 Solution

Use a common comparison for all of question2: retain the same number and height of fuel rods and approximately the same normalized rod-power shape; reduce fuel lattice pitch by making the active core cross-section smaller. Compare otherwise similar operating states of an initially under-moderated reactor, typical of a pressurized water reactor. Rod number, core height, reflector and control settings are not all fixed explicitly by the question, so these are stated assumptions. Keeping the external core envelope fixed while inserting more rods would be a different comparison.

Reactor reactivity normally decreases at unchanged controls and reference temperatures. Less water per fuel atom reduces neutron moderation and increases resonance absorption during slowing down. Reduced absorption in the water partly compensates, but on the under-moderated side the net lattice tendency is lower multiplication. This is the usual under-moderation argument, rather than a universal rule for arbitrary fuel lattices.^[ref]

Neutron leakage would usually tend to increase. The smaller transverse core extent increases the geometric tendency to leak, and reduced moderation generally increases the slowing-down distance. There are competing changes in fuel absorption, transport lengths and reflection. Thus a definite leakage fraction needs the actual core and reflector arrangement; the usual increase is a qualitative tendency, not a geometric theorem for every redesign.

The Doppler coefficient of reactivity remains negative under the ordinary uranium-fuel assumptions. Increasing fuel temperature broadens absorption resonances, increasing effective uranium-238 resonance capture and reducing reactor reactivity. This stabilizing fuel-temperature effect is distinct from moderator-density feedback.^[ref] Greater importance of resonance absorption can make this feedback more important in a tighter lattice, but its change in magnitude is not fixed by pitch alone: spectrum, resonance self-shielding, fuel composition, burnup and the reference fuel temperature all enter the derivative.

The fraction of fissions in uranium-238 tends to increase. The harder neutron spectrum gives more opportunity for fast fission before neutrons slow below the uranium-238 fission threshold. This concerns the fraction of the total nuclear fission rate, not an assertion that the total thermal power increases; that power is held fixed.