1. 1 Impedance and its link to vacuum chamber geometry T.F. Günzel Vacuum systems for synchrotron light sources 12 th september 2005

2. 2 Outline –Motivation –Basic properties of wakefields –Resistive wall impedance –Impedance budget –Comparison to measured single bunch thresholds –Incoherent tune shifts –Effective impedance under different aspects –Head-Tail instability –Conclusion

3. 3 Motivation ESRF particularly concerned by impedance-related instabilities, the single bunch transverse thresholds are low <0.7mA vertically, <1.7mA horizontally ESRF is an excellent case to study the link between vacuum chamber geometry and transverse impedance Verification of the impedance model by comparison of the calculated TMCI-thresholds to the measured ones Establishment of the impedance budget, identification of the elements with the largest detrimental effect

4. 4 Basic properties of wakefields long. wakefield excited by a particle at r 0 and observed by a test (witness) particle at r r0r0 origin test particle r0r0 transverse wakefield (2-dimensional vector field) :

5. 5 wakefield in a circular geometry with tapers axial-symmetric taper geometry in the (z,r)-planesource particle at offset r 0 to the origin r0r0 origin source particle all arrows point in the direction of the offset no dependence of the test particle position r0r0 if zero offset, the wake is also zero (everywhere!) this dipolar field contributes to the impedance

6. 6 Without offset W  only depends on the test particle position, it is locally quadrupolar However, this field does not contribute to the impedance wakefield in a rectangular chamber with vertical tapers and offset r 0 =0 magnification

7. 7 wakefield in a rectangular chamber with a vertical tapers and horizontal offset r 0  0 the wakefield is a sum of a quadrupolar field (depending on the test particle position) and a dipole field (depending on the offset of the exciting particle) source particle with a 1mm offset in x-direction r0r0 a vertical taper produces horizontal impedance ! magnification

8. 8 The role of the vacuum chamber cross section 2D calculation possible wakefield only depends on source particle position resistive wall impedance given by 3D calculation necessary wakefield depends on source as well as on test particle position resistive wall impedance (chamber is approximated by 2 parallel plates) a a b all quantities only depend on one geometric parameter: the vertical extension of the chamber a

9. 9 Resistive wall budget form-factors as well  -functions of vertical and horizontal plane taken into account standard ESRF vacuum chamber (33mm x 79mm)makes up two third of the vertical budget the horizontal RW-impedance budget even larger than the vertical one  VV horizontal  values are high in the straight sections, vertical  values are high in the dipoles.

10. 10 Geometrical Impedance budget 33 (39) different elements in the budget, 1775 elements in total (status of november 2004) 26 taper pairs (without in-vacuum) 8 invacuum undulators (all open) 293 RF-fingers 569 flanges 134 vertical pumps, 448 horizontal pumps 6 cavities and 3 cavity tapers 2 scrapers in operation position 277 BPM’s 7 kicker like chambers 1 septum Calculation of impedance with GdfidL (W.Bruns)

11. 11 Broadband Impedance budget (vertical and horizontal plane)

12. 12 Effective dipolar impedance budget (geometric and resistive wall) Vertical impedance distributed smoothly around the ring Horizontal impedance concentrated in the low-gap sections

13. 13 Coherent tune shift : dipolar impedance Thresholds and tune shifts in single bunch the thresholds only depend on the dipolar impedance vertical measured 0.65mA calculated 1.05mA horizontal measured 1.7mA (nov. 2004) calculated 1.10mA probably the horizontal (resistive wall) impedance is overestimated for tune shift : dipolar impedance + quadrupolar wakefield Incoherent tune shift deteriorates the operation in single bunch Incoherent tune shift : quadrupolar wakefield

14. 14 Coherent and incoherent tune shifts in single bunch Synonym with calculation of kick factors (effective impedance) of dipolar and quadrupolar wakefields This is what can be measured by the bump method Vertical incoherent and coherent kick factors add up positively Horizontal incoherent and coherent kick factors cancel out each other

15. 15 Incoherent tune shift in multi-bunch Explained by R. Nagoaka, PAC 2001 (Chigaco) 3531 horizontal vertical Measured in november 2004

16. 16

17. 17 Effective impedance map (dipolar + quadrupolar) (σ=40ps) Calculated values about 2/3 of the measured values apart from invacuum undulators + 10mm SS-chambers (larger deviations) Measurements from Th. Perron PhD thesis

18. 18 Head-Tail Instability (only general remarks) impedance not only has bad effects Head-Tail instability is driven by transverse impedance. P. Kernel (PhD thesis) showed that at the ESRF the vertical head-tail instability is damped by the incoherent synchrotron tune spread caused by longitudinal impedance. On the horizontal plane this finding has still to be checked The limit of stability is given by the Post-Head-Tail instability (P. Kernel, R. Nagaoka, JL Revol, EPAC 2000, Vienne)

19. 19 Conclusions The vertical impedance is determined by many different elements The horizontal impedance is mainly determined by the low gap chambers. The model explains 2/3 of the vertical mode detuning, but in the horizontal plane it predicts a too small threshold compared to the measured one Flat vacuum chambers give rise to an incoherent tune shift tune shift affects the beam in multibunch as well as in single bunch The Horizontal as well as vertical impedance are essentially created by the vertical walls : its geometrical variation and its resistivity

20. 20 Conclusions High impedance budget of the ESRF is mainly due to modularity of the vacuum system alternating vertical and horizontal β-function distribution stainless steel vacuum chambers (is evolving towards more aluminium chambers) flatness of the vacuum chambers

21. 21 Acknowledgements –discussions and explanations of R. Nagaoka –good user support from W. Bruns (GdfidL) –all participants to this work, in particular P Elleaume, L. Farvacque, T.Perron, JL Revol