Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions

A system of equations for the averaged phonon Green function determining the shapes of the lines in Raman scattering spectra is obtained in the coherent potential approximation. Conﬁgurationally split components of the Raman scattering are studied within a simple model. For the KH₂PO₄ and the CsH...

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Дата:	1999
Автори:	Stasyuk, I.V., Ivankiv, Ya.L.
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Мова:	English
Опубліковано:	Інститут фізики конденсованих систем НАН України 1999
Назва видання:	Condensed Matter Physics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/119929
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Цитувати:	Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions / I.V. Stasyuk, Ya.L. Ivankiv // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 155-171. — Бібліогр.: 18 назв. — англ.

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spelling	irk-123456789-1199292017-06-11T03:02:43Z Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions Stasyuk, I.V. Ivankiv, Ya.L. A system of equations for the averaged phonon Green function determining the shapes of the lines in Raman scattering spectra is obtained in the coherent potential approximation. Conﬁgurationally split components of the Raman scattering are studied within a simple model. For the KH₂PO₄ and the CsH₂PO₄-type ferroelectrics the possibility to observe the conﬁgurational splitting experimentally is discussed. В наближенні когерентного потенціалу одержана система рівнянь для усередненої фононної функції Гріна, що визначає профілі ліній в спектрах Рамана. На прикладі простої моделі проведено дослідження конфігураційно розщеплених компонент раманівських ліній. Для сегнетоелектриків типу KH₂PO₄ та CsH₂PO₄ обговорюється питання про можливий експериментальний вияв конфігураційного розщеплення. 1999 Article Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions / I.V. Stasyuk, Ya.L. Ivankiv // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 155-171. — Бібліогр.: 18 назв. — англ. 1607-324X DOI:10.5488/CMP.2.1.155 PACS: 77.84.Fa, 78.30.Ly http://dspace.nbuv.gov.ua/handle/123456789/119929 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A system of equations for the averaged phonon Green function determining the shapes of the lines in Raman scattering spectra is obtained in the coherent potential approximation. Conﬁgurationally split components of the Raman scattering are studied within a simple model. For the KH₂PO₄ and the CsH₂PO₄-type ferroelectrics the possibility to observe the conﬁgurational splitting experimentally is discussed.
format	Article
author	Stasyuk, I.V. Ivankiv, Ya.L.
spellingShingle	Stasyuk, I.V. Ivankiv, Ya.L. Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions Condensed Matter Physics
author_facet	Stasyuk, I.V. Ivankiv, Ya.L.
author_sort	Stasyuk, I.V.
title	Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions
title_short	Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions
title_full	Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions
title_fullStr	Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions
title_full_unstemmed	Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions
title_sort	conﬁgurational splitting in raman scattering spectra of crystals with the order-disorder type phase transitions
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1999
url	http://dspace.nbuv.gov.ua/handle/123456789/119929
citation_txt	Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions / I.V. Stasyuk, Ya.L. Ivankiv // Condensed Matter Physics. — 1999. — Т. 2, № 1(17). — С. 155-171. — Бібліогр.: 18 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT stasyukiv configurationalsplittinginramanscatteringspectraofcrystalswiththeorderdisordertypephasetransitions AT ivankivyal configurationalsplittinginramanscatteringspectraofcrystalswiththeorderdisordertypephasetransitions
first_indexed	2025-07-08T16:55:38Z
last_indexed	2025-07-08T16:55:38Z
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fulltext	Condensed Matter Physics, 1999, Vol. 2, No. 1(17), p. 155–171 Configurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions I.V.Stasyuk, Ya.L.Ivankiv Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine, 1 Svientsitskii Str., 290011 Lviv, Ukraine Received June 23, 1997 A system of equations for the averaged phonon Green function determin- ing the shapes of the lines in Raman scattering spectra is obtained in the coherent potential approximation. Configurationally split components of the Raman scattering are studied within a simple model. For the KH2PO4 and the CsH2PO4-type ferroelectrics the possibility to observe the configura- tional splitting experimentally is discussed. Key words: Raman scattering, configurational splitting, order-disorder type phase transitions, ferroelectrics PACS: 77.84.Fa, 78.30.Ly 1. Introduction A study of spectra of the Raman scattering of light in KH2PO4 (KDP) crys- tals is of specific interest for the discussion of phase transitions in these systems. Developments in the dynamical theory of phase transitions in KDP crystals have made the concept of a proton-phonon soft mode widely used. A frequency range related to phase transitions has become a subject of many investigations [1,2], because low-frequency lattice modes are most closely related to transition phenomena. Besides, it is of interest to study phonon modes at even higher frequencies, in particular those which are related to the internal oscillations of phonon groups. In crystals with order-disorder transitions, the specific features of these oscillations depend on the states or orientations of the structure elements that are ordered (the ionic groups and protons on hydrogen bonds). A new model for phase transitions in KDP crystals has been proposed [3-7] on the basis of the experimental data for ν3 and ν4 modes in scattering geometries for which the observation of such modes is forbidden by the selection rules for site symmetry S4 [8]. The model assumes that the main role in the transition belongs to the ordering c© I.V.Stasyuk, Ya.L.Ivankiv 155 I.V.Stasyuk, Ya.L.Ivankiv of PO4 dipoles which already existed in a high temperature phase rather than to the ordering of protons on H-bonds, i.e. the softening of a related proton-phonon mode. However, the application of the model encounters difficulties because the “ice” rule (one proton on bond and two protons close to the PO4 group) is violated. On the other hand, the idea about H2PO4 groups ordering is not confirmed by the NMR experimental data for KH2PO4 [9]. Thus, a clear interpretation of the available data has not been achieved. The results obtained can be explained satisfactorily even within the framework of the proton ordering model. It was shown in [10] that the description of the Raman spectra of internal oscillations in KH2PO4 clusters can be made if different realizations of proton configurations are taken into account. The model can easily be extended to allow for the short-range proton correlations [11]. In this respect it is interesting to study the peculiarities of phonon spectra of KDP crystals in more detail. In particular, the functions of spectral densities which determine the shape of lines in the Raman spectra are of importance. The subject of our study are the crystals in which phase transitions are related to the ordering of protons in one-dimensional chain structures. 2. A system of equations for the averaged Green function. Co- herent potential approximation An effective cross-section which is the main characteristic of the Raman scat- tering can be expressed as [12,13] d2σ dΩdω2 = 1 (4πε0)2 √ ε2 ε1 ω3 2ω1 ~2c4 ∑ αβ α ′ β ′ e1αe2βe1α′ e2β′ (2.1) × 1 2π +∞∫ −∞ dt ei(ω1−ω2)t〈P̂ β ′ α ′ k2,−k1 (−ω1, t)P̂ βα −k2,k1 (ω1, 0)〉, where e1, e2 are polarization vectors; ω1, ω2 are frequencies and k1, k2 are wave vectors of the incident and scattered light, respectively; α, β, α′, β ′ are coordinate indices; ε1,2 ≡ ε(ω1, ω2); P̂ is a polarizability operator: P̂ β′α′ k2,−k1 (−ω1, t) = ∑ nn′ ∑ ee′ eik2Rne−ik1Rn′ P̂ β′α′ ne,n′e′(−ω1, t), (2.2) (n, n′ are cell numbers; l, l′ are ion species in the cell) P̂ β′α′ ne,n′e′(−ω1, t) = δnn′δee′ +∞∫ −∞ dω′ 1P̃ β′α′ ne (ω′ 1,−ω1)e −i(ω1+ω′ 1 )t. (2.3) 156 Configurational splitting An expression for a scattering cross-section can be written in terms of the Green function constructed by the polarizability operators [10]: 2π ~ 〈〈P̃ βα l1 (ω2, ω1)\| P̃ +β′α′ l (ω2, ω1)〉〉ω,q = ~ 2e4〈Ĝβα,β′α′ l1l (ω2, ω1)〉ω,q (2.4) = ~ 2e4 ∑ iκi ∑ jκj R (j)κj l1βα (ω2, ω1) R ∗(i)κi lβ′α′ (ω2, ω1)〈ĝ (ji)κjκi l1l (ω,q)〉, where R (j)κj l1αβ (ω2, ω1) is a scattering tensor of the κj-mode of the l1-th ion complex having the j-th configuration; ĝ is the phonon Green function corresponding to internal vibrations of ion groups. The Green function ĝ (ji)κjκi l1l can be determined from equation [10]: ĝ = χ̂+ χ̂Φĝ. (2.5) Here, χ̂ (ij)κiκj nk = g κiκj ok δij B̂ ii nk is a phonon Green function of an isolated ion complex; g κiκj 0k = δ κiκj ω2 − Ω2 kκj , B̂ii nk is an occupation number of the i-th configuration of the (n,k)-th complex; Φ is a matrix of interactions between internal vibrations of the complexes. According to (2.4), the effective cross-section and the Raman scattering tensor are related to the averaged over configurations Green function (2.5). Averages of products of functions χ̂ can be expanded in cumulants. Thence, function 〈ĝ〉 is expressed in terms of full irreducible parts Σ [10] 〈ĝ〉 = (1− Σ Φ)−1Σ (2.6) or 〈g〉−1 = Σ−1 − Φ (q). Certain summation of diagrams for the (self-energy) irreducible part corre- sponds to equation (2.6). We introduce the diagrammatic notations: ✲ for χ̂nk, for Φkk′(nn ′), ✄✂ �✁✲ for averaging. Ovals surrounding n lines ✲ correspond to cumulants of the n-th order. An expression for Σ contains only irreducible diagrams (the diagrams which do not break down into separate parts if one interaction line is cut) Σ = ✲✞✝ ☎✆+ ✲ ✲ ✲✞✝ ☎✆✞✝ ☎✆ + ✲ ✲ ✞✝ ☎✆✞✝ ☎✆ ✲✞✝ ☎✆ ✲ 157 I.V.Stasyuk, Ya.L.Ivankiv + ✲ ✲ ✲✞✝ ☎✆✞✝ ☎✆✲ + ✲ ✲ ✲ ✲ ✲✞✝ ☎✆✞✝ ☎✆✞✝ ☎✆ + ✲ ✲ ✲ ✲ ✲✞✝ ☎✆✞✝ ☎✆ + ✲ ✲ ✲ ✲ ✲✞✝ ☎✆✞✝ ☎✆ ✞✝ ☎✆ + . . . In the spirit of a single-site approximation, the correlation between χnk is taken into account only if the site indices coincide. Then, we can sum up series (2.6) using a coherent potential approximation. Let us pick in Σ the diagrams the beginnings and ends of which correspond to the same site. In these diagrams, we separate out the ovals corresponding to the same site. As J we denote the sum of diagrams which start and end with lines of interaction on a given site and do not contain ovals with this site. In diagrammatic notations Jkk(nn) : We can calculate Jkk(nn) making use of the method described in [14,15]. Hence, we have Σ ⇒ Σ̃ = ✲✄✂ �✁+ ✲ ✲✄✂ �✁+ ✲ ✲ ✲ ✲✄✂ �✁✄✂ �✁✄✂ �✁ + ✲ ✲ ✲✄✂ �✁+ ✲ ✲ ✲✄✂ �✁✄✂ �✁ + ✲ ✲ ✲ ✲✄✂ �✁✄✂ �✁ + ✲ ✲ ✲ ✲✄✂ �✁✄✂ �✁ + ✲ ✲ ✲ ✲ ✄✂ �✁✄✂ �✁ + ✲ ✲ ✲ ✲✄✂ �✁✄✂ �✁ + ✲ ✲ ✲ ✲✄✂ �✁+ . . . (within J3). In a single site approximation we get the following equation for the Green function 〈ĝk〉: 〈ĝk〉∗ = Σ̃k + Σ̃kJkΣ̃k + Σ̃kJkΣ̃kJkΣ̃k + · · · (2.7) = Σ̃k + Σ̃kJk〈ĝk〉∗. 158 Configurational splitting In this approximation the self-energy part reads Σ̃k = 〈ĝk〉∗(1 + Jk〈ĝk〉∗) −1. (2.8) In the calculation of the Green function 〈ĝk〉 within a single site approximation, the correlation between unperturbed Green functions of the internal vibrations of the complex (n, k) having the i-th configuration is taken into account only if the site indices coincide. In this case 〈ĝ (ij)κiκj nk,nk 〉∗ = = 〈χ̂ (ij)κiκj nk 〉+ ∑ j1κj1 j2κj2 〈χ̂ (ij2)κiκj2 nk J (j2j1)κj2 κj1 kk (nn)χ̂ (j1j)κj1 κj nk 〉 (2.9) + ∑ j1κj1 j2κj2 ∑ j3κj3 j4κj4 〈χ̂ (ij2)κiκj2 nk J (j2j1)κj2 κj1 kk (nn)χ̂ (j1j4)κj1 κj4 nk J (j4j3)κj4 κj3 kk (nn)χ̂ (j3j)κj3 κj nk 〉 + · · · , where 〈χ̂ (ij)κiκj nk 〉 = δijδκiκj ω2 − Ω2 kκi 〈B̂ii nk〉, (2.10) 〈χ̂ (ij2)κiκj2 nk χ̂ (j1j)κj1 κj nk 〉 = δij2δκiκj2 δj1jδκj1 κj (ω2 − Ω2 kκi )((ω2 − Ω2 kκj ) 〈B̂ii nk B̂jj nk〉. (2.11) Taking into account the fact that 〈B̂ii nkB̂ jj nk〉 = 〈B̂ii nk〉δij, we find that 〈ĝ (ij)κiκj nk,nk 〉∗ = δij〈B̂ ii nk〉[(1− ĝokJ ii k (nn)) −1ĝok] κiκj . (2.12) Let us now evaluate Jk. We introduce the matrix J̃k′ which is calculated without the constraint that ovals with a given site are absent: J̃ = ✒✑ ✓✏ Σ̃ k k′ + ✒✑ ✓✏ ✒✑ ✓✏ Σ̃ Σ̃ k′ k′′ + ✒✑ ✓✏ ✒✑ ✓✏ ✒✑ ✓✏ Σ̃ Σ̃ Σ̃ + . . . Supposing that Σ̃ may correspond to the initial site as well, we can express J̃ in the form J̃ (j1j2)κj1 κj2 kk (nn) = 1 N ∑ q [∑ j′j′′ ∑ κj′κj′′ Φ (j1j′)κj1 κj′ kk′ (q) Σ̃ (j′j′′)κj′κj′′ k′ ×Φ (j′′j2)κj′′κj2 k′k (q) + (Φ Σ̃ Φ Σ̃ Φ) (j1j2)κj1 κj2 kk + . . . ] . (2.13) In matrix notations J̃k = 1 N ∑ q [ Φ(q) ( 1− Σ̃ Φ(q) ) −1] kk . (2.14) 159 I.V.Stasyuk, Ya.L.Ivankiv On the other hand, J = ✍✌ ✎☞ Σ̃ k′ + ✍✌ ✎☞ ✍✌ ✎☞ Σ̃ Σ̃ + . . . = (2.15) (here Σ̃ does not correspond to the initial site.) Therefore, J̃ can be presented as a series J̃ = + ❥̃Σ + ❥̃Σ ❥̃Σ + . . . (we use the notations of (2.15), or J̃ (j1j2)κj1 κj2 kk (nn) = J (j1j2)κj1 κj2 kk (nn) (2.16) + ∑ j3j4 κj3 κj4 J (j1j3)κj1 κj3 kk (nn) Σ̃ (j3j4)κj3 κj4 k J (j4j2)κj4 κj2 kk (nn) + . . . Hence, in a matrix form Jk = (1 + J̃kΣ̃k) −1J̃k. (2.17) Let us introduce an auxiliary function F̃k = 1 N ∑ q [〈ĝ〉−1 q ]kk. (2.18) Then equations (2.14) and (2.17) can be rewritten as Jk = Σ̃−1 k − F̃−1 k , (2.19) F̃k = 1 N ∑ q {[Σ̃−1 − Φ(q)]−1}kk. (2.20) Equations (2.8), (2.12), (2.19) and (2.20) comprise a system of equations for the averaged Green function 〈ĝ〉q. 3. Calculation of the averaged Green function for a simple model Let us consider, for example, crystals of the KH2PO4 and the CsH2PO4 families. Since the frequencies ν1, ν2, ν3, ν4 of internal vibrations of an isolated PO4 complex are fairly separated from each other, we can treat the coupled vibrations resulting from modes A1, E, F2, F2 independently. The κi takes only one value (pk = 1) in the case of A1 mode, two values (pk = 2) in the case of E mode and three values (pk = 3) in the cases of F2 modes. 160 Configurational splitting Let us consider, as an example, the case of A1 mode. We restrict our consider- ation to a simple model. We assume that there is only one group PO4 per unit cell (k = 1), and only two proton configurations around this group are allowed (r = 2). Configurations with protons close to upper or lower oxygen atoms (with respect to the c-axis of the crystal) (the case i = 1, 2 [9,10]) have the lowest energy. Then, Σ̃k = Σ̃, and Σ̃−1 = 〈ĝ〉−1 ∗ + J . (3.1) Let us denote the diagonal elements of the matrices J and Φ(q) as J11 and Φ11(q), respectively, and their non-diagonal elements as J12 and Φ12(q). Equations (2.8), (2.12), (2.19), (2.20) are transformed into the system of equations Σ̃ = g̃ D ( 1 + g̃(J11 − J12) 0 0 1 + g̃(J11 + J12) ) , (3.2a) 〈ĝ〉∗ = ( g̃ 0 0 g̃ ) , g̃ = 〈B11〉(1− g0J11) −1g0, (3.2b) F̃ = ( F + F ′ 0 0 F − F ′ ) = 1 N ∑ q ( g̃−1 + (J11 + J12)− (Φ11(q) + Φ12(q)) 0 0 g̃−1 − (J11 − J12)− (Φ11(q)− Φ12(q)) ) , (3.2c) J = ( J11 + J12 0 0 J11 − J12 ) = ( g̃−1 + J11 + J12 0 0 g̃−1 + J11 − J12 ) − − ( F + F ′ 0 0 F − F ′ ) −1 , (3.2d) where D = (1 + g̃J11) 2 − (g̃J12) 2. (3.3) From (3.2c) we get g̃ = 1 N ∑ q 1 (g̃−1 + J11 − Φ11(q))∓ (J12 − Φ12(q)) . (3.4) Within an isotropic crystal approximation, we take Φ12(q) = Φ′ 0/Φ0Φ11(q). Re- placing the sum in (3.4) by an integral with a half-elliptic density of states, ρ(t) = 2 πΦ2 0 √ Φ2 0 − t2, (3.5) we get the following expressions for the elements of matrix J : J11 = 1 2 g−1 0 [ 1− √ 1− g−1 0 (Φ2 0 +Φ ′2 0 )〈B〉 ] , (3.6) J12 = g−1 0 Φ0Φ ′ 0 Φ2 0 +Φ ′2 0 [ 1− √ 1− g−1 0 (Φ2 0 +Φ ′2 0 )〈B〉 ] . (3.7) 161 I.V.Stasyuk, Ya.L.Ivankiv Here we take into account the fact that 〈B11〉 = 〈B22〉 = 〈B〉, which corresponds to temperatures above Tc. Let us consider now the Green function (2.4) built on polarizability operators. Since the wave vector hardly changes at the Raman scattering, we assume that q ≈ 0. As one can see from (2.1), the effective cross-section of the Raman scattering is proportional to the imaginary part of the Green function 〈Ĝ〉. Using relation (2.4) we find that Im〈Ĝ〉q=0 = Im[R(1)R∗(1)〈q〉11q=0 +R(1)R∗(2)〈q〉12q=0 (3.8) +R(2)R∗(1)〈q〉21q=0 +R(2)R∗(2)〈q〉22q=0] for the model under consideration. As one can easily show, 〈q〉11q=0 = 1 2 ( 1 A+ C + 1 A− C ) , (3.9) 〈q〉12q=0 = 1 2 ( 1 A+ C − 1 A− C ) , (3.10) here A and C are elements of the matrix 〈ĝ〉−1 〈ĝ〉−1 = ( A C C A ) . Making use of (3.9) and (3.10) and taking into account the tensor symmetry which corresponds to the point symmetry of a crystal, we can write Im 〈Ĝ〉q=0 as for a KH2PO4 crystal [9,10]: (zz)-scattering Im〈Ĝ〉q=0 = 2\|g3\| 2Im 1 A+ C ; (3.11) (xy)-scattering Im〈Ĝ〉q=0 = 2\|h\|2Im 1 A− C ; (3.12) (xx) and (yy)-scattering Im〈Ĝ〉q=0 = [1 2 ( \|g1\| 2 + \|g2\| 2 ) + g ′ 1g ′ 2 + g ′′ 1g ′′ 2 ] Im 1 A+ C + [1 2 ( \|g1\| 2 + \|g2\| 2 ) − g ′ 1g ′ 2 − g ′′ 1g ′′ 2 ] Im 1 A− C . (3.13) for a quasi one-dimensional CsH2PO4 crystal1 (xx)-scattering Im〈Ĝ〉q=0 = 2\|c ′ 11\| 2Im 1 A+ C ; (3.14) 1Symmetry analysis of the Raman scattering from the CsH2PO4-type crystal when the config- urational splitting is taken into account is given in the Appendix 162 Configurational splitting (analogous expressions were also obtained for (yy), (zz), (xz)-scattering, with dif- ferent coefficients). (xy)-scattering Im〈Ĝ〉q=0 = 2\|c ′ 12\| 2Im 1 A− C . (3.15) A similar, but with a different coefficient, expression was obtained for the case of (yz)-scattering. Expressions (3.11)-(3.15) contain not only the quantities Im 1/(A + C) and Im 1/(A − C) but also the elements of the scattering tensor as “weight” factors defining the contribution of each item. Let us consider, for instance, 1/(A + C). With the help of (3.1), (3.6) and (3.7) we can rewrite its imaginary part as Im 1 A+ C = − (1 2 − Φ0Φ ′ 0 Φ2 0 + Φ ′2 0 )√1 2 (Φ2 0 +Φ ′2 0 )− (ω2 − Ω2)2 × [(3 2 + Φ0Φ ′ 0 Φ2 0 +Φ ′2 0 ) (ω2 − Ω2)− Φ0 − Φ ′ 0 ]2 (3.16) + (1 2 − Φ0Φ ′ 0 Φ2 0 +Φ ′2 0 )2[1 2 (Φ2 0 + Φ ′2 0 )− (ω2 − Ω2)2 ] −1 . For 1/(A− C) we have Im 1 A− C = − (1 2 + Φ0Φ ′ 0 Φ2 0 + Φ ′2 0 )√1 2 (Φ2 0 + Φ ′2 0 )− (ω2 − Ω2)2 × [(3 2 − Φ0Φ ′ 0 Φ2 0 + Φ ′2 0 ) (ω2 − Ω2)− Φ0 + Φ ′ 0 ]2 (3.17) + (1 2 + Φ0Φ ′ 0 Φ2 0 + Φ ′2 0 )2[1 2 (Φ2 0 + Φ ′2 0 )− (ω2 − Ω2)2 ] −1 . The region where the imaginary parts differ from zero is defined by the in- equalities ωlim(2) < ω < ωlim(1) where ωlim(1,2) = √ Ω2 ± √ 1 2 (Φ2 0 + Φ ′2 0 ). (3.18) As numerical calculations show, functions (3.16) and (3.17) have sharp peaks in the vicinity of ωlim(1). In this region the imaginary part of 1/(A + C) can be approximated as Im 1 A+ C = az a1 + a2z2 , (3.19) where z = √ ω2 lim(1) − ω2 is small. The coefficients a, a1, a2 are expressed in terms of the energies of interactions between internal vibrations of PO4 groups having different configurations. 163 I.V.Stasyuk, Ya.L.Ivankiv Using formula (3.19) and an analogous to it one for Im 1/(A − C) with the coefficients a′, a′1, a ′ 2, we can determine the values of the ratio ξ = Φ ′ 0 / Φ0, (3.20) for which the maxima of the quantities f1(ξ) = ( Im 1 A+ C ) max , (3.21) f2(ξ) = ( Im 1 A− C ) max , (3.22) occurring in the expressions of the averaged Green function 〈Ĝ〉 are of the same order of magnitude. The obtained dependencies are shown in figure 1. As one can see, the values of Im 1/(A± C) maxima slightly differ only at small values of ξ. The distance between the peaks is determined by the difference between fre- quencies corresponding to the maxima of Im 1/(A+ C) and Im 1/(A− C) ∆ω = ( Ω2 +Φ0 √ 1 2 (1 + ξ2)− a1 a2 )1/2 − ( Ω2 + Φ0 √ 1 2 (1 + ξ2)− a ′ 1 a ′ 2 )1/2 . (3.23) Frequency separation (3.23) depends on absolute values of the energies of in- teraction between equal and different configurational types of PO4 groups. In figures 2a and 2b we plot the curves for Im 1/(A ± C). The parameters values are chosen allowing for the condition of comparability of functions (3.21) and (3.22) and the condition of sufficient separation ∆ω of scattering peaks too. Figure 1. Dependence of the functions f1 = (Im 1/(A + C))max and f2 = (Im 1/(A−C))max defined the Raman scattering intensity for different geometries on constants of interaction of ion complexes internal vibrations. 164 Configurational splitting Figure 2. Configurational splitting in the Raman spectrum for the CDP crystal model for different constants of ion complexes internal vibrations. 4. Discussion The results obtained for a simple model confirm the possibility of configu- rational splitting which manifests itself as additional lines of Raman spectra at T > Tc. Depending on the scattering geometry, one or two split components may appear in the spectrum. In the former case, scattering in different geometries can be observed at different frequencies, that is some frequency shift takes place. To observe two lines in the case when they can appear simultaneously (for instance (3.13)), a sufficient frequency separation determined by (3.23) is necessary. Besides, as well as in the case of one component, these two lines must be intensive enough, which depends on absolute values of scattering tensor elements R (i)κi lαβ . As follows from the above analysis, the corresponding lines of Raman spectra may be of different width and intensity and can be observed in different scat- tering geometries. Simultaneous observation of split components is possible only at certain relations among the constants of interactions among various ion group configurations. 165 I.V.Stasyuk, Ya.L.Ivankiv The proposed method of the calculation of spectral density functions which determine line profiles in Raman spectra can also be applied to more sophisticated models. Appendix By means of structural and dielectric studies, the symmetry of the paraelectric CsH2PO4 (CDP) crystal was found to be monoclinic (the space group C2 2h [16]). The crystal structure of CDP essentially differs from that of KDP, mainly by the H-bond network configuration, positions of Cs+ ions and the order of PO3− 4 groups along the polar axis. Depending on the nature of the protons positions on hydrogen bonds, different configurations of HnPO4 groups (n = 1, 2, 3) can be realized. Among them the configurations presented in figure 1 are most probable. The electronic spectrum of PO4 groups is most sensitive to the positions of the nearest protons. Therefore, the scattering tensors R (j)κj lαβ are different for different HnPO4 configurations. The symmetry analysis allows one to elucidate the structure of these tensors for all the considered configurations (see figure A.1). Amongst the R (j)κj lαβ tensors components, the ones which describe scattering in the case of the isolated complexes of Td symmetry and those allowed by the local symmetry C1h must be the largest. The presence of lines νi in Raman spectra of the given scattering geometry is determined by the structure ofR (i)κi lαβ tensors, as well as by the symmetry of coupled vibrations. The geometry of the scattering of modes coming from different PO4 configurations at T > Tc is given in table A.1. Internal vibrations of an isolated PO4 complex (point symmetry Td) are clas- sified by irreducible representations A1 (frequency ν1), E (frequency ν2), and F2 (frequencies ν3 and ν4). In a crystal they transform into the modes of symmetries Ag; Ag, Bg; Ag, Ag, Bg in the case of i = 1 (l = 1) and i = 5 (l = 2) configurations, Ag, Bg; Ag, Bg, Ag, Bg; Ag, Bg, Ag, Bg, Ag, Bg in the case of i = 2, 3 (l = 1) and i = 6, 7 (l = 2) configurations, and Ag; Ag, Bg; Ag, Ag, Bg in the case of i = 4 (l = 1) and i = 8 (l = 2) configurations, respectively. Since the symmetry of HnPO4 (n = 1, 2, 3) configurations is lower than C1h of PO4 groups, in some instances there may appear the lines forbidden by both the site symmetry and configurational splitting. In particular, such a situation takes place in (xz) scattering (the z-axis is directed along the chain of H-bonds on which protons are ordered at all temper- atures; the y-axis coincides with the polar axis, and x ⊥ yz). From table A.1 it follows that mode ν2 of Ag symmetry may split1. Such a splitting is not allowed by the local symmetry C1h; moreover, in the case of Td symmetry this mode is for- bidden at all. These results confirm the experiment [11]. The observed split lines are of approximately equal(insignificant) intensity, the splitting itself is of about 83 cm−1. 1The frequencies values are given in [17] 166 Configurational splitting Configurational splitting may also take place in (xy)-scattering (frequencies ν2 and ν4 of Ag and Bg symmetries), (yz)-scattering (frequency ν4 of Bg symmetry), (xx), (yy), (zz), (xz)-scattering (frequency ν4 of Ag symmetry). The two former cases have been studied experimentally [18]. Table A.1 also describes the situation when, in addition to splitting, the ap- pearance of lines forbidden by the site symmetry is possible. It is found out that in addition to ν2, ν3 and ν4 frequencies, line ν1 of Bg symmetry may appear in the (xy) and (yz) spectra. However, in [18] neither the ν1 line in the scattering of this geometry nor the splitting of the ν4 mode in (xx), (yy), (zz), (xz)-scattering was observed. We may attribute that to a low intensity of the expected lines, for they originate from the i = 2, 3, 6, 7 configurations only. l = 1 l = 2 ✑ ✑ ✑✑ ◗ ◗ ◗◗ ✓✓ ✓✓ ❙❙ ❙❙ sss i =1 ◗ ◗ ◗◗ ✑ ✑ ✑✑ ❙❙ ❙❙ ✓✓ ✓✓ sssi =5 ✑ ✑ ✑✑ ◗ ◗ ◗◗ ✓✓ ✓✓ ❙❙ ❙❙ s s s i =2 ◗ ◗ ◗◗ ✑ ✑ ✑✑ ❙❙ ❙❙ ✓✓ ✓✓ s s si =6 ✑ ✑ ✑✑ ◗ ◗ ◗◗ ✓✓ ✓✓ ❙❙ ❙❙ s s s i =3 ◗ ◗ ◗◗ ✑ ✑ ✑✑ ❙❙ ❙❙ ✓✓ ✓✓ s s si =7 ✑ ✑ ✑✑ ◗ ◗ ◗◗ ✓✓ ✓✓ ❙❙ ❙❙ s s s i =4 ◗ ◗ ◗◗ ✑ ✑ ✑✑ ❙❙ ❙❙ ✓✓ ✓✓ s s s i =8 Figure A.1. The HnPO4 configurations (n = 1, 2, 3) for two sublattices of PO4 groups in a CsH2PO4 crystal; projection on the xy-plane. 167 I.V.Stasyuk, Ya.L.Ivankiv configurations C1h Td C2h Rxx Ryy Rzz Rxy Rxz Ryz 1, 5 2, 3, 6, 7 4, 8 ✍✌ ✎☞ A′ A1 3Ag ∗❣ ∗❣ ∗❣ ❥∗ Ag Ag Ag Bg ∗ ∗ Bg Au Au 3Bu Bu Bu Bu ✍✌ ✎☞ A′′ F2 Ag ∗ ∗ ∗ ∗ Ag 3Bg ∗❣ ❥∗ Bg Bg Bg 3Au Au Au Au Bu Bu ✍✌ ✎☞ A′ F2 3Ag ❥∗ ❥∗ ❥∗ ∗❣ Ag Ag Ag Bg ∗ ∗ Bg Au Au 3Bu Bu Bu Bu ✍✌ ✎☞ A′ F2 3Ag ❥∗ ∗❣ ∗❣ ❥∗ Ag Ag Ag Bg ∗ ∗ Bg Au Au 3Bu Bu Bu Bu ✍✌ ✎☞ A′ E 3Ag ∗❣ ∗❣ ∗❣ ❥∗ Ag Ag Ag Bg ∗ ∗ Bg Au Au 3Bu Bu Bu Bu ✍✌ ✎☞ A′′ E Ag ∗ ∗ ∗ ∗ Ag 3Bg ❥∗ ∗❣ Bg Bg Bg 3Au Au Au Au Bu Bu Table A.1. Geometry of the scattering of configurationally split modes of the CsH2PO4 crystal; T > Tc. First column – irreducible representations of a site PO4 group symmetry. Second column – irreducible representations of an isolated PO4 group. Third column – configurational splitting. Symbols ∗ denote non-zero components of scattering tensors; ∗ corresponds to the scattering of isolated complexes PO4; ❣∗ stands for the scattering allowed by a site C1h symmetries. Three last columns contain symmetry of the modes arising from different config- urations. 168 Configurational splitting κi = (1\|F2) κi = (2\|F2) κi = (3\|F2) i=1 ✞ ✝ ☎ ✆a12✞ ✝ ☎ ✆a12 ✞ ✝ ☎ ✆a23✞ ✝ ☎ ✆a23 ✞ ✝ ☎ ✆a11 ✞ ✝ ☎ ✆a13✞ ✝ ☎ ✆a22✞ ✝ ☎ ✆a13 ✞ ✝ ☎ ✆a33 ✞ ✝ ☎ ✆b11 ✞ ✝ ☎ ✆b13✞ ✝ ☎ ✆b22✞ ✝ ☎ ✆b13 ✞ ✝ ☎ ✆b33 (A1) (1\|E) (2\|E)✞ ✝ ☎ ✆c11 ✞ ✝ ☎ ✆c13✞ ✝ ☎ ✆c22✞ ✝ ☎ ✆c13 ✞ ✝ ☎ ✆c33 ✞ ✝ ☎ ✆d11 ✞ ✝ ☎ ✆d13✞ ✝ ☎ ✆d22✞ ✝ ☎ ✆d13 ✞ ✝ ☎ ✆d33 ✞ ✝ ☎ ✆d12✞ ✝ ☎ ✆d12 ✞ ✝ ☎ ✆d23✞ ✝ ☎ ✆d23 (1\|F2) (2\|F2) i=2 a′11 + a′′11 ☛ ✡ ✟ ✠a′12 + a′′12 a′13 + a′′13☛ ✡ ✟ ✠a′12 + a′′12 a′22 + a′′22 ☛ ✡ ✟ ✠a′23 + a′′23 a′13 + a′′13 ☛ ✡ ✟ ✠a′23 + a′′23 a′33 + a′′33 ☛ ✡ ✟ ✠a′11 − a′′11 a′12 − a′′12 ☛ ✡ ✟ ✠a′13 − a′′13 a′12 − a′′12 ☛ ✡ ✟ ✠a′22 − a′′22 a′23 − a′′23☛ ✡ ✟ ✠a′13 − a′′13 a′23 − a′′23 ☛ ✡ ✟ ✠a′33 − a′′33 (3\|F2) (A1)☛ ✡ ✟ ✠b′11 b′12 ☛ ✡ ✟ ✠b′13 b′12 ☛ ✡ ✟ ✠b′22 b′23☛ ✡ ✟ ✠b′13 b′23 ☛ ✡ ✟ ✠b′33 ☛ ✡ ✟ ✠c′11 c′12 ☛ ✡ ✟ ✠c′13 c′12 ☛ ✡ ✟ ✠c′22 c′23☛ ✡ ✟ ✠c′13 c′23 ☛ ✡ ✟ ✠c′33 (1\|E) (2\|E)☛ ✡ ✟ ✠d′11 + d′′11 d′12 + d′′12 ☛ ✡ ✟ ✠d′13 + d′′13 d′12 + d′′12 ☛ ✡ ✟ ✠d′22 + d′′22 d′23 + d′′23☛ ✡ ✟ ✠d′13 + d′′13 d′23 + d′′23 ☛ ✡ ✟ ✠d′33 + d′′33 d′11 − d′′11 ☛ ✡ ✟ ✠d′12 − d′′12 d′13 − d′′13☛ ✡ ✟ ✠d′12 − d′′12 d′22 − d′′22 ☛ ✡ ✟ ✠d′23 − d′′23 d′13 − d′′13 ☛ ✡ ✟ ✠d′23 − d′′23 d′33 − d′′33 (1\|F2) (2\|F2) i=3 a′11+a′′11 ☛ ✡ ✟ ✠−a′12 − a′′12 a′13+a′′13☛ ✡ ✟ ✠−a′12 − a′′12 a′22+a′′22 ☛ ✡ ✟ ✠−a′23 − a′′23 a′13+a′′13 ☛ ✡ ✟ ✠−a′23 − a′′23 a′33+a′′33 ☛ ✡ ✟ ✠a′11 − a′′11 −a′12+a′′12 ☛ ✡ ✟ ✠a′13 − a′′13 −a′12+a′′12 ☛ ✡ ✟ ✠a′22 − a′′22 −a′23+a′′23☛ ✡ ✟ ✠a′13 − a′′13 −a′23+a′′23 ☛ ✡ ✟ ✠a′33 − a′′33 (3\|F2) (A1)☛ ✡ ✟ ✠b′11 −b′12 ☛ ✡ ✟ ✠b′13 −b′12 ☛ ✡ ✟ ✠b′22 −b′23☛ ✡ ✟ ✠b′13 −b′23 ☛ ✡ ✟ ✠b′33 ☛ ✡ ✟ ✠c′11 −c′12 ☛ ✡ ✟ ✠c′13 −c′12 ☛ ✡ ✟ ✠c′22 −c′23☛ ✡ ✟ ✠c′13 −c′23 ☛ ✡ ✟ ✠c′33 169 I.V.Stasyuk, Ya.L.Ivankiv (1\|E) (2\|E)☛ ✡ ✟ ✠d′11 + d′′11 −d′12 − d′′12 ☛ ✡ ✟ ✠d′13 + d′′13 −d′12 − d′′12 ☛ ✡ ✟ ✠d′22 + d′′22 −d′23 − d′′23☛ ✡ ✟ ✠d′13 + d′′13 −d′23 − d′′23 ☛ ✡ ✟ ✠d′33 + d′′33 d′11 − d′′11 ☛ ✡ ✟ ✠−d′12+d′′12 d′13 − d′′13☛ ✡ ✟ ✠−d′12+d′′12 d′22 − d′′22 ☛ ✡ ✟ ✠−d′23+d′′23 d′13 − d′′13 ☛ ✡ ✟ ✠−d′23+d′′23 d′33 − d′′33 Figure A.2. Structure of scattering tensors R (i)κi lαβ . By rectangles and ovals we denote components allowed by the symmetry Td and the site symmetry C1h, respectively. References 1. Blinc R., Zeks B. Dynamics of order-disorder-type ferroelectrics and antiferroelectrics. // Adv. Phys., 1972, vol. 21, p. 693–757. 2. Bruce A.D. Cowley R.A. Structural phase transition. III. Critical dynamics and quasi- elastic scattering. // Adv. Phys., 1980, vol. 29, No. 1, p. 219–321. 3. Tominaga Y., Urabe H., Tokunaga M. Internal modes and local symmetry of PO4 tetrahedrons in KH2PO4 by Raman scattering. // Solid State Commun., 1983, vol. 48, No. 3, p. 265–267. 4. Tanaka H., Tokunaga M., Tatsuzaki I. Internal modes and the local symmetry of PO4 tetrahedra in K(H1−xDx)2PO4 by Raman scattering. // Solid State Commun., 1984, vol. 49, No. 2, p. 153–155. 5. Tokunaga M. Order-disorder model of PO4-dipoles for KH2PO4 type ferroelectric phase transition. // Progr. Theor. Phys. Suppl., 1984, No. 80, p. 156–162. 6. 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. 7. Tokunaga M., Tatsuzaki I. Light scattering spectra of polarization fluctuation and models of the phase transition in KDP type ferroelectrics. // Phase Transition, 1984, vol. 4, No. 1, p. 97–156. 8. Tokunaga M., Tominaga Y., Tatsuzaki I. Order-disorder model of PO4 dipoles in KDP based on recent Raman spectroscopic studies. // Ferroelectrics, 1985, vol. 63, No. 1/4, p. 171–178. 9. Blinc R. 170 NQR and the mechanism of the phase transition in KH2PO4 type H- bonded systems. // Z. Naturforsch., 1986, vol. 41a, p. 249–255. 10. Stasyuk I.V., Ivankiv Ya.L. Raman light scattering in crystals with ordered structure elements. // Preprint ITP–87–57R (Kyiv, 1987), 25 p. (in Russian). 11. Levitskii R.R., Sorokov S.I., Sokolovskii R.O. Correlations functions of quenched and annealed Ising systems. // Cond. Matt. Phys., 1995, No. 5, p. 81–104. 12. Cowley R.A. The lattice dynamics of an anharmonic crystal. // Adv. Phys., 1963, vol. 12, No. 48, p. 421–481. 13. Cardona M. Introduction. In: “Light scattering in Solids”. Ed. by M.Cardona, Berlin- Heidelberg-New York, Springer-Verlag, 1975, p. 11–37. 14. Elliot R.J., Krumhansl J.A., Leath P.L. Theory and properties of randomly disordered 170 Configurational splitting crystals and related physical systems. // Rev. Mod. Phys., 1974, vol. 46, p. 465–702. 15. Stasyuk I.V., Ivankiv Ya.L. Dynamic susceptibility of systems described by de Gennes model. In: Spectroscopy of Molecules and Crystals., Pt. 2, Kyiv, Naukova dumka, 1980, p. 176–179. 16. Uesu Y., Kobayashi J. Crystal structure and ferroelectricity of cesium dihydrogen phosphate // Phys. Status Solidi (a), 1976, vol. 34, No. 2, p. 475–481. 17. Martinez J.L., Calleja J.M., Gonzalo J.A. Temperature dependence of internal modes and local symmetry of PO4 tetrahedra in RbH2PO4 (RDP) and RbD2PO4 (DRDP) by Raman scattering. // Solid State Commun., 1984, vol. 52, No. 5, p. 499–501. 18. Kasahara M., Aoki M., Tatsuzaki I. Raman study of the phase transition in CsH2PO4 and CsD2PO4. // Ferroelectrics, 1984, vol. 55, No. 1–4, p. 715–718. Конфігураційне розщеплення в раманівських спектрах у кристалах з фазовими переходами типу лад-безлад І.В.Стасюк, Я.Л.Іванків Інститут фізики конденсованих систем НАН Укpаїни, 290011 Львів, вул. Свєнціцького, 1 Отримано 23 червня 1997 В наближенні когерентного потенціалу одержана система рівнянь для усередненої фононної функції Гріна, що визначає профілі ліній в спектрах Рамана. На прикладі простої моделі проведено досліджен- ня конфігураційно розщеплених компонент раманівських ліній. Для сегнетоелектриків типу KH2PO4 та CsH2PO4 обговорюється питання про можливий експериментальний вияв конфігураційного розщеп- лення. Ключові слова: комбінаційне розсіяння, конфігураційне розщеплення, фазові переходи типу лад-безлад, сегнетоелектрики PACS: 77.84.Fa, 78.30.Ly 171 172

Conﬁgurational splitting in Raman scattering spectra of crystals with the order-disorder type phase transitions

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