Muon spin spectroscopy

Muon spin spectroscopy (µSR) is the collective name for three measurement methods in nuclear solid-state physics . In the process, positively charged, spin-polarized muons are implanted into the material sample to be examined and come to rest there on interstitial spaces or atomic defects . The interaction of the muon spin with the magnetic moments of the host lattice leads to a characteristic temporal development of the spin polarization , which is observed and allows conclusions to be drawn about the local magnetic properties of the material sample being examined.

The acronym µSR stands for

• Muon spin rotation as the precession of the muon spin around a static, averaged magnetic field
• Muon spin relaxation as a decrease in the spin polarization of a muon ensemble over time
• Muon spin resonance as the response of the muon spin to an external high frequency pulse

history

In 1956, the two theoretical physicists Tsung-Dao Lee and Chen Ning Yang came to the conclusion in a groundbreaking work that, in contrast to the other three fundamental interactions, gravitation , electromagnetism and strong interaction, the law of conservation of parity is violated in the weak interaction and suggested several for verification Experiments. For this hypothesis , which was confirmed only a little later by the physicist Chien-Shiung Wu , in the Wu experiment named after her on ⁶⁰ co-nuclei, Lee and Yang received the Nobel Prize in Physics in 1957 the following year .

In the same year, the parity violation was detected by groups led by Leon Max Lederman , Richard Garwin and Vince Telegdi in the weak decay of the muon. In their article, Garwin et al. the trend-setting final remark : It seems possible that polarized positive and negative muons will become a powerful tool for exploring magnetic fields in nuclei [...], atoms, and interatomic regions : The beginning of muon spin spectroscopy.

Meaning and definition

With muon spin spectroscopy, the amount, direction, distribution and dynamics of internal magnetic fields can be measured.

Due to the characteristic properties:

• Probe in the interstitial area

• pure dipole moment (no quadrupole effects)
• big magnetic moment: ${\displaystyle \mu _{\mu }=3{,}18\ \mu _{p}=8{,}89\ \mu _{n}}$
• accessible fluctuation time of dynamic processes: ${\displaystyle 10^{-11}\ \mathrm {s} <t<10^{-5}\ \mathrm {s} }$

The muon as a magnetic probe is complementary to other methods of nuclear solid-state physics such as nuclear magnetic resonance spectroscopy and neutron scattering and is particularly suitable for investigation

• weak magnetic effects or moments down to / atom ${\displaystyle 10^{-3}\mu _{B}}$

• disordered or geometrically frustrated magnetic systems (e.g. spin glasses )
• short-range order or order transitions (complementary to neutron scattering).

Since pion production requires proton accelerators of a certain energy range, µSR measurements are only possible at four so-called meson factories :

• Tri University Meson Facility (TRIUMF) in Vancouver , British Columbia , Canada
• Rutherford Appleton Laboratory (RAL) near Didcot , Oxfordshire , England
• Paul Scherrer Institute (PSI) in Villigen and Würenlingen , Canton Aargau , Switzerland
• Japan Proton Accelerator Research Complex (J-PARC) at Tōkai , Ibaraki Prefecture , Japan

Physical background

The weak chain of decay ${\displaystyle \pi ^{+}\rightarrow \mu ^{+}\rightarrow e^{+}}$

The µSR measurement method is based on the parity violation in the weak decay chain :

${\displaystyle {\begin{array}{rll}\pi ^{+}\longrightarrow &\mu ^{+}+\nu _{\mu }&\quad {\text{mit}}\quad \tau _{\pi }=\ 26\ \mathrm {ns} \\&\downarrow &\\&\mu ^{+}\longrightarrow e^{+}+\nu _{e}+{\overline {\nu }}_{\mu }&\quad {\text{mit}}\quad \tau _{\mu }=2{,}2\ \mu \mathrm {s} \\\end{array}}}$

Taking into account the conservation laws for linear momentum and spin lead

• the clear left-handedness ( chirality , i.e. spin antiparallel to the linear momentum) of the two neutrinos involved and ${\displaystyle h=-1}$ ${\displaystyle \nu _{\mu },\nu _{e}}$
• the unequivocal right-handedness (chirality , i.e. spin parallel to the linear momentum) of the muon antineutrino ${\displaystyle h=+1}$ ${\displaystyle {\overline {\nu }}_{\mu }}$

on the two characteristics of this chain of decay:

1. Spin polarization of the muon during pion decay ${\displaystyle \mu ^{+}}$
2. Anisotropy of the emission probability of the decay positron during muon decay . ${\displaystyle e^{+}}$

Pion decay: production of spin-polarized muons

Positive pions ( ) are generated by the spallation reaction when light target nuclei (C, Be) are bombarded with high-energy protons (approx. 600 MeV). The two dominant nuclear reactions are: ${\displaystyle \pi ^{+}}$

${\displaystyle {\begin{array}{lll}p+p&\rightarrow &p+n+\pi ^{+}\\p+n&\rightarrow &n+n+\pi ^{+}\\\end{array}}}$

The free pions generated in this way decay into a positive muon and a muon neutrino after an average lifetime of . ${\displaystyle \tau _{\pi ^{+}}=26\,\mathrm {ns} }$

Depending on whether this decay takes place at rest or in flight, a distinction is made between surface or decay channel muons:

Pion decay at rest: surface muons

Decay of the pion in the rest system: Parity violation and conservation laws for linear momentum and angular momentum lead to helicity adjustment or spin polarization of the muon.

Surface myons come from the decay of low-energy pions that are stopped within the production target. Since the pion's decay is a two-body decay, the energy, momentum and angular momentum of the decay products are clearly defined. In the pion's rest system :

${\displaystyle {\begin{array}{rcccccl}&\pi ^{+}&\rightarrow &\mu ^{+}&+&\nu _{\mu }&\\{\text{Energieerhaltung}}&E_{\pi }=m_{\pi }c^{2}&=&E_{\mu }&+&E_{\nu }&\\{\text{Impulserhaltung}}&{\vec {p}}_{\pi }=0&\Rightarrow &{\vec {p}}_{\mu }&=&-{\vec {p}}_{\nu }&\\{\text{Drehimpulserhaltung}}&{\vec {\sigma }}_{\pi }=0&\Rightarrow &{\vec {\sigma }}_{\mu }&=&-{\vec {\sigma }}_{\nu }&\\{\text{Helizitätsanpassung}}&&\Rightarrow &H(\mu ^{+})&{\stackrel {!}{=}}&H(\nu _{\mu })&\equiv -1\\\end{array}}}$

Because of their low kinetic energy, only muons close to the surface of the target can leave it and have a low penetration depth. They are suitable for studying thin layers or light samples.

Pion decay in flight: decay channel myons

High-energy pions pass through a so-called decay channel. Due to the significantly shorter, mean lifespan of the pions, the beam mainly consists of muons after a certain flight distance. Since the pions decay in flight, these muons have a higher momentum, but are no longer 100% spin-polarized. Due to the transition from the pion's rest system to the laboratory system , two cases must be distinguished:

Muons emitted forwards (right) and backwards (left). is the momentum of the muon in the pion's rest system. relate to the laboratory system. ${\displaystyle p_{\mu _{0}}}$ ${\displaystyle p_{\mu },p_{\pi }}$

• Forward emitted muons : The muon is emitted in the direction of flight of the pion. In the pion's rest system, the impulses of the pion and muon add up. The muon impulse in the laboratory system is thus greater than that of the emitting pion and has the same direction as in the rest system. The orientation of the spin polarization is thus retained: forward-emitted muons are polarized backwards .

• Backward emitted muons : The muon is emitted against the direction of flight of the pion. The muon impulse in the laboratory system is thus lower than that of the emitting pion and opposite to that of the rest system. The spin polarization in the laboratory system is antiparallel to that of the rest system: backward-emitted muons are forward polarized .

Decay channel muons therefore have the following characteristic properties:

• The pulse and direction of polarization can be freely selected within certain limits.
• Their momentum is significantly larger than that of the surface muons: ${\displaystyle 210~\mathrm {MeV} /c>p_{\mu }>80~\mathrm {MeV} /c\gg 30~\mathrm {MeV} /c}$
• As a result, they have a greater penetration depth : ${\displaystyle 5~\mathrm {g} /\mathrm {cm} ^{2}>\rho >1~\mathrm {g} /\mathrm {cm} ^{2}\gg 0{,}1~\mathrm {g} /\mathrm {cm} ^{2}}$
• However, due to kinematic depolarization and finite angular resolution, the maximum achievable degree of polarization is only approx . ${\displaystyle H_{\mu }\approx 80\,\%}$

Muon decay: anisotropy of the decay positron

Muon decay: The two extreme cases are shown: emission of neutrino and antineutrino in the opposite direction (the positron remains at rest) or in the same direction opposite to the positron.

The muon then also decays via the weak interaction, after a relatively long mean life of into a positron, an electron neutrino and a muon antineutrino (three-body decay): ${\displaystyle \tau _{\mu ^{+}}=2{,}2\,\mathrm {\mu s} }$

${\displaystyle \mu ^{+}\rightarrow e^{+}+\nu _{e}+{\bar {\nu }}_{\mu }}$

Since it is a three-body decay, the energy and momentum of the decay products are not clearly defined, but depend on the mutual emission direction.

The following applies to the kinetic energy of the decay positron:

${\displaystyle 0\quad \leq \quad E_{\mathrm {kin} }\leq \quad E_{\mathrm {max} }=52{,}320\,\mathrm {MeV} }$

Due to the clear handedness of the (anti-) neutrinos involved, there is a correlation between the spin and the linear momentum of the decay positron, which is reflected in the anisotropy of the positron emission.

The µSR signal

Muon decay: Angular dependence of the emission probability for decay positrons of maximum or medium energy.

The interaction of the magnetic moments of a muon ensemble with the mean value and the distribution of the local magnetic field in the interstitial area results in a temporal change in the direction and degree of polarization, which can be observed via the anisotropy of the decay positron:

${\displaystyle W(\theta (t),t)=1+A_{0}\cdot {\vec {P}}(t)\cdot \cos(\theta )=1+A_{0}\cdot |{\vec {P}}(t)|\cdot \cos(\theta (t))}$

• ${\displaystyle A_{0}}$ denotes the initial asymmetry of the muon ensemble,
• ${\displaystyle \theta (t)}$ the angle between the emission and detection direction of the decay positron relative to the polarization direction of the muon ensemble.
• The decrease in the degree of polarization over time is called depolarization and is described by relaxation functions. ${\displaystyle |{\vec {P}}(t)|}$ ${\displaystyle G(t)}$

The basis of every time-differential µSR measurement is therefore the recording of the changes in the positron counting rate over time in one or more fixed spatial directions:

${\displaystyle N(t)/dt=B_{0}+N_{0}\cdot \exp(-t/\tau _{\mu })\cdot [\ 1+A_{0}\ G(t)\ \cos \theta (t)\ ]}$

• Signal background , normalization and initial asymmetry of the decay positron are determined by the sample geometry, beam and spectrometer properties. ${\displaystyle B_{0}}$ ${\displaystyle N_{0}}$ ${\displaystyle A_{0}}$
• ${\displaystyle N_{0}\cdot \exp(-t/\tau _{\mu })}$describes the free decay of the muons with the mean lifespan and, together with the signal background and defines the zero line. ${\displaystyle \tau _{\mu }\approx 2{,}2\ \mathrm {\mu s} }$
• The time behavior of interest is on this zero line . ${\displaystyle [1+A_{0}\ G(t)\ \cos \theta (t)]}$

Transverse field geometry

Muon spin rotation : Schematic structure and typical time course of the positron counting rate of a µSR measurement in an external magnetic field perpendicular to the muon polarization direction (transverse field geometry). ${\displaystyle {\vec {B}}_{\mathrm {ext} }\perp {\vec {P}}_{\mu }}$

Muon spin relaxation : Schematic structure and typical time course of the positron counting rate of a µSR measurement in zero field or longitudinal field geometry.

A polarized muon beam is collimated by a beam diaphragm and stopped in the material sample to be examined. The passage of a muon through the muon detector determines the zero time of the implantation. The decay positron is recorded with the aid of the positron counter arranged around the sample and provides the stop signal.

• In the absence of internal local magnetic fields, the muon spin precesses around the static external transverse field with the Larmor frequency :

${\displaystyle \omega _{\mu }=\gamma _{\mu }\cdot B_{\mathrm {ext} }}$

With ${\displaystyle \gamma _{\mu }/2\pi =135{,}5\;\mathrm {MHz/T} .}$

• If there are local magnetic fields inside the material sample, which also vary in time and / or space, the individual muon spins of an ensemble will have different precession frequencies. As a result, the muon spin rotation frequency reflects an averaging of the internal fields: ${\displaystyle B_{\mathrm {int} }\not \equiv 0}$

${\displaystyle {\overline {\omega }}_{\mu }=\gamma _{\mu }\cdot (B_{\mathrm {ext} }+\langle B_{\mathrm {int} }\rangle )}$

With

${\displaystyle \gamma _{\mu }/2\pi =135{,}5\;\mathrm {MHz/T} .}$

• Due to the field distribution, the original phase relationship of the individual spins is lost more and more. The rotation signal therefore has a temporal damping with the transverse relaxation function : ${\displaystyle G_{\perp }}$

${\displaystyle N(\phi ,t)/dt=B_{0}+N_{0}\cdot \exp(-t/\tau _{\mu })\cdot [\ 1+A_{0}\ G_{\perp }(t)\ \cos({\overline {\omega }}_{\mu }t+\phi )]}$

Zero field or longitudinal field geometry

• Without an external magnetic field, muon spin rotation occurs only in the ordered range of magnetic materials. However, the domain structure leads to a reduction in the rotation signal. For isotropic distribution of the domain magnetization, an amplitude ratio of and thus applies : ${\displaystyle A_{\perp }/A_{\parallel }=2/1}$

${\displaystyle {\begin{array}{rcl}N(\phi ,t)/dt&=&B_{0}+N_{0}\cdot \exp(-t/\tau _{\mu })\cdot f(\phi ,t)\\f(\phi ,t)&=&[\ 1+{\frac {A_{0}}{3}}\ (\ 2\cdot G_{\perp }(t)\ \cos({\overline {\omega }}_{\mu }t+\phi )+G_{\parallel }(t)\ )\ ]\\\end{array}}}$

• In the disordered, paramagnetic area, there is no magnetization without an external magnetic field and thus no muon spin rotation can be observed. The variable of interest here is the longitudinal relaxation function : ${\displaystyle G_{\parallel }(t)}$

${\displaystyle f(\phi ,t)=[\ 1+A_{0}\ G_{\parallel }(t)\ \cos \phi \ ]}$

Relaxation functions

Compared to the transverse relaxation function, the longitudinal relaxation function is characterized by a stronger, initial depolarization and a re-increase:

Comparison of the transverse and longitudinal relaxation function or for static field distributions. ${\displaystyle G_{\perp }(t)}$ ${\displaystyle G_{\parallel }(t)}$

• The stronger initial depolarization of results from the influence of two perpendicular components of the field distribution, instead of just one component for . ${\displaystyle G_{\parallel }(t)}$ ${\displaystyle G_{\perp }(t)}$
• The renewed increase is explained by the lack of influence of the longitudinal components of the field distribution.
  • The correlation time of slow fluctuations can be inferred from the magnitude of the rise again.
  • A complete rise is a clear indication of a static field distribution.
  • In the case of rapid fluctuations, the minimum of the static zero-field relaxation disappears completely and the relaxation function takes on an exponential course

• Longitudinal field measurements :
  • Since it is not possible to decouple rapidly fluctuating field distributions, longitudinal field measurements provide clear information as to whether the situation is static or dynamic.
  • By applying an external magnetic field parallel to the direction of polarization, it is possible to cancel the depolarization due to internal, static field distributions and to maintain the initial spin polarization. The strength of the magnetic field provides information about the interaction energy and the origin (electronic or nuclear moments) of the internal field distribution.
• Zero field measurements are crucial for the investigation of critical phenomena at the magnetic phase transition.

See also

• Nuclear solid state physics
• Electron spin resonance (ESR)
• Nuclear magnetic resonance (NMR)
• Disturbed gamma-gamma angle correlation

Web links

• Tri University Meson Facility (TRIUMF) in Vancouver , British Columbia , Canada
• Rutherford Appleton Laboratory (RAL) near Didcot , Oxfordshire , England
• Paul Scherrer Institute (PSI) in Villigen and Würenlingen , Canton Aargau , Switzerland
• Japan Proton Accelerator Research Complex (J-PARC) at Tōkai , Ibaraki Prefecture , Japan

literature

• Günter Schatz, Alois Weidinger: Nuclear solid-state physics: Nuclear physical measuring methods and their applications , Vieweg + Teubner Verlag, 4th, revised. Ed., 2010, ISBN 9783835102286
• Alexander Schenck: Muon Spin Rotation Spectroscopy: Principles and Applications in Solid State Physics , Adam Hilger, 1985, ISBN 0852745516
• Ernst Schreier: Muon spin rotation and relaxation to investigate the magnetic properties of the heavy rare earth metals , UFO-Verlag, Allensbach 1999, ISBN 9783930803736
• A. Yaouanc and P. Dalmas de Réotier: Muon Spin Rotation, Relaxation and Resonance: Applications to Condensed Matter , Oxford University Press, Oxford, 2011, ISBN 9780199596478

Individual evidence

1. ↑ TD Lee, CN Yang: Question of Parity Conservation in Weak Interactions . In: Physical Review . 104, 1956, pp. 254-258. doi : 10.1103 / PhysRev.104.254 .
2. ↑ CS Wu, E. Ambler, RW Hayward, DD Hoppes, RP Hudson: Experimental Test of Parity Conservation in Beta Decay . In: Physical Review . 105, 1957, pp. 1413-1415. doi : 10.1103 / PhysRev.105.1413 .
3. ↑ JI Friedman and VL Telegdi: Nuclear Emulsion Evidence for Parity Nonconservation in the Decay Chain π + → μ + → e + . In: Physical Review . 106, 1957, pp. 1290-1291. doi : 10.1103 / PhysRev.106.1290 .
4. ^ Richard L. Garwin, Leon M. Lederman, and Marcel Weinrich: Observations of the Failure of Conservation of Parity and Charge Conjugation in Meson Decays: the Magnetic Moment of the Free Muon . In: Physical Review . 105, 1957, pp. 1415-1417. doi : 10.1103 / PhysRev.105.1415 .