T. E. Sheridan and J. Goree

Department of Physics and Astronomy

The University of Iowa

Iowa City, Iowa 52242

We develop a model to predict the electric potential and ion energy in a plasma sheath, taking ion-neutral collisions into account. We improve on existing collisional sheath models in two ways. Firstly, our numerical and analytic results are valid not only in the collisionless and highly collisional (mobility-limited) regimes, but also in the transitional regime between them. Secondly, we allow a general power-law scaling for the energy dependence of the collision cross section. We derive useful analytic expressions for the sheath width, the ion energy at impact at the wall and the collision rate at which the transition takes place.

I. INTRODUCTION

When a plasma is in contact with an electrode or wall, a localized electric field appears near the wall in order to confine the electrons and expel the ions from the plasma. This forms a layer, called a sheath, where the electric charge is not neutral.^{[Chen, the book]} For many applications, such as plasma-surface interactions, one needs to know the average energy Eimpact of the ions when they strike the wall, and the electric potential [[phi]] as a function of position. The electric potential is parameterized by a sheath thickness, d.

For the simple case of a collisionless, planar, source-free sheath in a unmagnetized plasma, one can solve Poisson's equation and the ion equation of motion simultaneously to obtain the spatial dependence of the electric potential. This is done either numerically to yield an exact result^{[Sheridan and Goree IEEE 1989]}, or approximately to yield Child's law,^{[Child 1911]} [[phi]] [[proportional]] x^4/3. Child's law assumes that ions fall freely under the influence of the electric field, which is true if ion-neutral collisions are negligible.

On the other hand, if ion-neutral collisions are very strong, the ion motion becomes mobility-limited, d and Eimpact become smaller, and the electric potential adopts a different spatial dependence. Many authors have developed two special cases of this mobility-limited regime: when the cross section is independent of energy, the spatial dependence is [[phi]] [[proportional]] x^5/3,^{[ref. ???]} and when the mobility is independent of energy, [[phi]] [[proportional]] x^3/2.^{[ref. ???]}

In this paper, we improve on the existing literature by: (1) investigating the transitional regime between the collisionless and mobility-limited regimes, and (2) allowing a more general energy dependence in the cross section. We do this for a planar source-free unmagnetized sheath, with a collision rate that is parameterized by [[alpha]], the number of collisions experienced by an ion while traveling a distance of one Debye length.

In Sec. II B, numerical calculations of the sheath thickness d and energy at impact Eimpact are described. These useful results reveal the value of [[alpha]] where the transitional regime is centered. We find that the transition is centered on different values, [[alpha]]td and [[alpha]]tE, for collisional effects to appear in d and Eimpact, respectively.

These numerical results motivate the rest of the paper. In Sec. III we derive analytic expressions for the transitional values, [[alpha]]td and [[alpha]]tE. These are useful for determining the regime to which a given discharge belongs. For the reader's convenience in making practical application of our results, we summarize them in fully dimensional units in Sec. IV. In the Appendix, we derive an analytic expression, valid in an almost-collisionless sheath, for the ion energy lost in the sheath due to collisions.

II. STATEMENT OF THE GENERALIZED SHEATH PROBLEM

A. Governing equations

We consider a planar sheath in an unmagnetized two-component plasma consisting of positive ions and negative electrons, as sketched in Fig. 1. We adopt a spatial coordinate x that is measured from the plasma sheath interface toward the wall.

The governing equations are based on a two-fluid model. We assume that the electrons are thermalized so that they obey the Boltzmann relation,

ne = no exp [e [[phi]](x) / kBTe] , (eq. Boltzmann a)

where no and Te are the electron density and temperature measured in the neutral plasma outside the sheath. The electric potential [[phi]](x) must satisfy Poisson's equation,

[[gradient]]²[[phi]] = e (ne - ni ) / [[epsilon]]o , (eq. Poisson a)

where ni(x) is the ion density. The ions obey the steady-state fluid equations of motion,

ni (vi ^.[[gradient]])vi = - (ni / Mi) e[[gradient]][[phi]] - Fc , (eq. motion a)

and continuity,

[[gradient]] ^. (ni vi) = 0 , (eq. continuity a)

where the velocity and mass of the ions are given by vi(x) and Mi. As the ion fluid moves through the sheath, it experiences a collisional drag force Fc which is given by

Fc = ni vi [[nu]]

= ni vi nn [[sigma]] vi , (eq. motion)

where nn is the neutral gas density and [[sigma]] is the momentum transfer cross section for collisions between ions and neutrals. Two kinds of collisions, elastic and charge exchange, contribute to this cross section, which depends on the ion velocity, vi.

We will assume that the cross section has a power law dependence of the form

[[sigma]](vi) = [[sigma]]s (vi / cs)^[[gamma]], (eq. power law)

where cs = (kBTe / Mi)^1/2 is the ion-acoustic speed and [[sigma]]s is the cross section measured at that speed. The exponent [[gamma]] will in general lie in the range -1 <= [[gamma]] <= 0. Our use of this general power-law scaling is an improvement over the existing literature, where two special cases are generally treated: [[gamma]] = 0, the case of constant cross section, and [[gamma]] = -1, the case of constant collisional drag or mobility. The [[gamma]] = 0 case is generally the more accurate of the two. For example, it can be shown that for argon in the energy range of 3 to 300 eV, the power law [Eq. (eq. power law)] is most closely satisfied for [[gamma]] ~ -0.25. We give most of the following results three ways, first for an arbitrary value of [[gamma]], and then for [[gamma]] = 0 and [[gamma]] = -1.

B. Non-dimensional variables

We adopt a set of non-dimensional variables that is popular for sheath problems. ^{[Chen, the book]} We will summarize in Section IV all our major results, rewritten in practical dimensional variables.

All the relevant physical parameters are made non-dimensional by normalizing them. The electric potential [[phi]] is normalized by the electron temperature,

[[eta]] [[equivalence]] - e[[phi]] / kBTe , (eq. norm potential)

the distance x is normalized by the Debye length [[lambda]]D = ([[epsilon]]okBTe / noe²)^1/2,

[[xi]] [[equivalence]] x / [[lambda]]D , (eq. norm potential)

and the velocity vi is normalized by the acoustic speed,

u [[equivalence]] vi / cs . (eq. norm potential)

The ion kinetic energy E = [[Mu]]ivi² / 2 is normalized by the electron temperature,

[[epsilon]] [[equivalence]] E /(kBTe) ,

= u² / 2 . (eq. norm energy)

Since the velocity of the ions at the wall is uw, they strike the surface with an energy

[[epsilon]]impact [[equivalence]] uw² / 2 . (eq. def Eimpact)

The sheath thickness is denoted by the symbol d in this non-dimensional system.

In this paper we introduce collisions into the sheath problem. The fractional energy lost in the sheath due to collisions is

[[Delta]][[epsilon]] / [[epsilon]] [[equivalence]] ([[epsilon]]max - [[epsilon]]impact) / [[epsilon]]max , (eq. def frac energy loss)

where

[[epsilon]]max [[equivalence]] uo² / 2 + [[eta]]w (eq. emax)

is the value of [[epsilon]]impact if the ions lose no energy at all. To find expressions for [[Delta]][[epsilon]] / [[epsilon]] and for the sheath thickness d, we will perform derivations making use of the dimensionless collision rate, defined to be the number of collisions per Debye length,

[[alpha]] [[equivalence]] [[lambda]]D / [[lambda]]mfp , (eq. norm potential)

where [[lambda]]mfp = 1 / (nn [[sigma]]) is the mean-free-path. Note that [[alpha]] is proportional to the neutral gas density, nn. The collisionless case, [[alpha]] = 0, is the limit of zero gas density. If the gas density is high enough for the ions experience one collision per Debye length, then the collision rate is [[alpha]] = 1.

The governing equations, expressed in non-dimensional variables, are:

u u' = [[eta]]' - [[alpha]] u ^{(2+[[gamma]])} (eq. motion b)

and

[[eta]]'' = uo / u - e^-[[eta]] , (eq. Poisson b)

where the prime denotes differentiation with respect to the spatial coordinate, [[xi]].

To solve these equations, one needs the boundary conditions. At the wall, which is located at [[xi]] = d, the boundary condition is [[eta]](d) = [[eta]]w. At the plasma sheath boundary, located at [[xi]] = 0, the boundary conditions are [[eta]](0) = 0 and u(0) = uo, while the electric field [[eta]]' has a small positive value there.

C. Numerical solutions

We have solved the governing equations (eq. motion b) - (eq. Poisson b) exactly for the electric potential [[eta]]([[xi]]) and ion velocity u([[xi]]) by integrating them numerically with a Runge-Kutta^{[C.E. Roberts]} routine. The electric potential profile, [[eta]]([[xi]]), is characterized by its thickness, d. In Figures 2 and 3 we plot d and [[epsilon]]impact as functions of the collision rate, [[alpha]]. These plots demonstrate the three regimes of sheath collisionality. At low [[alpha]] the ions are collisionless and both d and [[epsilon]]impact are independent of [[alpha]], while at large [[alpha]] the sheath is the ions are mobility-limited. In between there is a transitional regime. For the collisionless and mobility-limited regimes, analytic expressions for d and [[epsilon]]impact have long been available. What makes our numerical results particularly valuable is their accuracy in the transitional regime.

The transitional regime is centered at a threshold value that is different depending on whether one examines the sheath thickness (Fig. 2) or the ion energy (Fig. 3). Accordingly, we introduce the notation [[alpha]]td and [[alpha]]tE to distinguish the thresholds for the sheath thickness d and ion energy [[epsilon]], respectively.

These threshold values decrease with the wall potential, [[eta]]w. In the next section we derive analytic expressions for this dependence of [[alpha]]td and [[alpha]]tE on [[eta]]w. We also find expressions for the fractional energy lost by ions in the sheath due to collisions, [[Delta]][[epsilon]] / [[epsilon]].

III. REGIMES OF SHEATH COLLISIONALITY

A. Collisionless regime

In the limit of no ion-neutral collisions, [[alpha]] = 0, one arrives at Child's Law, which we review here briefly. The equation of motion [Eq. (eq. motion b)] can be integrated once to yield a statement of the conservation of ion energy,

[[epsilon]] = u² / 2 = uo² / 2 + [[eta]], (eq. energy childs)

where uo is the initial velocity of the ions as they enter the sheath. Collisionless ions do not lose any energy as they fall through the sheath, so that

[[epsilon]]impact = uw² / 2 = uo² / 2 + [[eta]]w (eq. energy impact childs)

and

[[Delta]][[epsilon]] / [[epsilon]] = 0 . (eq. energy impact childs)

We improve on the latter result in the Appendix, where we find that when there is a small degree of collisionality, the fractional energy loss (for [[gamma]] = 0) is

[[Delta]][[epsilon]] / [[epsilon]] ~ (6/7) [[alpha]] d . (eq. almost coll frac en loss)

Inside the sheath, i.e., for 0 <= [[xi]] <= d, one can seek a power law scaling for the spatial dependence of the electric potential:

[[eta]] = a [[xi]]^b . (eq. power law scaling)

When the potential drop across the sheath is large, [[eta]]w [[equivalence]] - e[[phi]]w / kBTe >> 1, one can make two approximations. First, the electron term e^-[[eta]] in Poisson's equation [Eq. (eq. Poisson b)] can be neglected. Second, one can neglect the uo² / 2 term in comparison to the [[eta]] term in the conservation of energy [Eq. (eq. energy childs)]. To see how this latter simplification is justified, recall that the Bohm criterion^{[Chen, the book]}

uo >= 1 (eq. bohm criterion)

must be satisfied for the collisionless sheath. Generally, uo is only slightly larger than one and the uo² / 2 term is therefore small compared to [[eta]]w.

Making these two approximations yields Child's law:

. (eq. Child's law)

By evaluating Eq. (eq. Child's law) at the wall, where [[eta]] = [[eta]]w and [[xi]] = d, we obtain the sheath thickness:

d = 2^5/4 [[eta]]w^3/4 / (3uo^1/2) . (eq. collisionless sheath thick)

Equations (eq. bohm criterion) - (eq. collisionless sheath thick) are the familiar Child's law results. In another paper^{[Sheridan and Goree IEEE 1989]} we presented more accurate analytic expressions for the collisionless sheath, but in the present work we will use the Child's law results for simplicity.

B. Mobility-limited regime

In the limit of strong ion-neutral collisions, i.e., for the case of mobility-limited ion motion, the collision rate [[alpha]] is large. The equation of motion [eq. motion b] is simplified under these circumstances by neglecting the convective term on the left-hand side. The resulting equation is

u^2+[[gamma]] = [[eta]]' / [[alpha]]. (eq. vel in mobil lim sheath)

By inserting Eq. (eq. vel in mobil lim sheath) into Poisson's equation [Eq. (eq. Poisson b)], and neglecting the electron term e^-[[eta]], we again arrive at a power law solution of the form [Eq. (eq. power law scaling)]. The solution has an exponent

b = (5 + 2[[gamma]]) / (3 + [[gamma]]) , (eq. mobil lim exponent)

which is different from the value of 4/3 for the collisionless regime, and a coefficient:

. (mobil lim coeff)

The collision rate [[alpha]] appears explicitly in the coefficient, but not in the exponent.

One can also write an expression for the sheath thickness d by requiring that [[eta]] =[[eta]]w at the wall, [[xi]] = d. The result is:

(eq. mobil lim sh thick)

The reason that d decreases with the collision rate [[alpha]] can be understood by considering the governing equations. The frictional drag force reduces the ion velocity in the collisional sheath. In order to satisfy the conservation of particle flux [Eq. (eq. cont)], this smaller ion velocity requires an increase in the ion density. Through Poisson's equation [Eq. (eq. Poisson a,b)], this increase in the ion density leads to a stronger gradient in the electric field, [[gradient]]²[[phi]]. Having a larger gradient in the sheath means having a smaller scale length, i.e., a smaller sheath thickness, d. This reduced value of d for large [[alpha]] is revealed in Fig. 2.

The electric potential [[eta]] varies not only with [[xi]] and [[alpha]], but also with the energy dependence of the cross section, characterized by [[gamma]]. For the special case of constant cross section, [[gamma]] = 0, the results given above for the electric potential and the sheath thickness simplify to:

and

^{. (eq. [[gamma]] = 0 sheath thick)}

For the constant mobility case, [[gamma]] = -1 they simplify to:

and

^{. (eq. [[gamma]] = -1 sheath thick)}

We now find the ion energy at impact on the wall,

. (eq. energy mobil limited a)

Evaluating [[eta]]'w using Eq. (eq. power law scaling) and Eq. (eq. mobil lim exponent) - (mobil lim coeff) we obtain

. (eq. energy mobil limited b)

Note that the impact energy increases with the sheath potential [[eta]]w , but not linearly as it does for the collisionless sheath [Eq. (eq. energy impact childs)].

C. Transition between regimes

Between the collisionless and mobility-limited regimes discussed above there is a transition. One can always use the exact numerical solutions for the sheath thickness and ion energy presented in Sec. IIC for this transitional regime. Of course it is often more convenient to use the approximate analytic expressions derived above for the collisionless and mobility-limited regimes. The accurate use of these analytic expressions, however, requires that one confirm that the collision rate lies in a range where they are valid. To the best of our knowledge, there is no practical reference for the reader to find where this transition takes place. One of the principal aims of this paper is to supply this information.

The exact numerical solutions of the governing equations revealed that the transition takes place at different values of the collision rate, [[alpha]]td and [[alpha]]tE, for collisional effects to appear in the sheath thickness d and in the average ion energy [[epsilon]]impact at impact on the wall, respectively. In this section we find analytic expressions for [[alpha]]td and [[alpha]]tE .

1. Sheath thickness

Examining Fig. 2, it is clear that the sheath thickness d approaches different asymptotes for small and large values of [[alpha]]. These asymptotes, which we have sketched in Fig. 4, correspond to the collisionless and mobility-limited regimes discussed above. In the collisionless regime of small [[alpha]], the sheath thickness d is given by Eq. (eq. collisionless sheath thick). Because this expression is independent of [[alpha]], the corresponding asymptote in Fig. 4 is horizontal. In the mobility-limited regime of large [[alpha]] and high collisionality, d is given by Eq. (eq. mobil lim sh thick). The asymptote in this case slopes downward.

Between the two asymptotic regimes lies the transitional regime which is centered at the point where the two asymptotes cross. We define the transitional value, [[alpha]]td to be this point. Accordingly, [[alpha]]td can be found by equating expressions for the two asymptotes, Eqs. (eq. collisionless sheath thick) and (eq. mobil lim sh thick), and solving for [[alpha]]. For an arbitrary value of [[gamma]], this yields

^. (eq. alpha td)

For the constant cross section case, [[gamma]] = 0, this simplifies to [[alpha]]td = (125 / 2^17/4) uo^1/2 [[eta]]w^-3/4. Note that this has the same scaling on the wall potential as Child's law [Eq. (eq. Child's law)].

In Fig. 5 we have plotted [[alpha]]td as a function of [[eta]]w, using Eq. (eq. mobil lim sh thick). It shows that [[alpha]]td ~ 1, with the exact value depending on the wall potential [[eta]]w and the cross section scaling [[gamma]]. This means that about one collision per Debye length is required for collisions to manifest themselves by making the sheath thinner. This collision rate requires a fairly high neutral gas pressure.

In contrast to this result, we find in the next section that the ion energy at impact on the wall undergoes its transition at a much lower collision rate.

2. Ion energy at impact on the wall Here we perform the same procedure as above to find the transitional value [[alpha]]t, except that we examine the transition between the collisionless and mobility-limited regimes based on ion energy rather than on sheath thickness. Examining Fig. 3, it is clear that the ion energy on impact at the wall, [[epsilon]]impact, approaches two different asymptotes for small and large values of [[alpha]]. As before, we define the transitional value [[alpha]]tE to be the point at which the asymptotes cross.

The collisionless asymptote is given by energy conservation, Eq. (eq. energy impact childs), while the asymptote for the mobility-limited ion regime is given by Eq. (eq. energy mobil limited b). Equating the right-hand sides of these two results to find where the asymptotes cross gives

(eq. alpha tE a)

We can simplify Eq. (eq. alpha tE a) by neglecting uo² compared to 2[[eta]]w, provided that [[eta]]w >> 1, to yield

. (eq. alpha tE b)

For the constant cross section case, [[gamma]] = 0, this simplifies to These results for the threshold value of [[alpha]], based on ion energy, are the same as those based on sheath thickness [Eq. (eq. alpha td)] except that the leading numerical coefficient in brackets is smaller here.

IV. SUMMARY OF RESULTS IN PRACTICAL UNITS

All of our derivations in the preceding sections were done in a system of non-dimensional units. For the reader's convenience in making practical application of our results, we summarize our results here in fully dimensional SI units.

list

V. CONCLUSIONS

ACKNOWLEDGMENT

This work was supported by the Iowa Department of Economic Development.

APPENDIX

While the ion energy lost in the sheath can be computed numerically, it is sometimes more convenient to have a ready analytic expression. In Section III, we presented a result for [[Delta]][[epsilon]]/[[epsilon]] in the highly collisional regime, [[alpha]] >> [[alpha]]td [Eq. (eq. frac energy lost mobil)]. When collisions are neglected entirely, the ions do not lose any energy and [[Delta]][[epsilon]]/[[epsilon]] = 0. In this Appendix we improve on the latter result by deriving an analytic expression [Eq. (Aeq. almost coll frac en loss)] that is valid for the almost-collisionless sheath, i..e., in a sheath characterized by [[alpha]] < [[alpha]]tE and [[Delta]][[epsilon]] / [[epsilon]] << 1. For simplicity, we treat only the special case of constant collision cross section, [[gamma]] = 0.

Our approach here exploits the fact that [[alpha]] << [[alpha]]td in the almost collisionless sheath. Accordingly, neither the sheath thickness d nor the electric potential profile [[eta]]([[xi]]) is altered by collisions. This means that we can find the energy by integrating the equation of motion [Eq. (eq. motion b)] while using the electric potential predicted by the collisionless derivation of Child's law [Eq (eq. Child's law)].

Our objective here is to find [[epsilon]]impact so that we can insert it in Eq. (eq. def frac energy loss) to find the fractional energy lost [[Delta]][[epsilon]]/[[epsilon]]. To evaluate uw, we integrate the equation of motion [Eq. (eq. motion b)] from the edge of the sheath ([[xi]] = 0) to the wall ([[xi]] = d):

, (Aeq. step 1a)

so that

. (Aeq. step 1b)

where we have treated the special case [[gamma]] = 0. We insert this result into the definition of [[epsilon]]impact to find

, (Aeq. step 1c)

Now we make the critical approximation that the ion velocity u appearing in the integrand in Eq. (Aeq. step 1) is given by the collisionless velocity, which satisfies

[[epsilon]] = u² / 2 = uo² / 2 + [[eta]] . (Aeq. energy childs b)

This approximation limits the validity of the following results to the case where [[Delta]][[epsilon]] / [[epsilon]] << 1, which we call the almost collisionless sheath.

Doing this, we obtain

. (Aeq. step 2)

Performing the integral in Eq. (Aeq. step 2), and making use of Eq. (eq. collisionless sheath thick) for the collisionless sheath thickness d, we obtain

. (Aeq. almost coll im en)

Inserting this in Eq. (eq. def frac energy loss), which defines the fractional energy lost [[Delta]][[epsilon]] / [[epsilon]], yields:

. (Aeq. almost coll frac en loss a)

Note that the energy loss is proportional to [[alpha]]d, the product of the collision rate and the sheath thickness. Of course [[Delta]][[epsilon]] / [[epsilon]] must be less than one, so the above result is accurate only for [[alpha]] d << 1. This limit on the validity of Eq. (Aeq. almost coll frac en loss) comes about because we are treating the case of the almost-collisionless sheath.

We can simplify Eq. (Aeq. almost coll frac en loss a) by neglecting the term uo² / 2 compared to the [[eta]]w term in both the numerator and the denominator. This is a valid approximation when the potential drop across the sheath is large, [[eta]]w >> 1, as we explained earlier in deriving Eq. (eq. alpha tE b). Dropping the term uo² / 2, we obtain

. (Aeq. almost coll frac en loss b)

Recall that [[alpha]] is the number of collisions per Debye length and that d is the sheath width normalized by the Debye length, so that the product [[alpha]] d is the average number of collisions in one sheath width.

The principal results of this Appendix are that the ion energy loss is parameterized by the average number of collisions in the sheath, [[alpha]]d, and that the average fractional energy lost by ions is [[Delta]][[epsilon]] / [[epsilon]] ~ [[alpha]]d. These results apply only to the almost-collisionless sheath, where [[alpha]]d << 1 and [[Delta]][[epsilon]] / [[epsilon]] << 1. They are not valid in the mobility-limited regime where [[alpha]] > [[alpha]]tE ; in that regime one should use instead Eq. (eq. energy mobil limited a).

References

[child]. C. D. Child, "Discharge from Hot CaO," Phys, Rev., 32, 492 (1911).

[Chen, the book] Francis F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1, 2nd ed. (Plenum, New York, 1984) pp. 290-295.

[Sheridan & Goree IEEE] T. E. Sheridan and J. Goree, "Analytic Expression for the Electric Potential in the Plasma Sheath," IEEE Trans. Plasma Sci., in press 1990.

[Roberts] C. E. Roberts, Jr., Ordinary Differential Equations: A Computational Approach, (Prentice-Hall, Englewood Cliffs, 1979) pp. 111-116.

FIGURE CAPTIONS

Fig. 1. Model system for the plasma sheath. The potential, [[phi]], in the plasma sheath is sketched as a function of distance from the wall. Ions enter the sheath from the left as a mono-energetic beam with a velocity uo. Dimensionless quantities are shown to the right of their dimensional counterparts.

Fig. 2. Sheath thickness d as a function of the collision rate [[alpha]]. Here, [[alpha]] =[[lambda]]D/[[lambda]]mfp represents the number of collisions within one Debye length, [[lambda]]D, where [[lambda]]mfp is the mean-free-path. The results shown here and in Fig. 3 are exact numerical solutions of the governing equations, [Eq. (eq. motion b) - (eq. Poisson b)], assuming that the collision cross section is independent of energy, i.e, [[gamma]] = 0. Note that the sheath thickness is independent of collisionality below a value [[alpha]]t, and that it becomes thinner at higher collisionalities. The straight line is the prediction from the analytic theory derived in Section III C 1.[first fig entitled "comparison with exact solution" from TES seminar. in caption indicate that: To find d numerically, we define it to be the distance from the wall or wall to the point where the electric potential is -kBTe / e, i.e., where [[xi]] = 1]

Fig. 3. Average energy of ions at impact on the wall, [[epsilon]]impact, as a function of the collision rate [[alpha]]. Note that the ion energy is independent of collisionality below a value [[alpha]]tE, and that it becomes smaller at higher collisionalities. The straight line is the prediction from the analytic theory derived in Section III C 2. [second fig entitled "comparison with exact solution" from TES seminar.]

Fig. 4. Sketch of asymptotes in Fig. 2. We use the point where the asymptotes cross to define the transitional value [[alpha]]td between the collisionless and collisional regimes.

Fig. 5. Transitional value of collision rate. The transitional values of the collision rate, [[alpha]]td and [[alpha]]tE, are shown in panels (a) and (b), respecitively. These were determined from the point where the asymptotes cross, as sketched in Fig. 4. In both panels the two curves shown are plots of Eqs. (eq. alpha td) and (eq. alpha tE b) for the special cases [[gamma]] = 0 and [[gamma]] = -1. Here we have assumed that uo = 1.