# percent mille – pcm – unit of reactivity

## Units of Reactivity

Mathematically, reactivity is a dimensionless number, but it can be expressed by various units. 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 effective delayed neutron fraction, significantly changes with the fuel burnup. In LWRs the delayed neutron fraction decreases with fuel burnup (e.g. from βeff = 0.007 at the beginning of the cycle up to βeff = 0.005 at the end of the cycle). This is due to isotopic changes in the fuel. It is simple, fresh uranium fuel contains only 235U as the fissile material, meanwhile during fuel burnup the importance of fission of 239Pu increases (in some cases up to 50%). Since 239Pu produces significantly less delayed neutrons (0.0021 for thermal fission), the resultant core delayed neutron fraction of a multiplying system decreases (it is the weighted average of the constituent delayed neutron fractions).

βcore= ∑ Pii

## percent mille (pcm)

The unit of reactivity which is one-thousandth of a percent %ΔK/K (equal to 10-2x10-3 = 10-5 of keff). The unit of pcm is used at many LWRs because reactivity insertion values are generally quite small and units of pcm allows reactivity to be written in whole numbers. The operational changes such as control rods movement causes usually 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 βeff a response of a reactor on the same reactivity insertion (in units of pcm) is different at the beginning (BOC) and at the end (EOC) of the cycle.

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

(10 pcm = 1.43 cents for βeff = 0.007; 10 pcm = 2.00 cents for βeff = 0.005)

keff = 0.99      ρ = (keff – 1) / keff = -0.01      ρ = -0.01 * 105 = -1000 pcm

References:
Nuclear and Reactor Physics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.