CBZLIB and advanced Bondarenko modelThis material is a summary of the following paper:
G. Chiba, et al., “Advanced Bondarenko method for resonance self-shielding
calculations in deterministic reactor physics code system CBZ,” Ann. Nucl. Energy, 96,
p.277-286 (2016).

1

CBZ and CBZLIB- CBZ is a general-purpose reactor physics code system being
developed at Nuclear Reactor Engineering Laboratory of
Hokkaido University.
- CBZ is based on the deterministic method, so multi-grouprepresentations for neutron flux and reaction cross sections areadopted.
- CBZLIB is a multi-group library, and is used to calculate multigroup (effective) cross sections of medium in homogeneous andheterogeneous systems.
- CBZLIB is based on the Bondarenko-type library*; it consists ofinfinite-dilution cross sections, scattering matrices and selfshielding factors dependent on background cross sections.
* Bondarenko, I. I., “Group constants for nuclear reactor calculations,” Consultants Bureau,
New York (1964).

2

Concept of CBZLIB- Nowadays, advanced energy group structure like SHEM 281group structure and CASMO 586-group structure has beenadopted in many reactor physics code systems.
- When we use these fine-group libraries, energy groupcollapsing (or condensation) of cross sections is sometimes
required in assembly calculations or subsequent whole-corecalculations.
- The concept of CBZLIB is to improve calculation accuracy ofeffective cross sections WITHOUT increasing the number ofenergy groups; the SRAC-107 group structure is continuouslyutilized.

3

Basic theory of the Bondarenko model- Bondarenko-type libraries are applied to resonance selfshielding calculations based on the equivalence theory.
- The key point of the equivalence theory is how to accurately
evaluate background cross sections in arbitrary
heterogeneous systems.
- Escape probability from a single fuel pin is generally
approximated by the rational approximation.
- Lattice effect can be taken into account by the neutron
current method proposed by Yamamoto and Sugimura*.

* N. Sugimura, A. Yamamoto, “Evaluation of Dancoff factors in complicated geometry using
the method of characteristics," J. Nucl. Sci. Technol., 43, p.1182 (2006).

4

Escape probability representation in CBZLIB- Accuracy of the escape probability can be improved byincreasing the number of terms considered in the rational
approximation: multi-term rational approximation.
- CBZLIB adopts a different approach: Bell factor optimization.
ΣΣ

1

∝

1/̅

̅
Σ

- Optimized Bell factor can be calculated so as to satisfy thefollowing equation. The energy integration is carried out bythe sub-group method.

5

Further improvement in CBZLIB: resonance interference- Theoretically the equivalence theory cannot consider theresonance interference effect because when one resonant
nuclide is a target of calculation, other nuclides are assumednon-resonant.
- The TIMS code can provide self-shielding factors calculated
by solving neutron slowing-down equation with considering
two different resonant nuclides.
- With this capability, we can consider resonance interferencewith one-specific other nuclide. Generally resonanceinterference with uranium-238 are considered.

6

Further improvement in CBZLIB: resonance interference- In CBZLIB, a pair of nuclides where resonance interference isconsidered can be dependent on nuclide and energy group.
- If we need to consider important resonance interferences ofseveral nuclide pairs in the same energy group, multiple
correction is adopted:

7

Further improvement in CBZLIB: resonance interference(a) SRAC 48th group (61.4 - 78.9 eV)(b) SRAC 53th group (17.6 – 22.6 eV)

8Further improvement in CBZLIB: current-weight total XS- It has been well known that neutron current-weighted total crosssections should be considered in energy group collapsing in theneutron transport equation*.
- If we do not consider this, group collapsing errors affect numericalresults. To avoid this, the SPH factors are sometimes used.
- NJOY can calculate current-weighted total cross sections basedon the homogeneous P1 approximation and narrow resonanceapproximation, but this is applicable to limited problems and not
to light water reactor pin-cell problems.
- In CBZLIB, current-weight total cross sections are calculated withsome representative pin-cell problems with a fine-group library.
* T. Takeda, T. Kitada, “Direction and region dependent cross sections for use in MOX fuelanalysis,” J. Nucl. Sci. Technol., Supp.2, p.1057 (2002).

9

Further improvement in CBZLIB: current-weight total XS
Current-weight total cross sections of U-238 calculated with several different
LWR pin-cell models. Error bars show possible ranges.

Average values are implemented to CBZLIB as a correction factor. Thismeans that each of CBZLIB has its specific applicable problems.

10Further improvement in CBZLIB: current-weight total XS
In addition to U-238, total cross section of Pu-242 in the 61th group (2.38 –3.06 eV) is also corrected because of its large impact.

11Test of CBZLIB against LWR pin-cell problems
PWR

ID
12345678BWRID
12345678910111213141516U-235
Pu
Fuelenrichment enrichment Pu-vector [P8/P9/P0/P1/P2/A1]
type[%]
[%]
UO2
3.4
UO2
4.1
UO2
4.7
MOX
0.2
134.1/45.4/25.3/9.6/13.0/2.6
MOX
0.252.1/54.5/25.0/9.3/6.4/2.7
MOX
0.2
102.1/54.5/25.0/9.3/6.4/2.7
MOX0.2132.1/54.5/25.0/9.3/6.4/2.7
MOX
0.2
50.04/79.24/17.8/2.4/0.4/0.2
VoidU-235
Pu
Fuel
Step ratio enrichment enrichment Pu-vector [P8/P9/P0/P1/P2/A1]
type[%]
[%]
[%]
UO2
I03.0
UO2
I40
3.0
UO2
I70
3.0
UO2
II0
3.8
UO2
II40
3.8
UO2
II70
3.8
UO2
III
04.1
UO2
III
40
4.1
UO2
III
70
4.1
MOX
II00.2
41.5/58.7/26.6/8.3/4.0/0.8
MOX
II400.2
41.5/58.7/26.6/8.3/4.0/0.8
MOXII400.2
41.7/53.4/29.7/8.7/5.6/0.9
MOXII400.240.8/67.8/21.8/6.9/2.1/0.7
MOX
II
70
0.2
41.5/58.7/26.6/8.3/4.0/0.8
MOXII
70
0.2
8
1.5/58.7/26.6/8.3/4.0/0.8
MOX
II
70
0.2
13
1.5/58.7/26.6/8.3/4.0/0.8

12Test of CBZLIB against LWR pin-cell problems

Good agreement with continuous-energy Monte Carlo results
is observed in a wide range of LWR pin-cell problems.
13

Test of CBZLIB against LWR pin-cell problems

Each of the already-mentioned three techniques has large
influence, and current-weight total correction has the mostsignificant impact.
14

Test of CBZLIB against LWR pin-cell problems

Please be careful that multi-group library generated by NJOY
with the NR approximation gives not-bad results as the above.

15

Test of CBZLIB against BWR assembly problem

Good agreement with MVP-BURN result is obtained in BWR
assembly burnup problem.
16