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Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

The Vlasov formalism and the Circulant Matrix Model

Antoine Maillard

CERN, Geneva

July 2016

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

The Vlasov formalism

We want to evaluate transverse beam stability. ψ(t, Jy, θy, s, δ) : 4D-phase space distribution density.

dψ dt = 0 (Boltzmann) ∂ψ ∂t + ∂Jy ∂t ∂ψ ∂Jy + ∂θy ∂t ∂ψ ∂θy + ∂z ∂t ∂ψ ∂z + ∂δ ∂t ∂ψ ∂δ = 0

Assume that a mode is developing with frequency Ω = Qcω0. Perturbation of ψ : ψ = f0(Jy)g0(r) + f1(Jy, θy)g1(s, δ)e

jQc t R

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

The Vlasov formalism

We will use angular variables in the longitudinal plane : s = r cos φ δ = Qs ηR r sin φ η : slippage factor R : machine radius Qs : synchrotron frequency

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

The Vlasov formalism

After some calculations, one can rewrite Vlasov equation at first

• rder (no multiturn) :

(Qc−Qy(s, δ) − jQs∂φ)g1 = − 1 4πg0(r)

• d˜

sd˜ δW (s − ˜ s)g1(˜ s, ˜ δ)

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

The Circulant Matrix Model

Based on a discretization of the longitudinal phase space. Ns slices, Nr rings. Each cell is weighted by the free distribution g0(r).

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

The Circulant Matrix Model

The evolution of the system after one turn is described by the

• ne-turn map M.

We assume all weights of the cells equal to

1 NrNs .

If Qy = Qy,0, then M = M0(I + W ) M0 = 1 NrNs INr ⊗ PNsQs

Ns

⊗ B0(2πQy,0) B0(θ) is a 2x2 rotation matrix of angle θ. PNs ≡        1 · · · 1 · · · . . . . . . . . . ... . . . . . . · · · 1 1 · · ·       

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Vlasov vs CMM

We first investigate a simple case, with no impedance and no chromaticity. In the CMM, a few calculations lead to the eigensystem of M0 : −Ns − 1 2 ≤l ≤ Ns − 1 2 , ǫ ∈ {−1, 1} M0(Z r ⊗ X l ⊗ Y ǫ) = 1 NrNs e2πj(lQs+ǫQy,0)(Z r ⊗ X l ⊗ Y ǫ) (Z r)i = δi,r 1 ≤ i ≤ Nr (X l)k = 1 √Ns ej 2πl

Ns k

1 ≤ k ≤ Ns Y ǫ = 1 √ 2

• 1

ǫj

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Vlasov vs CMM

(Qc−Qy,0 − jQs∂φ)g1 = 0 We discretize the longitudinal phase space g1(r, φ) ≡

• i,j

αi,jθ∆φ(φ − φj)θ∆ri+1(r − ri) Eigenvalue equation on αa,b : Qcαa,b =

• a′,b′

M(a,b),(a′,b′)αa′,b′

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Vlasov vs CMM

M ≡ INr ⊗ (Qy,0INs + j Qs ∆φN) N≡        1 · · · −1 −1 1 · · · . . . . . . . . . ... . . . . . . · · · 1 · · · −1 1        Eigensystem : Qc = Qy,0 + e

2jπl Ns − 1

2jπ Ns

Qs M(Z r ⊗ X l) = Qc(Z r ⊗ X l) (Z r)i = δi,r X l(φ) = 1 √Ns e−jlφ

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

The wake function

Now we introduce the wake function. Vlasov : add a convolution term to the equation. CMM : multiply by a matrix I + W : W(a,b),(a′,b′) =

• W (s(a,b) − s(a′,b′))
• No longer closed form solutions for the eigensystem !

Our previous "free" eigenvectors were orthonormal ⇒ we can use perturbation theory as in quantum mechanics !

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

First order degenerate perturbation theory

Our "energies" Qc are degenerate in the r dimension. Quantum mechanics : diagonalize the matrix (X (0)

(r,l)|H1|X (0) (r′,l))r,r′

Degeneracy is lifted ! In Vlasov (after some calculation and simplifications), ∆(1)Qc(r, l) is the eigenvalue of : ∆l

r,r′ = −

1 4πN2

s Nr

• b,b′

W (s(r,b) − s(r′,b′))e−j 2πl

Ns (b−b′)

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

First order degenerate perturbation theory

In the CMM, the perturbation matrix is W × M0. If λ = e2jπQc is the eigenvalue of M, ∆(1)λǫ,r,l λǫ,r,l = 2πj(∆(1)Qc)ǫ,r,l After some calculation, (∆(1)Qc)ǫ,r,l eigenvalue of : Λr,r′ = − ǫ 4πNrN2

s

• b,b′

W (z(r,b) − z(r′,b′))e− 2jπl

Ns (b−b′)

ǫ = 1 ⇒ OK !

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Second order perturbation theory

In the Vlasov formalism : we need to diagonalize ∆(2),l

r,r′ ≡

• l′=l,r′′

Z r ⊗ X l|W |Z r′′ ⊗ X l′ Z r′′ ⊗ X l′|W |Z r′ ⊗ X l Q(0)

c (l) − Q(0) c (l′)

In the CMM : we need to diagonalize

• l′=l,r′′

Zr ⊗ X l ⊗ Y 1|WM0|Zr′′ ⊗ X l′ ⊗ Y 1 Zr′′ ⊗ X l′ ⊗ Y 1|WM0|Zr′ ⊗ X l ⊗ Y 1 e2πjQ(0)

c (l,1) − e2πjQ(0) c (l′,1)

+

• l′,r′′

Zr ⊗ X l ⊗ Y 1|WM0|Zr′′ ⊗ X l′ ⊗ Y −1 Zr′′ ⊗ X l′ ⊗ Y −1|WM0|Zr′ ⊗ X l ⊗ Y 1 e2πjQ(0)

c (l,1) − e2πjQ(0) c (l′,−1)

−

• r′′

Zr ⊗ X l ⊗ Y 1|WM0|Zr′′ ⊗ X l ⊗ Y 1 Zr′′ ⊗ X l ⊗ Y 1|WM0|Zr′ ⊗ X l ⊗ Y 1 e2πjQ(0)

c (l,1)

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Second order perturbation theory

One can prove that we recover the equivalence in the limit NsQs ≪ 1 Prevents us from going to Ns → ∞ But computationally very acceptable limit. However, numerical simulations of the TMCI show no sign of such a limit...

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Introduce first order chromaticity

Qy = Qy,0 + Q′

yδ

In Vlasov, shift of the free (no wake) eigenvectors. Free eigenvalues are unchanged ! In the CMM, the only way to obtain the same result (shifted eigenvectors, non-shifted eigenvalues) is to introduce chromaticity effects in the circulant matrix PNs itself ! PNs ⇒         e2πjǫ

Q′ y Ns Qs δ2

· · · . . . . . . ... . . . · · · e2πjǫ

Q′ y Ns Qs δNs

e2πjǫ

Q′ y Ns Qs δ1

· · ·        

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Introduce first order chromaticity

In particular, previous methods to introduce chroma (by the multiplication of the full one-turn matrix) in the CMM are not valid ! However, one can show that the previous method is valid in the limit : NsQs ≪ 1 Q′

yQs ≪ 1

Numerically tested :

When Ns ≥ 1/Qs, eigenvectors start to shift from the Vlasov

• nes.

When Q′

yQs ∼ 1, Qc starts to shift due to chromaticity.

Conclusion : be careful when introduce transverse detuning effects in the CMM !

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Introduce first order chromaticity

Free (no wake) eigenvectors stay orthonormal in Vlasov and in the CMM ! ⇒ One can apply the same degenerate perturbation theory techniques, at first and second order in the wake. Tedious calculation ⇒ Equivalence of the CMM and Vlasov ! Generalizes as well to a general transverse detuning Qy(s, δ)

Presentation of the models Free systems The wake function Chromaticity Landau damping and openings

Openings

Another thing done : Equivalence shown as well for an amplitude-dependent synchrotron tune. Still to do : Go to any order in perturbation theory in the wake ? Get rid of the condition NsQs ≪ 1 (seems numerically unfounded) ? Explain the appearance of Landau damping in the Vlasov equation ? How to introduce Landau damping in the CMM ?