LECTURE 15. Phase-amplitude variables. Non-linear transverse motion

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1 LETURE 5 Non-linea tansvese otion Phase-aplitude vaiables Second ode (quadupole-diven) linea esonances Thid-ode (sextupole-diven) non-linea esonances // USPAS Lectue 5 Phase-aplitude vaiables Although the petubation appoach discussed in the pevious lectue allows a geneal discussion of the conditions fo esonance, to analyze the otion in phase space nea a esonance in detail, we have to go back to the full equation of otion: Lectue, p. : B ξ + ξ β with the diving te given by B β [ i ] // USPAS Lectue 5 n, exp ψξψ n To solve this equation, we have to ake two oe changes of the phase space vaiables. The phase space vaiables associated with the Floquet coodinates ae (ξ, ξ ). Fo puely linea otion, the phase space is a cicle: ξ a -ψ We will change to pola co-odinates (,φ) in phase space, whee // USPAS Lectue 5 ξ ξ φ tan ξ ξ ξ + These ae soeties called phase-aplitude vaiables, because fo puely linea otion a the invaiant aplitude of the otion, and asinψ φ tan ψ Φ acosψ the betaton phase. // USPAS Lectue 5

2 ξ φ A paticle executing puely linea otion has constant, and φ advances by evey evolution. The convesion fo (,φ) to Floquet coodinates is ξ cos φ ξ sin φ We will now poceed to tansfo the equation of otion into phase-aplitude vaiables. Then we will identify the esonance ξ diving tes, and ignoe all othe tes. When we ae done, we will have an equation that we can integate to get the tajectoies in phase space. Fo the adial coodinate, we have d + d ξ + + ξ ξξ ξ ξ ξ ξ ξ ψ n ξ ξ ξ n, exp iψ ξ( ψ) + [ ] ξ n, exp[ iψ]ξ( ψ) n Eliinating the Floquet coodinates gives // USPAS Lectue 5 5 // USPAS Lectue 5 d n n cos φsinφ + exp[ iψ] Fo the pola angle vaiable, we have n, n+ exp, ξ φ ξ n [ iψ] d tanφ + ξ ξ + ξ n+ n + cos φ n, exp[ iψ] We ll now specialize to a paticula type of field eo. We ll stat with quadupole eos, fo which the otion eains linea. // USPAS Lectue 5 7 Second-ode (quadupole-diven) esonances A quadupole-diven esonance coesponds to n. The equations of otion ae d cosφsinφ exp[ iψ], + φ [ iψ] cos, exp Fo a single esonance, only one value of will be ipotant. Fo that value of, we have i ds s b s, exp [ ψ ] β( ) exp i ψ ψ // USPAS Lectue 5 8 [ ]

3 obining the positive and negative values of gives exp[ iψ]+ exp[ iψ] in which,, ds s b s β( ) cos ψ ψ A cos ψ + B sin ψ, A [ ] A ds β s B ds β s B s B s s cos Φ( ' ) s sin Φ( ' ) These ae the haonic coefficients that will dive the esonance. The equations of otion becoe d cosφsinφ Acosψ + Bsin ψ + cos φ( Acosψ + Bsin ψ) We expand out the tig functions: // USPAS Lectue 5 9 // USPAS Lectue 5 ( + ( + )) d A sin φ ψ sin φ ψ + B( cos( φ ψ) cos( φ+ ψ) ) Recall that a quadupole can only dive a second ode esonance: this is eflected in the te with aguent φ ψ ψ, which dives the second ode esonance at. (The tes with aguents φ+ ψ do not dive any esonances, since is always positive: they coespond to apidly oscillating tes that ay be neglected). So we have d Asin( φ ψ) + Bcos φ ψ [ ] // USPAS Lectue 5 A siila teatent of the equation fo φ gives A A B + + cosψ + sin ψ + [ Acos( φ ψ) Bsin( φ ψ) ] The tes with cosψ and sinψ will oscillate apidly, and can be neglected. The te coesponds to the quadupole-induced tune shift: + A A φ ψ B φ ψ + cos ( ) sin [ ] The tune shift is // USPAS Lectue 5

4 A B s ds β( s ) // USPAS Lectue 5 which we can ecognize fo ou pevious wok (Lectue 8, p 9) We need to ake one oe anipulation: we can siplify the aguents of the tig functions by intoducing the angle ψ φ φ Then the two equations fo phase and aplitude becoe ( + ) + A B [ cosφ sinφ ] d Asinφ + Bcosφ [ ] obining these equations gives us a diffeential equation fo the phase space tajectoies, that is, an equation fo as a function of φ A B d d d d [ sin φ + cos φ φ ] φ ( + ) + Acosφ Bsinφ d( φ ) A B d [ sinφ + cosφ ] ( + ) + [ Acosφ Bsinφ ] [ ] // USPAS Lectue 5 This equation can be integated elatively easily. The esult is A + a cosφ B sinφ + a Acos( φ ψ) B sin φ ψ + + whee a is a constant of integation; it can be intepeted as the value of fa fo the esonance, when the denoinato of the esonant te + is lage. To undestand this esult, we siplify it by taking B, and. Then, if we let δ +, a A + cos( φ ψ) δ This is a faily of ellipses in Floquet coodinate phase space, fo vaious values of a. Let s plot soe of these, fo δ., // USPAS Lectue 5 5 // USPAS Lectue 5

5 A., and fo a anging fo.5 to.5. The left figue is fo ψ, the ight fo ψ/. ẋ ÅÅÅÅÅ - - x x All the phase space tajectoies ae elliptical, even fo the sallest value of a. The elliptical shape eflects the fact that a quadupole // USPAS Lectue 5 7 ẋ ÅÅÅÅÅ - - petubation changes the lattice functions, and hence changes the Floquet tansfoation. We could estoe the cicula shapes in phase space if we edid the Floquet tansfoation, but used the new values of the lattice functions, afte the intoduction of the quadupole eos. This otion is linea and stable fo all aplitudes. Howeve, if we inspect the equation fo the phase spaces ellipses, we see that, fo a physical solution valid fo all φ, we ust have A + cos φ ψ δ δ A ds β s A δ // USPAS Lectue 5 8 B s cos [ Φ( s' )] This is the second-ode esonance stopband width. Note: if we had not assued B, we would have found fo the stopband width δ A + B If the tune shift is lage than this value, then the otion is unstable fo all aplitudes: The next two figues show the phase space just outside the stopband (bounded (stable) otion, left figue) and just inside the stopband (unbounded (unstable) otion, ight figue). ẋ ÅÅÅÅÅ - - x - - ẋ ÅÅÅÅÅ - - x - - // USPAS Lectue 5 9 // USPAS Lectue 5

6 Thid-ode (sextupole-diven) esonances A sextupole-diven esonance coesponds to n. The equations of otion ae d cos φsinφ exp[ iψ], + φ [ iψ] cos, exp Fo a single esonance, only one value of will be ipotant. Fo that value of, we have i ds s b s, exp [ ψ ] β( ) exp i ψ ψ // USPAS Lectue 5 [ ] obining the positive and negative values of gives exp[ iψ]+ exp[ iψ] in which,, ds s b s β( ) cos ψ ψ A cos ψ + B sin ψ A ds β s B ds β s // USPAS Lectue 5 [ ] B s B ρ s cos Φ( ' ) B s B s sin Φ( ' ) ρ These ae the haonic coefficients that will dive the esonance. The equations of otion becoe d cos φsinφ Acosψ + Bsin ψ + cos φ( Acosψ + Bsin ψ) We expand out the tig functions: d A B + ( + ) sin φ+ ψ sin φ ψ + sin( φ ψ) + sin( φ ψ) cos( φ+ ψ) + cos( φ+ ψ) cos( φ ψ) cos( φ ψ) The vaious tes in the expansion coespond to diffeent esonance odes. Recall that a sextupole can dive fist ode o thid ode esonances: this is eflected in the te with aguent φ ψ ψ, which dives the thid ode esonance at, and the te with aguent φ ψ ( ) ψ, which dives the fist ode esonance at. (The tes with aguents φ+ ψ and φ+ ψ do not dive any esonances, since is always positive: they coespond to apidly oscillating tes that ay be neglected). Since we ae only inteested in the tes that dive the thid ode esonance, we have // USPAS Lectue 5 // USPAS Lectue 5

7 d Asin( φ ψ) + Bcos φ ψ [ ] A siila teatent of the equation fo φ gives + Acos( φ ψ) Bsin φ ψ 8 [ ] As befoe, we siplify the aguents of the tig functions by intoducing the angle ψ φ φ Then the two equations fo phase and aplitude becoe + A B [ cosφ sinφ 8 ] // USPAS Lectue 5 5 d Asinφ + Bcosφ [ ] obining these equations gives us a diffeential equation fo the phase space tajectoies, that is, an equation fo as a function of φ A B d d d d d d [ sin φ + cos φ φ ] φ ψ ψ + Acosφ Bsinφ 8 [ ] This equation can be integated! The esult gives the phase space tajectoies in the vicinity of a thid-ode esonance: // USPAS Lectue 5 a A cosφ B sinφ + A cos( φ ψ) B sin( φ ψ) + whee a is a constant of integation; it can be intepeted as the value of the invaiant fa fo the esonance, when the denoinato of the esonant te is lage. To undestand this esult, we siplify it by taking B, and look at the point in the ing whee ψ. Then, if we let δ, a A cosφ + δ This is a faily of cuves in Floquet coodinate phase space, fo vaious values of a. Let s plot soe of these, fo δ., A., and fo a anging fo.5 to.9 // USPAS Lectue 5 7 // USPAS Lectue 5 8

8 x ÅÅÅÅ - - x bounday of stable otion, is the heavy tiangula line. Motion within the sepaatix is stable. Just outside the sepaatix, the otion tends to be chaotic: that is, sall changes in the initial conditions fo the otion can lead to lage changes afte any tuns. Nueical tun-by tun siulation of otion nea the thidode esonance: The cicula tajectoies in the cente coespond to linea otion with sall a. As the aplitude of the otion a inceases, the tajectoies distot fo cicula into tiangula shapes, chaacteistic of a thid-ode esonance. The sepaatix, the // USPAS Lectue 5 9 // USPAS Lectue 5 The cones of the tiangle ae the thee fixed points. φ sep The adial distance sep to the sepaatix is a easue of the axiu aplitude of stable otion. Fo geoety, on the vetical sepaatix, we have sep and a A cosφ sep +, whee a sep is the value of a cosφ δ coesponding to the sepaatix. So, a cos φ cosφ+ A cos φ δ sep sep sep Equating the coefficients of cos φ and cos φ gives asep 8 δ A Since a coesponds to the eittance of the paticle, we have an expession fo the thid-ode esonance width fo a paticle of eittance ε: // USPAS Lectue 5 // USPAS Lectue 5

9 A ε δ ε B s ds β s ( ) cos [ Φ( s' )] B ρ Note: we ignoed the B coefficient, fo siplicity. Including this coefficient, the esonance width is given by δ ε // USPAS Lectue 5 A + B The esonance widths ay be contolled though the aziuthal distibution of the sextupoles. With two failies of sextupoles at appopiate locations, both the A and B coefficients ay be iniized. Exaple:. 5 odel acceleato, with FODO lattice. Fo Lectue 8, p 7, we saw that we could copensate the natual choaticity by placing two sextupoles in the lattice: a sextupole of stength D at any D quad, whee β x,d.8, and a sextupole of stength F 59 - at the adjacent F quad, whee β x,f.8. What is the thid-ode esonance width poduced by these sextupoles, fo a bea of eittance ε -ad? The sextupoles had length L s.. Using δ x esonance width in the x-plane is ε L µ s βxff + βxd,, D cos B Bρ, the // USPAS Lectue 5 whee µ.78 is the phase advance pe cell. Plugging in the nubes gives δ x. 9. Note: this is an undeestiate, since we have ignoed the B coefficient. As fo the second ode esonance, the oientation of the phase space obits fo a thid ode esonance in Floquet coodinates depends on the aziuthal location at which the paticles ae obseved. The following figue shows phase space at ψ/ instead of ψ. // USPAS Lectue 5 5 x ÅÅÅÅ - - x - - In esonant extaction, the egion of stable phase space is gadually diven to zeo, typically by inceasing the stength of the nonlinea fields diving the esonance. This causes all paticles eventually to flow along the sepaatices. A agnetic o // USPAS Lectue 5

10 electostatic septu is placed at an appopiate aziuth to intecept the paticles flowing along the sepaatix, and they ae diveted into a agnetic extaction channel. // USPAS Lectue 5 7