# Control Rods in Nuclear Reactors

## James Grayson February 17, 2011

### Submitted as coursework for Physics 241, Stanford University, Winter 2011

 Fig. 1: Outline of the Uranium U-235 nuclear fission process. The small blue circles are neutrons. Source: Wikimedia Commons.

## Introduction

Control rods are an important technology for maintaining the desired state of fission reactions within a nuclear reactor. They constitute a real-time control of the fission process, which is crucial for both keeping the fission chain reaction active and preventing it from accelerating beyond control.

The nuclear fission chain reaction is the fundamental process by which nuclear reactors produce usable energy. Most commonly, Uranium U-235 is the fissionable material used in this chain reaction (as shown in Fig. 1), although the basic outline is applicable generally. In this process, a U-235 atom is struck by an incident neutron, causing the atom to fission into two smaller atoms (Krypton K-92 and Barium B-141) and also release an average of 2.5 new neutrons [1]. These new neutrons can then collide into more U-235 atoms, which undergo the same fission process, creating a chain reaction that releases substantial energy with each fission event.

Therefore, the key to sustaining the fission chain reaction is the amount of neutrons that propagate to the next generation of fissions. However, not all of the fission-produced neutrons trigger another fission event (some may simply escape the reactor or be absorbed by non-fissile isotopes, for example), so it is necessary to carefully engineer every parameter of the reactor to ensure that at least one neutron from each fission event is able to trigger another fission [1, 2]. Controls rods are one such adjustable parameter.

## Finding the Sweet Spot

The state of a fission chain reaction can be concisely summarized by the effective multiplication factor, k, which indicates the change in total number of fission events during successive generations of the chain reaction [2, 3]. It is defined as:

 k = total # of fission events in a given generation total # of fission events in the previous generation

A reactor that is in a steady state (i.e. each individual fission event triggers exactly one subsequent fission event) has k = 1, and the reactor is said to be critical. If k < 1, the reactor is subcritical and the chain reaction cannot be sustained. If k > 1, the reactor is supercritical and the reaction will grow exponentially.

 Fig. 2: Representative diagram of control rod usage. The left image shows the control rods (green) inserted fully into the reactor core, putting the reactor in a subcritical state. In the right image, the control rods are removed, allowing more neutrons to accelerate the fission chain reaction and go supercritical. Source: Wikimedia Commons.

The most important number for nuclear power reactors is therefore 1, as any other value of the multiplication factor k implies a very useless or very dangerous reactor. Maintaining precisely k = 1 is difficult, as this precise balance is influenced by a huge number of factors [2]. Some of these factors are inherent to the fissile fuel or reactor materials themselves, such as the number of neutrons produced in a fission event or the amount of neutron absorption due to fuel rod casings or moderators. However, even if engineered to perfect balance initially, the multiplication factor of a reactor will necessarily vary over time, as many byproducts of the fission reaction are neutron absorbers (referred to as poison) and will lower the overall neutron population as they accumulate.

Control rods thereby find their use as an effective method for combating these time-dependent changes in reactors. Control rods are essentially a highly effective neutron-absorbing mechanical structure, which can be actively inserted or withdrawn from the reactor core while the fission process is occurring. By controlling the portion of the control rod that interacts with the fission reaction, the multiplication factor can be finely tuned to maintain reactor criticality (see Fig. 2). In addition, control rods can be used to intentionally make rapid changes to the reactor state (i.e. turning the reactor on and off), especially as an emergency shut off feature by fully inserting the rods [2].

## Common Rod Materials and Design

As the functionality of a control rod depends on its ability to absorb neutrons from the fission chain reaction, the choice of highly neutron-absorbing material is crucial. The capability of a given substance to absorb neutrons is measured by its absorption cross section, σa, which is the target-area equivalent for an absorption interaction between an incident neutron and the substance. It is typically measured in barns, a unit of area equal to 10-28 square meters. Table 1 shows the absorption cross sections for several common control rod materials, as measured using thermal (20o C) incident neutrons.

B-10 B-11 Ag-105 Ag-107 Cd-113 In-115 Hf-174 Hf-176 Hf-177 Hf-178 Hf-179 Hf-180
σa
(barns)
3835 0.006 38 91 20600 202 561 24 373 84 41 13
Natural
Abundance
(%)
20 80 52 48 12 96 0.2 5 19 27 14 35
Table 1: Thermal neutron absorption cross sections and isotope abundance for several common control rod materials [4].
Cd-113 has a resonance here and is highly energy dependent in this regime.
 Fig. 3: A cluster control rod assembly. The individual rods are attached by the spider at the top. Source: Wikimedia Commons.

Most power reactors use thermal (low temperature and velocity) neutrons since U-235 is more fissile in this low energy regime [3]. Also, it is worth mentioning that at these low energies, the U-235 fission cross section remains roughly independent of energy, so these 20o C measurements are very similar to actual reactor temperatures (around 300o C) [2].

Of course, there are many more considerations than the absorption cross section alone in choosing a control rod material; the mechanical properties and cost are two important factors. As seen in Table 1, Boron B-10 is one of the best neutron absorbers. However, Boron's mechanical properties are less than desirable for building a control rod structure, as it is a brittle, salt-like material. Also, B-11 makes up the majority of natural Boron and has a negligible absorption, so Boron may need to be enriched to reach the necessary absorptivity. Some methods for getting around the mechanical issues are to use a steel alloy enriched with Boron, or to fill hollow, mechanically suitable rods with B-10 or Boron Carbide (B4C) powder [3]. Cadmium C-113 has a highly energy dependent cross section in the thermal energy regime, including the very high resonance shown in Table 1, so it is most commonly used as an alloy with Silver (Ag) and Indium (In), giving good mechanical properties and a more uniform absorption spectrum [3]. Hafnium (Hf) is unique in that its various isotopes' absorption cross sections are similar, even if only mediocre. Therefore, with its good mechanical properties as a metal, it is able to be used as a control rod material without combination with other metals [5].

Mechanical design of control rod assemblies comes in two common forms: cluster and cruciform. The cluster design is based on the realization that a single, large control rod in the nuclear reactor would create very nonuniform temperature and fission dynamics. By instead using a large number of evenly spaced, smaller control rods, uniform densities of neutrons and fissions can be achieved. Relatively thin rods, approximately the size of the fissile fuel rods, are attached on one end by a metal bracket (called a spider), as shown in Fig. 3. A typical power reactor might contain 50 such clusters with 20 rods each. Cruciform control rods approach the uniformity issue by instead using a crossed, double-blade design (i.e. an extrusion from a cross). This crossed blade structure provides good mechanical integrity, and can be fit into gaps between square sections of control rods. Like the cluster design, cruciform rods are also common in power reactors. [2]

© James Grayson. 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.

## References

[1] C. Elanchezhian, L. Saravanakumar and B. Vijaya Ramnath, Power Plant Engineering (I.K. International Publishing House, 2007).

[2] D. Bodansky, Nuclear Energy: Principles, Practices, and Prospects (Springer, 2004).

[3] J. Lamarsh, Introduction to Nuclear Engineering (Addison-Wesley, 1983).

[4] V. F. Sears, "Neutron Scattering Lengths and Cross Sections," Neutron News 3, No. 3, 26 (1992).

[5] J. Gambogi, Zirconium and Hafnium, USGS Minerals Yearbook (2010).