NH3D+ dominated proton spin-lattice relaxation in partly deuterated ammonium compounds

ARTICLE IN PRESS

Physica B 357 (2005) 456–471 www.elsevier.com/locate/physb

NH3D+ dominated proton spin-lattice relaxation in partly deuterated ammonium compounds E.E. Ylinena, P. Filipekb, M. Punkkinena,, Z.T. Lalowiczb b

a Wihuri Physical Laboratory, Department of Physics, University of Turku, FIN-20014 Turku, Finland H. Niewodniczanski Institute of Nuclear Physics of Polish Academy of Sciences, Radzikowskiego 152, 31-342 Krakow, Poland

Received 23 September 2004; accepted 10 December 2004

Abstract Proton spin-lattice relaxation is studied in partly deuterated ammonium compounds at low temperatures. A model is proposed for the NH3D+ related contribution, which increases the relaxation rate many times larger than in nondeuterated samples. The model introduces two kinds of level-crossing minima in T1, where some tunnel frequency is equal to a separation between the proton Zeeman levels in the external magnetic field. One kind of minimum involves a CH3-type tunnel splitting of NH3D+, when the deuteron is stationary at anyone of the four threefold axes of the ammonium ion. The corresponding relaxation rate is expected to depend on the experimental pulse sequence. The other kind of minima (there could be six of them) result from the NH3D+ rotations moving the deuteron from one threefold axis to another and back, which make the otherwise motionally independent AA part of the magnetic dipolar interaction of NH3D+ time dependent. The involved tunnel splitting is equal to (2/3) times the difference between the CH3-type tunnel splittings. In a level-crossing transition the tunnel energy is changed by that splitting, but the resulting energy imbalance is transferred fast to the lattice by spin-state preserving reverse deuteron jumps, removing any coupling to the tunnel energy reservoir. Thereafter another level-crossing transition is possible. Experiments on polycrystalline 4% and 10% deuterated ammonium hexachlorotellurate samples reveal two additional level-crossing minima in T1 below 20 K at the proton resonance frequencies 25 MHz and about 28 MHz, which are not found in the nondeuterated sample. The minima show different characteristics relative to the experimental pulse sequence and also agree otherwise well with the predictions of the model. r 2004 Elsevier B.V. All rights reserved. PACS: 7660E Keywords: Proton spin-lattice relaxation; Deuteration-enhanced relaxation; Partially deuterated (NH4)2TeCl6

Corresponding author. Tel.: +358 2 3335755; fax: +358 2 3335993.

E-mail addresses: eero.ylinen@utu.fi (E.E. Ylinen), matti.punkkinen@utu.fi (M. Punkkinen). 0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.12.012

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

457

1. Introduction Early experiments by Svare and Tunstall [1] showed that even a small partial deuteration of ammonium compounds exhibiting rotational tunnelling increases drastically, that is by orders of magnitude, the proton nuclear spin-lattice relaxation rate at low temperatures. Explanations of the increased rate remained quite qualitative, although it was understood that its origin lies in the less restricted mobility of the partly deuterated ammonium ions. Recently some papers were published on the deuteron NMR spectra in partly deuterated ammonium compounds which show a reasonable agreement between experiment and theory based on separate contributions, weighted according to the binomial distribution, of NH3D+, NH2D+ 2 , + + NHD+ 3 and ND4 ions [2,3]. At low deuteron concentrations the NH4 ions are most abundant while NH3D+ follow next. Therefore, NH3D+ ions should play the dominant role also in spin-lattice relaxation. An additional factor is its electric dipole moment, which via the interaction with the local electric field makes the equilibrium orientations of the four nitrogen–hydrogen bonds energetically nonequivalent in the case of NH3D+ ions [3]. Such a nonequivalence has also been observed by inelastic neutron scattering [4]. Furthermore, a somewhat similar phenomenon was observed with partly deuterated methyl groups [5]. In some samples, for example in partly deuterated NH4ClO4 and NH4PF6, the nonequivalence is so large that the deuterons occupy almost exclusively only one site or one threefold axis [4], while for example in (NH4)2S2O8 [3] and (NH4)2PdCl6 [6] such a preference is less pronounced. In spite of the explained progress in the understanding of the deuteron NMR spectra and tunnelling levels of partly deuterated ammonium compounds, the strong increase of the proton spin-lattice relaxation rate with deuteron concentration has remained a mystery. Possibly the dominant impetus for the present study was the careful experiments by Grabias and Pislewski [7] on proton relaxation in ammonium hexachlorotellurate with natural and increased deuteron concentrations. They observed a tunnel splitting equal to about 55 MHz at liquid helium temperatures in the sample with the natural deuteron concentration (the alternative expression ‘‘the nondeuterated sample’’ will also be used). According to the reasoning of Birczynski et al. [8] this is not the T–A splitting but some splitting between the T levels, although decisive experimental results are not yet available. Furthermore, Grabias and Pislewski [7] found in the sample with the 5% deuteration a minimum in the proton relaxation time T1 near 14 K at the proton resonance frequency 55 MHz, almost at the same temperature as the level-crossing minimum related to the 55 MHz tunnel frequency of NH+ 4 in the nondeuterated sample, but the minimum value of the 5% sample was about 10 times shorter than that of the sample with the natural concentration. In the latter sample the relaxation rate is dominated by the interactions and motion of NH+ 4 while the increased rate in the former sample is related to NH3D+. This gave us a reason to study if these minima differ in any way in their temperature and frequency dependence. A critical prerequisite for such a study is that only such experimental conditions are meaningful for which the relaxation rate in the sample with the increased deuteron concentration is clearly larger than in the nondeuterated sample. This limited most of our experiments to temperatures below 20 K. + In the following, the tunnel levels of NH+ are first briefly discussed. If the deuteron is 4 and NH3D + stationary at a certain site (a threefold axis of NH4 ), then the proton spin-lattice relaxation in NH3D+ resembles the relaxation in methyl compounds, the main features of which are also briefly reviewed. If the deuteron of NH3D+ changes site many times during the proton T1, a new very fast relaxation mechanism is possible. This mechanism also involves level-crossing transitions, but the relevant tunnel splitting is proposed to be equal to the difference between the single-axis tunnel splittings of NH3D+ multiplied by (2/3). The effectiveness of this relaxation channel is enhanced by a very fast, spin-independent transfer of the tunnel energy to lattice via such rotations of NH3D+, which take the deuteron back to the original site. Under the simultaneous influence of the fast spin diffusion between protons the considered NH3D+ ion is quickly returned also to its original spin state, whereafter another relaxation transition is possible. Experimental results on the proton relaxation rate, mainly as a function of the resonance frequency at a

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

458

constant low temperature, in polycrystalline ammonium hexachlorotellurate samples with the natural, 4% and 10% deuteron concentrations are finally presented. The results for the deuterated samples show two additional T1 minima, not present in the sample with the natural deuteron concentration, with clearly different characteristics. One of these minima agrees well with the methyl-like relaxation, while the other is consistent with the new fast relaxation mechanism.

2. Level structures The tunnel-split energy levels of the rotational ground state of NH+ 4 can be described in terms of the overlap matrix elements of the rotational Hamiltonian HR, for example h1 ¼ hHðuvwxÞjH R jHðuxvwÞi;

(1)

where H(uvwx) is a pocket-state function describing the motion of the four protons u, v, w and x near their equilibrium sites (u near the threefold axis 1 of NH+ 4 ) and H(uxvw) is a similar function obtainable from H(uvwx) by a 1201 rotation about the threefold axis 1. HR and h1 as well as other Hamiltonians and energy quantities are expressed in the units of angular frequency if not otherwise specified. The three other overlap matrix elements h2, h3 and h4 are defined similarly [9,10]. In the case, that the potential hindering the rotations of NH+ 4 is at least tetrahedrally symmetric, all hi’s are equal and three levels labelled as A, T (threefold degenerate) and E are obtained. (NH4)2TeCl6 is known to undergo a structural transition from cubic to trigonal symmetry at 88 K and no additional transitions have been observed with lowering temperature as far as the deuteron concentration is less than 15% [11]. If the hindering potential at the site of the ammonium ion has the trigonal symmetry, the overlap matrix elements obey h16¼h2 ¼ h3 ¼ h4 and the T levels should be split such that the levels T(1) and T(2) remain degenerate while T(3) has a somewhat lower energy (see Fig. 1a). A small additional distortion of lower symmetry in the hindering potential would finally lift the degeneracy of the T(1) and T(2) levels. If the deuteron of NH3D+ is stationary at the threefold axis i, then the rotational ground state of this ion has only two levels A and E with the energies E ¼ h0i and A ¼ 2h0i [12] (the overlap matrix elements hi and h0i have usually negative values). Experimental results on partly deuterated samples have shown that the replacement of a proton by deuteron modifies the hindering potential in such a way that the absolute value of any h0i is usually smaller than that of hi and hence different symbols are needed [3,4,6]. Thus depending

+ Fig. 1. Tunnel-split rotational ground state of (a) NH+ when the deuteron is 4 in a potential of trigonal symmetry and of (b) NH3D stationary at the threefold axis 1 or 2. The broken line shows the ground-state energy for no tunnelling.

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

459

on which threefold axis is occupied by the deuteron, the tunnel splitting of NH3D+ varies and can have four different values 3|hi|. In Fig. 1b two different tunnel splittings are shown.

3. Channels of relaxation 3.1. NH4 compounds The initial part of the proton spin-lattice relaxation curve in NH4 compounds can be described by the general expression X X An Jðot þ no0 ; tc Þ þ Bm Jðmo0 ; tc Þ: (2) 1=T 1 ¼ n

m

The possible values of the summation indices are n ¼ 71, 72 and m ¼ 1, 2. The spectral density functions are defined as usually Jðo; tÞ ¼ t=ð1 þ o2 t2 Þ:

(3)

The multipliers An and Bm contain often contributions from both intra (between the nuclei of the same ammonium ion) and inter (between the nuclei of two ammonium ions) parts of the magnetic dipolar interaction. In the case of NH+ 4 the intra contributions are dominant in An and Bm. Their expressions depend on which tunnel frequency ot =2p ¼ nt is of the same magnitude as the proton resonance frequency o0 =2p ¼ n0 : Explicit values have been derived for them for some special cases [10,13–14]. Later parts of the magnetisation recovery curve are described by exponentials with different exponents and weights, but their estimation remains usually somewhat qualitative. Only one correlation time tc is employed in the present study to describe the NH+ 4 motion by 1201 rotational jumps, which certainly is a great simplification, especially when the recent progress in the understanding of the CH3 motion, introducing new types of motion, is taken into account [15–17]. For limited temperature ranges a certain type of motion is usually dominant and therefore Eq. (2) should be a reasonable approximation. It should be noted that the correlation time we are discussing involves the effects of both reorientations and incoherent tunnelling. Recently a completely different kind of motion, so called limited jumps, was proposed as an important source of relaxation in some ammonium hexachlorometallates at low temperatures [18]. In this motion the ammonium ion jumps between a number of equilibrium orientations, separated from the tetrahedral orientations by a small angle, with a certain frequency q ¼ 1/ts, where ts is the corresponding correlation time. Also the contribution of such a motion can be described in terms of Eq. (2) by replacing tc by ts and An and Bm by the respective expressions. Actually, limited jumps were proposed to dominate the proton relaxation in (NH4)2TeCl6 near 15 K and below except at level crossings [18]. 3.2. Methyl-like relaxation of NH3D+ There are some experimental data showing that in the exact description of the CH3 motion two correlation times are needed, one for the AE and the other for the EaEb transitions (Ea and Eb refer to the one-dimensional irreducible representations of the point group C3, the corresponding energy levels are always degenerate) [15,16]. For simplicity we assume here only one correlation time, which can be called the rotational correlation time including the effects of reorientations and incoherent tunnelling. With these simplifications, Eq. (2) describes again the initial proton relaxation rate in methyl compounds. It should also be valid in the case of NH3D+ ions, when the deuterons are stationary and only protons rotate about the axis, where the deuteron is located. Now the multipliers An are dominated by intramolecular and Bm by intermolecular magnetic dipolar interactions.

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

460

There are, however, three complications. One is the presence of various kinds of ammonium ions. Provided the NH3D+ ions dominate the proton relaxation and the spin diffusion makes all the protons of the sample relax at the same rate, the average proton relaxation rate is obtained from Eq. (2) through multiplication by the factor 3c(1c)2. This factor equals the number of protons in NH3D+ ions divided by the number of all the protons in the sample as calculated from the binomial distribution (c is the deuteron concentration varying between 0 and 1. Another important fact is that the deuterons of NH3D+ ions can be at any of the four nitrogen–hydrogen bonds of the ammonium ion. Therefore, Eq. (2) should contain an additional summation over the deuteron sites i ¼ 1, 2, 3 and 4. For each i the tunnel frequency, correlation time and the multipliers An and Bm have their own value ot(i), tc(i), An(i) and Bm(i), respectively. This additional summation requires finally the normalizing division of the total relaxation rate, this time by the factor of about 4, if the deuteron sites have practically equal preference (if only one deuteron site is appreciably populated, then only that site is taken into account and no summation over i and no normalising procedure are required). If the ammonium ions are disordered so that the certain nitrogen–hydrogen bond can be parallel to eight different directions (corresponding to the eight corners of a cube, see Fig. 2), the situation becomes even more complicated. This possibility is not discussed in the present study. The third complication is the coupling between the relaxations of the proton magnetisation, tunnel energy, dipolar energy and the so-called rotational polarisation [19–21]. Below the temperature corresponding to the condition o0tc ¼ 1 (this is the temperature of the ‘‘conventional’’ T1 minimum) usually only the proton magnetisation and the tunnel energy are coupled. The coupling manifests itself for example in such a way that at the CH3-type level crossings ot ¼ no0 (n ¼ 1, 2) the experimentally observed relaxation rate depends on the pulse sequence used in the experiment. If before the ‘‘read’’ 901 RF pulse both the magnetisation and the tunnel reservoir are at least partly saturated, then the initial relaxation rate is observed to be slower than if only the magnetisation is saturated before the read pulse [21]. The same phenomenon was also found in our experiments on partly deuterated ammonium hexachlorotellurate at certain temperatures and resonance frequencies. The proton relaxation was fastest when, before each individual experiment, the sample was allowed to stay undisturbed so long that the perfect thermal equilibrium was reached before saturation by one 901 pulse and the subsequent read pulse (after a variable Z

u

2 d

1

Y

N

X

w 3

4 v

Fig. 2. The protons u, v and w and the deuteron d of NH3D+ at the apices of a cube defining the molecular frame XYZ. The hydrogen sites are numbered 1 to 4.

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

461

time). If the perfect thermal equilibrium was not reached before the saturating pulse, or if on purpose many saturating pulses, roughly at intervals of the initial relaxation time, were applied, then the relaxation was strongly slowered. Under special circumstances, for example if the saturation pulse of the next individual experiment is applied immediately after the transients of the induction signal created by the previous read pulse have decayed, it is possible that the tunnel reservoir becomes totally saturated and no level crossing is observed. It should be noted that a similar method dependence (although smaller) has been observed in NH4 compounds only if the tunnel splitting of the level crossing is the EA splitting (but not if the T levels are involved) [22]. 3.3. New fast relaxation channel in NH3D+ 3.3.1. Level-crossing transitions between A states When proton spin-lattice relaxation is considered in methyl compounds, the magnetic dipolar interaction between the three methyl protons is usually written in the symmetrised form. Here, we do the same for the three protons u, v and w of NH3D+ and ignore the dipolar interaction with the deuteron X   ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ HD ¼ BðmÞ vw Duv þ Buw Duw þ Bvw Dvw m¼1; 2

X   1

¼

3

 ðmÞ  ðmÞ ðmÞ ðmÞ ðmÞ BðmÞ uv þ Buw þ Bvw Duv þ Duw þ Dvw

m¼1; 2

1 ðmÞ  ðmÞ  ðmÞ ðmÞ ðmÞ ðmÞ 3 Buv þ Buw þ  Bvw Duv þ  Duw þ Dvw    ðmÞ  ðmÞ ðmÞ ðmÞ ðmÞ þ 13 BðmÞ uv þ  Buw þ Bvw Duv þ Duw þ  Dvw þ

ð4Þ

with 2 Bð0Þ uv ¼ Gð1  3cos yuv Þ;

Bð1Þ uv ¼ G sin yuv cos yuv expðifuv Þ; 2 Bð2Þ uv ¼ G sin yuv expði2fuv Þ;

ðmÞ ¼ BðmÞ Buv uv ;

1 Dð0Þ uv ¼ I uz I vz  4 ðI uþ I v þ I u I vþ Þ; 3 Dð1Þ uv ¼  2 ðI uz I vþ þ I uþ I vz Þ;  D2uv ¼  34 I uþ I vþ ;

¼ DðmÞ DðmÞ uv : uv

ð5Þ

The common multiplier of the B quantities is G ¼ g _/r , where g is the proton gyromagnetic ratio and r the proton–proton distance in NH3D+. The polar angles yuv and fuv define the direction of the proton–proton vector ruv in the laboratory frame with the z-axis parallel to the external magnetic field B0. The second and third lines of Eq. (4) contain those parts of HD which are of the type EaEb and EbEa, the former and latter symbols (Ea and Eb in EaEb, respectively ) representing the symmetry of the latticedependent and spin-dependent operator combinations. These operators are responsible for the relaxation discussed above in Section 3.2. If during NH3D+ rotation also the deuteron jumps between its four possible equilibrium positions (numbered 1–4 in Fig. 2), then the B operator combinations become additionally modulated. This additional time dependence is usually slower than the rotation of the protons about the axis containing the deuteron and thus no drastic changes in the EaEb and EbEa transition rates are expected. The first line of Eq. (4) is of the type AA and thus time independent in the rotation of the three protons if the deuteron is stationary at one site. Therefore, these terms do not give rise to any relaxation in CH3 2

3

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

462

compounds. However, if the deuteron of NH3D+ changes site many times during the proton T1, then they become time dependent. These terms are written here as X ðmÞ H DAA ¼ BðmÞ A DA m¼1; 2

¼

X   1 3

 ðmÞ ðmÞ ðmÞ ðmÞ ðmÞ BðmÞ uv þ Buw þ Bvw ÞðDuv þ Duw þ Dvw ;

ð6Þ

m¼1;2

 where the multiplier 13 is included in B(Am). Let us consider the time-dependence in the molecular frame defined by the cube with the hydrogens of the ammonium ion at its four apices (Fig. 2). If the deuteron is originally at the threefold axis 1 and jumps to the   axis 2, then the values of BA (the  index m is dropped for a while) before and after the jump are BA (1) ¼ 13 (B23+B24+B34) and BA (2) ¼ 13 (B13+B14+B34), which are clearly different. Quite often the transition rates are calculated from the time-dependent second-order perturbation theory by using the expression Z 1 hijH DAA ðtÞjfihfjH DAA ð0Þjii expðioif tÞ dt; (7) W i2f ¼ 1

where i and f refer to the initial and final states and oif corresponds to the energy difference between ð1Þ b a b1 1 1 them. The calculation reveals that the spin matrix elements hEa12jDð1Þ A jE  2i ¼ hE 2jDA jE  2i ¼ pffiffiffi ð1Þ 1 1 3 ð1Þ 1 matrix elements differ from zero: hA  2jDA jA  2i ¼ 3 3=2 hA2jDA jA  2i ¼ 0: Only the pffiffifollowing ffi ð2Þ 3 1 and hA  2jDA jAm2i ¼ 3 3=4: Thus only transitions between the A states are possible. The energy difference can be obtained as follows: The A level, when the deuteron lies at the axis 1, is 2jh01 j below the reference level, which is the energy of the rotational ground state without tunnelling (see Fig. 1b). If the deuteron is at the axis 2, then the A level is 2jh02 j below the reference level. The deuteron jump from site 1 to site 2 can create, for example, the relaxation transition A12 ! A32: Provided jh01 j is larger than jh02 j; then the corresponding energy change is 2jh01  h02 jo0. If a backward jump from the site 2 to 1 induces the same relaxation transition, the energy change is 2jh01  h02 jo0, which in the spectral density function is equivalent to 2jh01  h02 j+o0. Otherwise, if jh01 jojh02 j; the energy changes, produced by the deuteron jumps 1-2 and 2-1, are 2jh01  h02 jo0 and 2jh01  h02 jo0, respectively. Thus the same energy changes are obtained, independent of the relative magnitude of h01 and h02 ; although in the reversed order. These energy changes are consistent with the ‘‘effective’’ tunnel frequency 2jh01  h02 j; which appears in oif also with the negative sign depending on the direction of the deuteron jump and on the relative magnitude of h01 and h02 : By generalisation the deuteron jump from the axis i to the axis j (or backwards) corresponds to the effective tunnel splitting 2jh0i  h0j j: Thus there can be altogether 6 different effective tunnel splittings. Let us consider for simplicity such a situation, where the deuteron of NH3D+ has the same probability at the sites 1 and 2 but a very small probability to appear at the sites 3 and 4. Then the average value of BA in  Eq. (6) for any m is BA (av) ¼ 12 [BA(1)+BA(2)]. Because the relaxation transitions are induced only by time-dependent terms, the BA terms in Eq. (7) have to be replaced, in the described special situation, by BABA(av). Then for the transition A12 ! A32; induced by the deuteron jumps between 1 and 2, the corresponding rate is obtained from (7) Z 1 ð1Þ ð1Þ ð1Þ W A1=2!A3=2 ¼ ð27=4Þ ½Bð1Þ A ðtÞ  BA ðavÞ½BA ð0Þ  BA ðavÞ 1

 expðioif tÞ dt:

ð8Þ ð1Þ

ð1Þ ½BA ðtÞ

ð1Þ BA ðavÞ½Bð1Þ A ðtÞ

Bð1Þ A ðavÞ

  which will Eq. (8) contains the correlation function C ðtÞ ¼ be evaluated next. Since in the present example the deuteron is jumping between the two sites 1 and 2 (of course the protons undergo simultaneously a corresponding change of positions), we have to consider

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

463

the probabilities for the deuteron to appear at these sites. The probabilities are called p1 and p2. For simplicity also the jump rates r12 (from site 1 to 2) and r21 are assumed to be equal and the common value is marked briefly r (in reality the jump rates rij for different before-jump and after-jump sites can all have different values and thus also the occupation probabilities of the four sites may differ). If the deuteron is assumed to be at the site 1 at time t ¼ 0, then the two probabilities are p1 ¼

1 2

½expð2rtÞ þ 1;

p2 ¼

1 2

½ expð2rtÞ þ 1:

(9)

On average the deuteron is found at the sites 1 and 2 with the equal probability 0.5. Therefore the correlation function of Eq. (8) becomes ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ C ð1Þ ðtÞ ¼ 0:5½Bð1Þ A ð1Þ  BA ðavÞfp1 ðtÞ½BA ð1Þ  BA ðavÞ þ p2 ðtÞ½BA ð2Þ  BA ðavÞg ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ þ 0:5½BA ð2Þ  BA ðavÞfp2 ðtÞ½Bð1Þ A ð1Þ  BA ðavÞ þ p1 ðtÞ½BA ð2Þ  BA ðavÞg  1  ð1Þ ð1Þ ð1Þ ½BA ð1Þ  Bð1Þ ¼ 16 A ð2Þ½BA ð1Þ  BA ð2Þ ½2stationary deutron þ 1jump1!2 þ 1jump2!1  expð2rtÞ:

ð10Þ

The first line on the right side of Eq. (10) corresponds to the deuteron being initially at the site 1 and then either staying at that site with the probability p1(t) or jumping to the site 2 with the probability p2(t). The second line gives the similar contributions when the deuteron is originally at the site 2. The various motional steps of deuteron are expressed in the fourth line as subscripts of the corresponding relative weights, for example 1jump 1-2 means that the weight factor of that part of the correlation function, where the deuteron jumps from the site 1 to 2, equals 1. As explained above, the frequency change oif depends on the motional state of the deuteron. For the part of Eq. (10) representing the stationary deuteron oif equals—o0 in the considered transition. For the terms representing the jumps 1-2  and 2-1 oif obtains 0 0 0 0 0 0 the values 2 jh  h jo and 2 jh  h jo in the mentioned order jh j4jh 0 0 1 2 1 2 1 2 or in the reversed order  0  jh1 j4jh02 : Fortunately the weight factors for the jumps 1-2 and 2-1 are equal in Eq. (10), and therefore the transition rate (8) is independent of the relative magnitude of h01 and h02 : The angular dependence of the correlation function (10) can be relatively easily calculated in the molecular frame of Fig. 2, where the direction of B0 is defined by the polar angles y and f. The direction cosines of B0 are then sin y cos f, sin y sin f and cos y, for which we use a short notation a, b and c, respectively. By using the method described for example in [23], the angular dependence of the correlation function is obtained: ð1Þ ð1Þ ð1Þ ½BA ð1Þ  BA ð2Þ½Bð1Þ A ð1Þð1Þ  BA ð2Þ   ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ¼ 19 ½B23 þ B24  B13  B14 ½Bð1Þ 23 þ B24  B13  B14 

¼ ðG2 =9Þð1  3c2 þ 4c4 þ 2ab  8abc2 Þ

ð11Þ

When this is finally substituted into Eqs. (10) and (8), the discussed transition rate WA1/2-A3/2 and also the rate of the reverse transition A 32-A 12; turn out to be W 1 ¼ ð3G2 =32Þð1  3c2 þ 4c4 þ 2ab  8abc2 Þf2Jðo0 ; td Þ þ Jðð2jh01  h02 j  o0 Þ; td Þ þ Jðð2jh01  h02 j þ o0 Þ; td Þg

ð12Þ

with td ¼ 1=ð2rÞ: The first spectral density function J(o0, td) in Eq. (12) is related to that part of C(1)(t) that corresponds to a stationary deuteron, while the second and the third J-functions correspond to the deuteron jumps 1-2 and 2-1. The transition rate WA1/22A3/2 is also identical to (12).

ARTICLE IN PRESS 464

E.E. Ylinen et al. / Physica B 357 (2005) 456–471

The Dm ¼ 72 transition rates can be calculated exactly in the same way: W 2 ¼ W A1=22A3=2 ¼ W A1=22A3=2 ¼ ð3G2 =32Þð1  c4  2ab þ 2abc2 Þ½2Jð2o0 ; td Þ þ Jðð2jh01  h02 j  2o0 Þ; td Þ þ Jðð2jh01  h02 j þ 2o0 Þ; td Þ:

ð13Þ

It is interesting to realise that these rates are symmetric in the sense that the spectral density functions with 2 jh01  h02 jno0 and 2 jh01  h02 j+no0 have identical weights. The spin-lattice relaxation rate of NH3D+ protons can be calculated as described earlier [19–21]. When only the AA type transitions are taken into account, the following result is obtained for the initial relaxation rate: 1=T 1 ðAAÞ ¼ ðG2 =6Þð3W 1 þ 8W 2 Þ:

(14)

For quantitative comparisons with experimental data in partly deuterated ammonium compounds, result (14) has to be multiplied by the concentration-dependent factor 3c(1c)2 as described in the following section. Of course also the relaxation transitions described in the sections 3.1. and 3.2. have to be taken into account. In a general case the deuteron of NH3D+ can appear at each threefold axis with a comparable probability. Consequently, jumping between other equilibrium sites becomes important introducing transitions with rates similar to those of (12) and (13) but with different effective tunnel splittings 2 jh0i  h0j j and correlation times td. Furthermore, the effective tunnel splittings may not be determined exclusively by the overlap matrix elements and an energetic nonequivalence of the various deuteron positions (corresponding to different equilibrium orientations of NH3D+) could contribute. The nonequivalence is expected to make the reference energies (the broken line in Fig. 1b) different for different deuteron positions and thus affect the considered splittings. In addition, the probabilities of finding the deuteron at different axes can no longer be expressed by the simple formulas (9), and hence the numerical multipliers of the transition rates (12) and (13) are modified. Furthermore, the angular dependencies will be different. The transition rates can, nevertheless, be calculated as described above. It is seen that the rates W1 and W2 show maxima, when the effective tunnel frequency equals o0 or 2o0. Thus T1 minima can be expected when relaxation is studied either as a function of resonance frequency at some fixed low temperature or as a function of temperature at a certain resonance frequency. The former method is more difficult but, on the other hand, more reliable, since it can detect a level crossing also at such temperatures where the tunnel frequency is nearly temperature independent (this means often liquid helium temperatures). 3.3.2. Tunnel-energy transfer to the lattice + At low deuteron concentrations NH+ ions. It would be 4 ions are still much more numerous than NH3D then natural to ask how the described transition rates can speed up the proton relaxation rate by a decade or even more? For that to occur a process is needed which can transfer the imbalance in tunnel energy, created by the described level-crossing transitions, to the lattice. If such a channel does not exist, then NH3D+ would undergo only one relaxation transition without much effect on the average relaxation rate of all the protons. In the presence of deuteron jumps between various sites there is indeed a very effective transfer. Let us assume that one NH3D+ ion has undergone the fast level-crossing transition A 12-A 32 as a consequence of the deuteron jump from site 1 to site 2. In this transition some proton magnetisation was created, and the tunnel energy was increased by 2 jh01  h02 j: The partially grown magnetisation is transferred fast to neighbouring NH+ 4 ions via proton spin diffusion. After that the spin quantum number of NH3D+ is again 12: When the deuteron makes the reverse jump from site 2 to 1, the tunnel energy is decreased by 2 jh01  h02 j: If the spin state is not changed, the original state of NH3D+ is reached and a relaxation transition can occur again. It is reasonable to expect that deuteron jumps without a change in the

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

465

spin state are more numerous than the jumps producing spin-lattice relaxation. Therefore, the exchange of the tunnel energy with lattice should take place roughly at the rate r21, which is much faster than the transition rates (12) and (13). This tunnel energy equilibration is a nonresonant feature and thus does not depend in any way on the magnetic field. It removes effectively the coupling between the relaxations of the proton magnetisation and tunnel energy. Therefore no method dependence should be observed in the proton relaxation. It should be noticed that such a fast transfer is not possible in methyl compounds. Somewhat similar fast transfer between the T levels of NH+ 4 could be present [10,22] (but not as perfect as in the case of NH3D+), since method dependence has never been observed in the proton relaxation dominated by level-crossing transitions between T levels. Since the tunnel energy transfer to the lattice should be much faster than the rates W1 and W2 (see Eqs. (12) and (13)), the contribution of the AA type transitions to the proton relaxation rate at low deuteration is roughly equal to Eq. (14), multiplied by the concentration dependent factor 3c(1c)2. As mentioned above, a detailed consideration of the deuteron (and simultaneous proton) motion among the four threefold axes introduces other tunnel splittings 2 jh0i  h0j j; correlation times and angular dependencies. The presence of NH3D+ ions can also speed up the proton relaxation in nondeuterated ammonium ions. This seems to take place via a three-step process, where the first step is a normal level-crossing transition in an NH+ 4 ion, increasing the tunnel energy by ot, where ot is the relevant tunnel frequency. The second step is then the fast transfer of the tunnel energy (a tunnel-energy flip-flop) to NH3D+ via the inter part of the magnetic dipolar interaction, involving the deuteron jump. The third step is the fast transfer of the tunnel energy from NH3D+ to the lattice as explained above. To describe the tunnel flip-flop process the inter (0) interaction can be arranged in a symmetrised form, which will contain also terms like B(0) TADTA. There are also many other terms, which are responsible for the normal tunnel-energy flip-flops in nondeuterated ammonium and methyl compounds [24], but this term is the important contribution in partly deuterated ammonium compounds. Its first and second subindices show that the interaction is of the type T relative to + the NH+ ion. The superindex means that there is no net 4 ion and of the type A relative to the NH3D change in the proton magnetisation. Let us assume for example that the deuteron jumps from the site 1 to 2. Then the tunnel energy of NH3D+ increases by 2 jh01  h02 j while that of NH+ 4 decreases by ot. The corresponding transition rate is of the magnitude jH D ðinterÞj2 td =½1 þ ð2jh01  h02 j  ot Þ2t2d : jh01

(15)

h02 j

 and ot are nearly equal, the tunnel flip-flop rate (15) can be equal or even larger than Provided 2 the relaxation of the tunnel energy in nondeuterated compounds. Thus the NH+ 4 ion returns faster to the original state (via the interaction with a neighbouring NH3D+) than in nondeuterated compounds and another level-crossing transition is possible. Consequently, the level-crossing minima of the proton T1, observed in nondeuterated samples, may appear deeper in partly deuterated samples. Such a behaviour was observed already by Punkkinen et al. [25] and also our present results are consistent with this explanation. The proposed three-step process is, nevertheless, slower at low temperatures than the described fast relaxation, since it includes the additional process (15) related to the inter interaction, which makes it unavoidably slower than the processes induced by the intra part of the magnetic dipolar interaction.

4. Experimental results and discussion Experiments were carried out with the Bruker MSL pulsed spectrometer operated at the proton resonance frequencies between 20 and 30 MHz. The relatively low magnetic field was provided by a Bruker electromagnet. Near to the resonance coil was mounted a variable capacitor, which could be tuned by a long rod. The resonance circuit, with the sample inside, was surrounded by two concentric brass cylinders. These were immersed in liquid helium bath. The temperature of the inner cylinder and of the sample was

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

466

measured by a carbon-glass thermometer and controlled by regulating the heating current of the inner cylinder. Three samples of ammonium hexachlorotellurate were polycrystalline with the deuteron concentrations c ¼ 0.015% (natural concentration), 4% and 10%. They were obtained from the compound used in the previous study [26] by recrystallisation from appropriate mixtures of H2O and D2O. In most of our experiments on the proton spin-lattice relaxation the proton magnetisation was first saturated by one 901 pulse. Then after a variable time t, the 901 read pulse was applied to monitor the partly recovered magnetisation (this is called 901t901 method). The essential feature of this method is a long equilibration time between successive two-pulse sequences, otherwise the tunnel reservoir might become saturated. In some experiments a sequence of twenty 901 pulses, roughly at intervals of the initial relaxation time, was used for the saturation (called SSt901 method). This sequence saturated not only the proton magnetisation but also the tunnel energy reservoir, if it was coupled to the magnetisation. If no coupling existed, then the same magnetisation recovery should be observed with both the methods. First experiments were conducted on the 4% sample as a function of temperature at a constant resonance frequency of about 25 MHz (many frequencies were used). A broad minimum of T1 was observed between 15 K and 20 K, the temperature of the minimum varying with n0. Such results prove that the minimum is related to a level crossing, with the relevant tunnel frequency equal either to 25 MHz or 50 MHz. The study of Grabias and Pislewski [7] shows that the latter alternative is valid. These experimental results are not shown because more accurate data on the level crossing were obtained by experiments as a function of n0 at a constant temperature. The experimentally observed recovery curves of the proton magnetisation were at first fitted with a sum of three exponentials by the least squares method. The initial relaxation time T1,init was then determined from such fits by using the formula 1/T1,init ¼ wf /T1,f+wm/T1,m+ws/T1,s, where the subindices f, m and s refer to the fast, medium and slow components with the weights w and time constants T1. The good quality of the fits is demonstrated for example by the curves of Fig. 3, which show them and the experimental relaxation curves as observed by the 901t901 and SSt901 methods at n0 ¼ 25.5 MHz and T ¼ 16 K for the 10% sample. Usually the different exponentials have comparable weights, which vary somewhat with n0 and T. In most cases the time constants of the three exponentials, differing from each other by a

1.0

0.8

Mz / Mo

0.6

0.4 16 K

0.2

25.5 MHz 0.0 0.0

0.1

0.2

0.3

t (s) Fig. 3. The recovery curves of the proton magnetisation, observed by the methods 901t901 (circles) and SSt901 (triangles), for n0 ¼ 25.5 MHz and T ¼ 16 K in the 10% sample.

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

467

decade or more, behave similarly as functions of n0 and T. Unfortunately, the larger time constants show also larger scatter than the initial relaxation time. Therefore mainly T1,init is discussed in the following. Fig. 4 shows T1,init for the 10% and nondeuterated samples as the function of n0 at T ¼ 16 K. At this temperature the minimum in the proton relaxation time of the 10% sample at 25.5 MHz appears deepest in the plot of the smallest time constant T1,f as the function of n0. Hence also T1,f is shown in Fig. 4. T1,init data of similar experiments at T ¼ 6 K are given in Fig 5. Let us consider at first the T1,init data for the sample with the natural deuteron concentration. At the temperature of 6 K T1,init is roughly constant between 21 MHz and 25 MHz, then shows a minimum at about 28 MHz and increases again at higher frequencies (Fig. 5). At 16 K the initial relaxation time behaves quite similarly as the function of n0, although the numerical values are roughly ten times shorter (Fig. 4). Furthermore, the frequency of the minimum is shifted to 26.8 MHz. The T1 minima are clearly related to a level crossing. Our experiments alone cannot decide if the relevant tunnel frequency is about 27.5 MHz or 55 MHz. Since Grabias and Pislewski [7] observed a T1 minimum at 55 MHz at 14 K in their experiments on (NH4)2TeCl6 with the natural deuteron concentration, the tunnel frequency can be concluded to be about 55 MHz, with a small variation with temperature (Fig. 6). Actually, a closer examination of the considered minimum both at 6 K and 16 K shows that they could be double minima, the individual minima being about 0.5 MHz apart. Experiments were repeated to check this behaviour and the double minimum was again observed. The ‘‘easiest’’ explanation for the double minimum could be found in the level structure. As mentioned in the Section 2, a trigonal symmetry of the hindering potential would leave the T(1) and T(2) levels degenerate, while T(3) is somewhat lower (Fig. 1a). The energy difference in frequency units could be equal to the observed frequency 55 MHz. A small distortion of lower symmetry would lift also the degeneracy of T(1) and T(2). The distance between these levels is about 1 MHz according to our data. Next the results for the 10% deuterated sample are discussed. At 6 K the initial relaxation time is roughly constant between 20.5 MHz and 24 MHz but about a factor of ten shorter than in the nondeuterated sample. T1,init shows a shallow minimum at 25 MHz and a deeper, broad minimum between 28 MHz and 29 MHz and finally starts to increase again at higher frequencies (Fig. 5). There seems to be a slight 10 16 K

T1, init (s)

1

0.1

0.01 20

22

24

26 νo (MHz)

28

30

32

Fig. 4. T1,init, observed by the method 901t901, as the function of n0 for the nondeuterated (open circles) and 10% deuterated (filled circles) samples at 16 K. The grey circles and the filled triangle present T1,f by the same method (see text) and T1,init by SSt901, respectively, for the 10% sample.

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

468

6K

T1, init (s)

10

1

0.1 20

22

24

26 vo (MHz)

28

30

32

Fig. 5. T1,init, observed by the method 901t901, as a function of n0 for the nondeuterated (open circles) and 10% deuterated (filled circles) samples at 6 K. Result by the method SSt901 for the 10% sample is shown by triangle.

58

tunnel frequency (MHz)

56 54 52 50 48 46 6

8

10

12

14 16 T (K)

18

20

22

24

Fig. 6. Temperature dependence of the average tunnel frequency (see text), determined from the center of the broad minimum in 4% and 10% samples by temperature (filled squares) and frequency (filled circles) dependent experiments. The open circles show the average tunnel frequency of the nondeuterated sample.

maximum in the center of the broad minimum. The left-hand part of the minimum appears at the same frequency as the minimum of the sample with the natural concentration. At 16 K the behaviour is quite similar, but the numerical values are somewhat shorter (Fig. 4). Furthermore, the shallow minimum of 6 K has now about the same depth as the broad minimum. The frequency of the former minimum is slightly increased to 25.5 MHz while that of the broad minimum, between 26.5 MHz and 27.5 MHz, appears at somewhat lower frequency than at 6 K. The minimum near 25 MHz has the peculiar property that the results for T1,init depend on the experimental method. The points, measured by the SSt901 method, are

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

469

clearly above those obtained by the 901t901 method (see also Fig. 3). If the repetition time of the latter pulse sequence was too short for reaching thermal equilibrium, results were obtained somewhere between those of the two methods. Outside the range of this minimum, also at the broad minimum and around it, the two methods always gave practically the same result. Actually at the exact level crossing near 25 MHz the three exponentials are not completely sufficient to describe the relaxation curve of the 10% sample, observed with the 901t901 sequence for short waiting times, but a fourth exponential with a still smaller time constant is needed. Therefore, in the very vicinity of the 25 MHz minimum the corresponding experimental points (the filled circles of Figs. 4 and 5) would be shifted slightly lower. Simultaneously the computed relaxation curve (the upper curve of Fig. 3), although otherwise practically unchanged, would start from zero. This effect is probably related to a distribution of the tunnel frequencies and it was not observed to have any effect on the results with the SSt901 sequence. Thus the difference between the results by the two experimental sequences is in reality slightly larger than shown by Figs. 4 and 5 which strengthens the basis of the conclusions to be presented later on. Fig. 6 presents all our data on the temperature dependence of the tunnel splittings, both in the sample with the natural concentration and in the 4% and 10% samples (determined from the center of the broad minimum). The results were obtained partly from T1 vs. T curves (n0 constant), partly from T1,init vs. n0 curves (T constant). Actually the center of the broad minimum does not represent exactly any tunnel splitting, since the minimum caused by the tunnel splitting of NH+ 4 appears at a frequency 0.5 MHz smaller and that corresponding to the effective tunnel splitting 2 jh0i  h0j j at a frequency 0.5 MHz larger (see Figs. 4 and 5). Nevertheless, the shown temperature dependence should be valid for both these tunnel splittings, because the frequencies are determined practically by the same hindering potential. The tunnel frequencies of the 4% and 10% samples agree with each other but differ from those for the nondeuterated sample by about 1 MHz. The shallow minimum near 25 MHz is suggested to be a level-crossing minimum with the tunnel splitting 3 jh0i j; that is one of the methyl-like splittings of NH3D+. There are two facts which support this interpretation. One is that the corresponding T1 minimum is not found in the nondeuterated sample. Therefore, the minimum must be related to NH3D+. The other argument is the observed method dependence (Fig. 3). Such a dependence has been observed so far only with methyl compounds and since NH3D+ (when the deuteron is stationary) has properties similar to a methyl group (see the Section 3.2), it is natural to expect a method dependence also in samples containing NH3D+. [Some method dependence was observed at a level crossing of nondeuterated (NH4)2ZnCl4, when the relevant tunnel splitting was that between the E and A levels of NH+ 4 . For this feature to be the explanation in the present case, the corresponding T1 minimum should be observable also in the nondeuterated sample. Because this was not the case, the method dependence cannot be related to NH+ 4 ions.] The broad T1 minimum of the 10% sample is suggested to be a double minimum with the components related to (i) the same NH+ 4 tunnel splitting as that of the nondeuterated sample and (ii) one of the effective tunnel splittings 2 jh0i  h0j j of NH4D+, rotating in such a way that the deuteron jumps between the sites i and j. The NH+ 4 related component is quite natural, since about 66% of the ammonium ions of the 10% deuterated sample are still NH+ 4 ions. It also appears at the same frequency as the minimum of the nondeuterated sample. The fact that it is deeper than in the nondeuterated sample could be interpreted as a result of the three-step process discussed at the end of the Section 3.3.2. The additional processes of the tunnel diffusion (15) and the tunnel-to-lattice energy transfer, which are not present in the nondeuterated sample, might remove the double minimum character in partly deuterated samples. Also the presence of the other component (ii) may add to the depth of the discussed minimum. Two facts support the interpretation that the right-hand part of the broad minimum is related to the new relaxation channel with the effective tunnel frequency 2 jh0i  h0j j: One is its presence only in partly deuterated samples, which proves NH3D+ ions as its origin. The other is the absence of any method dependence in the proton relaxation curves. Thus it cannot be related to any methyl-like tunnel frequency 3

ARTICLE IN PRESS 470

E.E. Ylinen et al. / Physica B 357 (2005) 456–471

jh0i j (with stationary deuterons), since that would lead to the appearance of the method dependence. Hence the right-hand part of the broad minimum must originate from NH3D+ related relaxation involving deuteron jumping. For such a process there is only one model so far, that described in the Section 3.3, which seems to agree well with the experimental data. The ten times faster relaxation rate in the 10% sample than in the nondeuterated sample outside level crossings, for example between 21 MHz and 24 MHz in Fig. 5, should be mainly the result of the first terms of W1 (12) and W2 (13), which do not involve the effective tunnel frequency 2 jh01  h02 j It is not yet quite clear how these terms can be so much more effective than the corresponding terms related to NH+ 4 ions (2). Possibly the spin-state preserving jumps of NH3D+ ions, which are not possible in NH+ 4 ions, are an effective sink not only for the tunnel energy but also for the Zeeman energy, increasing the spectral density functions J(o0, td) and J(2o0, td) via the correlation time td. It would be interesting to compare the numerical values of Eq. (14) with the experimental data. Unfortunately, although the mathematical form of the level-crossing rates (12) and (13) is successful farther off from level crossings, it cannot be valid at the exact level crossing, since the rate increases with growing td without limits. There must be a limiting value roughly equal to the inverse proton line width or slightly larger. In any case T1,init of the 10% sample at the broad minimum at 29 MHz and 6 K, equal roughly 30 ms, is only about four times as long as the T1 minimum value of the nondeuterated sample, about 8.5 ms, at 24.6 MHz and 68 K (the conventional minimum) [7]. This factor agrees very well with the inverse of the concentration-dependent multiplier 3c(1c)2 for c ¼ 0.1. Some kind of estimate of the tunnel frequencies of NH3D+ can be done on the basis of the tunnelling matrix elements hi of (NH4)2TeCl6. Birczynski et al. [8] proposed that the tunnel splitting between the center of the T level and the A level corresponds to the frequency 240 MHz. Since the T(1),T(2)T(3) frequency is about 55 MHz and since the trigonal symmetry is consistent with h16¼h2 ¼ h3 ¼ h4, the two remaining matrix elements can be calculated (see Fig. 1a) and they are approximately h1/2p ¼ 43.7 MHz and h2/2p ¼ 25.4 MHz. These values give the smallest methyl-type frequency 3|h2|/2p ¼ 76.2 MHz and the effective tunnel frequency 2|h1h2|/2p ¼ 36.6 MHz. If the observed frequency of the methyl-type minimum, 25 MHz at 6 K, corresponds to 2n0 ¼ nt, then the tunnel frequency is 50 MHz, which is somewhat smaller than 76.2 MHz. Similarly, the observed effective tunnel frequency 58 MHz at 6 K is larger than the calculated result. The discrepancies can, in practice, be much smaller or even absent, when it is taken into account that the symmetry of the potential is lower than trigonal and the matrix elements of NH3D+ have been observed to be smaller (by absolute value) than those of NH+ 4 [3–4,6]. Also a slight energetic nonequivalence of the relevant deuteron positions could modify the effective tunnel frequency as described at the end of the Section 3.3.1.

5. Concluding remarks Although our model for NH3D+ related minima in the proton spin-lattice relaxation time in partly deuterated ammonium compounds agrees well with the present experimental data for ammonium hexachlorotellurate, the evidence for the validity of the model is not yet conclusive. This statement arises from the facts that (i) only one of the proposed four methyl-type minima, with the tunnel frequencies 3 jh0i j; and (ii) only one minimum of new kind out of six possible, with the effective tunnel frequencies 2 jh0i  h0j j; were observed. There is, nevertheless, no reason why these could not be found, if the studied frequency range is extended below 20 MHz and above 30 MHz. Such experiments, though somewhat slow to carry out, might anyway be worth trying. Of course it is possible that some of these minima are very near each other because of the approximate equality of the relevant elements h0i : Furthermore, all the minima might not be found due to the shortness of the relevant correlation time making the T1 minima too broad for observation. This does not look very probable in the case of ammonium hexachlorotellutate, since the

ARTICLE IN PRESS E.E. Ylinen et al. / Physica B 357 (2005) 456–471

471

temperature of the conventional T1 minimum is about 85 K [7]. Another factor preventing the observation of the proposed minima could be the nonequivalence of the deuteron sites so that the deuteron would have a nearly vanishing probability to occupy a certain threefold axis.

Acknowledgement Participation of PF and ZTL in this study was supported by the State Committee for Scientific Research Council (Poland) Grant no. 2 P03B 010 24 (2003–2005). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

I. Svare, D.P. Tunstall, J. Phys. C: Solid State Phys. 8 (1975) L559. Z.T. Lalowicz, A. Birczynski, Z. Olejniczak, G. Stoch, Mol. Phys. Rep. 31 (2001) 108. Z. Olejniczak, Z.T. Lalowicz, T. Schmidt, H. Zimmermann, U. Haeberlen, H. Schmitt, J. Chem. Phys. 116 (2002) 10343. H.G. Bu¨ttner, G.J. Kearley, B. Frick, Chem. Phys. 214 (1997) 425. A. Detken, H. Zimmermann, J. Chem. Phys. 108 (1998) 5845. M. Prager, P. Schiebel, M. Johnson, H. Grimm, H. Hagdorn, J. Ihringer, W. Prandl, Z.T. Lalowicz, J. Phys.: Condens. Matter 11 (1999) 5483. T. Grabias, N. Pislewski, Solid State NMR 12 (1998) 37. A. Birczynski, Z.T. Lalowicz, Z. Lodziana, Chem. Phys. 299 (2004) 113. A. Hu¨ller, Phys. Rev. B 16 (1977) 1844. A. Birczynski, Z.T. Lalowicz, Z. Olejniczak, M. Punkkinen, Solid State NMR 7 (1996) 105. J. Pelzl, C. Dimitropoulos, Z. Naturforsch. 49a (1994) 232. A.J. Horsewill, Prog. Nucl. Magn. Reson. Spectrosc. 35 (1999) 359. M. Punkkinen, J. Magn. Reson. 19 (1975) 222. A.J. Nijman, M. Sprik, N.J. Trappeniers, Physica B 98 (1980) 247. S. Szymanski, J. Chem. Phys. 111 (1999) 288. S. Szymanski, Z. Olejniczak, A. Detken, U. Haeberlen, J. Magn. Reson. 148 (2001) 277. G. Diezemann, Appl. Magn. Reson. 17 (1999) 345. M. Punkkinen, E.E. Ylinen, Physica B 337 (2003) 111. S. Emid, R.A. Wind, Chem. Phys. Lett. 33 (1975) 269. M. Punkkinen, S. Clough, J. Phys. C 7 (1974) 3403. M. Punkkinen, D.G. Taylor, P.S. Allen, Proceedings of the 18th AMPERE Congress, Nottingham, 1974, p. 383. L.P. Ingman, M. Punkkinen, A.H. Vuorima¨ki, E.E. Ylinen, J. Phys. C: Solid State Phys. 18 (1985) 5033. Z.T. Lalowicz, M. Punkkinen, A.H. Vuorima¨ki, E.E. Ylinen, A. Detken, L.P. Ingman, Solid State NMR 8 (1997) 89. S. Clough, Z.T. Lalowicz, J. Phys. C: Solid State Phys. 6 (1973) L452. M. Punkkinen, E.E. Ylinen, Yu.M. Bunkov, J. Phys. C: Solid State Phys. 10 (1977) 4149. A. Birczynski, Z.T. Lalowicz, Z. Olejniczak, G. Stoch, B. Petelenz, Chem. Phys. Lett. 339 (2001) 229.