research32 min studyLesson 988 in Physics

Muon spin rotation and local magnetic fields

Infer a local field from precession while separating static distributions, dynamics, background, and stopping fractions.

Before you begin

Nuclear magnetic resonance and spin relaxation

On this page

The question for today

What you’ll understand

Relate detector asymmetry to an ensemble polarization after calibration.
Convert a precession frequency into a local magnetic field.
State the assumptions behind static Gaussian relaxation and magnetic-volume estimates.

A polarized positive muon implanted in a solid precesses in its local magnetic environment before decaying. The emitted positron direction carries information about its spin orientation, allowing an ensemble time trace to be reconstructed from many decay events. This local probe can detect fields without a sharp magnetic Bragg peak, but its implantation site, stopping distribution, and observation timescale shape what it sees. A relaxation signal is evidence of a local field environment; it is not by itself a unique magnetic structure.

A detector asymmetry is a calibrated observable

Two opposite detector groups record decays with different angular sensitivity to polarization. Define their labels and an efficiency correction alpha, and form a normalized count difference after appropriate count-background treatment. The common muon-decay envelope largely cancels, leaving an asymmetry proportional to a polarization projection with an apparatus-dependent scale. Detector dead time, pileup, time-zero calibration, and unequal acceptance can distort this construction. An observed asymmetry below unity is expected even for a fully polarized beam.

Some muons stop in the sample, others in a holder or surrounding material; multiple sample sites can also contribute distinct signals. The measured asymmetry is then a weighted sum of polarization functions. A slowly relaxing background may resemble a nonmagnetic sample fraction, while a very rapidly relaxing component can be lost before the accessible time window. Converting fitted amplitudes into volume fractions therefore requires stopping and background calibration rather than simply dividing one visible oscillation amplitude by the total count rate.

\mathcal A(t)=\frac{N_F(t)-\alpha N_B(t)}{N_F(t)+\alpha N_B(t)},\qquad \mathcal A(t)=a_sP_s(t)+a_bP_b(t)

The detector labels here define the sign, alpha corrects relative efficiencies, and a_s and a_b are calibrated asymmetry amplitudes. They are not automatically identical to geometric sample and background volume fractions.↔ Scroll wide equations horizontally.

Precession measures a field at a stopping site

A uniform static field transverse to the initial polarization produces cosine precession with angular frequency equal to the muon gyromagnetic ratio times local field. The observable frequency in cycles per second contains a factor of two pi. The field can include dipolar, contact, applied, demagnetizing, and Lorentz contributions. A measured frequency therefore yields a local field magnitude under this model; deriving an ordered magnetic moment additionally needs a stopping site and a magnetic-field calculation.

A distribution of static fields causes different muons to accumulate different phases, damping the ensemble average even if no individual environment changes with time. For an isotropic zero-mean Gaussian distribution with independent Cartesian components, the zero-field longitudinal polarization has the static Gaussian Kubo–Toyabe form. Its long-time one-third tail follows orientational averaging. An anisotropic single crystal, a different distribution, multiple sites, or fluctuating fields need not follow that form; the tail is not a universal measure of an unmagnetized fraction.

P_\perp(t)=\cos(\gamma_\mu Bt+\phi),\qquad f_\mu=\frac{|\gamma_\mu|B}{2\pi},\qquad G_{\rm KT}(t)=\frac13+\frac23(1-\Delta^2t^2)e^{-\Delta^2t^2/2}

gamma_mu is an angular gyromagnetic ratio, f_mu is in hertz, and Delta=|gamma_mu| times the rms of each Cartesian field component, in inverse seconds. The KT expression assumes a static isotropic Gaussian field distribution and zero applied field.↔ Scroll wide equations horizontally.

Slow and static are timescale statements

Fluctuating local fields can also relax polarization, sometimes with an exponential-looking time dependence. A similar envelope can arise from a static broad distribution over a restricted time range. Applying a field parallel to the initial muon polarization can decouple sufficiently weak quasi-static transverse fields; its field dependence helps test a static model. It is not an absolute proof of time independence, because slow dynamics can look static within the muon window and dynamic relaxation also depends on applied field.

Implanted muons can locally perturb charge or structure, and thermally activated hopping can change the fields they sample. Compare zero-field, longitudinal-field, and transverse-field data with temperature dependence and independent magnetic probes. In superconductors, a vortex-state field distribution can constrain penetration-depth models only after nuclear broadening, disorder, geometry, and the vortex regime are treated. Neither a zero-field relaxation increase nor a fitted field width alone uniquely establishes a particular order parameter or broken symmetry.

\Delta B=\frac{\Delta}{|\gamma_\mu|},\qquad P_s(t)=\int dB\,p(B)\cos(\gamma_\mu Bt)

The second expression is an illustrative static transverse-field projection for a normalized scalar field distribution p(B) and a common field direction. Dynamics requires a temporal model; an arbitrary vector distribution cannot always be represented by this one-dimensional cosine integral.↔ Scroll wide equations horizontally.

Work it through

Calibrate polarization and local field separately

At one time bin of a hypothetical calibrated run let $N_F=600$ , $N_B=200$ , and $\alpha=2$ . The sample-only asymmetry scale is 0.25 with negligible holder contribution. A fitted precession frequency is $6.775\,\mathrm{MHz}$ . Use the rounded muon conversion $|\gamma_\mu|/(2\pi)=135.5\,\mathrm{MHz/T}$ .

The corrected asymmetry is $(600-400)/(600+400)=0.20$ . The unequal raw counts must not be interpreted before applying the detector-efficiency correction.
Dividing by the calibrated sample scale gives the polarization projection $P_s=0.20/0.25=0.80$ . This is an ensemble projection at the stated bin, not a magnetic volume fraction.
The frequency gives $B=6.775/135.5=0.050\,\mathrm T$ . No extra factor of two pi is inserted because the supplied conversion is already a frequency per field rather than an angular frequency per field.

The result

The corrected polarization projection is 0.80 and the fitted local static field is 50 millitesla under the precession model.

The living notebook

Explore the model

Interactive lab

Interactive lab · oscillator

A frequency changes phase accumulation

For fresh runs Reset and set damping ratio and natural frequency to $(0,1)$ and $(0,2)$ . Press Step once in each, reaching t=0.5 seconds. Predict displacements cos(0.5), about 0.878, and cos(1), about 0.540. This unit-mass oscillator with fixed initial displacement illustrates frequency and phase only; it is not a muon field distribution or relaxation simulator.

Displacement x(t)Current time

Displacement x(t): 1
Velocity v(t): 0
Regime: underdamped

Damping ratio ζ0.2Natural frequency ω₀1.4

t = 0 s

Unit mass, x(0) = 1 and v(0) = 0. The analytic solution includes underdamped, critically damped and overdamped cases; time is measured in seconds. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forA frequency calibration and a clock determine phase accumulation.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The one-third longitudinal component is geometric within the stated static isotropic model. It is not a direct nonmagnetic-volume measurement.

Your turn

Try it yourself

Evaluate the static Gaussian Kubo–Toyabe polarization at $\Delta t=1$ and in the long-time limit. Explain why either value alone cannot determine the magnetic volume fraction.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

At Delta t equal to one, the polynomial multiplying the exponential vanishes.

Compare with the solution

At $\Delta t=1$ the exponential term is multiplied by zero, giving $G_{\rm KT}=1/3$ . At very long times the exponential vanishes, again giving $1/3$ . These are projections of a normalized static isotropic ensemble. Extracting a sample fraction still requires asymmetry calibration, stopping fractions, background, site populations, and any unresolved fast relaxation.

Keep the idea

Take it with you

Muon precession gives a local field through an angular-frequency convention and stopping-site model.
Static dephasing and temporal fluctuations can produce similar-looking relaxation over limited windows.
Asymmetry amplitudes become volume estimates only after calibration and background modeling.

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