Physics of X ray radiation production and transport. Simulating - - PowerPoint PPT Presentation

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
• f 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‐

E0=20 keV E0=31 keV E0=40 keV E0 E0=20 keV E’=19 keV E’=8 keV e- E0

2

6 12000 0.511

e

E GeV m c MeV γ = = ≈

2 2 2 2 '2 2 2 1 1

( ) sin ( ) cot ( )

n n n n

P d P P e v d J J

π σ π

ω θ θ ν ξ θ ξ

∞ ∞ = =

⎡ ⎤ = Ω + = + ⎣ ⎦

∑ ∑ ∫ ∫

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/(mc2)]3~γ3

2 4 3 2

2 ( 1) 3 e v P n c R = =

Schott’s formula (1912)

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

• bserved the shrinking of the electron
• rbit 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)

3 2

6 2 2 53 3

3 ( ) ( ) 2

m c E

m c W W K x dx E

ν

ν ν π

∞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∫

2 2 2 2 3/2 2 2 3/ 2 2/3 1/3 2 3

( , ) ( ) cos ( ) 6 3 3 ce dW K K d R ν ν ν ν θ ε ε ε θ ε π ⎧ ⎫ = + Ω ⎨ ⎬ ⎩ ⎭

3 max

1 1 2 2

c

ω ω γ ω =

• Today, we can implement these functions in one line of code, e.g., Mathematica

BM – Emission by N incoherent e‐

• Monte Carlo (SHADOW)

– Energy (and polarisation) sampled from spectrum – Angular Distribution (1e‐, σ’x, σ’z) – Geometry (along the arc, σx, σz) – Limitation: Computer time and memory

• Typically: 103 ‐ 109 rays
• Desirable: one ray per photon, i.e., 1014 ‐1020

x y x x’ x y z Real Space (top) Phase Space (H)

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

• 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 N2.

• 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

2 2 * 1 1 N N N i i i j i i i j

P E E E E

= = ≠

= = +

∑ ∑ ∑

• 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

ITER

5 s

NIF

10‐9 s ∞ s

W MELTING POINT

T~20keV (200 million C). High densities, > 1020m-3, must be maintained to produce a sustainable reaction

X‐RAY PLASMA DIAGNOSTICS AT M.I.T. ALCATOR C‐MOD TOKAMAK X‐RAY PLASMA DIAGNOSTICS AT M.I.T. ALCATOR C‐MOD TOKAMAK

21

195 pixels for 60 mÅ 3×487 pixels×172 μm/pixel~25.13 cm

Ar16+ Crystal λ[Å] 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 (N2)

• 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

– Coherent scattering (Thompson) – Incoherent scattering (Compton)

• For many e‐ in atoms:

– Coherent&elastic scattering (Rayleigh) => f0(q) =>Fh(E,q) – Incoherent scattering (Compton)=>Shower – Photoelectric scattering (absorption, fluorescence)

1 n i δ β = − −

2

; 2 ;

e A

r N K f K A K f A λ ρ δ π ρ β ′ = = ′′ =

( , ) ( ) '( ) "( ) f Q E f Q f E if E = + +

• 2

8 0.6652448 barn 3

T

r π σ = =

2 2 2

1 cos ( ) 2

e

d r f d θ σ θ + = Ω

2 2 2

' ' sin 2 '

KN C

d r E E E d E E E σ θ ⎛ ⎞ ⎛ ⎞ = + − ⎜ ⎟ ⎜ ⎟ Ω ⎝ ⎠ ⎝ ⎠ ' 1 (1 cos ) E E γ θ = + −

Tabulations: DABAX, xraylib

The single interface (Fresnel)

• Structures in depth => playing with the reflectivity
• Structures along the surface =>playing with the direction

2 2 2 1 2

1 cos sin 2 2

c c

n n θ θ δ δ δ ⎛ ⎞ = ⇔ = − ≈ ⎜ ⎟ ⎝ ⎠

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?

Ki Kg g =>Dispersion in energy

Gratings

Zone plates/Lens

Amplitude FZP alternate zones - opaque Efficiency: 10.1 % (1st harmonic) 1.1 % (3rd harmonic) Phase FZP alternate zones - phase shifting Efficiency 40.5% Kinoform FZP (sawtooth profile) efficiency Up to 100% t Δ rn

Crystals

BRAGG or reflection LAUE or transmission

( ) ( )

2 2 2 2

1 for 1 ( ) 1 for 1 for 1 1 x x x R x x x x x ⎧ − − ≥ ⎪ ⎪ = ≤ ⎨ ⎪ ≤ − ⎪ + − ⎩

Darwin, Phil. Mag. 27 (1914) 315 & 675

Direction Reflectivity

Imaging vs Condensing Optical Systems

Imaging Optics NON Imaging Optics Demagnification M => Large objects (elephants) are more deformed than small objects (ants)

OK, but I always see Gaussians!

• Yes:

(Theorem of central limit)

• No: (plot it

in log scale!)

( / 2) 2 2ln(2) 2.35 RMS CL erf n FWHM σ σ σ σ = = = ≈

FWHM 76.1%

Imaging systems (grazing optics)

• In order for any optical system to form an image,

it must satisfy the "Abbé sine condition", at least approximately

• Two (or more) surfaces are needed
• E.g.: Wolter optics
• KB (1948): Good approximation

Non‐imaging system:

BL as a concentrator: which shape (in reflection)?

q p ρ 1 1 2sin p q θ ρ + = 1 1 2 sin p q R θ + =

• Point to point focusing (ellipsoid)
• Collimating (paraboloid)
• Notes:

– Focalization in two planes

• Tangential or Meridional (ellipse or parabola)
• Sagittal (circle)

– Demagnification: M=p/q – Easier:

• Only one plane => cylinder Ellipsoid => Toroid
• Parabola/Ellipse => circle
• Sagittal radius: constant (cylinder), linear (cone),

non‐linear (ellipsoid)

– Aberrations

ID20 Inelastic Scattering

• meV resolution (10‐3 times less than what you read in the spectra

ph/s/0.1%bw) => USE THE WHOLE BEAM (REDUCE THE LOSSES BY DIMENSIONS)

• High resolution=>Collimation in diffraction plane
• Hβ or Lβ? (Lβ has higher divergence, Hβ seems favourable)
• LBL (140m) or shorter?

energy in the 5 - 20 keV range focal spot size ≤ 10 μm minimal beam losses enough space (>20 cm) around the sample sub-eV resolution

Source

5 1 57x10 88x6 402x10 11x6.2 20 keV 400x10 10x10 ROUNDED 57x10 88x4 402x10 11x3.2 e‐ μm μrad 57x10 88x7.2 57x10 88x12 Low β μm 2 3 (L λ)1/2 μrad μm μrad 6 9 (L /λ)1/2 402x11 11x5.6 402x13 12x5.6 High β 10 keV 5 keV RMS

15 cm 19 cm 20 m 23 cm 28 cm 30 m 30cm 38 cm 40 m 38 cm 47 cm 50 m 61 cm 75 cm 80 m L(FWHM) p 76 cm 94 cm 100 m θ=3.1 mrad θ=2.5 mrad

1 mirror: How far?

M1xM2=100 LBL<50m θ=2.5mrad

10,2 130,47 10,2 17,0.1 10,2 185,22 10,7 166,27 10,10 100,100 M1 (H,V) M (H,V) Toroid + Ellipsoid Ellipsoid + Ellipsoid 14 mm 6 mm Toroid + Toroid Toroid + KB Rs (M1) Rs (M2) P1=3000 Q1=300 P2=1250 Q2=125 TARGET

NOT ON THE SAME SCALE!

BAD SHAPES!!

Sag Cyl Collimator+Ellipsoid*, q2=75cm

92,40 94,4 70m 79,40 81,3 60m 66,40 66,2.5 50m 186,40 207,7 140m M (H,V) M (ray tracing) 52,40 49,2 40m Rsag TOO SMALL (NEED Rs>2cm e.g. q2=300 @ 70m) , BUT OK IN H V H * Computed for point-to-point focusing, thus neglecting collimation

Towards final config

• Short BL
• Use of secondary source (M=M1*M2 MA=3.1*16 MB=2.4*23)
• First High Power Collimating mirror (sag/tan)
• KB: good optical performance (good approx to imaging system), tunability
• Mirror optimisation (toroid M~3, distances, astigmatism)
• Slope errors (0.5‐0.7 μrad RMS)
• Power Load
• Tolerances
• Monochromator(s) optimization

(1.8 x 15 μm2 without slope errors) 96%

BL transmitivity

4 6 8 10 12 14 16 18 20 1 2 3 4 5 6

Si(111) + Si(311)

Intensity [ 10

13 photons /s ]

Energy [ keV ] The angular distribution along M4 (Long mirror) implies that part of the mirror is not working

6 8 10 12 14 16 18 20 0,4 0,5 0,6 0,7 0,8

R Energy ( keV ) FM4: Rh 3.1 mrad FM4: Rh 2.5 mrad FM4: Pt 3.1 mrad

Si111@7keV or Si333@21keV

XRO software roadmap

RAY TRACING SR SOURCES WAVE OPTICS

KERNEL GUI

TOOLS SHADOW 3.0 SRW

XOP (ShadowVUI)

SHADOW 2.0

New Tool PANSOXTICS OpenSource Python+Qt?

Acknowledgements

• my colleagues
• Special thanks to Giulio Monaco and Lin Zhang)

References

Wikipedia X-ray Data Booklet Als-Nielsen & McMorrow, Elements of Modern X-ray Physics Attwood, Soft X-rays and Extreme UV radiation Michette (ed), X-ray science and technology Spiller, Soft X-ray Optics SPIE Press, 1994 Handbook of Optics (3rd Edition Volume 5)

Credits (figures and more)

http://www.nobelprize.org/ http://xkcd.com PENELOPE manual, A. Bielajew MC lecture http://www‐antenna.ee.titech.ac.jp/~hira/hobby/edu/em/dipole/ N.A. Dyson: X‐rays in Atomic and Nuclear Physics (2nd ed) http://hasylab.desy.de/science/studentsteaching/primers/synchrotron_radiation/ http://www.shimadzu.com/an/ftir/support/ftirtalk/letter9/mirror.html http://dx.doi.org/10.1107/S0021889806003232

• N. Pablant (Princeton U)

Viva la ciencia, Mingote & Sanchez Ron

Thank you!