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Dislocation model of strain anisotropy

Published online by Cambridge University Press:  29 February 2012

Tamás Ungár
Tamás Ungár*
Affiliation:
Department of Materials Physics, Eötvös University, Budapest, Hungary
*
Electronic mail: ungar@ludens.elte.hu
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Abstract

Dislocation model of strain anisotropy is presented. The dislocation theorem of strain broadening is suggested which means that strain broadening can only be caused by dislocation-type lattice defects. Based on this theorem strain anisotropy is modeled and accounted for by assuming that strain broadening is caused by dislocations or dislocation-type lattice defects. The effect of strain anisotropy is summarized in hkl dependent dislocation contrast factors, which can be either averaged over the permutations of hkl indices or are different for each different reflection. The dislocation model of strain anisotropy provides a powerful tool to analyze slip activity, Burgers vector populations, and plasticity on the basis of line profile analysis.

Keywords

strain anisotropydislocationscontrast factors
Type
X-Ray Diffraction
Information
Powder Diffraction , Volume 23 , Issue 2 , June 2008 , pp. 125 - 132
Copyright
Copyright © Cambridge University Press 2008

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References

Aqua, E. N. and Wagner, C. N. J. (1966). “X-ray diffraction study of deformation of Nb‐RE alloys,” Trans. Metall. Soc. AIME TMSAAB 236, 1085–1089.Google Scholar
Balogh, L., Ribárik, G., and Ungár, T. (2006). “Stacking faults and twin boundaries in fcc crystals determined by x-ray diffraction profile analysis,” J. Appl. Phys. JAPIAU 10.1063/1.2216195 100, 023512.CrossRefGoogle Scholar
Becker, K. (1927). “Der röntgenographische Nachweis von Kornwachstum und Vergütung in Wolframdraeten mittels des Debye-Scherrer-Verfahrens,” Z. Phys. ZEPYAA 42, 226–245.CrossRefGoogle Scholar
Borbély, A., Dragomir-Cernatescu, J., Ribárik, G., and Ungár, T. (2003). “Computer program ANIZC for the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889802021581 36, 160–162.CrossRefGoogle Scholar
Brindley, G. W. (1940). “Lattice distortion in cold-worked metals,” Proc. Phys. Soc. 52, 117–126.CrossRefGoogle Scholar
Caglioti, G., Paoletti, A., and Ricci, F. P. (1958). “Choice of collimators for a crystal spectrometer for neutron diffraction,” Nucl. Instrum. NUINAO 10.1016/0369-643X(58)90029-X 3, 223–228.CrossRefGoogle Scholar
Cordier, P., Ungár, T., Zsoldos, L., and Tichy, G. (2004). “Dislocations creep in MgSiO3 perovskite at conditions of the earth’s uppermost lower mantle,” Nature (London) NATUAS 10.1038/nature02472 428, 837–840.CrossRefGoogle ScholarPubMed
Dehlinger, U. and Kochendoerfer, A. (1939). “Roentgenographische Messung der Teilchengroesse und der verborgen elastischen Spannungen in kaltverformten Blechen,” Z. Metallkd. ZEMTAE 31, 231–234.Google Scholar
Dehlinger, U. and Schäcke, H. (1943). “Messung der reflektierten Röntgenintensität in Abhängigkeit vom Grad der plastischen Verformung bei Metallen,” Naturwiss. NATWAY 31, 548–548.CrossRefGoogle Scholar
Dragomir, I. C. and Ungár, T. (2002). “The contrast factors of dislocations in the hexagonal crystal system,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889802009536 35, 556–564.CrossRefGoogle Scholar
Eastabrook, J. N. and Wilson, A. J. C. (1952). “The diffraction of x-rays by distorted-crystal aggregates. III. Remarks on the interpretation of the Fourier coefficients,” Proc. Phys. Soc. London, Sect. B PPSBAP 10.1088/0370-1301/65/1/310 65, 67–75.CrossRefGoogle Scholar
Fischer, J. E., Bendele, G., Dinnebier, R., Stephens, P. W., Lin, C. L., Bykovetz, N., and Zhu, Q. (1995). “Structural analysis of fullerene and fulleride solids from synchrotron x-ray diffraction,” J. Phys. Chem. Solids JPCSAW 10.1016/0022-3697(95)00082-8 56, 1445–1457.CrossRefGoogle Scholar
Gaál, I. (1984). “Effect of dislocation distribution on the X-ray scattering from deformed metals,” in Proceedings of the 5th Riso International Symposium on Metallurgy and Material Science, edited by Andersen, N. Hessel, Eldrup, M., Hansen, N., Jensen, D. Juul, Leffers, T., Lilholt, H., Pedersen, O. B., and Singh, B. N. (Riso National Lab., Roskilde, Denmark), pp. 249–254.Google Scholar
Gillies, D. C. and Lewis, D. (1968). “Lattice strain in tungsten carbide,” Powder Metall. PWMTAU 11, 400–410.CrossRefGoogle Scholar
Hall, W. H. (1949). “X-ray line broadening in metals,” Proc. Phys. Soc., London, Sect. A PPSAAM 10.1088/0370-1298/62/11/110 62, 741–743.CrossRefGoogle Scholar
Jaswon, M. A. and Wheeler, J. A. (1948). “Atomic displacements in the austenite-martensite transformation,” Acta Crystallogr. ACCRA9 10.1107/S0365110X48000582 1, 216–224.CrossRefGoogle Scholar
Jones, F. W. and Sykes, C. (1937). “The superlattice in β brass,” Proc. R. Soc. London, Ser. A PRLAAZ 161, 440–446.Google Scholar
Klimanek, P. and Kužel, R. Jr. (1988). “X-ray diffraction line broadening due to dislocations in non-cubic materials. I. General considerations and the case of elastic isotropy applied to hexagonal crystals,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889887009580 21, 59–66.CrossRefGoogle Scholar
Kocks, U. F. and Scattergood, R. O. (1969). “Elastic interactions between dislocations in a finite body,” Acta Metall. AMETAR 10.1016/0001-6160(69)90093-5 17, 1161–1170.CrossRefGoogle Scholar
Kostorz, G. (1996). Physical Metallurgy, edited by Cahn, R. W. and Haasen, P. (North-Holland, Amsterdam), pp. 1115–1199.CrossRefGoogle Scholar
Krivoglaz, M. A. (1969). X-ray and Neutron Diffraction in Nonideal Crystals (Plenum, New York).Google Scholar
Krivoglaz, M. A. (1996). Theory of X-ray and Thermal Neutron Scattering by Real Crystals (Springer-Verlag, Berlin).Google Scholar
Kuhn, H.-A., Biermann, H., Ungár, T., and Mughrabi, H. (1991). “An x-ray study of creep-deformation induced changes of the lattice mismatch in the γ′ hardened monocrystalline Ni-base superalloy SRR 99,” Acta Metall. Mater. AMATEB 39, 2783–2794.CrossRefGoogle Scholar
Kužel, R. Jr. and Klimanek, P. (1988). “X-ray diffraction line broadening due to dislocations in non-cubic materials. II. The case of elastic anisotropy applied to hexagonal crystals,” J. Appl. Crystallogr. JACGAR 10.1107/S002188988800336X 21, 363–368.CrossRefGoogle Scholar
Kužel, R. Jr. and Klimanek, P. (1989). “X-ray diffraction line broadening due to dislocations in non-cubic crystalline materials. III. Experimental results for plastically deformed zirconium,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889889001585 22, 299–307.CrossRefGoogle Scholar
Kužel, R., Matěj, Z., Cherkaska, V., Pešička, J., Čižek, J., Procházka, I., and Islamgaliev, R. K. (2004). “Structural investigations of submicrocrystalline metals obtained by high-pressure torsion deformation,” J. Alloys Compd. JALCEU 378, 242–247.CrossRefGoogle Scholar
Lahrman, D., Field, R., Darolia, R., and Fraser, H. (1988). “Investigation of techniques for measuring lattice mismatch in a rhenium containing nickel base superalloy,” Acta Metall. AMETAR 10.1016/0001-6160(88)90283-0 36, 1309–1317.CrossRefGoogle Scholar
Leineweber, A. and Mittemeijer, E. J. (2006). “Anisotropic microstrain broadening due to compositional inhomogeneities and its parametrisation,” Z. Kristallogr. ZEKRDZ 2006, 117–122.CrossRefGoogle Scholar
Martinetto, P., Anne, M., Dooryhée, E., Drakopoulos, M., Dubus, M., Salomon, J., Simionovici, A., and Walter, Ph. (2001). “Synchrotron x-ray micro-beam studies of ancient egyptian make-up,” Nucl. Instrum. Methods Phys. Res. B NIMBEU 181, 744–748.CrossRefGoogle Scholar
Mughrabi, H., Ungár, T., Kienle, W., and Wilkens, M. (1986). “Long-range internal stresses and asymmetric x-ray line-broadening in tensile-deformed [001]-oriented copper single crystals,” Philos. Mag. A PMAADG 10.1080/01418618608245293 53, 793–813.CrossRefGoogle Scholar
Nabarro, F. R. N. (1952). “The mathematical theory of dislocations,” Adv. Phys. ADPHAH 10.1080/00018735200101211 1, 269–395.CrossRefGoogle Scholar
Noyan, I. C. and Cohen, J. B. (1987). Residual Stress: Measurement by Diffraction and Interpretation (Springer-Verlag, New York).CrossRefGoogle Scholar
Nye, J. F. (1957). Physical Properties of Crystals: Their Representation by Tensors and Matrices (Clarendon Press, Oxford).Google Scholar
Ribárik, G., Ungár, T., and Gubicza, J. (2001). “MWP-fit: A program for multiple whole-profile fitting of diffraction peak profiles by ab initio theoretical functions,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889801011451 34, 669–676.CrossRefGoogle Scholar
Ribárik, G., Gubicza, J., and Ungár, T. (2004). “Correlation between strength and microstructure of ball-milled Al‐Mg alloys determined by x-ray diffraction,” Mater. Sci. Eng., A MSAPE3 387–389, 343–347.CrossRefGoogle Scholar
Scardi, P. and Leoni, M. (2002). “Whole powder pattern modelling,” Acta Crystallogr., Sect. A: Found. Crystallogr. ACACEQ 10.1107/S0108767301021298 58, 190–200.CrossRefGoogle ScholarPubMed
Scardi, P., Dong, Y., and Leoni, M. (2001). “Line profile analysis in the Rietveld method and whole-powder-pattern fitting,” Mater. Sci. Forum MSFOEP 378–381, 132–141.CrossRefGoogle Scholar
Scardi, P., Leoni, M., and Delhez, R. (2004). “Line broadening analysis using integral breadth methods: A critical review,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889804004583 37, 381–390.CrossRefGoogle Scholar
Schonfeld, B., Reinhard, L., Kostorz, G., and Bohrer, W. (1997). “Diffuse scattering of Ni‐Al,” Acta Mater. ACMAFD 10.1016/S1359-6454(97)00164-X 45, 5187–5194.CrossRefGoogle Scholar
Shen, T. D., Schwarz, R. B., and Thompson, J. D. (2005). “Soft magnetism in mechanically alloyed nanocrystalline materials,” Phys. Rev. B PRBMDO 72, 14431–14438.CrossRefGoogle Scholar
Stokes, A. R. (1948). “A numerical Fourier-analysis method for the correction of widths and shapes of lines on x-ray powder photographs,” Proc. Phys. Soc. 61, 382–391.CrossRefGoogle Scholar
Stokes, A. R. and Wilson, A. J. C. (1942). “A method of calculating the integral breadths of Debye-Scherrer lines,” Proc. Cambridge Philos. Soc. PCPSA4 38, 313–322.CrossRefGoogle Scholar
Stokes, A. R. and Wilson, A. J. C. (1944). “The diffraction of x-rays by distorted crystal aggregates–I,” Proc. Phys. Soc. 56, 174–181.CrossRefGoogle Scholar
Stokes, A. R., Pascoe, K. J., and Lipson, H. (1943). “X-ray evidence of the nature of cold work in metals,” Nature (London) NATUAS 151, 137–137.CrossRefGoogle Scholar
Treacy, M. M. J., Newsam, J. M., and Deem, M. W. (1991). “A general recursion method for calculating diffracted intensities from crystals containing planar faults,” Proc. R. Soc. London, Ser. A PRLAAZ 433, 499–520.Google Scholar
Ungár, T. and Borbély, A. (1996). “The effect of dislocation contrast on x-ray line broadening: A new approach to line profile analysis,” Appl. Phys. Lett. APPLAB 10.1063/1.117951 69, 3173–3175.CrossRefGoogle Scholar
Ungár, T. and Tichy, G. (1999). “The effect of dislocation contrast on x-ray line profiles in untextured polycrystals,” Phys. Status Solidi A PSSABA 171, 425–434.3.0.CO;2-W>CrossRefGoogle Scholar
Ungár, T., Mughrabi, H., Rönnpagel, D., and Wilkens, M. (1984). “X-ray line broadening study of the dislocation cell structure in deformed [001]-oriented copper single crystals,” Acta Metall. AMETAR 10.1016/0001-6160(84)90106-8 32, 333–342.CrossRefGoogle Scholar
Ungár, T., Ott, S., Sanders, P. G., Borbély, A., and Weertman, J. R. (1998). “Dislocations, grain size and planar faults in nanostructured copper determined by high resolution x-ray Diffraction and a new procedure of peak profile analysis,” Acta Mater. ACMAFD 10.1016/S1359-6454(98)00001-9 46, 3693–3699.CrossRefGoogle Scholar
Ungár, T., Dragomir, I., Révész, Á., and Borbély, A. (1999a). “The contrast factors of dislocations in cubic crystals: the dislocation model of strain anisotropy in practice,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889899009334 32, 992–1002.CrossRefGoogle Scholar
Ungár, T., Leoni, M., and Scardi, P. (1999b). “The dislocation model of strain anisotropy in whole powder-pattern fitting: the case of an Li‐Mn cubic spinel,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889898012710 32, 290–295.CrossRefGoogle Scholar
Ungár, T., Gubicza, J., Borbély, A., and Ribárik, G. (2001). “Crystallite size distribution and dislocation structure determined by diffraction profile analysis: principles and practical application to cubic and hexagonal crystals,” J. Appl. Crystallogr. JACGAR 10.1107/S0021889801003715 34, 298–310.CrossRefGoogle Scholar
Ungár, T., Martinetto, P., Ribárik, G., Dooryhée, E., Walter, Ph., and Anne, M. (2002). “Revealing the powdering methods of black makeup in ancient Egypt by fitting microstructure based Fourier coefficients to whole x-ray diffraction profiles of galena,” J. Appl. Phys. JAPIAU 10.1063/1.1429792 91, 2455–2465.CrossRefGoogle Scholar
Ungár, T., Nyilas, K., and Skrotzki, W. (2008). “Dislocation densities in soft and hard oriented grains of compressed NiAl polycrystals,” Int. J. Mater. Sci. 3, (in press).Google Scholar
Ustinov, A. I., Olikhovska, L. O., Budarina, N. M., and Bernard, F. (2004). Diffraction Analysis of the Microstructure of Materials, edited by Mittemeijer, E. J. and Scardi, P. (Springer-Verlag, Berlin), pp. 309–362.Google Scholar
Warren, B. E. (1959). “X-ray studies of deformed metals,” Prog. Met. Phys. PMPHA7 10.1016/0502-8205(59)90015-2 8, 147–202.CrossRefGoogle Scholar
Warren, B. E. and Averbach, B. L. (1950). “The effect of cold work distortion on x-ray patterns,” J. Appl. Phys. JAPIAU 10.1063/1.1699713 21, 595–598.CrossRefGoogle Scholar
Wilkens, M. (1969). “Das Mittlere Spannungsquadrat 〈σ 2〉 Begrenzt Regellos Verteilter Versetzungen in Einem Zylinderfoermigen Koerper,” Acta Metall. AMETAR 10.1016/0001-6160(69)90092-3 17, 1155–1159.CrossRefGoogle Scholar
Wilkens, M. (1970a). “The determination of density and distribution of dislocations in deformed single crystals from broadened x-ray diffraction profiles,” Phys. Status Solidi A PSSABA 10.1002/pssa.19700020224 2, 359–370.CrossRefGoogle Scholar
Wilkens, M. (1970b). Fundamental Aspects of Dislocation Theory (National Bureau of Standards Special Publication, No. 317), edited by Simmons, J. A., de Wit, R., and Bullough, R. (National Bureau of Standards, Washington), Vol. II., pp. 1195–1221.Google Scholar
Wilkens, M. (1987). “X-ray line broadening and mean-square-strains of straight dislocations in elastically anisotropic crystals of cubic symmetry,” Phys. Status Solidi A PSSABA 10.1002/pssa.2211040137 104, K1–K6.CrossRefGoogle Scholar
Williamson, G. K. and Hall, W. H. (1953). “X-ray line broadening from filed aluminium and wolfram,” Acta Metall. AMETAR 10.1016/0001-6160(53)90006-6 1, 22–31.CrossRefGoogle Scholar
Wilson, A. J. C. (1949). “The diffraction of x-rays by distorted-crystal aggregates. II. Diffraction by bent lamellae,” Acta Crystallogr. ACCRA9 2, 220–222.CrossRefGoogle Scholar
Wilson, A. J. C. (1952). “The diffraction of x-rays by distorted-crystal aggregates. IV. Diffraction by a crystal with an axial screw dislocation,” Acta Crystallogr. ACCRA9 10.1107/S0365110X52000988 5, 318–322.CrossRefGoogle Scholar
Wollan, E. O. and Shull, C. G. (1948). “The diffraction of neutrons by crystalline powders,” Phys. Rev. PHRVAO 10.1103/PhysRev.73.830 73, 830–841.CrossRefGoogle Scholar