jeanflop/post_ocr_correction2 huggingface dataset:
local autorise des hétérogénéités plastiques induites (écart AFPz 3 %) nettement plus importantes que dans les matériaux microhétérogènes locaux. En conséquence, les contraintes internes résiduelles atteignent des valeurs considérables. Dans la présente simulation numérique, les contraintes équivalentes dans les parois sont proches du double de la limite d'écoulement macroscopique, en bon accord avec les mesures de Mughrabi [1] : Fig. 2. Essai de traction-compression. Fig. 2. Tension-compression test. Note remise le 29 mai 1992, acceptée après révision le 6 janvier 1993. RÉFÉRENCES BIBLIOGRAPHIQUES [1] H. MUGHRABI, Revue Phys. Appl., 23, 1988, p. 367-379. [2] J. D. ESHELBY, Prog. Sol. Mechanics II, 87, 1961. [3] E. KRÖNER, Acta Metall., 9, 1961, p. 155-165. [4] J. H. SCHMITT, Thèse de doctorat d'État, Grenoble, 1986. [5] T. TABATA et al., Phil. Mag. A, 46, 1982, p. 801-816. [6] A. ZAOUI, Thèse, Paris, 1970. [7] M. BERVEILLER et A. ZAOUI, J. Mech. Phys. Solids, 26, p. 325-344. [8] J. KRATOCHVIL, Proc. Int. Conf. on Nonlinear Mechanics, Shanghai, oct. 1985, p. 560-565. D. M. et M. B. Laboratoire de Physique et Mécanique des Matériaux, URA-CNRS n 1215, ENIM, Ile-du-Saulcy, 57045 Metz, France; J. K. : Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, 18040 Prague 8, Czechoslovakia. Physique de la matière condensée/ Condensed Matter Physics Étude des transitions de phase structurales par spectrométrie infrarouge des excitons vibrationnels : application au cristal de phénothiazine René GUERIN, Bernard COURTOIS, Abdellah REZZOUK et Alain LE ROY., Résumé La spectrométrie infrarouge des excitons vibrationnels est une technique d'étude particulièrement adaptée aux transitions de phase structurales du second ordre observées dans les cristaux moléculaires. Le cristal de phénothiazine (C12H9NS) présente une telle transition aux environs de 250 K, passant d'une phase « haute température » orthorhombique de groupe d'espace Pb n m (D2h) à une phase « basse température » monoclinique de groupe d'espace P21/n (C2h). Cette transition est étudiée par spectrométrie infrarouge et on montre que l'écart en fréquence Δv des composantes excitoniques des bandes infrarouges situées à 426 cm-1 (classe A") et à 925 cm-1 (classe A') varie, en fonction de la température, suivant une loi
18040 Prague 8, Czechoslovakia. Physique de la matière condensée/ Condensed Matter Physics Étude des transitions de phase structurales par spectrométrie infrarouge des excitons vibrationnels : application au cristal de phénothiazine René GUERIN, Bernard COURTOIS, Abdellah REZZOUK et Alain LE ROY., Résumé La spectrométrie infrarouge des excitons vibrationnels est une technique d'étude particulièrement adaptée aux transitions de phase structurales du second ordre observées dans les cristaux moléculaires. Le cristal de phénothiazine (C12H9NS) présente une telle transition aux environs de 250 K, passant d'une phase « haute température » orthorhombique de groupe d'espace Pb n m (D2h) à une phase « basse température » monoclinique de groupe d'espace P21/n (C2h). Cette transition est étudiée par spectrométrie infrarouge et on montre que l'écart en fréquence Δv des composantes excitoniques des bandes infrarouges situées à 426 cm-1 (classe A") et à 925 cm-1 (classe A') varie, en fonction de la température, suivant une loi en (Tc T)β, l'exposant critique β étant voisin de 0,45. Ce résultat est conforme à la théorie de Landau des transitions de phase du second ordre. A study of structural phase transitions by infrared spectroscopy of vibrational excitons: application to crystalline phenothiazine Abstract — Infrared spectroscopy of vibrational excitons is a good method to study the second order structural phase transitions which are observed in some molecular crystals. Crystalline phenothiazine (C12H9NS) undergoes such a transition at about Tc = 250 K from a high-temperature orthorhombic phase with a space group Pb n m (D2h) to a low-temperature monoclinic phase with a space group P 21/n (C2h). This transition is studied by infrared spectroscopy of two bands at 426 cm-1 (A" mode) and at 925 cm-1 (A' mode). The splitting of the exciton components of these bands is studied as a function of temperature and can be described by a (Tc T)β law with a critical exponent β of an approximate value of 0.45. This result agrees with Landau's theory of second order phase transitions. Abridged English Version This paper is devoted to the study of second order structural phase transitions in molecular crystals by means of infrared spectroscopy of vibrational
a good method to study the second order structural phase transitions which are observed in some molecular crystals. Crystalline phenothiazine (C12H9NS) undergoes such a transition at about Tc = 250 K from a high-temperature orthorhombic phase with a space group Pb n m (D2h) to a low-temperature monoclinic phase with a space group P 21/n (C2h). This transition is studied by infrared spectroscopy of two bands at 426 cm-1 (A" mode) and at 925 cm-1 (A' mode). The splitting of the exciton components of these bands is studied as a function of temperature and can be described by a (Tc T)β law with a critical exponent β of an approximate value of 0.45. This result agrees with Landau's theory of second order phase transitions. Abridged English Version This paper is devoted to the study of second order structural phase transitions in molecular crystals by means of infrared spectroscopy of vibrational excitons. The experimental results, obtained for the molecular crystal of phenothiazine, are given and analysed. In the theory of vibrational exciton ([1], [2]) the energy exchange between two molecules, one in the excited state, the other in the ground state, initiates the splitting of a nondegenerate excited state of the isolated molecule into several levels; their number is equal to the number of molecules in the unit cell of crystal. The splitting of spectroscopic levels depends on the position and the orientation of molecules in crystal and, consequently, measures the long-range ordering. The group theory enables simple determination of the symmetry of the components of the multiplet structure and specification of their infrared and Raman activity. In his theory ([3], [4]), Landau postulates that the transition occurs with a symmetry change to which a quantity 11 (T), labelled order parameter of the transition, is associated. For T < Tc, its expression is of the form 11 = 110 (Tc T)β where β is the critical exponent which generally lies between 0.25 and 0.50 [5]. For T > Tc, the order parameter is zero. This order parameter characterizes the long-range order and must be related to the splitting of the components
isolated molecule into several levels; their number is equal to the number of molecules in the unit cell of crystal. The splitting of spectroscopic levels depends on the position and the orientation of molecules in crystal and, consequently, measures the long-range ordering. The group theory enables simple determination of the symmetry of the components of the multiplet structure and specification of their infrared and Raman activity. In his theory ([3], [4]), Landau postulates that the transition occurs with a symmetry change to which a quantity 11 (T), labelled order parameter of the transition, is associated. For T < Tc, its expression is of the form 11 = 110 (Tc T)β where β is the critical exponent which generally lies between 0.25 and 0.50 [5]. For T > Tc, the order parameter is zero. This order parameter characterizes the long-range order and must be related to the splitting of the components of the excitonic structure. Then this splitting could characterize the phase transition of a molecular crystal. The molecular crystal of phenothiazine (molecular formula: C12H9NS, molecular symmetry: Cs) (fig. 1) undergoes a second order phase transition at Tc≈250 K [6]. The high Note présentée par Pierre-Gilles de GENNES. 0764-4450/93/03160441 $ 2.00 6 Académie des Sciences temperature phase is orthorhombic (spatial group: Pb n m) (fig. 2) [7] whereas the low-temperature phase is monoclinic (spatial group: P 21/n). In the two phases there are four molecules in the unit cell ([6], [8], [9]). The 63 fundamental molecular vibrations are shared between the A' and A" modes both active in infrared and Raman spectra. Their assignment is partly known ([10], [11]). Our experimental results show that the A" band at 426 cm-1 is resolved into two components in the low-temperature phase. This doublet disappears at the transition temperature to give a single absorption band in the high-temperature phase (fig. 3). In the low-temperature phase, the variation of the splitting Δv as a function of temperature is described by a (Tc — T)p law with P about 0.45 (fig. 4). The A' vibration at 925 cm-1 is resolved into two components in both crystalline
6 Académie des Sciences temperature phase is orthorhombic (spatial group: Pb n m) (fig. 2) [7] whereas the low-temperature phase is monoclinic (spatial group: P 21/n). In the two phases there are four molecules in the unit cell ([6], [8], [9]). The 63 fundamental molecular vibrations are shared between the A' and A" modes both active in infrared and Raman spectra. Their assignment is partly known ([10], [11]). Our experimental results show that the A" band at 426 cm-1 is resolved into two components in the low-temperature phase. This doublet disappears at the transition temperature to give a single absorption band in the high-temperature phase (fig. 3). In the low-temperature phase, the variation of the splitting Δv as a function of temperature is described by a (Tc — T)p law with P about 0.45 (fig. 4). The A' vibration at 925 cm-1 is resolved into two components in both crystalline phases (fig. 5). In the high-temperature phase, the splitting is a constant Δv0.5 cm-1. In the low-temperature phase, the variation of the splitting 5(Av) = Av — Av0 is represented by a (Tc -T)β law with β about 0.45 (fig. 6). The splitting Δv — or 5(Av) — of the exciton structure indicates a second order phase transition with a partial first order nature ([6], [9]). The order parameter could be the shear strain ε5 ([9], [12], [13], [14]) which is approximately equal to the shear angle θ ([15]). In the low-temperature phase, the energy levels of the crystal excited states are determined by a diagonalization of the resonance-interaction matrix and the splitting Δv — or δ (Δv) — can be expressed as a function of the matrix elements. For k = 0, it may be shown that Δv = 8 M12 where M12 represents the exciton coupling constant between the molecules I and 2. In sym-triazine the phase transition is generated by a rotation of the molecules as well as a shear of the unit cell ([16], [17]). By comparison, it may be shown that Δv is a quadratic function of the angle θ12 formed by the instantaneous dipole
exciton structure indicates a second order phase transition with a partial first order nature ([6], [9]). The order parameter could be the shear strain ε5 ([9], [12], [13], [14]) which is approximately equal to the shear angle θ ([15]). In the low-temperature phase, the energy levels of the crystal excited states are determined by a diagonalization of the resonance-interaction matrix and the splitting Δv — or δ (Δv) — can be expressed as a function of the matrix elements. For k = 0, it may be shown that Δv = 8 M12 where M12 represents the exciton coupling constant between the molecules I and 2. In sym-triazine the phase transition is generated by a rotation of the molecules as well as a shear of the unit cell ([16], [17]). By comparison, it may be shown that Δv is a quadratic function of the angle θ12 formed by the instantaneous dipole moments of the molecules I and 2. Consequently, the variation of the exciton structure as a function of temperature can be considered as an order parameter allowing a second order phase transition to be characterized. 1. INTRODUCTION. — This Note aims to show the interest of infrared exciton spectroscopy in the study of structural phase transitions of the second order. After having recalled the theories implemented in this work, we will present and analyze the experimental results obtained for the crystal of phenothiazine. 2. THEORIES MISES EN ŒUVRE. — 2. 1. Theory of the exciton vibration. In the model of the exciton molecular, one considers that there is an exchange of energy between two molecules, one in its excited state, the other in its fundamental state. The quantum of energy associated with this resonant transfer of the energy of vibration propagates throughout the crystal under the action of infrared radiation and carries the name of exciton vibration. This exchange of energy, within a crystal having n molecules per unit cell, provokes the decomposition of a non-degenerate level of the isolated molecule into subspectra. The separation of the components of the absorption band corresponding to this subspectra depends on the position
the study of structural phase transitions of the second order. After having recalled the theories implemented in this work, we will present and analyze the experimental results obtained for the crystal of phenothiazine. 2. THEORIES MISES EN ŒUVRE. — 2. 1. Theory of the exciton vibration. In the model of the exciton molecular, one considers that there is an exchange of energy between two molecules, one in its excited state, the other in its fundamental state. The quantum of energy associated with this resonant transfer of the energy of vibration propagates throughout the crystal under the action of infrared radiation and carries the name of exciton vibration. This exchange of energy, within a crystal having n molecules per unit cell, provokes the decomposition of a non-degenerate level of the isolated molecule into subspectra. The separation of the components of the absorption band corresponding to this subspectra depends on the position and orientation of the molecules in the crystal, therefore on the order at a great distance which reigns. 2.2. Theory of Landau's phase transitions. — In his theory, Landau ([3], [4]), considers that the phase transition is accompanied by a breaking of symmetry to which is associated a quantity η (T), function of the temperature, called the order parameter of the transition. This order parameter is, by definition, zero in the most symmetric phase (in general, the "high temperature" phase). It is different from zero in the least symmetric phase and decreases when the temperature increases to become zero at the transition temperature Tc, its expression being of the form η = η0 (Tc - T)β. According to Landau, the critical exponent of the order parameter β must be equal to 0.5, however, in practice, β varies between 0.25 and 0.50 [5]. The order parameter η describes the degree of establishment of the new order at a great distance resulting from the loss of elements of symmetry. However, the separation of the components of the absorption bands in infrared also depends, it too, on the order at a great distance reigning in the crystal and is therefore susceptible of being
(T), function of the temperature, called the order parameter of the transition. This order parameter is, by definition, zero in the most symmetric phase (in general, the "high temperature" phase). It is different from zero in the least symmetric phase and decreases when the temperature increases to become zero at the transition temperature Tc, its expression being of the form η = η0 (Tc - T)β. According to Landau, the critical exponent of the order parameter β must be equal to 0.5, however, in practice, β varies between 0.25 and 0.50 [5]. The order parameter η describes the degree of establishment of the new order at a great distance resulting from the loss of elements of symmetry. However, the separation of the components of the absorption bands in infrared also depends, it too, on the order at a great distance reigning in the crystal and is therefore susceptible of being a function of the order parameter. This Note aims to show how the evolution, as a function of temperature, of the structure of multiplets of an absorption band in infrared constitutes a simple procedure for studying second-order phase transitions. 2.3. Theory of groups. — The theory of groups allows one to determine, thanks to the correlation diagram, the rules of selection of the transitions excitonics of the molecular crystal, specifying the infrared and Raman activity of each component of the structure of multiplets. 3. INFRARED SPECTRUM AND PHASE TRANSITION OF THE CRYSTAL OF PHENOTHIAZINE. 3. 1. Preparation and experimental technique. — The experimental technique used is that of microcrystals of phenothiazine in potassium bromide with obtention, by compression, of a pastille of 250 mg containing 0.5% by weight of phenothiazine. The samples thus prepared are placed in a cryostat with circulating nitrogen whose temperature is controlled with an uncertainty of 0.2 K. The spectra are recorded on a Perkin-Elmer 125, double-beam, spectrometer of resolution better than 0.5 cm-1. 3.2. Crystal structure of phenothiazine. — The molecule of phenothiazine, of formula C12H9NS, belongs to the point group of symmetry Cs (fig. 1). The crystal of phenothiazine, whose melting point is located
to determine, thanks to the correlation diagram, the rules of selection of the transitions excitonics of the molecular crystal, specifying the infrared and Raman activity of each component of the structure of multiplets. 3. INFRARED SPECTRUM AND PHASE TRANSITION OF THE CRYSTAL OF PHENOTHIAZINE. 3. 1. Preparation and experimental technique. — The experimental technique used is that of microcrystals of phenothiazine in potassium bromide with obtention, by compression, of a pastille of 250 mg containing 0.5% by weight of phenothiazine. The samples thus prepared are placed in a cryostat with circulating nitrogen whose temperature is controlled with an uncertainty of 0.2 K. The spectra are recorded on a Perkin-Elmer 125, double-beam, spectrometer of resolution better than 0.5 cm-1. 3.2. Crystal structure of phenothiazine. — The molecule of phenothiazine, of formula C12H9NS, belongs to the point group of symmetry Cs (fig. 1). The crystal of phenothiazine, whose melting point is located at 495 K (186°C), presents a transition of phase of the second order, ferroelastic type, around 250 K (-23°C) [6]. The "high temperature" phase (T > 250 K) belongs to the orthorhombic system, the space group being Pbnm (~), with four molecules per unit cell occupying sites of symmetry Cs (fig. 2) [7]. The "low temperature" phase (T < 250 K) is monoclinic, of space group P21/a (C2h), with four molecules per unit cell localized in sites without symmetry C1 and presenting domains characteristic of a phase ferroelastic ([6], [8], [9]). Fig. 2. — Phase "haute température" orthorhombique du cristal de phénothiazine, d'après [9]. Fig. 2. — High-temperature orthorhombic phase of the phenothiazine crystal from [9]. 3.3. Spectre d'absorption infrarouge. — La molécule de phénothiazine appartenant au groupe ponctuel de symétrie Cs, ses 63 vibrations fondamentales se partagent entre les deux classes A' et A" actives l'une et l'autre en infrarouge et en Raman. Les attributions des bandes d'absorption infrarouge sont partiellement connues ([10], [11]). a. Règles de sélection des transitions excitoniques de la phase « haute température ». — La maille est orthorhombique, de groupe facteur D2h, les molécules étant localisées dans des sites Cs. Les règles de sélection
(fig. 2) [7]. The "low temperature" phase (T < 250 K) is monoclinic, of space group P21/a (C2h), with four molecules per unit cell localized in sites without symmetry C1 and presenting domains characteristic of a phase ferroelastic ([6], [8], [9]). Fig. 2. — Phase "haute température" orthorhombique du cristal de phénothiazine, d'après [9]. Fig. 2. — High-temperature orthorhombic phase of the phenothiazine crystal from [9]. 3.3. Spectre d'absorption infrarouge. — La molécule de phénothiazine appartenant au groupe ponctuel de symétrie Cs, ses 63 vibrations fondamentales se partagent entre les deux classes A' et A" actives l'une et l'autre en infrarouge et en Raman. Les attributions des bandes d'absorption infrarouge sont partiellement connues ([10], [11]). a. Règles de sélection des transitions excitoniques de la phase « haute température ». — La maille est orthorhombique, de groupe facteur D2h, les molécules étant localisées dans des sites Cs. Les règles de sélection des transitions optiques peuvent être déduites du diagramme de corrélation : Groupe de la molécule Groupe du site Groupe facteur Cs Cs D2h Ag (R) B1g (R) A'_2g (R) B3g (R) A" (Tz,Rx,Ry) —————————— XI BI u (I.R.), B2u (I.R.), B3u (I.R.) Ainsi, un niveau d'énergie A' de la molécule isolée présente quatre sous-niveaux dans le cristal dont deux sont actifs en infrarouge; un niveau d'énergie A" de la molécule donne, lui aussi, quatre sous-niveaux dans le cristal, un seul étant actif en infrarouge. b. Règles de sélection des transitions excitoniques de la phase « basse température ». — La maille est monoclinique, de groupe facteur C2h, les molécules occupant des sites sans symétrie C1. Le diagramme de corrélation est le suivant : Groupe de la molécule Groupe du site Groupe facteur Cs Ci C2h A' (Tx,Ty,Rz) -----------Ag (R) ------A Bg (R) ^===^=:=~=:::^zzr au(I.R.) A" (Tz,Rx,Ry) Bu (I.R.) Les vibrations de classe A' et A" se décomposent en quatre sous-niveaux, deux étant actifs en infrarouge. En résumé, les vibrations de classe A' présentent deux sous-niveaux actifs en infrarouge dans les deux phases cristallines tandis que les vibrations de classe A" donnent deux sous-niveaux actifs en infrarouge dans la phase «
isolée présente quatre sous-niveaux dans le cristal dont deux sont actifs en infrarouge; un niveau d'énergie A" de la molécule donne, lui aussi, quatre sous-niveaux dans le cristal, un seul étant actif en infrarouge. b. Règles de sélection des transitions excitoniques de la phase « basse température ». — La maille est monoclinique, de groupe facteur C2h, les molécules occupant des sites sans symétrie C1. Le diagramme de corrélation est le suivant : Groupe de la molécule Groupe du site Groupe facteur Cs Ci C2h A' (Tx,Ty,Rz) -----------Ag (R) ------A Bg (R) ^===^=:=~=:::^zzr au(I.R.) A" (Tz,Rx,Ry) Bu (I.R.) Les vibrations de classe A' et A" se décomposent en quatre sous-niveaux, deux étant actifs en infrarouge. En résumé, les vibrations de classe A' présentent deux sous-niveaux actifs en infrarouge dans les deux phases cristallines tandis que les vibrations de classe A" donnent deux sous-niveaux actifs en infrarouge dans la phase « basse température » mais un seul sous-niveau actif en infrarouge dans la phase « haute température ». 3.4. Résultats expérimentaux. L'étude expérimentale est consacrée à l'évolution, en fonction de la température et au voisinage du changement de phase, de deux bandes d'absorption infrarouge : l'une de classe A" à 426 cm-1, l'autre de classe A' à 925 cm-1. a. Vibration de classe A" à 426 cm 1. Ce résultat expérimental est particulièrement caractéristique; en effet ce niveau de vibration se décompose en deux sous-niveaux actifs en infrarouge dans la phase « basse température » alors qu'une seule bande d'absorption est observée dans l'autre phase. La figure 3 montre l'évolution, en fonction de la température, des deux composantes de la bande d'absorption. La variation, en fonction de la température, de l'écart en fréquence Av des deux sous-niveaux est représentée sur la figure 4. L'ajustement de ces résultats expérimentaux à une courbe de puissance de la forme Av = a (Tc T)-1 est obtenu en utilisant la méthode des moindres carrés et conduit à une valeur de voisine de 0,45; pour T = TC, Δv est nul. b. Vibration de classe A' à 925 cm-1. Ce niveau de vibration présente deux sous-niveaux