LONGITUDINAL INSTABILITIES

1) Inroducion ) Impedances and wake funcions 3) Longiudinal dynamics 4) Robinson insabiliy 5) Poenial well bunch lenghening LONGITUDINAL INSTABILITIES CAS 7, Daresbury; A Hofmann cas7li-1

1) INTRODUCTION Overview The single paricle moion is given by exernal guide fields (dipoles, quadrupoles, RF), iniial condiions and synchroron radiaion Beam wih many paricles induces currens in vacuum chamber impedance and creaes self fields acing back on i This collecive acion by many paricles can: give synchroron frequency shif due o modified focusing; increase iniial disurbance, insabiliy; change paricle disribuion, (bunch lenghening) Muli-urn effecs driven by narrow-band caviy wih memory build up insabiliy in many urns wih small self-fields reaed as perurbaion Sar a small disurbance from a saionary beam, calculae fields i produces in impedance, check if hey increase/decrease he iniial ampliude, give growh/damping rae Check his for orhogonal (independen) modes of disurbances cas7- Bunch induces fields in passive caviy, hey oscillae and ac back nex urn, in- or decrease iniial disurbance depending on phase bunch caviy V caviy () urn 1 urn Single raversal effecs driven by srong self-fields from broad impedances change disribuion, modify oscillaion modes and can couple hem Self consisen soluions are difficul o ge, bunch lenghening dl/ds I + + + + + + + + + + βc + + + + + + + + + + + + + + I w

Mechanism of single bunch, muli-urn insabiliy urn k: k I() σ urn k: ɛ İ k () I() k ɛ τ k σ τ E E I 1 T k + 1 T k + 1 ɛ τ ( + Q s ) (1 Q s ) E k + ( + Q s ) k + ɛ τ τ k I I 1 k + 3 k + 3 ɛ τ k + 4 k + 4 ɛ (1 Q s ) τ k τ k field k saionary bunch I k = I + I p cos(p ) I p = 1 T T I() cos(p )d oscillaing bunch, Q s = 1 4 ɛ k = ˆɛ sin(πq s k) phaseτ k = ˆτ cos(πq s k) space I = I + I p [cos(p )+ p ˆτ ( sin( + p ) + sin(p ) ) p ± = (p ± Q s ) 1 / E induced in Z r ( ± p ) acs on energy deviaion ɛ cas7li-3

4) IMPEDANCE, WAKE FUNCTION Resonaor Iw a Iw q image charges beam Beam induces wall curren I w = (I b I b ) E ḅunch İ R s I C L ~E v c V C s L R s Caviies have narrow band oscillaion modes which can drive coupled bunch insabiliies Each resembles an RCL - circui and can, in good approximaion, be reaed as such This circui has a shun impedance R s, an inducance L and a capaciy C In a real caviy hese parameers canno easily be separaed and we use ohers which can be measured direcly: The resonance frequency r, he qualiy facor Q and he damping rae α: r = 1 C, Q = R s LC L = R s = R s C r L r α = r Q, L = R s Q r, C = Q r R s cas7li-4

Driving his circui wih a curren I gives he volages and currens across he elemens Using L = R s /( r Q), C = Q/( r R s ) and α = r /(Q), r = 1/ LC gives diff eq I I R I C I L R s I C L V V R = I R R s V C = 1 I C d C V L = L di L d V + r Q V + rv = rr s I Q The soluion of he homogeneous equaion represens a damped oscillaion V R = V C = V L = V I R + I C + I L = I Differeniaing wih respec o gives I = I R + I C + I L = V R s + C V + V L V () = ˆV e α cos r 1 1 4Q + φ V () = e α A cos r 1 1 4Q +B sin r 1 1 4Q cas7li-5

Wake funcion Green funcion Response of RCL circui o a dela pulse I 1 r = C LC I R s L V C Q = R s L α = r Q V + r Q V + rv = rr s I Q Charge q brings he capaciy o a volage V ( + ) = q C = rr s Q q using C = Q r R s I() = qδ() Energy sored in C = energy los by q U = q C = rr s Q q = V (+ ) q = k pm q wih he parasiic mode loss facor k pm = r R s /(Q), given usually in [V/pC] Capacior discharges firs hrough resisor V ( + ) = q C = I R C = 1 V ( + ) C R s = rr s Q q = rk pm Q q Iniial condiions V ( + ), V ( + ) give from general soluion V () = e α A cos r 1 1 4Q + B sin r 1 1 4Q pulse response V () = qk pm e α cos r 1 1 4Q sin ( r 1 1 Q 1 1 4Q ) 4Q cas7li-6

G() = V () q = k pm e α cos r 1 1 4Q sin ( r 1 1 Q 1 1 G() is called Green or wake funcion G() k pm e α cos ( r ) for Q 1 This volage induced by charge q a = changes energy of a second charge q raversing caviy a by U = q V () = qq G() 1 G() k pm q q q E z () q wake funcion longiudinal field V () = G( )dq = I( )G( )d = qw () W () = V ()/q wake poenial cas7li-7 4Q ) 4Q, r = 1 LC G() is relaed o longiudinal field E z by an inegraion following he paricle wih v c and aking momenary field value V = Gq = z z 1 E z (z, )dz = f z z 1 E z (z)dz wih ransi ime facor f We use G() > where energy is los A paricle inside a bunch of charge q and curren I() going hrough a caviy a ime sees he wake funcion creaed by all he paricles passing a earlier imes < resuling in a volage I() dq

Impedance I() = Î cos() I I R s C L A harmonic exciaion of circui wih curren I = Î cos() gives differenial equaion V + r Q V +rv = rr s I Q = rr s Î sin() Q Homogeneous soluion damps leaving paricular one V () = A cos() + B sin() Pu ino diff-equaion, separaing cosine and sine ( r)a + r Q B = V ( r)b + r Q A = rr s Q Î Induced volage by he harmonic exciaion V () = ÎR cos() + Q r r sin() s 1 + Q ( r ) r has a cosine erm in phase wih exciing curren I absorbs energy, is resisive The sine erm is ou of phase, does no absorb energy, reacive Raio beween volage and curren is impedance as funcion of frequency 1 Z r () = R s 1 + Q ( r ) r Z i () = R s Q r r ) 1 + Q ( r r Resisive par Z r (), reacive par Z i () posiive below, negaive above r cas7li-8

Complex noaion We used a harmonic exciaion of he form I() = Î cos() = Î ej + e j wih I is convenien o use a complex noaion I() = Îe j wih giving compac expressions Using he differenial equaion V + r Q V + rv = rr s I Q wih I() = Î exp(j) and seeking a soluion V () = V exp(j), where V is in general complex, one ges + j r Q + r V e j = j rr s Q Îej The impedance, defined as he raio V/I becomes Z() = V Î = R s 1 + jq ( r ) r 1 jq r = R r s 1 + Q ( r ) = Z r + jz i r For Q 1 he impedance is only large for r or r / r = / r 1 and can be simplified 1 jq Z() R r s 1 + 4Q ( ) r Cauion: someimes I() = Îe i insead of I() = Îej is used, his reverses he sign Z i () cas7li-9

Properies of Green funcions and impedances G() k pm Green funcion 1 1 3 4 1 Impedance Z() R s 1 1 Z R () Z I () π/ r Z() = R s / r Q = 3 Q = 15 G() k pm Green funcion 1 1 3 4 π/ r 1 1 jq r r 1 + Q ( r r Impedance Z() R s 1 1 1 ) = Z r + jz i Z R () Z I () / r The resonaor impedance has some specific properies: a = r Z r ( r ) max, Z i ( r ) = < < r Z i () > (inducive) > r Z i () < (capaciive) and any impedance or wake poenial has he general properies Z r () = Z r ( ), Z i () = Z i ( ) Z() = G()e j d Z() Fourier ransform of G() for < G() =, no fields before paricle arrives, β 1 cas7li-1

Typical ring impedance E Z E fi 5 Aperure changes form caviy-like objecs wih r, R s and Q and impedance Z() developed for < r, where i is inducive 1 jq r Z() = R r s 1 + ( Q r ) jr s + Q r r Sum impedance a rk divided by mode number n = / is wih inducance L cas7li-11 Z n = k r Z i R sk = L = L βc Q k rk R Z r I depends on impedance per lengh, 15 Ω in older, 1 Ω in newer rings The shun impedances R sk increase wih up o cuoff frequency where wave propagaion sars and become wider and smaller A broad band resonaor fi helps o characerize impedance giving Z r, Z i, G() useful for single raversal effecs However, for muli-raversal insabiliies narrow resonances a rk mus be used

5) LONGITUDINAL DYNAMICS A paricle wih momenum deviaion p has differen orbi lengh L, revoluion ime T and frequency L = α p c p = α c E β E L T T = = α c 1 γ p p = η p c p wih momenum compacion α c = 1/γ T, slip facor η c A ransiion energy m c γ T he - dependence on p changes sign E > E T 1 γ < α c η c > 1, E < E < E T 1 γ > α c η c < 1, E > For γ 1 p/p E/E = ɛ, η c α c V () ^V U s =e s bunch RF-caviy of volage ˆV, frequency RF = h, SR energy loss U he energy gain or loss of a paricle in one urn δɛ = δe/e is δe = e ˆV sin(h ( s + τ)) U s = synchronous arrival ime a he caviy, τ= deviaion from i, synchronous phase φ s = h s For h τ 1 we develop δe = e ˆV sin(φ s ) + h e ˆV cos φ s τ U cas7li-1

For δe/e 1 use smooh approximaion Ė δe/t, τ = T/T = η c E/E Ė = e ˆV sin φ s + he ˆV cos φ s τ π π π U Use T = π/, relaive energy ɛ = E/E ɛ = e ˆV sin φ s + he ˆV cos φ s τ U πe πe π E Energy loss U may depend on ɛ and τ U(ɛ, τ) U + U U E + E τ giving for he derivaive of he energy loss ɛ = he ˆV cos φ s τ U πe π E ɛ U π τ τ = η c ɛ where we used ha for synchronous paricle ɛ =, τ = we have U = e ˆV sin φ s Combining hese ino a second order equaion τ + U π E τ + s + η c πe U τ =, s = hη c e ˆV cos φ s, α s = 1 U πe π E s1 = s αs + η c U πe s τ + α s τ + sτ = τ = ˆτe αs cos( s1 ), ɛ = ˆɛe αs sin( s1 ) From τ = η c ɛ we ge ˆɛ = sˆτ/η c To ge real s we need cos φ s above ransiion where η c > and vice versa To ge a sable (decaying) soluion we need an energy loss which increases wih E α s = U 4π E = U 4πE ɛ > cas7li-13

7) ROBINSON INSTABILITY Saionary bunch Specrum I K () I p () T I( + T ) σ I() ime domain T T I p () σ σ frequency domain I( T ) Symmeric bunch, I() = I( ), circulaes wih urns k of duraion T, is a periodic curren and expressed by a Fourier series Ĩ() = 1 π I() cos()d, I() single I k () = I( kt ) = I pe jp muli = I + I p cos(p ) raversal p=1 I p = 1 T / T T / I()ejp d = 1 T T I() cos(p )d = π Ĩ(p ) Wih I() = I( ), real I p, cosine erms only A low frequencies I p I Gaussian bunch: I() = q πσ e σ, I p = q p e σ, σ = 1 T σ cas7li-14

Volage induced by a saionary bunch Saionary bunch induces volage in impedance Z() = Z r () + jz i () I k () = p = I p e jp = I + p =1 Use Z() =, combine posiive/negaive frequencies wih Z r ( ) = Z r (), Z i ( ) = Z i () I p cos(p ) V k () = Z(p )I p e jp = I p [Z r (p ) cos(p ) Z i (p ) sin(p )] p= p=1 Energy loss of a saionary bunch Energy los by he whole bunch wih N b paricles per urn in impedance Z() is W b = T I k ()V k ()d W b = T p= I pz(p ) = T 1 I pz r (p ) has only Z r Loss U = W b /N b per paricle is U = T N b I 1 I pz r (p ) = e I p Z r I p 1 I pz r (p ) This conains inegrals T cos(p ) sin(p )d = T cos(p ) cos(p )d = T for p = p for p p cas7li-15

Robinson insabiliy Qualiaive reamen Z R () r h h r Z R () Imporan longiudinal insabiliy of a bunch ineracing wih an narrow impedance, called Robinson insabiliy In a qualiaive approach we ake single bunch and a narrowband caviy of resonance frequency r and impedance Z() aking only is resisive par Z r The revoluion frequency depends on energy deviaion E = η c E E While he bunch is execuing a coheren dipole mode oscillaion ɛ() = ˆɛ cos( s ) is energy and revoluion frequency are modulaed Above ransiion is small when he energy is high and is large when he energy is small If he caviy is uned o a resonan frequency slighly smaller han he RF-frequency r < p he bunch sees a higher impedance and loses more energy when i has an energy excess and i loses less energy when i has a lack of energy This leads o a damping of he oscillaion If r > p his is reversed and leads o an insabiliy Below ransiion energy he dependence of he revoluion frequency is reversed which changes he sabiliy crierion cas7li-16

Oscillaing bunch I K () Ĩ() T ime domain T T I( τ ) I( T τ 1 ) σ τ τ 1 τ I( T τ ) T T 3T σ frequency domain, > s I( 3T τ 3 ) τ 3 Bunch execuing synchroron oscillaion wih s = Q s and ampliude ˆτ modulaes passage ime k a caviy in successive urns k I k () = k= I( kt τ k ) wih τ k = ˆτ cos(πq s k) ˆτ cos( s ) giving curren wihou DC-par I k () = I p cos(p ( ˆτ cos( s ))) > Develop for p ˆτ 1 cos(p ˆτ) 1, sin(p ˆτ) p ˆτ I k () I p [cos(p ) + p ˆτ sin(p ) cos( s )] > = I p cos(p ) + p ˆτ > (sin((p + Q s) ) + sin((p Q s ) )) The modulaion by he synchroron oscillaion resuls in sidebands in he specrum They are ou of phase wih respec o carriers and increase firs wih frequency p cas7li-17

Volage induced by oscillaing bunch Abbreviae: + p = (p + Q s ), p = (p Q s ) I k () = I p cos(p ) + p ˆτ > (sin(+ p ) + sin(p )) We resric on resisive impedance Z r and ge volage V kr () = I p [Z r (p ) cos(p ) > + p ( ˆτ Zr ( p + ) sin( p + ) + Z r (p ) sin(p ) ) V kr () = I p [Z r (p ) cos(p ) > + p [ ˆτ Zr ( p + ) (sin(p ) cos( s ) + cos(p ) sin( s )) +Z r (p ) (sin(p ) cos( s ) cos(p ) sin( s )) ]] Z r () Ĩ() Ṽ r () r Z r () p p p Synchr moion, smoohed: τ k = ˆτ cos(πq s k) τ = ˆτ cos( s ) τ = sˆτ sin( s ) = η c ɛ V kr () = I p [Z r (p ) cos(p ) > + p Z r ( p + ) sin(p )τ cos(p ) τ + Z r ( p ) sin(p )τ + cos(p ) τ s s cas7li-18

Energy exchange Express facors differenly, use τ = η c ɛ I K ()= I p [cos(p ) + p sin(p )τ] > V kr ()= I p [Z r (p ) cos(p ) > + p [ (Zr ( p + ) + Z r (p )) sin(p )τ (Z r ( p + ) Z r (p )) cos(p ) ηɛ s The energy per paricle and urn exchanged beween bunch and impedance cas7li-19 U(τ, τ) = 1 N b T I K ()V K ()d, N b = πi e Neglec higher erms in τ, ɛ, use inegrals T cos(p ) cos(p )d = T cos(p ) sin(p )d = T if p = p if p p U = e I > I pz r (p ) e I > I pp (Z r ( p + ) Z r (p )) ηɛ s U ɛ = e I > I pp (Z r ( p + ) Z r (p )) η s Discussed sabiliy of phase oscillaion τ + α s τ + sτ =, τ = ˆτe α s cos( s1 ) α s = du 4πE dɛ = η c he ˆV cos φ s πe Ipp(Z r ( p + ) Z r (p ) I h ˆV cos φ s s α s = s pip(z r ( p + ) Z r (p )) > sable I h ˆV cos φ s < unsable

Narrow impedance, only one harmonic p r Z r p Damping if α s >, insabiliy if α s < ɛ = ˆɛe αs sin( s ) α s = spip(z r ( p + ) Z r (p )) > I h ˆV cos φ s Above ransiion: cos φ s <, sabiliy if: Z r (p ) > Z r ( p + ) Damping rae proporional o difference in Z r beween lower and upper sideband Imporan narrow-band impedance = RF-caviy:p = h, I p I α s I (Z r ( p + ) Z r (p )) s ˆV induced V cos φ s V RF slope cas7li- s Qualiaive undersanding I k() I() I 1() urn k τ T σ Oscillaing bunch (Q s = 5) Saionary bunch + Perurbaion I k() urn k+1 Caviy field induced by he wo sidebands E z r = ( + Q s ) E z r = ( Q s ) ɛ ɛ I() I 1() Phase moion of he bunch cener γ > γ T σ ɛ τ ɛ τ γ < γ T τ τ

6) POTENTIAL WELL BUNCH LENGTHENING dl/ds We ake a parabolic bunch form V () ˆV RF I + + + + + + + + + + + + + + + + + + + + + + + + V i I I w I() βc V i E z = dl di w dz V = d = dl dz di b d E z dz = L di b dz bunch I b (τ) = Î 1 τ ˆτ = 3πI 1 τ ˆτ ˆτ di b dτ = 3πI τ, I ˆτ 3 = I b, V = ˆV (sin φ s + h cos φ s τ) + 3πI Lτ, L ˆτ 3 = V = ˆV sin φ s + cos φ s h 1 + 3π Z/n I h ˆV cos φ s ( ˆτ) 3 s = hη c e ˆV cos φ s πe s = s 3π Z/n I 1 + h ˆV RF cos φ s ( ˆτ) 3 s = s s s 3π Z/n I h ˆV RF cos φ s ( ˆτ ) 3 Z n τ cas7li-1

V () ˆV RF bunch Reducion of s reduces longiudinal focusing and increases he bunch lengh s s = 1 + 3π Z/n I h ˆV RF cos φ s ( ˆτ) 3 s s = s 3π Z/n I s s h ˆV cos φ s ( ˆτ ) 3 Only incoheren frequency of single paricles is changed (reduced for γ > γ T, increased for γ < γ T ), bu no he coheren dipole (rigid bunch) mode This separaes he wo V () ˆV bunch ˆτ = ˆɛη c / s, ˆτ = ˆτ ˆɛη c / s = E s η c / s rel energy spread ˆɛ, long emi E s = ˆτ ˆɛ Proons: E s = consan, τ 1/ s small: ˆτ ˆτ 4 s s 3π Z/n I 4h ˆV cos φ s ( ˆτ ) 3, ˆτ or: + 3π Z/n I ˆτ ˆτ h ˆV cos φ s ( ˆτ ) 3 1 = ˆτ Elecrons: ˆɛ= cons by syn rad ˆτ 1/ s small: ˆτ ˆτ or: ˆτ ˆτ s s 3π Z/n I h ˆV cos φ s ( ˆτ ) 3, 3 ˆτˆτ + 3π Z/n I h ˆV cos φ s ( ˆτ ) 3 = cas7li-