Theoretical study of quantization of magnetic flux in a superconducting ring

Transcription

1 Thortial study of quantization of magnti flux in a supronduting ring DaHyon Kang Bagunmyon offi, Jinan , Kora -mail : samplmoon@hanmail.nt W rfind th onpts of ltri urrnt and fluxoid, and London s quation that spify quantum phnomna of moving ltrons and magnti flux in a losd iruit similar to a supronduting ring, so as not to violat th unrtainty prinipl. On this basi th rlation btwn th ltron motion and magnti flux in a suprondutor has bn thortially invstigatd by mans of Faraday s law and/or anonial momntum rlation. Th fat that minimum unit of th quantizd magnti flux is h/ dos not man th onurrnt motion of th two ltrons in a Coopr pair as is known so far. Howvr, it is shown to b rlatd with indpndnt motion of th ah ltron in a supronduting stat. Kywords : flux quantum, fluxoid, london s quation I. Rviw In 933, Missnr and Ohsnfld disovrd th phnomnon that th suprondutor xluds th magnti fild, and thn in 935, H. London and F. London thorizd Missnr Efft that ours in th suprondutor. v d th drift vloity of th ltron and E, th ltri fild ar rlatd as shown blow basd on th Nwton s sond law in th supronduting stat m dv d = E, () whr m and ar th ltron s mass and harg. If w substitut J = ρv d whr ρ is th numbr of ltrons pr unit volum, for v d th rarrangd formula is dj = ρ E. () m If w apply formula (), Ampr s Law ( B = 4π ) J and Faraday s Law w hav ( db ) = 4πρ db m ( E = db ) (3)

2 W know that B = 0, so Thrfor, answr for furmula (4) is db = 4πρ db m. (4) db = db 0 x λ B = B 0 x λ (5) whr x is th insid distan of th suprondutor, λ is London pnntration dpth, and B is th magnti fild. Formula (5) xplains that th suprondutor xluds th mangti fild, whr its strngth xponntially diminishs aording to th dpth of its prmiability into th magnti fild on th surfa of th suprondutor. Using formula (), A = B, and Faraday s Law, w hav J = ρ A, whih is known as m London Equation proposd by th London brothrs. In 950, F. London prsntd th onpt of fluxoid by prsnting P ds = mv d ds + A ds (6) whr th lft sid is floxoid and th first trm on th right is th ltri urrnt and th sond trm is th magnti flux. Bohr s quantum ondition an b applid to th lft sid of formula (6), thus P ds = nh, (7) whr n =,, 3,...and h is Plank s onstant. From this, th stimatd valu of th minimum unit of fluxoid is givn by φ = h. (8) Formula (6) an b indud in rlation to formula (). Th ltrons in th suprondutor rats th ltri urrnt whn alratd by th for gnratd by th ltri fild, E, thus, undr th assumption that th ltri fild E is produd by Faraday s Law, w hav m dv d = E = da. (9) If w rarrang this quation, w hav d [mv d + ] A = 0 (0) [mv d + ] A = P = onstant. ()

3 Formula (6) is indud by applying lin intgral on both sids of th formula (). Also, formula (6) shows that th xtrnal magnti flux boms quantizd going through th ring hol of suprondutor by applying th lin intgral on its insid path assuming that thr is no ltri urrnt in th dp insid of th rlativly thik suprondutor. nh =0+ A ds, () φ = h is indud from th formula (). Howvr, its masurd valu from th supronduting ring was φ = h = T m. (3) Currntly, w bliv that th suprurrnt is arrid by pairs of ltrons. II. Th problm of th fluxoid onpt ) From th fluxoid formula, P ds = mv d ds + A ds, P on th lft sid of th quation rprsnts th motion of ltrons and th first trm on th right sid, v d an b intrprtd as th drifting vloity of th ltron. This intrprtation is supportd by Bohr s Quantum Condition that is applid to P, whih oms from th appliation of th lin intgral on th momntum with its motion path. v d th ltri urnt of th suprondutor may b intrprtd as th drift vloity. Howvr, onsidring th drift vloity is mm pr sond and th vloity of th ltron is usually svral thousand km pr sond in th Frmi lvl rang, w an onlud that formula (6) an not haratriz th physial prinipl of th suprondutor du to th lak of orrspondn btwn two numbrs of both sids of th formula. ) Th formula, nh =0+ A ds an not provid authnti xplanation vn though it attmpts to indiat that th magnti flux boms quantizd undr th assumd ondition of no ltri urrnt in th dp insid of th suprondutor. To b spifi, by ombining J = ρ m A, Ampr s Law B = 4π J, and B = A, w hav ( A) = 4πρ m A. Rarranging this quation and using A = 0 yilds A = 4πρ m A, thn A = A 0 x λ. That is, from J 0 and A 0 in th dp insid of th suprondutor, th aurat dfinition of formula (3) is givn by nh = (4) 3

4 This formula shows that th xisting vidn that th xtrior magnti flux passing th ring hol of th suprondutor boms quantizd is inorrt. Sin thr is a flow of th xtrior magnti fild in th hol of suprondutor vn with J 0 and A 0 in th dp insid of th suprondutor, on an assrt that th sond trm on th right sid boms A ds = φ rathr than A ds =0. Howvr, this proof may show an altrnativ thory that th first trm on th right sid of th quation an not b mv d ds = 0 baus th magnti fild in th ring of th suprondutor rvals not only th xtrior magnti fild but also th magnti fild ratd by J, th suprfiial ltri urrnt. Thrfor, this shows that th vidn of th magnti flux quantum using th formula (3) aluating th xtrior magnti fild that passs th ring at th lin intgral path insid of th suprondutor is a fallay. 3) Currntly aptd fluxoid onpt dos not math th xprimntal rsult obtaind from Littl-park. To xplain th xprimntal rsult, n = 0 should b possibl on th lft sid of th formula (6). W an not dfin n = 0 as long as w us th Bohr s quantum ondition to dfin th fluxoid. If n =0,P = 0, whih ontradits th fat th any partil onfind in a rtain spa an not tak 0 valu aording to th Unrtainty Prinipl. III. A nw thory Th problm shown in th prsnt fluxoid onpt ssntially oms from th lak of undrstanding th Canonial Momntum Equation. Sin th partil onfind within limitd spa must tak othr than 0 valu, w nd to rstablish th onpt of th ltri urrnt of th suprondutor within th dfinition of th Unrtainty Prinipl. In normal irumstans, th ltri urrnt is onvrtd to th drift vloity of ltron within a ondutor. Th vloity sum boms 0 without th ltri urrnt. This mans thr is no th nt vloity or momntum. Th ltrons xist in multipl and gain its own vloity in as of no ltri urrnt in th ondutor. Howvr, If w total th vloity of all ltrons, th nt momntum is 0 as givn by mv k =0, (5) k whr th ltri urrnt in th solid is dlt undr this prmis lik fomula (5). Th ltrons mov as shown in formula (6) whn E, th onstant ltri fild is maintaind for th duration of t, th tim. mv f mv i = Et, (6) 4

5 whr mv f is th final momntum and mv i is th initial momntum, whih do not tak 0 valu aording to th Unrtainty Prinipl. If w sum up th vloity of all ltrons using th formula (6), (mv fk mv ik ) boms k mv fk mv ik, mv ik = 0 aording to th formula (5), thn, w hav th final mv fk = k k k k E k t. k If w sum it up, from formula (5) th sum of th bninning momntum boms 0, and rmain th sum of th vloity lmnt that ar in th sam dirtion in th ltri fild E within th ltron s nd momntum lmnts. Thrfor, w prsnt th drift vloity, v d that was usd by London brothrs lads to Using this formula, w hav Sin m dv i mv f mv i = mv d = Et. (7) m dv f mdv i = mdv d = E. (8) in formula (8) is not rlvant to th ltri fild, E, thn its valu boms m dv i = 0. Using Faraday s law E = m dv f da, w hav = m dv d = E = da. (9) Th diffrn btwn formula (9) and () is that thr ar mor alratd trms in th vloity of th ltron, ths trm ar thos of th drift vloity and quivaln. From th formula (9), w hav m dv f = da m dv d = (0) da. () Examining ths two formulas, w an spulat that F. London did not distinguish btwn (0) and () whil dvloping th fluxoid onpt. Formula (0) provids a formula that is similar to th rlational xprssion of th Canonial Momntum mv (initial) = mv(final) + A. () From formula (), w an find London Equation and a nw onpt of fluxoid onstant = mv d + A, (3) 5

6 whr th onstant inluds 0, v d is th drift vloity of th ltron, A is vtor potntial. Sin a onstant is aluatd by adding two trm, Bohr s quantum ondition an not b applid hr du to its irrlavan to th momntum of ltron. If a onstant is 0, it would oinid with London Equation, othrwis it indiats a nw fluxoid onpt. In formula (), mv and mv rprsnt th momntum that indiats th status of ah ltron, thus, Bohr s quantum ondition an b applid and an not tak 0 for thir vloity basd on th Unrtainty Prinipl. Th following is an attmpt to show that th magnti flux bom quantizd in th suprondutor with a hol lik th on in th ring of a suprondutor. For onvnin, w an substitut mv and mv for mv and mv into th formula (), so Apply dot produt on both sids of formula (4) with v, thn mv = mv + A. (4) mv v = mv v + A v. (5) By applying dot produt on both sids of formula (4) with v, thn w hav Th ombination of (5) and (6) boms mv v = mv v + A v. (6) mv v + mv v = mv v + A v + mv v + A v. (7) This shows that th sond trm of th lft sid of formula (7) and th first trm of th right ar idntial. If w rarrang (7), mv v = mv v + A v + A v. (8) Applying (intgral) alulus on formula (8), w hav mv v = mv v + A v + A v. (9) This an b simplifid to mv ds mv ds = A ds + A ds. (30) Bohr s quantum ondition an b applid to th two trm of th lft sid of formula (30) baus th momntum of ltrons boms quantizd. Thus, n h = n h + φ + φ, (3) 6

7 whr n and n ar intgr with vaus othr than 0. Sin th magnti flux in th ring of suprondutor is idntial rgardlss of th ltron s motion path, w know that (φ = φ = φ), so w hav φ = h (n n ). (3) I attmpt to alulat th xtrior magnti flux that passs th ring of a suprondutor by using Faraday s Law vn though it would b ssntially th sam proof as prsntd abov. Faraday s Law is E ds = dφ (33) W an substitut E = dp into th lft sid of th formula sin thr is no ltri rsistany in th suprondutor, thus, dp ds = dφ. (34) If w dvlop th formula (34), mv v mv v = dφ. (35) Applying alulus on both sids of (35) mv v mv v = mv ds mv ds = dφ (36) dφ (37) If magnti fild is dlivrd undr th ondition that magnti fild is missing in th ring of th suprondutor, w hav n h n h = (φ 0) (38) φ = h (n n ). (39) Th xtrior magnti flux boms quantizd in th ring of suprondutor as shown abov, thrfor, using formula () th fluxoid is givn by m dv d da = (40) d(mv d )= da. (4) 7

8 As shown in th xprimnt of Littl-park, assuming that vortx gts insrtd insid of th ring hol of th suprondutor, formula (4) volvs by using dfinit intgral [mv d 0] = [A A 0] (4) mv d + A = A 0. (43) Applying ontour intgral on both sids of formula (43) mv d ds + A ds = A 0 ds. (44) Using formula (39) w an rarrang th right sid of th formula (44) into m ρ J ds + A ds = nφ 0, (45) whr φ 0 = h and J = ρv d. As a nw fluxoid onpt, formula (45) oinids with th Littl-park xprimnt, whr n is intgr inluding 0 valu, and is fr of problms that may aus a onflit with th Unrtainty Prinipl. IV. Conlusion Th xisting fluxoid onpt ontaind a problm that ontradits th Unrtainty Prinipl by using th Canonial Momntum formula baus of no spifid distintion btwn th vloity and th drift vloity of th ltron. To rsolv this problm, w stritly dfind th ltri urrnt in th supronduting ring and th fluxoid onpt to dvlop an outom that two ltrons do not aompany in th supronduting stat. This rsult is basd on th fat that th minimum valu of xtrior magnti flux,φ is thortially h/ indud from th Canonial Momntum Equation and Faraday s Law and th prmiss of Bohr s Quantum Condition must b satisfid bfor and aftr th ltron s momntum in th suprondutor taks its fft on th magnti fild. This outom may b a usful rfrn in rsarh rlatd to th supronduting phnomnon. 8

9 Rfrns [] A. C. Ros-Inns and E. H. Rhodrik, Introdution to Suprondutivity, nd d. (prgamon, nwyork, 978), pp [] Mihal Tinkham, Introdution to Suprondutivity, nd d. (Dovr, nwyork, 004), p. 4 pp [3] J. B. Kttrson and S. N. Song, Suprondutivity, (ambrig univrity prss 999), pp [4] P. G. D Gnns, P. A. Pinus, Suprondutivity of Mtals and alloys, (W.A Bn jamin, in 966), pp