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Chapter 4 is about symmetry and crystal habit. It looks at thirty-two crystal classes; centres and inversion axes of symmetry; crystal symmetry and properties; translational symmetry elements; space groups; and Bravais lattices, space groups, and crystal structures. The chapter also examines crystal structures and space groups of inorganic compounds, close packing of organic molecules (Kitaigorodskii), long chain polymer molecules, and quasiperiodic crystals (Mackay). Finally, it considers icosahedral structures and transformations, which relates to crystalline (icosahedral packing of spheres).

Water is the most miraculous of fluids. As well as being ubiquitous on Earth and essential for life as we know it, it has remarkable properties which at first sight don’t seem to be consistent with its almost laughably simple chemical composition. Each molecule of water consists of a single oxygen atom (O) and two hydrogen atoms (H); its chemical formula is therefore, as just about everyone already knows, H2O. Here is one odd but hugely important anomalous property. A water molecule is only slightly heavier than a methane molecule (CH4; C denotes a carbon atom) and an ammonia molecule (NH3, N denotes a nitrogen atom). However, whereas methane and ammonia are gases, water is a liquid at room temperature. Water is also nearly unique in so far as its solid form, ice, is less dense than its liquid form, so ice floats on water. Icebergs float in water; methanebergs and ammoniabergs would both sink in their respective liquids in an extraterrestrial alien world, rendering their Titanics but not their Nautiluses safer than ours. Another very important property is that water is an excellent solvent, being able to dissolve gases and many solids. One consequence of this ability is that water is a common medium for chemical reactions. Once substances are dissolved in it, their molecules can move reasonably freely, meet other dissolved substances, and react with them. As a result, water will figure large in this book and this preliminary comment is important for understanding what is to come. You need to get to know the H2O molecule intimately, for from it spring all the properties that make water so miraculous and, more prosaically, so useful. The molecule also figures frequently in the illustrations, usually looking like 1, where the red sphere denotes an O atom and the pale grey spheres represent H atoms. Actual molecules are not coloured and are not made up of discrete spheres; maybe 2 is a better depiction, but it is less informative. I shall use the latter representation only when I want to draw your attention to the way that electrons spread over the atoms and bind them together.

The transport through a one-dimensional barrier is calculated within the tight-binding model. The surface Green’s functions are introduced as a method to invert the Green’s function matrix and to set-up convenient boundary conditions for simulations. The formalism is applied to calculate the transport properties of parallel stacked organic molecules. The extension to higher dimensions and multiband crystals is discussed. In this section we apply the GKB formalism to diffraction of electrons on a barrier. The system we study is a planar heterojunction of two ideal semi-infinite crystals or a surface of a crystal. As an initial condition we take a stream of electrons with a sharp momentum.