The Circular Current Loop as a Model of Fundamental Particles

Transcription

1 Th iul ut Loop s Modl of Fudtl Ptils D.-Ig. Güth Ldvogt Stuwg 9 Hug Gy -il: g.ldvogt@li-dsl.t Astt. Th pstd iul ut loop odl vls tht hgd fudtl ptils suh s th lto osist sstilly of lti d gti gy. Th gti poptis hv th s od of gitud s th lti os. Th ltogti fild gy is th oigi of th itil ss. Th Higgs oso, xistig o ot, is ot dd to xpli ptil ss. Th w d stog ul fos othig ls th gti fos. Th W d Z osos s wll s th gluos ot t ll od i gluig y ptils togth. Th gti ot of fudtl ptils is ot olous! Th oly idits th xist of sll dditiol out of iti gy. Thus fudtl ptils ot puly fild-li suh s photos d ot (sstilly) ss-li suh s tos, thy pst spil id of tt i tw. Thi iti gy is oviously ot du to y ltivisti fft, ut is ltd to idpdt physil lw tht povids, togth with th gti gy, th gul otu xtly to ћ/. Fudtl ptils (t lst) two-disiol. I th siplst s thi o osists of two oti, ly idtil ut loops. Thi ltiv dsig dtils, th oly fto, d th ottiol vloity of th uifoly distiutd lty hg follow fo th stility oditio, i.. lti d gti fo l, d do ot dpd o th ptil s st ss! Fudtl ptils ojts of lssil physis.. Physil Poptis of Fudtl Ptils. Sutoi ptils tht ot sudividd gdd to fudtl. This iluds tht thy stl. Typil fudtl ptils i this ss th lto d th posito. This ivstigtio is povisiolly stitd to th. Thy htid y th followig fou si titis: hg ± oly of gti ot st-ss o gul otu ћ/ As log s o dditiol titis of fudtl ptils ow, ths fou si os suffiit to dsi fudtl ptil. A ptl odl of fudtl ptil hs to pst its fou titis utly d osisttly. This sus tht th odl lso psts divd titis,.g. Boh gto B (.) Mgti ot B (.) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

2 opto wvlgth Rst-ss gy (.) (.) Stg ut tu: As lto d posito do ot show y itl stutu, utu his gds th to poit-ptils, i.. to hv o disios t ll. Suh ojt would totlly ulisti d o-physil. It ould ot hv y physil poptis: hg dsity d ss dsity would ifiit; gul otu ( spi ) d gti ot ould ot xist. Fotutly th is pioig xptio. Th ppoh of this igious ivstigtio is uit difft fo this o, ut th sults d olusios ly th s.. Th iul Loop s Modl of Fudtl Ptils. A o-disiol iul ut loop hvig fiit dius d yig th hg is th siplst odl tht fulfill th fou physil uits tiod ov. Th hg ust distiutd hoogously ov its iuf π, d it ust ott t ostt vloity v φ. Ths th oditios fo ostt ut d ostt gti fild. v I (.) This is th oly h tht th odl pst stl, o-ditig ptil. Th ut I ts gti ot I (.) wh v (.). Th Ntu of th Rst-Mss of Fudtl Ptils. As pstd h th ut loop odl is ltogti; silly it dos ot ow stss d st-ss gy. Th odl povids ltostti fild du to its hg ± d ostt gti fild du to th ottio of th hoogously distiutd hg t ostt vloity. osutly th gy ivtoy is xptd to puly ltogti s it is with ltogti wv. y Figu : O-disiol iul ut loop. x Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

3 Th hg ± dos ot xist s ottd ottig poit-hg, ut s ostt hg dsity wh (.) (.) is th opto dius of ptil hvig st-ss. y Id Idos ρ I x P(ρ,) Figu. Goti Dtils Th out of th lty hg is tity tht is foud to xtly th s with ll lodd ptils. Additiolly thy hv gti ot iludig stg oly. This s tht th hg (dsity) of ll ths ptils is ottig t ostt gul vloity. Th iul ut loop pstig ltogti ptil odl s tspt tht th st-ss is silly illusio sultig fo th ltostti (oulo) fild d fo th gti (Lot) fild of th ptil. Thotilly th ltogti fild is spd out ifiitly wid. Wh ptil is ltd i lti d/o gti fild, th du to th fiit vloity of light th ltogti fild of th ptil is distotd. Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

4 Th distotd fild tis to pull its spig tl pt. This stoig fo is itptd s ig du to th iti of ss. Th st-ss gy of th ptil is uivlt to th su of its lti gy l d gti gy g : l g (.) Th ptil dos ot possss lti gy l d gti gy g d dditiol stss gy. Th tu of th ptil gy is ltogti, d th st-ss is fititious tity i spit of th ft tht it sud.. Th ltogti Poptis of th iul ut Loop. Th xt thtil ttt of th iul ut loop is ot tivil. But th sults of th xt lultio vy usful wh th iul ut loop is gdd s odl of fudtl ptils o t lst si lt fo uildig up suh odl. O do i pop wy th sults y sv fov s uivsl pltfo fo ll posptiv lultios. At fist th goti ltioship tw il with dius, td t th oigi of othogol oodit syst, d poit P(ρ,) hs to lyd. I Figu th il is pld i th ρ-pl of ylid oodit f. I Figu, th dist of poit o th il hvig th oodits d to poit with th oodits ρ d = is giv y si os (.) As si os (.) (.) os: os (.) Th dist D of th poit o th il to poit P(ρ,) i th ρ--pl is th giv y o D os (.) D os (.5) It will tu out to usful to itodu th vitio (.5) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

5 Thus th dist D os D os (.6) Th ltostti situtio is htid y th otiuous d hoogous distiutio of th hg ov th iul otou: d d d (.7) A ptiul hg lt gts t th poit P(ρ,) th itl otiutio dv to th ltostti pottil V d dv d (.8) D os Th ltostti pottil V t th poit P(ρ,) is th su of ll ths otiutios oud th oplt il: d V (.9) os Th gti situtio is dsid y th vto pottil A tht is td y th ottio of th uifoly distiutd hg t ostt vloity v, thus odig to (.) d (.) pstig ostt ut v I (.) Aodig to th vto hvio of th ut, h lt Ids = Id is oly fftiv with its opot vtil to th x xis: I dsff I os d (.) Thus th vto pottil it da is Ids ff I os d da da (.) D os Th vto pottil A of th ut I log th il t th poit P(ρ,) is Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 5

6 Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 6 os os d I A A (.) Both th itgls i (.9) d (.) ot solvd i losd fo. It is wll-ow tht thy tsfod to oitios of th two stddid lliptil itgls. But this would oplit th situtio. Th supio thod is to dvlop th idtil doito of th two futios to itgtd ito pow sis: x x x x x x (.) wh os x (.5) Th offiits lultd odig to th usio foul (.6) Thus th two itgls witt s os os d d (.7) os os os d d (.8) Th followig suvy shows wht hpps: d (.9) si os d xd (.) os os x (.) si d x (.)

7 Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 7 os os os x (.) si si d x (.) Oly th ts with v xpots otiut to th solutios. Thus th itgl fo th ltostti pottil V slts fo th oth futio (.) th v ts d th itgl fo th vto pottil, s h t is ultiplid y os, slts th odd ts. This idits th los ltioship tw V d A, d it vls tht oth othogol to h oth. I gl, os d (.5) d ) ( ) )( ( os d (.6) Fo (.6) follows ) ( ) )( )( ( os d (.7) osutly th usio foul fo th itgtio sults is d d ) ( ) )( )( ( ) ( ) ( ) )( ( os os (.8) Aodigly th fist itgtio sults os d (.9) 8 6 os d (.)

8 Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg os 6 6 d (.) I od to sui th v ts of (.) it is usful to st. This tsfos (.6) d (.7) to (.) d os d (.) osutly th ltostti pottil odig to (.9) witt s V (.) wh (.5) o (.6) Th fist offiits (.7) Siilly th fil fo of (.) is hivd y sttig = +. This sults i d os os (.8) d I A A (.9)

9 Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 9 wh (.) o (.) Th fist offiits (.) 5. gis Btw Two iul ut Loops. As tt of syty ll poits o sod oti il hvig th dius ρ = d xil dist = fo th oigi hv th s vlus V d A gtd y piy iul ut loop positiod t th oigi. Th ltostti pottil of il is, V (5.) Its vto pottil is, I A A (5.) wh, (5.) Wh il with dius d xil dist is lodd with th uifoly distiutd hg th ltostti gy l tw th two hgd loops is ), ( V l (5.)

10 Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg o,, l (5.5) Wh th hg otts t vloity v th gti gy g is ), ( A v g (5.6) o,, I v g (5.7) wh v v I v (5.8) I od to th lti gy d th gti gy opl, d i od to o to ovit fo, th followig idtitis usd V (5.9) (5.) Thy tsfo (5.5) d (5.7) to,, l (5.) d,, g (5.)

11 Th stddid gis Figus d. l l g, d g is stddid odigly., pstd i Stddid gis, l Figu. g, d l g of two oti ut loops with R d t ltiv dist / R. Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

12 Figu : Stddid gis l, g, d l g of two oti ut loops with dii d t dist = = d. 6. Fos Btw Two iul ut Loops. Th lti fos F l i th ditio d F l ρ i th ρ ditio d V (, ) Fl (, ) (6.) V (, ) Fl (, ) (6.) wh V (, ) is giv y (5.) d (5.). With pplyig (5.9) th sults Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

13 Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg ), ( F l (6.) d ), ( F l (6.) Th gti fos F g i ditio d F g ρ i ρ ditio A v F g ), ( (6.5) d g A v F (6.6) wh A is giv y (5.) d (5.). By pplyig (5.8) d 5.9) th sults, F g (6.7) d, v v F g (6.8) Aodig to (5.) th fto / i (6.), (6.), (6.7), d (6.8) y pld fo (6.9) It will tu out tht sotis it is hlpful o ssy to dfi th vg dius

14 R (6.) Figu 5. Stddid xil fos tw two oti ut loops with R s futio of th ltiv dist /R t. Figu 6. Stddid dil fos tw two oti ut loops t dist = = s futio of th tio / t Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

15 Figus 5 d 6 giv suvy of th ig of th fou fo utios (6.), 6.), (6.7), d (6.8). Th two gy utios (5.) d 5.) d th fou fo utios pst th oplt st to lult ll itstig poptis of th twi-loop syst. Th lti d gti fild opots (fild stgths) ot xpliitly tiod us thy popotiol to th sptiv fo opots. Moov thy ot dd h. 7. Rs, olusios, d Spil Rsults. A sigl iul ut loop hs ltostti pottil V odig to (5.) d vto pottil A odig to (5.), ut o itisi lti d gti gis. gis odig to (5.) d (5.) oly xist tw (t lst) two lti d/o gti pottils. osutly th siplst ltogti odl of hgd ptil is twodisiol d osists of two iul ut loops. Thi totl lod o + is shd tw th. Thus th dil ltostti fos of fudtl ptil lwys pulsiv. I od to stlish th oppotuity of fo l th dil gti fos ust tttiv. This uis tht oth of th uts hv th s ditio. As th two ut loops of ptil otilly positiod i th s pl, oth of th xil fos lti d gti o. Th shig of th idivisil lty lod to two ut loops is o o i physil thoy us o loop is oly thti it d o physil ojt. Th siplst odl of stl fudtl ptil is twi-loop wh th twis ot physilly sptd. Nutl ptils (.g. th uto) osist of two fudtl ptils of opposit lti polity siil to th hydog to. I otst to hgd fudtl ptils th g of utl ptils ltogti fild d osutly thi si liitd. Th six gy d fo utios ovg s <. Wh is los to uity th ovg is vy poo. Fo xt lultios i th ost itstig g.999 o th 8 ts of th sis y dd. Wh = = d siultously = = R, i.. wh th two loops touh h oth t thi iuf, th xiu vlu = is hd. I this sigul s is v v, d th divd sis fo gis d dil fos do ot ovg. Thus th sptiv vlus ot lultd. But th diff d Fl Fg ovg d lultd t ly uliitd uy. Two fsitig 9 sults of th lultios with up to ts l,, g 9 9 l g (7.) wh = (7.) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 5

16 d wh F l F,, g (7.) (7.) I stio 9 will show tht th l odl slightly diffs fo this xt s. As to positiv o gtiv sigs of gis d fos it hs to phsid tht th gy d fo utios o thi outs, i.. gtiv sigs oittd. Wh sus o diffs lultd th ot sigs t. I th s of (7.) F is gtiv d F is positiv s. Wh, it is vi vs. Th ost ipott ssg of g / / (7.) is tht g is ot sstilly sll th l wh is los to uity. Figu shows tht i th ditio t dists >> R th lti gy l dss popotiol to /R whs th out of th gti gy g dss popotiol to /R. Th gti gy is highly ottd to th viiity of th iul ut loops, whs th lti gy is o distiutd ov th totl sp. I ts of ltogti fild gy th lto is ot ifiitly sll, ut ifiitly ig. Th disstous poit-ptil thoy suppsss th gti gy us it ot xist d ot lultd i suh odl. Figus 5 d 6 dostt th s sv ist with th fos. I lity th glly gltd gti fo is vy ipott wh th dist of th two ut loops is < R. Figu 5 ipssivly shows tht t /R =.9757 th su Fl Fg hs xiu vlu d dops dow to o t =. Th sptiv hvio of two poit hgs is puly ltostti. Th gti fo is opltly issig. Thus th lti fo d osutly th totl fo would ifiit t =. Figu 5 vls tht th w d stog ul fos othig ls th th tifiilly suppssd gti fos. Th W d Z osos s wll s th gluos ot t ll od i gluig y ptils togth. Fo th g o gluo, o sud vlus vill. This xpl shows how ossil thoy hllgs th ivtio of spiitulisti ffts d ptils. Two oti, idtil ( = = R), o-disiol ut loops pst th idlid odl of ojt tht odig to (7.) d (7.) is htid y l Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 6

17 l R, d R, l R R, R, g v v (7.5) g ot lultd fo th sptiv pow sis us thy do ot ovg. But th is sipl wy out tht ws ldy sthd i stio : Wh th l R, d twi loop is ltd th lti fild d th gti fild ospodig to R, g t stoig fo h. It is tditiolly lld itil fo d dfis th itil ss. Ths ptiul sss d l R, l (7.6) gr, g (7.7) Th oplt fild gy is l R, R, (7.8) g Wh th twi loop is to pst fudtl ptil its totl hg ust o, i th s of th posito l g (7.9) (7.) As th two loops (idlid) idtil, th (7.) d (7.) As ldy tiod th ptiul loops ot physil ojts. O loop lo ould ot ywy possss gy. lti d gti gis is fo th utul iflu of thi ltogti pottils. Oly fo thtil ttt thy y y th o-physil hg /. Wh th two loops ought togth fo ifiit dist to idtil positio, gy hs to spt i od to t lti gy gist thi utul pulsiv fo. O th oth hd, th gti fo ttts th loops d thus suppots th poss of ptil tio y otiutig gtiv gti gy. Fo (7.5) d (7.) follows Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 7

18 l g (7.) R Aodig to (7.8) is l (7.) g l g oitio of (7.) d (7.) sults i l (7.5) R d g (7.6) R Th ltil gy is puly stti d thus ot otiut to so dyi ffts suh s gti ot d gul otu. Thus ths ffts xlusivly ttiutd to th gti gy g sptivly to th gti ss g g (7.7) Dvid Bg foultd this ipott sttt ldy o th ys go. Si th it should l why th gul otu is / g R R (7.8) Fo (7.6), (7.7), d (7.8) follows o wh R R R (7.9) (7.) (7.) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 8

19 Th pts of th ut loop odl will igoously osidd i stio 9. With th dius odig to (7.) th gis odig to (7.5) d (7.6) o l (7.) g (7.) osutly is l g (7.) Th ut i th twi loop is odig to (7.) I (7.5) R Fo (7.5) follows th gti ot R I R (7.6) R o R (7.7) d filly B (7.8) wh B (7.9) is th Boh gto d -.6 (7.) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 9

20 is th ystious gti ot oly. All of ths sults d sttts d so o ldy giv i, ut othig hppd i th ouity of ist physiists. 8. Bl of Fos. Figu 6 illustts tht t i th whol g / th tttiv gti fo is gt th th pulsiv lti fo. This is th uiu h tht stl fudtl ptils y xist. I th totl g / fo h vlu of / two vlus of foud wh F g Fl (8.) This s tht fo l xists wh Fl (8.) F g At th low vlu odig to (8.) th fo l is stl. Th high vlu ig xtly los to / psts istl l. Th ost itstig gio is th g wh is los to d v (8.) is los to uity. Thfo it is ovit to osid th diffs d Fl (8.) F g (8.5) Figu 7 shows s futio of i th s of stl fo l. Th sll d, th o stps ssy to iv t ot vlus. This is iditd y th ft tht th sptiv uvs stt to d upwds s soo s th u of stps is o log suffiit fo full uy. Th oss t d Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg psts th fo l oditio of th lto. Its lultio would hv uid t lst 8 stps. I od to su, th xiu u of stps ws st t 9. Bsd o th sults of stio 9 th sultig fo Fl Fg odig to (6.) d (6.6) i th viiity of th lto fo l poit is show i Figu 8. I otst to Figu 6 th ipssiv uil vlus i MN/ lultd d ltd to th iuf of th iul ut loops. Th lultio is sd o (6.) d (6.9) d

21 Ws.7 77 N (8.6) 5 R (8.7) Th solut fos F g F l d ltd to th iuf N (8.8) F g R Fl R N/ (8.9) Figu 7. Vloity oditio fo dil fo l s futio of. Difft us of stps gig fo -7 to - idit how ovg dpds o. Th oss idits th lto positio. Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

22 Figu 8. Rdil fo tw th two ut loops of th lto s futio of / 9. A Modl of th lto. Th gol of th followig osidtios is to fid th siplst physil, osistt, ltogti odl of th lto. Th gy d fo utios will spilid to th s =, i.. wh th two loops positiod otilly i th s pl i od to pst stl fudtl ptil. Th dil vloitis v φ d v φ ssud to slightly sll th : v v v v (9.) (9.) (9.) Th itstig g of is wh th lti fo is gtiv (pulsiv) d th gti fo is positiv (tttiv). Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

23 Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg Aodig to (7.) th diff tw th lti gy l d th gti gy is giv y g l (9.) As ldy tiod i (.) th totl ltogti fild gy is g l (9.5) oitio of (9.) d (9.5) sults i ß ß ß l (9.6) d ß ß ß ß g (9.7) Rlvt fo futh vlutio is th diff ß ß ß ß g l (9.8) Now it s ti to liit so o isitpttios d os fo ptd physis: Th gti ot iss fo th iul ut I : R R v I (9.9) wh R d v odig to (6.) sptivly (8.) vg vlus. Th gti ot is R v R R I (9.)

24 o R (9.) Th xpitl fidig is (9.) Of ous, this is ot, ut th itpttio of s oly is oous. As th gti ot is du to th ut I i φ ditio th sult hs to ltd to th ss φ i ditio th th to th st ss. Wht lly is sud is (9.) Th ottiol gy opot of ottig ojt ot sud y s of ltio o gvittiol xpits. Th sult of suh suts is lwys. Th diff tw d hs to itptd s dditiol ottiol iti gy osutly is i V (9.) (9.5) d (9.6) Thus, th l situtio is s it hs to i ts of lssil physis: B (9.7) wh B is th Boh gto. Fo (9.) d (9.7) iss th lw of osvtio of gul otu: R (9.8) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

25 Th xist of th iti gy i odig to (9.) is ipott fo th gul otu J v / (9.9) Wh (9.9) is pplid to th lto it hs to osvd tht oly th ottig pt g i dy (9.) of th ss otiuts to th gul otu. Th oitio of g d i y pp stg, ut it is pov tht thy uivlt. Th xpitl vlu of J is /. Thus fo (9.7), (9.), (9.9), d (9.) follows ß ß ß ß R (9.) wh v i (9.9) is sustitutd fo (9.8) is usd i (9.) th sult is v d is sustitutd odig to (6.). Wh d filly ß ß ß ß ß ß ß ß (9.) (9.) If th iti gy i would ot hv t ito out th ight sid of (9.) would hv o. This would ui ß. So, it would hv ipossil to pst listi odl of th lto. Thus th sults giv i stio 7 d i f idtifid to ppoxitios. opiso of (9.) d (9.8) lds to th ipott sult l g (9.) As th ltogti fild gy is l g (9.5) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 5

26 th followig fudtl sttts foultd: l (9.6) g (9.7) dy g i (9.8) totl l dy (9.9) Now th si utios (9.8) d (9.8) xpli vy siply: J dy v R R (9.) o J (9.) Futh lysis d vlutio of (9.) ui so sol ssuptios. Assuptio. Th hg dsity o th two iul loops is ssud to th s: (9.) I (7.9) it ws ldy sttd tht th hg oditio of th lto is (9.) Fo (9.) d (9.) follows d (9.) (9.5) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 6

27 Th fto will tu out to vy los to uity. (9.6) Assuptio. Th gul vloitis d of th ottig hgs d ul: This s d (9.7) (9.8) (9.9) Wh this is pplid to (8.) th vg vlu os (9.) Togth with (9.8) d (6.) this lds to (9.) Th ssuptios sultig i (9.6), (9., d (9.) llow to tsfo (9.) to ß ß ß ß (9.) It tus out tht is xlusivly giv y th tio / us is lso dtid y / vi th fo l oditio odig to (8.). It llows lultio fo h vlu of / th sptiv vlu of. Th lultd odig to (9.). Figu 8 shows th sult i th itstig g of /( ) d Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 7

28 (9.) s futio of /. (9.) oly, Figu 9. Th itstig g of /( ) d s futio of /. Th oss s th positio of th lto. It is igful tht s.7 d tht s.7. By ittio th ossov poit foud t xtly high uy. At sptivly is (9.5) (9.6) / (9.7) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 8

29 Th is o igoous poof tht this sigifit poit dsis lso th physil lity, ut y oth hoi would o o lss ity. Th is o dditiol oditio tht i (9.) ould justify oth sttt th. Nvthlss th followig ppoxitio is giv s th oly sol lttiv: Wh i (9.) odig to (9.) is sustitutd fo, th it os As d it is ovious tht (9.8) osutly is d This ospods to d (9.9) (9.5) - (9.5) (9.5) / (9.5) Th xpitl vlu of th lto s gti ot oly is (9.5) Dspit th ft tht th vlus of (9.5) d (9.5) vy los togth, th solutio odig to (9.6) d (9.7) is pfd us it loos o tul d o systti, d it voids th itodutio of dditiol pt. Thus th hg d th gul Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg 9

30 otu / suffiit to div th w uivsl ostts / d. Thy dfi th si stutu of fudtl ptils d su thi stility (fo l). Th fist o is shp fto whil th sod o is ssy fo th gy ivtoy. Thy disiolss us siil to th fi-stutu ostt d fudtl physil ostts us thy hold fo ll sptiv ptils d ivit gist th vg dius R /( v ) sptivly th st ss. Moov it s tht fudtl ptil y widly is (ot ds!) its dius d ds (ot is!) its ottiol vloity v φ,.g. wh it jois oth hgd ptils, s f s odig to (9.8) th gul otu osvtio oditio R (9.55) v is fulfilld. A wll-ow xpl is th o-lto to with th ulus hg Z wh Z d R / Z / Z 5. Moov it ofis tht fudtl ptils show ss-li d ltivisti hvio, whil thi itl stutu is o-ltivisti d sstilly fild-li. O ipott ssg is tht ll f fudtl ptils hvig th hg d th gul otu / ust possss th s vlus of / d, gdlss of thi st ss. Th is stog vid tht this lso holds fo th f poto d th f tipoto 6. It is ipott disovy tht th gti ot oly idits th xist of th iti gy i. It is ssy to th pstd odl pft, d it vls tht fudtl ptils itlly ot puly fild-li suh s photos d ot (sstilly) ss-li suh s tos. Thy pst spil id of tt i tw. Nvthlss th fudtl ptil is opltly ojt of lssil physis iludig th o-olous gti ot /( ) v R /. Th iul ut loop odl is two-disiol. It is th siplst o tht is osistt d psts ll dtils of fudtl ptils t xtly high uy. Aodig to (9.5) th ottiol vloity v of th lto is giv y v (9.56) Th sptiv vlu of th pstd odl is v (9.57) Th xiu d iiu vlus of th twi loop odl (9.58) Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg

31 (9.59) Fo ptil us it is ot ssy to phsi th twi loop ht of fudtl ptil. It will suffiit to lult with th vg dius odig to (9.8) R (9.6) It is dil d ougig tht th pio of th ig odl is still sussfully woig t this sujt 7. Dditio. pulitio. This pp is dditd to Dv Bg who gously suppotd its Rfs. Bg, D. L., d Wsly, J. P., Spiig hgd Rig Modl of lto Yildig Aolous Mgti Mot, Glil ltodyis, vol., o. 5, pp (Spt./Ot.,99). Nul Ptil Goup, Joul of Applid Physis, Vol., p. 59 (July 6). v d Togt,., Th uivl of Mgti d Kiti gy, Glil ltodyis, Vol. 7, No. 6, pp.- (Nov. /D. 6). Physil Msut Lotoy of NIST, ODATA Ittiolly Rodd Vlus of th Fudtl Physil ostts. (6). 5 Mills, R. L., Th Gd Uifid Thoy of lssil Physis, p. - (). 6 Bg, D. L., Spiig hgd Rig Modl of lty Ptils. Glil ltodyis, vol., o., pp. - (Mh/Apil 99). 7 Bg, D. L., All, D.P., J., lto i th Goud gy Stt Pt, Foudtios of Si, Vol. 5, No., pp. - (F. ). Foudtios of Si Fuy, oo Ss Si Rpit/Itt Atil Pg