What determines whether a water-cooled nuclear reactor has a positive or negative void coefficient? If I'm not mistaken it is the case that in water-cooled reactors the water acts both as a neutron absorber (negatively contributing to the amount of fission) and as a moderator (positively contributing to the amount of fission). Thus when the water boils, this can both increase and decrease the amount of fission going on, and thus a water-cooled reactor can have both a positive void coefficient (such as the notorious RBMK-design) or negative void coefficient (often the case in boiling water reactors).
This makes me wonder: What determines whether a reactor has a positive or negative void coefficient? Does it have to do with the other moderators being used? In either case, what are the physics underlying this?
 A: I will restrict my answer to a few types of thermal reactors.  A thermal reactor uses a moderator to slow  fission neutrons down to the thermal energy range where the fission cross section is large.
Water provides both moderation and absorbs neutrons.  I only discuss light water , H2O, here, and not heavy water, D2O. (D2O is almost as good a moderator as H2O and absorbs fewer neutrons.)
By "coefficient" we mean the change in reactivity with change in temperature. Reactivity is essentially a measure of deviation from a critical state.  A positive reactivity indicates change from a critical to a supercritical state.
Chernobyl was a graphite moderated, water cooled, thermal reactor.  The water acted both as a moderator (to some extent) and a neutron absorber.  Chernobyl was “over moderated” in that with voiding of the water due to boiling, the graphite provided enough moderation to the extent that the dominant effect of voiding was to absorb fewer neutrons thereby providing positive feedback.  The water void coefficient was positive.
The currently operating reactors in the US are Pressurized Water Reactors (PWRs) and Boiling Water reactors (BWRs).  Both types use water, H2O, as the moderator and the coolant.  In the PWR the water in reactor vessel is pressurized to about 2200 psia and remains subcooled; hot water from the reactor vessel water is pumped to steam generators maintained at about 1000 psia, and the steam from the steam generators is passed to the turbine.  In the BWR, the pressure in the reactor vessel is about 1000 psia; the water boils in the reactor vessel and the steam is passed directly from the reactor vessel to the turbine.
A PWR uses chemical shim (borated water) to maintain a critical configuration (the control rods that drop from the top are basically for rapid shutdown).  The concentration of boron in the water is gradually decreased as fuel is burned up to offset loss of fissile material and buildup of neutron poisons in the fuel.  In operation, a BWR inserts control rods partially up the core from the bottom since in the vessel the quality of the water/steam mixture (mass fraction of steam) increases as the water moves up the core and is heated to produce steam; as neutron poisons build up in the fuel, the position of the rods is varied to maintain a critical configuration.  The BWR does not use chemical shim.
For both a PWR and  BWR the effect of decreased moderation with voiding is greater than the effect of less neutron absorption and the void coefficient is negative, but it is less negative in the PWR early in fuel life when the concentration of boron (for chemical shim) in the water is greatest.
The moderator coefficient is also considered in the reactor design: this is the change in reactivity with change in temperature of the water (no boiling).  This coefficient depends on water temperature and the moderator to fuel atom ratio in both the PWR and the BWR, and the amount of chemical shim in the PWR.  This coefficient varies as the reactor is heated up to operating temperatures (about 600 F); it can be positive at low temperatures in both the BWR and PWR, and positive at even high temperatures in the PWR early in core life when the chemical shim is the greatest.  However the magnitude of the moderator temperature coefficient when positive is small compared to the negative temperature coefficient of the fuel with increasing temperature.  The fuel temperature coefficient is called the “Doppler” coefficient due to the broadening of the U 238 neutron  resonance capture cross sections with temperature.  Fission neutrons are captured in the U 238 as they slow down to thermal energies, and with broadening of these resonances the absorption of neutrons is increased.
A good reference on these and other reactivity temperature coefficients is Nuclear Power Reactor Safety, by E. E. Lewis.
A: Your understanding is basically correct.
The coolant is partly a neutron absorber.  When the coolant absorbs a neutron, it will decrease the reactivity of the system.  Therefore, when you loose coolant, it will cause the reactivity to increase, which contributes to a positive reactivity coefficient.
The coolant can also act as a moderator to slow down neutrons.  Fast neutrons have a lower probability of fission than thermal neutrons, so when you slow neutrons down, you generally increase the reactivity.  Losing moderator would decrease the slowing down, and decrease the reactivity, which contributes to a negative reactivity coefficient.
This plays out differently in different reactor designs.
In a boiling water reactor (BWR), the void reactivity coefficient is always very negative.  Water boiling is a very negative feedback mechanism.
In a pressurized water reactor (PWR), the coolant doesn't void, but it does decrease density as it is heated up.  This almost always leads to a negative feedback mechanism.  However, there are some instances where the coolant has a lot of neutron absorber (soluble boron) in it, and the moderator reactivity coefficient can be positive, but it is small and the overall power coefficient is negative.
It gets a little more tricky in reactors that have separate coolant and moderators.  In CANDU reactors, the coolant acts as an absorber and heavy water acts as a moderator.  The heavy water is located in a different tank, so when you only lose the coolant, you still have the moderator, and it is a positive reactivity coefficient.
Likewise, in an RBMK reactor, the moderator is mainly graphite.  When you lose the coolant, the moderator is still there, so it is also a positive reactivity coefficient.
The exact details get more tricky, and you usually have to analyze the core with detailed calculation to find out what the magnitude of the reactivity coefficients are for a particular reactor type and temperature.
