Wulff construction - Revision history

213.100.234.146 at 14:21, 12 March 2014

2014-03-12T14:21:43Z

← Older revision		Revision as of 16:21, 12 March 2014
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	~~9 I'd like~~ to ~~first start~~ by ~~launching myself personally~~. I ~~am Kathryne but my husband doesn~~'~~t adore it anyway~~. ~~Auditing has become his occupation for~~ a ~~while~~. ~~Oklahoma~~ is ~~when~~ my ~~personal residence is~~. ~~Kayaking is just one~~ of the ~~circumstances i enjoy many~~. ~~Go to my personal website to discover more:~~ http://~~kimberlydoehnert~~.com/~~<br><br>Take~~ a ~~look at my blog post:~~ [http://kimberlydoehnert.com/ figure competition]		A great way to shed is integrate healthy fats into the diet plan. Many tend to be frightened of this phrase fat, exactly what people do not know is discover great fats which happen to be important for our overall health. A typical example of a fat origin is salmon.<br><br>Each pattern is 11 times.followed by 3 off times that can precede another 11 day pattern. ( if so when required). Of course, you happen to be able to continue shifting for as long as necessary to meet your goals! Some people require ONE pattern - other individuals repeat as until they've been happy with themselves, and their weight reduction!<br><br>In the ages of 22 decades she started doing work for the Secret provider,in 1995, and loved a 7 year career. During this period of her stretch together with the secret-service she became accredited making use of the Cooper Institute, in 1996, and began done quarterly fitness exams for uniformed officials inside the Secret Service. The actual assessments she administered provided pull-ups, sit-ups, push-ups, freedom and kilometer and a half run.<br><br>At this time, I'm in the process of starting over again, all over again. Going into that gymnasium three weeks hence, four-sizes heavier, incapable of operate above a kilometer, merely able to carry a portion of the metal I once could, had been challenging. It always is. That basic few days straight back my ego writhed with it's grave. And, I happened to be really unfortunate about most of the progress I'd lost. Was it truly lower than last year that I weighed 25 lbs less, with a set tummy and buff hands, in [http://browse.Deviantart.com/?q=training training] for a [http://kimberlydoehnert.com/ figure competition]? merely 90 days ago, I'd very little muscle, but no less than I became the things I would consider skinny and that I ended up being back to operating four miles, 3 times each week. Oh, the great have fallen.<br><br>Becausefinalweekcontacted, often calledmaximumfew days, she modestlyreducedthe womancalorie consumption and carbs, implemented a glycogen destructionwork out, and loweredherwater intake in a structuredmannerto eliminateextraliquid from under the womanskin. During topfew days she stated that she sawthe womanhuman bodychangingafter each and everydinner, acquiringmore challenging figure competition diet with an increase ofdetail, and fuller muscle mass bellies.<br><br>weight loss tea is not not used to us; truth be told it has got for ages been ingested in several nations like in India and Asia. Tea is actually common to its multiple health gains and soothing advantages. If you are searching for the best and the majority of great process or way to shed weigh; industry experts point out that having beverage every day ought to be considered. It will help you drop some weight, but additionally may also help the human body and may keep you nicely toned.<br><br>Second, does it raise the rate of your kcalorie burning? This is up for argument, plus my opinion, i believe this will depend to a certain degree throughout the person, their unique activity level, and their overall health prior to starting the green tea leaf diet plan. But, some individuals would declare that green tea extract really does raise the price regarding kcalorie burning.<br><br>Make an effort to take as numerous photographs of your self in your undies as you can. When you may suffer unpleasant achieving this, particularly at the start, it will really help that keep track of your progress aesthetically. This will act as a inspiration on those challenging times you will be certain to have.

86.139.155.39: /* Proof */

2014-02-24T19:29:44Z

Proof

← Older revision		Revision as of 21:29, 24 February 2014
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	A '''Peierls transition''' or '''Peierls distortion''' is a distortion of the periodic lattice of a one-dimensional crystal. Atomic positions oscillate so that the perfect order of the 1-D crystal is broken.		9 I'd like to first start by launching myself personally. I am Kathryne but my husband doesn't adore it anyway. Auditing has become his occupation for a while. Oklahoma is when my personal residence is. Kayaking is just one of the circumstances i enjoy many. Go to my personal website to discover more: http://kimberlydoehnert.com/<br><br>Take a look at my blog post: [http://kimberlydoehnert.com/ figure competition]

	~~== Peierls’ Theorem<ref>{{cite web \| last=Fowler \|~~ first~~=Michael \| title=Electrons in One Dimension: the Peierls Transition \| date=28 Feb 2007 \| url=http://galileo~~.~~phys.virginia.edu/classes/752.mf1i.spring03/PeierlsTrans.htm}}</ref> ==~~

	~~Peierls' Theorem states that ''a one-dimensional equally spaced chain with one electron per ion is unstable'~~'. ~~It was asserted in the 1930s by [[Rudolf Peierls]]. It can be proven using a simple model of the potential~~ for ~~an electron in~~ a 1-D crystal with lattice spacing a. The periodicity of the crystal creates energy band gaps in the E–k diagram at multiples of the value k = π/a (similar to the result of the [[Particle_in_a_one-dimensional_lattice_(periodic_potential)#Kronig–Penney_model\|Kronig–Penney model]], which helps to explain the origin of band gaps in semiconductors). ~~If the ions each contribute one electron, then the band will be half-filled, up to values of k = ±π/2a in the ground state.~~

	~~Imagine a lattice distortion where every other ion moves closer to one neighbor and further away from the other, the unfavourable energy of the long bond between ions~~ is outweighed by the energy gain of the short bond. The period has just doubled from a to 2a. In essence, the proof relies on the fact that doubling the period would introduce new band gaps located at multiples of k = π/2a. This would cause small energy savings, based on the distortion of the bands in the vicinity of the new gaps. Approaching k = π/2a from the left, the distortion due to the introduction of the new band gap will cause the electrons to be at a lower energy than they would be in the perfect crystal. Therefore, this lattice distortion becomes energetically favorable when the energy savings due to the new band gaps outweighs the elastic energy cost of rearranging the ions. Of course, this effect will be noticeable only when the electrons are arranged close to their ground state – in other words, thermal excitation should be minimized. Therefore, the Peierls transition should be seen at low temperature. This is ~~the basic argument for the occurrence of the Peierls transition, sometimes called dimerization.~~

	~~== Historical background ==~~

	~~Peierls’ discovery gained experimental backing during the effort to find new superconducting materials~~. In 1964, Dr. William Little of the Stanford University Department of Physics theorized that a certain class of polymer chains may experience a high T<sub>C</sub> superconducting transition.<ref>{{cite journal \| author = W. A. Little \| journal = [[Physical Review]] \| volume = 134 \| year = 1964 \| pages = A1416–A1424 \| doi = 10.1103/PhysRev.134.A1416 \| title = Possibility of Synthesizing an Organic Superconductor \| issue = 6A\|bibcode = 1964PhRv..134.1416L }}</ref> The basis for his assertion was that the lattice distortions which lead to pairing of electrons in the [[BCS theory]] of [[superconductivity]] could be replaced instead by rearranging the electron density in a series of side chains. This means that now electrons would be responsible for creating the Cooper pairs instead of ions. Because the transition temperature is ~~inversely proportional to the square root~~ of the ~~mass of the charged particle responsible for the distortions, the T<sub>C</sub> should be improved by a corresponding factor:~~

	~~<math>\frac{T}{T_i} = \sqrt{\frac{M_i}{m_e}}</math>~~

	~~The subscript ''~~i~~'' represents "ion," while ''e'' represents "electron~~.~~" The predicted benefit in superconducting transition temperature was therefore a factor of about 300.~~

	In the 1970s, various organic materials such as [[Charge-transfer_complex#Electrical_conductivity\|TTF-TCNQ]] were synthesized.<ref>{{cite journal \| author = P. W. Anderson, P. A. Lee, M. Saitoh \| journal = [[Solid State Communications]] \| volume = 13 \| year = 1973 \| pages = 595–598 \| doi = 10.1016/S0038-1098(73)80020-1 \| title = Remarks on giant conductivity in TTF-TCNQ \| bibcode=1973SSCom..13..595A \| issue = 5}}</ref> What was found is that these materials underwent an insulating transition rather than a superconducting one. Eventually it was realized that these were the first experimental observations of the Peierls transition. With the introduction of new band gaps after the lattice becomes distorted, electrons must overcome this new energy barrier in order to ~~become free~~ to ~~conduct. The simple model of the Peierls distortion as a rearrangement of ions in a 1-D chain could describe why these materials became insulators rather than superconductors.~~

	~~== Related physical consequences ==~~

	Peierls predicted that the rearrangement of the ion cores in a Peierls transition would produce periodic fluctuations in the electron density. These are commonly called [[charge density wave]]s, and they are an example of collective charge transport. Several materials systems have verified the existence of these waves. Good candidates are weakly coupled molecular chains, where electrons can move freely along the direction of the chains but motion is restricted perpendicular to the chains. NbSe<sub>3</sub> and K<sub>0.3</sub>MoO<sub>3</sub> are two examples in which charge density waves have been observed at relatively high temperatures of 145K and 180K, respectively.<ref>{{Cite news \| last=Thorne \| first=Robert \| title=Charge-Density-Wave Conductors \| magazine=[[Physics Today]] \| date = May 1996 \| url=http://~~pages~~.~~physics.cornell.edu~~/~~~rthorne/thorne_phystod_1996.pdf}}</ref>~~

	Furthermore, the 1-D nature of the material causes a breakdown of the [[Fermi liquid]] theory for electron behavior. Therefore, a 1-D conductor should behave as a [[Luttinger liquid]] instead. A Luttinger liquid is a [[Paramagnetism\|paramagnetic]] one-dimensional metal without Landau [[Quasiparticle\|quasi-particle]] excitations.

	~~== Research topics ==~~

	~~1-D metals have been the subject of much research. Here are a few examples of both theoretical and experimental research efforts to illustrate the broad range of topics:~~

	* Theory has shown that polymer chains that have been looped and formed into rings undergo a Peierls transition. These rings demonstrate a persistent current and the Peierls distortion can be modified by modulating the magnetic flux through the loop.<ref>{{cite journal \| author = S. D. Liang, Y. H. Bai, B. Beng \| journal = [[Physical Review B]] \| volume = 74 \| year = 2006 \| pages = 113304 \| doi = 10.1103/PhysRevB.74.113304 \| title = Peierls instability and persistent current in mesoscopic conducting polymer rings \| issue = 11 \|bibcode = 2006PhRvB..74k3304L }}</ref>

	* Density functional theory has been used to calculate the bond length alterations predicted in increasingly long chains of organic oligomers. The selection of which hybrid functional to use is paramount in obtaining an accurate estimate of the bond length alteration caused by Peierls distortions, as some functionals have been shown to overestimate the oscillation while others underestimate it.<~~ref~~>{{cite journal \| author = D. Jacquemin, A. Femenias, H. Chermette, I. Ciofini, C. Adamo, J. M. Andr, E. A. Perpte \| journal = [[Journal of Physical Chemistry A]] \| volume = 110 \| year = 2006 \| pages = 5952–5959 \| doi = 10.1021/jp060541w \| title = Assessment of Several Hybrid DFT Functionals for the Evaluation of Bond Length Alternation of Increasingly Long Oligomers \| pmid = 16640395 \| issue = 17 }}<~~/ref~~>

	* Gold deposited on a stepped Si(553) surface has shown evidence of two simultaneous Peierls transitions. The lattice period is distorted by factors of 2 and 3, and energy gaps open for nearly 1/2-filled and 1/3-1/4 filled bands. The distortions have been studied and imaged using [[Low-energy electron diffraction\|LEED]] and [[Scanning tunneling microscope\|STM]], while the energy bands were studied with [[ARPES\|ARP]].<ref>{{cite journal \| author = J. R. Ahn, P. G. Kang, K. D. Ryang, H.W. Yeom \| journal = [[Physical Review Letters]] \| volume = 95 \| year = 2005 \| pages = 196402 \| doi = 10.1103/PhysRevLett.95.196402 \| title = Coexistence of Two Different Peierls Distortions within an Atomic Scale Wire: ~~Si(553)-Au \| pmid=16384001 \| bibcode=2005PhRvL..95s6402A \| issue = 19}}</ref>~~

	* [[Luttinger liquid]]s have a power law dependence of resistance on temperature. This has been shown for purple bronze (Li<sub>0.9</sub>Mo<sub>6</sub>O<sub>17</sub>).<ref>{{cite journal \| author = C. A. M. dos Santos, M. S. da Luz, Yi-Kuo Yu, J. J. Neumeier, J. Moreno, B. D. White \| journal = [[Physical Review B]] \| volume = 77 \| year = 2008 \| pages = 193106 \| doi = 10.1103/PhysRevB.77.193106 \| title = Electrical transport in single-crystalline Li<sub>0.9</sub>Mo<sub>6</sub>O<sub>17</sub>: ~~A two-band Luttinger liquid exhibiting Bose metal behavior \| issue = 19\|bibcode = 2008PhRvB..77s3106D }}<~~/ref> Purple bronze may prove to be a very interesting material, since it has shown renormalization of the Luttinger-liquid density of states anomalous exponent,<ref>{{cite journal \| author = F. Wang, J.V. Alvarez, S.-K. Mo, J. W. Allen, G.-H. Gweon, J. He, R. Jin, D. Mandrus, H. Höchst \| journal = [[Physical Review Letters]] \| volume = 96 \| year = 2006 \| pages = 196403 \| doi = 10.1103/~~PhysRevLett.96.196403 \| title = New Luttinger-Liquid Physics from Photoemission on Li<sub>0~~.9</sub>Mo<sub>6</sub>O<sub>17</sub> \| bibcode=2006PhRvL..96s6403W \| pmid=16803117 \| issue = 19\|arxiv = cond-mat/0604503 }}</ref> which is one of the parameters that are used to describe Luttinger liquid behavior.<ref>{{cite arXiv \| last=Voit \| first=Johannes \| title=AIP Conference Proceedings \| date=5 May 2000 \| eprint=cond-mat/0005114 \| class=cond-mat.str-el \| doi=10.1063/1.1342524 \| chapter=A brief introduction to Luttinger liquids \| volume=544 \| pages=309}}</ref>

	* The dependence of resonant tunneling through island barriers in a 1-D wire has been studied, and is also found to be a power law dependence. This offers additional evidence of Luttinger liquid behavior.<ref>{{cite journal \| author = O. M. Auslaender, A. Yacoby, R. de Picciotto, K.W. Baldwin, L. N. Pfeiffer, K.W. West \| journal = [[Physical Review Letters]] \| volume = 84 \| year = 2000 \| pages = 1764–1767 \| doi = 10.1103/PhysRevLett.84.1764 \| title = Experimental evidence for resonant tunneling in a Luttinger liquid \| pmid=11017620 \| bibcode=2000PhRvL..84.1764A \| issue = 8\|arxiv = cond-mat/9909138 }}</ref>

	~~== See also ==~~
	* [[Charge density wave]]
	* [[Luttinger liquid]]

	~~== References ==~~
	~~<references/>~~

	~~[[Category:Superconductivity]]~~
	~~[[Category:Phase transitions]~~]

141.14.162.129 at 12:50, 9 January 2014

2014-01-09T12:50:19Z

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