Simulation Model for Creation of Hydrogen Atom Abhijit Biswas, Krishnan RS Mani
*
Department of Physics, Godopy Center for Scientific Research, Calcutta 700008, India
_______________________________________________________________________________ Abstract For this relativistic energy-based simulation model, the simple and most abundant hydrogen atom was chosen for simulation of the condition during the time of its creation, as depicted in the Big Bang scenario. In this model, the right creation situation could be achieved at about 3000oK, subject to the conditions that simultaneously the kinetic energy of the electron is precisely equal to the Rydberg energy at the ground state of the atom, as well as the calculated value of Mass deficit for the atom is precisely equal to the corresponding experimental value. The mass-energy balance calculations for bringing the created high-speed atom to the conditions at which the precision mass measurements are reported, were carried out for obtaining the calculated value of Mass deficit. Thus, it can be concluded that this model has been able to correctly and precisely simulate the creation condition of the hydrogen atom during the Decoupling Era.
Résumé Pour ce modèle simulé de l’énergie relativiste, l’atome choisi pour simuler ces conditions à l’époque de sa création (comme on les trouve dans le scénario du ‘Big Bang’) est celui de l’hydrogène puisque c’est simple et le plus abondant. Dans ce modèle, la situation exacte au moment de création pourrait être réalisée à environ 3000° K sous des conditions de réserve où l’ énergie cinétique de l’ électron est exactement égale à l’ énergie Rydberg à l’ état fondamental de l’atome ainsi que la valeur calculée de la masse déficit de l’atome est aussi toute égale à la valeur expérimentale correspondante. Les calculs sur les proportions de la masse et de l’énergie exigées (pour amener cet atome de haute-vitesse ainsi crée aux conditions auxquelles la mensuration précise de la masse était rapportée) étaient faits pour obtenir la valeur calculée de la masse déficit. Donc, on peut conclure que ce modèle a la capacité de simuler exactement et précisément les conditions de création de l’atome d’hydrogène pendant l’ère de découplage.
Key words: Big Bang; Rydberg energy; hydrogen atom; relativistic simulation; mass deficit; mass defect; binding energy * E-mail:
[email protected]. I.
Introduction:
Hydrogen (H) atom is the most abundant atom in the universe and quite expectedly, it played a major role in the creation of the universe. As the simplest atom known, H atom has been of much theoretical and experimental study. Because of its simple atomic structure, H atom together with its spectra has been central to the development of the theory of atomic structure. As the only neutral atom with an analytic solution to the Schrödinger equation, the study of its energetics and bonding played a very important role in the development of quantum mechanics. The H atom has special significance in quantum field theory and quantum mechanics as a physical system that is a model two-body problem, which has yielded many simple analytical solutions in closed-form. In view of the above, in the relativistic energy-based model presented here, the Decoupling Era situation in the Big Bang scenario, was simulated first by verifying it for the creation of a simple hydrogen atom. The first phase of such work culminated in this paper that presents the results of such numerical simulations that are in good agreement with the recent accurate experimental and/or recommended data. It may be mentioned here that no such work using similar approach for simulation of creation of atoms in the Big Bang scenario, could be found in the literature.
II. Discussions on the formulation of the problem and the equations used: In the Big Bang scenario, as the universe cooled further after the Nucleosynthesis Epoch, and the expansion continued over the next several hundred thousand years, the particles remained invisible; during this era of opaqueness, the photons intermixed with the H and Helium (He) nuclei had been so energetic that they destroyed any atoms that resulted from electron capture. Thus, the number of free or unattached electrons in the plasma was significantly high
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leading to their scattering effect on the photons that led to the opaqueness of this era. Thereafter, as the temperature came to about 3000oK during the Decoupling Era, the protons and He nuclei could start combining with electrons to form stable H and He atoms [1]. This paper presents a relativistic energy-based model for atom creation during the Decoupling Era, based on presently available high precision data. This model developed by the authors, has been applied for creation of H atom based on the following formulation and mathematical model.
A.
Formulation of the problem for Creation of H atom:
At about 3000 oK during the Decoupling Era, let us consider the case of a proton and a free electron that are moving in the expanding universe and that the electron speed corresponds to the Rydberg energy (ERy) of the H atom (the phrase “Rydberg energy” and the symbol ERy as used throughout this paper refers to Rydberg energy corresponding to the ground state of the H atom). At this condition, the proton could capture (if other conditions like the appropriate electron-proton distance, as well as momentum and energy conservation requirements etc. were fulfilled) the electron to form a H atom, while releasing an ultraviolet photon whose energy is equal to the Rydberg energy for H atom. This situation was simulated in our program.
B.
Mathematical Model
The one-electron atoms are the simplest case to begin with the quantum mechanical study. One may recall the historical importance of the one-electron H atom, because it was the first system on which Schroedinger applied and verified his theory of quantum mechanics. It may be mentioned that the reduced mass technique was applied here (i.e., the mass of electron me was replaced by its reduced mass, µ), as given by the following equation:
.m , m e + M p e Mp
µ = where
(1)
Mp = mass of proton.
Without going into the details of solving the time-independent Schroedinger equation, the only finite or acceptable solution has allowed value of total energy of the bound states of the atom as given [2] in the usual notations by
En =
where
e = Ze = h = ε0 =
− µ. Z 2 . e 4
,
(4. π.ε o ) . 2. h . n 2
2
(2)
2
electron’s charge, total charge of the nucleus, Reduced Planck's constant, permittivity constant, and
n, l, and ml are quantum numbers given by n = one of the integers (l + 1 ), (l + 2 ), (l + 3 ), …. l = one of the integers,
m , l
m + 1, l
m + 2, l
m + 3, …. l
ml = one of the integers, 0, 1, 2, 3, …. The acceptable solutions are most conveniently written [2] as:
− Z . r n .
Rnl (r ) = e where
a o
.
Z .r a o
l
. G nl Z .r
,
(3)
ao
r = distance between the electron and the nucleus, Z .r = polynomials in Z .r , with different forms for different values of n and l, and G nl ao ao
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ao =
the location of maximum in the radial probability density, given by
a =
4. π . ε o . h 2
o
µ .e
,
(4)
2
The eigenfunctions of one-electron atoms and associated eigenvalues lead to the following equation: Potential energy of the system, E
− e2
=
P
. . .
,
(5)
4 π ε o ao
From uncertainty principle, the above equation for the Potential energy of the system as well as the following equation for the kinetic energy of the electron at the moment of creation of H atom, can be derived. 2
EK =
h 2. µ . a o
2
,
(6)
For higher computational accuracy, the relativistic mass-energy concept need to be applied for the energy-based model presented here; hence, the reduced mass term in equation (6) above, needs to be replaced appropriately by reduced mass-energy term; however, for retaining the simple form of the equations presented here, the rearranged form of 2 equation (6) is given below by replacing µ only with the total mass-energy term ( ET / c ): ao = where
h.c 2. E K . E T
,
(7)
c = speed of light.
Based on the above, the potential and kinetic energy terms, and the total mass-energy term are calculated. Thereafter, a total mass-energy balance is done, and the kinetic energy and the Mass deficit values of the created atom are computed. Experimental values of mass deficit, mdE was calculated from
mdE = ( me + mp - mH ),
(8)
where the three rest-mass terms on the right-hand side of the equation correspond to that of electron, proton and H atom respectively. Computed values of mass deficit, mdC was calculated as follows:
mdC = ( ET - E0 - EK ) / c2 ,
(9)
where the three energy terms on the right-hand side of the equation correspond to the total mass-energy, rest massenergy and kinetic energy of H atom respectively.
III. Results: While starting the simulation work for creating a H atom, it was assumed that half of the electromagnetic potential energy (Ep) of the electron would be equal to the kinetic energy (EK) of the electron; but, at this condition, it was found that the atom could form and again after sometime could dissociate when a sufficiently energetic photon could rip the electron out; it was also found that both the kinetic energy (EK) and the Mass deficit (mdC) values differed considerably from the presently available high precision experimental data for the corresponding terms, that is, respectively the Rydberg energy (ERy) and the Mass deficit (mdE) values. These results have been reported in the second row of Tables 1 and 2 below, for the purpose of comparison. As the right creation situation could be simulated at 3000.127 oK, the kinetic energy of the electron (EK) became equal to the Rydberg energy (ERy), while simultaneously the right calculated value of Mass deficit could be achieved; the necessary mass-energy balance calculations were carried out for bringing the created high speed H atom to the conditions at which the precision rest-mass measurements are reported [3, 4, 5], and the value of mdC was computed and reported in Table 2 for comparison with mdE. The results of this fully relativistic and high precision case of creating a H atom have been reported in the first row of both Tables 1 and 2 below.
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Table 1. Computed Values of Rydberg Energies compared for two cases of H atom creation
Atom
Rydberg Energy: CODATA [3] Recommended cum-Prorated (*) values, ERy in eV
Uncertainty, in ppb
Kinetic Energy EK of e: (Computed results from simulation), in eV
Mismatch of Computed EK from Recommendedcum-Prorated values of ERy, in ppb
Remarks on the method of creating the H atom
Hydrogen
13.59828607*
25.01
13.59828609
1.5
The right creation condition for the H atom was simulated.
Hydrogen
13.59828607*
25.01
13.59864796
26613
Half of Ep of the electron was assumed to be equal to its EK.
Hypothe -tical atom
13.60569193 [3]
25
For the special case of an atom having a nucleus of infinite mass
Computed mass deficits for the two cases of H atom have been compared with those calculated from NIST-PML [4, 5] and CODATA [3] values of atomic and particle masses, in Table 2 below.
Table 2. Computed Values of the Mass deficits compared for two cases of H atom creation
Uncertainty, in ppb
Mass deficit: (Computed results from the Simulation Program), mdC in eV
Mismatch of experimental and calculated Mass deficits (mdE - mdC ), expressed as fraction (in ppb) of mdE
Remarks on the method of creating the H atom
13.26788663
25.01
13.26788659
3.1
The right creation condition for the H atom was simulated.
13.26788663
25.01
13.26823967
26609
Half of Ep of the electron was assumed to be equal to its EK.
Atom
Mass deficit: Calculated from NIST-PML [4, 5] and CODATA [3] values of atomic and particle masses, mdE in eV
Hydro -gen Hydro -gen
IV. Conclusion: In this model, the right creation situation could be achieved at 3000.127 oK, subject to the conditions that simultaneously the kinetic energy of the electron is precisely equal to the Rydberg energy at the ground state of the atom, as well as its calculated value of Mass deficit is precisely equal to the corresponding experimental value. The data presented in the first row of Table 1, shows that EK and ERy are comparing well with a mismatch of 1.5 ppb, whereas ERy is having an uncertainty of 25.01 ppb. The results of the subsequent mass-energy balance calculations presented in the first row of Table 2, show that mdE and mdC are having a mismatch of only 3.1 ppb, whereas mdE is having an uncertainty of 25.01 ppb. Such close match clearly proves that this model has been able to correctly and precisely simulate the creation condition of the H atom during the Decoupling Era, by extracting the signatures that the H atoms left during the moment of their creation, in their Rydberg energy and Mass deficit values, that are presently available at a high precision level.
V.
References:
[1] S. Weinberg, The First Three Minutes: A Modern View of the Origin of the Universe, Bantam Books, 1977. [2] R. Eisberg and R. Resnick: Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, John Wiley & Sons, N. Y., 1974. [3] P.J. Mohr, B.N. Taylor, and D.B. Newell, Rev. Mod. Phys. 80 (2008) 633. doi:10.1103/RevModPhys.80.633. http://physics.nist.gov/cuu/Constants/codata.pdf. [4] M.E. Wieser and M. Berglund: Atomic Weights of the Elements, NIST Physical Measurement Laboratory, 2007, http://www.nist.gov/pml/data/comp.cfm [5] Wieser, M. E., Pure Appl. Chem. 78 (2006), 2051.
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