Journal ArticleIn the presence of thermal fluctuations the incommensurate NMR line shape is a convolution of a static inhomogeneous with a dynamic homogeneous line shape which can be determined separately by 2D NMR. The form of the dynamic line shape and its variation over the  inhomogeneous NMR spectrum permit a separate determination of the relative sizes of the phason and amplitudon fluctuations, compared to the static part of the order parameter. 8 7Rb 2D spectra of ultrapure Rb2ZnCU agree with the above theory and show the existence of a temperature range where the incommensurate splitting induced  by the static part of the order parameter is averaged out by thermal fluctuations

Ailion, David Charles

Fajdiga, A. M.

English

The University of Utah: J. Willard Marriott Digital Library

VOLUME 69, NUMBER 18 PHYSICAL REVIEW LETTERS 2 NOVEMBER 1992 Thermal Fluctuations and NMR Spectra of Incommensurate Systems A. M. Fajdiga, T. Apih, J. Dolinsek, and R. Blinc 1. Stefan Institute, University of Ljubljana, Ljubljana, Slovenia A. P. Levanyuk and S. A. Minyukov Institute of Crystallography, Moscow, Russia, VIC D. C. Ailion Department of Physics, University of Utah, Salt Lake City, Utah 84112 (Received 20 July 1992) In the presence of thermal fluctuations the incommensurate NMR line shape is a convolution of a static inhomogeneous with a dynamic homogeneous line shape which can be determined separately by 2D NMR. The form of the dynamic line shape and its variation over the inhomogeneous NMR spec-trum permit a separate determination of the relative sizes of the phason and amplitudon fluctuations, compared to the static part of the order parameter. 87Rb 2D spectra of ultrapure Rb2ZnCl4 agree with the above theory and show the existence of a temperature range where the incommensurate splitting in-duced by the static part of the order parameter is averaged out by thermal fluctuations. PACS numbers: 76.60.-k Incommensurate insulators [I] are characterized by a modulation of the atomic positions with a periodicity that bears no simple rational relation to that of the underlying crystal lattice. For the simplest case of a one-dimensional modulation wave, the order parameter of the normal-to-incommensurate (N-I) transition is characterized by two components, the amplitude and the phase of the modula-tion wave. In contrast to the amplitude fluctuations (am-plitudons), which become large only near the N-/ transi-tion, phase fluctuations (phasons) are significant over the entire incommensurate phase. Whereas the static features of the incommensurate phase are reasonably well understood, this is not so for the dynamic aspects, especially in the vicinity of the tem-perature T/ of the N-/ transition [1]. In particular, the effect of thermal fluctuations on the onset of incommens-urate behavior and the possible floating of the modulation wave near T/ have been the subject of significant contro-versy. Furthermore, determination of the critical ex-ponents, for example, requires a precise determination of T/ as well as a separation of the dynamic and static con-tributions to the modulation wave. Because of its sensitivity to small displacements in the atomic positions, quadrupole-perturbed NMR, as well as nuclear quadrupole resonance (NQR), has been used ex-tensively in the past [1,2] for studying the modulation wave in both dielectric [2] and charge-density-wave in-commensurate systems [3,4]. Below T/ the NMR line becomes inhomogeneously broadened and acquires a specific shape that reflects the spatial distribution of the atomic positions in the modulation wave. Thermal fluc-tuations [5,6] significantly affect the shape of the NMR line. A number of experimental studies [5-11] have been devoted to the investigation of this effect. However, the experimental results do not agree with the theoretical cal-culations [5,6] predicting a motional narrowing of the NMR line but no other change in its Gaussian line shape. The disagreement may be due in part to an uncontrolled concentration of random impurities in the samples [11] and/or to certain approximations made in the calcula-tions, such as the substitution of the time averaging of the fluctuating resonant frequency by averaging over a sta-tionary Gaussian distribution of the phases of the modu-lation wave [5]. Here, we present a theory of the NMR line shape in in-commensurate systems in the presence of thermal fluctua-tions of the order parameter. This theory is free of the above-mentioned approximations and is valid within the Landau theory [1]. It shows that thermal fluctuations not only reduce the linewidth but also change the shape of the spectrum in a characteristic way. Within a scaling factor, the line shape depends on just two parameters which are determined by the relative sizes of the phason and amplitudon fluctuations compared to the static part of the order parameter. We show that the line shape is given by a convolution of a static incommensurate fre-quency distribution and a dynamic line shape determined by phason and amplitudon order-parameter fluctuations. We have also shown how these two contributions can be simultaneously independently determined by two-dimensional (2D) nuclear magnetic resonance. The theoretical results agree well with a 2D 87Rb NMR study of Rb2ZnCI4, obtained with high temperature resolution. The separate simultaneous determination of the static (inhomogeneous) and dynamic (homogeneous) line shapes allows a precise determination of T/ and clearly shows the presence of a region where thermal fluctuations of the modulation wave average out the quadrupolar splitting due to the static part of the order parameter. Let us consider the case of a one-dimensionally modu-lated incommensurate system with a two-component or-der parameter [1,21. Just below T/ where the plane wave © 1992 The American Physical Society 2721 VOLUME 69, NUMBER 18 PHYSICAL REVIEW LETTERS 2 NOVEMBER 1992 approximation [I] is valid, the incommensurate distortion wave 17 (x) can be expressed as 17 (x ) = 171 cos(kox) + 172 sin (kox ) , (1) where 171 and 172 are the two components of the order pa-rameter [1,12] describing the N-/ transition and ko is the wave vector of the incommensurate modulation. Thermal fluctuations of the order parameter result in 17 1 (x ,t ) = 17 1 £ + 17 i (x , t) , 1J2(X, t) == 172£ + 1J2(X,I) . (2a) (2b) For the equilibrium values 171£ and 172£ we can take [1] 1JtE a: (T/ - T) and 172£ =0. Let us now assume that the quadrupole perturbed NMR resonance frequency 11 of a nucleus at a site x is linearly related (1,2] to the nuclear displacement 1] (x ,t) at this site: (3) Here 1](x,t) is given by expressions (I), (2a), and (2b). The adiabatic NMR line shape /(ill) is now obtained [13] as (4a) where G(t} =e -inot \ \ exp [ - i 50t [0 (x ,I') - ooldl')) )x . (4b) The inner brackets () represent a thermodynamic ensem-ble average whereas <) x stands for an average over the in-homogeneous static distribution of resonance frequencies. Inserting expression (3) into (4b) we find the autocorre-lation function G (t) in the form (5) Here G 1 (t ) = ex p [ - i a 1] 1 £ t cos (k OX ) ] represents the well known [1,2] static inhomogeneous fre-quency distribution limited by two edge singularities, whereas G 2(r) =exp [ - ia 50t [1Ji (x, t')cos(koX) + 1J2{x ,t')sin (k OX )dt' ] (7) describes the effects of amplitudon [1]] (x ,t)] and phason [1J2(X, t)] fluctuations on the NMR line shape. Since the prod-uct of two functions in the time domain is equivalent to the convolution of their Fourier transforms in the frequency domain, the Fourier transform of G 2 represents the dynamic line shape which convolutes the static [1,2] one. After a straightforward [13] but somewhat lengthy calculation one finds the adiabatic line shape in the incommensurate phase as where X =cos(kox), 01 =a1]JE , a 2kBTrrc T w - a:-----loci - 4Jr82 (TI - T) 1/2 ' and [a 2kB Ty 1/2 ] 2/3 Wloc2 == 3/2 3/2 a: T 2/3 . 24Jr 8 Here 8 is the coefficient of the elastic term in the Landau free-energy expansion [11, y is the coefficient in front of the dissipation function [(r/2)(Tjf+Tji)1, and rc is the correlation radius [1] for the amplitude of the modulation wave. Expression (8) cannot be evaluated analytically because of the t 3/2 term, representing the phason contri-bution. Up to a scaling factor lew) depends on just two parameters, J= W)ocl T "='1== a: 01 T[- T (8) and Wloc2 T2/3 ~2= a:-----01 (T[-T)I/2 ' which measure the relative size of amplitudon and phason order-parameter fluctuations compared to the static order parameter. Far below T], ~ I and ~2 tend to zero, (G2(t»-4 I, and /(w) is reduced to the static incom-mensurate frequency distribution [I] which exhibits two edge singularities at + 0 I. It is important to note that 2D NMR allows a simul-taneous separate determination [14] of the inhomogene-ous line shape, represented by (G tG2), and the dynamic line shape, represented by (G 2). This is achieved by the application of a 180 0 refocusing pulse [14] in the middle of the evolution period. The 2D NMR line shape is obtained as [(WI, r.1 2)=J+ 1 dx rOOd rOOd i(W2- n O- O IX)t2 -[WIOCl t 2X 2+(W1oc2t 2)3/2(t-X2)] iw1tl -[COlocltlXl+(Wloc2tIP/2(1-X2)] LV' _ I ( 2) 1/2 J ( t 1 J ( t 2e e e e , 1 -x 0 0 (10) 2722 VOLUME 69, NUMBER 18 PHYSICAL REVIEW LETTERS 2 NOVEM BER 1992 ----..,. -v ___ fJL . · -4000 -2000 w,/2n: o (Hz) 2000 4000 40000 20000 0 -20000 -40000 -60000 w2/2n: (Hz) FIG. l. 20 Hseparation of interactions" NMR spectrum [14] of the 87Rb ~ -- - ~ transition in Rb2ZnC14 at an orientation a..LHo, {:c,Ho=122°, and T=291.2 K. Projections on both frequency axes are also shown. The 87Rb Larmor frequency is VL ( 87Rb) =88.34 MHz. where 1 I is the evolution and 12 the detection time [14] in the 2D experiment (Fig. 1). A high-temperature-resolution 2D NMR 87Rb ~ -+ - ~ transition study [14] of ultrapure Rb2ZnCl4 has been made in the vicinity of the N-/ transition in steps of 0.1 K. In the W2 domain we find the static inhomogene-ous line shape convoluted with the dynamic one (Fig. 1). In the (01 domain the static quadrupole interaction is el-iminated [14] and the homogeneous dynamic line shape is determined by the time-fluctuating part of the quadru-pole interaction (Fig. 1). The dynamic line shape varies over the inhomogeneous static incommensurate frequency distribution through its dependence on X =cos(kox) = (W2 - n)/ n I. In the center of the inhomogeneous line where X ==0, the dynamic line shape is determined by phason fluctuations (i.e., by the 1 3/2 term). At the edge singularities, where X = + I, the dynamic line shape is determined by amplitudon fluctuations (i.e., the t term), yielding a Lorentzian form. In Fig. 2 we show the temperature dependence of the positions of the edge singularities of the inhomogeneous line shape obtained in the (02 domain on the Rb line cen-tered around (02 ==0 (Fig. 1). The full width at half height of the dynamic line shape obtained in the (01 domain in the center of the inhomogeneous line shape (i.e., at (02 = n) is also shown. On the same plot we also show the temperature dependences of the soft mode as well as phason (TIll') and amplitudon (TIA) induced spin-lattice relaxation rates [1,2]. It is clearly seen that the width of the dynamic line shape exhibits a maximum at T/ == 304.4 K. At the same temperature the T I splits [1,2] into two branches, the temperature-independent phason contribution and the critically temperature-• 15 . Y 350 T,(ms) ./ ~VH (Hz) 10 300 5 ~o-.6vH 0 250 0 I I T 1= 304.4 K 303 304 I 305 306 307 3 I T (K ) v1NH(kHz) o~ I 2 I '" I 1 o . I .. I 0 o-ot-oo 0 0 0 -1 0" I -2 ~~ I Rb 2ZnCl 4 I -3 I -4 I T I303 304 305 306 307 T ( K) FIG. 2. The lower part of the figure shows the temperature dependence of the frequencies of the incommensurate edge singularities, VINH, of the inhomogeneous line shape. The upper part shows the temperature dependence of the full width at half height of the homogeneous dynamic line shape, llvH, in the center of the inhomogeneous spectrum. The temperature dependence of the soft mode (TI), phason (Tt"'), and amplitu-don (T IA) induced spin-lattice relaxation times is shown for companson. (b) -VI VI - 1 ·s -c: :::J ~ TI -T =OJK >-E L-a T1-T=O.3K - L--LJ :0 L- L-a ...9 >- >-- -~ VI c: OJ OJ -c: ----c: ...-1 -2 0 2 -1 - 0.5 0 0.5 1 w2/2Tt (kHz) w,/2n(kHz) FIG. 3. (a) Inhomogeneous and (b) homogeneous dynamic line shapes close to T/. The solid lines in (b) show the fit to ex-pression (10), whereas the dotted line is the fit to a Lorentzian. 2723 VOLUME 69, NUMBER 18 PHYSICAL REVIEW LETTERS 2 NOVEMBER 1992 - 2.5 -- 2.0 - N ::::r:: - 1.5 ~ ..--------.--------------~ 1.0 -0.4 0.0 T-Td~t _----2--2-----wloc 2 N 0.8 ~ C'J -0.6 ('J u o 3 C'J -r--U o 0.2 3" L-..-I.....--.J...._~~__.I. _ _L__~_L_____L....___'__..L.__J 0.0 - 0.5 - 0.4 - 0.3 - 0.2 - 0.1 0.0 T-TI (K) FIG. 4. Temperature dependence of the parameters {J)loc.!2Jr, {J)loc2i2Jr, and n .!2Jr obtained from a least-squares fit by com-paring experimental and theoretical inhomogeneous and homo-geneous line shapes (open squares). The additional n t points (solid squares) are obtained from the edge singularity data in the temperature range between T/ - T =0.4-1.5 K. The dashed line represents a least-squares fit n I ex: (T/ - T) Z with z =0.5. Inset: The normalized quantities ~I = (()Ioct! n I and ~2 =Wloc21 n I which measure the relative sizes of the fluctuating and static parts of the incommensurate order parameter. dependent amplitudon contribution. The temperature T/ is clearly the N-J phase transition temperature where the paraelectric soft mode condenses, resulting in a maximum width of the dynamic line shape. In contrast the inhomo-geneous line shape shows no splitting at this temperature. The incommensurate splitting starts to become observable only at T/ - T:> 0.4 K, i.e., outside the region where motional narrowing due to thermal fluctuations of the modulation wave is dominant. The inhomogeneous and the dynamical line shapes are shown in Fig. 3 at TJ - T =0.1, 0.3, and 0.5 K. At T/ - T =0.1 and 0.3 K the inhomogeneous line shape is still single peaked whereas the two edge singularities be-come clearly discernible at T/ - T = 0.5 K [Fig. 3 (a)]. The experimental and theoretical dynamic line shapes are 2724 compared in Fig. 3 (b). The deviations of the dynamic line shapes from a Lorentzian form show the importance of the phason fluctuations represented by the t 3/2 term. The comparison between the experimental and theoretical line shapes allows for a determination of the parameters ~l and ~2 from Wloc l=C[T/(T/-T)l/2], Wloc2=DT 3/2, and n I, which are shown in Fig. 4 as functions of temper-ature. We find that C=(5 +0.3)x 10- 3 s -I K -1/2 whereas D = 31 + 3 s -) K -2/3. n 1 varies with tempera-ture as n 1 ex: (T[- T) z, where z =0.5 as predicted by the Landau theory. It should be noted that n I, which is pro-portional to the amplitude of the frozen-in modulation wave, becomes comparable to Wlocl and Wloc2, which mea-sure the fluctuating parts of the order parameter about 0.1 K below T[. We can conclude that the adiabatic incommensurate NMR line shape is indeed given by the Fourier transform of <G> =(G IG2) and not just G 1 as tacitly assumed so far in all NMR studies of incommensurate phase transitions. The fluctuation correction, G2, which is somewhat similar to the Debye-Waller factor in x-ray scattering, deter-mines the form of the NMR spectrum close to TI. This work has been supported in part by MZT of Slo-venia and by the U.S. NSF Grant No. DMR 90-24196. [1] See Incommensurate Phases in Dielectrics, edited by R. Blinc and A. P. Levanyuk (North-Holland, Amsterdam, 1986), Vol. 1. [2] R. Blinc, Phys. Rep. 79, 333 (1981). [3] J. H. Ross, Z. Wang, and C. P. Slichter, Phys. Rev. Lett. 56, 633 (1986); B. H. Suits, S. Couturie, and C. P. Slichter, Phys. Rev. Lett. 45, 194 (1980). [4] A. Jannosy, C. Berthier, P. Slorasan, and P. Butand, Phys. Rev. Lett. 59, 2348 (1987). [5] R. Blinc, F. Milia, B. Topic, and S. Zumer, Phys. Rev. B 29,4173 (1984). [6] F. Milia, R. Blinc, and S. Zumer, Solid State Commun. 50, 1 0 1 9 (1 984) . [7] R. Blinc, D. C. Ailion, P. Prelovsek, and V. Rutar, Phys. Rev. Lett. 50,67 (1983). [8] B. Topic, U. Haeberlen, R. Blinc, and S. Zumer, Phys. Rev. B 43,91 (1991). [9] J. Emery, S. Hubert, and J. C. Fayet, J. Phys. (Paris) 46, 2099 (1985). [10] D. Michel, B. Muller, J. Petersson, A. Trumpert, and R. Walish, Phys. Rev. B 43,7505 (1991). (11] F. Milia, G. Papavassiliou, and A. Anagnostopoulos, Z. Naturforsch. 47a, 237 (1992). [12] R. A. Cowley and A. D. Bruce, J. Phys. C II, 3577 (1978); A. D. Bruce and R. A. Cowley, J. Phys. C t I, 3609 (1978). [13] A. Abragam, Principles of Nuclear Magnetism (Oxford Univ. Press, Oxford, 1986). [14] J. Dolinsek, J. Mag. Reson. 92, 312 (1991). 

Thermal fluctuations and NMR spectra of incommensurate systems

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Thermal fluctuations and NMR spectra of incommensurate systems

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