Simulation of direct space charge in Booster by using MAD program Y.Alexahin, A.Drozhdin, N.Kazarinov

Simulation of direct space charge in Booster

by using MAD program

Y.Alexahin, A.Drozhdin, 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. These ones give tune shift:

i

N

iiK

bb

14

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.

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)

)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.

Number of particle 2.5 1012

Kinetic energy, MeV 400

Initial momentum spread, p/p 2.75 10-4

rms emittance, mmmrad 1.278

RF voltage amplitude, MV 0.05 0.4

Transition energy, GeV 5.114

Tunes (x, y) (6.703, 6.812)

Harmonic number 84

Circumference, m 474.204


Beam and Booster Parameters

The simulation has been fulfilled for beam and booster parameters corresponding to ones during injection into the booster.


Bunching Factor and Momentum Spread

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. The dependence of the bunching factor on number of turn has been found by integrating the equations of longitudinal motion for 5000 particles during 250 turns after injection. The amplitude of the RF voltage increased linearly from 0.05 MV to 0.4 MV during 200 turns and after that kept constant.

- 3 - 2 - 1 0 1 2 3 , ra d

-2

-1

0

1

2

E ,

Me

V

- 3 - 2 - 1 0 1 2 3 , ra d

-2

-1

0

1

2

E ,

Me

V


Bunching Factor and Momentum Spread

0 50 100 150 200 250Tu rn

1

1.5

2

2.5

Bf

0 50 100 150 200 250Tu rn

1

1.5

2

2.5

3

3.5

p / p

0

/0fB 0/ pp

The final value of the bunching factor is in a good agreement with result of measurements /1/.

1. Xiaobiao Huang. Beam diagnosis and lattice modeling of the Fermilab booster. Ph.D. Thesis. Fermilab-thesis-2005-29.


Space Charge Tune Shifts

Lines 2x,y =13 – tune shifts corresponding to parametric resonances

Betatron tunes

The working point crosses the lines of the parametric resonance of the both horizontal and vertical betatron oscillations. Thereby the linear lattice functions are unstable while bunching factor Bf is greater than 1.3.

0 50 100 150 200 250Tu rn

-0.4

-0.32

-0.24

-0.16

x

y

2x = 13

2y = 13

6.56.25 6.75x

6.5

6.25

6.75

y

W .P .


Beta-functions

Beta functions at one period of booster without (dashed line) and with (solid line) space charge

0 4 8 12 16 20d is ta n ce , m

0

10

20

30

40

, m

x

y

0;1 ppfB


Magnet Nonlinearities

The measured nonlinearities of the bending magnets have been included into MAD file of the booster. Each bending magnet has been sliced up to five parts and thin multipole lens has been installed at its exit. The values of the nonlinearities bn and corresponding MAD parameters of thin multipole lens KnL are shown below.

F – magnet D – magnet n 104 bn/B0

*) KnL 104 bn/B0 *) KnL

3 0.0338836 1.755533410-2 -0.298606 -1.3154323 10-1 4 0.0673212 5.4921527 0.449709 3.1198078 10+1 5 -0.43997 -7.0655207 10+3 0.27762 3.7912577 10+3 6 -1.08279 -4.1075503 10+6 0.084459 2.7245578 10+5 7 0.644508 6.7380044 10+8 -0.213484 -1.8979281 10+8 8 1.17023 3.8532792 10+11 0.04472 1.2521965 10+10 9 0.192633 2.2474950 10+13 0.19317 1.9165457 10+13

*) B0 – bending magnetic field


Random Quadrupole Gradient Errors

As the parametric resonance driving term is defined by amplitude of 13-th harmonic of focusing field gradients the random errors have been introduced into the bending magnets quadrupole components K1. The errors have been distributed in accordance with Gauss law with K1/K1 = 5 10-4. The changes of tunes have been avoided with thin multipole lenses described in the previous section.

0 10 20 30 40 50B P M index

-0.4

-0.2

0

0.2

0.4

x,

ra

d

0 10 20 30 40 50B P M index

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

y,

ra

dThe quadrupole errors lead to additional phase advance between adjacent beam position monitor (BPM) as measured at booster /1/. The magnitudes of these ones are in rather good agreement with results of measurements.


Dynamic Aperture

To examine influence of space charge on the parametric resonance the horizontal tune has been moved to the value x = 6.52 closely to the resonance line. The dynamic aperture defined by particles dynamics during 400 turns have been computed for three cases.

0 1 2 3 4 5 6Ax/x

0123456

Ay/ y

0 1 2 3 4 5 6A x/x

0123456

Ay/ y

0 1 2 3 4 5 6Ax/x

0123456

Ay/ y

quadrupole errors space charge quadrupole errors + space charge

The space charge itself does not excite the resonance. The space charge prevents the resonance growth of almost all considered oscillation amplitudes.

The dynamic aperture for non-tuned working point in the presence also of the magnets and the chromaticity sextupoles nonlinearities coincides with one shown in central Figure.


Tracking

Initially 1000 particles have Gaussian distribution in the transverse phase space in accordance with the lattice Twiss functions at the beginning point of simulation and uniform distribution in longitudinal momentum

-0 .02 0 0.02x, m

-0.02

0

0.02

y, m

-0 .02 0 0.02x, m

-0.002

0

0.002

x', r

ad

-0 .02 0 0.02y, m

-0.002

0

0.002

y', r

ad

During tracking over 250 turns the parameters of the BB elements were recalculated at every turn in accordance with changing of the bunching factor Bf and momentum spread. Besides the strength of the orbit bump magnet which shift the orbit during injection went down to zero during 30 turns. All parameters variations have been done by using the Mathematica program. The final particle distributions are shown below. They don’t differ significantly from initial ones.

-0 .02 0 0.02x, m

-0.02

0

0.02

y, m

-0 .02 0 0.02x, m

-0.002

0

0.002

x', r

ad

-0 .02 0 0.02y, m

-0.002

0

0.002

y', r

ad

The particle losses take place mainly during initial 30 turns. This may be explained by influence of the strong magnetic field nonlinearities of the orbit bump magnets and multiple crossing the parametric resonance 2x = 13. Nearby turn 100 the losses may be explained by crossing the resonance 2y = 13. Total particle loss is less than 2%.


Tracking

0 50 100 150 200 250tu rn s

98

98.4

98.8

99.2

99.6

100

nu

mb

er

of p

art

icle

, %

Number of particles


Tracking

Both horizontal and vertical emittances of the beam slightly increased. The relative emittance growth is not greater than 15%.

0 50 100 150 200 250

1.2

1.3

1.4

1.5

x ,

mm

mra

d

0 50 100 150 200 250

1.2

1.3

1.4

1.5

y ,

mm

mra

d


Tracking

The emittance have been evaluated by fitting the integral of distribution function with:

)/exp(1)( IIF

where I – is action variable.

Simulation with changing from turn to turn emittance of the beam.

Fit at the first turn


Tracking


Nominal parameter of the beam as in previous case

Number of particleBeam emittance

Red – horizontal, blue – vertical


Tracking


Intensity increased twofold in 5 turns, then RF capture as in the previous cases

Number of particle Beam emittanceRed – horizontal, blue – vertical

Particle losses increased in two times in comparison with two previous case. Horizontal emittance is in two time greater than initial one.


Tracking


Fit at the last turn

Beam has non-Gaussian form


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 capture into the separatrix of longitudinal motion has given reasonable values of the bunching factor and the momentum spread. The magnetic field nonlinearities and quadrupole errors have been included into MAD file for the booster optics. The simulation of the beam dynamic in the presence of the beam space charge, magnetic field nonlinearities and quadrupole errors has been performed.

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Simulation of direct space charge in Booster by using MAD program Y.Alexahin, A.Drozhdin, N.Kazarinov