NMR determination of the Edwards-Anderson order parameter in the deuterated pseudo-spin glass Rb1-x(ND4)xD2PO4: anisotropy and concentration dependence of the ND4 deuteron second moment

We measured the rotationally averaged N D 4 + deuteron second moment in deuterated mixed single crystals Rb1_x(ND4 ) xD2P04 with x = 0.22, 0.44, 0.78, and pure ND4D2P04 . We found that the linewidths of the mixed systems show a strong dependence on the crystal's orientation with respect to the external magnetic field. We interpret the temperature dependence of the N D 4 + second moment as due to the slowing down of the O-D. . .O intrabond motions. We also show that, in the temperature region above the glass transition, the second moment may be proportional either to the first or to the second powers of the Edwards-Anderson order parameter, depending on the crystal orientation. We were able to fit our experimentally obtained qEA to a theoretical value obtained by Pirc, Tadic, and Blinc from the random-field random-bond Sherrington-Kirkpatrick Ising model. A comparison between similar measurements in pure and in mixed systems verifies that the strong orientational dependence of the linewidth is a property unique to the mixed systems.

P H Y S I C A L R E V I E W B V O L U M E 4 2 , N U M B E R 1 0 1 O C T O B E R 1 9 9 0 NMR determination of the Edwards-Anderson order parameter in the deuterated pseudo-spin-glass Rb, * (ND4)XD2P04: Anisotropy and concentration dependence of the ND4+ deuteron second moment Songhua Chen and David C. Ailion Department of Physics, University of Utah, Salt Lake City, Utah 84112 (Received 5 February 1990) We measured the rotationally averaged ND4+ deuteron second moment in deuterated mixed sin­gle crystals Rb1_x(ND4)xD2P04 with * = 0.22, 0.44, 0.78, and pure ND4D2P04. We found that the linewidths of the mixed systems show a strong dependence on the crystal's orientation with respect to the external magnetic field. We interpret the temperature dependence of the ND4+ second mo­ment as due to the slowing down of the O-D. . .O intrabond motions. We also show that, in the temperature region above the glass transition, the second moment may be proportional either to the first or to the second powers of the Edwards-Anderson order parameter, depending on the crystal orientation. We were able to fit our experimentally obtained qEA to a theoretical value obtained by Pirc, Tadic, and Blinc from the random-field random-bond Sherrington-Kirkpatrick Ising model. A comparison between similar measurements in pure and in mixed systems verifies that the strong orientational dependence of the linewidth is a property unique to the mixed systems. I. INTRODUCTION The mixed normal and isotropically substituted crystals, Rb1_x(NH4)xH2P04 (RADP) and Rb1_x(ND4)JCD2P04 (D-RADP), respectively, represent frustrated H-bond or D-bond systems with random com­peting ferroelectric (FE) and antiferroelectric (AFE) in­teractions due to random substitution of Rb+ ions for ammonium NH4+ or ND4+ ions.1,2 The temperature of the paraelectric (PE)-to-FE transition is lowered by the substitution of a small concentration of NH4+ or ND4+ ions (x^0.2). In contrast, for much larger NH4+ or ND4+ concentrations (x ^0.8) an AFE transition occurs. However, in the intermediate range of x, no long-range order is observed in the low-temperature phase, and the system forms a frozen proton or deuteron pseudo-spin- # ^ glass state somewhat similar to a magnetic spin glass. . The most important similarity between proton-spin- glass (PSG) and magnetic spin-glass (MSG) systems is that both MSG and PSG are quench-disordered sys­tems3,4 in which interactions between spins in the MSG (or the electric dipole moments in the PSG) prevent the formation of either a ferromagnetic or antiferromagnetic ground state in the MSG (or the formation of a FE or AFE ground state in the PSG) and thus lead to frustra­tion. However, there are also distinguishing features that are unique to the PSG. One fundamental difference arises from the larger concentration of substitutional im­purities in a proton PSG system. This feature results in an intrinsic random field that becomes the source of the smearing out of the temperatures of the glass transition.3 Thus the Edwards-Anderson order parameter, which is a unique characteristic of the spin-glass state, is nonzero even at temperatures substantially above the theoretical glass-transition temperature TG. Experimental evidence from dielectric and birefringence studies1,5 has shown that the dielectric susceptibility € has a gradual rather than a sharp singularity-type peak at T = TG. Such be­havior is in contrast to that of the magnetic spin-glass systems, which typically display a singularity in the low- field, low-frequency magnetic susceptibility X at Tg.4 Furthermore, other experimental techniques exhibit a similar smearing out of the glass transition. For instance, nuclear magnetic resonance (NMR) studies of 87Rb in D- RADP (Refs. 6 and 7) have shown that the Edwards- Anderson (EA) order parameter qEA is still nonzero at temperatures far above TG (e.g., room temperature). Recently, Blinc et a/.8 observed the deuteron reso­nance of ND4+ in Rb0 56(ND4)0. 44D2P04. In particular, they measured the temperature dependence of the NMR linewidth of ND4+ at one crystal orientation [<£c,H0) = 150°, where <J (c,H0) means the angle between the crystal c axis and the external magnetic field H0]. They found a nonzero qEA at temperatures much higher than TG, consistent with earlier 87Rb NMR experiments.6 Similar qEA studies have been reported recently for 87Rb and the O-D • • • O acid deuterons in an x = 0.5 pseudo- •*9 spin-glass crystal. In order to understand the roles of the ND4+ and the N-D-O bonds in forming a glassy state, and thus to better clarify the relationship between the order parameter and the linewidths, we extended this linewidth study to include measurements at other crystal orientations for various ND4+ concentrations. In partic­ular, we measured the orientation dependence of the second moment in four single-crystal samples [x =0.22, 0.44, 0.78, and 1.0 (pure D-ADP)]. We found that the second moment may be proportional to either first or second powers of the EA order parameter qEA, depend­ing on the crystal orientation with respect to the external field. This dependence of the linewidth on qEA is respon­sible for the observed ND4+ line broadening in the tem­perature region above TG. We were able to fit our experi­42 5945 ©1990 The American Physical Society5 9 4 6 S O N G H U A C H E N A N D D A V I D C . A I L I O N 4 2 mentally obtained qEA to a theoretical value obtained from the random-field random-bond Ising-model mean- field theory.3,9 A comparison between similar experi­ments in pure and in mixed systems verifies that the orientational dependence of the linewidth is a property unique to the mixed systems. II. THEORETICAL BACKGROUND Pirc, Tadic, and Blinc recently developed a theoretical model3 to describe proton pseudo-spin-glass systems. Their model is based upon the fact that occupancy of the double potential well of the O-H • • • O protons in hydrogen-bonded ferroelectrics is mathematically similar to the Ising problem for a spin-y nucleus. (The bonding proton is commonly called a pseudospin.) In particular, for PSG systems like RADP, Pirc et al.9 and Akseniv et al.10 employ a random-bond version of this model con­sisting of a transverse tunneling field plus an additional interaction term describing the local random longitudinal fields resulting from the random substitution of Rb+ for ND4+. The Hamiltonian of such a system is H=-}J,JijS*sf-n2,s*-2(E+fl )S?, (1) hj where H represents the tunneling frequency of the O-D • • • O deuterons9,11 and S( represents the pseudos­pins corresponding to the two equilibrium positions of the deuterons within the O-D • • • O bonds. The cou­pling constant represents the random bond (interac­tion) between the pseudospins, and E and f { are, respec­tively, the homogeneous and random local longitudinal fields at nuclear site i. The term /, is nonzero in the PSG system but vanishes in pure D-ADP or pure D-RDP be­cause of the absence of substitutional disorder. The in­dependent distributions of J{j and f { can be described by Gaussian probability densities3,9 1 J (2irJ) exp J; 2 J (2a) 1 v7 2ttA exp IL 2A (2b) where J2 = J2 /N= (Jfj) and A = A /N = (f2) are the variances of the random-bond and random-field distribu­tions, respectively. Furthermore, A depends on the ND4+ concentration x and is typically of the form3,12 Ax( 1 -x) . (3) This mean-field-theory model has been employed to pre­dict the glass transition in the RADP and RADA sys­tems and has also been successful in interpreting some PSG properties. By modeling the dueteron occupation within the O-D • • • O bonds as pseudospins, this model can be treated easily using spin-glass theories. Following the Sherrington-Kirkpatrick (SK) method,13 Pirc et al.3 ob­tained an integral equation to describe the Edwards- Anderson order parameter in the PSG system -f 00 oc exp Xtanh2 J kB T 1/2 dz (4) III. ELECTRIC-FIELD GRADIENT OF ND4+ IN D-RADP SYSTEMS In general, for nuclei with a nonzero quadrupole mo­ment (I > j), the NMR frequencies of individual nuclei reflect the values of the electric-field gradient (EFG) ten­sors at the particular nuclear sites. The observed inho- mogeneous line is a convolution (usually treated as Gaussian) of the individual NMR lines. In the case of ammonium ions in the D-RADP or D-ADP systems, ro­tational motion exchanges the four deuterons within an ND4+ group and averages out a large part of the EFG tensor.8,1214 Therefore, the observed ND4+ deuteron in- homogeneous linewidth is rotationally averaged, as is common to a number of ammonium compounds. This fast reorientational motion narrows the inhomogeneous line above 120 K in D-RADP systems. Hence, this rota­tionally averaged ND4+ deuteron spectrum does not reflect the values of the actual EFG tensors of the indi­vidual deuteron sites in the ND4+ ion; rather, it represents the EFG tensor averaged over the four deute­ron sites. The EFG tensor at an ND4+ site is determined mainly by the surrounding N-D-O bonds. There are eight surrounding oxygens of which four will form N-D-O bonds to a given ND4+ group.8,12 The formation of N-D-O bonds affects the ordering of the O-D-O bonds (i.e., the O-D • • • O deuteron occupancy) and is responsible for the formation of the AFE phase in D- ADP.15-17 An individual P043- oxygen may form one short N * • • D-O or one long N-D • • • O bond to an ND4+ group, corresponding to the ammonium deuteron's being closer to or further away from the oxy­gen, respectively. (Here we use N • • • D-O and N- D • • • O to represent the short and long bonds, respec­tively.) Any P043- oxygens forming a short N • • • D-O bond will tend to keep the acid deuterons away from this oxygen in order to minimize the potential energy, thereby resulting in polarization of the adjacent O-D • • • O bonds, as illustrated in Fig. 1. In contrast, formation of a long N-D • • • O bond to a P043- oxygen allows the acid deuteron to stay closer to the oxygen, thereby polarizing the O-D • • • O bonds in the opposite manner (Fig. 1). The formation of short and long bonds is accompanied by a shift in the ND4+ position, resulting in a small ND4+ tetrahedral distortion, which is the source of the ND4+ EFG. Because of the relatively fast O-D • • • O intrabond motion in the PE phase of D-ADP, the whole ND4+ ground experiences a thermally averaged EFG in which the short and long N-D-O bonds are also averaged. An x-ray study18 reveals that each ND4+ ion in the AFE4 2 N M R D E T E R M I N A T I O N O F T H E E D W A R D S - A N D E R S O N O R D E R . 5 9 4 7 NO, D O ACID D 1A N FIG. 1. Schematic structure of D-ADP. Each oxygen in a P043" group links one N-D • ■ • O and one O-D • • • O bond. The numbers (j,y) in the figure indicate the locations of the centers of the corresponding tetrahedra along the crystal z direction (perpendicular to plane of figure). earlier. The inhomogeneous linewidth can be viewed as a superposition of frequency-shifted lines arising from ND4+ ions in different clusters (having different polariza­tions). We consider the second moment a2 of the inho­mogeneous line of the ND4+ deuterons to consist of two terms: (1) a temperature-independent part g\ corre­sponding to the second moment in the high-temperature PE phase and (2) a temperature-dependent part (a')2 due to the freeze-out of the intrabond motions. To relate the temperature-dependent broadening (a')2 to the order parameter qEA, we need to take into account the four N-D-O bonds connecting to a particular ND4+. From Sec. Ill, we know that these N-D-O bonds (long or short) depend on the ordering of the four adjacent O-D • • • O bonds, since both types of bonds share the connecting oxygens. The NMR frequency shift of the deuterons for each ND4+ ion due to the O-D • * • O freeze-out is thus given by phase is shifted from its equivalent PE-phase lattice site to an off-center position. This shift results from the for­mation of two long and two short N-D-O bonds to the adjacent P043- groups, as mentioned earlier. The sublat­tice polarization results from the correction of the long and short N-D-O bonds with the O-D • • O bond ordering. The random substitution of Rb+ for ND4+ changes the above picture. In the mixed D-RADP systems, some oxygen atoms still form N-D-O bonds with accom­panying O-D • • • O order, while other oxygen atoms now have Rb+ neighbors, resulting in an absence of the N-D-O bonds. Since the change in the effective local polarization due to substitution of a Rb+ for an ND4+ is spatially random, the spatial average of this polarization will be undetectable in a microscopic observation. Nev­ertheless, this local polarization can be detected in a mac­roscopic measurement like NMR, since the EFG is very sensitive to changes in local electric fields or polariza­tions. At higher temperatures the mixed system becomes more like the PE system because of the rapid O-D • • • O intraband motions. At lower temperatures, however, O-D • • • O bond freezeout causes a glasslike state to appear. Since the ND4+ and N-D • • • O bonds have such important correlation with the O-D • • • O or­dering, it is clear that ND4+ NMR can serve as a useful probe in revealing the spin-glass properties of the D-RADP systems IV. DEPENDENCE OF ORDER PARAMETER ON ANISOTROPIC SECOND MOMENT a2 In the high-temperature region both pure D-ADP and mixed D-RADP are in the paraelectric phase. The inho­mogeneous line shape of the deuteron ND4+ reflects thermal averaging and is narrower. This paraelectric- phase linewidth can be described by a second moment a0. In the mixed systems the line broadening increases at lower temperatures due to the gradual freeze-out of the O-D • • • O intrabond motions and should be observable at temperatures substantially above TG, as mentioned + A va2=^{AiPi+BiPf+ ■■■), (5) 1 = 1 where a indicates the ath ND4+ ion, p( is the local bond order parameter representing the individual O-D • • • O polarizations, and i refers to the four different O- D • • • O bonds near a given ND4+ group. In general At and Bj are anisotropic with respect to crystal orientation. At most orientations the linear term is much bigger than the quadratic term, in which case Avf = 2 AiPi If we form an average over the whole crystal, we get (6) Av2 = Av" = 2 AiPj^O , 1=1 (7) since there is no bulk polarization (pt = 0) because of the absence of long-range order. The temperature-dependent part (a')2 of the second moment in this case is due to the cluster polarization and is given by (aY = ( Av?)2 = = 2 AiAjPiPj = '2,Afp? (8) ij = 1 Note, (Avf )2 is zero at high temperatures, where the rap­id thermal motions cause the individual cluster polariza­tions to average to zero. Replica symmetry requires all p} to be equal3 so that we may replace pf by a constant <7ea, the Edwards-Anderson order parameter,19 and take it out of the summation. Thus (ct')2-2^/9ea=^i9ea (9) Equation (9) indicates that (a')2 is directly proportional to the EA order parameter qEA, as has also been found for this linear case by Blinc et al.8 On the other hand, in some particular crystal orienta­ 5 9 4 8 S O N G H U A C H E N A N D D A V I D C . A I L I O N 4 2 tions (e.g., c||H0 for the rotationally averaged ND4+ case), the linear term AjPt almost vanishes due to crystal symmetry.11,20 In this case At «B( and we then need to consider the quadratic term, Av?=i 2 (10) / = 1 and Av2 = Avf= 2jBl9EA=cr2«EA - HI) / = 1 / = 1 which is nonzero. Therefore, in this orientation (cr')2=022q\K , (12) which is proportional to qEA. Since qEA « 1 at temperatures well above TG it follows that QEA •'>-> Q EA • (13) Even when the linear and quadratic coefficients are com­parable Eq. (13) guarantees that the linear term will dominate the second moment. V. EXPERIMENTAL RESULTS A. Experimental features We performed NMR studies at v0=12.5 MHz on the ND4+ deuterons in Rbx(ND4)1_;cD2P04 mixed single crystals with ND4+ concentrations x = 0.22, 0.44, 0.78, and 1.0 (pure D-ADP). The pure D-ADP crystal was ob­tained from a commercial source (Quantum Technolo­gies, Florida) with the deuteration level quoted to be about 90%. Moisture contamination was avoided by keeping the samples in a dry container before the experi­ments. The inhomogeneous linewidths were obtained from Fourier transforms of the NMR free induction decays (FID's). At lower temperatures the data was obtained from Fourier-transformed spin echoes rather than FID's, since the FID's are too short at the lower temperatures. The measurements were performed in a variable- temperature liquid-N2 cryostat. The temperature con­troller uses two sensors: one to control the current of a heating coil wound outside the sample chamber and another to read out the temperature near the sample. The temperature variation was typically less than ±0.5 K during each measurement. B. ND4+ quadrupole coupling constant We measured the orientational dependence of the quadrupolar line splitting for all four crystals and found, from the EFG tensor components, that the crystal c axis remains the symmetry axis at temperatures above the transition, thus indicating that the tetrahedral structure of the PE phase is not altered as the temperature de­creases. We measured the quadrupolar coupling con­stants of the ND4+ deuterons in all three mixed samples using VolkofFs method21 and found that they are nearly the same for all three mixed samples over a large temper­ature range. The value of e2qQ/h2 (^3.7 kHz) is ap­proximately equal to that for pure D-ADP in the high- temperature PE phase and is also close to the value for D-AD A.11 This uniformity in the quadrupolar coupling constants probably reflects a similarity in the underlying crystal structures in which the lattice parameters are comparable. Our linewidths, measured at v0= 12.5 MHz, are not much broader than those at v0 = 41.46 MHz (re­ported in Ref. 8) and certainly show no 1 /v0 dependence as would be expected if there were a significant second- order quadrupole effect on the rotationary averaged ND4+ deuteron spectrum. C. Orientation dependence of linewidths in Rb0.56( ND4 )0.44D2PO4 Figure 2 shows the temperature dependence of the second moment a2 of the ND4+ deuterons for the mixed pseudo-spin-glass sample D-RADP with x = 0.44. In contrast to the nearly-temperature-independent quadru­polar coupling constant, the linewidths of the mixed sam­ples increase significantly as the temperature decreases starting below about 240 K. For the two curves shown in the figure, the tetrahedral axis is perpendicular to the external magnetic field H0 but the orientation of the crys­tal c axis with respect to the H0 is different (0° and 30°). We chose these two orientations because they correspond to two different approximations [Eqs. (9) and (12)] in the dependence of the linewidth on the corresponding param­eter qEA. The c||H0 curve corresponds to the situation in which the linear term Af= 0 and the quadratic term B^0. As discussed in Sec. II the second moment for this orienta- T(K) FIG. 2. Temperature dependence of second moment cr2 for the x = 0.44 sample at two orientations. The solid curves are fits to the data points. Note that the c||H0 curve shows much less broadening than does the <J(c,H0) = 30° curve.4 2 N M R D E T E R M I N A T I O N O F T H E E D W A R D S - A N D E R S O N O R D E R . . . 5 9 4 9 tion is proportional to qEA [Eq. (12)]. In contrast, for <£ (c,H0) = 30o, the linear term is dominant so that the second moment is proportional to qEA [Eq. (9)] for this orientation. Since the order parameter is much less than unity at temperatures far above the glass transition tem­perature, we see that qEA»qEA at these temperatures. Therefore, we expect that the linewidth for c||H0 should be much less than for <£ (c, H0) = 30° and should start to broaden at much lower temperatures. The experimental confirmation of this expectation is displayed in Fig. 2. We performed similar experiments for several other orientations, and our results also show a similar broaden­ing, which increases as the temperature is lowered. For all these orientations, the measured linewidths are significantly larger than that observed for c||H0. Since the freeze-out of the O-D • * • O intrabond motion is a gradual process in structural deuteron spin-glass systems, the order parameter qEA, which is a measure of the strength of the local polarization within a cluster, is ex­pected to be nonzero well above TG. The gradually in­creasing linewidths observed in Fig. 2 reflect this feature. D. Evaluation of the order parameter qEA for x=0.44 Section V C contains experimental results for the tem­perature dependence of the second moment that agree qualitatively with our expectation for the temperature dependence of the order parameter. Since the tempera­ture dependence of the order parameter for the pseudo­spin- glass systems is theoretically predictable, as de­scribed in Eq. (4) for the random-bond random-field Ising model, it is possible to compare the experimentally ob­tained order parameter qEA with theory. Such compar­ison will provide confirmation of the validity of the theoretical model. Also, since the freeze-out of the O-D • • • O ordering is related to qEA, the comparison can verify the assumption that the freeze-out of the O- D • • • O bonds is the actual source of the temperature- dependent part of the second moment a2. We can derive the experimental order parameter qEA from our a2-versus-T curves in Fig. 2, according to Eqs. (9) and (12). Specifically, for c||H0, 9ea = [(°'')2/ct^)]1/2 , (14a) and for <£ (c, H0) = 30°, qEA = {(J> )2 /cr2 . (14b) To determine qEA requires a knowledge of a] or a\. In principle, these parameters can be determined by measur­ing the linewidth at temperatures much below TG> where the order parameter qEA is close to unity. However, there are two experimental difficulties in performing such measurements. First, at substantially lower temperatures the slowing down of the ND4+ rotational motion results in a substantial intramolecular contribution to the ND4+ linewidth, thus effectively prohibiting an accurate deter­mination of the contribution from O-D • * O bonds. Second, the ND4+ signal strength decreases rapidly as the temperature decreases below 130 K; this decrease is accompanied by a gradual increase in the O-D • • • O deuteron signal. (This phenomenon has been observed and reported earlier9 but is still unexplained.) Such weakening of the ND4+ signal substantially affects the accuracy of the linewidth determination. For these reasons we used an alternative approach to estimate a\ and o\ for the two orientations. Our approach is based on the recognition that the max­imum displacement of the individual ND4+ frequencies in the spin-glass system is approximately the same as the maximum quadrupolar splitting ND4+ in the AFE phase of the pure D-ADP, where all the O-D • • • O deuterons are ordered. At 206 K (a few degrees below the AFE transition occurring at Tc«215 K in our pure D-ADP sample) we measured in the pure D-ADP crystal max­imum quadrupole splittings 8v of approximately 8 kHz for c||H0 and 10 kHz for <£ (c, H0) = 30°. These splittings are much larger than the individual linewidths. Each of the coefficients <j\(30°) and cr^O0) is then estimated by equating it to the square of the corresponding 8v/2. This approach gives good precision provided that the temperature-dependent part of the linewidth is much larger than the temperature-independent part. The experimental results for qEA are plotted in Fig. 3. The individual data points are taken from Fig. 2 using Eq. (14), and the theoretical curve is determined from Eq. (4). To obtain numerically the q EA -versus- T curve we used an iteration method in which A/J 2 is varied and J /kB is held constant. From mean-field theory, the quantity J/kB in Eq. (4) equals TG and is a stable replica-symmetry-breaking solution.9 Even though the original second moments for the two orientations are far apart (Fig. 2), we found, using the value TG = 70 K, which is in the range (60-90 K) of theoretical values re­T( K) FIG. 3. Temperature dependence of the order parameter qEA of Rb0 56(ND4)0 44D2PO4. The experimental data are derived from the curves shown in Fig. 2 using Eq. (14). The theoretical curve is obtained from Eq. (4) using J /KB = 70 K and A//2 = 1.0. Note that the qEA values obtained from both crys­tal orientations lie on the same curve.5 9 5 0 S O N G H U A C H E N A N D D A V I D C . A I L I O N 4 2 ported in the literature,3,8,9 that the experimental data from both orientations can be approximately described by a single curve (that for which A//2 = 1.0). This result is in agreement with the fact that the Edwards-Anderson order parameter, which is dominated in mixed pseudo­spin- glass systems by the O-D • • • O freeze-out mecha­nism, should be independent of the crystal orientation. This feature becomes more evident if we make a compar­ison between the orientational dependences in Figs. 2 and 3. The temperature dependence for 0° and 30° in a2, shown in Fig. 2, certainly cannot be described by one curve. In contrast, the nearly-orientational-independent #ea f°r the two orientation shown in Fig. 3 is describable by a single curve. This large difference gives strong evi­dence that the dominant contribution to the ND4+ line broadening is from the O-D • • • O motions. The small discrepancy between the theoretical curve and the experimental data may arise from several sources. First, the experimental data in Fig. 3 appears to deviate slightly from the theoretical curve at tempera­tures above 220 K. This small discrepancy is not too surprising, since the integral equation for qEA [Eq. (4)] is an approximation whose validity decreases at tempera­tures far above the transition. Also, our precision in determining the experimental values for qEA is poor at these high temperatures, where the temperature- dependent portion of the linewidth is a small fraction of the total linewidth. Furthermore, there may be a small error due to approximations in estimating o\ and This error may not be significant at higher temperatures, where qEA is very small, but may become more significant as qEA increases at lower temperatures. In addition, slower ND4+ reorientations may make a small contribu­tion to the line broadening, especially at lower tempera­tures, since the ND4+ deuteron Tx minimum has been determined to occur at about 160 K.8 Nevertheless, it seems clear that the dominant contribution to the data of Figs. 2 and 3 arises from O-D • • • O freeze-out. E. Dependence of A on ND4+ concentration Figures 4(a) and 4(b) show the temperature dependence of the second moment a2 for the other two mixed sam­ples (x = 0.22 and 0.78). Both samples show an orienta­tional dependence similar to that of Fig. 2 for the x = 0.44 sample. From the fitting procedure for qEA described in Sec. V D, we can determine the value of the parameter A for x = 0.44. Since A depends on the ND4+ concentration, cvj N X CM 300 CM N I CM 2 50 300 200 T(K) FIG. 4. Temperature dependence of a2 for (a) jc = 0.22 and (b) x = 0.78. Both figures show an orientational dependence similar to that of Fig. 2. we may use this experimentally obtained A to provide a further check on the validity of the theory. Since A is proportional to x(l-x), [Eq. (3)], the x = 0.22 and 0.78 samples should have precisely the same value of A. Their A can be determined from the following formula, assum­ing J is known for each sample: A A A X x =0.44 J2 x =0.22 J2 x =0.78 J2 (J /kB )q ^ (J /kB )q 22 X 0.22X0.78 0.44X0.56 (15) The parameter J can be determined from a knowledge of Tg (J/kB = TG). Since mean-field theory predicts that the glass transition temperature should be 10-20 K higher for those concentrations (jc = 0.22 and 0.78) corre­sponding to the boundaries of the pseudo-spin-glass re­gion in the phase diagram9 than for *=0.44, we assigned the value 85 K to TG for the jc=0.22 and 0.78 samples, which is 15 K above the value (70 K) for the x = 0.444 2 N M R D E T E R M I N A T I O N O F T H E E D W A R D S - A N D E R S O N O R D E R 5 9 5 1 T(K) FIG. 5. Temperature dependence of order parameter qEA determined from x = 0.78 and 0.22 samples for (c,H0) = 30°. The experimental data points are obtained from Fig. 4. The theoretical curve, obtained from Eq. (4) with J/kB= 85 K and A//= 0.44, is identical for both x = 0.22 and 0.78. sample. Substituting these values into Eq. (15), we obtain A A J2 x =0.22 J2 x =0.78 = 0.44 . (16) We used these values and Eq. (4) to obtain a "theoretical" curve for qEA (the solid curve of Fig. 5). In Fig. 5 the x = 0.22 and 0.78 data are reasonably well described by this curve, in approximate agreement with the predic­tions of Eqs. (4) and (15). F. Second moments in pure D-ADP For comparison we also include here a plot of a versus T measured for two orientations in the pure D- ADP sample (Fig. 6). In contrast to the results for the mixed crystals (Figs. 2 and 4), the two sets of data are very close to each other, indicating that there is very lit­tle orientational dependence. The orientation, <J[(c,H0)=:27o, was chosen because individual lines are further apart and, thus, easier to resolve at this orienta­tion. The absence of an orientational dependence in the pure crystal and its presence in the mixed crystals suggest that the orientational dependence is a property of the pseudo-spin-glass. The temperature dependence of linewidths in this pure system may be explained by the slowing down of the thermal fluctuations of the N-D-O bonds, which are related to the sublattice po­larization. 22 VI. CONCLUSIONS AND DISCUSSIONS In this paper, we have reported our measurements of the anisotropy and temperature dependence of the rota­tionally averaged ND4+ second moment in mixed D- RADP as well as in pure D-ADP. From these measure- OJ N X -ac CM T (K) FIG. 6. Temperature dependence of cr2 in pure D-ADP for two orientations [<t (c,H0)=0°,27°]. The data show very little orientational dependence for the two orientations. ments we determined the Edwards-Anderson order pa­rameter qEA in the pseudo-spin-glass (PSG) phase. We interpret the temperature-dependent part of the second moment in the mixed crystals as arising from the gradual freezing of the O-D • * • O motions. We neglected the intramolecular contribution to the line broadening arising from the slowing down of the ND4+ reorientational motions. This treatment is justified, since the intramolecular interactions are motionally narrowed by the relatively rapid ND4+ reori­entations at all temperatures of our measurements. Furthermore, we found that the inhomogeneous linewidth of the ND4+ deuterons is several times larger than the homogeneous linewidth in all these samples at all the temperatures of our measurements, suggesting that the homogeneous linewidth is motionally narrowed.8 A comparison of our experimental results with the random-bond random-field SK Ising model proposed in Ref. 9 indicates that this mean-field-theory model is fairly good in describing the order parameter in these proton pseudo-spin-glass systems. One limitation of this model, however, is that it does not predict sharp variations in certain physical proper­ties, such as qEA, which would be expected to occur for ND4+ concentrations (x) corresponding to the boundary of the pseudo-spin-glass phase. As a result, knowledge of the ND4+ concentration x or of qEA in this region will not determine unambiguously whether a mixed sample is in a PSG or in a weakly ordered state. ACKNOWLEDGMENTS The authors would like to express their thanks to Pro­fessor Dr. R. Blinc for encouraging the present research and for providing the mixed single crystals of Rb;c(ND4)1_xD2P04. This research was supported in part by NATO under Grant No. RG 85/0081. One of the authors (S.C.) also thanks the University of Utah Research Committee for partial financial support.5 9 5 2 S O N G H U A C H E N A N D D A V I D C . A I L I O N 4 2 1E. Courtens, J. Phys. (Paris) Lett. 43, L199 (1982). 2R. Blinc and B. Zeks, Ferroelectrics 72, 193 (1987), and refer­ences therein. 3R. Pirc, B. Tadic, and R. Blinc, Phys. Rev. B 36, 8607 (1987). 4D. Chowdhury, Spin Glasses and Other Frustrated Systems (Princeton University Press, Princeton, 1986). 5E. Courtens, Ferroelectrics 72, 229 (1987), and references therein. 6R. Blinc, D. C. Ailion, B. Gunther, and S. Zumer, Phys. Rev. Lett 57, 2826(1986). 7R. Blinc, J. Dolinsek, R. Pirc, B. Tadic, B. Zalar, R. Kind, and O. Liechti, Phys. Rev. Lett. 63, 2248 (1989). 8R. Blinc, J. Dolinsek, V. H. Schmidt, and D. C. Ailion, Euro- phys. Lett. 6,55(1988). 9R. Pirc, B. Tadic, and R. Blinc, Z. Phys. B 61, 69 (1985). 10V. L. Akseniv, M. Bobeth, and N. M. Plakida, J. Phys. C 18, L519 (1985); Ferroelectrics 72, 257 (1987). UR. Blinc and B. Zeks, Soft Modes in Ferroelectrics and Antifer- roelectrics (North-Holland, Amsterdam, 1974), and references therein. 12R. Blinc, J. Slak, and M. Luzar, Phys. Status Solidi A 40, K121 (1977). 13D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, 1792 (1975). 14T. Chiba, Bull. Chem. Soc. Jpn. 38, 490 (1965). 15E. Matrushita and T. Matrubara, J. Phys. Soc. Jpn. 56, 200 (1987). 16W. Welke and E. Courtens, Ferroelectrics Lett. 5, 173 (1986). 17V. H. Schmidt, Ferroelectrics 72, 157 (1987); R. Kind, O. Liechti, and R. Bruschweiler, Phys. Rev. B 36, 13 (1987). 18R. O. Keelig, Jr. and R. Peplinsky, Z. Kristallogr. 106, 236 (1955). 19S. F. Edwards and P. W. Anderson, J. Phys. F 5, 965 (1975). 20R. Blinc, B. Gunther, and D. C. Ailion, Phys. Scr. T 13, 205 (1986). 21G. M. Volkoff, H. E. Petch, and D. W. L. Smellie, Can. J. Phys. 30, 270(1952). 22See, for example, W. Selke and E. Courtens, Ferroelectrics Lett. 5, 173 (1986).