Incommensurate Magnetic Ordering

Long-range magnetic ordering incommensurate with the lattice is generally expected to generate broad field distributions at each crystallographic muon site in the material. In theory, this can still result in quite distinctive muon spin relaxation functions, even though real-world observation of µSR in incommensurate magnets has generally been disappointing. Generally, for an interstitial muon site in a lattice with local moments in a long range ordered state, there should be a strict upper-limit cutoff field magnitude, so in some sense the broadest possible field-magnitude distribution is $P_c(\vert{\bf B}\vert)=const.$, for $0<\vert{\bf B}\vert<B_{\max }$. In spite of the broadness, this distribution causes a static ZF µSR relaxation function with clear oscillations. By direct integration

$$G_c(t)=\frac 13+\frac{2\sin (\gamma _\mu B_{\max }t)}{3(\gamma _\mu B_{\max }t)}.$$

Note that the fequency seen is not representative of some average of the fields in the distribution, but of the highest field involved in the distribution. For a simple amplitude-modulation spin density wave (SDW) magnetic ordering, if the local field is assumed to be proportional to the local amplitude of the SDW, $\vert{\bf B}\vert=B_{\max }\vert\cos (\phi )\vert$, with all values of $\phi$ equally likely.

$$P(\vert{\bf B}\vert)=\frac{P(\phi )}{\left( \frac{dB}{d\phi }\right) },$$

$$\Longrightarrow P_{SDW}(\vert{\bf B}\vert)=\frac 2{\pi \sqrt{B_{\max }^2-B^2}}\ ,\qquad 0<B<B_{\max }.$$

Peretto et al.,[5] considering perturbed angular correlation (PAC) in chromium metal, call this the "Overhauser" distribution, and discuss PAC relaxation functions (slightly different than for µSR) in terms of J₀, the zero-order Bessel function. The static ZF µSR function is

$$G_{SDW}(t)=\frac 13+\frac 23J_0(\gamma _\mu B_{\max }t).$$

After some initial relaxation, this also settles down to oscillation at the maximum field frequency of the distribution. To model µSR in a helical incommensurate magnetic ordered state, consider a single host moment to be closest to the muon site, and let the orientation of that moment point anywhere in a plane containing both the host moment and the muon positions, for fixed muon polarization direction. If there is only dipole coupling between host moment and muon spin,

$$\vert{\bf B}\vert=\frac m{r^3}\sqrt{1+3\cos ^2\theta },$$

where, in a simple case, m and ${\bf r}$ are fixed, and all values of $% \theta$ are equally probable. In this case $\vert{\bf B}\vert$ ranges between $% B_{\min }=m/r^3$ and $B_{\max }=2m/r^3$. If the plane of host moment orientation does not contain the muon, $B_{\max }$ is reduced. In the simplest "lone dipole" case, then,

$$P_{LD}(\vert{\bf B}\vert)=\frac 2{\pi \sqrt{\left( B^2-\left( . . . . . . ght) ^2-B^2\right) }},\qquad \frac m{r^3}<B<\frac{2m}{r^3}.$$

This distribution diverges as B goes to either $B_{\min }$ or $B_{\max }$. Numerical simulations[6] indicate that for muons dipole coupled to all the host moments in an incommensurate ordering in which the moments point anywhere in a plane the local field distribution has this form except that $B_{\min }$ and $B_{\max }$ are essentially independent parameters, and in particular, $B_{\min }$ can also be less than $B_{\max }/2$. This suggests that a generic incommensurate antiferromagnet muon-site field distribution might be

$$P_I(\vert{\bf B}\vert)=\frac 2{\pi \sqrt{\left( B^2-B_{\min } . . . . . . ft( B_{\max }^2-B^2\right) }},\qquad B_{\min }<B<B_{\max }.$$

Note that $P_{SDW}(\vert{\bf B}\vert)$ is the $B_{\min }\rightarrow 0$ limit of $% P_I(\vert{\bf B}\vert)$. This distribution is likely to generate a static ZF relaxation function with 2 frequencies, corresponding to the upper and lower limits of the distribution.

Figure 6: