Editing Drude model (section)

== Assumptions ==
Drude used the [[kinetic theory of gases]] applied to the gas of electrons moving on a fixed background of "[[ion]]s"; this is in contrast with the usual way of applying the theory of gases as a neutral diluted gas with no background. The [[number density]] of the electron gas was assumed to be 
<math display="block"> n  = \frac{N_\text{A} Z \rho_\text{m}}{A},</math>
where ''Z'' is the effective number of de-localized electrons per ion, for which Drude used the valence number, ''A'' is the atomic mass per mole,<ref name=":8" group="Ashcroft & Mermin" /> <math>\rho_\text{m}</math> is the mass density (mass per unit volume)<ref name=":8" group="Ashcroft & Mermin" /> of the "ions", and ''N''{{sub|A}} is the [[Avogadro constant]].
Considering the average volume available per electron as a sphere:
<math display="block">\frac{V}{N} = \frac{1}{n} = \frac{4}{3} \pi r_{\rm s}^3 .</math>
The quantity <math>r_\text{s}</math> is a parameter that describes the electron density and is often of the order of 2 or 3 times the [[Bohr radius]], for [[alkali metals]] it ranges from 3 to 6 and some metal compounds it can go up to 10.
The densities are of the order of 1000 times of a typical classical gas.<ref group="Ashcroft & Mermin" name=":1">{{harvnb|Ashcroft|Mermin|1976|pp=2–6}}</ref>

The core assumptions made in the Drude model are the following:
* Drude applied the kinetic theory of a dilute gas, despite the high densities, therefore ignoring electron–electron and electron–ion interactions aside from collisions.<ref group="Ashcroft & Mermin" name=":5">{{harvnb|Ashcroft|Mermin|1976|pp=4}}</ref>
* The Drude model considers the metal to be formed of a collection of positively charged ions from which a number of "free electrons" were detached. These may be thought to be the [[Electron shell|valence electrons]] of the atoms that have become delocalized due to the electric field of the other atoms.<ref group="Ashcroft & Mermin" name=":1">{{harvnb|Ashcroft|Mermin|1976|pp=2–6}}</ref>
* The Drude model neglects long-range interaction between the electron and the ions or between the electrons; this is called the independent electron approximation.<ref name=":1" group="Ashcroft & Mermin" />
* The electrons move in straight lines between one collision and another; this is called free electron approximation.<ref name=":1" group="Ashcroft & Mermin" />
* The only interaction of a free electron with its environment was treated as being collisions with the impenetrable ions core.<ref name=":1" group="Ashcroft & Mermin" />
* The average time between subsequent collisions of such an electron is {{mvar|τ}}, with a [[Memorylessness|memoryless]] [[Poisson distribution]]. The nature of the collision partner of the electron does not matter for the calculations and conclusions of the Drude model.<ref name=":1" group="Ashcroft & Mermin" />
* After a collision event, the distribution of the velocity and direction of an electron is determined by only the local temperature and is independent of the velocity of the electron before the collision event.<ref name=":1" group="Ashcroft & Mermin" /> The electron is considered to be immediately at equilibrium with the local temperature after a collision.

Removing or improving upon each of these assumptions gives more refined models, that can more accurately describe different solids:
* Improving the hypothesis of the [[Maxwell–Boltzmann statistics]] with the [[Fermi–Dirac statistics]] leads to the [[Drude–Sommerfeld model]].
* Improving the hypothesis of the Maxwell–Boltzmann statistics with the [[Bose–Einstein statistics]] leads to considerations about the specific heat of integer spin atoms<ref name="einstein24">{{cite journal|title=Quantum Theory of the Monatomic Ideal Gas|author=Einstein|journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse| year=1924 |pages=261–267}}</ref> and to the [[Bose–Einstein condensate]].
* A valence band electron in a semiconductor is still essentially a free electron in a delimited energy range (i.e. only a "rare" high energy collision that implies a change of band would behave differently); the independent electron approximation is essentially still valid (i.e. no electron–electron scattering), where instead the hypothesis about the localization of the scattering events is dropped (in layman terms the electron is and scatters all over the place).<ref>{{cite web | url=https://www.youtube.com/watch?v=e8BsQyafDh4&list=PLd9hKAUC3AZuo7is-aN45pmfDwJHOqKAj&index=17 | title=Solid State Physics, Lecture17: Dynamics of Electrons in Bands| website=[[YouTube]]}}</ref>