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Hyperfine structure

From Wikipedia, the free encyclopedia

Type of structure in atomic physics

Inatomic physics,hyperfine structure is defined by small shifts in otherwisedegenerate electronicenergy levels and the resultingsplittings in those electronic energy levels ofatoms,molecules, andions, due to electromagnetic multipole interaction between the nucleus and electron clouds.

In atoms, hyperfine structure arises from the energy of thenuclear magnetic dipole moment interacting with themagnetic field generated by the electrons and the energy of thenuclear electric quadrupole moment in theelectric 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 withfine structure, which results from the interaction between themagnetic moments associated withelectron spin and the electrons'orbital angular momentum. Hyperfine structure, with energy shifts typically orders of magnitude smaller than those of a fine-structure shift, results from the interactions of thenucleus (or nuclei, in molecules) with internally generated electric and magnetic fields.

Schematic illustration offine and hyperfine structure in a neutralhydrogen atom

History

The first theory of atomic hyperfine structure was given in 1930 byEnrico Fermi^[1] for an atom containing a single valence electron with an arbitrary angular momentum. TheZeeman splitting of this structure was discussed byS. A. Goudsmit andR. 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 ofeuropium,cassiopium (older name for lutetium),indium,antimony, andmercury.^[2]

Theory

The theory of hyperfine structure comes directly fromelectromagnetism, consisting of the interaction of the nuclearmultipole moments (excluding the electric monopole) with internally generated fields. The theory is derived first for the atomic case, but can be applied toeach 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 article:Dipole

The dominant term in the hyperfineHamiltonian is typically the magnetic dipole term. Atomic nuclei with a non-zeronuclear spin $\mathbf {I}$ have a magnetic dipole moment, given by: ${\boldsymbol {\mu }}_{\text{I}}=g_{\text{I}}\mu _{\text{N}}\mathbf {I} ,$ where $g_{\text{I}}$ is theg-factor and $\mu _{\text{N}}$ is thenuclear 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_I, placed in a magnetic field,B, the relevant term in the Hamiltonian is given by:^[3] ${\hat {H}}_{\text{D}}=-{\boldsymbol {\mu }}_{\text{I}}\cdot \mathbf {B} .$

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: $\mathbf {B} \equiv \mathbf {B} _{\text{el}}=\mathbf {B} _{\text{el}}^{\ell }+\mathbf {B} _{\text{el}}^{s}.$

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 –e at a positionr relative to the nucleus, is given by: $\mathbf {B} _{\text{el}}^{\ell }={\frac {\mu _{0}}{4\pi }}{\frac {-e\mathbf {v} \times -\mathbf {r} }{r^{3}}},$ where −r gives the position of the nucleus relative to the electron. Written in terms of theBohr magneton, this gives: $\mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{r^{3}}}{\frac {\mathbf {r} \times m_{\text{e}}\mathbf {v} }{\hbar }}.$

Recognizing thatm_ev is the electron momentum,p, and thatr ×p /ħ is the orbitalangular momentum in units ofħ,ℓ, we can write: $\mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{r^{3}}}{\boldsymbol {\ell }}.$

For a many-electron atom this expression is generally written in terms of the total orbital angular momentum, $\mathbf {L}$ , by summing over the electrons and using the projection operator, $\varphi _{i}^{\ell }$ , where ${\textstyle \sum _{i}\mathbf {\ell } _{i}=\sum _{i}\varphi _{i}^{\ell }\mathbf {L} }$ . For states with a well defined projection of the orbital angular momentum,L_z, we can write $\varphi _{i}^{\ell }={\hat {\ell }}_{z_{i}}/L_{z}$ , giving: $\mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{L_{z}}}\sum _{i}{\frac {{\hat {\ell }}_{zi}}{r_{i}^{3}}}\mathbf {L} .$

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_s, given by: ${\boldsymbol {\mu }}_{\text{s}}=-g_{s}\mu _{\text{B}}\mathbf {s} ,$ whereg_s is theelectron sping-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 incurrents in the opposite direction).

The magnetic field of a point dipole moment,μ_s, is given by:^[4]^[5] $\mathbf {B} _{\text{el}}^{s}={\frac {\mu _{0}}{4\pi r^{3}}}\left(3\left({\boldsymbol {\mu }}_{\text{s}}\cdot {\hat {\mathbf {r} }}\right){\hat {\mathbf {r} }}-{\boldsymbol {\mu }}_{\text{s}}\right)+{\dfrac {2\mu _{0}}{3}}{\boldsymbol {\mu }}_{\text{s}}\delta ^{3}(\mathbf {r} ).$

Electron total magnetic field and contribution

The complete magnetic dipole contribution to the hyperfine Hamiltonian is thus given by: ${\begin{aligned}{\hat {H}}_{\text{D}}={}&2g_{\text{I}}\mu _{\text{N}}\mu _{\text{B}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {1}{L_{z}}}\sum _{i}{\dfrac {{\hat {\ell }}_{zi}}{r_{i}^{3}}}\mathbf {I} \cdot \mathbf {L} \\&{}+g_{\text{I}}\mu _{\text{N}}g_{\text{s}}\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{S_{z}}}\sum _{i}{\frac {{\hat {s}}_{zi}}{r_{i}^{3}}}\left\{3\left(\mathbf {I} \cdot {\hat {\mathbf {r} }}\right)\left(\mathbf {S} \cdot {\hat {\mathbf {r} }}\right)-\mathbf {I} \cdot \mathbf {S} \right\}\\&{}+{\frac {2}{3}}g_{\text{I}}\mu _{\text{N}}g_{\text{s}}\mu _{\text{B}}\mu _{0}{\frac {1}{S_{z}}}\sum _{i}{\hat {s}}_{zi}\delta ^{3}{\left(\mathbf {r} _{i}\right)}\mathbf {I} \cdot \mathbf {S} .\end{aligned}}$

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 theFermi 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 ins-subshells). It has been argued that one may get a different expression when taking into account the detailed nuclear magnetic moment distribution.^[6] The inclusion of the delta function is an admission that the singularity in the magnetic inductionB owing to a magnetic dipole moment at a point is not integrable. It isB 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 componentA is the three-vector magnetic potential (such thatB =curl A), which for the point dipole is integrable.

For states with $\ell \neq 0$ this can be expressed in the form ${\hat {H}}_{\text{D}}=2g_{I}\mu _{\text{B}}\mu _{\text{N}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {\mathbf {I} \cdot \mathbf {N} }{r^{3}}},$ where:^[3] $\mathbf {N} ={\boldsymbol {\ell }}-{\frac {g_{s}}{2}}\left[\mathbf {s} -3(\mathbf {s} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}\right].$

If hyperfine structure is small compared with the fine structure (sometimes calledIJ-coupling by analogy withLS-coupling),I andJ are goodquantum numbers and matrix elements of ${\hat {H}}_{\text{D}}$ can be approximated as diagonal inI andJ. In this case (generally true for light elements), we can projectN ontoJ (whereJ =L +S is the total electronic angular momentum) and we have:^[7] ${\hat {H}}_{\text{D}}=2g_{I}\mu _{\text{B}}\mu _{\text{N}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {\mathbf {N} \cdot \mathbf {J} }{\mathbf {J} \cdot \mathbf {J} }}{\dfrac {\mathbf {I} \cdot \mathbf {J} }{r^{3}}}.$

This is commonly written as ${\hat {H}}_{\text{D}}={\hat {A}}\mathbf {I} \cdot \mathbf {J} ,$ with ${\textstyle \left\langle {\hat {A}}\right\rangle }$ being the hyperfine-structure constant which is determined by experiment. SinceI⋅J =1⁄2{F⋅F −I⋅I −J⋅J} (whereF =I +J is the total angular momentum), this gives an energy of: $\Delta E_{\text{D}}={\frac {1}{2}}\left\langle {\hat {A}}\right\rangle [F(F+1)-I(I+1)-J(J+1)].$

In this case the hyperfine interaction satisfies theLandé interval rule.

Electric quadrupole

Main article:Quadrupole

Atomic nuclei with spin $I\geq 1$ have anelectric quadrupole moment.^[8] In the general case this is represented by arank-2tensor, $Q_{ij}$ , with components given by:^[4] $Q_{ij}={\frac {1}{e}}\int \left(3x_{i}^{\prime }x_{j}^{\prime }-\left(r'\right)^{2}\delta _{ij}\right)\rho {\left(\mathbf {r} '\right)}\,d^{3}\mathbf {r} ',$ wherei andj are the tensor indices running from 1 to 3,x_i andx_j are the spatial variablesx,y andz depending on the values ofi andj respectively,δ_ij is theKronecker delta andρ(r) is the charge density. Being a 3-dimensional rank-2 tensor, the quadrupole moment has 3² = 9 components. From the definition of the components it is clear that the quadrupole tensor is asymmetric matrix (Q_ij =Q_ji) that is alsotraceless ( ${\textstyle \operatorname {tr} Q=\sum _{i}Q_{ii}=0}$ ), giving only five components in theirreducible representation. Expressed using the notation ofirreducible spherical tensors we have:^[4] $T_{m}^{2}(Q)={\sqrt {\frac {4\pi }{5}}}\int \rho {\left(\mathbf {r} '\right)}\left(r'\right)^{2}Y_{m}^{2}\left(\theta ',\varphi '\right)\,d^{3}\mathbf {r} '.$

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 ${\textstyle {\underline {\underline {q}}}}$ , another rank-2 tensor given by theouter product of thedel operator with the electric field vector: ${\underline {\underline {q}}}=\nabla \otimes \mathbf {E} ,$ with components given by: $q_{ij}={\frac {\partial ^{2}V}{\partial x_{i}\,\partial x_{j}}}.$

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, $T^{2}(q)$ , with:^[9] ${\begin{aligned}T_{0}^{2}(q)&={\frac {\sqrt {6}}{2}}q_{zz}\\T_{+1}^{2}(q)&=-q_{xz}-iq_{yz}\\T_{+2}^{2}(q)&={\frac {1}{2}}(q_{xx}-q_{yy})+iq_{xy},\end{aligned}}$ where: $T_{-m}^{2}(q)=(-1)^{m}T_{+m}^{2}(q)^{*}.$

The quadrupolar term in the Hamiltonian is thus given by: ${\hat {H}}_{Q}=-eT^{2}(Q)\cdot T^{2}(q)=-e\sum _{m}(-1)^{m}T_{m}^{2}(Q)T_{-m}^{2}(q).$

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 byQ_zz.^[8]

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 $I>0$ and an electric quadrupole term for each nucleus with $I\geq 1$ . The magnetic dipole terms were first derived for diatomic molecules by Frosch and Foley,^[10] 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.^[11]

Direct nuclear spin–spin

Each nucleus with $I>0$ 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 eachother magnetic moment gives the direct nuclear spin–spin term in the hyperfine Hamiltonian, ${\hat {H}}_{II}$ .^[12] ${\hat {H}}_{II}=-\sum _{\alpha \neq \alpha '}{\boldsymbol {\mu }}_{\alpha }\cdot \mathbf {B} _{\alpha '},$ whereα andα' 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 ${\hat {H}}_{II}={\dfrac {\mu _{0}\mu _{\text{N}}^{2}}{4\pi }}\sum _{\alpha \neq \alpha '}{\frac {g_{\alpha }g_{\alpha '}}{R_{\alpha \alpha '}^{3}}}\left\{\mathbf {I} _{\alpha }\cdot \mathbf {I} _{\alpha '}-3\left(\mathbf {I} _{\alpha }\cdot {\hat {\mathbf {R} }}_{\alpha \alpha '}\right)\left(\mathbf {I} _{\alpha '}\cdot {\hat {\mathbf {R} }}_{\alpha \alpha '}\right)\right\}.$

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,^[12] thus ${\hat {H}}_{\text{IR}}={\frac {e\mu _{0}\mu _{\text{N}}\hbar }{4\pi }}\sum _{\alpha \neq \alpha '}{\frac {1}{R_{\alpha \alpha '}^{3}}}\left\{{\frac {Z_{\alpha }g_{\alpha '}}{M_{\alpha }}}\mathbf {I} _{\alpha '}+{\frac {Z_{\alpha '}g_{\alpha }}{M_{\alpha '}}}\mathbf {I} _{\alpha }\right\}\cdot \mathbf {T} .$

Small molecule hyperfine structure

A typical simple example of the hyperfine structure due to the interactions discussed above is in the rotational transitions ofhydrogen cyanide (¹H¹²C¹⁴N) in its groundvibrational state. Here, the electric quadrupole interaction is due to the¹⁴N-nucleus, the hyperfine nuclear spin-spin splitting is from the magnetic coupling between nitrogen,¹⁴N (I_N = 1), and hydrogen,¹H (I_H =1⁄2), and a hydrogen spin-rotation interaction due to the¹H-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.^[13]

The dipoleselection rules for HCN hyperfine structure transitions are $\Delta J=1$ , $\Delta F=\{0,\pm 1\}$ , whereJ is the rotational quantum number andF is the total rotational quantum number inclusive of nuclear spin ( $F=J+I_{\text{N}}$ ), respectively. The lowest transition ( $J=1\rightarrow 0$ ) splits into a hyperfine triplet. Using the selection rules, the hyperfine pattern of $J=2\rightarrow 1$ transition and higher dipole transitions is in the form of a hyperfine sextet. However, one of these components ( $\Delta F=-1$ ) carries only 0.6% of the rotational transition intensity in the case of $J=2\rightarrow 1$ . This contribution drops for increasing J. So, from $J=2\rightarrow 1$ upwards the hyperfine pattern consists of three very closely spaced stronger hyperfine components ( $\Delta J=1$ , $\Delta F=1$ ) 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 ~ ${\tfrac {1}{2}}J^{2}$ (J is the upper rotational quantum number of the allowed dipole transition) the intensity of the entire transition. For consecutively higher-J transitions, there are small but significant changes in the relative intensities and positions of each individual hyperfine component.^[14]

Measurements and Applications

Hyperfine interactions can be measured, among other ways, in atomic and molecular spectra, and inelectron paramagnetic resonance spectra offree radicals andtransition-metal ions.

Astrophysics

The hyperfine transition as depicted on thePioneer 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 the21 cm line observed inH I regions ininterstellar medium.

Carl Sagan andFrank 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 thePioneer plaque and laterVoyager Golden Record.

Insubmillimeter astronomy,heterodyne receivers are widely used in detecting electromagnetic signals from celestial objects such as star-forming core oryoung stellar objects. The separations among neighboring components in a hyperfine spectrum of an observedrotational transition are usually small enough to fit within the receiver'sIF band. Since theoptical depth varies with frequency, strength ratios among the hyperfine components differ from that of their intrinsic (oroptically thin) intensities (these are so-calledhyperfine anomalies, often observed in the rotational transitions of HCN^[14]). Thus, a more accurate determination of the optical depth is possible. From this we can derive the object's physical parameters.^[15]

Nuclear spectroscopy

Hyperfine splitting in ferromagnetic cobalt at 3.5 K, observed by quasielastic neutron scattering.^[16]

Innuclear spectroscopy methods, the nucleus is used to probe thelocal structure in materials. The methods mainly base on hyperfine interactions with the surrounding atoms and ions. Important methods arenuclear magnetic resonance,Mössbauer spectroscopy,perturbed angular correlation, andhigh-resolution inelastic neutron scattering.

Nuclear technology

Theatomic vapor laser isotope separation (AVLIS) process uses the hyperfine splitting between optical transitions inuranium-235 anduranium-238 to selectivelyphoto-ionize only the uranium-235 atoms and then separate the ionized particles from the non-ionized ones. Precisely tuneddye lasers 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 amicrowave notch filter with very high stability, repeatability andQ factor, which can thus be used as a basis for very preciseatomic clocks. The termtransition frequency denotes the frequency of radiation corresponding to the transition between the two hyperfine levels of the atom, and is equal tof = ΔE/h, whereΔE is difference in energy between the levels andh is thePlanck constant. Typically, the transition frequency of a particular isotope ofcaesium orrubidium 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. Onesecond is nowdefined to be exactly9192631770 cycles of the hyperfine structure transition frequency of caesium-133 atoms.

On October 21, 1983, the 17thCGPM defined the meter as the length of the path travelled bylight in avacuum during a time interval of⁠1/299,792,458⁠ of asecond.^[17]^[18]

Precision tests of quantum electrodynamics

The hyperfine splitting in hydrogen and inmuonium have been used to measure the value of thefine-structure constant α. Comparison with measurements of α in other physical systems provides astringent test of QED.

Qubit in ion-trap quantum computing

The hyperfine states of a trappedion are commonly used for storingqubits inion-trap quantum computing. They have the advantage of having very long lifetimes, experimentally exceeding ~10 minutes (compared to ~1 s for metastable electronic levels).

The frequency associated with the states' energy separation is in themicrowave 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 oflaser 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 stimulatedRaman transition. In addition, near-field gradients have been exploited to individually address two ions separated by approximately 4.3 micrometers directly with microwave radiation.^[19]

See also

References

^E. Fermi (1930), "Uber die magnetischen Momente der Atomkerne". Z. Physik 60, 320-333.
^H. Schüler & T. Schmidt (1935), "Über Abweichungen des Atomkerns von der Kugelsymmetrie". Z. Physik 94, 457–468.
^^a ^bWoodgate, Gordon K. (1999).Elementary Atomic Structure. Oxford University Press.ISBN 978-0-19-851156-4.
^^a ^b ^cJackson, John D. (1998).Classical Electrodynamics. Wiley.ISBN 978-0-471-30932-1.
^Garg, Anupam (2012).Classical Electromagnetism in a Nutshell. Princeton University Press. §26.ISBN 978-0-691-13018-7.
^Soliverez, C E (1980-12-10)."The contact hyperfine interaction: an ill-defined problem".Journal of Physics C: Solid State Physics.13 (34):L1017 –L1019.doi:10.1088/0022-3719/13/34/002.ISSN 0022-3719.
^Woodgate, Gordon K. (1983).Elementary Atomic Structure. Oxford University Press, USA.ISBN 978-0-19-851156-4. Retrieved2009-03-03.
^^a ^bEnge, Harald A. (1966).Introduction to Nuclear Physics. Addison Wesley.ISBN 978-0-201-01870-7.
^Y. Millot (2008-02-19)."Electric field gradient tensor around quadrupolar nuclei". Retrieved2008-07-23.
^Frosch and Foley; Foley, H. (1952). "Magnetic hyperfine structure in diatomics".Physical Review.88 (6):1337–1349.Bibcode:1952PhRv...88.1337F.doi:10.1103/PhysRev.88.1337.
^Brown, John; Alan Carrington (2003).Rotational Spectroscopy of Diatomic Molecules. Cambridge University Press.ISBN 978-0-521-53078-1.
^^a ^bBrown, John; Alan Carrington (2003).Rotational Spectroscopy of Diatomic Molecules. Cambridge University Press.ISBN 978-0-521-53078-1. Retrieved2009-03-03.
^Ahrens, V.; Lewen, F.; Takano, S.; Winnewisser, G.; et al. (2002)."Sub-Doppler Saturation Spectroscopy of HCN up to 1 THz and Detection of $J={\ce {3 -> 2 (4 -> 3)}}$ Emission from TMC-1".Z. Naturforsch.57a (8):669–681.Bibcode:2002ZNatA..57..669A.doi:10.1515/zna-2002-0806.S2CID 35586070.
^^a ^bMullins, A. M.; Loughnane, R. M.; Redman, M. P.; et al. (2016)."Radiative Transfer of HCN: Interpreting observations of hyperfine anomalies".Monthly Notices of the Royal Astronomical Society.459 (3):2882–2993.arXiv:1604.03059.Bibcode:2016MNRAS.459.2882M.doi:10.1093/mnras/stw835.S2CID 119192931.
^Tatematsu, K.; Umemoto, T.; Kandori, R.; et al. (2004). "N₂H⁺ Observations of Molecular Cloud Cores in Taurus".Astrophysical Journal.606 (1):333–340.arXiv:astro-ph/0401584.Bibcode:2004ApJ...606..333T.doi:10.1086/382862.S2CID 118956636.
^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.
^Taylor, B.N. and Thompson, A. (Eds.). (2008a).The International System of Units (SI)Archived 2016-06-03 at theWayback Machine. Appendix 1, p. 70. This is the United States version of the English text of the eighth edition (2006) of the International Bureau of Weights and Measures publicationLe Système International d' Unités (SI) (Special Publication 330). Gaithersburg, MD: National Institute of Standards and Technology. Retrieved 18 August 2008.
^Taylor, B.N. and Thompson, A. (2008b).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.
^Warring, U.; Ospelkaus, C.; Colombe, Y.; Joerdens, R.; Leibfried, D.; Wineland, D.J. (2013). "Individual-Ion Addressing with Microwave Field Gradients".Physical Review Letters.110 (17): 173002 1–5.arXiv:1210.6407.Bibcode:2013PhRvL.110q3002W.doi:10.1103/PhysRevLett.110.173002.PMID 23679718.S2CID 27008582.

External links

The Feynman Lectures on Physics Vol. III Ch. 12: The Hyperfine Splitting in Hydrogen
Nuclear Magnetic and Electric Moments lookup—Nuclear Structure and Decay Data at theIAEA

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