Introduction to Quasicrystals. This page is meant to be an introduction to the field of Quasicrystals in order to educate the interested reader on some basic concepts in this relatively new branch of Crystallography. The more advanced reader may proceed to other sites and sources on quasicrystals. This page is intended for those having no prior knowledge in this field. In classical crystallography a crystal is defined as a threedimensional periodic arrangement of atoms with translational periodicity along its three principal axes. Thus it is possible to obtain an infinitely extended crystal structure by aligning building blocks called unit-cells until the space is filled up. Normal crystal structures can be described by one of the 230 space groups, which describe the rotational and translational symmetry elements present in the structure. Contents Since quasicrystals lost periodicity in at least one dimension it is not possible to describe them in 3D-space as easily as normal crystal structures.
Icosahedral QC contents. Quest.pdf (application/pdf Object) Quasicrystal. Potential energy surface for silver depositing on an aluminium-palladium-manganese (Al-Pd-Mn) quasicrystal surface. Similar to Fig. 6 in Ref.[1] Aperiodic tilings were discovered by mathematicians in the early 1960s, and, some twenty years later, they were found to apply to the study of quasicrystals.
The discovery of these aperiodic forms in nature has produced a paradigm shift in the fields of crystallography. Quasicrystals had been investigated and observed earlier,[2] but, until the 1980s, they were disregarded in favor of the prevailing views about the atomic structure of matter. In 2009, after a dedicated search, a mineralogical finding, icosahedrite, offered evidence for the existence of natural quasicrystals.[3] Roughly, an ordering is non-periodic if it lacks translational symmetry, which means that a shifted copy will never match exactly with its original. History[edit] A Penrose tiling The idea of a new structure was the necessary paradigm shift to break the impasse. Notes[edit] Bragg's law. Bragg diffraction (also referred to as the Bragg formulation of X-ray diffraction) was first proposed by William Lawrence Bragg and William Henry Bragg in 1913 in response to their discovery that crystalline solids produced surprising patterns of reflected X-rays (in contrast to that of, say, a liquid).
They found that these crystals, at certain specific wavelengths and incident angles, produced intense peaks of reflected radiation (known as Bragg peaks). The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction processes.[1] Both neutron and X-ray wavelengths are comparable with inter-atomic distances (~150 pm) and thus are an excellent probe for this length scale. X-rays interact with the atoms in a crystal. W. L. Bragg explained this result by modeling the crystal as a set of discrete parallel planes separated by a constant parameter d.
Bragg condition[edit] Bragg diffraction. Reciprocal space[edit] . Where is a reciprocal lattice vector and and . Scalar Superpotential Theory. A Simple Fractal Model of the Conscious Universe. Here's the definition of fractals from Wikipedia: “A fractal is 'a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole,' a property called self-similarity...
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that are approximated by fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, various vegetables (cauliflower and broccoli), and animal coloration patterns.” If you would like to see fractals in action, the NOVA documentary, “Fractals,: Hunting the Hidden Dimension," gives an excellent overview of fractals. • It has a fine structure at arbitrarily small scales. • It is too irregular to be easily described in traditional Euclidean geometric language. • It is self-similar (at least approximately or stochastically). • It has a simple and recursive definition. Fractals in Nature. Naturally Occurring Fractals (including plants, rivers, galaxies, clouds, weather, population patterns, stocks, video feedback, crystal growth, etc.)
The geometry of Fractals brings us a new appreciation for the natural world and the patterns we observe in it. Many things previously called chaos are now known to follow subtle subtle fractal laws of behavior. So many things turned out to be fractal that the word "chaos" itself (in operational science) had redefined, or actually for the FIRST time Formally Defined as following inherently unpredictable yet generally deterministic rules based on nonlinear iterative equations.
Fractals are unpredictable in specific details yet deterministic when viewed as a total pattern - in many ways this reflects what we observe in the small details & total pattern of life in all it's physical and mental varieties, too .... After a few dozen repetitions or ITERATIONS the shape we would recognize as a Perfect Fern appears from the abstract world of math. Fractal.
Figure 1a. The Mandelbrot set illustrates self-similarity. As the image is enlarged, the same pattern re-appears so that it is virtually impossible to determine the scale being examined. Figure 1b. The same fractal magnified six times. Figure 1c. Figure 1d. Fractals are distinguished from regular geometric figures by their fractal dimensional scaling. As mathematical equations, fractals are usually nowhere differentiable.[2][5][8] An infinite fractal curve can be conceived of as winding through space differently from an ordinary line, still being a 1-dimensional line yet having a fractal dimension indicating it also resembles a surface.[7]:48[2]:15 There is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Introduction[edit] The word "fractal" often has different connotations for laypeople than mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception.
History[edit] Figure 2. Fractal Wisdom - Fractal Wisdom.