September 27, 2014 Cheiron School 2014 X-ray Beamline Design 1 X-ray Monochromator Shunji Goto SPring-8/JASRI 1
1. Introduction 2. Light source 3. X-ray Monochromator Fundamental of Bragg reflection Dynamical theory DuMond diagram ~ extraction of x-rays from SR Double crystal monochromator Crystal cooling 4. Summary
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→ Vacuum protection of ring vacuum and beamline vacuum → Radiation safety Monochromator, mirror shutter, slit 3 Shielding and interlock pump,..
Check points to be considered for your SR application: - White or monochromatic - Energy range - Energy resolution - Flux & flux density - Beam size at sample (micro beam?,...) - Beam divergence/convergence at sample (Resolution in k-space) - Higher order elimination w/ mirror - Polarization conversion - Spatial coherency → Light source, monochromator, mirror, and other optical devices and components
Bending magnet: for wide energy range, continuous spectrum for wide beam application for large samples   Wiggler: for higher energy X-rays > 100 keV. Power, brilliance, flux density, partial flux,.. can be calculated using code. e.g. “SPECTRA” by T. Tanaka & H. Kitamura Brilliance for SPring-8 case 7
Angular divergence and band width → Core part we need Bending magnet Undulator
1 + K 2 / 2 nN ΔE E nN Bending Magnet 8
Kilowatt of SR power → mostly eliminated before/by monochromator Bending magnet Power distribution ( V , ≈ 1/ γ Y h ~ CONSt Undulator ≈ 1/Y
K: deflection parameter (K= 0.5~2.5) Bo ² sin ² (2πz/λ,, P tot [kW] = 1.27E² [GeV] − B o² [T]L[m]/[A] Bo= 0.87 T, L= 4.5 m → Ptot= 14 kW 9 Total power L Arc[m] P tot [kW] = 1.27E² [GeV]B² [T]R[m]ø[rad]/[A] E= 8 GeV, l= 0.1 A, B= 0.68 T, R= 39.3 m → Ptot= 0.15 kW/mrad
X-ray monochromator is key component for SR experiments: length gauge for structure analysis, A energy gauge for spectroscopy,... A
Principle of x-ray monochromator
Photon energy tuning ← Bragg’s law Energy resolution ← source divergence, Darwin width,.. Flux (throughput) ← related to Darwin width → Understanding the dynamical theory for large & perfect crystal
Practical of the monochromator
▷ Double-crystal monochromator for fixed-exit Single-bounce monochromator is for limited case ▷ Crystal cooling High heat load depending on light source → Mechanical engineering issues
Bragg’s law in real space 1)  Phase matching on the single net plane by mirror-reflection condition. 2) Phase matching between net planes. Laue condition (Kinematical) in reciprocal space K。− K。= h Reciprocal lattice vector h - Normal to net plane - Length = 1/d Scattered wave K s - net plane (spacing d )
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Ewald sphere: Radius = 1/ λ = Κ o Reciprocal lattice points → When a reciprocal lattice point is on the Ewald sphere, Bragg reflection occurs.
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d = a √h² + k² + l² Top view a = 5.431 Å d-spacing (400) : 1.3578 Å (111) : 3.1356 Å (311) : 1.6375 Å (511) : 1.0452 Å Side view [011] Diamond : a = 3.567 Å
Structure facrtor → Sum of atomic scattering with phase shift in the unit cell Atomic scattering factor f , (h, E )exp { 2πi ( kx , + ky , + lz , Position of atoms in the unit cell for diamond structure (xj, Yj, zj) = (0, 0, 0) 1, (1/4, 1/4) 2, (1/2, 1/2, 0) 3, (3/4, 3/4, 1/4) 4, For diamond structure h,k,l Mixture of odd and even numbers F = 0 (0, 1/2, 1/2) 5, (1/4, 3/4, 3/4) 6, (1/2, 0, 1/2)ך, (3/4, 1/4, 3/4) 8 h,k,l All odd, or, all even numbers, and m: integer, F = 8 f ← 8 atoms in phase h + k + l = 4m ± l F = 4(1 ± i ) f ← Half contribute with phase shift ± π /2 h + k + l = 4m ± 2 F = 0 ← Half cancel with π
(400), (220),... All in phase → F = 8 f (111), (311),... Half contribute with phase shift ± π /2 → F = 4(1 ± i ) f
→ Perfect single crystal: silicon, diamond,.. Photon energy tuning: ▷ Crystal & lattice plane ▷ Bragg angle range 12.3984 E [keV]= 2d hkl [Å]sin θ B e.g. for SPring-8 standard DCM Bragg angle: 3~27°
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3-dimensional periodic structure of unit cell with number N x , N x , N z Total scattering intensity becomes: I = I e |F(Q) 2 ··G(Q) 2
← 3-dim. Periodic structure h, k, l : integer→ Intense peaks → ( hkl ) reflection 120 One-dim. Laue function, N x = 10 sin 2 (πN h) sin 2 ( πh ) -1.5 -0.5 1.5 2 2.5 Peak intensity FWHM　Δh ≈ 0.8858/ N x ～ 1/ N x Crystal size becomes larger → narrower & higher, 17 approaching delta function N x Dynamical theory Two-beam approximation
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“Large & perfect” single crystal: 1) Multiple scattering w/ h & -h reflection 2) Extinction (Diffraction by “finite” number of net planes) Kinematical diffraction is invalid → Dynamical theory must be applied. Net plane
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Fundamental equation is derived using Maxwell’s equations and introducing Bloch wave for 3-dimensional periodic medium (= perfect single crystal): h, g,... : Reciprocal lattice points Eb, Eg : Fourier components of electric field Ko : Incident wave vector in vacuum kh : Wave vectors in the crystal Xh : Fourier components of the polarizability (Negative values, 10-6~10-5) P= (e h •e g ) : Polarization factor between h and g waves kh= ko + h : Momentum conservation
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Fundamental equation is reduced to the equation for two beams (waves) of incidence and “one” intense diffraction
Xo, X h, X-h  : Fourier components of the polarizability (Negative values, 10 -6~10 -5) P = (e o e h ) : Polarization factor ( σ :P = 1, π :P = cos 2θ B )
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Using two equations, we obtain following secular equation: (A) k h (B) Eh = PXhEo + XoEh K² Secular equation (k o ² − k² )k ,² − k² )= X h X − k P 2 K o 4 k² = (1+ χ o )K ² k : Mean wave number in the crystal Scheme of self-consistent wave field − K o − E o = X o E o + P X − h E h
We must consider connections of waves from vacuum into the crystal and from the crystal to vacuum, to solve the equations. Tangential component of wave vector must be continuous. Incident wave in vacuum → Refracted wave in the crystal → Bragg reflection in the crystal → Reflected wave in the crystal → Reflected wave in vacuum
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Laue case (Transmission geometry) Bragg case (Reflection geometry)
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Laue case: b>0
Bragg case: b< 0 Symmetric Bragg case: b= -1 n: normal vector to the surface
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- k ² )k , ² − k ² )= X h X − h P ² K o 4 and it gives four-point solution on the n-vector, Two dispersion surfaces show the gap near Bragg condition.

Excitation error
→ Geometrical deviation ζ 0 from Bragg condition: Distance between Ewald sphere and the reciprocal lattice point. ζo  is positive when H is inside the Ewald sphere (by S. Miyake).
Parameter W is related to the gap between two dispersion surfaces and total reflection occurs at −1<W<1 for Bragg case.
For symmetric Bragg case, sigma polarization:
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Angle deviation at fixed energy → direction change of wave vector Energy deviation at fixed angle → length change of wave vector
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Tie point moves by changing the incident angle at fixed photon energy (wavelength). (1)  Lower angle (2) Near Bragg condition    (3) Higher angle Total reflection Dominant branch for thick Bragg-case crystal is close to O-sphere. 30
ζが: Fourier component of polarizability → proportional to the structure factor
νς: unit cell volume Xh = Xhr + iXhi
Atomic form factor + real part of anomalous factor X hi $ f"(E) Imaginary part of anomalous factor For diamond structure
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From the solution of the fundamental equations, we obtain the ratio r = E n /E 0 (← reflection coefficient) as a function of parameter W. For Bragg case, no absorption, and thick crystal:

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Darwin curve (intrinsic reflection curve for monochromatic plane wave) for Bragg case, no absorption, and thick crystal:
(W < −1)
← Total reflection region
W: deviation parameter for s-polarization, symmetrical Bragg case Geometry for symmetrical Bragg case net plane 33
For Bragg case, no absorption, and thick crystal: W
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Reflectivity - symmetrical Bragg case, - s-polarization, - thick crystal
Note: No absorption  g = 0,κ = 0  ⇒  R → Darwin curve
Examples for symmetrical Bragg case, with absorption, s-polarization and thick crystal: Si 111 refl., 10 keV Xor = −9.78×10−6 Xoi = −1.48×10−7 X 111_r = −3.66×10−6 (1+ i ) Xııı i = −7.30×10−8(1+i) Si 333 refl., 30 keV Xor = −1.07×10−6 Xoi = −1.75×10−9 X333 r = −2.24×10−7(1+i) X333_i = −7.87×10−10(1+i) •Width of 0.1～100 µrad •Peak ~1 with small absorption
The diagram helps to understand how we can extract x-rays from SR source. Angular width (Darwin width) 2 X hr ← ΔW= 2
ΔE 2 ← Gaussian approximation for both light source and reflection curve Energy resolution E ΔE
Effective band width E sin² θ B Relative energy Relative energy Light source divergence φ
- Undulator ( N = 140) σ, ≈ 5 µrad Divergence of undulator radiation ~ diffraction width Natural divergence - Bending magnet σ,, ≈ 0.597 - Undulator 1 F ,  ～ 2NA,, For SPring-8 case: - Bending magnet σ,, ≈ 60 µrad
+Δθ, φ→ Angle-energy diagram (DuMond diagram) Ω: source divergence, ω: diffraction width For usual beamline : ΔE/E=10 -5～10 -3
Wider slit increases unused photons (power) on the monochromator !
SPring-8 standard undulator + 20 µrad slit + Si 111 DCM 10-keV photons → 1.3x10-4 lative energy (e
Angle → (A) Collimation using slit Angle → (B) Collimation using pre-optics w/ collimation mirror, CRL,.. Angle → (D) HR monochromator of π/2 reflection (~meV) (C) Additional crystal w/ (+,+) setting (B)~(D): restriction on photon energy Angle →
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(C) Additional crystal w/ (+,+) setting → HXPES Si 111 DCM Si nnn channel-cut mono. Si 333 refl. for 6 keV Si 555 refl. for 10 keV 0 100 FWHM= 18 meV 80 60 40 20 -20 -80 -100 0.8 0.6 0.4 0.2 0 1
Intensity (arb. units)

Angle → (D) HR monochromator of ~π/2 reflection (~meV) → Inelastic scattering Si 111 DCM Si nnn back-scattering mono. Si 999 refl. for 17.8 keV Si 11 11 11 refl. for 21.7 keV
Photon flux (throughput) after monochromator can be estimated using effective band width: Photon flux (ph/s) = Photon flux from light source (ph/s/0.1%bw) x 1000 x Effective band width of monochromator Throughput is estimated by overlapped area. Note difference from energy resolution.
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Starting with Darwin width in the energy axis
e.g. for Si 111 refl. DCM case Note relative energy width is constant. Independent of photon energy
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For single-bounce monochromator ΔE
E
For no absorption For double-crystal monochromator ΔE E
↑  For no absorption When you need flux →  Lower order (Si 111 refl.,..) When you need resolution → Higher order (Si 311, Si 511 refl,..) Effective band-width is obtained by integration of reflection curve. R(W)dW 47

Effective band width

Photon flux from light source (ph/s/0.1%bw) Photon x 1000 (A) x Effective band width of monochromator ,φ)² dEdφ buMond
This approach is valid.

(a) Total flux @ 0.1% b.w. (b) After frontend slit 1 × 1 mm² @30 m (c) Si 111 refl. @10 keV Effective b.w. = 1.3x10-4 (d) 3rd harmonics @30 keV Effective b.w. = 8.0x10-6 Higher harmonics elimination more → mirror or detuning of DCM We can obtain photon flux of 10¹³~10¹⁴ ph/s/100 mA/mm² using standard undulator sources and Si 111 reflections  at SPring-8 beamlines.
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Fixed-exit operation for usability at experimental station.んChoose suitable mechanism for energy range (Bragg angle range).んPrecision, stability, rigidity,... Fixed-exit operation using rotation (θ) + two translation mechanism
&= 5.7~72° (for lower energy range) =3~27°  for higher energy range SPring-8 std. DCM 51 Matsushita et al., NIM A246 (1986) Yabashi et al., Proc. SPIE 3773, 2 (1999) .

Why crystal cooling ?
Qin (Heat load by SR) = Qout (Cooling + Radiation,..) → with temperature rise ΔT → αΔΤ = Δd (d-spacing change) α: thermal expansion coefficient or → Δθ (bump of lattice due to heat load) Radiation & convection Heat conduction in crystal Heat transfer to coolant/holder
Miss-matching between 1st and 2nd crystals occurs:
→ Thermal drift, loss of intensity, broadening of beam, loss of brightness → Melting or limit of thermal strain → Broken !
We must consider:
- Thermal expansion of crystal: α, - Thermal conductivity in crystal: κ, - Heat transfer to coolant and crystal holder.
Figure of merit

Solutions: (S-1) κlα → Larger (S-2) Contact area between crystal and coolant/holder → larger (S-3) Irradiation area → Larger, and power density → smaller Figure of merit of cooling: Good for silicon (80 k) and diamond (300 K)
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<Bending magnet beamline> Power & power density: ~100 W, ~1 W/mm² Fin crystal direct-cooling - (S2) <Undulator beamline> Linear undulator, N= 140, λu= 32 mm Power & power density: 300~500 W , 300~500 W/mm² a)   Direct cooling of silicon pin-post crystal   – (S2) & (S3) b) Silicon cryogenic cooling - c) Ila diamond with indirect water cooling - (S1)
Key issues on the x-ray monochromator were shown, introducing the dynamical x-ray diffraction for large & perfect crystal, w/ several important points: 1) Total reflection occurs at the gap between dispersion surfaces. 2) Normalized deviation parameter W is related to the gap. 3) W is parameter of angular deviation and energy (wavelength) deviation. It gives DuMond diagram as a band of |W|<1. 4) By combination of light source and monochromator crystals, photon energy, energy resolution, photon flux,・・・  can be controlled / tuned. Double-crystal monochromator w/ crystal cooling is needed for practical use at the SR beamline. By understanding these, you will be approaching to good design/use of the beamline 54 for your SR science.
Ergebnisse der Exakt Naturwiss. 10 (1931) 133-158. Dover (1945) Theory of X-Ray W ILLIAM H. Z ACHARIASEN R. W. James “The Dynamical Theory of X-Ray Diffraction" Solod State Physics 15, (1963) 53-220. B. W. Batterman & H. Cole “Dynamical Diffraction of X- Rays by Perfect Crystals” Rev. Modern Phys. 36, (1964) 681-717. Oxford (2001) Springer (2004) [1] T. Matsushita & H. Hashizume, Handbook on Synchrotron Radiation Vol. 1, North-Holland (1983). [2] Y. Shvyd’ko, Springer (2004), for high energy resolution >For f [3] International Tables for X-ray Crystallography (IUCr). [4] Hubbell et al.: J. Phys. Chem. Ref. Data Vol. 4, No. 3, 1975. ▷For anomalous scattering factor f’, f’ [5] S. Sasaki, KEK report 88-14 (1989), [6] Tables of LBNL (B. L. Henke et al.), http://henke.lbl.gov/optical_constants/index.html, [7] Tables of NIST, http://physics.nist.gov/PhysRefData/FFast/html/form.htm Thank you for your attention.