Journal of Theoretics

The Threshold & Ionization Frequencies

in Relation to Light and Bohr Atomic Model

Author:?Pierre Lesbros  or


Abstract:? The threshold frequency and its relevant parameters and equations in reference to ionization energy in relation to the Bohr atom model and light, are presented in this paper.

Keywords:  threshold frequency, light, Bohr atom, atomic weight.



I am presenting these results to scientists in quantum physics, particle physics, and nuclear physics.?They are about the threshold frequency and its parameters, and about the ionization energy, both questions dealt on in their rapport to the Bohr's atom model and to light.


The Threshold Frequency Formula:

According to my research, the formula for the threshold frequency is given by:

F(t) = r2 x F(l) x 0.128 x (atw)?/span> /S(Bohr)


F(t) = threshold frequency

atw = atomic weight

S = surface of that element such as in PI x r2 = surface, and whereby n is the number of the  energy level corresponding the orbit of the valence electron(s) of that element.

F(1) = the frequency of light taken at 5x1014 Hz.

0.128 = an algebraically determined constant that was found at one point of differentiation in the parent formula.

r = 1.7x10-9m in its first power and corresponds to the distance traveled by an electron that is hit by a photon of energy equal to F(l) that has been converted into kinetic energy.

When all the constants have been calculated, the formula simplifies to:

F(t)= 1.849x10-4 x atw?/sup> / S(B)

Please note that the Bohr radius is taken at 4 significant digits, and that the result of this equation thus also has 4 significant digits too.  This formula shows the relationship between the atomic weight, the surface of an atom as calculated from the Bohr's atom model, and the threshold frequency.

Application to hydrogen (H):

?F(H) = 1,849x10-4 x 1.00794?/span> / (8.82x10-21)

?F(H) = 2.1x1016 Hz


Application to potassium (K):

?F(K) = 1.849x10-4 x 39.00983?/span>/ (2.26x10-18)

?F(K) = 5.1x1014 Hz

The Ionization Frequency:

My research work shows that the ionization frequency formula is given by:

F(i) = PI x  F(l) x ( m1- m2) /(m1 x m2) 

By convention, I will write the expression between brackets as the "module of m" (i.e.  [m]).

F(i) = the ionization frequency

In this formula, the constant parameters are:
PI = 3.14 ...
F(l) =
5x1014 Hz

The variables are m1 and m2, and they combine in the form of a quotient as noted above.  The whole value of their sum is in radians at the power -1.

Let's look at these m parameters separately:

Firstly, m1 is a per atom value that is inversely proportional to the atom's threshold frequency, all values in the numerator are constant ones:

m1 = PI x F(l) / F(t)

In such a way, m1 does not depend on m2 nor does it depend on n, the energy level number.
Its value is actually typical of the rest state of an atom but also of its excited state when the electron(s) reach the continuum levels of energy.  However, there is a defined value of m2 which depends on m1 and on n, in the following way:

m2 = n2 x m1 / (2m1 + n2)

Since m1 is a per atom value fixed value, m2 depends on n only and has a quantified value for every energy level.  Besides, if m1 has 4 significant digits because F(t) has 4 significant digits, m2 varies according to the integer n and its number of significant digits varies with the proportion of the importance of variable n against m1 in the final value of m2.  In that respect, m2 will be displayed with 5 digits, but the 5th one will be between brackets.

In its application to hydrogen (H) the calculation of m1 and of m2 to define [m] is shown below:

m1 = PI x F(l) / F(H) = PI x 5x1014 / (2.109 x 1016) = 0.0745

m2 = n2 x m1 / (2m1 + n2)

By putting these values of m2 and of [m] into a chart with the corresponding n levels of energy we get:

n          1              2            3              4             5                6                7              8
m2 0.065(2)    0.072   0.073(8)   0.074(3)   0.074(6)   0.074(7)    0.074(8)   0.0748
[m]    2            0.556     0.217       0.126        0.072        0.054         0.036       0.036

These values for [m] were calculated for hydrogen, but other atoms (with other m1 and m2 values) give same [m] values for the first levels generally.  For example, with potassium, m1=3.058 and m2 (n=3) = 1.820(6) hence [m] = 0.222.  For the same n, chlorine's m1 value would be : 3.192 and m2 would be 1.867, yielding a final value for [m] of 0.222 as well.  But separate calculations for these atoms also show that the continuum states are reached, for hydrogen, from n=7 and onwards, for potassium, from n=7 and onwards, and for chlorine that is when m2 tends towards the value of m1.

I can therefore calculate the ionization energy for every energy level of a given atom and the change in energy due to a "jump" of the electron between two levels.  For hydrogen, the results obtained from the formula given above, are always very close to those which are obtained from the Bohr's atomic model, which uses Coulomb's force law.  The discrepancies are within an average 7.5% margin from the results from the -13.6/n2 formula.


dE(4-2) is the change in energy of 1 electron jumping from the 4th to the 2nd level.
dE = (F4 - F2) x h ,whereby F4 and F2 are the ionization frequencies for the 4th and 2nd levels respectively, and h is Plank's constant.  

Reading from the above chart to find [m] for the two different levels, I can calculate F4 and F2 easily, and go through the calculation of dE.

(F4-F2) x h = (1.979x1014 - 8.734x1014)x h = -4.476 x 10-19 J = 2.79 eV

Here, the positive sign shows that energy is radiated.

The results shows a 10% discrepancy.  This is not invalidating my formula at all, as the Coulomb's force law utilizes the value of the permittivity of emptiness to the square, and there is no proof that the reference is not 10% in error when applying to the interspace between the nuclei and electrons in an atom.  In fact, I have calculated the energies of an electron in the Bohr's atom model without resorting to any of the fundamental equations in electro-magnetism.  Since the formula based upon [m] and F(l) renders the values of the change in energy due to the jumps between the Bohr's orbits, and since m2 is quantified, it belongs to quantum mechanics.

The Parent Formula:

It comes from my research work that the parent and macro scale formula is :

T = r2 x m / (S x F(l))

In this initial formula, T is in seconds and is the duration of a period of time witnessing the phased displacement of the m electrons in the numerator.  In the denominator, S is the surface in square meters, and r is the average spacing between the electrons of that given element.  In the understanding of how these charges are displaced under the action of light, F(l) has the value of
5x1014 Hz.


Both the threshold and the ionization frequencies formulae show the Bohr atomic model formulae being rewritten from a new ground that does not use any of the classical or electro-magnetism physics energy expressions.  Fundamental links have emerged such as that of atomic weight and
threshold frequency, in harmony with the Bohr's model and with the frequency of light at
5x1014 Hz.

Another link is fundamental, that between the macro and the micro scale through the differentiation of the parent formula.  The m module raises the interesting question of the wave-particle duality as  analytically, the wave particle duality can be apprehended through one single mathematical expression.  In my opinion, this is the main aspect and interest of the formulae, offering the opportunity for multiple applications and further theoretical developments.

This formula shows the relationship between atomic weight, the surface of an atom as calculated from Bohr's atom model, and the threshold frequency.?The result is, there seems to be here a more powerful tool providing the Bohr's atom model with the means to seize, quantify, and analyze experimental data or the atomic energy processing workings, in line with its postulates.



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