Though a plasma is globally neutral, it may well acquire local charge variations which establish an electric potential and give rise to an electric field. The thermal motion of ions and electrons will therefore be influenced by the consequent force effects. An indication of the spatial extent of such an effect is represented by the Debye length. The following analysis yields a useful explicit expression for this important parameter.

Consider a dominant positive charge or electrode inserted in a plasma, Fig. 3.4. Due to the mobility of electrons, a negatively charged cloud will immediately form around this point with its density decreasing with distance. Similarly, a negative charge will also create a positive ion cloud spherically symmetric about it. Obviously, a plasma tends to shield itself from applied electric fields; that is, if an electrode is inserted into a plasma it will affect only its immediate surroundings. If the plasma were cold, one would observe as many charges in the surrounding cloud as are required to neutralize the inserted charge, Fig. 3.4. However, due to the finite plasma temperature, the plasma particles possess a substantial kinetic energy of thermal motion so that some-particularly those at the edge of the cloud-will escape from the shielding cloud by surmounting the electrostatic potential well which, as is known, decreases with increased distance r.

Evidently, we need to determine some characteristic shielding range and consider for that-over a small distance r from an inserted charge-some little
perturbation of the electric charge density in the plasma. Beyond this range, a uniform neutral plasma continues to reside. The local electric field E thus established is related to the local charge density pc by Maxwell’s First Equation

V’E = — .

£o

With this conservative field, a scalar potential function Ф is identified and related to the electric field by

E = — VO. (3.17)

By substitution in Eq.(3.16) and specializing for the case of interest, Fig. 3.4., Poisson’s Equation takes on the form

PC

£o which, upon introducing the definition

pc = iN і >

J=e, l

is written for a plasma containing only one species of ions as у 2^_4eNe +4iNi _

£0

in which qi and qe are the ion and electron charge, and N; and Ne are the local ion and electron densities, respectively. Determining the potential function Ф requires a knowledge of N; and Ne as functions of position or of Ф.

	————- jr———
*77777	77777777777777. N-“ ++ґ	77777777777//У//7/ + . Plasma Ч++

+ + + +

Fig. 3.4: Local charge variation in a plasma upon insertion of two dominant point charges.

The particle densities Nj, however, at thermodynamic equilibrium and in the presence of a potential energy Ф are known to depend upon Ф, the specific charge and the equilibrium temperature Tj according to the Boltzmann relation

with the factors Cj being determined from the evident affinity Ф—>0 as r—to represent the undisturbed background densities Nj(r—>°°). Equation (3.21a) may then be expanded by a Taylor series to give

This refers to the region where I qjO / kTj I « 1, which is actually the dominant contributor to the thickness of the shielding cloud. Retaining only the linear terms of the expansion and substituting into Eq. (3.20) yields

where we have used the charge-neutrality boundary condition 4eNe (r °°) + <hNi (r °°) = 0 •

The defining equation for the electrical potential function Ф, Eq. (3.22), is evidently of the form

, Ф У2Ф — — = 0 Xd

and for screening electrons by ions by

Хйі=

we realize the relation

(3.27)

/y*De Di

Taking Te = Tj = T and assuming the presence of singly-charged ions only, that is Nj(r—>°°) = Ne(r—>°o) = N/2, notably simplifies Eq. (3.25) to the expression

The important dependence of Xd is therefore

(3.28b)

with typical values of interest to thermonuclear fusion being in the range of about

1 pm to 1 mm.

The role of A*d may be interpreted as a "shielding length" parameter for a plasma and becomes evident by solving Eq.(3.22) for the boundary conditions

Ф(г = 0)= ф„ and Ф(г —>°°) = 0. (3.29)

which may be compared to the free-space potential given by

Thus, the potential Ф associated with the imposed electrostatic perturbation is attenuated in a plasma according to the magnitude of A, D and is commonly said to be shielded to the distance of the Debye length.

This A*d parameter is thus a useful concept and application of it relates to the elimination of the cross section singularity of Sec.3.2 as we will show next.