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Geometric theory of defects

A description of dislocation and disclination defects in terms of the Riemann-Cartan geometry is given, with the curvature and torsion tensors interpreted as the surface densities of the Frank and Burgers vectors, respectively. A new free-energy expression describing the static distribution of defects is presented and equations of nonlinear elasticity theory are used to specify the coordinate system. Application of the Lorentz gauge leads to equations for the principal chiral SO(3) field. In the defect-free case, the geometric model reduces to elasticity theory for the displacement vector field and to a principal chiral SO(3)-field model for the spin structure. As illustrated by the example of a wedge dislocation, elasticity theory reproduces only the linear approximation of the geometric theory of defects. It is shown that the equations of asymmetric elasticity theory for Cosserat media can also be naturally incorporated into the geometric theory as gauge conditions. As an application of the theory, phonon scattering on a wedge dislocation is considered. The energy spectrum of impurities in the field of a wedge dislocation is also discussed.

Fulltext is available at IOP
PACS: 02.40.−k, 46.05.+b, 61.72.Lk, 63.20.Mt (all)
DOI: 10.1070/PU2005v048n07ABEH002027
URL: https://ufn.ru/en/articles/2005/7/b/
Citation: Katanaev M O "Geometric theory of defects" Phys. Usp. 48 675–701 (2005)
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:    « » 175 705–733 (2005); DOI: 10.3367/UFNr.0175.200507b.0705

References (100) ↓ Cited by (72)

  1. Kondo K in Proc. of the 2nd Japan National Congress for Applied Mechanics, Tokyo, Japan, 1952 (Tokyo, 1952) p. 41
  2. Nye J F Acta Metal. 1 153 (1953)
  3. Bilby B A, Bullough R, Smith E Proc. R. Soc. London A 231 263 (1955)
  4. Kröner E Kontinuumstheorie Der Versetzungen Und Eigenspannungen (Berlin: Springer, 1958)
  5. Sedov L I, Berditchevski V L In Mechanics Of Generalized Continua: Proc. Of The IUTAM-Symp. On The Generalized Cosserat Continuum Theory Of Dislocations With Applications, Germany, 1967 (Ed. E Kröner) (Berlin: Springer-Verlag, 1968) p. 214
  6. Kléman M In Dislocations In Solids Vol. 5 (Ed. F R N Nabarro) (Amsterdam: North-Holland Publ. Co., 1980) p. 243
  7. Kröner E In Physics Of Defects, Less Houches, Session XXXV, 1980 (Eds R Balian, M Kléman, J-P Poirier) (Amsterdam: North-Holland Publ. Co., 1981) p. 282
  8. Dzyaloshinskii I E, Volovick G E Ann. Physics 125 67 (1980)
  9. Kadić A, Edelen D G B A Gauge Theory Of Dislocations And Disclinations (Berlin: Springer-Verlag, 1983); Kadich A, Edelen D Kalibrovochnaya Teoriya Dislokatsii i Disklinatsii (M.: Mir, 1987)
  10. Kunin I A, Kunin B I Lecture Notes Phys. 249 246 (1986)
  11. Kleinert H Gauge Fields In Condensed Matter Vol. 2 (Singapore: World Scientific, 1990)
  12. Hehl F W et al. Phys. Rep. 258 1 (1995)
  13. Cartan E C.R. Acad. Sci. 174 593 (1922); Kartan E v Sb. Al’bert Einshtein i Teoriya Gravitatsii (M.: Mir, 1979) p. 535
  14. Malyshev C Ann. Physics 286 249 (2000)
  15. Lazar M Ann. Physik 9 461 (2000)
  16. Lazar M J. Phys. A 35 1983 (2002)
  17. Lazar M J. Phys. A 36 1415 (2003)
  18. Frank F C Discuss. Faraday Soc. 25 19 (1958)
  19. Dzyaloshinskii I E, Volovik G E J. Physique 39 693 (1978)
  20. Hertz J A Phys. Rev. B 18 4875 (1978)
  21. Rivier N, Duffy D M J. Physique 43 293 (1982)
  22. Katanaev M O, Volovich I V Ann. Physics 216 1 (1992)
  23. Katanaev M O Teor. Mat. Fiz. 135 338 (2003)
  24. Katanaev M O Teor. Mat. Fiz. 138 193 (2004)
  25. Furtado C, Moraes F Phys. Lett. A 188 394 (1994)
  26. Furtado C et al. Phys. Lett. A 195 90 (1994)
  27. Moraes F Mod. Phys. Lett. A 10 2335 (1995)
  28. Azevedo S, Moraes F Phys. Lett. A 246 374 (1998)
  29. Balachandran A P et al. Int. J. Mod. Phys. A 13 841 (1998)
  30. Furtado C, Moraes F Europhys. Lett. 45 279 (1999)
  31. Bausch R, Schmitz R, Turski L A Phys. Rev. Lett. 80 2257 (1998)
  32. Bausch R, Schmitz R, Turski L A Ann. Physik 8 181 (1999)
  33. Furtado C, Moraes F J. Phys. A 33 5513 (2000)
  34. Azevedo S, Moraes F J. Phys. Condens. Matter 12 7421 (2000)
  35. Azevedo S, Pereira J Phys. Lett. A 275 463 (2000)
  36. Furtado C, Bezerra V B, Moraes F Europhys. Lett. 52 1 (2000)
  37. de Lima Ribeiro C A, Furtado C, Moraes F Phys. Lett. A 288 329 (2001)
  38. Vieira S R, Azevedo S Phys. Lett. A 288 29 (2001)
  39. Azevedo S J. Phys. A 34 6081 (2001)
  40. Furtado C, Bezerra V B, Moraes F Phys. Lett. A 289 160 (2001)
  41. Marques G de A et al. J. Phys. A 34 5945 (2001)
  42. Furtado C, de Lima Ribeiro C A, Azevedo S Phys. Lett. A 296 171 (2002)
  43. Azevedo S Phys. Lett. A 293 283 (2002)
  44. Azevedo S Int. J. Quantum Chem. 90 1596 (2002)
  45. Azevedo S Mod. Phys. Lett. A 17 1263 (2002)
  46. Azevedo S Phys. Lett. A 307 65 (2003)
  47. Barbosa A de S, Bezerra de Mello E R, Bezerra V B Int. J. Mod. Phys. A 14 3565 (1999)
  48. Azevedo S, Furtado C, Moraes F Phys. Status Solidi B 207 387 (1998)
  49. Moraes F Phys. Lett. A 204 399 (1995)
  50. Bezerra de Mello E R, Bezerra V B, Grats Yu V Class. Quantum Grav. 15 1915 (1998)
  51. Azevedo S Phys. Lett. A 288 33 (2001)
  52. Furtado C, Bezerra V B, Moraes F Mod. Phys. Lett. A 15 253 (2000)
  53. Tanaka I Phys. Lett. A 277 262 (2000)
  54. De Lorenci V A, Moreira E S (Jr) Phys. Rev. D 65 085013 (2002)
  55. Azevedo S, Moraes F Phys. Lett. A 267 208 (2000)
  56. Miranda J A, Moraes F J. Phys. A 36 863 (2003)
  57. Moraes F Phys. Lett. A 214 189 (1996)
  58. de Padua A, Parisio-Filho F, Moraes F Phys. Lett. A 238 153 (1998)
  59. Katanaev M O, Volovich I V Ann. Physics 271 203 (1999)
  60. Miri M F, Rivier N J. Phys. A 35 1727 (2002)
  61. Landau L D, Lifshits E M Teoriya Uprugosti (M.: Nauka, 1987)
  62. Kosevich A M Fizicheskaya Mekhanika Real’nykh Kristallov (Kiev: Naukova dumka, 1981)
  63. Verçin A Int. J. Theor. Phys. 29 (1) 7 (1990)
  64. Teoriya Solitonov: Metod Obratnoi Zadachi (Pod red. S P Novikova) (M.: Nauka, 1980)
  65. Rajaraman R Solitons And Instantons: An Introduction To Solitons And Instantons In Quantum Field Theory (Amsterdam: North-Holland Publ. Co., 1982); Radzharaman R Solitony i Instantony v Kvantovoi Teorii Polya (M.: Mir, 1985)
  66. Takhtadzhyan L A, Faddeev L D Gamil’tonov Podkhod v Teorii Solitonov (M.: Nauka, 1986)
  67. Zakrzewski W J Low-Dimensional Sigma Models (Bristol: A. Hilger, 1989)
  68. Rybakov Yu P, Sanyuk V I Mnogomernye Solitony. Vvedenie v Teoriyu i Prilozheniya (M.: Izd-vo RUDN, 2001)
  69. Skyrme T H R Proc. R. Soc. London A 260 127 (1961)
  70. Landau L D, Lifshits E M Elektrodinamika Sploshnykh Sred (M.: Nauka, 1982)
  71. Faddeev L D "Nelokal’nye, nelineinye i nerenormiruemye teorii polya" Priprint D2-9788 (Dubna: OIYaI, 1977) p. 207
  72. Cosserat E, Cosserat F Théorie Des Corps Déformables (Paris: A. Hermann, 1909)
  73. Nowacki W Theory Of Asymmetric Elasticity (Oxford: Pergamon Press, 1986)
  74. Sandru N Int. J. Eng. Sci. 4 81 (1966)
  75. Staruszkiewicz A Acta Phys. Polon. 24 735 (1963)
  76. Clemént G Nucl. Phys. B 114 437 (1976)
  77. Deser S, Jackiw R, ’t Hooft G Ann. Physics 152 220 (1984)
  78. Holz A Class. Quantum Grav. 5 1259 (1988)
  79. Kohler C Class. Quantum Grav. 11 (1995)
  80. Kohler C Class. Quantum Grav. 12 2977 (1995)
  81. Landau L D, Lifshits E M Teoriya Polya (M.: Nauka, 1988)
  82. Guillemin V, Sternberg S Geometric Asymptotics (Providence, RI: American Mathematical Society, 1977); Giiemin V, Sternberg S Geometricheskie Asimptotiki (M.: Mir, 1981)
  83. Jahnke E, Emde F, Lösch F Tafeln Höherer Funktionen (Stuttgart: Verlagsgesellschaft, 1960); Yanke E, Emde F, Lesh F Spetsial’nye Funktsii (M.: Nauka, 1977)
  84. Aharonov Y, Bohm D Phys. Rev. 115 485 (1959)
  85. Feinberg E L Usp. Fiz. Nauk 78 (1) 53 (1962); Feinberg E L Sov. Phys. Usp. 5 753 (1963)
  86. Skarzhinskii V D Trudy FIAN 167 139 (1986)
  87. Katanaev M O J. Math. Phys. 38 946 (1997)
  88. Volovich I V, Katanaev M O Pis’ma ZhETF 43 212 (1986)
  89. Katanayev M O, Volovich I V Phys. Lett. B 175 413 (1986)
  90. Katanaev M O, Volovich I V Ann. Physics 197 1 (1990)
  91. Katanaev M O Dokl. Akad. Nauk SSSR 309 591 (1989)
  92. Katanaev M O J. Math. Phys. 31 882 (1990)
  93. Katanaev M O J. Math. Phys. 32 2483 (1991)
  94. Katanaev M O J. Math. Phys. 34 700 (1993)
  95. Vilenkin A, Shellard E P S Cosmic Strings And Other Topological Defects (Cambridge: Cambridge Univ. Press, 1994)
  96. Hindmarsh M B, Kibble T W B Rep. Prog. Phys. 58 477 (1995)
  97. Dubrovin B A, Novikov S P, Fomenko A T Sovremennaya Geometriya. Metody i Prilozheniya 4-e izd. (M.: Editorial URSS, 1998)
  98. Weyl H Berl. Ber. 465 (1918); Veil’ G v Sb. Al’bert Einshtein i Teoriya Gravitatsii (M.: Mir, 1979) p. 513
  99. Wolf J A Spaces Of Constant Curvature (Boston: Publish or Perish, 1974); Vol’f Dzh Prostranstva Postoyannoi Krivizny (M.: Nauka, 1982)
  100. Chern S S, Chen W H, Lam K S Lectures On Differential Geometry (Singapore: World Scientific, 1999)

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