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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.

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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

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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

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resulting effective Coulomb interaction potential

 

0 0

V r c

between a proton and a neighboring ion at a

distance r can be written, according to [4] as

   

0

0 0

2

0 2

1 2

2 1

2 2

0

0

0

:

2

exp

2

:

r

c

e

r

Ze r

V r dx x

r e

r

  

   

  

       

, (1)

where Z is the atomic number of the ion, e is the electron charge,

 

1 2

0 0   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]:

1

3 2

2

0 0

0

0 0 0

1 1 2

2 exp

2

nucl

nucl

r R

T E

 



                

           

(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 ),

nucl r

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

nucl r =

5

3 10

Å , 0

= 0.1 Å (corresponding to

0 = 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

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

0 N t  2

[6]:

   N 0

cosh g N   ,

0

2m0

  , (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

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):