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
Paul Percival
Mu+ = µ+
Ionization Energy = 13.54 eV
H: 13.60 eV
Bohr Radius = 0.532 Ǻ
H: 0.529 Ǻ
NUSC 341, November 2005
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The Periodic Table — Chemistry Department Quilt
http://www.sfu.ca/chemistry/news/pt_quilt/index.htm
Paul Percival
NUSC 341, November 2005
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The Simplest Atom
Paul Percival
NUSC 341, November 2005
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Muonium — a light isotope of hydrogen
Mu ≡ µ+ e−
e−
The properties of a single
electron atom are determined
by the reduced mass
µ+
1
1
1
1
=
+ ≈
mr mN me me
i.e. mr ≈ me
reduced mass of Mu = 0.995 mr(H)
ionization potential = 13.539 eV
Bohr radius = 0.532 Å
Paul Percival
NUSC 341, November 2005
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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:
π+ → µ + + ν µ
τ = 26 ns
In the pion rest frame:
Momentum is conserved:
Spin is conserved:
Spin and momentum are
related through helicity:
# #
⋅p
σ
hˆ = # #
σ. p
ν
µ
← ! 
→
""#
""#
pµ = − pν
""#
""#
σµ = −σν
hˆψ = ±1ψ
For the neutrino, h = -1, i.e. the spin and momentum are anti-parallel.
Therefore, muons are produced
with σ and p anti-parallel.
Paul Percival
σµ
σν
→
←
←  ! →
pν
pµ
NUSC 341, November 2005
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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.
Paul Percival
NUSC 341, November 2005
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Muon Decay
µ + → e+ + ν µ + ν e
The 3 particle decay results in a
spectrum of positron energies.
τ = 2.2 µs
N
Emax = 12 mµ
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 is emitted in the direction of the
muon spin at the moment of decay.
More generally,
= 52.8 MeV
Energy
σν
e+
dN ( E , θ)
= C [1 + D ( E ) cos θ]
dE d Ω
Paul Percival
νµ
e+
σe
νe
θ
NUSC 341, November 2005
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The Muon Lifetime Experiment
µ + → e+ + ν µ + ν e
M
µ+
beam
τ = 2.2 µs
F
S
e+
B
P
lifetime histogram
start
stop
N
CLOCK
0
time
N = N 0 e−t / τ + background
Paul Percival
NUSC 341, November 2005
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Muon Spin Rotation, µSR
M
µ+
beam
F
S
Apply a magnetic field
perpendicular to the initial
spin direction.
e+
B
P
µSR histogram
start
stop
N
CLOCK
time
N = N 0 e−t / τ [1 + aP (t ) cos (ωt + φ)] + background
Paul Percival
NUSC 341, November 2005
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The µSR Histogram
800
N 0 e−t / τ [1 + aP (t ) cos (ωt + φ)] + B
count
600
400
Subtract the constant background and
divide out the exponential decay …
200
0
0
1
2
3
4
5
6
time / µs
aP (t ) cos (ωt + φ)
muon asymmetry
… to get the muon asymmetry,
which represents the time
dependence of the muon spin
polarization.
0.2
0.1
0
-0.1
-0.2
0
1
2
3
4
5
6
time / µs
Paul Percival
NUSC 341, November 2005
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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
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Muonium in supercritical water
400°C, 245 bar
8G
0.2
Asymmetry
0.1
0.0
-0.1
-0.2
0.0
0.5
1.0
1.5
time / µs
2.0
2.5
3.0
Percival, Brodovitch, Ghandi et al., Phys. Chem. Chem. Phys. 1 (1999) 4999
Paul Percival
NUSC 341, November 2005
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Muonium in supercritical water
400°C, 245 bar, 8 G
Diamagnetic signal subtracted
Muon Asymmetry
0.15
AMue-λt
0.00
-0.15
0.00
0.50
1.00
1.50
2.00
2.50
3.00
time / µs
Paul Percival
NUSC 341, November 2005
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Muon Asymmetry
Muonium in supercritical water at 196 G
0.00
0.05
0.10
0.15
0.20
Fourier Power
time / µs
400°C
245 bar
200
220
240
260
280
300
320
340
Frequency / MHz
Paul Percival
NUSC 341, November 2005
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Energy levels of a two spin-½ system
Breit-Rabi diagram
e
αα
µ
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
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Muon spin precession in D2O crystal at 227 K
Muon Asymmetry
38 G
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
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Muonium precession frequencies in low magnetic field
isotropic hyperfine case
axial anisotropy
Energy
1
ν12
2
ν23
1
ν12
ν23
3
Magnetic Field
Paul Percival
2
3
Magnetic Field
NUSC 341, November 2005
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Muonium diffuses along the c-axis channels of ice-Ih
Mu
side view
Paul Percival
view along c channel
NUSC 341, November 2005
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µSR: Muon spectroscopic methods
Common features:
•
•
•
•
spin polarized muon beam
muons are implanted in the sample
muons decay:
µ+ → e + + ν µ + ν ε
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
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RF Transitions in Muonium at Low Magnetic Field
Energy
1
νrf
νrf
2
2νrf
3
Magnetic Field
Paul Percival
NUSC 341, November 2005
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RFµSR Spectrum of Mu@C60 in C60 Powder
120
130
140
150
160
170
Magnetic Field / G
Paul Percival
NUSC 341, November 2005
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Endohedral Muonium Mu@C60
Muonium in a universe of its own
Paul Percival
NUSC 341, November 2005
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Muoniated free radicals
Mu
R2
R1
C
C
R4
R3
µSR:
precession frequencies ⇒ muon hyperfine coupling
µLCR:
resonance fields ⇒ other nuclear hyperfine couplings
hyperfine couplings ⇒ distribution of unpaired electron spin
e.g. cyclohexadienyl
H Mu
0.33
-0.10
0.33
•
-0.10
0.48
Temperature dependence of hyperfine couplings
⇒ intramolecular motion
Paul Percival
NUSC 341, November 2005
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Transverse field muon spin rotation of radicals
Energy
Fourier Power
νR1
νD
νR2
0
50
100 150 200 250
300 350 400
Frequency / MHz
Aµ = νR2 – νR1
Magnetic Field
Paul Percival
NUSC 341, November 2005
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Fourier power µSR spectrum of tert-butyl
1 M tert-Butanol 280°C 250 bar 11.5 kG
CH3
Fourier Power
Aµ = 256 MHz
CH2Mu
CH3
0
Paul Percival
50
100
150 200 250
Frequency / MHz
300
350
400
NUSC 341, November 2005
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Avoided Muon Level Crossing Resonance
Energy
A+– A–
↑ e , ↑µ , ↓ X
6.8
7.0
7.2
7.4
7.6
7.8
8.0
8.2
Magnetic Field / kG
↑ e , ↓µ , ↑ X
1 Aµ − AX
Bres ≈
2 γµ − γ X
Magnetic Field
Paul Percival
NUSC 341, November 2005
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tert-Butyl radical in water
CH3
CH2Mu
CH3
200°C
A+-A-
250 bar
10.0
10.5
11.0
11.5
Field / kG
Paul Percival
NUSC 341, November 2005
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Hyperfine constants are used to map unpaired spin in radicals
Mu13C60 Avoided Level Crossing Resonance
Mu
1.5
3.3
30
1.4
14
5.4
2.0
% 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
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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
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
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Muonium Chemistry — areas of application
Muonium
(caged)
Mu as probe of local environment
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?
Free radicals: unpaired spin distribution
(needs e-spin)
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
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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
Paul Percival
NUSC 341, November 2005