Physics of Nuclear Reactors

Front Matter
Copyright
Contributors
Foreword
Preface
Acknowledgments
Introduction
	Introduction
	The atom and its nucleus
		The Bohr model of the atom
		The atomic energy levels
		Ionization
	The nucleus
		Nuclear dimensions and density
		Nuclear charge distribution
		Representation of a nucleus
		Isotopes, isotones, isobars
	Nuclide number density
	Nuclear stability
	Binding energy
		Binding energy per nucleon (BE/A)
		Binding energy of the last nucleon
		The drip lines
	Radioactivity
		Alpha decay
		Beta decay
		Positron decay or β+ decay
		Electron capture
		Gamma decay
		Rate of radioactive decay
		Mean life
		Half-life
		Units to express radioactivity emission
	Nuclear models
		The liquid drop model
		The shell model
	Nuclear energy levels
		Bound and excited energy levels
	Nuclear reactions
		Q-value of nuclear reactions
	Neutron interactions
	Neutron elastic scattering
		Kinematics of neutron elastic scattering
		The moderator
	Neutron inelastic scattering
		Discrete and continuum inelastic scattering
		Kinematics of neutron inelastic scattering
	Radiative capture
	(n, x) reaction
	Fission
	Probabilities of nuclear reactions
		Reaction rate and microscopic cross-section
		Flux, macroscopic cross-section, mean-free-path
		Total, capture, and absorption cross-sections
	Variation of cross-section with energy
	Compound nucleus reaction versus direct reaction
		Resonances
		Resolved and unresolved resonances
	Cross-section data representation and resonance formalism
	Fission mechanism
		Chance fissions
		Fission cross-sections
		Fission products
		Asymmetric fission
		Instability of the fission fragments
	Fission neutrons
		Prompt and delayed
		Fission neutron energies
	Energy released in fission
	Neutron chain reaction and generations
		Fission chain reaction
		Neutron generations and multiplication
	Concept of a nuclear reactor
		Criticality, critical size, and critical mass
		Basic design considerations
	Thermal reactor and fast reactor
		Energy region and fissile enrichment
		Energy distribution of neutrons in a reactor
	Spatial dependence of the neutrons in a reactor
	Basic physics of reactor design
	The reactor-power
	Summary
	Exercise problems
	References
	Further reading
Nuclear data
	Introduction: Nuclear data and its importance
		Application areas of nuclear data
	Nuclear data production
		Measurement of cross-sections
		Prediction of cross-sections
		Measurement of total, partial, and differential cross-sections
		Total cross-section estimation
		Partial cross-section estimation
		Use of cross-section standards
		EXFOR and CINDA
		Reaction model principles and cross-section predictions
		The spherical optical model
		The compound nucleus model
		Nuclear level densities
		The components of the potential
		Reaction model codes
		Validation of predicted data
		Error and co-variance data
	Nuclear data evaluation
		Evaluated nuclear data files
		The ENDF format
		Point-data representation in ENDF
		Resonance parameter data in ENDF
		The SLBW formalism
		The MLBW formalism
		The RM formalism
		Other RRR formalisms
		The unresolved resonance region
	Nuclear data processing
		Linearization
		Resonance reconstruction
		Doppler broadening
		Calculation of Doppler broadened resonance cross-sections
		Cross-section multigrouping
		Weighting function and self-shielding
		Problem-independent multigroup cross-section set
		Obtaining effective cross-sections in the resonance region
		Self-shielding in the URR
		Effect of Doppler-broadening on the reaction rate
		Group cross-sections in the resonance region for thermal neutron reactors
		Group-to-group transfer cross-sections
		Thermal scattering cross-sections
		Production cross-sections for particles other than neutrons
		Transport cross-sections
		Multigrouping the cross-sections from a linearized grid
		Cross-sections for shielding applications
		Displacement cross-sections
		Nuclear data processing codes
		Multigroup data validation
	Summary
	Exercise problems
	References
	Further reading
Types of nuclear reactors
	Introduction
	Classifications of nuclear reactors
		Based on materials used in reactors
		Other classifications
	Thermal neutron reactors
		Graphite moderated and water cooled reactors
		RBMK reactors
		Gas-cooled graphite-moderated reactors
		Boiling water reactors
		Pressurized water reactors
			Russian pressurized water reactors
		Pressured heavy water reactors
	Fast neutron reactors
		Breeding
		Sodium-cooled fast neutron reactors
	Future energy systems
	Fusion reactor systems
		Lawson criteria
		Inertial confinement
		Magnetic confinement
		Early fusion devices
	Fission-fusion hybrids
	Particle accelerators [11]
		LINACs and tandem accelerators
		Cyclotrons
		Synchrotron radiation sources
		Large hadron collider at CERN [13]
	Accelerator-driven subcritical systems (ADSS)
		Operational aspects of accelerator-driven system and its interface with reactor
		Overview of ADSS projects around the world
		The CERN energy amplifier
		The Los Alamos waste transmuter
		Other concepts
	One way coupled booster reactor concept in BARC
	More topics of interest
	Exercise problems
	References
	Further reading
Homogeneous reactor and neutron diffusion equation
	Introduction
	Neutron density and flux
	Neutron continuity equation
	Transport cross-section
	Fick's law of neutron diffusion
		Physical interpretation
		Fick's law limitations
	Diffusion equation
		Boundary conditions for the steady-state diffusion equation
	Neutron diffusion in nonmultiplying media
		A point source in an infinite medium
		Line source in an infinite media
		Plane source in a finite medium (bare slab)
		Physical meaning of diffusion length
	Neutron diffusion in a multiplying media
		Time-dependent flux in a slab reactor
		Finite bare cylindrical reactor
	Reflected infinite slab reactor
		Reflector savings
	Neutron life cycle in a thermal reactor
	Slowing down equation
		Moderation of neutrons in a medium having A>1
		Slowing down in an infinite medium of moderator and resonant absorber
	Neutron thermalization
		Monoatomic gas as moderator
		Bound moderator atoms in liquids and solids
			Water
			Heavy water
			Graphite
		Maxwellian distribution of neutron flux
		Reaction rate in a Maxwellian spectrum
		Comparison of PWR spectrum and FBR spectrum
	Two-group diffusion theory
	Multigroup diffusion equation
	Numerical solution of the diffusion equation in 1-D slab geometry
		Center mesh finite differencing scheme
			Boundary conditions
		Corner mesh finite differencing scheme
	Summary and more topics of interest
	Exercise problems
	References
	Further reading
Methods of solving neutron transport equation
	Introduction
	Assumptions in the neutron transport theory
	Neutron-nucleus interaction cross-sections
		Differential cross-section
	The neutron transport equation
		Formulation of the neutron transport problem
		Interface conditions
		Boundary conditions
		Time-independent transport equation: Fixed source problem
		Time-independent transport equation: k-eigenvalue problem
		The eigenvalues of neutron transport equations and their relations
	Solution to the neutron transport equation
		Multigroup approximation: Energy discretization
		The PN method
		P1 approximation
		The discrete ordinate method
	Numerical solution to the neutron transport equation
		The spatial discretization and sweeping scheme
		Limit on the size of spatial meshes and negative flux issue
		Source iteration method for multidimensions
		Discrete ordinate quadrature sets
			The double PN quadratures
		Discrete ordinate quadrature sets for multidimensional problems
			Level symmetric quadrature set
		The Ray effect
	The synthetic acceleration schemes
		The corrective equation
		The diffusion synthetic acceleration
		The transport synthetic acceleration
	The integral form of transport equation
	Solution to the integral transport equation
		Collision probability method
			Collision probability for slab geometry
			Collision probability for annular geometry
			Boundary condition
		Interface current method
			Discretized flux equation
			Properties of the collision probability matrices
			Calculation of collision probabilities in two-dimensional geometry
				Region-to-region collision probabilities
				Region-to-surface escape probability
				Surface-to-surface transmission probability
			Computation of collision probability integrals
			Use of boundary condition and solution to Collision Probability equations
	Method of characteristics approach
		MOC solution to one-dimensional slab geometry
		MOC solution for two-dimensional geometry
		Neutron tracking and quadrature weights
			Tracking of neutron trajectories
			Quadrature weights for polar and azimuthal angles
		Solution to the multigroup MOC equation
	Monte Carlo neutron transport
		Validity of Monte Carlo estimates
			Chebyshev inequality
			Law of large numbers
			The central limit theorem
		Random variable
			Uniform random variable
			Sampling from continuous distribution
		Simulating neutron transport with Monte Carlo method
			Simulating a neutron's history
			Sampling of source neutrons
			Sampling for neutron direction cosine from isotropic distribution
			Sampling for type of collision
		Tally computation
			Collision estimator
	More topics of interest
	Exercise problems
	References
Fuel burnup, fuel management, and fuel cycle physics
	Introduction
	Fuel burnup and its effects
		Energy production
		Changes in fuel composition and fission product buildup
			Units for measuring fuel burnup
		Fission product buildup
		Reactivity changes, effect of burnable poison, and absorber burnup
		Changes in kinetic parameters
	Bateman equation and solutions
		Solution to the Bateman equations
			Analytical methods
			Matrix exponential method
			Eigen-value methods
			Numerical recipes for solution of the Bateman equation
		Input data bases and reactor-specific cross-section sets
		Decay heat estimations
		Dose rates and radiotoxicities
	In-core fuel management
		Principles of continuous and batch refueling, in-core burnup and discharge burnup
			Out-to-in fueling
			In-to-out fueling
			Scattered fuel loading/checkerboard scheme
		Refueling aspects of BWR
		Refueling aspects of PHWRs
		Fuel management in PWRs
		Fuel management in FBRs
		Fuel doubling time
	Fuel management algorithms
	Long life cores
	Fuel utilization
		Components of a fuel cycle
		U-Pu fuel cycle
		Thorium fuel cycle
		FBR fuel cycle and closed fuel cycle
			System doubling time
	More topics of interest
	Exercise problems
	Additional exercises
	References
Nuclear reactor kinetics
	Introduction
	Definition of important kinetics parameters
	Prompt and delayed neutrons
		Delayed neutrons
		Delayed photo neutrons
	The point kinetics equations
		Derivation of point kinetic equations
		The solution of point kinetics equations
			Step change in reactivity
			Relation between reactor period and reactivity
				Importance of delayed neutrons
			Prompt jump/drop
			Reactivity unit as Inhour
			One group of delayed neutrons
		Ramp reactivity change
			Periodic change in reactivity
		Inverse point kinetics
	Kinetics of a subcritical reactor
		Subcritical reactor equilibrium power
		Power behavior with neutron source addition or withdrawal
		Power at criticality with ramp reactivity addition
		Instantaneous reactor period
		Safe approach to criticality
			Inverse count rate plot
			The maximum permissible reactivity addition rates during startup
	Numerical solution of point kinetics equations
		Simple example of determination of phi
		Solution of Eq. (7.77)
		Solution of Eq. (7.77) with A in two parts
	Reactivity feedback in nuclear reactors
		Factors causing reactivity change
		Thermal reactor feedback reactivity
		Fast reactor reactivity feedback
		Reactivity coefficients
			Static coefficients
			Dynamic reactivity coefficient
	Long irradiation reactivity effects in reactors
		Xenon poisoning effects
			Equilibrium xenon concentration
			Xenon poisoning after shutdown
		The buildup of samarium poison
		Other fission product poisons
	Reactor stability analysis
		Linear stability analysis
			Stability of zero power reactor
			Stability of reactor with feedback
		Frequency domain methods of stability analysis
			Zero power transfer function (ZPTF)
			Feedback transfer function (FBTF)
			Dynamic power coefficient of reactivity
			Response of reactor for sinusoidal reactivity input
			Nyquist stability criterion
				Contour mapping
			Information about relative stability
			Nyquist stability: ZPTF
			Nyquist stability: FBTF
		Nonlinear stability analysis
	Neutronic coupling in a reactor
		Size effect
		Shape effects
		Safety concerns
	Xenon oscillations
		Affected systems
		Controlling spatial oscillations
		Example: PWR axial control
	Space time kinetics
		Necessity of space-dependent modeling
		Evolution of space time kinetics methodology
			Solution of multigroup diffusion equations
		Flux factorization method
			Amplitude function
			Shape function
		Coupled neutron kinetics and thermal hydraulics [20]
		Case study [19-21]
	Summary
	Annexure
		Formal statements of stability and Lyapunov's direct method for global stability
			Lyapunov stability
			Asymptotic stability
			Exponential stability and rate of convergence
			Bounded-input-bounded-output stability
			Global stability of nonlinear systems
				Lyapunov's direct method for nonlinear stability
				Lyapunov's indirect theorem
	Exercise problems
	References
Nuclear reactor safety
	Introduction
		Broad safety objectives
		Defense in depth
		Safety during reactor operation
		The three Cs of nuclear safety
		Risk evaluation
		Safety assessment (regulatory authorities)
	Fundamental safety principles and safety framework [2, 9-11]
		Fundamental safety principles
		Safety framework
	Safety requirements and graded approach in safety
		Plant states
		Acceptance criteria
	Safety features in a NPP
		Inherent safety
		Reactor control system
		Reactor protection system
		Emergency core cooling system
		Reactor containment
		Active and passive safety features
		Analysis for safety assessment
	Deterministic safety assessment [12-22]
		The objective
		Evolution of DSA
		A case study of DSA (postulated rod withdrawal accident in a PWR) [18-22]
	Reliability analysis of safety systems [4, 23-30]
		Approach and aspects of reliability analysis
		Design and system function
		System boundary and system failure modes
		Common cause failures
			Beta (β) factor model
			Alpha factor model
		Human reliability analysis
			Human cognitive reliability model
		Data collection
		Reliability quantification
		Other issues
		Summary
	Probabilistic safety assessment: Level-1, Level-2, and Level-3. Case studies [31-54]
		Introduction approach to PSA
		Basis and approach
		Level-1 probabilistic safety assessment
			Management and organization
			Collection of information on design and operation of the plant
			Identification of sources of radioactivity and accident initiators
			Success criteria formulation and supporting analysis
			Accident sequence analysis
			Systems analysis
			Human reliability analysis
			Dependent failure analysis
			Data assessment and parameter estimation
			Model integration and core damage frequency quantification
			Sensitivity and uncertainty analysis
			Interpretation of results
			Quality assurance aspects of PSA
			Documentation of the analysis: Display and interpretation of results
			Computer programs available to perform Level-1 PSA
		Level-2 probabilistic safety assessment
			Interface between the Level-1 and Level-2 PSA
			Containment analysis and accident progression
			Quantification of the containment event tree and categorization of the endpoints into release categories
			Radiological source term analysis for the release categories
			Uncertainty analysis and sensitivity analysis
			Documentation of the analysis: Display and interpretation of results
			Codes available to perform Level-2 PSA
		Level-3 probabilistic safety assessment
			Interface between Level-2 analysis and Level-3 analysis
			Radionuclide release characterization
			Meteorological data and other information sources
			Environmental transport and deposition
			Radiological exposure and dose assessment
			Protective action (counter measure) modeling
			Economic consequences
			Uncertainty analysis and sensitivity analysis
			Documentation of the analysis: presentation and interpretation of results
			Programs available to perform Level-3 PSA
		Use and development of probabilistic safety assessment: Case studies
		Conclusion
	Uncertainty analysis in reliability and risk assessment [55-60]
		Introduction
		Types of uncertainty
			Input data (parametric) uncertainties
			Model uncertainty
		Uncertainty propagation [59, 60]
		Uncertainty methodology in PSA
			Monte-Carlo technique
			Discretization
			Method of moments
			Method of adjoint operator
		Importance and sensitivity analysis
			Importance measures
			Sensitivity analysis
	Major nuclear reactor accidents
		The three Mile Island-2 accident [2, 3]
		The Chernobyl accident [2, 3, 61, 62]
			Cause of accident
			The accident propagation sequence
				The event: April 25, 1986
				The event: April 26, 1986
				Actions initiated
		The Fukushima accident [3, 63]
	Brief history of the nuclear safety
	Summary
	Exercise problems
	References
Design methods and computer codes
	Introduction
	Methods of neutronics analysis in thermal reactors
		Neutron cross-section library for lattice and core calculations
			Resonance cross-section treatment
				Narrow resonance approximation
				Intermediate resonance approximation
		Heterogeneous system
		Fuel assembly analysis methods
			Supercell methods
			Direct transport methods
			Improvements in homogenization
		Description of lattice analysis codes
			Indian lattice analysis codes
			International lattice analysis codes
		Methods for 3-D whole-core analysis
			Methods using assembly or pin homogenization
				Finite difference method
				Nodal method
			Methods without homogenization or one-step methods
		Description of whole-core analysis codes
			Codes using assembly homogenization or pin-cell homogenization (India)
			A few examples of codes using fuel assembly homogenization or pin-cell homogenization (world)
			Codes using single-step methods
	Methods of neutronics analyses in fast reactors
		Self-shielding calculations using subgroup/probability table method
			Theory of probability table method
			Effective cross sections using subgroup parameters
			Heterogeneity correction to cross sections using subgroup/probability table method
		ABBN-90 group constants system
		ERANOS system
		Indian codes and methodology for fast reactor analysis
			Steady-state deterministic neutronics analysis using FARCOB system
			Steady-state stochastic neutronics analysis using OpenMC
			Safety analysis methods and codes used
	More topics of interest
	References
Experimental and operational reactor physics
	Introduction
	Neutron monitoring: Neutron detectors and instruments
		Modes of detector operation
			Pulse mode of operation
			Current mode of operation
		Detectors used for reactor start-up/power regulation
			Ion chambers
			Fission counter
			BF3 proportional counter
			10B-lined neutron detector
			Self-powered neutron detector
		In-core and out-core detectors
		Overlapping regimes for neutron monitoring during a reactor start-up
		Gamma sensitivity of neutron detectors
	Neutron flux measurement using activation method
		Characteristics of neutron spectrum in nuclear reactors
			Thermal neutron spectrum (Maxwellian spectrum)
			Intermediate neutron spectrum
			Fast flux-Fission neutron spectrum
		Foil activation methods
			Theory of activation method
				Self-shielding factor
				Flux depression factor:
			Activation detector properties
				Gold foil
				Copper foil
				Indium foil
				Manganese foil
			Thermal flux measurement and cadmium ratio
				Effective cadmium cut-off
			Measurement of fast neutron fluxes
		Measurement of Westcott spectrum parameters ``r´´ and ``T´´
		Neutron spectrum measurement
			One group case-total flux
			Two-group flux measurement
			Measurement of neutron spectrum in arbitrary number of groups
			Overview of SAND-II spectrum unfolding code
	Start-up or commissioning experiments in reactors
		Approach to criticality and subcritical multiplication measurements
		Nuclear instrumentation for start-up
		Neutron sources for reactor start-up
			Spontaneous fission neutrons
			Photo neutrons
			External neutron sources
		Neutron multiplication in a subcritical system
		Inverse multiplication (count rate) curves
		Reactivity addition rate and reactor period during first approach to criticality
	Reactivity measurements
		Asymptotic period method
		Rod drop method
		Subcritical counts method
		Inverse kinetics method
		Measurement of integral and differential worth of control rods
	Low power physics experiments in thermal reactors
		Calibration of the reactivity worth of regulation and protection devices
		Dynamic testing and measurement of worth of reactor protection systems
		Temperature coefficient measurement
		Absolute power calibration and thermal power measurement
		Measurement of xenon load
	Reactor start-up in sodium cooled fast reactors
		Initial fuel loading and FAC
		Control rod worth estimation
		Isothermal temperature coefficient measurement
		Power and flow coefficient measurements
		Power raising operation
	Failed fuel detection
		Failed fuel detection in thermal reactor systems
		Failed fuel detection in fast reactor systems
			Fast Breeder Test Reactor experiment of fuel pin clad rupture detection
	Regulatory aspects and reactor experimentation and operation
	Some critical/subcritical experimental facilities
		Examples of thermal reactor critical facilities
			ZERLINA
			PURNIMA I, II, and III
			KAMINI
			Advanced Heavy Water Reactor Critical Facility (AHWR-CF)
		Subcritical facilities with source
			MUSE
			YALINA
			BRAHMMA
		Some fast critical and experimental systems
			The zero power reactor (ZPR) and zero power physics reactor (ZPPR) facilities
			The Russian BFS critical facilities
			The MASURCA critical facility
			The experimental breeder reactor-I (EBR-I)
			The experimental breeder reactor-II (EBR-II)
			The fast breeder test reactor
	More topics of interest
	Exercise problems
	References
Radiation safety and radiation shielding design
	Introduction
	Basic radiation physics
		Fundamental definitions and concepts
		Alpha particles
			Bragg curve
		Beta particles
			Range of β particles
		Photons
			Photoelectric interactions
			Compton scattering
		Pair production
		Photon attenuation in materials
	Radiation dosimetry
		Radiation dose and units
		Radiation dose equivalent
		Radiation exposure
		External and internal exposures
		Radiation dose calculations
			Gamma radiation dose calculations by point kernel methods
			Point source in a vacuum
			Point source with a shield
			Extended sources
				Example: Line source
				Example: Disc source
			Buildup factor
			Dose rates with buildup factors
			Taylor's form
		Neutron radiation attenuation
			Boltzmann transport equation
			Discrete ordinates method
			Monte Carlo method
			Cross-sections for shielding applications
			Neutron irradiation effects in metals
			Displacement per atom
	Gamma shields
		Lead
		Bismuth
		Tungsten
		Iron and steel
		Concrete
		Half-value and tenth-value layers
	Neutron shields
		Hydrogenous materials
		Boron and its compounds
		Lithium
	Reactor sources of radiation
		Reactor core and vessel
		Reactor coolant system
		Steam and turbine system
		Radioactive waste system
	Radioactive sources in fuel cycle facilities
	Radiation dose limits for exposures
		Occupational dose limits
		Dose limits for members of the public
		Supervised areas
		Controlled areas
		Zone classification
	Shield design for reactors
		Interaction between shielding design and core design
			Low neutron leakage core design in LWR and FR
			Core monitoring and shield design in a pool type fast reactor
		Examples of shield design
			PHWR: TAPS 3 and 4 reactor
			Fast reactor: PFBR
			Secondary sodium activity computation in PFBR
	Complementary shielding
		Line of sight
			Cylindrical duct
			Rectangular duct
		Transport calculations
			Example: Streaming problem in AHWR top-shield
			Example: Apsara model for streaming through gaps in the transfer arm in PFBR
	Shield design methods in fuel cycle facilities
		Reprocessing plant
		Fuel fabrication
		Fuel assembly
		Waste disposal
			Example: Shield requirement estimation for a typical fuel pin storage room
	Summary and more topics of interest
	Exercise problems
	References
	Further reading
Nuclear reactors of the future
	Introduction
	Generation IV reactors
		Gen IV reactor types
			Very high temperature reactor
			Sodium-cooled fast reactor
			Lead-cooled fast reactors
			Gas-cooled fast reactor
			Molten salt reactor
			Super critical water-cooled reactor
			The Gen IV parameters
		Reactor physics challenges
		Indian high temperature reactors
			Compact high temperature reactor
			Innovative high temperature reactor
		Conclusions
	Small and modular reactors
		Introduction
		General features
			Size based
			Design based
		Classification of SMRs
			Temperature and power range
			LWR/PHWR water reactors
				A1. PHWR-220 (NPCIL, India)
				A2. AHWR-300 (BARC, India)
			Fast neutron spectrum reactor
				A1. GE-Hitachi (PRISM)
				A2. SVBR - 100
				A3. BREST-OD-300
			High temperature reactors
				HTR-PM (Tsinghua University, China)
			Molten salt reactors
				Integral molten salt reactor (Terrestrial Energy, Canada)
		The impact of SMRs
		Conclusions
	Traveling wave reactors
		Introduction
		The fuel burnup wave
			The genesis
			The mechanism
			Characteristics
		Requirements for sustained fuel burnup wave
			The necessary conditions
				A1. Feoktistov approach
				A2. The CANDLE fuel burnup
				A3. The van Dam approach
				A4. UC Berkeley approach
			The neutronics
			The constraints
		Mathematical methodology
			Mathematical approach
				A1. Steady state
				A2. Transient state
			General results
		Different types of B&B reactors
			Edward Teller
			HTGR TWR
			The CANDLE reactor
			The TerraPower standing wave reactor
			UC Berkeley SWR
		Conclusions
	Review questions
	References
	Further reading
Index
	A
	B
	C
	D
	E
	F
	G
	H
	I
	K
	L
	M
	N
	O
	P
	Q
	R
	S
	T
	U
	V
	W
	X
	Y
	Z