Since the discovery of cosmic rays in 1940, elementary particle muons have become fascinating and exotic particles which can be objectives and/or tools for fundamental physics and applied science. In particular, after intense muons became available by using particle accelerators in 1960, the field of scientific research using muons has been growing year by year.

From the author's viewpoint, three major unique features of muons have formed the basis of all muon-related scientific research: (1) unique mass, such as heavy electrons and light protons; (2) radioactivity with polarization phenomena; and (3) the electromagnetic interaction nature with matter without a strong interaction. These features have promoted the application of muons to: (1) muon catalyzed fusion for future atomic energy; (2) sensitive probes of the microscopic magnetic properties of new materials and biomolecules; and (3) radiography of a large-scale substance for preventing natural disasters, respectively.

The author made special efforts for this book to include a self-contained description of all physics principles required for muon applications. In these applications, we note that muons may be the key particles to provide answers to the basic problems associated with possible crises in human life in the twenty-first century, namely, a shortage of energy resources, the need for more information on the biological functioning of the human body, and the need to prevent natural disasters, such as volcanic eruptions and earthquakes. Complete descriptions are given of ways to apply elementary particle muons to these three major human problems.

The principle of the muon spin rotation/relaxation/resonance (μSR) method is based upon the laws of particle physics. As seen in Figure 1.5, the spin of the μ+(μ−), when it is born via the decay of the π+(π−, is completely polarized along the direction of its motion; once the μ+(μ−) are focused or collimated along one direction, the resulting beam is polarized along its direction of motion. During the slowing-down of the μ+(μ− inside the host material, as described in Chapter, the spin polarization is maintained in the long-lived form of diamagnetic μ+, paramagnetic Mu (ortho state with spin = 1), or ground state of a muonic atom in the case of μ−. After stopping at some specific microscopic location, the μ+(μ−) decays into e+(e−) and two neutrinos, as shown in Figure 1.5, with the e+ (e−) spatial distribution oriented preferentially along the μ+(μ−) spin direction. The decay e+(e−) energy ranges up to 50 MeV, and the direction of the μ+(μ−) spin can be observed in a time-resolved fashion by measuring these high-energy e+(e−) using detectors placed outside the target material to be investigated; measurements are carried out under variations of external conditions such as temperature, pressure, and applied magnetic or electric fields.

The μSR method can be considered as a sensitive magnetic “compass” to probe the microscopic magnetic properties of condensed matter.

The spatial profile of the depth or density times length of the substance can be known by observing the manner of penetration of the energetic particle determined by electromagnetic interaction between the incoming particle and the stopping substance, which is called radiography. The most popular radiography is X-ray photography of a human body. As summarized in Table 9.1 and Figure 9.1, among various possible particles, very-high-energy (≥ 100 GeV) muon is the most suitable for measuring the density profile of a large-scale (≥ 0.1 km) substance like a mountain.

As described in Chapter 9, GeV–TeV cosmic-ray muons are constantly irradiating every substance on the earth. Muons arriving vertically from the sky have an intensity of 1 muon/cm2 per min with a mean energy of a few GeV. The potential use of such high-energy muons to explore the internal structure of large-scale objects has been recognized in the past, with the prime example being the work done by Alvarez (1970), who studied the inside of an Egyptian pyramid in order to find a hidden chamber.

Muons arriving nearly horizontally along the earth's surface with a θz slightly less than 90° have a lower intensity on average, but have a higher intensity at energies higher than a few hundreds of GeV, as can be seen in Figure 2.10.

Application of μSR to studies of the intrinsic properties of condensed matter

The applications of muon spin rotation/relaxation/resonance (μSR) to condensed-matter studies can be roughly categorized into two types:

1. The probing of microscopic magnetic properties of the target material which are essentially unchanged by the muon's presence. In this case, the most important features of the experiment are the muon's capability to measure magnetic properties under zero external field, its unique response to spin dynamics, its sensitive detection of weak and/or random microscopic magnetic fields, and so forth. We may call this type of μSR “passive” probe.

2. The creation of a new microscopic condensed-matter system in the target material by the introduction of μ+, Mu, or μ−, and the study of the unique microscopic response of the material. Here, representative central topics are the presence or localization of the μ+ at the interstitial sites and its diffusion properties, electronic properties around μ+/Mu, the chemical reactions undergone by the hydrogen-like Mu center in semiconductors, transport phenomena of the electron brought in and probed by the μ+ in conducting polymers and biomolecules, and so forth. We may call this type of μSR “active” probe.

Activity in the field of μSR applications in condensed-matter studies has been increasing since the late 1970s, in parallel with the progress of intense proton accelerator facilities (known as meson factories) such as Los Alamos Meson Physics Facility (LAMPF), Paul Scherrer Institute (PSI), and Tri-University Meson Facility (TRIUMF).

The mean field theories of phase transitions considered in the previous chapter explicitly or implicitly neglect fluctuations, i.e. local deviations of the order parameter from its most probable, macroscopic value. This neglect is epitomized in the basic assumption underpinning Landau's theory, i.e. that the order parameter is uniformly replaced by the value which minimizes the phenomenological Landau free energy; fluctuations around this most probable value do not contribute to the partition function (4.36). The fact that local deviations of the order parameter from its average are not built into mean field considerations means, a fortiori, that correlations between fluctuations occurring in different subsystems are neglected. Correlations on the molecular scale, as quantified by the density correlation functions or static structure factors introduced in section 3.4, dominate first-order phase transitions like freezing, and can be accounted for quantitatively within standard structural theories of dense fluids, like the mean spherical approximation (MSA) and Percus–Yevick (PY) approximation mentioned in chapter 3. The range of these spatial correlations is typically a few molecular diameters. However, upon approaching the critical point of a second-order phase transition, e.g. the liquid–vapour critical point, density fluctuations become correlated over mesoscopic scales, while the macrosopic response of the system (e.g. the compressibility) diverges. A quantitative description of large-scale fluctuations is a highly challenging problem which has only been understood since the 1970s in terms of the idea of scale invariance, one of the key concepts of this chapter, which may also be applied to other, apparently unrelated, ‘critical’ objects like polymers.

One of the most remarkable observations in physical sciences or, for that matter, of everyday life, is that most substances, with a well defined chemical composition, can exist in one of several states, exhibiting very different physical properties on the macroscopic scale; moreover one can transform the substance from one state (or phase) to another, simply by varying thermodynamic conditions, like temperature or pressure. In other words, a collection of N molecules, where N is typically of the order of Avogadro's number NA, will spontaneously assemble into macroscopic states of different symmetry and physical behaviour, depending on a limited number of thermodynamic parameters. The most common states are either solid or fluid in character, and are characterized by qualitatively different responses to an applied stress. At ambient temperatures, the solid states of matter are generally associated with the mineral world, while ‘soft’ matter, and in particular the liquid state, are more intimately related to life sciences. In fact it is generally accepted that life took its origin in the primordial oceans, thus underlining the importance of a full quantitative understanding of liquids. However, even for the simplest substances, there are at least two different fluid states, namely a low density ‘volatile’ gas (or vapour) phase, which condenses into a liquid phase of much higher density upon compression or cooling. For more complex substances, generally made up of highly anisotropic molecules or of flexible macromolecules, the liquid state itself exhibits a rich variety of structures and phases, often referred to as ‘complex fluids’.

Probing dynamical properties at equilibrium

By definition, the macroscopic properties of a system at equilibrium do not change with time. This does not mean, however, that the system is dynamically inert. On the contrary, the equilibrium state is associated with permanent motion at the molecular level. Although this motion is sometimes described as random ‘thermal noise’, it is in fact quite well organized, and reflects the microscopic processes that govern the dynamics of the system. As these same processes will also determine the system response to an external perturbation, understanding their organization and time evolution is of primary importance for the determination of the material response, and of its relationship to the microscopic structure.

As a simple illustration of how seemingly random dynamical processes are organized in a coherent fashion, one may consider the case of vibrations in a crystalline solid. Each atom oscillates around its equilibrium position, in a way which is apparently very random. The well known harmonic analysis, however, shows that this motion is really caused by the superposition of well defined sound waves, the phonons, with different phases and directions of propagation. This organization of the atomic motions into excitations of well defined spatial and temporal structure determines many thermodynamic and transport properties of the crystal.

The harmonic crystal is of course particularly simple, since it is possible in that case to deduce analytically from the interaction potential the structure of the coherent excitations. In more complex, disordered systems, an analytical treatment is usually out of reach. Nevertheless, the information on the way atomic motions are organized is encoded in the correlation functions of atomic positions.