We are developing a positron–electron plasma trap based on a dipole magnetic field generated by a levitated superconducting magnet to investigate the physics of magnetized plasmas with mass symmetry as well as antimatter components. Such laboratory magnetosphere is deemed essential for the understanding of pair plasmas in astrophysical environments, such as magnetars and blackholes, and represents a novel technology with potential applications in antimatter confinement and the development of coherent gamma-ray lasers. The design of the device requires a preemptive analysis of the achievable self-organized steady states. In this study, we construct a theoretical model describing maximum entropy states of a collisionless positron–electron plasma confined by a dipole magnetic field and demonstrate efficient confinement of both species under a wide range of physical parameters by analyzing the effect of the three adiabatic invariants on the phase space distribution function. The theory is verified by numerical evaluation of spatial density, electrostatic potential, and toroidal rotation velocity for each species in correspondence with the maximum entropy state.

Pair plasmas consist of two species of charged particles having the same mass, but opposite electric charge. Such mass symmetry is in stark contrast with usual plasmas, where ions exhibit a significantly larger mass than the electron component, and it is the reason why pair plasmas are expected to possess peculiar turbulence, stability, and fluctuation properties.1 Positron–electron plasmas are one example of naturally occurring pair plasmas and can be found in astrophysical jets and magnetospheres of quasars,2–4 pulsars,5 and magnetars,6–8 where positrons and electrons are formed by pair production. More exotic pair plasmas, such as proton–antiproton plasmas, could occur as well as in suitable astrophysical environments.

Positron–electron plasma experiments based on magnetic mirrors have been considered in the past9 and efforts are under way for their realization,10 while pair plasmas involving oppositely charged fullerene ions have been produced in the laboratory.11,12 Furthermore, positron–electron pair production is expected to occur during the disruptive phase of tokamaks.13 At present, positron–electron plasma confinement schemes based on dipole magnetic fields represent an active area of research due to recent technological advancement in trap design,14–17 positron accumulation,18,19 and positron injection.20,21 Furthermore, the inhomogeneity of a dipole magnetic field makes these magnetic configurations suitable to confine both neutral and nonnetutral plasmas, in contrast with magnetic traps relying on a straight homogeneous magnetic field, which are appropriate for nonneutral plasmas.22–24 

If realized, a positron–electron plasma based on a dipole magnetic field has several potential applications ranging from the experimental study of astrophysical systems, such as astrophysical jets, quasars, and pulsars, to matter and antimatter confinement, potentially providing confinement to a large amount of charged particles (mainly depending on the properties of the matter/antimatter source) for a long time (300 s and beyond25) without requiring external electric fields and development of coherent gamma-ray sources (gamma-ray lasers). In the laser application envisioned here, the dipole trap would confine a dense low-temperature positron–electron plasma. The coherent gamma-ray source would then be obtained by the process of stimulated annihilation of positron–electron pairs through the action of impinging photons of suitable energy (annihilation laser26–28) rather than by stimulated emission.

The target parameters for a laboratory positron–electron plasma are expected to ensure a high degree of stability not seen in standard ion–electron plasmas.29,30 In such a confinement regime, the gyroradius r c = m v / | q | B is smaller than the Debye length λ D = ( ϵ 0 k B T / 2 n e 2 ) 1 / 2, which is smaller than the system size R, i.e., r c λ D R. Here, m denotes the particle mass, v is the velocity perpendicular to the magnetic field B, q is the electric charge, B is the modulus of the magnetic field, ϵ0 is the vacuum permittivity, kB is the Boltzmann constant, T is the temperature, n is the spatial density, and e = | q |. For a positron–electron plasma with B 1 T , n 10 12 m 3, k B T 10 eV, and R 1 m, one obtains r c 10 5 m and λ D 2 × 10 2 m. Furthermore, the plasma parameter Λ = 4 π n λ D 3 = ( 1 / 4 2 π ) ( r d / r C ) 3 / 2 10 8 1 is large, implying that the typical distance among particles r d = n 1 / 3 10 4 m is much larger than the distance r C = e 2 / 4 π ϵ 0 k B T 10 10 m at which the Coulomb energy becomes comparable with the average kinetic energy. In particular, the Coulomb energy at the typical distance rd is E C = e 2 / 4 π ϵ 0 r d 2 × 10 24 J. This value is small compared to the kinetic energy since E C / k B T 10 6. This makes the system weakly coupled, and diffusive (entropy maximizing) processes are dominated by collective electromagnetic fluctuations rather than localized Coulomb collisions.

At present, we are developing a positron–electron plasma trap based on a dipole magnetic field generated by a levitated superconducting coil operating in the plasma regime described above. This requires a preemptive analysis of the achievable plasma confinement. The aim of this study is, thus, to explore the nature of maximum entropy states that are self-organized by the plasma and assess the degree of confinement of both species. In this context, a maximum entropy state is defined as the distribution function of the largest entropy that is compatible with the macroscopic constraints affecting the system, such as conservation of total particle number, total energy, or total magnetic moment. Since a positron–electron system is formally analogous to any two-species plasma, such as an ion-electron plasma, thermal equilibria in a positron–electron plasma can be inferred from those of a two-species plasma. It is, therefore, important to stress that the core issue examined in the present paper is not the study of two-species thermal equilibria, which is an established matter, but the elucidation of the effect of the conservation of adiabatic invariants on self-organized maximum entropy states in two-species plasmas, a problem that is not discussed in the literature.

Once built, the positron–electron plasma trap will be used to study both waves and transport phenomena, which are expected to be in the 1 kHz and 1 Hz frequency range, respectively, based on previous experimental data from the RT-1 device.25 Hence, we aim at achieving confinement on time scales τc of the order of 1 s. Since in the present setting the Coulomb scattering frequency is of the order ν C 1 Hz, the plasma is expected to be quite collisionless, while the main mechanism increasing the entropy of the system is given by collective electromagnetic fluctuations. For example, a fluctuating electric field E can induce a random walk (diffusion process) across the magnetic field by exciting E × B drift motion. Hence, we shall develop the theory by assuming that Coulomb collisions can be neglected.

It should be emphasized that accumulating 1011–1012 positrons in a volume Ω 1 m 3 requires the development of positron accumulation and injection technologies. In the present trap design, positrons will be produced by a 10 eV pulsed positron source located at The National Institute of Advanced Industrial Science18 and progressively accumulated into a buffer-gas trap until a total positron number N p 10 11 is reached. The system above is also designed to minimize the positron energy spread. The positrons will then be released into the dipole trap over a time interval (injection timescale) τ i 10 μ s τ c 1 s, reaching a particle density in the target range 1011 10 12 m 3. Notice here that electrons “wait” inside the dipole field for the positron injection (electrons can be confined inside the dipole trap for time scales of the order 300 s). Efficient injection schemes are also being developed to minimize losses at the injection phase. Once injected, the main source of particle loss is not expected to be the interaction with the confining vessel or the current loop box, but charge exchange with neutral hydrogen impurities, resulting in the formation of positronium. The loss timescale for this process can be roughly estimated to be τ l 10 s. This value is sufficiently longer than the confinement timescale τc, which is itself longer than the relaxation timescale τr (the time needed by the system to approach the equilibrium state). In particular, experimental evidence from the RT-1 device31 suggests τr to be a small fraction of τc (although precise estimates for a positron–electron plasma would need experimental verification). The injection, relaxation, confinement, and loss time scales are, therefore, related according to τ i < τ r < τ c < τ l.

In addition to the technological hurdles listed above, other factors, such as particle loss associated with turbulent transport and plasma instability, may affect the quality of confinement. These aspects, which have been investigated in dipole geometry within the frameworks of MHD and gyrokinetics (see, e.g., Refs. 32–34) will not be addressed in the present paper since its scope is limited to establishing the existence of stable maximum entropy equilibria toward which the positron–electron plasma is expected to converge under ideal conditions.

In an inhomogeneous magnetic field such as a dipole magnetic field, the properties of the equilibrium states are strongly dependent on the presence of adiabatic invariants.35 When the timescale of collective electromagnetic fluctuations within the plasma is longer than the period of cyclotron gyration, bounce motion, or toroidal drift, the corresponding adiabatic invariant (magnetic moment μ, bounce action J , or magnetic flux Ψ) is conserved by the dynamics of a single charged particle. Indeed, the essential feature of adiabatic invariants is that they remain approximately constant as long as perturbations acting on a dynamical system are slow compared with the period of adiabatic motion. This results in a set of constraints on the equilibrium distribution function of the system, which departs from a standard Maxwell–Boltzmann distribution.36–38 To understand how this occurs, it is useful to consider the limiting case in which the value of one adiabatic invariant, say μ, is preserved exactly. Then, on each phase space submanifold corresponding to a level set of μ (a μ-leaf), the effective phase space measure is reduced to Bdxdydzd v , where ( x , y , z ) are Cartesian coordinates and v denotes the velocity along the magnetic field. If f μ ( x , y , z , v ) is the distribution function on a μ-leaf, entropy maximizing processes such as cross field diffusion lead to a flattening of f μ, with a corresponding relaxed spatial density n = B f μ d v d μ B, which is inhomogeneous for an inhomogeneous B (for further details, see Refs. 38–40). Therefore, a statistical description of maximum entropy states must take into account the nontrivial role played by adiabatic invariants in shaping the distribution function of each particle species.

The present paper is organized as follows. In Sec. II, we derive the maximum entropy distribution functions of positrons and electrons by taking into account the conservation of the first adiabatic invariant and obtain the corresponding form of the Poisson equation for the electrostatic potential. In Secs. III and IV, we study the effect of the second and third adiabatic invariants on positron–electron plasma maximum entropy states. In Sec. V, we report spatial densities and electrostatic potential obtained by numerical solution of the Poisson equation for the electrostatic potential and demonstrate efficient confinement of both species. In Sec. VI, we study the toroidal rotation velocity profile of the positron–electron plasma and find that it departs from the rigid rotation occurring in magnetic traps relying on a straight homogeneous magnetic field. Concluding remarks are given in Sec. VII.

We remark that the theory developed in the present paper holds provided that the macroscopic constraints defining each statistical ensemble (total particle number, total energy, total magnetic moment, and so on) are preserved throughout the relaxation of the system. In particular, the period of the slowest adiabatic motion must be considerably smaller than the typical timescale of the fastest turbulent fluctuations in the system. However, in real experiments, particles and energy will be lost via different mechanisms, such as by interaction with the vessel boundary, and the degree of conservation of adiabatic invariants will not be perfect. Therefore, the theory will be quantitatively accurate only if such losses are not too large. Nevertheless, we expect the theory to remain qualitatively consistent even in the presence of large losses since the surviving (trapped component) of the plasma should eventually converge toward the derived maximum entropy states. It is also worth observing that if one considers time scales longer than the typical interval required for a Coulomb collision to occur, the conservation of the adiabatic invariants of each charged particle is progressively broken due to cumulative changes in the particle energies. Nevertheless, we conjecture the theory to remain approximately valid even in such regime since it only relies on the hypothesis that the total value of each adiabatic invariant is preserved, and the total change due to multiple Coulomb scatterings should be, on average, negligible.

Finally, notice that the present theory does not make any predictions on the nature of electromagnetic fluctuations. It only provides information on the maximum entropy state that results from the action of these fluctuations, which act to increase the entropy of the system. Hence, although the model cannot describe transient phenomena such as waves, transport, or diffusion, it is sufficient to describe the “final” state of the system, i.e., the equilibrium toward which the plasma tends to settle over a sufficiently long time interval.

In the following, lower indexes p and e will be used to specify positrons and electrons, respectively. In a static equilibrium, the electrostatic potential Φ is determined by the Poisson equation,
Δ Φ = ρ ϵ 0 .
(1)
Here, ρ is the electric charge density and ϵ0 is the vacuum permittivity. In a positron–electron plasma, the charge density can be expressed as
ρ = e ( n p n e ) ,
(2)
where e is the positron charge, – e is the electron charge, np is the positron number density, and ne is the electron number density. Denoting with fp the probability distribution function of positrons, fe the probability distribution function of electrons, Np the total number of positrons, and Ne the total number of electrons, the number densities np and ne can be evaluated as
n p = N p 3 f p d 3 p , n e = N e 3 f e d 3 p .
(3)
In this notation, d 3 p denotes the volume element in momentum space.
Let Ω 3 denote the spatial volume occupied by the positron–electron plasma, Π = 3 × Ω the phase space of the system, and d Π = d 3 p d 3 x the phase space measure. Notice that the phase space measure d Π is invariant due to the Liouville's theorem arising from the underlying Hamiltonian structure and, therefore, serves as the natural measure with respect to which the ergodic hypothesis of statistical mechanics is enforced. Furthermore, it should be emphasized that the measure d Π remains invariant for any reduced subsystem of charged particle dynamics, such as guiding center dynamics. Next, we consider a guiding center plasma such that the first adiabatic invariant μ is a constant of motion of (isolated) charged particle dynamics. Here, the magnetic moment μ41–43 is defined as
μ = m | v v d | 2 2 B = m v c 2 2 B ,
(4)
where v is the particle velocity across the magnetic field, v d is the component of v independent of the cyclotron phase ϑ c and comprising the guiding center drifts, v c = v v d is the cyclotron velocity with modulus vc, B is the modulus of the magnetic field B, and m is the particle mass (which is the same for both positrons and electrons).
For the first adiabatic invariant to be preserved, the timescale Tf of fluctuations in particle energy caused by electromagnetic turbulence must be longer than the timescale of cyclotron motion Tc, i.e.,
T f T c = 2 π ω c = 2 π m e B ,
(5)
where ωc denotes the cyclotron frequency. In the following, we shall, therefore, assume that the condition (5) holds. We also recall that the conservation of the quantity (4), usually referred to as the lowest order magnetic moment, rests on the additional requirement that the wavelength λ of any fluctuation is longer than the Larmor radius ρc, i.e., ρ c λ. In this setting, the guiding center Hamiltonian functions for positron and electrons have expressions
H p = 1 2 m v 2 + μ B + e Φ , H e = 1 2 m v 2 + μ B e Φ .
(6)
Here, v is the velocity component along the magnetic field. Observe that the drift velocity v d does not contribute to the energies (6) under the assumption that the corresponding kinetic energy is small compared to the other terms. This is true, for example, when the potential energy e Φ / k B T ϵ, with ϵ = ρ c / L, where L is the characteristic scale length of the magnetic field, and kBT is a characteristic plasma temperature, represents a small perturbation of the particle energy (for additional details on this point, see discussion at the end of Sec. IV or Ref. 42). Then, the Shannon entropies Sp and Se, the total probabilities Pp and Pe, the total energies Ep and Ee, and the total magnetic moments Mp and Me of the system can be written as
S p = Π N p f p log ( N p f p ) d Π , S e = Π N e f e log ( N e f e ) d Π ,
(7a)
P p = Π f p d Π , P e = Π f e d Π ,
(7b)
E p = N p Π f p H p d Π , E e = N e Π f e H e d Π ,
(7c)
M p = N p Π f p μ d Π , M e = N e Π f e μ d Π .
(7d)
Under the ansatz (7a) for the entropy measure associated with each particle species, the thermodynamic equilibrium of the system can be obtained by the maximization of entropy under the constraint imposed by the constancy of Pp, Pe, Ep, Ee, Mp, and Me (for further details on this approach, see Refs. 37 and 38). Introducing Lagrange multipliers αp, αe, βp, βe, γp, and γe, we, therefore, define the target functional
F [ f p , f e ] = S p + S e α p N p P p α e N e P e β p E p β e E e γ p M p γ e M e .
(8)
The Euler–Lagrange equations obtained by variation of Eq. (8) with respect to fp and fe are
(9)
f p = A p exp { β p H p γ p μ } ,
(9a)
f e = A e exp { β e H e γ e μ } ,
(9b)
where A p = N p 1 exp { 1 α p } and A e = N e 1 exp { 1 α e } are normalization factors such that Π f p d Π = Π f e d Π = 1. In the same way βp and βe represent the characteristic inverse temperatures of the two particle species, the Lagrange multipliers γp and γe can be interpreted as chemical potentials describing the macroscopic energy changes β p d E p = γ p d M p and β e d E e = γ e d M e occurring when magnetic moments dMp, dMe are added to the system.

Next, it is useful to explain why Coulomb collisions have been neglected in the derivation of the collisionless equilibria (9). First, recall that, in the present setting, the (collective) electric potential Φ changes over a timescale Tf that is much longer than the cyclotron timescale Tc. This ensures that the individual magnetic moments of the particles remain constant, unless local Coulomb collisions occur. For the plasma regime under consideration, the frequency νC of such collisions can be estimated44 as ν C 5 × 10 11 nT 3 / 2 1 Hz, where n = 10 12 m 3 is the particle density and T = 10 eV is the temperature. Hence, over the confinement time scales τ c 1 / ν C 1 s considered in the model they are negligible, their effect being felt only over longer time intervals where particles undergo mostly small deflections with small changes in their kinetic energies. Indeed, recall that the distance r C 10 10 m at which Coulomb interactions are dominant is smaller than the typical particle distance r d 10 4 m.

We also remark that if a net transfer of total energy or total magnetic moment occurs between positrons and electrons, only the sums E p + E e and M p + M e are preserved. Such scenario can be taken into account by setting β = β e = β p and γ = γ p = γ e.

The expressions (9) for the distribution functions fp and fe can now be used to determine the spatial number densities np and ne according to Eq. (3). To evaluate the integral in Eq. (3), the phase space measure d Π must be expressed in a more convenient set of magnetic coordinates. To this end, we restrict our attention to magnetic fields of the type
B = Ψ × φ ,
(10)
where Ψ denotes the flux function and φ is the toroidal angle. Introducing a length coordinate along magnetic field lines with tangent vector = B / B, the functions ( , Ψ , φ ) define a set of curvilinear coordinates with Jacobian
· Ψ × φ = B · = B .
(11)
Next, decompose the particle velocity as
v = v + v c + v d ,
(12)
where v = v B / B is the velocity component along B, v c is the cyclotron velocity such that m v c 2 = 2 μ B [recall (4)], and v d = v d ( x ) is the particle drift velocity across B. Notice that Eq. (12) represents the velocity of a charged particle, and not the guiding center velocity. Furthermore, the drift velocity v d comprising E × B, gradient, and curvature drifts is treated as a spatial function, which is expected to be a valid approximation in a time-independent setting since in a vacuum field gradient and curvature drifts can be expressed as 2 k B T B × κ / q B 2, where κ is the field curvature, kBT is the temperature, and q = ± e is the relevant electric charge. Let r denote the cylindrical radius, ϑ c the phase of the cyclotron gyration, and A = Ψ φ the vector potential associated with the magnetic field (10). Then, the particle momentum p = m v + q A can be decomposed on the orthonormal set of basis vectors
( , Ψ | Ψ | , r φ ) ,
(13)
as
p = p + p Ψ Ψ | Ψ | + p φ r r φ ,
(14)
with
p = m v ,
(15a)
p Ψ = m ( v c sin ϑ c + v d · Ψ | Ψ | ) ,
(15b)
p φ = m r ( v c cos ϑ c + r v d · φ ) + q Ψ .
(15c)
It follows that at each point x Ω,
d 3 p = d p d p Ψ d ( p φ r ) = m 2 B d v d ϑ c d μ .
(16)
Combining (11), (15), and (16), we, thus, arrive at
d Π = m 2 d d v d φ d Ψ d ϑ c d μ .
(17)
Recalling (3) and assuming β p 0 , β e 0 , γ p 0, and γ e 0, the densities np and ne, therefore, have expressions
(18)
n p = m 2 N p 0 d μ 0 2 π d ϑ c d v A p B exp { β p ( 1 2 m v 2 + μ B + e Φ ) γ p μ } = σ p B exp { β p e Φ } γ p + β p B ,
(18a)
n e = m 2 N e 0 d μ 0 2 π d ϑ c d v A e B exp { β e ( 1 2 m v 2 + μ B e Φ ) γ e μ } = σ e B exp { β e e Φ } γ e + β e B ,
(18b)
where we defined the constants
σ p = ( 2 π m ) 3 2 A p N p β p , σ e = ( 2 π m ) 3 2 A e N e β e .
(19)
Substituting these expressions into Eq. (1), one, thus, obtains a second-order nonlinear partial differential equation,
Δ Φ = e ( 2 π m ) 3 2 ϵ 0 B ( A p N p β p exp { β p e Φ } γ p + β p B A e N e β e exp { β e e Φ } γ e + β e B ) ,
(20)
governing the spatial behavior of the electrostatic potential Φ within the magnetized positron–electron plasma under the effect of the magnetic field (10).
It is useful to consider equation (20) in the following limits. First, if N e N p, the system approaches a pure electron plasma. In such case, Eq. (20) reduces to
Δ Φ = e ( 2 π m ) 3 2 A e N e ϵ 0 β e B exp { β e e Φ } γ e + β e B .
(21)
Second, if N p = N e = N, the inverse temperature of the positron plasma equals that of the electron plasma β p = β e = β, and the Lagrange multipliers associated with the conservation of magnetic moment satisfy γ p = γ e = γ as well, Eq. (20) becomes
Δ Φ = e ( 2 π m ) 3 2 N ϵ 0 β B γ + β B ( A p exp { β e Φ } A e exp { β e Φ } ) .
(22)
Finally, if sufficiently long-time scales are considered, the conservation of the total magnetic moments Mp and Me is expected to break down. This scenario corresponds to the limit γ p = γ e = 0, which gives
Δ Φ = e ( 2 π m ) 3 2 ϵ 0 ( A p N p β p 3 2 exp { β p e Φ } A e N e β e 3 2 exp { β e e Φ } ) .
(23)
By further demanding N e N p, one recovers the Liouville equation,
Δ Φ = e A e N e ϵ 0 ( 2 π m β e ) 3 2 exp { β e e Φ } .
(24)
We conclude this section by observing that the departure from Maxwell–Boltzmann statistics occurring in equilibria such as Eq. (9) implies that the physical plasma temperatures (which differ from the temperatures Tp and Te associated with the Lagrange multipliers β p = ( k B T p ) 1 and β e = ( k B T e ) 1) are not spatially uniform. Indeed, integrals of the type,
k B T p ( x ) = 3 μ B f p d 3 p 3 f p d 3 p , k B T p ( x ) = 3 m 2 v 2 f p d 3 p 3 f p d 3 p ,
(25)
will generally exhibit a spatial dependence caused by the inhomogeneity of the magnetic field. Here, T p and T p are the perpendicular and parallel temperatures of the positron plasma. For example, using Eqs. (9a) and (16), one obtains
k B T p ( x ) = B ( x ) β p B ( x ) + γ p .
(26)
In this section, we consider a regime of plasma such that charged particles preserve both the first adiabatic invariant μ and the second adiabatic invariant (bounce action) J , which is defined by
J = m 2 π a b v d s .
(27)
Here, a and b denote the bouncing points along a field line with line element ds, while
v ( E , μ , , Ψ , φ ) = 2 m ( E μ B q Φ ) ,
(28)
represents the parallel velocity as a function of particle energy E, magnetic moment μ, and spatial position ( , Ψ , φ ). The conservation of the second adiabatic invariant (27) requires that timescale Tf of energy fluctuations caused by electromagnetic turbulence is longer than the timescale of bounce motion Tb, i.e.,
T f T b = 2 π ω b ,
(29)
where ωb denotes the bounce frequency. The (half) period Tb of the bounce oscillation can be written as
T b = a b d s v .
(30)
In the following, we shall, therefore, assume that both (5) and (29) hold.
Next, observe that the bounce averaged kinetic energy K b along the magnetic field can be evaluated as
K b = m 2 T b t a t b v 2 d t = m ω b 4 π a b v d s = 1 2 ω b J ,
(31)
where ta and tb are the instants at which the particle position reaches the bounce points a and b, while b denotes averaging over a bounce oscillation.

The actual magnetic field strength within the planned positron–electron trap will not be symmetric under vertical reflections z z due to a support coil placed at the top of the device to keep the main superconducting coil generating the dipole magnetic field levitated. Therefore, an accurate model of the trap would need to take into account such asymmetry. This problem will not be considered here to simplify the analysis, and a pure dipole magnetic field will be assumed. For completeness, we remark that the maximum field strength B supp of the support coil is expected to be less than 10% of the local dipole magnetic field B. In addition, this error is expected to be significantly smaller in the confinement region close to the superconducting coil. Since the particle density is roughly a linear function of the field strength [recall Eq. (18)], we, therefore, expect the support coil to introduce a relative error for the computed spatial density profiles of less than 10%.

Due to the axial symmetry and reflection symmetry of the dipole magnetic field strength B, one has a = b as well as v / φ = 0. Next, suppose that the number of positrons equals the number of electrons, Np = Ne. Then, we may assume the electric potential energy q Φ to be small compared to E μ B for the part of the bounce orbit close to the equatorial plane that contributes the most to J . In this case, Eqs. (27) and (30) can be simplified to
J = 2 m π 0 b ( E , μ , Ψ ) E μ B ( s , Ψ ) d s , T b = 2 m 0 b ( E , μ , Ψ ) d s E μ B ( s , Ψ ) ,
(32)
where the bouncing point b ( E , μ , Ψ ) > 0 is given as the positive solution of the equation E = μ B ( b , Ψ ), provided that such solution exists. Notice that both bounce action J and bounce frequency ω b = 2 π / T b are independent of particle charge q according to Eq. (32), and therefore, they have the same expression for both positrons and electrons. We also remark that, when N p N e, Eq. (32) is expected to remain a good approximation provided that the kinetic energy (temperature) kBT is sufficiently large compared to q Φ. Now observe that, using the identity (31) for the kinetic energy along the magnetic field, the positron energy and the electron energy can be approximated as
H ̃ p = 1 2 ω b J + μ B + e Φ , H ̃ e = 1 2 ω b J + μ B e Φ .
(33)
The errors H p H ̃ p and H e H ̃ e committed in replacing the energies Hp, He with H ̃ p , H ̃ e evidently go to zero when averaged over a bounce oscillation, H p H ̃ p b = H e H ̃ e b = 0. In the following, we shall be concerned with relaxation time scales τ longer than the bounce period, τ T b, and use the approximate expressions (33) for the particles energies. As it will be shown later, this approach simplifies calculations involving the particles distribution functions.
The conservation of the second adiabatic invariant gives rise to a macroscopic constraint on the statistical behavior of the system, which can be represented by adding the total bounce actions J p and J e given by
J p = N p Π f p J d Π , J e = N e Π f e J d Π ,
(34)
in the target functional (8) for the entropy principle. Introducing Lagrange multipliers ζp and ζe, the resulting expressions for the distribution functions fp and fe are
f p = A p exp { β p H ̃ p γ p μ ζ p J } ,
(35a)
f e = A e exp { β e H ̃ e γ e μ ζ e J } .
(35b)
Observe that Hp and He have been replaced with the approximated values H ̃ p and H ̃ e. Our next goal is to evaluate the spatial particle densities (3) so that Poisson's equation (1) can be applied to compute Φ. To this end, the phase space measure d Π must be expressed in a new set of coordinates that is appropriate to carry out integrals in momentum space. First, consider the change of variables ( , v , φ , Ψ , θ c , μ ) ( , E , φ , Ψ , θ c , μ ) with E = m v 2 / 2 + μ B + q Φ. Recalling (17), we, thus, have
d Π = m 2 d d v d φ d Ψ d ϑ c d μ = m v d dEd φ d Ψ d ϑ c d μ .
(36)
Next, perform the transformation ( , E , φ , Ψ , θ c , μ ) ( , J , φ , Ψ , θ c , μ ). In order to express the phase space measure in terms of the new coordinates, we must compute the partial derivative
J E = lim d E 0 m π d E [ 0 b ( E + d E , μ , Ψ ) v ( E + d E , μ , s , Ψ ) d s 0 b ( E , μ , Ψ ) v ( E , μ , s , Ψ ) d s ] = lim d E 0 m π d E [ d E 0 b v E ( E , μ , s , Ψ ) d s + d E b E ( E , μ , Ψ ) v ( E , μ , b , Ψ ) ] = 1 π 0 b d s v ( E , μ , s , Ψ ) = T b 2 π = 1 ω b .
(37)
Observe that here we used the fact that v ( E , μ , b , Ψ ) = 0. It follows that
d Π = m 2 d d v d φ d Ψ d ϑ c d μ = m ω b v d d J d φ d Ψ d ϑ c d μ .
(38)
To proceed, it is convenient to approximate the Jacobian v / m ω b with its bounce average,
v m ω b b = 2 m ω b T b 0 t b v d t = b m π ,
(39)
where we used the fact that by definition ω b = ω b ( E , μ , Ψ ) and, therefore, ω b / = 0. The bounce averaged phase space measure, thus, reads as
d Π ̃ = m π b d d J d φ d Ψ d ϑ c d μ = m π B b d J d ϑ c d μ d 3 x .
(40)
The spatial densities np and ne can now be computed as follows. For the positron component, we have
n p = m π A p N p B 0 d μ 0 d J 0 2 π d ϑ c b 1 exp { β p ( 1 2 ω b J + μ B + e Φ ) γ p μ ζ p J } = 2 m π 2 A p N p B 0 d μ 0 d J b 1 exp { β p ( 1 2 ω b J + μ B + e Φ ) γ p μ ζ p J } .
(41)
To simplify (41), we now follow the approach developed in Ref. 38. In general, the expression of the bounce frequency ωb is a function of J , μ, and Ψ. However, the dependence on the bounce action J disappears in the limit b / L 1 in which the bounce orbit size b is shorter than the characteristic magnetic field line length L. Indeed, in this limit, we may expand the magnetic field in powers of around the equatorial point = 0 of the dipole to obtain a second-order equation for the bounce position,
E = μ B ( 0 , Ψ ) + 1 2 μ 2 B 2 ( 0 , Ψ ) b 2 .
(42)
Here, we used the fact that B / = 0 at = 0. It is also worth observing that in a dipole field 2 B / 2 ( 0 , Ψ ) > 0 since B has a minimum at = 0 for each magnetic surface Ψ. Setting B 0 = B ( 0 , Ψ ) and B 0 = ( 2 B / 2 ) ( 0 , Ψ ), the positive bounce point b can, therefore, be approximated as
b = 2 ( E μ B 0 ) μ B 0 .
(43)
For a typical particle at = 0, the energy E can be decomposed into a perpendicular component E 0 = μ B 0 and a parallel component E 0 = α 0 p E 0 = α 0 p μ B 0, where α 0 p ( Ψ ) > 0 is the local positron temperature anisotropy at the equator. We may, therefore, estimate
1 b B 0 2 α 0 p B 0 .
(44)
Similarly, at second order, the bounce frequency becomes
1 ω b = 1 π 0 b d s 2 m ( E μ B 0 1 2 μ B 0 s 2 ) = m π μ B 0 0 b μ B 0 2 ( E μ B 0 ) d y 1 y 2 = m 2 μ B 0 .
(45)
Note that in the last passage Eq. (43) was used. Observe that the bounce frequency is now a function of μ and Ψ only. Recalling Eq. (41), we, thus, arrive at the following expression for the positron density:
n p = 2 m π 2 A p N p B B 0 2 α 0 p B 0 exp { β p e Φ } 0 d μ 0 d J exp { ( β p B + γ p ) μ ( β p μ B 0 m + ζ p ) J } , = 2 m π 2 A p N p B B 0 2 α 0 p B 0 exp { β p e Φ } 0 d μ exp { ( β p B + γ p ) μ } β p μ B 0 m + ζ p = 2 m π 2 A p N p B 2 α 0 p B 0 exp { β p e Φ } β p B + γ p 0 e y d y β p y m ( β p B + γ p ) + ζ p B 0 .
(46)
A similar expression can be obtained for the electron density ne by flipping the sign of the electric charge, and the Poisson equation (1) for the electrostatic potential Φ becomes
Δ Φ = 2 e m π 2 ϵ 0 B 2 B 0 [ A p N p exp { β p e Φ } α 0 p ( β p B + γ p ) 0 e y d y β p y m ( β p B + γ p ) + ζ p B 0 A e N e exp { β e e Φ } α 0 e ( β e B + γ e ) 0 e y d y β e y m ( β e B + γ e ) + ζ e B 0 ] .
(47)
The last integral in Eq. (46) can be written in terms of special functions. Furthermore, it gives a simple result in the limit ζ p / β p ω b 1. Indeed, one obtains
n p = 2 π ( m π ) 3 / 2 A p N p β p B exp { β p e Φ } 2 α 0 p B 0 ( β p B + γ p ) .
(48)
Here, it should be noted that although the Lagrange multiplier ζp has been neglected, and, thus, the conservation of the total bounce action J p is not felt by the ensemble, the effect of bounce motion on the particle distribution does not disappear. This is because bounce dynamics is encapsulated in the term ω b J / 2 of the Hamiltonian H ̃ p appearing in the distribution function fp. Observing that in the same limit ζ e / β e ω b 1 the electron density becomes
n e = 2 π ( m π ) 3 / 2 A e N e β e B exp { β e e Φ } 2 α 0 e B 0 ( β e B + γ e ) ,
(49)
the corresponding form of the Poisson equation (1) for the electrostatic potential Φ, therefore, reads
Δ Φ = 2 π e ( m π ) 3 / 2 ϵ 0 B 2 B 0 [ A p N p exp { β p e Φ } β p α 0 p ( β p B + γ p ) A e N e exp { β e e Φ } β e α 0 e ( β e B + γ e ) ] .
(50)
It is possible to develop an analogous model where the third adiabatic invariant (flux function) Ψ is also preserved by single particle dynamics and enforce conservation of total magnetic fluxes Ψ p = N p Π Ψ f p d Π and Ψ e = N e Π Ψ f e d Π in the corresponding entropy principle to obtain the distribution functions
f p = A p exp { β p H ̃ p γ p μ ζ p J η p Ψ } , f e = A e exp { β e H ̃ e γ p μ ζ e J η e Ψ } ,
(51)
where η p , η e are Lagrange multipliers associated with Ψ p , Ψ e. It is not difficult to verify that the density np of Eq. (46) is modified by an exponential factor exp { η p Ψ }. A similar modification applies to ne so that the Poisson equation (47) for the electrostatic potential Φ now reads
Δ Φ = 2 e m π 2 ϵ 0 B 2 B 0 [ A p N p exp { β p e Φ η p Ψ } α 0 p ( β p B + γ p ) 0 e y d y β p y m ( β p B + γ p ) + ζ p B 0 A e N e exp { β e e Φ η e Ψ } α 0 e ( β e B + γ e ) 0 e y d y β e y m ( β e B + γ e ) + ζ e B 0 ] .
(52)
However, it should be emphasized that for the third adiabatic invariant to be constant, the timescale Tf of energy fluctuations must be longer than the characteristic timescale Td of drift dynamics v d across the magnetic field B in the toroidal direction φ, i.e.,
T f T d = 2 π ω d ,
(53)
where ωd is the drift frequency. Since ωd is mainly determined by the E × B, curvature, and gradient drift velocities, this frequency is usually smaller than the cyclotron and bounce frequencies, and (53) poses a rather stringent condition on the allowed turbulent spectrum of electromagnetic fluctuations.

Finally, we remark that the contributions to the particle energy coming from the guiding center drifts are neglected in the guiding center energies H ̃ p , H ̃ e under the assumption that the corresponding kinetic energy is smaller than the other terms. In particular, this is true if the kinetic energy associated with the E × B drift v E = E × B / B 2 satisfies the ordering m | v E | 2 / 2 H ̃ p , H ̃ e. Such configuration can be achieved, for example, when the electric potential Φ is a first-order contribution in the ratio ϵ ρ c / L, where ρc is the Larmor radius and L is the characteristic scale length of the magnetic field. For a plasma with inverse temperature β, this implies β e | Φ | ϵ as well as β m 2 | v E | 2 ϵ 2, and the energies H ̃ p , H ̃ e are accurate at first order in ϵ. For additional details on the expression of the guiding center Hamiltonian, see Ref. 42.

The aim of this section is to numerically study the behavior of the positron–electron plasma as described by the model developed in Secs. II–IV for different values of the physical parameters involved.

For the spatial domain Ω occupied by the plasma, we consider a region r [ r 0 , r 0 + R ] , z [ R / 2 , R / 2 ] , φ [ 0 , 2 π ) mimicking an axially symmetric trap with radial size R, height R, and a physical axis (center stack) whose external boundary is located at r = r 0. In particular, we choose R = 1 m and r 0 = 0.1 m, which are values of the same order of the length parameters of the planned positron–electron trap. The dipole magnetic field used to confine the plasma is generated by a current loop of infinitesimal section and radius r l = 0.25 m enclosed in an axially symmetric toroidal box (coil) with squared cross section whose left side is located at r box = 0.2 m (see Ref. 45 for the expression of the magnetic field). The typical magnetic field generated by the current loop is around B 0.1 T and reaches B 2 T in close proximity of the bounding box. It is also convenient to introduce the characteristic magnetic field B = μ 0 I / 2 r l = 1.25 T, where μ0 is the vacuum permeability and I is the electric current flowing within the current loop. The setting described above is shown in Fig. 1.

FIG. 1.

(a) Contour plot of the magnetic field strength | B | within the domain Ω enclosed by the positron–electron trap. (b) Contour plot of | B | in the region surrounding the current loop. In both (a) and (b), dashed contours correspond to magnetic field lines, while white regions exceed the plotted range of values. The square centered at r = r l = 0.25 m represents the section of the toroidal box containing the current loop generating the dipole field.

FIG. 1.

(a) Contour plot of the magnetic field strength | B | within the domain Ω enclosed by the positron–electron trap. (b) Contour plot of | B | in the region surrounding the current loop. In both (a) and (b), dashed contours correspond to magnetic field lines, while white regions exceed the plotted range of values. The square centered at r = r l = 0.25 m represents the section of the toroidal box containing the current loop generating the dipole field.

Close modal

We will now examine the plasma equilibria corresponding to the statistical ensembles constructed in Secs. II–IV separately. To this end, we numerically solve the Poisson equation (1) for the electrostatic potential Φ by using the source term (2) obtained from the theoretical model. For example, the conservation of the first adiabatic invariant considered in Sec. II leads to the Poisson equation (20). Then, the distribution functions and the spatial densities can be evaluated according to Eqs. (9) and (18) by using the resulting value of Φ. In order to solve the Poisson equation (1), we use the NDSolveValue solver of Wolfram Mathematica 12.2. The specific parameter values and boundary conditions used in each simulation are described in the corresponding sections.

Consider the positron–electron plasma equilibrium (9) arising from the conservation of total magnetic moments Mp, Me. We shall refer to such equilibrium as a μ equilibrium. The condition for a μ equilibrium to hold is that the timescale of the electromagnetic fluctuations driving the system toward the relaxed state is longer than the timescale of cyclotron dynamics as described by Eq. (5). In practice, this means that the term q Φ occurring within the Hamiltonians Hp, He evolves over long time scales compared to the cyclotron timescale. This ensures that the constancy of the first adiabatic invariant μ is not broken.

For the system under consideration, the cyclotron frequency ωc is
ω c = e B m 10 10 Hz ,
(54)
where the typical value B 0.1 T has been used. Denoting with ω f = 2 π / T f the frequency of electromagnetic fluctuations, the condition (5) can, therefore, be written as
ω f ω c 10 10 Hz .
(55)
It is worth observing that for a comparable trap such as the RT-1 device,25 the spectrum of electromagnetic fluctuations is well below the 10 5 Hz range. We, therefore, expect the condition (55) to be easily fulfilled by the system under examination. We shall, therefore, assume that (55) holds and apply the equilibrium model developed in Sec. II.
In order to evaluate the densities (18), the Poisson equation (20) must be solved for the electrostatic potential Φ. The physical parameters Ap, Ae, Np, Ne, βp, βe, γp, and γe appearing on the right-hand side of equation (20) are determined as follows. First, the target temperature T for the positron–electron plasma is T T p T e 10 eV, where Tp and Te denote the temperatures of positrons and electrons, respectively. This fixes the inverse temperatures β p 1 = k B T p and β e 1 = k B T e, where kB is the Boltzmann constant. The target spatial density n is n n p n e 10 11 m 3 up to 10 12 m 3. Since the volume of the positron–electron plasma trap is of the order Ω 1 m 3, we consider a total particle number of the order N N p N e 10 11. Next, an estimate of the chemical potentials γp and γe can be obtained by observing that the changes in the total energies Ep, Ee caused by the addition of magnetic moments dMp, dMe to the system are given by β p d E p = γ p d M p and β e d E e = γ e d M e. The order of magnitude of γ p , γ e is, therefore, expected to be γ p β p H p / μ β p B and γ e β e H e / μ β e B. Finally, the parameters Ap, Ae, which represent measures of the volume Π occupied by the plasma in the phase space, can be estimated by observing that for a Maxwell–Boltzmann distribution
f B ( x , p ) = A exp { β p 2 2 m } ,
(56)
one has
A = 1 Π exp { β p 2 2 m } d 3 x d 3 p = 1 Ω ( β 2 π m ) 3 2 .
(57)
We, therefore, expect that A p Ω 1 ( β p / 2 π m ) 3 / 2 and A e Ω 1 ( β e / 2 π m ) 3 / 2.
The last ingredient needed to numerically solve Eq. (20) is the value of the potential Φ on the boundary Ω. Note that the boundary Ω consists of the vessel boundary Ω v and the coil surface Ω c. At the vessel boundary Ω v, the electrostatic potential Φ is grounded, i.e., Φ = 0 on Ω v. The coil generating the dipole magnetic field is levitated without mechanical contact through a secondary magnet located in the upper region of the trap. Therefore, the value Φ c of the electrostatic potential on the coil surface Ω c is determined by the amount of charged particles that hit it. In particular, electrons penetrating the coil surface push Φ c toward negative values, while positrons tend to increase Φ c (note that even if positrons annihilate with electrons after reaching the coil surface, this results in a positive increase in the coil charge). An upper bound to Φ c can be obtained from the Poisson equation (1) through the scaling,
Φ e n R 2 ϵ 0 10 3 V ,
(58)
where R is the radial size of the device and the value n 10 11 m 3 has been used. In practice, Φ c is expected to be much smaller than (58) since the charge separation e ( n p n e ) will be sensibly smaller than the characteristic charge density e n 10 8 C m 3 of each individual species. In summary, the boundary conditions that will be used to evaluate the Poisson equation (20) are
Φ = 0 on Ω v , Φ = Φ c on Ω c ,
(59)
with | Φ c | < 10 3 V.

Figure 2 shows the modification of spatial densities np, ne and electrostatic potential Φ obtained from numerical solution of Eq. (20) when varying the coil potential Φ c while keeping the other physical parameters fixed. As one may expect, a negatively charged coil attracts the positron component while repelling electrons. The converse occurs for a positively charged coil. The other essential feature of Fig. 2 is the effect of the first adiabatic invariant μ on the macroscopic equilibrium state of the system: both positrons and electrons tend to accumulate in regions of higher magnetic field strength. This feature combined with the dependence with respect to the coil potential implies that when Φ c > 0 positrons preferentially populate the interior region close to the center stack and enclosed by the coil, r < r box, while electrons surround the coil surface (first row in Fig. 2). The opposite occurs when Φ c > 0 (third and fourth rows in Fig. 2). Inspection of Eq. (20) also suggests that if the thermodynamic parameters of the two species are equal, i.e., Ap = Ae, Np = Ne, β p = β e, and γ p = γ e, and Φ c = 0 on Ω c, then, Φ vanishes throughout Ω while the spatial densities are identical, np = ne. This case is shown in the second row of Fig. 2.

FIG. 2.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ equilibrium. Each row corresponds to a different value of the coil potential Φ c: from top to bottom 20 , 0 , 10, and 20 V. The physical parameters used in this simulation are A p = A e = 3.8 Ω 1 ( β / 2 π m ) 3 / 2 with β 1 = k B T and T = T p = T e = 10 eV , N p = N e = 10 11, and γ p = γ e = 0.1 β B with B = 1.25 T the characteristic magnetic field. Dashed contours represent magnetic field lines. White regions exceed the plotted range of values.

FIG. 2.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ equilibrium. Each row corresponds to a different value of the coil potential Φ c: from top to bottom 20 , 0 , 10, and 20 V. The physical parameters used in this simulation are A p = A e = 3.8 Ω 1 ( β / 2 π m ) 3 / 2 with β 1 = k B T and T = T p = T e = 10 eV , N p = N e = 10 11, and γ p = γ e = 0.1 β B with B = 1.25 T the characteristic magnetic field. Dashed contours represent magnetic field lines. White regions exceed the plotted range of values.

Close modal

Taking the third row in Fig. 2 as reference case, Fig. 3 summarizes how spatial densities np, ne and electrostatic potential Φ change when physical parameters are modified. While equilibrium profiles are not too sensitive to changes in the chemical potentials γ p , γ e and in the inverse temperatures β p , β e, abrupt changes occur when there is asymmetry in the number of protons and electrons. The bottom row of Fig. 3 shows the case N e = 2 × 10 11 N p = 10 9. Note that positrons are almost completely expelled from the interior region r < r box and form a radiation belt-like structure on the exterior side of the coil, while electrons are mostly found in proximity of the center stack located at r = r0.

FIG. 3.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ equilibrium. Each row corresponds to a modification of one of the parameters used for the case shown in the third row of Fig. 2. In the first and second rows, the chemical potentials have been increased to γ p = γ e = 0.3 β B and γ p = γ e = 0.6 β B , respectively. In the third and fourth rows, the temperatures are T p = 100 and T e = 4 T p = 20 eV. In the fifth row, the particles numbers are N e = 2 × 10 11 and N p = 10 9. Parameter changes are also highlighted in each Φ plot. Dashed contours represent magnetic field lines. White regions exceed the plotted range of values.

FIG. 3.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ equilibrium. Each row corresponds to a modification of one of the parameters used for the case shown in the third row of Fig. 2. In the first and second rows, the chemical potentials have been increased to γ p = γ e = 0.3 β B and γ p = γ e = 0.6 β B , respectively. In the third and fourth rows, the temperatures are T p = 100 and T e = 4 T p = 20 eV. In the fifth row, the particles numbers are N e = 2 × 10 11 and N p = 10 9. Parameter changes are also highlighted in each Φ plot. Dashed contours represent magnetic field lines. White regions exceed the plotted range of values.

Close modal
We now consider the case in which individual particles preserve the second adiabatic invariant J . For the bounce action to be constant, the timescale Tf of electromagnetic fluctuations must be longer than the period Tb of a bounce oscillation as described by Eq. (29). An estimate of the bounce frequency ωb (and, thus, of Tb) can be obtained with the aid of Eq. (45), which rests on the hypothesis μ B e | Φ | or β p 1 , β e 1 e | Φ |. Therefore, assuming these conditions to hold, the typical bounce frequency is
ω b = 2 μ B 0 m 2 β 1 R m 10 6 Hz .
(60)
Here, the values β 1 = k B T with T = 10 eV and the trap scale length R = 1 m have been used. Hence, the condition (29) now reads
ω f ω b 10 6 Hz .
(61)
Notice that since ω b ω c [recall Eq. (55)] if the condition (61) is satisfied, then, the first adiabatic invariant μ is also preserved. Again, experimental evidence from the RT-1 device suggests that the requirement (61) is usually fulfilled since ω f < 10 5 Hz there.25 

In addition to the physical parameters Ap, Ae, Np, Ne, βp, βe, γp, and γe already encountered for the μ equilibrium, the μ- J equilibrium described by the Poisson equation (47) includes the temperature anisotropies α 0 p and α 0 e and the Lagrange multipliers ζp and ζe associated with the conservation of the total bounce actions J p and J e. In the following, we study a plasma with α 0 p = α 0 e = 1. Notice that if the plasma is isotropic α 0 p = α 0 e = 1 / 2 since perpendicular motion carries twice the degrees of freedom of parallel dynamics. Nevertheless, as long as α 0 p and α 0 e are constant, the spatial densities profiles remain unchanged, but they are only scaled by factors 1 / α 0 p and 1 / α 0 e, respectively [recall Eq. (46)]. We shall also consider the limiting case ζ p / β p ω b 1 and ζ e / β e ω b 1 to simplify the evaluation of the integrals on the right-hand side of Eq. (47). Then, the relevant Poisson equation is given by Eq. (50), where ζ p , ζ e do not appear explicitly. Notice that physically this implies that the total bounce actions J p , J e are not preserved during the relaxation of the system. However, the effect of bounce dynamics is still felt by the ensemble through the term ω b J / 2 appearing in the energies H ̃ p , H ̃ e and the bounce averaged phase space measure d Π ̃ of Eq. (40).

Figure 4 shows μ- J equilibria obtained by numerical solution of the Poisson equation (50) for different values of the coil potential Φ c when the other physical parameters are kept constant. As in the case of μ equilibria, a charged coil tends to push the species with charge of the same sign in the internal region r < r box while attracting the species with opposite charge. Next, observe that, as in the previous case, the conservation of the magnetic moment μ results in the tendency of particles to accumulate in regions of high magnetic field strength. However, there is a peculiar feature of μ- J equilibria: bounce dynamics squeezes spatial density profiles along the equatorial line z = 0. As a result, radiation belt-like structures are formed on both sides of the coil. This behavior is a consequence of the term 1 / B 0 appearing in the expression of the spatial densities np, ne [recall Eq. (48)], which is related to the characteristic bounce velocity vb given by
v b = 2 b T b = 1 π b ω b = 2 α p B 0 μ m ,
(62)
where Eqs. (44) and (45) were used. Since n p , n e 1 / B 0, and the characteristic bounce velocity v b B 0 is higher in regions of stronger magnetic field strength B 0 = B ( r , 0 ), the spatial densities have a minimum in correspondence of the maximum of B0, which is located at the radial position of the current loop r = r l. Physically, the formation of radiation belt-like structures can be understood as follows. Particles with a high characteristic bounce velocity are statistically less likely to occur, and the net result is the balance between the tendency of particles to populate regions of high magnetic field strength B as a result of the first adiabatic invariant, and the depletion effect caused by the second adiabatic invariant at those radial positions where B0, and thus vb, are higher.
FIG. 4.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ- J equilibrium. Each row corresponds to a different value of the coil potential Φ c: from top to bottom 20 , 1 , 10 ,, and 20 V. The parameters used in this simulation are A p = A e = 2.4 Ω 1 ( β / 2 π m ) 3 / 2 with β 1 = k B T and T = T p = T e = 100 eV , N p = 6 × 10 11 , N e = 8 × 10 11 , α 0 p = α 0 e = 1, and γ p = γ e = 0.05 β B with B = 1.25 T the characteristic magnetic field. Dashed contours represent magnetic field lines.

FIG. 4.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ- J equilibrium. Each row corresponds to a different value of the coil potential Φ c: from top to bottom 20 , 1 , 10 ,, and 20 V. The parameters used in this simulation are A p = A e = 2.4 Ω 1 ( β / 2 π m ) 3 / 2 with β 1 = k B T and T = T p = T e = 100 eV , N p = 6 × 10 11 , N e = 8 × 10 11 , α 0 p = α 0 e = 1, and γ p = γ e = 0.05 β B with B = 1.25 T the characteristic magnetic field. Dashed contours represent magnetic field lines.

Close modal

The dependence of spatial densities np, ne and electrostatic potential Φ on the various physical parameters is shown in Fig. 5. Each row corresponds to a variation of one of the parameters used in the example plotted in the second row of Fig. 4.

FIG. 5.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ J equilibrium. Each row corresponds to a modification of one of the parameters used for the case shown in the second row of Fig. 4. In the first and second rows, the chemical potentials have been modified to γ p = γ e = 0.3 β B and γ e = 10 γ p = 0.1 β B , respectively. In the third and fourth rows, the temperatures are T p = 10 and T e = 400 eV, respectively. In the fifth row, the positron number is N p = 4 × 10 11. Parameter changes are also highlighted within each Φ plot. Dashed contours represent magnetic field lines. White regions exceed the plotted range of values.

FIG. 5.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ J equilibrium. Each row corresponds to a modification of one of the parameters used for the case shown in the second row of Fig. 4. In the first and second rows, the chemical potentials have been modified to γ p = γ e = 0.3 β B and γ e = 10 γ p = 0.1 β B , respectively. In the third and fourth rows, the temperatures are T p = 10 and T e = 400 eV, respectively. In the fifth row, the positron number is N p = 4 × 10 11. Parameter changes are also highlighted within each Φ plot. Dashed contours represent magnetic field lines. White regions exceed the plotted range of values.

Close modal
If the frequency of electromagnetic fluctuations is sufficiently small, the third adiabatic invariant Ψ is preserved in addition to μ and J . For the system under examination, the drift frequency is determined by the frequency of the toroidal rotation around the vertical axis. At equilibrium, the toroidal drift velocity v φ = r v d · φ is given by the toroidal component of the guiding center drift velocity v d = v E + v k, which is the sum of E × B drift
v E = E × B B 2 = × Φ B ,
(63)
and gradient plus curvature drift
v k = 2 E + E q B × 2 = 1 + 2 α q β B × 2 ,
(64)
where 2 = · is the curvature of the magnetic field, and β and α = E / E are the inverse temperature and the temperature anisotropy of the particle species under consideration. In deriving (64), we used the fact that in a vacuum magnetic field the term × B and the curvature 2 are related by × ( × B ) = × ( B × ) B 2 = 0, which implies × B = B × 2. When the plasma temperature β 1 and the magnetic field curvature 2 are sufficiently small, the E × B drift velocity becomes the dominant contribution to v d. Then, the order of the drift frequency can be evaluated as
ω d E B R Φ B 10 4 Hz ,
(65)
where the estimate (58) for the electrostatic potential Φ, the typical magnetic field B 0.1 T, and the trap radial size R = 1 m were used. Unfortunately, experimental evidence25 suggests that the frequency of electromagnetic fluctuations within the trap will be comparable to Eq. (65), implying that the criterion for the existence of a third adiabatic invariant Ψ,
ω f ω d ,
(66)
will not be satisfied. Nevertheless, it is instructive to explore how the presence of the third adiabatic invariant Ψ modifies the spatial density profiles and the electrostatic potential at equilibrium.
Figure 6 shows the spatial densities np, ne and the electrostatic potential Φ obtained by numerical solution of Eq. (52) for different values of the chemical potentials η p , η e associated with the third adiabatic invariant Ψ. The effect of the preservation of total magnetic fluxes Ψ p , Ψ e is felt by the system only for large values of η p , η e, corresponding to large changes in energy β p d E p = η p d Ψ p 10 , β e d E e = η e d Ψ e 10 when magnetic fluxes d Ψ p , d Ψ e R 2 B 1 m 2 T are added to the system. The spatial densities are initially flattened at the equator (compare the spatial densities of first and second rows in Fig. 6), eventually splitting into upper and lower lobes separated by the equatorial line z = 0 (compare the spatial densities of first and third rows in Fig. 6). Physically, this behavior can be understood as follows. The kinetic energy associated with toroidal drift motion can be written as
K φ = 1 2 m v φ 2 = 1 2 m r 2 ( p φ q Ψ ) 2 .
(67)
FIG. 6.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ J Ψ equilibrium. Each row corresponds to a modification of one of the parameters used for the case shown in the second row of Fig. 4. In the first row, η p = η e = 1.6 / R 2 B . In the second row, η p = η e = 16 / R 2 B . In the third row, η p = η e = 47 / R 2 B . Here, R = 1 m is the radial size of the trap, while B = 1.25 T is the characteristic magnetic field. Parameter changes are also highlighted within each Φ plot. Dashed contours represent magnetic field lines. White regions exceed the plotted range of values.

FIG. 6.

Electrostatic potential Φ (left column), electron density ne (center column), and positron density np (right column) in the μ J Ψ equilibrium. Each row corresponds to a modification of one of the parameters used for the case shown in the second row of Fig. 4. In the first row, η p = η e = 1.6 / R 2 B . In the second row, η p = η e = 16 / R 2 B . In the third row, η p = η e = 47 / R 2 B . Here, R = 1 m is the radial size of the trap, while B = 1.25 T is the characteristic magnetic field. Parameter changes are also highlighted within each Φ plot. Dashed contours represent magnetic field lines. White regions exceed the plotted range of values.

Close modal

This term does not appear in the guiding center Hamiltonians (6) because it is usually smaller than the kinetic energies associated with parallel and cyclotron dynamics. In particular, this implies that p φ q Ψ, with p φ the canonical momentum of a charged particle in the φ direction and where we used the fact that in a dipole magnetic field the vector potential is A = Ψ φ. Since states with large deviations in the canonical momentum p φ q Ψ would break the conservation of the total magnetic flux Ψ tot, regions with large Ψ are penalized in the distribution functions through the factors e η p Ψ and e η e Ψ. If the chemical potentials η p , η e are sufficiently large, this effect prevails on the tendency caused by the first adiabatic invariant μ to concentrate particles in regions with strong magnetic field strength B, resulting in a preferential depletion of the equatorial region outside the coil (recall that along magnetic field lines, which correspond to level sets of Ψ, the magnetic field strength increases when approaching the coil from the outside).

In this section, we consider basic aspects pertaining to the macroscopic toroidal rotation properties of the positron–electron plasma. Using Eqs. (63) and (64) and recalling that B = Ψ × φ, the toroidal drift velocity has expression
v φ = r | Ψ | 2 ( Φ + 1 + 2 α q β 2 ) · Ψ = r Φ Ψ + r | Ψ | 2 ( Φ + 1 + 2 α q β 2 ) · Ψ .
(68)
It is well known that a single species plasma trapped in a straight homogeneous magnetic field B = B z z , B z relaxes to a self-organized rigidly rotating equilibrium with toroidal rotation velocity v φ = ω z r , ω z (see, e.g., Ref. 23) This result can be recovered from Eq. (68) as follows. First, observe that a straight magnetic field can be expressed as B = Ψ × φ with Ψ = B z r 2 / 2 + c Ψ and c Ψ . Furthermore, since = z and 2 = 0, the toroidal drift velocity reduces to
v φ = r Φ Ψ .
(69)
In addition, the homogeneity of the magnetic field implies that bounce motion is absent, while the conservation of the first adiabatic invariant μ does not affect the profile of the spatial density [recall Eq. (18)]. The only relevant constraint is, thus, that given by the third adiabatic invariant Ψ. Indeed, denoting with v = v + v E the velocity of a charged particle (other drifts are absent due to the homogeneity of the magnetic field), one has
d Ψ d t = v · Ψ = Φ φ = 0 ,
(70)
where we used the fact that the electrostatic potential is expected to be axisymmetric, Φ / φ = 0. The conservation of the total magnetic flux Ψ tot = N Π Ψ f d Π, where f is the particle distribution function and N is the particle number, then leads to a spatial density distribution
n = A exp { β q Φ η Ψ } ,
(71)
where A is a positive real constant and η is a Lagrange multiplier associated with the preservation of total magnetic flux. The Poisson equation for the electrostatic potential Φ in an infinite vertically symmetric Φ / z = 0 plasma column now reads
1 r r ( r Φ r ) = q A ϵ 0 exp { β q Φ 1 2 η B z r 2 η c Ψ } .
(72)
This equation admits the solution
Φ = η B z 2 β q r 2 + c Φ = η β q Ψ + η c Ψ β q + c Φ ,
(73)
such that Φ ( 0 ) = c Φ and Φ ( 0 ) = 0 with Φ = Φ / r, while the constant c Ψ is determined by the equation
exp { η c Ψ } = 2 η B z ϵ 0 β q 2 A exp { β q c Φ } .
(74)
Notice that the corresponding spatial density is constant, n = A exp { β q c Φ η c Ψ }. From Eqs. (69) and (73), it, therefore, follows that the plasma rigidly rotates around the z-axis with velocity
v φ = η β q r .
(75)
This result also implies that the frequency of the rotation is ω z = η / β q and that the magnetic field generated by the rotating plasma works to cancel the external magnetic field. Furthermore, the conservation of the total magnetic flux, which amounts to the conservation of the ensemble average of the squared radial position of charged particles, r 2 , provides radial confinement to a system with particles initially contained within a given radius. It is also worth noticing that this same confinement principle works even for a neutral plasma as long as the third adiabatic invariant holds. To see this, consider the simple case A p = A e = A , β p = β e = β, and η p = η e = η. Then, the Poisson equation for the electrostatic potential admits the trivial solution Φ = 0 such that n p = n e exp { η Ψ } = exp { η B z r 2 η c Ψ }, which results in radial confinement of the plasma. Unfortunately, it is known that standard neutral plasmas are poorly confined by a straight magnetic field due to the inherent fragility of the third adiabatic invariant, which is rapidly destroyed by symmetry-breaking electromagnetic perturbations, and the impossibility of containing the plasma at the vertical ends of the trap via electric fields.23,46 We remark that, however, we are not aware of positron–electron experiments in this context.
In a dipole magnetic field, the situation is essentially different because the confining mechanism is provided by the first adiabatic invariant μ, which forces particles in regions of high magnetic field strength, and not by the magnetic flux Ψ as in a straight magnetic field. As shown in Sec. II, this also implies that, in contrast with a straight magnetic field, a dipole magnetic field is suitable to trap both neutral and nonneutral plasmas. Notice also that the toroidal rotation velocity (68) now depends on both the electrostatic potential Φ and the underlying magnetic field geometry through Ψ , , and 2. Figure 7 shows the toroidal rotation velocity obtained by numerical evaluation of Eq. (68) for the case reported in the second row of Fig. 4. Both the positron toroidal velocity v φ p and the electron toroidal velocity v φ e increase with r but have opposite directions. This implies that, for the case considered, the charge-dependent drift v k is dominant with respect to v E. Furthermore, there is a net toroidal current density J φ = e ( n p v φ p n e v φ e ). Nevertheless, the resulting magnetic field B is negligible with respect to the dipole magnetic field. Indeed,
B μ 0 e R n v φ 10 9 T ,
(76)
where μ0 is the vacuum permeability and the characteristic values R = 1 m , n 10 11 m 3, and v φ 10 5 ms 1 were used.
FIG. 7.

Toroidal rotation velocity v φ in the μ J equilibrium. Left: electron toroidal rotation velocity v φ e. Right: positron toroidal rotation velocity v φ p. Dashed contours represent magnetic field lines.

FIG. 7.

Toroidal rotation velocity v φ in the μ J equilibrium. Left: electron toroidal rotation velocity v φ e. Right: positron toroidal rotation velocity v φ p. Dashed contours represent magnetic field lines.

Close modal

In this work, we studied the maximum entropy states of a collisionless positron–electron plasma trapped by a dipole magnetic field with the aim of elucidating the confinement properties of the system for different ranges of physical parameters. Such dipole magnetic field trap has several potential applications, including containment of pair and antimatter plasmas, experimental investigation of exotic and astrophysical plasmas, as well as technology development such as realization of coherent gamma ray lasers.

In a dipole magnetic field, the nature of plasma equilibria depends on the presence of adiabatic invariants. For such conserved quantities to hold, the timescale of electromagnetic fluctuations affecting the energy of a charged particle must be longer than the timescale of the periodic motion associated with each adiabatic invariant. Each adiabatic invariant constrains the maximum entropy state, resulting in a departure from standard Maxwell-Boltzmann statistics of an ideal gas.

Compared to a plasma trap with a straight magnetic field (such as a Penning–Malmberg trap) where radial confinement is provided by the conservation of the fragile third adiabatic invariant (the canonical momentum p φ q Ψ), in a dipole magnetic field containment is realized through the first adiabatic invariant (the magnetic moment μ), which results in a tendency of each charged species to move toward regions of high magnetic field strength B. For this reason, a dipole magnetic field is suitable to confine both neutral and nonneutral plasmas. The effect of the second adiabatic invariant (the bounce action J ) is to squeeze the spatial densities of both positrons and electrons along the equatorial line, with the formation of characteristic radiation belt-like structures around the coil.

By solving Poisson's equation for the electrostatic potential with the charge density obtained from the maximum entropy states as source term, we put the theoretical model to the test and showed efficient confinement of both species for a wide range of physical parameters. The equilibrium profiles appear to be mostly sensible to asymmetries in the number densities of the two species and to significant changes in the coil potential. This latter fact suggests that the capability to control the coil potential would be a desirable property of any experimental dipole magnetic field trap design.

The research of NS was partially supported by JSPS KAKENHI Grant Nos. 21K13851 and 22H04936. The author is grateful to H. Saitoh, who made helpful suggestions and critiqued a preliminary draft of the paper.

The authors have no conflicts to disclose.

Naoki Sato: Conceptualization (lead); Formal analysis (lead); Writing – original draft (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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