- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Influence of symmetry-resolved phonon dynamics on the temperature-dependent electric field gradients in solids

S. Q. Jin1,2,*,†, I. C. J. Yap3,4,*,‡, T. T. Dang3, D. Kviatkovskyi5,6, C. Noll7, R. Beck7, U. Köster8, H. C. Hofsäss4, D. C. Lupascu3 et al. (SSP ISOLDE-CERN Collaboration)

D. C. Lupascu3, A. Krawczuk2, and J. H. Schell3,5 (SSP ISOLDE-CERN Collaboration)

  • *These authors contributed equally to this work.
  • †Contact author: siqi.jin@mpinat.mpg.de
  • ‡Contact author: ian.chang.jie.yap@uni-due.de
PDF

Phys. Rev. B 114, 115105 – Published 10 August, 2026

DOI: https://doi.org/10.1103/xlpz-9gsw

Abstract

The temperature-dependent electric field gradient (EFG) tensor at a probe nucleus reflects the combined effects of local symmetry, bonding, and lattice dynamics. However, determining the primary atomic distortions responsible for this response can be difficult. Here, we combine electron-gamma (e-γ) time-differential perturbed angular correlation (TDPAC) measurements on Ta181 probes produced by the Hf181→Ta181 decay at substitutional Ti sites in rutile TiO2 with symmetry-resolved local frozen-phonon density functional theory (LFP-DFT) calculations. Atomic displacements of the TiO6 octahedron are decomposed into site-symmetry-adapted local modes, and their contributions to the EFG are thermally weighted via Boltzmann averaging. We find that, in pristine rutile TiO2, the temperature evolution of the EFG shape is primarily governed by two orthogonal B1g equatorial-oxygen shear distortions. For Ta substituting the Ti site, the same shear sector remains active but preserves a finite in-plane anisotropy, resulting in a comparatively weak temperature dependence of the asymmetry parameter η. Measurements between 30 and 470 K, together with earlier high-temperature gamma-gamma (γ−γ) data, show weak low-temperature curvature and high-temperature quasilinearity of the principal EFG component V33(T), while η(T) remains comparatively robust. The calculations reproduce these qualitative trends and provide a microscopic, symmetry-based interpretation of the thermal evolution of hyperfine tensors in solids.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (74)

  1. G. Schatz and A. Weidinger, Nuclear Condensed Matter Physics: Nuclear Methods and Applications (Wiley, Chichester, UK, 1996), Chaps. 2–5.
  2. R. M. Steffen and K. Alder, in The Electromagnetic Interaction in Nuclear Spectroscopy, edited by W. D. Hamilton (North-Holland, Oxford, 1975), pp. 583–643.
  3. T. Butz, Analytic perturbation functions for static interactions in perturbed angular correlation of γ rays, Hyperfine Interact. 52, 189 (1989).
  4. J. Schell, P. Schaaf, and D. C. Lupascu, Perturbed angular correlations at ISOLDE: A 40 years young technique, AIP Adv. 7, 105017 (2017).
  5. D. Torumba, K. Parlinski, M. Rots, and S. Cottenier, Temperature dependence of the electric-field gradient in hcp-Cd from first principles, Phys. Rev. B 74, 144304 (2006).
  6. K. Nishiyama, F. Dimmling, Th. Kornrumpf, and D. Riegel, Theory of the temperature dependence of the electric field gradient in noncubic metals, Phys. Rev. Lett. 37, 357 (1976).
  7. A. V. Nikolaev, N. M. Chtchelkatchev, A. V. Bibikov, D. A. Salamatin, and A. V. Tsvyashchenko, Ab initio based description of the unusual increase of the electric field gradient with temperature at Ti sites in rutile TiO2, Phys. Rev. B 102, 174305 (2020).
  8. K. Yagi, T. Taketsugu, K. Hirao, and M. S. Gordon, Direct vibrational self-consistent field method: Application to H2O and H2CO, J. Chem. Phys. 113, 1005 (2000).
  9. P. T. Panek and C. R. Jacob, Efficient calculation of anharmonic vibrational spectra of large molecules with localized modes, ChemPhysChem 15, 3365 (2014).
  10. T. Inui, Y. Tanabe, and Y. Onodera, Group Theory and Its Applications in Physics, Springer Series in Solid-State Sciences 78 (Springer, New York, 1990).
  11. M. S. Dresselhaus, G. Dresselhaus, and A. Jorio, Group Theory: Application to the Physics of Condensed Matter (Springer, Berlin, 2008).
  12. J. M. Adams and G. L. Catchen, Anomalous crystal chemistries of the In111→Cd111 and Hf181→Ta181 probes in rutile TiO2 studied using perturbed-angular-correlation spectroscopy, Phys. Rev. B 50, 1264 (1994).
  13. G. N. Darriba, L. A. Errico, P. D. Eversheim, G. Fabricius, and M. Rentería, First-principles and time-differential γ−γ perturbed-angular-correlation spectroscopy study of structural and electronic properties of Ta-doped TiO2 semiconductor, Phys. Rev. B 84, 239903(E) (2011).
  14. K. Freitag, A facility for ion implantation in samples colder than 0.5 K, Radiat. Eff. 44, 185 (1979).
  15. S. C. Wu, Nuclear data sheets for A=181, Nucl. Data Sheets 106, 367 (2005).
  16. M. B. Barbosa, J. G. Correia, K. Lorenz, R. Vianden, and J. P. Araújo, Studying electronic properties in GaN without electrical contacts using γ−γ vs e−−γ perturbed angular correlations, Sci. Rep. 9, 15734 (2019).
  17. I. C. J. Yap, J. Schell, T. T. Dang, C. Noll, R. Beck, U. Köster, R. Mansano, and H. C. Hofsäss, Room-temperature Ta181(TiO2): An e-γ TDPAC study, Crystals 12, 946 (2022).
  18. I. C. J. Yap, Part 1: A particular EFG temperature dependence for Ta181(TiO2): An electron–gamma TDPAC study, M.Sc. thesis, Georg-August Universität, Göttingen, 2024, https://repository.cern/records/gen6a-kc321.
  19. J. G. Marques, J. G. Correia, A. A. Melo, M. F. da Silva, J. C. Soares, and the ISOLDE Collaboration, A four-detector spectrometer for e−−γ PAC on-line with the ISOLDE-CERN isotope separator, Nucl. Instrum. Methods Phys. Res. Sect. B 99, 645 (1995).
  20. P. Kleinheinz, L. Samuelsson, R. Vukanović, and K. Siegbahn, A four-detector electron directional correlation spectrometer, Nucl. Instrum. Methods 32, 1 (1965).
  21. R. S. Raghavan and P. Raghavan, A new method for differential perturbed angular correlation measurements, Nucl. Instrum. Methods 92, 435 (1971).
  22. T. Butz and A. Lerf, Comment on “Mössbauer studies of the 6.2 keV γ-rays of Ta181 in Ta-dichalcogenides”, Phys. Lett. A 97, 217 (1983).
  23. J. Horta and D. Silva, Interlude, computer program (Leipzig University, Leipzig, Germany, 2002).
  24. J. G. Correia, GFIT19 (with N. P. Barradas for NNFIT), in PAC MANual (Lisbon, Portugal, 1992).
  25. N. P. Barradas, NNFIT, computer program, documented in the PAC MANual (Lisbon, Portugal, 1992); see also Ref. [26].
  26. N. P. Barradas, M. Rots, A. A. Melo, and J. C. Soares, Magnetic anisotropy and temperature dependence of the hyperfine fields of Cd111 in single-crystalline cobalt, Phys. Rev. B 47, 8763 (1993).
  27. See Supplemental Material at http://link.aps.org/supplemental/10.1103/xlpz-9gsw for the e-γ TDPAC formalism; conventions for the single-crystal perturbation function, static quadrupole Hamiltonian, construction of N(90∘,t) and N(180∘,t), and the extraction of V33 and η; comparison with non-single-crystal TiO2 TDPAC data; group- and representation-theoretical discussion of the rutile two-site EFG geometry; derivation of the local frozen-phonon method; self-consistent harmonic approximation and Padé-like temperature dependence; electronic structure of substitutional Ta in rutile TiO2; and cross-mode coupling check within the dominant B1g shear plane.
  28. M. Forker, K. Krien, and F. Reuschenbach, e−−γ TDPAC measurements of the magnetic and electric hyperfine interaction of Ta181 implanted in terbium, Hyperfine Interact. 9, 255 (1981).
  29. T. Wegner, Calculated perturbed angular correlations for In111 doped cubic single crystals, Hyperfine Interact. 23, 179 (1985).
  30. J. D. Rogers and A. Vasquez, Data reduction in perturbed angular correlation experiments, Nucl. Instrum. Methods 130, 539 (1975).
  31. R. Béraud, I. Berkes, J. Danière, G. Marest, and R. Rougny, Effect of finite time-resolution on perturbed angular correlation measurements, Nucl. Instrum. Methods 69, 41 (1969).
  32. S. K. Das, S. V. Thakare, and T. Butz, The nuclear quadrupole interaction at 111Cd and 181Ta sites in anatase and rutile TiO2: A TDPAC study, J. Phys. Chem. Solids 70, 778 (2009).
  33. D. Banerjee, S. K. Das, and S. V. Thakare, Simultaneous measurement of electric field gradient both at 111Cd and 181Ta sites in a single perturbed γ−γ angular correlation measurement, J. Radioanal. Nucl. Chem. 313, 677 (2017).
  34. D. Banerjee, S. K. Das, S. V. Thakare, P. Y. Nabhiraj, R. Menon, R. K. Bhandari, and K. Krishnan, Study of surface-bulk mass transport and phase transformation in nano-TiO2 using hyperfine interaction technique, J. Phys. Chem. Solids 71, 983 (2010).
  35. S. Schlabach, D. V. Szabó, D. Vollath, P. de la Presa, and M. Forker, Structure and grain growth of TiO2 nanoparticles investigated by electron and x-ray diffractions and Ta181 perturbed angular correlations, J. Appl. Phys. 100, 024305 (2006).
  36. T. Martucci, J. M. Ramos, A. W. Carbonari, A. S. Silva, and R. N. Saxena, Investigation of electric quadrupole interaction in TiO2 by means of perturbed gamma-gamma angular correlation spectroscopy, in Proceedings of the 2011 International Nuclear Atlantic Conference (INAC 2011), Belo Horizonte, Brazil (ABEN, Rio de Janeiro, Brazil, 2011).
  37. J. Schell, D. C. Lupascu, A. W. Carbonari, R. D. Mansano, I. S. Ribeiro, Jr., T. T. Dang, I. Anusca, H. Trivedi, K. Johnston, and R. Vianden, Ion implantation in titanium dioxide thin films studied by perturbed angular correlations, J. Appl. Phys. 121, 145302 (2017).
  38. J. Schell, Investigation of hyperfine parameters in pure and 3d transition metal doped SnO2 and TiO2 by means of perturbed gamma-gamma angular correlation spectroscopy, Ph.D. thesis, São Paulo University, Brazil, 2015.
  39. T. Butz and R. Vianden, The temperature dependence of the nuclear quadrupole interaction of Ti44(EC)Sc44 in rutile, Hyperfine Interact. 221, 99 (2013).
  40. W. Kohn and L. J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev. 140, A1133 (1965).
  41. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  42. G. Kresse and D. Joubert, From ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758 (1999).
  43. J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77, 3865 (1996).
  44. J. Heyd, G. E. Scuseria, and M. Ernzerhof, Hybrid functionals based on a screened Coulomb potential, J. Chem. Phys. 118, 8207 (2003).
  45. H. M. Petrilli, P. E. Blöchl, P. Blaha, and K. Schwarz, Electric-field-gradient calculations using the projector augmented wave method, Phys. Rev. B 57, 14690 (1998) .
  46. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  47. J. K. Burdett, T. Hughbanks, G. J. Miller, J. W. Richardson, Jr., and J. V. Smith, Structural-electronic relationships in inorganic solids: Powder neutron diffraction studies of the rutile and anatase polymorphs of titanium dioxide at 15 and 295 K, J. Am. Chem. Soc. 109, 3639 (1987).
  48. Ž. Kovačič, B. Likozar, and M. Huš, Electronic properties of rutile and anatase TiO2 and their effect on CO2 adsorption: A comparison of first principle approaches, Fuel 328, 125322 (2022).
  49. J. Pascual, J. Camassel, and H. Mathieu, Fine structure in the intrinsic absorption edge of TiO2, Phys. Rev. B 18, 5606 (1978).
  50. O. Kanert and H. Kolem, The unusual temperature dependence of the electric field gradient at titanium sites in rutile (TiO2), J. Phys. C: Solid State Phys. 21, 3909 (1988).
  51. E. T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106, 620 (1957).
  52. D. S. Dummit and R. M. Foote, Abstract Algebra, 3rd ed. (Wiley, Hoboken, NJ, 2004), Sec. 1.6.
  53. J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices (Oxford University Press, Oxford, 1985), Chaps. 1–2.
  54. F. Giustino, Electron-phonon interactions from first principles, Rev. Mod. Phys. 89, 015003 (2017).
  55. M. T. Dove, Introduction to Lattice Dynamics (Cambridge University Press, Cambridge, 1993).
  56. OriginLab, “InvsPoly Fit Function,” Origin Help (OriginLab, 2025), https://www.originlab.com/doc/Origin-Help/InvsPoly-FitFunc.
  57. M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wondratschek, Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups, Acta Cryst. A 62, 115 (2006).
  58. M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln, Ann. Phys. 389, 457 (1927).
  59. S. Baroni, S. de Gironcoli, A. D. Corso, and P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515 (2001).
  60. T. M. Apostol, Calculus, 2nd ed. (Wiley, Hoboken, NJ, 1991), Vol. 2.
  61. G. B. Folland, Advanced Calculus, 2nd ed. (University of Washington, Seattle, WA, 2023), Sec. 2.7 and Thm. 2.68.
  62. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, New York, 2006).
  63. M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Oxford University Press, Oxford, 1954).
  64. I. Errea, M. Calandra, and F. Mauri, Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmonic approximation: Application to platinum and palladium hydrides, Phys. Rev. B 89, 064302 (2014).
  65. N. R. Werthamer, Self-consistent phonon formulation of anharmonic lattice dynamics, Phys. Rev. B 1, 572 (1970).
  66. J.-P. Serre, Linear Representations of Finite Groups, Graduate Texts in Mathematics 42 (Springer, New York, 1977), Sec. 2.2.
  67. G. A. Baker Jr. and P. Graves-Morris, Padé Approximants, 2nd ed., Encyclopedia of Mathematics and its Applications 59 (Cambridge University Press, Cambridge, 1996).
  68. R. P. Feynman, Slow electrons in a polar crystal, Phys. Rev. 97, 660 (1955).
  69. L. A. Errico, Ab initio study of the temperature dependence of the EFG at Cd impurities in rutile TiO2, Hyperfine Interact. 158, 29 (2004).
  70. L. Isserlis, On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika 12, 134 (1918).
  71. I. C. J. Yap and S. Jin, PBE-calculated temperature dependence of the electric field gradient of the Ti site in pure TiO2 [Data set], Zenodo, 2026, 10.5281/zenodo.21215882.
  72. I. C. J. Yap and S. Jin, Coupled B1g shear-plane LFP–DFT polynomial-fit dataset for Ta-doped rutile TiO2, Zenodo, 2026, 10.5281/zenodo.20993965.
  73. I. C. J. Yap, J. Schell, H. Hofsäss, and T. T. Dang, Dataset for temperature-dependent Ta181(TiO2): An e-γ TDPAC study, version v1, Zenodo, 2026, 10.5281/zenodo.18685510.
  74. I. C. J. Yap and S. Jin, Materials used to prepare the figures for “Influence of symmetry-resolved phonon dynamics on the temperature-dependent electric field gradients in solids”, Zenodo, 2026, https://doi.org/10.5281/zenodo.21717066.

Outline

Information

Phys. Rev. B 114, 115105– Published 10 August, 2026
  • Received 19 February 2026
  • Revised 25 May 2026
  • Accepted 24 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by CERN.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation