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Friday, January 27, 2012

The dimensionality and geometry of the extra dimensions

We investigate possible signatures of black hole events at the LHC in the hypothesis that such objects will not evaporate completely, but leave a stable remnant. For the purpose of de fining a reference scenario, we have employed the publicly available Monte Carlo generator CHARYBDIS2, in which the remnant's behavior is mostly determined by kinematic constraints and conservation of some quantum numbers, such as the baryon charge. Our fi ndings show that electrically neutral remnants are highly favored and a signifi cantly larger amount of missing transverse momentum is to be expected with respect to the case of complete decay. See: Black Hole Remnants at the LHC by L. Bellagambab, R. Casadioa;by, R. Di Sipioa;bz and V. Viventiax 16 Jan 2012

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ATLAS Experiment © 2011 CERN  "Black Hole" event superimposed over a classic image of the ATLAS detector.
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If the fundamental Planck scale is of order a TeV, as the case in some extradimensions scenarios, future hadron colliders such as the Large Hadron Collider will be black hole factories. The non-perturbative process of black hole formation and decay by Hawking evaporation gives rise to spectacular events with up to many dozens of relatively hard jets and leptons, with a characteristic ratio of hadronic to leptonic activity of roughly 5:1. The total transverse energy of such events is typically a sizeable fraction of the beam energy. Perturbative hard scattering processes at energies well above the Planck scale are cloaked behind a horizon, thus limiting the ability to probe short distances. The high energy black hole cross section grows with energy at a rate determined by the dimensionality and geometry of the extra dimensions.See: High Energy Colliders as Black Hole Factories: The End of Short Distance Physics

Trackbacks for hep-ph/0106219

Wednesday, January 25, 2012

Quantum state


Probability densities for the electron of a hydrogen atom in different quantum states.(Click on Image)

In physics, a quantum state is a set of mathematical variables that fully describes a quantum system.

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Is there such a thing, as isometrical relations of orbitals, in cosmological designs? A Classical definition of the Quantum World perhaps?

The Physics of Reality

Roger Penrose discusses his experiment on the BBC (25 minutes in)
Melvyn Bragg examines the physics of reality. When Quantum Mechanics was developed in the early 20th century reality changed forever. In the quantum world particles could be in two places at once, they disappeared for no reason and reappeared in unpredictable locations, they even acted differently according to whether we were watching them. It was so shocking that Erwin Schrodinger, one of the founders of Quantum Theory, said "I don’t like it and I'm sorry I ever had anything to do with it." He even developed an experiment with a cat to show how absurd it was.

Quantum Theory was absurd, it disagreed with the classical physics of Newton and Einstein and it clashed with our experience of the everyday world. Footballs do not disappear without reason, cats do not split into two and shoes do not act differently when we are not looking at them. Or do they? Eighty years later we are still debating whether the absurd might actually be true See: The Physics of Reality

See Also:

Tuesday, January 24, 2012

SOHO Latest and Space Weather

EIT 304

A large solar flare yesterday triggered a coronal mass ejection travelling at 1400 km/s that will reach Earth today. An energetic eruption of this level can disrupt satellites, so operation teams at ESA and other organisations are closely monitoring the storm.

A coronal mass ejection (CME) is a huge cloud of magnetised plasma from the Sun's atmosphere – the corona – thrown into interplanetary space. They often occur in association with a solar flare. This ejection was detected by the ESA/NASA SOHO and NASA Stereo spaceborne solar observatories. 
See: Solar storm heading toward Earth


Center time of most recent polar pass measurement: 2012 Jan 25 0137 UTn = 2.16

ALICE EMCal installation with Peter Jacobs

Researcher Peter Jacobs explains what's happening as the final pieces of the ALICE experiment's electromagnetic calorimeter, or EMCal, are installed on Jan. 18 2012. Check out the symmetry breaking story for more.

  Jan 4, 2011 EMCAL Super Module installation at 40deg into ALICE slot 2 at CERN

 Nov 10, 2010 Loading an EMCAL SuperModule into the insertion tool, to prepare for the detector upgrade during the winter LHC shutdown.

Wednesday, January 18, 2012

Atlas Experiment Simulated Black Hole Photos

ATLAS Experiment © 2011 CERN  A new view of a black hole event. ATLAS collision events. In some theories, microscopic black holes may be produced in particle collisions that occur when very-high-energy cosmic rays hit particles in our atmosphere. These microscopic-black-holes would decay into ordinary particles in a tiny fraction of a second and would be very difficult to observe in our atmosphere.
The ATLAS Experiment offers the exciting possibility to study them in the lab (if they exist). The simulated collision event shown is viewed along the beampipe. The event is one in which a microscopic-black-hole was produced in the collision of two protons (not shown). The microscopic-black-hole decayed immediately into many particles. The colors of the tracks show different types of particles emerging from the collision (at the center).
Photo #: black-hole-event-wide

ATLAS Experiment © 2011 CERN  "Black Hole" event superimposed over a classic image of the ATLAS detector.

See: Atlas Photos

A Historical Look at Kaluza-Klein Particles?

In 1919, Kaluza sent Albert Einstein a preprint --- later published in 1921 --- that considered the extension of general relativity to five dimensions. He assumed that the 5-dimensional field equations were simply the higher-dimensional version of the vacuum Einstein equation, and that all the metric components were independent of the fifth coordinate. The later assumption came to be known as the cylinder condition. This resulted in something remarkable: the fifteen higher-dimension field equations naturally broke into a set of ten formulae governing a tensor field representing gravity, four describing a vector field representing electromagnetism, and one wave equation for a scalar field. Furthermore, if the scalar field was constant, the vector field equations were just Maxwell's equations in vacuo, and the tensor field equations were the 4-dimensional Einstein field equations sourced by an EM field. In one fell swoop, Kaluza had written down a single covariant field theory in five dimensions that yielded the four dimensional theories of general relativity and electromagnetism. Naturally, Einstein was very interested in this preprint (Link is now dead but what is said here is very important and may help with imagery needed?)

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After having formulated general relativity Albert Einstein did not immediately focus on the unification of electromagnetism and gravity in a classical field theory - the issue that would characterize much of his later work. It was still an open question to him whether relativity and electrodynamics together would cast light on the problem of the structure of matter [?]. Rather, in a 1916 paper on gravitational waves he anticipated a different development: since the electron in its atomic orbit would radiate gravitationally, something that cannot occur in reality", he expected quantum theory would have to change not only the "Maxwellian electrodynamics, but also the new theory of gravitation" [?]2. Einstein's position, however, gradually changed. From about 1919 onwards, he took a strong interest in the unification programme3. In later years, after about 1926, he hoped that he would and a particular classical unified field theory that could undercut quantum theory. Such a theory would have to contain the material objects -sources and fields- and their dynamics. He would even expect the distinction between these concepts to fade: \a complete field theory knows only fields and not the concepts of particle and motion" [?]. We will study how he wanted to realize these principles in classical Kaluza-Klein theory, and try to see what his objectives and results were.See: Einstein and the Kaluza-Klein particle

moving on further in the article toward the end...

Bergmann, now in Syracuse, wrote Einstein and asked if they could have a discussion sometime:

As anyone can only be a crank about his own ideas, and as you are someone who combines steadfastness with the ability to acknowledge his hypothesis could go wrong (usually one can only and just one of these qualities, mostly the latter) I would appreciate very much talking to you and hearing your observations; whether we appreciate the same or not, what we want is sufficiently related that we could easily come to an understanding." [?] 2

Einstein replies:
You are looking for an independent and new way to solve the fundamental problems. With this endeavor no one can help you, least of all someone who has somewhat fixed ideas. For instance, you know that on the basis of certain considerations I am convinced that the probability concept should not be primarily included in the description of reality, whereas you seem to believe that one should first formulate a field theory and subsequently 'quantize' it. This is in keeping with the view of most contemporaries. Your effort to abstractly carry through a field theory without having at your disposal the formal nature of the field quantities in advance, does not seem favorable to me, for it is formally too poor and vague."17 [?] 2

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Missing Energy Kicks New Physics Models Off The Board


The signature of large missing energy and jets is arguably one of the most important avenues for the study of potential new physics signatures at today's hadron colliders.

The above concept marks an interesting turn of events: the years of the glorification of charged leptons as the single most important tools for the discovery of rare production processes appears behind us. The W and Z discovery in 1983 by UA1 at CERN, or the top quark discovery by CDF and DZERO in 1995 at Fermilab, would have been impossible without the precise and clean detection of electrons and muons. However, with time we have understood that missing energy may be a more powerful tool for new discoveries.

Missing energy arises when a violent collision between the projectiles -protons against antiprotons at the Tevatron collider, or protons against protons at the world's most powerful accelerator, the LHC- produces an asymmetric flow of energetic bodies out of the collision point in the plane orthogonal to the beams: a transverse imbalance. This is a clear signal that something is leaving the detector unseen. And it turns out that there is a host of new physics signals which can do precisely that.

A large amount of missing transverse energy may be the result of the decay of a leptoquarks into jets and neutrinos, when the latter leave undetected; or from the silent escape of a supersymmetric neutral particle -the neutralino- produced in the chain of decays following the production of squarks and gluinos; or it may even be due to the escape of particles in a fourth dimension of space -an alternative dubbed "large extra dimensions".
(see more in linked title above)

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I have been slowly moving through the explanations for the extra-dimensions that are being explained by Matt Strassler:

In this article and the next, we will learn why extra dimensions lead to “Kaluza-Klein (KK) partner” particles (described in the previous article in this series, which you should read before this one.)  If a known type of  particle of mass m can travel in a dimension of which we are unaware — an “extra” dimension — then we will eventually discover many other types of particles, similar to the known one but heavier, with masses M>m.
See: Kaluza-Klein Partners — Why? Step 1

Sunday, January 15, 2012

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

Scientists' greatest pleasure comes from theories that derive the solution to some deep puzzle from a small set of simple principles in a surprising way. These explanations are called "beautiful" or "elegant". Historical examples are Kepler's explanation of complex planetary motions as simple ellipses, Bohr's explanation of the periodic table of the elements in terms of electron shells, and Watson and Crick's double helix. Einstein famously said that he did not need experimental confirmation of his general theory of relativity because it "was so beautiful it had to be true." See:2012 : WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION?
See which comments resonate with you. Some of my picks as I go through was by :

Raphael Bousso
Professor of Theoretical Physics, Berkeley



My Favorite Annoying Elegant Explanation: Quantum Theory .......General Relativity, in turn, is only a classical theory. It rests on a demonstrably false premise: that position and momentum can be known simultaneously. This may a good approximation for apples, planets, and galaxies: large objects, for which gravitational interactions tend to be much more important than for the tiny particles of the quantum world. But as a matter of principle, the theory is wrong. The seed is there. General Relativity cannot be the final word; it can only be an approximation to a more general Quantum Theory of Gravity.

But what about Quantum Mechanics itself? Where is its seed of destruction? Amazingly, it is not obvious that there is one. The very name of the great quest of theoretical physics—"quantizing General Relativity"—betrays an expectation that quantum theory will remain untouched by the unification we seek. String theory—in my view, by far the most successful, if incomplete, result of this quest—is strictly quantum mechanical, with no modifications whatsoever to the framework that was completed by Heisenberg, Schrödinger, and Dirac. In fact, the mathematical rigidity of Quantum Mechanics makes it difficult to conceive of any modifications, whether or not they are called for by observation.

Yet, there are subtle hints that Quantum Mechanics, too, will suffer the fate of its predecessors. The most intriguing, in my mind, is the role of time. In Quantum Mechanics, time is an essential evolution parameter. But in General Relativity, time is just one aspect of spacetime, a concept that we know breaks down at singularities deep inside black holes. Where time no longer makes sense, it is hard to see how Quantum Mechanics could still reign. As Quantum Mechanics surely spells trouble for General Relativity, the existence of singularities suggests that General Relativity may also spell trouble for Quantum Mechanics. It will be fascinating to watch this battle play out.



President, The Royal Society; Professor of Cosmology & Astrophysics; Master, Trinity...

Physical Reality Could Be Hugely More Extensive Than the Patch of Space and Time Traditionally Called 'The Universe' .....As an analogy (which I owe to Paul Davies) consider the form of snowflakes. Their ubiquitous six-fold symmetry is a direct consequence of the properties and shape of water molecules. But snowflakes display an immense variety of patterns because each is molded by its distinctive history and micro-environment: how each flake grows is sensitive to the fortuitous temperature and humidity changes during its growth.

If physicists achieved a fundamental theory, it would tell us which aspects of nature were direct consequences of the bedrock theory (just as the symmetrical template of snowflakes is due to the basic structure of a water molecule) and which cosmic numbers are (like the distinctive pattern of a particular snowflake) the outcome of environmental contingencies. .


Theoretical physicist

An Explanation of Fundamental Particle Physics That Doesn't Exist Yet.....What is tetrahedral symmetry doing in the masses of neutrinos?! Nobody knows. But you can bet there will be a good explanation. It is likely that this explanation will come from mathematicians and physicists working closely with Lie groups. The most important lesson from the great success of Einstein's theory of General Relativity is that our universe is fundamentally geometric, and this idea has extended to the geometric description of known forces and particles using group theory. It seems natural that a complete explanation of the Standard Model, including why there are three generations of fermions and why they have the masses they do, will come from the geometry of group theory. This explanation does not yet exist, but when it does it will be deep, elegant, and beautiful—and it will be my favorite.


Mathematician, Harvard; Co-author, The Shape of Inner Space

A Sphere....Most scientific facts are based on things that we cannot see with the naked eye or hear by our ears or feel by our hands. Many of them are described and guided by mathematical theory. In the end, it becomes difficult to distinguish a mathematical object from objects in nature.

One example is the concept of a sphere. Is the sphere part of nature or it is a mathematical artifact? That is difficult for a mathematician to say. Perhaps the abstract mathematical concept is actually a part of nature. And it is not surprising that this abstract concept actually describes nature quite accurately.



theoretical physicist; Professor, Department of Physics, University of California,...
 Gravity Is Curvature Of Spacetime … Or Is It?......We do not yet know the full shape of the quantum theory providing a complete accounting for gravity. We do have many clues, from studying the early quantum phase of cosmology, and ultrahigh energy collisions that produce black holes and their subsequent disintegrations into more elementary particles. We have hints that the theory draws on powerful principles of quantum information theory. And, we expect that in the end it has a simple beauty, mirroring the explanation of gravity-as-curvature, from an even more profound depth.



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.




Physicist, Harvard University; Author, Warped Passages; Knocking On Heaven's Door

The Higgs Mechanism......Fortunately that time has now come for the Higgs mechanism, or at least the simplest implementation which involves a particle called the Higgs boson. The Large Hadron Collider at CERN near Geneva should have a definitive result on whether this particle exists within this coming year. The Higgs boson is one possible (and many think the most likely) consequence of the Higgs mechanism. Evidence last December pointed to a possible discovery, though more data is needed to know for sure. If confirmed, it will demonstrate that the Higgs mechanism is correct and furthermore tell us what is the underlying structure responsible for spontaneous symmetry breaking and spreading "charge" throughout the vacuum. The Higgs boson would furthermore be a new type of particle (a fundamental boson for those versed in physics terminology) and would be in some sense a new type of force. Admittedly, this is all pretty subtle and esoteric. Yet I (and much of the theoretical physics community) find it beautiful, deep, and elegant.

Symmetry is great. But so is symmetry breaking. Over the years many aspects of particle physics were first considered ugly and then considered elegant. Subjectivity in science goes beyond communities to individual scientists. And even those scientists change their minds over time. That's why experiments are critical. As difficult as they are, results are much easier to pin down than the nature of beauty. A discovery of the Higgs boson will tell us how that is done when particles acquire their masses.



Professor of Quantum Mechanical Engineering, MIT; Author, Programming the Universe
 The True Rotational Symmetry of Space.....Although this excercise might seem no more than some fancy and painful basketball move, the fact that the true symmetry of space is rotation not once but twice has profound consequences for the nature of the physical world at its most microscopic level. It implies that 'balls' such as electrons, attached to a distant point by a flexible and deformable 'strings,' such as magnetic field lines, must be rotated around twice to return to their original configuration. Digging deeper, the two-fold rotational nature of spherical symmetry implies that two electrons, both spinning in the same direction, cannot be placed in the same place at the same time. This exclusion principle in turn underlies the stability of matter. If the true symmetry of space were rotating around only once, then all the atoms of your body would collapse into nothingness in a tiny fraction of a second. Fortunately, however, the true symmetry of space consists of rotating around twice, and your atoms are stable, a fact that should console you as you ice your shoulder.

Remember even though I pick some of these explanations does not mean I discount all others. It's just that some are picked for what they are saying in highlighted quotations. Lisi's statement on string theory is of course in my opinion far from the truth, yet,  he captures a geometrical truth that I feel exists.:) You sort of get the jest of where I am coming from in the summation of Paul Steinhardt

Friday, January 13, 2012

The Smoking Gun

One string theorist even went so far to conclude that a string theory calculation of Kaluza-Klein modes was the "smoking gun" that proved our theory was the same as the string theory that string theorists had already been studying.Warped Passages: Unraveling The Mysteries of the Universes Hidden Dimensions by Lisa Randall Pg 436, Para 4

Putting this together with what is real in our reality is of importance as well. While I may have my own metaphysical development and model building characteristics it was important that I learn the scientific one so that I could see where I may have been wrong in my own development scenario. Wrong in my own intuitions.

 Meanwhile I’m continuing to develop the Extra Dimensions series of articles, and I’ve now followed up my examples of extra dimensions with a next installment, a first discussion of what scientists would look for in trying to identify that our world actually has one or more extra dimensions .  The new article describes one of the key clues that would indicate their presence.  But this is far from the end of the story: I owe you more articles, explaining why extra dimensions would generate this clue, outlining how we try to search for this clue experimentally, and mentioning other possible clues that might arise.  All in due course…The Smoking Gun for Extra Dimensions by Theoretical Physicist Matt Strassler

Some may of not been forced to question them-self  with what it is that we have to ask of ourselves,  as we delve into the world of the sciences and philosophies. To ask ourselves whether we had always been dealing with the truth of our getting to the heart of things.

A professor may have asked what it is exactly what I wanted out of all of this,  and to him I have to relay a dream that has manifested because of his question.

In the dream I have been provided a forum for discussing my ideas.....but when it came to the time for speaking,  my preparations,  I felt lost as to where to begin. So it seems I have come to this point in time, as to "shit or get off the pot" as to what it is I wish to share of importance?

Giving these subjects the numbers of years since 2001, one would have thought  had served my own internship, but alas I remain ever the student with no classification. Yet it is the developing of the concepts with what is real in the push to experiment as to find what the real world examples are showing as attributes in the experimental processes as they unfold.

 In this example I’m going to map speed to the pitch of the note, length/postion to the duration of the note and number of turns/legs/puffs to the loudness of the note.See: How to make sound out of anything.


Who of us has the foresight to see where the process of the experiment had been developed to share an idea about what it was that we wanted to discover of nature? To see in the mind of the developers as to why the equipment has been superimposed from the schematics of theories to be tested as to discover what we may found in our model building.

Does all this prepare you to looking at the universe different?

 The Lagrange Points


In the above contour plot we see that L4 and L5 correspond to hilltops and L1, L2 and L3 correspond to saddles (i.e. points where the potential is curving up in one direction and down in the other). This suggests that satellites placed at the Lagrange points will have a tendency to wander off (try sitting a marble on top of a watermelon or on top of a real saddle and you get the idea). A detailed analysis (PDF link) confirms our expectations for L1, L2 and L3, but not for L4 and L5. When a satellite parked at L4 or L5 starts to roll off the hill it picks up speed. At this point the Coriolis force comes into play - the same force that causes hurricanes to spin up on the earth - and sends the satellite into a stable orbit around the Lagrange point. See: Space Travel and Propulsion Methods

I have to say who has not been touched as if we put on a pair of rose colored glasses to see the Lagrangian world as if the gravitons populated  locations of influence. As if they were descriptive as overlapping nodes of sound as to support some acoustical idea about levitation? Satellites that travel through space or held in position as our space station is.

 
Like different musical instruments, different types of stars produce different types of sound waves. Small stars produce a sound with a higher pitch than bigger stars, just like the 'piccolo' produces a higher sound than the cello

Thus it is as ones can see differently that I look upon the world as to discover what things we may not know of our own selves that we had missed in understanding our own physical evolution, that it is more then the matter with which we use and are made up of?


This recording was produced by converting into audible sounds some of the radar echoes received by Huygens during the last few kilometres of its descent onto Titan. As the probe approaches the ground, both the pitch and intensity increase. Scientists will use intensity of the echoes to speculate about the nature of the surface. Radar echos from Titan's surface

Thursday, January 12, 2012

Cross Pollination Works



"You really never know where your next great idea is going to come from."

Michelle Borkin of Harvard University combines astronomy and medical imaging to advance both fields and proves that interdisciplinary collaboration helps people develop great ideas otherwise undiscovered  . . . and save lives along the way.

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




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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:

The Belle B Factory Experiment

Existing standard hadrons and exotic hadrons. At the B Factory experiment, a series of new exotic mesons containing charm quarks (c) have been discovered. Unlike these exotic mesons, the newly discovered Zb particles contain bottom quarks (b) and have an electric charge. If only one bottom quark and one anti-bottom quark ( b ) are contained, the resulting particle is electrically neutral. Thus, the Zb must also contain at least two more quarks (e.g., one up quark (u) and one anti-down quark ( d )).



The Belle B Factory experiment, which began in 1999 with the aim of elucidating the origin of particle-anti-particle symmetry breaking (CP violation), has contributed to the Nobel Prize in Physics in 2008 awarded to Drs. Kobayashi and Maskawa. Moreover, data obtained from electron--positron collisions with the world's highest luminosity achieved at the KEKB accelerator have resulted in a series of unexpected discoveries of exotic hadrons, opening a new research frontier in particle physics. Data taking at the Belle Experiment has already been completed, but a vast amount of data is still awaiting detailed analysis. Moreover, an upgraded version of the KEKB/Belle Experiment, called SuperKEKB/Belle II is currently being prepared. Belle II aims to collect 50 times more data than the earlier experiment......... See: Belle Discovers New Heavy 'Exotic Hadrons'
Also See:

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The Belle experiment is a particle physics experiment conducted by the Belle Collaboration, an international collaboration of more than 400 physicists and engineers investigating CP-violation effects at the High Energy Accelerator Research Organisation (KEK) in Tsukuba, Ibaraki Prefecture, Japan

The Belle detector, located at the collision point of the ee+ asymmetric-energy collider (KEKB), is a multilayer particle detector. Its large solid angle coverage, vertex location with precision on the order of tens of micrometres (provided by a silicon vertex detector), good pionkaon separation at the momenta range from 100 MeV/c till few GeV/c (provided by a novel Cherenkov detector), and few-percent precision electromagnetic calorimetry (CsI(Tl) scintillating crystals) allow for many other scientific searches apart from CP-violation. Extensive studies of rare decays, searches for exotic particles and precision measurements of B mesons, D mesons, and tau particles have been carried out and have resulted in almost 300 publications in physics journals.

Highlights of the Belle experiment so far include

  • the first observation of CP-violation outside of the kaon system (2001)
  • observation of: B \to K^* l^+ l^- and b \to s l^+ l^-
  • measurement of ϕ3 using the B \to D K, D \to K_S \pi^+ \pi^- Dalitz plot
  • measurement of the CKM quark mixing matrix elements | Vub | and | Vcb |
  • observation of direct CP-violation in B^0 \to \pi^+ \pi^- and B^0 \to K^- \pi^+
  • observation of b \to d transitions
  • evidence for B \to \tau \nu
  • observations of a number of new particles including the X(3872)

The Belle experiment operated at the KEKB accelerator, the world's highest luminosity machine. The instantaneous luminosity exceeded 2.11×1034 cm−2·s−1. The integrated luminosity collected at the ?(4S) resonance mass is ~710 fb−1 (corresponds to 771 million BB meson pairs). Most data is recorded on the ?(4S) resonance, which decays to pairs of B mesons. About 10% of the data is recorded below the ?(4S) resonance in order to study backgrounds. In addition, Belle has carried out special short runs at the ?(5S) resonance to study B
s
mesons
as well as on the ?(3S) resonance to search for evidence of Dark Matter and the Higgs Boson.

The Belle II B-factory, an upgraded facility with two orders of magnitude more luminosity, has been approved in June 2010.[1] The design and construction work is ongoing.

 See also

 External links

 References

  1. ^ KEK press release

Monday, January 09, 2012

Loci: A Gallery of Ray Tracing for Geometers

The crystalline state is the simplest known example of a quantum , a stable state of matter whose generic low-energy properties are determined by a higher organizing principle and nothing else. Robert Laughlin


Figure 10 with linked animation: Five-fold rotational symmetry of the dodecahedron

Modern expositors of mathematics, with computers and open source ray tracing software at their disposal, have the tools necessary to create vivid and effective geometric constructions in 3-space. With a little additional effort, animations can be generated, adding the fourth dimension of time. The web provides a viable medium for enhancing mathematical presentations with photo-realistic graphics and video. A Gallery of Ray Tracing for Geometers
See Also: Creating Photo-realistic Images and Animations





Thursday, January 05, 2012

Control the Medium, You Control the Product?

The historical record, as I’ve also argued, is also quite unequivocal on the folly of allowing those who own the medium to control the message See: The Vertical Integration Elephant in the Room
 For differing points of view some might have a take on the process unfolding in our society today as a sign of the way things have always gone? Do you feel the same?

A Thomas Nast cartoon shortly after the Supreme Court affirmed Alexander Graham Bell's patents (library of congress)



Strike!

At least seven major cities adopted measured service in the 1880s: Boston; San Francisco; Buffalo; Pittsburgh; Indianapolis; Washington, DC; and Rochester, New York. The policy had a pro-consumer aspect; it expanded the market of any local exchange carrier to people who didn't want to pay a big monthly service fee, thus extending the circle of individuals any rate payer could call.
But while measured service initially succeeded in San Francisco and Buffalo, it failed everywhere else throughout the decade. Consumers could not shake the suspicion that its real purpose was to get them to pay more money to the telephone company. And so they resorted to what we would call boycotts and they called "strikes" to make their dissatisfaction known.

"The term 'strike' has come to be associated primarily with a work stoppage," John notes, but "in the 1880s its meaning was broader. A corporation could be struck not only by workers, but also by consumers and even lawmakers."

As early as 1881, for example, Washington, DC's "Telephone Subscribers Protective Association" launched a boycott of its exchange when the company adopted measured billing. It wasn't very long affair, just twelve days. But 300 out of the city's 700 phone subscribers participated. That was all it took for the firm to surrender and bring back flat rates. See: Mad about metered billing? They were in 1886, too

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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