Lecture 17. Physics Department Yarmouk University Irbid Jordan. Chapter V: Scattering Theory - Application. This Chapter:

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1 Lctu 17 Physics Dpatnt Yaouk Univsity 1163 Ibid Jodan Phys. 441: Nucla Physics 1 Chapt V: Scatting Thoy - Application D. Nidal Eshaidat This Chapt: 1- Intoduction - Ipotanc o th scatting thoy. - Coss-Sction. 3- Scatting by a Potntial. 4- Th Optical Tho. 5- Scatting Stationay Stats. 6- Pobability Cunts. 7- Applications. Cntal Potntial Rncs: 1. Quantu Mchanics, Vol. Chapt VIII C. Cohn-Tannoudji, B. Diu and F. Laloë. Quantu Physics, Chapt 19 Stphn Gasioowicz

2 Intoduction What is scatting? Scatting voks a sipl iag: Spaat objcts which a a apat and oving towads ach oth collid at so ti and thn tavl away o ach oth and vntually a a apat again. W don t ncssaily ca about th dtails o th collision xcpt insoa as w can pdict o it wh and how th objcts will nd up. This pictu o scatting is th ist on w physicists lan. 4

3 A bautiul xapl o th pow o consvation laws 5 In any cass th laws o consvation o ontu and ngy alon can b usd to obtain ipotant sults concning th poptis o vaious chanical pocsss. It should b notd that ths poptis a indpndnt o th paticula typ o intaction btwn th paticls involvd L.D. Landau Mchanics (1976) Gnal Sch An incidnt ba o paticls (1) stiks a tagt. Two dvics (dtctos) a placd in od to dtct th action sults. 6 Fig. 5-1

4 Elastic and Inlastic Scatting Th inal stat can b coposd o nw ntitis. In nucla physics th outgoing paticl b and th Y a th sult o a aangnt o nuclons and th pocss is calld aangnt collision Anoth ipotant cas is whn th inal stat is coposd o th sa paticls as in th initial stat. H w call th pocss lastic scatting. Othwis it is calld inlastic scatting 7 Undstanding th action Knowing th initial stat o th action and asuing th inal stat givs an ida about th natu o th action. 8 In quantu chanics languag th intaction potntial o th pocss can b undstood. In gnal, a scatting xpint ais to viy a thotical odl about th action.

5 Ipotanc o th scatting thoy Undstanding Matt 10 Sinc Ruthod s xplanation o Gig and Masdn sults, physicists undstood that o ngtic paticls than th α paticls w ncssay to pob th nuclus. In addition quantu chanics is ncssay in od to undstand what happns whn a nuclus is bobadd by such a pojctil. Collision o Scatting Thoy is th a usd by Physicists o this pupos.

6 Acclatos 11 On th xpintal lvl this ld to build acclatos o and o capabl o poviding high ngy paticls (lctons, potons o alpha s, havy ions ). On th thotical lvl this typ o xpints ld to ajo dvlopnts in quantu chanics. Acclatos S Supplnts 6

7 Elcton Scatting - Hostadt 13 Th ist pobs usd o invstigating th nucla stuctu w high ngy lctons. Hostadt usd th ist SLAC (Stanod Lina acclato) to invstigat th chag distibution in atoic nucli and atwads th chag and agntic ont distibutions in th poton and nuton. Th lctonscatting thod was usd to ind th siz and suac thicknss paats o nucli. Elcton Scatting and th Stuctu o th Nuclus S Supplnts 7

Hostadt In 1950 Hostadt lt Pincton to bco Associat Posso o Physics at Stanod Univsity wh h initiatd a poga on th scatting o ngtic lctons o th lina acclato, invntd by W.

8 Hostadt In 1950 Hostadt lt Pincton to bco Associat Posso o Physics at Stanod Univsity wh h initiatd a poga on th scatting o ngtic lctons o th lina acclato, invntd by W. W. Hansn, which was thn und constuction. At 1953 lcton-scatting asunts bca Hostadt's pincipal intst. With studnts and collagus h invstigatd th chag distibution in atoic nucli and atwads th chag and agntic ont distibutions in th poton and nuton. Th lcton-scatting thod was usd to ind th siz and suac thicknss paats o nucli. Many o th pincipal sults on th poton and nuton w obtaind in th yas Sinc 1957 phasis in th sach poga has bn placd on aking o pcis studis o th nuclon o actos. This wok is still in pogss Coss Sction S Supplnts 8

9 Lctu 18 Physics Dpatnt Yaouk Univsity 1163 Ibid Jodan Phys. 441: Nucla Physics 1 Scatting by a Potntial D. Nidal Eshaidat Elcton Scatting and Nucla Dnsity Rnc: Intoductoy Nucla Physics Chapt 3 (pags 54-48) K. S. Kan

10 Sipl Mathatical Tatnt W shall stat by a sipl athatical tatnt o th pobl in quantu chanics. On can consid th wav unction o th incidnt () paticls to b k ( ) i i ψ i = ( p i = h k i ) 1 Bcaus o th intaction potntial with th diacting (nucla) potntial, th wav unction psnting th scattd paticl can b wittn as: i k ψ ( ) = ( p = h k ) 19 Th Fo Facto

11 Th Fo Facto Th tansition pobability btwn th two stats is dind by: * λ F ki, k = ψ V ψi dv 3 F k i, k ( ) ( ) ( ) { is calld th o acto di. volu 1 Montu Tans Lt us call: q = k i k th chang o ontu o th paticls Th vcto q is siply calld th ontu tans (Th ontu that on paticl tanss to th oth on). Th o acto is wittn as a unction o th ontu tans, thus q. 3 is wittn as: + i q 4 F ( q) V ( ) In a standad pocdu, s th ollowing paagaph, th noalization constants o th wav unctions is chosn so that as F(0) = 1 = dv

12 Th Intaction Potntial Consid a dintial lnt o th nuclus chag dq (Fig. 5-). Consid also an abitay ntial syst. Lt & b th vctos psnting th position o th lcton and th chag dq spctivly. W shall coput h th cosponding potntial: Nuclus dq i ρ is th distibution o nucla chag thn ( ) v dv (, ) dq = Z ρ Z ρ ( ) dv = 4π ε 0 d And th intaction potntial is givn by: 4 Fig. 5-3 Fo Facto & Tansition Pobability Z ρ ( ) ( ) dv V = 4π ε 0 All dq' s in th nuclus 6 4 F ( q) = Z ( ) dv dv + i q sin θ 4πε 0 ρ 7 = + i q ρ ( ) dv 8 Th tansition pobability is thus latd to )( T ) which is th Foui Tanso o th dnsity ρ.

13 Th Fo Facto Cntal Potntial ρ ( ) i o siplicity dpnds only on (not on θ o φ ) thn: 4π F ( q) = q ρ ( ) d q sin and assuing an lastic scatting, i.. p i = p p α q = sin h 10 Masuing q (th ont tans) as a unction o α givs F(q) and by an invs Foui Tanso ρ is calculatd 9 5 Mont Tans - Goty 1 k i = p i h = p iˆ p i k k i α p q 1 k = p h = pcos α iˆ + psin α ˆj 6 q = Figu 5-3: Elastic scatting k i k q = q = k q p h = p h [( 1 cosα) iˆ + psinα ˆj ] ( 1 cosα) p α = sin h

14 4- Th Optical Tho Two Appoachs Two appoachs a usd to study what happns in a scatting pocss 1) Th wav packt appoach ) Th scatting stationay stats 8

15 Th Wav Packt Appoach 9 Th incidnt paticl is dscibd by a wav packt which should hav two pincipal chaactistics: It should b spatially lag so it dos not spad appciably duing th xpint, It ust b lag copad to th tagt dinsions but sall copad to th dinsions o th laboatoy wh w obsv th scatting. Th Wav Packt Appoach 30 Th latal dinsions a dtind by th ba siz. Th outgoing paticl is dscibd by a wav packt which lis in so diction dind by th scatting angl θ This is what w hav don in th pvious paagaph.

16 Schöding Equation H w shall siply stat acts you know o you pvious study in quantu chanics. Th solution o th Schöding quation in th absnc o a potntial, (i.. o a classical paticl), is th plan wav o wh: k is th wavnub o th associatd wav c K packt, i.. π p k = = = 11 λ h h c i k K is th kintic ngy o th paticl o st ass c. Fo a lativistic paticl w hav p k = = h K + c h c K 1 31 Th wavnub k In th xapls blow w coput th valu o k o a poton and an - o kintic ngy K = 50 MV. Fo a poton (c = MV) o kintic ngy K= 50 MV w hav: MV 50 MV 1 k = = 0.47 F 140 MV F π λ = = 5.44 F k Fo a lcton (c = MV) o kintic ngy K= 50 MV w hav: 3 ( 50 MV ) + 50 MV MV 1 k = 0.041F 140 MV F λ = π k 157.1F

17 Flux Th plan wav o dscibs a lux: j h = i h k ψ * ψ ψ ψ * = = j has th dinsion o vlocity. It psnts th total lux, i.. th nub o incidnt paticls p aa unit p ti unit volu Th classical quivalnc o this lux is th nub o incidnt paticls p aa unit p ti unit. Consvation o ngy-att iplis that this lux is consvd. p Th Outgoing Wav Packt k W usually choos to din th z axis (quantization axis). Thus th lag bhavio o this solution can b wittn as a lina cobination o an incoing sphical wav and an outgoing on, i.. i k i k l = 0 ( l + 1) - i ( k l π ) i ( k l π ) Pl ( cos θ) 14 34

18 Expintal Layout - Fluxs Th ollowing igu (5-4) shows a schatic layout o a scatting xpint. 35 Scattd lux j Incidnt lux j i Fig. 5-4 θ: th scatting angl is th laboatoy angl Ects o a Radial Potntial In th psnc o a adial potntial th pvious solution o Schöding quation is siply altd to a unction whos asyptotic o is: ( ) ( ) ψ( ) - i k l π i k l π i ( l + 1) Sl ( k) Pl ( cos θ) 14 k l = 0 36 Th asyptotic o ay b wittn in th o: ψ i k ( ) + ( l + 1) l = 0 ( k) Sl 1 Pl i k ( cos θ) i k 15 Excis: Chck Eq. 15

19 Th Phas Shit δ It is quid o Eq. 14 to b consistnt with th athatical a o wav chanics (S supplnt 9) that S l (k) = 1. S l (k) in a standad notation can b wittn as i l ( k ) δ wh th al unction δ(k) is calld th phas shit. It tanslats th act that th asyptotic o o th scattd wav dis o that o a paticl by a shit in phas o th agunt (Think wavs and intnc). 37 ψ = An incoing Plan Wav + An outgoing Sphical Wav i k i k Sl k 1 ψ( ) + ( l + 1) Pl ( cosθ) 15 l = 0 i k Eq. 15 psnts nothing ls but th su o a plan wav + an outgoing sphical wav! In th ollowing w shall us th unction with ( θ) = ( l + 1) ( k) ( cosθ) l = 0 16 ( k) l ( k) ( ) l P l Sl 1 = 17 i k 38

20 Flux o th Asyptotic Solution Flux o th Asyptotic Solution Th lux o th asyptotic o is: j = h i h = i j = i k h ψ * ψ ψ ψ * i * i k + i k + i k + i k ( θ) + ( θ) i k i k ( θ) + ( θ) i k * i k ( θ) + ( θ) i k i k i k * cc

21 Calculating j W shall skip so stps in od to calculat j and lav this as a howok. k = k cos θ 1) Fist using. )Thn calculating th gadint. 3) Nglcting th ts in 3 sinc thy a doinatd by th 1/ o lag. This is lagly justiid by th act that w a intstd in th lux a away o th intaction gion (wh th potntial is localizd), i.. at lag (s ig. 5-4) 41 Calculating j -Stps 4) Finally considing a scatting angl θ#0. This is also justiid by th act that th dtcto is always placd at an angl and in a asunt (Ruthod) w want th intgatd lux ov a sall but init solid angl (Ruthod scatting). 5) Finally using Riann-Lbsgu La whn intgating ov th sall but init solid angl. S Supplnt 9 and Pobl 7 o chapt 19 - in Gasioowicz 4

22 j 43 Th sult is th ollowing xpssion o j j = h k + h k iˆ ( θ) 18 î is th unit vcto in th diction o j - Discussion j = h k + h k iˆ ( θ) Fist ak: it is obvious that in th absnc o a scatting potntial, th nd t on th ight in q. 18 vanishs and w tiv th xpssion o th incidnt lux (q. 13) 18 44

23 Discussion Th Fist T 1 j = h k + h k iˆ In a wav-packt tatnt, Fo th adial lux, i. iˆ h k givs a contibution cosθ. k h ( θ) 18 would b ultiplid by a unction which dins th latal dinsions o th ba. h k th ist t 45 Discussion Th Fist T This contibution is only signiicant in th owad gion within a init gion o th z-axis. And sinc th dtcto is placd a outsid this gion w can consid that th ist t dos not contibut to th adial lux in th asyptotic gion. Asyptotic gion 46

24 Coss Sction Flux Nub o Paticls 48 Tanslating th asyptotic lux j into nub o outgoing scattd paticls (p ti unit) w wit: dn ( θ) k = j iˆ h da dω Ipotant ak: th acto vanishs and this is, a postioi, a justiication o having nglctd th ts in 3. 19

25 Dintial Coss Sction 49 Th dintial coss sction is dind by: dσ dω = dn j dω i h k ( θ) h k dω dω = ( θ) Copa with th classical dinition o th dintial coss sction in supplnt 8. 0 Spin Dpndnc 50 I th potntial has a spin dpndnc, as it is th cas o th nuclon-nuclon intaction thn th ay b an aziuthal dpndnc. Eq. 0 is gnalizd in this cas and w wit: dσ dω = ( θ,φ) 1

26 Total Coss Sction 51 Th total coss sction o th scatting pocss is obtaind by intgating ov all possibl valus o θ, i.. o dω σ tot dσ dω ( k) = dω = ( θ, φ) dω Total Coss Sction 5 On can show that, and w lav this as a howok, : σ tot 4π sin k ( k) = δ ( k) l = 0 l 3

27 Th Optical Tho 1 I on consids (0), (s qs. 16 & 17) i.: l = 0 ( k) Sl 1 ( 0) = ( l + 1) P l () 1 4 i k ( ) ( ) i δ k 1 0 = ( l + 1) Pl () 1 l = 0 i k 5 ( ) ( ) ( ) ( ) ( ) δ δ δ i k i k 1 i k 0 = l + 1 Pl () 1 k l = 0 i i δ ( ) ( ) ( k 0 = l + 1 ) sin δ( k) Pl () 1 7 k l = 0 Th Optical Tho Taking th iaginay pat o q. 7, w hav: I 1 8 k ( ( 0) ) = ( l + 1) sin δ( k) l = 0 Copaing with th xpssion o th total coss sction; 4π ( ) σtot k = sin δ ( ) l k 3 k l = 0 w hav: k I ( 0) = σtot ( k) 9 4π 54

28 Th Optical Tho 3 Equation 9 is known as th optical tho. I k = 4π ( 0) σ ( k) tot 9 55 It is tu vn whn inlastic pocsss can occu, as this ay b th cas in nucla actions. Th Optical Tho & Wav Mchanics Th optical tho appas claly whn using th wav (D Bogli) languag sinc it ollows th act that th total coss sction psnts th oval o lux o th incidnt ba.. In wav languag, this oval is th sult o dstuctiv intnc which occus in th owad gion btwn th incidnt and th lastically scattd wav. Finally and o copltnss on hav to ntion that th wav tatnt dos xplain th iaginay pat. This is gnally tu and xplaind in a o cophnsiv quantu scatting thoy. 56

29 Nxt Lctu 5-5 Scatting Stationay Stats Oth Appoach Rnc: Paagaphs B and C Quantu Mchanics, Vol. Chapt VIII C. Cohn-Tannoudji, B. Diu and F. Laloë End o Lctu 18

II.3. DETERMINATION OF THE ELECTRON SPECIFIC CHARGE BY MEANS OF THE MAGNETRON METHOD

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