QUANTUM LOWER LIMIT ON SCATTERING ANGLE IN THE CALCULATION OF MULTIPLE TOUSCHEK-EFFECT

1. QUANTUM LOWER LIMIT ON SCATTERING ANGLE IN THE CALCULATION OF MULTIPLE TOUSCHEK-EFFECT Sergei Nikitin BINP Russia IBS Mini Workshop, Cockcroft Institute, Daresbury 28-29 August 2007

2. Introduction Two definitions for a minimal scattering angle IBS parameters revision Method of Touschek effect calculation Account of the quantum lower limit on scattering angle Numerical examples for CLIC and VEPP-4M Conclusions CONTENT Sergei Nikitin IBS Workshop 28 August 2007

3. A value of impact parameter in IBS is bounded above (for instance, due to limitation of beam sizes). As consequence of the uncertainty principle, there exists a quantum minimal angle of particle scattering. Rutherford cross section used for a classical description of IBS is also valid in a quantum approach because of the Coulomb nature of scattering potential (Rutherford was pleased!). A quantum consideration of IBS (without spin effects) concerns mainly the limits of scattering parameters. Intoduction Sergei Nikitin IBS Workshop 28 August 2007

4. Questions: In what conditions a quantum lower limit on scattering angle is important? If the quantum limit is formally large, can this fact lead to a significant increase of IBS diffusion (beam sizes, energy spread) in comparison with a classical consideration? Sergei Nikitin IBS Workshop 28 August 2007

5. Two definitions for a minimal scattering angle(known from Plasma Physics) Classical Coulomb interaction Consequence of the uncerainty principle Classical definition validity violation:mV2>50 eV (ep), 40 keV (pp) A maximal impact parameter(Debye radius, a beam size etc.) Sergei Nikitin IBS Workshop 28 August 2007

6. Motivation • In typical electron-positron machines (SR sources, colliders) , at a GeV-order energy, the particle transverse momentum is up to 10-4 and over (>> 50 eV) • Widespread approach in the Touschek effect calculations uses a classical definition of • The role of Touschek effect and, therefore, the required calculation accuracy increase significantly in storage rings with an ultra-small beam emittance of nanometer-order Sergei Nikitin IBS Workshop 28 August 2007

7. Compare the classical and quantum approaches regarding a minimal scattering angle in the Touschek effect calculation Give the numerical examples Aims Sergei Nikitin IBS Workshop 28 August 2007

8. IBS parameters revision • In IBS theory the maximal impact parameter scale is denoted by • Coulomb logarithm Sergei Nikitin IBS Workshop 28 August 2007

9. Classical definition • Quantum definition - the transverse momentum spread , Sergei Nikitin IBS Workshop 28 August 2007

10. Typically, , bmax - in cm The classical parameter in the Coulomb logarithm is substituded by the quantum analog . At that, a coefficient ½ appears before the logarithm. Non-realistic superdense/superthin beams are required in order that ~1. Sergei Nikitin IBS Workshop 28 August 2007

11. Method of Touschek effect calculation D.Yu. Golubenko, S.A. Nikitin, PAC 2001 Proc., p.2845, 2001. • Modified function of distribution over momentum (p) in CMS coupling parameter in velocity space at (flat beam) at (“round” beam) Sergei Nikitin IBS Workshop 28 August 2007

12. Distribution function plot Coupling grows Note: the cross section (averaged over the spin states) is the same in the classical and quantum considerations! Sergei Nikitin IBS Workshop 28 August 2007

13. Co-Kinetics of the quantum fluctuation (Q) and multiple Touschek (T) processes the relative energy dispersion the radial phase volume Touschek Diffusion coefficients the classical lower limit the quantum lower limit the system of equations to determine the equilibrium values of u and v Sergei Nikitin IBS Workshop 28 August 2007

14. Diffusion factor B in a classical approximation at various values of the coupling Sergei Nikitin IBS Workshop 28 August 2007

15. Numerical examples Touschek increase of the CLIC emittance and energy spread vs. the emittance ratio: E=2.42 GeV, I=0.34 mA, Ex=2.8e-11 m, sigmaZ=1.1 mm Calculation in the classical approximation Sergei Nikitin IBS Workshop 28 August 2007

16. The results of Touschek calculation for CLIC made in the quantum approximation are not presented in the slide so they are practically coincide with the results obtained in the classical approximation. The same situation takes a place also in the analogical calculation done for VEPP-4M. Reason:the diffusion factors B of classical and quantum approximations appear to be close each to other in a value (with an accuracy of several percents) in spite of a huge difference between and . Sergei Nikitin IBS Workshop 28 August 2007

17. Diffusion factor vs. the in two approximations The CLIC example Bold points: CLIC E=2.24 GeV I=0.34 mA/bunch Ey/Ex=0.003 Sergei Nikitin IBS Workshop 28 August 2007

18. The VEPP-4M example Bold points: VEPP-4M E=0.9 GeV I=0.1 mA/bunch Ey/Ex=0.01 In this area the difference may become apparent Sergei Nikitin IBS Workshop 28 August 2007

19. The transformed CLIC example: is 3∙104 decreased Sergei Nikitin IBS Workshop 28 August 2007

20. Formally, a quantum lower limit on scattering angle must be included in consideration of the IBS processes.But … CLIC and VEPP-4M Touschek calculation examples show that an account of the quantum limit of minimal scattering angle instead of the classical one does not change notably the numerical results. This conclusion seems to be true for all existing and designed storage rings since an apparent difference between results of classical and quantum approximation may be only in the non-realistic case of super-dense/super-thin beams. Conclusions Sergei Nikitin IBS Workshop 28 August 2007