Muon Spin SpectroscopyUsing the Positive Muon to Probe Matter
Paul W. Percival
Department of ChemistrySimon Fraser University
and TRIUMF
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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
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
Positive muon µ+ a light proton (mass = 0.11 mp)
τ = 2.2 µs in vacuum and matter
spin I = ½
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The Periodic Table Chemistry Department Quilt
http://www.sfu.ca/chemistry/news/pt_quilt/index.htm
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The Simplest Atom
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Muonium a light isotope of hydrogen
Mu ≡ µ+ e−
reduced mass of Mu = 0.995 mr(H)
ionization potential = 13.539 eV
Bohr radius = 0.532 Å
µ+
e−
The properties of a single electron atom are determined by the reduced mass
r N e e
r e
1 1 1 1
i.e .
m m m m
m m
= + ≈
≈
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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:
µπ µ ν+ +→ + In the pion rest frame:
Momentum is conserved:
Spin is conserved:
ν µ← →!
p pµ ν
µ νσ σ
=−
=−
""# ""#
""# ""#
Spin and momentum are related through helicity: 1
.ph hp
σψ ψ
σ⋅= =±# ## #
For the neutrino, h = -1, i.e. the spin and momentum are anti-parallel.
Therefore, muons are produced with σ and p anti-parallel.
p pν
ν
µ
µσσ
←→ ← →!
τ = 26 ns
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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.
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Muon Decay
ee 2.2 sµµ ν ν τ µ+ +→ + + =
The 3 particle decay results in a spectrum of positron energies.
The spatial distribution of positrons is asymmetric and depends on the muon spin polarization.
Consider the decay pattern which results in the maximum positron energy:
The e+ is emitted in the direction of the muon spin at the moment of decay. More generally,
Energy
N
1max 2
52.8 MeV
E mµ==
e+νµ
νe
σeσν
[ ]( , ) 1 ( )cosdN E C D EdE d
θθ= +
Ωθ
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The Muon Lifetime Experiment
/0 bace kgroundtN N τ−= +
ee 2.2 sµµ ν ν τ µ+ +→ + + =
µ+
beamS
start stopCLOCK
M F
B
P
e+
time
N
lifetime histogram
0
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µ+
beamS
start stopCLOCK
M F
B
P
e+
time
N
µSR histogram
Muon Spin Rotation, µSR
( ) ( )[ ]/0 e 1 cos backgroundtN N aP t tτ ω φ−= + + +
Apply a magnetic field perpendicular to the initial spin direction.
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The µSR Histogram
-0.2
-0.1
0
0.1
0.2
0 1 2 3 4 5 6
time / µs
muo
n as
ymm
etry
0
200
400
600
800
0 1 2 3 4 5 6
time / µs
coun
t
( ) ( )[ ]/0 e 1 costN aP t t Bτ ω φ− + + +
Subtract the constant background and divide out the exponential decay
to get the muon asymmetry, which represents the time dependence of the muon spin polarization.
( ) ( )cosaP t tω φ+
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µSR: Muon Spectroscopy
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
Muon Spin Rotation
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Muonium in supercritical water
Percival, Brodovitch, Ghandi et al., Phys. Chem. Chem. Phys. 1 (1999) 4999
400°C, 245 bar
0.0 0.5 1.0 1.5 2.0 2.5 3.0time / µs
-0.2
-0.1
0.0
0.1
0.2
Asym
met
ry
8 G
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Muonium in supercritical water
-0.15
0.00
0.15
0.00 0.50 1.00 1.50 2.00 2.50 3.00
time / µs
Muo
n A
sym
met
ry400°C, 245 bar, 8 G
AMue-λt
Diamagnetic signal subtracted
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Muonium in supercritical water at 196 G
400°C
245 bar
0.00 0.05 0.10 0.15 0.20
time / µs
Muo
n A
sym
met
ry
200 220 240 260 280 300 320 340
Frequency / MHz
Four
ier P
ower
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Energy levels of a two spin-½ system
Breit-Rabi diagram
(αβ − βα)/√2
0.0 1.0 2.0 3.0
Magnetic Field / Hyperfine Constant
RelativeEnergy
(αβ + βα)/√2
αα
ββ
αα
ββ
αβ
βα
e µ
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Muon spin precession in D2O crystal at 227 K
38 G
0.0 0.5 1.0 1.5 2.0
time /µs
Muo
n A
sym
met
ry
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
+
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Muonium precession frequencies in low magnetic field
Magnetic Field
Ene
rgy
1
2
3
ν12
ν23
Magnetic Field
1
2
3
ν12
ν23
isotropic hyperfine case axial anisotropy
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Muonium diffuses along the c-axis channels of ice-Ih
Mu
side view view along c channel
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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
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RF Transitions in Muonium at Low Magnetic Field
Magnetic Field
Ene
rgy
1
2
3
νrf
νrf 2νrf
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RFµSR Spectrum of Mu@C60 in C60 Powder
120 130 140 150 160 170
Magnetic Field / G
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Endohedral Muonium Mu@C60
Muonium in a universe of its own
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Muoniated free radicals
e.g. cyclohexadienyl
µSR:precession frequencies ⇒ muon hyperfine couplingµLCR:resonance fields ⇒ other nuclear hyperfine couplings
hyperfine couplings ⇒ distribution of unpaired electron spin
Temperature dependence of hyperfine couplings⇒ intramolecular motion
R4R3
R2 CMu
R1
C
H Mu
0.33
-0.10
0.48
0.33
-0.10
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0 50 100 150 200 250 300 350 400Frequency / MHz
Four
ier P
ower
Transverse field muon spin rotation of radicals
Aµ = νR2 νR1
νR1
νR2
νD
Magnetic Field
Ene
rgy
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Fourier power µSR spectrum of tert-butyl
1 M tert-Butanol 280°C 250 bar 11.5 kG
0 50 100 150 200 250 300 350 400Frequency / MHz
Four
ier P
ower Aµ = 256 MHz CH3
CH3
CH2Mu
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Avoided Muon Level Crossing Resonance
µ Xres
µ X
12 γ γ
A AB
−≈
−
6.8 7.0 7.2 7.4 7.6 7.8 8.0 8.2
Magnetic Field / kG
A+ A
Magnetic Field
Ene
rgy
Xµe ,, ↑↓↑
Xµe ,, ↓↑↑
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NUSC 341, November 2005
tert-Butyl radical in water
10.0 10.5 11.0 11.5
Field / kG
A+ -A
-
CH3
CH3
CH2Mu
200°C
250 bar
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Hyperfine constants are used to map unpaired spin in radicals
Mu13C60 Avoided Level Crossing Resonance
15.010.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5
Magnetic Field /kG
Mu
3014
5.42.0
3.31.5
1.4
% Unpaired Spin Density
Percival, Addison-Jones, Brodovitch, Ji, et al., Chem. Phys. Lett., 245 (1995) 90
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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
because it is similar: tracer, spin label
because it is different : isotope effects
We study Mu as a substitute for H
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Muonium Chemistry areas of application
Muonium Mu as probe of local environment(caged) Fixed cage: fullerenes, silsesquioxanes
partial: icetransient: water
Kinetics How fast does it go?in gases: reaction dynamicsliquids: 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 productsTransfer of muon polarization
Dynamics Nature is floppy!intramolecular Temperature dependence of hyperfine constantsintermolecular LCR lineshape analysis
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Further Reading
http://www.sfu.ca/chemistry/news/pt_quilt/index.htm
http://musr.org/cmms.php
http://musr.org/intro/musr/muSRBrochure.pdf
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