Reactivity Determination and
Monte Carlo Simulation of
the Subcritical Reactor
Experiment – “Yalina”
Carl-Magnus Persson
Master of Science Thesis
Department of Nuclear and Reactor Physics
Royal Institute of Technology
Stockholm 2005
ISBN 91-7178-015-7
TRITA-FYS 2005:19
ISSN 0280-316X
ISRN KTH/FYS/--05:19--SE
© Carl-Magnus Persson, 2005
Printed by Universitetsservice US-AB, Stockholm 2005
2
Abstract
Accelerator-driven systems (ADS) has been proposed to transmute the long-lived
transuranic isotopes in the nuclear waste. This new type of reactor requires new
methods for reactivity determination, and this present study analyses experiments
concerning determination of subcriticality performed at the research facility “Yalina” in Minsk, Belarus. The methods investigated are the area method, the slope
fit method and the source jerk method. The pulsed source experiment, including
the area- and the slope fit methods, has also been investigated using the Monte
Carlo code MCNP. Experiments concerning spatial fission rate distributions were
also performed.
During the analysis it has become clear that the different methods have different advantages and drawbacks. The area method is straight forward to use, but the
results deviate from the other methods. The analysis using the slope fit method is
based on the same data as the area method, but is not definitely clear how to apply
to experimental data. The analysis of the source jerk method, on the other hand,
gives large errors.
The analysis shows that the area method underestimates the reactivity in comparison with the other methods. The values obtained by MCNP are in fairly good
agreement with the slope fit method, whereas the source jerk method overestimates the reactivity, relatively the area method and the slope fit method. Moreover, the source jerk method is subject to large uncertainty.
The fission rate distributions obtained by experiments are in good agreement
with the distributions obtained by MCNP. Concerning MCNP, it can be summarized that the code gives reliable results considering neutron distributions in space
and time but fails in predicting neutron time parameters.
3
4
Table of Contents
Acknowledgements............................................................................................................. 7
1
Introduction................................................................................................................. 9
1.1
Background ......................................................................................................... 9
1.2
Thesis Overview ............................................................................................... 10
2
Elementary Reactor Theory ...................................................................................... 13
2.1
Fission ............................................................................................................... 13
2.2
Neutron Interactions.......................................................................................... 14
2.3
Fission Chain Reactions.................................................................................... 16
2.4
Delayed Neutrons.............................................................................................. 17
2.5
Neutron Transport............................................................................................. 18
2.6
Definitions of Neutron Time Parameters.......................................................... 20
2.7
The Point Kinetic Equations ............................................................................. 21
3
Transmutation ........................................................................................................... 23
3.1
Fundamental Principals of ADS ....................................................................... 23
3.2
Design of ADS.................................................................................................. 25
4
Reactivity Determination Methods ........................................................................... 29
4.1
Area Method ..................................................................................................... 29
4.2
Slope Fit Method............................................................................................... 31
4.3
Source Jerk Method .......................................................................................... 32
5
Experimental Setup................................................................................................... 35
5.1
The Yalina Facility ........................................................................................... 35
5.2
Methodology of the Experiments...................................................................... 39
6
Monte Carlo Calculation Method ............................................................................. 41
6.1
MCNP ............................................................................................................... 42
6.2
Neutron Time Parameters in MCNP................................................................. 42
7
Monte Carlo Simulations of Yalina .......................................................................... 45
7.1
Fission Rate Simulation .................................................................................... 45
7.2
Criticality Calculation....................................................................................... 46
7.3
Delayed Neutron Fraction................................................................................. 46
7.4
Simulation of the Pulsed Neutron Source Experiment ..................................... 46
7.5
Investigation of the Validity of the Point Kinetic Approximation ................... 49
8
The Experiments ....................................................................................................... 51
8.1
Measurements of Fission Rate Distributions .................................................... 51
8.2
Measurements of the Pulsed Neutron Source Response................................... 53
8.3
Source Jerk Experiment .................................................................................... 58
9
Discussion of Results................................................................................................ 63
9.1
Comparison between Experiments and Simulations......................................... 63
9.2
Comparison between the Different Reactivity Determination Methods........... 65
9.3
Comparison with the MUSE-experiments ........................................................ 67
10
Conclusions........................................................................................................... 69
References......................................................................................................................... 71
5
6
Acknowledgements
I would like to express my gratitude to all people helping me with all work connected with this thesis. In random order:
• Professor Waclaw Gudowski for giving me the opportunity to do this work
and to visit countries very different from my own. This has really widened
my view of the world and different societies. His contribution to the discussions in reactor physics has been very valuable.
• My direct supervisor Per Seltborg for worthwhile discussions. The work
would have been very difficult to do without him.
• All staff at the Yalina facility in Sosny for the performance of the experiments, providing MCNP-input and for taking so good care of us. I would
like to specially mention Dr. Sergey Chigrinov for his contribution to make
the collaboration possible and Dr. Anna Kiyavitskaya for her enormous
kindness.
• Tania Kiyavitskaya for showing us Minsk and the Belarusian culture.
• Andrey, Sergey and Vitaly for nice evenings in Minsk.
• My collaborators Alexandra Åhlander and Thomas Stummer for nice company and discussions concerning physics as well as everyday life.
• Robin Klein Meulekamp at NRG, Petten, for helping me with calculation of
the effective delayed neutron fraction.
• All staff at the Reactor Physics group, KTH, for giving me a pleasing time
during my work.
Finally, I would like to thank my family for their assistance during my studies at
KTH, my friends making my time as student enjoyable and Elin for supporting me
in all situations.
Carl-Magnus Persson
Stockholm, 2005
7
8
1 Introduction
1.1 Background
Today nuclear power and fossil fuels are the most important contributors to the
world electricity supply together with hydro-power. The electricity consumption
shows no sign to decrease, and as the living standard in the developing countries
rises, the electricity consumption will increase further (Figure 1) [1]. Since the use
of fossil fuels contributes to the greenhouse effect and the global warming, nuclear
power will become more and more important in the future (Figure 2) [2]. On the
other hand, if the nuclear power will be used in the same way as it is used today, a
large amount of radiotoxic nuclear waste will be created. Several large repositories
have to be built and the radiotoxicity of the content will be harmful for humans
during several hundred thousands of years [3].
25000
23072
20688
20000
18453
TWh
16358
15000
13290
10000
5000
0
2001
2010
2015
2020
2025
Figure 1. Expected world net electricity consumption [1].
The elements that cause such long storage time are the transuranic elements
plutonium, americium and curium. Many of the isotopes of these elements are not
possible to fission in the reactors of today; instead they are created in increasing
amounts. This problem can be solved by constructing new kinds of nuclear reactors, which can handle also these isotopes. The most efficient way to handle the
transuranic isotopes originating from the fuel cycle and at the same time decrease
the waste stock is by the use of accelerator-driven systems (ADS) [4]. This reactor
type is currently being studied at different research institutes around the world. In
9
Figure 2. World energy sources according to a sustainable development vision scenario [2].
an ADS, a larger fraction of the potential nuclear energy in the fuel is used than in
conventional critical reactors, and the waste consists mostly of relatively shortlived fission products. With this method, named transmutation, the minimal required storage time of the waste in the final repository can be shortened to approximately two thousand years [5].
Transmutation by ADS seems promising, but there are still problems to be
solved. For example, new fuels, new cooling systems and a strong external neutron source have to be constructed. The ADS will operate in a subcritical state,
which means that the margin to criticality accidents (like in Chernobyl) decreases
[6]. In order to study phenomena connected with transmutation and ADS, a few
small-scale subcritical facilities have been constructed, such as MUSE (France)
[7] and Yalina (Belarus). Experiments performed at these facilities give information about neutron kinetic properties representative for a full-scale ADS. Other
projects under construction are SAD (Russia) [8] and RACE (USA) [9].
1.2 Thesis Overview
This study is devoted to different reactivity determination experiments performed
at the subcritical experimental reactor Yalina in Belarus. First, elementary reactor
physics is presented, needed in order to understand the basics of transmutation and
the derivation of the reactivity determination methods used. In chapter 3, the principals of accelerator driven systems are explained. In chapter 4, the different reactivity determination methods are derived from the theory in chapter 2. Then the
experimental setup of Yalina is presented (chapter 5) followed by a description of
10
the Monte Carlo calculation method (chapter 6). Up to this point, necessary information for starting the analysis is given. In chapter 7, the Yalina reactor system is
analyzed by Monte Carlo calculations and in chapter 8 the experiments are presented and analyzed. Chapter 9 contains discussions of the results from the experiments and the simulations, and the conclusions of the study are presented in
chapter 10.
11
12
2 Elementary Reactor Theory
2.1 Fission
Fission is the phenomenon that is the foundation stone in nuclear energy. When a
nucleus is divided in two parts by an incident neutron, the mass of the two fission
products is less than the mass of the original nucleus. The difference in mass is
transformed into energy according to the famous formula of Einstein [10]
E = ∆mc 2 ,
(1)
where E is the released energy, ∆m is the mass defect and c is the speed of light in
vacuum, approximately 300 000 km/s. Since c2 is a very large number, a small
mass difference causes an enormous amount of energy. This is in contradiction
with our idea of the everyday life. You would be very surprised if the bread got
toasted and decreased in mass while you were slicing it in the morning!
When a neutron is absorbed by a nucleus an excited compound state is created.
For some nuclei, with high mass number, this compound state is very unstable and
causes fission. This happens, even if the energy of the neutron only has thermal
energy (approximately 25 meV), for fissile isotopes such as 233U, 235U, 239Pu and
241
Pu. Other nuclei, such as 232Th, 238U and 240Pu, fission only if the neutron carries
enough energy to overcome a certain activation energy of the compound nucleus.
This requires often neutron energy more than 1 MeV. Fission can also occur spontaneously as a form of radioactive decay, so called spontaneous fission. Examples
are 244Pu and 252Cf.
Figure 3 summarizes the concept behind fission and fusion by regarding the
binding energy per nucleon as a function of the mass number. Physically everything strives to be in the most stable configuration. As is shown in Figure 3, 56Fe is
the most stable nucleus. The light elements can be more tightly bound if they are
merged together into heavier isotopes by fusion, and the heavy isotopes can be
transformed into more tightly bound isotopes by fission. In both cases, energy is
released. From the figure it can also be seen that the energy released per nucleon is
much larger in fusion than in fission, but in fission there are much more nucleons
involved so the total amount of energy released is larger [10].
13
10
Binding energy per nucleon [MeV]
56
Fe
9
8
4
235
U
He
7
Fission
6
5
Fusion
4
3
2
2
1
0
0
H
50
100
150
200
250
Mass number, A
Figure 3. Binding energy per nucleon [11]. The basic concept behind fission and
fusion.
2.2 Neutron Interactions
The quantity used to measure the probability of an interaction to occur is called
cross section, σ. It can be expressed by the reaction rate, R, for n neutrons traveling with speed v a distance dx in a material with N nuclides per unit volume:
σ=
R
nvNdx
(2)
The unit of the cross section is area, but since the cross sections constitute so small
areas, the unit 1 barn = 10-28 m2 is introduced. The cross section describes the interaction in a microscopic scale; therefore it is often called the microscopic cross
section. Sometimes it is more practical to regard the macroscopic effect of the microscopic cross section. This is called the macroscopic cross section, Σ, defined by
Σ = σN .
14
(3)
When a neutron (n) interacts with a nucleus (X), there are two basic possible
outcomes, scattering or absorption. Scattering can be described by the following
reaction formula
X + n → X’ + n’.
By scattering the neutron loses energy, which is transferred to the target nucleus.
This is an important process in reactors fuelled with 235U where the fission probability increases with lower neutron energy. This mechanism, when fast neutrons
are slowed down, is called moderation. It can be found that the average logarithmic energy loss in every scattering, ξ, is
2
(
A − 1) ⎛ A + 1 ⎞
ξ = 1−
ln
2A
⎜
⎟.
⎝ A −1⎠
(4)
Consequently, light elements slow down neutrons more efficiently than heavy
elements. The most common moderator used in nuclear reactors is light water. An
efficient moderator has a large scattering cross section but a small absorption cross
section. To be able to compare different moderators the moderating ratio is introduced, ξ Σ s / Σ a .
Table 1. Comparison of different moderators.
Number of colliModerator
ξ
sions (2 MeV to 1
eV)
H
1.0
14
D
0.725
20
0.920
16
H2O
0.509
29
D2O
He
0.425
43
Be
0.209
69
C
0.158
91
Na
0.084
171
Fe
0.035
411
238
U
0.008
1730
ξ Σs / Σa
71
5670
83
143
192
1134
35
0.0092
Absorption results in one of two events: fission or capture. Capture is when the
target nucleus absorbs the neutron without fissioning. This can be described as
A
XZ + n → A+1XZ,
15
where A is the mass number and Z the atomic number. This process steals neutrons from the system and produces new often unwelcome isotopes. This is the
underlying process that creates the transuranic elements in the nuclear waste.
The neutron capture is necessary for the operation of the reactor since the control rods are made of material with high neutron capture cross section and because
of the Doppler effect explained later [12].
2.3 Fission Chain Reactions
In a fission event, neutrons are produced, which can induce more fission. If there
is sufficient fissile material and if the neutrons are used in an efficient way, a chain
reaction can take place. The effective multiplication constant, keff, can be defined
as
k eff =
number of neutrons in generation i + 1
.
number of neutrons in generation i
(5)
If keff is equal to 1, there are enough neutrons to keep the chain reaction running at
a constant level and the reactor is said to be critical. If keff is less than one, the system is subcritical and the fission rate will die out. On the other hand, if keff is
higher than 1 the system is supercritical and the fission rate will increase. The neutrons that are created in fission have high energy. In order to induce new fissions
in a light water-cooled reactor where the fissile material is sensible mainly for
thermal neutrons, the neutrons must first be slowed down. Suppose that η neutrons
are produced per neutron absorbed in the fuel. f of these neutrons are absorbed in
the fuel and not in the moderator, cladding or construction materials. Some of
these neutrons, ε, contribute to the fission chain by fast fission. During the moderation process 1-p neutrons will be captured by the fuel without causing fission.
A fraction of PFNL fast neutrons and PTNL thermal neutrons will not leak out from
the system. In this way the effective multiplication constant can be defined as
keff = η f ε pPFNL PTNL .
(6)
In a critical reactor, the control rods and the thermal reactivity feedbacks control
the neutron flux in such a way that keff is always at or near unity.
From the effective multiplication factor, another quantity is derived, namely
the reactivity ρ;
ρ=
k eff − 1
k eff
16
(7)
The reactivity is used to quantify changes in the reactor. A critical reactor in equilibrium has zero reactivity. A subcritical reactor has negative reactivity and a supercritical reactor has consequently positive reactivity. The reactivity is often
measured in units of 10-5 or pcm (pro cent mille) [12].
2.4 Delayed Neutrons
The neutrons produced directly in the fission process are called prompt neutrons.
In a thermal reactor, these neutrons need typically about 0.1 ms to induce new fissions. If all neutrons were created promptly, it would be impossible to control the
reactor. However, some of the fission products emit neutrons after one or several
decays, and these neutrons contribute to the fission chains with a significant delay.
Precursors with similar half-lives are grouped together and usually six groups are
enough to describe their time behavior, see Table 2. The delayed neutrons introduce sluggishness in the system, which makes it possible to operate the reactor
safely.
Table 2. Delayed neutron data for thermal fission in 235U [13].
Group
Half-life [s] Energy [keV]
Fraction, βi
1
55.72
250
0.000215
2
22.72
560
0.001424
3
6.22
405
0.001274
4
2.30
450
0.002568
5
0.610
0.000748
6
0.230
0.000273
The total number of neutrons induced by fission,ν Tot , can be divided into
prompt neutrons, ν p , and delayed neutrons, ν d .
ν Tot = ν p + ν d
(8)
The fraction of delayed neutrons, β, is expressed as
β=
νd
ν Tot
and values for different isotopes are found in Table 3.
17
(9)
Table 3. Delayed neutron fraction
for different isotopes [14].
Isotope
β [pcm]
235
U
640
238
U
1480
239
Pu
200
241
Pu
540
241
Am
130
243
Am
240
242
Cm
40
Since the delayed neutrons are produced by decay of instable fission products,
and not by fission, they have a different energy spectrum. This influences the efficiency of the delayed neutrons, since they do not have to be slowed down as much
as the prompt neutrons. Therefore, the effective delayed neutron fraction, βeff, is
introduced, defined as the number of fissions induced by delayed neutrons, Nd,
compared to the total number of fissions induced in the same system, NTot [15]:
β eff =
Nd
.
N Tot
(10)
The reactivity defined in the previous section is often measured in fractions of the
effective delayed neutron fraction. One unit of ρ/βeff is called a dollar and one percent of one dollar is called a cent. Prompt supercriticality occurs if the inserted reactivity exceeds 1 $. In that case, the neutron flux increases very fast, since the fission chain reaction can rely on prompt neutrons only [12].
2.5 Neutron Transport
The time variation of the number of neutrons entering a volume in the reactor is
d
n(r , Ω, v, t )d rd 2 Ω = ((entering − exiting ) + (created − absorbed )
dt ∫∫∫
+(inscattered − outscattered ) + source)neutrons.
Mathematically this can be written as
18
(11)
1 ∂φ (r , v, Ω, t )
= −∇ Ωφ (r , v, Ω, t ) − Σt (r , v, Ω, t )φ (r , v, Ω, t )
v
∂t
(
∞
4π
0
0
)
+ ∫ dv′ ∫ d 2 Ω′Σ s (r , v′ → v, Ω′ → Ω, t )φ (r , v, Ω, t )
∞
4π
(
(12)
)
1
dv′ ∫ d 2 Ω′ ν (v)Σ f (r , v, Ω, t ) χ (v)φ (r , v, Ω, t ) + Sext (r , v, Ω, t )
+
∫
4π 0
0
which is the Boltzmann equation [16] [17]. In this equation
φ = nv
r
Ω
Σt
Σs
ν
Σf
χ
Sext
is the neutron flux (neutron density times velocity),
is the position of the neutron,
is the unit direction vector of the neutron velocity,
is the total cross section (absorption plus scattering)
is the scattering cross section,
is the number of neutrons per fission,
is the fission cross section,
is the energy distribution of the neutrons causing fission and
is the external neutron source.
Assuming that all neutrons have the same energy, a one-energy group, the neutron diffusion equation is obtained [14]. For a bare homogeneous reactor the timedependent diffusion equation is
1 ∂φ (r , t )
− D∆φ (r , t ) + Σ aφ ( r , t ) = νΣ f φ (r , t ) ,
v ∂t
(13)
where D is the diffusion constant. This equation can be solved for simple reactor
geometries when separated in a time-dependent part and a spatial part, with separation constant λ. The time-dependent part has the solution
T (t ) = T (0)e − λt
(14)
and the spatial part must satisfy
∆ϕ (r ) = − B g2ϕ (r )
19
(15)
where Bg is the geometric buckling depending on the geometry of the reactor. This
equation can be solved analytically for simple reactor geometries yielding the flux
profile and the geometric buckling [12].
2.6 Definitions of Neutron Time Parameters
Assuming a one-group model of a homogeneous reactor, the mean free path, lx , for
a certain reaction is given by the inverse of the macroscopic cross section for that
reaction, Σ −x 1 . Dividing this value by the velocity, the specific lifetime, defined as
the time from the birth of the neutron until it undergoes the reaction, is achieved.
In this manner it is possible to define lifetimes for all types of reactions, e.g. fission, absorption, removal etc. However, there exist some standard definitions,
which have their origin in the discussion above.
The mean time for one neutron to be removed from the reactor due to absorption
or leakage is called the neutron lifetime, l, defined by
l=
1
,
vΣ a (1 + L2 B 2 )
(16)
where v is the neutron velocity, Σa is absorption cross section, L is the diffusion
length and B is the buckling. The mean time for one neutron to cause fission is the
mean fission time, τ,
τ=
1
,
vΣ f
(17)
where Σf is the fission cross section. Since ν neutrons are produced in a fission
event, the mean time to produce one more neutron is given by
Λ=
1
,
ν vΣ f
(18)
which is the mean generation time, Λ. This time parameter plays and important
role in reactor kinetics, which will be shown later. In fact, the neutron mean generation time is the mean fission time per neutron,
Λ=
τ
.
ν
20
(19)
Moreover, the effective multiplication factor can be expressed as
keff =
νΣ f
Σ a (1 + L B )
2
2
=
l
.
Λ
(20)
From this we can conclude that for a critical system the neutron lifetime is equal to
the mean generation time, i.e. the time for neutron removal is equal to the time for
neutron creation. Therefore, in a subcritical system, the neutron lifetime is smaller
than the generation time, and vice verse in a supercritical system [18].
Note that the different time parameters described above are calculated averaged over the whole reactor and normalized per neutron present in the system, and
do not describe a single fission chain [19]. A common misunderstanding is that the
generation time is the average age of a neutron generation in the fission chain.
This misunderstanding would be avoided if the name “mean production time” was
used.
Since the time parameters described have large importance when analyzing
dynamical reactor experiments, it is of interest to find them by simulations. How
these quantities are treated in the Monte Carlo simulation code MCNP is described
in section 6.2.
2.7 The Point Kinetic Equations
Assuming that all neutrons in the reactor have the same energy, the timedependence of the neutron flux can be derived using the diffusion equation, Eq.
(13). The delayed neutron precursors, ci, described in section 2.4, satisfy
∂ci (r , t )
= β iνΣ f ( r , t )φ (r , t ) − λi ci (r , t ),
∂t
i = 1...6
(21)
where βi is the effective delayed neutron fraction for the delayed neutron group i,
ν is the number of neutrons released per fission, φ is the neutron flux and λi is the
decay constant for the delayed neutron group i. The neutrons emitted by the delayed neutron precursors can be regarded as a source in the diffusion equation.
6
1 ∂φ ( r , t )
− D∆φ ( r , t ) + Σ a (r , t )φ (r , t ) = (1 − β eff )νΣ f (r , t )φ ( r , t ) + ∑ λi ci ( r , t )
v ∂t
i =1
The solution to this equation is assumed to be separable in space and time.
21
(22)
φ (r , t ) = vn(t )ϕ (r ),
ci ( r , t ) = Ci (t )ϕ (r )
(23)
where ϕ (r ) is the fundamental mode solution to Eq. (15). When this assumption is
applied on Eq. (22) one obtains
6
∂n(t )
ϕ (r ) − Dvn(t )∆ϕ (r ) + Σ a vn(t )ϕ (r ) = (1 − β eff )νΣ f vn(t )ϕ (r ) + ∑ λi Ci (t )ϕ (r ) (24)
∂t
i =1
By using Eq. (15), Eq. (24) becomes
6
∂n
= (1 − β eff )νΣ f v − DvBg2 − Σ a v n + ∑ λi Ci
∂t
i =1
(
)
(25)
The following quantities are introduced:
L=
D
Σa
νΣ f − Σ a (1 + L2 Bg2 ) k eff − 1
ρ=
=
νΣ f
k eff
(26)
(27)
L is the diffusion length and ρ is the reactivity. Eq. (18), Eq. (26) and Eq. (27) inserted in Eq. (25) yields
6
dn(t ) ρ (t ) − β eff
=
n(t ) + ∑ λi Ci (t )
Λ
dt
i =1
dCi (t ) βi
= n(t ) − λi Ci (t ), i = 1,..., 6
dt
Λ
(28)
which are the point kinetic equations. These equations describe the kinetic behavior of the reactor and are the most fundamental relations from which all following
theory in this work is derived. When using the point kinetic equations, the assumptions made in the derivation must be kept in mind. The relations are valid if the
spatial flux distribution is constant in time and if a one-energy group model is sufficient to describe the system [12] [18].
22
3 Transmutation
3.1 Fundamental Principals of ADS
Nuclear reactors are grouped depending mainly on their characteristic material
composition. The most common reactor type is light-water reactors (LWR). These
reactors are divided into two groups: pressurized-water reactors (PWR) and boiling-water reactors (BWR). Heavy water has been used in the Canadian CANDU
reactor, and the reactors in Chernobyl were water-cooled graphite-moderated reactors (RMBK). Common for all reactor types mentioned so far is that they operate
with a thermal neutron spectrum, which means that the fuels of these reactors are
most sensitive to thermal neutrons. Since there always is a large amount of 238U in
the fuel, neutrons will be captured in the resonance region (see Figure 4) and after
successive captures and decays, Pu, Am and Cm will be produced. The reaction
chain starting with neutron capture in 238U and ending with 241Pu and 241Am is as
follows:
−
−
β (23.5 min)
β (2.4 d )
U + n ⎯⎯
→ 239U ⎯⎯⎯⎯⎯
→ 239 Np ⎯⎯⎯⎯
→ 239 Pu
238
239
240
Pu + n ⎯⎯
→ 240 Pu
−
β (14.35 y )
Pu + n ⎯⎯
→ 241Pu ⎯⎯⎯⎯
→ 241 Am
The capture cross sections of 238U and 241Am are plotted in Figure 4. Capture in
238
U is necessary for critical reactor operation. When the temperature in the fuel
increases, the probability for neutron capture increases due to broadening of the
absorption resonances, which decreases the reactivity. Without this negative temperature feedback, the Doppler effect, it would be difficult to stay critical [12].
A few reactors have been built with liquid sodium as coolant, for example Super-Phenix in France, MONJU in Japan and Beloyarsky-3 (BN-600) in Russia
[20]. The main purpose of using a coolant with relatively high mass number is that
the neutrons are less moderated and the process can rely on fission induced mainly
by fast neutrons. The same effect can also be achieved with for example pressurized helium or liquid lead as coolant, because of the low scattering and capture
cross sections. By using a fast neutron spectrum, isotopes acting as poisons in a
thermal spectrum, will instead act like fuel [6].
23
4
10
238
U
241
Am
2
σc [b]
10
0
10
-2
10
-4
10
-4
10
-2
10
0
2
10
10
4
10
6
10
Energy [eV]
Figure 4. Microscopic capture cross section for 238U and 241Am. Note
the high cross section for thermal neutrons and the deep dip after 1
MeV [21].
As explained in the introduction, accelerator-driven systems are required if the
transuranic elements from the nuclear waste stock and the fuel cycle are going to
be transmuted. The first thing that must be considered when discussing how to
minimize the amount of Pu, Am and Cm in the waste is that these isotopes must be
burnt and the capture in 238U must be reduced. Therefore, 238U is excluded from
the fuel and replaced by Pu, Am and Cm. The choice of fuel affects the effective
delayed neutron fraction. Table 3 shows how the delayed neutron fraction varies
with some isotopes relevant for transmutation. The small values for Am and Cm
indicates that the effective delayed neutron fraction for a fuel with low uranium
content will be very low.
A reason why a uranium-fueled reactor with high content of minor actinides,
even with 238U, cannot operate in critical mode is that when 241Am is present in the
fuel, the Doppler effect in 238U is strongly reduced [22]. This is due to the high
neutron capture cross section of 241Am in the region above the absorption resonances in 238U. The capture cross section is in the region 100 keV almost ten times
higher for 241Am than for 238U. No other fissile isotope shows such high Doppler
feedback as in 238U, which means that for uranium-free fuels the Doppler feedback
will be low.
Another condition that must be considered is the coolant void worth, which is
the reactivity feedback when there is a loss of coolant. Fast systems with liquid
metal coolant have a tendency to have high coolant void worth due to decreased
neutron absorption in the coolant and decreased neutron capture in the fuel [23].
The decreased Doppler feedback, the low delayed neutron fractions and the
positive coolant void worth indicates that a reactor with such parameters cannot be
24
operated in a critical mode with sufficient safety margins. A reactor with high content of minor actinides must therefore under all circumstances be subcritical.
3.2 Design of ADS
Since the core of an ADS will be designed to be subcritical, there will be no selfsustained fission chain reactions. In order to obtain a steady state neutron flux, an
external neutron source must be coupled to the reactor. This can be achieved by
accelerating protons on a target of heavy metal. Neutrons are released in spallation
reactions in the target and enter the surrounding subcritical core. The main components of an ADS are depicted in Figure 5.
Proton accelerator
Coolant
Fuel
Spallation target
Figure 5. Schematic view of an ADS.
3.2.1 The Neutron Source
The underlying process for the neutron source is the spallation reaction induced by
accelerated protons impinging on a target of lead or lead-bismuth eutectic. The
spallation process can be divided into two stages. First, the incident proton interacts with the individual nucleons in the nucleus, which can result in a head-on collision or a scattering reaction. This phase is called intranuclear cascade. Second,
the nucleus becomes excited and starts to evaporate particles such as neutrons,
protons, pions, and also fission products. A proton with energy 1 GeV will create
25
about 15 neutrons per proton. Some of these neutrons carry very high energy and
may induce further spallation reactions or (n,xn)-reactions, which increases the
total neutron yield in the target up to about 30 neutrons per proton [24].
This process requires first of all a powerful proton accelerator. The energy of
the protons should be in the order of 600 – 1000 MeV and the current at least 25 –
40 mA for an ADS of industrial scale. Accelerators of today can deliver proton
energies well above 1 GeV but the current is not sufficient. A full scale ADS operating at a power level of approximately 800 MWth must be able to run continuously for long time. If the reactor is shut down the core will be poisoned by neutron absorbing fission products, and coolant and target materials will be damaged.
This implies that the accelerator must be very reliable, which is not the case with
the accelerators of today. There are two main groups of accelerators: linacs (linear
accelerators) and cyclotrons. Linacs are most promising with respect to sufficient
accelerator current [6].
The spallation target is the physical barrier between the accelerator and the
core, and must withstand severe damage due to high-energy particles and thermal
load. A material with high neutron yield, good physical and chemical properties
and thermal-hydraulic performance must be chosen. The main candidate is liquid
lead-bismuth eutectic (LBE). Lead and bismuth has good neutron efficiency due to
(n,xn)-reactions and LBE has the melting point at 398 K and boiling point at 1943
K. The advantage of using LBE instead of pure lead is that the melting point is decreased with more than 200 K [15].
Radioactive spallation products will appear in the target and they must not be
released to the surroundings. There must be a safe confinement protecting the reactor building from these isotopes. In the accelerator tube, where the protons are
accelerated, is vacuum. The protecting interface between the target and the vacuum beam line is called window. The window must withstand the high radiation
and at the same time let the protons pass by. The lifetime of the window must be
sufficiently long to avoid repeated shutdowns of the ADS [6]. However, there exist also windowless solutions relying on the flow properties of the target material
[25].
3.2.2 The Core
The level of subcriticality of the core must be chosen with respect to sufficient
safety margins. A near criticality configuration requires low accelerator current
but increases the risk for approaching criticality during abnormal operation conditions. On the other hand, deep subcriticality requires a high accelerator current
which can be difficult and expensive to realize [6].
The ADS fuel will consist of a higher amount of plutonium and minor actinides (Np, Am and Cm) compared to regular reactor fuels used today. These fuels
will have high decay heat and high neutron and gamma emission rate, and must
26
have good thermal properties. Today mainly oxide fuels are used, but for ADS nitride fuels may be of interest due to four times higher thermal conductivity. The
drawback of nitride fuel is that it must be enriched to 15N to avoid production of
the radioactive isotope 14C through the reaction 14N(n,p)14C. Metallic fuels are also
of interest because of high thermal conductivity and achievement of harder neutron spectrum, but give problems with swelling [15] [22].
3.2.3 The Coolant
The coolant must be low-moderating with good thermal-hydraulic properties. A
good coolant candidate should also combine a low melting point and a high boiling temperature. Other important issues are corrosion and chemical activity with
air and water. The main focus lies on liquid metals, such as LBE or liquid sodium,
and gas coolants, primarily pressurized helium.
Liquid metal coolants allow operation at atmospheric pressure but involve corrosion and no possibility for visual core inspections. Sodium has excellent thermal
properties but is chemically very active in contact with water. LBE is chosen due
to its low melting point and high boiling point as mentioned above. LBE has lower
coolant void worth than sodium but lower heat removal capacity and limited flow
velocity due to erosion of the cladding material.
Gas-cooled cores must operate under high pressure to achieve sufficient heat
transfer. Helium-cooled reactors give the hardest spectrum and have no complications with cladding or construction materials. However, if the pressure drops there
is a risk of a loss of coolant accident due to the decay heat [15].
27
28
4 Reactivity Determination Methods
There are several different reactivity determination methods available and they can
be divided into three groups: static methods, dynamic methods and noise methods.
The static methods are based on the fact that the inverse of the neutron count
rate is proportional to keff. By moving a control rod or changing the source it is
possible to find the critical state and close to criticality states. The rod-drop
method is an example of a static method. These methods cannot be used for online reactivity monitoring.
The dynamic methods are all based on the point kinetic equations. The idea is
to investigate the time-dependent flux and to find different parameters derived
from the point kinetics. All methods applied in this work are dynamic methods and
they are described in detail below.
The noise methods are statistical methods used to analyze the inherent statistical behavior of the reactor. The number of radioactive decays emitted by a radioactive medium per unit time is Poisson distributed. However, when a neutron is
emitted through a fission process in a reactor, a number of neutrons will follow
due to the multiplicative properties of the fuel. This implies that the Poisson distribution is disturbed. By investigating the deviation from the pure Poisson distribution it is possible to obtain information about the subcriticality of the core. The
Rossi-α and the Feynman-α methods are examples of reactivity determination
methods where noise techniques are used [26]. These methods are often considered in subcritical systems, but they are not included in this study.
4.1 Area Method
The approach in the area method, also called the Sjöstrand method [27], is to investigate the response to a neutron pulse. The reactivity is achieved by investigation of the neutron flux decay during repeated injections of neutron pulses at constant frequency. After a large number of pulses an equilibrium level of delayed
neutron precursors is obtained. These precursors decay and cause a constant equilibrium level of delayed neutrons, whereas the prompt neutrons constitute a rapidly decaying time-dependent behavior. This is illustrated in Figure 6, where
40000 pulses have been added to each other. The delayed neutrons are represented
by the “delayed area”, Ad, and the prompt neutrons by the “prompt area”, Ap. If the
total amount of neutrons present at the same time in the reactor is divided into
prompt neutrons, np, and delayed neutrons, nd, according to
n (t ) = n p (t ) + nd (t ) ,
29
(29)
Counts/pulse
10
10
10
10
-1
-2
A
-3
p
-4
A
0
5
d
10
15
20
Time [ms]
Figure 6. Cumulative detector counts after
40000 pulse injections.
the point kinetic equations, Eq. (28), can be written as
dn p (t )
ρ − β eff
n p (t ) + soδ (t )
dt
Λ
6
dnd (t ) ρ − β eff
=
nd (t ) + ∑ λi Ci (t )
dt
Λ
i =1
dCi (t ) βi
= (n p (t ) + nd (t )) − λi Ci (t ),
Λ
dt
=
(30)
i = 1,..., 6
where soδ (t ) is the neutron pulse. It is assumed that prompt and delayed neutrons
multiply in the same way in the reactor, which means that they are released with
the same energy spectrum. Since the delayed neutron background is constant, the
last two equations equal zero. By introducing
30
∞
Ap = ∫ n p (t )dt
0
∞
Ad = ∫ nd (t )dt
(31)
0
∞
Ai = ∫ Ci (t )dt
0
the two last equalities of Eq. (30) can be written as
ρ − β eff
Λ
βi
Λ
6
Ad + ∑ λi Ai = 0
i =1
( Ap + Ad ) − λi Ai = 0,
(32)
i = 1,..., 6.
From these relations, the reactivity, expressed in dollars, can readily be obtained
as
Ap
ρ
=−
.
Ad
β eff
(33)
This method has the advantage that the involved parameters are integrals, which
reduces the statistical errors [28].
4.2 Slope Fit Method
When injecting a neutron pulse into a subcritical core, the neutrons will multiply
in the fuel in an exponentially decaying fission chain reaction . The solution of the
point kinetic equations gives the neutron flux after a neutron pulse as [7]
(
n(t ) = n0 β eff λ ′e − λ ′t + ρα eα t
)
(34)
where λ ′ = ρλ /( ρ − β eff ) , λ is the one-group delayed neutron precursor decay constant and α is the prompt neutron decay constant given by
31
α=
ρ − β eff
Λ
.
(35)
Assuming that λ ′t ≈ 0 , Eq. (34) can be written as a constant plus one term with
exponential time-dependence:
(
)
n(t ) = n0 β eff λ ′ + ρα eα t .
(36)
Thereby the decay of the delayed neutrons is neglected and only the prompt neutrons are taken into account. By measuring the neutron flux after a pulse injection
it is therefore possible to find the reactivity of the system by fitting an exponential
function with a decay constant α to the slope.
α can also be expressed as
α=
k p −1
l
(37)
where kp is the prompt multiplication factor, describing the multiplication of
prompt neutrons only and l is the mean lifetime [29]. kp is given by the relation
[30]
k p = keff (1 − β eff ).
(38)
Eq. (37) indicates that this method is in fact a measure of the prompt multiplication. In order to determine the total reactivity and keff, the effective delayed neutron
factor must be known.
It should be noted that the slope fit method is only useful if both the effective
delayed neutron fraction and the neutron generation time are known. These quantities can be determined either by other experiments or by simulations.
4.3 Source Jerk Method
The idea behind the source jerk method is to operate the subcritical reactor at
steady state, at flux level n0, and then suddenly remove the neutron source. At this
point the system will make a prompt jump to a lower level, n1, determined by the
delayed neutron background. This level is only quasistatic and will decay according to the decay rate of the delayed neutron groups [31]. The point kinetic equations describe the neutron level at equilibrium before the source jerk:
32
ρ − β eff
Λ
βi
6
n0 + ∑ λi Ci (t ) + S = 0
i =1
(39)
n0 − λi Ci (t ) = 0,
Λ
i = 1,..., 6
where S is the source strength. The neutron level is found from the first relation:
no =
Λ ⎛ 6
⎞
λi Ci + S ⎟ .
∑
⎜
β eff − ρ ⎝ i =1
⎠
(40)
Immediately after the source jerk the quasistatic level is obtained:
ρ − β eff
Λ
βi
Λ
6
n1 + ∑ λi Ci (t ) = 0
i =1
(41)
n1 − λi Ci (t ) = 0,
i = 1,..., 6
with the neutron level
n1 =
Λ
β eff − ρ
6
∑λ C .
i
i =1
(42)
i
From Eq. (39) we write
6
n
λi Ci = 0
∑
Λ
i =1
6
∑β
i =1
i
=
β eff
Λ
n0
(43)
and the source strength can be written as
S =−
ρ
Λ
n0 .
The ratio between the two different neutron levels is then
33
(44)
n0
= 1+
n1
ρ
S
6
∑λ C
i =1
i
i
= 1− Λ
β
Λ
n0
n0
= 1−
ρ
β eff
(45)
and the reactivity in dollars is obtained from
n −n
ρ
= 1 0.
n1
βeff
The neutron levels n1 and n0 can be obtained by experiment [31].
34
(46)
5 Experimental Setup
Recently, the comprehensive MUSE program (MUltiplication with an External
Source), performed at the MASURCA facility in Cadarache, France, was completed [7]. In the MUSE experiments, a neutron generator consisting of a deuteron
accelerator and a tritium target, was coupled to a subcritical core operating with a
fast neutron energy spectrum. A major part of the experiments was devoted to the
investigation of methods for reactivity determination. The pulsed neutron source
methods, the neutron noise methods and the source jerk method were studied.
Some neutron statics experiments were also performed, such as fission rate distributions and spectral distributions.
Parallel with the MUSE program another European ADS-related experiment
has been running at the Yalina facility outside Minsk, Belarus [32]. This facility
has the same basic construction with a neutron generator coupled to a subcritical
core. The major difference of the two experiments is that the neutron energy spectrum of the Yalina core is thermal. The investigations performed in the present
work are all based on a number of experiments performed at the Yalina facility in
October 2004. The experiments include measurements of fission rate distributions
and reactivity determination.
5.1 The Yalina Facility
The Yalina facility is located at the Joint Institute of Power and Nuclear Research
in Sosny outside Minsk, Belarus. Yalina does not fulfill the conceptual design of a
future ADS, but the neutronics of the subcritical core is an interesting feature since
the methods described is applicable in both fast and thermal systems independent
of the type of neutron source. The construction of facilities of this type is a necessary step towards a full-scale ADS, in order to understand the behavior of subcritical cores and the coupling between the main components; the accelerator, the target and the core.
5.1.1 The Neutron Generator
As the neutrons have no electric charge, they cannot be accelerated into the core.
Instead, the source neutrons in the Yalina experiments are produced through either
of the two following fusion reactions: [10].
35
t + d → 4He + n
Q = 17.6 MeV
d + d → 3He + n
Q = 3.3 MeV
The idea is to accelerate deuterons to an energy about 250 keV towards a replaceable target of either tritium ((d,t)-reaction) or deuterium ((d,d)-reaction). The generator can operate in continuous mode, producing a constant current of deuterons,
or in pulsed mode sending short deuteron pulses with a constant frequency. The
main properties of the neutron generator are found in Table 4.
Table 4. Main parameters of the neutron generator (NG-12-1) [32]
Deuteron energy
100 – 250 keV
Beam current
1 – 12 mA
Pulse duration
0.5 – 100 µs
Pulse repetition frequency
1 – 10000 Hz
Spot size
20 – 30 mm
Maximum neutron
~2.0·1012 ns-1
yield
(d,t)-target
Reaction Q-value
17.6 MeV
Maximum neutron
~3.0·1010 ns-1
yield
(d,d)-target
Reaction Q-value
3.3 MeV
5.1.2 The Subcritical Core
The target is surrounded by the fuel, which consists of uranium dioxide with 10%
enrichment of 235U. The fuel pins are situated in a lattice of quadratic geometry,
depicted in Figure 7. The region closest to the target is filled with lead in order to
obtain a more spallation like neutron spectrum, instead of a mono energetic
source. Outside the lead zone is a moderating region, filled with polyethylene
(C2H4). This is a rather efficient moderator with moderating ratio ( ξΣ s Σ −a1 ) 122
(compared to 71 for light water, see Table 1). The reflector is made of graphite
with a thickness of about 50 cm. Five experimental channels (EC) are placed at
different positions at different radial distances. The relative positions of the experimental channels are in order to minimize their influence on each other. As can
be seen in Figure 7, EC1 is close to the source, EC2 is in the lead zone, EC3 is in
the moderating thermal zone and EC5 and EC6 is situated in the reflector. There
are totally 280 fuel elements, each of them with a diameter of 11 mm. The spacing
between two adjacent elements is 20 mm and the total length of the active fuel is
500 mm.
36
Figure 7. Radial and axial cross section of the Yalina reactor. Dark blue color
represents lead, green color represents the polyethylene zone, yellow color
represents the graphite reflector and light blue color represents the experimental channels.
37
Figure 8. The author in front of the Yalina core.
5.1.3 Detectors
In the experiments, different fission chambers and a 3He-detector were used. The
functioning of each detector is based on the following principle; a neutron induces
charged particles inside the detector and these particles accumulate to a current in
an electric field. The size of the current is proportional to the neutron flux.
In a fission chamber, the charged particles are fission products from fissions
induced by the incoming neutrons. Since different fissile isotopes are sensitive to
different neutron energies, the detectors are sensitive to the energies characteristic
of its fissile component. The isotopes used in the following experiments are 232Th,
235
U and NatU (natural uranium). As is shown in Figure 9, the 235U-chamber is sensible mainly to thermal neutrons, the NatU-chamber is sensible to both thermal and
fast neutrons, whereas the 232Th-chamber is sensible mainly to neutrons with energy higher than about 1 MeV [21].
38
The charged particles created in the 3He-counter are protons from (n,p)reactions. In Figure 10 it appears that the detector is sensitive mainly to thermal
neutrons. The dead-time of the detector is 0.8 µs.
5
10
235
0
σf [b]
10
U
-5
232
10
238
Th
U
-10
10
-4
10
-2
10
0
2
10
10
Energy [eV]
4
10
6
10
Figure 9. Fission cross sections for the materials used in the three different
types of fission chambers (ENDF/B-VI.8) [21].
6
10
3
He
4
σ(n,p) [b]
10
2
10
0
10
-2
10
-4
10
-2
10
0
2
10
10
4
10
6
10
Energy [eV]
Figure 10. Cross section for (n,p)-reaction in 3He (ENDF/B-VI.8) [21].
5.2 Methodology of the Experiments
This study contains three different experiments: measurement of spatial fission
rate distributions, pulsed neutron source measurement and source jerk measurement. How these experiments were performed is explained below.
39
5.2.1 Fission Rate Distributions
The axial fission rate distribution was determined by using the different fission
chambers in EC1, EC2 and EC3. The detector is connected with a time analyzer
(Turbo MCS) with 16 384 channels. The time gate of each channel can be in the
interval from 5 ns to 65 535 s. The time analyzer collects all counts from the detector at each detector position until sufficient statistics is achieved. During this
measurement the neutron generator is working in continuous mode. The measurements performed were
1.
2.
3.
4.
235
U-fission chamber in EC2, (d,d)-source.
U-fission chamber in EC1, EC2 and EC3, (d,t)-source.
Nat
U-fission chamber in EC1, EC2 and EC3, (d,t)-source.
232
Th-fission chamber in EC1, EC2 and EC3, (d,t)-source.
235
5.2.2 Pulsed Neutron Source Measurements
The area method and the slope fit method were performed with the neutron generator operating in pulsed mode with a constant frequency of approximately 43 Hz
and the neutron flux was measured by the 3He-counter in each experimental channel. Every neutron pulse was registered by the time analyzer and all pulses were
added to each other. The final result is therefore a sum of 40 000 pulses.
5.2.3 Source Jerk Measurement
The source jerk method requires the neutron generator to operate in continuous
mode. A sudden source jerk was achieved by rapidly shutting down the neutron
generator. During this operation the neutron flux was measured by the 3He-counter
in EC2.
40
6 Monte Carlo Calculation Method
In order to solve the Boltzmann equation, Eq. (12), the problem can be divided
into discrete energy and space groups. The problem is then solved mathematically
for each group with proper boundary conditions. These types of computer codes,
which approximate the problem by solving the equations in discrete groups, are
called deterministic codes. To solve a problem with a deterministic code the problem must first be approximated with a simple geometry [14].
Another approach is to simulate the neutron transport as a stochastic process.
Given continuous interaction cross sections, physical description of reaction
events and geometry of the problem, a solution to the Boltzmann equation can be
provided by tracking the path of a large number of neutrons through the system. If
the problem is well modeled the average behavior of the neutrons represents the
solution. This technique is called Monte Carlo simulation and can solve problems
without making major approximations in geometry or energy dependence. Exactly
the same result will never occur twice, as with deterministic codes, and the results
always have a statistical error.
All events in Monte Carlo have its origin in random processes. Therefore it is
very important to have access to a true random number generator. It is preferable
to have the random numbers distributed [0,1], then these numbers are transformed
to any other distribution. In every moment an interaction can occur with a specific
probability. The probability density function (pdf), f, is defined as
b
∫ f ( x)dx = 1
(47)
a
and describes the probability density for an event over the interval a ≤ x ≤ b.
Sometimes it is more useful to use the cumulative density function (cdf)
x
F ( x) = ∫ f ( x ′)dx ′
(48)
a
which is the probability that the variable x takes on values less than or equal to x.
If k is a random number distributed between 0 and 1, the numbers F-1(k) are distributed as f(x). In this way the generated random numbers are transformed into
distributions that simulate each event in the reactor [12].
41
6.1 MCNP
One of the most frequently used Monte Carlo neutron transport codes is MCNP
(general Monte Carlo N-Particle transport code) [33], developed at Los Alamos
National Laboratory. MCNP reads the input file, where the geometry, the materials and the neutron source are described, and analyzes the problem. Results of interest can be scored by using tallies. A tally is a specification of what should be
included in the problem output, for example the neutron flux through a certain area
or the number of neutrons in a particular energy interval. In MCNP it is possible to
calculate integrals of the form
C ∫ φ ( E ) f ( E ) dE ,
(49)
where C is a multiplication constant, φ is the neutron flux and f can for instance
be a cross section. In this way, reaction rates with different materials can be determined.
Another powerful property of MCNP is the criticality calculation possibility.
MCNP starts a number of fission chains and when a steady state neutron population is achieved, the number of neutrons in each neutron generation is calculated.
This function is called KCODE and gives an estimation of keff.
6.2 Neutron Time Parameters in MCNP
In MCNP version 4B and later versions, two different types of lifetime parameters
are treated: lifespans and lifetimes. The lifespans describe the time from birth of a
neutron to a special event, e.g. absorption or leakage (birth-to-event), and the lifetimes describe the time between two events of the same type, e.g. fission-to-fission
(event-to-event).
The lifespans describe the time required for a neutron, destined to a specific
event, from the birth of a neutron to the occurrence of the particular event. This is
well suited for implementation in Monte Carlo, since the code tracks every single
neutron chain. The definition of lifespan, t, is
1
tx =
Nx
Nx
∑t
k =1
k
(50)
where N x is the number of neutrons destined to the event x and tk is the time required for the event to occur from the birth of the neutron.
The lifespans do not tell how often an event occurs or how large the probability
for the event is, only how long time it will take for the neutron to undergo the re42
action, if it happens. If also the probability for the event is considered, the lifetime
can be achieved. The removal lifetime, τr, is given by the sum of all lifespans
weighted with their probabilities, P, according to
τ r = ∑ Px t x = Pf t f + Pc tc + Pete ,
(51)
x
where f, c, and e refer to fission, capture and escape, respectively. The other lifetimes (fission, capture, and escape) are given by
τx =
τr
Px
.
(52)
Since the lifetimes also take into account the probability for each event, they describe the true behavior of the neutrons more accurately than the lifespans do.
Table 5 shows the corresponding parameters in reactor theory and in MCNP. The
neutron mean generation time, Λ, (see section 2.6) is neither a lifetime nor a lifespan and is not represented in MCNP. It has to be derived from Eq. (19) or
Eq. (20) [19][34].
Table 5. Time parameters in reactor theory and MCNP.
REACTOR THEORY
Parameter
Name
l
Neutron lifetime
τr
τ
Mean fission time
τf
-
-
τc
-
-
τe
-
-
tr
-
-
tf
-
-
tc
-
Mean generation
time
te
MCNP
Name
Prompt removal lifetime
Prompt fission lifetime
Prompt capture lifetime
Prompt escape lifetime
Prompt removal lifespan
Prompt fission lifespan
Prompt capture lifespan
Prompt escape lifespan
-
-
Λ
Parameter
Description
Removal-toremoval
Fission-to-fission
Capture-to-capture
Escape-to-escape
Birth-to-removal
Birth-to-fission
Birth-to-capture
Birth-to-escape
-
All parameters in reactor theory are adjoint-weighted. This means that each
neutron is weighted with its probability to cause fission, which is called importance. For instance, a neutron in the middle of the core generally has comparatively high importance, since it has high probability for inducing fission (i.e. low
43
probability for leaking out from the system), whereas the opposite is true for a
neutron close to the core periphery or in the reflector.
An important drawback of MCNP concerning the determination of the various
lifetimes, is that all quantities simulated are non-adjoint-weighted. All neutrons
contributing to a certain time parameter are assigned the same importance. This
means that a neutron that has been in the reflector for a long time and causes fission has equal importance when calculating the fission lifetime as a neutron that
lives a short time between two fissions in the fuel. If the flux and importance functions are similar, the differences between the adjoint-weighted lifetimes and the
non-adjoint-weighted lifetimes are small. But if there is a large reflector in the system it can be assumed that the non-adjoint-weighted lifetime estimation will be too
large. Therefore, MCNP can only estimate an upper limit for the neutron mean
generation time.
In bare reactors, where the importance function is more similar to the neutron
flux function, the non-adjoint-weighted mean generation time from MCNP agrees
better with the true value. However, for such cases the fission lifespan gives a better estimation. Since the fission probability is not considered when using fission
lifespan, this value is expected to underestimate the mean generation time. This
has been confirmed by Monte Carlo simulations of the Yalina core, with and without reflector.
44
7 Monte Carlo Simulations of Yalina
The fission rate distribution experiments and the pulsed neutron source experiment
have also been simulated with MCNP. The MCNP input file used in the present
study was prepared by the institute in Sosny.
7.1 Fission Rate Simulation
The fission rates with 235U, NatU and 232Th have been calculated with MCNP, according to relation (49) using data library ENDF/B-VI.8. The results are shown in
Figure 11. According to the diagrams, the thermal flux has a minimum in the middle of the core and two peaks outside the lead zone. On the other hand, the fast
neutron flux has a centered maximum, where the neutron source is located.
18
(d,d)-source, 235U-fission chamber
16
(d,t)-source, 235U-fission chamber
Fission rate/source neutron
Fission rate/source neutron
EC2
16
14
12
10
8
6
4
2
-300
-200
-100
0
100
200
12
10
8
6
4
2
-300
300
EC1
EC2
EC3
14
-200
-100
z [mm]
(d,t)-source, NatU-Fission Chamber
100
200
300
(d,t)-source, 232Th-fission chamber
0.11
0.012
0.09
Fission rate/source neutron
EC1
EC2
EC3
0.1
Fission rate/source neutron
0
z [mm]
0.08
0.07
0.06
0.05
0.04
0.03
EC1
EC2
0.01
0.008
0.006
0.004
0.002
0.02
0.01
-300
-200
-100
0
100
200
0
-300
300
z [mm]
-200
-100
0
100
200
z [mm]
Figure 11. Axial fission rate distributions obtained by MCNP simulation.
45
300
7.2 Criticality Calculation
MCNP offers, as mentioned earlier, an option called KCODE for determination of
keff. This option has been used for the input describing the experimental setup, see
Table 6.
Table 6. keff from KCODE simulation.
Library
keff ± 1σ
ENDF/B-VI.8
0.91803±0.00005
JEFF3.0
0.92010±0.00007
JENDL3.3
0.92114±0.00006
7.3 Delayed Neutron Fraction
The delayed neutron fraction, β, as well as the effective delayed neutron fraction,
βeff, was calculated by Monte Carlo simulations using a modified version of
MCNP [35]. The results are listed in Table 7.
Table 7. Delayed neutron fraction and effective delayed neutron
fraction for different libraries calculated by Monte Carlo technique.
Library
β ± 1σ [pcm]
βeff ± 1σ [pcm]
ENDF/B-VI.8
699.0 ± 5.9
788.4 ± 9.3
JEFF3.0
700.7 ± 5.9
792.5 ± 9.4
JENDL3.3
665.3 ± 5.8
742.1 ± 9.0
7.4 Simulation of the Pulsed Neutron Source Experiment
Neutron pulses and the subsequent flux in the reactor can be simulated by MCNP.
In the real experiment many pulses during several minutes are recorded and added
to each other, which results in a constant level of delayed neutrons. In MCNP,
only one pulse can be simulated and this is done without delayed neutrons.
It is of interest to perform this simulation for different reactivity levels approaching criticality. In MCNP, it can be done just by adjusting the 235U concentration in the fuel. Four different configurations with four different enrichments
have been investigated. The first configuration is the original Yalina configuration
and the fourth is a close-to-criticality configuration. The other two have reactivities in between (Table 8).
46
Table 8. Different reactivity levels of Yalina investigated by MCNP.
keff ± 1σ
Configuration # Enrichment (%)
(KCODE using ENDF/B-VI.8)
1
9.9
0.91803±0.00005
2
11.2
0.95164±0.00005
3
12.2
0.97254±0.00005
4
13.6
0.99764±0.00005
All these configurations have been simulated in a similar manner. A neutron pulse
with a duration of 2 µs is injected at time t = 0. The reaction rate with 3He, (n,p)reaction, is then tracked during the following 20 ms in the experimental channels
at z = 0. The output data can then be analyzed using Minuit from CERN [36] with
function minimization packages for function fitting.
The results from the MCNP simulations are plotted in Figure 12. It is easy to
see how the slope varies with reactivity, as predicted by theory. In the close-tocriticality configuration, the time for establishment of steady neutron flux decay is
faster than in the more subcritical states. Consequently, it is more evident how to
find the decay mode. The core channels have stabilized after only 2 ms. In the
most subcritical case, at least twice as long time is needed.
Figure 12. Neutron pulses simulated by MCNP for different reactivity levels (ENDF/B-VI.8).
47
From the plots in Figure 12, it is possible to determine α. Since the effective
delayed neutron fraction is known, it should be possible to calculate the reactivity
and the effective multiplication factor according to Eq. (35). keff is already calculated with a KCODE-run so this would be a good way to test if the method works
well. However, since the non-adjoint-weighted mean generation time, delivered
from MCNP, is a poor estimation of the true (adjoint-weighted) mean generation
time, this approach fails. Instead, the mean generation time can be calculated using
Eq. (35), Eq. (7) and the effective multiplication constants displayed in Table 8,
according to
Λ adj =
1⎛
1
− β eff
⎜⎜ 1 −
α ⎝ keff
⎞
⎟⎟ .
⎠
(53)
Assuming that the point kinetics is applicable, this value should give an estimation
of the true adjoint-weighted mean generation time. A weighted mean value of the
mean generation time for configuration 1 in Table 9 is about 141 µs. This value
should be representative for the experimental setup. In comparison, the nonadjoint-weighted mean generation time, calculated from the MCNP output, is approximately 370 µs for the most subcritical case and 330 µs for the near-critical
case.
48
Table 9. Results from the MCNP simulation of the pulsed neutron source
experiment, including 1σ statistical errors. ENDF/B-VI.8 is used overall.
β eff *
keff
Experimental
-1
Conf.
(KCODE)
[pcm]
1
0.91803 ±
0.00005
788.4 ±
9.3
2
0.95164 ±
0.00005
750 ±
100
3
0.97254 ±
0.00005
750 ±
100
4
0.99764 ±
0.00005
750 ±
100
channel
EC1
EC2
EC3
EC5
EC6
EC1
EC2
EC3
EC5
EC6
EC1
EC2
EC3
EC5
EC6
EC1
EC2
EC3
EC5
EC6
α [s ]
Λ adj [µs]
-726 ± 16
-713 ± 3
-716 ± 2
-690 ± 7
-679 ± 1
-548 ± 18
-573 ± 6
-584 ± 4
-563 ± 4
-540 ± 2
-414 ± 30
-407 ± 2
-411 ± 8
-406 ± 2
-398 ± 1
-136.4 ± 0.6
-135.2 ± 2.1
-135.0 ± 1.7
-136.2 ± 2.8
-134.2 ± 0.5
133.8 ± 3.0
136.3 ± 0.6
135.7 ± 0.5
140.8 ± 1.5
143.2 ± 0.3
106 ± 4
102 ± 2
100 ± 2
104 ± 2
108 ± 2
86 ± 7
88 ± 3
87 ± 3
88 ± 2
90 ± 3
72 ± 7
73 ± 7
73 ± 7
72 ± 8
74 ± 7
*Rough estimations for configuration 2, 3 and 4.
7.5 Investigation of the Validity of the Point Kinetic Approximation
Since all methods concerning reactivity determination in this study rely on the
point kinetic approximation, the validity of the approximation is of interest. As
mentioned in section 2.7, the point kinetic approximation is valid if the spatial flux
profile and the energy distribution are constant in time. This is difficult to measure
in reality because many measurements at different radial positions are needed to
achieve a good picture of the radial flux shape. However, in MCNP, the flux in an
arbitrary position can readily be calculated. The reaction rate with 3He was simulated at several radial positions at the z = 0 plane up to 8 ms after a neutron pulse.
The axial distribution is assumed to follow the same behavior as the radial distribution.
49
The radial flux profile after the neutron pulse was determined with an interval
of 0.5 ms up to 8 ms. There is a transient during the first milliseconds, followed by
a well established time independent flux profile. During the last milliseconds the
reliability in the data is lower due to bad statistics. In order to quantify the change
of the shape numerically the relative root-mean-square (RelRMS) deviation of the
local peak-to-average is defined as
RelRMS =
1
N
⎛ pi − p0,i
⎜
∑
⎜ p
i =1 ⎝
0,i
N
2
⎞
⎟⎟ ,
⎠
(54)
where N is the number of measurement points, pi is the local value divided by the
mean value and p0,i is the local value divided by the mean value for the initial
moment. The RelRMS-value as a function of time is plotted in Figure 13. Since the
RelRMS-value is approximately constant after 4 ms, the flux profile is approximately constant during the same time interval. This is an indication that the point
kinetic approximation is applicable after 4 ms. However, it must be kept in mind
that there is a one-energy group assumption behind the diffusion equation from
which the point kinetic equations are derived.
8
7
RelRMS [%]
6
5
4
3
2
1
0
0
1
2
3
4
5
6
Time [ms]
Figure 13. RelRMS as a function of time.
50
7
8
8 The Experiments
8.1 Measurements of Fission Rate Distributions
The results from the fission rate measurements are shown in Figure 14. The first
measurement was performed with the (d,d)-source. By comparing the count rates
in the different plots it can be seen that the intensity of this source is much lower
than that of the (d,t)-source. The double peaks come from the high thermal flux in
the moderating material outside the lead zone. In the last plot, the axial distribution of fast fission in 232Th is illustrated, which shows a fast neutron peak in the
middle of the core in EC1. In the center of the core, most of the neutrons are fast,
since they have only been slightly moderated by the lead. Outside the lead zone,
the neutrons are efficiently thermalized, which causes a large thermal flux. This is
the reason why the maximum of thermal neutron flux is located outside the lead
zone, whereas the fast neutron maximum is located in the centre. In the reflector,
almost all neutrons have been thermalized. All this can also be seen in the simulations in section 7.1.
When using NatU, both thermal and fast neutrons are detected. This is why
there is a triple peak in the third plot. The peak in the centre comes from fast fission, as for the 232Th-fission chamber, and the two other peaks come from thermal
fissions, as for the 235U-fission chamber. Of the two thermal peaks, one is slightly
higher, which is explained by the anisotropy of the spatial distribution of the neutrons escaping from the fusion reactions. The energy of the incident deuteron
causes a displacement of the spatial neutron distribution in the forward direction.
51
(d,d)-source, 235U-fission chamber
(d,t)-source, 235U-fission chamber
200
9000
8000
160
7000
Fission rate [s-1]
Fission rate [s-1]
EC2
180
140
120
100
6000
5000
4000
80
3000
60
2000
40
-300
-200
-100
0
100
200
1000
-300
300
EC1
EC2
EC3
-200
-100
z [mm]
2200
(d,t)-source, NatU-Fission Chamber
1800
100
200
300
(d,t)-source, 232Th-Fission Chamber
EC1
EC2
80
70
1600
Fission rate [s-1]
Fission rate [s-1]
90
EC1
EC2
EC3
2000
1400
1200
1000
800
60
50
40
30
20
600
10
400
200
-300
0
z [mm]
-200
-100
0
100
200
0
-300
300
z [mm]
-200
-100
0
z [mm]
Figure 14. Axial fission rate distributions from experiment.
52
100
200
300
8.2 Measurements of the Pulsed Neutron Source Response
The results from the pulsed neutron source measurement, presented in Figure 15,
have been analyzed first by the area method and then by the slope fit method.
EC1
EC2
EC3
EC5
EC6
-1
10
-2
Counts/pulse
10
-3
10
-4
10
0
5
10
15
20
Time [ms]
Figure 15. Neutron pulse responses from the pulsed neutron
source experiment.
8.2.1 Area Method
The data from the experiment, shown in Figure 15, has been dead time-corrected.
In order to analyze this information with the area method, the area under each
curve and the equilibrium level must be calculated. If the background is neglected,
Eq. (33) can be written as
A
ρ
= 1 − tot ,
βeff
Ld T
(55)
where Atot is the total area under the curve, Ld is the equilibrium level of the delayed neutrons and T is the period of the measurement (22.81 ms).
53
The area is calculated by trapezoidal numerical integration and is given by
N −1
ci + ci +1
∆t ⎛
⎞
∆t = ⎜ c1 + 2∑ ci + cN ⎟ ,
Atot = ∑
2
2⎝
i =1
i =2
⎠
N −1
(56)
where ci is the counts in time bin i, ∆t is the width of the time gate and N is the
number of time bins (channels). The number of counts in each time bin is assumed
to be Poisson distributed with the standard deviation
σ (ci ) = ci .
(57)
Using the formula for propagation of error,
2
⎛ ∂X
⎞
σ ( X ) = ∑ ⎜ σ ( xi ) ⎟ ,
i ⎝ ∂xi
⎠
(58)
the standard deviation of the total area is
2
N −1
⎛ ∂A
⎞
∆t
σ ( Atot ) = ∑ ⎜ tot σ (ci ) ⎟ =
c1 + 4∑ ci + cN .
2
i =1 ⎝ ∂ci
i=2
⎠
N
(59)
The equilibrium level, Ld, is calculated as the mean value of the counts from toptimal
to the end. toptimal is found as the time gate, Noptimal, where the quality of fitting a
constant level is best.
1
Ld =
N − N optimal
N
∑
i = N optimal
ci .
(60)
.
(61)
The standard deviation is [37]
N
σ ( Ld ) =
∑
i = N optimal
ci
N − N optimal
It is now possible to calculate the reactivity in dollar with the standard deviation
54
⎛ ∂ ( ρ / β eff )
⎞ ⎛ ∂ ( ρ / β eff )
⎞
⎜
⎟
⎜
σ ( ρ / β eff ) =
σ ( Atot ) +
σ ( Ld ) ⎟
⎜ ∂Atot
⎟ ⎜
⎟
∂Ld
⎝
⎠ ⎝
⎠
2
2
=
2
2
(62)
1 ⎛ σ ( Atot ) ⎞ ⎛ Atotσ ( Ld ) ⎞
⎜
⎟ +⎜
⎟ .
T ⎝ Ld ⎠ ⎝
L2d
⎠
The results are presented in Table 10. Note that only statistical error is taken under
consideration.
Table 10. Results from the area method.
Experimental ρ/βeff (Reactivity in $) ±
channel
1σ
EC1
-13.9 ± 0.1
EC2
-13.7 ± 0.1
EC3
-12.9 ± 0.1
EC5
-13.0 ± 0.1
EC6
-13.5 ± 0.1
The reactivity, ρ, and the effective multiplication constant, keff, can now be calculated since the effective delayed neutron fraction, βeff , is known (Table 7). The effective multiplication constant, calculated using Eq. (7), is found in Table 12. The
standard deviation is calculated using Eq. (58). All values for each experimental
channel are close to each other, within each library the differences are not more
than 0.7%.
Table 11. Reactivity, ρ, calculated from βeff (different libraries) and reactivity in
dollars (from the area method).
Experimental
ρ [pcm]
channel
ENDF/B-VI.8
JEFF3.0
JENDL3.3
EC1
EC2
EC3
EC5
EC6
-10960 ± 160
-10790 ± 170
-10180 ± 130
-10270 ± 140
-10620 ± 160
-11020 ± 160
-10840 ± 160
-10230 ± 130
-10320 ± 140
-10670 ± 160
55
-10310 ± 150
-10150 ± 160
-9580 ± 130
-9670 ± 130
-9990 ± 150
Table 12. keff calculated from βeff (different libraries) and reactivity (from the area
method).
Experimental
keff
channel
ENDF/B-VI.8
JEFF3.0
JENDL3.3
EC1
EC2
EC3
EC5
EC6
0.9012 ± 0.0013
0.9027 ± 0.0013
0.9076 ± 0.0011
0.9069 ± 0.0011
0.9040 ± 0.0013
0.9008 ± 0.0013
0.9022 ± 0.0013
0.9072 ± 0.0011
0.9064 ± 0.0011
0.9036 ± 0.0013
0.9065 ± 0.0013
0.9079 ± 0.0013
0.9126 ± 0.0011
0.9119 ± 0.0011
0.9091 ± 0.0012
8.2.2 Slope Fit Method
As described earlier, α is the logarithmic slope of the response function (Figure
15). However, the fact that the slope in each channel is not constant in time, complicates the analyze. During the first 2 ms, there is a very fast decay in the channels located in the core. This comes from the source neutrons, having a non-fission
spectrum, which enter the fuel first. At the same time, the neutron population detected by the detectors in the non-multiplicative reflector builds up from zero. After a few ms, the neutron population is established in the core and in the reflector,
and a fundamental decay mode is achieved. This decay mode should in principle
be visible in all detectors at the same time, but the time-dependent slope and the
disturbing delayed neutron background makes it difficult to extract one single decay constant. However, a first guess, just by looking at Figure 15 tells us that the
slope should be found after approximately 4 ms.
In order to calculate the reactivity from Eq. (35), the effective delayed neutron
fraction and the neutron generation time are required. These quantities were calculated with MCNP. The problem is now how to find α. As discussed above the decay rate is varying in time, giving rise to different harmonics. The idea is to find
these harmonics and the fundamental mode by fitting a series of exponentials. This
is done by using Minuit from CERN, including function minimization packages
[36]. A function of the form
4
f (t ) = ∑ Ai eαit .
i =1
(63)
is fitted to the measured data points. This method finds two fast harmonics, one
slower harmonic representing the fundamental decay mode, and finally a constant
term. The fitted functions are shown in Figure 16 and the values in Table 13.
56
Table 13. Values from Minuit. The amplitude constants, Ai, are per 40 000 pulses.
The reduced χ2-value describes the quality of the fit (equals unity for a perfect fit).
EC
A1
α1 [s-1]
A2
α2 [s-1]
A3
α3 [s-1]
A4 α4 [s-1] χ2
EC1
1983 -5627 3003 -1628 625 -675±13
9
0
1.13
EC2
935
-5939 2030 -1717 538 -722±19
6
4
1.18
EC3
1272 -6489 4614 -1666 1465 -711±11
17
3
1.04
EC5 -3458 -5576 2371 -946 2084 -634±21
16
0
1.02
EC6 -2838 -2527
0
0
2924 -653±2
9
0
1.08
Figure 16. Functions fitted by Minuit. The dashed line represents the fundamental
decay mode.
The fast constants, α1 and α2, are present only in EC1 to EC3 with positive
sign, and moreover, their high values would give unreasonable results if they were
used. Therefore α3 is chosen as the fundamental decay constant, since it is present
in all channels with similar values. The decay constant α3 gives the reactivities and
keff in Table 14.
57
Table 14. Reactivity and effective multiplication factor obtained by experiment
using the slope fit method and the MCNP-calculated ENDF/B-VI.8-values of Λ
and βeff.
EC1
EC2
EC3
EC5
EC6
-8240±
-9050±
-8870±
-8130±
-8560±
ρ [pcm]
260
260
160
310
30
0.9239±
0.9170±
0.9185±
0.9248±
0.9211±
keff
0.0022
0.0022
0.0013
0.0026
0.0002
The slope fit method in combination with the area method provides a way to
find the fraction Λ/βeff experimentally. By rewriting Eq. (35), Λ/βeff can be found
by the following expression:
Λ
β eff
=
⎞
1⎛ ρ
− 1⎟ .
⎜⎜
⎟
α ⎝ β eff
⎠
(64)
In this expression, α is found from above and ρ/βeff is found from the area method.
Using the MCNP-values of the effective delayed neutron fraction a “semiexperimental” value of the mean generation time is achieved, see Table 15. This
confirms that the value calculated before, approximately 141 µs, is a better estimation than the non-adjoint-weighted value of 370 µs.
Table 15. Experimental values of Λ/βeff. Effective delayed neutron
fraction from different libraries gives the mean generation time.
Λ [µs]
Λ [µs]
Λ [µs]
EC Λ/βeff [103 µs]
ENDF/B-VI.8
JEFF3.0
JENDL3.3
EC1
22.1±0.4
174.1±4.1
175.0±4.1
163.8±3.9
EC2
20.3±0.6
160.2±4.8
161.1±4.9
150.8±4.6
EC3
19.5±0.3
154.1±3.1
154.9±3.2
145.0±3.0
EC5
22.1±0.7
174.5±6.2
175.4±6.2
164.3±5.9
EC6
22.2±0.2
174.6±2.5
175.5±2.6
164.4±2.4
8.3 Source Jerk Experiment
The data from the source jerk experiment is shown in Figure 17. According to
Eq. (46), the source jerk method requires determination of two neutron flux levels.
The first flux level is evaluated as the end value of a function fitted to the data before the source jerk, and the second flux level is the start value of a function fitted
to the data after the source jerk. Before the jerk, the source is expected to be constant and a linear function is used:
58
f (t ) = a0t + b0
(65)
The constants a0 and b0 are found by minimizing the function
N
( yi − a0ti − b0 )
i =1
σ i2
S =∑
2
,
(66)
where N is the number of measurements, yi is the measured value at time ti, and σi
is the standard deviation of each measurement. Since the measured values are
Poisson-distributed, the standard deviation is
σ i = yi .
Figure 17. Data from source jerk experiment.
59
(67)
By introducing
N
A=∑
i =1
ti
σ
N
C=∑
i =1
N
E=∑
i =1
N
1
i =1
σ i2
N
ti2
B=∑
2
i
yi
D=∑
σ i2
i =1
N
ti yi
F =∑
σ i2
i =1
(68)
σ i2
yi2
σ i2
the result can be written as
a0 =
EB − CA
DB − A2
(69)
b0 =
DC − EA
DB − A2
(70)
with the standard deviations
σ (a0 ) =
B
BD − A2
(71)
σ (b0 ) =
D
.
BD − A2
(72)
Using this method the equilibrium neutron level before the source jerk is found as
n0 = a0tsource jerk + b0
σ (n0 ) =
(t
σ (a0 ) ) + σ (b0 ) 2
2
source jerk
.
(73)
Immediately after the removal of the source, there will be a slow decay due to
the decay of the delayed neutron precursors. This decay is better described by an
exponential function containing six exponentials, one for each delayed neutron
group. Unfortunately the low count rate makes it difficult to distinguish six different groups, so another approximate method must be used. The simplest method is
to make a linear fit, as before, but care must be taken since this implies the use of
only one delayed neutron group. Such approximation is only valid during a short
60
time after the source jerk. The validity of the method, or in other words, the quality of the fit, can be evaluated using the reduced χ2-value, given by
χ2
v
=
Smin
,
v
(74)
where v is the degree of freedom given by the number of measurements, N, minus
the number of extracted parameters, m.
v = N −m = N −2≈ N
(75)
The reduced χ2-value should be close to 1 for a good fit. Using a short interval
yields a reduced χ2-value close to one, since the data can be approximated by one
exponent. On the other hand, a short interval implies large statistical error in the
result. A longer interval gives smaller statistical error in the result, but decreases
the reduced χ2-value. There is obviously a balance between accuracy of the model
and size of the statistical error, see Figure 18. There seems to be an optimum after
almost 1200 points, which means 1.2 s after the source jerk.
This procedure gives the values in Table 16, neglecting the background. In
Table 17, the reactivity in pcm is given, together with keff and the mean generation
time, calculated from Eq. (7), Eq. (64) and the values in Table 7. The error bars
are very large, due to poor statistics in the lower level n1.
Table 16. Results from source jerk method.
Parameter
Value ± 1σ
-1
n0 [ms ]
99.3 ± 0.7
n1 [ms-1]
10.1 ± 1.7
-8.9 ± 1.7
ρ/βeff
Table 17. Reactivity, keff and mean generation time calculated from the
source jerk results.
Parameter
ENDF/B-VI.8
JEFF3.0
JENDL3.3
ρ [pcm]
-6990 ± 1330 -7020 ± 1340
-6580 ± 1250
0.935 ± 0.012 0.934 ± 0.012
0.938 ± 0.011
keff
Λ [µs]
114 ± 9
115 ± 9
108 ± 8
61
1.1
100
Reduced χ
2
90
σ(n1)
1.05
80
70
2
50
0.95
σ(n1) [%]
60
χ /v
1
40
30
0.9
20
10
0.85
200
400
600
800
1000
1200
1400
i, Number of fitted points
1600
1800
2000
2200
0
Figure 18. Reduced χ2-value and standard deviation of n1 as a function
of number of measurement points after source jerk taken into account in
the fitting process. The values chosen for the analysis is marked with
red circles.
62
9 Discussion of Results
9.1 Comparison between Experiments and Simulations
9.1.1 Fission Rate Distributions
By plotting the fission rate distributions from the experiment (Figure 14) and from
simulation (Figure 11) in the same diagram (Figure 19), it appears that the curves
fit each other well. This is a good indicator that MCNP describes the reality accurately. The comparatively small discrepancies are mainly due to low resolution in
the z-direction.
(d,d)-source, 235U-fission chamber
1.2
1.1
Exp, EC2
MCNP, EC2
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-300
Exp, EC1
Exp, EC2
Exp, EC3
MCNP, EC1
MCNP, EC2
MCNP, EC3
1
Relative Fission Rate
Relative Fission Rate
1.1
(d,t)-source, 235U-fission chamber
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
-200
-100
0
100
200
0.1
-300
300
-200
-100
z [mm]
(d,t)-source, NatU-fission chamber
Exp, EC1
Exp, EC2
Exp, EC3
MCNP, EC1
MCNP, EC2
MCNP, EC3
Relative Fission Rate
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
100
200
300
(d,t)-source, 232Th-fission chamber
Exp, EC1
Exp, EC2
MCNP, EC1
MCNP, EC2
1.2
1
0.8
0.6
0.4
0.2
0.2
0.1
-300
1.4
Relative Fission Rate
1.1
0
z [mm]
-200
-100
0
100
200
0
-300
300
z [mm]
-200
-100
0
100
200
300
z [mm]
Figure 19. Axial fission rate distributions normalized per channel from experiment
and simulation.
63
9.1.2 Pulsed Neutron Source Experiment
As mentioned before, no delayed neutron background is achieved when simulating
the pulsed neutron source experiment in MCNP. In order to compare the simulation with the experiment, this background must be subtracted from the experimental data. The results are depicted in Figure 20. This indicates that the mean generation time calculated by MCNP through the slope fit method should give an accurate value of the adjoint-weighted mean generation time. Figure 20 also illustrates
the difficulty to find the fundamental decay mode in EC1 to EC3, due to low statistics after 5 ms.
Figure 20. Comparison between data from experiment and MCNP (arbitrary
units). The delayed neutron background is removed from the experimental data.
64
9.2 Comparison between the Different Reactivity Determination
Methods
All results concerning determination of the effective multiplication factor are
summarized in Table 18 and Figure 21. The Monte Carlo option KCODE is well
validated and benchmarked and is supposed to give reasonable approximations of
the effective multiplication factor. Based on the values in Table 18, keff = 0.92 is
chosen as a reference value. When the experimental values are compared with this
reference level it can be concluded that the area method gives a lower value, the
source jerk method gives a higher value and the slope fit method gives a value
fairly close to the simulated value. Errors which not are originating from the statistical treatment, such as modeling approximations, systematic errors in the measurements etc, are not taken into account. These errors are expected to be large for
the slope fit method due to uncertainties in the determination of α.
Table 18. Comparison of all effective multiplication constants calculated.
ENDF/BVI.8
EC1
0.9012±
0.0013
0.9008±
0.0013
0.9065±
0.0013
0.9239±
0.0022
ENDF/BVI.8
-
JEFF3.0
-
JENDL3.3
-
ENDF/BVI.8
Area
JEFF3.0
JENDL3.3
Slope Fit
Method
Source
Jerk
KCODE
EC2
0.9027±
0.0013
0.9022±
0.0013
0.9079±
0.0013
0.9170±
0.0022
0.935±
0.012
0.934±
0.012
0.938±
0.011
ENDF/BVI.8
JEFF3.0
JENDL3.3
EC3
0.9076±
0.0011
0.9072±
0.0011
0.9126±
0.0011
0.9185±
0.0013
EC5
0.9069±
0.0011
0.9064±
0.0011
0.9119±
0.0011
0.9248±
0.0026
EC6
0.9040±
0.0013
0.9036±
0.0013
0.9091±
0.0012
0.9211±
0.0002
-
-
-
-
-
-
-
-
-
0.91803±0.00005
0.92010±0.00007
0.92114±0.00006
65
0.95
0.94
ENDF/B-VI.8
JEF3.0
JENDL3.3
Slope fit method
KCODE
keff
0.93
0.92
Area method
Source jerk
0.91
method
0.9
0.89
Figure 21. Comparison of all effective multiplication constants calculated. Note that only statistical errors are considered.
At this moment it is important to remember what exactly the different methods
measure. In the derivation of the area method, there was no distinction made between delayed and prompt neutrons. Since the delayed neutrons have a softer
spectrum than prompt neutrons, the area method should give a result deviating
somewhat from the point kinetics. When deriving the other two methods no such
approximations were made.
However, the slope fit method and the source jerk method have some other
drawbacks. For instance, the slope fit method is difficult to use for large subcriticalities, such as the Yalina configuration, due to the time-dependence of the decay
slope. According to the MCNP-simulations, this method is easier to apply for systems closer to criticality, as the fundamental decay mode is achieved faster. The
source jerk method, on the other hand, has the problem with bad statistics in the
lower neutron level, and in some cases it can be difficult to know where to find the
start of this level.
In Table 19, all values of the neutron mean generation time are summarized.
The different results from the reactivity determination are reflected in the estimation of the mean generation time, since the calculations are based on the same experiments. The results show the same pattern as the effective multiplication fac-
66
tors, but in the opposite direction. The area method in combination with the slope
fit method gives higher values than MCNP, and the source jerk method in combination with the slope fit method gives lower values.
Table 19. Comparison of all neutron mean generation times calculated [µs].
Area +
Slope
fit
ENDF/BVI.8
JEFF3.0
JENDL3.3
Method
Source
jerk +
Slope
fit
MCNP
EC1
174.1 ±
4.1
175.0 ±
4.1
163.8 ±
3.9
ENDF/BVI.8
-
JEFF3.0
-
JENDL3.3
-
ENDF/BVI.8
133.8 ±
3.0
EC2
160.2 ±
4.8
161.1 ±
4.9
150.8 ±
4.6
114 ±
9
115 ±
9
108 ±
8
136.3 ±
0.6
EC3
154.1 ±
3.1
154.9 ±
3.2
145.0 ±
3.0
EC5
174.5 ±
6.2
175.4 ±
6.2
164.3 ±
5.9
EC6
174.6 ±
2.5
175.5 ±
2.6
164.4 ±
2.4
-
-
-
-
-
-
-
-
-
135.7 ±
0.5
140.8 ±
1.5
143.2 ±
0.3
9.3 Comparison with the MUSE-experiments
Since the methods used in this study, were also investigated in the MUSE program, it is of interest to make a comparison. In Table 20, some results from MUSE
are summarized. It is interesting to note that, also in these experiments, the area
method gives a lower estimation of the criticality. This is the case in six of the
cases in Table 20; in configuration SC2 the results are similar. The same conclusion, that the area method gives lower values, can be drawn from MUSE, even
though the spectrum is fast rather than thermal, like in Yalina.
Table 20. Results from the MUSE-experiments [28]. The values represent negative
reactivities in $. (The source jerk method was not included in the study of ref.
[28].)
Configuration
Area method
Slope fit method
4SR↑PR↑
1.57 ± 0.02
1.48 ± 0.02
4SR↑PR↓
1.96 ± 0.04
1.90 ± 0.03
SC0
11.0 ± 0.8
10.0 ± 0.7
3SR↑SR1↓PR↓
3SR↑SR2↓PR↓
12.2 ± 1.0
11.4 ± 0.7
4SR↑PR↑
8.1 ± 0.4
8.3 ± 0.7
SC2
4SR↑PR↓
8.5 ± 0.4
8.7 ± 0.7
4SR↑PR↑
11.9 ± 0.9
11.1 ± 0.9
SC3
4SR↑PR↓
12.3 ± 1.0
11.7 ± 0.9
67
68
10 Conclusions
Three reactivity determination methods, based on the point kinetic equations, and
spatial fission rate distributions have been performed experimentally. Both the
spatial and the kinetic experiment have been simulated by MCNP. The simulations
provided parameters, such as effective delayed neutron fraction and mean generation time, which was necessary for the evaluation of the reactivity and the effective multiplication constant from the experiments.
From the measurements it can be concluded that:
• The area method underestimates the criticality, but gives low statistical error.
• The slope fit method is difficult to use, but seems to give reliable results.
The method gives better results closer to criticality.
• The source jerk method overestimates the criticality and is connected with
large uncertainties.
Moreover, from the simulations it can be concluded that:
• MCNP gives reliable results, which was shown by both the static and the
kinetic simulations in comparison with the experiments. However, care
must be taken when analyzing the time parameters.
• The point kinetic approximation describes the system well, which is also
confirmed by the experiments.
In this study, reactivity differences were not included. Even if the methods do
not predict the absolute value of the reactivity absolutely correct they may be able
to predict reactivity changes well.
None of these methods are able to measure the reactivity without disturbing a
running system. However, the methods can be used to estimate the subcriticality
during loading and for calibration of other possible measurement techniques. One
alternative way to measure reactivity changes is to compare the proton current, ip,
from the accelerator, with a neutron flux level in the ADS, φ . The flux-to-current
ratio obeys
φ
ip
∝
Zϕ *
ρ
,
(76)
where ρ is the reactivity, Z is the neutron yield and ϕ * is the neutron source efficiency [24]. The two quantities mentioned last are assumed to be constant during
69
short time perspectives, which means that a sudden change in the flux-to-current
ratio is caused by a reactivity perturbation.
Although the neutron spectrum of Yalina is thermal and has different kinetic
parameters than in MUSE, the results from the different experimental programs
are in agreement with each other.
70
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