Breit—Rabi formula

In intermediate fields the calculation is more complicated. For I or J = 1/2, the Breit-Rabi formula can be expressed in analogy with the fine structure case, see (2.31). For J = 1/2 we have... 

From the measured level-crossing positions the fine-structure splitting can be calculated using the Breit-Rabi formula for the fine structure, (2.31). 

In Chap. 7 we have discussed how hyperfine structure can be determined by level-crossing spectroscopy. Clearly, alkali atom states can readily be studied using this technique after stepwise excitation. We will here instead choose an example illustrating fine-structure measurements. In Fig. 9.17 the example of the inverted sodium 4d 5/2,3/2 state is given. From the measured level-crossing positions the fine-structiue splitting can be calculated using the Breit-Rabi formula for the fine structure, (2.31). 

The quadrupole interaction vanishes for J = - so that the Breit-Rabi formula gives an exact description of the hyperfine structure, in this case. If the small term gjVjjBM is... 

Fig.18.3. A plot of the Breit-Rabi formula for the case I = The abscissa is given by Cgj + gjin/M) PgB/hvjjpg where hvp pg is the energy difference between the levels F=2 and F=1 in zero field.

Finally, by solving the secular equation, determine the energies of the hyperfine sub-levels in fields of arbitrary strength. Verify the correctness of the results using the Breit-Rabi formula and Fig.18.16. 

Expand the Breit-Rabi formula up to terms of order x and thus obtain a precise expression for the frequency of the Zeeman magnetic resonance transitions AF=0,... 

By expanding the Breit-Rabi formula in the weak field approximation prove that the frequency of the AF= 1, Mp=0 Mp=0 transition is given by... 

AMj=0, AMj= 1 transitions at fields which were not quite large enough for the complete Paschen-Back formula, equation (18.37), to be applicable. By expanding the Breit-Rabi formula up to terras of order... 

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