Fundamental Quantities

Before proceeding further, it is helpful to define a few fundamental quantities.

Of course, quasi-neutrality demands that [18]

$\displaystyle n_i \simeq n_e.$ (2.16)

We can estimate typical particle speeds in terms of the so-called thermal speed [18],

$\displaystyle v_{t\,s} = \left(\frac{2\,T_s}{m_s}\right)^{1/2}.$ (2.17)

The typical gyro-radius of a charged particle gyrating in the magnetic field of a tokamak is given by

$\displaystyle \rho_s = \frac{v_{t\,s}}{\vert{\mit\Omega}_s\vert},$ (2.18)


$\displaystyle {\mit\Omega}_s = \frac{e_s\,B}{m_s}$ (2.19)

is the gyro-frequency associated with the gyration [18]. (Note that ${\mit\Omega}_e<0$, indicating that electrons gyrate around magnetic field-lines in the opposite direction to ions.)

The electron-ion and ion-ion collision times are written

$\displaystyle \tau_{e}$ $\displaystyle = \frac{6\!\sqrt{2}\,\pi^{3/2}\,\epsilon_0^{\,2}\,\sqrt{m_e}\,T_e^{3/2}}
{e^4\, n_e\,\ln{\mit\Lambda}},$ (2.20)
$\displaystyle \tau_{i}$ $\displaystyle = \frac{ 12\,\pi^{3/2}\,\epsilon_0^{\,2}\,\sqrt{m_i}\,T_i^{3/2}}
{e^4\, n_e\,\ln{\mit\Lambda}},$ (2.21)

respectively [18]. Here, $\ln{\mit\Lambda}\simeq 16$ is the Coulomb logarithm [42]. Note that $\tau_{e}$ is the typical time required for the cumulative effect of electron-ion collisions to deviate the path of an electron through $90^\circ$. Likewise, $\tau_{i}$ is the typical time required for the cumulative effect of ion-ion collisions to deviate the path of an ion through $90^\circ$.

The electron and ion collision frequencies are simply the inverses of the corresponding $90^\circ$ collision times:

$\displaystyle \nu_e$ $\displaystyle = \frac{1}{\tau_e},$ (2.22)
$\displaystyle \nu_i$ $\displaystyle = \frac{1}{\tau_i}.$ (2.23)

Finally, the mean-free-paths between collisions (i.e., $90^\circ$ scattering events) for electrons and ions are

$\displaystyle l_e$ $\displaystyle = v_{t\,e}\,\tau_e,$ (2.24)
$\displaystyle l_i$ $\displaystyle = v_{t\,i}\,\tau_i,$ (2.25)


Table 2.1 gives estimates for some of the fundamental plasma parameters defined in this section in a low-field and a high-field tokamak fusion reactor. Here, use has been made of the data shown in Table 1.2. It has also been assumed that $T_i=T_e$, for the sake of simplicity.

Table: 2.1 Fundamental plasma parameters in a low-field and a high-field tokamak reactor. Here, $B$ is the toroidal magnetic field-strength, $R_0$ the plasma major radius, $a$ the plasma minor radius, $n_e$ the electron number density, $T_e$ the electron temperature, $T_i$ the ion temperature, ${\mit \Omega }_e$ the electron gyro-frequency, ${\mit \Omega }_i$ the ion gyro-frequency, $\rho _e$ the electron gyro-radius, $\rho _i$ the ion gyro-radius, $\tau _e$ the electron collision time, $\tau _i$ the ion collision time, $l_e$ the electron mean-free-path, and $l_i$ the ion mean-free-path.
  Low-Field High-Field
$B({\rm T})$ 5.0 12.0
$R_0({\rm m})$ 7.6 3.2
$a({\rm m})$ 2.5 1.1
$n_e(10^{20}\,{\rm m}^{-3})$ 0.89 5.1
$T_e({\rm keV})$ 7.0 7.0
$T_i({\rm keV})$ 7.0 7.0
$\vert{\mit\Omega}_e\vert({\rm THz})$ 0.88 2.1
${\mit\Omega}_i({\rm GHz})$ 0.19 0.46
$\rho_e({\rm\mu m})$ 56 24
$\rho_i({\rm mm})$ 3.8 1.6
$\tau_e({\rm ms})$ 0.14 0.025
$\tau_i({\rm ms})$ 13.6 2.36
$l_e({\rm km})$ 7.0 1.2
$l_i({\rm km})$ 10 1.7