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Analysis of the Electron Pinch during a Bunch Passage

Analysis of the Electron Pinch during a Bunch Passage. Elena Benedetto , Frank Zimmermann CERN. Contents. Analytical calculation of the electron density evolution during the passage of a proton bunch through the electron cloud ( linear force approximation )

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Analysis of the Electron Pinch during a Bunch Passage

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  1. Analysis of the Electron Pinch during a Bunch Passage Elena Benedetto, Frank Zimmermann CERN

  2. Contents • Analytical calculation of the electron density evolution during the passage of a proton bunch through the electron cloud (linear force approximation) • Expression for the tune shift experienced by the protons into the bunch. • Simulations: extension to non-uniform, e.g. Gaussian, transverse beam profiles, which give rise to non-linear forces on the electrons. • Estimation of the tune spread from the simulations results. E.Benedetto, F.Zimmermann

  3. Electron Pinch along the bunch • The electrons are accumulated around the beam center during the bunch passage (pinch) • The aim is trying to understand the mechanism of the slow emittance growth, that is probably caused by the tune shift and tune spread due to the electron pinch. • For this reason we compute the electron cloud density evolution during a bunch passage. E.Benedetto, F.Zimmermann

  4. For any longitudinal bunch profile Uniform Gaussian Highlights of the analytical calculations Equation of motion of an electron in the bunch potential Linear force approximation via Liouville theorem Initially Gaussian electron distribution Time evolution of the electron density Tune shift experienced by the protons E.Benedetto, F.Zimmermann

  5. Electron density evolution • Electron distribution in the phase space (the density is obtained by integrating in the velocities). • In the linear force approximation, the horizontal and vertical planes are uncoupled → factorization • Liouville Theorem + assumption of Initial Gaussian Distribution • (x0, x’0) are obtained as a function of (x, x’) by inverting the solution of the Eq. of motion E.Benedetto, F.Zimmermann

  6. Equation of motion of an electron in the bunch potential (1) • Bunch distribution: • We consider t=0 when the bunch enters into the cloud: • Equation of motion ( the Electric field is obtained via Gauss Theorem): E.Benedetto, F.Zimmermann

  7. Equation of motion of an electron in the bunch potential (2) • Horizontal component of the Eq.of motion + approximation oflinear force (r«sr): • Solution in the form: • It can be inverted and inserted into the expression of the phase space density. E.Benedetto, F.Zimmermann

  8. Tune shift • The tune shift is obtained from the electron density,which iscomputed by integrating the phase space distribution: • The tune shift experienced by theprotons (as a function of r and z) is: • Where Eeis the field produced by the electrons is (→from Gauss theorem): E.Benedetto, F.Zimmermann

  9. Longitudinal Uniform Profile (1) • Eq. of motion→ harmonic oscillator : • The solution can be easily inverted: • The electron density is: E.Benedetto, F.Zimmermann

  10. Longitudinal Uniform Profile (2) • The tune shift is: • The maximum tune shift is inversely dependent on the electron initial temperature: • For s’0« wes0 it goes periodically to very high values when Keep in mind that this is only valid for the transverse linear force approximation !!! E.Benedetto, F.Zimmermann

  11. General longitudinal distribution (1) • Eq. of motion: • We look for a solution in the form: • WKB approximation: • The general solution can also be written as: E.Benedetto, F.Zimmermann

  12. General longitudinal distribution (2) • c1 and c2 are determined by the initial condition, so we obtain again a solution of the form: • that can be inverted in order to get: • The electron density is: • And the tune shift: In particular, we find the expression of D(t) for a longitudinal Gaussian distribution E.Benedetto, F.Zimmermann

  13. Simulations: Linear and Gaussian force (2) (Parametres of LHC @ inj) Vertical position vs. time for 6 electrons at different start amplitude, from 0.5sb to 3sb : linear forceapproximation (left) and Gaussian transverse profile (right). Gaussian bunch shape in z. vertical position [m] vertical position [m] z/sz z/sz E.Benedetto, F.Zimmermann

  14. Simulations: Linear and Gaussian force (1) r/r0 r/r0 Electron density vs. time at the centre of the pipe, during the passage of a bunch, assuming a linear transverse force (Left) and a Gaussian transverse beam profile (Right). In green, the analytical results. A Gaussian bunch profile is assumed in z. z/sz z/sz (The head of the bunch is on the right) E.Benedetto, F.Zimmermann

  15. Simulations: Linear and Gaussian force (3) z=- 3sz z=- 1.5sz z= 0sz z= 3sz z=- 3sz z=- 1.5sz z= 0sz z= 3sz Snap shot of radial distribution (r ∙r) at 4 different times during the bunch passage:linear forceapproximation (left) and Gaussian transverse profile (right). ec-density[a.u.] ec-density[a.u.] r/sb r/sb E.Benedetto, F.Zimmermann

  16. Density enhancement during the bunch passage (non linear force) r/r0 Ec-density vs. Time, during the passage of a Gaussian bunch • Inside the bunch the density enhancement is about a factor 50. t1 t3 t0 t2 z/sz E.Benedetto, F.Zimmermann

  17. Horizontal phase space at different time step: t0 = when the bunch enters into the cloud (z=-3sz) t1 = first peak t2 = first valley t3 = last peak Density enhancement during the bunch passage (non linear force) t0 t1 t2 t3 E.Benedetto, F.Zimmermann

  18. Estimated incoherent tune shift from the simulations (non linear force) • The density enhancement at the centre of the bunch is about a factor ~50. • A simple evaluation of the tune shift gives the value: ~ 0.13 Where the unperturbed electron density is: E.Benedetto, F.Zimmermann

  19. Estimated incoherent tune shift from the simulations (non linear force) Courtesy Papaphilippou • The tune shift expected from an unperturbed cloud is about ~0.0025. • The spread of the tune footprints computed via frequency map analysis from HEADTAIL simulations is ~20 times larger • In our estimate we got ~50 times larger E.Benedetto, F.Zimmermann

  20. Summary • Analytical approach to investigate the cause of a slow emittance growth due to the tune shift and tune spread from the electron pinch. • Analytical expression for the ec-density evolutionalong the bunch and for the tune shift induced on the protons (linear force approximation) • Numerical extension to non-linear force effects. • The simulations (with the parameters of LHC @ inj.) show that the density enhancement inside the bunch is about a factor 50. • First estimation gives a tune spread of DQ ≈ 0.13 (for an initial ec-density of 6e11 m-3. E.Benedetto, F.Zimmermann

  21. Ongoing and Future plans • Continue analytical approach to model electron cloud phenomena • Comparison with HEADTAIL simulations. • Produce instability diagrams for the electron cloud • Investigations about • what happensafter the bunch had passed • Longitudinal discontinuities in the electron plasma. E.Benedetto, F.Zimmermann

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