The Hydrogen Atomic Model Based on the Electromagnetic Standing Waves and the Periodic Classification of the Elements

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1 Appied Physics Research; Vo. 4, No. 3; 0 ISSN ISSN Pubished by Canadian Center of Science and ducation The Hydrogen Atomic Mode Based on the ectromagnetic Standing Waves and the Periodic Cassification of the ements Giuseppe Beotti 4/C, S. Gaudenzio Street, 005 Ivrea (To), Itay Correspondence: Giuseppe Beotti, 4/C, S. Gaudenzio Street, 005 Ivrea (To), Itay. -mai: giuseppe.beotti@ive.it Received: June 9, 0 Accepted: Juy 6, 0 Onine Pubished: Juy 7, 0 doi:0.5539/apr.v4n3p4 URL: Abstract By means of a new concept of the Hydrogen atomic structure [where the eectron of the Hydrogen is considered not as a non-dimensiona obect revoving around the atomic nuceus in a fied of centra forces, but as a three-dimensiona eectromagnetic spherica standing wave concentricay superimposed on the eectromagnetic standing wave of the proton] we can obtain a new periodic cassification of the eements. Up ti now, chemistry has considered atoms to be made up of sub-shes with, 6, 0, 4 eectrons and currenty, the great number of eectrons (4) in sub-shes f creates some probems in the formation of the sixth and seventh periods. Instead in this paper we wi ony consider sub-shes with, 6, 0 eectrons, to achieve a better ordering of the heavy metas. Hence, Ni, Pd, and Yb have been drawn up in coumns in the new periodic cassification of eements, and Ytterbium coud be proposed as a testing materia in the cod fusion experiments. Moreover many new theoretica (abeit unsteady) eectromagnetic wave-engths of Hydrogen Spectrum appear. Keywords: periodic cassification of eements, hydrogen atomic mode, eectromagnetic standing waves, hydrogen spectrum. Introducation A new concept of the Hydrogen atomic structure (where the Hydrogen eectron is considered not as a non-dimensiona obect revoving around the atomic nuceus in a fied of centra forces, but as a three-dimensiona eectromagnetic (e.m.) spherica standing wave concentricay superimposed on the e.m. standing wave of the proton) has been previousy considered in (Beotti, 0). Since the e.m. standing waves have a defined anguar frequency ω γ, then the wave equation can be written as: where: c t ( r,,, t) R( r) ( ) ( ) T( t) () ψ is a function defined in poar coordinates ( r, θ, φ ); t is time and c is the speed of ight. In (Beotti, 0) the foowing resuts were demonstrated: h k (3) Where: is the energy of the e.m. standing wave; ω γ is the anguar frequency of the e.m. standing wave and k is wave number; a 0 is the radius of the first noda surface of the eectron, considered to be an e.m. a0 spherica standing wave. ( ) sinm (4) 4 () 4

2 Appied Physics Research Vo. 4, No. 3; 0 where m Z is the magnetic quantum number. The functions Θ (θ) are : d x cos( ) P ( x) x for m 0! dx m m m d x P ( x) x P( x) for m 0 m dx The equations in 5 are not normaized. They can be normaized as per (Persico 97). (5) Tt () Ksin( t G) (6) Where: K = the ampitude G = the phase R0() r sin( kr) A0 cos( kr) r cos kr sin kr R( r) sin kr A cos kr r kr kr kr kr 3cos kr 3sin kr 3sin kr 3cos kr R () r sinkr A coskr r kr kr 6 cos kr 5sin kr 5 cos kr sin kr 3 kr kr kr R3 () r r 6 sin kr 5 cos kr 5sin kr A3 cos kr 3 kr kr kr 0 cos kr 45 sin kr 05 cos kr 05 sin kr sin kr 3 4 kr kr kr kr R4 () r r 0 sin kr 45 cos kr 05sin kr 05 cos kr A4 cos kr 3 4 kr kr kr kr with the foowing vaues of A ( where = 0,,, 3, 4) 3 A0 0 A A A A (7) (8) We can verify that for = 0 the R 0 ( r ) equation meets the boundary condition a 0 R 0 ( a 0 ) = 0 even when the wave number k is substituted by the wave numbers k with N. The corresponding graphs show a0 a0 that each a 0, = a 0 root of R 0 (r) is shifted to r = a 0 and we have n = noda spherica surfaces in the radia interva [ 0, a 0 ].. The Periodic Cassification of the ements Using the symbo ns the sub-shes s oined to the principa quantum number n can be defined. For every sub-she ns correspond the functions that vanish for r = a 0 and that affirm the condition = n. Then the R0 ( r) sin r r a0 4

3 Appied Physics Research Vo. 4, No. 3; 0 eigenvector R 0 (r) of the s sub-she (=) shows one noda surface within the interva of r [0, a 0 ] ( at r = a 0 ); the eigenvector R 0 (r) of s sub-she ( = ) shows two noda surfaces within [0, a 0 ]; the eigenvector R 0 (3r) of 3s sub-she ( =3), three; and so on. If we estabish a 0, to be the th-root of the function R 0 (r) shown in equation 7, then, from (3) and for the ns sub-shes, we can confirm that: 0, 0 0, a0 a0 k a a =,, 3, 4,... (9) and, according to equation of (Beotti, 009), the tota sub-she energy, that corresponds to each th-root a, of the R (r) shown in (7) is: a (0),, If a the w eigenvectors of a given (, ) sub-she have the same energy,, then from (0) we have: a a (),,,,, w w w with: 0 m0 w0 sub-shes s m0, w 3 m0,, w 5 sub-shes p sub-shes d 3 m0,,, 3 w3 7 sub-shes f 4 m0,,, 3, 4 w 9 sub-shes g Where w ( with = 0,,, 3,... ) are the number of equations present in the sub-shes. Now by substituting the function wr in pace of r, within the R (r) shown in (7), we obtain the functions wr. The roots a, of these functions wr wi be proportiona to the square root of the energy, of each eigenvector R (r) that has roots (or sometimes - roots, for f and g sub-shes) within the interva of r [0, a 0 ]. The eigenvectors R (r) are cacuated by substituting k = π/a 0 into the equations 7 instead of k = π/a 0 ; and by imposing the boundary condition a 0 R (a 0 )=0 in order to cacuate the corresponding A vaues. Then we can to draw the wr functions and obtain the graph shown in Figure where the energies, of the sub-shes increase in accordance with the foowing sequence (3): s (=), p (=), d (=3), p (=3), s (=), f (=5), g( = 6),... (3) Thus, each she associated to the principa quantum number n is formed by the sequence of the sub-shes ns, np, nd, np, ns, nf,...; where ns, np, nd are the first usefu root of the functions R0 wr, 0 R wr,..., that beong to the same she of the principa quantum number n.ns, np, etc., are the second usefu root of () R w r, R wr,..., that beong to the same she of the principa quantum number n. The root of the type d ( = 4), between p and s in Figure is to be omitted since the sub-she with minima energy where there are four noda surfaces within the interva [0, a 0 ] has to be the sub-she f. Since the eigenvector R 3 (r) of the sub-she f shows four noda surfaces within the interva [0, a ] ony when =5, then we have to consider the fifth root of R w r, whose associated eigenvector is R 3 ( 5r ). Now, sub-she f has an energy greater than the energy of sub-she s ; then aso sub-she d must have an energy greater than the energy of sub-she s. Hence, the fourth root of R w r does not have any corresponding energy vaue. function In accordance with the fractas form of the superposition of the e.m. standing waves (Beotti 0, 370), we presume a fiing up of each she reative to the principa quantum numbers n =,, 3,... in accordance with sub-she sequence (3) and in order to generate Tabe. In Tabe there are ony the s, p, d sub-shes. Therefore, we have to draw a graph (Figure ) in order that the sub-shes in Tabe and the roots of the functions wr correspond to each other. Now, if we consider the number of noda surfaces of the functions r (with = 0,, ) within the interva of r [0, a 0 ], then this number aso identifies the position of the th-root a, of the corresponding R wr. 43

4 Appied Physics Research Vo. 4, No. 3; 0 Tabe. First approach to the periodic cassification of the eements Principa quantum number Sub-shes coherent with the fiing uporder (3) n = = 0 s n = = 0 s = p n = 3 = 0 3s = 3p = 3d n = 4 = 0 4s = 4p = 4d = 4p n = 5 = 0 5s = 5p = 5d = 5p = 0 5s n = 6 = 0 6s = 6p = 6d = 6p = 0 6s n = 7 = 0 7s = 7p R w r 0,,,3, = 4 = = 3 Case of the kind d discussed in the artice = = r / a 0 s p d p s f g Figure. With this graph it is possibe to recognize the sequence of sub-shes with increasing energy, vaid for each she that corresponds to the principa quantum number n 44

5 Appied Physics Research Vo. 4, No. 3; 0 Given that the number of roots within the interva [0, a 0 ] for each eigenvector R 0 (nr) of the sub-she ns is equa to n, we have to consider a the ordered roots n =,, 3,, and 7 of the function R0 w0r. The eighth root of R0 wr 0 wi be the sub-she 5s, the ninth the sub-she 6s. On the other hand, for each she reated to the principa quantum number n, the number of roots of the functions R ( r ) within the interva [ 0, a 0 ] for sub-she np ( = ) must be equa to n; those for sub-she nd ( = ) must be equa to 3n. For sub-she 5s we have shown the position of the eighth root that coincides with 8s; sub-she 5p wi have five roots more when compared to 5s, (that is 3); sub-she 4p wi, instead, have roots [ in fact the hypothetica sub-she 4s ( that does not exist ) coincides with 7s and we have to consider = roots for sub-she 4p ]. Sub-she 6s coincides with the position 9s and so wi have 9 roots; instead sub-she 6p wi have six roots more, namey, 5 roots. Now it is possibe to generate Tabe, where to each th-root of the functions R 0 w0r, R wr and R wr corresponds an eigenvector,, R r R0 r R r and R r ; moreover each eigenvector R0 r R r and within the interva [ 0, a 0 ] gives the number of noda surfaces indicated (as roots) in Tabe II, and a these numbers of roots coincide with the th-root a, position of the functions wr. Now we can order the sub-shes shown in Tabe according to the increasing energies, shown in Figure and begin buiding Tabe III. Through tests, we can check that, for Hydrogen atom, the frequencies ν A,B that correspond to the transitions B A [ where the status A is characterized by the anguar quantum number A and by the A position of the considered root of the function A A position of the considered root of the function B B w r ; and status B is characterized by the anguar quantum number B and by the B R w r ] are obtained in the foowing Formua 4 (White, 934): (3) 0 Hz B A 5 AB, Rc h na nb na nb (4) where: R = Rydberg constant = () 0 7 m - Tabe. Second approach to the periodic cassification of the eements. The number of roots within the r R w r function interva [0, a 0 ] of the R ( r) eigenvector coincides with the th-root position of the Principa quantum number Sub-shes coherent with the fiing up order (3) n = = 0 s root n = = 0 s roots = p 4 roots n = 3 = 0 3s 3 roots = 3p 6 roots = 3d 9 roots n = 4 = 0 4s 4 roots = 4p 8 roots = 4d roots = 4p roots n = 5 = 0 5s 5 roots = 5p 0 roots = 5d 5 roots = 5p 3 roots = 0 5s 8 roots n = 6 = 0 6s 6 roots = 6p roots = 6d 8 roots = 6p 5 roots = 0 6s 9 roots n = 7 = 0 7s 7 roots = 7p 4 roots c = speed of ight = (0) 0 8 m / s 45

6 Appied Physics Research Vo. 4, No. 3; 0 A the vaues of the Physica constants are taken from (Jordan, 988). n A, n B = numerica data (shown in Tabe 3) proportiona to the square of the corresponding root vaues of the functions w A r and A B B R w r, where we have considered the minimum vaue of n A = n =0, = = quation (4) is in accordance with the foowing equation (5), that was obtained from (9), (0) and (), appied to ns sub-shes ( = 0): 0, k 0, w0 w0 a0 a0 a0, n0, =,, 3, 4,... (5) where n 0, has the same meaning as n A, n B in (4), for = 0. Now we can assume that the sub-she energy,, that corresponds to each th-root a, of (7), is:,, w a, n, wr obtained from (6) where the n, in (6) correspond to either the n A or n B in (4), according to the ( A, A ) or ( B, B ) vaues. Both the n A or n B vaues are proportiona to the square of the corresponding root vaues a,, A a of the functions A B, B R w r. w A r and A B B Now, by putting n A = and n B into (4), we can cacuate the maximum energy A = R c h that in a B A transition can be emitted as a ight spectrum row; this energy is haf the energy mc e of an eementary e.m. standing wave of radius a 0 (Beotti, 0). On the other hand this resut is aso vaid for a the s, p, d sub-shes. In fact we can substitute the equivaent energy referred to the ast coumn of Tabe 3, in pace of n A, in 4; and we can substitute a vaue tending towards infinity in pace of n B ; and in accordance with (6) (and its marking), we can obtain two possibiities for emitting a ight ray for each eigenvector (an e.m. standing wave) that forms each sub-she. In fact: R ch m c (7), e n, n, Where: h = Panck constant; α = fine structure constant and m e = eectron mass. By returning to () and the definition of w (with = 0,,, 3,...) being the equation number present in each sub-she, we can assume that in the sub-shes s ( = 0 ) we have w 0 = possibiities of ight emissions (and this vaue of w 0 can repace the number of equations w 0 = ). In a simiar way, we can obtain w = 6 instead of w = 3 for the sub-shes p ( = ); w = 0 for the sub-shes d ( = ); and so on. The energy property of this eigenvector brings to mind the Paui principe. So the sequence of the sub-shes shown in Tabe 3 faithfuy reproduces the properties of the eements in their periodic cassification in a natura way. Now we can buid Tabe 4. Up ti now, chemistry has considered atoms to be made up of sub-shes with, 6, 0, 4 eectrons and, currenty, the great number of eectrons, 4, in sub-shes f creates some probems in the formation of the sixth and seventh periods. Instead, Tabe 3 and Tabe 4 show that we can consider ony sub-shes with, 6, 0 eectrons, to achieve a better ordering of the heavy metas. Hence, Ni, Pd, and Yb have been drawn up in coumns in the new periodic cassification of eements, and Ytterbium coud be proposed as an optimum materia for cod fusion. The ast coumn of Tabe 3 shows some rea numbers. This is in accordance with Mian Perkovac s considerations on equations (00) - (04) of (Perkovac, 00). It is possibe to go deeper into Perkovac s mode in (Perkovac, 00; Perkovac, 003). 46

7 Appied Physics Research Vo. 4, No. 3; 0 Tabe 3. Sequence of the sub-shes fiing up, according to the increase in energy Sub-shes Transitions Number Atomic Numbers of the ements in each sub-she ements of the sub-she nergy n a A, of the B= A Transition s Z = H He s Z = 3 4 Li Be p 6 Z = 5 0 B C N O F Ne 3s Z = Na Mg p 6 Z = 3 8 A Si P.380 S C Ar 4s Z = 9 0 K Ca d 0 Z = 30 Sc Ti V Cr Mn Fe Co Ni Cu Zn 4p 6 Z = 3 36 Ga Ge As.794 Se Br Kr 5s Z = Rb Sr d 0 Z = Y Zr Nb Mo Tc Ru Rh Pd Ag Cd 5p 6 Z = In Sn Sb Te I Xe 6s Z = Cs Ba p 6 Z = 57 6 La Ce Pr Nd Pm Sm 5d 0 Z = 63 7 u Gd Tb Dy Ho r Tm Yb Lu Hf 6p 6 Z = Ta W Re Os Ir Pt 7s Z = Au Hg p 6 Z = 8 86 T Pb Bi 57.9 Po At Rn 5s Z = Fr Ra p 6 Z = Ac Th Pa U Np Pu 6d 0 Z = Am Cm Bk Cf s Fm Md No Lw Ku 6p 6 Z = 05 0 Ha s Z =

8 Appied Physics Research Vo. 4, No. 3; 0 Tabe 4. New periodic cassification of the eements. It is interesting to notice the bue coumn-sequence Ni Pd Yb (No) and to consider Ytterbium for cod fusion experiments H He Li Be B C N O F Ne Na Mg A Si P S C Ar K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Tc I Xe Cs Ba La Ce Pr Nd Pm Sm u Gd Tb Dy Ho r Tm Yb Lu Hf Ta W Re Os Ir Pt Au Hg T Pb Bi Po At Rn Fr Ra Ac Th Pa U Np Pu Am Cm Bk Cf s Fm Md No Lw Ku Ha 3. Cacuation of the Spectra Wave Lengths of Hydrogen Now we can cacuate a the wave-engths of the Hydrogen spectrum by using the foowing formua (8) of first approximation, derived from (4) (White, 934): nn 0 0 A B AB, R nb na (8) where: R = () 0 7 m -. n A, n B are the vaues shown in the ast coumn of Tabe 3. They correspond to the B A transition. Tabe 5 shows the most important waveength that coud form the Hydrogen spectrum. But in equation (8) a greater number of Hydrogen spectrum rows were computed compared to the fundamenta ones obtained in Quantum Mechanics. This resut has to be confirmed by means of a dynamic experimenta check. In fact, in a induced mechanica vibrations, the contribution of high order eigenvectors vanishes quicky and ony the main eigenvectors, that correspond to the spectrum rows oined to =0 (s sub-shes), are present in the stationary state. As in Quantum Mechanics, now the whoe vaues of n A (see Tabe 3) correspond to = 0. Hence, we can obtain the norma rows of emission spectrum of Hydrogen by equation (8). On the other hand, spectrum rows oined to = ( p sub-shes) and to = (d sub-shes), that correspond to the non-whoe n A vaues, are present ony in the transient of Hydrogen atoms excitation. But the presence of either a transient or a stationary state can ony be ustified when it is possibe to consider each Hydrogen atom as a continuous function of energy in a space domain; in other words: when each Hydrogen atom can be considered as an e.m. standing wave. And this condition compies to the fundamenta hypothesis of this artice. Of course this approach aows for the quantization of energy, since the wave equation eigenvectors can take on different discrete vaues of energy (see Figure ). Then it seems that the traditiona idea, i.e. the rows of atomic emission spectra are produced by quantum umps of the eectrons between two different energy eves, has to be formuated in a different way. Tabe 5. The most important rows of Hydrogen spectrum expressed in nm 6s s p s d s p s s s p s s s p s d s p s s s p s d s s s p s

9 Appied Physics Research Vo. 4, No. 3; 0 3d s s s p s s s p s.407 s s.503 6s s p s d s p s s s p s s s p s d s p s s s p s d s s s p s d s s s p s s p p p d p p p s p p p s p p p d p p p s p p p d p s p s s p p d p s p p p s 3s p 3s d 3s p 3s s 3s p 3s s 3s p 3s d 3s p 3s s 3s p 3s d 3s s 3s s p p s s 3p p 3p d 3p p 3p s 3p p 3s p 3p s 3p p 3p d 3p p 3p s 3p d 3s p 3p s 4s

10 Appied Physics Research Vo. 4, No. 3; 0 R wr 0,, = = = r / a 0 s s p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4p 5d 6p 7s 5p 5s 7p 6d 6p Figure. Sequence according to the increase in energy in the sub-shes fiing up, in order to reproduce the Periodic Cassification of the ements 4. Concusions In this paper, we have considered the eectron of Hydrogen as a three-dimensiona e.m. spherica standing wave concentricay superimposed onto the e.m. standing wave of the proton. By hypothesizing a fractas form for the Hydrogen structure, it was possibe to obtain a new periodic cassification of the eements. Up ti now, chemistry has considered atoms to be made up of sub-shes with, 6, 0, 4 eectrons and, currenty, the great number of eectrons, 4, in sub-shes f creates some probems in the formation of the sixth and seventh periods. Instead, in this paper we considered ony sub-shes with, 6, 0 eectrons, to achieve a better ordering of the heavy metas. Then Ni, Pd, and Yb have been drawn up in coumns in the new periodic cassification of the eements. So it is possibe to propose Ytterbium for the cod fusion experiments. Moreover, in accordance with Mian Perkovac s anaysis (Perkovac, 00), we obtained many fast decaying new theoretica e.m. wave-engths of Hydrogen Spectrum. Do any appropriate tests exist for vaidating this? If so, the proposed approach to Hydrogen atomic structure (a superposition of the e.m. spherica standing waves of eectron and proton) wi have made a worthy experimenta contribution. Furthermore, it is interesting to note that equations (7) are soutions of either the wave equation () (being appied to the Hydrogen atom) or the dynamic bi-lapacian equation (Beotti, 009) that anayses the Hydrogen nucear properties. Coud this mathematica ink be used in estabishing a theory of cod fusion? References Beotti, G. (009). The dynamic bi-lapacian equation in poar coordinates and the magic numbers of atomic nuceus. Phys. ssays, (68). Beotti, G. (0). The hydrogen atomic mode founded on the eectromagnetic standing waves. Phys. ssays, 4(364). 50

11 Appied Physics Research Vo. 4, No. 3; 0 Jordan,. C. (985). Reference Data for ngineers (VII - 988). Indianapois, IN: Howard W. Sams & Company Perkovac, M. (00). Quantization in Cassica ectrodynamics. Phys. ssays, 5(4). Perkovac, M. (003). Absorption and mission of Radiation by an Atomic Osciator. Phys. ssays, 6(6). Perkovac, M. (00). Statistica test of Duane-Hunt s aw and its comparison with an aternative aw. Retrieved from arxiv: v Persico,. (939). Fondamenti dea Meccanica Atomica (VI-97). Boogna: Zanichei White, H.. (934). Introduction to Atomic Spectra. New York, NY: Mc Graw-Hi - Kögakusha. 5

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