Non-fission neutrons are a significant part of the core neutron population only during reactor startup prior to attaining criticality, and during the latter stage of reactor shutdown when power has decayed into the lower range of the nuclear instruments, i.e. into the lower Intermediate Range or into the Source Range. In effect, this is the power range below the minimum power at which criticality can be established. Herein, this power range is referred to as the Sub-Critical region. Note that it is possible for a reactor to be subcritical without being in the Sub-Critical Region.
It is in the Sub-Critical region that the analytic treatment of subcritical source multiplication applies, with the most notable steady-state condition being known as "equilibrium subcritical multiplication". In the Sub-Critical region, non-fission source neutrons are multiplied by the production of fission neutrons. And, although most texts do address subcritical source multiplication, few, if any, answer a simple question that occurs to many students, namely "Does the non-fission source have a reactivity worth?". Or stated somewhat differently, "Does the presence of a non-fission neutron source introduce a change in reactivity of the core, i.e. a delta-rho?". Now, although many may consider the answer to be obvious, it is useful to seek out the reason that the question arises so frequently.
To do so, three aspects of the issue will be discussed: the mathematical formulation for source multiplication, the definition of k-effective, and actual reactor behavior.
MATHEMATICAL FORMULATION
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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48.1 |
Thus, the subcritical neutron population, N, is a function of two parameters, the non-fission source strength, S, and k-effective.
With simple mathematical manipulation, this transforms to:
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48.2 |
In fact, Equation 48.2 is a misleading formulation. It is Equation 48.1 that represents the physical process of source multiplication. In the physical process, the neutron population, N, is determined by two unrelated factors, the non-fission source strength and the k-effective value. Mathematically, N is said to be a dependent variable, while S and k-effective are said to be independent variables. N, is strictly and entirely determined by the value of the non-fission source strength, S, and on the reactors nuclear status, as defined by k-effective. The source strength is defined independently of any other parameter. And, k-effective is independent of any other parameter (more on this below).
In Equation 48.2, the independent variable cannot be changed without making a corresponding change to the dependent variable. If S is doubled, (k - 1) does not double because N in the denominator must also be doubled. Therefore, it cannot be concluded from this equation that (k - 1) is a function of the non-fission source strength. The lesson here is that manipulation of equations must be done with caution because the results can sometimes be interpreted incorrectly.
But continuing, if both S and N on the right-hand-side of Equation 48.2 are doubled because of dependency, delta-k maintains a constant value. If delta-k is constant regardless of the strength of the non-fission source, then k-effective cannot be affected by non-fission source strength. A neutron source cannot introduce reactivity change due to the emission of non-fission neutrons.
K-effective DEFINITION
K-effective is a measure of the ability of a reactor to regenerate neutrons by the fission process. It is a potential that exists whether neutrons are present or not. K-effective, as defined by the six-factor formula, is based on the life cycle model that ignores non-fission neutrons completely. Investigation of each of the six factors one by one, from top to bottom, inside and out will reveal no factor that incorporates non-fission neutrons into the life cycle, or into the value of k-effective. The fact that the neutron source emits neutrons is not relevant to the value of k-effective because the neutrons emitted are non-fission neutrons and not fission neutrons. Non-fission neutrons are not part of fission neutron regeneration in the life cycle.
Each of the six factors is defined by specific properties related to reactor composition and physical size, including cross sections, other nuclear properties, neutron diffusion lengths, and reactor size and shape. These are all physical properties of the particular environment and are independent of any entity emitting neutrons by some process other than fission. If the artificial neutron source itself does not displace appreciable moderator, and does not have a significant macroscopic absorption or scattering cross section, then k-effective will not be altered by the physical presence of the artificial neutron source. For k-effective to exist in a particular environment requires the presence of fissile atoms, but not the presence of neutrons.
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48.3 |
| THE SIX FACTOR FORMULA |
A second important result from the life cycle model is the definition of k-effective in terms of the fission neutron populations in successive life cycles:
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48.4 |
This representation of k-effective does not mean that neutrons need be present for a k-effective to exist. The right-hand-side of Equation 48.4 is the effect of k-effective, which here too is an independent variable. But by similar false reasoning as occurs with Equation 48.2, it can be concluded that value of k-effective is determined by the number of fission neutrons in the a particular generation. Not true!
ACTUAL REACTOR BEHAVIOR
Observation of power decay at a stable negative rate after reactor scram provides obvious evidence that the reactor is subcritical. However, as the power decays into the lower regions of the Intermediate Range, or into the Source Range, decay begins to decelerate and power finally settles to a constant value. This behavior was discussed in NUKEFACT #33 - The Anatomy of a Reactor Scram as item #5 and was illustrated in Figure 33.1 of that essay.
This behavior resembles what might be expected if a very slow positive reactivity rate were introduced to reestablish criticality. However, this is not the only possibility. When the neutron population reaches very low levels, the weak source of extraneous non-fission neutrons can make up for losses in the fission process and create steady-state power. No doubt, this too raises question as to whether the non-fission neutrons must not somehow equate to reactivity change. This creates a condition called equilibrium subcritical multiplication. The neutron population is constant with time. It appears just as if at criticality. But, it is not criticality. The reactor is still subcritical at whatever the initial subcritical k-effective. The non-fission neutrons did not raise k-effective to 1.0000!
If any doubt exists as to whether the non-fission source neutrons somehow introduces positive reactivity change to attain criticality, then the neutron response to an incremental withdrawal of control rods from an equilibrium subcritical multiplication condition proves otherwise. A positive delta-rho introduction does not produce a stable rate. Equilibrium subcritical multiplication does not equate to criticality.
As a analogy, a balloon has lost its buoyancy and is slowly descending. The law of gravity is working. Then, someone places a fan under the balloon and its descent slows. Does this mean that the fan has introduced a change in the value of the gravitational constant? Obviously not! An entirely separate force is influencing the behavior of the balloon. And, so it is with a non-fission source in a reactor with a given k-effective.
Since the advent of the scientific method, all fields of science have employed experiments to verify theory. Reactor experiments have shown, consistently, that if the source strength is doubled, the total neutron population doubles. By Equation 48.1, if both N and S doubled, then k-effective must remain constant. The only possible conclusion is that S has no influence on k-effective.
SUMMARY
A neutron source does not possess reactivity worth because of the emission of non-fission neutrons.