Merge pull request #2 from jhillairet/DispersionRelation

Dispersion relation

Dispersion_Relation.tex

@@ -1,5 +1,5 @@
 \chapter{Dispersion Relation}
-\section{Vacuum dispersion relation}
+\section{Vacuum Dispersion Relation}
 In a non-dispersive isotropic homogeneous non-lossy dielectric medium, such as vacuum (or air), the wavevector $\mathbf{k}=k\mathbf{\hat{k}}$ direction is given by:
 \begin{equation}
  \mathbf{\hat{k}}\cdot\mathbf{E}=0
@@ -13,6 +13,60 @@ \section{Vacuum dispersion relation}
 
 The condition for that system of 3 equations and 3 unknowns $(E_{x},E_{y},E_{z})$ to have a non-trivial solution, consists in solving the \emph{dispersion relation} between the wavevector $\mathbf{k}$ and the angular frequency $\omega$, i.e. $\mathbf{k}(\omega)$. In this medium, this relation is simple (TODO demo): $k=\sqrt{\mu\varepsilon}\omega=\frac{c}{n}\omega$\parencite[(7.4)]{Jackson1998}\footnote{While in a lossy medium $k^{2}=\mu\varepsilon\omega^{2}-j\omega\mu\sigma$ \parencite[sec. 8.2]{Bladel2007}.}.
 
+\subsection{Cold Plasma Dispersion Relation}
+Starting from Maxwell equations expressed in the $k-\omega$ domain (\ref{eq:k-omegaMaxwellEquations}) and assuming that the dispersion relations are of the form (\ref{eq:k-omegaDispersionRelation}), with the supplementary assumption that the medium is non magnetic (i.e. $\boldsymbol{\mu}=\mu_0$), one has thus:
+\begin{subequations}
+	\begin{align}
+	\mathbf{k} \times \boldsymbol{\mathcal{E}} (\mathbf{k}, \omega) 
+	=& 
+	\omega \mu_0 \boldsymbol{\mathcal{H}}(\mathbf{k}, \omega) 
+	\\
+	\mathbf{k} \times \boldsymbol{\mathcal{H}} (\mathbf{k}, \omega) 
+	=& 
+	\left(
+	-\omega \boldsymbol{\varepsilon} (\mathbf{k}, \omega) 
+	+ 	
+	j\boldsymbol{\sigma} (\mathbf{k}, \omega) 
+	\right) \cdot 
+	\boldsymbol{\mathcal{E}}(\mathbf{k}, \omega)
+	\\
+	\mathbf{k}  \cdot \boldsymbol{\varepsilon}(\mathbf{k}, \omega)  \cdot \boldsymbol{\mathcal{E}} (\mathbf{k}, \omega) 
+	=& jq_v(\mathbf{k}, \omega) 
+	\\
+	\mathbf{k}  \cdot \boldsymbol{\mathcal{H}} (\mathbf{k}, \omega) 
+	=& 0	
+	\end{align}
+\end{subequations}
+combining the first two equations leads to the propagation equation:
+\begin{equation}
+\mathbf{k} \times \mathbf{k} \times \boldsymbol{\mathcal{E}} (\mathbf{k}, \omega) 
++
+	\left(
+	\omega^2 \mu_0 \boldsymbol{\varepsilon} (\mathbf{k}, \omega) 
+	- 	
+	j\omega\mu_0\boldsymbol{\sigma} (\mathbf{k}, \omega) 
+	\right) \cdot 
+	\boldsymbol{\mathcal{E}}(\mathbf{k}, \omega)
+	=
+	\boldsymbol{0}
+\end{equation}
+Let's define the equivalent relative dielectric tensor $\mathbf{K}$ such as:
+\begin{equation}
+\mathbf{K}(\mathbf{k},\omega)
+=
+\mathbf{I}
+- 	
+\frac{j}{\omega\varepsilon_0}\boldsymbol{\sigma} (\mathbf{k}, \omega)
+\end{equation}
+then,
+\begin{equation}
+\mathbf{k} \times \mathbf{k} \times \boldsymbol{\mathcal{E}} (\mathbf{k}, \omega) 
++ \frac{\omega^2}{c^2}
+\mathbf{K}(\mathbf{k}, \omega) \cdot 
+\boldsymbol{\mathcal{E}}(\mathbf{k}, \omega)
+=
+\boldsymbol{0}
+\end{equation}
 
 % Dans un plasma froid magnétisé, la situation est radicalement différente,
 % car les courants de polarisation générés par les mouvements électroniques

Figures/Fusion Cross Section Plots/Fusion Cross Section.py

@@ -0,0 +1,69 @@
+# -*- coding: utf-8 -*-
+"""
+Created on Thu Jan  8 15:27:56 2015
+
+@author: JH218595
+"""
+
+from pylab import *
+
+""" 
+Plot the Reaction rates in cm^3 s^-1 as a function of
+ E, the energy in keV of the incident particle 
+ [the first ion of the reaction label]
+
+ Data taken from NRL Formulary 2013.
+"""
+E,DD,DT,DH=loadtxt('reaction_rates_vs_energy_incident_particle.txt', 
+                   skiprows=1, unpack=True)
+
+figure(num=1)
+clf()
+loglog(E, DD, E, DT, E, DH, lw=2)
+grid()
+xlabel('Temperature [keV]', fontsize=16)
+ylabel('Reaction Rates $<\sigma v>$ [$cm^3.s^{-1}$]', fontsize=16)
+xticks(fontsize=14)
+yticks(fontsize=14)
+ylim([1e-26, 2e-15])
+legend(('D-D', 'D-T', 'D-He$^3$'), loc='best', fontsize=18)
+
+"""
+Plot the total cross section in m^2 for various species vs incident energy in keV
+
+"""
+
+def cross_section(E, A):
+    """
+    The total cross section in barns (1 barns=1e-24 cm^2) as a function of E, 
+    the energy in keV of the incident particle.
+    
+    Formula from NRL Formulary 2013.
+    """
+    sigma_T = (A[4]+((A[3]-A[2]*E)**2+1)**(-1) * A[1])/(E*(exp(A[0]/sqrt(E))-1))
+    return(sigma_T)
+
+A_DD_a = [46.097, 372, 4.36e-4, 1.220, 0]
+A_DD_b = [47.88, 482, 3.08e-4, 1.177, 0]
+A_DT   = [45.95, 50200, 1.368e-2, 1.076, 409]
+A_DHe3 = [89.27, 25900, 3.98e-3, 1.297, 647]
+A_TT   = [38.39, 448, 1.02e-3, 2.09, 0]
+A_THe3 = [123.1, 11250, 0, 0, 0]
+
+E = logspace(0, 3, 501)
+barns2SI = 1e-24 * 1e-4 # in m^2
+
+sigma_DD = barns2SI*(cross_section(E, A_DD_a) + cross_section(E, A_DD_b))
+sigma_DT = barns2SI*cross_section(E, A_DT)
+sigma_DHe3 = barns2SI*cross_section(E, A_DHe3)
+
+figure(num=2)
+clf()
+loglog(E, sigma_DD, E, sigma_DT, E, sigma_DHe3, lw=2)
+grid()
+xlabel('Deuteron Energy [keV]', fontsize=16)
+ylabel('Cross-section $\sigma$ [$m^2$]', fontsize=16)
+legend(('D-D', 'D-T', 'D-He$^3$'), loc='best', fontsize=18)
+ylim([1e-32, 2e-27])
+xticks(fontsize=14)
+yticks(fontsize=14)

Figures/Fusion Cross Section Plots/reaction_rates_vs_energy_incident_particle.txt

@@ -0,0 +1,11 @@
+Temperature [keV]	D�D	D-T	D-He3						
+1.0	1.5e-22	5.5e-21	1.0E-26						
+2.0	5.4e-21	2.6e-19	1.4e-23						
+5.0	1.8e-19	1.3e-17	6.7e-21						
+10.0	1.2e-18	1.1e-16	2.3e-19						
+20.0	5.2e-18	4.2e-16	3.8e-18						
+50.0	2.1e-17	8.7e-16	5.4e-17						
+100.0	4.5e-17	8.5e-16	1.6e-16						
+200.0	8.8e-17	6.3e-16	2.4e-16						
+500.0	1.8e-16	3.7e-16	2.3e-16						
+1000.0	2.2e-16	2.7e-16	1.8e-16