# Hyperfine splitting of deuterium

1. Jan 24, 2008

### Aaronse_r

[SOLVED] Hyperfine splitting of deuterium

1. The problem statement, all variables and given/known data
Calculate the wavelength of the photon emitted under a hyperfine transition in the ground state (n=1) of deuterium. Deuterium is a proton and a neutron in the nucleus, but still one electron. The spin of deuterium is 1.

2. Relevant equations

H prime=(magnetic moment)* B-field

mag moment of electron = (g-factor*e) / m_e
mag moment of deuterium = (g-factor*e) / (2m_d) g-factor for deut = 1.71

3. The attempt at a solution

I was able to solve up to the part where you get the expectation values...something like <S^2 - S(d)^2 - S(e)^2>. [the S(e) and S(d) are spin of electron and deuterium, sorry it's hard to read]

I dont know how to add the spins for the total spin vector though. My attempt at the spin states was this.

m = 1+1/2
m = 1/2
m = -1/2
m = -3/2

So S can be any of these values time hbar, and S^2 is hbar^2 *m(m+1)

Last edited: Jan 24, 2008
2. Jan 25, 2008

### malawi_glenn

Use clebsh gordan tables.

3. Jan 25, 2008

### Aaronse_r

Thank you i will try to make some sense of that thing!

4. Jan 25, 2008

### Aaronse_r

S= 3/2 and 1/2

Last edited: Jan 25, 2008
5. Jan 25, 2008

### pam

You need s.S for s= spin 1/2 and S= spin 3/2.
Since J=s+S, you get 2s.S=J^2-S^2-s^2=j(j+1)-2-3/2.