Where E{Δ} is the energy difference between the initial and final energy state of the electron plasma oscillation, and h is Planck's constant. E{Δ} can be determined using equation 3.10, as E{Δ} is equivalent to the work done on the electron by radiative forces, equal to W= –bvA. Therefore,
E{Δ} = –bvA
E{Δ} = –(ae[2] / 6πε{0}m{e}c[3]Av)vA
E{Δ} = –(ae[2] / 6πε{0}m{e}c[3] )
Here a is the acceleration, or dv/dt of the system, which is equal to -A(E1/4 / (256me1/4)), making:
E{Δ} = ((A(E[1/4] / (256me[1/4]))) e[2] / 6πε{0}m{e}c[3] )
ω= ((A(E[1/4] / (256m{e}[1/4]))) e[2] / 6πε{0}m{e}c[3] ) / h
Equation 3.13 therefore gives the frequency of emitted radiation for a transfer of energy E into plasma with electron plasma oscillation amplitude A. As this radiative damping continues, the frequency of emitted radiation will decrease at a rate proportional to a. In the case that the transfer of energy was kinetic from an incident projectile, it is clear that the plasma would have the effect of slowing the projectile by "Bleeding away" its kinetic energy. However, it is likely that for any real-world projectile velocity, the density of the plasma would have to be extreme to have much effect before the projectile passed completely through the plasma. However it may be possible to multiply the "Kinetic Bleeding" effect, which will be investigated in the next section.
Section 4: Stimulated Emission in Non-Equilibrium Plasmas
Stimulated emission is possible in plasmas where there is a population inversion, a non-equilibrium condition where more quantum systems are in the higher excited state than the lower state of a particular transition. Rapidly cooled plasma can produce a population imbalance in which there are more ions in an upper energy level than in the lower. Of course, this "Cooling" need not necessarily be thermal; a bleeding mechanism similar to that described in the previous section may function just as well. The plasma will inevitably "jump down" into a lower energy state, either triggered by an outside force or spontaneously. Spontaneous emission can be considered a stimulated emission reaction, but induced by virtual photons from the Quantum-Electrodynamic vacuum fluctuations that compose the zero-point field. However, if triggered by an (externally produced) incident photon, an ion in the plasma can make a transition to the lower energy state, creating another photon in the process. To satisfy conservation of energy, the energy difference between the upper and lower states exactly matches the photon energy, so the frequency of emitted radiation is identical to the incident. This can also be explained in that the temporal evolution of the coherent superposition of the lower and upper quantum wave functions act much like an oscillating dipole antenna; emitting electromagnetic radiation. The emitted photon then continues to propagate at velocity c, leaving the de-excited ion in its lower energy state. The alternating electric field creates a dynamic stark effect perturbation which increases the transition probability, making an encounter with another atom probable. Due to this and the relatively high density of the plasma, the photon will encounter another excited ion of the same species and stimulate a transition identical to that which formed it. The probability of this transition depends on the strength of the electric field produced by the emitted photons, and is directly proportional to the number of photons already present in the environment—Meaning that more times such an encounter takes place, the more likely it is to again. This second transition creates a second photon with exactly identical properties to the emitted photon as well as the incident photon. The two photons are identical; they have the same phase and wavelength. They form a Bose-Einstein condensate where all the constituent photons contribute to create a macroscopic quantum state, whose amplitude is much greater than the random spontaneous emissions occurring in ordinary plasma—and also identical to the initial incident photon. This "Photon Chain Reaction" increases exponentially per unit distance, eventually resulting in a quantum-coherent release of electromagnetic radiation exponentially more energetic than the initial incident photon, but exactly in phase and of matching frequency. This is, incidentally, an exact analog to the working principle of a laser. Per work done by Einstein, it is known that on thermodynamic grounds the probability for spontaneous emission, α, is related to the probability of stimulated emission, β, by the relationship:
α /β = 8πhω3/c3
From the development of the theory behind blackbody radiation, it is known that the equilibrium radiation energy density per unit volume per unit frequency was equal must be equal to this ratio α : β. As one would expect, the probability of spontaneous emission is greater if the energy density of the plasma is greater. This makes the energetic state more unstable, and limiter mechanisms of some sort would be required to prevent a random discharge of the plasma's energy.
Section 5: Prospects for Energetic Shielding
(FIGURE 5.1)
The concept presented in figure 5.1 may be a workable model for energetic shielding. It consists of two layers of plasma, of depth l1 and l2 , separated by a distance d and backed by a mirrored surface. The outer layer is a Glow-Discharge "Cold Plasma", like that described in Section 3, of ion density ρ1. The inner layer is Non-Equilibrium plasma with ion density ρ2 and an inverted ion energy population, like that described in Section 4.
If an incident projectile with mass m, velocity v, and cross-sectional area A impacts the cold plasma layer at a right angle, the energy of the projectile is:
K = ((½ mv[2] ) Sin (π/2)
K = ½ mv[2]
Therefore the applied kinetic energy per unit area is K/A,
E{app} = (½ mv[2] ) / A
Considering at period of time t after the projectile makes contact with the surface of the plasma, the distance penetrated into the plasma layer will be simply vt. We will consider only times for which vt < l1. The number of perturbed electrons will therefore be the volume of projectile penetration multiplied by the ion density, ρ1—
n{e} = vtAρ1
Hence the energy applied to each ion will be;
E{ion} = K / n{e} = K / vtAρ1
E{ion} = (mv) / (2tAρ1)
Where m is the mass of the projectile. In the process described in Section 3, this energy is transferred to the far less massive electrons, inducing electron plasma oscillations of frequency,
ω{pe} = (E{ion}[1/4] / (256m{e}[1/4]))
ω{pe} = (mv[1/4] / ((256m{e}[1/4])( 2tAρ1) [1/4]))
The period of these oscillations is then simply 2π /ω, which will radiate energy at a rate of;
dE/dt = ω{pe}[2]Ae[2] /(6πε{0}m{e}c[3])
dE/dt = mv[1/2]Ae[2] / ((6πε{0}m{e}c[3])((256m{e}[1/4])( 2tAρ1) [1/4]))
Just as before, the frequency of emitted radiation is then::
f= ((A(E[1/4] / (256m{e}[1/4]))) e[2] / 6πε{0}m{e}c[3] ) / h
f= ((A((mv) / (2tAρ1)[1/4] / (256m{e}[1/4]))) e[2] / 6πε{0}m{e}c[3] ) / h
Therefore n photons with energy hf will propagate outward from the cold plasma. This will cause a "Photon Chain Reaction" in the second layer of plasma, as discussed in Section 4. Such a reaction increases exponentially per unit distance, eventually resulting in a quantum-coherent release of electromagnetic radiation exponentially more energetic than the initial incident photon, but exactly in phase and of matching frequency. If the photons emitted from the cold plasma arrive over cross-sectional area a and the initial probability of stimulated emission is S, there will be n{react} reactions—
n{react} = n{p}aTS
T is the opacity of the plasma. If the rate of reaction probability increase is s(x), the total number of photons emitted over the depth of the plasma l2 is-
n{photon} = (n{react} ) s(x)l
n{photon} = (n{p}aTS ) s(x)l
Having now traveled the entire depth of the plasma, distance x is now simply l2. The total energy exiting the plasma is then--
E{incident} = n{photon} (hf)
E{incident} = (npaTS ) s(x)l (hf)
E{incident} = (npaTS ) s(x)l ((A((mv) / (2tAρ1)[1/4] / (256m{e}[1/4]))) e2 / 6πε{0}m{e}c[3] )
These photons will be reflected off the mirrored surface and travel back through the second section of plasma. This will of course cause a second round of stimulated emission reactions, however this can be accounted for by changing l2 to 2(l2).
This discharge of coherent photons will reach the incident projectile after time t = 2(d + l2 ) / c. The projectile will have traveled distance 2v(d + l2 ) / c. The energy lost to the plasma will be equal to-
