|
|
| Fig. 1: U-235 Fission Cross section. (Image Source: A. Mukunda, after Lubitz. [5] |
When a nuclear reactor is turned on, a neutron source typically Cf-252 made at Oak Ridge labs in Tennessee for reactors in the United States is inserted into the reactor core, and this material, due to its innate physical properties, starts releasing neutrons. [1] These neutrons fly around the reactor and crash into things: the reactor walls, the water that fills up the reactor core, the fuel rods, the control rods, and very rarely even each other.
Most importantly, when a neutron crashes into the nucleus of an atom of U-235 contained in the fuel rods, it sometimes gets absorbed into that U-235, causing it to turn into the highly unstable U-236 which then immediately breaks into two slightly uneven parts. This process, known as fission, releases a tremendous amount of energy, and importantly also releases more neutrons.
On average each fission releases 2.43 neutrons. [2] These travel at roughly 2.0 × 107 m/s. [3] By comparison, the fastest ever fighter jet developed by NASA reached a top speed of about 2.0 × 103 m/s. So fission neutrons are going 10,000 times faster than the fastest fighter jet. [4]
While the high speed of fission neutrons is impressive, it actually presents a challenge to nuclear engineers. Fast neutrons are very unlikely to crash into U-235 atoms and cause more fission. [4] (See Fig. 1) This means that if you want to design a nuclear reactor with a sustainable chain reaction (a fission occurs, releases neutrons, those neutrons crash into U-235 to cause more fission, and this process repeats itself) you have to find some way to slow the neutrons down to increase the chance of fission. This process of slowing down neutrons is known as neutron thermalization. [4]
Let us now address some of the basic questions about neutron thermalization. First, why do we call it thermalization instead of just slowing down? Second, how and why is water used to thermalize neutrons? Third, how long does it take for a neutron to become thermal in water? Finally, what are the best resources and datasets to learn more about the journey of neutrons in a reactor?
 |
| Fig. 2: Fission neutron energy distribution. (Image Source: A. Mukunda after Grundel. [4]) |
The reason why slowing-down is called thermalization can be best understood by examining the relationship between two foundational concepts of physics and chemistry: kinetic energy and temperature. The kinetic energy Ei of an object of mass m traveling at speed vi is
| Ei | = | 1 2 |
m vi2 |
When two such objects collide, they bounce off each other in complex ways, but the sum of their kinetic energies remains exactly the same. Thus if we have N such objects colliding off each other, then the sum of their kinetic energies
| ETotal | = | N ∑ i=1 |
Ei |
remains constant. The energy distribution of neutrons released from fission of U-235 is plotted in Fig. 2. Eventually the velocities of a population of N such neutrons, will scramble through this process of collisions. This randomization is called "thermalization". When N is very large, a neutron picked at random will then have a probability P(v) to have its velocity between v and v+dv of
|
(1) |
where kB = 1.38 × 10-23 J °K-1 is Boltzmann's constant and T is the absolute temperature in °K. This expression is plotted in Fig. 3. It effectively defines the temperature T, which has no meaning until the motions have thermalized. Then since
 |
| Fig. 3: Maxwell Boltzmann distributions of particles with molar mass of 18, as described by Eq. (1). (Image source A.Mukunda) |
| ∫0∞ P(v) dv = 1 |
as required of a probability, a series of experiments measuring a neutron's kinetic energy will average to the value
| ∫0∞ ( | 1 2 |
M v2) P(v) dv | = | 3 2 |
kB T |
This then relates the temperature T to ETotal through
| ETotal | = | 3 2 |
N kBT |
Thus, the reason that slowing down is called thermalization is that the neutrons are slowed down such that their energy distribution roughly coincides to the energy distribution of particles who are in thermal equilibrium with the surrounding material.
One of the main reasons why water is used for neutron thermalization is that water molecules contain hydrogen and hydrogen atoms have roughly the same mass as neutrons. This enables hydrogen to be very effective at slowing neutrons down.
In order to understand why this is the case we must first consider a neutron of mass m traveling at speed v hitting a target of mass M. Let us assume for simplicity that the velocity is in the x-direction. We transform to the center-of-mass frame, which travels with speed v0 defined by
| m (v-v0) - M v0 = 0 | v0 | = | ( | m M+m |
) v |
In this center-of-mass frame the neutron colliding with the target scatters through some angle θ that depends on details of how it impacts. But, regardless of these details, momentum and energy will be conserved in the scattering. This requires the center-of-mass speed of the neutron after the collision be the exactly same as it was before the collision. The x- and y-components of the neutron velocity in the center-of-mass frame after the collision are thus
| v'x(CM) | = | (v-v0) cos(θ) |
| v'y(CM) | = | (v-v0) sin(θ) |
hus the x- and y- components of the neutron velocity in the original laboratory frame after the collision are
 |
| Fig. 4: Neutron average fractional energy retained as a function of target mass M/m given by Eq. (2) . (Image Source: A. Mukunda) |
| v'x | = | (v-v0) cos(θ) + v0 |
| v'y | = | (v-v0) sin(θ) |
The kinetic energy of the neutron in laboratory frame after the collision is then
| E'(θ) | = | 1 2 |
m [ v'x2 + v'y2 ] |
| = | 1 2 |
m [ (v-v0)2 + v02 + 2 (v-v0) v0 cos(θ) ] |
The scattering of the neutron in a real situation is "s-wave", which means all scattering directions are equally probable. Averaging over all scattering angles (in 3 dimensions) we obtain
| E' | = | 1 2 |
∫0π E'(θ) sin(θ) dθ |
| = | 1 2 |
m [ (v-v0)2 + v02] | |
| = | 1 2 |
m v2 (m2 + M2) / (m + M)2 |
Thus on average each collision diminishes the energy E by the factor
|
(2) |
Fig. 4 shows a plot of this final equation which clearly demonstrates that as the mass of the target particle increases the slowing down effectiveness decreases rapidly. This shows one of the main reasons why H-1 H-2 in light and heavy water respectively are well suited moderators.
 |
| Fig. 5: Rough Comparison of Total and Elastic Microscopic Cross-sections for H-1 and O-16. (Image Source: A. Mukunda after Horsley and Ray et al. [6,7]) |
In order to understand and calculate the thermalization time constant in light water we must first understand two key concepts: mean free path and scattering cross-sections. When we as human beings look at water, we don't see any empty space. You cant put your finger in a cup of water without touching water molecules and getting wet. At the atomic scale however, this is not the case. All materials are in fact mostly empty space because an atom has a relatively small nucleus and large area around it in which the even smaller electrons are orbiting. So, if our fast neutron is speeding through water, it can pass right through the Hydrogen and Oxygen atoms without ever colliding with the nucleus. Mean free path is the average distance a particle (in this case a neutron) travels before crashing into something (in this case the nucleus of an atom).
From the known mass density of water and the atomic weights of H-1 and O-16 we obtain for the number density of the nuclei for each type of atom in water
| nH | = | 1000 kg m-3 × | 6.023 × 1023 mole-1 0.018 kg mole-1 |
× 2 |
| = | 6.68 × 1028 m-3 | |||
| nO | = | 3.34 × 1028 m-3 | ||
Assuming the neutron starts with the average energy of fission neutrons (1.935 MeV) the cross sections for H-1 and O-16 are approximately σH = 1.83 barns and σO=1.57 barns (see Fig. 5), we obtain our macroscopic crossection (Σ) for a neutron in water
| Σ | = | σH nH + σO nO |
| = | (6.68 × 1022 m-3) × (1.83 × 10-28 m2) + (3.34 × 1022 m-3) × (1.57 × 10-28 m2) | |
| = | 17.5 × m-1 |
 |
| Fig. 6: Estimated neutron mean free path as a function of energy. (Image Source A.Mukunda using cross section data from Horsley [6] and Grochowski [7] |
The mean free path then for this first fission neutron is then
| ℓ | = | 1 Σ |
= | 1 17.5 m-1 |
|
| = | 5.71 × 10-2 m = 5.71 cm | ||||
Because cross-sections have an energy dependence as shown in Fig. 5, ℓ is not constant. The mean free path of a neutron ranging in energy from .02 MeV to 1.935 MeV were calculated using the same method above but substituting in the estimated cross-sections at various neutron energies. The results are are plotted in Fig. 6. The energy after the kth collision is then
| Ek | = | ηk E0 |
For the case of water η = roughly 0.398. [1] The number of collisions required to thermalize the neutron from EFission= 1.935MeV to EThermal = 0.26 eV is then calculated using the following method
| kThermal | = | ln(EThermal/E0) ln(η) |
= | ln(0.026 eV / 1.935 × 106 eV) ln(0.398) |
= | 19.67 |
The above value for the number of collisions to thermalize a neutron in water mirrors published values of roughly 19.2 collisions. [1] Since the speed of the neutron with energy Ek is
| vk | = | ( | 2 Ek m |
)1/2 |
the total time required to thermalize the neutron is
|
(3) |
Evaluating this expression with a neutron mass of m = 1.67 × 10-27 kg , E0 = 1.935MeV, 19 collisions, and the table of mean free paths provided in Fig. 6, we obtain
| tThermal | = | 2.97 × 10-9 sec + 3.00 × 10-9 sec + 5.53 × 10-9 sec + ... + 1.55 × 10-6 sec | |||
| = | 4.538 × 10-6 sec | ||||
 |
| Fig. 7: Average Time to Collision in Light Water for Each of 19 Collisions as given by Eq (3). (Image Source A. Mukunda) |
This calculated neutron thermalization time closely mirrors experimental and theoretical results that place neutron thermalization time in light water at roughly 4 - 5.3 microseconds. [8] The time for each collision is plotted in Fig. 7 and it is notable that despite the shortening mean free path, as the neutrons thermalize their decreasing speed ensures that the latter collisions take far more time than the earlier collisions. The incredibly short time of thermalization is a very important factor in understanding the importance and engineering complexity of implementing control mechanisms to prevent prompt criticality in nuclear reactors.
Fundamentally the design and operation of a nuclear reactor is all about managing a neutron population in the reactor to ensure the rate of fissions stays relatively constant. If too many of the neutrons reach thermal speeds and cause fissions the reactor will overheat or worse explode. If not enough reach thermal speeds, the reaction will slow and eventually stop. This seemingly complex process can however actually be understood and broken down into a few relatively simple parts. We need to first know the distribution of speeds at which neutrons are being produced. We need to know all the materials inside our reactor: what are all the elements and isotopes and how will the amount of each isotope change over time (chances that they are produced in a fission and their half lives). For each of those elements and isotopes, we also need to know their absorption (fission, capture, 1 in i out, 1 in 1 out/inelastic scattering) and elastic scattering cross-sections at all neutron speeds. These enable us to understand the speed and probability of neutron thermalization as well as the main ways in which neutrons are captured.
© Amar Mukunda. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] "DOE Fundamentals Handbook: Nuclear Physics and Reactor Theory, Vol. 1," U.S. Department of Energy, DOE-HDBK-1019/1-93, January 1993.
[2] E. Barnard et al., Time-of-Flight Measurements of Neutron Spectra From the Fission of U-235, U-238 and Pu-239," Nucl. Phys. 71, 228 (1965).
[3] "Nuclear Science - A Guide to the Nuclear Science Wall Chart," Contemporary Physics Education Project, 2019, Chapter 14.
[4] J. A. Grundel, "Fast-Neutron Spectra: Macroscopic and Integral Results," in Neutron Standards and Flux Normalization, ed. by A. B. Smith, U.S. Atomic Energy Commission, CONF-701002, August 1971, p. 417.
[5] C. Lubiz, "Neutron Cross Sections for Uranium-235 (ENDF/B-IV Release 3)," Knolls Atomic Power Laboratory, "KAPL-4825, September 1996.
[6] A Horsley, "Neutron Cross Sections of Hydrogen in the Energy Range 0.0001 eV - 20 MeV," Nucl. Data Sheets A 2, 243 (1966).
[7] J. H. Ray, G. Grochowski, and E. S.; Troubetzkoy, "Neutron Cross Sections Of Nitrogen, Oxygen, Aluminum, Silicon, Iron, Deuterium, And Beryllium," U.S. Army Nuclear Defense Arsenal, AD UNC-5139, November 1965.
[8] M. Fujino and K. Sumita, "Measurements of Neutron Thermalization Time Constant of Light Water by Pulsed Neutron Method," J. Nucl. Sci. Technol. 7, 277 (1970).