Zeeman Effect - Intermediate Field For J = 1/2

Intermediate Field For J = 1/2

In the magnetic dipole approximation, the Hamiltonian which includes both the hyperfine and Zeeman interactions is

To arrive at the Breit-Rabi formula we will include the hyperfine structure (interaction between the electron's spin and the magnetic moment of the nucleus), which is governed by the quantum number, where is the spin angular momentum operator of the nucleus. Alternatively, the derivation could be done with only. The constant is known as the zero field hyperfine constant and is given in units of Hertz. is the Bohr magneton. and are the electron and nuclear angular momentum operators. and can be found via a classical vector coupling model or a more detailed quantum mechanical calculation to be:

As discussed, in the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the basis. In the high field regime, the magnetic field becomes so large that the Zeeman effect will dominate, and we must use a more complete basis of or just since and will be constant within a given level.

To get the complete picture, including intermediate field strengths, we must consider eigenstates which are superpositions of the and basis states. For, the Hamiltonian can be solved analytically, resulting in the Breit-Rabi formula. Notably, the electric quadrapole interaction is zero for, so this formula is fairly accurate.

To solve this system, we note that at all times, the total angular momentum projection will be conserved. Furthermore, since between states will change between only . Therefore, we can define a good basis as:

We now utilize quantum mechanical ladder operators, which are defined for a general angular momentum operator as

These ladder operators have the property

as long as lies in the range (otherwise, they return zero). Using ladder operators and We can rewrite the Hamiltonian as

Now we can determine the matrix elements of the Hamiltonian:

Solving for the eigenvalues of this matrix, (as can be done by hand, or more easily, with a computer algebra system) we arrive at the energy shifts:

where is the splitting (in units of Hz) between two hyperfine sublevels in the absence of magnetic field ,

is referred to as the 'field strength parameter' (Note: for the square root is an exact square, and should be interpreted as ). This equation is known as the Breit-Rabi formula and is useful for systems with one valence electron in an level.

Note that index in should be considered not as total angular momentum of the atom but as asymptotic total angular momentum. It is equal to total angular momentum only if otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with different but equal (the only exceptions are ).