143
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
IX. Low-Energy (Thermal) Neutrons
Introduction
The vast majority of today’s reactors are thermal reactors. In a thermal reactor, most fissions are
caused by “thermal” neutrons –– neutrons that are approximately in thermal equilibrium with the
reactor materials. In this chapter we study the physics of low-energy neutron interactions, the
energy distribution (“spectrum”) of neutrons in the thermal range, and thermal reaction rates.
Infinite Medium, No Sources or Absorption –– the Maxwellian
Consider an infinite medium in which nothing depends on position. In such a medium the exact
balance equation, which by now we know very well, is:
[∞-medium balance] (1)
Now consider the case in which there is no source and no absorption:
( ) ( ) ∫0 dE ' Σ s ( E ' → E )φ ( E ') .
Σs E φ E =
∞
[∞-medium balance, no source or abs.] (2)
Note that our neutrons will behave just like one (dilute) component in a mixture of gases –– they
will just bounce around forever. What distribution will they attain? The answer is the
distribution, which is
.
[ Maxwellian for energy-dependent flux] (3)
Here
ntot = thermal-neutron density [n/cm3] ,
m
= neutron mass ,
k
= Boltzmann’s constant
≈
T
≈
= temperature (absolute) .
The rather amazing answer, Eq. (3), is completely
This result comes from statistical mechanics, and its proof is beyond the scope of our study. (In
a nutshell: this is the maximum-entropy distribution, to which the particles must eventually
relax.)
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
144
Do not forget the result (3), which we now restate:
In the absence of ________________________
neutrons attain the _____________________________
which is given by Eq. (3).
Here we plot the Maxwellian for two different temperatures:
You can observe in the figure something that is easy to prove: The peak of the Maxwellian
scalar-flux distribution is at the following energy:
[You can prove this by setting dφM/dE = 0 and solving for E.] That is, the “most probable”
energy is simply kT. Recall that at room temperature, kT is approximately
A neutron with this energy has approximately the following speed:
When we wish to compute thermal-neutron reaction rates, we will of course need thermalneutron cross sections. These cross sections depend on the relative speed between the neutron
and nucleus, of course. It turns out (as we shall see) that in many cases all we really need to
know is the cross section evaluated at only one relative speed. In the beginning days of nuclear
engineering, the speed 2200 m/s was chosen to be the relative speed at which to tabulate cross
sections. The chart of the nuclides provides 2200-m/s cross sections, and most books about
reactor theory contain appendices that tabulate such cross sections for nuclides that are
commonly used in reactors.
145
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
More Statistical Mechanics: Principle of Detailed Balance
Recall that our exact equation (1) simply states particle balance:
rate at which particles
scatter out of dE
rate at which particles
scatter into dE .
=
Statistical mechanics gives us another interesting tidbit: given no source and no absorption, then
particle balance holds in much more detail:
rate at which particles
scatter out of dE
rate at which particles
scatter into dE
=
This is true for all dE and dE′. Mathematically, this means
,
or
.
(4)
This is called the
It is not at all obvious, but it is a law of nature. Note that it rules out a balance maintained by
cyclic process such as:
Note that it also places quite a restriction on the form of the scattering kernel –– nature’s
scattering kernels all have this relation in common! That is, every differential scattering cross
section satisfies Eq. (4)! The origin of this is the microscopic reversibility of each collision.
That is, the physics of an individual scattering event works the same way forward and backward.
Infinite Medium, General
Consider now an infinite medium in which there is a small amount of absorption and a small
source. [Here “small” means relative to scattering.] We expect the neutron energy distribution
to be “close” to Maxwellian, and we ask how it will differ.
Effect of Absorption
Consider first the effect of absorption. To understand this effect, we must know something about
absorption cross sections. We know that cross sections depend on vr, the
between the neutron and nucleus. It turns out that for small vr, the absorption cross section of
many materials is proportional to 1/vr:
146
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
σa(vr) =
(5)
Here v0 is just some relative velocity at which the cross section is known. [As we mentioned
above, the v0 at which cross sections are tabulated is 2200 m/s. There is nothing magical about
this value – people had to pick something, and the most-probable speed at room temperature is as
convenient as anything else.]
Because of this 1/vr dependence, we see that absorption is going to preferentially deplete the
low-energy part of the neutron distribution. As a result, the spectrum is
shifted to higher energies,
or made
“harder.”
We often refer to such a shift as
or “absorption heating.” If the effect is not too large, we can model the shifted spectrum
reasonably well by using a Maxwellian at an increased temperature. This is called the
Below is Figure 8-4 of Lamarsh’s Introduction to Nuclear Reactor Theory. This figure gives an
idea of how much larger Tn is than the material temperature (which is called Tw in the figure
because the background was water). You will see that Tn can be larger by quite a bit –– tens of
percent – if significant absorption is present.
Figure: Fractional change in the “neutron temperature” relative to the background temperature
as a function of the absorption cross section at 2200 m/s. The absorber is assumed to be 1/v.
The funny units of barns per hydrogen atom actually make sense if you think about it.
The bottom line is that absorption can significantly shift the thermal spectrum.
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
147
Effect of Sources
Consider next the effect of a source. The “source” in the thermal energy range in a reactor is due
to neutrons scattering down from higher energies. The source tends to fill in the low spots in the
Maxwellian distribution, especially the high-energy “tail.”
The result of one particular calculation, given absorption and a downscattering source from
higher energies, is shown in Figure 8-3 of Lamarsh’s Introduction to Nuclear Reactor Theory
and is reproduced below. Note that below ≈ 0.1eV, the spectrum looks much like a shifted
Maxwellian. Above 0.1eV the spectrum deviates significantly from a Maxwellian shape and
blends into a 1/E shape, as we know it should. (Is this cool or what?)
Figure: Energy-dependent scalar flux for a mixture of H2O and a 1/v absorber (5.2 b/H-atom) at
23 °C. Also shown is the Maxwellian flux at the same temperature.
Incidentally, calculations like the one that produced this figure require knowledge of the
scattering kernel Σs(E′→E) and the absorption cross section Σa(E). In general, the scattering
kernel can be a very complicated function, for it must take into account the fact that atoms are
bound to one another in molecules and crystals. It depends on more than just the nuclides
present and their number densities:
1)
because molecular binding energies may not be negligible, each nucleus does not
interact alone –– there are molecular effects.
2)
because the neutron wavelength is large, there can be wave effects such as
diffraction.
With today’s computers, thermal spectra in infinite media are very easy to compute (using the
multigroup method, for example), given all the needed cross-section data. But be aware that in
general, this data may be difficult to obtain!
Finite Medium, General
In a finite medium, we must account for
leakage effects.
148
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
In commercial reactors, thermal leakage is negligible, so this is not an important effect. In some
small research reactors, it can be important.
What is the effect of leakage on the thermal spectrum? Intuitively, we might imagine that
are more likely to leak than are
slower ones.
This is, in fact, the case, as we can show by a variety of arguments. (See for example pp. 249250 of Lamarsh’s Introduction to Nuclear Reactor Theory.)
The net effect of this is just the opposite of the effect of absorption. (Remember, absorption
preferentially depletes the low end of the spectrum.) As a result of leakage, the spectrum is
shifted to lower energies,
or made
“softer.”
We often refer to such a shift as
We note again that this is not a large effect in most reactors. The effect of absorption is usually
much more important.
Absorption & Fission Rates for Thermal Neutrons
Warning: In this section we shall ignore molecular effects and crystal effects, pretending that
each neutron interacts with only one nucleus at a time. This is not strictly true for extremely
low-energy neutrons, but if we consider only absorption reactions (which include fission), it is
very close to the truth for essentially all of the neutrons in a reactor.
Consider a distribution of neutrons in an infinite medium, and consider how many of them
have velocities in the “velocity box” dvxdvydvz around the velocity v ≡ vxex + vyey + vzez:
n/cm3 in “box” = n(v)dvxdvydvz ≡ n(v)d3v .
(6)
Here
n(v) = velocity-dependent neutron number density [n/(cm3-(cm/s)3)].
Consider also a distribution of nuclei that have velocities in the “velocity BOX” dVxdVydVz
about the velocity V ≡ Vxex + Vyey + Vzez:
nuclei/cm3 in “BOX” = N(V)dVxdVydVz ≡ N(V)d3V .
Here
N(V) = velocity-dependent nuclei density [nuclei/(cm3-(cm/s)3)].
(7)
149
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
The rate at which the nuclei in “BOX” absorb the neutrons in “box” is just
Abs. rate/cm3
of neutrons in d3v
by nuclei in d3V
=
(8)
where
vr
=
≡
relative speed between neutron and nucleus
[ (vx–Vx)2 + (vy–Vy)2 +(vz–Vz)2 |1/2 =
(9)
The total absorption rate density is therefore
Abs. rate density =
(10)(a)
where vr depends on both v and V, as seen in Eq. (9), and thus must be inside all six integrals.
This is a very general result; our only assumption is that neutrons interact with one nucleus at a
time.
The fission rate is similar:
Fission rate/cm3 =
.
(10)(b)
Remark/Reminder
We have often used the expression
x-reaction rate density at r
=
,
(11)
where “x” could be scattering, fission, absorption, or any other kind of reaction, and where E is
the neutron kinetic energy in the lab frame. This expression seems to imply that the cross section
depends only on the neutron’s lab-frame speed. But we know that in reality, the cross section for
a given reaction depends fundamentally on the
relative speed, vr,
between the neutron and the nucleus, not on the lab speed of the neutron. Thus, when we write
something like Eq. (11), we are tacitly assuming that Σx(r,E) has been appropriately averaged
over the velocity distribution of the nuclei. See the section “Temperature Dependence” in
Chapter II of these notes for more discussion.
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
150
It is this averaging, by the way, that produces Doppler-broadened cross sections. We discussed
the effects of Doppler broadening in Chapters II and VII of these notes.
1/v Absorbers
Now consider the special (but common) case of a “1/v absorber.” In this case σa is:
σa(vr) =
.
(12)
If we insert this into our general expression for absorption rate we find that all dependence on vr
vanishes and we are left with a very simple result:
Abs. rate/cm3
=
=
=
=
(13)
where E0 is the neutron energy corresponding to the speed v0, ntot is the neutron density [n/cm3],
and Ntot is the nucleus density [nuclei/cm3].
Equation (13) is an important and remarkable result:
Given a 1/v cross section, the reaction rate is
of the velocity distribution of the neutrons, and
of the velocity distribution of the nuclei!!!
Given a 1/v absorber, we can get the thermal absorption rate if we know only a few simple
things:
1)
the neutron density ntot [n/cm3],
2)
the nucleus density Ntot [nuclei/cm3],
3)
the cross section at some specific relative speed v0.
It is common practice to tabulate cross sections at this specific speed:
v0 =
(14)
It is also conventional to define
φ0 ≡ “2200 m/sec flux” ≡
(15)
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
Important note:
φ0 = “2200 m/s flux” ≠ φ(E0) and ≠ φTh !!!!
151
(16)
Observe:
φ0 = ntotv0 =
φ(E0) =
Also note that φ0 is not the “thermal flux”, and it does not have the physical meaning of pathlength rate per unit volume. It is simply a convenient product of two numbers!
Example
A few cm away from some H2O-pool-type reactor, the density of thermal neutrons is found to be
105 per cm3. At that point what is the thermal absorption rate density?
Solution:
Since hydrogen and oxygen are 1/v absorbers, all we need to know is:
•
Σa evaluated at some known relative speed,
•
the neutron density.
The neutron density is given. Cross sections evaluated at 2200 m/sec are tabulated in an
appendix of your textbook, and also on the chart of the nuclides. From there we find
ΣaH20(2200 m/s) = 0.0222 cm–1 .
(This assumes a water density of approximately 1 gram per cc.) Thus,
Thermal abs. rate density
=
=
=
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
152
Cross Sections of Interesting Nuclides
Following are some cross section plots that highlight 1/v and non-1/v behavior for a few nuclides
of interest to nuclear engineers. A good place to put these is right after this page in your notes.
A few comments, plot by plot:
H-1. Like all light nuclei (if they absorb at all), hydrogen is a 1/v absorber. Note that for E>1eV
the absorption cross section is ~1000 times smaller than scattering, and it continues to drop like
1/v and become even more negligible.
H-2. For neutrons with energy less than 10 MeV, the only interaction with deuterium is elastic
scattering. There’s no capture, because deuterium has all the neutrons (1) it wants. Above 10
MeV, the n,2n reaction becomes noticeable – the incident neutron basically knocks the other
neutron out, leaving a proton (hydrogen). Note that the scattering cross section is much smaller
than that of hydrogen.
He-4. The only interaction that neutrons (with reactor-relevant energies) have with He-4 is
elastic scattering. He-4 is quite happy with its 2 protons and 2 neutrons, and no one else is
invited to its tightly-bound party. Note that the scattering cross section is quite small.
He-3. Another 1/v absorber, with a pretty high cross section. For E<10000eV, the dominant
interaction is (n + He3) → (p + H3), which is production of hydrogen and tritium. For higher
energies, elastic scattering becomes dominant.
B-10. Another very strong 1/v absorber. For E<10000eV, the dominant interaction is (n + B10)
→ (alpha + Li7), which is production of helium-4 and lithium-7. For higher energies, elastic
scattering becomes dominant. On the second plot you can clearly see that an energy change of a
factor of 1E4 produces a cross section change of a factor of 1E2: perfect 1/v behavior.
Cd. Cadmium’s cross section jumps by a factor of 1000 from E>1eV to E<1eV. This huge lowlying resonance makes Cd a non-1/v absorber (although you can see that even this becomes 1/v
at very low energies). This huge jump makes Cd a very useful material for screening out thermal
neutrons. Cd covers are often placed on foils for this purpose in neutron-absorption experiments.
In-113. Indium-113 has a big fat resonance at 1.45 eV, which makes it good for absorbing
neutrons that are almost thermal. If you cover an In foil with Cd, the In will not see thermal
neutrons and will absorb mostly neutrons of energies close to 1.45 eV.
Xe-135. The most striking feature of Xenon-135 is how enormous its cross section is for slow
neutrons – more than a million barns! Xe-135 is a fission product, and it is also the decay
product of another fission product (Iodine-135). In fact, >6% of fissions ultimately produce this
neutron-hungry nuclide. We must design our reactors to have enough “excess reactivity” to stay
critical even after Xe-135 builds up.
U-235. This is the workhorse nuclide for fission reactors today. Note the 1/v behavior at low
energy, the resonances at intermediate energy, and the drop-off of the capture cross section at
high energy. On the second plot you can see that the 2200-m/s total cross section is a bit below
700b and that the capture cross section is only ~100b – fission is by far the most likely
interaction for a thermal neutron with U-235.
U-238. Compare against U-235. The U-238 cross section is much lower for thermal neutrons,
and fission does not become important until E > ~1E6.
Pu-239. Looks a lot like U-235! Main differences: its low-lying resonance is much stronger
and its cross section is significantly higher for thermal neutrons. Its capture/fission ratio is also
higher. It is easy to see that this could be an excellent substitute for U-235 as a fissile fuel.
Pu-240. Check out the size of that low-lying resonance! Note that fission is almost a non-event
until E gets above 100 keV. Note also that Pu-240 is always produced if Pu-239 is present.
Pu-238. Note that for E<1 keV, Pu-238 mainly captures neutrons and thus produces Pu-239.
10
2
Hydrogen-1 total, elastic, and capture
ENDF-VI (n,total) xsec
ENDF-VI (n,elastic) xsec
ENDF-VI (n,gamma) xsec
Cross Section (barns)
101
100
10-1
10-2
10-5
10-4
10-3
10-2
10-1
100
101
102
Energy (eV)
103
104
105
106
107
108
Deuterium total, elastic, (n, n+n+p): ENDF pointwise
103
Cross Section (barns)
(n,total) xsec
(n,elastic) xsec
(n,2n) xsec
10
2
101
100
10-1
10-5 10-4 10-3 10-2 10-1
100
101
102
Energy (eV)
103
104
105
106
107
He-4 total and elastic, ENDF pointwise
102
Cross Section (barns)
(n,total) xsec
(n,elastic) xsec
10
1
100
10-1
10-5 10-4 10-3 10-2 10-1
100
101
102
Energy (eV)
103
104
105
106
107
He-3 total, elastic, (n, proton+triton); ENDF pointwise
106
Cross Section (barns)
105
10
(n,total) xsec
(n,elastic) xsec
(n,p) xsec
4
103
102
101
100
10-1
10-5 10-4 10-3 10-2 10-1
100
101
102
Energy (eV)
103
104
105
106
107
Boron-10 total, (n,alpha), and elastic; ENDF pointwise
106
5
Cross Section (barns)
10
(n,total) xsec
(n,a) xsec
(n,elastic) xsec
4
10
103
102
101
100
10-5 10-4 10-3 10-2 10-1
100
101
102
Energy (eV)
103
104
105
106
107
Boron-10 total, (n,alpha), and elastic; ENDF pointwise
105
Cross Section (barns)
(n,total) xsec
(n,a) xsec
(n,elastic) xsec
4
10
103
102
10-4
10-3
10-2
Energy (eV)
10-1
100
Natural Cadmium
105
4
Cross Section (barns)
10
(n,total) xsec
(n,gamma) xsec
103
102
1
10
100
10-5 10-4 10-3 10-2 10-1
100
101
102
Energy (eV)
103
104
105
106
107
Natural Cadmium
104
Cross Section (barns)
(n,total) xsec
(n,gamma) xsec
3
10
102
101
100
10-1
100
101
Energy (eV)
102
103
In-115 total and capture, JENDL-3.2-pointwise
105
(n,total) xsec
(n,gamma) xsec
4
Cross Section (barns)
10
103
102
1
10
100
10-5 10-4 10-3 10-2 10-1
100
101
102
Energy (eV)
103
104
105
106
107
In-115 total and capture, JENDL-3.2-pointwise
105
(n,total) xsec
(n,gamma) xsec
4
Cross Section (barns)
10
103
102
1
10
100
10-1
100
101
Energy (eV)
102
103
Xe-135, JENDL-3.2-pointwise
108
7
Cross Section (barns)
10
(n,total) xsec
(n,gamma) xsec
6
10
5
10
104
103
102
1
10
100
10-5 10-4 10-3 10-2 10-1
100
101
102
Energy (eV)
103
104
105
106
107
U-235 total, fission, and capture; Pointwise ENDFB-V
105
Cross Section (barns)
104
10
(n,total) xsec
(n,fiss) xsec
(n,gamma) xsec
3
102
101
100
10-1
10-2
10-5 10-4 10-3 10-2 10-1
100
101
102
Energy (eV)
103
104
105
106
107
U-235 total, fission, and capture; Pointwise ENDFB-V
104
Cross Section (barns)
(n,total) xsec
(n,fiss) xsec
(n,gamma) xsec
3
10
102
101
100
10-4
10-3
10-2
10-1
Energy (eV)
100
101
U-238 total, fission, and capture; Pointwise ENDFB-V
Cross Section (barns)
104
10
3
10
2
(n,total) xsec
(n,fiss) xsec
(n,gamma) xsec
101
100
10-1
10-2
10-5 10-4
10-3
10-2
10-1
100
101
102
Energy (eV)
103
104
105
106
107
U-238 total, fission, and capture; Pointwise ENDFB-V
104
Cross Section (barns)
10
(n,total) xsec
(n,fiss) xsec
(n,gamma) xsec
3
102
101
10
0
10-1
10-4
10-3
10-2
10-1
Energy (eV)
100
101
Pu-239 total, fission, and capture; Pointwise ENDFB-V
105
(n,total) xsec
(n,fiss) xsec
(n,gamma) xsec
Cross Section (barns)
104
10
3
102
101
100
10-1
10-2
10-5 10-4
10-3
10-2
10-1
100
101
102
Energy (eV)
103
104
105
106
107
Pu-239 total, fission, and capture; Pointwise ENDFB-V
104
Cross Section (barns)
(n,total) xsec
(n,fiss) xsec
(n,gamma) xsec
3
10
102
101
100
10-4
10-3
10-2
10-1
Energy (eV)
100
101
Pu-240 total, fission, and capture; Pointwise ENDFB-V
Cross Section (barns)
106
10
5
10
4
10
3
(n,total) xsec
(n,fiss) xsec
(n,gamma) xsec
102
101
100
10
-1
10-2
10-5 10-4
10-3
10-2
10-1
100
101
102
Energy (eV)
103
104
105
106
107
Pu-238 total, fission, and capture; Pointwise ENDF
105
(n,total) xsec
(n,fiss) xsec
(n,gamma) xsec
Cross Section (barns)
104
10
3
102
101
100
10-1
10-2
10-5 10-4
10-3
10-2
10-1
100
101
102
Energy (eV)
103
104
105
106
107
172
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
Non–1/v Absorbers
A survey of cross sections shows that
However, some intermediate and heavy nuclei are not. We have an incredibly simple expression
for absorption and fission rate densities for nuclei that are 1/v absorbers – nature is very kind to
nuclear engineers in this case – but what do we do about the others?
The first thing to note is that even intermediate and heavy nuclei have absorption cross sections
that are 1/v or nearly 1/v over much of the thermal range. This suggests that an approximate
correction to our simple 1/v expression for absorption or fission rate density might be all we
need.
Such a correction, called a
was in fact pursued in the early days of nuclear engineering. In 1962 C. H. Westcott published a
set of non-1/v factors under the following assumptions:
-
neutrons have Maxwellian distribution at the “neutron temperature” Tn,
-
nucleus motion in lab frame is negligible.
While both of these seem a bit severe, especially the second one, it is important to remember
that:
-
the cross sections are almost 1/v;
-
when cross sections are 1/v the velocity distributions of the neutrons and nuclei
don’t matter at all;
-
the correction we need is not very big.
Also note that it is particle energies, not speeds, that are comparable given thermal equilibrium.
Heavier thermal particles are therefore slower, on average, than lighter ones. A neutron with the
same energy as a uranium nucleus is more than 15 times as fast.
So Westcott proceeded as follows. The assumptions stated above cause the expression for
absorption rate density to simplify a lot:
Abs. rate/cm3
=
[Eth = limit of thermal range] (17)
For each nuclide of interest, Westcott performed this integral (numerically) using real data for σa
and compared it against the result from a 1/v σa. The ratio is the “non-1/v factor”:
=
(18)
173
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
This is the “simple correction” factor we were after. If we know this correction factor for a
given nuclide then we can calculate absorption rate densities in non-1/v absorbers almost as
easily as in 1/v absorbers:
Abs. rate dens.
=
(19)
There is a similar expression for fission:
Fisn. rate dens.
=
(20)
Note that the non-1/v factors for absorption and fission are different in general!
Table: Non-1/v Factors
T, °C
Cd
In
Xe135
Sm149
ga
ga
ga
ga
U233
ga
U235
gf
ga
U238
gf
ga
Pu239
ga
gf
20
1.3203
1.0192
1.1581
1.6170 0.9983 1.0003 0.9780 0.9759 1.0017 1.0723 1.0487
100
1.5990
1.0350
1.2103
1.8874 0.9972 1.0011 0.9610 0.9581 1.0031 1.1611 1.1150
200
1.9631
1.0558
1.2360
2.0903 0.9973 1.0025 0.9457 0.9411 1.0049 1.3388 1.2528
400
2.5589
1.1011
1.1864
2.1854 1.0010 1.0068 0.9294 0.9208 1.0085 1.8905 1.6904
600
2.9031
1.1522
1.0914
2.0852 1.0072 1.0128 0.9229 0.9108 1.0122 2.5321 2.2037
800
3.0455
1.2123
0.9887
1.9246 1.0146 1.0201 0.9182 0.9036 1.0159 3.1006 2.6595
1000
3.0599
1.2915
0.8858
1.7568 1.0226 1.0284 0.9118 0.8956 1.0198 3.5353 3.0079
Table based on C. H. Westcott, “Effective Cross Section Values for Well-Moderated Thermal Reactor Spectra,”
AECL-1101, January 1962. Xe135 data based on E. C. Smith, et al., Phys. Rev. 115, 1693 (1959).
Thermal Cross Sections and Diffusion Coefficients
The previous section addresses how to calculate thermal absorption and fission rates. However,
this section does not address how to obtain some key terms that appear in many results from
previous chapters:
We begin by recalling that in previous chapters, the thermal absorption rate density has been
expressed as
Abs. rate dens. =
(21)
We also know that
φth =
(22)
174
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
and
Abs. rate dens. =
.
(23)
For all these to be true, we must have
Σa,th =
.
(24)
We would really like to know how this thermal absorption cross section compares to the 2200m/s absorption cross section. We can get a good idea of this by invoking the same assumptions
that Westcott used to generate non-1/v factors: neutrons in a Maxwellian and nuclei with
negligible speeds. In this case we can evaluate the integrals in Eq. (24), and we find:
[if Maxwellian flux] (25)
Here T0 is the temperature such that kT0 = E0. (For E0 = 0.0253, T0 = 293.15 K.) Similarly,
[if Maxwellian flux] (26)
Given a Maxwellian flux, we can easily show that:
φTh
=
=
=
.
[if Maxwellian at Tn] (27)
Thus, assuming φ(E) is Maxwellian, we have
Abs. rate/cm3
=
φTh
=
=
(28)
= Σf(E0)φ0 gf(Tn) .
(29)
Similarly,
Fisn rate/cm3
175
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
Example
In an imaginary research reactor there is a homogeneous mixture of 235U and H2O, with 1
atom of 235U per 1000 molecules of H2O. The reactor is operating at 100 °C.
At some point in this reactor, the density of thermal neutrons is 106 per cm3. At that point
what is the thermal absorption rate density? What is the thermal fission rate density?
Solution:
Abs. rate density
= ΣaH2O(v0)ntotv0 + Σa235(v0)ntotv0 ga235(Tn)
We obtain our cross sections from an appendix in a textbook (or other sources), and we obtain
our non-1/v factors from the table a few pages back:
ΣaH2O
= 0.0222 /cm (assuming water density of 1 g/cm3, which is probably a bit high)
σa235
= 680 b
σf235
= 580 b
ga235
= 0.961 (assuming that Tn ≈ Twater)
gf235
= 0.958 (“)
We need the number density of the
molecules:
235U.
It is 1/1000 times the number density of the water
N235 ≈ 0.001 [ 1 g/cm3][1mole/18g][0.6 × 1024 /mole] = 3.33 ×x 1019 / cm3 .
⇒ Σa235 =
and
Σf235 =
3.33 x 1019 × 680 x 10–24 /cm ≈ 0.0227 /cm
3.3 x 1019 × 580 x 10–24 /cm ≈ 0.0193 /cm
Thus,
Abs. rate density
= [(0.0222) + (0.0227)(0.961)](cm–1)(106 cm–3)(220000 cm/s)
= 9.64×109 per cm3 per sec .
Similarly,
Fisn. rate density
= (0.0193)(0.958)(cm–1)(106 cm–3)(220000 cm/s)
= 4.07×109 per cm3 per sec .
176
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
Summary
In this chapter we studied thermal neutrons –– neutrons whose energies are comparable to the
energies of the background nuclei. We first studied the neutrons’ energy distribution. We
found:
1)
In an ∞ medium with Σa=S=0, φ(E) = Maxwellian at T, where T = temperature of
medium.
2)
With small absorption, φ(E) ≈ Maxwellian at “neutron temperature” Tn>T.
3)
In finite medium, “diffusion cooling” can slightly lower Tn. (Small effect in large
reactors).
4)
In finite media with large absorption, φ(E) in the thermal range can sometimes
deviate noticeably from a Maxwellian, but qualitatively it always looks something
like a Maxwellian.
We also studied thermal interaction rates. We found that scattering is too complicated for us
in this course, but we got some results for absorption & fission rates:
5)
In general,
Abs. rate density =
6)
.
We defined some terms:
φ0
≡ “2200 m/s flux” ≡ ntot v0 ,
φTh
≡ “thermal flux” ≡
,
≡
⇒
7)
φTh.
Abs. rate density =
Given 1/v absorbers,
Abs. rate density = Σa(E0) ntotv0 = Σa(E0)φ0 ,
independent of neutron or nuclei velocity distributions.
8)
Given non-1/v absorbers,
≈
9)
.
Given Maxwellian φ at Tn,
φTh
=
.
NUEN 301 Course Notes, Marvin Adams, Fall 2009
Ch. IX. Low-Energy (Thermal) Neutrons
10) The absorption rate density of thermal neutrons by a non-1/v absorber is
approximately:
where Σa(E0) = absorption cross section evaluated at a relative speed of 2200 m/s,
and ga(Tn) is the absorption non-1/v factor for a neutron distribution that is
approximately a Maxwellian at temperature Tn. The precise formula for a non-1/v
absorber is really more complicated –– it depends in general on the actual velocity
distributions of the neutrons and nuclei –– but for this course we will go with the
formula given above, which is usually a reasonable approximation.
Finally, remember:
For every absorption formula there is a similar fission formula.
177