# Zeeman Effect

1. Jul 17, 2013

### knowLittle

1. The problem statement, all variables and given/known data
• Consider a hydrogen atom in its ground level,
placed in a magnetic field of 0.7 T along the z axis.
(a) What is the energy difference between the spin-up
and spin-down states? (b) An experimenter wants to
excite the atom from the lower to the upper state by
sending in photons of the appropriate energy. What
energy is this? What is the wavelength? What kind of
radiation is this? (Visible? UV? etc.)

This is more a theoretical question.
2. Relevant equations

3. The attempt at a solution
a.) By theory I have read, supposedly there is no spin for s-levels. In other words, In ground state there is no spin-up and down and therefore no splitting of Energies by a magnetic field.
So, There is no energy difference, because there is no spin-up or down on the ground state.

b I guess it would be En=Eo/(n^2)

2. Jul 17, 2013

### rude man

The electron can have spins of +/- 1/2.

Use the Sommerfeld expression for the energy level of the atom, with n =1, l = 0, and so j = l +/- 1/2 = +/- 1/2. Then the delta energy between j = +/- 1/2 is your answer.

Photon energy is of course hf.

Hint: use the Rydberg constant to facilitate the math.

3. Jul 17, 2013

### knowLittle

So, in other words. At the ground state, there is still spin of the electrons, but they don't produce a splitting in the energy due to the magnetic field?

Then, a.) would be ΔE= +-1/2 But, +-1/2 what?
(1/ λ)= R( (1/1)- (1/4) ) (Lyman series)
λ=91nm (UV light)

b.) E=h*c/λ

Would this be correct?

4. Jul 17, 2013

### janhaa

In NMR for example, I would say:

$$\Delta E(proton)=g\mu B_o$$

g : g-factor
$$\mu:$$ nuclear magneton
$$B_o:$$external magnetic field

5. Jul 17, 2013

### knowLittle

I don't know what NMR or g-factor is.

Can anyone answer my previous post?

6. Jul 17, 2013

### rude man

Look at https://en.wikipedia.org/wiki/Hydrogen_atom

for the Sommerfeld energy expression. I think I told you wrong; l = 1 otherwise the expression gives zero as the energy difference between the two electron spin orientations.

So E = R{1 + α2(1/(j + 1/2) - 3/4)}
with R = Rydberg constant ~ 13.6 eV and α the fine structure constant ~ 1/137.

I myself am not sure why (or that) l = 1 in the H ground state. If l = 0 you would be right in that the spin difference gives zero energy difference. Maybe one of our experts can lend a hand.

For b) you have the correct formula.

7. Jul 17, 2013

### janhaa

And in ESR (EPR)

$$\Delta E(proton)=g\mu_b B_o$$
g : g-factor
$$\mu_b:$$ Bohr magneton$$B_o:$$external magnetic field

where $$\mu_b:$$ is related to the Zeeman effect

8. Jul 17, 2013

### knowLittle

Janhaa, I realize that you are talking about some applied spetroscopy properties, but I should be able to use "basic" concepts of Modern Physics to solve this problem. It's an introductory class to Modern Physics.

9. Jul 18, 2013

### janhaa

$$\Delta E(electron)=g\mu_b B_o$$
g : g-factor
$$\mu_b:$$ Bohr magneton$$B_o:$$external magnetic field

where $$\mu_b:$$ is related to the Zeeman effect

10. Jul 20, 2013

### knowLittle

Ok, so I have an update on the theory. I misinterpreted it.

Normal Zeeman effect doesn't take into account spin up or down of electrons. So, ground state or s-levels have no energies' splittings.

Anomalous Zeeman effect shows splitting in s-levels:
$\Delta E= \pm \mu _{B} B$
Separation of levels is $= 2 \mu_{B} B$

For part a:
So, about the problem s-levels means that l=0
Energy difference between spin up and down is
$\Delta E= \pm \mu _{B} B = \pm (5.79) 10^{-5} \dfrac {eV}{T} 0.7 T= \pm 4.053 10^{-5} eV$

In part b, I am not sure anymore.
Any help?

Last edited: Jul 20, 2013
11. Jul 20, 2013

### knowLittle

Ok. So, I guess that to excite an atom to an upper level we need
$E_{\gamma}= + 4.053 10^{-5} eV= \dfrac{hc}{\lambda}$
Solving for λ:

$\lambda =\dfrac{hc}{4.053 10^{-5} eV} = \dfrac {1.24 10^{-6} eV m} {4.053 10^{-5} eV}= .0305946213m = 30 594 621.3 nm$

What is wrong with this?

12. Jul 20, 2013

### TSny

For part (a) you are asked to find the energy difference between the spin down and spin up states. So, you need to find the difference between -4.05 x 10-5 eV and +4.05 x 10-5 eV.

In part (b) you are asked to find the energy of a photon that will cause the electron to flip from spin down to spin up. Once you have the correct value for this energy, then you can find λ using the method that you have already shown.

13. Jul 20, 2013

### knowLittle

a.) Separation of levels $(4.05- (-4.05))* 10^{-5} eV = 8.10 * 10^{-5} eV$

b.) $\lambda= 15308 * \mu m$

14. Jul 20, 2013

### TSny

I think that's right. Still need to state the type of radiation (i.e., region of the electromagnetic spectrum).

15. Jul 22, 2013

### knowLittle

Note that for part b.), I am using the energy difference. In this way, I am using energy from part a.) to solve the De Broglie relation.

16. Jul 22, 2013

### TSny

OK, radar. Sometimes called "microwaves". See this chart.

[Minor point: The relationship $E = hf = hc/\lambda$ for a photon is due to Einstein rather than de Broglie.]

Last edited: Jul 22, 2013
17. Jul 22, 2013

### knowLittle

18. Jul 22, 2013

### TSny

OK. As you can see, different people will divide the spectrum somewhat differently. The chart you are using incorporates "radar" within a broader heading of "radio waves". "Radar" or "microwaves" is more specific.

19. Jul 22, 2013

### knowLittle

I like my chart better.
Thank you so much for your help TSny!