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Diffstat (limited to 'source/know/concept/spitzer-resistivity/index.md')
| -rw-r--r-- | source/know/concept/spitzer-resistivity/index.md | 52 |
1 files changed, 26 insertions, 26 deletions
diff --git a/source/know/concept/spitzer-resistivity/index.md b/source/know/concept/spitzer-resistivity/index.md index f9fd969..283b7fd 100644 --- a/source/know/concept/spitzer-resistivity/index.md +++ b/source/know/concept/spitzer-resistivity/index.md @@ -9,17 +9,17 @@ layout: "concept" --- If an [electric field](/know/concept/electric-field/) -with magnitude $E$ is applied to the plasma, the electrons experience -a [Lorentz force](/know/concept/lorentz-force/) $q_e E$ +with magnitude $$E$$ is applied to the plasma, the electrons experience +a [Lorentz force](/know/concept/lorentz-force/) $$q_e E$$ (we neglect the ions due to their mass), -where $q_e$ is the electron charge. +where $$q_e$$ is the electron charge. However, collisions slow them down while they travel through the plasma., -This can be modelled as a drag force $f_{ei} m_e v_e$, -where $f_{ei}$ is the electron-ion collision frequency -(we neglect $f_{ee}$ since all electrons are moving together), -$m_e$ is their mass, -and $v_e$ their typical velocity relative to the ions in the background. +This can be modelled as a drag force $$f_{ei} m_e v_e$$, +where $$f_{ei}$$ is the electron-ion collision frequency +(we neglect $$f_{ee}$$ since all electrons are moving together), +$$m_e$$ is their mass, +and $$v_e$$ their typical velocity relative to the ions in the background. Balancing the two forces yields the following relation: $$\begin{aligned} @@ -27,7 +27,7 @@ $$\begin{aligned} = f_{ei} m_e v_e \end{aligned}$$ -Using that the current density $J = q_e n_e v_e$, +Using that the current density $$J = q_e n_e v_e$$, we can rearrange this like so: $$\begin{aligned} @@ -37,12 +37,12 @@ $$\begin{aligned} = \eta J \end{aligned}$$ -This is Ohm's law, where $\eta$ is the resistivity. -From our derivation of the [Coulomb logarithm](/know/concept/coulomb-logarithm/) $\ln(\Lambda)$, -we estimate $f_{ei}$ to be as follows, -where $n_i$ is the ion density, -$\sigma$ is the collision cross-section, -and $\mu$ is the [reduced mass](/know/concept/reduced-mass/) +This is Ohm's law, where $$\eta$$ is the resistivity. +From our derivation of the [Coulomb logarithm](/know/concept/coulomb-logarithm/) $$\ln(\Lambda)$$, +we estimate $$f_{ei}$$ to be as follows, +where $$n_i$$ is the ion density, +$$\sigma$$ is the collision cross-section, +and $$\mu$$ is the [reduced mass](/know/concept/reduced-mass/) of the electron-ion system: $$\begin{aligned} @@ -52,15 +52,15 @@ $$\begin{aligned} \approx \frac{1}{2 \pi} \frac{Z q_e^4}{\varepsilon_0^2 m_e^2} \frac{n_e}{v_e^3} \ln(\Lambda) \end{aligned}$$ -Where we used that $\mu \approx m_e$, -and $q_i = -Z q_e$ for some ionization $Z$, -and as a result $n_e \approx Z n_i$ due to the plasma's quasi-neutrality. -Beware: authors disagree about the constant factors in $f_{ei}$; +Where we used that $$\mu \approx m_e$$, +and $$q_i = -Z q_e$$ for some ionization $$Z$$, +and as a result $$n_e \approx Z n_i$$ due to the plasma's quasi-neutrality. +Beware: authors disagree about the constant factors in $$f_{ei}$$; recall that it was derived from fairly rough estimates. This article follows Bellan. -Inserting this expression for $f_{ei}$ into -the so-called **Spitzer resistivity** $\eta$ then yields: +Inserting this expression for $$f_{ei}$$ into +the so-called **Spitzer resistivity** $$\eta$$ then yields: $$\begin{aligned} \boxed{ @@ -70,10 +70,10 @@ $$\begin{aligned} } \end{aligned}$$ -A reasonable estimate for the typical velocity $v_e$ +A reasonable estimate for the typical velocity $$v_e$$ at thermal equilibrium is as follows, -where $k_B$ is Boltzmann's constant, -and $T_e$ is the electron temperature: +where $$k_B$$ is Boltzmann's constant, +and $$T_e$$ is the electron temperature: $$\begin{aligned} \frac{1}{2} m_e v_e^2 @@ -85,8 +85,8 @@ $$\begin{aligned} Other choices exist, see e.g. the [Maxwell-Boltzmann distribution](/know/concept/maxwell-boltzmann-distribution/), -but always $v_e \propto \sqrt{T_e/m_e}$. -Inserting this $v_e$ into $\eta$ then gives: +but always $$v_e \propto \sqrt{T_e/m_e}$$. +Inserting this $$v_e$$ into $$\eta$$ then gives: $$\begin{aligned} \eta |