# One-dimensional physics in organic conductors (TMDTDSF)2X, X = PF6, Re04: 77Se-NMR experiments

Denis Jerome, Pascale Auban-Senzier, B. Gotschy, Claude Bourbonnais, K.

Bechgaard, Enric Canadell, Rui T. Henriques, I. Johansen, A. Farrall

► To cite this version:

Denis Jerome, Pascale Auban-Senzier, B. Gotschy, Claude Bourbonnais, K. Bechgaard, et al., One-dimensional physics in organic conductors (TMDTDSF)2X, X = PF6, Re04: 77Se-NMR experiments. Journal de Physique I, 1992. hal-04996721

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74.70K — 75.30F — 76.60E — 71.45

# One-dimensional physics in organic conductors
(TMDTDSF)₂X, X = PF₆, ReO₄ : ⁷⁷Se-NMR experiments

B. Gotschy (¹, *). P. Auban-Senzier (¹), A. Farrall (¹), C. Bourbonnais (¹, **), D. Jérome (¹), E. Canadell (²), R. T. Henriques (³), I. Johansen (⁴) and K. Bechgaard (⁴)

(Received 16 October 1991, accepted in final form 23 January 1992)

**Résumé.** — Nous présentons une étude RMN (spectres et mesures du temps de relaxation $T_1$) sur le noyau ⁷⁷Se, pour les composés (TMTSF)₂ReO₄, (TMDTDSF)₂ReO₄ et (TMDTDSF)₂PF₆. Pour tous ces composés, la dépendence en température du facteur d'augmentation de la relaxation ($TT_1$)⁻¹ suit le carré de la susceptibilité statique uniforme $χ_s(T)$ dans le régime paramagnétique. Des écarts à cette variation sont observés pour (TMDTDSF)₂PF₆ en dessous de la température de localisation $T_ρ$ qui sont expliqués en termes de corrélations antiferromagnétiques. La susceptibilité montre une divergence en racine carrée de la température au voisinage de la transition de phase vers un état onde de densité de spin. La théorie d'échelle pour le modèle de gaz d'électrons quasi-unidimensionnels décrit parfaitement le comportement RMN de ces systèmes organiques mixtes soufre-sélénium. Bien que la densité de charge électronique sur les sites de sélénium déterminée par un calcul de type Hückel étendu suggère une influence non négligeable du désordre, les résultats ne permettent pas de décrire de façon satisfaisante la forme des spectres observés pour le noyau ⁷⁷Se.

**Abstract.** — We present an NMR analysis (spectra and relaxation data) of ⁷⁷Se nuclei for (TMTSF)₂ReO₄, (TMDTDSF)₂ReO₄ and (TMDTDSF)₂PF₆. In all compounds the temperature dependence of the relaxation enhancement ($TT_1$)⁻¹ follows the square of the temperature dependent uniform static susceptibility $χ_s$ in the paramagnetic regime. Deviations from this behaviour are visible in (TMDTDSF)₂PF₆ below the charge localization temperature $T_ρ$ and are explained in terms of antiferromagnetic correlations. The staggered susceptibility follows a square root temperature divergence in the vicinity of the phase transition towards a spin density wave state. The scaling theory for the quasi-one-dimensional electron gas model accounts very well for the NMR behaviour of these mixed molecule systems. Although the electron charge density on Se sites determined using an extended Hückel type calculation suggests a non negligible influence of disorder on the ⁷⁷Se Knight shifts, the results cannot account in a satisfactory way for the observed shape of the ⁷⁷Se spectra.

<footnote>(¹) Laboratoire de Physique des Solides, Bât. 510, Université Paris-Sud, F-91405 Orsay, France</footnote>

<footnote>(²) Laboratoire de Chimie Théorique, Université Paris-Sud, F-91405 Orsay, France</footnote>

<footnote>(³) Laboratorio Nacional de Engenharia e Tecnologia Industrial, Departamento de Quimica, P-2686 Sacavem, Portugal</footnote>

<footnote>(⁴) H. C. Oersted Institute, Universitetsparken 5, DK-2100 Copenhagen, Denmark</footnote>

<footnote>(*) Permanent address : Physikalisches Institut der Universitaet Bayreuth, Germany.</footnote>

<footnote>(**) Permanent address : Centre de Recherche en Physique du Solide, Département de Physique, Université de Sherbrooke, Québec, Canada J1K-2R1.</footnote>
**Introduction.**

Organic conductors based on molecules of tetramethyl-tetrathiafulvalene (TMTTF) or tetramethyl-tetraselenafulvalene (TMTSF) and with the general structure $(TMTTF)_2X$ or $(TMTSF)_2X$ have been intensively studied recently [1]. The nature of the inorganic anion $X$, governs in a crucial way especially the low temperature properties of these salts. Various phase transitions between competing ground states have been identified in this series of materials : spin-Peierls (SP) to spin density wave (SDW) states in $(TMTTF)_2X$ salts [2] and SDW state to superconductivity in $(TMTSF)_2X$ materials [3]. All members of the $(TM)_2X$ series are isostructural and there exists no major difference at first sight between a compound such as $(TMTTF)_2PF_6$ which exhibits an insulating behaviour below room temperature and $(TMTSF)_2ClO_4$ which displays a metal-like conduction and superconductivity below 1.2 K.

The ability to span a wide variety of physical properties in the $(TM)_2X$ series either by considering salts with different anions or by using high pressure has strongly stimulated the development of a theoretical framework for the understanding of quasi-one-dimensional (Q-1-D) conductors.

A new organic conductor based on the hybrid molecule of TMTTF and TMTSF, the so-called tetramethyl-dithiadiselenafulvalene (TMDTDSF), has provided a series of intermediate compounds whose physical properties lie in between those of $(TMTTF)_2X$ and $(TMTSF)_2X$ series. This picture is supported by the fact, that the unit cell parameters of $(TMDTDSF)_2PF_6$ are nearly midway between those of $(TMTTF)_2PF_6$ and $(TMTSF)_2PF_6$ [4].

In a previous paper, transport properties and ¹H-relaxation data of $(TMDTDSF)_2PF_6$ have been reported [4]. The divergence of the relaxation rate at 7 K, has been attributed to a SDW ordering establishing at the same temperature. However, protons in $(TMDTDSF)_2PF_6$ belong exclusively to the methyl groups and as such may not be appropriate to probe the temperature dependence of the electronic degrees of freedom over an extended temperature regime since the quantum or thermally activated rotation of those groups provides an additional channel of spin-lattice relaxation (in particular above 30 K [5]). Thus, the nuclear relaxation caused by hyperfine interaction with the conduction electrons, a property which yields a valuable information about the electronic system, is completely obscured at high temperatures. Furthermore, the protons are located at the outer extremities of the TMDTDSF molecule. Though the spatial part of the electronic wave function is usually spread in organic conductors over the whole molecule, we expect only a small fraction of the conduction electron spin density on $CH_3$ groups. This can be concluded from spin density maps in $(TMTSF)_2X$ compounds [6]. Thus, the hyperfine interaction which for protons simply scales with the spin density will be small, resulting in long relaxation times if the coupling to conduction electrons is the main source of nuclear relaxation and implies time consuming experiments. A third relaxation mechanism, though experimentally not yet confirmed, might be caused by a rotation of the $PF_6$ anions [7]. Therefore, selenium atoms ($^{77}$Se, $I = 1/2$) which are located on the central region of the TMDTDSF molecule, being free from extrinsic relaxation channels, could be considered as good nuclei to study the relaxation of electronic origin.

Detailed $^{77}$Se-NMR experiments have already been carried out in $(TMTSF)_2PF_6$ [8-10]. The aim of our measurements was to establish a complete temperature dependent profile of the nuclear spin lattice relaxation time $T_1$ in $(TMDTDSF)_2PF_6$ and to compare our results with $^{77}$Se relaxation data of $(TMTSF)_2PF_6$. The temperature dependence of the NMR data will be discussed in the frame of a scaling theory for the Q-1-D electron gas model [9, 11, 12]. It was previously found that it is precisely from the analysis of the temperature profile of the relaxation that a great deal of information about the statics, the dynamics and the dimensionality of electronic correlations can be extracted. In this theory the temperature
dependent electronic susceptibility $\chi_S$ enters in an essential way. Thus, in order to allow a quantitative comparison with the theory, a new experimental study of $\chi_S$ for (TMDTDSF)₂PF₆ is also presented in this paper.

A second point of interest concerns the reported orientational disorder of the TMDTDSF molecules in the stacks. The unsymmetrical TMDTDSF molecule can have two orientations in the unit cell, depending whether S or Se atoms are closest to the anions. On the other hand, though disorder is a common feature of the (TMDTDSF)₂X family, the degree of disorder seems to depend on the nature of the anion [13]. In crystals with small anions like PF₆ the disorder is complete, whereas for ReO₄ as an anion the degree of disorder seems to be smaller. So, it is interesting to compare ⁷⁷Se NMR measurements in (TMDTDSF)₂PF₆ and (TMDTDSF)₂ReO₄. In the latter compound the tetrahedral anion ReO₄ exhibits an anion ordering phase transition (AO) into an insulating ground state at T AO = 163 K [14]. Again, T AO for the mixed molecule compound is intermediate between the AO transition of (TMTTF)₂ReO₄ (T AO = 156 K [15]) and that of (TMTSF)₂ReO₄ (T AO = 180 K [16]). Thus, ⁷⁷Se NMR data in (TMTSF)₂ReO₄ and (TMDTDSF)₂ReO₄ are also reported and discussed.

## Experimental.

All crystals examined in this study were grown electrochemically. The crystals had typical dimensions of 5 × 0.5 × 0.2 mm³.

Magnetic susceptibility results are obtained from two experimental techniques : a Faraday balance and an ESR spectrometer (9.4 GHz), in the range of temperatures (4.2 K, 300 K). Using the Faraday balance, static susceptibility was measured on powdered samples of (TMDTDSF)₂ReO₄ and (TMDTDSF)₂PF₆ with respective weights 6.8 mg and 20.6 mg. We checked on the first compound the good linearity of the magnetization with the magnetic field, at room temperature, between 0 and 7 T. Magnetization data give the static susceptibility from which it is possible to reach the spin susceptibility $\chi_s(T)$ by removing the T-independent core diamagnetism $\chi_d$, calculated from the Pascal's constants. We used $\chi_d = -3.67 \times 10^{-4}$ emu/mole and $\chi_d = -3.71 \times 10^{-4}$ emu/mole respectively for the (TMDTDSF)₂ReO₄ and (TMDTDSF)₂PF₆ compounds [17].

Because of the low intensity of the sample signal, the contribution of the teflon sample-holder to the total signal could not be neglected and fortunately could be derived from the temperature dependence of the signal obtained with (TMDTDSF)₂ReO₄. Below the anion ordering transition, the spin susceptibility of this salt goes to zero exponentially while entering a semi-conducting phase [14]. At low temperature, the only remaining magnetization comes from the sample holder and from the sample core diamagnetization. We have fitted this contribution with a law of the type : $M = A/T + B$. After removing this contribution, we obtained the genuine temperature dependence of the spin susceptibility for (TMDTDSF)₂ReO₄ with a room temperature value, $\chi_s = 5.58 \times 10^{-4}$ emu/mole. These Faraday results are in fair agreement with the spin susceptibility data obtained from ESR measurements performed on a single crystal. Faraday and ESR data are displayed together in figure 1. ESR data were normalized to the room temperature value obtained with the Faraday balance technique. Furthermore, a calibration of the room temperature ESR susceptibility against (TMTSF)₂PF₆ leads to $\chi_s = 5.3-5.8 \times 10^{-4}$ emu/mole (see Tab. I). The temperature dependence of $\chi_s$ is roughly linear above the anion ordering transition and can be fitted below 163 K by a law of the type $\chi_s \sim T^{-1} \exp(-\Delta/T)$ (activated paramagnetism), with an activation gap $\Delta \approx 970$ K (see Tab. I and Fig. 1).
<figure><img src="image_2.png" /><figcaption>Fig. 1. — Top: Temperature dependence of the spin susceptibility for (TMDTDSF)₂ReO₄, showing the anion ordering transition at T AO = 163 K, bottom: A plot of the type Log (Tχs) versus 1/T for the same data gives, below 163 K, an activation gap Δ ≈ 970 K. These results come from Faraday balance (*) and ESR measurements (O).</figcaption></figure>

<table><caption>Table I.</caption><thead><tr><th></th><th>Δ<sub>σ</sub> = Δ<sub>ESR</sub> (K)</th><th>Δ<sub>NMR</sub> (K)</th></tr></thead><tbody><tr><td>(TMTSF)₂ReO₄</td><td>1 000 (°)</td><td>1 150</td></tr><tr><td>(TMDTDSF)₂ReO₄</td><td>970</td><td>1 050</td></tr></tbody></table>

<table><thead><tr><th rowspan="2"></th><th colspan="3">χs (300 K) [10<sup>-4</sup> emu/mole]</th><th rowspan="2">T<sub>ρ</sub> (K)</th></tr><tr><th>Faraday</th><th>ESR Calibration<br>(Ref: (TMTSF)<sub>2</sub>PF<sub>6</sub>)</th><th>NMR Calibration<br>(T<sub>1</sub> Tχ<sub>s</sub><sup>2</sup> = C<sub>0</sub>)</th></tr></thead><tbody><tr><td>(TMTTF)<sub>2</sub>PF<sub>6</sub></td><td>6.4 (*)</td><td></td><td></td><td>220</td></tr><tr><td>(TMTSF)<sub>2</sub>PF<sub>6</sub></td><td>3.06 (**)</td><td></td><td></td><td></td></tr><tr><td>(TMTSF)<sub>2</sub>ReO<sub>4</sub></td><td>2.73 (**)</td><td>3.6</td><td>2.95</td><td></td></tr><tr><td>(TMDTDSF)<sub>2</sub>ReO<sub>4</sub></td><td>5.58</td><td>5.3-5.8</td><td>5.09</td><td>&gt; 300</td></tr><tr><td>(TMDTDSF)<sub>2</sub>PF<sub>6</sub></td><td></td><td>4.35</td><td>3.7-3.9</td><td>100</td></tr></tbody></table>

<footnote>(°) TOMIC S., Thèse, Université Paris-Sud (1986).</footnote>

<footnote>(*) COULON C., Thèse, Université de Bordeaux I (1982).</footnote>

<footnote>(**) See references [35] and [36].</footnote>
The direct measurement of the sample holder's contribution is no longer possible for (TMDTDSF)₂PF₆ since no ordering of the centrosymmetric anions occurs in that system. Therefore, Faraday spin susceptibility results are reliable only in the high temperature range when the holder's signal is small and constant. In figure 2, we have combined these last results with the spin susceptibility derived from ESR measurements previously published [4]. The room temperature value, $χ_s = 4.35 × 10^{-4}$ emu/mole, was obtained from an ESR calibration against (TMTSF)₂PF₆ [4]. The same linear temperature dependence is observed down to 60 K associated with a reduction of $χ_s$ by a factor 2 between 300 K and 60 K. This decrease is actually larger than what is currently observed in compounds such as (TMTTF)₂Br where $χ_s(60 K)/χ_s(300 K) = 0.67$ [18].

<figure><img src="image_3.png" /><figcaption>Fig. 2. — Temperature dependence of the spin susceptibility for (TMDTDSF)₂PF₆. These results come from Faraday balance (*) and ESR measurements (O). The insert shows the charge localization on the resistivity below $T_ρ = 100$ K.</figcaption></figure>

The temperature dependence of the longitudinal resistivity is displayed in the insert of figure 2. These recent data reveal a resistivity minimum at $T_ρ = 100$ K at variance with earlier data showing a shallow minimum around 180 K [4]. We tend to believe that the new data are the more reliable ones since they have been obtained with the clamped-contacts technique [19], which allows to get $ρ(T)$ data without any crack. It is interesting to notice from figure 2, that the spin degrees of freedom are not affected by the charge localization occurring below $T_ρ$, as it is usual when long wavelength spin and charge excitations are decoupled in a 1D conductor, for instance in (TMTTF)₂X compounds [20, 11, 12, 21]. The behaviour of the susceptibility below 60 K is however very different from what is observed in neighbour compounds such as (TMTTF)₂Br. Instead of a saturation at low temperature, $χ_s(T)$ drops significantly below 60 K down to 10 K (see Fig. 2). This effect is likely to be related to the existence of 1D 2 $k_F$-CDW fluctuations detected in X-ray diffuse scattering experiments [13], in the same range in temperature. The maximum intensity of this diffuse scattering is located around 20 K which corresponds to a sharper decrease of the EPR susceptibility attributed to the onset of a SP transition [4]. However, below this temperature the diffuse scattering does not condensate into Bragg's spots so that this SP state is only short range ordered.
The NMR experiments were performed with a commercial Fourier transform pulsed spectrometer, operated at a frequency of 75.6 MHz (stationary field $B_0$ = 9.5 T). For the measurements of the spin lattice relaxation the nuclear magnetization was inverted. After a variable time delay $\tau$ an echo sequence was applied and the echo was sampled as a function of the delay. Extensive pulse phase cycling was used to extract the z-component of the magnetization. One half of the echo was Fourier-transformed and the imaginary part integrated. At high temperatures the recovery of the magnetization was observed to be exponential over more than one decade in $\tau$.

Single crystal spectra were recorded using a standard echo sequence. Figures 3 and 4 give some examples of NMR lineshapes obtained when the static field is perpendicular to the stacking axis for $(TMTSF)_2X$ and $(TMDTDSF)_2X$ series.

The four selenium compounds display a well resolved spectrum showing four resonance lines according to the four magnetically non equivalent sites in the unit cell (see Fig. 4 for $(TMTSF)_2PF_6$). Figure 3 (left) shows the temperature dependence of $^{77}$Se NMR of $(TMTSF)_2ReO_4$. This compound undergoes a phase transition towards an insulating ground

<figure><img src="image_4.png" /></figure>

Fig. 3.—Left: $^{77}$Se NMR spectra at different temperatures for a single crystal of $(TMTSF)_2ReO_4$ ($a \parallel B_0$). From top to bottom: 285 K, 186 K, 178 K, 165 K, 152 K, 135 K. $T_{AO} = 180$ K. Right: $^{77}$Se NMR spectra at different temperatures for a single crystal of $(TMDTDSF)_2ReO_4$ ($a \parallel B_0$). From top to bottom: 230 K, 200 K, 180 K, 150 K, 130 K, 120 K. $T_{AO} = 163$ K. Frequency axis is in ppm against the $^{77}$Se resonance of liquid $H_2SeO_4$.
<figure><img src="image_5.png" /><figcaption>Fig. 4. — Top : ⁷⁷Se NMR spectra for a single crystal of (TMDTDSF)₂PF₆ (a || B₀) at room temperature. Bottom : ⁷⁷Se NMR spectra for few aligned crystals of (TMDTDSF)₂PF₆ (a || B₀) at T = 165 K. Frequency axis is in ppm against the ⁷⁷Se resonance of liquid H₂SeO₄.</figcaption></figure>

state at T AO = 180 K, driven by an ordering of the non-centrosymmetric anions. The volume of the unit cell doubles below T AO and the resolved spectrum can be attributed to locally resolved chemical shifts as the shifts related to χs (Knight shifts) vanish exponentially in the semiconducting ground state.

As far as the ⁷⁷Se-NMR of mixed S-Se compounds are concerned, a broad structureless resonance line is observed with a width of about 1 000 ppm near room temperature. The ⁷⁷Se-NMR lineshape of (TMDTDSF)₂PF₆ is nearly temperature independent down to low temperatures. A broadening of the line at very low temperature can be attributed to fluctuating precursors of the AF ordering at 7 K. The temperature dependence of (TMDTDSF)₂ReO₄ NMR spectra is however different, figure 3 (right), since the ⁷⁷Se lineshape is resolved in the anion ordered phase at low temperature. Four non-equivalent selenium sites are observed in agreement with an alternate packing of TMDTDSF molecules along the *a*-axis.

It is tempting to attribute the origin of the non-resolved spectrum at high temperature to a distribution of local Knight shifts due to the existence of some residual disorder in the molecular packing. These data tend to suggest that the strong disorder evidenced by Laue scattering experiments in PF₆ salts is also present with ReO₄ anions [13]. A better understanding of the effect of disorder on the lineshape will await an improved knowledge of the Knight shift and Knight shift anisotropy in these low dimensional conductors.
To improve the signal to noise ratio especially at high temperatures for relaxation experiments, several crystals aligned along the stacking (a-) axis were used. ⁷⁷Se measurements in (TMTSF)₂PF₆ seem to support the idea that the hyperfine interaction is mainly isotropic in the plane perpendicular to the stacking axis. In this compound the anisotropy of T₁ was about 15% [8]. If we assume that the same is valid for (TMDTDSF)₂PF₆, the error made by using aligned needles due to the anisotropy of T₁ is somewhat within the experimental error of about ± 10%.

<figure><img src="image_6.png" /></figure>

Fig. 5. — Temperature dependence of $T_1^{-1}$ for ⁷⁷Se in a polycrystalline sample of (TMDTDSF)₂PF₆. At $T_ρ$ = 100 K the change in the curvature shows the onset of 2 $k_F$ fluctuations. Near $T_N$ data were fitted to $T_1^{-1} \sim |T - T_N|^{-1/2}$. The insert shows a log-log plot of the data between 7 K and 25 K. The slope of the line is 1/2.

Figure 5 shows $T_1^{-1}$ as a function of temperature $T$ in (TMDTDSF)₂PF₆. Four different regimes can be distinguished. First, at high temperature, $T_1^{-1}$ shows a slight upward curvature. This is the paramagnetic, metallic phase with a quasi linear increase of $χ_s$. The temperature $T_ρ$ ≈ 100 K corresponds to the resistivity minimum with a gap opening in the charge degrees of freedom driven by a 4 $k_F$ Mott-Hubbard localization of the carriers [4]. $T_ρ$ is not accompanied by any 3D ordered lattice distortion. So, although the Fermi surface vanishes, the spin degrees of freedom are not affected as shown by the absence of any anomaly in measurements of the magnetic susceptibility. However, below $T_ρ$, the relaxation is faster than expected from the extrapolation of the high temperature data and finally a plateau is observed below about 60 K. Below 25 K the recovery of the magnetization becomes nonexponential, and is accompanied by a strong broadening of the ⁷⁷Se resonance. These effects can be regarded as antiferromagnetic (AF) precursors above the onset of a SDW state at 7 K and the square root temperature dependence of $T_1^{-1}$ is the signature of critical fluctuations above a phase displaying 3D long range magnetic order [8-10, 22]. The modulation of the internal magnetization in the SDW phase leads to an inhomogeneous local field at the Se sites and causes $T_2^*$ to decrease with the concomitant broadening of the NMR line [10]. The nonexponentiality of the magnetization recovery is probably due to the critical slowing down of the magnetic fluctuations when the correlation time $τ_c$ of these fluctuations satisfies the relation $ω_n τ_c > 1$.

At 20 K, a loss of the spin degrees of freedom is evidenced from a kink in the temperature dependence of the EPR spin susceptibility [4]. However, the dimerization, below 20 K of the uniform 1D antiferromagnetic chain of localized spins (above 20 K) remains incomplete still
allowing the onset of magnetic ordering at 7 K and the establishment of 3D magnetic critical
fluctuations up to 25 K as shown by the square root divergence of $T_1^{-1}$. This regime will be
discussed more extensively in the next section. Figures 6 and 7 show the dependence of
$T_1^{-1}$ as a function of temperature for (TMDTDSF)₂ReO₄ and (TMTSF)₂ReO₄, respectively.
The behaviour of $T_1^{-1}$ versus T is similar to (TMDTDSF)₂PF₆ with the same slight upward
curvature at high temperature. However, the values of $T_1^{-1}$ at room temperature are quite
different. As will be shown below, this can be attributed to different values of $\chi_s$. Both
systems undergo a transition into a semiconducting ground state driven by an ordering of the
non centrosymmetric anions at T AO = 180 K for (TMDTDSF)₂ReO₄ and T AO = 163 K for
(TMTSF)₂ReO₄. The susceptibility becomes activated, leading to a sharp drop of
$T_1^{-1}$

<figure><img src="image_7.png" /></figure>

Fig. 6. — Temperature dependence of $T_1^{-1}$ for ⁷⁷Se in a polycrystalline sample of (TMDTDSF)₂ReO₄. The clear change in the curvature of $T_1^{-1}$ is the signature of anion ordering at T AO = 163 K. For details of the fit see text. The insert shows $T_1^{-1}$ versus $\chi_s^2 T$ above T AO.

<figure><img src="image_8.png" /></figure>

Fig. 7. — Temperature dependence of $T_1^{-1}$ for ⁷⁷Se in a polycrystalline sample of (TMTSF)₂ReO₄. The clear change in the curvature of $T_1^{-1}$ is the signature of anion ordering at T AO = 180 K. For details of the fit see text. The insert shows $T_1^{-1}$ versus $\chi_s^2 T$ above T AO $\cdot \chi_s$ was taken from reference [35].
Discussion.

In order to test the possible influence of disorder on the ⁷⁷Se Knight shift of (TMDTDSF)₂X we decided to carry out model molecular orbital calculations. The simplest way to tackle the problem is by considering a TMDTDSF molecule in the vicinity of two nearest neighbours along the chain. As mentioned above, once the dimerization along the chain is taken into account, eight different units of this type can be generated (Fig. 8). Because of the stoichiometry the mean charge per TMDTDSF donor in (TMDTDSF)₂X is + 1/2 e. Consequently it should be possible to gain some insight concerning the influence of disorder on the selenium Knight shift by calculating the selenium electron-spin density associated with the highest occupied molecular orbital (HOMO) of the central (TMDTDSF)⁺¹/² in the eight [((TMDTDSF)⁺¹/²)₃]₃ trimeric units schematically shown in figure 8.

<figure><img src="image_9.png" /><figcaption>Fig. 8. — Left, illustration of the overlap of the π orbitals in the (TMDTDSF)₂X stacks. Right, schematic representation of the eight different environments of a TMDTDSF donor when nearest neighbour interactions along the chain are considered.</figcaption></figure>

An effective one-electron Hamiltonian of the extended Hückel type [23] and a basis set of single – ξ Slater type orbitals were used in the calculations. All valence electrons were taken into account. The exponents (ξ) and the atomic parameters (H<sub>μμ</sub>) used are summarized in table II. The off-diagonal matrix elements of the Hamiltonian were calculated according to the modified Wolfsberg-Helmholz formula [24]. The geometry of the trimeric units 1-8 was fixed in the following way. First, an ideal DTDSF molecule (i.e., hydrogen atoms were used instead of the methyl groups) was built on the basis of accurate structures for other (TMTSF)₂X and (TMTTF)₂X salts. The geometrical parameters for this ideal DTDSF were
<table><caption>Table II. — Exponents and parameters used in the calculations.</caption><thead><tr><th>Atom</th><th>Orbital</th><th>H<sub>μμ</sub> (eV)</th><th>ξ</th></tr></thead><tbody><tr><td rowspan="2">Se</td><td>4s</td><td>-20.5</td><td>2.440</td></tr><tr><td>4p</td><td>-13.2</td><td>2.070</td></tr><tr><td rowspan="2">S</td><td>3s</td><td>-20.0</td><td>1.817</td></tr><tr><td>3p</td><td>-13.3</td><td>1.817</td></tr><tr><td rowspan="2">C</td><td>2s</td><td>-21.4</td><td>1.625</td></tr><tr><td>2p</td><td>-11.4</td><td>1.625</td></tr><tr><td>H</td><td>1s</td><td>-13.6</td><td>1.300</td></tr></tbody></table>

chosen to be: $C_{\alpha} = C_{\alpha} \cdot 1.357 \AA$, $C_{\alpha}$-Se: $1.876 \AA$, $C_{\beta}$-Se: $1.896 \AA$, $C_{\beta} = C_{\beta} \cdot 1.332 \AA$, $C_{\alpha}$-S: $1.736 \AA$, $C_{\beta}$-S: $1.745 \AA$, $C_{\beta'} = C_{\beta'} \cdot 1.343 \AA$, Se-C$_{\alpha}$-Se: $114.2^{\circ}$, $C_{\alpha}$-Se-C$_{\beta}$: $94.2^{\circ}$, S-C$_{\alpha}$-S: $114.0^{\circ}$, $C_{\alpha}$-S-C$_{\beta'}$: $96.3^{\circ}$, where $C_{\alpha}$ and $C_{\beta}$ refer to carbon atoms of the inner and outer double bonds respectively. Second, the center and the direction of the inner C=C double bond of the three molecules were assumed to be in the same position as in the average structure of Thorup *et al.* [25]. Since electron repulsions are not explicitly considered in extended Hückel calculations, we will assume that the relative change of electron and spin densities along the series of trimers 1-8 are very similar. The calculated selenium HOMO electron densities for the central donor ($D^{+1/2}$) of the eight trimeric units [$D^{+1/2}$]$_3$ of figure 8 are reported in table III. These electron densities spread over 0.0128 e, i.e. around 10% of the total selenium HOMO electron density. Disorder clearly has a noticeable effect on the selenium HOMO electron density. In addition, two remarks should be placed here. First, the values of table III reflect the influence of different electron transfer integrals on the selenium electron density of the central donor. It is well known, that single-$\xi$ type calculations underestimate the magnitude of these transfer integrals [26], so that the electron density difference of table III are likely to be too small. Second, the electrostatic

<table><caption>Table III. — Selenium HOMO electron density calculated for the central donor molecule ($D^{+1/2}$) of different trimeric units. See figure 6 for labeling.</caption><thead><tr><th>Unit</th><th>Electron density</th><th>Unit</th><th>Electron density</th></tr></thead><tbody><tr><td>1</td><td>0.1332</td><td>5</td><td>0.1340</td></tr><tr><td>2</td><td>0.1344</td><td>6</td><td>0.1405</td></tr><tr><td>3</td><td>0.1358</td><td>7</td><td>0.1277</td></tr><tr><td>4</td><td>0.1368</td><td>8</td><td>0.1400</td></tr></tbody></table>

interaction with the anions has not been considered. Since there are two possible ways to place a TMDTDSF donor with respect to the acceptor, there are twice as much different environments for a selenium atom. This is likely to induce a considerable additional spread. With these two observations in mind it is clear, that the 10% spread of the selenium HOMO electron density should only be considered as a lower limit. A more quantitative estimation would require calculations of the spin densities including explicitly both electron repulsion
and donor-acceptor interactions. Although the 10% spread does not account for the observed shape of the ⁷⁷Se spectra, we believe these results suggest quite a sizeable control of the ⁷⁷Se Knight shifts by disorder.

Electronic degrees of freedom modulate the hyperfine interaction in conductors. Especially in the (TM)₂X salts and their derivatives this turns out to be the dominant mechanism for nuclear relaxation. However, the individual properties of each salt are well reflected in the details of the nuclear relaxation, though the general behaviour can be described by an uniform theory.

We first look at the $T_1^{-1}$ versus T data of (TMDTDSF)₂PF₆ in the non ordered phase well above $T_N$ where the lattice softening effect are sufficently small to be ignored. From the EPR data of figure 2 and X-ray measurements [13], this corresponds to the temperature domain $T > 30$ K. Within this paramagnetic domain figure 3 shows, that at sufficiently high temperatures, $T_1^{-1}$ presents an upward curvature which is typically found in Q-1-D conductors [8, 10, 27]. This behaviour is well known to result from the uniform ($q \approx 0$) spin fluctuations which dominate the relaxation. Previous calculations [9, 12] have shown, that in the presence of spin fluctuations characterized by harmonic paramagnon dynamics, the small $q$ integration of the basic expression for $T_1^{-1}$ [28], namely:

$$T_1^{-1} = 2 \gamma_N^2 (1/2 \pi)^D T \int d^D q |A_q|^2 \chi_{\perp}''(q, \omega_N)/\omega_N \quad (1)$$

where $A_q$ is the hyperfine coupling constant and $\chi_{\perp}''$ is the imaginary part of the transverse retarded spin response function, can be uniquely expressed in terms of the temperature dependent static and uniform magnetic susceptibility $\chi_s$, that is

$$ \begin{aligned} T_1^{-1}[q \approx 0] &= 2 \gamma_N^2 (1/2 \pi)^D |A_0|^2 T \int_{q \approx 0} d^D q \chi_{\perp}''(q, \omega_N)/\omega_N \\ &= C_0 T[\chi_s(T)]^{(S-D)/2} \end{aligned} \quad (2) $$

In one dimension this reduces for $q \approx 0$ to

$$T_1^{-1}[q \approx 0] = C_0 T \chi_s^2(T) \quad (3)$$

with $C_0 = \pi \gamma_N^2 |A_0|^2$. We want to emphasize, that this expression is obtained in the so-called collisionless (non-diffusive) limit where there is no field dependence as long as $T > \omega_e$ with $\omega_e$ as the electronic Larmor frequency. This limit is consistent with the absence of field dependence of $T_1$ up to 6 T in systems such (TMTTF)₂PF₆ or (TMTTF)₂Br and it provides an indication for the validity of equation (3) [29]. From the $\chi_s$ and $T_1^{-1}$ data we see, that the plot of figure 9 shows, that the relation (3) for $D = 1$ is indeed well satisfied for $T > 150$ K. So, this result is of interest since it shows, that both $T_1^{-1}$ and $\chi_s$ are influenced by 1D paramagnons which are decoupled from long wavelength charge excitations as can be observed in the resistivity measurements.

From figure 9 deviations to the $T_1^{-1} \sim T\chi_s^2$ law become clearly visible below $T_\rho$. These deviations come from the antiferromagnetic spin fluctuations centered at $q \approx 2 k_F$ in one dimension and which are expected to grow as the temperature is lowered. The corresponding contribution of this large $q$ value to $T_1^{-1}$ is well known [12, 30]. Indeed at small $\omega$, one has:

$$T_1^{-1}(q \approx 2 k_F) \approx \gamma_N^2/(4 \pi) [N(E_F)]^2 |A_{Q_0}|^2 T \bar{X}(2 k_F, T) = C_1 \quad (4)$$
<figure><img src="image_10.png" /><figcaption>Fig. 9. — $T_1^{-1}$ versus $\chi_s^2 T$ of $^{77}$Se in a polycrystalline sample of (TMDTDSF)$_2$PF$_6$. The straight line is a fit to equation (1), which determines the $q \approx 0$ contribution. Deviations from this linear behaviour are caused by the $q \approx 2 k_F$ contribution. The insert shows the 2 $k_F$ part of $T_1^{-1}$ versus $\chi_s^2 T$. It was calculated by subtracting the $q \approx 0$ contribution from $T_1^{-1}$.</figcaption></figure>

where in the presence of a correlation gap $\Delta_\rho$ the auxiliary susceptibility can be written in the following scaling form :

$$ \bar{\chi}(2 k_F, T) = \bar{\chi}(T_\rho/E_F) \bar{\chi}(T/T_\rho). \quad (5) $$

Here $\bar{\chi}(T_\rho/E_F) \approx (T_\rho/E_F)^{-\gamma_0}$ gives the power law contribution for energy scales above $T_\rho$ where $0 < \gamma_0 < 1$. At lower temperatures one has $\bar{\chi}(T/T_\rho) \approx (T/T_\rho)^{-\gamma}$. It turns out, that below $T_\rho$ elaborate calculations show, that $\gamma$ reaches the maximum value $\gamma = 1$ which is exact in absence of magnetic anisotropy [21]. It should be mentioned, that the scaling form (5) neglects all transients when the electronic system evolves to the regime of strong electronic Umklapp processes near $T_\rho$. Therefore $C_1$ is a temperature independent quantity as long as the strong coupling regime prevails for the uniform charge degrees of freedom and the model for the temperature dependence of $T_1$ becomes

$$ T_1^{-1} = C_0 T \chi_s^2(T) + C_1. \quad (6) $$

From figure 9 one can see, that the low temperature deviation to the $T_1^{-1} \sim T\chi_s^2$ law does extrapolate to a temperature independent contribution like (4). One can now single out the deviations to the uniform contribution (see insert of Fig. 9). The quantity $(T_1 T)^{-1}$ near $2 k_F$ is according to (4) directly proportional to $\bar{\chi}(2 k_F, T)$. Figure 10 shows the resulting $\bar{\chi}(2 k_F, T)$ versus T variation and it gives nice confirmation of the power law behaviour of $\bar{\chi}(2 k_F, T)$ together with the relevance of 1D Umklapp processes and the value of $\gamma = 1$ in this compound. From the same figure one can observe that the strong coupling regime of correlations which leads to $\gamma = 1$ seems to be fully established only below 80 K or so, i.e. below the resistivity minimum at $T_\rho \approx 100$ K. There exists thus a sizeable temperature domain, between 80 and 100 K, dominated by transients associated to the crossover from the weak to the strong Umklapp coupling regime. These effects are non-universal and depend on microscopic details of the system. In this respect, the comparison with previous observations
<figure><img src="image_11.png" /><figcaption>Fig. 10. — Temperature dependence of the 2 k<sub>F</sub> part of the electronic auxiliary susceptibility in (TMDTDSF)<sub>2</sub>PF<sub>6</sub>. The vanishing 2 k<sub>F</sub> part above T<sub>p</sub> is consistent with the power law dependence : X̄(2 k<sub>F</sub>, T) ~ (T/E<sub>F</sub>)<sup>-γ</sup> (γ ≪ 1). Low temperature data were fitted with γ = 1 (strong Umklapp scattering regime).</figcaption></figure>

made for the sulfur series (TMTTF)<sub>2</sub>X is also of interest. Indeed, in the compound (TMTTF)<sub>2</sub>PF<sub>6</sub>, a localization is observed at a higher temperature (T<sub>p</sub> ≈ 220 K) [32] than for the mixed salts and therefore, the strong coupling regime with a temperature independent C<sub>1</sub> parameter in (4) is already established in the vicinity of 220 K [9, 31]. Since T<sub>p</sub> is known to be a measure of the dimerization in the solid state this is consistent with the fact that salts of (TMTTF)<sub>2</sub>X series are more strongly dimerized (i.e. half-filled band) than the salts of the (TMDTDSF)<sub>2</sub>X family. Indeed, localization effects are absent in the TMTSF family which exhibits hardly any structural dimerization. So, the study of (TMDTDSF)<sub>2</sub>PF<sub>6</sub> provides the unique opportunity to follow this 1D crossover between weak and strong Umklapp coupling regimes. Above 100 K or so, C<sub>1</sub> is weak and temperature dependent (γ<sub>0</sub> < 1), while at lower temperatures C<sub>1</sub> becomes appreciable and temperature independent (γ = 1).

If we proceed now to the analysis of critical ordering effect we first observe, that a divergence of the form T<sub>1</sub><sup>-1</sup> ~ |T - T<sub>c</sub>|<sup>-1/2</sup> is invariably obtained in other similar compounds near AF ordering [8, 10, 22, 31], and is well known to be the signature of a singular AF correlation length ξ<sub>AF</sub> at the Neel temperature T<sub>N</sub> which is here 7 K. As shown in reference [9] the exponent 1/2 results from the 3D critical effects described by Gaussian type of exponents. These effects seem to be strong enough to overcome the short range order, introduced by the disorder. We want to mention, that it is the disorder that may prevent the system to become superconducting under pressure [4]. Neglecting for a moment the influence of the SP ordering on T<sub>N</sub> and adopting the scaling picture of reference [9, 12] where the relevant critical quantities of T<sub>1</sub><sup>-1</sup>, namely χ'', ω and d<sup>D</sup>q are all dimensioned in terms of ξ<sub>AF</sub> for the integration sector near the AF modulation vector Q<sub>0</sub>, one gets the critical form :

$$T_1^{-1}(q \approx Q_0) \sim C(Q_0) \xi_{AF} \quad (7)$$

where C(Q<sub>0</sub>) is a constant independent of ξ<sub>AF</sub>.

Within the Gaussian or the RPA approximation for the AF critical indices, the correlation length verifies ξ<sub>AF</sub> ~ |T - T<sub>N</sub>|<sup>-1/2</sup> The resulting 1/2 exponent in D = 3 however, tells us nothing about the mechanisms of propagation of long range ordering. In order to understand
the interplay between the SP and the AF ordering it is necessary to specify such a mechanism. Microscopic calculations for the quasi 1D electron gas in the presence of non-zero interchain single electron hopping $t_⊥$ and a 1D correlation gap $Δ_ρ > t_⊥$ have shown, that the transverse single electron band motion is frozen and has no chance to develop at low temperature [11]. Transverse one-particle virtual motion being always possible, it will lead to an effective AF interchain exchange interaction $J_⊥ \sim 2 \pi v_F (t'_⊥ / Δ_ρ)^2$, where $t'_⊥ < t_⊥$ is renormalized by 1D many-body effects (see [11]). This coupling is two-particle like and leads to an interchain electron-hole pair propagation which is necessary to the transverse ordering of AF correlations. According to the microscopic results of reference [11] the critical parameter $r_{AF}(\xi_{AF} \sim r_{AF}^{1/2})$ vanishes at the critical point and is given by

$$r_{AF} = \delta - (2 \gamma)^{-1} \tilde{J}_{\perp} [(T/T_{\rho})^{-\gamma} - 1] \approx \gamma \delta (T - T_{N})/T_{N} \quad (T \to T_{N}) \quad (8)$$

where $\tilde{J}_⊥ = J_⊥ / \pi v_F$ and $0 < \delta < 1$ is a positive constant, that gives the contribution of the exchange to the transition above $T_ρ$. Owing to the spin-charge separation of the 1D electron gas problem [21], the $2 k_F$ susceptibility exponent $γ$ namely

$$\gamma = 2 - \gamma_{\sigma}^{-1} - \gamma_{\rho} \qquad (9)$$

contains two independent contributions, one linked to the spin ($γ_σ$) degrees of freedom and the other for the charge ($γ_ρ$). In presence of strong Umklapp effects and a correlation gap one has $γ_ρ = 0$. As for the spin part, the bosonization technique tells us that $γ_σ$ is directly connected to the spin compressibility $K_σ$ [33], that is

$$\gamma_{\sigma} = 2 \pi v_{\sigma} K_{\sigma} \qquad (10)$$

where $v_σ$ is the long wavelength spin degrees of freedom velocity introduced previously. It turns out, that $K_σ = (2 \pi v_σ)^{-1}$ coincides with the zero temperature susceptibility per spin of the mode. Under the influence of SP fluctuations which favor the formation of spin singlet dimers, $χ_s$ does not saturate to the temperature independent value $2 K_σ$ but is sizeably depressed at low temperature as it is clearly seen from the EPR data of figure 2 below 40 K or so. As suggested in reference [34] in the context of precursors to the SP ordering of the (TMTTF)₂PF₆ compound, $γ_σ$ can be taken as a temperature dependent quantity which can be directly related to the observed depression in $χ_s$. That is

$$\gamma(T) = 2 - \gamma_{\sigma} (T_{\text{SP}}^{0})^{-1} \chi_{s}(T_{\text{SP}}^{0}) / \chi_{s}(T) \qquad (11)$$

where $T_{SP}^0$ is the 1D energy scale for the SP lattice softening. From X-ray and EPR data it corresponds roughly to 40 K. Taking $γ_σ(T_{SP}^0)^{-1} = 1$ in absence of SP effects, the ratio $χ_s(T_{SP}^0)/χ_s(T)$ can then be estimated for the observed depression in $χ_s(T)$ in figure 2. From this semi-phenomenological approach, it is clear that as long as $γ(T)$ is positive, spin degrees of freedom are still present and they can bring the critical parameter $r_{AF}$ to zero at a finite value for $T_N$. From (8) the latter is given by

$$T_N = T_\rho [\tilde{J}_\perp (\delta \gamma(T))^{-1} [1 + \tilde{J}_\perp (\delta \gamma(T))^{-1}]^{1/\gamma(T)}] \qquad (12)$$

This is clearly a non vanishing quantity as long as $γ(T)$ is positive. From (11) a rough estimate would predict, that if the depression of the susceptibility due to SP ordering remains less than 50 %, the onset of AF ordering is still possible. From the $T_1^{-1}$ versus T data of figure 5, AF critical effects are observed up to 20 K or so, which is consistent with less than 50 % of
reduction of $\chi_s(T)$ down to 20 K and with the AF ordering arising in the same temperature range. It is worthwhile to note at this point that the depression of the susceptibility seen in $(TMDTDSF)_2PF_6$ is similar to the one already reported for the $(TMTTF)_2SbF_6$ compound in the same range of temperature [32]. For the latter, it was proposed that even if 1D precursors effects are neglected, the kinetic interchain coupling can play a role in the interplay between the SP and the AF ground states.

Now let us turn to $(TMDTDSF)_2ReO_4$ and $(TMTSF)_2ReO_4$. At high temperatures, above $T_{AO}$, both systems relax according to equation (3). In this temperature region the electronic susceptibility shows a linear increase with temperature ($\chi_s \approx A + BT$). The constants A and B were adapted to give a good approximation of the real high temperature $\chi_s$ data (data of $(TMDTDSF)_2ReO_4$ as published here, data for $(TMTSF)_2ReO_4$ were taken from reference [35]). Thereafter the calculated susceptibility was normalized to its value at room temperature ($\chi_s(RT) = 1$). The idea of this treatment is to get rid of possible troubles encountered by the calibration of $\chi_s$. On the other hand there is usually no doubt about the temperature dependence of $\chi_s$, as long as only relative values are concerned. To account for a better comparison of all systems the normalization was also made for $\chi_s$ of $(TMDTDSF)_2PF_6$. Later in this paper, we will show how the $^{77}$Se relaxation can provide the absolute values of $\chi_s$ for $(TMDTDSF)_2PF_6$, $(TMDTDSF)_2ReO_4$ and $(TMTSF)_2ReO_4$. Following the above discussion, $C_1$ in equation (4) should be set to zero in the high temperature regime (metallic phase, weak coupling). Using the above approximations in equation (3) one gets a polynomial of degree 3 in $T$, which can be fitted easily to the NMR data of figure 5. The agreement between the fit and the data is quite good. The inserts in figure 6 and figure 7 show $T_1^{-1}$ versus $\chi_s^2 T$ above $T_{AO}$ for both compounds. The linear dependence with a zero intercept for $T_1^{-1}$ at zero temperature is obvious. Below the anion ordering temperature, both components show a semiconducting behaviour. Low temperature data were fitted to a law of the form :

$$T_1^{-1} = CT^{-1}\exp(-2\Delta/T) \quad (13)$$

were C is a constant. This is of course still the simple relation of equation (3), where for $\chi_s$ an activated paramagnetism was assumed. $\Delta$ is of the order of 1000 K. Even a small error in the temperature measurement for $T_1^{-1}$ or $\chi_s$ has a strong influence on the analysis and therefore the direct fit seems to be more reliable. The values of $\Delta$ as extracted from the fit are given in table I.

So far, we have made no use of the coefficient $C_0$, which is the slope of $T_1^{-1}$ versus $\chi_s^2 T$ in the high temperature regime. $C_0$ contains the details of the $^{77}$Se hyperfine interaction (hfi) and, since our $\chi_s$ data are normalized to unity, the value of $\chi_s$ at room temperature. So, from an analysis of $C_0$ an absolute calibration of $\chi_s$ should be possible. In other words, a measure of $T_1^{-1}$ can be used to probe the electronic susceptibility [8]. However, the conduction electrons are in $\pi$-orbitals and the observed hyperfine interaction is caused by a dipolar interaction together with a polarization of lower orbitals (core polarization), which makes it impossible to calculate correctly. Thus for the following analysis we made the assumptions, that the hyperfine interaction is the same for both $(TMTSF)_2X$ and $(TMDTDSF)_2X$. This assumption is supported by very similar molecular and cristalline geometries in the two compounds. Furthermore, only values from powdered samples will be used. We want to remind the reader, that the resonance of $(TMDTDSF)_2X$ is « powderlike » as a consequence of the intrinsic disorder which leads to a continuous distribution of Knight shifts. We hope to include this effect, at least to some extent, in the powder mean values of the hfi. So, if $C_0$ and the absolute value of $\chi_s$ are known for one compound, $\chi_s$ for other Se-containing materials in which $T_1^{-1}(T)$ has been measured, can be derived by a
simple scaling argument. The values for $\chi_s$, thus found by this procedure, have been summarized in table I. $C_0$ was taken from measurements in (TMTSF)₂PF₆ in reference [8] and $\chi_s$ from reference [36]. The determination of $\chi_s$ via $T_1$ measurements has also been used in an other context, namely under pressure [8, 29].

## Summary.

The magnetic and NMR studies of the mixed S-Se salts presented and discussed in this article complete earlier works performed on sulfur or selenium compounds. (TMDTDSF)₂PF₆ is a unique material in the sense that all regimes predicted by the theory can be observed experimentally. A 1D regime with dominant $q \approx 0$ spin fluctuations is observed between 300 and 180 K ($T_I^{-1} \sim T\chi_s^2(T)$) whereas the contribution to the relaxation coming from 2 $k_F$ spin fluctuations takes over gradually below 180 K and follows the strong Umklapp scattering temperature dependence ($\chi_{SDW}(2 k_F) \sim T_I^{-1}$) only below $T_\rho = 100$ K. The 2 $k_F$ spin-phonon coupling contributes to a further lowering of the uniform susceptibility below 60 K. However, the tendency of the 1D antiferromagnetic chain towards dimerization is not strong enough to stabilize a SP ground state. Instead, SDW ordering is achieved below 7 K with 3D 2 $k_F$-SDW fluctuations detectable via $T_I^{-1}$ measurements up to 25 K or so, ($T_I^{-1} \sim |T - T_N|^{-1/2}$).

## Acknowledgements.

One of us (B.G.) wants to thank the Deutsche Forschungsgemeinschaft for financial support during his stay at Orsay. We acknowledge P. Wzietek, E. Barthel, S. Ravy and J. P. Pouget for several discussions.

This work has been partly supported by the ESPRIT-Basic Research Action MOL-COM 3121 and the DRET Contract n° 88/198.

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