Transition of Calcium Hydrogen-like Ion

[TerraForce469]

Homework Statement

A hydrogen‑like ion of calcium emits a photon with energy E = 756 eV. What transition was involved?

Homework Equations

The energy equation: E=Z^2*E_R*(1/n^2-1/n'^2)

The Attempt at a Solution

First, Z=20 for a calcium ion, and E_R is the Rydberg energy 13.6 eV. Then I plugged everything in assuming that the electron which emits the photon goes back to the ground state, i.e. n=1.

However, I get n' ≈ 1.07, which does not seem sensible to me.

Any ideas or suggestions, please? Much appreciated.

Note: How do you write down the equations in a neat format on the forums? I'm a newbie so please go easy on me!

 

[mfb]

If your result is wrong, one of the assumptions is wrong. In this case, the electron does not go back to the ground state.

Formula formatting -> LaTeX

 

[TerraForce469]

If your result is wrong, one of the assumptions is wrong. In this case, the electron does not go back to the ground state.

Formula formatting -> LaTeX

Thank you for the formula formatting guidelines!

In that case, how do I deduce the transition involved? Because otherwise the problem doesn't state much, unless I'm missing another given assumption.

 

[TerraForce469]

Attempt at the problem

Ok so I separated the transition term of the equation ##E=Z^2E_R(\frac{1}{n^2}-\frac{1}{n'^2})##: $$\frac{E}{Z^2E_R}=\frac{1}{n^2}-\frac{1}{n'^2}$$

and knowing that the right hand side's value, I started plugging in values for n for which $$\frac{1}{n'^2}= \frac{1}{n^2}-\frac{E}{Z^2E_R}$$ will still be come out positive.

I tried for a transition back to the n=2 state and got a close approximation to an integer value n'=9.

Could someone please check this for me?

 

[mfb]

Did you consider the square for n'?
I get a different result with smaller numbers.

 

[TerraForce469]

I did.

If I consider a transition back to the ground state, I get n' to be some non-integer close to 1, which does not make sense.

Did you consider the square for n'?
I get a different result with smaller numbers.

What was your result?

 

[mfb]

n=2 is right, it fits perfectly to n'=3 and not to 9.

 

[TerraForce469]

n=2 is right, it fits perfectly to n'=3 and not to 9.

Yes, you are right about that! That was my mistake.

The transition is from n=3 to n'=2. Guess for this problem they wanted you make a reasonable guess.