This essay is both an explanation of the neutron life cycle model, as to what it represents and what it fails to represent, and a critique of how unique aspects of the model are being inappropriately invoked in explaining important concepts of reactor behavior. To the beginning student, the neutron life cycle model provides valuable early insight by bringing together all of the physical interactions that neutrons undergo within a reactor. Unfortunately, the neutron life cycle model also serves as the unrecognized source of garbled terminology and false concepts found in later commercial reactor behavior studies. In this sense, the model is both a blessing and a curse.
The neutron life cycle model, was discussed briefly in NUKEFACT #31 - The Many Models of Reactor Behavior. Typically, the life cycle model is represented as a closed loop block diagram involving six specific neutron interactions. In Figure 47.1 the sixth interaction is displayed in two parts.
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The neutron life cycle model is one of a series of models used in proceeding from an over simplified representation of the reactor to a final surprisingly realistic one-delay group model. The life cycle model is used to introduce the concept of keffective, both in terms of the six-factor formula and in terms of the ratio of neutrons in successive neutron generations. Although it is a primitive model that makes no distinction between prompt and delayed neutrons, its popularity undoubtedly stems from the ease of understanding it presents by breaking the life cycle into discrete and sequential steps, where the graphical depiction of the physical process has direct correlation to a mathematical representation.
From interaction 6.b. in Figure 47.1, the principal result from the life cycle model is the six-factor formula. Each of the six factors represent a step in the sequence and all are defined by specific parameters of the system, i.e. by compositions, cross sections and other nuclear properties, neutron diffusion lengths, and system size.
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47.1 |
| THE SIX FACTOR FORMULA |
A second important result from interaction 6.b. in Figure 47.1 is the definition of k-effective in terms of the fission neutron populations in successive life cycles:
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47.2 |
It should be noted that Equations 47.1 and 47.2 differ in that Equation 47.1 is like input which represents cause, by expressing k-effective in terms of the physical properties of the reactor, while Equation 47.2 is like output which represents effect, by expressing k-effective as a ratio of neutrons in successive generations. Unfortunately, this latter definition leads to misunderstanding of reactor behavior, as discussed later. Since the number of surviving neutrons in a generation diminishes after each interaction, it is important that the ratio in Equation 47.2 is for neutrons at the same point in the life cycle.
In any case, these results are certainly impressive for such a simple model. Unfortunately, father time has shown that the neutron life cycle model exhibits a Jekyll and Hyde personality. Paradoxically, its dark side is that it is too good. It is so good that the concepts learned at this early stage of the subject development resist being discarded as new and better models supersede it. A principal reason for this reverse-Alzheimers syndrome is that the later improved models retain some features of the life cycle model, albeit they are subtly modified to better represent the actual situation. In particular, keff, which derives from the life cycle model, is carried into the very realistic one-delay group model. Reactivity, which is derived from keff, appears in the one-delay group model equations. And since both keff and reactivity are part of the everyday nuclear vocabulary for defining the reactor's nuclear status, there can't possibly be anything wrong here. All of this just validates the life cycle model. Well, don't be so sure!
Despite the life cycle model's value as an instructional tool, it is still a primitive representation of the reactor because of several inherent assumptions, that are usually not clearly identified when the life cycle model is introduced. These assumptions include:
Fictitious Fission Neutrons: The lifecycle model incorporates neutrons that are neither prompt neutrons nor delayed neutrons. The neutrons in the lifecycle model are "fictitious" neutrons that never have and never will exist. The reason that the lifecycle neutrons are fictitious is that they represent an "average" neutron. These fictitious neutrons ALL exhibit the same variable generation time that represents neither the average lifetime of the prompt neutron nor of the delayed neutron. To illustrate, one only needs compare typical lifetimes for prompt and delayed neutrons with the generation time for the "fictitious" neutrons at the condition of criticality:
mean lifetime of delayed neutrons = 12.5 seconds
lifetime of "fictitious" neutrons = 5x10-2 seconds
Energy of Fast "Fictitious" Fission Neutrons: The birth energy of the fictitious fission neutrons is the same for all neutrons in a generation. This presumption is built into the model because all neutrons starting a generation encounter the same probabilities for each of the six interactions. But, even though delayed neutrons are not explicitly represented in the life cycle model, their much lower emission energy should be taken into account by use of beta-effective in determining the average lifetime of the fictitious fission neutrons (see following discussion on Generation Time).
Omission of Non-Fission Source Neutrons: The life cycle model does not account for non-fission neutrons, which typically act as source neutrons that initiate chain reactions. And, as with the omission of delayed neutrons, without the non-fission source neutrons, the concept of source multiplication in the subcritical region cannot exist. In short, there is no explanation for equilibrium subcritical multiplication in the life cycle model.
At the condition of equilibrium subcritical multiplication, where precursor production and loss are in balance, source multiplication takes its simplest form. Here, the presence of a non-fission neutron source in a subcritical multiplying medium creates an equilibrium neutron population, where the fission neutron losses are balanced by the non-fission neutron production:
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47.3 |
The neutron balance equation for the one-delay group model treats both non-fission source neutrons and delayed source neutrons to yield a more general definition of source multiplication as:
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47.4 |
| GENERAL EQUATION FOR SOURCE MULTIPLICATION |
where:
P = n/(nu x 3.1x1010 ), power in watts
n = thermal neutrons causing fission, sec-1
nu = neutrons/fission
S-bar = S/(nu x 3.1x1010 ), watts
Lambda = single precursor group effective decay constant, seconds-1
C-bar = C/(nu x 3.1x1010 )
C = core precursor inventory, atoms
beta = precursor yield fraction
rho = reactivity = (k - 1)/k (<+.0040)
Equation 47.4 defines power from shutdown to rated conditions, as generated by source multiplication. Without source neutrons, the life cycle model fails to exhibit the all important underlying physical process of source multiplication, which derives from multiple ongoing chain reactions, all of which are initiated by source neutrons.
Neutron Generation: The life cycle model invariably starts with a large set of newly created fast fission neutrons. There is good reason for this. The interactions that a neutron undergoes during a life cycle are all probabilistic. As such neutrons must be treated mathematically in large groups. For the life cycle model, this large group is called a "generation" of neutrons.
The definition of what constitutes a neutron generation has been badly corrupted over the years. A "generation of neutrons" has its origin in the life cycle model, nowhere else. And, the terminology of a "generation of neutrons" has meaning only in the context of the life cycle model. In the life cycle model it is assumed that a group of "fictitious" fast neutrons start a cycle and proceed through each of the sequential steps in the cycle as a group. This group of "fictitious" neutrons, which includes ALL of the neutrons starting the lifecycle, is a single generation of neutrons. ALL of the fission neutrons in the "fictitious" group are born at the same time, the group is diminished by each step of the life cycle, and the remnant of the "fictitious" group that cause fission at the end of the lifecycle, expire at the same time. This "fictitious" group of fission neutrons is a single generation of neutrons. The group of fission neutrons makes up the entire neutron population for a life cycle. It can be likened to a generation in the human population, which consists of a specific set of individuals. The "fictitious" group of neutrons existing in one life cycle are designated as a GENERATION of neutrons, and they have an average life time which has been given the name "generation time". The dictionary defines "generation" as "the entire body of individuals born and living at about the same time." Read "individuals" as "fictitious neutrons", with emphasis on the "entire body." This explanation, or definition, of what constitutes a generation of neutrons is consistent with neutron generation as used in the definition of k-effective in Equation 47.2.
There is no generation of neutrons that consists of a partial group, such as prompt neutrons or delayed neutrons. There is no such thing as a "generation" of prompt neutrons. Likewise, there is no such thing a "generation" of delayed neutrons.