Path: santra!tut!draken!kth!mcvax!hp4nl!htsa!fransvo
From: fransvo@htsa.uucp (Frans van Otten)
Newsgroups: alt.fusion
Subject: Limits on Cold Fusion paper
Message-ID: <848@htsa.uucp>
Date: 21 Apr 89 14:21:33 GMT
Organization: AHA-TMF (Technical Institute), Amsterdam, The Netherlands
Lines: 461

Yesterday the FOM (Dutch research institute) gave a talk about
hot fusion in tokamaks.  Cold fusion was mentioned, but the tone
was very sceptical.  In their cold fusion experiment, all they
have observed are bubbles (oxygen and D2).

Anyway, I got a copy of the paper "Limits on Cold Fusion in
Condensed Matter: A Parametric Study".  As I haven't seen this
one in ascii yet, I decided to type it out.  The paper I got
is a copy of a fax of a fax of a ... so there might be errors.
Also, as ascii doesn't support Greek characters, I wrote them
(where used) fully out (like sigma, beta etc).



      Limits on Cold Fusion in Condensed Matter:

	          a Parametric Study


	  J. Rafelski, M. Gajda and D. Harley

	        Department of Physics

     University of Arizona, Tucson, Arizona 85721


		         and


		    S. E. Jones

		Department of Physics

	       Brigham Young University

      Brigham Young University, Provo, Utah 84602


		    March 27, 1989



The rate of nuclear fusion of d-d hydrogen isotopes is
established quantitatively as function of relative energy in
the range of 500 eV; as function of maximum allowed hydrogen
separation; as function of the effective mass of the electron
and as function of the effective charge of the electron. It
is shown that a neutron rate of 10E-33 /s/atom can arise from
a combination of these effects within a perhaps plausible
range of parameters.



1.  Introduction

In view of the recent interest generated within the scientific
community regarding the possibility of cold fusion of hydrogen
nuclei within metallic hydrides [1], we wish to present along
with our experimental results, a few theoretical observations
arising in part from our work in muon catalyzed fusion.  It is
well known that the fusion of hydrogen isotopes can be made to
occur within 10E-8 to 10E-11 s at room temperature, in the
presence of a muon [2].  Together with two hydrogen nuclei, the
muon forms a muo-molecule of about 250 fm in size, squeezing
the hydrogen nuclei together and allowing them to penetrate the
Coulomb barrier that usually strongly inhibits their fusion.
In an ordinary hydrogen molecule, which is about 0.74 angstrom
in size, the fusion rate resulting from the electron binding is
expected to be in the order of 10E-74 /s [3], or about 85 orders
of magnitude less than in the case of the muomolecule.  This
extraordinary variation in magnitude, resulting from a decrease
in the molecular size by a factor of 2000, leads one to suspect
that even a small pertuberation of the hydrogen molecule's
wavefunction could result in a dramatic change in the spontaneous
fusion rate.  This was the motivation leading to the first
theoretical study undertaken [3] and to the experimental work [1]
to which this note directly refers.

We report here on the detailed but schematic results arising from
variation of a few of the physical conditions arising in solid
state, hydride environment, that could lead to an observable
fusion rate of about 10E-23 /s.  Our results are consistent with
the experimental finding [1] of cold fusion in metal hydrides.
In the next section, we examine a nuclear reaction picture of
fusion resulting from a current of deuterons passing through a
hydride and establish the values of the nuclear reaction constant.
In section three, we examine the consequences of imposing boundary
conditions on the hydrogen molecule's nuclear wave function,
hoping to simulate in this way the confinement of nuclei within a
lattice cell.  In section four, we examine in turn the effect of
introducing a "pseudo-mass", and "pseudo-charge", that would form
a more tightly bound hydrogen molecule.  We should mention here
that none of this discussion in intended to form the basis of a
realistic model for cold fusion; instead, we intend to provide an
order of magnitude lesson for ourselves and to demonstrate the
surprising sensitivity of the fusion rate to external conditions.
In the theoretical methods employed we follow closely the
established techniques developed for muon catalyzed fusion research.



2.  A reaction description of dd fusion in a hydride

The first analysis one might perform of fusion at low relative
deuteron energy, is to simply compute the event rate resulting
from a current of deuterons I, passing through a hydride
containing a density of deuterons ro.  The event rate resulting
from such a system would the be:

        dN/dT = I/I0 * ro*v*sigma       (1)

Here, I0 is the current of a single particle, v is the velocity
of the incident current, and sigma id the energy dependant cross
section for dd-fusion.  s can be easily computed from the S
function, which is a slowly varying function of energy and is
related to the fusion cross section by:

               2S(E)  -beta/sqrt(E)
	s(E) = ----- e                (2)
		 E

For energies less than a few keV, the fusion neutron S-function
is expected to be about 53 keV barn, and beta = 44.4021 sqrt(keV)
[4].  The resulting event rate for an incident current of 1 ampere
impinging on a hydride of hydrogen density 4*10E22 atoms/cm3 is
given in figure 1, as a function of energy.  One finds that it is
possible to obtain an event rate of one fusion per second, if the
incident current of deuterons has an energy of about 380 eV.  It
is unlikely that any mechanism for the generation of such energies
can be found in a cold solid exposed to potentials of O(10 V).
We can therefore conclude that any standard d-d collision
description of an observable nuclear fusion process is unlikely
to succeed.

At this point let us introduce the reaction constant K0 related
to the S function by:

		       2S
	K0 = lim   ----------     (3)
	     E->0  pi*alfa*mu

where alfa is the file structure constant (=1/137), mu is the
nuclear reduced mass.  K0 is defined in such a way, that the
fusion rate in a static environment where the relative wave
function of two deuterons is PSI, is given by:

                           2
	labda = K0 |PSI(0)|     (4)

K0 is the fusion constant describing the fusion from the l = 0
partial wave as the relative nuclear velocity goes to zero.
For the d(d,n)3He reaction, of particular interest to us here,
                                              2
K0 is found to be 1.48*10E-16 cm3/s.  |PSI(0)|  is the probability
amplitude that the two hydrogen nuclei come close together; in
the following calculations we shall use a nuclear interaction
range of 3 fm for the d-d nuclear fusion.



3.  The effect of boundary conditions on the (dde)+ ion

In this section, we confine two hydrogen nuclei to be a sphere,
on which we impose the condition that the nuclear wavefunction
be zero on the boundary.  As the sphere is decreased in size,
the nuclei are squeezed together and the ground state energy is
raised, as the amplitude of the wave function increases at the
origin.  In our calculation, we included the electron binding by
obtaining the adiabatic effective ground state energy of a single
electron as a function of the nuclear separation, and including
it as an addition to the Coulomb repulsion in the calculation of
the relative nuclear wavefunction.  The Schrodinger equation for
the nuclear wave function was solved numerically subject to the
boundary conditions that the wavefunction be zero at some finite
radius Rmax, and be regular near the origin.  The result,
presented as the fusion rate against the boundary size Rmax, is
presented in fig. 2.  The boundary size is given in units of
electronic Bohr radius a0, which is equal to 0.529 angstrom.
This can be compared to the equilibrium nuclear separation in a
D2 molecule, which is 0.74 angstrom, and the lattice spacing in
metallic hydrides, which is about 2.5 angstrom.  Through this
presentation we do not wish to imply that such "confined" hydrogen
structures are actually present as static objects in hydrides,
indeed our calculations show that such systems are typically
unbound, thus allowing the hydrogens to move freely.  When the
d-d system is confined to regions of about 0.3 a0, we find that
the fusion rate is of the order of 10E-24 /s.  Although this is
somewhat smaller than the lattice cell size, one should note that
Coulomb forces between the deuterons and ionic centers in the
metallic lattice could lead to such a configuration in a
_dynamical_ process of hydrogen flow, leading to a change in the
fusion rate by fifty orders of magnitude.

The confinement of the d-d motion is of course only achieved at
the cost of raising the system energy.  The total energy cost of
achieving a fusion rate of 10E-24 by confinement turns out to be
about 70 eV in our schematic approach, consisting of relative
nuclear energy of about 150 eV, less the electron binding.  The
relative nuclear energy is considered less than the estimate of
about 400 eV in the previous section, _and_some_of_this_energy_
can_be_provided_by_the_electron_binding_to_the_confined_d-d_system.
The exact amount would depend on the details of the hydride
environment, a point we have not considered in this first
calculation.



4.  The effect of the electron mass and charge on the fusion rate

It is interesting to speculate what the effect of a larger
electronic mass and charge would be on the fusion rate in a
true (dd-mu)+ molecule.  It is known, from theory and
experiment, that the fusion rate in a (dde)+ ion is about
10E11 /s, an increase of 85 orders of magnitude as compared
to the fusion rate in the (dde)+ ion [3].  What is perhaps
not so widely realized is that the last 33 orders of magnitude
in the fusion rate are achieved at a cost of 103 MeV in the
binding particle's mass; simply increasing the mass of the
electron from 0.5 to 3.0 MeV results in an increase in the
fusion rate by 52 orders of magnitude !  This is illustrated
in fig. 3, which gives the fusion rate in a (dd-emeff)+ ion,
"emeff" being at present a particle of charge -e and mass meff.
As one can see, a fusion rate of 10E-23 /s is achieved for an
effective mass of about 2.6 MeV, that is 5 times the regular
electron mass.

This result can be understood theoretically quite easily: the
penetration probability (the Gamov factor) below the Coulombic
barrier scales as sqrt(mu/m).  Thus a change by factor 5 in the
electron mass changes the suppressing exponent by more than a
factor 2, resulting in the fusion rate quoted.  Changing the
mass by a factor 200 then makes the penetration integral much
smaller, but other factors of the exact wavefunction lead to
the fusion rates found in the field of muon catalyzed fusion,
which are of the magnitude 10E11 /s.  We have verified our
calculations, that when the mass of the "effective" electron
reaches that of the muon, we recover the expected result.  At
this point it should be noted that a similar gain in the fusion
rate as due to the electron mass can be accomplished by reducing
the relative nuclear reduced mass mu, e.g. by using the proton
as one of the nuclear fusion particles.

A second quantum mechanics exercise in playing with the
fundamental constants is to vary the electron's charge.  This
was achieved by computing the effective potential between two
d nuclei resulting from the presence of the electron, and then
multiplying this potential by a factor eeff/e.  The resulting
fusion rate in a (dd-eeeff) ion is presented in fig. 4, as a
function of the effective charge, where the mass is that of the
electron's.  The effective charge is varied between e and 2e,
resulting in a variation in the fusion rate by 22 orders of
magnitude.  It is worth noting that an effective charge of
eeff = 1.06 used in the (dde)+ ion reproduces the D2 molecules
binding energy of -4.6 eV, with a corresponding fusion rate of
2*10E-73 /s.  We therefore do not expect the addition of a
second electron to a (dde)+ ion to significantly influence the
fusion rate.  This conclusion has also been reached in ref. [3].

The effect of the increased effective electron mass and charge
is to bring the (dd-eeff) system closer together; again we see
the high sensitivity of the response to the precise nature of
the physical parameters governing the inter-nuclear properties.



5.  The combined effects of confinement, mass and charge

The final question we wish to address is what the combination
of all of the above effects has on the fusion rate.  To
illustrate that the above effects are _cumulative_, we have
selected particular combinations of Rmax, meff and eeff and
computed the resulting fusion rates.  These we then compare to
the individual contributions, which are collected in table 1.
We fist note that fpr Rmax = 6 a0 we recover the spontaneous
fusion rate of the regular molecular ion, with meff/me = 1,
and eeff/e = 1.  For the set of parameters Rmax = 0.5 a0,
meff = 2 me, and eeff = 1.5 e, we find the combined effect
is to increase the fusion rate by no less than 16 orders of
magnitude over and above any of the _individual_ contributions,
as shown in table 1.


	meff/me   efff/e   Rmax (a0)   labda fus /s

	   1        1.0       6.0        8.3*10E-77
	   1        1.5       6.0        2.3*10E-61
	   2        1.0       6.0        1.1*10E-47
	   1        1.0       0.5        5.7*10E-37
	   1        1.5       0.5        1.4*10E-36
	   2        1.5       1.0        2.8*10E-33
	   2        1.0       0.5        7.1*10E-21
	   2        1.5       0.5        1.4*10E-20
	   2        1.2       0.4        2.4*10E-16
	   2        1.5       0.3        1.6*10E-12


		        Table 1:

        The fusion rate resulting from the simultaneous
	variation of boundary conditions, effective
	electronic mass and charge, as compared to the
	fusion rate resulting from the individual
	variations of these parameters.

              [ Note: the exponents are hard
                to read, some might be wrong. ]



6.  Conclusion

While it is difficult to imagine how a collective effect in a
metallic hydride could confine a d-d system to 0.5 angstrom,
or/and increase the electron's effective mass to 1 MeV or
effective charge to 1.5 it's [sic] usual value, the foregoing
experimental result [1] and the encouraging order of magnitude
considerations presented in this work suggest that one should
investigate with great care the various systems of interest.
Our numerical exercises have highlighted the extraordinary
sensitivity of the (dde)+ fusion rate to pertuberations in the
constants and the enviroment that determine the fusion rate.
We wish to emphasize that the actual physical explanation of
the experimental results of ref. [1] must account for the
dynamical effect associated with the infusion of hydrogen
isotopes into metal lattices, a point that has been nearly
totally ignored in this work.  Furthermore, our free use of
the mass and charge of the electron as parameters in our
calculations should not be construed to imply that we know of
any means by which such changes can be arrived at in our
context.

Acknowledgements:  We thank M. Danos, B. Muller and H. Rafelski
for stimulating and interesting comments.  This work was
supported by the Advanced Energy Projects Division of the US
Department of Energy.



Figures (approximate).


                    |  Event rate for 1 ampere
  dN           10E9 +
  -- (/s)           |                           *
  dt                |                       *
	       10E6 +                   *
                    |                *
                    |             *
	       10E3 +           *
                    |          *
                    |        *
	       10E0 +       *
                    |      *
                    |    *
	      10E-3 +   *
                    |  *
                    +---+---+---+---+---+---+---+---
			   0.5                 1.0

					E (KeV)

		Figure 1:  The neutron fusion event rate dN/dt
		for 1 ampere of deuterons impinging on a deuteron
		sample at density 4*10E22 /cc at energy E.



                       |
log (labda fus)        | *     Fusion rate vs. box size
(labda fus in /s)  -20 +  *
                       |   *
                       |    *
                   -30 +     *
		       |       *
                       |         *
                   -40 +            *
                       |               *
                       |                   *
                   -50 +                         *
                       |
                        -+--+--+--+--+--+--+--+--+-
		        0.2   0.4   0.6   0.8   1.0

				       Rmax (a0)

		Figure 2: Neutron fusion rate as function
		of maximum allowable separation Rmax (a0)
		between two deuterons.



                       |
log (labda fus)        |  Fusion rate vs. effective mass
(labda fus in /s)  -20 +                                *
                       |                           *
                       |                      *
                       |                  *
                   -40 +               *
                       |            *
                       |         *
                       |       *
                   -60 +     *
                       |   *
                       | *
                       |*
                   -80 +--+----+---+--+--+--+-+-+-+-+++++
				   1            2      3

                                               meff (MeV)

		Figure 3: Neutron fusion rate as function of
		the electron effective mass in the (dde)+
		structure.



                   -50 +  Fusion rate vs. effective charge
log (labda fus)        |                                *
(labda fus in /s)      |                            *
                       |                        *
                       |                    *
                   -60 +                *
                       |             *
                       |          *
                       |        *
                       |       *
                   -70 +     *
                       |    *
                       |   *
                       |  *
			--+--+--+--+--+--+--+--+--+--+--+-
			 1.0            1.5            2.0

                                            theta eff / e

		Figure 4: Neutron fusion rate as function of the
		electron effective charge in the (dde)+ structure.



References.

[1] Jones, S.E., Palmer, E.P., Czirr, J.B., Decker, D.L.,
    Jensen, G.L., Thorne, J.M., Taylor, S.F. and Rafelski, J.
    "Observation of Cold Nuclear Fusion in Condensed Matter",
    submitted to Nature.

[2] Jones, S.E. Nature 321, 127-133 (1986); Rafelski, J. and
    Jones, S.E., Scientific American 257, 84-89 (July 1987).

[3] Van Siclen, C.D. and Jones, S.E., Journal of Physics G,
    Nucl. Phys. 12, 213-221 (1986).

[4] Jarmie, N. and Brown, R.E., Nuc. Inst. Meth. b10/11, 405-
    410 (1985), and private communication.

-- 
	Frans van Otten
	Algemene Hogeschool Amsterdam
	Technische en Maritieme Faculteit
	fransvo@htsa.uucp