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diff --git a/source/know/concept/cylindrical-parabolic-coordinates/index.md b/source/know/concept/cylindrical-parabolic-coordinates/index.md index 766c9b6..58358dd 100644 --- a/source/know/concept/cylindrical-parabolic-coordinates/index.md +++ b/source/know/concept/cylindrical-parabolic-coordinates/index.md @@ -8,82 +8,97 @@ categories: layout: "concept" --- -**Cylindrical parabolic coordinates** are a coordinate system -that describes a point in space using three coordinates $$(\sigma, \tau, z)$$. -The $$z$$-axis is unchanged from the Cartesian system, -hence it is called a *cylindrical* system. -In the $$z$$-isoplane, however, confocal parabolas are used. -These coordinates can be converted to the Cartesian $$(x, y, z)$$ as follows: +**Cylindrical parabolic coordinates** extend parabolic coordinates $$(\sigma, \tau)$$ to 3D, +by describing a point in space using the variables $$(\sigma, \tau, z)$$. +The $$z$$-axis is the same as in the Cartesian system, (hence the name *cylindrical*), +while the coordinate lines of $$\sigma$$ and $$\tau$$ are confocal parabolas. + +[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$ +and this system $$(\sigma, \tau, z)$$ are related by: $$\begin{aligned} \boxed{ - x = \frac{1}{2} (\tau^2 - \sigma^2 ) - \qquad - y = \sigma \tau - \qquad - z = z + \begin{aligned} + x + &= \frac{1}{2} (\tau^2 - \sigma^2) + \\ + y + &= \sigma \tau + \\ + z + &= z + \end{aligned} } \end{aligned}$$ -Converting the other way is a bit trickier. -It can be done by solving the following equations, -and potentially involves some fiddling with signs: +Conversely, a point given in $$(x, y, z)$$ can be converted +to $$(\sigma, \tau, z)$$ using these formulae: $$\begin{aligned} - 2 x - = \frac{y^2}{\sigma^2} - \sigma^2 - \qquad \qquad - 2 x - = - \frac{y^2}{\tau^2} + \tau^2 + \boxed{ + \begin{aligned} + \sigma + &= \sgn(x) \sqrt{\sqrt{x^2 + y^2} - x} + \\ + \tau + &= \sqrt{\sqrt{x^2 + y^2} + x} + \\ + z + &= z + \end{aligned} + } \end{aligned}$$ Cylindrical parabolic coordinates form an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), -so we would like to find its scale factors $$h_\sigma$$, $$h_\tau$$ and $$h_z$$. -The differentials of the Cartesian coordinates are as follows: +whose **scale factors** $$h_\sigma$$, $$h_\tau$$ and $$h_z$$ we need. +To get those, we calculate the unnormalized local basis: $$\begin{aligned} - \dd{x} = - \sigma \dd{\sigma} + \tau \dd{\tau} - \qquad - \dd{y} = \tau \dd{\sigma} + \sigma \dd{\tau} - \qquad - \dd{z} = \dd{z} + h_\sigma \vu{e}_\sigma + &= \vu{e}_x \pdv{x}{\sigma} + \vu{e}_y \pdv{y}{\sigma} + \vu{e}_z \pdv{z}{\sigma} + \\ + &= - \vu{e}_x \sigma + \vu{e}_y \tau + \\ + h_\tau \vu{e}_\tau + &= \vu{e}_x \pdv{x}{\tau} + \vu{e}_y \pdv{y}{\tau} + \vu{e}_z \pdv{z}{\tau} + \\ + &= \vu{e}_x \tau + \vu{e}_y \sigma + \\ + h_\sigma \vu{e}_\sigma + &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z} + \\ + &= \vu{e}_z \end{aligned}$$ -We calculate the line segment $$\dd{\ell}^2$$, -skipping many terms thanks to orthogonality: - -$$\begin{aligned} - \dd{\ell}^2 - &= (\sigma^2 + \tau^2) \:\dd{\sigma}^2 + (\tau^2 + \sigma^2) \:\dd{\tau}^2 + \dd{z}^2 -\end{aligned}$$ - -From this, we can directly read off the scale factors $$h_\sigma^2$$, $$h_\tau^2$$ and $$h_z^2$$, -which turn out to be: +By normalizing the **local basis vectors** +$$\vu{e}_\sigma$$, $$\vu{e}_\tau$$ and $$\vu{e}_z$$, +we arrive at these expressions, +where we have defined the abbreviation $$\rho$$ for convenience: $$\begin{aligned} \boxed{ - h_\sigma = \sqrt{\sigma^2 + \tau^2} - \qquad - h_\tau = \sqrt{\sigma^2 + \tau^2} - \qquad - h_z = 1 + \begin{aligned} + h_\sigma + &= \rho + \equiv \sqrt{\sigma^2 + \tau^2} + \\ + h_\tau + &= \rho + \equiv \sqrt{\sigma^2 + \tau^2} + \\ + h_z + &= 1 + \end{aligned} } -\end{aligned}$$ - -With these scale factors, we can use -the general formulae for orthogonal curvilinear coordinates -to easily to convert things from the Cartesian system. -The basis vectors are: - -$$\begin{aligned} + \qquad\qquad \boxed{ \begin{aligned} \vu{e}_\sigma - &= \frac{- \sigma}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_x + \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_y + &= - \vu{e}_x \frac{\sigma}{\rho} + \vu{e}_y \frac{\tau}{\rho} \\ \vu{e}_\tau - &= \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_x + \frac{\sigma}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_y + &= \vu{e}_x \frac{\tau}{\rho} + \vu{e}_y \frac{\sigma}{\rho} \\ \vu{e}_z &= \vu{e}_z @@ -91,13 +106,54 @@ $$\begin{aligned} } \end{aligned}$$ -The basic vector operations (gradient, divergence, Laplacian and curl) are given by: +Thanks to these scale factors, we can easily convert calculus from the Cartesian system +using the standard formulae for orthogonal curvilinear coordinates. + + + +## Differential elements + +For line integrals, +the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows: + +$$\begin{aligned} + \boxed{ + \dd{\vb{\ell}} + = \vu{e}_\sigma \: \rho \dd{\sigma} + \: \vu{e}_\tau \: \rho \dd{\tau} + \: \vu{e}_z \dd{z} + } +\end{aligned}$$ + +For surface integrals, +the normal vector element $$\dd{\vb{S}}$$ for a surface is given by: + +$$\begin{aligned} + \boxed{ + \dd{\vb{S}} + = \vu{e}_\sigma \: \rho \dd{\tau} \dd{z} + \: \vu{e}_\tau \: \rho \dd{\sigma} \dd{z} + \: \vu{e}_z \: \rho^2 \dd{\sigma} \dd{\tau} + } +\end{aligned}$$ + +And for volume integrals, +the infinitesimal volume $$\dd{V}$$ takes the following form: + +$$\begin{aligned} + \boxed{ + \dd{V} + = \rho^2 \dd{\sigma} \dd{\tau} \dd{z} + } +\end{aligned}$$ + + + +## Common operations + +The basic vector operations (gradient, divergence, curl and Laplacian) are given by: $$\begin{aligned} \boxed{ \nabla f - = \frac{\vu{e}_\sigma}{\sqrt{\sigma^2 + \tau^2}} \pdv{f}{\sigma} - + \frac{\vu{e}_\tau}{\sqrt{\sigma^2 + \tau^2}} \pdv{f}{\tau} + = \vu{e}_\sigma \frac{1}{\rho} \pdv{f}{\sigma} + + \vu{e}_\tau \frac{1}{\rho} \pdv{f}{\tau} + \vu{e}_z \pdv{f}{z} } \end{aligned}$$ @@ -105,78 +161,135 @@ $$\begin{aligned} $$\begin{aligned} \boxed{ \nabla \cdot \vb{V} - = \frac{1}{\sigma^2 + \tau^2} - \Big( \pdv{(V_\sigma \sqrt{\sigma^2 + \tau^2})}{\sigma} + \pdv{(V_\tau \sqrt{\sigma^2 + \tau^2})}{\tau} \Big) + \pdv{V_z}{z} + = \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\sigma V_\sigma}{\rho^3} + + \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\tau V_\tau}{\rho^3} + + \pdv{V_z}{z} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \times \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \bigg) + \\ + &\quad\: + \vu{e}_\tau \bigg( \pdv{V_\sigma}{z} - \frac{1}{\rho} \pdv{V_z}{\sigma} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} + \frac{\sigma V_\tau}{\rho^3} + - \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\tau V_\sigma}{\rho^3} \bigg) + \end{aligned} } \end{aligned}$$ $$\begin{aligned} \boxed{ \nabla^2 f - = \frac{1}{\sigma^2 + \tau^2} \Big( \pdvn{2}{f}{\sigma} + \pdvn{2}{f}{\tau} \Big) + \pdvn{2}{f}{z} + = \frac{1}{\rho^2} \pdvn{2}{f}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{f}{\tau} + \pdvn{2}{f}{z} } \end{aligned}$$ + + +## Uncommon operations + +Uncommon operations include: +the gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$, +the gradient of a vector $$\nabla \vb{V}$$, +the advection of a vector $$(\vb{U} \cdot \nabla) \vb{V}$$ with respect to $$\vb{U}$$, +the Laplacian of a vector $$\nabla^2 \vb{V}$$, +and the divergence of a 2nd-order tensor $$\nabla \cdot \overline{\overline{\vb{T}}}$$: + $$\begin{aligned} \boxed{ \begin{aligned} - \nabla \times \vb{V} - &= \vu{e}_\sigma \Big( \frac{\vu{e}_1}{\sqrt{\sigma^2 + \tau^2}} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \Big) + \nabla (\nabla \cdot \vb{V}) + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \mpdv{V_\tau}{\sigma}{\tau} + + \frac{1}{\rho} \mpdv{V_z}{\sigma}{z} + \\ + &\qquad\qquad + \frac{\tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{\sigma}{\rho^4} \pdv{V_\tau}{\tau} + + \frac{\rho^2 - 3 \sigma^2}{\rho^6} V_\sigma - \frac{3 \sigma \tau V_\tau}{\rho^6} \bigg) \\ - &+ \vu{e}_\tau \Big( \pdv{V_\sigma}{z} - \frac{1}{\sqrt{\sigma^2 + \tau^2}} \pdv{V_z}{\sigma} \Big) + &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \mpdv{V_\sigma}{\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + + \frac{1}{\rho} \mpdv{V_z}{\tau}{z} \\ - &+ \frac{\vu{e}_z}{\sigma^2 + \tau^2} - \Big( \pdv{(V_\tau \sqrt{\sigma^2 + \tau^2})}{\sigma} - \pdv{(V_\sigma \sqrt{\sigma^2 + \tau^2})}{\tau} \Big) + &\qquad\qquad - \frac{\tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{\sigma}{\rho^4} \pdv{V_\sigma}{\tau} + - \frac{3 \sigma \tau V_\sigma}{\rho^6} + \frac{\rho^2 - 3 \tau^2}{\rho^6} V_\tau \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \mpdv{V_\sigma}{z}{\sigma} + \frac{1}{\rho} \mpdv{V_\tau}{z}{\tau} + \pdvn{2}{V_z}{z} + + \frac{\sigma}{\rho^3} \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} \pdv{V_\tau}{z} \bigg) \end{aligned} } \end{aligned}$$ -The differential element of volume $$\dd{V}$$ -in cylindrical parabolic coordinates is given by: - $$\begin{aligned} \boxed{ - \dd{V} = (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} \dd{z} + \begin{aligned} + \nabla \vb{V} + &= \quad \vu{e}_\sigma \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\tau V_\tau}{\rho^3} \bigg) + + \vu{e}_\sigma \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} - \frac{\tau V_\sigma}{\rho^3} \bigg) + + \vu{e}_\sigma \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\sigma} + \\ + &\quad\: + \vu{e}_\tau \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\sigma V_\tau}{\rho^3} \bigg) + + \vu{e}_\tau \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\sigma V_\sigma}{\rho^3} \bigg) + + \vu{e}_\tau \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\tau} + \\ + &\quad\: + \vu{e}_z \vu{e}_\sigma \pdv{V_\sigma}{z} + + \vu{e}_z \vu{e}_\tau \pdv{V_\tau}{z} + + \vu{e}_z \vu{e}_z \pdv{V_z}{z} + \end{aligned} } \end{aligned}$$ -The differential elements of the isosurfaces are as follows, -where $$\dd{S_\sigma}$$ is the $$\sigma$$-isosurface, etc.: - $$\begin{aligned} \boxed{ \begin{aligned} - \dd{S_\sigma} &= \sqrt{\sigma^2 + \tau^2} \dd{\tau} \dd{z} + (\vb{U} \cdot \nabla) \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{U_\sigma}{\rho} \pdv{V_\sigma}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\sigma}{\tau} + U_z \pdv{V_\sigma}{z} + + \frac{\tau}{\rho^3} U_\sigma V_\tau - \frac{\sigma}{\rho^3} U_\tau V_\tau \bigg) \\ - \dd{S_\tau} &= \sqrt{\sigma^2 + \tau^2} \dd{\sigma} \dd{z} + &\quad\: + \vu{e}_\tau \bigg( \frac{U_\sigma}{\rho} \pdv{V_\tau}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\tau}{\tau} + U_z \pdv{V_\tau}{z} + + \frac{\sigma}{\rho^3} U_\tau V_\sigma - \frac{\tau}{\rho^3} U_\sigma V_\sigma \bigg) \\ - \dd{S_z} &= (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} + &\quad\: + \vu{e}_z \bigg( \frac{U_\sigma}{\rho} \pdv{V_z}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_z}{\tau} + U_z \pdv{V_z}{z} \bigg) \end{aligned} } \end{aligned}$$ -The normal element $$\dd{\vu{S}}$$ of a surface and -the tangent element $$\dd{\vu{\ell}}$$ of a curve are respectively: - $$\begin{aligned} \boxed{ - \dd{\vu{S}} - = \vu{e}_\sigma \sqrt{\sigma^2 + \tau^2} \dd{\tau} \dd{z} - + \vu{e}_\tau \sqrt{\sigma^2 + \tau^2} \dd{\sigma} \dd{z} - + \vu{e}_z (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} + \begin{aligned} + \nabla^2 \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\tau} + \pdvn{2}{V_\sigma}{z} + + \frac{2 \tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{2 \sigma}{\rho^4} \pdv{V_\tau}{\tau} - \frac{V_\sigma}{\rho^4} \bigg) + \\ + &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \pdvn{2}{V_\tau}{z} + - \frac{2 \tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{2 \sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{V_\tau}{\rho^4} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho^2} \pdvn{2}{V_z}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_z}{\tau} + \pdvn{2}{V_z}{z} \bigg) + \end{aligned} } \end{aligned}$$ $$\begin{aligned} \boxed{ - \dd{\vu{\ell}} - = \vu{e}_\sigma \sqrt{\sigma^2 + \tau^2} \dd{\sigma} - + \vu{e}_\tau \sqrt{\sigma^2 + \tau^2} \dd{\tau} - + \vu{e}_z \dd{z} + \begin{aligned} + \nabla \cdot \overline{\overline{\mathbf{T}}} + &= \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{T_{\sigma \sigma}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \sigma}}{\tau} + \pdv{T_{z \sigma}}{z} + + \frac{\sigma T_{\sigma \sigma}}{\rho^3} + \frac{\tau T_{\sigma \tau}}{\rho^3} + + \frac{\tau T_{\tau \sigma}}{\rho^3} - \frac{\sigma T_{\tau \tau}}{\rho^3} \bigg) + \\ + &+ \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{T_{\sigma \tau}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \tau}}{\tau} + \pdv{T_{k \tau}}{z} + - \frac{\tau T_{\sigma \sigma}}{\rho^3} + \frac{\sigma T_{\sigma \tau}}{\rho^3} + + \frac{\sigma T_{\tau \sigma}}{\rho^3} + \frac{\tau T_{\tau \tau}}{\rho^3} \bigg) + \\ + &+ \vu{e}_z \bigg( \frac{1}{\rho} \pdv{T_{\sigma z}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau z}}{\tau} + \pdv{T_{zz}}{z} + + \frac{\sigma T_{\sigma z}}{\rho^3} + \frac{\tau T_{\tau z}}{\rho^3} \bigg) + \end{aligned} } \end{aligned}$$ + ## References 1. M.L. Boas, *Mathematical methods in the physical sciences*, 2nd edition, diff --git a/source/know/concept/cylindrical-polar-coordinates/index.md b/source/know/concept/cylindrical-polar-coordinates/index.md index fe7d7c1..cf227a6 100644 --- a/source/know/concept/cylindrical-polar-coordinates/index.md +++ b/source/know/concept/cylindrical-polar-coordinates/index.md @@ -12,9 +12,9 @@ layout: "concept" by describing the location of a point in space using the variables $$(r, \varphi, z)$$. The $$z$$-axis is unchanged from the Cartesian system, -hence it is called *cylindrical*. +hence the name *cylindrical*. -Cartesian coordinates $$(x, y, z)$$ +[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$ and the cylindrical system $$(r, \varphi, z)$$ are related by: $$\begin{aligned} diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md index b175e64..c4faf81 100644 --- a/source/know/concept/drude-model/index.md +++ b/source/know/concept/drude-model/index.md @@ -9,124 +9,117 @@ categories: layout: "concept" --- -The **Drude model** classically predicts -the dielectric function and electric conductivity of a gas of free charge carriers, +The **Drude model**, also known as +the **Drude-Lorentz model** due to its analogy +to the *Lorentz oscillator model* +classically predicts the [dielectric function](/know/concept/dielectric-function/) +and electric conductivity of a gas of free charges, as found in metals and doped semiconductors. + ## Metals -An [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) -has an oscillating [electric field](/know/concept/electric-field/) -$$E(t) = E_0 \exp(- i \omega t)$$ -that exerts a force on the charge carriers, -which have mass $$m$$ and charge $$q$$. -They thus obey the following equation of motion, -where $$\gamma$$ is a frictional damping coefficient: +In a metal, the conduction electrons can roam freely. +When an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/) +passes by, its oscillating [electric field](/know/concept/electric-field/) +$$\vb{E}(t) = \vb{E}_0 e^{- i \omega t}$$ exerts a force on those electrons, +so the displacement $$\vb{x}(t)$$ of a particle from its initial position +obeys this equation of motion: $$\begin{aligned} - m \dvn{2}{x}{t} + m \gamma \dv{x}{t} - = q E_0 \exp(- i \omega t) + m \dvn{2}{\vb{x}}{t} + = q \vb{E} - \gamma m \dv{\vb{x}}{t} \end{aligned}$$ -Inserting the ansatz $$x(t) = x_0 \exp(- i \omega t)$$ -and isolating for the displacement $$x_0$$ yields: +Where $$m$$ and $$q < 0$$ are the mass and charge of the electron. +The first term is Newton's third law, +and the last term represents a damping force +slowing down the electrons at rate $$\gamma$$. -$$\begin{aligned} - - x_0 m \omega^2 - i x_0 m \gamma \omega - = q E_0 - \quad \implies \quad - x_0 - = - \frac{q E_0}{m (\omega^2 + i \gamma \omega)} -\end{aligned}$$ - -The polarization density $$P(t)$$ is therefore as shown below. -Note that the dipole moment $$p$$ goes from negative to positive, -and the electric field $$E$$ from positive to negative. -Let $$N$$ be the density of carriers in the gas, then: +Inserting the ansatz $$\vb{x}(t) = \vb{x}_0 e^{- i \omega t}$$ +and isolating for the displacement $$\vb{x}$$, we find: $$\begin{aligned} - P(t) - = N p(t) - = N q x(t) - = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} E(t) + \vb{x}(t) + = \vb{x}_0 e^{- i \omega t} + = - \frac{q \vb{E}}{m (\omega^2 + i \gamma \omega)} \end{aligned}$$ -The electric displacement field $$D$$ is thus as follows, -where $$\varepsilon_r$$ is the unknown relative permittivity of the gas, -which we will find shortly: +The polarization density $$\vb{P}(t)$$ is therefore as shown below. +Note that the dipole moment vector $$\vb{p}$$ is defined +as pointing from negative to positive, +whereas the electric field $$\vb{E}$$ goes from positive to negative. +Let $$N$$ be the metal's electron density, then: $$\begin{aligned} - D - = \varepsilon_0 \varepsilon_r E - = \varepsilon_0 E + P - = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) E + \vb{P}(t) + = N \vb{p}(t) + = N q \vb{x}(t) + = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} \vb{E}(t) \end{aligned}$$ -The parenthesized expression is the desired dielectric function $$\varepsilon_r$$, -which depends on $$\omega$$: +The electric displacement field $$\vb{D}$$ is then as follows, +where the parenthesized expression is the dielectric function +$$\varepsilon_r$$ of the material: $$\begin{aligned} - \boxed{ - \varepsilon_r(\omega) - = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} - } + \vb{D} + = \varepsilon_0 \vb{E} + \vb{P} + = \varepsilon_0 \bigg( 1 - \frac{N q^2}{\varepsilon_0 m} \frac{1}{\omega^2 + i \gamma \omega} \bigg) \vb{E} + = \varepsilon_0 \varepsilon_r \vb{E} \end{aligned}$$ -Where we have defined the important so-called **plasma frequency** like so: +From this, we define the **plasma frequency** $$\omega_p$$ +at which the conductor "resonates", +leading to so-called **plasma oscillations** of the electron density +(see also [Langmuir waves](/know/concept/langmuir-waves/)): $$\begin{aligned} + \varepsilon_r(\omega) + = 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} + \qquad\qquad \boxed{ \omega_p \equiv \sqrt{\frac{N q^2}{\varepsilon_0 m}} } \end{aligned}$$ -If $$\gamma = 0$$, then $$\varepsilon_r$$ is -negative $$\omega < \omega_p$$, -positive for $$\omega > \omega_p$$, -and zero for $$\omega = \omega_p$$. -Respectively, this leads to -an imaginary index $$\sqrt{\varepsilon_r}$$ (high absorption), -a real index tending to $$1$$ (transparency), -and the possibility of self-sustained plasma oscillations. -For metals, $$\omega_p$$ lies in the UV. - -We can refine this result for $$\varepsilon_r$$, -by recognizing the (mean) velocity $$v = \idv{x}{t}$$, -and rewriting the equation of motion accordingly: - -$$\begin{aligned} - m \dv{v}{t} + m \gamma v = q E(t) -\end{aligned}$$ +Suppose that $$\gamma = 0$$, +then we can identify three distinct scenarios for $$\varepsilon_r$$ here: -Note that $$m v$$ is simply the momentum $$p$$. -We define the **momentum scattering time** $$\tau \equiv 1 / \gamma$$, -which represents the average time between collisions, -where each collision resets the involved particles' momentums to zero. -Or, more formally: +* $$\omega < \omega_p$$, so $$\varepsilon_r < 0$$, + so the refractive index $$\sqrt{\varepsilon_r}$$ is imaginary, + meaning high absorption and high reflectivity + (due to the large complex index difference between media). +* $$\omega = \omega_p$$, so $$\varepsilon = 0$$, + allowing for self-sustained plasma oscillations. +* $$\omega > \omega_p$$, so $$\varepsilon_r > 0$$, + so the index $$\sqrt{\varepsilon}$$ is real and asymptotically goes to $$1$$, + leading to high transparency and low reflectivity from air. -$$\begin{aligned} - \dv{p}{t} - = - \frac{p}{\tau} + q E -\end{aligned}$$ +For most metals $$\omega_p$$ is ultraviolet, +which explains why they typically appear shiny to us. +In reality $$\gamma > 0$$, reducing the reflectivity somewhat when $$\omega < \omega_p$$. -Returning to the equation for the mean velocity $$v$$, -we insert the ansatz $$v(t) = v_0 \exp(- i \omega t)$$, -for the same electric field $$E(t) = E_0 \exp(-i \omega t)$$ as before: +The Drude model also lets us calculate the metal's conductivity. +We already have an expression for $$\vb{x}(t)$$, +which we differentiate to get the velocity $$\vb{v}(t)$$: $$\begin{aligned} - - i m \omega v_0 + \frac{m}{\tau} v_0 = q E_0 - \quad \implies \quad - v_0 = \frac{q \tau}{m (1 - i \omega \tau)} E_0 + \vb{v}(t) + = \dv{\vb{x}}{t} + = - i \omega \vb{x} + = \frac{i \omega q \vb{E}}{m (\omega^2 + i \gamma \omega)} + = \frac{q \vb{E}}{m (\gamma - i \omega)} \end{aligned}$$ -From $$v(t)$$, we find the resulting average current density $$J(t)$$ to be as follows: +Consequently the average current density $$\vb{J}(t)$$ is found to be: $$\begin{aligned} - J(t) - = - N q v(t) - = \sigma E(t) + \vb{J}(t) + = N q \vb{v}(t) + = \sigma \vb{E}(t) \end{aligned}$$ Where $$\sigma(\omega)$$ is the **AC conductivity**, @@ -134,57 +127,76 @@ which depends on the **DC conductivity** $$\sigma_0$$: $$\begin{aligned} \boxed{ - \sigma - = \frac{\sigma_0}{1 - i \omega \tau} + \sigma(\omega) + = \frac{\gamma \sigma_0}{\gamma - i \omega} } - \qquad \quad + \qquad\qquad \boxed{ \sigma_0 - = \frac{N q^2 \tau}{m} + \equiv \frac{N q^2}{\gamma m} } \end{aligned}$$ -We can use these quantities to rewrite -the dielectric function $$\varepsilon_r$$ from earlier: +Recall that $$\gamma$$ measures friction. +Specifically, Drude assumed that the electrons often collide with obstacles, +each time resetting their momentum to zero; +in that case $$\vb{v}$$ should be interpreted as the average "drift" +of many electrons in an ensemble. +The mean time between those collisions is +the **momentum scattering time** $$\tau \equiv 1 / \gamma$$, so: + +$$\begin{aligned} + \sigma(\omega) + = \frac{\sigma_0}{1 - i \omega \tau} + \qquad\qquad + \sigma_0 + = \frac{N q^2 \tau}{m} +\end{aligned}$$ + +After defining all those quantities, +the dielectric function $$\varepsilon_r(\omega)$$ can be written as: $$\begin{aligned} \boxed{ - \varepsilon_r(\omega) - = 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega} + \begin{aligned} + \varepsilon_r(\omega) + &= 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} + \\ + &= 1 + \frac{i \sigma(\omega)}{\varepsilon_0 \omega} + \end{aligned} } \end{aligned}$$ + ## Doped semiconductors Doping a semiconductor introduces -free electrons (n-type) -or free holes (p-type), -which can be treated as free particles -moving in the bands of the material. - -The Drude model can also be used in this case, -by replacing the actual carrier mass $$m$$ -by the effective mass $$m^*$$. +free electrons (n-type doping) or free holes (p-type doping), +which can be treated as free charge carriers moving through the material, +so the Drude model is also relevant in this case. + +We must replace the carriers' true mass $$m$$ with their *effective mass* $$m^*$$ +found from the material's electronic band structure. Furthermore, semiconductors already have -a high intrinsic permittivity $$\varepsilon_{\mathrm{int}}$$ -before the dopant is added, -so the diplacement field $$D$$ is: +a high intrinsic dielectric function $$\varepsilon_{\mathrm{int}}$$ +before being doped, so the displacement field $$\vb{D}$$ becomes: $$\begin{aligned} - D - = \varepsilon_0 E + P_{\mathrm{int}} + P_{\mathrm{free}} - = \varepsilon_{\mathrm{int}} \varepsilon_0 E - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} E + \vb{D} + = \varepsilon_0 \vb{E} + \vb{P}_{\mathrm{int}} + \vb{P}_{\mathrm{free}} + = \varepsilon_0 \varepsilon_{\mathrm{int}} \vb{E} - \frac{N q^2}{m^* (\omega^2 + i \gamma \omega)} \vb{E} + = \varepsilon_0 \varepsilon_r \vb{E} \end{aligned}$$ -Where $$P_{\mathrm{int}}$$ is the intrinsic undoped polarization, -and $$P_{\mathrm{free}}$$ is the contribution of the free carriers. +Where $$\vb{P}_{\mathrm{int}}$$ is the intrinsic polarization before doping, +and $$\vb{P}_{\mathrm{free}}$$ is the expression we calculated above for metals. The dielectric function $$\varepsilon_r(\omega)$$ is therefore given by: $$\begin{aligned} \boxed{ \varepsilon_r(\omega) - = \varepsilon_{\mathrm{int}} \Big( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \Big) + = \varepsilon_{\mathrm{int}} \bigg( 1 - \frac{\omega_p^2}{\omega^2 + i \gamma \omega} \bigg) } \end{aligned}$$ @@ -194,29 +206,28 @@ to include $$\varepsilon_\mathrm{int}$$: $$\begin{aligned} \boxed{ \omega_p - = \sqrt{\frac{N q^2}{\varepsilon_{\mathrm{int}} \varepsilon_0 m^*}} + \equiv \sqrt{\frac{N q^2}{\varepsilon_0 \varepsilon_{\mathrm{int}} m^*}} } \end{aligned}$$ The meaning of $$\omega_p$$ is the same as for metals, -with high absorption for $$\omega < \omega_p$$. -However, due to the lower carrier density $$N$$ in a semiconductor, -$$\omega_p$$ lies in the IR rather than UV. +but the free carrier density $$N$$ is typically lower in this case, +so $$\omega_p$$ is usually infrared rather than ultraviolet. -However, instead of asymptotically going to $$1$$ for $$\omega > \omega_p$$ like a metal, -$$\varepsilon_r$$ tends to $$\varepsilon_\mathrm{int}$$ instead, -and crosses $$1$$ along the way, -at which point the reflectivity is zero. -This occurs at: +Furthermore, instead of $$\varepsilon_r \to 1$$ +for $$\omega \to \infty$$ like a metal, +now $$\varepsilon_r \to \varepsilon_\mathrm{int}$$. +Along the way, there is a point where $$\varepsilon_r = 1$$ +and the reflectivity becomes zero. This occurs at: $$\begin{aligned} \omega^2 = \frac{\varepsilon_{\mathrm{int}}}{\varepsilon_{\mathrm{int}} - 1} \omega_p^2 \end{aligned}$$ -This is used to experimentally determine the effective mass $$m^*$$ -of the doped semiconductor, -by finding which value of $$m^*$$ gives the measured $$\omega$$. +If $$N$$ and $$\varepsilon_\mathrm{int}$$ are known, +this can be used to experimentally determine $$m^*$$ +by finding which value of $$\omega_p$$ would lead to the measured zero-reflectivity point. diff --git a/source/know/concept/orthogonal-curvilinear-coordinates/index.md b/source/know/concept/orthogonal-curvilinear-coordinates/index.md index c7299ee..669358c 100644 --- a/source/know/concept/orthogonal-curvilinear-coordinates/index.md +++ b/source/know/concept/orthogonal-curvilinear-coordinates/index.md @@ -21,7 +21,8 @@ where the coordinate surfaces are always perpendicular. Examples of such orthogonal curvilinear systems include [spherical coordinates](/know/concept/spherical-coordinates/), [cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/), -and [cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/). +[cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/), +and (trivially) [Cartesian coordinates](/know/concept/cartesian-coordinates/). @@ -690,12 +691,9 @@ When this index notation is written out in full, it becomes: $$\begin{aligned} \nabla^2 f - = \frac{1}{h_1 h_2 h_3} - \bigg( - \pdv{}{c_1}\Big(\! \frac{h_2 h_3}{h_1} \pdv{f}{c_1} \!\Big) - + \pdv{}{c_2}\Big(\! \frac{h_1 h_3}{h_2} \pdv{f}{c_2} \!\Big) - + \pdv{}{c_3}\Big(\! \frac{h_1 h_2}{h_3} \pdv{f}{c_3} \!\Big) - \bigg) + = \frac{1}{h_1 h_2 h_3} \bigg( \pdv{}{c_1} \Big( \frac{h_2 h_3}{h_1} \pdv{f}{c_1} \Big) + + \pdv{}{c_2} \Big( \frac{h_1 h_3}{h_2} \pdv{f}{c_2} \Big) + + \pdv{}{c_3} \Big( \frac{h_1 h_2}{h_3} \pdv{f}{c_3} \Big) \bigg) \end{aligned}$$ This is trivial to prove: $$\nabla^2 f = \nabla \cdot (\nabla f)$$, diff --git a/source/know/concept/spherical-coordinates/index.md b/source/know/concept/spherical-coordinates/index.md index 7f6d111..1607b61 100644 --- a/source/know/concept/spherical-coordinates/index.md +++ b/source/know/concept/spherical-coordinates/index.md @@ -22,8 +22,8 @@ Note that this is the standard notation among physicists, but mathematicians often switch the definitions of $$\theta$$ and $$\varphi$$, while still writing $$(r, \theta, \varphi)$$. -Cartesian coordinates $$(x, y, z)$$ and the spherical system -$$(r, \theta, \varphi)$$ are related by: +[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$ +and the spherical system $$(r, \theta, \varphi)$$ are related by: $$\begin{aligned} \boxed{ @@ -114,8 +114,6 @@ using the standard formulae for orthogonal curvilinear coordinates. - - ## Differential elements For line integrals, @@ -169,7 +167,7 @@ $$\begin{aligned} $$\begin{aligned} \boxed{ \nabla \cdot \vb{V} - = \pdv{V_r}{r} + \frac{2}{r} V_r + = \pdv{V_r}{r} + \frac{2 V_r}{r} + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_\theta}{r \tan{\theta}} + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi} } @@ -216,15 +214,15 @@ $$\begin{aligned} \begin{aligned} \nabla (\nabla \cdot \vb{V}) &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\theta}{r}{\theta} + \frac{1}{r \sin{\theta}} \mpdv{V_\varphi}{\varphi}{r} - + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta} \\ - &\qquad\qquad - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi} + &\qquad\qquad + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta} + - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{1}{r \tan{\theta}} \pdv{V_\theta}{r} - \frac{2 V_r}{r^2} - \frac{V_\theta}{r^2 \tan{\theta}} \bigg) \\ &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r} \mpdv{V_r}{\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta} - + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} + + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi} \\ - &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta} + &\qquad\qquad + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta} - \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg) \\ &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r \sin{\theta}} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\theta}{\varphi}{\theta} @@ -275,23 +273,22 @@ $$\begin{aligned} \begin{aligned} \nabla^2 \vb{V} &= \quad\: \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_r}{\varphi} - + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta} \\ - &\qquad\qquad - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi} + &\qquad\qquad + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta} + - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi} - \frac{2 V_r}{r^2} - \frac{2 V_\theta}{r^2 \tan{\theta}} \bigg) \\ &\quad\: + \vu{e}_\theta \bigg( \pdvn{2}{V_\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta} - + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r} + + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi} \\ - &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta} + &\qquad\qquad + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta} - \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg) \\ &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\theta} - + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi} + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} + + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi} \\ - &\qquad\qquad + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi} - + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\varphi}{\theta} - - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg) + &\qquad\qquad + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi} + + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\varphi}{\theta} - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg) \end{aligned} } \end{aligned}$$ |