IFM – The Department of Physics, Chemistry and Biology
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Note to students: (refs. 21 and 22) : Before this labbration it is strongly suggested that you read the two first chapters in : Kittel C. Introduction to Solid
State Physics about crystals and diffraction.
During the 20th century several techniques were developed for structure determination. The most important of these is x-ray diffraction. Max von Laue obtained (1912) the first diffraction pattern of a single crystal, one of zinc sulfide. Later Von Laue, William H. Bragg, and his son, William L. Bragg, showed how these diffraction patterns could be used to determine the arrangement of atoms in crystals. Max von Laues experiment became both an evidence for the wave character of x-rays, as well as, an evidence for the structural quality of crystalline materials at the atomic level. His simple experiment, that you will perform during this laboration, will show the important theoretical base for the solid state physics as well as material science. This laboration will deal with concepts as x-rays, reciprocal space as well as crystals.
Figure 1: X-ray equipment (tube) in the old days.

Early in the 20th century, scientists speculated that crystalline solids are composed of periodically arrayed atoms. In 1913, Max von Laue, that was working near the laboratories of Wilhelm Röentgen, the discoverer of x-rays, realized that such arrays should be suitable diffraction gratings for x-rays, which were presumed to have wavelengths useful at the atomic scale. Laue was able to demonstrate that x-rays were, in fact, diffracted by crystals, a discovery followed by that of the British father -and-son team of the Bragg family. The two developed a simple relation, now called Bragg's law, relating the wavelength of x-rays to their glancing angle of reflection. This relation enabled them to determine both the grating spacing and the wavelength of x-rays.
They and many others proceeded to determine the crystal structures of numerous compounds.
Knowledge of a crystal's atomic structure, by disclosing the nature of its atomic bonding, provides a basis for understanding its physical and mechanical properties. This knowledge has in turn accounted for such matters as why a diamond crystal is exceptionally hard and why most salts dissolve in water.
Experimental proof of the regularity of crystal structures also gave rise to the quantum-mechanical band theory of solids. The theory that energies of electrons in solids exist in restricted ranges, or bands. And this led to the invention of transistors, lasers, and other solid-state devices. The determination of the crystal structure of penicillin enabled the purification of that important antibiotic.
Within one year of Laue's discovery a fairly complete theoretical basis for the diffraction of x-rays by crystals had been developed by Charles Galton Darwin, grandson of Charles Darwin. He showed that the intensity of the x-ray beams diffracted at the angles determined by Bragg's law could be related mathematically to the scattering power of the electrons in a crystal's atoms.
The contributions of each atom may be combined to give a structure factor for the crystal. Darwin showed that the diffracted intensity is proportional to this structure factor squared. Thus, by measuring the intensities of the diffracted beams, it is possible to determine the magnitude of the structure factor. It is not possible, however, to determine if it is a positive or negative quantity. This difficulty is known as the phase problem in x-ray crystallography.
A mathematically more elegant theory of x-ray diffraction was published by
German crystallographer P. P. Ewald in 1917. He developed the basis for what became known as the reciprocal lattice concept. That is, x-ray diffraction by a single crystal produces a discrete set of beams. Their disposition in space is possible to relate to a lattice with a shape reciprocal to that of the crystal being irradiated. Such a reciprocal lattice has dimensions inversely proportional to those of the crystal lattice. The importance of this concept is in the interpretation of the diffracted beams recorded on, for example, a photographic film.
The film, set to intercept the diffracted beams as spots (beam cross- sections), contains a photographic record of the reciprocal lattice from which the crys-

tal's lattice may be deduced directly. The intensity of these reciprocal-lattice points on the film is the intensity of the diffracted beams, so their magnitude may be used to determine the atomic arrays within the unit cells of that same lattice, if the intensity of the x-ray beam is known.
With our present knowledge of matter and how it is built up of atoms, a crystal is described as a periodically repeated atom, or arrangement of atoms, in space. The physical lattice is the repeat of a pattern of dots, identically preserved, when moved by the integer vector a
1
, a
2
, a
3
. The unitcell , defined by the three base vectors, is thus repeated throughout the whole crystal. The true arrangement of atoms and electrons within the unitcell is called the base . The crystal is thus formed with a certain base repeated in a certain lattice structure.
14 Bravais lattices makes up the main symmetries, that can be observed macroscopically among crystals in the seven crystal systems . The Base of the crystal can reduce the symmetry of the lattice in a way that can be seen macroscopically. This makes it possible to divide the crystals into 32 point groups or symmetry classes . The final, complete picture, including the atomic configuration will define a total of 230 space groups . For those that are interested, there is a lot to discover in books about solid state physics, crystallography and mineralogy.
An object or pattern has symmetry if it is seemingly unchanged by an operation such as rotation about an axis or reflection across a plane. If an object possesses rotational symmetry, it will appear unmoved after rotation about a rotation axis. For example, a hexagon looks unmoved after rotation of 60 degrees about a perpendicular axis through its center, and a cube appears unmoved after rotation of 120 degrees about an axis through a body diagonal. If a pattern possesses mirror symmetry, its two halves, separated by an imaginary plane, appear to be mirror images of one another--for example, the letter
M.
Crystallographers demonstrated that only certain angles of rotational symmetry are possible for crystals: 60 degrees , 90 degrees , 120 degrees , 180 degrees , and 360 degrees. The associated symmetry elements they termed 6fold, 4-fold, 3- fold, 2-fold, and 1-fold rotation axes , respectively, based on the number of rotations required to equal 360 degrees. They further demonstrated that, for crystals, the possible combinations of symmetry operations involving rotation and reflection are limited. The combinations define the 32 crystal classes, which were grouped into the 7 crystal systems: isometric (cubic), hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic. Further subdivision into the 230 space groups follows from the added consideration of translation, that is, repetition in space by shifting, as in the building of a pattern by repeating a unit cell.

Figure 2: The seven crystal systems and the fourteen Bravais lattices.
Crystals with well-developed faces may show some of the internal symmetry of the pattern. For example, crystals of beryl commonly have 6 sides meeting at 120 degrees interfacial angles, consistent with the 60 degrees rotation axis of the beryl crystal structure. For crystals without good external form, physical tests involving light, electricity, force, sound, and so on can reveal some of

their internal symmetry. If a crystal possesses rotational symmetry, the physical properties of the crystal, such as the speed of sound or light within it, will be the same along directions that can be rotated into one another by a rotation axis. For example, simple optical tests using polarized light can help identify the crystal system of transparent crystals based on the symmetry of the refraction (bending) of the light rays. Irregularly shaped crystals of galena (PbS) can be broken along 3 mutually perpendicular planes into tiny cubes, consistent with the isometric symmetry of the galena structure. But the best and most versatile way of measuring crystals has been shown to be x-ray diffraction measurements
Crystallographers today continue to use x-ray diffraction to determine and better understand crystal structures. Diffraction experiments are greatly aided by the use of computer-controlled diffractometers and powerful data analysis software. Information about the intensity and position of diffracted beams is used to compute three-dimensional maps of the electron density in the unit cell. Studies at high temperatures or pressures, or both, are being undertaken in order to learn how crystal structure is modified as the crystals are heated or stressed. X-ray diffraction, commonly by powdered crystals, is used for quality control in the metals and ceramics industries, for identification of minerals by geologists, and for characterization of asbestos by health scientists.
Scientists also continue to find new methods of crystal study. In the 1960s, transmission electron microscopes (TEM) were developed that made possible the study of electron diffraction by crystals. Furthermore, TEM images of very thin crystals provided visual evidence of some of the "mistakes" (defects) in crystal patterns, such as missing atoms (vacancies) and layer offsets (dislocations). Scanning tunnelling microscopy (STM), developed in the 1980s, uses a very -fine-tipped tungsten stylus to map the atomic topography of crystal surfaces. Because crystals interact with their surroundings across their surfaces,
STM studies are providing important information about crystal growth and dissolution as well as adsorption of substances onto crystal surfaces.
Research biologists trying to identify and characterize important molecules in plants and animals commonly attempt to concentrate and crystallize the molecules. If this can be done, their structure can be determined by x-ray diffraction. For instance, x-ray studies of DNA crystals by Rosalind Franklin (1920-
58) and others led James D. Watson and Francis Crick to propose the double helix structure of the DNA molecule in 1952. Since then the structures of many other important organic molecules have been determined in this way. A great many molecular biologists and organic chemists are therefore crystallographers or make use of crystallographic data.
Clearly, crystals and crystallography are at the center of considerable academic and industrial research today. There are several related topics of current interest, as well. Among them are the quasicrystals, solids with 5-fold rotation axes that cannot have a structure that includes repetition of a unit cell. Liquid crystals are liquids that have long-range order in one or two dimensions but

not three; they are used in liquid crystal displays (LCDs) in watches and other devices. Crystal whiskers--needle like crystals of substances such as silicon or carbon add strength to special composite materials. Finally, crystal spectroscopy is an enormous field of study involving many techniques that provide information about atoms and bonding in crystals.
An x-ray beam hits a crystal fixed in space. Now, we can imagine an infinite number of sets of parallel planes, on which the atoms are situated within the crystal. Depending on the orientation of the crystal, compared to the x-ray beam, each set of parallel planes, with the repeat distance d, will form a fixed angle

with the x-ray beam. The concept of x-ray diffraction emanates from the fact that the wavelength

of the radiation is comparable with the distance between atoms in the material. This can be described by Braggs law in the following way: reflection in atomic planes occur when eq. (1.0) is fulfilled.
2 d sin
  n

(1)
Figure 3: Bragg scattering condition
One very common method to analyze material compositions is powder diffraction. During powder diffraction there is only one wavelength

of x-ray radiation sent towards a collection of small crystals, so finely grained that they are a powder. Braggs law will then give all d-spacings that are present in the powder by register reflections at certain angles

with a movable sensor.
The Laue method involves sending many different wavelengths of x-rays towards a single fixed crystal. The resulting reflexes are registered on a piece of film and then analyzed regarding orientation of planes, as well as, the orientation of the crystal itself. Theoretically one would like to send a continuous spectra of x-rays towards the crystal. Then, there would be one wavelength
 available for each set of planes, with the spacing d, in the crystal. Thus Braggs law 2dsin(

)=n

would be fulfilled for a set of angles

for all set of planes. In reality there is a lower limit of the wavelengths available, and thus also a lower limit of d-spacings possible to measure.

Figure 4: Example of beam path in the case of reflection in the
(130) and (120) planes.
The reciprocal lattice is the fourier transform of the physical lattice , so that all the points present in reciprocal space represents all the different sets of lattice planes in physical space. During a Laue measurement the reciprocal space will be expressed in the position of the spots on the film. Thus, it is only the angular direction to a reciprocal lattice point that is preserved on the film, as in any photography, while it is the length G of the reciprocal lattice vectors that is detected during powder diffraction measurements.

The well known geometry from optics is always valid at reflection:
1.
The exit and entrance beams are both confined within in a plane.
2.
The plane is at a right angle compared to the material that reflects the beam.
3.
The angle 2

between the beams is divided in two separate equal angles

4.
The reflective plane normal is the divider of entrance and reflective angle (both =

).
However, Braggs law for reflection in atomic planes is formulated using the
 angle between the x-ray beam and the reflective plane.

is the complementary angle to

. An alternative to the formulation above can be done with the wave vector k and the reciprocal lattice vector G . G has the absolute value 2

/d and points in the normal direction of the set of atomic planes in question. G forms also a triangle with k e out
and k e in
see fig. 5.
Figure 5: Law of reflection.
In this lab we are going to use back reflection, the other option is transmission but requires very thin samples. Back reflection means that the film is placed in front of the crystal with its light sensitive surface towards the crystal. The xrays are sent towards the crystal, trough the film, at a right angle, to then be backscattered by the different planes in the crystal towards the film, see fig. 6.
Only some of the back reflected beams will hit the film. in our case the distance L is 3 cm and the film size is 10x6 cm. Thus, if the angle is too large the beam will miss the film, but also if the angle is too low there will be no detection. In this latter case the beam will not be distinguishable from the background produced by the primary beam passing the film on the way to the crystal. Therefore we will have a slightly limited view of reciprocal space.

Figure 6: Experimental set-up.
One distortion that one has to accept is that the Laue photo is enlarged compared an ordinary photo. This is since the detected angle is 2

an not

which is the true angle towards the spot in reciprocal space. Another distortion is the same problem as all cartographers face. Shall one represent land and sea on a globe or on a flat paper, and what will the distortion be? In our case it would have been better to reproduce our Laue pattern on a film shaped like a half sphere. Now, the film is flat, and therefore larger angles from the normal will be represented by an abnormally long length on the flat film, see fig. 7.

Figure 7: Different type of film and the resulting patterns (4 fold patterns).
The simplest crystal system to start with is the cubic system, so this will be a system for us to take a closer look at. When one would like to determine the orientation of a cubic crystal there is help to get from the symmetry operations. In a cubic lattice there is one inversion center (origo), 3 fourfold turning axes with mirror planes, 6 twofold turning axes with mirror planes, and 4 threefold turning axes. All in all called 4/m 3 2/m, or even shorter m3m. If the x-ray beam is parallel with one of the turning axes there will be a pattern generated on the film that has the same rotational symmetry as that axis. Thus, it is possible to solve the orientation of the crystal. The only part left to do is to determine the Miller index of at least one Laue spot and then the rotational angle is possible to measure in the photo. In a more complex case it is always possible to turn the crystal to a turning axis, while keeping track of the angle and then determine the symmetry at that point.

1.
Load the sample onto the holder and make sure the height is correct then zero both the tilt and rotational angles on the holder and finally set the distance to 3 cm.
2.
Make sure the exposure handle is in the LOAD or down position.
3.
Turn slowly on the voltage to 40 kV and current to 40 mA. (This is maximum do not overload ! )
4.
Insert fully a Polaroid 57 professional film with the

mark towards the x-ray source, and pull it back out again. (Now, the negative still stay inside the camera).
5.
Pull the Exposure handle in the direction of the arrows. (towards you)
6.
Close the cabinet and open shutters for approx. 1min and 15 sec. And then close shutter again.
7.
Push the Exposure handle back and then push the film back into the camera.
8.
Turn the exposure handle to the up position.
9.
Now, this is the critical moment, pull the film straight out in a steady pace. (If not, the film opens up and developer contaminates the camera
(cleaning 40 min))
10.
After 15 to 20 sec. pull the Polaroid film case open and take the picture out.

(To be done before the laboration)
0.
Verify to yourself that eq. (1.0) is valid.
1.
Calculate the angle between the (100) and (110) vectors as well as the
(110) and (111). (This type of calculation will be done a lot during the lab so memorize the eq.)
2.
a) Determine the maximum angle

visible in a Laue photo of size
90x115 mm with the set-up described in this text (see fig. 6). b) Will the (513) spot be visible in four fold symmetry i.e. with the (100) axis in the center of the photo?
3.
find out, from some book, the general shape of the Intensity vs. wavelength for a typical lab x-ray source, and what it is that determine the low wavelength limit.

Questions
(To be solved during the laboration)
0.
Familiarise yourself with the Carine software. Try to look at a cubic crystal in both real space, reciprocal space, and as a pole-figure.
1.
Use the Carine software to print out pole-figures for the three symmetry axes (100), (110), and (111). (To be used later on.)
2.
Find out how to validate homework 1 with the software.
3.
Measure the first crystal (A) with the Laue camera, and determine the orientation.
4.
Measure crystal (B) and determine the rotation and tilt necessary to find the third, no yet measured, symmetry axis with crystal (B).
5.
Measure the third symmetry axis with crystal (B).
6.
In one Laue photo there is a very bright spot. Calculate miller indices and wavelength for this spot, and explain the brightness.
Before leaving, you are required to do a 1-2 page laboratory report. You can work as a team and submit one copy from the group. This report will be evaluated by the laboratory instructor. Any comments or suggestions you have about your preparation (the lecture and laboration write-up) or delivery of the lab. instructions/direction can also be made at this time.
Your lab write-up should summarize your lab. activities and findings and be organized in a formal way according to the following subject headings:
Purpose, Equipment, Experiment, Results, Discussion and Conclusions. You may group 1 or more of these headings together as seems appropriate.
Purpose: A 1-2 sentence statement on the purpose of your experiment, i.e. what are you trying to measure and determine.
Equipment: State the major piece of equipment using a model number.
Under what conditions did you use it (e.g. V, A settings, mode)?
What sample did you study (e.g. 2000 Å V film on a MgO <111> substrate)?
Experiment: What did you do?
Results: raw data, calculations, graphs
Discussion: explain your results
Conclusions: one statement usually.
Comments: Were you lacking background information? Was the lab. too easy, difficult, long, short, boring, ... ? Did you learn something? Do you want more hands-on? more theory?

1.
Giacovazzo, C., Fundamentals of Crystallography (1992)
2.
Holden, A., and Morrison, P. S., Crystals and Crystal Growing (1982)
3.
Hurle, D. T., Handbook of Crystal Growth (1993)
4.
Janot, C., Quasicrystals (1992)
5.
Mullin, J. W., Crystallization, 3d ed. (1993)
6.
Rhodes, G., Crystallography Made Crystal Clear (1993).
7.
Berry, L., Mineralogy: Concepts, Descriptions, Determinations, 2d ed. (1983)
8.
Blackburn, W. H., and Demmen, William H., Principles of Mineralogy (1988)
9.
Carmichael, R. S., CRC Handbook of the Physical Properties of Rocks and Minerals
(1988)
10.
Deer, W. A., et al., An Introduction to the Rock-Forming Minerals (1966)
11.
Desautels, P. E., Rocks and Minerals (1982)
12.
Frye, K., ed., The Encyclopedia of Minerals (1982)
13.
Klein, C., Minerals and Rocks (1989)
14.
O'Donoghue, M., American Nature Guide to Rocks and Minerals (1990)
15.
Pough, F. H., A Field Guide to Rocks and Minerals, 4th ed. (1976)
16.
Roberts, W. L., The Encyclopedia of Minerals, 2d ed. (1989)
17.
Sinkankas, J., Mineralogy (1975).
18.
Abruna, H. D., The Study of Solid/Liquid Interfaces with x-rays, Science, Oct. 5,
1990
19.
Elter, M. C. et al., Solid State NMR and x-ray Crystallography (1988)
20.
Taylor, C. R., Diffraction (1987).
21.
P. E. J. Flewitt and R. K. Wild, physical methods for materials characterisation
(1994)
22.
C. Kittel, Introduction to Solid State Physics about crystals and diffraction. (1986)

(Not necessary to read for the lab, only for the interested.)
The German physicist Max Theodor Felix von Laue, b. Oct. 9, 1879, d. Apr.
24, 1960, received the 1914 Nobel Prize for physics for his discovery (1912) of the diffraction of x-rays by crystals. He showed that x-rays are very short electromagnetic waves that can be used to study the structure of materials. He thus helped establish the field of x-ray structural analysis, an important branch of physics and chemistry. Laue, who played an essential role in the golden age of German physics, was professor of theoretical physics at Berlin from 1919 until his retirement in 1943, and was instrumental in rebuilding German science after World War II.
A mineral is a natural, homogeneous, inorganic solid with a crystalline atomic structure. Crystallinity implies that a mineral has a definite and limited range of composition, and that the composition is expressible as a chemical formula.
Some substances that do not satisfy all these conditions, such as metallic liquid mercury, are commonly considered in the mineral realm but should more properly be designated mineraloids. The word mineral may have different meanings in nonmineralogical sciences. In nutrition, it may mean any non organic element. In economics and economic geology, minerals may be practically anything of value extracted from the Earth, including petroleum and natural gas (which are not minerals according to the above geological definition, being neither inorganic nor solid).
Minerals comprise the vast majority of the material of the solid Earth. Aside from air, water, and organic matter, practically the only non minerals in the
Earth as a whole are molten rocks (magmas) and their solid glassy equivalents.
Crystalline rocks themselves, and even soils for the most part, consist of aggregates of minerals. Almost all inorganic substances that are used by or of value to humans are derived from minerals. Over 3,000 minerals are currently known, and about 50 new ones are now discovered each year. Most Gems are minerals, though some, such as opal, are mineraloids.
Mineralogy is the study of the nature and origin of minerals. Although it is not one of the fundamental sciences, it was one of the first scientific fields to be developed, and curiosity about minerals led to many discoveries in physics and chemistry. Georgius Agricola (Latinized name of Georg Bauer), sometimes considered the founder of mineralogy, summarized most early knowledge of minerals in his works De Natura Fossilium (On the Nature of
Fossils, 1546; Eng. trans., 1955) and De Re Metallica (On Metals, 1556; Eng. trans., 1912). The discovery by Nicolaus Steno, Arnould Carangeot (1742-
1806), and Jean Baptiste Rome de l'Isle (1736-90) in the 17th and 18th centuries that interfacial angles of crystals of a given mineral are constant, laid the

basis for the science of crystallography. The founder of crystallography, however, is considered to be Rene Just Hauy, whose most important works were published in 1784 and 1801. He proposed correctly that crystals are formed by stacking of identical structural blocks, now called unit cells, and he showed that, as a consequence, the intercepts that the crystal faces make on a set of carefully chosen axes are always rational numbers when divided by an appropriate common factor. Other early-19th-century scientists who contributed to the development of crystallography include Johann F. C. Hessel (1796-1872),
William H. Miller (1801-80), and Auguste Bravais (1811-63). Hauy's contemporary Abraham Gottlob Werner and his followers developed methods for identifying crystals by physical characteristics.
The science of chemistry developed rapidly in the early 19th century, and the determination of the compositions of minerals proceeded apace, especially through the efforts of Jöns Jakob Berzelius and his followers. The first lasting classification of minerals according to chemical composition was published by
James Dwight Dana in 1837.
The elucidation of the optical properties of crystals took place in the early 19th century, with noteworthy work being done by Jean Baptiste Biot and Sir David Brewster. The invention of the polarizing microscope and the use of thin sections by William Nicol (1768-1851) made it possible to identify minerals by means of their optical properties, and Henry Clifton Sorby, who is considered to be the founder of petrography, later applied these methods to the study of rocks.
The simultaneous discovery in the late 19th century by Arthur M. Schoenflies
(1853-1928), William Barlow (1845-1934), and E. S. Fedorov (1853-1919) of the 230 space groups (see below) essentially completed the development of classical crystallography. With the discovery (1912) of x-ray diffraction by
Max von Laue, Walter Friedrich, and Paul Knipping, the determination of the internal atomic structures of crystals became possible. Sir William H. Bragg,
Sir William L. Bragg, and others immediately began to use this tool to determine crystal structures of minerals, and this work continues to the present. xray diffraction also provided a rapid and objective means for identifying minerals.
The chemical analysis of minerals advanced importantly with the invention
(1949) of the electron microprobe by R. Castaing. The development of highspeed electronic computers has greatly facilitated the determination of the crystal structures of minerals and other substances.
A very important part of mineralogy in the 20th century has been the reproduction of minerals in the laboratory, with the principal objective of ascertaining the conditions of temperature and pressure and the nature of liquid or gaseous phases present during the formation of natural minerals. Other reasons for synthesizing minerals are to obtain pure end-member specimens for determination of physical properties, and to supply substitutes for scarce minerals of economic importance.

Synthesis has traditionally been achieved using furnaces with tungsten- or platinum-wire heating elements, and large steel presses or hydraulic apparatus for attaining high pressure. The sample is taken to high pressure and temperature, held for a time until the reaction is complete, and then quenched and examined.
Extremely high pressures corresponding to those in the Earth's mantle have recently been attained with the diamond anvil, a compact apparatus in which the sample is placed between the faces of two diamond crystals and compressed with a thumbscrew. The device is small enough to fit on a microscope stage, and the sample can be observed through the transparent diamond faces.
A laser beam can be projected through the microscope to heat the sample.