The influence of Hively and Bosch-Hale reactivities on hot ion mode in deuterium/helium-3 fuel
Subject Areas : Journal of Theoretical and Applied Physics
Armin Taghipour
1
,
S. Mohammad Motevalli
2
,
Fereshteh Fadaei
3
1 - Department of Physics, Faculty of Science, University of Mazandaran, Babolsar, Iran
2 - Department of Physics, Faculty of Science, University of Mazandaran, Babolsar, Iran
3 - Department of Physics, Payam Noor University, Tehran, Iran
Keywords:
Abstract :
The Influence of Hively and Bosch-Hale Reactivities on Hot Ion Mode in Deuterium/Helium-3 Fuel
A Taghipour1, S M Motevalli1,* and F Fadaei2
1 Department of Physics, Faculty of Science, University of Mazandaran, P. O. Box 47415-416, Babolsar, Iran
2 Department of Physics, Payam Noor University, P. O. Box 19395-3697, Tehran, Iran
Author to whom any correspondence should be addressed
E-mail: motavali@umz.ac.ir
Abstract
Nowadays, there is much extensive research investigating nuclear fusion reaction with D-3He fuel as one of the most essential advanced fusion fuels. One of the most important quantities in fusion is the reactivity. In this work, with consideration of different temperatures for ion and electron (hot ion mode), we intend to study the effects of two different reactivities (Hively and Bosch-Hale) on D-3He fusion reaction in spherical tokamak. Accordingly, by writing the system of particle and energy balance equations for this reaction in hot ion mode, we will investigate the effects of reactivities on plasma parameters in spherical tokamak.
Keywords: D-3He fusion, spherical tokamak, particle and energy equation, reactivity.
1. Introduction
One way to achieve nuclear fusion is to confine the plasma by using a magnetic field. Up to now, the tokamak system is recognized as one of the most successful magnetic confinement fusion devices [1]. In this system, the effect of the magnetic field on charged particles is used to confine the hot plasma. Most of the experiments performed in the field of nuclear fusion were on D-T fuel, which, one of the most important reasons for this matter is to perform this reaction at a lower temperature (compared to other fusion reactions) and having the highest reactivity in the intended temperature range in fusion reactors. However, disadvantages include high radiation, the possibility of a meltdown in the reactor, and the lack of natural resources of tritium (which, to solve this problem, lithium blankets are used to produce tritium in the reactor), have attracted attention to other fuels. In the meantime, D-3He fuel has been considered by a large number of researchers because of its advantages such as direct energy conversion and fewer dangers due to radiation, the impossibility of meltdown, higher efficiency, and lower cost (more straightforward structure due to no need to use blankets) [2-10]. Another noteworthy point about D-3He fuel is the issue of tokamak structure. High aspect ratio tokamak reactors are unsuitable for D-3He fusion operation due to their high toroidal magnetic field and low beta. After investigating the ignition capabilities of two models of spherical tokamak reactors and high aspect ratio tokamaks for D-3He fuel, the researchers found that in a spherical tokamak, the power fraction required for ions in the hot ion mode (Ti >Te) is smaller due to lower synchrotron radiation loss. Therefore, spherical tokamak has been considered due to its low aspect ratio [11-18]. In spherical tokamaks, unlike conventional tokamaks, with a generally round donut-like confinement area with large cavities in the middle, the size of the cavities is reduced as much as possible, resulting in an almost spherical plasma shape [19-25]. On the other hand, a significant quantity to consider in the study of fusion fuels is reactivity (the 
parameter). This quantity is defined as the probability of reacting per time unit per the target nuclear density unit, and it is a function of plasma temperature (in units of keV) which in the simplest case is specified only by the parameter. The parameter is often referred to as the reactivity parameter [26-30].
2. Fusion Power and Reactivity
If only one type of fusion process occurs at the rate of density Rfu with the unit of released energy on the Qfu reaction, then the fusion power produced per volume unit is given by the following equation
(1)
If the unit of Rfu is based on m-3.s-1 and the Qfu unit is based on MeV for each reaction, the unit of fusion density Pfu will be MeV.m-3.s-1 and the total energy released for a uniform distribution of power over the time interval τ and volume V is thus
(2)
and for each time t
(3)
The energy produced in a fusion reaction, Qfu, is called the "Q value" of the reaction, and can be obtained experimentally or through existing tables. To determine the expression for the fusion reactivity density, we consider two particles, a and b, the number densities of them are Na and Nb, respectively. If vr is the relative velocity of the two particles at the intersection point and consider the cross-section as 
, the explicit equation of the fusion reactivity density will be as follows [31]
(4)
3. Reactivity of Nuclear Fusion Reactions (
Parameter)
A vital quantity to consider in the study of fusion fuels is the reactivity. This quantity is defined as the probability of reacting per time unit on the target nuclear density unit, and it is a function of plasma temperature (in units of keV). This quantity is specified by the parameter. This parameter is called the "Sigma-Vi" parameter (reactivity parameter). In general, with consideration of the relative velocity v for each pair of nuclei that collide, the motion of the target nucleus is different. In this case, we will get an average value, which is specified in the following equation.
(5)
In equation (5), f(v) is a scattering function related to relative velocities, which are normalized as 
. The effectiveness of the fusion fuel is determined by its reactivity () [32].
4. Parameter Calculations
The Maxwell velocity scattering equation is specified by the following statement
(6)
index j identifies the species. T illustrates temperature, and kB illustrates the Boltzmann constant. The expression for the average reactivity can be rewritten in the form of the following equation
(7)
which 
and integrals are calculated on the three-dimensional velocity space. In equation (7), to get a suitable form to integrate, the velocities v1 and v2 are written by the center of mass relative velocities. Therefore, we will have
(8)
which 
and indicates that integration over dv1dv2 can be replaced by integration over dvcdv. If we write the volumetric element in velocity space as 
, by using the definition of the energy of the center of mass as 
, we will finally reach the following equation.
(9)
The fusion cross-section equation for all fusion reactions is shown in the following equation [33].
(10)
The values of C1, C2, and C3 are determined by the type of reaction [33]. Also ma and mb are the mass number of the striking nucleus and the target nucleus, respectively (for example, for deuterium as the colliding nucleus, there is ma=2). Elab (fuel energy in the laboratory system) is in keV unit, and σ is in barn unit [32-35].
5. Effects of Different 
on Plasma Parameters
In the previous section, we briefly explained how to calculate the parameter based on the Maxwell distribution. Now, by using the equations calculated for the parameter by Hively and Bosch-Hale, we write the system of particle and energy balance equations for the D-3He reaction. It should be noted that due to the high temperature required for the D-3He fusion reaction, there is a possibility of D-T reaction and two D-D reaction channels. Accordingly, we rewrite the particle and energy balance equations considering four fusion reactions (D-3He, two channels of D-D, and D-T) [10, 36].
Before expressing the system of particle and energy balance equations, we will introduce the equations used for the parameter. The temperature-dependent equation of Hively reactivity is expressed as follows.
(11)
The values of ai and r for D-3He reaction and other reactions are given in the tables in the reference [37].
The Bosch-Hale reactivity equation is also temperature-dependent and is shown by the following equation.
(12)
(13)
(14)
Numerical values of the constants C1 to C7 and also the values of mrc2 (keV) and BG (
) for the D-3He fusion reaction and other side reactions are given in the tables in reference [34].
Now we rewrite the system of particle and energy balance equations considering the four reactions mentioned, and then, for each one of the mentioned equations (eq. 11 and eq. 12), we solve these equations numerically separately and examine the results. It is important to note that this set of equations is written in zero-dimensional model [38-40].
The system of equations includes the differential equations of the change rate of the density of deuterium, helium-3, tritium, proton, and alpha particles relative to time and the temperature changes of ions and electrons concerning the hot ion mode relative to time [38-40]. Generally, for the particles balance equations we have
(15)
where i and j represent suitable species for 
reaction. The term of 
is the diffusion loss term and Sk is the external particle supply of the k nucleus. 
is the fusion reactivity for all production and consumption particles (i, j, k = D, 3He, T, p, α). The particle balance equations are mentioned in detail in reference [14].
As well as, for the ion temperature equilibrium equation, we have
(16)
and for the electron temperature equilibrium equation.
(17)
In equations (16) and (17), 
and the j index represent the D, 3He, p, α, and T particles. ni(t) and ne(t) represent the ion and the electron density, V0 is the plasma volume αn and αT are the density and temperature profiles, Pfu, PEXT, Poh, Pb, Ps, Pie are the fusion product heating power density, the external heating power, the ohmic heating power density, the bremsstrahlung loss power density for electron-electron interactions, the synchrotron radiation loss power density, the density of transferred power by collisions due to the ion and electron temperature difference, respectively. Also, PLe and PLi are the electron and ion conduction loss, respectively. All used parameters in equations (16) and (17), are described in detail in reference [14, 41-44].
The bremsstrahlung loss power (Pb) and synchrotron radiation loss (Ps), with consideration of the relativistic effects, are expressed as follows.
(18)
(19)
(20)
(21)
Reff is the wall reflectivity, fH is the hole fraction for divertor and ports (
), 
is the effective charge, and, β(t) is the central toroidal beta that is given by [45, 46]
(22)
where 
is the average toroidal beta[14].
Now, by using the system of particle balance equations and equations (16) and (17), we can rewrite the particle and energy balance equations, taking into account the hot ion mode, as a system of differential equations of temporal evolution of particles density and ion and electron temperatures, and then, by numerically solving this system for two different (Hively and Bosch-Hale), we examine the results.
In figure 1, Hively and Bosch-Hale reactivities have been demonstrated as a function of temperature in the range of 0 to 200 keV.
At first, we will examine the results obtained for the system of particle and energy balance equations based on the Hively equation.
Figure 2 shows that after ~100 seconds, the densities of deuterium and helium-3 particles will achieve steady state. In the following, we will study the ion and electron temperatures as a function of the density of the reaction fuel, i.e., deuterium and helium-3 particles. Figures 3 and 4 show the ion and electron temperature as a function of deuterium and helium-3 particle density changes, respectively. These diagrams are also drawn taking into account the hot ion mode.
What can be seen in figures 3 and 4 is that by increasing particle density, ion and electron temperatures decrease due to loss powers. Another noteworthy point is that the electron temperature in both diagrams has decreased more than the ion temperature, which indicates the effects of the bremsstrahlung loss power and synchrotron radiation loss which can also be seen in the written equations for ion and electron temperatures (equations 16 and 17).
secondly, we examine the results obtained for the system of particle and energy balance equations based on the Bosch-Hale equation. Figure 5 shows the temporal evolution of the densities of deuterium and helium-3 particles for 500 seconds.
The exciting point about figure 5 is that when we use the Bosch-Hale reactivity for the particle and energy balance equation, the graphs of the deuterium and helium-3 particle density variations fit precisely together. It is also observed that after ~100 seconds, the densities of deuterium and helium-3 particles achieve steady state.
In the following, we will examine the ion and electron temperatures as a function of the density of the reaction fuel, i.e., deuterium and helium-3 particles variations. However, this time we use the Bosch-Hale reactivity. Figures 6 and 7 show the ion and electron temperatures as a function of deuterium and helium-3 particle densities variations, respectively.
Figures 6 and 7 show that with increasing particle density, ion and electron temperatures decrease due to loss powers. We also see a further decrease in the temperature of the electrons in both figures 6 and 7, which shows the effects of the bremsstrahlung loss power and synchrotron radiation loss (equations 16 and 17).
Now we want to investigate the hot ion mode for both Hively and Bosch-Hale reactivities by comparing the graphs obtained for the ion and electron temperatures as a function of time and the densities of deuterium and helium-3 particles. First, we study the temporal evolution of fusion product densities (proton and alpha particles) for two different reactivities of Hively and Bosch-Hale. Figure 8 shows the temporal evolution of proton and alpha particle densities for both the Hively and Bosch-Hale reactivities.
Figure 8 shows that the densities of the D-3He fusion reaction products (proton and alpha particles) will reach steady state after ~200 seconds. We should note that the proton and alpha particle densities are more significant when using the Bosch-Hale reactivity than when using the Hively reactivity. Then in figure 9, we see the temporal evolution of ion and electron temperatures for two different reactivities (Hively and Bosch-Hale).
Here we introduce a parameter called the δ parameter as an indicator of the temperature ratio of ions to electrons. After ~100 seconds, when the ion and electron temperatures reach steady state for both reactivities, we compute the δ parameter for two different Hively and Bosch-Hale reactivities:
Figure 9 shows that the temperatures obtained for ions and electrons using the Hively reactivity, when achieved to steady state, are higher than those obtained for ions and electrons achieved by using the Bosch-Hale reactivity. However, the ion to electron temperature ratio (δ parameter) is almost the same value for both reactivities. In the following, figures 10 and 11 show the ion and electron temperature as a function of deuterium and helium-3 densities changes for both Hively and Bosch-Hale reactivities, respectively.
7. Conclusion
With consideration of hot ion mode, we intend to investigate the effects of Hively and Bosch-Hale reactivities on D-3He fusion reaction in spherical tokamak. Then, by writing the system of particle and energy balance equations for this reaction, we studied the effects of two different reactivities on plasma parameters in spherical tokamak. Eventually, we saw that the value of the δ parameter is almost the same for both reactivities (Hively and Bosch-Hale). In other words, for both Hively and Bosch-Hale reactivities, in the hot ion mode, the ion to electron temperature ratio will not change so much. Also, in figures 10 and 11, it can be seen that ion and electron temperatures as a function of density based on the Bosch-Hale reactivity have a steeper decreasing gradient than the Hively reactivity. Another critical point is that when we used the Hively reactivity to calculate the system of particle and energy balance equations, ion and electron temperatures compared to variations in fusion fuel density of D-3He (deuterium and helium-3 particles) had more stability than using the Bosch-Hale reactivity to calculate the particle and energy balance equations.
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Figure 1- The D-3He fusion reactivity as a function of temperature.
Figure 2 – The temporal evolution of deuterium and helium-3 particle densities based on Hively reactivity
Figure 3 – The ion and electron temperatures as a function of deuterium density changes based on Hively reactivity for 500 seconds
Figure 4 – The ion and electron temperatures as a function of helium-3 density changes based on Hively reactivity for 500 seconds
Figure 5 – The temporal evolution of deuterium and helium-3 particle densities based on Bosch-Hale reactivity
Figure 6 – The ion and electron temperatures as a function of deuterium density changes based on Bosch-Hale reactivity for 500 seconds
Figure 7 – The ion and electron temperatures as a function of helium-3 density changes based on Bosch-Hale reactivity for 500 seconds
Figure 8 – The temporal evolution of proton and alpha particle densities (fusion reaction products) for both Hively and Bosch-Hale reactivities
Figure 9 – The temporal evolution of ion and electron temperatures based on Hively and Bosch-Hale reactivities
Figure 10 – The ion and electron temperatures as a function of deuterium density changes based on Hively and Bosch-Hale reactivities for 500 seconds
Figure 11 – The ion and electron temperatures as a function of helium-3 density changes based on Hively and Bosch-Hale reactivities for 500 seconds
