Key words: condensed matter, Dislocations of the ions within the metal, Coherence Theory, low energy nuclear reactions (LENR)

Transcription

1 Theoetical Stuy of the Phenomenon of Col Fusion within Conense Matte an Analisys of Phases (,, ) Theoetical Moel Between Low Enegies of Deuteium Nuclei Col Fusion FULVIO FRISONE Depatment of Physics, Univesity of Catania Via Santa Sofia, Catania (Italy) Phone , Fax , fisone@ct.infn.it Abstact: The aim of this wok is to explain the euteon-euteon eactions within pallaium lattice by means of the coheence theoy of nuclea an conense matte. The coheence moel of conense matte affims that within a euteon-loae pallaium lattice thee ae thee iffeent plasmas: electons, ions an euteons plasma. Then, accoing to the loaing pecentage x=d/p, the euteium ions can take place on the octaheical sites o in the tetaheal on the (1,0,0)-plane. Futhe, the pesent wok is concentate on Pallaium because, when subjecte to themoynamic stess, this metal has been seen to give esults which ae inteesting fom both the theoetical an expeimental points of view. Moeove in P lattice we can coelate the euteium loaing with D-P system phases (i.e., an ) by means of theoy of Conense Matte. Futhe, This pape seek to emonstate that, at oom tempeatue, the efomation of the cystalline lattice can influence the pocess of inteaction of euteons intouce within it. Calculations of this pobability, in fact, showe an incease of at least -3 oes of magnitue with espect to the pobability of fusion on the suface of the lattice. These phenomena open the way to the theoetical hypothesis of a kin of chain eaction, as a esult of the euteium loaing an catalyse by mico-cacks fome in the stuctue by mico-explosions, can favou the pocess. In the secon section we will iscuss the poblem of inteaction of Deuteon-Plamon. Key wos: conense matte, Dislocations of the ions within the metal, Coheence Theoy, low enegy nuclea eactions (LENR) I. Intouction The Coheence Theoy of Conense Matte is a geneal theoetical famewok, which is wiely accepte by most scientists woking on col fusion phenomena. Accoing to this coheence theoy of conense matte [5], it is assume that the electomagnetic (e.m.) fiel ue to elementay constituents of matte (i.e. ions an electons) plays a vey impotant ole on system ynamics. Consieing a coupling between e.m. equations ue to chage matte an the Scöinge equation of fiel matte opeato, it is inee possible to emonstate that in poximity of an e.m. fequency 0, the matte system featues a coheent ynamics. Thus it is possible to efine coheence omains, whose length is about CD =/ 0. Obviously, the simplest moel of matte with a coheence omain is a plasma system. In the common plasma theoy, a plasma fequency p must be consiee, as well as the Debye length measuing the Coulomb foce extension, i.e. the coheence omain length. Fo a system with N chage Q of m mass within a V volume, the plasma fequency can be witten as: Q N p (1) m V In this pesent wok, the nuclea envionment has been stuie, which supposely exists within a D -loae pallaium lattice at oom tempeatue, as in accoance with the Coheence Theoy. Taces of nuclea eactions have been obseve in a pallaium lattice when this is loae with euteium gas [1,, 3]. Fo this eason, Low Enegy Reaction Nuclea (LERN) has been efine by many physicists. Moe obust expeiments have shown that in the case of D -loae pallaium the following nuclea eactions ae moe fequent [3, 4]: ISSN: Volume, 017

2 3 D D H p 4. 03MeV (1) 4 D D He 3. 85MeV () In this pesent wok, a coheence moel is also popose, by means of which the occuence of eactions 1 an can be explaine in accoance with moe eliable expeiments, as well as thei pobability. Fistly, an analysis of the envionment has been caie out though the coheence theoy of matte, i.e. of plasmas which ae pesent within pallaium (-electons, s-electons, P-ions an D-ions); then, the potential epote in ef. [6, 7] has been consiee, aing the ole of lattice petubations. Thus, a D-D tunneling pobability has been compute. plasma fequency epens on loaing atio (D/P pecentage), by means of the following fomula [5]: e N x se (6) m V a whee N a 1 Vp V (7) an V p is the volume actually occupie by the P-atom. As epote in efeence [5], 1/ x 15.eV / (8) se As an example, fo x=0.5, se ~10.7 ev/ħ.. Plasmas within non loae pallaium Accoing to the Coheence Theoy of Conense Matte, electon shells ae in a coheent egime within a coheent omain in a P cystal at oom tempeatue. Inee, they oscillate in tune with a coheent e.m. fiel tappe in coheent omains. Fo this eason, plasmas of s- electons an -electons must be taken into account in oe to escibe the lattice envionment..c. Plasmas of P ions Finally, plasmas ceate by Pallaium ions foming the lattice stuctue must be consiee. In this case, fequency can be emonstate as being [5] p 0. 1eV (9).A. Plasmas of -electons They ae fome by electons of pallaium -shell. Computing: e n N (3) m V -electons plasma fequency is obtaine. But accoing to the coheence theoy of matte, this plasma fequency must be ajuste of a facto This coection can be easily unestoo by obseving that fomula (3) is obtaine assuming a unifom -electons chage istibution. But of couse the -electon plasma is localize in a shell of R aius (that is about 1Å), so the geometical contibution is (4) Labeling a enomalize -electon plasma fequency with e [ 5], e 41.5eV / (5) an the maximum oscillation amplitue is about 0.5 Å..B. Plasmas of elocalize s-electons The s-electons ae those neutalizing the asobe euteons ions in a lattice. They ae elocalize an thei 3. Plasmas Within D-Loae Pallaium It is known that euteium is asobe when place nea to a pallaium suface. This loaing can be enhance using electolytic cells o vacuum chambes woking at appopiate pessue [8, 9]. By means of the Theoy of Conense Matte by Pepaata, it is assume that thee phases exist concening the D -P system, accoing to a x=d/p atio: 1) phase fo x<0.1 ) phase fo 0.1<x<0.7 3) phase fo x > 0.7 In the phase, D is in a isoee an non coheent state (D is not chage!). Concening the othe phases, the following ionization eaction takes place on the suface, ue to lattice e.m.: D lattice D e (10) Accoing to the x=d/p loaing pecentage, euteium ions can take position on octaheal sites (fig.1) o in the tetaheal ones (fig.) in the (1,0,0)-plane. Accoing to the coheence theoy, a euteons plasma in an octaheal site is efine as -plasma, wheeas a euteons plasma in a tetaheal one is efine as a -plasma. ISSN: Volume, 017

3 It is possible to state that fequency of a -plasma is given by [5]: 1/ ( x 0.05 (11) 0 ) whee 1/ e N ev / (1) 1/ 1/ m V D a As an example, fo a =0.4 an x=0.5, =0.168 ev/ħ. In tetaheal sites, D + can occupy the thin isk encompassing two sites (fig 3), thus foming a baie to D + ions. Notice that the electons of the -shell oscillate past an equilibium istance y 0 (about 1.4 Å), thus embeing ions in a static clou of negative chage which can sceen the Coulomb baie. As epote in [5], 4Z eff 0.65eV / (13) md y0 This fequency obviously epens also on the chemical conitions of pallaium (impuities, tempeatue etc ). Due to a lage plasma oscillation of -electons, a high ensity negative chage conenses in the isk-like tetaheal egion whee the -phase D + ae locate, giving ise to a sceening potential W(t) whose pofile is epote in fig. 4. It must be highlighte that the -phase epen on the x value an that this new phase has been expeimentally obseve [11]. The new phase is a vey impotant one in LERN investigation. In fact, many col fusion scientist claim that the main point of col fusion potocol is that the D/P loaing atio must be highe than 0.7, i.e. that euteium must take position in tetaheal sites. Fig. 1. The octaheal sites of a P lattice whee euteons take position Fig.. The thin isks of tetaheal sites of a P lattice whee euteons take place a Fig. 3. Possible -electon plasma oscillation in a P lattice Fig.4. The pofile of the electostatic potential in a abitaily iection η 4. D-D potential As shown in efeence [6], the phenomena of fusion between nuclei of euteium in a cystalline lattice of a metal is conitione by stuctual featues, by the ynamic conitions of the system, an also by the concentation of impuities which ae pesent in the examine metal. A stuy has been hel of the cuves of the inteaction potential between euteons (incluing a euteon-plasmon contibution) in the case of thee typical metals (P, Pt an Ti). A thee-imensional moel showe that the height of the Coulomb baie eceases on vaying the total enegy an the concentation of impuities which ae pesent in the examine metal. A potential accounting fo both the ole of tempeatue an impuities is given by the following expession [6]: V () k 0 In (14), the q M V M J kt R V M Mose potential is given by: V M J/ exp exp (14) 0 0 (15) Hee paametes φ an 0 epen on the ynamic conitions of the system, is a paamete epening on the stuctual featues of the lattice, i.e. numbe of ban electons an type of lattice symmety, vaying between an Obviously the Mose potential is use in an inteval incluing an inne tuning point a an continuing towas ISSN: Volume, 017

4 =0, whee it appoaches the Coulomb potential (fig 5). In efeence [6], a fusion pobability is obtaine by means of the following fomula ( is the zeo cossing -value of potential): P exp K (16) 0 whee: K V / (17) This fusion pobability is obtaine using the easonable value of 10 1 min -1 fo the nuclea ate, an it is nomalize to a numbe of events pe minute of 10-5 fo =0.34Å, E=50 ev, T=300K an J=0.75 (high impuities case). Many expeiments confime these fusion ate values concening eactions 1 an [10]. In this pesent wok, the ole of potential (14) is stuie in accoance with the coheence theoy of conense matte, in the thee iffeent phases, an. In this theoetical famewok, two key points nee a claification: 1) what KT is. ) what the ole of electons an ions plasmas is. Concening the fist point an accoing to iffeent euteon-lattice configuations, KT can be: i) the lattice tempeatue if we consie euteons in the -phase. ii) if we consie euteons the in -phase. iii) if we consie euteons in the -phase The secon point is a moe contovesial issue. In fact, the lattice envionment is a mixtue of coheent plasmas (P ions, electons an euteons plasmas) at iffeent tempeatues, ue to iffeent masses. Thus, escibing an emeging potential is a vey ha task. The metho popose in this pesent wok is that of consieing the total contibution of lattice envionment at D-D inteaction (i.e. V tot ) as a anom potential Q(t). In accoance with this moel, V tot ( t) V ( ) Q( t) (18) obviously assuming that: V tot ( t) 0 (19) t That is, a secon oe potential contibution Q(t) is suppose to be a peioic potential whose fequency will be labele by Q, oscillating between the maximum value Q max an 0. The ole of potential Q(t) is that of inceasing o eceasing the baie. The plot of potential V tot fo two iffeent value of Q(t) is epote in figue 6. This means that the following main cases may occu in accoance with Q an with the enegy of incoming paticles to the baie: 1) the paticle cosses the baie in the point ) the paticle cosses the baie in the point Scenaio ) can be egae as a wost case to obtain a high tunneling pobability, an scenaio 1) as a best case. To etemine the moel paametes, some hypothesis must be suggeste on Q(t) an Q. In this pesent wok, an appoximation is mae of Q(t) as equivalent to a sceening potential W(t) ue to -electons, as epote in fig 5. This means that Q ~. Obviously, thee is a stong epenence between the scenaio an the euteon phase, since Q(t) is only the -electons sceening potential, at fist oe. To esume, the following cases may occu in a pallaium lattice accoing to the loaing atio: i) -phase In the phase, euteons ae in a molecula state an themal motion is about: 0.0 ev < ħ < 0. ev This phase takes places when x is lowe than 0.1. Since W(t) is zeo, the D-D potential is: q J R V ( ) const M VM ( ) (0) The expession (0) has been patially evaluate in a pevious pape [6], only focusing on the epenence of a tunneling pobability on impuities which ae pesent within the lattice. In this pesent wok, a coelation is mae between potential featues an loaing atio. Numeical esults ae shown in paagaph 6. ii-phase When x is highe than 0.1 but lowe than 0.7, phase happens. The inteaction takes place between euteon ions oscillating by the following enegy values: 0.1 ev < ħ < 0. ev In this case W(t) is zeo, so the potential is given by expession (1): q J R V ( ) const M V M ( ) (1) Compaing expessions 0 an 1, it seems vey clea that the weight of impuities is moe impotant in the -phase. This conclusion is obviously in accoance with pevious papes [6, 7], whee the ole playe by tempeatue in the tunneling effect was stuie. iii)-phase Finally, the euteon-pallaium system is in the phase when the loaing atio is highe than 0.7. Accoing to a synchonism between phase oscillations of euteon an - electons plasmas, the following two cases must be consiee: Case 1: Q(t)=0 In this case, the potential is a natual extension of fomula (14), an can be witten as: q J R V ( ) const M V M ( ) () Case : Q(t)0 This is the moe inteesting case. It happens when is about Q an obviously when the espective oscillations ae in phase. Deuteons unego a sceening ue to the - ISSN: Volume, 017

5 electons shell, so a supposition is mae that D-D potential must be compute assuming a isappeaance of the well which is pesent in potential (14), ue to Mose contibution. Inee, using a classical plasma moel whee D + ions ae the positive chage an -electons ae the negative one, it is extemely easonable to suppose that the following potential must be use: q J R D V (, t) const M e +Q(t) (3) whee V D C ( ) e (4) an D is the Debye length of this classical plasma. Notice that Q(t) is an unknown petubative potential. About this, it can only be state that: Wmax Q( t) t (5) As peviously sai, this is suppose to be a sceening potential ue to -electons an its ole is suppose to be that of eucing the epulsive baie. D Fig. 5. D-D potential featues using a Mose potential Fig. 6. Potential featues fo two iffeent abitay values of Q(t). 5. The baie cossing teatment A iscussion follows on how to hanle the cossing of the baie in the -phase an when Q(t) is iffeent fom zeo. The stating point in any case is the Schöinge equation: ( ) E Vtot (, t) ( ) 0 (6) Nevetheless, this is a ifficult poblem to solve. To hanle this topic in a simple way, it can be obseve that using V tot (,t)=v()+w max /, this poblem concens fou main enegy values E 1, E, E 3 an E 4 (check fig. 7). This poblem is equivalent to the teating of a ouble baie case. Fom efeence [1], E a few ev; (7) 1 E D 1 me E E 3 M N ~ 1 D ~ ev (8) (9) E4 D' (30) is the constant of metal anhamonicity an is the vibational constant. Anothe impotant quantity is D, which is the epth of the potential well. Accoing to the Mose potential (15), this is J/. Now builing an enegy tenso E ij : E 11 =E 1 E =E E 33 =E 3 E 44 =E 4 E ij =E i -E j E ij =-E ji a squae quaatic enegy value can be efine: te ij ij E E (31) 4 an a ispesion: i j ij E ij E (3) 4 If we neglect the tem Q(t) an consie only the anom featue of euteon enegy, the following coul be a easonable value fo K(): 1 TE ( ) ji ij E K V 4 (33) An finally: P( ) exp K (34) 0 ISSN: Volume, 017

6 But accoing to statistical teatment, P P, E (35), whee =[Q(t)] (36) as aleay seen concening the phase. Since the geate contibution to Q(t) is supposely ue to a sceening effect of -electons in the -phase (i.e. of anom potential), any othe case can be neglecte an only the following two cases must be consiee (i.e. featuing a ouble baie appoximation): 1) Q(t) =0 = 0.34 Å. ) Q(t) 0 =0.16 Å Of couse, case ) is the moe avantageous to obtain a high tunneling pobability. Fig. 7. Featues of V() an V()+W max / 6. Results an iscussion A pesentation follows of the D-D fusion pobability nomalize to numbe of events pe minute concening D-D inteaction in all iffeent phases. Moe exactly, fusion pobability in phases, an ae compae, using a easonable squae aveage value of 00 ev an a value of 50 ev, in oe to coss potential (14) in all fou points E 1, E, E 3 an E 4. The ole of -electon sceening is also consiee as a petubative lattice potential. This teatment only concens the case when Q(t) is iffeent fom zeo, an implies changing the time-epenent poblem of a tunneling effect into a ouble baie poblem. To esume, the emeging of a ouble baie in the -phase is a new physics fact. Notice that col fusion scientists built up thei expectations about a new phase, because sceening enhances the fusion pobability. Fom an expeimental point of view, it is possible to state that thee typologies of expeiments exist in the phenomenology of col fusion [13]: 1) those that have given negative esults. ) those that have given some esults (little signs of etection with espect to backgoun, fusion pobability of about 10-5 ), using a vey high loaing atio. 3) those that have given clea positive esults, like Fleischmann an Pons expeiments. Nevetheless, ou opinion is that the expeiment like in point 3) ae lacking in accuacy fom an expeimental point of view. Fo this eason, we believe that this theoetical moel of the contovesial phenomenon of col fusion must only explain the expeiments like in point 1) an ). In this case, the ole of loaing atio must be consiee in the expeimental esults. Results about the -phase ae shown in Table 1. In this case, it can be obseve that the theoetical fusion pobability is lowe than 10-74, which is vey small. It is possible to state that if the euteium is loae with a x < 0. pecentage, no fusion event is obseve! The same absence of nuclea phenomenon is compatible with a loaing atio of about 0.7 (Table ), since the peicte fusion pobability is less than 10-4 in this case. These peictions ae obviously in ageement with the expeimental esults. But fo x>0.7, a full ange of vali expeiments on col fusion has epote some backgoun spikes (check efeence [10] as an example). A emakable esult of ou moel is shown in Table 3: some backgoun fluctuations can be obseve in the -phase, since we peict a fusion pobability about 10-5 ue to a vey high loaing atio. This epesents a new esult with espect to efeences [6, 7], since in those cases the fusion pobability was inepenent of loaing atio. In oe to peict a vey notewothy nuclea evience (about ), must be compaable with Q (Table 4). Only une this conition can the sceening potential enhance the tunneling pobability an the D-D inteaction become a like-debye potential. The conition allowing this equivalence esult of to Q will be iscusse in anothe pape. Hee, we only uneline that it is a vey unlikely conition. TABLE I Fusion pobability has been compute fo impue P (J 0.75% ), using a -potential (potential 0), an nomalize to a numbe of event/min fo iffeent values of enegy (=±50 ev). Pallaium J 0.75%, Å, E 00eV 0.05 ev 0.1 ev 0.15 ev 0. ev -50 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P ISSN: Volume, 017

7 TABLE II Fusion pobability has been compute fo impue P (J 0.75% ), using a -potential (potential 1), an nomalize to a numbe of event/min fo iffeent values of enegy (=±50 ev). 10 P P P P P P P P P P P P P P P P P P P P10-17 Pallaium J 0.75%, Å, E 00eV 0.33 Å 0.68 Å 1.03 Å 1.38 Å -50 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P10-4 TABLE III Fusion pobability has been compute fo impue P (J 0.75% ), using -potential with Q(t) =0 (potential ), an nomalize to a numbe of event/min fo iffeent values of enegy (=±50 ev). Pallaium J 0.75%, Å, E 00eV 0.6 ev 0.65 ev 0.7 ev 0.75 ev 150 P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P10-5 TABLE IV Fusion pobability has been compute fo impue P (J 0.75% ), using a Debye potential (potential 4), an nomalize to a numbe of event/min fo iffeent values of enegy (=±50 ev). Pallaium J 0.75%, Å, E 00eV D Å D Å D Å D 1.7 Hz Å -50 P P P P P P P P P P P P P P P P P P P P P P P P10-37 The pincipal theoetical objective of this section is that of emonstating whethe thee is a elationship between low enegies, if lattice efomation at oom tempeatue can influence the pocess of fusion. In paticula, the pobability of fusion within a micocack was calculate to evience a possible enhancement of the tunneling effect. Futhe, the aim is to evaluate theoetically the influence of the concentation of impuities using the ten of the cuve of potential. A vey high baie is foun within the pue lattice (J=0.5% appox.). While fo the impue metal (J=0.75% appox.), maintaining the same themoynamic conitions fo the system, thee may be a highe pobability of fusion, with a lowe total potential enegy so that the tunneling effect is enhance. 7. Defomation This communication epots an analysis of the influence of vaiations in the themoynamic conitions which coul, as a esult of efomation, pouce islocations in the lattice an micocacks which may be able to concentate in thei vicinity a significant faction of the euteons pesent in the metal, catalysing a chain eaction which coul favou the pocess. Futhe, the stuy eseaches a elation between low enegies to confim the hypothesis egaing micocacks by means of theoetical quantitative estimations of the coefficient of stuctual efomation[1] of the petube cystalline lattice, inepenent of time. Moe pecisely, the pobability of fusion within a micocack was calculate an compae with that calculate on the suface, to evience a possible enhancing effect, also taking into consieation the vibational states of the euteons. Theoetical inications which we consie inteesting. In ageement with the hypothesis of chain eaction popose in efeence[1], it was foun that the appeaance of micocacks, in effect, inceases the ate of euteon fusion within the lattice. We also analyse the ten of the cuve of potential of euteon-plasmon inteaction within the Pallaium lattice, fom which it can be euce that the thickness of the Coulomb baie is euce on vaying the total enegy an the pecentage of impuities pesent in the metal. The tunneling effect was again calculate in oe to evience any possible enhancement of this effect within the micocack. 8. The thee-imensional moel This section consies whethe, an within what limits, the numbe of fusions within a geneic cubic lattice coul be ISSN: Volume, 017

8 conitione o influence by both extensive lattice efects an othe chaacteistics an themoynamic conitions. In fact, in the case of intenal petubation, the inteaction between the impuities pesent an the islocations which ae pouce in the metal uing efomation can significantly moify the electical popeties of the mateial. Possible efomations pouce in the cystalline lattice by vaiations in the tempeatue ae also consiee. Futhe, une conitions fa fom satuation, the ate of fusion within the metal epens on the numbe of euteium nuclei absobe pe unit of time, an also this coul epen on the efomation of the lattice. If this effectively occus, it is not ifficult to hypothesise that the enegy pouce by the mico-explosions within the micocacks pesent coul favou the fomation of new micocacks, which in tun woul captue futhe euteons by the same mechanism. In a thee-imensional moel, the pobability of fusion between euteium nuclei (whee no micocack exists, that is in a zone on the suface of the cystalline lattice), is equal to the pobability of penetation in a Coulomb potential baie V, given by: 1 P exp EVeff ( ) su (34) su 0 is appoximately 0.11 Å, E is the total initial enegy in ev, pincipally themal in natue, is Planck s constant, an is the euce mass of the euteons. The pocess of fusion within cystalline lattices can be schematise supposing that the electical chage is unifomly istibute on a thin spheical shell an is equal to the ange of effective inteaction between the nucleons, which can be escibe in tems of an effective Coulomb potential: V eff q () const (35) whee q is the euteon chage. It is known that in the pesence of inteactions between euteium nuclei an collective phonic excitation in the metal, the numbe of fusions f in a gas consisting of euteons with ensity is given by: f 4 1 (36) p whee is the euce mass of the euteium nuclei, p is thei impulse, an whee the paentheses epesent the themal mean. Examining, fo convenience, a CFC lattice stuctue subjecte to efomation, it is possible to calculate the pobability of fusion,, within a micocack, on vaying the tempeatue. Denoting the volume of a single cell by, the efomation of the entie lattice is given by: ~ J ( ) ibhl F k Dk exp a kt ktj (37) J is the concentation of impuities, is a paamete which epens on the lattice an electonic stuctue of the metal une consieation,, ( ) is the numbe of islocations; i bhl epesents the inepenence of the intenal stess fom the extenal conitions an the hypothesis, vali in this appoximation, is that the enegy of the baie in iffeent states of nea equilibium of nuclei within the lattice, which inclues the stess une conitions of non-equilibium, is vey small if not almost zeo. i efes to the stess within the lattice; bh ae two paametes which epen on the islocations; L is the length of the islocation; a is the position of equilibium of the islocation coe, sepaate along a split in the cystalline lattice with symmety, in this case, of CFC; kt is the themal enegy to which the metal is submitte, which at oom tempeatue is appoximately 0.05eV); F k is the Helmholtz fee enegy; D k epesents the point of minimum appoach, within a micocack, between the euteium nuclei with enegy kt; J is the concentation of impuities aoun a islocation. We have aleay suggeste that a geate numbe of events coul occu within a micocack, une appopiate conitions the possible consequence of a islocation, than on the lattice suface. To emonstate this, appoximate calculations wee mae, taking into account the lattice efomation an the epth of the micocack. Taking the cente of mass as the efeence system, the pobability of fusion can be witten as [1] : P exp K int (38) int 0 whee is appoximately 0.11Å, by: int / int V (39) is given E is the total initial enegy in ev, pincipally themal in natue, is Planck s constant, is the euce euteon mass. Equations (38) - (39) efe to the pocess of fusion within a micocack. The Coulomb potential V, containing the tempeatue contibution, is given by the expession: V () k q J kt R 0 M V M (40) In (41), V M is the Mose potential, given by: / exp exp V M J 0 0 (41) ISSN: Volume, 017

9 J inicates the concentation of impuities pesent in the metal, while the paametes, 0 epen on the ynamic conitions of the system. is a paamete which epens on the stuctual chaacteistics of the lattice, the numbe of ban electons an the type of lattice symmety, vaying between an If (38) is ivie by (36) an multiplie by (37), it follows that: exp int 0 (4) 4 1 f p Expession (4) epesents the pobability of fusion of euteons within a micocack: this is invesely popotional to the numbe of nuclei absobe by the metal. In the context of the appoximations applie, the pobability of fusion so calculate is equal to the coefficient of efomation of the cones pe unit of total efomation of the entie lattice. Using (4) an aopting both the Mose an effective potentials, the pobability of fusion, nomalise to the numbe of events pe minute, was calculate using a simulation pogamme. V() k q Veff () J R (43) whee is the mean kinetic of the gas, an is the vibational enegy, typically of the oe of some ev fo the quantum states consiee. Fig. 1. Compaison between the potentials (41) an (43), fo J 0.75% an J 0.5%, espectively, at tempeatue T = 80 K. In the fist case. both the height an thickness of the potential baie ae euce. 9. Conclusions As a conclusion, we have shown that the moel popose in this pesent pape can explain some anomalous nuclea taces in solis, but efinitely jeopaizes any hope about the possibility of contolle fusion eactions in matte. In the fist pat of the wok we cosieato vaious paametes type the potential function of time, the numbe of electons, the epth D of the Coulomb 'baie, the vaious enegy levels an thei tensos. The effective inteaction between the euteons insie the metal: it in fact shows that the coupling between plasmons an euteons, in the pesence of impuities, is able not only to ecease the thickness, but also to lowe the height of the baie Coulomb K in vaious types of euteate lattices. We also assume that instea of a static Coulomb baie ae two togethe that oscillate an fo this eason that the phenomenon of col fusion is open to any oa both theoetical an expeimental. Refeences [1] IWAMURA et al., Elemental Analysis of P Complexes: Effects of D Gas Pemeation. Jpn. J. Appl. Phys. A,Vol. 41: p (00.) [] O. REIFENSCHWEILER, Reuce aioactivity of titium in small titanium paticles. Phys. Lett. A, Vol.184: p. 149 (1994.) Physics Lettes A,Vol. 184, p.149 (1994). [3] O. REIFENSCHWEILER, Some expeiments on the ecease of titium aioactivity. Fusion Technol., Vol. 30: p. 61. (1996) [4] Rothwell, J., Intouction to the Col Fusion Expeiments of D. Melvin Miles. Infinite Enegy,. Vol.3(15/16): p. 7. (1997) [5] G..PREPARATA, QED Coheence in Matte, Wol Scientific Publishing (1995). [6] F. FRISONE, Theoetical Moel of the Pobability of Fusion Between Deuteons Within Defome Cystalline Lattices with Micocacks at Room Tempeatue Fusion Technology,Vol. 40, N. pp , (001). [7] F. FRISONE, Deuteon inteaction within a micocack in a lattice at oom tempeatue, Fusion Technology, Vol. 39, N.,pp.60-65, (001). [8] M. FLEISCHMANN an Pons, Etalonnage u systeme P-DO: effets e potocole et fee-back positif. ["Calibation of the P-DO system: potocol an positive fee-back effects"]. J. Chim. Phys., Vol.93, p. 711(1996) (in Fench). [9] A.DE NINNO et al. Evience of emission of neutons fom a titanium-euteium system". Euophys. Lett.,. Vol.9 p. 1. (1989) [10] Heui Kyeong An, et al. Analysis of Defome Pallaium Cathoes Resulting Fom Heavy Wate Electolysis. Fusion Technology Vol. 7 pp ,(1995) ISSN: Volume, 017

10 [11] J.MENGOLI et al. The obsevation of titium in the electolysis of DO at pallaium sheet electoes. J. Electoanal. Chem.,.Vol. 304: p. 79. (1991) [1] C.DeW. VAN SICLEN an S.E.,Jones, Piezonuclea fusion in isotopic hyogen molecules. J. Phys. G: Nucl. Pat. Phys.,. Vol.1 p. 13 (1986). [13] D.MORRISON, The Rise An Decline of Col Fusion. Physics Wol, p. 35,(1990). ISSN: Volume, 017