Landau damping in the transverse plane N ic ol as M ou n et, C E RN / B E -AB P-H SC A c k n o w l e d g e m e n t s : S e rg e y A r s e n y e v , X a v i e r B u ff a t , G i o v a n n i I a d a ro l a , Ke v i n L i , E l i a s M t r a l , Ad r i a n O e ft i g e r , G i o v a n n i Ru m o l o Landau damping in transverse Alex W. Chao: [] there are a large number of collective instability mechanisms acting on a high intensity beam in an accelerator []. Yet the beam as a whole seems basically stable, as evidenced by the existence of a wide variety of

working accelerators[]. One of the reasons for this fortunate outcome is Landau damping, which provides a natural stabilizing mechanism against collective instabilities if particles in the beam have a small spread in their natural [] frequencies. N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 2 Landau damping from frequency spread: a first sketch Following closely A. W. Chao (Physics of Collective Beams Instabilities in High Energy Accelerators, John Wiley and Sons 1993, chap. 5): Lets consider a beam particle following Hillslongitudinal equation coordinate along the accelerator

(smooth approximation) with an additional external force: beam velocity betatron frequency Lets now imagine this force is proportional to the beam average position (e.g. due to impedance), itself assumed to be a complex exponential (i.e. a damped or growing oscillation), of frequency : Solution: N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 3 Landau damping from frequency spread: a first sketch Now, lets imagine we have collection of particles with various betatron frequencies , forming a continuous

distribution . The beam average position is then simply given by the continuous superposition But is also the source of the external force and given by , so to get any non-trivial solution, itself must self-consistently obey the dispersion relation If is real, this integral looks divergent... N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 4 A detour through plasma physics When a similar integral was found first by Vlasov in the context of plasma waves [A. A. Vlasov, Russ. Phys. J. 9, 1 (1945) 25], Vlasov took its principal value to solve the problem. Then Landau found a mathematically robust way to compute

the integral. L. D. Landau, J. Phys. USSR 10, 25 (1946) N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 5 Dispersion integral: Landaus approach The dispersion integral for any complex can be obtained by analytical continuation: the idea is to replace the integration along the real axis in the complex plane, by an integration along a Landau contour which avoids the singularity: Courtesy A. W. Chao, Physics of Collective Beams Instabilities in High Energy Accelerators, John Wiley and Sons (1993), chap. 5

Another, equivalent way (as far as the value for real is concerned) to compute this, is to consider with a small vanishing imaginary part. N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 6 From the dispersion relation to Landau damping Considering a coherent frequency with a vanishing imaginary part such that beam is at the onset of instability): keeping in mind that , i.e. the bare coherent frequency shift in the absence of betatron frequency spread. What does this mean? Even with real, there are both a real and imaginary part between the square brackets.

This means the equation can hold even when is complex and the final coherent frequency is real! An instability that would be present when no tunespread is there, can turn out into a stable coherent motion. This is Landau damping. N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 7 Is this theory general enough? First, we have assumed the betatron frequency does not depend on the amplitude of the position itself (otherwise Hills equation itself gets modified ). This would work only for e.g. the indirect term of the octupolar detuning, but not the direct one: This is still solvable within ~ same formalism [H. G. Hereward CERN

69-11 (1969)] ( action) Nevertheless, there are still open questions, in particular, does it make sense that the coherent frequency without spread is computed completely separately and without taking into account N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 frequency spread? Stability diagrams

modes with a tuneshift inside the diagram are stable (here an LHC example) 8 Distribution of particles in phase space In a classical (i.e. not quantum-mechanical) picture, each beam particles has a certain position and momentum for each of the three coordinates (x, y, z). For a 2D distribution, in e.g. vertical, such a distribution of particles can be easily pictured in phase space (y,py ): py y Uniform density

Gaussian fall-off the distribution function represents the density of particles in phase space Total number of particles N. MOUNET VLASOV SOLVERS I CAS 21/11/2018 9 Liouville theorem Vlasov equation is based on Liouville theorem (or equivalently, on the collisionless Boltzmann transport equation), which expresses that the local phase space density does not change when one follows the flow (i.e. the trajectory) of particles.

=0 In other words: local phase space area is conserved in time: Red particles at time t become the orange ones at time t + dt, and the black square becomes the grey parallelogram which contains the same number of particles. N. MOUNET VLASOV SOLVERS I CAS 21/11/2018 10 Vlasov equation 9, 25 (1945)] [A. A. Vlasov, J. Phys. USSR Vlasov equation was first written in the context of plasma physics. The idea is to integrate the collective, selfinteraction EM fields into the Hamiltonian, instead of

writing them as a collision term. Assumptions: conservative & deterministic system (governed by Hamiltonian) no damping or diffusion from external sources (no synchrotron radiation), particles are interacting only through the collective EM fields (no short-range collision), there is no creation nor annihilation of particles. The inclusion of amplitude detuning into a Vlasov theory of coherent modes was performed first by Y. Chin, CERN/SPS/ 85-09 (1985). We consider here detuning only from the same plane as N. MOUNET VLASOV I CAS 21/11/2018 11 theSOLVERS instability. (easy to extend to detuning from the other transverse

Vlasov equation with Hamiltonians and Poisson brackets being the Hamiltonian of the system:Poisson brackets ( positions, momenta): Knowing the stationary distribution of the system without any collective effects (governed by the Hamiltonian ): Then, we are looking for first order perturbations of both the Hamiltonian and the distribution: After expansion, to first order Vlasov equation becomes linearized Vlasov equation Many thanks to Kevin Li N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 12

Hamiltonian Lets consider the coordinates in a lattice without coupling, and use the smooth approximation. With the following actionangle variables transverse: longitudinal: , , the unperturbed Hamiltonian reads and its perturbation, in the case of a dependent force: : slippage factor, : machine radius, : unperturbed transverse tune,

dipolar, : angular revolution frequency, : synchrotron frequency, : beam velocity, : relativistic mass factor : particle rest mass : chromaticity, : detuning coefficient N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 13 Stationary distribution We assume that the unperturbed Hamiltonian

admits as stationary distribution keeping the same definition of the actions as for a linear machine. In other words, the chromaticity and detuning are considered to have a negligible effect on the stationary distribution. Note: normalization of these distributions are such that N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 14 Linearized Vlasov equation Reminder: , , We get: Note: we do the typical assumption that the transverse plane does not affect the longitudinal one, hence we neglected and

as in Chaos book. N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 15 Writing the perturbation We assume a single mode of angular frequency close to , and we introduce for convenience Then we decompose this mode using a Fourier series of the angle and another one for the angle : Additional phase factor (that we put here without loss of generality) headtail phase factor N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019

16 Getting the perturbed distribution Injecting the perturbation into Vlasov equation, we can simplify it even more: This is where we use the headtail phase factor to get rid of the term within the brackets. Term by term identification leads to and the assumption , gives (see Chaos book) N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 17 Getting the perturbed distribution

Pushing further the computation gives: Depends only on and the ratio must be a constant. Depends only on and N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 18 Getting the perturbed distribution This gives the transverse shape of the perturbative distribution: Putting the proportionality constant inside : N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 19

Force from impedance The collective force due to a dipolar impedance is obtained as a 4D integral over the perturbed distribution, convoluted with the multiturn wake: Disp. integral N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 2

[( 0 ~ + 0 ) 20 ] Sacherer equation equation with detuning Plugging the force back into the linearized Vlasov equation,

identifying term by term the Fourier series in , and doing the standard approximation in the impedance and Bessel functions, we get an integral equation in : Equation obtained first by Y. Chin, CERN/SPS/85-09 (1985). N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 21 Strategy to solve the equation Decompose and over a basis of orthogonal polynomials such as Laguerre polynomials and compute the integrals involving Bessel functions analytically, as in codes MOSES and DELPHI: , and constants to be adjusted

One gets an equation of the form where each coefficient of the matrix can be computed analytically. Non trivial solution are found if and only if Dispersio n integral N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 Y. Chin, CERN/SPS/85-09 (1985). 22 Limiting cases of the determinant equation If there is no frequency spread, the dispersion integral becomes such that the equation becomes the usual eigenvalue problem:

and the coherent frequency shifts are obtained from a diagonalization. N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 23 Limiting cases of the determinant equation If there is no coupling, the determinant equation becomes Only diagonal such that we recover the stability diagram theory: terms of the matrixshifts frequency the are proportional to the coherent of the pure diagonal modes (weak headtail modes, but

also radial modes without coupling), we get a set of equations of the form (for each and ): which gives one possible coherent Dispersion integral frequency for each we can consider separately the coherent frequency shift and the dispersion integral, as in the stability diagram theory. N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 24 The determinant equation

To solve the general problem one has to find the that are the roots of a very non-linear expression: There is a priori no general strategy to find all the roots. this is a very difficult equation to solve. Also, a crucial aspect is the smoothness of the dispersion integral as a function of the vanishing imaginary part strategy typically fails here, as the integral will be smooth on only one side of the complex plane its better to use Landau contours, which involves computing some residues and having some fun with branch cuts N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 25 Is there a stability diagram still in the general case?

We can try to map the complex plane of unperturbed tuneshifts, thanks a broad scan of the phases and gains of a damper (inspired by the experimental study by S. Antipov et al, CERNACC-NOTE-2019-0034) without impedance Making a fine mesh of phases and gains, we can cover a large area in the complex plane: Case Q=0 N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 26 Can we recover the stability diagram theory? Strategy: for each gain/phase of the damper, we compute the determinant along the real tune shifts when it touches the

stability diagram, the minimum of this 1D curve should go to zero Case Q=0 27 Generalized stability diagrams The color represents the minimum of the previous 1D curves: Usual stability diagram Case Q=0 N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 28

Effect of chromaticity The color represents again the minimum of the 1D curves: Usual stability diagrams around 0, -Qs and -2Qs Case Q=5 N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 29 Effect of chromaticity The color represents again the minimum of the 1D curves: Usual stability diagrams around 0, -Qs

and -2Qs Case Q=15 Generalized stability diagrams, different from the usual ones and chromaticity dependent. N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 30 Transverse Landau damping - Summary Landau damping is one of the main mitigation of all kinds of instabilities in the transverse plane. We have sketched the standard approach to Landau damping, leading to the stability diagram theory. We have reviewed a generalization of the approach using

the Vlasov formalism, from which we can find the stability diagram theory as a limiting case. but the resulting non-linear determinant equation is extremely difficult to solve. Generalized, chromaticity-dependent, stability diagrams could be obtained using the general formalism were presented (still preliminary). N. MOUNET TRANSVERSE LANDAU DAMPING MCBI WORKSHOP 24/09/2019 31