Particle Studio simulations of the resistive wall impedance of copper cylindrical and rectangular beam pipes C. Zannini E. Metral, G. Rumolo, B. Salvant (CERN – BE-ABP-LIS) GSI/CERN collaboration meeting - Feb 19 th 2009 – GSI Darmstadt 1 Special acknowledgement: O. Sebastia (AB desktop)

Overview Context and Objectives Definition of the detuning, driving and general wake First simulations – Rectangular shape – Cylindrical shape New boundary condition in CST 2009 Form factor studies Conclusions Open questions Future Plans 2

3 Context High intensity in the CERN complex for nominal LHC operation, and foreseen LHC upgrade Need for a good knowledge of the machines beam impedance and their main contributors To obtain the total machine impedance, one can: – Measure the quadrupolar oscillation frequency shift (longitudinal) or the tune shift (transverse) with the SPS beam – obtain the impedance of each equipment separately and sum their contributions: Analytical calculation (Burov/Lebedev, Zotter/Metral or Tsutsui formulae) for simple geometries Simulations for more complicated geometries RF Measurements on the equipment  available impedance and wake data compiled in the impedance database ZBASE In this talk, we focus on the benchmark of theory and time domain simulations of the wakes of simple structures with finite conductivity

Objectives Separation of the dipolar and quadrupolar terms of the rectangular shape with Particle Studio simulations, and comparison with theory. Simulation of the wake form factor in a rectangular shape Analysis of the nonlinear term in the wake of the rectangular shape 4

5 Broader objectives for the “impedance team”: 1) Which code should we trust to obtain the wakes for Headtail? (Headtail needs the dipolar and quadrupolar terms disentangled) 3) Should we include coupled or higher order terms of the Resistive Wall impedance in the Headtail code?

Overview Context and Objectives Definition of the detuning, driving and general wake First simulations – Rectangular shape – Cylindrical shape New boundary condition in CST 2009 Form factor studies Conclusions Open questions Future Plans 6

simulated Detuning and driving terms of the transverse wake 7 x y x y x y x y calculatedsimulated calculatedsimulated

Why do we want to separate the dipolar and quadrupolar contribution? The general wake has an impact on the transverse betatron tune shift measured in the machine The driving wake has an impact on the transverse instability threshold Therefore, in machines with flat chambers: - no negative horizontal tune shift (or even positive one) - but existence of a horizontal instability threshold

Overview Context and Objectives Definition of the detuning, driving and general wake First simulations – Rectangular shape – Cylindrical shape New boundary condition in CST 2009 Form factor studies Conclusions Open questions Future Plans 9

Simulation Parameters Geometric parameters Thickness Copper = 0.2cm Length = 1m Vacuum Chamber : Rectangular shape : height=2cm; width= 6cm Particle Beam Parameters σ bunch = 1cm Charge = 1e-9 β=1 10

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12 Horizontal wake in a rectangular shape  In the horizontal plane, W general =0, and W driving =- W detuning

13 Vertical wake in a rectangular shape Detuning General Driving (calc.)  In the vertical plane, W general =3*W detuning, and W driving = 2* W detuning

14 Finally, W y detuning =W x driving, and all relative values of these wakes are consistent with the theory Yokoya (Part. Acc. 1993) and Gluckstern, Zotter, Zeijts (Phys Rev 1992) Summary plot for the rectangular shape: Vertical and horizontal wakes Wx driving (calc.) Wx detuning Wy driving (calc.) Wy general Wx general Wy detuning

Overview Context and Objectives Definition of the detuning, driving and general wake First simulations – Rectangular shape – Cylindrical shape New boundary condition in CST 2009 Form factor studies Conclusions Open questions Future Plans 15

Simulation Parameters Geometric parameters Thickness Copper = 0.2cm 1cm Length = 1m 0.2m Vacuum Chamber : Cylindrical shape : radius=2cm Particle Beam Parameters σ bunch = 1cm 0.8cm 0.5cm Charge = 1e-9 β=1 16

Cylindrical shape 17 Detuning terms are nonexistent, as expected. However, unphysical ripple observed for the cylindrical shape Wy driving = Wxdriving Wy detuning = Wxdetuning

Overview Context and Objectives Definition of the detuning, driving and general wake First simulations – Rectangular shape – Cylindrical shape New boundary condition in CST 2009 and comparison with theory Form factor studies Conclusions Open questions Future Plans 18

19 New boundary condition in CST 2009 Modelling a lossy metal without the conducting wall condition in CST 2009 The lossy metal is explicitly modelled around the vacuum The lossy metal is only modelled through a boundary condition (background material has to be changed to loss metal too)

Boundary condition conducting wall 20 The conducting wall boundary condition allows to simulate easily also the cylindrical shape. To simulate explicitly the cylindrical copper layer without ripple, an unmanageable number of mesh cells has to be used.

Number of mesh ~ 10 6 Device length = 20 cm b=1cm Rms bunch length = 1 cm Displacement =0.1*b Boundary conditions: conducting wall in x and y open in z Normalization at device of 1m Comparison of the simulated wake potential with the theoretical wake potential of a point charge 21 Theory: from Palumbo, Vaccaro, Zobov, INFN, 1994 But we are comparing the simulated wake of a gaussian bunch with the theoretical wake of a point charge. We need to convolute the theoretical wake with the source bunch

Comparison of the simulated wake potential with the theoretical wake potential of a Gaussian bunch 22  Theoretical and simulated wake potential are very similar  Short range wakes are subject to more noise in simulations  Also the theory is not valid at high frequencies

Overview Context and Objectives Definition of the detuning, driving and general wake First simulations – Rectangular shape – Cylindrical shape New boundary condition in CST 2009 Form factor studies Conclusions Open questions Future Plans 23

Simulations with MWS 2008 form factor studies 2 b 2 h Form factor q: 24

Simulation parameters Number of mesh Device length = 20cm b=1cm Displacement = 0.1*b,h Boundary conditions: electric in x and y open in z Normalization at device of 1m All wakes (including the driving term) are now simulated 25

Rectangular shape with form factor q=0.5 2 b 2 h q=0.5  h=3b 26 All the results simulated are normalized by the factor

2 b 2 h q=0.33  h=2b Rectangular shape with form factor q= All the results simulated are normalized by the factor

2 b 2 h q=0.1  h ~ 1.22 b Rectangular shape with form factor q= All the results simulated are normalized by the factor

29 Comparison of the theoretical and simulated wake form factor Theory: from Gluckstern, Ziejts, Zotter, Phys. Rev., 1992

Overview Context and Objectives Definition of the detuning, driving and general wake First simulations – Rectangular shape – Cylindrical shape New boundary condition in CST 2009 Form factor studies Conclusions Open questions Future Plans 30

Conclusion A factor 4.4 (probably ) is observed between the amplitude of simulated wakes and theoretical wakes. This amplitude factor aside, we have separated the dipolar and quadrupolar terms in the rectangular shape, and they agree with the theory. The simulated wakes obtained for several rectangular shape form factors also agree with the theoretical curve. 31

Open questions 32 Factor 4.4 between theory and simulations  most likely a difference of convention. Issues with cylindrical shape Particle Studio outputs the wake potential (gaussian bunch source), but Headtail expects the wake function (point charge source).  should we simulate short bunches for high frequency applications (e.g. multi bunch effects), and long bunches for low frequency applications (single bunch effects)?

33 In Headtail, the wake is assumed to have linear uncoupled dependance on the source particle and the test particle. This linear approximation should be valid for small particle amplitudes. If the amplitude grows, do we have to include higher order terms? At what displacement? Besides, are there coupled terms between planes? Future plans: coupling terms and non linear terms

bxbx δxδx Test beam Source beam x y the displacement is along the diagonal of the rectangular shape and the wake is normalized to the displacement 34 First results of simultaneously moving x and y location of the source beam These first results are difficult to explain without involving non linear higher order dependance of the wake on the transverse location. The threshold for the onset of a nonlinear dependance seems very low (  ~0.1 b)

Thank you for your attention! 35

Number of mesh ~ Device length = 2.5cm Displacement =0.0333*b,h Boundary conditions: electric in x and y open in z Normalization at device of 1m 2 b 2 h q=0.5  h=3b 36

Different boundary conditions 37

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