Accelertor Phyic G. A. Krfft Jefferon L Old Dominion Univerity Lecture 5 ODU Accelertor Phyic Spring 15
Inhomogeneou Hill Eqution Fundmentl trnvere eqution of motion in prticle ccelertor for mll devition from deign trjectory d d y 1 B 1 p d B p 1 B 1 p y d y B y p ρ rdiu of curvture for end, B' trnvere field grdient for mgnet tht focu (poitive correpond to horizontl focuing), Δp/p momentum devition from deign momentum. Homogeneou eqution i nd order liner ordinry differentil eqution. ODU Accelertor Phyic Spring 15
Diperion From theory of liner ordinry differentil eqution, the generl olution to the inhomogeneou eqution i the um of ny olution to the inhomogeneou eqution, clled the prticulr integrl, plu two linerly independent olution to the homogeneou eqution, whoe mplitude my e djuted to ccount for oundry condition on the prolem. = = A B y y A y B y p 1 p y 1 y Becue the inhomogeneou term re proportionl to Δp/p, the prticulr olution cn generlly e written p D p yp Dy p where the diperion function tify = = 1 1 1 1 d B d B dd B ddy B D Dy y y p p ODU Accelertor Phyic Spring 15
M 56 In ddition to the trnvere effect of the diperion, there re importnt effect of the diperion long the direction of motion. The primry effect i to chnge the time-ofrrivl of the off-momentum prticle compred to the on-momentum prticle which trvere the deign trjectory. = d z D p p d d p z D d p d D p p Deign Trjectory Dipered Trjectory M 56 1 D D y y d ODU Accelertor Phyic Spring 15
Dipole Solution Homogeneou Eqn. co / in / i i i d d in / / co / i i i d d Drift i 1 i d d 1 i d d ODU Accelertor Phyic Spring 15
Qudrupole in the focuing direction k B/ B i i co k in k / k i d d k in k co i i k i d d Thin Focuing Len (limiting ce when rgument goe to zero!) 1 d d 1/ f 1 d d Thin Defocuing Len: chnge ign of f ODU Accelertor Phyic Spring 15
Trnfer Mtrice Dipole with end Θ (put coordinte of finl poition in olution) fter efore co in d in / co d fter efore d d Drift fter efore 1 Ldrift d 1 d fter efore d d ODU Accelertor Phyic Spring 15
Qudrupole in the focuing direction length L Qudrupole in the defocuing direction length L fter co k L in k L / k efore d in co d fter k k L k L efore d d fter coh k L inh k L / k efore d inh co d fter k k L k L efore d d Wille: pg. 71 ODU Accelertor Phyic Spring 15
Thin Lene f f Thin Focuing Len (limiting ce when rgument goe to zero!) len len 1 d d 1/ f 1 len len d d Thin Defocuing Len: chnge ign of f ODU Accelertor Phyic Spring 15
Compoition Rule: Mtri Multipliction! Element 1 Element 1 Rememer: Firt element frthet RIGHT 1 M1 1 More generlly 1 M 1 MM 1 M M M... M M tot N N 1 1 ODU Accelertor Phyic Spring 15
Some Geometry of Ellipe Eqution for n upright ellipe y y 1 In em optic, the eqution for ellipe re normlized (y multipliction of the ellipe eqution y ) o tht the re of the ellipe divided y π pper on the RHS of the defining eqution. For generl ellipe A By Cy D ODU Accelertor Phyic Spring 15
The re i eily computed to e Are AC D B Eqn. (1) So the eqution i equivlently y y AC A B, AC B B, nd AC C B ODU Accelertor Phyic Spring 15
When normlized in thi mnner, the eqution coefficient clerly tify 1 Emple: the defining eqution for the upright ellipe my e rewritten in following uggetive wy y β = / nd γ = /, note, m y m ODU Accelertor Phyic Spring 15
Generl Tilted Ellipe Need 3 prmeter for complete decription. One wy y y= y where i lope prmeter, i the mimum etent in the -direction, nd the y-intercept occur t ±, nd gin ε i the re of the ellipe divided y π 1 y y ODU Accelertor Phyic Spring 15
ODU Accelertor Phyic Spring 15 Identify,, 1 Note tht βγ α = 1 utomticlly, nd tht the eqution for ellipe ecome y y eliminting the (redundnt!) prmeter γ
Ellipe Dimenion in the β-function Decription y, y== α / β /, A for the upright ellipe m, y m Wille: pge 81 ODU Accelertor Phyic Spring 15
Are Theorem for Liner Optic Under generl liner trnformtion ' M y' M 11 1 y n ellipe i trnformed into nother ellipe. Furthermore, if det (M) = 1, the re of the ellipe fter the trnformtion i the me tht efore the trnformtion. M M 1 Pf: Let the initil ellipe, normlized ove, e y y ODU Accelertor Phyic Spring 15
Let the finl ellipe e y y Effect of Trnformtion, y y y The trnformed coordinte mut olve thi eqution. M 1 M y, The trnformed coordinte mut lo olve the initil eqution trnformed. y y ' M11 M1 y' M1 M y 1 1 M11 M 1 1 1 y M1 M y ODU Accelertor Phyic Spring 15
Becue 1 1 M M 11 1 ' y 1 1 M M y' 1 The trnformed initil ellipe i y y 1 1 1 1 M M M M 11 11 1 1 M M M M M M M M 1 1 1 1 1 1 1 1 11 1 11 1 1 1 1 1 1 1 M M M M 1 1 ODU Accelertor Phyic Spring 15
Becue (verify!) 1 1 1 1 M M M M 1 1 M M 1 1 M M 1 1 11 11 1 1 1 det M the re of the trnformed ellipe (divided y π) i, y Eqn. (1) Are det M 1 det M ODU Accelertor Phyic Spring 15
ODU Accelertor Phyic Spring 15 Tilted ellipe from the upright ellipe In the tilted ellipe the y-coordinte i ried y the lope with repect to the un-tilted ellipe y y 1 1 ' ',, 1 1,,, M Becue det (M)=1, the tilted ellipe h the me re the upright ellipe, i.e., ε = ε.
Phe Advnce of Unimodulr Mtri Any two-y-two unimodulr (Det (M) = 1) mtri with Tr M < cn e written in the form M 1 co 1 in The phe dvnce of the mtri, μ, give the eigenvlue of the mtri λ = e ±iμ, nd co μ = (Tr M)/. Furthermore βγ α =1 Pf: The eqution for the eigenvlue of M i M M 1 11 ODU Accelertor Phyic Spring 15
Becue M i rel, oth λ nd λ* re olution of the qudrtic. Becue Tr M i 1 TrM / For Tr M <, λ λ* =1 nd o λ 1, = e ±iμ. Conequently co μ = (Tr M)/. Now the following mtri i trce-free. M 1 co 1 M 11 M M 1 M M 1 M 11 ODU Accelertor Phyic Spring 15
Simply chooe M M in, M1, in 11 M 1 in nd the ign of μ to properly mtch the individul mtri element with β >. It i eily verified tht βγ α = 1. Now M M n 1 nd more generlly 1 co 1 co 1 in n inn ODU Accelertor Phyic Spring 15
Therefore, ecue in nd co re oth ounded function, the mtri element of ny power of M remin ounded long Tr (M) <. NB, in ome em dynmic literture it i (incorrectly!) tted tht the le tringent Tr (M) enure oundedne nd/or tility. Tht equlity cnnot e llowed cn e immeditely demontrted y counteremple. The upper tringulr or lower tringulr ugroup of the two-y-two unimodulr mtrice, i.e., mtrice of the form 1 1 1 1 clerly hve unounded power if i not equl to. or ODU Accelertor Phyic Spring 15