Felix Bloch
"Whether we like it or not, whetherwe intend it or not, we are destined to have authoritarian order." Alexander Solzenitsyn
Felix Bloch
We all share a fascination for crystals, the beautiful regular stones of nature which can come in different shapes and colors. They were collected by cavemen just like they are collected today. What distinguishes them from shapeless rocks, and what determines their shape and color? Why are some transparent while others are dark, some shaped as a cube while others have hexagonal faces? These questions were raised millenea ago, characteristically by the Greeks--the name itself comes from crystallos, 'clear ice' in Greek- but we still do not have the answer to many such puzzles. One fact, however, is clear-- their attributes, color and form are due to the highly regular, periodic arrangement of the atoms which form the crystals, as was first suggested by Plato.
The arrangement of atoms reflects a particular kind of symmetry, and is also intimately related to order and repetition. These concepts are widely employed in architecture and the decorative arts, and are also often used to convey a message.
To a scientist the reason for such periodicity is fundamental and well understood. It comes from the fact that atoms attract, but, at close distances, repel each other. This leads to an equilibrium distance between the neighbors and thus to a periodic array. This periodicity has fundamental consequences. For a physicist who wants to understand and describe the properties of these solids, the fact that each element of this structure is the same, and that each has the identical environment leads to enormous simplification. Consequently, we can calculate many properties of crystals in spite of the enormous number of atoms involved. Our lives would be different without such periodicity; the material that is the basis of a computer chip is a crystal, and so are many other materials used in electronics and communication, just to mention two important examples.
All crystals have a fundamental, so-called translational symmetry: displacing the pattern by the fundamental distance between the units (or by any multiple of this unit) leads to the identical structure; we call this translational symmetry. There are other symmetries involved which allow the rigorous classification of the various crystal structures using a field called group theory. For example, the drawings by Mauritius Escher display an evident periodic pattern. (Figure 5.1), (Figure 5.2), (Figure 5.3) We can make a copy, displace it, and place it on the original drawing, and we will not notice a difference. Try, however, to rotate the drawing by a certain angle to achieve this; the different drawings will behave in a different way. These drawings have a translational symmetry, common to all crystals, and displacing the drawing by a certain distance, called the 'lattice constant' in crystallography, leads to the original pattern. We can rotate different drawings by different degrees to get the original pattern. The first drawing when rotated by 180 degrees results in the original pattern, and the second rotation by the same degree leads by virtue of a full rotation to the original position. (Figure 5.1) This Figure has a two-fold symmetry. The following drawing can be rotated by the angles 120, 240 and finally 360 degrees, and all lead to the same arrangement. (Figure 5.2) We call this three-fold symmetry. Other drawings such as the one shown on the next figure have four-fold symmetry; four rotations all lead to the same identical arrangement. (Figure 5.3) Some of these drawings are also invariant to reflection; they look the same when we perform a mirror reflection, sort of flipping over the image and putting it down again.
Using these three types of operations--translation, rotation and reflection--the various structures can be classified into groups (hence the name group theory) which have the same symmetrical properties. In two dimensions, we have five possibilities. (Figure 5.4) In three dimensions, the situation is more complicated and fourteen distinct different situations occur. (Figure 5.5)
Note that there is no five-fold symmetry; we can not build up a periodic array where five-fold symmetry prevails. This has been known by artists and also scientists for centuries; a drawing by Durer makes this point. (Figure 5.6) Similarly, a drawing by Kepler illustrates this same point. (Figure 5.7)
There are various ways in which such patterns can be seen and studied. We can scatter waves on such patterns, and the waves will interfere leading to their own pattern, related to the original arrangement. X-ray scattering works in this way. There are many such scattering methods, but recently these structures have been seen directly by a method utilizing a scanning tunneling microscope, an elegant little device. (Figure 5.8)
So far so good, but we still do not understand why crystals have such beautiful regular faces, parallel to each other although millions and millions of atoms apart. The structure of the crystals is related to the strusture of the atomic arrangements. This has been known for a long time--since the 17th century. (Figure 5.9) What makes them grow so regularly? How does one side know that it has to grow parallel to the opposite side? We do not know exactly, but it appears that layers upon layers of atoms are used to cover a surface in a self-healing fashion, and this type of growth preserves the direction of growth. Obviously all this would not happen if the atoms themselves were not organized with distances between them set to an incredible precision.
Among the various crystals, snowflakes are probably the subject of most fascination, because of their myriad of shapes, all with one constant attribute: they all have hexagonal symmetry. (Figure 5.10) This has been known for ages, and by now we know the microscopic origin: it lies in the hexagonal arrangement of the atoms in an ice crystal.
The hexagonal pattern is very common and occurs under various circumstances. The honeycomb lattice derives its name from the arrangement produced by bee colonies. (Figure 5.11) This but it can also be observed on a microscopic scale: atoms often arrange themselves in this fashion. (Figure 5.12) We can understand this as the simple consequence of the most economic arrangement of objects, such as spheres (as the simplest atoms with closed electronic shells may be visualized) into a pattern which, appropriately is called "close packing." (Figure 5.13)
The fact that crystals have a periodic pattern leads to important consequences. The equation by Felix Bloch quoted at the beginning of this chapter casts this periodicity into a simple and elegant mathematical form which also allows us to calculate the various properties of materials. These properties may be based on whether or not they conduct current, whether they are magnetic, or whether or not they are transperent to visible light. But more important, this fact leads to discoveries and possibilities we would not have otherwise. We could not use silicon chips in our computers--in fact maybe computers would not be possible to build; and the beautiful transparent diamonds also would not have this attribute without such periodic arrangement.
To an artist (or philosopher) this periodicity has a message; the concept of returning to the origin, and the notion of perfect order are both appealing. We add to this the economy of construction in using identical elements, and we arrive at important features of architecture, rugs or tilings. (Figure 5.14), (Figure 5.15)
This periodicity has also been used to express monotony, so prevalent in our modern life, where we are bombarded by the same message over and over again. Andy Warhol used this repeatedly to convey this message. (Figure 5.16) There are, of course, minor variations from image to image. The work is more than a trivial mechanical repetition, and after a short while we focus on these differences and variations.
The periodic arrays we discussed are in clear contrast to the chaotic, random arrays we are also accustomed to seeing. People strolling down a street assume this formation, and so do atoms in the liquid or in the gaseous state. The two different states are displayed in a simple one dimensional space. (Figure 5.17a) The upper part shows an ordered array, the middle a disordered array. What about the array on the next Figure? (Figure 5.17b) It clearly appears disordered, just like the ?upper? part on the previous figure. However, this structure can be resolved into two periodic patterns as the two lower arrays, the sum of the two leads to the uppermost part of the Figure - demonstrate.
Such systems are between ordered and disordered structures; they are quasi-periodic. The same thing also happens in two dimensions. It was first suggested as an elegant mathematical construction done by Penrose a few decades ago. Several such structures exist, and one example of such a two dimensional tiling is displayed in the next image, with some coloring enhancing the effect of quasi-periodicity. (Figure 5.18) As it turns out, such structures have five-fold symmetry; this was discovered by using X-ray scattering. We have also discussed how the external apperance of crystals mirrors their molecular arrangements; and indeed these materials produced in "crystalline" form have pentagonal faces with five-fold symmetry. (Figure 5.19) This astonishing fact, seemingly against all rules of crystallography, is by now understood. The classic theorem that long-range order cannot develop with five-fold symmetry is still valid; the structures do not have full long-range order. They are not real crystals--they are quasi-crystals.
Periodic patterns may also emerge as the consequence of an instability. Such patterns are triggered by a small disturbance in the system. We are all familiar with the images of sand dunes or snow moguls. In both cases the underlying system is periodic. For example, the uniform surface of pristine snow powder just after the snowfall collects small imprints, which are then enhanced by subsequent skiers. The pattern is different after each snowfall, but the periodicity is the same. The same pattern may emerge in time; the ocean waves arriving in a periodic fashion to the beach is a familiar example.
We would think that these ordered patterns are the most symmetric forms of nature. To a physicist, however, this is not the case. And in order to appreciate this we have to go back and see how these states develop. Taking the snow moguls as an example--just after the snowfall, the hill is evenly covered by snow; we call this the perfectly symmetric case. After a day's skiing this symmetry is broken, and we have places where the snow is deep and places where the snow cover is shallow. While we have a pattern--the periodic array of snowmoguls--the uniform symmetry is broken. Something similar happens in the case of a collection of atoms. In the gas or liquid form the atoms or molecules execute a chaotic random motion, and for the sake of argument let us assume that we have two kinds of atoms, black and white. If we look at a certain point in space, wait for a time, and count how many black or white atoms we encounter, we will find that they are equal in number; the place looks gray; and again we say the state is symmetric. Look, however, when these atoms form a crystal (probably crystallizing in a way that white and black atoms alternate), and we will find that at every point there is either a black or a white atom; not all the places are the same anymore, and the symmetry is broken. The transition between these two states occurs in a fashion we described in Chapter 1.1.