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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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