Deuteron spin-lattice relaxation in partly and fully deuterated (NH4)2PdCl6

ARTICLE IN PRESS Solid State Nuclear Magnetic Resonance 34 (2008) 77– 85

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Deuteron spin-lattice relaxation in partly and fully deuterated ðNH4 Þ2 PdCl6 A. Birczyn´ski a, E.E. Ylinen b, M. Punkkinen b, A.M. Szymocha a, Z.T. Lalowicz a, a b

´ ski Institute of Nuclear Physics of PAS, ul. Radzikowskiego 152, 31– 342 Krako ´w, Poland H. Niewodniczan Wihuri Physical Laboratory, Department of Physics, University of Turku, FI– 20014 Turku, Finland

a r t i c l e in fo

abstract

Article history: Received 21 March 2008 Accepted 13 May 2008 Available online 27 June 2008

Deuteron spin-lattice relaxation and spectra were studied in partially and fully deuterated ðNH4 Þ2 PdCl6 in the temperature range 5–300 K. The relaxation rate maximum was observed at 45 K in ðND4 Þ2 PdCl6 . Its value is reduced due to limited jumps by about 33% relative to the theoretical value expected for threefold reorientations. Limited jumps correspond to an N–D vector jumping between six directions on a cone around a Pd–N vector, the angle between the N–D and Pd–N vectors being denoted D. This motion makes a part of the quadrupole interaction ineffective in relaxation thus reducing the maximum rate at 45 K. The observed reduction leads to the value D ¼ 21 . Limited jumps are quenched to a large extent at the order–disorder phase transition and consequently a decrease is observed in the rate. Below the transition NDþ 4 ions reorient between the tetrahedral orientations of the ordered phase, therefore the quadrupole interaction has the full relaxing efficiency. In the 10% deuterated sample the temperature of the rate maximum is shifted to 35 K and below 20 K the rate itself is one order of magnitude larger than in ðND4 Þ2 PdCl6 . The increase is related to (1) the absence of the order–disorder phase transition and (2) to the enhanced mobility of NH3 Dþ because of its electric dipole moment. Limited jumps are claimed to be the dominant relaxation mechanism below 20 K. The relaxation in the disordered 30% deuterated sample is quite similar to that in 10% sample. The 50% and 70% deuterated samples undergo a transition to the ordered phase. The relaxation is biexponential with the characteristic rates somewhat smaller than those in ðND4 Þ2 PdCl6 , but approaching them with increasing deuteration. This variation can be explained with different mobilities and varying relative numbers of the various isotopomers NH4n Dþ n , n ¼ 12 4. & 2008 Elsevier Inc. All rights reserved.

Dedicated to Professor Hans-Heirich Limbach on the occasion of his 65th birthday. Keywords: Deuteron NMR NMR relaxation Molecular dynamics NDþ 4 Tunnelling Phase transition Incoherent tunnelling Limited jumps Ammonium ion isotopomers

1. Introduction Nuclear spin-lattice relaxation of protons has been studied in many ammonium compounds, some of which show features related to quantum mechanical tunnelling at low temperatures. The number of papers on deuteron relaxation in corresponding fully deuterated samples is already much smaller. Besides the studies mentioned in [1] there is a recent work on ðND4 Þ2 PtCl6 [2]. And if our interest is focused on the deuteron relaxation in partly deuterated ammonium compounds, there are rather a few published papers, only partly deuterated samples of ðNH4 Þ2 SnCl6 [3,4], NH4 ClO4 [4] and ðNH4 Þ2 TeCl6 [5] have been studied. As far as the deuteron spin-lattice relaxation in partly deuterated compounds is concerned, the most important recent result is the observation that the relaxation rate is orders of magnitude larger than in fully deuterated samples below roughly 20 K [4,5]. This behaviour was explained by a greatly enhanced

 Corresponding author. Fax: +48 12 66 28 458.

E-mail address: [email protected] (Z.T. Lalowicz). 0926-2040/$ - see front matter & 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ssnmr.2008.05.002

rotation rate of NH3 Dþ ions relative to that of NDþ 4 ions, which in turn was attributed to the electric dipole moment of NH3 Dþ [6,7]. The electric dipole moment feels the agitation of its surroundings much more stronger than the NDþ 4 ions, which do not have the dipole moment. Another prominent feature was the low-temperature maxima in the deuteron relaxation rate of NH3 Dþ ions, when presented as a function of temperature [3,5]. These were interpreted by resorting to methyl-like tunnel splittings otj , corresponding to a stationary deuteron at the equilibrium position j [5]. If the deuteron jumps from the position i to j, the tunnel energy changes either by ð2=3Þðotj  oti Þ or ð1=3Þðotj  oti Þ depending on the rotational state (A or E, respectively) of the three NH3 Dþ protons. If this deuteron jump induces simultaneously a relaxation transition including changes in the deuteron spin state and in the interaction energy of the spin with the external magnetic field, then it may be possible to observe maxima in the deuteron relaxation rate when o0 ¼ ðg=3Þjotj  oti j. Here o0 is the deuteron resonance radial frequency and the symbol g can have the values 12, 1 and 2. This study deals with the deuteron relaxation in partly and fully deuterated ðNH4 Þ2 PdCl6 . These belong, together with

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´ ski et al. / Solid State Nuclear Magnetic Resonance 34 (2008) 77–85 A. Birczyn

ðNH4 Þ2 PtCl6 , ðNH4 Þ2 SnCl6 and ðNH4 Þ2 TeCl6 mentioned above, to the group of ammonium hexachlorometallates, which all have the fcc cubic structure at high temperatures. The potential hindering ammonium rotations is smallest for the Pd compound and increases monotonously for Pt, Sn and Te [8–10]. Therefore, ammonium ions are freest to rotate in the palladium and 2 2 platinum compounds. Also PdCl6 and PtCl6 ions can move so fast that the motion starts to dominate proton relaxation roughly above 250 K [8,11]. On the other hand it is understandable that NHþ 4 rotational tunnelling splittings at liquid helium temperatures are largest in the palladium and smallest in the tellurium compound; the T–A splitting nTA in the former was found to be 13.5 GHz [12] and the estimate for the latter is 240 MHz [10]. Thus also the tunnel splittings in partly and fully deuterated Pd and Pt compounds are clearly larger than in the corresponding Sn and Te compounds [10], a fact which has important consequences concerning the deuteron relaxation. An important factor in the deuteron relaxation in some ammonium metal hexachlorides is limited jumps (LJ) [2,5,13,14]. These are related to the fact that the minimum potential energy of the NDþ 4 ion does not correspond to the N–D and Pd–N vectors being parallel to each other, but each N–D vector has a number of equilibrium orientations making a certain angle D with the nearest metal–N vector. These orientations are obtained by rotations by an angle j about the fourfold axes of the hightemperature unit cell (j is somewhat larger than D [14]. Thus there are 24 equilibrium orientations, which exist in groups of six for an N–D vector near each Pd–Cl direction, between which ammonium ions undergo LJ. Since LJ occur much more often than the large-angle rotations about the threefold symmetry axes of NDþ 4 , they produce a maximum in the deuteron relaxation rate at a low temperature [5,14]. Furthermore, they affect the amplitude of the relaxation rate maximum related to large-angle rotations, which is observed between 45 and 70 K in ammonium metal hexachlorides [2–5]. A motional model was presented, which takes both these motions into account simultaneously. The decrease in the amplitude was consistent with the deviation angle j ¼ 32 in ðND4 Þ2 PtCl6 , corresponding to the angle D ¼ 26 between the Pt–N and N–D vectors [2]. Another important point is the structure of the deuteron NMR spectrum at low temperatures; in fully deuterated tin and tellurium compounds it is rather broad and shows a detailed structure [15–17] while in platinate compound the spectrum is structureless and quite narrow [2]. This means that the relevant tunnelling frequencies are larger than 10 MHz in the latter compound and the observed spectrum is mainly related to the A and E species ammonium ions. The T species spectral components are to a large extent spread out beyond observation, although they contribute somewhat to the broader component of the observed spectrum. A theory of deuteron NMR spectra for NH3 Dþ ions, taking into account dipolar interactions between all five spins in the ion, was worked out [18]. The structure of single crystal spectra makes possible to distinguish between cases of rigid, tunnelling or reorienting protons. Moreover, the spectra indicate that NH3 Dþ ions exhibit diverse mobility in partially deuterated ammonium perchlorate at helium temperature. There exist ions with immobile deuterons, which populate one position in the crystal unit cell, while other deuterons reorient about a C 3 axis parallel to the direction of the N–D bond of the rigid ones. More cases of diverse mobility of ammonium ion isotopomers were found in a study of powder spectra of partially deuterated ammonium compounds at low temperatures [17]. In ammonium hexachlorþ otellurate both NH2 Dþ 2 and about 50% NH3 D ions are rigid, while þ remaining NH3 D perform LJ. In ammonium hexachlorostannate NH3 Dþ ions perform either jumps about a C 2 axis or LJ, while

remaining isotopomers undergo rotational tunnelling. Very low values of the activation energy derived for all the related spectral components, from the temperature dependence of their contribution to the spectra, indicate incoherent tunnelling involved in the observed dynamic processes. Still another relevant factor is the order–disorder phase transition, in which also the structure is changed. ðND4 Þ2 PdCl6 was observed to undergo a transition to the monoclinic space group P21=n at 30.2 K [12,19], while the corresponding transition in ðND4 Þ2 PtCl6 is to theorthorhombic space group Fmmm at 27.2 K [20,21]. The transition temperature decreases with decreasing deuteration degree to 18 K in a 50% deuterated Pd compound and no transition was observed at the 30% deuteration [6]. In the next section we review theoretical models derived for the deuteron relaxation in fully and partly deuterated ammonium compounds. Then experimental data on deuteron spectra and relaxation are presented for ammonium hexachloropalladate with deuteron concentrations ranging from 10% to 100%. Dominant features of the experimental results are well explained in terms of the described models.

2. Theoretical 2.1. Deuteron relaxation in NDþ 4 ions with negligible tunnelling splittings Although the tunnelling splitting at low temperatures may be larger than the resonance frequency o0, they usually decrease fast above about 20 K. This fact together with the present experimental data shows that the tunnelling frequency nTA is smaller than o0 at the temperature of the relaxation-rate maximum near 45 K in ðND4 Þ2 PdCl6 . Then we can use a model in which largeangle reorientations and LJ dominate the relaxation in nontunnelling NDþ 4 ions. Such a model was recently presented and used to interpret the deuteron relaxation in ðND4 Þ2 PtCl6 [2]. For the initial relaxation rate in a powder sample the following expression was derived:  1 1 ð1Þ ð1Þ ¼ o2Q ½3Sð1Þ 0 þ 3S1  S2 powder T1 6 ½Jðo0 ; W 3 Þ þ 4Jð2o0 ; W 3 Þ    3 ð1Þ þ 4½5Sð1Þ 0  S1 powder J o0 ; W 3 þ r 4   3 þ 4J 2o0 ; W 3 þ r . 4

(1)

Here the strength of the quadrupole interaction is given by oQ ¼ e2 qQ =_, W 3 =24 is the rate for an approximately 120 reorientation taking the considered deuteron from one of the 24 equilibrium positions to another, and r=6 is the rate of LJ transferring the deuteron between the six equilibrium positions near a Pd–N direction. The spectral density functions are defined by Jðo; WÞ ¼ W=ðW 2 þ o2 Þ. The two orientation-dependent functions Bð1Þ ¼ ð3=8ÞoQ sin yij cos yij expð{fij Þ and Bð2Þ ¼ ð3=16ÞoQ ij ij sin2 yij expð{2fij Þ determine the sums SðmÞ 0 ¼

4 X 6 1 X jBðmÞ j2 , 24 i¼1 j¼1 ij

SðmÞ 1 ¼

4 X 6 1 X ðmÞ BðmÞ ij Bij0 , 24 i¼1 0 j;j ¼1 jaj0

SðmÞ 2 ¼

4 X 6 1 X BðmÞ BðmÞ , ij i0 j 24 0 0 i;i ¼1 j;j ¼1 iai0

(2)

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where the indices i and i refer to the directions of the Pd–N vectors and j and j0 to the six deuteron equilibrium positions near each Pd–N vector. The polar angles yij and fij define the direction of the N–D vector ij in the laboratory frame with the z axis parallel to the external magnetic field. The first line of Eq. (1) represents the contribution of largeangle reorientations while the second line shows the effect of LJ ð1Þ ð1Þ for r4W 3. The powder average ½3Sð1Þ 0 þ 3S1  S2 powder decreases ð1Þ when the deviation angle j grows while ½5Sð1Þ 0  S1 powder increases as shown in [2].

2.2. Deuteron relaxation in NDþ 4 ions in the presence of tunnelling At low temperatures the tunnelling splittings can be equal to or larger than o0 and hence they have to be taken into account. Unfortunately the existing models for relaxation do not include simultaneously the effect of LJ and large tunnelling frequencies, therefore we ignore LJ in the following. In the case of a strong hindering potential each nitrogen–hydrogen vector, which is a threefold axis of the NDþ 4 ion, is oriented practically parallel to one of the four crystal-fixed equilibrium directions. When the potential is weak, the hydrogens can tunnel through the hindering potential so that the nitrogen–hydrogen bonds become parallel to the four equilibrium directions but in a different order. This introduces tunnel splittings to the rotational ground state of the ammonium ion. Mathematically the situation can be described in terms of the tunnel matrix elements hj of NHþ 4 or  NDþ rotations of the ammonium ion 4 , corresponding to 120 about one of its threefold axes. As an example h1 ¼ hHðuvwxÞjHR jHðuxvwÞi, where HR is the rotational Hamiltonian and HðuvwxÞ the pocket-state function describing ammonium oscillation when the hydrogen u is near the equilibrium position 1, the hydrogen v near the equilibrium position 2, etc. These matrix elements split the rotational ground state into A, 3T, Ea and Eb levels, the labels being the irreducible representations of the tetrahedral point group. The energies of the A and E (representing both Ea and Eb ) levels are 2ðh1 þ h2 þ h3 þ h4 Þ and ðh1 þ h2 þ h3 þ h4 Þ, respectively. The energies of the three T levels depend generally on a third-order equation [22]. If the symmetry of the hindering potential is nearly cubic, the T levels are near each other, while the E–T and T–A splittings oET and oTA are much larger. For NHþ 4 the total spin of the four protons is related to the symmetry of the rotational wave function so that the total spins 2, 1 and 0 can appear only with the A, T and E type symmetries of the rotational function, respectively. For NDþ 4 such a restriction is not valid but different total spins can be connected with the rotational wave functions also in many ways [23]. Models for the deuteron spin-lattice relaxation of NDþ 4 ions in solids have been derived in the presence of both small [24,25] and large [13,26] tunnel splittings relative to o0 . The latter ones give the initial relaxation rate in terms of contributions from T–T, T–A and E–T transitions. When tunnel splittings are much smaller than the thermal energy, the upwards and downwards transition rates are practically equal. In such a case the initial deuteron relaxation rate for NDþ 4 ions in a polycrystalline material equals [13,26] 2 X 1 1 o2 ¼ n2 ½14Dðno0 ; tc Þ T1 480 Q n¼1

þ 5DðoTA þ no0 ; tc Þ þ 5DðoTA  no0 ; tc Þ þ 6DðoET þ no0 ; tc Þ þ 6DðoET  no0 ; tc Þ,

(3)

where tc is the rotational correlation time and the spectral density functions are now defined by Dðo; tÞ ¼ t=ð1 þ o2 t2 Þ. The validity of

79

Eq. (3) requires that the T–T tunnel splittings are much smaller than o0 . It can be expected that the spin–diffusion rate between the deuterons of neighbouring NDþ 4 ions is slower than the initial relaxation rate in Eq. (3). Then the deuteron relaxation during longer waiting times after saturation will proceed nonexponentially. Some of the nonexponentiality comes from the angular dependence of the various transition rates. Since this variation is rather small, roughly within a factor of two [4,13], such nonexponentiality is very difficult to observe in powder samples. There is, however, more pronounced nonexponentiality possible in the case of large tunnel frequencies oET ; oTA bo0 . Then the T type ammonium ions relax fast via the terms 14Dðo0 ; tc Þþ 56Dð2o0 ; tc Þ. The remaining terms of Eq. (3) are much smaller and therefore the A and E type ions should relax much more slowly than the T type ions. Detailed expressions for the relaxation rates of the T and A+E magnetisations are given in [27]. Here a somewhat qualitative approach is considered sufficient, details will be discussed in connection with the experimental data for the fully deuterated compound. 2.3. Relaxation of NH3 Dþ deuterons When the ammonium compound is only slightly deuterated, then almost all deuterons belong to NH3 Dþ ions. These ions can also undergo LJ, and therefore Eq. (1) can be used under some circumstances. Of course the rates r and W 3 probably differ from those of NDþ 4 . But there is another basic difference related to tunnelling, which causes in NH3 Dþ (protons tunnel) much larger þ shifts in energy levels than in NDþ 4 (deuterons tunnel). If NH3 D ions rotate about the threefold symmetry axis, on which the deuteron lies, its quadrupole coupling is not modulated. Because there is no time-dependence, there are no relaxation transitions, either. Therefore, only such rotations cause relaxation, which change the deuteron position. Such rotation gives rise to transitions between the energy levels (corresponding to a stationary deuteron at the equilibrium position j) 0

EmsMj ¼ mo0  MoH þ ks hj ,

(4)

where m and M are the spin quantum numbers for the deuteron and three protons, respectively, oH is the proton resonance 0 frequency, hj is the tunnel matrix element for NH3 Dþ when the deuteron is at the position j and the numerical factor ks equals 2 (when the three protons are at the methyl-like tunnelling state A) or 1 (for the methyl-like tunnelling states Ea and Eb ). Deuteron relaxation was considered mathematically for the described level structure in the recent work [5]. Unfortunately the effect of LJ was not included. It was assumed that three of the four crystal-fixed equilibrium directions (which are practically parallel to the threefold axes of the ammonium ion) are equivalent. This 0 0 0 means that corresponding tunnel matrix elements h2 , h3 and h4 0 are equal, while the last element, h1 , may be different. Then also the corresponding rotation rates have to obey analogous conditions. In some samples one of the deuteron equilibrium positions (position 1 was taken as the preferred one) may have a larger occupancy than the other [4,18,28]. In such a case the rate R03 =8 of the 120 rotations about axes 2–4, which move the NH3 Dþ deuteron from any of the positions 2–4 to the preferred position 1, can differ from the rate R3 =8 for inverse rotations about corresponding axes moving the deuteron from the position 1 to any of the positions 2–4. The rotation rate about the axis 1, which transfers the deuteron among the positions 2–4, is denoted W 03 =8. Finally a fourth rate R3a =8 is needed to define the frequency of rotations about axes 2–4, which transfer the deuteron between the positions 2–4. By calculating the relevant transition

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probabilities from the time-dependent perturbation theory, a result was obtained for the initial relaxation rate of NH3 Dþ deuterons [5]. For polycrystalline samples it reduces to 1 R3 R03 ¼ o2Q T1 80ð3R3 þ R03 Þ2 X  m2 ½10Jðr 1 ; mo0 Þ þ 3Jðr 1 ; o12 þ mo0 Þ m¼1;2

þ 3Jðr 1 ; 2o12 þ mo0 Þ þ

R3 5ð3R3 þ R03 Þ

o2Q ½Jðr 2 ; o0 Þ þ 4Jðr 2 ; 2o0 Þ.

(5) R03 Þ

and The characteristic rotation rates are r 1 ¼ ð1=4Þð3R3 þ r 2 ¼ ð1=8Þð3W 03 þ 2R03 þ 3R3a Þ. The tunnel splitting o12 is defined in terms of the tunnel matrix elements of NH3 Dþ as 0 0 o12 ¼ jh1  h2 j ¼ ð1=3Þjot2  ot1 j, where otj is the methyl-like tunnel splitting of NH3 Dþ , when the deuteron is at the equilibrium position j. Only one tunnel frequency is needed since the assumed symmetry requires that o12 ¼ o13 ¼ o14 and o23 ¼ o24 ¼ o34 ¼ 0. If the hindering potential at the ammonium site is relatively strong, then o12 is much smaller than o0 ; in which case Eq. (5) is significantly simplified [5]. If in addition the rotation rates R03 , R3 and R3a are equal, it finally reduces to Eq. (17) of [24] 1 1 2 o ½Jðr 1 ; o0 Þ þ 2Jðr 2 ; o0 Þ þ 4Jðr 1 ; 2o0 Þ þ 8Jðr 2 ; 2o0 Þ, ¼ T 1 40 Q

(6)

where r 1 ¼ R3 and r 2 ¼ ð1=8Þð3W 03 þ 5R3 Þ. The NH3 Dþ splittings otj , which are related to proton tunnelling, can be larger than o0 in the case of low potential. This is especially possible for partly deuterated ammonium hexachloropalladate and -platinate and Eq. (5) is applicable. þ In more heavily deuterated samples also NH2 Dþ 2 , NHD3 and NDþ ions have an important effect on the deuteron relaxation. 4 Fully deuterated ions were already discussed in Sections 2.1 and 2.2. For deuteron relaxation in NHDþ 3 ions models could also be worked out in principle. Two channels are possible depending on ammonium rotation. If the proton is stationary and only the three deuterons move then the deuterons behave like a deuterated methyl group and earlier derivations on deuteron relaxation can be used. If rotation changes also the proton position, then the method applied on the proton relaxation in NH3 Dþ ions can be used [29].  Tunnel splittings of NH2 Dþ 2 ions are related to 180 overlaps of the pocket state functions and therefore they are believed to be much smaller than o0 . Thus the decisive factor in the relaxation of NH2 Dþ 2 deuterons is the frequency of rotation and of LJ.

was followed by 2 ms read pulse after a variable time delay. The amplitude of the magnetisation, recovered during the delay, was determined by recording the free induction decay (FID). This was reasonable since the deuteron resonance spectrum remained narrow down to liquid helium temperatures, the width at the half height being only a few kHz at 5 K. A satisfactory S=N ratio was achieved by accumulating a number of signals. After the Fourier transform of the FID, phase and baseline corrections, the signal intensity was determined by integration of the whole spectrum. Typically 25 delays, covering the range from 0 to about 3T 1 , were used to determine the magnetisation recovery. An additional data point was measured at a time longer than 5T 1 , in order to determine precisely the equilibrium magnetisation. This approach was found to improve the accuracy of the three-parameter singleexponential fit to the data. Fits with two exponentials were necessary at lower temperatures, which lead to two characteristic rates and the corresponding weights.

3.2. Experimental results and discussion—spectra for ðND4 Þ2 PdCl6 The deuteron resonance spectra remained rather narrow down to liquid helium temperatures, but two components could be separated. Fig. 1 shows the temperature dependence of the width, measured at half height, for both the components in ðND4 Þ2 PdCl6 . They behave quite similar to the corresponding ones observed in ðND4 Þ2 PtCl6 [2]. The motional narrowing condition tc d1, where we have a product of a correlation time tc and the half width d of the spectrum, refers to homogeneous spectra of dipolarly coupled nuclei. As the deuteron spectrum is composed of many doublets with highly different positions from the center, the parameter d refers now to a doublet separation. It means that doublets with a small separation will undergo narrowing first, i.e. at lower temperatures with correspondingly longer correlation times. The narrowing process of doublets leads to the absorption spectrum IðoÞ: IðoÞ ¼

ðo  oÞ½2ðo  oÞk þ D1 d2 þ D2 d1   ð2k þ 12dÞðD1 D2  dk  d1 d2 Þ ðD1 D2  dk  d1 d2 Þ2 þ ½2ðo  oÞk þ D1 d2 þ D2 d1 2

,

(7) where o1 ¼ a and o2 ¼ a define the frequencies of a symmetric doublet, while d1 and d2 are their natural line widths, respectively.

5 4.5 4

3. Experimental

Deuteron relaxation and spectra were measured in five polycrystalline samples of ammonium hexachloropalladate deuterated at 10%, 30%, 50%, 70% and 100%. Some samples were previously used in INS experiments [6,12]. The NMR experiments were performed with the APOLLO (Techmag, USA) spectrometer, operating on the 7.04 T/89 mm superconducting magnet and thus at the deuteron resonance frequency 46 MHz. A low-temperature NMR probe was placed inside the Oxford CF1200 continuous flow cryostat, and the temperature was regulated by the Oxford CT503 temperature controller. The temperature accuracy and stability were within 0:1 K in the whole temperature range. The spin-lattice relaxation time was measured by the saturation-recovery method. An aperiodic 10-pulse saturating sequence

h1/2 [kHz]

3.1. Experimental setup and procedure

TPT

3.5 3 2.5 2 1.5 1 0.5 0 0

5

10

15

20

25

30 35 T [K]

40

45

50

55

60

65

Fig. 1. Temperature dependence of the half width of the narrow (.) and broad (m) spectral components. The half widths calculated from Eq. (7) are shown for the doublet separations 2a ¼ 20 kHz ðþÞ, 40 kHz (), 100 kHz ( ), 135 kHz (}) and 200 kHz (&) in the temperature range between 25 and 50 K.

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1000 0.65

TPT

0.6 100 0.5

1/T1 [s-1]

Sn, Sb

0.55

0.45

10

0.4 0.35

1

0.3 0.001

0.01

0.1

1

10

100

1000

10000

τ [s] Fig. 2. Recovery of the narrow (.) and broad (m) spectral components after saturation at 13 K. Continuous line shows a fit with parameters given in the text.

The remaining quantities are d ¼ d1 þ d2 , D1 ¼ o  o1 , D2 ¼ o  o2 , o ¼ ðo1 þ o2 Þ=2 and k is the exchange rate [30]. Reorientation modulates the spin-dependent interaction. The corresponding correlation time 1=W 3 , defined as the mean time between jumps, has the meaning of a lifetime of a given orientation. We will use formula (7) in the following with the exchange rate understood as the lifetime of respective deuteron positions. We may also treat the spectrum as motionally narrowed classical one. For some simulations we take Eq. (7) and use for the exchange rate k the reorientation frequency W 3 as obtained from the fit of Eq. (1) to the relaxation rates. From the broad deuteron spectrum of rigid NDþ 4 ions, spreading over the range 135 kHz, we have selected some representative doublet positions. The calculated half widths for different doublet separations for the temperature range 30–50 K fall above (large separation) and below (small separation) the experimental points in Fig. 1. This is understandable because the experimental data represent the powder average of the half width for many doublets with separations between 1 and 135 kHz. To study the origin of the spectral components we determined their relative weights as the function of the delay between the saturation and the read pulse (Fig. 2). The weight of the narrow component is seen to decrease with decreasing delay while that of the broad component increases. The variation with the delay of both the weights is nonexponential and two characteristic rates, equal to those obtained in the relaxation experiments, are needed for its description. This indicates that both the fast and slowly relaxing ammonium ions contribute to the narrow and broad spectral components. Since the relative weight of the narrow component decreases with decreasing delay, it is mainly related to the slowly relaxing ammonium ions. Similar reasoning shows that the broad component is mainly related to the fast relaxing ammonium ions. 3.3. Experimental results and discussion—relaxation in ðND4 Þ2 PdCl6 Fig. 3 shows the spin-lattice relaxation rate of deuterons in ðND4 Þ2 PdCl6 as a function of inverse temperature. A prominent maximum is seen in the rate near 45 K. This maximum is related to large-angle rotations, the correlation time obeying the condition o0 tc 1. On the high-temperature side the rate at first decreases until T is about 200 K but starts then to increase at higher temperatures. The magnetic dipolar and spin-rotational interactions are more effective in relaxing protons than deuterons. Our numerical

0.1 0

10

20

30

40

50

60 1000/T [K-1]

70

80

90

100

Fig. 3. Experimental deuteron relaxation rates in polycrystalline ðND4 Þ2 PdCl6 . The dotted curve and broken line below the phase transition describe the relaxation exclusively via 120 reorientations, while the continuous curve includes also the 2 effect of limited jumps. The increase above 200 K arises from the motion of PdCl6 anions.

estimates show that these interactions are too weak to cause the observed increase in the deuteron relaxation rate above 200 K. Therefore, the most likely explanation is the electric field gradient, 2 created by the PdCl6 chlorines at the deuteron sites and modulated by the anion motion [8,11]. The value of the relaxation rate maximum at 45 K is about 33% smaller than the theoretical value 470 s1 for the resonance frequency 46 MHz, based on the coupling frequency oQ ¼ 2p  1:80  105 s1 . In ðND4 Þ2 PtCl6 the reduction was about 50% [2], while in ðND4 Þ2 SnCl6 , which remains cubic even at 4.2 K without any order–disorder transition, the theoretical and experimental maximum values are equal within the experimental accuracy [4]. In ðND4 Þ2 PtCl6 the reduction was explained by LJ and the corresponding terms in Eq. (1), leading to the value j ¼ 32 for the deviation angle [2]. This means that the angle D between the Pt–N and N–D vectors is 26 . The same explanation for the reduction of the maximum value in ðND4 Þ2 PdCl6 leads to j ¼ 26 and D ¼ 21 . The data calculated from Eq. (1) are described by the continuous curve in Fig. 3 and they are based on the activation energies and pre-exponential factors for the reorientations and LJ, E3a ¼ 3:5 kJ=mol and W 30 ¼ 7  1012 s1 ; Era ¼ 1:5 kJ=mol and r 0 ¼ 1012 s1 , respectively. On the low-temperature side of the discussed maximum the relaxation again decreases and shows a step-like reduction at the phase-transition temperature near 30 K, although the step does not appear so distinct as in ðND4 Þ2 PtCl6 [2]. Since in crossing this temperature downwards the ammonium ions become ordered [18,28], the contribution of LJ to relaxation is reduced practically discontinuously causing the observed decrease in the relaxation rate. Thus the rate maximum, which is predicted near 20 K by using Eq. (1) and the mentioned parameters for LJ, is not observed. This behaviour can be explained by assuming that the rate of LJ to the preferred position, r pr , becomes much larger than the rate of jumps to any nonpreferred position, r op [2]. In the range 25–30 K the deuteron relaxation is most likely dominated by tetrahedral reorientations by 120 between ammonium equilibrium orientations of the ordered state. Then the entire quadrupole interaction is effective in the relaxation [2], and the corresponding rate is described by the broken line in Fig. 3. At still lower temperatures a clear nonexponentiality is observed and two exponentials were used to describe the magnetisation recovery. The corresponding rates become practically independent of temperature below 20 K, which can be

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1000

1 0.9 0.8

100

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explained if incoherent tunnelling dominates over thermally activated rotation. As mentioned in Section 2.2, the T-type ammonium ions should relax fast and the A- and E-type ions much more slowly because the latter process involves large tunnel splittings oTA and oET . Thus the rate of the fast relaxing component, about 10 s1 below 20 K in ðND4 Þ2 PdCl6 , should be determined by the term proportional to 14Dðo0 Þ þ 56Dð2o0 Þ in Eq. (3). It seems natural to explain the rate of the slow component, about 0:15 s1 , by relating it to the remaining spectral density functions DðoTA  no0 Þ and DðoET  no0 Þ. In the slow motion limit the ratio of the rates of the slow and fast components should be equal to the ratio of the discussed spectral density functions, which is DðoTA  no0 Þ=Dðo0 Þ ðo0 =oTA Þ2 . The estimated tunnel frequency is oTA =2p ¼ 520 MHz for the palladate compound [10]. This leads to the density-function ratio 0.0078. Experimentally the ratio was observed to be 0.015, which is of the expected magnitude. The described explanation requires that the spin diffusion rate between the deuterons of neighbouring NDþ 4 ions must not exceed the rate of the slow component, otherwise the latter rate would be dominated by spin diffusion. Additional support for the described interpretation is provided by the saturation experiments of Fig. 2, which show that the broad spectral component relaxes faster than the narrow component. If all the deuterons in the 100% deuterated sample were observed, the relative contribution of the T species ions to the observed signal in thermal equilibrium would be 58% and that of the A and E species ions 42%. The experimentally observed weights of the fast and slowly relaxing components below 20 K are about 0.30 and 0.70, respectively (Fig. 4). This seems to mean that only about 30% of all the T species ions are observed. Quite similar nonexponentiality was observed in the deuteron relaxation in ðND4 Þ2 PtCl6 below 20 K [2]. Two possible explanations for it were discussed, one based on the different relaxation rates of T and A þ E species ions, while the other explanation resorted to domains and domain boundaries in the ordered structure. The final conclusion remained somewhat uncertain, the reasons being that saturation experiments of Fig. 2 were not carried out and that only estimates were available for oTA. 3.4. Relaxation in 10% and 30% deuterated samples Our results on the deuteron relaxation rate in 10% deuterated ammonium hexachloropalladate are shown in Fig. 5. The temperature dependence of the relaxation rate on the high-

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1000/T [K-1] Fig. 5. Temperature dependence of the relaxation rate for 10% deuterated ammonium hexachloropalladate (on the background are the results for the 100% deuterated compound—open symbols). The dotted curve describes the combined effect of the reorientations of the various isotopomers. The continuous line exhibits the total relaxing power of the reorientations and limited jumps of all the isotopomers. See text for used parameters.

temperature side of the maximum is very similar to that in fully deuterated salts. However, the maximum itself is somewhat changed. Its position is shifted to about 35 K, which is about 10 K lower than in ðND4 Þ2 PdCl6 . The maximum appears also broader than in the fully deuterated sample. Furthermore, there is only a small decrease in the relaxation rate at lower temperatures and the practically constant value, 100 s1 , is reached already near 20 K. This is a factor of 10 larger relaxation rate than in ðND4 Þ2 PdCl6 . No appreciable nonexponentiality was observed. In 10% deuterated samples 73% of the deuterons belong to NH3 Dþ and 24% to NH2 Dþ 2 ions. In the following we consider nearly exclusively the contribution of NH3 Dþ and say something about the latter ions at the end. The increased relaxation rate is believed to demonstrate at least two effects. At first there is no phase transition, therefore the LJ terms of Eq. (1) have their full power, consistent with the value of the deviation angle j. Another factor is related to the frequencies of rotation and LJ, which can be clearly larger for NH3 Dþ than NDþ 4 ions as observed in ammonium perchlorate, hexachlorostannate and -tellurate samples [4,5]. The NH3 Dþ ions have the electric dipole moment and therefore their rotational mobility is larger than that of NDþ 4 [6]. The increase in the mobility in the four studied hexachlorometallates with 5% deuteration is the largest in Sn and Te compounds, by about four orders of magnitude, while in hexachloropalladate the increase is about one order of magnitude. The decrease in the relaxation rate below 35 K occurs smoothly and slowly, this behaviour is usually explained by incoherent tunnelling becoming more efficient than reorientations. This explanation might be possible even now, although it is difficult to imagine what incoherent tunnelling is like in the case of NH3 Dþ ions. Only the three protons can tunnel coherently and incoherently and thus increase the apparent frequency of rotation about the threefold axis of NH3 Dþ . But this rotation does not modulate the deuteron quadrupole coupling, it modulates only the magnetic dipolar interaction between the deuteron and the three protons. And this latter interaction has only a minor effect on the deuteron relaxation because of its weakness. This is proved by the experimental fact that in the 10% deuterated sample the relaxation rate below 20 K is only by a factor of 3 smaller than at the maximum at 35 K. Since the overall effectiveness of the deuteron–proton dipolar interaction is by the factor 190 smaller than that of the deuteron quadrupole coupling in ammonium

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compounds [4,5], the former cannot dominate below 20 K in the 10% deuterated sample. It seems to us that the relaxation in this temperature range is dominated by LJ. Eq. (1) with the motional parameters of ðND4 Þ2 PdCl6 predicts, besides the rate maximum at 45 K another maximum near 23 K. Experimentally no maximum is observed, rather a plateau. However, the motional parameters for NH3 Dþ and NDþ 4 ions are probably different. Therefore, we have used Eq. (1) again, by using the same deviation angle j ¼ 26 as for ðND4 Þ2 PdCl6 but varying the motional parameters, to fit the experimental data between 10 and 200 K for the 10% sample. Actually two sets of rotational and LJ parameters were used, one þ þ set for NH3 Dþ ions and the other for NH2 Dþ 2 , NHD3 and ND4 ions. We obtained the continuous curve shown in Fig. 5 by using the activation energies and pre-exponential factors E3a ¼ 2:1 kJ=mol, W 30 ¼ 1:4  1012 s1 , Era ¼ 0:2 kJ=mol and r 0 ¼ 5  109 s1 for the former set with the relative weight 0.729; and E3a ¼ 3:3 kJ=mol, W 30 ¼ 1:2  1013 s1 , Era ¼ 1:5 kJ=mol and r 0 ¼ 2  1012 s1 for the latter set with the relative weight 0.271. The agreement is quite satisfactory. The dotted curve represents the total relaxation rate of the mentioned isotopomers via the 120 reorientations without the reducing effect of LJ. The activation energies E3a and Era for NH3 Dþ are clearly smaller than the corresponding parameters for NDþ 4 . That can be explained by the smaller moment of inertia of NH3 Dþ and by its electric dipole moment [6,7]. E3a for NH3 Dþ is even smaller than the activation energy 2.55 kJ/mol of the NHþ 4 ions in ðNH4 Þ2 PdCl6 [8], which have no electric dipole moment. A similar fitting procedure was carried out for the 30% deuterated sample. We used the same deviation angle j and the same motional parameters as above for NH3 Dþ and NH2 Dþ 2 , but for NHDþ and NDþ the following parameters were used 3 4 E3a ¼ 3:5 kJ=mol, W 30 ¼ 5  1012 s1 , Era ¼ 1:7 kJ=mol and r 0 ¼ 2  1012 s1 . The corresponding relative weights are 0.343 for þ þ NH3 Dþ , 0.441 for NH2 Dþ 2 and 0.216 for NHD3 and ND4 . These produce the continuous curve in Fig. 6. In fitting we used the condition that the activation energy of both reorientations and LJ increases with the mass of the ammonium ion. If the relaxation rates of the various isotopomers differ more than by a factor of about 3, the relaxation should appear nonexponential because of the slow spin diffusion. Nevertheless, the observed initial relaxation rate would not be changed. Therefore, the present approach should be applicable as far as no nonexponentiality is observed. It has to be noted that in the fitting we assumed that the tunnelling splitting o12 (see Eq. (5)), it takes into account the

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indirect effect of the proton tunnelling on deuteron relaxation is smaller than o0 . For the case o12 4o0 the theoretical curve should be by about 6% lowered below 25 K, actually in a better agreement with experiment (Fig. 5). Experimentally two or three tunnelling splittings were observed for NH3 Dþ below 10 K [6], which seem to be consistent with the case o12 4o0 . Theoretical calculations lead to three large splittings [31], also in agreement with o12 4o0 . In any case the incoherent tunnelling of NH3 Dþ does not have the þ same meaning as that of NHþ 4 and ND4 . The most important type of reorientation in deuteron relaxation of NH3 Dþ ions is such, which changes the deuteron position. These rotations are not related to any tunnelling, they just rotate the axis of NH3 Dþ tunnelling by 120. 3.5. Relaxation in more heavily deuterated ðNH4 Þ2 PdCl6 Fig. 7 shows the results for 50% deuterated sample. The data remind those for lower deuteration, the only additional feature is the appearance of the slowly relaxing component with the rate slightly less than 1 s1 below 20 K. In the 70% deuterated sample the relaxation rates of the fast and slow components, about 40 and 0:2 s1 , are remarkably smaller than in the 50% sample (Fig. 8). Still they are somewhat larger than in the fully deuterated compound. At still higher deuterations these two rates probably decrease smoothly towards those of the 100% deuteration. Eq. (1) was also used to describe the data of the 50% and 70% samples by using the same deviation angle j and the same motional parameters as for the 30% sample. Only the relative weights were changed to correspond to the higher deuteration degree. The obtained variation with temperature is shown by continuous curves in Figs. 7 and 8. The curves describe only the exponential relaxation above 25 K, because we do not know which isotopomers or which portion of each isotopomer relaxes at the faster and slower rates in the range of nonexponentiality. Also the phase transition occurs soon below 25 K and causes additional difficulties in estimating the mentioned rates. The dotted curves represent the total contribution via 120 reorientations of all the differently deuterated ions. The decrease in the relaxation rate of the fast component and the appearance of the slowly relaxing component are inevitably related to changes in the relative abundance of NH3 Dþ , NH2 Dþ 2 and NHDþ 3 . These isotopomers have different tunnel splittings and probably also different rotational mobilities. It would be natural to expect some variation in the weights of the fast and slow component with the deuteration degree. Some variation was found but a comparison with theory is rather demanding because

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of the presence of many isotopomers. To find out in detail their contribution would require more experiments especially on samples with deuterations only slightly less than 100%. The difficulty is that then most deuterons belong to NDþ 4 ions and mask to a large extent the contribution of other isotopomers. Another possibility would then be to study protons. No significant change in the relaxation rate was observed for 50% and 70% deuterated ammonium hexachloropalladate at the order–disorder phase transition temperature (Figs. 7 and 8). The transformation from the LJ dominated relaxation (believed to prevail in the 10% and 30% deuterated samples below about 25 K) to the incoherent tunnelling dominated relaxation (in the 100% sample) depends on the ratio r op =r pr , but also on the deuteration degree and the relative contributions to the relaxation rate of the 120 reorientations and LJ. In differently deuterated ammonium hexachlorotellurate samples such a change was seen rather clearly [5]. Another important factor is the frequency of these two motions, which might be increased by incoherent tunnelling, þ þ different from that of NHþ 4 and ND4 . If for example NH3 D would undergo incoherent tunnelling transitions equivalent to 120 rotations changing the deuteron position, then analogous transitions should also be possible between the six equilibrium positions near each Pd–N vector [32]. Because the potential barrier for LJ is much smaller than for the 120 rotations, the increase in the corresponding motional rate via such incoherent tunnelling should be clearly larger in the case of LJ than 120 rotations.

ðND4 Þ2 SeCl6 , ðND4 Þ2 PtCl6 and ðND4 Þ2 PtBr6 , 20:4 , 20:9 and 27:4 , respectively, are of the same magnitude [32]. The present method gave for D the value 26 in ðND4 Þ2 PtCl6 [2]. At the order–disorder phase transition near 30 K one of the six directions near each Pd–N direction becomes exclusively populated. This causes a drastic decrease in the effectiveness of LJ in relaxation. However, NDþ reorientations are still occurring 4 between the tetrahedral orientations in the ordered phase and they dominate below 30 K. Below 25 K nonexponential relaxation is observed. The nonexponentiality is believed to become observable when the tunnelling frequencies become larger than the resonance frequency. The fast relaxing component is believed to represent the T species and the slow component the A þ E species NDþ 4 ions. Additional support for this interpretation is provided by the saturation experiments showing that the broad spectral component relaxes faster than the narrow spectral component. Ammonium hexachloropalladate deuterated at 10% remains in the disordered phase in the entire temperature range of the present study. The deuteron relaxation is exponential down to 4.2 K and the maximum of the relaxation rate is shifted to about 35 K. This shift is related to the larger mobility of NH3 Dþ relative to that of NDþ 4 . The electric dipole moment of the former ion feels the vibrations of the surrounding atoms much more sensitively than NDþ 4 , which does not have that moment. Partly from the same reason the relaxation rate below 20 K is about one order of magnitude larger than that of the completely deuterated, ordered sample. The decrease in the rate below 35 K is quite slow. Such a behaviour is usually explained by resorting to incoherent tunnelling. Here we offer another explanation based on LJ, which have their full effectiveness, consistent with the actual value of the deviation angle j, in the disordered structure. The temperature dependence of the relaxation rate in the 30% deuterated sample is quite similar to that in the 10% sample. This is understandable since both samples remain disordered down to 4.2 K. In the 50% sample a clear nonexponentiality is observed but the characteristic rates of the fast and slow component are considerably larger than in ðND4 Þ2 PtCl6 . In the 70% sample the characteristic rates are already much nearer to those of the fully deuterated sample. This behaviour is easily explained in terms of the varying relative numbers and different mobilities of the þ þ isotopomers NH3 Dþ , NH2 Dþ 2 , NHD3 and ND4 . Acknowledgments We are grateful to M. Prager of Institut fu¨r Festko¨rperforschung des Forschungszentrums Ju¨lich for kind donation of samples used in this study.

4. Conclusion References Experimental data on the deuteron relaxation rate in polycrystalline ðNH4 Þ2 PdCl6 show a maximum related to large-angle (about 120 ) reorientations at 45 K. Its value is reduced from the theoretical value due to LJ which commute the ammonium deuterons between positions near the four Pd–N directions. These positions are obtained by rotating NDþ 4 by an angle j about the fourfold axes of the high temperature unit cell. They appear in groups of six near each Pd–N direction. Since LJ occur much more often than large-angle reorientations, they render a part of the deuteron quadrupole interaction ineffective in relaxation and cause a reduction in the relaxation rate. In ðND4 Þ2 PdCl6 this reduction is about 30% at 45 K, which is consistent with the deviation angle j ¼ 26 . The corresponding angle between the N–D and Pd–N directions, D, is 21 . Although D has not been determined by neutron diffraction in this sample, its values for

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