Simulation of direct space charge in Booster by using MAD program

Simulation of direct space charge in Booster

by using MAD program

Y.Alexahin, N.Kazarinov

Direct space charge in Booster

The calculation of the direct space charge in MAD program may be fulfilled by using set of BEAMBEAM (BB) elements, which give tune shift:

i

N

iiK

bb

18

1

Where β – the Twiss beta-function at BB location, K – kick acting on the particle, Nbb – number of BB elements. For Gaussian beam kick can be expressed in terms of complex error function W:

)(2

)()exp()(2

2221

221

221

yxy

x

x

y

cyx

rr

yi

r

xz

r

yi

r

xz

zWzzzWr

NriKK

As the space charge force is continuous function of distance s the sum in formula (1) has to represent the integral over distance - - and give correct expression for linear tune shift. Thereby the number of particle N’ in fictitious colliding beam must be set as:

(1)

(2)

)1( 2

C

NLBN if (3)

Here Bf – bunching factor, N – number of particle, C – circumference, Li – distance between successive BB elements, – relativistic factor.


Number of BB elements

For correct representation of space charge effect the distance between two successive BB elements should have minimum value as possible. For linear lattice function calculation the BB elements have been placed just after every optical elements of the ring. Long elements such as dipole magnets have been sliced up so that the length of each element was less than 60 cm. Resulting number of BB elements is equal to 1015. Unfortunately TRACK module of MAD does not permit to have more than 200 BB elements. Thereby the particle tracking has been fulfilled with 197 BB elements with average distance between these ones equal to 2.4 m. Comparison of computational results for two sets of BB elements for the same beam parameters gives the following result:

Nbb = 1015 Δx = -0.11354 Δy = -0.299186

Nbb = 197 Δx = -0.11403 Δy = -0.300516

These results are in rather good agreement.


Bunching factor

Bunching factor – the ratio of the beam peak current to the average one - is not constant and changes due to longitudinal oscillation of the particles. Its maximum value may be evaluated with help of simple model.

2tm

2t i

2

i

2

m

in jected beam

beam after quarter oscilla tion period

separatrix

ΔEm – bucket energy height

ΔEi – injected beam energy spread

i

m

m

if

E

E

t

tB


Injected beam parameters

Number of particles 2.5 1012

Kinetic energy 400 MeV

Energy spread 10-3, 3.3 10-4

Emittance 1.278 mmmrad


Bunching factor, tunes

0 0.1 0.2 0.3 0.4 0.5R F vo lta g e p e r re so n a to r, M V

0

2

4

6

8

10

12

Bu

nch

ing

fac

tor,

Bf

1

2

6.25 6.5 6.75 7Qx

6.25

6.5

6.75

7

Qy

W .P.

Bunching factor for two values of injected energy spreadCurve 1 – ΔEi / E = 10-3

Curve 2 – ΔEi / E = 3.310-4

Betatron tunes for various values of bunching factor(curve 1)

V < 0.25 MV

Stable solutions for lattice functions do not exist for amplitude of RF voltage greater than 0.25 MV: Parametric resonance for vertical oscillations.


Tune shifts, beta-functions

0 0.1 0.2 0.3

R F vo ltage, M V

-0.3

-0.2

-0.1

0

Qx,

y

Q x

Q y

3 6 4 0 4 4 4 8 5 2 5 6 6 0d i s t a n c e , m

0

1 0

2 0

3 0

4 0

x,y

, m

x

y

Tune shifts (curve 1, V ≤ 0.25 MV) Twiss β-functions without (black) and with (red) space charge for V = 0.2 MV.

Horizontal tune shift is almost constant due to contribution of the momentum spread to the horizontal beam size.


Estimating of the beam emittance growth

The vertical tune shifts are inverse proportional to beam emittance. Thereby a small increase in the vertical emittance could keep the tune at the boundary of the parametric resonance stopband and stabilize the oscillations.

0 0.1 0.2 0.3 0.4 0.5

R F vo ltage, M V

-0.4

-0.3

-0.2

-0.1

0

Qx,

y

Q x

Q y

0 0.1 0.2 0.3 0.4 0.5R F vo lta g e , M V

0

5

10

15

20

25

y/ y

, %

Vertical tune shift is kept constant by emittance growth for V0.25 MV Corresponding emittance growth

Not very big (< 25%) emittance growth may stabilize vertical oscillations.


Tracking

Tracking have been fulfilled for the following parameters:

Number of particles 2.5 1012

Kinetic energy 400 MeV

Energy spread 10-3

Emittance 1.278 mmmrad

RF voltage 0.05 MV

Bf 1.15

Qx 6.58179 Qy 6.65291

Number of particle in simulation 1000


Tracking

1000 particles have Gaussian distribution in the transverse phase space in accordance with the lattice Twiss functions at the initial point of simulation and uniform distribution in longitudinal momentum. The initial particle distributions in various phase planes is shown below.

-30 -20 -10 0 10 20 30x, m m

-10

-8

-6

-4

-2

0

2

4

6

8

10

y, m

m

-30 -20 -10 0 10 20 30x, m m

-0.8

-0.4

0

0.4

0.8

x',

mra

d

-10 -5 0 5 10y, m m

-2

-1

0

1

2

y', m

rad

(x,y) plane (x,x’) plane (y,y’) plane

The simulation have been fulfilled for three cases: – the influence on the beam dynamics space charge only – the influence on the beam dynamics nonlinearities caused by damping magnets– the influence on the beam dynamics both space charge and nonlinearities


Tracking

0 40 80 120 160 200Turn

0.8

0.85

0.9

0.95

1N

/N0

space charge

non linearities

space charge+non linearities

Particle losses

In the third case the number of lost particle is smaller than in second case.


Dynamic aperture

The computation of the dynamic aperture have been performed in all above cases

0 1 2 3 4 5 6A x/x

0

1

2

3

4

5

6

Ay/ y

0 1 2 3 4 5 6A x/x

0

1

2

3

4

5

6

Ay/ y

0 1 2 3 4 5 6Ax/x

0

1

2

3

4

5

6

Ay/

y

Space charge Nonlinearities Space charge + nonlinearities

Space charge increases the dynamic aperture reduced by nonlinearities


Conclusion

Within the framework of the MAD program the algorithm of computation of the direct space charge has been created. The simulation of the beam dynamic in the presence of the beam space charge has been performed The space charge may provide a beneficial influence on beam dynamics

Near future plans

To simulate the beam dynamics with variable from turn to turn space charge taking into account the changing of the bunching factor due to longitudinal oscillation and possible emittances growth. The software for this simulation have been created. To include into simulation the measured nonlinearities of the magnetic elements.

Simulation of direct space charge in Booster by using MAD program

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