The Effect of Energy Leakage Probability on Burn Propagation in an Optically Thick Fusion Plasma

In an optically thick plasma, the mean free path of bremsstrahlung photons is smaller than the plasma radius, and radiation can be treated as a photon gas in thermal equilibrium. In these conditions, the black body radiation spectrum exceeds the number of hot photons, and reabsorption processes such as inverse bremsstrahlung radiation and inverse Compton scattering become important. It has been shown that a dense fusion plasma like the one being used in ICF method is initially optically thick. When the fuel pellet is burning, the temperature of its electrons rises approximately greater than 90KeV , and the pellet becomes rapidly optically thin. In this paper, we have shown that the energy leakage probability makes electron temperature remain low approximately smaller than 55KeV , and as a result the fuel pellet remains optically thick during burning.


Introduction
Research in laser driven inertial confinement fusion ICF began soon after the invention of the laser see the historical notes in 1, 2 .In recent years, because of the development of high intensity lasers especially in NIF: National Ignition Facility, USA 3 , LMJ: Laser Megajoule, France 4 , the ICF method is in the forefront of energy production from fusion process.The main components of the ICF fuel pellets are the hydrogen isotopes, that is, deuterium and tritium DT .The DT mixture is the fuel with the lowest ignition temperature and has the highest specific yield.The problem is that tritium is an unstable artificial isotope, decaying to 3 He with a half-life of 12.3 years and, as a result, needs to be produced within the DT fuel cycle.Therefore, the tritium production has the most significant radiological problem in the future of DT fusion reactors.Furthermore, another problem is energetic neutrons 14 MeV that are yielded from DT reaction.
One of the solutions to overcome the problem of DT fuel utilization is using Proton-Boron-11 P 11 B → 3α 2.7 MeV and deuterium-helium-3 D 3 He → p 14.7 MeV α 3.6 MeV fuel configuration.These configurations are called advanced fusion fuels.In classical plasmas, self-burning of advanced fuels is unlikely to occur 5 , because at high temperatures the bremsstrahlung loss may exceed the fusion power produced.However, If the final temperature reached is low enough, the electrons of the plasma can be degenerated 6 .If this is the case, bremsstrahlung emission is strongly suppressed 7-12 , and ignition temperature becomes lower than in the classical plasmas.In addition, several research showed that in Fermi degenerate plasmas, the properties of the aneutronic fuel burning can be very different from those of classical plasmas, due to the reduction of ion-electron i-e collisions, both of which allow the ion temperature to exceed the electron temperature and reduce the bremsstrahlung loss 13-17 .Nevertheless, low gain is the main failure of using advanced fusion fuels 15 .
Another solution has been proposed to be adding a small amount of tritium to the deuterium fuel in the form of DT x , where x is the ratio of tritium to deuterium in initial fuel 18, 19 .It has been shown that in x ∼ 0.01 and with a given initial condition, namely, density ρ 5000 g/cm 3 , areal density ρR 12.5 g/cm 2 , and temperature T 10 KeV , internal breeding of tritium takes place.Internal breeding of tritium means that the remaining amount of tritium after the burning of fuel is equal to initial content.However, the low gain of fuel pellet is still the main failure of this fuel configuration.
The other solution for the above problem is adding 3 He to DT x fuel to form a new configuration as DT x 3 He y , where x is the ratio of tritium to deuterium, and y is the ratio of helium-3 to deuterium in the initial pellet.It has been shown that the configuration fuel as DT x 0.0112 3 He y 0.0399 , and in the same initial conditions as those described for the case just mentioned, internal breeding of tritium and helium-3 takes place.Furthermore, the energy gain is increased and neutrons produced are decreased due to the reduction in the initial content of deuterium and tritium.
When the size of the fusion fuel pellet is comparable with the thermalization range of fusion products, a fraction of the energy will escape from the burning pellet.This fraction was calculated for DT x 0.0112 , with the same initial conditions but was not applied for burn propagation simulation 20 .In an earlier study, we inserted the energy leakage probability to the relevant calculations to obtain an ignition condition for configuration fuel DT x 3 He y in different ratios and densities 21 .In the present study, we applied this effect to recalculate the burn simulation in more detail.
The organization of this paper is as follows.The one-dimensional power balance equations are introduced in Section 2. Energy deposition of charged particles and energy leakage probability are given in Section 3, optical depth are discussed in Section 4, and finally, conclusion is placed in Section 5.

One-Dimensional Power Balance Equations
Here we consider thermonuclear reaction starts in a hot spot with radius 2.5 × 10 −3 cm, density of 5000 g/cm 3 , and temperature of 10 keV which has been produced inside the surrounding shell after compression phase.The equation of energy balance for ions, electrons, and radiation are given by where T i T e is the ion electron temperature, n i n e is the ion electron number density, η j k is the energy leakage probability of the product k created in the reaction j, and f j k is the fraction of the energy of the product k created in the reaction j that is deposited into the plasma ions.P j fk is the fusion power of product k created in the reaction j, P mi , and P me are the mechanical expansion loss power of ion and electron, respectively, and P e is the thermal conduction of electrons 1 .T r is the radiation temperature, and P r is the photon energy loss power 19 .
For the optically thick plasma the mean free path of bremsstrahlung photons is smaller than the plasma radius and radiation can be treated as a photon gas in thermal equilibrium.Thus, a black body radiation spectrum is obtained, and the radiation temperature can be defined.In this case, the reabsorption process, such as inverse bremsstrahlung and inverse Compton effect are more important.Then, we can write the bremsstrahlung loss, P B , as in the following form: From the Planck distribution for the photons, we can easily estimate the inverse Compton scattering contribution.Provided the photons are of low energy, hν/m e c 2 1, and the electron temperature moderate, kT e /m e c 2 1, the energy lost by the electrons in inverse Compton scattering is given by 23 as follows: where 8/3 πr 2 e is the Thomson cross-section, and r e is the classical electron radius.In a burning plasma, the fusion energy will be deposited mainly into the ions 2, 20 .Therefore, the ion temperature will be higher than the electron temperature, and an energy flow from the ions to the electrons by collisions is expected.In a dense plasma scattering is predominantly quantum mechanical and entails not only the first Fermi-Dirac correction, ISRN High Energy Physics but also the first classical correction when the electron density starts to increase.P ie is the ion-electron energy exchange term that is achieved by 24 as follows: where ln Λ QM BPS is the leading term in the quantum limit together with the Δ correction that exhibited by, ln Λ ΔC BPS is the first classical correction, and ln Λ FD BPS is the first correction when Fermi-Dirac statistics start to become important.The numerical values of the zeta-function and its derivative are and ω b is the plasma frequency for spices b, where ω 2 I i ω 2 i .The energy gain of burning region is defined as the ratio between the total energy deposited by fusion products and the energy contained in the originally heated plasma, The catalytic regime of tritium is obtained when the final amount of tritium is slightly higher than the initial amount.In this case, external tritium breeding is avoided and is replaced by internal tritium breeding.The internal tritium breeding ITB is defined as

Energy Deposition of Charged Particles and Energy Leakage Probability
When two ions fuse together, the energy E j is yielded from fusion reaction j.The reaction product k carries with itself a fraction of w j k of energy E j .Next, the reaction product, k, with an energy equal to w j k E j slows down and deposits a fraction of f j k of its energy to the ions.This fraction is calculated through the following: where dE/dx is the stopping power of charged particle with an initial energy of w j k E j which starts to slow down to the final energy of 3/2T i , 26 .Subscripts i and e represent ions and electrons, respectively.
The distance traversed by a charged particle, x E 0 , E , which loses energy from energy E 0 to E is obtained as

3.2
We can now calculate the fraction of the energy the charged particle releases from the pellet of radius R. The parameter x can be expressed in terms of the radial distance from the center, r, the pellet radius, R, and the angle between the path and the line passing through the center, μ cos θ, as shown in Figure 1:

3.3
If the charged particle is produced with uniform probability in the sphere, R, the probability of escaping can be calculated as where E 0 is the initial energy of the charged particle, and E r, μ is the energy of the same particle after traversing a distance x from the point of birth to the spherical surface.In Figure 2, we illustrate energy leakage probability of charged particles for various reactions during the burning of fuel pellet.

Optical Depth
Optical depth is a parameter that determines whether the plasma is optically thick or thin.Since hot photons with hν > T e are important for the reabsorption process, this parameter is defined by 22 as follows: where E BB ν is the radiation energy density of a black body, and E Br ν is the bremsstrahlung radiation energy density 27 .When this parameter is greater than unity, the black

Figure 4:
The ratio between the number of hot photons with hν > T e obtained from the black body radiation spectrum E BB ν , and from bremsstrahlung radiation spectrum, E Br ν .body radiation spectrum exceeds the number of hot photons and, as a result, plasma becomes optically thick.Earlier studies have shown that the plasma is initially optically thick but rapidly becomes optically thin transparent as the electron temperature rises greater than ∼90 KeV 28, 29 .As shown in Figure 3, the electron temperature cannot be greater than ISRN High Energy Physics 9 55 KeV if energy leakage probability is taken into consideration.As a result, burning fusion pellet will not be optically thin during burning time Figure 4 .

Conclusion
In the present study, the effects of burn propagation of a pellet with the configuration DT x 0.0112 3 He y 0.0399 with the initial conditions, namely, density ρ 5000 g/cm 3 , areal density ρR 12.5 g/cm 2 , and temperature T 10 KeV -taking into consideration energy leakage probability-has been reinvestigated.The results indicate that the electrons' temperature does not exceed 55 KeV, and the fuel pellet remains optically thick during the burning process.Also, the effects of leakage probability resulted in the reduction of energy gain can be reduced to a gain of 70. Figure 5 illustrates internal tritium breeding during the burning process.As Figure 5 shows, energy leakage probability effect brings about a reduction in ITB, and consequently, the final ratio of tritium is less than the primary ratio.

Figure 1 :
Figure 1:The relation between distance, x, with radial distance from the center, r, the pellet radius, R, and the angle between the path and the line passing through the center μ cos θ.

Figure 2 :Figure 3 :
Figure 2: Energy leakage probability of charged particles for various reactions during the burning of fuel pellet.

2 ITBFigure 5 :
Figure 5: Ratio of the number of tritium particles at time t to the initial number of tritium particles as a function of burning time.