International Tables for Crystallography (2013). Vol. D. ch. 3.1, pp. 358-396
https://doi.org/10.1107/97809553602060000915

Chapter 3.1. Structural phase transitions

Contents

3.1. Structural phase transitions (pp. 358-396) | html | pdf | chapter contents |
J.-C. Tolédano, V. Janovec, V. Kopský, J. F. Scott and P. Boček
- 3.1.1. Introduction (pp. 358-360) | html | pdf |
  J.-C. Tolédano
- 3.1.2. Thermodynamics of structural transitions (pp. 360-369) | html | pdf |
  J.-C. Tolédano
  - 3.1.2.1. Introduction (p. 360) | html | pdf |
  - 3.1.2.2. Basic ideas of Landau's theory of phase transitions (pp. 360-364) | html | pdf |
    - 3.1.2.2.1. Description of a prototype example (p. 360) | html | pdf |
    - 3.1.2.2.2. Basic assumptions and strategy (p. 361) | html | pdf |
    - 3.1.2.2.3. Symmetry constraints and form of the free energy (p. 361) | html | pdf |
    - 3.1.2.2.4. Reduction of the number of relevant degrees of freedom: order parameter (pp. 361-362) | html | pdf |
    - 3.1.2.2.5. Stable state below T_c and physical anomalies induced by the transition (pp. 362-363) | html | pdf |
    - 3.1.2.2.6. Symmetry considerations (pp. 363-364) | html | pdf |
      - 3.1.2.2.6.1. Order-parameter symmetry (p. 363) | html | pdf |
      - 3.1.2.2.6.2. Degeneracy of the low-symmetry phase (pp. 363-364) | html | pdf |
  - 3.1.2.3. Free-energy models for discontinuous transitions (pp. 364-365) | html | pdf |
  - 3.1.2.4. Generalization of the approach (pp. 365-367) | html | pdf |
    - 3.1.2.4.1. Description of the phase transition (p. 366) | html | pdf |
    - 3.1.2.4.2. Order parameter (pp. 366-367) | html | pdf |
    - 3.1.2.4.3. Stable states and symmetry in the vicinity of (p. 367) | html | pdf |
  - 3.1.2.5. Application to the structural transformation in a real system (pp. 367-369) | html | pdf |
    - 3.1.2.5.1. Nature of the groups and of their irreducible representations (p. 367) | html | pdf |
    - 3.1.2.5.2. The example of gadolinium molybdate, Gd₂(MoO₄)₃ (pp. 367-369) | html | pdf |
      - 3.1.2.5.2.1. Experimental identification of the order-parameter symmetry (pp. 367-368) | html | pdf |
      - 3.1.2.5.2.2. Construction of the free energy and stable states (pp. 368-369) | html | pdf |
      - 3.1.2.5.2.3. Macroscopic behaviour of GMO (p. 369) | html | pdf |
- 3.1.3. Equitranslational phase transitions. Property tensors at ferroic phase transitions (pp. 370-381) | html | pdf |
  V. Janovec and V. Kopský
  - 3.1.3.1. Equitranslational phase transitions and their order parameters (p. 370) | html | pdf |
  - 3.1.3.2. Property tensors at ferroic phase transitions. Tensor parameters (pp. 370-375) | html | pdf |
  - 3.1.3.3. Tables of equitranslational phase transitions associated with irreducible representations (pp. 375-380) | html | pdf |
    - 3.1.3.3.1. Explanation of Table 3.1.3.1 (pp. 378-380) | html | pdf |
  - 3.1.3.4. Examples (pp. 380-381) | html | pdf |
- 3.1.4. Example of a table for non-equitranslational phase transitions (p. 381) | html | pdf |
  J.-C. Tolédano
- 3.1.5. Microscopic aspects of structural phase transitions and soft modes (pp. 381-392) | html | pdf |
  J. F. Scott
  - 3.1.5.1. Introduction (p. 381) | html | pdf |
  - 3.1.5.2. Displacive phase transitions (pp. 381-392) | html | pdf |
    - 3.1.5.2.1. Landau–Devonshire theory (pp. 381-382) | html | pdf |
    - 3.1.5.2.2. Soft modes (pp. 382-383) | html | pdf |
    - 3.1.5.2.3. Strontium titanate, SrTiO₃ (pp. 383-385) | html | pdf |
    - 3.1.5.2.4. Lanthanum aluminate, LaAlO₃ (p. 385) | html | pdf |
    - 3.1.5.2.5. Potassium nitrate, KNO₃ (p. 385) | html | pdf |
    - 3.1.5.2.6. Lanthanum pentaphosphate (pp. 385-386) | html | pdf |
    - 3.1.5.2.7. Barium manganese tetrafluoride (pp. 386-387) | html | pdf |
    - 3.1.5.2.8. Barium sodium niobate (p. 387) | html | pdf |
    - 3.1.5.2.9. Tris-sarcosine calcium chloride (TSCC) (pp. 387-388) | html | pdf |
    - 3.1.5.2.10. Potassium dihydrogen phosphate, KH₂PO₄ (pp. 388-389) | html | pdf |
    - 3.1.5.2.11. Sodium nitrite, NaNO₂ (pp. 389-390) | html | pdf |
    - 3.1.5.2.12. Fast ion conductors (p. 390) | html | pdf |
    - 3.1.5.2.13. High-temperature superconductors (pp. 390-392) | html | pdf |
  - 3.1.5.3. Low-temperature ferroelectric transitions (p. 392) | html | pdf |
- 3.1.6. Group informatics and tensor calculus (pp. 392-394) | html | pdf |
  V. Kopský and P. Boček
- 3.1.7. Glossary (p. 394) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 3.1.2.1. Model of a structural transition (p. 360) | html | pdf |
  - Fig. 3.1.2.2. (a) Variation of the free energy as function of the amplitude of the displacement of the central ion in Fig. 3.1.2.1 (p. 360) | html | pdf |
  - Fig. 3.1.2.3. Plots representative of the equations $[\alpha_1(p, T)=0]$ and $[\alpha_2(p, T)=0]$ (p. 362) | html | pdf |
  - Fig. 3.1.2.4. Plots of the Landau free energy as a function of the order parameter, for values of the temperature above or below or coincident with (p. 362) | html | pdf |
  - Fig. 3.1.2.5. (a) Qualitative temperature dependence of the specific heat at a continuous transition (p. 363) | html | pdf |
  - Fig. 3.1.2.6. (a) Plots of the free energy as a function of the order parameter for various temperature values in the framework of the model of a discontinuous transition associated with equation (3.1.2.13) (p. 365) | html | pdf |
  - Fig. 3.1.2.7. Plots of the free energy as a function of the order parameter for various temperatures in the framework of the model of a discontinuous transition associated with equation (3.1.2.16) (p. 365) | html | pdf |
  - Fig. 3.1.2.8. Schematic representation of the displacement associated with the order parameter in a crystal having trigonal (rhombohedral) symmetry (p. 365) | html | pdf |
  - Fig. 3.1.2.9. Rotations/reflections belonging to the point group of gadolinium molybdate (p. 368) | html | pdf |
  - Fig. 3.1.2.10. Temperature dependence of the macroscopic susceptibility (or elastic compliance, $[s_{ij}]$ ) in gadolinium molybdate (p. 369) | html | pdf |
  - Fig. 3.1.3.1. Lattice of subgroups of the group $[m\bar3m]$ (p. 379) | html | pdf |
  - Fig. 3.1.3.2. Lattice of subgroups of the group (p. 379) | html | pdf |
  - Fig. 3.1.5.1. Free energy and order parameter from the Landau–Devonshire theory [equation (3.1.5.1a)] for a continuous second-order ferroelectric phase transition (p. 383) | html | pdf |
  - Fig. 3.1.5.2. Free energy and order parameter from the Landau–Devonshire theory [equation (3.1.5.1a)] for a discontinuous first-order ferroelectric phase transition (p. 383) | html | pdf |
  - Fig. 3.1.5.3. Three-dimensional graph of phase boundaries as functions of temperature T, pressure p and applied electric field E, showing a tricritical point where three continuous phase boundaries intersect (p. 383) | html | pdf |
  - Fig. 3.1.5.4. Structure of strontium titanate above (undisplaced ions) and below (arrows) its anti-ferrodistortive phase transition at ca. 105 K (p. 384) | html | pdf |
  - Fig. 3.1.5.5. Rotation angle versus temperature for the oxygen octahedron distortion below 105 K in strontium titanate described in Fig. 3.1.5.4 (p. 384) | html | pdf |
  - Fig. 3.1.5.6. Raman spectra of strontium titanate below its cubic–tetragonal phase transition temperature (p. 384) | html | pdf |
  - Fig. 3.1.5.7. Temperature dependence of phonon branches observed in the Raman spectra of tetragonal strontium titanate (p. 384) | html | pdf |
  - Fig. 3.1.5.8. (a) Structure of lanthanum aluminate above (undistorted) and below (arrows) its cubic–rhombohedral phase transition near 840 K (p. 385) | html | pdf |
  - Fig. 3.1.5.9. Phase diagram of potassium nitrate, KNO₃ (p. 385) | html | pdf |
  - Fig. 3.1.5.10. (a) `Soft' optical phonon frequency versus temperature in LaP₅O₁₄, showing displacive character of the phase transition (p. 386) | html | pdf |
  - Fig. 3.1.5.11. (a) Structure of barium metal fluoride BaMF₄ (M = Co, Mn, Mg, Zn, Ni) at ambient temperature (300 K) (p. 386) | html | pdf |
  - Fig. 3.1.5.12. Structure of the tungsten bronze barium sodium niobate Ba₂NaNb₅O₁₅ in its highest-temperature phase above 853 K (p. 387) | html | pdf |
  - Fig. 3.1.5.13. Sequence of phases encountered with raising or lowering the temperature in barium sodium niobate (p. 387) | html | pdf |
  - Fig. 3.1.5.14. Structure of tris-sarcosine calcium chloride, (CH₃NHCH₂COOH)₃CaCl₂ (p. 388) | html | pdf |
  - Fig. 3.1.5.15. `Soft' optical phonon frequencies versus temperature in both ferroelectric and paraelectric phases of tris-sarcosine calcium chloride (p. 388) | html | pdf |
  - Fig. 3.1.5.16. The structure of potassium dihydrogen phosphate, KH₂PO₄, showing the O⋯H⋯O hydrogen bonds (p. 388) | html | pdf |
  - Fig. 3.1.5.17. Double-well models (p. 389) | html | pdf |
  - Fig. 3.1.5.18. Pressure dependence of the `soft' optical phonon branch Raman spectra in potassium dihydrogen phosphate (p. 389) | html | pdf |
  - Fig. 3.1.5.19. Structure of sodium nitrite, NaNO₂ (p. 389) | html | pdf |
  - Fig. 3.1.5.20. Raman spectra of sodium nitrite, showing diffusive Debye-like response due to large-amplitude flopping over of nitrite ions (p. 390) | html | pdf |
  - Fig. 3.1.5.21. Phase diagram for sodium nitrite for `conjugate' electric fields applied along the polar b axis, showing triple point, tricritical point and critical end point (p. 390) | html | pdf |
  - Fig. 3.1.5.22. Phase diagram for sodium nitrite for electric fields applied perpendicular to the polar b axis (p. 390) | html | pdf |
  - Fig. 3.1.5.23. (a) Crystal structure of silver iodide tungstate (Ag₁₃I₉W₂O₈); (b) showing conduction paths for Ag ions (p. 391) | html | pdf |
  - Fig. 3.1.5.24. Evidence for three phase transitions in silver iodide tungstate: (a) dielectric and conductivity data; (b) specific heat data; (c) Raman data (p. 391) | html | pdf |
  - Fig. 3.1.5.25. Raman spectra of YBa₂Cu₃O_7−x below an apparent phase transition at ca. 235 K (p. 392) | html | pdf |
- Tables
  - Table 3.1.1.1. Ferroic classification of structural parameters (p. 359) | html | pdf |
  - Table 3.1.2.1. Transformation of the components of $[\bf d]$ under the symmetry operations of group (p. 363) | html | pdf |
  - Table 3.1.2.2. Matrices defining the irreducible representations of for $[{\bf k}={\bf a}_1^*+{\bf a}_2^*]$ (p. 368) | html | pdf |
  - Table 3.1.2.3. Action of the generators of on the order parameter and on the polarization and strain components (p. 369) | html | pdf |
  - Table 3.1.3.1. Point-group symmetry descents associated with irreducible representations (pp. 372-377) | html | pdf |
  - Table 3.1.3.2. Symmetry descents $[G\Downarrow F_1]$ associated with two irreducible representations (p. 378) | html | pdf |
  - Table 3.1.3.3. Important property tensors (p. 380) | html | pdf |
  - Table 3.1.4.1. Possible symmetry changes across transitions from a parent phase with space group , , , , or (p. 382) | html | pdf |
  - Table 3.1.5.1. Low-temperature ferroelectrics (p. 392) | html | pdf |