Muon Knight Shift: Probing Electronic Magnetism

• Muon Knight shift is defined as the relative shift in the local magnetic field at a muon site compared to an applied field, arising from hyperfine interactions with conduction electrons and localized f-moments.
• It is measured using transverse-field μSR techniques to investigate correlated electron behavior, including Kondo screening, magnetic clustering, and quadrupolar excitations in complex materials.
• Applications in systems like UTe₂ and Pr₁₋ₓNdₓOs₄Sb₁₂ demonstrate its ability to diagnose electronic phase transitions, relocalization effects, and unconventional superconductivity.

The muon Knight shift is a quantitative local probe of electronic magnetism, defined as the relative shift of the local magnetic field at the positive muon ($\mu^+$) site from the externally applied magnetic field. Originating from hyperfine interactions with both conduction electrons and local $f$-electron moments, the muon Knight shift provides insight into spin and orbital polarization phenomena, hybridization effects, and electronic phase transitions in metals and intermetallics. Recent $\mu$SR studies on materials such as UTe$_2$ and Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$ have leveraged this probe to investigate Kondo-lattice physics, clustering phenomena, and the impact of quadrupolar excitations (Azari et al., 2023, Ho et al., 2014).

1. Definition and Physical Origin

The muon Knight shift ($K$) quantifies the relative change in the local magnetic field at the muon site ($\mathbf{B}_\mu$) compared to the applied field ($\mathbf{H}$):

$K^* = \frac{B_\mu - H}{H}$

After correcting $K^*$ for demagnetization and Lorentz-field effects, the total shift separates into two terms:

$K = K_0 + K_{f}$

• $K_0$: "chemical" or orbital shift from the Pauli polarization of conduction electrons (Fermi-contact term).
• $K_{f}$: shift due to local $f$-electron moments (e.g., U $5f$ or rare-earth $4f$), comprising both dipolar coupling and indirect (RKKY-mediated) polarization of conduction electrons by the local moments.

In metals with dominant local moment magnetism and a temperature-independent hyperfine coupling ($A$), the Knight shift is frequently parameterized as:

$K = A\chi + K_0$

where $\chi$ is the (bulk or local) magnetic susceptibility and $A$ the hyperfine coupling constant. In cubic crystals such as PrOs$_4$Sb$_{12}$, dipolar contributions to the average shift vanish by symmetry, yielding a pure contact-dominated shift.

2. Relation to Susceptibility and Knight-Shift Anomalies

In the simple (high-$T$) regime, $K$ versus $\chi$ yields a linear "Clogston–Jaccarino" relation if $A$ is $T$-independent and the magnetism is solely due to the local moments. However, in Kondo-lattice or correlated electron systems, a generalized "two-fluid" model is more appropriate:

$$\begin{aligned} K(T) &= A\chi_{cc}(T) + (A+B)\chi_{cf}(T) + B\chi_{ff}(T) \ \chi(T) &= \chi_{cc}(T) + 2\chi_{cf}(T) + \chi_{ff}(T) \end{aligned}$$

where $\chi_{cc}$ (conduction band), $\chi_{ff}$ (localized $f$-moments), and $\chi_{cf}$ (mutual polarization) have distinct $T$-dependencies. When $A \ne B$, the Knight-shift–susceptibility linearity breaks down below a crossover temperature due to hybridization and the onset of Kondo coherence; this nonlinearity is termed the "Knight-shift anomaly."

In Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$, the pure Nd endmember exhibits a consistent linear $K(\chi)$ behavior across a wide $T$ range, whereas Pr-rich alloys and PrOs$_4$Sb$_{12}$ itself display an anomalous saturation (collapse of the hyperfine coupling) for $T \lesssim 15$ K, not attributable to crystal field depopulation, but rather to the emergence of non-magnetic quadrupolar excitations that modify the indirect RKKY-type muon–$4f$ coupling (Ho et al., 2014). In UTe$_2$, the breakdown and subsequent restoration of $K$–$\chi$ linearity bracket the temperature range for coherent Kondo liquid formation and relocalization transitions (Azari et al., 2023).

3. Experimental Methodology and Component Analysis

Muon Knight shift measurements are realized via transverse-field $\mu^+$SR (TF-$\mu$SR), where the Larmor precession frequency of the muon is monitored as a function of temperature, field, and composition. The generic TF-$\mu$SR signal is fit by oscillatory functions accounting for different muon stopping sites or magnetic environments:

$$A(t) = \sum_i a_i\, e^{-\sigma_i^2 t^2} \cos\left(\gamma_\mu B_i t + \phi_i\right)$$

where $a_i$ are volume fractions, $\sigma_i$ Gaussian depolarization rates, and $B_i$ mean local fields. The number and characteristics of components in $A(t)$ reflect underlying crystallographic or mesoscopic inhomogeneity.

In UTe$_2$, TF-$\mu$SR reveals three distinct Knight shift components for $\mathbf{H} \parallel c$, corresponding to site and cluster inhomogeneities:

Component	$a_i$ (%)	Scaling	$A$ or $A'$ (Oe/$\mu_B$)	$K_0$ (ppm)
$K_1$	27	$\chi_c$	587(12)	$-590(27)$
$K_2$	55	$\chi_c$	587(12)	$-590(27)$
$K_3$	18	$\chi_a$ (above $T_r$)	1994(6)	$-2700$

Here $K_3$ is identified with a minority magnetic cluster phase, two orders of magnitude larger than the "bulk" $K_{1,2}$, with atypical temperature and directional scaling (Azari et al., 2023). In Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$, single-component shifts from macroscopically averaged sites suffice due to cubic symmetry (Ho et al., 2014).

4. Temperature Dependence and Anomalous Behaviors

The temperature dependence of the muon Knight shift encodes key correlations:

UTe$_2$ (Azari et al., 2023)

• $T > T^* \approx 30$ K: $K_{1,2}$ scale linearly with $\chi_c$, indicative of independent $5f$-local moment behavior.
• $12\,\mathrm{K} < T < 30\,\mathrm{K}$: Kondo coherence emerges; $K_{1,2}$ deviate from linearity with $\chi_c$ (Knight-shift anomaly). $K_3$, tracking $\chi_a$, signals cluster moment freezing.
• $T < T_r \approx 12$ K: Linear scaling of $K_{1,2}$ vs $\chi_c$ is gradually restored—"relocalization" of $5f$ moments occurs, interpreted as partial transfer of spectral weight from itinerant heavy-electron fluid back to local $5f$ moments. $K_3$ begins to track $\chi_c$, reflecting reorientation of cluster moments.

Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$ (Ho et al., 2014)

• $15\,\mathrm{K} \lesssim T \lesssim 200\,\mathrm{K}$: $K$ exhibits textbook linear dependence on $\chi$ in both Nd- and Pr-rich alloys, with alloy-specific slopes $A_c$.
• $T \lesssim 15$ K, Pr-containing alloys: Linear $K(\chi)$ dependence collapses, saturating at $K_{\text{sat}} \simeq -0.5\%$, indicating an abrupt reduction of the effective hyperfine coupling $A_{hf}$. This is absent in NdOs$_4$Sb$_{12}$ and is correlated with the onset of low-lying nonmagnetic quadrupolar excitations known from neutron and thermodynamic probes.

5. Interpretations: Relocalization, Cluster Physics, and Quadrupolar Effects

Relocalization in Kondo Lattices:

In UTe$_2$, the temperature window bounded by $T^*$ and $T_r$ encapsulates the gradual formation of a correlated Kondo liquid and its partial relocalization. Above $T^*$, moments behave locally. Below $T^*$, Kondo screening induces a collapse of Knight shift linearity. Sub-$T_r$, relocalization returns the system towards single-ion local-moment character, consistent with Ce-based precedents. The restoration of linearity below $T_r$ is ascribed to partial reversal of Kondo hybridization, leading to increased spectral weight in localized $5f$ channels and competing with the development of an itinerant heavy-electron fluid necessary for spin-triplet superconductivity.

Magnetic Clusters:

The third Knight shift component $K_3$ in UTe$_2$ is associated with spatially inhomogeneous magnetic clusters. Above 12 K, $K_3$ correlates with $\chi_a$ (despite field alignment along $c$), indicating $a$-axis locking of cluster moments. Below 12 K, cluster moments become itinerant and track $\chi_c$, and the depolarization rate $\sigma_3$ increases sharply, producing significant field broadening. This cluster fraction, about 18%, evidences phase coexistence of uniform Kondo lattice and locally clustered magnetic moments.

Quadrupolar Excitations:

In Pr-rich Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$, the saturation of $K(\chi)$ below $\sim$15 K is interpreted as suppression of the indirect contact hyperfine coupling by the emergence of itinerant Pr$^{3+}$ quadrupolar excitons. These nonmagnetic excitations introduce charge-spin correlations which can interfere destructively with the RKKY-mediated muon–$4f$ interaction. This behavior is unique to Pr-containing samples and not present in the Nd endmember, supporting the role of itinerant quadrupolar modes as opposed to simple crystalline electric field depopulation or static disorder.

6. Implications for Unconventional Superconductivity and Correlated Electron Systems

These studies extend the muon Knight shift from a conventional probe of static magnetism to a precise discriminator of correlated-electron behavior:

• In UTe$_2$, the Knight shift exposes the interplay between itinerant heavy-electron fluid and local $5f$-moment fluctuations in the approach to unconventional spin-triplet superconductivity. A plausible implication is that the observed relocalization at $T_r$ signifies a crossover to an underscreened Kondo-lattice regime, impacting the superconducting pairing mechanism (Azari et al., 2023).
• In Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$, the suppression of Knight shift at low $T$ indicates that collective quadrupolar excitations can strongly renormalize and even quench the hyperfine coupling that underpins many local-probe techniques, cautioning against naive linear interpretations of $K(\chi)$ (Ho et al., 2014).

These results demonstrate the diagnostic capacity of the muon Knight shift for detecting not only conventional magnetic ordering, but also hybridized-electron fluids, magnetic clustering, and exotic collective (non-dipolar) modes in correlated systems. Care is warranted in interpreting nonlinearities in Knight-shift data, as they may reflect both electronic reorganization and multipolar fluctuations, not solely magnetic order or multiplet depopulation.

7. Representative Numerical Parameters and Comparative Summary

UTe$_2$ (H$\parallel c$) (Azari et al., 2023):

• $K_2 = A\,(\chi_c/0.55\,\text{emu/mol}) + K_0$ with $A = 587(12)\,\text{Oe}/\mu_B$, $K_0 = -590(27)$ ppm.
• $K_3 = A'(\chi_a/0.18\,\text{emu/mol}) + K_0'$, $A' = 1994(6)\,\text{Oe}/\mu_B$, $K_0' = -2.7\times 10^3$ ppm.
• Volume fractions: $a_1 = 27\%$, $a_2 = 55\%$, $a_3 = 18\%$.

Pr$_{1-x}$Nd$_x$Os$_4$Sb$_{12}$ (Ho et al., 2014):

• NdOs$_4$Sb$_{12}$: $K_0 = 0.019(7)\%$, $A_c = -0.031(5)\,\text{mole/cm}^3$
• PrOs$_4$Sb$_{12}$: $K_0 = 0.06(2)\%$, $A_c = -0.072(9)\,\text{mole/cm}^3$
• Coupling collapse onset: $$\chi_{\text{onset}} \approx 0.05\,\text{cm}^3/\text{mole}\, (T_{\text{onset}}\approx 15\,\text{K})$$
• Saturation: $K_{\text{sat}}\approx -0.5\%$ (for $x \leq 0.75$)

Such results confirm the muon Knight shift as an incisive microscopic probe of complex correlated-electron materials, able to distinguish between uniform local-moment magnetism, hybridization crossovers, clustering, and higher-order multipolar phenomena.