# What is the nuclear spin of this atom?

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1. Jun 7, 2015

### unscientific

1. The problem statement, all variables and given/known data

A singly ionized atom exhibits hyperfine structure in a $^3P_1 \rightarrow ^3S_1$ transition. Assuming $I$ is greater than $J$ and given that the $^3S_1$ levels are split by $66~GHz$ and $54~GHz$, find the nuclear spin $I$.

2. Relevant equations

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

3. The attempt at a solution

Since $J=1$, the $^3S_1$ level is split by the higher level $F=I+1$ and lower level $F = I$.

$$\frac{(I+1)(I+2) - I(I+1) - J(J+1)}{I(I+1) - I(I+1) - J(J+1)} = \frac{66}{54}$$
$$\frac{2I}{2} = 1.22$$

I get $I = 1.22$, which is strange as shouldn't nuclear spin be either in multiples of $\frac{1}{2}$?

2. Jun 7, 2015

Solved.