#
Theoretical Study of the Phonon Energy and Specific Heat of Ion-Doped LiCsSO_{4}—Bulk and Nanoparticles

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−1}and 72 cm

^{−1}, and their damping of the ferroelastic LiCsSO

_{4}compound, are calculated within Green’s function technique. It is observed that the first mode increases whereas the second one decreases with increasing temperature T. This different behavior is explained with different sign of the anharmonic spin–phonon interaction constant. At the ferroelastic phase transition temperature ${T}_{C}$, there is a kink in both modes due to the spin–phonon interaction. The phonon damping increases with T, and again shows an anomaly at ${T}_{C}$. The contributions of the spin–phonon and phonon–phonon interactions are discussed. ${T}_{C}$ is reduced by decreasing the nanoparticle size, and can be enhanced by doping with K, Rb and NH

_{4}ions at the Cs site. ${T}_{C}$ decreases by doping with Na, K or Rb on the Li site. The specific heat ${C}_{p}$ also shows a kink at ${T}_{C}$. ${C}_{p}$ decreases with decreasing nanoparticle size and the peak disappears, whereas ${C}_{p}$ increases with increasing K ion doping concentration.

## 1. Introduction

_{4}(LCS) is of interest as a material undergoing a phase transformation and having ferroelastic properties at low temperatures [1]. At room temperature, LCS crystals exhibit an orthorhombic pseudo-hexagonal symmetry and belong to the space group Pcmn. LCS undergoes a second-order structural phase transition at ${T}_{C}$∼202 K, shifting from the paraelastic phase to the ferroelastic monoclinic structure without altering the unit cell content [2,3,4]. This transition is of the order–disorder type. The mechanism driving the ordered phase involves rotations of the SO

_{4}tetrahedra within the $ab$ plane [3,5]. But, the transition mechanism in LCS from the para- to the ferroelastic phase remains unclear.

_{1g}, B

_{2g}, and B

_{3g}symmetries. The Raman lines in LCS are categorized into three distinct frequency regions: 0 to 200 cm

^{−1}, 360 to 660 cm

^{−1}, and 1000 to 1200 cm

^{−1}[15]. The lowest frequency range encompasses the translational vibrations of Li

^{+}ions, while the intermediate frequency range is attributed to the librational motions of sulfate ions. The highest frequency range includes bands identified as modes derived from the stretching vibrations, specifically ${\nu}_{1}$ at $\omega $ = 1016 cm

^{−1}, and ${\nu}_{3}$, at frequencies between 1110 and 1200 cm

^{−1}. The behavior of the surface phonons in the vicinity of the phase transition temperature ${T}_{C}$ was studied by Trzaskowska et al. [18,19] using Brillouin spectroscopy. Recently, the size effects of the linear permittivity ${\u03f5}^{\prime}$ in ferroelastic LCS nanoparticles (NPs) were investigated by Milinskiy et al. [20]. The measurements were carried out by linear and non-linear methods of dielectric spectroscopy. The phase transition temperature ${T}_{C}$ is reduced compared to that in the bulk LCS, as reported by Borisov et al. [21].

_{4}, Cu, Mn, etc., on the phase transition temperature ${T}_{C}$ in the bulk LCS are reported by Czaja [22], Zapart et al. [23], Lima et al. [24], and Misra et al. [25]. They observed tuning of the ${T}_{C}$. It is expected that, as a result of substituting Cs

^{+}ions with other ions, a modification of the ferroic properties will take place. A strong increase in the ${T}_{C}$ is reported by Czaja et al. [22] for the NH

_{4}-doped LCS, from 202 to 230.8 K for the doping concentration x = 0.15. Zapart et al. [23] determined that, in Rb-doped LCS, the phase transition temperature is ${T}_{C}$ = 215 K, i.e., 13 K above that in pure LCS. Lima et al. [24] have investigated temperature-dependent Raman scattering studies in Rb-doped LCS for x = 0.35 in the temperature range of 7-295 K. They have shown that the doped compound undergoes a phase transition at a ${T}_{C}$ of about 275 K. Misra et al. [25] have performed EPR studies on Mn

^{+}-doped LCS in the temperature range of 3.8–301 K, as well as on Cu

^{+}-doped LCS at room temperature.

## 2. Model and Method

## 3. Numerical Results and Discussion

^{−1}, A = 6.7 cm

^{−1}, B = −3.1 cm

^{−1}.

#### 3.1. Temperature Dependence of the A_{1g} Phonon Modes ${\omega}_{0}$ = 32 and 72 cm^{−1} in Bulk LCS

_{1g}modes ${\omega}_{0}$ = 32 and 72 cm

^{−1}were evaluated. They are connected with the translational Li

^{+}vibrations. Let us note that we can also investigate the other phonon modes within our model and method. The results are shown in Figure 1. It can be seen that the phonon energy for the ${\omega}_{0}$ = 0.32 cm

^{−1}mode increases with an increase in temperature T (curve 1), whereas for the other mode $\omega $ = 72 cm

^{−1}, it decreases with T (curve 2). In order to explain this different temperature behavior, for the first case, we must chose a positive anharmonic spin–phonon interaction constant, $R>0$ (curve 1), and a positive one for the second case, $R<0$ (curve 2) [27]. At the ferroelastic phase transition temperature ${T}_{C}$ = 202 K, both curves show a kink in agreement with Refs. [17,18,19], which is due to the spin–phonon interaction. Above ${T}_{C}$, the phonon energy slightly decreases, in agreement with Refs. [4,17]. It must be noted that, at low temperatures, the anharmonic spin–phonon interaction plays an important role, whereas above ${T}_{C}$, there remain only the anharmonic phonon–phonon interactions. We have calculated the phonon energy for different relation $\mid {J}_{2}/{J}_{1}\mid $ values. ${\omega}_{0}$, and the phase transition temperature ${T}_{C}$ at which the kink appears, increase with an increasing $\mid {J}_{2}/{J}_{1}\mid $, i.e., with an increase in the magnetization. This shows the influence of the magnetic exchange interaction constants on the phonon energy, and the existence of a strong spin–phonon interaction. Our results are in good qualitative agreement with the experimental data of Kaczmarski and Wiesner [17]. It must be noted that an increase in ${T}_{C}$ with increases in the $\mid {J}_{2}/{J}_{1}\mid $ values was reported by Arnalds et al. [10], where the authors have theoretically studied the temperature dependence of the magnetization in an hcp Ising model. Unfortunately, we have not observed the two additional transitions at ${T}_{1}$≈ 180 K and ${T}_{2}$≈ 100 K [17]. Therefore, in our next paper, we will additively consider the temperature dependence of the dielectric constant, so as to obtain a better understanding of the structural changes in LCS. It must be noted that the phonon energy $\omega $ and the phase transition temperature ${T}_{C}$ (see Figure 3) decrease with a decrease in the NP size.

#### 3.2. Temperature and Size Dependence of the Damping of the A_{1g} Phonon Modes ${\omega}_{0}$ = 32 and 72 cm^{−1} in Bulk LCS

^{−1}(curve 1) and ${\omega}_{0}$ = 72 cm

^{−1}(curve 2), with a fixed ratio $\mid {J}_{2}/{J}_{1}\mid $ = 0.3. It can be seen that both damping curves increase with an increase in temperature T, for both cases $R>0$ and $R<0$, because $\gamma $ is proportional to ${R}^{2}$. This means that the Raman peaks are broader by higher temperatures. Let us emphasize that the experimentally obtained broadened peaks in the Raman spectra of NPs, and especially of LCS NPs, cannot be understood within the random phase approximation (RPA) for small particles. We go beyond the RPA, taking into account all correlation functions, using the method of Tserkovnikov [26], and calculate the phonon damping effects in LCS NPs, including anharmonic spin–phonon and phonon–phonon interactions. At the phase transition temperature ${T}_{C}$, there is again a kink. Above ${T}_{C}$, the damping begins to decrease because the anharmonic spin–phonon contribution vanishes, and there remain only the anharmonic phonon–phonon interactions. A similar experimental behavior for the full width at half-maximum (FWHM), which corresponds in our model to the phonon damping for the second mode ${\omega}_{0}$ = 72 cm

^{−1}, is observed by Kaczmarski and Wiesner [17].

#### 3.3. Size Dependence of the Ferroelastic Phase Transition Temperature ${T}_{C}$

_{3}and PbTiO

_{3}, ferromagnetic NPs ${T}_{C}$ also decrease with a decrease in NP size d but, in MnO for example, BiFeO

_{3}or other antiferromagnetic NPs the Neel temperature increase with a decrease in d [28,29]. It depends on the strain that appears in the compound by changing the size.

#### 3.4. Temperature Dependence of the Specific Heat ${C}_{p}$ in Bulk LCS

_{4}SO

_{4}, LiKSO

_{4}, and Ru-doped LiKSO

_{4}, as meticulously documented by Polomska et al. [31], Kassem et al. [32], and Yurtseven et al. [33], respectively. A noticeable trend is the reduction in Cp with diminishing NP size, where the peak at TC diminishes in magnitude and shifts towards lower temperature values. This effect is accentuated in very small NPs, ultimately leading to the disappearance of the peak, as illustrated in Figure 4, curve 1a. Unfortunately, experimental data for Cp(d) in LCS are not available.

#### 3.5. Ion Doping Dependence of the Phase Transition Temperature ${T}_{C}$ and the Specific Heat ${C}_{p}$

^{+}ion with N${\mathrm{H}}_{4}^{+}$ ion (see Figure 5, curve 2), which is in agreement with the experimental data of Czaja [22]. Curve 3 in Figure 5 presents the increase in ${T}_{C}$ in LCS after doping with Rb

^{+}ions, in coincidence with the result of Zapart et al. [23]. It must be noted that the ${T}_{C}$ of LiKSO

_{4}is 708 K [33], whereas of LiRbSO

_{4}, it is 477 K [34]. We would also observe an increase in ${T}_{C}$ and ${C}_{p}$ by doping with Sm

^{3+}or Dy

^{3+}ions on the Cs site, as reported by Kassem et al. [35], when doped with the last two ions LiRbSO

_{4}. Tuszynski et al. [13] and Melo et al. [4] reported an increase in ${T}_{C}$ as a function of an applied uniaxial stress.

^{+}ion with Na

^{+}, K

^{+}or Rb

^{+}ions, which is characterized by larger ionic radii (0.97, 1.33, and 1.47 Ȧ, respectively), compared to the host Li ion (0.9 Ȧ), induces a tensile strain [36]. This means that we must choose the relation ${J}_{d}<{J}_{b}$ that would lead to reduction in the phase transition temperature ${T}_{C}$ compared to that of pure LCS. Thus, our model can explain the dependence of ${T}_{C}\left(x\right)$ on a microscopic level. Unfortunately, there are no experimental data for this behavior.

## 4. Conclusions

_{1g}modes ${\omega}_{0}$ = 32 cm

^{−1}and 72 cm

^{−1}are calculated. It is observed that both modes have different temperature dependences. The first mode increases, whereas the second one decreases with an increase in temperature T. This behavior is explained with the different sign of the anharmonic spin–phonon interaction constant R. At the ferroelastic phase transition temperature ${T}_{C}$∼202 K is a kink in both curves, due to a strong spin–phonon interaction in LCS. Above ${T}_{C}$, the phonon energies slightly decrease. The influence of the exchange interaction constants J on the phonon modes is shown. The phonon modes increase with an increase in the $\mid {J}_{2}/{J}_{1}\mid $-value. The phonon damping for both phonon modes increases with the temperature T, and shows a kink at ${T}_{C}$. The contribution of the anharmonic spin–phonon and phonon–phonon interactions in different temperature intervals is discussed. ${T}_{C}$ decreases with a decrease in NP size. Substituting the Cs ion with K, NH

_{4}, or Rb ions enhances the ${T}_{C}$, whereas replacing the Li ion with Na, K or Rb reduces the ${T}_{C}$. The specific heat ${C}_{p}$ increases with increases in temperature and K ion doping concentration, and shows a kink at ${T}_{C}$. ${C}_{p}$ is reduced in LCS NPs compared to the bulk case.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Salje, E.K.H. Ferroelastic Materials. Annu. Rev. Mater. Res.
**2012**, 42, 265. [Google Scholar] [CrossRef] - Hidaka, T. Isotope effects on structural phase transitions in several sulfates. Phys. Rev. B
**1992**, 45, 440. [Google Scholar] [CrossRef] [PubMed] - Asahi, T.; Hasebe, K. X-Ray Study of LiCsSO
_{4}in Connection with Its Ferroelastic Phase Transition. J. Phys. Soc. Jpn.**1988**, 57, 4184. [Google Scholar] [CrossRef] - Melo, F.E.A.; Saip, J.A.B.; Guedes, I.; Freire, P.T.C.; Mendes-Filho, J.; Chaces, A.S. Inhibition of the phase transition in CsLiSO
_{4}induced by uniaxial pressure. Ferroelectrics**1999**, 233, 57. [Google Scholar] [CrossRef] - Niwata, A.; Itoh, K. Structural Study of Ferroelastic CsLiSO
_{4}in the High-Temperature Phase. J. Phys. Soc. Jpn.**1995**, 64, 4733. [Google Scholar] [CrossRef] - Lehmann-Szweykowska, A.; Wojciechowski, R.J.; Kurzynski, M.; Wiesner, M.; Mroz, B. Soliton theory of incommensurate phases in LiCsSO
_{4}crystals. J. Phys. Conf. Ser.**2010**, 213, 012034. [Google Scholar] [CrossRef] - Lehmann-Szweykowska, A.; Kurzynski, M.; Wojciechowski, R.; Wiesner, M.; Mroz, B. Anomalous Phase Transitions in LiCsSO
_{4}in the Compressible hcp Ising Model. Acta Phys. Pol. A**2012**, 121, 1108–1110. [Google Scholar] [CrossRef] - Lukyanchuk, I.; Jorio, A.; Pimenta, M.A. Basal-plane incommensurate phases in hexagonal-close-packed structures. Phys. Rev. B
**1998**, 57, 5086. [Google Scholar] [CrossRef] - Hoang, D.-T.; Diep, H.T. Hexagonal-Close-Packed Lattice: Phase Transition and Spin Transport. arXiv
**2011**, arXiv:1112.5724v1. [Google Scholar] - Arnalds, U.B.; Chico, J.; Stopfel, H.; Kapaklis, V.; Baerenbold, O.; Verschuuren, M.A.; Wolff, U.; Neu, V.; Bergman, A.; Hjoervarsson, B. A new look on the two-dimensional Ising model: Thermal artificial spins. New J. Phys.
**2016**, 18, 023008. [Google Scholar] [CrossRef] - Hasebe, K.; Asahi, T. Phenomenological Theory of the Ferroelastic Phase Transitionin LiCsSO
_{4}. J. Phys. Soc. Jpn.**1991**, 60, 4199. [Google Scholar] [CrossRef] - Zhou, G.; Bai, X.; Wei, Y.; Guo, M. Ferroelastic phase transition of LiCsSO
_{4}crystal. Ferroelectrics**2016**, 502, 221. [Google Scholar] [CrossRef] - Tuszynski, J.A.; Mroz, B.; Kiefte, H.; Clouter, M.J. Comments on the hysteresis loop in ferroelastic LiCsSO
_{4}. Ferroelectrics**1988**, 77, 111. [Google Scholar] [CrossRef] - Shashikala, M.N.; Chandrabhas, N.; Jayaram, K.; Jayaraman, A.; Sood, A.K. High pressure Raman spectroscopic study of LiCsSO
_{4}: Pressure induced phase transitions and amorphization. J. Phys. Chem. Solids**1994**, 55, 107. [Google Scholar] [CrossRef] - Lemos, V.; Silveir, E.S.; Melo, F.E.A.; Ilho, J.M.; Pereir, J.R. Raman study of LiCsSO
_{4}. Phys. Status Solidi B**1991**, 164, 577. [Google Scholar] [CrossRef] - Morell, G.; Devanarayanan, S.; Katiyar, R.S. Temperature-dependent Raman scattering studies in ferroelastic LiCsSO
_{4}. J. Raman Spectr.**1991**, 22, 529. [Google Scholar] [CrossRef] - Kaczmarski, M.; Wiesner, M. Temperature-dependent low wavenumber Raman scattering studies in LiCsSO
_{4}crystal. J. Raman Spectr.**2010**, 41, 1765. [Google Scholar] [CrossRef] - Trzaskowska, A.; Mielcarek, S.; Mroz, B. Behaviour of surface phonons in LiCsSO
_{4}crystal at phase transition. Cryst. Res. Technol.**2005**, 40, 449. [Google Scholar] [CrossRef] - Trzaskowska, A.; Mielcarek, S.; Mroz, B.; Andrews, G.T. Surface Phonons and Soft Bulk Modes in LiCsSO
_{4}Crystals Studied by the High Resolution Brillouin Scattering. Ferroelectrics**2008**, 363, 168. [Google Scholar] [CrossRef] - Milinskiy, A.Y.; Baryshnikov, S.V.; Charnaya, E.V.; Nguyen, H.T. Size effects in the ferroelastic LiCsSO
_{4}. Ferroelectrics**2019**, 543, 12. [Google Scholar] [CrossRef] - Borisov, B.F.; Charnaya, E.V.; Baryshnikov, S.V.; Pirozerskii, A.L.; Bugaevd, A.S.; Tien, C.; Lee, M.K.; Michel, D. Ferroelastic phase transition in LiCsSO
_{4}embedded into molecular sieves. Phys. Lett. A**2010**, 375, 183. [Google Scholar] [CrossRef] - Czaja, P. Detection of a ferroelastic phase transition in Csx(NH4)1-xLiSO
_{4}with the use of the DSC method. J. Therm. Anal. Calorim.**2013**, 113, 91. [Google Scholar] [CrossRef] - Zapart, M.B.; Zapart, W.; Czaja, P.; Mila, T.; Solecki, J. EPR Spectroscopy and Ferroelastic Domain Structure in the LiRbSO
_{4}-LiCsSO_{4}System. Ferroelectrics**2011**, 417, 70. [Google Scholar] [CrossRef] - Lima, R.J.C.; Freire, P.T.C.; Sasaki, J.M.; Ayala, A.P.; Melo, F.E.A.; Filho, J.M.; Hernandes, A.C. Temperature-dependent Raman scattering studies in CsLiSO
_{4}and RbxCs1–xLiSO_{4}(x = 0.35). J. Raman Spectr.**2001**, 32, 1046–1051. [Google Scholar] [CrossRef] - Misra, S.K.; Misiak, J.E. Electron-paramagnetic-resonance study of Cu2- and Mn2+-doped LiCsSO
_{4}single crystals: LiCsSO_{4}phase transitions. Phys. Rev. B**1993**, 48, 13579. [Google Scholar] [CrossRef] [PubMed] - Tserkovnikov, Y.A. Decoupling of chains of equations for two-time Green’s functions. Teor. Mat. Fiz.
**1971**, 7, 511. [Google Scholar] [CrossRef] - Wesselinowa, J.M.; Apostolov, A.T. Anharmonic effects in ferromagnetic semiconductors. J. Phys. Condens. Matter
**1996**, 8, 473. [Google Scholar] [CrossRef] - Golosovsky, I.V.; Mirebeau, I.; Andre, G.; Kurdyukov, D.A.; Kumzerov, Y.A.; Vakhrushev, S.B. Magnetic Ordering and Phase Transition in MnO Embedded in a Porous Glass. Phys. Rev. Lett.
**2001**, 86, 5783. [Google Scholar] [CrossRef] [PubMed] - Wesselinowa, J.M. Size and anisotropy effects on magnetic properties of antiferromagnetic nanoparticles. J. Magn. Magn. Mater.
**2010**, 322, 234–237. [Google Scholar] [CrossRef] - Delfino, M.; Loiacono, G.M.; Smith, W.A.; Shaulov, A.; Tsuo, Y.H.; Bells, M.I. Thermal and dielectric properties of LiKSO
_{4}and LiCsSO_{4}. J. Solid State Chem.**1980**, 31, 131. [Google Scholar] [CrossRef] - Polomska, M.; Wolak, J.; Szczesniak, L. High temperature phase transition of β-LiNH4SO
_{4}single crystal. Ferroelectrics**1994**, 159, 179. [Google Scholar] [CrossRef] - Kassem, M.E.; El-Wahidy, E.F.; Kandil, S.H.; El-Gamal, M.A. Thermal anomaly in LiKSO
_{4}crystals in the temperature range 300–800 K. J. Therm. Anal.**1984**, 29, 325. [Google Scholar] [CrossRef] - Yurtseven, H.; Tirpanci, D.V.; Karacali, H. Analysis of the Specific Heat of Ru Doped LiKSO
_{4}Close to Phase Transitions. High Temp.**2018**, 56, 462. [Google Scholar] [CrossRef] - Kassem, M.E.; El-Muraikhi, M.; AL-Houthy, L.; Mohamed, A.A. Dielectric dispersion in pure and doped lithium rubidium sulphate. Radiat. Eff. Defects Solids
**1996**, 138, 285. [Google Scholar] [CrossRef] - Kassem, M.E.; El-Muraikhi, M.; Al-Houty, L.; Mohamed, A. A Specific heat of pure and doped LiRbSO
_{4}crystals. Thermochim. Acta**1992**, 206, 107. [Google Scholar] [CrossRef] - Choi, J.H.; Kim, N.H.; Lim, A.R. Nucleus-phonon interactions of MCsSO
_{4}(M = Na, K, or Rb) single crystals studied using spin–lattice relaxation time. J. Korean Magn. Res. Soc.**2014**, 18, 15. [Google Scholar] [CrossRef]

**Figure 1.**Temperature dependence of the phonon energy in bulk LCS for two modes ${\omega}_{0}$ = 32 cm

^{−1}, $R>0$ (1), and 72 cm

^{−1}, $R<0$ (2), with $\mid {J}_{2}/{J}_{1}\mid $ = 0.3, and for ${\omega}_{0}$ = 32 cm

^{−1}with $\mid {J}_{2}/{J}_{1}\mid $ = 0.15 (1a) and 0.5 (1b).

**Figure 2.**Temperature dependence of the phonon damping $\gamma $ in bulk LCS for two phonon modes ${\omega}_{0}$ = 32 cm

^{−1}, $R>0$ (1), and 72 cm

^{−1}, $R<0$ (2), with $\mid {J}_{2}/{J}_{1}\mid $ = 0.3.

**Figure 3.**Size dependence of the ferroelastic phase transition temperature ${T}_{C}$ in LCS; N is the number of NP shells.

**Figure 4.**Temperature dependence of the specific heat ${C}_{p}$ in (1) pure bulk LCS and (2) K-doped bulk LCS, x = 0.15; (1a) a LCS NP with N = 5 shells.

**Figure 5.**The phase transition temperature ${T}_{C}$ in bulk LCS as a function of different ion doping concentration x at the Cs site: (1) K; (2) NH

_{4}; (3) Rb; and at the Li site (4) K.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Apostolov, A.T.; Apostolova, I.N.; Wesselinowa, J.M.
Theoretical Study of the Phonon Energy and Specific Heat of Ion-Doped LiCsSO_{4}—Bulk and Nanoparticles. *Materials* **2024**, *17*, 2845.
https://doi.org/10.3390/ma17122845

**AMA Style**

Apostolov AT, Apostolova IN, Wesselinowa JM.
Theoretical Study of the Phonon Energy and Specific Heat of Ion-Doped LiCsSO_{4}—Bulk and Nanoparticles. *Materials*. 2024; 17(12):2845.
https://doi.org/10.3390/ma17122845

**Chicago/Turabian Style**

Apostolov, Angel T., Iliana N. Apostolova, and Julia Mihailowa Wesselinowa.
2024. "Theoretical Study of the Phonon Energy and Specific Heat of Ion-Doped LiCsSO_{4}—Bulk and Nanoparticles" *Materials* 17, no. 12: 2845.
https://doi.org/10.3390/ma17122845