Clifford E Carnicom
 Jan 22 2004


This page contains the mathematics of the atmospheric plasma - kinetic energy model under consideration. This page is subject to revision.

The primary basis for the model is the equation for the kinetic energy density of a plasma that results from the the slow modulation of the amplitude of a high frequency wave1:

KEdensity = (1 / 4 ) * eo * ( wp2 / w2 ) * (E) . (E*)


KEdensity is the kinetic energy density of the plasma in joules per cubic meter.

eo is the permittivity of free space2; eo = 8.85E-12 C2 / N m2.

wp is the plasma frequency in radians per second.

w is the modulating frequency.

E is the electrical field strength in volts per meter, a vector quantity, and E* is the conjugate complex of E, and (E) . (E*) represents the dot product of E and E*.

(E) . (E*) is further defined as3:

(E) . (E*) = E1 . E1 + E2 . E2

where E1 and E2 are respectively the real and imaginary parts of E.

In this problem we are concerned only with the real parts of E, therefore:

(E) . (E*) = E1 . E1

and since E for our problem will be determined only in the radial direction:

(E) . (E*) =( E(h) )2


KEdensity = (1 / 4 ) * eo * ( wp2 / w2 ) * ( E(h) )2

let us revert to letting E = E(h), therefore:

KEdensity = (1 / 4 ) * eo * ( wp2 / w2 ) * ( E )2

A reasonable model for the electrical field strength of the earth can be determined by an exponential regression from physical data sources4,5 as:

E ( radial direction approx.) = 120 * e-3.5E-4*h

A check on the validity of the model can be made by also determining the total potential of the electrical field of the earth as:

Vearth =120 * e-3.5E-4*hdh

where V represents the total potential of the electrical field of the earth as measured in a radial direction. Solution of this equation for limits of 0 to 65 kilometers (km) will lead to a value of Vearth = 3.43E5 Volts, which agrees quite well with the tabulated sources referenced. Feynman6 estimates a total potential of 4E5 Volts at 50 km.

The next step in the problem is to formulate the total energy within a shell volume of atmosphere that surrounds the earth. We are therefore dealing with a variable energy density integrated with respect to volume, or:

Total Energy = Energy Density * Volume

Therefore we have:

Total Energy (KE) = [ (1 / 4 ) * eo * ( wp2 / w2 ) * ( E )2 ] * Volume

since E is now a function of elevation above the earth only:

KE = [ (1 / 4 ) * eo * ( wp2 / w2 ) * ( E(h) )2 ] dV

For the time being, let us regard wp and w as constants. Therefore,

KE = (1 / 4 ) * eo * ( wp2 / w2 ) ( E(h) )2 dV

Since our model is considering the variation in kinetic energy with respect to elevation only, and using the symmetry of the sphere, we have:

KE = (1 / 4 ) * eo * ( wp2 / w2 )* ( 4 / 3 ) * pi * ( ( R + upper)3 - (R + lower)3 ) ( E(h) )2 dh

where upper is the upper elevation limit of the atmospheric shell in meters, and lower is the lower elevation limit of the atmospheric shell in meters, and R is the mean radius of the earth in meters.

Next, let us improve our estimate of the modeling of the plasma frequency, since it is not a constant, and it varies primarily as to the square root of the electron density. The square of the plasma frequency is given as7:

wp2 = ( qe2 * n ) / (eo * me)

where qe is the charge of an electron in coulombs8, qe = -1.6E-19Coulombs (C),

and n is the electron density (electrons / cubic meter),

and me is the mass of an electron in kilograms9, me = 9.11E-31kg.

The electron density of the atmosphere is a complex phenomenon. The majority of sources of electron density information appear to be of a classified nature, and of not wide access to the public. Some publicly available sources have been located over the past several years, primarily from foreign countries. Regardless of source, electron density measurements at lower elevations appear to be essentially non-existent, as most historical applications of electron density are associated with the ionosphere. Current aerosol operation research indicates that this interest is warranted across the entire range of elevations of the atmosphere of our earth, including the lower elevations. Any information on detailed electron density profiles from lower to upper elevations will be beneficial to refine this study.

A first approximation for the electron density of the normal atmosphere can also be developed from an exponential regression of data that has been available from the University of Leicester29, leading to:

n (approx.) = 1E6 * e4.605E-5*h

with a maximum range from 0 to 300km considered in this evaluation.

We therefore have as a first approximation for the typical electron density:

wp2 (approx.) = ( qe2 * 1E6 * e4.605E-5*h ) / (eo * me)

Since we are dealing with an atmospheric shell of variable upper and lower limits, it is beneficial to formulate the plasma frequency square in an integral form. Therefore:

wp2 (approx.) = [( qe2 * 1E6 * e4.605E-5*h ) / (eo * me)] * 4.605E-5 * e4.605E-5*hdh

Our model for the kinetic energy of the atmospheric shell is therefore now of the following form:




KE =

(1 / 4 ) * eo * ( 4 / 3 ) * pi * ( ( R + upper)3 - (R + lower)3 )


( E(h) )2 dh




wp2 (approx.) = [( qe2 * 1E6 * e4.605E-5*h ) / (eo * me)] * 4.605E-5 * e4.605E-5*hdh

The evidence further indicates that we must strongly consider and incorporate the photoelectric effect of an atmosphere modified with metallic aerosols. The photoelectric effect is important as it provides a further source of electrons to the atmosphere, therefore increasing the plasma frequency of the medium. It can be seen that increasing the plasma frequency will cause an increase in the energy density of the plasma. It is conceivable that this influence is especially important and is not to be disregarded.

A further description of the photoelectric effect is given by Max Born, Nobel Laureate of 1954:

"The most direct transformation of light into mechanical energy occurs in the photoelectric effect (Hertz 1887), Hallwachs, Elster and Geitel, Ladenburg). If short-wave (ultra-violet) light falls on a metal surface (alkalies) in a high vacuum, it is it observed in the first place that the surface becomes positively charged; it is therefore giving off negative electricity, which issues from it in the form of electrons.....The number of electrons expelled is equal to the number of incident light quanta, and this is given by the intensity of the light falling on the metal.... Evidence even more patent for the existence of light quanta is given by the classic experiments of E. Meyer and W. Gerlach (1914) on the photoelectric effect with the small pieces of metal dust; by irradiation of these with ultra-violet light photoelectrons are again liberated, so that the metallic particles become positively charged10."

Readers will also wish to investigate the characteristic of metals known as the work function, and the relationship of the work function to the release of electrons. Readers may also wish to refer to an earlier paper entitled Ionization Apparent11 for a further discussion of the work function in relation to specific elements under consideration. Additional information on the photoelectric effect is available from numerous sources on quantum physics12,13, 14, 15, 16, 17. Experiments with metallic aerosols are frequently referred to in the discovery of this fundamental principle of modern physics.

The next question that arises is, how many photons are striking the earth's surface, and by projection, how many photons are striking the upper atmosphere? It is these photons that serve as a source for producing electrons in combination with metallics aerosols of a low work function. It is commonly stated that an approximate value for the energy of the sun at the earth's surface is on the order of 1000 watts / square meter18. We wish to find how many photons pass through a square meter per second at the earth's surface.

Let us assume the wavelength (lambda) of sunlight is 500 nanometers.

The energy of a photon is given as19:

Ep = h * f

where h is Planck's constant20, h = 6.62E-34,

and f is the frequency of the radiation in cycles per second.

Since c = f * lambda

where c is the speed of light21, c = 3E8 meters /sec.

and lambda is the wavelength of sunlight, then

f = c / lambda


Ep = h * (c / lambda)


Ep = 6.62E-34 (joules * sec) * ( 3E8 (m/sec) / 500E-9m ) = 3.97E-19 joules

for the energy level of a single photon per square meter at the frequency of sunlight.

The number of photons per square meter can be determined from

Np = I / Ep

where Np is the number of photons per square meter,

I is the intensity of the incoming radiation in Watts per square meter ( or joules / (sec * m2) )

and Ep is the energy level of a single photon.

Solving this equation, we are led to

Np = (1000 joules / sec * m2 ) / 3.7E-19 joules

Np = 2.70E21 photons / m2 * sec.

The next question of interest is, what density of aerosols in the atmosphere is required if all photons are converted to electrons? Is this density feasible compared to observation reports, visibility and density studies that have accrued over the past several years?

If we assume one free electron per atom22,23 of the element and if we choose barium as the element of consideration, we require 2.70E21 atoms per column of atmosphere that is 1 meter square in area.

In terms of mass of the element barium, the number of moles is:

2.70E21 atoms / 6.02E23 (atoms/mole) = 4.48E-3 moles of barium per atmospheric column 1 meter square in area.

Since the molar mass of barium24 is 137.3 grams, the number of grams required for full photon conversion is:

137.3 grams * 4.48E-3 moles = .615 grams per atmospheric column 1 meter square in area.

The density level required is:

.615 grams / Volume of atmospheric column

which is

.615 grams / (1 meter * 1 meter * height of column)

Let us choose the examples of 300km, 100km and 10km for a column height:

Height of column in meters
(1 meter square)

Density required for
full photon conversion to electrons
(in micrograms per cubic meter)

% of EPA standards for Particulate Matter (PM10)
50 ugms per cubic meter

300,000 meters (186 miles)

2.05 ugms / m3


100,000 meters (62 miles)

6.15 ugms / m3


10,000 meters (6.2 miles)

61.5 ugms / m3


This result leads to some apparently realistic estimates of the impact upon the environment from the injection of metallic aerosols of low work function in sufficient density to produce electron conversion from sunlight. One earlier study25, entitled Air Quality Data Requires Public Scrutiny, dated Aug 27, 2001 derived an estimate of a concentration level of 59ugms /m3 based upon visibility studies at lower elevations. An additional study26, entitled Microscopic Particle Count Study New Mexico 1996-1999 (March 23, 2000) of particulate matter from the highest quality air monitoring stations of New Mexico (PM2.5 capability) produces similar results that raise issue with the particulate trends within the atmosphere. The satisfaction and enforcement of EPA air quality standards, or the lack thereof, exists as an obvious and current environmental issue. The particulate density estimates of this section are used indirectly within the model under development, and they are used as a means of feasibility analysis.

The model now assumes the presence of additional free electrons that exist as a result of photon to electron conversion as a result of metallic aerosol injection. These free electrons exist above and beyond the normal electron density of the atmosphere. The normal atmosphere has an increase in electron density with altitude; this increase is dramatic and results in the unusual electrical properties of the ionosphere. The introduction of electrons at high altitude from the photoelectric effect within the model will have an opposing effect; electron density will be high at altitude and decrease significantly in the direction of the earth. The rationale for this decrease lies in the exponential decay law of light intensity27, as has also been discussed in visibility studies presented on this site28.

The model chosen for the electron density as a product of the estimated photoelectric effect will be of the form:

np = c * e-b*(300E3 - h)

where np represents the electron density per cubic meter resulting from the estimated photoelectric effect.

c and b are coefficients to be determined, and 300E3 represents the upper limit of the atmospheric shell in meters under consideration.

h represents the elevation above ground level.

The coefficient c will be determined as follows:

c = ( 2.5E21 photons / m2 ) / 300E3m3 = 8.3E15 electrons / m3 at 300,000 meters. This will be a maximum value for np.

This represents the most conservative value that can be established at this point in the analysis, and it represents a maximum that will decrease exponentially towards the earth's surface.

Setting the condition that np = 1E5 electrons / m3 at ground level,

np = 8.3E15 * e-9.15E-5*( 300E3 - h )

Therefore the influence of np upon the plasma frequency is now:

wp 2(photoelectric effect) = ( qe2 * 8.3E15 * e-9.15E-5 * ( 300E3 - h) ) / (eo * me)

and again developing this into an integral form due to the shell region under consideration,

wp 2(photoelectric effect)

wp2 (photoelectric effect) = [( qe2 * 8.3E15 * e-9.15E-5 * ( 300E3 - h) ) / (eo * me)] * 9.15E-5 * e-9.15E-5 * ( 300E3 - h) dh

we are now led to a revised estimate of the plasma frequency as:

wp* = (wp2 + wp2(photoelectric effect) )1/2

Which leads to a final model form of:




KE =

(1 / 4 ) * eo * ( 4 / 3 ) * pi * ( ( R + upper)3 - (R + lower)3 )


( E(h) )2 dh




wp* = (wp2 + wp2(photoelectric effect) )1/2


wp2 (approx.) = [( qe2 * 1E6 * e4.605E-5*h ) / (eo * me)] * 4.605E-5 * e4.605E-5*hdh


wp2 (photoelectric effect) = [( qe2 * 8.3E15 * e-9.15E-5 * ( 300E3 - h) ) / (eo * me)] * 9.15E-5 * e-9.15E-5 * ( 300E3 - h) dh

integrated with respect to the upper and lower limits of the atmospheric shell in meters.

This model remains subject to revision.

1. Charles Herach Papas, Theory of Electromagnetic Wave Propagation, (Dover Publications, 1988), 182.
2. Gordon Coleman, The Addison-Wesley Science Handbook, (Addison-Wesley, 1997), 3.
3. Papas, 14.
4. David R. Lide, CRC Handbook of Chemistry and Physics, (CRC Press, 2001), 14-34.
5. Dwight E. Gray, PhD, American Institute of Physics Handbook, (McGraw-Hill, 1963), 5-280.
6. Richard Feynman, The Feynman Lectures on Physics, Vol II, (Addison-Wesley, 1964), 9-3.
7. Feynman, 7-7.
8. Coleman, 2.
9. Coleman, 2.
10. Max Born, Atomic Physics, (Dover, 1969), 83-84.
11. Clifford E Carnicom, Ionization Apparent, (, 03/01/2001).
12. Nouredine Zettili, Quantum Mechanics, (Wiley and Sons, LTD, 2003), 10-12.
13. Ronald Gautreau, Modern Physics, (McGraw-Hill, 1999), 60-61.
14. George Joos, Theoretical Physics, (Dover, 1986), 431-433.
15. David Bohm, Quantum Theory, (Dover, 1989), 23-26.
16. Albert Messiah, Quantum Mechanics, (Dover, 1999), 11-13, 16, 41, 760, 1007-9, 1052.
17. Michael Mansfield, Understanding Physics, (Wiley and Sons, 1998), 362-365.
18. Alvin Halpern, Beginning Physics II, (McGraw-Hill, 1998), 464.
19. Halpern, 460.
20. Coleman, 3.
21. Coleman, 3.
22. Feynman, 32-12.
23. Born, 237.
24. Coleman, 126.
25. Clifford E Carnicom, Air Quality Data Requires Public Scrutiny, (, 03/01/2001).
26. Clifford E Carnicom, Microscopic Particle Count Study New Mexico, (, 03/23/2000).
27. Gautreau, 63.
28. Clifford E Carnicom, Air Quality Data Requires Public Scrutiny.
29. University of Leicester, Ionospheric Physics, (

Back to Aerosol Operations Main Page