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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. - PowerPoint PPT Presentation
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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
1s
2p2s
3s3p
4s
3d4p
Energy Electron orbitals of an atom
(Na)
(K)
Molecular Spectroscopy
spectroscopy allows study of different
energy levels
Na K
(R)
Internuclear separation
Experimental setup
M
MM
Moveable Mirror
LL
NaK Heat Pipe Oven
Ti-Sapphire Laser
Dye Laser
Green Fluor. PMT
Red fluor. PMT
M- Mirror
L- Lens
Electronic State Notation13nS+1
•Numeric label
•S-electron spin:
2 electron molecules have parallel
(S=1, triplet) or anti-parallel (S=0,
singlet) spins
•-orbital angular momentum along internuclear axis:
Whole integer numbers (
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 NaKEnergy 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
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
LNa K
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+SF=J+I
F=|J-I|,…,J+IFor 13 of NaK, I=3/2 soF = J-3/2, J-1/2, J+1/2, J+3/2
Hyperfine structure
N = rotational angular momentum
J = total angular momentum (excluding the nuclear spin)
F=total angular momentum (including nuclear spin)
F=18.5
N=16
J=15
J=16
J=17
F=14.5F=15.5F=16.5F=17.5
F=15.5F=16.5F=17.5
F=13.5F=14.5F=15.5F=16.5
N=17
J=16
J=17
J=18
F=15.5F=16.5F=17.5F=18.5
F=16.5F=17.5F=18.5F=19.5
F=14.5F=15.5F=16.5F=17.5
Experimental Data
N=15
N=45
N=38N=26
As N becomes larger, the spacing between the groups of peaks becomes less.
N=86
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
F=15.5F=16.5F=17.5F=18.5
F=17.5F=18.5
N=17
G=5/2
G=3/2
G=1/2
F=14.5F=15.5F=16.5F=17.5
F=13.5
F=18.5N=17
J=16
J=17
J=18
F=15.5F=16.5F=17.5F=18.5
F=16.5F=17.5F=18.5F=19.5
F=14.5F=15.5F=16.5F=17.5
Case 1 Case 2
Model Hamiltonian for NaK (3)H = Hspin-orbit + Hrotation + Hhyperfine + Hspin-rotation
Hspin-orbit = AvLS
Hrotation= Bv [(N(N+1) - 2 ] - Dv [(N(N+1) - 2 ] 2
Hhyperfine= bIS
Hspin-rotation= RS
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
Intermediate CaseF=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
Rel
ativ
e E
nerg
y
0.0
-1.0
2.0
1.0
-2.0Case 1 limit Case 2 limit
N=15F=J+I F=N+G
Hyperfine coupling strength
Rel
ativ
e E
nerg
y
Case 1 limit Case 2 limit
0.0
-1.0
2.0
1.0
-2.0
N=38F=J+I F=N+G
Hyperfine coupling strength
Rel
ativ
e E
nerg
y
0.0
-1.0
2.0
1.0
-2.0Case 1 limit Case 2 limit
N=86F=J+I F=N+G
Hyperfine coupling strength
Rel
ativ
e E
nerg
y
0.0
-1.0
2.0
1.0
-2.0Case 2 limitCase 1 limit
Comparison of Experiment and Theory
Hyperfine coupling strength
N=15
N=26
N=38 & 45 N=86 & 87
Red
uced
Ene
rgy
Case 1 limit Case 2 limit
Conclusions
1) The intermediate angular momentum coupling cases explain data.
2) The coupling scheme changes with N.
3) 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