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Presentation · April 2015 DOI: 10.13140/RG.2.2.16280.19207
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The user has requested enhancement of the downloaded file. Introduction into the dynamical theory of X-ray diffraction for perfect crystals
Jürgen Härtwig
European Synchrotron Radiation Facility, BP 220, 38043 Grenoble Cedex, France
1 Outline
1. Introduction 2. Some results of the kinematical (geometrical) theory of X-ray diffraction 3. Dynamical theory of X-ray diffraction Short theoretical background Basic results and helpful tools (dispersion surfaces) The one “beam” case – refraction and reflection The two “beam” case 4. Some effects of dynamical X-ray diffraction Pendellösung length (Laue case), Pendellösung effect Anomalous transmission – Borrmann effect Bragg case and a bit X-ray optics Plane waves, monochromatic waves
2 1. Introduction
What is/means “dynamical” diffraction theory? Why do we need it? Which are the differences to other, simpler theories? May I quantify if a simple theory is sufficient or not? Where it is applied?
3 Dynamical Diffraction – Applications today
X-ray topography strain and defects in (single) crystals, e.g., dislocations, precipitates Crystal growth electronic & micro- electronic developement Bragg-diffracting optical elements for synchrotron radiation monochromators, phase plates High resolution x-ray diffraction single crystals, epitaxial films, superlattices ... Grazing incidence methods (reflection, GID, GISAXS) thin films & interfaces X-ray standing wave method (XRSW) secondary effects (photoelectrons, x-ray fluorescence etc), excited by a standing wave in a diffracting crystal Three-beam diffraction determination of phases of the structure factors Dynamical diffraction of light (λ ~ 1.5 mm) in a two-dimensional or three dimensional array of holes Optical photonic crystals
4 Some history
The discovery of X-ray diffraction and first theories
5 The story of X-ray diffraction and of the dynamical theory (-ies) of X-ray diffraction started with Paul Peter Ewald (1888-1985) and Max Laue (1879-1960)
6 It started in 1910 with … … Paul Peter Ewalds thesis project
Ewald asks Arnold Sommerfeld for a topic for a dissertation
Proposal: to find out whether a lattice-like anisotropic arrangement of isotropic resonators might be capable of exhibiting light-optical birefringence (double refraction)
Result: theory that relates macroscopic properties of dispersion and refraction in a crystal to the interaction of propagating waves with a microscopic distribution of resonators
This was already a dynamical theory, only that is was formulated for visible light
7 January 1912 Ewald wanted to discuss his results with Max Laue
During the presentation of his subject he mentioned:
“ … 3-dimensional lattice …” ??!!!
Laue
8 Laue: What are the distances between the lattice points ? Ewald: Maybe about one thousandth of the wavelength of light!
Max Laue got the crucial idea: If crystals were indeed constructed like 3-D-lattices (with distances mentioned by Ewald), and if X-rays had the properties of waves (with distances estimated by Arnold Sommerfeld, ~0.6nm) then X-rays should be diffracted when passing a crystal!!!
9 around Easter 1912: Experiment by W. Friedrich, P. Knipping & M. Laue
Friedrich’s and Knipping’s set-up to check Laue’s idea
M. Laue, W. Friedrich, P. Knipping Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften, June 1912, M. Laue, W. Friedrich, P. Knipping, Annalen der Physik (Leipzig), 41 971 (1913)
10 One of the first exposures, The famous exposure “Fig. 5”, taken with a CuS crystal adjusted ZnS crystal. plate (Friedrich et al. 1912) (Deutsches Museum, München)
11 1. X-rays are waves! 2. Crystals have a discrete, 3-D-periodic (lattice) structure!
Nobel prize 1914 for Max von Laue "For his discovery of the diffraction of X-rays by crystals"
“Laue technique”
Crystal orientation
(SR)
Structure analysis X-ray topography (oscillation method) (white beam topography)
12 Nowadays: Laue image of an AlPdMn icosahedral quasicrystal showing five-fold symmetry
“Laue spots” (position, intensity) information about crystal symmetry In general white beam used, sample position fixed Integrated intensity recorded
13 ID19 ~ 40 cm 18 cm
Typical Laue pattern from a small silicon sample. Δω=0° and λ=“infinite” (white beam).
Typical “Laue pattern” from a tetragonal HEW lysozyme. Higher harmonics “contamination” Δω=1° and λ=0.8Å. for white beam λ/n = d(hkl)/n · sin θ(hkl)
14 First X-ray diffraction theories promptly followed the discovery of X-ray diffraction:
1912/13 Laue’s geometrical (or kinematical) theory, Laue equations 1913 Bragg’s law 1914 1916/17… Diffraction by a 3D-lattice (crystal)
a (sina - sina0) = h l
b (sinb - sinb0) = k l
c (sing - sing0) = l l Laue equations
h, k, l – “Miller” (Laue) indices
15 William Lawrence Bragg (son, left) and Sir William Henry Bragg (father, right) (Courtesy Edgar Fahs Smith Memorial Collection, Department of Special Collections, University of Pennsylvania Library.)
Nobel Prize in Physics 1915: "For their services in the analysis of crystal structure by means of X-rays"
16 W H Bragg, W L Bragg, Proc Roy Soc A88, 428 (1913) Bragg’s law - 1913 W L Bragg - 22 years old student! Equivalent to Laue equation real space reciprocal space incident wave, l diffracted wave
q q Khkl d h d· sinq K 0
l = 2dhklsinqhkl Kh = K0 + h 0 scalar form vectorial form of Bragg’s law of Bragg’s law Ewald’s construction
17 Determination of the first crystal structures
NaCl (rock-salt) fcc, KCl (Sylvine) pc, ZnS (Zincblende) fcc, CaF2,
(Fluorspar) fcc, CaCO3 (Calcite) rhombohedral W H Bragg, W L Bragg, Proc Roy Soc A89, 248 (1913), submitted 21 June, accepted 26 June !!!
Diamond W H Bragg, W L Bragg, Nature 91, 557 (1913), submitted 28 July, publishing date 31 July Proc Roy Soc A89, 277 (1913), submitted 30 July!!!
18 First X-ray diffraction theories promptly followed the discovery of X-ray diffraction:
1912/13 Laue’s geometrical (or kinematical) theory 1913 Bragg’s law
1914 Darwin’s geometrical and dynamical theories C. G. Darwin, The theory of X-ray reflection, Phil. Mag. 27, 315, 675 (1914)
1916/17 Ewald’s extension of his theory to X-rays (fully dynamical theory) P. P. Ewald, Ann. Phys. (Leipzig) 49, 1, 117 (1916); 54, 519 (1917) (English translations exist!)
all for perfect crystals!
19 2. Kinematical (geometrical) theory of X-ray diffraction
Amplitude of the diffracted wave derived by: - adding the amplitude of the waves diffracted by each scatterer - by simply taking into account the optical path differences ( “geometrical”) - neglecting the interaction of the propagating waves with matter (only one scattering process, no absorption, no refraction, energy conservation law violated!!!)
20 Repetition of some results to better see the differences to the dynamical theory
Scalar waves, single scattering P h- scattering vector, k h k 0 h (not necessary a crystal)
h = 2 sinB/l, k0 = kh Amplitude in point P r r r
r h k h r source k 0 0
21 Optical path difference: r r kh r k 0 r 1 (kh k 0 )r l hr k 0 k 0 k 0 k 0 r r
k 0
r k 0 h k h k r h rsource 0 k h
22 Used approximations: Fraunhofer approximation: rsource r and r r elastic (coherent) scattering: k 0 k h
Amplitude in point P:
3 (r) = 0(r) + 1(r) = 0(r) + G(r|r’) V(r’) 0(r’) d r’ exp(2ik 0 r r ) Exact solution with: with: Gr | r 4r r (r’) instead of 0(r’) Vr 4reNr - for photon scattering on electrons
(re– classical electron radius, N(r) – electron density) 2 Vr VFermi r b(2 /m)r - for thermal neutron scattering on the nuclear potential (b – scattering length) 2m Vr U r - for electron scattering on the Coulomb potential 2 C
23 Example – X-rays 0(r) – plane wave electrical field
exp(2ik r) Scattered wave: (r) E r 0 F(h) 1 h r
3 Scattering amplitude: F(h) reE0P N(r)exp(2ihr)d r
1 for s-polarization P = cos(2B) for -polarization polarization factor for amplitudes
Rayleigh scattering Special cases: 1. Scattering on 1 electron: N(r) (r)
2 2 r 1 IT (r) ET e C I C [1 cos2 (2 )] h h r 2 0 2 B polarization factor for intensities
Thomson scattering 2. Scattering on 1 atom: N(r) Natom (r) Eatom (h) f – atom form factor h N(r)exp(2ihr)d3r f(h) T (amplitude) Eh
Iatom IT f 2 Remark: Fourier h h transform of N(r)!!!
3. Scattering on a crystal: N(r) Ncrystal (r)
Icrystal IT G 2 F 2 Like in classical optics – Fraunhofer h h h diffraction by a grating
G - lattice amplitude Fh - structure amplitude (factor) (factor) Fourier transform of N(r)! 3 Fh (h) N(r)exp(2ihr)d r unitcell Structure analysis Kinematical scattering on a crystal:
Icrystal IT G 2 F 2 Like in classical optics – Fraunhofer h h h diffraction by a grating
G - lattice amplitude Fh - structure amplitude (factor) (factor)
Icrystal ~ F 2 Such dependence only in h h kinematical approximation
Fourier transform of N(r)! 3 Fh (h) N(r)exp(2ihr)d r unitcell Structure analysis
27 Valid only under the following conditions:
1. Only one scattering process (no multiple scattering)
1(r) 1 << 0 k0 2 negligible 0(r) 1st Born approximation 2(r)
2. Plane wave approximation (Fraunhofer approximation): rsource r and r r 3. Elastic scattering, no absorption k 0 k h “geometrical” theory = kinematical scattering theory theory in 1st Born approximation Fraunhofer approximation
28 Only with these approximations:
crystal 2 Ih ~ |Fh| and convenient use of:
Fh FT N(r’)
29 When these approximations could be valid?
Comparison with Thomson scattering: Intensity of the wave scattered by 1 electron 1 C [1 cos2 (2 )] 2 B r2 IT (r) e C I polarization factor for h r2 0 intensities re – classical electron radius
-15 Estimation: re 3·10 m T 2 -21 1 electron, for r = 0.1mm Ih /I0 (re/r) 10 cubic crystal: a = 0.5nm, Z = 30 in (1mm)3 ~ 3·1011 electrons single scattering OK in (1mm)3 ~ 3·1020 electrons single scattering?!?!?
30 Some questions: - Multiple scattering negligible for “small” crystals, but how small? What is small? - How to describe “large”, perfect (or even imperfect) crystals? (e.g. X-ray optics!!!) - What happens close to or within the crystals? (no Fraunhofer approximation)
Other theory needed: dynamical theory of diffraction
31 3. Dynamical theory of X-ray diffraction
First dynamical X-ray diffraction theories promptly followed the discovery of X-ray diffraction:
32 Basic idea of dynamical diffraction
dynamical theory means - theory including multiple scattering
1st Born approximation n-th Born approximation with n
In principle we have to change from: 3 (r) = 0(r) + G(r|r’) V(r’) 0(r’) d r’ to: 3 (r) = 0(r) + G(r|r’) V(r’) (r’) d r’ 1914 Darwin’s dynamical theory C. G. Darwin, The theory of X-ray reflection, Phil. Mag. 27, 315, 675 (1914)
C. G. Darwin (1887-1962) grandson of C. Darwin
1916/17 Ewald’s extension of his theory to X-rays (fully dynamical theory) P. P. Ewald, Ann. Phys. (Leipzig) 49, 1, 117 (1916); 54, 519 (1917)
Nowadays mostly used: 1931 Laue’s dynamical theory, solution of Maxwell equations in a periodic medium (crystal) in the form of Bloch-waves (wave fields) M. Von Laue, Ergeb. Exakt. Naturwiss. 10, 133 (1931)
34 Darwin’s approach
Diffracted wave is calculated as a superposition of plane waves atomic planes reflected from and transmitted through individual atomic planes, multiple reflection is included.
C. G. Darwin, Phil. Mag. 27, 315 (1914); 27, 675
t0 = 1-i q0 rh = -i qh
qo, qh - transmission and reflection coefficient t0, rh - transmissivity and reflectivity Summing up all contributions Improved version in use today for layered systems
35 The surprising result Bragg case or reflection case Darwin curve Si 111, 60keV, 10mm thick plate 1.0 Region of “interference total reflection” in a 0.8 finite angular range
0.6 Full width at half q wh maximum (FWHM)
reflectivity 0.4 Shift from the kinematical Bragg position due to 0.2
kinematical Braggkinematicalposition refraction q0 0.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 angle - (arc seconds) q qB
HERCULES Grenoble, March 2009 36 Si 111, 60keV, 500mm thick plate
1,0
0,8
Effects of absorption 0,6
and crystal thickness reflectivity 0,4
0,2
0,0 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 angle q - q (arc seconds) Prins-Darwin or reflectivity curve B Si 111, 8keV, 5cm thick plate 1,0
Si 111, 60keV, 500mm thick plate 0,8 1,0
0,6 0,8
0,6
reflectivity 0,4
0,4
transissivity 0,2 0,2
0,0 -5 0 5 10 15 20 0,0 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 angle q - q (arc seconds) angle q - q (arc seconds) B B
Bragg case or reflection case Laue case or transmission case
37 By the way … Those calculations were done with the program (program package) XOP http://www.esrf.eu/computing/scientific/xop2.1/documentation.html
38 Few more typical results (and differences) of kinematical and dynamical theory
Full widths at half maximum of reflectivity curves
Kinematical theory: symB symmetrical Bragg case: wh (dhkl/t)tanqB (=Scherrer formula) symL symmetrical Laue case: wh dhkl/t f(l) t – crystal thickness Dynamical theory: symB symmetrical Bragg case: wh (dhkl/0)tanqB symL symmetrical Laue case: wh dhkl/0 new parameter: 0 - Pendellösung length symB symm.Bragg case: 0 = VUC / (2 r0 |P| Fhkl dhkl) f(l) symL symm. Laue case: 0 = VUC / (2 r0 |P| Fhkl dhkl tanqB)
39 The Pendellösung length 0 allows to define what are “small”/thin or “large”/thick crystals:
with A = t / 0
A<<1 “thin” crystal kinematical and dynamical theory are valid
A>>1 “thick” crystal only dynamical theory applies
40 Area under reflectivity curve – integrated reflectivity
q Ri Rh (q)dq
No absorption, thick crystal (dynamical theory):
q q q Laue case: RiL ~ | Fh | Bragg case: RiB 2RiL
No absorption, thin crystal (kinematical limit):
q 2 Both cases: Ri,thin ~ V | Fh |
41 Integrated reflectivity
7 kinematical q Ri 6 1 5 dynamical (Bragg case)
4
1/23
2 dynamical (Laue case)
1
0 200 400 600 800 1000 1 t/2 3 t / 0
42 In the Bragg case (thick crystal) the form and width of the reflectivity curve (as well as the integrated reflectivity) depend anymore on the crystal thickness! K 0 Kh
damping with depth - extinction extinction length (for q = qB + q0, centre of the reflection curve)
tex = 0 / 2 ~ micrometers
Ih(z=tex) = I0/e
43 Which was the first direct experimental verification of the dynamical theory???
Darwin’s well-known top-hat shaped reflectivity curve??? Or the asymmetric (absorption!) Prins-Darwin one???
44 First published measurements of reflectivity curves: (not rocking curves!)
1962 nearly 50 years after Darwin’s results!!!
45 1st experimental reflection curves
a=-4.48
a=0
a=4.48
111-reflections of Ge, different asymmetries
CuKa1-radiation, triple crystal set-up R. Bubáková, Czech. J. Phys. B12, 776 (1962)
46 Why so late?
Not many very good (natural) crystals available starting from the 50ies: intensive efforts to grow high quality semiconductor crystals for the electronic industry (later also for optics and opto-electronics)
No plane, monochromatic waves available – X-ray optics also solved with high quality semiconductor crystals
47 First direct experimental evidence of a dynamical theory related effect anomalous transmission of X-rays through a thick crystal “Borrmann effect”
discovered in 1941 G. Borrmann, Physikalische Zeitschrift 9/10, 157-162 (1941)
theoretical interpretation: M. von Laue 1949 with the existing dynamical theory
48 Short theoretical background
P.P. Ewald Many excellent theoreticions participated in the C.G. Darwin development and completion of the dynamical theory of X-rays diffraction M. von Laue W.H. Zachariasen Variants for: P. James “conventional” and “extended” theory A. Authier B. W. Batterman perfect crystals; plane wave, spherical wave approaches A. R. Lang G. Borrmann deformed crystals; geometric and wave optical (even QFT) approaches U. Bonse P. Penning & D. Polder Layered systems N. Kato Two- and many beam cases S. Takagi D. Taupin M. Kuriyama ... 49 As already mentioned, in principle we have to solve:
3 (r) = 0(r) + G(r|r’) V(r’) (r’) d r’ solution of the inhomogeneity of homogeneous equ. the equation special solution of the inh. equ.
But this is nothing else as the integral representation of a solution of an equation of the type: inhomogeneous wave equation
50 We shall not start with the integral representation, but from the differential equation itself and use the special property of our samples:
Periodicity
To describe the propagation of the waves outside and inside the crystal we use:
Schrödinger equation Maxwell equations Time free Schrödinger and monochromatic wave equations: 2m 2mE 2m (r) [E U(r)] (r) 0 or (r) (r) U(r) (r) E 2 E E 2 E 2 E and rot rotE(r,) 42K2[1 e (r,)]E(r,) 0 or rot rotE(r,) 42K2E(r,) 42K2e (r,)E(r,) with K=1/l, l-vacuum wave length using rot rot = grad div - and divE = 0 we obtain one type of equation for both cases: (r) a(r) V(r)(r)
The same type of equation for X-rays, electrons, neutrons, ions, atoms, etc.! e For a crystal V(r), VFermi(r), N(r), (r), UC(r) are periodic in 3-D!!!
We look for solutions in the form of Bloch waves.
for the electrical field vector: E(r,) exp(2ik r)E (r,) 0 k0 E (r,) where k0 is triply periodic
k0 – wave number in the crystal
This allows to use Fourier expansions for the solution of the differential equation
53 e e For the electrical susceptibility: (r,) h ()exp(2ihr) h
For the triply periodic amplitude E (r,) E exp(2ihr) k0 h of the electrical field vector: h we obtain: E(r,) exp(2ik 0r)Eh exp(2ihr) Eh exp(2ikhr) h h Here kh k 0 h holds for the wave vectors within the crystal This does not imply that Kh K0 h holds for the vacuum!!!!
Two equivalent images: 1. “plane” wave with a modulated (3-D) amplitude (Bloch w.) 2. Wave field consisting of h plane waves with different
wave vectors kh
54 Fourier component of the electrical susceptibility is directly related to the structure factor
e e (r,) h ()exp(2ihr) h
2 e r0l h Fh; VUC
with r0 – classical electron radius
and VUC – unit cell volume
55 Schematic way of solution inhomogeneous differential equation put Fourier expansions of all 3-D-periodic functions into it
basic equation of dynamical theory homogeneous system of algebraic equations of the order h
condition for non-trivial solutions
dispersion relation
all possible solutions for the infinite crystal boundary conditions for wave vectors and amplitudes
realised solutions for the finite crystal
56 Basic results and helpful tools (dispersion surfaces) The one “beam” case – refraction and reflection One reciprocal lattice point close to the Ewald sphere
“Classical” results contained in the theory Ewald’s construction This is kinematical theory sphere radius K vacuum wave number! K 0
0
modified Ewald’s construction
57 But the crystal is matter with nmatter nvacuum
The theory provides Basic equation: 2 2 2 e (K – k0 ) E0 + K 0 E0 = 0 Dispersion equation: K 2 e 2 2 2 0 K K (1+0 ) – k0 = k - k0 = 0
mean wave number 0 in the crystal 1 k k nK K(1 e ) K 2 0 complex value - absorption 15keV, Si dispersion surface n = 1 – 2.2 10-6 –i 1.5 10-8 for the one beam case
58 Let’s introduce a boundary reciprocal space direct space
K 0 k 0 K n vacuum k k t nmatter k r K m
n
- boundary condition for the wave vectors – Continuity of the tangential components of the wave vectors
59 Flat surface/interface – translational symmetry – Continuity/conservation of the tangential components of the wave vectors
Mechanical analogy: A rolling sphere and a surface-step along the x-direction. Translation symmetry along the y-direction
Conservation of the momentum py
60 In the one beam case of crystal diffraction and in classical optics: 1 incident, 1 reflected, 1 refracted plane wave
reciprocal space direct space
K 0 k 0 K nvacuum k nmatter k r K m
n
61 “External” total reflection
reciprocal space direct space
K 0
K nvacuum c k k 0 nmatter n K m
Nothing really new. No difference if crystalline n n or non-crystalline matter.
range of external total reflection
HERCULES Grenoble, March 2009 62 The two “beam” case Two reciprocal lattice points close to the Ewald sphere
The theory provides Basic equations: 2 2 2 e (k – k0 ) E0 + K P -h Eh = 0 2 e 2 2 K P h E0 + (k – kh ) Eh = 0 Dispersion equation: 2 2 2 2 4 2 e e (k – k0 ) (k - kh ) = K P h -h 0
Remarks and conclusions: 1. Known problem - electron in periodic potential (e.g. Kittel) 2 2 2 2 2 2 2 2 2. It never holds that k0 = k , kh = k , k0 = K , kh = K . The endpoints of vectors k0 and kh are never ON the Ewald sphere, nor on the sphere with diameter k. 3. The dispersion relation provides “sets” of 4 solutions, but an infinite number of
them (direction of k0 is free – boundary condition necessary for selection) 4. Polarisation - two independent solution sets for P=1 and P=cos2qB
63 modified Ewald’s construction
K K h 0 K h h 0 k
The spheres with diameter k are not yet the dispersion surfaces!!!
HERCULES Grenoble, March 2009 64 Dispersion surfaces in the two “beam” case (cut), one polarization
La
Lo K 0 K h
h h 0 k 0i
n
khi vectors not drawn for simplicity
HERCULES Grenoble, March 2009 65 One wave vector K0 and the boundary condition for the wave vectors (related surface normal n) selects one “set” of 4 solutions (for one polarization state)
Dispersion equation – 4th order equation: 2 2 2 2 4 2 e e (k – k0 ) (k - kh ) = K P h -h 0
4 wave fields (consisting of two plane waves each) may be excited the related wave vectors start at the 4 “tie points” (intersection point of the dispersion surface with the surface normal)
“Extended” dynamical theory, needed in techniques/cases like:
K0, or Kh, or both with “small” angle to surface (~ Θc, strongly asymmetric cases), GID, SAXRD
“Extended” dynamical theory is the non-approximated one!
66 Gracing Incidence Diffraction
The incident angle below or close to αc (Θc)
67 Dispersion surfaces in the two “beam” case (cut), one polarization
K 0 K h This is the physically most h h 0 important part
n One wave vector K0 and the related surface normal n selects one “set” of 4 solutions (for one polarization state), but only two may be of physical importance 2 wave fields (consisting of two plane waves each) out of the 4 excited ones are considered the related wave vectors start at the 2 “tie points”, close to the Laue- and Lorentz points
In this approximation: dispersion equation – 2nd order equation: 2 2 e e (k – k0) (k - kh) = ¼K P h -h 0
“Conventional” dynamical theory (this is an approximation!!!)
69 Dispersion surfaces in the two “beam” case – physically most important part
1 Ke La 2 0
K Lo k
K 0 n K h
k k h1 k h2 k 02 01 h h 0
70 Now we have Bragg diffraction – two diffracted beams/waves outside the crystal. What happens inside?
If Bragg condition is fulfilled (maximum of the reflectivity curve): Wave length of the k k 0 h standing wave pattern h equals spacing of the reflecting lattice planes reflecting lattice planes Position of nodes/antinodes of one standing wave field At the maximum of the Bragg position
71 The dynamical effects are related to the existence of coherent wave fields in the crystal and are due to different interferences between their plane wave components
72 Some effects of dynamical X-ray diffraction Pendellösung length (Laue case), Pendellösung effect g1Kn K K h gKn K h La h 0 0 k 0i K0 giKn Lo khi k 0i h 1 e K0 2 | k01 k02 || kh1 kh2 | 1/ K n 0 Coherently excited waves interfere Kh beating frequency is
(relative lengths differences k k kh1 kh2 02 01 e -5 4 -1 |0 | K ~ 10 10 mm ~ 10mm)
HERCULES Grenoble, March 2009 73 Amplitude of the E-field in the crystal:
(C) E (r) EWF 1(r) EWF 2 (r) E01 exp(2ik 01r) Eh1 exp(2ik h1r) E02 exp(2ik 02r) Eh2 exp(2ik h2r) (C) E (r) E01 exp(2ik 01r) E02 exp(2ik 02r) Eh1 exp(2ik h1r) Eh2 exp(2ik h2r) E02 E01 exp(2ik 01r){1 exp[2i(k 02 k 01 )r)]} E01 Eh2 Eh1 exp(2ik h1r){1 exp[2i(k h2 k h1 )r)]} Eh1 | k01 k02 || kh1 kh2 | 1/ nr z - depth in the crystal E z E(C) (r) E [1 02 exp(2i )]exp(2ik r) 01 E 01 01 Two “plane” waves with E z h2 amplitudes “slowly” Eh1[1 exp(2i )]exp(2ik h1r) Eh1 depending on z E0 (z)exp(2ik 01r) Eh (z)exp(2ik h1r)
74 If Bragg condition is fulfilled (maximum of the reflectivity curve, n intersects Lorentz point) maximum value of Pendellösung length 0 La 0 2 gh / | gh | Lo
gh = sin(a- qB) - normalized angular coordinate
Modulation is maximum K 0 n for = 0
Examples for 0: 1mm Si, 220-reflection k k k k h1 h2 02 01 8keV: 22.4μm 60keV: 127μm
HERCULES Grenoble, March 2009 75 Pendellösung effect
Boundary condition for amplitudes at the entrance surface:
E0 E01 E02 , 0 Eh1 Eh2 = 0, no absorption At the surface z = 0
Amplitude of wave E0: E0(z=0) = E0
and of wave Eh: Eh(z=0) = 0
In depth z = 0/2
E0(z=0/2) = 0
Eh(z=0/2) = E0
In depth z = 0
E0(z=0) = E0
Eh(z=0) = 0 etc.
76 Planes where the diffracted 2 wave |Eh| has K 0 maxima
0
Planes where the forward diffracted wave 2 |E0| has maxima K h
2 |Eh|
x
HERCULES Grenoble, March 2009 77 Experimental example – Pendellösung fringes in X-ray diffraction topography ~ equal thickness fringes in optics
h
500mm
Double crystal X-ray diffraction topograph of a silicon sample with wedge-shaped borders. (crystal: 100 orientation, t450mm, asymmertrical 111-reflection in transmission; monochromator: symmetrical 111-reflection;
energy E=30keV; 0 46mm
78 Origin of the term “Pendellösung”
One pendulum may rest without motion and the other one moves and vive versa, with a continuous energy transition.
Two coupled pendula. The two "Eigenstates" of the coupled pendula (in-phase oscillation and anti-phase oscillation) are associated with different frequencies. A frequency gap exists.
79 Possible applications: - measurements of structure factors (thickness known) symL symm. Laue case: 0 = VUC / (2 r0 |P| Fhkl dhkl tanqB)
- thickness measurements for “inaccessible” objects (e.g. growing dendrites in a crucible)
80 Al-3.5wt%Ni A B C (d2) A 500 mm « Free » A B (d1) growth B
g black white 200 white black
C C 1 mm 1 mm 1 mm “confined” growth (d3) (d3) Al-7.0wt%Si
g200 g2-20
81 Dendrites in 3D (« free »)
3D rendering of the dendrite tip Al-7.0wt%Si
« Free » growth
200 mm [010] 200 um projection along the growth direction [010]
82 3D structure of dentrites
83 First direct experimental evidence of a dynamical theory related effect anomalous transmission of X-rays through a “thick” (strongly absorbing) crystal “Borrmann effect”
discovered in 1941 G. Borrmann, Physikalische Zeitschrift 9/10, 157-162 (1941)
theoretical interpretation: M. von Laue 1949 with the existing (his) dynamical theory
84 Anomalous transmission – Borrmann effect
Observation: In Bragg condition it is possible that X-rays pass a good quality crystal for combinations of wavelength λ and crystal thickness t, where normal absorption is so high that
I = I0 exp(-μ0t) 0
85 Anomalous transmission – Borrmann effect
Assumptions: - two beam case, coplanar diffraction
- no extreme condition (extreme asymmetry, qB 90º) - symmetrical Laue (transmission) case - s-polarisation
Two wave fields in the crystal
2ik01r 2ikh1r 2ik02r 2ikh 2r E(r) E01e Eh1e E02e Eh2e
wave field 1 wave field 2 khi k 0i h
86 Once more - Laue case in reciprocal space
La M A Symmetrical Laue 1 case: M = La Lo maximum of reflection
A2 K h K 0 n kh1 k01 k k h2 02 h h 0
All waves coherent interferences
87 Two standing wave fields are excited within the crystal first wave field
k 0 k h h
reflecting lattice planes Position of nodes/antinodes of one standing wave field At the maximum of the Bragg position
88 Two standing wave fields are excited within the crystal second wave field
k 0 k h h
reflecting lattice planes
Position of nodes/antinodes At the maximum of of one standing wave field the Bragg position
89 Intensity of one wave field in the crystal
2 (i) 2ik 0ir 2ikhi r 2 E (r) | E0ie Ehie |
2 2 E E E emz 1 hi 2 hi cos(2hr) 0i 2 E E0i 0i
For simplicity at the maximum: Ehi/E0i = 1 -1 WF1; 1 WF2
2 E(i) (r) ~ 1 cos(2hr)
hr n; n 0, 1, 2, .... equation of lattice planes
90 Intensities of the standing wave fields lattice planes
wave field 1
khi k0i
h
wave field 2
Principle of the Borrmann effect: Two standing wave patterns are produced by two times two coherent, travelling plane waves (with wave vectors k0i and khi). At the exact Bragg position one has its maxima between (the red; WF1) and one on (the blue; WF2) the lattice planes. This leads to large differences in the photoelectric absorption.
HERCULES Grenoble, March 2009 91 Effect on absorption coefficients
31.34 30
)
-1 m 25 2
20 m /g 0 0 15 15.92
10 0.500 m 5 1
absorption coefficients (mm coefficients absorption
0 -5 -4 -3 -2 -1 0 1 2 3 4 5 normalised angle numerical example: 1 mm Si plate, 220-reflection, 8 keV -7 exp(- m0t/g0) = 1.2 10 , exp[-min(-m1) t] = 0.606
92 Reflection- and transmission curves, zero absorption (mt = 0) Both wave fields contribute equally
1,0
0,8 T 0,6
0,4 R 0,2
reflectivity, transmissivity reflectivity,
0,0 -5 -4 -3 -2 -1 0 1 2 3 4 5 normalised angle
93 Reflection- and transmission curves, high absorption (mt = 14.6) only wave field 1 “survives”
0.20
0.15
0.10 T R
0.05
reflectivity, transmissivity reflectivity,
0.00 -4 -3 -2 -1 0 1 2 3 4 angular deviation (arc sec)
1 mm Si plate, 220-reflection, 8 keV
94 Similar effects for other particles, e.g. ion channeling Allows to pass through rather thick samples Strongest effect for s-polarized wave may be used as polariser Very sensitive to perfection detection of defects in bulky crystals
We used it to check the degree of perfection of icosahedral Al-Mn-Pd quasicrystals
95 Laue image of an AlPdMn icosahedral quasicrystal showing five-fold symmetry
Courtesy of J. Gastaldi
96 X-ray diffractometry - widths of Bragg peaks
24keV 250000 24 keV 225000
200000
175000 I 0I0 150000
125000
(ab. units) (ab. 100000
75000
counting counting I I 50000 h h
25000
0 mt 1.6 11.136 11.138 11.140 11.142 11.144 11.146 omega (degrees)
Reflection curves for the diffracted (blue, Ih) and
forward-diffracted (red, I0) beams, taken at 24 keV.
Measured half width of Ih : 5.2” deconvoluted FWHM: 1.4” (-0.4”, +1.6”)
97 1111keV keV 3500
3000
2500 I h Ih 2000
(ab. units) (ab. 1500 I I0 0 1000
counting counting
500
0 25.764 25.766 25.768 25.770 25.772 omega (degrees) mt 12.3
Reflection curves for the diffracted (blue, Ih) and
forward-diffracted (red, I0) beams, taken at 11 keV. The Borrmann effect is clearly visible.
J. Härtwig, S. Agliozzo, J. Baruchel, R. Colella, M. de Boissieu, J. Gastaldi, H. Klein, L. Mancini, J. Wang, Anomalous transmission of X-rays in quasicrystals, J. Phys. D: Appl. Phys. 34, A103-A108 (2001) 98 Borrmann topography
X-ray diffraction topography under strong absorption conditions (μ0t >> 1)
μ0t < 1 InP μ0t >> 1
99 Interference total reflection (Bragg case)
Darwin curve Si 111, 60keV, 10mm thick plate 1.0
0.8
0.6 q wh
reflectivity 0.4
0.2 q0 0.0 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 angle - (arc seconds) q qB
100 Bragg case – most important case for monochromators
direct space
K 0 K h
n
HERCULES Grenoble, March 2009 101 Dispersion surfaces in the two “beam” case – physically most important part
q h k h1 k h2 K h
La Lo h
K 0 k 01 n k 02
0 Reciprocal space
102 Range of interference total reflection
q h
La Lo h
(2) (1) K 0 K 0
n 1 n2 0 range of interference total reflection
103 Mathematics very simple to calculate a reflectivity curve for a semi-infinite crystal (Darwin solution)
e 2 h 2 Rh[(q)] e ( 1) h
Normalised angular coordinate or deviation parameter:
1 2 2(q q ) g (q q )sin 2q g sin(a q ) 0 0 0 B 0 B wq g e e 1 2 g sin(a q ) h h P hh h B
Full width at half maximum of the reflection curve:
e e 1 2 1 2 q 2P hh gh wh 2; wh sin 2qB g 0
Refraction correction (middle of the reflection domain): e 0 g 0 q0 1 2sin 2qB gh
104 To be complete – Pendellösung length (general form):
1 2 (g 0 gh ) 0 e e 1 2 K P(hh )
– other expression for the full width at half maximum of the reflection curve:
q 2dhkl gh wh 0 cosqB
105 Integrated reflectivity
q Ri Rh (q)dq
No absorption, thick crystal (dynamical theory):
e e 1 2 q P hh q q Laue case: RiL 1 2 ~ | Fh | Bragg case: RiB 2RiL 2b sin 2qB
asymmetry factor: b g0 gh sin(a qB ) sin(a qB )
No absorption, thin crystal (kinematical limit):
2 2 e e q P Khht 2 Both cases: Ri,thin ~ V | Fh | g 0 sin 2qB
106 Integrated reflectivity
7 kinematical q Ri 6 1 5 dynamical (Bragg case)
4
1/23
2 dynamical (Laue case)
1
0 200 400 600 800 1000 1 t/2 3 t / 0
107 Some properties of asymmetrical reflections
Up to now we looked at symmetrical cases of Bragg diffraction
a 0 a 90 symmetrical Bragg case symmetrical Laue case (reflection case) (transmission case)
a – angle between lattice planes and surface Asymmetrical cases of Bragg diffraction Δθ in K 0 Lin K h Δθout Lout
a 0 This example: >0, |b|<1: a Asymmetry factor: L > L and Δθ < Δθ in out in out sin(q a) b B sin(qB a) Δθin Lin = Δθout Lout = constant For a<0 – less divergence Δθout = |b| Δθin possible to obtain
Page 109 I Leiden l February 2015 l Jürgen Härtwig Of course, the same works also other way around. Δθ out K h Lout K 0 Δθin Lin
a 0
So we have possibilities to act on the beam dimension (expansion, compression), as well as on the beam divergence (smaller, larger)
Page 110 I Leiden l February 2015 l Jürgen Härtwig But all is related and things have their price
in wh K L 0 in Asymmetry factor: K h out wh Lout sin(q a) g b B 0 sin(qB a) gh a 0
This example: a>0, |b|<1: w out = |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 (a<0) – less wh = |b| wh out 1/2 sym divergence possible to obtain wh = |b| wh Range of interference total reflection – asymmetric reflection
q h
(1) K h
out wh (2) Kh La Lo h in wh
(2) K 0 direct space
(1) K 0 0 n1 n2
112 EXAMPLE – ASYMMETRIC BRAGG DIFFRACTIONS BUT …
Δθin Lin = Δθout Lout = constant
This is too simple, hand-waving “derivation”. We need an additional dimension for the phase space.
Besides sizes (widths Lx) and angles (Δθx) also wavelength, or energy, or wave number (ΔK) is needed
Page 113 I Leiden l February 2015 l Jürgen Härtwig EXAMPLE – ASYMMETRIC BRAGG DIFFRACTIONS
From electrodynamics (and dynamical diffraction theory) we know that:
For the wave vectors outside the crystal: Kh K0 h
But for the tangential components - continuity: Kht K0t ht
And remember, wave vectors depend on wavelength: K f(l)
Page 114 I Leiden l February 2015 l Jürgen Härtwig EXAMPLE – ASYMMETRIC BRAGG DIFFRACTIONS
Possibility to change the divergence (qin , qout) or the energy band pass (K) h t a 0 = qB + a - /2 qin n h = qB - a + /2 qout qB qB
K K h gKn K h h 0 0 K 0 h K h 0
Kh t (K0 h gKn)t K0t ht with nt 0
Page 115 I Leiden l February 2015 l Jürgen Härtwig EXAMPLE – ASYMMETRIC BRAGG DIFFRACTIONS
For small qin, qout and K we obtain (for Bragg and Laue case!):
cos(qB a) cos(qB a) K | qout | b qin sinqB a K
The divergence qin and polychromaticity K/K of the incoming beam contribute to the divergence qout of the outgoing beam
An increased divergence qout of the outgoing beam (with respect to that if the incoming beam) means: the source is virtually closer, or the source size is virtually larger, or the angular source size is virtually larger.
Special case 1 Monochromatic, divergent incoming radiation
K = 0, qin 0 qout = |b| qin
Page 116 I Leiden l February 2015 l Jürgen Härtwig EXAMPLE – ASYMMETRIC BRAGG DIFFRACTIONS Special case 2 Polychromatic, “parallel” (plane wave) incoming radiation cos(qB a) cos(qB a) K K 0, qin = 0 qout sin(qB a) K Remember, “our (SR)” beams often are rather close to plane waves, but rather polychromatic
2.1. qout = 0 if cos(qB - a) = cos(qB + a) if a 0 (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 qout 0 for all other cases A divergent, polychromatic beam is transformed in a even more divergent, polychromatic beam!
Page 117 I Leiden l February 2015 l Jürgen Härtwig 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
Geometrical demagnification, Diffraction limited focusing: source size limit: ρDL = 1.22 λ / sinα
ρG = q s/p = q δ s q a ρ
p q (graph: J. Susini) Literature books W. H. Zachariasen, Theory of X-ray diffraction in crystals New York: John Wiley 1945 Dover Publications, Inc., New York 1994 (unchanged edition 1945) M. Fatemi, NRL Report 7556 (1973) Explanatory Notes on “W. H. Zachariasen’s “Theory of X-ray Diffraction in Ideal Crystals” M. von Laue, Röntgenstrahl-Interferenzen Frankfurt am Main: Akademische Verlagsgesellschaft 1960 L. V. Azaroff, X-ray Diffraction New York: McGraw-Hill Book Company 1974 Z. G. Pinsker, Dynamical scattering of X-rays in crystals Berlin: Springer Verlag 1978 A. Authier, Dynamical theory of X-ray diffraction Oxford Univ. Press 2001, 2005 (2nd ed.) the most complete monograph U. Pietsch, V. Holy, T. Baumbach, High Resolution X-ray Scattering, Springer, Berlin 2004 (thin films, nanostructures)
121 classic works (first and most current dynamical theories) C. G. Darwin, The theory of X-ray reflection, Phil. Mag. 27, 315, 675 (1914) P. P. Ewald, Zur Begründung der Kristalloptik Teil I Dispersionstheorie Teil II Theorie der Reflexion und Brechung Teil III Die Kristalloptik der Röntgenstrahlen Ann. Phys. (Leipzig) 49, 1, 117 (1916); 54, 519 (1917) (English translations exist!) M. von Laue, Die Theorie der Röntgenstrahlinterferenzen in neuer Form Ergebn. Exakt. Naturwiss., 10, 133 (1931) review articles R. W. James, The Dynamical Theory of X-ray Diffraction Solid State Physics 15, 55 (1963) B. W. Batterman & H. Cole, Dynamical diffraction of X-rays by perfect crystals Rev. Mod. Phys. 36, 681 (1964) A. Authier, Ewald waves in theory and experiment Adv. Struct. Res. Diffr. Methods 3, 1 (1970) A. Authier, Dynamical Theory of X-ray Diffraction International Tables for Crystallography, Volume B, Part 5.1, 464 (1993) A. M. Afanas'ev, R. M. Imamov & E. K. Mukhamedzanov, Asymmetric X-ray Diffraction Cryst. Rev. 3, 157 (1992)
122 schools International Summer School on X-ray Dynamical Theory and Topography August 18-26, 1974, Limoges, France X-ray and Neutron Dynamical Diffraction - Theory and Applications April 9-21, 1996, Erice, Italy ed. A. Authier, S. Lagomarsino, B. K. Tanner, Plenum Press, New York 1996 further classic works (deformed crystals, QFT) P. Penning, D. Polder, Anomalous Transmission of X-rays in Elastically Deformed Crystals Philips Res. Reports 16, 419 (1961) P. Penning, Theory of X-ray Diffraction in Unstrained and Lightly Strained Perfect Crystals Thesis, Univ. Delft 1966, in Philips Res. Repts Suppl. 5,1-109 (1967) N. Kato, Pendellösung Fringes in Distorted Crystals I. Fermats Principle for Bloch Waves II. Application to the Two-Beam Case III. Application to Homogeneously Bent Crystals J. Phys. Soc. Japan 18, 1785 (1963), 19, 67, 971 (1964) S. Takagi, Dynamical Theory of Diffraction Applicable to Crystals with any Kind of Small Distortion Acta Cryst. 15, 1311 (1962) S. Takagi, A Dynamical Theory of Diffraction for a Distorted Crystal J. Phys. Soc. Japan 26, 1239 (1969) D. Taupin, Théorie dynamique de la diffraction des rayons X par les cristaux déformés Bull. Soc. Franc. Miner. Crist. 87, 469 (1964), Thesis: Paris 1964 M. Kuriyama, Theory of X-ray Diffraction by a Distorted Crystal J. Phys. Soc. Japan 23, 1369 (1967)
123 Thank you for your attention!
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