Nuclear reactions and Q-values

The fourth session starts with a design review comment. The instrumentation team at Meridian Nuclear Engineering has proposed boron-lined neutron detectors for the startup range of the Clinch River reactor, and a reviewer has asked why boron-10 and not helium-3 or lithium-6, and whether the oxygen in the reactor water will produce a gamma hazard in the steam lines. Both questions turn on nuclear reactions: what happens when a particle strikes a nucleus, and how much energy the reaction releases or needs.

Writing nuclear reactions

A nuclear reaction has a projectile \(a\) striking a target nucleus \(X\), producing a residual nucleus \(Y\) and an outgoing particle \(b\). The long form is \(a + X \to Y + b\); the compact form used throughout nuclear data is \(X(a,b)Y\). For example, \(^{10}\text{B}(n,\alpha)^{7}\text{Li}\) means a neutron striking boron-10 and producing an alpha particle and lithium-7. Reactions conserve the total number of nucleons, total charge, energy, momentum and angular momentum, so the residual nucleus can always be found by bookkeeping.

Reactions are classified by what comes out. In elastic scattering, \(X(n,n)X\), the target is left in its ground state and kinetic energy is conserved. In inelastic scattering, \(X(n,n')X^*\), the target is left excited and later emits a gamma ray. In radiative capture, \(X(n,\gamma)\), the projectile is absorbed and the excitation leaves as gamma rays; this is how reactors make cobalt-60 from cobalt-59 and plutonium-239 from uranium-238. In transmutation reactions such as \((n,p)\) and \((n,\alpha)\), a charged particle leaves. In fission, \((n,f)\), the nucleus splits.

The Q-value

The Q-value is the energy released by a reaction, equal to the decrease in rest mass. A positive Q means the reaction releases energy (exothermic); a negative Q means kinetic energy must be supplied (endothermic).

As in Lesson 1, atomic masses are used so that electron masses cancel; when a proton appears in a reaction, the hydrogen-1 atomic mass is used for it, and for an alpha particle the helium-4 atomic mass. The second form of the equation is the same statement in terms of kinetic energies: Q is the gain in total kinetic energy of the particles. Because the masses of nuclei are known to better than a millionth of a percent, Q-values computed from mass tables are extremely precise.

# Nuclear reactions: Q-values and threshold energies from atomic masses.
U = 931.494
m = {'n': 1.008665, 'H-1': 1.007825, 'H-2': 2.014102, 'H-3': 3.016049, 'He-3': 3.016029,
     'He-4': 4.002603, 'Li-6': 6.015123, 'Li-7': 7.016003, 'B-10': 10.012937, 'C-12': 12.000000,
     'N-14': 14.003074, 'O-17': 16.999132, 'O-16': 15.994915, 'N-16': 16.006102}

def q_value(reactants, products):
    return (sum(m[x] for x in reactants) - sum(m[x] for x in products)) * U

reactions = [
    ('B-10(n,alpha)Li-7', ['B-10', 'n'], ['Li-7', 'He-4']),
    ('Li-6(n,alpha)H-3', ['Li-6', 'n'], ['H-3', 'He-4']),
    ('He-3(n,p)H-3', ['He-3', 'n'], ['H-3', 'H-1']),
    ('N-14(alpha,p)O-17', ['N-14', 'He-4'], ['O-17', 'H-1']),
    ('O-16(n,p)N-16', ['O-16', 'n'], ['N-16', 'H-1']),
    ('H-2(d,n)He-3', ['H-2', 'H-2'], ['He-3', 'n']),
]
for name, r, p in reactions:
    Q = q_value(r, p)
    line = f"{name:20s} Q = {Q:+8.3f} MeV"
    if Q < 0:   # endothermic: threshold in the lab with a projectile hitting a target at rest
        proj, targ = (r[1], r[0])
        Eth = -Q * (m[proj] + m[targ]) / m[targ]
        line += f"   threshold {Eth:.2f} MeV"
    print(line)

B-10(n,alpha)Li-7    Q =   +2.791 MeV
Li-6(n,alpha)H-3     Q =   +4.784 MeV
He-3(n,p)H-3         Q =   +0.764 MeV
N-14(alpha,p)O-17    Q =   -1.192 MeV   threshold 1.53 MeV
O-16(n,p)N-16        Q =   -9.638 MeV   threshold 10.25 MeV
H-2(d,n)He-3         Q =   +3.270 MeV

The program answers the reviewer's first question. Boron-10, lithium-6 and helium-3 all absorb slow neutrons in exothermic reactions that release charged particles with hundreds of keV to several MeV of energy, easily detected by the ionization they produce. Boron-10's reaction releases 2.79 MeV to the ground state, though about 94 percent of reactions leave lithium-7 in an excited state, releasing 2.31 MeV as kinetic energy plus a 0.48 MeV gamma. Lithium-6 releases 4.78 MeV. Helium-3 gives the cleanest signal but is scarce and expensive, because it comes mainly from the decay of tritium. Boron-10 is abundant (about 20 percent of natural boron), easy to coat on tube walls or use as boron trifluoride gas, and has a very large cross section, which is why it is the usual choice.

Thresholds for endothermic reactions

A negative Q-value is not the whole story. Because momentum must be conserved, some of the projectile's kinetic energy must remain as motion of the products, so the projectile needs more than \(|Q|\). For a projectile striking a target at rest, the threshold kinetic energy is

For neutrons on oxygen-16, the \((n,p)\) reaction has \(Q = -9.64\) MeV and a threshold of about 10.2 MeV, so only the fastest fission neutrons can cause it. Yet it happens often enough in a reactor core to matter. The product, nitrogen-16, has a half-life of only 7.1 seconds and emits gamma rays of 6.1 and 7.1 MeV, among the most energetic in any reactor. In a boiling water reactor such as the BWRX-300 planned at Clinch River, nitrogen-16 travels with the steam to the turbine, so turbine buildings in BWRs need shielding and access controls during power operation. The short half-life means the activity vanishes within minutes of shutdown. That is the answer to the reviewer's second question.

Energy in the products

For exothermic reactions with slow neutrons, the projectile brings almost no kinetic energy, so the products share \(Q\) in inverse proportion to their masses, just as in alpha decay. In boron-10 capture, the alpha particle gets about 1.47 MeV and the lithium-7 nucleus 0.84 MeV (for the main branch). Both stop within a few micrometers in solid material, depositing their energy locally. This is the basis of boron neutron capture therapy for some cancers: boron compounds that collect in tumor cells are irradiated with neutrons, and the short-range products damage only the cells containing boron.

For reactions with a fast projectile, the energy of the outgoing particle depends on its angle. The full relation, from conservation of energy and momentum, is the Q-equation; engineers use it to calculate the energies of neutrons from accelerator sources such as deuterium-tritium generators, which produce 14.1 MeV neutrons used in oil-well logging and materials testing.

Using conservation laws as a check

Conservation laws are the fastest way to catch mistakes in reaction bookkeeping. Write the mass number and atomic number of each particle as superscript and subscript, then check that both sums balance across the arrow. A neutron is \(^1_0 n\), a proton \(^1_1 p\), an alpha \(^4_2\alpha\), a gamma ray \(^0_0\gamma\). For \(^{6}_{3}\text{Li} + ^1_0 n \to ^{3}_{1}\text{H} + ^{4}_{2}\alpha\), the mass numbers give \(6 + 1 = 3 + 4\) and the charges \(3 + 0 = 1 + 2\). This reaction is how tritium is bred in proposed fusion blankets (Lesson 8), and it also explains why lithium-6 is used in neutron detectors and in shielding that must avoid the high-energy capture gamma rays of hydrogen.

Charge and nucleon number fix the residual nucleus, but energy and momentum fix what can happen. A reaction that balances on paper may still be impossible at a given energy if its Q-value is too negative, and even an allowed exothermic reaction with charged particles can be extremely unlikely if the particles must tunnel through a Coulomb barrier, as Lesson 8 shows for fusion. Neutrons carry no charge and face no such barrier, which is why they can cause reactions at almost any energy.

The compound nucleus

Niels Bohr proposed in 1936 that many reactions proceed in two steps. The projectile is absorbed, forming a compound nucleus in which its energy is quickly shared among all the nucleons. The compound nucleus then lives for a time long compared with the transit time of the projectile, about \(10^{-16}\) seconds, and decays in a way that does not depend on how it was formed. A neutron absorbed by uranium-235 forms uranium-236 with about 6.5 MeV of excitation (Lesson 1); that excited uranium-236 then fissions about 85 percent of the time and emits gamma rays (capture) the rest of the time, whatever the neutron's direction.

The compound nucleus picture explains resonances. A compound nucleus has discrete excited levels, and when the projectile's energy plus the separation energy matches a level, the formation probability rises sharply. Cross sections of heavy nuclei for neutrons between about 1 eV and a few keV show thousands of these resonances. Lesson 5 shows how they appear in cross-section plots and Lesson 11 how they shape neutron slowing down in reactor fuel.

At higher projectile energies, reactions can also proceed directly, with the projectile interacting with only one or a few nucleons on a fast timescale. Direct reactions dominate above about 20 MeV and in reactions that pick up or strip off single nucleons. For reactor neutrons, nearly all reactions of interest are compound nucleus or potential scattering processes.

Activation

Neutron capture turns stable nuclides into radioactive ones, a process called activation. It is useful when deliberate, such as producing cobalt-60 for sterilization or molybdenum-99 for medicine, and a design concern when not. Activation of structural steel, concrete and coolant impurities sets the doses that maintenance workers receive and the amount of radioactive waste a plant produces at decommissioning.

The activity produced by activation follows a simple balance. Atoms of the product are created at the reaction rate \(R = N\sigma\phi\) (Lesson 5) and decay at \(\lambda N_p\). Starting from zero, the product activity grows toward a saturation value equal to the production rate.

After one half-life of irradiation the activity reaches half of saturation, and after several it is essentially saturated. This is why short irradiations are enough for short-lived products, while cobalt-60 in reactor components keeps building up for years. When irradiation stops, the activity decays with the product's own half-life.