2210:
1618:
2205:{\displaystyle {\begin{aligned}{\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{aligned}}}
4993:
5278:
5745:
4676:
107:
1606:
5013:
4988:{\displaystyle {\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\}.}
1398:
2756:
3647:
4120:
3316:
5273:{\displaystyle {\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} .}
1277:
771:
2595:
2214:
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
1286:
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,
77:
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
5319:), and a hydrogen spin-rotation interaction due to the 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.
1601:{\displaystyle \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} ).}
2383:
4672:
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
4659:
3866:
913:
636:
518:
4516:
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
3474:
3146:
1131:
649:
527:
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
4403:
1378:
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
428:
3053:
284:
2258:
1041:
1361:
3822:
2825:
2751:{\displaystyle {\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}}}.}
3826:
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,
2499:
3738:
2221:
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
6057:
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
818:
541:
1126:
443:
4563:
4407:
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
4223:
5755:
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.
3871:
3461:
1623:
2862:
6483:
3381:) is the charge density. Being a 3-dimensional rank-2 tensor, the quadrupole moment has 3 = 9 components. From the definition of the components it is clear that the quadrupole tensor is a
4230:
4115:{\displaystyle {\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}}}
3681:
5823:, often observed in the rotational transitions of HCN). Thus, a more accurate determination of the optical depth is possible. From this we can derive the object's physical parameters.
5664:) 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 ~
4479:. The magnetic dipole terms were first derived for diatomic molecules by Frosch and Foley, and the resulting hyperfine parameters are often called the Frosch and Foley parameters.
2560:
5701:
5397:
969:
6663:. 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 publication
5604:
5572:
5508:
5476:
5444:
3468:
368:
4558:
2926:
345:
229:
3642:{\displaystyle 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} '.}
6763:
3311:{\displaystyle 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} ',}
138:
later that year. In 1935, H. SchĂŒler and
Theodor Schmidt proposed the existence of a nuclear quadrupole moment in order to explain anomalies in the hyperfine structure of
1272:{\displaystyle \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} .}
937:
766:{\displaystyle \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 }}.}
311:
224:
5540:
3651:
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
2253:
5662:
5633:
5353:
1303:
6070:. In addition, near-field gradients have been exploited to individually address two ions separated by approximately 4.3 micrometers directly with microwave radiation.
3861:
4477:
3743:
3099:
4514:
4451:
3141:
2763:
5295:. Here, the electric quadrupole interaction is due to the N-nucleus, the hyperfine nuclear spin-spin splitting is from the magnetic coupling between nitrogen, N (
2388:
3694:
99:. Hyperfine structure, with energy shifts typically orders of magnitudes smaller than those of a fine-structure shift, results from the interactions of the
6692:
Warring, U.; Ospelkaus, C.; Colombe, Y.; Joerdens, R.; Leibfried, D.; Wineland, D.J. (2013). "Individual-Ion
Addressing with Microwave Field Gradients".
974:
5780:
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
2226:-subshells). It has been argued that one may get a different expression when taking into account the detailed nuclear magnetic moment distribution.
5948:
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
4427:
The molecular hyperfine
Hamiltonian includes those terms already derived for the atomic case with a magnetic dipole term for each nucleus with
4125:
2378:{\displaystyle {\hat {H}}_{D}=2g_{I}\mu _{\text{B}}\mu _{\text{N}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {\mathbf {I} \cdot \mathbf {N} }{r^{3}}},}
130:
for an atom containing a single valence electron with an arbitrary angular momentum. The Zeeman splitting of this structure was discussed by
174:(excluding the electric monopole) with internally generated fields. The theory is derived first for the atomic case, but can be applied to
5707:
is the upper rotational quantum number of the allowed dipole transition) the intensity of the entire transition. For consecutively higher-
5711:
transitions, there are small but significant changes in the relative intensities and positions of each individual hyperfine component.
1055:
354:
There is an energy associated with a magnetic dipole moment in the presence of a magnetic field. For a nuclear magnetic dipole moment,
432:
In the absence of an externally applied field, the magnetic field experienced by the nucleus is that associated with the orbital (
5906:
denotes the frequency of radiation corresponding to the transition between the two hyperfine levels of the atom, and is equal to
5292:
5287:
A typical simple example of the hyperfine structure due to the interactions discussed above is in the rotational transitions of
908:{\displaystyle \mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{r^{3}}}\mathbf {\ell } .}
631:{\displaystyle \mathbf {B} _{\text{el}}^{\ell }={\frac {\mu _{0}}{4\pi }}{\frac {-e\mathbf {v} \times -\mathbf {r} }{r^{3}}},}
6790:
6406:
6376:
6293:
6255:
6189:
6164:
6134:
5860:
5839:
in materials. The methods mainly base on hyperfine interactions with the surrounding atoms and ions. Important methods are
6656:
513:{\displaystyle \mathbf {B} \equiv \mathbf {B} _{\text{el}}=\mathbf {B} _{\text{el}}^{\ell }+\mathbf {B} _{\text{el}}^{s}.}
4654:{\displaystyle {\hat {H}}_{II}=-\sum _{\alpha \neq \alpha '}{\boldsymbol {\mu }}_{\alpha }\cdot \mathbf {B} _{\alpha '},}
17:
3408:
2830:
6682:(Special Publication 811). Gaithersburg, MD: National Institute of Standards and Technology. Retrieved 23 August 2008.
6667:(Special Publication 330). Gaithersburg, MD: National Institute of Standards and Technology. Retrieved 18 August 2008.
6433:
6112:
H. SchĂŒler & T. Schmidt (1935), âĂber
Abweichungen des Atomkerns von der Kugelsymmetrieâ. Z. Physik 94, 457â468.
6396:
6245:
6084:
5720:
5510:
transition and higher dipole transitions is in the form of a hyperfine sextet. However, one of these components (
198:
3654:
5875:
only the uranium-235 atoms and then separate the ionized particles from the non-ionized ones. Precisely tuned
5815:
varies with frequency, strength ratios among the hyperfine components differ from that of their intrinsic (or
917:
For a many-electron atom this expression is generally written in terms of the total orbital angular momentum,
6046:. They have the advantage of having very long lifetimes, experimentally exceeding ~10 minutes (compared to ~1
5848:
178:
in a molecule. Following this there is a discussion of the additional effects unique to the molecular case.
6310:
6079:
2529:
5667:
5358:
4482:
In addition to the effects described above, there are a number of effects specific to the molecular case.
942:
5799:
are widely used in detecting electromagnetic signals from celestial objects such as star-forming core or
6043:
5840:
3464:
2508:
5577:
5545:
5481:
5449:
5410:
4524:
4398:{\displaystyle {\hat {H}}_{Q}=-eT^{2}(Q)\cdot T^{2}(q)=-e\sum _{m}(-1)^{m}T_{m}^{2}(Q)T_{-m}^{2}(q).}
2217:
323:
5844:
5796:
5606:
upwards the hyperfine pattern consists of three very closely spaced stronger hyperfine components (
96:
39:
5010:
is the internuclear displacement vector), associated with the bulk rotation of the molecule, thus
5002:
The nuclear magnetic moments in a molecule exist in a magnetic field due to the angular momentum,
920:
289:
207:
6785:
6206:
6019:
5954:
5792:
5719:
Hyperfine interactions can be measured, among other ways, in atomic and molecular spectra and in
5513:
2232:
74:
62:
5638:
5609:
5329:
6023:
5808:
3830:
3058:
5804:
5800:
5785:
4456:
3078:
4493:
4430:
3116:
6711:
6617:
6558:
6496:
6341:
5832:
78:
nuclear magnetic moments and the magnetic field generated by the rotation of the molecule.
8:
5767:
1615:
The complete magnetic dipole contribution to the hyperfine
Hamiltonian is thus given by:
423:{\displaystyle {\hat {H}}_{\text{D}}=-{\boldsymbol {\mu }}_{\text{I}}\cdot \mathbf {B} .}
6715:
6621:
6562:
6500:
6345:
3048:{\displaystyle \Delta E_{\text{D}}={\frac {1}{2}}\left\langle {\hat {A}}\right\rangle .}
279:{\displaystyle {\boldsymbol {\mu }}_{\text{I}}=g_{\text{I}}\mu _{\text{N}}\mathbf {I} ,}
58:, due to electromagnetic multipole interaction between the nucleus and electron clouds.
6735:
6701:
6633:
6607:
6576:
6548:
6514:
6282:
5478:) splits into a hyperfine triplet. Using the selection rules, the hyperfine pattern of
1380:
1371:
314:
6222:
6103:
E. Fermi (1930), âUber die magnetischen
Momente der Atomkerneâ. Z. Physik 60, 320-333.
5803:. The separations among neighboring components in a hyperfine spectrum of an observed
4521:
magnetic moment gives the direct nuclear spinâspin term in the hyperfine
Hamiltonian,
6727:
6637:
6580:
6402:
6372:
6289:
6251:
6226:
6185:
6160:
6130:
530:
171:
155:
6757:
The
Feynman Lectures on Physics Vol. III Ch. 12: The Hyperfine Splitting in Hydrogen
6739:
6518:
1036:{\textstyle \sum _{i}\mathbf {\ell } _{i}=\sum _{i}\varphi _{i}^{\ell }\mathbf {L} }
642:
gives the position of the nucleus relative to the electron. Written in terms of the
6723:
6719:
6625:
6566:
6504:
6349:
6218:
6067:
5728:
5288:
3382:
2503:
If hyperfine structure is small compared with the fine structure (sometimes called
804:
348:
167:
6756:
6062:
pulses can be used to drive the transition, by having their frequency difference (
1356:{\displaystyle {\boldsymbol {\mu }}_{\text{s}}=-g_{s}\mu _{\text{B}}\mathbf {s} ,}
103:(or nuclei, in molecules) with internally generated electric and magnetic fields.
6660:
6332:
6330:
Frosch and Foley; Foley, H. (1952). "Magnetic hyperfine structure in diatomics".
6066:) equal to the required transition's frequency. This is essentially a stimulated
5934:
5872:
5836:
5323:
3370:
1287:
which is the source of a magnetic field. An electron with spin angular momentum,
131:
88:
3817:{\displaystyle q_{ij}={\frac {\partial ^{2}V}{\partial x_{i}\,\partial x_{j}}}.}
2864:
being the hyperfine-structure constant which is determined by experiment. Since
6651:
5781:
5749:
2820:{\displaystyle {\hat {H}}_{\text{D}}={\hat {A}}\mathbf {I} \cdot \mathbf {J} ,}
2523:
111:
100:
83:
66:
31:
6779:
6537:"Radiative Transfer of HCN: Interpreting observations of hyperfine anomalies"
6230:
5971:
cycles of the hyperfine structure transition frequency of caesium-133 atoms.
5812:
3684:
1043:. For states with a well defined projection of the orbital angular momentum,
643:
135:
115:
92:
43:
27:
Small shifts and splittings in the energy levels of atoms, molecules and ions
6677:
6353:
5863:(AVLIS) process uses the hyperfine splitting between optical transitions in
6731:
6571:
6536:
5899:
5891:
5724:
3688:
3106:
2494:{\displaystyle \mathbf {N} ={\boldsymbol {\ell }}-{\frac {g_{s}}{2}}\left.}
202:
127:
6509:
6429:
6022:α. Comparison with measurements of α in other physical systems provides a
5744:
5542:) carries only 0.6% of the rotational transition intensity in the case of
3733:{\displaystyle {\underline {\underline {q}}}=\nabla \otimes \mathbf {E} ,}
6612:
6430:"Sub-Doppler Saturation Spectroscopy of HCN up to 1 THz and Detection of
5868:
5864:
5777:
6428:
Ahrens, V.; Lewen, F.; Takano, S.; Winnewisser, G.; et al. (2002).
106:
5773:
5763:
5759:
3102:
3070:
143:
70:
6053:
The frequency associated with the states' energy separation is in the
6766:
6535:
Mullins, A. M.; Loughnane, R. M.; Redman, M. P.; et al. (2016).
6054:
5888:
5879:
are used as the sources of the necessary exact wavelength radiation.
5876:
3402:
201:
is typically the magnetic dipole term. Atomic nuclei with a non-zero
126:
The first theory of atomic hyperfine structure was given in 1930 by
6629:
6553:
5942:
5895:
2570:. In this case (generally true for light elements), we can project
939:, by summing over the electrons and using the projection operator,
151:
139:
51:
6706:
5937:. Typically, the transition frequency of a particular isotope of
5407:
is the total rotational quantum number inclusive of nuclear spin (
46:
and the resulting splittings in those electronic energy levels of
6015:
5938:
1610:
1121:{\displaystyle \varphi _{i}^{\ell }={\hat {\ell }}_{z_{i}}/L_{z}}
6691:
6594:
Tatematsu, K.; Umemoto, T.; Kandori, R.; et al. (2004). "N
6003:
5983:
5949:
3110:
192:
147:
6762:
6059:
6039:
6009:
5979:
6427:
61:
In atoms, hyperfine structure arises from the energy of the
6770:
6207:"The contact hyperfine interaction: an ill-defined problem"
5975:
47:
6311:"Electric field gradient tensor around quadrupolar nuclei"
5882:
4227:
The quadrupolar term in the
Hamiltonian is thus given by:
6035:
5978:
defined the meter as the length of the path travelled by
5887:
The hyperfine structure transition can be used to make a
2923:
is the total angular momentum), this gives an energy of:
55:
4218:{\displaystyle T_{-m}^{2}(q)=(-1)^{m}T_{+m}^{2}(q)^{*}.}
6593:
2592:
is the total electronic angular momentum) and we have:
6679:
6534:
6029:
5807:
are usually small enough to fit within the receiver's
5672:
4453:
and an electric quadrupole term for each nucleus with
3657:
3411:
2833:
977:
166:
The theory of hyperfine structure comes directly from
6436:
6329:
5898:, which can thus be used as a basis for very precise
5670:
5641:
5612:
5580:
5574:. This contribution drops for increasing J. So, from
5548:
5516:
5484:
5452:
5413:
5361:
5332:
5282:
5016:
4706:
4679:
4566:
4527:
4496:
4459:
4433:
4233:
4128:
3869:
3833:
3746:
3697:
3477:
3149:
3119:
3081:
3057:
In this case the hyperfine interaction satisfies the
2929:
2766:
2715:
2679:
2655:
2598:
2532:
2391:
2342:
2318:
2261:
2235:
1739:
1710:
1686:
1621:
1539:
1401:
1306:
1134:
1058:
945:
923:
821:
652:
544:
446:
371:
326:
292:
232:
210:
6598:
H Observations of Molecular Cloud Cores in Taurus".
6394:
6366:
365:, the relevant term in the Hamiltonian is given by:
3456:{\textstyle \operatorname {tr} Q=\sum _{i}Q_{ii}=0}
6477:
6281:
5695:
5656:
5627:
5598:
5566:
5534:
5502:
5470:
5438:
5391:
5347:
5272:
4987:
4653:
4552:
4508:
4471:
4445:
4397:
4217:
4114:
3855:
3816:
3732:
3675:
3641:
3455:
3310:
3135:
3093:
3047:
2857:{\textstyle \left\langle {\hat {A}}\right\rangle }
2856:
2819:
2750:
2554:
2493:
2377:
2247:
2204:
1600:
1355:
1271:
1120:
1035:
963:
931:
907:
765:
630:
522:
512:
422:
339:
305:
278:
218:
6541:Monthly Notices of the Royal Astronomical Society
87:, which results from the interaction between the
69:generated by the electrons and the energy of the
6777:
4422:
6650:Taylor, B.N. and Thompson, A. (Eds.). (2008a).
6478:{\displaystyle J={\ce {3 -> 2 (4 -> 3)}}}
5927:is difference in energy between the levels and
3105:. In the general case this is represented by a
1281:
170:, consisting of the interaction of the nuclear
1611:Electron total magnetic field and contribution
6398:Rotational Spectroscopy of Diatomic Molecules
6390:
6388:
6369:Rotational Spectroscopy of Diatomic Molecules
1386:The magnetic field of a point dipole moment,
181:
6767:Nuclear Magnetic and Electric Moments lookup
6323:
5894:with very high stability, repeatability and
5748:The hyperfine transition as depicted on the
5386:
5371:
5326:for HCN hyperfine structure transitions are
3326:are the tensor indices running from 1 to 3,
6018:have been used to measure the value of the
6014:The hyperfine splitting in hydrogen and in
5945:atoms is used as a basis for these clocks.
4485:
6530:
6528:
6385:
6120:
6118:
6010:Precision tests of quantum electrodynamics
5835:methods, the nucleus is used to probe the
3676:{\textstyle {\underline {\underline {q}}}}
6769:âNuclear Structure and Decay Data at the
6705:
6611:
6570:
6552:
6508:
6469:
6462:
6454:
6447:
6360:
6308:
6211:Journal of Physics C: Solid State Physics
6204:
6150:
6148:
6146:
3794:
3615:
3284:
226:have a magnetic dipole moment, given by:
6243:
6237:
6182:Classical Electromagnetism in a Nutshell
6124:
5743:
5446:), respectively. The lowest transition (
4997:
105:
38:is defined by small shifts in otherwise
6676:Taylor, B.N. and Thompson, A. (2008b).
6665:Le SystÚme International d' Unités (SI)
6525:
6154:
6115:
5883:Use in defining the SI second and meter
5826:
4618:
2401:
1564:
1520:
1469:
1309:
399:
235:
14:
6778:
6653:The International System of Units (SI)
6275:
6273:
6143:
3463:), giving only five components in the
3064:
538:relative to the nucleus, is given by:
6395:Brown, John; Alan Carrington (2003).
6367:Brown, John; Alan Carrington (2003).
6050:s for metastable electronic levels).
5861:atomic vapor laser isotope separation
5854:
5403:is the rotational quantum number and
3683:, another rank-2 tensor given by the
2555:{\displaystyle {\hat {H}}_{\text{D}}}
440:) angular momentum of the electrons:
114:and hyperfine structure in a neutral
6279:
6179:
5696:{\displaystyle {\tfrac {1}{2}}J^{2}}
5392:{\displaystyle \Delta F=\{0,\pm 1\}}
964:{\displaystyle \varphi _{i}^{\ell }}
6270:
6184:. Princeton University Press. §26.
6173:
6030:Qubit in ion-trap quantum computing
5819:) intensities (these are so-called
2562:can be approximated as diagonal in
197:The dominant term in the hyperfine
81:Hyperfine structure contrasts with
24:
6034:The hyperfine states of a trapped
5642:
5613:
5517:
5362:
5333:
5283:Small molecule hyperfine structure
3795:
3781:
3767:
3716:
3467:. Expressed using the notation of
3212:
3197:
2930:
2255:this can be expressed in the form
186:
71:nuclear electric quadrupole moment
25:
6802:
6750:
5065:
755:
6761:
6250:. Oxford University Press, USA.
5599:{\displaystyle J=2\rightarrow 1}
5567:{\displaystyle J=2\rightarrow 1}
5503:{\displaystyle J=2\rightarrow 1}
5471:{\displaystyle J=1\rightarrow 0}
5439:{\displaystyle F=J+I_{\text{N}}}
5263:
5244:
5178:
4948:
4925:
4889:
4871:
4843:
4828:
4633:
3723:
3691:with the electric field vector:
3628:
3535:
3297:
3271:
2810:
2802:
2727:
2719:
2706:
2698:
2691:
2683:
2473:
2456:
2445:
2431:
2393:
2354:
2346:
2191:
2183:
2167:
2020:
2012:
1993:
1982:
1961:
1950:
1790:
1782:
1588:
1505:
1486:
1404:
1346:
1262:
1137:
1029:
925:
824:
750:
732:
655:
608:
597:
547:
492:
472:
457:
448:
413:
269:
212:
6685:
6670:
6644:
6587:
6421:
6284:Introduction to Nuclear Physics
6085:Electron paramagnetic resonance
5739:
5734:
5721:electron paramagnetic resonance
5714:
4553:{\displaystyle {\hat {H}}_{II}}
523:Electron orbital magnetic field
340:{\displaystyle \mu _{\text{N}}}
6724:10.1103/PhysRevLett.110.173002
6470:
6463:
6456:
6448:
6401:. Cambridge University Press.
6371:. Cambridge University Press.
6302:
6198:
6106:
6097:
6038:are commonly used for storing
5974:On October 21, 1983, the 17th
5758:Hyperfine structure gives the
5590:
5558:
5494:
5462:
5024:
4952:
4893:
4687:
4574:
4535:
4389:
4383:
4362:
4356:
4332:
4322:
4300:
4294:
4278:
4272:
4241:
4203:
4196:
4169:
4159:
4153:
4147:
4083:
4051:
4031:
4025:
3958:
3952:
3895:
3889:
3850:
3844:
3499:
3493:
3039:
3036:
3024:
3015:
3003:
2994:
2982:
2976:
2966:
2844:
2795:
2774:
2606:
2540:
2477:
2466:
2460:
2441:
2269:
2135:
1997:
1965:
1904:
1748:
1633:
1592:
1584:
1509:
1490:
1229:
1084:
379:
361:, placed in a magnetic field,
63:nuclear magnetic dipole moment
13:
1:
6205:Soliverez, C E (1980-12-10).
6090:
5849:perturbed angular correlation
4423:Molecular hyperfine structure
3469:irreducible spherical tensors
6791:Foundational quantum physics
6244:Woodgate, Gordon K. (1983).
6125:Woodgate, Gordon K. (1999).
6080:Dynamic nuclear polarisation
3143:, with components given by:
2760:This is commonly written as
1383:in the opposite direction).
1282:Electron spin magnetic field
932:{\displaystyle \mathbf {L} }
306:{\displaystyle g_{\text{I}}}
219:{\displaystyle \mathbf {I} }
7:
6247:Elementary Atomic Structure
6223:10.1088/0022-3719/13/34/002
6129:. Oxford University Press.
6127:Elementary Atomic Structure
6073:
5535:{\displaystyle \Delta F=-1}
3352:depending on the values of
2248:{\displaystyle \ell \neq 0}
146:(older name for Lutetium),
10:
6807:
6044:ion-trap quantum computing
5986:during a time interval of
5841:nuclear magnetic resonance
5657:{\displaystyle \Delta F=1}
5628:{\displaystyle \Delta J=1}
5348:{\displaystyle \Delta J=1}
3740:with components given by:
3465:irreducible representation
3340:are the spatial variables
3103:electric quadrupole moment
3068:
2507:-coupling by analogy with
785:is the electron momentum,
190:
182:Atomic hyperfine structure
121:
110:Schematic illustration of
6157:Classical Electrodynamics
6155:Jackson, John D. (1998).
1291:, has a magnetic moment,
161:
6309:Y. Millot (2008-02-19).
6280:Enge, Harald A. (1966).
4486:Direct nuclear spinâspin
3856:{\displaystyle T^{2}(q)}
3075:Atomic nuclei with spin
97:orbital angular momentum
6694:Physical Review Letters
6354:10.1103/PhysRev.88.1337
6020:fine-structure constant
5793:submillimeter astronomy
5302:= 1), and hydrogen, H (
4472:{\displaystyle I\geq 1}
3094:{\displaystyle I\geq 1}
2526:and matrix elements of
75:electric field gradient
6479:
5845:Mössbauer spectroscopy
5752:
5697:
5658:
5629:
5600:
5568:
5536:
5504:
5472:
5440:
5393:
5349:
5274:
4989:
4655:
4554:
4510:
4509:{\displaystyle I>0}
4473:
4447:
4446:{\displaystyle I>0}
4399:
4219:
4116:
3857:
3818:
3734:
3677:
3643:
3457:
3312:
3137:
3136:{\displaystyle Q_{ij}}
3095:
3049:
2858:
2821:
2752:
2556:
2495:
2379:
2249:
2206:
1602:
1357:
1273:
1122:
1037:
965:
933:
909:
767:
632:
514:
424:
341:
307:
280:
220:
118:
6600:Astrophysical Journal
6510:10.1515/zna-2002-0806
6480:
6180:Garg, Anupam (2012).
6024:stringent test of QED
5805:rotational transition
5801:young stellar objects
5786:Voyager Golden Record
5747:
5698:
5659:
5630:
5601:
5569:
5537:
5505:
5473:
5441:
5394:
5350:
5275:
4998:Nuclear spinârotation
4990:
4656:
4555:
4511:
4474:
4448:
4400:
4220:
4117:
3858:
3819:
3735:
3678:
3644:
3458:
3313:
3138:
3096:
3050:
2859:
2822:
2753:
2557:
2496:
2380:
2250:
2207:
1603:
1358:
1274:
1123:
1038:
966:
934:
910:
768:
633:
515:
425:
342:
308:
281:
221:
109:
65:interacting with the
6572:10.1093/mnras/stw835
6485:Emission from TMC-1"
6434:
5904:transition frequency
5833:nuclear spectroscopy
5827:Nuclear spectroscopy
5797:heterodyne receivers
5668:
5639:
5610:
5578:
5546:
5514:
5482:
5450:
5411:
5359:
5330:
5291:(HCN) in its ground
5014:
4677:
4564:
4525:
4494:
4457:
4431:
4231:
4126:
3867:
3831:
3744:
3695:
3655:
3475:
3409:
3147:
3117:
3079:
2927:
2831:
2764:
2596:
2530:
2389:
2259:
2233:
1619:
1399:
1304:
1132:
1056:
975:
943:
921:
819:
650:
542:
444:
369:
324:
290:
230:
208:
6716:2013PhRvL.110q3002W
6622:2004ApJ...606..333T
6563:2016MNRAS.459.2882M
6501:2002ZNatA..57..669A
6346:1952PhRv...88.1337F
6217:(34): L1017âL1019.
5821:hyperfine anomalies
5768:interstellar medium
5127:
4818:
4733:
4382:
4355:
4195:
4146:
4024:
3951:
3888:
3585:
3492:
3216:
3201:
3065:Electric quadrupole
3059:Landé interval rule
1933:
1777:
1418:
1258:
1151:
1073:
1027:
960:
838:
669:
561:
506:
486:
95:and the electrons'
36:hyperfine structure
18:Hyperfine splitting
6700:(17): 173002 1â5.
6659:2016-06-03 at the
6475:
6288:. Addison Wesley.
5855:Nuclear technology
5753:
5693:
5681:
5654:
5625:
5596:
5564:
5532:
5500:
5468:
5436:
5389:
5345:
5270:
5105:
5099:
4985:
4796:
4766:
4744:
4719:
4651:
4615:
4550:
4506:
4490:Each nucleus with
4469:
4443:
4395:
4365:
4341:
4321:
4215:
4178:
4129:
4112:
4110:
4007:
3934:
3874:
3853:
3814:
3730:
3711:
3707:
3673:
3671:
3667:
3639:
3571:
3478:
3453:
3433:
3308:
3202:
3187:
3133:
3091:
3045:
2854:
2817:
2748:
2743:
2712:
2676:
2552:
2491:
2375:
2370:
2339:
2245:
2202:
2200:
2127:
1919:
1894:
1779:
1763:
1737:
1726:
1707:
1598:
1560:
1402:
1353:
1269:
1244:
1219:
1135:
1118:
1059:
1033:
1013:
1012:
987:
961:
946:
929:
905:
822:
763:
653:
628:
545:
510:
490:
470:
420:
337:
303:
276:
216:
119:
6408:978-0-521-53078-1
6378:978-0-521-53078-1
6295:978-0-201-01870-7
6257:978-0-19-851156-4
6191:978-0-691-13018-7
6166:978-0-471-30932-1
6136:978-0-19-851156-4
5680:
5433:
5293:vibrational state
5240:
5174:
5128:
5079:
5077:
5062:
5033:
5027:
4955:
4896:
4819:
4746:
4743:
4726:
4690:
4595:
4577:
4538:
4312:
4244:
4049:
3915:
3911:
3809:
3700:
3699:
3660:
3659:
3520:
3519:
3424:
3174:
2969:
2954:
2940:
2847:
2798:
2783:
2777:
2742:
2711:
2675:
2651:
2641:
2615:
2609:
2549:
2543:
2480:
2463:
2423:
2369:
2338:
2314:
2304:
2272:
2138:
2118:
2116:
2088:
2078:
2068:
2058:
2049:
2000:
1968:
1934:
1907:
1885:
1883:
1866:
1843:
1833:
1823:
1813:
1778:
1751:
1728:
1725:
1706:
1682:
1672:
1662:
1636:
1571:
1559:
1527:
1512:
1493:
1476:
1452:
1411:
1342:
1316:
1259:
1232:
1210:
1208:
1191:
1168:
1144:
1087:
1003:
978:
895:
878:
855:
831:
775:Recognizing that
758:
746:
726:
709:
686:
662:
623:
585:
554:
499:
479:
464:
406:
388:
382:
334:
300:
265:
255:
242:
172:multipole moments
16:(Redirected from
6798:
6765:
6744:
6743:
6709:
6689:
6683:
6674:
6668:
6648:
6642:
6641:
6615:
6613:astro-ph/0401584
6591:
6585:
6584:
6574:
6556:
6547:(3): 2882â2993.
6532:
6523:
6522:
6512:
6484:
6482:
6481:
6476:
6474:
6473:
6425:
6419:
6418:
6416:
6415:
6392:
6383:
6382:
6364:
6358:
6357:
6340:(6): 1337â1349.
6327:
6321:
6320:
6318:
6317:
6306:
6300:
6299:
6287:
6277:
6268:
6267:
6265:
6264:
6241:
6235:
6234:
6202:
6196:
6195:
6177:
6171:
6170:
6152:
6141:
6140:
6122:
6113:
6110:
6104:
6101:
6068:Raman transition
6049:
6001:
5999:
5998:
5995:
5992:
5970:
5969:
5966:
5963:
5932:
5926:
5919:
5811:band. Since the
5729:transition-metal
5710:
5706:
5702:
5700:
5699:
5694:
5692:
5691:
5682:
5673:
5663:
5661:
5660:
5655:
5634:
5632:
5631:
5626:
5605:
5603:
5602:
5597:
5573:
5571:
5570:
5565:
5541:
5539:
5538:
5533:
5509:
5507:
5506:
5501:
5477:
5475:
5474:
5469:
5445:
5443:
5442:
5437:
5435:
5434:
5431:
5406:
5402:
5398:
5396:
5395:
5390:
5354:
5352:
5351:
5346:
5318:
5317:
5313:
5289:hydrogen cyanide
5279:
5277:
5276:
5271:
5266:
5258:
5254:
5253:
5252:
5247:
5241:
5239:
5238:
5237:
5224:
5223:
5222:
5213:
5212:
5211:
5197:
5192:
5191:
5190:
5181:
5175:
5173:
5172:
5163:
5162:
5161:
5160:
5147:
5146:
5136:
5129:
5126:
5121:
5120:
5101:
5098:
5097:
5078:
5076:
5068:
5064:
5063:
5060:
5054:
5053:
5040:
5035:
5034:
5031:
5029:
5028:
5020:
4994:
4992:
4991:
4986:
4981:
4977:
4976:
4972:
4971:
4970:
4969:
4957:
4956:
4951:
4946:
4939:
4938:
4937:
4928:
4917:
4913:
4912:
4911:
4910:
4898:
4897:
4892:
4887:
4880:
4879:
4874:
4857:
4856:
4855:
4846:
4837:
4836:
4831:
4820:
4817:
4812:
4811:
4795:
4794:
4793:
4792:
4779:
4778:
4768:
4765:
4764:
4745:
4742:
4734:
4732:
4727:
4724:
4718:
4717:
4707:
4701:
4700:
4692:
4691:
4683:
4670:
4660:
4658:
4657:
4652:
4647:
4646:
4645:
4636:
4627:
4626:
4621:
4614:
4613:
4588:
4587:
4579:
4578:
4570:
4559:
4557:
4556:
4551:
4549:
4548:
4540:
4539:
4531:
4515:
4513:
4512:
4507:
4478:
4476:
4475:
4470:
4452:
4450:
4449:
4444:
4418:
4404:
4402:
4401:
4396:
4381:
4376:
4354:
4349:
4340:
4339:
4320:
4293:
4292:
4271:
4270:
4252:
4251:
4246:
4245:
4237:
4224:
4222:
4221:
4216:
4211:
4210:
4194:
4189:
4177:
4176:
4145:
4140:
4121:
4119:
4118:
4113:
4111:
4104:
4103:
4082:
4081:
4066:
4065:
4050:
4042:
4023:
4018:
4002:
4001:
3983:
3982:
3950:
3945:
3929:
3928:
3916:
3907:
3906:
3887:
3882:
3862:
3860:
3859:
3854:
3843:
3842:
3823:
3821:
3820:
3815:
3810:
3808:
3807:
3806:
3793:
3792:
3779:
3775:
3774:
3764:
3759:
3758:
3739:
3737:
3736:
3731:
3726:
3712:
3682:
3680:
3679:
3674:
3672:
3648:
3646:
3645:
3640:
3635:
3631:
3625:
3624:
3614:
3610:
3609:
3598:
3584:
3579:
3570:
3569:
3564:
3560:
3547:
3546:
3542:
3538:
3521:
3515:
3507:
3506:
3491:
3486:
3462:
3460:
3459:
3454:
3446:
3445:
3432:
3400:
3383:symmetric matrix
3317:
3315:
3314:
3309:
3304:
3300:
3294:
3293:
3283:
3282:
3278:
3274:
3260:
3256:
3255:
3254:
3242:
3241:
3236:
3232:
3215:
3210:
3200:
3195:
3175:
3167:
3162:
3161:
3142:
3140:
3139:
3134:
3132:
3131:
3100:
3098:
3097:
3092:
3054:
3052:
3051:
3046:
2975:
2971:
2970:
2962:
2955:
2947:
2942:
2941:
2938:
2922:
2908:
2882:
2881:
2877:
2863:
2861:
2860:
2855:
2853:
2849:
2848:
2840:
2826:
2824:
2823:
2818:
2813:
2805:
2800:
2799:
2791:
2785:
2784:
2781:
2779:
2778:
2770:
2757:
2755:
2754:
2749:
2744:
2741:
2740:
2731:
2730:
2722:
2716:
2713:
2710:
2709:
2701:
2695:
2694:
2686:
2680:
2677:
2674:
2666:
2665:
2656:
2653:
2652:
2649:
2643:
2642:
2639:
2633:
2632:
2617:
2616:
2613:
2611:
2610:
2602:
2591:
2561:
2559:
2558:
2553:
2551:
2550:
2547:
2545:
2544:
2536:
2500:
2498:
2497:
2492:
2487:
2483:
2482:
2481:
2476:
2471:
2465:
2464:
2459:
2454:
2448:
2434:
2424:
2419:
2418:
2409:
2404:
2396:
2384:
2382:
2381:
2376:
2371:
2368:
2367:
2358:
2357:
2349:
2343:
2340:
2337:
2329:
2328:
2319:
2316:
2315:
2312:
2306:
2305:
2302:
2296:
2295:
2280:
2279:
2274:
2273:
2265:
2254:
2252:
2251:
2246:
2229:For states with
2211:
2209:
2208:
2203:
2201:
2194:
2186:
2181:
2180:
2176:
2175:
2170:
2159:
2158:
2149:
2148:
2140:
2139:
2131:
2126:
2117:
2115:
2114:
2102:
2100:
2099:
2090:
2089:
2086:
2080:
2079:
2076:
2070:
2069:
2066:
2060:
2059:
2056:
2050:
2042:
2037:
2032:
2028:
2024:
2023:
2015:
2007:
2003:
2002:
2001:
1996:
1991:
1985:
1975:
1971:
1970:
1969:
1964:
1959:
1953:
1935:
1932:
1927:
1918:
1917:
1909:
1908:
1900:
1896:
1893:
1884:
1882:
1881:
1869:
1867:
1865:
1857:
1856:
1847:
1845:
1844:
1841:
1835:
1834:
1831:
1825:
1824:
1821:
1815:
1814:
1811:
1802:
1797:
1793:
1785:
1780:
1776:
1771:
1762:
1761:
1753:
1752:
1744:
1740:
1736:
1727:
1724:
1723:
1711:
1708:
1705:
1697:
1696:
1687:
1684:
1683:
1680:
1674:
1673:
1670:
1664:
1663:
1660:
1649:
1644:
1643:
1638:
1637:
1629:
1607:
1605:
1604:
1599:
1591:
1583:
1582:
1573:
1572:
1569:
1567:
1561:
1555:
1554:
1553:
1540:
1534:
1530:
1529:
1528:
1525:
1523:
1514:
1513:
1508:
1503:
1500:
1496:
1495:
1494:
1489:
1484:
1478:
1477:
1474:
1472:
1453:
1451:
1450:
1449:
1433:
1432:
1423:
1417:
1412:
1409:
1407:
1362:
1360:
1359:
1354:
1349:
1344:
1343:
1340:
1334:
1333:
1318:
1317:
1314:
1312:
1278:
1276:
1275:
1270:
1265:
1260:
1257:
1252:
1243:
1242:
1234:
1233:
1225:
1221:
1218:
1209:
1207:
1206:
1194:
1192:
1190:
1182:
1181:
1172:
1170:
1169:
1166:
1150:
1145:
1142:
1140:
1127:
1125:
1124:
1119:
1117:
1116:
1107:
1102:
1101:
1100:
1099:
1089:
1088:
1080:
1072:
1067:
1051:
1042:
1040:
1039:
1034:
1032:
1026:
1021:
1011:
999:
998:
993:
986:
970:
968:
967:
962:
959:
954:
938:
936:
935:
930:
928:
914:
912:
911:
906:
901:
896:
894:
893:
881:
879:
877:
869:
868:
859:
857:
856:
853:
837:
832:
829:
827:
815:, we can write:
805:angular momentum
802:
772:
770:
769:
764:
759:
754:
753:
748:
747:
744:
735:
729:
727:
725:
724:
712:
710:
708:
700:
699:
690:
688:
687:
684:
668:
663:
660:
658:
637:
635:
634:
629:
624:
622:
621:
612:
611:
600:
588:
586:
584:
576:
575:
566:
560:
555:
552:
550:
519:
517:
516:
511:
505:
500:
497:
495:
485:
480:
477:
475:
466:
465:
462:
460:
451:
429:
427:
426:
421:
416:
408:
407:
404:
402:
390:
389:
386:
384:
383:
375:
349:nuclear magneton
346:
344:
343:
338:
336:
335:
332:
312:
310:
309:
304:
302:
301:
298:
285:
283:
282:
277:
272:
267:
266:
263:
257:
256:
253:
244:
243:
240:
238:
225:
223:
222:
217:
215:
168:electromagnetism
91:associated with
89:magnetic moments
21:
6806:
6805:
6801:
6800:
6799:
6797:
6796:
6795:
6776:
6775:
6753:
6748:
6747:
6690:
6686:
6675:
6671:
6661:Wayback Machine
6649:
6645:
6597:
6592:
6588:
6533:
6526:
6455:
6443:
6435:
6432:
6431:
6426:
6422:
6413:
6411:
6409:
6393:
6386:
6379:
6365:
6361:
6333:Physical Review
6328:
6324:
6315:
6313:
6307:
6303:
6296:
6278:
6271:
6262:
6260:
6258:
6242:
6238:
6203:
6199:
6192:
6178:
6174:
6167:
6153:
6144:
6137:
6123:
6116:
6111:
6107:
6102:
6098:
6093:
6076:
6047:
6032:
6012:
5996:
5993:
5990:
5989:
5987:
5967:
5964:
5961:
5959:
5935:Planck constant
5928:
5921:
5907:
5885:
5871:to selectively
5857:
5837:local structure
5829:
5742:
5737:
5717:
5708:
5704:
5687:
5683:
5671:
5669:
5666:
5665:
5640:
5637:
5636:
5611:
5608:
5607:
5579:
5576:
5575:
5547:
5544:
5543:
5515:
5512:
5511:
5483:
5480:
5479:
5451:
5448:
5447:
5430:
5426:
5412:
5409:
5408:
5404:
5400:
5360:
5357:
5356:
5331:
5328:
5327:
5324:selection rules
5315:
5311:
5310:
5308:
5301:
5285:
5262:
5248:
5243:
5242:
5230:
5229:
5225:
5218:
5214:
5204:
5203:
5199:
5198:
5196:
5183:
5182:
5177:
5176:
5168:
5164:
5153:
5152:
5148:
5142:
5138:
5137:
5135:
5134:
5130:
5122:
5113:
5109:
5100:
5090:
5083:
5069:
5059:
5055:
5049:
5045:
5041:
5039:
5030:
5019:
5018:
5017:
5015:
5012:
5011:
5000:
4962:
4958:
4947:
4945:
4944:
4943:
4930:
4929:
4924:
4923:
4922:
4918:
4903:
4899:
4888:
4886:
4885:
4884:
4875:
4870:
4869:
4868:
4864:
4848:
4847:
4842:
4841:
4832:
4827:
4826:
4825:
4821:
4813:
4804:
4800:
4785:
4784:
4780:
4774:
4770:
4769:
4767:
4757:
4750:
4735:
4728:
4723:
4713:
4709:
4708:
4705:
4693:
4682:
4681:
4680:
4678:
4675:
4674:
4668:
4638:
4637:
4632:
4631:
4622:
4617:
4616:
4606:
4599:
4580:
4569:
4568:
4567:
4565:
4562:
4561:
4541:
4530:
4529:
4528:
4526:
4523:
4522:
4495:
4492:
4491:
4488:
4458:
4455:
4454:
4432:
4429:
4428:
4425:
4417:
4409:
4377:
4369:
4350:
4345:
4335:
4331:
4316:
4288:
4284:
4266:
4262:
4247:
4236:
4235:
4234:
4232:
4229:
4228:
4206:
4202:
4190:
4182:
4172:
4168:
4141:
4133:
4127:
4124:
4123:
4109:
4108:
4096:
4092:
4074:
4070:
4058:
4054:
4041:
4034:
4019:
4011:
4004:
4003:
3994:
3990:
3975:
3971:
3961:
3946:
3938:
3931:
3930:
3921:
3917:
3905:
3898:
3883:
3878:
3870:
3868:
3865:
3864:
3838:
3834:
3832:
3829:
3828:
3802:
3798:
3788:
3784:
3780:
3770:
3766:
3765:
3763:
3751:
3747:
3745:
3742:
3741:
3722:
3698:
3696:
3693:
3692:
3658:
3656:
3653:
3652:
3627:
3626:
3620:
3616:
3602:
3591:
3590:
3586:
3580:
3575:
3565:
3553:
3549:
3548:
3534:
3533:
3529:
3528:
3508:
3505:
3487:
3482:
3476:
3473:
3472:
3438:
3434:
3428:
3410:
3407:
3406:
3401:) that is also
3398:
3391:
3386:
3371:Kronecker delta
3368:
3338:
3331:
3296:
3295:
3289:
3285:
3270:
3269:
3265:
3264:
3247:
3243:
3237:
3225:
3221:
3220:
3211:
3206:
3196:
3191:
3183:
3179:
3166:
3154:
3150:
3148:
3145:
3144:
3124:
3120:
3118:
3115:
3114:
3080:
3077:
3076:
3073:
3067:
2961:
2960:
2956:
2946:
2937:
2933:
2928:
2925:
2924:
2910:
2879:
2875:
2874:
2865:
2839:
2838:
2834:
2832:
2829:
2828:
2809:
2801:
2790:
2789:
2780:
2769:
2768:
2767:
2765:
2762:
2761:
2736:
2732:
2726:
2718:
2717:
2714:
2705:
2697:
2696:
2690:
2682:
2681:
2678:
2667:
2661:
2657:
2654:
2648:
2644:
2638:
2634:
2628:
2624:
2612:
2601:
2600:
2599:
2597:
2594:
2593:
2579:
2546:
2535:
2534:
2533:
2531:
2528:
2527:
2524:quantum numbers
2472:
2470:
2469:
2455:
2453:
2452:
2444:
2430:
2429:
2425:
2414:
2410:
2408:
2400:
2392:
2390:
2387:
2386:
2363:
2359:
2353:
2345:
2344:
2341:
2330:
2324:
2320:
2317:
2311:
2307:
2301:
2297:
2291:
2287:
2275:
2264:
2263:
2262:
2260:
2257:
2256:
2234:
2231:
2230:
2199:
2198:
2190:
2182:
2171:
2166:
2165:
2161:
2160:
2154:
2150:
2141:
2130:
2129:
2128:
2122:
2110:
2106:
2101:
2095:
2091:
2085:
2081:
2075:
2071:
2065:
2061:
2055:
2051:
2041:
2036:
2030:
2029:
2019:
2011:
1992:
1990:
1989:
1981:
1980:
1976:
1960:
1958:
1957:
1949:
1948:
1944:
1940:
1936:
1928:
1923:
1910:
1899:
1898:
1897:
1895:
1889:
1877:
1873:
1868:
1858:
1852:
1848:
1846:
1840:
1836:
1830:
1826:
1820:
1816:
1810:
1806:
1801:
1795:
1794:
1789:
1781:
1772:
1767:
1754:
1743:
1742:
1741:
1738:
1732:
1719:
1715:
1709:
1698:
1692:
1688:
1685:
1679:
1675:
1669:
1665:
1659:
1655:
1650:
1648:
1639:
1628:
1627:
1626:
1622:
1620:
1617:
1616:
1613:
1587:
1578:
1574:
1568:
1563:
1562:
1549:
1545:
1541:
1538:
1524:
1519:
1518:
1504:
1502:
1501:
1485:
1483:
1482:
1473:
1468:
1467:
1466:
1462:
1458:
1454:
1445:
1441:
1434:
1428:
1424:
1422:
1413:
1408:
1403:
1400:
1397:
1396:
1395:, is given by:
1394:
1368:
1345:
1339:
1335:
1329:
1325:
1313:
1308:
1307:
1305:
1302:
1301:
1299:
1284:
1261:
1253:
1248:
1235:
1224:
1223:
1222:
1220:
1214:
1202:
1198:
1193:
1183:
1177:
1173:
1171:
1165:
1161:
1146:
1141:
1136:
1133:
1130:
1129:
1112:
1108:
1103:
1095:
1091:
1090:
1079:
1078:
1077:
1068:
1063:
1057:
1054:
1053:
1052:, we can write
1049:
1044:
1028:
1022:
1017:
1007:
994:
989:
988:
982:
976:
973:
972:
955:
950:
944:
941:
940:
924:
922:
919:
918:
897:
889:
885:
880:
870:
864:
860:
858:
852:
848:
833:
828:
823:
820:
817:
816:
803:is the orbital
790:
781:
749:
743:
739:
731:
730:
728:
720:
716:
711:
701:
695:
691:
689:
683:
679:
664:
659:
654:
651:
648:
647:
617:
613:
607:
596:
589:
587:
577:
571:
567:
565:
556:
551:
546:
543:
540:
539:
525:
501:
496:
491:
481:
476:
471:
461:
456:
455:
447:
445:
442:
441:
412:
403:
398:
397:
385:
374:
373:
372:
370:
367:
366:
360:
331:
327:
325:
322:
321:
297:
293:
291:
288:
287:
268:
262:
258:
252:
248:
239:
234:
233:
231:
228:
227:
211:
209:
206:
205:
195:
189:
187:Magnetic dipole
184:
164:
124:
28:
23:
22:
15:
12:
11:
5:
6804:
6794:
6793:
6788:
6786:Atomic physics
6774:
6773:
6759:
6752:
6751:External links
6749:
6746:
6745:
6684:
6669:
6643:
6630:10.1086/382862
6606:(1): 333â340.
6595:
6586:
6524:
6495:(8): 669â681.
6489:Z. Naturforsch
6472:
6468:
6465:
6461:
6458:
6453:
6450:
6446:
6442:
6439:
6420:
6407:
6384:
6377:
6359:
6322:
6301:
6294:
6269:
6256:
6236:
6197:
6190:
6172:
6165:
6142:
6135:
6114:
6105:
6095:
6094:
6092:
6089:
6088:
6087:
6082:
6075:
6072:
6031:
6028:
6011:
6008:
5958:to be exactly
5884:
5881:
5856:
5853:
5828:
5825:
5817:optically thin
5782:Pioneer plaque
5750:Pioneer plaque
5741:
5738:
5736:
5733:
5716:
5713:
5690:
5686:
5679:
5676:
5653:
5650:
5647:
5644:
5624:
5621:
5618:
5615:
5595:
5592:
5589:
5586:
5583:
5563:
5560:
5557:
5554:
5551:
5531:
5528:
5525:
5522:
5519:
5499:
5496:
5493:
5490:
5487:
5467:
5464:
5461:
5458:
5455:
5429:
5425:
5422:
5419:
5416:
5388:
5385:
5382:
5379:
5376:
5373:
5370:
5367:
5364:
5344:
5341:
5338:
5335:
5306:
5299:
5284:
5281:
5269:
5265:
5261:
5257:
5251:
5246:
5236:
5233:
5228:
5221:
5217:
5210:
5207:
5202:
5195:
5189:
5186:
5180:
5171:
5167:
5159:
5156:
5151:
5145:
5141:
5133:
5125:
5119:
5116:
5112:
5108:
5104:
5096:
5093:
5089:
5086:
5082:
5075:
5072:
5067:
5058:
5052:
5048:
5044:
5038:
5026:
5023:
4999:
4996:
4984:
4980:
4975:
4968:
4965:
4961:
4954:
4950:
4942:
4936:
4933:
4927:
4921:
4916:
4909:
4906:
4902:
4895:
4891:
4883:
4878:
4873:
4867:
4863:
4860:
4854:
4851:
4845:
4840:
4835:
4830:
4824:
4816:
4810:
4807:
4803:
4799:
4791:
4788:
4783:
4777:
4773:
4763:
4760:
4756:
4753:
4749:
4741:
4738:
4731:
4722:
4716:
4712:
4704:
4699:
4696:
4689:
4686:
4650:
4644:
4641:
4635:
4630:
4625:
4620:
4612:
4609:
4605:
4602:
4598:
4594:
4591:
4586:
4583:
4576:
4573:
4547:
4544:
4537:
4534:
4505:
4502:
4499:
4487:
4484:
4468:
4465:
4462:
4442:
4439:
4436:
4424:
4421:
4413:
4394:
4391:
4388:
4385:
4380:
4375:
4372:
4368:
4364:
4361:
4358:
4353:
4348:
4344:
4338:
4334:
4330:
4327:
4324:
4319:
4315:
4311:
4308:
4305:
4302:
4299:
4296:
4291:
4287:
4283:
4280:
4277:
4274:
4269:
4265:
4261:
4258:
4255:
4250:
4243:
4240:
4214:
4209:
4205:
4201:
4198:
4193:
4188:
4185:
4181:
4175:
4171:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4144:
4139:
4136:
4132:
4107:
4102:
4099:
4095:
4091:
4088:
4085:
4080:
4077:
4073:
4069:
4064:
4061:
4057:
4053:
4048:
4045:
4040:
4037:
4035:
4033:
4030:
4027:
4022:
4017:
4014:
4010:
4006:
4005:
4000:
3997:
3993:
3989:
3986:
3981:
3978:
3974:
3970:
3967:
3964:
3962:
3960:
3957:
3954:
3949:
3944:
3941:
3937:
3933:
3932:
3927:
3924:
3920:
3914:
3910:
3904:
3901:
3899:
3897:
3894:
3891:
3886:
3881:
3877:
3873:
3872:
3852:
3849:
3846:
3841:
3837:
3813:
3805:
3801:
3797:
3791:
3787:
3783:
3778:
3773:
3769:
3762:
3757:
3754:
3750:
3729:
3725:
3721:
3718:
3715:
3710:
3706:
3703:
3670:
3666:
3663:
3638:
3634:
3630:
3623:
3619:
3613:
3608:
3605:
3601:
3597:
3594:
3589:
3583:
3578:
3574:
3568:
3563:
3559:
3556:
3552:
3545:
3541:
3537:
3532:
3527:
3524:
3518:
3514:
3511:
3504:
3501:
3498:
3495:
3490:
3485:
3481:
3452:
3449:
3444:
3441:
3437:
3431:
3427:
3423:
3420:
3417:
3414:
3396:
3389:
3364:
3360:respectively,
3336:
3329:
3307:
3303:
3299:
3292:
3288:
3281:
3277:
3273:
3268:
3263:
3259:
3253:
3250:
3246:
3240:
3235:
3231:
3228:
3224:
3219:
3214:
3209:
3205:
3199:
3194:
3190:
3186:
3182:
3178:
3173:
3170:
3165:
3160:
3157:
3153:
3130:
3127:
3123:
3090:
3087:
3084:
3069:Main article:
3066:
3063:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2974:
2968:
2965:
2959:
2953:
2950:
2945:
2936:
2932:
2852:
2846:
2843:
2837:
2816:
2812:
2808:
2804:
2797:
2794:
2788:
2776:
2773:
2747:
2739:
2735:
2729:
2725:
2721:
2708:
2704:
2700:
2693:
2689:
2685:
2673:
2670:
2664:
2660:
2647:
2637:
2631:
2627:
2623:
2620:
2608:
2605:
2542:
2539:
2490:
2486:
2479:
2475:
2468:
2462:
2458:
2451:
2447:
2443:
2440:
2437:
2433:
2428:
2422:
2417:
2413:
2407:
2403:
2399:
2395:
2374:
2366:
2362:
2356:
2352:
2348:
2336:
2333:
2327:
2323:
2310:
2300:
2294:
2290:
2286:
2283:
2278:
2271:
2268:
2244:
2241:
2238:
2197:
2193:
2189:
2185:
2179:
2174:
2169:
2164:
2157:
2153:
2147:
2144:
2137:
2134:
2125:
2121:
2113:
2109:
2105:
2098:
2094:
2084:
2074:
2064:
2054:
2048:
2045:
2040:
2035:
2033:
2031:
2027:
2022:
2018:
2014:
2010:
2006:
1999:
1995:
1988:
1984:
1979:
1974:
1967:
1963:
1956:
1952:
1947:
1943:
1939:
1931:
1926:
1922:
1916:
1913:
1906:
1903:
1892:
1888:
1880:
1876:
1872:
1864:
1861:
1855:
1851:
1839:
1829:
1819:
1809:
1805:
1800:
1798:
1796:
1792:
1788:
1784:
1775:
1770:
1766:
1760:
1757:
1750:
1747:
1735:
1731:
1722:
1718:
1714:
1704:
1701:
1695:
1691:
1678:
1668:
1658:
1654:
1651:
1647:
1642:
1635:
1632:
1625:
1624:
1612:
1609:
1597:
1594:
1590:
1586:
1581:
1577:
1566:
1558:
1552:
1548:
1544:
1537:
1533:
1522:
1517:
1511:
1507:
1499:
1492:
1488:
1481:
1471:
1465:
1461:
1457:
1448:
1444:
1440:
1437:
1431:
1427:
1421:
1416:
1406:
1392:
1372:electron spin
1366:
1352:
1348:
1338:
1332:
1328:
1324:
1321:
1311:
1295:
1283:
1280:
1268:
1264:
1256:
1251:
1247:
1241:
1238:
1231:
1228:
1217:
1213:
1205:
1201:
1197:
1189:
1186:
1180:
1176:
1164:
1160:
1157:
1154:
1149:
1139:
1115:
1111:
1106:
1098:
1094:
1086:
1083:
1076:
1071:
1066:
1062:
1047:
1031:
1025:
1020:
1016:
1010:
1006:
1002:
997:
992:
985:
981:
958:
953:
949:
927:
904:
900:
892:
888:
884:
876:
873:
867:
863:
851:
847:
844:
841:
836:
826:
779:
762:
757:
752:
742:
738:
734:
723:
719:
715:
707:
704:
698:
694:
682:
678:
675:
672:
667:
657:
646:, this gives:
627:
620:
616:
610:
606:
603:
599:
595:
592:
583:
580:
574:
570:
564:
559:
549:
534:at a position
524:
521:
509:
504:
494:
489:
484:
474:
469:
459:
454:
450:
419:
415:
411:
401:
396:
393:
381:
378:
358:
330:
296:
275:
271:
261:
251:
247:
237:
214:
191:Main article:
188:
185:
183:
180:
163:
160:
132:S. A. Goudsmit
123:
120:
84:fine structure
67:magnetic field
32:atomic physics
26:
9:
6:
4:
3:
2:
6803:
6792:
6789:
6787:
6784:
6783:
6781:
6772:
6768:
6764:
6760:
6758:
6755:
6754:
6741:
6737:
6733:
6729:
6725:
6721:
6717:
6713:
6708:
6703:
6699:
6695:
6688:
6681:
6680:
6673:
6666:
6662:
6658:
6655:
6654:
6647:
6639:
6635:
6631:
6627:
6623:
6619:
6614:
6609:
6605:
6601:
6590:
6582:
6578:
6573:
6568:
6564:
6560:
6555:
6550:
6546:
6542:
6538:
6531:
6529:
6520:
6516:
6511:
6506:
6502:
6498:
6494:
6490:
6486:
6466:
6459:
6451:
6444:
6440:
6437:
6424:
6410:
6404:
6400:
6399:
6391:
6389:
6380:
6374:
6370:
6363:
6355:
6351:
6347:
6343:
6339:
6335:
6334:
6326:
6312:
6305:
6297:
6291:
6286:
6285:
6276:
6274:
6259:
6253:
6249:
6248:
6240:
6232:
6228:
6224:
6220:
6216:
6212:
6208:
6201:
6193:
6187:
6183:
6176:
6168:
6162:
6158:
6151:
6149:
6147:
6138:
6132:
6128:
6121:
6119:
6109:
6100:
6096:
6086:
6083:
6081:
6078:
6077:
6071:
6069:
6065:
6061:
6056:
6051:
6045:
6041:
6037:
6027:
6025:
6021:
6017:
6007:
6005:
5985:
5981:
5977:
5972:
5957:
5956:
5951:
5946:
5944:
5940:
5936:
5931:
5925:
5918:
5914:
5910:
5905:
5901:
5900:atomic clocks
5897:
5893:
5890:
5880:
5878:
5874:
5870:
5866:
5862:
5852:
5850:
5846:
5842:
5838:
5834:
5824:
5822:
5818:
5814:
5813:optical depth
5810:
5806:
5802:
5798:
5794:
5789:
5787:
5783:
5779:
5775:
5771:
5769:
5765:
5761:
5756:
5751:
5746:
5732:
5730:
5726:
5725:free radicals
5722:
5712:
5688:
5684:
5677:
5674:
5651:
5648:
5645:
5622:
5619:
5616:
5593:
5587:
5584:
5581:
5561:
5555:
5552:
5549:
5529:
5526:
5523:
5520:
5497:
5491:
5488:
5485:
5465:
5459:
5456:
5453:
5427:
5423:
5420:
5417:
5414:
5383:
5380:
5377:
5374:
5368:
5365:
5342:
5339:
5336:
5325:
5320:
5305:
5298:
5294:
5290:
5280:
5267:
5259:
5255:
5249:
5234:
5231:
5226:
5219:
5215:
5208:
5205:
5200:
5193:
5187:
5184:
5169:
5165:
5157:
5154:
5149:
5143:
5139:
5131:
5123:
5117:
5114:
5110:
5106:
5102:
5094:
5091:
5087:
5084:
5080:
5073:
5070:
5056:
5050:
5046:
5042:
5036:
5021:
5009:
5005:
4995:
4982:
4978:
4973:
4966:
4963:
4959:
4940:
4934:
4931:
4919:
4914:
4907:
4904:
4900:
4881:
4876:
4865:
4861:
4858:
4852:
4849:
4838:
4833:
4822:
4814:
4808:
4805:
4801:
4797:
4789:
4786:
4781:
4775:
4771:
4761:
4758:
4754:
4751:
4747:
4739:
4736:
4729:
4720:
4714:
4710:
4702:
4697:
4694:
4684:
4671:
4664:
4648:
4642:
4639:
4628:
4623:
4610:
4607:
4603:
4600:
4596:
4592:
4589:
4584:
4581:
4571:
4545:
4542:
4532:
4520:
4503:
4500:
4497:
4483:
4480:
4466:
4463:
4460:
4440:
4437:
4434:
4420:
4416:
4412:
4405:
4392:
4386:
4378:
4373:
4370:
4366:
4359:
4351:
4346:
4342:
4336:
4328:
4325:
4317:
4313:
4309:
4306:
4303:
4297:
4289:
4285:
4281:
4275:
4267:
4263:
4259:
4256:
4253:
4248:
4238:
4225:
4212:
4207:
4199:
4191:
4186:
4183:
4179:
4173:
4165:
4162:
4156:
4150:
4142:
4137:
4134:
4130:
4105:
4100:
4097:
4093:
4089:
4086:
4078:
4075:
4071:
4067:
4062:
4059:
4055:
4046:
4043:
4038:
4036:
4028:
4020:
4015:
4012:
4008:
3998:
3995:
3991:
3987:
3984:
3979:
3976:
3972:
3968:
3965:
3963:
3955:
3947:
3942:
3939:
3935:
3925:
3922:
3918:
3912:
3908:
3902:
3900:
3892:
3884:
3879:
3875:
3847:
3839:
3835:
3824:
3811:
3803:
3799:
3789:
3785:
3776:
3771:
3760:
3755:
3752:
3748:
3727:
3719:
3713:
3708:
3704:
3701:
3690:
3686:
3685:outer product
3668:
3664:
3661:
3649:
3636:
3632:
3621:
3617:
3611:
3606:
3603:
3599:
3595:
3592:
3587:
3581:
3576:
3572:
3566:
3561:
3557:
3554:
3550:
3543:
3539:
3530:
3525:
3522:
3516:
3512:
3509:
3502:
3496:
3488:
3483:
3479:
3470:
3466:
3450:
3447:
3442:
3439:
3435:
3429:
3425:
3421:
3418:
3415:
3412:
3404:
3399:
3392:
3384:
3380:
3376:
3372:
3367:
3363:
3359:
3355:
3351:
3347:
3343:
3339:
3332:
3325:
3321:
3305:
3301:
3290:
3286:
3279:
3275:
3266:
3261:
3257:
3251:
3248:
3244:
3238:
3233:
3229:
3226:
3222:
3217:
3207:
3203:
3192:
3188:
3184:
3180:
3176:
3171:
3168:
3163:
3158:
3155:
3151:
3128:
3125:
3121:
3112:
3108:
3104:
3088:
3085:
3082:
3072:
3062:
3060:
3055:
3042:
3033:
3030:
3027:
3021:
3018:
3012:
3009:
3006:
3000:
2997:
2991:
2988:
2985:
2979:
2972:
2963:
2957:
2951:
2948:
2943:
2934:
2921:
2917:
2913:
2906:
2902:
2898:
2894:
2890:
2886:
2872:
2868:
2850:
2841:
2835:
2814:
2806:
2792:
2786:
2771:
2758:
2745:
2737:
2733:
2723:
2702:
2687:
2671:
2668:
2662:
2658:
2645:
2635:
2629:
2625:
2621:
2618:
2603:
2590:
2586:
2582:
2577:
2573:
2569:
2565:
2537:
2525:
2521:
2517:
2513:
2511:
2506:
2501:
2488:
2484:
2449:
2438:
2435:
2426:
2420:
2415:
2411:
2405:
2397:
2372:
2364:
2360:
2350:
2334:
2331:
2325:
2321:
2308:
2298:
2292:
2288:
2284:
2281:
2276:
2266:
2242:
2239:
2236:
2227:
2225:
2220:
2219:
2218:Fermi contact
2212:
2195:
2187:
2177:
2172:
2162:
2155:
2151:
2145:
2142:
2132:
2123:
2119:
2111:
2107:
2103:
2096:
2092:
2082:
2072:
2062:
2052:
2046:
2043:
2038:
2034:
2025:
2016:
2008:
2004:
1986:
1977:
1972:
1954:
1945:
1941:
1937:
1929:
1924:
1920:
1914:
1911:
1901:
1890:
1886:
1878:
1874:
1870:
1862:
1859:
1853:
1849:
1837:
1827:
1817:
1807:
1803:
1799:
1786:
1773:
1768:
1764:
1758:
1755:
1745:
1733:
1729:
1720:
1716:
1712:
1702:
1699:
1693:
1689:
1676:
1666:
1656:
1652:
1645:
1640:
1630:
1608:
1595:
1579:
1575:
1556:
1550:
1546:
1542:
1535:
1531:
1515:
1497:
1479:
1463:
1459:
1455:
1446:
1442:
1438:
1435:
1429:
1425:
1419:
1414:
1391:
1390:
1384:
1382:
1377:
1375:
1369:
1350:
1336:
1330:
1326:
1322:
1319:
1298:
1294:
1290:
1279:
1266:
1254:
1249:
1245:
1239:
1236:
1226:
1215:
1211:
1203:
1199:
1195:
1187:
1184:
1178:
1174:
1162:
1158:
1155:
1152:
1147:
1113:
1109:
1104:
1096:
1092:
1081:
1074:
1069:
1064:
1060:
1050:
1023:
1018:
1014:
1008:
1004:
1000:
995:
990:
983:
979:
956:
951:
947:
915:
902:
898:
890:
886:
882:
874:
871:
865:
861:
849:
845:
842:
839:
834:
814:
810:
806:
801:
797:
793:
788:
784:
778:
773:
760:
740:
736:
721:
717:
713:
705:
702:
696:
692:
680:
676:
673:
670:
665:
645:
644:Bohr magneton
641:
625:
618:
614:
604:
601:
593:
590:
581:
578:
572:
568:
562:
557:
537:
533:
532:
520:
507:
502:
487:
482:
467:
452:
439:
435:
430:
417:
409:
394:
391:
376:
364:
357:
352:
350:
328:
319:
317:
294:
273:
259:
249:
245:
204:
200:
194:
179:
177:
173:
169:
159:
157:
153:
149:
145:
141:
137:
133:
129:
117:
116:hydrogen atom
113:
108:
104:
102:
98:
94:
93:electron spin
90:
86:
85:
79:
76:
72:
68:
64:
59:
57:
53:
49:
45:
44:energy levels
41:
37:
33:
19:
6697:
6693:
6687:
6678:
6672:
6664:
6652:
6646:
6603:
6599:
6589:
6544:
6540:
6492:
6488:
6423:
6412:. Retrieved
6397:
6368:
6362:
6337:
6331:
6325:
6314:. Retrieved
6304:
6283:
6261:. Retrieved
6246:
6239:
6214:
6210:
6200:
6181:
6175:
6156:
6126:
6108:
6099:
6063:
6052:
6033:
6013:
5973:
5953:
5947:
5929:
5923:
5916:
5912:
5908:
5903:
5902:. The term
5892:notch filter
5886:
5873:photo-ionize
5858:
5830:
5820:
5816:
5790:
5772:
5762:observed in
5757:
5754:
5740:Astrophysics
5735:Applications
5718:
5715:Measurements
5321:
5303:
5296:
5286:
5007:
5003:
5001:
4666:
4662:
4518:
4489:
4481:
4426:
4414:
4410:
4406:
4226:
3825:
3689:del operator
3650:
3394:
3387:
3378:
3374:
3365:
3361:
3357:
3353:
3349:
3345:
3341:
3334:
3327:
3323:
3319:
3074:
3056:
2919:
2915:
2911:
2904:
2900:
2896:
2892:
2888:
2884:
2870:
2866:
2759:
2588:
2584:
2580:
2575:
2571:
2567:
2563:
2519:
2515:
2509:
2504:
2502:
2228:
2223:
2216:
2213:
1614:
1388:
1387:
1385:
1373:
1364:
1300:, given by:
1296:
1292:
1288:
1285:
1045:
916:
812:
808:
807:in units of
799:
795:
791:
786:
782:
776:
774:
639:
535:
529:
526:
437:
436:) and spin (
433:
431:
362:
355:
353:
315:
203:nuclear spin
196:
176:each nucleus
175:
165:
136:R. F. Bacher
128:Enrico Fermi
125:
82:
80:
60:
35:
29:
5997:299,792,458
5869:uranium-238
5865:uranium-235
5778:Frank Drake
5764:H I regions
5723:spectra of
5322:The dipole
789:, and that
199:Hamiltonian
42:electronic
6780:Categories
6554:1604.03059
6414:2009-03-03
6316:2008-07-23
6263:2009-03-03
6091:References
5877:dye lasers
5784:and later
5774:Carl Sagan
5760:21 cm line
3071:Quadrupole
1128:, giving:
144:Cassiopium
40:degenerate
6707:1210.6407
6638:118956636
6581:119192931
6464:⟶
6449:⟶
6231:0022-3719
6159:. Wiley.
6055:microwave
5889:microwave
5643:Δ
5614:Δ
5591:→
5559:→
5527:−
5518:Δ
5495:→
5463:→
5381:±
5363:Δ
5334:Δ
5260:⋅
5250:α
5232:α
5220:α
5206:α
5185:α
5170:α
5155:α
5144:α
5115:α
5111:α
5092:α
5088:≠
5085:α
5081:∑
5074:π
5066:ℏ
5057:μ
5047:μ
5025:^
4964:α
4960:α
4953:^
4941:⋅
4932:α
4905:α
4901:α
4894:^
4882:⋅
4877:α
4859:−
4850:α
4839:⋅
4834:α
4806:α
4802:α
4787:α
4776:α
4759:α
4755:≠
4752:α
4748:∑
4740:π
4721:μ
4711:μ
4688:^
4640:α
4629:⋅
4624:α
4619:μ
4608:α
4604:≠
4601:α
4597:∑
4593:−
4575:^
4536:^
4464:≥
4371:−
4326:−
4314:∑
4307:−
4282:⋅
4257:−
4242:^
4208:∗
4163:−
4135:−
4068:−
3985:−
3969:−
3796:∂
3782:∂
3768:∂
3720:⊗
3717:∇
3709:_
3705:_
3669:_
3665:_
3604:φ
3593:θ
3526:ρ
3523:∫
3513:π
3471:we have:
3426:∑
3416:
3403:traceless
3262:ρ
3245:δ
3218:−
3213:′
3198:′
3177:∫
3086:≥
3019:−
2998:−
2967:^
2931:Δ
2845:^
2807:⋅
2796:^
2775:^
2724:⋅
2703:⋅
2688:⋅
2672:π
2659:μ
2646:μ
2636:μ
2607:^
2541:^
2522:are good
2512:-coupling
2478:^
2461:^
2450:⋅
2436:−
2406:−
2402:ℓ
2351:⋅
2335:π
2322:μ
2309:μ
2299:μ
2270:^
2240:≠
2237:ℓ
2188:⋅
2152:δ
2136:^
2120:∑
2093:μ
2083:μ
2063:μ
2017:⋅
2009:−
1998:^
1987:⋅
1966:^
1955:⋅
1905:^
1887:∑
1863:π
1850:μ
1838:μ
1818:μ
1787:⋅
1749:^
1746:ℓ
1730:∑
1703:π
1690:μ
1677:μ
1667:μ
1634:^
1576:δ
1565:μ
1547:μ
1521:μ
1516:−
1510:^
1491:^
1480:⋅
1470:μ
1439:π
1426:μ
1337:μ
1323:−
1310:μ
1230:^
1227:ℓ
1212:∑
1188:π
1175:μ
1163:μ
1156:−
1148:ℓ
1085:^
1082:ℓ
1070:ℓ
1061:φ
1024:ℓ
1015:φ
1005:∑
991:ℓ
980:∑
957:ℓ
948:φ
899:ℓ
875:π
862:μ
850:μ
843:−
835:ℓ
756:ℏ
737:×
706:π
693:μ
681:μ
674:−
666:ℓ
605:−
602:×
591:−
582:π
569:μ
558:ℓ
483:ℓ
453:≡
410:⋅
400:μ
395:−
380:^
329:μ
260:μ
236:μ
52:molecules
6740:27008582
6732:23679718
6657:Archived
6519:35586070
6074:See also
6064:detuning
5943:rubidium
5920:, where
5896:Q factor
5399:, where
5235:′
5209:′
5188:′
5158:′
5118:′
5095:′
4967:′
4935:′
4908:′
4853:′
4809:′
4790:′
4762:′
4643:′
4611:′
3863:, with:
3633:′
3607:′
3596:′
3558:′
3540:′
3302:′
3276:′
3230:′
3101:have an
2973:⟩
2958:⟨
2851:⟩
2836:⟨
1381:currents
971:, where
152:Antimony
140:Europium
6712:Bibcode
6618:Bibcode
6559:Bibcode
6497:Bibcode
6342:Bibcode
6016:muonium
6000:
5988:
5955:defined
5952:is now
5939:caesium
5933:is the
5314:⁄
4122:where:
3687:of the
3369:is the
2909:(where
2878:⁄
2578:(where
2385:where:
1376:-factor
1370:is the
638:where â
347:is the
318:-factor
313:is the
156:Mercury
122:History
101:nucleus
73:in the
6738:
6730:
6636:
6579:
6517:
6405:
6375:
6292:
6254:
6229:
6188:
6163:
6133:
6048:
6040:qubits
6004:second
5984:vacuum
5950:second
5847:, and
5731:ions.
4661:where
3318:where
3111:tensor
1363:where
286:where
193:Dipole
162:Theory
148:Indium
54:, and
6736:S2CID
6702:arXiv
6634:S2CID
6608:arXiv
6577:S2CID
6549:arXiv
6515:S2CID
6060:laser
6002:of a
5982:in a
5980:light
4669:'
4519:other
2827:with
2574:onto
48:atoms
6771:IAEA
6728:PMID
6403:ISBN
6373:ISBN
6290:ISBN
6252:ISBN
6227:ISSN
6186:ISBN
6161:ISBN
6131:ISBN
5976:CGPM
5867:and
5859:The
5776:and
5727:and
4665:and
4501:>
4438:>
3373:and
3356:and
3348:and
3333:and
3322:and
3107:rank
2566:and
2518:and
320:and
154:and
134:and
112:fine
56:ions
6720:doi
6698:110
6626:doi
6604:606
6567:doi
6545:459
6505:doi
6493:57a
6350:doi
6219:doi
6042:in
6036:ion
5968:770
5965:631
5962:192
5941:or
5911:= Î
5831:In
5791:In
5766:in
3109:-2
2514:),
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6782::
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6225:.
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6213:.
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6117:^
6026:.
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5851:.
5843:,
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5788:.
5770:.
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5309:=
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2873:=
2587:+
2583:=
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830:el
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798:/
794:Ă
661:el
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150:,
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6742:.
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3903:=
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3040:]
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2619:=
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1978:(
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