# Multiplication of Fast Neutrons Source Flux by Using Deuterium-Helium-3 Plasma

Authors: Mohammad Mahdavi; Maryam Shahbahrami
Journal: ISRN High Energy Physics
 (2013)

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## Abstract

The production of fast neutrons source is examined by
using a thermal neutron flux insideD/He3 plasma. In order to
reach a favorable yield of fast neutrons flux, the parameters
such as energy loss rate, reaction probability, and neutron
absorption length are calculated. The nuclear
conversion efficiency, ζ, of thermal neutron to fast neutrons is obtained to be 2×10-3 by calculating the physical parameters for the D/He3 plasma designed.


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## Body

## 1. Introduction

With considering the importance of widespread application of neutrons’ flux in various branches of science and technology, numerous efforts have always been undertaken to improve the efficiency of neutrons’ sources. A special attention in this field is now focused on developing fast neutron sources. Unlike neutron beams with less energy, the choice of sufficient tools for the production of fast neutrons is restricted. In fact, the most natural way of producing an appreciable source of these neutrons is based on employing theD(t,n)4He nuclear reaction. Different concepts of D/T neutron sources have been proposed. However, in spite of differences in design suggested for fast neutron sources, all of them have a common feature; that is, radioactive tritium must be initially in the ion beams or in the targets. In fact, operation with tritium materials is known to be prohibited in many research centers. Thus, a developmental study of fast (14.1 MeV) neutrons is still urgent. Some studies have been performed in this field such as production of fast neutrons by using thermal neutrons with the help of the LiD combination [1]. One of these applicable methods that was examined in this paper is the production of fast neutron sources in the D/He3 plasma using a thermal neutrons flux. Avoiding the implementation of initial tritium due to its radioactive property is one of the main factors in using of D/He3 plasma. To provide such conditions one should use deuterium-containing materials. Secondary tritium particles are generated from the interaction of thermal neutron with the helium nucleus,He3(n,p)T in D/He3 plasma. Tritium particles fuse with target deuterons and release fast neutrons 14.1 MeV as follows:
(1)D+T⟶He4(3.5MeV)+n(14.1MeV).In this work, we have examined the theory of the fast neutrons’ sources production using theD/He3 plasma. The result of these calculations led to the attainment of the most desirable efficiency for fast neutrons flux.


## 2. Theory

In aD/He3 mixture irradiated by a flux of thermal neutrons, the nuclei interaction with thermal neutrons leads to three nuclear reactions:
(2)n+He3⟶T+p+763KeV,n+He3⟶γ+He4,n+D⟶T+γ+6.26MeV.Among these three processes, theHe3(n,p)T reaction fully dominates because of a very large cross-section (see Figure 1) [2].Figure 1
Cross-sections forn+D and n+He3 nuclear reactions at neutron energy En≤1MeV [2].At the first step, tritium ions are produced from interaction of thermal neutron with helium nuclei:(3)nth+He3⟶T+p+763KeV.The contribution of tritium and proton particles energies according to energy and momentum conservation is191KeV and 572KeV, respectively. At the second step, tritium ions fuse with plasma deuterons as follows:
(4)T+D⟶n+α+17.6MeV.The image of thermal neutrons interaction withD/He3 plasma that leads to production of fast neutrons is shown in Figure 2.Figure 2
Image of collision of thermal neutron and helium nuclei inD/He3 plasma.There are two important factors of the reactionHe3(n,p)T. Firstly, this reaction contains a very large cross- section of about ~5300b [3]. Secondly, the initial energy of the tritium particles produced, ET0=191KeV, is very close to resonance energy, ETres=160KeV, of the D(t,n)He4 reaction [4]. The nuclear conversion efficiency, ξ, of thermal neutrons to fast neutrons can be described according to
(5)ξ=Fn3He×FTD,
where Fn3He is the thermal neutron absorption probability by helium nuclei and FTD is the probability of tritium interaction with deuterons plasma. The amount of FTD can be defined by analyzing the slowing-down of the 191keV tritium in the D/He3 plasma. Principally, fast charged particles slow down in plasma through Coulomb and nuclear elastic scattering caused by plasma electrons and ions. The contribution of (4) is negligible since the initial tritium kinetic energy is 191KeV. Tritium Coulomb slow-down in the D/He3 plasma can be calculated according to the conventional binary-collision model [5]. According to the above, the average instantaneous rate of tritium energy loss, 〈dET/dt〉, in Maxwellian D/He3 plasma can be represented as
(6)〈dETdt〉=∑j〈dETdt〉j,
where the term 〈dET/dt〉j defines tritium energy loss in T-j scattering (j=e,D,He3). In order to calculate tritium energy loss, we should estimate the average rate of the particles energy loss. With the assumption that the ion of mass M and charge Ze with velocity v in the laboratory system collides with a field particle of mass m and charge ze with velocity w, the cross-section for scattering through an angle θ in the c.m. system is;
(7)σ(θ)=(Zze2(1/2)[Mm/M+m](v-w)2)2×1[4sin2(θ/2)]2.[
m
/
M
+
m
]
(
v
-
w
) is the velocity of the ion in the c.m. system. In the laboratory system, the change ΔE of the kinetic energy of the ion is [6]
(8)ΔE=-mM(M+m)2×{2sin2(θ2)[(v-w)(Mv+mw)]+sinθcosϕ|(v-w)×(Mv+mw)|θ2};
here θ is the angle of scattering in the c.m. system and ϕ is the azimuthal angle of scattering in the c.m. system. The average value of sin2(θ/2) for such a collision is obtained from (7)
(9)(sin2(θ2))av=π2σt(Zze2(1/2)[Mm/(M+m)](v-w)2)2lnΛ,
where σt is the total scattering cross-section and the second term of (8) averages to zero. Thus, the average energy change of the particle from such a collision is simply
(10)ΔE=-4πσt(Zze2)2m×lnΛ×((v-w)[v(1+(m/M))-(m/M)(v-w)](v-w)4).The velocity distribution of the field particles under consideration isf(w)dw per unit volume. The number of collisions with field particles having velocities between w and w+dw in a range of time, δt, is σt|v-w|f(w)dwδt. Thus, the rate of change of energy of the ion determined by plasma particles with velocity w is
(11)(dEdt)=-4πm(Zze2)2×lnΛ×[(v-w)[v(1+(m/M))-(m/M)(v-w)](v-w)3]×f(w)dw.If we take a Maxwellian distribution,(12)f(w)=f0exp[-(wwt)2],
where wt=(2kBTj/m)2 is the thermal speed of the plasma particles. kB and Tj are Boltzmann constant and temperature particles (energy), respectively. After solving the above integral, we obtain [7]
(13)〈dETdt〉j=-4πe4(2mjTj)1/2(ZTZj)2njlnΛjPhi(ζj)ζjΦ(ζj)=erf(ζj)-2π1/2(1+mjmT)ζjexp(-ζj2)ζj=(mjETmTTj)1/2,
where nj is the density of plasma particles species j having Maxwellian temperature Tj. It is assumed that TD=THe3=Ti for the homogenous temperature plasma. The Coulomb logarithm in classical, and quantum mechanical limits are defined as [5]
(14)lnΛjclassical=ln(1+D2ρj2)1/2,lnΛjquantum=ln4πDλj-12,
where D is the Debye length and ρj and λj are the cut-off impact parameter and the scattering wave length in the (T-j) system, respectively. During the slowing down of the 191KeV tritium particle, its kinetic energy transfers to plasma electrons and ions. The fraction of energy ηj transferred to plasma species j is introduced as follows:
(15)ηj=1ET0-Eth∫EthET0(〈dET/dt〉j〈dET/dt〉)dET,
where Eth is the ion thermal energy, (3/2)kBTi. The fractions of energy transferred to electrons, nuclei 3He and D are calculated using (15) as 0.6, 0.3, and 0.1, respectively. Therefore, tritium energy is mainly deposited to the plasma electrons. FTD is an important parameter that is effective in nuclear conversion efficiency calculation and is defined as fusion probability of slowed-down tritium ions and background deuterons of D/He3 plasma. The differential reaction probability, dFTD(ET), at tritium energy, ET, is given in the following form [8]:
(16)dFTD(ET)=(1-FTD(ET))nDσ(ET)dET〈dET/dx〉dET,
where σ(ET) is the D-T reaction cross-section. The probability, FTD(E), of reaction during tritium slowing from the initial energy, ET0, to an energy E equals:
(17)FTD(E)=1-exp×(∫EET0‍(2ETmT)1/2nDσ(ET)〈dET/dt〉dET).


## 3. Conclusion

The probabilityFTD(Eth) of the D(t,n)He4 reaction, when tritium decreases to thermal energy, Eth, is illustrated in Figure 3. As it is seen, the value of FTD(Eth) rises with increasing Te. Also, the change of D/3He plasma density does not significantly affect the value of FTD(Eth). Figure 4 shows the reaction probability in terms of plasma ion density ni=nD+n3He in the 1019–1023 cm−3 range. Another parameter which can determine the size of the plasma is the length of neutron absorption (mean free path) that is indicant of an average distance traversed by the neutron for the creation of first collision. If the generator system does not include a neutron reflector, the plasma size should be comparable with the length of neutron absorption, Ln via the He3(n,p)T reaction. The absorption length is Ln=(n3Heσ-np)-1 which is the He3(n,p) reaction cross-section for the most probable relative velocity of the colliding particles in the plasma with the amount of σ-np≈150b. By considering the 3He density order of 1021cm-3, the neutron absorption length is Ln~7 cm. Therefore, plasma with 2-3 times the size of neutron absorption length of (10–20 cm) would be sufficient for the present concept of a fast neutron source. By considering plasma size and the He3 density in the order of 1021cm-3, the neutron absorption probability, Fn3He, is close to unity (Fn3He~1). Therefore, ξ=FTD(Eth), at electron temperature, Te=1KeV, the nuclear conversion efficiency, ξ, is obtained 2×10-3. This should be sufficient for a well-yield flux of fast neutrons. For instance, if the flux of thermal neutrons by order 1012–1015(n/cm2s) as modern neutron sources [9] is injected to plasma, finally, this can lead to the production of 14.1 MeV neutrons by order 109–1012(n/cm2s). In this paper, the production of fast neutrons via thermal neutrons is investigated. It is shown that by designing the plasma dimensions, the neutrons are absorbed to helium nuclei at about 90 percent when thermal neutrons enter the considered plasma. By considering the changes of the reaction probability of tritium ions and deuteron in the D/He3 plasma with respect to electron temperature and ion density, it can be seen that, with the injection of thermal neutron fluxes into the designed D/He3 plasma in the electron temperature of Te=1KeV, a fast neutron flux with a nuclear conversion efficiency of 2×10-3 will be resulted. Therefore it will be enough for the fast neutron flux efficiency.Figure 3
The probabilityFTD(Eth) of the in-flight D(t,n)He4 reaction in the D/3He plasma induced by the 191keV tritium in terms of electron temperature. Density nD=nHe3=1021cm-3 and ion temperature Ti=100eV.Figure 4
The probabilityFTD(Eth) of D+T reaction in the D/He3 plasma in terms of plasma ion density ni=nD+nHe3. Ion temperature Te=1KeV and it is assumed that nD=nHe3.



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*Source: 689739-2013-05-30.xml*