Raman scattering in crystals with order-disorder type phase transition

The theory based on the “operator” two-time Green functions formalism applied for the construction of the polarizability operators entering the general expression for the effective cross-section of Raman scattering is developed. The presented microscopic scheme is applied to the investigation of t...

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Datum:	1999
1. Verfasser:	Ivankiv, Ya.
Format:	Artikel
Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 1999
Schriftenreihe:	Condensed Matter Physics
Online Zugang:	https://nasplib.isofts.kiev.ua/handle/123456789/120544
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Zitieren:	Raman scattering in crystals with order-disorder type phase transition / Ya. Ivankiv // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 523-529. — Бібліогр.: 15 назв. — англ.

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spelling	nasplib_isofts_kiev_ua-123456789-1205442025-02-09T17:55:32Z Raman scattering in crystals with order-disorder type phase transition Раманівське розсіяння у кристалах з фазовими переходами типу лад-безлад Ivankiv, Ya. The theory based on the “operator” two-time Green functions formalism applied for the construction of the polarizability operators entering the general expression for the effective cross-section of Raman scattering is developed. The presented microscopic scheme is applied to the investigation of the ferroelectrics with hydrogen bonds. На основі формалізму двочасових функцій Гріна розвинуто теорію побудови оператора поляризованості, що входить у загальний вираз для ефективного поперечного перерізу раманівського розсіяння. Представлена мікроскопічна схема використовується для дослідження сегнетоелектриків з водневими зв’язками. 1999 Article Raman scattering in crystals with order-disorder type phase transition / Ya. Ivankiv // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 523-529. — Бібліогр.: 15 назв. — англ. 1607-324X DOI:10.5488/CMP.2.3.523 PACS: 77.84.Fa, 78.30.Ly https://nasplib.isofts.kiev.ua/handle/123456789/120544 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	The theory based on the “operator” two-time Green functions formalism applied for the construction of the polarizability operators entering the general expression for the effective cross-section of Raman scattering is developed. The presented microscopic scheme is applied to the investigation of the ferroelectrics with hydrogen bonds.
format	Article
author	Ivankiv, Ya.
spellingShingle	Ivankiv, Ya. Raman scattering in crystals with order-disorder type phase transition Condensed Matter Physics
author_facet	Ivankiv, Ya.
author_sort	Ivankiv, Ya.
title	Raman scattering in crystals with order-disorder type phase transition
title_short	Raman scattering in crystals with order-disorder type phase transition
title_full	Raman scattering in crystals with order-disorder type phase transition
title_fullStr	Raman scattering in crystals with order-disorder type phase transition
title_full_unstemmed	Raman scattering in crystals with order-disorder type phase transition
title_sort	raman scattering in crystals with order-disorder type phase transition
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1999
url	https://nasplib.isofts.kiev.ua/handle/123456789/120544
citation_txt	Raman scattering in crystals with order-disorder type phase transition / Ya. Ivankiv // Condensed Matter Physics. — 1999. — Т. 2, № 3(19). — С. 523-529. — Бібліогр.: 15 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT ivankivya ramanscatteringincrystalswithorderdisordertypephasetransition AT ivankivya ramanívsʹkerozsíânnâukristalahzfazovimiperehodamitipuladbezlad
first_indexed	2025-11-29T03:31:19Z
last_indexed	2025-11-29T03:31:19Z
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fulltext	Condensed Matter Physics, 1999, Vol. 2, No. 3(19), pp. 523–529 Raman scattering in crystals with order-disorder type phase transition Ya.Ivankiv Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 79011 Lviv, Ukraine Received October 14, 1998 The theory based on the “operator” two-time Green functions formalism applied for the construction of the polarizability operators entering the gen- eral expression for the effective cross-section of Raman scattering is de- veloped. The presented microscopic scheme is applied to the investigation of the ferroelectrics with hydrogen bonds. Key words: Raman scattering, order-disorder phase transition PACS: 77.84.Fa, 78.30.Ly 1. Raman scattering tensor The main characteristics of Raman scattering is the effective cross-section which can be expressed by Raman tensor [1] 1 2π +∞ ∫ −∞ dtei(ω1−ω2)t〈P̂ β′α′ ~k2,− ~k1 (−ω1, t)P̂ βα − ~k2, ~k1 (ω1, 0)〉 = Hβ′α′,βα(ω1ω2) (1.1) where P̂ β′α′ ~k2,− ~k1 (−ω1, t) = i t ∫ −∞ ds [ M̂β′ (~k2, t), M̂ α′ (−~k1, s) ] eiω1(s−t) (1.2) is the polarizability operator; ω1, ω2 are frequencies; ~k1, ~k2 are wave vectors of the incident and the scattered light, respectively; α, β, α ′, β ′ are coordinate indices. Here, M̂β(~k) = ∑ n ei ~k ~RnM̂β n are the Fourier transforms of the dipole momenta operators of the unit crystal cell (n is the cell number). ~̂Mn = ∑ k (e ~̂Dnk + zk ~Unk) ≡ ∑ k ~̂Mnk, (1.3) c© Ya.Ivankiv 523 Ya.Ivankiv where ~̂Dnk are the electronic operators; ~Unk are the ionic displacements. The sum- mation in (1.3) is fulfilled over ionic species in the cell (z k is the ionic charge, k is the ionic type index); in (1.2) the operators are given in Heisenberg representation. In the semiphenomenological description of the Raman scattering the polarizability operators are presented as the expansions in vectors of ionic displacements. Equa- tion (1.2) enables us to obtain the explicit form for the operator P̂ βα −~k2,~k1 ; its specific features are determined by the Hamiltonian of the system. In the following we use the polarizability operator presented in terms of the “operator” Green functions [2]: ĜÂB̂(t, s) ≡ {{ Â(t)\|B̂(s) }} = −iθ(t− s) [ Â(t), B̂(s) ] . (1.4) where Â(t), B̂(s) are the operators written in Heisenberg representation. The sta- tistical averaging of the ĜÂB̂(t, s) operator leads to the retarded two-time Green function 〈〈 Â(t)\|B̂(s) 〉〉 . After the Fourier transformation we have h̄ω1 {{ Â\|B̂ }} ω1,ω2 = h̄ 2π [ Â, B̂ ] ω1−ω2 + {{[ Â, Ĥ ] \|B̂ }} ω1,ω2 . (1.5) In the presence of small parameters, the equation of the (1.5) type makes it possi- ble to construct the operator expansions for the Fourier transformants {{ Â\|B̂ }} ω1,ω2 . By means of (1.3) and (1.4) the expression (1.2) for the polarizability operator may be presented as follows P̂ β′α′ ~k2,−~k1 (−ω1, t) = −2π ∑ nn′ ∑ ll′ ei ~k2 ~Rne−i~k1 ~Rn′ · +∞ ∫ −∞ dω′ 1 {{ M̂ β′ nl \|M̂ α′ n′l′ }} ω′ 1 ,−ω1 e−i(ω1+ω′ 1 )t. (1.6) Thus, the problem of the explicit expression for the polarizability operator is reduced to determining the Fourier transformant {{ M̂ \|M̂ }} ω′ 1 ,−ω1 . 2. Raman scattering on internal vibrations of ionic groups in crystals of KH2PO4 In the problem of Raman scattering by the crystals with a complicated structure, we dwell upon the scattering by the internal vibrations of ionic groups which can be in different configurations. Configurations are considered to mean different loca- tion of the structural elements which form an ionic group including the structural elements which can be ordered at phase transition. In the case when the vibrations of internal and external (translational and librational) modes differ essentially they may be considered independently. Consider the case when characteristics of vibrations and related electron states depend on the configurations of some of the ionic complexes in the unit cell. Then 524 Raman scattering in crystals with order-disorder type phase transition the Hamiltonian is written: Ĥ = ∑ nk ∑ i ( ∑ si λ (i) ksi X̂sisi nk ) N ii nk − ∑ nk να ∑ i    ∑ sis ′ i θ sis ′ i kναX̂ sis ′ i nk U ′iα nkν   N ii nk + Ĥph. (2.1) Here, N ii nk is the occupation number of the i-th configuration of the complex (n,k); X̂ sis ′ i nk are the Hubbard operators (see [3]) acting based on the electron states of the groups; λ (i) ksi denotes energy of the state si of the group k. We confine ourselves to the account of electron-vibrational interaction within individual ionic groups; U ′ nkv are the vectors of ion displacement from equilibrium positions (ν is the number of the ion in a group); θ sis ′ i kνα is the interaction matrix. Beside the terms in (2.1) the total Hamiltonian includes the interaction energy of ionic groups (dependent on their configurations) which is responsible for the transition to the order state (in the case of the crystal possessing the order-disorder type phase transition). Expression for the scattering cross-section can be written in terms of the Green function constructed of the polarizability operators [4]. In order to describe the one phonon Raman scattering by internal vibrations of ionic groups in the ferroelectric crystals with hydrogen bonds we apply configura- tional averaging and the approach of mean field type. As an example we consider the crystals of KH2PO4 (KDP)-type. In such crystals the various configurations of HnPO4 groups (n=0, 1,...,4) can be realized depending on the character of proton location in the double potential wells on the hydrogen bonds. The most probable between them are the configurations with two protons in the neighbourhood of the given complex PO4 [5,6]. Confine ourselves to the account for such configurations only; their number is six, more preferable energetically are two of them with protons near the upper or lower (with respect to the crystal c-axis) oxygen atoms (figure 1). We consider the case T > Tc. i=1 i=2 i=3 i=5i=4 i=7 i=6 i=8 i=12i=11i=10i=9 l=2 l=1 Figure 1. H2PO4 configurations for two sublattices of PO4 groups in KDP crystal (projection to xy-plane). 525 Ya.Ivankiv For isolated PO4 complex of the point symmetry Td the internal vibrations are classified in terms of irreducible representations A1 (ν1 frequency); E (ν2 frequency); F2 (ν3 and ν4 frequencies). In the case of configurations i=1,2 (l=1), i=7,8 (l=2) they reduce to the modes of the symmetry A1, B1, B2; 2A1, 2B1, 2B2; A1, B1, B2, 4E respectively, and for configurations i=3,...,6 (l=1) and i=9,...,12 (l=2) into the modes with symmetry A1, B1, B2, 2E; 2A1, 2B1, 2B2, 4E; 3A1, 3B1, 3B2, 6E. Splitting of the frequencies ν2, ν3, ν4 is insignificant when symmetry lowers and its value is not established experimentally, although it is quite possible that the splitting of the mode ν3 observed in [7] is configurational. The electronic spectrum of PO4 groups is more sensitive to the location of the nearest proton. The configurationally averaged expression for the effective Raman scattering cross-section contains contributions from different configurations of HnPO4 groups. If the correlation corrections are neglected the specific weights of such contributions are proportional to the mean occupation numbers 〈N ii l 〉 and to the scattering tensor elements and are determined by the proton distribution on the hydrogen bonds [8]. The possible situations for H2PO4 groups with account of the contributions of different configurations are presented on table 1; the symmetry of phonon modes arising due to both configurational and Davydov splitting is shown. Since H2PO4 configurations possess lower symmetry as compared to the S4 sym- metry of the position of PO4 groups in the crystal, it is possible that the lines may appear which are forbidden by the site symmetry. In particular, such a situation takes place at the (zz) - scattering (e1oz, e2oz) [9]. As it follows from table 1, at a given scattering geometry, besides the lines with ν1 and ν2 frequencies, the lines ν3 and ν4 should appear too (that are missing both for Td symmetry and for S4). This result agrees with experimental data [10]; ν3 and ν4 lines that are observed in Raman scattering spectra are considerably weaker in comparison with the principal ones and appear at the temperatures near Tc only. The lines forbidden by the symmetry of PO4 groups may appear also in the cases of (xy) – scattering (ν1 and ν2 frequencies), of (xz) – or (yz) – scattering (ν1 and ν2 frequencies) and of (xx) – or (yy) – scattering (ν3, ν4 frequencies). The former of these cases for ν1 frequency is investigated experimentally in [10]. The second one is of interest, because “lateral” H2PO4 configurations contribution are only (i=3,...6, 9,...,12). This is investigated experimentally in [11]. One of the peaks of Raman intensity, observed in this work, may correspond to the frequency ν1. However authors of this work connect this peak to the splitted mode ν3 and think, that these “lateral” configurations of the symmetry C1 do not manifest itself in the Raman scattering spectra. As to the possible usage of the proton ordering model for describing the ferro- electric phase transition in crystals of KDP group the attention should be paid to a more thorough experimental investigation of the cases (see table 1) when “lateral” configurations of protons, possessing C1 symmetry, appear in the scattering since the model given allows for the existence of this kind of configuration (and configurations with different proton number). This problem is of importance in connection with the possibility of another interpretation of the phase transition in KH2PO4 which 526 Raman scattering in crystals with order-disorder type phase transition Table 1. Scattering geometry of the configurationally splitted modes (T > Tc case). First column is the Davydov splitting. Fourth column is the configurational splitting. Symbols * denote the presence of non-zero components of scattering tensors; ⊗ correspond to the scattering in the case of isolated PO4 complexes of Td symmetry. Two last columns show the symmetry of modes arising from different configurations. D2α S4 Tα D2α (xx) (yy) (zz) (xy) (xz) (yz) i=1, i=3,..., { k x̄ } {x̄} {x̄} { k(i) x̄i } 2,7,8 6,9, ...,12 A1 A A1 2A1 ⊗ ⊗ ⊗ A1 A1 + 2A2 A2 A2 A2 2B1 * * B1 B1 2B2 * B2 B2 2E * * 2E 2E E F2 2A1 * * * 2A1 2A2 2A2 2B1 * * 2B1 2B2 * 2B2 8E ⊗ ⊗ 4E 4E B1 B F2 2A1 * * * A1 A1 + 2A2 A2 A2 B2 2B1 * * B1 B1 2B2 ⊗ B2 B2 2E * * 2E A1 A E 2A1 ⊗ ⊗ ⊗ A1 A1 + 2A2 A2 A2 A2 2B1 * * B1 B1 2B2 * B2 B2 2E * * 2E B1 B E 2A1 * * * A1 A1 + 2A2 A2 A2 B2 2B1 ⊗ ⊗ B1 B1 2B2 * B2 B2 2E * * 2E 527 Ya.Ivankiv has been discussed recently – as a transition related with ordering of H2PO4 groups with protons near the “upper” or the “lower” oxygen ions (with C2 symmetry). This point of view is presented in [10, 12]. We note that practical realization of such a model causes some difficulties due to the violations in the case of disordered state of the “ice rule” (one proton on the bond and two protons near PO4 group). On the other hand, the idea of ordering H2PO4 groups was not confirmed in the experi- ments for investigating the nuclear quadrupole resonance in KH2PO4 [13] as well as the isotopic frequency shift in K(DxH1−x)2PO4 [14]; the data obtained support the proton ordering model. Finally, the approach proposed in this paper may be applied in the analysis of Raman scattering in other ferroelectric crystals with hydrogen bonds as well as in the crystals of another type with the phase transition of the order-disorder type. In the framework of the theory developed the interactions between electrons and lattice anharmonisms [15] may be taken into account. More detailed investigation of phonon spectrum of the crystals considered is of interest. In particular, the systems of equations for the averaged phonon Green functions determining the shapes of the lines in Raman scattering spectra are obtained in the coherent potential approximation [4]. Configurationally splitted components of the Raman scattering lines are studied using a simple model. For KH 2PO4 and CsH2PO4 type ferroelectrics, the possibility to observe experimentally the configu- rational splitting is discussed. References 1. Cowley R.A. The lattice dynamics of an anharmonic crystal. // Adv. Phys., 1963, vol. 12, No. 18, p. 421–481. 2. Stasyuk I.V., Ivankiv Ya. The specific features of Raman scattering in the order- disorder type ferroelectrics. – In: “Sovremennye problemy statisticheskoj fiziki. Trudy Vsesoyuznoj Konferencii, L’vov, 3–5 fevralja 1987, vol. 2”. Kyiv, “Naukova Dumka”, 1989, p. 261–265 (in Russian). 3. Slobodyan P., Stasyuk I.V. The diagram technique for Hubbard operators. // Theor. Math. Phys., 1974, vol. 19, p. 423–429 (in Russian). 4. Ivankiv Ya., Stasyuk I.V. Configurational splitting in Raman scattering spectra in crystals with order-disorder type phase transitions. Preprint of the Institute of theo- retical physics, ITP–88–80R, Kyiv, 1988, 24 p. (in Russian). 5. Slater Y.C. Theory of the transition in KH2PO4. // J. Chem. Phys., 1941, vol. 9, p. 16–33. 6. Ishibashi Yo., Suzuki I. Detailed analyses and an extension of the Slater theory of the phase transitions in KH2PO4. // J. Phys. Soc. Jap., 1984, vol. 53, No. 1, p. 244–249. 7. Tanaka H., Tokunaga M., Tatsuzaki I. Internal modes and the local symmetry of PO4 tetrahedra in K(H1−xDx)2PO4 by Raman scattering. // Sol. Stat. Commun., 1984, vol.149, No. 2, p. 153–155. 8. Stasyuk I.V., Ivankiv Ya. Raman light scattering in crystals with ordered structure elements. Preprint of the Institute of theoretical physics, ITP–87–57R, Kyiv, 1987, 25 p. (in Russian). 528 Raman scattering in crystals with order-disorder type phase transition 9. Tokunaga M., Tominaga Y., Tatsuzaki I. Order-disorder model of PO4 dipoles in RDP based on recent Raman spectroscopic studies. // Ferroelectrics, 1985, vol. 63, No. 1/4, p. 171–178. 10. Tominaga Y., Tokunaga M., Tatsuzaki I. Scattering wave vector dependence of Raman intensity of the ν1 internal mode in KH2PO4. // Sol. Stat. Commun., 1985, vol. 54, No. 11, p. 979–980. 11. Tominaga Y., Tokunaga M., Tatsuzaki I. Dynamical mechanism of ferroelectric phase transition in KH2PO4 by Raman scattering study. // Jap. J. Appl. Phys., 1985, Pt. 1, vol. 24, Suppl. 24–2, p. 917–919. 12. Tokunaga M. Order-disorder model of PO4 dipoles for KDP and ADP. // Jap. J. Appl. Phys., 1985, Pt. 1, vol. 24, Suppl. 24–2, p. 908–910. 13. Blinc R. 17O NQR and the mechanism of the phase transition in KH2PO4 type H- bonded systems. // Z. Natur forsch., 1986, vol. 41a, p. 249–255. 14. Anachkova E., Socvatinova I. Isotopic shift and mode interaction in K(DxH1−x)2PO4 crystals. // Phys. status solidi (b), 1985, vol. 131, No. 2, p. K101–K105. 15. Ivankiv Ya. Polarizability operator of the dielectric crystals. // Fiz. electronika, 1985, No. 30, p. 43–47 (in Russian). Раманівське розсіяння у кристалах з фазовими переходами типу лад-безлад Я.Іванків Інститут фізики конденсованих систем НАН Укpаїни, 79011 Львів, вул. Свєнціцького, 1 Отримано 14 жовтня 1998 р. На основі формалізму двочасових функцій Гріна розвинуто теорію побудови оператора поляризованості, що входить у загальний ви- раз для ефективного поперечного перерізу раманівського розсіян- ня. Представлена мікроскопічна схема використовується для до- слідження сегнетоелектриків з водневими зв’язками. Ключові слова: раманівське розсіяння, фазові переходи типу лад-безлад PACS: 77.84.Fa, 78.30.Ly 529 530

Raman scattering in crystals with order-disorder type phase transition

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