Distribution Parameters

It is most important to recognize that while the particles possess a range of velocities, speeds, and energies, the temperature T describes a particular distribution function and is a fixed parameter for a given thermal state; changing the temperature of the medium will alter the various moments of the function but its characteristic shape is retained, Fig.2.2.

Further, in a volume domain containing a mixture of particles-as in the case of a plasma containing electrons, various ion species, and neutrals-each particle species may possess a different distribution function characterized by a different temperature. Then, however, the entire plasma is not in thermodynamic equilibrium. Indeed, in the presence of a magnetic field, even the same species may have a different temperature in, say, the direction parallel to the magnetic field lines than in the perpendicular direction. Several methods or devices used to obtain fusion energy involve plasmas that are just that-not in thermodynamic equilibrium. Most that will be considered herein, however, are not so and thus we will rely on Maxwell-Boltzmann distributions to characterize many of the plasmas that will be discussed in subsequent chapters.

Having a sufficiently accurate distribution function is of considerable utility.

For example, the most probable value % — that is the peak of the distribution-is found by differentiating and finding the root of

Подпись: = 0.Подпись:

Подпись: and Подпись: dE image044

Э M(Z)

In Eq. (2.17a), the subscript x is to suggest any one component of the vector v; hence v = 0.

Average values can similarly be found based upon the formal definition of

_ f

£ = 2————— . (2.18)

J M(Z№

Thus, for the three cases of interest here we get

Подпись: v = J vM( v )d о image046 Подпись: (2.19b)

Ух= jv* M(y)dx = 0 (i. e. v = 0) (2.19a)


E = J EM(E)dE = jkT (2.19c)


with the particles possessing three degrees of freedom.

The analysis leading to the depictions of Fig. 2.2 makes it clear that the temperature T-here in units of degrees Kelvin, К-is an essential characterization of a Maxwellian distribution; hence, the numerical value of T uniquely specifies an equilibrium distribution. It has also become common practice to multiply T by the Boltzmann constant к and to call this product the kinetic temperature, which is obviously expressed in units of energy, either Joules (J) or electron volts (eV) with the latter generally preferred. Using this product kT, a Maxwellian population at T = 11,609 К may be said to possess a kinetic temperature of 1 eV; similarly, a 3 keV plasma in thermodynamic equilibrium has an absolute temperature of 3.48ХІ07 K.

The convention of interchangeably using energy and temperature, wherein the adjective "kinetic" and Boltzmann’s constant in kT are commonly suppressed, may seem peculiar, but expressing a physical variable in related units is a very common practice. For example, travelers often use time as a measure of distance (s = vt) if the speed of transport is understood, test pilots often speak of a force of so many g’s (F = mg), and physicists often quote rest masses in units of energy (E = me2).

This convention of using the product kT leads to a number of uses which need to be distinguished; we note here several common cases: kT = (kinetic) temperature of a plasma; f kT = average energy of Maxwellian-distributed particles;


kT = most frequently occurring particle energy of Maxwellian- distributed particles;

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