|
|
| Line 1: |
Line 1: |
| [[File:Electrona in crystallo fluentia.svg|thumb|300 px|right|Drude Model electrons (shown here in blue) constantly bounce between heavier, stationary crystal ions (shown in red).]]
| | Spend money on a rechargeable battery rrn your wireless gaming controller. You can buy regular power supplies for each and every controller. If you want to play video games regularly, you will be diet through a small fortune in the batteries was considered to run your controllers. If you have any questions concerning wherever and how to use clash of clans hack cydia ([http://circuspartypanama.com Read Significantly more]), you can speak to us at our page. A rechargeable battery can help you save a lot of profit in the long run.<br><br>Yet somehow Supercell, by allowing the actual illusion on the multi player game, taps into instinctual male drive as a way to from the status hierarchy, and even though it''s unattainable to the surface of your hierarchy if there's no need been logging in each and every because the game became available plus you invested honest money in extra builders, the drive for obtaining a small bit further forces enough [http://Www.Google.de/search?q=visitors visitors] to spare a real income in relation to virtual 'gems'" that hobby could be the top-grossing app within the App Store.<br><br>Present is a patch game button that you need click after entering each of our desired values. When you check back on to the game after their late twenties seconds to a minute, you will already make the items. On that point is nothing wrong by making use of secrets. To hack is truly the best way in enjoy clash of clans cheats. Make use of unquestionably the Resources that you have, and take advantage pertaining to this 2013 Clash for Clans download! The actual reason why pay for coins maybe gems when you has the potential to get the needed pieces with this tool! Hurry and get your incredible very own Clash for Clans hack tool right now. The needed items are just a quantity clicks away.<br><br>There are no fallout in the least on the way to attacking other players in addition to losing, so just tackle and savor it. Win or lose, clients may lose the a great deal troops you have associated with the attack since they are going to are only beneficial on one mission, nevertheless, somebody can steal more resources with the enemy town than it cost to make the troops. And you just form more troops within your new barracks. It''s an important good idea to get them queued up until now you decide to onset and that means the customer are rebuilding your troopers through the battle.<br><br>Stay for game of these season editions of worthwhile titles. These traditionally come out per 12 month or higher after the initial headline, but feature a lot of the down-loadable and extra content material material which was released all the way through steps once the first basic headline. These ball game titles supply a tons more bang for the buck.<br><br>In which to access it into excel, copy-paste this continued menu into corpuscle B1. If you again access an majority of second in abnormal in corpuscle A1, the bulk over treasures will arise while in B1.<br><br>It is undoubtedly a helpful component of the diversion as fantastic. When one particular player has modified, the Conflict of Clan Castle shambles in his or the woman village, he or she will successfully start or sign up for for each faction using diverse gamers exactly where they can take a look at with every other while giving troops to just another these troops could get in touch either offensively or protectively. The Clash with regards to Clans cheat for free of cost additionally holds the leading district centered globally conversation so gamers could laps making use of exceptional players for social couples and as faction entering.This recreation is a have to perform on your android instrument specially if you continue to be employing my clash having to do with clans android hack product. |
| | |
| The '''Drude model''' of [[electrical conduction]] was proposed in 1900<ref>
| |
| {{cite journal
| |
| |last= Drude |first= Paul
| |
| |year= 1900
| |
| |title= Zur Elektronentheorie der metalle
| |
| |url= http://www3.interscience.wiley.com/cgi-bin/fulltext/112485959/PDFSTART
| |
| |journal= [[Annalen der Physik]]
| |
| |volume= 306 |issue=3 |pages=566
| |
| |doi= 10.1002/andp.19003060312
| |
| |bibcode = 1900AnP...306..566D }}</ref><ref>
| |
| {{cite journal
| |
| |last= Drude |first= Paul
| |
| |year= 1900
| |
| |title= Zur Elektronentheorie der Metalle; II. Teil. Galvanomagnetische und thermomagnetische Effecte
| |
| |url= http://www3.interscience.wiley.com/cgi-bin/fulltext/112485893/PDFSTART
| |
| |journal= [[Annalen der Physik]]
| |
| |volume= 308 |issue=11 |pages=369
| |
| |doi= 10.1002/andp.19003081102
| |
| |bibcode = 1900AnP...308..369D }}</ref> by [[Paul Karl Ludwig Drude|Paul Drude]] to explain the transport properties of [[electron]]s in materials (especially metals). The model, which is an application of [[kinetic theory]], assumes that the microscopic behavior of electrons in a solid may be treated classically and looks much like a [[pinball]] machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.
| |
| | |
| The two most significant results of the Drude model are an electronic equation of motion,
| |
| :<math>\frac{d}{dt}\mathbf{p}(t) = q\left(\mathbf{E}+\frac{\mathbf{p}\times\mathbf{B}}{m}\right) - \frac{\mathbf{p}(t)}{\tau},</math>
| |
| and a linear relationship between [[current density]] <math>J</math> and electric field <math>E</math>, | |
| :<math>\mathbf{J} = \left( \frac{n q^2 \tau}{m} \right) \mathbf{E}.</math>
| |
| Here <math>t</math> is the time and <math>p</math>, <math>q</math>, <math>n</math>, <math>m</math>, and <math>\tau</math> are respectively an electron's momentum, charge, number density, mass, and mean free time between ionic collisions. The latter expression is particularly important because it explains in semi-quantitative terms why [[Ohm's Law]], one of the most ubiquitous relationships in all of electromagnetism, should be true.<ref name="Ashcroft6-7">
| |
| {{cite book
| |
| | author = Neil W. Ashcroft
| |
| | coauthors = N. David Mermin
| |
| | year = 1976
| |
| | title = Solid State Physics
| |
| | publisher = [[Saunders College]]
| |
| | pages = 6–7
| |
| | isbn = 0-03-083993-9
| |
| }}</ref><ref>
| |
| {{cite book
| |
| | author = Edward M. Purcell
| |
| | year = 1965
| |
| | title = Electricity and Magnetism
| |
| | publisher = [[McGraw-Hill]]
| |
| | pages = 117–122
| |
| | isbn = 978-0-07-004908-6
| |
| }}</ref><ref>
| |
| {{cite book
| |
| | author = David J. Griffiths
| |
| | year = 1999
| |
| | title = Introduction to Electrodynamics
| |
| | publisher = [[Prentice-Hall]]
| |
| | pages = 289
| |
| | isbn = 978-0-13-805326-0
| |
| }}</ref>
| |
| | |
| The model was extended in 1905 by [[Hendrik Antoon Lorentz]] (and hence is also known as the '''Drude–Lorentz model''') and is a [[Classical physics|classical]] model.
| |
| Later it was supplemented with the results of quantum theory in 1933 by [[Arnold Sommerfeld]] and [[Hans Bethe]], leading to the
| |
| [[Free electron model|Drude-Sommerfeld model]].
| |
| | |
| == Assumptions ==
| |
| | |
| The Drude model considers the metal to be formed of a mass of positively-charged ions from which a number of "free electrons" were detached. These may be thought to have become delocalized when the valence levels of the atom came in contact with the potential of the other atoms.<ref name="ashcroft mermin 2-6" >
| |
| {{cite book
| |
| | author = Neil W. Ashcroft
| |
| | coauthors = N. David Mermin
| |
| | year = 1976
| |
| | title = Solid State Physics
| |
| | publisher = [[Saunders College]]
| |
| | pages = 2–6
| |
| | isbn = 0-03-083993-9
| |
| }}</ref>
| |
| | |
| The Drude model neglects any long-range interaction between the electron and the ions or between the electrons.
| |
| The only possible interaction of a free electron with its environment is via instantaneous collisions. The average time between subsequent collisions of such an electron is <math>\tau</math>, and the nature of the collision partner of the electron does not matter for the calculations and conclusions of the Drude model.<ref name="ashcroft mermin 2-6"/>
| |
| | |
| == Explanations ==
| |
| | |
| === DC field ===
| |
| | |
| The simplest analysis of the Drude model assumes that electric field <math>\mathbf{E}</math> is both uniform and constant, and that the thermal velocity of electrons is sufficiently high such that they accumulate only an infinitesimal amount of momentum <math>d\mathbf{p}</math> between collisions, which occur on average every <math>\tau</math> seconds.<ref name="Ashcroft6-7"/>
| |
| | |
| Then an electron isolated at time <math>t</math> will on average have been traveling for time <math>\tau</math> since its last collision, and consequently will have accumulated momentum
| |
| :<math>\Delta\langle\mathbf{p}\rangle= q \mathbf{E} \tau.</math>
| |
| During its last collision, this electron will have been just as likely to have bounced forward as backward, so all prior contributions to the electron's momentum may be ignored, resulting in the expression
| |
| :<math>\langle\mathbf{p}\rangle = q \mathbf{E} \tau.</math>
| |
| Substituting the relations
| |
| ::<math>\langle\mathbf{p}\rangle = m \langle\mathbf{v}\rangle,</math>
| |
| ::<math>\mathbf{J} = n q \langle\mathbf{v}\rangle,</math>
| |
| results in the formulation of [[Ohm's Law]] mentioned above:
| |
| :<math>\mathbf{J} = \left( \frac{n q^2 \tau}{m} \right) \mathbf{E}.</math>
| |
| | |
| === Time-varying analysis ===
| |
| | |
| The dynamics may also be described by introducing an effective drag force. At time <math>t=t_0+dt</math> the average electron's momentum will be | |
| :<math>\langle\mathbf{p}(t_0+dt)\rangle=\left( 1 - \frac{dt}{\tau} \right) \left(\langle\mathbf{p}(t_0)\rangle + q\mathbf{E}dt\right),</math>
| |
| because, on average, a fraction of <math>( 1 - dt/\tau )</math> of the electrons will not have experienced another collision, and the ones that have will contribute to the total momentum to only a negligible order.<ref>
| |
| {{cite book
| |
| | author = Neil W. Ashcroft
| |
| | coauthors = N. David Mermin
| |
| | year = 1976
| |
| | title = Solid State Physics
| |
| | publisher = [[Saunders College]]
| |
| | pages = 11
| |
| | isbn = 0-03-083993-9
| |
| }}</ref>
| |
| | |
| With a bit of algebra, this results in the differential equation
| |
| :<math>\frac{d}{dt}\langle\mathbf{p}(t)\rangle = q\mathbf{E} - \frac{\langle\mathbf{p}(t)\rangle}{\tau},</math>
| |
| | |
| where <math>\langle\mathbf{p}\rangle</math> denotes [[average]] momentum and q the charge of the electrons. This, which is an inhomogeneous differential equation, may be solved to obtain the general solution of
| |
| :<math>\langle\mathbf{p}(t)\rangle = q \tau \mathbf{E} + \mathbf{C} e^{-t/\tau}</math>
| |
| for p(t). The [[steady state]] solution (<math>\frac{d}{d t}\langle\mathbf{p}\rangle = 0</math>) is then
| |
| :<math>\langle\mathbf{p}\rangle = q \tau \mathbf{E}.</math>
| |
| | |
| As above, average momentum may be related to average velocity and this in turn may be related to current density,
| |
| ::<math>\langle\mathbf{p}\rangle = m \langle\mathbf{v}\rangle,</math>
| |
| ::<math>\mathbf{J} = n q \langle\mathbf{v}\rangle,</math>
| |
| and the material can be shown to satisfy [[Ohm's Law]] with a [[Direct current|DC]]-conductivity <math>\, \sigma_0</math>:
| |
| | |
| :<math>\mathbf{J} = \left( \frac{n q^2 \tau}{m} \right) \mathbf{E}.</math>
| |
| | |
| [[File:DrudeModelComplexConductivity.png|thumb|300 px|right|Complex conductivity for different frequencies assuming that <math>\tau=10^{-5}</math> and that <math>\sigma_0=1</math>.]]
| |
| The Drude model can also predict the current as a response to a time-dependent electric field with an angular frequency <math>\, \omega</math>, in which case
| |
| | |
| :<math>\sigma(\omega) = \frac{\sigma_0}{1 + i\omega\tau}= \frac{\sigma_0}{1 + \omega^2\tau^2}- i\omega\tau\frac{\sigma_0}{1 + \omega^2\tau^2}.</math>
| |
| | |
| Here it is assumed that
| |
| | |
| :<math>E(t) = \Re(E_0 e^{i\omega t});</math>
| |
| :<math>J(t) = \Re(\sigma(\omega) E_0 e^{i\omega t}).</math>
| |
| | |
| In other conventions, <math>\, i</math> is replaced by <math>\, -i</math> in all equations. The imaginary part indicates that the current lags behind the electrical field, which happens because the electrons need roughly a time <math>\, \tau</math> to accelerate in response to a change in the electrical field. Here the Drude model is applied to electrons; it can be applied both to electrons and holes; i.e., positive charge carriers in semiconductors. The curves for <math>\sigma(\omega)</math> are shown in the graph.
| |
| | |
| === Drude response in real materials ===
| |
| | |
| The characteristic behavior of a Drude metal in the time or frequency domain, i.e. exponential relaxation with time constant <math>\tau</math> or the frequency dependence for <math>\sigma(\omega)</math> stated above, is called Drude response. In a conventional, simple, real metal (e.g. sodium, silver, or gold at room temperature) such behavior is not found experimentally, because the characteristic frequency <math>1/\tau</math> is in the infrared frequency range, where other features that are not considered in the Drude model (such as [[Electronic band structure|band structure]]) play an important role.<ref name="Dressel">{{cite journal|author=M. Dressel, M. Scheffler|year=2006|title=Verifying the Drude response|journal=[[Annalen der Physik|Ann. Phys.]]|volume=15|issue=7–8|pages=535–544|doi=10.1002/andp.200510198|bibcode = 2006AnP...518..535D }}</ref>
| |
| But for certain other materials with metallic properties, frequency-dependent conductivity was found that closely follows the simple Drude prediction for <math>\sigma(\omega)</math>. These are materials where the relaxation rate <math>1/\tau</math> is at much lower frequencies.<ref name="Dressel"/> This is the case for certain [[Doping (semiconductor)|doped semiconductor]] single crystals,<ref>{{cite journal|author=M. van Exter, D. Grischkowsky|year=1990|title=Carrier dynamics of electrons and holes in moderately doped silicon|journal=[[Physical Review|Phys. Rev. B]]|volume=41|issue=17|pages=12140–12149|doi=10.1103/PhysRevB.41.12140|bibcode = 1990PhRvB..4112140V }}</ref> high-mobility [[2DEG|two-dimensional electron gases]],<ref>{{cite journal|author=P. J. Burke, I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, K. W. West|year=2000|title=High frequency conductivity of the high-mobility two-dimensional electron gas|journal=[[Applied Physics Letters|Appl. Phys. Lett.]]|volume=76|issue=6|pages=745–747|doi=10.1063/1.125881|bibcode = 2000ApPhL..76..745B }}</ref> and [[Heavy fermion|heavy-fermion metals]].<ref>{{cite journal|author=M. Scheffler, M. Dressel, M. Jourdan, H. Adrian|year=2005|title=Extremely slow Drude relaxation of correlated electrons|journal=[[Nature (journal)|Nature]]|volume=438|pages=1135–1137|doi=10.1038/nature04232|issue=7071|bibcode = 2005Natur.438.1135S }}</ref>
| |
| | |
| == Accuracy of the model ==
| |
| | |
| Historically, the Drude formula was first derived in an incorrect way, namely by assuming that the charge carriers form an [[ideal gas]]. We know now that they follow [[Fermi-Dirac distribution]] and have appreciable interactions, but amazingly, the result turns out to be the same as the Drude model because, as [[Lev Landau]] derived in 1957, a gas of interacting particles can be described by a system of almost non-interacting '[[quasiparticle]]s' that, in the case of electrons in a metal, can be well modelled by the Drude equation.
| |
| | |
| This simple classical Drude model provides a very good explanation of DC and AC conductivity in [[metal]]s, the [[Hall effect]], and [[thermal conductivity]] (due to electrons) in metals near room temperature. The model also explains the [[Wiedemann-Franz law]] of 1853. However, it greatly overestimates the electronic heat capacities of metals. In reality, metals and insulators have roughly the same heat capacity at room temperature. The model can be applied to positive (hole) charge carriers, as demonstrated by the Hall effect.
| |
| | |
| One note of trivia surrounding the theory is that in his original paper Drude made a conceptual error, estimating electrical conductivity to in fact be only half of what it classically should have been.<ref>
| |
| {{cite book
| |
| | author = Neil W. Ashcroft
| |
| | coauthors = N. David Mermin
| |
| | year = 1976
| |
| | title = Solid State Physics
| |
| | publisher = [[Saunders College]]
| |
| | pages = 23
| |
| | isbn = 0-03-083993-9
| |
| }}</ref>
| |
| | |
| ==See also==
| |
| *[[Free electron model]]
| |
| *[[Arnold Sommerfeld]]
| |
| *[[Classical and quantum conductivity]]
| |
| *[[Electrical conductivity]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| | |
| {{Atomic models}}
| |
| | |
| [[Category:Condensed matter physics]]
| |
Spend money on a rechargeable battery rrn your wireless gaming controller. You can buy regular power supplies for each and every controller. If you want to play video games regularly, you will be diet through a small fortune in the batteries was considered to run your controllers. If you have any questions concerning wherever and how to use clash of clans hack cydia (Read Significantly more), you can speak to us at our page. A rechargeable battery can help you save a lot of profit in the long run.
Yet somehow Supercell, by allowing the actual illusion on the multi player game, taps into instinctual male drive as a way to from the status hierarchy, and even though its unattainable to the surface of your hierarchy if there's no need been logging in each and every because the game became available plus you invested honest money in extra builders, the drive for obtaining a small bit further forces enough visitors to spare a real income in relation to virtual 'gems'" that hobby could be the top-grossing app within the App Store.
Present is a patch game button that you need click after entering each of our desired values. When you check back on to the game after their late twenties seconds to a minute, you will already make the items. On that point is nothing wrong by making use of secrets. To hack is truly the best way in enjoy clash of clans cheats. Make use of unquestionably the Resources that you have, and take advantage pertaining to this 2013 Clash for Clans download! The actual reason why pay for coins maybe gems when you has the potential to get the needed pieces with this tool! Hurry and get your incredible very own Clash for Clans hack tool right now. The needed items are just a quantity clicks away.
There are no fallout in the least on the way to attacking other players in addition to losing, so just tackle and savor it. Win or lose, clients may lose the a great deal troops you have associated with the attack since they are going to are only beneficial on one mission, nevertheless, somebody can steal more resources with the enemy town than it cost to make the troops. And you just form more troops within your new barracks. Its an important good idea to get them queued up until now you decide to onset and that means the customer are rebuilding your troopers through the battle.
Stay for game of these season editions of worthwhile titles. These traditionally come out per 12 month or higher after the initial headline, but feature a lot of the down-loadable and extra content material material which was released all the way through steps once the first basic headline. These ball game titles supply a tons more bang for the buck.
In which to access it into excel, copy-paste this continued menu into corpuscle B1. If you again access an majority of second in abnormal in corpuscle A1, the bulk over treasures will arise while in B1.
It is undoubtedly a helpful component of the diversion as fantastic. When one particular player has modified, the Conflict of Clan Castle shambles in his or the woman village, he or she will successfully start or sign up for for each faction using diverse gamers exactly where they can take a look at with every other while giving troops to just another these troops could get in touch either offensively or protectively. The Clash with regards to Clans cheat for free of cost additionally holds the leading district centered globally conversation so gamers could laps making use of exceptional players for social couples and as faction entering.This recreation is a have to perform on your android instrument specially if you continue to be employing my clash having to do with clans android hack product.