D-Cluster Dynamics and Fusion Rate by Langevin Equation

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1 D-Clust Dynamics an Fusion at by Langvin Equation kito Takahashi** an Noio Yabuuchi High Scintific sach Laboatoy Maunouchi-4-6, Tsu, Mi, Japan **Osaka Univsity STCT Conns matt nucla ffct, spcially 4D-clust fusion, in mtal-utium complx systms, has bn stui by applying Langvin quations. On imnsional Langvin quations fo solving tim-pnnt - istanc t fo uton-clusts un th Platonic symmty w fomulat fo D- atom, D molcul, D + ion, D 3 + ion, 4D/TSC an 6D - /OSC. Establish valus of goun stat - istancs gs w pouc by xpctation-valu quations, which w obtain by nsmbl avaging with wight of quantum mchanical wav functions Gaussian wav functions, fo D-atom, D, D +, an D 3 + molcul. In analogy to th abov Langvin quations, th Langvin quation fo 4D/TSC un th ttahal oubl Platonic symmty was iv an numically solv by th Vlt tim-stp mtho. It was shown that only 4D/TSC among 5 D- systms xcpt D-atom coul conns ultimatly fom t==74 pm to vy small chag nutal ntity with about fm aius at TSC-min stat aft about.4 fs connsation tim. Th 6D - /OSC systm convg at gs =about 4 pm, namly convg on th way of connsation fom t==74 pm. Tim-pnnt Coulomb bai pntation pobabilitis bai factos fo connsing 4D/TSC w calculat by th Havy Mass Elctonic Quasi-Paticl Expansion Mtho HMEQPET. 4D fusion at p TSC gnation was obtain bas on th Fmi s fist goln ul to sult in almost % 4D fusion action p 4D/TSC gnation. Fusion ats w compa with thos of muonic molcul, D molcul an *, Coop pai molcul to mt goo consistncy. Majo nucla poucts of 4D fusion a two 3.8 MV α-paticls. 4H/TSC shoul conns in th sam way until whn TSC-min stat with classical lcton aius.8 fm coms, but no stong intaction xists among potons an will mak p to 4p captu tansmutations with host mtal nucli whn 4H/TSC has sufficint ift CMS momntum. Kywos: D-clust fusion, ynamics, connsation, 4D/TSC, D-molculs, Langvin quation, bai factos, 4D fusion, hlium-4 36

2 . Intouction To xplain appant ha-aiation-lss xcss hat with hlium-4 ash in CMNS conns matt nucla scinc xpimnts, spcially in ynamic PDx systms, w hav on a long sis of stuy fo moling D-clust o multi-boy uton fusion action mchanisms to ach at ou latst thoy bas on Langvin quations,. This pap scibs th basics of fomulation of Langvin quations fo D-clust ynamics, spcially fo D-atom, D molcul, D + ion, D 3 + ion, 4D/TSC ttahal symmtic connsat an 6D - /OSC octahal symmtic connsat. Fist on-imnsional Langvin quations fo D-clusts with th - istanc a fomulat un th Platonic symmty,3 of multi-paticl D-clust systms with utons an quantum-mchanical lcton cnts. Un th othogonally coupl Platonic symmty fo a Platonic uton-systm an a Platonic lcton systm, ynamic quations fo so-many-boy systm of utons an lctons with mtal atoms mo than 4 utons plus 4 s lctons of utium atoms plus 4 4-shll lctons of 4 P atoms in fcc lattic plus suouning lattic atoms un D-phonon xcit stats shoul b consi in ou moling 7, a simpl on-imnsional Langvin quation fo th int-nucla - istanc can b fomulat, as w will show in this pap. y th nsmbl avaging of on-imnsional Langvin quation with th wight of quantum mchanical wav-functions fo lctons an utons, w can futh iv a tim pnnt quation fo xpctation valu < >, which is nonlina, but can b solv by th Vlt tim stp mtho. W show in this pap that only 4DoH/TSC can conns ultimatly to b finally vy small chag nutal ntity with about - fm aius. t th final stag of 4D/TSC connsation in about x - s, 4D fusion with 4 H poucts taks plac with almost % pobability, accoing to ou HMEQPET calculation fo bai factos an fusion at fomula by th Fmi s fist goln ul. In th nxt sction, w show th ivation of Langvin quations fo known systms as D-atom, D molcul, D + ion, an D 3 + ion. This pocu givs th basics fo fomulating Langvin quations of complx D-clust systms as 4D/TSC an 6D/OSC. In analogy, w apply th mthoology an iv Langvin quations fo 4D/TSC an 6D/OSC connsation motions in th following sction.. Langvin Equations fo Known D-Systms.. Langvin quation in gnal Th Langvin quation is usful to tat ynamic motion of paticls un fiction o constaint an anom fluctuation of foc-fil. 37

3 m F c t t f ' t H m is th paticl mass, is paticl position, F c is th Coulombic foc, ς is th cofficint fo fiction o constaint an f t is th anomly fluctuat foc tm whit nois, fo ou utons plus lctons systm... Langvin quation fo D-atom In Fig., simpl quantum mchanical imag of D-atom is awn. In th viw of Platonic symmty, D-atom is th othogonal coupling of cntal point uton an sph lcton-wav. Th Langvin quation is givn as balanc of th cntiptal foc of Coulombic attaction btwn plus-chag uton an minus-chag lcton an th cntifugal foc of lcton otation aoun cntal point uton; m t [ ] mv f t H, m is th lcton mass, is th - istanc, is th unit chag an v is D-tom: Point-Sph Coupling a 3 / a 4 4 Pak at =a= 5.9 pm Wight of lcton fo D-atom Squa of *Wav-function Wight 5 5 aius pm Elcton Wight Pak Localizs at = a = =5.9 pm Figu.. Quantum mchanical imag of D-atom 38

4 th lcton vlocity. W hav no fiction in this cas. y taking nsmbl avag of Eq. with th wight of squa -s wav function, w obtain, m t m v 3 Th ight si bcoms zo, bcaus of th avag kintic ngy EKE m v 3. 6V an th avag Coulomb ngy EC 7. V as wll known fo hyogn o utium atom. W can intgat Eq. 3 ov tim to gt, m t F T T f t t f t 4 Th tim-avag intgal of anom fluctuation ft is qual to th nsmbl avag <ft> u to th goic pocss. W intgat Eq. 4 to obtain th wll known sult as, t 5. 9 pm 5 Namly, xpctation valu < > of aial lcton obit is constant to b oh aius = 5.9 pm..3. Quantum Mchanical Ensmbl vag fo D-Clust Sinc both positions of lctons an utons fluctuat quantum mchanically fo D-clust systms, w n to avag with both wights of wav functions fo lctons an utons. pplying th on-oppnhim aiabatic appoximation fo total wav function, w can mak stp-wis avaging fo lcton-wavs an thn fo utonwavs. Th aiabatic wav function fo D molcul is; ;,,, X D Th lcton wav function of D molcul is givn 4 by, X S, D s S

5 3 n th wav function fo a - pai is appoximat by th Gaussian wav function as, witing X with Ψ an putting =, / ' xp ', 8 Quantum-mchanical nsmbl avag of obsvabl G is givn by, G G nsmbl 9.4. Langvin quation fo D molcul Elcton localization wight istibution of D molcul is oughly unstoo by th nomalization quation of wav function,,,, 4 4 D,,, 4 4,,, Sinc is th lmnt of paticl nsity localization function, localiz paks appa at = = = =a= 5.9 pm; namly th awn sph with oh aius is a goo masu of lcton localization.

6 D Molcul Elcton Localization: / Platonic Systm: Di-Con Dipol-Cicl Coupling t = an = : 44 4,,, t = = = = a Maximum appas a, a, a, a a a a 4 = a = 5.9 pm Tous of lcton cnt = = 74. pm Classical viw: Elctons otat aoun axis. Figu. Localization of lcton wav an smi-classical imag of D molcul In th viw of Platonic symmty, D molcul is an othogonal coupling of - lin ipol an cicl tous of lcton cnt to fom a icon. Fom of lcton motion is constain by th xistnc of countpat uton an lcton to fom th tous of lcton cnt, but avag cntifugal foc xists as th otation of lctons aoun axis. S Fig.. Th Langvin quation fo D molcul bcoms as, m t mv Vs 4 / ;, f t 3 H th Coulomb foc tm un Platonic symmty is iv by ivativ minus sign of Coulomb ngy, E C 4 4 3

7 E C 4 5 y taking QM-nsmbl avag with wight of squa lcton wav function, m t 5.6 mv Vs 4 ;, f t 6 Th fist an scon tm of Eq. 6 ight si cancls ach oth 4, an w obtain, m t Vs ;, f t 7 y taking nsmbl avag with th Gaussian wav function of - pai, th scon tm of Eq. 7 <ft> bcoms zo, sinc w hav no istotion in - ipol lin by QM fluctuation to viat fom th Platonic symmty. Thus, Eq. 7 bcoms wll known Nwtonian mchanical quation, with constaint by molcula tapping potntial V s ;,. Mathmatical fomulas fo tapping shil potntials of D an D + systms a givn in ou pvious paps, 5. Figu of plott ata fo two potntials a shown in Fig. 3. Tapping Potntials fo D an D+ Potntial Hight V gs=3 pm fo D + gs=74 pm fo D Vs Vs pm Figu 3. Tapping potntial of - pai fo D molcul an D + ion 3

8 s unstoo by potntial shap fo D molcul givn in Fig. 3, th Langvin quation fo xpctation valu < > givs always convgnc to gs =74 pm aft timpnnt motion stating fom abitay position t=. If > gs, w hav acclation foc. If < gs, w hav clation foc..5. Langvin Equation of D + Ion In analogy to th D molcul, Langvin quation fo D + ion stabl in vacuum is givn by, m t mv Vs ;, f t 8 y taking QM-nsmbl avag, m v 9 m t Vs ;, Potntial cuv is shown in Fig. 3. In th viw of Platonic symmty, D + ion molcul is an longat icon with =3 pm, with otating tiangl of -- typ fac aoun th axis. Dynamic motion of uton by Eq. givs convgnc to = gs =3 pm. In th following fo complx D-clust systms, Eq. 7 an Eq. with thos potntials will povi intinsic componnts of fiction constaint by QM lcton wavs with D-clust connsation..6. Langvin Equations of Expctation Valus fo Complx D-Clusts In complx D-clust systms un Platonic symmty, avag otation motion ov whol systm is pohibit by constaints of many paticl aangmnts. This fom 3 of slf-oganization maks simpl tatmnt to iv on-imnsional Langvin quation possibl. Th tm fom is mta-physical concpt. Th QM-nsmbl avag on lcton wav function can b subivi as multipl constaint function of -- typ o --- typ potntial ivativ as, Consta V ;, si int N lcton wav f 33

9 H N f is th numb of facs of Platonic polyhon fo a D-clust, an i= fo th - typ D + typ fac an i= fo th --- typ D typ fac. Th Langvin quation fo a D-clust un Platonic symmty with N numb of - gs an N f numb of facs is wittn fo N >, N m t k N f Vs f t H k is constant.8 fo 4D/TSC. Th QM-nsmbl avag on - wav function assuming Gaussian fom is givn by N m N f k, ', ', ' t Vs, ', ', ', ' f t, ' 3 with a Gaussian wav function fo - pai of D-clust,, ' xp ' / 4 W iv a Langvin quation fo xpctation valu < > = <> as, N m k t N f Vs f t 5 Fo complx D-clust, <ft> valu os not always zo bcaus of viation of D- clust systm fom ial Platonic symmty, u to th quantum mchanical fluctuation of -positions which may istot th Platonic symmty. Th ptub foc componnt by this QM istotion is appoximatly givn by th nxt fomula, which is th chang of systm Coulomb ngy ivativ, as, 34

10 k k E C k k k 6 y using a Gaussian squa wav function fo - pai fluctuation, w wit, 7 Th chang of Coulomb foc by istotion is givn by k <Chang of Coulombic Foc> = 8.7. Langvin Equation of D 3 + Ion Molcul It is wll known that ti-atomic hyogn molcula ion D 3 + o H 3 + is gnat in ion souc an glow ischag plasma an vy stabl in vacuum. Howv, quantum molcula physics fo th systm is of ifficult poblm to solv an stuis a bing continu in astophysics ns. Th analogy of th psnt mthoology fo D o H-clust un othogonal coupling of Platonic symmtis fo lctons an utons o potons can povi ath simpl way of moling its ynamics. pplying Eq. 5, w obtain, 6.3 3m t Vs 6 ;, f t 9 H th foc is givn with unit of [kv/pm]. Imag of D 3 + ion is givn in Fig

11 3D + Ion ; Smi-classical viw of paticl aangmnt uton Elcton cnt <>: oh aius sph a Top Viw b i viw Tigonal Dipyami : Tiangl-Dipol Coupling Figu 4. Ti-atomic hyogn utium molcula ion an Platonic aangmnt Th systm Coulomb ngy an its ivativ can b calculat by simpl gomty xcis fo th Platonic symmty systm of tigonal i-pyami which is th othogonal coupling of th 3 gula tiangl an th <> - <> lin ipol. H two lcton cnts o lcton balls appa in th systm, an systm-avag otation of lctons is pohibit no avag cntifugal foc. y istotion of ial Platonic symmty with QM fluctuations of uton positions, w hav positiv <ft> bias. s w hav 3 - gs in th systm, 3 tims of Eq. 8 bcoms th bias about 3% of main Coulomb acclation foc. Thfo th total potntial of th systm bcoms in xpct valu quation, as, V 6.3 main 6V s ;, + <ft> componnt 3 3D Th calculat cuv of this potntial is shown in Fig

12 Tapping Potntial of 3D+ V 3D+ Potntial V gs=8 pm V3D+ V pm 6.3 3m t Vs 6 ;, f t f t E C ' ;, t :Distotion of Coulomb foc fom 3D gula tiangl aangmnt about 3% Figu 5. Tapping potntial of D 3 + ion molcul with Langvin quation fo xpctation valu of - istanc of 3 gula tiangl Th ti-atomic hyogn ion is thus stabl an has its goun stat at gs = 8 pm. s a fnc, Hlm t al Fibug Univsity, 3; googl tiatomic hyogn ion an Hlm gav about gs = 85 pm 8 which ags consiably wll with ou sult taking into account that appopiat sigma-valu of wav function is about 3 % of. W can conclu that ou appoach with on-imnsional Langvin quations fo D- clust systms look succssful. 3. Langvin Equation fo 4D/TSC an Numical Solution 3.. Doubl Platonic Symmty In Fig. 6, w show fatu of lcton clou fo 4D/TSC t=, compa with thos of D-atom an D molcul. 37

13 Fatu of QM Elcton Clou = 53 pm Elcton cnt; <>= + / oh obit of D H Ψ Duton a D atom stabl osoniz lcton Cnt tous fo + Obit of osoniz Elcton coupling Fo + c 4D/TSC lif tim about 6 fs 73 pm b D molcul stabl: ΨD =+Δ -/ [Ψ Ψ+ Ψ Ψ]ΧsS,S Figu 6. Fatu of QM lcton clous fo 4D/TSC t=, compa with thos of D-atom an D molcul Th fom 3 of 4D/TSC t= wav function is givn,3,4 as, Wav Function fo 4D/TSC t= Ψ4D ~a [Ψ Ψ + Ψ Ψ]XsS,S +a [Ψ ΨD4 + Ψ4 ΨD]XsS,S4 +a3 [Ψ ΨC4 + Ψ4 ΨC]XsS,S4 +a4 [Ψ ΨD3 + Ψ3 ΨD]XsS,S3 +a5 [Ψ ΨC3 + Ψ3 ΨC]XsS,S3 +a6 [ΨC3 ΨD4 + ΨC4 ΨD3]XsS3,S4 6-ons of osonoz lcton-pais +, which foms gula Ttahon P 4-Elcton-Cnts at Vtxs of gula Ttahon P us = Ψ = /π / /a 3/ xp-/a Top quation 3 This TSC systm has oubl symmty of gula ttahons fo utons an lcton-cnts, namly th oubl Platonic symmty which is th most ial systm in 38

14 3-imnsional connsation squzing into th cntal focal point Cnt-of-Mass; CMS. 3.. Langvin Equation fo 4D/TSC Th systm Coulomb ngy an its ivativ a givn in ou pvious wok,. W wit h sulting final Langvin quation fo Mont-Calo calculation. t.85 Vs 6m 6 t [ t] t;, t f t f ' t 3 with f ' t f t f t 33 Ec f t [ ]mo[ X ' ; t] 34 ' ; t xp[ ' t / ] 35 Fo QM-nsmbl avag quation, w obtain, 6m t.85 Vs 6 ; m, Z 6.6 ' 4 36 Tim-pnnt potntial fo this quation was givn as, V tsc ':.85 t 6V s t t; m, Z. ' [ t t] H w fix m= an Z= fo V s *, potntial fo numical calculation. Th thi tm of ight si of Eq. 36 givs about 5% positiv bias to main Coulomb foc fist tm, an was mg in th fist tm by multiplying facto.85 in th numical calculation by th Vlt mtho. In Fig. 7, w show th calculat tapping potntial of 4D/TSC, compa with that of 6D - /OSC shown lat. 4Do H/TSC kps in avag th always acclating foc in its connsation motion, hnc it can conns ultimatly until whn TSC-min stat 39

15 about - fm aius coms, as illustat in Fig. 8. On th contay, 6D - /OSC convgs to =4 pm on th way of connsation w iv quation lat. Within th psntly stui 5 kins of D-clusts, only 4D/TSC can conns ultimatly to vy small chag nutal ntity. Main Tapping Potntial of 4D/TSC an 6D/OSC 5 Vosc V Vtsc V Potntial V gs =4 pm fo 6D/OSC pm Figu 7. Tapping potntial of 4Do H/TSC, always attactiv, compa with 6D - /OSC potntial which has convging point 4 pm at its goun stat 3

16 TSC foms Elcton fs % Elcton p o 4 = 4x.8 fm Minimum TSC 4 H 4 H Duton 5 fm 3 8 * fomation 4 ak up Figu 8. Connsation of 4D/TSC an 4D-fusion to two 4 H-paticls bak-up Numical solution of Eq. 36 was obtain by a comput co bas on th Vlt tim-stp mtho, a stana sult is shown in Fig. 9. TSC Stp vag <ft>, pm o E kv fs - Tim fs pm E kv Figu 9. Numical sults of 4D/TSC connsation motion; tim-vaiation of < > an man uton kintic ngy <E > 3

17 Th connsation tim of 4D/TSC is vy fast as.47 fs. s w show in th nxt sction, 4D fusion action taks plac with almost % pobability in th final stag of connsation within th tim intval of about x - s. Fo oth tails of timpnnt bhavio of TSC ynamics with iffnt conitions, s ou pvious pap Estimation of Fusion ats Th Gaussian wav function was quit usful fo making QM-nsmbl avaging of Langvin quation, but unfotunatly th accuacy in its tails fo vy small valus is not high nough to apply fo th stimation of bai facto of - pai an 4 clust, as w illustat th lation btwn tapping shil Coulomb potntial, wav function an th vy shot within = about 5 fm ang of nucla stong intaction, in Fig.. To obtain usabl accuacy in bai facto, w intouc th Havy Mass Elctonic Quasi-Paticl Expansion Mtho HMEQPET to povi quivalnt tim-pnnt potntials fo th squzing 4D/TSC systm. Dtail sciption is givn in ou pvious pap. iabatic Potntial fo Molcul * an its goun stat squa wav function Stong F. ' ; t xp[ ' t / ] Scn Engy a Coulomb Potntial t gs b Vsmin ~5fm Vs * goun stat Scn Tapping Coulomb Potntial -V Figu. lation btwn - tapping shil Coulomb potntial, wav function an th vy shot within about 5 fm ang of nucla stong intaction. y calculating Gamow intgal fom gs to, w obtain bai facto fo fusion at stimation. Tim-pnnt o quivalntly -pnnt bai factos w calculat as w copy fom ou pvious pap as givn in Tabl-. TSC stats at =74 pm an 3

18 connss vy apily to ach at th final TSC-min stat with =.6 pm.6 fm in this cas. On th way of connsation, TSC passs th quivalnt stat with that of -muon molcul fo which w hav fnc ata 6 to show goo agmnt with ou calculation. ai factos fo an 4 fusion w calculat using th WK appoximation with Gamow intgal. Fusion ats fo stay molculs w thn calculat bas on th Fmi s fist goln ul, n W Pn 3.4 Pn W 38 H P n is bai facto fo nd-clust an <W> is th avag valu of imaginay pat of nucla optical potntial 7. Th xtapolation of <W> valu to 4 fusion was ma 5 by using th scaling law W PEF with PEF-valu which is givn in unit of ivativ of on pion xchang potntial OPEP simpl cas of Hamaa-Johnston potntial fo pion xchang mol givn by OnPEF V OPEP x VOPEP x,.43 x, xp x V OPEP x v S x x x 4 m c H x fm.43 an S is th tnso opato S 3 3 f m c c. an v 3. 65MV n is th isospin opato fo n-p chag pion xchang an is th spin opato fo nuclon stat. Th tabl of <W> valus is givn in ou pvious pap. Tim-intgat fusion yil p TSC gnation was givn by th following fomulas, 4 xp 4 t t t c 4 33

19 3 4 t 3.4 W P4 ; t.88 P4 ; t tc t 4 3 c t t.88 P4 ; t t 4 43 tc P 4 : t t Macoscopic fusion at is givn by Y4 Q tsc 4 46 W hav obtain that 4D fusion may tak plac with almost % yil p a TSC gnation, so that macoscopic 4 fusion yil is givn by simply with TSC gnation at Q tsc in th xpimntal conitions of CMNS. Howv, whn w consi that on uton has spin-paity + an combination of 4 has total spin stat 4, 3,, an, th 4 fusion with out-going channl to two 4 H +:gs paticls is fobin, by spin-paity consvation fo S-wav in/out channls, xcpt fo th + spin-paity stat T= of 4 combination, to b xplain tail analysis incluing P-wav an D-wav stats with isospin lswh. 34

20 Tabl-: bai factos un 4D/TSC connsation ai Factos of 4D/TSC un connsation, Calculat by HMEQPET Co =gs pm.6 TSC-min μ *, 74. D molcul P ; D ai Facto 4.44E-.6E-.43E E-5 9.4E-7.E E- 6.89E E-.6E-3.3E-46.E-85 P4; 4D ai Facto.98E-3.E-4.5E-6.E-9.6E-3.E-8 9.4E- 4.75E E E-64.69E-9.E-7 Tabl-: Fusion ats by Fmi s goln ul fo stay molculs Molcul =gs pm Pn ; - Facto <W> MV λ f/s 74..E E E E-7.85.E-9.8.4E+..98E-3 6 D *, μ 4D/TSCmin λ4 f/s 3.7E+ Th ultimat connsation is possibl only whn th oubl Platonic symmty of 4D/TSC is kpt in its ynamic motion. Th sufficint incas sup scning of bai facto is also only possibl as fa as th Platonic symmtic 4D/TSC systm is kpt. Thfo, th shoul b always 4 utons in bai pntation an fusion pocss, so that 4 simultanous fusion shoul tak plac pominantly. Th potion of D usual fusion at is consi to b ngligibl. Majo nucla poucts of 4D fusion a two 3.8 MV α-paticls 5,7. 4H/TSC shoul conns in th sam way until whn TSC-min stat with classical lcton aius.8 fm coms, but no stong intaction xists among potons an will mak p to 4p captu 35

21 tansmutations with host mtal nucli whn 4H/TSC has sufficint ift CMS momntum. CM 6D-/OSC =5.9 pm = =.44 -coss=.44 -CM =/.73 = /.44= -CM = /.449 =.73/.44 Elcton-Cnt Figu. Illustation of 6D - /OSC systm 4. Langvin Equation fo 6D - /OSC To fulfill th othogonally coupl Platonic symmty fo 6D-clust, 8 lcton cnts shoul appa on th cnt lins of 8 gula tiangl facs of 6 octahon; s Fig.. Thfo, th Platonic OSC shoul b with - ngativ ion stat. Th Langvin quation fo 6D - /OSC bcoms as, t 9.3 m t [ t] Vs 4 t;, t f t f ' t 47 Th ffctiv tapping potntial of this systm was alay givn in Fig. 7, which tlls us that 6D - /OSC os not mak ultimat connsation. Howv, in tansint connsation pocss, w may hav small pobability that - istanc woul appoach in 36

22 shot - istancs than 4 pm of its goun stat an 6D fusion at may b somhow nhanc. W n numical stuy fo this. W n a iffnt stuy on if th xists a connsing systm of nutal 6D-clust fac-cnt ocahon by coupling of two octahons on of 6 utons an th oth of 6 lcton-cnts. 5. Conclusions Platonic Symmtic angmnt alizs Engy-Minimum Stat of Many-oy Systm. Platonic symmty appas in D-atom, D, D +,D 3 + molcul, an 4D/TSC. 3 Platonic symmty appas in CMNS of 4D/TSC fo both of th Coulombic intaction an th stong intaction. 4 Dynamic Platonic symmty is of ky fo sup-scning of Coulomb pulsion an 4D Clust Fusion. 5 W hav obtain goo solutions of molcula ynamics with Langvin Equations, fo Platonic symmtic systms as, D, D, D 3 + an 4Do H/TSC. 6 It was shown that about % 4D-fusion p TSC gnation is possibl, by th psnt wok, in th conns matt nucla ffcts. 7 Only 4D o H/TSC can conns ultimatly to a vy small chag nutal ntity with - fm aius siz, as fa as 5 kins of D-clusts stui in this wok. 8 osoniz /+-/ coupling fo th --- systm maks D typ facs of 4Do H/TSC to hlp its ultimat connsation. 9 6D - /OSC convgs its connsation at about =4 pm, but clos - istanc in tansint may appa with small pobability. Singl <- >-cnt stats fo th -- D + typ facs of D 3 + ion molcul an 6D - /OSC nhanc constaint fiction fo thi connsation. 37

23 cknowlgmnts Th authos a gatful to D. Pt Haglstin, D. Talbot Chubb, D. Ttsuo Sawaa, D. Jacqu Dufou, D. kia Kitamua an D. Pt Gluck fo thi kin commnts an citiqus fo impoving this wok. fncs. Takahashi, N. Yabuuchi: Conns matt nucla ffcts un platonic symmty, submitt to Poc. ICCF3, Sochi, 7. Takahashi, N. Yabuuchi: Stuy on 4D/TSC connsation motion by non-lina Langvin quation, submitt to Poc. Nw Engy Tchnologis, mican Chmical Socity, 7 to b publish fom Oxfo Univsity Pss 3 N. Yabuuchi,. Takahashi: Fom of nucla fusion in soli cystals, submitt to Poc. ICCF3, Sochi, Jun, Takahashi, N. Yabuuchi: On connsation foc of TSC, Poc. sti6 Wokshop, J. Conns Matt Nucla Scinc, Vol., No. 7, to b issu 5. Takahashi: Duton clust fusion an ash, Poc. sti5 mting, 4, J. Conns Matt Nucla Scinc, Vol., No. 7, to b issu 6 G. Hal, T. Tally: Tans. Fusion Tchnology, Takahashi, N. Yabuuchi: Fusion ats of bosoniz connsats, Poc. sti6 Wokshop, J. Conns Matt Nucla Scinc, Vol., No. 7, to b issu 8 I. D. Ptsalakis, t al.: J. Chm. Phys., Takahashi : TSC-Inuc Nucla actions an Col Tansmutations, J. Conns Matt Nucla Scinc, Vol, No. 7, to b issu T. Hamaa, I. Johnston: Nucla Physics, Takahashi: TSC-inuc nucla actions an col tansmutations, J. Conns Matt Nucla Scinc, Vol, No. 7, to b issu Multipl sonanc Scatting T. Toimla 38

The angle between L and the z-axis is found from

The angle between L and the z-axis is found from Poblm 6 This is not a ifficult poblm but it is a al pain to tansf it fom pap into Mathca I won't giv it to you on th quiz, but know how to o it fo th xam Poblm 6 S Figu 6 Th magnitu of L is L an th z-componnt