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Showing posts with label Quasicrystals. Show all posts
Showing posts with label Quasicrystals. Show all posts

Friday, February 07, 2014

Unus Mundus-One World

Unus mundus, Latin for "one world", is the concept of an underlying unified reality from which everything emerges and to which everything returns.

The idea was popularized in the 20th century by the Swiss psychoanalyst Carl Jung, though the term can be traced back to scholastics such as Duns Scotus[1] and was taken up again in the 16th century by Gerhard Dorn, a student of the famous alchemist Paracelsus.

The striving  for me was to dig deeper into our very natures.  It always the quest to understand the  patterns that reside in us. The very idea for me was that in  this quest to unify,  the objective world(matter) with the world that resides in a center place. To me that place was the source from which all things manifest.

 Jung, in conjunction with the physicist Wolfgang Pauli, explored the possibility that his concepts of the archetype and synchronicity might be related to the unus mundus - the archetype being an expression of unus mundus; synchronicity, or "meaningful coincidence", being made possible by the fact that both the observer and connected phenomenon ultimately stem from the same source, the unus mundus.[2]

So while there was this objective striving to see how such formations emerged as materiality of such expression,  was a final construct that existed in that external world. For me this was something no one could quite explain to me, yet,  as I moved forward  I began to find such correlates as to others who tried to map that expression.

 It was this psychoid aspect of the archetype that so impressed Nobel laureate physicist Wolfgang Pauli. Embracing Jung's concept, Pauli believed that the archetype provided a link between physical events and the mind of the scientist who studied them. In doing so he echoed the position adopted by German astronomer Johannes Kepler. Thus the archetypes which ordered our perceptions and ideas are themselves the product of an objective order which transcends both the human mind and the external world.[2]

This as the idea emerged,  I looked for what emergence might mean, as an example of a beginning,  and the subsequent model that may emerge from that source. This then became know as the "arche,"  and the tendency to form"(type)" as a movement forward in the solidifying of that expression. This was a matter bound expression, fully recognizing the need for a spiritual recognition of this opposition as a struggle in with consciousness to seek balance with materiality. Polarity,  as the world of the real.

One of Duchamp's close friends Man Ray (1890–1976) was also one of Duchamp's collaborators. His photograph 'Dust Breeding' (Duchamp's Large Glass with Dust Notes) from 1920 is a document of The Large Glass after it had collected a year's worth of dust while Duchamp was in New York. See:
Dust Breeding (Man Ray 1920)


Such histrionically values were tied to such expressions to have found that the inner world and the outer-world were extremely connected. The observance not seen until it was understood that this psychology was topological interpreting itself from an inductive/deductive stance,  as to the question, and with regard to the nature of the question.

 Jung interpreted the practice of alchemy as the symbolic projection of psychic processes. In Psychology and Alchemy and Mysterium Coniunctionis (1955/56), Jung’s empirical exploration and rediscovery of the objective psyche led him to recognise that the basis of the alchemist’s endeavour was the archetypal union of opposites by means of the integration of opposing polarities: conscious and unconscious, reason and instinct, spiritual and material, masculine and feminine. In the last summaries of his insights on the subject, influenced in part by his collaboration with the Nobel Prize winning physicist Wolfgang Pauli, the old Jung envisions a great psycho-physical mystery to which the old alchemists gave the name of unus mundus (one world). At the root of all being, so he intimates, there is a state wherein physicality and spirituality meet. See:Reflections On Duchamp, Quantum Physics, And Mysterium Coniunctionis

This would place myself in the position of questioning this causal nature to have said that "will" was deeply connected to our psyche,  to have not understood this deeper perception of a reality connection. Also,  that such unification was deeper embedded in this practice of unification,  so as to strive to form,  as a example of an idea into expression.

Betrayal of Images" by Rene Magritte. 1929 painting on which is written "This is not a Pipe"

This alchemy valuation of that work toward expression was based on a fundamental reality of joining the objectified world with the nature of the source. This forming process,  the constructs,  as a fundamental structure of the reality given.

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Sunday, June 02, 2013

Two Paul Steinhardt Projects: "Cyclic Universe" and "Quasicrystals"



Two Paul Steinhardt Projects: "Cyclic Universe" and "Quasicrystals"






Albert Einstein Professor in Science, Departments of Physics and Astrophysical...
Quasi-elegance....As a young student first reading Weyl's book, crystallography seemed like the "ideal" of what one should be aiming for in science: elegant mathematics that provides a complete understanding of all physical possibilities. Ironically, many years later, I played a role in showing that my "ideal" was seriously flawed. In 1984, Dan Shechtman, Ilan Blech, Denis Gratias and John Cahn reported the discovery of a puzzling manmade alloy of aluminumand manganese with icosahedral symmetry. Icosahedral symmetry, with its six five-fold symmetry axes, is the most famous forbidden crystal symmetry. As luck would have it, Dov Levine (Technion) and I had been developing a hypothetical idea of a new form of solid that we dubbed quasicrystals, short for quasiperiodic crystals. (A quasiperiodic atomic arrangement means the atomic positions can be described by a sum of oscillatory functions whose frequencies have an irrational ratio.) We were inspired by a two-dimensional tiling invented by Sir Roger Penrose known as the Penrose tiling, comprised of two tiles arranged in a five-fold symmetric pattern. We showed that quasicrystals could exist in three dimensions and were not subject to the rules of crystallography. In fact, they could have any of the symmetries forbidden to crystals. Furthermore, we showed that the diffraction patterns predicted for icosahedral quasicrystals matched the Shechtman et al. observations. Since 1984, quasicrystals with other forbidden symmetries have been synthesized in the laboratory. The 2011 Nobel Prize in Chemistry was awarded to Dan Shechtman for his experimental breakthrough that changed our thinking about possible forms of matter. More recently, colleagues and I have found evidence that quasicrystals may have been among the first minerals to have formed in the solar system.

The crystallography I first encountered in Weyl's book, thought to be complete and immutable, turned out to be woefully incomplete, missing literally an uncountable number of possible symmetries for matter. Perhaps there is a lesson to be learned: While elegance and simplicity are often useful criteria for judging theories, they can sometimes mislead us into thinking we are right, when we are actually infinitely wrong. See:

2012 : WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION?



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Thursday, July 19, 2012

Process Fractal vs Geometry Fractals

Let proportion be found not only in numbers, but also in sounds, weights, times and positions, and whatever force there is.Leonardo Da Vinci
The Mandelbrot set, seen here in an image generated by NOVA, epitomizes the fractal. Photo credit: © WGBH Educational Foundation

 "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." So writes acclaimed mathematician Benoit Mandelbrot in his path-breaking book The Fractal Geometry of Nature. Instead, such natural forms, and many man-made creations as well, are "rough," he says. To study and learn from such roughness, for which he invented the term fractal, Mandelbrot devised a new kind of visual mathematics based on such irregular shapes. Fractal geometry, as he called this new math, is worlds apart from the Euclidean variety we all learn in school, and it has sparked discoveries in myriad fields, from finance to metallurgy, cosmology to medicine. In this interview, hear from the father of fractals about why he disdains rules, why he considers himself a philosopher, and why he abandons work on any given advance in fractals as soon as it becomes popular. A Radical Mind

As I watch the dialogue between Bruce Lipton and Tom Campbell here, there were many things that helped my perspective understand the virtual world in relation to how the biology subject was presented. It is obvious then why Bruce Lipton likes the analogies Tom Campbell has to offer. The epiphanies Bruce is having along the road to his developing biological work is very important. It is how each time a person makes the leap that one must understand how individuals change, how societies change.



Okay so for one,  the subject of fractals presents itself and the idea of process fractals and Geometry Fractals were presented in relation to each other. Now the talk moved onto the very thought of geometry presented in context sort of raised by ire even though I couldn't distinguish the differences. The virtual world analogy is still very unsettling to me.

So ya I have something to learn here.

I think my problem was with how such iteration may be schematically driven so as toidentify the pattern. Is to see this process reveal itself on a much larger scale. So when I looked at the Euclidean basis as a Newtonian expression the evolution toward relativity had to include the idea of Non Euclidean geometries. This was the natural evolution of the math that lies at the basis of graduating from a Euclidean world. It is the natural expression of understanding how this geometry can move into  a dynamical world.

So yes the developing perspective for me is that even though we are talking abut mathematical structures here we see some correspondence in nature . This has been my thing so as to discover the starting point?

A schematic of a transmembrane receptor


It the truest sense I had already these questions in my mind as  I was going through the talk. The starting point for Bruce is his biology and the cell. For Tom, he has not been explicit here other then to say that it is his studies with Monroe that he developed his thoughts around the virtual world as it relates to the idea of what he found working with Monroe.

So it is an exploration I feel of the work he encountered and has not so far as I seen made a public statement to that effect. It needs to be said and he needs to go back and look over how he had his epiphanies. For me this is about the process of discovery and creativity that I have found in my own life. Can one feel so full as to have found ones wealth in being that you can look everywhere and see the beginnings of many things?

This wealth is not monetary for me although I recognized we had to take care of or families and made sure they were ready to be off on their own. To be productive.

The Blind Men and the Elephant
John Godfrey Saxe (1816-1887)
 So for me the quest for that starting point is to identify the pattern that exists in nature as much as many have tried various perspective in terms of quantum gravity. Yes, we are all sort of like blind men trying to explain the reality of the world in our own way and in the process we may come up with our epiphanies.

These epiphanies help us to the next level of understanding as if we moved outside of our skeletal frame to allow the membrane of the cell to allow receptivity of what exist in the world around as information. We are not limited then to the frame of the skeleton hardened too, that we cannot progress further. The surface area of the membrane then becomes a request to open the channels toward expansion of the limitations we had applied to ourselves maintaining a frame of reference.

Monday, July 16, 2012

Where is Our Starting Point?



"The end he (the artist) strives for is something else than a perfectly executed print. His aim is to depict dreams, ideas, or problems in such a way that other people can observe and consider them." - M.C. Escher


Can you trace the patterns in nature toward matter manifestations?


To them, I said,
the truth would be literally nothing
but the shadows of the images.
-Plato, The Republic (Book VII)

The idea here is about how one's observation and model perceptions arises from some ordered perspective. Some use a starting point as an assumption of position. Do recognize "the starting point" in the previous examples?

 Cycle of Birth, Life, and Death-Origin, Indentity, and Destiny by Gabriele Veneziano


In one form or another, the issue of the ultimate beginning has engaged philosophers and theologians in nearly every culture. It is entwined with a grand set of concerns, one famously encapsulated in an 1897 painting by Paul Gauguin: D'ou venons-nous? Que sommes-nous? Ou allons-nous? "Where do we come from? What are we? Where are we going?"


The effective realization that particle constructs are somehow smaller windows of a much larger perspective fails to take in account this idea that I am expressing as a foundational approach to that starting point.




If you do not go all the way toward defining of that "point of equilibrium" how are you to understand how information is easily transferred to the individual from a much larger reality of existence? One would assume information is all around us? That there are multitudes of pathways that allow us to arrive at some some probability density configuration as some measure of an Pascalian ideal.

Of course there are problems with this in terms of our defining a heat death in individuals?

That's not possible so one is missing the understanding here about equilibrium. I might have said we are positional in terms of the past and the future with regard to memory and the anticipated future? How is that heat death correlated? It can't.

So you have to look for examples in relation to how one may arrive at that beginning point. Your theory may not be sufficiently dealing with the information as it is expressed in terms of your approach to the small window?

There are mathematical inspections here that have yet to be associated with more then discrete functions of reality as expressive building blocks of interpretation. The basic assumption of discrete function still exists in contrast to continuity of expression. This is the defining realization in assuming the model that MBT provides. I have meet the same logic in the differences of scientific approach toward the definition of what is becoming?

On the one hand, a configuration space as demonstrated by Tom that is vastly used in science. On the other, a recognition of how thick in measure viscosity is realized and what the physics is in this association. Not just the physical manifestation of, but of what happens when equilibrium is reached. Hot or very cold. Temperature, is not a problem then?

See my problem is that I can show you levitation of objects using superconductors but I cannot produce this in real life without that science. Yet, in face of that science I know that something can happen irregardless of what all the science said, so I am looking as well to combining the meta with the physical to realize that such a conditions may arise in how we as a total culture have accepted the parameters of our thinking.

So by dealing with those parameters I too hoped to see a cultural shift(paradigm and Kuhn) by adoption of the realization as we are with regard to the way in which we function in this reality. So if your thinking abut gravity how is this possible within the "frame work" to have it encroach upon our very own psychological makeup too?

Friday, June 29, 2012

A Inherent Pattern of Consciousness


This image depicts the interaction of nine plane waves—expanding sets of ripples, like the waves you would see if you simultaneously dropped nine stones into a still pond. The pattern is called a quasicrystal because it has an ordered structure, but the structure never repeats exactly. The waves produced by dropping four or more stones into a pond always form a quasicrystal.

Because of the wavelike properties of matter at subatomic scales, this pattern could also be seen in the waveform that describes the location of an electron. Harvard physicist Eric Heller created this computer rendering and added color to make the pattern’s structure easier to see. See: What Is This? A Psychedelic Place Mat?
See Also: 59. Medieval Mosque Shows Amazing Math Discovery





A CG movie inspired by the Persian Architecture, by Cristóbal Vila. Go to www.etereaestudios.com for more info.






Circle Limit III, 1959




In 1941, Escher wrote his first paper, now publicly recognized, called Regular Division of the Plane with Asymmetric Congruent Polygons, which detailed his mathematical approach to artwork creation. His intention in writing this was to aid himself in integrating mathematics into art. Escher is considered a research mathematician of his time because of his documentation with this paper. In it, he studied color based division, and developed a system of categorizing combinations of shape, color and symmetrical properties. By studying these areas, he explored an area that later mathematicians labeled crystallography.

Around 1956, Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's works
Circle Limit I–IV demonstrate this concept. In 1995, Coxeter verified that Escher had achieved mathematical perfection in his etchings in a published paper. Coxeter wrote, "Escher got it absolutely right to the millimeter."


Snow Crystal Photo Gallery I
 
If you have never studied the structure of Mandala origins of the Tibetan Buddhist you might never of recognize the structure given to this 2 dimensional surface?  Rotate the 2d surface to the side view. It becomes a recognition of some Persian temple perhaps? I mean,  the video really helps one to see this,  and to understand the structural integrity is built upon.

So too, do we recognize this "snow flake"  as some symmetrical realization of it's individuality as some mathematical form constructed in nature?

I previous post I gave some inclination to the idea of time travel and how this is done within the scope of consciousness. In the same vein, I want you to realize that such journeys to our actualized past can bring us in contact with a book of Mandalas that helped me to realize and reveals a key of symmetrical expressions of the lifetime, or lifetimes.

Again in relation how science sees subjectivity I see that this is weak in expression in terms of how it can be useful in an objective sense as to be repeatable. But it helps too, to trace this beginning back to a source that while perceived as mathematical , shows the the mathematical relation embedded in nature.




See: Nature = Mathematics?

Wednesday, January 11, 2012

The Nobel Prize in Physics 1914 Max von Laue

Laue diagram of a crystal  See: Experimental diffraction

The Laue method in transmission mode

The Laue method in reflection mode




There are two different geometries in the Laue method, depending on the crystal position with regard to the photographic plate: transmission or reflection.

Concerning the Detection of X-ray Interferences




***


Max Theodor Felix von Laue (9 October 1879 – 24 April 1960) was a Germanphysicist who won the Nobel Prize in Physics in 1914 for his discovery of the diffraction of X-rays by crystals. In addition to his scientific endeavors with contributions in optics, crystallography, quantum theory, superconductivity, and the theory of relativity, he had a number of administrative positions which advanced and guided German scientific research and development during four decades. A strong objector to National Socialism, he was instrumental in re-establishing and organizing German science after World War II.
Max von Laue

Laue in 1929

Contents

  1 Biography




See:

Also See:

Thursday, January 05, 2012

Crystal Structure

In mineralogy and crystallography, crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The symmetry properties of the crystal are embodied in its space group.

A crystal's structure and symmetry play a role in determining many of its physical properties, such as cleavage, electronic band structure, and optical transparency.

Contents 

Unit cell

 

The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a small box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a three-dimensional shape. The unit cell is given by its lattice parameters, which are the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi  , yi  , zi) measured from a lattice point.




 Miller indices

 


Planes with different Miller indices in cubic crystals
Vectors and atomic planes in a crystal lattice can be described by a three-value Miller index notation (ℓmn). The , m, and n directional indices are separated by 90°, and are thus orthogonal. In fact, the component is mutually perpendicular to the m and n indices.

By definition, (ℓmn) denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it simply means that the planes do not intersect that axis (i.e., the intercept is "at infinity").

Considering only (ℓmn) planes intersecting one or more lattice points (the lattice planes), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula:



d = 2\pi / |\mathbf{g}_{\ell m n}|

 

Planes and directions

 

The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Likewise, the crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:
  • Optical properties: Refractive index is directly related to density (or periodic density fluctuations).
  • Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes.
  • Surface tension: The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.
  •  

Dense crystallographic planes
  • Microstructural defects: Pores and crystallites tend to have straight grain boundaries following higher density planes.
  • Cleavage: This typically occurs preferentially parallel to higher density planes.
  • Plastic deformation: Dislocation glide occurs preferentially parallel to higher density planes. The perturbation carried by the dislocation (Burgers vector) is along a dense direction. The shift of one node in a more dense direction requires a lesser distortion of the crystal lattice.
In the rhombohedral, hexagonal, and tetragonal systems, the basal plane is the plane perpendicular to the principal axis.

Cubic structures

 

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similarly for the reciprocal lattice. So, in this common case, the Miller indices (ℓmn) and [ℓmn] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent (ℓmn) lattice planes is (from above):




d_{\ell mn}= \frac {a} { \sqrt{\ell ^2 + m^2 + n^2} }


Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:
  • Coordinates in angle brackets such as <100> denote a family of directions that are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.
  • Coordinates in curly brackets or braces such as {100} denote a family of plane normals that are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.
For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.

 Classification

 

The defining property of a crystal is its inherent symmetry, by which we mean that under certain 'operations' the crystal remains unchanged. For example, rotating the crystal 180° about a certain axis may result in an atomic configuration that is identical to the original configuration. The crystal is then said to have a twofold rotational symmetry about this axis. In addition to rotational symmetries like this, a crystal may have symmetries in the form of mirror planes and translational symmetries, and also the so-called "compound symmetries," which are a combination of translation and rotation/mirror symmetries. A full classification of a crystal is achieved when all of these inherent symmetries of the crystal are identified.[1]

Lattice systems

 

These lattice systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each lattice system consists of a set of three axes in a particular geometrical arrangement. There are seven lattice systems. They are similar to but not quite the same as the seven crystal systems and the six crystal families.


The 7 lattice systems
(From least to most symmetric)
The 14 Bravais Lattices Examples
1. triclinic
(none)
Triclinic
2. monoclinic
(1 diad)
simple base-centered
Monoclinic, simple Monoclinic, centered
3. orthorhombic
(3 perpendicular diads)
simple base-centered body-centered face-centered
Orthorhombic, simple Orthorhombic, base-centered Orthorhombic, body-centered Orthorhombic, face-centered
4. rhombohedral
(1 triad)
Rhombohedral
5. tetragonal
(1 tetrad)
simple body-centered
Tetragonal, simple Tetragonal, body-centered
6. hexagonal
(1 hexad)
Hexagonal
7. cubic
(4 triads)
simple (SC) body-centered (bcc) face-centered (fcc)
Cubic, simple Cubic, body-centered Cubic, face-centered

The simplest and most symmetric, the cubic (or isometric) system, has the symmetry of a cube, that is, it exhibits four threefold rotational axes oriented at 109.5° (the tetrahedral angle) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal, tetragonal, rhombohedral (often confused with the trigonal crystal system), orthorhombic, monoclinic and triclinic.

 

 Atomic coordination


By considering the arrangement of atoms relative to each other, their coordination numbers (or number of nearest neighbors), interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing then.



HCP lattice (left) and the fcc lattice (right).

 Close packing

 

The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer was placed directly over plane A, this would give rise to the following series :




...ABABABAB....
 
This type of crystal structure is known as hexagonal close packing (hcp).

If however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises:



...ABCABCABC...
 
This type of crystal structure is known as cubic close packing (ccp)


The unit cell of the ccp arrangement is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the {111} planes of the fcc unit cell. There are four different orientations of the close-packed layers.

The packing efficiency could be worked out by calculating the total volume of the spheres and dividing that by the volume of the cell as follows:




 \frac{4 \times \frac{4}{3} \pi r^3}{16 \sqrt{2} r^3} = \frac{\pi}{3\sqrt{2}} = 0.7405


The 74% packing efficiency is the maximum density possible in unit cells constructed of spheres of only one size. Most crystalline forms of metallic elements are hcp, fcc, or bcc (body-centered cubic). The coordination number of hcp and fcc is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74. The APF of bcc is 0.68 for comparison.

 Bravais lattices

 

When the crystal systems are combined with the various possible lattice centerings, we arrive at the Bravais lattices. They describe the geometric arrangement of the lattice points, and thereby the translational symmetry of the crystal. In three dimensions, there are 14 unique Bravais lattices that are distinct from one another in the translational symmetry they contain. All crystalline materials recognized until now (not including quasicrystals) fit in one of these arrangements. The fourteen three-dimensional lattices, classified by crystal system, are shown above. The Bravais lattices are sometimes referred to as space lattices.

The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the 14 Bravais lattices. The characteristic rotation and mirror symmetries of the group of atoms, or unit cell, is described by its crystallographic point group.

Point groups

The crystallographic point group or crystal class is the mathematical group comprising the symmetry operations that leave at least one point unmoved and that leave the appearance of the crystal structure unchanged. These symmetry operations include
  • Reflection, which reflects the structure across a reflection plane
  • Rotation, which rotates the structure a specified portion of a circle about a rotation axis
  • Inversion, which changes the sign of the coordinate of each point with respect to a center of symmetry or inversion point
  • Improper rotation, which consists of a rotation about an axis followed by an inversion.
Rotation axes (proper and improper), reflection planes, and centers of symmetry are collectively called symmetry elements. There are 32 possible crystal classes. Each one can be classified into one of the seven crystal systems.

 Space groups

 

The space group of the crystal structure is composed of the translational symmetry operations in addition to the operations of the point group. These include:
  • Pure translations, which move a point along a vector
  • Screw axes, which rotate a point around an axis while translating parallel to the axis
  • Glide planes, which reflect a point through a plane while translating it parallel to the plane.
There are 230 distinct space groups.

Grain boundaries

 

Grain boundaries are interfaces where crystals of different orientations meet. A grain boundary is a single-phase interface, with crystals on each side of the boundary being identical except in orientation. The term "crystallite boundary" is sometimes, though rarely, used. Grain boundary areas contain those atoms that have been perturbed from their original lattice sites, dislocations, and impurities that have migrated to the lower energy grain boundary.

Treating a grain boundary geometrically as an interface of a single crystal cut into two parts, one of which is rotated, we see that there are five variables required to define a grain boundary. The first two numbers come from the unit vector that specifies a rotation axis. The third number designates the angle of rotation of the grain. The final two numbers specify the plane of the grain boundary (or a unit vector that is normal to this plane).

Grain boundaries disrupt the motion of dislocations through a material, so reducing crystallite size is a common way to improve strength, as described by the Hall–Petch relationship. Since grain boundaries are defects in the crystal structure they tend to decrease the electrical and thermal conductivity of the material. The high interfacial energy and relatively weak bonding in most grain boundaries often makes them preferred sites for the onset of corrosion and for the precipitation of new phases from the solid. They are also important to many of the mechanisms of creep.

Grain boundaries are in general only a few nanometers wide. In common materials, crystallites are large enough that grain boundaries account for a small fraction of the material. However, very small grain sizes are achievable. In nanocrystalline solids, grain boundaries become a significant volume fraction of the material, with profound effects on such properties as diffusion and plasticity. In the limit of small crystallites, as the volume fraction of grain boundaries approaches 100%, the material ceases to have any crystalline character, and thus becomes an amorphous solid.

Defects and impurities

 

Real crystals feature defects or irregularities in the ideal arrangements described above and it is these defects that critically determine many of the electrical and mechanical properties of real materials. When one atom substitutes for one of the principal atomic components within the crystal structure, alteration in the electrical and thermal properties of the material may ensue.[2] Impurities may also manifest as spin impurities in certain materials. Research on magnetic impurities demonstrates that substantial alteration of certain properties such as specific heat may be affected by small concentrations of an impurity, as for example impurities in semiconducting ferromagnetic alloys may lead to different properties as first predicted in the late 1960s.[3][4] Dislocations in the crystal lattice allow shear at lower stress than that needed for a perfect crystal structure.[5]

Prediction of structure

 


Crystal structure of sodium chloride (table salt)
The difficulty of predicting stable crystal structures based on the knowledge of only the chemical composition has long been a stumbling block on the way to fully computational materials design. Now, with more powerful algorithms and high-performance computing, structures of medium complexity can be predicted using such approaches as evolutionary algorithms, random sampling, or metadynamics.

The crystal structures of simple ionic solids (e.g., NaCl or table salt) have long been rationalized in terms of Pauling's rules, first set out in 1929 by Linus Pauling, referred to by many since as the "father of the chemical bond".[6] Pauling also considered the nature of the interatomic forces in metals, and concluded that about half of the five d-orbitals in the transition metals are involved in bonding, with the remaining nonbonding d-orbitals being responsible for the magnetic properties. He, therefore, was able to correlate the number of d-orbitals in bond formation with the bond length as well as many of the physical properties of the substance. He subsequently introduced the metallic orbital, an extra orbital necessary to permit uninhibited resonance of valence bonds among various electronic structures.[7]

In the resonating valence bond theory, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": 1/2, 1/3, 2/3, 1/4, 3/4, etc. The choice of structure and the value of the axial ratio (which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.[8][9]

After postulating a direct correlation between electron concentration and crystal structure in beta-phase alloys, Hume-Rothery analyzed the trends in melting points, compressibilities and bond lengths as a function of group number in the periodic table in order to establish a system of valencies of the transition elements in the metallic state. This treatment thus emphasized the increasing bond strength as a function of group number.[10] The operation of directional forces were emphasized in one article on the relation between bond hybrids and the metallic structures. The resulting correlation between electronic and crystalline structures is summarized by a single parameter, the weight of the d-electrons per hybridized metallic orbital. The “d-weight” calculates out to 0.5, 0.7 and 0.9 for the fcc, hcp and bcc structures respectively. The relationship between d-electrons and crystal structure thus becomes apparent.[11]

Polymorphism

 


Quartz is one of the several thermodynamically stable crystalline forms of silica, SiO2. The most important forms of silica include: α-quartz, β-quartz, tridymite, cristobalite, coesite, and stishovite.

Polymorphism refers to the ability of a solid to exist in more than one crystalline form or structure. According to Gibbs' rules of phase equilibria, these unique crystalline phases will be dependent on intensive variables such as pressure and temperature. Polymorphism can potentially be found in many crystalline materials including polymers, minerals, and metals, and is related to allotropy, which refers to elemental solids. The complete morphology of a material is described by polymorphism and other variables such as crystal habit, amorphous fraction or crystallographic defects. Polymorphs have different stabilities and may spontaneously convert from a metastable form (or thermodynamically unstable form) to the stable form at a particular temperature. They also exhibit different melting points, solubilities, and X-ray diffraction patterns.

One good example of this is the quartz form of silicon dioxide, or SiO2. In the vast majority of silicates, the Si atom shows tetrahedral coordination by 4 oxygens. All but one of the crystalline forms involve tetrahedral SiO4 units linked together by shared vertices in different arrangements. In different minerals the tetrahedra show different degrees of networking and polymerization. For example, they occur singly, joined together in pairs, in larger finite clusters including rings, in chains, double chains, sheets, and three-dimensional frameworks. The minerals are classified into groups based on these structures. In each of its 7 thermodynamically stable crystalline forms or polymorphs of crystalline quartz, only 2 out of 4 of each the edges of the SiO4 tetrahedra are shared with others, yielding the net chemical formula for silica: SiO2.
Another example is elemental tin (Sn), which is malleable near ambient temperatures but is brittle when cooled. This change in mechanical properties due to existence of its two major allotropes, α- and β-tin. The two allotropes that are encountered at normal pressure and temperature, α-tin and β-tin, are more commonly known as gray tin and white tin respectively. Two more allotropes, γ and σ, exist at temperatures above 161 °C and pressures above several GPa.[12] White tin is metallic, and is the stable crystalline form at or above room temperature. Below 13.2 °C, tin exists in the gray form, which has a diamond cubic crystal structure, similar to diamond, silicon or germanium. Gray tin has no metallic properties at all, is a dull-gray powdery material, and has few uses, other than a few specialized semiconductor applications.[13] Although the α-β transformation temperature of tin is nominally 13.2 °C, impurities (e.g. Al, Zn, etc.) lower the transition temperature well below 0 °C, and upon addition of Sb or Bi the transformation may not occur at all.[14]

Physical properties

 

Twenty of the 32 crystal classes are so-called piezoelectric, and crystals belonging to one of these classes (point groups) display piezoelectricity. All piezoelectric classes lack a centre of symmetry. Any material develops a dielectric polarization when an electric field is applied, but a substance that has such a natural charge separation even in the absence of a field is called a polar material. Whether or not a material is polar is determined solely by its crystal structure. Only 10 of the 32 point groups are polar. All polar crystals are pyroelectric, so the 10 polar crystal classes are sometimes referred to as the pyroelectric classes.

There are a few crystal structures, notably the perovskite structure, which exhibit ferroelectric behavior. This is analogous to ferromagnetism, in that, in the absence of an electric field during production, the ferroelectric crystal does not exhibit a polarization. Upon the application of an electric field of sufficient magnitude, the crystal becomes permanently polarized. This polarization can be reversed by a sufficiently large counter-charge, in the same way that a ferromagnet can be reversed. However, it is important to note that, although they are called ferroelectrics, the effect is due to the crystal structure (not the presence of a ferrous metal).

See also

 

For more detailed information in specific technology applications see Materials science, Ceramic engineering, or Metallurgy.

 

References

 

  1. ^ Ashcroft, N.; Mermin, D. (1976) Solid State Physics, Brooks/Cole (Thomson Learning, Inc.), Chapter 7, ISBN 0030493463
  2. ^ Nikola Kallay (2000) Interfacial Dynamics, CRC Press, ISBN 0824700066
  3. ^ Hogan, C. M. (1969). "Density of States of an Insulating Ferromagnetic Alloy". Physical Review 188 (2): 870. Bibcode 1969PhRv..188..870H. doi:10.1103/PhysRev.188.870.
  4. ^ Zhang, X. Y.; Suhl, H (1985). "Spin-wave-related period doublings and chaos under transverse pumping". Physical Review a 32 (4): 2530–2533. Bibcode 1985PhRvA..32.2530Z. doi:10.1103/PhysRevA.32.2530. PMID 9896377.
  5. ^ Courtney, Thomas (2000). Mechanical Behavior of Materials. Long Grove, IL: Waveland Press. pp. 85. ISBN 1-57766-425-6.
  6. ^ L. Pauling (1929). "The principles determining the structure of complex ionic crystals". J. Am. Chem. Soc. 51 (4): 1010–1026. doi:10.1021/ja01379a006.
  7. ^ Pauling, Linus (1938). "The Nature of the Interatomic Forces in Metals". Physical Review 54 (11): 899. Bibcode 1938PhRv...54..899P. doi:10.1103/PhysRev.54.899.
  8. ^ Pauling, Linus (1947). Journal of the American Chemical Society 69 (3): 542. doi:10.1021/ja01195a024.
  9. ^ Pauling, L. (1949). "A Resonating-Valence-Bond Theory of Metals and Intermetallic Compounds". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934-1990) 196 (1046): 343. Bibcode 1949RSPSA.196..343P. doi:10.1098/rspa.1949.0032.
  10. ^ Hume-rothery, W.; Irving, H. M.; Williams, R. J. P. (1951). "The Valencies of the Transition Elements in the Metallic State". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934-1990) 208 (1095): 431. Bibcode 1951RSPSA.208..431H. doi:10.1098/rspa.1951.0172.
  11. ^ Altmann, S. L.; Coulson, C. A.; Hume-Rothery, W. (1957). "On the Relation between Bond Hybrids and the Metallic Structures". Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences (1934–1990) 240 (1221): 145. Bibcode 1957RSPSA.240..145A. doi:10.1098/rspa.1957.0073.
  12. ^ Molodets, A. M.; Nabatov, S. S. (2000). "Thermodynamic Potentials, Diagram of State, and Phase Transitions of Tin on Shock Compression". High Temperature 38 (5): 715–721. doi:10.1007/BF02755923.
  13. ^ Holleman, Arnold F.; Wiberg, Egon; Wiberg, Nils; (1985). "Tin" (in German). Lehrbuch der Anorganischen Chemie (91–100 ed.). Walter de Gruyter. pp. 793–800. ISBN 3110075113.
  14. ^ Schwartz, Mel (2002). "Tin and Alloys, Properties". Encyclopedia of Materials, Parts and Finishes (2nd ed.). CRC Press. ISBN 1566766613.

 

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