As a member of the ferroelectrics A₂BX₄ compounds, Rb₂ZnBr₄ crystals have attracted a strongly increasing attention for the last several years. The existence of incommensurate phases of the crystal is highly intriguing[1]. These phases are reported to create two or more periodicity simultaneously, which are incommensurate to each other. This incommensurate phase occurs as an intermediate phase between a low-temperature commensurate and a high-temperature normal phase, both having ordinary crystal structures. The incommensurability implies a loss of periodicity in one or more directions of the crystal lattice[2].

The crystals undergo a sequence of phase transitions (PT) under varying temperature[3-7]. When cooled, Rb₂ZnBr₄ crystals undergo a second-order phase transition from a paraelectric orthorhombic to an incommensurate orthorhombic phase at TC2 = 347 K, and a first-order transition from the incommensurate to the ferroelectric commensurate phase at TC1 = 200 K[8-11]. Besides these phase transitions, Rb₂ZnBr₄ crystals undergo three more phase transitions at TC3 = 108 K, TC4 = 80 K, and TC5 = 50 K at lower temperatures. In case of Rb₂ZnBr₄, however, there are reports of inconsistencies concerning the phase transition temperatures and even the existence itself for certain transitions[12, 13]. Rb₂ZnBr₄ has orthorhombic symmetry from the melting point down to liquid helium temperature. However, below 347 K, the lattice is deformed by incommensurate displacement waves, and these incommensurable structures have gained considerable interest over the past years[14].

The space group of the high temperature paraelectric phase is D2h16-Pmcn and the space group of the low temperature ferroelectric phase is C2v9-P21cn. Rb₂ZnBr₄ has been found to show antiferroelectricity along the b-axis in the phase from 80 K to 108 K and ferroelectricity along the a-axis below 200 K[15, 16]. The properties of crystals with incommensurate phases are very sensitive to impurities and other lattice defects. This is especially true for molecular and ionic crystals of Rb₂ZnBr₄, where interatomic forces in the ZnBr₄ tetrahedron are strong, while the chemical bonds between Rb​+ ions and the ZnBr₄ tetrahedra are weak and sensitive in the crystal lattice[17].

In the presence of nuclear quadrupole interactions, NMR has proven to be both sensitive and accurate in investigating incommensurate phases. It investigates the local environment of the nucleus through the interactions of quadrupole moments and electric field gradients at individual lattice sites[18]. The numbers of nuclear quadrupole resonance (NQR) frequency for ​81Br nuclei in Rb₂ZnBr₄ can be different depending on the phase of Rb₂ZnBr₄[13]. The statics and dynamics in incommensurate Rb₂ZnBr₄ has been investigated by ⁸⁷Rb NMR[18]. Blinc et al.[1, 19-21] reported the ⁸⁷Rb NMR line shape and the distributions of electric field gradient (EFG) tensor of Rb nuclei in the paraelectric, incommensurate, and ferroelectric phases of Rb₂ZnBr₄ crystals. In addition to this, several more papers studying the ⁸⁷Rb nuclei in Rb₂ZnBr₄ have been published[22-24].

NMR measurements on nuclei is very helpful for investigation of local structure and the dynamics of the incommensurate phase. In this paper, two sets of ⁸⁷Rb NMR spectra in Rb₂ZnBr₄ crystal are analyzed. The temperature dependence of the ⁸⁷Rb nuclei in Rb₂ZnBr₄ crystal and spin-lattice relaxation time (T1) within the laboratory coordinate system is studied in the paraelectric, incommensurate, and ferroelectric commensurate phases of Rb₂ZnBr₄. The chemical shifts and the relaxation mechanism of the ⁸⁷Rb NMR spectra are also discussed in Rb₂ZnBr₄ crystal.

When the temperature decreases, the paraelectric phase, in the space group Pmcn, transforms into the incommensurate phase at TC1 = 347 K, and further into the ferroelectric phase, in the space group P21cn, at TC2 = 200 K. The lattice constants in the paraelectric phase have been reported as a = 7.679 Å, b = 13.386 Å, and c = 9.753 Å at 373 K[25]. The structure of Rb₂ZnBr₄ is shown in Fig. 1. The average unit cell of the crystal is orthorhombic and has four formula units with two sets of chemically inequivalent Rb atoms and four ZnBr₄ tetrahedral groups considered as rigid entities. The incommensurate modulation occurs along the c-axis in the incommensurate phase, and the commensurate superlattice is recovered in the ferroelectric phase with a period of 3c. Spontaneous polarization occurs along the a-axis. The unit cell in the paraelectric phase has 8 Rb ions belonging to two chemically inequivalent sites: Rb​I and Rb​II. The Rb​I sites lie on the pseudohexagonal and Rb​II sites on the pseudotrid axes. All Rb sites have site symmetry m. Rb₂ZnBr₄ melts at 753 K[26].

Figure 1. (Color online) Structure of Rb₂ZnBr₄ single crystal in orthorhombic phases. The interlattice vacancies between the isolated tetrahedral ZnBr₄ units are filled with Rb cations. The white balls represent Zn cations and the black balls designate Br anions.

Rb₂ZnBr₄ single crystals were grown through slow evaporation of aqueous solutions by making RbBr and ZnBr₂ into aqueous solutions at a molar ratio of 2:1. The crystals were colorless though not completely transparent and were selected for our experiment. The directions of the single crystal axes were determined through an X-ray spectrometer.

The NMR spectra of the ⁸⁷Rb nuclei in Rb₂ZnBr₄ crystals were obtained using the Bruker FT NMR spectrometers (DSX 400) at the Korea Basic Science Institute. A 9.4 T magnetic field was applied, and the rf frequency of the ⁸⁷Rb nucleus was set to 130.93 MHz. The spin-lattice relaxation time was observed using π–t–π/2 pulse sequences. The nuclear magnetization of the ⁸⁷Rb nucleus was obtained from the inversion recovery sequence at time t after the π pulse. NMR data were obtained for ⁸⁷Rb nuclei in Rb₂ZnBr₄ with temperature variations of 180 to 420 K. The inversion-recovery methods related to spin-echo sequences and phase cycling are applied. The width of the π/2 pulse is set to 2 μs, and in particular, close to Ti, the repetition time is selected as about 2 s to avoid heating the sample.

Figure 2 shows the inversion recovery traces for ⁸⁷Rb nuclei in Rb₂ZnBr₄ when the delay time t varies from 1000 ns to 2 s at 300 K. The y-axis shows the relative magnetization according to the delay time. The typical NMR spectra of ⁸⁷Rb in a Rb₂ZnBr₄ crystal are obtained at 300 K and represented in Fig. 3 when the static magnetic field is in parallel to the crystallographic c-axis. The spectrum is a Fourier transform (FT) of free-induction decay for the ⁸⁷Rb NMR. The ⁸⁷Rb NMR spectrum is shifted from the operating frequency ω0/2π = 130.93 MHz due to the chemical shift. Not all but only some of the resonance field lines can be excited to measure T1, as it is impossible to homogeneously excite all resonant lines due to the large quadrupole splitting between the lines.

Figure 2. (Color online) The inversion recovery traces with delay time variation for ⁸⁷Rb nucleus at 300 K.

Figure 3. (Color online) Typical NMR spectra of ⁸⁷Rb in the Rb₂ZnBr₄ crystal at 180 K. The resonance spectra consist of two physically inequivalent ⁸⁷Rb​I and ⁸⁷Rb​II nuclei sites.

Energy levels of a ⁸⁷Rb nucleus (I = 3/2) can be split into 4 energy levels (3/2, 1/2, -1/2, -3/2) because nuclear quadrupole separation is due to nuclear quadrupole interactions with electric field gradients. The electric quadrupole moment of a ⁸⁷Rb nucleus is Q=+0.132×1024 cm​2. Then, three ⁸⁷Rb nuclear magnetic resonance lines in Rb₂ZnBr₄ crystal can be obtained between the adjacent energy levels. One central resonance line (-1/2 ↔ +1/2) shows a strong intensity and two satellite lines (-3/2 ↔ -1/2, +1/2 ↔ +3/2) show a relatively weaker intensity. If the Rb₂ZnBr₄ crystal has only a single type of chemically equivalent ⁸⁷Rb nucleus, a central resonance line and two satellite lines should be observed. However, 6 resonance lines of the ⁸⁷Rb nuclei (natural abundance: 27.835%) were observed in the Rb₂ZnBr₄ crystal as shown in Fig. 3. The following discussion suggests a reason why 6 resonance lines have been obtained instead of the usual 3 resonance lines.

The unit cell in a Rb₂ZnBr₄ crystal contains eight Rb ions. The eight Rb nuclei can be distinguished into two chemically inequivalent sets, Rb​I and Rb​II. Each Rb​I and Rb​II nuclei have four magnetically equivalent Rb nuclei, respectively. We thus may expect to see six Rb nuclear magnetic resonance lines of Rb nuclei in Rb₂ZnBr₄; 3 lines from Rb​I and another 3 lines from Rb​II considering the Rb₂ZnBr₄ crystallography. The six resonance lines obtained in our experiments have definitely originated from the two chemically inequivalent ⁸⁷Rb nuclei, not from the quadrupole splitting of ⁸⁷Rb nucleus in Rb₂ZnBr₄ crystal. The six resonance lines are divided into three resonance lines named the Rb​I site and three resonance lines named the Rb​II site. The intensity of the resonance line is not perfectly symmetric because it is practically impossible to excite all of the 6 resonance lines equally. The zero point of frequency axis in Fig. 3 shows the resonance frequency ω0/2π = 130.93 MHz, of the bare ⁸⁷Rb nucleus.

The temperature variation data of the ⁸⁷Rb NMR spectrum in the Rb₂ZnBr₄ crystal are measured at 25 different temperatures between 180 K and 420 K. Only the NMR spectra of 320 K – 350 K, at around TC2 = 347 K, for Rb​I and Rb​II nuclei in the Rb₂ZnBr₄ crystal are shown in Fig. 4. The y-axis shows the relative intensity of the Rb NMR spectra. We obtained six nuclear magnetic resonance lines of the Rb nuclei in Rb₂ZnBr₄; three lines from Rb​I and another three lines from Rb​II below TC2. The line splitting between the satellite and central resonance lines of both Rb​I and Rb​II sites was found to decrease as temperature increases over all temperature ranges as shown in Fig. 4. This means that the quadrupole interaction of the ⁸⁷Rb nucleus decreases and the electric field gradient of the ⁸⁷Rb atoms changes with increasing temperature. The central resonance line, coming from the transition between 1/2 and -1/2, is not affected by nuclear quadrupole interaction, unlike the satellite resonance lines. The decreasing quadrupole interaction with increasing temperature observed in our empirical results can be predicted by the Bayer theory[27-29] because the vibration amplitude of a rigid body increases as temperature increases.

Figure 4. (Color online) Temperature variation of NMR spectra for ⁸⁷Rb in Rb₂ZnBr₄ crystal.

For increasing temperatures, three resonance lines for each Rb​I and Rb​II sites combine into a single resonance line for each site. In the paraelectric phase (T>TC1), only two sets of lines for the ⁸⁷Rb nucleus, one line from Rb​I and another line from Rb​II, are observed. When the crystal is cooled below TC1, however, the splitting of these two sets of lines is observed in each of the Rb​I and Rb​II sites. The NMR line splitting between the central resonance lines of the Rb​I and Rb​II nuclei in the Rb₂ZnBr₄ crystal at 320 K, 330 K, 340 K and 350 K are 11.719 kHz, 11.523 kHz, 11.313 kHz, and 11.035 kHz, respectively. Namely, the splitting of the resonance lines between Rb​I and Rb​II nuclei slightly decreases as temperature increases. This reveals that the difference between the Rb​I and Rb​II sites in the Rb₂ZnBr₄ crystal decreases with increasing temperature. It is predictable that lower crystal symmetry goes to the higher symmetry at higher temperatures. The line widths for the ⁸⁷Rb​I and ⁸⁷Rb​II nuclei in the Rb₂ZnBr₄ crystal were 2.02 kHz and 2.37 kHz at 350 K, respectively. The spectra of the ⁸⁷Rb​I and ⁸⁷Rb​II sites in Fig. 4 are shifted by 2.873 kHz and 13.908 kHz from the operating frequency ω0/2π = 130.93 MHz, respectively, because of the chemical shift.

Chemical shifts of the ⁸⁷Rb nucleus in the Rb₂ZnBr₄ crystal are displayed in Fig. 5 in the temperature change range of 180 to 400 K. The nuclear magnetic resonance frequency of the ⁸⁷Rb nucleus embedded in the Rb₂ZnBr₄ crystal is different from that of the ‘bare’ nucleus because of the diamagnetic effect of the electronic charge around ⁸⁷Rb and polarization of the electronic shells when a magnetic field of 9.4 T is applied. This chemical shift shows different values for different chemical compounds and also depends on the surrounding electrons around the nucleus[30, 31]. The frequency shift (blue circles) of the central line for Rb​I NMR spectra in Fig. 5 continuously increases when temperature is increased up to 420 K. However, the frequency shift (red circles) of the central line for the Rb​II NMR spectra continuously decreases with increasing temperature. The discrepancy between the chemical shifts of the Rb​I and Rb​II NMR Spectra is reduced as temperature increases as displayed in Fig. 5.

Figure 5. (Color online) The NMR chemical shifts for both ⁸⁷Rb​I and ⁸⁷Rb​II nuclei in Rb₂ZnBr₄ crystal with temperature variation. The reference frequency was set to ω0/2π = 130.93 MHz for ⁸⁷Rb.

In order to analyze the spin-lattice relaxation time for the ⁸⁷Rb nuclei (I = 3/2) in the Rb₂ZnBr₄ crystal, only the central resonance line for ⁸⁷Rb​II in a Rb₂ZnBr₄ crystal was excited due to the large line splitting. Spin-relaxation time was calculated from the magnetization of the ⁸⁷Rb​II nuclei in the temperature range 180 K – 420 K. The T1 values obtained by excitation of only one line and excitation of all lines are theoretically equivalent. When only the central transition of ⁸⁷Rb​II in the Rb₂ZnBr₄ crystal is considered, the saturation recovery function would have been given by[32, 33]:

where W1 and W2 represent the ⁸⁷Rb nucleus transition probabilities, and S the nuclear magnetization after saturation.

The magnetization recovery trace for ⁸⁷Rb is set from the time evolution of magnetic resonance spectra for the delay time and was obtained by an inversion recovery method, where inversion recovery traces depend on the delay time. The recovery traces of the ⁸⁷Rb​II resonance line with dominant quadrupole relaxations in Rb₂ZnBr₄ crystal can be expressed by the two exponential functions. The relaxation times of the ⁸⁷Rb​II nuclei in the Rb₂ZnBr₄ crystal were obtained from Eq. (1). The temperature variation T1 for ⁸⁷Rb in the Rb₂ZnBr₄ crystal is displayed in Fig. 6.

Figure 6. (Color online) Temperature variation of ⁸⁷Rb​II NMR spin-lattice relaxation times in Rb₂ZnBr₄ crystal.

The relaxation times of the ⁸⁷Rb​II nucleus in the Rb₂ZnBr₄ crystal undergo drastic changes at about 200 K and 347 K, indicating abrupt changes in the spin dynamics at those particular temperatures in Fig. 6. This sharp discontinuity of T1 at about 200 K and 347 K agrees very well with the phase transition TC1 (200 K) between the ferroelectric and the incommensurate phase and the phase transition TC2 (347 K) between the incommensurate and the paraelectric phase, respectively. As shown above, the temperature dependent variation of the spin-lattice relaxation time reflects the phase transition phenomenon in a Rb₂ZnBr₄ crystal well.

One of the main contributions for spin-lattice relaxation mechanism for a nucleus with a nuclear spin of 1 or greater than 1 could be the interaction of lattice vibrations and nuclear quadrupole moments. The coupling can generally be represented by a spin-lattice Hamiltonian[30]. At temperatures significantly beneath the melting point of the crystal, it can be reasonably assumed that the thermal stress must be small and the first few parts are important. The first-order term shows a direct process and the next second-order term is Raman process. The Raman processes show that the reciprocal of the spin relaxation time is in proportion to T2[34].

The T1 values of ⁸⁷Rb in Rb₂ZnBr₄ decrease with increasing temperature between TC1 and TC2 as displayed in Fig. 6. The tendency is reasonable because the lattice oscillations become more active as the temperature of the crystal increases, resulting in shorter relaxation times. The T1 values for the ⁸⁷Rb nuclei can be explained by the relaxation mechanism caused by the lattice vibrations associated with the quadrupole moments. The relaxation rate (1/T1) of the ⁸⁷Rb​II nuclei in the Rb₂ZnBr₄ crystal is proportional to T2. Therefore, the dominant relaxation mechanism of the ⁸⁷Rb nuclei in the Rb₂ZnBr₄ crystal is due to the coupling of thermal fluctuations of the local EFG with quadrupole moments via Raman process. Since our measurement used crystal samples with no appreciable impurities, it can be said that our experimental results show an intrinsic relaxation process in the Rb₂ZnBr₄ crystal. In conclusion, the spin lattice relaxation of the ⁸⁷Rb nuclei in the Rb₂ZnBr₄ crystal seems to be originated mainly from Raman process in incommensurate phase between TC1 And TC2.

The nuclear magnetic resonance of ⁸⁷Rb nuclei in the Rb₂ZnBr₄ crystal has been explored in the temperature range 180 K – 420 K by a FT NMR spectrometer. Six nuclear magnetic resonance lines of the ⁸⁷Rb nuclei (natural abundance: 27.835%) in the Rb₂ZnBr₄ crystal are obtained as a function of temperature below TC2 in our experiment. The 6 resonance lines turned out to be originated from the Rb​I site having 3 lines (1 central and 2 satellite nuclear magnetic resonance lines) and the Rb​II site having another 3 resonance lines in the Rb₂ZnBr₄ crystal. These two sets of ⁸⁷Rb NMR spectra have definitely originated from the two chemically inequivalent ⁸⁷Rb sites, not from the quadrupole splitting of ⁸⁷Rb nuclei in the Rb₂ZnBr₄ crystal.

The six NMR lines of Rb​I and Rb​II sites in Rb₂ZnBr₄ merge into two NMR lines (one line from the Rb​I site and the other from the Rb​II site) above TC2. However, one line from the Rb​I site and the other from the Rb​II site are split into three lines each in the incommensurate (below TC2) and ferroelectric phase (below TC1). The quadrupole interaction of ⁸⁷Rb nucleus decreases and the EFG at the ⁸⁷Rb atom changes with increasing temperature. The decreasing quadrupole interaction of ⁸⁷Rb nucleus with increasing temperature observed in our empirical results can be supported by the Bayer theory. Chemical shifts of the ⁸⁷Rb nucleus in the Rb₂ZnBr₄ crystal are observed with temperature variations of 180 K – 400 K. The frequency shift from the reference frequency depends on the sample temperature and moves continuously.

Spin-relaxation time of ⁸⁷Rb​II nuclei in the Rb₂ZnBr₄ crystal was calculated according to the temperature change between 180 K – 420 K. The relaxation time of the ⁸⁷Rb nuclei undergoes abrupt changes near both 200 K and 347 K, which indicates the occurrence of phase transitions. These two sharp discontinuities at 200 K and 347 K in our study agree well with the phase transition temperatures TC1 (between ferroelectric and incommensurate phases) and the phase transition temperature TC2 (between incommensurate and paraelectric phases) of the Rb₂ZnBr₄ crystal. The dominant relaxation mechanism of ⁸⁷Rb nuclei can be explained by the combination of the lattice oscillations to the nuclear electric quadrupole moments via the Raman spin-phonon process between 200 K and 347 K.