Simplified Step-by-Step Nonlinear Static Program Investigating Equilibrium Conditions of Electrons in Atom and Ionization Energies : Case Study on Argon

For investigation of equilibrium conditions of electrons in an atom, and Ionization Energies of Elements, a simplified deterministic static model is proposed. The electrons are initially uniformly and sparsely arranged on the outer surface of nucleus. Then, by taking into account the nucleus-electron interaction (attractive and repulsive) and the mutual electron-electron repulsions, and by a simple step-by-step nonlinear static analysis program, all the electrons are found to equilibrate on the outer surface of the same sphere, which is concentric and larger than nucleus. In a second stage, starting from an equilibrium sphere of electrons, one of the electrons is subjected to gradual forced removal, radially and outwards with respect to nucleus. Within each removal step, the produced work increment is determined and the increments are summed. When no more significant attraction is exerted by nucleus to removed electron, the total work gives the Ionization Energy. After removing of single electron, the remaining electrons fall on a lower shell, that is, they equilibrate on the outer surface of a smaller concentric sphere. For nucleus-electron interaction, an L-J (Lennard-Jones) type curve, attractive and repulsive, is adopted. When the parameter of this curve is n > 1.0, the Ionization Energy exhibits an upper bound. As parameter n increases from 1.0 up to 2.0, the attractive potential of L-J curve is gradually weakened. The proposed How to cite this paper: Papadopoulos, P.G., Koutitas, C.G., Dimitropoulos, Y.N. and Aifantis, E.C. (2018) Simplified Step-by-Step Nonlinear Static Program Investigating Equilibrium Conditions of Electrons in Atom and Ionization Energies: Case Study on Argon. Open Journal of Physical Chemistry, 8, 33-56. https://doi.org/10.4236/ojpc.2018.82003 Received: March 29, 2018 Accepted: May 13, 2018 Published: May 16, 2018 Copyright © 2018 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access P. G. Papadopoulos et al. DOI: 10.4236/ojpc.2018.82003 34 Open Journal of Physical Chemistry model is applied on Argon. It is observed that, as the number of electrons increases, the radius of equilibrium sphere increases, too, whereas the attractive nucleus-electron potential is reduced; thus the Ionization Energy is reduced, too. Particularly, as the number of electrons and the radius of equilibrium sphere exceed some critical values, the above two last quantities exhibit abrupt falls. A regular polyhedron is revealed, which can accommodate Elements up to atomic number Z = 146, that is 28 more than Z = 118 of existing last Element, as guide for initial locations of electrons in the above first program.


Introduction
Aim of present work is to develop a simplified deterministic static model investigating the equilibrium conditions of electrons in an atom and the Ionization Energies of the Elements [1] [2].
The concept of Quantum Mechanics, used in Computational Chemistry, is stochastic; so, it is accurate, but complicated [3]- [8].Recently, efforts have been made to use alternatively the Finite Element Method [9] [10].For the moment, the proposed F.E.M models seem somewhat complicated, too.
It is recognized that a symmetric deterministic model can give results similar to those of a corresponding stochastic model [11] [12].Such a symmetric deterministic model is proposed here.
The electron-electron electrostatic repulsion and the nucleus-electron interaction (attractive and repulsive), described by a Lennard-Jones type curve [13], are taken into account.And a simple step-by-step nonlinear static analysis algorithm is used [14].
Main finding of the proposed model is that all the electrons equilibrate on the outer surface of the same sphere, concentric and larger than nucleus.When one electron is removed from an equilibrium sphere, the remaining electrons fall to a lower shell, that is, they equilibrate on the outer surface of a sphere concentric and smaller than the previous equilibrium sphere and larger than nucleus.
The proposed model is applied on Argon [15], the noble gas of the third Period of Periodic Table [16], because, only for the first three Periods, complete data exist for Atomic Radii and Ionization Energies of Elements [17] [18] [19], for calibration and comparison.
Based on the results of the Application, observations are made on the variations, compared to each other, of main quantities of present problem: number of electrons, radius of equilibrium sphere of electrons, attractive potential of Lennard-Jones curve, Ionization Energy.

Estimation of Equilibrium Positions of Electrons by Use of a Step-by-Step Nonlinear Static Program
The surroundings of an atom are assumed as a dielectric medium, where the electrostatic laws hold.The nucleus is assumed motionless.
In order to find the equilibrium positions of electrons in an atom, the electrons are initially located on the outer surface (assumed spherical) of the nucleus, as shown in Figure 1(a), by using as guide a rhombic dodecahedron [20], which will be described in Section 2.5.The electrons are initially uniformly and sparsely arranged, so that to avoid very small distances between neighboring electrons, which create numerical problems.
Then, by taking into account nucleus-electron interactions (attractive and repulsive), described by a Lennard-Jones curve [13], which will be described in Section 2.4, and mutual electron-electron repulsions, as shown in Figure 1(b), and by using a simple and short computer program [14], described by the flow-chart of Figure 2, we find that all the electrons reach in equilibrium on the outer surface of the same sphere, concentric and larger than nucleus, as shown in Figure 1(c).
According to the flow-chart of Figure 2, the proposed computer program of step-by-step nonlinear static analysis for finding the equilibrium positions of electrons, in an atom, is as follows: First, the general input data are read, which are: the atomic number Z of the Element, equal to the number of its protons [1], the nuclear radius, given by the formula , where 37 pm H r = nuclear radius of hydrogen [1], which is reasonable for assumption of homogeneous spherical nuclei.The number ν of electrons and the steplength Δu of the algorithm, e.g.Δu = 0.001 pm, for which information will be given in Section 2.4.
Then, the initial locations { }, 1, , of the electrons are read.In every step of the algorithm, as shown in Figure 2, for every electron i, first its distance r, from nucleus center O is determined and, by calling subroutine NUCLEL, the force i F (attractive or repulsive), by which the nucleus acts on the electron, is found.
Then, the distance ik r of electron i from every other electron k is determined and, by calling the subroutine ELEL, the repulsive force ik F , by which electron k acts on electron i is found.
The forces ik F , from all the other electrons k, on the electron i are summed to its force i F from nucleus and the resultant force i F on any electron i is determined.
If the maximum absolute value of the out-of-balance forces of all the electrons is greater than the desired accuracy, e.g. if max 0.01 mdyn , as in the program of [14], and we go to the next step of the algorithm.As the steplength u ∆ of the algorithm is very small, e.g.0.001 pm u ∆ = , millions of steps of the algorithm are performed; however, this happens in a few seconds of computing time.

Estimation of Ionization Energy
A second computer program of step-by-step nonlinear static analysis is proposed, for estimation of Ionization Energies, which can be connected to the first program of previous Section 2.1, as a continuation of it.
This second program starts from the electrons being located on the outer surface of an equilibrium sphere, as shown in Figure 3(a), and one of these electrons is subjected to gradual forced displacement, radially and outwards with respect to center O of nucleus.In reality, the displacement increment u ∆ , of this single electron, is very small; however, in Figure 3(a), it is presented with expanded scale, so that to become visible.After every forced displacement increment of this electron, we wait for the remaining electrons to equilibrate on the outer surface of a slightly smaller concentric sphere, by performing, again for them, the first program described in Section 2.1.
Within each forced displacement increment, the produced work increment is easily determined according to Figure 3 ← + ∆ , to obtain the total work E. As this single electron is gradually removed, the attractive force F diminishes and tends to zero, 0 F → .So, 0 E ∆ → , too, thus the total work = ∑ no more signifi- cantly increases and tends to an upper bound u E , which is the Ionization Energy u IE E = and the procedure is interrupted.After the removal of this single electron, the remaining electrons equilibrate on a lower shell, that is, on the outer surface of a sphere smaller than the previous equilibrium sphere, but larger than nucleus, as shown in Figure 3(b).
In Figure 3(c), is shown how the total work E gradually increases and tends asymptotically to an upper bound u E , which is the Ionization Energy, This is usually achieved for 100 , where r is the distance of removed electron from the center O of nucleus and n r is the radius of nucleus sphere.

Electrostatic Forces in the Atom. Mutual Repulsive Forces in a Couple of Electrons
The mutual repulsive force F, in a couple of electrons, is given by the following formula, with respect to the distance r between the two electrons, where the permittivity constant is The derivative of function F(r), with respect to r, according to Equation ( 1), which can be called stiffness K, is The definite integral of function F(r), from 1 r up to ∞, is

Lennard-Jones Curve for Nucleus-Electron Interaction (Attractive and Repulsive)
For the nucleus-electron interaction (attractive and repulsive), a Lennard-Jones function [13] is adopted, ( ) where n x r r = , r distance of electron from nucleus center and n r nuclear ra- dius.The first above term is repulsive and the second one attractive.
In Figure 5 The complete equation, giving the force F with respect to the distance r, is represented by the curve of Figure 6, where the maximum value of ( )   , n = 1.057.
The derivative dF/dr of the above function F(r), which can be called stiffness K, is also represented in Figure 6.
By observing the maximum values of F, in Figure 4 and Figure 6, maxF = 6.680 and 11.035 mdyn, respectively, it is obtained that an accuracy of 0.01 mdyn is sufficient in calculations.Also, by observing the maximum stiffnesses maxK = 0.2273 and 0.4811 mdyn/pm, in the same Figure 4 & Figure 6, we can choose a steplength Δu, for the step-by-step nonlinear static analysis algorithm of Section Open Journal of Physical Chemistry So, a steplength Δu = 0.02 pm is sufficiently small.However, an even smaller steplength Δu can be used, up to Δu = 0.001 pm, for more accuracy, because millions of steps of the proposed algorithm are performed in a few seconds of computing time.
The definite integral of the L-J function ( ) which, for n > 1.0, exhibits an upper bound So, in the specific example under consideration, in the last 18 th IE (Ionization Energy) of Argon, with only one electron, for the removal of this one electron in every atom of Argon, from n r up to infinity, for1.0 mole of Element Argon, with the accurate value of parameter n = 1.05727, we have And the upper bound of produced work u E , that is the Ionization Energy, is that is, it agrees accurately with the value of database [19].

Rhombic Dodecahedron as Guide for Initial Locations of Electrons on Outer Spherical Surface of Nucleus
As mentioned in Section 2.1, in the proposed step-by-step program of nonlinear static analysis, a rhombic dodecahedron [20] is used as a guide, in order to determine the initial locations of the electrons on the outer spherical surface of nucleus, so that the electrons be uniformly and sparsely arranged and avoid very small distances between neighboring electrons, which create numerical problems.
In Figure 7, is shown how a rhombic dodecahedron is formed from a regular octahedron and a cube.In Figure 8, the plan and elevation of rhombic dodecahedron are shown and the order by which, one-by-one, the electrons are added, first to the 6 vertices of regular octahedron, then to the 8 vertices of cube, projected on the circumscribed sphere and the centers of 12 faces of rhombic dodecahedron, coinciding with the middles of edges of regular octahedron, also projected on the outer surface of circumscribed sphere.Attention is paid to the location of the fourth electron, so that not to lie on the same plane with the first three electrons.
In order to extend the use of rhombic dodecahedron to Elements with higher atomic numbers, Z > 26, we can also use the 12 × 4/2 = 24 middles of edges, and the 12 × 4 = 48 middles of distances of every face center from the four vertices of Open Journal of Physical Chemistry corresponding rhomb.All, projected on the outer surface of circumscribed sphere, can be used as guide for initial electrons locations.
So, according to the above, the total number of electrons, which can be accommodated by the rhombic dodecahedron, is 6 8 12 24 48 98 20 less than the 118 Electrons of the last existing Element.
In the present work, the applications will not exceed the 18 electrons.So, the rhombic dodecahedron, which is very simple, is enough here and will be used as guide for the initial locations of electrons.

New Regular Polyhedron Accommodating Elements with High Atomic Number up to Z = 146
In order to accommodate Elements with very high atomic number Z > 98, another regular polyhedron [21] is revealed, which is obtained by the projections of the 8 cube vertices on the circumscribed sphere of rhombic dodecahedron.
The plan view of this regular polyhedron is shown in Figure 9 The above revealed new regular polyhedron has the 6 vertices of regular octahedron and the 8 vertices of cube, totally 6 + 8 = 14 vertices, 6 × 4 = 24 triangular faces and 24 × 3/2 = 36 edges.Vertices, gravity centers of triangular faces and middles of edges, all projected on the circumscribed sphere, can be used as guide for initial electrons locations.
Also, the 24 × 3 = 72 middles of distances of gravity centers of triangular faces from corresponding triangle vertices, projected on the circumscribed sphere, can be used.
According to the above, the maximum number of electrons, which can be accommodated by the new regular polyhedron, is

Case Study on Argon
The proposed model, for finding equilibrium spheres of electrons, in an atom, and estimating Ionization Energies, will be applied on the Element Ar (Argon) [15], the noble gas of the third Period of Elements [16].The Argon has been chosen, because complete data for Atomic Radii and Ionization Energies, for calibration and comparison, exist only for the three first Periods of the Periodic Table [17] [18] [19].
The Element Argon has atomic number Z = 18, that is, its nucleus has 18 Open Journal of Physical Chemistry In studying the IEs (Ionization Energies) of the Argon, the inverse order will be followed, that is from the last 18 th IE, with only one electron, by gradually adding one-by-one the electrons, up to the 1st IE, with 18 electrons.

Investigation of the Two Last Ionization Energies by Hand Calculator
The two last Ionization Energies of Argon, the 18th and 17th IE, can be investigated by a hand calculator having the option of fractional exponent, on the basis of Figure 11  previous Equation ( 6) where Z = 18 atomic number of Argon, according to previous Equation ( 5).
The database [19] gives, for the last 18th IE of Argon, the value 427,066 kJoules/mole.
By substituting the above values in Equation ( 7), is obtained ( )  12, the value of parameter n, of L-J curve, is obtained graphically, for the last 18th IE of Argon, which is n = 1.05727.In the diagram n n ϕ − of Figure 12(a), are found the values of parameter n of L-J curve, for the last IE of 12 Elements of 3 first Periods, from Argon (Z = 18) up to Nitrogen (Z = 7), all in the same way as for Argon, with help of database [19].The same diagram n n ϕ − is continued to Figure 12(b), with contracted n scale, where the values of parameter n of L-J curve, for the last IEs of the remaining 6 Elements of 3 first Periods, from Carbon (Z = 6) up to Hydrogen (Z = 1), are found in the same way.
Based on Figure 11(b), we can estimate the 17th IE of Argon, too, by subtracting, from 18th IE, the negative work produced by the separation of the two electrons, which is very small, as follows, according to Equation (2) Open Journal of Physical Chemistry Thus, for the 17th IE of Argon, the value 427,066 − 708.9 = 426,357 kJoules/mole results.The database [19] gives a somewhat smaller value, 17th IE = 397,605 kJoules/mole for Argon, which is, however close to the last 18th IE = 427,066 kJoules/mole, too.
By substituting the value of parameter n = 1.05727 of L-J function, found for the last 18th IE of Argon, in the Equation (4) of definite integral of L-J function ( ) ( ) ( ) we find how the work produced by the removal of a single electron in every atom, referred to 1.0 mole of the Element, varies, as the electrons are gradually removed from the atoms. (

Running of All the Ionizations Energies of Argon by the Proposed Step-by-Step Nonlinear Static Program
All the Ionization Energies of Argon, from last 18th IE, with 1 electron, up to first IE, with 18 electrons, have run by the proposed two step-by-step nonlinear static analysis programs of Sections 2.1, 2.2, and the results are represented in Figure 14 up to 16, and in Table 1.
In Figure 14, according to the results of first program, the radius of equilibrium sphere varies from 1 97.0 pm r = , which is the nuclear radius (18th IE with 1 electron), up to 122.8 pm, which corresponds to 1st IE, with 18 electrons ( ) ; so, this is the atomic radius predicted by the first program.Also, an intermediate radius is shown, with value 110.4 pm, which corresponds to 10th IE with 9 electrons.More details for radii of equilibrium spheres of electrons can be found in Table 1.15, that between 9th and 8th IE, that is for number of electrons ν between 10 and 11, an abrupt fall occurs, in these curves, with the value of u IE E = falling from about 40,000 to about 14,000 kJoules/mole.
In Table 1, for all the Ionization Energies, from last 18th IE with only 1 electron, up to 1st IE with 18 electrons, the corresponding values of the following quantities are given, obtained by the proposed two step-by-step programs of nonlinear static analysis and by the database [19].
1) Radius r ν , for 1, ,18 ν =  , of the equilibrium sphere of electrons, ob- tained by the first program, increasing gradually from 97.0 up to 122.8 pm.[19], as well as of proposed model, occur, from about 40,000 to about 14,000 kJoules/mole.
On the basis of three diagrams of Figure 16, the following can be observed: As the number ν of electrons increases, these, by their mutual repulsive forces, are pushed to larger equilibrium spheres, which has as consequence the wea-Open Journal of Physical Chemistry kening of attractive potential of the nucleus-electron interaction, manifested by an increase of parameter n of L-J curve, as mentioned in Section 2.4, see Figure 5. So, the IEs (Ionization Energies) values are reduced, the electrons are more loosely connected to the nucleus and can be more easily removed.
Particularly, when the number ν of electrons and the radius r of equilibrium sphere exceed some critical values, here ν = 10 and r = 110.0pm (average of two extreme values (97.0 + 122.8)/2 = 109.9pm), then an abrupt fall of attractive nucleus-electron potential occurs, manifested by a jump of L-J parameter n from 1.28 to 1.68, consequently an abrupt fall of IE (Ionization Energy) value, both that of database [19], as well as that of second proposed step-by-step nonlinear static analysis program, from about 40,000 to about 14,000 kJoules/mole.

Comparison with Published Data for First Ionization Energies and Atomic Radii of Second Period Elements
The values of First IEs (Ionization Energies), as well as the Atomic Radii of the Elements of Second Period, Lithium (Z = 3) up to Neon (Z = 10), obtained by the proposed model, are compared with the corresponding ones from published data, based on Experiments, [1] (Chapter 7, Table 7.2, page 310, Figure 7.25, page 308).By controlling the value of L-J (Lennard-Jones) exponent n, from n = 2.4 for Lithium, with a smooth variation, up to n = 1.915 for Neon, a satisfactory approximation , in the values of First Ionization Energies, between published data [1] and proposed model, is achieved.Whereas, in the values of Atomic Radii, only in the first two Elements, Lithium and Beryllium, large differences are observed between published data and proposed model.
In Table 2, in the first column the symbols of Elements of Second Period are noted, from Li up to Ne.In the second column, the atomic number Z of the element,

Figure 1 .
Figure 1.(a) Electrons initially located on the assumed spherical outer surface of the nucleus; (b) Attraction and repulsion from nucleus center O to electron i.Mutual repulsion between electrons i and k; (c) All electrons equilibrate on the outer surface of the same sphere concentric and larger than nucleus.

Figure 2 .
Figure 2. Flow-chart of the proposed step-by-step nonlinear static analysis algorithm for finding the equilibrium positions of electrons in an atom.

Figure 3 .
Figure 3. (a) One electron is subjected to gradual forced removal, radially and outwards, from the outer surface of an equilibrium sphere; (b) As the single electron is gradually removed, the remaining electrons equilibrate on smaller spheres; (c) The produced work of electron removal gradually increases and tends to an upper bound u E , which is the Ionization Energy let us consider the Element Argon with atomic number Z = 18 and nuclear radius 97.0 pm n r = .According to following Section 2.5, can be proved that the minimum distance between two electrons, in the proposed model, for

∫
and, by referring to 1.0 mole of the Element, for Avogadro constant 23 6.022 10 mole A C = × , the work produced for separation of couples of atoms Open Journal of Physical Chemistry

Figure 4 .
Figure 4. Mutual repulsive force F, in a couple of electrons, versus their distance r.Example: For min 58.77pm, max 6.680 mdyn, max d d 0.2273 mdyn pm r F K f r = = = = − .
, curves ( ) x ϕ are shown, corresponding to the above function of Equation (3), for three different values of the parameter n: 1.05, 1.5, 2.0.It is observed, in this Figure 5, that, as the value of parameter n increases from 1.05 up to 2.0, the attractive potential of the L-J function ( ) x ϕ is gradually weakened.

Figure 8 .
Figure 8. Plan and elevation of rhombic dodecahedron.Order by which the electrons are initially located on outer spherical surface of nucleus.Up to 26 electrons are considered here.
(b), in comparison with the very simple plan view of corresponding rhombic dodecahedron, shown in Figure 9(a).In both Figure 9(a) & Figure 9(b), only the visible faces are shown.

6 8
24 36 72 146 ν = + + + + = , that is 28 more than the 118 electrons of last existing Element.These 146 electrons are demonstrated in Figure 10(b), in comparison with the 98 electrons that can be accommodated by the rhombic dodecahedron, shown in Figure 10(a).

Figure 9 .Figure 10 .
Figure 9. (a) Plan view of rhombic dodecahedron.Only visible faces.The projections of cube vertices on circum-scribed sphere are noted; (b) Plan view of the new revealed regular polyhedron.Only visible faces.
(a) & Figure 11(b).For the last 18th IE (Figure 11(a)), the following formula is used, according to Open Journal of Physical Chemistry

Figure 11 .
Figure 11.Data for investigating the two last Ionization Energies, 18th and 17th IE, of Argon.
upper bound u ϕ of L-J function integral, for parameter

Figure 12 .
Figure 12.(a) Diagram for finding the value of parameter n of L-J function from given value of its upper bound 1 1 1 2 1 u n n ϕ = − − − , for the last IEs of 12 Elements of 3 first Periods, from Argon (Z = 18) up to Nitrogen (Z = 7); (b) Continuation of diagram of figure (a), with contracted n scale, for the remaining 6 Elements of 3 first Periods, from Carbon (Z = 6) up to Hydrogen (Z = 1).
of the above function E(x) is shown in Figure 13, for n x r r = up to 10 8 , that is for r up to about 1.0 cm.The produced work reaches up to about 280,000 kJoules/mole and tends asymptotically to the upper bound 427,066 kJoules/mole.The produced work E(x), corresponding to the 17th IE, is slightly lower than that of last 18th IE, according to what mentioned above, and almost coincides with it.

Figure 13 .
Figure 13.Variation of work produced by removal of a single electron from every atom, for one mole of the Element, corresponding to the last 18th IE, with 1 electron, of Argon, obtained by hand calculator, up to 8 10 n x r r = =, that is for r up to about 1.0 cm.

Figure 14 .
Figure 14.Gradual increase of radius r of equilibrium sphere of electrons obtained by the first proposed step-by-step nonlinear static analysis program.For one electron, 1 ν = , nuclear radius 1 97.0 pm n r r = = .For 10 ν = electrons, 10 110.4 pm r = .For 18 Z ν = =

Figure 15 .
Figure 15.Variation of the work produced by removal of a single electron in every atom, for 1.0 mole of the Element Argon, from the last 18th IE with 1 electron, up to 1st IE with 18 electrons, for n x r r = increasing up to 100, that is for r up to about

Figure 16 .
Figure 16.Based on data of Table 1, the following diagrams are obtained, for number ν of electrons varying from 1 up to 18, that is, for all the IEs of Argon, from last 18th up to 1st IE.(a) Gradual increase of radius r ν of equilibrium sphere of electrons from 97.0 up to 122.8 pm; (b) Increase of parameter n of L-J (Lennard-Jones) curve, from n = 1.057 up to 1.93.Between values of number of electrons ν = 10 and 11, jump of n from 1.28 to 1.68, occurs.The increase of parameter n means weakening of attractive L-J curve potential; (c) Decrease of IE (Ionization Energy) of database[19] from values about 90,000 up to about 1,500 kJoules/mole.The values of last two IEs, 18th and 17th, are very high, about 400,000 kJoules, as shown in Table1, so they are not presented in this diagram.The IEs obtained by proposed model, decreasing from about 81,000 to about 1500 kJoules/mole, are also

Table 1 .
[19]all the IEs (Ionization Energies) of Argon, last 18th up to 1st IE, that is, for number ν of electrons increasing from 1 up to 18, variations of main quantities of present problem are shown: 1) Gradual increase of radius r ν in pm of equilibrium sphere of electrons, from 97.0 up to 122.8 pm; 2) Gradual increase of ratio of r ν to nuclear radius Increase of exponent n of L-J curve, from n = 1.057 up to 1.93.Between values of number of electrons ν = 10 and 11, jump of n, from 1.28 to 1.68, occurs.Increase of n means weakening of attractive potential of L-J curve, as mentioned in Section 2.4; 4) Decrease of IE (Ionization Energy) of database[19], from 427,066 kJoules/mole (corresponding to very large value of

Table
1, so they are not presented in this diagram.The IEs obtained by proposed model, decreasing from about 81,000 to about 1500 kJoules/mole, are also

Table 2 .
[1]mic radii and first ionization energies of second period elements.1)Elementsymbol.2) Atomic number Z. 3) Lennard-Jones exponent n. 4) Nuclear radius r n in pm. 5) Atomic radius r a in pm of published data[1].6)Atomicradius r a in pm of proposed model.7)Firstionization energy in kJoules/mole of published data[1].8) First ionization energy in kJoules/mole of proposed model.