J. Condensed Matter Nucl. Sci. 24 (2017) 1–12
Research Article
Catalytic Mechanism of LENR in Quasicrystals based on Localized
Anharmonic Vibrations and Phasons
V. Dubinko∗
NSC “Kharkov Institute of Physics and Technology,” Kharkov, Ukraine
D. Laptev
B. Verkin Institute for Low Temperature Physics and Engineering, Kharkov, Ukraine
K. Irwin
Quantum Gravity Research, Los Angeles, USA
Abstract
Quasicrystals (QCs) are a novel form of matter, which are neither crystalline nor amorphous. Among many surprising properties
of QCs is their high catalytic activity. We propose a mechanism explaining this peculiarity based on unusual dynamics of atoms
at special sites in QCs, namely, localized anharmonic vibrations (LAVs) and phasons. With the vibrations, one deals with a large
amplitude (∼ fractions of an angstrom) time-periodic oscillations of a small group of atoms around their stable positions in the
lattice, known also as discrete breathers, which can be excited in regular crystals as well as in QCs. On the other hand, phasons are
a specific property of QCs, which are represented by very large amplitude (∼ angstrom) oscillations of atoms between two quasi-
stable positions determined by the geometry of a QC. Large amplitude atomic motion in LAVs and phasons results in time-periodic
driving of adjacent potential wells occupied by hydrogen ions (protons or deuterons) in case of hydrogenated QCs. This driving may
result in the increase of amplitude and energy of zero-point vibrations (ZPV). Based on that, we demonstrate a drastic increase of
the D–D or D–H fusion rate with increasing number of modulation periods evaluated in the framework of Schwinger model, which
takes into account the suppression of the Coulomb barrier due to lattice vibrations. In this context, we present a numerical solution of
Schrodinger equation for a particle in a non-stationary double well potential, which is driven time-periodically imitating the action
of a LAV or phason. We show that the rate of tunneling of the particle through the potential barrier separating the wells is drastically
enhanced by the driving, and it increases strongly with increasing amplitude of the driving. These results support the concept of
nuclear catalysis in QCs that can take place at special sites provided by their inherent topology. Experimental verification of this
hypothesis can lead to new ways of engineering materials containing nuclear active environments based on QC catalytic properties.
c⃝ 2017 ISCMNS. All rights reserved. ISSN 2227-3123
Keywords: Quasicrystals, Localized anharmonic vibrations, Low energy nuclear reactions, Nuclear active sites, Phasons, Tunneling
c⃝ 2017 ISCMNS. All rights reserved. ISSN 2227-3123
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V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
1. Introduction
The tunneling through the Coulomb potential barrier during the interaction of charged particles presents a major
problem for the explanation of low energy nuclear reactions (LENR) observed in solids [1–3]. Corrections to the
cross section of the fusion due to the screening effect of atomic electrons result in the so-called “screening potential,”
which is far too weak to explain LENR observed at temperatures below the melting point of solids. Nobel laureate
Julian Schwinger proposed that a substantial suppression of the Coulomb barrier may be possible at the expense of
lattice vibrations [4,5]. The fusion rate of deuteron–deuteron or proton–deuteron oscillating in adjacent lattice sites
of a metal hydride, according to the Schwinger model, is about 10−30 s−1 [6], which is huge as compared to the
conventional evaluation by the Gamov tunnel factor (∼10−2760). However, even this is too low to explain the observed
excess heat generated, e.g. in Pd cathode under D2O electrolysis. The fusion rate by Schwinger is extremely sensitive
to the amplitude of zero-point vibrations (ZPV) of the interacting ions, which has been shown to increase under the
action of time-periodic driving of the harmonic potential well width [6]. Such a driving can be realized in the vicinity
of localized anharmonic vibrations (LAVs) defined as large amplitude (∼ fractions of an angstrom) time-periodic
vibrations of a small group of atoms around their stable positions in the lattice. A sub-class of LAV, known as discrete
breathers, can be excited in regular crystals by heating [1–3,7] or irradiation by fast particles [8]. Based on that, a
drastic increase of the D–D or D–H fusion rate with increasing number of driving periods has been demonstrated in
the framework of the modified Schwinger model [6,8].
One of the most important practical recommendations of the new LENR concept is to look for the nuclear active
environment (NAE), which is enriched with nuclear active sites, such as the LAV sites. In this context, a striking site
selectiveness of LAV formation in disordered structures [9] allows one to suggest that their concentration in quasicrys-
tals (QCs) may be very high as compared to regular crystals where discrete breathers arise homogeneously, and their
activation energy is relatively high. Direct experimental observations [10] have shown that in the decagonal quasicrys-
tal Al72Ni20Co8, mean-square thermal vibration amplitude of the atoms at special sites substantially exceeds the mean
value, and the difference increases with temperature. This might be the first experimental observation of LAV, which
has shown that they are arranged in just a few nanometers from each other, so that their average concentration was
about 1020 per cubic cm that is many orders of magnitude higher than one could expect to find in periodic crystals
[1–3,7]. Therefore, in this case, one deals with a kind of ‘organized disorder’ that stimulates formation of LAV, which
may explain a strong catalytic activity of quasicrystals [11].
In addition to the enhanced susceptibility to the LAV generation, QCs exhibit unique dynamic patterns called pha-
sons, which are represented by very large amplitude (∼ angstrom) quasi time-periodic oscillations of atoms between
two quasi-stable positions determined by the geometry of a QC. It is natural to expect that the driving effect of phasons
can exceed that of LAVs due to the larger oscillation amplitude in phasons. The main goal of the present paper is
to develop this concept to the level of a quantitative comparison between the driving/catalytic action of LAVs and
phasons, which could be used to suggest some practical ways of catalyzing LENR.
The paper is organized as follows. In the next section, the Schwinger model [4,5] and its extension [6] are shortly
reviewed to demonstrate an importance of time-periodic driving of potential wells in the LENR triggering.
In Section 3, we extend our analysis beyond the model case of infinite harmonic potential (the tunneling from
which is impossible) and obtain numerical solution of Schrödinger equation for a particle in a non-stationary double
well potential, which is driven time-periodically imitating the action of a LAV or phason. We show that the rate of
∗E-mail: xxx
V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
3
tunneling of the particle through the potential barrier separating the wells is enhanced drastically by the driving, and
it increases strongly with increasing amplitude of the driving. In Section 4, we present some examples of dynamic
patterns in QCs and their clusters and discuss the ways of experimental verification of the proposed concept. The
summary and outlook is given in Section 5.
2. Schwinger Model of LENR in an Atomic Lattice Modified with Account of Time-periodic Driving
According to Schwinger [4], the effective potential of the deuteron–deuteron (D–D) or proton–deuteron (P–D) inter-
actions is modified due to averaging 0 ⟨ ⟩0 related to their zero-point vibrations (ZPV) in adjacent harmonic potential
wells, where 0 ⟨ ⟩0 symbolizes the phonon vacuum state. This means that nuclei in the lattice act not like point-like
charges, but rather (similar to electrons) they are “smeared out” due to quantum oscillations in the harmonic potential
wells near the equilibrium positions. The resulting effective Coulomb interaction potential0 ⟨Vc (r)⟩0between a proton
and a neighboring ion at a distance r can be written, according to [4] as
0 ⟨Vc (r)⟩0 =
Ze2
r
√
2
π
r/Λ0
∫
0
dx exp
(
−12x
2
)
≈
{
r ≫ Λ0 : e
2
r
r ≪ Λ0 :
(
2
π
)1/2 e2
Λ0
,
(1)
where Z is the atomic number of the ion, e is the electron charge, Λ0 = (}/2mω0)1/2 is the ZPV amplitude, } is
the Plank constant, m is the proton mass, and ω0 is the angular frequency of the harmonic potential. A typical value
of Λ0∼0.1 Å, which means that the effective repulsion potential is saturated at approximately several hundred eV as
compared to several hundred keV for the unscreened Coulomb interaction. Schwinger estimated the rate of fusion as
the rate of transition out of the phonon vacuum state, which is reciprocal of the mean lifetime T0 of the vacuum state,
which can be expressed via the main nuclear and atomic parameters of the system [5,6]:
1
T 0
≈ 2πω0
(
2π}ω0
Enucl
)1/2 (
rnucl
Λ0
)3
exp
[
−1
2
(
R0
Λ0
)2]
,
(2)
where Enucl is the nuclear energy released in the fusion, which is transferred to the lattice producing phonons (that
explains the absence of harmful radiation in LENR), rnucl is the nuclear radius, R0 is the equilibrium distance between
the nuclei in the lattice.
For D–D ⇒ He4 fusion in PdD lattice, the mass difference Enucl = 23.8 MeV. Assuming rnucl = 3 × 10−5Å,
Λ0 = 0.1 Å (corresponding to ω0 = 320 THz) and R0 = 0.94 Å as the equilibrium spacing of two deuterons placed in
one site in a hypothetical PdD2 lattice, Schwinger estimated the fusion rate to be ∼10−19 s−1 [5]. For a more realistic
situation, with two deuterons in two adjacent sites of the PdD lattice, one has R0 = 2.9 Å. Even assuming a lower
value of ω0 = 50 THz corresponding to larger Λ0 = 0.25 Å [6], Eq. (2) will results in the fusion rate of ∼10−30 s−1,
which is too low to explain the observed excess heat generated in Pd cathode under D2O electrolysis.
The above estimate is valid for the fusion rate between D–D or D–H ions in regular lattice sites. However, the
ZPV amplitude can be increased locally under time-periodic modulation of the potential well width (that determines
its eigenfrequency) at a frequency that exceeds the eigenfrequency by a factor of ∼2 (the parametric regime). Such
regime can be realized for a hydrogen or deuterium atom in metal hydrides/deuterides, such as NiH or PdD, in the
vicinity of LAV [2,3]. Under parametric modulation, ZPV amplitude increases exponentially fast (Fig. 1a) with
increasing number of oscillation periods N = ω0t/2π [6]:
ΛN = Λ0
√
cosh (gωπN), Λ0 =
√
}
2mω0
,
(3)
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V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
where gω ≪ 1 is the amplitude of parametric modulation, which is determined by the amplitude of LAV. For example,
gω = 0.1 corresponds to the LAV amplitude of ∼0.3 Å in the PdD lattice with R0 = 2.9 Å, which is confirmed
by molecular dynamic simulations of gap discrete breathers in NaCl type crystals [7]. Substituting Eq. (3) into the
Schwinger Eq. (2) one obtains a drastic enhancement of the fusion rate with increasing number of oscillation periods
N (Fig. 1b):
1
T N
≈ 2πω0
(
2π}ω0
Enucl
)1/2 (
rnucl
ΛN
)3
exp
[
−1
2
(
R0
ΛN
)2]
,
(4)
The parametric driving considered above requires rather special conditions similar to those in gap breathers in
diatomic crystals [7], while in many other systems, e.g. in metals [12], oscillations of atoms in a discrete breather have
different amplitudes but the same frequency. This case is closer to the driving of the potential well positions with the
frequency equal to the potential eigenfrequency. Such driving does not increase the ZPV amplitude since the wave
packet dispersion remains constant, however, the mean oscillation energy grows with time as [13]:
⟨E⟩ = }ω0
2
+
g2x}ω0
16
[
ω20t
2 + ω0t sin 2ω0t+ sin
2 ω0t
]
,
(5)
where gx is the relative amplitude of the position driving. Accordingly, one could expect an acceleration of the escape
from a potential well of a finite depth similar to the parametric driving.
In reality, one is interested in the effect of potential well driving on the tunneling through the barrier of finite
height between the wells as a function of the driving frequency and strength (amplitude). An analytical solution of
the non-stationary Schrödinger equation even for the simplest case of a double well potential cannot be obtained. In
the following section, we will analyze a numerical solution of Schrödinger equation for a particle in a double well
potential, which is driven time-periodically imitating the action of a LAV or phason.
Figure 1.
(a) Zero-point vibration amplitude of deuterium ions vs. N in the parametric regime [6] for different ω0 according to Eq. (3) at g = 0.1.
(b) D–D fusion rate D–D ⇒ He4 + 23.8 MeV in PdD lattice according to Eq. (4) for deuterium ions in PdD lattice oscillating near equilibrium
positions in one site (R0 = 0.94 Å) or in two neighboring lattice sites (R0 = 2.9 Å).
V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
5
3. Tunneling in a Periodically Driven Double Well Potential
Consider the Schrödinger equation for a wave function ψ (x, t)of a particle with a massmin the non-stationary double-
well potential V (x, t):
i}
∂
∂t
ψ (x, t) = − }
2
2m
∂2
∂x2
ψ (x, t) + V (x, t)ψ (x, t) ,
(6)
V (x, t) =
mω20
2
[
a (t)
x20
x4 − b (t)x2
]
,
x0 =
√
}
mω0
,
(7)
where a (t) and b (t) are the dimensionless parameters that determine the form and the driving mode of the potential
shown in Fig. 2.
a (t) =
1
2
√
α
[α− β cos (Ωt)] ,
b (t) =
1
2
√
α
√
α− β cos (Ωt),
(8)
Ω is the driving frequency of the eigenfrequencies ωeigen and positions xmin of the potential wells in the vicinity of the
minima given by
ωeigen
ω0
=
√
2b =
4
√
1− β
α
cos (2ω0t) ≈
[
1− β
4α
cos (2ω0t)
]
, gω =
β
2α
≪ 1,
(9)
xmin
x0
= ±
√
b
2a = ±
1√
2
1
4
√
α−β cos(2ω0t)
= 1√
2
1
α1/4 4
√
1− βα cos(2ω0t)
≈ xmin(0)
x0
(
1 + β4α cos (2ω0t)
)
, gx ≡ β4α ≪ 1
,
(10)
Figure 2. Double-well potential given by Eq. (7) at α = 0.0005, β = 0.0001, which corresponds to the ratio of the potential depth to ZPV
energy given by 1/8
√
α ≈ 5.6.
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V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
Figure 3.
(a) Initial wave function ψ (x, t0 = 0) and (b) the probability distribution to find the particle at the point x : ρ (x, t0 = 0) =
|ψ (x, t0 = 0)|2 in the left potential well shown in Fig. 1.
From Eqs. (9) and (10) it follows that the driving under consideration results in a simultaneous time-periodic
modulation of the potential well positions and eigen frequencies with amplitudes gx and gω , respectively. Therefore,
we are dealing here with a synergetic effect of the two mechanisms considered separately for a harmonic oscillator in
the previous section and in [13].
Initial state of the system is described by a wave function of the Gaussian form placed near the first energy minimum
(Fig. 2a):
ψ (x, t0 = 0) =
1
4
√
πx20
exp
(
−
(x− xmin)20
2x20
)
.
(11)
The probability distribution of finding the particle at the point x is given by ρ (x, t0 = 0) = |ψ (x, t0 = 0)|2, which is
shown in Fig. 3b. It can be seen that the probability density is concentrated at xmin ≈ 4.73, which means the particle
spends most of its time at the bottom of the potential well.
At the selected parameters, the potential depth to ZPV energy ratio is given by 1/8
√
α ≈ 5.6, which is a typical
ratio for solid state chemical reactions. This means that the particle energy is 5.6 times lower than the energy required
to ‘jump’ over the barrier into another well. The mean time of tunneling through the barrier from a stationary potential
well is very large, as can be seen from Fig. 4 showing the probability distribution of the particle at different moments
of time t = 2π/ωeigen, measured in the oscillator periods. For example, t = 1000 corresponds to 1000 “attempts”
to escape from the left well. However, one can see that the probability to find the particle in the right well is still
negligibly small. Only at t = 10 000, does it become higher than the probability to find the particle in the left well.
The situation becomes dramatically different in the case of time-periodically driven wells, as demonstrated in Fig. 5
for the two driving frequencies Ω = ωeigen; 2ωeigen. In both cases, already at t = 100, the probability of finding the
particle in the right well becomes comparable with the probability of finding the particle in the left well. This means
that the mean escape (tunneling) time has decreased by∼2 orders of magnitude due to the driving with a comparatively
small driving amplitude gω = 2gx = 0.1 ≪ 1.
V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
7
Figure 4. The probability distribution of the particle at different moments of time t = 2π/ωeigen in stationary potential wells: α = 0.0005;
β = 0.
The driving frequency effect is different from that obtained for a harmonic oscillator [13], where two sharp peaks
were observed at resonant frequencies Ω = ωeigen and Ω =2ωeigen. Due to a simultaneous time-periodic modulation of
the potential well positions and eigenfrequencies, the accelerating effect of driving depends non-monotonously on the
driving frequency with a several maximums lying between ωeigen and 2ωeigen.
Stationary well
Driving with Ω = 2π/ωeigen
Driving with Ω = ωeigen
Figure 5. The probability distribution of the particle at different moments of time t = 2π/ωeigen in stationary potential wells (α = 0.0005;
β = 0) and under the potential driving (α = 0.0005; β = 0.0001) corresponding to gω = β/2α = 0.1; gx = β4α = 0.05. The driving
frequency Ω is indicated in the figure.
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V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
gω, gx
t = 10
t = 50
t = 100
gω = 2, gx = 0.05
gω = 2, gx = 0.1
Figure 6. The probability distribution of the particle at different moments of time under the potential driving at Ω = ωeigen, α = 0.0005;
β = 0.00005÷ 0.0002, corresponding to different driving amplitudes gω , gx as indicated in the figure.
Finally, dependence of the tunneling time on the driving amplitude is shown in Fig. 6. It appears that increasing
the amplitude by a factor of two results in decreasing the mean tunneling time by an order of magnitude. This example
demonstrates the importance of the time-periodic driving of the potential wells in the vicinity of LAVs and phasons in
the reactions involving quantum tunneling.
In the following section, we consider some characteristic examples of LAVs and phasons in quasicrystals.
4. LAVs and Phasons in Nanocrystals and Quasicrystals
The fact that the energy localization manifested by LAV does not require long-range order was first realized as early
as in 1969 by Ovchinnikov, who discovered that localized long-lived molecular vibrational states may exist already
in simple molecular crystals (H2, 02, N2, NO, CO) [14]. He realized also that stabilization of such excitations was
connected with the anharmonicity of the intramolecular vibrations. Two coupled anharmonic oscillators described by
a simple set of dynamic equations demonstrate this idea:
ẍ1 + ω
2
0x1 + ελx
3
1 = εβx2,
ẍ2 + ω
2
0x2 + ελx
3
2 = εβx1,
(12)
where x1 and x2 are the coordinates of the first and second oscillator, ω0 are their zero-point vibrational frequencies,
ε is a small parameter, and λ and β are parameters characterizing the anharmonicity and the coupling force of the
two oscillators, respectively. If one oscillator is displaced from the equilibrium and starts oscillating with an initial
V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
9
amplitude, A, then the time needed for its energy to transfer to another oscillator is given by the integral:
T =
ω0
εβ
π/2
∫
0
dφ
√
1−
(
A2γ
/
4
)2
sin2 φ
,
γ =
3λ
β
,
(13)
from which it follows that the full exchange of energy between the two oscillators is possible only at sufficiently small
initial amplitude: A2γ/4 < 1. In the opposite case, A2γ/4 > 1, the energy of the first oscillator will always be larger
than that of the second one. And for sufficiently large initial amplitude, A ≫
√
4/γ, there will be practically no
sharing of energy, which will be localized exclusively on the first oscillator.
Thus, Ovchinnikov has proposed the idea of LAV for molecular crystals, which was developed further for any
nonlinear systems possessing translational symmetry; in the latter case, LAVs have been named discrete breathers
(DBs) or intrinsic localized modes (ILMs). Now, we are coming back to the idea of LAV arising at “active sites”
in defected crystals, quasicrystals and nanoclusters. As noted by Storms, “Cracks and small particles are the Yin
and Yang of the cold fusion environment”. A physical reason behind this phenomenology is that in topologically
disordered systems, sites are not equivalent and band-edge phonon modes are intrinsically localized in space. Hence,
different families of LAV may exist, localized at different sites and approaching different edge normal modes for
vanishing amplitudes [9]. Thus, in contrast to perfect crystals, which produce DBs homogeneously, there is a site
selectiveness of energy localization in the presence of spatial disorder, which has been demonstrated by means of
atomistic simulations in biopolymers [9], metal nanoparticles [15] and, experimentally, in a decagonal quasicrystal
Al72Ni20Co8 [11].
The crystal shape of the nanoparticles (cuboctahedral or icosahedral) is known to affect their catalytic strength
[16], and the possibility to control the shape of the nanoparticles using the amount of hydrogen gas has been demon-
strated both experimentally by Pundt et al. [17], and by means of atomistic simulations by Calvo et al. [18]. They
demonstrated that above room temperature the icosahedral phase should remain stable due to its higher entropy with
respect to cuboctahedron. And icosahedral structure is one of the forms quasicrystals take, therefore one is tempted to
explore further the link between nanoclusters and quasicrystals.
Figure 7 shows the structure of Pd147H138 cluster containing 147 Pd and 138 H atoms having minimum free energy
configuration, replicated using the method and parameters by Calvo et al. [18]. In particular, Fig. 9(b) reveals the
Figure 7.
(Left) Structure of PdH cluster containing 147 Pd and 138 H atoms having minimum free energy configuration, replicated using the
method and parameters by Calvo et al. [18]; (Right) H–H–H chains in the nanocluster, which are viable sites for LAV excitation [19].
10
V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
presence of H–H–H chains aligned along theI-axis of the cluster. This ab initio simulation points out at the possibility
of excitation of LAVs in these chains, with a central atom performing large-amplitude anharmonic oscillations and its
neighbors oscillating in quasi-harmonic regime [19], which is similar to that considered in [7] for regular diatomic
lattice of NaCl type. Such oscillations have been argued to facilitate LENR [2,3], and in the present paper we develop
this concept further.
Let us consider phasons observed in a decagonal quasicrystal Al72Ni20Co8 [11] and a possible link between LAVs
and phasons.
Abe et al. [11] have measured by means of high resolution scanning transmission microscope (STEM) temper-
ature dependence of the so-called Debye–Waller (DW) factor in decagonal quasicrystal Al72Ni20Co8. DW factor is
determined by the mean-square vibration amplitude of the atoms. The vibrations can be of thermal or quantum nature
depending on the temperature. The authors demonstrated that the anharmonic contribution to Debye–Waller factor
increased with temperature much stronger than the harmonic (phonon) one. This was the first direct observation of a
“local thermal vibration anomaly”, i.e. LAVs, in our terms. The experimentally measured separation between LAVs
was about 2 nm, which meant that their mean concentration was about 1020 per cm3 that is many orders of magnitude
higher than one could expect to find in periodic crystals [7].
The LAV amplitude dependence on temperature fitted by two points at 300 and 1100 K has shown that the max-
imum LAV amplitude at 1100 K = 0.018 nm (Fig. 8a). What is more, it appears that LAVs give rise to phasons
at T >990 K, where a phase transition occurs, and additional quasi-stable sites β arise near the sites α. The phason
amplitude of 0.095 nm (Fig. 8b) is an order of magnitude larger than that of LAVs. Thus, on the one hand, the driving
amplitude induced by phasons is larger than that by LAVs, but on the other hand, phason oscillations may be less time-
periodic (more stochastic), which requires more detailed investigations of the driving stochasticity effect on tunneling,
as discussed in the following section.
5. Discussion and Outlook
In the present paper, we presented a numerical solution to the Schrödinger equation for a particle in a non-stationary
double well potential, which is driven time-periodically imitating the action of a LAV or a phason on the reaction
(a)
(b)
Figure 8.
(a) LAV amplitude dependence on temperature in Al72Ni20Co8, fitted by two points at 300 and 1100 K, according to Abe et al. [11].
The maximum LAV amplitude at 1100 K is 0.018 nm. (b) LAVs give rise to phasons at T > 990 K, where a phase transition occurs, and additional
quasi-stable sites β arise near the sites α. The phason amplitude of 0.095 nm is an order of magnitude larger than that of LAVs.
V. Dubinko et al. / Journal of Condensed Matter Nuclear Science 24 (2017) 1–12
11
cite in their vicinity. We have shown that the rate of tunneling of the particle through the potential barrier separating
the wells can be enhanced by orders of magnitude with increasing number of driving periods. This effect is novel,
since it differs qualitatively from a well-studied effect of resonance tunneling [20–22], a.k.a. Euclidean resonance (an
easy penetration through a classical nonstationary barrier due to an under-barrier interference). In the latter case, the
tunneling rate has a sharp peak as a function of the particle energy when it is close to the certain resonant value defined
by the non-stationary field. Therefore, it requires a very specific parametrization of the tunneling conditions. In contrast
to that, the time-periodic driving of the potential wells considered above, results, first of all, in a sharp and continuous
(not quantum) increase of the ZPV amplitude and energy [6,13], which in its turn increases the tunneling rate. This
result is closely related to the correlation effects, proposed by Dodonov et al. [23] and analyzed in details by Vysotskii
et al. [24] who predicted a giant increase of sub-barrier transparency (up to hundreds of orders of magnitude) during
the increase of the so-called correlation coefficient at special periodic action on a quantum system. This prediction
was based on the numerical calculation of the time dependence of the correlation coefficient r (t) for the case of a
periodically driven harmonic potential, in which case it was shown that
|r (t)| −→
t→∞
1
at any initial energy of the oscillator, which was argued to result in a tunneling through the potential barrier of any
height after sufficient number of driving periods. However, as shown more recently by Dubinko and Laptev [6], the
oscillator energy also increases exponentially with time in the parametric regime considered in [24], which poses a
limit to the correlation coefficient increase in a double well system (which is more relevant for the tunneling analysis
than the infinite parabolic well considered in [6,24]).
As shown in the present paper, the tunneling probability increases strongly with increasing strength of the driving,
which is related to the amplitude of the non-linear dynamic phenomenon that causes the driving. As we have demon-
strated in the previous section, the driving amplitude induced by phasons may be larger than that induced by LAVs by
an order of magnitude, which implies that phasons may be stronger catalysts than LAVs. However, further research
is needed in order to make more definite conclusions, since the phason dynamics itself is an activated process driven
by thermal or quantum fluctuations. Therefore, phasons can hardly induce a strictly time-periodic driving considered
in the present paper. The tunneling rate through a fluctuating barrier in the presence of a periodically driving field has
been shown to decrease with increasing fluctuation strength [25]. One may expect similar effects due to fluctuations in
the cases of LAV and phason driven tunneling, which requires further investigation.
In conclusion, the present results support the concept of nuclear catalysis in QCs taking place at special sites
caused by their inherent topology. This makes QCs a logical model to explain the structure of the microscopic nuclear
active environments, or hot-spots observed by experimentalists in TEM analysis.
Acknowledgements
The authors would like to thank Dmitry Terentyev for designing Fig. 7 and Dan Woolridge – LAV animation [19]. VD
and DL gratefully acknowledge financial support from Quantum Gravity Research.
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