![Schematic structural diagram of the Pd147Hx cluster in the icosahedral, cuboctahedral and liquid phases, after [18]. Inset: heat capacities of three clusters, in units of kB per atom, versus canonical temperature. Icosahedral phase is predicted to be more stable above room temperature.](https://www.researchgate.net/profile/V-Dubinko/publication/308412310/figure/fig3/AS:669700056969245@1536680344956/Schematic-structural-diagram-of-the-Pd147Hx-cluster-in-the-icosahedral-cuboctahedral-and_Q320.jpg)
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1
Catalytic mechanism of LENR in quasicrystals based on localized anharmonic
vibrations and phasons
# Volodymyr Dubinko 1, Denis Laptev 2,Klee Irwin 3,
1 NSC “Kharkov Institute of Physics and Technology”, Ukraine
2 B. Verkin Institute for Low Temperature Physics and Engineering, Ukraine
3Quantum Gravity Research, Los Angeles, USA
E-mail: vdubinko@hotmail.com
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. In the former case, 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 suppression of the
Coulomb barrier due to lattice vibrations.
In this context, we present 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 enhanced drastically 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 open the new ways towards engineering of nuclear active environment based on the QC
catalytic properties.
Keywords: quasicrystals, localized anharmonic vibrations, phasons, low energy nuclear reactions, nuclear active sites.
__________________________________________________________________________________________
Content
1. Introduction ................................................................................................................................................................. 1
2. Schwinger model of LENR in an atomic lattice modified with account of time-periodic driving .............................. 2
3. Tunneling in a periodically-driven double well potential ............................................................................................ 4
4. LAVs and phasons in nanocrystals and quasicrystals ................................................................................................ 11
5. Conclusions and outlook ........................................................................................................................................... 15
1. Introduction
The tunneling through the Coulomb potential barrier during the interaction of charged particles pre-
sents 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 suppres-
sion 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
V. Dubinko, D. Laptev, I. Klee
2
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 am-
plitude (~ 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 quasicrystals (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 quasicrystal 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 nm 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 pat-
terns called phasons, 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 practi-
cal 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 tun-
neling from which is impossible) and obtain 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 enhanced drastically by the driving, and it increases strongly with increasing amplitude of the
driving.
In section 4, we present some examples of dynamical 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-deu-
teron (P-D) interactions 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. It 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
V. Dubinko, D. Laptev, I. Klee
3
resulting effective Coulomb interaction potential
00
c
Vr
between a proton and a neighboring ion at a
distance r can be written, according to [4] as
0
00
2
0
22
11
22
2
000
:
2exp 2
:
r
c
e
rr
Ze
V r dx x
re
r
, (1)
where Z is the atomic number of the ion, e is the electron charge,
12
00
2m
is the ZPV amplitude,
is the Plank constant, m is the proton mass, and
0
is the angular frequency of the harmonic potential.
A typical value of
0
~0.1 Å , which means that the effective repulsion potential is saturated at ~ 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]:
132
2
00
0
000
2
11
2 exp 2
nucl
nucl
rR
TE
(2)
where
nucl
E
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 ),
nucl
r
is the nuclear radius,
0
R
is the
equilibrium distance between the nuclei in the lattice.
For D-D => He4 fusion in PdD lattice, the mass difference
nucl
E
= 23.8 MeV. Assuming
nucl
r
=
5
3 10
Å ,
0
= 0.1 Å (corresponding to
0
= 320 THz) and
0
R
=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
0
R
=2.9 Å. Even assuming a lower value of
0
= 50 THz corresponding to larger
0
= 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. How-
ever, 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
02Nt
[6]:
0cosh
NgN
,
00
2m
, (3)
where
g
<<1 is the amplitude of parametric modulation, which is determined by the amplitude of
LAV. For example,
g
= 0.1 corresponds to the LAV amplitude of ~ 0.3 Å in the PdD lattice with
0
R
=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):
V. Dubinko, D. Laptev, I. Klee
4
132
2
00
02
11
2 exp 2
nucl
Nnucl N N
rR
TE
, (4)
Figure 1. (a) Zero-point vibration amplitude of deuterium ions vs. N in the parametric regime [6] for
different
0
according to eq. (3) at
0.1g
. (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
(
0
R
=0.94 Å) or in two neighboring lattice sites (
0
R
=2.9 Å).
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 more close 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, how-
ever, the mean oscillation energy grows with time as [13]:
22 2 2
00
0 0 0 0
sin2 sin
2 16
x
g
E t t t t
, (5)
where
x
g
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). Ana-
lytical 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 Schrö-
dinger equation for a particle in a double well potential, which is driven time-periodically imitating the
action of a LAV or phason.
3. Tunneling in a periodically-driven double well potential
Consider Schrödinger equation for a wave function
,xt
of a particle with a mass m in the non-
stationary double-well potential
,V x t
:
V. Dubinko, D. Laptev, I. Klee
5
22
2
, , , ,
2
i x t x t V x t x t
t m x
, (6)
242
02
0
,2
at
m
V x t x b t x
x
,
00
xm
(7)
where
at
and
bt
are the dimensionless parameters that determine the form and the driving mode
of the potential shown in Fig.2.
1cos
2
a t t
,
1cos
2
b t t
, (8)
Figure 2. Double-well potential
given by eq. (7) at
= 0.0005,
= 0.0001, which correspond to
the ratio of the potential depth to
ZPV energy given by
15.6
8
is the driving frequency of the eigenfrequencies
eigen
and positions
min
x
of the potential wells in
the vicinity of the minima given by
400
0
2 1 cos 2 1 cos 2 , 1
42
eigen b t t g
, (9)
min
41/4
0040
min 0
0
1 1 1 1
222
cos 2 1 cos 2
01 cos 2 , 1
44
x
xb
xa tt
xtg
x
, (10)
V. Dubinko, D. Laptev, I. Klee
6
From eqs (9), (10) it follows that the driving under consideration
[https://www.dropbox.com/s/eczwm6ny8f0939t/Potential%20driving.gif?dl=0] results in a simultane-
ous time-periodic modulation of the potential well positions and eigenfrequencies with amplitudes
x
g
and
g
, 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 ref. [13].
Initial state of the system is described by a wave function of the Gaussian form placed near the first
energy minimum (Fig. 2a):
2
min 0
02
2
40
0
1
, 0 exp 2
xx
xt x
x
, (11)
The probability distribution of finding the particle at the point x is given by
2
00
, 0 , 0x t x t
, which is shown in Fig. 3b. It can be seen that the probability density is
concentrated at
min 4.73x
, which means the particle spends most of its time at the bottom of the
potential well.
a
b
Figure 3. (a) Initial wave function
0
,0xt
and (b) the probability distribution to find the particle
at the point x:
2
00
, 0 , 0x t x t
in the left potential well shown in Fig. 1.
Figure 4. The probability distribution of the particle at different moments of time
2eigen
t
in
stationary potential wells:
= 0.0005;
= 0.
V. Dubinko, D. Laptev, I. Klee
7
Stationary well
Driving with
= 2
eigen
Driving with
=
eigen
Figure 5. The probability distribution of the particle at different moments of time
2eigen
t
in sta-
tionary potential wells (
= 0.0005;
= 0) and under the potential driving (
= 0.0005;
= 0.0001)
corresponding to
2 0.1g
;
4 0.05
x
g
. The driving frequency
is indicated in
the figure.
At the selected parameters, the potential depth to ZPV energy ratio is given by
1 8 5.6
, which
is a typical ratio for solid state chemical reactions. It 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 proba-
bility distribution of the particle at different moments of time
2eigen
t
, measured in the oscillator
periods. For example, t = 1000 corresponds to 1000 ‘attempts’ to escape from the left well. However,
V. Dubinko, D. Laptev, I. Klee
8
one can see that the probability to find the particle in the right well is still negligibly small. Only at t =
10000, it becomes higher than the probability to find the particle in the left well.
The situation become dramatically different in the case of time-periodically driven wells
[https://www.dropbox.com/s/eczwm6ny8f0939t/Potential%20driving.gif?dl=0], as demonstrated in Fig.
5 for the two driving frequencies
=
eigen
; 2
eigen
. In both cases, already at t = 100, the probability
to find the particle in the right well becomes comparable with the probability to find the particle in the
left well. This means that the mean escape (tunneling) time has decreased by ~ two orders of magnitude
due to the driving with a comparatively small driving amplitude
g
= 2
x
g
= 0.1 <<1.
Fig. 6 demonstrates dependence of the driving effect on the driving frequency, which is different from
that obtained for a harmonic oscillator [13], where two sharp peaks were observed at resonant frequen-
cies
=
eigen
and
=2
eigen
. 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
eigen
and 2
eigen
.
Finally, dependence of the tunneling time on the driving amplitude is shown in Fig. 7. It appears that
increasing the amplitude by a factor of 2 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 quasicrys-
tals.
Non-resonant driving with various frequencies
eigen
t=10
t = 50
t = 100
0.5
0.8
V. Dubinko, D. Laptev, I. Klee
9
1.2
1.5
1.8
2.1
V. Dubinko, D. Laptev, I. Klee
10
2.2
2.5
Figure 6. The probability distribution of the particle at different moments of time under the potential
driving (
= 0.0005;
= 0.0001) corresponding to
2 0.1g
;
4 0.05
x
g
. The
driving frequency
eigen
= 0.5; 0.8; 1.2; 1.5; 1.8; 2.1; 2.2; 2.5 is indicated in the figure.
Effect of the driving strength/amplitude, g
g
,
x
g
t=10
t = 50
t = 100
g
=2
x
g
=0.05
V. Dubinko, D. Laptev, I. Klee
11
g
=2
x
g
=0.1
g
=2
x
g
=0.2
Figure 7. The probability distribution of the particle at different moments of time under the potential
driving at
=
eigen
,
= 0.0005;
= 0.00005÷0.0002, corresponding to different driving amplitudes
g
,
x
g
as indicated in the figure.
4. LAVs and phasons in nanocrystals and quasicrystals
4.1 LAVs 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 Alexander Ovchinnikov who discovered that localized long-lived mo-
lecular 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 intramo-
lecular vibrations. Two coupled anharmonic oscillators described by a simple set of dynamic equations
demonstrate this idea:
23
1 0 1 1 2
23
2 0 2 2 1
x x x x
x x x x
, (12)
where x1 and x2 are the coordinates of the first and second oscillator,
0
are their zero-point vibrational
frequencies,
is a small parameter, and
and
are parameters characterizing the anharmonicity and
the coupling force of the two oscillators, respectively. If one oscillator is displaced from the equilibrium
and start oscillating with an initial amplitude, A, then the time needed for its energy to transfer to another
oscillator is given by the integral:
2
02
22
01 4 sin
d
T
A
,
3
, (13)
V. Dubinko, D. Laptev, I. Klee
12
from which it follows that the full exchange of energy between the two oscillators is possible only at
sufficiently small initial amplitude:
24A
< 1. In the opposite case,
24A
> 1, the energy of the first
oscillator will always be larger than that of the second one. And for sufficiently large initial amplitude,
4A
, 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 fur-
ther 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 striking 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 demonstrated both experimentally by Pundt et al [17], and by means of atomistic simula-
tions 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 (Fig. 8). And icosahedral structure
is one of the forms quasicrystals take, therefore one is tempted to explore further the link between
nanoclusters and quasicrystals.
Figure 8. Schematic structural diagram of the
Pd147Hx cluster in the icosahedral, cuboctahe-
dral and liquid phases, after [18]. Inset: heat
capacities of three clusters, in units of kB per
atom, versus canonical temperature. Icosahe-
dral phase is predicted to be more stable above
room temperature.
Fig. 9 shows the structure of Pd147 H138 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 presence of H-H-H chains aligned along the I-axis of the cluster. This ab initio
simulation points out at the possibility of excitation of LAVs in these chains, with a central atom per-
forming 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.
V. Dubinko, D. Laptev, I. Klee
13
a
b
Figure 9. (a) 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]; (b) H-H-H chains in the
nanocluster, which are viable sites for LAV excitation [19].
In the following section, we will consider phasons observed in a decagonal quasicrystal Al72Ni20Co8
[11] and a possible link between LAVs and phasons.
4.2 LAVs vs. phasons in quasicrystals
Abe et [11] has measured by means of high resolution scanning transmission microscope (STEM)
temperature 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 vibra-
tions 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 (Fig. 10). 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 K and 1100 K has shown
that the maximum LAV amplitude at 1100K = 0.018 nm (Fig. 11a). 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. 11b) 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.
V. Dubinko, D. Laptev, I. Klee
14
Figure 10. STEM images of LAVs of the decagonal Al72Ni20Co8 at (a) 300 K and (b) 1100 K, according
to Abe et al[11]. Connecting the center of the 2 nm decagonal clusters (red) reveals significant temper-
ature-dependent contrast changes, a pentagonal quasiperiodic lattice (yellow) with an edge length of 2
nm can be seen in (b).
a
Figure 11. (a) LAV amplitude dependence on temperature in Al72Ni20Co8, fitted by two points at 300 K
and 1100 K, according to Abe et al [11]. The maximum LAV amplitude at 1100K = 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, D. Laptev, I. Klee
15
5. Conclusions and outlook
In the present paper, we presented 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 a
phason on the reaction 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 increas-
ing number of driving periods. This effect is novel, since it differs qualitatively from a well-studied
effect of resonance tunneling [20-24], a.k.a. Euclidean resonance (an easy penetration through a classical
nonstationary barrier due to an under-barrier interference) [20-23]. 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 nonstationary field. Therefore, it requires a very specific parametrization of the tunneling condi-
tions. 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 energy [6, 13], which in its turn
increases the tunneling rate. It 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 larger than that 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. Tunneling rate through a fluctuating
barrier in the presence of a periodically driving field has been shown to decrease with increasing
fluctuation strength [24]. One may expect similar effects due to fluctuations in the cases of LAV
and phason driven tunneling, which requires further investigations.
In conclusion, the present results support the concept of nuclear catalysis in QCs that can take place
at special sites provided by their inherent topology, which makes QCs a promising nuclear active envi-
ronment.
Acknowledgements
The authors would like to thank Dmitry Terentyev for designing Fig. 8 and Dan Woolridge – LAV
animation [19]. VD and DL gratefully acknowledge financial support from Quantum Gravity Re-
search.
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