THE MOMENT OF MOMENTUM AND THE PROTON RADIUS

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Anna McLaughlin
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1 ussian Phsis Journal Vol 45 No 5 pp () THE OENT OF OENTU AND THE POTON ADIUS S Fdosin and A S Kim UDC 59 Th thor of nular gravitation is usd to alulat th momnt of momntum of th gravitational fild of a proton whih is ompard to th orrsponding momnt of momntum of th ltromagnti fild As a rsult th proton radius is stimatd and a rlation for th momnt of momntum of th fild is stablishd whih oinids in form with th prssion of th virial thorm for nrg A proton as a quantum objt posssss its inhrnt magnti momnt ltri harg spin mass and othr haratristis whih ar masurd with a high aura in numrous primnts in lmntar partil phsis Obviousl man paramtrs of a proton ma b rlatd to on anothr b som prssions that follow from th phsial natur of intrations Charatristi ampls ar th proportionalit btwn th magnti momnt th spin and th spifi harg obsrvd for th majorit of lmntar partils and th proportionalit btwn th spin and th squard mass for partils on gg trajtoris In quantum hromodnamis it is supposd that th intgrit of a proton is providd b th strong intration btwn its thr onstitunt quarks and th fild quanta gluons With anothr approah nular gravitation [] is introdud b analog with onvntional gravitation whr th intgrit of osmi objts is du to th balan of th attrating gravitational fors and th rpulsing ltromagnti fors of mattr partils In this papr in trms of gravitational fild thor a ondition is plad on th momnt of momntum of a proton and its radius is stimatd Th salar gravitational potntial insid a proton for th as of a homognous dnsit distribution of th mattr is prssd with propr boundar onditions as πγρ( r ) Γ ψ ( r) = for ψ ( ) = ψ( ) = Hr Γ is th nular gravitation onstant; ρ is th dnsit of th proton mattr; r is th moving radius and and ar th proton mass and radius rsptivl In th stati as th alration of a mattr undr th ation of a fild is dtrmind in trms of th potntial gradint: Γ ( r) ρ = ψ = r = r () r whr (r) is th mass of th mattr within th radius r For a rotating proton th gravitational fild aquirs a momnt of momntum whos volumtri dnsit aording to [] is found b th formula whr Ω is th torsion of th gravitational fild g = Ω () 4 πγ Prm Stat Univrsit Translatd from Izvstia Vsshikh Uhbnkh Zavdnii Fizika No 5 pp 9 97 a Original artil submittd Novmbr //455-54$7 Plnum Publishing Corporation

2 To stimat th torsion insid th proton w prform an instantanous Lorntz transformation of th gravitational fild tnsor whos omponnts ar th omponnts of th vtors / and Ω from a rsting fram of rfrn S into a fram of rfrn S whih movs with vloit V along th -ais Sin th torsion Ω in S is qual to zro nglting th Lorntz fator w find for th fram of rfrn S = + V Ωz = V z z V Ω = z V () Ω = Vz Vz Ω Ω = V Ω + Ω z = V z V V In virtu of th rlativit of a motion in th fram of rfrn S th proton also movs with th vloit V but in th rvrs dirtion If th transformation is prformd from th fram of rfrn S into S at ah point insid th proton th linar vloit V an b prssd in trms of angular rotational vloit w and sphrial oordinats r θ and ϕ as V = w rsinθ and () and () an b rwrittn: 4πΓρrsinθ osϕ = 4πΓρrsinθ sinϕ = z 4πΓρrosθ = 4πΓρ wr sinθ osθosϕ Ω = (4) 4πΓρ wr sinθ osθsinϕ Ω = 4πΓρ wr sin θ Ω z = For not vr grat vloitis w ma nglt th additions to th fild omponnts (4) that appar du to th fat that th transformations should b prformd in fat in frams of rfrn rotating along th z-ais rathr than in loall inrtial frams of rfrn Aording to (4) if th proton rotats ountrlokwis th intrnal torsion Ω z is dirtd vrwhr opposit to th z-ais whil th projtions of th torsion Ω on th plan z = onst ar dirtd awa from th z-ais With prssion () w find th omponnts of th momntum dnsit vtor of th gravitational fild insid th proton: 4π Γw ρ r sinθ sinϕ g = ( Ωz zω ) = 9 4π Γw ρ r sinθ osϕ g = ( zω Ω z) = 9 gz = ( Ω Ω ) = Th vtor g points in th sam dirtion as th linar vloit of rotation of unit volums of th proton mattr To alulat th momnt of momntum of th gravitational fild insid th proton it is nssar to multipl th modulus of th vtor g b th distan from th z-ais that is b r sin θ and thn intgrat th rsult ovr th proton volum: 55

3 π π 7 w 6 w 4πΓ ρ π Γ ρ L = r dr sin θdθ dϕ = 9 89 In viw of th prssions for th proton mass distribution of th mattr th quantit L an b writtn 4 π = ρ and spin I = 4 w for th as of a uniform 5 Γ I L = (5) th vtor L bing dirtd along th spin I W assum that thr is onl on tp of dgnrat objts in vr gravitational fild whih hav th maimum mattr dnsit and aordingl th highst gravitational and ltromagnti filds For nular gravitation and onvntional gravitation objts of this tp ar rsptivl nulons and nutron stars Thn it should b ptd that in (5) th momnt of momntum L of th gravitational fild is qual to th proton spin I Atuall if it wr th as that L > I thn th gravitational fild would start th rotation of th proton thrb inrasing its spin Similarl th ltromagnti prssur on th mattr is dirtd along th momntum dnsit vtor of th ltromagnti fild and is proportional to th fild nrg absorbd b th mattr B rduing th quantitis L and I in (5) w an stimat th proton radius: 5 Γ 6 = = 67 m (6) radius Th nular gravitation onstant Γ in (6) is found from th ondition that in a hdrogn atom within th Bohr th gravitational for is qual to th ltrostati on: B Γ B 4 B = πε Γ= = 54 m kg s 4 πε 9 (7) whr is th lmntar ltri harg; ε is th diltri onstant and and ar th proton and th ltron mass rsptivl For omparison with th rsult (6) w find th proton radius b othr mthods [] A nutron and a proton form togthr an isotopi dublt and ar vr similar to on anothr in proprtis Th diffrn btwn th masss of an ltriall nutral nutron and a proton of harg an b asribd to th mass-nrg of th ltri fild of th proton: n K = 4 πε Putting K = 6 as for a uniforml hargd ball and substituting th nutron mass n and th vloit of light 6 w find th proton radius: = 668 m In th planation of th d Brogli wavs aompaning moving partils in trms of th ltromagnti fild a ondition plad on th partil sizs has bn found For protons w obtain h 6 = = 66 m whr h is Plank s onstant Eprimntall dtrmind valus of th proton radius ar rathr los to th valu givn b (6) In this as as a rul th man-squar harg radius is dtrmind whih an b gratr than Thus in primnts on ltron q 6 sattring b protons [] it has bn found that q = 75 m Aording to [] th ross stion for th intration of nulons with on anothr that is stablishd at nrgis ovr V is 8 mb In th lassial limit this ross stion an 56

4 b assumd to b los to th gomtrial ross stion of olliding partils that is to 6 < 78 m π Thn w hav Equalit (7) allows us to rlat th gravitational and th ltromagnti nrg in a proton For ths nrgis w an writ KΓ U = K W = 4 πε whr K and K ar offiints dpnding on th mass and th harg distribution rsptivl; for a uniform distribution w hav K = K = 6 Putting K K with th hlp of (7) w gt U W = 865 that is th gravitational-to-ltrostati nrg ratio is approimatl qual to th proton-to-ltron mass ratio Lt us now rturn to rlation (5) to support th onlusion that th momnt of momntum of th gravitational fild insid a proton is qual to its spin Assum that th proton harg is uniforml distributd ovr a volum of radius and th magnti momnt P m is onntratd at th ntr and dirtd along th z-ais To alulat th momntum dnsit of th ltromagnti fild outsid th proton for r > w us th following onvntional prssions: g out =ε E B µ ( Pm r) r Pm E = r B = 4 πε 5 r 4 π r r whr E is th ltri fild strngth; B is th magnti fild indution and µ is th magnti onstant Th vtor g out lis vrwhr in th plans z = onst is normal to th z-ais and rotats ountrlokwis For its modulus w an writ in sphrial oordinats: µ P m sinθ g out = 5 6 π r Th momnt of momntum of th ltromagnti fild outsid th proton is dtrmind b th volum intgral from r = to infinit: L out µ P = = m r g out dv V 6 π Th ltri fild strngth insid a uniforml hargd proton th modulus of th momntum dnsit vtor and th momnt of momntum of th fild ar givn b r () E = r 4 πε r g µ P m sin θ in = 6 π r µ in P L = m π whr (r) is th harg within th radius r W hav obtaind that th momnt of momntum of th ltromagnti fild insid a proton is half that outsid th proton: 57

5 Lin = L out (8) In addition th law of onsrvation of momnt of momntum rlats th mhanial momnt of momntum L q of th hargs moving insid a proton whih rat th magnti fild of th proton and th total momnt of momntum of th ltromagnti fild L : f Lf = L + L Lq + L f = (9) in out If w assum that th magnti momnt of a proton is onntratd at its ntr and is dirtd along th z-ais from (8) and (9) it follows that L q and P m ar oppositl dirtd and th magnti fild of th proton is suh as if it had bn formd du to th motion of ngativ hargs lokwis about th z-ais In this as th following qualit should b fulfilld: L q = out L In anothr opposit as th magnti momnt is not loalizd at th ntr of a proton but is uniforml distributd ovr its volum Among th wll-known objts th losst analog of a proton is a nutron star whos magnti fild mattr dnsit and dgr of dgnra ar not muh lss than thos of a proton In a nutron star th magnti fild should b frozn in th mattr bing supportd b th ordrd stat of th magnti momnts of th nutrons whos magnti momnt and th spin ar ountrdirtd Lt us imagin that th magnti momnt of a proton whih w arlir onsidrd to b loatd at th ntr now oupis th whol of th volum so that th amplitud of th magnti fild both insid and outsid th proton ould b onsidrd invariabl Thn th magnti fild pattrn outsid th proton will b th sam; howvr th magnti fild insid th proton will hang its sign and instad of (8) and (9) w shall gt = Lf = L + L = L = L Lq = Lin = L out () Lin L out in out out in Irrsptiv of th haratr of th magnti momnt distribution ovr th volum of a proton its total magnti momnt appars to b opposit in dirtion to th mhanial momnt of momntum of th partils rating th magnti momnt This is also valid for nutron stars so that thr is on mor indiation of similarit If w ompar in pairs th ltrial and gravitational quantitis L q and I and L in and L thn from () just follows th qualit of th momnt of momntum of th rotating masss of a proton or its spin to th momnt of momntum of th gravitational fild insid th proton: I = L and this has bn usd in (5) to stimat th proton radius In aordan with th forgoing for th as whr th magnti fild or mass sours ar distributd uniforml th intrnal momnt of momntum of th ltromagnti or rsptivl gravitational fild is qual aurat to th sign to half th momnt of momntum of th fild outsid th objt lation () rmarkabl has somthing in ommon with th wll-known virial thorm aording to whih th work of irrlvant fors on th ration of an objt is utd so that on half th pndd nrg gos into th kinti nrg of th objt partils whil th othr gos into th nrg of th fild and gnrall is arrid awa b radiation EFEENCES S Fdosin Phsis and Phlosoph of Similarit from Prons and tagalatis [in ussian] Stil- Prm (999) Hofstadtr Eltron Sattring and Nular and Nulon Strutur W A Bnjamin NY (96) V S Barashkov Cross Stions for th Intration of Elmntar Partils [in ussian] Nauka osow (966) 58

Lecture 14 (Oct. 30, 2017)

Lecture 14 (Oct. 30, 2017) Ltur 14 8.31 Quantum Thory I, Fall 017 69 Ltur 14 (Ot. 30, 017) 14.1 Magnti Monopols Last tim, w onsidrd a magnti fild with a magnti monopol onfiguration, and bgan to approah dsribing th quantum mhanis