# Fine-structure constants

• John Greger
In summary, the conversation discusses the identification of frequencies and transitions in a hyperfine structure measurement in 115In (I = 9/2). The goal is to draw an energy diagram and determine the hyperfine constants using the Landé interval rule. Steps to solve the problem include identifying transitions, taking differences between transitions to get energy differences between hyperfine levels, and using the Landé interval rule to solve for the constants.

#### John Greger

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.

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?

mjc123 said:
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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Philip Land said:
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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Closed for cleanup

Edit: cleaned up and reopened.

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## 1. What is the fine-structure constant?

The fine-structure constant, also known as the Sommerfeld constant, is a dimensionless physical constant that characterizes the strength of the electromagnetic interaction between charged particles. It is denoted by the symbol α and has a value of approximately 1/137.

## 2. How is the fine-structure constant calculated?

The fine-structure constant is calculated by taking the square of the elementary charge (e), divided by Planck's constant (h) multiplied by the speed of light in a vacuum (c). This gives a value of approximately 1/137. This constant is a fundamental part of the theory of quantum electrodynamics, which describes the interactions between charged particles and electromagnetic fields.

## 3. What is the significance of the fine-structure constant?

The fine-structure constant is significant because it is a fundamental constant of nature that governs the strength of the electromagnetic interaction between particles. It is also a dimensionless constant, meaning that its value is the same in all systems of units, making it a universal constant. It is also important in understanding the structure and behavior of atoms and molecules.

## 4. Has the fine-structure constant ever been observed to change?

Although there have been some studies that suggest a possible variation in the fine-structure constant over time, these have not been conclusively proven. The current consensus among scientists is that the fine-structure constant is a constant value and does not change over time.

## 5. How does the fine-structure constant relate to other physical constants?

The fine-structure constant is related to other fundamental constants, such as the speed of light, Planck's constant, and the elementary charge. It also has connections to other physical phenomena, such as the strength of the strong nuclear force and the ratio of the masses of the electron and the proton. Understanding the fine-structure constant is crucial to our understanding of the laws of nature and the behavior of the universe.