Muon spin spectroscopy
Encyclopedia
Muon spin spectroscopy is an experimental technique based on the implantation of spin-polarized
muon
s in matter and on the detection of the influence of the atomic, molecular or crystalline surroundings on their spin motion. The motion of the muon spin
is due to the magnetic field experienced by the particle and may provide information on its local environment in a very similar way to other magnetic resonance
techniques, such as electron spin resonance (ESR or EPR) and, more closely, nuclear magnetic resonance
(NMR).
around a still magnetic field
), or a relaxation towards an equilibrium direction, or, again, a more complex dynamics dictated by the addition of short radio frequency
pulses. The intention of the mnemonic acronym was to draw attention to the analogy with NMR
and ESR
. More generally speaking, the abbreviation covers any study of the interactions of the muon's magnetic moment with its surrounding when implanted into any kind of matter.
and diffraction
techniques. The NMR technique which is closest parallel to µSR is "pulsed NMR", in which one observes time-dependent transverse nuclear polarization or so-called "free induction decay
" of the nuclear
polarization. However, a key difference is the fact that in µSR one uses a specifically implanted spin (the muon's) and does not rely on internal nuclear spins.
In addition, and due to the specificity of the muon, the µSR technique does not require any radio-frequency technique to align the probing spin. On the other hand, a clear distinction between the µSR technique and those involving neutrons or x-rays is that scattering is not involved. Neutron diffraction
techniques, for example, use the change in energy and/or momentum of a scattered neutron
to deduce the sample properties. In contrast, the implanted muons are not diffracted but remain in a sample until they decay. Only a careful analysis of the decay product (i.e. a positron
) provides information about the interaction between the implanted muon and its environment in the sample.
As 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 Seth Neddermeyer
and Carl D. Anderson in 1936, pioneer experiments on its properties were performed with cosmic rays. Indeed, with one muon hitting each square centimeter of the earth's surface every minute, the muons constitute the foremost constituent of cosmic rays arriving at ground level. However, µSR experiments require muon fluxes of the order of
muons per second and square centimeter. Such fluxes can only be obtained in high-energy particle accelerators which have been developed during the last 50 years.
) via the possible reactions:
From the subsequent decay of the pions (mean lifetime
= 26.03 ns) positive muons (
) are formed via the two body decay:
Since the neutrino has an helicity
= -1, both the neutrino and the 
have their spin antiparallel to their momentum in the pion rest frame. According to the value of the pion momentum at the decay-time, different types of 
-beams are available for µSR measurements.
In the laboratory frame the polarization of a high-energy muon beam is limited to about 80% and its energy is of the order of ~40-50MeV. Although such a high energy beam requires the use of suitable moderators and samples with sufficient thickness, it guarantees an homogeneous implantation of the muons in the sample volume. Such beams are also used to study specimens inside of recipients, e.g. samples inside pressure cells.
Such muon beams are available at PSI
, TRIUMF
, J-PARC
and RIKEN-RAL.
). Here muons are used that arise from pions decaying at rest still inside, but near the surface, of the production target. Such muons, which are 100 % polarized, ideally monochromatic and have a very low momentum of 29.8 MeV/c, which corresponds to a kinetic energy of 4.1 MeV, have a range width in matter of the order of 180 mg/cm2. Hence the paramount advantage of this type of beam is the possibility to use relatively thin samples.
Such muon beams are available at PSI
(Swiss Muon Source SµS), TRIUMF
, J-PARC
, ISIS
and RIKEN-RAL.
Up to now, PSI
is the only Facility where such low-energy muon beam is available on a regular basis. Technical developments have been also conducted at RIKEN-RAL, but with a strongly reduced low-energy muons rate. J-PARC
is projecting the development of a high-intensity low-energy muon beam.
For continuous (or cw= continuous wave
) muon sources no dominating time structure is present. By selecting an appropriate muon incoming 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 sources: (i) a non-negligible random background signal; (ii) a rather limited muon incoming rate. Only the background problem can be overcome by using electrostatic deflectors to ensure that no muons enter the sample until the decay of the precedent muon has been observed.
PSI
and TRIUMF
host the two cw muon sources available for µSR experiments.
For pulsed muon sources protons hitting the production target are bunched into pulses which is also reflected on the secondary muon beam. The advantages of a pulsed muon sources are that in principle, one can use the entire incoming muon intensity and there is almost no background due to accidental coincidences between the incoming muons and decay positrons. The absence of background allows the extension of the time window for measurements up to about ten times the muon mean lifetime. The reverse of the medal is that the width of the muon pulse limits the time resolution.
ISIS
and J-PARC
are the two pulsed muon sources available for µSR experiments.
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 interstitial sites of the crystallographic lattice. In most metallic samples the muon's positive charge is collectively screened
by a cloud of conduction electrons. Thus, in metals, the muon is in a so-called diamagnetic state and behave like a free muon. In insulators or semiconductors a collective screening cannot take place and the muon will usually pick-up one electron and form a so-called muonium
(Mu=μ++e-), which has similar size (Bohr radius
), reduced-mass
and ionization energy
to the hydrogen
atom.
τμ = 2.197034(21) μs:
The parity violation in the weak interaction leads 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 asymmetry parameter which depends on the polarization of the muon beam and positron energy. This anisotropic emission constitutes in fact the basics for the μSR technique.
The asymmetry of
is given by 
, where
is the beam polarization (of the order of ~1) and
is an intrinsic asymmetry parameter determined by the weak decay mechanism. Theoretically, an average of 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 time scale on which the spin motion may be exploited is that of the muon decay
, i.e. a few mean lifetimes, each roughly 2.2 µs (2.2 millionths of a second). Both the production of muon beams with nearly perfect alignment of the spin to the beam direction (what was referred to above as spin polarization
and caused by the spontaneous symmetry breaking), and the ability to detect the muon spin direction at the instant of the muon decay rely on the violation of parity
, which takes place whenever weak forces are at play.
In short this means that certain elementary events happen only when including clockwise (or only when including counter-clockwise) rotations. For instance, the positive muon decays into a positron
plus two neutrino
s and the positron is preferentially emitted in the direction of the muon spin. Therefore it would most often see the spin
as a counter-clockwise rotation while flying away from the decay point.
Spin alignment allows the production of a muon beam with an aligned magnetic moment. Muons are injected into the material under investigation as short-lived probes sending information from the interior back out to the experimental apparatus. These muons are able to send a message from inside the crystal about the local magnetic field in their surroundings. After some time (mean lifetime 2.2 µs) these spies decay and emit positrons. A beam of aligned muons produces asymmetric positron radiation. The asymmetry of positron radiation contains information about the direction of local magnetic field in the moment of muon decay. Taking into consideration the initial direction of muon magnetic moment and the time interval between the moment of injection and moment of muon decay we can calculate the precession frequency (how rapidly the muon's magnetic moment rotates). The frequency of magnetic moment precession depends on the local magnetic field. Larmor precession
is appeared with z-direction magnetic field and only decay in 2.2 µs. But when x-direction magnetic field is applied in muon, the rate of decay is enhanced by gaussian with depolarization rate.
Since 1987 this method was used to measure internal magnetic fields inside high-temperature superconductors
. High-temperature superconductors are Type II superconductors, in which the local magnetic fields inside the superconductor depend on the superconducting carrier density—one of the significant parameters of any superconductor (see for example the Bardeen–Cooper–Shrieffer theory
of superconductors).
s at the atomic scale inside matter, such as those produced by various kinds of magnetism
and/or superconductivity
encountered in compounds occurring in nature or artificially produced by modern material science.
The London penetration depth is one of the most important parameters characterizing a superconductor
because its inverse square provides a measure of the density ns of Cooper pairs. The dependence of ns on temperature and magnetic field directly indicates the symmetry of the superconducting gap. Muon spin spectroscopy provides a way to measure the penetration depth, and so has been used to study high-temperature cuprate superconductors since their discovery in 1986.
Other important fields of application of µSR exploit the fact that positive muons capture electrons to form muonium
atoms which behave chemically as light isotope
s of the hydrogen
atom. This allows investigation of the largest known isotope effect
in some of the simplest types of chemical reactions, as well as the early stages of formation of radical
s in organic chemicals. Muonium is also studied as an analogue of hydrogen in semiconductor
s, where hydrogen is one of the most ubiquitous impurities.
for the production of a muon beam. This is presently achieved at few large scale facilities in the world: the CMMS continuous source at TRIUMF
in Vancouver, Canada; the SµS continuous source at the Paul Scherrer Institut (PSI) in Villigen, Switzerland; the ISIS
and RIKEN-RAL pulsed sources at the Rutherford Appleton Laboratory
in Chilton, United Kingdom; and the J-PARC
facility in Tokai, Japan, where a new pulsed source is being built to replace that at KEK
in Tsukuba, Japan.
Muon beams are also available at the Laboratory of Nuclear Problems, Joint Institute for Nuclear Research
(JINR) in Dubna, Russia.
The International Society for µSR Spectroscopy (ISMS) exists to promote the worldwide advancement of µSR. Membership in the society is open free of charge to all individuals in academia, government laboratories and industry who have an interest in the society's goals.
Spin polarization
Spin polarization is the degree to which the spin, i.e., the intrinsic angular momentum of elementary particles, is aligned with a given direction. This property may pertain to the spin, hence to the magnetic moment, of conduction electrons in ferromagnetic metals, such as iron, giving rise to...
muon
Muon
The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...
s in matter and on the detection of the influence of the atomic, molecular or crystalline surroundings on their spin motion. The motion of the muon spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
is due to the magnetic field experienced by the particle and may provide information on its local environment in a very similar way to other magnetic resonance
Magnetic resonance
Magnetic resonance can mean:*Nuclear magnetic resonance*Electron spin resonance*Magnetic resonance imaging *Functional magnetic resonance imaging *Muon spin spectroscopy...
techniques, such as electron spin resonance (ESR or EPR) and, more closely, nuclear magnetic resonance
Nuclear magnetic resonance
Nuclear magnetic resonance is a physical phenomenon in which magnetic nuclei in a magnetic field absorb and re-emit electromagnetic radiation...
(NMR).
Acronym
In analogy with the acronyms for these previously established spectroscopies, the muon spin spectroscopy is also known as µSR, which stands for muon spin rotation, or relaxation, or resonance, depending respectively on whether the muon spin motion is predominantly a rotation (more precisely a precessionPrecession
Precession is a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
around a still magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
), or a relaxation towards an equilibrium direction, or, again, a more complex dynamics dictated by the addition of short radio frequency
Radio frequency
Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...
pulses. The intention of the mnemonic acronym was to draw attention to the analogy with NMR
NMR
NMR may refer to:Applications of Nuclear Magnetic Resonance:* Nuclear magnetic resonance* NMR spectroscopy* Solid-state nuclear magnetic resonance* Protein nuclear magnetic resonance spectroscopy* Proton NMR* Carbon-13 NMR...
and ESR
Electron paramagnetic resonance
Electron paramagnetic resonance or electron spin resonance spectroscopyis a technique for studying chemical species that have one or more unpaired electrons, such as organic and inorganic free radicals or inorganic complexes possessing a transition metal ion...
. More generally speaking, the abbreviation covers any study of the interactions of the muon's magnetic moment with its surrounding when implanted into any kind of matter.
Introduction
µSR is a relatively new nuclear method. Roughly speaking it can be classified in between NMRNMR
NMR may refer to:Applications of Nuclear Magnetic Resonance:* Nuclear magnetic resonance* NMR spectroscopy* Solid-state nuclear magnetic resonance* Protein nuclear magnetic resonance spectroscopy* Proton NMR* Carbon-13 NMR...
and diffraction
Diffraction
Diffraction refers to various phenomena which occur when a wave encounters an obstacle. Italian scientist Francesco Maria Grimaldi coined the word "diffraction" and was the first to record accurate observations of the phenomenon in 1665...
techniques. The NMR technique which is closest parallel to µSR is "pulsed NMR", in which one observes time-dependent transverse nuclear polarization or so-called "free induction decay
Free induction decay
In Fourier Transform NMR, free induction decay is the observable NMR signal generated by non-equilibrium nuclear spin magnetisation precessing about the magnetic field ....
" of the nuclear
polarization. However, a key difference is the fact that in µSR one uses a specifically implanted spin (the muon's) and does not rely on internal nuclear spins.
In addition, and due to the specificity of the muon, the µSR technique does not require any radio-frequency technique to align the probing spin. On the other hand, a clear distinction between the µSR technique and those involving neutrons or x-rays is that scattering is not involved. Neutron diffraction
Neutron diffraction
Neutron diffraction or elastic neutron scattering is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material: A sample to be examined is placed in a beam of thermal or cold neutrons to obtain a diffraction pattern that provides information of...
techniques, for example, use the change in energy and/or momentum of a scattered neutron
Neutron
The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...
to deduce the sample properties. In contrast, the implanted muons are not diffracted but remain in a sample until they decay. Only a careful analysis of the decay product (i.e. a positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
) provides information about the interaction between the implanted muon and its environment in the sample.
As 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 Seth Neddermeyer
Seth Neddermeyer
Seth Henry Neddermeyer was an American physicist who co-discovered the muon, and later championed the implosion design of the plutonium atomic bomb, at the Manhattan Project....
and Carl D. Anderson in 1936, pioneer experiments on its properties were performed with cosmic rays. Indeed, with one muon hitting each square centimeter of the earth's surface every minute, the muons constitute the foremost constituent of cosmic rays arriving at ground level. However, µSR experiments require muon fluxes of the order of
muons per second and square centimeter. Such fluxes can only be obtained in high-energy particle accelerators which have been developed during the last 50 years.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 decay of the pions (mean lifetime
= 26.03 ns) positive muons () are formed via the two body decay:Since the neutrino has an helicity
= -1, both the neutrino and the have their spin antiparallel to their momentum in the pion rest frame. According to the value of the pion momentum at the decay-time, different types of -beams are available for µSR measurements.High-energy beam
The first type of muon beam is formed by the pions escaping the production target at high energies. They are collected over a certain solid angle by quadrupole magnets and directed on to a decay section consisting of a long superconducting solenoid with a field of several Tesla. If the pion momentum is not too high, a large fraction of the pions will have decayed before they reach the end of the solenoid.In the laboratory frame the polarization of a high-energy muon beam is limited to about 80% and its energy is of the order of ~40-50MeV. Although such a high energy beam requires the use of suitable moderators and samples with sufficient thickness, it guarantees an homogeneous implantation of the muons in the sample volume. Such beams are also used to study specimens inside of recipients, e.g. samples inside pressure cells.
Such muon beams are available at PSI
Psi
-Alphabetic letters:* Psi 23rd letter of the Greek alphabet* Psi , a letter of the early Cyrillic alphabet, adopted from Greek-Mathematics:* Tangential angle of a curve*Chebyshev function*Dedekind psi function*Digamma function...
, TRIUMF
TRIUMF
TRIUMF is Canada’s national laboratory for particle and nuclear physics. Its headquarters are located on the south campus of the University of British Columbia in Vancouver, British Columbia. TRIUMF houses the world's largest cyclotron, source of 500 MeV protons, which was named an IEEE Milestone...
, J-PARC
J-PARC
J-PARC is a high intensity proton accelerator facility. It is a joint project between KEK and JAEA and is located at the Tokai campus of JAEA. J-PARC aims for the frontier in materials and life sciences, and nuclear and particle physics...
and RIKEN-RAL.
Surface beam
The second type of muon beam is often called the surface or Arizona beam (recalling the pioneer works of Pifer et al. from the University of ArizonaUniversity of Arizona
The University of Arizona is a land-grant and space-grant public institution of higher education and research located in Tucson, Arizona, United States. The University of Arizona was the first university in the state of Arizona, founded in 1885...
). Here muons are used that arise from pions decaying at rest still inside, but near the surface, of the production target. Such muons, which are 100 % polarized, ideally monochromatic and have a very low momentum of 29.8 MeV/c, which corresponds to a kinetic energy of 4.1 MeV, have a range width in matter of the order of 180 mg/cm2. Hence the paramount advantage of this type of beam is the possibility to use relatively thin samples.
Such muon beams are available at PSI
Psi
-Alphabetic letters:* Psi 23rd letter of the Greek alphabet* Psi , a letter of the early Cyrillic alphabet, adopted from Greek-Mathematics:* Tangential angle of a curve*Chebyshev function*Dedekind psi function*Digamma function...
(Swiss Muon Source SµS), TRIUMF
TRIUMF
TRIUMF is Canada’s national laboratory for particle and nuclear physics. Its headquarters are located on the south campus of the University of British Columbia in Vancouver, British Columbia. TRIUMF houses the world's largest cyclotron, source of 500 MeV protons, which was named an IEEE Milestone...
, J-PARC
J-PARC
J-PARC is a high intensity proton accelerator facility. It is a joint project between KEK and JAEA and is located at the Tokai campus of JAEA. J-PARC aims for the frontier in materials and life sciences, and nuclear and particle physics...
, ISIS
ISIS
ISIS is an industry standard interface for image scanning technologies, developed by Pixel Translations in 1990 ....
and RIKEN-RAL.
Low-energy muon beam
Finally, 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 using moderators, as a thin layer of a van der Waals gas frozen on a substrate. The tunable energy range of such muon beams corresponds to implantation depths in solids of less than a nanometer up to several hundred nanometers. Therefore the study of magnetic properties as a function of the distance from the surface of the sample is possible.Up to now, PSI
Psi
-Alphabetic letters:* Psi 23rd letter of the Greek alphabet* Psi , a letter of the early Cyrillic alphabet, adopted from Greek-Mathematics:* Tangential angle of a curve*Chebyshev function*Dedekind psi function*Digamma function...
is the only Facility where such low-energy muon beam is available on a regular basis. Technical developments have been also conducted at RIKEN-RAL, but with a strongly reduced low-energy muons rate. J-PARC
J-PARC
J-PARC is a high intensity proton accelerator facility. It is a joint project between KEK and JAEA and is located at the Tokai campus of JAEA. J-PARC aims for the frontier in materials and life sciences, and nuclear and particle physics...
is projecting the development of a high-intensity low-energy muon beam.
Different types of muon sources: continuous vs. pulsed
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 (or cw= continuous wave
Continuous wave
A continuous wave or continuous waveform is an electromagnetic wave of constant amplitude and frequency; and in mathematical analysis, of infinite duration. Continuous wave is also the name given to an early method of radio transmission, in which a carrier wave is switched on and off...
) muon sources no dominating time structure is present. By selecting an appropriate muon incoming 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 sources: (i) a non-negligible random background signal; (ii) a rather limited muon incoming rate. Only the background problem can be overcome by using electrostatic deflectors to ensure that no muons enter the sample until the decay of the precedent muon has been observed.
PSI
Psi
-Alphabetic letters:* Psi 23rd letter of the Greek alphabet* Psi , a letter of the early Cyrillic alphabet, adopted from Greek-Mathematics:* Tangential angle of a curve*Chebyshev function*Dedekind psi function*Digamma function...
and TRIUMF
TRIUMF
TRIUMF is Canada’s national laboratory for particle and nuclear physics. Its headquarters are located on the south campus of the University of British Columbia in Vancouver, British Columbia. TRIUMF houses the world's largest cyclotron, source of 500 MeV protons, which was named an IEEE Milestone...
host the two cw muon sources available for µSR experiments.
For pulsed muon sources protons hitting the production target are bunched into pulses which is also reflected on the secondary muon beam. The advantages of a pulsed muon sources are that in principle, one can use the entire incoming muon intensity and there is almost no background due to accidental coincidences between the incoming muons and decay positrons. The absence of background allows the extension of the time window for measurements up to about ten times the muon mean lifetime. The reverse of the medal is that the width of the muon pulse limits the time resolution.
ISIS
ISIS
ISIS is an industry standard interface for image scanning technologies, developed by Pixel Translations in 1990 ....
and J-PARC
J-PARC
J-PARC is a high intensity proton accelerator facility. It is a joint project between KEK and JAEA and is located at the Tokai campus of JAEA. J-PARC aims for the frontier in materials and life sciences, and nuclear and particle physics...
are the two pulsed muon sources available for µSR experiments.
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 (ionizationIonization
Ionization is the process of converting an atom or molecule into an ion by adding or removing charged particles such as electrons or other ions. This is often confused with dissociation. A substance may dissociate without necessarily producing ions. As an example, the molecules of table sugar...
of atoms, electron scattering
Electron scattering
Electron scattering is the process whereby an electron is deflected from its original trajectory. As they are charged particles, they are subject to electromagnetic forces.-Phenomena:...
, electron capture
Electron capture
Electron capture is a process in which a proton-rich nuclide absorbs an inner atomic electron and simultaneously emits a neutrino...
) 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 interstitial sites of the crystallographic lattice. In most metallic samples the muon's positive charge is collectively screened
Screening effect
In solids, especially in metals and semiconductors, the electrostatic screening or screening effect reduces the electrostatic field and Coulomb potential of an ion inside the solid...
by a cloud of conduction electrons. Thus, in metals, the muon is in a so-called diamagnetic state and behave like a free muon. In insulators or semiconductors a collective screening cannot take place and the muon will usually pick-up one electron and form a so-called muonium
Muonium
Muonium is an exotic atom made up of an antimuon and an electron, which was discovered in 1960 and is given the chemical symbol . During the muon's lifetime, muonium can enter into compounds such as muonium chloride or sodium muonide . Due to the mass difference between the antimuon and the...
(Mu=μ++e-), which has similar size (Bohr radius
Bohr radius
The Bohr radius is a physical constant, approximately equal to the most probable distance between the proton and electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom...
), reduced-mass
Reduced mass
Reduced mass is the "effective" inertial mass appearing in the two-body problem of Newtonian mechanics. This is a quantity with the unit of mass, which allows the two-body problem to be solved as if it were a one-body problem. Note however that the mass determining the gravitational force is not...
and ionization energy
Ionization energy
The ionization energy of a chemical species, i.e. an atom or molecule, is the energy required to remove an electron from the species to a practically infinite distance. Large atoms or molecules have a low ionization energy, while small molecules tend to have higher ionization energies.The property...
to the hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
atom.
Detecting the Muon Polarization
The decay of the muon into an electron and two neutrinos occurs via the weak interaction process after an average lifetime ofτμ = 2.197034(21) μs:
The parity violation in the weak interaction leads 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 asymmetry parameter which depends on the polarization of the muon beam and positron energy. This anisotropic emission constitutes in fact the basics for the μSR technique.The asymmetry of
is given by , whereis the beam polarization (of the order of ~1) and
is an intrinsic asymmetry parameter determined by the weak decay mechanism. Theoretically, an average of 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 time scale on which the spin motion may be exploited is that of the muon decay
Particle decay
Particle decay is the spontaneous process of one elementary particle transforming into other elementary particles. During this process, an elementary particle becomes a different particle with less mass and an intermediate particle such as W boson in muon decay. The intermediate particle then...
, i.e. a few mean lifetimes, each roughly 2.2 µs (2.2 millionths of a second). Both the production of muon beams with nearly perfect alignment of the spin to the beam direction (what was referred to above as spin polarization
Spin polarization
Spin polarization is the degree to which the spin, i.e., the intrinsic angular momentum of elementary particles, is aligned with a given direction. This property may pertain to the spin, hence to the magnetic moment, of conduction electrons in ferromagnetic metals, such as iron, giving rise to...
and caused by the spontaneous symmetry breaking), and the ability to detect the muon spin direction at the instant of the muon decay rely on the violation of parity
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...
, which takes place whenever weak forces are at play.
In short this means that certain elementary events happen only when including clockwise (or only when including counter-clockwise) rotations. For instance, the positive muon decays into a positron
Positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. The positron has an electric charge of +1e, a spin of ½, and has the same mass as an electron...
plus two neutrino
Neutrino
A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...
s and the positron is preferentially emitted in the direction of the muon spin. Therefore it would most often see the spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
as a counter-clockwise rotation while flying away from the decay point.
Spin alignment allows the production of a muon beam with an aligned magnetic moment. Muons are injected into the material under investigation as short-lived probes sending information from the interior back out to the experimental apparatus. These muons are able to send a message from inside the crystal about the local magnetic field in their surroundings. After some time (mean lifetime 2.2 µs) these spies decay and emit positrons. A beam of aligned muons produces asymmetric positron radiation. The asymmetry of positron radiation contains information about the direction of local magnetic field in the moment of muon decay. Taking into consideration the initial direction of muon magnetic moment and the time interval between the moment of injection and moment of muon decay we can calculate the precession frequency (how rapidly the muon's magnetic moment rotates). The frequency of magnetic moment precession depends on the local magnetic field. Larmor precession
Larmor precession
In physics, Larmor precession is the precession of the magnetic moments of electrons, atomic nuclei, and atoms about an external magnetic field...
is appeared with z-direction magnetic field and only decay in 2.2 µs. But when x-direction magnetic field is applied in muon, the rate of decay is enhanced by gaussian with depolarization rate.
Since 1987 this method was used to measure internal magnetic fields inside high-temperature superconductors
High-temperature superconductivity
High-temperature superconductors are materials that have a superconducting transition temperature above . From 1960 to 1980, 30 K was thought to be the highest theoretically possible Tc...
. High-temperature superconductors are Type II superconductors, in which the local magnetic fields inside the superconductor depend on the superconducting carrier density—one of the significant parameters of any superconductor (see for example the Bardeen–Cooper–Shrieffer theory
BCS theory
BCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...
of superconductors).
Applications
Muon Spin Rotation and Relaxation are mostly performed with positive muons. They are well suited to the study of magnetic fieldMagnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
s at the atomic scale inside matter, such as those produced by various kinds of magnetism
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...
and/or superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
encountered in compounds occurring in nature or artificially produced by modern material science.
The London penetration depth is one of the most important parameters characterizing a superconductor
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
because its inverse square provides a measure of the density ns of Cooper pairs. The dependence of ns on temperature and magnetic field directly indicates the symmetry of the superconducting gap. Muon spin spectroscopy provides a way to measure the penetration depth, and so has been used to study high-temperature cuprate superconductors since their discovery in 1986.
Other important fields of application of µSR exploit the fact that positive muons capture electrons to form muonium
Muonium
Muonium is an exotic atom made up of an antimuon and an electron, which was discovered in 1960 and is given the chemical symbol . During the muon's lifetime, muonium can enter into compounds such as muonium chloride or sodium muonide . Due to the mass difference between the antimuon and the...
atoms which behave chemically as light isotope
Isotope
Isotopes are variants of atoms of a particular chemical element, which have differing numbers of neutrons. Atoms of a particular element by definition must contain the same number of protons but may have a distinct number of neutrons which differs from atom to atom, without changing the designation...
s of the hydrogen
Hydrogen
Hydrogen is the chemical element with atomic number 1. It is represented by the symbol H. With an average atomic weight of , hydrogen is the lightest and most abundant chemical element, constituting roughly 75% of the Universe's chemical elemental mass. Stars in the main sequence are mainly...
atom. This allows investigation of the largest known isotope effect
Isotope effect
Isotope effect can refer to:* Kinetic isotope effect* Magnetic isotope effect* Superconductive transition temperature varying by isotope atomic weight; See BCS theory#Successes of the BCS theory...
in some of the simplest types of chemical reactions, as well as the early stages of formation of radical
Radical (chemistry)
Radicals are atoms, molecules, or ions with unpaired electrons on an open shell configuration. Free radicals may have positive, negative, or zero charge...
s in organic chemicals. Muonium is also studied as an analogue of hydrogen in semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
s, where hydrogen is one of the most ubiquitous impurities.
Facilities
µSR requires a particle acceleratorParticle accelerator
A particle accelerator is a device that uses electromagnetic fields to propel charged particles to high speeds and to contain them in well-defined beams. An ordinary CRT television set is a simple form of accelerator. There are two basic types: electrostatic and oscillating field accelerators.In...
for the production of a muon beam. This is presently achieved at few large scale facilities in the world: the CMMS continuous source at TRIUMF
TRIUMF
TRIUMF is Canada’s national laboratory for particle and nuclear physics. Its headquarters are located on the south campus of the University of British Columbia in Vancouver, British Columbia. TRIUMF houses the world's largest cyclotron, source of 500 MeV protons, which was named an IEEE Milestone...
in Vancouver, Canada; the SµS continuous source at the Paul Scherrer Institut (PSI) in Villigen, Switzerland; the ISIS
ISIS neutron source
ISIS is a pulsed neutron and muon source. It is situated at the Rutherford Appleton Laboratory on the Harwell Science and Innovation Campus in Oxfordshire, United Kingdom and is part of the Science and Technology Facilities Council...
and RIKEN-RAL pulsed sources at the Rutherford Appleton Laboratory
Rutherford Appleton Laboratory
The Rutherford Appleton Laboratory is one of the national scientific research laboratories in the UK operated by the Science and Technology Facilities Council . It is located on the Harwell Science and Innovation Campus at Chilton near Didcot in Oxfordshire, United Kingdom...
in Chilton, United Kingdom; and the J-PARC
J-PARC
J-PARC is a high intensity proton accelerator facility. It is a joint project between KEK and JAEA and is located at the Tokai campus of JAEA. J-PARC aims for the frontier in materials and life sciences, and nuclear and particle physics...
facility in Tokai, Japan, where a new pulsed source is being built to replace that at KEK
KEK
, known as KEK, is a national organization whose purpose is to operate the largest particle physics laboratory in Japan, which is situated in Tsukuba of Ibaraki prefecture. Established in 1997. The term "KEK" is also used to refer to the laboratory itself, which employs approximately 900 employees...
in Tsukuba, Japan.
Muon beams are also available at the Laboratory of Nuclear Problems, Joint Institute for Nuclear Research
Joint Institute for Nuclear Research
The Joint Institute for Nuclear Research, JINR , in Dubna, Moscow Oblast , Russia, is an international research centre for nuclear sciences, with 5500 staff members, 1200 researchers including 1000 Ph.D.s from eighteen member states The Joint Institute for Nuclear Research, JINR , in Dubna, Moscow...
(JINR) in Dubna, Russia.
The International Society for µSR Spectroscopy (ISMS) exists to promote the worldwide advancement of µSR. Membership in the society is open free of charge to all individuals in academia, government laboratories and industry who have an interest in the society's goals.
External links
- introduction to µSR
- µSR Brochure (a 3.2 MB PDF file)