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The influence of the presence or absence of a reflector on the of

Maryam Noorani Department of physics, University of Mohaghegh Ardabili, Ardabil, Iran [email protected]

Farhad Zolfagharpour Department of physics, University of Mohaghegh Ardabili, Ardabil, Iran [email protected]

Sara Azimkhani Department of physics, University of Mohaghegh Ardabili, Ardabil, Iran [email protected]

Abstract

To increase productivity and improve the neutron economy in the reactor, neutron reflection is used. This reflectors cause lack of neutron leakage and reactor fuel can reach critical mass faster. Various materials such as , neutron, carbon, water, lead and so on could satisfy this property. In this paper, by using the MCNP code, critical mass of uranium in the term of percent of enrichment for different reflectors is obtained. The results show beryllium, carbon and deuterium are an excellent choice for this purpose. Also increasing the thickness of the neutron reflector causes an increase in neutron multiplication factor. Keywords: Neutron reflector, Critical mass, Uranium.

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Introduction

Initial calculations of critical mass or the amount of needed to continue refers to Heisenberg works on it (Logan, 1995). Heisenberg expresses that by using fissile material and calculating the critical mass can manufacture (Serber, 1992) but calculations of nuclear reactors for energy production in 1950s was pursued with greater intensity. Nowadays the calculation of critical mass takes place by computing codes and simulation and codes for (Kastanya, 2015). For a that is working with a constant potential over the time, it is needed that neutron chain reaction takes place inside with the multiplication factor K=1, it means that

number of produced by fission is equal to the total number of neutrons absorbed and leaked out of the reactor core (Krane, 1995). Whatever the neutrons leaking out to be decreased, the efficiency of the reactor will be increased and the reactor can work with the fuel that is enriched less. In fact, installing neutron reflectors around a reactor improves the economy and can increase productivity (Krane, 1995; Lamarsh, 1990). For example, Miniature Reactor (MNSR) in Esfahan Research and Production Center has beryllium as reflector (Daozhu, 1990).

Neutron multiplication factor and critical mass

At first, large amount of uranium is intended that it will be natural isotopic composition U235 0.72% and U238 99.28%. On average, every single fission event provides about 2.5 neutrons, which at birth have 2 Mev average energy (Knoll, 2000) and each of these second-generation neutrons are able to create another fission event and again produce more neutrons and so on. This process is a chain reaction of fission events and each fission event releases 200Mev in the form of kinetic energy of heavy parts as heat and radiation. Also, the amounts of absorbed neutrons not lead to fission and are absorbed by processes such as (n, γ) and -β decay. Assume that neutron multiplication factor is infinite for the environment K∞. If ν is the average number of neutrons which produced per each fission factor η is defined by the average number of produced neutrons from fission per initial heat neutrons (It is clear thatη>ν ).

σ η=ν f (1) σf+σa

235, Also η can be defined as the probability that a neutron causes fission. For U σf= 584b and σa= 238 97b and for U σf= 0b and σa= 2.75b. For mixture of natural uranium η obtained 1.33. If 3.5% is using, η will be obtained 1.88 and critical condition still remains. If the number of N thermal neutrons are absorbed and the rest of them produce fission, so the number of fast neutrons that we have is Nη that which must be slowing up to thermal energy. When fast neutrons in a chain reaction start motion some of them may be faced with U235cores and the fast neutrons are slightly increased so the fast fission factor ε is considered(That from the collision of fast neutrons obtained ε=1.03), So multiplication factor will be the equation (2).

K∞ =εηN (2)

Because the neutron resonance absorption area for the U238 is a distance of about 10-100 eV and the cross-sectional area is about 1000b (larger the cross-section of U235) If you want to change all

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neutrons into thermal, We must find a way to escape from the neutron resonance trap and If uranium and graphite are periodically mixed While slowing in intensification areas absorption of neutrons will be increased. Because the will not be totally eliminated this effect will be internd by resonance escape probability into our problem so:

K∞ =ηpε (3)

That P (probability of escape from resonant absorption) in practice is considered to be about 90%. If the percentage of neutrons that is not absorbed equivalent to coefficient f (Coefficient of thermal or fraction of the thermal neutron that are absorbed in U235and U238) Neutron

multiplication factor that is known as the four-factor formula, will be obtained as equation (4) for an infinite medium:

K∞ =ηεpf (4)

In actual reactor design, optimization by three parameters P and ε and f plays a major role in the effectiveness of a reactor. Uranium’s very large pieces tend to have a lower P because resonant absorption occurs mainly on the surface and neutrons with 10-100eV energy that interns into uranium don’t have chance to encounter inside the piece then whatever piece of uranium is larger, its surface can act as a central uranium shield to absorb neutrons. On the other hand, if the pieces are too big, the same effect happens for thermal neutron that causes fission it means more fission occurs near the surface and as we move forward toward the center of uranium piece, thermal neutron density decreases (Derringh, 1990; Reed, 1996). Using four coefficient estimated for natural uranium and graphite reactor multiplication rate is calculated K∞=1.11. Here neutron leakage is not negligible and effects in the process. If 푙푓 and 푙푡 are neutrons’ dissipative fractions due to leakage of fast neutrons and thermal neutrons for K we get:

((1-lt)(1-lf))K=η ε P f (5)

2 2 ½ If R is dimension of reactor and by defining M= (ld + ls ) as small length that ld is thermal neutrons’ diffusion length and ls is slowing length so we will have:

M2 K∞-K∝ (6) R2

And critical size of reactor will be commensurate with (Chyba and Milne, 2014):

M RC∝ (7) √K∞−1

For spherical graphite uranium natural reactor that is ls= 18.7cm and ld=50.8cm we can get R=5m to reach to K≈1.

Computational method of MCNP code

MCNP code (Hendricks, 1996) is one of the most important multipurpose computational code that is in the base of the Monte Carlo method. The distinguishing feature of this code is the possibility of three-dimensional simulation of different issues, particles transportation of neutrons, electrons, photons, or a combination of them. We choose to simulate spherical

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geometry because this geometry quickly makes the material critical (Kastanya, 2015) and using the data obtained from the last section we do simulation in MCNP code. Substances such as beryllium, C2H4, carbon, D2O, steel, water and lead are good reflectors of neutron. Critical mass in terms of enrichment percent of uranium in the presence and absence of the neutron reflector for these substances is given in Figure (1).

Figure 1. Critical mass in terms of enrichment percent in the presence and absence of the neutron reflector

According to Figure (1) can be seen that the reflector of neutrons in a nuclear reactor is essential. For beryllium, carbon and deuterium critical mass almost reaches to one quarter when there was no reflector. As for C2H4and steel critical mass almost reaches to half, and for water to three quarters when no reflector. To select the optimal reflector in a reactor in Figure (2) the reflectors are compared with each other.

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Figure 2. Critical mass in terms of present of uranium enrichment for different neutron reflectors

Conclusion

In economic recovery of neutron in reactors, guards around the reactor play an important role. These guards are doing two tasks at the same time. First, stop the leakage of gamma and neutron and second, return thermal and fast neutrons into the reactor’s core by neutron reflection that this phenomenon will improve the neutron economy. The simulation results in Figure (1) shown that reflector of neutrons in a reactor are essential. Beryllium and carbon-deuterium are excellent choices as a reflection of neutrons in nuclear reactors. In Figure (2) we see that beryllium is better neutron reflector than C2H4, carbon, D2O, steel, water and lead and at the same condition, for a variety of different neutron reflectors, uranium more quickly reaches critical mass and beryllium is very convenient choice for miniature neutron source reactor MNSR in Isfahan.

References

Chyba and Milne. (2014). Simple calculation of the critical mass for highly enriched uranium and -23. American jounal of physics 82: 977-979.

Daozhu. (1990). Iran Miniator Reactor Pure Water Production System. New york University:Larchmont, China Institute of Atomic Energy.

Derringh. (1990). Estimate of the critical mass of a fissionable isotope. Am.J. Phys 58: 363–364.

Hendricks. (1996). MCNPX 239. 1 Manual, 2111. Los Alamos National Laboratory, Tech. Report, LA-CP-07-1473.

Kastanya. (2015). Critical mass calculations using MCNP: An academic exercise. Annals of Nuclear Energy 75: 228 -231.

Knoll. (2000). Radiation Detection and Measurment. 1st edition, John Wiley and Sons.

Krane. (1995). Introductory Nuclear Physics.1st edition. John Wiley and Sons, 813 P.

Lamarsh. (1990). Introduction to nuclear reactor theory. New york University:Larchmont. 2nd edition, 926 P.

Logan. (1995). The Critical Mass. American Scientist, 84: 263 –272.

Reed. (1996). Estimating the critical mass of a fissionable isotope. J. Chem. Educ 73:162–164.

Serber. (1992). The los alamos primer. The First Lectures on How To Build an Atomic Bomb, 98 p.

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