UMEÅ UNIVERSITY
DEPARTMENT OF PHYSICS
2004-04-01
Agnieszka Iwasiewicz
Sylvia Benckert
Leif Hassmyr
Ludvig Edman
SOLID STATE PHYSICS
CRYSTAL ORIENTATION
1. THE TASK
Use the Laue method of X-ray diffraction in order to:
 study the symmetry characteristics of a cubic single crystal
 determine the orientation of unit cells is a given crystal
 find the Miller indices of planes from which the X-rays are reflected (according to
your experimental results)
2. ADDITIONAL LITERATURE
1.
2.
3.
4.
5.
N.W. Ashcroft and N.D. Mermin, “Solid State Physics”, chapters 4 – 6
C. Kittel, “Introduction to Solid State Physics”, chapters 1 – 2
J. R. Hook and J. E. Hall, “Solid State Physics”, chapters 1.1 – 1.4 and 11.2
H. M. Rosenberg, “The Solid State”, chapters 1 – 2
B. D. Cullity, “Elements of X-ray diffraction”, pages 89–91, 138–144, 215–229
3. AIM OF THE LAB
X-ray diffraction is a powerful method for studying crystal structures, widely used in
solid state physics and material science research. The lab is designed for you to learn the basic
principles of the method, get to know how to analyze the diffraction images and extract the
interesting data. The Laue method bases on the crystal symmetries, and therefore you will
have to implement the knowledge about different crystal structures and symmetry operations
which you gained during the lectures. During this lab you will gain a more thorough
understanding of crystal structures, because it is necessary for completing the task. You will
be asked to exercise your three dimensional imagination, and by help of crystal models to
solve the problem of determining the orientation of a given crystal.
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4. THEORY
4.1 Lattices, planes and Miller indices
The description of all crystalline solids bases on the fundamental concept of the
Bravais lattice. The lattice points represent the repeatable units within the crystal structure –
single atoms, molecules, groups of atoms, ions, etc. The Bravais lattice carries the information
about the geometry of a crystal. In a three dimensional case, all the lattice points have the
positions given by R  n1a1  n2a 2  n3a 3 , where ni are integers. A three dimensional lattice is
therefore uniquely determined by the three primitive vectors: a1, a2, and a3, provided that not
all these vectors are in the same plane. The primitive vectors span a so-called direct space,
and set a convenient coordinate system for the description of the crystal. The length of each
primitive vector sets the unit length along the corresponding coordinate axis.
In the same way as the direct space is a space of the position vectors R, the reciprocal
space is the momentum space, with the k-vectors. Because of the symmetries in the crystal
lattice not all of the momentum values are allowed. The points in the reciprocal space
corresponding to the allowed k vectors also form a repeatable pattern, the reciprocal lattice. It
is again possible to determine the set of primitive vectors for the reciprocal lattice (b1, b2, and
b3), so that every allowed k vector is given by k  k1b1  k2b 2  k3b3 .
Any three lattice points (in the direct space) which are not situated along one line
constitute a plane within the crystal, called a lattice plane. Every lattice plane can be
described in a convenient way by the use of Miller indices. It is important to realize that even
though the Miller indices are used for the description of the lattice planes in the direct space,
they are in fact the coordinates of a vector in the reciprocal space (k1 = h, k2 = k, and k3 = l),
which is perpendicular to the (h k l) plane.
4.1.1 How to determine the Miller indices for a given plane?
There is an easy method to determine the Miller indices:
A. Check whether the plane is intersecting the axes of a coordinate system. Write down
the coordinates of the crossing with a1, a2 and a3 (let us call them x, y and z) axes. If the
plane is parallel to an axis, infinity is to be taken in the place of the corresponding
intersection coordinate.
B. Take the reciprocals of the coordinates.
C. We aim for a set of integers as the Miller indices, so if needed - multiply the numbers
obtained in the above step by the smallest possible number, bringing them to the
integer form. This result is a set of Miller indices for our plane.
Example:
Let us consider a plane which intersects the x-axis in point 3, the z-axis at point 2 and is
parallel to the y-axis.
A. The intersection coordinates are: (3, ∞, 2)
1 1
B. The reciprocals:  ,0, 
3 2
C. We can multiply the whole parenthesis above by 6, to obtain: (2, 0, 3). The Miller
indices are therefore (2 0 3)
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Note:
The inverse of the above procedure can be implemented if one wants to determine the
plane’s orientation in space by knowing its Miller indices.
Exercise 1
Consider a general, two-dimensional Bravais lattice, as shown below. For a two dimensional
case the lattice planes reduce to lines, and they will therefore be determined by two Miller
indices each. Determine the Miller indices for the three lattice “planes”: A, B, and C:
A:
(……)
B:
(……)
C:
(……)
Do not hesitate to use the crystal models available in the lab whenever needed. They might
come handy for testing your ideas.
Exercise 2
What can we say about the lattice planes with the same Miller indices?
Exercise 3
Consider a simple cubic lattice. What are the Miller indices for the planes perpendicular to:
a) the x-axis: (… … …)
b) the y-axis: (… … …)
c) the z-axis: (… … …)
d) the surface diagonal x = y : (… … …)
e) the surface diagonal x = z : (… … …)
f) the surface diagonal y = z : (… … …)
g) the space diagonal x = y = z : (… … …)
Looking at the Miller indices which you have just determined in the above exercise, one can
notice that the planes of similar kind have similar form of the notation. Sometimes it is not
necessary to refer to one of the planes only, but rather to the whole set of planes,
symmetrically equivalent. They might be considered as a family of the equivalent planes.
It is useful not only to talk about the planes within a crystal, but also the directions, for
example in order to describe the direction of X-rays propagation. One can define a direction
in the space, by giving the coordinates of the vector pointing in that direction. Some of the
directions are symmetrically equivalent, and we can talk about the family of directions.
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The notational conventions for planes, directions and Miller indices are given below:
SUMMARY OF THE NOTATIONAL CONVENTIONS:





(h k l) – a lattice plane with the Miller indices h, k and l, or a set of parallel planes.
{h k l} – a family of the equivalent planes, e.g. {100}= (100) and (010) and (001)}
[n1 n2 n3] – a direction
n1 n2 n3 – a family of directions
-h is denoted by h
4.2 Symmetry
Symmetry considerations are a major part of crystallographic studies. There are many
different symmetry operations, e.g. mirror transformation, inversion, rotation around a given
axis, etc. Today we will mainly focus on the rotational and mirror symmetries, which are
important for the correct interpretation of the X-ray diffraction pictures obtained with the Laue
method.
4.2.1 How to check if a given structure (pattern) is symmetric with respect to
the rotation?
First of all, we have to define the rotational axis. In the next step we check if it is possible to
turn the pattern around the given axis by a certain angle α, in such a way, that the result looks
2
exactly the same as the initial pattern. Usually the value of the angle α can be defined as
,
n
and then the pattern is told to have an n-fold rotational symmetry. The axis around which the
pattern was turned is called an n-fold rotational axis.
Exercise 4
Look at the patterns given below. Assume that the rotational axes are perpendicular to the
plane of the figure and the points of intersection are marked with a black dot.
What kind of rotational symmetry does each pattern have?
A.
B.
... - fold
C.
... - fold
D.
... - fold
... - fold
In the case of a crystal lattice, n-fold rotational symmetry means, that after every rotation by
2
an angle
a resulting lattice is identical to the initial one.
n
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Exercise 5
Some of the symmetry axes of a simple cubic lattice are shown in the figure below. Count
how many symmetry axes of each type are there in total and mark the symmetry axes in the
figure with the corresponding symbols:
2-fold symmetry axes:
mark with: ◊
number of: ........
3-fold symmetry axes:
mark with: Δ
number of: ........
4-fold symmetry axes:
mark with: □
number of: ........
Mirror symmetry is another feature of some crystallographic groups. Cubic lattices
have the mirror symmetry with respect to mirror planes.
4.2.2 How to recognize a mirror plane?
Let us assign a line perpendicular to the “candidate” plane and passing through a
lattice point. Now, let us translate the lattice point along this line to the other side of the plane,
so that the eventual absolute distance from the plane is same as the initial distance. If this final
position is also a lattice point for all lattice points, the plane is a mirror plane.
Exercise 6
Mark two arbitrary mirror planes of :
a) a cubic lattice:
b) a tetragonal lattice (unit cell with two
edges of the same length and one different):
Note :
If there is an n-fold rotational axis in a mirror plane, then n mirror planes cut each other
along this axis.
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Exercise 7
How many mirror planes are there in a cubic lattice? Sketch them :
4.3 X-ray diffraction
4.3.1 X-ray radiation
X-rays are electromagnetic radiation with very short wavelengths (0.01 to 100 nm).
They are used in the studies of crystals, because the wavelength of radiation matches the
lattice spacing, and therefore a crystal acts as a diffraction grating.
X-rays can be generated for example by some of astrophysical objects, but they can be
also produced in the lab. The X-ray generator used in this lab will described in the
“Experimental setup” section.
4.3.2 Bragg’s law
Bragg considered a crystal to be made out of parallel planes of ions, lattice planes,
separated by distance d. In case of a cubic lattice this distance can be easily calculated
knowing the Miller indices of the plane and the lattice constant a:
a
d
h2  k 2  l 2
Exercise 8
Calculate the distance between two adjacent planes in a cubic lattice for:
a) (1 0 0) planes : ...............................................................................................................
b) (0 1 1) planes : ...............................................................................................................
c) (1 1 1) planes : ...............................................................................................................
d) (1 2 3) planes : ...............................................................................................................
For the X-ray diffraction pattern to occur, Bragg’s law must be fulfilled. The diffracted
beam will be observed when the Bragg condition:
n  2d sin 
is satisfied. The Bragg condition is nothing else than a constructive interference condition
(path difference has to be equal to the integer multiple of wavelengths λ). The integer n is
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known as the order of reflection. For the given wavelength and lattice plane, an angle θ for
which the Bragg condition is satisfied is called the Bragg angle.
Note:
The Bragg angle is the angle between the incident beam and the lattice plane, not the normal
to this plane (as it is conventionally used in geometrical optics).
The Bragg condition tells us about the allowed geometry of the X-ray diffraction, but
does not determine the intensity of each diffracted beam. The intensities will be discussed
below.
The rays are reflected specularly (the angle of incidence equals the angle of reflection).
4.3.3 The method
In the solid state lab there is a possibility of performing experiments with two X-ray
diffraction methods:
a) The Debye-Scherrer method (powder method) – uses a monochromatic X-ray beam
on a powder sample (small crystal grains). The sample is rotated during the exposure,
to ensure that all the incidence angles are present. The detected diffraction pattern is
the effect of the Bragg’s law (not all the reflections are allowed). One can use this
method for determination of the crystal structure.
b) The Laue method – which you will be soon implementing – bases on the symmetries
of a crystal structure. The diffraction pattern obtained by irradiating a single crystal
fixed in its position has the same symmetry as the crystal itself, while looking along
the incident beam direction. “White” X-rays (continuous spectrum) are used to
increase the possibility of the Bragg’s condition to be fulfilled (for the constant angle
of incidence, but different wavelengths). In this way the amount of diffracted light is
large enough to be detected, and the information about the crystal orientation
interesting for us can be extracted from the diffraction data.
4.3.4 Diffracted beam’s intensity, geometrical structure factor and atomic form
factor
Not all the diffraction peaks allowed by the Bragg condition appear in the diffraction
pattern with the same intensity. The intensity of each scattered beam depends on the
arrangement of atoms in a unit cell and the scattering capacity of each atom species. It is
possible to determine the ratios of intensities of different diffraction peaks.
Each atom within the unit cell can be regarded as a scatterer localized in position dj.
The scattered X-rays change their wave vector by K  k ' k . The amplitudes of rays scattered
at each atom are proportional to eiKd j , scaled by the “scattering ability” of each atom species.
The scattering capacity of each atom type is included in a so called atomic form factor f j  K 
. The total amount of the scattered X-rays in one unit cell is the sum of contributions of all
individual rays, so its amplitude is proportional to:
n
SK  å f j  K  e
j 1
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iK d j
where SK is called a geometrical structure factor, and the summation is over all the atoms in
the unit cell.
In case if all the atoms are of the same type, the geometrical structure factor reduces to
n
SK  å e
iKd j
j 1
,
because the overall multiplication by a constant atomic form factor is of no interest for the
intensity ratios.
A RECIPE FOR CALCULATION OF GEOMETRICAL STRUCTURE FACTOR FOR A GIVEN LATTICE:

DETERMINE THE BASIS. Let us take a b.c.c. (body-centered cubic) lattice as an example. A
b.c.c. lattice is effectively equivalent to a simple cubic lattice with two atoms in the
basis. These atoms’ coordinates inside the unit cell are given by: d1  0xˆ  0yˆ  0zˆ and
a
a
a
d 2  xˆ  yˆ  zˆ in the coordinate system spanned by the primitive vectors axˆ , ayˆ ,
2
2
2
and azˆ .
 RECIPROCAL LATTICE. Now we consider our crystal to have the simple cubic lattice with a
basis. Simple cubic direct lattice with a lattice constant a has a reciprocal lattice also
2
being simple cubic, but with a lattice constant equal to
.
a
We know that the reciprocal lattice of a b.c.c. crystal has a f.c.c.(face-centered cubic)
structure. If this structure is recovered as a result of our considerations, we will have a
proof of the validity of this approach.
 GENERAL K VECTOR. In a simple cubic reciprocal lattice a scattering wave vector has a
general form
2
K
 n1xˆ  n2yˆ  n3zˆ 
a
 SUM EVALUATION. Let us now calculate the geometric form factor using the formula
given above:
iK d1
SK  e
iK d 2
e
e
0
iK a xˆ yˆ zˆ 
e 2
i n n n
n n  n
 1 e  1 2 3  1   1
1

2
i 2  n1xˆ n2yˆ  n3zˆa  xˆ yˆ zˆ
2
 1 e a
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PHYSICAL MEANING. The value of a geometrical structure factor depends on coordinates
of the scattering wave vector. If n1  n2  n3 is an odd number, the structure factor is
equal to 0 and no scattered light will be observed. For n1  n2  n3 even, SK  2 and the
diffraction peak will occur. In this way, the simplifying assumption of a simple cubic
structure with a basis will be leading to the right solution.
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Exercise 9
Use the figure (of a simple cubic reciprocal lattice) below, for indicating the wave vectors
(color the corresponding points in the reciprocal lattice) for which the geometrical structure
factor is nonzero in case of a b.c.c. crystal, as considered above. What kind of reciprocal
lattice do we obtain? Is that what we expected ? Draw a correct coordinate system and label
the visible lattice sites.
Exercise 10
As a practice, perform the geometrical structure factor calculation for a f.c.c. crystal. Follow
the same procedure as was presented for a b.c.c. crystal.
The result of this calculation will be helpful in more efficient analysis of your experimental
data.
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Let us now focus on the symmetry characteristics of the X-ray diffraction on a crystal.
The following exercises deal with the behavior of an X-ray beam in a crystal. We will
consider the detection plate (X-ray sensitive Polaroid film) to be situated between the X-ray
source and the crystal. Such geometry, allowing to detect the back scattered X-rays, is
implemented in our experimental setup.
Exercise 11
When an X-ray beam is incident onto a crystal along one of the crystal’s mirror planes, the
resulting diffraction pattern will also exhibit the mirror symmetry. Complete the schematic
image below in such a way, that it illustrates the origin of the mirror symmetry of the
diffraction pattern:
Exercise 12
Let the X-ray be incident along the [1 0 0] direction in a cubic lattice, and let the diffracted
radiation be detected with a plate situated as shown on a cross-section sketch below:
a) What kind of symmetry will the diffraction image have?
b) Construct the path of radiation reflected against the (1 0 0), (1 1 0), (2 1 0) and (3 1 0)
planes.
c) Calculate the Bragg angles and the diffraction angles α (angles between the incident and
diffracted rays):
 100  ......................
 100  ......................
 110  ......................
 110  ......................
 210   ......................
  210  ......................
 310   ......................
 310  ......................
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Exercise 13
Keeping in mind the geometry of the detection in our experiment (back-scattered rays)
determine the dependence of the Bragg angles θ on :
a) the lattice constant a
b) the wavelength of X-ray radiation λ
Hint: Make use of the formulas which appeared earlier in the instructions and think of the
range of possible θ angles.
At this point you should be able to solve the experimental task of the lab, so we can move on
to the experimental part.
If something is still not clear – ask the lab supervisor for an explanation before you move on
to do the experiment. You have to understand the underlying theory in order to complete the
task of this lab.
5. EXPERIMENTAL SETUP
5.1 Crystal
A single crystal of LiF is to be examined. LiF has the same structure as NaCl (check
the model at the laboratory desk), the Li+ and F– ions are each forming a f.c.c. lattice,
displaced with respect to each other by half a lattice vector.
Since the atomic form factor for F– ions is roughly five times bigger than for Li+, the
X-rays will only see the f.c.c. structure of F– ions. Keeping in mind our considerations about
the geometrical structure factor of a f.c.c. lattice one has to remember that not all the
reflections will be present.
5.2 X-rays
X-rays used in our experiment are generated from a copper anode by accelerated
electrons. The X-ray tube containing the above mentioned copper anode is hidden inside of
the X-ray generator, but there is also a copy of the same tube present in the lab as a model.
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The X-rays are generated and sent out in a following way:
 A high voltage is applied between the anode and the cathode. As a result, electrons are
emitted from the cathode, and accelerated on the way to the anode
 Inside the anode the electrons decelerate, causing a so-called deceleration radiation, an
emission of continuous spectrum of X-rays. The intensity is directly proportional to
the atomic number of the anode material. Usually in the Laue method experiments the
anode is made out of tungsten, which produces higher intensity radiation than copper.
In our experiment we use an extra sensitive Polaroid detection film, so there is no need
for a very high X-ray intensity.
 The X-ray tube is enclosed in a metal cover, equipped with four apertures, so that Xrays can only be let out in the four given directions. The radiation is able to leave the
shield only if one or more apertures are opened.
 In front of each aperture there is a diaphragm, a filter selector and a shutter. The role of
these elements will be explained by the lab supervisor directly before the use of the Xray setup.
 X-rays on the way to the crystal pass a Laue camera located between the shutter and
the crystal. The camera should be loaded with a Polaroid film sheet before starting the
X-ray generator. The back-scattered beams will also be visible on the photo.
 A LiF single crystal is fixed to a rotating holder, allowing for the X-ray exposure at
different directions of incidence. Only the rotation around the vertical axis will be
used, the other rotation screws should not be moved. For the correct positioning of the
crystal, a small angle correction of the holder rotation scale might be necessary –
check for the sticker next to the scale.
Do not touch the crystal! A single fingerprint might destroy it. The cost of cleaning of
the crystal is about 10 000kr per fingerprint.
Whenever the X-ray exposure is to be performed – cover the Laue camera and the
crystal with a protecting shield. Follow the instructions of operation for the X-ray
generator, available in the lab. All the exposures should take place under the supervision
of your lab instructor.
6. EXPERIMENTAL TASKS
IMPORTANT!
X-ray radiation is very dangerous! Special precautions must be taken while working with
the X-ray generators. Read carefully the “Radiation Safety Rules” brochure (available in the
lab) and make sure that you have your dose-meter on you at all times. Follow the safety
instructions! Plan your experiment carefully, and consult your plan with the lab supervisor
before starting. Check the shields each time before you run the generator.
The experiment is completely safe, provided that the safety rules are obeyed.
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The task of this experiment is to determine the orientation of the given crystal at the level
of the unit cell. We use the X-ray diffraction method, since it is not possible to determine the
unit cells’ orientation just by looking at the macroscopic shape of the crystal. It splits easily
along many planes, in our case {1 0 0}, {1 1 0}, and {1 1 1} planes. A good illustration of that
fact can be for example the beautifully shaped jewelry cut out of diamond crystals.
1. FIRST EXPOSURE – FACING PLANE
 Take a picture of the diffraction pattern when the crystal is in the position 0° at the
holder scale (mind the angle correction).
 From the symmetry of the obtained picture determine which lattice plane is facing the
X-ray incidence direction
2. SECOND EXPOSURE – CRYSTAL ORIENTATION
 Turn the crystal by 90° and take another photo
 With help of the experimental data obtained by now, determine the orientation of
[1 0 0], [0 1 0], and [0 0 1] directions in the crystal
 What is the orientation of the unit cells?
3. THIRD EXPOSURE – CONFIRMATION OF YOUR RESULT
 Calculate the angle on the crystal holder that corresponds to the X-ray exposure along
[1 1 1] direction
 What kind of symmetry should the last picture exhibit?
 Check your prediction by adjusting the crystal’s position (with the angle correction)
and taking one more photo. Were you right?
4. DATA ANALYSIS – VISIBLE REFLECTIONS
 Recall which lattice planes reflections should be visible on the first two photos you
have taken.
 State the symmetry axes and planes.
 Measure the distance between the crystal and the film plane.
 Ask the lab supervisor for the lattice plane checklist.
 For the given set of planes calculate the Bragg angles and the expected position of
corresponding diffraction peaks on the photo.
 Compare the experimental and theoretical results in order to determine which lattice
planes contributed to label the diffraction peaks. Label the visible peaks.
 Make a good oral report.
GOOD LUCK!
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