NUKEFACTS #47 Continued

Generation Time: Generation time is the unique lifetime of the fictitious fission neutrons starting a life cycle as fast neutrons. It is the time for a generation of fictitious neutrons to traverse one complete life cycle and produce a new set of fictitious fission neutrons. The first set of fast neutrons can be designated as generation-1 and the new set of fast neutrons can be designated as generation-2. None of the generation-1 neutrons survive into generation-2. All fictitious fission neutrons starting a life cycle are lost by the completion of the lifecycle.

Generation time is not the life time of ALL neutrons in the generation. The neutrons that are lost during slowing down and thermal diffusion naturally have lifetimes that are shorter than the generation time. Neutrons that are lost prior to causing fission are of no further interest because they do not contribute to the continuation of the process. The generation time is the average lifetime of those neutrons starting the generation that survive to produce fissions and fast neutrons to start the next generation.

In order to develop a mathematical representation of generation time it is necessary to define several components of the total neutron lifetime. The average lifetime of a prompt neutron is made up of three contributing intervals. These are:

t_pe = prompt neutron emission time: the time between the fission event and the ejection of a prompt neutron by fission fragments, ~1x10^-14 seconds.

t_sd = slowing down time: the average time for fast neutrons to reach thermal energy (relatively constant over core life), ~1x10^-5 seconds

t_td = thermal diffusion time: the average time that thermal neutrons undergo thermal diffusion before causing fission (a function of core composition and life), ~1x10^-4 seconds in U.S. commercial reactors.

l_p = prompt neutron lifetime: the average time between the fission event and the resultant prompt neutrons causing fission, 1x10^-4 seconds.

	47.5
PROMPT NEUTRON LIFETIME

Thus, the prompt neutron lifetime is the sum of the three components, the prompt neutron emission time, the slowing down time, and the thermal diffusion time. Since the thermal diffusion time is much longer than the other two components, the prompt neutron lifetime can be approximated as being equal to the thermal diffusion time, t_td. The value of the prompt neutron lifetime may vary slightly over core life.

The average lifetime of a delayed neutron is also made up of three contributing intervals, two of which are the same as for the prompt neutrons. The first component of the delayed neutron lifetime is the time for emission of delayed neutrons after the fission event and is determined by the mean lifetime of the precursors as:

t_m = mean lifetime of precursors: the mean time from the fission event to the emission of a delayed neutron by radioactive decay of the precursor, 1/lambda-effective, seconds.

l_d = delayed neutron lifetime: the average time between the fission event and the resultant delayed neutrons causing fission, 12.5 seconds at critical (a function of reactivity and core age).

	47.6
DELAYED NEUTRON LIFETIME

Thus, the delayed neutron lifetime is the sum of the three components, the mean lifetime of the precursor atoms, the slowing down time, and the thermal diffusion time. Since the mean lifetime of the precursor atoms is much longer than the other two components, the delayed neutron lifetime can be approximated as being equal to the mean lifetime of the precursor atoms, t_m. And, since lambda-effective depends on the precursor mix, which is a function of power change, which is function of reactivity, the lifetime of the delayed neutrons is not a constant value.

Since a generation of fictitious average neutrons is made up of all the prompt neutrons and all the delayed neutrons traversing a particular life cycle, the lifetime of the fictitious average neutrons, called the generation time, must be determined by weighting the lifetimes of the prompt and delayed neutrons by the fraction of each that is in the generation. A frequently used erroneous definition of generation time is based on the incorrect belief that beta always represents the fraction of delayed neutrons in the total neutron population. Generation time is then determined by weighting the lifetimes of the prompt and delayed neutrons by their respective fractions, as follows:

	47.7
A COMMON INCORRECT DEFINITION OF NEUTRON GENERATION TIME

where:
l_g = neutron generation time, seconds
(1 - beta) = prompt neutron yield fraction
beta = precursor yield fraction

Equation 47.7 indicates that, for a given time in core life, the generation time is a fixed value, a constant. This is not correct. Beta does represent the delayed neutron fraction for conditions where the neutron level is constant with time, namely for criticality and for equilibrium subcritical multiplication. For any transient power situation, the delayed neutron population differs from beta based on the value of reactivity. For increasing power, the delayed neutron population is less than beta. And for decreasing power, the delayed neutron population is greater than beta.

The correct general expression for the delayed neutron population fraction is (beta - rho), as discussed in NUKEFACT #1 - Beta is NOT The Delayed Neutron Population Fraction. Using (beta - rho) for weighting, the correct expression for neutron generation time is then:

	47.8
CORRECT DEFINTION OF NEUTRON GENERATION TIME

where:
[1 - (beta - rho)] = prompt neutron population fraction
(beta - rho) = delayed neutron population fraction

Since the delayed neutron component is much larger than the prompt neutron component, the neutron generation time can be approximated as being equal to the delayed neutron component, [l_d x (beta - rho)] for operational transients. Generation time must be based on a correct weighting of the mix of prompt and delayed neutrons in the existing neutron population, or in a particular neutron generation, and on a precursor mean life, t_m, that is a function of the variable lambda-effective. In common practice, the definition of beta has been corrupted. Beta is not the fraction of the population that consists of delayed neutrons. Beta is the precursor yield fraction.

Also, the generation time applies only to those "fictitious" average neutrons of the life cycle model. The term generation time was never intended to apply to prompt neutrons alone and it was never intended to apply to delayed neutrons alone. There is no such thing as a "generation time" for prompt neutrons. Likewise, there is no such thing as a "generation time" for delayed neutrons.

And since the life cycle model is a primitive model that does not accurately represent reactor behavior, it is necessary to dispel another myth. Generation time does not account for reactor controllability because generation time is a property of a fictitious average neutron. How can a fictitious neutron account for anything? What provides for reactor controllability is the lifetime of the delayed neutrons as determined by the decay properties of the precursor atoms in the precursor mix.

Sequential Interactions: In the life cycle model there are six discrete interactions. In reality, the individual interactions are not occurring in sequence but occur simultaneously. Sequencing is an artificial concept introduced into the model for accounting purposes, and one which does not affect the results of the process. This simplification is also used in the one-delayed group model, albeit for a much shorter time interval.

The order, or sequence of these interactions is important, because it is not arbitrary. This is so because the number of neutrons undergoing interaction in any one of the six steps depends on the number of neutrons surviving the previous steps. For example if the order of appication of the thermal utilization factor and thermal non-leakage factor is reversed, the number of neutrons lost in each interaction will change. And, obviously, it would not make sense for the first interaction applied to the fast neutrons to be thermal fission via the reproduction factor, eta. In fact, an unwritten rule, or assumption, is that thermal fission must be the final interaction. The sequence needs to correspond to the deceleration of the neutrons as they are moderated to thermal energy. On this basis, the sequence should be such that the fast interactions occur first, the intermediate energy interactions occur second, and the thermal interactions occur last. However, having said all of this, the overall mathematical result, in terms of the product of the six factors, is not dependent on the order of the six factors.

The logical starting point of the life cycle is with a new generation of fast neutrons, and this is usually where examples of numeric tracking are begun. From this point the sequence is as follows:

1. Fast fission: the energy range over which fast fission can occur is narrow because the high threshold energy for fast fission, being greater than 1.0 Mev. Therefore, fast fission is the first interaction.

2. Fast Non-Leakage: the energy range for fast non-leakage is large, from fission energy to thermal energy. Therefore, fission energy is the second interaction.

3. Resonance Escape: the energy range for resonances is from the start of the epithermal region to thermal energy, which has overlap with the lower part of the energy range for fast leakage. Therefore, this interaction follows fast leakage.

4. Thermal Leakage: occurs at thermal energy and is usually applied to all thermal neutrons. Therefore, themal leakage occurs before thermal utilization.

5. Thermal Utilization: occurs at thermal energy and accounts for neutrons being lost by capture.

6. Reproduction: occurs at thermal energy and applies to the neutron remnant that survives the other five interactions. Fission must be the final interaction of the sequence.

Chain Reaction: However, there is a quick and easy way to apply the results of the life cycle model to a large number of neutron generations. This is done by using k-effective itself as the multiplier for each completed life cycle, i.e. by applying the effect of all six interactions in a single stroke rather than continuing to plod on, one interaction at a time.

To illustrate this procedure, the first generation must start with a large number of fictitious fast neutrons, which will be designated as N_o. Since the life cycle model has no neutron source, the origin of these initial neutrons must remain as a great mystery. Be satisfied, these neutrons are just in the right place at the right time. As explained under Neutron Generation, a large group of neutrons is required because each of the interactions that the neutrons undergo during the life cycle is probabilistic. As such, neutrons must be treated mathematically in large groups. And, since k-effective represents the overall net interaction, k-effective too is a probabilistic term.

Then, by multiplying N_o by k-effective each time a life cycle is completed, we have:

This sequence of life cycles can be represented as a chain reaction:

And, in equation form this becomes:

	47.9
FISSION NEUTRONS STARTING GENERATION-i

The chain reaction shown above, although it consists of many neutrons, is but a single chain. All neutrons started the generation sequence at the same time, all are multiplied by k-effective at the completion of each generation, and all neutron histories are alike. There are no independent chains because there is no neutron source. All of the fictitious average neutrons are propagators of the chain because none act as source neutrons.

It is from this single chain representation in the life cycle model that the serious error of believing that chain reactions are self-sustaining at criticality arises. At criticality, in the primitive life cycle model that certainly has to be the case. But in the real world of actual reactor behavior it doesn't happen. This topic was discussed in NUKEFACT #4 - Chain Reactions Are Not Self-Sustaining At Criticality.

For the more realistic one-delay group model the chain reactions are NOT self-sustaining at criticality because the delayed neutrons are treated as source neutrons that initate chains and not as neutrons that propagate chains. In removing the delayed neutrons from chain propagation, the operational reactor is always subcritical on prompt neutrons and therefore, the chains are never self-sustaining, including at criticality. The more accurate one-delay group model consists of multiple parallel chains. A new chain is initiated by delayed neutron emissions at the start of each prompt neutron lifetime. The chains are not multiplied by k-effective, but rather by k(1 - beta) because it is the prompt neutrons that propagate the chains. This model is appears as follows:

where:
C' = C x l_p
l_p = the prompt neutron lifetime

Thus, the product (Lambda x C') represents the number of delayed neutrons introduced into each prompt lifetime interval. These delayed neutrons act as source neutrons, initiating chain reactions at the start of each time unit. The chains are propagated by prompt neutrons, whose lifetime in commercial reactors is of the order of 1x10^-4 seconds. Since delayed neutrons initiate the chains, they cannot also be counted in the propagation of the chains. Hence, the effective multiplication factor is multiplied by (1 - beta), the fraction of fission neutron production that is prompt, to remove the delayed neutrons from the propagation. This bracketed factor, [k_eff(1 - beta)], is known as the prompt multiplication factor, k_p. For criticality, with k_eff = 1.0000 and beta = 0.0065, k_p = 0.9935. And, since k_p < 1, each set of chains is finite, i.e. each set of chains contracts from lifetime to lifetime, and eventually expires. The individual chain reactions are not self-sustaining at criticality. The overall process is self-sustaining on fission neutrons alone (prompts + delays) because new chains initiated by delayed neutrons exactly make up for the expiring chains.

Constant k-effective: The life cycle model is limited to illustrating only conditions where k-effective is held constant. This is unfortunate since it is vital that the reactor operator understand reactor behavior under transient reactivity conditions. The problem with this limitation is compounded by the fact that the resultant behavior for constant k-effective is taken to be valid for all situations, even those where k-effective is changing with time.

For an off-critical condition, the life cycle model will produce exponential power change with time because it produces a process of repetitive multiplication by a constant k-effective. On this basis, Equation 47.2 is used to establish the following relationships between the value of k-effective and the neutron population response:

It is these relationships for constant k-effective that seem to endure and that muddle the understanding of reactor behavior. These relationships are not supported by later more realistic models which show that the value of k-effective, by itself, does not determine the direction of change in neutron population with time. Unfortunately, this later refinement escapes many, who continue to take the above listed relationships as sacrosanct. In fact, since the great majority of U.S. Control Rooms do not provide a reactivity meter, the reactor operators are expected to evaluate off-critical nuclear status by interpreting neutron level behavior with time. As it turns out, this is an impossibility !

Stable Reactor Rate in the Life Cycle Model: So, you may ask, with all of this, is it possible that the stable reactor rate for the life cycle model is in agreement with the reactor rate for the one-delay group model? Surprisingly, the answer is a qualified yes, if the correct formulation for generation time is employed. First it is easily shown that Equation 47.9 can be expressed in real time as:

	47.10
NEUTRON POPULATION WITH TIME FOR LIFE CYCLE MODEL

where:
N(t) = fast neutrons at time "t"
N(o) = fast neutrons at time zero
e = 2.718
delta-k = k - 1
t = time in seconds

The coefficient of the exponent, (delta-k/l_g), is a form of reactor rate. Using the approximation of generation time from Equation 47.8, this ratio can be expressed as:

	47.11

Then, inverting both sides of this equation gives:

	47.12
REACTOR PERIOD FOR LIFE CYCLE MODEL

For the one-delay group model the conventional expression for stable reactor period is:

	47.13
REACTOR PERIOD FOR ONE-DELAY GROUP MODEL

On comparison, it can be seen that the two representations for reactor period are remarkably similar. The reactor period for the primitive life cycle model contains delta-k in the denominator instead of the correct term of reactivity. However, for typical off-critical operational transients this difference will be insignificant. Thus, the correct stable rate expression for the primitive life cycle model is essentially in agreement with the conventional expression for stable reactor rate. But, even though this agreement exists for the stable rate, the life cycle model cannot accurately represent other transient conditions.

SUMMARY: It is quite likely that the life cycle model gains unwarranted legitimacy from the fact that it is the model known for development of k-effective. However, the life cycle model is nothing more than a means to an end. It is one of several stepping stones to the ultimate objective, the one-group precursor model, which provides an excellent representation of actual reactor behavior. It is the one-delay group model that provides the basis for explaining and understanding actual reactor behavior. The one-delay group model stands on its own as the best representation of reactor behavior, and this is where the final emphasis should be placed. But instead, the continued insistence on regression to terminology and concepts that are in the domain of the primitive life cycle model is inexcusable, unacceptable, confusing to the student, and absolutely counterproductive. The following is a summary of the inappropriate terminology and false concepts arising from the life cycle model:

1. A "generation" of fictitious average neutrons includes all neutrons traversing a particular life cycle as a group. A generation of neutrons exists only in the context of the primitive life cycle model, which is a model that cannot accurately represent transient reactor behavior. The concept of a generation of neutrons does not exist in the one-delay group model.

2. "Generation time" applies only to the entire generation of fictitious average neutrons in the life cycle model. Lack of understanding of generation time has led to corruption of the term as in prompt neutron generation time and delayed neutron generation time. Generation time, even when developing the primitive life cycle model, is rarely defined correctly.

3. Source multiplication is a process that cannot occur in the primitive life cycle model because non-fission source neutrons are absent. The all important underlying physical process of source multiplication extends from shutdown to full power and is crucial to understanding reactor behavior. The one-delay group model treats source multiplication in its entirety.

4. "Controllability" of a reactor is not due to the lifetime of the fictitious average neutrons, called the generation time. The mean lifetime of the precursor atoms provides for controllability, as exhibited in the one-delay group model.

5. "Off-critical reactor behavior" cannot be accurately defined by a primitive model that is limited to a constant k-effective. Understanding of off-critical power behavior must address reactivity change with time, as does the one-delay group model.

6. "K-effective" is in everyday usage, even though it derives from the primitive life cycle model. However, it is loosely applied, along with reactivity, with few realizing its true origin. Fewer still can accurately state the time interval over which this factor of change in fission neutron population occurs, failing to recognize that it varies as a function of reactivity. In the one-delay group model k-effective is modified for separate accounting of prompt and delayed neutrons.

7. "Self-sustaining chain reactions" at criticality are a fiction strictly associated with the primitive life cycle model. In actuality, the operational reactor is always subcritical on prompt neutrons, which are the only neutrons that propagate chain reactions, as correctly represented in the one-delay group model.

As an analogy, consider the two well known models of the universe, the first by Ptolemy and the second by Copernicus. Ptolemy's primitive model incorrectly placed the earth at the center of the universe, with the heavenly bodies moving about it. Copernicus's later realistic, and correct model, had the planets revolve around the sun. No current day expert in astronomy would ever attempt to explain, or teach, about planetary movements by combining parts of these two models. Yet, that is essentially what is happening in reactor behavior training today.

Now, having said all of the above, we are not advocating the elimination of the primitive life cycle model. It serves a useful purpose in training. However, the limitations of this model should be well defined when it is presented. Then, when the more realistic one-delay group model is developed, that is where all future explanation of reactor behavior should be focused. Once, the one-delay group model is made available, there is absolutely no reason to retain features of the life cycle model for explaining reactor behavior.

This error is contained in many training resources, such as fundamentals texts, handouts, and lesson plans. However, the root cause is that the error incorporated in to NRC and INPO test item exam banks. It is long past time for corrective action on this issue.