Hyperfine structure

Hyperfine structure splitting of the energy levels using the example of the hydrogen atom (not to scale);
Designation of the fine structure levels s. Term symbol , explanation of the remaining formula symbols in the text

The hyperfine structure is an energy split in the spectral lines of the atomic spectra . It is about 2000 times smaller than that of the fine structure split. The hyperfine structure is based on the one hand on the interaction of the electrons with magnetic ( dipole ) and electrical ( quadrupole ) moments of the nucleus and on the other hand on the isotopy of the elements .

Nuclear spin effect

In a narrower sense, hyperfine structure is understood as the splitting of the energy levels of an atom - compared to the levels of the fine structure - due to the coupling of the magnetic moment of the nucleus with the magnetic field that the electrons generate at its location: ${\ displaystyle V _ {\ mathrm {HFS}}}$ ${\ displaystyle {\ vec {\ mu}} _ {I}}$ ${\ displaystyle {\ vec {B}} _ {J}}$

{\ displaystyle \ Delta V _ {\ mathrm {HFS}} = - {\ vec {\ mu}} _ {I} \ cdot {\ vec {B}} _ {J}}

The indices mean:

${\ displaystyle I}$ : Nuclear spin
${\ displaystyle J}$ : Envelope angular momentum .

The greatest hyperfine structure splitting is shown by s-electrons , because only they have a greater probability of being at the location of the nucleus.

In a weak external magnetic field, the energy levels split further according to a very similar formula according to the magnetic quantum number of the hyperfine structure ( Zeeman effect ). In a strong external magnetic field, the angular momentum of the nucleus and the envelope decouple, so that a splitting according to the magnetic quantum number of the nucleus can be observed ( Paschen-Back effect ). The Breit-Rabi formula can be used for any field strengths (in the case of vanishing orbital angular momentum) . ${\ displaystyle m_ {F}}$ ${\ displaystyle m_ {I}}$

Mathematical formulation

The coupling causes the total angular momentum of the atom, which is the sum of the envelope angular momentum and the nuclear spin , to be quantized : ${\ displaystyle {\ vec {F}}}$ ${\ displaystyle {\ vec {J}}}$ ${\ displaystyle {\ vec {I}}}$

{\ displaystyle | {\ vec {F}} | = \ hbar {\ sqrt {F (F + 1)}} = | {\ vec {J}} + {\ vec {I}} |.}

The quantum number is half ( Fermi-Dirac statistics ) or an integer ( Bose-Einstein statistics ) and can take values at a distance of 1. ${\ displaystyle F}$ ${\ displaystyle \ {| JI |, \ dots, J + I \}}$

The interaction energy is

{\ displaystyle {\ begin {aligned} \ Delta V _ {\ mathrm {HFS}} & = - {\ vec {\ mu}} _ {I} \ cdot {\ vec {B}} _ {J} \\ & = g_ {I} \ cdot \ mu _ {\ mathrm {K}} \ cdot B_ {J} \, {\ frac {F (F + 1) - [J (J + 1) + I (I + 1) ]} {2 {\ sqrt {J (J + 1)}}}} \\ & = {\ frac {A} {2}} \, \ left (F (F + 1) - [J (J + 1 ) + I (I + 1)] \ right). \ End {aligned}}}

It is

{\ displaystyle g_ {I}}

the Landé factor of the core

${\ displaystyle \ mu _ {\ mathrm {K}} = {\ frac {e \ hbar} {2m _ {\ mathrm {p}}}}}$ the nuclear magneton

${\ displaystyle e}$ the elementary electric charge

${\ displaystyle \ hbar}$ the reduced Planck quantum of action

${\ displaystyle m_ {p}}$ the proton mass

{\ displaystyle A = {\ frac {g_ {I} \ mu _ {k} B_ {J}} {\ sqrt {J (J + 1)}}}}

the hyperfine structure constant.

The magnetic moment and the angular momentum of the nucleus are related as follows:

{\ displaystyle {\ vec {\ mu}} _ {I} = {\ frac {g_ {I} \ mu _ {\ mathrm {K}}} {\ hbar}} {\ vec {I}}.}

To determine , one needs the sizes and . The value of can be determined by nuclear magnetic resonance measurements, the value of from the wave function of the electrons, which can only be calculated numerically for atoms with an atomic number greater than 1. ${\ displaystyle V _ {\ mathrm {HFS}}}$ ${\ displaystyle g_ {I}}$ ${\ displaystyle B_ {J}}$ ${\ displaystyle g_ {I}}$ ${\ displaystyle B_ {J}}$

Applications

Transitions between hyperfine states are used in atomic clocks because their frequency (like that of all atomic transitions) is constant. In addition, it can be generated and measured very precisely with relatively simple means, since it is in the radio frequency or microwave range. Since 1967 the physical unit second has been determined by means of transitions between the two hyperfine structure levels of the ground state of the cesium isotope ¹³³ Cs.

The frequency for the transition of the ground state of the hydrogen atom between and ( spin-flip ) is 1.420 GHz , which corresponds to a wavelength of 21 cm. This so-called HI line (H-one line) is of great importance for radio astronomy . By measuring the Doppler shift of this line, the movement of interstellar gas clouds relative to the earth can be determined. ${\ displaystyle F = 1}$ ${\ displaystyle F = 0}$

Isotope effects

There are also the isotope effects. Unlike nuclear spin, these do not provide level splitting within a single atom. Rather, there is a shift in the spectral lines for different isotopes of the same element, the so-called isotope shift . As a result, a splitting of the lines can be observed with an isotope mixture .

Nuclear mass effect

The nuclear mass effect is based on the movement of the atomic nucleus. This manifests itself in a lower effective mass of the electron. Since the nuclei of different isotopes have different masses, the effective mass of their electrons is also slightly different, which is expressed in a corresponding shift of all states towards higher energy. Since the movement of the nucleus decreases as the mass of the nucleus increases, this effect is particularly important for light atomic nuclei.

Core volume effect

The nuclear volume effect is based on the finite expansion of the atomic nucleus. Electrons in s-states (i.e. with orbital angular momentum 0) have a non-negligible probability of being in the nucleus, where the potential no longer has the pure Coulomb form . This deviation means an increase in the energies of the states, which depends on the volume of the nucleus. In absolute terms, this effect is greatest with heavy atoms, since these have the largest atomic nuclei. However, the splitting is again greater with the smaller atomic nuclei, since the ratios of the nucleus volumes of different isotopes are greater here.

Hyperfine interaction in molecules and crystals

Electric and magnetic fields of the neighboring atoms in molecules and crystals as well as the atomic shell itself influence the splitting of the spin states into the observed hyperfine structure. In solid-state physics and solid-state chemistry , methods of nuclear solid-state physics are used to investigate the local structure in solids (metals, semiconductors, insulators). These methods, such as nuclear magnetic resonance spectroscopy (NMR), Mössbauer spectroscopy and disturbed gamma-gamma angle correlation (PAC spectroscopy) make it possible to explore structures on an atomic scale with high sensitivity using the atomic nucleus as a probe. In biochemistry , NMR is used for structural analysis of organic molecules.

literature

Stephanus Büttgenbach: Hyperfine structure in 4d- and 5d-shell atoms. Springer, Berlin 1982, ISBN 0-387-11740-7
Lloyd Armstrong: Theory of the hyperfine structure of free atoms. Wiley-Interscience, New York 1971, ISBN 0-471-03335-9