## Units of Reactivity

Mathematically, reactivity is a **dimensionless number**, but various units can express it. The most common units for **research reactors** are units normalized to the **delayed neutron fraction (e.g.,, cents and dollars)** because they exactly express a departure from prompt criticality conditions.

The most common units for **power reactors** are units of **pcm** or **%ΔK/K**. The reason is simple. Units of **dollars are difficult to use** because the normalization factor, **the effectively delayed neutron fraction**, significantly **changes with the fuel burnup**. In LWRs, the delayed neutron fraction decreases with fuel burnup (e.g.,, from **β _{eff} = 0.007** at the beginning of the cycle up to

**β**at the end of the cycle). This is due to isotopic changes in the fuel. It is simple.

_{eff}= 0.005**Fresh uranium fuel**contains only

^{235}U as the fissile material. Meanwhile, during fuel burnup, the importance of fission of

^{239}Pu increases (in some cases up to 50%). Since

^{239}Pu produces significantly fewer delayed neutrons (

**0.0021**for thermal fission), the resultant core delayed neutron fraction of a multiplying system decreases (the weighted average of the constituent delayed neutron fractions).

β_{core}= ∑ P_{i}.β_{i}

## percent mille (pcm)

The reactivity unit is **one-thousandth** of a percent **%ΔK/K** (equal to 10^{-2}x10^{-3} = **10 ^{-5}** of k

_{eff}). The unit of

**pcm**is used at many

**LWRs**because reactivity insertion values are generally quite small, and units of pcm allow reactivity to be written in

**whole numbers**. The operational changes such as control rods movement usually cause reactivity insertion of the order of units of pcm per one step. The fact that the effective delayed neutron fraction changes with the

**fuel burnup**have an important consequence. Due to the difference in

**β**a response of a reactor on the same reactivity insertion (in units of pcm) is different at the beginning (

_{eff}**BOC**) and the end (

**EOC**) of the cycle.

For example,** one step** of control rods causes a **greater response** at EOC than at BOC. Even though we assume that one step causes the same reactivity insertion (e.g.,, +10pcm) in both cases, this assumption is not always correct because the control rod’s worth increases with fuel burnup.

(10 pcm = 1.43 cents for **β _{eff} = 0.007**; 10 pcm = 2.00 cents for

**β**)

_{eff}= 0.005**keff = 0.99** ρ = (keff – 1) / keff = -0.01 ρ = -0.01 * 105 = **-1000 pcm**