**Chapter 3 - Preliminary analysis of the energetic ** **meshing for the neutronics calculations **

**3.1 Introduction **

The main part of this research work was performed in collaboration with NRG (Petten - The Netherlands) and TUD (Delft – The Netherlands). Particularly it was studied an improvement of the pebble mixing neutronics model to be useful for the PANTHERMIX code. The present and the following two chapters describe the work performed on this topic.

In order to carry out this task it has been necessary a preliminary theoretical analysis (both from mathematical and physical point of views). This research work has requested a lot of tools for investigating the phenomena of interest.

In this chapter some of the used methodologies are analyzed. Particularly the
theoretical background of the SHEM-281 energy group structure and another
method (own developed in order to find out how to predict properly the
neutronic behaviour of pebble-bed systems) are considered. Aiming at
solving the mixing problem in the PANTHERMIX code, results obtained with
MCNP5^{©} (see Chapter 5) are considered, as a preliminary approach and a
starting point for this part of the thesis.

**3.2 SHEM (Santamarina Hfaiedh Energy Mesh) **

**SHEM (Santamarina Hfaiedh Energy Mesh) [3-1] is a refined energy mesh, **
which is well-suited for reducing errors in multi-group studies of spectrum
and cross-sections of LWR systems loaded with UOX or MOX fuels as well as
of sodium-cooled FR systems. SHEM has been developed to avoid self-
shielding and mutual shielding calculations below 23 eV: resonances of
actinides, main fission products and absorbers are described using an
optimized energy mesh. In the intermediate and fast range, the resonances
of coolants and structural materials (O^{16}, Na^{23}, Al^{27}, Fe^{56}, Ni^{58}, Mn^{55}) are
accounted for, as well as the cross-section variation of threshold reactions
such as U^{238} (n,n’), U^{238} (n,f), U^{238} (n,2n).

The SHEM has been developed to reduce errors obtained by using XMAS 172 neutron-energy groups mesh. This latter was developed in the framework of a CEA-UKEA collaboration[3-2], and aimed mainly at improving pressurized water reactor (PWR) calculation performances, particularly mixed-loading cores using MOX assemblies. Unfortunately, the XMAS-172g mesh was constructed by simply joining the 99-group and 69-goup meshes of the APOLLO1 and WIMS data libraries. Consequently XMAS presents abnormal irregular groups. In addition, it has not reached the goal of eliminating self-

shielding errors in the thermal range (E

cut-off=4.0eV), especially in the case of
the Pu^{242} resonance at 2.7 eV. Therefore, using the XMAS multigroup library
in APOLLO2[3-2] requires the use of AUTOP (APOLLO2 self-shielding module)
for this 2.7 eV resonance. Indeed the APOLLO2 space-dependent self-
shielding formalism reduces the Pu^{242} thermal resonance calculation
discrepancy.

Therefore, it was needed to develop a new optimized mesh to eliminate self-
shielding calculation of thermal resonances. This mesh should also avoid the
use of self-shielding model in the U^{238} first large resonances, E

R = 6.7 eV and
20.9 eV. This new mesh should also take account for the mutual shielding
effect: for instance U^{238}/U^{235} resonance overlap (E

U238=6.7 eV with E

U235=6.4 eV and 7.1 eV ; E

U238=20.9 eV with E

U235=21.1 eV), U^{236}/U^{234} resonance
overlap in enriched reprocessed uranium (E

U236 = 5.4 eV and E

U234 = 5.2 eV),
U^{238}/Pu^{240} resonance overlap at 20.9 eV in MOX fuels, etc.

As anticipated above, SHEM was developed to avoid self-shielding and mutual shielding calculations below 23 eV.

In the framework of SHEM, it has been performed:

1. Optimization in the Thermal range (0 – 0.25 eV (Figure 3.1).

2. Optimization of the Epithermal resonance range (0.25 – 4.0 eV) (Figure 3.2).

3. Optimization in the Large Resonance Range (4.0 – 23.0 eV) (Figure 3.3).

**Figure 3.1: Resonances in the range of 0 eV – 0.25 eV**

**Figure 3.2: Resonances in the range of 0.25 eV – 4 eV**

**Figure 3.3: Resonances in the range of 4 eV – 23 eV**

In SHEM almost all the main resonances of the actinide elements and of the most important FPs are subdivided into the previously mentioned 3 groups.

These resonance peaks are listed below (Par. 3.3.1).

3.2.1 Actinide Resonances

The nuclear fuel is composed of actinide nuclides, then actinides are the most important isotopes as far as the definition of the neutronic behaviour of mixtures of different pebbles is concerned. Indeed, this difference depends on the ratio fertile nuclides Vs. fissile nuclides in the considered mixture. Odd nuclides are often fissile, while even nuclides are generally fertile (although this is not a rule!).

The SHEM resonance peaks are shown in the following Table 3.1:

**Table 3.1: Resonances of Actinides Nuclides **
Actinides Nuclides

Nuclides Energy Peaks (eV)

U^{234} ^{5.2 }

U^{235} 2.0 2.8 3.6 4.8 5.4 6.4 7.1 8.8 9.3 11.7 12.4 14.0 16.1 16.7 19.3 21.1 23.59

U^{236} ^{5.4 }

U^{238} ^{6.7 20.9 }

Np^{237} 0.5 1.5 3.8 5.8

Pu^{238} ^{18.6 }

Pu^{239} 0.30 7.8 10.9 11.9 14.3 14.7 17.7 22.3

Pu^{240} ^{1.0 20.5 }

Pu^{241} 0.26 4.3 8.6 13.4 14.8 17.9

Pu^{242} ^{2.7 }

Am^{241} 0.31 0.58 1.3 5.4 5.9

Am^{243} ^{1.36 }

Cm^{243} ^{2.3 }

Cm^{244} ^{7.7 }

3.2.2 Fission Products

In the following Table 3.2 the resonance peaks of the most important fission
products are collected. Some of these fission products are short-living, as
Xe^{135}, but they show high resonance peaks for some energy values.

Consequently, as well known, they are non negligible from the reactor behaviour point of view.

**Table 3.2: Resonances of Fission Product Nuclides **
Fission Product Nuclides

Nuclides Energy Peaks (eV)

Mo^{095} ^{44.9 }

Tc^{099} ^{5.6 20.3 }

Rh^{103} ^{1.26 }

Cs^{133} ^{5.9 }

Xe^{131} ^{14.4 }

Xe^{135} ^{0.08 }

Nd^{145} ^{4.3 }

Pm^{147} ^{5.4 }

Sm^{149} ^{0.10 }

Sm^{152} ^{8.1 }

Eu^{153} ^{2.3 }

Eu^{154} ^{0.60 }

Eu^{155} ^{0.20 }

3.2.3 Mat ix and Bond Materials r

Matrix and bond materials represent the major part of the material inside the core, but their absorption cross sections are generally quite small (i.e. 1÷50 barns). However, they show resonance peaks for a few energy values, as shown in Table 3.3:

**Table 3.3: Resonances of Matrix and Bond Material Nuclides **
Matrix and Bond Material Nuclides

Nuclides Energy Peaks (MeV)

He^{4} ^{1.15 }

O^{16} 0.434 1.0 1.312

C^{Nat.} ^{8.55 }

Si^{28} 0.0577 0.0677 0.0867 0.18

Materials for control rods and burnable poisons are considered in SHEM as well, but they will not be considered here since they do not are requested for the current analysis of the PANTHERMIX mixing problem (see Chapter 4 and 5). For the sake of completeness, Appendix C shows all the 281-SHEM groups[3-2].

**3.3 Grouping Methodology **

PANTHERMIX (see Chapter 4) can only handle up to 12 energy groups. So a method, similar to SHEM-281, was developed to obtain directly cross-sections of pebbles characterized by different burn-up levels as a function of the neutron spectrum. First of all, it is important to find out which of these energy group mesh permits us to achieve good results sampling a large cell (PANTHERMIX mesh typically containing a few to a few tens of pebbles). We performed few calculations on simple cases (1÷2 types of pebble in the considered cell), and thereafter on 8-9 pebbles in different positions in order to obtain the flux spectrum variation (see Figure 3.4).

A A

A A

+

Φ AverageΣ2 Groups Φ (in the Centre)

Φ AverageΣ2 Groups

Φ (in the Centre)

B

B B

B

**Figure 3.4: Example of tally calculation in 8-9 groups**

That gives us an idea of grouped cross-section differences as a function of the mesh chosen.

It is interesting to show the energy boundaries considered (6÷12 groups):

• Energy 0-0.01 eV: 1\v range for light elements (like matrix and bond elements)

• Energy 0.01-0.04 eV: This group shows properties that are similar to those of the previous one, but for light element the cross-section trend is constant.

• Energy 0.04-0.21 eV: the upper limit corresponds to the large resonance of 135Xe (0.08 eV, 107 b), and is close to the resonance of 149Sm (0.1 eV, 105 b).

In this range, the following approximation is valid (large resonance approximation):

0 0

### 2

### 1 *E*

*p*

### α

### >> − Γ

Where:

o

### 2

^{0}

^{0}

### 1 − α *E*

is the average energy lost per collision.

o represents the width at half-height of the resonance.

*II*
*I*

*p*

## = *E*

_{0}

## + *E*

_{0}

## Γ

• Energy 0.21-4.0 eV: this range contains the first large resonances of most of actinides (Pu239, Pu241, Am241 and others). These peaks are 104÷105 b high, see Figure 3.5. The energy of 0.21 eV has an important meaning. It represents the start of large resonances (without Samarium and Xenon) and it is the separation boundary between thermal and epithermal-fast groups.

• Energy 4.0-10 eV: The upper limit of this group is a particular energy point: indeed there are no resonance peaks close to it (see Figure 3.2 and Figure 3.3). The peaks inside this group have values which are around 103 – 105 b.

• Energy 10-24 eV: This group shows properties that are similar to those
of the previous one, but the peaks are less high (10^{2}-10^{4}). Also this
group represents the upper boundary of the large resonance ranges.

Thereafter, the “narrow” resonance approximation can be used. The next formula [3-3] represents the meaning of “narrow” resonance:

0 0

### 2

### 1 *E*

*p*

## α

### << −

### Γ

Where:

o

### 2

^{0}

^{0}

### 1 − α *E*

is the average energy lost per blow.

o

## Γ

*p*

## = *E*

_{0}

^{I}## + *E*

_{0}

*represents the width of resonance.*

^{II}**Figure 3.5: Pu**^{239}** first large resonance[3-4] (Dark Blue line is σ**_{t}**) **

• Energy 24-6·10^{2} eV: it is the upper limit of a series of small resonances
(thus, it is useful to group them together). It is interesting to see that
there is a large resonance of Mo95 in the middle of this small
resonance group. It would be interesting to check if it causes a
deviation in the general behaviour of small resonances [3-1].

• Energy 6·10^{2}-10^{4} eV: the latter represents the upper limit of the second
small resonance group, where all resonances of medium-heavy and
heavy elements finish. After that there is only an average 1\E behaviour
(non-resolved range).

• Energy 10^{4}-10^{7} eV: it is the upper limit of our groups and it represents
an “infinite” energy.

The typical structure shown in Table 3.1 underlines the similarities in the behaviour of different nuclides. Hence, it is possible to define this simple structure by means of a small number of groups. It is therefore interesting to note the hole between 4 and 10 eV, probably due to a similar nuclear structure. In fact, similar energy levels mean similar nuclear structure (Table 3.4). The Table is interesting because it represents the meaning of the previous description in upper limits. The colours show the progressive improvement of the importance of minor actinides. The second group isn’t necessary in the case of the fresh fuel and in the upper limit of the 6 group, there is the possibility to use sometime 45 eV, if we want the contribution of the last large resonance of Mo95.

**Table 3.4: Upper Limits for groups at different burn-up levels (eV) **

**Type Burn-Up ****(GWd/ton) **

Value 0 25 50 100 200 300

Material 1 Group 2 Group 3 Group 4 Group 5 Group 6 Group 7 Group 8 Group 9 Group

Matrix 1,00E-02 - - - - - 1,00E+04 1,00E+07

Bond 1,00E-02 - - - 1,00E+04 1,00E+07

Fission Product - - -

Heavy Metals - - 2,10E-01 4 10 24 6,00E+02 1,00E+04 1,00E+07 Fission Product - 4,00E-02 - 4 10 - 6,00E+02 1,00E+04 1,00E+07 Heavy Metals - - 2,10E-01 4 10 24 6,00E+02 1,00E+04 1,00E+07 Fission Product - 4,00E-02 - 4 10 - 6,00E+02 1,00E+04 1,00E+07 Heavy Metals - - 2,10E-01 4 10 24 6,00E+02 1,00E+04 1,00E+07 Fission Product - 4,00E-02 - 4 10 - 6,00E+02 1,00E+04 1,00E+07 Heavy Metals - - 2,10E-01 4 10 24 6,00E+02 1,00E+04 1,00E+07 Fission Product - 4,00E-02 - 4 10 45 6,00E+02 1,00E+04 1,00E+07 Heavy Metals - - 2,10E-01 4 10 24 6,00E+02 1,00E+04 1,00E+07 Fission Product - 4,00E-02 - 4 10 45 6,00E+02 1,00E+04 1,00E+07 Heavy Metals - - 2,10E-01 4 10 24 6,00E+02 1,00E+04 1,00E+07

The objective of the Table 3.4 is to simplify the SHEM group structure, giving the upper limits to set the energy groups. As an example, if we have only matrix, structural and fresh fuel material, we will be able to use only 8- groups. In contrast, if differently burnt pebbles are in the cell, it will be useful to consider 9 groups (see Table 3.5). The nuclides chosen for this study are the same of the SHEM [3-1] theory (Paragraph 3.2).

Finally the results show that we can approximate the behaviour of a pebble with only 8 or 9 groups, which enable us to obtain a qualitative shape of the flux spectrum and of cross-sections.

**Table 3.5: Nine Groups Upper Limits Table (eV)**
Energy

Groups 10E-02 4.0E-02 2,10E-01 4 10 24 6,00E+02 1,00E+04 1,00E+07

**3.4 Conclusions **

The SHEM theory and group structure allows us to perform a detailed neutronics study and it explains how to identify groups able to represent qualitatively both flux and grouped macroscopic cross-sections shapes as a function of energy.

This methodology is going to be applied in this work in order to study the recirculation problem (see Chapter 5). Additionally, SHEM groups are useful to take into account the differences among minor actinides, fission products and structural materials cross-sections. The use of SHEM method can be considered just as the initial part for the study. Indeed further developments of this research, more suitable for the PANTHERMIX code, will be described in the Chapter 5.