SAC-PUB-8 Al 6 Anl of -Dn Pcl Acclon fo Pln Tnn Boun * T. Pln.. Gnon oo Snfo Un Snfo CA 945 Ac Th cl lo h ncon wn onochoc ln w l n lc lcon n h nc of hn lcc nn oun. I foun h h gn of h ncon wn h l n h lcon n h own c n h ocl h l of h l cu h oun n h cn o o cncl h ncon n h u c. Boh h nnon on cu n h lcc fl h ngl ho how h ul. * Wo uo Dn of ng gn D-FG-97R476 Wo uo n Dn of ng conc D-AC-76SF55 Snfo n Acclo Cn Snfo CA 945
Inoucon SAC-PUB-8 Al 6 Th hn nfn oun go fo l-n cl cclon n ffc n l ho o f h won-woow ho n w uccfull lo n h f oof-of-ncl n fo l cclon n cuu []. Alhough h ncon wn h l n h lcon n h go oo u o lc h nfn hn oun ch l fo ung h c hcl chn fo l-n cl cclon n cuu. In nc ffn l fo h hn oun llow u o lo h ffn oun conon fo h lcognc fl: Rflc Ao Tnn Fgu llu h h c of n. Fo h oof-of-ncl n flc gol-co Kon w lo oun coonng o Fgu (. ( ( (c l oun l oun l oun Fg. : l cclon fo ( flc oun ( o lc oun n (c lol nn oun C ( n ( h n lo n ou cl [] n w foun h fo lc lcon h c ng gn fo h l qλ α Δ U co ρ nϕ π α γ wh q h lcc chg of h cl h lu of h lcc fl of h l ln w hng h fo ( Pˆ ( ρ co( ϕ λ h l wlngh α h l-lcon cong ngl ρ h olon ngl of h l n ϕ h nl ocl h of h l. I w foun h fo h flc oun h In Ron ho n h Ph Ingl ho c h lcon ng gn of quon gl of h oun onon. Fo h c of h ong oun how h In Ron cu fo clculng ng gn o no l.
SAC-PUB-8 Al 6 In h cl w nl h c ng gn fo lc lcon fo nn oun oh h In Ron ho n h Ph Ingl ho. Fo lc w wll focu oun on n-bw ngl fo h l ln w uch h h no flc l. Th Ph Ingl Mho Fo h nl w wll u h go c n Fgu. θ ψ θ θ ˆ α chg q l lcc oun ŷ ẑ Fg. : Th l-lcon ncon n h nc of h ffn lcc l Th h l ln w n n gn ( ( co( n ( nθ coθ φ ( ( co( n( nθ coθ φ ( ( co( n ( nθ coθ φ wh h lu of h lcc fl n cuu n χ h l lcc. H h nc of fcon n ( coonl on I u h h gnc l of h lcc l no ffn fo h n cuu; μ μ. A hown n Fgu θ h ngl of h ln w n u wh c o h -. fn π λ c wh λ h wlngh n cuu. Fo connu of h fl h oun w qu h φ φ n
SAC-PUB-8 Al 6 w qu h n coθ φ n coθ φ. h nl l h φ φ. Th h of h ln w n u co ( n θ θ φ co n co φ In ou c w wll u cuu oh u n own of h l. Hnc n n θ θ θ n fn n n θ θ. Thfo h fl cn ( co( ( nθ coθ φ ( ( co( n( nθ coθ ( co( ( nθ coθ φ φ 4 Incon n h u c Fo Fgu w cn h h lcc fl coonn lll o h lcon co nα wh α θ ψ. Thfo ( ( nα co( ( nθ θ φ co 5 Th lcon o c ( fl coonn c nψ. Thfo h lll lcc ( ( nα co( ( nθ nψ θ φ co 6 Aung unfo lcon loc h cn ( Th g βccoψ. coα β 7 ( ( nα co φ coψ Thfo h wo on on h cl h fl ΔU q qnα ( ( ( q ( coψ coα β co φ coψ coψ 8 4
SAC-PUB-8 Al 6 Ung l co u ( u φ u nφ w fn nα nφ q ΔU 9 ( coα β In h lc l wh β γ n << α ΔU co qλ α Δ U nφ π α γ whch no ungl h ng gn h on clcul fo h flc o o oun. A h ou cong ngl of α ± γ h ng gn h u lu of qλγ Δ U π Incon n h u In h u h lcc fl coonn lll o h lcon ( ( nα co( n( nθ θ φ co wh α θ ψ n θ l o θ Snll lw. B nlog o h u c h wo on on h cl n h gon Δ U q ( qnα whch lf o coψ ncoα β co φ coψ coψ qλ nα ncoα Δ β U nφ n φ 4 π ( ncoα β coψ W u h h l cong ngl o fo h ncon n h cuu α γ. Thn h ncon n h u Δ U h u lu of 5
SAC-PUB-8 Al 6 qλ nα qλ Δ U f ( n π ncoα β 5 π A hown n Fgu f(n ll hn n o wh ncng n n. A cn n fo h lo fo γ 5 γ n γ f(n how onl w nnc on γ..9 f(n.8.7.6.5.4... γ γ γ 5..4.6.8..4.6.8 n of fcon n Fg. : f(n u n fo ou lu of γ Thfo un h ccunc Δ U ΔU γ n h u ncon n h u uch ll hn h u ncon n h cuu. Noc h h lu fo h ng gn nnn of h hcn of h lcc l. Incon n h own c H h lll lcc fl coonn gn ( ( nα co( ( nθ θ φ co 6 n h wo on on h cl 6
SAC-PUB-8 Al 6 ΔU qnα q ( ( ( qnα ( coα β qnα ( coα β coα β co φ coψ coψ A n co ( u φ ( A φ u coα A coψ β q λ α coα β n ncoθ coθ φ π α γ coψ 7 Th ol ng gn h u of h ncon n h h gon ΔU ΔU ΔU Δ. How o fo ΔU << Δ n h U U conuon fo h ng gn n h u cn nglc. Hnc h ol ng gn ol Δ U ΔU ΔU. ΔU q λ α coα β nφ n ncoθ coθ φ 8 π α γ coψ Noc h h ng gn ll h h ln nnc on h wlngh λ n on h lcc fl lu n how h nnc on h l-cong ngl h ng gn fo h flc o o oun of quon. Th ffnc l n h hown n c n quon 8. Th how h nfnc of wo h on of whch n on h ocl h on cu h oun. To llu h connn o w quon 7 ΔU qλ α { nφ n( φ φ } 9 π α γ wh h ocl h on φ φ coα β ncoθ coψ coθ Noc h φ ooonl o h l l hcn wh c o h wlngh φ λ. Whn α γ h f ll n cn nglc. φ ( ncoθ coθ 7
SAC-PUB-8 Al 6 U/U.8.6.4..8.6.4..5.5.5.5 4 4.5 5 π π π 4π ocl h l Fg. 4: Coon wn h ng gn fo flc oun (h ln n h ng gn fo nn oun (ol ln funcon of h ocl h l Fgu 4 llu h ffc of h ocl h on on h ng gn fo nn oun. Whn h gn of h fl h ng gn n h u n n h own gon conucl. Th u lu fo h ol ng gn qγλ Δ U ( α γ π Th In-Ron ho A cu lwh [] h ho on Ponng Tho n u no nf of ng fo o o h u lf. Un h ccunc h ng gn cn hown o [4] ΔU πz S * ( ( TR ( wh h l fl n TR h cl w fl. In h nnc TR coon o h non on fo lcc l. Th cul non on ol h n nl ou uho n h [5]. In ou cul c w no n n h o gnl on of TR n cn lf fo fo h fl. Ung h och n [5] fong ln w cooon of h fl o fn h non on fl. F w fn h ln w cu of h fl of unfol ong cl of chg q. In u h onl h o f 8
SAC-PUB-8 Al 6 A μ A μj μqδ Φ μ Φ ρ q δ ( ( 4 W fo Fou nfoon on quon 4 n fn h Ung πqμ A δ μ π q Φ δ μ ( ( A Φ w fn fo h ln w cu of h cl lcc fl c πqz c δ ( c wh χ n Z μc h cuu nc. n h ln w cu of h cl fl n h u. To ch h oun conon h fon n c nfc of h lcc l w of ln w (h hoognou oluon o quon (4 hown n Fgu 5 5 6 ψ Fg.5: Tnon on fo l; Pln w fl coonn Th ln w f o lnl. Fuho h lu wll chon uch h h lcc fl oun conon n f. Th 9
SAC-PUB-8 Al 6 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 7 In of h ln w coonn h oun conon co ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 8 W n n h ln w h wll ol wh h l fl h hoonll ol n lng n h - ln. Hnc w no n n h lu of n. Fuho onl h hoonll lng ln w coonn fo whch cn ol wh h l n h f fl. Thfo fo h cl of ln w h oun conon lf o ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 9 Th ln w of h fo ( ( c f δ. Hnc ngng o ln h l funcon ng n h fl coonn n w on
( ( ( ( ( ( ± ( ( ( ( ( ( ( ( ± ( ( ( ( ( ( ± ( ± ( ( ( ( ( ( ( ± ( ± ( ( ( SAC-PUB-8 Al 6 wh n c h lu wh h l funcon δ ( c o. Snc w u h h ln w fl n ong o h lf ( Fgu 5 w onl h ng oo whl fo h fl n h own c n h ong o h gh w h o oo. Fo h fl n h l w h o u h h coonn lng n oh con. Thfo wll connn o ll h ln w coonn n u l ( ; ( ( ; ( ( ; n( ( ; ( n Th of quon n h o l fo ll n. Th f whn h gun of h Fou nfoon qul fo ll n. Hnc ( ( ( ( ( n( ( ( ( ( n( ( ( ( n ( ( ( ( ( ( ( ( ( ( n Snc h hoognou ln w oluon f w cn l o
SAC-PUB-8 Al 6 Now w u h cfc lu of ± o h ln fl coonn n c of h gn of null nng on h con of h ln w coonn. W h n n 4 I connn o fn h followng c c h c c qz π 5 wh ( fn fo. Wh h fnon n h lon of h coonn of quon 4 h oun conon of quon co ( ( ( ( n n n n h h h h 6 W now h of fou ln quon n fou unnown. Ung h f w cn n n n of ( ( ( ( n h h h h 7
SAC-PUB-8 Al 6 W cn o l hng wh h con of oun conon n n n of. n n ( ( ( ( h h ( ( ( ( h h Conng quon 6 n 7 n lnng n w cn fn h lu n. Fo w fn ( ( q n ( { } ( q { } 9 wh ± ± Scl c n q ( h h W n n cl c wh h ngl of h l n h lcon co n Bw ngl. Th nu full non of h l fl o h own c n hnc h ng gn of h lcon whn h ul h conon. Alo lhough quon 8 gnl w wll focu on h lc l fo h cl loc n wll u ll ngl ffnc wn h l ogon con n h lcon co. W u h noon fo h ngl of n n Fg.. A Bw ngl n ung h u n cuu n θ n n n θ n. W nouc nw ngl θ ψ n nc w n n h fl olng wh h l w n n h gon wh α <<. Snc w fn c h - coonn of h -co n h ffn n nθ wh n h n of fcon of h u n quon. Wh h fnon w lo fn h -coonn n coθ.. 8 n nθ nθ nθ nθ n 4
SAC-PUB-8 Al 6 Wh h fnon w fn h { } { } ( ( nθ nθ ( nθ nθ ( q ( ( nθ nθ ( nθ nθ ( q W cn fn ll-ngl oon fo n q. Th 4 q πz β πz β nθ γ γ ( h h nθ 4 wh βc h of h cl n h cuu nc. Ung h Bw ngl conon w cn uu n θ ± nθ n n n h lu of n q n coθ w on fo quon 4 Z ± n n { } ( ( ( πz β { } ( ( γ n 4 whch lf o πz coθ β γ ( ( { } 44 Fo lc cl wh β n fo ll l cong ngl θ ψ α α << h coffcn co nψ coθ coθ 45 β coψ wh u lwh n h c h wnu n cuu. No h n h u cuu coθ nθ. Snc h ol ln w lu ngl θ coθ quon 44 co 4
πz β γ ( { } SAC-PUB-8 Al 6 46 Th cuou ul h h Bw ngl h no flc non on coonn. Inul on woul c h nc h -ol ln w coonn fo h cl fl houl uff no flcon fo h oun Bw ngl. In h f fl locon ( α φ R h lcc fl l o h ln w cu of quon 46 TR R ( φ 47 R ( ( π lung quon 47 w g { } R qz ( ( TR 48 R π γ A n h ou con w fn h l-cong ngl α θ ψ n u h α <<. Fo h ln w l fl n h own gon gn quon h f-fl πλ R ( ( ( ( ( R ϕ ( ( R ϕ δ α ξ δ δ 49 Now w cn u h ol ngl of quon o fn h ng gn c h n-on cu n fn h Δ U πz qz α π α γ ( φ φ ( πλ ( φ ( φ ( { } 5 ncoθ coθ φ ( quon n φ φ φ ( quon u nc ( ( ΔU ( φ φ ( φ φ φ ( φ φ ( φ φ φ { } q α 5 π α γ λ whch co 5
SAC-PUB-8 Al 6 ΔU λq α { nφ n( φ φ } 5 π α γ whch n gn wh h ng gn foul of quon 9 fo h h ngl ho. Rfnc. T. Pln R.. B. Col B. Cown C.M.S. S J.. Snc R.H. Snn Poof-of-ncl n fo l-n cclon of lc lcon n -nfn cuu Ph. R. ST Accl. B 8 (5. T. Pln Anl of -Dn Pcl Acclon fo Pln Infn Conuc Boun SAC-PUB-67 (6. M. X A Funnl Tho on Pcl Acclon Pocng of h Pcl Acclo Confnc ( 4. Z. Hung G. Suo n M. Zoloo Clculon n Oon of Acclon n Vcuu Ph. R. Scl Toc - Acclo n BVol. 7 (4 5. M.. T-Mln Hgh-ng lcognc Poc n Conn M Incnc Tc on Phc n Aono John Wl&Son Inc. 6