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{{Quantum book backlink|Atomic and spectroscopy}}
'''Hyperfine structure''' refers to small shifts and splittings of otherwise degenerate electronic energy levels in atoms, molecules, and ions due to interactions between the [[Physics:Atomic nucleus|nucleus]] and the surrounding electron cloud.

These shifts are typically much smaller than those of [[Physics:Fine structure|fine structure]] and arise from electromagnetic multipole interactions, primarily involving nuclear magnetic dipole and electric quadrupole moments.

[[File:Fine hyperfine levels yellow.png|thumb|400px|Fine and hyperfine structure in the hydrogen atom]]

==Overview==

In atomic systems, hyperfine structure originates from:

* interaction between the [[Physics:Nuclear magnetic moment|nuclear magnetic dipole moment]] and magnetic fields produced by electrons
* interaction between the [[Physics:Quadrupole|nuclear electric quadrupole moment]] and the electric field gradient

In molecules, additional contributions arise from:

* nuclear spin–spin interactions
* nuclear spin–rotation coupling

Hyperfine structure is fundamentally weaker than fine structure and reflects the coupling between nuclear and electronic degrees of freedom.

==Magnetic dipole interaction==

For a nucleus with spin <math>\mathbf{I}</math>, the magnetic dipole moment is

<math>\boldsymbol{\mu}_I = g_I \mu_N \mathbf{I}</math>

where <math>g_I</math> is the nuclear ''g''-factor and <math>\mu_N</math> the nuclear magneton.

The interaction Hamiltonian is:

<math>\hat{H}_D = -\boldsymbol{\mu}_I \cdot \mathbf{B}</math>

where <math>\mathbf{B}</math> is the magnetic field generated by electrons.

In the effective angular momentum form, this becomes:

<math>\hat{H}_D = \hat{A}\,\mathbf{I} \cdot \mathbf{J}</math>

leading to the hyperfine energy shift:

<math>\Delta E = \frac{1}{2}A\left[F(F+1) - I(I+1) - J(J+1)\right]</math>

where:

* <math>\mathbf{J}</math> = total electronic angular momentum
* <math>\mathbf{F} = \mathbf{I} + \mathbf{J}</math> = total angular momentum

This interaction satisfies the [[Physics:Landé interval rule|Landé interval rule]].

==Electric quadrupole interaction==

For nuclei with spin <math>I \ge 1</math>, an electric quadrupole moment exists.

The quadrupole Hamiltonian is:

<math>\hat{H}_Q = -e\,T^2(Q)\cdot T^2(q)</math>

where:

* <math>T^2(Q)</math> describes the nuclear quadrupole moment
* <math>T^2(q)</math> describes the electric field gradient

This interaction reflects deviations from spherical nuclear charge distributions.

==Molecular hyperfine structure==

In molecules, hyperfine structure includes additional contributions:

===Spin–spin interaction===
Magnetic coupling between nuclei:

<math>\hat{H}_{II} = -\sum_{\alpha \ne \alpha'} \boldsymbol{\mu}_\alpha \cdot \mathbf{B}_{\alpha'}</math>

===Spin–rotation interaction===
Coupling between nuclear spins and molecular rotation:

<math>\hat{H}_{IR} \propto \mathbf{I} \cdot \mathbf{T}</math>

These effects are important in rotational spectroscopy.

==Experimental observation==

Hyperfine structure is observed in:

* atomic spectra
* molecular spectroscopy
* [[Physics:Electron paramagnetic resonance|electron paramagnetic resonance]]
* [[Physics:Nuclear magnetic resonance|nuclear magnetic resonance]]

A key example is the 21 cm hydrogen line in astrophysics.

==Applications==

===Atomic clocks===
The SI second is defined via the hyperfine transition of caesium-133:

<math>f = \frac{\Delta E}{h}</math>

One second equals exactly:

9192631770 cycles of this transition.

===Astrophysics===
Hyperfine transitions probe the interstellar medium and molecular clouds.

===Quantum computing===
Hyperfine states serve as long-lived qubits in trapped-ion systems.

===Precision physics===
Measurements of hyperfine splitting provide tests of quantum electrodynamics.

==History==

Hyperfine structure was first described theoretically by [[Biography:Enrico Fermi|Enrico Fermi]] in 1930.<ref>E. Fermi (1930), "Über die magnetischen Momente der Atomkerne". Z. Physik 60, 320–333.</ref>

The nuclear quadrupole moment was introduced in 1935 by H. Schüler and T. Schmidt.<ref>H. Schüler & T. Schmidt (1935), "Über Abweichungen des Atomkerns von der Kugelsymmetrie". Z. Physik 94, 457–468.</ref>

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

==References==
{{reflist|3}}

{{Author|Harold Foppele}}
[[Category:Atomic physics]]
[[Category:Quantum mechanics]]

{{Sourceattribution|Physics:Quantum Hyperfine structure|1}}

Physics:Quantum Hyperfine structure - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template