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# 17O NMR anisotropic properties of the planar oxygen sites in an YBa2Cu3O6.5 single crystal

F. Barriquand, Philippe Odier, Denis Jerome

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F. Barriquand, Philippe Odier, Denis Jerome. 17O NMR anisotropic properties of the planar oxygen sites in an YBa2Cu3O6.5 single crystal. Physica C, 1991. hal-05000437

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$^{17}$O NMR anisotropic properties of the planar oxygen sites in an
$YBa_2Cu_3O_{6.5}$ single crystal

F. Barriquand a,b, P. Odier c and D. Jérome b

a Laboratoires de Marcoussis, DEM, route de Nozay, 91 Marcoussis, France

b Laboratoire de Physique des Solides, Université Paris-Sud, 91405 Orsay, France

c CRPHT-CNRS, 1D Avenue de la Recherche Scientifique, 45071 Orléans Cedex 2, France

Received 6 January 1991

Revised manuscript received 20 March 1991

We have measured the Knight shift and the spin-lattice relaxation of the O(2, 3) planar sites in a single crystal of $YBa_2Cu_3O_{6.5}$.
We report the anisotropy of these quantities, together with their striking temperature dependences. Our data are not easy to
understand in any theoretical model presently available. One approach is to consider the $T_1$ data to indicate that, for temperatures
greater than $T_c$, some electronic fluctuations develop preferentially on the O(2p) orbitals, and not on the Cu(3d) orbitals. How-
ever, such an analysis seems incompatible at first sight with our Knight shift data, because the planar oxygen Knight shift tensor
appears to result from a single electronic susceptibility and not two different susceptibilities. We, however, propose an alternative
"two-fluid" model analysis which may account qualitatively for the NMR features observed. The originality of our experimental
work, if compared with some other, lies to a large extent in our ability to take advantage of the possibilities offered by a $^{17}$O-
enriched single crystal study.

# 1. Introduction

Since the discovery of the high-$T_c$ superconducting oxides, the nature of the electronic structure of these com-
pounds has been widely debated, and the coupling between the copper and the oxygen orbitals in the $CuO_2$
planes still remains a matter of controversy. A very important point is to know whether thermal electronic
fluctuations (i.e. low-energy electronic excitations) in the planes can be accurately described by a single elec-
tronic "fluid" or not.

Indeed, the theoretical situation is so confused that it seems that all possible physical models have been ad-
vocated: some theories [1], using the "one-band" Hubbard Hamiltonian as a start, state that two different
kinds of low-energy electronic excitations exist, namely "holons" and "spinons". Completely opposite is the
point of view defended by Emery and Reiter [2], proposing that even by using an "extended" (i.e. "multi-
band") Hubbard Hamiltonian, it is possible to reduce the electronic degrees of freedom to a single one at the
Fermi level. Lastly, other theoretical works [3] have suggested that by the use of a similar "extended" Hubbard
Hamiltonian, two different electronic degrees of freedom could be derived at finite temperatures, with a dif-
ferent microscopic origin for each.

"One-fluid" and "two-fluid" models have different implications for the NMR properties of high-$T_c$ superconductors. This fact has motivated a lot of NMR work in all the high-$T_c$ compounds, with the aim of establishing by experiment what kind of theoretical frame could properly describe the high-$T_c$ oxides in the metallic state. Of these works many [4-6] have looked at the strange behaviour of the susceptibility in the metallic state, which strongly depends both on stoichiometry and temperature (see also refs. [8,9]), and have provided quite strong arguments in favour of the "one-fluid" picture, although one NMR group [7] has reported contrary results.

All authors agree that the temperature-dependent part of the susceptibility has its origin in the Cu(3d, $x^2-y^2$)
orbitals of the planes, and also marginally in the planar O(2p, σ) orbitals. The most striking feature of this susceptibility is its regular, downward slope from room temperature to $T_c$, which is probably indicative of some unknown critical mechanism extending over a very broad range of temperatures. However, very few theoretical works have ventured to propose any quantitative understanding of this temperature behaviour, although Trugman [10], and also quite recently Li et al. [11], have tackled this problem. The question upon which most efforts have concentrated up to now has been rather more simply to decide whether a single electronic "fluid" (in other words a single degree of freedom) could be sufficient to describe the electronic fluctuations in the high-$T_c$ compounds, or not.

In order to possibly settle the question, we have investigated very carefully the temperature dependence of the Knight shift tensor and mostly the anisotropy of $1/T_1$ on the planar oxygen sites of a particular $YBa_2Cu_3O_{7-\delta}$ single crystal, with $\delta \approx 0.5$ ($35 \text{ K} < T_c < 55 \text{ K}$). This stoichiometry is precisely one for which the susceptibility displays strong variations above $T_c$.

The $1/T_1$ anisotropy data that we derive represent, to our knowledge, a new piece of information that must be added to the already impressive amount of published NMR results in the field.

In the first part of this paper, we review some aspects of the chemistry employed, together with the essentials of our NMR technique. We next focus on the discussion of our NMR data, which we develop in two successive steps. We first examine our Knight shift data. We then develop a tentative interpretation of our $1/T_1$ data in the frame of a "two-fluid" model which we try, step by step, to elaborate in a consistent way with all the NMR data presently available in similar compounds. Our conclusions suggest the needs for refinements of all current theories.

## 2. Experimental

The sample, with its Meissner characterization ($35 \text{ K} < T_c < 55 \text{ K}$) and room temperature $^{17}$O NMR spectra, has already been described in a former publication [23]. Let us first digress a bit on some of its special chemical features.

Because $YBa_2Cu_3O_{7-\delta}$ melts incongruently, methods such as the floating zone have been useless up to now for growing $YBa_2Cu_3O_{7-\delta}$ crystals. On the other hand, some early experiments [12–14] have demonstrated that $YBa_2Cu_3O_{7-\delta}$ crystals could be obtained from molten compositions comprised in the subsystem $YBa_2Cu_3O_{7-\delta}/BaCuO_2/CuO$.

Thus an accurate knowledge of the primary crystallisation field of the $YBa_2Cu_3O_{7-\delta}$ phase and in general of the liquidus relations in the compositional region $YBa_2Cu_3O_{7-\delta}/BaCuO_2/CuO$ is essential both to optimize the many experimental parameters involved and to understand the mechanisms of the crystal growth. Extended studies [16–21] have been undertaken to complete the preliminary data of Roth et al. [15]. The best compositions for crystal growth seem to be achieved when $YBa_2Cu_3O_{7-\delta}$ is mixed with compounds having Ba/Cu ratios in the range of 7 mol BaO/18 mol CuO, owing to the presence of fixed points in this area (one binary eutectic, one ternary eutectic and one ternary peritectic).

However, the flux is very abrasive with respect to most of the crucible materials investigated up to now, e.g. alumina [14]. For this reason we have searched for a new technique [21,22] avoiding any contamination from the crucible by using a $Y_2O_3$ crucible containing a mixture of 7 mol BaO/18 mol CuO. The resulting process is particularly elaborate since the flux interacts with the crucible, the microstructure of which is important. Our best results have been obtained for well-sintered $Y_2O_3$. In appropriate conditions, plate-like $YBa_2Cu_3O_{7-\delta}$ crystals ($a-b$ face) were obtained ($2 \times 2 \times 0.2 \text{ mm}^3$) containing a small fraction of inclusions from the flux. X-ray diffraction, thermogravimetric analysis and electron microprobe analysis (providing the means for detection of the light elements) all agreed on the oxygen content. The electron microprobe determinations displayed a systematic departure of the copper atomic fraction from that of the stoichiometric $YBa_2Cu_3O_{7-\delta}$ compound,
thus indicating the composition YBa$_{2.05}$Cu$_{2.94}$O$_{6.35-6.45}$. Strontium impurities (6 at.‰) were found in some samples and were attributed to a contamination from the barium source.

The $^{17}$O oxygenations were performed in a closed quartz tube attached to a vacuum system. The end part of the tube was inserted into the furnace. Several samples (up to 5) could be treated simultaneously, lying vertically on a zirconia crucible introduced into the tube. The latter was first evacuated to 100 Pa for a few minutes at room temperature and then filled up with a $^{17}$O/$^{16}$O mixture (51%/49%) at a pressure of 9×10$^4$ Pa. After this the tube was closed and the temperature was raised up to the exchange temperature.

Several trials were performed with the aim of optimising both the oxygen content and its distribution within the sample. The resulting oxygen contents were estimated by the measurement of transition temperatures in a DC magnetisation experiment and the degree of homogeneity was estimated qualitatively from the $T_c$ distribution. Our best homogeneity [23] was obtained after a first treatment at 800°C for 36 h and cooling down to room temperature, followed by a further annealing at 600°C for 6 days and a subsequent quench at room temperature.

Under these conditions the electron microprobe analysis of one of the samples pointed to a composition of YBa$_{2.02}$Cu$_{2.94}$O$_{6.48}$, in good agreement with the occurrence of an onset superconducting transition at 54 K. The Meissner effect indicated a distribution of $T_c$ extending over 20 K.

Here we take the opportunity to mention a rather fancy scheme that we devised in order to obtain three-dimensionally oriented powders of YBa$_{2}$Cu$_{3}$O$_{7}$. The interest lies in the possibility for us to carry out NMR studies on bigger homogeneous samples, so as to obtain better NMR signal to noise ratios.

Our idea was to improve the experimental set-up of Farrell et al. [24] by adding a second magnetic field $H_1(t)$ which could be continuously rotated around an axis perpendicular to the first field $H_0$ (fig. 1). The order of magnitude of $H_1(t)$ should be the same as that for $H_0$. If the rotation of $H_1(t)$ does not exceed the orientation speed of the crystal powder in a single static magnetic field of the same magnitude, the c-axis of the powder should remain oriented along the sum of the two magnetic fields $H_0+H_1(t)$ at any time, which is clearly not our purpose.

At higher rotating speeds, however, it can be seen that the a-axis of the powder (which is the least paramagnetic axis, because of the Van Vleck paramagnetism of the chain copper atoms) should become oriented perpendicularly to the fixed plane containing $H_0$ and $H_1(t)$, since there would never be any magnetic field in that direction. The c-axis of the powder, which is the most paramagnetic axis, should still remain aligned in the $H_0$ direction, so that all the powder would be three-dimensionally oriented!

From copper NMR results [25] for YBa$_{2}$Cu$_{3}$O$_{7-δ}$ with $δ \approx 0$, we know that

$$ (\chi_{cc} - \chi_{bb}) \approx 2(\chi_{bb} - \chi_{aa}). \qquad (1) $$

This comes from the fact that only the Van Vleck paramagnetism is expected to be anisotropic at room temperature. As can be seen, all three eigenvalues of the bulk susceptibility are quite different from one another, so that our scheme does not seem unrealistic. Unfortunately, because we still cannot synthesize monocrystalline untwinned powders, we have not been able to put our proposal to practice. So we let the reader speculate further on this matter. All the NMR experiments were performed in a 9.3 T magnet with a Fourier Transform com-

<figure><img src="image_3.png" /><figcaption>Fig. 1. Schematic diagram for the possible three-dimensional orientation of monocrystalline powders.</figcaption></figure>
mercial spectrometer. The $^{17}$O NMR frequency for H$_{2}$$^{17}$O in that field was 53.73 MHz.

The signal acquisitions were based on the principle of the $(\pi/2-\tau-\pi)$ echo, which fits in well with cases of inhomogeneous line broadenings.

The NMR delays could thus read:

$$ \pi/2-\tau'-\pi-\tau'-\text{acquisition} \qquad (2) $$

for the Knight-shift measurements, and

$$ \pi-\tau-\pi/2-\tau'-\pi-\tau'-\text{acquisition} \qquad (3) $$

for the $1/T_1$ measurements. $\tau'$ is a short refocusing delay, and $\tau$ is the variable recovery time.

For (2), we used the following intricate phase cycling:

$$ \pi/2: \quad +X+X+X+X-Y-Y-Y-Y-X-X-X-X+Y+Y+Y+Y, $$

$$ \pi: \quad +X-Y-X+Y-Y-X+Y+X-X+Y+X-Y+Y+X-Y-X, $$

$$ \text{receiver}: -Y+Y-Y+Y-X+X-X+X+Y-Y+Y-Y+X-X+X-X. $$

Such phase cycling permits suppression of the accumulation of spurious signals coming from the ringing of the probe, because the receiver averages the phases of each pulse $(\pi/2$ and $\pi)$ to zero. The sequence is thus very suitable for the detection of weak signals.

# 3. Results and discussion

## 3.1. Knight shift

It has already been shown [26,27] that the Knight shift tensor on the oxygen planar sites can be approximately decomposed into an isotropic part $K_{\text{iso}}(T)$, and a traceless, axially symmetric part $K_{\text{ax}}(T)$.

$$ K(T) = K_{\text{iso}}(T) + K_{\text{ax}}(T). \qquad (4) $$

We show our conventions for the quadrupolar shift, Knight shift and $1/T_1$ anisotropies in fig. 2.

As an illustration of eq. (4), the Knight shift tensor of our sample at 300 K amounts to [23]

$$ (K_{\parallel}, K_{\perp}, K_c)(300 \text{ K}) \approx (0.252, 0.143, 0.128)(\%) , \qquad (5) $$

which can be decomposed as:

$$ (K_{\parallel}, K_{\perp}, K_c)(300 \text{ K}) = 0.1743 (1, 1, 1) + 0.0413 (2, -1, -1) + 0.005 (-1, 2, -1), \qquad (6) $$

<figure><img src="image_4.png" /><figcaption>Fig. 2. Conventions for the quadrupolar shift, Knight shift and $1/T_1$ anisotropies. (a) $\varphi=(H_0, c)=0^\circ$, (b) $\varphi=(H_0, c)=90^\circ$, with $H_0\|a$ or $H_0\|b$, respectively, for the two twin domains of the crystal.</figcaption></figure>
and where the last term on the right side describes the slight departure from axial symmetry. A priori, we may decompose $K_{iso}(T)$ as:

$$K_{iso}(T) = K_{iso Cu}(T) + K_{iso O(2s,3s)}(T) + K_{iso core polarisation O(1s, 2s)/O(2p)}(T), \quad (7)$$

where $K_{iso Cu}(T)$ comes from the hybridisation of the Cu(3d, $x^2-y^2$) orbitals with the O(2s, 3s) orbitals, $K_{iso O(2s,3s)}(T)$ from some non-zero density of states directly based on the O(2s, 3s) orbitals, and $K_{iso core polarisation O(1s, 2s)/O(2p)}(T)$ from the core polarisation of the O(1s, 2s) orbitals by the O(2p) orbitals.

In fact, $K_{iso core polarisation O(1s, 2s)/O(2p)}(T)$ is quite negligible. We known from the literature [28] that the core hyperfine field $A_{iso core polarisation O(1s, 2s)/O(2p)}$ can be expected to be one order of magnitude smaller than the dipolar hyperfine field $A_{ax O(2p)}$ coming from the O(2p) orbital itself, which should imply that $K_{iso}(T) \ll K_{ax}(T)$ if $K_{iso core-polarisation O(1s, 2s)/O(2p)}(T)$ were the most important contribution to $K_{iso}(T)$. However, the experimental value for $K_{iso}(T)$ is even larger than that for $K_{ax}(T)$. Our results for the $1/T_1$ anisotropy, which strongly diminishes at low temperatures, indirectly confirm that $K_{iso core polarisation O(1s, 2s)/O(2p)}(T)$ is very small. This is because the core relaxation must always have the same temperature dependence as the relaxation coming from the O(2p) orbitals, thus leaving the anisotropy of $1/T_1$ unchanged.

Thus, $K_{iso}(T)$ is only attributable to $K_{iso Cu}(T)$ or to $K_{iso O(2s, 3s)}(T)$. One may also imagine, but quite less tentatively, that a small part of $K_{iso}(T)$ comes from the hybridisation of the four nearest-neighbouring O(2p) orbitals with the O(2s, 3s) orbitals.

$K_{ax}(T)$, on the other hand, originates mainly from the dipolar interaction of the O(2p) orbitals with the oxygen nuclei. An additional dipolar contribution, however, is provided by the neighbouring Cu(3d) orbitals. Both contributions have the same symmetry. In a first step, we must therefore formally divide $K_{ax}(T)$ into two parts:

$$K_{ax}(T) = K_{ax Cu}(T) + K_{ax O(2p)}(T). \quad (8)$$

The contribution $K_{ax Cu}(T)$ has been neglected by other authors. Their approximation, however, needs to be taken more carefully. Assuming that the Cu(3d) orbitals are sufficiently far away from the oxygen atoms, we can approximate the distance between these orbitals and the O nuclei by the Cu–O distance, thus writing:

$$\langle 1/r_{\text{elect(Cu3d)-nuclear(17O)}}^3 \rangle \approx 1/r_{\text{nucleus(Cu)-nucleus(17O)}}^3 \approx 0.02 \text{ a.u.}, \quad (9)$$

which is to be compared to the value estimated for instance by Yoshinari et al. [6]:

$$\langle 1/r_{\text{elect(O2p)-nucleus(17O)}}^3 \rangle \approx 3.72 \text{ a.u.}, \quad (10)$$

The difference between eqs. (9) and (10) illustrates how magnetic electron/nucleus interactions are short ranged in general. However, the smallness of $\langle 1/r_{\text{elect(Cu3d)-nucleus(17O)}}^3 \rangle$ is partly compensated by the difference between the spin density on the Cu(3d) orbitals and the O(2p) orbitals, since experimental results [6] have established that the spin density on copper is nearly 20 times larger than what it is on the oxygen. On the whole, we therefore estimate that the contribution of the Cu(3d) orbitals to $K_{ax}(T)$ is only five times smaller than the contribution of the O(2p) orbitals themselves, and that

$$A_{ax Cu} \gtrsim \frac{1}{2} \left( \frac{1}{3} A_{ax O(2p)} \right) \quad (11)$$

(each oxygen has two copper neighbors).

The actual contribution of $A_{ax Cu}$ to $K_{ax}$ is presumably even larger, because of the rough approximation performed in eq. (9).

We guess, however, that certainly $K_{ax Cu}(T) < K_{ax O(2p)}(T)$. This is justified by the weak anisotropy of the oxygen $1/T_1$ at low temperatures. The details of the explanation will be given later. We may also mention that the present understanding of the yttrium Knight-shift data requires that $K_{ax O(2p)}(T)$ be rather strong [29,30].

Horvatić, Berthier et al. [27,7] were the first to discuss the interpretation of the Knight-shift anisotropy on
the O(2, 3) planar sites. These authors have devoted particular attention to the temperature dependence of this quantity, and their original motives may be summarised essentially as follows.

If the electronic system for the planes could be described by a “one-fluid” system, a single susceptibility $\chi(q, \omega, T)$ could be ascribed to the latter. Thus $K_{\text{iso}}(T)$ and $K_{\text{ax}}(T)$, which would merely represent two different microscopic probes of the same single susceptibility $\chi(0, 0, T)$, could be scaled linearly one against the other as a function of temperature. As a matter of fact, shifts of orbital origin also contribute in a small proportion to $K_{\text{iso}}(T)$ and $K_{\text{ax}}(T)$; however, because this orbital contribution is usually considered to be temperature independent, the “one-fluid” picture actually implies that we should find a way to determine two temperature-independent parameters $\alpha$ and $\beta$ such that:

$$K_{\text{iso}}(T) = \alpha K_{\text{ax}}(T) + \beta, \qquad (12)$$

$\beta \approx (K_{\text{iso orbital}} - \alpha K_{\text{ax orbital}})$ being a very small quantity. Our conventions are

$$K_{\text{iso}}(T) = \frac{1}{3}(K_{\parallel}(T) + 2K_{\perp}(T)) \qquad (13)$$

and

$$K_{\text{ax}}(T) = \frac{1}{3}(K_{\parallel}(T) - K_{\perp}(T)). \qquad (14)$$

If, on the contrary, the “one-fluid” picture were not valid, the idea of Horvatić et al. [27] was that the O(2p) orbitals should possess their own susceptibility $\chi_h(q, \omega, T)$ which would be somewhat independent from the main susceptibility $\chi(q, \omega, T)$ coming from the copper atoms. Hence no simple scaling such as that of eq. (12) should be expected in that case.

The temperature dependences of $K_{\text{iso}}(T)$ and $K_{\text{ax}}(T)$ have been studied experimentally by Takigawa et al. [5], Berthier et al. [7] and Yasuoka et al. [6] in materials for which the macroscopic susceptibility presents a strong temperature variation.

The experimental determination of $K_{\text{iso}}(T)$ and $K_{\text{ax}}(T)$ from oriented powders is not an easy matter, and indeed the results of Berthier et al. for $K_{\text{ax}}(T)$, which do not support the “one-fluid” picture, completely differ from those of the other authors. The latter evidence a very good scaling between $K_{\text{iso}}(T)$ and $K_{\text{ax}}(T)$ for temperatures ranging from $T_c$ to 300 K.

We now present our Knight shift data. Our own determination of $K_{\text{iso}}(T)$ and $K_{\text{ax}}(T)$ has been greatly facilitated by the use of a single crystal instead of oriented powders.

In fig. 3, we first plot $v_{\parallel}(T)$, with:

$$|v_{\parallel}(T)| \approx \|v_{\text{exp}\parallel(3/2,1/2)}\| - |v_{\text{exp}\parallel(-1/2,-3/2)}(T)\|/2. \qquad (15)$$

The precision of these measurements is very poor because the satellite lines are very broad (ca. 50 kHz);

<figure><img src="image_5.png" /><figcaption>Fig. 3. Temperature dependence of $v_{\parallel}(T)$ for the planar oxygen sites, measured on the (3/2, 1/2) and (-1/2, -3/2) NMR transitions.</figcaption></figure>
nevertheless, we can estimate that $v_‖$ decreases roughly by 2% between room temperature and 60 K. The variation of $v_⊥(T)$ and $v_c(T)$ has been found to be of the same order of magnitude; this fact is not surprising because the trace of the quadrupolar tensor is zero [31] i.e.

$$v_‖(T) + v_⊥(T) + v_c(T) = 0. \qquad (16)$$

The temperature variation of these quadrupolar eigenvalues is so small that it does not affect the study of the NMR shifts for the central planar transitions, since the second-order quadrupolar shifts of these (1/2, -1/2) transitions are already nearly ten times smaller than the Knight shifts at 300 K, when the magnetic field of the experiment is 9.3 T.

We thus focus on $K_‖(T)$ and $K_⊥(T)$ (fig. 4), which we have measured with the crystal oriented as in fig. 2(b). Even at 100 K (fig. 5), the two NMR lines corresponding to the parallel and perpendicular orientations could be clearly distinguished.

As a result, we find that we have nearly (see fig. 4)

$$(K_‖(T) + 2K_⊥(T)) = 5 (K_‖(T) - K_⊥(T)). \qquad (17)$$

thus following relation (12), with $α=5$ and $β=0$.

This is an additional confirmation of the results obtained by Takigawa et al. [5] and Yoshinari et al. [6].

The scaling between $K_{iso}(T)$ and $K_{ax}(T)$ is very accurate indeed, and apparently supports the “one-fluid” picture.

<figure><img src="image_6.png" /><figcaption>Fig. 4. ($K_‖(T)$ and $K_⊥(T)$ for the planar oxygen sites, measured on the (1/2, -1/2) transitions with the crystal placed as in fig. 2(b). We also include $K_{iso}$ and $5K_{ax}$ in the drawing, where $K_{iso} = \frac{1}{3}(K_‖(T) + 2K_⊥(T))$ and $K_{ax} = \frac{1}{3}(K_‖(T) - K_⊥(T))$. Lines are guides to the eye.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Central line spectrum of the planar oxygen sites at 100 K. The signals corresponding to $K_‖(T)$ and $K_⊥(T)$ give rise to the observed splitting. A very small signal corresponding to the oxygen apical sites (which relax too slowly for our repetition rate) gives rise to the observed shoulder on the right side of the spectrum.</figcaption></figure>
Now, let us examine our $1/T_1$ measurements, and see if they confirm the same theoretical view.

## 3.2. Relaxation processes

Our $1/T_1$ measurements come from an inversion-recovery sequence, fitted respectively for the central, first satellite and second satellite transitions by the different formulas [33]

$$ (M_z(\infty) - M_z(t))/M_z(\infty) \propto 2/35e^{-2Wt} + 16/45e^{-12Wt} + 100/63e^{-30Wt}, \quad (18) $$

$$ (M_z(\infty) - M_z(t))/M_z(\infty) \propto 1/35e^{-2Wt} + 3/56e^{-6Wt} + 1/40e^{-12Wt} + 25/56e^{-20Wt} + 25/65e^{-30Wt}, \quad (19) $$

$$ (M_z(\infty) - M_z(t))/M_z(\infty) \propto 1/35e^{-2Wt} + 3/14e^{-6Wt} + 2/5e^{-12Wt} + 2/7e^{-20Wt} + 1/14e^{-30Wt}. \quad (20) $$

Here, W is equal to $1/2T_1$.

Such equations correspond to the case where the relaxation is purely magnetic. Quite different relations can be expected in general if the origin of the relaxation is quadrupolar [34].

At room temperature, we have obtained (fig. 14):

$$ (1/T_{1\parallel}, 1/T_{1\perp}, 1/T_{1c}) (300 \text{ K}) = (62, 100, 94) \text{ s}^{-1}. \quad (21) $$

For the $1/T_{1\perp}$ (300 K) value, we measured all three nuclear transitions ($-\frac{5}{2}, -\frac{3}{2}$), ($-\frac{3}{2}, -\frac{1}{2}$), ($-\frac{1}{2}, \frac{1}{2}$) and we found that the experimental $1/T_1$ values derived from our fits were all the same within only 2% error. The ($-\frac{5}{2}, -\frac{3}{2}$) transition was the most difficult to measure, and nearly two days of signal accumulation were necessary to obtain the required accuracy. We show the signals for this ($-\frac{5}{2}, -\frac{3}{2}$) transition in fig. 6 and the corresponding fit in fig. 7. Let us emphasize that for a magnetic relaxation mechanism, the satellites relax much more slowly than the central line, which is what observe. This only happens in some fortuitous cases with a quadrupolar relaxation process [33]. So the fact that the computer fits performed on all the transitions gave similar results fully supports the idea that our planar oxygen relaxation is purely magnetic. Also the fact that the relaxation of the O(4) apical sites measured by Hammel et al. [35] on YBa$_2$Cu$_3$O$_7$ is more than one order of magnitude smaller than our planar $1/T_1$ values is quite reassuring: this is easiest to understand in terms of the magnitude of the hyperfine coupling, because the O(4) Knight shift is very small if compared to the Knight shifts of the planar oxygen sites.

A similar agreement of 2% has been obtained between the ($-\frac{3}{2}, -\frac{1}{2}$) and ($-\frac{1}{2}, \frac{1}{2}$) transitions for $1/T_{1\parallel}$. One might argue that because $1/T_{1\parallel}$ is smaller than $1/T_{1\perp}$, quadrupolar relaxation processes could contribute to a significant part of $1/T_{1\parallel}$ more easily than for $1/T_{1\perp}$. However, the difference between $1/T_{1\perp}$ and $1/T_{1\parallel}$ is not that large, and at any rate we can be sure that the difference between the magnetic contributions to $1/T_{1\parallel}$ and $1/T_{1\perp}$ is at least as big as the difference we effectively measure between $1/T_{1\parallel}$ and $1/T_{1\perp}$ (if, as already stated, $1/T_{1\perp}$ is purely magnetic).

So we are confident that the anisotropy of the magnetic $1/T_1$ in the CuO$_2$ planes follows very closely the ratio

$$ \frac{1/T_{1\perp} - 1/T_{1\parallel}}{1/T_{1/vq}} (300 \text{ K}) = 61\%. \quad (22) $$

This large $1/T_1$ anisotropy at room temperature is one of the most important results of our study, since we shall see very shortly that it is very difficult to justify theoretically.

In principle, we may derive the anisotropy of $1/T_1$ from the anisotropy of the hyperfine tensors. In order to understand how we can do this, we shall first examine the simpler case of a single 2p orbital coupled by dipolar coupling to an oxygen nucleus. If we call $\theta$ the angle between this 2p orbital and the magnetic field $H_0$ (fig. 8), the dipolar coupling may be written as [31]
<figure><img src="image_8.png" /></figure>

Fig. 6. Inversion-recovery NMR sequence applied on the second-satellite transition of one of the oxygen planes (giving 1/T₁⊥) at 300 K. The crystal is oriented as in fig. 2(b). The irradiation frequency ν is roughly νₐₗ ≈ (1+Kₐ)ν₀ + 2|νₐ|, where ν₀ is the resonance frequency of H₂¹⁷O in our field. νₐₗ thus corresponds to the middle of the peak under study. The right sides of all the displayed spectra are affected by the incursion of a first-satellite NMR transition from the other oxygen planar site, whose frequency is ν ≈ (1+Kₐ)ν₀ + |νₐ|. Electronic filtering has removed noise except on the central 40% of the plotted spectral width. This is enough to estimate our signal to noise ratio. Within the NMR sequence described in eq. (3), we used π/2=2.7 µs, τ' =24 µs, and the delay between each sequence was fixed at 12 ms. The variable recovery times τ are indicated in the figure. Each spectrum represents 600000 accumulations.

<figure><img src="image_9.png" /></figure>

Fig. 7. Three-parameter fit for 1/T₁⊥ (300 K), obtained with the signal shown in fig. 6 and formula (20). (a) linear scale fit, (b) the same data plotted on a semi-logarithmic scale.
<figure><img src="image_10.png" /><figcaption>Fig. 8. Definition of $\theta$ in eq. (24).</figcaption></figure>

$$ \mathcal{H}_d = \hbar\gamma_N A_{2p} \{D_0 + D'_0 + D_{-1} + D_1 + D_2 + D_{-2}\}, \quad (23) $$

where

$$ \begin{aligned} D_0 &= (3\cos^2\theta - 1)S_z I_z, & D'_0 &= -\frac{1}{4}(3\cos^2\theta - 1)(S_+ I_- + S_- I_+), \\ D_{\pm 1} &= \frac{3}{2}\sin\theta \cos\theta (S_{\pm} I_z + S_z I_{\pm}), & D_{\pm 2} &= \frac{3}{4}\sin^2\theta S_{\pm} I_{\pm}. \end{aligned} \quad (24) $$

The nuclear relaxation rate, which is twice the spin-flip probability per unit time, may thus be expressed in terms of local spin correlation functions as

$$ \frac{1}{T_1}(\theta) = 2\gamma_N^2 A_{2p}^2 \{[\frac{1}{4}(3\cos^2\theta - 1)]^2 J_+(\omega_N) + (\frac{3}{2}\sin\theta \cos\theta)^2 J_z(\omega_N) + (\frac{3}{4}\sin^2\theta)^2 J_-(\omega_N)\}, \quad (25) $$

where $\omega_N$ corresponds to the Larmor frequency of the oxygen nucleus in the magnetic field of the experiment, and where we have

$$ J_{\pm}(\omega) = \int_{-\infty}^{+\infty} \langle S_{\pm}(0)S_{\mp}(t) \rangle e^{i\omega t} dt $$

and

$$ J_z(\omega) = \int_{-\infty}^{+\infty} \langle S_z(0)S_z(t) \rangle e^{i\omega t} dt \quad (26) $$

The brackets $\langle \rangle$ in eq. (26) mean a statistical average [32].

For $\theta=0^\circ$ or $\theta=90^\circ$, we recover the more classical result [35] that $1/T_1$ is proportional to the squares of the hyperfine field eigenvalues which are perpendicular to the magnetic field $H_0$.

Equation (25) corresponds to the dynamics of the electrons in the magnetic field of the experiment. However, we are in fact interested in the dynamics of these electrons in zero magnetic field, and eq. (25) may be rewritten as
$$1/T_1(\theta) = 2\gamma_N^2 A_{2p}^2 \{ (\frac{1}{4}(3\cos^2\theta - 1))^2 J_{0\perp}(\omega_N - \omega_e) + (\frac{3}{2}\sin\theta \cos\theta)^2 (2J_{0\parallel}(\omega_N)) \\ + (\frac{3}{4}\sin^2\theta)^2 J_{0\perp}(\omega_N + \omega_e) \}, \quad (27)$$

where $J_{0\perp}$ and $J_{0\parallel}$ correspond to the electronic correlation functions in zero field. Because $\omega_N + \omega_e \approx \omega_e$ and $\omega_N - \omega_e \approx -\omega_e$, we may also simplify eq. (27) as:

$$1/T_1(\theta) \approx 2\gamma_N^2 A_{2p}^2 \{ [ (\frac{1}{4}(3\cos^2\theta - 1))^2 + (\frac{3}{4}\sin^2\theta)^2 ] J_{0\perp}(\omega_e) + (\frac{3}{2}\sin\theta \cos\theta)^2 (2J_{0\parallel}(\omega_N)) \}. \quad (28)$$

The distinction between $\omega_N$ and $\omega_e$ we made in eqs. (27) and (28), which is often ignored in the literature, might appear a little confusing: at first sight, one might be tempted to think that because the energy that is required to relax a nuclear spin is always $\hbar\omega_N$, $\omega_N$ should be the only frequency appearing in eq. (28). However, the frequencies needed in eq. (28) do not correspond to the nuclear spin-flip frequency, but rather to the change in the kinetic energy of the electrons involved in the relaxation. Whenever an electronic spin is flipped during the latter relaxation (which corresponds to the terms $J_\pm$ in eq. (25)), the kinetic energy of the corresponding electron needs to be changed by nearly $\hbar\omega_e$, whereas when the electronic spin is conserved (which corresponds to the term with $J_z$), the kinetic energy of the same electron is only changed by $\pm \omega_N$.

The appearance of the frequency $\omega_e$ in eq. (27) requires, however, at a minimum, that the electronic quasiparticles do effectively possess a well-defined Larmor frequency in the magnetic field $H_0$. This is not at all the case if these quasiparticles are similar to spin waves, or if their spin, by any chance, happens to be zero. On the other hand, $\omega_e$ in principle should represent a useful quantity if the quasiparticles consist of more conventional spin $\frac{1}{2}$ interacting fermions.

Now, the fluctuation-dissipation theorem allows for the interpretation of the spin correlation functions $J(\omega)$ in terms of a complex electronic susceptibility [32] $\chi = \chi' + i\chi''$, with:

$$\chi''_{\text{trans}}(\omega) = \omega\mu_B^2/k_B T \int_{-\infty}^{+\infty} \langle S_+ S_- \rangle e^{i\omega t} dt$$

and

$$\chi''_{\text{parall}}(\omega) = 2\omega\mu_B^2/k_B T \int_{-\infty}^{+\infty} \langle S_z S_z \rangle e^{i\omega t} dt \quad (29)$$

$\chi''_{\text{parall}}$ corresponds to the electronic fluctuations in the $z$ direction, and $\chi''_{\text{trans}}$ to the fluctuations in the other two directions.

It is also often convenient to convert eqs. (22–27) in the reciprocal lattice, which may be done easily with the Fourier transforms

$$S_{\pm}(\mathbf{k}, t) = 1/\sqrt{N} \sum_{\mathbf{R}} S_{\pm}(\mathbf{R}, t) e^{-i\mathbf{k}\mathbf{R}}$$

and

$$S_z(\mathbf{k}, t) = 1/\sqrt{N} \sum_{\mathbf{R}} S_z(\mathbf{R}, t) e^{-i\mathbf{k}\mathbf{R}} \quad (30)$$

so that eq. (25) may eventually be rewritten as

$$1/T_1(\theta) \approx \sum_{\mathbf{k}} 2\gamma_N^2 A_{2p}^2 k_B T / \mu_B^2 \{ [ (\frac{1}{4}(3\cos^2\theta - 1))^2 + (\frac{3}{4}\sin^2\theta)^2 ] \chi''_{\text{trans}}(\mathbf{k}, \omega_e) / \omega_e \\ + 2(\frac{3}{2}\sin\theta \cos\theta)^2 \chi''_{\text{parall}}(\mathbf{k}, \omega_N) / \omega_N \}. \quad (31)$$
Let us now come to the more elaborate expression of $1/T_1$ in our YBa$_2$Cu$_3$O$_{6.5}$ compound. In the frame of a “one electronic fluid”, we might write

$$ 
\begin{aligned}
\frac{1}{T_1(T, \theta)} ={}& \gamma_N^2 k_B T / 2 \mu_B^2 \sum_q \{ (A_{Cu\text{iso}}(\mathbf{q}))^2 |1 + \exp(i\mathbf{qa})|^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \\
& - 2A_{Cu\text{iso}}(\mathbf{q})A_{ax\text{Cu}}(\mathbf{q}) |1 + \exp(i\mathbf{qa})|^{2\frac{1}{2}} (3\cos^2\theta - 1) \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \\
& + (A_{ax\text{Cu}}(\mathbf{q}))^2 |1 + \exp(i\mathbf{qa})|^{2\frac{1}{4}} [(3\cos^2\theta - 1)^2 + 9\sin^4\theta] \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \\
& + (A_{ax\text{Cu}}(\mathbf{q}))^2 |1 + \exp(i\mathbf{qa})|^{2\frac{1}{4}} [18\sin^2\theta \cos^2\theta] \chi_{\text{parall}}''(\mathbf{q}, \omega_N, T) / \omega_N \\
& + (A_{ax\text{O}(2p)}(\mathbf{q}))^{2\frac{1}{4}} [(3\cos^2\theta - 1)^2 + 9\sin^4\theta] \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \\
& + (A_{ax\text{O}(2p)}(\mathbf{q}))^{2\frac{1}{4}} [18\sin^2\theta \cos^2\theta] \chi_{\text{parall}}''(\mathbf{q}, \omega_N, T) / \omega_N \}.
\end{aligned} 
\quad (32) $$

In particular, the results of eq. (32) for $\theta=0^\circ$ and $\theta=90^\circ$ may be simplified to

$$ 
\begin{aligned}
\frac{1}{T_1(T, 0^\circ)} ={}& \gamma_N^2 k_B T / 2 \mu_B^2 \sum_q \{ (A_{Cu\text{iso}}(\mathbf{q}))^2 |1 + \exp(i\mathbf{qa})|^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \\
& - 2A_{Cu\text{iso}}(\mathbf{q})A_{ax\text{Cu}}(\mathbf{q}) |1 + \exp(i\mathbf{qa})|^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \\
& + (A_{ax\text{Cu}}(\mathbf{q}))^2 |1 + \exp(i\mathbf{qa})|^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e + (A_{ax\text{O}(2p)}(\mathbf{q}))^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \}
\end{aligned} 
\quad (33) $$

and

$$ 
\begin{aligned}
\frac{1}{T_1(T, 90^\circ)} ={}& \gamma_N^2 k_B T / 2 \mu_B^2 \sum_q \{ (A_{Cu\text{iso}}(\mathbf{q}))^2 |1 + \exp(i\mathbf{qa})|^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \\
& + A_{Cu\text{iso}}(\mathbf{q})A_{ax\text{Cu}}(\mathbf{q}) |1 + \exp(i\mathbf{qa})|^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \\
& + \frac{5}{2}(A_{ax\text{Cu}}(\mathbf{q}))^2 |1 + \exp(i\mathbf{qa})|^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e + \frac{5}{2}(A_{ax\text{O}(2p)}(\mathbf{q}))^2 \chi_{\text{trans}}''(\mathbf{q}, \omega_e, T) / \omega_e \}.
\end{aligned} 
\quad (34) $$

In the frame of a “two-fluid” model, $1/T_1(T)$ should rather be expressed as the sum of two independent contributions,

$$ \frac{1}{T_1(T)} = \frac{1}{T_1(T)}\chi_1 + \frac{1}{T_1(T)}\chi_2, $$

where $1/T_1(T)\chi_1$ and $1/T_1(T)\chi_2$ should be written in the same way as $1/T_1(T)$ in eq. (32).

In eq. (32) we have supposed that $K_{\text{iso}}(T) = K_{\text{iso Cu}}(T)$, thus neglecting $K_{\text{iso O}(2s, 3s)}(T)$ and $K_{\text{iso core-polarisation O}(1s, 2p)/O(2p)}(T)$ (cf. eq. (7)). Hence $2A_{\text{Cu iso}}(O)$ corresponds to all the isotropic hyperfine field on the oxygen site. The reason to neglect $K_{\text{iso O}(2s, 3s)}(T)$ is not a priori straightforward. However, this matter is of little importance in the frame of the “one-fluid” model, nor is it important in the frame of the “two-fluid” model if, by any chance, $K_{\text{iso Cu}}(T)$ and $K_{\text{iso O}(2s, 3s)}(T)$ originate from the same fluctuations.

$\theta$ is the angle between the $\sigma_{\text{O}(2p)}$ orbital under study and the magnetic field $H_0$ (fig. 7).

(1 + exp(iqa)) is the form factor which filters the antiferromagnetic fluctuations coming from the copper atoms. We have applied no form factor to the fluctuations of the O(2p) orbitals. The reason is simply that our formalism must allow for the existence of antiferromagnetic fluctuations between the oxygen atoms themselves, and not just between the copper atoms. This point is essentially academic, because certainly for the high-$T_c$ superconductors the antiferromagnetism between the oxygen atoms does not exist, and we must reckon that it would be quite natural to say that as far as the O(2p) orbitals contribute to the super-exchange coupling between the copper atoms, the same form factor as for $A_{\text{Cu iso}}(\mathbf{q})$ and $A_{\text{ax Cu}}(\mathbf{q})$ should be applied to $A_{\text{ax O}(2p)}(\mathbf{q})$. However, we suggest that in this case the reader would rather consider the form factor to be implicitly included in $A_{\text{ax O}(2p)}(\mathbf{q})$.

It is important to note that we have put no cross term between $A_{\text{Cu iso}}(\mathbf{q})$ and $A_{\text{ax O}(2p)}(\mathbf{q})$, nor between $A_{\text{ax Cu}}(\mathbf{q})$ and $A_{\text{ax O}(2p)}(\mathbf{q})$ in eq. (32). This does not mean that any underlying coherence between these terms is nec-
essarily absent, but rather that we could not detect it anyway by NMR because our NMR spectroscopy has
merely access to an average of the electronic fluctuations over all the thermally accessible states: we cannot
observe each fluctuation taken separately. Let us examine this point more closely.

Basilly $1/T_1$ may always be written as a sum over “initial” $|i\rangle$ and “final” $|f\rangle$ states in a way such as:

$$
1/T_1 \propto \sum_{i,f} |\langle f | \mathcal{H}_{\text{int}} | i \rangle|^2, \qquad (35)
$$

where the sum over i and f should respect the statistics of the particles and also conservation of energy.

Now, thanks to the symmetric positions of the O(2p) orbitals in the YBa₂Cu₃O₇-δ lattice, we can see that for any planar quasiparticle |f⟩, it is possible to construct a very similar quasiparticle |f'⟩ with the same energy, by a simple symmetric inversion over all the sites. In fig. 9, because we have not inverted the signs in the O(2p) orbital itself in the drawing, the weighting of this orbital in |f⟩ and |f'⟩, which we call β<sub>f</sub> and β<sub>f'</sub>, must have the opposite sign:

$$
\beta_f = -\beta_{f'}. \tag{36}
$$

On the contrary, the weightings of the O(2s) orbital do not change:

$$
\alpha_f = \alpha_{f'}. \tag{37}
$$

The change of sign for $\beta$ comes from the odd symmetry of the 2p orbital with respect to its main axis.

Now, the Hamiltonian for $|f\rangle$ and $|f'\rangle$ can be decomposed as

$$
\mathcal{H} = \mathcal{H}_0 + \mathcal{H}_c + \mathcal{H}_d, \tag{38}
$$

where $\mathcal{H}_0$ is the sum of kinetic and electrostatic energies, and $\mathcal{H}_c$ and $\mathcal{H}_d$ the contact and dipolar interactions with the $^{17}$O nucleus, respectively.

Thus

$$
1/T_1(^{17}\text{O}) \propto \sum_{i,f} \{ |\langle f | \mathcal{H}_c | i \rangle|^2 + |\langle f | \mathcal{H}_d | i \rangle|^2 + \langle i | \mathcal{H}_c | f \rangle \langle f | \mathcal{H}_d | i \rangle + \langle i | \mathcal{H}_d | f \rangle \langle f | \mathcal{H}_c | i \rangle \}, \quad (39)
$$

where the cross-terms may also be rewritten as:

$$
\sum_{i,f} \{ \langle i | \mathcal{H}_c | f \rangle \langle f | \mathcal{H}_d | i \rangle + \langle i | \mathcal{H}_d | f \rangle \langle f | \mathcal{H}_c | i \rangle \} = \frac{1}{2} \sum_{i,f} \{ \langle i | \mathcal{H}_c | f \rangle \langle f | \mathcal{H}_d | i \rangle + \langle i | \mathcal{H}_d | f \rangle \langle f | \mathcal{H}_c | i \rangle \} \\
+ \frac{1}{2} \sum_{i,f} \{ \langle i | \mathcal{H}_c | f' \rangle \langle f' | \mathcal{H}_d | i \rangle + \langle i | \mathcal{H}_d | f' \rangle \langle f' | \mathcal{H}_c | i \rangle \}. \quad (40)
$$

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 9. Diagram for the oxygen/copper coherent effects cancellation in formula (32). The drawings correspond to two quasiparticles $|f\rangle$ and $|f'\rangle$, which we have arbitrarily chosen to extend over 11 copper sites only. $|f\rangle$ and $|f'\rangle$ have the same energy, and also the same density probability on corresponding copper site numbers.</figcaption></figure>
Because of eqs. (36) and (37) we have

$$ \mathcal{H}_c |f\rangle = \mathcal{H}_c |f'\rangle, \qquad (41) $$

where

$$ \mathcal{H}_d |f\rangle = -\mathcal{H}_d |f'\rangle, \qquad (42) $$

and eq. (40) is equal to zero.

Mathematically, this is just equivalent to saying that: $(x+y)^2 + (x-y)^2 = 2x^2 + 2y^2$.

The same argument would hold for the core relaxation of the planar oxygen sites, if we had not neglected it: no coherent terms would appear between this relation and any of the other processes. Such considerations about coherent and non-coherent effects are much along the lines of the work of Yafet and Jaccarino on core relaxation mechanisms in transition metals [37].

A posteriori, the analysis of the anisotropy of our room temperature $1/T_1$ data tells us that the orbital relaxation mechanism is negligible. This is quite an important statement, because orbital processes are known to be sometimes much more efficient than dipolar mechanisms for relaxing nuclei [38].

We know quite generally [39] that orbital fluctuations always involve the transition between two adjacent levels of the operator $L \cdot H_0$, where $L$ is the electronic orbital momentum and $H_0$ is the magnetic field of the experiment. In the case of 2p orbitals, the transitions between the orbital state $m=0$ and the states $m = \pm 1$ are the only ones permitted.

When the electronic quasiparticles are fermions with kinetic momentum wavevectors $k$, the orbital relaxation may read [39]:

$$ 1/T_{1\text{orb}} = (4\pi/\hbar)(g\beta\gamma_N\hbar)^2 \sum_{k,k'} \frac{1}{4} |\langle\psi_k|L_+/r^3|\psi_{k'}\rangle|^2 \delta(E_k - (E_{k'} - \hbar\omega_N)) n(E_k) (1 - n(E_{k'})). \qquad (43) $$

In the $1/T_{1\parallel}$ configuration, the $\sigma_{O(2p)}$ orbital under study is in the $m=0$ state, whereas in the $1/T_{1\perp}$ and $1/T_{1c}$ configurations this orbital is half in the $m=-1$ state, and half in the $m=1$ state [36]. Thus, if this $\sigma_{O(2p)}$ orbital is the only O(2p) orbital which is significantly occupied at the Fermi level, $1/T_{1\text{orb}}$ is equal to zero in the $1/T_{1\parallel}$, $1/T_{1\perp}$ and $1/T_{1c}$ configurations. We known already, from the near axial symmetry of the oxygen Knight shift (see eq. (5)), that this should not be far from the truth. If, however, any of the two other O(2p) planar orbitals, which we shall call O(p)' and O(2p)", are sparsely populated at the Fermi level, the orbital relaxation will not be strictly zero in the $1/T_{1\parallel}$, $1/T_{1\perp}$ and $1/T_{1c}$ configurations.

We have in fact the only following possibilities:

<table><thead><tr><th></th><th>σ<sub>O(2p)</sub></th><th>O(2p)'</th><th>O(2p)"</th></tr></thead><tbody><tr><td>configuration 1/T<sub>1||</sub></td><td>(m=0)</td><td>(m=+1 and m=-1)</td><td>(m=+1 and m=-1)</td></tr><tr><td>configuration 1/T<sub>1⊥</sub></td><td>(m=+1 and m=-1)</td><td>(m=+1 and m=-1) or (m=0)</td><td>(m=0) or (m=+1 and m=-1)</td></tr><tr><td>configuration 1/T<sub>1c</sub></td><td>(m=+1 and m=-1)</td><td>(m=0) or (m=+1 and m=-1)</td><td>(m=+1 and m=-1) or (m=0)</td></tr></tbody></table>

Thus it can be seen that the orbital contribution to the oxygen relaxation must be at a maximum in the $1/T_{1\parallel}$ configuration, and not at a minimum.

Furthermore, it is very likely that its contribution to both $1/T_{1\perp}$ and $1/T_{1c}$ is effectively quite different, (i.e., if O(p)' and O(2p)" are differently occupied at the Fermi level), whatever its order of magnitude, whereas the experimental values for $1/T_{1\perp}$ (300 K) and $1/T_{1c}$ (300 K) are nearly the same.

Incidentally, the preceding arguments also prove that the theory of Ikeda [40], proposing that an oxygen $2p\pi$ band should exist at the Fermi level, is certainly not accurate, precisely because the electronic density on any of the oxygen $2p\pi$ orbitals is necessary extremely low at the Fermi level.
Now, if the “one fluid” model (cf. eq. (32)) were correct, the numerical values for the hyperfine tensor that we possess from the oxygen Knight shift would allow us to calculate the anisotropy of $1/T_1$. We shall first suppose that $A(\mathbf{q})$ is proportional to $A(\mathbf{0})$ for any $\mathbf{q}$ value. We shall also approximate $(1+\exp(i\mathbf{qa}))$ by 2 (i.e. $\mathbf{q}\approx\mathbf{0}$), except for those $\mathbf{q}$ values for which $\chi''(\mathbf{q}, \omega)$ exhibits a strong peaking, namely in the vicinity of $(\pi, \pi)$, where we shall take $(1+\exp(i\mathbf{qa}))=0$. Also, we shall neglect $A_{\text{ax Cu}}(\mathbf{q})$, and we shall take $\chi''_{\text{trans}}=\chi''_{\text{parall}}$ in eq. (32).

From eq. (32), we thus obtain:

$$ 
\begin{aligned}
& \frac{1/T_{1\perp} - 1/T_{1\parallel}}{1/T_{1\parallel}} (T) \\
&= \frac{\{[(2A_{\text{ax O}(2p)})^2 + (2A_{\text{iso}})^2] + [A_{\text{ax O}(2p)}^2 + (2A_{\text{iso}})^2]\} - \{[A_{\text{ax O}(2p)}^2 + (2A_{\text{iso}})^2] + [A_{\text{ax O}(2p)}^2 + (2A_{\text{iso}})^2]\}}{2(A_{\text{ax O}(2p)})^2 + 2(2A_{\text{iso}})^2} \\
&= \frac{3A_{\text{ax O}(2p)}^2}{2(A_{\text{ax O}(2p)})^2 + 2(2A_{\text{iso}})^2} = \frac{3K_{\text{ax}}^2}{2(K_{\text{ax}})^2 + 2(K_{\text{iso}})^2} = 5.8\%.
\end{aligned}
\quad (44) $$

The result is temperature independent since all the Knight shift components show the same temperature behaviour.

However, the assumption that: $(1+\exp(i\mathbf{qa}))\approx 2$ for all the pertinent $\mathbf{q}$ values is not necessarily valid. For instance, in a magnetically very localised picture where all the copper spins fluctuate incoherently (i.e., the magnetic fluctuations are $\mathbf{q}$ independent), it can be seen that the relaxation induced by the isotropic hyperfine field $2A_{\text{Cu iso}}$ should read, in the real space:

$$ 1/T_{1\text{Cu iso}}(T) = \gamma_{\text{N}}^2/2(A_{\text{Cu iso}}^2 + A_{\text{Cu iso}}^2) \int_{-\infty}^{+\infty} \langle S_+(\text{Cu}, 0)S_-(\text{Cu}, t) \rangle e^{i\omega t} dt, \quad (45) $$

instead of

$$ 1/T_{1\text{Cu iso}}(T) = \gamma_{\text{N}}^2/2(2A_{\text{Cu iso}}^2) \int_{-\infty}^{+\infty} \langle S_+(\text{Cu}, 0)S_-(\text{Cu}, t) \rangle e^{i\omega t} dt \quad (46) $$

for coherent fluctuations. So the result for incoherent fluctuations would change eq. (44) into

$$ 
\begin{aligned}
& \frac{1/T_{1\perp} - 1/T_{1\parallel}}{1/T_{1\parallel}} (T) \\
&= \frac{\{[(2A_{\text{ax O}(2p)})^2 + (2A_{\text{iso}})^2] + [A_{\text{ax O}(2p)}^2 + (2A_{\text{iso}})^2]\} - \{[A_{\text{ax O}(2p)}^2 + (2A_{\text{iso}})^2] + [A_{\text{ax O}(2p)}^2 + (2A_{\text{iso}})^2]\}}{2(A_{\text{ax O}(2p)})^2 + 2(2A_{\text{iso}})^2} \\
&= 11.1\%.
\end{aligned}
\quad (47) $$

This is still very far from the room temperature experimental ratio (which is 61%).

In an additional effort to increase the expected theoretical ratio, one might suppose that each neighbouring Cu(3d) orbital, and also all four neighbouring O(2p) orbitals would contribute equally to $K_{\text{iso}}$, and also that all these orbitals would fluctuate incoherently one from each other. Nevertheless, this would only increase the ratio up to:

$$ \frac{1/T_{1\perp} - 1/T_{1\parallel}}{1/T_{1\parallel}} = 29\% \quad (48) $$

which is still a long way from the experimental finding.
Our last attempt to explain the experimental anisotropy of the oxygen 1/T₁ at room temperature in the frame of the “one-fluid” model will be to attribute it to the anisotropy of χ''.

We have in fact already suggested in a former publication [23] that the electronic susceptibility might well be anisotropic in high fields if it were in any way connected to T_c, because T_c itself is anisotropic in high fields. However, the remark is irrelevant in the present case because 1/T₁⊥ (300 K) and 1/T₁‖ (300 K) have been measured with the same orientation of the crystal, namely that of fig. 2(b).

We should also expect $\chi''(q, \omega, T)$ to be anisotropic for two supplementary reasons:

- firstly, the spin orbit coupling on the oxygen site. This contribution has the symmetry of the $\sigma_{O(2p)}$ orbital. However, it is expected to be very weak, because the oxygen atom is a very small atom.

- secondly, the spin orbit coupling on the copper sites, together with the dipole-dipole interaction between the Cu(3d) planar electrons. This contribution has the symmetry (in plane/out plane), and is expected to be less negligible than the former (in particular, it is responsible for the specific direction of the copper spins in the antiferromagnetic phase of YBa₂Cu₃O₇-δ when δ is close to one). The two limiting cases would be:

$$ \chi''(q, \omega, T)_{\text{out plane}} = \chi''(q, \omega, T)_{\text{in plane}}, \qquad (49) $$

where the coupling between the copper planar spins would be of the Heisenberg type, and

$$ \chi''(q, \omega, T)_{\text{out plane}} \ll \chi''(q, \omega, T)_{\text{in plane}}, \qquad (50) $$

where the coupling between the copper planar spins would be of the XY type.

However, it is easy to see that even in the case of eq. (50), we still cannot account for the 1/T₁ anisotropy at room temperature. In order to prove this, we shall keep the other assumptions used to obtain formula (44) (namely: A(q) proportional to A(0) for any q value, (1+exp(iqa)) ≈ 2 or (1+exp(iqa)) ≈ 0 for all the q values concerned, and also A_{ax Cu}(q) ≈ 0). These assumptions are quite reasonable. Then we write

$$ \frac{1/T_{1\perp} - 1/T_{1\parallel}}{1/T_{1\parallel}} (300 \text{ K}) = \frac{\{[(2A_{\text{ax}})^2 + A_{\text{iso}}^2]\} - \{[A_{\text{ax}}^2 + A_{\text{iso}}^2]\}}{A_{\text{ax}}^2 + A_{\text{iso}}^2} = 11.6\%, \qquad (51) $$

which is again very far from the experimental value (61%).

Let us make some final comments on the values of 1/T₁⊥, 1/T₁⊥ and 1/T₁c at 300 K. As we notice, 1/T₁⊥ (300 K) is slightly larger than 1/T₁c (300 K). We might attribute this difference to the anisotropy of the g factor, or to supplementary dipolar relaxation via some other O(2p) orbital than the O(2p) σ orbital. However, if we consider the slight departure from axial symmetry of the central-line magnetic shift tensor and if we suppose that it is metallic in origin (cf. eq. (6)), we must expect that 1/T₁⊥ < 1/T₁c, instead of 1/T₁⊥ > 1/T₁c. Thus, more tentatively, the difference between 1/T₁⊥ (300 K) and 1/T₁c (300 K) may be attributed to either the anisotropy of the electronic correlation functions in the magnetic field, or to some orbital contribution to the relaxation. Both latter interpretations allow us to understand once more that the orders of magnitude they involve are just that small.

Now we come to the analysis of the temperature dependence of 1/T₁. Below room temperature, the 1/T₁ acquisitions were performed on the first satellite oxygen planar transitions only, which, thanks to quadrupolar splitting, gave us the possibility to study separately 1/T₁⊥ and 1/T₁‖. We show the signal of these two satellites at 55 K in fig. 10.

What about the experimental temperature dependence of the ratio $(1/T_{1\perp} - 1/T_{1\parallel})/(1/T_{1\parallel})$ (T)?

We see in fig. 11 that this ratio decreases monotonically when we lower the temperature, to reach a value (≈10%) quite similar to the expected ones with either coherent or incoherent fluctuations at temperatures close to T_c.

The weak anisotropy of the 1/T₁ observed at low temperatures is interesting because it allows us to justify the assumption previously made that the contribution of A_{ax Cu} to the oxygen relation is not so important (see in particular eqs. (11) and (32)). We can estimate an upper limit for A_{ax Cu} by simply supposing that A_{ax O(2p)} (55 K) = 0, thus calculating the value of A_{ax Cu} that accounts for all the experimental anisotropy of 1/T₁ at 55 K.
<figure><img src="image_13.png" /><figcaption>Fig. 10. First-satellite spectra used to measure the 1/T₁ anisotropy at 55 K.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 11. Temperature dependence of the 1/T₁ anisotropy. We plot the quantity 1/T₁⊥(T) - 1/T₁‖(T)/1/T₁‖(T) vs. the temperature. Lines are guides to the eye.</figcaption></figure>

Let us also suppose again that the assumptions used to derive eq. (44) are valid. We thus first write

$$ \frac{1/T_{1\perp} - 1/T_{1\parallel}}{1/T_{1\parallel}} (55 K) = \frac{\{(2A_{\text{iso}} + 4A_{\text{ax Cu}})^2 + (2A_{\text{iso}} - 2A_{\text{ax Cu}})^2\} - \{(2A_{\text{iso}} - 2A_{\text{ax Cu}})^2 + (2A_{\text{iso}} - 2A_{\text{ax Cu}})^2\}}{2(2A_{\text{iso}} - 2A_{\text{ax Cu}})^2} \quad (52) $$

Assuming that the latter ratio is less than 10% at 55 K then leads to the result

$$ A_{\text{ax Cu}} \lesssim \frac{1}{32} A_{\text{iso}}. \quad (53) $$

In fact we could already estimate from eq. (11) that

$$ A_{\text{ax Cu}} \gtrsim \frac{1}{30} A_{\text{iso}}, \quad (54) $$

which we obtain from eqs. (11) and (17), respectively, $A_{\text{ax Cu}} \gtrsim \frac{1}{10} A_{\text{ax O(2p)}}$ since $2A_{\text{ax Cu}} + A_{\text{ax O(2p)}} \simeq \frac{1}{5}(2A_{\text{iso}})$.

Both inequalities eqs. (53) and (54) are slightly contradictory. However, the uncertainties of the 1/T₁ interpretation, together with the imprecision of the experiment, may easily account for an even larger misfit. On the whole we can say that presumably $A_{\text{ax Cu}}$ has just the order of magnitude given by both eqs. (53) and (54).

Consequently, the contribution of $A_{\text{ax Cu}}$ to the oxygen 1/T₁ is not quite negligible. This comes only from the cross terms between $A_{\text{ax Cu}}$ and $A_{\text{iso}}$ in eq. (32), since $A_{\text{iso}} \gg A_{\text{ax Cu}}$. At high temperatures, however, the experimental anisotropy of the oxygen 1/T₁ increases, so that the contribution of $A_{\text{ax Cu}}$ becomes less important. Let us bear in mind that the temperature dependence of $K_{\text{ax Cu}}$ is the same as that of $K_{\text{iso Cu}}$, so that the contribution of $A_{\text{ax Cu}}$ to the anisotropy ratio $(1/T_{1\perp} - 1/T_{1\parallel})/(1/T_{1\parallel}) (T)$ cannot exceed its upper estimate at 55 K (which is 10%), even at room temperature.

Thus, for the sake of simplicity, and as already done in eq. (44) in particular, $A_{\text{ax Cu}}$ with be neglected for all temperatures through the rest of our paper. This approximation is basically incorrect at temperatures close to $T_c$, but even in that particular region the qualitative understanding of the temperature dependence of the 1/T₁ anisotropy cannot be really affected.

The most naive way to account for the temperature dependence of the 1/T₁ anisotropy is to forget the scaling between $K_{\text{iso}}(T)$ and $K_{\text{ax}}(T)$, and say that for temperatures higher than $T_c$, we have found evidence for some new, rather unsuspected electronic fluctuations which develop on the oxygen sites and not the copper sites; in other words, at temperatures close to $T_c$ the “one-fluid” picture is essentially correct, whereas at temperatures
higher than $T_c$, only the "two-fluid" picture is correct. Let us go on with this simple scheme for a while. We may also suppose that $\chi_1(T)$ alone accounts for the Knight shifts of all the nuclei involved, and that $\chi_2(T)$ is essentially based on the oxygen atoms, even if, for some reason that we ignore, its contribution to $^{17}$O $K_{ax}(T)$ is too small to be detected. We shall also suppose that formula (44) accounts correctly for the anisotropy of $1/T_1(\chi_1)$.

We do not want to justify these assumptions at present, but rather want to show that the results we deduce from them are interesting.

In our case, $\chi_1(T)$ leads to a nearly isotropic relaxation

$$1/T_{1\perp}(\chi_1) = 1.058 \times 1/T_{1\parallel}(\chi_1)(T), \qquad (55)$$

whereas $\chi_2(T)$ leads to a relaxation for which

$$1/T_{1\perp}(\chi_2)(T) = \frac{5}{2} \times 1/T_{1\parallel}(\chi_2)(T), \qquad (56)$$

as can be expected for a single 2p orbital ($\frac{5}{2} = (2^2 + (-1)^2)/((-1)^2 + (-1)^2)$.

We can thus algebraically separate the relaxations coming from $\chi_1$ and $\chi_2$ in a very simple way by writing:

$$1/T_{1\parallel\text{experiment}}(T) = 1/T_{1\parallel}(\chi_1)(T) + 1/T_{1\parallel}(\chi_2)(T), \qquad (57)$$

and

$$1/T_{1\perp\text{experiment}}(T) = 1.058(1/T_{1\parallel}(\chi_1)(T)) + \frac{5}{2}(1/T_{1\parallel}(\chi_2)(T)). \qquad (58)$$

As we can see in fig. 12, the derived temperature behaviours of $1/T_{1\perp}(\chi_1)(T)$ and $1/T_{1\perp}(\chi_2)(T)$ are very different. We have actually:

$$1/T_{1\perp}(\chi_1)(T)T \approx c^{\prime e} \qquad (59)$$

and something much more complicated happens to $1/T_{1\perp}(\chi_2)(T)T$.

It is possibly an encouraging feature for the decomposition performed in fig. 12 that $1/T_1\chi_2$ becomes very close to zero near $T_c$.

This fact might explain why no visible effect of $T_c$ has been measured by Takigawa et al. [5] on the $^{17}$O $1/T_1$ of an YBa$_2$Cu$_3$O$_{6.63}$ compound, contrary to what they saw in an YBa$_2$Cu$_3$O$_7$ compound. Maybe $1/T_1\chi_2$ also exists in YBa$_2$Cu$_3$O$_7$, but drops suddenly at $T_c$, whereas it has already smoothed down to zero just above $T_c$ in the YBa$_2$Cu$_3$O$_{6.63}$ compound.

As another side remark, we may also speculate that, should $\chi_2$ vanish, the temperature dependence of the planar oxygen $1/T_1$ would be nearly linear, just as it is for $1/T_{1c}$ in the YBa$_2$Cu$_3$O$_7$ compound above $T_c$ [5]. We could thus imagine that either the $^{17}$O $1/T_1$ for the YBa$_2$Cu$_3$O$_7$ concentration is nearly isotropic (which

<figure><img src="image_15.png" /><figcaption>Fig. 12. Hypothetical decomposition of $1/T_1(T)$ in two different magnetic contributions $1/T_{1(\chi_1)}(T)$ and $1/T_{1(\chi_2)}(T)$. Lines are guides to the eye. $1/T_{1(\chi_2)}(T)$ is supposed to give no contribution to the static susceptibility $\chi(0, 0, T)$, and to come only from the $\sigma_{O(2p)}$ orbitals, so that $1/T_{1\perp(\chi_2)}(T) = \frac{5}{2}1/T_{1\parallel(\chi_2)}(T)$, whereas the anisotropy of $1/T_{1(\chi_1)}(T)$ is only 5.8% (cf. eq. (40)). With our conventions, $1/T_{1\perp}(T) = 1/T_{1\perp(\chi_1)}(T) + 1/T_{1\perp(\chi_2)}(T)$ and $1/T_{1\parallel}(T) = 1/T_{1\parallel(\chi_1)}(T) + 1/T_{1\parallel(\chi_2)}(T)$. At the end of the paper, we attribute $1/T_{1\chi_2}(T)$ to the fluctuations of Zhang and Rice singlets.</figcaption></figure>
is contrary to our preceding speculation), or that maybe there is a reason to suppose that $\chi_2$ is proportional to $\chi_1$ in that latter compound.

Other authors [42] have in fact already partly measured the anisotropy of the $^{17}$O(2, 3) $1/T_1$ in oriented powders of YBa$_2$Cu$_3$O$_7$ ($T_c$=92 K), at 100 K and 77 K. In both cases they found nearly

$$ \frac{1/T_{1c} - 1/T_{1\parallel}}{1/T_{1\parallel}} \approx 66\% \qquad (60) $$

with an uncertainty of about ±20% at 100 K, and ±60% at 77 K. Their ratio is comparable to our room temperature anisotropy ratio, which is quite interesting, and which proves that certainly $1/T_1\chi_2 \neq 0$ in YBa$_2$Cu$_3$O$_7$. The number of available data, however, is rather limited at present, so that we feel any more elaborate conclusions concerning the YBa$_2$Cu$_3$O$_7$ compound should wait for further experimental results.

Another conclusion may be drawn from the temperature dependence of the anisotropy of $^{17}$O $1/T_1$. In previous works, some authors [5,6] were tempted to find a scaling between $1/T_{1c}(T)$ and $\chi(T)$ and arrived at

$$ 1/T_{1c}(T)T \propto \chi(T). \qquad (61) $$

We can actually recover a similar scaling between $1/T_{1\perp\exp}(T)T$ and $\chi_1(T)$ (fig. 13), but not between $1/T_{1\parallel\exp}(T)T$ and $\chi_1(T)$. So very likely the fit for $1/T_{1c}(T)T$ should be taken with a grain of salt.

The preceding and rather straightforward analysis of our $1/T_1$ data in the frame of a “two-fluid” model, which has led to the $1/T_1$ decomposition of fig. 12, has been very pleasant up to now because it has given us the possibility to juggle with two different susceptibilities $\chi_1(q, \omega, T)$ and $\chi_2(q, \omega, T)$ which could have two different temperature behaviours and two different microscopic origins. Unfortunately, what seems to be wrong with the “two-fluid” model at this stage is that, because of the usual sum rules over the $q$ and $\omega$ dimensions, the temperature variations of $\chi_1(0, 0, T)$ and $\chi_2(0, 0, T)$ should always be closely related to those of $\chi_1(q, \omega, T)$ and $\chi_2(q, \omega, T)$ for more general $q$ and $\omega$ values. It is very difficult to imagine that, as if by magic, $\chi_1(0, 0, T)$ and $\chi_2(0, 0, T)$ could have the same temperature dependence if $\chi_1(q, \omega, T)$ and $\chi_2(q, \omega, T)$ for $(q, \omega) \neq (0, 0)$ had not.

So, if from the $1/T_1$ data we known that $\chi_2(0, 0, T)/\chi_1(0, 0, T)$ is not temperature independent, only the possibility that

$$ \chi_2(0, 0, T) \ll \chi_1(0, 0, T) \qquad (62) $$

remains to be explored in order to explain our Knight shift data.

However, we shall see shortly that relation (62) is not very easy to explain, especially if $\chi_2$ corresponds to a “simple” metallic band formed of oxygen 2p orbitals.

If, as stated, $\chi_1$ is essentially based on the copper atoms, and $\chi_2$ on the oxygen atoms, one cannot think of

<figure><img src="image_16.png" /><figcaption>Fig. 13. “linear fit” of $1/T_{1\perp\exp}(T)$ versus $\chi_1(T)$. The corresponding values for $1/T_{1\parallel\exp}(T)$ are also indicated.</figcaption></figure>
having $\chi_1(0, 0, T) \ll \chi_2(0, 0, T)$ instead of eq. (62), first because the anisotropy of the $^{17}$O Knight shift data is rather small, and second because of the temperature dependence of the transverse copper Knight shift data [5]: the latter data can only be accounted for by the polarisation of the Cu(3d) orbitals themselves and not by the hybridisation of the O(2p) orbitals with the Cu(4s) orbitals (about this latter hybridisation, see ref. [41]) because the coupling of the Cu(4s) orbitals with the Cu nucleus is isotropic, whereas the c-axis copper Knight shift is known to be temperature independent.

In the frame of the “two-fluid” model, the relation (62) may only be justified by two different reasons.

The first one would lie in the frequency dependence of $\chi_2(0, \omega, T)/\omega$.

However, the measurements that we show in fig. 14 have not provided a positive answer to this question.

Let us first come back to eq. (32). The terms with $\chi''(q, \omega_N, T)/\omega_N$ probe the electronic fluctuations at the nucleus Larmor frequency $\omega_N$ which, in a 9.3 T magnetic field, amounts to 53.7 MHz. The other terms with $\chi''(q, \omega_c, T)/\omega_c$ probe the electronic fluctuations at the electronic Larmor frequency, which, in the same magnetic field, should be about 260 GHz if the electronic quasiparticles were made of a kind of usual metallic band. Under this latter hypothesis, a mere angular study of the $^{17}$O $1/T_1$ can reveal whether the $^{17}$O $1/T_1$ is frequency dependent or not.

We see in fig. 14 that the overall precision is enough to support strongly that

$$ \chi''(q, 260 \text{ GHz}, T)/260 \text{ GHz} \approx \chi''(q, 53 \text{ MHz}, T)/53 \text{ MHz}. \quad (63) $$

Incidentally, we indicate that another theory, the “paramagnon theory”, predicts that for the case of a bidimensional diffusive electron gas, the electronic correlation functions should display a logarithmic frequency dependence, at least in a certain energy range [43]. For the $^{17}$O nucleus, $\ln \omega_c/\ln \omega_N \approx 1.7$, so that the effect is clearly not observed.

The other reason to have $\chi_2(0, 0, T) \ll \chi_1(0, 0, T)$ would be that $\chi_2(q, 0, T)$ is very weak for $q \approx 0$. A somewhat natural way to understand this would be to suppose that $\chi_2(q, 0, T)$ is a contrario enhanced around some $q$ value which is far from zero. However, we already know that the electronic susceptibility, let us say tentatively $\chi_1(q, \omega, T)$, is peaked around $q=(\pi, \pi)$, and yet $\chi_1(0, 0, T)$ is important. Moreover, some neutron inelastic measurements performed by Shirane et al. [44] on three YBa$_2$Cu$_3$O$_{6.4}$, YBa$_2$Cu$_3$O$_{6.45}$ and YBa$_2$Cu$_3$O$_{6.5}$ single crystals show no trace of any other singularity in $\chi(q, \omega, T)$ than for $q=(\pi, \pi)$ even at room temperature, with $\omega$ such as $\hbar\omega=3$ meV.

There is, nevertheless, a second reason why $\chi_2(q, 0, T)$ is very weak for $q \approx 0$. It consists in attributing $1/T_1\chi_2$ to the fluctuations of spin-zero particles, such as the singlets of Zhang and Rice [45]. Precisely because these particles are spin singlets, their susceptibility may remain negligible at any temperature, so that they should remain invisible in Knight shift measurements.

As a side remark, we note that the polarisations of the Cu(3d) and O(2p) orbitals are both positive. If the

<figure><img src="image_17.png" /><figcaption>Fig. 14. Angular dependence of $1/T_1(\varphi)$ at room temperature. $\varphi=0^\circ$ corresponds to the orientation of fig. 2(a), $\varphi=90^\circ$ to that of fig. 2(b). We have drawn the theoretical curves corresponding to $\{1/T_{1c}... 1/T_{1\parallel}\}(\varphi)$ for three different values of the ratio $J_{2p}(\omega_N)/J_{2p}(\omega_c)$, where $J(\omega)$ is the widely spread notation for a correlation function. $1/T_{1c}... 1/T_{1\parallel}$ means that the measured $1/T_1$ varies continuously from $1/T_{1c}$ to $1/T_{1\parallel}$ when we rotate the crystal from $\varphi=0^\circ$ to $\varphi=90^\circ$. We also plot our results for $\{1/T_{1c}... 1/T_{1\perp}\}$ (the other planar site), which are nearly independent of $\varphi$.</figcaption></figure>
Zhang and Rice singlets were paramagnetic or diamagnetic, we think it is possible to imagine that if the polarisation of the Cu(3d) orbitals was positive, the polarisation of the O(2p) orbitals should be negative and vice versa. We have no proof for this non-trivial statement, which certainly deserves to be debated more seriously than is done here. In any case we shall assume from that point that $\chi(0, 0, T)=0$ for any singlet such as the singlets constructed by Zhang and Rice.

The next step is to consider that because the oxygen 2p orbitals are as much involved as the copper orbitals in the formation of the Zhang and Rice singlets, the relaxation these singlets would induce on the oxygen nuclei should have the expected $\frac{5}{2}$ anisotropy.

Actually it might seem a little bit surprising to hear that zero-spin particles may contribute to the $1/T_1$ of $^{17}$O nuclei at all. We may, however, convince the reader that our suggestion is not a fake by writing

$$ \begin{array}{c} \text{typical terms in the dipolar} \\ \text{interaction Hamiltonian between the} \\ \text{O(2p) orbital and the } ^{17}\text{O nucleus} \\ \overbrace{(S_{(2p \text{ orb.})}^{\dagger} I_{17\text{O}} + S_{(2p \text{ orb.})}^{-}} \\ \text{Zhang and Rice singlet} \\ \overbrace{|(Cu\uparrow O(2p)\downarrow - Cu\downarrow O(2p)\uparrow)\times} \\ \overbrace{(^{17}\text{O}\downarrow)>} \\ \text{average magnetisation} \\ \text{of the } ^{17}\text{O nuclei} \\ \text{after a } \pi \text{ NMR pulse} \\ \overbrace{= |Cu\downarrow O(2p)\downarrow\rangle \times (^{17}\text{O}\downarrow)>} \\ \text{which is non-zero and also has a non-zero overlap with a neighbouring Zhang and Rice singlet.} \end{array} $$

We prefer to call such singlets “Zhang and Rice singlets” rather than “t-J singlets” because two successive steps were performed in the original paper of Zhang and Rice [45], namely, first, the construction of these singlets, and second, the derivation of their fluctuations which, still according to Zhang and Rice, followed those of a one-band Hubbard “t-J” model. We think that our measurements may provide experimental evidence for the existence of Zhang and Rice singlets. We have no idea, however, whether the fluctuations of these singlets are given by the “t-J” model or not.

What we observe is that their fluctuations, as probed by NMR, should correspond to what we have drawn in fig. 12 as $1/T_1\chi_2(T)$.

What about $\chi_1$? We know many theoretical evidences that the Zhang and Rice singlets do perturb the antiferromagnetic background lattice so as to introduce some magnetic fluctuations at $q \neq (\pi, \pi)$ in the lattice. We thus attribute $1/T_1\chi_1(T)$ to these latter fluctuations, and $\chi_1(0, 0, T)$ to the ferromagnetic part of such fluctuations at zero frequency. The fact that $\chi_1(0, 0, T)$ and $1/T_1\chi_1(T)$ do not display the same temperature behaviour is not very easy to understand, but it must be kept in mind that all the components of the wavevector $q$ can contribute to $1/T_1\chi_1(T)$, whereas the Knight shift probes only the electronic properties at $q=0$ and $\omega=0$.

On the other hand, the fact that both $\chi_1(0, 0, T)$ and $1/T_1\chi_2(T)$ display some kind of critical behaviour around $T_c$ which are perhaps in some way connected to one another is not so surprising in our view: while the fluctuations of the Zhang and Rice singlets increase with temperature as they become more mobile, the antiferromagnetic background gets more and more disturbed by these singlets, thus developing paramagnetic fluctuations which must increase the bulk susceptibility of the material. We guess that this could also explain why no counterpart of the sudden drop in the Cu $1/T_1$ around 130 K [5] can be found in the copper transverse Knight shift data, as well as in any Knight shift data, because the Cu $1/T_1$ probes the copper antiferromagnetic fluctuations, whereas the Knight shift probes the copper ferromagnetic fluctuations.

Eventually, we wish to emphasize again that the comparison between the Knight shift intensities on the O(2, 3) nuclei and the copper nuclei, which has been discussed in great detail by Yoshinari et al. [6], indicates that the electronic planar spin polarisation resides on the Cu(3d) orbitals, and only marginally on any other planar orbital, including the oxygen 2pσ orbitals. In the paper of Yoshinari et al., this fact is presented as a supplementary argument in favour of the “one-fluid” picture. This conclusion, however, is not so easily made compatible with experimental results of photoemission studies (as an example, see ref. [46]) which have for long been taken to lead to the conclusion that high-$T_c$ superconductors possess a Fermi surface where the oxygen 2pσ orbitals contribute approximately just as much as the copper 3d orbitals.

Our $1/T_1$ measurements are particularly welcome in that context because they may provide a way to rec-
oncile the current NMR and photoemission points of view. If we attribute the $1/T_1\chi_2$ of fig. 12 to the fluctuations of charged spin-zero fermions (such as the singlets of Zhang and Rice, which possess and equal density on copper and oxygen orbitals), and if we suppose that photoemission Fermi-level studies probe that kind of particles, we may take the liberty to attribute $1/T_1\chi_1$ and the Knight shifts of all $CuO_2$ planar orbitals (copper, oxygen and also yttrium) to the one-half spins of the copper sites, whose statistics (are they fermions or bosons?) we cannot guess.

### 4. Conclusion

We have shown here that the temperature dependence of the $^{17}$O Knight shift anisotropy might seem to support the “one-fluid” picture, but that the anisotropy of the $^{17}$O $1/T_1$ in $YBa_2Cu_3O_{6.5}$ necessarily rules out the former point of view.

In fact, the high anisotropy of the room temperature oxygen $1/T_1$ alone may not be explained either by quadrupolar relaxation, or by orbital relaxation, or by the anisotropy of a single electronic susceptibility, or by the anisotropy of the hyperfine couplings, at least in a “one electronic fluid” picture.

We thus believe that two different electronic excitations contribute to the relaxation of the oxygen planar nuclei.

The problem then becomes to explain why we observe two different contributions to the $^{17}$O $1/T_1$, and only one to the $^{17}$O Knight shift. The answer to this problem presumably does not lie in the frequency dependence of the anisotropic part of the $^{17}$O $1/T_1$.

We suggest that the existence of zero-spin particles such as Zhang and Rice singlets, possessing specific fluctuations, may well account for the strange NMR features observed. At any rate, the mixture of the Knight shift and $1/T_1$ oxygen data is not easy to explain in any model currently available, and is likely to dislodge quite a number of theoretical pictures.

As a motive for further studies, we may say that our tentative understanding of the temperature dependence of NMR data is basically just qualitative, and we hope that a thorough quantitative description of all these data, including their striking temperature behaviours, will appear in the future. Also, the possible physical consequences of our study for the nature of the superconducting mechanism are outside the scope of this publication.

Presumably, the experimental results for the anisotropy of $^{17}$O $1/T_1$ in $YBa_2Cu_3O_{7-\delta}$ for other $\delta$ values will be quite interesting. Experiments are now in progress in order to increase the oxygen content and to improve the homogeneity of our crystals.

### Acknowledgements

We greatly thank B. Gotschy and P. Wzietek for their computer help, G. Quirion for the graphics and J.C. Ameline and F. Rethore for various punctual assistance. We also wish to thank C. Bourbonnais for many stimulating discussions during the work, A. Le Mehaute for generous support, and lastly H. Alloul, H. Schulz and V.J. Emery for pertinent remarks concerning the manuscript. This work has received the financial support of the DRET contract n° 88/34/198. One of us (FB) has benefited from a CIFRE convention with the Laboratoires de Marcoussis, Corporate Research Center of the Compagnie Générale d'Electricité.

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