The Polarization Potential as a Probe into Interstellar Matter

By simple arguments, it is shown how alkaline types of atoms can be seen in dense molecular clouds depending on the average temperature of such regions of the Universe. This paper predicts infrared lines of such atoms that could be seen in molecular clouds. The theory is developed in the first part and the predicted experimental results will follow. It results that the alkaline atoms modelized using the short-range polarization potential are found in quantum states


Introduction
It is established for a long time (1926) that one can model atomic structures of alkaline species adding to the Coulomb potential an attractive short-range potential: ( ) Our main purpose is to show that these corrections coming from the modification of the core structure in their neutral states lead to observable radiation energy in the infrared domain.
When one considers the polarization potential, it could be detectable in low-temperature parts of the universe that contain atoms, molecules and dust, it is known that these temperatures vary between 50 K 500 K

Dealing with Realistic Molecular Clouds Environment
We consider the classical value of the 4 1 r value as is given by Born 1960 K, whose values are: It is easy to define the polarization term for hydrogen atom, that 1 Z = , and the hydrogenic atom is obtained with: 2 Z ≥ that is:  Table 1). This implies: Thus the maximal value of R is obtained for: This quantity is a measure of the polarization potential for alkaline atoms, in units of the polarization potential for hydrogen. max R is a ratio of two energies, it is easy to transform it:

Adapting the Alkaline Structure to Cold Interstellar Matter
The purpose is to show how atomic structure effects (these atoms for which exist quantum defects [5]), could be found in clouds or HI regions.
These equations are applied to any atoms once their static polarizability: D α ′ is inserted: It is easy to consider the H atom, for the same temperature 100 K T = , and consider for such value the principal quantum number n obtained by solving: The principal quantum number for the H atom is rounded to 3 n = .
The static polarizability of the atoms that we consider are always: 9 2 D α ′ ≥ , thus for a fixed temperature, clouds T the state n of the hydrogen atom will be such: A simple rule can be set from Equation (5)  It can be generalised for any static polarizability:

Transitions to Be Seen
Interpreting the results in Figure 1 shows that could exist alkaline atoms, in their not so high or low states, there will be a difference with hydrogen in HI regions, Figure 1. Effective quantum numbers * n as a function of low temperatures T in clouds

A. de Kertanguy
where hydrogen lines exist. In fact, the abundance of elements different from hydrogen is around 1% Depending on the wavelengths seen in molecular clouds emitted by these al- for a p s → transition. If some of these atoms are in thermal equilibrium, it is possible to calculate the fraction of these atoms having the energy: Here the Z charge parameter can be defined for all neutral alkaline elements, It is possible to produce with such equilibrium distribution the number of such states. These equations need some comment, when one wants to calculate these partition functions we will use the factor β defined in [6] using: These equations are useful for the conversion of the thermal energy B k T in eV and in atomic units.
Tables of quantum defects or effective quantum numbers for such atoms or ions can be found in Topbase database: [5]. It is necessary to take into account the level distribution of alkaline atoms in such a way that the partition function is not a divergent sum, (the thing that happens when a negative sign exists in the energy expression): It is also important to take into account the n sub levels degeneracy: for each level from * n . The assumption implied here is that the po-Journal of Applied Mathematics and Physics larization potential does not affect the number of sub levels that exists for pure Coulomb states. n g with the spin two states for electron 2 e g = .
The Maxwell-Boltzmann distribution of such atoms is then:

Interstellar Matter
It is possible to use the cell radiating power for quantum processes such as absorption/emission of light by atoms. This has been done in a very exhaustive and general matter for the Ly-α line [8].
This is interesting to use this formula valid for one excited atom in an identi- The total radiating power is obtained when the oscillator strengths: of the i j → is calculated, such atomic data are now available (with a 10% agreement) in [5] and [7] these data take into account atomic structure: NZ number of protons and NE number of electrons for each atomic species Mg, Na, Li, Ca, and fortunately atomic data exists for high states of atoms

Links to Oscillator Strengths of Mg, Na, Li, Cs, K, Ca
The oscillator strength ij f and of the line strengths: ij S formulae are given using:

Special Treatment for Element K Potassium and Cs Caesium
It is possible to use data of NIST database for lines 3 , and to use the output for oscillator strengths: ik gf and ik S line strengths for known transitions as for instance (Table 7): It is possible to evaluate quantum defects s δ , and p δ , for the K element. This is impossible for the Cs element because it has so many electrons although there exist observed and identified transitions provided with the oscillator strengths ik gf [5], this enables anyhow to give an estimate of the emitted power by Cs atoms if of course, these atoms exist in the molecular cloud.
The way to get there is quite simple and easily performed using Mathematica software, the NIST database gives transitions observed and calculated. One 3 Transitions and gf atomic data exist for K element and Cs element in NIST database [5]. Journal of Applied Mathematics and Physics needs a least two transitions, with the same kinetic moment change 1 L ± , that is ns np → , these transitions are defined in Table 8. Two identified transitions for K elements suffice to evaluate the , s p δ δ quantum defects, more than two will reinforce the following equation solutions.

Links to the Surrounding Medium
Once atomic parameters are obtained, the following statistic of the emitters is possible, it has two parts: The first ( ) Here we shall use data for the number of hydrogen existing in a molecular cloud, the fact that the cosmic abundances of the elements Ca and Na are nearly the same is commonly accepted.

Emitted Power Estimate from GMC Molecular Cloud for Mg, Na, K, Li Elements
It is accepted that millimetre wave or submillimetre wave in emission from a molecular cloud, meet an optically thin medium.
It is possible to construct, in a purely theoretical way many lines with high quantum numbers ( * 5 n ≥ ), when the quantum defects are known.
The power emitted at the edge of the molecular clouds, depends on the nature of the atom (Li, Na, K, Mg), through the ij S line strength and of the related

Telescope Detection of GMC Emitters
It is admitted that the molecular clouds in infrared or near-infrared wavelengths are optically thin where these high quantum states are optically are to be found (the emitted photons of these alkaline atoms).
The receiving device, being at a distance max 640 D = light-years, that is 18 max 6.054 10 m D = × , it is necessary to define the solid angle of the molecular cloud for the observer device that is: , the angle θ is defined as follows:  max h is the maximal spatial extension of the cloud.    4 The received light emitted from the molecular cloud should be i R P , each index i, 1, 2,3, 4 i = for subsequent elements Mg, Ca, Li, Na. R is the radius of the telescope (Table 11 & Table 12).
It is assumed that for the following wavelengths in μm, the emitting medium is optically thin, it is remarkable that the photons flux is proportional to where at the maximum, max h h = , the following data give an estimate for: 4 The model GMC given in the table yields a field in the squared degree of 0.0019 nearly 500 times less than a squared degree. Journal of Applied Mathematics and Physics

Physical Data of Giant Molecular Clouds
Here are some data to be used for receiving the alkaline nS mP → transitions to an observer on Earth (or even to the future infrared space telescope JWST) of such emissions from giant molecular clouds.

Conclusions
It is shown how alkaline atoms transitions whose structures are described by these set of quantum numbers: * n n δ = − with quantum defects δ such as 0 1.9 δ ≤ ≤ and principal quantum numbers 5 9 n ≤ ≤ can exist in cold molecular clouds where it is found neutral H atoms and many molecular compounds.
Under the reasonable assumptions of an optically thin media, at the rather high wavelength, and with an abundance of the considered atoms in accordance with cosmological data, it is given for experimental device (such as a large telescope), the number of photons, by second, for each different line whose line strengths are known.
It is obvious to see in the last figure that the photon flux N t Φ Φ = is directly proportional to the length of the spatial extent of the clouds.
To the author's knowledge, these infrared lines are not yet detected but are seen in a future survey of molecular clouds, these should be coming from LEA, little excited atoms. It is probably possible that these radiating neutral atoms could be part of vortices, as the CO molecules, and then the information to track