Simulating transition crossing in the PS with HeadTail E. Métral, G. Rumolo, B. Salvant LIS meeting - April 7th, 2008

Agenda Context of transition crossing in the PS HeadTail Simulations at constant energies HeadTail Simulations with acceleration Longitudinal aspects Crossing transition with a broadband impedance Perspectives

Context of PS transition crossing Elias presentation in January : Instabilities appear at transition crossing in the PS

Context of PS transition crossing Question from PS2 project:  how fast should we cross transition in PS2 to avoid instabilities ?  Analytical calculations made by Elias (see 29/01/08 LIS meeting presentation )  HeadTail was modified to enable acceleration during simulation

Simulation parameters No broadband impedance No space charge impedance nTOF PS parameters (E. Métral) Transition Energy is constant (t ~ 6.08) Higher orders of momentum compaction factor are not taken into account Beam momentum is increased linearly with time: Here

Agenda Context of transition crossing in the PS HeadTail Simulations at constant energies HeadTail Simulations with acceleration Longitudinal aspects Crossing transition with a broadband impedance Perspectives

BB – no SC – no acceleration – simulations at various energies Bunch lengths and momentum spread were adjusted at each energy to get a linearly matched bunch in the PS bucket with a longitudinal emittance of 2 eVs. Bunch Length Emittance Getting closer to transition transition = 6.08 The bunch is longer far from transition  linear matching equations do not hold

BB – no SC – no acceleration – simulations at various energies Analytical calculations of the longitudinal motion of a particle at the edge of a bunch with 2 eVs longitudinal emittance in a static bucket. For comparison, the accelerating bucket is also shown Stationary bucket 2 eVs for a linear restoring force 2 eVs for non linear restoring force Accelerating bucket (the area of the (z,) ellipse is not constant, but these are not canonical variables)

BB – no SC – no acceleration – simulations at various energies Instabilitity threshold is at  ~ 5.2

With BB no acceleration -  = 5.2 -  = -0.010

With BB no acceleration -  = 5.4 -  = -0.007

With BB no acceleration -  = 5.6 -  = -0.005

With BB no acceleration -  = 5.8 -  = -0.003

With BB no acceleration -  = 6.0 -  = -0.0008

Comparison with analytical estimates

Agenda Context of transition crossing in the PS HeadTail Simulations at constant energies HeadTail Simulations with acceleration Longitudinal aspects Crossing transition with a broadband impedance Perspectives

Longitudinal ellipse shape while crossing transition Elias’s plot of the tilt of the ellipse in the longitudinal phase space Case of the nTOF bunch in the PS WITHOUT space charge (2/2)

Checking simulated longitudinal motion during transition crossing HeadTail simulation : observing 10% of the macroparticles in the longitudinal phase space What about the shape of the ellipse?  follow some particles over 10 turns…

Initial distribution Only the blue particles will be displayed to assess the tilt of the ellipse

Tracking macroparticles with l ~ 2 eVs

No BB – no SC – comparing HeadTail with analytical calculations from Elias Reasonable agreement

Agenda Context of transition crossing in the PS HeadTail Simulations at constant energies HeadTail Simulations with acceleration Longitudinal aspects Crossing transition with a broadband impedance Perspectives

BB impedance – no SC Initial  = 5.65 Unstable from t=0

BB impedance – no SC  Linear scale <y> Log scale <y> Initial  = 4.6 Stable at t = 0 Unstable at  = 5.25  Relativistic gamma Linear scale <y> Vertical position of Centroid [m] Log scale <y> Vertical position of Centroid [m] Growth rate ~ 60 µs

Agenda Context of transition crossing in the PS HeadTail Simulations at constant energies HeadTail Simulations with acceleration Longitudinal aspects Crossing transition with a broadband impedance Perspectives

Perspectives Good agreement between analytical calculations and HeadTail simulations In HeadTail simulations Include possibility of t jump Include transverse space charge Take into account the non linear momentum factor 1 Include possibility of longitudinal space charge Simulate PS2 parameters and assess the maximum time allowed for transition crossing