Antiparticle cloud temperatures for antihydrogen experiments

A simple rate-equation description of the heating and cooling of antiparticle clouds under conditions typical of those found in antihydrogen formation experiments is developed and analyzed. We include single-particle collisional, radiative, and cloud expansion effects and, from the modeling calculations, identify typical cooling phenomena and trends and relate these to the underlying physics. Some general rules of thumb of use to experimenters are derived.

Figure 1

Comparison of the time dependencies of the change in temperature $\mathrm{\Delta }T$ [which are $\mathrm{\Delta }{T}_e\left(t\right)$ and $\mathrm{\Delta }{T}_i\left(t\right)$ as appropriate; see text] of the positron (dashed lines) and antiproton (full lines) clouds for the Spitzer (upper panes) and Hurt et al. (lower panes) ion cooling terms. The densities, $n_e$, investigated are $7\times {10}^{13}$ (black), $7\times {10}^{14}$ (red), and $7\times {10}^{15}{\mathrm{m}}^{-3}$ (green), here all shown for $d{r}_e/dt=0$, and with $\epsilon ={10}^{-4}$ and $E_i\left(0\right)=15$ eV.

Figure 2

The time dependencies of $\mathrm{\Delta }T$ [which are $\mathrm{\Delta }{T}_e\left(t\right)$ and $\mathrm{\Delta }{T}_i\left(t\right)$ as appropriate; see text] for antiprotons and positrons using the term of Hurt et al. for ${\tau }_i$. The curve coding and parameters are as for Fig. 1, except that here $u\ne 0$ since $d{r}_e/dt=2\times {10}^{-5}{\mathrm{m}\mathrm{s}}^{-1}$.

Figure 3

Variation of the maximum positron temperature, $\mathrm{\Delta }{T}_e^{max}$, versus antiproton injection kinetic energy for various values of $\epsilon$, and for $u=0$: left panel, $\epsilon ={10}^{-4}$, central panel, $\epsilon ={10}^{-2}$, and right panel, $\epsilon ={10}^{-1}$. Key: red, $B_z=2\mathrm{T}$ and $n_e={10}^{12}{\mathrm{m}}^{-3}$; green, $B_z=3\mathrm{T}$ and $n_e={10}^{12}{\mathrm{m}}^{-3}$; blue, $B_z=2\mathrm{T}$ and $n_e={10}^{14}{\mathrm{m}}^{-3}$; orange, $B_z=3\mathrm{T}$ and $n_e={10}^{14}{\mathrm{m}}^{-3}$; pink, $B_z=2\mathrm{T}$ and $n_e={10}^{16}{\mathrm{m}}^{-3}$; cyan, $B_z=3\mathrm{T}$ and $n_e={10}^{16}{\mathrm{m}}^{-3}$.

Figure 4

Variation of the maximum positron temperature, $\mathrm{\Delta }{T}_e^{max}$, versus $\epsilon$ at various input kinetic energies, $E_i\left(0\right)$, and for $u=0$. The lines shown are indicative to draw attention to data trends. The lower line is for $E_i\left(0\right)=1.5$ eV, while the upper corresponds to 30 eV. The intermediate data sets are for 7.5 and 15 eV. For clarity, see Fig. 5 for the key.

Figure 5

The time to reach equilibrium (see text) versus $\epsilon$ for $E_i\left(0\right)=15$ eV and $u=0$. Key: cyan $◯ B_z=2$ T and $n_e={10}^{13}{\mathrm{m}}^{-3}$; $\square B_z=3$ T and $n_e={10}^{13}{\mathrm{m}}^{-3}$; $△ B_z=2$ T and $n_e=5\times {10}^{13}{\mathrm{m}}^{-3}$; $◊ B_z=3$ T and $n_e=5\times {10}^{13}{\mathrm{m}}^{-3}$; $+ B_z=2$ T and $n_e={10}^{14}{\mathrm{m}}^{-3}$; $★ B_z=3$ T and $n_e={10}^{14}{\mathrm{m}}^{-3}$; $▿ B_z=2$ T and $n_e=5\times {10}^{14}{\mathrm{m}}^{-3}$; $+ B_z=3$ T and $n_e=5\times {10}^{14}{\mathrm{m}}^{-3}$; $* B_z=2$ T and $n_e={10}^{15}{\mathrm{m}}^{-3}$; red $◯ B_z=3$ T and $n_e={10}^{15}{\mathrm{m}}^{-3}$.

Figure 6

The time to reach maximum positron temperature $t_{max}$ versus $\epsilon$ for various $E_i\left(0\right)$ and with $d{r}_e/dt=2\times {10}^{-7}{\mathrm{m}\mathrm{s}}^{-1}$. Key: as for Fig. 5.

Figure 7

The time to reach maximum positron temperature $t_{max}$ versus $\epsilon$ for various $E_i\left(0\right)$ and with $d{r}_e/dt=2\times {10}^{-6}{\mathrm{m}\mathrm{s}}^{-1}$. Key: as for Fig. 5.