null  null
Advanced labcourse
F69 - Laue X-ray diffraction
INF 501 room 103
May 2015 - C. Neef
TABLE OF C ONTENTS
1. Introduction and general safety instructions
3
2. Preparation and literature
5
3. Generation and detection of X-rays
3.1. Generation by X-ray tube . . . . . . . . . . .
3.1.1. Generation by microfocus tubes . . .
3.2. Detection of X-rays . . . . . . . . . . . . . .
3.2.1. Chemical processes and image plates
3.2.2. Counting tube . . . . . . . . . . . . .
3.2.3. CCD cameras . . . . . . . . . . . . .
.
.
.
.
.
.
7
7
9
9
10
10
10
4. Single crystal X-Ray diffraction
4.1. Electron density of periodic lattices . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Specifics of a Laue experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
12
14
15
5. Crystal structure and stereographic projections
5.1. Definition of zones and lattice planes . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. Stereographic projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
16
17
18
6. Experimental setup
6.1. Sample mounting . . . . . . .
6.2. Automatic Psi-Circle operation
6.3. Tube and generator operation .
6.4. Detector and Software . . . .
6.5. Pattern simulation . . . . . . .
.
.
.
.
.
19
20
21
21
23
25
.
.
.
.
28
28
28
29
29
A. Addendum
A.1. Systematic distinction of reflexes . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2. Guide for the generation of stereographic projections . . . . . . . . . . . . . . . . .
30
30
31
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
7. Experimental scope and analysis
7.1. Identification of the orientation of a cubic crystal and sample-detector distance .
7.2. Crystal shape and crystal lattice . . . . . . . . . . . . . . . . . . . . . . . . .
7.3. Crystal orientation and cutting . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4. Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
I NTRODUCTION AND GENERAL SAFETY INSTRUC TIONS
How do atoms arrange in a material in order to form a crystal? This and many other fundamental
questions on condensed matter are answered by crystallography and X-ray structure determination,
which are inevitable tools not only for solid state physics but many other sciences. Whenever crystalline samples are the subject of an experiment, the structure analysis will be the first step prior to
the determination of any other physical property. It can provide information regarding structure and
symmetry, purity and defects and relation between external shape and internal crystal lattice. The
Laue X-ray diffraction method is particularly used for the analysis of single crystalline samples. Often, the scope of such experiments is to find the crystallographic orientation of a sample with respect
to the laboratory system, determine grain sizes or find out about defects. The method’s applications
range from scientific questions to industrial testing procedures like quality assurance of semiconductor wavers or determination of material fatigue of turbine blades.
The set up of this lab course experiment is appropriate to investigate small samples with mm-sized
dimensions. The first step of characterising unknown or new samples is to determine the symmetry
in which atoms organize in a crystallographic unit cell. Then the sample’s orientation must be determined in order to prepare it for further physical experiments. Whenever physical properties are
expected to show anisotropy, the alignment of crystal and laboratory system is required. In practice
this means for example aligning a crystal parallel to the [001]-axis by means of the Laue setup, which
then enables to measure a physical property along the c-axis in another experiment.
The task of this lab course experiment is to understand the relation between crystal structure (invisible
to the eye) and X-ray diffraction patterns on a screen (visible to the eye) which are a direct image of
the elsewise abstract k-space. In fact, XRD experiments have motivated to introduce the concept of
k-space, which is a key issue for understanding modern physics. The strength of a Laue experiment
thereby is the simultaneous visibility of many different diffraction spots which reproduce the symmetry of the crystal in a single picture. Furthermore, the handling of an X-ray source and the software
supported diffraction-data analysis should be learned. In the process, different minerals are available
which will be investigated and whose structure and symmetry will be determined.
3
1. Introduction and general safety instructions
Radiation protection and tube handling
The participation of a radiation protection course is absolutely mandatory for the execution of this
experiment.
Within the lab course at hand, an X-ray tube will be used to generate X-rays with an energy of up to
50 keV. Exposition to this kind of ionising radiation can cause severe short and long term damages to
biological tissue. Although the setup is shielded within a protective cabinet, special attention should
be paid to the correct and safe execution of the experiment. This can only be guaranteed if the
protective cabinet is closed during use of the X-ray tube.
As a precaution, the X-ray generator is coupled to several safety switches which react on opening and
closing the cabinet doors and will be automatically shut down if the doors are opened. This procedure
is however very rough and might cause damage to the tube and generator. For safe operation of
the system, the waiting times for tube-power increase and decrease need to be followed absolutely.
Unnecessary starting or stopping of the high voltage need to be desisted in any case.
Toxicity of samples
Depending on availability and choice, irritant or harmful samples might have to be processed within
the lab course. Direct handling of sample crystals should therefore only be carried out in accordance
with the course supervisor. If necessary, protective gloves have to be used to avoid direct skin contact
with these samples. Do not swallow or inhale any components of the experiment.
Experimental setup and detector
The complete setup is already pre-adjusted, meaning that source, collimator, detector and goniometer
are on a straight line. Non of these components except for the goniometer should be moved since the
adjustment might easily be lost. The re-adjustment might take several days.
The X-ray detector used in this experiment is a very sensitive device. It should not be touched roughly
or moved in position. Special care has to be taken of the scintillating layer on the outside of the device.
The layer must not be touched! Scratching or even touching might cause damages which make the
detector unusable.
4
2
P REPARATION AND LITERATURE
The following manual does not give a global introduction to the science of crystallography and X-ray
diffraction. For a successful participation in the lab course, more sources should be consulted. A
selection of books and websites is given below. The previous participation in the condensed matter
physics lecture is strongly recommended. If this is not the case, please contact the supervising tutor
in advance.
You should be able to answer the following questions before you start the lab course.
Crystal structure:
• Why do atoms arrange in a given way (crystal), what is translational symmetry?
• What kind of lattice systems do exist and how do the Bravais lattices look like?
• What are the required symmetries for the cubic system?
• What is described by the term "crystal habit"? How is it influenced by the crystal structure?
X-ray experiments:
• What are X-rays and how can they be generated?
• How can they be detected? How are they interacting with matter?
• Why are they used to study crystal structures?
• Which kinds of X-ray experiments do you know? What are they used for?
• What kind of safety precautions need to be taken while dealing with X-rays?
Data analysis and evaluation:
• What are stereographic projections? What does "angle conserving" mean?
• What is a Wulff net?
• Why does the diffraction angle (angle between incident and scattered beam) not depend on the
lattice parameters in the cubic system? (Contradiction to Bragg’s law?)
5
2. Preparation and literature
The following literature can be reviewed in the university library:
• Kristallographie - Eine Einführung für Naturwissenschaftler - 8. Auflage; Walter BorchardtOtt, Heidrun Sowa; Springer Verlag 2013.
• Moderne Röntgenbeugung - Röntgendiffraktometrie für Materialswissenschaftler, Physiker und
Chemiker - 2. Auflage; Lothar Spieß et al.; Vieweg + Teubner 2009.
• Festkörperphysik - 1. Auflage; Siegfried Hunklinger; Oldenbourg Wissenschaftsverlag 2007.
• Basic Concepts of Crystallography - 1st edition; Emil Zolotoyabko; Wiley-VCH 2011.
• Basic Concepts of X-Ray Diffraction - 1st edition; Emil Zolotoyabko; Wiley-VCH 2014.
• Introduction to solid state physics - 8th edition; Charles Kittel; Wiley 2011.
Helpful simulations and information can be found on the internet:
• Overview of stereographic projections along with some interactive apps:
http://www.doitpoms.ac.uk/tlplib/stereographic/index.php
• "Clip" software for the simulation of Laue patterns:
http://clip4.sourceforge.net
• Space group tables and sketches of all groups:
http://img.chem.ucl.ac.uk/sgp/large/sgp.htm
6
3
G ENERATION AND DETECTION OF X- RAYS
X-rays are electromagnetic waves with photon energies of several keV. Their wavelength is in the
range between 0.1 Å and 100 Å making them ideal candidates for diffraction experiments in solid
state physics. Amongst many possible experiments, structure determination of crystals is one of the
main applications of X-rays, since their wavelength is in the same range as crystal lattice constants.
A structure determination can be achieved by the detection of specific X-rays after interaction with
the electronic shells of molecules or atoms inside a lattice. Depending on the physical problem
at hand, the requirements regarding X-ray energy, spectral distribution and intensity can be very
diverse. While high quality diffraction experiments rely on high intensity X-rays with a high degree
of monochromatization, the Laue experiment is carried out with polychromatic (i.e. "white") X-ray
light. The following chapter will describe the most common ways of X-ray generation and detection
with specific attention to the techniques used in this experiment.
3.1
G ENERATION BY X- RAY TUBE
The use of so called X-ray tubes is a common and rather inexpensive way of X-ray generation. The
principle is based on electrons which are accelerated by a dc voltage and strike a metal target. During
the latter process they loose their energy by interaction with the atomic cores in the metal. A typical
setup consists of a cathode (C) and anode (A), which are located inside an evacuated tube (see fig.
3.1). The heated cathode (Uh ) produces free electrons by thermoelectric emission, which are then
accelerated towards the anode by a given dc voltage (Ua ). The fast deceleration of the electrons
during collision with the anode causes the emission of electromagnetic radiation, which can than be
used for the desired X-ray experiment. This kind of radiation is called "Bremsstrahlung".
The spectrum of Bremsstrahlung is given by Kramer’s rule and illustrated in fig. 3.2.
I(λ) ∝ Z · (
1
λ
− 1) · 2
λmin
λ
(3.1)
With I being the spectral intensity, Z the atomic number and λmin the minimum wavelength (and thus
maximum energy) of a photon which is determined by the kinetic energy of the electrons and thus the
acceleration voltage Ua :
photon
Eelectron
= e ∗ U = Emax
kin
=
2π~c
λmin
(3.2)
It can be seen, that the intensity increases with increasing acceleration voltage Ua and atomic number
7
3. Generation and detection of X-rays
Figure 3.1.: Principle of X-ray generation by an X-ray tube (from: wikipedia): Thermally (Uh , C) released
electrons are accelerated (Ua ) onto a watercooled (Win , Wout ) metal target (A). The deceleration of the
highly energetic electrons leads to the generation of X-ray photons (X).
of the target material.
Figure 3.2.: X-ray spectra generated by an X-ray tube. A: Intensity trend in dependence of the targets
atomic number Z; B: intensity and shortest possible wavelength dependence of the acceleration voltage
Ua ; C: Schematic representation of a real spectrum consisting of Bremsstrahlung and characteristic radiation.
In addition to the continous intensity distribution of Bremsstrahlung, narrow lines of high intensity
with a certain wavelength can appear in a spectrum, which are called charateristic lines. They appear
if the acceleration energy is high enough to kick out an electron of the shell of the target materials
atoms into vacuum, leaving a hole (A) in the shell. After this kind of excitation, the atom will fall
back to the ground state when the hole is filled up again by an electron of a higher shell (B). The
energy difference between both states EB - EA can be released by emission of an X-ray photon.
Characteristic lines with high intensity are generated if the hole is located in the inner K-shell of an
atom. Specific lines are labelled with respect to the shell origin of the electron which is filling the
hole: L-shell corresponds to Kα, M-shell: Kβ, etc. Tab. 3.1 shows the wavelength of specific lines
8
3.2. Detection of X-rays
for some common anode materials.
Element
Co
Cu
Mo
W
Z
27
29
42
74
Kα Å(keV)
1.79 (6.92)
1.54 (8.05)
0.71 (17.46)
0.21 (59.0)
Kβ Å(keV)
1.62 (7.65)
1.39 (8.92)
0.63 (19.68)
0.18 (68.9)
Table 3.1.: Characteristic lines
As can be deduced from simple atomic models, the energy intervals between different atomic shells
scale proportional to Z2 , meaning that X-Ray energies increase quadratic with the atomic number.
Fig. 3.2 B shows a simple spectrum consisting of characteristic lines and Bremsstrahlung. The
relative linewidth of characteristic radiation is of the order of ∆E/E ≈ 10(−4) . Note, that a real
spectrum can be very complex and may show various overlapping lines due to the number of possible
electronic states and multi-step relaxation processes. In many diffraction experiments, only the Kα1
line, corresponding to an electronic transition from 2p3/2 state to 1s1/2 exhibitting the highest intensity
of all lines, is used. For that purpose, special monochromators must be applied to filter out other
characteristic lines and the continuum of Bremsstrahlung.
Besides the right choice of materials and acceleration voltage, tube design and electron beam focus
have great influence on the quality and intensity of the generated X-rays. Generally, the efficiency of
conventional tubes is of the order of 1%, meaning that almost all of the cathode’s energy transforms
to heat and needs to be cooled off the tube again. Additionaly, most experiments will need a beam
with small angular divergence, which reduces the efficiency even further, since only a small part of
the distributed X-rays can be used. The power input of a typical tube is some kW, while the X-ray
output will be of the order of one Watt.
3.1.1
G ENERATION BY MICROFOCUS TUBES
A drastic increase of efficiency and thus decrease of input power could be achieved by application of
so called "microfocus X-ray tubes". Thereby the electron beam is focused on a spotsize of 10 µm to
100 µm leading to a much higher power density on the anode. This can increase the brightness of the
tube by a factor of 100 compared to conventional tubes. The input power of such a system lies in the
range of several W. Note however, that due to the high thermal load on the focus spot, the lifetime of
micro focus tubes is relatively short.
3.2
D ETECTION OF X- RAYS
The detection of X-rays can be achieved by fotographic ex-situ techniques as well as in-situ cameras.
Depending on the experiment and physical problem, direction, energy and intensity of these photons
need to be measured. In the following, the most common methods will be presented.
9
3. Generation and detection of X-rays
3.2.1
C HEMICAL PROCESSES AND IMAGE PLATES
X-rays can be made visible by using films which have either fluoreszent properties or change their
chemical composition during irradiation. A prominent example for the latter option is the use of ionic
AgBr crystals which are pasted on a photo plate. The Ag+ Br− bond can be broken by an X-ray
photon, which leads to the reduction of Ag to atomic silver. Thereby a crystal defect is generated,
that will darken the film. A broad variety of other materials and techniques with different X-ray sensitivity and resolution exist, however all have in common, that a seperate visualization or digitalization
process is neccesary making it impossible to observe an experiment in situ.
3.2.2
C OUNTING TUBE
The counting tube is based on the ability of X-rays to ionize atoms in a gas by building pairs of
electrons and ions. Typical ionization energies of gas molecules are in the range of 10 eV, which
means that an X-ray photon with an energy of several keV will ionize a great number of atoms. If
a high dc voltage is applied over the gas volume in the tube, ions and electrons can be seperated an
lead to a current in the system, which is used as signal for measurement. In an ideal case, the strength
of this signal is proportional to the energy of the X-ray photon. This principle can be extended by
arranging several circuits in an array to form an area detector.
3.2.3
CCD CAMERAS
For direct digital processing of the Laue data, the combination of a szintillating screen and a CCD
camera can be utilized. The conversion of X-ray photons into detectable photons in the visible range
is carried out by the szintillator material Gadox (Terbium doped Gd2 O2 S) in the case of the CCD
detector used in this experiment. The underlying process is the generation of electron hole pairs inside
the material by the high energetic X-Ray photons that can than recombine by emission of visible light
at an Terbium activator. In the case of Gadox, the wavelength of emitted light is in the range between
382 nm and 622 nm with a maximum in the green region. The advantage of this material arises from
it’s high density and the high atomic number of Gadolinium which makes it a good interactor with
X-ray photons.
CCD detectors can thereby yield a high sensitivity due to a high quantum efficiency and the small
formation energy of electron hole pairs as compared to the ionisation energy of gas molecules. A
high spatial resolution given by the small pixel size and a comparably low cost can be achieved due
to the industrial availability of CCD chips.
10
4
S INGLE CRYSTAL X-R AY DIFFRACTION
The following chapter will give a short introduction on elastic scattering of X-rays by interaction with
electronic shells in a periodic lattice. The concept of an incident monochromatic plane wave Ei will
be used, which is scattered on a certain electron density ρ(r). This density is of course strongly related
to the lattice and atomic positioning of the crystal. The final step of an diffraction experiment will
then be to figure out the lattice from the measured X-ray intensity I(R).
If a plane wave Ei is ellastically scattered by an electron at position r, the electron becomes the starting
point of a spherical wave Es . Both amplitudes can be described by functions as follows:
Ei (r) = E0 · exp(−i(ω0 t − k0 r)
Es (r∗) =
Ẽ
· exp(−i(ω0 t − k0 r∗)
r∗
(4.1)
with ω being the angular frequency, k0 the wave wector and E0 , Ẽ the amplitudes. In the case of
several scattering centres or a electron density ρ(r), the total scattered intensity follows from the
overlap of all scattered waves. As can be seen from fig. 4.1 these waves differ by a certain phase
shift, which is given by their position of origin r. If an arbitrary point of origin "‘0"’ is chosen,
the scattered wave from a volume element at position r will have a path length difference of δs =
∆S1 + ∆S2 = (k · r)/k − (k0 · r)/k0 , when seen from a detector in direction of k. In the elastic limit,
k = k0 and δs becomes (k − k0 )/k · r.
Figure 4.1.: Principle of a diffraction experiment (from: Festkörperphysik - S. Hunklinger, 2007, Oldenbourg). An elastically scattered plane wave k0 → k has a path length different δs(r) which will lead to
constructive interference or extinction measured by a detector.
The contribution from the volume element dV at position r to the amplitude seen at the detector r∗
11
4. Single crystal X-Ray diffraction
gets thereby:
dEs = ρ(r)Es dV =
Ẽ
ρ(r) · exp(−i(ω0 t − kr ∗ +(k − k0 )r)dV
r∗
(4.2)
And the integrated scattering amplitude of sample volume V gets:
Ẽ
E = · exp(−i(ω0 t − kr∗)
r∗
Z
ρ(r) · exp(−i((k − k0 )r)dV
(4.3)
V
The volume integral can be identified with the Fourier transformation of the electron density ρ(r). If
the scattering amplitude is measured, it should be possible to carry out a backwards Fourier transformation in order to calculate this density and thus the atomic structure of the sample volume. This
is however not possible, since only the intensity I ∝ |E|2 and not the amplitude itself can be measured.
The accompanying loss of phase information is known as the "’phase problem"’ in crystallography.
Luckily one can still gather a lot of information from the intensity itself. Since the problem can not
be solved by direct calculation, it is usefull to compare a measured intensity with the calculated intensity I0 derived from known or calculated structures ρ0 (r). One possibility is thereby to search for a
matching intensity pattern in crystal structure databases. Similar structures can offer a good starting
point to construct a model that describes the own data. Another possibility is to try to solve the crystal
structure just by trial and error methods. Thereby complex numerical algorithms like the "‘charge flip
procedure"’ are used to find a matching structure starting from a completely random density ρ0 (r).
The convergence of such a procedure is of course particullarly depending on the data quality and good
single crystals are needed.
4.1
E LECTRON DENSITY OF PERIODIC LATTICES
After discussing general scattering at an unspecific charge density ρ(r), the special case of scattering
at an ordered crystal lattice will be considered in the following. The periodicity and symmetry of the
lattice ρ(r) = ρ(r + R), with R = ua1 + va2 + wa3 being a lattice vector will be used to expand the
charge density in a Fourier series:
ρ(r) =
X
·exp(iGhkl r)
(4.4)
h,k,l
with G being a reciprocal lattice vector and h,k,l integers. From translation invariance exp(iGhkl r) =
exp(iGhkl r + R) follows directly Ghkl ·R = 2πp, with p being an integer, which is exactly the definition
for the reciproke lattice.
If a scattering vector K = k0 − k is introduced, the scattering intensity can be written as (compare
with equation: 4.3):
2
Z
X
ρhkl h,k,l
exp(i(G − K)r)
|A(K)|2 = h,k,l
V
(4.5)
It can be see that the integral over an infinitely large volume V equals 0 because of the oscillating
12
4.1. Electron density of periodic lattices
character of the exponential function. This means a cancellation of the scattering intensity in all
directions that do not fullfill the condition G − K = 0. In the case of crystal volumes which are large
compared to the unit cell, a finite scattering intensity will thus only be measured if X-ray beam (k0 ),
crystal (Ghkl ) and position of detector (direction of k) are matching. Of course the scattered intensity
will be influenced by additional parameters in real experiments. First of all the source will show some
degree of polychromaticity (k , const) and beam divergence, secondly small crystallite sizes or the
finite penetration depths of x-rays as well as thermal movement of atoms will lead to a broadening of
scattered reflection peaks or decrease of intensity.
The scattering condition can be visualized nicely with the so called "’Ewald-sphere"’, see fig. 4.2 A.
Thereby all matching reciproke vectors can be found, that fit to an elastic scattering with |k| = |k0 |
which is represented by the radius of the "‘Ewald-sphere"’. In the picture shown, the condition is only
fullfileld for TWO reciprocal vectors (310), (5-60) and intensity could be measured in the direction
of k and k0 only.
Figure 4.2.: Ewald-sphere representing the scattering condition G−K = 0. A: For monochromatic X-rays
k0 ; B: for polychromatic X-rays k0 ∈ [kmin , kmax ]
Additionaly, the scattering condition can be transformed into famous Bragg’s law with:
dhkl = 2π/ |Ghkl |
(4.6)
with dhkl being the distance between two lattice planes (hkl).
|K| = |k − k0 | = 2k0 sin(θ) = 4πsin(θ)/λ
(4.7)
with λ being the wavelength of the incident X-ray beam and θ half the angle between k and k0 .
Introducing into the condition G − K = 0 yields:
2dhkl sin(θ) = λ
(4.8)
An equivalent description can be given by the so called Laue conditions, which is another transform-
13
4. Single crystal X-Ray diffraction
ation of the scattering condition G − K = 0. The scalar product of G = hb1 + kb2 + lb3 with the
lattice vectors ai yields due to the relation bi · aj = 2πδi j :
(k − k0 )/k · a1 = λh
(k − k0 )/k · a2 = λk
(4.9)
(k − k0 )/k · a3 = λl
Since k/k and k0 /k are known or measured during the experiment, h,k,l can be deduced by the knowledge of the lattice constants. Each of the three Laue conditions can be geometrically interpreted in
the picture of Laue cones. Therefore we consider the condition for k for which constructive interference will be observed. At fixed wavelength and setup geometry, k0 and the ai are constant. For a
particular lattice plane (hkl), each equation can be written as:
kai = const.
(4.10)
cos(α) · k · ai = const.
(4.11)
or
,with α being the angle between k and ai . Constructive interence can thus be possible if the scattered
X-ray beam k lies within a cone with specific opening angle α with respect to a lattice vector ai (see
therefore fig. 5.1 A in section 5). Note however that all three Laue conditions need to be fullfilled
at the same time. This means that constructive interference will only be observed if k points to the
intersection of three Laue cones generated by the three Laue conditions.
4.2
S TRUCTURE FACTOR
While the general scattering condition K = G could be explained as a property following from the
translation symmetry of the lattice, the influence of further symmetries and atomic properties will
show up in the intensity and disappearance of certain diffraction peaks. These information were
absorbed into the Fourier coefficients ρhkl in equation 4.5, which are given by:
Z
ρhkl = 1/VZ ·
ρ(r)exp(−iGhkl r)dV
(4.12)
VZ
with VZ being the unit cell volume. The electronic density ρ(r) needs to be considered as a function
of the atomic positions inside the base on the one hand, and the internal structure, e.g. atomic shells of
these atoms on the other hand. The latter is called atomic structure factor fatom and can be calculated
or deduced from collision experiments. In practical diffraction analysis, these factors will only be
taken from tables but not refined from the data, since the approximation of free atoms or ions will
often be sufficient for the interpretation of diffraction patterns. The positioning of atoms inside the
base however will have crucial influence on these patterns, since it gives an additional interference
between X-rays scattered by different atoms. If this kind of distinction between atomic positions and
atomic stucture factor is done, equation 4.12 can be written as:
14
4.3. Specifics of a Laue experiment
ρhkl
Z
1 X
1 X
·
exp(−iGratom )
·
fatom (G) · exp(−iGratom )
=
ρatom (r∗ )exp(−iGr∗ )dV =
VZ
VZ
V
atom
base
base
(4.13)
with ratom being the discrete atomic positions and the sum being executed over all atoms in the base.
Equation 4.13 can be simplified if ratom is written in terms of the unit cell vectors ratom = ua1 + va2 +
wa3 and the relations between R and G are used:
ρhkl = 1/VZ ·
X
fatom (G) · exp(−2πi(hu + kv + lw))
(4.14)
base
Depending on the atomic position and symmetries of the crystal, ρhkl can become 0 if the terms inside
the sum cancel each other out. An overview for several crystal symmetries and cancellation of reflexes
is given in the appendix.
4.3
S PECIFICS OF A L AUE EXPERIMENT
While for the derivation of this scattering theory, the assumption of an monochromatic X-ray source
(|k0 | = const.) was made, the situation at a Laue experiment is different. Here we are using a polychromatic or "‘white"’ X-ray source and thus the scattering condition is fullfilled for many different
reflexes at once. The picture of "‘Ewald-sphere"’ can however still be used to explain the outcome
of the experiment if k0 is replaced by an interval [kmin , kmax ] while kmax is given by the acceleration
voltage and kmin by the decreasing trend of X-ray efficiency for lower wavelengths (see section 3.1).
Instead of one Ewald sphere we now find a volume between the Ewald spheres corresponding to kmin
and kmax (see fig. 4.2 B). It can be seen that many reflexes will be found under different directions.
The benefit of this method is obvious: if an area detector is used, a lot of different reflexes will be seen
at once and the orientation and symmetry of the crystal under investigation can be deduced imidiately.
It should on the other hand be clear, that an advanced crystal structure analysis can not be carried out
by this method, since all the information, which is carried by the reflex intensity, is nearly lost because
of the wavelength dependent beam intensity.
15
5
C RYSTAL STRUCTURE AND STEREOGRAPHIC PRO JECTIONS
The basic concepts of crystal structure and lattice symmetries should be known and can be obtained
from standard literature. The following section will only discuss specific aspects which are particullarly important for the Laue experiment.
5.1
D EFINITION OF ZONES AND LATTICE PLANES
Real space straight lines inside the lattice can be represented as linear combinations of the unit cell
vectors ai : x = ua1 + va2 + wa3 as lattice vector [uvw]. Note that [uvw] does not only represent
the straight line between 0 and x but also any parallel line within the translation invariance of the
lattice. A lattice plane is spanned by two non parallel lattice vectors [u1 v1 w1 ] and [u2 v2 w2 ] and can
be defined by the straight line normal to this plane with Miller indices (hkl). These indices follow
from the zone euqations:
h · u1 + k · v1 + l · w1 = 0
h · u2 + k · v2 + l · w2 = 0
(5.1)
Note again that (hkl) does not only represent one plane, but any possible parallel lattice plane with
distance n · dhkl . Conversely the intersection of two planes (h1 k1 l1 ) and (h2 k2 l2 ) will give a lattice
vector [uvw] following:
u · h1 + v · k1 + w · l1 = 0
u · h2 + v · k2 + w · l2 = 0
(5.2)
Since the assignment of lattice vectors and plains is not unique, the term "‘zone axis"’ [uvw] is used
to describe the family of planes (hi ki li ) that all contain the axis [uvw]. It is important to note that a
zone [uvw] is represented as a vector in real space but as a plane (containing all reciprocal vectors
(hi ki li )) in reciprocal space. Vice versa a real space lattice plane (hkl) containing all perpendicular
vectors [ui vi wi ] is represented as vector in reciprocal space.
16
5.2. Projections
5.2
P ROJECTIONS
One of the main problems that need to be faced for the interpretation of Laue images is the distortion,
which occurs when the spherical diffraction pattern is recorded on a flat, two-dimensional detector.
In fact, the imaging situation is called "’gnomonic projection"’ since the center of projection is also
the center of the object to be projected. All the information coming from a Laue experiment are the
angles between different reflected rays (corresponding to different (hkl)) which are transformed into
distances between different spots once they hit the screen. Fig. 5.1 A shows the projection of one
cone corresponding to one zone axis [uvw] on the screen. One can see, that this zone axis has an
inclination of φ away from the plane of the screen as well as α in the plane of the screen causing the
cone to be projected as a tilted hyperbola.
Figure 5.1.: A: Principle of a back scattering Laue experiment: scattered intensity can be found on specific
cones (from: Structure of Metals - C.S. Barrett, 1943, Mc Graw Hill Book); B: Back scattering Laue
image of an arbitrarily oriented Li(Mn,Ni)PO4 single crystal; C: Laue image of an Si-crystal oriented in
111 direction.
The hyperbola becomes a straight line if the zone axis is parallel to the screen and a centered spot
if it is perpendicular to the screen. According to the Laue equations (4.9), constructive interference
only occurs if three cones intersect in one line. An example is given in fig. 5.1 B which shows the
back scattering image of an arbitrarily oriented Li(Mn,Ni)PO4 single crystal. As can be seen, all
intensity spots lie on intersections of hyperbolas or parabolas. Note, that the Laue equations are only
fullfilled for an arbitrarily oriented crystal, because the incident radiation is white. This means that
not only discrete Laue cones are possible, as would be the case for monochromatic radiation, but a
17
5. Crystal structure and stereographic projections
continuum. Hence a lot of cone intersections and thus spots on the screen are possible for any kind of
crystal orientation. Fig. 5.1 C shows the back-scattering diffraction image of a Silicon crystal which
is oriented in 111-direction. The intensity spots are symmetric with respect to this axis and a threefold
symmetry can be observed.
There are several options of how to find the angles between different zone axis and thus the orientation of the crystal under investigation. One is simply by calculation and transformation of the
zone dependend reflections from the laboratory system into the crystal system. While processing this
"’by hand"’ can be very time consuming, pen and paper methods have been developed to solve this
problem in a graphical way. The "’Greninger Net"’ is an easy tool to obtain meridian and parallel
coordinates of spots, similar to coordinates on a globe. It will be used during the lab course to find
the orientation of a simple cubic crystal.
Another approach comes from the opposite direction: If the crystal structure is known, a simulated
Laue pattern can be generated and compared with the measured pattern. Especially for complex
lattices, the use of suitable software can be very helpfull to obtain the orientation in a short time.
5.3
S TEREOGRAPHIC PROJECTIONS
Plotting the diffraction spots as a stereographic projection is a helpfull tool to find the crystal orientation. Stereographic means projecting a sphere onto a plane as it is for example done with two
dimensional images of the globe (see fig. 5.2 A). A stereographic projection is also angle preserving,
this means, that angular differences between diffraction spots will not change if the direction of projection is changed. Fig. 5.2 B shows the standard projection of a cubic crystal in (100) direction
(corresponding to 0◦ ). The (010) and (001) faces are perpendicular to the viewing direction and thus
can be found on the outer great circle (90◦ ).
Figure 5.2.: A: Stereographic projection of the globe (from Encyclopædia Britannica, 2010); B: Standard
projection of a cubic crystal with 100 direction corresponding to 0◦ longitude and latitude (from Laue
Atlas - E. Preuss et al, 1974, Wiley).
Similar to the globe, the position on a stereographic projections can be given by a longitude (Längengrad, δ) and latitude (Breitengrad γ). A so called Wulff net will be used during the lab course to find
the relative angles between different lattice planes of a crystal.
18
6
E XPERIMENTAL SETUP
A schematic sketch of the setup is given in fig. 6.1. The collimated X-rays from the tube pass the
detector through a centered hole and are scattered by the sample. The diffraction image is taken by
the detector in back scattering geometry.
Figure 6.1.: Schematic sketch of the experimental setup. The tube and beamline are positioned inside a
safety cabinet.
Generally a Laue experiment consists of 4 steps:
1. Adjustment of the sample on a goniometer and positioning in the beam line.
2. Starting of the X-ray source and data acquisition.
3. Stopping of X-ray source.
4. Data analysis.
Depending on the scientific question, this procedure than needs to be repeated several times, for
example to find the right sample orientation. While operation of tube and detector are straight forward,
as will be described below, the deal of a Laue experiment is the data analysis (e.g. determination of
19
6. Experimental setup
orientation, symmetry, etc.) and the correct translation of these information into the laboratory system
if, for example, the goniometer should be used to orient the sample in a specific direction.
6.1
S AMPLE MOUNTING
In general, single crystalline samples may not have a regular shape since they might be broken or cut
of bigger entities (e.g. rocks, polycrystalline fabric). Without any hints by visible external faces, the
sample will have to be rotated around three different axes to get into the desired orientation.
Several fixed samples will be provided which can directly be mounted on the goniometer column.
New samples can easily be glued on a goniometer head.
Figure 6.2.: Goniometer system. A: Coordinate transformation by z, y’, x” convention (1. Ψ, 2. θ, 3. Φ
rotation) (from: wikipedia); B: Coordinate system of the used goniometer.
In this experiment a goniometer will be used which allows for the orientation of the sample using three
Euler’s angles on the one hand and for a shift of the sample in three directions in order to hit a specific
spot with the X-ray beam on the other hand. The angular system used for this kind of goniometer
follows the z, y’, x” convention and is called yaw (Gier), pitch (Nick), roll (Roll) system. These
names origin from automotive engineering. Fig. 6.2 A shows the procedure of achieving a specific
orientation. The starting coordinate system (x,y,z) corresponds with the lab system (see fig. 6.2 B).
First, the sample, which is positioned at the origin of the system, is rotated around the z-axis (Ψrotation). Thereby the x- and y- axis are transformed into x’- and y’ axis. The next rotation is made
around the new y’-axis (θ- or R y -rotation) giving new x”- and z”-axis. The last rotation (Φ or Rx )
around the x”-axis gives the final coordinate system (red X=x”,Y,Z in fig. 6.2 A).
The option of linear shifting is implemented in the upper part of the goniometer. This means that it
does not influence the angular changes made by the Ψ,R y ,Rx rotations since it is also rotated. It can
be considered as a shift within the final "red" coordinate system X,Y,Z→X+∆X,Y+∆Y,Z+∆Z.
The last free parameter is the distance between sample and detector which can be adjusted on the
20
6.2. Automatic Psi-Circle operation
linear rail by moving the whole goniometer column. For all experiments however a fixed distance of
30 mm should be kept in order to ease the graphical and software supported data analysis. For this
purpose a special distance piece is available which enables a quick adjustment. Nevertheless special
care needs to be taken not to touch the scintillating layer on the detector’s surface. The metallic
distance piece might easily scratch and destroy this layer!
6.2
AUTOMATIC P SI -C IRCLE OPERATION
The goniometer base is rotatable by an automatic step drive. This corresponds to a Ψ-rotation of the
goniometer around the z-axis. The step drive can be operated even when the X-ray tube is running,
which enables to take images of different orientations without having to actually touch the sample and
goniometer. The drive can be controlled on the panel next to the pc. After the controler was turned on
(red on/off switch), the electronics should be reseted by switching the 3-position I/0/II switch to "0"
for few seconds and then back to position "II". The drive can now be activated by pressing the enable
switch. Pushing the direction switch will invert the direction.
It is absolutely neccesary to ensure that no part of the goniometer can collide with the detector due
to rotation, before using the automatic drive! Any collision or scratch might destroy the sensitive
detector surface.
6.3
T UBE AND GENERATOR OPERATION
The X-ray tube can and must only be operated if the protection cabinet is properly closed. Opening
of the cabinet during X-ray operation must be avoided.
The generator used in this experiment is a "Kristalloflex 710 X-ray Generator" by Siemens. It is
capable of applying a high voltage between 20 kV and 50 kV at a tube current of 5 mA to 40 mA
with a total output power of 2 kW. The cathode is heated with a power of up to 50 W. As mentioned
in chapter 3, most of the input power is converted to heat since the efficiency of the X-ray tubes used
in this experiment is very low. To cool off the waste heat, both generator and tube are connected to
cooling water. In order to operate the generator, the water valve needs to be completely opened. The
flow can be checked at the flow meter. A miminum flow rate of 3.5 l/min must be kept.
To protect the tube from damages, the high voltage needs to be set slowly. Depending on the tube’s
history, there are certain delay times during which the voltage needs to be kept constant. Tab. 6.1
Shows the retention times in dependence of the time that the tube has not been used. As an example,
if the tube has not been used for 5 days and a voltage of 35 kV shall be applied, it needs to be powered
up in 4 steps: Starting at the minimum voltage of 20 kV and minimum current of 5 mA followed by
120 s of waiting; Slow increase of voltage to 25 kV followed by 120 s of waiting; Slow increase of
voltage to 30 kV followed by 180 s of waiting; Slow increase to 35 kV followed by 180 s of waiting.
After this procedure, the current can be adjusted to the desired level. This should also be done in steps
of 10 mA with a retention time of 120 s.
Before switching off the tube, the current needs to be set back to the minimum of 5 mA. After that,
the voltage is to be set back to the minimum of 20 kV and the tube can be shut down. The values for
21
6. Experimental setup
Stop
(days)
0.5 to 3
3 to 30
>30
20kV
2
2
2
25kV
2
2
2
Retention time (min)
30kV 35kV 40kV 45kV
2
2
3
3
3
3
5
5
3
3
5
10
50kV
3
10
15
55kV
3
10
15
Table 6.1.: Tube retention time
high voltage and tube current can be read by pressing the respective black button "kV" and releasing
the button on the front of the generator. The heating current can be measured by pressing the button
"A".
Figure 6.3.: Front side of the hV generator.
The starting procedure is described in the following list step by step (see also fig. 6.3:
1. Turn on external cooling water supply and check water flow.
2. Switch on main switch on the cabinet panel and make sure cabinet doors are properly closed.
3. Set control potentiometers for high voltage (kV) and current (mA) to left minimum (20 kV, 5
mA).
4. Set keyswitch (generator main) to "ON"
22
6.4. Detector and Software
5. Press the red "Heating" key and wait for green "Operate" key (hV on) to light. The tube heating
is now activated. Wait for 120 s.
6. Press green "Operate" key (hV on). The tube is now running and emits X-rays.
7. Adjust the high voltage to the desired level in accordance to tab. 6.1 using the control potentiometer (kV). (max. 0.2 kV/s)
8. Adjust the current to the desired level using the control potentiometer (mA). Maximum steps
are 10 mA with 120 s retention time. (max. 0.1 mA/s)
The tube is now running and detector images can be taken using the software as is described below.
The shut down procedure is described in the following list:
1. Decrease the current to the minimum of 5 mA using the control potentiometer (mA). (max. 0.1
mA/s)
2. Slowly decrease the high voltage to the minimum of 20 kV using the control potentiometer
(kV). (max. 0.2 kV/s)
3. Press the red key (Heating). The high voltage is now off and it is safe to open the X-ray cabinet.
4. In order to switch off the heating, press the yellow button (off). This should only be done at the
end of the experiment.
At the end of the day or experiment, the generator and complete system (PC, detector) should be shut
down by using the key-switch and main-switch. The cooling water should be switched off by closing
the valve again.
6.4
D ETECTOR AND S OFTWARE
The CCD detector used in this experiment consists of two seperate CCD chips with 1392 x 1040
pixels and an active area of 85 mm · 110 mm each. The incident X-ray beam can pass the detector
through a middle hole between both chips. Each pixel is read out by an A/D converter giving a
12 bit intensity value. To minimize noise by stray radiation, both CCDs are placed inside a black
box. The front side is however covered with a light impermeable but X-ray transparent capton layer.
The szintillating Gadox layer and following optics are placed directly in front of the CCD chips.
Furthermore, thermal noise is reduced by cooling the chips with two peltier elements to -10 Â◦ C. To
ensure proper cooling, the detector should be switched on 20 minutes prior to image acquisition. The
power supply is activated automatically by turning the main-switch on (see fig. 6.3).
The read out and analysis software can be started with the "PSLView" icon on the computer’s desktop.
Before this is done, the detector should be running. To activate the data transfer, the detector can be
started by choosing the NTXLaue option in the camera menu of PSLView (see fig. 6.4). Fig. 6.5
shows the main screen of the software which consists of a toolbar on the left and the image display
on the right.
Several instances of the main screen can be opened, e.g. for comparison of different images. Acquisition of a new image however is only possible in the window which was started first.
Before acquiring an image, all options can be set in the area marked red. The measurement can then
be started by clicking the "play" button marked green. To enhance contrast, the maximum intensity
23
6. Experimental setup
Figure 6.4.: Start-up screen of the PSLViewer software. The Laue camera can be chosen in the "Camera"
menu.
corresponding to "white" can be set in the area marked magenta. The image option are explained in
more detail below.
Figure 6.5.: Main window of the PSLViewer software. The toolbar on the left will open several submenus. Acquired images are shown on the right
• Exposure: The image and spot intensity is mainly given by the exposure time. More integrated
intensity is always beneficial since it will allow a better interpretation of the measured patterns.
This is however limited by the maximum photon count of the CCD’s pixels. An overload
of intensity will firstly become noticeable in the center of an image since the back scattered
intensity is strongest for small angles. Furthermore, very long integration times (t > 600s) will
also amplify the CCDs characteristics e.g. dark current, hot pixels, etc. which then requires the
use of backround subtraction techniques.
• Acquisition mode: Besides the option of taking a single image, several consecutive images can
be taken. The respective intesities can either be summed up or averaged which reduces the
overall image noise.
• Binning: Using the hardware binning option will effectively combine the measured intensity
of several pixels, resulting in a higher sensitivity however smaller resolution. Since a higher
sensitivity will always allow smaller integration times, binning should be used up unto a level
24
6.5. Pattern simulation
that still resolves diffraction spots well enough. This means: perceptibility of the spot position
should not be affected and distinctness of neighboring spots should still be given. As the spotsize mainly depends on the size of the collimated beam, the binning should also be chosen with
respect to the used collimator.
Images can be saved by right clicking on the image area and choosing "SaveAs...". The standard
image format is *.tif which features lossless data treatment in a greyscale format. Note however
that the intensity or greyscale is not saved as an absolute value. To display an image correctly, the
maximum value corresponding to white and minimum value corresponding to black must be given in
addition. If needed, all images can also be saved as *.jpeg or *.bmp. Note however, that it is strongly
recommended to save every image in *.tif format since mathematical operations can only be executed
with this format.
Besides the task of image acquisition, the software is also capable of executing mathematical operations and image enhancement techniques as well as, more importantly, finding and refining the crystal
orientation of a sample with known lattice constants and symmetry.
6.5
PATTERN SIMULATION
With PSLViewer software
The peak symbol in the PSLViewer toolbar (marked magenta in fig. 6.6) offers the possibility of automatic peak detection. Depending on the parameters given, the image will be evaluated and possible
peaks will be marked with red circles. The threshold parameter in particular determines how big the
contrast needs to be between backround and peak in order to be recognized as such. In case a peak
detection is not desired, the threshold can be set to a high value, e.g. 1000.
The crystal structure symbol (marked blue in fig. 6.6) will open the symulation and fitting option. In
sub-menu "Crystal data", the lattice constants, angles and space group need to be entered in order to
simulate the theoretical pattern of such a crystal. The sub-menu "Instrument data" contains all the
parameters associated with the detector (e.g. area of detector and tilting angles). The only adjustable
parameter however is the sample detector distance. All other parameters should be kept at their values.
In order to simulate a specific direction in terms of hkl, the sub-menu "Find <hkl> spot" will rotate
the pattern into the right direction after pressing the "Find" button. The pattern can than be rotated
or shifted by pressing left or right mouse button and moving the mouse. Alternatively the orientation
can directly be set by entering orientation angles in the lower part of the menu. A detailed written
instruction is available in the laboratory.
With CLIP software
Alternatively to the techniques offered by PSLViewer, other software is available for the simulation
of patterns. The Cologne Laue Indexation Programme (CLIP) is free of charge and might be more
suitable to find a specific orientation, since the system of rotations is more user friendly. The software
can be started from the desktop. A detailed manual is available in the laboratory. Since CLIP has no
25
6. Experimental setup
Figure 6.6.: Main window of the PSLViewer software with shape detection and orientation menus.
26
6.5. Pattern simulation
direct link to the detector, images need to be imported either as *.png or *.jpeg. To do this, first adjust
contrast and brightness with PSLViewer and import the image in CLIP using the folder-like button
in the main window. It is neccessary to input the experiments geometry (detector width and height,
distance) in the configuration section (press screw-wrench button) for a correct simulation. Input of
all crystal data is analog to PSLViewer.
27
7
E XPERIMENTAL SCOPE AND ANALYSIS
Depending on the availability of single crystals, the scope of this lab course can be adapted. Task 7.1
is however mandatory and the second task should comprehend the relation between crystal structure,
symmetry and observed Laue images.
7.1
I DENTIFICATION OF THE ORIENTATION OF A CUBIC CRYSTAL AND
SAMPLE - DETECTOR DISTANCE
• Take several Laue images of one of the cubic crystals in arbitrary orientation and find suitable
values for binning and integration time of the detector at 30 kV, 20 mA tube power.
• Take an image of high quality which shows a spot of high symmetry. Therefore you might need
to re-orient the sample. Make sure the sample almost touches the spacer to have a well defined
sample-detector distance.
• Use PSLViewer’s simulation option to find the orientation of the crystal.
• Refine the sample-detector distance. This value will be needed for all further experiments.
7.2
C RYSTAL SHAPE AND CRYSTAL LATTICE
• Choose one of the regularly shaped minerals and take Laue images perpendicular to the visible
crystal faces with low binning and long integration time. Fix the sample to the goniometer head
in a way that allows to use the automatic rotation option. Make sure the samples centerline
coincides with the axis of rotation.
• Deduce the corresponding crystallographic planes using the PSL orientation software and make
a sketch of your sample that shows the visible faces and their crystallographic directions.
• What kind of symmetries can be found in the patterns?
• Explain the crystal shape with respect to the unit cell, atomic positions and symmetries.
• Find the positions of pregnant diffraction spots and create an ascii file with the pixel positions
(x [TAB] y). Make sure to quote enough spots so that the symmetries are still indentifiable
by these spots (around 50 per image should be sufficient if distributed well over the image).
You can also put the hkl index in a third column (x [TAB] y [TAB] hkl) (see example.txt on
desktop).
• Generate a stereographic projection by using the mathematica file "LaueVersuch.nb"
28
7.3. Crystal orientation and cutting
• Compare with a given standard projection. Explain the absence of specific diffraction spots by
the crystal’s symmetry.
7.3
C RYSTAL ORIENTATION AND CUTTING
• Find the crystallographic main directions (100), (010), (001) of an unknown sample and cut the
sample to a cuboidal shape.
• Explain the crystal’s symmetry and the resulting symmetry of the Laue images.
Important:
Whenever you take an image, write down all the parameters, such as: binning, integration
time, number of pictures and all the goniometer angles (Ψ of auto-drive, 2 angles of
goniometer head). Save all your images as *.tif and *.png files. If neccessary, save your
analysis/fits as screenshot. If you found an orientation, write down the position and indizes of
important diffraction spots and most importantly write down the three angles (α, β, γ), given
by the software, that define your orientation.
7.4
E VALUATION
Your experiment’s evaluation and discussion should contain print-outs of the important Laue images
as well as the stereographic projection you developed. To understand those images, a sketch of the
experimental setup containing the beam line, sample and detector should be added which shows the
geometrical arrangement of those objects and the meaning of longitude and latitude of a diffraction
spot. Furthermore the trigonometric functions should be given which transform a position on the
detector (e.g. x,y-pixel) into those angles.
To understand your diffraction images, you should explain your experimental approach as well as the
patterns themselves. Therefore, index the diffraction spots and add lines depicting any mirror planes
or other symmetry axes/planes. In case of the regularly shaped minerals, you should also make a
sketch of the sample and index the visible faces with their crystallographic directions. You should
also have a look at the crystal structure (see *.cif files) in order to explain the crystal’s morphology
in terms of atomic bonds or distances within the lattice. Therefore compare the real space crystalstructure with the shape of the mineral you investigated. Can you convey reasons from the atomic
positioning and bonds why it might be favorable to develop the visible crystal faces?
Calculate the maximum values hkl that are theoretically visible for the acceleration voltage you applied. (Compare minimum wavelength and minimum lattice distance dhkl ). Is this in agreement with
your experimental observations?
Discuss the possible measurement errors. Which parameters of the geometry are unknown? How will
a tilted detector / poorly collimated beam or variable sample-detector distance effect the pattern? Did
you experience any effects of crystal imperfection and defects?
29
A
A DDENDUM
A.1
S YSTEMATIC DISTINCTION OF REFLEXES
Systematic distinction of reflexes by the Bravais type lattice
centering
primitive
A-centered
B-centered
C-centered
reflex
-
condition
no cancellation
k+l = 2n
h+l = 2n
h+k = 2n
centering
reflex
condition
body centered
face centered
rhomboedric
hkl
hkl
hkl
h+k+l = 2n
h+k=2n, h+l=2n, k+l=2n
-h+k+l = 3n
Zonal distinction of reflexes by mirror glide planes
glide plane
a
b
n
d
a
c
n
d
b
c
n
d
position
(001)
(001)
(001)
(001)
(010)
(010)
(010)
(010)
(100)
(100)
(100)
(100)
glide vector
a/2
b/2
a/2 + b/2
a/4 +/- b/4
a/2
c/2
a/2 + c/2
a/4 + c/4
b/2
c/2
b/2 + c/2
b/4 +/- c/4
reflex
hk0
hk0
hk0
hk0
h0l
h0l
h0l
h0l
0kl
0kl
0kl
0kl
condition
h = 2n
k = 2n
h+k = 2n
h+k = 4n, h = 2n, k = 2n
h = 2n
l = 2n
h+l = 2n
h+l = 4n, h=2n, l=2n
k = 2n
l = 2n
k+l = 2n
k+l = 4n, k = 2n, l = 2n
Serial distinction of reflexes by screw axes
screw axis
21 ; 42 ; 63
31 ; 32 ; 62 ; 64
41 ; 43
61 ; 65
21 ; 42 ; 63
31 ; 32 ; 62 ; 64
30
position
[100]
[100]
[100]
[100]
[010]
[010]
reflex
h00
h00
h00
h00
0k0
0k0
condition
h = 2n
h = 3n
h = 4n
h = 6n
k = 2n
k = 3n
screw axis
41 ; 43
61 ; 65
21 ; 42 ; 63
31 ; 32 ; 62 ; 64
41 ; 43
61 ; 65
position
[010]
[010]
[001]
[001]
[001]
[001]
reflex
0k0
0k0
00l
00l
00l
00l
condition
k = 4n
k = 6n
l = 2n
l = 3n
l = 4n
l = 6n
A.2. Guide for the generation of stereographic projections
A.2
G UIDE FOR THE GENERATION OF STEREOGRAPHIC PROJECTIONS
The provided mathematica script (LaueVersuch.nb on the desktop) offers a basic method of calculating longitude (δ) and latitude (γ) from the pixel position of a diffraction spot. The input format is
an ascii textfile formated as "x-pixel-position" [TAB] "y-pixel position". Multiple diffraction spots
are written in seperate lines. (As an example: see example.txt on the desktop). If needed, additional
information as the associated (hkl) indizes can be written in a third column. Experimental parameters
as detector-sample distance and resolution (number of pixels) are given in the first block and need to
be adjusted. The second block concerns the processing of pixel lists. Therefore you need to change
all the directories in the script to your data directory which contains your lists. Running the script
will output several *.txt files containing corresponding latitude and longitude of the spots. The third
block outputs a graphic of the stereographic projection as well as a list of positions projected on a
2D-plane. Before altering the script, please save a copy in your data directory and leave the original
script unchanged.
In case you want to merge several diffraction images in order to get one big stereographic map of your
system, it might be convenient to start with one centered image and arrange all other images around
it. It is recommended to follow these steps:
1. Process all spot-lists with the mathematica script in order to get longitude/latitude files.
2. Choose one image and one crystallographic main direction, that you want to have in the center of your projection. Find out the longitude and latitude of this particular spot. Since the
stereographic projection is angle preserving, you are free to shift your spots by any angle. It
will not change the angular differences between individual spots. To center your chosen crystallographic direction, subtract the longitude and latitude of the corresponding spot from every
coordinate in your list. Mathematica allows to directly subtract lists from each other. One way
is to generate a list with arbitrary length, only containing the angles you want to subtract, using
the "Table" command: NewList = OldList − Table[{latitude, longitude}, {Length[OldList]}].
The "Length" command gives the number of entries in a list.
3. In order to merge two or more data sets from Laue images, it is most convenient to find one
diffraction spot, which is present in two images respectively. One of the data sets can then be
corrected by the angular difference of this spot in both images.
{lat1, long1}: angles in image 1 of one spot, {lat2, long2}: angles in image 2 of same spot.
An arbitrary spot can thereby be transformed as:
{lat, long}image1 = {lat, long}image2 − {lat2 , long2 } + {lat1 , long1 }
You can use similar operations on lists to calculate the new spot positions as given above.
4. The merging step can either be done by manually copying all corrected values from several
lists into one list (text editor) or by using the "Join" command, which will do the merging step
within Mathematica in the form: Join[list1, list2, ...].
5. Plot a stereographic projection of the merged images by using the third block in the mathematica script.
This method however only allows for the merge of images that have been taken in the same coordinate
system. In practice this refers to images of a sample that has only been re-oriented by changing
31
A. Addendum
it’s longitude and latitude with respect to the beam-line. Use of the automatic Ψ-drive for example
corresponds to a change of longitude of the sample. Hence images can easily be merged. If the sample
was however rotated around the beam-line direction, two pictures can not be merged by modulating
their latitude and longitude (see fig. A.1. Moreover the additional rotation needs to be taken into
account. This can be done by re-rotating an image in the first part of the mathematica program
around the beam axis by an angle µ (korrMu).
Figure A.1.: Consequences of γ-, δ-rotation and µ-rotation on the orientation of Laue images.
32
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement