The energy partitioning of non-thermal particles in a plasma: or the Coulomb logarithm revisited
The charged particle stopping power in a highly ionized and weakly to moderately coupled plasma has been calculated exactly to leading and next-to-leading accuracy in the plasma density by Brown, Preston, and Singleton (BPS). Since the calculational techniques of BPS might be unfamiliar to some, and since the same methodology can also be used for other energy transport phenomena, we will review the main ideas behind the calculation. BPS used their stopping power calculation to derive a Fokker-Planck equation, also accurate to leading and next-to-leading orders, and we will also review this. We use this Fokker-Planck equation to compute the electron-ion energy partitioning of a charged particle traversing a plasma. The motivation for this application is ignition for inertial confinement fusion — more energy delivered to the ions means a better chance of ignition, and conversely. It is therefore important to calculate the fractional energy loss to electrons and ions as accurately as possible, as this could have implications for the Laser Mégajoule (LMJ) facility in France and the National Ignition Facility (NIF) in the United States. The traditional method by which one calculates the electron-ion energy splitting of a charged particle traversing a plasma involves integrating the stopping power . However, as the charged particle slows down and becomes thermalized into the background plasma, this method of calculating the electron-ion energy splitting breaks down. As a result, the method suffers a systematic error of order , where is the plasma temperature and is the initial energy of the charged particle. In the case of DT fusion, for example, this can lead to uncertainties as high as 10% or so. The formalism presented here is designed to account for the thermalization process, and in contrast, it provides results that are near-exact.
pacs:52.20.-j, 52.25.Dg, 52.57.-z
When a fast charged particle with initial energy traverses a plasma, it looses energy at a rate per unit of distance, and it comes into thermal equilibrium after depositing its energy into the electrons and ions that make up the plasma. Using the formalism of Brown, Preston, and Singleton (BPS) (1), which provides a means of regulating the kinetic equations at short and long distances in a consistent manner while treating quantum mechanical effects exactly, we shall calculate this energy splitting in a uniform plasma to leading and next-to-leading order in the plasma coupling. Our motivation is the thermonuclear burn of deuterium and tritium in inertial confined fusion experiments where a fast particle is born with an energy MeV. The more efficiently the ions are heated, the easier it will be to initiate the bootstrap heating process that triggers ignition and burn.
In this paper we shall always work with a plasma whose components have a single temperature (measured in energy units). We are currently generalizing our formalism to a plasma in which the electrons and ions are at different temperatures (2). The energy partitioning is usually computed within the context of a fast charged particle traversing the plasma until coming to a complete stop, in which case the energy partition into ions and electrons is given by
Here and are the stopping power contributions from the ions and electrons,
and thus . This is only an approximate description because the fast charged particle does not simply come to rest within the plasma, but rather it becomes thermalized at the temperature . One should not extend the integrals in Eqs. (1) and (2) down to zero, but rather a lower limit on the order of the thermal plasma energy, . Consequently the systematic error in the above calculation of and is of order , and as we shall see, the correct electron-ion energy partition relation reads
Before examining the energy splitting, we need to discuss the calculational framework within which it appears. As we shall see, the correct expression for the energy splitting arises from a Fokker-Planck equation derived in BPS; however, before jumping into details, it will be useful to briefly review some salient features of the stopping power calculation.
Ii The BPS Formalism
Calculating Coulomb energy exchange processes in a hot plasma is notoriously difficult because of the subtleties of the Coulomb interaction, which produce logarithmic divergences at both long and short distance scales. This problem was first spelled out and solved to leading order by Landau and then Spitzer in the context of electron-ion temperature equilibration, and later by Corman et al. for the charged particle stopping power (3); (4); (5). Since the divergences are only logarithmic, one introduces ad hoc short and long distance cutoffs and , and the rate of energy loss of some process (such as temperature equilibration or stopping power) can be cast in the form
The prefactor is easy to compute exactly. The
logarithmic term, conventionally called the Coulomb logarithm,
can only be approximated within the above scheme.
The long distance scale is set by the relevant Debye
screening length, while the short distance scale is
determined by either the Landau length or the thermal de Broglie wave
length (or some interpolation between them). As such, this method
suffers a systematic uncertainty in the argument of the Coulomb
Reference (6) discusses at length the manner by which one can expand thermodynamic quantities such as as a perturbation series in powers of a small parameter , the plasma coupling defined by
where is the Debye wave number. This is just the ratio of the potential energy of two electrons a Debye distance apart to the thermal kinetic energy of the plasma, and it is related to the usual plasma parameter by . Quantities expand in integer powers of , except for possible terms, and the rate of energy exchange for the stopping power takes the form
We have indicated the leading order (LO) and the next-to-leading order (NLO) terms in the -expansion. Here , with defined by , and therefore knowing the next-to-leading order coefficient is equivalent to knowing the exact coefficient under the logarithm. To get a feel for the numbers, at the center of the sun , and the error term in Eq. (7) is consequently small.
The problem with directly calculating and is that the kinetic equations diverge and must be regularized in the appropriate manner. Furthermore, to find the coefficient under the logarithm, this regularization procedure must preserve the delicate balance between the long and short distance physics. Indeed, the BPS calculation (1) includes both short distance physics and dynamic collective long distance physics, joined together exactly and unambiguously (and this is the rub), systematized by a power series expansion in the plasma coupling constant . The coefficients are also calculated to all orders in the dimensionless quantum two-body scattering parameter , thereby providing an exact interpolation between the extrme classical and quantum regimes.
The rigorous starting point is the BBGKY hierarchy (or its quantum generalization), which is finite and well defined and does not suffer from the aforementioned divergences. One must of course truncate this vast number of equations to something manageable, such as the Boltzmann or Lenard-Balescu equations, and it is this truncation process that renders the various three dimensional integrals divergent. However, as shown in Ref. (7), in spatial dimensions these divergences become simple poles of the form . In spatial dimensions the BBGKY hierarchy reduces to the Boltzmann equation (BE) to leading order in (the BE is finite and does not have the usual long distance divergence for ). Calculating the rate of energy loss using the -dimensional BE gives a result of the form
The “greater-than” superscript is to remind us that the calculation has been performed in dimensions . In a similar manner, to leading order in the BBGKY hierarchy reduces to the Lenard-Balescu equations (LBE) for (the LBE is finite and does not suffer from short distance divergences when ). A calculation of the energy rate with the LBE gives a form
Note that both rates are of order in three dimensions, and they both suffer from a divergent simple pole. The coefficients and can be expanded in powers of , with
The leading terms must be equal, . This arises from the calculation itself and is not imposed by hand, and it makes the short- and long-distance poles cancel, thereby giving a finite result.
Since the rates and were calculated in mutually exclusive dimensional regimes, one might think that they cannot be compared. However (and this is perhaps the most crucial step in the method, and certainly the most subtle), we can analytically continue the quantity to dimensional values , after which we can directly compare the rates (8) and (9) in a common dimension , and the limit may then be taken. Upon writing the -dependence of Eq. (9) as , when we see that the rate (9) is indeed higher order in than Eq. (8) since :
The individual pole-terms in Eqs. (8) and (11) will cancel giving a finite result when the leading and next-to-leading order terms are added. Summing terms (8) and (11), using the relation , and taking the limit gives
with . This is in agreement with Eq. (7). In this way, BPS has calculated the charged particle stopping power accurate to leading order and next-to-leading order in .
BPS also derived a Fokker-Planck equation accurate to leading and next-to-leading order in the plasma coupling (1). Denoting the phase space density for the dilute collection of charged particles by , this equation reads
where the sum runs over the plasma components , , the vector is the velocity of a particle with momentum , and the summation convention is used for repeated vector indices. The symmetric tensor has longitudinal and transverse components,
where and , and . We denote the sum of the ion components of the -coefficients by and the electron component by , with . Expressions for the can be found in BPS (1). With our conventions, the number and kinetic energy densities of the charged particles is given by
We can derive a relation between the stopping power and the -coefficients in the following manner. For a single particle at moving with velocity , the distribution function takes the form , and the Fokker-Planck equation gives the particle’s rate of energy loss as
Upon substituting the decomposition (14) for the scattering tensor and dropping the projectile subscript , the contribution from species appears as
As gets large () note that .
Iii Formulation of the Problem
With this background in hand, we turn now to the energy splitting problem. Rather than tracking an individual charged particle slowing down in the plasma, it is much simpler to examine a homogeneous and isotropic source of charged particles of a single energy . The distribution function will therefore depend only upon the energy and time, . Furthermore, the homogeneity and isotropy conditions will greatly simplify the form of the Fokker-Planck equation: only the transverse coefficients will enter the diffusion kernel on the right-hand side of Eq. (13), while the convective term on the left-hand side will of course vanish. Thus, we now consider the inhomogeneous Fokker-Planck equation with a source, which, after some algebra reads
We find (2) that the relevant solution is of the form
We proceed now to motivate the structure of the solution (20) and the nature of the time independent function . The particles will eventually thermalize to a Maxwell-Boltzmann distribution, and the first term of Eq. (20) merely represents the buildup and subsequent thermalization of the particles produced by the source. The normalization factor is chosen so that is the number density of the produced particles once they have thermalized into the Maxwell-Boltzmann distribution . The time-independent piece describes the steady state of nonthermal particles losing energy to the plasma, i.e. particles cascading down “energy bins” from the initial energy to the final thermal energy. The situation described here can be pictured as the flow of water over a rocky waterfall that slows the motion of the water as it descends. The initial rate of flow of the river corresponds to the rate of produces particles; the height of the waterfall to the initial energy . The energy dissipated in the fall corresponds to the energy lost to the ions and electrons and is determined by . The final flow into a horizontal lake corresponds to the build up of the particles into their final thermal equilibrium state. This is illustrated in Fig. 1.
We now specialize to the relevant case in which the source slowly turns on and attains a constant value. In this case, particles become produced at a constant rate per unit volume , and from Eq. (16) the rate of change in energy density becomes
The fraction of energy lost to the ions and electrons is now identified as (2)
In the steady state, the energy density build up of final particles is their thermal energy per particle times the increase in number density, . And so, in view of Eqs. (22) and (23), the energy balance expression (21) now appears as
which gives expression (4): the original energy of a produced particle is lost to the ions and electrons with the remainder being the thermal energy of a free particle.
where is the error function. Since for large energies, and the error function approaches unity, at high energies we see that Eqs. (25) and (26) approach the same form as the more intuitive but less accurate results (1) and (2). The primary differences occur for .
Let us compare these near-exact results with the less precise but well known result of Fraley et al. (FR) (12). Starting with a phenomenological model of the stopping power, these authors show that the simple rule
provided a good fit to their results. Figure 2 shows the fraction energy loss to ions for BPS and FR for a DT plasma with electron number density . We see that FR somewhat underestimates the energy deposited to ions for temperatures up to around 100 keV, and FR slightly over estimates for larger energies. In Fig. 3 we compare the percent difference between BPS and FR over a wide range of densities.
- preprint: LA-UR-08-3394
- The constant under the logarithm sometimes varies by an order of magnitude from paper to paper within the literature, depending upon the choices of and .
- L. S. Brown, D. L. Preston, and R. L. Singleton Jr., Phys. Rep. 410 (2005) 237-333, arXiv:physics/0501084.
- L. S. Brown, D. L. Preston, and R. L. Singleton Jr., LA-UR-07-8096.
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- L. S. Brown, Phys. Rev. D 62 (2000) 045026, arXiv:physics/9911056.
- R.L. Singleton Jr., BPS Explained I: Temperature Relaxation in a Plasma, arXiv: 0706.2680; BPS Explained II: Calculating the Equilibration Rate in the Extreme Quantum Limit, arXiv: 0712.0639.
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