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 , J y , θ y , s , δ ) : 4D-phase space distribution density. d ψ d t = 0 (Boltzmann) ∂ t + ∂ J y ∂ J y + ∂θ y ∂ψ ∂ψ ∂θ y + ∂ z ∂ψ ∂ψ ∂ z + ∂δ ∂ψ ∂δ = 0 ∂ t ∂ t ∂ t ∂ t Assume that a mode is developing with frequency Ω = Q c ω 0 . Perturbation of ψ : jQc t ψ = f 0 ( J y ) g 0 ( r ) + f 1 ( J y , θ y ) g 1 ( s , δ ) e 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 = Q s η R r sin φ η : slippage factor R : machine radius Q s : 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 order (no multiturn) : ( Q c − Q y ( s , δ ) − jQ s ∂ φ ) g 1 = − 1 � s d ˜ s , ˜ 4 π g 0 ( r ) d ˜ δ W ( s − ˜ s ) g 1 (˜ 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. N s slices, N r rings. Each cell is weighted by the free distribution g 0 ( 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 one-turn map M . 1 We assume all weights of the cells equal to N r N s . If Q y = Q y , 0 , then M = M 0 ( I + W ) 1 I N r ⊗ P N s Q s M 0 = ⊗ B 0 ( 2 π Q y , 0 ) N s N r N s B 0 ( θ ) is a 2x2 rotation matrix of angle θ . 0 1 0 · · · 0 0 0 0 1 · · · 0 0 . . . . . ... . . . . . P N s ≡ . . . . . 0 0 0 · · · 0 1 1 0 0 · · · 0 0
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 M 0 : − N s − 1 ≤ l ≤ N s − 1 , ǫ ∈ {− 1 , 1 } 2 2 1 M 0 ( Z r ⊗ X l ⊗ Y ǫ ) = e 2 π j ( lQ s + ǫ Q y , 0 ) ( Z r ⊗ X l ⊗ Y ǫ ) N r N s ( Z r ) i = δ i , r 1 ≤ i ≤ N r 1 e j 2 π l ( X l ) k = Ns k √ N s 1 ≤ k ≤ N s 1 � � 1 Y ǫ = √ ǫ j 2
Presentation of the models Free systems The wake function Chromaticity Landau damping and openings Vlasov vs CMM ( Q c − Q y , 0 − jQ s ∂ φ ) g 1 = 0 We discretize the longitudinal phase space � g 1 ( r , φ ) ≡ α i , j θ ∆ φ ( φ − φ j ) θ ∆ r i + 1 ( r − r i ) i , j Eigenvalue equation on α a , b : � Q c α a , b = M ( a , b ) , ( a ′ , b ′ ) α a ′ , b ′ a ′ , b ′
Presentation of the models Free systems The wake function Chromaticity Landau damping and openings Vlasov vs CMM 1 0 0 · · · 0 − 1 − 1 1 0 · · · 0 0 M ≡ I N r ⊗ ( Q y , 0 I N s + j Q s . . . . . ... . . . . . ∆ φ N ) N ≡ . . . . . 0 0 0 · · · 1 0 0 0 0 · · · − 1 1 Eigensystem : 2 j π l Ns − 1 Q c = Q y , 0 + e Q s 2 j π N s M ( Z r ⊗ X l ) = Q c ( Z r ⊗ X l ) ( Z r ) i = δ i , r 1 X l ( φ ) = e − jl φ √ N s
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 : � � 0 0 W ( a , b ) , ( a ′ , b ′ ) = W ( s ( a , b ) − s ( a ′ , b ′ ) ) 0 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" Q c are degenerate in the r dimension. Quantum mechanics : diagonalize the matrix ( � X ( 0 ) ( r , l ) | H 1 | X ( 0 ) ( r ′ , l ) � ) r , r ′ Degeneracy is lifted ! In Vlasov (after some calculation and simplifications), ∆ ( 1 ) Q c ( r , l ) is the eigenvalue of : 1 W ( s ( r , b ) − s ( r ′ , b ′ ) ) e − j 2 π l Ns ( b − b ′ ) � ∆ l r , r ′ = − 4 π N 2 s N r 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 × M 0 . If λ = e 2 j π Q c is the eigenvalue of M , ∆ ( 1 ) λ ǫ, r , l = 2 π j (∆ ( 1 ) Q c ) ǫ, r , l λ ǫ, r , l After some calculation, (∆ ( 1 ) Q c ) ǫ, r , l eigenvalue of : ǫ W ( z ( r , b ) − z ( r ′ , b ′ ) ) e − 2 j π l Ns ( b − b ′ ) � Λ r , r ′ = − 4 π N r N 2 s 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 � Z r ⊗ X l | W | Z r ′′ ⊗ X l ′ � � Z r ′′ ⊗ X l ′ | W | Z r ′ ⊗ X l � ∆ ( 2 ) , l � r , r ′ ≡ Q ( 0 ) c ( l ) − Q ( 0 ) c ( l ′ ) l ′ � = l , r ′′ In the CMM : we need to diagonalize � Z r ⊗ X l ⊗ Y 1 | WM 0 | Z r ′′ ⊗ X l ′ ⊗ Y 1 � � Z r ′′ ⊗ X l ′ ⊗ Y 1 | WM 0 | Z r ′ ⊗ X l ⊗ Y 1 � � e 2 π jQ ( 0 ) ( l , 1 ) − e 2 π jQ ( 0 ) ( l ′ , 1 ) l ′� = l , r ′′ c c � Z r ⊗ X l ⊗ Y 1 | WM 0 | Z r ′′ ⊗ X l ′ ⊗ Y − 1 � � Z r ′′ ⊗ X l ′ ⊗ Y − 1 | WM 0 | Z r ′ ⊗ X l ⊗ Y 1 � � + e 2 π jQ ( 0 ) ( l , 1 ) − e 2 π jQ ( 0 ) ( l ′ , − 1 ) l ′ , r ′′ c c � Z r ⊗ X l ⊗ Y 1 | WM 0 | Z r ′′ ⊗ X l ⊗ Y 1 � � Z r ′′ ⊗ X l ⊗ Y 1 | WM 0 | Z r ′ ⊗ X l ⊗ Y 1 � � − e 2 π jQ ( 0 ) ( l , 1 ) r ′′ c
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 N s Q s ≪ 1 Prevents us from going to N s → ∞ 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 Q y = Q y , 0 + Q ′ y δ 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 P N s itself ! Q ′ y e 2 π j ǫ Ns Qs δ 2 0 · · · 0 . . . ... . . . . . . P N s ⇒ Q ′ y e 2 π j ǫ Ns Qs δ Ns 0 0 · · · Q ′ y e 2 π j ǫ Ns Qs δ 1 0 · · · 0
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 : N s Q s ≪ 1 Q ′ y Q s ≪ 1 Numerically tested : When N s ≥ 1 / Q s , eigenvectors start to shift from the Vlasov ones. When Q ′ y Q s ∼ 1, Q c 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 Q y ( 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 N s Q s ≪ 1 (seems numerically unfounded) ? Explain the appearance of Landau damping in the Vlasov equation ? How to introduce Landau damping in the CMM ?
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