With material power production, temperatures and other phenomena change the effective multiplication constant K. Higher temperatures increase the vibrations of component nuclei and decrease their densities. One effect is to widen the effective resonance absorption bands of U-238. Because neutrons are slowed down by scattering in energy steps many times larger than these resonance widths, their non­fissile capture rates by U-238 increase with fuel temperature [58]. This negative reactivity feedback mechanism is called the Doppler Effect,[33] and it is an important re-stabilizing influence on the neutron populations of both fast and thermal reactors. Indeed fast reactor designs in particular have progressively evolved with greater U-238 content and less energetic neutrons so as to exploit the effect. Denoting the mean absolute temperature of the fuel by T, then in terms of the effective multiplication factor it is found that under normal operation

—T = d with d > 0 (2.30)

dT

where the Doppler Constant d is specific to a reactor design. Equation (2.30) shows that due to the Doppler Effect the effective multiplication factor is inversely proportional to the logarithm of the mean fuel-
temperature ratio. Other temperature feedback effects on reactor dynamics are generally associated with

— variations in coolant-density either by thermal expansion or by vaporization that alter the absorption or moderation of neutrons

— thermal expansion of the fuel and control rods similarly alters their macroscopic cross-sections

— thermal expansion of a moderator leading to faster neutrons (i. e., a “harder spectrum”) with more resonance absorptions by U-238 nuclei

— thermal expansion of the fuel cladding (zircalloy or stainless steel tubing) whose “bowing” excludes coolant.

These interactions and their associated time delays are significant features of nuclear reactor dynamics.

Neutron absorption is also significantly affected by the in-pile dwell time of the fuel, and its preceding 7 to 47 h power history due to developing concentrations of Xe-135 and Sm-149. Glasstone and Edlund [58] quantify the former as the more dominant “neutron poison,” and it is the daughter of the fission product I-135 whose half-life is 9.17 h [76]. Whilst a reactor is at power, Xe-135 is transmuted (“burned up”) faster by neutron capture than by its natural decay rate. However, in the event of a complete operational shutdown (trip), its concentration increases progressively even for as long as 12 h because of the relatively faster disintegration of I-135. As a result, it could be impossible to restart power production in this period unless sufficient latent reactivity has been held in reserve (i. e., by pre-trip control rod insertions).

The direct cycle RMBK reactor at Chernobyl was moderated by both graphite and a light water coolant, which was partially converted to steam for electric power production. Unlike heavy water, light water is both an effective moderator and absorber,12 so its conversion to less dense steam reduces both neutron moderation and absorption. In the unauthorized fateful incident, the night-shift operators cancelled trip settings and withdrew all 211 control rods [12]. Progressive reductions in inlet-water flow then resulted in a growing volume of steam in the reactor core, and a net reduction in neutron absorptions eventually

See Section 1.8.

occurred because moderation by its graphite was still sufficient. In this way a progressive positive feedback process was initiated that produced an exponentially increasing reactor power with the prompt time con­stant [58,80,117] of around 1ms, and some hundred times [12] full­rated power resulted. Subsequently by

i. the contact of molten fuel with liquid water [59,212],

ii. the accretion of then more mobile fission products into sizeable bubbles, whose external pressures increase as the surface ten­sion effect is reduced [210], and

iii. hydrogen production [12] from the chemical reduction of steam by graphite,

there was the recorded explosive destruction of the site. A nuclear explosion was not involved.

Due to astute design of core-lattice geometry [61] and fuel enrich­ment, light water reactors outside Russia have always been designed to become under-moderated with increasing steam production in order to stifle such potentially explosive events. Indeed a negative power- reactivity coefficient is a necessary prerequisite for licensing by European Regulatory Authorities.