Breit-Rabi formula

The Breit-Rabi formula (after Gregory Breit and Isidor Isaac Rabi (1931)) describes the hyperfine structure splitting of the hydrogen atom and hydrogen-like atoms (with valence electrons in the s-shell) as a function of an external magnetic field in atomic physics . Its main benefit is that it is also quantitatively valid in the transition area between weak ( Zeeman effect ) and strong field strengths ( Paschen-Back effect ). This is of particular importance in the case of the hydrogen atom because its core and shell angular momentum decouple even at low flux densities in the area . ${\displaystyle B\approx 0{,}05\,\mathrm {T} }$

The Breit-Rabi formula is an expression for the energy shift of a level with general nuclear spin and magnetic quantum number of the total angular momentum , but a given envelope angular momentum . It is: ${\displaystyle I}$ ${\displaystyle m_{F}}$ ${\displaystyle J={\frac {1}{2}}}$

${\displaystyle {W_{I\pm {\frac {1}{2}},m_{F}}=-{\frac {A}{2(2I+1)}}+g_{I}m_{F}\mu _{\mathrm {K} }B\pm {\frac {A}{2}}{\sqrt {1+{\frac {4m_{F}\left(g_{J}\mu _{\mathrm {B} }-g_{I}\mu _{\mathrm {K} }\right)B}{(2I+1)A}}+\left({\frac {\left(g_{J}\mu _{\mathrm {B} }-g_{I}\mu _{\mathrm {K} }\right)B}{A}}\right)^{2}}}}}$

It is the nuclear specific hyperfine coupling constant, the Bohr and the nuclear magneton . and are the Landé factors of the envelope angular momentum or nuclear spin . ${\displaystyle A}$ ${\displaystyle \mu _{\mathrm {B} }}$ ${\displaystyle \mu _{\mathrm {K} }}$ ${\displaystyle g_{J}}$ ${\displaystyle g_{I}}$ ${\displaystyle J}$ ${\displaystyle I}$

Derivation for the ground state of the hydrogen atom

The angular momentum is described here with the angular momentum quantum number , which corresponds to the amount of an angular momentum in units of Planck's reduced quantum of action . The hydrogen atom has a nuclear spin . The only electron in the ground state ( ) has only one spin angular momentum, which is also the entire envelope angular momentum . Nuclear spin and envelope angular momentum couple according to the angular momentum algebra to form the total angular momentum . The derivation that follows for this simplest case can be strongly generalized for different values ​​of and . The basic procedure is clearly evident in the form presented here. ${\displaystyle \hbar }$ ${\displaystyle I={\frac {|{\vec {I}}|}{\hbar }}={\frac {1}{2}}}$ ${\displaystyle l=0}$ ${\displaystyle J={\frac {|{\vec {J}}|}{\hbar }}={\frac {1}{2}}}$ ${\displaystyle {\vec {F}}={\vec {I}}+{\vec {J}}}$ ${\displaystyle J}$ ${\displaystyle I}$

The Hamilton operator of the hyperfine structure with a B-field in the z-direction is:

${\displaystyle {\hat {H}}_{\mathrm {HFS} }=A{\frac {{\vec {I}}\cdot {\vec {J}}}{\hbar ^{2}}}+\left(g_{J}\mu _{\mathrm {B} }{\frac {J_{z}}{\hbar }}-g_{I}\mu _{\mathrm {K} }{\frac {I_{z}}{\hbar }}\right)B}$

This Hamilton operator is now diagonalized in a suitable basis , which is composed of "good quantum numbers"; with the projection of the angular momentum onto the direction of the magnetic field (magnetic quantum number). The first term of the above Hamiltonian is diagonal in this basis and can be expressed as ${\displaystyle |JIFm_{F}\rangle }$ ${\displaystyle {\vec {F}}}$ ${\displaystyle m_{F}=m_{J}+m_{I}}$

${\displaystyle A{\frac {{\vec {I}}\cdot {\vec {J}}}{\hbar ^{2}}}={\frac {A}{2}}\left(F(F+1)-I(I+1)-J(J+1)\right)}$

The components and can also be represented in matrix form using the Wigner-Eckart theorem . The rows and columns are provided with indices on the left and above, which are to be read as. Off the diagonal, almost all entries are zero, except for those that mix. ${\displaystyle z}$ ${\displaystyle I_{z}}$ ${\displaystyle J_{z}}$ ${\displaystyle (F|m_{F})}$ ${\displaystyle m_{F}=0}$

${\displaystyle {\frac {\langle JIF'm_{F}'|J_{z}|JIFm_{F}\rangle }{\hbar }}=\left({\begin{array}{c|cccc}&\left(0|0\right)&\left(1|-1\right)&\left(1|0\right)&\left(1|1\right)\\\hline \left(0|0\right)&0&0&{\frac {1}{2}}&0\\\left(1|-1\right)&0&-{\frac {1}{2}}&0&0\\\left(1|0\right)&{\frac {1}{2}}&0&0&0\\\left(1|1\right)&0&0&0&{\frac {1}{2}}\end{array}}\right)}$

Analogously it follows for the component of the nuclear spin: ${\displaystyle z}$

${\displaystyle {\frac {\langle JIF'm_{F}'|I_{z}|JIFm_{F}\rangle }{\hbar }}=\left({\begin{array}{c|cccc}&\left(0|0\right)&\left(1|-1\right)&\left(1|0\right)&\left(1|1\right)\\\hline \left(0|0\right)&0&0&-{\frac {1}{2}}&0\\\left(1|-1\right)&0&-{\frac {1}{2}}&0&0\\\left(1|0\right)&-{\frac {1}{2}}&0&0&0\\\left(1|1\right)&0&0&0&{\frac {1}{2}}\end{array}}\right)}$

If you add up all three terms, which are individually represented in a matrix, and insert as well for the hydrogen atom, the result for the Hamiltonian is: ${\displaystyle I=J={\frac {1}{2}}}$ ${\displaystyle g_{J}\approx 2}$

${\displaystyle {\hat {H}}_{\mathrm {HFS} }=\left({\begin{array}{c|cccc}&\left(0|0\right)&\left(1|-1\right)&\left(1|0\right)&\left(1|1\right)\\\hline (0|0)&-{\frac {3A}{4}}&0&\left(\mu _{\mathrm {B} }+{\frac {g_{I}}{2}}\mu _{\mathrm {K} }\right)B&0\\(1|-1)&0&{\frac {A}{4}}-\left(\mu _{\mathrm {B} }-{\frac {g_{I}}{2}}\mu _{\mathrm {K} }\right)B&0&0\\(1|0)&\left(\mu _{\mathrm {B} }+{\frac {g_{I}}{2}}\mu _{\mathrm {K} }\right)B&0&{\frac {A}{4}}&0\\(1|1)&0&0&0&{\frac {A}{4}}+\left(\mu _{\mathrm {B} }-{\frac {g_{I}}{2}}\mu _{\mathrm {K} }\right)B\end{array}}\right)}$

The eigenvalues ​​of this matrix result in neglecting quadratic terms in for general values ​​for and especially the above-mentioned Breit-Rabi formula. ${\displaystyle \mu _{\mathrm {K} }}$ ${\displaystyle I,F}$ ${\displaystyle m_{F}}$

Individual evidence

1. ^ Gregory Breit, Isidor Isaac Rabi: Measurement of Nuclear Spin . In: Physical Review Letters . 38, No. 11, November 1931, pp. 2082-2083. doi : 10.1103 / PhysRev.38.2082.2 .
2. ↑ Florian Scheck: Quantum Physics . Springer, 2013, ISBN 9783642345630 , p. 284.
3. ^ Blair, BE and Morgan, AH: Frequency and Time . US Government Printing Office, 1972, ISBN 9783642345630 , pp. 13-14.
4. ↑ Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 - Atomic Physics and Basics of Spectroscopy . 1st edition. Springer, Berlin, Heidelberg 2008, ISBN 978-3-540-30613-9 , pp. 362 .
5. ↑ Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 - Atomic Physics and Basics of Spectroscopy . 1st edition. Springer, Berlin, Heidelberg 2008, ISBN 978-3-540-30613-9 , pp. 367 ff .