Accelerator Physics. G. A. Krafft Jefferson Lab Old Dominion University Lecture 7

Transcription

1 Aeerator Phyi G. A. Krafft Jefferon Lab Od Dominion Univerity Leture 7 ODU Aeerator Phyi Spring 05

2 Soution to Hi Equation in Ampitude-Phae form To get a more genera expreion for the phae advane, onider in more detai the inge partie oution to Hi equation d x d K x 0 From the theory of inear ODE, the genera oution of Hi equation an be ritten a the um of the to ineary independent peudo-harmoni funtion here x Ax Bx x i e ODU Aeerator Phyi Spring 05

3 ODU Aeerator Phyi Spring 05 are to partiuar oution to Hi equation, provided that and, 3 d d K d d and here A, B, and are ontant (in ) That peifi oution ith boundary ondition x( ) = x and dx/d ( ) = x' ha ' ' ' x x e i e i e e B A i i i i Eqn. (3)

4 ODU Aeerator Phyi Spring 05 Therefore, the unimoduar tranfer matrix taking the oution at = to it oordinate at = i,,,,,,, ' in ' o o ' ' in ' ' in in ' o ' x x x x here d,

5 Cae I: K() periodi in Suh boundary ondition, hih may be ued to deribe iruar or ring-ike aeerator, or periodi fouing attie, have K( + L) = K(). L i either the mahine irumferene or period ength of the fouing attie. It i natura to aume that there exit a unique periodi oution () to Eqn. (3a) hen K() i periodi. Here, e i aume thi to be the ae. Later, it i be hon ho to ontrut the funtion expiity. Ceary for periodi L L ith L d i ao periodi by Eqn. (3b), and μ L i independent of. ODU Aeerator Phyi Spring 05

6 The tranfer matrix for a inge period redue to o L ' ' ' ' 0 0 o in in L L in L o here the (no periodi!) matrix funtion are L L in L in L ',, By Thm. (), thee are the eipe parameter of the periodiay repeating, i.e., mathed eipe. ODU Aeerator Phyi Spring 05

7 Genera formua for phae advane In term of the β-funtion, the phae advane for the period i L L 0 d and more generay the phae advane beteen any to ongitudina oation and ' i ', ' d ODU Aeerator Phyi Spring 05

8 M Tranfer Matrix in term of α and β ', Ao, the unimoduar tranfer matrix taking the oution from to ' i ' o ', in ', ' ' in ', o ', ' ' o ' ', in ' ', in Note that thi fina tranfer matrix and the fina expreion for the phae advane do not depend on the ontant. Thi onuion might have been antiipated beaue different partiuar oution to Hi equation exit for a vaue of, but from the theory of inear ordinary differentia equation, the fina motion i unique one x and dx/d are peified omehere. ', ODU Aeerator Phyi Spring 05

9 Method to ompute the β-funtion Our previou ork ha indiated a method to ompute the β- funtion (and thu ) direty, i.e., ithout oving the differentia equation Eqn. (3). At a given oation, determine the one-period tranfer map M +L, (). From thi find μ L (hih i independent of the oation hoen!) from o μ L = (M +M ) /, and by hooing the ign of μ L o that β() = M () / in μ L i poitive. Likeie, α() = (M -M ) / in μ L. Repeat thi exerie at every oation the β-funtion i deired. By ontrution, the beta-funtion and the apha-funtion, and hene, are periodi beaue the inge-period tranfer map i periodi. It i traightforard to ho =(β()) / atifie the enveope equation. ODU Aeerator Phyi Spring 05

10 Courant-Snyder Invariant Conider no a inge partiuar oution of the equation of motion generated by Hi equation. We ve een that one a partie i on an invariant eipe for a period, it mut tay on that eipe throughout it motion. Beaue the phae pae area of the inge period invariant eipe i preerved by the motion, the quantity that give the phae pae area of the invariant eipe in term of the inge partie orbit mut ao be an invariant. Thi phae pae area/π, x xx' x' x x' x / i aed the Courant-Snyder invariant. It may be verified to be a ontant by hoing it derivative ith repet to i zero by Hi equation, or by expiit ubtitution of the tranfer matrix oution hih begin at ome initia vaue = 0. ODU Aeerator Phyi Spring 05

11 x dx d give Peudoharmoni Soution o in 0 0,0 0 x dx 0 in,0 0 o,0 in,0 d 0 o 0,0,0 in x x' x / x x' x 0,0 / Uing the x() equation above and the definition of ε, the oution may be ritten in the tandard peudoharmoni form x x' x o,0 here tan x0 The the origin of the terminoogy phae advane i no obviou. 0 ODU Aeerator Phyi Spring 05

12 Cae II: K() not periodi In a ina or a reiruating ina there i no oed orbit or natura mahine periodiity. Deigning the tranvere opti onit of arranging a fouing attie that aure the beam partie oming into the front end of the aeerator are aeerated (and ometime deeerated!) ith a ma beam o a i poibe. Therefore, it i imperative to kno the initia beam phae pae injeted into the aeerator, in addition to the tranfer matrie of a the eement making up the fouing attie of the mahine. An initia eipe, or a et of initia ondition that omeho bound the phae pae of the injeted beam, are traked through the aeeration ytem eement by eement to determine the tranmiion of the beam through the aeerator. The deign are uuay made up of eundertood modue that yied knon and undertood tranvere beam optia propertie. ODU Aeerator Phyi Spring 05

13 here Definition of β funtion No the peudoharmoni oution appie even hen K() i not periodi. Suppoe there i an eipe, the deign injeted eipe, hih tighty inude the phae pae of the beam at injetion to the aeerator. Let the eipe parameter for thi eipe be α 0, β 0, and γ 0. A funtion β() i impy defined by the eipe tranformation rue M 0 M M 0 M M 0M 0M / 0 M M M,0 M M 0 ODU Aeerator Phyi Spring 05

14 One might think to evauate the phae advane by integrating the beta-funtion. Generay, it i far eaier to evauate the phae advane uing the genera formua, M tan ', ', M ', M ', here β() and α() are the eipe funtion at the entrane of the region deribed by tranport matrix M ',. Appied to the ituation at hand yied tan,0 M 0 M M 0 ODU Aeerator Phyi Spring 05

15 Beam Mathing Fundamentay, in iruar aeerator beam mathing i appied in order to guarantee that the beam enveope of the rea aeerator beam doe not depend on time. Thi requirement i one part of the definition of having a tabe beam. With periodi boundary ondition, thi mean making beam denity ontour in phae pae aign ith the invariant eipe (in partiuar at the injetion oation!) given by the eipe funtion. One the partie are on the invariant eipe they tay there (in the inear approximation!), and the denity i preerved beaue the inge partie motion i around the invariant eipe. In ina and reiruating ina, uuay different purpoe are to be ahieved. If there are region ith periodi fouing attie ithin the ina, mathing a above enure that the beam ODU Aeerator Phyi Spring 05

16 enveope doe not gro going don the attie. Sometime it i advantageou to have peifi vaue of the eipe funtion at peifi ongitudina oation. Other time, re/mathing i done to preerve the beam enveope of a good beam oution a hange in the attie are made to ahieve other purpoe, e.g. hanging the diperion funtion or hanging the hromatiity of region here there are bend (ee the next hapter for definition). At a minimum, there i uuay a mathing done in the firt part of the injetor, to take the phae pae that i generated by the partie oure, and hange thi phae pae in a ay toard agreement ith the nomina tranvere fouing deign of the ret of the aeerator. The eipe tranformation formua, oved by omputer, are eentia for performing thi proe. ODU Aeerator Phyi Spring 05

17 Diperion Cauation Begin ith the inhomogeneou Hi equation for the diperion. dd d K D Write the genera oution to the inhomogeneou equation for the diperion a before. Here D p an be any partiuar oution, and e uppoe that the diperion and it derivative are knon at the oation, and e ih to determine their vaue at. x and x are ineary independent oution to the homogeneou differentia equation beaue they are tranported by the tranfer matrix oution M, aready found. = p D D Ax Bx x M D M D, ;,, ;, x M D M D, ;,, ;, ODU Aeerator Phyi Spring 05

18 To buid up the genera oution, hooe that partiuar oution of the inhomogeneou equation ith homogeneou boundary ondition D D p,0 p,0 0 Evauate A and B by the requirement that the diperion and it derivative have the proper vaue at (x and x need to be ineary independent!) 0 M, A B 0 D D M D M D p,0,, D D M D M D p,0,, ODU Aeerator Phyi Spring 05

19 3 by 3 Matrie for Diperion Traking M, M, D p,0 p D D D M, M, D,0 D 0 0 Partiuar oution to inhomogeneou equation for ontant K and ontant ρ and vanihing diperion and derivative at = 0 K < 0 K = 0 K > 0 D p,0 () oh K K o K K D' p,0 () inh K K in K K ODU Aeerator Phyi Spring 05

20 M 56 In addition to the tranvere effet of the diperion, there are important effet of the diperion aong the diretion of motion. The primary effet i to hange the time-ofarriva of the off-momentum partie ompared to the on-momentum partie hih travere the deign trajetory. = d z D p p d d p z D d p d D p p Deign Trajetory Dipered Trajetory M 56 Dx x D y y d ODU Aeerator Phyi Spring 05

21 Caia Mirotron: Veker (945) 6 Extration Magneti Fied y RF Cavity x ODU Aeerator Phyi Spring 05

22 Synhrotron Phae Stabiity Edin MMian diovered phae tabiity independenty of Veker and ued the idea to deign firt arge eetron ynhrotron. V (t) h / frf f RF t t / f RF h Lf / RF Harmoni number: # of RF oiation in a revoution ODU Aeerator Phyi Spring 05

23 Tranition Energy Beam energy here peed inrement effet baane path ength hange effet on aeerator revoution frequeny. Revoution frequeny independent of beam energy to inear order. We i auate in a fe eek Beo Tranition Energy: Partie arriving EARLY get e aeeration and peed inrement, and arrive ater, ith repet to the enter of the bunh, on the next pa. Appie to heavy partie ynhrotron during firt part of aeeration hen the beam i non-reativiti and aeeration ti produe veoity hange. Above Tranition Energy: Partie arriving EARLY get more energy, have a onger path, and arrive ater on the next pa. Appie for eetron ynhrotron and heavy partie ynhrotron hen approah reativiti veoitie. A een before, Mirotron operate here. ODU Aeerator Phyi Spring 05

24 ODU Aeerator Phyi Spring 05 Phae Stabiity Condition Synhronou eetron ha Phae o ev E E o Differene equation for differene after paing through avity pa + : Beaue for an eetron paing the avity before after ev E E o o E E M ev E 0 in 0 56

25 ODU Aeerator Phyi Spring 05 Phae Stabiity Condition ) / ( E E 56 0 o / D M d d E E ev ev E E in 4 in 4 / i i K,,0 o / 0 x p i i i D

26 Phae Stabiity Condition Have Phae Stabiity if Tr M ev in E ev frfev frf in o tan tan E fm f i.e., 0 tan ODU Aeerator Phyi Spring 05