Editing Hyperfine structure

{{Short description|Small shifts and splittings in the energy levels of atoms, molecules and ions}}
In [[atomic physics]], '''hyperfine structure''' is defined by small shifts in otherwise [[degenerate energy levels|degenerate]] electronic [[energy levels]] and the resulting [[energy level splitting|splittings]] in those electronic energy levels of [[atom]]s, [[molecule]]s, and [[ion]]s, due to electromagnetic multipole interaction between the nucleus and electron clouds.

In atoms, hyperfine structure arises from the energy of the [[nuclear magnetic moment|nuclear magnetic dipole moment]] interacting with the [[magnetic field]] generated by the electrons and the energy of the [[quadrupole|nuclear electric quadrupole moment]] in the [[electric field gradient]] due to the distribution of charge within the atom. Molecular hyperfine structure is generally dominated by these two effects, but also includes the energy associated with the interaction between the magnetic moments associated with different magnetic nuclei in a molecule, as well as between the nuclear magnetic moments and the magnetic field generated by the rotation of the molecule.

Hyperfine structure contrasts with ''[[fine structure]]'', which results from the interaction between the [[magnetic moment]]s associated with [[electron spin]] and the electrons' [[azimuthal quantum number|orbital angular momentum]]. Hyperfine structure, with energy shifts typically orders of magnitudes smaller than those of a fine-structure shift, results from the interactions of the [[atomic nucleus|nucleus]] (or nuclei, in molecules) with internally generated electric and magnetic fields.

[[File:Fine hyperfine levels.svg|thumb|right|Schematic illustration of [[fine structure|fine]] and hyperfine structure in a neutral [[hydrogen atom]] ]]

==History==
The first theory of atomic hyperfine structure was given in 1930 by [[Enrico Fermi]]<ref>E. Fermi (1930), "Uber die magnetischen Momente der Atomkerne". Z. Physik 60, 320-333.</ref> for an atom containing a single valence electron with an arbitrary angular momentum. The [[Zeeman splitting]] of this structure was discussed by [[Samuel Goudsmit|S. A. Goudsmit]] and [[Robert Bacher|R. F. Bacher]] later that year.

In 1935, H. Schüler and Theodor Schmidt proposed the existence of a nuclear quadrupole moment in order to explain anomalies in the hyperfine structure of [[europium]], [[cassiopium]] (older name for lutetium), [[indium]], [[antimony]], and [[Mercury (Element)|mercury]].<ref>H. Schüler & T. Schmidt (1935), "Über Abweichungen des Atomkerns von der Kugelsymmetrie". Z. Physik 94, 457–468.</ref>

==Theory==
The theory of hyperfine structure comes directly from [[electromagnetism]], consisting of the interaction of the nuclear [[multipole moments]] (excluding the electric monopole) with internally generated fields. The theory is derived first for the atomic case, but can be applied to ''each nucleus'' in a molecule. Following this there is a discussion of the additional effects unique to the molecular case.

===Atomic hyperfine structure===

====Magnetic dipole====
{{Main|Dipole}}

The dominant term in the hyperfine [[Hamiltonian (quantum mechanics)|Hamiltonian]] is typically the magnetic dipole term. Atomic nuclei with a non-zero [[nuclear spin]] <math>\mathbf{I}</math> have a magnetic dipole moment, given by:
<math display="block">\boldsymbol{\mu}_\text{I} = g_\text{I}\mu_\text{N}\mathbf{I},</math>
where <math>g_\text{I}</math> is the [[g-factor (physics)|''g''-factor]] and <math>\mu_\text{N}</math> is the [[nuclear magneton]].

There is an energy associated with a magnetic dipole moment in the presence of a magnetic field. For a nuclear magnetic dipole moment, '''μ'''<sub>I</sub>, placed in a magnetic field, '''B''', the relevant term in the Hamiltonian is given by:<ref name="Woodgate">{{cite book |title=Elementary Atomic Structure |last=Woodgate |first=Gordon K. |year=1999 |publisher=Oxford University Press |isbn=978-0-19-851156-4}}</ref>
<math display="block">\hat{H}_\text{D} = -\boldsymbol{\mu}_\text{I}\cdot\mathbf{B}.</math>

In the absence of an externally applied field, the magnetic field experienced by the nucleus is that associated with the orbital ('''ℓ''') and spin ('''s''') angular momentum of the electrons:
<math display="block">\mathbf{B} \equiv \mathbf{B}_\text{el} = \mathbf{B}_\text{el}^\ell + \mathbf{B}_\text{el}^s.</math>

=====Electron orbital magnetic field=====
Electron orbital angular momentum results from the motion of the electron about some fixed external point that we shall take to be the location of the nucleus. The magnetic field at the nucleus due to the motion of a single electron, with charge –''[[elementary charge|e]]'' at a position '''r''' relative to the nucleus, is given by:
<math display="block">\mathbf{B}_\text{el}^\ell = \frac{\mu_0}{4\pi}\frac{-e\mathbf{v} \times -\mathbf{r}}{r^3},</math>
where −'''r''' gives the position of the nucleus relative to the electron. Written in terms of the [[Bohr magneton]], this gives:
<math display="block">\mathbf{B}_\text{el}^\ell = -2\mu_\text{B} \frac{\mu_0}{4\pi}\frac{1}{r^3} \frac{\mathbf{r} \times m_\text{e}\mathbf{v}}{\hbar}.</math>

Recognizing that ''m''<sub>e</sub>'''v''' is the electron momentum, '''p''', and that {{nowrap|'''r''' × '''p''' / ''ħ''}} is the orbital [[angular momentum]] in units of ''ħ'', '''ℓ''', we can write:
<math display="block">\mathbf{B}_\text{el}^\ell = -2\mu_\text{B}\frac{\mu_0}{4\pi}\frac{1}{r^3}\boldsymbol{\ell}.</math>

For a many-electron atom this expression is generally written in terms of the total orbital angular momentum, <math>\mathbf{L}</math>, by summing over the electrons and using the projection operator, <math>\varphi^\ell_i</math>, where <math display="inline">\sum_i\mathbf{\ell}_i = \sum_i\varphi^\ell_i\mathbf{L}</math>. For states with a well defined projection of the orbital angular momentum, {{math|''L<sub>z</sub>''}}, we can write <math>\varphi^\ell_i = \hat{\ell}_{z_i}/L_z</math>, giving:
<math display="block">\mathbf{B}_\text{el}^\ell = -2\mu_\text{B} \frac{\mu_0}{4\pi} \frac{1}{L_z}\sum_i\frac{\hat{\ell}_{zi}}{r_i^3}\mathbf{L}.</math>

=====Electron spin magnetic field=====
The electron spin angular momentum is a fundamentally different property that is intrinsic to the particle and therefore does not depend on the motion of the electron. Nonetheless, it is angular momentum and any angular momentum associated with a charged particle results in a magnetic dipole moment, which is the source of a magnetic field. An electron with spin angular momentum, '''s''', has a magnetic moment, '''μ'''<sub>''s''</sub>, given by:
<math display="block">\boldsymbol{\mu}_\text{s} = -g_s\mu_\text{B}\mathbf{s},</math>
where ''g<sub>s</sub>'' is the [[G-factor (physics)|electron spin ''g''-factor]] and the negative sign is because the electron is negatively charged (consider that negatively and positively charged particles with identical mass, travelling on equivalent paths, would have the same angular momentum, but would result in [[electrical current|currents]] in the opposite direction).

The magnetic field of a point dipole moment, '''''μ'''''<sub>s</sub>, is given by:<ref name="Jackson">{{cite book |title=Classical Electrodynamics |last=Jackson |first=John D. |year=1998 |publisher=Wiley |isbn=978-0-471-30932-1}}</ref><ref name="Garg">{{cite book |title=Classical Electromagnetism in a Nutshell |last=Garg |first=Anupam |year=2012 |publisher=Princeton University Press |isbn=978-0-691-13018-7 |at = §26}}</ref>
<math display="block">\mathbf{B}_\text{el}^s = \frac{\mu_0}{4\pi r^3} \left(3\left(\boldsymbol{\mu}_\text{s} \cdot \hat{\mathbf{r}}\right) \hat{\mathbf{r}} - \boldsymbol{\mu}_\text{s}\right) + \dfrac{2\mu_0}{3} \boldsymbol{\mu}_\text{s}\delta^3(\mathbf{r}).</math>

=====Electron total magnetic field and contribution=====
The complete magnetic dipole contribution to the hyperfine Hamiltonian is thus given by:
<math display="block">\begin{align}
  \hat{H}_D ={} &2g_\text{I}\mu_\text{N}\mu_\text{B} \dfrac{\mu_0}{4\pi}\dfrac{1}{L_z} \sum_i \dfrac{\hat{\ell}_{zi}}{r_i^3} \mathbf{I} \cdot \mathbf{L} \\
                & {}+ g_\text{I}\mu_\text{N}g_\text{s}\mu_\text{B} \frac{\mu_0}{4\pi} \frac{1}{S_z}\sum_i \frac{\hat{s}_{zi}}{r_i^3} \left\{ 3\left(\mathbf{I} \cdot \hat{\mathbf{r}}\right)\left(\mathbf{S}\cdot\hat{\mathbf{r}}\right) - \mathbf{I}\cdot\mathbf{S}\right\} \\
                & {}+ \frac{2}{3} g_\text{I}\mu_\text{N}g_\text{s}\mu_\text{B}\mu_0 \frac{1}{S_z}\sum_i\hat{s}_{zi}\delta^3{\left(\mathbf{r}_i\right)} \mathbf{I}\cdot\mathbf{S}.
\end{align}</math>

The first term gives the energy of the nuclear dipole in the field due to the electronic orbital angular momentum. The second term gives the energy of the "finite distance" interaction of the nuclear dipole with the field due to the electron spin magnetic moments. The final term, often known as the ''[[Fermi contact interaction|Fermi contact]]'' term relates to the direct interaction of the nuclear dipole with the spin dipoles and is only non-zero for states with a finite electron spin density at the position of the nucleus (those with unpaired electrons in ''s''-subshells). It has been argued that one may get a different expression when taking into account the detailed nuclear magnetic moment distribution.<ref>{{Cite journal |last=Soliverez |first=C E |date=1980-12-10 |title=The contact hyperfine interaction: an ill-defined problem |url=https://iopscience.iop.org/article/10.1088/0022-3719/13/34/002 |journal=Journal of Physics C: Solid State Physics |volume=13 |issue=34 |pages=L1017–L1019 |doi=10.1088/0022-3719/13/34/002 |issn=0022-3719|url-access=subscription }}</ref> The inclusion of the delta function is an admission that the singularity in the magnetic induction '''B''' owing to a magnetic dipole moment at a point is not  integrable. It is '''B''' which mediates the interaction between the Pauli spinors in non-relativistic quantum mechanics. Fermi (1930) avoided the difficulty by working with the relativistic Dirac wave equation, according to which the mediating field for the Dirac spinors is the four-vector potential (V,'''A'''). The component  V is the Coulomb potential. The component '''A''' is the three-vector magnetic potential (such that '''B''' = '''curl A'''), which for the point dipole is integrable. 

For states with <math>\ell \neq 0</math> this can be expressed in the form
<math display="block">\hat{H}_D = 2 g_I \mu_\text{B} \mu_\text{N} \dfrac{\mu_0}{4\pi} \dfrac{\mathbf{I}\cdot\mathbf{N}}{r^3},</math>
where:<ref name="Woodgate"/>
<math display="block">\mathbf{N} = \boldsymbol{\ell} - \frac{g_s}{2} \left[\mathbf{s} - 3(\mathbf{s}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}\right].</math>

If hyperfine structure is small compared with the fine structure (sometimes called ''IJ''-coupling by analogy with [[Russell−Saunders state|''LS''-coupling]]), ''I'' and ''J'' are good [[quantum number]]s and matrix elements of <math>\hat{H}_\text{D}</math> can be approximated as diagonal in ''I'' and ''J''. In this case (generally true for light elements), we can project '''N''' onto '''J''' (where {{math|1='''J''' = '''L''' + '''S'''}} is the total electronic angular momentum) and we have:<ref name="Woodgate2">{{cite book |url=https://books.google.com/books?id=nUA74S5Y1EUC&q=woodgate+atomic+structure |title=Elementary Atomic Structure |access-date=2009-03-03 |last=Woodgate |first=Gordon K. |isbn=978-0-19-851156-4 |year=1983 |publisher=Oxford University Press, USA}}</ref>
<math display="block">\hat{H}_\text{D} = 2 g_I \mu_\text{B} \mu_\text{N} \dfrac{\mu_0}{4\pi} \dfrac{\mathbf{N}\cdot\mathbf{J}}{\mathbf{J}\cdot\mathbf{J}} \dfrac{\mathbf{I}\cdot\mathbf{J}}{r^3}.</math>

This is commonly written as
<math display="block">\hat{H}_\text{D} = \hat{A}\mathbf{I}\cdot\mathbf{J},</math>
with <math display="inline">\left\langle\hat{A}\right\rangle</math> being the hyperfine-structure constant which is determined by experiment. Since {{math|1='''I'''⋅'''J''' = {{1/2}}{'''F'''⋅'''F''' − '''I'''⋅'''I''' − '''J'''⋅'''J'''}<nowiki/>}} (where {{math|1='''F''' = '''I''' + '''J'''}} is the total angular momentum), this gives an energy of:
<math display="block">\Delta E_\text{D} = \frac{1}{2}\left\langle\hat{A}\right\rangle[F(F + 1) - I(I + 1) - J(J + 1)].</math>

In this case the hyperfine interaction satisfies the [[Landé interval rule]].

====Electric quadrupole====
{{Main|Quadrupole}}
<!--look at Brown-Carr. p.568-->

Atomic nuclei with spin <math>I \ge 1</math> have an [[quadrupole|electric quadrupole moment]].<ref name="Enge">{{cite book |title=Introduction to Nuclear Physics |last=Enge |first=Harald A. |year=1966 |publisher=Addison Wesley |isbn=978-0-201-01870-7 |url-access=registration |url=https://archive.org/details/introductiontonu0000enge}}</ref> In the general case this is represented by a [[tensor order|rank]]-2 [[tensor]], <math>Q_{ij}</math>, with components given by:<ref name="Jackson"/>
<math display="block">Q_{ij} = \frac{1}{e} \int\left(3x_i^\prime x_j^\prime - \left(r'\right)^2 \delta_{ij}\right)\rho{\left(\mathbf{r}'\right)} \, d^3\mathbf{r}',</math>
where ''i'' and ''j'' are the tensor indices running from 1 to 3, ''x<sub>i</sub>'' and ''x<sub>j</sub>'' are the spatial variables ''x'', ''y'' and ''z'' depending on the values of ''i'' and ''j'' respectively, ''δ''<sub>''ij''</sub> is the [[Kronecker delta]] and ''ρ''('''r''') is the charge density. Being a 3-dimensional rank-2 tensor, the quadrupole moment has 3<sup>2</sup> = 9 components. From the definition of the components it is clear that the quadrupole tensor is a [[symmetric matrix]] ({{math|1=''Q<sub>ij</sub>'' = ''Q<sub>ji</sub>''}}) that is also [[traceless]] (<math display="inline">\operatorname{tr} Q = \sum_i Q_{ii} = 0</math>), giving only five components in the [[irreducible representation]]. Expressed using the notation of [[irreducible spherical tensor]]s we have:<ref name="Jackson"/>
<math display="block">T^2_m(Q) = \sqrt{\frac{4\pi}{5}} \int \rho{\left(\mathbf{r}'\right)} \left(r'\right)^2 Y^2_m\left(\theta', \varphi'\right) \, d^3\mathbf{r}'. </math>

The energy associated with an electric quadrupole moment in an electric field depends not on the field strength, but on the electric field gradient, confusingly labelled <math display="inline">\underline{\underline{q}}</math>, another rank-2 tensor given by the [[outer product]] of the [[del operator]] with the electric field vector:
<math display="block">\underline{\underline{q}} = \nabla\otimes\mathbf{E},</math>
with components given by:
<math display="block">q_{ij} = \frac{\partial^2V}{\partial x_i \, \partial x_j}.</math>

Again it is clear this is a symmetric matrix and, because the source of the electric field at the nucleus is a charge distribution entirely outside the nucleus, this can be expressed as a 5-component spherical tensor, <math>T^2(q)</math>, with:<ref>{{cite web |url=http://www.pascal-man.com/tensor-quadrupole-interaction/EFG-tensor.shtml |title=Electric field gradient tensor around quadrupolar nuclei |access-date=2008-07-23 |author=Y. Millot |date=2008-02-19}}</ref>
<math display="block">\begin{align}
  T^2_0   (q) &= \frac{\sqrt{6}}{2} q_{zz} \\
  T^2_{+1}(q) &= -q_{xz} - i q_{yz} \\
  T^2_{+2}(q) &= \frac{1}{2}(q_{xx} - q_{yy}) + iq_{xy},
\end{align}</math>
where:
<math display="block">T^2_{-m}(q) = (-1)^mT^2_{+m}(q)^*.</math>

The quadrupolar term in the Hamiltonian is thus given by:
<math display="block">\hat{H}_Q = -e T^2(Q) \cdot T^2(q) = -e\sum_m (-1)^m T^2_m(Q) T^2_{-m}(q).</math>

A typical atomic nucleus closely approximates cylindrical symmetry and therefore all off-diagonal elements are close to zero. For this reason the nuclear electric quadrupole moment is often represented by {{math|''Q''<sub>''zz''</sub>}}.<ref name="Enge"/>

===Molecular hyperfine structure===
The molecular hyperfine Hamiltonian includes those terms already derived for the atomic case with a magnetic dipole term for each nucleus with <math>I > 0</math> and an electric quadrupole term for each nucleus with <math>I \geq 1</math>. The magnetic dipole terms were first derived for diatomic molecules by Frosch and Foley,<ref name="Frosch-Foley">{{cite journal |author=Frosch and Foley |title=Magnetic hyperfine structure in diatomics |year=1952 |journal=[[Physical Review]] |volume=88 |issue=6 |pages=1337–1349 |doi=10.1103/PhysRev.88.1337 |bibcode=1952PhRv...88.1337F |last2=Foley |first2=H.}}</ref> and the resulting hyperfine parameters are often called the Frosch and Foley parameters.

In addition to the effects described above, there are a number of effects specific to the molecular case.<ref name="BrownCarr">{{cite book |title=Rotational Spectroscopy of Diatomic Molecules |last=Brown |first=John |author2=Alan Carrington |year=2003 |publisher=Cambridge University Press |isbn=978-0-521-53078-1}}</ref>

====Direct nuclear spin–spin====
Each nucleus with <math>I > 0</math> has a non-zero magnetic moment that is both the source of a magnetic field and has an associated energy due to the presence of the combined field of all of the other nuclear magnetic moments. A summation over each magnetic moment dotted with the field due to each ''other'' magnetic moment gives the direct nuclear spin–spin term in the hyperfine Hamiltonian, <math>\hat{H}_{II}</math>.<ref name="BrownCarr2">{{cite book |url=https://books.google.com/books?id=TU4eA7MoDrQC&q=brown+carrington+diatomic |title=Rotational Spectroscopy of Diatomic Molecules |access-date=2009-03-03 |last=Brown |first=John |author2=Alan Carrington |isbn=978-0-521-53078-1 |year=2003 |publisher=Cambridge University Press}}</ref>
<math display="block">\hat{H}_{II} = -\sum_{\alpha\neq\alpha'} \boldsymbol{\mu}_\alpha \cdot \mathbf{B}_{\alpha'},</math>
where ''α'' and ''α{{'}}'' are indices representing the nucleus contributing to the energy and the nucleus that is the source of the field respectively. Substituting in the expressions for the dipole moment in terms of the nuclear angular momentum and the magnetic field of a dipole, both given above, we have
<math display="block">
  \hat{H}_{II} =
  \dfrac{\mu_0\mu_\text{N}^2}{4\pi} \sum_{\alpha\neq\alpha'} \frac{g_\alpha g_{\alpha'}}{R_{\alpha\alpha'}^3} \left\{
      \mathbf{I}_\alpha\cdot\mathbf{I}_{\alpha'} -
      3\left(\mathbf{I}_\alpha \cdot \hat{\mathbf{R}}_{\alpha\alpha'}\right)\left(\mathbf{I}_{\alpha'} \cdot \hat{\mathbf{R}}_{\alpha\alpha'}\right)
    \right\}.
</math>

====Nuclear spin–rotation====
The nuclear magnetic moments in a molecule exist in a magnetic field due to the angular momentum, '''T''' ('''R''' is the internuclear displacement vector), associated with the bulk rotation of the molecule,<ref name="BrownCarr2"/> thus
<math display="block">
  \hat{H}_\text{IR} =
  \frac{e\mu_0\mu_\text{N}\hbar}{4\pi}\sum_{\alpha \neq \alpha'} \frac{1}{R_{\alpha\alpha'}^3}
    \left\{\frac{Z_\alpha g_{\alpha'}}{M_\alpha} \mathbf{I}_{\alpha'} +
  \frac{Z_{\alpha'}g_\alpha}{M_{\alpha'}} \mathbf{I}_\alpha\right\}\cdot\mathbf{T}.
</math>

====Small molecule hyperfine structure====
A typical simple example of the hyperfine structure due to the interactions discussed above is in the rotational transitions of [[hydrogen cyanide]] (<sup>1</sup>H<sup>12</sup>C<sup>14</sup>N) in its ground [[rotational–vibrational spectroscopy|vibrational state]]. Here, the electric quadrupole interaction is due to the <sup>14</sup>N-nucleus, the hyperfine nuclear spin-spin splitting is from the magnetic coupling between nitrogen, <sup>14</sup>N (''I''<sub>N</sub> = 1), and hydrogen, <sup>1</sup>H (''I''<sub>H</sub> = {{frac|1|2}}), and a hydrogen spin-rotation interaction due to the <sup>1</sup>H-nucleus. These contributing interactions to the hyperfine structure in the molecule are listed here in descending order of influence. Sub-doppler techniques have been used to discern the hyperfine structure in HCN rotational transitions.<ref>{{cite journal |last1=Ahrens |title= Sub-Doppler Saturation Spectroscopy of HCN up to 1 THz and Detection of <math chem>J = \ce{3 -> 2 (4 -> 3)}</math> Emission from TMC-1 |journal=Z. Naturforsch. |year=2002 |volume=57a |issue = 8|pages= 669–681 |first1=V. |last2=Lewen |first2=F. |last3=Takano|first3=S. |last4=Winnewisser |first4=G. |display-authors=etal |bibcode= 2002ZNatA..57..669A |doi= 10.1515/zna-2002-0806|s2cid = 35586070|doi-access=free}}</ref>

The dipole [[selection rules]] for HCN hyperfine structure transitions are <math>\Delta J = 1</math>, <math>\Delta F = \{0, \pm 1\}</math>, where {{mvar|J}} is the rotational quantum number and {{mvar|F}} is the total rotational quantum number inclusive of nuclear spin (<math>F = J + I_\text{N}</math>), respectively. The lowest transition (<math>J = 1 \rightarrow 0</math>) splits into a hyperfine triplet. Using the selection rules, the hyperfine pattern of <math>J = 2 \rightarrow 1</math> transition and higher dipole transitions is in the form of a hyperfine sextet. However, one of these components (<math>\Delta F = -1</math>) carries only 0.6% of the rotational transition intensity in the case of <math>J = 2 \rightarrow 1</math>. This contribution drops for increasing J. So, from <math>J = 2 \rightarrow 1</math> upwards the hyperfine pattern consists of three very closely spaced stronger hyperfine components (<math>\Delta J = 1</math>, <math>\Delta F = 1</math>) together with two widely spaced components; one on the low frequency side and one on the high frequency side relative to the central hyperfine triplet. Each of these outliers carry ~<math>\tfrac{1}{2} J^2</math> ({{mvar|J}} is the upper rotational quantum number of the allowed dipole transition) the intensity of the entire transition. For consecutively higher-{{mvar|J}} transitions, there are small but significant changes in the relative intensities and positions of each individual hyperfine component.<ref name="mullins">{{cite journal |last1=Mullins |title=Radiative Transfer of HCN: Interpreting observations of hyperfine anomalies |journal=Monthly Notices of the Royal Astronomical Society |year=2016 |volume=459 |issue=3 |pages=2882–2993 |doi=10.1093/mnras/stw835 |bibcode=2016MNRAS.459.2882M |first1=A. M. |last2=Loughnane |first2=R. M. |last3=Redman |first3=M. P. |doi-access=free |display-authors=etal |arxiv=1604.03059 |s2cid=119192931}}</ref>

==Measurements and Applications==

Hyperfine interactions can be measured, among other ways, in atomic and molecular spectra, and in [[electron paramagnetic resonance]] spectra of [[free radical]]s and [[transition metal|transition-metal]] ions. 

===Astrophysics===
[[File:Pioneer plaque hydrogen.svg|thumb|The hyperfine transition as depicted on the [[Pioneer plaque]]]]
As the hyperfine splitting is very small, the transition frequencies are usually not located in the optical, but are in the range of radio- or microwave (also called sub-millimeter) frequencies.

Hyperfine structure gives the [[21 cm line]] observed in [[H I region]]s in [[interstellar medium]].

[[Carl Sagan]] and [[Frank Drake]] considered the hyperfine transition of hydrogen to be a sufficiently universal phenomenon so as to be used as a base unit of time and length on the [[Pioneer plaque]] and later [[Voyager Golden Record]].

In [[submillimeter astronomy]], [[superheterodyne receiver|heterodyne receiver]]s are widely used in detecting electromagnetic signals from celestial objects such as star-forming core or [[young stellar objects]]. The separations among neighboring components in a hyperfine spectrum of an observed [[rotational transition]] are usually small enough to fit within the receiver's [[intermediate frequency|IF]] band. Since the [[optical depth]] varies with frequency, strength ratios among the hyperfine components differ from that of their intrinsic (or ''optically thin'') intensities (these are so-called ''hyperfine anomalies'', often observed in the rotational transitions of HCN<ref name="mullins"/>). Thus, a more accurate determination of the optical depth is possible. From this we can derive the object's physical parameters.<ref>{{cite journal |last1=Tatematsu |title=N<sub>2</sub>H<sup>+</sup> Observations of Molecular Cloud Cores in Taurus |journal=Astrophysical Journal |year=2004 |volume=606 |issue=1 |pages=333–340 |doi=10.1086/382862 |bibcode=2004ApJ...606..333T |arxiv=astro-ph/0401584 |first1=K. |last2=Umemoto |first2=T. |last3=Kandori |first3=R. |s2cid=118956636 |display-authors=etal}}</ref>

===Nuclear spectroscopy===

[[File:Cobalt QENS.pdf|thumb|Hyperfine splitting in ferromagnetic cobalt at 3.5 K, observed by quasielastic neutron scattering.<ref>Adapted from T Chatterji, M Zamponi, J Wuttke, Hyperfine interaction in cobalt by high-resolution neutron spectroscopy, J Phys: Condens Matter  31, 0257801 (2019), Fig 1.</ref>]]

In [[nuclear spectroscopy]] methods, the nucleus is used to probe the [[local structure]] in materials. The methods mainly base on hyperfine interactions with the surrounding atoms and ions. Important methods are [[nuclear magnetic resonance]], [[Mössbauer spectroscopy]], [[perturbed angular correlation]], and [[quasielastic neutron scattering|high-resolution inelastic neutron scattering]].

===Nuclear technology===
The [[atomic vapor laser isotope separation]] (AVLIS) process uses the hyperfine splitting between optical transitions in [[uranium-235]] and [[uranium-238]] to selectively [[photoionization|photo-ionize]] only the uranium-235 atoms and then separate the ionized particles from the non-ionized ones. Precisely tuned [[dye laser]]s are used as the sources of the necessary exact wavelength radiation.

===Use in defining the SI second and meter===
The hyperfine structure transition can be used to make a [[microwave]] [[notch filter]] with very high stability, repeatability and [[Q factor]], which can thus be used as a basis for very precise [[atomic clock]]s. The term ''transition frequency'' denotes the frequency of radiation corresponding to the transition between the two hyperfine levels of the atom, and is equal to {{math|1=''f'' = Δ''E''/''h''}}, where {{math|Δ''E''}} is difference in energy between the levels and {{math|''h''}} is the [[Planck constant]]. Typically, the transition frequency of a particular isotope of [[caesium]] or [[rubidium]] atoms is used as a basis for these clocks.

Due to the accuracy of hyperfine structure transition-based atomic clocks, they are now used as the basis for the definition of the second. One [[second]] is now ''[[2019 revision of the SI|defined]]'' to be exactly {{val|9192631770}} cycles of the hyperfine structure transition frequency of caesium-133 atoms.

On October 21, 1983, the 17th [[CGPM]] defined the meter as the length of the path travelled by [[light]] in a [[vacuum]] during a time interval of {{sfrac|299,792,458}} of a [[second]].<ref>Taylor, B.N. and Thompson, A. (Eds.). (2008a). [http://physics.nist.gov/Pubs/SP330/sp330.pdf ''The International System of Units (SI)''] {{Webarchive|url=https://web.archive.org/web/20160603215953/http://physics.nist.gov/Pubs/SP330/sp330.pdf |date=2016-06-03}}. Appendix&nbsp;1, p.&nbsp;70. This is the United States version of the English text of the eighth edition (2006) of the International Bureau of Weights and Measures publication ''Le Système International d' Unités (SI)'' (Special Publication 330). Gaithersburg, MD: National Institute of Standards and Technology. Retrieved 18 August 2008.</ref><ref>Taylor, B.N. and Thompson, A. (2008b). [http://physics.nist.gov/cuu/pdf/sp811.pdf ''Guide for the Use of the International System of Units''] (Special Publication 811). Gaithersburg, MD: National Institute of Standards and Technology. Retrieved 23 August 2008.</ref>

===Precision tests of quantum electrodynamics===
The hyperfine splitting in hydrogen and in [[muonium]] have been used to measure the value of the [[fine-structure constant]] α. Comparison with measurements of α in other physical systems provides a [[precision tests of QED|stringent test of QED]].

===Qubit in ion-trap quantum computing===
The hyperfine states of a trapped [[ion]] are commonly used for storing [[qubit]]s in [[ion-trap quantum computing]]. They have the advantage of having very long lifetimes, experimentally exceeding ~10 minutes (compared to ~1{{nbsp}}s for metastable electronic levels).

The frequency associated with the states' energy separation is in the [[microwave]] region, making it possible to drive hyperfine transitions using microwave radiation. However, at present no emitter is available that can be focused to address a particular ion from a sequence. Instead, a pair of [[laser]] pulses can be used to drive the transition, by having their frequency difference (''detuning'') equal to the required transition's frequency. This is essentially a stimulated [[Raman transition]]. In addition, near-field gradients have been exploited to individually address two ions separated by approximately 4.3 micrometers directly with microwave radiation.<ref>{{cite journal |last1=Warring |title=Individual-Ion Addressing with Microwave Field Gradients |journal=Physical Review Letters |year=2013 |volume=110 |issue=17 |pages=173002 1–5 |arxiv=1210.6407 |first1=U. |last2=Ospelkaus |first2=C. |last3=Colombe |first3=Y. |last4=Joerdens |first4=R. |last5=Leibfried |first5=D. |last6=Wineland |first6=D.J. |bibcode=2013PhRvL.110q3002W |doi=10.1103/PhysRevLett.110.173002 |pmid=23679718 |s2cid=27008582}}</ref>

==See also==
* [[Dynamic nuclear polarization]]
* [[Electron paramagnetic resonance]]

==References==
{{reflist}}

==External links==
* [https://feynmanlectures.caltech.edu/III_12.html The Feynman Lectures on Physics Vol. III Ch. 12: The Hyperfine Splitting in Hydrogen]
* [[File:Queryensdf.jpg]] [http://www-nds.iaea.org/queryensdf Nuclear Magnetic and Electric Moments lookup]—Nuclear Structure and Decay Data at the [[IAEA]]

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[[Category:Atomic physics]]
[[Category:Foundational quantum physics]]