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Brazilian Journal of Physics
Online version ISSN 16784448
Braz. J. Phys. vol. 27 n. 4 São Paulo Dec. 1997
http://dx.doi.org/10.1590/S010397331997000400014
Catalitically Induced DD Fusion in Ferroelectrics
V.D. Dougar Jabon^{1}, G.V. Fedorovich^{2} and N.V. Samsonenko^{3}
^{1}Escuela de Fisica, Universidad Industrial de Santander,
Bucaramanga, A.A.678, Colombia
^{2}Theoretical Problem Department, Russian Academy of Sciences,
Moscow, 121002, Russia
^{3}Department of Theoretical Physics, Russian Friendship University,
Moscow, 117198, Russia
Received March 15, 1997
A model of deuteron acceleration in ferroelectrical crystals under the process of domain polarization reversal is proposed. Experimental verfication of the model with LiTaO_{3} crystals saturated with deuterium was fulfilled. It was shown that in the 75 kV/cm a.c. field the neutron emission attributed to DD fusion is two order magnitude higher the Jones level.
I. Introduction
There are strong grounds (both theoretical [1,2] and experimental [3]) to believe that the phenomenon of enhancement of the DD fusion rate during electrolytic infusion of deuterons into metallic Ti or Pd electrodes connected with the crack and break formations in the cathode material. Also, a neutron emission from a crushing process of Lithium Niobate crystal in deuterium gas atmosphere was observed [4].
To put this another way, the physical mechanism of cold fusion in electrolytic cells [5,6] is the same as the one in the case of a destruction of deutered crystals [7,8]. A model of that mechanism (the stochastic acceleration of particles in the field of intensive oscillations that are generated at the crack boundary) has been proposed in [8,9]. The analysis of the model allows to make some conclusions which are of interest for following investigations. It has been found that, for deuterium contained crystals, the emissions of high energy electrons and electromagnetic radiation (in the range from visual light to Xrays) are that indicator of creating conditions for enhancement of the nuclear fusion rate. It was noted [9] in connection with this that the phenomenon of light [10] and high energy electron [11,12] emission are involved in the process of polarization reversal of ferroelectrics. The direct evidence are found for the nuclear fusion in deuterated ferroelectrics during changes of their crystalline structure in the process of phase transition [13] and polarization reversal [14,23,24].
The foregoing is a good reason to consider the phenomenon of the stochastic acceleration of dions in the field of dominant mode oscillations, which are generated during the polarization reversal, more closely. This will allow an revealing the parameters of crystals that determine the DD fusion rate. An understanding of the effect will guide the way to improvements in the design of experiments. This is the main purpose of the paper.
II. Model of a domain wall motion
Let us assume that for every polarization of the individual cell we can introduce certain quantitative characteristics, coordinate s, which is defined in such a manner that it is equal zero in a nonpolarized cell and it has positive or negative in variously polarized cells. It will be reasonable to assume that temporal variations of s are described by the Hamiltonian (see also [15])

where M and U(s) are effective mass and potential, g is the forced constant. In order to describe the possibility of the crystal polarization reversal, let us define the doubleminimum potential energy function by the formula: U(s) = (a/2)s^{2}+(b/4)s^{4}. This potential is minimal when s = s_{eq} º ±(a/ b)^{1/2}. One can see that if E = 0 the system is in a symmetric potential well. As the field E increases to the critical value E_{0} = (2a/ 3e)(a/ 3b)^{1/2}, the left hand part of the well disappears. The summary potential transforms into the broad asymmetric well in which the system can oscillate near the right hand equilibrium position.
To get an equation for the dominant mode oscillations, which are generated during the polarization reversal of ferroelectrics, it is convenient to use a continuum approximation. Then by defining k º gd^{2} and introducing the coordinate x along the crystal, we obtain

The eq.(2) is classed as a quasilinear hyperbolic equation. All of the perturbations, described by this equation, extend with the velocity C = (k / M)^{1/2} in both directions of the xaxis. In order to determine the dependence s(x, t) it is necessary to add the corresponding initial and boundary conditions to the eq.(2). Let us suppose that the field E = E_{0} 0 is applied to the crystal. The system of each cell is characterised by the svalue near the left equilibrium (unstable) position. For turning the system out of this state, the disturbing force f @ f_{o}d(x)d(t) is added in the righthand side of the eq.(2). As a result, the breaks of the function s(x,t) will extend along both directions x = ±Ct. The magnitude of s(x,t) varies from s_{0} = (a/ 3b)^{1/2} (before a break) to s_{1} = s_{0} + C f_{0}/2 (immediately behind a break). Let us consider variations of s in the area  Ct < x < +Ct. The boundary conditions

is sufficient for the definition of s(x, t) over the whole area.
The solution of the problems (1)(3) was obtained numerically. It brings us to the description of the domain polarization by the following function
where the function S(n) is shown in Fig. 1a (characteristic points of the diagram are S_{m} @ 1,66, S_{eq} @ 1,16) and P_{s} is the value of the spontaneous polarization of ferroelectrics. The amplitude and the frequency of the oscillations relative to the equilibrium value are

Figure 1. Left side (a) represents the results of numerical solution for U(s).The characteristic points: S_{m} = 1,66 and S_{eq} = 1,15; right side (b) is the illustration of the Dion acceleration area.
III. Fermi acceleration of deuterons
A model of a ferroelectrical crystal consisted of a sequence of layers of electrical dipoles p_{j} (arrows in Fig.1b) which are spaced d apart is studied. The integral index j enumerates layers of dipoles. We suppose that a deuteron moves along the domain axis. The straight line represents the deuteron parth. It crosses sequential layers of dipoles. If the crystal is found between electrodes then the depolarization field E = 4pp/d^{3} exist in the crystal (here p is the average value of cell dipoles p_{j}). The field compensates the summary potential difference on the bounds of the crystal.
The variation of the potential energy along the path of the movement of the Dion is represent in Fig. 1b (here D is the effective size of dipoles). The oscillations of atoms forming dipoles bring to variation of the dipole moments with respect to the average value p. In general, the value of difference p¢_{j} = p_{j}  p can be represented as the sum of the normal waves. However, we assume that the dominating oscillation exists in the crystal (see above). It can be described by the function p¢(t) = p_{0} coswt. It is convenient to consider the limiting case where dipole sizes are sufficiently small (D << d). In this case the Dion trajectory x(t) can be determined from the equation

where M is the deuteron mass. This equation describes the uniformly accelerated movement of the Dion on intervals jd < x < ( j + 1)d and the dampening shocks on the interval boundaries.
Let at t = t_{n} the Dion is in the point x = jd  0, it has the kinetic energy T_{n} > 4pep_{j}/d^{2} its velocity is oriented in the positive direction of the xaxis. At t = t_{n+1}, where t_{n+1} is determined by the relation

the Dion will be in the point x = dj_{n+1}  0 (where j_{n+1} = j_{n} + 1) with the kinetic energy T = T_{n+1} determined by the relation

The set of relations (5),(6) represents a mapping [16] which defines the Dion dynamics with a sufficiently large positive velocity. Similar relations can be determined for the case of a small positive velocity (if T_{n} < 4pep_{j} / d^{2}) and also for the case of a negative velocity. For dimensionless variables W = T / T_{o}, ( T_{o} º 4pe p / d^{2}) and y = wt the set of mappings, which entirely defines the dynamics of the Dion, has the form


If W_{n} < 1+q(y_{n}) then at t = t_{n} +0 the Dion velocity becomes negative. Two possibilities exist in this case:




where

If the value p_{j} depends arbitrarily on t, the mapping (7) has the form of a radial twist mapping. For q µ cos(y) eqs.(7) becomes the standard mapping (also known as ChirikovTaylor mapping). This mapping has been well investigated lately in connection with the problem of the Fermi acceleration (see [1618]). It was shown, in particular, that a stochastic acceleration regime exists if P_{j}(t) is an oscillating function. Then it results that the particle energy W can noticeably exceed the value 1 + Amp{q(y)}.
For W >> 1 stochastic motion of the deuteron (at least strong stochastic motion) disappears. In this case the inequality Amp{q(y)} < 1 seems to be reasonable so that variations of W are small (DW << W) for each steps of mapping. By assuming that Q is small too, we obtain that the phase y of a deuteron movement varies only slightly (Dy << 1).
The fulfilment of these conditions makes possible the transformation of difference equations (7) into differential ones and we can turn to the continuous variations of n. For definiteness we consider particles which move in the positive direction of xaxis. The equations

follows from (7a) after this transformation. The system (8) has the first integral

Let the energy W exceeds 1 at some moment when the phase of its movement is y_{0}. Then the variations of W in the course of moving are described by the implicit function W^{3/2} 1  (W  1)^{3/2} = G[siny_{0}  siny] where G º 3q / 2Q. Since the value of Q is small, the value of G is great.
Under appropriate choice of the initial phase y_{0}, the value of W may be as MaxW = (4G/ 3)^{2} >> 1. It suggests that the effect of the deuteron acceleration in ferroelectrical crystals is quite actual. It is interesting to note that the considered phenomenon is the combination of the two radically different mechanisms of acceleration: when the energy is small (W < 1), the stochastic acceleration takes place, this mechanism is similar to the Fermi acceleration. In the region W > 1 the Dion energy increases (if particles enter into this region with the appropriate phase) at the expense of the resonance acceleration in the field of the dominate mode oscillations.
IV. Energy losses and limiting energy accelerated deutrons
The energy losses of accelerated deuterons are determined by a combination of individual collision processes. The systematic description of the processes is given in [19] (the quantitative data see in [20]). Stopping power is maximal at velocities of approximately V_{0} Z^{2/3}_{A}, where V_{0} is the Bohr velocity and Z_{A} is the charge of the incident particle, corresponding to 25 keV/nucleon. At lower energies the inelastic electronic interactions and elastic recoils can completely dominate the stopping process.
The theoretical treatment in this low velocity region is largely due to Lindhard and his collaborators [21]. They give simple expressions for electronic stopping power S (S º n^{1}(dT/dx), where n is the density of absorber material ) based on the ThomasFermi model of the atom. The stopping power is the function of the velocity of the incident particle A and the absorber material consisting of particles B with charge Z_{B}. For V < V_{0} the expression for S has the form

In this expression c = Z^{1/6}_{A}, Z = (Z_{A}^{2/3}+Z_{B}^{2/3})^{3/2}, a_{0} is the Bohr radius. The molecular effects are small in the total energy loss process.
Using eq.(9), the value of the dimensionless (see Sec.III) energy loss DW at one collision can be estimated by the formula

The accounting of the energy losses brings to the variation of the form of the mapping (7). We have W_{n+1} = W_{n}  q(y_{n})  aW_{n}^{1/2} instead of the first relation of eq.(7a). The other relations of the mapping (7) must be changed similarly. The modified system (8) has the form (for W >> 1 )

The system (11 ) has the first integral

where

The second term in eq.(12) is the exponentially damped contribution of the initial condition.
To make a quantitative estimate of the maximal energy, we use parameters of some ferroelectrics listed in [22,23]. The results are collected in the Table 1.
One can see that some oxide ferroelectrics with the large magnitude of the spontaneous polarization have turned out to be interesting from the viewpoint of DD fusion in cold samples, so that it may be worth devoting some space to them.
V. Experiment on polarization reversal
To check the theoretical prognosis of catalytic enhancement of the DD fusion reactions in ferroelectrics, experiments on warming up of Ddopant subsystem in a field of the dominant mode oscillations are performed. Possible evidence of DD catalysis under the domain polarization reversal of some ferroelectrics in an a.c. field was looked for.
Choice of ferroelectrics
The advantage of this experimental program is available theoretical model of the catalytic induced fusion reactions. The model dictate ways of the expected effect intensification by choosing ferroelectric with the next properties:
(1) large value of spontaneous plarization P_{s},
(2) low energy losses of Dions at movement in a crystal,
(3) coercive force less than the dielectric strength of a crystal,
(4) capability of dissolving of a sufficient number of deuterium.
There are no ferroelectrics which are fit completely all of these requirements.
For LiNbO_{3}, as example, a large value of the spontaneous polarization P_{s} is a characteristic property. This ferroelectric satisfies the condition (2) but not (3). It is common knowledge that the problem (4) is not exist for some ferroelectrics as the deuterium forms part of its chemical composition. An example is KD_{2}PO_{4} which answers the requirement (3) but has moderate spontaneous polarization (P_{s} = 5 mC/cm^{2}) and does not meet the demand (2).
To perform experiments, LiTaO_{3} (LT) and Ba_{0.4}Sr_{0.6}Nb_{2}O_{6} (SNB) of perovskitelike structure were used. This choice is a certain compromise between the requirements (1)(4). The polarization reversal takes place at actual electric fields E_{c} @ 50 kV/cm for LT and at E_{c} = 5 kV/cm for SNB [23]. The additional point to be emphasised is that the both ferroeiectrics dissolve a great number N_{D} of deuterium (N_{D} » 10^{22} cm^{3} for LT, N_{D} » 10^{21} cm^{3} for SNB) without the ferroelectric properties degradation.
Samples of LT cut from a monocrystal were annealed at temperature of 1200^{o}C and were subjected to monodominization process in an electric field. The crystal samples to be used are of two types. Ones have a form of parallelepiped with dimensions of 10 × 10 mm in a plane perpendicular to the ferroelectrical axis and 5 mm along this axis (thick samples). The other ones are discs of 5 mm in diameter and 1 mm of thickness (thin samples). The axis of the discs coincides with the ferroelectrical one. The opposite wide sides of samples were covered with nickel thin films by a method of vacuum deposition in order to use them as electrodes. Samples of SBN crystals were prepared essentially similar to LT ones with the only difference that the thickness of these samples along the ferroelectrical axis was 2 mm.
Experimental device
Experiments were performed on a device which is shown schematically in Fig.2. The main part of the device is a processing ss chamber which is equipped with a pipe for pumping and a pipe for gas injection from a container with deuterium under pressure of 2,5 at. The chamber is evacuated up to pressure of 1.10^{9} at by a turbomolecular pump. A pressure in the chamber is controlled by a manometer. The lower part of the chamber is a quartz cylindrical vessel 39 cm in long and 4 cm in diameter. A crystal samples are placed between two plate nickel electrodes connected through two high voltage leadin with the oscillator generating sinusoidal voltage 50 Hz and regulated up to 10 kV amplitude. The samples are oriented in such a way that the electric field is directed along the ferroelectric axis.. From the outer side of the quartz tube there is an electric heater to raise a sample temperature to 600^{o}C which is controlled by a thermocouple.
Figure 2. Scheme of experimental device.
It seems reasonable to say that the most adequate manner of searching the DD reactions is a record of neutrons. This is at least under assumption of classical channels of DD fusion with a low rate. A block consisting of 10 proportional He^{3} counters surrounded by paraffin moderator is used for neutron registration. In order to diminish the neutron background, the processing chamber is surrounded by a Boron polyethylene wall of 20 cm in thickness. Detector signals were registered by a 1024 channel pulseinheight discriminator and a digital type. The total efficiency on neutron registration is estimated as 3% in accordance with measurements made with a 600 neutron/s Cf^{252} calibrated source. The registration system is equipped with a special earth contour to suppress an electrical interference.
Preparation of crystals
Preparation of selected samples for experiments was consisted in degassing of the samples by a method of annealing at temperature of 400450^{o}C under a vacuum of 10^{6} at. This process was followed by saturation of the samples with deuterium at an initial pressure of 0,61,2 at during time intervals from a few hours to 5 days. A quantitative characteristic of deuterium absorption was reduction of pressure in the chamber [25].
It was found that a LT sample absorbs detachable quantity of deuterium during the first 10 hours leading to pressure drop in the chamber from 0,7 at to 0,5 at. After 30 hours of a saturation procedure the pressure decreased to 0,3 at. By this meant that each lattice of LT crystal absorbed one atom of deuterium. For the SNB samples the limiting density of absorbed deuterium was slightly below, approximately 5.10^{20} cm^{3}, that was reached after the exposure time of 2530 hours.
Experimental results
As mentioned above the experiments was fulfilled with the samples subjected to an alternating electric field. Each experimental cycle was divided into 10 min intervals during which total neutron yield was recorded. The control experiments were consisted in a neutron registration at absence of a sample in the chamber but an electric field between the electrode was left to be switched on. The neutron background was registered during 70 hours.
Based on this experiments, a mean background level M of 12,0 ±3,4 neutrons per 10 minutes interval and the 90% confidence level H and L were found (see. Fig.3). It is necessary to note that during 58 cycles a tendency for sequential weakening of the neutron emission of LT samples was observed. This peculiarity can be attributed to depletion of deuterium in the active zones where the domain walls propagates (see above Sec.II). In this zones an intensive warming of Ddopant sublattice takes place which followed by the diffusion of deuterium atoms into neighbouring regions of crystal as well as releasing them from a sample. So, the diffusion of deuterium is responsible for a reduction of the total neutron emission from samples.
Figure 3. An example of experiments with a thick (a) and a thin (b) samples.
In order to equalise the deuterium density in a crystal body, an additional annealing of the samples in the deuterium atmosphere at the temperature of 450500^{o}C was made. Then the crystal activity partly reestablished. On the other hand after 57 experimental cycles the samples lost their capability to produce excess of neutrons in comparison with the background data.
Typical results on neutron detection taken with the LT samples is shown in Fig.3. Each column shows a number of neutrons detected during the 6.10^{4} polarization reversals. As it is easy see, there is essential difference between the distribution which was obtained for a thick sample (Fig.3a) at the electric field of 25 kV/cm and the thin ones (Fig.3b) at 75 kV/cm. It must be emphasised that the number of neutrons recorded during all operating intervals for case B is above the middle level for the case A although the volume of the thick samples is approximately seven times larger. But it is reasonable to note that the value of electric field was insufficient for the effective domain reversal polarization of the thick ferroelectrics so as the field of 25 kV/cm is less than the coercive force.
The differences between the background level and the average level through exposures are (0,53 ±0,21) neutrons/min for the thick sample and (0,89±0,32) neutrons/min for the thin one. On the assumption that this difference is due to a neutron emission owing to DD fusion reactions, the activity of a LT sample to be used can be estimated as Q = (0,30±0,12) neutrons/s for the thick sample and (0,49±0,17) neutrons/s for the thin one. This gives the fusion rate (6,0 ±2,4)10^{22} s^{1} per deuteron pair for the thick crystal and (7,8±1,4)10^{21} s^{1} for the thin one. The last value is practically two orders of magnitude higher the Jones level. This results permit us to suppose that the process of the reversal polarization of domains results the acceleration of the deuterium nuclei to energy of 200 eV (which corresponds to a temperature of 10^{6} K) and larger in an cold crystal sample.
For SNB samples, any excess of neutron emission above the background level was not found in our experiments. There are some reason which can explain this negative result.
Conclusing remarks
From the experimental results it is possible conclude that the mechanism of Fermi acceleration in the field of the wave generated during the domain polarization reversal of LT crystals can raise the Dnuclei energy up to value enough to realise the fusion reactions in ferroelectrics with the rate which is significantly higher the one determined through the Jones level.
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