BLOND AT INJECTION IN THE CERN PS BOOSTER D. Quartullo, V. Forte - - PowerPoint PPT Presentation

LONGITUDINAL SPACE CHARGE SIMULATIONS WITH BLOND AT INJECTION IN THE CERN PS BOOSTER

• D. Quartullo, V. Forte

Acknowledgments: E. Benedetto, A. Lasheen, G. Rumolo, E. Shaposhnikova, H. Timko,

• L. Wang, C. Zannini

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Motivation (1/2)

 The plans of the LHC Injectors Upgrade (LIU) project for the CERN PS Booster:

• Increase of injection and extraction energy to 160 MeV and 2 GeV

respectevely (with Linac4)

• Replacement of the existing RF cavities with FINEMET cavities
• Analysis of longitudinal beam stability with new RF system using

realistic impedance model  Since the space charge impedance is dominant in the longitudinal plane it’s very important to have:

• A reliable and fast code to simulate longitudinal dynamics
• A good evaluation of the longitudinal space charge impedance

 Studies of the new injection scheme from Linac4 to reduce the transverse space charge

• Longitudinal simulations needed for transverse beam dynamics

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Motivation (2/2)

 PyOrbit has been used extensively to study longitudinal dynamics for transverse purposes (see slides by V. Forte at LIU-PSB Injection Working Group, CERN) .  PyOrbit (full 6D tracking code) can take one day to simulate 10000 turns with longitudinal space charge and PTC tracking using parallelization with 48 cores.  BLonD is a pure longitudinal code recently developed at CERN; it needs 20 minutes with one core for the same simulation.  The benchmark between the two codes was aimed to check if the two software give the same results  Expected target parameters for LHC standard beams from Linac4 have been taken as reference:

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CONTENTS

 Motivation  Part I: Benchmark between BLonD and PyOrbit (with V. Forte)  Part II: SC impedance calculation with LSC code  Conclusions Bibliography

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Starting scenario

• Longitudinal space charge simulations done with PyOrbit to study the injection
• f Linac4 bunches in the PS Booster.
• Time frame considered: the first 10 ms of the injection.

uniform waterbag like t = 0 ms t = 10 ms

• Evolution in phase space of 5000

particles distributed uniformly between –π and π without space charge.

• Acceleration implies bucket shrinkage

and synchronous phase displacement.

• Double RF.

Good compromise, initial distribution used for the last benchmark.

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Injection optimisation with PyOrbit

• From single Linac4 bunches it is possible to create arbitrary injection structures

The goal was to optimize:  bunch length to minimize the number of injected turns;  energy spread to minimize filamentation and peaks in line density.

Peak line density too high (bad for transverse space charge) in the first 200 turns. Short bunch length (many turns needed for given N).

𝚬𝒒 𝒒 = 𝟐. 𝟒𝟕 ∙ 𝟐𝟏−𝟒 𝚬𝒒 𝒒 = 𝟏. 𝟒𝟗 ∙ 𝟐𝟏−𝟒 𝚬𝒒 𝒒 = 𝟑 ∙ 𝟐𝟏−𝟒

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• Movie of bunch profile and distribution in phase space evolutions from 0 to 10 ms.
• Space charge is included, N = 295 ∙ 1010, # macro particles = 5 ∙ 105.

Injection to CERN PS Booster at 160 MeV (with Linac4)

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Single particle tracking

• Acceleration, one RF and no space charge

In BLonD the phase space trajectories for small amplitudes are not symmetric with respect to ∆𝐹 = 0, in PyOrbit with PTC-tracking they are. synchronous phase starting point of a particle at ∆𝐹 = 0

• BLonD equations of motion (impulsive system)
• 1. Kick in energy
• 2. Drift

with 𝜀 ≐ Δ𝑞

𝑞𝑇 = Δ𝐹 𝛾𝑇

2𝐹𝑇

Δ𝐹(𝑜+1) = Δ𝐹(𝑜) + 𝑊 sin 𝜄(𝑜) − 𝛾𝑇

𝑜+1 (𝑞𝑇 𝑜+1 − 𝑞𝑇 𝑜 )

𝜄(𝑜+1) = 𝛾𝑇

(𝑜+1)

𝛾𝑇

(𝑜) 𝜄(𝑜) + 2 𝜌 𝜃(𝑜+1)𝜀(𝑜+1)

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𝛾𝑇

(𝑜+1)

𝛾𝑇

(𝑜) = 1

But the derivation of the theta equation is clear: 𝑢 𝑜+1 = 𝑢 𝑜 + … Ω𝑆𝐺

(𝑜+1) 𝑢 𝑜+1 = Ω𝑆𝐺 (𝑜+1)

Ω𝑆𝐺

(𝑜) Ω𝑆𝐺 (𝑜)𝑢(𝑜) + ⋯

Ω𝑆𝐺

(𝑜+1)

Ω𝑆𝐺

(𝑜)

= 𝛾𝑇

(𝑜+1)

𝛾𝑇

(𝑜)

𝜄(𝑜) = Ω𝑆𝐺

(𝑜) 𝑢(𝑜)

𝜄(𝑜+1) = 𝛾𝑇

(𝑜+1)

𝛾𝑇

(𝑜) 𝜄(𝑜) + ⋯

Single particle tracking

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Why in BLonD the beta ratio implies an asymmetry in phase space at least when we are near the synchronous phase? Δ𝐹(1) = 𝑊 𝑡𝑗𝑜 𝜄(0) − 𝑊 𝑡𝑗𝑜 𝜄𝑇

(0)

We can even estimate a priori the energy offset: 𝜄(0) → 𝜄𝑇

(0)

Δ𝐹(1) → 0 𝜄(1) → 𝛾𝑇

(𝑜+1)

𝛾𝑇

(𝑜) 𝜄(0) > 𝜄(0)

∆𝐹𝑝𝑔𝑔𝑡𝑓𝑢= − 𝛾𝑇 𝛾𝑇 𝜄𝑇 𝜃 𝜕0 𝐹𝑇 We will investigate further this discrepancy between the codes. Possible risk: divergenge could increase for long simulations. formula: ∆𝐹𝑝𝑔𝑔𝑡𝑓𝑢 = 53.99 eV simulated: ∆𝐹𝑝𝑔𝑔𝑡𝑓𝑢 = 57.59 eV the ellipse is shifted up

Single particle tracking

Let’s consider the equations of motion for : Red: BLonD, Blu: PyOrbit

Space charge in BLonD and PyOrbit

• Different ways to calculate the space charge kick in BLonD and PyOrbit:

𝐽𝑤 𝑇𝐷 𝑢 = −𝑓 ℱ−1 𝑎 𝑇𝐷 𝜏 = − 𝑓 2 𝜌

−∞ ∞

𝑎 𝑇𝐷 𝜏 𝑓𝑗 𝜕 𝑢 𝑒𝜕 = 𝑓 2 𝜌

−∞ ∞

𝑗 |𝑎 𝑇𝐷| 𝑜 𝑜 𝜏 𝑓𝑗 𝜕 𝑢 𝑒𝜕 = 𝑓 2 𝜌 |𝑎 𝑇𝐷| 𝑜 1 𝜕𝑠𝑓𝑤

−∞ ∞

𝑗 ω 𝜏 𝑓𝑗 𝜕 𝑢 𝑒𝜕 = 𝑓 2 𝜌 𝜕𝑠𝑓𝑤 |𝑎 𝑇𝐷| 𝑜

−∞ ∞ 𝑒

𝑒𝑢 𝑓𝑗 𝜕 𝑢 𝜏 𝑒𝜕 = 𝑓 2 𝜌 𝜕𝑠𝑓𝑤 |𝑎 𝑇𝐷| 𝑜 𝑒 𝑒𝑢

−∞ ∞

𝑓𝑗 𝜕 𝑢 𝜏 𝑒𝜕 = 𝑓 𝜕𝑠𝑓𝑤 |𝑎 𝑇𝐷| 𝑜 𝑒 𝑒𝑢 ℱ−1 𝜏 = 𝑓 𝜕𝑠𝑓𝑤 |𝑎 𝑇𝐷| 𝑜 𝑒 𝑒𝑢 𝜇(𝑢) BLonD Specific routine for constant imaginary

|𝑎 𝑇𝐷| 𝑜

PyOrbit ℱ−1 𝑎 𝑇𝐷 𝜏 is calculated after the beam spectrum computation as if it were a general impedance.

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Benchmark N1: realistic case no space SC

• 6 different longitudinal distributions tracked with acceleration, double RF and no

space charge, # macro particles ≈ 5 ∙ 105 good qualitative agreement. PyOrbit BLonD

No significant shifts up to 𝑂 = 1010

• Parabolic bunch at 160 MeV in the PSB, single RF, no acceleration.
• The bunch was not matched with space charge effects.
• Simulation parameters:
• ℎ = 1, 𝑔

𝑠𝑓𝑤 = 991968.3 Hz, 𝑊 = 8 kV, 𝜃 = -0.67, 𝛾𝑇 = 0.52, 𝐹𝑇 = 1098.2 MeV

• 𝑓 = proton charge, 𝑂 = number of particles, 𝐷 = 157 m, 𝑚𝑗𝑜𝑗𝑢 = 148.16 ns (~23m)
• 𝑎

𝑜 𝑇𝐷 = 795.8 W same value for the two codes,

• # macro particles = 5 ∙ 105
• Syncrotron frequency of the syncronous particle:

𝑔

𝑡0 =

ℎ 𝑔

𝑠𝑓𝑤 2

𝑊 |𝜃| 2 𝜌 𝛾𝑇

2 𝐹𝑇

𝑔

𝑡 = 𝑔 𝑡0

1 − 3 𝑓 𝑂𝑔

𝑠𝑓𝑤

𝜌2 ℎ 𝑊 𝐷 𝑚

3

𝑎 𝑜

𝑇𝐷

No space charge. With space charge (for a matched parabolic bunch below transition). 𝑔

𝑡 < 𝑔 𝑡0

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𝑡[𝐼𝑨]

Benchmark N2: parabolic bunch

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Max θ [rad]

𝑂 = 0

Max θ [rad]

𝑂 = 2.95 ∙ 109

Max θ [rad]

• Good quantitative correspondence between the codes and the analytical formula

up to 𝑂 = 2.95 ∙ 1010, where space charge effect is negligible in comparison to the RF voltage and there is no blow-up.

𝑚𝑗𝑜𝑗𝑢 𝑚𝑗𝑜𝑗𝑢 𝑚𝑗𝑜𝑗𝑢 𝑔

𝑡

𝑔

𝑡

𝑔

𝑡

𝑂 = 2.95 ∙ 1010

Benchmark N2: synchronous frequency diagram

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Max θ [rad] Max θ [rad]

• For 𝑂 > 2.95 ∙ 1010 blow up occurs and the formula is not anymore applicable.
• However the codes give practically the same values for the bunch lengthening and the

synchronous particle frequency shift.

𝑚𝑗𝑜𝑗𝑢 𝑚𝑗𝑜𝑗𝑢 𝑔

𝑡

𝑂 = 2.95 ∙ 1011 𝑂 = 2.95 ∙ 1012

Benchmark N2: synchronous frequency diagram

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• Parabolic bunch evolution in phase space, PyOrbit (up), BLonD (down).
• Blow-up does not occur up to 𝑂 = 2.95 ∙ 1010.

𝑂 = 0 𝑂 = 2.95 ∙ 109 𝑂 = 2.95 ∙ 1010

Benchmark N2: phase space

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• Bunch blow-up.

𝑂 = 2.95 ∙ 1011 𝑂 = 2.95 ∙ 1012

Benchmark N2: phase space

• Double RF with acceleration, space charge with

𝑎 𝑜 𝑇𝐷 = 795.8 W.

• Realistic PSB initial distribution:

bunch length max-min = 474 ns,

𝚬𝒒 𝒒 = 𝟐. 𝟒𝟕 ∙ 𝟐𝟏−𝟒

# macro particles ≈ 5 ∙ 105

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𝜄[𝑠𝑏𝑒] Δ𝐹[𝑁𝑓𝑊]

• Tangent method introduced to

measure the bunch length before and after having applied a low- pass filter to the profile.

𝜄[𝑠𝑏𝑒] Total bunch length

Points of minimum and maximum profile derivative

Benchmark N3: realistic case

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Benchmark N3: results (1/3)

• Moving average filter, window size = 1 (left), window size = 5 (right)
• Normalised top line density, PyOrbit(left) and BLonD(right)

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Benchmark N3: results (2/3)

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• BLonD shows bigger blow up due to space charge in terms of min-max bunch length with

respect to PyOrbit; the tangent method shows less divergence between the codes.

• PyOrbit seems not to feel the space charge in this particular example.
• In any case this relatively slight difference in the final benchmark should be investigated:
• the divergence could increase for very long simulations.
• First attempt to investigate this discrepancy:
• compare the different methods to calculate the space charge kick.

Benchmark N3: results (3/3)

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• In the benchmark between BLonD and PyOrbit no importance has been given to

the numerical method used to compute the derivative. The simplest first order difference method has been used.

• The Savitzky-Golay filter is very known in literature and it has been used in PSB

studies to smooth profiles and calculate their derivatives.

SC kick computation: intro

PyOrbit BLonD 𝜇 𝑢 unsmoothed profile ℱ−1 𝑎 𝑇𝐷 ℱ 𝜇 𝑎 𝑇𝐷 𝑜 𝑒 𝑒𝑢 𝜇(𝑢)

• At a given turn:

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• Comparison of different methods using the same conditions of the last benchmark, one
• f them being slicing the entire ring into 300 parts.
• In addition the case of 1000 slices is presented: the over-slicing implies a lot of noise.
• ‘diff’ = first order difference, ‘gradient’ = second order central difference,

‘Sav-Gol 2order’ = second order Savitzky-Golay filter with three points, ‘Sav-Gol 4order’ = fourth order Savitzky-Golay filter with five points, ‘frequency domain’ = space charge kick calculated in frequency domain (as in Pyorbit).

SC kick computation : results (1/4)

300 slices in (-π, π) 1000 slices in (-π, π)

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300 slices in (-π, π) 1000 slices in (-π, π)

SC kick computation : results (2/4)

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300 slices in (-π, π) 1000 slices in (-π, π)

[ns] [ns] [ns] [ns] [ns] [ns]

SC kick computation : results (3/4)

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• If 300 slices are used as in the last benchmark the various methods give slight

different results and these discrepancies are comparable to what obtained in the benchmark itself.

• With 1000 slices the calculation in frequency domain is completely different from

the computations in time domain: we have a significant amount of particle lost while for most of the derivative methods the losses are practically zero.

• This could be explained by the fact that with 1000 slices we introduce a lot of

noise that is amplified in frequency domain by the multiplication of a linear increasing space charge impedance; instead in time domain the derivative acts

• nly on independent portions of the bunch profile.

 In frequency domain a filter on the profile is needed.

• In any case the choice of the more accurate way to calculate the space charge kick

remains delicate.

SC kick computation : results (4/4)

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CONTENTS

 Motivation  Part I: Benchmark between BLonD and PyOrbit (with V. Forte)  Part II: SC impedance calculation with LSC code  Conclusions Bibliography

Different estimations for Z/n (1/3)

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• In the longitudinal studies with PyOrbit (and in its benchmark with BLonD) it has been

assumed that: 𝑎𝑇𝐷 𝑜 = 𝑎0 𝑕 2 𝛾 𝛿2 = 𝑎0 2 𝛾 𝛿2 1 + 2 log 𝑐 𝑏 = 795.8 Ω Impedance free space At 160 MeV 𝑐 = radius chamber = 30 mm 𝑏 = radius beam = 11 mm

• This g factor is exact for round uniform beam in circular chambers.
• In the PSB the chamber type varies a lot along the ring.
• 30 mm is approximately the lowest half-height of all the chambers.
• The transverse distributions are not uniform but resemble a Gaussian.
• For this reason in the above formula, supposing that 𝜏 ≈ 5.5 𝑛𝑛 for standard LHC beams

at injection, the radius of the beam is approximated as 𝑏 = 2 𝜏.

• Longitudinal field on axis, so worst case scenario. But the smallest possible value for 𝑐

compensates it.

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• It has been suggested by Baartman in 1992 that it’s more realistic to consider the average

potential instead of the on-axis potential; if the beam radius 𝑏 of a Gaussian distribution can be interpreted as twice the rms width of the distribution itself then: 𝑎𝑇𝐷 𝑜 = 𝑎0 2 𝛾 𝛿2 0.5 + 2 log 𝑐 2 𝜏 = 663.7 Ω

• But still we have strong hypotheses: neither the beam nor the beam pipe is round nor of

constant dimension along the ring.

• However the last formula has already been used to calculate g in the PSB at 100 MeV,

g(100 MeV) = 2. The transverse size was calculated through the beamscope.

• Using this value for g we can have a third estimation for 𝑎𝑇𝐷/n at 160 MeV.

Different estimations for Z/n (2/3)

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𝑎𝑇𝐷 𝑜 = 𝑕 𝑎0 2 𝛾 𝛿2 𝑕(𝐹𝑙) = 0.5 + 2 ln 𝑐 𝑏(𝐹𝑙) 𝑏 𝐹𝑙 = 𝑑𝑝𝑜𝑡𝑢 𝛾 𝛿

does not depend on 𝐹𝑙

𝑕(100 𝑁𝑓𝑊) = 2 ln 𝑐 𝑏(100 𝑁𝑓𝑊) = 0.75 𝑏 𝐹𝑙 = 𝑏(100 𝑁𝑓𝑊)

𝛾 𝛿(100 𝑁𝑓𝑊) 𝛾 𝛿

ln 𝑐 𝑏(𝐹𝑙) = ln 𝑐 𝑏(100 𝑁𝑓𝑊) + 1 2 ln 𝛾 𝛿 𝛾 𝛿(100 𝑁𝑓𝑊) 𝑎𝑇𝐷 𝑜 = 𝑎0 𝛾 𝛿2 1 + 1 2 ln 𝛾 𝛿 𝛾 𝛿(100 𝑁𝑓𝑊) = 595.5 Ω We have an uncertainty from 600 Ω to 800 Ω at least. Considering that: 𝐽𝑤 𝑇𝐷 ∝ 𝑂 𝑎𝑇𝐷 𝑜 and that the space charge is dominant in the PSB then we have a factor 1.33 of uncertainty

• n the number of particles we can have in stability thresholds studies.

We need a code aimed at the calculation of the longitudinal space charge.

Different estimations for Z/n (3/3)

• LSC (Longitudinal Space Charge) code developed by L. Wang at SLAC.
• It solves internally the wave equation for 𝐹𝑨:
• Here 𝜍 and 𝑲 are the charge and current densities, respectively.

MAIN INPUTS:

• 𝜏𝑌, 𝜏𝑍;
• chamber cross section

type;

• transverse distribution

type;

• beam energy.

CALCULATION OUTPUT:

• 𝑎/𝑀 averaged over 1 𝜏;
• 𝑎/𝑀 on axis.

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LSC code

∇2𝑭 − 1 𝑑2 𝜖𝑭 𝜖𝑢 = ∇𝜍 𝜁0 + 𝜈0 𝜖𝑲 𝜖𝑢

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• The same distribution used in the last benchmark has been considered.
• To calculate the tranverse beam sizes we use the known formulae:

𝜏𝑦(𝑡) = 𝜁𝑦 𝛾𝑦 𝑡 + 𝐸𝑦

2 𝑡 𝜀2

𝜏𝑧(𝑡) = 𝜁𝑧 𝛾𝑧(𝑡) 𝜁𝑦,𝑧: not normalised emittances 𝑡: position along the ring 𝛾𝑦,𝑧: beta functions 𝐸𝑦: dispersion function 𝜀 = 𝛦𝑞

𝑞 = 1.36e-3

Inputs for LSC (1/5)

𝜄[𝑠𝑏𝑒] Δ𝐹[𝑁𝑓𝑊]

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• To derive 𝜁𝑦,𝑧 we can look at the brightness curve with Linac4 at PSB extraction:
• To translate at injection we

have to consider that the beam losses and emittance growth budgets are 5% for both. 𝑂𝑓𝑦𝑢𝑠 = 1 − 5 100 𝑂𝑗𝑜𝑘 𝜁𝑓𝑦𝑢𝑠

𝑜𝑝𝑠𝑛 = 0.0059 𝑂𝑓𝑦𝑢𝑠

𝜁𝑗𝑜𝑘

𝑜𝑝𝑠𝑛 =

1 − 5 100 𝜁𝑓𝑦𝑢𝑠

𝑜𝑝𝑠𝑛

𝑂𝑗𝑜𝑘 = 295 ∙ 1010

• Then in our case at 160 MeV:

𝜁𝑗𝑜𝑘

𝑜𝑝𝑠𝑛 = 1.57 𝜈𝑛

𝜁𝑦,𝑧 = 𝜁𝑗𝑜𝑘

𝑜𝑝𝑠𝑛

𝛾 𝛿 = 2.58 𝜈𝑛

Inputs for LSC (2/5)

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• The beta and dispersion functions depend on the position along the ring:
• The PSB can be divided in 211 parts. Each part differs from the two adjacent ones from

the cross section shape or dimensions or both.

• Using MAD-X tables and by interpolation we get the values of 𝛾𝑦,𝑧 and 𝐸𝑦 for the 211

elements and then, applying the formulae, we have 𝜏𝑦,𝑧 for every part.

Inputs for LSC (3/5)

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• Four different shapes of chambers in the PSB: round for drift tubes (117 parts) and then:
• ‘rectangular’ for dipoles

(47 parts);

• ‘diamond’ for quadrupoles

(44 parts);

• ‘oblong’ for septum

magnets at injection and extraction (3 parts).

• The ‘rectangular’ and ‘oblong’ shapes are very similar. In rather good approximation we

can consider them all as ‘oblong’.

Inputs for LSC (4/5)

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• LSC needs the grids of the various chambers. L. Wang calculated the grids for the

characteristic ‘diamond’ and ‘oblong’ shapes after having given him the boundaries.

• The grids are not banally

uniform but they depend on the shapes themselves to have a better resolution.

Inputs for LSC (5/5)

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• The bunch spectrum at 160 MeV is in blu; each line refers to one of the 211 parts.
• We can assume that Z/(L n) is constant with very good approximation where the bunch

spectrum is sitting.

• But we can see that the constant values Z/(L n) vary significantly for the 211 parts.

LSC results (1/2)

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• If we want to obtain the SC induced voltage over one revolution turn we need to sum:

𝑎 𝑜 =

𝑗=1 211

𝑀𝑗 𝑎 𝑜 𝑀 𝑗 = 633.14 Ω

• This value averaged over 1 𝜏 is quite close to the measured-rescaled one of 595.5 Ω. On

the other hand the Z/n on axis obtained with LSC is 662.49 Ω and consequently the value used in the benchmarks seems to be too pessimistic.

LSC results (2/2)

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CONTENTS

 Motivation  Part I: Benchmark between BLonD and PyOrbit (with V. Forte)  Part II: SC impedance calculation with LSC code  Conclusions Bibliography

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Conclusions

 PyOrbit and BLonD are quite different in their structure: one is a full 6D tracking code and the

• ther is a pure longitudinal code.

 Different longitudinal simulations have been run for benchmark studies at injection in the PS Booster from Linac4: the obtained results are qualitatively in agreement with a discrepancy in the last realistic benchmark.  It has been showed that the method chosen to calculate the space charge kick could be significant when in presence of noise induced by over-slicing; in this case particular care should be taken.  Even taking into account the presence of few noise using 300 slices, the two dynamics in the last benchmark remain qualitatively different: PyOrbit doesn’t seem to feel at all the space charge unlike BLonD.  Longitudinal space charge impedance in the PSB is dominant: the LSC code has been used to estimate more accurately the value of Z/n. NEXT STEPS:

• Effort to try to reduce even more the divergences between the two codes.
• Benchmark against measurements (with current Linac2).
• Convergence studies should be done fixing the number of slices and increasing the number of

macro particles (this could be a time-related limitation for PyOrbit).

• Phase and radial loops are very important in the PS Booster: they will be introduced in BLonD as

well as the entire impedance model (kickers, cables, cavities, resistive wall...).

• The entire PSB ramp will be simulated in BLonD with an estimated execution time of one-two

days for a single simulation using 5 ∙ 105 macro particles.

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CONTENTS

 Motivation  Part I: Benchmark between BLonD and PyOrbit (with V. Forte)  Part II: SC impedance calculation with LSC code  Conclusions Bibliography

 BLonD can be downloaded at http://blond.web.cern.ch/  PyOrbit can be downloaded at http://code.google.com/p/py-orbit/  LSC can be downloaded at https://www.slac.stanford.edu/~wanglf/LSC/

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Bibliography

 LHC INJECTORS UPGRADE, Technical Design Report, Volume I: Protons, 2014  R. Baartman, TRIUMF Technical Report No. TRI-DN-92-K206, 1992.  S. Burger, C. Carli, M. Ludwig, K Priestnall, U. Raich, The PS Booster fast wire scanner, 2003  V. Forte and D. Quartullo, presentations at LIU-PSB Injection Meeting, CERN, 8/12/2014, 24/02/2014  S. Hancock and M. Lindroos, Longitudinal phase space tomography with space charge, 2000  S. Hansen et al., Effects of space charge and reactive wall impedance on bunched beams, IEEE Transactions on nuclear Science, 1975  L. Wang and W. Li, Analysis of the longitudinal space charge impedance of a round uniform beam inside parallel plates and rectangular chambers, 2015

Resources Contacts: danilo.quartullo@cern.ch, vincenzo.forte@cern.ch