Calculation of matched beams under space-charge conditions

Transcription

1 Calculation of matched beams under space-charge conditions Jürgen Struckmeier Vortrag im Rahmen des Winterseminars Aktuelle Probleme der Beschleuniger- und Plasmaphysik des Instituts für Angewandte Physik der Johann Wolfgang Goethe-Universität Frankfurt am Main Riezlern, März 007

2 Outline 1. Review of the K-V envelope equations, reasons for beam matching. Generalized Newton method for vector functions 3. Calculation of the Jacobi matrix of the envelope mapping through a periodic structure 4. Iteration prescription to obtain periodic solutions of the K-V envelope equations 5. Example: Heidelberg Test Storage Ring (TSR) 6. Conclusions

3 1. K-V equations In linear approximation, the transverse motion of a single particle along the reference trajectory under non-negligible space-charge conditions is described by the Hill equations K ei x i + kx () s xi = 0, K = X( X Y + ) 4πε mc,ext β γ K y i + ky,ext() s yi = 0 Y ( X Y + ) k x,y,ext (s) describe the s-dependent strengths of the linear external forces acting on the beam particles. K denotes the scaled beam current ( generalized perveance ). X(s),Y(s) denote the beam envelopes. These functions must be known in advance if K 0 collective effect.

4 1. K-V equations From the single particle equations, we can directly derive the equations for (see Lecture 6.). We thus obtain the K-V equations for the beam envelopes X(s) and Y(s). This closed set of equations describes the transformation of an unbunched beam under non-negligible space-charge conditions in linear approximation d K ε Xs () + k x () s X = 0 x,ext 3 ds X + Y X d K εy () +,ext() = 0 Ys k y sy 3 ds X + Y Y ε x,y denote the beam emittances, which are constants of motion under the given assumptions of both linear external and linear self fields. K 0 induces coupling between X(s) and Y(s) envelope functions.

5 1. K-V equations This is a non-linear system of two ordinary second order differential equations. It is furthermore non-autonomous, since the coefficients k x,y,ext (s) depend explicitly on the independent variable, s. No analytic solutions X(s), Y(s) exist in general. Condition for a matched beam: If k x (s), k y (s) are S-periodic functions, find initial values X(0), X (0), Y(0), Y (0) so that the envelopes X(s) and Y(s) are also S-periodic Xs ( + S) = Xs ( ), X ( s+ S) = X ( s), Ys ( + S) = Ys ( ), Y ( s+ S) = Y ( s) Matching ensures: The maximum beam extents X max, Y max are minimized. Resonances between the particle betatron oscillation and the beat wave of the envelope mismatch oscillation modes are avoided.

6 1. K-V equations Mismatched beam in a periodic quadrupole channel (mixed mode)

7 1. K-V equations Mismatched beam in a periodic quadrupole channel (even mode)

8 1. K-V equations Mismatched beam in a periodic quadrupole channel (odd mode)

9 1. K-V equations Matched beam in a periodic quadrupole channel

10 1. K-V equations Cell Emittance plots (top: x,x, bottom: y,y ) of a PIC simulation of a matched beam propagating through a quadrupole channel at σ 0 = 60, σ = 15

11 1. K-V equations Cell Emittance plots (top: x,x, bottom: y,y ) of a PIC simulation of a mismatched beam propagating through a quadrupole channel at σ 0 = 60, σ = 15

12 . Newton s method for the K-V equations Question: How can we determine the initial conditions X(0), X (0), Y(0), Y (0) for given S-periodic external force functions k x,y,ext (s), scaled beam current K, and given emittances ε x and ε y that yield S-periodic beam envelopes X(s), Y(s)? Answer: We will make use of Newton s iteration method, adapted for the case of a vector function. Problem: We cannot deal with analytic functions and its derivatives, as is usually the case when we apply Newton s method. Instead, we only have functions that are given as numerical solutions of the K-V equations.

13 . Newton s method for the K-V equations We define the abbreviation Xs () Ys () X() s = X () s Y () s Integrating the K-V equations through on focusing period from s = 0 to s = S means to perform the non-linear mapping X(0) X( S), X( S ) = MX(0) M Beam matching thus means to find the particular X m (0) = MX (0) m X m (0) with

14 . Newton s method for the K-V equations Primitive method to determine X m (0) : we perform the iteration ( ) ( S ) t+ 1 1 t t 1 t t X (0) = X (0) + MX (0) = X (0) + X ( ) This method works sometimes, but converges very slowly if at all. Better way: Newton s method. For a function f of one variable x, we determine a zero x 0 of f via x t t 1 t fx ( ) + = x t f ( x ) A zero at x 0 of the function f is a fixed point of a function g, defined as Def gx () = x fx (), fx ( ) = 0 gx ( ) = x 0 0 0

15 . Newton s method for the K-V equations A fixed point of the function g is thus obtained by the iteration x t t 1 = t gx ( ) x x t g ( x ) 1 t + In the case of functions of multiple variables, Newton s method is then 1 ( J E) ( ) X + = X MX X t 1 t t t t with E denoting the unit matrix, and J the Jacobi matrix of M at X( S). Question: How do we determine the Jacobi matrix of a mapping that is only given as the numerical solution of a differential equation? Answer: The Jacobi matrix J is the solution matrix of the system of linear perturbations that is induced by the mapping M.

16 3. Jacobi matrix of K-V solutions With the K-V equations given, we can set up the linear system of equations for the perturbations x(s), y(s) that are defined by Xs () = X() s + xs (), Ys () = Y() s + ys (), x(0) X(0), y(0) Y(0) Herein, X 0 (s) and Y 0 (s) denote reference envelopes, and X(s), Y(s) neighboring envelopes. We insert X(s) and Y(s) into the K-V equations and apply the approximations 1 3 x + y x + y x x 1+ 1, X Y X Y X + + X This yields a linear coupled set of equations for the perturbations, i.e. for the small deviations x(s) and y(s) of reference and neighboring envelopes.

17 3. Jacobi matrix of K-V solutions ε K 3 x K x () s kx () s x() s + = 4 ( + ) y() s 0 X Y X0 ( X + Y ) ε K 3 y K y () s ky () s y() s + = 4 ( + ) x() s 0 X Y Y0 ( X + Y ) Again, the coupling is induced by a non-vanishing K. We can rewrite these two second-order equations as a linear system of four firstorder equations: z1() s z1() s d z() s z() s As (), As () = = ds z3() s z3() s a1() s a0() s 0 0 z4() s z4() s a0() s a() s 0 0 As X 0 (s) and Y 0 (s) must be known, this system can only be integrated together with K-V envelope equations.

18 In vector form, we get with 3. Jacobi matrix of K-V solutions z () s = As () z() s z x, z y, z x, z y a () s = K [ X () s + Y () s ] ε a () s k () s a () s 1 x = x X0 () s 0 3ε a () s k () s a () s = y y Y0 () s 0 The general 4 4 solution matrix Z(s) has the property Z () s = As () Z() s

19 4. Iteration prescription The Jacobi matrix J of solutions X(s), Y(s) of the K-V equations is the solution matrix of the system for the perturbations x(s) and y(s) J = Z( S), Z ( s) = As ( ) Z( s), Z(0) = E Recall: The matrix Z(s) maps a perturbation (x(0), y(0), x (0), y (0)) into (x(s), y(s), x (s), y (s)). Its initial value is thus the unit matrix: Z(0) = E. The iteration prescription to find a fixed point of the K-V equations, i.e., to determine the matched beam parameters under space charge conditions for a periodic structure thus finally writes: t+ t t 1 t t X 1 (0) = X (0) Z ( S ) E X ( S ) X (0) ( ) ( )

20 4. Iteration prescription We quantify the residual deviation of the actual beam parameters from an accurate fixed point in terms of the dimensionless mismatch factor: [ α α ] [ β β ][ γ γ ] FMM = ( S) (0) ( S) (0) ( S) (0) with α, β, and γ denoting either the x,x or the y,y CLS ellipse parameters that are related to the beam envelopes and their derivatives. Example: Quadrupole channel at σ 0 = 90 with depressed tune σ = 15. Initial values: matched zero current beam parameters. Iteration cycle 1: Mismatch factor x,x = E+01, Mismatch factor y,y = E+01 Iteration cycle : Mismatch factor x,x = 9.364E-01, Mismatch factor y,y = 3.68E+00 Iteration cycle 3: Mismatch factor x,x = E-03, Mismatch factor y,y = E-03 Iteration cycle 4: Mismatch factor x,x = E-09, Mismatch factor y,y =.0786E-09 Iteration cycle 5: Mismatch factor x,x = E-0, Mismatch factor y,y = E-0

21 5.Example: Heidelberg Test Storage Ring Matched beam in the Heidelberg TSR

22 6. Conclusions 1. In order to maintain maximum beam quality, beam matching to periodic lattices (e.g. storage rings, RFQs) is necessary.. If the self fields of the beam cannot be neglected, we must resort to the K-V envelope equations to find the beam transformation (to first order, i.e. for the linear approximation of all forces). 3. Newton s method to find fixed points of a mapping can be applied even if the mapping cannot be expressed in analytic form. 4. For Newton s method, we need the (linear) differential map that is induced by the original generally nonlinear mapping. 5. For mappings that cannot be expressed in analytic form, the differential map is given by the solution matrix of the system of perturbations.