ELECTRODYNAMIC CONFINEMENT

Patent Application PCT/SE96/00966

ABSTRACT
       Electrodynamic confinement of extremely energetic, charged plasmas for use in energy production and energy storage is achieved by exploiting nonlinearities in the electrodynamic equations to effect an oscillatory motion of the charges in the plasma, whereby the charges repetitiously "bounce against the electric field and thereby instantaneously reverse their direction of motion, thus composing a dynamic state in which there is a net electric field and acceleration, and yet no net expansion.

Front page WO 97/04626

                        ELECTRODYNAMIC CONFINEMENT

FIELD OF THE INVENTION
       The present invention relates to a method for electrodynamic confinement of charged plasmas, or ensembles of charged particles in general, and applications thereof in energy production and energy storage, and potentially also as building blocks for new types of materials. The invention comprises but is not limited to a method for electrodynamic self-confinement, whereby the ensemble of charges is being confined by its own forces only and without external means.

BRIEF DESCRIPTION OF PRIOR ART
       Electrodynamic self-confinement was first suggested in 1973 as a conceivable mechanism for strong interactions, possibly manifesting itself on a macroscopic scale in ball lightning (A Bergström, Phys. Rev. D 8, 4394 (1973)). Although the basic equations are essentially equivalent to those used below, the calculations in the cited article involved considerable simplifications, and the peculiar confinement mechanism of the present invention was not understood at the time. The cited article also does not consider how to circumvent fundamental theoretical objections of the type studied below in the discussion of the present invention. Thus, despite the fact that a connection between electromagnetic and weak interactions has since been established, the proposed electrodynamic origin of strong interactions is far from being generally accepted. Nevertheless, the cited article influenced experimental work, most prominently by K Shoulders who in 1989 filed patents for an experimental apparatus which produces entities which can be interpreted as microscopic ball lightnings (US Patent 5,018,180, US Patent 5,054,047). A type of electromagnetic self-confinement somewhat similar to the above work from 1973 but involving a self-generated rotating magnetic field in the confinement mechanism has been proposed by G Arnhoff (European Transactions on Electrical Power Engineering 2, 137 (May/June 1992)) as an explanation of ball lightning.

SUMMARY OF THE INVENTION
       The confinement mechanism of the present invention is based on the electrodynamic properties of charged, gaseous configurations with extreme energy contents, and involves primarily the electron population. The mechanism is governed only by electric fields, and it should be emphasized that no magnetic fields are necessarily involved in the mechanism. Indeed, in the following essentially only the case of spherical symmetry will for simplicity be considered, in which case Maxwell's equations do not allow any magnetic field. One objective of the present invention is thus to provide a method of plasma confinement not requiring the bulky and expensive paraphernalia of normal magnetic confinement, which constitutes a tremendous structural overhead in any magnetic confinement device. In a tokamak, for example, which is the most conceivable candidate for a commercial, magnetic confinement device for thermonuclear power production (cf R W Conn, Scientific American Oct 1983, p 44), a typical weight of the magnets and supporting structure in the magnetic container is of the order of hundreds of tons compared to hectograms for the contained gas producing the energy.
       Another advantage of electrodynamic confinement versus magnetic confinement is that electrodynamic confinement involves only short-range motion of the charges, in contrast to the long-range motion (typically meters) of the charges required in magnetic confinement. A second objective of the invention is thus to employ electrodynamic confinement instead of magnetic in order to construct devices with very small over-all dimensions, in addition to the minimal structural overhead.
       In contrast to magnetic fields, electric fields are easily produced within a gaseous medium itself by charge separation. A third objective of the invention is thus to provide a method of electrodynamic self-confinement, i e a state in which a charge configuration becomes stable as a result of its own internal forces without requiring external fields for its subsistence.
       The energy involved in the chemical bonds in molecules is typically of the order of eV, whereas the total mass of the accompanying atoms is typically of the order of GeV or more. This means that if we are limited to the materials nature provides us with for energy storage and structural materials, then less than one part in a billion of the total mass is actually used for the purpose in question, the rest is just dead weight. All the heavier elements in nature, such as aluminum, iron, titanium or lead, are slag left over from supernova explosions and not necessarily optimal for using as structural materials or energy storage as we do. A further, long-term objective of the invention is thus to provide a method for producing a self-confined, gaseous state of matter with high energy density and structural stability, and in which the energy involved in the confinement mechanism consitutes a considerable fraction of the total mass, thus providing a potential for creating radically new, light-weight materials for energy storage and mechanical structures.
       According to the invention therefore there is provided a method and/or device for electromagnetic confinement of an ensemble of highly energetic, charged particles, and/or for extraction of energy from such an ensemble, characterized in that fundamental nonlinearities inherent in the electrodynamics of said ensemble, and/or the spacetime dynamics of said ensemble, with respect to accelerations and deceleration of said particles in said ensemble are utilized to confine said ensemble and/or extract energy from said ensemble.
       Preferably, a nonlinearity in the mutual electrodynamic interactions between the electromagnetic field and said accelerating charged particles in said ensemble at strong fields is utilized to achieve confinement of said ensemble by producing a violent, radially oscillating motion of said particles in which, due to a repetitive discontinuity in the velocity of said particles when - analogously to bouncing - their direction of motion is instantaneously reversed, the average spatial displacement of each particle is essentially zero even though its average acceleration is not zero.
       According to the invention there is provided a method and/or device characterized in that said electrodynamic confinement is utilized to store electric energy, thus providing extremely light-weight energy storage. Preferably also the energy content per volume of said energy storage is increased by compartmentalization of the confinement structure into a large number of subunits, and said compartmentalization of the confinement structure is used to create extremely light-weight, mechanical structure elements.
       As will be further discussed below, there exists a relativistic Carnot process in which it is not quite clear that energy in its normal definition is conserved. This relativistic Carnot process involves accelerations and decelerations which are simultaneous in different frames of reference, resulting in a possible excess energy after each cycle in the process which may be due to extraction of energy from vacuum fluctuations. The possibility of extracting energy from the zero-point fluctuations of the vacuum field has recently been seriously considered in the scientific literature (R L Forward, Phys. Rev. B 30, 1700 (1984), D C Cole and H E Puthoff, Phys. Rev. E 48, 1562 (1993)). The confinement mechanism of the present invention involves accelerations and decelerations of the type required in the above-mentioned Carnot process. Indeed, an excess energy production has actually been reported (K R Shoulders, PCT/US90/02368), albeit on a microscopic scale. A further feature of the invention is thus to provide macroscopic conditions where it is possible to test experimentally whether excess energy can be extracted from a relativistic Carnot process of the type disclosed below, and if so, extract such energy.

BRIEF DESCRIPTION OF THE DRAWINGS
       Fig 1 shows the velocity as a function of time for a nonsymmetric, periodic motion, and illustrates the fact that it is impossible to achieve self-confinement by tailoring an oscillatory motion so that there is zero average displacement but still a nonzero average acceleration, and with the velocity and acceleration assumed to be continuous functions of time.
       Fig 2 depicts the acceleration (Fig 2A), velocity (Fig 2B), and displacement (Fig 2C) as functions of time in a harmonic oscillator with two identical particles bouncing against each other, and where each particle has a zero average displacement even though its average acceleration is nonzero, this being possible due to a discontinuity in the velocity caused by the bounces as shown in Fig 2B.
       Fig 3 depicts a complete solution of the electrodynamic equations in the case of extremely high fields when the mutual interactions between the field and the moving charges dominate over collisional effects. Shown are the acceleration (Fig 3A), velocity (Fig 3B), and displacement (Fig 3C) as functions of time for a charge configuration with extreme excess charge, illustrating that there exist solutions which exhibit the type of bouncing oscillations which is the characteristic of self-confinement.
       Fig 4 depicts the deformation, as observed from a stationary spacetime frame xt and calculated by the Lorentz transformation, of a frame x't' which is accelerating with respect to frame xt by a constant acceleration simultaneous in x't' (i e a constant acceleration as observed from an inertial frame which momentarily coincides with x't'). The x't' frame exhibits the hyperbolic form expected from a frame in constant acceleration, but closer scrutiny reveals that the time t' is observed from the xt frame to go in different directions for positive and negative times t. The inserted perspective figure which displays a picture of the x't' frame in xtx'-space clearly illustrates the topology involved.

DISCUSSION OF THE INVENTION
       Intuitively, the idea of attaining electrodynamic self-confinement of, say, a cloud of electrons would seem out of the question. To be more specific, the following very fundamental theoretical objections can be raised against the idea of electrodynamic self-confinement.
       Suppose first that we have a static, spherically symmetric charge configuration with a density of excess charge given as a function q(r) of the distance r to the centre of symmetry. For simplicity, and also because electrons are more mobile than ions, we consider the excess charge to consist of an excess or deficit of electrons against a background of positive ions. The static excess charge density q(r) in the assumed charge configuration may be negative (indicating an excess of electrons) in some spherical shells, and positive (deficit of electrons) in others. As a result, the entire charge configuration may have a total charge  0.
       Regardless of the distribution of shells with different charge, there must always be some innermost sphere, extending from radius r = 0 to some radius r = a, within which the charge density has only one sign, say positive, and a total charge 0. Any possible stability of the entire charge configuration rests on the possibility of the charges in this innermost sphere being stable and not blowing apart due to their own mutual Coulomb repulsion.
       In a medium with electric permittivity , the relationship (or definition) of the charge density q(r) to the electric field E(r) is given by the Maxwell equation

                      ^. E(r) = q(r) /  .                                                                                         (1)

       Using Gauss' theorem, the radial component of the electric field at radius r then becomes in the case of spherical symmetry

                     E(r) = Q(r) / (4   r²),                                                                                   (2)

where Q(r) is the total charge inside radius r, i e

            Q(r) =   4  r² q(r) dr.                                                                                 (3)

       According to Newton's second law, the acceleration a(r) of a charged particle with charge-to-mass ratio e/m (positive for positive charges, negative for negative charges) due to the electrostatic Coulomb field in Eq (2) then becomes

                    a(r) = e/m E(r) = e/m Q(r) / (4   r²).                                                             (4)

       In the case studied with an assumed deficit of electrons in the innermost sphere, all quantities in Eq (4) are positive. Thus the acceleration a(r) must also be positive, i e directed outwards. If we had, instead, a negative excess charge in the innermost sphere, then e/m and Q(r) would both be negative but their product and thus the acceleration a(r) would again be positive, and thus again directed outwards. From these very fundamental considerations there seems no way for anything but an outward motion of the charges in the innermost sphere, and thus eventually for the entire charge configuration.
       The case for self-confinement becomes more promising, at least at first sight, if we consider a dynamic instead of static charge configuration. In this case, a special kind of motion could perhaps be achieved in which the average spatial displacement is zero even though the average acceleration as discussed above is not zero. Specifically, we here study quasi-stationary motion such as, e g, radial oscillations of an electron gas with respect to a background of heavy positive ions, which can be regarded as essentially stationary due to their much higher mass. Considering such radial oscillations of the innermost sphere discussed above, we would have a charge density which is negative everywhere inside the sphere during one half-period and positive everywhere inside the sphere during the next half-period (both states being due to minute radial displacements of each electron to the inside of or to the outside of the radius vector). This means that an electron in the innermost sphere would experience an outward acceleration during the first half-period and an inward acceleration during the second half-period. Conceivably, this acceleration might then be tailored so that the time-averaged displacement of the electron due to the acceleration is zero even though the time-averaged acceleration at radius r (and thus the average charge inside r) is not, as will be attempted below.
       Also in the case of a dynamic charge configuration there is, however, a very fundamental theretical objection to the possibility of attaining self-confinement by tailoring an oscillatory motion as was proposed above. This objection will now be discussed in some detail.
       Suppose that the acceleration of an electron in the charge configuration at time t can be described by a function a(t), assumed to be continuous and periodic (with period T) in t. If we assume the initial velocity of the electrons at time t=0 to be zero, then from the definition of acceleration their velocity at a later time t is given as

                    v(t) =  ₀ ^t a(t)dt,                                                                                             (5)

where v(t) is also continuous and periodic (with period T) in t, and obviously has a continuous first derivative.
       Fig 1 gives an example of what the function v(t) may look like in a typical situation where an assumed oscillating field has produced an oscillating, periodic velocity of the electrons of the type given in Eq (5). The asymmetric oscillation in Fig 1 is the sum of two harmonic oscillations with angular frequencies and 2, respectively, and in phase with one another. We will now investigate whether it is possible to tailor the asymmetry of the oscillation in such a way that there is an average acceleration 0 and still no net displacement.
       The velocity in Eq (5) results in a radial displacement s(t) of the electrons, which after a period is

                    s(T) =  ₀^T v(t)dt,                                                                                            (6)

corresponding to the area between the curve and the horisontal axis, with positive sign above the axis and negative sign below. In the example in Fig 1, the integral in Eq (6) is constant (= 0), so that there is no net radial displacement and the required state of quasi-stationary motion is thus at hand.
       We now address the question of the time-averaged charge inside radius r corresponding to the assumed acceleration a(t). As discussed above we have, due to Newton's second law and Gauss' theorem, a proportionality between electric field, acceleration and charge, i e for the charge inside r averaged over a period we have

                    Q(r)    ₀^T a(t) dt / T = [v(T) - v(0)]/T.                                                        (7)

       In order for Q(r) to be compatible with an assumed excess charge, we must thus require that v(T) v(0). On the other hand, in order for the motion to be quasi-stationary, we must require that it return to its initial state after a period, i e that v(T) = v(0). Since an excess charge Q(r) 0 is thus possible only if the velocity averaged over a period is nonvanishing, v 0, this is incompatible with a quasi-stationary motion of the type assumed above.
       A crucial part of the invention is thus how to find a way around the theoretical objections just discussed, and this can be achieved by observing that the above objection to dynamic stability is based on the assumption that v(t) is a continuous function of t. For motions containing cusps at which the function v(t) is discontinuous (like, e g, the common cycloid or the bouncing harmonic oscillator discussed below), the above objection is circumvented. Now cusps are typically a nonlinear effect, and fortunately - from this point of view - the electrodynamical equations form a nonlinear system. The invention exploits nonlinearities inherent in the electrodynamic interaction between the field and the moving charges to circumvent the above objection to dynamical stability by effecting a motion which contains discontinuities in v(t). In fact, precisely the common cycloid appears as a solution to the nonlinear electrodynamic equations.
       In the disclosure of the invention below, we will first illustrate confinement according to the invention in a different context - a bouncing harmonic oscillator - which will be shown to exhibit a type of confinement in which the above theoretical objections are circumvented. With this in mind, we then give a complete analytical solution for the case of extremely high fields when the interactions back on the field from the velocities and accelerations of the charges caused by the field completely dominate over collisional effects. Indeed, this case turns out to exhibit the same type of confinement - and with the bouncing harmonic oscillator reappearing as a close approximation.

DISCLOSURE OF THE INVENTION - "BOUNCING HARMONIC OSCILLATOR"
       Consider an adiabatic, harmonic oscillator with two identical particles moving in response to an attractive force proportional to the distance d to their common center-of-mass (as if they were connected by a rubber band), and with their common center-of-mass d=0 at rest. When the particles meet at d=0, we can either assume that they bounce against each other in a perfectly elastic collision, or pass through each other without frictional loss, depending on whether we choose to define the particle going outwards after the collision to be the same as the one going inwards or not. We here assume the particles to bounce against each other so that Fig 2C in Fig 2 describes the trajectory of particle 1. The trajectory of particle 2 is then described by a mirror image above the t-axis of the trajectory of particle 1 in Fig 2C.
       Since the time-dependent restoring force T(t) and acceleration a(t) in a harmonic oscillator is assumed to be proportional to the displacement d(t), we can (with suitable scaling of t) express the equation of motion for particle 1 of the harmonic oscillator in the form

                    d²T(t)/dt² = - |T(t)|,                                                                                        (8)

which (with suitable scaling of T(t) and choice of t=0) has the solution

                    T(t) = |cos(t)|,                                                                                                 (9)

and the first derivative

                    dT(t)/dt = ± (1-T(t)²),                                                                               (10)

which is discontinuous when T(t) = 0.

       Obviously, the net displacement of particle 1 (and particle 2) after a period is zero, and it experiences an average acceleration outwards as depicted in Fig 2A. This is thus a case where we have indeed achieved a state of motion with a net acceleration but with zero net displacement - a situation which seemed to be ruled out from the discussion above. Why this is possible at all is because the assumption above of v(t) being a continuous function of t is no longer valid in the motion we are now discussing; there is a discountinuity in v(t) at the time of collision, when v(t) instantaneously returns to its starting value by reversing direction, as is shown in Fig 2B.
       Suppose now for illustration that the particles are separated from each other at d=0 by a large, rigid sphere which transfers momentum between the particles (so that the motion of particles looks somewhat like two balls bouncing against the surface of the earth at two antipodes). Then obviously both particles have net accelerations outwards (or both inwards depending which one we put where) and at the same time zero net displacements. As was discussed above, these are the requirements for quasistationary motion of the charges in a stable configuration with an excess charge.
       Summarizing the results so far, Fig 2 thus depicts the acceleration (Fig 2A), velocity (Fig 2B), and displacement (Fig 2C) as functions of time for a bouncing harmonic oscillator, for which the velocity of one particle is repetitiously and instantaneously reflected inwards by collision with something, and where the recoil momentum at each collision with this 'something' is somehow taken up by a similar reflection of the velocity inwards of a second particle opposite with respect to the center of symmetry of the system. For this type of oscillatory motion with a discontinuity in v(t) (in this case a reflection of the direction of the velocity), it would thus be possible to have a quasistationary charge configuration despite an excess electric charge. What will be shown in the following is that a motion of this type can indeed be achieved in a charge configuration by the field itself, and with the field playing the role of the 'something' mentioned above with which the particles collide and which transfers momentum between them.
       The algebra in the derivation below is with necessity extensive. For this reason the algebraic calculations have been performed using a computer program for symbolic mathematics (Maple V.2, see B W Char et al, First Leaves: A Tutorial Introduction to Maple V (Springer, 1992)). In the text below only a summary of the algebraic calculations is given. For the detailed derivation the reader is referred to the output from the computer program in Appendices A-C in the priority application.
       A method and/or device according to the invention thus preferably incorporates an ensemble of charges which is essentially spherically-symmetric, and the electromagnetic field is an essentially spherically-symmetric electric field. In one embodiment external electric fields are used to produce the periodic, radially-oscillating motion of the charged particles and preferably then said electric field is created by a central, preferentially spherical electrode supplied by an external high-frequency source connected to said electrode by a shielded connector.

DISCLOSURE OF THE INVENTION - ELECTRODYNAMIC BOUNCING
       We define extremely high electric fields as a field regime where the interactions of the field on the charges - and the corresponding interactions back on the field from the accelerations and velocities of the charges - completely dominate over effects of collisions between the charges or with the background medium. To be more specific, it will be shown that under special circumstances the charges will experience "bouncing" against the field due to nonlinearities in the electrodynamic equations. In extremely high fields this "bouncing" will be the dominant effect for changing the velocities of the charges, and appear on a time-scale which is much shorter than the collisional relaxation time, so that velocity changes due to collision effects are negligible in comparison. In this case, energy losses due to Ohmic heating can also be assumed to be negligible in comparison with the energies involved in the mutual interactions between the charges and the field, and the particle dynamics thus be assumed to be essentially adiabatic.
       In the extremely high electric fields discussed here, the inertial mass corresponding to the electric field energy may become comparable with or higher than the mass of the charges themselves which take part in the mechanism. Thus the conditions described here may enter into the regime within the "classical particle radius"

                    r_c = Q² / (4  ₀ M c²),                                                                                 (11)

where Q and M are, respectively, the total charge and mass of the particles constituting the excess charge (normally electrons). The "classical particle radius" is normally regarded to be of the order of fermis (10-13 cm) or less, and to be a regime where classical electrodynamics breaks down and has to be replaced by a quantum description. However, conditions corresponding to those within the "classical particle radius" as defined by Eq (12) can be realized on a macroscopic scale at comparatively modest fields, voltages and excess charges, and it is not quite clear why or what quantum description should be used for a system with "classical particle radius" of the order of decimeters. Assume, for instance, that the molecules in a sphere of air at normal density become slightly deformed by a radial electric field so that all electrons (approx. 4 10²³ electrons/dm³) are being slightly displaced radially inwards in the sphere relative to their unperturbed positions by 10^-12 m (i e 1/100 of the radius of the atom), and that this displacement of the electron population relative to the background nuclei constitutes the charge configuration in the present study. For a sphere with radius 1 dm we then have an excess charge Q equal to the electron charge times the number of atoms in a 10^-12 m shell with radius 1 dm, i e Q 0.8 10^-5 As. With Q/Mc² = electron charge/electron mass-energy 2 10^-6 V^-1, Eq (11) then gives r_c 1.5 dm for the "classical particle radius" for this system. The voltage is 500 kV and the electric field at the surface of the sphere is 5 10⁶ V/m, i e of the order of the breakdown field strength in air. The charges, voltages and electric fields at macroscopic "classical particle radii" of the order of decimeters are thus far from exotic, and in a range easily attainable in the laboratory.
       The electric field E(r,t) is derived from the basic electrodynamic equations (Maxwell's equations plus conservation of charge, mass/energy and momentum):

1) Relationship between electric field E, electric permittivity , and charge density q as described by Maxwell's equation  ^. E = q/, Eq (1), which in spherical symmetry becomes

                              q =  [2 E/r + E/r].                                                                         (12)

2) Relationship between electric field E, electric permittivity , and current density j as described by Maxwell's equation c²  × B = j/ + E/t, which with j = q v and in spherical symmetry, when the magnetic field B 0, becomes

                              0 = q v / + E/t.                                                                              (13)

3) Continuity equation for charge density q, i e q/t +  ^. j = 0, which with j = q v and in spherical symmetry becomes

                              q/t + 1/r² (r² q v)/r = 0.                                                               (14)

4) Continuity equation for mass/energy density , i e /t +  ^. ( v) = 0, which in spherical symmetry becomes

                              /t + 1/r² (r²  v)/ = 0.                                                              (15)

5) Relationship between electric field E, charge density q, mass density , and "convective derivative" dv/dt = v/t + (v·) v as described by Newton's second law dv/dt = q E/, which in spherical symmetry, when dv/dt = v/t + v v/r, becomes

                              v/t + v v/r = q E/.                                                                      (16)

In addition to the 'physical' assumptions Eqs (12) through (16), the following 'mathematical' assumption is necessary in order to be able to solve the equations:

6) The electric field E(r,t) is assumed to be separable into one factor R(r) which is a function of radius r alone, and one factor T(t) which is a function of time t alone,

                             E(r,t) = R(r) T(t).                                                                                (17)

       The general solution of the coupled electrodynamic equations Eqs (12) - (17) is performed using the symbolic mathematics program Maple V.2. It is interesting to note that the electrodynamic equations in spherical symmetry is one of the rare cases where nonlinear partial differential equations possess exact analytical solutions, viz

                    E(r,t) = Q(r,t) / (4 ₀ r²),                                                                            (18)

                    v(r,t) = - ½ r C sin / (  ( ½ cos + ½ )²),                                                 (19)

                    q(r,t) = - Q(r,t)  / (4 r³),                                                                           (20)

                    (r,t) = Q(r,t)^{2 3} ( ½ cos + ½ )³ / (8 ² ₀³ C² r⁶),                                   (21)

where the time-dependence is implicit through the variable , defined through

                    t =  ( ½ sin + ½ ) / C,                                                                            (22)

C and  being constants, and where Q(r,t) is the total charge inside radius r at time t, given by the expression (A is positive or negative real-valued scaling constant)

                    Q(r,t) = 4  ₀ A  r ^-  ( ½ cos + ½ ),                                                     (23)

       The solutions in Eqs (18) through (22) can be verified by back substitution into the basic electrodynamical equations, Eqs (12)-(16). The most important characteristic of the solutions is that (for  >0 when t is real) they exhibit a recurring discontinuity in the velocity at times corresponding to = , 3, 5, etc, when the velocity abruptly changes direction. This behaviour is a crucial characteristic of the nonlinear electrodynamic equations for sufficiently steep charge distributions (viz  >0), and is the essence of the confinement mechanism according to the invention as discussed above in connection with the analysis of the "bouncing harmonic oscillator".
       As seen from Eqs (18), (22) and (23) the time-dependence of the electric field E(r,t) for  >0 is of a general cycloid form, which when approaches 0 becomes broader and broader around the peak, and eventually approaches a rectangular shape with E(t) constant, interrupted by narrow spikes when E(t) dips down to zero. In the asymptotic case   ^_> 0, the solution to the electrodynamic equations thus reduces to the simple static solution

                    E(r,t) = const / r².                                                                                         (24)

       Thus the Coulomb field 1/r² is singled out as a peculiar case when there is a static solution to the electrodynamic equations. Isolated, however, there is no clue why the Coulomb field should correspond to a static charge configuration, nor how a static solution could exist at all in view of the discussion above of theoretical obstacles to stability. It is only when considered as a limiting case of "bouncing" fields as depicted in Fig 3 and explained in terms of the confinement mechanism of the invention that des Pudels Kern of its static nature reveals itself.
       Furthermore, the Coulomb field also has the following "boot-strapping" property. If we calculate the charge density q(r) from the Maxwell equation Eq (12), we find that a Coulomb field corresponds to a charge density q(r) 0. Or stated in an alternative way: Any infinitesimal charge density of a form approaching that which gives a Coulomb field will produce an arbitrarily large field. For exemplification, consider a charge density q(r) of the form

                    q(r) = A r ^-3,                                                                                                (25)

where is a small number <<1 (for simplicity here assumed to be positive). In accordance with Eq (3), the charge Q(r) within a sphere of radius r is then

                    Q(r) = 4 (A/) r,                                                                                      (26)

which according to Eq (2) corresponds to an electric field E(r) at radius r,

                    E(r) = (A/) r ^-2 /  = A (r/) / ( r ²).                                                          (27)

       When   ^_> 0, the electric field E(r) approaches the Coulomb field A /( r ²) but with a factor r/ which tends to infinity as  ^_> 0, so that no matter how small the factor A may be in the charge density in Eq (25), a finite field may still result from Eq (27). The Coulomb field is thus an asymptotic limit when an arbitrarily large field can be produced by an infinitesimally small charge distribution of the form in Eq (25).
       In a method and/or device according to the invention thus the electric field is preferentially created internally by the electrodynamics of the ensemble of charges itself, thus producing an electrodynamic self-confinement of said ensemble.
       Preferably, also said self-confined ensemble of charges is created from an initially cylindrical discharge, which is caused to pinch due to the magnetic field of the discharge current, characterized in that said pinched discharge is allowed to deform by inherent instabilities, preferentially of 'sausage' type, leaving - after the current is switched off - at least one highly energetic, essentially spherically symmetric charge configuration which is arranged through the current distribution in the initial cylindrical discharge to get a radial charge density distribution proportional to 1/r^N, where r is the distance from the center of symmetry and with N greater than 3 (or approaching 3 infinetely close from above), corresponding to a radial electric field distribution in said charge configuration which is (possibly only infinitesimally) steeper than a Coulomb field, i e proportional to 1/rⁿ with n greater than 2 (or approaching 2 from above).

ONE MODE FOR CARRYING OUT THE INVENTION - NUCLEAR FUSION
       The confinement mechanism of the invention involves high-frequency oscillations of an electron population with respect to a background of positive ions. In order to conserve momentum the ions must also oscillate, and with typical velocities depending on their mass. With a sufficient excess charge in the charge configuration, extremely high energies may be involved in the oscillations. This is also necessary in order to overcome losses and recombinations, and for the system to be in the field regime where confinement according to the invention can take place. Extrapolating from estimates of energy contents in ball lightnings (which as discussed above are considered to be natural embodiments of the confinement mechanism of the invention) energies of the order of tens to perhaps hundreds of keV per particle can be assumed to be attainable in a confinement according to the invention. With a plasma containing a mixture of ions with different mass, say hydrogen and deuterium, the proton population would oscillate with higher velocities than the deuteron population. This constitutes an ideal situation for thermonuclear fusion both because thermonuclear energies are easily attained but also because the proton and deuteron populations oscillate relative to each other. This means that thermonuclear reactions between the populations are greatly enhanced relative to reactions within each population. In the case discussed with a mixture of hydrogen and deuterium, the reaction

                    D² + H^{1 _}> He³ + + 5.5 MeV,

which involves ions with different mass would thus be greatly enhanced compared to the competing reactions

                    D² + D^{2 _}> He³ + n + 3.3 MeV,

                    D² + D^{2 _}> T³ + H¹ + 4.0 MeV,

                    H¹ + H^{1 _}> D² + e⁺ + _e + 1.4 MeV - 0.3 MeV,

which involve particles with the same mass. Here the two branches of the D-D reaction occur with about equal probability, and the negative energy contribution in the last formula represents energy which is being carried away by the neutrino.
       The D-H reaction is the main energy source in medium-sized stars. The cross section for the D-H reaction is much lower than for the D-D reaction, for which reason it is the D-D reaction (and to some extent the deuterium-tritium reaction D-T) that is being utilized both in nuclear fusion weapons and proposed fusion reactor designs. However, in any viable long-term scheme for power production from nuclear fusion, it is probably necessary to suppress the D-D reaction (and the D-T reaction) since this reaction is a heavy polluter: One branch produces neutrons which penetrate exponentially deep into materials and easily cause induced radioactivity, as is all too apparent in ordinary nuclear fission reactors. The other branch produces tritium which is a serious health hazard due to its beta radiation. In contrast to this, the charged He³ particles from the enhanced H-D reaction discussed above can be completely stopped even by very thin radiation shields from which power is easily converted by conventional methods or directly into usable form, and helium-3 is also a stable, inert and completely harmless substance. A further reduction of the polluting D-D reaction can be obtained by lowering the deuterium/hydrogen ratio of the mixture.
       Similarly, in a mixture of D² and He³ the following reaction would be enhanced since it involves nuclei with different mass

                    D² + He^{3 _}> He⁴ + H¹ + 18.3 MeV,

in comparison to the competing reactions

                    D² + D^{2 _}> He³ + n + 3.3 MeV,

                    D² + D^{2 _}> T³ + H¹ + 4.0 MeV,

                    He³ + He^{3 _}> He⁴ + 2 H¹ + 12.9 MeV,

which involve nuclei with the same mass. This embodiment of the invention thus provides a promising candidate for future clean nuclear power.
       According to the invention there is thus provided a method and/or device characterized in that violent, high-frequency oscillations of the positive background ions in response to oscillations of the electron population in the confined ensemble of charges are employed to produce energy through nuclear fusion reactions. Preferably also said oscillations of said background ions, due to differences in their mass, enhance such nuclear fusion reactions which produce essentially only charged particles and no neutrons as end products.
       Preferably also said background ions are protons (H¹) with a small fraction of deuterons (D²), and which produce fusion energy through the reaction D² + H¹ with a negligible proportion of the reaction D² + D², or alternatively, said background ions are helium-3 ions (He³) with a small fraction of deuterons (D²), and which produce fusion energy through the reaction D² + He³ with a negligible proportion of the reaction D² + D².

FURTHER MODE OF THE INVENTION - EXTRACTION OF ZERO-POINT ENERGY
       As mentioned above in the summary of the invention, there has recently been serious discussions in the scientific literature on the possibility of extracting energy from the zero-point fluctuations of the vacuum field. One example where the zero-point fluctuations are observable is the Casimir effect, in which parallel plates placed close together are attracted towards one another by a (normally minute) force caused by the zero-point fluctuations (cf, e g, P W Milonni and M-L Shih, Contemporary Physics 33, 313 (1992)). Reference is also made to the experiments by K Shoulders, documented in PCT/US90/02368 as cited above in the summary. A further, independent indication is the remarkable efficiency in conversion of pulsar rotational energy to synchrotron radiation which occurs in some supernova remnants, notably the Crab nebula, and which might possibly be indicative of a natural occurrence of energy extraction in accordance with the ring-shaped electron accelerator method described below. Considering the paramount importance of these indications - frail though they are - both from the theoretical and practical point of view, an analysis will here be made of the feasibility of using the confinement mechanism according to the invention as a test-bed to test whether energy could possibly be extracted from the vacuum fluctuations in a sustained manner in accordance with the conjecture of the relativistic Carnot process disclosed below, and if so, extract such energy on a useful scale.
       Zero-point energy should not be understood as a new physical concept providing an easy, alternative source of energy. Rather, zero-point energy has been an accepted concept in physics, in particular quantum mechanics, for a long time but previously regarded essentially as a mathematical entity without physical relevance. First the Lamb shift, then other effects like spontaneous emission and the Casimir effect eventually drew the attention to the physical reality of the zero-point fluctuations. Zero-point fluctuations can be regarded as an enormous heat bath permeating the entire universe, and with which normal physical objects are in thermal equilibrium at absolute temperature T=0. Vacuum fluctuations provide an alternative, and in some respect more comprehensive description of many phenomena in nature like inertia and gravitation (H E Puthoff, Phys. Rev. A 39, 2333 (1989); B Haisch, A Rueda and H E Puthoff, Phys. Rev. A 49, 678 (1994)). It should be emphasized, however, that it is an alternative description, not an alternative theory (somewhat analogously to the kinetic theory of gases being an alternative description of the classical, phenomenological gas laws). Thus, if energy is to be extracted from the vacuum field, this must be in obeyance with the basic laws in physics, and it is against this background that the case for energy extraction from the vacuum field will be analysed in the following.
       Of the thermodynamical laws, the second law (the total entropy never decreases) was early put on a firm theoretical foundation by Clausius, Carnot and others, and connected to irreversibility. Stephen Hawking (S W Hawking, A Brief History of Time, Bantam Press (1988), Ch 9) considers the second law as an almost tautological consequence of our awareness of time with a past which is ordered into our memories and a future which then becomes more disordered. Except in some elementary and simple processes, the total entropy is always increasing. In most macroscopic systems, the behaviour of the system is infinitely sensitive to the initial conditions and any attempt to produce a reversible process will only result in a new, less ordered state and a higher total entropy (R P Feynman, The Character of Physical Law, MIT Press (1967), Ch 5). Even a Maxwellian demon is expected to have to wait infinitely long time between the rare situations when he could extract work from heat in violation of the second law (C M Caves, Phys Rev Lett 64, 2111 (1990)).
       Concerning the first law of thermodynamics (conservation of energy), the original, empirical nature of this law has now been supplemented with a firm theoretical foundation based on the understanding that conservation laws are connected to symmetries according to Noether´s theorem (Emmy Noether, Nachrichten Gesell. Wissenschaft. Göttingen 2, 235 (1918), cf also H Goldstein, Classical Mechanics, 2 Ed (Addison-Wesley, 1980), Sect 12-7). Thus symmetry with respect to a translation in space implies conservation of momentum. Symmetry with respect to a rotation in space implies conservation of angular momentum. Similarly, symmetry with respect to a translation in time implies conservation of energy and thus leads to the first law of thermodynamics. (It should be emphasized that Noether's theorem states that symmetry implies conservation laws. The inverse is, however, not true. Lack of symmetry does not necessarily imply lack of conservation; there may be conservation laws without corresponding symmetry).
       Of the three conservation laws for momentum, angular momentum, and energy, the last one has a somewhat different character than the other two since it is based on symmetry with respect to time. As discussed above, we know that - save for some very simple phenomena - time is not symmetrical, there is always an increase in total entropy with time. For this reason it does not seem necessarily true that a symmetry with respect to a time translation is always maintained. Viewed in this way, the second law of thermodynamics thus erodes the foundation for the first law.
       On the other hand, there is a symmetry which is not included in the list above, viz symmetry with respect to Lorentz transformations. This symmetry requires conserved quantities to be Lorentz invariant. Specifically, if momentum and energy form a four-vector, then conservation of momentum also implies conservation of energy. In this way we have thus retrieved the first law of thermodynamics without having to invoke time symmetry.
       The above rederivation of the first law of thermodynamics from symmetry with respect to the Lorentz transformation is, however, valid only if momentum and energy transform like components of a four-vector. Now, it is known that in some extended systems this requirement is not fulfilled (cf C MÆller, Mat. Fys. Medd. Dan. Vid. Selsk. 36, 1 (1967); N G van Kampen, Phys. Rev. 173, 295 (1968)). Thus, if energy can be extracted from the vacuum field in seeming violation of the first law, then the above argument indicates that the only allowed regime for such occurrences would be relativistic phenomena involving accelerations of extended systems.
       A closely related observation is that there is - surprisingly enough - no uniquely defined relativistic transformation of temperature such as there is for, e g, velocity, energy, pressure etc ("the Planck-Ott imbroglio", cf W Israel and J M Stewart, Progress in Relativistic Thermodynamics and Electrodynamics of Continuous Media, in A Held (Ed), General Relativity and Gravitation (Plenum 1980), Vol 2, p 508. Cf also J Shaffer, Il Nuovo Cimento 103B, 259 (1989)). Defining temperature as usual by a Carnot process, an acceleration to a specific relativistic velocity leads to a temperature of the system which is different if, say, the acceleration has taken place simultaneously in the stationary or in the moving frame of reference. The above observations, and the fact that vacuum fluctuations are Lorentz invariant and only manifest themselves at accelerations, lend support to the conjecture disclosed below that a relativistic Carnot process involving accelerations and decelerations which are simultaneous in different frames of reference might be a way to extract energy from the vacuum field in a sustained manner, and which is not in conflict with basic thermodynamical or conservation laws.
       In contradistinction to the case of accelerations at classical velocities, and due to the difference in simultaneity in systems at relativistic velocities, accelerations at relativistic velocities are not unique. For relativistic accelerations of extended objects, it is necessary to specify in which frame the acceleration is simultaneous, i e constant over the object. If we have a stationary frame xt and a second frame x't' corresponding to an extended object in accelerated motion with respect to frame xt, we must specify if the acceleration is simultaneous in the frame xt, or in the frame x't', or in any other frame. For accelerations due to a constant force and simultaneous in the xt frame, such as due to a static field, the x't' frame forms a family of hyperbolas in the xt frame, as is well known from the literature. For accelerations due to a constant force and simultaneous in the x't' frame, such as corresponding to a rigid object, which case does not seem to have been studied in detail before, we can derive the following differential equation for the relative velocity v(t') between the xt and x't' frames.
       From the relativistic addition of velocities we get (with velocity of light c = 1) for the increase in relative velocity between the xt and x't' frames due to a constant acceleration a in the x't' frame during a short time dt'

                    dv(t') = (v(t') + a dt')/(1+v(t') a dt') - v(t'),                                                     (28)

which for dt' ^_> 0 gives the ordinary differential equation

                    dv(t')/dt' = a (1 - v(t')²).                                                                                (29)

       With suitable choice of t'=0, Eq (29) has the solution v(t') = tanh (t'a), which inserted in the Lorentz transformation connecting the xt and x't' frames, gives

                    x = x' cosh(a t') + t' sinh(a t'),                                                                        (30)

                    t = x' sinh(a t') + t' cosh(a t').                                                                         (31)

       In Fig 4 the x't' frame is plotted in the xt frame. Superficially, the x't' frame also in this case forms a family of hyperbolas. Closer scrutiny, however, reveals the fact, as seen most clearly by the inserted 3D separation of the lines, that the time t' in the accelerating system will be observed from the stationary system xt to go forwards for t > 0 and backwards t < 0. This remarkable fact could thus make it possible to design a Carnot process in which in principle, say, a heat engine/pump employed by the observer in the x't' frame whose time goes backwards, as observed by a stationary observer in the xt frame, is used by the observer in the xt frame to extract energy. The oscillations described above in an apparatus according to the invention alternate between acceleration phases due to a field, i e involving accelerations with are simultaneous in the stationary frame, and deceleration phases at the bouncing, involving decelerations which are simultaneous in the moving frame. The confinement mechanism according to the invention thus provides an apparatus to implement on a macroscopical scale a Carnot process as described above, involving extraction of energy from vacuum fluctations, or whatever other source may supply the energy in the Carnot process described.
       More specifically, in accordance with the invention an asymmetry in the spacetime dynamics of an extended system with respect to relativistic accelerations and decelerations which are essentially simultaneous in different frames of reference is utilized in a relativistic Carnot process to extract energy. Preferably, said extended system is a device as mentioned above, in which phases of relativistic accelerations which are essentially simultaneous in the stationary frame alternate with phases of relativistic decelerations which are essentially simultaneous in the co-moving frame of the charges.
       Preferably, also said violent, radially oscillating motion includes expansion phases in which the mass/energy in a subsystem of said extended system is negative, corresponding to a thermal energy of the zero-point energy fluctuations of the vacuum in said subsystem which is lower than the ambient thermal energy of the vacuum fluctuations, resulting in a flow of energy from the ambient zero-point field into said subsystem, and which energy during the compression phases of the oscillation is dissipated to the immediate surroundings of said subsystem, thus providing a pumping of energy from the infinetely-extending vacuum field to said immediate surroundings of said subsystem for purposes of power generation.
       Preferably, also said dissipation of energy from said subsystem to its immediate surroundings is achieved by slight inhomogeneities in said subsystem, which are arranged so that the dissipation of energy during the compression phases matches the energy inflow from the ambient vacuum field during the expansion phases, resulting in a steady-state power generation. Preferably, also said extended system is a ring-shaped electron accelerator in which electron bunches are cyclicly accelerated and decelerated by separate processes such that the acceleration and deceleration processes are simultaneous in different frames of reference, and from which energy is extracted in the of form synchrotron radiation.
       In summary, the confinement method of the invention thus uses a reflection of the direction of the velocity in order to achieve a state of motion in which the average spatial displacement is essentially zero even though the average acceleration may not be zero. Although discussed above in relation to electrodynamic confinement or self-confinement, the confinement method is not limited to those cases, but may also find use in other types of confinement, such as magnetic confinement.

CLAIMS
       1. A method and/or device for electromagnetic confinement of an ensemble of highly energetic, charged particles, and/or for extraction of energy from such an ensemble, characterized in that fundamental nonlinearities inherent in the electrodynamics of said ensemble, and/or the spacetime dynamics of said ensemble, with respect to accelerations and deceleration of said particles in said ensemble are utilized to confine said ensemble and/or extract energy from said ensemble.
       2. A method and/or device according to claim 1, characterized in that a nonlinearity in the mutual electrodynamic interactions between the electromagnetic field and said accelerating charged particles in said ensemble at strong fields is utilized to achieve confinement of said ensemble by producing a violent, radially oscillating motion of said particles in which, due to a repetitive discontinuity in the velocity of said particles when - analogously to bouncing - their direction of motion is instantaneously reversed, the average spatial displacement of each particle is essentially zero even though its average acceleration is not zero.
       3. A method and/or device according to claim 1, characterized in that an asymmetry in the spacetime dynamics of an extended system with respect to relativistic accelerations and decelerations which are essentially simultaneous in different frames of reference is utilized in a relativistic Carnot process to extract energy.
       4. A method and/or device according to claim 3, characterized in that said extended system is a device according to claim 2, in which phases of relativistic accelerations which are essentially simultaneous in the stationary frame alternate with phases of relativistic decelerations which are essentially simultaneous in the co-moving frame of the charges.
       5. A method and/or device according to claims 1 and 2, characterized in that said violent, radially oscillating motion includes expansion phases in which the mass/energy in a subsystem of said extended system is negative, corresponding to a thermal energy of the zero-point energy fluctuations of the vacuum in said subsystem which is lower than the ambient thermal energy of the vacuum fluctuations, resulting in a flow of energy from the ambient zero-point field into said subsystem, and which energy during the compression phases of the oscillation is dissipated to the immediate surroundings of said subsystem, thus providing a pumping of energy from the infinetely-extending vacuum field to said immediate surroundings of said subsystem for purposes of power generation.
       6. A method and/or device according to claims 1 and 2, characterized in that said dissipation of energy from said subsystem to its immediate surroundings is achieved by slight inhomogeneities in said subsystem, which are arranged so that the dissipation of energy during the compression phases matches the energy inflow from the ambient vacuum field during the expansion phases, resulting in a steady-state power generation.
       7. A device according to claim 3, characterized in that said extended system is a ring-shaped electron accelerator in which electron bunches are cyclicly accelerated and decelerated by separate processes such that the acceleration and deceleration processes are simultaneous in different frames of reference, and from which energy is extracted in the of form synchrotron radiation.
       8. A method and/or device according to claim 2, characterized in that said ensemble of charges is essentially spherically-symmetric, and said electromagnetic field is an essentially spherically-symmetric electric field.
       9. A method and/or device according to claim 2, characterized in that external electric fields are used to produce said periodic, radially-oscillating motion of the charged particles.
       10. A device according to claim 9, characterized in that said electric field is created by a central, preferentially spherical electrode supplied by an external high-frequency source connected to said electrode by a shielded connector.
       11. A method and/or device according to claim 8, characterized in that said electric field is created internally by the electrodynamics of said ensemble of charges itself, thus producing an electrodynamic self-confinement of said ensemble.
       12. A method and/or device according to claim 11, in which said self-confined ensemble of charges is created from an initially cylindrical discharge, which is caused to pinch due to the magnetic field of the discharge current, characterized in that said pinched discharge is allowed to deform by inherent instabilities, preferentially of 'sausage' type, leaving - after the current is switched off - at least one highly energetic, essentially spherically symmetric charge configuration which is arranged through the current distribution in the initial cylindrical discharge to get a radial charge density distribution proportional to 1/r^N, where r is the distance from the center of symmetry and with N greater than 3 (or approaching 3 infinetely close from above), corresponding to a radial electric field distribution in said charge configuration which is (possibly only infinitesimally) steeper than a Coulomb field, i e proportional to 1/rⁿ with n greater than 2 (or approaching 2 from above).
       13. A method and/or device according to claim 2, characterized in that violent, high-frequency oscillations of the positive background ions in response to said oscillations of the electron population in the confined ensemble of charges are employed to produce energy through nuclear fusion reactions.
       14. A method and/or device according to claim 13, characterized in that said oscillations of said background ions, due to differences in their mass, enhance such nuclear fusion reactions which produce essentially only charged particles and no neutrons as end products.
       15. A method and/or device according to claim 14, characterized in that said background ions are protons (H¹) with a small fraction of deuterons (D²), and which produce fusion energy through the reaction D² + H¹ with a negligible proportion of the reaction D² + D².
       16. A method and/or device according to claim 14, characterized in that said background ions are helium-3 ions (He³) with a small fraction of deuterons (D²), and which produce fusion energy through the reaction D² + He³ with a negligible proportion of the reaction D² + D².
       17. A method and/or device according to claim 2, characterized in that said electrodynamic confinement is utilized to store electric energy, thus providing extremely light-weight energy storage.
       18. A method and/or device according to claim 17, characterized in that the energy content per volume of said energy storage is increased by compartmentalization of the confinement structure into a large number of subunits.
       19. A method and/or device according to claims 2 and 18, characterized in that said compartmentalization of the confinement structure is used to create extremely light-weight, mechanical structure elements.

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