Lab #2 Spatial Distribution of Neutrons Introduction

Engineering Physics, McMaster University
Lab #2
4U04 Lab
Spatial Distribution of Neutrons
by Barry Diacon and Mike Butler
Introduction
A knowledge of neutron flux distributions is important in the design of experiments and to
the utilization of the reactor. Knowing the neutron flux at different positions in the reactor allows
researchers and reactor operators to determine how long and where in the core samples must be
irradiated in order to achieve a desired activity. This is especially important at McMaster’s Nuclear
Reactor, as the sample irradiation facilities are frequently used.
Flux distributions are a function of both position and energy. The range of energies varies
from several mega-electron volts (MeV) at “birth” in fission to a few milli-electron volts (meV) at
eventual absorption. As a very good approximation, the distribution of neutrons can be determined
using the diffusion equation. (Ref: Lamarsh)
In this experiment, experimenters will examine the spatial distribution of neutrons inside and
near the McMaster Nuclear Reactor core. The results will then be compared to diffusion theory.
In particular, this experiment will illustrate how mono-energetic diffusion theory breaks down near
boundaries between different materials.
Theory
The neutron flux around a rectangular parallelepiped reactor can be shown to have a cosine
distribution. This flux distribution is valid when all neutrons are assumed to have the same energy
and are in a homogeneous medium. In reality, the distribution of neutrons is greatly affected by
different media and boundaries; the neutron distribution in the core and reflector is not so simple.
This will be demonstrated using one-group and multi-group diffusion theory.
One Group Theory
Mono-energetic, or “one-group”, diffusion theory assumes that all neutrons have the same
speed (i.e. the same energy). In the steady state production of neutrons (through fission) and loss
of neutrons (through absorption and leakage) are equal; thus,
 Neutrons Lost
∂ n  Neutrons   Neutrons Lost
= 
 − 
 − 
 = 0
∂t
 Produced  by Absorption
 by Leakage 
Using mathematical expressions the above can be written:
− D∇ 2 φ + Σ a φ = S
where:
− D∇ 2φ
(2.1)
is the leakage term given by the neutron diffusion coefficient times
the second spatial derivative of the flux;
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∑
a
φ
4U04 Lab
is the absorption term given by the macroscopic absorption crosssection times the flux;
S
is the production term, otherwise known as the source term.
Equation (2.1) can also be re-written as:
∇ 2φ −
where L2 =
1
−S
φ
=
D
L2
(2.2)
D
is called the diffusion area and has units of cm2. (Lamarsh)
Σa
By separating the neutron flux into functions of spatial variables, equation (2.2) becomes:
∂ 2φ ∂ 2φ ∂ 2φ 1
−S
+
+
−
φ
=
D
∂ x 2 ∂ y 2 ∂ z 2 L2
(2.3)
Solving for a rectangular parallelepiped with sides X, Y, and Z, the flux is
 πx
 πy
 πz
φ ( x , y , z ) = φ o cos  cos  cos 
 X
 Y
 Z
(2.4)
The above equation is an approximation, as it is based on the assumption that production,
leakage and absorption occur at a single energy. In reality neutrons have a range of energies.
Neutrons are constantly changing in energy as a result of collisions; high energy neutrons slow
down through scattering collisions with atomic nuclei until they are thermalized, while thermal
neutrons can exchange energy with moderator atoms and gain energy. (Glasstone)
Two Group Theory
In order to more accurately determine the distribution of neutrons, neutrons of different
energy ranges are divided into a finite number of discrete groups. The greater the number of
groups used, the more accurate the distribution will be, with the drawback that the computations
become more complex. (Glasstone)
For any steady state multi-group system, the neutron balance in any group energy “E g” can
be represented by: (Glasstone)
 Leakage from  Absorption  Scattering out  Scattering   Production 
−
 −
 −
 +
 +
 = 0
group g   in group g   of group g   into group g  in group g

In this lab, we will consider two-group theory. In two-group theory, neutrons produced by
fission are assumed to be in either a low-energy region (thermal neutrons) or in a high-energy
region (fast neutrons). E1 and E2 are defined to be the energies of fast and thermal neutrons,
respectively.
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4U04 Lab
For this case, the diffusion equations are:
and
− D1∇ 2 φ 1 + Σ a1φ 1 = ν Σ a 2 φ 2
(2.5)
− D2 ∇ 2φ 2 + Σ a 2φ 2 = ν Σ R1 φ 1
(2.6)
where:
− D∇ 2 φ 1
Σ a1φ 1
ν Σ a 2φ
− D2 ∇ 2 φ 2
Σ a 2φ 2
− Σ R1φ
is the leakage term from group 1 (fast);
is the absorption term from group 1;
is the source term, given by the average number of neutrons
produced per fission reaction ( ν ) times the rate of absorption of
neutrons from group 2;
is the leakage term from group 2 (thermal);
is the absorption term from group 2;
is the source term, which is the result of thermal neutrons produced
when fast neutrons undergo scattering and lose energy.
A very good illustration of two-group theory occurs at the boundaries of the core, particularly
near the reflectors. Here thermal neutrons increase in number as fast neutrons slow down and are
thermalized. Equations (2.5) and (2.6) can be solved to give the results shown in Fig. 2.1.
(Glasstone)
Figure 2.1:
Thermal and fast neutron flux
distribution in core and reflector.
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Engineering Physics, McMaster University
4U04 Lab
This lab is divided into two sections. First, the in-core neutron distribution will be measured
in the vertical, “y”, axis. This is followed by a study of the out-of-core distribution in the “z” axis.
The “x” axis – i.e. horizontally across the face of the core – is not measured in this study.
In-Core Flux Measurement
For the in-core distribution, a self-powered flux detector is used to measure the vertical flux
distribution inside a graphite reflector. Although the reflectors are outside the fuelled section of the
core, this should be a region of peak thermal neutron flux (see Fig. 2.1). This detector is made of
rhodium. Natural rhodium is composed 100% of Rh-103 which absorbs a neutron ( σ c = 134 ) to
form Rh-104. This decays by beta emission with a half-life of 42.3 seconds. Rh103 can optionally
form Rh-104m ( σ c = 11 ), a metastable isotope which decays to Rh-104 with a half-life of 260.4
seconds. The emitted beta particles result in a small current that can be measured with a sensitive
ammeter.
Activity Analysis
Measurements of the out-of-core flux are slightly more complicated. The objective is to
measure the flux distribution along the “z” axis starting from the edge of the core, midway between
top and bottom, leading perpendicularly away from the core. This is achieved by “storing” the
thermal neutron flux intensity in a copper wire placed in the relevant field. A copper wire placed
for a short time in a neutron flux is activated. Natural copper has two isotopes, Cu-63 (69.17%)
and Cu-65 (30.83%). Two isotopes are produced: Cu-64 and Cu-66. Cu-66 has a half-life of 5.1
min. which is too short for the purposes of this analysis. Cu-64 has a half-life of 12.7 hours. This
is the activated isotope which contains the neutron flux “memory”.
The induced activity can be related to the flux as follows. Activity, A, is equal to the decay
constant, λ , times the number of decaying atoms, N:
A = λN
where
λ
is the decay constant.
λ =
(2.7)
ln 2
T1/ 2
The change in the number of decaying atoms is equal to the production by neutron capture
minus the loss by decay. Mathematically this is,
dN
= Σ aφ − λ N
dt
where
Σ a is the macroscopic absorption cross section and
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(2.8)
φ
is the neutron flux.
Engineering Physics, McMaster University
4U04 Lab
Assuming that all quantities, except N, are constant in time, and further assuming that at
t = 0 the number of N atoms is zero; then,
N (t ) =
Σ aφ
(1 − e − λt )
λ
(2.9)
Thus, the activity is given by:
A(t i ) = Σ a φ (1 − e λti ) = Ai
(2.10)
where t i is the irradiation time. At a time td after the irradiation ends, the activity will be
Ad = Ai e − λtd
(2.11)
Combining equations (2.10) and (2.11) and rearranging gives:
Ad e λtd
φ =
Σ a (1 − e − λti )
(2.12)
Experiment
In-core Distribution
1.
The reactor operator will place the self-powered flux detector within one of the MNR
graphite reflectors (see Fig. 2.6) in the lowest vertical position – i.e. the “A” position (see
Table 2.1).
2.
Once the detector is placed in the core, the current rises as neutrons activate rhodium
atoms. Eventually fresh activation reaches equilibrium with decay. This initial point is
reached within no more than fifteen minutes. Successive measurements will only take
about 5 minutes. When the ammeter stops fluctuating, record the current. The vertical
positions to be measured are shown in Table 2.1. Record the current at every position.
3.
From the values obtained, determine the neutron flux at every position using the fact that
Neutron Flux ~= 2 x 1020 I, where I is the detector current in amperes.
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Table 2.1: In core flux density
measurement positions
Vertical Position
Height (cm)
O
70.2 ± 0.1
N
65.2 ± 0.1
M
60.3 ± 0.1
L
55.4 ± 0.1
K
50.5 ± 0.1
J
45.6 ± 0.1
I
40.6 ± 0.1
H
35.7 ± 0.1
G
30.8 ± 0.1
F
25.9 ± 0.1
E
21.0 ± 0.1
D
16.0 ± 0.1
C
12.2 ± 0.1
B
7.3 ± 0.1
A
2.5 ± 0.1
Current (nA)
Out-of-core Distribution
Irradiation of Copper Wire
Cu-64 decays according to the scheme shown in Fig. 2.2. The half-life of 12.70 hours
means that statistically significant counts can be obtained before the activity is seriously depleted.
The copper wire is positioned using an aluminum device called a “flux mapper”. See Fig. 2.3 for
a diagram. The aluminum provides structural stiffness, while the aluminum has a low thermal
neutron cross-section and a short half-life.
A copper wire is inserted in the flux mapper. The flux mapper is then lowered to the core
and positioned perpendicular to the side of the core face so that one end of the wire touches the
core and the other end is far away from the core.
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4U04 Lab
Figure 2.2: Decay scheme for Cu-64 (Cu-63 + n). Energies are in MeV.
The decay mode of primary interest is the 19% probability positron which
annihilates to produce a 511 KeV gamma photon.
1.
Although the MNR pool is filled with very pure de-ionized water, it is still radioactive.
Therefore, experimenters must wear gloves and use tongs and tweezers when handling the
flux mapper because it is stored in the pool.
2.
Cut a piece of copper about 22” long. The reactor operator will hold the flux mapper near the
edge of the pool. Insert the copper wire in the open end of the tube until it emerges from the
end which intersects the rod. At the end which intersects the rod twist a bit of wire around the
tube to keep it from moving. Cut off any excess wire from the open end.
3.
The reactor operator will then move the device close to the core. Once the tube is positioned
correctly, it will be held fixed in that spot for 1 minute. When the minute has elapsed, the
device is moved away from the core.
4.
The apparatus is left in the pool for one day so that short-lived active isotopes can decay.
The primary examples are Al-28 (t1/2 = 2.24 min) and Cu-66 (t1/2 = 5.1 min).
Counting Activity of the Wire
Note: Do not touch active material at any time
5.
On the day after irradiation, the reactor operator will lift the flux mapper apparatus out of the
pool. Use a radiation detector to monitor the fields around the device to maintain a safe
working environment. Note both gamma and beta doses. Wearing gloves, using tongs and
extreme caution, remove the wire from the aluminum tube.
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Engineering Physics, McMaster University
4U04 Lab
Figure 2.3: “Flux mapper” apparatus.
6.
Attach the copper wire to the bed of the activity measuring apparatus (see Fig. 2.4). This
is in the low level counting room.
Figure 2.4: Apparatus on which irradiated copper wire is mounted for automated counting.
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7.
4U04 Lab
A Sodium Iodide detector is mounted in a lead shield above the copper wire. A narrow gap
allows activity in a short section of the wire to be “visible” to the detector, as seen in Fig. 2.5.
Measure the thickness of the lead blocks and the width of the gap between the blocks.
8.
Figure: 2.5: Schematic of lead shield and detector setup.
9.
The bed of the apparatus can
be moved by a stepper motor in
increments of approximately
0.51 cm. The Multi-Channel
Analyzer (MCA) acquires the
gamma signal resulting from
positron annihilation in the wire
for a fixed time interval. The
MCA has a supervisor batch
mode which allows it to acquire
a sample, save that to disk, and
send an instruction to advance
the stepping motor. This cycle
is repeated many times until the
entire wire has been measured.
This produces a data set of flux
intensity versus distance from
the core.
A small program is used to extract the region of interest data from the multiple spectrum files
into a single text file.
Analysis and Discussion
1.
The measured Cu-64 activation intensities have to be normalized to account for decay which
occurs over the course of 2 to 2.5 hours of data acquisition. The acquisition time for each
sample is recorded in the above file.
2.
Plot the obtained flux maps with error bars and compare with one group theory and two-group
theory.
3.
Why is multi-group theory much more realistic than one-group theory in describing the flux
inside a reactor?
4.
Explain why one group theory breaks down at material boundary points.
5.
Discuss how well or how poorly equation (2.4) might describe the McMaster core.
6.
Using the results from the in-core flux measurement, fit a cosine curve to the data. Discuss
deviations in terms of energy, material boundaries and properties.
7.
Discuss all sources of error in the experiment. For instance, what percentage of 511 KeV
gammas is transmitted through the thickness of lead that is used in the shield? How effective
is the lead shield in blocking 511 KeV gammas from parts of the wire not being measured?
What does beam collimation mean? What effect would imperfect collimation have on the
error in the measurements?
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Engineering Physics, McMaster University
4U04 Lab
Figure 2.6: McMaster Nuclear Reactor core plan.
Note: Make sure to make a rough sketch of the Core Loading map to use in the analysis, noting
any differences between that map and Fig. 2.6.
References
1.
Lamarsh, J.R., Introduction to Nuclear Engineering, Addison-Wesley Pub. Co., 1975.
2.
Foster, A.R,. and Wright, R.L., Basic Nuclear Engineering, 3rd. ed., Allyn and Bacon, 1977.
3.
Meem, J.L., Two Group Reactor Theory, Gordon and Breach, 1964.
4.
Duderstadt, J.J., Nuclear Reactor Analysis, Wiley, 1976.
5.
Murray & LeRoy, Raymond, Introduction to Nuclear Engineering, Prentice-Hall, NY.
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