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4 Complicating Factors
4.1 Boron Contamination in Graphite
The presence of impurities in the graphite used as a moderator in the CP-1 and
Hanford reactors was a matter of serious concern in the Manhattan Project. Since
the purpose of the graphite was to slow and scatter neutrons without capturing them,
it was important for it to be as free as possible of any neutron-capturing impurities. Some neutron capture was inevitable as carbon itself has a small capture
cross-section for thermal neutrons (3.53 millibarns), but the greater danger was that
commercially-produced graphite at the time often contained trace amounts of boron,
which has a voracious appetite for capturing neutrons. Indeed, it was unappreciated
boron contamination of graphite that led German researchers to conclude that only
heavy water could serve as an adequate moderator, a decision that was at least in part
responsible for their failure to achieve a self-sustaining chain-reaction during World
War II (Reed 2020). In this section we examine the severity of this effect.
Two isotopes of boron occur naturally:
10 B (19.9%) and
11 B (80.1%). Boron-10
has a minute thermal-neutron capture cross-section, but that of boron-10 is enormous,
about 3840 barns. Weighting by abundance, this gives boron a bulk capture crosssection of about 760 barns: A single “average” atom of boron has a neutron capture
effect equivalent to that of over 200,000 carbon atoms. The culprit reaction involved
is neutron capture to produce an alpha particle, an (n, α) process:
1
0 n +
10
5 B →
4
2 He +
7
3 Li.
(4.1)
As will be seen, the presence of even a small amount of boron-10 can quickly
suppress the desired chain reaction.
The approach used in Sect. 3.1 to investigate reactor criticality can be modified to
account for capture effects due to carbon and boron. As in that section, let σ f 5 , σ c5 ,
and σ c8 designate the cross-sections for fission and capture by
235 U and capture by
238 U. To these, add symbols for capture by carbon and boron: σ cC and σ cB .
Imagine the pile idealized as a homogeneous mixture of uranium, carbon, and
boron atoms, with thermalized neutrons flying about. The minimal graphite-tocarbon ratio necessary to sustain a chain reaction is a matter of nuclear engineering,
but I make an estimate here by assuming knowledge of a graphite pile that is known
to have just achieved criticality: Enrico Fermi’s CP-1. From figures given in a paper
published by Fermi on the tenth anniversary of that achievement [Fermi (1952)], it can
be determined that CP-1 contained ~37,700 kg of pure uranium and ~349,700 kg of
graphite, which gives a C:U mass ratio of ~9.3:1. The molecular weight of uranium
is about 19.8 times that of carbon, so this corresponds to a C:U number ratio of
~180:1. Call this ratio R. Analogous numbers can be computed for the Oak Ridge
X-10 and Hanford reactors, and indicate R ~ 120, but those devices are complicated
by the presence of fuel channels and coolant. I will stick with R = 180; the results
which follow are not, however, wildly sensitive to reasonable changes in R.
4 Complicating Factors
4.1 Boron Contamination in Graphite
The presence of impurities in the graphite used as a moderator in the CP-1 and
Hanford reactors was a matter of serious concern in the Manhattan Project. Since
the purpose of the graphite was to slow and scatter neutrons without capturing them,
it was important for it to be as free as possible of any neutron-capturing impurities. Some neutron capture was inevitable as carbon itself has a small capture
cross-section for thermal neutrons (3.53 millibarns), but the greater danger was that
commercially-produced graphite at the time often contained trace amounts of boron,
which has a voracious appetite for capturing neutrons. Indeed, it was unappreciated
boron contamination of graphite that led German researchers to conclude that only
heavy water could serve as an adequate moderator, a decision that was at least in part
responsible for their failure to achieve a self-sustaining chain-reaction during World
War II (Reed 2020). In this section we examine the severity of this effect.
Two isotopes of boron occur naturally:
10 B (19.9%) and
11 B (80.1%). Boron-10
has a minute thermal-neutron capture cross-section, but that of boron-10 is enormous,
about 3840 barns. Weighting by abundance, this gives boron a bulk capture crosssection of about 760 barns: A single “average” atom of boron has a neutron capture
effect equivalent to that of over 200,000 carbon atoms. The culprit reaction involved
is neutron capture to produce an alpha particle, an (n, α) process:
1
0 n +
10
5 B →
4
2 He +
7
3 Li.
(4.1)
As will be seen, the presence of even a small amount of boron-10 can quickly
suppress the desired chain reaction.
The approach used in Sect. 3.1 to investigate reactor criticality can be modified to
account for capture effects due to carbon and boron. As in that section, let σ f 5 , σ c5 ,
and σ c8 designate the cross-sections for fission and capture by
235 U and capture by
238 U. To these, add symbols for capture by carbon and boron: σ cC and σ cB .
Imagine the pile idealized as a homogeneous mixture of uranium, carbon, and
boron atoms, with thermalized neutrons flying about. The minimal graphite-tocarbon ratio necessary to sustain a chain reaction is a matter of nuclear engineering,
but I make an estimate here by assuming knowledge of a graphite pile that is known
to have just achieved criticality: Enrico Fermi’s CP-1. From figures given in a paper
published by Fermi on the tenth anniversary of that achievement [Fermi (1952)], it can
be determined that CP-1 contained ~37,700 kg of pure uranium and ~349,700 kg of
graphite, which gives a C:U mass ratio of ~9.3:1. The molecular weight of uranium
is about 19.8 times that of carbon, so this corresponds to a C:U number ratio of
~180:1. Call this ratio R. Analogous numbers can be computed for the Oak Ridge
X-10 and Hanford reactors, and indicate R ~ 120, but those devices are complicated
by the presence of fuel channels and coolant. I will stick with R = 180; the results
which follow are not, however, wildly sensitive to reasonable changes in R.