1. science
2. mathematics
3. geometry

International Conference
Nuclear Energy for New Europe 2009
Bled / Slovenia / September 14-17
1BOUPOF#MVF$
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Calculations of Effective Sample Mean Chord Length for
Anisotropic Neutron Flux
ˇ
G. Zerovnik,
A. Trkov, L. Snoj
Joˇzef Stefan Institute
Jamova 39, SI-1000 Ljubljana, Slovenia
[email protected], [email protected], [email protected]
ė#
ABSTRACT
The mean chord length parameter defined by the equivalence principle in reactor physics has
been generalized for arbitrary angular neutron flux. Analytical expressions for cylindrical samples and realistic neutron sources have been derived and verified by Monte Carlo simulations.
The extension of the MATSSF code for self-shielding factor calculations in neutron activation
analysis (NAA) by including generalization of the mean chord length estimator has been justfied
by comparing calculated self-shielding factors to the ones obtained by Monte Carlo method.
1
INTRODUCTION
Probability of a neutron absorption, scattering, or escape event in a sample is closely related
to its geometry, more specifically the average length of the sample ’as seen by the neutrons’,
i.e. the sample mean chord length. It can be shown [1] that the sample mean chord length
is a function of sample surface area and volume, and does not depend on the shape of the
sample. However, the expression holds only for convex samples and more importantly, isotropic
neutron flux distribution. Thus, we introduced the so-called generalized mean chord length of
the sample, taking into account the anisotropy of the flux distribution.
We formulated an expression for neutron generalized mean chord length in sample for arbitrary angular flux and sample geometry. We showed that the generalized expression reduces to
the ordinary mean chord length in case of isotropic flux. Additionaly, for samples of cylindrical
shape, we derived analytical expressions for generalized mean chord length for different angular flux distributions. We verified our results by Monte Carlo neutron transport code MCNP [2]
calculations.
The purpose of our derivations was to improve and generalize the MATSSF code [3], which
is used to calculate the self-shielding factors of samples for neutron activation analysis (NAA).
The numerical model in MATSSF unites all spatial effects in one parameter, mean chord length.
By including analytical expressions of generalized mean chord length, the usage of the code
can be extended to different (and more realistic) sample - neutron source configurations without
losing computational efficiency.
118.1
118.2
2
MATSSF CODE
The MATSSF code [3] was developed for calculations of sample thermal and epithermal
self-shielding factors for NAA. Self-shielding factors can be calculated for arbitrary cylindrical samples and different neutron source configurations assuming Maxwellian spectrum below
0.55 eV and 1/E spectrum above 0.55 eV.
A library for the MATSSF code containing the epithermal self-shielding factors was generated from the ENDF/B-VII evaluated nuclear data library [4] using the NJOY data processing
system [5]. The integration limits are 0.55 eV to 2 MeV. The self-shielding factors are tabulated as a function of the microscopic dilution cross section σ0 = Σ0 /N , where N is the number
density of the absorber. The MATSSF code calculates the dilution cross section according to the
sample composition and geometry and obtains the self-shielding factor for the single resonance
absorber by interpolation from the tables.
There are cases when other resonance nuclides admixed with the absorber nuclides under
investigation cause interference. Therefore, a multigroup library of cross sections in 640 energy
groups was generated in order to solve the neutron slowing-down equation for an arbitrary
mixture of nuclides. The group structure is the same as used in the IRDF-2002 dosimetry library
[6]. The revised self-shielding factor takes into account the flux depression (or an increase
because of scattering resonances) due to other isotopes present in the sample.
The calculation method for thermal self-shielding factors follows mainly the work of De
Corte [7]. However, in this paper we concentrate on the epithermal self-shielding, mainly the
generalization of the mean chord length introduced by equivalence principle.
2.1
Resonance self-shielding
Strong resonances deplete the neutron spectrum at the resonance energy due to absorption
and scattering, therefore the reaction rate is reduced because of the dip in the neutron spectrum.
This is the so-called resonance self-shielding effect [8]. For an infinite homogeneous system of
absorber k and moderator m it is described by the integral slowing-down balance equation with
removal reaction rates on the left and the source terms (including downscattering) on the right:
Z E/αm
Z E/αk
Σs,m
Σs,k (E 0 ) ∗ 0
0
∗
φ (E )dE +
φ∗ (E 0 )dE 0 , (1)
[Σt,m + Σt,k (E)] φ (E) =
0
0
(1
−
α
)E
(1
−
α
)E
k
m
E
E
where α = [(A − 1)/(A + 1)]2 for a nuclide of mass number A, Σt is the total and Σs is the
scattering macroscopic cross section, while φ∗ is the perturbed neutron spectrum within the
sample [5]. The first integral on the right handside corresponds to the scattering on the absorber
nuclide whereas the second represents the moderator and/or the background.
If the moderator is ideal, non-absorbing, with a constant scattering cross section Σm =
Σs,m = Σt,m , the effect of the flux perturbation on the second integral on the right handside is
small and can be neglected. Assuming φ∗ (E) ' φ(E) = 1/E, the second integral on the right
handside can be substantially simplified:
Z
E/αm
E
Σs,m
Σm
∗
0
0
φ
(E
)dE
'
= Σm φ(E).
(1 − αm )E 0
E
(2)
Based on the equivalence principle (Ref. [5]), the effects of geometrical self-shielding can
be added to the moderator cross section to form a single macroscopic background (or dilution)
Proceedings of the International Conference Nuclear Energy for New Europe 2009, Bled, Slovenia, Sept.14-17, 2009
118.3
cross section Σ0 . In its simplest form, for cylindrical samples of radius r and height d in
isotropic neutron flux the following expression can be derived:
Σ0 = Σm +
a∗
,
l
l=
4V
2rd
=
,
S
r+d
(3)
where l denotes the mean chord length of the sample cylinder and a∗ is the Bell factor. Commonly adopted value of the Bell factor is a∗ = 1.16; in this paper Bell factor is used to tune the
results for specific sample-source configuration, as discussed later. Also, generalization of the
parameter l in MATSSF is described below.
2.2
Generalization of the mean chord length
The latest version of MATSSF uses a generalized expression for the parameter l used in
Eq. (3). Let us define l as the length of the sample, ’seen’ by the average neutron
R
~ Ω
~
V Φ(Ω)d
l=R
,
(4)
~
~ Ω
~
S⊥ (Ω)Φ(
Ω)d
~ is the angular flux, and S⊥ (Ω)
~ component of the surface of the sample, normal to
where Φ(Ω)
~ As will be explained in further discussion, the mean chord length is just an
the direction Ω.
example of the length l according to the above definition.
Eq. (4) implicitly assumes that the flux at the position of the sample is not space dependent.
In other words, the sample is small enough to neglect the effects of finite dimensions of the
neutron source.
Eq. (4) is valid for arbitrary (neutron) source and sample. The only requirement is that the
sample is of convex shape.
In the MATSSF code, the sample body is restricted to cylinders, which is sufficient for most
realistic cases. Thus, in Eq. (4), V = πr2 d and
S⊥ = 2rd| cos α| + πr2 | sin α|,
(5)
where α is the angle relative to the normal plane of the axis of the cylinder (same as the polar
coordinate in spherical coordinate system).
2.2.1
Isotropic neutron source
~ = 1. In spherical coorFor isotropic neutron source, the angular flux is constant: Φ(Ω)
dinates (if we take the sample cylinder’s axis perpendicular to the azimuthal coordinate ϕ),
S⊥ (α) → S⊥ (ϑ), where ϑ is the polar coordinate. Taking the symmetries of the cylinder and
the sphere into account, Eq. (4) yields
R π/2 R π/2 2
πr d cos ϑdϑdϕ
0
l = R π/2 R π/2 0
(2rd cos ϑ + πr2 sin ϑ) cos ϑdϑdϕ
0
0
2rd
4V
=
=
.
(6)
r+d
S
Thus, in the case of isotropic neutron source, the generalized mean sample length l leads to the
well-known expression 4V /S for the mean chord length of the sample.
Proceedings of the International Conference Nuclear Energy for New Europe 2009, Bled, Slovenia, Sept.14-17, 2009
118.4
2.2.2
Cylindrical source
A voided irradiation channel passing through the reactor can be approximated by an isotropic
cylindrical source with radius R and height H.
For a uniform infinite source of cylindrical shape, the angular flux in the axis of the cylinder
is proportional to:
1
,
(7)
cos α
where α is the angle relative to the normal plane of the axis of the cylinder.
Angular flux in the middle of the channel (at the expected position of the sample) is proportional to:
H
H
,
arctan
1/
cos
α,
α
∈
−
arctan
2R
2R
~ =
(8)
Φ(Ω)
0,
elsewhere.
~ =
Φ(Ω)
Furthermore, we considered two different examples of sample orientation.
First, let us consider the sample cylinder (foil of wire) with its axis parallel to the source
cylinder. If we define spherical coordinate system where the equatorial plane is normal to the
cylinder axis, then in Eq. (5) we just have to replace the angle α with the polar coordinate ϑ.
Obviously, the system source-sample is axially symmetric. Regarding all other symmetries, the
parameter l in Eq. (4) reduces to
R ϑmax 2 1
πr d cos ϑ cos ϑdϑ
πr2 d ϑmax
0
=
l = R ϑmax
(2rd sin ϑmax + πr2 (1 − cos ϑmax ))
(2rd cos ϑ + πr2 sin ϑ) cos1 ϑ cos ϑdϑ
0
q
H2
H
πrd 1 + 4R
2 arctan 2R
q
.
=
(9)
H
H2
d R + πr
1 + 4R2 − 1
Second, if the sample cylinder (usually wire) is lying in the irradiation channel so that the
axes of the source and the sample cylinders are perpendicular, Eq. (5) requires a transformation.
The result (in spherical coordinates) is not very simple and intuitive:
q
S⊥ = 2rd cos2 ϑ cos2 ϕ + sin2 ϑ + πr2 | cos ϑ sin ϕ|.
(10)
The integration of Eq. (4) yields
H
π 2 r2 d arctan 2R
,
l=
2H
H
πr
2 2rd g arctan 2R + q H 2
2R
1+
(11)
4R2
where g(x) is a non-elementary uniformly continuous real function, defined for x ∈ [0, π/2]:
Z π/2
Z xq
g(x) =
dϕ
cos2 ϑ cos2 ϕ + sin2 ϑ dϑ.
(12)
0
0
For our purposes, approximation by 8th degree polynomial in the given range
g(x) ' 0.999105x + 0.029183x2 + 0.427788x3 − 0.456087x4 +
+ 0.339182x5 − 0.185970x6 + 0.059748x7 − 0.008226x8
is sufficient, since the relative error of (13) is within 0.1 % on its domain.
Expressions (9) and (11) are incorporated as an option in the MATSSF code [3].
Proceedings of the International Conference Nuclear Energy for New Europe 2009, Bled, Slovenia, Sept.14-17, 2009
(13)
118.5
2.3
Generalization of the Bell factor
The equivalence principle takes into account only first order interactions with neutrons.
When the wire is lying flat in the channel (perpendicular to the channel axis), most of the neutrons intersect the sample along the short dimension and therefore the generalized mean chord
length is relatively short. If the wire contains a significant amount of material with strong
scattering resonances (such as Ni), many neutrons are deflected along the wire axis and consequently ’see’ a longer flight-path, what increases the reaction rate and reduces the self-shielding.
This can be accounted for by a suitable first order adjustment of the Bell factor. The values of
the Bell factor adopted for different sample-source configurations are given in Table 1. Note
that the described Bell factor adjustment is valid for long (H R) irradiation channels only.
Table 1: Bell factor for different source-sample configurations. Σs and Σt are full sample onegroup scattering and total cross sections, respectively.
sample-source configuration
a∗
Wire along channel axis (parallel)
1.16
Wire lying flat in the channel (perpendicular) 1.30 + 0.5 Σs /Σt
Spherical source (isotropic)
1.16
3
MCNP MODEL
The most rigorous treatment of self-shielding that takes the energy-dependence of cross sections and geometry into account is to directly apply the Monte Carlo technique. The MCNP5
code (Ref. [2], pp. VIII. 9-13) and the associated library based on ENDF/B-VII.0 data [4] were
used in the calculations. The irradiation facility was modelled by a cylindrical isotropic surface source, emitting neutrons with a pure 1/E distribution in epithermal range (from 0.55 eV
to 2 MeV). The obtained reaction rates were proven to be insensitive (within the reasonably
small statistical error) to small deviations from the 1/E spectrum, e.g. spectrum of light water
reactors. The radius of the source surface was equal to the radius of the irradiation channel in
TRIGA Mark II reactor [9]. The height of the surface source was taken equal to the effective
core height. The monitor material samples were modelled explicitly. Two separate runs were
made for each sample, one with full geometry including the sample material, and the other with
void in the sample material cell (zero density). The true self-shielding factor was calculated
as the ratio of the perturbed and the unperturbed (zero density) reaction rates. The number of
particle histories was sufficiently large so that the statistical uncertainty (in this paper always
stated as one standard deviation) in the ratio was reasonably small.
4
VERIFICATION OF THE MATSSF MEAN CHORD LENGTH ESTIMATOR
The MATSSF mean chord length estimators (9) and (11) were verified by comparison with
Monte Carlo mean-free-path-in-cell calculation for realistic source-sample configurations. In
Fig. 1, cylindrical source of height H = 30 (note: default unit is cm but in this discussion units
are intentionally omitted due to the dimensionlessness of the problem) was taken as a function
of its radius R for different sample sizes and orientations. Examples shown in Fig. 1 are: thick
foil (r = 0.4, d = 0.025), thin foil (r = 0.4, d = 0.002), thick wire (r = 0.1, d = 1), and thin
wire (r = 0.01, d = 1), for two orientations: along the source cylinder axis, and perpendicular
Proceedings of the International Conference Nuclear Energy for New Europe 2009, Bled, Slovenia, Sept.14-17, 2009
118.6
to the axis. The source and sample are always centred in the same point. Generally, the results
agree within the statistical error of the Monte Carlo calculation.
Figure 1: Generalized mean chord length l of the cylindrical sample as a function of the cylindrical source radius R for source height H = 30, and different sample radius r and height d. Red
dashed curves and corresponding data points with error bars represent l according to Eq. (11)
and Monte Carlo calculation, respectively, both for perpendicular orientation of the cylinders.
On the contrary, green dotted curve indicates parallel orientation of the sample with respect to
the source. Grey line shows the reference mean chord length 4V /S for isotropic source.
5
SELF-SHIELDING FACTORS
Detailed analysis was performed for a realistic sample of a 1 mm thick wire, about 3.8 mm
in length, made of nickel-based alloy with the following (mass percent) composition:
Ni
Mo
W
Mn
Au
80.93 % 15.16 % 2.76 % 0.41 % 0.29 %
The remainder (0.45 %) was arbitrarily assigned to Fe. The density of the material was 9.21 g/cm3 .
The source was placed on a cylindrical surface of radius R = 1.2 cm and effective height
H = 30 cm, representative of an irradiation channel in TRIGA reactor [9], or placed on a
spherical surface (Fig. 2).
First, we compared the mean chord length as calculated in MATSSF and MCNP for all
three source-sample configurations (Table 2). The MCNP calculated values agree very well
with the corresponding MATSSF values again verifying the expressions (9) and (11), which
differ significantly for different sample configurations, justifying the use of MATSSF mean
chord length estimator.
Epithermal self-shielding factors of the nickel alloy were calculated for all nuclides of interest for NAA for different neutron sources and orientations of the sample and are given in
the Table 2. A single-isotope analysis was performed, assuming that the number density of
Proceedings of the International Conference Nuclear Energy for New Europe 2009, Bled, Slovenia, Sept.14-17, 2009
118.7
Figure 2: Three different source-sample configurations: cylindrical source with sample wire
along channel axis (left), cylindrical source with sample lying flat (middle), and spherical source
(right). Note: source and sample dimensions are not to scale.
Table 2: Comparison of generalized mean chord length l and resonance self-shielding factors
for the constituents of the nickel-alloy wire for different source-sample configuration, calculated
by MATSSF and MCNP computer codes. Isotopes are treated separately.
code
MATSSF
MCNP
MATSSF
MCNP
MATSSF
MCNP
source
cylindrical parallel
spherical
cylindrical perpendicular
l [cm]
0.0988
0.0988
0.0884
0.0883
0.0853
0.0851
isotope
resonance self-shielding factors
55
0.9934
0.992 ± 0.001
0.9940
0.996 ± 0.000
0.9963
0.997 ± 0.000
Mn
58
0.9911
0.980 ± 0.001
0.9918
0.995 ± 0.000
0.9945
0.999 ± 0.000
Ni
64
Ni
0.9997
1.000 ± 0.001
0.9997
1.000 ± 0.000
0.9998
1.000 ± 0.000
92
0.9919
0.994 ± 0.008
0.9928
0.996 ± 0.002
0.9954
0.995 ± 0.002
Mo
98
0.9290
0.939 ± 0.006
0.9349
0.950 ± 0.002
0.9546
0.953 ± 0.002
Mo
100
0.9337
0.946 ± 0.009
0.9396
0.953 ± 0.002
0.9594
0.957 ± 0.002
Mo
184
W
0.9629
0.966 ± 0.005
0.9664
0.976 ± 0.001
0.9770
0.979 ± 0.001
186
W
0.7864
0.767 ± 0.004
0.8011
0.814 ± 0.002
0.8445
0.829 ± 0.002
197
Au
0.9158
0.927 ± 0.004
0.9236
0.942 ± 0.001
0.9344
0.943 ± 0.001
the nuclide under investigation was the same as in the real material, but the number densities
of all other components were zero, thus ignoring the resonance interference between different
isotopes. The values obtained by MATSSF and MCNP are in separate columns. Also, three
different source-sample configurations are considered.
The agreement between the two methods (MATSSF and MCNP) is generally very good
for all nuclides and source-sample configurations. For isotopes with significant self-shielding
effect (e.g. 186 W) partucularly note the deviations in factors for different angular fluxes. The
difference in self-shielding factors for different realistic source-sample configurations is usually
larger than the difference between the two methods. Even if there is some systematic deviation
between MCNP and MATSSF (e.g. 197 Au), the differences in self-shielding factors between
different configurations is very similar for both methods.
Therefore, the extension of the MATSSF code by generalization of the mean chord length
estimator has been justfied. The method has also been tested on several other realistic samples
leading to similar conclusions, but the results are omitted here due to extensiveness.
Proceedings of the International Conference Nuclear Energy for New Europe 2009, Bled, Slovenia, Sept.14-17, 2009
118.8
6
CONCLUSIONS
The Monte Carlo method is rather cumbersome and is not suitable for routine calculation of
self-shielding factors in neutron activation analysis. Therefore, a simple method of MATSSF
has been developed to determine first-order self-shielding corrections. Also, source and sample
geometry are successfully analytically treated in the first order via generalization of the mean
chord length parameter and Bell factor. If still higher accuracy is needed, it is best to calculate
the self-shielding factors directly by the Monte Carlo technique.
REFERENCES
[1] G. I. Bell, and S. Glasstone, Nuclear Reactor Theory, Litton Educational Publishing, Inc.,
1970, pp. 115-120
[2] X-5 Monte Carlo Team, MCNPTM - A General Monte Carlo N-Particle Transport Code,
Version 5, Manual, LA-UR-03-1987, Los Alamos National Laboratory, 2004
[3] A. Trkov, MATSSF Program
Available online: http://www-nds.iaea.org/naa/matssf/ (August 2009)
[4] M. B. Chadwick, et. al., ENDF/B-VII.0: Next Generation Evaluated Nuclear Data Library for Nuclear Science and Technology, Special Issue on Evaluated Nuclear Data File
ENDF/B-VII.0, Nuclear Data Sheets, 2006
[5] NJOY99.0, Code System for Producing Pointwise and Multigroup Neutron and Photon
Cross Sections from ENDF/B Data, RSICC Peripheral Shielding Routine Collection, Oak
Ridge National Laboratory, 2006
[6] O. Bersillon, et. al., International Reactor Dosimetry File 2002 (IRDF-2002), Technical
Reports Series No. 452, International Atomic Energy Agency, Vienna, 2006
[7] F. De Corte, The k0 -standardization method, a move to the optimization of neutron activation analysis, PhD thesis, University of Gent, Belgium, 1987
[8] L. Snoj, M. Ravnik, ”Effect of fuel particles’ size and position variations of multiplication
factor in pebble-bed nuclear reactors”, Kerntech. (1987), 72, 2007, pp. 251-254
[9] M. Ravnik, R. Jeraj, ”Research reactor benchmarks”, Nucl. Sci. and Eng., 145, 2003, pp.
145-152
Proceedings of the International Conference Nuclear Energy for New Europe 2009, Bled, Slovenia, Sept.14-17, 2009