Muon Spin Spectroscopy Using the Positive Muon to Probe Matter

Paul W. Percival

Department of Chemistry Simon Fraser University and TRIUMF 2 Muon and Muonium Properties

− Negative muon µ a heavy electron (mass = 207 me) τ = 2.2 µs in vacuum … … less in matter because of nuclear capture spin I = ½

+ Positive muon µ a light proton (mass = 0.11 mp) τ = 2.2 µs in vacuum and matter spin I = ½

Muonium Mu = µ+e− a light hydrogen atom Mu+ = µ+ Ionization Energy = 13.54 eV H: 13.60 eV Bohr Radius = 0.532 Ǻ H: 0.529 Ǻ

Paul Percival NUSC 341, November 2005 3 The Periodic Table — Chemistry Department Quilt

http://www.sfu.ca/chemistry/news/pt_quilt/index.htm

Paul Percival NUSC 341, November 2005 4 The Simplest Atom

Paul Percival NUSC 341, November 2005 5

Muonium — a light isotope of hydrogen

Mu ≡ µ+ e− e− The properties of a single electron atom are determined µ+ by the reduced mass 1111 =+≈ mmrNee mm

i.e. mmre≈

reduced mass of Mu = 0.995 mr(H) ionization potential = 13.539 eV Bohr radius = 0.532 Å

Paul Percival NUSC 341, November 2005 6 Pions Decay to Give Muons

At TRIUMF, pions are produced by bombardment of a Be or C production target with 500 MeV protons. We take the positive pions, which decay to positive muons: ++ ν µ πµν→ + µ In the pion rest frame: ←! → ""#""# τ = 26 ns Momentum is conserved: ppµ = − ν ""#""# Spin is conserved: σµ = −σν

## Spin and momentum are σ⋅ p related through helicity: hhˆˆ==±## ψ 1ψ σ.p For the neutrino, h = -1, i.e. the spin and momentum are anti-parallel. σ σ Therefore, muons are produced →ν ←µ with σ and p anti-parallel. ←→! ppν µ

Paul Percival NUSC 341, November 2005 7 Muon Beams are Spin Polarized

There are two commonly used types of muon beam: Surface Muon Beams collect and transport muons which are created from the decay of pions at or near the surface of the pion production target.

select these muons to get these spins:

beam line

Decay Muon Beams collect muons from pions which decay in flight.

In the π rest frame only backward {}and forward muons are transported.

In the laboratory frame, both momentum bites travel in the forward direction. The forward (momentum) muons (p~180 MeV/c) have backward spin; the backward muons (p~80 MeV/c) have forward spin.

Paul Percival NUSC 341, November 2005 8 Muon Decay

++ µνντµ→ e2.2++µ e = s

The 3 particle decay results in a N spectrum of positron energies. 1 Emmax = 2 µ The spatial distribution of positrons is = 52.8 MeV asymmetric and depends on the muon spin polarization. Energy Consider the decay pattern which σ νµ σ results in the maximum positron energy: ν e+ e νe The e+ is emitted in the direction of the muon spin at the moment of decay. More generally, dN(,) E θ =+CDE[]1()cosθ θ dE dΩ

Paul Percival NUSC 341, November 2005 9

The Muon Lifetime Experiment

++ µ → e2.2++ννµ e τ = µs

M F µ+ S beam e+ B

P lifetime histogram

start stop N CLOCK

0 time

−t / τ NN= 0 e + background

Paul Percival NUSC 341, November 2005 10

Muon Spin Rotation, µSR

Apply a magnetic field M F perpendicular to the initial µ+ S spin direction. beam e+ B

P µSR histogram

start stop N CLOCK

time

−t / τ NN=+0 e1[ aPt( ) cos(ωφ t +)]+ background

Paul Percival NUSC 341, November 2005 11 The µSR Histogram

800

600 −t / τ NaPttB0 e1[ +++( ) cos(ω φ)]

400 count Subtract the constant background and

200 divide out the exponential decay …

0 0123456

time / µs 0.2

… to get the muon asymmetry, 0.1 which represents the time dependence of the muon spin 0 polarization.

aP( t)cos(ωφ t + ) muon asymmetry -0.1

-0.2 0123456 time / µs

Paul Percival NUSC 341, November 2005 12

µSR: Muon Spectroscopy

Muon Spin Rotation

• spin polarized muon beam • muons are implanted in the sample

+ + • muons decay: µ → e + νµ + νe τ = 2.2 µs • angular distribution of e+ has maximum in muon spin direction • precession of muon spins in transverse magnetic field • equivalent to free induction decay in pulsed NMR or ESR

Paul Percival NUSC 341, November 2005 13 Muonium in supercritical water

400°C, 245 bar 8 G 0.2

0.1

0.0 Asymmetry -0.1

-0.2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 time / µs

Percival, Brodovitch, Ghandi et al., Phys. Chem. Chem. Phys. 1 (1999) 4999

Paul Percival NUSC 341, November 2005 14 Muonium in supercritical water

400°C, 245 bar, 8 G

Diamagnetic signal subtracted

0.15

-λt AMue

0.00 Muon Asymmetry -0.15 0.00 0.50 1.00 1.50 2.00 2.50 3.00 time / µs

Paul Percival NUSC 341, November 2005 15 Muonium in supercritical water at 196 G Muon Asymmetry

0.00 0.05 0.10 0.15 0.20 time / µs

400°C 245 bar Fourier Power

200 220 240 260 280 300 320 340 Frequency / MHz Paul Percival NUSC 341, November 2005 16 Energy levels of a two spin-½ system

Breit-Rabi diagram e µ αα αα (αβ + βα)/√2 αβ ββ

Relative Energy

ββ

(αβ − βα)/√2 βα

0.0 1.0 2.0 3.0 Magnetic Field / Hyperfine Constant

Paul Percival NUSC 341, November 2005 17

Muon spin precession in D2O crystal at 227 K

38 G Muon Asymmetry

0.0 0.5 1.0 1.5 2.0 time /µs

The high frequency precession is due to “triplet” (F=1) muonium. The low frequency is due to muons in diamagnetic environments, + such as MuOD and MuOD2

Paul Percival NUSC 341, November 2005 18 Muonium precession frequencies in low magnetic field

isotropic hyperfine case axial anisotropy

1 1

ν12 2 ν12 2 Energy

ν23 ν23

3 3

Magnetic Field Magnetic Field

Paul Percival NUSC 341, November 2005 19 Muonium diffuses along the c-axis channels of ice-Ih

side view view along c channel

Paul Percival NUSC 341, November 2005 20

µSR: Muon spectroscopic methods

Common features: • spin polarized muon beam • muons are implanted in the sample + + • muons decay: µ → e + νµ + νε τ = 2.2 µs • angular distribution of e+ has maximum in muon spin direction Muon Spin Rotation µSR (TF-µSR) • precession of muon spins in transverse magnetic field • equivalent to free induction decay in pulsed NMR or ESR Muon Spin Resonance RF-µSR • muon spin polarization along magnetic field • transitions induced by rf field as in conventional NMR Muon Level Crossing Resonance µLCR (ALCR) • mixing of spin levels causes avoided level crossing • polarization is lost at magnetic fields where level crossing occurs

Paul Percival NUSC 341, November 2005 21 RF Transitions in Muonium at Low Magnetic Field

νrf 2 Energy ν rf 2νrf

Magnetic Field

Paul Percival NUSC 341, November 2005 22

RFµSR Spectrum of Mu@C60 in C60 Powder

120 130 140 150 160 170 Magnetic Field / G

Paul Percival NUSC 341, November 2005 23

Endohedral Muonium Mu@C60

Muonium in a universe of its own

Paul Percival NUSC 341, November 2005 24

Muoniated free radicals

R2 C C R R 4 1 R µSR: 3 precession frequencies ⇒ muon hyperfine coupling µLCR: resonance fields ⇒ other nuclear hyperfine couplings hyperfine couplings ⇒ distribution of unpaired electron spin H Mu

e.g. cyclohexadienyl 0.33 0.33

-0.10 • -0.10 0.48 Temperature dependence of hyperfine couplings ⇒ intramolecular motion

Paul Percival NUSC 341, November 2005 25

Transverse field muon spin rotation of radicals

νR1 νD

νR2 Fourier Power Energy 0 50 100 150 200 250 300 350 400 Frequency / MHz

Aµ = νR2 – νR1

Magnetic Field

Paul Percival NUSC 341, November 2005 26

Fourier power µSR spectrum of tert-butyl

1 M tert-Butanol 280°C 250 bar 11.5 kG

CH3 Aµ = 256 MHz

CH2Mu

CH3 Fourier Power

0 50 100 150 200 250 300 350 400 Frequency / MHz

Paul Percival NUSC 341, November 2005 27

Avoided Muon Level Crossing Resonance

↑e ,↑µ ,↓X – – A + A

Energy 6.87.07.27.47.67.88.08.2 Magnetic Field / kG

↑ ,↓ ,↑ e µ X 1 Aµ − AX Bres ≈ 2 γµ −γX Magnetic Field

Paul Percival NUSC 341, November 2005 28 tert-Butyl radical in water

CH3

CH2Mu

CH3

200°C 250 bar - -A + A

10.0 10.5 11.0 11.5 Field / kG

Paul Percival NUSC 341, November 2005 29

Hyperfine constants are used to map unpaired spin in radicals

13 Mu C60 Avoided Level Crossing Resonance Mu 1.5 3.3 1.4 30 14

2.0 5.4

% Unpaired Spin Density

10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 15.0 Magnetic Field /kG

Percival, Addison-Jones, Brodovitch, Ji, et al., Chem. Phys. Lett., 245 (1995) 90

Paul Percival NUSC 341, November 2005 30

H atoms and free radicals in liquids

H is the simplest atom ⇒ fundamental chemistry H is a constituent of water, hydrocarbons, carbohydrates ⇒ essential chemistry

Chemically speaking, Mu ≡ H

We study Mu as a substitute for H

because it is similar: tracer, spin label

because it is different : isotope effects

Paul Percival NUSC 341, November 2005 31

Muonium Chemistry — areas of application

Muonium Mu as probe of local environment (caged) Fixed cage: fullerenes, silsesquioxanes partial: ice transient: water Kinetics How fast does it go? in gases: reaction dynamics liquids: solvent effects; diffusion vs. activation control Structure Where does it end up? (needs e-spin) Free radicals: unpaired spin distribution Mechanism How does it get there? Identification of radical products Transfer of muon polarization Dynamics Nature is floppy! intramolecular Temperature dependence of hyperfine constants intermolecular LCR lineshape analysis

Paul Percival NUSC 341, November 2005 32 Further Reading

http://www.sfu.ca/chemistry/news/pt_quilt/index.htm

http://musr.org/cmms.php

http://musr.org/intro/musr/muSRBrochure.pdf

Paul Percival NUSC 341, November 2005