An approximate treatment of neutron slowing by elastic collisions with nuclei of a moderating material in a nuclear reactor is developed on the basis of a compact treatment of elastic collisions previously published in this journal [B. C. Reed, Am. J. Phys. **86**(8), 622 (2018)]. The analysis is straightforward enough to be presented to lower-level students and gives results in good accord with more sophisticated treatments.

A time-honored application of the theory of elastic collisions in nuclear physics is to estimate the average number of scatterings *N* that will be necessary to bring the kinetic energy *K* of a neutron born in a fission (*K* ∼2 MeV) down to the thermal range (*K* ∼1 eV or less) as it collides with nuclei of a moderating material.^{1} As *N* is involved in dictating how far apart fuel elements will need to be spaced within a reactor in order to achieve effective thermalization while minimizing neutron loss, it is an important parameter for nuclear engineers.

Elastic collisions are a staple of lower-level courses, but students often do not get to see this technologically relevant topic because many introductory-level texts restrict their treatments of collisions to purely head-on situations. Formal analyses of scattering are usually restricted to higher-level or specialized courses and involve concepts such as analyzing collisions in the center-of-mass reference frame of the neutron/nucleus system and dealing with integrals whose integrands involve a product of a logarithmic function of the scattering angle and its probability distribution.

The purpose of this Note is to give an approximate development for estimating *N* based on a very compact treatment of elastic collisions published previously in this journal.^{2}

Before entering the derivation, a brief description of why moderators are necessary and how they function is offered to help orient readers who are not specialists in reactor physics. More extensive details can be found in any text on reactor engineering.^{3}

Nuclear chain reactions propagate themselves via neutrons emitted in fissions of nuclei of uranium-235 (^{235}U) striking other nuclei of ^{235}U and inducing yet more fissions. ^{235}U nuclei are inherently fissile when struck by neutrons, but in practice, there is a serious competing effect that has to be overcome to realize a chain reaction. The more common isotope of uranium, ^{238}U, tends to capture neutrons without fissioning, hence removing them from circulation. In natural uranium, ^{238}U nuclei outnumber those of ^{235}U by a ratio of about 138:1, which renders a self-sustaining chain reaction impossible unless the uranium has been substantially enriched in ^{235}U, a difficult and expensive task. However, Nature offers a loophole in that reaction cross sections can be very sensitive functions of the energy of bombarding particles. Neutrons liberated in fissions emerge with very great kinetic energies, equivalent to about two million electron-volts (eV). However, for neutrons that have been slowed to energies typical of room-temperature thermal motions, ∼1/40 eV, the fission cross section for ^{235}U is large enough to overcome the capture effect of ^{238}U to the extent that fission of ^{235}U nuclei is actually somewhat more probable than neutron capture by ^{238}U nuclei. A chain reaction with unenriched uranium is possible if *slow* neutrons are used. (In fact, in slow-neutron environments, it is necessary to introduce control rods of the neutron-capturing material into the uranium to keep the chain reaction in a steady-state condition.)

The trick to utilizing this loophole is that is it necessary to slow fission-liberated neutrons before they can be captured by nuclei of ^{238}U. To do this, the uranium fuel in a reactor is configured not as a single massive block but rather in rods or lumps separated by a moderating material whose purpose is to slow neutrons without capturing them. The idea is to have neutrons escape from the fuel elements before they encounter another uranium nucleus, scatter around, and become slowed within the moderator and then encounter another fuel element and have an enhanced chance at fissioning a nucleus of ^{235}U.

Aside from having small neutron-capture cross sections of their own, the best moderating materials will be light elements. The ideal case is hydrogen, where the nuclei are lone protons. The rationale for this is that, since neutrons and protons have essentially identical masses, a collision will be like that between two billiard balls, and an initially fast neutron will be slowed after a very few collisions. For practical purposes, water is commonly used as a moderator because it can also serve as a heat-transporting coolant. The drawback with water, however, is that hydrogen does have some neutron capture cross section; to counteract this, commercial water-moderated and cooled power-producing reactors use uranium enriched to a level of a few percent in ^{235}U. Heavy water, on the other hand, makes for an excellent moderator and coolant as it has a small neutron-capture cross section since the hydrogen is in deuterated form with a neutron already present. Heavy water moderated and cooled reactors can economically use natural uranium as their fuel; the Canadian CANDU (Canada Deuterium Uranium) series of reactors operate in this way. Beyond light or heavy water, the next practical moderator is crystallized carbon in the form of graphite. During the World War II Manhattan Project, graphite-moderated reactors were used to synthesize plutonium for use in nuclear weapons (which use unmoderated chain reactions); these reactors used rods of natural uranium fuel and were cooled by very thin films of water, which flowed rapidly over the rods. Some designs of graphite-moderated reactors can be intrinsically unstable; at Chernobyl in 1986, it was a graphite reactor that underwent a catastrophic steam explosion due to vaporized cooling water. The story of the Chernobyl disaster is dramatically related in Adam Higginbotham's *Midnight in Chernobyl.*^{4}

Since neutrons slow by random-walking their way between successive collisions in a moderator, an important design consideration will be how many collisions are required to thermalize them. If only a few collisions are needed, fuel rods can be placed close together, making the overall structure more compact. The random-walk distance to thermalization will also depend on the scattering cross section of the moderator. In Enrico Fermi's first graphite pile in late 1942, lumps of natural uranium fuel were placed in a cubical lattice arrangement with a side length of 21 cm.

My concern in this paper, however, is strictly with estimating the number of scatterings to thermalization. To picture a scattering, see Fig. 1, which depicts a collision between two smooth, non-deformable, non-rotating spheres or disks of masses *m* and *M*; the former is taken to be the neutron and the latter the struck moderator nucleus. At the moment of collision, the two exert equal and opposite forces on each other, with $\Delta p$ being the change in momentum of *m*. Eventually, we will set *m *=* *1 and *M *=* A* mass units as is customary in nuclear analyses.

In Ref. 2, it is shown that, quite generally,

where $e\u0302M\u2192m$ is a unit vector from the center-of-mass of *M* to the center-of-mass of *m*, that is, in the same direction as $\Delta p$. The magnitude of the momentum exchange is given by

where *p*_{M} and *p*_{m} are the momenta of the two disks before the collision (velocities *v*_{M} and *v*_{m}).

Now, suppose that *M* is stationary before the collision; this will certainly be a reasonable approximation for thermal environments. Then, *v*_{M} = **0**, and we can write

where *ψ* is the angle between the original direction of motion of *m* and $\Delta p$. Note that *ψ* is *not* the traditional “scattering angle” of nuclear physics, which is defined to be that between the initial and final velocity vectors of the neutron.

The momentum of *m* after the collision is then *p*_{m} + $\Delta p$, and its post-collision kinetic energy will be $(pm+\Delta p)\u2022(pm+\Delta p)/2m$. With this and the help of Eq. (3), the ratio of the post-collision to pre-collision kinetic energies of mass *m* can be derived as follows: First expand out the scalar product,

From Fig. 1, we have $pm\u2022\Delta p=pm\Delta p\u2009\u2009\u2009cos\u2009\psi $. Using Eq. (3) for *Δp* with *mv _{m}* =

*p*gives

_{m}On substituting Eqs. (5) and (6) into Eq. (4), a common factor of $pm2$ appears, which, upon extraction and combination with the factor of 2*m* in the denominator, gives the pre-collision kinetic energy. This transforms Eq. (4) to

The term in square brackets can be simplified; the final result is

where

By sketching a few collisions of the form of Fig. 1, it will be seen that *π* /2 ≤ *ψ* ≤ *π*; that cos*ψ* must be negative can also be seen from the absolute value in Eq. (3). Hence, (1 − *β*) ≤ (*K _{post}/K_{pre}*) ≤ 1, a standard result. For light moderator nuclei such as hydrogen or deuterium,

*β*will be near unity, and the neutron can lose anywhere from none to practically all of its current kinetic energy in any one collision.

For *N* successive collisions, the final energy of the neutron will be given by a product of terms of the form of Eq. (8), each with its own value of *ψ*. To get an *approximate* estimate for the value of *N* necessary for the neutron to reach some final energy *K _{final}* after starting with initial energy

*K*, I adopt the average value $\u27e8\u2009cos2\psi \u27e9=1/2$ over the range (

_{initial}*π*/2 ≤

*ψ*≤

*π*), to apply for all collisions. Then, the logarithm of Eq. (8) is taken to give

A more elaborate estimate could be had by examining the probability distribution of *ψ*, but the idea here is an introductory-level approach.

Table I shows results for common moderating materials, taking *K _{initial}* = 2 MeV and

*K*= 1 eV; to consider lower final energies would be questionable because moderator nuclei certainly cannot be regarded as being at rest relative to the neutrons as the average neutron energy gets close to being thermal. The last column of the table shows results for the same energy decrement adopted from the nuclear-power.net website.

_{final}^{5}It is reassuring that the present approach gives results in reasonable accord with these, although this must be somewhat a matter of canceling errors in view of the approximations involved.

. | Mass . | . | Collisions to thermalize . | |
---|---|---|---|---|

Element . | A
. | β . | (This paper) . | (Ref. 5) . |

H | 1 | 1 | 21 | 15 |

D | 2 | 0.8888 | 25 | 20 |

He | 4 | 0.64 | 38 | 34 |

Be | 9 | 0.36 | 73 | 70 |

C | 12 | 0.2840 | 95 | 92 |

O | 16 | 0.2215 | 124 | 121 |

. | Mass . | . | Collisions to thermalize . | |
---|---|---|---|---|

Element . | A
. | β . | (This paper) . | (Ref. 5) . |

H | 1 | 1 | 21 | 15 |

D | 2 | 0.8888 | 25 | 20 |

He | 4 | 0.64 | 38 | 34 |

Be | 9 | 0.36 | 73 | 70 |

C | 12 | 0.2840 | 95 | 92 |

O | 16 | 0.2215 | 124 | 121 |

In conclusion, this model, despite its simplifications of assuming isotropic collisions and equal scattering probabilities for different moderator materials, gives credible results when compared to more sophisticated treatments. Students need to be made aware of these limitations, but this approach can serve as an example of how basic mechanics bears on a technological issue of current importance. If time permits, the discussion could be supplemented by a description of the concept of the mean free path between collisions in terms of scattering cross sections and the random-walk distance a neutron will cover between its birth and thermalization.

## ACKNOWLEDGMENTS

The author is grateful to three anonymous reviewers whose comments and suggestions resulted in significant improvements to this paper.