Introduction

Muon spin spectroscopy (or Muon Spin Rotation, μSR) is a technique commonly used to investigate the magnetic states within the material. Magnetic states of the material include diamagnetism, paramagnetism and ferromagnetism, where they differ in the ordering of magnetic spins within a material. μSR uses the rotation of muons in response to the magnetic field of the solid to obtain information about the material^{[1, p.394-398]}. Muons act as a volume averaged probe to measure local magnetic fields, giving insights into the magnetic phenomena observed in solids^[2]. 

Muons

Muons are leptons and can exist as positively charged (μ⁺) or negatively charged (μ^-) particles^[3]. When a negatively charge muon is implanted in a solid, the muon will be captured by the positively charged nucleus and cascades into the lowest muonic orbital, undergoing nuclear capture ^{[3][4, p.149-178]}. Hence, in muon spin spectroscopy, we mostly work with μ⁺ particles ^[1]. The following table shows the properties of muons (Table 1).

Table 1: Properties of muons 

Production of Muons 

When cosmic rays collide with the upper atmosphere, muons are produced. These muons produced naturally can be used to study very large objects with radiography ^[2]. However, these muons have very low flux and cannot be used for atom-scale measurements.

Hence, for μSR, muons are generated by the irritation of graphite or beryllium production targets with high energy protons ^[1][3]. These high energy protons interact with the protons and neutrons in the graphite target to produce pions. This is followed by the decay of pions via a weak interaction into muon and neutrinos.

Based on conservation of momentum and energy and assuming the pion decays at rest, the neutrino and muons will be emitted in opposite directions. Pions are particles with a spin of zero, while both muons and neutrinos have a spin of ½. Hence, based on the conservation of angular momentum, the angular momentum of the neutrino and muon are opposite of each other. Additionally, based on violation parity (in quantum mechanics, parity conservation is defined by two physical systems, where one of which is a mirror image of the other, must behave identically), only neutrinos with spin antiparallel to linear momentum exists. Therefore, muons produced will have a spin ½ antiparallel to its linear momentum^[4].

Figure 1 illustrates the overall process of muon production.

Figure 1. Schematic illustration of muon production (Figure: Audrey Kwan (inspired by ^[4])

Muon Decay

A muon decays after 2.2 x10^-6s into a positron (e⁺), neutrino (ν_μ), and antineutrino(ν_e). This decay is a three-body decay that occurs by a process of weak interaction .

As soon as the muons are implanted into the sample, the muon interacts with the internal magnetic field with frequency \( ω_μ=γ_μ B \)  and decays, emitting a positron, neutrino, and antineutrino. Based on parity violation, antineutrino will have a spin parallel to its linear momentum, while both the neutrino and positron will possess a spin antiparallel to its linear momentum. Since the decay occurs via a weak interaction, the positron emitted will have a range of energies that correlates with the orientation of the muon’s spin at the time of decay ^[1][3][4]. The spatial distribution of the emitted positron energy can be described in the following mathematical equation and is illustrated in Figure 2. At maximum positron energy, the decay is a collinear case where both the neutrino and antineutrino is emitted in opposite direction from the positron, as shown in Figure 3.

where  \( E(x)= 2x^2 (3-2x) \) and  \( A(x)= \frac {2x-1}{3-2x}\ \) and x is the reduced positron energy and is defined by the ratio of positron energy/ maximum positron energy

Figure 2. Polar diagram of the angular distribution W(θ) of positrons from muon decay: maximum positron energy when W(θ)=1.  Figure from Wikipedia (License: CC-BY-SA-4.0) 

Figure 3. Schematic illustrating the collinear case of muon decay (Figure: Audrey Kwan, inspired by ^[4])

Principle

Emitted muons trigger the clock that defines time t₀, which is defined by the signal produced when the muon crosses the muon counter in front of the sample^{[5, p.1119-1180]}. The implanted muon will then interact with internal magnetic fields of the material and precess around the field, rendering the polarisation of muons (P_m) to be time dependent.

The rate of positrons emitted from the muon decay will be recorded by the array of detectors around the samples ^[1]. Therefore, by detecting the direction of positron emission, it is possible to ascertain the polarisation of muon’s spin as a function of time.  The number of positrons detected as a function of time is represented by the following equation:

where B: time-independent background due to uncorrelated events 

           N₀: initial intensity and  \( N_0=\frac{N_{μ,0}∆Ω_ℇ}{4πτ_μ}\ \)

           τ_μ: muon lifetime 

           A₀: asymmetry term 

            P(t): time-dependent muon polarisation

The detector will record the number of positrons emitted at a specified time interval, plotting a graph of rate of positron emission against time. After correcting for the background and exponential decay of muon, we can obtain a graph for the μSR signal (asymmetry signal) that reflects the time dependent muon polarisation.

Experimental Set-up

There are different experimental set-ups depending on the material properties you are investigating. We will cover some of the more common techniques used.

Zero-field Technique

Zero-field technique (ZF) is used to investigate the magnetic system in the material by measuring the effects of muon polarisation produced by the internal field of the sample ^[5]. Figure 4 is a schematic of the set-up for conducting zero-fields experiments. The muon spin is usually antiparallel to the muon beam direction.

Figure 4. Schematic of zero-field (ZF) experimental set-up (Figure: Audrey Kwan, inspired by ^[5])

Figure 5 shows arbitrary results that can be obtained from the backward detector in a zero-field experiment. For a paramagnetic object, the magnetic spins are randomly ordered in the material. Hence, the muon polarisation does not show a change with time and a horizontal line will be observed for the μSR signal. On the other hand, for a ferromagnetic object, with magnetic spins that are aligned with each other, muon spin precess around the local field with frequency corresponding to Lamour frequency and can be observed from the μSR signal^[4][5].

Figure 5: Arbitrary results from a ZF experiment, where the graphs on the left (a) corresponds to the results obtained from paramagnetic objects and the graphs on the right (b) corresponds to the results obtained from ferromagnetic objects (Figure: Audrey Kwan, inspired by ^[5])

Moreover, ZF techniques can be used to determine sample inhomogeneity of magnetic ordering. A set of arbitrary results are shown in Figure 6. Both samples are assumed to have similar magnetisation values but one of the samples is inhomogeneous. For the inhomogeneous sample, the paramagnetic portions of the sample will not contribute to the μSR signal, resulting in a weaker signal detected. Moreover, as the asymmetry obtained for the inhomogeneous sample, we are also able to determine the volume fraction of the paramagnetic portions since the muon signal will be made of two components- the paramagnetic and the ferromagnetic components.

Figure 6. Arbitrary results for distinguishing magnetic homogeneity (Figure: Audrey Kwan, inspired by ^[5])

Amitsuka et al conducted a zero-field experiment to determine the inhomogeneous antiferromagnetic phase of a heavy-fermion superconductor URu₂Si₂ at low temperatures^[6]. This experiment was conducted at high pressures. Based on the asymmetry graphs, they were able to obtain the volume fraction of the antiferromagnetic phase based on the following formula: v_AF=A_osc/(A−A_cell)

where A_cell is the asymmetry component from the background

           A_osc is the asymmetry from oscillating component from the muon

           A is the total asymmetry estimated from transverse field measurements.

_{Transverse Field Technique}

Transverse-field (TF) experiments are conducted with an external magnetic field applied transverse to the initial direction of the muon beam^[4]. The experimental set-up is shown in Figure 7. Figure 8 shows some arbitrary results of the graphs you can expect to obtain from the detector.

Figure 7. Schematic of transverse field experimental set-up (Figure: Audrey Kwan, inspired by ^[5])

Figure 8. Arbitrary results from a transverse-field experiment (Figure: Audrey Kwan, inspired by ^[5])

Yamauchi et al used both ZF and TF techniques to investigate the local spin structure of α-RuCl3 honeycomb-lattice magnet. By conducting the zero-field experiments, he was able to observe the magnetic transitions at the critical temperature, 7K, where α-RuCl₃ transitioned from paramagnetic to antiferromagnetic. Furthermore, he applied an external magnetic field in the paramagnetic phase to deduce the muon stopping sites. External magnetic field was applied parallel to the honeycomb planes. From the frequency shifts observed, with decreasing temperatures, the frequency peaks decreases and broadens. Thus, they inferred that the stopping sites were in between the honeycomb planes. In this experiment, muon spin spectroscopy was used since muons are local probes and can be implanted into the material to analyse the magnetic properties at specific interfaces within the bulk material.^[7].

Advantages and Disadvantages of Muon Spin Spectroscopy

Unlike nuclear magnetic resonance, there is no surface effect in muon spin rotation as nuclear magnetic resonance requires a small external field whereas in muon spin rotation it can be studied in a zero magnetic field with no external probe field ^[1]. Furthermore, this technique has high sensitivity as muons are able to detect very small magnetic moments compared to other spectroscopy techniques. Moreover, as the probe can be implanted in the material, muons are not susceptible to surface effects and the measurements can represent the local magnetic properties ^[2]. 

However, muon spin spectroscopy requires a large accelerator for muon beam production, limiting this technique to large laboratories ^[1]. In addition, there is a short time window to conduct the experiment, limited by the decay of muons. Another disadvantage of using this technique is the inability to precisely determine the site where muons stop, making it challenging to ascertain its behaviour in the sample ^[3].

References