Supplemental material for the 92Zr neutron capture cross section paper



Resonance parameters

The resonance parameters of n_TOF paper-1 [1] and of the present paper (paper-2) are provided here in computer-readable format. The parameters of Boldeman et al. (1976) are available here as well.

In addition, ENDF-6 formatted files are provided with the resolved resonance parameters (RRP) from the n_TOF paper 1 and 2. Note that the RRP energy range covered in paper 1 is 0 - 40 keV, while the results of paper 2 covers resonance parameters up to 81 keV. Above these energies, the cross section data from evaluated data libraries are adopted.

resonance parameters reference
Boldeman et al. (1976) txt file [2]
JENDL-4.0 txt file [3]
ENDF/B-VIII.0 txt file [4]
n_TOF paper-1 txt file [1]
n_TOF paper-2 txt file [this work]
resonance parameters + evaluations ENDF-6 formatted file capture cross section
n_TOF paper-1 + ENDF/B-VIII.0 (En > 40 keV) endf6 file pointwise txt file
n_TOF paper-1 + JENDL-4.0 (En > 40 keV) endf6 file pointwise txt file
n_TOF paper-2 + ENDF/B-VIII.0 (En > 81 keV) endf6 file pointwise txt file
n_TOF paper-2 + JENDL-4.0 (En > 81 keV) endf6 file pointwise txt file

Capture kernel

plot kernel.png
Capture kernel
plot kernel-0-cumul.png
Cumulative capture kernel for s-waves
plot kernel-1-cumul.png
Cumulative capture kernel for p-waves

Ratios

plot kernel-rats12.png
Capture kernel ratio paper-2/paper-1
plot kernel-rats2boldeman.png
Capture kernel ratios to Boldeman et al. (1976).
plot kcomp.png
Capture kernel ratio paper-2/paper-1

Average values

s-waves
source # of resonances Γγ ± std. [meV] capture kernel ± std. [meV]
Boldeman et al. (1976) [2] 12 141 ± 94 137 ± 86
JENDL-4.0 15 144 ± 86 136 ± 79
ENDF/B-VIII.0 15 109 ± 36 107 ± 30
n_TOF Paper 1 11 138 ± 74 131 ± 62
n_TOF Paper 2 11 128 ± 54 128 ± 54

p-waves
source # of resonances Γγ ± std. [meV] capture kernel ± std. [meV]
Boldeman et al. (1976) [2] 60 383 ± 309 423 ± 523
JENDL-4.0 86 315 ± 21 421 ± 427
ENDF/B-VIII.0 86 318 ± 24 486 ± 420
n_TOF Paper 1 27 203 ± 98 262 ± 136
n_TOF Paper 2 55 238 ± 145 297 ± 165

Capture cross section and MACS

The capture cross section in the full energy range, from 10-5 eV to 20 MeV, are calculated assuming RRP from the n_TOF paper-1 and paper-2. The capture width of the negative resonance (parameters from JENDl-4.0) has been slightly adjusted to reproduce the thermal cross section of 230 mb [3]. The upper energy ranges, from 40 keV and from 81 keV, respectively for paper-1 and paper-2 parameters, from JENDL-4.0 have been adopted. The pointwise cross sections are available in the files linked above (pointwise txt file for paper-1 and pointwise txt file for paper-2).

plot xssqrt.png
Reduced capture cross sections from 100 eV to 1 MeV
plot Sigma.png
Cumulative MACS fractions for different thermal energies

The maxwellian averaged capture cross section, is calculated using the different sets of parameters obtained in the n_TOF paper 1 and paper 2, complemented with the ENDF/B-VIII.0 and JEDNL-4.0 evaluated cross section, respectively above 40 keV and above 81 keV.

plot macs.png
MACS vs thermal energy
plot macs3.png
MACS vs thermal energy
plot macs-compare.png
MACS vs thermal energy in paper 1 and paper 2, with different energy ranges

The data are available in tabular form in paper 2 and in computer-readable text file here >>. A comparison of the MACS in tabular form for the results of paper 1 and paper 2 is available in the computer-readable text file here >>.

The contribution of single resonances to the total MACS are shown here below (n_TOF paper 2 resonances + JENDL-4.0 above 81 keV assumed).

plot macs-fract-5keV.png
resonance contribution to the MACS for kT = 5 keV
plot macs-fract-30keV.png
resonance contribution to the MACS for kT = 30 keV
plot macs-fract-50keV.png
resonance contribution to the MACS for kT = 50 keV
plot macs-fract-100keV.png
resonance contribution to the MACS for kT = 100 keV

Direct Radiative Capture contribution

The neutron radiative capture process can proceed, through E1 transitions, to positive-parity bound states in Zr-93 by p-wave direct capture (DRC). There are states in Zr-93 with a strong single-particle component for configurations of Zr-92 ground-state coupled to both, s1/2 and d5/2 orbitals (see the ENSDF data base), which favour the DRC process.

The s-wave DRC, which could contribute to the 1/v component of the capture strength from the thermal region, is unlikely to take place as there are very few negative-parity states in Zr-93. Usually, as it is the case here, a negative (bound) state is introduced to account for the thermal cross section and 1/v component of the capture cross section. In the case of Zr-92, all the evaluated nuclear data files introduce a negative resonance in order to reproduce a thermal cross section of ~230 mb.

For the keV neutron energy region, the p-wave component of the DRC can be estimated considering transitions to two bound states in Zr-93 with the strongest single-particle configuration

#    Ex [MeV]   Bind (MeV)    J   Par     C2S       spo
      0.0         6.7344     2.5   +1    0.6400    2d5/2  
      0.9470      5.7874     0.5   +1    0.9200    3s1/2  
A Woods-Saxon shape for the n + 92Zr interaction is considered [5]. The resulting capture cross section can be averaged over a maxwellian distribution of neutron energies to produce a MACS which can be compared to the experimental results.

plot macs-DRC.png
DRC contribution to the MACS

As can be seen, the estimated DRC contribution to the MACS is negligible for low thermal energies, while becoming significant (up to 20%) for higher temperatures.


References

  1. G. Tagliente et al. (The n_TOF Collaboration), Phys. Rev. C 81, 055801 (2010). doi
  2. J.W. Boldeman, A. R. de L. Musgrove, B. J. Allen, J. A. Harvey, and R. L. Macklin, Nucl. Phys. A269, 31 (1976). doi
  3. K. Shibata et al., J. Nucl. Sci. Technol. 39, 1125 (2002). doi
  4. D.A. Brown et al., ENDF/B-VIII.0: The 8th Major Release of the Nuclear Reaction Data Library with CIELO-project Cross Sections, New Standards and Thermal Scattering Data, Nuclear Data Sheets 148, 1-142 (2018). doi
  5. A. Mengoni, T. Otsuka, and M. Ishihara, Direct radiative capture of p-wave neutrons, Phys. Rev. C 52, R2334(R). doi

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