## Abstract

The nuclear reaction of deuterium–tritium (D–T) fusion by the usual magnetic or inertial confinement suffers from a number of difficulties and problems caused by tritium handling, neutron damage to materials and neutron-induced radioactivity, etc. The study of the nuclear synthesis reaction of deuterium–helium-3 (D–^{3}He) at low collision energies (below 1 keV) is of interest for its applications in nuclear physics and astrophysics. Spherical tokamak (ST) reactors have a low aspect ratio and can confine plasma with *β*≈1. These capabilities of ST reactors are due to the use of the alternative D–^{3}He reaction. In this work, the burn condition of D–^{3}He reaction was calculated by using zero-dimensional particles and power equations, and, with the use of the parameters of the ST reactor, the stability limit of D–^{3}He reaction was calculated and then the results were compared with those of D–T reaction. The obtained results show that the burn conditions of D–^{3}He reaction required a higher temperature and had a much more limited temperature range in comparison to those of D–T reaction.

## 1 Introduction

Deuterium–tritium (D–T) fuel for a fusion reactor is the dominant fuel in use in today’s fusion research project [1–3]. The D–T fuel cycle has the largest cross section of all the fusion reactions, and it burns at the lowest temperature. In the D–T fuel cycle, 80% of the fusion energy is in the form of 14-MeV neutrons. These neutrons activate the reactor wall and induce radiation damage and radioactivity in the reactor structure. Furthermore, a breeding and storage system for a radioactive substance, tritium, that needs extra complexity, cost and radial space for a lithium blanket is required for the D–T reactor. The D–^{3}He reaction generates no neutrons, so a fusion reactor with this fuel is very attractive from this viewpoint. Neutron activation levels and radiation damage rates by a factor of 17–20 relative to the D–T fuel cycle are reduced by the low neutron production of the D–^{3}He fuel cycle [4]. In the D–^{3}He reaction, if the D–D reactions are disregarded, all fusion energy is released in charged particles that can be directly converted to electricity at high efficiency in some configurations. The other advantages of D–^{3}He over D–T are full lifetime materials, easier maintenance and proliferation resistance [5–9]. But this reaction has many problems. The D–^{3}He fusion cross section is lower than that of D–T fuel. Consequently, D–^{3}He requires a density-confinement time product that is 50 times higher and a fusion power density in the plasma that is 80 times higher than that for D–T [5, 6]. Burning D–^{3}He fuel thus requires substantial, continued progress in plasma physics. There has been only relatively small progress beyond the progress already accomplished in the historically well-funded tokamak program. The crucial physics issues for advanced fusion configuration are the confinement and control of the resulting fusion ash buildup [10]. In innovative confinement concepts such as the spherical tokamak (ST), the key physics issues have been identified, but resources to test issues adequately have not been available [10]. The fact that the terrestrial supply of the helium isotope ^{3}He is severely limited is another impediment to competitive D–^{3}He fusion, while the hydrogen isotope deuterium is found in plentiful supply in nature. Fusion can be applied to space propulsion. Research projects address the application of fusion to generate electrical power in space, as well as propulsion. As mentioned previously, the standard D–T fusion reaction has some fundamental limitations; moreover, the lack of ^{3}He on the earth is the main problem for the D–^{3}He fusion reaction, but ^{3}He is found all throughout the universe and thus the reactor could be refueled at any number of locations such as the moon. For theses reasons, it seems that D–^{3}He fusion reaction is an appropriate fuel for space propulsion. The first study on D–^{3}He fusion reactors for space propulsion was published in 1962, and one of the noted studies to use D–^{3}He fusion reaction for propulsion was done by the British Interplanetary Society called Daedalus [11].

As mentioned previously, a spherical tokamak is capable of operating with the alternative D–^{3}He fuel because the fact that ST reactors can confine plasma with *β*≈1 allows plasma to be confined at the same pressure as in classical tokamak reactors but at a significantly lower toroidal magnetic field [12]. The physics key to the attractiveness of the ST approach is in the order unity *β* values expected to be achieved by the combination of low aspect ratio. A low aspect ratio makes it possible to create a relatively compact ST reactor, while a high *β* value allows the reactor to operate with the alternative D–^{3}He reaction [12]. The advantages of the ST approach have been discussed for many years [13, 14]. In recent years, interest in the ST approach has grown rapidly [15, 16]. The low cost of the ST approach in comparison with other conventional tokamaks allows ST reactors to continue to be considered as a potential option for producing thermonuclear burn. The possibility of D–^{3}He fueled ST reactors has been indicated already [17].

The aim of this paper was to study the burn condition of D–^{3}He nuclear fusion reaction in comparison to that of D–T fusion reaction. In these reactions, investigation of the burn criterion is very important. Several attempts have been made to extend the original burn criterion. By using *τ*_{R}, the time scale for reaction kinetics, which replaced *τ*_{E}, the energy confinement time, Maglich and Miller [18] introduced burn criterion in 1975. This approach was questioned by Chen et al. [19] in 1977. Burn criterion was defined by Chen et al. [19] as the product of plasma density and energy confinement time in which all loss processes can be included realistically in the calculation of the energy confinement time. In this work, *τ*_{E} has been applied in the calculation of the triple product *n*_{e}*τ*_{E}*T* for the D–^{3}He reaction. This parameter was obtained from the work of Rebhan and Oost [20] on D–T reaction, and it was also calculated in this work for the D–^{3}He reaction and for comparison with the D–T reaction.

Zero-dimensional particles and power equations have been used for nuclear fusion reactions. In this model, ions and electrons are assumed to have Maxwellian distributions and share the same temperature at all time, *T*_{i}=*T*_{e}. Using the global power equation and scaling ansatz brings about the closed ignition curves for D–^{3}He nuclear fusion reaction, which are compared with D–T reaction results [21], then, with the use of energy confinement scaling and the parameters of the ST reactor, plasma instability limits were obtained and compared with D–T reaction results [20]. In this work, in addition to D–^{3}He nuclear fusion reaction, the two branches of D–D reaction and D–T reaction were taken into consideration in our calculations. Synchrotron radiation power was also taken into account in the D–^{3}He reaction, while it was ignored in the D–T reaction.

## 2 Theoretical Calculation

The main fusion reactions in this work are

and for the D–^{3}He cycle

all the aforementioend reactions were applied in this work. The reactions (D–D, D–^{3}He) for the D–T reaction were ignored due to their small fusion cross section.

The equations to be solved in this paper were the general particle balance and general global plasma power balance equations. The particle balance equation, which describes the diffusion loss, fuel injection, production and consumption of a given species, is of the form

where *i* and *j* represent the appropriate species for reaction *i*+ *j*→*k*. The term *S*_{k} is the external particle supply of nucleus *k*. 〈*σν*〉_{ij} is the fusion reaction rate for all particle production and consumption (*i*, *j*, *k*= *D*, *T*, *p*, ^{3}He, *α*), and for all reactions, 〈*σν*〉_{ij} was taken from [22]. For like particles, *i*= *j*; otherwise, *i*≠ *j*.

In this work, the particle confinement time *τ*_{p} was coupled to the energy confinement time *τ*_{E} by this relation [20],

The general global plasma power balance equation is

where *P*_{f} is the fusion power density, which is calculated by considering all fusion reactions, which are given by

where *i*+ *j*→*k*, and for like particles, *i*= *j*; otherwise, *i*≠ *j*.

*P*_{aux} is the auxiliary power density; it is useful to express the auxiliary power in multiples of fusion power by

*P*_{br} is the bremsstrahlung radiation power loss and is defined as [23]

*Z*_{eff} is the effective charge of the plasma ions and is defined as

and *A*_{b} is the bremsstrahlung radiation coefficient [24]. The electron density is governed by the neutrality condition.

*P*_{syn} is the synchrotron radiation loss and is defined as [23]

where

where *A* is the aspect ratio, which is *R*/*a*; *B* is the toroidal magnetic field on axis; and *R*_{w} is the reflection factor of the radiation from the vacuum wall. The synchrotron power depends on the plasma parameters, i.e., aspect ratio, *β* value, magnetic field, minor radius of plasma, wall reflectivity and density. In this work, it is used by the parameters of the ST for the D–^{3}He reaction [25]. Synchrotron radiation is important at high temperature; at low temperature, however, bremsstrahlung radiation dominates the losses, so, in this work, synchrotron radiation was ignored in the D–T reaction.

*P*_{ie} is the equilibration power between ions and electrons

where

The parameters *α*_{n}, *α*_{T} represent the parabolic profile exponents. In this work, it was assumed that *T*_{i}=*T*_{e}, so *P*_{ie}=0.

*P*_{oh} is the ohmic power density, which, for steady-state operations, is negligible. The last parameter in (8) is the power density of the transport losses, where *W* is the plasma thermal energy content defined as

where *n*_{tot}=*n*_{i} + *n*_{e} and *n*_{i} is the ion density.

Now, the *n*_{e}*τ*_{E}*T* parameter of the ideal ignition and the breakeven values of the D–T and D–^{3}He reaction should be calculated. To do this, in steady state, the d/d*t* term equals zero, so, in this state, we shall neglect the presence of impurities as well as the helium and proton ash [20]. For breakeven, the *f*= 1 in (10) and *f*= 0 for ignition were set. By solving (8), considering (9), (10), (11) and (14), the ideal fusion product *n*_{e}*τ*_{E}*T* for the D–T and D–^{3}He reaction in the state of ignition and breakeven was obtained [20]. Its temperature dependence for both of the states and two reactions is shown in Figure 1. Comparison of these two graphs shows that the minimum temperature required for ignition in the D–T reaction was 14 keV and for breakeven was 13 keV. The minimum for D–^{3}He reaction was 65 and60 keV for ignition and breakeven, respectively. The graph shows that the minimum value of *n*_{e}*τ*_{E}*T* for the D–^{3}He reaction was 26 and 68 times larger than that for the D–T reaction for ignition and breakeven, respectively.

We shall consider the influence of ash particle and calculate the *n*_{e}*τ*_{E}*T* parameter, but this time we need the particle balance equations with regard to (6) and the scaling ansatz (7) [21, 26]. One can derive an equation for *n*_{e}*τ*_{E}*T* where it is a function of *T* and *ρ* and can be put into the form *ρ*(*n*_{e}*τ*_{E}*T, T*) [21]. The dependence of this parameter on temperature in the D–T and D–^{3}He reactions is shown in Figure 2. This time it is a function of *ρ* and *T*.

Figure 2 shows the ignition curves (*ρ* = const) for the D–T and D–^{3}He reactions. This figure shows the closed ignition curves that exist for the D–T reaction for *ρ* < 15 [20]; these curves exist for D–^{3}He fuel for *ρ* = 5.5 and *ρ* = 6, and for larger values of *ρ* they disappear. The largest value of *ρ* in the D–^{3}He reaction was 6.21. It is obvious from Figure 2 that the burn condition of the D–^{3}He reaction in comparison to that of the D–T reaction is at a higher temperature. The minimum *n*_{e}*τ*_{E}*T* for the state *ρ* = 6 in the D–^{3}He reaction in comparison with *ρ* = 6 in the D–T reaction was approximately 40 times larger.

The energy confinement time is a quantity of central importance in the design of next-generation tokamak reactors, which is equal to the plasma stored energy divided by the total power loss from the plasma in equilibrium. Because tokamak transport is problematic, it is very difficult to predict the energy confinement time theoretically. Empirical scaling laws are needed. Empirical scaling has been derived from databases for global plasma parameters collected from different devices. The energy confinement time depends on the plasma parameters. In this work, the energy confinement scaling used for the D–T reaction is ITER 89-P [27].

This scale with plasma parameters is

There are several confinement regimes; the H-mode is the most important regime for reactor applications. The currently recommended H-mode scaling is known as IPB98 (y,2), which is used for the D–^{3}He reaction

where *P*, the net plasma heating power in megawatts, is given by

In the steady state, it can be replaced by

H_{h} is the enhancement factor; the recent NSTX experiments show that an enhancement factor of *H*_{h}>1.5 is possible. However, we conservatively set *H*_{h}=1.0. At this point, by using the parameters in Table 1 for D–T reaction and those in Table 2 for D–^{3}He reaction, the ignition contours can be translated into an *n*_{e}, *T* plane.

Parameter | Unit | Value |
---|---|---|

Toroidal current, I_{P} | MA | 22 |

Magnetic field, B_{t} | T | 4.85 |

Elongation, κ | 2.2 | |

Major radius, R | m | 6 |

Minor radius, a | m | 2.15 |

Plasma volume, V | m^{3} | 1100 |

Troyon coefficient, g_{f} | 2.5 |

Parameter | Unit | Value |
---|---|---|

Toroidal current, I_{P} | MA | 128 |

Magnetic field, B_{t} | T | 2.7 |

Elongation, κ | 3 | |

Major radius, R | m | 8 |

Minor radius, a | m | 6.15 |

Plasma volume, V | m^{3} | 1.792×10^{4} |

Wall reflection, R_{w} | 1 | |

Troyon coefficient, g_{f} | 4.2 |

By substituting the energy confinement time in the equation *n*_{e}*τ*_{E}*T*, this equation can be transformed to an *n*_{e}, *T* plane. Using this transformation, we obtained the previous ignition curves for *ρ* =0, 3, 5 for the D–T reaction and *ρ* =5, 5.5, 6 for the D–^{3}He reaction in the *n*_{e}, *T* diagram shown in Figure 3. In this figure, the *n*_{e}, *T* plane for the D–^{3}He reaction in comparison to that for the D–T reaction [20] is shown. The fact that the minimum temperature range of 6 keV in the D–T reaction shifted to 32 keV in the D–^{3}He reaction was obtained from the comparison of the ignition curve in the state *ρ* =5 in Figure 3. The other finding of this comparison is that the minimum of electron density for the D–T reaction at a temperature of about 14 keV was 4.7×10^{19} (m^{–3}) for the ignition curve in the state *ρ* =5, while the temperature for the D–^{3}He reaction was about 50 keV and 1×10^{20} (m^{–3}).

And using the *β* parameter as

the ignition curves can be transformed to the *β*, *T* plane.

The normalised *β*_{N} is given by

The following operation limits were mainly implied from the tokamak operation window. The *β* limit is expressed as

where *g*_{f} is the Troyon coefficient and identified in Tables 1 and 2 for the D–T reaction and D–^{3}He reaction, respectively. The Greenwald density limit is

where *k*_{G}=0.85 for the D–^{3}He reaction [25]. In Figure 4, the *β*_{N}, *T* plane for the D–^{3}He reaction is shown in comparison to that for the D–T reaction [20]. The importance of representing this ignition curve as *n*_{e}(*T*) or *β*(*T*) is that the impact of plasma stability limits like the *β* limit and the density limit can be seen immediately.

## 3 Conclusions

In this work, using zero-dimensional particles and power equations for D–T and D–^{3}He reactions, the parameter *n*_{e}*τ*_{E}*T* was first obtained in ideal conditions where the presence of impurities as well as of helium and proton ash was ignored and then its temperature dependence was shown for two states. Then, with regard to particle equations, the temperature dependence of this parameter was both calculated and shown again.

The results show that the burn conditions of the D–^{3}He reaction required a higher temperature in comparison that of the D–T reaction. In addition, the burn condition of the D–^{3}He reaction in comparison to that of the D–T reaction has a much more limited temperature range. Afterwards, using and substitding the energy confinement time and plasma parameter for the D–^{3}He and D–T reactions in the equation *n*_{e}*τ*_{E}*T*, the equation can be transformed to the *n*_{e}, *T* plane. It is worth noting that the parameters of ST reactor were used for the D–^{3}He reaction, which can be seen in Table 2.

Regarding the present progress in building reactors, it seems unlikely that D–T will be replaced by D–^{3}He reaction as fuel for fusion reactors. Because D–^{3}He reaction requires a much higher plasma temperature, density and *n*_{e}*τ*_{E}*T* parameter than a comparable D–T reactor, it seems challenging and unlikely to consider D–^{3}He reaction as a suitable fuel in a fusion reactor. Lack of a reliable and suitable source of ^{3}He is another problem in D–^{3}He reaction. The results show that the burn conditions of D–^{3}He reaction required a higher temperature in comparison to those of D–T reaction. In addition, the burn condition of D–^{3}He reaction in comparison to that of D–T reaction has a much more limited temperature range.

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**Received:**2014-4-29

**Accepted:**2015-1-7

**Published Online:**2015-2-7

**Published in Print:**2015-2-1

©2015 by De Gruyter