# Energy differences

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Without hyperfine splitting:
The energy levels are calculated as follows:
E = -gm_Bm_JB_0 with B_0 = 2T, g = 1, m_B = 5.7883818012(26)×10^−5 eV·T^−1. We only have transitions if the m_j difference is 1, so we have:
deltaEwithout = abs(-gm_Bdeltam_JB_0)=m_B B_0 = 11.57676*10^-5 eV=115.768 µeV.

If a hyperfine interaction is applied with a Bhf of 10 T, we have an extra energy splitting:
1)
between m_J=-3/2 and -1/2:
Here A = µBhf/IJ = gµ_NBhf/J = µ_N10/3/2=20µ_N/3 T=3.1524512550*10^-8eV*20/3 = 0.21 µeV.

green: m_I = +1/2
deltaE = deltaEwithout + abs(Am_Idm_J)=deltawithout+A/2=115.768 µeV+0.105 µeV=115.873µeV

yellow: m_I = -1/2
deltaE = deltaEwithout – A/2 = 115.663µeV

2)
between m_J = -1/2 and m_J = 1/2:
yellow: m_I = -1/2
deltaE = deltaEwithout-abs(AdmJm_I) = deltaEwithout – A/2 = 115.663µeV
green: m_I = +1/2, zo
deltaE = 115.873 µeV

So we see two peaks, as is indicated by I=1/2, one on 115.873 µeV and one on 115.663 µeV.

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