I–V analysis for nuclear-excited low-temperature plasmas in porous metals

We demonstrate how a high surface-to-volume ratio porous electrode structure within an ionizing radiation field generates a nuclear-excited low-temperature plasma source. In a foundational experiment, we compared the I–V characteristics of two electrode systems comprised of: (1) open-cell reticulated copper foam discs (pore radius ∼500 µm) and (2) solid copper discs. Both systems were held at 2 atm neon and irradiated in a nuclear reactor under steady-state operation. The primary source of plasma ionization was electrons derived from reactor γ-rays; the secondary source was electrons from the β − decay of 64Cu and 66Cu formed by the capture of thermal neutrons. The two electrode systems exhibited identical I–V proportionality for applied voltages between 2 V and 5 V, evidencing a sheath structure evolving within the copper foam discs. The energy fraction absorbed in the gas per unit electrode mass was 20%–70% greater for the porous electrode than for the solid electrode, corresponding to a factor of 3.5 increase in the specific ion output current [A g−1]. The plasma densities achieved in the porous and solid electrode systems were estimated to be (3.2−3.9)×109  cm−3 and 7.1×109  cm−3 respectively, assuming a spatially independent density profile and approximately ambient electron temperature.


Introduction
Nuclear-excited low-temperature plasmas (NLTPs) occur in gaseous media when they absorb ionizing radiation from nuclear reactions at sufficient power densities.Typically, an NLTP system consists of (1) a solid component in which charged particle-producing nuclear reactions occur and (2) a gaseous component that absorbs energy from the charged particles escaping the solid component.To date, only simple and relatively low specific surface area A [cm 2 g −1 ] solid-gas interfaces have been considered for these systems, such as solid parallel plate and annular configurations (e.g.[1,2]).Initial NLTP research pursuits beginning in the 1960s considered broad applications to direct energy conversion (DEC)-based nuclear power [3] and nuclear pumped lasers (NPLs) [4]; the only practical device that resulted was the ionization chamberbased neutron sensor-a crucial but rather niche use case [5,6].By the 1990s, NLTP research was essentially abandoned (outside Russia) [7][8][9].
In this work we studied a more complex surface-to-gas configuration consisting of an open-cell reticulated metal foam electrode.The small (∼1 mm), randomized pores in electrically conductive foams strongly attenuate low-energy electrostatic and electromagnetic fields within a few pore depths from its surface.Thus, when using electrostatic or electromagnetic LTP ionization sources, it would be counterintuitive to envisage decisive advantages for using porous electrodes: the LTP size within the substrate would be limited to a few pore depths from its surface.In contrast, high-energy nuclear radiation (e.g.neutrons and γ-photons) can penetrate several centimeters into solid media -far beyond the pore depth of a conductive foam.This nuclear radiation drives high energy charged particle-producing reactions within the substrate; exit products with sufficient energy escape the substrate and ionize the gaseous media within the porous electrode.Thus, provided a sufficient power density, NLTPs can form within metal foam electrodes at substrate size scales that would otherwise be inaccessible by non-nuclear ionization sources.Therefore, using an open-cell reticulated metal foam to increase an electrode's A may lead to an increased specific ion current I i [A g −1 ] extraction through an NLTP produced within it.This concept is summarized diagrammatically in figure 1.
NLTPs produced in high A electrodes could enhance existing technologies and revive dormant ones.Numerous applications for NLTPs in porous electrodes to ion-current driven devices include but are not limited to surface modification, etching, biomedical, and other industrial and manufacturing processes.The proportionality for the specific ion current is [1]: where S + [cm −3 s −1 ] is the ionization rate and µ + [cm 2 V −1 s −1 ] ion mobility (both assumed constant).
The most compelling immediate use however may be thermionic energy conversion devices for compact nuclear energy sources.For electron current-driven devices, e.g.separately excited plasma diode-type thermionic energy converters (PTECs) [10][11][12], the electrical power output is proportional to the random electron current I e [A g −1 ] at the emitter and collector which enables the transport and subsequent power conversion of thermionic electrons, i.e.
where n e [cm −3 ] is the electron density and T e [K] is electron temperature-both are assumed to be constant between cathode (emitter) and anode (collector).
For PTECs, we note that the maximum power output is also dependent on both the electrode contact potential ∆ϕ [eV] and plasma potential V p [eV].However, where ∆ϕ and V p might range from zero to several eV in a given irradiated environment, the A and consequently I e for porous electrodes under the same conditions may be several orders of magnitude greater than that of conventional electrodes used in previous NLTP configurations.This latter use case motivates our study.
Here, we demonstrate the basic experimental and modeling and simulation (M&S) methods for determining the expected properties of NLTPs within porous electrodes.Using a floating electrode configuration, we measured the specific currentvoltage (I-V) characteristic of two electrode types, consisting of (1) solid copper and (2) open-cell reticulated copper foam electrodes (see figures 2 and 3).Copper is a suitable material for experimental purposes because of its responsiveness to both thermal neutrons and γs, ease of radioactive waste disposal, and availability and manufacturability in both solid and porous forms.It was used in this experiment to demonstrate the basic physical concepts we propose in a reproducible fashion that is both simple and economical.Further studies that explore practical devices should employ correspondingly appropriate isotopic compositions.
The probes were placed in the Mark II TRIGA nuclear reactor beam port experimental facility at the Nuclear Engineering Teaching Laboratory at the University of Texas at Austin, where they received identical neutron and gamma dose rates.Comparing the solid and foam electrode I-V characteristic curves to one another, we observed significant differences in both the power deposition and specific current of the electrodes.These results are validated by M&S methods; analytical models predict n e and T e for the configurations based on the observed power deposition, whereas I e and I i scaling extrapolate the results observed in laboratory systems to practical devices.

Analytical and experimental considerations for NLTPs
The basic analytical methods used to calculate the energy deposited in two-component media and resulting plasma parameters are presented in the following sections.The physical quantities of interest can be deduced using experimental and M&S methods.The former involves current-voltage (I-V) curve measurements in high radiation environments whereas the latter requires a radiation transport code that treats high energy (>1 keV) charged particle and photon sources.

Charged particle transport through matter and its ionization of gases
High-energy charged particles deposit their energy in gases by ionization and excitation collisions [13].This energy deposition per unit path length dE/dx [eV cm −1 ] is a complicated function of the particle type, energy, and charge state [14,15].While previous deterministic methods assessed such transport in simple geometries, for example infinite planes and cylinders for heavy ions [3,[16][17][18], probabilistic methods are applicable to arbitrary geometries.For this, computational approaches are usually appropriate for incorporating charged particle transport into physical models.By contrast, the 'W' value w i,a [eV ion −1 ], a historical term defining the energy cost per ionization collision, assumes a constant value over a wide range of energies (for a given charged particle and gaseous species) [13]; all energies concerning low-energy (<20 MeV) nuclear energy sources.w i,a to good approximation is 1.71 times the gaseous species' ionization potential V i,a [eV] when the ionizing species is electrons [13]; agreement between the experimental data and modeling of w i,a is within 10% [4].w i,a and V i,a are listed in table 1 and are energy independent for initial electron energies >500 eV [4,13,19].With knowledge of dE/dx and w i,a , the number of ionizations per source particle occurring in a gas can be determined.

Energy deposition in two-component media
The reaction rate density R ijl [cm −3 s −1 ], of streaming particles i into target media j creating reaction type l is where is the cross section of reaction l between streaming particles i and target media j, and n j [cm −3 ] is the particle density of media particles j.The sum of power densities p tot [W cm −3 ] is: where E(E 0 ) l,j→k [J] is the energy released by process l (of initial particle energy E 0 ) originating in media j and depositing in media k, V j and V k are the volumes [cm 3 ] of media j and k, respectively.
For example, consider the energy deposition of one streaming particle species producing one reaction type in a two component media, where the first component is solid and the second, gaseous.Expanding equation ( 4): ) . ( In many instances, the energy deposited from reactions originating from the gaseous component is negligible because n 2 ≪ n 1 .Defining the power density of the solid media where is the ratio of the energy deposited in the gaseous and solid media and ν = V 1 /V 2 is the ratio of the solid to gaseous volume in the two component media.The product νβ(E 0 ) allows the direct calculation of an NLTP's p a via p solid , which is particularly useful when considering the NLTP parameters achievable given a media's p solid tolerance.β(E 0 ) is readily calculated using Monte Carlo-based radiation transport codes for arbitrary combinations of material compositions and geometries, source particles and energies, and reaction types.We note that these power density relations may be adapted to radioactive sources by taking ϕ ij σ ijl → λ jl , which is the decay constant [s −1 ] of decay type l in media component j.

Determining ne and Te for NLTPs
In the high-pressure NLTPs under consideration, a flat density n e [cm −3 ] and Maxwellian temperature profile T e [K] for the electrons are assumed as in [18, chapter 2] and [1].n e and T e are then solved simultaneously via electron particle and power balance equations, respectively.In section 4 we will discuss the global model's limits of applicability.

Particle balance for ne.
The particle balance equations-analogous to the treatment in [4, chapter 4]observing the main ion production and loss rate mechanisms for a gaseous species a (summarized in table 2) are: ) where p a (ew i,a ) −1 ≡ S + is the ionization rate density; ) −ξ (8) where the scaling constants α 0 , η, and ξ are tabulated in table 3. From this point forward, it is referred to as k dr,a + 2 (T a , T e ).Owing to the strong temperature dependence of collisional-radiative recombination and three body recombination (T −4.5 e and T −2.5 e respectively) compared to the temperature independent ion conversion process, the last two terms of equation (7a) may be neglected.Under steady state conditions equation (7a) becomes: which can be substituted into equation (7b) along with equation (7c) to give a quadratic form for [a + 2 ], For implying that plasma recombination processes are dominated by the dissociative recombination of molecular ion species.For the conditions under consideration (T e ∼ T a = 300 K, n a = 5 × 10 19 cm −3 ), k ic = (0.6-3.5) × 10 −31 cm 6 s −1 [4, chapter 4], and k dr,Ne + 2 = 1.8 × 10 −7 cm 3 s −1 [20], thus p a < 1 kW cm −3 for the n e = [a + 2 ] approximation is valid, which is satisfactory for even most pulsed nuclear reactor conditions.The solution to equation (9) is which relates the T a -and T e -dependent equilibrium electron density n e to the input power p a .

Power balance for Te.
A power balance equation is derived from the electron Boltzmann equation in the Lorentzian limit (electron-electron collision terms neglected): where S(f ) is the sum of the sources and sinks of electrons belonging to distribution f (v) and I ea (f) is the electron-atom collision integral [21] where v [cm s −1 ] is the electron velocity, σ ea [cm 2 ] is the electron-atom elastic scattering cross section and m e and m a [kg] are the electron and atomic species masses, respectively.Multiplying each term of equation ( 11) by its average energy gain (or loss) during the process, then integrating over the electron velocity space, we obtain a power balance equation as a function of T e : where v ε is the electron velocity at the energy of the first excited state of species a. [18, chapter 2] demonstrated that for the conditions described here, where T e ≪ 30 000 K, f (v) is Maxwellian.Integrating the RHS of equation ( 12) by parts and substituting equation (10) for n e , a direct relation between the steady state electron temperature and charged particle power deposition density is found to be where Finally, to be considered a plasma vice ionized gas, the system's size must be much larger than the Debye length λ D [cm], given by the following formula: This dimensional constraint can be recast in terms of p a by manipulating equation (10).Considering a spherical pore with radius R [cm] R ≫ 6.9 For instance, neon under roughly ambient temperatures T e ∼ T a = 300 K, p a = 10µW cm −3 yields R ≫ 20µm.Equations ( 10) and ( 13) are valid as long as equation ( 14) is satisfied across a wide range of power densities (10 −6 -10 3 W cm −3 ) and for all single component noble gases except for He.Their accuracy mainly depends on k dr,a + 2 (T e , T a ), which is experimentally characterized for the T a and T e ranges in table 3.

I-V characteristics for NLTPs produced by two-component media
Characterizing NLTPs using I-V curves has been previously considered [1,2].These experiments used floating electrode configurations analogous to those in figure 1; [1] derived the relationship between the ion current I i and supplied voltage V s [V], where C = (4 × 10 −2 µ a + 2 ϵ 0 e 3 ) 1/4 is a constant value comprised of the ion species mobility, µ a + 2 , permittivity constant ε 0 [C V −1 m −1 ] and electric charge e [C]; we have replaced S + with νβp solid (ew i,a ) −1 .Equation ( 16) is valid for an applied voltage V s ≫ kT e between the electrodes and where the sheath dimension, s [cm] is less than the distance between the electrode surfaces.By using the matrix sheath approximation [25], and by substituting equation ( 10) (for T a = T e ∼ 300), . Assuming 2 V ⩽ V s ⩽ 5 V and 10 µW cm −3 ⩽ p a ⩽ 50 µW cm −3 , s ranges from (1.85-4.40)× 10 −2 cm.Equation (17) implies that the physical relationship between I and s, i.e. the current collected by the biased electrode, is directly influenced by the expansion (or contraction) of the associated sheath region.
The previous efforts of [1,2] both validated equation (16) for V s ≫ kT e and quantified the effects of reactor power and gaseous composition on n e and T e .This study aims to illuminate the effects of electrode A on I i and p a .From equation (16) for V s ≫ kT e and assuming the same source particle flux, microscopic cross section, and electrode volume, a V s -independent I i ratio evolves between the foam electrode of mass m f [g] and area A f [cm 2 ] and solid electrode of mass m s [g] and area A s [cm 2 ] : The electrode area and mass are readily measured as is the total current drawn through them.Thus, ] can be directly measured and compared with results obtained by M&S methods.

Experimental and M&S descriptions and results
To establish the proposed phenomena experimentally, I-V curves were measured on two electrode assemblies inserted into the neutron beam port cavity of a Mark II TRIGA nuclear reactor at the Nuclear Engineering Teaching Laboratory at the University of Texas at Austin (figure 2).For M&S validation, the assemblies were modeled in MCNP6.2 [26], an industry-standard radiation transport code for electron and photon transport.The M&S results were compared with those from the experimental I-V curves.

Description of experimental facility and assemblies
The assemblies were designed and fabricated identically (besides the electrode assemblies), were both held under approximately 2 atm neon and irradiated by the same neutron flux (thermal, epithermal, fast) (4.05, 2.06, 1.69) × 10 12 n cm −2 s −1 and gamma dose rate (1.99 × 10 −2 W g −1 ) in Si.The electrodes consisted of (1) solid copper and (2) open-cell reticulated copper foam electrodes; additionally, one of the solid copper electrodes was coated with a thin layer (100 nm) of a dissimilar metal, molybdenum, to demonstrate the principle of converting the ion current into electrical power via electrodes with dissimilar work functions.The thin layer contributes negligibly to the total mass and surface area of the electrode and is assumed not to affect A s or νβ.
The reactor was operated at full power (950 kW) with the assemblies placed in the beam port cavities for 30 min before I-V measurements were taken to ensure that 66 Cu (t 1/2 = 5.12 min) had come into secular equilibrium; 64 Cu, with its much longer t 1/2 = 12.7 h, was not in secular equilibrium, but varied negligibly over the I-V measurement time (approximately 3 min).The I-V measurements were taken using a Keithley SMU 2450, which simultaneously sources user-defined voltages to and measures the corresponding current from the electrode assembly under test.The voltages were swept from −5 V to 5 V.

M&S description
The complex structure of the open-cell reticulated copper foam was approximated by a cubic array of unit cells of length 940 µm containing spherical 'pores' of radius 500 µm centered within the unit cell, which is representative of the 38% density foam electrode used in the experiment.
The energy deposition tally for electrons (F6:e), which ionize the gaseous media, tracks the fraction of electron energy deposited in the solid and gaseous media per source particle,  16), where the value pa is determined by a least squares regression on the data for Vs ⩾ 2 V. enabling a direct calculation of β.Source electrons originate from the β − decay of 64 Cu (E β,ave = 0.191 MeV) and 66 Cu (E β,ave = 1.06 MeV) upon thermal neutron capture and source photons, from reactor generated γs.The former is represented by a volumetric electron source within the electrode system and the latter by a volumetric photon source encapsulating the electrode system.Source particle energies from 0.1 MeV to 2 MeV in 0.1 MeV increments were simulated.

Experimental and simulation results
The I-V data are presented in figure 4, where the solid electrode curve is shifted by the observed contact potential difference, ϕ Mo − ϕ Cu , of −0.25 eV.This demonstrates a power producing ion current from the Mo to Cu electrode for V s = 0-0.25 V, roughly 20 nW at maximum.
The experimental s data are presented in figure 5, where the 'High A f ' and 'Low A f ' scenarios account for the 15% uncertainty in the surface areas of the copper foams.This range of ) overlays the energy-and source particle-dependent obtained by M&S in of figure 6.From figure 5, a constant foam to solid electrode current ratio arises for 2 V < V s < 5 V thereby satisfying the conditions V s ≫ kT e and s less than the smallest system dimension (R for the foam electrode and interelectrode gap distance for the solid electrode system) for which equations ( 16) and ( 18) are valid.This observation strongly indicates that a matrix sheath structure occurred within the foam electrode.
A least squares regression in the applicable voltage range directly infers p a according to equations ( 6) and ( 16) where A is given by values in table 4 and µ Ne + 2 = 3.2 cm 2 V −1 s −1 [27].p a was determined to be (12.8-19.3)µW cm −3 and 70.1 µW cm −3 for the foam and solid electrodes, respectively.Directly substituting these values into equation (10) gives the corresponding n e values (3.2-3.9)× 10 9 cm −3 and 7.1 × 10 9 cm −3 .For both electrodes, T e ∼ T a = 300 K. Uncertainty in experimental νβ/m mainly arises from the uncertainty in the foam electrode surface area, A f , which for the copper foams used is known to approximately ±15%.K comprises values that are known to a sufficiently high precision so that they are not included in the error bar.
s simulated in MCNP for photons (blue) and electrons (red) compared to the average experimentally observed value (shaded blue).At lower source particle energy (E 0 < 1 MeV), simulations indicate that s is significantly larger for source electrons originating within the electrodes than source photons originating outside the electrodes.
Given the uncertainty in A f , the νβ/m ratio was determined to be within the range of 1.2-1.7;these results are displayed in figure 5.The corresponding specific ion current ratio, which was directly measured, was 3.5.The established range of νβ/m is shown in relation to the energy-and source particle-dependent β f (E 0 )/β s (E 0 ) values obtained by modeling and simulation in figure 6.Interestingly, simulations predict an average νβ/m that is 50% higher for source electrons than for source photons over the 0.1-1.0MeV range and 25% higher from 1.0-2.0MeV.However, this cannot be discerned experimentally due to the uncertainty in A f .Despite the uncertainty in A f , the M&S results reflected those obtained experimentally.Although our global model appears to be applicable to the electrodes in this study, its appropriateness for foam electrodes with smaller pore dimensions and/or lower power densities is less certain.

Scaling laws
With the experimental and analytical methods now demonstrated, it is possible to extrapolate NLTP characteristics expected in porous media, given the solid and gaseous composition of the media and electrode p solid .

ne and Te.
Equation ( 13) (via substitution p a = νβp solid ) gives a direct relation between T e and p solid ; n e as a function of p solid is obtained by tabulating k dr (T e , T a ) values when solving equation (7a), and then directly inserting these values into equation (6).n e (p solid ) and T e (p solid ) are plotted in figure 7 using νβ = 1.4 × 10 −4 n a = 5.0 × 10 19 cm −3 and T a = 300 K: representative values for this study's experimental conditions.We emphasize that this parameter space does not include the effects of using:  The first four would increase n e or T e (or both).These effects should be investigated in future studies.

Ie and I i .
Revisiting equations ( 1) and (2) now having drawn connections between the electrode and NLTP properties, the specific ion and electron currents scale For example, when inserting the values from table 4 into equation (18) to evaluate the foam-to-solid I i ratio (A foam ∼ 55 cm 2 ), one obtains the experimentally observed value of 3.5.
For solid parallel plate (equal thickness) and coaxial configurations: where ρ ave [g cm −3 ] is the electrode system's average solid density, t is the plate thickness [cm], and r o [cm] is the radius of the outer electrode.Assuming that the characteristic pore dimension is much smaller than the total substrate dimension, the specific surface area for porous materials A p is constant.Typical values for A p for a 100 pores per inch foam with 10% density material coating [28] are more than 1.5 × 10 2 cm 2 g −1 .
For typical ρ ave values (5-10 g cm −3 ), solid configurations must achieve t ∼ (0.66-1.3) × 10 −3 cm and r 0 ∼ (1.3-2.6)× 10 −3 cm to attain values similar to A p .For nuclear applications, neither r o nor t fall below 5.0 × 10 −2 cm, more than and order of magnitude the size required to meet either condition A par = A p or A ann = A p .
4.2.3.Devices: ionization chambers and PTECs.Equations ( 18) and (19a) can be used to compare the specific current extraction of previously mentioned devices when using solid and porous electrodes, namely ionization chambers and PTECs.
For ionization chambers with equal spatial dimensions, noting p solid ∝ σ ijl from equation ( 3), the ratio of specific ion current drawn from a porous electrode I i,p to solid electrode I i,s is proportional to For a prototypical fission chamber (an ionization chamber whose active material is fissile typically used for nuclear reactor instrumentation), r o = 2.35 × 10 −1 cm and ρ ave ≈ 10 g cm −3 .Assuming A p = 1.5 × 10 2 cm 2 g −1 and active material mass m p(s) scales with the total electrode surface area, I i,p /I i,s ≈ 2.5 × 10 3 , implying a greater than three orders of magnitude increase in sensor sensitivity per unit mass (proportional to specific ion current).
I e,dem = 0.11 A g −1 is calculated directly by using the values given in [29].Assuming a 25% mass density nuclear fuel and the same thermal heat per unit fuel mass, p solid(β − ,γ) = 2 W cm −3 (the power density due to β − and γ heating), νβ = 10 −4 , T e = T a = 2000 K, I e,p /I e,dem = 50 implying a porous electrode's increased-by a factor of 50-ability to transport thermionic current in PTEC devices.It is important to note that the comparison between I e,p and I e,dem does not imply a proportional electrical power output increase, which depends principally on the heat transfer (via radiation and conduction) between cathode and anode; these properties are not yet known for porous electrode-based PTEC devices.The scaling laws above suggest that porous electrodes can enable significantly higher specific currents from NLTPs compared to traditional configurations.However, a better understanding of both νβ scaling and heat transfer properties under various system conditions is necessary to determine porous electrodes' ultimate level of performance in practical devices.Several factors including but not limited to electrode composition (solid and gaseous components), ionizing charged particle type, and A por will all affect I i(e) appreciably; only continued experimental and M&S simulation efforts will begin elucidating this new design paradigm.

Conclusion
This work demonstrates an experimental and M&S basis for producing and analyzing NLTPs within complex electrode structures for A ∼ 10 cm 2 g −1 .More optimized porous electrodes have the potential to facilitate dramatically higher specific powers and current extractions in nuclear DECs and sensors.Such devices can use available manufacturing techniques to dispense neutron(γ)-active materials or radioisotopes within porous structures, enabling the construction of large-scale systems (potentially m 3 ) [28].Previously unexplored, NLTPs of this nature could have broad applications to energy systems offering opportunities for cross-pollination between the nuclear and LTP science and engineering communities.Through continued interest in and investigation of these NLTPs by a diverse research community, more uses will undoubtedly be discovered.

Figure 1 .
Figure 1.Conceptual diagram illustrating (a) a metal foam electrode system (b) the relative penetration depths of LTP ionization sources for system (a) and (c) the NLTP ionization process within system (a).

Figure 2 .
Figure 2. Experimental apparatus for measuring NLTP plasma I-V curves using (left) solid and (right) open-cell reticulated copper foam electrodes.

Figure 3 .
Figure 3. Schematic and electrical circuit diagram of the experiment.The electrode assemblies shown in figure 2 were contained in an aluminum 'ampoule' enclosure in a 2 atm Ne gas and placed into the reactor beam port facility.Electrical leads extended from the facility to a Keithley Source Measuring Unit (SMU) 2450 with voltage Vs applied to the floating system.

Figure 4 .
Figure 4. I-V curve for solid and foam copper electrodes.The corresponding curves (solid lines) are computed according to equation (16), where the value pa is determined by a least squares regression on the data for Vs ⩾ 2 V.

4. 1 .
Validity of the global model assumptionA key assumption for a global model, where n e is flat, is that the characteristic time of dissociative recombination, τ dr,a + 2 = [k dr,a + 2 (p solid )n e (p solid )] −1 is much less than the ambipolar diffusion time, τ D = Λ 2 /D a where Λ [cm] is the characteristic length of the system and D a [cm 2 s −1 ], the ambipolar diffusion coefficient.For high pressure systems and where T e ∼ T i , D a = 2µ a + 2 T e .By approximating the solid electrode configuration as infinite parallel planes, separated by distance d (Λ = d/π) and the average pore of the foam electrode, a sphere of radius R (Λ = R/π), the characteristic time of the respective systems differ ∝ (R/d) 2 .For the observed p a in this experiment, τ dr,a + 2 /τ D ∼ 0.001 and 0.1 for the solid and porous electrode configurations, respectively.
(i) Light and heavy ion-based NLTP ionization.(ii) Penning gas mixtures.(iii) Increased operating temperatures.(iv) Pulsed reactor operation.(v) Diffusion-dominated density profile for small pore dimensions or low p solid .

Table 1 .
Table of w i,a values for high energy electrons.

Table 2 .
Rate constants for pertinent ion production and loss processes.

Table 3 .
Ta-and Te-dependent dissociative recombination constants and their temperature ranges of applicability.

Table 4 .
Electrode properties under irradiated conditions.
[29]g fuel element dimensions from[29], r o = 1.17 cm, ρ ave = 10 g cm −3 and assuming a 25% mass density for the porous electrode, I e,p /I e,s ≈ 4.4 × 10 2 .Since the only deployed PTEC-based nuclear reactors[29]did not use a radiation-ionized LTP, a more illustrative comparison would be with the demonstrated plasma current density I e,dem :