# Simple Integrated Fission Gas Release and Swelling (Sifgrs) for UOvar element = document.getElementById("moose-equation-b027f834-c35e-4631-8041-6aae6692e22f");katex.render("_2", element, {displayMode:false,throwOnError:false});

Recommended fission gas model to account for generation of fission gasses in nuclear fuel

## Description

The processes induced by the generation of the fission gases xenon and krypton in nuclear fuel have a strong impact on the thermo-mechanical performance of the fuel rods. On the one hand, the fission gases tend to precipitate into bubbles resulting in fuel swelling, which promotes pellet-cladding gap closure and the ensuing pellet-cladding mechanical interaction (PCMI). On the other hand, fission gas release (FGR) to the fuel rod free volume causes pressure build-up and thermal conductivity degradation of the rod filling gas.

The fundamental physical processes, which control the kinetics of fission gas swelling and release in irradiated fuel, may be summarized as follows. Fission gas atoms generated in the fuel grains diffuse towards the grain boundaries through repeated trapping in and irradiation-induced resolution from nanometre-size intra-granular gas bubbles. Although a part of the gas atoms that reach the grain boundaries is dissolved back to the grain interior by irradiation, the majority of the gas diffuses into grain-face gas bubbles, giving rise to grain-face swelling. Bubble growth brings about bubble coalescence and inter-connection, eventually leading to the formation of a tunnel network through which a fraction of the gas is released to the fuel rod free volume.

In Bison, fission gas behavior is computed for each integration point in the fuel finite element mesh. The gas produced at each integration point is computed by a numerical time integration of the gas production rate, given as the product of the fission rate and fractional yield of gas atoms per fission.

## Physics-Based Model

The Simple Integrated Fission Gas Release and Swelling (Sifgrs) model is intended for consistently evaluating the kinetics of both fission gas swelling and release in UO. The model incorporates the fundamental features of fission gas behavior, among which are gas diffusion and precipitation in grains, growth and coalescence of gas bubbles at grain faces, thermal, athermal, steady-state, and transient gas release. Through a direct description of the grain-face gas bubble development, the fission gas swelling and release are calculated as inherently coupled processes, on a physical basis. The level of complexity of the model is consistent with reasonable computational cost and the uncertainties inherent in engineering-scale fuel analysis. The Sifgrs model draws on and extends the approach described in Pastore et al. (2013).

## Intra-granular Gas Behavior

Fission gas transport from within the fuel grains to the grain faces is computed through numerical solution of the relevant diffusion equation in one-dimensional spherical geometry (1) where (m) is the intra-granular gas concentration, (s) the time, (ms) the effective gas diffusion coefficient, (m) the radial co-ordinate in the spherical grain, and (ms) the gas generation rate. The effective diffusion coefficient, which accounts for the effects of repeated trapping in and irradiation-induced resolution from intra-granular bubbles, is calculated based on White and Tucker (1983) and Speight (1969). Eq. 1 is solved using dedicated numerical algorithms. Both the algorithm from Hermansonn and Massih (2002) and the more recent one from Pizzocri et al. (2016) are available in Bison.

In alternative to the single diffusion equation with the effective diffusion coefficient , it is possible to solve the coupled equations describing gas diffusion and exchange of gas between the grain matrix and bubbles (2)

(3) where and (m) are the single gas atom concentration and the gas in bubble concentration, respectively, (ms) the single atom diffusion coefficient, (s) the trapping rate, and (s) the re-solution rate. This system of equations is solved in time-varying conditions by a dedicated algorithm, extending the one from Pizzocri et al. (2016). The solution of this system of equations is expected to be needed during fast transients (e.g., RIA). Nevertheless, Eq. 1 provides a good description of the intra-granular gas behavior for the wide majority of temperature and power conditions.

An empirical model (Lassmann et al., 1995) is included to consider intra-granular gas depletion in the high burnup structure (HBS). No specific model for gas release from the HBS is considered.

### Evolution of Intra-grannular Bubbles

The contribution of intra-granular bubbles to fission gas swelling (intra-granular swelling), which is generally less important than the swelling due to grain-face bubbles, at least for burnup below about 45 GWd/t (Kashibe et al., 1969), is considered. In fact, a model is available to calculate the evolution during irradiation of intra-granular bubble density and radius, based on Pizzocri et al. (2018). Nucleation and re-solution may occur by different mechanisms, i.e., heterogeneous and homogeneous (Olander and Wongsawaeng, 2006). Heterogeneous nucleation and re-solution refer to the creation of new bubbles nuclei as a direct consequence of the interaction of fission fragments with the lattice and the bubbles destruction occurring en-bloc by passing fission fragments, respectively. The homogeneous mechanisms accounts for the nucleation of bubbles by diffusion-driven interactions of dissolved gas atoms and re-solution occurring gradually by ejection of individual atoms. The dominant mechanisms depend upon the nature of the interactions between fission fragments and lattice (electronic or phononic). Based on experimental and theoretical findings, the heterogeneous mechanisms is assumed as dominating in UO. The equations for the evolution of the intra-granular gas bubble number density and gas atom concentrations are: (4)

(5)

(6) where (m) is the number density of intra-granular bubbles, is the number of gas atoms per bubble, and (m) are the intra-granular gas concentration in the matrix and in the bubbles, respectively, (s) the time, (ms) the single-atom gas diffusion coefficient, (m) the radial coordinate in the spherical grain, (ms) the gas generation rate, (s) the trapping rate, (s) the re-solution rate, and () the nucleation rate, which is calculated as: (7) being (bubble/fisson-fragment) the average number of bubbles nucleated per fission fragments (e.g., Turnbull (1971)) and the fission rate.

Intra-granular bubbles radius is calculated, assuming spherical shape of the bubbles, multiplying the number of atoms per intra-granular bubble, i.e. , by the equivalent radius associated to atoms in intra-granular bubbles (derived from experimental measurements, e.g. Olander and Wongsawaeng (2006)).

## Grain-face Gas Behavior

Numerical solution of Eq. 1 allows estimating the arrival rate of gas at the grain faces, thus providing the source term for the grain-face gas behavior module. The latter computes both the fission gas swelling and release through a direct description of the grain-face bubble development, including bubble growth and coalescence (which are reflected in fuel swelling), and eventual inter-connection (leading to thermal FGR). In outline:

• Peculiarities related to the presence of grain edges (where three grains meet) are neglected (e.g., Kogai (1997) and Massih and Forsberg (2008)).

• The flux of gas atoms dissolved from the grain faces back to the grain interior by irradiation is neglected (Rest, 2003).

• An initial number density of grain-face bubbles (nucleation centers) is considered, and further nucleation during the irradiation is neglected (one-off nucleation, e.g., White (2004)).

• The absorption rate of gas at the grain-face bubbles is assumed to equal the arrival rate of gas at the grain faces (White, 2004; Olander and Uffelen, 2001).

• All grain-face bubbles are considered to have, at any instant, equal size and equal lenticular shape of circular projection (e.g., Veshchunov (2008)); hence, the fractional volume grain-face fission gas swelling is given by

(8) where is the number density of grain-face bubbles per unit surface, the grain radius, the bubble semi-dihedral angle, the geometric factor relating the volume of a lenticular-shape bubble to that of a sphere, which is , and the bubble radius of curvature. The factor 1/2 is introduced in Eq. 8 because a grain-face bubble is shared by two neighboring grains.

### Bubble Growth

Bubble growth is treated using the model from Speight and Beere (1975), which describes the growth (or shrinkage) of grain-face bubbles as proceeding by absorption (or emission) of vacancies in grain boundaries, induced by the difference between the pressure of the gas in the bubble, (Pa), and the mechanical equilibrium pressure, (Pa). The vacancy absorption/emission rate at a bubble is given by

(9) where (-) is the number of vacancies in the bubble, (m s) the vacancy diffusion coefficient in grain boundaries, (m) the thickness of the diffusion layer in grain boundaries, and the parameter (-) may be calculated as (White, 2004) (10) with being the fraction of grain faces covered by bubbles (fractional coverage). The mechanical equilibrium pressure, , of the gas in a lenticular bubble of circular projection is given by (11) where (Jm) is the gas specific surface energy, (m) the bubble radius of curvature, and (Pa) the hydrostatic stress (considered to be negative if the solid medium is under compression). For describing the bubble thermodynamic state, the Van der Waals' equation of state is adopted in the following form: (12) where (-) is the number of fission gas atoms per bubble, (JK) the Boltzmann constant, (K) the temperature, (m) the bubble volume, and (m) the Van der Waals' volume of a fission gas atom. Given that each bubble consists of vacancies and gas atoms, the volume of a bubble comprising fission gas atoms and vacancies is given by (13) where (m) is the atomic (vacancy) volume in the bubble. The combination of Eq. 12 and Eq. 13 gives for the pressure of the gas in the bubble (14) The above approach allows computing the bubble growth rate from the rate of inflow of gas atoms along with the rate of absorption (emission) of vacancies at the bubble. The combined effects of gas atom inflow and vacancy absorption (emission) are interactive, since the addition of fission gas atoms gives rise to a change in the bubble pressure via Eq. 14. The bubble pressure affects the propensity of the bubble to absorb (or emit) vacancies through Eq. 9. Given the volume, , of a lenticular bubble of circular projection, the bubble radius of curvature is calculated as (15)

### Grain-Face Bubble Coalescence

The process of grain-face bubble coalescence, which leads to a progressive decrease of the bubble number density throughout irradiation, is described using an improved model of White (Pastore et al., 2013; White, 2004). According to this model, the rate of loss of bubbles by coalescence is given by

(16) where and represent the number density and projected area of grain-face bubbles, respectively.

### Bubble Number Density Saturation Condition

The release of fission gas to the fuel rod free volume following inter-connection of grain-face bubbles and consequent formation of pathways for gas venting to the fuel exterior (thermal release) is modeled based on a principle of grain face saturation. More specifically, a saturation coverage concept is adopted, namely, it is considered that once the fractional coverage, , attains a saturation value, , the bubble number density and projected area obey the saturation coverage condition (17) where is the bubble number density and is the bubble projected area on the grain face. The commonly accepted value for is 0.5. Eq. 17 implies that, after attainment of the saturation coverage, a fraction of the gas reaching the grain faces is released to the fuel exterior to compensate for continuing bubble growth.

## Transient Gas Behavior

Experimental observations relative to both in-reactor irradiation and post-irradiation annealing of oxide nuclear fuel indicate that substantial fission gas release can occur on a small time scale during temperature transients (burst release). The rapid kinetics of the process cannot be interpreted as purely diffusion-controlled. From the available experimental evidence (Rothwell, 1962; Une and Kashibe, 1990; Sartori et al., ; Ducros et al., 2013), the following main aspects of transient fission gas behavior emerge:

• Burst release occurs through grain-face separation (micro-cracking) which entails gas depletion of a fraction of the grain faces.

• Release bursts are triggered by temperature variations, both heating and cooling.

• The rate of gas release during bursts is a peaked function of temperature with the maximum at a 'central' temperature, which is dependent on the burnup.

An extension (transient model) of the treatment of grain-face gas behavior described in the section "Grain-face gas behavior" is available in Bison, which introduces the effect of micro-cracking on fission gas behavior (Pastore et al., 2014). According to the Bison transient model, gas depletion of a fraction of the grain faces is modeled as a reduction of the fractional coverage, . In particular, is scaled by a factor, , corresponding to the fraction of non-cracked (intact) grain faces. The reduction of the fractional coverage effectively leads to a decrease of the amount of gas retained in the fuel – consequently, of fission gas swelling – and to a corresponding increase of FGR. This contribution to thermal FGR supplements the diffusion-interconnection mechanism considered in the basic model (See section "Grain-face gas behavior"). Also, the lost gas storing capacity of cracked grain faces is represented by scaling the saturation coverage, , by the factor f. Moreover, the healing process of cracked grain faces is considered as a progressive restoration of the grain-face gas storing capacity. Therefore, the fractional coverage and saturation coverage obey (18)

(19) where stands for diffusion-controlled processes (basic model in the section "Grain face gas behavior"), stands for micro-cracking, and for micro-crack healing. The value for the maximum (initial) saturation coverage (corresponding to all intact grain faces) is . The calculation of the term representing the effects of micro-cracking is detailed hereinafter.

We simplify the micro-cracking process into a purely temperature-dependent behavior, characterized by a micro-cracking parameter, . We also observe that the process can only affect intact grain faces, and write (20) where is the reduction rate due to micro-cracking of the fraction of intact grain faces, . The micro-cracking parameter is taken as a function of the sole temperature, hence (21) Then, Eq. 20 can be written as (22) implying (23) which conforms to the experimentally observed characteristic of burst release as triggered by temperature variations. Under the condition expressed by Eq. 21, the analytic solution of Eq. 20 with initial conditions and is (24) Based on the available experimental evidence, the functional form of is chosen as a temperature-dependent sigmoid function (25) where (K) is the central temperature, (K) is a measure of the temperature-domain width of the phenomenon, (-) is a parameter, and is defined as (26) so that increases during both heating and cooling transients. The following values are used for the parameters: = 5 K, = 33. The micro-cracking parameter, , and the parameter derivative, , are plotted in Figure 1 for a value of K. According to Eq. 25 the absolute value of the temperature derivative for is maximum at (see also Figure 1), thus the combined Eq. 20 and Eq. 25 reproduce the maximum rate of burst release at as temperature varies in time.

For the calculation of the parameter in the model, the following correlation from Barani et al. (2017) is used (27) where K, K, GWdt, and (GWdt) is the burnup at current time step. Eq. 27 is derived from the best-estimate fit of quantitative experimental data, Figure 2.

Figure 1: Micro-cracking parameter, , and derivative, , as a function of temperature, considering a central temperature equal to 1773 K.

Figure 2: Experimental data for the temperature of maximum burst release rate from Une and Kashibe (1990) and Baker and Killeen (1987) as a function of burnup and best-estimate fitting curve.

A simple burnup-dependent model is used for micro-crack healing, which is not described here for brevity. Details can be found in Pastore et al. (2014). The above treatment of transient fission gas behavior preserves the continuity in both time and space as well as the consistent coupling of the calculated fission gas release and swelling. Extensive validation has indicated that the model is capable of consistently representing the kinetics of FGR during transient fuel irradiations (Pastore et al., 2014; Pizzocri et al., 2015; Barani et al., 2017).

## Athermal Gas Release

At low temperature, the fission gas in the matrix of the solid is relatively immobile. Only the gas formed at the external surface of the solid is capable of escape, with an emission rate that is independent of temperature. This athermal contribution to FGR arises from the surface-fission release mechanisms of recoil (direct release of a fission fragment due to its high kinetic energy) and knockout (ejection of a gas atom following elastic interaction with either a primary fragment or energetic particle created in a collision cascade) (Lewis, 1987). These release mechanisms affect only the outer layer of the fuel (within about from the surface). The rate of gas atom release per unit fuel volume due to recoil and knock-out, (ms), may be calculated as (Lewis, 1987) (28) where (-) is the fractional yield of fission gas atoms, the fission rate density (ms), (m) the volume of fuel, (m) the geometrical surface area of fuel, (m) the total surface area of fuel (including cracked surface), (m) the fission fragment range in the fuel, and (m) the range of the higher order uranium knock-on in UO.

In line with Koo et al. (2000), the number and length of cracks in each fuel pellet is estimated in a simple way. First, radial cracks are considered to cross the outer, brittle region of the fuel pellet with a temperature lower than 1200 C (Olander, 1976). Second, the number of pellet cracks is considered to increase linearly with fuel linear power (Oguma, 1983). Then, once the linear power and pellet dimensions are given, the total pellet surface area available for athermal gas release can be calculated.

## Grain Growth and Grain Boundary Sweeping

Being the fission gas behavior physically dependent on the granular structure of the fuel, the Sifgrs model is coupled with the grain growth model (See section on Grain Growth in the Theory Manual Overview of Ceramic Fuels). The grain growth phenomenon affects the fission gas release in three ways. First of all, due to the low solubility of the fission gas, the moving grain boundary does not redeposit any gas in the newly-formed crystal behind it, thus acting as a filter and contributing to the collection of gas at the grain faces (_grain boundary sweeping_). This effect is taken into account in Sifgrs by adding a supplementary fractional release term () from within the grains to the grain faces that is equal to the volume fraction of the fuel swept by the moving boundaries: (29) where the indices and refer to the previous and current time, respectively. Secondly, the diffusion distance for the fission gas atoms created in the grains increases as the grains grow. Unlike the first consequence this tends to reduce the release rate. Thirdly, grain growth reduces the capacity of the grain boundaries to store fission gas, as it results in a decrease of the total grain surface-to-volume ratio.

## Example Input Syntax


[./fission_gas_release]
type = Sifgrs
temp = temp
fission_rate = fission_rate
[../]
(test/tests/element_integral_power/fission_gas_error_check_1D.i)

## Input Parameters

• trap_param_option0Select trapping parameter

Default:0

C++ Type:short

Description:Select trapping parameter

• temperature_scalef1scaling factor for temperature

Default:1

C++ Type:double

Description:scaling factor for temperature

• gbdiffcoeff_scalef1scaling factor for grain boundary diffusion coefficient

Default:1

C++ Type:double

Description:scaling factor for grain boundary diffusion coefficient

• eff_diff_coeff_option0Select effective diffusion coefficient

Default:0

C++ Type:short

Description:Select effective diffusion coefficient

• diff_coeff_option3Select diffusion coefficient

Default:3

C++ Type:short

Description:Select diffusion coefficient

• pellet_brittle_zoneThe name of the UserObject that computes the width of the brittle zone in the fuel pellet

C++ Type:UserObjectName

Description:The name of the UserObject that computes the width of the brittle zone in the fuel pellet

• burnup_functionBurnup function

C++ Type:BurnupFunctionName

Description:Burnup function

• ig_diff_algorithm0Select intra-granular diffusion algorithm

Default:0

C++ Type:short

Description:Select intra-granular diffusion algorithm

Default:1

C++ Type:double

• resolutionp_scalef1scaling factor for resolution parameter

Default:1

C++ Type:double

Description:scaling factor for resolution parameter

• hydrostatic_stress_const0constant hydrostatic stress (Pa)

Default:0

C++ Type:double

Description:constant hydrostatic stress (Pa)

Default:5e-06

C++ Type:double

• ig_bubble_model0Select bubble evolution model

Default:0

C++ Type:short

Description:Select bubble evolution model

• effdiffcoeff_scalef1scaling factor for intragranular effective diffusion coefficient

Default:1

C++ Type:double

Description:scaling factor for intragranular effective diffusion coefficient

• boundaryThe list of boundary IDs from the mesh where this boundary condition applies

C++ Type:std::vector

Description:The list of boundary IDs from the mesh where this boundary condition applies

• nuclerate_scalef1scaling factor for nucleation rate

Default:1

C++ Type:double

Description:scaling factor for nucleation rate

• fract_yield0.3017fractional yield of fission gas atoms (Xe + Kr) (/)

Default:0.3017

C++ Type:double

Description:fractional yield of fission gas atoms (Xe + Kr) (/)

• axial_power_profileaxial power peaking function.

C++ Type:FunctionName

Description:axial power peaking function.

• hydrostatic_stressCoupled Hydrostatic Stress

C++ Type:std::vector

Description:Coupled Hydrostatic Stress

• rod_ave_lin_powlinear power function.

C++ Type:FunctionName

Description:linear power function.

• res_param_option0Select resolution parameter

Default:0

C++ Type:short

Description:Select resolution parameter

• transient_option0Select transient release model

Default:0

C++ Type:short

Description:Select transient release model

• ig_fully_coupled0Solving diffusion coupled to bubble evolution

Default:0

C++ Type:short

Description:Solving diffusion coupled to bubble evolution

• percolation_to_surface1.0Optional AuxVariable that indicates whether the local position is connected by a percolated path to a free surface to allow gas release. If this parameter is not set, path to free surface is not considered in gas release calculation.

Default:1.0

C++ Type:std::vector

Description:Optional AuxVariable that indicates whether the local position is connected by a percolated path to a free surface to allow gas release. If this parameter is not set, path to free surface is not considered in gas release calculation.

• igdiffcoeff_scalef1scaling factor for intragranular diffusion coefficient

Default:1

C++ Type:double

Description:scaling factor for intragranular diffusion coefficient

C++ Type:std::vector

• pellet_idCoupled Pellet ID

C++ Type:std::vector

Description:Coupled Pellet ID

• computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

Default:True

C++ Type:bool

Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

• tempCoupled Temperature

C++ Type:std::vector

Description:Coupled Temperature

• fission_rateCoupled Fission Rate

C++ Type:std::vector

Description:Coupled Fission Rate

• initial_porosity0.05initial fuel porosity (/)

Default:0.05

C++ Type:double

Description:initial fuel porosity (/)

• saturation_coverage0.5saturation coverage (/)

Default:0.5

C++ Type:double

Description:saturation coverage (/)

• fission_gas_concCoupled Fission Gas Concentration

C++ Type:std::vector

Description:Coupled Fission Gas Concentration

• burnupCoupled Burnup

C++ Type:std::vector

Description:Coupled Burnup

• nucleation_option0Select intragranular bubble nucleation model

Default:0

C++ Type:short

Description:Select intragranular bubble nucleation model

• trappingp_scalef1scaling factor for trapping parameter

Default:1

C++ Type:double

Description:scaling factor for trapping parameter

• blockThe list of block ids (SubdomainID) that this object will be applied

C++ Type:std::vector

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Default:False

C++ Type:bool

Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

• hbs_modelFalseActivates the high burnup structure model

Default:False

C++ Type:bool

Description:Activates the high burnup structure model

• control_tagsAdds user-defined labels for accessing object parameters via control logic.

C++ Type:std::vector

Description:Adds user-defined labels for accessing object parameters via control logic.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

• gbs_modelFalseActivates the grain-boundary sweeping model

Default:False

C++ Type:bool

Description:Activates the grain-boundary sweeping model

• testing_outputFalseProvides an analytic reference for the value of the intra-granular fission gas release

Default:False

C++ Type:bool

Description:Provides an analytic reference for the value of the intra-granular fission gas release

• skip_bdr_modelFalseSkips the grain-boundary model

Default:False

C++ Type:bool

Description:Skips the grain-boundary model

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

• ath_modelFalseActivates the athermal release model

Default:False

C++ Type:bool

Description:Activates the athermal release model

• constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

Default:NONE

C++ Type:MooseEnum

Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

• output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

C++ Type:std::vector

Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

• outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object

Default:none

C++ Type:std::vector

Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

## References

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2. T. Barani, E. Bruschi, D. Pizzocri, G. Pastore, P. Van Uffelen, R.L. Williamson, and L. Luzzi. Analysis of transient fission gas behaviour in oxide fuel using \mbox BISON and \mbox TRANSURANUS. Journal of Nuclear Materials, 486:96–110, 2017.[BibTeX]
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5. S. Kashibe, K. Une, and K. Nogita. Formation and growth of intragranular fission gas bubbles in UO$_2$ fuels with burnup of 6-83 GWd/t. Journal of Nuclear Materials, 206:22â€“34, 1969.[BibTeX]
6. T. Kogai. Modelling of fission gas release and gaseous swelling of light water reactor fuels. Journal of Nuclear Materials, 244:131â€“140, 1997.[BibTeX]
7. Y.-H. Koo, B.-H. Lee, and D.-S. Sohn. Analysis of fission gas release and gaseous swelling in UO$_2$ fuel under the effect of external restraint. Journal of Nuclear Materials, 280:86â€“98, 2000.[BibTeX]
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10. A.R. Massih and K. Forsberg. Calculation of grain boundary gaseous swelling in UO$_2$. Journal of Nuclear Materials, 377:406â€“408, 2008.[BibTeX]
11. M. Oguma. Cracking and relocation behavior of nuclear-fuel pellets during rise to power. Nuclear Engineering and Design, 76(1):35â€“45, 1983.[BibTeX]
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16. G. Pastore, D. Pizzocri, S. R. Novascone, D. M. Perez, B. W. Spencer, R.L. Williamson, P. Van Uffelen, and L. Luzzi. Modelling of transient fission gas behaviour in oxide fuel and application to the BISON code. In Enlarged Halden Programme Group Meeting, RÃ¸ros, Norway, September 7â€“12, volume. 2014.[BibTeX]
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20. J. Rest. The effect of irradiation-induced gas-atom re-solution on grain-boundary bubble growth. Journal of Nuclear Materials, 321:305â€“312, 2003.[BibTeX]
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