Fundamentals of Plasma and Its Diagnostics

Abstract

This chapter briefly reviews the fundamentals of plasma physics with diversified diagnostics, emphasizing essential subjects for beginners and researchers who are not so familiar with plasma. Further, it may be beneficial for those studying plasma in the past, regardless of their fields of specialty. Besides the main stories on RF studies presented in Chaps. 3 and 4, this chapter is helpful as a general textbook on fundamental plasma.

Appendices

Appendix

1.1  Fundamental Parameters (SI Unit)

2.2.1 1.1.1  Physical Constants

Symbol	Quantity	Value
c	Velocity of light	2.998 × 108 (m/s)
e	Elementary charge	1.602 × 10−19 (C)
$\varepsilon_{0}$	Permittivity in vacuum	8.854 × 10−12 (F/m)
$\mu_{0}$	Permeability in vacuum	1.257 × 10−6 (H/m)
h	Planck constant	6.626 × 10−34 (Js)
$k_{{\text{B}}}$	Boltzmann constant	1.381 × 10−23 (J/K)
$N_{{\text{A}}}$	Avogadro’s number	6.022 × 1023 (1/mol)
$m_{{\text{e}}}$	Electron mass	9.109 × 10−31 (kg)
m p	Proton mass	1.673 × 10−27 (kg)
$m_{\rm{p}} /m_{{\text{e}}}$ $\left( {m_{\text{p}} /m_{{\text{e}}} } \right)^{1/2}$	$\text{Mass ratio}$ $\text{Root mass ratio}$	${1.836 \times 10^3(-)}$ $42.85 (-)$
eV	Electron volt	1.602 × 10−19 (J)
	Corresponding temperature	1.160 × 104 (K)
	Corresponding wavelength	1.240 × 10−6 (m)
Neutral atom density @ 1 mTorr and 300 K		3.219 × 1019 (m−3)

2.2.2 1.1.2  Handy Formulas

Symbol	Quantity	Formula	Handy formula (SI unit unless otherwise specified)
f pe	Electron plasma frequency	$(n_{{\text{e}}} e^{2} /m_{{\text{e}}} \varepsilon_{0} )^{1/2} /2\uppi$	$2.8 \times 10^{10} (n_{{\text{e}}} /10^{19} )^{1/2} \ \ ({\text{s}}^{ - 1} )$
f ce	Electron cyclotron frequency	$eB/2\pi m_{{\text{e}}}$	$2.8 \times 10^{10} \,B\ \ ({\text{s}}^{ - 1} )$
f ci	Ion cyclotron frequency	$eB/2\pi m_{{\text{i}}}$	$1.5 \times 10^{7} \,B\ \ ({\text{s}}^{ - 1} )\,({\text{H}}^{ + } )$ $3.8 \times 10^{5} \,B\ \ ({\text{s}}^{ - 1} )\,({\text{Ar}}^{ + } )$
f LH	Lower hybrid frequency	$\sim (f_{{{\text{ce}}}} f_{{{\text{ci}}}} )^{1/2}$ (if $f_{{{\text{pi}}}}^{\ 2} > > f_{{{\text{ce}}}} f_{{{\text{ci}}}}$)	$6.5 \times 10^{8} \,B\ \ ({\text{s}}^{ - 1} )\,({\text{H}}^{ + } )$ $1.0 \times 10^{8} \,B\ \ ({\text{s}}^{ - 1} )\,({\text{Ar}}^{ + } )$
f UH	Upper hybrid frequency	$(f_{{{\text{ce}}}}^{\ \ 2} + f_{{{\text{pe}}}}^{\ \ 2} )^{1/2}$	(Use the above)
$\lambda_{{\text{D}}}$	Debye length (electron)	$(\varepsilon_{0} k_{{\text{B}}} T_{{\text{e}}} /n_{{\text{e}}} e^{2} )^{1/2}$	$2.4 \times 10^{ - 6} \,\left[ {T_{{\text{e}}} \,({\text{eV}})/(n_{{\text{e}}} /10^{19} )} \right]^{1/2} \ ({\text{m}})$
$\rho_{{\text{e}}}$	Electron Larmor radius	$v_{{{\text{the}}}} /\omega_{{{\text{ce}}}} \sim (m_{{\text{e}}} k_{{\text{B}}} T_{{\text{e}}} )^{1/2} /eB$ (if isotropic VDF)	$2.4 \times 10^{ - 6} \,\left[ {T_{{\text{e}}} \ ({\text{eV}})} \right]^{1/2} /B\ ({\text{m}})$
$\rho_{{\text{i}}}$	Ion Larmor radius	$v_{{{\text{thi}}}} /\omega_{{{\text{ci}}}} \sim (m_{{\text{i}}} k_{{\text{B}}} T_{{\text{e}}} )^{1/2} /eB$ (if isotropic VDF)	$1.0 \times 10^{ - 4} \,\left[ {T_{{\text{i}}} \ ({\text{eV}})} \right]^{1/2} /B\,({\text{m}})\,({\text{H}}^{ + } )$ $6.4 \times 10^{ - 4} \,\left[ {T_{{\text{i}}} \ ({\text{eV}})} \right]^{1/2} \ ({\text{m}})\,({\text{Ar}}^{ + } )$
$v_{{{\text{the}}}}$	Electron thermal velocity	$\sim (k_{{\text{B}}} T_{{\text{e}}} /m_{{\text{e}}} )^{1/2}$	$4.2 \times 10^{5} \,\left[ {T_{{\text{e}}} \ ({\text{eV}})} \right]^{1/2} \ \ ({\text{m}}/{\text{s}})$
$v_{{{\text{thi}}}}$	Ion thermal velocity	$\sim (k_{{\text{B}}} T_{{\text{i}}} /m_{{\text{i}}} )^{1/2}$	$9.8 \times 10^{3} \,\left[ {T_{{\text{i}}} \ ({\text{eV}})} \right]^{1/2} \ \ ({\text{m}}/{\text{s}})\,({\text{H}}^{ + } )$ $1.5 \times 10^{3} \,\left[ {T_{{\text{i}}} \ ({\text{eV}})} \right]^{1/2} \ \ ({\text{m}}/{\text{s}})\,({\text{Ar}}^{ + } )$
$v_{{\text{A}}}$	Alfvén velocity	$\sim B/(\mu_{0} m_{{\text{i}}} n_{{\text{i}}} )^{1/2}$ (if $\mu_{0} m_{{\text{i}}} n_{{\text{i}}} > > B^{2} /c^{2}$)	$6.9 \times 10^{6} \,B/(n_{{\text{e}}} /10^{19} )^{1/2} \ \ ({\text{m}}/{\text{s}})\,({\text{H}}^{ + } )$ $1.1 \times 10^{6} \,B/(n_{{\text{e}}} /10^{19} )^{1/2} \ \ ({\text{m}}/{\text{s}})\,({\text{Ar}}^{ + } )$
$C_{{\text{s}}}$	Ion sound (or ion acoustic, or Bohm) velocity	$(k_{{\text{B}}} T_{{\text{e}}} /m_{{\text{i}}} )^{1/2}$ (if Ti = 0)	$9.8 \times 10^{3} \,\left[ {T_{{\text{e}}} \ ({\text{eV}})} \right]^{1/2} \ \ ({\text{m}}/{\text{s}})\,({\text{H}}^{ + } )$ $1.5 \times 10^{3} \,\left[ {T_{{\text{e}}} \ ({\text{eV}})} \right]^{1/2} \ \ ({\text{m}}/{\text{s}})\,({\text{Ar}}^{ + } )$
$\nu_{{{\text{en}}}}$	Electron-neutral collision frequency	$n_{{\text{n}}} < \sigma (v )\, v >$ (depending on gas species)	$\sim 1.4 \times 10^{6} \ \ T_{{\text{e}}} \ ({\text{eV}})\ ({\text{s}}^{ - 1} )$ $(T_{{\text{e}}} < 10\ {\text{eV,}}\,{\text{Ar}}\,1\ \ {\text{mTorr}})$
$\nu_{{{\text{ei}}}}$	Electron-ion Coulomb collision frequency	$\dfrac{{Z^{2} e^{4} n_{{\text{e}}} \ln \varLambda }}{{51.6\,\pi^{1/2} \epsilon_{0}^{\ \ 2} m_{{\text{e}}}^{\ 1/2} (k_{{\text{B}}} T_{{\text{e}}} )^{3/2} }}$	$1.5 \times 10^{8} \,(n_{{\text{e}}} /10^{19} )Z^{2} /[T_{{\text{e}}} \ \ ({\text{eV}})]^{3/2} \ ({\text{s}}^{ - 1} )$ $(\ln \varLambda = 10)$
$\eta_{{{\text{sp}}}}$	Spitzer specific resistivity	$\dfrac{{Ze^{2} m_{{\text{e}}}^{\ 1/2} \ln \varLambda }}{{51.6\,\pi^{\ 1/2} \epsilon_{0}^{\ 2} (k_{{\text{B}}} T_{{\text{e}}} )^{3/2} }}$	$5.2 \times 10^{ - 4} \,Z/\left[ {T_{{\text{e}}} \ \ ({\text{eV}})} \right]^{3/2} \ (\Omega \ {\text{m}})$ $(\ln \varLambda = 10)$
$p_{{\text{n}}}$	Neutral gas pressure	$k_{{\text{B}}} n_{{\text{n}}} T_{{\text{g}}}$	${\text{N}}/{\text{m}}^{2} = {\text{Pa}} = 7.5 \times 10^{ - 3} \ {\text{Torr}}$
$p = p_{{\text{e}}} + p_{{\text{i}}}$	Plasma pressure	$k_{{\text{B}}} (n_{{\text{e}}} T_{{\text{e}}} + n_{{\text{i}}} T_{{\text{i}}} )$	$1.6\,(n_{{\text{e}}} /10^{19} )\left[ {T_{{\text{e}}} \ ({\text{eV}}) + T_{{\text{i}}} \ ({\text{eV}})} \right]\ ({\text{N/m}}^{2} )$
$p_{{\text{B}}}$	Magnetic pressure	$B^{2} /2\mu_{0}$	$4.0 \times 10^{5} \,B^{2} \ ({\text{N/m}}^{2} )\,\left[ {3.9\,B^{2} \,({\text{atm}})} \right]$
$\beta$	Ratio of plasma pressure to magnetic one	$k_{{\text{B}}} (n_{{\text{e}}} T_{e} + n_{{\text{i}}} T_{i} )/(B^{2} /2\mu_{0} )$	(Use the above)
$N_{{\text{d}}}$	Particle number within Debye sphere	$(4{\uppi }/3)n_{{\text{e}}} \lambda_{{\text{d}}}^{\ 3}$	$5.5 \times 10^{2} \,\left[ {T_{{\text{e}}} \ ({\text{eV}})} \right]^{3/2} /\left[ {(n_{{\text{e}}} /10^{19} )} \right]^{1/2}$ (–)
$\varGamma$	Coupling coefficient	$\left[ {(Ze)^{2} /a} \right]/4{\uppi }\varepsilon_{0} k_{{\text{B}}} T_{{\text{e}}}$	$5.0 \times 10^{ - 3} \,Z^{2} \,\left[ {(n_{e} /10^{19} )} \right]^{1/3} /\left[ {T_{{\text{e}}} \ ({\text{eV}})} \right]$ (–) $\left[ {{\text{if}}\,\,(4\pi /3)a^{3} n_{{\text{e}}} = 1} \right]$
$D_{{\text{B}}}$	Bohm diffusion	$(1/16)(kT_{{\text{e}}} /eB)$	$6.3 \times 10^{ - 2} \,T_{{\text{e}}} \ ({\text{eV}})/B\ ({\text{m}}^{2} /{\text{s}})$

1.2  Useful Formulas

2.3.1 1.2.1  Vector Relations

A, B, and C are vectors, and $\phi$ is a scalar variable.

$$ {\mathbf{A}} \cdot \left( {{\mathbf{B}} \times {\mathbf{C}}} \right) = {\mathbf{B}} \cdot \left( {{\mathbf{C}} \times {\mathbf{A}}} \right) = {\mathbf{C}} \cdot \left( {{\mathbf{A}} \times {\mathbf{B}}} \right) $$

$$ {\mathbf{A}} \times \left( {{\mathbf{B}} \times {\mathbf{C}}} \right) = {\mathbf{B}}\left( {{\mathbf{A}} \cdot {\mathbf{C}}} \right) - {\mathbf{C}}\left( {{\mathbf{A}} \cdot {\mathbf{B}}} \right) $$

$$ \nabla \cdot \left( {\phi {\mathbf{A}}} \right) = {\mathbf{A}} \cdot \nabla \phi + \phi \nabla \cdot {\mathbf{A}} $$

$$ \nabla \times \left( {\phi {\mathbf{A}}} \right) = \nabla \phi \times {\mathbf{A}} + \phi \nabla \times {\mathbf{A}} $$

$$ \nabla \cdot \left( {{\mathbf{A}} \times {\mathbf{B}}} \right) = {\mathbf{B}} \cdot \left( {\nabla \times {\mathbf{A}}} \right) - {\mathbf{A}} \cdot \left( {\nabla \times {\mathbf{B}}} \right) $$

$$ \nabla \times \left( {{\mathbf{A}} \times {\mathbf{B}}} \right) = {\mathbf{A}}\left( {\nabla \cdot {\mathbf{B}}} \right) - {\mathbf{B}}\left( {\nabla \cdot {\mathbf{A}}} \right) + \left( {{\mathbf{B}} \cdot \nabla } \right){\mathbf{A}} - \left( {{\mathbf{A}} \cdot \nabla } \right){\mathbf{B}} $$

$$ \nabla \cdot \left( {{\mathbf{A}} \times {\mathbf{B}}} \right) = {\mathbf{B}} \cdot \left( {\nabla \cdot {\mathbf{A}}} \right) - {\mathbf{A}} \cdot \left( {\nabla \times {\mathbf{B}}} \right) $$

$$ \nabla \times \left( {\nabla \times {\mathbf{A}}} \right) = \nabla \left( {\nabla \cdot {\mathbf{A}}} \right) - \nabla^{2} {\mathbf{A}} $$

$$ \nabla \times \nabla \phi \equiv 0 $$

$$ \nabla \cdot \left( {\nabla \times {\mathbf{A}}} \right) \equiv 0 $$

2.3.2 1.2.2  Vector Integral

Here, dl, dS, and dV represent line, surface, and volume integrals, respectively. A represents a vector, and A_n indicates a component normal to S.

$$ \iint {\nabla \times {\mathbf{A}} \cdot d{\mathbf{S}}} = \oint {{\mathbf{A}} \cdot d{\mathbf{l}}} \quad \left( {{\text{Stokes}}\,{\text{theorem}}} \right) $$

$$ \iiint {\nabla \cdot {\mathbf{A}} dV} = \iint {{\mathbf{A}} \cdot d{\mathbf{S}}} = \iint {A_{{\text{n}}} \,dS}\quad \left( {{\text{Gauss}}\,{\text{theorem}}} \right) $$

2.3.3 1.2.3  Partial Differentiation in Cylindrical Geometry

Here, e_r, ${\mathbf{e}}_{\uptheta }$, and e_z represent unit vectors, A denotes a vector, and $\phi$ represents a scalar variable.

$$ \nabla \cdot {\mathbf{A}} = \frac{1}{r}\frac{\partial }{\partial r}(rA_{{\text{r}}} ) + \frac{1}{r}\frac{\partial }{\partial \theta }A_{{\uptheta }} + \frac{\partial }{\partial z}A_{{\text{z}}} $$

$$ \begin{aligned} \nabla \times {\mathbf{A}} & = \left( {\frac{1}{r}\frac{\partial }{\partial \theta }A_{{\text{z}}} - \frac{\partial }{\partial z}A_{{\uptheta }} } \right){\mathbf{e}}_{\text{r}} + \left( {\frac{{\partial A_{{\text{r}}} }}{\partial z} - \frac{{\partial A_{{\text{z}}} }}{\partial r}} \right){\mathbf{e}}_{\uptheta } \\ \end{aligned} $$

$$+ \left[ {\frac{1}{r}\frac{\partial }{\partial r}\left( {rA_{{\uptheta }} } \right) - \frac{1}{r}\frac{{\partial A_{{\text{r}}} }}{\partial \theta }} \right]{\mathbf{e}}_{\rm{z}} $$

$$ \nabla \phi = \frac{\partial \phi }{{\partial r}}{\mathbf{e}}_{\rm{r}} + \frac{1}{r}\frac{\partial \phi }{{\partial \theta }}{\mathbf{e}}_{\uptheta } + \frac{\partial \phi }{{\partial z}}{\mathbf{e}}_{\rm{z}} $$

$$ \nabla^{2} \phi = \frac{1}{r}\frac{\partial }{\partial r}\left( {r\frac{\partial \phi }{{\partial r}}} \right) + \frac{1}{{r^{2} }}\frac{{\partial^{2} \phi }}{{\partial \theta^{2} }} + \frac{{\partial^{2} \phi }}{{\partial z^{2} }} $$

2.3.4 1.2.4  Maxwell’s Equations

$$ \nabla \cdot {\varvec{D}} = \rho $$

$$ \nabla \cdot {\mathbf{B}} = 0 $$

$$ \nabla \times {\mathbf{E}} = - \frac{{\partial \,{\mathbf{B}}}}{\partial \,t} $$

$$ \nabla \times {\mathbf{H}} = {\mathbf{j}} + \frac{{\partial \,{\mathbf{D}}}}{\partial \,t} $$

2.3.5 1.2.5  Bessel Functions

Bessel’s differential equation

Here, z denotes a function of x, and m represents a real number.

$$ x^{2} \frac{{d^{2} z}}{{dx^{2} }} + x\frac{dz}{{dx}} + \left( {x^{2} - m^{2} } \right) = 0 $$

The solutions have the first and the second kind of Bessel functions, $J_{{\text{m}}} \left( x \right)$ and $Y_{{\text{m}}} \left( x \right)$, respectively.

Recursion relations

$$ \begin{aligned} & J_{{{\text{m}} - 1}} \left( x \right) + J_{{{\text{m}} + 1}} \left( x \right) = \frac{2 m}{x}J_{{\text{m}}} \left( x \right) \\ {}\\& J_{{{\text{m}} - 1}} \left( x \right) - J_{{{\text{m}} + 1}} \left( x \right) = 2\frac{{dJ_{{\text{m}}} \left( x \right)}}{dx} \\ \end{aligned} $$

Typical waveforms of J ₀ ( x ), J ₁ ( x ), and J ₂ ( x )

Roots $\lambda_{{{\mathbf{k}},{\mathbf{m}}}}$ of the first kind of Bessel function J _m ( x )