Physics of the electron beam source: beam size, shape and lifetime and the relationship to the x ‐ ray radiation properties Boaz Nash Accelerator Source Division, ESRF
Outline • Schematic History of synchrotron radiation • Radiation brightness and beam size • Physics of one electron in the storage ring • Electron beam emittance, size and shape • Beam lifetime
Schematic History of Synchrotron Radiation • Accelerators were built for nuclear and particle physics. (~1930 onward) • Synchrotron radiation was discovered (observed 1947) and seen as a problem: it limits acceleration! • Schwinger, Ivanenko and Pomeranchuk, others already described synchrotron radiation (1944 and earlier) • radiation is useful! • Machines built to optimize its production • Generations… 1 st , 2 nd , 3 rd ,4 th , (5 th ?)…
The big picture: photons come from electrons • electrons:gun ‐ >linac ‐ > booster ‐ > storage ring • xrays: source ‐ >front end ‐ > beamline ‐ > experiment xray properties determined by electron beam properties Focus on this! x-ray Storage ring source therefore this! light source schematic (ESRF) beamline booster linac sample photons e- gun
Brightness/brilliance of x ‐ ray beam F ( ω ) B = 4 π 2 Σ x Σ x ' Σ z Σ z ' (photon beam sizes/divergence at source) Where F is the photon flux in the central cone of a given harmonic for a given frequency bandwidth (e.g. .1%). The photons are created in the bending magnets and undulators. More about undulators, radiation and brightness: -- � J. Chavanne next month.
Photon beam size and divergence is determined by a combination of electron beam and single electron emission Σ = σ + σ 2 2 2 x x , elec x , photon Σ = σ + σ 2 2 2 x ' x ', elec x ', photon Σ = σ + σ 2 2 2 z z , elec z , photon Σ = σ + σ 2 2 2 z ' z ', elec z ', photon Σ + Σ 2 2 2 D These are at source. A distance D away, beam size become: x , 0 x ' , 0
The relationship between electron beam and x ‐ ray beam may be more complex. An example: The standard formula for the source divergence due Wavelength of to undulator radiation is given by λ radiation σ = x ' 2 L Length of However, there are important corrections to this formula undulator due to the electron beam energy spread, particularly at higher ( See Tanaka et. al., Journal of Synch. Rad. 16, 380-386 (2009) and talk by J. Chavanne next month ) harmonics. As we look at smaller and smaller electron beam sizes, vertical and horizontal, we ought to revisit many questions regarding the interaction between electron beam and x-ray beam. Like any relationship… there are two parties involved. So, let’s consider the electron beam.
ESRF electron beam synopsis. What is going on behind this? Some basic questions: The current is 196.87 mA. What does this really mean? What is uniform multibunch. What is lifetime and emittance? More difficult questions: What determines the value of the emittances? What determines the value of the lifetime? How much control do we have over these parameters? How to answer these questions? Start with the basics. Try Wikipedia? No luck.
What is an electron? Spin ½ elementary particle − = e = − × 19 charge 1 . 6 10 Coulombs = = × − 31 mass m e 9 . 11 10 kg What do free electrons do? They move and get pushed around by electric and magnetic fields. r r r r = + × F e ( E v B ) (Lorentz force law) They radiate when they turn or accelerate. ⎛ ⎞ dp dp 2 0 r ⎜ μ μ ⎟ = • (relativistic Larmour equation) P ⎜ ⎟ τ τ ⎝ ⎠ 2 m c d d e = π β 4 4 4 r c E For circular motion: − = × P = 0 5 3 C 8 . 846 10 m /( GeV ) C γ γ π ρ 2 3 2 3 ( m c ) 2 e Bending radius
How to store a high energy electron • Accelerate to high energy (E=6.04 GeV for ESRF) in linac and booster, then inject into ring. • Use dipole magnets to create circular trajectory. • Use quadrupoles to confine the beam transversely. • Use sextupoles to fix chromatic aberration caused by the quadrupoles. • Use an RF cavity to replenish energy and confine longitudinally.
Components needed to store electrons dipole quadrupole sextupole RF cavity
Electron closed orbit dipole The dipole magnets have constant vertical magnetic fields that bend the electron into a big circular trajectory. There is an orbit that closes on itself that is called the ideal orbit. e- Bending magnet y x ρ ρ = B [ Tm] 3.3357 p [ GeV/c] s B=.86 T use perpendicular coordinates, x, y p/c=6.04 GeV and coresponding angles x’=px/P0, y’=py/P0 ρ = 23 . 4 m Transverse phase space: (x,x’,y,y’) For recent work on orbit correction, see: E. Plouviez et. al. “Fast Orbit Correction for the (controlled by dipoles+correctors) ESRF Storage Ring”, ipac ‘11 (MOPO002 )
Quadrupole focusing quad + = x ' ' k ( s ) x 0 x (Hill’s equation) + = ' ' ( ) 0 z k s z z + = k ( s C ) k ( s ) x , z x , z The fact that we use quadrupole magnetic fields for focusing implies: > ⇒ < k 0 k 0 x z < ⇒ > k 0 k 0 x z Cristofilos (1949) and Courant-Snyder (1952)discovered that a combination of focusing and defocusing quadrupoles leads to a net focusing effect. (principle of strong focusing ).
Harmonic oscillator in phase space Twiss Parameters measuring the position ε = γ + α + β 2 2 x 2 xx ' x ' over time, it will oscillate invariant with position! x ' ε x γ x ε x β x turn 1 x turn 2 α − slope: turn 3 β tune is defined by number of oscillations about closed orbit over 1 turn This is at one position in the ring.
Transverse dynamics turn 1 turn 2 turn 3 s 3 s 2 s 1 x ' tune = phase advance per turn x
Tune measurement Shake the beam at different frequencies and measure the response. Intensity of response
Tune Resonance Diagram we want to avoid tunes near resonances, i.e. n nux + m nuy = k for some integers, n, m, k.
Energy dependence of transverse motion • electrons have an average energy of 6.04 GeV, but will have a spread about this value (energy spread) σ δ E / E = 10 − 3 • The transverse oscillation vary with energy, since the bending (dipoles) and focusing (quadrupoles) effect does so. • Orbit shift with energy ‐ > dispersion. • Tune shift with energy ‐ > chromaticity. • This tune shift would be unacceptable. • It is fixed with the sextupoles.
Lattice The layout of dipoles, quadrupoles and sextupoles is called a lattice. The lattice is chosen to try to satisfy many constraints, and to optimize important parameters. The strengths of the quadrupole and sextupole magnets are controllable. This gives a certain amount of flexibility in setting the beta functions and dispersion and improving the non-linear dynamics with the sextupoles even with the magnets fixed in place. ESRF Lattice is a double bend achromat, but with distributed dispersion.
ESRF Optical Functions BetaX BetaZ Dispersion 60.00 0.400 0.350 50.00 0.300 (high beta straight) 40.00 0.250 (low beta straight) 30.00 0.200 0.150 20.00 0.100 10.00 0.050 0.00 0.000 -5.000 0.000 5.000 10.000 15.000 20.000 25.000 30.000 -10.00 -0.050 recall: beta describes variation of beam envelope around ring. Dispersion is the off energy orbit function. This is one half of a super period which is repeated mirror symetrically. There are 16 super periods in the ring to create the full circumference of 844.39 meters.
RF longitudinal motion • energy loss from radiation would cause particles to be lost. • RF cavity gives this energy back. • RF cavity also causes longitudinal focusing! V_RF f_RF = h f_0 h=992 t synchrotron oscillations and synchrotron tune. nus=6e-3 dp/p or 1 oscillation every 166 turns. ct RF buckets
Non ‐ linearities from S sextupoles • sextupole fixes chromaticity, but introduces cubic term in potential. unstable = + Stable dynamic aperture
Stability over many turns: how to predict/control? for ESRF, 10 hour lifetime = 13 billion turns! Age of earth = 4.5 billion years = 4.5 billion turns around the sun. Our solar system seems quite stable… but in fact… its chaotic and may be unstable in the long run! Jacques Laskar and his colleague Mickaël Gastineau in 2009 took a more thorough approach [to studying the solar system evolution] by directly simulating 2500 possible futures. Each of the 2500 cases has slightly different initial conditions: Mercury's position varies by about 1 metre between one simulation and the next.[13] In 20 cases, Mercury goes into a dangerous orbit and often ends up colliding with Venus or plunging into the sun. Moving in such a warped orbit, Mercury's gravity is more likely to shake other planets out of their settled paths: in one simulated case its perturbations send Mars heading towards Earth.[14] http://en.wikipedia.org/wiki/Stability_of_the_Solar_System (30/11/2011) [13] New Scientist, “ Solar system's planets could spin out of control “, 10 June 2009 [14] J. Laskar 1 & M. Gastineau, “Existence of collisional trajectories of Mercury, Mars and Venus with the Earth ”, Nature 459 , 817-819 (2009)
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