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Ch. VIII. Neutron Moderation and the Six Factors
VIII. Neutron Moderation and the Six Factors
Introduction
We continue our quest to calculate the multiplication factor (keff) and the neutron distribution (in
position and energy) in nuclear reactors. In the previous chapter we found that we could solve a
diffusion k-eigenvalue equation for the thermal flux in bare homogeneous reactors. We obtained
expressions for the multiplication factor and the thermal-flux spatial distribution (fundamental
mode) for several simple geometries. We found that:
,
(1)
but to get to this point we employed special definitions of some of the terms:
D means Dth, the thermal diffusion coefficient.
Σa means Σa,th – Σs,th→fast, the thermal absorption + upscattering cross section.
Σf means Σf,th, the thermal fission cross section.
ν means
Thus, Eq. (1) really says the following:
(2)
This is consistent with the six-factor formula as it was derived in Chapter III, because:
and
.
(3)
If we recall the definition of the fast-fission factor (ε), we can rewrite our expression for the
multiplication factor as follows:
keff = PFNL p
1
ν th Σ f ,th
1 + L2th Bg2 Σ a,th
(
)
+ PFNL 1 − p u F η F .
(4)
In this chapter we develop useful approximate expressions for the factors that are related to the
neutron slowing-down process:
PFNL (fast non-leakage probability),
p (resonance-escape probability), and
uF ηF (fast utilization times fast reproduction factor) .
In the next chapter we shall address the factors related to thermal neutrons (those that have
already slowed down).
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Ch. VIII. Neutron Moderation and the Six Factors
Some Physics
In thermal reactors, neutron energies range from almost zero to more than 107 eV (10 MeV). We
can divide this energy range into three segments, each characterized by different kinds of
neutron-nucleus interactions. (The boundaries between segments are fuzzy.)
Energy
Range
0 to ≈1 eV
⋅ Upscattering
Interaction ⋅ Molecular Effects
Physics:
⋅ Quantum-Mech.
Effects
(diffraction)
Interaction
Results:
≈1 eV to ≈105 eV
⋅ No Upscattering
⋅ Elastic Scattering from
free nuclei; isotropic in
CM frame (s-wave)
⋅ Resonance Absorption
(resolved resonances)
Neutron
Neutron
(slowing down)
≈105 eV to ≈107 eV
⋅ Fission Source
⋅ Elastic Scattering
from free nuclei;
anisotropic in CM
frame (p-wave)
⋅ Inelastic Scattering
⋅ Resonance
Absorption
(unresolved
resonances)
Fast Fission;
Moderation
In this chapter we will focus mostly on the middle (slowing-down) regime and somewhat un the
upper (highest-energy) regime.
Some Math
Recall the continuous-energy diffusion equation and consider the k-eigenvalue problem:
(5)
Given a homogeneous reactor (one in which the material properties are the same at every spatial
location), the D and the cross sections do not depend on position, and we have:
(6)
If we consider a bare reactor (one that is surrounded by vacuum), then the following
extrapolated boundary condition is perhaps reasonable:
φ(rs + den, E) = 0, rs on boundary.
(7)
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Ch. VIII. Neutron Moderation and the Six Factors
In reality, it would be best to let the extrapolation distance, d, be different for different neutron
energies. But to keep the math tractable we shall declare it independent of energy for purposes
of our explorations in this chapter.
Let us try separation of variables for the energy-dependent scalar flux:
(8)
If we insert this guess into the energy-dependent diffusion equation and divide through by the
product (Dfψ), we obtain
(9)
Observe that
the right-hand side
and
the left-hand side
Thus, each side must equal a constant. In keeping with previous chapters, we shall call this
constant “B2.” We therefore obtain two separate problems:
Spatial problem:
(10)
Energy problem:
(11)
We recognize that the problem given by Eqs. (10) is the “Helmholtz” eigenvalue problem that
we have solved previously. We know that
the
B2 eigenvalue is the one associated with the
and with the
NUEN 301 Course Notes , Marvin Adams, Fall 2009
Ch. VIII. Neutron Moderation and the Six Factors
133
This smallest B2 eigenvalue is the
of the reactor. Recall that
geometric buckling =
and depends very little on its material properties. (The dependence on material properties is only
through the extrapolation distance, d, in the boundary condition. We have noted that a good
value to use for this distance is 2D, where D is the diffusion coefficient of the reactor material.
But for a large reactor, the value used for the extrapolation distance has very little effect on the
multiplication factor or on the fundamental-mode distribution.)
If we put these pieces together, we see that the spatial part of fundamental-mode solution
satisfies:
.
(12)
The energy-dependent equation, Eq. (11), is exactly like an
k-eigenvalue equation except for the
term. This term plays the same role in the equation as
It accounts for outleakage from the reactor, and can be thought of as a
(Recall that we are considering bare reactors, which means that net outleakage through the
reactor surface is the same as outleakage, because inleakage is zero.)
Note that if f is a solution of Eq. (11), then so is any constant times f. That is, this equation says
nothing about f’s amplitude. It gives information only about f’s shape (which is the energy
distribution of the neutrons in the reactor). Thus, without losing any information, we can
normalize f(E) such that the following is true:
.
(13)
This allows us to rewrite our equation:
(14)
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Ch. VIII. Neutron Moderation and the Six Factors
Solving for the Energy Distribution
Hydrogen as the main neutron slower
Consider a water-moderated reactor. The vast majority of neutrons’ energy loss in such a reactor
comes from scattering off of
There are three main reasons for this:
1)
The scattering cross section for hydrogen is much larger than that for oxygen or for
fissionable nuclides.
2)
There is more hydrogen than anything else.
3)
The average energy loss per scattering collision is much greater for scattering from
hydrogen than from heavier nuclei. (On average, a fast neutron loses half of its
energy in a collision with hydrogen, but only around 10% in a collision with oxygen
and less than 1% in an elastic collision with uranium.)
We are fortunate that we can solve Eq. (14) analytically if we ignore the energy loss from
collisions with all nuclei except hydrogen. In this case, the differential scattering cross section
becomes that for hydrogen, which has an exceptionally simple form:
[for hydrogen, H-1] (15)
The analytic solution of Eq. (14), given the differential scattering cross section (15), is
, [H-1] (16)
where we have defined
= probability that a neutron born with energy E′
(17)
pnl(E,E') ≡
= probability that a neutron that scatters at energy E′
(18)
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Ch. VIII. Neutron Moderation and the Six Factors
pre(E,E') ≡
= probability that a neutron that scatters at energy E′
(19)
Here E0 is an energy above which there are essentially no neutrons in the reactor (something in
the range of 10 - 12 MeV would be a good number for this).
We can use this solution to generate approximate expressions for the factors that we seek. Recall
that these factors are:
PFNL (fast non-leakage probability),
p (resonance-escape probability), and
uF ηF (fast utilization times fast reproduction factor) .
Before we do this, though, we generalize the solution to allow for slowing down caused by
nuclides other than hydrogen.
Including scattering from nuclides with A>1
This case is too difficult to solve analytically, but it is not terribly difficult to create an accurate
approximate solution. This is:
,
(20)
where we have defined
= probability that a neutron born with energy E′
scatters before it is absorbed or leaks.
(21)
pnl(E,E') ≡
= probability that a neutron that scatters at energy E′
does not leak before it reaches energy E.
(22)
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Ch. VIII. Neutron Moderation and the Six Factors
pre(E,E') ≡
= probability that a neutron that scatters at energy E′
does not get absorbed before reaching energy E.
Note that the only difference in the solutions is the inclusion of the factor
of several terms. This factor is
(23)
in the denominator
≡ average “logarithmic energy decrement” per scattering event
=
(24)
= average “lethargy” gain per scattering event
The symbol here is the Greek letter “xi.” We call it “worm-bar.” (Some unsophisticated
nuclear engineers have been known to call it “squiggle.” We shall not stoop to such a level.)
If you compare to the previous solution, which said that hydrogen was the only nuclide that
could cause neutrons to lose energy, you would conclude that.
for pure H-1 must be
(25)
This is correct. Only hydrogen has such a large value of worm-bar. Worm-bar for some other
moderators is given in Table 1 below, along with some other interesting measures of moderator
performance.
Moderator
H2 0
D2 0
He
C (graphite)
U238
Table 1. Slowing-down parameters for common moderators.
Avg. number
A
of collisions
Σs
from 2 MeV
to 1 eV
1 and 16
0.920
1.35
2 and 16
0.509
0.176
4
0.425
––
12
0.158
0.06
238
0.008
0.003
Σs / Σa
Obtaining the Factors
Now we shall use what we have learned to create useful approximate expressions for the factors
that we need.
Fast Non-Leakage Probability
We are interested in the probability that a neutrons slows to “thermal” without leaking or getting
absorbed. We shall define an energy,
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Ch. VIII. Neutron Moderation and the Six Factors
to be the thermal “cutoff” energy. If a neutron slows to Eth, we say that it has become “thermal.”
From Eq. (22) we have:
pnl(Eth,E') ≡
= probability that a neutron that scatters at energy E′
does not leak before it reaches energy Eth.
(26)
We shall not prove the following, but it turns out that the integral in the exponent is related to a
“slowing-down distance”:
(27)
where
τ(E′→E)
≡ (1/6) of the average squared distance
from the point where a neutron
to the point where
We are often very interested in the distance from the neutron’s birthpoint (from a fission event)
to the point where it becomes thermal. We define:
τth
≡ (1/6) of the average squared distance
from the point where
to the point where
(28)
With this definition and the previous equations, we obtain our useful approximate formula for
the fast non-leakage probability:
PFNL ≈
[approximation for fast non-leakage prob.] (29)
This is the approximation you should use for the fast non-leakage probability in a large,
bare, homogeneous reactor!
If you use this expression (correctly) and it tells you that PFNL is far from unity (say, less than
0.8), then your reactor is probably not large enough for this to be a very accurate approximation.
For historical reasons (going back to Fermi and a very clever approximation that he made to do
the first nuclear reactor analysis),
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Ch. VIII. Neutron Moderation and the Six Factors
and
Remember that “Age” is proportional to a mean-squared slowing-down distance, and thus has
units of
“Age to thermal” characterizes distance from fast-neutron birth point to thermalization point, in
the same way that Lth2 characterizes distance from thermal-neutron birth point to absorption
point.
One more point about PFNL: If PFNL is close to 1, then its exponent is small, and
PFNL ≈
.
[if Bg2τth << 1] (30)
Resonance-Escape Probability
We found earlier that
pre(Eth,E0) ≡
(31)
This is the probability of a neutron’s escaping absorption all the way from birth at energy E0 to
slowing down to the “thermal” energy Eth. Because of the presence of E in the denominator of
the integrand and because of the relatively low value of the absorption cross section for highenergy neutrons, this probability is relatively insensitive to the value chosen for E0. We can
therefore pick a value for E0 and then use the resulting expression as the resonance-escape
probability for all neutrons born from fission.
We now take a closer look at the denominator of the integrand that appears in the exponent. If
we multiplied Σt by the scalar flux, we would obtain the collision-rate density. If we multiplied
DBg2 by the scalar flux, we would obtain the net outleakage rate density. It follows that
=
for a neutron of energy E. In a large reactor, this ratio will be
Thus, we shall ignore the leakage term in the denominator and use the approximation:
(32)
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Ch. VIII. Neutron Moderation and the Six Factors
p ≈
.
(33)
We recall now that in the slowing-down energy range, the dominant interaction between
neutrons and light nuclei (which includes nuclei in all moderators) is
This is elastic scattering off of the “potential” of the nucleus, which means the neutron does not
actually penetrate the nucleus. The potential scattering cross section is essentially independent
of the energy of the neutron, and we give it the symbol Σ p.
We multiply and divide by this cross section, recognizing that it is independent of energy, to
obtain:
p ≈
.
(34)
We define the resonance integral for absorption for nuclide i in some given mixture of
nuclides as follows:
.
(35)
(There is a corresponding resonance integral for fission.) In the general case of an arbitrary
mixture of fuel and moderator nuclides, this integral would need to be evaluated on a case-bycase basis, because it would depend upon the details of the mixture. However, there is a limiting
case that simplifies things considerably:
in which there are lots, lots more moderator nuclei than fuel (or other absorbing) nuclei. In this
limit, note that
Σt(E′′) =
(36)
In this case, the resonance integral becomes independent of the details of the mixture, and we
have:
=
(37)
Note the “infinity” superscript that denotes the “infinitely dilute” limit.
Thus, in the dilute limit (lots more moderating atoms than absorbing atoms), the resonance
integral can be replaced by the infinitely-dilute resonance integral, and our expression for the
resonance-escape probability becomes:
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Ch. VIII. Neutron Moderation and the Six Factors
p ≈
[dilute limit](38)
If there are lots more moderator atoms than absorber atoms, this is the approximation to
use for resonance-escape probability.
If you use this expression (correctly) and it tells you that p is very small (say, less than 0.5), then
your mixture is not dilute enough for it to be very accurate.
Fast Utilization times Fast Reproduction Factor
The product of these two factors is
the number of fission neutrons emitted
That is,
uFηF =
.
(39)
Most of the absorptions and fissions will take place at energies at which most of the neutrons
present are neutrons that have scattered, not neutrons that are freshly born from fission. (Most
fission neutrons are born with energies above 100 keV; most absorption takes place below this
energy.) For these energies, the direct contribution of c(E) to f(E) is small, and our approximate
solution for f(E) can be further approximated as:
,
(40)
where A is a constant. (Here we have used the fact that D(E)Bg2 << Σt(E), as discussed above.)
We insert this into Eq. (39) to obtain:
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Ch. VIII. Neutron Moderation and the Six Factors
uFηF ≈
.
(41)
We now assume that:
1)
prepnl can be replaced in each integral with an average value, and
2)
this average value is the same for both integrals.
We also multiply numerator and denominator by the potential scattering cross section of the
moderator. We obtain:
uFηF ≈
≈
(42)
Note that the resonance integrals have appeared again!
Recall that the only resonance integrals we have readily available to us are the
resonance integrals, in which the ratio of Σp for the moderator to Σt for the mixture is taken to be
unity. Recall that the resonance integrals approach this limit when the ratio of fuel atoms to
moderator atoms is very small.
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Ch. VIII. Neutron Moderation and the Six Factors
Summary
In this chapter we have developed several useful formulas that will help us estimate the
multiplication factor for a bare homogeneous reactor. They are summarized here.
PFNL ≈
.
(43)
p ≈
uFηF ≈
.
.
(44)
(45)