How many electrons in a circle does it take to create an electron at the center?

In the model used here, electrons are spherical wave structures. Since an electron is made up of only waves, it takes only waves to make one. On the vacuum energy to electrical energy conversion theory page, it was shown how this might be done.

Assuming we formed a circle of electrons as in the diagram below, how many electrons would have to be on the circumference of that circle to have the total amplitude of all waves at the center of the circle equal the amplitude at the center of an electron? The following diagram doesn`t contain the answer but illustrates the idea.

Figure 1. Adding amplitudes of a ring of electrons.

To get the answer I started with the wave formulas for a stationary electron, the same equations that were used to produce the animation at right. out-wave formula in trig. form:    Converted from:    in-wave formula in trig. form:    Converted from:    scalar wave formula:   in-wave - out-wave or where:   k = 2.589604676x10^12 /meter, calculated from (2xpi)/   w = 7.763439511x10^20 radians/second, calculated from (2xpixc)/ = kc    = 2.4263102175x10^-12 meters, Compton wavelength for electron   c = 299792458 meters/second   max = 1x10^-12 meters   Starting value for t = 3.5x10^-18 seconds, delta for each iteration was dt = 1x10-22 seconds.   r was in increments of 0.25x10-12 meters for the animation. Note: I didn't know what value to use for max so one was selected that resulted in a value suitably scaled for the animation.

Figure 2. 1-dimensional animation of waves for electron.

I next modified the program for the above animation to capture the maximum values for the amplitudes for one side of the scalar/standing wave. The following chart was produced from those captured values.

Figure 3. Captured maximum amplitudes of electron scalar/standing wave.

To get finer and finer values for the maximum amplitudes I took radius values from either side of the captured maximum amplitudes and plugged them back into the program as endpoints for new sets of captures, thus zeroing in closer to the actual locations of the maximum amplitudes. Three iterations of this resulted in data with accuracy I felt good about, the following of which is a sample taken from that data. Note that since I don't know what value to use for _max, the amplitudes obtained are not the actual values for electrons but are to scale.

Figure 4. Sample of maximum amplitudes and additional calcuations.

The first two columns are the captured data, Radius (distance from electron center) and Maximum amplitude at the peaks at that distance. Only the amplitudes at the peaks of the waves in Figure 3 are included in the table.

The Distance between full waves is a sanity check and should be close to the Compton wavelength, 2.42631x10^-12 meters. It is the Radius on that line subtracted from the Radius two lines above it i.e. one wavelength of distance.

The fourth column, Required number of electrons, is the result I was after. It's calculated as 5.17827 divided by the Maximum amplitude. The 5.17827 is taken directly from the bottom left of the table and is an amplitude very near the center of the electron. Remember, the objective is to find out how many electrons are needed in a circle such that the total amplitude at the center of the circle is the same as the amplitude near the center of an electron. So as an example, for the first line in the table, with a radius for the circle of 2.4863x10^-11, 64.39 electrons need to be on the circumference of the circle for the amplitude at the center of the circle to be 5.17827, the amplitude near the center of an electron.

The last column was an unexpected result. If you take the number of required electrons, 64.39 for the first line, and reduce the radius of the circle by a half-wavelength i.e. the second line, then you need pi less electrons, 61.25. So you need pi extra electrons for each half wavelength bigger that you make the radius of the circle.

So the final formula is:

The reason for multiplying by 2 is because the distance between two adjacent peaks is actually a half wavelength. The peak number is found by dividing the radius at the peak by Compton's wavelength and is how many peaks away a given peak is from the center of the electron. So in more broken down form it is:

And here is a table of values obtained using the above formula. Comparison with the above table containing values obtained from raw data leaves no doubt that the formula is correct.

Figure 5. Number of electrons needed calcuated using the above formula.