A. The History and Theory of Crystals
Let us look at the history of the study of crystals. If you research the history you find out that the story is not as clear cut as the short introductions make it appear. An important idea may have been mentioned by a person but he may not have given it much emphasis at the time and instead have been concerned about some other part of his theory which might even have been wrong.
The study of crystals has been the concern of mineralogists. In fact, one of the most accessible sources is Dana’s
Manual of Mineralogy,
you will find it in most public libraries. James Dwight Dana (1813-1895) and his son Edward Salisbury Dana (1849-1935) were Yale professors. In 1837 the father wrote the standard mineralogic reference
A System of Mineralogy,
the son kept it up to date and it is still being kept up to date by others. It is available in your local library. The Manual is a good source of illustrations. Hopefully, this unit will be an introduction of crystallography to readers who will want to find out more.
Another major source is Martin J. Buerger the inventor of the precession method of x-ray diffraction who has written a number of books on crystallography. The biographical notes on authors in the collection
Fifty Years of X-ray Diffraction
tell us that his field of study was geology and mineralogy.
Crystallography in many respects is another example were the theory or mathematics existed long before the “reality”. In 1912 with the birth of x-ray diffraction the demonstration of the theory became visible.
In 1669 Nicolaus Steno in his Latin publication
De Solido intra solidum naturaliter contento
Dissertationis Prodromus
described how he took sections of quartz crystals and measured the angles. He found that the angles were the same no matter how long or short the sides. He did this by cutting the crystal perpendicular to an edge to get a wafer which he then traced onto paper. If he did this to different quartz crystals, he got the same angles when he cut wafers at corresponding edges. For this he gets credit for the law of constancy of angle also known as Stensen’s Law from his name in his mother tongue. This was not the main point of his book, he did not say anything about crystals of other substances.
The law of constancy of angle was stated by Rome de I’Lsle in 1783 and says that for all crystals of the same substance the angles between corresponding faces are congruent. Can you propose a reason why nature works this way? What makes it happen?
The next story is like Newton’s Apple; there is controversy that it ever happened. The story goes that one day in 1784 while examining some crystals Rene Just Hauy dropped a specimen and one of the larger crystals broke off. He noticed that it had broken to form a face. He tried, and succeeded, to cleave it in other directions. The piece that remained was not the same shape as the original crystal. He claimed the original crystal was made of building blocks shaped like the nucleus achieved by cleavage. Read the
Origins of the Science of
Cystals by John G. Burke to find out more. If you read Burke you will find that many people worked on the subject and said many things that turned out to be true later. Do they deserve credit for stating the principle? Did they really know what they were saying?
The text books that give historical notes surely make the history sound certain. It is not so clear when you try to read the whole story. Often times the “discoverer” was arguing about something else. Crystals are ice that is too cold to melt. Crystals grow like plants. The history seems to be evidence that civilized people search for ways to explain nature by logical theories.
So Hauy wrote some articles and books and showed how large crystals could be built out of little cubes to get shapes that were not cubes. He did this by stacking the cubes at certain slopes. Put down a layer, then go in 2 cubes before starting the next layer, and so forth. Different faces call for different steps. See the mineralogy books for pictures of Hauy’s models.
From this time crystallography became an exact science. The interfacial angles of crystals were accurately measured. The device to measure the angles is called a goniometer. There are various types. The simplest goniometer, called a contact goniometer, is a semicircular protractor with an arm pivoted at the center. Place the base against one face and the arm against another and read the angle off the protractor.
(figure available in print form)
In 1809 William H. Wollaston invented the reflecting goniometer which allowed the measuring of angles on much smaller crystals, even ones with rough faces.
So angles became the big issue with crystallographers. The length of the edges did not matter it was the angles. The shape of the crystal as far as lengths are concerned is called the habit of the crystal. Crystallographers visualized the crystal at the center of a sphere with lines radiating out perpendicular to its faces to intersect the sphere at points which could be mapped like longitude and latitude. Then they decided to map the sphere onto a plane using the technique called stereographic projection. One place we all have seen stereographic projections is the trade mark of the National Geographic Society. The portion of an astrolabe that does not move is also a stereographic projection. The stereographic projection is a topic of study unto itself.
If crystals are made of bricks all stacked next to each other what can we say? We can explain the three laws of crystallography. The bricks will be so small that the steps will not be visible to the naked eye thus giving flat faces. The stacking of the bricks will give fixed slopes which in turn give fixed angles. Notice that a stacking schedule of one over and two up will have a tangent twice that of a stacking schedule one over and one up. The tangents will double but not the angles. The tangents will be integral multiples of a fundamental value, but not the angles. The edges of the bricks will line up to give a three dimensional coordinate system and the dimensions of the brick will give the units to use along each axis.
What shape will the bricks have? If they were cubes then our coordinate system would have three axes at 90° to each other with equal unit lengths along each axes, just like analytic geometry. Crystals are not so simple. There are crystals with cubical building blocks, they form the cubic or isometric system of crystals. Other building blocks form other systems of crystals. Can you determine the possible shapes of the other building blocks? There are only five more.
We could have traditional rectangular bricks where the lengths of the edges are three distinct values. Such bricks make the orthorhombic system. We could tip our traditional brick at an angle so one dihedral angle was not 90° this makes the monoclinic system. We could tip it in three directions to make the triclinic system. We could take the cube and change the length of one dimension to make the tetragonal system. We could use bricks that did not have all rectangular faces, we could use hexagonal tiles to make the hexagonal system. That makes the six crystal systems.
(figure available in print form)
(figure available in print form)
The reader might wonder why triangular prisms do not make a crystal system. We said that crystals are bricks stacked next to each other, we meant touching along faces without turning the brick. This idea is called translation. If we were to place triangles along a line there would be empty “upside down” triangular spaces between them. To fill those places we would have to turn the triangle as we moved it. So we put two triangles together to get a parallelogram and the monoclinic or triclinic systems.
Knowing the system of a crystal scientists can predict how physical properties will behave. For example, a cubic crystal will expand equally in all direction when heated, while a tetragonal crystal will expand a different amount in one direction.
Let us think some more about the bricks. If you were at any corner of any brick you would not be able to tell it from any other brick. The bricks would have to be of proper shape. You might be able to define directions, some points could be closer to you than others, but then again maybe not. You might be able to look at the system in certain ways and still see the same arrangement as before, stand on your head, turn and look back, maybe do both. These tricks lead to the concept of transformations: translations, reflections, inversions, rotations, which in turn lead to the mathematical structure called a group. A translation is when you slide from one brick to the next ending up in the same configuration you started at. If you continue in that direction each time you go the same distance you will end up in the same configuration. Reflections are like mirrors. Two points are reflections of each other if the mirror is the perpendicular bisector of the line segment connecting them.
Inversion is the hardest to describe. You need a center of inversion, a point. To find the inversion image of a point you draw a line segment from the point to the center of inversion, then you continue the line on to a point as far from the center as the original point. It is as if the mirror had been shrunk to a point. The center of inversion is the midpoint of the line segment joining the original point and its image.
A rotation is the easiest to describe. You pick a line to serve as if it were the axis of a top and rotate the configuration about it. All the transformations can be described in terms of the coordinates of the points. You start with one point and you get another point called the image of the first.
All these transformations may be expressed in terms of what they do to the coordinates of any point. My experience was that the coordinate explanation for inversion was much easier to understand than the verbal: point (x,y,z) becomes (-x,-y,-z).
So transformations lead to groups, so let us tell _____
