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Raman process studied by 87Rb spin-lattice relaxation in a Rb2ZnCl4 single crystal at low temperature
PERGAMON
Solid State Communications 118 (2001) 453±457
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Raman process studied by 87Rb spin-lattice relaxation in a Rb2ZnCl4 single crystal at low temperature Ae Ran Lim a,*, Jae Kap Jung b, Se-Young Jeong c b
a Department of Physics, Jeonju University, Jeonju 560-759, South Korea BK21 Physics Research Division, Department of Physics, Sungkyunkwan University, Suwon 440-746, South Korea c Department of Physics, Pusan National University, Pusan 609-735, South Korea
Received 12 January 2001; accepted 18 March 2001 by A. Pinczuk
Abstract 87 Rb 1/2 $ 2 1/2 spin-lattice relaxation time in the Rb2ZnCl4 single crystal was measured in the temperature range of 4.2± 300 K. The dominant relaxation of this crystal in the whole temperature range investigated here is due to quadrupolar interaction. The changes in the 87Rb spin-lattice relaxation rate near 75 and 192 K correspond to the phase transitions in the crystal. These suggest that the phase IV to III transition is of second-order, and the phase III to II transition is of ®rst-order. The temperature dependence of the relaxation rates in phases II, III, and IV is in accordance with the Raman process dominated by the phonon mechanism. q 2001 Published by Elsevier Science Ltd.
PACS: 76.60.2k; 77.80.Bb; 77.80.2e Keywords: A. Ferroelectrics; B. Crystal growth; D. Phase transitions; E. Nuclear resonances
1. Introduction Rb2ZnCl4 belongs to a large family of crystals with the general formula A2BX4, where A is an alkali metal and BX4 is a tetrahedral group (SO4, SeO4, CrO4, ZnCl4, and so on) [1,2]. This family of crystals has been attracting the attention of investigators for many decades. These substances are of interest due to the great diversity of successive structural phase transitions. Incommensurate phases have been discovered in some members of this family of crystals. The numerous studies that have been carried out on Rb2ZnCl4 crystals have shown that this compound can be found in at least four phases depending on temperature [3±6]. Rb2ZnCl4 undergoes successive phase transitions from a paraelectric phase (N phase) to an incommensurate phase (IC phase) at Tc3 303 K, to a ferroelectric commensurate phase (C phase) at Tc2 192 K [3,4,7], and to a lowest-temperature phase at Tc1 75 K [8]. The high-temperature phase belongs to the ortho* Corresponding author. Tel.: 182-63-220-2514; fax: 182-63220-2362, 182-42-483-6155. E-mail address: [email protected] (A.R. Lim).
rhombic space group Pmcn with four formula units per unit cell. The intermediate phase is incommensurate and 9 the low-temperature commensurate phase C2v exhibits improper ferroelectricity with the lattice parameter tripled along the pseudohexagonal axis. This crystal undergoes a second-order phase transition from a paraelectric (N phase) to an incommensurate (IC phase) at about Tc3 303 K and a ®rst-order phase transition from the IC phase to ferroelectric commensurate at Tc2 192 K [9]. Previously, a variety of experimental methods have been applied to verify the concept of phase solitons or discommensurations in structurally incommensurate systems. Particularly, it has been shown that nuclear magnetic resonance techniques can be used to quantitatively determine the fractional part of the volume of a crystal that already has a nearly commensurate structure in the incommensurate phase near the lock-in transition. Among these systems, Rb2ZnCl4 was probably most often investigated by nuclear magnetic resonance. Many researchers have reported on soliton density near Tc3 ( 303 K) using 87Rb NMR technique [10±17]. The information regarding the structure and internal motion of solids can be obtained by nuclear magnetic
0038-1098/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0038-109 8(01)00145-4
454
A.R. Lim et al. / Solid State Communications 118 (2001) 453±457
resonance techniques. From the relaxation time measurements, it was determined that the slope of relaxation time plotted as a function of temperature undergoes an abrupt change in the neighborhood of the phase transition temperature. The interest in this research lies in the links between changes in spin-lattice relaxation times and the structural phase transitions. The current investigation is to study the nuclear spinlattice relaxation time, T1, near 75 and 192 K for 87Rb in a Rb2ZnCl4 single crystal in order to obtain more information about the characteristic spin dynamics. The aim of this investigation is to study in more detail the phase transition mechanism at low temperature. The measured relaxation rate was found to be proportional to the T 2 in each phase. Furthermore, the relaxation time of 87Rb in a Rb2ZnCl4 single crystal at low temperature is a new observation.
2. Crystal structure The phase diagram of Rb2ZnCl4 is shown schematically in Table 1, with the transition temperatures separating phases IV, III, II, and I, de®ned as Tc1, Tc2, and Tc3 as indicated in Table 1. Phase I in the paraelectric state is centrosymmetrical. The space group is orthorhombic D16 2h Ê , b 12.730 A Ê , and with lattice constants a 7.285 A Ê [18,19]. An incommensurate modulation of c 9.265 A the superlattice in phase II takes place along the c-axis, and the commensurate superlattice is recovered in phase III with a period of 3c [20]. Phase IV is also ferroelectric and belongs to a monoclinic space group Cs4 . The structure of Rb2ZnCl4 is shown in Fig. 1. The two Rb 1 ions and two of the Cl 2 ions from each molecule occupy inequivalent sites on a mirror plane, as does the Zn 21 ion [21]. The other two Cl 2 ions occupy equivalent sites Cl(3) on opposite sides of the mirror plane [22,23]. The paraelectric unit cell contains eight Rb ions, which belong to two inequivalent sets: Rb(1) and Rb(2). The Rb(1) sites lie on the pseudohexagonal and Rb(2) sites on the pseudotriad axes. All Rb sites have site symmetry m and lie in the mirror plane perpendicular to the crystal a-axis.
Fig. 1. Projections of the atomic positions of Rb2ZnCl4: (a) in the ab-plane; (b) in the ac-plane.
3. Experimental procedure The single crystal Rb2ZnCl4 samples used in this work were grown by evaporation at room temperature from an aqueous solution containing 2:1 stoichiometric amounts of RbCl and ZnCl2. Samples with a size of approximately 8 £ 4 £ 1.5 mm 3 were prepared. Nuclear magnetic resonance signals of 87Rb in the Rb2ZnCl4 single crystal were measured using a homemade pulse NMR spectrometer. The static magnetic ®eld was 4.7 T, and the central rf frequency was set at v o/ 2p 65.477 MHz for 87Rb nucleus. The spectra were taken by using a solid echo sequence, (p/2 2 t 2 p/2), to eliminate artefacts due to probe ringing. The p/2 pulse width was 6 ms and the pulse separation t was 60 ms. The sample temperatures were constantly maintained by
Table 1 Phase transition sequence of Rb2ZnCl4 Transition temperature Phase Space group Structure Lattice parameters
IV Ferroelectric Commensurate C1c1 (Cs4 ) Monoclinic Ê a 7.201 A Ê b 12.542 A Ê c 27.439 A Z 24
Tc1 (75 K)
Tc2 (192 K) III Ferroelectric Commensurate 9 P21cn (C2v ) Orthorhombic a0 a b0 b c 0 3c Z 12
Tc3 (303 K) II Incommensurate Pmcn Orthorhombic Ê a 7.253 A Ê b 12.646 A Ê c 9.221 A
I Paraelectric Normal Pmcn (D16 2h ) Orthorhombic Ê a 7.285 A Ê b 12.730 A Ê c 9.265 A Z4
A.R. Lim et al. / Solid State Communications 118 (2001) 453±457
controlling helium gas ¯ow and heater current, giving an accuracy of ^0.1 K. 4. Experimental results and analysis Rb2ZnCl4 have two inequivalent Rb(1) and Rb(2) nuclei. Because of the presence of the center of symmetry, two Rb sites can be distinguished for each chemically inequivalent set. For a general direction of the magnetic ®eld in the ac-plane, all four lines should be seen [24]. For a general direction of the magnetic ®eld in the ab- and cb-planes, the two spectra of each inequivalent site coincide due to symmetry. The NMR spectrum of 87Rb (I 3/2) consists of two central lines at room temperature as shown in Fig. 2. It consists of two lines displaced to the higher frequency side relative to the reference signal. The reference signal was given by 87Rb line measured in an aqueous solution of RbNO3. The signals of these two Rb represent the transition of 87Rb central NMR lines due to Rb(1) and Rb(2). With lowering temperatures, the lines broaden
Fig. 2. The NMR spectra for Rb(1) and Rb(2) at room temperature.
455
signi®cantly and their intensity decreases. As the crystal is cooled, the splitting of resonance lines is observed. At lowering temperatures below 192 K, new lines are detectable in addition to the initial two lines. The spin-lattice relaxation times of 87Rb in Rb2ZnCl4 were measured over a temperature range of 4.2±300 K. The spin-lattice relaxation times were taken by saturation recovery using solid echo sequence, [(p/2 2 t 2 (p/2 2 t 2 p/2)]. The echo amplitudes were measured with varying times, t, in the above pulse sequence. The recovery trace for the central line of 87Rb with dominant quadrupole relaxation can be represented by a linear combination of two exponential functions [25,26]; 1 2 Mz t=M1 1=2exp 22W1 t 1 exp 22W2 t
1
where W1 and W2 are the transition probabilities for uDmu 1, 2. The recovery data for the temperature investigated here ®ts well into Eq. (1). Thus the relaxation times are given by [25,26]: 1=T1 2=5 W1 1 4W2
2
We measured the relaxation times of the central lines for Rb(1) and Rb(2) with decreasing temperature. The temperature dependence of the nuclear spin-lattice relaxation rate, T121 , for 87Rb is shown in Fig. 3. This is the relaxation rate for Rb(1). However, the trend of T121 for Rb(2) is very similar, and its value is consistent in the error range. The relaxation rate for the 87Rb nucleus exhibits remarkable changes near 75 and 192 K, indicating a drastic alteration of spin dynamics at the transition temperatures. These means that the phase IV to III transition is a second-order type, and the phase III to II is a ®rst-order phase transition. The relaxation rate increases with increasing temperature in phases III and IV, while in phase II it decreases with increasing temperature. The spin-lattice relaxation times
Fig. 3. Temperature dependence of the 87Rb spin-lattice relaxation rate in a Rb2ZnCl4 single crystal. The solid line represents a ®t due to Raman process.
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A.R. Lim et al. / Solid State Communications 118 (2001) 453±457
are T1 5.41 ms at 291 K (phase II), T1 104 ms at 121 K (phase III), and 50 s at 4.2 K (phase IV), respectively. The interaction of the nuclear quadrupole moment with lattice vibrations is a vital relaxation mechanism for nuclear spin I ^ 1 in many crystals. The coupling can generally be written as a spin-lattice Hamiltonian [27]: X q q H F A 3 where F (q) and A (q) are the lattice and spin operators, respectively, of order q. The lattice operators F (q) (hereafter we will omit the index q, for brevity) can be expanded as a function of stress tensor s : F F 0 1 F1 s 1 F 2 s 2 1 F3 s 3 1 ¼
4
At temperatures far below the melting temperature of the crystal, we can expect the thermal stress to be small, and only the ®rst few terms in Eq. (4) are important. The term F1s represents the absorption or emission of a single phone (direct procss). The next term F2s 2 indicates the emission or absorption of two phonons, or absorption of one phonon followed by emission of another one (Raman process). In the direct process, the spin-lattice relaxation rate T 121 is proportional to the square of the frequency v 0 and to the absolute temperature T for kBT/Év 0 q 1. On the other hand, the Raman process allows for a relaxation rate proportional to T 2. Our experimental results show that the temperature dependences of spin-lattice relaxation rate, T121 , for 87Rb are proportional to the T 2 in phases II, III and IV, which are shown by the solid line in Fig. 3. Therefore, the relaxation times undergo the Raman process. In simple NMR theory, the general behavior of the spin-lattice relaxation rate for random motions of the Arrhenius type with a correlation time t c is described in terms of three regimes, including both, fast and slow motion regimes. The fast motion regime can be described as v ot c p 1, T121 , expEa =RT; and the slow motion regime as v ot c q 1, T121 , v22 o exp2Ea =RT; where v o is Larmor frequency and Ea is the activation energy. It is known that different limits are satis®ed for v ot c in each of the three temperature ranges separated by Tc1 75 K and Tc2 192 K. Speci®cally, the limit v ot c q 1 would apply for T . Tc2, and the limit v ot c p 1 for both T , Tc1 and Tc1 , T , Tc2. The activation energies obtained from the distinct slopes in the spin-lattice relaxation rates (T121 ) vs. the inverse temperature (1000/T ) are 0.17 kJ/mol in the low temperature phase below Tc1, 2.05 kJ/mol between Tc1 and Tc2, and 3.72 kJ/mol in the high temperature phase above Tc2. 5. Conclusion The relaxation processes of 87Rb for the Rb2ZnCl4 crystal were studied below room temperature. The changes in the curve of the spin-lattice relaxation rate near 75 and 192 K
correspond to a second-order and ®rst-order phase transition in the crystal, respectively. The relaxation rate slowly increases in phases III and IV as the temperature is increased, while in phase II it decreases with increasing temperature. The anomalous decrease in T1 around Tc1 is due to the critical slowing down of the soft mode as it is usually observed in a structural phase transitions from the monoclinic to the orthorhombic phase in this crystal. The jump in T121 at Tc2 and the behaviour of T121 throughout phase II are, however, rather unusual. The abrupt change in T121 at Tc2 can be considered a shortening in the c-direction as a result of a phase transition from the commensurate to the incommensurate; above Tc2, the short T1 means that the energy transfer rate from the nuclear spin system to the surrounding environment is easier than below Tc2. The linear dependences in Fig. 3 are described with the simple power law T121 A 1 BT k : Least squares ®t for our data at lower temperatures gave the value of k 2 for the 87 Rb nucleus. Among the phonon processes, the Raman process with k 2 is considered to be more effective than the direct process for nuclear quadrupole relaxation. The temperature dependence of the relaxation rate is in accordance with the Raman process of the nuclear spinlattice relaxation in each phase. Acknowledgements This work was supported by a Korean Research Foundation Grant (KRF-99-015-DP0130). References [1] N.G. Zamkova, V.I. Zinenko, JETP 80 (1995) 713. [2] Y. Koyama, T. Nagata, K. Koike, Phys. Rev. B 51 (1995) 12157. [3] S. Sawada, Y. Shiroishi, A. Yamamoto, M. Takashige, M. Matsuo, J. Phys. Soc. Jpn. 43 (1977) 2099. [4] K. Gesi, M. Iizumi, J. Phys. Soc. Jpn. 46 (1979) 697. [5] E. Francke, M.L. Postollec, J.P. Mathieu, H. Poulet, Solid State Commun. 33 (1980) 155. [6] I. Noiret, Y. Guinet, A. Hedoux, Phys. Rev. B 52 (1995) 13206. [7] M. Wada, A. Sawada, Y. Ishibashi, J. Phys. Soc. Jpn. 47 (1979) 1185. [8] M. Wada, A. Sawada, Y. Ishibashi, J. Phys. Soc. Jpn. 50 (1981) 531. [9] H.M. Lu, J.R. Hardy, Phys. Rev. B 45 (1992) 7609. [10] E. Schneider, Solid State Commun. 44 (1982) 885. [11] R. Blinc, I.P. Aleksandrova, A.S. Chaves, F. Milia, V. Rutar, J. Seliger, B. Topic, S. Zumer, J. Phys. C: Solid State Phys. 15 (1982) 547. [12] R. Blinc, F. Milia, V. Rutar, S. Zumer, Phys. Rev. Lett. 48 (1982) 47. [13] V. Rutar, F. Milia, B. Topic, R. Blinc, T. Rasing, Phys. Rev. B 25 (1982) 281. [14] R. Blinc, P. Prelovsek, R. Kind, Phys. Rev. B 27 (1983) 5404.
A.R. Lim et al. / Solid State Communications 118 (2001) 453±457 [15] J. Petersson, E. Schneider, Z. Phys. B-Condens. Matter 61 (1985) 33. [16] D.C. Ailion, S. Chen, Solid State Commun. 92 (1994) 835. [17] R. Walisch, J. Petersson, D. Schuûler, U. Hacker, D. Michel, J.M. Perez-Mato, Phys. Rev. B 50 (1994) 16192. [18] S.R. Andrews, H. Mashiyama, J. Phys. C: Solid State Phys. 16 (1983) 4985. [19] F. Parisi, H. Bonadeo, Acta Cryst. A 53 (1997) 286. [20] A. Hedoux, D. Grebille, J. Jaud, G. Godefroy, Acta Cryst. B 45 (1989) 370. [21] A.S. Secco, J. Trotter, Acta Cryst. C 39 (1983) 317.
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[22] P.J. Edwardson, V. Katkanant, J.R. Hardy, L.L. Boyer, Phys. Rev. B 35 (1987) 8470. [23] P.J. Edwardson, J.R. Hardy, Phys. Rev. B 38 (1988) 2250. [24] V. Rutar, J. Seliger, B. Topic, R. Blinc, I.P. Aleksandrova, Phys. Rev. B 24 (1981) 2397. [25] A. Avogadro, E. Gavellus, D. Muller, J. Petersson, Phys. Stat. Solidi (b) 44 (1971) 639. [26] M. Igarashi, H. Kitagawa, S. Takahashi, R. Yoshizaki, Y. Abe, Z. Naturforsch 47a (1992) 313. [27] A. Abragam, The Principles of Nucleus Magnetism, Oxford University Press, Oxford, 1961.
Solid State Communications 118 (2001) 453±457
www.elsevier.com/locate/ssc
Raman process studied by 87Rb spin-lattice relaxation in a Rb2ZnCl4 single crystal at low temperature Ae Ran Lim a,*, Jae Kap Jung b, Se-Young Jeong c b
a Department of Physics, Jeonju University, Jeonju 560-759, South Korea BK21 Physics Research Division, Department of Physics, Sungkyunkwan University, Suwon 440-746, South Korea c Department of Physics, Pusan National University, Pusan 609-735, South Korea
Received 12 January 2001; accepted 18 March 2001 by A. Pinczuk
Abstract 87 Rb 1/2 $ 2 1/2 spin-lattice relaxation time in the Rb2ZnCl4 single crystal was measured in the temperature range of 4.2± 300 K. The dominant relaxation of this crystal in the whole temperature range investigated here is due to quadrupolar interaction. The changes in the 87Rb spin-lattice relaxation rate near 75 and 192 K correspond to the phase transitions in the crystal. These suggest that the phase IV to III transition is of second-order, and the phase III to II transition is of ®rst-order. The temperature dependence of the relaxation rates in phases II, III, and IV is in accordance with the Raman process dominated by the phonon mechanism. q 2001 Published by Elsevier Science Ltd.
PACS: 76.60.2k; 77.80.Bb; 77.80.2e Keywords: A. Ferroelectrics; B. Crystal growth; D. Phase transitions; E. Nuclear resonances
1. Introduction Rb2ZnCl4 belongs to a large family of crystals with the general formula A2BX4, where A is an alkali metal and BX4 is a tetrahedral group (SO4, SeO4, CrO4, ZnCl4, and so on) [1,2]. This family of crystals has been attracting the attention of investigators for many decades. These substances are of interest due to the great diversity of successive structural phase transitions. Incommensurate phases have been discovered in some members of this family of crystals. The numerous studies that have been carried out on Rb2ZnCl4 crystals have shown that this compound can be found in at least four phases depending on temperature [3±6]. Rb2ZnCl4 undergoes successive phase transitions from a paraelectric phase (N phase) to an incommensurate phase (IC phase) at Tc3 303 K, to a ferroelectric commensurate phase (C phase) at Tc2 192 K [3,4,7], and to a lowest-temperature phase at Tc1 75 K [8]. The high-temperature phase belongs to the ortho* Corresponding author. Tel.: 182-63-220-2514; fax: 182-63220-2362, 182-42-483-6155. E-mail address: [email protected] (A.R. Lim).
rhombic space group Pmcn with four formula units per unit cell. The intermediate phase is incommensurate and 9 the low-temperature commensurate phase C2v exhibits improper ferroelectricity with the lattice parameter tripled along the pseudohexagonal axis. This crystal undergoes a second-order phase transition from a paraelectric (N phase) to an incommensurate (IC phase) at about Tc3 303 K and a ®rst-order phase transition from the IC phase to ferroelectric commensurate at Tc2 192 K [9]. Previously, a variety of experimental methods have been applied to verify the concept of phase solitons or discommensurations in structurally incommensurate systems. Particularly, it has been shown that nuclear magnetic resonance techniques can be used to quantitatively determine the fractional part of the volume of a crystal that already has a nearly commensurate structure in the incommensurate phase near the lock-in transition. Among these systems, Rb2ZnCl4 was probably most often investigated by nuclear magnetic resonance. Many researchers have reported on soliton density near Tc3 ( 303 K) using 87Rb NMR technique [10±17]. The information regarding the structure and internal motion of solids can be obtained by nuclear magnetic
0038-1098/01/$ - see front matter q 2001 Published by Elsevier Science Ltd. PII: S 0038-109 8(01)00145-4
454
A.R. Lim et al. / Solid State Communications 118 (2001) 453±457
resonance techniques. From the relaxation time measurements, it was determined that the slope of relaxation time plotted as a function of temperature undergoes an abrupt change in the neighborhood of the phase transition temperature. The interest in this research lies in the links between changes in spin-lattice relaxation times and the structural phase transitions. The current investigation is to study the nuclear spinlattice relaxation time, T1, near 75 and 192 K for 87Rb in a Rb2ZnCl4 single crystal in order to obtain more information about the characteristic spin dynamics. The aim of this investigation is to study in more detail the phase transition mechanism at low temperature. The measured relaxation rate was found to be proportional to the T 2 in each phase. Furthermore, the relaxation time of 87Rb in a Rb2ZnCl4 single crystal at low temperature is a new observation.
2. Crystal structure The phase diagram of Rb2ZnCl4 is shown schematically in Table 1, with the transition temperatures separating phases IV, III, II, and I, de®ned as Tc1, Tc2, and Tc3 as indicated in Table 1. Phase I in the paraelectric state is centrosymmetrical. The space group is orthorhombic D16 2h Ê , b 12.730 A Ê , and with lattice constants a 7.285 A Ê [18,19]. An incommensurate modulation of c 9.265 A the superlattice in phase II takes place along the c-axis, and the commensurate superlattice is recovered in phase III with a period of 3c [20]. Phase IV is also ferroelectric and belongs to a monoclinic space group Cs4 . The structure of Rb2ZnCl4 is shown in Fig. 1. The two Rb 1 ions and two of the Cl 2 ions from each molecule occupy inequivalent sites on a mirror plane, as does the Zn 21 ion [21]. The other two Cl 2 ions occupy equivalent sites Cl(3) on opposite sides of the mirror plane [22,23]. The paraelectric unit cell contains eight Rb ions, which belong to two inequivalent sets: Rb(1) and Rb(2). The Rb(1) sites lie on the pseudohexagonal and Rb(2) sites on the pseudotriad axes. All Rb sites have site symmetry m and lie in the mirror plane perpendicular to the crystal a-axis.
Fig. 1. Projections of the atomic positions of Rb2ZnCl4: (a) in the ab-plane; (b) in the ac-plane.
3. Experimental procedure The single crystal Rb2ZnCl4 samples used in this work were grown by evaporation at room temperature from an aqueous solution containing 2:1 stoichiometric amounts of RbCl and ZnCl2. Samples with a size of approximately 8 £ 4 £ 1.5 mm 3 were prepared. Nuclear magnetic resonance signals of 87Rb in the Rb2ZnCl4 single crystal were measured using a homemade pulse NMR spectrometer. The static magnetic ®eld was 4.7 T, and the central rf frequency was set at v o/ 2p 65.477 MHz for 87Rb nucleus. The spectra were taken by using a solid echo sequence, (p/2 2 t 2 p/2), to eliminate artefacts due to probe ringing. The p/2 pulse width was 6 ms and the pulse separation t was 60 ms. The sample temperatures were constantly maintained by
Table 1 Phase transition sequence of Rb2ZnCl4 Transition temperature Phase Space group Structure Lattice parameters
IV Ferroelectric Commensurate C1c1 (Cs4 ) Monoclinic Ê a 7.201 A Ê b 12.542 A Ê c 27.439 A Z 24
Tc1 (75 K)
Tc2 (192 K) III Ferroelectric Commensurate 9 P21cn (C2v ) Orthorhombic a0 a b0 b c 0 3c Z 12
Tc3 (303 K) II Incommensurate Pmcn Orthorhombic Ê a 7.253 A Ê b 12.646 A Ê c 9.221 A
I Paraelectric Normal Pmcn (D16 2h ) Orthorhombic Ê a 7.285 A Ê b 12.730 A Ê c 9.265 A Z4
A.R. Lim et al. / Solid State Communications 118 (2001) 453±457
controlling helium gas ¯ow and heater current, giving an accuracy of ^0.1 K. 4. Experimental results and analysis Rb2ZnCl4 have two inequivalent Rb(1) and Rb(2) nuclei. Because of the presence of the center of symmetry, two Rb sites can be distinguished for each chemically inequivalent set. For a general direction of the magnetic ®eld in the ac-plane, all four lines should be seen [24]. For a general direction of the magnetic ®eld in the ab- and cb-planes, the two spectra of each inequivalent site coincide due to symmetry. The NMR spectrum of 87Rb (I 3/2) consists of two central lines at room temperature as shown in Fig. 2. It consists of two lines displaced to the higher frequency side relative to the reference signal. The reference signal was given by 87Rb line measured in an aqueous solution of RbNO3. The signals of these two Rb represent the transition of 87Rb central NMR lines due to Rb(1) and Rb(2). With lowering temperatures, the lines broaden
Fig. 2. The NMR spectra for Rb(1) and Rb(2) at room temperature.
455
signi®cantly and their intensity decreases. As the crystal is cooled, the splitting of resonance lines is observed. At lowering temperatures below 192 K, new lines are detectable in addition to the initial two lines. The spin-lattice relaxation times of 87Rb in Rb2ZnCl4 were measured over a temperature range of 4.2±300 K. The spin-lattice relaxation times were taken by saturation recovery using solid echo sequence, [(p/2 2 t 2 (p/2 2 t 2 p/2)]. The echo amplitudes were measured with varying times, t, in the above pulse sequence. The recovery trace for the central line of 87Rb with dominant quadrupole relaxation can be represented by a linear combination of two exponential functions [25,26]; 1 2 Mz t=M1 1=2exp 22W1 t 1 exp 22W2 t
1
where W1 and W2 are the transition probabilities for uDmu 1, 2. The recovery data for the temperature investigated here ®ts well into Eq. (1). Thus the relaxation times are given by [25,26]: 1=T1 2=5 W1 1 4W2
2
We measured the relaxation times of the central lines for Rb(1) and Rb(2) with decreasing temperature. The temperature dependence of the nuclear spin-lattice relaxation rate, T121 , for 87Rb is shown in Fig. 3. This is the relaxation rate for Rb(1). However, the trend of T121 for Rb(2) is very similar, and its value is consistent in the error range. The relaxation rate for the 87Rb nucleus exhibits remarkable changes near 75 and 192 K, indicating a drastic alteration of spin dynamics at the transition temperatures. These means that the phase IV to III transition is a second-order type, and the phase III to II is a ®rst-order phase transition. The relaxation rate increases with increasing temperature in phases III and IV, while in phase II it decreases with increasing temperature. The spin-lattice relaxation times
Fig. 3. Temperature dependence of the 87Rb spin-lattice relaxation rate in a Rb2ZnCl4 single crystal. The solid line represents a ®t due to Raman process.
456
A.R. Lim et al. / Solid State Communications 118 (2001) 453±457
are T1 5.41 ms at 291 K (phase II), T1 104 ms at 121 K (phase III), and 50 s at 4.2 K (phase IV), respectively. The interaction of the nuclear quadrupole moment with lattice vibrations is a vital relaxation mechanism for nuclear spin I ^ 1 in many crystals. The coupling can generally be written as a spin-lattice Hamiltonian [27]: X q q H F A 3 where F (q) and A (q) are the lattice and spin operators, respectively, of order q. The lattice operators F (q) (hereafter we will omit the index q, for brevity) can be expanded as a function of stress tensor s : F F 0 1 F1 s 1 F 2 s 2 1 F3 s 3 1 ¼
4
At temperatures far below the melting temperature of the crystal, we can expect the thermal stress to be small, and only the ®rst few terms in Eq. (4) are important. The term F1s represents the absorption or emission of a single phone (direct procss). The next term F2s 2 indicates the emission or absorption of two phonons, or absorption of one phonon followed by emission of another one (Raman process). In the direct process, the spin-lattice relaxation rate T 121 is proportional to the square of the frequency v 0 and to the absolute temperature T for kBT/Év 0 q 1. On the other hand, the Raman process allows for a relaxation rate proportional to T 2. Our experimental results show that the temperature dependences of spin-lattice relaxation rate, T121 , for 87Rb are proportional to the T 2 in phases II, III and IV, which are shown by the solid line in Fig. 3. Therefore, the relaxation times undergo the Raman process. In simple NMR theory, the general behavior of the spin-lattice relaxation rate for random motions of the Arrhenius type with a correlation time t c is described in terms of three regimes, including both, fast and slow motion regimes. The fast motion regime can be described as v ot c p 1, T121 , expEa =RT; and the slow motion regime as v ot c q 1, T121 , v22 o exp2Ea =RT; where v o is Larmor frequency and Ea is the activation energy. It is known that different limits are satis®ed for v ot c in each of the three temperature ranges separated by Tc1 75 K and Tc2 192 K. Speci®cally, the limit v ot c q 1 would apply for T . Tc2, and the limit v ot c p 1 for both T , Tc1 and Tc1 , T , Tc2. The activation energies obtained from the distinct slopes in the spin-lattice relaxation rates (T121 ) vs. the inverse temperature (1000/T ) are 0.17 kJ/mol in the low temperature phase below Tc1, 2.05 kJ/mol between Tc1 and Tc2, and 3.72 kJ/mol in the high temperature phase above Tc2. 5. Conclusion The relaxation processes of 87Rb for the Rb2ZnCl4 crystal were studied below room temperature. The changes in the curve of the spin-lattice relaxation rate near 75 and 192 K
correspond to a second-order and ®rst-order phase transition in the crystal, respectively. The relaxation rate slowly increases in phases III and IV as the temperature is increased, while in phase II it decreases with increasing temperature. The anomalous decrease in T1 around Tc1 is due to the critical slowing down of the soft mode as it is usually observed in a structural phase transitions from the monoclinic to the orthorhombic phase in this crystal. The jump in T121 at Tc2 and the behaviour of T121 throughout phase II are, however, rather unusual. The abrupt change in T121 at Tc2 can be considered a shortening in the c-direction as a result of a phase transition from the commensurate to the incommensurate; above Tc2, the short T1 means that the energy transfer rate from the nuclear spin system to the surrounding environment is easier than below Tc2. The linear dependences in Fig. 3 are described with the simple power law T121 A 1 BT k : Least squares ®t for our data at lower temperatures gave the value of k 2 for the 87 Rb nucleus. Among the phonon processes, the Raman process with k 2 is considered to be more effective than the direct process for nuclear quadrupole relaxation. The temperature dependence of the relaxation rate is in accordance with the Raman process of the nuclear spinlattice relaxation in each phase. Acknowledgements This work was supported by a Korean Research Foundation Grant (KRF-99-015-DP0130). References [1] N.G. Zamkova, V.I. Zinenko, JETP 80 (1995) 713. [2] Y. Koyama, T. Nagata, K. Koike, Phys. Rev. B 51 (1995) 12157. [3] S. Sawada, Y. Shiroishi, A. Yamamoto, M. Takashige, M. Matsuo, J. Phys. Soc. Jpn. 43 (1977) 2099. [4] K. Gesi, M. Iizumi, J. Phys. Soc. Jpn. 46 (1979) 697. [5] E. Francke, M.L. Postollec, J.P. Mathieu, H. Poulet, Solid State Commun. 33 (1980) 155. [6] I. Noiret, Y. Guinet, A. Hedoux, Phys. Rev. B 52 (1995) 13206. [7] M. Wada, A. Sawada, Y. Ishibashi, J. Phys. Soc. Jpn. 47 (1979) 1185. [8] M. Wada, A. Sawada, Y. Ishibashi, J. Phys. Soc. Jpn. 50 (1981) 531. [9] H.M. Lu, J.R. Hardy, Phys. Rev. B 45 (1992) 7609. [10] E. Schneider, Solid State Commun. 44 (1982) 885. [11] R. Blinc, I.P. Aleksandrova, A.S. Chaves, F. Milia, V. Rutar, J. Seliger, B. Topic, S. Zumer, J. Phys. C: Solid State Phys. 15 (1982) 547. [12] R. Blinc, F. Milia, V. Rutar, S. Zumer, Phys. Rev. Lett. 48 (1982) 47. [13] V. Rutar, F. Milia, B. Topic, R. Blinc, T. Rasing, Phys. Rev. B 25 (1982) 281. [14] R. Blinc, P. Prelovsek, R. Kind, Phys. Rev. B 27 (1983) 5404.
A.R. Lim et al. / Solid State Communications 118 (2001) 453±457 [15] J. Petersson, E. Schneider, Z. Phys. B-Condens. Matter 61 (1985) 33. [16] D.C. Ailion, S. Chen, Solid State Commun. 92 (1994) 835. [17] R. Walisch, J. Petersson, D. Schuûler, U. Hacker, D. Michel, J.M. Perez-Mato, Phys. Rev. B 50 (1994) 16192. [18] S.R. Andrews, H. Mashiyama, J. Phys. C: Solid State Phys. 16 (1983) 4985. [19] F. Parisi, H. Bonadeo, Acta Cryst. A 53 (1997) 286. [20] A. Hedoux, D. Grebille, J. Jaud, G. Godefroy, Acta Cryst. B 45 (1989) 370. [21] A.S. Secco, J. Trotter, Acta Cryst. C 39 (1983) 317.
457
[22] P.J. Edwardson, V. Katkanant, J.R. Hardy, L.L. Boyer, Phys. Rev. B 35 (1987) 8470. [23] P.J. Edwardson, J.R. Hardy, Phys. Rev. B 38 (1988) 2250. [24] V. Rutar, J. Seliger, B. Topic, R. Blinc, I.P. Aleksandrova, Phys. Rev. B 24 (1981) 2397. [25] A. Avogadro, E. Gavellus, D. Muller, J. Petersson, Phys. Stat. Solidi (b) 44 (1971) 639. [26] M. Igarashi, H. Kitagawa, S. Takahashi, R. Yoshizaki, Y. Abe, Z. Naturforsch 47a (1992) 313. [27] A. Abragam, The Principles of Nucleus Magnetism, Oxford University Press, Oxford, 1961.