Understanding Closed Orbit Distortions in Accelerators

CLOSED ORBIT DISTORTIONS Eric Prebys, UC Davis

2 USPAS Fundamentals, June 4-15, 2018 E. Prebys, Accelerator Fundamentals: Closed Orbit Distortions Closed Orbit Distortion We have considered the effect of a small quadrupole perturbation We will now discuss effect of a small dipole perturbation, which is generally referred to as a closed orbit distortion Note that a misaligned quadrupole has the same effect as the addition of a small dipole term.

3 USPAS Fundamentals, June 4-15, 2018 E. Prebys, Accelerator Fundamentals: Closed Orbit Distortions Closed Orbit Distortion ( cusp ) We place a dipole at one point in a ring which bends the beam by an amount . The new equilibrium orbit will be defined by a trajectory which goes once around the ring, through the dipole, and then returns to its exact initial conditions. That is 0 x 0 0 x x x ( ) 0 0 0 + = = M I M 0 0 x x x 0 ( ) 1 0 = I M 0 x Recall that we can express the transfer matrix for a complete revolution as ( ( 1 2sinpnJe-Jpn= 1 2sinpn 1 2sinpn =Icos2pn+Jsin2pn =eJ2pn cos2pn+a(s)sin2pn -g(s)sin2pn )=1-eJ2pn=eJpne-Jpn-eJpn -1= -2sinpnJ ( b(s)sin2pn cos2pn -a(s)sin2pn )=-eJpn2sinpnJ M(s+C,s)= ( ( ( ) I-M ) ) ) -1 -1eJpn J I-M 1 ( ) = 2sinpnJ Icospn-Jsinpn ) = 2 J I ( 1 = J J = Jcospn+Isinpn acospn+sinpn -gcospn bcospn -acospn+sinpn = ? ? ?

4 USPAS Fundamentals, June 4-15, 2018 E. Prebys, Accelerator Fundamentals: Closed Orbit Distortions Plug this back in cos + cos cos sin cos 0 x 1 0 = 0 + sin x 2 sin cos 0 cos 0 = sin 2 sin We now propagate this around the ring b(s) b0 ( ) cosDy +a0sinDy b0b(s)sinDy = b0cospn sinpn -a0cospn q x(s) x (s) 2sinpn b0 b(s) 1 ( ) ( )cosDy - 1+a0a(s) ( )sinDy ( ) a0-a(s) cosDy -a(s)sinDy b0b(s) q b(s) b0 ( )b0cospn + b0b(s)sinDy sinpn -a0cospn ( ) x(s)= cosDy +a0sinDy 2sinpn =q b0b(s) 2sinpn =q b0b(s) 2sinpn ( ) cosDy cospn +sinDy cospn cos Dy -pn ( )

5 USPAS Fundamentals, June 4-15, 2018 E. Prebys, Accelerator Fundamentals: Closed Orbit Distortions Local Correction Recall our generic transfer matrix b1 b0 ( ) cosDy +a0sinDy b0b1sinDy = x1 x1 x0 x0 b0 b1 1 ( ) ( )cosDy - 1+a0a1 ( )sinDy ( ) a0-a1 cosDy -a1sinDy b0b1 If we use a dipole to introduce a small bend at one point, it will in general propagate as b(s) b0 ( ) cosDy +a0sinDy b0b(s)sinDy = x(Dy) x (Dy) 0 q b0 b(s) 1 ( ) ( )cosDy - 1+a0a(s) ( )sinDy ( ) a0-a(s) cosDy -a(s)sinDy b0b(s) x(Dy)=q b0b(s)sinDy b0 b(s) Remember this one forever ( ) x (Dy)=q cosDy -a(s)sinDy

6 USPAS Fundamentals, June 4-15, 2018 E. Prebys, Accelerator Fundamentals: Closed Orbit Distortions Three Bump Consider a particle going down a beam line. By using a combination of three magnets, we can localize the beam motion to one area of the line 2 12 1 23 2, 3 2 y13=y12+y23 3, 1, We require 3 1 Local Bumps are an extremely powerful tool in beam tuning!! x3=q1b1b3siny13+q2b2b3siny23=0 b1 b2 siny23 siny13 q2=-q1 b1 b3 b2 b3 ( )+q2 ( ) Cancel out angle from first two bends q3=- q1 cosy13-a3siny13 cosy23-a3siny23 b1 b3 b1 b2 siny13 siny23 b2 b3 ( )- ( ) =-q1 cosy13-a3siny13 cosy23-a3siny23 sin y23-y13 ( siny23 ) b1 b3 cosy13-siny13 b1 b3 siny23cosy13-cosy23siny13 siny23 b1 b3 =-q1 cosy23 =-q =-q1 siny23 b1 b3 siny12 siny23 q3=q1

7 USPAS Fundamentals, June 4-15, 2018 E. Prebys, Accelerator Fundamentals: Closed Orbit Distortions Controls Example From Fermilab Acnet control system The B:xxxx labels indicate individual trim magnet power supplies in the Fermilab Booster Defining a MULT: N will group the N following magnet power supplies Placing the mouse over them and turning a knob on the control panel will increment the individual currents according to the ratios shown in green