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1. Accelerator Physics:Synchrotron radiation Lecture 2 Henrik Kjeldsen – ISA

2. Synchrotron Radiation (SR) • Acceleration of charged particles • Emission of EM radiation • In accelerators: Synchrotron radiation • Our goals • Effect on particle/accelerator • Characterization and use • Litterature • Chap. 2 + 8 + notes

3. General Electric synchrotron accelerator built in 1946, the origin of the discovery of synchrotron radiation. The arrow indicates the evidence of arcing.

4. Emission of Synchrotron Radiation • Goal • Details (e.g.): Jackson – Classical Electrodynamics • Here: Key physical elements • Acceleration of charged particles: EM radiation • Lamor: Non-relativistic, total power • Angular distribution (Hertz dipole)

5. Relativistic particles • Lorenz-invariant form • Result

6. Linear acceleration • Using dp/dt = dE/dx: • Energy gain: dE/dx ≈ 15 MeV/m • Ratio between energy lost and gain: • h = 5 * 10-14 (for v ≈ c) • Negligible

7. Circular accelerators • Perpendicular acceleration: • Energy constant... • dp = pda → dp/dt = pw = pv/R • E ≈ pc, g = E/m0c2 • In praxis: Only SR from electrons

8. Energy loss per turn • Max E in praxis: 100 GeV (for electrons)

9. Angular distribution I • Similar to Hertz dipole in frame of electron • Relativistic transformation

10. Spectrum of SR • Spectrum: Harmonics of frev • Characteristic/critical frequency • Divide power in ½

11. ASTRID2 • Horizontal emittance [nm] • ASTRID2:12.1 • ASTRID: 140 • Diffraction limit:

12. Storage rings for SR • SR – unique broad spectrum! • 0th generation: Paracitic use • 1st generation: Dedicated rings for SR • 2nd generation: Smaller beams • ASTRID? • 3rd generation: Insertion devices (straight sections), small beam • ASTRID2 • 4th generation: FEL

13. Insertion devices

14. Wigglers and undulators(Insertion devices) • The magnetic field configuration • Technical construction • Equation of motion • Wigglers vs. Undulators • Undulator radiation • The ASTRID undulator

15. Coordinate system

16. Magnetic field • Potential: • Solution: • Peak field on axis:

17. Magnetic field on axis Construction a) Electromagnet; b) permanet magnets; c) hybrid magnets

18. Insertion devices • Single period, strong field (2T / 6T) • Wavelength shifters • Several periods • Multipole wigglers • Undulators • Requirement • no steering of beam

19. Example (ASTRID2):Proposed multi-pole wiggler (MPW) • B0 = 2.0 T • l = 11.6 cm • Number of periods = 6 • K = 21.7 • Critical energy = 447 eV

20. Summary – multi-pole wiggler(MPW) • Insertion device in straight section of storage ring • Shift SR spectrum towards higher energies by larger magnetic fields • Gain multiplied by number of periods

21. Equation of motion Set Bx = 0, vz = 0 → coupl. eq.

22. Undulator/wiggler parameter: K • K – undulator/wiggler parameter • K < 1: Undulator • Qw< 1/g • K > 1: Wiggler • Qw > 1/g • Equation of motion: s(t)

23. Undulator radiation I • Coherent superposition of radiation produced from each periode • Electron motion in lab frame: • Radiation in co-moving frame (cb*): • Radiation in lab:

24. Undulator radiation II • If not K << 1: Harmonics of Ww

26. Insertion devices: Summary • Wiggler (K > 1, Q > 1/g) • Broad broom of radiation • Broad spectrum • Stronger mag. field: Wavelength shifter (higher energies!) • Several periods: Intensity increase • Undulator (K < 1, Q < 1/g) • Narrow cone of radiation: Very high brightness • Brightness ~ N2 • Peaked spectrum (adjustable) • Harmonics if not K<<1 • Ideal source!

27. Use of SR • Advantage: broad, intense spectrum! • Examples of use: • Photoionization/absorption • e.g. hn + C+ → C++ + e- • X-ray diffraction • X-ray microscopy • ...

28. Optical systems for SR I • Purpose • Select wavelength: E/DE ~ 1000 – 10000 • Focus: Spot size of 0.1∙0.1 mm2

29. Optical systems for SR II • Photon energy: few eV’s to 10’s of keV • Conventional optics cannot be used • Always absorption • UV, VUV, XUV (ASTRID/ASTRID2) • Optical systems based on mirrors • X-rays • Crystal monochromators based on diffraction

30. Mirrors & Gratings • Curved mirrors for focusing • Gratings for selection of wavelength • r and r’ – distances to object and image • Normally q ~ 80 – 90º • Reflectivity!

31. Mirrors: Geometry of surface: Plane, spherical, toriodal, ellipsoidal, hypobolic, ... • Plane: No focusing (r’ = -r) • Spherical: simplest, but not perfect... • Tangential/meridian • Saggital • Toriodal: Rt ≠ Rs • Parabola: Perfect focusing of parallel beam • Ellipse: Perfect focusing of point source

32. Focusing by mirrors: Example

33. Gratings • kNl = sin(a)+sin(b) • NB: b < 0 • N < 2500 lines/mm • Optimization • Max eff. for k = (-)1 • Min eff. for k = 2, 3 • Typical max. eff. ≈ 0.2

34. Design of ‘beamlines’ • Analytically • 1st order: Matrix formalism • Higher orders: Taylor expansion • Optical Path Function Theory (OPFT) • Optical path is stationary • Only one element • Numerically • Raytracing (Shadow)

35. Useful equations • Bending radius • Critical energy • Total power radiated by ring • Total power radiated by wiggler • Undulator/wiggler parameter • Undulator radiation • Grating equation • Focusing by curved mirror (targentical=meridian / saggital)