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             \lower0.6ex\hbox{$\sim$}}}
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             \lower0.6ex\hbox{$\sim$}}}
 
\baselineskip=24pt plus .5pt minus .5pt
 
\overfullrule=0pt
 
\centerline{\bf Enhancement of cold fusion rates by fluctuations}
\medskip
\centerline{S. E. Koonin${}^*$ }
\medskip
\centerline{\it Institute for Theoretical Physics}
\centerline{\it University of California}
\centerline{\it Santa Barbara, CA  93106}
\centerline{(Submitted to {\it Physical Review Letters}, April 19, 1989)}
 
\bigskip
 
{\narrower\narrower\smallskip
The rate at which two nuclei tunnel through the coulomb barrier inhibiting
their fusion can be enhanced significantly by modest fluctuations in their
environment.  Such enhancements might play a role in recent experiments
claiming to observe cold fusion.
\smallskip}
 
\bigskip
\goodbreak
 
Recent reports have suggested the observation of fusion of deuterium nuclei
introduced by electrolysis into solid Palladium~[1] and Titanium~[2].  The
fusion rates
implied by these results are 40--50 orders of magnitude larger than might
be expected naively~[3,4], although the temperatures
involved are far too low to achieve such rates by the conventional
thermonuclear mechanism of surmounting the coulomb barrier~[5]. Thus, if these
experiments are indeed revealing nuclear processes, alternative explanations
must be sought.  In this Letter, I suggest that modest fluctuations in the
environment of the fusing nuclei can lead to the required enhancements. The
mechanism is closely related to an explanation of large sub-barrier
enhancements of heavy-ion fusion cross sections~[6].
 
The rate at which two nuclei will fuse depends upon the extent to which they
tunnel through the potential $V({\bf r})$ describing their
interaction as a function
of their separation $\bf r$.  In an isolated diatomic molecule, the electronic
time scale $\tau_e$ is far shorter than the fusion
time scale $\tau_f$ (i.e., the ``time'' the system is in a classically
forbidden region), and
$V({\bf r})$ can be taken to be
the usual Born-Oppenheimer potential.  Indeed, a
simple estimate is $\tau_f / \tau_e \approx \mu^{1/2}$, where $\mu$ is the
reduced mass of the fusing nuclei.  [Note that I am using atomic units where
$\hbar = m_e = e = 1$, so that $\mu = M_n A_1 A_2 / (A_1 + A_2)$, where
$A_{1,2}$ are the mass numbers of the fusing nuclei and
$M_n =1833$ is the ratio of the nucleon and electron masses.
Lengths are therefore
measured in Bohr radii ($0.53 \times 10^{-8}\;{\rm cm}$), energies are
measured in Hartrees (27.2 eV), and times are measured in $\tau_e = \hbar
/(27.2 {\rm eV}) \approx 2.5 \times 10^{-17}\; {\rm s}$.]
 
For two hydrogen nuclei in a solid, I must consider the effect of all other
degrees of freedom beyond $\bf r$.  The electronic coordinates are still safely
adiabatic, but the coordinates of the other
nuclear degrees of freedom are likely not. For a solid composed
of nuclei of mass number $A_L$, the lattice (phonon) time scale is $\tau_L
\approx A_L^{1/2} M_n$, the time scale for the
center-of-mass motion of the fusing
pair is $\tau_{CM} \approx (A_1 + A_2 )^{1/2}M_n$, and the time scale for motion
 
of the other hydrogen nuclei is $\tau_H \approx A_{1,2}^{1/2} M_n$.  Typically,
 
I expect $\tau_L \gg \tau_f$ ($A_L = 106$ for Pd), and $(\tau_{CM} , \tau_H )
\gtorder \tau_f$.
 
Although methods for treating tunneling in general multi-dimensional situations
have been developed using functional-integral methods~[7], the sudden limit is
one plausible and tractable approach to the present problem.  The tunneling of
the fusing nuclei must be calculated in an instantaneous potential $V({\bf r};
\xi
)$ that depends parametrically upon the non-adiabatic coordinates, which we
have denoted collectively by $
\xi$.  Thus, $V({\bf r};
\xi )$ as a function of
$
\xi$ embodies variations of the interaction potential between the fusing pair
associated with their location within the lattice, the shape of the
interstitial cavity in which they might be contained, the possible presence of
a third hydrogen nucleus nearby, etc.  For $r \ll 1$, I expect the
inter-nuclear coulomb repulsion to dominate, so that
$$
V({\bf r};
\xi ) \approx {1 \over r} + V_0 (
\xi)\;,
\eqno(1)
$$
while for distances smaller than the electronic screening length, $V({\bf r};
\xi)$
can be expanded in terms of the instantaneous local electric field, quadrupole
field, etc.
 
The coordinates $
\xi$ are not fixed, but rather fluctuate on time scales
longer than $\tau_f$ due to zero-point and thermal motion, as well as
non-equilibrium conditions.  If the instantaneous fusion rate of the
pair is $\Lambda (\xi)$, and if $P(\xi)$ denotes the normalized
probability distribution of the non-adiabatic coordinates, then the effective
fusion rate is
$$
\Lambda_{\rm eff} = \int d
\xi P(\xi )\Lambda ( \xi )\;.
\eqno(2)
$$
This expression is analogous to the usual one for thermonuclear rates in which
the fusion cross section is averaged over a Maxwellian distribution of relative
velocities~[6].  Here, however, the average is over the potential through which
the nuclei tunnel, rather than their asymptotic kinetic energy.
 
The coordinates $
\xi$ can be defined such that $
\xi = 0$ is the most probable
configuration (e.g., as might be assumed for an interstitial pair in a fixed
lattice), so that I can take
$$
P(\xi) = {1 \over {(2 \pi \sigma^2)^{1/2}}} e^{-\xi^2 /2 \sigma^2}\;,
\eqno(3)
$$
with $\sigma$ characterizing the scale of the fluctuations.  Further, if we
write
$$
\Lambda (
\xi) = A e^{-2S(
\xi)}
\eqno(4)
$$
with $A$ the nuclear rate constant and $e^{-2S(
\xi)} = |\Psi (r=0;
\xi )|^2$
the probability to find the fusing pair at $r=0$, then the change in the fusion
rate due to fluctuations is
$$
E = {\Lambda_{\rm eff} \over {\Lambda (
\xi = 0)}} = \int {{d
\xi} \over {(2 \pi
\sigma^2)^{1/2}}} e^{-
\xi^2/2\sigma^2 -2[S(
\xi)-S(0)]}\;.
\eqno(5)
$$
 
It is easy to see that most typically $E>1$, so that fluctuations enhance
the effective fusion rate.  For example, if a Taylor expansion of $S(\xi)$
about $\xi =0$ is valid,
$$
S(\xi)-S(0) \approx  S'
\xi + {1 \over 2} S''
\xi^2 + O(\xi^3)
\eqno(6)
$$
where the derivatives are evaluated at $
\xi=0$, then
$$
E \approx (1 + 2 S'' \sigma^2)^{-1/2} e^{2S^{\prime 2}\sigma^2}\;.
\eqno(7)
$$
Apart from the prefactor, which is unimportant for small $\sigma$, $E>1$ and
increases exponentially with $\sigma^2$.  Moreover, as $S$ is generally
large, even small fluctuations in $V({\bf r};
\xi)$ (i.e., small $\sigma$) can
have a significant effect.
 
The magnitude of the fluctuations required to produce a given enhancement is
then the crucial question.  A full evaluation of $E$ requires a realistic
specification of $V({\bf r};
\xi)$ and a numerical integration of the Schroedinger
equation to find $S(
\xi)$.  However, for the present illustrative
discussion, it is
sufficient to adopt a shifted Coulomb potential characterized by a single
non-adiabatic parameter
$$
V({\bf r};
\xi) = \biggl[ {1 \over r} - U (1 +
\xi)\biggr] \Theta \bigl(r^{-1}-U(1+
\xi)\bigr)\;,
\eqno(8)
$$
($\Theta$ is the unit step
function) and the simple WKB approximation for the zero-energy penetration,
$$
S(
\xi) = (2\mu)^{1/2} \int_0^{[U(1+
\xi)]^-1}\, dr \biggl[{1 \over r} - U (1+
\xi)
\biggr]^{1/2} = \pi \biggl[ {\mu \over {2U(1+
\xi)}} \biggr]^{1/2}\;.
\eqno(9)
$$
Here, the parameter $U$ describes the average of the constant term
$V_0(
\xi)$ appearing in Eq. (1), and the fluctuations of this term are given
by $
\xi U$. A rough estimate of $U$ can be had by considering tunneling in
the molecular Hydrogen potential~[4], which results in $S = 2.07 \mu^{1/2}$, so
that $U = 1.15$, or about 31 eV.  I adopt this value for the purposes of
concreteness in the following discussion.
 
With the assumptions above, the enhancement is given by
$$
E = \int {{d
\xi} \over {(2\pi\sigma^2)^{1/2}}} e^{-
\xi^2/2\sigma^2 - \pi (2\mu/U)^{1/2}
[(1+
\xi)^{-1/2}-1]} \;.
\eqno(10)
$$
A saddle-point evaluation of this integral proceeds by finding $
\xi^*$, the
most effective value of $
\xi$, as a root of the equation
$$
- {\xi^* \over \sigma^2} + {\pi (\mu/2U)^{1/2} \over
(1+\xi^*)^{3/2}} =0
\eqno(11)
$$
and then
$$
E \approx [1+ {{3\pi} \over 4} (2\mu U)^{1/2} (1 +
\xi^*)^{-5/2} ]^{-1/2}
e^{-
\xi^{*2}/2\sigma^2 - \pi (2\mu/U)^{1/2}
[(1+
\xi^*)^{-1/2}-1]} \;.
\eqno(12)
$$
In Table~I, I show the enhancement expected for the $\rm d + d$ and $\rm
p + d$ fusion rates
for various values of $\sigma$.  Note that even relatively small fractional
fluctuations ($\sigma \ltorder 0.5$, or a fluctuation
in $V_0$ of 15~eV) can produce enhancements of some 30 orders of magnitude.
The most effective values of $\xi$ generally lie in the extreme wings of
$P(\xi)$.  Thus, even very improbable configurations can have a significant
effect, as the tunneling rate is very sensitive to the potential.
Also note that the precise relation between $E$ and $\sigma$
depends upon what I've assumed for $S(0)$ and the way in which fluctuations
perturb the potential.
 
Are these fluctuations reasonable?  A proton at a distance of 1~Bohr radius
generates a potential of 1 in atomic units, so that given the possibility of
multiple occupation of the interstitial sites or large distortions of
the lattice, $\sigma \sim 0.5$ is perhaps not too implausible.
Further, it seems likely that the $\rm p + d$ system, with its
non-vanishing electric dipole moment operator, will be influenced by
fluctuations more than the $\rm d + d$ system,
which couples only to the electric quadrupole field.  Finally, I note that any
conditions in the system that enhance fluctuations will also enhance
fusion rates.  These might include heating to increase the number of
phonons, high hydrogen fractions to increase the multiple vacancy probability,
and the flow of a current to induce gross motion of the hydrogen nuclei.
 
In conclusion, I have shown that fluctuations in the environment of a fusing
pair of nuclei within a solid can significantly enhance the rate at which they
fuse.  This is because of the extreme sensitivity of the fusion rate to the
effective potential barrier inhibiting fusion.  In a schematic calculation for
a shifted coulomb potential, I showed that a fractional {\it rms}
fluctuation of the potential by only 0.1 (about 3 eV) will enhance the fusion
rate by some 8 orders of magnitude and that fractional rms fluctuations of 0.5
will lead to enhancements of more than 30 orders of magnitude.  While these
results are suggestive, a more detailed calculation of the interaction between
hydrogen nuclei in a solid (and their tunneling in the presence of
non-adiabatic degrees of freedom) would be required to establish the relevance
of this mechanism to cold fusion.
 
 
\bigskip
 
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{[1]} M. Fleischmann, S. Pons,
and M. Hawkins, J. Electroanal. Chem. {\bf 261} (1989) 301.
 
\item{[2]} 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 (submitted to {\it Nature}).
 
\item{[3]} C. D. Van Siclen and S. E. Jones, Journal of Physics G {\bf
12} (1986) 213.
 
\item{[4]} S. E. Koonin and M. Nauenberg, Santa Barbara Institute
for Theoretical Physics preprint NSF-ITP-89-48, April, 1989.
 
\item{[5]} D. D. Clayton, {\it Principles of Stellar Evolution and
Nucleosynthesis}, (McGraw-Hill, New York, 1968), Ch. 4.
 
\item{[6]} H. Esbensen, Nucl. Phys. {\bf A352} (1981) 147.
 
\item{[7]} A. D. Caldeira and A. J. Leggett, Ann. Phys. (NY) {\bf 149} (1983)
374; R. P. Feynman and F. L. Vernon, Ann. Phys. (NY) {\bf 24} (1963) 118.
\vfill
\eject
\def\mystrut{\vrule height 18pt depth 6pt width 0pt}
\centerline{Table 1:  Enhancement factors and most efective
configurations}
\centerline{for the truncated Coulomb potential, Eq. (8).}
$$
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& &\omit\hfil\hidewidth$\sigma=0.1$\hidewidth\hfil&\omit\hfil 0.2\hfil&
&\omit\span\hfil 0.3\hfil&\omit\span\hfil 0.4\hfil&\omit\span\hfil 0.5\hfil&
\omit\span\hfil 0.6\hfil&\omit\span\hfil 0.7\hfil\cr
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&\xi^*     &0.44& 1&05& 1&62& 2&17& 2&69& 3&20& 3&69\cr\noalign{\medskip}
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\bye

