2.1. Levitated dipole and stellarator

A device for the confinement of a low-density pair plasma must be able to confine particles with both signs of charge. Although excellent confinement of either positrons or electrons can be achieved in a Penning–Malmberg trap, this configuration cannot simultaneously confine both positively and negatively charged particles with a spatial overlap greater than the Debye length (Yoshida et al. Reference Yoshida, Ogawa, Morikawa, Himura, Kondo, Nakashima, Kakuno, Iqbal, Volponi and Shibayama1999; Andresen et al. Reference Andresen, Bertsche, Boston, Bowe, Cesar, Chapman, Charlton, Chartier, Deutsch and Fajans2007; Saitoh et al. Reference Saitoh, Mohri, Enomoto, Kanai and Yamazaki2008; Pedersen et al. Reference Pedersen, Danielson, Hugenschmidt, Marx, Sarasola, Schauer, Schweikhard, Surko and Winkler2012; Kuroda et al. Reference Kuroda, Ulmer, Murtagh, Van Gorp, Nagata, Diermaier, Federmann, Leali, Malbrunot and Mascagna2014). Nevertheless, Greaves & Surko (Reference Greaves and Surko1995), as well as Gilbert et al. (Reference Gilbert, Dubin, Greaves and Surko2001), observed collective effects involving sufficiently dense populations of both species. They passed an electron beam through a positron plasma confined in a Penning–Malmberg trap and excited a two-stream instability. Pair plasma confinement in a magnetic mirror device is also being explored (Boehmer, Adams & Rynn Reference Boehmer, Adams and Rynn1995; Higaki et al. Reference Higaki, Sakurai, Ito and Okamoto2012), although this approach suffers from a loss-cone instability. Higaki et al. are pursuing this approach and have simultaneously trapped 10$^7$ electrons and 10$^5$ positrons for times as long as 70 ms, although the density was too low for collective effects to emerge (Higaki et al. Reference Higaki, Kaga, Fukushima, Okamoto, Nagata, Kanai and Yamazaki2017). As discussed in § 3, mirror confinement is also being developed to trap laser produced electron–positron plasmas. Paul traps are used to trap charged plasmas and have the potential to trap electron–positron plasma (Hicks, Bowman & Godden Reference Hicks, Bowman and Godden2019), although heating of the particles due to coupling with the radiofrequency (RF) fields that form the trapping potential must be overcome. The combination of a Penning trap and a Paul trap was first suggested by Greaves to study pair plasmas (Greaves & Surko Reference Greaves and Surko2002), but this also has issues controlling the subsequent heating from the trapping RF fields.

In contrast to conventional linear configurations, where axial confinement is realized by plugging electrostatic fields, toroidal configurations have no open ends. Some toroidal geometries can also confine charged particles of both signs together, as a plasma, at any degree of non-neutrality. The two configurations that have this property and are being pursued are the magnetic dipole and the stellarator. The dipole concept has been the focus of recent work. The idea of dipole confinement itself dates back to early fusion studies in the 1960s and 1970s (Freeman et al. Reference Freeman, Johnson, Okabayashi, Pacher, Schmidt and Yoshikawa1971). Besides its application to fusion science, the dipole field is one of the most fundamental magnetic configurations found in the universe. Motivated by satellite observations of high-pressure flowing plasmas in the Jovian magnetosphere (Krimigis et al. Reference Krimigis, Armstrong, Axford, Bostrom, Fan, Gloecker, Lanzerotti, Keath, Zwickl and Carbary1979), the dipole confinement concept has attracted a renewed interest in recent years. In a laboratory, use of a levitated coil to produce the dipole field is essential, as otherwise mechanical support structures would impair the magnetic confinement and cause unacceptable particle losses. Two experiments, the Ring Trap 1 (RT-1) at The University of Tokyo (Yoshida et al. Reference Yoshida, Saitoh, Morikawa, Yano, Watanabe and Ogawa2010) and the Levitated Dipole Experiment (LDX) at MIT (Boxer et al. Reference Boxer, Bergmann, Ellsworth, Garnier, Kesner, Mauel and Woskov2010), have achieved high-performance plasma confinement in levitated dipole configurations.

The dipole magnetic field is characterized by a strong field gradient. As observed in planetary magnetospheres and experiments in RT-1 and LDX, charged particles in an inhomogeneous dipole field exhibit interesting self-organization and complex nonlinear dynamics (Yoshida et al. Reference Yoshida, Saitoh, Yano, Mikami, Kasaoka, Sakamoto, Morikawa, Furukawa and Mahajan2013). The high compressibility of the dipole field provides remarkable stability for plasmas even in the presence of a strong pressure gradient. Contrary to the naive expectation, diffusion in such plasmas leads to a plasma density profile strongly peaked towards the dipole magnet (Yoshida et al. Reference Yoshida, Saitoh, Yano, Mikami, Kasaoka, Sakamoto, Morikawa, Furukawa and Mahajan2013). These unique self-organization and inward-transport processes play important roles in penetration of solar wind particles and structure formation in magnetospheres (Schulz & Lanzerotti Reference Schulz and Lanzerotti1974). Moreover, we can use such properties of the dipole field for scientific applications. The strong inward particle pinch and the very high plasma pressure attainable in a dipole makes it feasible to consider advanced fusion fuels for future power production (Hasegawa, Chen & Mauel Reference Hasegawa, Chen and Mauel1990), as intensively studied in RT-1 and LDX.

Dipole confinement of non-neutral plasmas has also been studied at RT-1 (Ogawa et al. Reference Ogawa, Yoshida, Morikawa, Saito, Watanabe, Yano, Mizumaki and Tosaka2009). In pure electron plasma experiments, detailed measurements of the internal potential structure and electrostatic fluctuations revealed a remarkable self-organization process for dipole non-neutral plasmas (Saitoh et al. Reference Saitoh, Yoshida, Morikawa, Yano, Watanabe and Ogawa2010). After turbulence-induced inward-diffusion transported the injected electrons into a strong field region, a rigidly rotating equilibrium state was spontaneously generated in the dipole field. This relaxed state was so robust that, of order $10^{10}$ electrons were stably trapped for more than 300 s, which is a world record for confinement of a non-neutral plasma in a toroidal geometry.

Helander & Connor have examined the stability of low-density, quasineutral pair plasmas in a magnetic dipole to both low-frequency, magnetic curvature driven electrostatic instabilities and interchange modes (Helander & Connor Reference Helander and Connor2016). The stability diagram shown in figure 2 is adapted from their paper. In this figure the abscissa and ordinate axes are, respectively, the inverse density gradient scale length (normalized to the scale length for variation of the magnetic flux) and the inverse of the normalized temperature gradient scale length. The unstable regions for interchange and low-frequency electrostatic modes are labelled. Despite the exceedingly low plasma $\beta$ (likely to be in the range of $< 10^{-6}$), Helander & Connor predict the possibility of interchange instability, which is possible because the field lines close on themselves and do not need to be bent. The interchange mode in question is, in fact, electrostatic in nature. There is, however, a large region in this parameter space that is linearly stable to both of these types of modes. This is the region of flatter profiles, especially for the density. These theoretical predictions have recently been confirmed by direct gyrokinetic simulations (Kennedy et al. Reference Kennedy, Mishchenko, Xanthopoulos, Helander, Navarro and Görler2020). Moreover, one can show that a subset of the linearly stable region is even nonlinearly stable (Helander Reference Helander2017). To set the scale of the axes, the scale length for the variation of the magnetic flux in a point dipole varies as the first power of $r$, the radial coordinate. Experiments in LDX and RT-1 observed self-generated peaking of the density profile in the strong field region near the levitating coil. However, in the absence of the onset of turbulence, it is not clear that such a mechanism would be operative in a low-density pair plasma. Even if the profiles are up to three times steeper than the fall-off in the magnetic flux, theory predicts the system should be stable (depending on the relative temperature and density gradients). If the confinement is very good, cyclotron cooling may be effective at flattening (or even inverting) the temperature gradient as this process is more efficient in the strong field region which could destabilize electrostatic modes.

Figure 2. The stability landscape for pair plasma in a magnetic dipole (adapted from figure 1 in Helander & Connor Reference Helander and Connor2016) showing the stability boundaries for interchange and low-frequency electrostatic modes in the space of normalized inverse temperature gradient scale length versus normalized inverse density gradient scale length. The dashed line with unit slope indicates the contour where the temperature and density gradients are equal.

The stellarator is another suitable configuration for trapping toroidal plasmas, both quasi-neutral and non-neutral (Wakatani Reference Wakatani1998), that is being pursued as a concept for electron–positron plasma studies. The stellarator is one of the promising magnetic configurations for nuclear fusion power production, and is one in which closed magnetic surfaces are generated solely by external coils. The Columbia Non-neutral Torus (CNT) (Pedersen & Boozer Reference Pedersen and Boozer2002) was designed and operated for the purpose of studying pure electron plasmas in a stellarator, with the ultimate goal of paving the way for the creation of an electron–positron plasma. In CNT, stable equilibrium (Kremer et al. Reference Kremer, Pedersen, Lefrancois and Marksteiner2006) and relatively long confinement of an electron plasma for more than 90 ms were demonstrated (Brenner & Pedersen Reference Brenner and Pedersen2012), which is much longer than the time scales of most plasma phenomena. The stellarator has the advantage that the confinement configuration can be generated by mechanically supported coils. This is in marked contrast to the levitated dipole experiment. The stellarator is, in several respects, a complementary magnetic confinement device to a levitated dipole for the purposes of understanding the basic plasma physics behaviour. Some of the complementary features of the two configurations are listed in table 1.

Table 1. Comparison of stellarator and levitated dipole for confining electron–positron plasma.

2.2. Target parameters for magnetically confined pair plasma experiments

The APEX (a positron–electron experiment) and EPOS (electrons and positrons in an optimized stellarator) (Stenson Reference Stenson2019) projects aim to create electron–positron plasmas in levitated dipole and stellarator configurations, respectively. As explained below, in order to observe self-generated collective behaviour of the charged particles, the Debye length ($\lambda _D=\sqrt {\epsilon _0 k_BT/(2n_e e^2)}$, where $\epsilon _0$ is the vacuum permittivity, $k_B$ is Boltzmann's constant, $n_e=n_p$ is the density, and the factor of 2 reflects the equal contribution to screening effects due to each species) must be smaller than the spatial scale length of the plasma. These experiments target the creation of plasmas with number densities in the range of $10^{11}\text {--}10^{13}\ \textrm {m}^{-3}$ and typical temperatures of $k_BT =0.1\text {--}1\ \textrm {eV}$. This corresponds to Debye lengths in the range of $\lambda _D$ of 0.1–1 cm, which is significantly smaller than the plasma size of approximately 8 cm. These parameters are shown in the orange region of figure 3(a) in comparison with the planned parameter regime for mirror confined relativistic plasma (discussed below) and with results of pure electron plasma experiments in RT-1 (a levitated dipole) (Saitoh et al. Reference Saitoh, Yoshida, Morikawa, Yano, Watanabe and Ogawa2010) and CNT (a stellarator) (Brenner & Pedersen Reference Brenner and Pedersen2012). The target density range is comparable to the electron densities realized in the previous non-neutral plasma experiments, indicating that this aim is realistic in view of the confinement performance of the trapping configuration. The confinement volume of APEX or EPOS will be approximately $10^{-2}\ \textrm {m}^3$ (10 l), which is smaller than those of the previous two experiments (1000 l for RT-1 and 100 l for CNT). Thus, the small Debye length criterion can be satisfied with a smaller number of charged particles (Pedersen et al. Reference Pedersen, Danielson, Hugenschmidt, Marx, Sarasola, Schauer, Schweikhard, Surko and Winkler2012); effective injection of order $10^{10}\text {--}10^{11}$ positrons in APEX or EPOS will be needed (along with the same number of electrons). Figure 3(b) shows anticipated ranges for the plasma size divided by the Debye length (ordinate) versus the plasma size divided by the skin depth (discussed further in § 3).

Figure 3. $(a)$ Target parameters (temperature and density) for pair plasma experiments discussed in this paper: low-density magnetically confined plasmas (orange) and relativistic laser-produced plasmas (blue). Also shown are achieved parameters for non-neutral plasmas in a levitated dipole (RT-1) and a stellarator (CNT). The contours indicate the Debye length (long dash) and skin depth (short dash) at each combination of density and temperature. $(b)$ Planned operating regimes of the relativistic pair mirror (blue) and low-density (low-temperature) magnetically confined pair plasma (orange) in terms of the plasma size, $L$, as a ratio of the Debye length, $\lambda _D$, versus as a ratio of the skin depth, $\lambda _s$.

At typical pressures in ultra-high vacuum systems ($10^{-8}\ \textrm {Pa}$) the density of neutrals will be comparable to those of the charged species, nevertheless, unless the plasma temperature exceeds approximately 1 eV, ion contamination is not expected to be a significant effect. Diffusion due to elastic scattering of charged particles with neutrals and positron annihilation are effects to consider. Given cross-sections for expected residual (or introduced) gases, we estimate diffusion time scales for the planned experiments to be more than 1000 s. Annihilation effects are treated in the subsequent section.

2.3. Annihilation of positrons does not rule out long confinement

Before considering accumulation and injection methods for positrons it is worth noting that annihilation effects neither impede the production nor the study of long-lived electron–positron plasmas for realistically achievable parameters (Greaves & Surko Reference Greaves and Surko2002; Saitoh et al. Reference Saitoh, Pedersen, Hergenhahn, Stenson, Paschkowski and Hugenschmidt2014). These considerations are summarized in figure 4 for magnetically confined low-density pair plasmas. One might expect that direct annihilation would be a serious obstacle to the experimental efforts described in this paper. However, the cross-section for direct annihilation of positrons with electrons is determined by the classical electron radius, which is very small. Due to Coulomb focusing at low energies, direct annihilation is enhanced for lower plasma temperatures (Crannell et al. Reference Crannell, Joyce, Ramaty and Werntz1976; Gould Reference Gould1989), however, the limitations on plasma lifetime based on this process, which scales inversely with the plasma density, is many days for the target densities and temperatures (purple lines). Positronium (Ps) formation via radiative recombination is a faster process than direct annihilation for temperatures below 59 eV (Gould Reference Gould1989); the lifetime associated with this process also scales inversely with density and is many hours for the anticipated experimental densities (green lines). As far as the authors are aware, there is no complete theory for the rate of three-body recombination in a positron–electron plasma. For electron–proton (or positron–antiproton) plasmas, the rate of three-body recombination has been calculated and found to scale as $n^2/T^{9/2}$ (Glinsky & O'Neil Reference Glinsky and O'Neil1991). We anticipate a similar rate for Ps formation, which is expected to be negligible compared with radiative recombination unless the plasma temperature becomes very low (${<}0.01\ \textrm {eV}$); the blue lines in figure 4 use the results of Glinsky & O'Neil (Reference Glinsky and O'Neil1991) but assume no magnetic field (approximately 11 times faster than the strong field rate) and an additional factor of two enhancement to account for the presence of twice the number third particles (both positrons and electrons) compared with the comparable process in an electron–ion plasma.

Figure 4. Lifetimes associated with annihilation of positrons versus plasma density (at temperatures of 0.1 eV and 1 eV) set by direct annihilation with plasma electrons (purple), Ps formation via radiative recombination (green) and via three-body recombination (blue); lifetimes for positrons set by direct annihilation on atomic/molecular electrons (in red, for two gas composition and pressure values), and by Ps formation via charge exchange on atomic hydrogen at various plasma temperatures (yellow). The anticipated experimental range for low-density magnetically confined plasma experiments is indicated in grey. This figure is adapted from similar figures in Greaves & Surko (Reference Greaves and Surko2002) and Saitoh et al. (Reference Saitoh, Pedersen, Hergenhahn, Stenson, Paschkowski and Hugenschmidt2014).

Additional relevant effects involve interactions with neutral gas. In addition to residual gases associated with imperfect vacuum conditions (e.g. $N_2$), gas may be intentionally introduced as part of the process for cooling the superconducting coils (He), for the purposes of cooling the plasma (Natisin, Danielson & Surko Reference Natisin, Danielson and Surko2014), and to serve as annihilation target for diagnostic purposes (probably using noble gases (Kurz, Greaves & Surko Reference Kurz, Greaves and Surko1996)). Direct annihilation with the bound electrons in neutral atoms and molecules (Iwata, Greaves & Surko Reference Iwata, Greaves and Surko1997) depends on the neutral pressure and composition, but is independent of plasma density; this process has a weak temperature dependence and the red lines in figure 4 use the low-temperature limit (Kurz et al. Reference Kurz, Greaves and Surko1996). For base pressures in the range of $10^{-6}\ \textrm {Pa}$, this process could limit positron lifetimes to of order an hour and motivates the desire to achieve ultra-high vacuum conditions. Positronium formation via charge-exchange collisions have relatively large, but energy dependent cross-sections (Zhou et al. Reference Zhou, Li, Kauppuila, Kwan and Stein1997) and could limit the positron lifetime; in particular, charge-exchange reactions have a threshold energy that is the difference in the binding energies of the electron with the atom/molecule and that of Ps (6.8 eV). For atomic hydrogen, which is a residual gas in ultra-high vacuum systems and has a relatively small binding energy, the threshold energy is 6.8 eV, and even at plasma temperatures of 1 eV and partial pressures of $10^{-8}\ \textrm {Pa}$ could lead to substantial positron losses (yellow lines). The lifetime limit based on charge-exchange reactions in figure 4 is not likely to be as pessimistic as this calculation indicates since energetic positrons in the tail of the velocity distribution would be preferentially lost, leading to rapid cooling and a reduction in the strongly temperature-dependent loss rate. In fact, $H_2$ might be used to cool the plasma via inelastic collisions that excite vibrational and rotational states. Molecular hydrogen has a similar maximum cross-section to atomic hydrogen, but a somewhat higher threshold energy (8.6 eV). The main conclusion here is that annihilation processes do not forbid the study of positron–electron plasma in the parameter range of interest for time scales of the order of minutes, which is much longer than the time scales of most plasma phenomena and would be a record long-confinement time for a quasi-neutral plasma.

2.4. The importance of a small Debye length

Collective plasma behaviour is expected when the Debye length is smaller than the size of the plasma (Chen Reference Chen1984; Bittencourt Reference Bittencourt2004; Hazeltine & Waelbroeck Reference Hazeltine and Waelbroeck2004; Piel Reference Piel2010). For example, self-generated electrostatically driven dynamics lead to the drift-wave type turbulence described in § 1. The same drivers are strong candidates for causing a whole host of other complex phenomena, such as the anomalously fast rate of magnetic reconnection, the larger than expected rate of inward transport in accretion disks, and the larger than expected rates of outward transport in fusion experiments. Such phenomena occur if the electrostatic potential is large enough that it substantially affects the motion of many plasma particles. This is generally true if $e\phi /k_BT$ is not negligibly small, with $\phi$ the electric potential and $e$ the electron charge. A requirement is therefore that the absolutely largest conceivable potential perturbation that can be created by a plasma satisfies $e\phi _{\max }/k_BT \approx 1$.

The maximal space charge that a plasma with density $n$ can create, results if one species entirely leaves the other species behind. In this case, the space charge electrostatic potential can be estimated from Poisson's equation. We assume for simplicity a spherical plasma with radius $L$ and constant density of the remaining species, which is singly charged:

(2.1)\begin{equation} {\epsilon_0}{\nabla^2}{\phi} = en \Longleftrightarrow {\epsilon_0}{\frac{1}{r^2}\frac{\partial}{\partial{r}}\left(r^2\frac{\partial\phi}{\partial{r}}\right) = en \Longrightarrow |\phi_{\max}|=\frac{1}{6\epsilon_0}enL^2}. \end{equation}

Combining this with our requirement that the maximum normalized potential perturbation be close to unity, we get

(2.2)\begin{equation} \frac{e}{k_BT}\frac{enL^2}{6\epsilon_0} \approx 1 \Longleftrightarrow L^2 \approx 6\frac{\epsilon_0k_BT}{ne^2} \Longleftrightarrow L^2 \approx 6{\lambda_D}^2 \Longleftrightarrow \lambda_D \approx \frac{L}{\sqrt{6}}. \end{equation}

That is, the Debye length should be smaller than the plasma size, otherwise the charge cloud is too thin or too hot for electrostatic dynamics to play a role. If one resorts to a more realistic model, in which one species of particles is not entirely separated from the other, then a stricter upper bound on the Debye length results. Most textbooks require $\lambda _D \ll L$ without specifying what that exactly means; we take it to mean

(2.3)\begin{equation} \lambda_D < \frac{L}{10}. \end{equation}

The creation of electron–positron plasmas with Debye lengths satisfying this criteria have not yet been achieved on Earth for a cloud of simultaneously confined electrons and positrons, although numerous theoretical and numerical studies of pair plasmas show that they behave very differently from traditional plasmas regarding such self-generated collective dynamics (e.g. microinstabilities) (Zank & Greaves Reference Zank and Greaves1995; Pedersen et al. Reference Pedersen, Boozer, Dorland, Kremer and Schmitt2003; Helander Reference Helander2014). A more comprehensive discussion of Debye-length physics of pair plasmas can be found in the paper by Stenson et al. (Reference Stenson, Horn-Stanja, Stoneking and Pedersen2017).

2.5. Plans for the APEX levitated dipole and the EPOS optimized stellarator

The biggest obstacle to creating electron–positron plasmas is the difficulty of obtaining sufficient quantities of positrons. This reality argues for construction of a small experiment. At fixed density, the number of positrons required would scale as the cube of the system size, $L$. However, as discussed above, to realize collective plasma effects, there must be many Debye lengths in the system. To maintain a fixed number of Debye lengths as $L$ is varied (at fixed temperature), the density must vary as the inverse square of the size and the total number of required positrons therefore varies linearly with spatial scale (Pedersen et al. Reference Pedersen, Danielson, Hugenschmidt, Marx, Sarasola, Schauer, Schweikhard, Surko and Winkler2012). So, a smaller experiment requires fewer positrons, but the effective scaling is much weaker than the cube of $L$. The competing considerations of wanting to produce closed magnetic field traps with strong fields and system sizes that can be spatially resolved with annihilation gamma detectors placed remotely, lead to the choice of a tabletop experiment with confinement volumes of approximately 10 l.

To achieve the target parameters discussed above (and displayed in figure 3), the minimum number of required positrons is approximately $10^{10}$. Typical radioactive sources used to produce non-neutral positron plasmas can generate up to $10^7$ positrons per second (Danielson et al. Reference Danielson, Dubin, Greaves and Surko2015). Higher positron fluxes can be obtained from nuclear reactor based sources (up to $10^9$ per second), although at these fluxes, the positrons typically require further remoderation to produce brightness enhanced beams with the properties suitable for efficient injection and trapping. Nevertheless, the higher flux available from the neutron-induced positron source at Munich (NEPOMUC) makes it the source of choice for the APEX and EPOS projects (Hugenschmidt et al. Reference Hugenschmidt, Piochacz, Reiner and Schreckenbach2012). The NEPOMUC beam has been characterized at the beam port where positrons are delivered to the APEX experiment and tailored for the injection and confinement experiments described below (Stanja et al. Reference Stanja, Hergenhahn, Niemann, Paschkowski, Pedersen, Saitoh, Stenson, Stoneking, Hugenschmidt and Piochacz2016).

The APEX/EPOS plan for producing magnetically confined pair plasmas is shown schematically in figure 5 (Stoneking et al. Reference Stoneking, Saitoh, Singer, Stenson, Horn-Stanja, Pederson, Nil, Hergenhahn, Yanagi and Hugenschmidt2018). The positron beam from the NEPOMUC source is sent to the buffer gas trap system (BGTS) (Danielson et al. Reference Danielson, Dubin, Greaves and Surko2015). Such devices are the standard apparatus in positron research and have also been adapted for use in the creation of antihydrogen (Gabrielse et al. Reference Gabrielse, Bowden, Oxley, Speck, Storry, Tan, Wessels, Grzonka, Oelert and Schepers2002; Andresen et al. Reference Andresen, Bertsche, Boston, Bowe, Cesar, Chapman, Charlton, Chartier, Deutsch and Fajans2007). In these traps, positrons suffer inelastic collisions with an N$_2$ buffer gas and a cooling gas (typically SF$_6$ or CF$_4$) in a differentially pumped trap. After cooling and compression, positrons are transferred to an ultra-high vacuum trap where multiple cycles of buffer-gas/cooling pulses can be stacked and up to $10^9$ positrons accumulated. The intense positron pulse source (IPPS) is a multicell trap that makes use of a high field (5T) superconducting magnet (Surko & Greaves Reference Surko and Greaves2003). This component employs additional techniques from the field of non-neutral plasma physics to manipulate the positron pulses coming from the accumulator, shift them off-axis and load them into one of a set of up to 21 storage cells. The high field magnet ensures that cyclotron cooling will maintain low temperatures and long storage times (Danielson, Weber & Surko Reference Danielson, Weber and Surko2006; Hurst et al. Reference Hurst, Danielson, Baker and Surko2019). Using this multicell trap, more than $10^{10}$ positrons will be delivered in a controlled sequence of pulses to the dipole or stellarator trap. Electrons can either be injected before injecting positrons, since both the dipole and the stellarator are suitable configurations for confining non-neutral as well as quasi-neutral plasma, or in a sequence of pulses alternating with positron injection.

Figure 5. The APEX/EPOS project overview. Up to $10^9$ positrons from a reactor-based source (NEPOMUC) will be accumulated in a buffer-gas trap and accumulator (BGTS) and then transferred to a high capacity multicell trap (IPPS) that can store more than $10^{10}$ positrons for injection into one of two magnetic traps: a levitated dipole (APEX) or an optimized stellarator (EPOS).

An alternative technique for formation of positron–electron plasma involves directing the intense positron pulses from the multicell trap onto a mesoporous silica film to form a neutral positronium beam (Liszkay et al. Reference Liszkay, Corbei, Perez, Desgardin, Barthe, Ohdaira, Suzuki, Crivelli, Gendotti and Rubbia2008). Laser excitation of the positronium into highly excited Rydberg states extends their lifetime and allows them to drift across the confining magnetic field of the trap where they can subsequently be photoionized or field-ionized due to the motional Stark effect (Deller et al. Reference Deller, Alonso, Cooper, Hogan and Cassidy2016). There are substantial challenges associated with development of such a Rydberg positronium source with high enough flux, but it does offer significant potential advantages: injection of neutral particles that can drift across the confining magnetic field, intrinsic charge density symmetry of the plasma created with this technique, and the ability to control the initial energy of the particles using tuned laser ionization.

A schematic design for the APEX levitated dipole experiment is shown in figure 6. The commercial availability of rugged high-$T_c$ superconducting tape with high critical currents in the presence of substantial magnetic fields makes construction of a compact levitated dipole feasible. The APEX levitated dipole makes use of such tape to construct a small circular coil (${\approx }7.5\ \textrm {cm}$ in radius) that weighs less than 2 kg and carries more than 30 kA-turns of current without exceeding the critical current at temperatures below 50 K. The magnetic field produced by such a coil is in the range of 1 T on the inboard side near the coil. Levitation is provided by a normal water-cooled coil placed outside the vacuum chamber that is feedback-controlled to provide stable levitation (Saitoh, Stoneking & Pederson Reference Saitoh, Stoneking and Pederson2020). Laser range sensors provide the position signal that is the input for an analogue proportional-integral-derivative (PID) feedback circuit. The closed-lead floating coil must be inductively charged by (i) preloading the required magnetic flux when the coil is normally conducting; (ii) cooling to the superconducting state; and (iii) turning off the source of the charging flux to induce the required current. The charging flux is provided by a second, open-lead superconducting coil that is fixed to a coldhead. The temperature cycling of the floating coil is achieved by controlled introduction of helium gas to a small chamber that forms part of the cooling assembly and that can be opened to permit lifting of the coil into its equilibrium levitation position. The anticipated levitation time, limited by radiation heating and/or current decay due to residual resistance is of order 1 h.

Figure 6. Schematic diagram of the APEX levitated dipole experiment with representative field lines and, in blue, the surface of outermost field lines that do not intersect material surfaces.

Annihilation gamma rays provide a convenient and powerful signal for diagnosing electron–positron plasmas. An array of bismuth germanate (BGO) scintillator detectors with photomultiplier tubes will be used to diagnose both volumetric annihilation (and hence the plasma density) as well as localized positron losses. High-purity germanium detectors exhibiting a high-energy resolution (i.e. more than a factor of 30 higher compared with BGO detectors) will be used to put upper limits on the plasma temperature (Iwata et al. Reference Iwata, Greaves and Surko1997). Introduction of micropellets (via laser blow-off) (Pedersen et al. Reference Pedersen, Danielson, Hugenschmidt, Marx, Sarasola, Schauer, Schweikhard, Surko and Winkler2012) or gas jets can be used to provide localized enhancement of annihilation signal to measure spatial profiles of the plasma density, and perhaps the temperature due to the energy dependence of the annihilation rate (Kurz et al. Reference Kurz, Greaves and Surko1996).

A prototype experiment has been operated to test injection and confinement of positrons from the NEPOMUC positron beam using a supported permanent magnet. Those experiments succeeded in achieving lossless injection of a 5 eV positron beam using a combination of ExB drift, electrostatic reflection and magnetic mirroring (Stenson et al. Reference Stenson, Nil, Hergenhan, Horn-Stanja, Singer, Saitoh, Pederson, Danielson, Stoneking and Dickmann2018). By switching off the potentials used to inject positrons, good confinement of single particle orbits was confirmed despite substantial residual asymmetries. Confinement times exceeding 1 s were observed, and this was determined to be limited by diffusion associated with elastic collisions with residual gas neutrals (Horn-Stanja et al. Reference Horn-Stanja, Nil, Hergenhahn, Pederson, Saitoh, Stenson, Dickmann, Hugenschmidt, Singer and Stoneking2018). Longer confinement is, therefore, expected in future experiments operated at lower base neutral pressure. Detailed comparison of positron injection and confinement data in this prototype trap with simulations demonstrate a thorough understanding of the single-particle regime trajectories and neutral transport and promise predictive capabilities for upcoming experiments (Nißl et al. Reference Nißl, Stenson, Hergenhahn, Horn-Stanja, Pederson, Saitoh, Hugenschmidt, Singer, Stoneking and Danielson2020).

Creating and studying electron–positron plasmas in a stellarator offers the opportunity for synergy in the use of resources as well as complementarity in the landscape of physics studies (see table 1). The stellarator as a confinement concept for fusion research has its origins in the very earliest efforts to magnetically confine plasma (Spitzer Reference Spitzer1958). It remains a forefront concept in that effort with recent results from the W7-X stellarator demonstrating its continuing promise (Wolf et al. Reference Wolf, Alonso, Äkäslompolo, Baldzuhn, Beurskens, Beidler, Biedermann, Bosch, Bozhenkov and Brakel2019). Unlike the tokamak concept which requires a large net current carried by the plasma to create the confining magnetic field, the magnetic field of the stellarator is largely determined by the current in external coils.

Historically, stellarator confinement has been challenging because of the existence of poorly confined particle orbits, but the past twenty years have seen a revolutionary theoretical and computational effort that has resulted in new optimization approaches to designing stellarators. The HSX stellarator is an early example (Anderson et al. Reference Anderson, Almagri, Anderson, Matthews, Talmadge and Shohet1995) and the W7-X device is the latest device to benefit from such optimization. There are a large number of optimization targets for stellarator design that include collisional transport, instability minimization and fast particle orbits, among others. To confine a low-density pair plasma, the choice of optimization parameters differs from those of a fusion-relevant device. Such an optimization and experimental verification project provides a unique opportunity to test stellarator theory in a hitherto unexplored parameter regime.

The EPOS project is designed to take up this challenge; EPOS makes use of the NEPOMUC positron beam and the positron buffer-gas trap and accumulator to provide positron pulses for a small, high-field stellarator (figure 5). The target plasma volume, density and temperature are similar to those of the APEX dipole experiment. The design of the experiment is in the early stages, but the use of readily available high-temperature superconductors (as in the APEX levitated dipole experiment) and 3-D printing technology for producing the required coil forms are anticipated.

3.1. The importance of the plasma skin depth

We now compare the target properties for magnetically confined electron–positron pair plasmas with an alternative experimental approach, namely production of a pair plasma by laser-based techniques. The laser-produced electron–positron plasmas described in this section are highly complementary to the low-density magnetically confined plasmas just described, which will have a small Debye length but a large plasma skin depth, $\lambda _s=c/\omega _p$ (where $c$ is the speed of light and $\omega _p$ is the plasma frequency, which for a pair plasma with equal densities of the two species is $\omega _p = \sqrt {(2n_e e^2)/(\epsilon _0 \gamma m_e)}$ with $m_e$ the electron/positron mass and $\gamma$ the Lorentz factor). At the expected densities of the APEX/EPOS experiments’ magnetically confined pair plasmas, that is around $10^{13}\ \textrm {m}^{-3}$, $\lambda _s$ will be of order 5 m, whereas the plasma size will be of order 0.1 m. For laser-produced relativistic pair plasmas this is not true: the skin depth and Debye length approach equality (Stenson et al. Reference Stenson, Horn-Stanja, Stoneking and Pedersen2017). The skin depth is the characteristic size a charged cloud must have to significantly affect externally launched electromagnetic radiation. An electromagnetic wave is reflected from an unmagnetized plasma if the plasma frequency exceeds the electromagnetic wave frequency, $\omega _p > \omega$. However, the wave fields will still penetrate a distance $\lambda _s$. If $\lambda _s$ is significantly greater than $L$ (the plasma size) then the plasma is not large and dense enough to reflect the wave, nor will it substantially be able to change the amplitude or the phase of the wave. Thus, to study how a plasma affects the propagation of an electromagnetic wave, the skin depth must be small. And indeed, just as electrostatic turbulence is uniquely different in small Debye-length pair plasmas compared with otherwise equivalent electron–ion plasmas, electromagnetic wave propagation is uniquely different in small-skin-depth pair plasmas compared with otherwise equivalent electron–ion plasmas (Stenson et al. Reference Stenson, Horn-Stanja, Stoneking and Pedersen2017). If laser-produced pair plasmas can be magnetically confined, it may be possible to simultaneously reach small Debye length and small skin depth conditions. We discuss efforts in this direction below.

3.2. Parameters for laser-produced pairs

The target properties of electron–positron pair plasmas produced via laser–matter interactions differ significantly from those of magnetically confined electron–positron pair plasmas produced from reactor-based beams. Intense laser-plasma interaction can accelerate electrons to MeV and GeV energies. These electrons then interact with a high-Z target (e.g. Au, Pb or Pt) where the electrons are deflected by nuclei, producing Bremsstrahlung photons. These photons can create electron–positron pairs through several processes, the dominant one being the Bethe–Heitler process, which involves a photon interacting with a second nucleus (Heitler Reference Heitler1954).

Laser-pair production techniques differ in how the electrons are generated and accelerated. One-stage laser-pair production (Cowan et al. Reference Cowan, Perry, Key, Ditmire, Hatchett, Henry, Moody, Moran, Pennington and Phillips1999; Chen et al. Reference Chen, Wilks, Bonlie, Liang, Myatt, Price, Meyerhofer and Beiersdorfer2009, Reference Chen, Wilks, Meyerhofer, Bonlie, Chen, Chen, Courtois, Elberson, Gregori and Kruer2010, Reference Chen, Fiuza, Link, Hazi, Hill, Hoarty, James, Kerr, Meyerhofer and Myatt2015b; Liang et al. Reference Liang, Clarke, Henderson, Fu, Lo, Taylor, Chaguine, Zhou, Hua and Cen2015) irradiates the high-Z target directly with the laser. The prepulse of the laser produces a preformed plasma on the surface which is then accelerated by the main laser through $J \times B$ forces (Wilks et al. Reference Wilks, Kruer, Tabak and Langdon1992) and stochastic processes (Kemp et al. Reference Kemp, Fiuza, Debayle, Johzaki, Mori, Patel, Sentoku and Silva2014). In two-stage laser-positron production (Gahn et al. Reference Gahn, Tsakiris, Pretzler, Witte, Thirolf, Habs, Delfin and Wahlström2002; Sarri et al. Reference Sarri, Schumaker, Di Piazza, Vargas, Dromey, Dieckmann, Chvykov, Maksimchuk, Yanovsky and He2013, Reference Sarri, Poder, Cole, Schumaker, Di Piazza, Reville, Dzelzainis, Doria, Gizzi and Grittani2015; Williams et al. Reference Williams, Pollock, Albert, Park and Chen2015; Xu et al. Reference Xu, Shen, Xu, Li, Yu, Li, Lu, Wang, Wang and Liang2016) the laser first ionizes a neutral gas jet and accelerates electrons through laser wakefield acceleration (known as LWFA) towards the target. Both methods produce diverging beams of relativistic electrons and positrons. The largest positron yield per shot, $10^{12}$ relativistic (4–20 MeV) positrons, has been achieved with the one-stage method (Chen et al. Reference Chen, Link, Sentoku, Audebert, Fiuza, Hazi, Heeter, Hill, Hobbs and Kemp2015a). The two-stage method produces orders of magnitude fewer positrons with higher energy (${>}100\ \textrm {MeV}$).

These short-pulse laser-produced pairs approach conditions at which it is possible to study physics relevant to energetic astrophysics such as GRB (Chen et al. Reference Chen, Fiuza, Link, Hazi, Hill, Hoarty, James, Kerr, Meyerhofer and Myatt2015b). For example, instabilities involving electromagnetic phenomena such as the Weibel instability are believed to play an important role in the formation of relativistic collisionless shocks in GRBs (Gruzinov Reference Gruzinov2001; Chang, Spitkovsky & Arons Reference Chang, Spitkovsky and Arons2008; Pe'er Reference Pe'er2019). To study these instabilities in plasma it is necessary for the scale length of the plasma to be greater than the skin depth $\lambda _S$ (as discussed above) and the lifetime of the plasma to be greater than the instability growth rate $\tau$, which is related to the inverse of the plasma frequency ($\tau \propto 1/\omega _p)$ (Chen et al. Reference Chen, Fiuza, Link, Hazi, Hill, Hoarty, James, Kerr, Meyerhofer and Myatt2015b). To achieve these conditions, work has been done to improve the positron yield (Chen et al. Reference Chen, Link, Sentoku, Audebert, Fiuza, Hazi, Heeter, Hill, Hobbs and Kemp2015a; Liang et al. Reference Liang, Clarke, Henderson, Fu, Lo, Taylor, Chaguine, Zhou, Hua and Cen2015), to reduce the pair beam divergence (Liang et al. Reference Liang, Clarke, Henderson, Fu, Lo, Taylor, Chaguine, Zhou, Hua and Cen2015; Sarri et al. Reference Sarri, Poder, Cole, Schumaker, Di Piazza, Reville, Dzelzainis, Doria, Gizzi and Grittani2015), to collimate the beam with magnetic lenses (Chen et al. Reference Chen, Fiksel, Barnak, Chang, Heeter, Link and Meyerhofer2014), or as we will address here in detail, to increase the lifetime of the pair plasma through magnetic confinement.

3.3. Magnetic confinement of laser-produced pairs

Here, we discuss the feasibility of trapping a pair plasma in a volume of ${\sim }1\ \mathrm {cm}^3$ ($10^{-6}\ \textrm {m}^3$) with densities of $10^{16}\text {--}10^{18}\ \mathrm {m}^{-3}$ and energies of 1–10 MeV for a few nanoseconds (${\sim }10$ times the plasma oscillation period), which is long enough for collective behaviour to develop, and approaches the small Debye length and skin depth conditions. Assuming a Maxwellian velocity distribution, the targeted relativistic pair plasma is in the temperature regime where the Debye length and skin depth approach each other due to energy dependence of the relativistic mass in the skin depth. This is depicted in figure 3(a) which shows the short (skin depth) and long (Debye length) dashed contours converging at high temperature, while at low temperatures the skin depth exceeds the Debye length (by a factor of $715$ at 1 eV). In terms of the number of Debye lengths versus the number of skin depths in the system, the mirror-trapped relativistic pair plasmas described below are complementary to the pair plasmas discussed in § 2 (figure 3b).

At short-pulse laser facilities, magnetic coils have been driven with lasers and pulsed power supplies. Nanosecond pulse lasers with energies of hundreds of joules can drive laser-diode powered coils generating ${\sim }500\ \textrm {T}$ magnetic fields (Korobkin & Motylev Reference Korobkin and Motylev1979; Santos et al. Reference Santos, Bailly-Grandvaux, Giuffrida, Forestier-Colleoni, Fujioka, Zhang, Korneev, Bouillaud, Dorard and Batani2015); however, the mechanisms driving the current in the laser-driven diode are not well understood (Fiksel et al. Reference Fiksel, Fox, Gao and Ji2016; Tikhonchuk et al. Reference Tikhonchuk, Bailly-Grandvaux, Santos and Poyé2017). In addition, utilizing such laser-driven coils would necessitate using two short-pulse lasers, one for positron generation and one for the coil, introducing complexities in aligning the pulses and potential laser crosstalk. Recently developed, inductively coupled, pulsed-power driven coils can generate up to 30 T for microseconds (Barnak et al. Reference Barnak, Davies, Fiksel, Chang, Zabir and Betti2018; Fiksel et al. Reference Fiksel, Backhus, Barnak, Chang, Davies, Jacobs-Perkins, McNally, Spielman, Viges and Betti2018; Shapovalov et al. Reference Shapovalov, Brent, Moshier, Shoup, Spielman and Gourdain2019). Although these pulsed-power driven coils have an order of magnitude weaker magnetic fields, the physics of pulsed power circuits is well understood and the fields are steady-state over the targeted plasma lifetime.

Here, we discuss a magnetic mirror design utilizing pulsed-power-driven coils. The 30 T fields are strong enough to trap particles with MeV but not 100 MeV energies. This energy constraint makes the one-stage production method with its 4–20 MeV positrons preferable to the two-stage approach with its 100 MeV positrons. In addition, the positron yield and positron energy in one-stage pair production method can be independently controlled. Chen et al. (Reference Chen, Wilks, Meyerhofer, Bonlie, Chen, Chen, Courtois, Elberson, Gregori and Kruer2010) demonstrated that the sheath responsible for the target-normal sheath acceleration (known as TNSA) of the positrons can be manipulated without affecting the positron yield.

Two pulsed power coils can be arranged in a magnetic mirror configuration. A mirror made of two 5 mm radius, 15 T coils spaced 15 mm apart has a field at the midpoint along the axis between the two coils of 6 T. Figure 7(a) shows simulated paths of 3 MeV electrons and positrons launched from a 0.5 mm diameter, 0.25 mm thick cylindrical target. The nominal Larmor radius for such particles is 1 mm, which is of the order of the system size. The particle paths are found by integrating the relativistic equation of motion with the Lorentz force term but without any collective effects. Charged particles complete three motions in a magnetic mirror: they gyrate around the magnetic field lines, they bounce back and forth between the strong magnetic field at the coils and they drift azimuthally around the mirror. Each of these motions is constrained by an adiabatic invariant which is conserved if the particle does not experience too large a change in magnetic field during a single cycle of the motion. The relativistic pair mirror field is not sufficient to confine the high-energy pairs adiabatically. Instead the particles follow chaotic orbits (Saitoh et al. Reference Saitoh, Yoshida, Yano, Nishiura, Kawazura, Horn-Stanja and Pedersen2016). Losses through the mirror ends are due to particles moving non-adiabatically through phase space into the loss cone (Porazik et al. Reference Porazik, Johnson, Kaganovich and Sanchez2014). Particle collisions with the target are assumed to lead to loss, a pessimistic assumption as not all scattering will lead to loss (in fact, this would likely moderate the particles to lower energy, making them easier to trap (Mills Reference Mills1980; Gerola, Waeber & Shi Reference Gerola, Waeber and Shi1995; Gerchow et al. Reference Gerchow, Braccani, Carzaniga, Cooke, Döbeli, Kirch, Köster, Müller, Van der Meulen and Vermeulen2018)). The confinement time is of the order of $1$ ns, which is approximately 10 bounce periods (figure 7b). Due to this short confinement time, even fast-growing kinetic loss-cone instabilities are not expected to be able to establish themselves at these time scales, so there is less need to remove the loss cone from the relativistic pair plasma confinement scheme than from the schemes for low-temperature pair plasma.

Figure 7. $(a)$ Shown here are single particle paths for a 3 MeV electron (blue) and a 3 MeV positron (red) in a magnetic mirror generated by two 15 T 5 mm inner radius coils separated by 15 mm. The paths originate at a target placed in the centre of the mirror. $(b)$ The time evolution of 100 positrons in the mirror. The total positron number in the mirror (blue), losses due to target collisions (green) and losses due to (non-adiabatic) outflows (red).

Placing the target inside the mirror has the advantage that the pairs are ‘born’ inside the mirror with a phase space suitable for confinement. Disadvantages are the surplus of electrons produced and impurity protons escaping the target. The surplus electrons may be removed by moving the target off the central mirror axis so that the electrons are preferentially lost in the first gyro-orbit. Placing the target off axis also results in improved positron confinement – by up to a factor of three – due to the increased physical space for the particle trajectories. Protons, which will eventually exit the laser target and the scattering foil, may be separated by canting the mirror, as such bent mirrors have been shown to separate particles by mass (Gueroult & Fisch Reference Gueroult and Fisch2012). The target could also be placed outside the mirror, and one of the mirror coils could be used as a magnetic lens to transport the pairs into the mirror, focusing at a scattering thin foil to mix up the phase space and trap the pair plasma.

This concept, a (several tens of teslas) magnetic mirror, produced by recently developed pulsed power coils, suffices to trap $10^{16}\text {--}10^{18}\ \mathrm {m}^{-3}$, 1–10 MeV pairs for a few nanoseconds in a $1\ \mathrm {cm}^3$ magnetic mirror configuration. Future increases in the magnetic field could increase the energy of pairs that can be trapped. Although the size of a mirror loss cone in phase space only depends on the mirror ratio rather than the magnitude of the magnetic field, increasing the magnetic field would allow a decrease in the mirror volume, increasing the pair density for a given number of pairs injected.