In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate electronic energy levels and the resulting splittings in those electronic energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucleus and electron clouds. In atoms, hyperfine structure arises from the energy of the nuclear magnetic dipole moment interacting with the magnetic field generated by the electrons and the energy of the nuclear electric quadrupole moment in the electric field gradient due to the distribution of charge within the atom. Molecular hyperfine structure is generally dominated by these two effects, but also includes the energy associated with the interaction between the magnetic moments associated with different magnetic nuclei in a molecule, as well as between the nuclear magnetic moments and the magnetic field generated by the rotation of the molecule. Hyperfine structure contrasts with fine structure, which results from the interaction between the magnetic moments associated with electron 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 the nucleus (or nuclei, in molecules) with internally generated electric and magnetic fields.
History The first theory of atomic hyperfine structure was given in 1930 by Enrico Fermi 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 europium, cassiopium (older name for lutetium), indium, antimony, and mercury.
Theory The theory of hyperfine structure comes directly from electromagnetism, consisting of the interaction of the nuclear multipole moments (excluding the electric monopole) with internally generated fields. The theory is derived first for the atomic case, but can be applied to each nucleus in a molecule. Following this there is a discussion of the additional effects unique to the molecular case.
Atomic hyperfine structure
Magnetic dipole
The dominant term in the hyperfine Hamiltonian is typically the magnetic dipole term. Atomic nuclei with a non-zero nuclear spin I {\displaystyle \mathbf {I} } have a magnetic dipole moment, given by:
μ I = g I μ N I , {\displaystyle {\boldsymbol {\mu }}_{\text{I}}=g_{\text{I}}\mu _{\text{N}}\mathbf {I} ,}
where g I {\displaystyle g_{\text{I}}} is the g-factor and μ N {\displaystyle \mu _{\text{N}}} is the nuclear magneton. There is an energy associated with a magnetic dipole moment in the presence of a magnetic field. For a nuclear magnetic dipole moment, μI, placed in a magnetic field, B, the relevant term in the Hamiltonian is given by:
H ^ D = − μ I ⋅ B . {\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:
B ≡ B el = B el ℓ + B el s . {\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:
B el ℓ = μ 0 4 π − e v × − r r 3 , {\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:
B el ℓ = − 2 μ B μ 0 4 π 1 r 3 r × m e v ℏ . {\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 mev is the electron momentum, p, and that r × p / ħ is the orbital angular momentum in units of ħ, ℓ, we can write:
B el ℓ = − 2 μ B μ 0 4 π 1 r 3 ℓ . {\displaystyle \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, L {\displaystyle \mathbf {L} } , by summing over the electrons and using the projection operator, φ i ℓ {\displaystyle \varphi _{i}^{\ell }} , where ∑ i ℓ i = ∑ i φ i ℓ L {\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, Lz, we can write φ i ℓ = ℓ ^ z i / L z {\displaystyle \varphi _{i}^{\ell }={\hat {\ell }}_{z_{i}}/L_{z}} , giving:
B el ℓ = − 2 μ B μ 0 4 π 1 L z ∑ i ℓ ^ z i r i 3 L . {\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, μs, given by:
μ s = − g s μ B s , {\displaystyle {\boldsymbol {\mu }}_{\text{s}}=-g_{s}\mu _{\text{B}}\mathbf {s} ,}
where gs is the electron spin g-factor and the negative sign is because the electron is negatively charged (consider that negatively and positively charged particles with identical mass, travelling on equivalent paths, would have the same angular momentum, but would result in currents in the opposite direction). The magnetic field of a point dipole moment, μs, is given by:
B el s = μ 0 4 π r 3 ( 3 ( μ s ⋅ r ^ ) r ^ − μ s ) + 2 μ 0 3 μ s δ 3 ( r ) . {\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:
H ^ D =
2 g I μ N μ B μ 0 4 π 1 L z ∑ i ℓ ^ z i r i 3 I ⋅ L
+ g I μ N g s μ B μ 0 4 π 1 S z ∑ i s ^ z i r i 3 { 3 ( I ⋅ r ^ ) ( S ⋅ r ^ ) − I ⋅ S }
+ 2 3 g I μ N g s μ B μ 0 1 S z ∑ i s ^ z i δ 3 ( r i ) I ⋅ S . {\displaystyle {\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 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. The inclusion of the delta function is an admission that the singularity in the magnetic induction B owing to a magnetic dipole moment at a point is not integrable. It is B which mediates the interaction between the Pauli spinors in non-relativistic quantum mechanics. Fermi (1930) avoided the difficulty by working with the relativistic Dirac wave equation, according to which the mediating field for the Dirac spinors is the four-vector potential (V,A). The component V is the Coulomb potential. The component A is the three-vector magnetic potential (such that B = curl A), which for the point dipole is integrable. For states with ℓ ≠ 0 {\displaystyle \ell \neq 0} this can be expressed in the form
H ^ D = 2 g I μ B μ N μ 0 4 π I ⋅ N r 3 , {\displaystyle {\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:
N = ℓ − g s 2 [ s − 3 ( s ⋅ r ^ ) r ^ ] . {\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 LS-coupling), I and J are good quantum numbers and matrix elements of H ^ D {\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:
H ^ D = 2 g I μ B μ N μ 0 4 π N ⋅ J J ⋅ J I ⋅ J r 3 . {\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
H ^ D = A ^ I ⋅ J , {\displaystyle {\hat {H}}_{\text{D}}={\hat {A}}\mathbf {I} \cdot \mathbf {J} ,}
with ⟨ A ^ ⟩ {\textstyle \left\langle {\hat {A}}\right\rangle } being the hyperfine-structure constant which is determined by experiment. Since I⋅J = 1⁄2{F⋅F − I⋅I − J⋅J} (where F = I + J is the total angular momentum), this gives an energy of:
Δ E D = 1 2 ⟨ A ^ ⟩ [ F ( F + 1 ) − I ( I + 1 ) − J ( J + 1 ) ] . {\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 I ≥ 1 {\displaystyle I\geq 1} have an electric quadrupole moment. In the general case this is represented by a rank-2 tensor, Q i j {\displaystyle Q_{ij}} , with components given by:
Q i j = 1 e ∫ ( 3 x i ′ x j ′ − ( r ′ ) 2 δ i j ) ρ ( r ′ ) d 3 r ′ , {\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, xi and xj are the spatial variables x, y and z depending on the values of i and j respectively, δij is the Kronecker delta and ρ(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 a symmetric matrix (Qij = Qji) that is also traceless ( tr Q = ∑ i Q i i = 0 {\textstyle \operatorname {tr} Q=\sum _{i}Q_{ii}=0} ), giving only five components in the irreducible representation. Expressed using the notation of irreducible spherical tensors we have:
T m 2 ( Q ) = 4 π 5 ∫ ρ ( r ′ ) ( r ′ ) 2 Y m 2 ( θ ′ , φ ′ ) d 3 r ′ . {\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 q _ _ {\textstyle {\underline {\underline {q}}}} , another rank-2 tensor given by the outer product of the del operator with the electric field vector:
q _ _ = ∇ ⊗ E , {\displaystyle {\underline {\underline {q}}}=\nabla \otimes \mathbf {E} ,}
with components given by:
q i j = ∂ 2 V ∂ x i ∂ x j . {\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, T 2 ( q ) {\displaystyle T^{2}(q)} , with:
T 0 2 ( q ) = 6 2 q z z T + 1 2 ( q ) = − q x z − i q y z T + 2 2 ( q ) = 1 2 ( q x x − q y y ) + i q x y , {\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:
T − m 2 ( q ) = ( − 1 ) m T + m 2 ( q ) ∗ . {\displaystyle T_{-m}^{2}(q)=(-1)^{m}T_{+m}^{2}(q)^{*}.}
The quadrupolar term in the Hamiltonian is thus given by:
H ^ Q = − e T 2 ( Q ) ⋅ T 2 ( q ) = − e ∑ m ( − 1 ) m T m 2 ( Q ) T