From: system@cc.usu.edu
Newsgroups: sci.physics.fusion
Subject: electrostatic fusion
Date: 29 Dec 92 11:28:35 MDT
Organization: Utah State University

Introduction:  Researchers at Hill Air Force Base have recently
submitted a patent application concerning the use of electrostatic
devices for creating nuclear fusion.  On Dec 7, an announcement
appeared on the network about using electrostatics for controlling
nuclear fusion.  We are making this disclosure in order to clarify
our contribution to these developments.



                    Obtaining Nuclear Fusion
                   Using Electrostatic Devices

                     Lloyd G. Allred, Ph. D.
                    Software Support Division
                    Hill Air Force Base, Utah

1.   Executive Summary.

1.1  Overview.  The announcement of cold fusion by Stanley Pons and
Martin Fleishmann is probably the most controversial scientific
claim of this century.  Pons and Fleishmann felt that their
experience in electrochemistry gave them the unique ability to
concentrate electric fields, and that fusion was the result of this
claim.  While Pons and Fleishmann were widely acclaimed in the
field of electrochemistry, mainstream physicists were quick to
condemn Pons and Fleishmann's work, even though their experimental
results have not yet been explained.  This document presents
underlying physical principals which will explain the results of
Pons and Fleishmann, and describe how to build practical devices to
produce large amounts of power from nuclear fusion on a repeatable
basis.  This document also explains why prior researchers have been
unable to get repeatable results.  

It is interesting to note that the main thrust of the experiments
and apparatus of Pons and Fleishmann was directed more toward
charging up the palladium rods with positive deuterium ions.  The
issue of electric field focusing was ignored.  In particular,
electrostatic fields are focused by putting sharp points on
electrostatic objects.  The sharp points produce enormous fields. 
In the phenomena known as Saint Elmo's fire, a corona is observed
at the edges of leaves where the charge on the leaves leaks off
into the surrounding atmosphere.  The observed corona is caused by
the ionization of the air caused by the electric field
concentration.  Returning to the experiments of Pons and
Fleishmann, the sharp points on the palladium rods were smoothed
off when an explosion occurred.  The two experimenters may have
thus eliminated the very effect they were trying to create.

1.2  Approach.  Some two years ago, I made the conjecture that cold
fusion could be induced by focusing electric fields using
electrostatic devices, in particular, that electric fields at the
corners of such devices could be made sufficient to induce
thermonuclear fusion.  It seemed to me that it would be relatively
simple to solve for the electrical fields surrounding a pointed
object, such as a cone.  After researching this field, I quickly
came to the conclusion that the general solution of electrostatic
fields is one of the unsolved problems of modern physics.  While
the study of electrostatic fields is a required subject in most
college physics classes, the published solutions, using Gauss's
law, have been solved only for very simplified cases.  The
published solutions for a charged wire and a cylinder violate the
underlying assumptions, and are essentially wrong! 

1.3  Summary.  It has taken me some two years to find a method
which would solve this problem.  The results indicate that electric
fields, sufficient to cause nuclear fusion, can be obtained using
a cone with a half angle of 15 to 20 degrees, and using rather
moderate positive
voltages.

1.4  Claims.  I claim the right to all thermonuclear fusion devices
which employ charging a metal with deuterium and which apply an
electrostatic charge (e.g. positive voltage) to the metal, and
which use a pointed tip to focus the electrostatic fields.  I also
claim the rights to the process of fuzzy genetic algorithms for
solving the charge distribution associated with a arbitrary
electrostatic objects and for applications of such solutions.  I
also offer the U.S. American government free use of this idea, but
reserve all applications in the private sector.

2.   Theory.  This document is provided without references.  The
underlying physics can be found in any college physics textbook or
physics handbook.

2.1  Electrostatic Charges.  Coulomb's law portrays the force, F,
exerted between two electronic charges, q and q0, at spacial
positions X and X0 and can be computed by 

       - q q0 (X - X0) 
  F  =  ________________                                        (1)
                     3
         e0  X - X 

Coulombs law can be extended to calculate the electric potential or
voltage of any point in space.  

        1                   - rho(X) dS  
  V  = ________ Integral   ________________                     (2)
                   S
       4 pi e0                 X - X 

Electric potential is closely related to potential energy.  In
particular, it expresses the amount of potential energy per unit of
charge for at any point in space.  The electric field is defined as
the gradient of the electric potential, 
       partial V       1               rho(X) (X-X0) dS
  E =  _________ =  _______  Integral  ________________         (3)
                                                  3
       partial X    4 pi e0     S        X - X0 

In the presence of a non-zero electric field, a charged particle
will be accelerated in the direction of the field, inducing an
electric current.  At any point in the interior of a conductive
device, the electric field must therefore be zero, otherwise, any
electron at that point would be accelerated, thus changing the
distribution of the electric charges and the corresponding electric
fields and the electric fields would not therefore be static.  At
the surface of a conductive device, an electric field can be
resolved into two components, one parallel to the surface, and one
perpendicular to the surface.  The parallel component must be zero
(otherwise a current would be generated parallel to the surface of
the conductor), hence the electric field must be perpendicular to
the surface of the charged conductive object. 

Gauss's law relates the integral of an electrostatic field over an
arbitrary surface to the enclosed charge.  
    1
 _______ Integral  E . dA = Q                                 (4)
 4 pi e0    A

This equation can be employed to produce some very strange results. 
Electric charges in a charged conductor tend to repel each other. 
As a result, they get as far away from each other as is possible. 
Taking a surface just inside the charged device, the above equation
implies that the enclosed charge is zero!  As a consequence, the
charge must be distributed entirely on the surface of the object. 
Additionally, the charge density at the surface is directly
proportional to the electric field over the surface.  If one
considers a surface consisting of two connected parallel plates,
both parallel to the surface with one above the surface, and one
below.  Because the one below the surface is inside the metal, the
electric field is zero.  It follows that 
    1
 _______ Integral  E . dA = Q = rho dA                        (5)
 4 pi e0    A

Consequently

  E = rho 4 pi 0                                              (6)


For an electrostatic object of a cone, we intend to show that the
electric field approaches infinity as one approaches the tip of the
cone.  Because the charge density at the tip is proportional to the
electric field at the tip, then, byt the above equation, the charge
density also approaches infinity.  If the charges consist of
positively charged deuterium ion,s then nuclear fusion will occur.


2.2  Supportive Evidence.  In the phenomena known as Saint Elmo's
fire, a corona is observed at the edges of leaves where the charge
on the leaves leaks off into the surrounding atmosphere.  The
observed corona is caused by the ionization of the air caused by
the electric field concentration.  

Some metals, such as platinum, palladium, titanium, and nickel have
the ability to absorb large amounts of hydrogen and deuterium. 
After absorption, the hydrogen nucleus and electron separate.  An
ionized hydrogen nucleus, having no shell, is free to wander
throughout the metal in the same manner as a free electron.  If the
conductive metal becomes positively charged, the hydrogen nuclei
will congregate at the sharp points of the surface, and be ejected
from the surface.  It was trace amounts of tritium and helium,
observed by Russian scientists in nickel deposits, which first
prompted the initial investigations into cold fusion. 

2.3  "Antimatter".  In the past, electric field confinement has
been rejected as a method for obtaining fusion.  In particular, a
theorem exists that a charged particle cannot be confined by an
electrostatic field.  In practice, confinement is not required. 
One example is the production of fusion obtained by imploding
spheres of frozen hydrogen.  While confinement is not achieved in
this process, the imploding deuterium nuclei come crashing into
each other.  The resulting fusion has been observed in the
laboratory; unfortunately, however, the resulting fusion has not
been not sufficient to achieve a positive energy return.

3.0  Proposed Mechanics.  I fully understand that infinite
electrical fields are an impossibility.  As a consequence, static
conditions cannot be achieved on an object with a tip because the
charge leaks off into empty space.  The charge at such a tip will
be less than
infinite.  However, as the subsequent discussions will show, the
electric fields at the tip are caused by the focusing of the
electric fields of charges away from the tip.  As a result, the
charged particles within the device will be accelerated toward the
tipthese accelerations approach infinity as the charges move
toward the tip.  Inasmuch as fusion has been produced by
accelerating protons together by implosion, it would seem that
fusion could be obtained by accelerating protons together by
focusing electric fields. In the case of the cone, the charges are
all concentrated on the surface of the cone.  As a consequence, the
electric fields are all
directed towards the tip of the cone.  If the metal contains
positively charged hydrogen ions, they will be accelerated toward
the tip.  Electric currents move at the speed of light.  While the
individual electrons and protons will move at speeds somewhat less
than the speed of light, I suspect the accelerations, even at short
distances, are enormous.  I would be greatly surprised
if fusion did not occur.

3.1  Caveats.  Surface conditions would be very critical to this
process.  Rough surfaces would tend to leak the hydrogen off. 
Smooth surfaces seem to be a rudimentary requirement in existing
cold fusion experiments.  Metals such as titanium and aluminum form
an oxide coatingI suspect that protons may not be the principal
charge carriers on the surface of these metals, perhaps explaining
the failure to produce positive cold fusion results using these
metals.  While hydrogen absorption and the corresponding metal
embrittlement is observed in many metals, much better absorption
characteristics are obtained using nickel, palladium, and platinum.

4.   Problem Solution.

4.1  Traditional Analytical Methods.  College textbooks usually
employ Gauss's equation to solve for the distribution of charge on
a surface.  For a sphere, for example, one can argue, by symmetry,
that the electric field is equal for all points of the sphere. 
Gauss's equation reduces to
    1                       E 4 pi R R
 _______ Integral  E . dA   __________  =    Q                  (7)
 4 pi e0    A                4 pi e0

Solving for the electric field, E,
      Q e0
  E = ____                                                      (8)
         2
        R

For applications to objects such as an electrical wire, the
textbooks say, "assume that the charge density is distributed
equally along the length of the wire".  We could equally assume
that the world is flat, or that we have a spherical cow.  

If one had such a charge distribution along a wire, compute the
electric field at a point on the left side of the wire.  Applying
Equation (3) to our example of a wire of length extending in x
direction,
          1        1      rho(x) (x-x0) dS
  E =  ________ Integral  ________________                     (9)
                                      3
       4 pi e0     0        x - x0 



The integral splits into two parts, one for x < x0 and x > x0. 
Substituting x = t + x0,

   x0     rho(x0-t) dt       1-x0      rho(x0+t) dt
 Integral ____________   =  Integral  _____________           (10)
   0           2              0             2
              t                            t

The above equations pointedly illustrate the problems with
attempting to deal with charged particles using traditional math. 
The above two integrals are both infinite!  Solution of Equation
(9) to get a zero electric field requires the subtraction of two
infinities.  To obtain practical solutions to Equation (11)
requires that we treat the electric densities in terms of
individual charges, and that the electric field, acting on each
charge, approaches zero.  Taking this point of view, individual
charges are separated by a minimal subatomic distance, m.  The
above equation becomes

   x0     rho(x0-t) dt       1-x0      rho(x0+t) dt
 Integral ____________   =  Integral  _____________           (11)
   m           2              m             2
              t                            t


For uniform density, the above equation cannot be met unless x0 =
1 - x0, or x0 = 0.5, at the midpoint on the wire.  For a point on
the left of the midpoint of the wire, there is more total charge on
the right side of the wire, so the electric field points to the
left.  The bottom line is that the charge distribution along a
fixed wire cannot be uniform.  

What is the charge distribution along a wire?  As x0 decreases, the
integration distance of the left integral decreases, so the
corresponding values of  must increase to compensate. Conversely,
the integral on the right, Ir must increase.  Not only does the
integration distance increase, but as x0 increases, the integral on
the right begins to incorporate some of those increased  values
which were created to compensate for increases in Ir for slightly
larger values of x0 and so forth.  Applying the mean value theorem
of integral calculus to the above equation, there exist some point
xL, 0  xL  x0.  

   x0     rho(x0-t) dt      
 Integral ____________   =  Ir                                (12)
   m           2            
              t             

Integrating,

  rho(XL) (-1/x0 + 1/m) = Ir                                  (13)

Collecting terms,

  rho(xL) = m x0 Ir / (x0 - m)

As x0 decreases toward m, the right side of the above equation
approaches infinity.  The mean value theorem of integral calculus
states that (xL) can be interpreted as an average value of the
density function rho between m and x0.  As x0 decrease to m, this
average density approaches infinity!  Similar arguments can be
presented for charges near the tip of the cone.  Unfortunately, the
above mathematical arguments do not provide quantitative analysis
for the solution of the charge densities, nor can qualitative
mathematical arguments be employed to arrive at a fixed hardware
design.

4.2  Finite element approximations.  My attempts at solving various
forms of Equation (11) over a two year time-span met with dismal
failure.  The reasons for this failure have some interesting
physical ramifications.  Suppose that one attempts some finite
approximation to Equation (11) with a number of charges at fixed
positions, then the failure to achieve zero in Equation (11) can be
interpreted as a force on the individual charge.  I attempted to
solve the equations through gradient search techniques, equivalent
to move each charge according to the applied force.  As one moves
toward the tip of the wire, the forces become infinite, and the set
of simultaneous equations becomes unstable and will not converge! 
The tracking of the individual charges is equivalent to solving for
the standing waves of the individual electrons on the wire. 
Standing waves in a conductive media are real physical phenomena;
unfortunately, the standing waves have little to do with the
electrostatic solution we seek.  

After many attempts at solving the electrostatic fields surrounding
a pointed cone, I was able to obtain a solution by modifying a
method proposed by a Dr. Magdy K. Iskander of the University of
Utah [Electromagnetic Fields, Prentice Hall].  Iskander's method
approximates the electric potential in Equation (2) by a finite
sum, by evaluated at a finite number of points, X1,...,Xn.


By definition, the voltage potential, V, is constant for any point
on the conductor.  For a fixed selection of n points,
(Y1),(Y2),...(Yn), the above formula will generate a linear
equation for
any point Xi within the conductive object.  The resulting equations
are linear in the (Yj) and

          1      n     - rho(Yj) dSj  
  V  = ________ Sum   ________________                     (2)
                j=1
       4 pi e0             X - X 

can be solved by traditional linear equation methods.  The
resulting solution for a charged wire, published by Iskander,
conforms to my solution, Equation (14).  Unfortunately, the method
is highly sensitive to the selection of the X's and the Y's.  In
particular, the X's and the Y's cannot coincide, otherwise
numerical division errors occur.  In addition, finite
approximations are not exact, so the  voltages resulting from the
finite approximation are not uniform across the object.  As one
increases the number of elements, the equations become singular,
and cannot be solved. While able to reproduce Iskander's results,
I was not able to refine the results by taking smaller step sizes
for a charged wire.  I was also unable to apply the technique to
surfaces of revolution, such as spheres and cones.  

5.0  Solution Using Self-Organizing Systems.  Over the past four
years, my software group, the Neural Engineering Research and
Development Section (NERDS) have been solving difficult problems
using self-organizing systems.  One such method employs a fuzzy
differential genetic algorithm.  During a simulation, individuals
are created with genetic code which is represented as binary bits. 
The binary bits are interpreted as the distribution of charges
along the cone.  Using this charge distribution, the voltage is
then calculated at various points on the cone, and an individual's
fitness is evaluated based on how uniform the resulting voltage
potential is.  Using simulated natural selection individuals are
selected to breed (survival of the fittest), and using simulated
sexual reproduction succeeding generations of individuals are
generated.  Through the process of simulated natural selection, an
individual is found which solves the charge distribution along the
cone.  

The fuzzy genetic algorithm was able to obtain solutions to the
electrostatic charge for a variety of physical shapes.  Equilibrium
is achieved as the voltage approaches uniformity across the shape's
surface.  (See attached figures).  In roughly 15 minutes, the
program will solve for the charge distribution for an arbitrary
surface of revolution.  Validation of the software was achieve by
solving for the known charge distribution for a sphere. 

For objects containing tips, such as wires and cones, the electric
fields approached infinite as one moves toward the tips.  While
mathematical techniques may prove that the charge density
approaches infinity as one approaches the tip of the cone, the
mathematical techniques offer no clue toward finding an optimal
cone design.  The fuzzy genetic algorithm was employed to solve for
the charge distribution for a variety of cone angles, giving
effective delivered proton compression for each shape.  Simulations
for an optimized cone shape are presented in the next section.

6.0  Engineering Considerations for Building a Prototype.

Some metals, such as platinum, palladium, titanium, and nickel have
the ability to absorb large amounts of hydrogen and deuterium. 
After absorption, the hydrogen nucleus and electron separate.  The
lone proton (or paired proton/neutron) separate.  An ionized
hydrogen nucleus, having no shell, is free to wander throughout the
metal in the same manner as a free electron.  Titanium is not
recommended because of surface conditions involving titanium oxide. 
Metal oxides are generally poor conductors of electricity, and the
oxide will probably limit the ejection of protons from the cone's
surface.

An electrolysis process is involved (shades of Pons and Fleishmann)
to saturate the metal of the cone with hydrogen.  During this
process, a slight negative voltage should be applied to the
conductor to insure that the protons will not leave the cone.  Once
the metal is charged, a high positive voltage will be applied to
the cone.  It is recommended that the cone be mounted so that tip
of the cone penetrates the tub of the water chamber into a second
chamber, or be pointed upward, out of the water.  We also recommend
that the second chamber be evacuated.  This is done to insure that
the discharging of the cone be accomplished by ejection of protons,
and not by the ionization of molecules which happen to reside at
the tip of the cone.  Obtaining nuclear fusion requires more than
intense electric fields.  It also requires that two protons be
force toward each other with sufficient velocity to overcome the
repulsion of the two particles.  In the sun, it is achieved by
temperatures of millions of degrees.  It has also been achieved in
particle accelerators, and by imploding frozen hydrogen pellets.
   
Suppose that one has a cone charged up with deuterium ions. 
Applying a positive voltage pulse to the cone, deuterium ions will
be accelerated toward the tip.  When the particle densities exceed
critical values, deuterium ions will be ejected from the tip,
causing disequilibrium.  Protons inside the cone will be
accelerated toward the tip at enormous velocities.  As they
approach the tip, the magnitude of the acceleration will approach
infinity because the electric field at he tip are caused by the
focusing of the electric fields of charges away from the tip.  The
protons will therefore implode at the tip of the cone. 

Consider two deuterium ions, being ejected from the cone at the
same time from opposite directions.  If the cone is pointed up,
then the vertical velocities of each deuterium ion will be the
same, and it would appear, relative to each ion, that the other ion
is headed directly toward it.  It is a fairly straight forward
application of classical physics that if the accelerations are
strong enough, then sufficient velocities can be achieved to cause
nuclear fusion, similar to that which occurs in more traditional
particle accelerators.

It is here that we deviate from traditional attempts of cold
fusion.  To increase the acceleration of the charged particles
requires that we increase the magnitude of the applied voltage. 
The applied voltage can be scaled up considerably using
conventional voltage amplification devices or electronic pulsers. 
(This has not been attempted in traditional cold fusion
applications).  It is also important that material be evacuated
from the tip of the cone either by penetrating the side of the
water bath, or by pointing the cone upward out of the water bath. 
In particular, the ionization of the water will provide electrons,
which when attracted toward the cone, neutralizing the positive
charge of the cone, and mitigating the effect we are trying to
produce.  This has not been attempted in traditional cold fusion
applications.  In addition, we are reconfiguring the geometry by
using pointed shapes as opposed to smoothed shapes.  The angle of
the point must be optimized to produced the maximal result.

Although the electric fields become very large for a variety of
cone shapes, the delivery of an infinite electric field will be
mitigated by errors in cone geometry and by charge leakage at the
tip.  To compare the effectiveness of the various geometries, we
compute a delivered electric field as the electric field at a point
a slight distance from the tip, namely at a distance 1% of the
length of the cone.  Returning our attention to the two deuterium
ions being ejected from the cone at the same time, the
accelerations on the two ions have two componentsone component
accelerates them in the same direction, the other accelerates them
toward each other.  The effective implosive force per charge is the
effective acceleration per charge toward each other is the
delivered electric field times the cosine of the cone angle.  A
plot of effective implosive force for a variety of cone shapes is
presented in the figure.  The optimal half-cone angle is about 15
deg.

Compromises for practical device construction will involve rods
made of nickel or palladium which will have one end spherically
rounded and the other end to be ground down to a 15 deg point.  Our
results with wire simulations show that increasing the length of
the rod increases the electric field at the end.

At present we have many ideas for improvement of the basic design,
ideas for energy recovery, and so forth.  These ideas can wait
until we prove the basic feasibility.


Lloyd G. Allred
26 Oct 92






Lloyd G. Allred
OO-ALC/TISAA
Hill AFB, Utah 84056

Electrostatic Field Computations


Don Baker
Los Alamos National Laboratory

1.   Thank you for your review and comments regarding electrostatic
fields.  You raise some interesting points.  I do not claim to have
all of the answers.  What I am hoping to do is shed some light on
some obscure aspects of physics and suggest some experiments which
might have some unusual outcomes.  With this in mind, I will
attempt to address some of the points you have made, and suggest
some further investigations.

2.   Since the electrostatic fields are normal to the surface won't
the particles be emitted in the high field region before reaching
the point?  I am reasonably confident that particle emissions will
be a significant factor in these processes.  This is just as well,
as we are not trying to make a 20 megaton bomb.  The issue here is
will the density at the tip be sufficient to induce nuclear fusion? 
I think the hypothetical equilibrium solution for a static
conditions sheds some light on this subject.  The solution predicts
infinite density for static conditions.  You and I both know that
infinite density cannot be achievedother processes, such as
emission, etc., will intervene to mitigate these factors.  The
question I raise is as follows:  during the attempt of the system
to achieve equilibrium, will the induced fusion be sufficient to
achieve a positive energy return?  

3.   ... fusion cross-sections have not been measured below 5
Kev...Since the particles get their energy from an electrostatic
field, they must fall through potentials equal to the kinetic
energies required.  How do you explain the positive results in
previous cold fusion results using low voltages?  What you analysis
does suggest, however, is that we include in our design the ability
to crank up to 20 kilovolts.  I should point out that when
inductances are considered, that net particle accelerations can
exceed the applied voltages.  A similar effect can be observed with
the earth's tides which raise the sea level some six feet.  There
are certain bays and fjords which focus the tidal energy, producing
tides in excess of 20 feet.  This effect is observed
electronically, when a voltage pulse is sent down a wire.  When the
pulse reaches the end of the wire, it is reflected, thus doubling
the voltages.  When one considers a collection of positive ions
converging on the tip, and the associated reflections, the
resulting energies will exceed the amount computed by falling to
the potential of the nominal applied voltage.

4.   ...one must overcome ... the Coulomb barrier ... before ...
tunneling becomes appreciable.  My experience with electronic
response of zener diodes has taught me that tunneling is probably
the controlling factor in most barrier phenomena.  I do not see why
Coulomb barrier potential should be exceptional.  I suspect that
the traditional M.I.T. sledge-hammer approach to fusion may have 
overlooked the more subtle effects of tunnelingincluding the
initial energy of Helium 4 nucleas and the resulting decay.  This
could be a whole new ball game.

5.   Is it clear that the hydrogen isotopes can be highly
accelerated through the lattice?  Won't they be impeded by
collisions?  In Rutherford's seminal work on particle physics, he
bombarded a thin metal foil with a stream of protons.  Proton
deflections were extremely rare events.  These experiments showed
that matter consists of mostly empty space.  To a positively
charged proton, the interior of the metal appears to be so much
empty space.  Won't the protons be impeded by collisions?  Oh God!
I hope so.  May be we'll get some fusion when they collide!  

6.   I would like to know more about the calculation method.  In
the classical electrostatic case, the problem is represented as a
problem in genetics.  Individuals are created at random; their
genetic material is interpreted as a potential solution to the
electrostatic problem.  Through a process of simulated natural 
selection and simulated reproduction, succeeding generations
ofindividuals are created which better solve the problem until a
solution is reached.  We have developed a tool which we call a
differential fuzzy genetic algorithm which an optimize general
problems of this kind.  In many respects, the details of the
differential fuzzy genetic algorithm are transparent to the user. 
The user need only write a procedure which evaluates how well a
hypothetical problem solutions works.

In the electrostatic problem, finite charges are distributed around
the surface of the electrostatic device, and the electronic
potential is calculated at points centered between the charges. 
The process is repeated until the variance of the electronic
potential is minimized, creating electrostatic equilibrium.  Our
software program can solve the charge distribution problem for a
general surface of revolution such as a wire, a cone, or a sphere. 
The program produces uniform distribution for a sphere, and the
solution for the wire agrees with the published solutions.  Once
the placement of charges has been determined, one then calculates
the electric field at the points centered between the charges.  On
obtain a set of points, x1,...,xn, with corresponding electric
fields E1,...,En.  This points can be smoothed to obtain a curve,
y = f(x).  To obtain a more accurate representation of the curve,
one places more points in the simulation.  I am not intending to
patronizeI only want this point to be made perfectly clear.  This
fill-in-the-points process is employed in all computer-based
representations of continuous processes.  As the number of points
approaches infinity, one approaches the continuous curve.  In
application to a 15 degree cone, the curve is y = k / x0.6.  This
curve approaches infinity as one approaches the tip of the cone. 

7.   A real difficulty is that the jump from ideal field 
calculations treating the conducting particles as a continuous
fluid in a perfect conductor to a realistic quantum-mechanical
many-body problem calculation is a long way off.  No kidding.  To
my understanding, the many body problem has not been generally
solved for simple moleculesthe solutions which do exist (such as
diatomic hydrogen) can take weeks to run on a Cray using
traditional techniques.  It might interest you to know that the
technique which we used to solve the electrostatic problem may
prove equally applicable to the quantum mechanical problem.  In
particular, both problems involve solving the probability
distributions of particle placement.  Using a similar method, one
can compare various hypothetical distributions, and through a
process of natural selection, find the one which best describes the
problem.  My experience with Schroedinger's equation shows that
excellent solutions can be obtained by fitting the solution with
piece-wise linear solutions to the wave equations.  A similar 
approach could be obtained in application to the multi-body
problem, or to the distribution of charges along a cone.  The
Schroedinger equation is usually solved for application where the
electric potential is known (such as a barrier well); however, one
could extend the method to simultaneously solve for the electric
potential for a hypothetical placement of particles in a multi-body
problem.  While I do extra curricular work in support of students,
various universities, and various research projects, I also have a
large family, and I cannot afford to undertake such a task on my
spare time.  In my career plans, I never anticipated researching in
fundamental particle physics.  For a nominal fee, we could be
persuaded.  Send money.  Our organization is industrially funded,
which means that we have to find a customer to pay for our time. 
My going rate is about $200,000 per year.  This includes support
staff and equipment.  These algorithms we are investigating are
directly amenable to parallel processing; routinely solving these
kind of problems warrants the purchase of a massively parallel
machine. 

8.   One question, not using your accelerator ideas.  How high a
density can one pack the fusion particles in the pointed region in
the static case?  Good question.  As I understand the process of
hydrogen adsorption, hydrogen can be regularly distributed
throughout the metal lattice.  I have read descriptions of enormous
adsorption capabilities of platinum, but do not have the sources at
my disposal.  I would be interested to see the actual figures. 



DR LLOYD G ALLRED, Technical Lead
Neural Engineering Research and Development Support (NERDS) Team
Aircraft Software Development Branch
Software Engineering Division 
Newsgroups: sci.physics.fusion