# Dollar – \$ – 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

## dollar (\$)

The unit of reactivity which has been normalized to the delayed neutron fraction. Reactivity in dollars = ρ / βeff. The cent is 1/100 of a dollar. This is very useful unit, because the reactivity in dollars (rather in cents) determines exactly the response of the reactor on the reactivity insertion. Conversion of dollars to pcm depends on βeff. For reactor core with βeff = 0.006 (0.6%) one dollar is equal to about 600 pcm. It is very important amount of reactivity, because if the reactivity of the core is one dollar, the reactor is prompt critical.

BOC and βeff = 0.006

keff = 0.99      ρ = (keff – 1) / keff = -0.01       ρ = -0.01 / 0.006 = -1.67 \$ = -167 cents

EOC and βeff = 0.005

keff = 0.99      ρ = (keff – 1) / keff = -0.01       ρ = -0.01 / 0.005 = -2.00 \$ = -200 cents

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.