Journl of Modrn Physics, 5, 6, 648-659 Publishd Onlin Aril 5 in Scis. htt://www.scir.org/journl/jm htt://dx.doi.org/.436/jm.5.657 About Ntur of Nuclr Forcs Boris V. Vsiliv Pontcorvo strt, 7-48, Dubn, Moscow District, 498, ussi Emil: bv.vsiliv@yndx.com civd 3 Fbrury 5; cctd Aril 5; ublishd 3 Aril 5 Coyright 5 by uthor nd Scintific srch Publishing Inc. This work is licnsd undr th Crtiv Commons Attribution Intrntionl Licns (CC BY). htt://crtivcommons.org/licnss/by/4./ Abstrct A nw roch to th roblm of nuclr forc ntur is considrd. It is shown tht n ttrction in th roton-nutron ir cn occur du to th xchng of rltivistic lctron. Th stimtion of this xchng nrgy is in grmnt with th xrimntl vlus of th binding nrgy of som light nucli. At tht, nutron is rgrdd s comosit coruscul consisting of roton nd rltivistic lctron tht llows rdicting th nutron mgntic momnt, its mss nd th nrgy of its dcy. Kywords Proton, Nutron, Mgntic Momnt, Bohr Atom, ltivistic Elctron, Dutron. Introduction Th hydrogn tom is on of th simlst quntum systms. This is th only systm for th dscrition of which n xct solution of th Schrodingr qution cn b found []: m ψ = ( U ) ψ, () h whr ψ is th lctron wv function, m is mss of lctron, nd U r th full nrgy nd th otntil nrgy of lctron. If w nglct th motion of th nuclus, ccording to th solution of th Schrodingr qution, th lctron in th ground stt of th hydrogn tom hs nrgy: whr =, () B How to cit this r: Vsiliv, B.V. (5) About Ntur of Nuclr Forcs. Journl of Modrn Physics, 6, 648-659. htt://dx.doi.org/.436/jm.5.657
B. V. Vsiliv B = (3) m is th Bohr rdius. Du to th fct tht th motion of th lctron in th s-shll is nonrltivistic, th chrctristic rmtrs of th sttionry stt of th hydrogn tom cn b obtind from siml condition of quilibrium of forcs cting on th lctron in this shll. Assuming tht s-shll hs rdius r nd th otntil nrgy of th lctron in this shll in th Coulomb fild of roton Tking into ccount th Bohr ostult =. (4) r m vr =, (5) (whr v = αc is th lctron vlocity on th first Bohr orbit) s th rsult of siml substitutions w find tht th rdius of th s-orbit is qul to th Bohr rdius B, nd totl nrgy of th lctron is givn by th Eqution ().. Nutron.. Mgntic Momnts of th Proton nd Nutron Th min hysicl rortis of roton nd nutron wr scrutinizd. Thr r msuring of thir mss, chrg, sin, tc. Of rticulr intrst r thir mgntic momnts which r msurd with vry high ccurcy. In units of th nuclr mgnton thy r [] ξ =.79847337 (6) ξ =.9347.. Th Elctromgntic Modl of Nutron n For th first tim ftr th discovry of th nutron, hysicists wr discussing whthr or not to considr it s n lmntry rticl. Exrimntl dt, which could hl to solv this roblm, did not xist thn. And soon th oinion ws formd tht th nutron is n lmntry rticl lik roton [3]. Howvr, th fct tht th nutron is unstbl nd dcys into roton nd n lctron (+ntinutrino) givs rson to considr it s nonlmntry comosit rticl. Is it ossibl to now on th bsis of xrimntlly studid rortis of th nutron to conclud tht it is lmntry rticl or it is not? Lt s considr th comosit coruscl, in which round roton with sd v c sinning lctron with mss m nd chrg. Th rsnc of th intrinsic mgntic momnt of th rotting rticl dos not mttr bcus of th rticulritis of th rsulting solutions (). Btwn th ositivly chrgd roton nd ngtivly chrgd lctron thr must b forc Coulomb ttrction: It is cusd by xisting of th Coulomb intrction nrgy: F =. (7) =. (8) whr is th rdius of n orbit of th rotting rticl. Th mgntic fild gnrtd by th lctron orbitl motion crts forc which is oosing to th Coulomb forc nd tnds to brk th orbit. 649
B. V. Vsiliv According to th Biot-Svrt lw n lmnt of orbit dl with th currnt J crts t distnc th mgntic fild: Th forc cting on n lmnt coil dl nd tnding to rnd th coil is Th ntir coil will rutur by th forc Th ction of this forc t v c blncs th Coulomb ttrction. Intgrting (), w find tht th lmnt coil dl cquirs th nrgy J dh = 3 [ dl,] 4 [ dl, ] (9) c π v dl d Fm = [ v,d H ], () 3 c c π v F m =, () c v d m = d. l () c π At thus s it follows from Eqution (), th nrgy of ring tring t v c will tnd to m, (3) tht togthr with th Coulomb nrgy (8) will rovid th stdy-stt of th currnt ring. As th rsult, th Coulomb forc nd th mgntic forc will b comnstd. Only th Lorntz s forc rising from th intrction of moving chrgd lctron nd mgntic momnt of th roton µ rmins uncomnstd. An obsrvr moving in mgntic fild H y ss in his rfrnc systm n lctric fild ([4], 4, (4.)): v whr β =. c Th Lorntz forc conforming to this fild is: E z = β H y β, (4) F = E = β L z H y β. (5) If th rottion is in th ln of th qutor of th roton, th mgntic fild is: In quilibrium, th Lorntz forc is blncd by th cntrifugl forc: H y µ =. (6) 3 F c mv =. β (7) Tht llows us to dtrmin th rdius of th lctron quilibrium orbit (t v c): 65
B. V. Vsiliv αξ 4 = 9. cm. c mm (8) whr α = is th fin structur constnt, ξ.79 is th nomlous momnt of roton, m nd c 37 M r msss of lctron nd roton (in th rst)..3. Sin of th Currnt ing Th ngulr momntum (sin) of th currnt ring is crtd by th gnrlizd momntum of lctron [, ] S = (9) mc β = A β c () nd dnds on th mgntic momnt of th roton A = µ,. β 3 () Aftr th substitution of vlu of th vctor-otntil A from Eqution () nd th vlu of rdius of currnt ring from Eqution (8) to Eqution (), t β w obtin nd rsctivly S. () At zro sin of th lctron ring thr is no rfrrd dirction long which would b orintd own sin of th lctron. Thrfor, own mgntic momnt of th lctron dos not mnifst itslf in stblishing quilibrium in th systm..4. Accounting for th Effct of th Prcssion of th Orbit Th rottion of th lctron must b chrctrizd by two intgrls of motion. At this moving, th nrgy of rotting rticl W nd its momnt of rottion K must b kt constnt. If β, on cn writ mc µ W = = const β r β (3) nd mr dθ K = =. (4) β dt whr w tk th roton s th origin of th olr coordint systm (r nd Θ ), nd v v β = c dr dθ = + r dt dt If to rmov β nd t from ths qutions, w obtin:. (5) 65
B. V. Vsiliv dr Wr r + =, dθ µ C mc r (6) whr C = is th Comton rdius. mc Aftr rlcing of vribl nd tking th drivtiv d, w obtin dθ u =, (7) r d u u + ( ϑ ) = dθ. (8) Thr w tk into ccount tht th drivtiv nd w indict ϑ =. π Th solution of th qution dθ c =Ω= is th ngulr vlocity of th rticl rottion π dt is th llis d u u + = dθ ( ε ) (9) u = const + cos Θ. (3) Th Eqution (8) dscribs th lmost lliticl trjctory, which rcsss round th roton: r rvolution of th lctron, th orbit rotts on π ϑ. Thus, this rcssion of th llis with th frquncy ω is surimosd on th rottion of th rticl on th lliticl orbit with th frquncy Ω : ω π ϑ = =. Ω π 4π To tk into ccount th ffct of this rcssion, instd of Eqution (8), w introduc th ffctiv rdius. Du to th fct tht this rdius is dtrmind by th rtio of univrsl constnts only, it cn b clcultd with vry high ccurcy: (3) αξ = mm c 4π 4 9.8849 cm. (3).5. Th Mgntic Momnt of th Nutron Attmts to clcult th mgntic momnt of th nutron hv bn md bfor [5] [6]. In th frm of th constructd lctromgntic modl, th nutron mgntic momnt cn b clcultd with vry high ccurcy. Th currnt J in ring with rdius crts mgntic momnt tht is roortionl to th squr of th ring: 65
B. V. Vsiliv µ =. W cn rwrit it in units of nuclr Bohr mgnton ( µ N =, whr M n is nutron mss). In ths cm units th mgntic momnt of th ring is quil µ M αξ ξ = = 4π n 4.7637. µ N mm Th rsulting mgntic momnt of th nutron is qul to th sum of th roton mgntic momnt nd th mgntic momnt of th ring: n (33) (34) ξn = ξ + ξ =.7985 4.7637.935, (35) tht vry wll grs with th msurd vlu of th mgntic momnt of th nutron (6):.6. Th Enrgy of Nutron Dcy ξ ξ n ( clc) ( ms) n.935 =.5.934 Th dnding on th rltivistic fctor v c trms of nrgy of th currnt ring form th intgrl of motion Eqution (3). At substituting in Eqution (3) of th obtind vlu of th quilibrium orbit rdius r =, w cn sily s tht t quilibrium th rltivistic trms of nrgy blnc ch othr nd W =. At th sm tim th Coulomb nrgy of th ring (Eqution (8)) nd its mgntic nrgy (Eqution ()) r indndnt on th rltivistic cofficint v c. Thir sum is not qul to zro: (36) = + m = 797 kv. (37) At th dcy of nutron, this nrgy must go into th kintic nrgy of th mittd lctron (nd ntinutrinos). Tht is in quit stisfctory grmnt with th xrimntlly dtrmind boundry of th sctrum of th dcy lctrons, qul to 78 kv..7. Discussion This consnt of stimts nd msurd dt indicts tht th nutron is not n lmntry rticl. It should b sn s rltivistic nlog of th Bohr hydrogn tom. With th diffrnc: non-rltivistic lctron in th Bohr tom forms shll by mns of Coulomb forcs nd in nutron th rltivistic lctron is hld by th mgntic intrction [7]. This must chng our roch to th roblm of nuclon-nuclon scttring. Th nuclr rt of n mlitud of th nuclon-nuclon scttring should b th sm t ll css, bcus in fct it is lwys roton-roton scttring (th only diffrnc is th rsnc or bsnc of th Coulomb scttring). It crts th justifiction for hyothsis of chrg indndnc of th nuclon-nuclon intrction. According to th rincil which ws dvlod by W. Gilbrt nd G. Glilo mor thn 4 yrs go, thorticl construct cn b ttributd to rlibly stblishd if it is confirmd by xrimntl dt. This rincil is th bsis of modrn hysics nd thrfor th msurmnt confirmtion for th discussd bov lctromgntic modl of nutron is th most imortnt, rquird nd comltly sufficint rgumnt of its crdibility. Nvrthlss, it is imortnt for th undrstnding of th modl to us th stndrd thorticl rtus t its construction. It should b notd tht for th scintists who r ccustomd to th lngug of rltivistic quntum hysics, th mthodology usd for th bov stimts dos not contribut to th rction of th r- 653
B. V. Vsiliv sults t surficil glnc. It is commonly thought tht for th rlibility, considrtion of n ffction of rltivism on th lctron bhvior in th Coulomb fild should b crrid out within th Dirc thory. Howvr tht is not ncssry in th cs of clculting of th mgntic momnt of th nutron nd its dcy nrgy. In this cs, sin of th lctron in this stt is qul to zro nd ll rltivistic ffcts dscribd by th trms with cofficints v c comnst ch othr nd comltly fll out. Th nutron considrd in our modl is th quntum objct. Its rdius is roortionl to th Plnck constnt. But it cn not b considrd s rltivistic rticl, bcus cofficint v c is not includd in th dfinition of. In th rticulr cs of th clcultion of th mgntic momnt of th nutron nd th nrgy of its dcy, it llows to find n quilibrium of th systm from th blnc of forcs, s it cn b md in th cs of non-rltivistic objcts. Anothr cs xits t th vlution of th nutron liftim. Th rltivism ffcts on this rmtr rntly nd on cn not obtin vn corrct stimtion of th ordr of its vlu. 3. Th On-Elctron Bond btwn Two Protons Lt us considr quntum systm consisting of two rotons nd on lctron. If rotons r srtd by lrg distnc, this systm consists of hydrogn tom nd th roton. If th hydrogn tom is t th origin, thn th ortor of nrgy nd wv function of th ground stt hv th form: H =, = m r 3 π ( ) r ϕ r (38) If hydrogn is t oint, thn rsctivly H =, = m r 3 π ( ) r ϕ r In th ssumtion of fixd rotons, th Hmiltonin of th totl systm hs th form: H = m r r + r (39) (4) At tht if on roton rmovd on infinity, thn th nrgy of th systm is qul to th nrgy of th ground stt E, nd th wv function stisfis th sttionry Schrodingr qution: (,) H ϕ = E ϕ (4),, W sk zro-roximtion solution in th form of linr combintion of bsis functions: ψ = t ϕ + t ϕ (4) whr cofficints ( t ) nd ( ) dndnt Schrodingr qution: ( ) ( ) t r functions of tim, nd th dsird function stisfis to th nrgy- (,) ( ), dψ i = H + V ψ, (43) dt whr V, is th Coulomb nrgy of th systm in on of two css. Hnc, using th stndrd rocdur of trnsformtion, w obtin th systm of qutions i + is = E + Y + S + Y {( ) ( ) } {( ) ( ) } is + i = E S + Y + + Y, whr w hv introducd th nottion of th ovrl intgrl of th wv functions (44) * * φφ d φφ d (45) S = v = v 654
B. V. Vsiliv nd nottions of mtrix lmnts Givn th symmtry Y = φ Vφdv * E Y = φ V φ dv * E Y = φ Vφdv * E Y = φ V φ dv * E (46) Y = Y Y = Y, (47) ftr th dding nd th subtrcting of qutions of th systm (44), w obtin th systm of qutions i( + S)( + ) = α ( + ) i S = β whr As rsult, w gt two solutions whr From hr nd ( )( ) ( ) {( ) } {( ) } α = E + S + Y + Y β = E S + Y Y E E x x + = Cx i t x i t = C i t i t Y = E Y = E + Y ( + S ) Y ( S ) iωt = +. i t i t iωt = i t i t cos t = + cos t = (48) (49) (5) (5) (5) (53) 655
B. V. Vsiliv As with th initil conditions nd or Y SY = E (54) S ( ) ( ) = = (55) C C C = = (56) C = = (57) w obtin th oscillting robbility of lcing of lctron nr on or othr roton: = ( + cos ωt) (58) = ( cos ωt) Thus, lctron jums into dgnrt systm (hydrogn + roton) with frquncy ω from on roton to nothr. In trms of nrgy, th frquncy ω corrsonds to th nrgy of th tunnl slitting rising du to lctron juming (Figur ). As rsult, du to th lctron xchng, th mutul ttrction riss btwn rotons. It dcrss thir nrgy on ω = (59) Th rising ttrction btwn rotons is urly quntum ffct, it dos not xist in clssicl hysics. Th tunnl slitting (nd th nrgy of mutul ttrction btwn rotons) dnds on two rmtrs: = E Λ, (6) Figur. Th schmtic rrsnttion of th otntil wll with two symmtric stts. In th ground stt, lctron cn b ithr in th right or in th lft hol. In th unrturbd stt, its wv functions r ithr ϕ or ϕ with th nrgy E. Th quntum tunnling trnsition from on stt to nothr lds to th slitting of nrgy lvl nd to th lowring of th sublvl on. 656
B. V. Vsiliv whr E is nrgy of th unrturbd stt of th systm (i.., th lctron nrgy t its ssocition with on of roton, whn th scond roton rmovd on infinity), nd function of th mutul distnc btwn th rotons Λ. This dndnc ccording to Eqution (54) hs th form: Y Λ= SY ( S ) Th grhic stimtion of th xchng slitting indicts tht this ffct dcrss xonntilly with incrsing distnc btwn th rotons in full comlinc with th lws of th rticls ssing through th tunnl brrir. 4. Th Molculr Hydrogn Ion Th quntum-mchnicl modl of simlst molcul th molculr hydrogn ion ws first formultd nd solvd by Wltr Hitlr nd Fritz London in 97 [8]-[]. At tht, thy clcult th Coulomb intgrl (Eqution (46)): (6) th intgrl of xchng (Eqution (46)) nd th ovrl intgrl (Eqution (45)) ( x) x Y = +, ( ) Y = x + x x x x S = + x+. 3 (6) (63) (64) whr x = is th dimnsionlss distnc btwn th rotons. B Th otntil nrgy of hydrogn tom nd with tking into ccount Eqution (6)-Eqution (64) At vrying th function ( x) ( x) Λ = x = (65) B x x( + x) + x+ ( + x) 3 x x + x + 3 x ( ) Λ w find tht th nrgy of th systm hs minimum t x.3 whr Λ x=.3.43. As rsult of rmuttions of ths vlus w find tht in this minimum nrgy th mutul ttrction of rotons rchs mximum vlu mx 9.3 rg. (66) (67) This rsult grs with msurmnts of only th ordr of mgnitud. Th msurmnts indict tht th quilibrium distnc btwn th rotons in th molculr hydrogn ion x nd its brking nrgy on roton nd hydrogn tom is clos to 4.3 rg. Th rmrkbl mnifsttion of n ttrction rising btwn th nucli t lctron xchng is showing himslf in th molculr ion of hlium. Th molcul H dos not xist. But nutrl hlium tom togthr with singly ionizd tom cn form stbl structur th molculr ion. Th bov obtind comuttionl vlution is in ccordnc with msurmnt s for both hydrogn tom nd hlium tom th rdius of 657
B. V. Vsiliv s-shlls is qul to B, th distnc btwn th nucli in th molculr ion of hlium, s in cs of th hydrogn molculr ion, must b nr x nd its brking nrgy nr 4. rg. In ordr to chiv bttr grmnt btwn clcultd rsults with msurd dt, rsrchrs usully roduc vrition of th Schrodingr qution in th dditionl rmtr th chrg of th lctron cloud. At tht, on cn obtin th quit wll consnt of th clcultions with xrimnt. But tht is byond th sco of our intrst s w wr nding th siml considrtion of th ffct. 5. Dutron Th lctromgntic modl of nutron, discussd bov, givs ossibility on nw look on th mchnism of th roton-nutron intrction. According to this modl nutron is roton surroundd by rltivistic lctron cloud. Thrfor dutron consists of th sm rticls s th molculr ion of hydrogn. Thr is diffrnc. In th cs of dutron, th rltivistic lctron cloud hs th linr dimnsion 3 cm (Eqution (8)). On might think tht ftur occurs t such smll siz of th lctron cloud. Whn n lctron jums from on roton to nothr, stil ovrl of th wv functions will not ris nd thrfor th ovrl intgrl S Eqution (45) cn b st qul to zro. In ccordnc with th viril thorm nd Eqution (37), th otntil nrgy of this systm t th unrturbd stt is =. (68) Th function Λ ( x) (Eqution (6)) t S = nd tking into ccount Eqution (63) obtins th form ( x) x( x) x Λ = + (69) (whr x = is dimnsionlss distnc btwn th rotons.) Whn vrying this xrssion w find its mximum vlu Λ mx =.8399 t x =.68. Aftr substituting ths vlus, w find tht t th minimum nrgy of th systm du to xchng of rltivistic lctron, two rotons rduc thir nrgy on Λ mx 6.3 rg. (7) To comr this binding nrgy with th msurmnt dt, lt us clcult th mss dfct of dutron whr M d is mss of dutron. This mss dfct corrsonds to th th binding nrgy 7 Md = M + m Md 3.9685 g, (7) 6 d = Md c 3.567 rg. (7) Thus th quntum mchnicl stimtion of th bonding nrgy of dutron Eqution (7), s in th cs of th hydrogn molculr ion, consistnt with th xrimntlly msurd vlu Eqution (7), lthough thir mtch is not vry ccurt. 6. Conclusions Th good grmnt btwn th clcultd binding nrgy of th nutron-roton ir nd th msurd dutron binding nrgy suggsts tht nuclr forc hs rlly th xchng chrctr dscribd bov. Ths forcs ris s rsult of th quntum-mchnicl xchng nd hv no clssicl xlntion. For th first tim th ttntion on th ossibility of xlining th nuclr forcs bsd on th ffct of lctron xchng rntly drw I. E. Tmm [] bck in th 3s of th lst cntury. Howvr, ltr th modl of 658
B. V. Vsiliv th π -mson xchng bcoms th dominnt in nuclr hysics. Th rson for tht is clr. To xlin th mgnitud nd rng of th nuclr forcs nd rticl with smll wvlngth. Non-rltivistic lctrons do not fit it. Howvr, on th othr hnd, th modl π -mson xchng ws not roductiv: it givs not ossibility to clcult th binding nrgy of vn such siml nuclus s dutron. Thrfor, th siml ssssmnt of th binding nrgy givn bov nd consistnt with msurmnts is th clr roof tht th so-clld strong intrction (t lst in th cs of th dutron) is mnifsttion of th quntum-mchnicl ffct of ttrction btwn rotons roducd by th rltivistic lctron xchng. frncs [] Lndu, L.D. nd Lifshitz, E.M. (965) Quntum Mchnics (Volum 3 of A Cours of Thorticl Physics). Prgmon Prss, Nw York. [] Bringr, J., t l. () Physicl viw D, 86, Articl ID:. htt://dx.doi.org/.3/physvd.86. [3] Bth, H.A. nd Morrison, P. (956) Elmntry Nuclr Thory. John Wily & Sons, Inc., Nw York. [4] Lndu, L.D. nd Lifshitz, E.M. (97) Th Clssicl Thory of Filds (Volum of A Cours of Thorticl Physics). Prgmon Prss, Nw York. [5] Zldovich, J.B. (965) UFN, 86, 33-34. [6] Ioff, B.L. nd Smilg, A.V. (984) Nuclr Physics B, 3, 9-4 htt://dx.doi.org/.6/55-33(84)9364-x [7] Vsiliv, B.V. (4) Is Nutron n Elmntry Prticl. Print JIN, 3-4-77, Dubn. (In ussin) [8] Hitlr, W. nd London, F. (97) Zitschrift fr Physik, 44, 455-47. [9] Glsston, S. (948) Thorticl Chmistry. Vn Nostrnd inhold Inc., Nw York. [] Flügg, S. (973) Prcticl Quntim Mchnics I. Sringr, Brlin-Nw York. [] Tmm, I.E. (934) Ntur, 34, -. htt://dx.doi.org/.38/34c 659