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Subject: Dharma-wardana CNF Paper
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Date: 12 Jul 89 05:18:37 GMT
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Notation:

	X[y] = X subscript y
	X{y} = X superscript y
	integral:a:b = definite integral from a to b
	x ~ z = x is approximately z
	r^ = vector r

Notes:
	The use of D in the preprint for the diffusion rate has been
	replaced by d to avoid confusion with the notation for the
	atomic deuteron.

	I have slightly reorganized some of the equations for better
	readability in ASCII form.

	a.u. in the text presumably means "Angstrom unit".

------------------------------------------------------------------------
Submitted to Chem. Phys. Letters, 24 April 1989
Revised version 15 June 1989


		A model for the cold fusion of deuterium in
			deuterated palladium systems

				 by

		M. W. C. Dharma-wardana and G. C. Aers
		          Division of Physics
		National Research Council of Canada
			Ottowa, Canada K1A OR6

		(Bitnet: Chandre at NRCVM01, FAX: (613) 957-8734)

We have estimated the enhancement of the nuclear fusion rate of Pd-D
type systems and the D[2]{+}-muonium molecule in comparison with the
fusion rate in a D[2]-molecule at room temperature. Electronic
screening present in the Pd leads to an insufficient enhancement of
the fusion rate to account for the claimed results. The enhancement
factor _decreases_ as the D{+}-concentration increases. However we
argue that observable fusion could occur if ionic-screening from
non-equilibrium D{+}-concentrations could come into play. The screening
dependence of the D{+}-diffusion under a driving potential could
generate bursts of non-equilibrium D{+}-ion densities, presenting some
conditions when cold fusion might happen.

Recently claims of observation of room temperature fusion of high
density deuterium electrolytically impregnated into palladium and
titanium have been reported [1,2]. Further experimental investigations
have so far not confirmed these claims except for some sample dependent
calorimetric anomalies, tritium concentrations and neutron emissions
reported at the Santa Fe meeting [3]. However, the possibility of cold
fusion is of sufficient importance to warrant an examination of the
theoretical rate of room temperature fusion, and how it could be 
enhanced, taking into account some of the special features found in
hydrogenated metal systems.

Palladium can absorb large amounts of hydrogen or deuterium forming
at first a solid solution and then condensed phases alpha and beta
of increasing density [4]. The beta phase is rishest in deuterium
and occurs for concentrations in excess of about 0.6 deuteriums per
atom of Palladium. The deuterium, in the form of deuterons, occupy 
the octahedral sites associated with the Pd-ions which form a slightly
expanded FCC-lattice with a lattice constant of about 4 Angstroms. The
potential wells at the octahedral sites are relatively shallow [4],
being about 0.2 eV deep for low deuterium impregnation as may be
estimated from the activation energies for diffusion. The protons
(or deuterons) have a high mobility in the lattice and at least that
fraction of the protons which are sufficiently energetic (E > 0.2 eV)
may be regarded as a "hydrogenic plasma" since the hydrogens are fully
ionized.

Since two deuterium nuclei held together by a muon (D[2]{+}-mu) have
a fusion rate which is many orders of magnitude larger than for the
D[2]-molecule, it has been suggested [5] that the heavy effective mass
of electrons (d-electrons) in palladium may be playing a role similar
to that of the muon, and that there is some quasi-confinement of two
deuterium nuclei. Unlike in D[2]{+}-mu, there are no bound states
associated with the deuterons in Pd. The hydrogen (or deuterium) on
entering the Pd metal gives up its electron to the 4d and 5s bands and
becomes a proton (deuteron). This is because the high electron
density in a metal provides sufficient screening to reduce the Coulomb
potential of the protons to such an extent that even the deep 1s bound
state of atomic hydrogen is pushed into the continuum [6]. At low
proton densities the electrons given up by the protons go into the 4d
band. These d-electrons have a higher effective mass and a high density 
of states N(E[F]) at the Fermi energy and strongly screen the protons.
Since screening is strong the rigid band picture is a good approxi-
mation [4]. Increasing the proton density eventually leads to a filling
of the 5s band and then the electronic screening reduces to values
comparable to that in silver.

The rate of a thermonuclear reaction for a pair of nuclei, with relative
velocity v is given by sigma*v where sigma is the standard cross-
section [7]. The cross section itself has a form (s(v)/v)*exp(-w(v))
where s(v) is a function weakly dependent on energy and hence treated
as a constant sbar. The exponential term exp(-w(v)) arises from the
tunneling process and was first described by Gamow [8]. If the number
of pairs of particles is N[pair], the reaction rate is given by:

		R = N[pair]*sbar*<p(v)*exp(-w(v))>                 (1)

The exponent

	w(v) = 2*sqrt(m)*integral:r[N]:r[c] sqrt(U(r) - E(v)) dr   (2)

arises from the WKB description of the tunneling process. Atomic units
(e = hbar = m[e] = 1) will be used throughout, with energies in
Hartrees (27.21 eV) unless otherwise stated. The average in (1) is taken
over the velocity distribution p(v). The energy E(v) is simply the
kinetic energy of relative motion, viz, .5*m*v**2 where m is the mass
of a deuteron. The classical turning radius r[c] is such that
u(r[c]) = E(v). The lower limit of integration r[N] is the nuclear radius
and will be taken as zero.

If we consider N[D[2]] molecules of deuterium and calculate the reaction
rate R[D[2]] for the gas, and then proceed to calculate R[p] for the
plasma of deuterons confined in Pd, the ratio will give us the enhancment
factor f = R[p]/R[D[2]]. Clearly, if N[D[2]] molecules of D[2] are
converted to 2*N[D[2]] atoms of D, which become D{+} ions in Pd, the
number of possible D{+}-D{+} pairs is 2*N[D[2]]**2. However, only a
fraction x ~ exp(-beta*V[0]), where beta is the inverse temperature
1/(k[B]*T), and V[0] is the well depth for localization at the octahedral
sites, will be energetic enough to move about and achieve collisions.
Hence the effective number of pairs is:

		N[pair] = 2*(N[D[2]]*x)**2                         (3)

Hence, for a gram mole of D[2] compressed into the metal, with
V[0] = 0.2 eV, there is already an enhancement of the reaction rate
by a factor of about N[D[2]]*x**2 which is of the order of 10**16. In
the following analysis locality, with its _local_ concentration of
deuterons, and its local distribution of well depths V[0].

[[The previous sentence appears here exactly as it does in the preprint.
It doesn't make any sense to me either. - DWM]] 

The Pd-lattice constant for the beta-phase is about 4 Angstroms, and
increases slightly with increasing deuterium concentration. On this
basis, the D-D nearest neighbor distance for two octahedral sites is
5.38 a.u., leading to a Wigner-Seitz radius r[ws] = 2.69 a.u. for the
D{+}-lattice in the regions where all the octahedral sites are fully
occupied. Even if we use the localization well depth of V[0] = 0.2 eV
valid at the low deuteron density limit, at room temperature (0.026 eV)
a small but significant fraction of the deuterons will be mobile. The
Coulomb interaction between these deuterons will be screened by the
conduction electrons in the system. In the simplest approximation the
electron-screening momentum k[se] is given by:

		k[se]**2 = 4*pi*(e**2)*N(E[F])                     (4)

where N(E[F]) is the density of states per unit volume. Band structure
calculations [9] show that N(E[F]) = 2.31 states per eV per Pd atom.
Coherent potential approximation (CPA) calculations [10] for the Pd-H
system gives N(E[F]) = 0.24. These two values, for Pd and Pd-H hence
give k[se] = 2.68 /a.u. and 0.74 /a.u. respectively. Thus the electronic
screening has decreased and become similar to that of Ag where a similar
calculation gives k[se] = 0.92 /a.u. Some of the D{+}-ions (a fraction
exp(-beta*V[0]) ) are not lodged in the potential wells but are mobile,
and can produce ion density fluctuations and hence screening effects. 
The screening due to mobile deuterons can be estimated using the classical
Debye-screening formula:

		k[si]**2 = 4*pi*n*(e**2)*beta                      (5)

Here n is the number of _mobile_ protons. Since the total number density
N = 3/(4*pi*r[ws]**3), taking the deuteron localization energy
V[0] = 0.2 eV, and estimating n we find that k[si] = 0.42 /a.u. in Pd-H.
Hence the combined screening momentum k[sc] = sqrt(k[se]**2 + k[si]**2)
becomes 0.85 /a.u. The ionic screening reflects the existence of density
fluctuations in the mobile D{+}-subsystem and is in no way an overcounting
of screening effects. If the rigid band picture is replaced by a more exact
description an effective charge Z close to unity (rather than unity) will
enter into Eq. (5) but for our purpose we take Z = 1 as the charge on
the mobile deuterons.

We write the screened Coulomb potential between two deuterons in Pd as:

		U(r)[p] = exp(-k[sc]*r)/r                          (6)

where the suffix "p" indicates plasma-like deuterons in palladium.
Eq. (6) defines the U(r) occurring in Eq. (2).

In order to directly compare the fusion rate calculated for a pair
of deuterons (D{+}D{+}) in the Pd-lattice with that for a D[2] molecule
in the gas, it would be useful to formulate the fusion problem in D[2]
in a manner analogous to the screened potential description of Eq. (6).
If we regard the D[2] molcule, with nuclear separation R = 1.36 a.u.,
as consisting of two Wigner-Seitz spheres, each of radius r[s] = 0.68 a.u.,
and having an average of one electron each, the D[2] ions may be regarded
as being contained in an electron gas of density n such that
n = 3/(4*pi*r[s]**3). Hence a screening constant

		k[sc]**2 = 4/(pi*alpha*r[s]), 

		alpha = (4/9*pi)**(1/3)

can be derived from Thomas-Fermi theory [11]. This gives k[sc] = 1.87.
We will use this value of k[sc], together with the form (6) to define
the potential U(r)[D[2]] to be used in Eq. (2) for the calculations of
D[2] gas. Such a procedure ignores the molecular binding energy
effects but correctly includes the all-important screening effects on
the Coulomb barrier. The simple approach of using two spheres to
estimate an effective r[s] is consistent with the usual "muffin-tin"
approach and recognizes the existence of two nuclear centers in the
system. We could have chosen to use a different procedure in which it is
assumed that there are two electrons in a large sphere of radius
R = 1.36 a.u., the screening constant would then be decreased by about
20%. Hence the two-sphere treatment given here will give a somewhat
higher rate for the D[2] molecules, making the calculated enhancement
factor f more conservative.

In the case of the molecule D[2]{+}-mu, the deuterium atoms are held
very close together by the 1s-type wave function of the muon, viz,
psi[1s] ~ exp(-m[mu]*r) where r is in ordinary atomic units, and the
muon mass m[mu] = 200. Hence the muon (a heavy electron) can be regarded
as forming an "electron" gas occupying a sphere of radius 1/200 a.u., 
i.e., r[s] = 0.005. This gives us a screening constant k[sc] = 22.1 /a.u.
This will be used to form a screened potential seen by the deuterons
in D[2]{+}-mu in our calculation of the fusion rate using Eq. (2).

The calculation of the fusion rate also requires the velocity distribution
p(v). Since the energy E(v) = .5*m[r]*v**2, where m[r] is the reduced
mass, is equal to u(r[c]), the average over the velocity distribution
can be replaced by an integration over r[c] with the weight factor
exp(-beta*u(r[c])). A better approximation (not needed for our purpose)
is obtained via the pair distribution function g(r[c]). In either case
the averaging over the velocities can be replaced by an averaging over
r[c], with suitable modifications (for details see ref. 12), in doing the
numerical calculations.

In Table 1, we show the results of calculations for D[2], several Pd-D
systems, and for the D[2]{+}-mu molecule using the screened potential
approach outlined in the above. For the case PdD, x = 1, if we assume
that only a thermal distribution exp(-beta*V[0]) (i.e., 0.00046) of the
D{+} ions are delocalized, the screening is due mainly to the electronic
contribution and we get k[sc] = 0.85. The fusion rate is thus actually
_reduced_ by a factor of ~10**-38 from that of the D[2] molecule rate
(since the absolute rate for a D[2]-molecule is ~ 10**-74 fusions/sec,
we need an enhancement f ~ 10**50 to achieve the sort of rates discussed
by Jones et. al., ref. 2). The values k[sc] = 8, 10 correspond to about 
40% and 60% delocalization of the D{+} ions in the interstitial volume.
Such large delocalized (mobile) populations are strongly off equilibrium.
Hence we consider a scenario whereby some form of non-equilibrium
D{+}-concentrations could arise.

Consider the PdD[x] system for _small x_, where electronic screening k[sc]
is large. The diffusion constant d(x), and the mobility mu(x) of
D{+}-ions in the lattice are all influenced by electronic screening
since the latter moderates all the interactions. The flux j of D{+}-ions
is given by an equation of the form:

		j = -N*d*grad(x) + N*x*mu*grad(phi)                   (7)

where phi is a driving potential (due to an applied electric field,
stress field, etc.) N is the number of PD{+} ions per unit volume so
that N*x is the D{+} density. For strong driving potentials, and in the
neighborhood of a critical point in the metal-deuterium phase diagram,
when small forces can cause large concentration shifts, equation (7) has
to be modified to take note of the fact that d and mu are strongly
dependent on the D{+}-loading concentration profile x(r^). This makes
Eq. (7) extremely non-linear and also dependent on the boundary conditions
imposed, i.e., strongly sample dependent. The dependence of d and mu
on x(r^) arises from the strong sensitivity of screening to the D{+}
concentration as discussed previously. Hence we can envisage the
following scenario. As D{+}-loading is initiated screening is high and
the ions diffuse into the sample rapidly, increasing the value of x.
But as x(r^) increases in some locality r^, the local Fermi level moves
to energies with a lower density of states and hence the electronic
screening constant rapidly decreases. This is accompanied by a strong
decrease in the local diffusion constant and the local mobility, leading
to a bottle-necking effect, generating pockets of significant off-
equilibrium concentrations of D{+}-ions. However, as soon as such pockets
are formed, ion screening comes into play and weakens all interactions
and hence once again the D{+}-mobility, rate of diffusion (as well as
the rate of fusion) etc. are strongly enhanced and the bottle-neck is
cleared. The driven system is once again ready for a repeat burst of
build up and decay. Such a scenario is consistent with the production 
of bursts of neutrons if nuclear fusion could take place. The non-linear
nature of the diffusion equation would also explain the sensitivity of
the system to specific sample features. In this connection we remark
that "critical slowing down" and sample specific diffusion effects are
well known, especially for hydrogen in Niobium [13]. The time constants
associated with the build up and decay of non-equilibrium concentration
bumps in a PdD[x] electrode, and the extent of nuclear fusion (if any)
during such a time period etc. cannot be estimated without detailed
numerical solutions.

In Rafelski et al [5], it is estimated that when the D{+}-D{+} system
is confined to regions of about 0.3 a.u. an enhancement factor f ~ 10**50
is obtained (they quote the rate 10**-24 /sec for the 0.3 a.u. system
and the rate 10**-74 /sec for the hydrogen molecule). Our estimate for
300K, taking k[s] = 8 for r[s] = 0.3 gives an enhancement factor of
10**46, in general agreement with Rafelski et al. But Rafelski provides
no model for "confining" or bringing the two D{+}-ions as close as
0.3 a.u. In our model this is achieved by the reduction of the Coulomb
barrier due to ion-screening. In fact, if almost all the deuterium in 
the octahedral sites of PdD could be delocalized, then the _ion screening
alone_ gives k[si] = 12.7 /a.u. This is in excess of the value of
k[sc] = 8 required. Thus observable bursts of nuclear fusion may be
achievable in these systems IF such delocalized densities could be
formed. Calculations for Ti and V also lead to similar conclusions.

The estimated enhancement factor for the D[2]{+}-mu molecule, i.e.,
10**64 - 10**68 is in fair agreement with the somewhat uncertain value
of about 10**80 quoted in the literature. This suggests that the model of
the screened Coulomb potentials used here provides a conservative order
of magnitude estimate. We also note that increasing the temperature from
300K to 1000K raises the enhancement by about 10**7 in the palladium
system.

Improved calculations of enhancement factors will require, say, a
density functional type calculation for the D{+}-D{+} potentials and
pair distribution functions in the system under study, together with
solutions of the non-linear driven diffusion equation for suitable
sample shapes.

In conclusion, we showed that the electronic screening alone does not
give enough enhancement of the D{+} fusion rate in PdD[x] to have
observable effects. In fact, contrary to an apparently widely held
point of view, as x increases, the electronic screening and the rate of
enhancement actually _decrease_. We also pointed out that since 
ion-screening due to D{+}-ion density fluctuations in the mobile fraction
of deuterons could occur, then there is a possibility that fusion could
also occur in short bursts, due to bottle-necking effects arising from
the non-linear nature of the concentration-dependent loading of D{+}-ions
under a driving potential. The actual extent to which this might happen
has not been investigated.

Table 1.	Calculated Enhancmenet factors f = R/R[D[2]] for a pair of
		deuterons in various deuterium systems. Eq. (1) and the
		screened Coulomb potential model, Eq. (6) are used.
		Screening in the beta-phase involves exp(-beta*V[0])
		(~ 0.05%), 5%, 40% and 60% delocalized D{+} ions (see
		text) for k[sc] = 0.85, 2.93, 8, and 10 /a.u. respectively.

		--------------------------------------------------

		 System			k[sc]		f(300K)

		--------------------------------------------------

		 D[2]-molecule		1.87		1

		--------------------------------------------------

		 PdD[x], x -> 0		2.68		10**14

		 beta-phase, x = 1	0.85		10**-38

					2.93		10**18

					8.00		10**46

					10.0		10**51

		--------------------------------------------------

		 D[2]{+}-mu molecule	22.00		10**64

		--------------------------------------------------

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