Calculation of the Zeeman-Fine Energies and the Spectrum with Doppler-Shift Correction of Atomic Lithium ()
1. Introduction
The investigation of hydrogen-like atoms with their low ionization potential and relative simplicity of their outer shell structure, have attracted a lot of attention. During the last two decades with the advent of laser cooling [1-4] and magnetic trapping [5], as well as spin polarization related to quantum entanglement [6], lithium and sodium have been the focus of a number of theoretical and experimental studies. Moreover with the realization of Bose Einstein condensation [7,8], they have attracted further attention.
Recently Saglam et al. [9] calculated the Zeeman-fine energy expression of hydrogen-like atoms given by:
corresponding to the eigenstates
. Here the constant 
is a characteristic for each atom and determined from the ionization energy, 
is the cyclotron angular frequency corresponding to the effective magnetic field, 
inside the atom and 
is the effective Landé g-factor, which is treated as a varying parameter. As was discussed by Saglam et al. [9] the effective magnetic field, 
can be very high so that this leads to the spin-flip energies of the order of (eV). For the case of atomic Cesium, the spin-flip energy was shown to be 1.38 eV. Saglam et al. [9] defined a dimensionless function:
which takes the form:
and depends on 
and 
directly and depends on 
indirectly as the range of 
is determined by
. They used the plots of
(as a function of
) to study the
(
) 
(
)
(
)
transitions in hydrogen-like atoms and showed that the entanglements of 
and 
states, occur at
. The aim of the present study is to calculate the Zeeman-fine energies and the spectrum of the atomic Lithium by using the above mentioned varying effective Landé g-factor method. The outline of the present study is as follows: In Section 2.1 the energy levels of Hydrogen-like atoms in the presence of a uniform magnetic field is studied. In Section 2.2 we calculate the Zeeman-fine energies of Li atom. In Section 2.3 we establish the connection between the Zeeman-fine energies, effective Landé-g factors, and the quantum flux of both photon and the electronic orbits corresponding to the entangled states. Section 2.4 gives the detailed calculation of the Zeeman-fine energies of Li atom. The calculation of the Doppler shift is given in Section 2.5. Section 3 is the conclusions.
2. Formalism
2.1. Energy Levels of Hydrogen-Like Atoms in the Presence of a Uniform Magnetic Field
As was discussed by Saglam et al. [9], when an atom is subject to a laser beam, because of the photon’s magnetic moment [10] and hence the large intrinsic magnetic field along the propagation direction [11], we will have diamagnetic and paramagnetic effects which is associated with a large magnetic field inside the atom. This field is called the effective field,
. The Landé-g factor is also replaced by the effective value, 
which is treated as a varying parameter [12,13]. With these replacements [9] the energy eigenvalues corresponding to the eigenstates 
reads:
(1)
Here the constant 
is the characteristic of each atom and determined from the ionization energy. Substituting the value of 
and 
in Equation (1), we find:
(2)
where 
is the cyclotron angular frequency corresponding to
. As was discussed by Saglam et al. [9] the effective magnetic field, 
inside the atom can be very high so that this leads to the spin flip energies at the order of a few electron volts (eV). For the case of atomic Cesium, the spin flip energy is taken to be 1.38 eV. To proceed further, following Saglam et al. [9] we define a dimensionless function, 
which is given by the relation:
(3a)
or
(3b)
which depends on 
and 
directly and depends on 
indirectly as the range of 
is determined by
. The plot of these 
functions with respect to 
gives us the possible 
values in the energy expression given by (1). At the first glance we see that the crossing of these lines correspond to integer values of 
such as:
. However in the following section we will see that for Li atom the allowed values of 
are restricted only with the three odd integers. These are:
. In the present study we will consider only 
values up to 10. We will see that in the range of (
) we get 26 different energy values which produce 325 wavelengths some of which are the same. For each 
value we also calculate the Doppler shift and find that it is equal to: 
. The Doppler shift-corrected wavelengths are in perfect agreement with the observed (NIST) values [14] for atomic Li.
2.2. Calculation of the Effective Landé G-Factors of the Zeeman-Fine Energies of Li Atom
In Li atom we have 3 electrons altogether. Therefore it will be easy to study 
as a function of
. The plots for 
and 
are given in Figures 1(a) and 2(a) respectively; For 
we have 
and hence
, so from Equation (3) we get two lines crossing at
. Therefore the first two electrons occupy the entangled state, with the energy corresponding to the crossing energies of 
and 
states, which is the
entanglement at 
(Figure 1(a)). For 
we have 
and hence
, so from Equation (3), the plots of 
gives us the diamond shaped parallelogram whose corners correspond to 
entanglement at
, 
and 
entanglements at 
and 
entanglement at 
(Figure 2(a)). Therefore the third electron (the so called 2s electron) occupies the entangled state at crossing of 
and 
states [(
) entanglements at
] which corresponds to the lowest energy, the ground state energy, for
. The plots of energy expression (2) for 
and 
are given in Figures 1(b) and 2(b) respectively. In order to find the excited states we go to the higher values of
. The plots of 
and
for 
and 
are given in Figures 3 and 4 respectively. We see that although the crossings of these lines occur at the integer values of 
such as:
, however, as far as the photonic transitions are concerned, in the following section we will see that for Li atom the allowed values of 
are restricted only with the three odd integers. These are:
. The reason for this is that: First, The energy values given by (2) are limited to the range:
. Secondly, the value of the spin-flip energy which is equal to 
for Li atom and finally, the photonic transitions occur between the points satisfying the condition: (
even integer). In passing we note that for a given
, although the maximum value of 
is equal to (
), any energy value corresponding to 
can be obtained by the values in the range: 
as well.
Therefore the 
values in this reduced zone (
) will give us all the possible energy values.
2.3. Flux Quantization Argument
Let us assume that the ground state electron [the electron occupying the (
) entangled state at
] is excited to a higher level by the absorption of a single photon. Now we ask: What is quantum flux difference between the final and initial states? Recently Saglam and Sahin [10,11] showed that, depending on its helicity photon carries an intrinsic magnetic moment and hence a quantum flux of 
where 
is the flux quantum. Therefore in the end of a one photon absorption process the quantum flux difference between the final and initial quantum states of electron must be equal to the intrinsic quantum flux of the absorbed photon which is equal to
. On the other hand Saglam et al. [15]
also calculated the quantized magnetic flux through the electronic orbits of Dirac hydrogen atom corresponding to the
. It is shown that the quantum flux is given by:
. For the present case we have the entangled states with the energies given by (1) where we have (
) which stands for
. Therefore for the present case the quantum flux through the orbits corresponding to the entangled states in hydrogen-like atoms will be given by:
(4)
We have stated that when the ground state electron is
excited to a higher level with an absorption of one photon (or more photons) the flux difference between two quantum orbits must be equal to integer multiples of (
). Therefore the difference of the flux in (4) must be equal to integer multiples of (
). In (4), since we have
, the above requirement is possible
only when the change of 
between two states is equal to an even integer number:
. (5)
We have said that in Li atom the ground state electron (2s electron) occupies the entangled state of 
and 
at
. Therefore a photonic transition occurs either to the entangled states with 
or
. In the present study we will consider only 
values up to 10. We will see that in this range (
) we get 26 different energy values which produce 325 wavelengths some of which are the same.
2.4. Detailed Calculation of the Zeeman-Fine Energies of Li Atom
From the (NIST) values, the ionization energy of Li atom is
, while the smallest amount of energy that allows a transition from the ground state to the nearest excited state(
) is equal to 
which corresponds to 6707.91 Å. In this transition the initial and the final value of 
is the same and equal to unity:
. Therefore we can write 
Therefore from (2) the energies of the initial and final states are:
(6)
and
(7)
respectively. The solutions of (6) and (7) gives us that 
and
. Substitution of these values in (2) gives the Zeeman-fine energies of Li atom:
(8)
In the present study we will take 
values in the range (
). For each value of 
substituting the values of 
and taking 
we get 26 different energy values which are given in Table 1. These 26 different energy values give us 325 wavelengths some of which are the same. The Doppler shift-corrected wavelengths are in perfect agreement with the observed (NIST) values [14] for atomic Li (Figure 5). In Table 2, we give the comparison between the observed values and the corresponding calculated values which have Doppler shift-correction as well. The Doppler shift-corrected wavelengths are in perfect agreement with the observed (NIST) values for atomic Li.
2.5. Calculation of the Dopplershift for Li Atom
The above calculations are based on the assumption that the center of mass of the Li atom is at rest, but only the outermost electron is moving. But since the experimental
results are taken from the moving Li atom, there will be a small difference coming from the Doppler shift. To calculate the Doppler shift we take two different repeated wavelengths from the (NIST) database. For example if we 
and
. The corresponding calculated values are 
and 
respectively. Now we can calculate the average Doppler shift by using the above values:
(9)
So the magnitude of the Doppler shift is found to be
Figure 5. A comparison between Doppler-shift corrected wavelengths (
) and observed values.
Table 2. Comparision between the observed and Dopplershift-corrected calculated wavelengths.
. (10)
3. Conclusions
We have calculated the effective Landé g-factors, 
, related to the Zeeman-fine energies and the spectrum of the atomic Lithium (Li) by using the varying effective Landé g-factor method. It is shown the allowed values of 
are restricted only with three odd integers (
) and this result is independent of the limit of the principle quantum number,
. In the present study we take the principle quantum number in the range; (
). For this range we find 26 different energy values and 325 wavelengths some of which are the same. The present calculations are based on the assumption that the center of mass of the Li atom is at rest, but only the outermost electron is moving. But since the experimental results are taken from the moving Li atom, there will be a small difference coming from the Doppler shift. The Doppler shift is found to be
. The Doppler shift-corrected wavelengths are in perfect agreement with the observed (NIST) values for atomic Li. The present results suggest new wavelengths such as
, 
and 
should be observed. Applications of the above treatment can give access to the production of new laser lights from atomic Lithium as well. Extending the range of 
to (
) will allow us to calculate all the observed wavelengths for Li atom. A more detailed study will be presented in the future.