Practical course B41
Investigations on crystals using a 2-circle reflection goniometer in addition with x-ray techniques
Introduction
There exists a strong coupling between inner structure of periodically assembled condensed matter, its symmetry and thus resulting anisotropy of the physical properties. The morphology of a crystal mirrors, within
certain limits, the inner structure of a crystal as a three-dimensional periodic lattice. Natural or artificial grown
crystal facets are in general not of the same size and quality. Nevertheless, the angular orientation of the
plane normal of the facets by an individual single crystal is an invariant dimension within one sort of crystal.
This relationship was found in 1669 by the Danish scientist Niels Stensen (Nicolaus Steno) and is called
"Steno's law of constant angles". It allows us to characterize the symmetry of one type of single crystal independent from its morphological shape.
Our practical course is divided into two parts. At the first part we perform a crystallographic description of a
crystal polyhedron by getting information about its crystallographic point group. This is done by a stereographic projection of all angular relationships of crystal facet normals of one single crystal, measured by a 2circle reflection-goniometer. Because of the strong coupling of morphology and inner fine structure, we can
get distinct information about the lattice metric by the use of the “equation of axial sections”.
In the second part, we have a closer look at the inner structure of the crystal. This will be done by using x-ray
Bragg-scattering with a distinct wavelength at lattice planes of the crystal. This gives us information about the
symmetry, the crystal metric by lattice parameters (edge length and angles) and finally, by taking into account
extinctions, about the space group of our crystal.
Preliminary works to practical course B41
As a physicist you should have detailed knowledge about production of x-rays and the resulting spectrum of
an x-ray tube. Furthermore, absorption processes of x-rays by materials should be well understood.
Our ambition is to show you how the inner ordering of crystalline material can be extracted by using different
x-ray experiments and how it is related to the macroscopic morphology of a crystal. The aim of the course is
to demonstrate the specific properties of the ordered state of crystalline material in different experiments.
The main problem is that there is no method to visualize the internal order of the crystal directly, but we have
to detect its ordering from different physical effects. For that we design a mathematical model - which is
accepted after long experience - but our task will be to test it experimentally.
The model is the threefold periodical pattern of the atoms. For your preparation we put some questions about
which you should have thought about resulting in terms for which we need some precise definitions:
1.
2.
3.
4.
5.
6.
7.
8.
What is a crystal lattice and how can it be described?
What is symmetry and what kind of symmetry elements exist?
What is a crystallographic point group, what a crystal class?
In which way are Weiss Indices and Miller Indices connected to each other?
Which association exists between the reciprocal and the direct crystal lattice?
Which physical phenomena is described by the "Braggs Law"and "Ewald construction"?
What is the definition of the "structure factor"?
What's the meaning of systematic extinctions in terms of x-rays and why does they exist for different types of symmetry-elements.
If you thought successfully about all these questions and having at hand a written note on it (please do not
copy the handout of the practical course) you are sufficiently prepared for the course.
What you couldn't find out clearly, please prepare in written form for the initial discussion.
Part 1: Macroscopic consideration
In the first part of the experiment the angular relations among the morphological facets of a crystalline polyhedron are determined by means of a reflection-goniometer using visible light. A quantitative projection in
form of a stereogram using Wulff's net is used to find out the point group symmetry of the polyhedron. The
symmetry elements observed in the stereogram allow to find a crystal adapted coordinate system and with
the help of the axial section equation important relations for the lattice metric can be found.
1.1
The two-circle goniometer
Figure 1 shows a drawing of the principle arrangement of a two-circle reflection goniometer, the photo presents the apparatus used in the course. The crystal K placed on a goniometer head (G, cf. Fig. 2) can be observed by a microscope if it was moved into the light beam. Using the rotation circles a crystal facet can be
brought into the reflection position. For fine adjustment a parallel light beam coming from the light source Q
is used. If the normal of a crystal facet bisects the angle between the incoming light beam and the observation
telescope direction (O) the crystal facet is in observation position. This can be observed as a bright light reflection dependent on the quality of the crystal facet.
Fig. 1:
Principle drawing (left) and photo (right) of a two circle goniometer. F: micro adjustment, G: goniometer head, K: crystal, O: ocular/telescope,
Q: light source, R: button for insertion of the micro adjustment, Dφ, ρ: rotation circles, Z: button for the additional optics.
To determine the angles of the reflection positions for all facets, the crystal is arranged in the intersection
point of two orthogonal rotation axes. The table of the apparatus contains the rotation circle Dρ for the rotation in the horizontal plane. The scale at the circle shows the
angle ρ (pole distance). The circle Dρ is coupled with a further rotation circle Dφ, the
vertical circle which allows to set the azimuth angle. Combined rotation about Dφ and
Dρ each crystal facet can be brought into a reflection position, which is described by
a pair of angles (φ, ρ).
Both rotation circles have a micro adjustment button (F) on the side. After arresting
the button using a screw at the opposite position the angle can be adjusted in steps
of a hundredth degree. While adjusting a reflecting facet it can be recognized that the
crystal may be turned slightly without changing the light reflection noticeably. For the
exact fixing of the reflection position in front of the bulb a cruciform aperture is
Fig. 2: Goniometer head
mounted on which is focused by an additional optic built within the telescope and to
with glass filament and
glued crystal
be put in the light path by a small button (Z) at the side of the telescope. If the normal
of a polyhedron facet is adjusted correctly the crystal maps the cross in the middle of
the ocular (O). If necessary, the crystal and the mapped cross can be re-adjusted by the two micro adjustment
buttons.
Mount the goniometer head with the crystal carefully on the reflection goniometer (Don't lose the crystal!).
By the crossed tilt mechanisms (K) and the translation slides (T) (to re-center the crystal) of the goniometer
head (Fig. 3) it is possible to adjust the crystal such that a macroscopic visible important zone axis (edge
direction) is parallel to the rotation axis Dφ, the facet normals of all facets of the zone then will have a ρ-value
of 90°; otherwise some rolling of the stereographic projection may be necessary.
The (φ, ρ) angle pairs for all for all facet normals can now be determined.
Using Wulff's net (a stereographic projection of the geographic net of a sphere into a plane, containing the
north-south pole axis, see Fig. 4 and 5) all facet poles given by their angle pairs can directly be turned into a
stereographic projection of the crystal polyhedron.
Fig. 3 and 4: Principle of stereographic projektion, Wulff's Net with 2° angular scaling of mayor- and minor-circles
1.2 Stereographic projection and Wulff’s net
A clearly arranged presentation of the angular relations between the facet normals is given by the stereographic projection. Figure 5 shows the idea of this construction: The crystal is put in the midpoint of a unit
sphere. The penetration point P of a facet normal is connected by a straight line with the pole of the opposite
hemisphere. The intersection point of this line with the equatorial plane P' is the projection of P. The stereographic projection is an isogonal and circle-related mapping of the sphere surface onto the plane. For a quantitative presentation of the angle relations of the facet normals as
a stereogram helps Wulff's net (see Fig.4). It is generated by a
stereographic projection of the geographic net (meridians and
latitudes) of a sphere into a plane. If the two poles are projected
on one point we will get a net of concentric circles and a star of
straight lines from the midpoint. If the projection contains the
north-south pole axis (N'-S') we have Wulff's net (Fig.5), the advantage of which is the projection of a set of great circles by which
any two facet poles of a hemisphere can be connected. For the
measured (φ,ρ) values a stereogram of the crystal polyhedron can
be constructed using Wulff's net. To do so, a transparent paper is
Fig.5: Prinziple of Wullf's net
pinned at the midpoint of Wulff's net and as an origin of the azimuth angle φ the intersection of the line E'W' with the equator circle. To plot the facet poles the transparent
paper is rotated about the midpoint of Wulff's net by the azimuth angle φ and then starting from the midpoint
the pole distance ρ is taken.
1.3 Determination of a crystal-adapted coordinate system
After plotting all measured facet poles the first step is to look for zones, i.e. sets of facet poles which lie on
one great circle, and for symmetry directions. This information may help to choose a crystal-adapted basis.
The main task is to recognize the point symmetry of the crystal polyhedron. Possible symmetry elements are
rotation axes (twofold, threefold fourfold and sixfold) and the corresponding roto-inversion axes (the inversion center 1 and the mirror plane m inclusive). Note that only symmetry elements which pass the projection
midpoint can be observed directly. To identify symmetry elements in other directions the projection has to
be turned over.
By symmetry axes six lattice symmetry groups can be identified. A further symmetry group (triclinic = anorthic)
has no symmetry axis but only a center of symmetry. So we have altogether seven lattice symmetry groups.
They are listed in Table 1. Note well: The metrical conditions of the axial systems are the result of the lattice
symmetry, not conversely!
The symmetry directions are merged to classes of symmetrical equivalent symmetry directions, the so-called
“Blickrichtungen”. Each symmetry group of a polyhedron, has at most three “Blickrichtungen”. All symmetry
direction of one “Blickrichtung” contain equivalent symmetry operations. It is thus sufficient to list the rotation axes and reflections for each “Blickrichtung”. Rotations are described by their order (2, 3, 4, 6). An over
lined order means a roto-inversion operation generated by a rotation of this order multiplied by the inversion
-3, -4, -6. The rotation axis -2 corresponds to a mirror plane and is notated by m. If a rotation axis exists
parallel to a normal of a mirror plane this is marked by a slash, e.g. 2/m for a mirror plane orthogonal to a
twofold axis.
The crystal class is thus fixed by symmetry elements along well defined directions. Crystals need not have the
full lattice symmetry; if this complete symmetry occurs as given in Table 1 we call this the full form of the
crystal system (holohedry). A complete list of the 32 point groups (point symmetries, crystal classes) shows
Table 2.
Examples:
1) The triclinic axis system is compatible with trivial symmetry element 1 and the inversion 1. Additional symmetry elements lead immediately to an axial system of higher symmetry. If the crystal is bounded by pairs of
parallel facets (pinacoid), the resulting point group is -1. Does it show only singular facets (pedion), it belongs
to the point group 1.
2) Due to the monoclinic angle, the monoclinic crystal family shows only one symmetry relevant direction and
the holoedry is 2/m. According to a time-honored convention this axis is named as b-axis or [010]. If only 2 or
m in one direction occur, a monoclinic axis system has to be chosen as well. The monoclinic family thus consists of the three point groups 2/m, 2 and m in [010]-direction.
3) If a crystal polyhedron shows a twofold and a threefold axis which are not perpendicular to each other, the
cubic system is present (23 is the least cubic point group). The twofold axis is used as a-axis ([100]) application
of the threefold axis on this direction gives the b- and c-axis ([010] and [001]). The threefold axis then points
in the space-diagonal direction ([111]). The application of the three twofold axes on the [111]-direction gives
the directions of the further three space diagonals of the cube.
Table 1: The six crystal families / seven crystal systems
Crystal system
Short description Symmetry elements of the holohedry Restrictions on Axial relametrical pationship
1. direction 2. direction 3. direction
rameters
Triclinic = anorthic (a) no symmetry direction
[-1]
a,b,c
α,β,γ
monoclinic (m)
One 2-fold r.ax.
a/o mirror plane
[010]
2/m
a,b,c α=γ=90°, a/b : 1: c/b
β>90°
orthorhombic (o)
2-fold r.ax. a/o
[100]
mirror plane in
2/m
three orthogonal
directions
[010]
2/m
tetragonal (t)
one 4-fold rotation axis
[001]
4/m
[100]; [010] [110], [110] a=b ≠c
2/m
2/m
α=β=γ=90°
c/a
trigonal/
rhombohedral (r)
one 3-fold rotation axis
[001]
-3
[100];[010],
[110]
2/m
a=b ≠c
α=β=90°,
γ=120°
c/a
hexagonal (h)
one 6-fold rotation axis
[001]
6/m
[100];[010], [110]; [210], a=b ≠c
[110]
[120]
α=β=90°,
2/m
2/m
γ=120°
c/a
cubic (c)
four 3-fold rotation axes under
109.5°
[001]
2/m
a,b,c
α=β=γ=90°
[100]; [010]; [111], 111], [110]; [110]; a=b=c
[001]
[111], [111] [101]; [101]; α=β=γ=90°
4/m
-3
[011]; [011]
2/m
r.ax. = rotation axis, a/o = and /or
a/b : 1: c/b
a/b : 1: c/b
1
Note: As each rhombohedral lattice contains a hexagonal lattice and each hexagonal lattice contains a rhombohedral lattice, the hexagonal and the
rhombohedral crystal system (defined by their lattice symmetry groups) are in a special relation and both together are considered as a crystal family.
The subgroups of the rhombohedral symmetry can occur in combination with a rhombohedral lattice or a hexagonal lattice. In the latter case the
name “trigonal” crystal is in use. Usually, even in the rhombohedral case a hexagonal basis is chosen and the lattice vectors pointing to the one and
two third points of the long space diagonal of the hexagonal cell are considered as “R-centering” of the hexagonal cell. These additional centering
points reduce the hexagonal point symmetry to a rhombohedral group. In this case the second or third hexagonal symmetry direction loses its twofold
symmetry. To mark this case a “1” is introduced in the symbol, e.g. 3m1 or 31m. For the comparison with cubic crystals a rhombohedral primitive
description with a basis characterized by a=b=c, α=β=γ is used, but the determination of these basis vectors is difficult in an early stage of the crystal
investigation, as the basis vectors do not coincide with symmetry axes.
Table 2: Point groups and crystal systems
Crystal system
Point groups
holohedry
(short symbol)
subgroups
triclinic
-1
1
monoclinic
2/m
m, 2
orthorhombic
2/m 2/m 2/m
(mmm)
mm2, 222
tetragonal
4/m 2/m 2/m
(4/mmm)
-42m, 4mm, 422, 4/m, -4, 4
rhombohedral
-3 2/m
(-3m)
3m, 32, -3, 3
hexagonal
6/m 2/m 2/m
(6/mmm)
-6m2, 6mm, 622, 6/m, -6, 6
cubic
4/m -3 2/m
(m-3m)
-43m, 432, 2/m-3 (m-3), 23
1.4 Axial relations and Miller indices
Once having chosen a basis, the analytic description of the crystal facets can be done, i. e. the determination
of the linear form:
L (xyz) = h · x + k · y + l · z.
Each special value of the linear form L (xyz) = const. fixes a plane. But, according to the law of constant angles,
the exact position of the plane is insignificant for our investigation. Only the coefficients (hkl) are important
to characterize the crystal polyhedron and because of the arbitrary constant, the ratio h : k : l is relevant,
alone. The law of rational proportions, given by Ch. S. Weiss, tells that the coefficients of the linear form have
integral proportions h : k : l if the coordinates are described with respect to a symmetry related basis (see
above). (This law does not hold in general, if the linear form is described with respect to an orthonormal
basis!) These integral co-prime coefficient triplets are called Miller indices and the procedure of their determination is called the “indexing” of the crystal.
The first step in the indexing procedure is to identify the type of a facet: If a facet (with const.≠ 0) does not
intersect an axis, this Miller index has to be zero (e. g. a facet which does not intersect the b axis is of type
(h0l)). Facets which are parallel to two basis vectors are already completely indexed (e. g. (100) for a facet
parallel to b and c. NB (200), (300) … are no meaningful Miller indices, but (100) is. Why?). The Miller indices
of all further facets can be determined using the axis intercept equation:
h : k : l = a cos(ϕa) : b cos(ϕb) : c cos(ϕc)
(1)
It combines the Miller indices (hkl) with their axis intercepts, i. e. intersection points of the plane with the
axes of a symmetry adapted coordinate system (see Table 1). φa, φb and φc denote the angles between the
facet normal and the corresponding axes along the base vectors. Their connection with the angles φ and ρ of
the goniometer measurement has to be considered below. Further the metrical parameters a, b and c have
to be fixed, i. e. the lengths of the basis vectors. From equation (1) only their proportions can be found, so
arbitrarily b=1 is fixed by convention (“virtus in medio”). If the classification of facets into types (see above)
produces a well measured facet of type (hkl) without 0, this facet is supposed to be a (111)-facet and is used
to determine the proportion c/a. If there exists no such facet, two facets of special position have to be chosen
(e. g. a (hk0) → (110) and a (0kl) → (011) facet). From the axis intercept equation (1) we get:
ℎ
𝑘
=
𝑎 cosϕ𝑎
1∙cosϕ𝑏
𝑎=
ℎ cosϕ𝑏
k∙cosϕ𝑎
=
cosϕ𝑏
cosϕ𝑎
𝑙
𝑘
=
𝑐 cosϕ𝑐
1∙cosϕ𝑏
𝑐=
𝑙 cosϕ𝑏
k∙cosϕ𝑐
=
cosϕ𝑏
cosϕ𝑐
The relations between the angles φa, φb and φc of the facet normal and the goniometric results (φ, ρ) can be
determined by the “spherical Pythagoras” (The cosine of the hypotenuse
in a spherical rectangular triangle is equal to the product of the cosines of
the cathetus). Figure 5 demonstrates the relationship for the monoclinic
case.
Considering the triangle ① we get the following term (2):
cos ϕ𝑐 = cos (90° − 𝜌𝐹 ) · cos (φ𝐹 − φ𝑐 ) = sin 𝜌𝐹 · cos (φ𝐹 − φ𝑐 )
and from triangle ② we have (3):
cos ϕ𝑎 = cos(90° − 𝜌𝐹 ) · cos(φ𝑎 − φ𝐹 ) = sin 𝜌𝐹 · cos(φ𝑎 − φ𝐹 )
Fig. 5:
Determination of the axial
intercepts.
The angle ϕ𝑏 coincides with the angle ρ. The angle  between the coordinate axes depends on the choice of the basis and can be determined easily.
So we get the axis intercepts in this case:
𝑎=
𝑐𝑜𝑠 ϕ𝑏
cos 𝜌𝐹
=
𝑐𝑜𝑠 ϕ𝑎 sin 𝜌𝐹 ∙ cos(φ𝑎 − φ𝐹 )
𝑐=
𝑐𝑜𝑠 ϕ𝑏
cos 𝜌𝐹
=
𝑐𝑜𝑠 ϕ𝑐
sin 𝜌𝐹 ∙ cos(φ𝐹 − φ𝑐 )
After the determination of a and c and the equations (2) and (3) with ϕ𝑏 = ρ for any facet the ratio h : k and
k : l can be calculated and finally the Miller indices can be developed.
In the experiment a practical procedure is helpful to determine the angles φa, φb and φc of a facet directly. For
each axis which coincides with a symmetry direction the equivalent facet poles must form the same angle
with the axis. Such a direction can be fixed in the midpoints of the corresponding facet poles. For other directions the connection between the facet normals of the facets (100), (010) and (001) and the axes may be
helpful. The facet (100) is spanned by the b- and c-axis, analogously for the two other basic facets. After fixing
of the three basic facets the direction of the three basis vectors a, b and c can be found by drawing three a
great circles centered by these facet poles (90° distance). The intersection points of the three great circles
define the directions of the basis vectors.
The drawing using Wulff's net helps to recognize the symmetry of the crystal. The (weak) precision of the
drawing makes it not advisable to use it for the evaluation of the axial ratios except for a first approximation.
(The angles are measured with a precision of about a tenth of a degree and the drawing will allow one degree
at most.) The evaluation of the axial ratios and the determination of the Miller indices thus should be done
numerically. For that it is helpful to transform the polar coordinates (ρ, φ) of the facet poles into vectors of
Cartesian coordinate system:
xc
yc
zc
=
=
=
cos(φ)sin(ρ)
sin(φ)sin(ρ)
cos(ρ)
If the crystal has a rotation axis each symmetrical equivalent facet pole must have the same angle with the
rotation axis. Its direction can be fixed as the midpoint of the corresponding facet poles. If there are more
sets of symmetrically equivalent poles with respect to this axis, the procedure may be repeated and the results may be averaged.
If there are no rotation axes the directions of the basis vectors can be derived as given above, but at first the
facets (100), (010) and (001) have to be chosen. They should belong to two well occupied zones. The facet
(100) is spanned by the b- and c-axis, analogously for the two other basic facets. After fixing of the three basic
facets the direction of the three basis vectors a, b and c can be found by drawing three a great circles centered
by these facet poles (90° distance). The intersection points of the three great circles define the directions of
the basis vectors:
a
b
c
⊥
⊥
⊥
(010)
(001)
(100)
and
and
and
(001)
(100)
(010)
These geometrical conditions allow to use the vector product to determine the directions of the base vectors
a, b, c from the facet poles.
a
b
c
=
=
=
(010)
(001)
(100)
×
×
×
(001)
(100)
(010)
Note that the vector product is not commutative! The vectors calculated in this procedure are unit vectors
parallel to the basis vectors. Now the angles φa, φb and φc can be calculated from the scalar product between
the vectors of the facet poles and the basis vectors.
As all vectors are unit vectors the denominators are 1.
Tasks for part 1. (Summary):








Acquaint yourself with the different mechanical parts of the reflection goniometer.
Measure the (φ, ρ) angle pairs for all reflecting facets of the crystal.
Draw a stereographic projection of the facet normals of the crystal using Wulff's net.
Determine the symmetry elements of the crystal from the stereographic projection and subsequent
the crystal class to fix the coordinate axes.
Mark the sphere-intersection points of the a-, b- and c-axes.
Having chosen the basis, using important zones if necessary, determine the axial ratio a : 1 : c and
the Miller indices of all facets.
Calculate the angles φa, φb and φc for all facet poles.
Perform the a-, b- and c-axes evaluation numerically.
Part 2:
X-ray investigations on a microscopic scale
In this part we will examine the crystal using x-ray diffraction. “Diffraction means any deviation of the light
from its straightforward path of rays as far as it is not to be considered as reflection or refraction.” (A. Sommerfeld, Optik, p.156) As the refraction index differs about some per mill only from 1.0, in the case of lighting
of crystals by x-rays we can consider the reflection and refraction effects as per-mill effects and attend completely to the diffraction and interference effects.
2.1 Basics of x-ray scattering
2.1.1 Fourier space – reciprocal space
Interaction between photons and matter causes a change in propagation direction of the incident primary
wave. By the electric field of the incident electromagnetic wave the electrons are enforced to vibrations.
After the principles of classical electrodynamics those vibrating electrons irradiate electromagnetic waves by
themselves (dipole radiation). Those secondary waves have the same frequency as the stimulating ones, they
will interfere each other. Every atom itself is acting as a scattering center for the incident beam and the
overlay of the coherent secondary waves yields the scattering pattern. Taken into account the "Fraunhofer
approximation", the diffraction pattern corresponds to the Fourier transform of the diffracting object. In
terms of a crystal, the “diffracting object” is a periodic electron density the resulting interference pattern is
a three dimensional “reciprocal lattice”. The position of the reciprocal lattice points can be derived by the
vectorial Laue equations. The reciprocal lattice can be described by a set of base vectors a*, b*, c* within the
same crystal coordinate system as used for the description of the direct lattice.
2.1.2 Bragg-Equation
Maxima of intensity exist in the scattering space if the path difference of the scattering waves is an integral
multiple j of the wavelength (see fig. 6). The geometrical relationship between possible scattering maxima
and the scattering lattice planes is given by the Bragg equation:
2 dhkl sin Θ = j λ
dhkl
λ
Θ
distance of lattice planes
wavelength
incident angle of x-rays, according to the lattice
planes (not to the surface!)
To fulfill the Bragg condition, lambda has to
be smaller than double of the lattice plane
distance. This factor limits visible reflections for an experimental setup using a
fixed wavelength. According to this, there
is no appearance of Bragg reflection by using visible light as a source.
Fig 6: X.-ray scattering on a set of lattice planes, characterized by the reciprocal
lattice vector G0. Direction of G0 is defined by the normal of the lattice planes, their
distance by d = 2π/G0.
2.1.2 Laue-Equations and the Ewald construction
If we describe (as in fig. 6) the incident wave vector as k and the outgoing wave vector as k', a scattering
vector ∆k = k' – k can be defined. For getting maxima within the interference pattern, all 3 Laue equations
have to be fulfilled for integer values of h, k, l. This applies only in case of ∆k is correlata⋅∆k = 2π h
ing with a translational vector G of the reciprocal lattice.
b ⋅∆k = 2π k
c ⋅∆k = 2π l
(Laue- Equations)
Fig. 7: Ewald – construction in 2D for a (-230)Reflection in scattering condition
A descriptive geometrical representation of this issue in reciprocal- or Fourier space is given by the Ewald construction (Fig.
7). As the incoming- and the scattered wave vector have the
same length, the radius of the Ewald sphere has the value of
2/the orientation of the sphere is given by the orientation
of the incoming wave vector. The angle between the primary
beam and the diffracted beam is conventionally named 2θ. If
the crystal and thus the reciprocal lattice is rotated in a way that
the beginning and the end of a reciprocal lattice vector simultaneously lay on the surface of the sphere, scattering condition is
fulfilled and k' can be measured under the angle of 2θ with respect to the incoming beam.
The position of all Bragg-reflections in reciprocal space are
solely specified by the translation vectors of the crystal lattice
(Laue equations, Bragg equation).
G hkl 
2
d hkl
The knowledge of all Ghkl allows us to define the metric (lattice constants) of the smallest building block of
the crystal, the so called "unit cell".
The intensity of the scattering maxima is proportional to the square of the absolute value of all scattered
waves from all unit cells within the crystal. Whether a reciprocal lattice point is decorated with intensity
solely depends from the phase relationship of the scattered waves. Exist, because of symmetry conditions
(centering, gliding planes, screw axis), an additional lattice plane which has a phase shift of 180° with respect
to the original lattice plane, we get destructive interference. A 21 screw axis parallel to the b-axis as an example maps each atom with the coordinates (x, y, z) on an atom with the coordinates (x, ½+y, z) or formally:
0
𝑥
−1 0 0
(R, T) ∙ 𝑋 = ( 0 1 0 ) ∙ (𝑦) + (1⁄2).
𝑧
0 0 −1
0
The symmetrically equivalent atom (sub-lattice) lies just in the middle between the (010) net planes passing
(x, y, z). So all waves diffracted at these atoms interfere such that they extinct each other. The intensity of
the Bragg reflection 010 will be zero in this case.
These extinction rules are tabulated and are the main device for the determination of space-group symmetry.
2.2 Oscillation- and rotating crystal procedure
A part of the lattice constants is measured by the oscillation- respectively rotating crystal pro-cedure.
By rotation of the crystal about a pre-orien-tated axis (see part 1) with high rotational symmetry during
illumination, the net planes of the reciprocal lattice perpendicular to this rotation axis are mapped on a coaxial
cylindrical imageplate (Fig. 8).
Fig. 8: Schematic diagram of an oscillation- respectively rotating crystal procedure with a picture, showing the typical layer lines
All reciprocal lattice points appear pairwise on strait lines per-pendicular to the cylinder axis of the film.
The parallel lines have different distances but appear in pairs with the same distance to the midline. This
follows the "orthogonality con-dition" of the reciprocal lattice. By assuming an alignment of the crystal along
the basis vector b, the reciprocal basis vectors a* and b* and the reciprocal
net plane spanned by them are perpendicular to b (Fig. 9). By rotating the
crystal around the b-axis, whole reciprocal lattice can be decomposed into
net planes …, hk1̅, hk0, hk1, … perpendicular to the b axis. For each net plane
there is an intersection circle with the Ewald sphere and if the crystal and its
reciprocal lattice is turned about b the reciprocal lattice points hit the surface
of the Ewald sphere in such a circle. This means that all diffracted beams
coming from reciprocal lattice points of one net plane lie on a cone with the
cone end in the midpoint of the Ewald sphere and point to the contour of
the intersection circle. The cone axis is parallel to the rotation axis (Fig. 8). If
n·d* [Å-1] is the distance of the n-th net plane of the reciprocal lattice from
the zero plane hk0, the translation period b of the lattice line in the Fig. 9: Schematic sketch of the
rotation crystal procedure
corresponding crystal lattice is b = 1/d*. The following conditions result from
this geometry:
n: layer line number
radius of the Ewald sphere
While during the recording the crystal is turned about 360° each reciprocal lattice point of a layer hits its
Ewald circle twice, once on the right side of the film at a rotation angle φ and once again on the left side with
rotation angle φ’ (Fig. 10). Each reciprocal lattice point hkl generates two reflections which are located mirror
symmetric to the plane formed by the rotation axis and the primary beam. Further in any lattice with the
lattice point hkl there exists the inverse lattice point ̅̅̅̅
ℎ𝑘𝑙 with the same scattering angle θ. It is situated in the
(-l)-th layer and hits the Ewald circle after a rotation of 180° compared to hkl. On the whole we find four spots
with rotation angles φ, φ ', φ +180° and φ '+180° which show the symmetry mm2 on the unrolled imageplate.
If the crystal is turned only about a small angle (ca. 10°-20°) not all four reflection positions can be observed.
Such a recording is called an oscillation photograph. Compared to the rotation crystal method the symmetry
of the spot pattern which is caused from the operation technique is reduced! But if the crystal lattice has a
mirror plane perpendicular to the rotation axis, the zero-layer line will be a mirror line in the photograph.
(Example: a crystal with the monoclinic point group 2 or m. Both, its lattice and its reciprocal lattice will have
the symmetry 2/m.) This is helpful to detect a symmetry direction along the rotation axis.
2.2.1 Description of the experiment with the oscillation camera
For the oscillation photograph we use the Weissenberg camera with fixed film cylinder and dismounted
Weissenberg aperture. The oscillation angle is ± 20°. The oscillation axis is the crystal setting direction. The
wavelength of the (filtered) CuKα-radiation 1.5418 Å (weighted mean value). The diameter of the film cylinder
is 114.2 mm. 2 crystals with different main axis orientation will be investigated.
2.2.2 Evaluation:
The translation period along the oscillation axis can be found using the following relations:
t = translation period of lattice in direct space
λ = wavelength of monochromatic radiation
n = net plane number
n = outer angle of the n-th scattering cone with respect to the primary beam direction
R = Diameter of the film cylinder (114,2 mm)
ln = distance of -n-th to n-th net plane line
Any measurable variable can only be fixed within certain tolerances. The error of t is determined by the
uncertainty of the wavelength λ and the diameter R of the film cylinder and the inaccuracy in measuring ln.
As we have a minor influence on the first two variables, we try to minimize the uncertainty of the last
parameter. Thus we measure the values of 6 reflection pairs of all n net plane distances ln and calculate a
mean value out of them.
with
ΔR/R is estimated by ΔR/R ≈ 2·10-3. In comparison to the uncertainty of ΔR/R und Δl/l, the uncertainty of λ is
negligible.
So we get for every net plane line a value t(n) ± Δt(n) with
The finite value is given by a weighted mean value
the weighted uncertainty is
Remark: The detailed error analysis for the oscillation recording can be found in the appendix.
Keep in mind:
By an oscillation photograph the translation period of a crystal along the swing axis can be determined;
further it can be recognized if the reciprocal lattice has a mirror plane perpendicular to this axis, i.e. the crystal
has a twofold axis and/or a mirror plane in this direction.
2.3 Weissenberg technique
The disadvantage of the oscillation procedure is that a whole net plane of the reciprocal lattice is mapped on
one line, the layer line, i.e. different reflection spots can be mapped onto each other and there exists no
unique assignment between rotation angle and reflection. A complete resolution can be achieved if each
angle position can be assigned to a definite reflection. A method which allows for this is e.g. the Weissenberg
method.
A single layer line is selected by
sliding in two metal cylinders
between film and crystal which
absorb all but the diffracted
rays coming from the selected
layer line. Then the film
cylinder is coupled with a
spindle which moves the film
synchronic with the rotation of
the crystal with the result that
each
rotation
angle
corresponds to a unique film
position parallel to the rotation
axis (Fig. 10). Each reflection of
the layer line is thus uniquely
Abb. 10: Prinzipskizze einer
Weissenberg-Aufnahme. Die Blenden
determined by the pair of
sind so eingestellt, dass eine Schicht
angles φ and 2θ (Fig. 11). The
höherer Ordnung auf der Detektordistribution of the reflection
folie abgebildet wird
spots on the film in an image of
the net plane of the reciprocal lattice, distorted because of the
film translation. However due to the method, we don't see an
Fig. 11: Geometrical condition for a Weissenundistorted reciprocal lattice out of the picture, applying the
berg-plot of the zero layer
geometry of the setup an undistorted lattice can be easily
calculated (Fig. 12).
The calculation of the undistorted
lattice is done as follows:
Skala is an arbitrary scale factor
(~ 8 cm), i marks the translation
from a chosen 0-point to the
reciprocal lattice point. It
depends on the film translation.
After the redrawing of all
reflections, a suitable base which
allows us to index all lattice points
with integral coordinates can be
chosen and a proper indexation
can be done again.
Fig. 12:
Part of an undistorted Weissenberg-picture of a a* - c* lattice plane
2.3.1 Reconstructive drawing and indexing
At the first glimpse, the arrangement of the bragg reflections within a Weissenberg plot looks, due to the
distortion, a bit strange. But by detailed observation we find that the distribution of reflection spots obeys
well defined rules (see Fig. 13).
Reflections with only one miller index (h, k or l) belonging to a main
axes rod form a straight line on the plot (see fig. 13, 14). Their
identification can be done by looking after the shortest distance t
between two scattering maxima with miller index +/- h, +/- k or
+/- l (see chapter 2.2.2). They are of major importance for the
determination of lattice constants. In the case of an axial extinction
the first miller index is not appearing. This can generally be
recognized by sketching in the parabolic like garlands which are
described in the following text passage. If there is a reflection spot
missing which can be seen in the zenith at the following garlands, it
is a clear hint for the existence of an extinction.
Please remark that, due to the rotation angle major to 180° one of
the two main axes rods appears twice in the picture.
Ewald construction and resulting plot for
the appearance of the main axis rods
within a Weissenberg picture
Reflections with mixed miller indices (h and k, h and l, k and l) lay on one of the clearly visible parabolic like
garlands (zones). Sketching these garlands in the Weissenberg picture allow us to index all diffraction spots.
Due to the fact that the diffraction spots all lay at the crossing point between two of the garlands, the miller
index can be determined by taking the number of the index the garland is cutting the main axes rod in its
zenith. The index of the mixed hkl which lays directly at the intersection point of the garland with h = 2 and
the one with l = 3 gets the value (203) as miller index.
2.3.2 Determination of lattice parameters
One Weissenberg shot gives access to three parameters of the reciprocal lattice, two lattice constants (base
vector lengths) and their included angle. First we have to find the main axes rods. To recognize them we look
for the scattering points with the shortest path difference between the upper and the lower part of the
Weissenberg picture (they are separated by the shading of the beam stop). Taking at all main axes rods half
of the shortest distance allows us to define a zero 2θ line. Taking the distance between two different main
axes rods on this line, the included angle can be calculated.
Determination of the base vector lengths is done by using the 2Θ-Values, which have been calculated out of
the measured distance t (see fig. 15) for the first three main axes maxima. Remembering that the Weissenberg
camera was developed in a time when calculations were done with a slide-rule the film cylinder got a
perimeter of 114.2 mm and the film translation was chosen such that the angle 2Θ can be measured with
1°/mm and φ in the film plane with 2°/mm.
It is advantageous to measure the distance t between pairs of + and - 2θ
as this improves the accuracy considerably (see Fig. 15). The inclination of
the line between the two reflections is – fixed by the film translation arctan 4 ≡ 75.96. So we get for 2θ:
2Θi = t /2sin(arctan(4)) = t/2sin(75.96)
Measuring values:
ti
Fig. 15: Evaluation of angular pairs out
of the Weissenberg picture
distance between the pairs of spots in mm
Derived values:
Rotation angle for the oscillation of the reciprocal
lattice point with pi = translation during the crystal-rotation
Θi. Bragg angle of the reciprocal lattice point
The reciprocal a* value can be calculated out of the 2θi values from different pairs of refections n * (100) using
the Bragg’s equation:
By the same way you get the value of the other main axes rod.
2.3.3 Performance of the experiment:
With the crystals used in the oscillation camera a Weissenberg photograph of the zero order layer has to be
produced. With the Weissenberg aperture cylinder the zero order layer line is selected (front blade value
should be 3 mm, back blade value 57.5 mm). For the translation movement of the film cylinder along the axis
the gear drive is coupled with the driving spindle. The oscillation angle has to be set to +/-60° such that the
maximal translation range is used.
2.3.4 Evaluation:
Sketch within the + part of the Weissenberg plot all main axes rods and all garlands till n = 3. Evaluate all
lattice parameters (see chapter 2.3.2 Determination of lattice parameters). For the base vector lengths use
the – h00 and h00 respectively 00-l and 00l reflections till n = 3 and calculate first the reciprocal base vector
lengths a* and c*. The incident angle can be calculated from the distance of the h00 axes rod to the 00l axes
rod on the zero line. Afterwards you can convert those values in real space lattice parameters.
With help of a second Weissenberg photograph of the a* b* plane zero order one should try to observe
systematic extinctions in the data set which are generated by screw axes, glide planes and centering
translations. From a single photograph usually only serial extinctions can be recognized, i.e. systematically
missing reflection along one straight line of the reciprocal net. The interpretation of a face-centered net is not
unique: it may result from a glide plane or an integral extinction from a centering. For the identification at
least a recording of the first layer is necessary.
Keep in mind:
From a Weissenberg photograph of the zero order layer three lattice parameters (one angle and two base
vector lengths) of the reciprocal net plane perpendicular to the straight net line of the setting direction can
be determined. Further the effect of the point symmetry of the crystal in this plane can be observed. Finally,
it is possible to recognize the effect of screw axes, glide planes and centerings from systematic extinctions.
For their complete identification recordings of higher layers or zero order layers of other directions are
necessary.
2.3.5 Error analysis
While in the error analysis of experiment no. 2 a weighting scheme with the order of the lattice line was
applied because of the with the order growing error caused by the beam divergence, in the Weissenberg
photograph all reflections have the same divergence error. Thus the weighting scheme with tan[θ] following
Bragg’s equation is adequate:
The weighting scheme for the averaging must be transferred to the error calculation. Because of the error in
the measurement of β resp. β*for the inaccuracy of a, the error propagation formula has to be applied:
So we have:
Literature:
Giacovazzo et al. Fundamentals of Crystallography, S 245-254.
Ladd & Palmer: Structure Determination by X-ray Crystallography, S. 147-166.
Stout & Jensen: X-Ray Structure Determination, S. 98-109, 115-122.
Wölfel: Theorie und Praxis der Röntgen-Strukturanalyse S. 105 ff.
International Tables for X-Ray Crystallography Vol B, S. 185.
Kaelble: Handbook of X-Rays 24-2.
Woolfson: X-Ray Crystallography S. 141 ff.
Appendix: Detailed error analysis for the oscillation photograph
The formulae for the evaluation of an oscillation photograph are:
 R
n
l
tan[a n ]  n and t 
combined t  n 1   
R
sin[a n ]
 ln 
t =
λ =
n =
n =
R =
ln =
2
translation period of the crystal lattice (direct space)
wavelength of the used monochromatised radiation
layer line number
external angle of the n-th diffraction cone with the direction of the primary beam
diameter of the film cylinder (here 114.2 mm)
distance from the -n-th to the +n-th layer line
For the discussion of the error propagation the partial derivations for the error prone parameters are
calculated:
2
 R
t
t
 n 1    


 ln 
R
l n2
t
t
1
 n
 
2
2
R
R
 ln 
 R
1  
1   
 R
l
 n
R2
l n3
t
t
1
 n  
 
2
2
l n
ln
 ln 
 R
1  
1   
 R
l
 n
Using the Gauss rule for the error propagation we get the following error estimate:




2
t
1
   

 
 
l
t
 n 
  
1   
 R

2
  R  2  l  2 

  n  
  R   l n  


The measured values for the characteristic wavelengths come from precision measurements and can be
estimated for the weighted mean value of the CuKα1 - and CuKα2 wavelengths by Δλ/λ ≈ 10-5. The fact that we
cannot separate α1 and α2 is included in the error Δdn.
The errors ΔR/R and Δln/ln can be estimated by 3·10-3 for R = 114.2 mm and ΔR ≈ Δln about 0.1 mm. Compared
with this value the wavelength part of the error can be neglected. This allows to reduce the error formula:
t  t 
  R  2   l  2 

  n  
  R   l n  


1
l 
1  n 
 R
2
Substituting in the root ln = R · tan (μn) and ΔR ≈ Δln ≈ Δ, gives:
t 
t

R

l 
1  n 
 R
2
1
1
tan [ n ]
2
The n-dependent terms can be separated from the n-independent ones:
t 1

t

1
t
 

 

2
2
t
R 1  tan [ n ] n R n(1  tan [ n ]) 
The term in the middle alone is dependent from n. If we want to develop a weighting scheme for the
dependence of the error Δt, according to the derivation up to here, we would have to use w (n) = n · (1 + tan2
[αn]) / c. A look at the oscillation photograph shows immediately that the high orders will be overestimated
by this scheme because with growing order the diffraction cones intersect the camera cylinder under a more
and more flat angle such that the reflections become indefinite. This is a systematic error which is not covered
by our formula because the divergence of the primary beam is not introduced in the formula. This shall be
caught up now. If we observe a beam which deviates from the middle line deviates about an angle ± δ we can
estimate the n-dependence of the divergence error Δ div by the difference
div  tan[ n   ]  tan[ n   ] 
tan[ n ]  tan[ ]
tan[ n ]  tan[ ]

1  tan[ n ] tan[ ] 1  tan[ n ] tan[ ]
tan[ ]  tan 2 [ n ] tan[ ]
2
1  tan 2 [ n ]  tan 2 [ ]
.
As αn ≤ 45° and δ « μn, we have tan2 (αn) · tan2 (δ) « 1, and thus Δdiv = 2· tan(δ)·(1+ tan2 (αn))
This shows that the error Δdiv grows just with the factor (1 + tan2 (αn)) which appears in the denominator of
the error for Δt. If we use the weight n only, instead of n · (1 + tan2 (αn)) we just compensate the influence of
this error. From this discussion we get, after normalization of the weights to the sum one, the weighting
scheme:
w ( n) 
n
N




2n
N  ( N  1)
1
if N is the number of the observed layer lines.
The error Δ ln can be roughly estimated by ≈ 0.1 mm which we can reduce by a number of K repeated
measurements along the layer lines. By these measurements also a systematic tendency may be observed
(resulting from misalignment of the crystal axis against the rotation axis). We have:
1 K
l n   l n
k  1
 ln 
1 K
( l n  l n )2

K  1  1
At least six measurements should be present because the application of statistical formulae otherwise cannot
be justified. Except experimental care for ΔR we have no possibility to get some information how precise the
inserted film meets the distance of 114.2/2 mm from the rotation axis. So we must be content with the
estimation by ΔR / R ≈ 2·10-3. For each layer line n by the formulae given above, we determine a value t (n) ±
Δt (n), where
2
t ( n )
2
 R    l n 
 t( n)  
.
 
 R   l n 
The weighting factor 1 / (1 + tan2 [αn]) is omitted because of the divergence error. The final measured value
we get using the weighted mean value:
Analogical we find the mean weighted error:
For comparison with this error we can consider the standard deviation for the series of the t(n). As the result
was evaluated by the weighted mean value, the same procedure has to be used for the standard deviation.
In a well-done evaluation Δt and S (t) should have the same order of magnitude where the standard deviation
a smaller value offers, but which should be considered with care as the number S is still very unreliable.