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

Outline Molecular Spectroscopy Energy levels of NaK Angular Momentum Coupling Conclusions

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

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

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

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=0 S, L=1 P, L=2 D)

Different Electronic States

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

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

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

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

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

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)

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.

More Angular Momentum Coupling F= N+S+I Case 1 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

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

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 1 Case 2 BvN >> Av >> b BvN >> b >>Av

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

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

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

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

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

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.

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

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

Electron Transition