreed well with experiment.

Electron spin

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Main article:Spin (physics)

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

‖S‖=ℏs(s+1)=32 ℏ .{\displaystyle \|{\mathbf {S}}\|=\hbar {\sqrt {s(s+1)}}={\tfrac {\sqrt {3}}{2}}\ \hbar ~.}

The hydrogen spectrumfine structure is observed as a doublet corresponding to two possibilities for thez-component of the angular momentum, where for any given directionz:sz=±12ℏ .{\displaystyle s_{z}=\pm {\tfrac {1}{2}}\hbar ~.}

whose solution has only two possiblez-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 tomagnetic moment, which was a requisite for the fourth quantum number.

The magnetic moment vector of an electron spin is given by:

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

where−e{\displaystyle -e} is theelectron charge,m{\displaystyle m} is theelectron mass, andgs{\displaystyle g_{\text{s}}} is theelectron spin g-factor, which is approximately 2.0023.Itsz-axis projection is given by the spin magnetic quantum numberms{\displaystyle m_{\text{s}}} according to:

μz=−ms gs μB=±12 gs μB {\displaystyle \mu _{z}=-m_{\text{s}}\ g_{\text{s}}\ \mu _{\mathsf {B}}=\pm {\tfrac {1}{2}}\ g_{\text{s}}\ \mu _{\mathsf {B}}\ }

where μB {\displaystyle \ \mu _{\mathsf {B}}\ } is theBohr 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 inelectron spin resonance.

Nuclear spin

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Atomic nuclei also have spins. The nuclear spinI is a fixed property of each nucleus and may be either an integer or a half-integer. The componentmI of nuclear spin parallel to thez-axis can have (2I + 1) valuesI,I−1, ..., −I. For example, a14N nucleus hasI = 1, so that there are 3 possible orientations relative to thez-axis, corresponding to statesmI = +1, 0 and −1.[14]

The spinsI of different nuclei are interpreted using thenuclear shell model.Even-even nuclei with even numbers of both protons and neutrons, such as12C and16O, have spin zero. Odd mass number nuclei have half-integer spins, such as⁠3/ 2 ⁠ for7Li,⁠ 1 /2⁠ for13C and⁠5/ 2 ⁠ for17O, usually corresponding to the angular momentum of the lastnucleon added. Odd-odd nuclei with odd numbers of both protons and neutrons have integer spins, such as 3 for10B, and 1 for14N.[15] Values of nuclear spin for a given isotope are found in the lists of isotopes for each element. (Seeisotopes of oxygen,isotopes of aluminium, etc. etc.)

Electron paramagnetic resonance

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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 ofelectron paramagnetic resonance (EPR) or electron spin resonance (ESR), used to studyfree 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 themicrowave region.

Relation to spin vectors

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For a solution of either the nonrelativisticPauli equation or the relativisticDirac equation, the quantized angular momentum (seeangular momentum quantum number) can be written as:‖s‖=s(s+1)ℏ{\displaystyle \Vert \mathbf {s} \Vert ={\sqrt {s\,(s+1)\,}}\,\hbar }where

Given an arbitrary directionz (usually determined by an external magnetic field) the spinz-projection is given by

sz=msℏ{\displaystyle s_{z}=m_{s}\,\hbar }

wherems is themagnetic spin quantum number, ranging from −s to +s in steps of one. This generates 2 s + 1 different values ofms.

The allowed values fors are non-negativeintegers orhalf-integers.Fermions have half-integer values, including theelectron,proton andneutron which all haves =⁠++ 1 /2⁠ .Bosons such as thephoton and allmesons) have integer spin values.

Algebra

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The algebraic theory of spin is a carbon copy of theangular momentum in quantum mechanics theory.[16]First of all, spin satisfies the fundamentalcommutation relation: [Si,Sj]=i ℏ ϵijk Sk ,{\displaystyle \ [S_{i},S_{j}]=i\ \hbar \ \epsilon _{ijk}\ S_{k}\ ,} [Si,S2]=0 {\displaystyle \ \left[S_{i},S^{2}\right]=0\ }where ϵijk {\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 theuncertainty principle.

Next, theeigenvectors of S2 {\displaystyle \ S^{2}\ } and Sz {\displaystyle \ S_{z}\ } satisfy: S2 |s,ms⟩=ℏ2 s(s+1) |s,ms⟩ {\displaystyle \ S^{2}\ |s,m_{s}\rangle ={\hbar }^{2}\ s(s+1)\ |s,m_{s}\rangle \ } Sz |s,ms⟩=ℏ ms |s,ms⟩ {\displaystyle \ S_{z}\ |s,m_{s}\rangle =\hbar \ m_{s}\ |s,m_{s}\rangle \ } S± |s,ms⟩=ℏ s(s+1)−ms(ms±1) |s,ms±1⟩ {\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 S±=Sx±iSy {\displaystyle \ S_{\pm }=S_{x}\pm iS_{y}\ } are theladder (or "raising" and "lowering") operators.

Total spin of an atom or molecule

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For someatoms thespins of severalunpaired electrons (s1,s2, ...) are coupled to form atotal spinquantum numberS.[17][18] This occurs especially in light atoms (or inmolecules formed only of light atoms) whenspin–orbit coupling is weak compared to the coupling between spins or the coupling between orbitalangular momenta, a situation known asL S coupling becauseL andS areconstants of motion. HereL is the totalorbital angular momentum quantum number.[18]

For atoms with a well-definedS, themultiplicity 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 momentumJ for a given (L,S) combination, provided thatS ≤L (the typical case). For example, ifS = 1, there are three states which form atriplet. Theeigenvalues ofSz for these three states are+1ħ, 0, and−1ħ.[17] Theterm symbol of an atomic state indicates its values ofL,S, andJ.

As examples, the ground states of both theoxygen atom and thedioxygen molecule have two unpaired electrons and are therefore triplet states. The atomic state is described by the term symbol3P, and the molecular state by the term symbol3Σ−
g where the superscript "3" indicates the multiplicity.

See also

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References

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  1. ^Pauling, Linus (1960).The nature of the chemical bond and the structure of molecules and crystals: an introduction to modern structural chemistry. Ithaca, N.Y.: Cornell University Press.ISBN 0-8014-0333-2.OCLC 545520.{{cite book}}:ISBN / Date incompatibility (help)
  2. ^"ISO 80000-10:2019".International Organization for Standardization. Retrieved2019-09-15.
  3. ^Atkins, Peter; de Paula, Julio (2006).Atkins' Physical Chemistry (8th ed.). W.H. Freeman. p. 308.ISBN 0-7167-8759-8.
  4. ^Banwell, Colin N.; McCash, Elaine M. (1994).Fundamentals of Molecular Spectroscopy. McGraw-Hill. p. 135.ISBN 0-07-707976-0.
  5. ^abPerrino, Charles T.; Peterson, Donald L. (1989). "Another quantum number?".J. Chem. Educ.66 (8): 623.Bibcode:1989JChEd..66..623P.doi:10.1021/ed066p623.ISSN 0021-9584.
  6. ^Petrucci, Ralph H.; Harwood, William S.; Herring, F. Geoffrey (2002).General Chemistry (8th ed.). Prentice Hall. p. 333.ISBN 0-13-014329-4.
  7. ^Whitten, Kenneth W.; Galley, Kenneth D.; Davis, Raymond E. (1992).General Chemistry (4th ed.). Saunders College Publishing. p. 196.ISBN 0-03-072373-6.
  8. ^Karplus, Martin, and Porter, Richard Needham. Atoms and Molecules. United States, W.A. Benjamin, 1970.
  9. ^Manjit Kumar, Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality, 2008.
  10. ^Langmuir, Irving (1919)."The arrangement of electrons in atoms and molecules".Journal of the Franklin Institute.187 (3):359–362.doi:10.1016/S0016-0032(19)91097-0.
  11. ^Giulini, Domenico (September 2008)."Electron spin or "classically non-describable two-valuedness"".Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics.39 (3):557–578.arXiv:0710.3128.Bibcode:2008SHPMP..39..557G.doi:10.1016/j.shpsb.2008.03.005.hdl:11858/00-001M-0000-0013-13C8-1.S2CID 15867039.
  12. ^Wolfgang Pauli.Exclusion principle and quantum mechanics Nobel Lecture delivered on December 13th 1946 for the 1945 Nobel Prize in Physics.
  13. ^Halpern, Paul (2017-11-21)."Spin: The quantum property that should have been impossible".Forbes. Starts with a bang. Archived fromthe original on 2018-03-10. Retrieved2018-03-10.
  14. ^Atkins, Peter; de Paula, Julio (2006).Atkins' Physical Chemistry (8th ed.). W.H. Freeman. p. 515.ISBN 0-7167-8759-8.
  15. ^Cottingham, W.N.; Greenwood, D.A. (1986).An introduction to nuclear physics. Cambridge University Press. pp. 36, 57.ISBN 0-521-31960-9.
  16. ^David J. Griffiths,Introduction to Quantum Mechanics (book), Oregon, Reed College, 2018, 166 p.ISBN 9781107189638.
  17. ^abMerzbacher, E. (1998).Quantum Mechanics (3rd ed.). John Wiley. pp. 430–431.ISBN 0-471-88702-1.
  18. ^abAtkins, P.; de Paula, J. (2006).Physical Chemistry (8th ed.). W.H. Freeman. p. 352.ISBN 0-7167-8759-8.

External links

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Quantum numbers
Ground-state configurations
Electron filling
Electron pairing
Bonding participation
Electron counting rules
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