Quantum states of deuterons in palladium

Transcription

1 Tsuchda K. Quantum states of deuterons n palladum. n Tenth Internatonal Conference on Cold Fuson Cambrdge MA: LENR-CANR.org. Ths paper was presented at the 10th Internatonal Conference on Cold Fuson. It may be dfferent from the verson publshed by World Scentfc Inc (003) n the offcal Proceedngs of the conference. Quantum states of deuterons n palladum Ken-ch Tsuchya Department of Chemcal Scence and Engneerng Tokyo Natonal College of Technology 10- Kunugda Hacho Tokyo E-mal: tsuchya@tokyo-ct.ac.p Bose-Ensten condensaton (BEC) s one of the canddates to nduce the nuclear fusons n solds because d-d repulsons are screened by conducton electrons and deuterons can be condensed at defects n solds. In ths work d-d fuson rate n Pd nduced by BEC s estmated. The equvalent lnear two-body method whch s based on an approxmate reducton of many-body problems by varatonal prncple s used for the calculaton. Thomas-Ferm and non-lnear screenng potentals are used as d-d nteractons. 1 Introducton Usng equvalent lnear two-body (ELTB) method to the many-body problems of charged bosons trapped n an on trap 1 Y.E.Km and A.L.Zubarev derved the ground-state wave functon and the nucleus-nucleus fuson rate. In ths work Km-Zubarev theory s modfed n the followng two ponts. Frstly vacances n sold are regarded as harmonc on traps and the frequency of ths potental s estmated by usng sphercal approxmaton. The ELTB soluton s obtaned numercally and also the rate of d-d nuclear fuson n Pd lattce defect s obtaned. Secondly the crtcal temperature of ths phenomenon s ntroduced. Applcaton of Km-Zubarev Method to the Phenomenon n Solds In Km-Zubarev method an sotropc harmonc potental s used for the on trap potental. Then the Hamltonan of the system ncludng N charged partcle s N N h mω e H = + + r m (1) r r r = 1 = 1 < s the poston of the partcle. Usng ELTB method the ground state of ths manybody problem s wrtten as The wave functon φ n eq.() satsfes φρ ( ) Ψ ( r r... r ) () 1 N ( 1) / ρ N = 1 1/ ρ = r. (3) h d h ( 1)( 3) mωρ NΓ( ) e φ( ρ) = Eφ( ρ). (4) 3( N 1) mdρ m 4ρ 3 πγ ( ) ρ

2 The fourth term s the translated form of the summaton v( r r ) n eq.(1) nto ρ < space by Usng x = mω h r and ε E 5 Γ( ) ρ r. (5) 3( 1) 3 0 ρ N( N 1) V( ρ) = drr v( r) 1 πγ ρ N ( ) = /h ω eq.(4) s rewrtten as d p q x ( x) ( ) dx + + x + x φ = εφ x (6) N ( ) 3 ( 1)( 3) mc 4NΓ p = and q = α 4 hω 3 π Γ 3( N 1) ( ) α s the fne structure constant. Now the applcaton of Km-Zubarev method to the phenomenon n sold s shown here. In eqs.(4) and (6) harmonc term s the electro magnetcally nduced potental n the on trap devce. 1 In crystallne solds ths term corresponds to the nteracton between host ons and mpurty deuterons. The Hamltonan of ths system s wrtten as N h Ze exp( K ) e exp( k ) H = + R r + r r (8) m = 1 R r < r r R s the Bravas lattce vector and Z s the effectve charge of a host on. By ntutve estmaton the second term n eq.(8) s approxmately harmonc at the center of the defects n the crystallne solds. Ths can be explaned as followng. The -th component of the second term n eq.(8) can be expanded nto sphercal harmoncs as Ze exp( K R r ) = A ( r) Y ( θ φ). (9) lm lm R r lm If ths s approxmately sphercal functon domnant term n the expanson s the l = m =0 component whch s wrtten as snh Kr exp( KR ) exp( ) ( Kr) KR A ( r) Y ( θφ ) = Ze Ze (10) Kr R 6 R Therefore f we defne ω as Ze K exp( KR ) 1 mω (11) 6 R the second term n eq.(8) becomes 1 onstant + c mω r for small Kr. Ths means that transformed form of the second term n eq.(8) nto ρ space s smlar to the thrd term n eq.(4). On the other hand transformed form of the thrd term n eq.(8) nto ρ space s not smlar to the fourth term n eq.(4) because of the exstence of the screenng factor exp( k r r ). If we transform eq.(8) nto x space t s wrtten as f d p q + x + + f( x) φ( x) = εφ( x) (1) dx x x s a screenng functon. For Thomas-Ferm potental t s gven by π 4 h f ( x) = 3( N 1) dθ snθcos θ exp k xsnθ TF. (13) 0 mω (7)

3 3 Non-Lnear Screenng Potental It s well known that deuterons provde very strong potentals to the electron gas. Ths effect s ntroduced by usng the densty functonal formalsm of Hohenberg-Kohn-Sham. 34 From the varatonal prncple they have derved the one body equaton; h +Φ ( r) + V xc ( r) ψ k ( r) = ε k ψ k ( r ) (14) m Φ and Vxc are electrostatc and exchange-correlaton potental respectvely. From the self consstent solutons of eq.(14) the densty of the non-lnear screenng cloud nduced around a deuteron n the electron gas can be obtaned. The non-lnear screenng d-d nteracton s obtaned by consderng the change n energy due to the embeddng of two deuterons n electron gas. It s wrtten as e V () r = v() r + d n( ){v( ) + φ ( )} NL s s xc r r r r r r r (15) n v s and φxc are devaton of electron densty from mean densty nduced statc potental and exchange-correlaton potental respectvely. They are defned as e n( r r ) 1 v( r1 r ) = dr (16) s r r r d( ε n) xc φ ( r) = v ( n + n( r)) v ( n ) and v = (17) xc xc 0 xc 0 xc dn s the poston of a deuteron and εxc s the exchange-correlaton energy per one electron. The calculated results of d-d par potental usng non-lnear and Thomas-Ferm screenngs are plotted n Fgure 1. Fgure1. Screened d-d par potental The screenng functon for non-lnear screenng potental VNL s gven by π 4 h 1 h f ( x) = 3( N 1) dθsnθcos θ V xsnθ xsn θ. NL NL 0 mω e mω (18) 4 Nuclear Reacton Rate The ELTB soluton gves the nuclear reacton rate by F < Ψ ImV < Ψ> R = h <Ψ Ψ> (19)

4 F ImV = Ahδ( r r ) s magnary part of Ferm pseudopotental. The short-range nteractons of nuclear forces between two bose nucle are ntroduced by usng δ functon. The constant A = Sr B / π h s determned by the S factor of the nuclear reacton between two nucle. If the ELTB soluton s obtaned the crtcal temperature of BEC s estmated by 3 / h n Tc = (0) 3 πmk ζ( ) B n s the number densty of bose partcles. The probablty of the ground-state occupaton s gven by Ω=1-(T/Tc) /3 for T<Tc. If the ground state occupaton s accounted the fuson rate s gven by RΩ. 5 Results and Dscussons In ths work ELTB solutons for N deuterons trapped n Pd defects have been obtaned and d-d nuclear reacton rate has been estmated. The calculatons were performed wthn the followng condtons. (a) The octahedral vod constructed by 6 vacances (VacO) s adopted as an on trap n sold. The radus Rv s 3.37 Å The frequency ω s 0.86x10 14 sec -1. (b) In eq.(11) the convergences of the lattce summatons are kept to be smaller than 1%. (c) Thomas-Ferm and non-lnear screenng potentals are used to descrbe d-d nteractons. The screenng constant n eq.(11) s /(1 st NN dstance). (d) The effectve charge of a host Pd s one. (e) The S factor s 110kevb. Ths s consstent wth Km and Zubarev. The ELTB solutons are plotted n Fgures and 3. The results for nuclear reacton rates are gven n Table 1. In eq.(3) f all the partcles exst at the same radal component r ρ would become N r. Therefore f a poston of a sharp peak s smaller than NR v Rv s the radus of the defect the condensed deuterons are trapped n the defect. These values are also gven n Table 1. Seeng the ELTB ground state solutons n Fgures and 3 sharp peak exsts. For the non-lnear screenng n Fgure 3 the peak exsts n the negatve regon of the total potental. The peak poston n Fgure 3 s smaller than that n Fgure. They are the results from the dfference between two potentals plotted n Fgure 1. Seeng Table1 postons of the sharp peak are completely ncluded n the defects. For Thomas- Ferm screenng Tc s are lower than the room temperature. In contrast to them for nonlnear screenng they are hgher than room temperature. Nuclear reacton rates are extremely hgh. If a nuclear fuson happens mmedately temperature becomes hgher than Tc. Then Ω becomes zero. And the reacton wll be stopped. These results lead us to the concluson that BEC of condensed deuterons trapped n the Pd defect nduces cold and calm fuson.

5 Fgure. The ELTB soluton for the system ncludng 5 deuterons n VacO n fcc Pd. Thomas-Ferm screenng potental s used as the d-d nteracton. The nondmensonal quantty x s defned as x = mω h r ω 14 1 = sec. The screenng constant n eq.(13) s defned as k=1/(rdd) Rdd (=0.74Å) s the d-d separaton of D molecule. The sold lne means the ELTB soluton. The dashed lnes mean each potental n eq.(1) normalzed by ε. Fgure 3. The ELTB soluton for the system ncludng 5 deuterons n VacO n fcc Pd. Non-lnear screenng potental s 14 1 used for the d-d nteracton. The nondmensonal quantty x s defned as x = mω h ρ ω = sec. The sold lne means the ELTB soluton. The dashed lnes mean each potental n eq.(1) normalzed by ε. Table 1. Nuclear reacton rates as a functon of N for trapped deuterons n VacO n Pd. Thomas-Ferm screenng non-lnear screenng N N R v ρ max ρ T c R ρ max ρ T c R N : the number of the trapped deuterons Rv : radus of the defect [Å] ρmax : poston of a peak n ELTB soluton [Å] Tc : crtcal temp. of BEC [K] ρ : poston of the rght sde foot of the peak [Å] R : nuclear reacton rate [10 7 sec -1 ] Acknowledgements The author wshes to thank Professors Y.E.Km (Purdue Unv.) and H.Yamada (Iwate Unv.) for helpful dscussons and encouragements. References 1. P.K.Ghosh "Ion Traps" Oxford Clarendon P ess 1995 r. Y.E.Km and A.L.Zubarev Fuson Technology37151(000) 3. H.Hohenberg and W.Kohn Phys.Rev.136B864(1964) 4. W.Kohn and L.J.Sham P hys.rev.140a1133(1965)