XX--rayray optics Crystal optics

Jürgen Härtwig

ESRF X-ray Optics Group, Crystal Laboratory

X OOG ESRF X-ray Optics Group What was already presented (among others)?

Physics of the electron beam source (Boaz Nash)

Physi cs of X-ray radia tion prodtiduction and transpor t (Ml(Manuel Sánchez del Río) Multilayers in synchrotron optics (Christian Morawe)

Energy resolving detectors for X-ray spectroscopy (John Morse)

So we continue today with the X-ray optics Outline

1. Introduction 2. Monochromators 3. Some properties of asymmetr ica l refltiflections 4. Shortly about high energy resolution 5. Crystal quality and how to measure it (6. Plane or divergent, monochromatic or polychromatic waves in our experiments?) 1. Introduction Some questions I plan to discuss and maybe to answer: Which kind of monochromators are used? How may I change the energy resoltilution, beam divergence, beam dii?dimension? May the Bragg diffraction geometry have an influence on the coherence? RlRole of source size an d angu lar source size. IflInfluence on transversa l coherence, resolution etc. ? Are “Imaging quality”, “focusing properties”, “coherence preservation” related? What is a “highly perfect” crystal? What are the lowest strains that we can measure? Is there a “highly parallel (monochromatic)” beam? Is there a “nanometric parallel” beam? We need to define what a “plane” or a “monochromatic” wave could be in the real exppypperimental life. How may we approximate them? etc. Optical system / experimental set-up: Source  optical elements  sample  optical elements  detector

The task of the optics: To transform the beam to obtain the best matching with the experiment; not loosing the good properties of the beam after its creation.

It acts on: - shape - wavelength/energy - divergence - polarisation - coherence “No optics is the best optics”!? Yes, btbut … In principle possible - all in vacuum, working with non-modified “white” or “pink” beam. However, not very useful. We may need e.g. monochromatisation, focussing, ... . A whole zoo of optical elements • slits, pinholes • filters, windows • mirrors (reflectivity based) • beam splitter monochromators (crystals) • monoch/ll/lhromators/collimators/analysers (crystals, multi-layers – often also called “mirrors”) •phase plates ((hphase retard er, pol li)arizer) (crystals) • lenses, zone plates •combine d e lemen ts (ML gra tings, Bragg-FlFresnel-l)lenses) • etc.

Mirrors and monochromators, collimators, analysers ffatlat, but asoalso bent for collimat ion, focussing, imag e magnification, ... Main physical effects used in X-ray optics were discovered in the first years starting from the discovery of X-rays by C. W. Röntgen: absorption (Röntgen 1895/96  filters) Bragg-diffraction (Laue 1912  monochromators, etc.) specular reflection (Compton 1922  mirrors) refraction (Larsson, Siegbahn, Waller 1924  later lenses)

PtiProperties of dbldouble and many-crystltal set-ups: Jesse W. M. DuMond, Physical Review, 52, 872-885 (1937) DuMond graphs, dispersive and non-dispersive set-ups, channel cut crystals (later invented as “Bonse-Hart-camera”), four crystal spectrometer (later invented as “ Barthels monochromator”), etc.

But -many newer ddlevelopments Quite a lot of literature

Overviews:

Tadashi Matsushita, X-ray Monochromators, in Handbook on Synchrotron Radiation, Vol. 1, ed. E. E. Koch, North-Holland Publi s hing Company, 1983

Dennis M. Mills, X-Ray Optics for Third-Generation Synchrotron Radiation Sources, in Third-Generation Hard X-ray Sources, ed. Dennis M. Mills, John Wiley & Sons Inc., New York, 2002 Short remark concerning beam dimensions

Few years ago – micro-beams were modern, now – nano-beams are in vogue.

But - we need all kind of beams: large beams (()decimetre sized) and small ones (nanometre sized), “parallel”, divergent and focussed beams. Beam dimensions – examppple: paleontology

Examples from Paul Tafforeau

Nearly 4 orders of magnitude in dimension Without scanning Without magnification

Multi scale experiments! Further large scale objects: the monochromator crystals

For their tests we like to use: wide beams, if possible at least 10 x 45 mm2 (V x H), with a “good” spatial resolution ~ 1 μm

The above field of view and resolution needs sensors with: 10,000 x 45,000 pixels, this is 450 Mega pixels

Not yet on the market (?) 2. Monochromators but also by: filters + 100 source spectrum + scintillator screen E = 8 keV response spectrum 10-1 (Pau l Ta fforeau/ID19)

forbidden area 10-2

R = 1

-3 rs) rs) reflectivity reflectivity oooo dddd 10 (Mirr(Mirr High-Z

Integrate Integrate Ge -4 111 ML's ML's 10 ML's ML's Low-Z dddd Si nnnn 111 ML's * C 111

-5 Crystals 10 -resolutio-resolutio pth-grade pth-grade eeee hhhh Be dd 110 highig

10-6 10-6 10-5 10-4 10-3 10-2 10-1 100  High resolution E/E 10-8 and below Ch. Morawe Bragg diffracting X-rayyp optical elements like monochromators, analysers, etc.

MftdManufactured from diltidislocation free crys tltals. Mostly used: Si, Ge, C* (locally dislocation free diamond).

We mainly use silicon.

They must be tailored into monochromators etc. Orientation, cutting, lapping, polishing and etching. Strain free crystal preparation. Accurate and stable mounting. Adequate cooling scheme. Crystal laboratory:

Manufacturing of nearly all perfect Si (and few Ge) crystal monochromators & analysers, etc for all ESRF beam lines, CRG blibeamlines and externa l laboratories. Silicon pieces are made from float zone silicon ingots with 100 mm diameter (Wacker). More than 1.5 tons of silicon single crystal material has been processed in more than twenty years.

Example manufacturing crystal monochromators and analysers Which types of monochromators are used? Single crystal monochromators - beam splitter monochromators

white beam white beam white beam

monochr. beam 2 monochr. beam 1

Double crystal monochromators

monochr. beam Reflection (Bragg) and white beam tra nsmissio n (Laue ) geometry used Single crystal monochromators - beam splitter monochromators

white beam white beam white beam

monochr. beam 2 monochr. beam 1

Double crystal monochromators

Reflection (Bragg) and white beam monochr. beam tra nsmissio n (Laue ) geometry used Few theory and definitions

Reflectivity (and transmissivity) curve of a crystal

plane R Theory monochromatic Plane & monochromatic wave Darwin incoming wave, width varying the angle of incidence counting the diffracted photons  RRTA1+T+A=1 Real situation - experiment Rocking curve Convolution (autocorrelation) of reflectivity (or transmissivity) curve any wave with other reflectivity curves, or/and wavelengg(th (energy)  distribution, or/and divergence distribution (instruments/apparatus f unction) Numerical example (using XOP): dbldouble monochromator non-dispers ive (+,-) double crystal set-up in Bragg case two 111-Si reflections

Darwin curve two Si 111 crystals Si 111,Si 111,60keV, 60keV, 5cm 10mm thick thick plateplate 1.0 1,0

g 

080.8 gg wh  080,8  0.6 wh 0,6 position reflectivity reflectivity 0.4 0,4 matical Bra ee

0.2 kin 0,2 0 0.0 0,0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 angle  -  (arc seconds) angle - (arc seconds) B  B

known since: C. G. Darwin, However, in operation The theory of X-ray reflection, – mostly fixed angle Phil. Mag. 27, 315, 675 (1914) Reflectivity curves, with stronggper absorption

Prins-Darwin or reflectivity curve Si 111, 8keV, 5cm thick plate 1,0

080,8  wh  

y 0,6 ivit tt

reflec 0,4

0,2

0,0 -5 0 5 10 15 20 angle - (arc seconds)  B

How to measure it? Not so easy! See later. Double crystal monochromators

Incident and exit beams have the same direction

R() In working position: R

fixed angle (mostly); 2 R multiplication of two reflectivity curves  Two vari ants

Fixed exit monochromator - more than one movement necessary Two vari ants

Fixed exit monochromator - more than one movement necessary Two vari ants

Fixed exit monochromator - more than one movement necessary

Channel-cut monochromator - NOT fixed exit - naturally aligned - weak link plus piezo movement for detuning etc. Two vari ants

Fixed exit monochromator - more than one movement necessary

Channel-cut monochromator - NOT fixed exit - naturally aligned - weak link plus piezo movement for detuning etc. Two vari ants

Fixed exit monochromator - more than one movement necessary

Channel-cut monochromator - NOT fixed exit - naturally aligned - weak link plus piezo movement for detuning etc. “Generic” cryogenically cooled channel-cut double crystal monochromator no fixed exit high heat-load applications Pusher mechanism

Steel flexure

Channel-cut crystal

Crystal assembly (simplified for visibility) Possible problem – “higher harmonics”

if (h,k,l)=(nh’,nk’,nl’) 2dnh’,nk’,nl’ sinB = n

Si (2n,2n,0),  = 1.6Å,  = 0.8Å, etc.

Refraction correction (middle of the reflection domain): r 2F    0 0  h  0  1    0  sin  B , h  sin  B  2Vuc sin 2B   0  Re duc tion of hig her harm oni cs (if no mirrors may be us ed) monochromator detuning

At dddetuned position (slightly misaligned)

SllSmaller bdband in angle and energy Synchrotron optics: Multilayer high flux monochromators • Two bounce optics • 100x larger bandwidth compared with Si(111) • Harmonics suppression due to refraction and filling factor

Ch. Morawe – SPring-8 11.09.07 How mayyg I change the beam dimension, the beam divergence, the energy resolltiution? 3. Some ppproperties of asy mmetrical reflections

Up to now we looked at symmetrical cases of Bragg diffraction

  symmetrical Bragg case symmetrical Laue case (reflection case) (transmission case)

– angle between lattice planes and surface With asymmetric reflections - ppgossibilities to change the beam width and the divergence for a single crystal reflection

w in h  K L 0 in  K h out wh Lout

 Of course, the same works also other way around.

So we have possibilities to act on the beam dimension (expansion, compression), as well as on the beam divergence (ll(smaller, larger) w out h  L Kh out  K0 in wh Lin

 But all is related and things have their price w in h  K L 0 in  Asymmetry factor: K h out wh Lout sin(    )  b  B  0 si(in(  B   )  h 

This example: >0, |b|<1: w out =|b|w= |b| w in in out h h Lin > Lout and wh < wh Relation to symmetrical in out wh Lin = wh Lout = constant reflections in -1/2 sym right asymmetry (<0) – less wh = |b| wh out 1/2 sym diverggpence possible to obtain wh = |||b| wh Some “pppyhilosophy” With flat crystals we may change the divergence. We mayy( decrease (or increase) it. Not “more p p,arallel”, but less divergent! This is collimation*). Focussing needs convergent beams. We can not focus wihith flat crystal s in a cl assi cal way. It work s wi ihth bent crystals. Often other X-ray optical elements are more efficient for this (future lectures?!).

*) The word " collimate" comes from the Latin verb collimare, which originated in a misreading of collineare, "to direct in a straight line". (Wikipedia) BUT …

in out wh Lin = wh Lout = constant

This is too simple, hand-waving “derivation”. We need an additional dimension for the phase space. Besides size and angle also wavelength (or energy) is needdded

LtLet us st art from the basi cs  0 = B +  -   in K 0   h h = B -  + out t  n

 B From electrodynamics (and B dynamical diffraction theory)   K K we know that: 0 h h 0

For the wave vectors outside the crystal:    K h  K 0  h

But for the tangential components - continuity:    Kht  K0t  ht

And remember,,pg wave vectors depend on wavelength: K  f() For small in, out and K we obtain (for Bragg and Laue case!):

sinB   cos(B  )  cos(B  ) K | out |  in  sinB   sinB   K

The divergence in and polychromaticity K/K of the incoming beam contribute to the divergence out of the outgoing beam

An increased divergence out of the outgoing beam (with respect to that if the iiincoming b)beam) means: the source is virtually closer, or the source size is virtually large r, or the angular source siz e is virtuall y larger.

Special case 1 Monochromatic, divergent incoming radiation

K = 0, in  0 out = |b| in out in or wh = |b| wh Special case 2 PlPolych romati c, paralllllel iiincoming radiditiation

K  0, in = 0 cos(B  )  cos(B  ) K out  sin(B  ) K

Remember, “our” beams often are rather close to plane waves, but rather polychromatic

2.1. out = 0 if cos(B - ) = cos(B + ) if  (sym. Bragg case!) Only in the symmetric Bragg case the beam divergence is conserved for a polychromatic beam!!! Only in that case coherence is conserved!!! Only in that case focussing is not perturbed! Only in that case highest geometrical resolution possible!

2.2 out  0 for all other cases A divergent, polychromatic beam is transformed in a even more divergent, po lhlychromatic b!beam! Source size and angular source size are crucial parameters with respect to the character of the wave “seen” by the sample. angular source size  ( = s/p) s p δ not source divergence!!

The angular source size is important for further physical properties: the geometrical resolution for imaging, the transversal coherence length, the demagnification limit of a “lens”. Image blurring due to non-zero source size angular source size: δ = s/p

s δ  p q

geometrical resolution: ρ = q s/p = q δ

s δ  p q

Spatial coherence

Transversal coherence length: lT = ½ λ p/s = ½ λ/δ Magnification, demagnification, focussing properties/quality

Geomet ri cal demagn ifica tion, Diffrac tion li m ite d focus ing: source size limit: ρDL = 1.22 λ / sinα

ρG = q s/p = q δ

s   ρ

p q (graph: J. Susini) 4. Shortly about high energy resolution

One way: Sophisticated many crystal set-ups.

Example ID18: A two-step collimation with low-index asymmetric reflections followed by a two-step angular analysis with high-index asymmetric reflections.

Idea: M.Yabashi, K.Tamasaku, S.Kikuta, and T.Ishikawa, Review of Scientific Instruments, 72, 4080 (2001). Alexander Chumakov ID18 ESRF ID18/ESRF High-resolution optics for Nuclear Resonance Scattering “0.5 meV” monochromator (∆E/E ≈ 3.5·10-8) (theoretically expected performance)

acceptance: 130 nano-rad divergence: 65 nano-rad

31 mm 3 2

vertical size: 0.6 mm vertical size: 0.6 mm divergence: 2 rad divergence: 2 rad

4 1

1st crystal: 2nd crystal: 3rd crystal: 4th crystal: Si(400) Si(400) Si(12 2 2) Si(12 2 2)

B = 18.469º B = 18.469º B = 77.533º B = 77.533º asymmet|b|018try: |b| = 0.18 asymmet|b|018try: |b| = 0.18 asymmet|b|98try: |b| = 9.8 asymmet|b|32try: |b| = 3.2

in = 5.4º in = 5.4º in = 27.6º in = 35.4º out = 31.6º out = 31.6º out = 2.7º out = 10.4º angular acceptances: angular acceptances: angular acceptances: angular acceptances:

in = 20 rad in = 20 rad in = 0.72 rad in = 1.3 rad out = 3.6 rad out = 3.6 rad out =7 rad out = 4 rad footprint: 6.4 mm footprint: 33 mm footprint: 41 mm footprint: 3.3 mm Other possibility – back scattering geometry (Bragg angle close to 90 deg) with high order reflections (large h k l)

 = 2dhkl sinB ∆ = ∆  · 2dhkl cosB

∆= ∆E/E = ∆/tanB

a)  → a f10few 10-8 radb)d b) → d[t(deg [cotg() → 0]

Silicon @@, 20KeV, h=k=l=13, ,( (E/E) ~10-8E ≈ 0.5 meV, ID16, ID28 analyser Sample

Detector source monochromator 5. How higgyh crystal q qyuality and how to measure it?

Crystal quality - limit in high energy resolution, …(?)

IflInfluence on coherence preservati on, image qua lity, focussing efficiency, ...

∆d/d= 5x10-9 Crystal interferometers Diamonds at the ESRF From the very beginning of the ESRF used as phase plltates and monochthromators.

Now locally dislocation- and stacking fault free material available.

Which is the level of local residual strains? Future MX BL ID30A (MASSIVE)

~100 μm2 ~15 μm2

56 X-ray diffraction topography White beam topograph in transmission (work with Fabio Masiello) 110-oriented plate supplier: Paul Balog/Element Six

Dislocation free areas of 11. 3 mm 6x4mm2 and more!!!

Locally crystal quality close to that of silicon, quantitatively confirmed by double crystal topography

But are we able to measure weak strains quantitatively? QQyyuantitative analysis strain analysis 110-oriented crystal plate effective misorientation map basing on one topograph

20keV, Si [880] C* [660], detection limit: δθ > 8·10-9

The effective misorientation is of the order of 4 × 10−8 for a region of interest of -8 050.5 × 050.5 mm2 and × 10 1 × 10−7 in a region of 1 × 1 mm2

Sample is slightly bent due to the non-homogeneous dislocation distribution! work with Fabio Masiello 6. Plane or divergent, monochromatic or polhlychromati c waves Twwquo basic questions: What are “plane” or “divergent” waves? What are “hi”“monochromatic”, “lh“polychromati i”c”, “hi”“white” beams/waves? For our monochromator and/or single crystalline sample! Reminder of basic physics: Plane wave – infinite extend, wave front is plane, one wave vector perpendicular to it, delta-function in k-space. Monochromatic wave – wave train of infinite length, infinitely sharp spectral line, delta-function in ω-space. They do not exist in nature!!! The full width at half maximum of the reflectivityy, curve, good reference for our sample, monochromator, ... , to define the character of a wave

FWHM in the angular space: R()

 wh  2 P h h  h wh  sin2B  0 

FWHM in the wavelength space: R()

 wh ww    cot  hh B  Example of a typical crystal/monochromator reflection: silicon, 111 reflection, Bragg case, thick crystal

  energy wh wh / 8 keV 7.6 arcsec 1.5·10-4 20 keV 292.9 arcsec 151.5· 10-4

Those are to be compared with source properties: wave length spread of the source  ()

angular source size  ( = s/p) s p δ not source divergence!! relative wave length spread of the source (E/E)

 << FWHM in the wavelength space wh

“h“monochromatic” wave

angular source size  = s/p

 << FWHM in the angular space wh

“plane” wave Practical examples: relative wave length or energy spread of the source 

 source  energy w / h

-4 -4 Si 111 double mono 2·10 8 keV 1.5·10

-4 -4 laboratory (e.g. CuK1) 3·10 20 keV 1.5·10

white beam 1···10

 SR–white beams:  >>> wh really polychromatic

”monochromatic” beam (SR or laboratory):   wh often not monochromatic (for all reflections narrower than the 1.5·10-4 for silicon 111)

sppyppyecial effort is necessary to approximate “monochromaticity” Practical examples: angular source size  ( = s/p)

source source size ssource dist. D δ

-3  class. lab tube 400 µm 0.4 m  1·10  3.5 arcmin energy wh

µ-focus tbtube 5 µm 1 m  1·10-5  1 arcsec 8 keV 767.6 arcsec

SR 100 µm 150 m  6.7·10-7  0.14 arcsec 20 keV 2.9 arcsec

SR 50 µm 75 m  6.7·10-7  0.14 arcsec

 laboratory:  >< wh divergent & quasi plane waves possible

 SR - EEFSRF:  < wh qqpuasi plane wave (mostl y)

Source divergggences > angular source sices For yyyour curiosity: First published (direct) measurements of reflect iv ity curves, (Not rocking curves! Those were measured earlier), the ones that we are able to calculate since Darwin’s first dynamical theory published in 1914, was in??? 1962

nearly 50 years after Darwin’s results!!!

C. G. Darwin, The theory of X-ray reflection, Phil. Mag. 27, 315, 675 (1914) Charles Galton Darwin (1887-1962) grandson of Charles Darwin How can we measure a reflectivity curve???

Not trivial. One needs two ingredients.

1. Narrow instruments function in anggpular space – asymmetrical ref lections in wavelength space – high-resolution mono.

2. Crystals of good quality. They became available with development of electronics, later micro-electronics and opto-electronics Double (or triple) crystal monochromator-collimator with one asymmetrical Bragg reflection

 1st experimental reflection curves

=-4.48

=0

=4.48

111-reflections of Ge, different asymmetries

CuK1-radiation, triple crystal set-up, silicon double monochromator R. Bubáková,,Jy Czech. J. Phys. B12,,( 776 (1962) By the way – the prob abl y fir st publish e d measurements in a double crystal set-up with an asymmetric monochromator/collimator crystal were about X-ray diffraction topography in 1952.

W. L. Bond J. Andrews, Structural Imppyerfections in Quartz Crystals, Am. Mineral. 37, 622-632 (1952)

This technique was later developed further to the “Plane” Wave Topography, to dtdetec t extreme ly small stitrain fildfields in nearly perfect crystals. Thank you for your atten tion !