The Planck blackbody radiator equation (w is wavelength);
# = (2*p*h*c^2)/(w^5*((EXP((h*f)/(k*t)))-1))
A multiplier is necessarily attached to the end of the equation to align
the result with the desired units. # * 1.0572E-10 does the trick for watts
per meter squared at 1 meter radius.
If EXP(e^1) is permitted to vary, the possible values for h are endless. Planck's constant is dependent entirely on e being logically sound for the role it plays. But there's really no reason why it should be better suited than some other constant which is directly linked to e. pi is apparently one such constant. Another is the Compton wavelength, which is supposedly derived from h (being fundamentally linked to e).
This is a list of alternative h and e combinations which were each
automatically generated and plugged into Planck's equation and run for the
500 incrementing wavelength steps in a program I use to generate blackbody
curves.
# = (2*pi*h?*c^2)/(w^5*((e1?^((h?*f)/(k*t)))-1)). The multiplier aligns
the results with a scale in watts per m^2 at 1 meter radius. The multiplier
alters inversely to h?, while e^1? alters at an inverse squaring rate to
h?.
(halving values of h) h? e^1? Multiplier 6.626E-34 2.71828 1.0572E-10 3.313E-34 7.389047 2.1144E-10 1.6565E-34 54.59801 4.2288E-10 8.2825E-35 2980.943 8.4576E-10 4.14125E-35 8886020 1.69152E-09 2.070625E-35 7.896135E+13 3.38304E-09 1.035312E-35 6.234896E+27 6.76608E-09
This next set was generated for doubling values of h. Each of these two sets push the limits of an 8 digit calculator. The equation covers a h? range between 1.035312E-35 and 6.947864E-28
h? e^1? Multiplier 6.626E-34 2.71828 1.0572E-10 1.3252E-33 1.648721 5.286E-11 2.6504E-33 1.284025 2.643E-11 5.3008E-33 1.133148 1.3215E-11 1.06016E-32 1.064494 6.6075E-12 2.12032E-32 1.031743 3.30375E-12 4.24064E-32 1.015748 1.651875E-12 8.48128E-32 1.007843 8.259375E-13 1.696256E-31 1.003914 4.129688E-13 3.392512E-31 1.001955 2.064844E-13 6.785024E-31 1.000977 1.032422E-13 1.357005E-30 1.000488 5.162109E-14 2.71401E-30 1.000244 2.581055E-14 5.428019E-30 1.000122 1.290527E-14 1.085604E-29 1.000061 6.452637E-15 2.171208E-29 1.000031 3.226318E-15 4.342415E-29 1.000015 1.613159E-15 8.684831E-29 1.000008 8.065796E-16 1.736966E-28 1.000004 4.032898E-16 3.473932E-28 1.000002 2.016449E-16 6.947864E-28 1.000001 1.008225E-16And this set, which isn't taken to any limit, was generated from this equation;
h? e^1? Multiplier 1 1.0145 3.959E-26 .5 1.02921 7.918E-26 .25 1.059274 1.5836E-25 .125 1.122061 3.1672E-25 .0625 1.259022 6.3344E-25 .03125 1.585136 1.26688E-24 .015625 2.512655 2.53376E-24 .0078125 6.313437 5.06752E-24The possible combinations are enormous.
This is the final image after all of the above combinations had been cycled through the program and overlaid on top of each other. For obvious reasons a different color was chosen for each set. The blackbody temperature was 6000 K.
18000 data points tested OK. And that applies for any temperature radiator.
It's assumed that the Compton wavelength is derived from h;
h/(mc) = w (w is the Compton wavelength). m is the electron rest mass. But
the assumption that h is derived from the Compton wavelength is equally as
valid; h = wmc. The constant e is linked to w, m and c through Planck's
constant.
This is the modified equation for a Compton wavelength replacement for
Planck's constant. The replacement for e is necessarily 3.9e+8.
# = 2.426e-12 /(w^5*(( 3.9E+08^ (( 2.426e-12 *f)/t))-1)). The relevant
power multiplier is 1.6326E-14. The numbers, 2, pi and c^2 can be included
if the multiplier is decreased by their combined value, but the removal of
Boltzmann's constant is essential to bring the equation within the range
of a normal calculator.
Using the Compton wavelength photon, the fine structure constant
becomes,
e^2/(2*e0*w*m*c^2): e is the electron charge in this case. But
such a photon would carry 3.66e+21 times the energy of a Planck photon and
would have trouble explaining the Compton or photoelectric effects.
The rest of this page is dedicated to demonstrating why the Compton wavelength is the proper constant relating to all aspects of E/M radiation and that Planck's constant is a consequence of that, having very little relevance other than to link e to the Compton wavelength.
This animation depicts an E/M wave being formed by the oscillatory motion of an electron relative to its local charge structure, and the wave's effect on the charge structure of the receiver at the base. The red circles attached to each wave phase indicate the peak of each half wave and the motion direction of the electron which was recorded in each transmitted phase. The receiver electron will obviously react as shown unless the receiver electron's resonant frequency within its local charge structure is the same as the incident ray frequency. In that case, the wave-electron relationship is shifted by 180 degrees.
Compare this with any radio transmitter and receiver.
In the next simple animation, the oscillating electron generates waves which diverge into the plane of dimension across the screen as they move outward from the source and the power carried outward to any point necessarily reduces at a linear rate per distance. Including a diverging z plane, received power at any point reduces at a squaring rate per distance. But whatever the case, the total accumulated power received over the area of each arc is always the same.
The motion of the electron in the y plane (left-right) has encompassed a specific distance in a specific time and although the recording of that event is expanding, it still carries exactly the original information to every single point around any of the plotted arcs.
From the viewpoint of an electron located on the top arc between the blue vertical lines, which is oscillating back and forth in apparent synch with its counterpart at the base, its path will be constantly adjacent with the path of its counterpart, and it will measure the travel time and distance of its counterpart as being exactly the same as its own. The recorded time-distance obviously cannot diverge along with the wavefront otherwise the total power received over each arc would increase with distance. Intensity only, has diminished.
In the same way, the time and travel distance of an oscillating electron which generates an E/M wavetrain remains permanently fixed in the wave until the recording interacts with matter somewhere.
No matter how many common waves reinforce in one place, the maximum rate that an electron can be shifted is permanently fixed in each wave.
IT CAN NEVER CHANGE.
In generating one phase of a wave from a thermal energy source, or anything that is driving the energy field, an electron is driven by fluctuations in the field and is necessarily accelerated from zero to a maximum speed and back to zero again according to the cycle time and energy carried in the fluctuation. If the energy carried in a fluctuation is directly proportional to frequency, the electron will always be driven by the same distance before the next phase sends it back in the opposite direction. The travel distance per cycle is constant (motion constant).
Example:
If 1 energy unit applied for 1 second is 1*1 = 1 unit/sec, then 2 energy
units applied for .5 seconds is 2*.5 = 1 unit/sec. It will always be the
same.
Alternately; constant = (c / f) * (m * f): (c/f) is time: For (m*f), f gives the relative driving power applied to the electron. m is the electron rest mass. The result is always (inconsequentially?) 2.7327e-22.
In order to generate the required maximum ejection speeds which compare with the curve generated by both equations listed below, the constant distance over which the electron will be driven is necessarily 2.426e-12 meters, or the Compton wavelength. Nothing else fits the evidence.
The two curve plots in the next image were generated simultaneously from these two entirely different equations. The blue dots inside the small red circles (if you can see them) were generated from # = sqr(2*(h*f-h*w)/m) being a variant of Einstein's photoelectric equation : # = photoelectron speed : w = work function : m = electron mass, while the equation which generated the blue circles is # = sqr(f-w)*(t*ex) : t = time over a half wavelength (c/f/2 sec). Each half cycle will eject an electron in opposite directions. ex is the electron motion constant (explained above). The result from each is multiplied by sqr(1-#^2/c^2) to limit the maximum ejection speed to that of light.
The chosen work function is 3E+14 hz. The initial frequency is 1E+16 hz.
For a zero work function, photoelectric emission begins at the lowest possible frequency. Electrons would then be driven from the surface in all directions perpendicular to the incident E/M ray direction. If the ray is polarized, the emission direction would be likewise polarized. That's impossible to test of course, but the same effect should be noted when the incident ray frequency is i.e. 1/10th that of the Compton wavelength frequency, where the material work function is overwhelmed by the applied force.
When the incident ray frequency approaches the Compton wavelength frequency, the emission direction would abruptly shift forward toward the incident ray direction. That is testable.
In such scattering, the wavelength increase is attributed to a photon colliding with an electron and deflecting it along the incident path to a degree dependent on the ratio between the energy of an electron at rest and the energy of the photon carried at the speed of light. When the energy of each is equal, the electron is deflected along the incident path by half the speed of the photon, while the photon reduces energy by doubling its wavelength. Its speed of course remains unchanged.
The consequence of a single photon collision with an electron which leads to the regenerated lower energy scattered photon is described thus; the lower frequency photon is emitted as the scattered electron, due to the collision, oscillates at a lower frequency than the original photon.
How on earth can the electron do that? Where's the other half of the dipole that's essential in any other kind of oscillating mechanism? It doesn't oscillate within itself, surely?
The relationship between the incident photon and the electron can only involve Coulomb forces. There is nothing else. As was demonstrated in the previous chapters, those forces are acting perpendicular to the incident path, driving the electron to oscillate back and forth over a straight line path for a constant distance in one complete oscillation cycle, in the time span of a wavelength. The perpendicular orientation of the path around the incident ray was set when the wave was created.
Image (1) of the set of three, shows the electron scatter depth (green wiggle curve) as being the Compton wavelength. The down motion was initiated by the first phase of the wavetrain shown. If the electron had traveled the straight line path, it would have averaged 2/3 the speed of light for the journey. If the electron had remained on the surface, in order to be driven over the Compton wavelength distance during the full cycle of the passing wave, its average speed would be twice that of light. That is of course not possible. In the surface oscillation cycle, the absolute maximum average speed is 1/2 the speed of light.
Image (2) shows the same scatter depth, but the wavelength is eight times that for image (1). Clearly, the electron would oscillate on the surface because its average speed there would be only 1/4 the speed of light, reaching a maximum speed of 1/2 c.
The final image of the set depicts a Compton wavelength wavetrain passing by the down scattered electron. If the electron remained on the surface, its average speed would be the speed of light, with a maximum speed twice that of light. It will be driven down with the passing wave until the maximum speed becomes that of light.
It would be expected that some wiggle in the path would always remain. Which is another problem. The wiggle will reduce as wavelength shortens and the electron is compelled to follow an increasingly straighter path, becoming dead straight only when the wavelength is zero.
This is a somewhat exaggerated example of the problem.
In this case, only those wavelength increases viewed close to the incident ray path would be detectable. The detectable range would expand to include angles closer to 90 degrees as wavelength shortens and the wiggle decreases.
One of the two curves that have been overlaid on each other in the next graph was generated according to 2.426e-12*(1-cos(angle)) while the other is according to cos(angle) * 2.426e-12. The result of the latter is subtracted from the line representing the constant scatter depth. It may be the same thing in the end, but it better represents the point I'm trying to make, that the curve is a consequence of doppler effect only. The total scatter depth is only effective in the scattered rays very near the incident ray path, with zero effect at 90 degrees. The frequency generated as the electron moves downward would be lower relative to the down moving electron, but would be exactly the same as the incident ray from a fixed point of observation at the base along the line of the incident ray.
