Migma Fusion

Migmacell: A Low-Gain "Driven" Fusion Power Amplifier as an Interim Energy Source

Appendix A
Appendix B
Appendix C
Biography of Bogdan Maglich

Appendix A

A1. The main advantage of the migma orbit configuration is that it renders it possible to store and collide MeV ions. All other advantages follow from it. Referring to Figure A1, we see that in the MeV range, the fusion cross sections are at the maximum, while all the loss cross sections are at the minimum. High energy fusion is advantageous already in a simple mirror. An example is given in the theoretical computation of McNally (Ref. 75), shown in Fig. A2. However, it is not clear how MeV ions, which exhibit random motion in a plasma can be confined with the magnetic field technology available and projected today. Migma is a system in which ion motions are ordered and MeV confinement appears possible.

A2. What is Migma? The idea of using colliding beams for fusion is old, but the problems of low fusion reaction rates and high multiple coulomb scattering rates appeared, until fairly recently, insurmountable. If two hypothetical intersecting storage rings of deuterium ions, shown in Fig. A3, were constructed, multiple coulomb scattering at their collision point would destroy the beam before any significant fusion had occurred. We have demonstrated that these problems can be overcome by storing the two colliding beams into a single enclosure placed in a magnetic field, instead of two storage rings (Fig. A3, right). The ions scattered at the intersection would now return to the scattering point instead of hitting the beam tube. In our design, the strength of the magnetic field decreases with the radius. This makes the ions precess about the center in the horizontal plane. This results in an orbital configuration shown in Fig. A4. Since it looks like a mixture of orbits, we call it migma (Greek word for mixture). The central region of the migma is geometrically equivalent to an infinite number of colliding beams intersecting at all crossing angles, from 0 degrees to 180 degrees.

A3. Simple Mirror. The flux lines of the mirror magnetic field needed to form a migma configuration are shown in Fig. A5. This is a B-maximum field, also known as the "simple mirror". The adiabatic (plasma) orbits remain attached to the flux lines and gyrate about them. They "bounce" back to the z = 0 region from the high B region at high z (if the mirror ratio and their pitch angle are right) typically in 105 to 107 gyrations. "Adiabatic" orbits imply zero magnetic gradient across the orbit. There can be no axial or radial focusing with the zero gradient.

A4. Semi-Open Mirror (Focusing Mirror). When an ion’s energy becomes so large that the diameter if its orbit equals the radius of the magnetic field, the magnetic gradient across the orbit become ~ 50% and the orbit becomes highly non-adiabatic. In this case the ions cut through the magnetic lines of force from r = 0 to r = rmax and bounce back to the z = 0 typically in one or two turns. The focusing force, acting on non-adiabatic orbits confines them to the lines of effective potential, rather than by the flux lines, shown in Fig. A5. One and the same physical magnetic field acts differently on plasma orbits than on migma orbits. As seen in Fig. A5, the mirror is about one-half less open ended, or semi-open mirror (focusing mirror).

Semi-Open (Focusing Mirror as Power Source. Migmacell is nothing but a Focusing Mirror. It has been shown (e.g., App. A2) that, with the presently available superconductive magnet technology, the losses from the open-ended migmacell are sufficiently suppressed that it can have a favorable energy balance, i.e., a fusion power source. This is due to a combination of factors:

(1) Although open-ended, a focusing mirror is about half as open-ended to migma ions as the simple mirror is to plasma ions; the rate of the angular (and energy) spread due to multiple scattering is suppressed by a factor of 2 from that in a simple mirror at the same ion energy;

(2) Additional improvement comes from the fact that focusing mirror can confine 100 times higher ion energy than a simple mirror; since the reactivity for multiple scattering is inversely proportional to kinetic energy 3/2, the importance of this loss mechanism is further reduced;

(3) Another loss mechanism that is substantially reduced by higher ion energies is the charge transfer; its reactivity decreases exponentially with the ion kinetic energy (See Fig. A1)

The synchrotron and bremsstrahlung radiation losses are discussed by R.A. Miller in App. B.

Additional Improvement in the Controlled Fusion Conditions (Eq. A8):

A5. Semi-Closed Mirror. When the densities in excess of 10 13 ions/cm3 are reached and exceeded, the diamagnetic field of the migma ions reshapes the overall magnetic field and makes it a minimum-B field (smallest in strength near the center of the cell). Thus, the "mirror ratio", which is already higher for the migma orbits, becomes even higher. The end losses are further suppressed by the positive ambipolar potential of migma. The "semi-open" mirror becomes "semi-closed".

A6. Plugged Mirror. In our design named Trigma, we went one step further; both loss cones are to be closed to ions by two electrostatic accelerators placed on the z-axis and facing one another through the migmacell. The accelerators act as decelerators to the leaking positive ions. They are set at positive voltages greater than the maximum energy of the migma fuel ions and, therefore, represent electrostatic plugs to the migmacell ions. This configuration named Trigma, consisting of a migmacell, injection accelerator and two plug accelerators (decelerators), is shown in Fig. 1 of the main text. The Trigma concept is possible only by the focusing action of the migmacell, to transmit the ions from the migmacell (where ions gyrate about the z-axis) to decelerator (where ion motion should be in the z-direction). This process of "beaming" is improved by the insertion of the magnetic cusp between the migmacell and accelerators conceptually drawn in Fig. A6, Let us follow the path of an ion that "leaked out" of the migmacell through the loss cone. It will undergo two magnetic and one electrostatic focusing actions:

(1) magnetic loss effect at one of the z max ends of the focusing mirror;

(2) magnetic cusp effect at one of the magnetic field inversions between the mirror and accelerator;

(3) electrostatic lens focusing within the electrostatic accelerator from the last to the first electrode.

All three actions (1-3) are time reversible and the ion trajectories are the same going from left to right, or from right to left. Since all the beam transport elements in +z and z direction are exactly antisymmetric, the ions exhibit oscillatory motion between the two accelerators after deceleration, the oscillatory process would result in a maintenance of the migma density.

A fraction of the leaking migma ions will not return to migmacell, but for various reasons such as collisions, remain within the accelerator after having been brought to a full stop by deceleration. While these ions are lost as fuel, their energy is not lost. It has been given to the accelerator, thus making a positive contribution to the energy balance. Since the cost of ions as fuel is less than the cost of the energy it carries, the trigma system can be considered "closed" to this type of ion loss too.

Finally, a fraction of ions will be lost and not return to the migmacell before it gives all or some of its energy to the accelerator.

The last two processes will result in a slow decrease in the migmacell density, which in turn, has to be maintained by continuous replenishment.

To electrons, the loss cone is partially closed by two different mechanisms: focusing and triode oscillator action inside the migmacell. While in principle the positive electrodes of the accelerators would have an attractive sucking action on the electrons, this is not so in the trigma design. The first electrode "seen" by the leak electrons is on the ground potential.

The electrons that succeeded in leaking together with the ions in the direction of the accelerators are removed from the ion beam by the magnetic cusp. Their magnetic field will bend electrons much more than the massive ions. These electron loses from migmacell are not nearly as harmful to the energy balance as those of ions: electron energy in a migmacell is typically 1% of the energy. Part of  the energy conversion studies will investigate the possibility of electron energy recovery in a manner similar to that in thermoelectric diodes.

The electron leaks are to be recovered by electron guns coupled to a sensor-connected feedback circuit, responding in a matter of 10-4 seconds to an electron content change of 0.1%.

Direct Linear Conversion of Ion Energy into Electricity. The two accelerators act, at the same time, as converters of the ion kinetic energy into electric energy by deceleration. Energies of both the leaking ions and the fusion products are converted into electricity directly by this process, named "linear" conversion because the ion orbits remain nearly parallel to the z-axis until brought to a halt.

To a single ion orbit, the process of deceleration is in every way equal and opposite the well-known electrostatic acceleration in, say, van de Graaf accelerators. The focusing action of the accelerator lenses is direction independent in the same way a symmetrical optical lense is to light rays.

In practice, the ions will not behave as single ion orbits because of (1) space charge in the decel-accelerated beam; and (2) collision with residual gas in vacuum. Effects (1-2) will increase somewhat the emittance of the beam:

(A1)   Emittance = ( Delta R ) ( Delta Phi ) T ½ mr cm MeV ½

in every traversal, so that eventually the beam will "blow up".

The lifetime of the oscillating beam is a sensitive function of the vacuum. Taking into account the suppression of the multiple scattering spread due to focusing, we estimate the lifetime of the oscillating beam to be of the order of the fusion time.

Condition for Controlled Fusion. The problem if achieving controlled fusion in an open-ended system reduced itself, to the first approximation, to making the fusion time, tF, about equal to or less than the time constants of each of the four principal loss mechanisms: multiple scattering, t1, charge exchange, t2, synchrotron radiation losses, t3, and bremsstrahlung radiation losses, t4. If we denote the overall loss time by t:


the "controlled" fusion condition is given by:


The tF is given by the reactivity <ov> and density n[cm-3]:


The time constants for migma are evaluated by R. A. Miller in Appendix B

References cited in Appendix A are to be found after the text of the main paper.

Figure A1. Thermonuclear versus High-Energy Fusion

The diagram shows reaction probabilities  <ov12> for various gain and loss processes operative in a fusion device as a function of ion "temperature", Tion.

Fusion reaction probabilities leading to energy gains increase sharply with energy for all fusion reactions. Curves for d-t, d-d, d-He3, and He3-He3 fusion reactions are shown.

Charge transfer (CT), multiple scattering (MS), and synchrotron radiation ("synch") probabilities leading to energy losses, all decrease sharply at higher temperatures.

Since the interior of the hottest stars (White dwarfs) is 100 KeVm we designate this temperature the dividing line between Thermonuclear and High-Energy Fusion. It is seen that below 100 KeV energy loss processes are generally orders of magnitude more probable than fusion gains, while above 1 MeV the energy losses are clearly less than the fusion gains.

Figure A2. Parametric requirements for producing B ~ 1 plasma in 2/1 mirror geometry. The trapped current is that actualy trapped in the plasma and may be of order Io at 10 KeV and of order 10-2 Io at MeV injection energies. I+ = n2V/nt a B4V/E7/2o using kT ~ Eo/2. Neglects beta effect (From Ref. 75).

Figure A2

Figure A3. Illustration (left) of the fact that two colliding beams in storage rings could not make a fusion reactor because the cross section for the multiple coulomb scattering, oMS, through an angle greater than, say 10° is so much greater than that for fusion, o fusion, that the beam would spread out and hit the walls of the rings, and cease to be a "beam" before any significant number of ions fuse. The scattering time is shorter than the fusion time.

(Right) If, however, two rings are removed and two beams are made to collide in a single vacuum chamber placed in a uniform magnetic field, scatterings in the center that change either the angle or energy of the ions will not spread the beam in the center of the magnetic field. Liousville’s Theorem requires that after any perturbation of the ion orbit in the center, the ion must return to the center. This becomes less true, the farther away from the center the scattering takes place. Since most of the scatterings take place at or neat the center the orbits will be concentrated in the central zone for a relatively long time, i.e., time comparable to or longer than fusion time. This is why intersecting orbits in migmacell can generate fusion energy and intersecting storage rings.

Figure A3

Figure A4. If the strength of the magnetic field decreases towards the periphery, orbit will exhibit precession, in addition to revolutions. Time sequence of a precessing orbit (for simplicity, drawn as a circle) in a magnetic field of a migmacell is shown in three consecutive "snapshots". At one half the precession period, ½ tp, another orbit was injected. The orbits continue precessing in the head-on collision configuration at the center, with the multiple traversal factor of about 103/s. If the injection is continuous, the mixture of multitude of orbits, similar to D is formed. This is migma.

Figure A4

Figure A5. An example if a plasma orbit (adiabatic); and the lines of effect potential that confine migma ion orbit (non-adiabatic). The higher the energy of the ion, the better it confines the ion because of the focusing force. As seen, the focusing mirror is one half less open-ended.

Figure A5

Figure A6.

Appendix B

Time Constants at High Energies

Robert A. Miller

In order to demonstrate the important qualitative point that the important loss processes become more manageable at high energies, in the vicinity of 1 MeV, we calculate the time constants for fusion, multiple scattering end losses, charge transfer, bremsstrahlung, and synchrotron radiation.

We perform the calculations for a 2 MeV deuteron migma with an effective average density of 10 14 cm-3, which is approximately the asymptotic average density due to diamagnetism (Ref. 32).

Fusion. The fusion time constant is taken to be:

(1) tF = 1 / n<ov>


(2) n = 1014 cm-3
<ov> = 2 x 10-16 cm3/s.

The value of <ov> is taken from Reference 62. Then:

(3) tF = 50 s.

Multiple Scattering End Losses. The dominant effect is known to be ion-ion scattering perpendicular to the velocity vector. We find the following expression for the ion-ion, perpendicular collision frequency in Reference 79, page 54:


We take:


and we find:


If we identify vii| with d <O2>/dt, which is given in FEC-01-75, the respective formulas give results which agree.

The collision frequency (6) is roughly the rate of randomization of the initial direction of an ion. If Delta Omega is the fraction of the total solid angle through which an ion can be lost, then the time constant for multiple scattering end losses is:


For a loss cone of 60° , measured from the mid-plane, Delta Omega = 0.1


(8) tMS = 169 s.

Charge Transfer. The charge transfer time is given by


where no is the density of background gas, assumed to be D2.

The charge transfer cross section for a 2 MeV deuteron is taken to be the same as that for a 1 MeV proton (that is, dependent on velocity alone), which we find in Reference 80,


If we assume a D2 pressure of 10-9 torr, the background gas density is no = 3.5 x 107 cm-3. we find:

(11) tCT = 1.0 x 105 s.

Note that at high energies <ov> decreases as Td-5 so that tCT increases as T5d

Synchrotron Radiation. At high (reactor) densities, synchrotron radiation is reabsorbed with high probability before escaping from the migma. An accurate treatment of this process is exceedingly difficult, and so far only preliminary results are available showing that a positive power balance can be achieved in the face of synchrotron radiation for fusion power in the range of a megawatt per cell.

We do the following calculation placing the uncertainty in the reabsorption coefficient R.

Including the reabsorption coefficient in Equation 2 of Reference 50 we find the sycnchrotron power radiated by a single electron:


Define a time constant:


The ion energy is used because ultimately it is the ion energy loss through dynamic friction and electron radiation that is important:


where B is in kG.

Note that the average magnetic field will be reduced dramatically by diamagnetism (Ref. 32) We take:


so that

(16) tsynch = 256 s.

It should be remembered, however, that this is a rough estimate to be replaced by a more elaborate calculation which properly handles energy balance, diamagnetism, and reabsorption.

Bremsstrahlung. From Reference 50, Equation 4


We define


We overestimate Pbrem by only a factor of 2 if we take *** . Since nd = 1014 cm-3, we find


Summary. Using (3), (8), (11), (16) and (19), we can calculate the following ratios:


All of these are for a highly diamagnetic deuteron migma of effective average density 1014 cm and energy 2 MeV.

References to Appendix B are to be found after the text of the main paper.

Appendix C

Remarks on Some Stability Problems for Migma Configuration

1. The Migma Configuration and Its Physical Parameters ~

Since the proposal of the migma concept (Ref. 1-3), questions have been raised concerning the stability of the system, in particular at the densities of the order of 108 cm-3, and above, where other machines have encountered difficulties. For a stability evaluation, migma can be briefly described, at low and moderate densities, as a simple mirror magnetic field around 3-6 x 104 Gauss, filled by injection of monoenergetic high energy ions (-0.5 - 1MeV) through the symmetry axis. That is, with nearly zero canonical angular momentum (  )  being the magnetic flux enclosed in a circle of  radius r). The geometry of the device is compact, the axial extent being about 8 cm for the present Migma IV state for the experiments. The chamber walls are about 50 cm in radius. Ideally the ion distribution function is the product of two delta functions, , although scattering processes will produce spread in energy and momentum of a magnitude which depends on the density. A Monte Carlo Transport Code was developed at FEC to study a steady state distribution of the ions (in the absence of instabilities) and the following characteristics were found to hold over a large range of densities ( ~ 108-1012 cm-3 ). The density distribution is strongly peaked in the center with a radial scale length  (for instance in the Migma IV experiment,  ). The radial extend of the plasma , the radial scale length of the magnetic field  is larger than RLi and much greater than a. The central density peak is followed by a "plateau" of lower density (for Migma IV estimated to be 1/6 of the central density) until the density decreases to zero at r = 2RLi (the plateau extends up to 8 cm and density drops to zero at 10 cm for Migma IV).

The features described above correspond to a physical regime for which no adequate results for plasma stability exist at the present. Expansions in power of RLi/Ro used for plasma systems are not satisfactory here. The strong decrease in density along the ion orbits implies drastic changes of the dielectric properties over a Larmor Radius, and this effect has not yet been included with any generality to be useful for the migma stability analysis, in the existing theories. In the current trend of thermonuclear research the assumptions a >> RLi and a >> Lambda (the wave length of a perturbation) are generally justified and so the WKB or the local approximations are largely used (See Ref 5 for instance). These techniques cannot be applied to the Migma case (see for instance an attempt done in this sense in the Robson Panel Report, Ref. 6, for the ion-counterstreaming instability and the FEC demonstration of the shortcomings of their method, Ref. 7, Appendix.

Summing up, no results from the existing literature are applicable with confidence to Migma because of (i) zero canonical momentum distribution, (ii) 2RLi ~ Ro, (iii) a << 2RLi, and (iv) short axial extent typical of the Migma configuration. Properties (ii) and (iv) imply that the two usual adiabatic invariants of the plasma mirror theory --- the ion magnetic moment and the longitudinal invariant --- are absent in the non-adiabatic migma orbits, New instability calculations are needed. Some have been started at FEC with encouraging results (See Ref. 8) and more work is under way.

2. Comments on the Instability Predictions ~

The goal of the migma experiments to demonstrate the feasibility of a self-colliding orbit type of compact containment of ions in the MeV range, where charge exchange losses become negligible, important gains in n are expected (if the density is Coulomb Scattering limited) relative to lower injection energies, and large ion orbits plus high electron temperature provide some stabilizing properties.

The maximum density achievable practically with this scheme is unknown and this point cannot be decided confidently without actually performing the experimental research. We have noted already the shortcomings of the existing theories. However, even more refined analysis should be tested in the laboratory. In general, plasma collective behavior is so complex as to often defy a purely theoretical stability forecast. In this field, as in others involving many-body behavior, theory and experiments must follow each other closely in order to develop reliable knowledge.

Consider the case of the drift-cone instability. B. Kadomtsev (Ref. 9) noted, "As is well known this instability in experimental mirror devices was found to be much less dangerous than the theoreticians had predicted". In relation to this see also Ref. 5. This instability may be a matter of serious concern for Migma at high densities, in view of the small value of a. But again, on the basis of present knowledge, the value of any definite statement about its relevance is dubious, and perhaps only the experiment can clarify the extent of this threat.

3. Negative Mass Instability ~

The decisive identification of the negative mass instability in the DCX-1 experiments is reported in Ref. 8. Since the DCX-1 magnetic field is quite similar to the migma one, although the investigations were aimed differently, negative conclusions have been drawn sometimes about the Migma project on the basis of the DCX experience. The relevance of the DCX-1 data for migma has been analyzed before in Sect. 3.3 (See also Ref. 9). The main points are the following: the DCX-1 was injected at a rather high energy producing ions of 0.3 MeV (comparable with the 0.55 MeVof Migma IV); no flute instability was observed (Ref. 10); the density regime was such that w2pe > w2ci >w2pi. The Migma III experiment was operated in the same regime and no flute instabilities were observed, or any other for that matter (Ref. 22). We believe this is because both experiments were performed at large ion Larmor radii, which has a stabilizing effect on the flute disruptive instability, as we shall review in the next point. The DCX-1 experiment identified without any doubt the negative mass effect, when the concentric orbit (also called "circular orbit") was populated with ions. In fact, the negative mass collective behavior is a property of the concentric orbit, but is reduced and tends to cancel out for non-concentric orbits (zero canonical momentum). This was shown in the migma experiment and theoretically in Ref. 6. We quote the conclusions from the abstract:

"Negative mass instability via a 2-dimensional, electrostatic numerical simulation was investigated for two quite dissimilar orbit geometries in a simple magnetic mirror field. For the case of a high energy ion beam circulating about the magnetic axis, it is found that the system is highly unstable and the beam clumps azimuthally --- in agreement with theory and experiment. However, if the beam is displaced so as to pass through the magnetic axis, forming a migma-like configuration, the system remains highly stable and there is no tendency towards particle clumping even for w2pi/wci >1. Analytical results support the conclusions reached by the simulation study".

In fact, about two orders of magnitude improvements in the density threshold of DCX-1 were reported (Ref. 10) when the injection was shifted from the standard mode to the central mode (migma-like mode). The density was still assumed to be limited by the instability at 2 x 108 cm-3. However, we must note that: (a) at this point the experiment may have been within observational error of the maximum density that was possible, allowing for the given injection current, the vacuum and the orbital properties; and (b) there was an important difference in the beam cross section between DCX and Migma (which can be traced back to the different aims in the research). While in Migma the beam was focused to 0.3 cm diameter, in DCX-1 it was about 5 cm (formed by the passage of a H+ beam through a water vapor cell). Hence the circular orbit was populated even when the injection was aimed at the center (migma-like mode) because of the broad beam. The DCX experiments were discontinued probably because the main interest was then attracted by the magnetic well concept.

In the Migma III experiment the unstable orbit was avoided and the density exceeded the instability threshold of DCX-1 by an order of magnitude without any instability observed (Ref. 22). The density was limited only because of limitations in the current and vacuum available.

Another experiment of injection through the axis of a simple mirror field which presented remarkable stability properties was the MMII at Fontenay-aux-Roses (Ref. 11), although the energy was lower (70-80 KeV) then the previously mentioned ones. Unfortunately, the experiment was discontinued in the course of policy-making priority decision about the French Fusion Program (Ref. 12), apparently for reasons not related to the scientific merits of the project.

4. The Flute Instability ~

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of density limitation in simple mirrors. However, the existence of a great variety of devices and physical regimes of operation does not allow for such a simple experiment, During the 60s, much was written on the simple mirrors and many effects, like the finite Larmor radius stabilization, the influence of metalwallsm the presence of uncompensated space charge, etc., were studied in connection with the stabilization of flutes. Some  results of the time may have been left in the shadow, so to speak, by the new trend in mirror research, the minimum-B concept, that has dominated since the middle 60s up to the present. In fact, even the most unstable m = 1 mode is influenced by finite Larmor radii effects. We will presently review the relevant information.

Rosenbluth, Krall, and Rostoker (Ref. 13), using a finite albeit small RLi theory, found that all the flute modes except the m = 1 were stable for cylindrical geometry. The fact that, in their radially unbounded plasma model, the m = 1 has a uniform electric field associated with it is the main reason for the persistence of the flute instability. As a consequence, the electric drift is the same for both species, ions and electrons, and is not affected by the size of orbits. However, it is precisely the non-uniformity of the electric field along the ion’s orbits that provides the finite Larmor radius stabilization effect. This was shown in  a clear physical way by F. Hoh (Ref. 4). Afterwards many authors have shown that the electric field of the perturbation can be non-uniform even for the m = 1 mode in bounded plasmas. Hence the finite Larmor radius stabilization can be found also for the m = 1 mode, contrary to a widespread belief. See for instance L. Kuo, et al. (Ref. 15, 16), A. Futch, et al., (Ref. 17), K. Jungwirth, and M. Seidl (Ref. 18), to quote a few of the most relevant works. The non-uniformity of the electric field in these more realistic models is due to the finite radial size of the plasma and the presence of the chamber walls.

4.1 Flutes at Low Density, w2pi << w2ci

The calculations of Kuo, et al. (Ref. 15; see also Futch, et al, Ref. 17) were intended to explain the results of the Phoenix Mirror experiment, This device operated at 30 KeV and at densities such that w2pi << w2ci. Flute instabilities were observed at densities above a certain threshold, but for low values of the magnetic field (large Larmor radius) the stability improved as the density threshold was shifted toward the highest densities achievable with the apparatus. The theory was always about one order of magnitude below the observed data in its prediction of the density threshold, probably because it was not accurate enough at very large ion orbits, i.e., RLi comparable with Rp, and also because the axial inhomogeneity was ignored in the model. In spite of these shortcomings, the enhanced stability effect at large ion orbits was brought in evidence and qualitatively reproduced by the model. In the paper of Futch, et al. (Ref. 17), space charge effects (deviation from neutrality in the plasma) on the flute modes were also computed. A more recent study of this stabilizing effect was given by Putvinsky (Ref. 20). Although this effect contributes,it is doubtful that it may provide full stabilization acting alone, since intolerably high space charge would be needed to quench the m = 1 mode.

The Kuo, et al. results (Ref. 15) can be compared with the Migma III experiments. The average density of Migma III was near the threshold and the central density well above it. However, in view of the differences between the density distribution in Migma and the Kuo, et al., model and the approximate nature of their results, infinite conclusions cannot be drawn here. The absence of flute in Migma III (and also in DCX-1) may be due to the stabilization mechanism incorporated in the above mentioned model and consistent with the Phoenix experience at low magnetic fields.

4.2 Flutes at High Densities, w2pi >> w2ci

In another work, Ku, et al. (Ref. 16) demonstrated that the m = 1 mode can be stabilized in the high density regime w2pi >> w2ci, at all densities (above n >> B2/4 pi mic2 ) compatible with the electrostatic approximation (i.e., low Beta). To that effect the following condition must be satisfied:

Uo being the ion velocity and omega the ion precession frequency die to themagnetic field gradient. This result is very interesting for the proposed Migma IV experiment where the regime w2pi >> w2ci is expected. With the following numerical values adequate for Migma IV, wci = 1.4 x 10 8 sec-1, Uo = 7.2 x 108 cm/sec (D+, 0.55 MeV) Omega = 3 x 106 sec-1 and Rp = 2RLi, we found that the above criterium is satisfied with A ~ 30. Referring to the Phoenix program, Kuo, et al., commented that "An A of 10 requires a field of 5 Kgauss which gives RLi / Rp =  ½ . Such a large ratio of Larmor radius to the plasma radius is of course not satisfactory but there are several means to improve it". What was unsatisfactory in a thermonuclear context, that would need many RLi to fit into the plasma,is instead quite natural in a high energy device such as a migmacell where RLi / Rp = ½ is an intrinsic feature. If this theoretical result is confirmed by the Migma IV experiment, a major step forward in the compact fusion concept will be achieved, as the flute is a dangerous disruptive instability.

4.3 More Remarks about Flutes: Intermediate Densities ~

Complete calculation for the m =1 mode in cylindrical geometry at intermediate densities around wpi = wci and for arbitrary values of A are not available as far as we know. These extensions of the Kuo, et al. work must form a part of the research for Migma IV. A plane slab geometry model (with small RLi and large inhomogeneity scale length) has been worked out and may be used to give qualitative orientations, bearing in mind that the translation of results from a plane to a cylindrical geometry is unsatisfactory. In the plane model the flute mode shows an unstable gap between two stable regions at high and low densities respectively. From Mikhailovskii’s general review (Ref. 21, Chap. 6, p. 127, vol II), it can be seen that the unstable gap is defined by

approximately, for lambday = RLi, i.e., translating crudely from the plane to the cylindrical geometry for the m = 1 mode.

This is ignoring the axial inhomogeneity (plasma infinite in z). When the short axial extent of devices like migma is taken into account the stabilizing effect of metal conducting ends (which enforce finite values on the axial wavemnumber) appears. For the same model (Mikhailovskii, Ref. 21, Ch. 6, p. 126, vol II, and taking into account that (wpe a/c)2 < 1, in Migma IV) the stability condition, when conducting ends play a role on the flute mode, is

which holds well for the migma parameters.

Another question is raised in the literature of the 60s regarding the flutes is that some models show that ion viscosity dissipation can be destabilizing, even when the finite Larmor radius effect is operating (see for instance Ref. 21, p. 131). The effect was not observed in low density experiments. For migma we estimate its growth rate of the order of the relaxation frequency of the ions, which due to the MeV range ion injection energy, is extremely low. This may explain why this effect was not observed.

Thus the Migma IV experiment will provide an opportunity to test critically the several stabilization mechanisms of the flute modes and will contribute to answering the many unsettled questions in the stability of large ion orbits.

References ~

(1) B. Maglich, et al.: Phys. Rev. Letters 27: 909 (1971)

(2) B. Maglich, et al.: IEEE Trans. Nucl. Sci. 22(3): 1790 (1975)

(3) B. Maglich: Nucl. Instr. And Meth. 144: 33 (1 July 1977)

(4) J. E. Golden: "Migma IV Parameters" in "Analysis of the Final Report of the ERDA-Appointed Panel (Robson Panel) to Evaluate the Migma Program of Controlled Fusion"; Also, Abstract in the 1977 Spring Meeting of the Amer. Physical Soc. APS Bull. 22: P 649 (April 1977)

(5) D. E. Baldwin: Rev. Modern Physics 49: 317 (1977)

(6) M. Emery: "Analytical Study of Negative Mass" and "Numerical; Study of Negative Mass" in FEC Rebuttal, March 1977 as appendixes. Also: Abstract in the Spring 1977 Meeting of the Amer. Physical Soc., APS Bull. 22: 649 (April 1977)

(7) B.B. Kadomtsev: Nucl. Fusion 15(2): 319 (1975)

(8) B. Postma, et al.: Phys. Rev. Letters 16: 265 (1966)

(9) J. Golden, et al.: FEC Report (November 1976); "The Relevance of DCX Data to Migma"

(10) J.L. Dunlap, et al.: Proc. Culham Conf. On Plasma Phys. And Controlled Thermonuclear Reactions, 1965., p. 67.

(11) F. Prevot, et al.: Proc. Culham Conf. On Plasma Phys. And Controlled Thermonuclear Reactions, 1965., p. 135

(12) F. Prevot, et al.: Pivate communication to B. Maglich, 1977.

(13) M. Rosenbluth, et al.: Nucl. Fusion Suppl. 1: 143 (1962)

(14) F. Hoh: Phys. Fluids 6: 1359 (1963)

(15) L. G. Kuo, et al.: Phys. Fluids 7: 988 (1964)

(16) L. Kuo, et al.: J. Nucl. En. C 6: 505 (1964)

(17) A. Futch, et al.: Proc. Culham Conf. On Plasma Phys. And Controlled Thermonuclear Reactions, 1965., p.3.

(18) K. Jungwirth, M. Seigl: J. Nucl. En. C 7: 563 (1965)

(19) . Schmidt: Physics of High Temperature Plasmas; Academic Press, 1966.

(20) S. V. Putvinskij: Nucl. Fusion 15: 78 (1975)

(21) A. B. Mikhailovskii: Theory of Plasma Instabilities, Vol. 1 and 2; Consultants Bureau, NY 1974.

(22) J. Ferrer, et al.: "Observation of Higly Localized Trapping of MeV Deuterons into Self-Colliding Orbit Distribution (Migma) and Measurement of its Density and Confinement Time"; Submitted to J. Nucl. Fusion, 1977.

Biography of Bogdan Maglich

Bogdan Maglich was born in 1928 in Yugoslavia. His father was a lawyer and an electrd member of the Yugoslav (Royal) Parliament. At age 12m he was, with his mother, imprisoned in a Nazi concentration camp for Serbs, then escaped.

Maglich graduated in Physics from the University of Belgrade in 1951; obtained a MS Degree at the Univ. of Liverpool, England in 1955; PhD degree at MIT in 1959. He joined the research staff of Dr Louis Alvarex at Lawrence Berkeley Lab in 1959. In 1961 he discovered the omega meson for which he and the entire team received a White House Citation by President Kennedy. In 1962 he invented and patented the sonic spark chamber.

In the period 1963-67 he was senior staff member of CERN European Organization for Nuclear Research in Geneva, Switzerland. He was Deputy Group Leader of the British team, then Group Leader of a predominantly Swiss team of 30 physicists. In 1964 he invented "Missing Mass Spectrometers" and discovered a number of new particles.

In 1967 he became professor at the Univ. of Pennsylvania; visiting faculty at Princeton Univ.; and in 1969, Professor and Principal Investigator for High Energy Physics at Rutgers Univ. until 1974, when he resigned to form Fusion Energy Corp.

He was Group Leader in a number of experiments at Brookhaven National Lab, Princeton-Pennsylvania Accelerator, Argonne National Lab and Fermilab (National Accelerator Lab).

He was a member of the US delegation to the International Conference on High Energy Physics in Vienna 1968); Kiev (1970); course lecturer at international postdoctoral schools of physics in Erice (Italy), Madras (India), Erevan (USSR); invited lecturer at a number of scientific institutions such as American Physical Society, British Physical Society, W. German Physical Society, Yukawa Hall (Japan), Nobel Institute (Sweden), Tata Univ. (India) and most of the US, Canada and Western European universities and laboratories; Resident Scientist at a UN Seminar on the Economic Development of East Africa in Kenya. In 1973, nominated by the President of Switzerland, he became an honorary citizen of that country.

In 1976 he became Chairman of the Yugoslav-American Bicentennial Committee. In the same year, President Ford sent him as Special Representative of the President to a Yugoslav National Celebration.

Maglich is President of Fusion Energy Corporation and Chairman of Migma Institute of Clean Fusion, its research laboratory. He is also Editor and Publisher of World Science Education, Inc., which publishes the series Adventures in Experimental Physics.

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