Keywords

1 Introduction

The radiation source term for the fuel stacks and the structure materials is one of the important inputs for the radiation shielding design. Accurate prediction of the source term could support the design of nuclear facilities and improve economic efficiency. Nuclide inventory is the basis of the source term calculation, which involves the depletion calculation of the fuel stacks and the activation of the non-fissile materials. The existing codes such as ORIGEN-2 [1] and FISPACT [2] suffer from low accuracy and bad robustness of old algorithms. Nevertheless, the in-house modules do not open interfaces to other organizations, including ORIGEN-S in SCALE [3] and son on.

In this work, a new depletion, activation and radiation source term calculation code TIST is developed and verified with the high-fidelity code NECP-X [4] and FISPACT. The content is organized as follows. In Sect. 2, the theory of TIST is presented, including the library of TIST, depletion and source term models. The accuracy demonstration of the depletion calculation, activation calculation and the source term calculation are shown in Sect. 3. Conclusions are summarized in Sect. 4.

2 Theory

The development of depletion, activation and radiation source term calculation functions in TIST include the generation of the depletion and activation chain library, the estimation of nuclide inventories and the radiation source term calculation from different radioactive sources. This section can be divided into three parts. Firstly, the high-fidelity depletion and activation chain library is generated for TIST, and then the basic governing equation of depletion and activation calculation and its solution are introduced. Finally, the method of calculating radiation source term in TIST is described in detail.

2.1 Generation of the Depletion and Activation Chain Library for TIST

The depletion and activation chain library provides nuclides’ property, transmutation and decay information for TIST, including the half-life, the energy per disintegration, transmutation paths and branching ratios, etc. A full-fidelity depletion and activation chain library is developed in TIST ENDF/B-VII.0 evaluated nuclear data library [5] and the neutron-induced activation library EAF-2010 [6] in this work. Nuclides in TIST are grouped into three categories: actinides, fission products and activations. The major characteristics of libraries of TIST are presented in Table 1, including number of three kinds of nuclides and information of transmutation and decay paths.

Table 1. Main characteristics of the depletion and activation chain of TIST
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2.2 The Solution of the Bateman Equations in TIST

The Bateman equations are solved for a set of time steps, which can be written in a matrix form as follows:

$$ \frac{{d\overrightarrow {N} }}{dt} = A \cdot \overrightarrow {N} (t) $$
(1)

The general solution of Eq. (1) is given in Eq. (2):

$$ \overrightarrow {N} (t2) = e^{tA} \cdot \mathop{N}\limits^{\rightharpoonup} (t1) $$
(2)

where \(A\) is the transmutation and decay matrix, which is a large-scale and stiff system. \(\mathop{N}\limits^{\rightharpoonup} (t1)\) is the initial number densities of all nuclides of previous time step.

The Chebyshev rational approximation method (CRAM) [7] is used for calculating Eq. (2) in TIST. The eigenvalues of the coefficient matrix of Bateman equations are clustered around the negative real axis. In this situation, exponential function for the interval (−∞, 0] can adopt Chebyshev rational approximation. When the approximation is applied to the matrix exponential, Eq. (2) can be presented as following:

$$ \overrightarrow {N} (t2) = \alpha_{0} \overrightarrow {N} (t1) + 2\left[ {{\rm{Re}}\sum\limits_{i = 1}^{k/2} {\alpha_{i} (A\Delta t + \;\theta_{i} I)^{ - 1} } } \right]\overrightarrow {N} (t1) $$
(3)

where \(\alpha_{0}\) is the limiting value of the approximation at infinity, \(\alpha_{i}\) and \(\theta_{i}\) is the residues and poles which depend on order \(k\) and thus they can be pre-calculated and tabulated against expansion order k in the code. \(t1\) is time of the beginning of the step, \(t2\) is the time of the end of the step, \(\Delta t\) equals \(t2 - t1\).

Due to the sparse structure of the coefficient matrix, Eq. (3) can be solved accurately and efficiently with the non-zero storage function in and the Jacobian iterative method in TIST.

2.3 Radiation Source Term Calculation Methods

Based on the isotopic inventory from the transmutation calculation, and the decay constants, the radioactivity of every isotope can be explicitly tracked:

$$ A_{inuc} = N_{inuc} .\lambda_{inuc} $$
(4)

where \(A_{inuc}\) is the radioactivity, \(\lambda_{inuc}\) is the decay constant of isotope inuc, and \(N_{inuc}\) is the number of isotope inuc.

The decay heat of every isotope relies on the radioactivity and the energy per disintegration can be obtained:

$$ DH = \sum\limits_{inuc}^{nnuc} {A_{inuc} .} E_{inuc}^{decay} $$
(5)

where \(DH\) is the decay heat, \(E_{inuc}^{decay}\) is the energy per disintegration of isotope \(inuc\), \(nnuc\) is the total number of isotopes.

The photon source term model implemented in TIST considers gammas arising from X-rays, gamma-rays, bremsstrahlung, spontaneous fission, and (α, n) reactions. ENSDF-2011 [8] is adopted to provide the photon yields. The photon source term calculation method is the same as ORIGEN’S method. When the line-energy data is transformed to multi-group photon yields, the adjusted group yield per disintegration of the emitter \(inuc\) is given as following based on the conservation of energy:

$$ yield(ig,inuc) = \sum\limits_{i = 1}^{n} {yield_{i,inuc} .E_{i} /E_{ig} } $$
(6)

where \(yield(ig,inuc)\) is the photon yield per disintegration of group ig, \(yield_{i,inuc}\) is the actual photon yield at photon energy \(E_{i}\), \(E_{ig}\) is the mean energy of the group and n is the total number of photon yield in the group \(ig\) given in ENSDF-2011.

The neutron source term of TIST includes neutrons produced from spontaneous fission and (α, n) reactions. The calculation of the neutron source term includes the calculation of strengths and the spectra. The methods are adopted from the SOURCE 4C code [9].

The spontaneous fission reaction can be considered as a decay reaction, and its strength \(N_{sf}\) can be directly calculated with the activity of the precursor \(A_{sf}\), and the number of neutrons per fission \(\upsilon\).

$$ N_{sf} = A_{sf} .\;\upsilon $$
(7)

As for the normalized spontaneous neutron spectrum, the Watt fission spectrum [10] is applied:

$$ N(E) = ce^{ - E/a} \sinh \sqrt {bE} = ce^{ - E/a} \frac{{\left( {e^{{\sqrt {bE} }} - e^{{ - \sqrt {bE} }} } \right)}}{2} $$
(8)

where a and b are evaluated nuclide-dependent constants, c and E are the normalization constant and neutron energy.

The calculation of neutron source strength from the (α, n) reaction is based on the neutron yield and the activity of alpha decay. It is assumed that all alpha particles are stopped within the medium, and react with the target nuclides in the medium, not transporting to adjacent regions [9].

In the medium, the α particles have multiple origins and the (α, n) reactions happen with different target nuclides. For one specific α particle and a target nuclide i, the calculation of the neutron yield from (α, n) reaction requires the stopping power \(S(E)\), microscopic (α, n) reaction cross-section \(\sigma_{i}^{(\alpha ,\;n)} (E)\) of the target nuclide i, and the alpha particle energy E.

The stopping power \(S(E)\) is defined as the loss energy \(dE\) per unit path length \(dx\):

$$ S(E){ = - }\frac{dE}{{dx}} $$
(9)

The yield of the neutron production from the alpha partial with the target nuclide i is given as follows:

$$ Y_{i} (E_{\alpha } ) = \int_{{E_{\alpha } }}^{0} {P_{i}^{{_{(\alpha ,\;n)} }} (E)} dE = N_{i} \int_{0}^{{E_{\alpha } }} {\frac{{\sigma_{(\alpha ,\;n)} (E)}}{S(E)}} dE $$
(10)

where Ni is the number density of target nuclide in the mediums, Eα is the energy of the alpha particle.

3 Numerical Results

Three cases were used to verify TIST. The results provided by TIST were compared with FISPACT and the high-fidelity code NECP-X. Firstly, the second international activation calculation benchmark [11] was presented to demonstrate the accuracy of the activation calculation. Secondly, a pin cell problem from VERA depletion benchmark was used to test the depletion calculation in TIST. Finally, the radiation source term calculation based on the spent fuel is conducted by TIST and NECP-X, respectively.

3.1 The Second International Activation Calculation Benchmark

The activation benchmark was completed under the coordination of the IAEA Nuclear Data Section in 1994. Two problems were developed in this benchmark, including the activation of 50Cr and natural Fe. Besides, a 100-group fusion flux was also provided for the calculations. FISPACT-2007 is chosen to offer the reference results, which are publicly available in the literature [12].

The results are shown in Table 2 and Table 3 for the activation of 50Cr and natural Fe, respectively (T for TIST, N for NECP-X and F for FISPACT2007). It is found that the differences between FISPACT2007, TIST and NECP-X are small. The errors of nuclide inventory for all isotopes between TIST and FISPACT are within 0.7% and the maximum error is −0.63% for 50V.

Table 2. Comparison results for the activation of 50Cr
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Table 3. Comparison results for the activation of natural Fe
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3.2 The VERA Depletion Benchmark

In order to assert the accuracy of the depletion capability of TIST, a pin cell problem from the VERA depletion benchmark [13] is tested.

VERA depletion benchmark problems were developed based on the VERA progression problems [14] by Oak Ridge National Laboratory in 2015. It contains detailed information, including the geometry, temperature and the materials of fuel, moderator, guide/instrument tubes and burnable poisons. Pin cell problems cove the enrichment of fuel from 2.1% to 4.6%, and fuel temperature from 565 K to 1200 K. More detailed pin cell configurations, materials, depletion options may refer to the benchmark report [13]. VERA-1C problem is chosen for the test in this work, the cross sections are provided by NECP-X and important isotopes at 60 GWd/tU are compared, which is presented in Table 4. It is found that the differences between TIST and NECP-X are small. The errors of nuclide inventory for all isotopes are within 0.4% and the maximum error is 0.37% for 243Am.

Table 4. Comparison results for the actinides and fission products
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3.3 Radiation Source Term of the Spent Nuclear Fuel

The accuracy of TIST for the source term calculation is asserted by comparison of code-to-code. The comparison of TIST to NECP-X is conducted here. To eliminate the errors introduced by the depletion calculation, the same compositions of SNF are provided for both codes, which come from the nuclide inventory of irradiated nuclear fuel of VERA-1C in this work. Note that ‘0 year’ means the shut-down time.

Figure 1 and Fig. 2 present the comparison of the radioactivity and the decay heat results between TIST and NECP-X, respectively. In the figures, the results of NECP-X are taken as the reference. As shown in Fig. 1 and Fig. 2, the maximum relative difference of decay heat and radioactivity of two codes for all decay time range is less than 0.15%.

Figure 3 and Fig. 4 shows the relative difference of photon spectra and neutron spectra between TIST and NECP-X. The maximum relative difference of photon spectra and neutron spectra between TIST and NECP-X is 0.44% and -0.37%, respectively, whose strength is pretty low in this situation.

Fig. 1.
figure 1

Comparison of the radioactivity between TIST and NECP-X

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Fig. 2.
figure 2

Comparison of the decay heat between TIST and NECP-X

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Fig. 3.
figure 3

Comparison of photon spectra between TIST and NECP-X

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Fig. 4.
figure 4

Comparison of neuron spectra between TIST and NECP-X

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4 Conclusions

The depletion, activation and radiation source term calculation code TIST is developed and verified in this work. The theoretical models of TIST are introduced briefly in this paper, including the depletion and activation chain library, depletion model and the source term calculation model. The accuracy of TIST for the depletion calculation, the activation calculation and the source term calculation are verified by a set of benchmarks, and the results show good consistency for TIST, FISPACT and the high-fidelity code NECP-X. More work is under going to compare the results of TIST to the measurements.