Angle-dependent Knight shift

This chapter is a worked example of an angle-resolved Knight-shift measurement: rotating a single crystal in a transverse field and following the muon Knight shift \(K(\theta)\) of each site as the orientation changes. It is the most direct μSR route to the muon stopping site, because the anisotropy of \(K(\theta)\) is fixed by the dipolar coupling tensor at the site (Amato & Morenzoni 2024, Ch. 5). The data here are synthetic, but the workflow — per-orientation fitting, conversion to the Knight shift, and a joint \(K(\theta)\) fit that resolves the component labelling through crossings — is exactly the one used on real data. The full feature reference lives at Knight shift; the per-orientation grouped fit that feeds it is documented in Grouped time-domain fitting.

Physical motivation

A muon in an applied field \(B\) precesses at a frequency shifted from the bare Larmor value by the local hyperfine field. The fractional shift is the muon Knight shift,

\[K = \frac{\nu - \nu_{\mathrm{ref}}}{\nu_{\mathrm{ref}}},\]

the muon analogue of the NMR Knight shift (Knight 1949). It splits into an isotropic contact term and a traceless dipolar term; for a crystal rotated about a principal axis the latter carries the orientation dependence,

\[K(\theta) = K_{\mathrm{iso}} + K_{\mathrm{ax}}\,\frac{3\cos^2\theta - 1}{2}.\]

The contact part \(K_{\mathrm{iso}}\) reports the local spin susceptibility; the axial part \(K_{\mathrm{ax}}\) and its sign report the dipolar geometry, and so pin down where in the unit cell the muon sits. The axial factor \((3\cos^2\theta - 1)/2\) vanishes at the magic angle (\(\theta \approx 54.7^\circ\), and again at \(125.3^\circ\)), where every site shows only its contact shift — which is exactly where two sites’ branches can cross and their labels become ambiguous.

The data

The example is a synthetic orientation series of transverse-field runs at \(\theta = 0, 15, \ldots, 165^\circ\), with two inequivalent muon sites. The two sites are given the same contact shift (\(K_{\mathrm{iso}} = 0.40\%\)) and equal-but-opposite axial shifts (\(K_{\mathrm{ax}} = \mp 0.30\%\)), so their \(K(\theta)\) branches cross twice — at both magic angles — the case the joint fit exists to handle.

Walkthrough

Load and tag the orientation. Load the runs and give each one an Angle (°) value in a logbook column (Columns: metadata and custom). This is the column the trend panel will use as its x-axis.
Fit each orientation. Fit the precession frequency at every angle. For a site-resolved measurement this is the individual-groups time-domain fit (Grouped time-domain fitting), which fits each detector group separately so the per-site lines are kept apart. Seed each fit from its neighbour (chained batch seeding) so each component keeps a stable label through the scan and the trend follows one site at a time.
Convert to the Knight shift. With the fitted frequencies trended, open the Knight shift analysis window — the Knight shift window… button in the Derived parameters section of the Fit Parameters panel, or Analysis → Knight shift analysis… — and reference against the Applied field in the Conversion section of its sidebar. Each frequency trace becomes a branch in the Branches section, converted live as you edit the reference or unit. This yields the directly measured shift \(K_{\mathrm{exp}}\); if the sample’s shape and bulk susceptibility are known, tick Lorentz/demag correction, pick the Shape (or Custom N) and enter χ (SI), to recover the intrinsic \(K_\mu\) — see Reading the result, below, for the caveat this correction carries under rotation.

The Knight shift analysis window on a two-site angle scan. The sidebar reads top to bottom as the pipeline — Source (the fitted series supplying the frequencies), Conversion (reference and unit), Branches (one \(K\) trace per converted component, with a count of the crossings flagged along the scan), Model fit (the joint \(K(\theta)\) fit covered in a later step, shown here already run), and Suggest next angle (covered further below, shown here already expanded with a computed Refine parameters suggestion); the plot shows both branches against angle with their fitted curves, Crossing markers on (dashed at the scan intervals where the raw component labels can swap), and the suggestion’s utility band with the suggested angle marked. Converting here does not touch the trend table — press Send K columns to trend table in the footer to publish the \(K[\ldots]\) columns for plotting and export (the joint fit runs here in the window and does not need them published first).
Plot against orientation. Back in the trend panel, select Angle (°) as the trend x-axis to see the published \(K[\ldots]\) traces. If the scan wraps past one period, the Fold control overlays equivalent orientations onto a single \(180^\circ\) period, doubling the effective angular sampling — the same fold the analysis window offers as its own Fold 180° view toggle for inspecting the branches before publishing.
Resolve and fit with a joint \(K(\theta)\) fit. Back in the Knight shift analysis window, use its Model fit sidebar section: pick a model (KnightAnisotropy for the axial dipolar form used here) and press the footer’s Run joint K(θ) fit button. This fits one \(K(\theta)\) curve per site at once and, at each angle, assigns that angle’s points one-to-one to the curves they best match (a Hungarian matching), iterating until both the curves and the assignment settle. The plotted branches realign so each follows a single physical site continuously through the crossings, with the per-curve fits overlaid and swap markers at the angles where the assignment changes. KnightAnisotropy also fits a per-site \(\theta_0\), the goniometer/mount misalignment between the scale’s zero and the crystal’s principal axis. A large reduced \(\chi^2\) with \(\theta_0\) pinned at zero was the old failure mode here — a mount that is even slightly off-axis pushes the residual misalignment into \(K_{\mathrm{iso}}\) and \(K_{\mathrm{ax}}\) instead, biasing exactly the parameters that identify the site; fitting \(\theta_0\) absorbs it. If Scale errors by √χ²ᵣ is ticked and the fit’s reduced \(\chi^2\) still exceeds one after that, the quoted uncertainties are inflated accordingly. The Knight shift analysis window captured above already shows the result of this step — both branches converted and the joint fit run, with swap markers at the angles where the assignment changes so each trace continues through the crossings along its own site.
Which angle next? With the joint fit in hand, the window’s own collapsible Suggest next angle section (Knight shift) plans the next run rather than just fitting the ones already taken. Leave Mode on Refine parameters and Target on All parameters (D-optimal) to see where a new angle would tighten both sites’ fitted curves at once — for this scan, still coarsely sampled every \(15^\circ\), the suggestion lands away from the two magic-angle crossings, where each site’s own curve has the steepest slope (the screenshot above shows this suggestion already computed).

It is also worth asking whether the rotation axis is really aligned rather than assuming it, especially given how cleanly the two sites came out equal-and-opposite: switch Mode to Test misalignment and click Suggest again. This fits the first-harmonic AngularFourier2 alternative in the background and reports which model the data currently prefer — here, correctly, the aligned KnightAnisotropy fit, by a decisive Akaike weight, since the synthetic scan was built from a perfectly aligned axis.

Finally, Resolve assignment targets the crossings themselves: near \(54.7^\circ\) and \(125.3^\circ\) the classification-EM step above had a competing, near-equally-good labelling available (the envelope assignment this whole workflow exists to avoid), and this mode ranks candidate angles by how well a new run would tell the winning assignment apart from that runner-up — typically somewhere between the crossings, where the two labellings genuinely disagree, rather than at a crossing itself.

Reading the result

The joint fit recovers the two branches cleanly: \(K_{\mathrm{iso}} \approx 0.40\%\) for both sites, with \(K_{\mathrm{ax}} \approx -0.30\%\) and \(+0.30\%\) — the values the scan was built from. The point of the assignment is visible at the two magic angles: without it, a fit to the raw label order would track the envelope of the two branches (the upper curve, then the lower) and report a spurious near-flat shift; with it, each trace continues through the crossing along its own site, and the fitted \(K_{\mathrm{ax}}\) carries the real sign and magnitude. That sign, together with the magnitude of the anisotropy, is what constrains the candidate stopping site.

For the absolute shift, note that the conversion by itself yields the directly measured \(K_{\mathrm{exp}}\); recovering the intrinsic \(K_\mu\) needs the Lorentz/demag correction step above, with the sample geometry and bulk susceptibility as inputs. For a rotating sample that is not itself spheroidal, remember that the correction assumes a fixed demagnetisation factor \(N\) along the field — exact for a sphere, an approximation as the sample turns otherwise. To turn a \(K\)–\(\chi\) pair into a hyperfine coupling, see the Clogston–Jaccarino discussion at Knight shift.

References

A. Amato and E. Morenzoni, Introduction to Muon Spin Spectroscopy: Applications to Solid State and Material Sciences, Lecture Notes in Physics Vol. 961 (Springer, Cham, 2024).
1. 1. Knight, Phys. Rev. 76, 1259 (1949).