Skip to main content
SpringerLink
Log in

Gauge invariance and relativistic effects in X-ray absorption and scattering by solids

  • Regular Article
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

There is an incompatibility between gauge invariance and the semi-classical time-dependent perturbation theory commonly used to calculate light absorption and scattering cross-sections. There is an additional incompatibility between perturbation theory and the description of the electron dynamics by a semi-relativistic Hamiltonian. In this paper, the gauge-dependence problem of exact perturbation theory is described, the proposed solutions are reviewed and it is concluded that none of them seems fully satisfactory. The problem is finally solved by using the fully relativistic absorption and scattering cross-sections given by quantum electrodynamics. Then, a new general Foldy-Wouthuysen transformation is presented. It is applied to the many-body case to obtain correct semi-relativistic transition operators. This transformation considerably simplifies the calculation of relativistic corrections. In the process, a new light-matter interaction term emerges, called the spin-position interaction, that contributes significantly to the magnetic X-ray circular dichroism of transition metals. We compare our result with the ones obtained by using several semi-relativistic time-dependent Hamiltonians. In the case of absorption, the final formula agrees with the result obtained from one of them. However, the correct scattering cross-section is not given by any of the semi-relativistic Hamiltonians.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. C. Brouder, J. Phys.: Condens. Matter 2, 701 (1990)

    ADS  Google Scholar 

  2. Y. Joly, S. DiMatteo, O. Bunau, Eur. Phys. J. Special Topics 208, 21 (2012)

    Article  ADS  Google Scholar 

  3. D.H. Kobe, A.L. Smirl, Am. J. Phys. 46, 624 (1978)

    Article  ADS  Google Scholar 

  4. K.H. Yang, J. Phys. A: Math. Gen. 15, 437 (1982)

    Article  ADS  Google Scholar 

  5. N. Bouldi, N.J. Vollmers, C.G. Delpy-Laplanche, Y. Joly, A. Juhin, Ph. Sainctavit, Ch. Brouder, M. Calandra, L. Paulatto, F. Mauri et al., Phys. Rev. B 96, 085123 (2017)

    Article  ADS  Google Scholar 

  6. J.W.E. Lamb, Phys. Rev. 85, 259 (1952)

    Article  ADS  Google Scholar 

  7. I.P. Grant, J. Phys. B: At. Mol. Phys. 7, 1458 (1974)

    Article  ADS  Google Scholar 

  8. I.P. Grant, A.F. Starace, J. Phys. B: At. Mol. Phys. 8, 1999 (1975)

    Article  ADS  Google Scholar 

  9. K.H. Yang, Ann. Phys. 101, 6 (1976)

    Google Scholar 

  10. R. Zeyher, H. Bilz, M. Cardona, Solid State Commun. 19, 57 (1976)

    Article  ADS  Google Scholar 

  11. J.J. Forney, A. Quattropani, F. Bassani, Nuovo Cimento 37B, 78 (1977)

    Article  ADS  Google Scholar 

  12. Y. Aharonov, C.K. Au, Phys. Rev. A 20, 1553 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  13. S.T. Epstein, Chem. Phys. Lett. 65, 417 (1979)

    Article  ADS  Google Scholar 

  14. G. Grynberg, E. Giacobino, J. Phys. B: At. Mol. Phys. 12, L93 (1979)

    Article  ADS  Google Scholar 

  15. K. Haller, R.B. Sohn, Phys. Rev. A 20, 1541 (1979)

    Article  ADS  Google Scholar 

  16. S. Olariu, I. Popescu, C.B. Collins, Phys. Rev. D 20, 3095 (1979)

    Article  ADS  Google Scholar 

  17. D.H. Kobe, K.H. Yang, J. Phys. A: Math. Gen. 13, 3171 (1980)

    Article  ADS  Google Scholar 

  18. D.H. Kobe, C.T. Wen, Phys. Lett. 80A, 121 (1980)

    Article  ADS  Google Scholar 

  19. C. Leubner, P. Zoller, J. Phys. B: At. Mol. Phys. 13, 3613 (1980)

    Article  ADS  Google Scholar 

  20. Y. Aharonov, C.K. Au, Phys. Lett. 86A, 269 (1981)

    Article  ADS  Google Scholar 

  21. M.I. Shirokov, Sov. Phys. JETP 54, 645 (1981)

    Google Scholar 

  22. D.H. Kobe, Int. J. Quant. Chem. 21, 685 (1982)

    MathSciNet  Google Scholar 

  23. D.H. Kobe, E.C.T. Wen, K.H. Yang, Phys. Rev. D 26, 1927 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  24. Y. Aharonov, C.K. Au, Phys. Lett. 88A, 491 (1982)

    Google Scholar 

  25. K.H. Yang, Phys. Lett. 92A, 71 (1982)

    Article  ADS  Google Scholar 

  26. K.H. Yang, J. Phys. A: Math. Gen. 15, 1201 (1982)

    Article  ADS  Google Scholar 

  27. T.E. Feuchtwang, E. Kazes, H. Grotch, P.H. Cutler, Phys. Lett. 93A, 4 (1982)

    Article  ADS  Google Scholar 

  28. E. Kazes, T.E. Feuchtwang, P.H. Cutler, H. Grotch, Ann. Phys. 142, 80 (1982)

    Article  ADS  Google Scholar 

  29. Y. Aharonov, C.K. Au, Phys. Lett. 95A, 412 (1983)

    Article  ADS  Google Scholar 

  30. D. Lee, A.C. Albrecht, J. Chem. Phys. 78, 5373 (1983)

    Article  ADS  Google Scholar 

  31. K.H. Yang, J. Phys. A: Math. Gen. 16, 919 (1983)

    Article  ADS  Google Scholar 

  32. C.K. Au, J. Phys. B: At. Mol. Phys. 17, L59 (1984)

    Article  ADS  Google Scholar 

  33. T.E. Feuchtwang, E. Kazes, P.H. Cutler, J. Phys. A: Math. Gen. 17, 1157 (1984)

    Article  ADS  Google Scholar 

  34. T.E. Feuchtwang, E. Kazes, P.H. Cutler, H. Grotch, J. Phys. A: Math. Gen. 17, 151 (1984)

    Article  ADS  Google Scholar 

  35. D.H. Kobe, K.H. Yang, Phys. Rev. A 32, 952 (1985)

    Article  ADS  Google Scholar 

  36. D. Adu-Gyamfi, J. Phys. A: Math. Gen. 19, 3443 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  37. K.H. Yang, D.H. Kobe, Ann. Phys. 168, 104 (1986)

    Article  ADS  Google Scholar 

  38. D.H. Kobe, S.M. Golsham, J. Phys. A: Math. Gen. 20, 2813 (1987)

    Article  ADS  Google Scholar 

  39. J.W.E.Lamb, R.R. Schlicher, M.O. Scully, Phys. Rev. A 36, 2763 (1987)

    Article  ADS  Google Scholar 

  40. G.P. Arrighini, C. Guidotti, N. Durante, Gazz. Chim. Ital. 118, 703 (1988)

    Google Scholar 

  41. N. Durante, G.P. Arrighini, C. Guidotti, Z. Phys. D 8, 63 (1988)

    Article  ADS  Google Scholar 

  42. K.H. Yang, Ann. Phys. 186, 209 (1988)

    Article  ADS  Google Scholar 

  43. K. Haller, E. Lim-Lombridas, Found. Phys. 24, 217 (1994)

    Article  ADS  Google Scholar 

  44. R.G. Woolley, Mol. Phys. 94, 409 (1998)

    Article  ADS  Google Scholar 

  45. R.G. Woolley, Int. J. Quant. Chem. 74, 531 (1999)

    Article  Google Scholar 

  46. R.G. Woolley, Proc. R. Soc. Lond. A 456, 1803 (2000)

    Article  ADS  Google Scholar 

  47. I.M. Savukov, W.R. Johnson, Phys. Rev. A 62, 052506 (2000)

    Article  ADS  Google Scholar 

  48. A.M. Stewart, J. Phys. A: Math. Gen. 33, 9165 (2000)

    Article  ADS  Google Scholar 

  49. S.W. Qian, Z.Y. Gu, Commun. Theor. Phys. 38, 267 (2002)

    Article  Google Scholar 

  50. R.G. Woolley, On gauge invariance and molecular electrodynamics, in New trends in quantum systems in chemistry and physics. Volume 2. Advanced problems and complex systems. Paris, France, 1999. Progress in theoretical chemistry and physics, edited by J. Maruani, C. Minot, R. McWeeny, Y. Smeyers, S. Wilson (Kluwer Academic Publishers, Dordrecht, 2002), Vol. 7

  51. K. Rzazewski, R.W. Boyd, J. Mod. Opt. 51, 1137 (2004)

    Article  ADS  Google Scholar 

  52. S.W. Qian, J.S. Wang, Commun. Theor. Phys. 49, 308 (2008)

    Article  ADS  Google Scholar 

  53. A. Stokes, J. Phys. B: At. Mol. Opt. Phys. 46, 145505 (2013)

    Article  ADS  Google Scholar 

  54. A.D. Bandrauk, F. Fillion-Gourdeau, E. Lorin, J. Phys. B: At. Mol. Opt. Phys. 46, 153001 (2013)

    Article  ADS  Google Scholar 

  55. A. Mandal, K.L.C. Hunt, J. Chem. Phys.144, 044109 (2016)

    Article  ADS  Google Scholar 

  56. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Photons and atoms. Introduction to quantum electrodynamics (Wiley-VCH, New York, 1997)

  57. M. Blagojević, F.W. Hehl, Gauge theories of gravitation (Imperial College Press, London, 2013)

  58. A. Bohm, Quantum mechanics: foundations and applications, 3rd edn. (Springer, New York, 1993)

  59. T.E. Feuchtwang, E. Kazes, J. Phys. A: Math. Gen. 18, 2859 (1985)

    Article  ADS  Google Scholar 

  60. T.E. Feuchtwang, E. Kazes, P.H. Cutler, Found. Phys. 16, 1263 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  61. T. Goldman, Phys. Rev. D 15, 1063 (1977)

    Article  ADS  Google Scholar 

  62. E. Kazes, T.E. Feuchtwang, H. Grotch, P.H. Cutler, Phys. Rev. D 27, 1388 (1983)

    Article  ADS  Google Scholar 

  63. S. Weinberg, The quantum theory of fields. I. Foundations (Cambridge University Press, Cambridge, 1995)

  64. N. Nakanishi, I. Ojima, Covariant operator formalism of gauge theories and quantum gravity (World Scientific, Singapore, 1990)

  65. M.E. Peskin, D.V. Schroeder, An introduction to quantum field theory (Addison-Wesley, Reading, 1995)

  66. O. Steinmann, Perturbative quantum electrodynamics and axiomatic field theory (Springer, Berlin, 2000)

  67. F. Strocchi, An introduction to non-perturbative foundations of quantum field theory (Oxford Science Publications, Oxford, 2013)

  68. I. Bialynicki-Birula, Phys. Rev. D 2, 2877 (1970)

    Article  ADS  Google Scholar 

  69. R.E. Kallosh, I.V. Tyutin, Sov. J. Nucl. Phys. 17, 98 (1973)

    Google Scholar 

  70. E.B. Manoukian, Int. J. Theor. Phys. 27, 787 (1988)

    Article  MathSciNet  Google Scholar 

  71. S. Weinberg, The quantum theory of fields. II. Modern applications (Cambridge University Press, Cambridge, 1995)

  72. S. Hollands, Rev. Math. Phys. 20, 1033 (2008)

    Article  MathSciNet  Google Scholar 

  73. I.A. Batalin, P.M. Lavrov, I.V. Tyutin, Int. J. Mod. Phys. A29, 1450166 (2014)

    Article  ADS  Google Scholar 

  74. P.Y. Moshin, A.A. Reshetnyak, Int. J. Mod. Phys. A 30, 1550021 (2015)

    Article  Google Scholar 

  75. H. Matsuda, R. Kubo, Prog. Theor. Phys. 63, 275 (1980)

    Article  ADS  Google Scholar 

  76. B.L. Voronov, P.M. Lavrov, I.V. Tyutin, Sov. J. Nucl. Phys. 36, 292 (1982)

    Google Scholar 

  77. F. Lenz, H.W.L. Naus, K. Ohta, M. Thies, Ann. Phys. 233, 17 (1994)

    Article  ADS  Google Scholar 

  78. T. Kashiwa, N. Tanimura, Fortschr. Phys. 45, 381 (1997)

    Article  MathSciNet  Google Scholar 

  79. D. Grigore, Ann. Phys. (Leipzig) 10, 439 (2001)

    Article  ADS  Google Scholar 

  80. A.K. Das, R.R. Francisco, J. Frenkel, Phys. Rev. D 88, 085012 (2013)

    Article  ADS  Google Scholar 

  81. K. Haller, L.F. Landovitz, Phys. Rev. 171, 1749 (1968)

    Article  ADS  Google Scholar 

  82. K. Haller, L.F. Landovitz, Phys. Rev. D 2, 1498 (1970)

    Article  ADS  Google Scholar 

  83. K. Haller, Acta Phys. Aust. 42, 163 (1975)

    Google Scholar 

  84. K. Haller, L.F. Landovitz, Phys. Rev. Lett. 22, 245 (1969)

    Article  ADS  Google Scholar 

  85. K. Ohtaka, Y. Tanabe, Rev. Mod. Phys. 62, 929 (1990)

    Article  ADS  Google Scholar 

  86. E. Klevak, J.J. Kas, J.J. Rehr, Phys. Rev. B 89, 085123 (2014)

    Article  ADS  Google Scholar 

  87. I.P. Grant, Relativistic quantum theory of atoms and molecules (Springer, New York, 2007)

  88. W. Greiner, J. Reinhardt, Field quantization (Springer, Berlin, 1996)

  89. I. Bialynicki-Birula, The Hamiltonian of quantum electrodynamics, in Quantum electrodynamics and quantum optics. NATO ASI series B, edited by A.O. Barut (Plenum Press, New York, 1984), Vol. 110, pp. 41–61

  90. M.J. Esteban, M. Lewin, E. Séré, Bull. Am. Math. Soc. 45, 535 (2008)

    Article  Google Scholar 

  91. W. Kutzelnigg, Chem. Phys. 395, 16 (2012)

    Article  ADS  Google Scholar 

  92. C. Brouder, G. Panati, G. Stoltz, Ann. Henri Poincaré 11, 1285 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  93. W. Heitler, The quantum theory of radiation, 3rd edn. (Dover, New York, 1984)

  94. C. Brouder, G. Panati, G. Stoltz, Phys. Rev. Lett. 103, 230401 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  95. B.A. Veklenko, Sov. Phys. J. 30, 555 (1987)

    Article  Google Scholar 

  96. S. Hassing, E.N. Svendsen, J. Raman Spectrosc. 35, 87 (2004)

    Article  ADS  Google Scholar 

  97. P.W. Milonni, R. Loudon, P.R. Berman, S.M. Barnett, Phys. Rev. A 77, 043835 (2008)

    Article  ADS  Google Scholar 

  98. S. Mukamel, Phys. Rev. A 76, 021803 (2007)

    Article  ADS  Google Scholar 

  99. S.T. Epstein, Phys. Rev. 98, 196 (1955)

    Article  ADS  MathSciNet  Google Scholar 

  100. B. Hatfield, Quantum field theory of point particles and strings (Addison-Wesley, Reading, 1992)

  101. P. Strange, Relativistic quantum mechanics (Cambridge University Press, Cambridge, 1998)

  102. W.R. Johnson, D.R. Plante, J. Sapirstein, Relativistic calculations of transition amplitudes in the helium isoelectronic sequence, in Advances in atomic, molecular, and optical physics (Academic Press, New York, 1995), Vol. 35, pp. 255–329

  103. D. Baye, Phys. Rev. C 86, 034306 (2012)

    Article  ADS  Google Scholar 

  104. H. Ebert, P. Strange, L. Gyorffy, J. Appl. Phys. 63, 3055 (1988)

    Article  ADS  Google Scholar 

  105. O. Šipr, J. Minár, H. Wende, H. Ebert, Phys. Rev. B 84, 115102 (2011)

    Article  ADS  Google Scholar 

  106. L.L. Foldy, S.A. Wouthuysen, Phys. Rev. 78, 29 (1950)

    Article  ADS  Google Scholar 

  107. W. Greiner, Relativistic quantum mechanics, 3rd edn. (Springer, Berlin, 2000)

  108. A.J. Silenko, Phys. Rev. A 93, 022108 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  109. A.L. Fetter, J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, Boston, 1971)

  110. E.K.U. Gross, E. Runge, O. Heinonen, Many-particle theory (Adam Hilger, Bristol, 1991)

  111. C. Cohen-Tannoudji, B. Diu, F. Laloë, Mécanique Quantique (Hermann, Paris, 1973)

  112. J.D. Walecka, Advanced modern physics. Theoretical foundations (World Scientific, Singapore, 2010)

  113. C. Brouder, M. Alouani, K.H. Bennemann, Phys. Rev. B 54, 7334 (1996)

    Article  ADS  Google Scholar 

  114. C. Gougoussis, Excitations électroniques et magnétisme des matériaux: calcul ab initio de l’absorption X et du dichroïsme circulaire magnétique au seuil K, Ph.D. thesis, Université Pierre et Marie Curie, Paris 6, 2009

  115. J. Als-Nielsen, D. McMorrow, Elements of modern X-ray physics (Wiley, New York, 2000)

  116. M. Blume, J. Appl. Phys. 51, 3615 (1985)

    Article  ADS  Google Scholar 

  117. M.M. Nieto, Phys. Rev. Lett. 38, 1042 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  118. M. Blume, SSRL Report 83/02, 1983, p. 126

  119. M. Blume, Magnetic effects in anomalous dispersion, in Resonant anomalous X-ray scattering, edited by G. Materlik, C.J. Sparks, K. Fischer (North-Holland, Amsterdam, 1994), pp. 495–512

  120. L.L. Foldy, Phys. Rev. 87, 688 (1952)

    Article  ADS  MathSciNet  Google Scholar 

  121. C. Itzykson, J.B. Zuber, Quantum field theory (McGraw-Hill, New York, 1980)

  122. J.D. Bjorken, S.D. Drell, Relativistic quantum fields (McGraw-Hill, New York, 1965)

  123. W.E. Caswell, G.P. Lepage, Phys. Lett. 167B, 437 (1986)

    Article  ADS  Google Scholar 

  124. G. Paz, Mod. Phys. Lett. A 30, 1550128 (2015)

    Article  ADS  Google Scholar 

  125. M. Takahashi, N. Hiraoka, Phys. Rev. B 92, 094441 (2015)

    Article  ADS  Google Scholar 

  126. S.D. Matteo, Private communication, 2016

  127. R. Loudon, The quantum theory of light, 3rd edn. (Clarendon Press, Oxford, 2000)

  128. H.A. Bethe, E.E. Salpeter, Quantum mechanics of one- and two-electron atoms (Plenum, New York, 1977)

  129. L.M. Brown, Stud. Hist. Philos. Mod. Phys. 33, 211 (2002)

    Article  Google Scholar 

  130. J.J. Sakurai, Advanced quantum mechanics (Addison-Wesley, Reading, 1967)

  131. H. Baum, Spin-Strukturen und Dirac-Operatoren über pseudoriemannschenMannigfaltigkeiten (Teubner, Leipzig, 1981)

  132. C. Brouder, N. Bizi, F. Besnard, arXiv:1504.03890 (2015)

  133. E. Eriksen, Phys. Rev. 111, 1011 (1958)

    Article  ADS  MathSciNet  Google Scholar 

  134. E. Eriksen, M. Kolsrud, Nuovo Cimento (Suppl.) 18, 1 (1960)

    Article  ADS  Google Scholar 

  135. E. Prodan, H. Schulz-Baldes, Bulk and boundary invariants for complex topological insulators (Springer, Berlin, 2016)

  136. M. Pourahmadi, Foundations of time series analysis and prediction theory (Wiley, New York, 2001)

  137. S. Lang, Linear algebra. Undergraduate graduate texts in mathematics, 3rd edn. (Springer-Verlag, Berlin, 1987)

  138. L.J. Wallen, Mich. J. Math. 16, 153 (1969)

    Article  Google Scholar 

  139. A.A. Joye, Dynamical localization of random quantum walks on the lattice, in XVIIth International Congress on Mathematical Physics, edited by A. Jensen (World Scientific, Singapore, 2014), pp. 486–494

  140. F.A. Berezin, M.A. Shubin, The Schrödinger equation (Kluwer, Dordrecht, 1991)

  141. Z.V. Chraplyvy, Phys. Rev. 91, 388 (1953)

    Article  ADS  MathSciNet  Google Scholar 

  142. Z.V. Chraplyvy, Phys. Rev. 92, 1310 (1953)

    Article  ADS  MathSciNet  Google Scholar 

  143. K. Pachucki, Phys. Rev. A 71, 012503 (2005)

    Article  ADS  Google Scholar 

  144. M. Moshinsky, A. Nikitin, Rev. Mex. Fis. 50 (Suppl. 2), 66 (2004)

    Google Scholar 

  145. W. Weiglhofer, Am. J. Phys. 57, 455 (1989)

    Article  ADS  Google Scholar 

  146. J.D. Morrison, R. Moss, Mol. Phys. 41, 491 (1980)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sorbonne Universités, UPMC Univ Paris 06, UMR CNRS 7590, Muséum National d’Histoire Naturelle, IRD UMR 206, Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, 4 place Jussieu, 75005, Paris, France

    Nadejda Bouldi & Christian Brouder

  2. Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, 91192, Gif-sur-Yvette Cedex, France

    Nadejda Bouldi

Authors
  1. Nadejda Bouldi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Brouder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Brouder.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bouldi, N., Brouder, C. Gauge invariance and relativistic effects in X-ray absorption and scattering by solids. Eur. Phys. J. B 90, 246 (2017). https://doi.org/10.1140/epjb/e2017-80266-5

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2017-80266-5

Keywords

Search

Navigation