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atomic physics Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
s,
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and b ...
s, 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 In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...
'', which results from the interaction between the
magnetic moment In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electroma ...
s associated with electron spin and the electrons' orbital angular momentum. Hyperfine structure, with energy shifts typically orders of magnitudes smaller than those of a fine-structure shift, results from the interactions of the
nucleus Nucleus ( : nuclei) is a Latin word for the seed inside a fruit. It most often refers to: * Atomic nucleus, the very dense central region of an atom *Cell nucleus, a central organelle of a eukaryotic cell, containing most of the cell's DNA Nucl ...
(or nuclei, in molecules) with internally generated electric and magnetic fields.


History

In 1935, H. Schüler and Theodor Schmidt proposed the existence of a nuclear quadrupole moment in order to explain anomalies in the hyperfine structure.


Theory

The theory of hyperfine structure comes directly from
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, 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 In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe ...
\mathbf have a magnetic dipole moment, given by: :\boldsymbol_\text = g_\text\mu_\text\mathbf, where g_\text is the ''g''-factor and \mu_\text 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: :\hat_\text = -\boldsymbol_\text\cdot\mathbf. In the absence of an externally applied field, the magnetic field experienced by the nucleus is that associated with the orbital (ℓ) and spin (s) angular momentum of the electrons: :\mathbf \equiv \mathbf_\text = \mathbf_\text^\ell + \mathbf_\text^s. 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: :\mathbf_\text^\ell = \frac\frac, where −r gives the position of the nucleus relative to the electron. Written in terms of the
Bohr magneton In atomic physics, the Bohr magneton (symbol ) is a physical constant and the natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum. The Bohr magneton, in SI units is defined as \mu_\m ...
, this gives: :\mathbf_\text^\ell = -2\mu_\text \frac\frac \frac. Recognizing that ''me''v is the electron momentum, p, and that r×p/''ħ'' is the orbital
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
in units of ''ħ'', ℓ, we can write: :\mathbf_\text^\ell = -2\mu_\text\frac\frac\mathbf. For a many-electron atom this expression is generally written in terms of the total orbital angular momentum, \mathbf, by summing over the electrons and using the projection operator, \varphi^\ell_i, where \sum_i\mathbf_i = \sum_i\varphi^\ell_i\mathbf. For states with a well defined projection of the orbital angular momentum, ''Lz'', we can write \varphi^\ell_i = \hat_/L_z, giving: :\mathbf_\text^\ell = -2\mu_\text\frac\frac\sum_i\frac\mathbf. The electron spin angular momentum is a fundamentally different property that is intrinsic to the particle and therefore does not depend on the motion of the electron. Nonetheless it is angular momentum and any angular momentum associated with a charged particle results in a magnetic dipole moment, which is the source of a magnetic field. An electron with spin angular momentum, s, has a magnetic moment, μ''s'', given by: :\boldsymbol_\text = -g_s\mu_\text\mathbf, 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 dipole moment, μ''s'', is given by: :\mathbf_\text^s = \frac \left(3\left(\boldsymbol_\text \cdot \hat\right)\hat - \boldsymbol_\text\right) + \dfrac\boldsymbol_\text\delta^3(\mathbf). The complete magnetic dipole contribution to the hyperfine Hamiltonian is thus given by: :\begin \hat_D = &2g_\text\mu_\text\mu_\text\dfrac\dfrac\sum_i\dfrac \mathbf \cdot \mathbf \\ & + g_\text\mu_\textg_\text\mu_\text \frac \frac\sum_i \frac \left\ \\ & + \frac g_\text\mu_\textg_\text\mu_\text\mu_0 \frac\sum_i\hat_\delta^3\left(\mathbf_i\right)\mathbf\cdot\mathbf. \end 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. For states with \ell \neq 0 this can be expressed in the form :\hat_D = 2g_I\mu_\text\mu_\text\dfrac\dfrac, where: :\mathbf = \mathbf - \frac\left mathbf - 3(\mathbf\cdot\hat)\hat\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 \hat_\text 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: :\hat_\text = 2g_I\mu_\text\mu_\text\dfrac\dfrac\dfrac. This is commonly written as :\hat_\text = \hat\mathbf\cdot\mathbf, with \left\langle\hat\right\rangle being the hyperfine-structure constant which is determined by experiment. Since I·J = (where F = I + J is the total angular momentum), this gives an energy of: :\Delta E_\text = \frac\left\langle\hat\right\rangle (F + 1) - I(I + 1) - J(J + 1) In this case the hyperfine interaction satisfies the
Landé interval rule In atomic physics, the Landé interval rule Landé, A. Termstruktur und Zeemaneffekt der Multipletts. Z. Physik 15, 189–205 (1923). https://doi.org/10.1007/BF01330473 states that, due to weak angular momentum coupling (either spin-orbit or spin-s ...
.


Electric quadrupole

Atomic nuclei with spin I \ge 1 have an electric quadrupole moment. In the general case this is represented by a rank-2
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
, \underline, with components given by: :Q_ = \frac\int\left(3x_i^\prime x_j^\prime - \left(r^\prime\right)^2\delta_\right)\rho\left(\mathbf^\prime\right) \, d^3r^\prime, 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, ''δ''''ij'' 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''i''''Qii'' = 0), giving only five components in the irreducible representation. Expressed using the notation of irreducible spherical tensors we have: :T^2_m(Q) = \sqrt \int \rho\left(\mathbf^\prime\right)\left(r^\prime\right)^2 Y^2_m\left(\theta^\prime, \varphi^\prime\right) \, d^3r^\prime. 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 \underline, another rank-2 tensor given by the outer product of the del operator with the electric field vector: :\underline = \nabla\otimes\mathbf, with components given by: :q_ = \frac. Again it is clear this is a symmetric matrix and, because the source of the electric field at the nucleus is a charge distribution entirely outside the nucleus, this can be expressed as a 5-component spherical tensor, T^2(q), with: :\begin T^2_0(q) &= \fracq_ \\ T^2_(q) &= -q_ - iq_ \\ T^2_(q) &= \frac(q_ - q_) + iq_, \end where: :T^2_(q) = (-1)^mT^2_(q)^*. The quadrupolar term in the Hamiltonian is thus given by: :\hat_Q = -eT^2(Q) \cdot T^2(q) = -e\sum_m (-1)^m T^2_m(Q) T^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''.


Molecular hyperfine structure

The molecular hyperfine Hamiltonian includes those terms already derived for the atomic case with a magnetic dipole term for each nucleus with I > 0 and an electric quadrupole term for each nucleus with I \geq 1. The magnetic dipole terms were first derived for diatomic molecules by Frosch and Foley, 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.


Direct nuclear spin–spin

Each nucleus with I > 0 has a non-zero magnetic moment that is both the source of a magnetic field and has an associated energy due to the presence of the combined field of all of the other nuclear magnetic moments. A summation over each magnetic moment dotted with the field due to each ''other'' magnetic moment gives the direct nuclear spin–spin term in the hyperfine Hamiltonian, \hat_. :\hat_ = -\sum_\boldsymbol_\alpha\cdot \mathbf_, 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 : \hat_ = \dfrac \sum_ \frac \left\.


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, thus : \hat_\text = \frac\sum_ \frac \left\\cdot\mathbf.


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 = ), 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. The dipole selection rules for HCN hyperfine structure transitions are \Delta J = 1, \Delta F = \, where is the rotational quantum number and is the total rotational quantum number inclusive of nuclear spin (F = J + I_\text), respectively. The lowest transition (J = 1 \rightarrow 0) splits into a hyperfine triplet. Using the selection rules, the hyperfine pattern of J = 2 \rightarrow 1 transition and higher dipole transitions is in the form of a hyperfine sextet. However, one of these components (\Delta F = -1) carries only 0.6% of the rotational transition intensity in the case of J = 2 \rightarrow 1. This contribution drops for increasing J. So, from J = 2 \rightarrow 1 upwards the hyperfine pattern consists of three very closely spaced stronger hyperfine components (\Delta J = 1, \Delta F = 1) together with two widely spaced components; one on the low frequency side and one on the high frequency side relative to the central hyperfine triplet. Each of these outliers carry ~\tfrac J^2 ( is the upper rotational quantum number of the allowed dipole transition) the intensity of the entire transition. For consecutively higher- transitions, there are small but significant changes in the relative intensities and positions of each individual hyperfine component.


Measurements

Hyperfine interactions can be measured, among other ways, in atomic and molecular spectra and in
electron paramagnetic resonance Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spin ...
spectra of free radicals and
transition-metal In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that can ...
ions.


Applications


Astrophysics

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
21 cm line The hydrogen line, 21 centimeter line, or H I line is the electromagnetic radiation spectral line that is created by a change in the energy state of neutral hydrogen atoms. This electromagnetic radiation has a precise frequency of , w ...
observed in
H I region An HI region or H I region (read ''H one'') is a cloud in the interstellar medium composed of neutral atomic hydrogen (HI), in addition to the local abundance of helium and other elements. (H is the chemical symbol for hydrogen, and "I" is the R ...
s in interstellar medium. 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 submillimeter astronomy,
heterodyne receiver A superheterodyne receiver, often shortened to superhet, is a type of radio receiver that uses frequency mixing to convert a received signal to a fixed intermediate frequency (IF) which can be more conveniently processed than the original carr ...
s are widely used in detecting electromagnetic signals from celestial objects such as star-forming core or
young stellar objects Young stellar object (YSO) denotes a star in its early stage of evolution. This class consists of two groups of objects: protostars and pre-main-sequence stars. Classification by spectral energy distribution A star forms by accumulation of mate ...
. The separations among neighboring components in a hyperfine spectrum of an observed rotational transition are usually small enough to fit within the receiver's IF band. Since the
optical depth In physics, optical depth or optical thickness is the natural logarithm of the ratio of incident to ''transmitted'' radiant power through a material. Thus, the larger the optical depth, the smaller the amount of transmitted radiant power throug ...
varies with frequency, strength ratios among the hyperfine components differ from that of their intrinsic (or ''optically thin'') intensities (these are so-called ''hyperfine anomalies'', 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.


Nuclear spectroscopy

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
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
notch filter with very high stability, repeatability and
Q factor In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy ...
, which can thus be used as a basis for very precise
atomic clock An atomic clock is a clock that measures time by monitoring the resonant frequency of atoms. It is based on atoms having different energy levels. Electron states in an atom are associated with different energy levels, and in transitions betwe ...
s. The term ''transition frequency'' denotes the frequency of radiation corresponding to the transition between the two hyperfine levels of the atom, and is equal to , where is difference in energy between the levels and is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
. Typically, the transition frequency of a particular isotope of
caesium Caesium (IUPAC spelling) (or cesium in American English) is a chemical element with the symbol Cs and atomic number 55. It is a soft, silvery-golden alkali metal with a melting point of , which makes it one of only five elemental metals that a ...
or rubidium atoms is used as a basis for these clocks. Due to the accuracy of hyperfine structure transition-based atomic clocks, they are now used as the basis for the definition of the second. One second is now ''defined'' to be exactly cycles of the hyperfine structure transition frequency of caesium-133 atoms. On October 21, 1983, the 17th CGPM defined the meter as the length of the path travelled by
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
in a
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often ...
during a time interval of of a second.


Precision tests of quantum electrodynamics

The hyperfine splitting in hydrogen and in
muonium Muonium is an exotic atom made up of an antimuon and an electron, which was discovered in 1960 by Vernon W. Hughes and is given the chemical symbol Mu. During the muon's lifetime, muonium can undergo chemical reactions. Due to the mass diff ...
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 ion are commonly used for storing qubits in ion-trap quantum computing. They have the advantage of having very long lifetimes, experimentally exceeding ~10 minutes (compared to ~1s for metastable electronic levels). The frequency associated with the states' energy separation is in the
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ra ...
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
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The ...
pulses can be used to drive the transition, by having their frequency difference (''detuning'') equal to the required transition's frequency. This is essentially a stimulated Raman transition. In addition, near-field gradients have been exploited to individually address two ions separated by approximately 4.3 micrometers directly with microwave radiation.


See also

* Dynamic nuclear polarisation *
Electron paramagnetic resonance Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a method for studying materials that have unpaired electrons. The basic concepts of EPR are analogous to those of nuclear magnetic resonance (NMR), but the spin ...


References


External links


The Feynman Lectures on Physics Vol. III Ch. 12: The Hyperfine Splitting in Hydrogen
*
Nuclear Magnetic and Electric Moments lookup
��Nuclear Structure and Decay Data at the
IAEA The International Atomic Energy Agency (IAEA) is an intergovernmental organization that seeks to promote the peaceful use of nuclear energy and to inhibit its use for any military purpose, including nuclear weapons. It was established in 195 ...
{{Authority control Atomic physics Foundational quantum physics