Simulation of a high proton temperature plasma toroidal magnetic trap to be used in proton-B fusion

Several tokamaks-like toroidal magnetic confinement structures have been simulated using 500keV protons to be used in P-B11 fusion. In order to find the optimal confinement configuration, the simulation was carried out by an evolutionary algorithm running 145,000 simulations whose results are presented in this document including the feasibility to reach ignition with some of them by accelerating and colliding 11 B and proton ions.


I. Introduction
After extensive testing of the Pulsotron-2A zpinch device and several ion accelerators using the Pulsotron-2B device it was found that it is necessary to design fusion devices that reduce the electron temperature to decrease loses [14] [15] [16]. The Pulsotron II 2A device consists of a small Z-pinch device and the Pulsotron II 2B represents a modification of a capacitor bank to be used in several tests related to plasma confinement. The Z-pinch successfully achieved the required pressure but with a low plasma temperature. In order to design a boron-proton fusion reactor it was designed an electrostatic and a magnetic ion accelerator but the design of a confinement chamber is still needed. In the present work the results reached after simulating 145,000 different magnetic toroidal Tokamak-like design are presented. The chamber is designed to sustain boron-proton fusion. The Boron fuel state will be Ions with one or more positive charges B.
The Larmor radius of the protons is taken to be the major radius R instead of the minor one that would require much larger magnetic fields. Boron can be injected at low speed but must be confined within the higher concentration of protons. In this first simulation campaign, only protons are used in order to determine which reactor configuration is the best candidate. The losses are calculated from the ion acceleration and a Larmor radius equal to chamber major radius for a confinement time of 0.59 ms for a small 200 mm reactor, so losses are about 5.47 x 10 -24 watts according to the excel table found in [17], formula 41, so less than one eV is lost by the protons during their confinement. Defining the Larmor radius as the major radius of the chamber the reactor chamber dimensions can be greatly reduced as, for example, a 500 keV proton whose mass is 1.66 x 10 -27 kg has an approximate speed of 9.8222 x 10 6 m/s, so when submitted to a 1.5 Tesla magnetic field (easily achievable in a small reactor), the Larmor radius decreases to: = = 6.787 10 −3 . Hence, a proton plasma can theoretically be confined in a small 135 mm diameter torus. In the simulation the particles leaving the torus are taken into account as losses.
In real toroidal systems, coils would be installed to recover electricity from lost ions and to slow them down to non-harmful kinetic energy levels.
The main coils would receive some proton flux, but this was not calculated in this round of simulations. In the simulation copper wires are used instead of superconductive coils that would suffer less from ion bombardment.
high ion speed a new configuration design is used. Additionally, a new parameter A eff gives the effective cross section of the reactor that can be used to obtain the percentage of reacting particles inside the reactor in a direct way. C++ source code used in the simulation is also provided which equations are described in more detail in [2,3,5]. The main objective was to find the feasibility of using a tokamak-like configuration to allow the fusion reactions inside. The configuration must allow the installation of direct electricity recovery coils to be used to extract currents from the generated alpha particles.
Particles energy was assumed to be 500+12 keV which is close to the maximum cross section [5] with angle distribution of + 20º obtained from elastic scattering [9][10][11]6].

II. Equations and algorithms used
The percentage of the particles that react can be obtained using the following formula: Where: N is the number of particles (protons or borons) σ=cross section of the P-11 B reaction, A eff =Effective cross section of the reaction, t = average laps number of particles paths. All areas are taken in square meters.
The maximum cross section for proton- 11  The exponential part of the reactor equation (1) must be about 1 in order to allow the reactions between most of the particles, so Eq. (1) can be used to calculate how many particles must be injected into the chamber. In order to have highly accurate and speedy simulations the C++ multithread technology was used to run high speed algorithms that apply elliptic integrals. As reference equations and codes collected from the book by J. D. Jackson [3] were used. The mutual inductance and inductance calculations uses Maxwell's Method [2] and the equations can be can be found at [13]. The obtained results of inductance and magnetic field were checked against the general equations of magnetic fields for standard devices, such as loops and wires. Furthermore, test devices were built and tested using standard magnetic field sensors, oscilloscopes and current sensors. With only one hundred simulations the achieved A eff was too low, so an increase in the number of simulations to more than 10 thousands of tokamak-like configurations. To do that an initially low particle number of 16 was simulated and subsequently increased to 196 particles in those toroidal configurations that achieved longer confinement time and lower A eff. Also the genetic algorithm was used to generate vectors of data input in the more promising reactors.

II.a)
Validity of Eq. (1) The validity of Eq. (1) exponential part can be easily demonstrated using a small excel table using as example Nσ/A=0.02 where the remainder particles reacts each other every turn. As can be seen the exponential equation simplifies results:

III. Assumptions
The particles are injected when the magnetic field is saturated so there are only slow magnetic field variations. According to the simulations the maximum B-field is reached between 5 milliseconds and some seconds to values that ranges from 0.43 teslas for larger coils to 1.6 teslas for smaller ones.
The particles energy is 500+12keV according to the Miranda particles injector specifications. The particles angle distribution is +20º, according to our simulations taking into account kinetics scattering data obtained from [6,10,11]. The magnetic field was simulated in different configurations by using Biot-Savart's law integrating along every coil. In order to calculate the inductance with high accuracy during short time elliptic integrals were applied along with the multithread technology. Using fixed time and length steps generates large errors in the coils proximity so the spatial increment in the proximity to the coils has to be narrowed down from one millimetre to less than 1 micrometre.

IV. Simulation Setup
In order to have high accurate simulations and speed the C++ multithread technology was used to run the algorithm, which works on the basis of elliptic integrals.

IV.a)
Input data The main input data are the dimensions of coils, number of turns t, energy injected and proton energy (550 keV). Simulations were performed with 2 to 20 toroidal coils and up to 4 parallel coils (but more coils can be set). V.
Performance of the simulator Using low particle simulations allows a reduction in simulation time from some minutes per reactor to 10 tokamak configurations per second. If a particle passes close to a coil, the magnetic field gradient is higher so more calculation points are taken and the calculus is slower. One software thread per particle was used, so when a particle escapes to the reactor wall its thread is terminated. Hence, a better confinement in the reactor increases the escape time for particles and the simulation runs slower. The time consumption for one simulation is directly proportional to the particle number and the confinement time of the particles. It turns out that with only 9, 16 or 25 particles one can have similar results as with 196 or more particles. Hence, such preliminary results can be used for comparison purposes to filter out the best reactors before using a larger number of particles as can be seen in following graphs: By using more particles the simulation result stabilizes: Where L med is the average particle path length of a proton before it escapes to the container walls. The best reactors have particles travelling up to 800 meters (see Fig 8).

VI. Simulation results
Initial results demonstrate that the A eff was so high that very few particles could react within the confinement region. As can be seen from the following data in Table 2, A eff was larger than 2 x 10 -3 m² for tokamaks of 0.32 meters major radius, so the exponent part was so low that only one out of every 5 x 10 8 particles reacts with another after injecting 100 kJ of input energy.  Only after 6000 simulations the cross section went lower than 3 x 10 -4 as shown in the following Fig. 6: Figure 6: A eff as a function of number of simulations before improvements.
In order to achieve better results algorithms that are used in Artificial Intelligence as genetic algorithms were applied. They give major improvements as the "23 fellows system" that consists basically of a selection of the 23 best matches and makes variations over them. The algorithm does not use random input data but lowers A eff from a run to the next linearly. The results improved A eff from 2 x 10 -4 to 10 -6 m 2 , which was a 200-fold improvement but was still not good enough. The trajectories of the particles were simulated and it turns out that in some reactors they pass through very well confined areas as depicted in the following Fig. 7: Figure 7: Plot of particles trajectory inside a reactor chamber obtained from a simulation run. Thus, the cross section of the particles in 3 dimensions was included into the simulations. This increases the computational time but giving very good results, even when using only 16 particles. Afterwards 196 particles were used to obtain a highly accurate results improving from cross sections between 10 -7 to 2 x 10 -3 to cross sections between 4.32 x 10 -12 to 10 -24 m². With those simulations of 1660 reactors were performed. Out of these 1660 reactor configurations 1466 obtained a net energy gain of more than 4 times the input power. 784 reactor configurations obtained almost the maximum energy gain of 12.64. The reactor dimension was 0.13 to 0.47 meters in diameter and the energy range for confinement from 75 J to 360 kJ. The output power depends on how many times per minutes the reactor is fired. In the following plot (Fig. 8) the improvement of the simulations with respect to the number of simulations using the real cross section and also using the "23 fellow" genetic algorithm is summarized: These results can be used for other types of aneutronic fusion reactions but a very accurate design must be done.

VI.a)
Simulation results as a function of the toroidal coil number In the following Fig. 9 A eff is depicted as a function of the number of toroidal coils: Figure 9: A eff (m²) as a function of toroidal coils number. It was tried from 3 to 19 coils/reactor but only between 5 and 11 coils reach ignition conditions

VI.b)
Reactor cross section as a function of chamber dimensions The following Fig. 10 shows A eff as a function of the external radius of the reactor. The dimensions of the reactors selected by the genetic algorithm showed that the best reactor have specific dimensions. It is not necessary to build large reactors: Figure 10: A eff as a function of the external chamber radius. In the vertical axis there is A eff (m 2 ) and in the horizontal axis the radius (m). The algorithm was commanded to try between 35 and 250mm radius tokamaks, finding the best of them at three discrete chamber radius.

VI.c)
Reactor cross section as a function of injected energy The performance of A eff as a function of the optimal energy injected. As it can be seen from Fig. 11 some reactors need very low energy levels. It is not shown in Table 2, but the simulations prove that low energy reactors have a very low path length with very thin particle trajectory cross sections so a very accurate reactor design must be implemented.

VII. Conclusion
We were able to show in this paper with extensive computer simulations that our Pulsotron tokamak design for aneutronic protonboron fusion can reach energetic break-even. Furthermore, our numerical simulations reveal that it is also possible to construct small scale fusion devices by using hot protons with kinetic energies around 500 keV and comparably small electron temperatures.

IX. Annex: Code samples
The following represents the C++ code sample of the main functions used to calculate the magnetic field in the simulations. The code uses the formula obtained using the Law of Biot Savart, integrated over a circular current loop to obtain the magnetic field at any point in space. Its result was compared using the on axis formula result (as shown in Fig. 16).
= 0 √ [ ( ) Α=r/a β=x/a γ=z/r (4) Where B 0 is the magnetic field at the coil centre: B 0 =μ o I/(2a) K(k) is the complete elliptic integral function, of the first kind E(k) is the complete elliptic integral function, of the second kind.