Physics of X ‐ ray radiation production and transport. Simulating photons and waves from the X ‐ ray sources to the samples Manuel Sánchez del Río AAM, ISDD, ESRF
Outline • Evolution of x ‐ ray science • Sources: single particles (easy) and sets of particles (bunches, materials…) • Optics: why X ‐ ray optics is different? Concepts – Calculations • Example: ID20 (UPBL6) • Next generation of simulation tools
At the beginning • Röntgen 1895 • Revolutionary rays
X ‐ rays are “only” light (optics)
But also “particles” (photons) “I could have done it in a much more complicated way" said the red Queen, immensely proud. Lewis Carroll
“What I cannot create, I do not understand.” “I calculate everything myself.” If you cannot calculate… Just simulate it! It may be a good starting point.
Light (EM radiation) emission by moving e ‐
E 0 =40 keV E 0 =31 keV E 0 =20 keV e - E 0 E 0 E’=8 keV E’=19 keV E 0 =20 keV
E 6 GeV γ = = ≈ 12000 2 m c 0.511 MeV e
Schott’s formula (1912) π ∞ ∞ ∑ ∑ ∫ ∫ ⎡ ⎤ = Ω + = ω θ θ ν ξ + θ ξ 2 2 2 2 '2 2 2 P d ( P P ) e v d sin J ( ) cot J ( ) ⎣ ⎦ σ π 0 n n = = n 1 n 1 0 2 4 2 e v = = P n ( 1) 3 2 3 c R This formula is valid for all values of the velocity v. • In the non-relativistic limit, v<<c •the argument is small and we can only consider the first term in the series expansion of the Js. •The main contribution to the total radiated power is provided by the first harmonic (n=1, dipole radiation). •The radiation of the higher harmonics (n > 1) is suppressed in a power fashion: P(n)/P(1)<<1. •Most interesting is the ultrarelativistic case, v/c~1. The maximum of the radiation spectrum lies in the region of high harmonics, n~[E/(mc 2 )] 3 ~ γ 3
1912: Schott’s formula 1944 Ivanenko and Pommeranchuck theorized that maximum attainable energy is limited by the radiation losses 1946 Blewett at the General Electric Labs observed the shrinking of the electron orbit at the highest energy of 100 MeV in “a manner consistent with the predictions of the theory”. They failed in detecting the emitted radiation because they searched it in the energy range close to the first harmonic, whereas the maximum of the frequency spectrum lies in the region close to the critical energy. 1947, April 24th, Pollock, Langmuir, Elder, and Gurewitsch saw the bluish ‐ white light emerging from the transparent tube of their new 70MeV synchrotron at General Electric's Laboratory: Synchrotron radiation had been seen.
SR Formulation (1 e ‐ ) Ivanenko and Sokolov (1948) derived an asymptotic formula for the spectral distribution of the radiation intensity. The same result was also obtained by Schwinger (1949) 6 ⎛ ⎞ ν 2 3 ∞ m c ⎠ ∫ ν = 0 ⎜ ⎟ W ( ) W K ( ) x dx 3 ⎛ ⎞ ν 2 π 2 m c 53 ⎝ ⎜ ⎟ 0 2 E ⎜ ⎟ 3 E ⎝ ⎠ ν ν ν ⎧ ⎫ 2 2 ce ν θ = ε ε + ε θ ε Ω ⎨ 2 2 3/2 2 2 3/ 2 ⎬ 0 dW ( , ) K ( ) cos K ( ) d π 2/3 1/3 ⎩ ⎭ 2 3 R 6 3 3 1 1 ω ω = γ ω � 3 max c 0 2 2 Today, we can implement these functions in one line of code, e.g., Mathematica
BM – Emission by N incoherent e ‐ z Phase Space (H) Real Space (top) x x’ y x y x • Monte Carlo (SHADOW) – Energy (and polarisation) sampled from spectrum – Angular Distribution (1e ‐ , σ ’ x , σ ’ z ) – Geometry (along the arc, σ x , σ z ) – Limitation: Computer time and memory • Typically: 10 3 ‐ 10 9 rays • Desirable: one ray per photon, i.e., 10 14 ‐ 10 20
Wiggler: Like BM, but a bit more complex
Undulator: 1e ‐ emission interferes with itself = θγ K For a single energy (odd harmonic)
Codes • XOP: Urgent (Walker), US+WS (Dejus), Xwiggler, BM • SRW (Chubar and Elleaume) • SPECTRA (Tanaka and Kitamura) • SynchrSim (Grimm) http://flash.desy.de/sites/site_vuvfel/content/e403/e1642/e2308/e2310/infoboxContent2357/TESLA ‐ FEL2008 ‐ 05.pdf
Ligh emitted by tight ‐ bunched beams 2 N N N ∑ ∑ ∑ = = 2 + * P E E E E i i i j = = ≠ i 1 i 1 i j • The second term, due to the random position of the electrons, it is randomly positive and negative, its average value is zero. • This is not true if the electron bunch length is shorter than the wavelength, and the power is proportional to the square of the stored current. … This is the basis of the Free Electron Laser (FEL). In practice, the spectral flux observed is proportional to a number between N and N 2 . • In practical cases, the coherent radiation is weak and hidden by the incoherent emission. To make it dominate, a very long undulator of several tens of meters must be installed in a special by ‐ pass section of the ring. This is quite demanding from an accelerator point of view. It requires the highest peak current, the smallest emittance, the smallest energy spread, and very long undulators. The spectral range of the emitted radiation is limited to VUV or soft x ‐ rays. • Codes like GENESIS (http://pbpl.physics.ucla.edu/~reiche/) are used to calculate XFEL emisison
• XOP (W,Mo,Rh, Boone et al. ) • Monte Carlo particle transport (MCNP, EGS, GEANT4, PENELOPE, …)
Radiation scattered from optical elements E. Secco and M. Sanchez del Rio, SPIE Proc 8141, 81410Z (2011) W. Salah and M. Sanchez del Rio JSR 18 (2011) 512
Plasmas • 99% of the visible matter in the Universe is in form of plasma • Plasmas emit X ‐ rays due to various effects (thermal radiation, accelerated charged particles, transitions in ions, nuclear reactions) • On Earth we found natural plasmas (e.g., lighting in thunderstorms) • Artificial plasmas (electric discharges [pinches], laser generated plasmas) may be used as X ‐ ray source • X ‐ rays are a very useful diagnostic tool for artificial plasmas 5 s 10 ‐ 9 s ITER NIF ∞ s T~20keV (200 million C). W High densities, > 10 20 m -3 , must be maintained MELTING POINT to produce a sustainable reaction
X ‐ RAY PLASMA DIAGNOSTICS X ‐ RAY PLASMA DIAGNOSTICS AT M.I.T. ALCATOR C ‐ MOD TOKAMAK AT M.I.T. ALCATOR C ‐ MOD TOKAMAK Ar 16+ λ [Å] 3 × 487 pixels × 172 μ m/pixel~25.13 cm Crystal 195 pixels for 60 m Å 21 Courtesy: PPPL
Coherence and Incoherence • If the source is incoherent, we add the intensities of the emission of each e ‐ at the observation plane (typically in ray ‐ tracing) (N) • If the source is coherent (such as a point monochromatic source at infinity=>Plane wave) we add E at the observation plane and square it to get the intensity => Wave optics propagation. E.g., Fresnel ‐ Kirchhoff propagator in free space (N 2 ) • If the source is incoherent but small, there is still some coherence observed (van Citter ‐ Zernike) • But one cannot see a source too small, because there is a limit (diffraction limit) • Moreover, fully coherence or fully incoherence do not exist =>partial coherence • The source is complicated, and this is only the beginning….
Photon ‐ Matter interaction (before optics) • For 1e ‐ at E<1MeV π 8 σ = = 2 r 0.6652448 barn – Coherent scattering (Thompson) T 0 3 σ 2 E ⎛ ⎞ ⎛ ⎞ KN 2 d r E ' E ' E = + – Incoherent scattering (Compton) = + − θ E ' 2 C 0 ⎜ ⎟ ⎜ sin ⎟ γ − θ Ω 1 (1 cos ) ⎝ ⎠ ⎝ ⎠ d 2 E E E ' • For many e ‐ in atoms: + θ 2 – Coherent&elastic scattering (Rayleigh) 1 cos σ = θ 2 Ω 2 d r f ( ) d e 2 => f 0 (q) =>F h (E,q) – Incoherent scattering (Compton)=>Shower – Photoelectric scattering (absorption, fluorescence) � � = + + f Q E ( , ) f Q ( ) f '( ) E if "( ) E 0 = − δ − β 1 n i λ ρ 2 r N ′ δ = = e A K f ; K π A 2 ρ ′′ β = K f ; A
Tabulations: DABAX, xraylib
The single interface (Fresnel) 2 ⎛ ⎞ n = θ ⇔ θ = δ − δ ≈ δ ⎜ 1 ⎟ 2 2 1 cos sin 2 2 c c ⎝ n ⎠ 2 •Structures in depth => playing with the reflectivity •Structures along the surface =>playing with the direction
Multilayers • no reflection from the back of the substrate • compute recurrently the reflectivity of each layer from bottom (substrate) to top
What happens to the direction if the interface is not plane? K i K g g =>Dispersion in energy
Gratings
Zone plates/Lens Amplitude FZP Phase FZP Kinoform FZP alternate zones - opaque alternate zones - (sawtooth profile) phase shifting Efficiency: Efficiency efficiency 10.1 % (1 st harmonic) 40.5% Up to 100% 1.1 % (3 rd harmonic) Δ r n t
Crystals Direction BRAGG or reflection LAUE or transmission Reflectivity ( ) 2 ⎧ − 2 − 1 x x ⎪ for ≥ 1 x ⎪ = ≤ ( ) 1 for 1 ⎨ R x x ⎪ ( ) ≤ − 2 for 1 x ⎪ + 2 − 1 x x ⎩ Darwin, Phil. Mag. 27 (1914) 315 & 675
Imaging vs Condensing Optical Systems Imaging Optics Demagnification M NON Imaging Optics => Large objects (elephants) are more deformed than small objects (ants)
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