Beam-Beam Stability in Electron-Positron Storage Rings

Transcription

1 Beam-Beam Stability in Electron-Positron Storage Rings Bjoern S. Schmekel, Joseh T. Rogers Cornell University, Deartment of Physics, Ithaca, New York 4853, USA Abstract At the interaction oint of a storage ring each beam is subject to erturbations due to the electromagnetic field of the counter-rotating beam. These erturbations lead to a limit of the achievable luminosity in the storage ring. We investigate this limit in the framework of the strong-strong icture. Motion is considered only in the vertical direction and the beams are resumed to be one-dimensional. Based on this model [] we try to find stability criteria for the beam in an electron-ositron storage ring taking into account the daming of the betatron oscillations by synchrotron radiation but neglecting the discrete nature of the radiation rocess and resuming a Gaussian distribution. We analyze the instabilities by solving a linearized Fokker-Planck equation without the quantum excitation term. BEAM EVOLUTION Aside of diole induced bending the beam is subject to focusing, daming by synchrotron radiation and beam-beam kicks at the interaction oint. The beam-beam kick from the first (second) beam on the second (first) one is given by 4ßNr e L x fl dy sgn(y y) y ; = dy ψ ; (y; y ) () N is the number of articles in a bunch and r e the classical radius of the electron. The distributions of the beams ψ and ψ are normalized to unity, i.e. dy dy ψ ; (y; y )= () and are assumed to be one-dimensional with horizontal width L x. Motion is considered only in the vertical direction. Using the small-angle aroximation we obtain for synchrotron radiation and for focusing, resectively y rad = dy fl dt dx dt dx dy K = K(s)y (3) ds dt m = y fl y (4) is the momentum of the article and fl the total momentum change after each turn. We neglect the discrete nature of the radiation rocess. The dioles have no influence on the betatron oscillations. In a damed system the hase sace density increases. The equations describing the motion of ψ ; are given + ; + ; = L ψ ; (5) They differ from the Fokker-Planck equations by a missing quantum excitation term and from the Vlasov equations by the fact that the hase sace density is not a constant anymore. This gives us L ψ ; ; + L y + 4ßNr ; L x ffi (s) dy sgn(y y) dy ψ ; (y; y ;s) (6) and a similar equation with the first and second beam being interchanged ffi (s) denotes a eriod delta function that has singularities at all interaction oints. Because the system is damed an equilibrium does not exist. We define the distribution ψ (s) such that it satisfies L y L y + 4ßNr L x ffi (s) dy sgn(y y) dy ψ (y; y ;s) (7) with ψ being a function of s if daming is resent. We can think of ψ (s) being a temorary equilibrium distribution if the characteristic daming time is much longer than the eriod of betatron oscillations or erturbations. We are not interested in finding the actual distributions. All we want to know is whether the beam gets unstable or not. Thus, we choose a erturbative ansatz ψ ; = ψ ± ψ ; (8) Substituting eqn. 8 into eqn. 6, subtracting eqn. 7 and neglecting the term made u of two erturbations we find

2 @ ψ ; + ψ @y L y + F (y; s) 4ßNr e L x fl dy sgn(y y) dy ψ ; (y; y ;s)= L ψ ; (9) F (y; s) =K(s)y + 4ßNr e L x fl ffi (s) dy sgn(y y) dy ψ (y; y ) () With the aroximation F (y; s) ß F (s)y () we can treat the erturbation as a art of the erturbed focusing function F (s). The drawback is that this ste makes the study of henomena like Landau daming imossible. In the next ste we transform eqn. 9 to action-angle coordinates y = s J fi y = fij cos ffi () sin ffi fi cos ffi (3) Only the hase of the two beams at the interaction oint is of imortance for our roblem, but not the articular lattice design. Therefore, we can set fi =. This gives us an exlicit reresentation of J J = y fi + (4) fi fi denotes the erturbed instead of the unerturbed betatron function now. Forming the linear combinations y f ± = ψ ± ψ (5) and using the enveloe equation eqn. 9 can be rewritten in action-angle coordinates as L f ± ± J fil sin + L cos ffi sin ± fij sin ffi 4ßNr e L x fl dy sgn(y y) dy f ± (y; y ;s) (6) assuming that ψ = ψ (J). In the following discussion we omit the label ±. SOLVING THE EQUATIONS OF MOTION The distribution for electron bunches is aroximately Gaussian. ψ (J) = ßffß e J ff (7) Note that the Jacobi determinant of the transformation 3 is. Although our ψ as it is used in eqn. 7 deends on s we choose an s-indeendent ψ. This can be done if the daming after each turn is small and if we kee adjusting ff or the beam-beam arameter ο, resectively, which will be introduced later. In the final result we simly have to consider ο in an aroriate range. Considering the roblem on a turn-by-turn basis also justifies the usage of 3 J is a constant of motion. f must be eriodic in ffi. Thus, we choose the ansatz f (J;ffi;s)=Je J ff X l= g l (s)e ilffi (8) Substituting eqn. 7 and eqn. 8 into eqn. 6 and making use of the symmetry of the integrand we obtain X + il fi g l g k (i) ß Nr e fi L x flß 3= ff ffi (s) 5= dffi ß dffi μ dj μ sin(ffi) sin(lffi)cos(k μ ffi) μ Je μ J ff sgn( ff cos(ffi) μ J cos( μ ffi)) = L g l (9) having averaged out the sin ffi cos ffi-term which is due to synchrotron radiation (slow rocess). We evaluate the equation at J = ff we exect the erturbations to be largest since Jψ (J) has a maximum at J = ff. However, doing so removes all radial modes. The d μ J - integration can be evaluated by integrating by arts. The Heaviside ste function is denoted by H. μje J μ ff sgn( ff cos ffi J μ cos ffi)= μ ff sgn(cos ffi) ff sgn(cos ffi)h μ cos ffi cos ffi μ e cos 4 ffi cos 4 ffi μ () Thus, the equation determining the g l s becomes + il fi g l 4Nr e fi X ß ßL x fl ff ffi (s) k= g k M lk = L g l ()

3 ß ß M lk i dffi dffi μ sgn(cos ffi)h μ cos ffi cos ffi μ sin ffi sin lffi cos kffie μ cos 4 ffi cos 4 ffi μ () The matrix M has the following roerties M lk = 3ß= 3ß= i dffidffi μ sin ffi sin lffi cos kffie μ cos 4 ffi cos 4 ffi μ ß= ß= for l + k = even and M lk =for l + k = odd M l;k = M lk M l;k = M lk (3) No attemt is made to solve the remaining double integral analytically. Introducing a beam-beam strength arameter r ο 4Nr e fi Λ ßL x fl ff (4) fi Λ denotes the beta function at the interaction oint one obtains the following relation for the g l s immediately before and immediately after the interaction oint by integrating through the interaction oint. g l ( + ) g l ( )=± ο ß X k= M lk g k ( ) (5) There is no couling among different Fourier comonents between collisions. In this case eqn. simlifies + which is solved by il g l (L )=g l ( + )e il R L fi(s) g l = L g l (6) ds fi(s) = g l ( + )e ilffi (7) 3 DYNAMIC TUNE We calculate the tune ν in terms of the unerturbed tune ν which allows us to exress our final result in terms of ν. This is more convenient since we usually neglect beambeam effects when doing the lattice design. The tune shift due to synchrotron radiation is many orders of magnitude lower and is neglected. Eqn. 8 exresses the erturbed betatron tune ν in terms of the unerturbed tune ν and the difference in the focusing structure. I cos ßν =cosßν sin ßν fi(s)(f (s) K(s))ds (8) Evaluating the integral in eqn gives dμy sgn(y y) dy e J μ ff = ßffß y fi ßffß dye y4 y 4 6fi ff K =4 y 6fi ff = 3=4 ßfiff( 3 4 ) y + O(y ) (9) we have used the linearization eqn. and exanded the integrand keeing only the constant term. K ν (x) denotes the modified Bessel function. Thus, cos ßν =cosßν 3=4 ß 3= ( 3 4 ) ο sin ßν (3) 4 COHERENT BEAM-BEAM INSTABILITY We can summarize the solution of the equations of motion by introducing a matrix that acts on a coloumn vector G which contains all g l s. Therefore, G(L) =TG() (3) T = R ± ο M ß (3) and R is a diagonal matrix which has entries e ßilν (with ffi =ßν) for all l on its diagonal. Beam-beam instability occurs if one of the eigenvalues of T has eigenvalues whose absolute value is larger than. 5 RESULTS In the following lots we have calculated the matrix to the indicated order for both signs and drawn a oint at (ν ;ο) if all eigenvalues of T have an absolute value smaller or equal. 6 DISCUSSION The coarse structure of all lots is shown in fig. 4 we lotted the region in which the dynamic tune becomes comlex. In this region the accelerator cannot maintain a stable beam. All lots have this basic structure in common. Including resonances u to higher and higher orders the lots

4 Figure : Stability diagrams for = :. Resonances u to second, fourth and sixth order, resectively, have been included. The horizontal axis refers to and Figure : Stability diagrams for = :5. Resonances u to second, fourth and sixth order, resectively, have been included. The horizontal axis refers to and

5 Figure 4: Area in which the dynamic tune is real. The horizontal axis refers to and. get more and more comlicated. Small joints enter the diagrams which disaear again when synchrotron radiation daming is increased. As a rule of thumb the beam is stable if the oerating oint lies in the shaded area of fig. 4 and satisfies <. However, this rule is a bit too restrictive ACKNOWLEDGMENTS We would like to exress our thanks to Alex Chao from SLAC for his helful criticism. This work was suorted by the National Science Foundation. 8. REFERENCES [] A. W. Chao, R. D. Ruth, Coherent Beam-Beam Instability in Colliding-Beam Storage Rings, Particle Accelerators, 985, Vol. 6,. -6, Gordon and Breach Figure 3: Stability diagrams for = :. Resonances u to second, fourth and sixth order, resectively, have been included. The horizontal axis refers to and