Impedance and Component Heating

Abstract

The impedance is a complex function of frequency, which represents, for the plane under consideration (longitudinal, horizontal or vertical), the force integrated over the length of an element, from a "source" to a "test" wave, normalized by their charges. In general, the impedance in a given plane is a nonlinear function of the test and source transverse coordinates, but it is most of the time sufficient to consider only the first few linear terms. Impedances can influence the motion of trailing particles, in the longitudinal and in one or both transverse directions, leading to energy loss, beam instabilities, or producing undesirable secondary effects such as excessive heating of sensitive components at or near the chamber wall, called beam-induced RF heating. The LHC performance limitations linked to impedances encountered during the 2010-2012 run are reviewed and the currently expected situation during the HL-LHC era is discussed.

Analysis of transverse beam stabilization with radio frequency quadrupoles

“…Section IV contains the numerical proof for beam stabilization with an rf quadrupole. The study is based on a slow head-tail instability that is observed in the LHC and proven to be Landau damped by means of the magnetic octupoles both experimentally and in simulations [11]. Here, tracking studies show that an rf quadrupole manages to stabilize the beam in a similar manner.…”

“…The numerical validation of the stabilizing effect of an rf quadrupole is based on a horizontal single-bunch instability originally observed experimentally in the LHC at 3.5 TeV during machine commissioning in 2010 [11]. Dedicated studies in the LHC demonstrated that this particular instability is characterized as a slow head-tail mode with an azimuthal mode number m ¼ −1.…”

Design of an rf quadrupole for Landau damping

“…It consist of an optimizer that creates and distributes elliptical designs over a number of parallel running tuners used for the frequency adjustment. The optimizer uses a constrained BFGS algorithm 3 to minimize an arbitrary chosen quality function uðxÞ with the special feature that the entries of the Jacobian ∂u=∂x i (with x i as free design variables) are calculated in parallel. Using a five-point differentiation in each variable the parallelization reduces the evaluation by a factor of 10 on an Intel i7 processor of the third generation.…”

“…The incoherent frequency or tune spread is caused by nonlinearities in the machine. To ensure this mechanism dedicated nonlinear elements, typically octupole magnets, also known as Landau octupoles (LO) are installed in the accelerator [3,4]. However, adiabatic damping and increased beam rigidity reduce their efficiency at higher energies.…”

Nested head-tail Vlasov solver

“…According to Refs. [20,21], the incoherent tune shifts δν ≡ ðδν x ; δν y Þ T introduced by the LHC octupoles fed with current I o ¼ þ100 A, with both emittances ε ¼ 2 μm, are as follows:…”