## Conditions for CriticailityEdit

The degree of criticality of a reactor is usually measured by the term $ k_{eff} $, which is the effective neutron gain per fission cycle [Bennet and Thompson 1989, Glasstone and Sesonske 1981]. The neutron population is categorised into two main groups, prompt neutrons and delayed neutrons. A simplified expression (i.e. one which assumes one group of delayed neutrons instead of six subgroups) for the rate of change of the neutron population is defined by the equation:

- $ <br />\tfrac{dn}{dt}\ = k_{eff}(\rho - \beta)\tfrac{n}{l_p}\ + \lambda c $ .......... (a)

The neutron population, $ n $, changes in time, $ t $, as a function of the delayed neutron fraction, $ \beta $, the prompt neutron lifetime, $ l_p $, the delayed neutron precursor population, $ c $, the decay constant of precursor nuclei, $ \lambda $, the effective neutron multiplication factor, $ k_{eff} $ and the reactivity, $ \rho $, which is defined: $ \rho = (k_{eff}-1)/k_{eff} $. The rate of change of the concentration of delayed-neutron precursor nuclei is defined as:

- $ \tfrac{dc}{dt}\ = k_{eff}\beta\tfrac{n}{l_p}\-\lambda c $ .......... (b)

If the reactor is critical and generating a steady power output then $ k_{eff} = 1 $, $ \rho=0 $, $ dn/dt=0 $ and $ dc/dt=0 $, therefore both equations (a) and (b) give:

- $ 0 = - \beta \tfrac{n}{l_p}\ + \lambda c $

If there is a step (instantaneous) increase in reactivity, but $ \rho < \beta $, then the first term on the right-hand side of equation (a) is negative (i.e. prompt neutrons are still creating less than one prompt neutron for fission per fission cycle; delayed neutrons are still required to reach criticality); $ \lambda c $ therefore determines the rate of change in neutron population. If there is a step increase such that $ \rho = \beta $, the reactor reaches criticality through prompt neutrons alone (this is the condition for terming a reactor ‘prompt-critical’), fission through prompt neutrons will go on to cause one more fission through prompt fission, therefore the period over which its neutron population will rise is defined solely by the delayed neutrons, $ \lambda c $. Finally, if $ \rho > \beta $ the first term on the right-hand side of equation (a) is positive (i.e. prompt neutrons create more than one prompt neutron for fission per fission cycle), causing an exponential increase in the neutron population. The time period over which this takes place is primarily determined by the prompt neutron lifetime, $ l_p $.

For a prompt-critical reactor the delayed neutron fraction is no longer important and can be neglected. Therefore, for a prompt-critical reactor equation (a) reduces to:

- $ \tfrac{dn}{dt}\ = k_{eff} \rho \tfrac{n}{l_p}\ $

Rearranging for $ dt $ and integrating over the ranges $ 0 $ and $ t $ and also $ n_0 $ and $ n_t $ (the initial neutron population and the population after time, $ t $, respectively) gives:

- $ t = \tfrac{l_p}{(k_{eff}-1)}\ (ln(n)-ln(n_0)) $

which can be re-written:

- $ n = n_0 exp(t/T) $

The time period, $ T $, is defined: $ T = l_p / ((k_{eff}-1)) $. For a prompt critical reactor with a positive reactivity the neutron population grows by a factor of $ e $ after time $ t = T $. Assuming the lifetime of a prompt neutron is 0.1 μs (this is typical of fast reactors) and the neutron gain coefficient is $ k_{eff} = 1.001 $, the neutron population would grow by $ e $ every 0.1 ms. It is thus impossible to control a prompt critical fast reactor. In a thermal light water reactor the prompt neutron lifetime is approximately 0.2 ms, if this reactor reaches prompt criticality its neutron population would then grow by $ e $ every 0.2 seconds, three orders of magnitude slower.

Because delayed neutrons are emitted following the β^{-}decay of precursor fission fragments, the lifetime between their fission generations, $ l_d $, is orders of magnitude longer than for prompt neutrons. For ^{235}U the mean lifetime of a neutron emitting fission fragment precursor is 12.5 seconds. The mean lifetime of prompt and delayed neutrons combined, $ \bar{l} $, is:

- $ \bar{l} = (1- \beta)l_p + \beta (\bar{l_d}+l_p) \cong l_p + \beta \bar{l_d} \cong \beta \bar{l_d} $

For ^{235}U $ \beta = 0.0065 $ and therefore the mean lifetime of prompt and delayed neutrons is $ \bar{l} \cong 0.008 $ seconds. When including the delayed neutron fraction, the equations for the time period, $ T $, and the neutron population, $ n $, change. The period is now defined:

- $ T = l_d \bigg(\tfrac{\beta - \rho}{\rho}\ \bigg) $

The equation for the neutron population is significantly more complex when including delayed neutrons; however, providing $ \rho \leq \beta/2 $, the change in population can be approximated to be a function of two exponents:

- $ n = n_0 \bigg\{ \tfrac{\beta}{\beta-\rho}\ exp\bigg( \tfrac{\lambda \rho}{\beta - \rho}\ t \bigg) - \tfrac{\rho}{\beta-\rho}\ exp \bigg(- \tfrac{\beta-\rho}{l_p}\ t \bigg) \bigg\} $ .......... (c)

- $ n = n_0 \tfrac{\beta}{\beta-\rho}\ exp\bigg( \tfrac{\lambda \rho}{\beta - \rho}\ t \bigg) = n_0 \tfrac{\beta}{\beta - \rho}\ exp (t/T) $ .......... (d)

Considering once again a reactor with properties similar to a ^{235}U fuelled thermal light water reactor ($ \beta = 0.0065 $, $ l_p = 0.1 $ ms and $ \lambda = 0.08 $ s^{-1}) and the same step increase in the neutron gain coefficient $ k_{eff}= 0.001 $, the neutron population would grow by $ e $ in ∼ 780 seconds. In a fast reactor the response would be nearly identical, as the only parameter that is significantly different is $ l_p $, which is eliminated when making the approximation from equation (c) to equation (d). In a thermal or a fast reactor and for a given change in $ k_{eff} $ the delayed neutron fraction is the determining factor of the time period for a growth of $ e $ in the neutron population. The response of fast and thermal reactors to a fixed reactivity increase is the same, however their reactivity feedback will differ.

In prompt-critical reactors the neutron population can grow at a rate that is impossible to control in any type of reactor. Prompt criticality should be avoided. A large delayed neutron fraction and/or an external neutron source are desirable for reactor control; both of these features reduce the likelihood of a reactivity increase being so large that it causes the reactor to go prompt critical. If a reactor should go prompt critical, in a fast reactor the neutron population will escalate orders of magnitude more rapidly than in a thermal reactor.

Bennet, D.J. and Thompson, J.R., 1989 “The Elements of Nuclear Power”, 3rd edition, John Wiley and Sons, New York, USA

Glasstone, S., Sesonske, A., 1981. “Nuclear Reactor Engineering”, 3rd edition, Van Nostrand Reinhold Ltd., Princeton NJ, USA.

## Negative Doppler Reactivity FeedbackEdit

The parity of reactivity feedbacks are termed such that when there has been a change in reactivity a positive feedback will exacerbate the change and a negative feedback will act in the opposite direction to the change. With regard to safety, negative reactivity feedbacks are desirable following increases in reactivity, as they use the laws of physics to passively slow or even entirely suppress runaway reactivity increases, which otherwise have potentially hazardous consequences.

The kinetic energy of a nuclear reaction (specifically neutron absorptions) is not solely dictated by the neutron kinetic energy, the kinetic energy of the nucleus is also significant. The mean kinetic energy of the nuclei inside a reactor is defined by the operating temperature of the core. The nuclei are distributed by the Maxwell-Boltzmann curve. An increase in reactivity will increase the core temperature, and therefore the mean kinetic energy of the core nuclei. More importantly, the deviation in their kinetic energy is broader. Reactivity changes (approximately) do not affect the energy of the neutron flux. Because there is greater deviation in the kinetic energy of core nuclei, there is also greater deviation in the centre of mass energy of neutrons and nuclei during reactions. This effectively broadens the epithermal resonance capture peaks (see next two paragraphs), which increases the total rate of neutron capture. For fissile nuclei this is balanced by a corresponding increase in the fission rate. For non-fissile nuclei the increase in neutron absorption is an increase in neutron captures only. The temperature increase therefore reduces the number of neutrons available to be absorbed by fissile nuclei and the reactivity returns back to its equilibrium value [Jevremovic 2009, Bodansky 2008].

Resonance peak broadening can be understood through the following: consider a stationary nucleus with a neutron capture resonance of finite width whose capture cross-section is at its maximum for neutrons with kinetic energy $ E_0 $. Now consider the same nucleus but with some kinetic energy, the maximum probability for neutron capture will occur when the centre of mass energy (not the neutron energy) is $ E_0 $. When the nucleus has kinetic energy the angle of the collision is important. Considering a flux of neutrons of all energies crossing the nucleus in all directions. The neutron energy width for neutron captures will be narrow for a static nucleus. As the kinetic energy of the nucleus increases a broader range of neutron energies will result in collisions that have a centre of mass energy at or near the resonance energy, $ E_0 $. For increasing kinetic energy of the nucleus in a uniform flux of neutrons of all energies the absolute capture cross-section remains constant, but the neutrons energies over which the capture occurs broadens, see the figure to the right.

In a real reactor the reason that there is a net increase in neutron absorption for non-fissile nuclei in the core is because the mean free path of neutrons of energies near the centre of resonances (where the probability of capture has been reduced) is still significantly shorter than the geometry of the reactor, they are thus still captured. However the mean free path of neutron energies at the wings of resonances has been reduced and so more absorption of these nuclei occurs.

In uranium fuelled reactors it is ^{238}U that it responsible for causing the majority of the reactivity feedback. 238U will not be present in large quantities in a thorium fuelled ADSR (or one fuelled with minor actinides). The reactivity feedback due to ^{232}Th and other nuclear species has to be considered for these fuels. In a fast reactor the absence of moderating the prompt neutrons means that the vast majority of neutrons are absorbed before they lose enough energy to reach the epithermal resonance region, fast reactors are therefore less sensitive to Doppler affects than thermal reactors. Voiding takes over as the dominant passive mechanism by which reactivity changes are controlled.

Bodansky, D., 2008, “Nuclear energy: principles, practices, and prospects”, 2nd edition Springer, New York, USA.

Jevremovic, T., 2009, “Nuclear Principals in Engineering” 2nd Edition Springer, New York, USA.

## VoidingEdit

A void usually refers to the occurrence of bubbles within the moderator or coolant. This change in material density has multiple impacts on the reactivity of the reactor, the effects are material dependent. The overall void coefficient is defined as the ratio of the change in reactivity to the change in the void fraction. A negative void coefficient reduces the reactivity and a positive coefficient increases it. Implications of a void vary by reactor design, material choice and operating conditions. Some example changes are: (1) the decrease in coolant/moderator density increases the neutron mean free path, thus increasing the number of neutrons that leak out of the core. This has a negative reactivity affect. (2) The fraction of neutrons absorbed by the fuel and not the moderator/coolant is increased. In a thermal light water reactor this increases the number of neutrons captured by 238U resonances and thus has a negative impact on the reactivity, it would not necessarily be negative for all reactor types.