1. Theoretical Analysis of the Hyperfine Structure of NaK
Angela Wilkins
Advisors : Dr. Hickman of Lehigh U.
& Dr. Semak of UNC

2. Outline Molecular Spectroscopy Energy levels of NaK
Angular Momentum Coupling
Conclusions

3. Alkali Molecular Structure
Each successive orbital has a higher energy and lower energy orbitals are filled first
Alkali atoms have 1 valence electron
NaK acts like a 2 electron molecule 3d 4p
(K) 4s 3p
(Na) 3s 2p 2s 1s
Energy
Electron orbitals of an atom

4. Molecular Spectroscopy spectroscopy allows study of different
energy levels Na K
(R)
Internuclear separation

5. Experimental setup Moveable Mirror M- Mirror L- Lens M Dye Laser L L
Green Fluor. PMT
Red fluor. PMT
NaK Heat Pipe Oven
Ti-Sapphire Laser M M

6. Electronic State Notation
13D n2S+1L
•Numeric label
•S-electron spin:
2 electron molecules have parallel
(S=1, triplet) or anti-parallel (S=0,
singlet) spins
•L-orbital angular momentum along internuclear axis:
Whole integer numbers (L= S, L= P, L= D)

7. Different Electronic States

8. Energy Levels of a Diatomic Molecule
Electronic State (i.e. 13)
Vibrational levels (v)
Rotational levels (N)
Fine Structure
Hyperfine Structure

9. Energy Levels of NaK
Energy levels are labeled by the angular momentum quantum numbers: R,L,S, and I.
rotation of nuclei
R is the nuclear orbital angular momentum
L is the electronic orbital angular momentum
S is the electron spin momentum
I is the nuclear spin momentum Na K

10. Fine Structure N=+R J=N+S J=|N-S|,…, N+S
L precesses rapidly about the inter- nuclear axis,  is a component of L.
N=+R
J=N+S
J=|N-S|,…, N+S
For the triplet NaK cases, S=1,
So J= N-1, N, N+1 Na K L

11. Fine Structure Levels N = rotational angular momentum
J = total angular momentum (excluding the nuclear spin)
N=17
J=16
J=17
J=18
N=16
J=15
J=16
J=17
N=15
J=14
J=15
J=16

12. Hyperfine Structure (Includes Nuclear Spin)
N=+R
J=N+S
F=J+I
F=|J-I|,…,J+I
For 13 of NaK, I=3/2 so
F = J-3/2, J-1/2, J+1/2, J+3/2

13. Hyperfine structure N = rotational angular momentum
J=15
J=16
J=17
F=14.5
F=15.5
F=16.5
F=17.5
F=13.5
N=17
J=16
J=17
J=18
F=15.5
F=16.5
F=17.5
F=18.5
F=19.5
F=14.5
N = rotational angular momentum
J = total angular momentum (excluding the nuclear spin)
F=total angular momentum (including nuclear spin)

14. Experimental Data N=38 N=15 N=26 N=86 N=45
As N becomes larger, the spacing between the groups of peaks becomes less.

15. More Angular Momentum Coupling
F= N+S+I
Case Case 2
F= [N+S] + I F=N + [S+I]
J=N+S G=S+I
 F=J+I  F=N+G
Recall: For 13 of NaK, S=1 and I=3/2
G=|S-I|,…,S+I
  G=1/2, 3/2, 5/2

16. Energy Levels for Limiting Cases
J=16
J=17
J=18
F=15.5
F=16.5
F=17.5
F=18.5
F=19.5
F=14.5
F=15.5
F=16.5
F=17.5
F=18.5
N=17
G=5/2
G=3/2
G=1/2
F=14.5
F=13.5

17. Model Hamiltonian for NaK (3)
H = Hspin-orbit + Hrotation + Hhyperfine + Hspin-rotation
Hspin-orbit = AvLS
Hrotation= Bv [(N(N+1) - 2 ] - Dv [(N(N+1) - 2 ] 2
Hhyperfine= bIS
Hspin-rotation=  RS
The 12 energy levels are the eigenvalues of this Hamiltonian.
We adjusted Av, b, and  to fit the experimental energies.
Case Case 2
BvN >> Av >> b BvN >> b >>Av

18. Hyperfine coupling strength
Intermediate Case
F=J+I
F=N+G
J=N-1
J=N
J=N+1
G=5/2
G=3/2
G=1/2
Hyperfine coupling strength
Relative Energy
0.0
-1.0
2.0
1.0
-2.0
Case 1 limit
Case 2 limit

19. Hyperfine coupling strength
F=J+I
F=N+G
Hyperfine coupling strength
Relative Energy
Case 1 limit
Case 2 limit
0.0
-1.0
2.0
1.0
-2.0

20. Hyperfine coupling strength
F=J+I
F=N+G
Hyperfine coupling strength
Relative Energy
0.0
-1.0
2.0
1.0
-2.0
Case 1 limit
Case 2 limit

21. Hyperfine coupling strength
F=J+I
F=N+G
Hyperfine coupling strength
Relative Energy
0.0
-1.0
2.0
1.0
-2.0
Case 1 limit
Case 2 limit

22. Comparison of Experiment and Theory
Reduced Energy
Case 1 limit
Hyperfine coupling strength
Case 2 limit

23. Conclusions
The intermediate angular momentum coupling cases explain data.
The coupling scheme changes with N.
Plan to work further and continue analysis on data at N values > 86 to check agreement with limiting cases and include other electronic states.

24. Acknowledgements Dr A. Peet Hickman Dr Matthew Semak Dr. Huennekens
Laurie Sibbach & Catherine Deibel
NSF for funding

25. Transition from LS to jj coupling
Light atoms tend to exhibit LS coupling, and heavy atoms tend to exhibit jj coupling. The transition from one to the other can be seen as one goes down a column in the periodic table.
Diagram adapted from Condon and Shortley

26. Electron Transition