Transcript 2 nd Lecture: Introduction into the dynamical theory of X-ray

Introduction into the
dynamical theory of X-ray diffraction
for perfect crystals
Jürgen Härtwig
ESRF - The European Synchrotron
CS 40220, 38043 Grenoble Cedex 9, France
[email protected]
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, pointing vector
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 is it applied?
3
Dynamical Diffraction – Applications today

X-ray topography  strain and defects in (single) crystals, e.g.,
dislocations, precipitates  Crystal growth  electronic & microelectronic development  photovoltaics …

Bragg-diffracting optical elements for synchrotron radiation 
monochromators, phase plates (X-ray optics in general)

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
and Max Laue
(1888-1985)
(1879-1960)
6
It started in 1910 with …
… Paul Peter Ewald’s thesis project
Ewald asks Arnold Sommerfeld for a topic for a
dissertation (remember – at that time it was not sure
what exactly were X-rays and what a crystal)
Proposal: Find out whether a lattice-like (3D) anisotropic
arrangement of isotropic resonators might be capable of
exhibiting light-optical birefringence (double refraction)
Result: A theory that relates macroscopic properties of
light 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,
taken with a CuS crystal
plate
The famous exposure “Fig. 5”,
adjusted ZnS crystal.
(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
(oscillation method)
X-ray topography
(white beam topography,
Lang 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
~ 40 cm
ID19
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.
Δω=1° and λ=0.8Å, Δλ/λ~10-4.
Higher harmonics “contamination”
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
1913
1914
Laue’s geometrical (or kinematical) theory,
Laue equations
Bragg’s law
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
Bragg’s law - 1913
W H Bragg, W L Bragg, Proc Roy Soc A88, 428 (1913)
W L Bragg - 22 years old student!
Equivalent to Laue equation
real space
reciprocal space
diffracted wave
incident wave, l
q
q

K hkl
d

h
d· sinq
l = 2dhklsinqhkl
Kh = K0 + h
scalar form
of Bragg’s law
vectorial form
of Bragg’s law
(here no nl !!!)

K0
0
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
1913
Laue’s geometrical (or kinematical) theory
Bragg’s law
1914
Darwin’s geometrical and dynamical theories
1916/17
Ewald’s extension of his theory to X-rays
(fully dynamical theory)
C. G. Darwin, The theory of X-ray reflection, Phil. Mag. 27, 315, 675 (1914)
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

 

h- scattering vector, k h  k 0  h
h = 2 sinB/l,
(not necessary a crystal)
P
k0 = kh
 
r  r
Amplitude in point P

r

k0

rsource

r

h

k0

kh
0
21
Optical path difference:
   
 
 
k h  r k 0  r 1 

 (k h  k 0 )  r   l h  r 
k0
k0
k0

k0

rsource
 
r  r

r

h

k0

r
 
k 0  r
k0

kh
0
 
k h  r
kh
22
We use the following conditions:
1. Plane wave approximation (Fraunhofer approximation):




rsource  r and r  r
2. Elastic scattering, no absorption


k0  kh
23
3. Only one scattering process (no multiple scattering)
1(r)
k0
0(r)
2(r)
1 << 0
2 negligible
1st Born approximation
“geometrical” theory = kinematical scattering theory
 theory in 1st Born approximation
 Fraunhofer approximation
24
A little bit of theory - Amplitude in point P:
(r) = 0(r) + 1(r) = 0(r) +  G(r|r’) V(r’) 0(r’) d3r’
with:
 
exp(
2

ik
r
 r )
 
0
G r | r ) 
 
4 r  r 
This is an approximation!
Exact solution with:
 (r’) instead of 0(r’)


Vr )  4re Nr ) - for photon scattering on electrons
(re– classical electron radius, N(r) – electron density)



Vr )  VFermi r )  b( 2 2 / m)r ) - for thermal neutron scattering on the
nuclear potential (b – scattering length)
 2m

Vr )  2 U C r ) - for electron scattering on the Coulomb potential

25
Example – X-rays
0(r) – plane wave
electrical field
Scattered wave:

 exp(2ik 0r ) 
1 (r )  Eh r ) 
F(h )
r
Scattering amplitude:

  3

F(h)  reE0 P  N(r) exp(2ihr)d r
re  3·10-15m

1
for s-polarization
P=
 cos(2B) for -polarization
polarization factor for amplitudes
Rayleigh scattering
29
Special cases:
1. Scattering on 1 electron:
re2

T 2
I (r )  Eh  2 C I 0
r
T
h
re 
3·10-15m



N(r )  (r)
1
C  [1  cos2 ( 2 B )]
2
polarization factor for intensities
Thomson scattering
30
2. Scattering on 1 atom:
E

(h )
atom
h
T
h
E


N(r)  Natom (r)
  3


  N(r) exp(2ihr)d r  f (h)
Remark: Fourier
transform of N(r)!!!
T
I atom

I
h
h f
f – atom form factor
(amplitude)
2
31
T
I crystal

I
h
h G Fh
2
G - lattice amplitude
(factor)
I
crystal
h
~ Fh
2
Like in classical optics – Fraunhofer
diffraction by a grating
Fh - structure amplitude
(factor)
2
Such dependence only in
kinematical approximation

  3

Fh (h )   N(r) exp(2ihr)d r
unitcell
Fourier transform of N(r)!
Structure analysis
29
Only with these approximations:
Ihcrystal ~ |Fh|2
and convenient use of:
Fh  FT  N(r’)
30
When these approximations could be valid?
Comparison with Thomson scattering:
Intensity of the wave scattered by 1 electron
2

re
T
I h (r )  2 C I 0
r
Estimation:
1
C  [1  cos2 ( 2 B )]
2
polarization factor for
intensities
re – classical electron radius
re  3·10-15m
1 electron,
for r = 0.1mm  IhT/I0  (re/r)2  10-21
cubic crystal:
a = 0.5nm, Z = 30
in (1mm)3

~ 3·1011 electrons single scattering OK
in (1mm)3
 ~ 3·1020 electrons single scattering?!?!?
31
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
32
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, pointing vector
Bragg case and a bit X-ray optics
Plane waves, monochromatic waves
33
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
34
3. Dynamical theory of X-ray
diffraction
First dynamical X-ray diffraction theories promptly
followed the discovery of X-ray diffraction:
35
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:
(r) = 0(r) +  G(r|r’) V(r’) 0(r’) d3r’
to:
(r) = 0(r) +  G(r|r’) V(r’) (r’) d3r’
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)
37
Darwin’s approach
Diffracted wave is calculated as a
superposition of plane waves
reflected from and transmitted
through individual atomic planes,
multiple reflection is included.
Reflection geometry (“Bragg case”)
atomic planes
C. G. Darwin, Phil. Mag. 27, 315 (1914); 27, 675
t0 = 1-i q0
rh = -i qh
[hkl] [-h-k-l]
qo, qh - transmission and reflection coefficient
t0, rh - transmissivity and reflectivity
Summing up all contributions
Improved version in use today for layered systems
38
The surprising result
Bragg case or reflection case
Darwin curve
Si 111, 60keV, 10mm thick plate
1.0
kinematical Bragg position
reflectivity
0.8
0.6
0.4
0.2
0.0
-1.0
Region of “interference
total reflection” in a
finite angular range
-0.5
Full width at half
maximum (FWHM)
whq
q0
0.0
0.5
1.0
Shift from the kinematical
Bragg position due to
refraction
1.5
2.0
2.5
angle q - qB (arc seconds)
HERCULES Grenoble, March 2009
39
Si 111, 60keV, 500mm thick plate
1,0
reflectivity
Effects of absorption
and crystal thickness
0,8
0,6
0,4
0,2
0,0
-2,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
2,0
1,5
2,0
angle q - qB (arc seconds)
Prins-Darwin or reflectivity curve
Si 111, 8keV, 5cm thick plate
1,0
Si 111, 60keV, 500mm thick plate
1,0
0,6
0,8
transissivity
reflectivity
0,8
0,4
0,6
0,4
0,2
0,2
0,0
-5
0
5
10
15
20
angle q - qB (arc seconds)
Bragg case or reflection case
0,0
-2,0
-1,5
-1,0
-0,5
0,0
0,5
1,0
angle q - qB (arc seconds)
Laue case or transmission case
40
By the way …
Those calculations were done with the program
(program package)
XOP
http://www.esrf.eu/computing/scientific/xop2.4/documentation.html
41
Few more typical results (and differences)
of kinematical and dynamical theory
Full widths at half maximum (FWHM)
of reflectivity curves
Kinematical theory:
symmetrical Bragg case: whsymB  (dhkl/t)tanqB
symmetrical Laue case:
Dynamical theory:
(=Scherrer formula)
whsymL  dhkl/t  f(l)
t – crystal thickness
symmetrical Bragg case: whsymB  (dhkl/0)tanqB
symmetrical Laue case:
whsymL  dhkl/0
new parameter:
0 - Pendellösung length
symm.Bragg case: 0symB =  VUC / (2 r0 |P| Fhkl dhkl)  f(l)
symm. Laue case: 0symL =  VUC / (2 r0 |P| Fhkl dhkl tanqB)
42
Area under reflectivity curve – integrated reflectivity
R qi   R h (q)dq
No absorption, thick crystal (dynamical theory):
Laue case:
R qiL ~ | Fh |
Bragg case: R qiB  2R qiL
No absorption, thin crystal (kinematical limit):
Both cases:
R qi ,thin ~ V | Fh |2
V – illuminated crystal volume
43
Integrated reflectivity
Riq
kinematical
7
6
15
dynamical (Bragg case)
4
1/23
dynamical (Laue case)
2
1
0
400
200
1
600
t/ 2
1000
800
3
t / 0
44
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!
K0
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
45
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
46
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???
47
First published measurements of
reflectivity curves:
(not rocking curves!)
1962
nearly 50 years after Darwin’s results!!!
48
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)
49
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 like silicon
50
large silicon ingots
dislocation free
51
Giant Crystals of Gypsum Naica mine (Chihuahua, Mexico)
giant, faceted, and transparent single crystals of gypsum (CaSO4·2H2O)
as long as 11 m. Prof. Garcia Ruiz, Universidad de Granada, Spain
Photos by Javier
Trueba and Paolo
Petrignani, La
Venta & S/F
Archives, et al.
52
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 (his) existing dynamical theory
53
Short theoretical background
Many excellent theoreticions participated in the
development and completion of the dynamical
theory of X-rays diffraction
P.P. Ewald
C.G. Darwin
M. von Laue
W.H. Zachariasen
Variants for:
P. James
“conventional” and “extended” theory
A. Authier
perfect crystals; plane wave, spherical wave
approaches
A. R. Lang
deformed crystals; geometric and wave optical
(even QFT) approaches
Layered systems
Two- and many beam cases
B. W. Batterman
G. Borrmann
U. Bonse
P. Penning & D. Polder
N. Kato
S. Takagi
D. Taupin
M. Kuriyama ...
54
As already mentioned, in principle we have to solve:
(r) = 0(r) +  G(r|r’) V(r’) (r’) d3r’
solution of the
homogeneous equ.
inhomogeneity of
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
55
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


E (r )  2 [E  U(r )]E (r )  0

and

or  E (r ) 
2mE
 2m 


(
r
)

U
(
r
)

(
r
)
E
E
2
2


 
 
2
2
e 
rot rotE(r , )  4 K [1   (r , )]E(r , )  0
or
 
 


2
2
rot rotE(r , )  4 K E(r , )  4 2 K 2  e (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.!
For a crystal V(r), VFermi(r), N(r), e(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(2ik 0r )Ek (r , )
0
where
 
Ek 0 (r , ) is triply periodic
k0 – wave number in the crystal
This allows to use Fourier expansions for the solution of the
differential equation
58
For the electrical susceptibility:
ε=
(1+xe)


e
 (r , )    h () exp(2ihr )
e
h
ε0
For the triply periodic amplitude
of the electrical field vector:
 


Ek (r , )   Eh exp(2ihr )
0
h
 
  


 
we obtain: E(r , )  exp(2ik 0r ) Eh exp(2ihr )   Eh exp(2ik h r )
h
h



Here k h  k 0  h holds for the wave vectors within the crystal



This does not imply that K h  K 0  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
59
Fourier component of the electrical susceptibility
is directly related to the structure factor


e
 (r , )    h () exp(2ihr )
e
h
2
r
l
 eh    Fh ;   0
VUC
with r0 – classical electron radius
and VUC – unit cell volume
60
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
61
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, pointing vector
Bragg case and a bit X-ray optics
Plane waves, monochromatic waves
62
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!

K0
0
modified Ewald’s construction
63
But the crystal is matter with nmatter  nvacuum
The theory provides
Basic equation:
(K2 – k02) E0 + K2 0e E0 = 0
Dispersion equation:

K0
K
0
k
dispersion surface
for the one beam case
K2 (1+0e) – k02 = k2 - k02 = 0
mean wave number
in the crystal
1
k  nK  K (1   e0 )  K
2
complex value
- absorption
15keV, Si
n = 1 – 2.2 10-6 –i 1.5 10-8
64
Let’s introduce a boundary
reciprocal space
direct space

K0

k0

k t
kr

K0
K
k
nvacuum

nmatterk t

Km

kt 
k0

n

Km

n
- boundary condition for the wave vectors –
Continuity of the tangential components of the wave vectors
65
In the one beam case of crystal diffraction
and in classical optics:
1 incident, 1 reflected, 1 refracted plane wave
reciprocal space
direct space

K0

k0

kr

K0
K
nvacuum
k
nmatter

Km

k0

n

Km

n
66
“External” total reflection
reciprocal space
direct space

K0

K0

k0
c
nvacuum
K
k
nmatter

Km

n

n

Km

k0

n
Nothing really new. No
difference if crystalline
or non-crystalline matter.
range of external total reflection
HERCULES Grenoble, March 2009
67
The two “beam” case
Two reciprocal lattice points close to the Ewald sphere
The theory provides
Basic equations:
(k2 – k02) E0 + K2 P -he Eh = 0
K2 P he E0 + (k2 – kh2) Eh = 0
Dispersion equation:
(k2 – k02) (k2 - kh2) = K4 P2 he -he  0
Remarks and conclusions:
1. Known problem - electron in periodic potential (e.g. Kittel)
2. It never holds that k02 = k2, kh2 = k2, k02 = K2, kh2 = K2.
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
68
modified Ewald’s construction

K0

Kh
h

h
0
K
k
The spheres with diameter k are
not yet the dispersion surfaces!!!
HERCULES Grenoble, March 2009
69
Dispersion surfaces in the two “beam” case (cut), one polarization
La
Lo

Kh

h
h

K0
 0
k 0i

n
khi vectors not drawn for simplicity
HERCULES Grenoble, March 2009
70
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:
(k2 – k02) (k2 - kh2) = K4 P2 he -he  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!
71
Gracing Incidence Diffraction
The incident angle below or close to αc (Θc)
72
Dispersion surfaces in the two “beam” case (cut), one polarization

K0

Kh

h
h

n
0
This is the
physically most
important part
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:
(k – k0) (k - kh) = ¼K2 P2 he -he  0
“Conventional” dynamical theory (this is an approximation!!!)
74
Dispersion surfaces in the two “beam” case – physically
most important part
La
1
 K e0
2
Lo
K
k

K0

n

Kh

k h1
h

k h2

h

k 02

k 01
0
75
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):

k0

kh
Wave length of the
standing wave pattern

h
reflecting
lattice planes
equals
spacing of the
reflecting lattice
planes
Position of nodes/antinodes
of one standing wave field
At the maximum of
the Bragg position
76
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
77
Some effects of dynamical X-ray diffraction
Pendellösung length (Laue case), Pendellösung effect

g1Kn




 
K h  K 0  h  gKn  K 0  h



k 0 i  K 0  g i Kn



k hi  k 0 i  h
La
Lo
1
 K e0
2

Kh


k h1 k h2

K0

n




| k 01  k 02 || k h1  k h 2 | 1 / 
Coherently excited waves
interfere
beating frequency is 

k 02

k 01
(relative lengths differences
 |0e| K ~ 10-5 104mm-1
 ~ 10mm)
HERCULES Grenoble, March 2009
78
Amplitude of the E-field in the crystal:



E( C ) (r )  E WF 1 (r )  E WF 2 (r ) 
 
 
 E01 exp(2ik 01r )  Eh1 exp(2ik h1r ) 
 
 
 E02 exp(2ik 02r )  Eh 2 exp(2ik h 2r )
 
 

E( C ) (r )  E 01 exp(2ik 01r )  E 02 exp(2ik 02r ) 
 
 
 Eh1 exp(2ik h1r )  Eh 2 exp(2ik h 2r ) 
 

 
E
 E 01 exp(2ik 01r ){1  02 exp[2i(k 02  k 01 )r )]} 
E 01
 

 
Eh 2
 Eh1 exp(2ik h1r ){1 
exp[2i(k h 2  k h1 )r )]}
Eh1





| k 01  k 02 || k h1  k h 2 | 1 / 
nr  z - depth in the crystal
E
(C)
 
E 02
z

(r )  E 01 [1 
exp(2i )]exp(2ik 01r ) 
E 01

 
Eh 2
z
 Eh1 [1 
exp(2i )]exp(2ik h1r ) 
E h1

 
 
 E 0 ( z ) exp(2ik 01r )  Eh ( z ) exp(2ik h1r )
Two “plane” waves with
amplitudes “slowly”
depending on z
79
If Bragg condition is fulfilled (maximum of the reflectivity curve, n
intersects Lorentz point)  maximum value of   Pendellösung length 0
La

Lo
0
2  g h / | g h |
gh = sin(a- qB)
 - normalized
angular coordinate

K0

n
Modulation is maximum
for  = 0
Examples for 0:

k h1

k h2

k 02

k 01
1mm Si, 220-reflection
8keV: 22.4μm 60keV: 127μm
HERCULES Grenoble, March 2009
80
Pendellösung effect
Boundary condition for amplitudes at the entrance surface:
E0  E01  E02 , 0  Eh1  Eh 2
 = 0, no absorption
At the surface z = 0
Amplitude of wave E0:
and of wave Eh:
In depth z = 0/2
In depth z = 0
etc.
E0(z=0) = E0
Eh(z=0) = 0
E0(z=0/2) = 0
Eh(z=0/2) = E0
E0(z=0) = E0
Eh(z=0) = 0
81
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.
82
Planes where
the diffracted
wave |Eh|2 has
maxima

K0
0
Planes where the
forward
diffracted wave
|E0|2 has maxima

Kh
|Eh|2
x
HERCULES Grenoble, March 2009
83
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, t450mm, asymmertrical 111-reflection
in transmission; monochromator: symmetrical 111-reflection;
energy E=30keV; 0  46mm
84
Defect images in an X-ray white beam topograph
stacking fault
strain field of inclusion
dislocations
Highly pure HPHT
type IIa
100-oriented
diamond
(Element Six SA)
4mm
“Perfect” parts:
Homogeneous local
intensity/grey level
equal thickness
fringes
surface scratch
transmission geometry
85
Possible applications:
- measurements of structure factors
(thickness known)
symm. Laue case: 0symL =  VUC / (2 r0 |P| Fhkl dhkl tanqB)
- thickness measurements for “inaccessible”
objects (e.g. growing dendrites in
a crucible)
86
A
Al-3.5wt%Ni
(d2)
(d1)
A
A 500 mm
B
B
C
« Free »
growth
white
black
black
white
g200
C
B
C
1 mm
1 mm
1 mm
“confined”
growth
(d3)
(d3)
Al-7.0wt%Si
g200
g2-20
87
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]
88
3D structure of dentrites
89
Short break for discussion
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, pointing vector
Bragg case and a bit X-ray optics
Plane waves, monochromatic waves
91
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
92
Repetition
Penetration depth, extinction depth, absorption depth
Silicon, 333 reflection, symmetrical Bragg case CuKα1 (Handbook - Authier)
93
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
94
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
 
 
 
 

2 ik 01r
2 ik 02r
2 ik h 1r
2 ik h 2r
E(r )  E01e
 Eh1e
 E02e
 Eh 2e
wave field 1
wave field 2



k hi  k 0 i  h
95
Once more - Laue case in reciprocal space
La M
Symmetrical Laue
case: M = La
maximum of
reflection
A1
Lo
A2

Kh

k h1

n

k h2
h
All waves coherent

h

k 02

K0

k 01
0
interferences
96
Two standing wave fields are excited within the crystal
first wave field

k0

kh

h
reflecting
lattice planes
Position of nodes/antinodes
of one standing wave field
At the maximum of
the Bragg position
97
Two standing wave fields are excited within the crystal
second wave field

k0

kh

h
reflecting
lattice planes
Position of nodes/antinodes
of one standing wave field
At the maximum of
the Bragg position
98
Intensity of one wave field in the crystal
 
 
 2
2 ik 0 i r
2 ik hir 2
E (r ) | E0i e
 Ehi e
| 
(i )
2

 
E
Ehi
2 mz
hi
 E0i e 1 
2
cos(2hr )
2
E0i
E0i


For simplicity at the maximum:
Ehi/E0i = 1
-1  WF1; 1  WF2

 2
E (r ) ~ 1  cos(2hr)
(i )

hr  n; n  0,  1,  2, ....
equation of lattice planes
99
Intensities of the standing wave fields
lattice planes
wave field 1
k0i
khi
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
100
Effect on absorption coefficients
31.34
-1
absorption coefficients (mm )
30
m2
25
20
m0/g0
15.92
15
10
0.500
m1
5
0
-5
-4
-3
numerical example:
-2
-1
0
1
2
3
4
5
normalised angle
1 mm Si plate, 220-reflection, 8 keV
exp(- m0t/g0) = 1.2 10-7, exp[-min(-m1) t] = 0.606
101
Reflection- and transmission curves, zero absorption (mt = 0)
Both wave fields contribute equally
reflectivity, transmissivity
1,0
0,8
T
0,6
0,4
R
0,2
0,0
-5
-4
-3
-2
-1
0
1
2
3
4
5
normalised angle
102
Reflection- and transmission curves, high absorption (mt = 14.6)
only wave field 1 “survives”
reflectivity, transmissivity
0.20
0.15
T
0.10
R
0.05
0.00
-4
-3
-2
-1
0
1
2
3
4
angular deviation (arc sec)
1 mm Si plate, 220-reflection, 8 keV
103
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
104
Laue image of an AlPdMn icosahedral quasicrystal
showing five-fold symmetry
Courtesy of J. Gastaldi
105
X-ray diffractometry - widths of Bragg peaks
24keV
24 keV
250000
225000
counting (ab. units)
200000
I0
175000
150000
I0
125000
100000
75000
Ih
50000
Ih
25000
0
11.136
mt  1.6
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”)
106
11keV
11
keV
3500
counting (ab. units)
3000
2500
Ih
Ih
2000
1500
I0 I
0
1000
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)
107
Borrmann topography
X-ray diffraction topography under strong
absorption conditions (μ0t >> 1)
μ0t < 1
InP
μ0t >> 1
108
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, pointing vector
Bragg case and a bit X-ray optics
Plane waves, monochromatic waves
109
Directions of wave field energy flows (pointing vectors)
La
Lo

n
Perpendicular to the
dispersion surface in
the tie points

k h1

K0

k h2

k 02

k 01
110
No plane wave but narrow beam in one direction
h
Section topography - A.R. Lang,
Acta metallurgica, 5, 358-364 (1957)
Perfect region
no homogeneous intensity
“Kato fringes”
Quartz, t = 1.8mm, slit width 10 μm, 20-20 and 20-2-2 reflections
550 μm
111
Interference total reflection (Bragg case)
Darwin curve
Si 111, 60keV, 10mm thick plate
1.0
reflectivity
0.8
0.6
whq
0.4
0.2
0.0
-1.0
q0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
angle q - qB (arc seconds)
112
Bragg case – most important case for monochromators
direct space

K0

Kh

n
HERCULES Grenoble, March 2009
113
Dispersion surfaces in the two “beam” case – physically most important part
h
q

k h1

k h2
La

Kh

h
Lo

k 01

n

K0

k 02
Reciprocal space
0
114
Range of interference total reflection
h
q

h
Lo
La

K (01)

n1

n2
 ( 2)
K0
range of interference total reflection
0
115
Mathematics very simple to calculate a reflectivity curve for a
semi-infinite crystal (Darwin solution)
he
2
1/ 2 2
R h [ (q)]  e   (  1)
h
Normalised angular coordinate or deviation parameter:
2( q  q 0 )  g 0 


 
q

wh
 gh 
12
( q  q 0 ) sin 2q B
P  eh  eh )
12
g 0  sin(a  qB )
g h  sin(a  qB )
Full width at half maximum of the reflection curve:
2 P   )
w h  2; w hq 
sin 2q B
e
h
e 12
h
 gh 


 g0 
12
Refraction correction (middle of the reflection domain):
 e0 
g 
 1  0 
q 0  
2 sin 2q B 
gh 
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To be complete – Pendellösung length (general form):
( g 0 g h )1 2
0 
K P ( eh  eh )1 2
– other expression for the full width at half
maximum of the reflection curve:
w hq 
2d hkl g h
 0 cos q B
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Integrated reflectivity
R qi   R h (q)dq
No absorption, thick crystal (dynamical theory):
Laue case: R qiL 
 P  
e
h
)
e 12
h
2 b ) sin 2q B
12
~ | Fh |
Bragg case: R qiB  2R qiL
asymmetry factor: b  g 0 g h  sin(a  qB ) sin(a  qB )
No absorption, thin crystal (kinematical limit):
 2 P K eh  eh t

~ V | Fh |2
g 0 sin 2q B
2
Both cases:
R qi ,thin
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Integrated reflectivity
Riq
kinematical
7
6
15
dynamical (Bragg case)
4
1/23
dynamical (Laue case)
2
1
0
400
200
1
600
t/ 2
1000
800
3
t / 0
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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
Δθin
Lin

K0

Kh
Lout
Δθout
a0
This example: a>0, |b|<1:
Lin > Lout and Δθin < Δθout
Δθin Lin = Δθout Lout = constant
For a<0 – less divergence
possible to obtain
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Asymmetry factor:
b
sin( q B  a )
sin( q B  a )
Δθout = |b| Δθin
Of course, the same works also other way around.
Δθout
Lout

Kh

K0
Lin
Δθin
a0
So we have possibilities to act on the beam
dimension (expansion, compression), as well as on
the beam divergence (smaller, larger)
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But all is related and things have their price
whin
Lin

K0

Kh
Lout
whout
Asymmetry factor:
sin(q B  a ) g 0
b

sin(q B  a ) g h
a0
This example: a>0, |b|<1:
Lin > Lout and whin < whout
whout = |b| whin
whin Lin = whout Lout = constant
Relation to symmetrical
reflections
right asymmetry (a<0) – less
divergence possible to obtain
whin = |b|-1/2 whsym
whout = |b|1/2 whsym
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Range of interference total reflection – asymmetric reflection
h
q
 (1)
Kh
whout 
K (h2 )
La
Lo

h
whin
 ( 2)
K0
direct space

n1
 (1)
K0
0

n2
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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
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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:



K ht  K 0t  h t
And remember, wave vectors depend on wavelength: K  f (l )
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EXAMPLE – ASYMMETRIC BRAGG DIFFRACTIONS
Possibility to change the divergence (qin , qout)
or the energy band pass (K)

K0

t
0 = qB + a - /2  qin
h = qB - a + /2  qout




 
K h  K 0  h  gKn  K 0  h
 


 
K h t  (K 0  h  gKn ) t  K 0t  h t
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
h
a

n
qB qB

K0
h
0
with

nt  0

Kh
EXAMPLE – ASYMMETRIC BRAGG DIFFRACTIONS
For small qin, qout and K we obtain (for Bragg and Laue case!):
| q out |  b qin 
cos(qB  a)  cos(qB  a) K
sinqB  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 (see later) is virtually larger.
Special case 1
Monochromatic, divergent incoming radiation
K = 0, qin  0
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qout = |b| qin
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
We show later: “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!
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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)
not source divergence!!
s
p
δ
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”.
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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
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Magnification, demagnification, focussing properties/quality
Geometrical demagnification,
source size limit:
Diffraction limited focusing:
ρDL = 1.22 λ / sinα
ρG = q s/p = q δ
q
s
p
ρ
a
q
(graph: J. Susini)
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AT THE END – TWO BASIC QUESTIONS
Two basic questions:
What are “plane” or “divergent” waves?
What are “monochromatic”, “polychromatic”, “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!!!
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AT THE END – TWO BASIC QUESTIONS
The full width at half maximum of the reflectivity curve,
is a good reference for our sample to define the
character of a wave
FWHM in angular space:
w
h
2 P h h

sin2 B
R(q)
whq
gh
g0
FWHM in wavelength space:
q
R(l)
whl
w lh  w 
h l cot  B
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l
AT THE END – TWO BASIC QUESTIONS
Example of a typical crystal/monochromator reflection:
silicon, 111 reflection, Bragg case, thick crystal
energy
wh
whl/l
8 keV
7.6 arcsec
1.5·10-4
20 keV
2.9 arcsec
1.5·10-4
Those are to be compared with source properties:
wavelength spread of the source l (l/l)
angular source size 
( = s/p)
not source divergence!!
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s
p
δ
AT THE END – TWO BASIC QUESTIONS
relative wave length spread of the source l/l (E/E)
<<
FWHM in wavelength space w hl
“monochromatic” wave
angular source size  = s/p
<<

FWHM in angular space w h
“plane” wave
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AT THE END – TWO BASIC QUESTIONS
Silicon
111-oriented
source
energy
w h
8 keV
7.6 arcsec
20 keV
2.9 arcsec
source size s source dist. L0
For the angular scale
to be compared with:
angular source size 
( = s/L0)
δ
class. lab tube
400 µm
0.4 m
 1·10-3  3.5 arcmin
µ-focus tube
5 µm
1 m
 1·10-5  1 arcsec
neutrons
50 mm
30 m
 1.7 10-3  5.7 arcmin
SR
100 µm
150 m
 6.7·10-7  0.14 arcsec
laboratory, n:  >< wh divergent & quasi plane waves possible
SR - ESRF:
 < wh quasi plane wave (mostly)
Reduction of angular source size  possible by X-ray optics
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AT THE END – TWO BASIC QUESTIONS
relative wavelength or energy spread of the source l/l
source
l/l
Si 111 double mono
energy
whl/l
2·10-4
8 keV
1.5·10-4
laboratory (e.g. CuKa1)
3·10-4
20 keV
1.5·10-4
white beam
1···10
SR–white beams: l/l >>> whl really polychromatic
”monochromatic” beam (SR or laboratory):
l/l <?> whl often not monochromatic
(for all reflections narrower than the 1.5·10-4 for silicon 111)
special effort is necessary to approximate “monochromaticity”
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LITERATURE
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 (earlier 2 editions in Russian)
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)
139
LITERATURE
classic works (first and most current dynamical theories – for perfect crystals!!!)
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)
140
LITERATURE
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)
141
Thank you for your
attention!
HERCULES Grenoble, March 2009
142