Every student of reactor behavior is familiar with two important reactor conditions, namely "criticality", more precisely referred to as "delayed criticality", and "prompt-criticality". And even though both conditions use the descriptive "criticality", the difference between the conditions could hardly be more pronounced, delayed-criticality being a steady-state condition and prompt-criticality being a transient-state of extremely rapid power increase. In Nukefact #4, Chain Reactions Are Not Self-Sustaining At Criticality, a serious flaw in the understanding of "criticality" was identified. In this essay, the classic In-Hour equation is used to expose error in the concept of "prompt criticality".

**INTRODUCTION**

Prompt criticality is neither a constant power condition nor an operational condition. It is an excessive positive reactivity condition, where rho = +beta, that creates an extremely rapid power excursion, with high potential for causing core damage. Conventional wisdom holds that the chain reactions at prompt criticality are self-sustaining on prompt neutrons alone, making the delayed neutrons of no further significance to the chain reaction process. Nothing could be further from the truth!

**SPECIFIC EXAMPLES OF CONVENTIONAL WISDOM**

The following quotations concerning the prompt critical condition are representative of the error promulgated. Every statement is an excerpt from widely distributed text books, Nuclear Training Center lesson plans, contractor training manuals, and government handbooks. Statements that contradict this prevailing error are not known to exist.

**General Electric-BWR Academic Series - Reactor Theory; also Nuclear Training Center Lesson Plan**

"If enough reactivity is added to the reactor, a point will be reached where the reactor is producing enough prompt neutrons to sustain itself without benefit of any delayed neutrons. Under these circumstances (rho equal to or exceeding the value of beta), the reactor is said to be Prompt Critical, and reactor power would increase extremely rapidly. Without delayed neutrons the reactor is uncontrollable by ordinary methods."

**General Electric BWR Technology**

"Prompt critical is defined as the point when the reactor is critical on prompt neutrons alone. ... rho = beta ... It can be seen from the period equation that when rho = beta, T = l*/rho. Thus, the period is dependent upon the prompt neutron lifetime, and it is a very short period."

**Nuclear Training Center Lesson Plan**

"... for reactivity equal to or greater than beta ... the reactor is critical on prompt neutrons alone ... power increases as fast as prompt neutrons are produced and utilized ... by a factor of k_{eff}^{20000} in one second"

**Robert Reed - Introduction to Nuclear Reactor Operations**

"If (1 - beta) x k_{eff} is made equal to or greater than unity, self sustaining neutron multiplication occurs due to prompt neutrons alone. The reactor is said to be prompt critical. ...The delayed neutron fraction is not actually zero, but can be viewed as being zero because delayed neutrons are not necessary to maintain reactor criticality and, in this prompt critical condition, they have no impact on reactor control. ... T = l*/(delta-k) is the reactor period in a prompt critical reactor in terms of prompt generation time, l*, and delta-k"

**General Physics - Reactor Theory Course Manual**

"When rho is equal to or greater than beta, the reactor is critical on prompt neutrons alone (prompt critical) without the aid of delayed neutrons. For this condition, the period equation is not valid because the delayed term breaks down. (Actually, it becomes negative, which has no physical meaning for a positive reactivity insertion.) The prompt term gives: T = l*/rho."

**M. A. Schultz - Control of Nuclear Reactors and Power Plants**

"When the effective multiplication factor of a reactor is 1.0075, the reactor is said to be prompt critical. This statement means that the reactor would be capable of sustaining a chain reaction without the use of the delayed neutrons. If k is greater than 1.0075, extremely rapid exponential multiplication of reactor power level results."

**Samuel Glasstone and Milton Edlund - The Elements of Nuclear Reactor Theory.**

"Hence, for reactivities, or excess multiplication, larger than beta, that is, larger than 0.0075, the reactor period becomes very small, approaching that for the case in which there are no delayed neutrons."

**Samuel Glasstone - Nuclear Reactor Engineering**

"the contribution of delayed neutrons at prompt critical is negligible."

**DOE Fundamentals Handbook -Nuclear Physics and Reactor Theory**

"If can be readily seen from Equation (4-7) that if the amount of positive reactivity added equals the value of beta_{eff}, the reactor period equation becomes the following:

In this case, the production of prompt neutrons alone is enough to balance neutron losses and increase the neutron population. The condition where the reactor is critical on prompt neutrons, and the neutron population increases as rapidly as the prompt neutron generation lifetime allows is known as prompt criticality. ... Prompt critical is ... a convenient condition for marking the transition from delayed neutron to prompt neutron time scales."

Clearly, the common theme among this sampling of expert opinion about prompt criticality is to emphasize the prompt neutron's importance and to dismiss the delayed neutron's role. This consensus rests on three false and unsupportable propositions, namely that:

- the prompt neutron production rate exactly equals the neutron loss rate
- delayed neutron production rate is so small as to be insignificant
- the reactor rate is ten times greater than the reactor rate determined by the IN-HR equation

**ON THE PRODUCTION AND LOSS OF NEUTRONS**

The long standing scenario put forth to explain the so-called condition of "prompt criticality" is defective. If the prompt neutron production rate is equal to the neutron loss rate, and the delayed neutrons are insignificant, then from whence come the excess neutrons necessary to sustain the rapid increase in reactor power? Equating production and losses does nothing other than produce a constant neutron level and is not supportive of a rapidly escalating transient. That prompt neutrons alone are capable of maintaining a constant neutron level while the neutron level itself is rapidly increasing is not particularly impressive. A review of both the neutron balance equation and the IN-HR equation shows clearly that a condition where the reactor functions on prompt neutrons alone does not and cannot exist. For the reactor to function on prompt neutrons alone, the burden on prompt neutron production is not just to make up for neutron losses. It is necessary that the prompt neutrons also be in appropriate excess so as to account for the existing stable rate. Unfortunately, there is no positive reactivity at which this occurs. The prompt neutrons are always deficient because delayed neutrons are alway present and contributing to the reactor period.

Both the neutron balance equation and the IN-Hr equation confirm that reactor behavior exhibits a smooth transition from far subcritical to far beyond the so-called prompt critical condition. There is no discontinuity, no anomaly, nor a singularity at rho = +beta. There is no transformation in the physical process from just below rho = +beta to just beyond. In fact, there is no particular importance associated with the prompt neutron production rate being exactly equal to neutron losses when rho = +beta. The so-called condition of "prompt criticality" is pure fiction based on flawed logic that ignores the message contained in the two principal equations describing reactor transient behavior.

**THE IMPORTANCE OF DELAYED NEUTRONS AT PROMPT CRITICALITY**

The IN-HR equation dramatically illustrates that delayed neutrons are very significant at prompt criticality. As presented in Nukefact #31 the IN-HR equation is:

67.1 |

THE IN-HOUR EQUATION |

rho = reactivity

l* = l

l

k

T = reactor period, seconds

beta-i = the yield fraction of the i

lambda-i = the decay constant of the i

Reactivity is the accepted method for measuring worth in the reactor. The IN-HR equation provides the relationship between reactivity and stable reactor rate for any off-critical reactivity condition, from far subcritical to far beyond prompt criticality. But, it does much more than this. It provides a break down the total reactivity into reactivity contributions from the prompt neutrons and the six delayed neutron groups, as follows:

67.2 |

reactivity on left-hand-side of IN-HR equation |

first term on right-hand-side of IN-HR equation, reactivity contributionof prompt neutrons |

second through seventh terms on right-hand-side of IN-HR equation, reactivity contributionof delayed neutron groups, n = 1 to 6 |

Thus, the IN-HR equation is perfectly suited for either verifying or disproving delayed neutron significance at prompt criticality.

With rho = beta = +0.0065, l_{p} = 0.0001 seconds, and using a standard set of precursor yield and decay characteristics for U-235 fission, the IN-HR equation yields:

- the prompt neutron reactivity contribution from the IN-HR equation is:
**rho**_{prompt neutrons}= +0.00047 - while the sum of reactivity contributions of the six delayed groups is:
**rho**_{delayed neutrons}= +0.00603

**THE REACTOR PERIOD AT PROMPT CRITICALITY**

Since the IN-HR equation provides the definitive relationship between reactivity and reactor period, it offers the ideal means for demonstrating that the reactor period at prompt criticality is not represented by T = l*/rho.

Again, taking rho = beta = +0.0065, l_{p} = 0.0001 seconds, and using a standard set of precursor yield and decay characteristics for U-235 fission, the IN-HR equation yields:

- the advertised reactor period for the so-called prompt critical condition is:
**T = (0.0001/1.0065)/0.0065 = 0.015 seconds (1737 DPM)** - while the IN-HR equation gives a value for the reactor period of:
**T = 0.21 seconds (124 DPM)**

Clearly, this is a glaring and unacceptable discrepancy. T = l*/rho is not representative of the reactor period at rho = +beta. The explanation for this nonconfirmation is not hard to find. The delayed neutrons, with their long mean lifetime, are present in small, but sufficient, quantity to greatly slow the stable rate at prompt criticality from what it would be with prompt neutrons alone. Thus, the IN-HR equation again proves that any contention about delayed neutrons being insignificant at prompt criticality is totally, and completely, wrong.

**SUMMARY**

There is only one state of criticality, and that is "delayed criticality." The condition of "prompt criticality" is a fiction that defies the fundamental equations of reactor behavior. The three propositions supporting conventional wisdom about the prompt criticality are all seriously flawed.

*Other than the prompt neutron production rate being equal to the rate of neutron losses, which is not at all noteworthy, there is nothing special about the so-called condition of prompt criticality.*

Yet, the price we pay for this bit of trivia is the spewing forth of much misinformation that is completely counterproductive to a student's understanding of reactor behavior. The subject of reactor theory would have been better served if the condition of prompt criticality were never put forth as a concept. "Prompt criticality" should be removed from the reactor theory vocabulary. It is of no importance whatsoever.

One major ramification of properly accounting for the delayed neutrons is that one finds the physical process underlying reactor behavior is __always__ source multiplication, with delayed neutrons acting as the source. This important concept will be discussed in a forthcoming Nukefact.