**7. References**

Arsenlis, A., B. D. Wirth, et al. (2004). "Dislocation density-based constitutive model for the mechanical behaviour of irradiated Cu." Philosophical Magazine 84(34): 3617-3635.

Multiscale Materials Modeling of Structural Materials for Next Generation Nuclear Reactors 281

Deo, C. S., M. A. Okuniewski, et al. (2007). "Helium bubble nucleation in bcc iron studied by kinetic Monte Carlo simulations." Journal of Nuclear Materials 361(2-3): 141-148. Deo, C. S., S. G. Srinivasan, et al. (2007). "Kinetics of the Migration and Clustering of

Deo, C. S., S. G. Srivilliputhur, et al. (2006). Kinetics of the nucleation and growth of helium bubbles in bcc iron. Materials Research Society Symposium Proceedings. Domain, C., C. S. Becquart, et al. (2004). "Simulation of radiation damage in Fe alloys: An

Evans, J. H. (2004). "Breakaway bubble growth during the annealing of helium bubbles in

Fichthorn, K. A. and W. H. Weinberg (1991). "Theoretical foundations of dynamical Monte

Finnis, M. W. and J. E. Sinclair (1984). "A SIMPLE EMPIRICAL N-BODY POTENTIAL FOR

Friedel, J. (1955). "On the linear work hardening rate of face-centred cubic single crystals."

Fu, C.-C. and F. Willaime (2005). "Ab initio study of helium in alpha -Fe: dissolution,

Fu, C.-C., F. Willaime, et al. (2004). "Stability and mobility of mono- and di-interstitials in

Gao, F., D. J. Bacon, et al. (1999). "Kinetic Monte Carlo annealing simulation of damage

Hasegawa, A., H. Shiraishi, et al. (1994). "Behavior of helium gas atoms and bubbles in low activation 9Cr martensitic steels." Journal of Nuclear Materials 212/215: 720-724. Hayward, E. and C. Deo (2010). "A Molecular Dynamics Study of Irradiation Induced

Heinisch, H. L. and B. N. Singh (2003). "Kinetic Monte Carlo simulations of void lattice formation during irradiation." Philosophical Magazine 83(31/34): 3661-3676. Heinisch, H. L., B. N. Singh, et al. (2000). "Kinetic Monte Carlo studies of the effects of

Johnson, R. A. and D. J. Oh (1989). "Analytic embedded atom method model for bcc metals."

Carlo simulations." Journal of Chemical Physics 95(2): 1090-1096.

Structure Defects and Mechanical Properties 50(1): 45-55.

alpha -Fe." Physical Review Letters 92(17): 175503-175504.

Simulation of Materials - Beyond pair potentials": 219-222

interstitial clusters." Journal of Nuclear Materials 276: 59-64.

Journal of Materials Research 4(5): 1195-1201.

metals." Journal of Nuclear Materials 334(1): 40-46.

Philosophical Magazine 46: 1169-1186.

and Materials Physics) 72(6): 64117-64111.

100613.

145.

16(2): 101-116.

Extrinsic Gas in bcc Metals." Journal of ASTM International 4(9): 100698, 100691-

object kinetic Monte Carlo approach." Journal of Nuclear Materials 335(1-3): 121-

TRANSITION-METALS." Philosophical Magazine a-Physics of Condensed Matter

migration, and clustering with vacancies." Physical Review B (Condensed Matter

produced by cascades in alpha-iron." Materials Research Society Symposium - Symposium on Microstructural Processes in Irradiated Materials 540: 703-708. Ghoniem, N. M., E. P. Busso, et al. (2003). "Multiscale modelling of nanomechanics and micromechanics: an overview." Philosophical Magazine 83(31-34): 3475-3528. Harrison, R. J., A. F. Voter, et al. (1989). Embedded atom potential for bcc iron. "Atomic

Cascades in Iron Containing Hydrogen." Cmc-Computers Materials & Continua

Burgers vector changes on the reaction kinetics of one-dimensionally gliding


Bacon, D. J., A. F. Calder, et al. (1995). "COMPUTER-SIMULATION OF DEFECT

Bacon, D. J., F. Gao, et al. (2000). "The primary damage state in fcc, bcc and hcp metals as seen in molecular dynamics simulations." Journal of Nuclear Materials 276: 1-12. Barashev, A. V., D. J. Bacon, et al. (1999). "Monte Carlo investigation of cascade damage

Barashev, A. V., D. J. Bacon, et al. (2000). "Monte Carlo modelling of damage accumulation in metals under cascade irradiation." Journal of Nuclear Materials 276(1): 243-250. Becquart, C. S., C. Domain, et al. (2005). "The influence of the internal displacement cascades

Bortz, A. B., M. H. Kalos, et al. (1975). "A new algorithm for Monte Carlo simulation of Ising

Bringa, E. M., B. D. Wirth, et al. (2003). "Metals far from equilibrium: From shocks to

Calder, A. F. and D. J. Bacon (1993). "A molecular dynamics study of displacement cascades

Calder, A. F. and D. J. Bacon (1994). "MD MODELING OF DISPLACEMENT CASCADES IN

Caturla, M. J., N. Soneda, et al. (2000). "Comparative study of radiation damage accumulation in Cu and Fe." Journal of Nuclear Materials 276: 13-21. Caturla, M. J., N. Soneda, et al. (2006). "Kinetic Monte Carlo simulations applied to

Cirimello, P., D. G., et al. (2006). "Influence of metallurgical variables on delayed hydride carcking in Zr-Nb prssure tubes." Journal of Nuclear Materials 350: 135-146. Cottrell, G. A. (2003). "Void migration, coalescence and swelling in fusion materials." Fusion

de la Rubia, T. D., H. M. Zbib, et al. (2000). "Multiscale modelling of plastic flow localization

Dederichs, P. and K. Schroeder (1978). "Anisotropic diusion in stress fields." Physical

Deo, C., C. Tom, et al. (2008). "Modeling and simulation of irradiation hardening in

Deo, C. S., M. Baskes, et al. (2007). "Helium Bubble Nucleation in BCC Iron studied by kinetic Monte Carlo simulations." Journal of Nuclear Materials To be published.

structural ferritic steels for advanced nuclear reactors." Journal of Nuclear Materials

B: Beam Interactions with Materials and Atoms 202(SUPPL.): 56-63.

spin systems." Journal of Computational Physics 17(1): 10-18.

in alpha iron." Journal of Nuclear Materials 207: 25-45.

growth." Journal of Nuclear Materials 351(1-3): 78-87.

in irradiated materials." NATURE 406(6798): 871-874.

Engineering and Design 66-8: 253-257.

Review B 17(6): 2524-2536.

377(1): 136-140.

Materials and Atoms 102(1-4): 37-46.

Materials and Atoms 228: 181-186.

in Solids 129(1-2): 65-68.

709-714.

PRODUCTION BY DISPLACEMENT CASCADES IN METALS." Nuclear Instruments & Methods in Physics Research Section B-Beam Interactions with

effects in metals under low temperature irradiation." Materials Research Society Symposium - Symposium on Microstructural Processes in Irradiated Materials 540:

structure on the growth of point defect clusters in radiation environment." Nuclear Instruments & Methods in Physics Research Section B-Beam Interactions With

radiation damage." Nuclear Instruments and Methods in Physics Research, Section

BCC IRON USING A MANY-BODY POTENTIAL." Radiation Effects and Defects

irradiated materials: The effect of cascade damage in defect nucleation and


Multiscale Materials Modeling of Structural Materials for Next Generation Nuclear Reactors 283

Seeger, A., J. Diehl, et al. (1957). "Work-hardening and Work-softening of face-centred cubic

Sencer, B. H., G. M. Bond, et al. (2001). "Microstructural alteration of structural alloys by low

International Symposium on Effects of Radiation on Materials 1045: 588-611. Sencer, B. H., F. A. Garner, et al. (2002). "Structural evolution in modified 9Cr-1Mo

spectra at low temperatures." Journal of Nuclear Materials 307: 266-271. Simonelli, G., R. Pasianot, et al. (1994). "Point-Defect Computer Simulation Including

Singh, B. N., S. I. Golubov, et al. (1997). "Aspects of microstructure evolution under cascade

Singh B. N. and H. Trinkaus (1992). "An analysis of the bubble formation behaviour under different experimental conditions." Journal of Nuclear Materials 186(2): 153-165. Stoller, R. E., G. R. Odette, et al. (1997). "Primary damage formation in bcc iron." Journal of

Terentyev, D., C. Lagerstedt, et al. (2006). "Effect of the interatomic potential on the features

Theis, U. and H. Wollenberger (1980). "Mobile interstitials produced by neutron irradiation in copper and aluminium." Journal of Nuclear Materials 88(1): 121-130. Thompson, L., G. Youngblood, et al. (1973). "Defect retention in copper during electron

Tome, C. N., H. A. Cecatto, et al. (1982). "Point-Defect Diffusion in a strained crystal."

Trinkaus, H., B. Singh, et al. (1996). "Microstructural evolution adjacent to grain boundaries

Vitek, V. (1976). "Computer simulation of screw dislocation-motion in bcc metals under

Was, G. (2007). Fundamentals of Radiation Materials Science (Metals and Alloys), Springer-

Wen, M., N. M. Ghoniem, et al. (2005). "Dislocation decoration and raft formation in

Wirth, B. D. and E. M. Bringa (2004). "A Kinetic Monte Carlo Model for Helium Diffusion

Wirth, B. D., O. G.R., et al. (2004). "Multiscale Modeling of Radiation Damage in Fe-based Alloys in the Fusion Environment." Journal of Nuclear Materials 329-333: 103. Zinkle, S. J. (1987). "Microstructure and properties of copper alloys following 14-MeV

irradiated materials." Philosophical Magazine 85(22): 2561-2580.

neutron irradiation." Journal of Nuclear Materials 150(2): 140-158.

under cascade damage conditions and helium production." Journal of Nuclear

effect of external shear and uniaxial stresses." Proceedings of the Royal Society of London Series a-Mathematical Physical and Engineering Sciences 352(1668): 109-

of displacement cascades in alpha-Fe: A molecular dynamics study." Journal of

Angular Forces in BCC Iron." Physical Review B 50(3): 727-738.

damage conditions." Journal of Nuclear Materials 251: 107-122.

irradiation at 80K." Radiation Effects 20(1/2): 111-134.

Nuclear Materials 251: 49-60.

Nuclear Materials 351(1-3): 65-77.

Physical Review B 25(12): 7428-7440.

in Fusion Environments." Physica Scripta T108: 80-84.

Materials 237: 1089-1095.

124.

Verlag.

and Clustering

temperature irradiation with high energy protons and spallation neutrons." American Society for Testing and Materials Special Technical Publication, 20th

ferritic/martensitic steel irradiated with mixed high-energy proton and neutron

metal crystals." Philosophical Magazine 2: 323-350.


Kinchin, G. H. and R. S. Pease (1955). "The displacement of atoms in solids by radiation."

Kocks, U. F. (1977). "The theory of an obstacle-controlled yield strength-report after an international workshop." Material Science and Engineering 27(3): 291-298. Koppenaal, T. J. and R. J. Arsenault (1971). "Neutron-irradiation-strengthening in f.c.c. single

Kroupa, F. and P. B. Hirsch (1964). "Elastic interaction between prismatic dislocation loops and straight dislocations." Discussions of the Faraday Society(38): 49-55. Lebensohn, R. A. and C. N. Tome (1993). "A self-consistent anisotropic approach for the

Malerba, L. (2006). "Molecular dynamics simulation of displacement cascades in alpha-Fe: a

Marksworth, A. J. (1973). "Coarsening of gas filled pores in solids." Metallurgical

Matthews, J. R. and M. W. Finnis (1988). "Irradiation creep models - an overview." Journal of

Morishita, K., R. Sugano, et al. (2003a). "MD and KMC modeling of the growth and

Morishita, K., R. Sugano, et al. (2003b). "Thermal stability of helium-vacancy clusters in

Muroga, T., H. L. Heinisch, et al. (1992). "A comparison of microstructures in copper

Norgett, M. J., M. T. Robinson, et al. (1975). "Proposed method for calculating displacement

Odette, G., B. D. Wirth, et al. (2001). "Multiscale-Multiphysics Modeling of Radiation-

Olander, D. (1981). Fundamental Aspects of Nuclear Reactor Fuel Elements, National

Osetsky, Y. N., D. J. Bacon, et al. (1999). Atomistic simulation of mobile defect clusters in

Phythian, W. J., R. E. Stoller, et al. (1995). "A comparison of displacement cascades in copper

Rice, P. M. and S. J. Zinkle (1998). "Temperature dependence of the radiation damage

shrinkage mechanisms of helium-vacancy clusters in Fe." Journal of Nuclear

iron." Nuclear Instruments & Methods in Physics Research Section B-Beam

irradiated with fission, fusion and spallation neutrons." Journal of Nuclear

Damaged Materials: Embrittlement of Pressure Vessel Steels." MRS Bulletin 26:

metals. Microstructural Processes in Irradiated Materials. S. J. Zinkle, G. E. Lucas,

and iron by molecular dynamics and its application to microstructural evolution."

microstructure in V-4Cr-4Ti neutron irradiated to low dose." Journal of Nuclear

neutron irradiation." Philosophical Magazine 7(74): 285-299.

critical review." Journal of Nuclear Materials 351(1-3): 28-38.

Interactions With Materials and Atoms 202: 76-81.

dose rates." Nuclear Engineering and Design 33(1): 50-54.

simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys." Acta Metallurgica et Materialia 41(9): 2611-2624. Makin, M. J., A. D. Whapman, et al. (1962). "Formation of dislocation loops in copper during

Reports on Progress in Physics 18: 1-51.

Transactions 4(11): 2651-2656.

Nuclear Materials 159: 257-285.

Materials 323(2/3): 243-250.

176.

Materials 191/194(pt B): 1150-1154.

Technical Information Service.

R. C. Ewing and J. S. Williams. 540: 649-654.

Journal of Nuclear Materials 223(3): 245-261.

Materials 258/263(pt B): 1414-1419.

crystals." Metallurgical Reviews(157): 175-196.


**1. Introduction**

part thereof.

and appreciate the central concepts.

Group theory is a vast mathematical discipline that has found applications in most of physical science and particularly in physics and chemistry. We introduce a few of the basic concepts and tools that have been found to be useful in some nuclear engineering problems. In particular those problems that exhibit some symmetry in the form of material distribution and boundaries. We present the material on a very elementary level; an undergraduate student well versed in harmonic analysis of boundary value problems should be able to easily grasp

**Application of Finite Symmetry Groups to** 

**Reactor Calculations** 

*1NRC, Washington DC* 

Yuri Orechwa1,\* and Mihály Makai2

*2BME Institute of Nuclear Techniques, Budapest* 

**13**

*1USA 2Hungary* 

The application of group theory to the solution of physical problems has had a curious history. In the first half of the 20th century it has been called by some the "Gruppen Pest" , while others embraced it and went on to win Noble prizes. This dichotomy in attitudes to a formal method for the solution of physical problems is possible in light of the fact that the results obtained with the application of group theory can also be obtained by standard methods. In the second half of the 20th century, however, it has been shown that the formal application of symmetry and invariance through group theory leads in complicated problems not only to deeper physical insight but also is a powerful tool in simplifying some solution methods. In this chapter we present the essential group theoretic elements in the context of crystallographic point groups. Furthermore we present only a very small subset of group theory that generally forms the first third of the texts on group theory and its physical applications. In this way we hope, in short order, to answer some of the basic questions the reader might have with regard to the mechanical aspects of the application of group theory, in particular to the solution of boundary value problems in nuclear engineering, and the benefits that can accrue through its formal application. This we hope will stimulate the reader to look

The main illustration of the application of group theory to Nuclear Engineering is presented in Section 4 of this chapter through the development of an algorithm for the solution of the neutron diffusion equation. This problem has been central to Nuclear Engineering from the very beginning, and is thereby a useful platform for demonstrating the mechanics of bringing group theoretic information to bear. The benefits of group theory in Nuclear Engineering are

\*The views expressed are those of the authors and do not reflect those of any government agency or any

more deeply into the subject is some of the myriad of available texts.

