Here is a preprint of a paper on cold nuclear fusion in TeX. Please feel
free to distribute it to anyone who might be interested.
Charles J Horowitz Bitnet: Charlie@IUCF
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\centerline{\bf Cold Nuclear Fusion in Metallic Hydrogen}
\centerline{\bf and Normal Metals}
\bigskip
\centerline{Charles J. Horowitz$^*$}
\medskip
\centerline{Physics Department and Nuclear Theory Center}
\centerline{Indiana University}
\centerline{Bloomington, IN 47405}
\medskip
\centerline{Submitted to Physical Review C}
\bigskip
\centerline{\bf ABSTRACT}
\medskip
{\narrower\smallskip The rate of nuclear fusion from tunnelling
in very dense metallic hydrogen in the core of Jupiter is
calculated to be ten to the minus fifty four ($10^{-54}$) per
hydrogen-deuterium pair per second. It is estimated that the
width of the fusion barrier for deuterium in Palladium or a similar
metal must be reduced to, of order, 0.125 Angstroms for the fusion
rate to be ten to the minus twenty three ($10^{-23}$) per deuterium
per second. If this scale is achieved, the ratios of various
nuclear reaction rates will be very different for cold versus
thermonuclear fusion. \smallskip}
\bigskip
\bigskip
There is great interest in cold nuclear fusion induced by quantum
mechanical tunnelling of the zero point motion of Deuterium in a
solid. Recently, two groups have claimed to observe neutrons [1]
and possibly heat [2] from such reactions. Furthermore, there
could be geophysical and astrophysical implications of cold fusion.
For example, cold fusion in the dense metallic hydrogen core of
Jupiter could contribute to the planets heating [1].
We first consider fusion in metallic hydrogen because of its very
simple structure. In addition to direct applications to Jupiter,
our results may provide insight for fusion in more complicated
materials. Next, we estimate the change in scale of the fusion
barrier required in a normal metal such as Palladium to produce
the claimed fusion rate of ten to the minus twenty three ($10^{-
23}$) fusions per deuterium per second [1].
Cold fusion, if it can be achieved, will involve a much larger
tunneling exponential then in conventional thermonuclear fusion.
This will strongly favor reactions with light reduced masses such
as deuterium plus proton going to ${}^3$He plus $\gamma$. We will
discuss the very different ratios of reaction rates expected for
cold versus thermonuclear fusion.
In the limit of very high density, the electrons in metallic
hydrogen should become a Fermi gas. Therefore, we model metallic
hydrogen as a Fermi gas of electrons and a crystal of nuclei
interacting via screened coulomb potentials. [Our results are not
expected to be qualitatively changed if the protons are in a liquid
phase.] The effective potential between two nuclei $V(r)$\ which
includes the effects of electron screening is given, in a simple
Thomas Fermi model [3], by
$$V(r) ={e^2\over r} {\rm exp}[-{r\over \lambda(n)}].\eqno(1)$$
The density dependent Thomas Fermi screening length $\lambda$\ is,
$$\lambda(n)=\bigl({\pi a_0\over 4 k_F}\bigr)^{1/2}\, =
\Bigl[\bigl({3\over \pi}\bigr)^{1/3} 4e^2 m_e\Bigr]^{-1/2} n^{-
1/6}.\eqno(2)$$
Here $a_0$\ is the Bohr radius, $m_e$\ the electron mass and $k_F$\
is the electron Fermi wave-number which is related to the electron
density $n=k_F^3/(3\pi^2)$. We note that $\lambda$\ decreases only
as the one sixth power of the density. More sophisticated electron
screening calculations may modify eq (1) somewhat at large
distances. However, eqs (1-2) are expected to be qualitatively
correct at short distances and it is the short distance behavior
of V(r) that will be important for fusion rate calculations.
As an example, we consider a density of $n=3.15\ {\rm\AA}^{-3}$\
which corresponds to a density parameter $r_s$\ ($n^{-1}={4\over
3}\pi a_0^3 r_s^3$) of 0.8. At this density, the pressure is
estimated to be 73 Mbar [4]. This compares to the roughly 60 Mbar
pressure expected at the center of Jupiter [5]. The screening
length is
$$\lambda = 0.30 {\rm \AA},\eqno(3)$$
which is shorter then the inter-particle spacing of $\approx$0.68
\AA. Therefore electron screening reduces the width of the Coulomb
barrier substantially and this should increase the
fusion rate.
We will need the vibrational frequency $\nu$\ of the crystal's zero
point motion. This is easily estimated from the classical energy
of a crystal lattice using the two body interaction in eq (1).
This gives,
$$h\nu \approx 1 eV,\eqno(4)$$
which agrees well with the frequency estimated from the Lindemann
ratio (of the amplitude of zero point motion to the inter-particle
spacing) calculated in ref [4]. The frequency is relatively low
because the nuclei are weakly interacting given that the screening
length is smaller then the average separation.
It is now a simple matter to make a WKB estimate of the fusion
rate. The ratio of the square of the wave function $\psi^*\psi$\
at some small distance $r_n\approx 5$\ Fm compared to $\psi^*\psi$\
at the classical turning point $r_0\approx 0.68$\ \AA\ is,
$$P=\bigl|{k(r_n)\over k(r_e)}\bigr|{\rm exp}[-
\alpha(r_n,r_0)].\eqno(5)$$
Here the local wave vector is $k(r)=[2M(V_{eff}(r)-E)]^{1/2}$, M
is the reduced mass of the two nuclei and the tunneling exponential
is
$$\alpha(r_n,r_0)=2\int_{r_n}^{r_0} dr' [2M(V_{eff}(r)-
E)]^{1/2}.\eqno(6)$$
In eq (5) we have approximated the WKB connection of the wave
function across the classical turning point by simply evaluating
$k(r_e)$\ at the equilibrium distance $r_e$. Thus $k(r_e)$\ is the
wave number of the zero point oscillation
($\hbar^2k(r_e)^2/2m_e=h\nu$).
The fusion rate R is calculated by multiplying P in eq (5) by the
frequency of attacks on the Coulomb barrier (which is just the
vibrational frequency $\nu$) and the probability of a nuclear
reaction $P_n$\ (once the nuclei have made it to $r_n$). The
probability $P_n$\ is about 0.1 to 1 for a strong interaction
process such as $D+D\rightarrow {\rm {}^3He} + n$. However, $P_n$\
is about ten to the minus six ($10^{-6}$) for the electromagnetic
reaction $D+p\rightarrow {\rm {}^3He} + \gamma$. (See the
discussion of S factors below). $$R=\nu P_n \bigl|{k(r_n)\over
k(r_e)}\bigr|{\rm exp}(-
\alpha)\eqno(7)$$
This equation will serve as our ``generic" estimate of a fusion
rate in the remainder of this paper.
The Born Oppenheimer potential energy surface $V_{eff}$\ includes
the interaction of the nuclei with all of their neighbors.
However, because the screening length is short, the total potential
energy surface for two nuclei as the move together in the crystal
is essentially just the interaction, eq (1), between the fusing
nuclei, $V_{eff}(r)\approx V(r)$. It is now a simple matter to
evaluate eqs (1, 6 and 7). For the p +
D$\rightarrow {}^3{\rm He} + \gamma$\ reaction using $P_n=10^{-6}$\
(ten to the minus six) we get,
$$R_{pD}\approx 10^{-54}\, {\rm (ten\ to\ the\ minus\ fifty\
four)\, sec^{-1}},\eqno(8)$$
per H, deuterium pair. {\it This fusion rate is much too small
to contribute to the planets heating}.
The rate for the D+D$\rightarrow {\rm {}^3He} + n$\ reaction (using
$P_n=1$) is,
$$R_{DD}\approx 10^{-63}\, ({\rm ten\ to\ the\ minus\ sixty\
three)\, sec^{-1} },\eqno(9)$$
even smaller because of the larger reduced mass in eq (6).
Increasing the density beyond n=3.15 \AA$^{-3}$\ will increase the
rate. However, we note the small 1/6 exponent in eq (2). For
example at n=25 \AA$^{-3}$\ we estimate $R_{pD}\approx 10^{-35}$\
and
$R_{DD}\approx 10^{-39}$\ per second. At this density the pressure
is about ten to the third ($10^3$) Mbar.
We have used a simple Fermi gas model for metallic hydrogen.
However, in the limit of very high density this model is expected
to become increasingly valid. We conclude that fusion is unlikely
to be important in Jupiter. In a latter paper we will present
fusion rates at higher densities for other astrophysical objects
such as white dwarfs.
We turn now to fusion of deuterium dissolved in Palladium. The
interaction $V_{eff}$\ is assumed to be the Born Oppenheimer
potential energy function for this complicated system. However,
we will be interested in $V_{eff}$\ primarily at short distances
where it can be approximated, $$V_{eff}(r) \approx {e^2\over r} -
Const.\eqno(10)$$
For example in $H_2$\ at small distances, the electronic energy is
very close to that of
an isolated He atom (E=-79.0eV). Hence, the constant would be just
the difference between this and the 27.2 eV binding energy of two
H atoms, (Const.=-51.8 eV). Replacing eq (10) with the full
potential energy surface will not change our results very much at
short distances.
Using eq. (10) the integral in eq (6) can be easily evaluated.
This gives for the fusion rate,
$$R\approx \nu P_n\bigl|{k(r_n)\over k(r_e)}\bigr| {\rm
exp}\bigl\{-\pi\bigl[{2M\over m_e}{r_0\over
a_0}\bigr]^{1/2}\bigr\}.\eqno(11)$$
Here, $r_0$\ is the width of the fusion barrier to the classical
turning point. The vibrational frequency is estimated to be about
that of an isolated $H_2$\ molecule, $h\nu\approx 0.5$\ eV. Rates
from eq (11)
are collected in table I for different isotopes. The tunneling
greatly prefers a smaller reduced mass. Indeed the weak
interaction process
$p+p\rightarrow D+e^++\nu$\ has a larger rate (at large $r_0$) then
$D+D$\ fusion despite its very small reaction probability
$P_n\approx 10^{-23}$\ (ten to the minus twenty three).
If hydrogen is dissolved in a normal metal such as Palladium the
width of the fusion barrier $r_0$\ can be reduced both by forcing
the equilibrium position of the atoms closer together and through
electron screening of the repulsive coulomb interaction. Note, the
fusion rate is very sensitive to $r_0$. Using eq (11) we estimate
that the scale of the fusion barrier must change by about a factor
of five (compared to $r_0\approx$0.7 \AA\ in $H_2$) until,
$$r_0\approx 0.125{\rm \AA,}\eqno(12)$$
in order for the fusion rate to be near the claimed ten to the
minus twenty three ($10^{-23}$) per second [1]. This factor of
five is much smaller then the factor of 200 in muonic fusion.
Nevertheless, it is a major change. It is not at all clear how to
obtain an electronic configuration with such a small length scale.
Thus the fusion observations are surprising and must be carefully
confirmed.
If the experiments are confirmed and this length scale can be
achieved (perhaps by a combination of effects including a large
effective electron mass [6]) then the
relative rates of various reactions will be quite different for
cold compared to thermonuclear fusion. This is because the much
larger tunneling exponential in cold fusion is extremely sensitive
to the reduced mass. In table II we evaluate the ratio of the
exponential factors, eq (5), for various reactions compared to the
D+D reaction. This enhancement factor depends only on the reduced
mass. [We assume the same fusion barrier for all reactions.] As
an example we consider cold fusion with $r_0=0.125$\ \AA\ and
thermonuclear (hot) fusion with $r_0=144$\ Fm (which corresponds
to an energy of 10 keV). The relative reaction rate (per isotope
pair) is the product of this enhancement factor and the ratio of
the cross section factors S for the basic nuclear reactions. The
cross section at an energy E, $\sigma (E)$, is commonly expressed
in terms of an S factor.
$$\sigma={S\over E} {\rm exp}(-e^2\pi(2M)^{1/2}/E^{1/2})\eqno(13)$$
We collect the experimental S factors in table II. For hot fusion
the larger S factor suggests that the $D+T$\ reaction will
dominate. However, for cold fusion the smaller reduced mass will
favor the $p + D$\ reaction even though this has a small S factor.
Clearly, more attention should be focused on the $p+D\rightarrow
{}^3$He + $\gamma$\ reaction.
It is interesting to note that the small mass favors the weak
$p+p$\ reaction by twelve orders of magnitude. If a cold fusion
``reactor" could be set up with a large fusion rate then even this
weak interaction might be observable at a rate only some ten to the
minus twelve of the $D+D$\ reaction.
It is important to emphasize that table II assumes the same
solubility, chemical environment, etc. for the different hydrogen
isotopes. However, these could be different. For example,
hydrogen is expected to have a larger zero point motion then
deuterium because of its smaller mass and Fermi statistics.
Can cold fusion be qualitatively different from hot fusion? One
experiment claimed to see heat corresponding to a fusion rate of
ten to the thirteen per second [2]. Furthermore, the observed
neutron flux was some nine orders of magnitude too small for the
heat to be from the $D+D\rightarrow {}^3{\rm He} + n$\ reaction.
The authors claim that the heat is from unknown
nuclear reactions which are very different from those in hot
fusion.
Clearly, one possibility is that the heat is from the $p+D$\
reaction which can not produce neutrons. Thus one should control
even trace amounts of hydrogen in the apparatus and perform
experiments with different H to D ratios. It is possible that the
$p+D$\ reaction rate is even higher then that estimated in table
II because of a previously unmeasured pair production branch,
$$p+D\rightarrow {}^3{\rm He} + e^+ + e^-.\eqno(14)$$
Pair production can compete with the ${}^3$He+$\gamma$\ reaction
because the gamma emission is highly suppressed.
The cross section for $D+D\rightarrow {}^4{\rm He} + \gamma$\ is
very small because of the $1^+$\ spin and parity of the deuteron.
Given the $0^+$\ ${}^4$He and two D in a relative s state one needs
an electric quadrupole ($2^+$) photon. Therefore, the rate is
lower then in an electric dipole ($1^-$) transition. Furthermore,
since the photon couples most strongly to the nuclear orbital
motions the reaction probably proceeds through a small d state
admixture in either a D or the ${}^4$He [7]. The pair production
reaction could proceed through a virtual $0^+$\ Coulomb monopole
photon and will not suffer either suppression.
The situation for the $p+D\rightarrow {}^3{\rm He} + \gamma$\
reaction is very similar because of the $1/2^+$\ spins and parities
of the p and ${}^3$He. Thus, pair production may be important for
this reaction also. Indeed muon capture (a similar process) is
significant in muon induced p+D fusion. About seven percent of the
time a 5.4 MeV muon is ejected [8].
In conclusion, we have examined cold fusion in metallic hydrogen
and other materials. Fusion is not expected to be important in
the metallic hydrogen core of Jupiter. The fusion rate is very
sensitive to the width of the fusion barrier. With a conventional
mechanism, there should not be a substantial fusion rate until the
barrier width has been reduced to about 0.125 \AA. Because cold
fusion involves a large tunneling exponential it is very sensitive
to the nuclear reduced mass. This will favor the p+D and hinder
the D+T reaction. Finally, we speculate that reactions involving
pair production could be important.
\bigskip
\bigskip
\centerline{\bf References}
\bigskip
\noindent
* Bitnet Charlie@IUCF
\bigskip
\noindent
Supported in part by Department of Energy Contract DF-FG02-
87ER40365.
\bigskip
\noindent
1. S. E. Jones et al, ``Observation of Cold Nuclear Fusion in
Condensed Matter", Brigham Young University Preprint.
\medskip
\noindent
2. Martin Fleischmann and Stanley Pons, ``Electrochemically Induced
Nuclear Fusion of Deuterium", Submitted to Journal of
Electroanalytical Chemistry.
\medskip
\noindent
3. A. L. Fetter and J. D. Walecka, ``Quantum Theory of Many-
Particle Systems", (McGraw Hill 1971 N.Y.)
\medskip
\noindent
4. K. K. Mon et al., Phys. Rev. {\bf B21}, 2641 (1980).
\medskip
\noindent
5. C. DeW Van Siclen and S. E. Jones, J.Phys. G {\bf 12}, 213
(1986).
\medskip
\noindent
6. J. Rafelski et al., ``Limits on Cold Fusion in Condensed Matter:
a Parametric Study", Univ. of Arizona preprint.
\medskip
\noindent
7. J. Piekarewicz and S. E. Koonin, Phys. Rev. {\bf C36}, 875
(1987).
\medskip
\noindent
8. L. Bracci and G. Fiorentini, Phys. Reports {\bf 86}, 169 (1982).
\vfill\eject
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\next% This \next must be outside of the local values, because now
% we want those troublesome macros in the \let's above to have
% their normal actions.
}% End of macro \makeTABLE.
%
\def\makePREAMBLE#1{% This macro generates the necessary preamble for a
% ruled table with #1 primary columns.
% (Primary columns means the number of columns NOT
% counting those used for vertical rules.)
\ncols=#1% Get the number of columns desired.
\begingroup% Start local parameter definitions.
\let\ARGS=0% This is the key to the whole thing; it prevents \ARGS
% from being expanded in the following \edef's.
\edef\xtp{\widevline\ARGS\tabskip\tabskipglue%
&\ctr{\ARGS}\tstrut}% A 1-column preamble. Gets the sizing right.
\advance\ncols by -1% One column has been generated; decrement the
% counter.
\loop% Append as many further columns as needed to the preamble.
\ifnum\ncols>0 %
\advance\ncols by -1%
\edef\xtp{\xtp&\vrule width\thinsize\ARGS&\ctr{\ARGS}}%
\repeat
\xdef\preamble{\xtp&\widevline\ARGS\tabskip0pt%
\crnorm}% Adds the last \vrule.
\endgroup% End of local parameters.
}% End of macro \makePREAMBLE.
%
\def\countROWS#1\into#2{% This counts the number of rows in #1 by
% looking for control sequences that end a row,
% e.g., \cr, \crthick, etc., and puts the result
% into count register #2.
\let\countREGISTER=#2%
\countREGISTER=0%
% \out{In countROWS: tokens are [\the#1]}%
\expandafter\ROWcount\the#1\endcount%
}%
%
\def\ROWcount{%
\afterassignment\subROWcount\let\next= %
}%
\def\subROWcount{%
% \out{In subROWcount: next is [\meaning\next]}% Debugging aid.
\ifx\next\endcount %
\let\next=\relax%
\else%
\ncase=0%
\ifx\next\cr %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\endrow %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\crthick %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\crnorule %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\crthickneg %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\crnoruleneg %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\crneg %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\header %
% \out{In subROWcount: next=header, ncase set=1}%
\ncase=1%
\fi%
% \out{In subROWcount: ncase is [\the\ncase]}%
\relax%
\ifcase\ncase %
\let\next\ROWcount%
% \out{subROWcount---> ncase=\the\ncase}%
\or %
\let\next\argROWskip%
% \out{subROWcount---> ncase=\the\ncase}%
\else %
\fi%
\fi%
% \out{subROWcount---> NEXT=\meaning\next}%
\next%
}% End of macro \subROWcount.
%
\def\counthdROWS#1\into#2{%
\dvr{10}%
\let\countREGISTER=#2%
\countREGISTER=0%
\dvr{11}%
% \out{In counthdROWS: tokens are [\the#1]}%
\dvr{13}%
\expandafter\hdROWcount\the#1\endcount%
\dvr{12}%
}%
%
\def\hdROWcount{%
\afterassignment\subhdROWcount\let\next= %
}%
\def\subhdROWcount{%
%\out{In subhdROWcount: next is [\meaning\next]}%
\ifx\next\endcount %
\let\next=\relax%
\else%
\ncase=0%
\ifx\next\cr %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\endrow %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\crthick %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\crnorule %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next\header %
%\out{In subhdROWcount: next=header, ncase set=1}%
\ncase=1%
\fi%
%\out{In subhdROWcount: ncase is [\the\ncase]}%
\relax%
\ifcase\ncase %
\let\next\hdROWcount%
%\out{subhdROWcount---> ncase=\the\ncase}%
\or%
\let\next\arghdROWskip%
%\out{subhdROWcount---> ncase=\the\ncase}%
\else %
\fi%
\fi%
%\out{subhdROWcount---> NEXT=\meaning\next}%
\next%
}%
%
{\catcode`\|=13\letbartab
\gdef\countCOLS#1\into#2{%
% \out{In countCOLS: tokens are [\the#1]}
\let\countREGISTER=#2%
\global\countREGISTER=0%
\global\multispancount=0%
\global\firstrowtrue
\expandafter\COLcount\the#1\endcount%
\global\advance\countREGISTER by 3%
\global\advance\countREGISTER by -\multispancount
% \out{countCOLS-->[\the\countREGISTER]}
}%
%
\gdef\COLcount{%
\afterassignment\subCOLcount\let\next= %
}%
{\catcode`\&=13%
\gdef\subCOLcount{%
%\out{In subCOLcount: next is [\meaning\next]}
\ifx\next\endcount %
\let\next=\relax%
\else%
\ncase=0%
\iffirstrow
\ifx\next& %
\global\advance\countREGISTER by 2%
\ncase=0%
\fi%
\ifx\next\span %
\global\advance\countREGISTER by 1%
\ncase=0%
\fi%
\ifx\next| %
\global\advance\countREGISTER by 2%
\ncase=0%
\fi
\ifx\next\|
\global\advance\countREGISTER by 2%
\ncase=0%
\fi
\ifx\next\multispan
\ncase=1%
\global\advance\multispancount by 1%
\fi
\ifx\next\header
\ncase=2%
\fi
\ifx\next\cr \global\firstrowfalse \fi
\ifx\next\endrow \global\firstrowfalse \fi
\ifx\next\crthick \global\firstrowfalse \fi
\ifx\next\crnorule \global\firstrowfalse \fi
\ifx\next\crnoruleneg \global\firstrowfalse \fi
\ifx\next\crthickneg \global\firstrowfalse \fi
\ifx\next\crneg \global\firstrowfalse \fi
\fi% End of \iffirstrow.
\relax%\out{subCOL--> ncase=[\the\ncase]}
% \out{subCOL--> next=\meaning\next}
\ifcase\ncase %
\let\next\COLcount%
\or %
\let\next\spancount%
\or %
\let\next\argCOLskip%
\else %
\fi %
\fi%
% \out{subCOL--> countREGISTER=[\the\countREGISTER]}
\next%
}%
\gdef\argROWskip#1{%
% Deletes the next balanced, undelimited argument from a
% token list.
% \out{---> Entering argROWskip <---}
% \out{In argROWskip: deleted arg is [#1]}%
\let\next\ROWcount \next%
}% End of macro \argskip.
\gdef\arghdROWskip#1{%
% Deletes the next balanced, undelimited argument from a
% token list.
% \out{---> Entering arghdROWskip <---}
% \out{In arghdROWskip: deleted arg is [#1]}%
\let\next\ROWcount \next%
}% End of macro \arghdROWskip.
\gdef\argCOLskip#1{%
% Deletes the next balanced, undelimited argument from a
% token list.
% \out{---> Entering argCOLskip <---}
% \out{In argCOLskip: deleted arg is [#1]}%
\let\next\COLcount \next%
}% End of macro \argskip.
}% End of active &'s.
}% End of active |'s.
\def\spancount#1{%\out{spancount--->\meaning#1}
\nspan=#1\multiply\nspan by 2\advance\nspan by -1%
\global\advance \countREGISTER by \nspan
% \out{number spancount--->\the\nspan; \the\countREGISTER}
\let\next\COLcount \next}%
%
%\def\dvr#1{\vrule width 1.0pt depth 0pt height 12pt$_{#1}$}
\def\dvr#1{\relax}%
% \omit\hfil%
% \parindent=0pt\hsize=1.1in\valign{%
% \vfil#\vfil&\vfil#\vfil\cr\hfil\hbox{\ Added to\ }\hfil&%
% \hfil\hbox{\ empty events\ }\hfil\cr}\hfil%
\def\header#1{%
\dvr{1}{\let\cr=\@mpersand%
\hdtks={#1}%
%\out{In header: hdtks=[\the\hdtks]}%
\counthdROWS\hdtks\into\hdrows%
\advance\hdrows by 1%
\ifnum\hdrows=0 \hdrows=1 \fi%
%\out{In header: Nhdrows=[\the\hdrows]}%
\dvr{5}\makehdPREAMBLE{\the\hdrows}%
%\out{In header: headerpreamble=[\headerpreamble]}%
\dvr{6}\getHDdimen{#1}%
%\out{In header: hdsize=[\the\hdsize]}%
%\striplastCR{#1}%
{\parindent=0pt\hsize=\hdsize{\let\ifmath0%
\xdef\next{\valign{\headerpreamble #1\crnorm}}}\dvr{7}\next\dvr{8}%
}%
}\dvr{2}}% End of macro \header.
%\def\striplastCR#1\cr{\xdef\headerbody{#1}}%
\def\makehdPREAMBLE#1{%This macro generates the necessary preamble for a
\dvr{3}%
% ruled table with \ncols primary columns.
% (Primary columns means the number of columns NOT
% counting those used for vertical rules.
\hdrows=#1% Get the number of columns desired.
{% Start local parameter definitions.
\let\headerARGS=0%
% This is the key to the whole thing; it prevents \ARGS
\let\cr=\crnorm%
% from being expanded in the followin \edef's.
\edef\xtp{\vfil\hfil\hbox{\headerARGS}\hfil\vfil}%
\advance\hdrows by -1% One row has been generated; decrement the
% counter.
\loop% Append as many further rows as needed to the preamble.
\ifnum\hdrows>0%
\advance\hdrows by -1%
\edef\xtp{\xtp&\vfil\hfil\hbox{\headerARGS}\hfil\vfil}%
\repeat%
\xdef\headerpreamble{\xtp\crcr}%
}% End of local parameters.
\dvr{4}}% End of \makehdPREAMBLE.
%
\def\getHDdimen#1{%
%\out{In getHDdimen: Arg 1=[#1]}%
\hdsize=0pt%
\getsize#1\cr\end\cr%
}% End of macro getHDdimen.
\def\getsize#1\cr{%
%\out{In getsize: Arg 1=[#1]}%
% Here we have to check arg#1 and see if the first token in #1 is an
% \end; if so, we stop, else we check the width of arg#1.
% We recall that each arg#1 will be terminated with a \cr token.
\endsizefalse\savetks={#1}%
%\out{In getsize: the savetks = [\the\savetks]}%
\expandafter\lookend\the\savetks\cr%
%\out{In getsize: ifendsize = [\meaning\ifendsize]}%
\relax \ifendsize \let\next\relax \else%
\setbox\hdbox=\hbox{#1}\newhdsize=1.0\wd\hdbox%
\ifdim\newhdsize>\hdsize \hdsize=\newhdsize \fi%
%\out{In getsize: hdsize=[\the\hdsize]}%
%\out{In getsize: newhdsize=[\the\newhdsize]}%
\let\next\getsize \fi%
\next%
}%
\def\lookend{\afterassignment\sublookend\let\looknext= }%
\def\sublookend{\relax%
%\out{In sublookend: looknext = [\looknext]}%
\ifx\looknext\cr %
%\out{In sublooknext: looknext=cr}%
\let\looknext\relax \else %
%\out{In sublooknext: looknext/=cr}%
\relax
\ifx\looknext\end \global\endsizetrue \fi%
\let\looknext=\lookend%
\fi \looknext%
}%
%
% Allow the user to make his own names for crthick, etc.
%
\def\tablelet#1{%
\tableLETtokens=\expandafter{\the\tableLETtokens #1}%
}%
\catcode`\@=12% Change @'s back to their normal category code.
%
\begintable
$r_0$\ (\AA) | $R_{DD}$\ ($P_n=1$) & $R_{pD}$\ ($P_n=10^{-6}$) &
$R_{pp}$\ ($P_n=10^{-23}$)
\crthick
0.5 | $10^{-64}$ & $10^{-55}$ & $10^{-63}$ \nr
0.25 | $10^{-40}$ & $10^{-36}$ & $10^{-46}$ \nr
0.125 | $10^{-23}$ & $10^{-22}$ & $10^{-35}$ \nr
0.1 | $10^{-19}$ & $10^{-19}$ & $10^{-32}$
\endtable
\vskip .1in
\centerline {\bf TABLE I}
\vskip .04 in
Fusion rates (per second) versus the width of a ``generic" fusion
barrier $r_0$\ calculated from eq (11) for D+D, p+D and p+p
reactions. Here $P_n$\ is the nuclear reaction probability once
the nuclei have reached a separation of 5 Fm. [These are estimated
from the S values in table II.]
\bigskip
\bigskip
\begintable
|\multispan{2} |\multispan{2} Kinematic
Enhancement|\multispan{2} Relative Rate \nr
Reaction | Mass & S | Cold & Hot | Cold & Hot \crthick
$D+D\rightarrow {}^3{\rm He}+n$\ | 1 & $5.2 \times 10^{-2}$ | 1 &
1 | 1 & 1 \nr
$p+D\rightarrow {}^3{\rm He} + \gamma$\ | 2/3 & 2.5 x 10$^{-7}$ |
2 x 10$^{7}$ & 6 | 100 & $10^{-5}$ \nr
$D+T\rightarrow {}^4{\rm He} + n$\ | 6/5 & 11 | $10^{-4}$ & .4 |
$10^{-2}$ & 100 \nr
$p+p\rightarrow D + e^++\nu$\ | 1/2& 4 x $10^{-25}$ | 6 x $10^{11}$
& 18 | $10^{-12}$ & $10^{-22}$
\endtable
\vskip .1in
\centerline{\bf Table II}
\vskip 0.04 in
Relative fusion rates compared to the $D+D\rightarrow {}^3{\rm He}
+ n$\ reaction. The S factors are in Mev-b while the kinematic
factors describe the increase in the tunneling exponential in eq
(11) for systems with a lighter reduced mass. The hot fusion
numbers assume an energy of 10 keV which corresponds to a barrier
width of $r_0$=144 Fm.
\vfill
\end
Alternatively, we speculate that there is a small
breakdown in the Born Oppenheimer approximation for those very rare
configurations which lead to fusion. A fluctuation which increases
the electron density between the two nuclei will enhance the
probability that they fuse. Thus a correction to the Born
Oppenheimer approximation could increase the fusion rate.
Furthermore, if this is correct, the effective electron density
(near the two nuclei) only for those very rare configurations which
lead to fusion could be very high. This high electron density may
make electron capture reactions important. For example, instead
of $p+D\rightarrow {}^3{\rm He}+\gamma$\ one could have internal
conversion of the photon on an electron leading to,
$$e+p+D\rightarrow {}^3{\rm He} + e(5.5 MeV),$$
or,
$$e+D+D\rightarrow {}^4{\rm He} + e(23.8 MeV),\eqno(14)$$
with the ejection of a high energy electron of the indicated
energy.
--