6.4 Inferences about the Magnetic Structure in A₁C₆₀

In typical well-ordered magnets, oscillations are often observed in the ZF $\mu {\cal SR}$ spectra (Chapter 5). The frequency of these oscillations is a measure of the static internal field, and thus of the magnetization, or in the case of a SDW, the frequency may be a measure of the SDW amplitude [23]. Interpretations based on the temperature dependence of the oscillation frequency and its relaxation can then be made. In A₁C₆₀, as has been presented in the previous section, the ZF spectra do not admit such an obvious analysis, except possibly at the lowest temperature in Cs₁C₆₀ (Fig.6.42). Recall that other measurements on these systems with samples from different sources find similar behaviour. Thus there is no clear interpretation of the parameters from fits to the phenomenological form (Eq. 6.1) above.

Let us at this point backtrack, and ask the questions: What are the qualitative features of the ZF relaxation in the magnetic phase, and what, if anything, do these features tell us about the magnetic structure? To which the answer is: The most prominent characteristic feature of the observed ZF $\mu {\cal SR}$ relaxation in the all samples of the two magnetic phases (A = Cs and Rb) is its two component nature. The second most apparent feature is that the relaxation seems to develop quite gradually with temperature, and both the amplitudes and relaxation rates of the two components are temperature dependent. These answers contain relatively little information, but something important may still be gleaned from them.

Further information can be added from other techniques. In the NMR of Cs₁C₆₀, the ¹³C lineshape is distinctly two component, whereas the ¹³³Cs line is not. The close analogy between the ¹³C and the $\mu {\cal SR}$ suggests that the the muon is sampling the static field distribution in the same way as the ¹³C. While the sites of the $\mu^+$ in these materials are currently a matter of speculation, the positions of the carbon atoms are not. It is possible, perhaps even likely, that the interstitial $\mu^+$ is closely associated with the outer surface of the (distorted) C₆₀^-, lending geometric support to the idea that the muon and carbon sample the same volume of the the spatial distribution of magnetic fields. In addition both the AFMR and the NMR suggest some sort of antiferromagnetically (AF) ordered state, but whatever, the structure, it must be consistent with the lack of magnetic neutron scattering[66]. Furthermore, if the two component nature is characteristic of the magnetic structure, and if the conclusion (found in NMR AND AFMR) that the AF state is spin-flopped in fields exceeding $\approx2.7$T is correct, then the occurrence of two components must be robust to the application of an external field, since the $\mu {\cal SR}$ is done in zero field.

If we assume that the two component nature is not due to different muon sites, but, rather, it reflects the distribution of static fields in the sample which are sampled in a way closely analogous to the ¹³C. The existence of two components, thus, indicates inhomogeneity of the sample, but macroscopic inhomogeneity is ruled out by the one component nature of the magnetic broadening of the ¹³³Cs NMR line, so the inhomogeneity must be microscopic. One possibility is that a fraction of the chains are magnetic, while the remainder are not, and the two kinds of chains are ``intimately mixed''[208]. Variation with temperature of the amplitudes in this model could be due to a distribution of transition temperatures in the magnetic chains. For example, alkali occupancy and imperfect polymerization will lead to chains with a distribution of lengths (N), and perhaps T_c depends sensitively on N. However, it is not clear why the amplitude of the fast component in this model would saturate at a value less than the full sample amplitude at low temperature. The amplitudes for A = Rb, appear to saturate at with $A_F \approx A_S$ in high TF. In ZF the saturation may not be complete by 2.5K, but for A = Cs, it appears to be by 3K (with $A_F \approx A_S$). Furthermore, one would expect significant sample variation in the relative amplitudes. The similar temperature dependence of the amplitudes in another study (on A = Cs) with saturation to equal amplitudes at low temperature[204] suggests that this is an intrinsic property of the magnetic state. It is perhaps possible that the magnetic structure is inhomogeneous in such a way that, for example, alternating chains are magnetically ordered and completely non-magnetic, but perhaps there exists another obvious, appealing, geometric explanation that could be related to the well-established crystal structure of these materials.

In order to attempt to find such an explanation, consider the following. There are two inequivalent regions of the distorted C₆₀ molecule in the polymer, call them the pole and equator by analogy with the earth. The axis is just the chain direction, and the poles are where the isolated C₆₀ molecular structure is most strongly modified (recall the carbons at the poles are sp³). The Magic Angle Spinning NMR[60] indicates that there are as many as seven inequivalent carbons, but they can be grouped into the two categories above. Because the lattice constants are so large, the magnetic field in the magnetic state at any position in the unit cell will be dominated by the nearest few moments (because of the r^-3 fall off of the diploar field). In an antiferromagnet, cancellation of the internal fields between neighbouring oppositely aligned moments, leads to a region of low field. Perhaps this region of low field can explain the presence of the slowly relaxing component. To address this possibility consider the following models. Assume that the AF ordering wavevector lies along the chain. Consider two Bohr magneton (largest reasonable moment for this situation) moments in neighbouring positions along the chain, i.e. separated by 9.1Å, the chain lattice constant. This might be the structure if the conduction electrons localize at the poles, a situation which may not be particularly likely but is the simplest place to start. First assume that the moments are completely localized. There are two cases: moments collinear with the ordering wavevector and moments transverse to the wavevector. As an indication of the minimum fields in this structure, the magnitude of the magnetic field on the plane directly between the moments is plotted for these two cases in Figures 6.48 and 6.49. From these two figures, it is clear that the region of zero field is very small, and that the region the ball (which would be the equatorial region for the moments at the poles) which samples this region of the lowest field, actually sees rather high fields. We now consider the more likely case of localization at the equator. The sp³ carbon at the polymer bond is satisfied, so it will not contribute strongly to the molecular orbitals which make up the conduction band. These orbitals have some complex structure as their t_1u symmetry has been broken by polymerization, but consider a simple model, with the antiferromagnetically aligned moments delocalized in rings at the equator of the balls. The region of lowest field (at the plane halfway between the rings) then corresponds to the pole region of two neighbouring molecules. The magnetic field in this plane for the two cases of transverse and collinear moments is shown in Figures 6.50 and 6.51. In order to make quantitative comparisons, much more detailed calculations of the full spatial field distributions would have to be made, but these simple calculations suggest that this might be a fruitful avenue for further investigation. However, it is not clear that such an explanation would explain the observed temperature dependence of the two amplitudes. Furthermore it is not clear that this region of reduced field would be robust to a spin-flop along an arbitrary direction (in the powder samples). Further data from other techniques are likely required before a final conclusion a regarding the magnetic structure is reached.