Modelling the behavior of the positron plasma temperature in antihydrogen experimentation

Antihydrogen is now routinely produced at CERN by overlapping clouds of positrons and antiprotons. The mechanisms responsible for antihydrogen formation (radiative capture and the three-body reaction) are both dependent on the temperature of the positrons (Te), though with a different weight. Here we present a simple model of the behavior of the positron temperature based on the main processes involved during antihydrogen synthesis, namely: antiproton–positron collisions, positron heating due to plasma expansion and cooling via the emission of synchrotron radiation. The time evolution of Te has been simulated by changing the relevant parameters of the mechanisms involved in order to highlight the importance of the different (competing) effects.

features of the physical processes between them, even though antimatter study is usually more difficult than that of ordinary matter. For example, the measurement of thep mass achieved from high precision spectroscopy of the antiprotonic system (pHe + ) [1] can be used to increase the precision of the knowledge of the mass of the proton [2].
From the first detections of coldH in 2002 [3,4], several achievements have been made even if many questions remain with no clear answer. One topic to be clarified is the role of the different mechanisms inH production when e + andp clouds are merged together in dedicated experiments.
The expected mechanisms are spontaneous radiative recombination: and a three-body recombination (see e.g. [5] for a review): p + e + + e + →H + e + .
The outcomes of both mechanisms depend on the density (n e ) and the temperature (T e ) of the e + plasma and produce markedly differentH binding energies. For spontaneous radiative recombinationH is produced in low-lying quantum states and the rate varies as n e × T −0.63 e . In the case of three-body recombination the favoredH states are weakly bound and the production rate depends strongly on the e + plasma parameters, since in equilibrium it is proportional to n 2 e × T −4.5 e . The observation of reactions 1 and 2 is complicated by the multi-step nature of antihydrogen formation [6] and by the difference between produced and detectedH s [7]. The experimental results seem to indicate that three-body recombination plays a role inH production in some cases [4,[8][9][10][11][12][13][14], while in other instances [15][16][17][18][19] this is less clear.
Due to the influence of the e + plasma temperature inH production, we have developed a simple model to determine the behavior of T e during antihydrogen synthesis.

The model
The physical processes considered in the model to be responsible for the modification of the e + temperature are: cooling due to the radiation emitted by the cyclotron motion of the e + s (synchrotron radiation); heating due to the energy lost by theps injected into the e + plasma; heating due to the decrease of the electrostatic energy of the e + plasma following its expansion.
Actually, at high magnetic fields and low plasma temperatures it is necessary to consider two different temperatures: one for the component of the motion along the magnetic field (T e ) and one for the perpendicular direction (T e⊥ ). This happens when the positronpositron collision rate is reduced so much that the energy transfer from the motion along the magnetic field (B) to those in the perpendicular direction, where it is dissipated via synchrotron radiation, is ineffective.
It is possible to speak about only one temperature (T e ) when T e = T e⊥ and this occurs when the condition λ coll τ e 1 is satisfied, where λ coll is the positron-positron collision frequency and τ e is the synchrotron radiation time. To check the above condition we use the following formulae [20]: Herev = 2k B T e /m e is the mean velocity parallel to the magnetic field (k B is Boltzmann's constant, m e is the positron mass),b = 2e 2 /4πε 0 k B T e is twice the distance of closest approach (e is the positron charge, ε 0 is the vacuum permittivity with c the speed of light). The quantity I (k) is a function ofk = eBb m ev and in the region wherek 1 (i.e. in the magnetised plasma regime) the following analytical expression is valid [21]: In Fig. 1 the quantity λ coll τ e is plotted for the small T e where the parameterization of (4) can be used (i.e, whenk 1). The condition λ coll τ e 1 (equivalent to considering only one temperature) is satisfied with the exception of very low temperatures T e and densities with relatively high magnetic fields. In the present paper we will only discuss the case where a single value for the e + temperature can be used.
In a standard antihydrogen experiment using a Penning trap, a cloud of N i antiprotons is injected into a plasma of N e positrons. Due to Coulomb collisions the two particles species are thermalized and their rate equations can be written as: Here T i and T e represent the temperatures of antiprotons (with mass m i ) and positrons, respectively. The rate τ i is the Spitzer relaxation time given by: where the so-called Coulomb logarithm is: In (5) and (6) the factors L and K (with 0 < L ≤ 1 and 0 < K ≤ 1) are introduced to consider the partial overlap between antiprotons and positrons along the axial and radial directions, respectively. In particular the parameter K is introduced to describe the effect occurring when the antiproton cloud has a larger radius than the positron plasma [22]. The antiprotons which do not radially overlap the positron plasma are assumed, in this model, to not interact with those that do.
In the above equations only the effect ofp energy loss has been considered. In order to take into account the heating due to the e + plasma expansion and the cooling due to the synchrotron radiation, (6) has to be modified. The former effect is a consequence of the torque exerted on the e + plasma by collisions with the residual gas or by the lack of perfect cylindrical symmetry of the electric and magnetic fields of the trap [23,24]. When the e + plasma expands, the electrostatic energy of its charge decreases and is converted into internal energy thereby heating the plasma. By modeling the plasma as an infinite cylinder with an increasing radius r e , the term e 2 6ε 0 k B n e r e 2 0 1 r e dr e dt must be added to the right side of (6). The subscript 0 indicates that the the quantity n e r e 2 is assumed time-independent and dr e dt is the plasma expansion rate (v e ). When we consider also the effect of the synchrotron radiation which depends on the parameter T res (that is the ambient temperature surrounding the e + plasma) the new rate equation can be written as: The e + plasma andp temperatures, T e and T i , can be determined by numerically solving the system of differential equations (5) and (9). The solutions will depend also on the initial temperature of the positrons (T 0 e ), the initial kinetic energy of antiprotons (E 0 ) and the initial e + plasma radius (r 0 ) which accounts for the effective value at the antiproton injection time.

Simulations
Since a general solution of (5) and (9) cannot be determined due to the large number of parameters involved, particular cases have to be considered individually. Here we give some examples by varying the parameters that influence the behavior of T e . Since (5) and (9) are linear in respect of T res and of T e , we will consider the quantity T = T e − T res whose behavior is the same as T e . We want to point out that some of the simulated conditions may be far from real experimental situations since the aim of these evaluations is to highlight the relative importance of the mechanisms responsible for the time evolution of T e . In the following we consider two broad examples, namely, (i) when the number of positrons is much greater than the number of antiprotons and (ii) when the relative numbers are much closer. For v e = 0.2 μm s −1 the maximum increase of T e (25 K) is the same as for v e = 0 while the T e values at equilibrium are some kelvin over T res . It must be noted that the effects on H production are negligible as far as the T values are low in respect to T res .
For v e = 2 μm s −1 and v e = 20 μm s −1 the maximum T ranges from some tens to hundreds of kelvin. In addition the equilibrium values are achieved very late for the extreme e + plasma density (70 × 10 14 m −3 ) with T remaining above several tens of kelvin for  Fig. 3. As for Fig. 2 the plots on the upper panel are for v e = 0. Here the maximum of T varies from 1000 K, for the highest N e value, to 3500 K, for the lowest N e value. These large values are due to the strong heating from the numerous antiprotons injected into few positrons. The T res limit is reached slowly in all cases, for instance in 3-4 s for N e = 3 × 10 7 , rising to 30 s for N e = 3 × 10 5 .
Due to large T variations, the plots on the lower panel for v e > 0 are in logarithmic scale, also to appreciate the trends at late times. When v e = 1μm s −1 the results are very similar to the v e = 0 case. This means that the dominant heating effect is still the antiproton energy loss. The only difference is that at equilibrium the T e values are somewhat larger than T res .
For v e = 10 μm s −1 and 100 μm s −1 the expansion effect starts to become relevant: not for the maximum T e values, which are unchanged compared to v e = 0 case, but for the behavior at late times, especially for the lowest selected number of positrons (N e = 3 × 10 5 ) where, for v e = 100 μm s −1 , T is around 500 K even 30 s after the start of mixing.
In addition it is worth pointing out that the naive expectation that with more positrons the heating is lower is not always correct: at the equilibrium the T e values tend to be higher for larger numbers of positrons. This is a consequence of the e + plasma expansion which converts the electrostatic energy into thermal energy, and this is increased when (n e r e 2 ) 0 is large (see (9)).

Conclusions
We have developed a simple model to assess the behavior of the positron plasma temperature in antihydrogen experiments. It is based on the heating from antiproton-positron collisions, from the positron plasma expansion and on the cooling from the positron synchrotron radiation emission. It is the first time that these mechanisms have been included together in a model.
The solutions depend upon several parameters and not all of them are usually known from experiment. The results indicate when the heating of the positron plasma could be relevant in reducing the antihydrogen production rates. The proposed model can also be used to help the interpretation of the experimental data not only about antihydrogen production [25] but also for the recent analysis on protonium formation [26,27].