Model of the Pulsing Atom

Transcription

1 Advances in Materials Physics and Chemistry, 017, 7, ISSN Online: ISSN Print: X Mdel f the Pulsing Atm Heinrich Ehrlich Chemical Department, Lmnsv Mscw State University, Mscw, Russia Hw t cite this paper: Ehrlich, H. (017) Mdel f the Pulsing Atm. Advances in Materials Physics and Chemistry, 7, Received: April 15, 017 Accepted: May 15, 017 Published: May 18, 017 Cpyright 017 by authr and Scientific Research Publishing Inc. This wrk is licensed under the Creative Cmmns Attributin Internatinal License (CC BY 4.0). Open Access Abstract In this wrk, we reanalyzed the mvement f an electrn in the electrstatic field f nucleus. The trajectry f the electrn s mtin is an ellipse with a minr semiaxis, tending twards zer. Frm a mathematical pint f view the mvement f an electrn in such an rbit will be equivalent t the scillatin f an electrn. The actin prduced by electrns in mvement between statinary pints is discrete and prprtinal t a Planck cnstant. This cnditin sets the allwable values f the electrn energy and the radius f their rbit. Electrns n the same shell perfrm symmetric synchrnus scillatins. Their frequency is f the rder f Hz. Mst f the time the electrns are lcated n the periphery f the atm, peridically they simultaneusly rush t the nucleus, the atm rapidly cmpresses and immediately decmpresses, i.e. pulsates. The mdel gives Bhr frmula fr the energy f single-electrn atm and suitable values f inizatin ptentials f the atms f the secnd perid f the Peridic Table. Keywrds Atm Structure, Quantum Thery, Oscillatin, Bhr-Smmerfeld Mdel, Inizatin Ptentials 1. Intrductin A little ver ne hundred years ag, Niels Bhr [1] intrduced the planetary mdel f the atm with electrns revlving in circular rbits arund a nucleus. Arnld Smmerfeld extended this mdel t include elliptic rbits []. The main cntradictin f these mdels was that in line with Maxwell thery the electrn mving with acceleratin in an electrstatic field f nucleus must emit electrmagnetic waves, and thus lse energy. Bhr by passed this restrictin by pstulating the existence f certain special rbits, alng which the electrn can mve withut electrmagnetic radiatin, and intrduced the actin quantizatin rule fr mvement alng these rbits. This pstulate lked (and lks) insufficiently substantiated, and it has been rejected alngside the Bhr mdel, with nset f DOI: /ampc May 18, 017

2 the era f wave mechanics. Yet ne mre ptin has remained unanalyzed. Within the ambit f the classical Kepler prblem, reducing electrn s angular mmentum leads t the cntractin f elliptic rbit s at M 0 the trajectry f the electrn will aspire t a straight line. At the pint f maximum distance frm the nucleus, the speed f the electrn equals zer. This allws taking a new lk at the radiatin prblem during electrn s accelerating mvement in the field f nucleus. Preliminary estimates shw that the actin, A, made by the electrn during mvement at atmic distances has the rder f a Planck cnstant, h. Accrding t Max Planck s classic quantum thery there exists a minimal value f actin, h. S the actin f the electrn n the trajectry between tw statinary pints cannt nt be less than h. Thus, cnditin A = h sets electrn rbit parameters with a minimal distance frm the nucleus. The electrn mves in such an rbit with acceleratin, nevertheless, it cannt emit radiatin, because this wuld lead t reductin in its energy and speed, and thus reducing its actin belw minimum values practically pssible. This lgical cnclusin lks like the Bhr pstulate n the existence f special rbits f an electrn. Yet they differ essentially. While cntinuusly mving in circular rbit, an electrn gradually prduces mre and mre actin, and in excess f the minimum value (h) gains the ability t emit energy in the frm f electrmagnetic waves. The fact that this desn't ccur shuld certainly be pstulated. In ur mdel, mvement happens between statinary pints, while the lack f radiatin is a direct cnsequence f its minimal actin. Reasning in the same way, we reach a cnclusin abut the existence f a secnd rbit f an electrn, and mvement in such rbit between statinary pints satisfies cnditin A = h. In this case the actin A = h can transfrm int electrmagnetic radiatin, the residual actin f an electrn will be equal t the minimal value, i.e. h. Cntinuing with this line f reasning, we reach a cnclusin abut the existence f a set f rbits f an electrn in an atm meeting the general cnditin A = n * h. Fr single-electrn systems (hydrgen-like atms) the levels with n > 1 crrespnd t the excited states f atm. Experimental data shw that the lifetime f such excited states many times exceeds the rbital perid f the electrn (see belw fr a mre detailed explanatin). This indicates a critical feature f the actin f the Maxwell electrmagnetic thery at the micr level. The imperative nature f its actin peculiar t macr wrld, at the micr level is transfrmed t a prbable ne. An electrn mving with acceleratin and actin exceeding the minimum value h generates a quantum f electrmagnetic radiatin nly with a certain prbability which is significantly smaller 1. We can assume that the emissin f a phtn ccurs at the time f passing f an electrn at the minimum distance frm a nucleus, the ver flight f the nucleus, when the electrn s speed and acceleratin reach maximum values. And there the absrptin f a phtn takes place. This mechanism permits t remve ne mre cntradictin f the Bhr mdel which cannt explain hw the elec- 189

3 trn skips frm ne rbit t anther at the emissin r absrptin f a phtn. The electrn which is at the maximum distance frm a nucleus cannt absrb r emit a phtn. It can be remved frm its rbit mechanically, fr example, after impact with an external electrn r ther particle, including cllisin with ther atms. In cnclusin, let s discuss the prblem f multielectrn atms, in which electrns settle int rbits which have varius values f actin. Thugh the prbability f emissin f a phtn at the mvement f an electrn with A > h is very small, sner r later it will ccur, and as a result all electrns have t pass int an rbit with A = h. But as we knw frm experience, electrnic cnfiguratins f multielectrn atms in their basic state are steady, and in the absence f external influences it is pssible t speak, in principle, abut their infinite stability. This can be explained thrugh the peratin f the fllwing mechanism. A phtn is emitted when an electrn mving n a far rbit passes near a nucleus is absrbed by the electrn mving n a near rbit and als being at this mment near a nucleus. As a result, these electrns seemingly exchange rbits, and at the same time the electrnic cnfiguratin f the atm in general remains invariable. The time which an electrn spent near a nucleus is insignificantly small in cmparisn with the rbital perid, and during this time the vectr f speed changes the directin t the ppsite. S, frm a mathematical pint f view the mvement f an electrn in such an rbit will be equivalent t the scillatin f an electrn alng the straight line cnnecting it t a nucleus with a perfectly elastic reflectin frm the nucleus. Here after fr brevity s sake we will speak abut the scillatins f electrns.. Hydrgen Atm and Hydrgen-Like Atms Let s nw cnsider the mtin f an electrn in the nucleus field alng the line cnnecting their centers. Since the masses f the nucleus and electrn differ by three rders, it is pssible t disregard the nucleus mtin in the first apprximatin. Furthermre, we disregard the nucleus s wn dimensins and the nes f the electrn, and treat them as material pints. Accrding t Newtn s secnd law: d x az m = d t x where x distance t the nucleus, z nuclear charge. Multiply by dx/dt and integrate: 4π a = e ε0 m dx az = E dt x where E integratin cnstant, which has the dimensin f energy. dx az E dt = m x (1) 190

4 The actin f the electrn during mvement frm x = R1 = az E t the nucleus ( x = 0 ): az A = pdx = d d R mv x = m 1 R E x 1 R1 x az Ex 0 = m x az Ex + arcsin E az R az π mze = m = E 8 ε E Fr the unidirectinal scillatry mtin f the electrn its actin shuld be a divisible f h/: mze h = n 8ε E 1 () Frm this: 4 me z E = hn (3) 8ε which cincides with the Bhr equatin fr the energy spectrum f an electrn in a hydrgen-like atm. Maximum distance f an electrn frm the nucleus is given by: ε R hn n = (4) πme z Fr the first level f the hydrgen atm R 1 = 106 pm, which is twice the Bhr radius; and the same as the diameter f a hydrgen atm accrding t Bhr, which is lgical fr this methd f calculatin. In ur mdel it s meaningless t talk abut the radius f the hydrgen atm, but fr multielectrn atms the term acquires a clear physical meaning. Multielectrn atms pssess the shape clse t spherical ne with the radius equal t the utmst limit f the electrns mvement. The equatin f electrn mtin is btained frm the differential Equatin (1). Separating the variables and integrating, we btain: t m m az arc sin 1 Ex = x ax Ex 3 E + E (5) az The half-perid f scillatin we btain substituting in (5) extreme values x = R1 = az E and x = 0 : 3 ε τ 1 = h = h 4 ezm 4E Fr the first level f the hydrgen atm this value equals s. In general case: 3 3 ε hn 4 τ n = (6) ezm This implies that mst f the time an electrn is lcated near the brder f the atm: ne-third f the time in the range f (0.93 1) R 1 and a half in the range 191

5 f (0.84 1) R 1. Similar with the apprach f quantum mechanics, we can talk abut the prbability f finding the electrn at a certain distance frm the nucleus. On the ther hand, the lcatin f the electrn can be characterized by sme average distance frm the nucleus: x τ1 xt d 0 av = = 0.75 τ1 This rati can prve useful in calculating the characteristics f atms and mlecules because it permits apprximatin f sme electrns in an atm r an atm in the whle by a static mdel, in which electrns are lcated at sme fixed distances frm the nucleus. The phase f maximum distance frm the nucleus is als very interesting. R n ~n, τ n ~n 3. Frm this it fllws that the electrn in highest excited levels is lcated very far away frm the nucleus and fr an extremely lng time, as cmpared t the basic level. If we mnitr such an electrn, we can easily take it fr a free electrn that is nt assciated with any particular atm, mving with a slw speed and exchanging energy with ther free electrns. And suddenly this electrn takes ff and begins t mve with ever-increasing speed t its wn nucleus; then pushes away frm it, and after sme time again ges int a free mvement state. It seems t us that this physical picture is very similar t the quantum entanglement. 3. Helium Atm and Tw-Electrn (e) Atms The stable scillatins f tw electrns must be synchrnized. There are tw pssible mdes f scillatins: symmetric, when the electrns are mving simultaneusly t/frm the nucleus; and asymmetric, in which the lcatin f ne electrn near the nucleus crrespnds t the maximum distance frm the nucleus f the secnd electrn. Fr symmetric scillatins f tw electrns in e-atms: R 1 (7) The prblem is slved in explicit frm: m d x az a = + d t x x (8) ( ) m dx az a a = E = ( z 0.5) E dt x 4x x Hence, fr helium and e atms we have: an atm radius R R ( z ) 1 = H 0.5, where R H the radius f the hydrgen atm, z the nuclear charge; Energy f an electrn, in ev, E ( z 0.5) =, where ev energy f electrn detachment frm the lwest level f the hydrgen atm. The energy f the system, cnsisting f tw electrns (energy f e-atm), 19

6 equals t: ( z ) E = The energy required fr the detachment f the first electrn, r the first inizatin ptential, I 1 = E E 1, where E 1 the energy f a crrespnding hydrgen-like atm, E 1 = z. Fr helium we have E = 83.4 ev, I 1 = 8.83 ev, which exceeds the experimental data by 5.4% and 17.%, respectively. With increasing nuclear charge deviatin frm the literature data decreases and fr nen, fr example, it is +1.0% and +.% respectively. Evaluatin f electrns energy at the asymmetric scillatins gives the fllwing values fr helium: E = ev, I 1 = ev, which is higher than the experimental data by 14.5% and 46.5%, respectively. Thus, an ptin f the symmetric scillatins lks preferable. In general, the prpsed mdel f the atm can be called pulsing. Mst f the time the electrns are lcated n the periphery f the atm, peridically they simultaneusly rush t the nucleus, the atm rapidly cmpresses and immediately decmpresses, i.e. pulsates. The accurate slutin f the differential equatin can be btained fr tw-electrn atms. In principle, this can be dne als fr multielectrn systems, but it generates sme difficulties and incnveniencies. Therefre, belw we have used numerical slutin methds. 4. Multielectrn Atms Only tw electrn scan ccupy the first energy level. Perhaps in the future it will be pssible t justify this statement, yet we accept it as given, as a pstulate which fllws frm the ttality f experimental data n the structure f atms. Thus, all the electrns f the atms f the secnd perid, with the exceptin f tw electrns, are in the secnd energy level, fr which A = h.these electrns make synchrnus symmetric scillatins, that is, have the same scillatin frequency, and apprximately the same energy and the same range f scillatins (see belw).in the prpsed mdel there is n reasn fr the separatin t s- and p-electrns, and all electrns f this energy level are equivalent. T calculate the energy f an electrn in a multielectrn atm f the secnd perid f the Peridic table we have used the equatin: where Ee e, in 1 e e, in e e, ex j d 0 za 1 1 h R E E E x = x m (9) E j the energy f an electrn in the secnd level cntaining j electrns; energy f interactin f the electrn with ther electrns f the secnd level; Ee e, ex energy f interactin f the electrn with the first level electrns. The calculatin algrithm is as fllws. In the first step we calculate the energy f interactin between the uter electrns, which depends n their relative psitin. It is cnvenient t characterize 193

7 this interactin by parameter z, s that 1 Ee e, in = z ar. Assessment f the interactin energy with the first level electrns is carried ut using a static mdel with an arrangement f these electrns at a distance f 0.75 R 1 frm the nucleus and their apprximating by a sphere with a charge f. Then, in accrdance with the laws f electrstatics we can divide the integral (9) int tw: R1 ( z 1) a za 0 za za a h Ej dx+ Ej dx = (10) x x x x 0.75R1 m R 0.75R1 where R 1 radius f the first shell. Calculatins shwed that the R 1 value is nt dependent n the presence and the number f electrns at the secnd level, and cincides with the value R 1 f the crrespnding helium-like (e) atm. Substituting the values z and R 1 int Equatin (10), we find numerically the value E j and the value R crrespnding t it. In the secnd step we calculate the energy f the first level electrns, which unlike the radius depends n the number and lcatin f the electrns at the uter level. Here again we use the static mdel and apprximate the uter electrns by a sphere with radius 0.75 R and a charge j. Calculatins are carried ut accrding t the equatin, which is a cmbinatin f (9) and (10): 0 R1 j a za z a 1 h E dx = x x 0.75R m where z = 0.5. In the third step, we calculate the ttal energy f (je, e)-atm: 1 E = E + j E j 1 j In the furth step, we calculate the inizatin energy f (je, e)-atm, r (n j + 1)-the atm stepwise ptential cntaining n electrns n the filled uter level. I = E E n j+ 1 j j 1 This is t say, we avid adiabatic apprximatin and assume that the energy state f the inner level electrns als changes in the inizatin prcess. The inizatin energy is calculated as the energy difference between the equilibrium states f the atm. The main uncertainty in this mdel arises frm the chice f mutual spatial arrangement f electrn rbits an electrn cnfiguratin. A cnfiguratin with the maximum pssible distance between the electrns lks natural, since it prvides a minimum energy f electrn repulsin and thus a minimum f the ptential energy f the atm. E.g., fr 4, 6 and 8 electrns such ptimal cnfiguratins are the tetrahedrn, the ctahedrn and the cube, respectively. Hwever, calculatin f inizatin energy ptential f varius atms fr such cnfiguratins shws substantial differences frm the published (experimental and theretical) data. We have cnsidered varius ptins fr the lcatin f the electrns in space. 194

8 The best results were btained with a cubic system: eight cells are lcated in the crners f a cube and electrns cnsistently fill them with the frmatin f the fllwing cnfiguratins. This apprach may lk rather artificial: but, firstly, it is very similar t the ideas f Gilbert Lewis [3] and Irving Langmuir [4] abut the structure f electrn shells. Secndly, if we lk at the cnfiguratin with five electrns, we see smething like tday s accepted cnfiguratin fr the nitrgen atm with tw paired s-electrns and three unpaired p-electrns, and s n. Of curse, this scheme is very idealized and in reality these cnfiguratins may be highly distrted (even beynd recgnitin), but it is useful, in particular, t cnsider chemical bnding. When analyzing the cnfiguratins presented in Figure 1 we see that the electrns there are nt equivalent (beside and 8 electrns cnfiguratins), in particular, they have different energies f interactin with neighbring electrns. On the ther hand, in the prpsed mdel f a pulsing atm, all the electrns n the same shell shuld scillate with the same frequency, i.e. shuld be equivalent in this sense. This can be achieved by assigning different energy t different electrns and, accrdingly, a different radius f rbit. But in this case different electrns will prduce different actins during ne mvement cycle. Hw will this affect the actin quantizatin rule? We prpse t use here the principle f summed actin: in a system f interacting particles the summed actin f particles is quantized. This is in tune with Planck s idea f quantizatin in the system f interacting resnatrs [5]. As fr calculatins, this principle permits t use average energy f electrn repulsin and characterize the electrns n the same shell by the average energy values and the radius f their rbit - which greatly simplifies the calculatins. It shuld be nted that differences in the characteristics f different electrns in the atm are nt as great as they might seem at a first glance at the electrn cnfiguratin. Fr example, fr nitrgen atms, the half-perid f scillatins f uter shell electrns equals s, average energy ev, radius f the rbit-85.5 pm. Each electrn has the energy f 76.08, 78.34, 78.34, 78.34, and ev with the rbit radius 84.3, 85.5, 85.5, 85.5, and 86.6 pm, crrespndingly, that is, the deviatin frm the average values des nt exceed 3% fr energy and 1.5% fr radius. Figure 1. The rder f filling f the secnd electrn shell f an atm. 195

9 We have calculated the ttal energy f the electrn shells f all the elements f the secnd perid f the Peridic table fr different cnfiguratins f shells. Their deviatin frm published values des nt exceed 1%. Fr inizatin ptentials (Table 1) the discrepancy frm the published data is much higher, but verall is quite acceptable. We nte als that the inizatin ptential is calculated as the difference between tw large numbers, e.g., the first inizatin ptential f nen is nly 0.5% f the ttal energy f the electrn shell, s the errr in its calculatin must be inevitably large. 5. Cnclusins Thus, the prpsed mdel f a pulsing atm can be summarized in the fllwing theses. 1) An electrn in an atm is cnsidered as a material bject mving alng a particular trajectry. ) The trajectry f the electrn s mtin is an ellipse with a minr semiaxis, tending twards zer. Frm a mathematical pint f view the mvement thrugh such an rbit is equivalent t scillatin with a perfectly elastic repulsin frm the nucleus. 3) At the pint f maximum distance frm the nucleus the speed f the electrn drps t zer (statinary pint). This distance crrespnds t the majr semiaxis f the elliptic trajectry. We call this the radius f the rbit. Fr uter shell electrns this is equivalent t the radius f the atm. 4) The actin prduced by electrns in mvement between statinary pints is discrete and prprtinal t a Planck cnstant, A = n h, where n = 1,, 3,. This cnditin sets the allwable values f the electrn energy and the radius f their rbit. Table 1. The calculated inizatin ptentials (in ev) f the atms f the secnd perid f the Peridic Table and their deviatin frm the literature [6] values (in%). Element I 1 I I 3 I 4 I 5 I 6 I 7 I 8 Li 5.7. Be B C N O F Ne

10 5) An electrn mving in an rbit with A = h, des nt emit electrmagnetic waves. When mving in rbits with A > h, the emissin f a phtn is pssible, and it ccurs at the maximum speed f an electrn s mtin, near the nucleus. Emissin is prbabilistic in nature. A phtn emitted by an electrn can be absrbed by anther electrn which is als mving clse t the nucleus. 6) Mst f the time the electrn is lcated n the periphery f the rbit (the atm), and peridically rushes t the nucleus and rebunds back. The average distance f the electrn frm the nucleus is 0.75 R, where R radius f the rbit. The frequency f scillatin f the electrn is f the rder f Hz. 7) Electrns in multielectrn atms are lcated n the electrn shells crrespnding t A = n h, where n = 1,, 3,. It is pstulated that the first shell can be ccupied by electrns, the secnd by 8 electrns, etc. 8) Electrns n the same shell perfrm symmetric synchrnus scillatins. 9) The frequency f their scillatin is the same, and in this sense all the electrns n ne shell are equivalent. Separatin int s-, p-, etc., electrns is excluded. 10) T calculate the energy f multielectrn atms, the principle f summed actin is used: in a system f interacting particles the summed actin f particles is quantized. References [1] Bhr, N. (1913) On the Cnstitutin f Atms and Mlecules. Philsphical Magazine, 6, [] Smmerfeld, A. (1916) Zur Quantentherie der Spektrallinien. Annalen der Physik, 51, [3] Lewis, G.N. (1916) The Atm and the Mlecule. Jurnal f the American Chemical Sciety, 38, [4] Langmuir, I. (1919) The Arrangement f Electrns in Atms and Mlecules. Jurnal f the American Chemical Sciety, 41, [5] Planck, M. (1901) Uber das Gesetz der Energieverteilungim Nrmalspektrum. Annalen der Physik, 4, [6] Sansnetti, J.E., Martin, W.C. and Yng, S.L. (004) Handbk f Basic Atmic Spectrscpic Data. Natinal Institute f Standards and Technlgy, Gaithersburg, USA. 197

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