Hyperfine Structure of NaK (13)
Angela Wilkins
Department of Physics, University of Northern Colorado
Advised by:
A. P. Hickman
Department of Physics, Lehigh University
Mathew Semak
Department of Physics, University of Northern Colorado
Abstract
The hyperfine structure of the 13 excited state of NaK was analyzed, using
spectroscopic data collected by Prof. Huennekens’ group at Lehigh University. A 12 12
Hamiltonian matrix was constructed that included terms for rotational energy, centrifugal
distortion, spin-orbit, spin-spin, and spin-rotation energy. Constants Bv and Dv describing
the rotational energy levels have been determined previously by Prof. Huenenkens’
group; in the present work a least squares fitting program was used to determine values of
the spin-orbit constants Av, spin rotation constant , and the constant b for the Fermi
contact term in the nuclear spin interaction that best fit about 1200 experimentally
measured energy levels. The analysis shows that the data illustrate a transition between
Hund’s angular momentum coupling cases bJ and bS. Low values of the rotational
quantum number N(~15) correspond to bJ, and higher values N(~45) approach the bS
case. A theory based on the techniques of angular momentum algebra enables one to
calculate the smooth transition between the limiting cases and to match closely the
behavior of the data. Further analysis of recent experimental data for N~87 will be
required.
Theoretical Analysis of the Hyperfine Structure of 13 NaK
Introduction:
I began this project last summer at Lehigh University under the guidance of Dr.
A. Peet Hickman. Dr. Hickman and the University of Northern Colorado have allowed
me to continue this research as my senior thesis. We worked on the theoretical analysis
of the hyperfine structure of the 13 of the diatomic molecule Sodium-Potassium. Our
purpose was to take previously found data and fit it empirically to find the spin-orbit,
hyperfine, and spin-rotation coupling constants. We also investigated the transition
between different Hund’s coupling cases as the rotational number increases.
The molecule NaK is made up of two alkali atoms. The alkali atoms each have
one valence electron, which causes NaK molecule to act like two-electron system.
Molecular spectroscopy allows us to analyze the different energy levels of a molecule.
Dr. Huennekens and his atomic group at Lehigh University took the data that was used in
the analysis. They used perturbation-facilitated optical-optical double resonance
technique (PFOODR) to study the energy levels of the molecule. Data was taken at
varying vibrational numbers (3v33) and rotational numbers 15, 26, 38, and 45. About
1200 experimentally measured energy levels were included in the analysis.
Theory:
I. Born-Oppenheimer approximation
The time independent Schrödinger’s equation can be used to calculate the
electronic energy states of atoms. With it, we can calculate how the wave function varies
as a function of distance from the nucleus to be able to describe the position of the
electron. The time independent Schrödinger’s equation is stated in equation (1).
H= E
(1)
where H is the Hamiltonian operator, E represents the energy and is the eigenvalue of H.
 is the wave function or eigenfunction. Unfortunately, Schrödinger’s equation can only
be solved exactly for a one-electron system. In order to solve the equation for any system
larger than an atom with one electron, various approximations need to be made. One of
the approximations used is the Born-Oppenheimer approximation. It separates electron
and nuclear motion based on the idea that nuclear mass is much larger than the electron
mass and, therefore, the nuclei are almost fixed particles. Electrons react instanteously to
changes in the position of the nuclei. This assumption allows us to construct a potential
energy curve showing the dependence of the energy on the length of the inter-nuclear
axis, R, of a diatomic molecule (see figure 1). It defines the electronic energy as a
function of distance between the two atoms in the molecule. Re is the equilibrium
distance and De is the dissociation energy.
Figure 1. Potential well
The Hamiltonian can be separated into two parts as seen in equation (2):
H= Ho+ H’
(2)
where Ho represents the electronic Hamiltonian and H’ represents the smaller interactions
in the diatomic molecule. Some of the smaller interactions, including the vibrational
energy and rotational energy, will be discussed later in this paper. The electronic
Hamiltonian operator consists Tnuclear, Telectron and V.
Ho=Tnuclear + Telectron + V
(3)
In equation (3), Tnuclear represents the nuclear kinetic energy operator of the molecule.
The Born-Oppenheimer Approximation allows us to neglect the nuclear kinetic energy
operator. Telectron represents the electronic kinetic energy operator. V is the Coulomb
potential energy operator of the molecule. A complete derivation of the Born-
Oppenheimer Approximation is beyond the scope of this discussion. Please refer to
Spectra of Atoms and Molecules by Peter F. Bernath for further reading.
H’ represents the smaller interactions in the diatomic molecule. Eo are the
eigenvalues of the operator H’ (shown in formula (4)). The total energy due to the smaller
interactions, Eo, can be written as the sum of the energies of the smaller interactions
(vibrational energy, rotational energy…), as shown in formula (6).
H’= Eo
(4)
H’ = Hvibrational + Hrotation + + Hspin-rotation + Hhyperfine + Hspin-rotation +…
(5)
Eo = G(v) + Bv [ N(N+1) - 2 ] - Dv [ N(N+1) - 2 ]2 +…
(6)
G(v) represents the vibrational energy which has a dependence on v, the vibrational
quantum number (v= 0, 1, 2…). Bv [ N(N+1) - 2 ] represents the rotational energy and
Dv [ N(N+1) - 2 ]2 represents the centrifugal distortion correction to the rotational
energy. Bv and Dv are empirically found constants with a dependence on vibrational
number, v. The dependence of vibrational energy is due to as the vibrational level
increases the internuclear separation increases, which would cause the inertia of the
molecule to be different. A change in the inertia affects the rotational energy. Bv can be
written as follows:
Bv = B0 +B1(v + 0.5) +B2(v + 0.5)2 + …
(7)
where B0, B1, B2, … are empirically found constants. Dv can be written as follows:
Dv = D0 +D1(v + 0.5) +D2(v + 0.5)2 + …
(8)
where D0, D1, D2, … are empirically found constants.
Smaller interactions that contribute to the total energy must be considered. The
spin-orbit interaction, which is due to the coupling between the electronic orbital angular
momentum and the electron spin angular momentum, causes the fine structure. The spinorbit coupling is written as:
Hspin-orbit = AvLS
(9)
where Av is:
Av = A0 +A1(v + 0.5) +A2(v + 0.5)2 + …
(10)
where A0, A1, A2, … are empirically found constants. Then there is the interaction
between the nuclear spin and the electron spin angular momenta which causes the
hyperfine structure. The hyperfine coupling is written as follows:
Hhyperfine = bIS
(11)
Finally there is the spin-rotation interaction between the nuclear orbital angular
momentum and the electron spin angular momentum. The spin-rotation coupling is
written as shown in formula (12).
Hspin-rotation = RS
(12)
II. Energy Levels of a Diatomic Molecule
Energy levels of a molecule can be represented by angular momentum quantum
numbers. R is the nuclear orbital angular momentum. Nuclear orbital angular
momentum is caused by the rotation of the two atoms around one another. L is the
electronic orbital angular momentum. S is the electron spin angular momentum. I is the
nuclear spin angular momentum.
In molecular physics, electronic states have the notation 2S+1.  is the
component or the projection of the electronic orbital angular momentum on the internuclear axis. The electric field due to the nuclei couples the orbital angular momentum
of the electron to the inter-nuclear axis and only the component of momentum around
that axis is well defined. If =0 then it is a  state. If =1 then it is a  state. If =2
then it is a  state. The subject of this paper will be the  state.
Since NaK is an alkali diatomic molecule there are two electrons determining the
total electron spin. The spin of these two electrons will either be parallel or anti-parallel.
If anti-parallel, the electron spin is equal to 0 and is referred to as a singlet state. If
parallel, the electron spin is equal to 1 and is referred to as a triplet state.
In the previous section, we discussed different contributions to the total energy of
a molecule. Now we will examine the energy levels pictorially using the potential well.
Diatomic molecules have several different electronic states. These electronic states can be
illustrated by use of potential curves. Each electronic state has several different
vibrational levels. The vibrational levels are labeled with the quantum number v. At low
v the vibrational energy levels are evenly spaced, but as v increases the spacing between
the energy level decreases. Each vibrational level has several different rotational levels.
Rotational levels are represented by quantum number N. There is a splitting of the
rotational levels, which is referred to as the fine structure. The fine structure is due to the
interaction between the electronic orbital angular momentum and the spin angular
momentum. If S=0 (singlet state), then there is no fine structure. There is also a splitting
of the fine structure called the hyperfine structure. The hyperfine structure is due to the
interaction between the spin angular momentum and the nuclear spin angular momentum.
Figure 2 shows these different levels.
Vibrational Levels
Hyperfine Structure
SSSSSSSSSSSStru
ctureStructure
Fine Structure
Rotational
Levels
Internuclear axis
Figure 2. Energy levels of a molecule: Electronic State (i.e.13)
III. Fine Structure and Hyperfine Structure
There are four sources of angular momentum in a diatomic molecule. They are the
electron angular momentum spin (S), the electronic orbital angular momenta (L), the
nuclear orbital angular momenta (R), and the nuclear spin angular momentum (I). There
are interactions that couple these different momenta together various ways. In the 1920s,
Hund attempted to give order to the discussion of all these possible coupling schemes by
focusing attention on five basic types of coupling referred to as Hund coupling cases a, b,
c, and d. Real molecules do not have a structure that corresponds to just one of the Hund
cases, but most of the time one coupling scheme describes the structure more closely than
any other. In the case of the 13 state of NaK, case b can explain the spectra. So this
discussion will focus only on case b and in particular sub-cases bJ and bS. Coupling
scheme b for fine structure is shown figure 3.
First we will discuss the fine structure. L, the electron orbital angular momentum,
precesses rapidly about the inter-nuclear axis.  is the component, or the projection, of
the electronic orbital angular momentum on the inter-nuclear axis.
Figure 3. Hund coupling case b
Na
K
L
 and R, the nuclear orbital angular momentum, couple to form N, the rotational
angular momentum (N=+R). N and S, the electron spin angular momentum, couple to
form J the total angular momentum excluding nuclear spin (J=N+S). Quantum number J
is equal to |N-S|,…, N+S in integer steps. In the case of a triplet state the S=1, J will
equal N-1, N, and N+1. This interaction between the rotational angular momentum and
the electron spin angular momentum is referred to as the fine structure. At each
rotational energy level there will be three fine structure levels.
The hyperfine structure takes in account the nuclear spin. J and I, the nuclear spin
angular momentum, couple to form F, the total angular momentum including nuclear spin
(F=J+I). Quantum number F will be F=|J-I|, …, J+I in integer steps. In the case of a 13Δ
state the nuclear spin is 3/2, so F will equal J-3/2, J-1/2, J+1/2 and J+3/2. At each fine
structure energy level there will be four hyperfine structure energy levels. This coupling
scheme is shown in figure 4 and is referred to as Hund case bJ.
Figure 4. Hund coupling case bJ
Real molecules do not have a structure that corresponds completely to one of the
Hund cases, although normally one scheme describes the structure more closely than
another. In most cases, some of the characteristics of the other cases will be apparent in
the spectrum of the molecule. The tendency of one scheme to be contaminated by another
is called decoupling of the momenta. Decoupling often increases as N, the rotational
number, increases, because the electrons start to be unable follow the nuclear motion. In
the case of the 13 NaK, at higher N coupling scheme bS begins to apply (refer to figure
5).
Figure 5. Hund coupling case bS
In Hund coupling case bS, again  and R couple to form N, the rotational angular
momentum (N=+R). J is no longer a good quantum number in this coupling scheme. S,
the electron spin angular momentum, and I, nuclear spin angular momentum couple to
form G, the total spin angular momenta, the total spin angular momentum (G=I+S).
Quantum number G is equal to |I-S|,…, I+S in integer steps. In the case of a triplet state
the S=1 and I =3/2, G will equal 1/2, 3/2, and 5/2. ). N and G, the nuclear spin angular
momentum, couple to form F the total angular momentum including nuclear spin
(F=N+G). In the case of a G=1/2, F will equal N-1/2 and N+1/2. At G=1/2, there will be
two hyperfine structure energy levels. At G=3/2, there will be four hyperfine structure
energy levels and G=5/2 will have six hyperfine structure energy levels.
Figures 6 and 7 show the differences between the energy levels of the Hund
coupling cases bJ and bS. In the case of Hund coupling case bJ, Figure 6 shows at each
rotational level there is splitting into three fine structure energy levels. Each fine structure
energy level has four hyperfine structure energy levels, which give a total of twelve
energy levels for each rotational energy level.
N=17
J=16
F=14.5
F=15.5
F=16.5
F=17.5
J=17
F=15.5
F=16.5
F=17.5
F=18.5
J=18
F=16.5
F=17.5
F=18.5
F=19.5
Figure 6. Hund coupling case bJ
Figure 7 shows the splitting of energy levels in the case of Hund coupling case
bS. Again at each rotational energy level there is three fine structure levels, and twelve
hyperfine structure energy levels. The difference lies in how the energy levels split off in
groups of peaks of two, four, and six instead three groups of four shown on figure 6.
G=5/2
N=17
G=3/2
G=1/2
F=13.5
F=14.5
F=15.5
F=16.5
F=17.5
F=18.5
F=15.5
F=16.5
F=17.5
F=18.5
F=17.5
F=18.5
Figure 7. Hund coupling case bS
Figures 8 through 11 are examples of the data taken by Dr. Huennekens group at
various values of rotational number. In figure 8 and 9, the graph shows the fine and
hyperfine structure at N=15 and N=26. The center peak is actually four peaks, but due to
the closeness of peaks can’t be resolved experimentally. Figure 10 and 11 begin to show
the change in coupling scheme.
N=15
N=26
J=15
J=26
J=14
J=16
Figure 8. N=15 data
J=27
J=25
J=25
Figure 9. N=26 data
N=38
J=45
N=45
J=38
J=46
J=39
J=44
J=37
Figure 10. N=38 data
Figure 11. N=45 data
Analysis and Conclusions:
A model Hamiltonian for NaK (3) was used to analyze the data taken by Dr.
Huennekens and his group. The Hamiltonian is a 1212 matrix based on formulas (13)
through (17). The matrix is diagonalised to give twelve eigenvalues. The twelve energy
levels of the spectra are the eigenvalues of this Hamiltonian. For the sake of clarity,
earlier relations are regrouped as follows:
H = Hspin-orbit + Hrotation + Hhyperfine + Hspin-rotation
(13)
Hspin-orbit = AvLS
(14)
Hrotation = Bv[ N(N+1) - 2] - Dv[ N(N+1) - 2 ]2
(15)
Hhyperfine = bIS
(16)
Hspin-rotation = RS
(17)
Dr. Huennekens and his group previously determined Bv and Dv. Table 1 shows the
values used for Bv and Dv.
Table 1
Constant
Value
B0
0.0773826E-5
B1
-5.0162E-4
B2
4.5E-7
B3
-1.920E-7
D0
-2.30E-7
D1
1.9E-9
D2
-1.8E-10
We adjusted Av, b, and  to fit the experimental energies. Av is the spin-orbit
constant, b is the hyperfine constant, and  is the spin rotation constant. Table 2 shows the
values found for Av, b, and . These results are based on about 1200 experimentally
measured energy levels.
Constant
Value
A0
-2.475062E-1
A1
4.587068E-4
A2
-4.674143E-5
b
0.0126745

8.317761E-5
Table 2
The data can be explained by the intermediate angular momentum coupling cases
bJ and bS. The coupling scheme changes with rotational number. We plan to continue
analysis on data at rotation number (N) higher than 86 to check agreement with limiting
cases and include other electronic states.
Figure 12 shows the experimental data on the theoretical curves given by the
Hamiltonian. The data corresponding to N=86 and N=87 is included on the graph to show
the agreement with theory but was not included in the analysis.
F=J+I
F=N+G
GG
N=38 & 45
N=15 N=26
N=86 & 87
G=5/2
J=N-1
Reduced
Energy
J=N
G=3/2
G=1/2
J=N+1
Case 1 limit
Hyperfine coupling strength
Figure 12. Comparison of Experiment and Theory
Case 2 limit
Acknowledgements
I would like to thank Dr. Hickman for involving me in this project and for the continuous
guidance and mentoring over the last year. I also thank Dr. Semak for his patience and
support. I would like to acknowledge Dr. Huennekens and his atomic group. Also Laurie
Sibbach wrote a wonderful senior thesis that has been invaluable resource for me.
1. G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic
Molecules (Krieger, Malabar, 1989)
2. P. F. Bernath, Spectra of Atoms and Molecules (Oxford University Press, New
York, 1995)
3. L.L. Sibbach, Experimental Studies of the NaK Triplet States, Undergraduate
Honors Thesis, Moravian College, 2001, unpublished.