Scott R. Chubb and Talbot A. Chubb Research Systems, Inc., 9822 Pebble Weigh Ct., Burke. VA USA

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1 Cubb, S.R. and T.A. Cubb. Teoecal Famewok fo Anomalous Hea and 4He n Tanson Meal Sysems. n 8 Inenaonal Confeence on Cold Fuson Lec (La Speza), Ialy: Ialan Pyscal Socey, Bologna, Ialy. Teoecal Famewok fo Anomalous Hea and 4 He n Tanson Meal Sysems Sco R. Cubb and Talbo A. Cubb Reseac Sysems, Inc., 98 Pebble Weg C., Buke. VA USA Inoducon Cold Fuson as been plagued w msconcepons abou wa s and s no possble, based on e Laws of Quanum Mecancs. An mpoan eason fo s s e seemngly mpossbly lage dffeence n leng-scale beween nuclea- and aomc- pocesses. In convenonal fuson, ese scales eman so fa apa a ey effecvely don alk o eac oe, usually. Howeve, elecomagnec neacons (MI s) ave nfne ange. Fo s eason, s possble a MI s can explan ow s appaen poblem can be elmnaed. A somewa supsng feaue of MI s also s ey can do s n e seemngly mpossble suaon n wc eac of e pacles a s nvolved effecvely as vansngly small momenum. A key pon n undesandng ow s can become possble s assocaed w ow momenum and leng-scale ae elaed o eac oe. Aloug a g enegy, e elaonsp beween ese quanes can be vewed as beng localzed, s s because wen e assocaed Debogle wavelengs (λ Debogle s) of ndvdual pacles do no ave appecable ovelap, e momenum p of an ndvdual pacle can be eaed n ems of a classcal pcue, n wc p=mv, wee m s e mass of e pacle, and v s s velocy. As p-> 0, s pcue beaks down because unceanes n p and poson (x) become newned w e elecomagnec feld. Fo s eason, bounday effecs and symmey, oug (mplc and explc) MI s lead o foms of coeence a explan well-known effecs (Mossbaue effec, supe-conducvy, Bagg-scaeng, ea and eleccal conducvy n solds) n wc momenum can be saed by many pacles nsananeously. Because MI s ae also esponsble fo non-sepaable foms of couplng beween elecomagnec and nuclea pocesses (n wc e coodnaes assocaed w ese foms of foces depend on eac oe) n one fom of eacon (D+D-> 4 He), s eoecally possble a e wo foms of neacon can become coupled. Fo s eason, s also plausble a s fom of neacon can become domnan n suaons n wc non-local, coeen effecs a occu as p->0 (o wen elaed lms, assocaed w lage values of λ Debogle ) become domnan. An mpoan pon s a wen e λ Debogle s of many pacles become suffcenly lage (o ae consaned by symmey o pacula values oug e usual ules of Quanum Mecancs), coeen couplng beween caged pacles and an elecomagnec feld can occu even n e lm n wc e combned momenum of e pacles becomes vansngly small. Te assocaed effec can explan ow momena can be ansfeed fom an solaed locaon o many locaons, all a once, wou g enegy beng ansmed o any ndvdual locaon. We exploe e assocaed mplcaons of s on Cold Fuson-elaed penomena. Insde and Ousde e Box and e Oganzng Pncples of Convenonal Fuson Logcal oug eques ules. In pyscs, e logcal ules follow fom Newon s laws of moon, Maxwell s quaons, Quanum Mecancs, and Relavy. Because ese ules povde a famewok, ofen ey can be self-lmng. Fo example, somemes pyscss msnepe e ules, smply because ey ae condoned o look a em n a pacula way. Tey become used o a pacula woldvew. Te woldvew can be oug of as a knd of box a defnes a comfo zone. Ofen, e box s ed o e way we ave leaned a pacula subjec. Dffeen people vew e box n dffeen ways. Kun[] efes o, absacly, as elaes o scence, as a paadgm. Oes ave no been as open-mnded[].

2 Fg. sows a pcoal epesenaon of convenonal fuson eacons supe-posed on an dealzed epesenaon of e box, assocaed w wa s commonly vewed as convenonal (labeled nsde e box ) and unconvenonal (labeled ousde e box ) scence. In s scemac, all eacons ognae fom a confguaon n wc wo deueons (sown as poonneuon pas) ovelap w eac oe n a manne a foms a confguaon (sown n e cene of e plo) a esembles an exced sae of a 4 He nucleus. Te wo, domnan eacons (D+D- > 3 He+n, and D+D-> 3 H+p) a occu n fee space ae essenally blnd o e pesence of e elecomagnec neacon (MI). Fo s eason, s possble o ea ese eacons wn a famewok n wc e dependence of e eacon on elecomagnec neacons s ndependen of s dependence on e nuclea (song foce) neacon. Ts means a n ese eacons, e assocaed wave funcons descbng e nal and fnal saes do no couple e nuclea and elecomagnec neacons. As a esul, e geneal eacon ae expesson (wc s descbed below) effecvely pecludes e song foce fom alkng o e elecomagnec foce, by consucon. Te fgue scemacally llusaes s pon oug e labels ( gnoe. M. ), Fg. : Pcoal epesenaon of convenonal fuson eacons. Dakened and lgened ccles, especvely epesen neuons and poons. nex o e aows a ae sown n e g poon of e fgue. Also sown s e emanng fuson eacon (D+D-> 4 He). Ts eacon occus aely n convenonal fuson. Fo s eason, n e fgue s sown as occung a e bounday of e box. A second eason we ave dawn a e bounday s a volaes a paadgm a many nuclea pyscss beleve o be vald: n convenonal fuson, e song and elecomagnec neacons eman uncoupled. Fo s eason, s wdely beleved a e fnal (D+D-> 4 He) eacon sould aely occu and e wo emanng

3 eacons sould occu w ougly e same pobably. Howeve, e D+D-> 4 He does occu, and e eason a s no fequenly obseved s well-undesood: volaes enegy-momenum consevaon unless a g enegy momenum gamma ay s emed, and e assocaed MI nvolves a complcaed (quadupola) couplng beween nucleon spns (a occus as a second ode elecomagnec pocess). Two mpoan pons ae:. aloug s fnal eacon occus nfequenly elave o e oes, wen occus, e nuclea and elecomagnec neacons do alk o eac oe, and. occus aely because e assocaed pocesses nvolve ovelap beween wo pacles a a sngle locaon. Movaonal Pyscs fo Geng Ousde e Box Pa of e confuson w e box assocaed w convenonal nuclea pyscs nvolves e defnon of momenum p: fo a sngle caged pacle, p does no equal mass (m) mes velocy (v); e ules of e box ae: fo a pacle possessng cage q, mv=p-q/ca, wee A (e veco poenal) s assocaed w e elecomagnec neacon, and c s e speed of lg. Aloug s ule s based on classcal pyscs, ow and wee apples seems o ave been a souce of confuson. Te ule follows fom e box defned by classcal pyscs. (Mss-assumpons abou s ule no only appea o ave led o confuson abou Cold Fuson bu o moe seous poblems.) An example of e mpoance of s dsncon occus n e p->0 lm, wen many pacles sae a common densy ρ o. Wen s occus, mv, wc s popoonal o e cuen J (povded ρ o s unfomly consan[3]), becomes popoonal o A. Bu A, wc s defned by e 4πJ 4πq ρ A sac wave equaon ( A ), en obeys a Helmolz equaon[3] ( A o ) c mc a esuls n A asympocally vansng beyond a ccal coeence leng, wee J appoaces a consan value. Ts occus even n e absence of an appled elecomagnec feld (MF). Te esulng pcue explans e penomenon of supe-conducvy. I also explans ow as p->0, supe-conducvy no only s pesen, bu because e cuen vanses a some bounday, suoundng e egon wee supeconducvy occus, e effecs of boundaes may esul n e expulson of magnec flux wen p=0 (e Messne effec) o flux quanzaon[3], wen p does no vans bu akes on values a ae conssen w e assocaed ules (defned by e box ) assocaed w e equemens of quanum mecancs[3]. Te bass of bo penomena s a p does no equal mv ; n suaons wee e DeBogle wavelengs of pacles become suffcenly lage, pacles become wavelke. In s knd of suaon, e aveage value of e gaden of e pase of e assocaed collecon of waves (wc s descbed by e many-body wave funcon) defnes e momenum. Te mpoan pon s a e pase of e many-body wave funcon, as opposed o a quany elaed ee decly o e cuen o o mass x velocy defnes ow e momenum beaves. Wen p->0, s quany can be affeced n ways a ae non-local n caace. Ts may occu because non-local canges n A can sgnfcanly ale e value of e pase. Because a po s no possble o pedc f a sold s a es o n moon, fo example, s cene-of-mass wave funcon can be aleed by an abay complex numbe. Ts noduces e possbly of an abay gauge ansfomaon n e defnon of e A a apples nsde and ousde a sold. Because n e p->0 lm, becomes possble o deemne f e sold s n moon o a es, e assocaed abaness n gauge s emoved. No only does s mean a e assocaed gauge symmey becomes boken, bu pyscal effecs (fo example, e expulson of magnec flux, o sponaneous lace ecol [as n e Mossbaue effec]) can occu. Te esulng coeence can be vewed n dffeen ways, wn e famewok (e box ) assocaed w a pacula dscplne. 3

4 In smla ways, effecs of peodc ode and oe symmees can become mpoan n suaons n wc e wave-lke caace assocaed w lage DeBogle wavelengs becomes mpoan. Te mpoan pon s a because momenum s assocaed w wave-lke beavo, can cange suddenly, n unexpeced ways, on abaly so me-scales. Tese canges can esul n nsananeous canges n wc lage amouns of momenum coeenly ae sfed o many pacles, and vce-vesa. How o f s occus s dcaed by e dynamcs of e many-body sysem. Moe Pecse Pyscs: Mulple Scaeng Te exac poblem s fomdable. I nvolves solvng e eacon ae poblem fo e (many-body) poenal V a neacs w all caged pacles. To undesand ow eacon aes n nuclea (as well as oe) pocesses can be affeced ee oug coeence a esuls as p->0 o (n a less escve sense) wen symmees consan ow p may vay dung pacula foms of neacon, s no necessay o addess all aspecs of s poblem. In pacula, e assocaed coeence occus as a esul ee of degeneacy o oug e neacons beween nealy degeneae saes. Te sang pon of e analyss s geneal: we consde e poblem of evaluang eacon aes, fo a many-body sysem n wc a peubng poenal V s pesen a asympocally s assumed o ave vansed n e dsan pas. Assocaed w e evaluaon of eacon ae s e me evoluon of e ovelap < - + > beween an ouwadly popagang manybody scaeng sae + > (defned by e assympoc lm of e exac many-body sae Ψ ( = me) > : + > Ψ ( ) > ) w any of e possble nal, nwadly lm lm Ψ' popagang saes - > ( ( = me) > lm of e well-known Lppmann-Scwnge equaon a: ). I follows fom e asympoc ( ) < ' + >=< ' (0) ( ' + ε) e (0) > + ' + ε < ' (0) V Ψ (0) >.() Hee, mplcly,, w e nfnesmal vaable ε appoacng zeo, consaned by /ε >. In sngle-pacle scaeng eoy, q. s an negal epesenaon of e Scoednge equaon. In many-body pyscs, s equvalen o e Dyson equaon, povded a suable defnon of V s employed. In fac, n paccal applcaons nvolvng many-body pyscs, V s neve known exacly and s usually appoxmaed usng nfomaon povded by an appoxmae epesenaon. Specfcally, an mpoan eason fo usng an appoxmae fom fo V s a usually nee V o Ψ > s known explcly. Howeve, o addess e moe geneal poblem of undesandng ow non-local neacons can evolve n many-body sysems, s no necessay o solve fo ee of ese quanes. Insead, an alenave pocedue, nvolvng nfomaon assocaed w e undelyng bounday condons (and sngula beavo of e wave funcon) may be employed. In s alenave fomulaon, V s eplaced fomally, usng e knec enegy opeao Tˆ assocaed w e undelyng many-body Scoednge equaons fo Ψ > and >. Ts fomal consucon, wc s a genealzaon of a sandad, mulple-scaeng ecnque[4,5], a as been appled n aomc, molecula, and sold sae pyscs poblems, n essence, elaes e fomal poblem assocaed w scaeng fom a pacula poenal (as n q. ) o a non-local momenum-balance poblem. In s alenave poblem, e max elemen < () V Ψ ( ) > s ansfomed no a quany nvolvng Ψ >, > and e 4

5 devaves of ese quanes, evaluaed a boundaes of e egon, wee V ee becomes sngula o vanses. Specfcally, wen Ψ > sasfes e many-body Scoednge equaon assocaed w e many-body poenal U, and enegy, follows a ( H Tˆ) Ψ >= ( Tˆ) Ψ > " " U Ψ >, () wee, n e coodnae epesenaon, Tˆ = (m s e mass of e pacle m (=,,n)) s e many-body knec enegy opeao, and we ave used e condon a Ψ > s an egensae of H. Snce a smla elaonsp olds wen < ' obeys a compaable Scoednge equaon assocaed w poenal U, follows a n a fomal sense snce < ' V Ψ > < ' U U ' Ψ >, a < ' V Ψ >= ( ' ) < ' Ψ > * * { ˆ + d...,, )} (,...,, ) (,...,, ){ ˆ d... d n TΨ ' (, n Ψ n Ψ ' n TΨ (,..., n, )} wee Ψ (,,..., n, ) <,,..., n Ψ >, and Ψ ' (,,..., n, ) <,,..., n ' > ae coodnae epesenaons of Ψ > and ' >. Supefcally, mg appea a e second em n q. 3 vanses. In fac, s s no e case because of e mplc bounday condons assocaed w qs. and. In pacula, because bo e unpeubed and peubed Hamlonans ae bo Hemean, and ' ae fne and eal. Bu s means a a pons wee U and/o U' become sngula, compaable sngulaes occu n e ems a nvolve Tˆ on e lef-sdes of qs. and 3. As a consequence, suface ems (fom dsconnues n one o moe componens of e gadens of one o bo of e many-body wave funcons) occu n e second em fom egons a bound e locaons wee sngulaes n U and/o U' ae pesen. Also, because Ψ > and ' > ae no equed o vans (and may ave appecable ovelap) n egons wee U=U', also follows a e second em on e gsde of q. 3 educes o a sum of suface ems (oug Geens eoem) a e boundaes of s egon. To undesand e possble aes of eacon n Cold Fuson, we examne e suaons assocaed w q. n wc e many-body saes assocaed w - > and - > ae dffeen. Because ese saes ae dffeen, e ovelap max elemen < - (0) - (0)> vanses. Ts means a ε + e < ' > = < ' (0) V Ψ (0) >,(4) ( ' + ε)( ' ε) fom wc follows a < ' + > ε = lm < ' ε 0 ( ' + ε)( ' ε) π = δ ( ') < ' (0) V Ψ (0) > (0) V Ψ (0) >., (3) 5

6 π * * ( ') 3n ˆ { δ } d n Ψ Ψ Ψ Ψ ' ' boundaes m π δ ( ' ) < ' Ψ > n.(5) Te oal eacon ae R s consuced by summng e g sde of q. 5 ove all possble fnal (many-body) saes. Te effecs of coeence may ene wen e nal sae s fomed fom saes a possess degeneacy. In pacula, fo example, s can occu wen ' n ε '( { k} k k k ),(6) wee ε (k) = enegy of sngle-pacle sae possessng egenvalue k; nk = wegng faco a accouns fo s degeneacy/occupaon. In a many-body sysem, nvolvng a macoscopc numbe of pacles, {k} can be que lage, and, n geneal, wen a sum s aken ove all fnal saes, many lage and small ems can be nvolved on e g-sde of q. 5. Howeve, n suaons wee lage degeneacy occus n e nal sae (and a small numbe of values of n k can become lage ), a sngle em o a small numbe of ems can become domnan. An example of s knd of suaon, fo example, occus wen peodc ode causes a lage numbe of egenvalues o be peodc funcons w espec o e se of wave-vecos G (ecpocal lace vecos) a defne e Foue ansfom of e undelyng (peodc) poenal because s fom of symmey can cause a sgnfcan numbe of saes ε (k) o become degeneae: fo example, ε (k) can = ε (k+g) fo a lage numbe N of values of G; n pacula, N can ~ numbe of un cells=n cell. Fo llusave puposes, suppose =Nε (k). Te esulng expesson fo e oal eacon ae R s R P (0) (0) ' ( '') anyng= π k < {' k} n Ψ{ k'}' > = ( '( k) ) ' anyng δ ε k'' N N Wen dsconnues n < ' (0) Ψ (0) > can be eaed as f e assocaed { k} n { k' '} saes beave as band saes, e g-sde (R.S.) peseves peodc ode. Ten, e R.S. nvolves small, equal canges n momenum (o enegy) fom eac un cell. Wen e nal sae concenaon c of band sae deueons s suffcenly small, eac max elemen scales as c ; wle e dela funcon, mulpled by /N (wee N~Ncell ) scales as e concenaon c f of fnal sae, eacan poducs ( e.g., band-sae 4 He o 3 He), pe un cell. Tus, wen c * c f s suffcenly small n magnude, e R.S. can become compaable o aomc mescales. (Ts occus as a esul of e non-local naue of e max elemen.) Ten, wen N~ un cells (nea empeaue T=0), nuclea eacons can poceed a aes a can couple o aomc-scale pocesses, wou g enegy pacles. Fo geae T, ese knds of eacons can occu a smalle values of N. Acknowledgemen We ank Robe Bass and e efeees fo povdng ccal commens concenng e pesen manuscp and Roy Swanson fo s pevous conbuons. Refeences [] T. S. Kun, Te Sucue of Scenfc Revoluons. (Unv. of Ccago Pess, Ccago, 96) Vol II, 0. []Pak, Voodoo Scence: e Road fom Foolsness o Faud (Oxfod Unv. Pess, Oxfod, 000). [3]R.P. Feynman, R. B. Legon, M. Sands, Te Feynman Lecues on Pyscs. (Addson Wesley Publsng, Inc., New Yok, 965), v 3, c, pp [4]A. Gons and W.H. Bule, Mulple Scaeng n Solds. (Spnge-Velag, N.Y., 000), 350 pp. [5] J. Konga, Pysca 3,39 (947). W. Kon and N. Rosoke, Pys. Rev. 94, (954). (7) 6

Field due to a collection of N discrete point charges: r is in the direction from

Field due to a collection of N discrete point charges: r is in the direction from Physcs 46 Fomula Shee Exam Coulomb s Law qq Felec = k ˆ (Fo example, f F s he elecc foce ha q exes on q, hen ˆ s a un veco n he decon fom q o q.) Elecc Feld elaed o he elecc foce by: Felec = qe (elecc