Fission product poisoning: xenon and samarium

On Hana's third week, one of TVA's units trips from full power on a turbine fault and is ready to return to power seven hours later. The control room calls the core physics engineer on duty. "What boron do we need for the restart?" the shift manager asks. Teodora checks the time since the trip. "Xenon is near its peak right now," she says. "We can restart, but the critical boron will be much lower than before the trip, and as the xenon burns out after restart we will be diluting and then borating to follow it." Hana spends the afternoon working out why.

Two poisons that behave differently

Among the hundreds of fission products, two have thermal absorption cross sections so large that they dominate the short-term reactivity behavior of every thermal reactor. Xenon-135 has a thermal absorption cross section of about 2.6 million barns, the largest of any known nuclide. Samarium-149 has about 41 000 barns. Both are produced mainly by the decay of other fission products, and both are removed mainly by absorbing neutrons. The difference is that xenon-135 is radioactive, with a half-life of about 9.1 hours, while samarium-149 is stable.

Xenon at equilibrium

In a reactor at steady power, iodine-135 and xenon-135 reach equilibrium concentrations within about two days. Setting the time derivatives to zero gives, for a thermal flux φ and macroscopic fission cross section Σ_f:

The equilibrium xenon concentration rises with flux but saturates: at high flux, xenon is destroyed by absorption as fast as it is produced. The reactivity worth of equilibrium xenon in a critical reactor can be written, using νΣ_f ≈ Σ_a for the core:

The flux φ_X is about 0.8 × 10¹³ n/cm²·s. In power reactors, where the thermal flux is a few times larger, equilibrium xenon worth approaches its limiting value of about (γ_I + γ_X)/ν, a little under 3 percent in reactivity. That is about 2 000 to 3 000 pcm, or 20 to 30 milli-k, a substantial part of the excess reactivity that a core must carry.

Xenon after shutdown

When a reactor shuts down, the flux falls to nearly zero. Xenon production from iodine decay continues for hours, because the iodine inventory decays with a 6.6-hour half-life, but xenon removal by neutron absorption stops. Xenon therefore builds up after shutdown, reaching a peak some hours later before decaying away. The concentration after shutdown, starting from equilibrium, is:

The higher the flux before shutdown, the larger the iodine inventory relative to the xenon, and the larger and later the peak. The program below calculates equilibrium xenon worth and the post-shutdown peak for two thermal fluxes, and the equilibrium samarium worth.

# Xenon-135 and samarium-149 in a thermal reactor (point model).
# Yields and decay constants are standard textbook values for U-235 fission.
import math
gI, gX, lI, lX, sX = 0.0639, 0.00237, 2.87e-5, 2.09e-5, 2.65e-18     # sX in cm^2
gP, lP, sS = 0.0113, 3.63e-6, 4.1e-20
nu = 2.43
def xe_eq_rho(phi):
    """Equilibrium xenon reactivity, using rho = -(gamma_I+gamma_X)/nu * phi/(phi+phi_X)."""
    phiX = lX / sX
    return -(gI + gX) / nu * phi / (phi + phiX)
def sm_eq_rho():
    return -gP / nu                     # independent of flux at equilibrium
def after_shutdown(phi, hours):
    """Xenon reactivity after a trip from equilibrium at flux phi.
    Concentrations are per unit Sigma_f, so rho = -sigma_X * X / nu."""
    I0 = gI * phi / lI
    X0 = (gI + gX) * phi / (lX + sX * phi)
    t = hours * 3600
    X = X0 * math.exp(-lX * t) + I0 * lI / (lX - lI) * (math.exp(-lI * t) - math.exp(-lX * t))
    return -X * sX / nu
for name, phi in (("PWR-like, phi = 3e13", 3e13), ("CANDU-like, phi = 7e13", 7e13)):
    r = xe_eq_rho(phi)
    print(f"{name}: equilibrium xenon {r*1e5:7.0f} pcm ({r*1e3:5.1f} mk)")
    peak_t, peak = 0, 0.0
    for h in range(0, 61):
        val = after_shutdown(phi, h)
        if val < peak: peak, peak_t = val, h
    eq = after_shutdown(phi, 0)
    print(f"   after a trip the xenon worth peaks near {peak_t} h at {peak*1e5:7.0f} pcm, "
          f"{peak/eq:.2f} times its equilibrium value")
print(f"samarium-149 at equilibrium: {sm_eq_rho()*1e5:.0f} pcm, independent of flux; it does not decay, so it stays after shutdown")

PWR-like, phi = 3e13: equilibrium xenon   -2159 pcm (-21.6 mk)
   after a trip the xenon worth peaks near 9 h at   -4138 pcm, 1.92 times its equilibrium value
CANDU-like, phi = 7e13: equilibrium xenon   -2451 pcm (-24.5 mk)
   after a trip the xenon worth peaks near 10 h at   -8370 pcm, 3.41 times its equilibrium value
samarium-149 at equilibrium: -465 pcm, independent of flux; it does not decay, so it stays after shutdown

The higher-flux case peaks at more than three times its equilibrium worth. This is the case for reactors with high thermal flux, and it is why the time just after a trip matters so much in practice.

In a pressurized water reactor, soluble boron gives the operators a large and flexible way to compensate. Early in the cycle, when the boron concentration is high, there is enough boron to remove to overcome the xenon peak, and the unit can usually restart at any time after a trip. Late in the cycle, when the boron concentration is already low, there may be too little left to dilute, and a restart near the xenon peak may not be possible until the xenon decays. Operators plan the dilution and boration that follow a restart, because xenon burns out quickly once the flux returns.

Samarium

Samarium-149 builds up more slowly, through the 53-hour decay of promethium-149, and its equilibrium worth does not depend on flux:

This gives about −460 pcm, or −4.6 milli-k, for uranium-235 fission. Because samarium is stable, it does not decay after shutdown. Instead, the promethium inventory continues to decay to samarium, and the samarium worth rises after shutdown to a new, higher level that remains until the reactor operates again and burns it off. This post-shutdown samarium increase is smaller than the xenon peak but permanent until restart, and core designers include it in shutdown margin and restart calculations.

Xenon oscillations

Xenon also affects the spatial power distribution. In a large core, if power rises in one region, xenon is burned out there at first, raising reactivity and power further; hours later, extra iodine decays to extra xenon, which lowers reactivity there. Meanwhile the opposite happens in the region where power fell. The result can be a slow oscillation of power between regions, with a period of roughly a day. Whether such oscillations grow or die out depends on the size of the core in migration lengths, the power level, the power coefficient and the burnup.

Large PWR cores are susceptible mainly to axial xenon oscillations, in which power shifts between the top and bottom halves of the core. Operators control them by monitoring the axial flux difference, the difference between the top and bottom ex-core detector readings, and moving control banks to keep it within a target band. Boiling water reactors such as the BWRX-300 have strong void feedback, which tends to damp xenon oscillations in the axial direction.

Xenon after power changes

Xenon also changes after any change in power, not only after a trip. When power is reduced, xenon builds up for several hours, as it does after a shutdown but to a smaller extent, and then settles at a new, lower equilibrium. When power is raised, the stronger flux burns out xenon faster than the iodine can replace it, so xenon falls at first, adding reactivity, and then rises to its new, higher equilibrium. The operators must compensate for these swings with control rods, boron or zone controllers. A load-following plant that changes power daily must plan its xenon carefully, which is one reason many power reactors are run at constant full power when they can be.

Iodine and xenon in the core model

Core simulators do not use the point model of this lesson. They track iodine and xenon in every node, using the local flux, so that changes in the power distribution after a control action or a refueling are followed in space. The equilibrium and post-shutdown relations derived here are still the basis of those calculations, written for each node with its own flux and fission rate. In a CANDU core, where the flux is high and the core is large, the local xenon follows the local power closely, and the zone controllers respond to the combined effect. In a PWR, the axial distribution of xenon after a power change can shift power toward the top or bottom of the core and must be watched through the axial flux difference.

Operators plan around these effects every day. Before a planned power reduction, the physics group predicts the xenon transient that will follow, so that the control room knows how much the control devices or boron will need to move, and when the return to full power is possible. After an unplanned trip, the time available to restart before xenon builds beyond the available excess reactivity is a key consideration, and procedures give that time for each unit. During extended low-power operation, the slow approach of xenon to its new, lower equilibrium also needs to be followed, because the reactivity released by that decrease must be absorbed by the control system.