Muon spin spectroscopy
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Muon spin spectroscopy, also known as µSR, is an experimental technique based on the implantation of
Introduction
More generally speaking, muon spin spectroscopy includes any study of the interactions of the muon's magnetic moment with its surroundings when implanted into any kind of matter. Its two most notable features are its ability to study local environments, due to the short effective range of muon interactions with matter, and the characteristic time-window (10−13 – 10−5 s) of the dynamical processes in atomic, molecular and condensed media. The closest parallel to µSR is "pulsed NMR", in which one observes time-dependent transverse nuclear polarization or the so-called "free induction decay" of the nuclear polarization. However, a key difference is that in µSR one uses a specifically implanted spin (the muon's) and does not rely on internal nuclear spins.
Although particles are used as a probe, µSR is not a diffraction technique. A clear distinction between the µSR technique and those involving neutrons or
As with many of the other nuclear methods, µSR relies on discoveries and developments made in the field of particle physics. Following the discovery of the muon by
Muon production
The collision of an accelerated proton beam (typical energy 600 MeV) with the nuclei of a production target produces positive pions () via the possible reactions:
From the subsequent weak decay of the
Parity violation in the weak interactions implies that only left-handed neutrinos exist, with their spin antiparallel to their linear momentum (likewise only right-handed anti-neutrino are found in nature). Since the pion is spinless both the neutrino and the are ejected with spin antiparallel to their momentum in the pion rest frame. This is the key to provide spin-polarised muon beams. According to the value of the pion momentum different types of -beams are available for µSR measurements.
Energy classes of muon beams
Muon beams are classified into three types based on the energy of the muons being produced: high-energy, surface or "Arizona", and ultra-slow muon beams.
High-energy muon beams are formed by the pions escaping the production target at high energies. They are collected over a certain solid angle by
The second type of muon beam is often called the surface or Arizona beam (recalling the pioneering work of Pifer et al.
Positive muon beams of even lower energy (ultra-slow muons with energy down to the eV-keV range) can be obtained by further reducing the energy of an Arizona beam by utilizing the energy-loss characteristics of large
Continuous vs. pulsed muon beams
In addition to the above-mentioned classification based on energy, muon beams are also divided according to the time structure of the particle accelerator, i.e. continuous or pulsed.
For continuous muon sources no dominating time structure is present. By selecting an appropriate incoming muon rate, muons are implanted into the sample one-by-one. The main advantage is that the time resolution is solely determined by the detector construction and the read-out electronics. There are two main limitations for this type of source, however: (i) unrejected charged particles accidentally hitting the detectors produce non-negligible random background counts; this compromises measurements after a few muon lifetimes, when the random background exceeds the true decay events; and (ii) the requirement to detect muons one at a time sets a maximum event rate. The background problem can be reduced by the use of electrostatic deflectors to ensure that no muons enter the sample before the decay of the previous muon. PSI and TRIUMF host the two continuous muon sources available for µSR experiments.
At pulsed muon sources
Spectroscopic technique
Muon implantation
The muons are implanted into the sample of interest where they lose energy very quickly. Fortunately, this deceleration process occurs in such a way that it does not jeopardize a μSR measurement. On one side it is very fast (much faster than 100 ps), which is much shorter than a typical μSR time window (up to 20 μs), and on the other side, all the processes involved during the deceleration are Coulombic (ionization of atoms, electron scattering, electron capture) in origin and do not interact with the muon spin, so that the muon is thermalized without any significant loss of polarization.
The positive muons usually adopt
Detection of muon polarization
The decay of the positive muon into a positron and two neutrinos occurs via the weak interaction process after a
Parity violation in the weak interaction leads in this more complicated case (three body decay) to an anisotropic distribution of the positron emission with respect to the spin direction of the μ+ at the decay time. The positron emission probability is given by
where is the angle between the positron trajectory and the μ+-spin, and is an intrinsic asymmetry parameter determined by the weak decay mechanism. This anisotropic emission constitutes in fact the basics for the μSR technique.
The average asymmetry is measured over a statistical ensemble of implanted muons and it depends on further experimental parameters, such as the beam spin polarization , close to one, as already mentioned. Theoretically =1/3 is obtained if all emitted positrons are detected with the same efficiency, irrespective of their energy. Practically, values of ≈ 0.25 are routinely obtained.
The muon spin motion may be measured over a time scale dictated by the muon decay, i.e. a few times τμ, roughly 10 µs. The asymmetry in the muon decay correlates the positron emission and the muon spin directions. The simplest example is when the spin direction of all muons remains constant in time after implantation (no motion). In this case the asymmetry shows up as an imbalance between the positron counts in two equivalent detectors placed in front and behind the sample, along the beam axis. Each of them records an exponentially decaying rate as a function of the time t elapsed from implantation, according to
with for the detector looking towards and away from the spin arrow, respectively. Considering that the huge muon spin polarization is completely outside thermal equilibrium, a dynamical relaxation towards the equilibrium unpolarized state typically shows up in the count rate, as an additional decay factor in front of the experimental asymmetry parameter, A. A magnetic field parallel to the initial muon spin direction probes the dynamical relaxation rate as a function of the additional muon Zeeman energy, without introducing additional coherent spin dynamics. This experimental arrangement is called Longitudinal Field (LF) μSR.
A special case of LF μSR is Zero Field (ZF) μSR, when the external magnetic field is zero. This experimental condition is particularly important since it allows to probe any internal quasi-static (i.e. static on the muon time-scale) magnetic field of field distribution at the muon site. Internal quasi-static fields may appear spontaneously, not induced by the magnetic response of the sample to an external field They are produced by disordered nuclear magnetic moments or, more importantly, by ordered electron magnetic moments and orbital currents.
Another simple type of μSR experiment is when implanted all muon spins precess coherently around the external magnetic field of modulus , perpendicular to the beam axis, causing the count unbalance to oscillate at the corresponding
Since the Larmor frequency is , with a gyromagnetic ratio Mrad(sT)−1, the frequency spectrum obtained by means of this experimental arrangement provides a direct measure of the internal magnetic field intensity distribution. The distribution produces an additional decay factor of the experimental asymmetry A. This method is usually referred to as Transverse Field (TF) μSR.
A more general case is when the initial muon spin direction (coinciding with the detector axis) forms an angle with the field direction. In this case the muon spin precession describes a cone which results in both a longitudinal component, , and a transverse precessing component, , of the total asymmetry. ZF μSR experiments in the presence of a spontaneous internal field fall into this category as well.
Applications
Muon spin rotation and relaxation are mostly performed with positive muons. They are well suited to the study of
The London penetration depth is one of the most important parameters characterizing a
Other important fields of application of µSR exploit the fact that positive muons capture electrons to form muonium atoms which behave chemically as light isotopes of the hydrogen atom. This allows investigation of the largest known kinetic isotope effect in some of the simplest types of chemical reactions, as well as the early stages of formation of radicals in organic chemicals. Muonium is also studied as an analogue of hydrogen in semiconductors, where hydrogen is one of the most ubiquitous impurities.
Facilities
µSR requires a
See also
Notes
- ^ Resonance techniques are often characterized by the use of resonant circuits, which is not the case for muon spin spectroscopy. However the true resonant nature of all these techniques, muon spectroscopy included, lies in the very narrow, resonant requirement upon any time dependent perturbation in order for it to effectively influence the probe's dynamics: for every excitation interacting with the muon (lattice vibrations, charge and electronic spin waves) only those spectral components very closely matching the muon precession frequency in the specific experimental condition can cause a significant muon spin motion.
References
- .
- doi:10.1063/1.881018.
- PMID 10033111.
- PMID 9942727.