Interactions of charged particles on surfaces

LENR-CANR.org ^ | 7 December 2009 | Nabil M. Lawandya

[Kevmo]

Interactions of charged particles on surfaces
Nabil M. Lawandya_
Department of Physics and Division of Engineering, Brown University, Providence,
Rhode Island 02903, USA


_Received 5 October 2009; accepted 11 November 2009; published online 7 December 2009_


Charges of the same polarity bound to a surface with a large dielectric contrast exhibit an attractive
long-range Coulomb interaction, which leads to a two-particle bound state. Ensembles of like
charges experience a collective long-range interaction, which results in compacted structures with
interparticle separations that can be orders of magnitude smaller than the equilibrium separation of
the pair potential minimum. Simulations indicate that ensembles of surface bound nuclei, such as D
or T, exhibit separations small enough to result in significant rates of fusion. © 2009 American
Institute of Physics. _doi:10.1063/1.3270537_



Forty years ago, it was predicted that electrons could be
trapped above metallic and dielectric surfaces by image
forces.1–4 Single electrons would be expected to result in an
infinite number of bound image states, which exhibit a Rydberg
series similar to hydrogenic atoms.3,4 Since this pioneering
work, many such systems have been identified and studied
extensively using a variety of realistic crystal potentials
and various particle scattering and optical techniques.5–7 In
addition to planar surfaces, work on clusters, droplets, and
carbon nanotubes has also been undertaken.8–10
In an attempt to exploit the properties of electrons above
liquid helium, a body of work has emerged on multielectron
systems confined to the surface of liquid helium for quantum
computing applications.11–14 This work has focused on the
weakly interacting limit that allows for the creation of qubit
states by switching voltages on separated electrodes. For
quantum computing applications, electrons at typical surface
densities of 108 cm−2 or less behave classically and are
trapped within the potential of each electrode with only repulsive
interactions between them.
When the dielectric constant is much higher, the interaction
of each real charge with the other charge’s image can
result in a sizable attractive component. This attractive force
is simply the result of satisfying the boundary conditions for
the Poisson equation with two charges above a plane and is a
result of the superposition of the resulting surface charge
densities at the interface.
When two like charges, whether they are electrons, positrons,
ions, muons, or deuterium nuclei, are bound by image
charges to a surface, as shown in Fig. 1, the energy governing
their relative interaction is given by __1=_2=__
U =
_Z1e__Z2e_
4__ _1
R
+
2_
S _, _1_
where q1=Z1e and q2=Z2e are the real charges, _ and _s are
the permittivities of the space the charges reside in and the
substrate respectively, and _ =__−_s_ __+_s_.
In the limit that both charges are at the same height _
above the ideal interface, the potential exhibits a local minimum
at a charge separation given by15
Rmin
2 =
4_2
_2__23 − 1
. _2_
Equation _2_, as well as the minimization of the force equation,
shows that when __−12, there will be a bound state,
as shown in Fig. 2. Clearly, for experiments with liquid
helium, this was not the case since for that system,
_ =−0.027. For two like charges of magnitude Z1e and Z2e,
respectively, residing in free space above a high dielectric
constant substrate, the binding energy in electron volts between
the two positively or negatively charged particles is
given by
U_eV_ = −
7.2_Z1Z2_
_
__2__23 − 1_32, _3_
where _ is given in angstroms. The pair interaction energy
for two like charges is roughly half of the classical binding
energy of a single charge above an ideal classical surface
with an infinite dielectric constant difference __ =−1 limit_.
The potential between two like charges results in a bound
two-dimensional state on a high dielectric constant surface
with several degrees of freedom including rotation in the
plane, rocking on the surface, and vibration with angular
frequency of __1015 s−1 for electrons and __1013 s−1 for
more massive particles such as D nuclei. The accurate description
of these degrees of freedom will depend on the
band structure of the solid surface and the resultant effective
masses. Including the zero point vibrational energy of the
two electrons __0.53 eV_ results in a ground state binding
energy, which is approximately three times the ground state
energy for an electron trapped above the surface in an ideal
image state. For electrons, this two-particle bound state can
a_Electronic mail: nlawandy@spsy.com.
FIG. 1. Interaction between two like charges of magnitude q1 and q2 at an
interface between two media with dielectric constants _ and _S, respectively.


APPLIED PHYSICS LETTERS 95, 234101 _2009_



0003-6951/2009/95_23_/234101/3/$25.00 95, 234101-1 © 2009 American Institute of Physics
Downloaded 28 Jun 2010 to 98.191.56.230. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp
be short lived due to the lifetime of the surface states, particularly
the n=0 states, which penetrate into the bulk.16
Electrons’ lifetimes, however, are particularly short relative
to other particles, such as more massive and positively
charged nuclei and ions.
The pair of bound like charges is analogous to Cooper
pairs and will have a ground singlet state of zero spin, thus
creating a bosonic quasiparticle for a large number of like
charge systems including electrons, muons, nuclei, and ions.
The accurate binding energy of pairs of identical particles
will include exchange interactions, which may become large
when the separations are small. Bound states, however, need
not be between like particles and can result in new forms of
two-dimensional ions such as electrons bound to negative
muons, where exchange forces are not in effect. It should
also be noted that, for two oppositely charged particles, the
potential is attractive at short separations but can exhibit a
potential barrier at larger separations, preventing oppositely
charged particles from forming bound states, such as hydrogen
on the surface except through tunneling or thermal effects.
This barrier has a height equal to the relative interaction
binding energy for the like charge case but of opposite
sign.
Equation _2_ shows that the classical equilibrium separation
scales as the distance from the surface. Typically, _ is of
the order of a few angstroms, depending on the details of the
band structure of the substrate, the properties of the external
charge, and where the vacuum level lies in relation to the
various electronic bands. This set the bound-pair interparticle
equilibrium separation at a distance of about 2.61_ in the _
=−1 limit.
For a classical interface, the solution for the most probable
distance above the surface obtained from the
Schrödinger equation for the wave functions of the image
problem are, like the Bohr radius, determined by the mass of
the particle, its charge, and the value of _. When the charged
particle is a deuteron nucleus above a metallic or high dielectric
constant surface, Rmin assumes a value of 10−13 m, a
distance scale where the combination of tunneling and
nuclear forces begins to play a significant role. However, this
is not the case, as the surface band structure and the extent of
electron orbitals limit how small the most likely distance the
hydrogenic wave function predicts for much more massive
particles.
On the surface, charges are free to move in two dimensions
above the interface and dissipation results in various
equilibrium symmetry configurations. When the number of
charges is large, close packing dominates and hexagonal
symmetry prevails, as is often the case in two-dimensional
systems.17 Since the attractive component of the two-charge
potential is of a long-range nature, it is expected that interactions
far beyond nearest neighbors would play a significant
role in determining the surface structure parameters.
Simulations that maintain the hexagonal symmetry of
the system of particles but allow for displacements along
symmetry vectors show that this two-dimensional system of
interacting charges results in closest separations between certain
particles within neighboring hexagonal shells with much
smaller distances than the two-particle minimum. For the
case of _ =−1, the scaling law obtained is given by
Rmin = _4.976__N_−0.7926_, _4_
where in this case, N represents the number of shells in the
hexagonal arrangement. For N=106 and _ =1 Å, Rmin
_10−14 m. Figure 3 shows the minimum separation as a
function of the shell position for the hexagonal arrangement
under the forces of the entire ensemble for 330 particles.
Mirroring this compaction and deformation of the inner
shells is a full array linear dimension _L_, which scales as
L = 2.75_N_0.2829_. _5_
The scaling of the array size with the number of shells predicts
that compaction would result in 106 bare charges such
as D or T nuclei occupying an area _10−15 m2 with a minimum
separation of ten Fermi. In the limit of Rmin___L and
large N, the binding energy of a single unit charge to the
surface is approximately given by
Ub_eV_ _
e
8__0_
N_0.44_. _6_
For the case of N=106, this leads to a value of Ub_3 keV.
The predicted small separations of the charges when N is
large suggests that this system could lead to enhanced fusion
rates when an ensemble of charged D, T, or D-T mixtures are
created on a surface with a high dielectric constant, even in
the presence of other negatively charged and neutral species.
Clustering and segregation of the positive nuclei is expected
since the forces only act on the charged species and oppositely
charged particles experience a potential barrier. In order
to estimate the maximum fusion rate, an estimate of the
wave function probability of the closest nuclei being separated
by distances of the order of the alpha particle diameter
_R0=3.22 F_ is required. With this estimate, the fusion rate
for the specific pair located at shells j and j+2 is given by
_ = A_
_R0__2, _7_
where the rate constant A, determined from the low energy
limit of the nuclear S-factor for D-D fusion, is given by18
FIG. 2. Attractive potential between two charges as a function of the distance
above the surface ___ for the case of infinite dielectric contrast.
FIG. 3. Position of the minimum separation between two hexagonal shells
in a 55-shell hexagonal structure distorted by long-range image interactions.
234101-2 Nabil M. Lawandy Appl. Phys. Lett. 95, 234101 _2009_



Downloaded 28 Jun 2010 to 98.191.56.230. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp
A = 1.478 10−22 m3 s−1. _8_
Much work has been performed on other variants of this
problem, beginning with Jackson’s seminal calculations for
muon catalyzed fusion.19 Since this work, various calculations
using Wentzel-Kramers-Brillouin approximation
_WKB_ and other methods to evaluate the fusion rate in systems
where fusion might occur at temperatures far below
tens of millions of degrees have been undertaken.20,21
In the system of many particles described, interacting
pairs behave as a one-dimensional system capable of vibration
and supporting phonons of the entire ensemble of
charges forming the structure. In this limit, the system is
approximately describable as a nucleon trapped in a potential
created by the array whose collective long-range interactions
have forced the two charges in the j and j+2 shells to a
separation where their Coulomb repulsion is preventing further
compression. This potential is well estimated by the sum
of two terms along the symmetry axis _r_ of the pair and is
given by
Uj,j+2 =
e2
4__0
1
_r + Rmin_
+
1
_Rmin − r_. _9_
This potential has a ground state harmonic oscillator solution
with a zero point energy given by
E0 = h
e2
8_3_0mRmin
3 12
, _10_
where m is the deuteron mass. At very close separations, this
energy is large enough to have an effect on the turning point
and the tunneling probability.
In the limit that no condition is placed on the wave function
to assume a zero value at r=_Rmin, the Langer correction
term is not required and the Gamow factor _F_ for tunneling,
including zero point vibrational motion in the ground
state, is found in terms of incomplete elliptic integrals of the
second kind.22
For a nearest neighbor array separation of Rmin=5
10−13 m, F=84, a value which is several orders of magnitude
smaller than those obtained for angstrom scale separations
of nuclei but much closer to the values in muon catalyzed
fusion calculations. The Gamow factor along with an
estimated volume for the localization of the deuteron wave
function of V__Rmin
2 _, yields an estimated fusion rate per
pair of
_ _
Ae−2F
_Rmin
2 _
. _11_
At Rmin=510−13 m, the closest pair fusion rate is _
_105 s−1.
For an ensemble of 830 charges, Rmin=10−11 m, F
=37.6, and the highest pair fusion rate expected is
_10−23 s−1. Increasing the number of particles from 830 to
1000 increases the fusion rate by 19 orders of magnitude to
_1.310−4 s−1.
Creating large ensembles of charged nuclei on atomically
smooth surfaces for a sufficiently long period is not
trivial and would require energy input of the order of 1 MeV
for _35 000 D nuclei. This energy input would in turn release
approximately 35 MeV or more when the six closest set
of pairs react in approximately 10 _s ___105 s−1_. One potentially
efficient approach to this problem is the use of infrared
driven Keldysh ionization processes, which are locally
enhanced using phonon-polariton resonances in nano- and
microcrystalline materials as the substrates. SiC, for example,
has a large dc dielectric constant _9.66–10.03, depending
on crystalline orientation_ and exhibits a strong localized
phonon-polariton mode for particles or pores as large
as one micron at frequencies resonant with highly efficient
pulsed CO2 lasers.23–25
In conclusion, it has been shown that a system of like
charges can bind on the surface of a high dielectric constant
interface leading to new two-dimensional charged species or
ions with the possibility of having bosonic properties in the
ground state. In addition, when the larger ensembles of
charges are present, the long-range nature of the attractive
image forces results in compressions of the interparticle
spacing leading to high local surface charge densities and to
separations where light nuclei are expected to exhibit high
fusion rates even in the presence of other neutral species.
The predicted multiple-charge bound states are also expected
to have implications for surface reactions, catalysis, and biological
processes which depend on local surface charge density.
Future work will focus on refining the modeling of the
solid using density functional methods to better model the
dielectric response of the solid at various length scales and
the inclusion of exchange interactions in the binding energies
of a system of identical charges.
The author is grateful to Mr. Srihari Sritharan for assistance
with simulations and to Ms. Chena Immel for assistance
with the preparation of the manuscript, and to Dr. Andrei
Smuk for critical reading of the manuscript.
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234101-3 Nabil M. Lawandy Appl. Phys. Lett. 95, 234101 _2009_



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