Reactivity

Reactivity is the measure of the departure of a reactor from criticality. In the nuclear reactor, the control rods are being used to control reactivity.

For example, we take Uranium-235 that is being used as fuel in a nuclear reactor. When the nucleus of U-235 are struck by a slow-moving neutron, they will under go fission reaction. As by products of the reactions, it will release fragments, radiation, and neutrons. If these neutrons are slowed down  and hit another U-235 nucleus, that nucleus will also fission and  continues the chain reaction.

 Figure showing the fission of U-253 nucleus

For the chain reaction to be self-sustaining, each generation of fission events has to produce enough neutrons so that there are enough are left to cause just as many fission events in the next generation.

Positive reactivity causes power to rise exponentially proportional to the reactivity. Negative reactivity causes power to decrease. To change power in a planned manner, reactivity is adjusted by moving the control rods, either manually or by means of automatic controls. Partially removing a control rod is expected to increase the reactivity, causing the power to rise to a new level.

 

Figure above shows the effect of a relatively large initial reactivity leading to a rapid rise to a blowup.

The effective neutron multiplication factor, k, is the average number of neutrons from one fission that cause another fission. The value of k determines how a nuclear chain reaction proceeds: 

if k < 1 (subcriticality): The system cannot sustain a chain reaction, and any beginning of a chain reaction dies out over time.  

if k = 1 (criticality): Every fission causes an average of one more fission, leading to a fission (and power) level that is constant. Nuclear power plants operate with k = 1 unless the power level is being increased or decreased.

if k > 1 (supercriticality): The result is that the number of fission reactions increases exponentially.  

reference: www.wordiq.com/definition/Nuclear_chain_reaction 
reference: www.sizes.com/properties/reactivity