Journal of Fusion Energy

, Volume 31, Issue 4, pp 341–345

Dependence of Potential Well Depth on the Magnetic Field Intensity in a Polywell Reactor

  • F. Kazemyzade
  • H. Mahdipoor
  • A. Bagheri
  • S. Khademzade
  • E. Hajiebrahimi
  • Z. Gheisari
  • A. Sadighzadeh
  • V. Damideh
Original Research

DOI: 10.1007/s10894-011-9474-4

Cite this article as:
Kazemyzade, F., Mahdipoor, H., Bagheri, A. et al. J Fusion Energ (2012) 31: 341. doi:10.1007/s10894-011-9474-4


Using OOPIC-Pro assisted-two dimensional simulation we have considered the dependencies of the electron and ion densities, as well as the central electric potential on the magnetic-field intensity in the Polywell fusion reactor. It is shown that the potential well depth increases with decreasing the magnetic intensity, while much narrower well width (thus more effective deuteron trapping) is achieved with increasing the magnetic field intensity. The results obtained can be employed to adjust the magnetic field intensities at which more effective electron confinement, thus more effective ion-flux convergence, is expected. Furthermore, this study can be used to reach the optimized conditions of the reactor operation as well as to relate to the next generation fusion fuels.


Polywell fusion reactor Particle-in-cell code Negative potential well (NPW) Magnetic field intensity 



Negative potential well


Polywell is an inertial electrodynamics confinement fusion reactor with an easy conceptual design registered by Bussard [1]. This reactor had been designed based on electrostatic confinement fusion (IECF) device and was an improved version of it. Virtual cathode formation at the reactor center (formation of a dens electron cloud), due to confining electrons by strong-enough external magnetic field, is an essential mechanism through operation of this device. Potential well, created by the high-density electron cloud, accelerates the produced ions from all direction towards the center, where they meet each other, and as a result fusion reactions occur, or the ions may be scattered. If the ions are scattered, then they return to the center due to potential well attraction, and finally incorporate in the fusion reactions after multiple round trips. Effective saving of lost electrons due to magnetic mirror effect, establishing electron (input and loss) fluxes balance, and efficient self-stabilizing of magnetic geometry (as a result of its convex nature) are among many advantages brought by Polywell fusion reactor.

Several experimental and simulation studies, so far, have been carried out toward improving the electrostatic confinement reactors. Increasing in neutron yield via additional electrode inside the cathode, which suppresses the virtual anode formation at high currents, is reported by Noborio et al. [2]. Takamatsu et al. [3] reported a ten-times increase in number of fusion reactions as well as an increase in overall efficiency, all when ion sources were employed in the setup. The control of the ion motions during the device operation by applying an axial magnetic field was simulated by Tomiyasu et al. [4]. Following these studies, Kurilenkov et al. [5] investigated the effects of virtual cathode, and single- and double-well potentials on the IECF reactor operation when an electric discharge is applied. Finally, the most recent experimental work by Damideh et al. [6] on IECF reactor (as a continuous fusion reaction device), which operated at -140 kV voltage and 70 mA current, resulted in 2 × 107 D-D reactions per second.

Nevertheless, there are a few reports on the electrodynamics confinement devices (e.g., Polywell reactor). Besides the experimental and theoretical reports made by Bussard and his coworkers at EMC2 (Energy Matter Conversion Corporation), there are limited literature for this research field. Recently, Rogers has provided insights towards obtaining scaling rule, estimating an exact potential-well depth, and optimizing the device operation [7]. Moreover, the experimental studies on Polywell reactors have been carried out only on the small size setup. The investigation of the effects of variations in the coil current and background gas pressure on the virtual cathode behavior, which was done out by Carr et al. [8] is among these studies. To the best of our knowledge, there is no report on potential well behavior under magnetic-field-strength variation. So, in this work, the effect of magnetic field strength on the potential well depth is investigated using OOPIC-Pro two-dimensional (2D) code, which has been broadly used by many authors [9, 10, 11, 12, 13]. Moreover, PIC simulation was used by Bussard but the results were not published as he mentioned in his paper at IAC conference [14].

This paper is organized in this fashion. In Sect. “Simulation Model” the geometry of the device simulated is described. The simulation results are presented and discussed in Sect. “Results and Discussion”. The results obtained are summarized and finally the article concludes with an outlook for future research.

Simulation Model

The geometry of Polywell fusion reactor is presented in Fig. 1. The 1.4 m length-tank walls are biased up to a Vw DC voltage, while the magnet boxes (containing the magnetic coils) are biased to Vc DC voltage. Each of four Electron guns (ion guns) is working with current Ie (Ii) and voltage Ve (Vi). The electron (hydrogen-ion) guns are located out of (in) the magnetic coil configuration (magrid) and directed along the magnet coil axes. The magnetic field of the coils is simulated with the magnetic field produced by eight straight wires perpendicular to the geometry plane, each passing through the coil cross sections. The current directions in two opposite coils (or wires) are different so that the magnetic field in the center becomes zero (see Fig. 1b). The magnetic field is stronger near the magnets and weaker toward the center. Using this magnet configuration, the line cusps, which appear when only two magnets are used, have been changed to point cusps as a result of the squeezing magnetic field lines. More details of the main parameters of devices used in Polywell fusion reactor are summarized in Table 1.
Fig. 1

The geometry of simulation model of the Polywell reactor cross section (a) and the magnetic field surface (b). The positions of the electron and ion guns are indicated by EG and IG notations, respectively in (a). The face and corner cusps of the magnetic field geometry are indicated by arrows in (b)

Table 1

Main simulation parameters



Vacuum tank dimensions

1.4 × 1.4 m2

Tank potential

0 V

Electron gun current

20 A

Electron gun potential

100 V

Ion gun current

1.5 A

Ion gun potential

100 V

The simulation starts with an empty tank and as time goes on the tank is filled with electrons, being emitted from electron guns (directed along the face cusps, see Fig. 1b), which pass through the face cusps and are converged radially by the magnetic field to the center region, the so-called electron confinement. This magnetic field assisted-electron confinement leads to formation of a high density electron cloud, so that the electron kinetic energies are completely transformed into a negative potential energy-appearance of a deep potential well-which is becoming deeper(due to electron density increase) as time passes. In the same time, the Hydrogen ions produced (and derived) by ion guns (within the coil configuration) are repulsed towards the center by the positive coil potential. This is why the ion guns are located between the coils, otherwise it may be impossible to direct ions towards the center. This ion acceleration (towards the center) also becomes more effective via ion trapping in the potential well so that, finally, all deuterons become mono-energetic species. As a result, the ion density increases at the center, which in turn the fusion reaction rate becomes higher. The scattered ions pass through the electron cloud formed and are slowed down, again are attracted towards the center, so that they finally hit other trapped ions and fuse.

The electron and ion densities, electron and ion energies, as well as the potential well behavior are displayed by the OOPIC-Pro diagnostic plots, which change as the simulation time advances. The ion density in three-dimension is given by [15]
$$ n_{\text{i}} = \left( {\frac{1}{{7.4 \times 10^{2} }}} \right)^{2} \frac{{\rho_{\text{i}}^{2} }}{{E_{\text{i}} }} $$
where ni and ρi are ion densities in three- and two-dimensions, respectively, and Ei is the ion energy at the center. Finally, the Wiffleball condition is established when we have [16]:
$$ n_{\text{i}}^{ * } = \frac{{B^{2} }}{{16\pi {\kern 1pt} E_{\text{i}} }} $$
where, B is the external-magnetic field intensity and ni* is the critical 3D ion density below which the net-power operation [14] will not take place.

Results and Discussion

In this section we present the results of the simulation of the electron cloud formation and its effects on the potential and ion density profiles for two cases of low and high magnetic strengths.

Low Magnetic Field Condition

Let us first consider the case of low magnetic strength, when the currents in the wires (coils) generate a magnetic field of 0.25 T (2,500 G) at the face cusps and 0.15 T at the corner cusps. Figure 2 shows the potential profile which vary with injecting electrons into the center of the reactor. It is clearly seen that as time goes on more electron are populated at center leading to the potential to be dropped to a negative value, here −4 kV (see Fig. 2b). Although, the electrons are repelled by each other, but the magnetic field confines them. As a result, the electron cloud is formed at the center region. The ions produced are driven by the strong electric force towards the center, being trapped within the potential well. By increasing the ion velocity (towards the center) more ion are populated at the Polywell fusion-reactor center (thus more fusion reactions are likely to happen), and as a result a positive-potential energy is generated, leading to an increase in the center potential. This is what clearly shown in Fig. 2c for the time 9 μs, when the center potential becomes positive, and as a result the potential well is doubled.
Fig. 2

The contours of the potential inside the coil configuration at (a) t = 0, (b) t = 0.8, and (c) t = 9 μs. The magnetic field strength is 0.25 T at the face cusps and 0.15T at the corner cusps

The contour of electron density, shown in Fig. 3a, clearly represents the electron cloud formation (at the center of Polywell fusion reactor) as a consequence of the electron injection (through the face cusps) and the magnetic field confinement. Here, the electrons have been distributed in the broader region, at the center as well as through the center to the corners. Due to magnetic field-assisted confinement, the electron density (ρe) reaches to 6.3 × 1011 cm−2 at the center and becomes lower at the corners, where the electrons can escape from the machine. The ion density (ρi) at center is 7 × 1011 cm−2 (see Fig. 3b). Using (1), for Ei = 38 keV, the ion density of 23 × 1012 cm−3 obtained at the center. For the case of low magnetic intensity, ni* = 15.7 × 1012 cm−3, so ni is well above the critical value, which clearly indicates that the Polywell fusion reactor can operate at the net power.
Fig. 3

Contours of the (a) electron, ρe, and (b) ion, ρi, densities

Here, the electrons have been distributed in the wide central region and the electron density at the magrid corners is remarkable, indicating an effective electron loss at the corner. The ion density increase at the center, which is clearly shown in Fig. 3b, describe the efficient ion trapping due to ion dropping at the center of the negative potential well (NPW).

High Magnetic Field Condition

Now we consider the case of the high magnetic field, when the strength of magnetic field at the corner cusps and face cusps is 10 times higher. In Figs. 4a–c, the contours of the electric potential inside Polywell reactor are shown for three different simulation times, here, t = 0, 3.8, and 14 μs, corresponding to the startup, maximum potential depth, and Wiffleball conditions, respectively. From Fig. 4b, one can clearly see that, in the case of high magnetic field intensity, the magnitude of the negative potential (at the center) −1.2 kV is less than the case of the low magnetic field (−4 kV, see Fig. 2b), which is in good agreement with experimental results for high-magnetic intensity range [8]. By applying a strong magnetic field, the electron confinement becomes more effective, so that more ions are trapped in the center region (due to efficient trapping by high density electron cloud). This results in increasing the positive potential energy, which in turn reduces the magnitude of the net-NPW (due to electron cloud formation). From (2) one can obtain the corresponding ion density for the Wiffleball condition, ni* = 16.2 × 1013 cm−3. The ion density at 14 μs is ρi = 3.1 × 1012 cm−2 (or ni = 3.5 × 1014 cm−3 obtained from (1)), thus Polywell fusion-reactor operation with net power is expected. By looking at Figs. 4a–c and 2a–c, it is found that the potential-well width becomes smaller as the magnetic strength is increased. This behavior can be explained based on effective electron trapping (confinement), which leads to more effective hydrogen ion trapping and eventually increase in the fusion reaction rate [16].
Fig. 4

The contours of the potential inside the coil configuration at (a) t = 0, (b) t = 3.8, and (c) t = 14 μs. The magnetic field strength is 2.5 T at the face cusps and 1.5 T at the corner cusps

We have also plotted the electron and ion density contours in the case of high magnetic intensity. Figures 5a and b display the electron and ion density distributions, respectively. One easily can notice that the ions, as well as electrons, are distributed in a narrower region at the center, indicating that ions are also magnetized, while the electrons are strongly magnetized. Very efficient electron confinement can be clearly observed by looking at Fig. 5a. It is seen that very small number of electrons (indicated by white followed by red color to the center) can reach the corner cusps (where the charged particles can leave the magrid). It is because, by increasing the magnetic field strength, the magnetic lines are more squeezed at the (face and corner) cusps, so that the point cusps become much narrower, so that the charged particles cannot escape.
Fig. 5

Contours of the electron (ρe) (a) and ion (ρi) (b) densities. Here, the magnetic field strength is 2.5 T at the face cusp

By applying such high magnetic field, the electron density ρe tends to much higher values (2.9 × 1012 cm−2) at the center. In response to this electron density increase (and as a result much deeper NPW expected), more effective convergence concentration of hydrogen ions takes place in much smaller region at the Polywell reactor center, which in turn leads to much higher positive energy, and thus decreasing the magnitude of the negative potential.


Thus, we have simulated the plasma environment of the Polywell fusion reactor using OOPIC-Pro program. Using the program, we have shown that the potential-well behavior strongly depends on the strength of the applied external magnetic field. The magnitude of NPW surprisingly becomes lower with increasing the strength, while the well width decreases, confirming more effective confinement of the electrons (due to much higher magnetic strength generated). At high magnetic field intensities, an efficient electron confinement and less effective electron loss happen, leading to an effective radial convergence of hydrogen ions to the center, which is impossible with a low-density electron cloud (created by weak magnetic fields). The results obtain can be used to optimize the operation of the future Polywell fusion reactors towards net-power generation. More importantly, the main conclusions are not restricted to the D–D fuel and may be relevant to the broader range of the fusion fuels (D-T and p-B11) will be used in Polywell fusion reactor plans, which may be the aim of our future works.

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • F. Kazemyzade
    • 1
  • H. Mahdipoor
    • 1
  • A. Bagheri
    • 1
  • S. Khademzade
    • 1
  • E. Hajiebrahimi
    • 1
  • Z. Gheisari
    • 1
  • A. Sadighzadeh
    • 1
  • V. Damideh
    • 1
  1. 1.Plasma Physics and Nuclear Fusion Research SchoolNuclear Science and Technology Research Institute, AEOITehranIran

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