Some authors put quartz into the hexagonal crystal system. Note: All figures that have been presented so far show renderings of a right-handed quartz structure. for quartz the numeber of SiO2 formulae) it ca be obtained by the formula Z = (N.V.D)/M where N = Avogadro number Also note that while the relative positions of the atoms are correct, this is probably not an accurate model of quartz surface structure (I do not have any empirical data on that). In theory, there are almost as many different twinning laws as crystal forms, but in nature only very few can actually be observed in a mineral species. If the finger moves clockwise, the helix is right-handed. Polymorphism is the occurrence of multiple crystalline forms of a material. If Z is the number of unit formulae contained in the unit cell (e.g. Figure 5.08 is just a copy of Fig.4.04 and shows both motifs a and b as basic elements of the crystal structure. The following table lists morphological, structural and optical features of left- and right-handed quartz. Some authors put quartz into the hexagonal crystal system. Any textbook on mineralogy will discuss this topic extensively. 3 We can calculate a number of atoms/molecules and ions in a unit cell easily by In a real crystal, the oxygen atoms on the surface each have an additional small hydrogen atom attached. Again, I would recommend the very nice explanation that can be found at this website: Introduction to Crystallography and Mineral Crystal Systems. But the helices are a geometrical feature of quartz that has important implications for its symmetry. Thus quartz is said to have a macromolecular structure. The left helix in Fig.5.03 is left-handed, the right one is right-handed. Hexagons can cover a plane completely to form a honeycomb pattern, but their edges will not join to form the straight axes of a coordinate system. They describe the crystal structure quantitatively and look for measures that help to relate the structure to the outer shape of the crystal. After a 180° turn they point downwards, of course, and look a bit like leaves hanging down in the helix to the right. So far we have only determined the dimension of the unit cell, but not its position relative to the crystal lattice, and accordingly there are several candidates for unit cells shown as examples at arbitrary positions in the rendering. There is nothing wrong with that, it′s just that they choose to distinguish 6 crystal systems and regard the trigonal system as a special case of the hexagonal system. The relationship between the handedness of the threefold virtual helices, the morphology and the physical properties is straightforward. In theory, there are almost as many different twinning laws as crystal forms, but in nature only very few can actually be observed in a mineral species. In a penetration twin (Fig.10, bottom) the border between the subindividuals has an irregular shape. The yellow horizontal lines that extend to the right demonstrate that these planes are evenly spaced along the vertical axis (or c-axis). For example, the dominance of certain forms on a quartz crystal might cause a cube-like look that is accordingly called "pseudo-cubic habit". The handedness of quartz crystal structure is not only expressed in the geometry of quartz crystals, it also plays a role in its optic properties. Another interesting feature are black holes, gaps in the structure. This figure is nice but not very helpful for understanding the underlying structure and we need to change the perspective to gain more insight. These notions are in turn all based on the concepts of the unit cell and the point lattice. There is a mathematical terminology for describing crystallographic forms, the so called Miller indices for crystal systems with three axes and the extension for crystal systems with four axes, the Miller-Bravais indices. The central SiO4 tetrahedra in Fig.5.09 are members of two helices and the whole structure looks a bit like a chain of twisted rings. 5 Parallelly intergrown crystals might look like twins to most people, but in most cases they are not twins. A geometrical body that is not congruent with itself after a rotation by 180° is called polar. Of course that is so because it is displayed as an isolated molecule. 2 last modified: Friday, 06-Jan-2017 01:36:00 CET, Document status: usable, section on twinning missing. Crystals that look very different (e.g. The shadows in the rendering indicate that the three SiO4 tetrahedra do not form a closed triangle - this would be an impossible geometrical figure. Fig.10.09: Unit cell (blue) and hexagonal crystallographic axes projected into a quartz crystal. Although they contain the same types of tetrahedra, the two helices differ in polarity (compare to Fig.5.06). You will note that the pattern has been shifted, and both halves of the slice don't match anymore. Typical terms in that context are "druzy", "compact", "dendritic", "fibrous", or "granular". To determine the handedness of a helical structure (a screw for example) you place it upright in front of you and follow the helix downward with your finger. 4 It does not help to shift or rotate both helices relative to each other, Fig.5.06 shows the best possible match. For example, the tetrahedra in the upper left top views of both figures point in different directions. It is sufficient to take three motifs to form the inner ring of SiO4 tetrahedra around central channel. This is obviously not a hexagonal structure, it is trigonal and has a threefold rotational symmetry. Here the two arrows, designated c and a1 (the one already shown in the upper part of the figure), define another parallelogram, in this case a rectangle. This is just a matter of convenience for me, as the basic data set that has been used to render all figures on this page uses the coordinates of the left unit cell model. To demonstrate it in a quartz lattice is very confusing because the structure is very complex and for the same reason beyond my technical possibilities. 6 The sixfold double helix and the threefold helices are of opposite handedness. Now the ring structure shown in Fig.6.01 is only three tetrahedra high, so a sixfold helix around the central channel cannot be complete. If one could actually see the atoms, they would perhaps look more like a cloudy sphere with no clear border or surface, the clouds consisting of quickly moving electrons. By placing the unit cell at that position, the tetrahedra have been ripped apart. A form does not have to be a three-dimensional body, it can be a simple two-dimensional plane. This is not simply a slice of the crystal with a layer of atoms, it is a top view of the crystal structure, and the atoms in it lie in different planes.    -    Source: http://www.quartzpage.de/crs_terms.html, English Wikipedia Article on Miller Indices, German Wikipedia Article on Miller Indices, Introduction to Crystallography and Mineral Crystal Systems. The top view looks mirror symmetric at a first glimpse, but if you take into account the three-dimensional structure you can see it is not. A sixfold double helix surrounding the large central channel. But on a small scale the symmetry is broken. According to Gibbs' rules of phase equilibria, these unique crystalline phases are dependent on intensive variables such as pressure and temperature. In a penetration twin (Fig.10, bottom) the border between the subindividuals has an irregular shape. The point is that one just needs to know the position of the atoms in unit cell coordinates and the unit cell dimensions to construct the entire crystal structure with all its features. Since this is a top view of a three-dimensional crystal, these gaps are actually channels that run through the entire crystal, parallel to the c-axis. both aspects differ in orientation and position of the tetrahedra (that is to be expected after a rotation). Both helices are connected by the outer gray tetrahedra.   a   0.49133 nm or 4.9133 Å So the task is to identify the characteristic element that defines the crystal structure of quartz. This model is preferred because it shows the most important structural element of quartz: the SiO4 tetrahedron. 2 The unit cell of quartz is not trigonal, but hexagonal. In the top view we can see what is causing the twist in the ring: the unit cell encloses two neighboring threefold helices. Both helices are left-handed so the handedness is indeed preserved just as expected. In certain crystal classes some forms are not "complete" bodies, but open at one or two ends, and are called open forms (e.g. And another fact that can be found when inspecting Fig.6.06: Each SiO4 tetrahedron is member of 2 threefold and 2 sixfold helices.