# Hyperfine Interaction Splitting

## Homework Statement

(a)Find splitting between F=0 and F=1 in hydrogen
(b) Find the constant ##A## and nuclear magnetic moment

## The Attempt at a Solution

Hyperfine splitting is given by:

$$H_{hf} = \frac{A}{2} \left[ F(F+1) - I(I+1) - J(J+1) \right]$$

The intervals are given by:
$$\Delta E_{hf} = E_F - E_{F-1} = AF$$

Part(a)

Using ##G_I = 5.6##, ##I= \frac{1}{2}## for a single proton, ##n = 1, Z = 1## I get splittings as ## 1.42~ MHz## and ##474.6 ~cm^{-1}##. Could somebody verify this?

Part (b)

For ##J = \frac{9}{2}##, it implies that ##I = \frac{1}{2}## for the energy levels to be split into 6 levels, ##F= I + J = 0,1,2,3,4,5##.

Using the interval rule, the energy intervals are ##A, 2A, 3A, 4A, 5A##.

Taking the maximum energy - minimum energy observed = energy spacing between F=0 and F=5.

$$15 A = 10^9 h (9568.19-2312.87)$$

This gives ##A = 3.21 \times 10^{-22}## and ##g_i \mu_N = 3.65 \times 10^{-23}##.

However, this means that ##g_I = 7200##. Is this reasonable?

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mfb
Mentor
Your answers in (a) are inconsistent by a factor of 10, and both are wrong by some power of 10.
Taking the maximum energy - minimum energy observed = energy spacing between F=0 and F=5.
I don't think that makes sense. Those photon energies are differences between states already, not the absolute energy of something.
Also, where does the factor of 109 come from?

Your answers in (a) are inconsistent by a factor of 10, and both are wrong by some power of 10.
I don't think that makes sense. Those photon energies are differences between states already, not the absolute energy of something.
Also, where does the factor of 109 come from?
I'm thinking consider a transition from the upper level to the F=5 level vs the transition from the upper level to the F=0 level. The difference in energy observed would be the spacing between F=0 and F=5 levels, which is 15A.