# Fine-structure constants

Okey so I think this question or a similar one was here recently but I can't find it so creating a new.

## Homework Statement

The figure below shows the hyperfine structure in the transition 6s ##^2S_{1/2} - 8p ^2P_{3/2}## in 115In (I = 9/2). The measurement is made using a narrow-band tunable laser and a collimated atomic beam; hence the Doppler width is greatly reduced. The 6 components shown have the following frequencies 31, 112, 210, 8450, 8515 and 8596 MHz. Draw a schematic figure of the energy levels with the appropriate quantum numbers and show the allowed transitions. Determine the hyperfine constants, in MHz, for the two fine structure levels.

## Homework Equations

[/B]
$$A=\frac{2E}{F(F+1)-I(I+1)-J(J+1)}$$

## The Attempt at a Solution

Now, since I have I and J I have F and can draw an energy diagram with all the allowed transitions.##F(^2P) = (6,5,4,3)## and .# and#F(^2s) = (5,4).## So I can draw 6 allowed transitions.

I can even take it one step further and express the two A's in terms of the energy and vise versa.

But how exactly do I solve for A? However I twist and bend I just can't get anything solvable. Neither can I connect the given frequencies to the energy differences, have no idea what so ever how they relate, see or know no pattern.

mfb
Mentor

The transitions clearly have two different groups. It makes sense to assume that one goes to F(^2s) = 5 and one to F(^2s) = 6.
Differences between lines within a group are then differences between energy levels in ^2P. There is one difference that appears in both groups. What does that tell you?

From the figure, try to identify what frequency correspond to what transition. And while doing this, have in mind that the energy increase between each hyperfine energy level, as F increases that is. Then you can easily take differences between transitions to get energy differences between the hyperfine levels.

Then use Landé interval rule, not the formula above. With this information you can figure out how to solve the problem.

So, identify transitions -> relevant subtractions to get energy between hyperfine levels -> use land'e interval rule.

Let me know if anything wasn't clear and I'll give you better hints (rather then just repeating myself and expect you to understand, as others do...).

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From the figure, try to identify what frequency correspond to what transition. And while doing this, have in mind that the energy increase between each hyperfine energy level, as F increases that is. Then you can easily take differences between transitions to get energy differences between the hyperfine levels.

Then use Landé interval rule, not the formula above. With this information you can figure out how to solve the problem.

So, identify transitions -> relevant subtractions to get energy between hyperfine levels -> use land'e interval rule.

Let me know if anything wasn't clear and I'll give you better hints (rather then just repeating myself and expect you to understand, as others do...).

Thanks a lot! Thats very helpful! I actually forgot to use landé interval rule but following the steps it should be pretty straight forward. I starred myself blind on the equation above to solve for A.

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Dale
Mentor
2021 Award
Closed for cleanup

Edit: cleaned up and reopened.

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