## 1. Introduction

Nuclear fuels are used to generate electricity in nuclear power plants, and finally are discharged after full use, which are called spent fuels (SFs). SF contains actinides, fission products, and activation products [1]. Since the radioactivity of SF is strongly dependent on its burnup, the burnup of SF should be well estimated for the safe management, storage, and final disposal of SF [2]. Nowadays more accurate burnup estimation of SF is needed for the applicability of burnup credit. Based on 10 CFR part 71 and 72, transport (and storage) casks must be guaranteed criticality safety. As the SF capacity in storage casks increases, the critical margin in the casks must be guaranteed accurately using burnup credit [3-5]. Hence, the burnup estimation of SF becomes important, and it should also be verified accurately. The amount of burnup of SF can be obtained by the data of local powers within a reactor measured during the nuclear reactor operation. However, the burnup from the operational data must be verified or confirmed by other methods to safely manage SF. There are two types of radiations emitted from SF, i.e., neutrons and gamma rays, to be used for verification purposes. Most gamma rays in SF are emitted from the fission products produced during the burnup of fuels. Hence, gamma rays from SF are a good index for the estimation or verification of the burnup of SF [6-11]. Neutrons from SF are also a good index for verification. Measurements using active and passive neutrons can give valuable information for the estimation of the burnup and the number of fissile nuclides in SF [12-14].

The burnup of a SF can be estimated by the detector measurements of gamma-rays and neutrons emitted from SF. Important radioactive sources for gamma rays are ^{137}Cs, ^{134}Cs, and ^{154}Eu, and one of the important neutron sources is ^{244}Cm [15-17]. However, the methodology for the burnup estimation from the known activities of important radioactive sources has not been published openly so far. The reason may be the easy use of tools for the burnup estimation such as ORIGEN code [18]. The activities of important radioactive sources for gammas and neutrons can be easily obtained using ORIGEN at a given input which includes the initial fissile amount, fuel composition, irradiation option, cooling time, etc. But the burnup of SF cannot be directly obtainable from the known activities of important sources. A cut-and-try method is needed for the burnup estimation from the known activities of radioactive sources. In this study, we analyzed the dependency of the burnup on the important source activities using ORIGEN-ARP, and suggested simple correlations that relate the burnup and the important source activities directly. The important sources of gamma rays are ^{137}Cs, ^{134}Cs, and ^{154}Eu, and total neutron source intensity (TNSI) from SF is used as an index for the burnup estimation in this study.

## 2. Methodology

### 2.1 Cases Used in the Analysis

The initial enrichment and the burnup of SFs generated in Korean pressurized water reactors up to 2019 were summarized recently [19] by Korea Hydro and Nuclear Power Co. Ltd (KHNP) which is shown in the Fig. 1. Generally, the burnup of SF increases with the initial enrichment. Based on this figure, the following equation relating the average burnup and the initial enrichment of SFs was suggested for the analysis purpose in this study (red line in Fig. 1).

where BU is the burnup (GWD·MTU^{−1}) and IE is the initial enrichment (%).

The actinide nuclides for neutron sources and the fission products for gamma sources in SF were analyzed using ORIGEN-ARP, and the values of IE and BU for the calculation were assumed to follow the eq. (1). Fig. 2 shows important actinide nuclides producing neutrons or contributing TNSI (neutrons/sec-MTU) (left side), and the activities (Ci/MTU) of important gamma sources (right side) in SF in a case (IE=4%, BU=42 GWD·MTU^{−1}, CE16×16 fuel assembly) with respect to cooling time. Generally, most neutrons in SF are originated from spontaneous fissions rather than alpha-n reactions up to 100 years of cooling time. And the dominant neutron source is ^{244}Cm. ^{244}Cm is shown to be the main source for spontaneous fissions from 5 years up to about 60 years of cooling time. The important gamma sources are shown to be ^{137}Cs, ^{134}Cs, and ^{154}Eu after 5 years of cooling time. The decay curves of the gamma sources show clear exponential decay with time, which indicates no additional decay chains exist for the main gamma sources after about 5 years of cooling time. Based on these observations, we made a model to estimate the burnup of PWR fuels under the conditions which are; 2.4%<IE<4.8%, 0.7 BU_{IE}<BU<1.3 BU_{IE} where BU_{IE} is obtained by eq. (1) at a value of IE, and 5 years<cooling time<60 years. These conditions cover the major region of the database as shown in Fig. 1. We selected 80 cases (16 IEs, 5 BUs per IE, i.e., 16×5=80) which are shown as the red points in Fig. 1. Table 1 shows the 80 cases we selected. And three types of nuclear fuels were considered, which are CE16×16, WH14×14, and WH17×17, making total 240 cases. ORIGEN-ARP was used to get the data of each case (total 240 cases) for the relations among the burnup, the important gamma source activities, and TNSI in SF with respect to cooling time.

##### Table 1

BUs (GWD MTU^{–1}) |
1.3 BU_{IE} |
1.15 BU_{IE} |
BU_{IE}^{*} |
0.85 BU_{IE} |
0.7 BU_{IE} |
---|---|---|---|---|---|

IEs (%) | |||||

2.4 | 33.8 | 29.9 | 26 | 22.1 | 18.2 |

2.6 | 36.4 | 32.2 | 28 | 23.8 | 19.6 |

2.9 | 40.3 | 35.65 | 31 | 26.35 | 21.7 |

3.1 | 42.9 | 37.95 | 33 | 28.05 | 23.1 |

3.2 | 44.2 | 39.1 | 34 | 28.9 | 23.8 |

3.4 | 46.8 | 41.4 | 36 | 30.6 | 25.2 |

3.5 | 48.1 | 42.55 | 37 | 31.45 | 25.9 |

3.6 | 49.4 | 43.7 | 38 | 32.3 | 26.6 |

3.7 | 50.7 | 44.85 | 39 | 33.15 | 27.3 |

3.8 | 52 | 46 | 40 | 34 | 28 |

4.1 | 55.9 | 49.45 | 43 | 36.55 | 30.1 |

4.2 | 57.2 | 50.6 | 44 | 37.4 | 30.8 |

4.4 | 59.85 | 52.9 | 46 | 39.1 | 32.2 |

4.5 | 61.1 | 54.05 | 47 | 39.95 | 32.9 |

4.65 | 63.05 | 55.775 | 48.5 | 41.225 | 33.95 |

4.8 | 65 | 57.5 | 50 | 42.5 | 35 |

### 2.2 Modeling

Neutron sources in SF are generated either by spontaneous fissions or alpha-n reactions. Up to 100 years of cooling time, spontaneous fission produces most neutrons from the actinide sources. And ^{244}Cm turned out to be the major source. ^{244}Cm is produced from the neutron absorption of ^{238}U. ^{238}U transforms to ^{239}Pu, and it transforms to ^{243}Pu by continuous absorption of neutrons, and ^{243}Pu decays to ^{243}Am, then finally becomes ^{244}Cm after neutron absorption and decay. Briefly speaking, the formation of ^{244}Cm depends on the absorption of neutrons and the amount of ^{238}U. Hence, the amount of ^{244}Cm formation is dependent mainly on burnup and initial enrichment. If ^{244}Cm is the main source of neutrons in SF, the burnup (BU) of SF can be proposed as shown in the following equation.

where S_{N} is the TNSI (n/sec-MTU), IE is initial enrichment, and t is cooling time.

Since gamma-ray sources in SF are fission products, the amount of fission products is strongly dependent on the burnup of SF. Major gamma sources in SF (5 years<cooling time<60 years) are ^{137}Cs, ^{134}Cs, and ^{154}Eu. And these gamma sources simply decay just after the end of the burnup. If the activity (A) of a gamma source at the cooling time, t is known, then the burnup of SF can be obtained. We propose the following relation.

where A is the activity (Ci/MTU) of a gamma source at the cooling time, t.

Based on the database of 240 cases obtained from ORIGEN- ARP, the constants, a, b, c, and d in eq. (2) for the burnup estimation using neutrons were obtained by the linear regression method using Excel program. And the constants, a and b in eq. (3) for the burnup estimation using the gamma sources, ^{137}Cs, ^{134}Cs, and ^{154}Eu were also obtained using the Excel program, respectively.

## 3. Results and Discussion

### 3.1 Correlations for the Burnup Estimation

The burnup estimation equation using neutrons, eq. (2) can be written as a logarithm form such as,

The database of 240 cases after applying the linear regression method in Excel gives the following results,

And the coefficient of determination, R^{2} was 99.8%.

Hence, the burnup estimation equation using TNSI, IE, and cooling time, t (eq. (2)) becomes as follows.

where the units are; *S _{N}*: n/sec-MTU,

*IE*:%,

*t*: year.

Fig. 3 shows the comparison between data points and the calculation values. At four different initial enrichments, 2.4%, 3.5%, 4.1%, and 4.65%, TNSI values were compared with calculated values with respect to burnup and cooling time. The fuel assembly types, CE16×16, WH14×14, and WH17×17 show almost the same values indicating negligible effects of fuel types on the burnup estimation. The burnup estimation equation, eq. (5) well describes the values from the database. Slight difference can be detected in the case of IE=2.4% at lower burnup (18.2 GWD·MTU^{−1}) at longer cooling time (t >50 year). Otherwise, eq. (5) well describes the burnup estimation using TNSI, IE, and cooling time.

The burnup estimation equations (eq. (3)) from the gamma-ray emission nuclides ^{137}Cs, ^{134}Cs, and ^{154}Eu were also obtained from the database of 240 cases, respectively. Fig. 4 shows the activities (Ci/MTU) of the main gamma sources at discharge at a given burnup. The activities of the gamma sources are proportional to the amount of burnup, and the differences in discharge activities among fuel types are negligible. Based on the activities at discharge after burnup, the burnup estimation equations from the activities of major gamma sources (eq. (3)) were obtained as follows.

where A is the activity (Ci/MTU) and t is cooling time in year.

### 3.2 Source Neutron Spectrum and Difference in Fuel Type

The neutron spectrums produced in SF in 3 different fuel types (CE16×16, WH14×14, WH17×17) are shown in Fig. 5 at 40.2 GWD·MTU^{−1} after 20-year cooling. The peak of neutron energy was observed at about 0.9 MeV. There is negligible difference in the neutron spectrum among different fuel types. But the amount of source neutrons is slightly less in the case of WH17×17 than the other 2 types. As cooling time increases, the amount of source neutrons decreases due to the decay of ^{244}Cm. But the shape of spectrum remains the same since the main source of neutrons is still ^{244}Cm. Based on the similar neutron spectrums among different fuel types, it is concluded that the fuel type does not affect burnup estimation correlation (eq. (5)). And the burnup estimation from the activities of the major gamma sources ^{137}Cs, ^{134}Cs, and ^{154}Eu (equations (6), (7), and (8)) is not dependent on fuel types because the fission products only depends on the burnup.

### 3.3 Initial Enrichment Effect on Burnup Estimation

^{244}Cm is the major neutron source in SF, and the production of ^{244}Cm is affected by the initial enrichment of uranium. The production of ^{244}Cm starts from the neutron absorption of ^{238}U. Hence, initial enrichment affects TNSI in SF. In addition to the amount of ^{238}U, heavily dependence on Pu fissile nuclides (^{239}Pu, ^{241}Pu and ^{243}Pu) during burnup for the lower enriched fuel results in more production of ^{244}Cm than the higher enriched fuel. Fig. 6 shows the dependency of TNSI and major gamma sources, ^{137}Cs and ^{134}Cs on the initial enrichment of FA at the same burnup. Major gamma sources are fission products, the amounts of which are proportional to burnup. Hence, there is no effect of initial enrichment on the amounts of major gamma sources. However, TNSI is dependent on initial enrichment since neutrons in SF are mainly originated from spontaneous fission of ^{244}Cm. Lower enriched fuel (3.1% in Fig. 6) makes more ^{244}Cm during irradiation than higher enriched fuel (4.65%) resulting in higher TNSI for the lower enriched fuel at the same burnup. Since the major neutron source is ^{244}Cm, the shape of neutron spectrums in SF is the same regardless of initial enrichment. Only total amount, i.e., TNSI is dependent on initial enrichment.

### 3.4 Specific Power Dependence and Impurity Effects on Burnup Estimation

Nuclear fuels can be burned in different specific powers. We analyzed three different cases to see the effect of the specific power on burnup estimation- 12-month (37.2 MW·TU^{−1}), 16-month (27.9 MW·TU^{−1}), and 18-month (24.8 MW·TU^{−1}) operations, all with three cycles. The results indicated the effect of specific power was not noticeable enough if cooling time was longer than 5 years. Fig. 7 shows the neutron spectrum of the three different specific powers at the burnup of 40.2 GWD·MTU^{−1} after 20-year cooling. High power operation (12 Month) shows slightly higher value of TNSI than lower power operation; however, the difference looks negligibly small [20]. Based on the results, it can be concluded that the specific power difference during fuel burnup contributes almost negligible effects on fuel burnup estimation.

There are many impurities in UO_{2} fuel such as C, P, Na, Al, Zn, Fe, Cd, etc. The level of impurities is about ppm. These impurities are not actinides, so they cannot affect TNSI. However, they can be activated during irradiation, and may become gamma sources in SF. Fig. 8 shows the gamma ray spectrums with and without impurities. As shown in the figure, there is no difference between two cases, and they are almost identical. Hence, it can be concluded that the impurities in the fuel do not affect gamma radiation spectrum in SF.

## 4. Conclusion

Neutrons and gamma-ray activities are important tools for estimating the burnup of spent nuclear fuels. In this study, it was checked whether the burnup could be estimated by the known values of TNSI (n/sec-MTU), IE(%), cooling time (year) in the case of neutrons and by the known values of activities of major gamma sources (^{137}Cs, ^{134}Cs, ^{154}Eu) after cooling time (year) in the case of gamma radiations. The database to get the correlations for the burnup estimation was made by ORIGEN-ARP, and total 240 cases were studied. The burnup estimation equation of PWR fuels for neutrons using TNSI, IE, and cooling time, t was obtained as,

where the units are; *S _{N}* (or TNSI): n/sec-MTU,

*IE*: %,

*t*: year.

The burnup estimation equations for PWR fuels from the activities of major gamma sources were obtained as,

where A is the activity (Ci/MTU) and t is cooling time in year.

Different PWR fuel assembly types (CE16×16, WH14×14, WH17×17) negligibly affect the burnup estimation equations. Hence, eq. (5)–eq. (8) can be used regardless of fuel types for the burnup estimation of SF. The specific power of a fuel also negligibly affects the burnup estimation. The difference in the burnup estimation due to different specific power becomes negligible after 5 years of cooling. Impurities in the fuel do not make any contribution in gamma ray spectrum and have no effect on burnup estimation equations. However, initial enrichment affects the value of TNSI at the same burnup. Hence, for the burnup estimation using neutrons, initial enrichment should be considered as an important input variable as shown in eq. (5). The burnup estimation equations, (5), (6), (7), and (8) should be used in the conditions which are; 2.4%<IE<4.8%, 0.7 BU_{IE}<BU<1.3 BU_{IE} where BU_{IE} is obtained by eq. (1) at a value of IE, and 5 years<cooling time<60 years.