April 7, 1989
 
There follows the TeX script of a preprint on Cold Fusion that might be
of interest to you.  Please feel free to distribute it (either electronically
or in hard copy) to others who might be interested.
                                        Steve Koonin [KOONIN@SBITP.BITNET]
 
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\centerline{\bf Cold fusion in isotopic hydrogen molecules}
\medskip
\centerline{S. E. Koonin${}^*$ and M. Nauenberg${}^{**}$}
\medskip
\centerline{\it Institute for Theoretical Physics}
\centerline{\it University of California}
\centerline{\it Santa Barbara, CA  93106}
\centerline{Submitted to {\it Nature}, April 7, 1989}
 
\bigskip
 
{\narrower\narrower\smallskip
We have calculated cold fusion rates in diatomic hydrogen molecules
involving various isotopes.  An accurate Born-Oppenheimer potential was
used to calculate the ground state  wave functions.  We find that the
rate for $\rm d+d$ fusion is $3 \times10^{-64}\,\rm s^{-1}$, some 10
orders of magnitude faster than a previous estimate.  We also find that
the rate for $\rm p+d$ fusion is $10^{-55}\,{\rm  s}^{-1}$, which is
larger than $\rm d+d$ due to the enhanced tunneling in the lighter
system.  Enhancements of the electron mass by factors of 5--10 would be
required to bring cold fusion rates into the range of recently claimed
observations.
\smallskip}
 
\bigskip
\goodbreak
 
``Cold fusion'' (CF) occurs when two nuclei with very small relative
energy tunnel through their mutual coulomb barrier to initiate a nuclear
reaction. The phenomenon is well-studied in muon catalyzed
fusion~[1,2,3], where the large mass of the muon relative to an electron
results in tightly-bound diatomic muo-molecules of Hydrogen (e.g.,
d-$\mu$-t) with cold fusion rates of order $10^{12}\;{\rm s}^{-1}$. It
is also believed to occur as pycno-nuclear reactions in certain
astrophysical environments~[4].  Recent reports of CF between hydrogen
isotopes embedded in Palladium~[5] and Titanium metal~[6] have prompted
us to reconsider previous estimates of the CF rates for free diatomic
isotopic hydrogen molecules.
 
Consider a free diatomic molecule composed of two hydrogen nuclei (which
might be different isotopes).  In the Born-Oppenheimer approximation the
fusion rate $\Lambda$
is proportional to the probability that the nuclei are very
close together:
$$
\Lambda = A |\Psi ( \rho )|^2\;,
\eqno(1)
$$
where $\Psi$ is the normalized wave function describing their relative
motion The inter-nuclear separation $\rho$ is a typical distance where
nuclear interactions occur, approximately 10~fm.
 
The nuclear rate constant $A$ for a given pair of nuclei is related to
the low-energy behavior of the corresponding fusion cross section.  If
the variation of the cross section $\sigma(E)$  with relative energy $E$
is parameterized in terms of the usual S-factor,
$$
\sigma (E) = {S(E) \over E} e^ {-2 \pi \eta }; \ \ \
\eta={e^2 \over {(2E\hbar^2 /\mu)^{1/2}} }\;,
\eqno(2)
$$
with $\mu$ the reduced mass of the two nuclei,
then
$$
A = {S(E=0) \over {\mu c^2}} \  {c \over {\pi \alpha }}
\eqno(3)
$$
with $\alpha=e^2/\hbar c \approx 1/137$.  Table~1 shows the nuclear rate
constants for four possible interactions between two hydrogen
nuclei~[7].
 
We restrict ourselves only to s-wave nuclear motion (for which the
fusion rate will be largest), so that the wave function can be written
as
$$
\Psi (r) =   {\psi(r) \over {4\pi r}}
\eqno(4)
$$
with the normalization $\int_0^\infty \psi^2 \,dr = 1$. The radial wave
function $\psi$ then satisfies the Schroedinger equation
$$
[- {\hbar^2 \over {2 \mu}} {d^2 \over dr^2}  + V(r)] \psi(r) = \epsilon
\psi(r)
\eqno(5)
$$
where $\epsilon$ is the eigenvalue for relative motion.  Unless
otherwise specified, we will hereafter work in atomic units $(e^2 =
\hbar = m_e = 1)$, so that all energies are measured in Hartrees
($\approx 27.2 \;{\rm eV}$) and all distances are measured in Bohr radii
($a \approx 0.53 \times 10^{-8}\;{\rm cm}$).
 
Simple considerations determine the general features of $V(r)$.  If
energies are measured relative to the energy of two isolated Hydrogen
atoms $(-1)$, $V$ vanishes at large $r$.  Further, it must have a
minimum of the proper depth and separation to support the observed
molecular bound states.  At small separations, the electronic structure
is that of the He atom  with an energy of $V_0 = - 1.9037$ relative to
two isolated hydrogen atoms, so that
$$
V(r \to 0) \to {1 \over r} + V_0\; .
\eqno(6)
$$
 
An estimate of the supression of the fusion rate by tunneling is given
by the barrier penetration factor obtained from the WKB approximation to
Eq. (5):
$$
               B=e^{-2 \int_0^{r_0} k(r) dr}
\eqno(7)
$$
where the local wavenumber is $k(r) = [2\mu (V(r) - \epsilon )]^{1/2}$
and the integral extends to the classical turning point, $r_0$.
 
To estimate the barrier penetration integral (7), we have taken for the
diatomic molecular potential $V(r)$ the current best available numerical
calculation in the Born-Oppenheimer approximation done by Kolos and
Wolniewicz (K--W)~[8,9]. For $1.1 < r <3$ this potential is well
approximated by the Morse potential
$$
         V(r)=0.1745 [e^{-2.08(r-1.4)} -2 e^{-1.04(r-1.4)}]
\eqno(8)
$$
For smaller values of $r$ we fitted the calculated values of $V - 1/r$ to a
seven term Lagrange interpolation formula. Upon numerical evaluation of
the integral using the exact $d + d$ eigenvalue and turning point,
we find
$$
           B = e^{-4.13 \sqrt\mu}
\eqno(9)
$$
The numerical coefficient of 4.13 is to be compared with the estimate of
3.0--3.3 made by Zeldovich and Gershtein [2]; the difference leads
to a penetration factor which is 15-21 orders of magnitude
smaller.
 
An accurate evaluation of the CF rates can be obtained by a direct
numerical integration of the Schroedinger equation~(5) with the K--W
potential, as shown in the second column of Table~2. The nuclear radius
was taken as $\rho = 2 \times 10^{-4} (\approx 10\,{\rm fm}$). The
numerical methods of~[10] were used, with the wave function being treated
explicitly only for $r>0.005$; the regular s-wave Coulomb function was
used to extrapolate this solution to $r=\rho$.
 
The exact dependence of the barrier factor on reduced mass that we
extract from these results is good agreement with the WKB approximation
(7), but disagrees with references [2] and [11]. The estimate of the
$\rm d+d$ fusion rate made in [11] is too low by about 10 orders of
magnitude, because in the calculation of the WKB penetration integrals
an unshifted coulomb potential was used at small separations (i.e., our
Eq.~(6) with $V_0 = 0$). The effective energy with which the nuclei
assault the coulomb barrier is therefore lower than it should be, and
hence the calculated fusion rate is smaller. We note that the ${\rm
p+d}$ fusion rate is 7.5 orders of magnitude larger than the $\rm d+d$ rate.
Although the nuclear rate constant for $\rm p+d$ is some 5.5 orders of
magnitude smaller than for $\rm d + d$, the smaller reduced mass
enhances the tunneling probability more than enough to compensate for
this.
 
It is interesting to ask by how much the internuclear separation must be
decreased in order to reach the fusion rates claimed in [5,6].  Although
the precise answer depends upon the details of the internuclear
potential, a simple way of quantifying the problem is to endow the
electron with a larger mass $m^*$ than it actually has.  The equilibrium
internuclear separation then scales as $m_e/m^*$, while Eq.~(9) above
allows the enhancement to be estimated as
$$
\log_{10} \,[\Lambda(m^*)/\Lambda(m_e)] = 3 \log_{10} \,(m^*/m_e) -79
(\mu/M_n)^{1/2} [(m_e/m^*)^{1/2}-1]\;,
\eqno(10)
$$
where $M_n$ is the nucleon
mass and the logarithmic variation with $m^*$ is due to the scaling of
$\Psi (\rho)$. More accurate estimates can be had by numerical
integration of the Schroedinger equation (5) with the K--W potential
[8,9], as shown in Table~2.
 
Note that a mass enhancement of $m^* \approx 5m_e$ would be required to bring
the CF rates into the range claimed by [6] while $m^* \approx 10m_e$ is
required by the results of [5].  These should be taken as only a rough
guide, however, as Hydrogen in Palladium is dissociated into atoms and
ionized to bare nuclei [12].
 
It is worth remarking on the validity of the Born-Oppenheimer
approximation we have used in our calculations.   Since there is
a large difference between the potential and total energies in the
classically forbidden region, one might naively expect a failure of the
adiabatic assumption. However, more careful reflection suggests
otherwise. Sytematic corrections to the adiabatic approximation are
possible by considering the full coupled-channels generalization of
Eq.~(5) [13].  The largest coupling terms are of order
$$
{1 \over \mu} \langle n| {\partial \over \partial r} |m\rangle
{d\Psi_m (r) \over dr } \approx {k(r) \over \mu} \Psi_m (r) \;,
\eqno(11)
$$
where $n,m$ label the electronic states, which we assume to vary on the
scale of the Bohr radius. The local wavenumber at small distances is
$k(r) \approx (2 \mu /r)^{1/2} $. This term is to be compared with the
effect of the
diagonal potential at short distances, $\approx \Psi_m /r$, giving a
correction to adiabaticity of order $(k/\mu)/(1/r) \approx (r
/\mu)^{1/2} \ll 1$.
 
In summary, we have calculated cold fusion rates in isotopic Hydrogen molecules.
 
We find that the rate for $\rm d+d$ fusion in the free molecule
($\approx 3 \times 10^{-64}\,{\rm s}^{-1}$) is some 10~orders of
magnitude larger than the most recent previous estimate [11], but that
the rate for $\rm p+d$ is faster yet by some 8 orders of magnitude.
This latter remains true for rates slower than $\approx 3 \times
10^{-17}\, {\rm s}^{-1}$  ($m^* \approx 6 m_e$). Hence, if refs. [5,6] are
seeing any nuclear process at all, it is more likely the neutron-free
$\rm p+d$ reaction rather
than $\rm d+d$. We also find that hypothetical enhancements of the
electron mass by factors of 5--10 are required to bring CF rates into
the range of values claimed experimentally. However, we know of no
plausible mechanism for achieving such enhancements.
\bigskip
 
We would like to thank D.~Eardley and many of our other colleagues at
the ITP for fruitful discussions.  We are also grateful to B. Kirtman
for a numerical calculation of the diatomic potential at $r=0.1$.  This
work was support in part by National Science Foundation grant
PHY82-17853 at Santa Barbara, supplemented by
NASA funds,  and by National Science Foundation grants
PHY86-04197 and PHY88-17296 at Caltech.
\vfill\eject
\centerline{\bf References}
\frenchspacing
\medskip
\item{*} Permanent address: W. K. Kellogg Radiation Laboratory, Caltech
106-38, Pasadena, CA 91125
\item{**} Permanent address: Physics Dept. and Institute of Nonlinear
Sciences, University of California, Santa Cruz, CA 95064
 
\item{[1]} J. D. Jackson,  Phys. Review {\bf106} (1957) 330.
 
\item{[2]} Ya. B. Zel'dovich and S. S.Gershtein, Soviet Physics Uspekhi
{\bf3} (1961) 593.
 
\item{[3]} J. Rafelski and S. E. Jones, Scientific American {\bf 257}
(July, 1987) 84.
 
\item{[4]} S. L. Shapiro and S. L. Teukolsky, Black Holes, White Dwarfs and
Neutron Stars (Wiley, New York, 1983) p.~72.
 
\item{[5]} M. Fleischmann and S. Pons, University of Utah preprint, March, 1989.
 
 
 
\item{[6]} S. E. Jones, E. P. Palmer, J. B. Czirr, D. L. Decker, G. L. Jensen,
J. M. Thorne, S. F. Taylor, and J. Rafelski, University of Arizona preprint
AZPH-TH/89-18, March, 1989.
 
\item{[7]} W. A. Fowler, G. R. Caughlan, and B. A. Zimmerman, Annual Reviews of
Astronomy and Astrophysics {\bf 5} (1967) 525.
 
\item{[8]} W. Kolos and L. Wolniewicz, Journal of Chemical Physics,
{\bf41} (1964) 3663; {\bf49} (1968) 404.
 
\item{[9]} W. Byers-Brown and J.D. Power, Proc. Roy. Soc. Lond. {\bf A.
317} (1970) 545.
 
\item{[10]} S. E. Koonin, {\it Computational Physics}\/,
(Addison-Wesley, Menlo Park, 1985) Ch. 3.
 
\item{[11]} C. D. Van Siclen and S. E. Jones, Journal of Physics G {\bf
12} (1986) 213.
 
\item{[12]} F. A. Lewis, Platinum Metal Reviews {\bf 26} (1982) pp.
20, 70, 121.
 
\item{[13]} A. Messiah, Quantum Mechanics (North Holland, 1965) vol. II, p. 786.
 
 
\vfill
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\centerline{\bf Table 1:  Rate constants for fusion of hydrogen
isotopes}
$$
\vbox{\tabskip=0pt\offinterlineskip
\halign to \hsize{
\mystrut#&\tabskip=1em plus 2em&
$\rm #$\hfil&
\hfil$#$\hfil&
\hfil$#$\hfil\tabskip=0pt\cr
&\omit\hfil \hbox{Reaction}\hfil&S(E=0) ({\rm Mev-b})&A (\rm cm^3\,
s^{-1})\cr
&p+d \to {}^3{\rm He} + \gamma & 2.5 \times 10^{-7}&5.2\times 10^{-22}\cr
%
&p+t \to {}^4{\rm He} + \gamma & 2.6 \times 10^{-6}&4.8 \times 10^{-21}\cr
%
&d+d \to {}^3{\rm He} + n \oplus {}^3\rm H + p&1.1 \times 10^{-1}&1.5 \times 10^
{-16}\cr
%
&d+t \to {}^4\rm He + n  &1.1 \times 10^1 & 1.3 \times 10^{-14}\cr
}}
$$
\bigskip
\bigskip
\centerline{\bf Table 2: CF rates in isotopic hydrogen molecules}
\centerline{(Entries are $\log_{10}$ of the rate is $\rm s^{-1}$)}
$$
\vbox{\offinterlineskip\tabskip=0pt
\halign to \hsize{
\mystrut#&
\hfil$\rm #$\hfil\tabskip=1em plus 2em&
\hfil$#$\hfil&
\hfil$#$\hfil&
\hfil$#$\hfil&
\hfil$#$\hfil\tabskip=0pt\cr
&\ &m^*/m_e =  1\hfil&2 &5&10\cr
%
&p + d&-55.0&-36.0&-19.0&-10.4\cr
%
&p + t&-57.8&-37.7&-19.7&-10.5\cr
%
&d + d&-63.5&-40.4&-19.8&-9.1\cr
%
&d + t&-68.9&-43.5&-20.9&-9.4\cr
}}
$$
\bye

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