Chaos of energetic positron orbits in a dipole magnetic field and its potential application to a new injection scheme

We study the behavior of high-energy positrons emitted from a radioactive source in a magnetospheric dipole field configuration. Because the conservation of the first and second adiabatic invariants is easily destroyed in a strongly inhomogeneous dipole field for high-energy charged particles, the positron orbits are nonintegrable, resulting in chaotic motions. In the geometry of a typical magnetospheric levitated dipole experiment, it is shown that a considerable ratio of positrons from a ${}^{22}\mathrm{Na}$ source, located at the edge of the confinement region, has chaotic long orbit lengths before annihilation. These particles make multiple toroidal circulations and form a hollow toroidal positron cloud. Experiments with a small ${}^{22}\mathrm{Na}$ source in the Ring Trap 1 (RT-1) device demonstrated the existence of such long-lived positrons in a dipole field. Such a chaotic behavior of high-energy particles is potentially applicable to the formation of a dense toroidal positron cloud in the strong-field region of the dipole field in future studies.

Figure 1

(a) The poloidal cross section and (b) top view of RT-1. Thin lines in (a) show lines of force and strength of a magnetic field generated by a superconducting dipole field magnet. Projections of typical orbits of 1-keV and 100-keV positrons are plotted.

Figure 2

Temporal variation of (1) magnetic moment $\mu$, (2) longitudinal invariant $J$, and (3) magnetic flux $\mathrm{\Psi }$, (4) radial position of a positron, and (5) its frequency power spectrum. Positrons were injected from $r=80$ cm and $z=0$ cm with kinetic energy of (a) $E_k=1\mathrm{keV}$, (b) 20 keV, and (c) 100 keV.

Figure 3

The Poincaré maps of positron orbits in the dipole field of RT-1 injected from $r=80$ cm and $z=0$ cm with a kinetic energy of (a) $E_k=1$ keV, (b) 20 keV, and (c) 100 keV. The orbits were traced for 50 $\mu \mathrm{s}$, and 11 orbits with different initial pitch angles ($\sin \theta =0.1n$, $0\le n\le 10$) are superposed in each of the panels.

Figure 4

(a) A conservation map of $\mu$ for different kinetic energy and pitch angle of a positron injected from the edge confinement region at $r=80$ cm and $z=0$ cm. Circles are when $\mu$ is conserved and positron motions are periodic. Squares are when $\mu$ is not conserved and positron motions are chaotic. Triangles are marginal cases, as shown in Fig. 2.

Figure 5

(a) Fraction of confined positrons as a function of flight length for three different $E_k$. (b-1) Averaged flight lengths as a function of $E_k$. The radius of the source was $r_{\mathrm{source}}=5$ cm. (b-2) An emission spectrum of positrons from ${}^{22}\mathrm{Na}$ is shown for comparison.

Figure 6

(a) Fraction of monoenergetic positrons spatially isotropically injected with $E_k=100$ keV as functions of flight length for different source size $r_{\mathrm{source}}$. (b) The averaged flight length of positrons as a function of $r_{\mathrm{source}}$. The best power-law fitting was obtained for $\propto r_{\mathrm{source}}^{-1.96}$.

Figure 7

Fraction of positrons injected from a ${}^{22}\mathrm{Na}$ source spatially isotropically, as functions of (a) flight time and (b) flight length for different $r_{\mathrm{source}}$. Circles show averaged values of flight times and lengths for all particles.

Figure 8

Annihilation $\gamma$-ray energy spectra recorded for 1200 s (a) when the target plate was located at $r_{\mathrm{target}}=72$ cm, (b) when the target plate was located at outside the chamber, and (c) when the dipole field magnet was turned off while $r_{\mathrm{target}}=72$ cm. The ${}^{22}\mathrm{Na}$ source was positioned at $r_{\mathrm{source}}=80$ cm (Fig. 1) for all of the cases.

Figure 9

Counts of annihilation 511-keV $\gamma$ rays from the target plate as a function of $r_{\mathrm{target}}$. Counts from the supporting rod of the target were subtracted. Positrons were injected from $r_{\mathrm{source}}=80\mathrm{cm}$ at the SE port. The detector was located at $r_{\mathrm{detector}}=72$ cm using the NW port. The thin line is a guide to the eye.

Figure 10

Typical calculated example of the ratio of positrons lost at each of the channels. The ${}^{22}\mathrm{Na}$ source and the target plate were located at $r_{\mathrm{source}}=r_{\mathrm{target}}=72$ cm.

Figure 11

Comparison of coincidence counts of annihilation $\gamma$ rays and numerical results of the ratio of positrons lost at the target plate and the target supporting rod inside the view of detectors. The target plate and the detectors were located at $r_{\mathrm{target}}=r_{\mathrm{detector}}=72$ cm.

Figure 12

The Poincaré maps of a positron of $E_k=100$ keV for a different time step $h$. The calculation conditions are same as the case of Fig. 3 and $\sin \theta =0.5$.

Figure 13

Calculation results on fraction of injected positrons for a different time step $h$, normalized by the inverse of the typical cyclotron frequency of positrons. The calculation conditions are same as the case of $r_{\mathrm{source}}=5$ cm in Fig. 6.