Hyperfine structure

History

The first theory of atomic hyperfine structure was given in 1930 by Enrico Fermi ^[1] for an atom containing a single valence electron with an arbitrary angular momentum. The Zeeman splitting of this structure was discussed by S. A. Goudsmit and 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

The dominant term in the hyperfine

${\displaystyle \mathbf {I} }$

$${\displaystyle {\boldsymbol {\mu }}_{\text{I}}=g_{\text{I}}\mu _{\text{N}}\mathbf {I} ,}$$

${\displaystyle g_{\text{I}}}$

${\displaystyle \mu _{\text{N}}}$

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]

$${\displaystyle {\hat {H}}_{\text{D}}=-{\boldsymbol {\mu }}_{\text{I}}\cdot \mathbf {B} .}$$

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:

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

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 position r relative to the nucleus, is given by:

$${\displaystyle \mathbf {B} _{\text{el}}^{\ell }={\frac {\mu _{0}}{4\pi }}{\frac {-e\mathbf {v} \times -\mathbf {r} }{r^{3}}},}$$

where −r gives the position of the nucleus relative to the electron. Written in terms of the Bohr magneton, this gives:

$${\displaystyle \mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{r^{3}}}{\frac {\mathbf {r} \times m_{\text{e}}\mathbf {v} }{\hbar }}.}$$

Recognizing that m_ev is the electron momentum, p, and that r×p/ħ is the orbital angular momentum in units of ħ, ℓ, we can write:

$${\displaystyle \mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{r^{3}}}\mathbf {\ell } .}$$

For a many-electron atom this expression is generally written in terms of the total orbital angular momentum, ${\displaystyle \mathbf {L} }$, by summing over the electrons and using the projection operator, ${\displaystyle \varphi _{i}^{\ell }}$, 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 ${\displaystyle \varphi _{i}^{\ell }={\hat {\ell }}_{z_{i}}/L_{z}}$, giving:

$${\displaystyle \mathbf {B} _{\text{el}}^{\ell }=-2\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{L_{z}}}\sum _{i}{\frac {{\hat {\ell }}_{zi}}{r_{i}^{3}}}\mathbf {L} .}$$

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:

$${\displaystyle {\boldsymbol {\mu }}_{\text{s}}=-g_{s}\mu _{\text{B}}\mathbf {s} ,}$$

The magnetic field of a point dipole moment, μ_s, is given by:^[4][5]

$${\displaystyle \mathbf {B} _{\text{el}}^{s}={\frac {\mu _{0}}{4\pi r^{3}}}\left(3\left({\boldsymbol {\mu }}_{\text{s}}\cdot {\hat {\mathbf {r} }}\right){\hat {\mathbf {r} }}-{\boldsymbol {\mu }}_{\text{s}}\right)+{\dfrac {2\mu _{0}}{3}}{\boldsymbol {\mu }}_{\text{s}}\delta ^{3}(\mathbf {r} ).}$$

Electron total magnetic field and contribution

The complete magnetic dipole contribution to the hyperfine Hamiltonian is thus given by:

$${\displaystyle {\begin{aligned}{\hat {H}}_{D}={}&2g_{\text{I}}\mu _{\text{N}}\mu _{\text{B}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {1}{L_{z}}}\sum _{i}{\dfrac {{\hat {\ell }}_{zi}}{r_{i}^{3}}}\mathbf {I} \cdot \mathbf {L} \\&{}+g_{\text{I}}\mu _{\text{N}}g_{\text{s}}\mu _{\text{B}}{\frac {\mu _{0}}{4\pi }}{\frac {1}{S_{z}}}\sum _{i}{\frac {{\hat {s}}_{zi}}{r_{i}^{3}}}\left\{3\left(\mathbf {I} \cdot {\hat {\mathbf {r} }}\right)\left(\mathbf {S} \cdot {\hat {\mathbf {r} }}\right)-\mathbf {I} \cdot \mathbf {S} \right\}\\&{}+{\frac {2}{3}}g_{\text{I}}\mu _{\text{N}}g_{\text{s}}\mu _{\text{B}}\mu _{0}{\frac {1}{S_{z}}}\sum _{i}{\hat {s}}_{zi}\delta ^{3}{\left(\mathbf {r} _{i}\right)}\mathbf {I} \cdot \mathbf {S} .\end{aligned}}}$$

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 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.^[6]

For states with ${\displaystyle \ell \neq 0}$ this can be expressed in the form

$${\displaystyle {\hat {H}}_{D}=2g_{I}\mu _{\text{B}}\mu _{\text{N}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {\mathbf {I} \cdot \mathbf {N} }{r^{3}}},}$$

where:^[3]

$${\displaystyle \mathbf {N} ={\boldsymbol {\ell }}-{\frac {g_{s}}{2}}\left[\mathbf {s} -3(\mathbf {s} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}\right].}$$

If hyperfine structure is small compared with the fine structure (sometimes called IJ-coupling by analogy with

and matrix elements of ${\displaystyle {\hat {H}}_{\text{D}}}$ can be approximated as diagonal in I and J. In this case (generally true for light elements), we can project N onto J (where J = L + S is the total electronic angular momentum) and we have:

$${\displaystyle {\hat {H}}_{\text{D}}=2g_{I}\mu _{\text{B}}\mu _{\text{N}}{\dfrac {\mu _{0}}{4\pi }}{\dfrac {\mathbf {N} \cdot \mathbf {J} }{\mathbf {J} \cdot \mathbf {J} }}{\dfrac {\mathbf {I} \cdot \mathbf {J} }{r^{3}}}.}$$

This is commonly written as

$${\displaystyle {\hat {H}}_{\text{D}}={\hat {A}}\mathbf {I} \cdot \mathbf {J} ,}$$

${\textstyle \left\langle {\hat {A}}\right\rangle }$

$${\displaystyle \Delta E_{\text{D}}={\frac {1}{2}}\left\langle {\hat {A}}\right\rangle [F(F+1)-I(I+1)-J(J+1)].}$$

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

Electric quadrupole

Atomic nuclei with spin ${\displaystyle I\geq 1}$ have an

, ${\displaystyle Q_{ij}}$, with components given by:

$${\displaystyle Q_{ij}={\frac {1}{e}}\int \left(3x_{i}^{\prime }x_{j}^{\prime }-\left(r'\right)^{2}\delta _{ij}\right)\rho {\left(\mathbf {r} '\right)}\,d^{3}\mathbf {r} ',}$$

where i and j are the tensor indices running from 1 to 3, x_i and x_j are the spatial variables x, y and z depending on the values of i and j respectively, δ_ij is the

${\textstyle \operatorname {tr} Q=\sum _{i}Q_{ii}=0}$

$${\displaystyle T_{m}^{2}(Q)={\sqrt {\frac {4\pi }{5}}}\int \rho {\left(\mathbf {r} '\right)}\left(r'\right)^{2}Y_{m}^{2}\left(\theta ',\varphi '\right)\,d^{3}\mathbf {r} '.}$$

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 the

$${\displaystyle {\underline {\underline {q}}}=\nabla \otimes \mathbf {E} ,}$$

$${\displaystyle q_{ij}={\frac {\partial ^{2}V}{\partial x_{i}\,\partial x_{j}}}.}$$

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, ${\displaystyle T^{2}(q)}$, with:^[9]

$${\displaystyle {\begin{aligned}T_{0}^{2}(q)&={\frac {\sqrt {6}}{2}}q_{zz}\\T_{+1}^{2}(q)&=-q_{xz}-iq_{yz}\\T_{+2}^{2}(q)&={\frac {1}{2}}(q_{xx}-q_{yy})+iq_{xy},\end{aligned}}}$$

where:

$${\displaystyle T_{-m}^{2}(q)=(-1)^{m}T_{+m}^{2}(q)^{*}.}$$

The quadrupolar term in the Hamiltonian is thus given by:

$${\displaystyle {\hat {H}}_{Q}=-eT^{2}(Q)\cdot T^{2}(q)=-e\sum _{m}(-1)^{m}T_{m}^{2}(Q)T_{-m}^{2}(q).}$$

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 by Q_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 ${\displaystyle I>0}$ and an electric quadrupole term for each nucleus with ${\displaystyle I\geq 1}$. 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 ${\displaystyle I>0}$ 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 each other magnetic moment gives the direct nuclear spin–spin term in the hyperfine Hamiltonian, ${\displaystyle {\hat {H}}_{II}}$.^[12]

$${\displaystyle {\hat {H}}_{II}=-\sum _{\alpha \neq \alpha '}{\boldsymbol {\mu }}_{\alpha }\cdot \mathbf {B} _{\alpha '},}$$

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

$${\displaystyle {\hat {H}}_{II}={\dfrac {\mu _{0}\mu _{\text{N}}^{2}}{4\pi }}\sum _{\alpha \neq \alpha '}{\frac {g_{\alpha }g_{\alpha '}}{R_{\alpha \alpha '}^{3}}}\left\{\mathbf {I} _{\alpha }\cdot \mathbf {I} _{\alpha '}-3\left(\mathbf {I} _{\alpha }\cdot {\hat {\mathbf {R} }}_{\alpha \alpha '}\right)\left(\mathbf {I} _{\alpha '}\cdot {\hat {\mathbf {R} }}_{\alpha \alpha '}\right)\right\}.}$$

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

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

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 (¹H¹²C¹⁴N) in its ground vibrational state. Here, the electric quadrupole interaction is due to the ¹⁴N-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 the ¹H-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 dipole

${\displaystyle \Delta J=1}$

${\displaystyle \Delta F=\{0,\pm 1\}}$

${\displaystyle F=J+I_{\text{N}}}$

${\displaystyle J=1\rightarrow 0}$

${\displaystyle J=2\rightarrow 1}$

${\displaystyle \Delta F=-1}$

${\displaystyle J=2\rightarrow 1}$

${\displaystyle J=2\rightarrow 1}$

${\displaystyle \Delta J=1}$

${\displaystyle \Delta F=1}$

${\displaystyle {\tfrac {1}{2}}J^{2}}$

Measurements

Hyperfine interactions can be measured, among other ways, in atomic and molecular spectra and in

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

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

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, and perturbed angular correlation.

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 photo-ionize only the uranium-235 atoms and then separate the ionized particles from the non-ionized ones. Precisely tuned dye 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 a

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 defined to be exactly 9192631770 cycles of the hyperfine structure transition frequency of caesium-133 atoms.

On October 21, 1983, the 17th

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 stringent test of QED.

Qubit in ion-trap quantum computing

The hyperfine states of a trapped

The frequency associated with the states' energy separation is in the

• Dynamic nuclear polarisation
• Electron paramagnetic resonance

References