Fundamental Quantities

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

Of course, quasi-neutrality demands that [18]

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

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

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

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

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

where

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

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

$$\tau_{i} = \frac{ 12\,\pi^{3/2}\,\epsilon_0^{\,2}\,\sqrt{m_i}\,T_i^{3/2}} {e^4\, n_e\,\ln{\mit\Lambda}},$$ (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:

$$\nu_e = \frac{1}{\tau_e},$$ (2.22)

$$\nu_i = \frac{1}{\tau_i}.$$ (2.23)

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

$$l_e = v_{t\,e}\,\tau_e,$$ (2.24)

$$l_i = v_{t\,i}\,\tau_i,$$ (2.25)

respectively.

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