10 - THE HYDROGEN ATOM.
Some features of the hydrogen atom can be described with the aether of aetherinos presented in this work. As said at the end of the Annex A, for the case of an electron orbiting a proton, the net force acting on the electron at a given epoch t, is the vector sum of the electron "autoforce", whose components are given in [A-34,35] , and of the "centripetal" force of the proton, whose components may be deduced from [A-20...].
Previous test calculations have been made of the electron’s autoforce using "plausible" residual distributions for the electron. It has been found that the numerical values taken by the integrals [A-34] and [A-35] do not change sensibly if the lower limit of integration is assigned to an epoch corresponding to a few orbits before the epoch of observation of the force instead of going back to tE = - ¥. But for the small time interval taken by the electron to complete a few orbits it can be admitted that the IO speed of the electron does not decrease sensibly and therefore the approximation f(t) =1 can safely be made in the equations giving the autoforce and the proton force. This means that in the analysis of the hydrogen atom it will be supposed that the rate of advance of the Ideal clock is approximately the same to that of the Official clock during the small time intervals that have influence in the pertinent forces. It also implies that it can be assumed in the calculations that, in the reference frame of the nucleus, an electron in a circular stable orbit moves at constant speed, not only for OO but also very approximately for IO, while assuming at the same time that the aetherinos move in straight lines at constant speeds. The letter t will be used to represent ‘time’ assuming therefore that the results of this section do also describe in a straightforward way what the Official observer sees.
The analysis will start with the supposition that the electron travels in a circular orbit with the proton in its centre. A posteriori it will be examined under what conditions the forces involved are actually able to maintain this situation.
Fig [10-1]
Considering that the force suffered by the electron at a given epoch t is the result of the vector addition of the force exerted by the proton with the electron’s "autoforce" (due to the retarded influence of the electron on itself as explained in Annex A), the sub index P will be used for the proton force and the sub index A for the autoforce.
These forces suffered by the electron at a given epoch t will be analyzed into two components that are taken respectively tangent and perpendicular to the orbit. Without loss of generality it will be supposed that at the epoch t of observation of the forces the electron is passing the position shown in Fig[10-1] for which the tangent to the orbit proceeds along the axis X of the lab reference frame XY. Therefore the subindex X will be used to label the vector components ‘tangent’ to the orbit and subindex Y to refer to the components perpendicular to the orbit (i.e. centripetal component).
The stability conditions for a circular orbit can be expressed by the simultaneous fulfilment of the following 2 equations:
[10-1]
[10-2]
Equation [10-1] states in the usual way the equilibrium between the centripetal and the centrifugal force.
Equation [10-2] states that the tangential force must be zero (in the present supposition of circular orbits) or otherwise the electron would increase or decrease its speed moving therefore to another orbit.
Numerical evaluations have next been made for the 4 pertinent force components FPX , FPY , FAX , FAY (using the equations of Annex A) for different values of the electron speed Ve and for different radii R of the electron. The proton and the electron redistributions that have been used in these evaluations are of the simplified type presented at the end of Annex D (see D-23):
[10-3]
where r(u) is the so called canonical speed distribution of the aether:
where N is a multiplicative constant related with the total number of aetherinos per unit volume and VM is the speed of the more abundant aetherinos (i.e. the speed for which the distribution reaches its maximum).
The redistributions of the proton rP(v) and of the electron re(v) will be supposed to differ only by the value of the constant k appearing in the expression (10-3).
As an example, taking for the constants the values N =1 , ke = 0.9 , VM = 1 yields the following graphic for the electron redistribution re(u):
Fig [10-5]
Fig[10-5] represents a "heating" redistribution. The aetherinos after colliding with an electron have on the average higher emerging speeds than the incident ones. For the proton, taking for example kP = 1.1 the opposite happens (i.e. the proton redistribution is approximately equal to -re(u) and therefore of a "cooling" kind). But the use of the words "heating" and "cooling" is just for intuition purposes and does not imply that the standard concept of temperature is applied to the aether, since the aetherinos are basic entities that have no mass (but are responsible of its appearance as a physical measurable magnitude).
NOTE 10-1
It may seem that the proton and electron redistributions are chosen ad hoc in different sections of this work. These sometimes different elections try to give the maximum simplicity without shadowing description features of interest for the phenomenon being studied. But it can be checked in the corresponding calculations that, as long as the function representing the electron redistribution behaves qualitatively like the one shown in Fig[10-5] and the proton redistribution approximately as its negative, the exact mathematical function chosen gives the same qualitative predictions and has only a very small quantitative influence on the values of the forces deduced from it. Furthermore, as has been said before, the purpose of the present version of this work is only to give a qualitative feeling of the descriptive capabilities of an aether of aetherinos. The final expressions proposed for the electron and proton redistributions are left for a future version of this work where, G.W., a more precise quantitative description of several phenomena will be pursued.
______
A numerical evaluation of these aetherinical forces has been made with the initial supposition that both the proton and the electron redistributions are isotropic and independent of the absolute velocity of these particles relative to the local aether. The above mentioned equations of Annex A, (A-20, A-34, A-35) have been used for the evaluations.
The evaluations show that the centripetal components of both the proton force FP and the electron autoforce FA depend on the radius R of the electron orbit in the usual 1/R2 way. It can also be seen that both forces depend also on the orbital speed Ve of the electron.
The force FAY ( centripetal component of the autoforce) acts in the same semidirection as the standard proton attractive force, i.e. tends to pull the electron towards the proton. But for small values of the electron speed ve , i.e. for ve << VM this component of the autoforce is much smaller than the corresponding component of the proton force, i.e. FAY << FPY and they only become of the same strength for ve of the order of VM (see Fig(10-8) below).
The following 4 graphics represent the evaluations of the perpendicular to the orbit
component (Y component for the situation of Fig 10-1) of these forces in the range of ve
between 0 and VM.
Fig [10-6] Fig [10-7]
Fig (10-8) Fig (10-9)
For graphics (10-6) to (10-9) it has been taken:
VM = 1, ke = 0.9, kP = 1.1, R = 1,
in which case the electrostatic force (ve = 0 ) suffered by the electron due to the attraction of the proton takes (in arbitrary units) the value FPY = - 0.06
Fig (10-8) represents (in the same arbitrary units) both the proton force FPY (upper curve) and the autoforce FAY (lower curve).
Fig (10-9) shows that the speed dependence of the net centripetal force FPY + FAY is small up to electron speeds of the order of VM/2 . The increasing dependence at higher speeds is mainly due to the autoforce as can be seen in Fig(10-8).
____________
NOTE 10-2
The exact integrations for FPY and FAY could not be obtained but with the help of numerical integrations approximate expressions have been guessed for these forces giving:
* An approximate expression for the perpendicular to the orbit component of the proton force:
[10-6]
where k1 is a positive constant that depends on the constant kP of the proton redistribution,
and where k2 is also a positive constant that seems to depend only on the constant kP of the proton redistribution and can be approximated by
k2 @ kp - 0.1 [10-7]
* An approximate expression for the perpendicular to the orbit component of the (electron) Autoforce:
[10-8]
where kA is a positive constant that depends on the constant ke of the electron redistribution in roughly the following way:
[10-9]
that for ke = 0.9 gives kA @ 0.054
____________
Numerical evaluations have next been made of the tangential components FPX and FAX of both the proton force and the electron autoforce. The following 3 graphics (Figs 10-11, 10-12 and 10-14) represent the evaluations of the tangential to the orbit component (X component for the situation of Fig 10-1) of these forces in the range of ve between 0 and VM. Positive values of Fx indicate that the force acts along the semidirection of the electron’s velocity tending to increase its speed.
Fig (10-11) Fig (10-12)
Fig (10-14)
For graphics (10-11) to (10-14) it has also been taken:
VM = 1, ke = 0.9, kP = 1.1, R = 1,
Fig (10-14) shows that (FPX + FAX) < 0 for all ve . So there is always a net dragging force that tends to slow down the electron and the condition Eq(10-2) is never fulfilled.
___________
NOTE 10-3
Other elections of the redistributions constant k (for example ke = 0.8, kP = 1.1 ) produce a faster increase of FAX with ve , and an electron speed vo is reached for which (FPX + FAX) = 0. But for ve > vo the net tangential force remains now (FPX + FAX) > 0. So in this case there is a single orbit (ve = vo ) for which the stability condition Eq(10-2) is fulfilled. This is obviously insufficient to explain the spectral lines of the hydrogen atom.
It is not passing unnoticed that the election of the constants ke and kP has a priori important consequences in the electrical neutrality of the atom. Furthermore, recalling that the aetherinical forces between material particles described in this work are 'relative velocity' dependent, if a hydrogen atom is composed of a rest proton and an electron with speed v1 the redistribution constants can be adjusted ad hoc so that 'on the average' the atom is electrically neutral. But if later the electron is imagined to be in another orbit with a different speed v2 the atom will no longer be (on the average) electrically neutral. This problem together with the model's interpretation of the electrical charge will be treated in a next version of this work.
(Note: In this context the expression 'on the average' can be interpreted as the mean force of a full sample of spatial arrangements in which a test charge is placed in presence of a hydrogen atom of a given state (and not subject to magnetic nor electric external fields) ).
___________
Although evident, it must be remarked here that, due to the isotropy of the proton and the electron redistributions assumed above, for a given electron speed ve and a given orbital radius R all the forces studied above FPX, FPY , FAX , FAY are constant in time.
_________
A new evaluation of these forces has been made with the supposition that both the proton and the electron redistributions are anisotropic (but otherwise independent of the absolute velocity of these particles relative to the local aether) together with the supposition that these anisotropies rotate in space at a constant angular speed.
These suppositions of the proton and the electron redistributions can (for example) be implemented as follows:
(1) It will be supposed that both the proton and the electron have a preferred symmetry axis in what respects the intensity of its aetherinical redistributions. Calling j the angle of emergence of some given redistributed aetherinos relative to the redistribution axis of the particle, it will be supposed that the aetherinos speed’s redistribution varies in strength according to the relation:
[10-15]
where r(u) is the redistribution defined in Eq [10-3].
(2) It will be supposed that a proton’s redistribution axis rotates in space with angular speed wP and with a spin axis perpendicular to the redistribution axis. (The flows of aetherinos emerging the proton somehow ressemble a lighthouse beacon).
Note: The rotation of a redistribution axis will be called spin to avoid confusion with the rotation of the electron along its orbit. Furthermore the word ‘axis’ will be avoided as far as possible when referring to the ‘spin’ axis to avoid confusion with the redistribution axis. The ‘spin vector’ will be used instead and the word ‘axis’ will be used as far as possible only in the context of the ‘redistribution axis’.
(3) It will similarly be supposed that an electron’s redistribution axis spins with angular speed we and with a spin vector perpendicular to its redistribution axis.
(4) It will be supposed that, for an electron bound to a proton, the electron’s and the proton’s spin vectors remain parallel to each other and perpendicular to the electron’s orbit (i.e. the proton’s and the electron’s redistribution axis remain confined to the orbital plane).
Fig [10-17]
Two possibilities have been found of interest in what respects the assignments of the spin vectors wp and we. These possibilities are shown in Fig(10-17). In case (1) wp and we are equal to each other and parallel to the angular orbital rotation vector of the electron. In case (2) wp and we have also been taken equal to each other but anti parallel to the angular orbital rotation vector of the electron.
w
p = we = w [10-17]In what follows it is case (1) that will be assumed.
New numerical evaluations of the proton force and of the autoforce have been made assuming such anisotropy and such spin in the redistributions:
It can be checked that now, for a given electron speed ve
and a given orbital radius R, both the proton force and the autoforce oscillate in time.Consider first FPX (i.e. the tangential component of the proton force).
For a given ve and R this force oscillates with a period of repetition TPX that has been found in the evaluations to be given by:
[10-18]
Eq [10-18] is a quite evident relation considering that, in good logic, for given ve and R, the phase of the force FPX detected by the electron at a given epoch t should only depend on the angle c that the proton’s redistribution axis makes with the direction P_E (joining the proton with the electron) at that epoch t, because if the orbital angular speed ve/R and the proton’s spin w have been constant in the past, all the aetherinos of a given speed coming from P and arriving at E at an epoch of observation in which the redistribution axis of P makes the same given angle c with P_E , must have emerged with the same angle j relative to the proton’s redistribution axis at that epoch of emergence and therefore the net force suffered by the electron at these epochs of the same c , is integrated by the same contributions when adding for all the aetherinos (speeds) that contribute to the force. The repetition period TPX is therefore the time elapsed between 2 consecutive situations in which the redistribution axis of the proton makes the same angle with the direction P_E. Considering that this direction P_E is itself advancing at an angular speed ve/R and considering that the redistribution axis is by definition a symmetry axis (with the redistribution repeating itself on both sides of its ‘equatorial plane’), a redistribution axis tilt of p radians is undistinguishable from the first and therefore TPX may be deduced from the condition:
[10-18b]
Fig [10-18]
Fig[10-18] illustrates the repetition period of the proton force on the electron. An electron of constant speed in a circular orbit around a proton suffers at time t0 the same force as at time t0+TPX , because in this example at both epochs there is one end of the redistribution axis of the proton pointing to the electron so in both cases the aetherinos arriving at each given speed v have departed the proton at an epoch the same amount earlier of their respective arrival and therefore in both cases they departed the proton making the same angle with the proton’s redistribution axis. The same is true for all the speeds v of the aetherinos arriving at the electron at these two observation epochs t0 and t0+TPX.
Consider next FAX (i.e. the tangential component of the electron’s autoforce).
It has also been checked in the corresponding evaluations of this force that for a
given ve and R this autoforce
oscillates with a period of repetition TAX equal to TPX
and therefore given by: 
[10-19]
Eq [10-19] is again a quite evident relation (see now Fig[10-19] ) due to the same reasoning applied above to FPX.
Fig [10-19]
But all that has just been said in relation with the tangential components of the proton force and of the autoforce is also true for the centripetal components of these forces. I.e. for a given radius and electron speed, FPY and FAY do also oscillate in time with the same repetition period.
Summarizing: due to the equality of the spins given in Eq[10-17] the repetition periods of FPX , FAX , FPY and FAY are equal and they will all be simply called TF :
[10-20]
Although in Fig[10-19] the redistribution axis of the proton and of the electron have been drawn parallel to each other this is not a necessary condition to obtain the results of this section (see below). They may differ by a constant angle f that, due to Eq[10-17], will be the same at all epochs.
The numerical evaluations show that the forces FPX , FAX , FPY and FAY are made of the addition of a pure harmonic function with a non oscillating function. They can be expressed in a somewhat general way as:
[10-21]
[10-22]
[10-23]
[10-24]
An interesting result of the evaluations has been to observe that, admitting that the angle f (between the proton’s and the electron’s redistribution axis) does not vary when the electron performs an orbit change, the phase difference qPX - qAX between the sine functions of FPX and FAX depends on R. And similarly, the phase difference qPY - qAY between the sine functions of FPY and FAY does also depend on R. It has been observed in the evaluations that this is due to the fact that the phases qPX and qPY of the proton force components show a strong dependence on the radius R while the electron’s autoforce phases qAX and qAY do not. Or more precisely:
If the phase Q(FP) of the proton force FP is evaluated at an epoch in which the proton’s redistribution axis makes a given angle c with the direction P_E, then this phase Q(FP) depends not only on c but, for a given c, changes ‘quickly’ with R in a monotonous way as this radius R is increased. But on the other hand, for a given c, the phase Q(FA) of the electron’s autoforce changes instead in comparison very ‘slowly’ as the orbital radius R is increased
Notice that letter Q is being used for the net phase of a force at a given
epoch and letter q for that part of the phase that does not depend on the
epoch t of observation of the force. I.e.
looking at the sine argument of Eqs [10-21…24] it can be assumed that:
Q = 2p t /TF + q
[10-25]
Note: The above assertion that for a given c, the electron’s autoforce q-phase changes very slowly with R seems to be valid only for orbits in which R is not too small or more precisely for orbital angular speeds ve/R smaller than 'say' 2w/3 . But in any case, for a given w it is easy to find a wide interval of orbital radii along which the proton’s force q-phase changes by many integer multiples of 2p while at the same time the electron’s autoforce q-phase changes by some quantity smaller than p.
The different evolution of the phases of the proton’s force and of the autoforce as R increases has the consequence that there can be found a discrete set of radii Rn for which those phases are in opposition. For example ‘several’ values of R have been found for which:
Q
(FPX) = Q(FAX) + p [10-26]and for which
Q
(FPY) = Q(FAY) + p [10-27]This phase opposition represented in Eqs[10-26] and [10-27] is of course valid for all epochs t considering that all these forces oscillate with the same repetition period TF .
Note: It is suspected that, using the above suppositions, not ‘several’ but ‘infinite’ values of R can be found for which the phase of the proton force is in opposition with that of the autoforce. But it must be recalled that the evaluation of the pertinent forces has been made with numerical integrations, since the exact integrals (giving the exact expressions for the forces) could not be obtained. But for high values of R, the numerical evaluations become either unreliable or too slow.
The thesis here is that a circular orbit can only be stable if the phase of the proton force is in opposition with the phase of the autoforce.
This phase opposition is considered a necessary condition but not a sufficient one. It seems reasonable at first sight that the stability of an orbit requires the exact fulfilment at all epochs of Eq[10-2]. But for this: (1) the amplitudes of the sine parts of FPX and FAX should be equal, and (2) the non oscillating parts should also be equal in magnitude and of opposite sign. I.e. using Eqs[10-21] and [10-22] the stability condition of Eq[10-2] requires:
[10-28]
Furthermore it seems reasonable (at first sight) that the stability of a circular orbit requires that the net centripetal force FPY+FAY does not oscillate. I.e.
[10-29]
But with the present suppositions about the redistributions, it has been so far impossible to adjust the constants so as to fulfil quantitatively these requirements [10-28] and [10-29]. For all reasonable elections of the constants (ke , kp, w, …) the proton force is stronger (by several orders of magnitude) than the autoforce...
For the proton force FP it has also been found in the evaluations that :
-Whatever radius R, speed ve and spin w, the phase of the tangential and the centripetal components of the proton force differ by p , i.e.
Q
(FPY) = Q(FPX) + p [10-31](notice that both phases in [10-31] correspond to the proton force and therefore this observed relation should not be confused with Eqs [10-26] and [10-27] which occur only for some radii).
-The time average of FPY is proportional to 1/R2
-The time average of FPX is approximately zero (whatever the radius R, speed ve and spin w).
-The amplitude of the time oscillations of FPY , i.e. the function BPY(R,ve), decreases with R faster than 1/R2
-The amplitude of the time oscillations of FPX , i.e. the function BPX(R,ve), decreases with R faster than 1/R2
- For a given t (or equivalently for a given angle c), the phases of the proton force components Q(FPY) and Q(FPX) (related by Eq[10-31]) have been found to change in a monotonous way as R is increased. This phase change with a radius increase R, has been found to be given by:
[10-32]
where VM is as usual (see for example [10-4]) the speed for which the aether canonical distribution reaches its maximum, and where qPK is a constant. The relation [10-32] holds as long as R is "not too small".
Stable orbits
It was said above that the evaluations show that the proton’s force q-phase changes by many integer multiples of 2p while at the same time the electron’s autoforce q-phase changes by some quantity smaller than p. Admitting that for the orbital radii R of interest the electron’s autoforce q-phase does not change at all when R is increased, then for a given t (or equivalently for a given angle c), the phase of the x component of the autoforce can be expressed as:
qAX(R) = qAK [10-34]
where qAK is a constant.
The phase difference between the proton force and the autoforce is therefore:
[10-35]
at all epochs (since, see [10-25], the time dependent term 2p t /TF is common to both forces FPX and FAX ).
It has also been found in the evaluations that "supposing that the proton redistribution axis forms at all epochs an angle p/4 with the electron redistribution axis" (instead of being parallel like in Fig 10-19), then qPK - qAK is equal to p (or equivalently to –p) and [10-35] can be written as:
[10-36]
Eq [10-36] means that each time R takes a value such that
[10-37]
where n is an integer then the forces FPX and FAX are in phase opposition at all epochs which is one of the conditions considered necessary for the stability of an orbit. It is expected (see Note 10-45 below) that the other necessary stability conditions [10-29] (e.g. the equality of the amplitudes of the oscillating terms) will also be predicted in a revised version of the hydrogen atom model. The analysis is continued admitting the possibility of success of such revision.
Eq [10-37] means that the radii Rn for which a circular orbit is stable are given by:
[10-38]
Supposing now that, according to the classical non relativistic point of view, the electron orbiting speed ve is related with the orbital radius R by the relation deduced from equating the Coulomb centripetal force with the Newtonian centrifugal force:
[10-39]
where K is a constant and e is the charge of the proton, (the charge of the electron being –e). The frequency of revolution n of an orbit of radius R can be obtained from [10-39] dividing both members by R, taking the square root and dividing by 2p:
[10-40]
and substituting R by its allowed values given in [10-38]:
[10-41]
Supposition 10-42
The radiation emitted when the electron of an hydrogen atom jumps from an orbit of radius Rm to an orbit of radius Rn corresponds to the superposition of the "radiations" emitted by both harmonic orbital movements. But the superposition of 2 harmonic waves of frequencies nm and nn is well known to originate amplitude pulses of a frequency given by: nmn = nn - nm and therefore
[10-43]
in analogy with the experimental Balmer formula.
NOTE 10-44
It remains to show that according to the description of light made by the model (see some of this description in Section 6), the different speed "flows" of aetherinos (necessary to explain the constancy in the speed of light) produce at long distances from the source a destructive interference with themselves when the emission is originated by a finite pure harmonic oscillation. But, when the emission consists of a long lasting harmonic oscillation followed in time by another long lasting one of different frequency, the interference of the aetherino flows is no longer destructive at long distances of the source. (This result is being subject to preliminary tests and can not be presented yet).
Notes
:Eq[10-43] leans on a mixture of ‘classic Bohr Physics concepts’ with ‘aether model concepts’. Again, the main intention has been to give a qualitative feeling of the possibilities of the model. For this purpose it has been thought that it would be simpler to make a preliminary description using as the centripetal force the simple classic Coulomb expression instead of using the model magnitudes describing such centripetal force suffered by the electron. But, as said above, the evaluations show that the centripetal force predicted by the model (by addition of FPY and FAY) does not behave as simply as the Coulomb force; because, although the time average of the proton part of the centripetal force has been seen to change as 1/R2 the autoforce part diminishes faster than that. But the new description paradigm presented in Note 10-45 is believed able to give model predictions of the centripetal force in accordance with 1/R2.
NOTE 10-45
It has recently been estimated that there is not much hope to meet all the above stability conditions (in particular Eqs 10-28 & 10-29 ) in a quantitative way by just introducing small changes in the electron and proton redistributions and/or in the aetherino’s speed distribution of the local aether. It seems that for increasing atom radii the autoforce can not catch up in strength with the proton force. This does not mean that the autoforce is just an ad hoc concept that must be thrown away since it has not been able to account by itself with the adequate prediction of stable orbits. By the contrary, the autoforce is a clear unavoidable prediction of the model but whose strength becomes significant only for small orbital radii. Perhaps it will prove more interesting in the domain of nuclear forces. But in the new study that is now in progress to fully account for the stability conditions of electron orbits in the hydrogen atom it will be supposed:
- For the radii pertinent to the problem the electron autoforce is negligible compared with the proton force.
- The spin paradigms of both the proton and the electron are maintained in a much similar way as explained above. I.e. the proton produces an anisotropic redistribution that spins at angular speed w in analogy with a lighthouse. The electron also has an internal spin of the same angular speed w.
- But now, the important feature of the electron spin to be accounted for is not its lighthouse kind redistribution that shines on itself (or on other bodies) but the fact that such spin is a manifestation of an inner structure that can be modelled by an internal movement of simple particle components of periodicity 1/(2p w).
- The aetherinos implementing the proton force act directly on the internal moving parts of the electron. But since the internal movement of the electron parts has the same periodicity 1/(2p w) as the lighthouse kind proton redistribution, then, an interference is predicted that will reinforce the time-oscillation amplitude of the proton force for given radii and cancel such oscillation for other (stable) radii.
(Although this section needs a lot more work to be satisfactory enough for the author, specially in relation with the implementation of Note 10-45, the interested reader will have to wait a few months or try the calculus by himself).