G4-STORK: Stochastic Calculations of Reactor Kinetics · G4-STORK: Stochastic Calculations of...

G4-STORK: Stochastic Calculations of Reactor KineticsGeant4 neutron mini workshop

Adriaan Buijs

McMaster University

July 10, 2014

Adriaan Buijs (McMaster University) G4-STORK: Stochastic Calculations of Reactor Kinetics July 10, 2014 1 / 27

Staff

Adriaan Buijs:

1982-1986: Grad student PEP9/SLAC

1986-1994: Fellow/staff OPAL/CERN

1994-2001: Professor Utrecht Univ. L3/ALICE/LHC-B/

2002-2008: Reactor Physics Atomic Energy of Canada Limited

2009- Now: Professor Engineering Physics McMaster University

At McMaster:

Liam Russell: Original code developer (now with AMEC NSS)

Salma Mahzooni: ZED-2 Subcritical experiment

Andrew Tan: SLOWPOKE rod-extraction transient

Wesley Ford: Super-Critical Water Reactor design

Work supported by National Science and Engineering Research Council(NSERC)


Outline

Background, why apply GEANT4 to nuclear reactors?

Method, implemented in G4-STORKI Combing;I Criticality calculation;I Time evolution.

Some examples, results

Issues, future workI Nuclear data libraries;I (On the fly) Doppler broadening;I Delayed neutrons;I Feedback to geometry;I Adjoint flux.


Background

Nuclear reactors:

Nuclear reactors are all about neutrons. And photons. And electrons.Not neutrinoes.

In Canada, we have CANDU reactors.

About as big as, say the OPAL experiment.

Has many detectors, too: > 100.

Simulation of nuclear reactors:

Traditionally, neutrons have been treated as a gas:I Neutrons move freely;I Don’t interact;

Balance between absorption of neutrons and creation of new neutronsin fission reactions: critical reactor

keff = 1 or ρ = 1− 1

keff= 0. (1)

Use diffusion equations to calculate neutron populations;


1

2

3

4

5

6

7

970667-2�

1CALANDRIA

2CALANDRIA END SHIELD

3SHUT-OFF AND CONTROL RODS

4POISON INJECTION

5FUEL CHANNEL ASSEMBLIES

6FEEDER PIPES

7VAULT

CANDU 6 Reactor Assembly

Use of Monte Carlo in Reactor Physics

Obviously, Monte Carlo is the most precise method to simulate areactor.

Obviously, computing power is limiting the usefulness of Monte Carlo.

Has been used for small assemblies, of the exploding kind.

For example, MCNP. Also used for dose calculations in LHC.

Recently, also used for static full-core calculations.


Monte Carlo Procedure for Static Calculations

Define geometry

Generate, say, 1,000,000 neutrons;

Follow them until they die.

Say 41% of them cause a fission reaction in, say, U-235.

Then the keff = 0.996 for this cycle; it is subcritical.1

The distribution of fission sites is now the source of neutrons for thenext cycle.

In the next cycle, 40% cause a fission reaction, keff = 0.972, and soon.

You need many cycles, to achieve convergence on:I Source distribution: where is the power in the reactor produced?I keff : is the reactor critical?

1each fission produces 2.43 new neutrons...Adriaan Buijs (McMaster University) G4-STORK: Stochastic Calculations of Reactor Kinetics July 10, 2014 12 / 27

Combing

If multiplication constant not unity (keff 6= 1), then

If keff > 1: exponential growth from cycle to cycle; run out ofcomputer memory soon;

If keff < 1: exponential decline from cycle to cycle; run out ofstatistics;

Remedy by combing:

If keff > 1: randomly remove fission sites;

If keff < 1: randomly add fission sites;

In both cases, preserve the fission source distribution.Can be tricky, needs to be done efficiently.Sidenote: when parallellizing, neutrons are independent, but each cycleconstitutes a barrier.


Fule Bundle Model in Geant-4


Results for CANDU Fuel Bundle


Why GEANT-4?

Why GEANT-4? Because:

It’s there.

Traditional MC’s work with cycles and generations. The concept oftime is lost because some neutrons live longer than others;

Geant-4 tracks particles in ”real” time.

.


Time Evolution of Neutron Population

G4-STORK: A Stochastic (Monte Carlo) Reactor Kinetics Simulation CodeL. Russell, A. Buijs and G. Jonkmans (AECL-CRL)Department of Engineering Physics

AbstractThe G4-STORK code (Geant4 Stochastic Reactor Kinetics) is a time-dependent Monte Carloparticle physics code for reactor physics applications. It was built using the Geant4 MonteCarlo toolkit and is designed to model the continuous evolution of a population of neutronsin space and time. From this evolution, various important reactor physics quantities can becalculated, including the reactivity of the system. System properties, such as the temperatureof a material, can be changed incrementally to approximate time-dependence. Thus, the G4-STORK code can be used to model reactor kinetics, and was used to simulate a system thatunderwent an instantaneous increase in temperature (without delayed neutrons).

IntroductionA nuclear reactor, even when critical (i.e. has a stable neutron population), experiencessmall fluctuations in the neutron population that are compensated for by the control system.In addition, power manoeuvres and accidents introduce large dynamic effects that elicit atime-dependent response from the neutron population in the reactor. This time-dependentbehaviour, known as reactor kinetics, has been traditionally modeled using time-dependentdeterministic approaches such as the point kinetics approximation (with the improved qua-sistatic correction) and various space-energy dynamics solutions. Since these approaches aredeterministic, they are subject to the same limitations as static deterministic nuclear codes:approximations through discretization of energy, space, out-going angles, and time [1]. Assuch, the accuracy and efficiency of these codes depend on the number of discrete energygroups, spatial regions, etc.This discretization is not necessary for a stochastic (Monte Carlo) approach. Moreover,Monte Carlo particle physics simulations follow individual particles in space and time, sothey possess an innate time-dependence [2]. Many Monte Carlo particle physics codes eitherignore temporal data until all particles in the simulation have been lost, or simulate time-independent events (e.g. neutron fission generations). However, the innate temporal trackingcan be used to create a simulation where the overall properties of the system (e.g. reactivity,spatial distribution of neutrons) can be determined at any time during the simulation. Thus,the evolution of these properties with respect to time can be quantified.We used the Geant4 Monte Carlo toolkit to develop a new code for stochastic, time-dependentreactor simulations called G4-STORK (Geant4 Stochastic Reactor Kinetics) [2].

Background

Geant4 Monte Carlo Toolkit

Geant4 is an open-source, particle physics modeling package written in C++ that providesend-users a framework for developing Monte Carlo simulations for unique applications. Geant4has been used extensively in high-energy particle physics research and detector design forprojects such as the Large Hadron Collider, as well as other fields such as radiotherapy anddosimetry [2, 3, 4, 5, 6]. However, the Geant4 toolkit does not encompass a simulationcode in the traditional sense; it is not a program that can be executed. Instead it providesend-users with

Basic simulation framework (organization and data transfer)Particle tracking algorithmsPhysics modelsGeometric, material and particle descriptorsProgramming hooks for tallying and analysisMany other helpful functions and classes

These allow to adapt Geant4 to many different unique applications, including reactor physics.It also satisfies Geant4’s goals of being adaptable, extensible and transparent.The end-user is required to add C++ code defining the following:

Simulation geometry and material compositionState of initial primary particlesList of applicable physics models

All of this code is written using the classes and functions from the toolkit, and it defines aunique simulation. Besides these mandatory additions, most classes in the toolkit may bemodified and/or overridden using user-added code (e.g. polymorphism) [7]. This includesthe physics models, particle definitions and simulation management processes [2].

Criticality Calculations

The criticality of a reactor is represented by the neutron multiplication factor, keff

keff =Neutron Production Rate

Neutron Loss Rate =P(t)

L(t)(1)

This ratio is difficult to calculate directly in practise, so various approximations are made.This leads to two different types of criticality calculations: dynamic (time-dependent) andstatic (time-independent) calculations.In a Monte Carlo code, the dynamic criticality can be calculated using the total neutronproduction and loss over a time period T (NP(T ) and NL(T ), respectively). That is

keff =P(t)× TL(t)× T =

NP(T )

NL(T )(2)

Equation 2 is true for any simulation geometry as long as the simulation geometry does notchange during T .The static criticality may be calculated by comparing the total number of neutrons producedthrough fission in subsequent generations. That is

keff =Nf (i + 1)

Nf (i) (3)

where Nf (i) is the number of neutrons produced by fission in generation i . Since all neutrons inthe current generation have a unique lifetime, this definition of criticality is time-independent.Static calculations, including Equation 3 are only accurate for near-critical systems (keff ≈ 1).

G4-STORK Implementation

Neutron Population Renormalization and Run Management

Timet0 t0 + T

LEGEND G4-STORK Run First Fission Neutron Fission Capture Elastic Inelastic Inelastic Terminated by

Boundaries Generation (n,n') (n,2n) Renormalization

Figure 1: Population renormalization and division of the simulation into runs in G4-STORK.The second generation of neutrons in the run is shown in blue.

Anytime a system is not critical, the neutron population will grow (supercritical) or shrink(subcritical) exponentially. So as not to run out of computing resources or neutrons, thepopulation must be constrained. This is done by periodically renormalizing the neutronpopulation at regular intervals, as shown in Figure 1. The intervals between renormalizationare referred to as runs.The following steps are taken after the mth run to renormalize the neutron population and toinitialize the next run:

1. Stop and save all neutrons (survivors) at tf (m) = t0(m) + T

2. Calculate important quantities:krun - Population change over the run

krun =N(tf (m))

N(t0(m))(4)

H - Shannon entropy to quantify neutron spatial distribution

H = − Nb∑

i=1Pi log2(Pi) (5)

keff - Dynamic criticality (Equation 2)

3. Make changes to the simulation geometry (see below)

4. Duplicate/delete individual survivors until N → N(t0(m))

After this process has finished, the renormalized survivors are used to begin the next run. Forcontinuity, the neutrons begin in the exact same state they were in at the end of the currentrun.

Dynamic Simulation Geometry

Between runs, material or geometric properties of the simulation geometry may be changed.Since the end user of the G4-STORK code will define their own simulation geometry, anyproperty could be time-dependent. Example properties include

Material temperature or densityIsotopic concentrationsControl rod positionsEtc.

To best approximate a continuous change in one of these properties, the incremental changesbetween runs should be relatively small and the run duration should be relatively short.

Verification and Validation

Simulation Geometries for Testing

Table 1: Simulation geometries used in the validation casesSpheres U235 UHWMaterial U235 (100%) U235 (0.077%)

U238 (10.653%)D2O (89.270%)

Radius (cm) Variable VariableDensity (g/cm3) 18.75 3.0143Temperature (K) 293.6 293.6Infinite Lattice CANDU 6 Lattice CellMaterial Fresh fuel (Standard materials)Pitch (cm) 28.575 (Variable)Density (g/cm3) 10.554

0.80741.0888

(Fuel)(Coolant)(Moderator)

Temperature (K) 859.99561.29336.16

(Fuel)(Coolant)(Moderator)

These three simulation geometries were used in the verification and validation of G4-STORK [8]. Each features variable parameters that can be adjusted to achieve differentlevels of reactivity. End users of G4-STORK can create any simulation geometrythat can be defined in Geant4.

Verification and Validation Cont’d

Simulation Codes and General Parameters

Table 2: List of codes and data libraries used in the verification and validation

Software Data Library ENDF/BVersion

EvaluationTemperature (K)

G4-STORK G4NDL4.0C6-ENDF6

76

0.0293.6

MCNP5 endf66 6 293.6DRAGON 3.06J IAEA WIMS-D4 7 MultipleTART 2005 Continuous Energy 6 0.0

The G4-STORK simulations were generally performed with at least 100,000 initial neutronswith energies normally distributed about 1.0 MeV. Since the run duration should be chosensuch that the size of the neutron population remains manageable (a good rule of thumb iskrun ∈ [0.5, 2]), T will vary depending on the reactivity of the system and the average promptneutron lifetime.The code-to-code validation was performed using the codes and data libraries listed in Ta-ble 2 [8, 9, 10]. The simulation parameters for the other codes were set to match theG4-STORK simulations as closely as possible. For example, all of the Monte Carlo codesused the same initial neutron spatial and energy distributions.

Spatial and Energy Distributions

0 5 10 154

5

6

7

8

9

10

11

12

13

14

Distance from centre of fuel bundle (cm)

Neu

tro

ns p

er

cm

−2

5 us

25 us

50 us

125 us

250 us

1000 us

10−15

10−10

10−5

100

105

10−1

100

101

102

103

104

105

106

Energy (MeV)

Neu

tro

ns p

er

bin

5 us

25 us

50 us

125 us

250 us

1000 us

Figure 2: Convergence of the neutron spatial (left) and energy (right) distributions in theCANDU 6 lattice cell.

The spatial and energy distributions of theCANDU 6 lattice cell are shown in Fig-ure 2 as they converge in time. Addition-ally, the accuracy of the spatial distribu-tion was validated using DRAGON. Theflux estimates for each region in DRAGONwere converted to neutron densities (ni)using

ni =Ng∑

g=1

φig

vg(6)

vg =

√√√√√√√E g

max + E gmin

mn(7)

Figure 3 shows that DRAGON andG4-STORK generally agree within thestochastic variation of the G4-STORKneutron density estimates.

0 5 10 150.005

0.01

0.015

0.02

0.025

Distance from centre of fuel bundle (cm)

Ne

utr

on

De

ns

ity

(c

m−

2)

G4−STORK

DRAGON

Figure 3: Comparison of neutron spatialdistributions in G4-STORK and DRAGON forthe CANDU 6 lattice cell.

Criticality Comparisons

4 6 8 10 12 140.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Radius (cm)

ke

ff

4 6 8 10 12 14

−20

−15

−10

−5

0

5

10

15

20

Erro

r TA

RT

vs G

4−

ST

OR

K (m

k)

G4−STORK

MCNP

TART

Error T−G4

7 7.5 8 8.5 9 9.5 10

0.9

0.95

1

1.05

1.1

Radius (cm)

ke

ff

7 7.5 8 8.5 9 9.5 10

−20

−15

−10

−5

0

5

10

15

20

Erro

r MC

NP

vs G

4−

ST

OR

K (m

k)

G4−STORK

MCNP

TART

Error M−G4

Figure 4: Comparison of criticality estimates in G4-STORK, MCNP and TART for U235spheres with varying radii (delayed neutrons were ignored).

Criticality comparisons between G4-STORK, MCNP and TART for U235spheres with varying radii are shown inFigure 4. G4-STORK and TART agree atall points within 10 mk (left); however,MCNP only agrees with the other twocodes in the near-critical region (right)because MCNP calculates the staticcriticality, while G4-STORK and TARTboth calculate the dynamic criticality.Additionally, criticality estimates betweenG4-STORK and DRAGON are shown fora CANDU 6 lattice cell at different latticepitches in Figure 5. Again, the codesagree within 10 mk.

20 25 30 35 40 45 50 551.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

Lattice Pitch (cm)

kin

f

G4−STORK

DRAGON

Figure 5: Comparison of criticality estimates inG4-STORK and DRAGON for a CANDU 6lattice cell at different lattice pitches.

Verification and Validation Cont’d

Reactor Kinetics Simulations

The ability of G4-STORK to model time-dependent simulation geometry is shown in Fig-ures 6 and 7. Both simulations feature an increase in the temperature of an 87.5 cm radiusUHW sphere from 293.6 to 1000 K. In Figure 6, the temperature increase is instantaneous. Tocapture the response of the neutron population to this change, the run duration was chosen tobe 100 ns (approximately one-fifth of the prompt neutron lifetime), and the simulation startedfrom a pre-converged neutron distribution. The reactivity of the system gradually decreasedas the thermal neutrons were up-scattered, which increased the resonance absorption.

200

300

400

500

600

700

800

900

1000

1100

Tem

pera

ture

(K)

0 5 10 15 20 25 300.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Time (us)

ke

ff

keff

Dynamic T

keff

Static T

Temperature

Figure 6: Reactivity response of an 87.5 cm radius UHW sphere to an instantaneoustemperature change from 293.6 to 1000 K (delayed neutrons were ignored).

In Figure 7, the temperature increase occurs linearly over 50 ms. Since the run durationwas 1000 times longer (100 µs), the neutrons had sufficient time in a single run to respondto the incremental increases in temperature. This is shown by the agreement between thereactivity of the simulation with time-dependent simulation geometry (dynamic temperature)and reactivity estimates at discrete (static) temperatures.

200

300

400

500

600

700

800

900

1000

1100

Tem

pera

ture

(K)

0 20 40 60 80 1000.88

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Time (ms)

ke

ff

keff

Dynamic T

keff

Static T

Temperature

Figure 7: Reactivity response of an 87.5 cm radius UHW sphere to a linear temperature changefrom 293.6 to 1000 K over 50 ms(delayed neutrons were ignored). Reactivity estimates at statictemperatures are also shown.

ConclusionsThe G4-STORK code was developed using the Geant4 Monte Carlo toolkit to simulate thecontinuous evolution of a neutron population in time. Through periodic renormalization, sub-or supercritical environments can be simulated over any time period. Before the renormal-ization occurs at the end of each run, reactor physics quantities are calculated (e.g. keff) andincremental changes are made to simulation geometry (if necessary). The combination ofthese abilities allows G4-STORK to model the response of a dynamic system.These capabilities were verified and validated against DRAGON, MCNP and TART usingthree simulation geometries, including a CANDU 6 lattice cell. These tests showed agree-ment between the codes for the accuracy of spatial distributions and criticality calculations.Additionally, the dynamic response of a 87.5 cm radius UHW sphere to a 700 K increase intemperature was shown to be a decrease in reactivity of approximately 100 mk overall, andwas also shown to lag the temperature change on sufficiently small time scales.

References[1] K.O. Ott and R. J. Neuhold, Introductory Nuclear Reactor Dynamics. American Nuclear Society, 1985.[2] S. Agostinelli et al., “Geant4 - a simulation toolkit,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers,

Detectors and Associated Equipment, vol. 506, no. 3, pp. 250 – 303, 2003.[3] J. Allison et al., “Geant4 developments and applications,” IEEE Transactions on Nuclear Science, vol. 53, pp. 270 –278, 2006.[4] J. Apostolakis et al., “Progress in hadronic physics modelling in Geant4,” in Journal of Physics: Conference Series, vol. 160, 2009.[5] E. Mendoza, D. Cano-Ott, C. Guerrero, and R. Capote, “New evaluated cross section libraries for the GEANT4 code,” Tech. Rep. INDC(NDS)-0612,

International Atomic Energy Agency, 2012.[6] J. Wellisch, “Geant4 physics validation for large hep detectors,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators,

Spectrometers, Detectors and Associated Equipment, vol. 502, no. 2 - 3, pp. 669 – 672, 2003.[7] B. Stroustrup, The C++ Programming Language. Addison-Wesley Publishing Company, 3 ed., 2000.[8] G. Marleau, A. Hebert, and R. Roy, A User Guide for DRAGON 3.06. Ecole Polytechnique de Montreal, 2012. IGE-174 Rev. 10.[9] X-5 Monte Carlo Team, MCNP - A General Monte Carlo N-Particle Transport Code, Version 5. Los Alamos National Laboratory, 2003. LA-UR-03-1987.[10] D.E. Cullen, “Tart 2005 a coupled neutron-photon 3-d, combinatorial geometry time dependent monte carlo transport code,” Tech. Rep.

UCRL-SM-218009, Lawrence Livermore National Laboratory, 2005.


Reactivity Calculations with GEANT-4

Neutrons are now followed in ”runs” rather than ”cycles”;

One run is typically a few ms.

For parallellisation, fixed point in time is now the barrier.

Since neutrons are now followed in time, the geometrical propertiesmay be changed in time as well: feedback between number of fissions(power), causing fuel to heat up, causing absorption to increase,causing the number of fissions to decrease, etc. (self-terminatingpower excursion).

Essential ingredient: On the fly Doppler broadening.


Simple Temperature Change U-235 Sphere

M.A

.Sc.T

hesis-Liam

RussellM

cMaster

-EngineeringPhysics

0 10 20 30 40 50 60 70 80 90

0.9

0.95

1

1.05

1.1

Time (ms)

ke

ff

200

300

400

500

600

700

800

900

1000

1100

Tem

pera

ture

(K)

keff

Temperature

Figure 5.22: Transient for a 87.5 cm UHW sphere where the temperature rises from 293.6 to 1000 K (delayedneutrons not simulated).

136


Application: SLOWPOKE Rod Extraction

16

new G4PVPlacement(0, G4ThreeVector(0, 0, 53.042*cm-reactorDim[2]/2), D2OContainerLogical, "D2OPhysical", cellLogical, 0, 0); D2OLogical = new G4LogicalVolume(theSolids->GetSolid("D2O"), matMap["D2O"],"D2OLogical"); new G4PVPlacement(0, G4ThreeVector(0, 0, 0.25375*cm), D2OLogical, "D2OPhysical", D2OContainerLogical, 0, 0);

The resulting volume looks as follows:

Figure 10 Reactor with added D2O thermal column (On the left is a top view and on the right is a 45 degree view)

Inner Irradiation Tubes

Geometry Explanation These are modelled as aluminum tubes filled with air. There are five irradiation tubes in the

beryllium annulus reflector and they are evenly spread throughout the angles.

Coding The dimensions are defined as well as the position of the aluminum tubes where the first entry is

the distance from the center. The second entry is the angle in radian between adjacent tubes and the

third entry is the angle off-set in radian of the first tube from the x-axis (Can be used to rotate the tubes).

G4double smallAlumTubeDim[3] = {0.0*cm, 1.56718*cm, 515.332*cm};

G4double smallBTubeDim[3] = {0.0*cm, 1.40208*cm, 515.332*cm};

G4double alumTubePos[3]={14.56182*cm, 0.4*CLHEP::pi, 0.};

The aluminum solid and the air solid are then defined: new G4Tubs("smallAlumTube", smallAlumTubeDim[0], smallAlumTubeDim[1], smallAlumTubeDim[2]/2, 0., 2.0*CLHEP::pi);

new G4Tubs("smallBeamTube", smallBTubeDim[0], smallBTubeDim[1], smallBTubeDim[2]/2, 0., 2.0*CLHEP::pi);

The aluminum tube logical volume is create first; then the air logical volume is created and placed

inside the aluminum volume as a daughter. Finally, five aluminum tubes instances are created. (Note:

holePos is simply a G4ThreeVector)

insAlumLogical = new G4LogicalVolume(theSolids->GetSolid("smallAlumTube"),matMap["AlAlloy3"],"insAlumLogical");

insBeamLogical = new G4LogicalVolume(theSolids->GetSolid("smallBeamTube"), matMap["Air"], "insBeamLogical");

new G4PVPlacement(0, G4ThreeVector(0., 0., 0.), insBeamLogical, "insBeamTubePhysical", insAlumLogical, 0, 0);

G4int copy =0;

for (G4int i=0; i<5; i++){

Aluminum Container

Heavy Water


Application: SLOWPOKE Power Transient

Design Philosophy

UNRESTRICTED / ILLIMITÉ 5

• Inherently safe– Maximum excess

reactivity less than prompt critical

– Core lattice under-moderated

– Negative temperature and void coefficients

– Self-limiting power excursion

• No electromechanical safety devices


Purpose of the Application

Validation of fuel temperature coefficient (Doppler broadening);important for reactor licencing.

Crucial ingredient: delayed neutrons. Fission products emit additionalneutrons after a certain time. These neutrons affect the reactivity ofthe core as a function of time.

Delayed neutrons must be combed, too.


Sub/super-Critical Systems

MCNP inherently unsuited for calculation of deep sub-critical andsuper-critical systems.Comparison of MCNP and G4-STORK for U-235 sphere of differentsizes:


Other Applications: ZED-2 Subcritical Experiment

ZED-2 (Zero Energy Deuterium) Reactor in Chalk River, Canada:


ZED-2 Subcritical Experiment

Moderator drain experiment. Measurement of sub criticality.


Summary and Conclusions

We have produced a user code for GEANT4 (G4-STORK), which:I Follows neutron population in fissioning materials;I Maintains population size through a combing algorithm;I Includes delayed neutrons;I Calculates converged fission source distribution and keff ;I Allows for feedback between neutronics and material properties;

Issues and challenges:I Nuclear data format. Reverse engineered the format to be able to

generate files from MCNP data;I On-the-fly Doppler broadening. Essential part of the code, but rather

slow up to now.I Proper treatment of delayed neutrons;I Increase time span of the calculation;I Adjoint flux calculation.


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