# How can nuclear reactors control the exponential cascade of neutrons fast enough to prevent an explosion?

The Wiki article on nuclear reactor physics states that

the average neutron lifetime in a typical core is on the order of a millisecond, the exponential factor is as small as 0.01, in one second the reactor power will vary by a factor of (1 + 0.01)^1000, or more than ten thousand.

These are called prompt neutrons. It also states that

so-called delayed neutrons (typically <1% of created neurons) increase the effective average lifetime of neutrons in the core, to nearly 0.1 seconds, so that a core with average neutron lifetime of 0.01 would increase in one second by only a factor of (1 + 0.01)^10, or about 1.1: a 10% increase. This is a controllable rate of change.

Most nuclear reactors are hence operated in a prompt subcritical, delayed critical condition: the prompt neutrons alone are not sufficient to sustain a chain reaction, but the delayed neutrons make up the small difference required to keep the reaction going. This has effects on how reactors are controlled: when a small amount of control rod is slid into or out of the reactor core, the power level changes at first very rapidly due to prompt subcritical multiplication and then more gradually, following the exponential growth or decay curve of the delayed critical reaction.

This still seems to imply that a very fast control loop is needed since not acting on a criticality increase within a few seconds still leads to an exponential increase, after about 100 seconds it would again be a factor of 10000. So how fast does the feedback have to be and how is it applied? Only through the control rods, moving a few millimeters?

How is it even possible to measure the state of the core so quickly?

How much does the reactor's power vary around the steady state, where one neutron causes exactly one other neutron to be absorbed? Does it look like a cycle? Does the output power vary by a magnitude or less?

• You are asking to explain reactor control and reactor kinetics in a single answer, which is a big ask. The answer in the end is that reactor power does not change as fast as you think. When a reactor is changing power levels, a somewhat normal rate of change is a factor of 2 in two minutes, which is very controllable. A fast rate of change would be a factor of 2 in 30 seconds. An operating power reactor can take several days to go from 0 to 100% power. Commented Apr 28, 2022 at 20:19

Water-cooled nuclear reactors can be designed to have two fundamentally different modes of operation: one in which a sudden and temporary thermal output excursion cause the core to produce less power (which is called a negative void volume coefficient) or to produce more power (positive void volume coefficient).

How is this done? Note that the water used to cool the core also acts both as an absorber of neutrons (which reduces the core output by taking neutrons out of circulation) and a moderator of (fast) neutrons, slowing them down and making them more available to trigger fission.

Artful design of the core and the cooling system can therefore yield a case in which either the absorption effect or the moderation effect is dominant. This determines how the reactor will respond to a loss of coolant accident, in which the water boils out of the cooling channels, leaving steam voids behind. If the moderation effect is dominant, then losing the coolant causes the core to produce less power and the system "fails safe* i.e., it shuts itself down if it loses coolant. This is the negative void volume case.

If the absorption effect is dominant by design, then having boiled-out voids in the coolant pipes causes the core output to rise, which boils out more water, which makes more void volume, and the system "runs away" and explodes.

The only way to make a reactor with a positive void volume coefficient safe is by equipping it with multiple, redundant automatic control systems that can act far faster than any human to prevent a power excursion from blowing up the core.

The Russian RBMK-1000 reactor (as used in Chernobyl) had a positive void volume coefficient and multiple redundant automatic control systems to keep it stable. These were disabled on purpose while the reactor operators were running tests on the reactor, and by chance they got an excursion which blew up the core on a timescale of ~milliseconds- far too fast for the operators to "catch" it on time.

If generation time was so low, indeed it would not be possible to control the reactor with just mechanical measures such as rods.

But apart from those neutrons you mention reactor has plenty of others sources of neutrons, that have much higher delay, from fission products for example. It is worth considering all of them, if you want to find how fast the reactor controls need to be. In practice it is minutes or even more.

Here this idea is described in more details: https://www.nuclear-power.com/nuclear-power/fission/delayed-neutrons/mean-generation-time-with-delayed-neutrons/

Mostly the reactor is thermally stable once it is settled in. Nuke plants provide base power. They don't like to vary their thermal load - hence power output - very much. And they do it gradually when they do adjust it. If the reaction starts to accelerate, the core begins to heat up. That changes the density of everything and the neutron efficiency drops a tiny bit.

In boiling water reactors, The water turns to steam in a bit more of the core, and that reduces the number of high efficiency thermal neutrons.

The control rods are mostly for startup, shutdown, and to make adjustments as the fuel is burned up and neutron poisons increase.

moderator temperature coefficient

The moderator temperature coefficient – MTC is defined as the change in reactivity per degree change in moderator temperature.

αM = dρ⁄dTM

It is expressed in units of pcm/°C or pcm/°F. The value of moderator temperature coefficient usually ranges from 0 pcm/°C to -80 pcm/°C. The moderator temperature coefficient’s magnitude and sign (+ or -) is primarily a function of the moderator-to-fuel ratio.