Polarized muon spins in pulsed magnetic fields

Dipartimento di Fisica e Istituto Nazionale per la Fisica della Materia
Università di Parma, Parco Area delle Scienze 7a, I-43100 Parma, Italy

Low energy (4.2 MeV) positive muons produced at particle accelerators provide a very useful tool for studying condensed matter. They are I=1/2 leptons with a mean lifetime of ~2.2 µs, which may be implanted (i.e. brought to rest) into any material. Both their birth - from the decay of a pion - and their own decay into a positron and two neutrinos, are governed by the parity violating weak interactions.

At their birth, parity violation means that all the 4.2 MeV muons have their spin (i.e. their magnetic moment) anti-parallel to their velocity, rather than distributed randomly.
At their decay, it provides a high correlation (absent when parity is conserved) between the muon spin direction and the direction in which the positron is emitted in the muon decay.

In a typical µSR experiment [1, 2, 3] (the acronym generally stands for Muon Spin Rotation) an ensemble of muons are implanted in a sample and stop in equivalent sites - either an interstice of the crystal lattice or a chemically bound state in a molecule. Since all muon spins are initially aligned, the direction of enhanced positron emission coincides with and follows coherently the muon-spin direction. Any internal magnetic field is reflected in a modulation (Larmor precession) of the positron count rate from the decay of the ensemble, just like the beam of a lighthouse.

Hence µSR provides a powerful local microscopic magnetic probe, whose use, on the par with NMR [4, 5], is by no means limited to magnetic materials.

A very important feature of µSR relies on the chemical behaviour of the muon as a light isotope of hydrogen. For both muons and protons the chemical state ranges from nearly free, in a metal - due to strong conduction electron shielding - to bound in a diamagnetic molecule, and finally to the formation of paramagnetic radical states - the prototype of which are atomic hydrogen and its counterpart, muonium (Mu = e^- + µ⁺). These states are directly revealed by the spin precessions, which in Mu, for example, are largely determined by the huge hyperfine field due to the unpaired electron magnetic moment.

A largely debated question regards muonium and its radicals formation, which can take place with a negligible formation time, typically in an epithermal regime, or according to a given rate equation, with a long formation time, as it may happen in a thermal regime. When studying the last case, the precession coherence is lost in a constant magnetic field and practically no direct evidence is so far available for such delayed processes.

Our work was inspired by the possibility of a straightforward investigation of delayed processes, which we show to be experimentally possible when using a pulsed source of polarized muons (ISIS at RAL) together with a suddenly applied external magnetic field properly delayed with respect to the muon pulses [7].

Intuitively speaking, in a delayed formation process a delayed applied magnetic field recovers the precession phase coherence that could otherwise be lost if the magnetic field were constantly present at all times. It is like observing the light of a large number of lighthouses, which are turned on (lit and set in rotation) at times spread over an interval: if this interval is shorter than the beam period one still observes the sweep of the total beam, while if the interval is much larger than the sweeping period the single lights merge into an average illumination. Our pulsed field solution is analogous to inhibiting the beam rotation until all lighthouses are lit.

Conventional µSR detects the muon spin precession directly as a modulation signal in the time domain, times being measured from the instant of the muon implantation. The pulsed field technique introduces a second time domain, that of the variable delay. The availability of these two times, which gives quite naturally access to correlation functions, <S(t)·S(t+dt)>, is well known and fully exploited in pulsed NMR.
The prototype examples are the Inversion Recovery and Spin Echo techniques, introduced by Hahn [6] in 1950, where the delayed application of a radio frequency pulse yields direct measures of the relaxation rates.
Pulsed field µSR is just an introductory example which we have devised to explore the corresponding concept for the muons.

The technical feasibility of such experiments depends critically on two characteristic times, namely the time scale during which the magnetic field varies and the typical precession period of the muon (muonium) spin. The muon spin behaviour is determined by the ratio of these two typical times. This means that depending on how suddenly the field is changed the spins either follow it adiabatically (adiabatic regime) or precess around it (sudden regime). An accurate measurement of the cross-over from sudden to adiabatic regimes is the prerequisite for the manipulation of the initial spin state, which allows a whole new range of experiments in µSR.

The pulsed magnetic field is achieved by using a flat-loop device [8] in which spatially uniform pulsed currents flow in opposite directions in two parallel slabs with the sample sandwiched in between. We use a quartz disk sample in which 90% of muons form muonium and the rest stays in a diamagnetic µ⁺ form.

Fig. 1: µSR signal for two different delays of a pulsed transverse field. Magnetic field is switched on before (a), and after (b) the muon pulse arrival.
(a) Muonium atoms precess out of phase and only the slow precession of µ⁺ is observed.
(b) No dephasing occurs and the coherent muonium precession signal can be observed.
The initial variable frequency signal due to the gradual increase of the transverse field is shown in the insert.

These two species have very different gyromagnetic ratios which offer the possibility of testing the adiabatic-to-sudden cross-over in two extreme limits. Fig. 1.a shows what happens when the field is applied before the muons pulse arrival. It is equivalent to a constantly applied field and no muonium is observed since its fast signal is lost due to dephasing. Only the slow precession of free muons can be observed. However, when the pulsed field is applied after the muons implantation, it is the field itself which sets the initial time for the precession, so the full muon and muonium signals can both be measured Fig. 1.b).

For technical reasons we introduced also a steady longitudinal field, inessential to the concepts thus far explained, which creates a rotating field at an adjustable rate. Fig. 2.a shows that, when the field rotates at high rates relative to the muonium frequency, both muonium and muon respond by precessing around the field with a large cone. In the opposite case, i.e. for lower field rotation rates (fig. 2.b), the muonium spins are fast enough to adiabatically follow its variations. Bare muons with their ~ 100 times lower gyromagnetic ratio (not shown) would still continue to precess as in the sudden regime.

Fig. 2: The initial spin s₀ and field B_|| lie along x-axis. When the transverse B_t(t) is switched on, the spin precession around the total field B(t) is indicated by the solid curve generated by the tip of the spin arrow.
a) and b) show typical sudden and adiabatic behaviours, respectively.

For an introduction to the µSR spectroscopy and its applications see for example: