Space Charge in Isochronous Regime (IR)

E. Pozdeyev, BNL*

*work partly done at Michigan State University in 2001-2003 (experiments and simulations)

together with J.A. Rodriguez

08/27/08 E. Pozdeyev, HB08 2

Isochronous regime

• Several types of machines operate / run into IR:– rings for precise nuclear mass spectrometry – some isochronous-optics light sources. – hadron synchrotrons during transition crossing– cyclotrons (FFAG?)

• Studies of beam dynamics of intense beams around transition have been conducted and documented (including text books, K. Ng)

• Effect of space charge (SC) on transverse motion and coupling of radial and longitudinal motion must be included in consideration in IR (usually omitted)

08/27/08 E. Pozdeyev, HB08 3

Longitudinal impedance at short λ

2,ln

21)( 2

00||

kbaRZikkZ -Long wavelength approximation

(includes image charges)

kaKka

kaRZikZ 12

00|| 12)( -Short wavelength approximation

(no image charges)

5/ ab

SC impedance peaks at short wavelength: λm~2.5

08/27/08 E. Pozdeyev, HB08 4

Transverse SC fieldLinear charge density modulation => Energy modulation => Radius modulation => Radial electric field

x

z Er

Er

The radial field comes from the snaky shape and can come from image charges.(We neglect images assuming flat vacuum chamber (like in a cyclotron).)

))(1(2 1 kaKkaxEr The radial field due to snaky shape

Ez

xEE zr 2

Field from a slice

Er

08/27/08 E. Pozdeyev, HB08 5

Dispersion function and slip factor

2220

20 1

cmeE

ppx

Rx r

ppkaKkaRx

SCss

))(1(21 120

0Steady state solution

Exactly at the transition 012 tr

ps

If there is dispersion function error, the slip factor is

0Rs

))(1(2 1 kaKkaSC

ps

Negative Mass instability below γtr ?! Sure…

08/27/08 E. Pozdeyev, HB08 6

Growth rate with SC in Isoch. Reg.

The growth rate for the microwave instability:EZkReI

ik s2

||000

1

2)(

0

11

))(1(24)(

T

kaKkak SC

1824

08/27/08 E. Pozdeyev, HB08 7

Beam H2+

Energy 10-30 keV

x, y 1.14, 1.11

p – 1/ 2 1.7e-4

T 5 sec

C 6.6 m

Nturns 100

Ipeak 0-25 A

FFC

Small Isochronous Ring (SIR), Circa end of 2003

08/27/08 E. Pozdeyev, HB08 8

Beam dynamics simulations in SIR

X

Z

Turn

CYCO simulations (Np=3105) Bunch breaks up within a few turns throughout the bunch

Contour plot ofbunch charge density in median plane for turns 0 to 75

J.A. Rodriguez Ph.D. dissertation, MSU

Ipeak=10 A

08/27/08 E. Pozdeyev, HB08 9

Experimental results:Longitudinal beam dynamics

Measured longitudinal bunch profileTurn# 10 (fixed), Current increases

Vertical axis: peak currentmeasured byFaraday Cup

Horizontal axis: arrival time to the Faraday Cup (equivalent to Z)

08/27/08 E. Pozdeyev, HB08 10

Comparison Experiments to Simulations

# of clusters as a function of # of turns for 5, 10 and 20 A

Simulations

Experiments

J.A. Rodriguez, Ph.D. dissertation, MSU

Simulations included only SC and image chargeson the vacuum chamber

08/27/08 E. Pozdeyev, HB08 11

Experimental Results:Scaling with Beam Current

I = 5 μAI = 9.9 μAI = 19.9 μA

)( AITTEQ

J.A. Rodriguez Ph.D. dissertation, MSU

Growth rate depends linearly on the beam current!!! Not assqrt(I)!!!

I0(μA) / (turn)+ SC

(turn)NO SC

5 -0.011 4.9 51.3

9.9 -0.023 2.5 36.2

19.9 -0.045 1.3 26.5

08/27/08 E. Pozdeyev, HB08 12

Simulation results:Dependence on Emittance

Simulations: number of clusters vs. turn # for different beam emittance

J.A. Rodriguez Ph.D. dissertation, MSU

1~~)(1

SC

k

08/27/08 E. Pozdeyev, HB08 13

Experimental Results:Scaling with Bunch Length

TBUNCH = 600 ns

TBUNCH = 300 ns

TBUNCH = 150 ns

J.A. Rodriguez, Ph.D. dissertation, MSU

Breakup happens throughoutthe bunch (no roll-up from the ends as thought by some).

Size of clusters and number per unit length does not depend on current.

08/27/08 E. Pozdeyev, HB08 14

7.1

-

+

E1

E2

E2 > E1

y0

1.72 0

y

Experimental results:Transverse beam dynamics

I=20 A, E=17 keV, T=1 sec

%52 216

EE

Energy spread grows from 0 to 5% in 10-20 turns

08/27/08 E. Pozdeyev, HB08 15

Conclusions

• Effect of space charge (SC) on transverse motion and coupling of radial and longitudinal motion can play a crucial role at IR (usually omitted)

• This can drive Negative Mass Instability at and below γtr

• Simulation results (CYCO) and experimental data (SIR) agree remarkably well

• They show that – the instability causes very fast beam fragmentation and energy

spread growth– the growth rate is proportional to the beam current and inversely

proportional to the beam emittance

• Landau damping most likely exist through modification of the dispersion function non-coherently

08/27/08 E. Pozdeyev, HB08 16

Acknowledgements

Special thanks: J.A. Rodriguez, F. Marti, R.C. York

J. Bierwagen, D. Cole, D. Devereaux, R. Fontus, S. Hitchcock, D. Lawton, D. Pedtke,

D. Sanderson, J. Wagner, A. Zeller, R. Zink

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