Investigation of materials
using μSR
Adrian Hillier
ISIS Muon Group
Warwick University Feb ‘14
Lecture Plan
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Introduction
Setting the muon in its historical context
Properties
Production of muons
Sources
Techniques
Applications of μSR to materials
Who ordered that!
I. Rabi (1898-1988) – on the discovery of the muon
Father Theodor Wulf (1911)
Experiments and theory
agree!
It’s a dangerous thing when experiments
agree with theory
Yukawa (1907-1981)
• p+ and e- interact by the exchange
of virtual photons (QED)
• => if the strong force in the nucleus is
mediated by exchange of ‘virtual
mesotrons’, then need to borrow energy
• ΔEΔt~h/2π
• This give ΔE~200me
Marcello Conversi (1917-1988)
Results
• Implant cosmic ray muons in matter
and measure their lifetime
• μ+ in anything 2μs
• μ- in C
2 μs
• μ- in Pb
0.07 μs
• mesotrons are actually muons
Properties
Charge
Spin
electron, e ±e 1/2
muon, μ
±e 1/2
proton, p ±e 1/2
Mass
(me)
1
Moment
(μp)
γ/2π
(MHz/T)
657
2.8x103 ∞
207 3.18
1836 1
135.5
42.6
Lifetime
2.19
∞
Cosmic Rays
Average energy 4 GeV
1 muon cm-2 min-1
ISIS
Where we do muon experiments…
UK:
ISIS
Canada:
TRIUMF
Japan:
JPARC
Switzerland:
PSI
Muon Production
Made of Graphite
Low Z less proton scatter
~ 900 K
Takes 5% of the proton beam
Pions
Pions are produced by p+p
π++p+n
Pions decay in 26 ns
π+
pμ
μ+
Sμ
μ++νμ
π+
Sν
νμ
pν
Surface Muons
Some pions stop in the target
They decay into muons, which escape if formed near the target surface
Muons collected into the beamline
100% spin polarised
Intense beam, though low momentum (E=4.1 MeV, p=29 MeV/c)
Beamline
The µSR technique…
High energy protons
(800 MeV at ISIS)
collide with carbon nuclei
producing pions
Implantation,
(stopped in ~1mm
water)
Muons interact with
local magnetic
environment
π+ → μ+ + νμ
4 MeV muons are
100% spin
polarised
Decay, lifetime 2.2μs
μ+ → e+ + νe + νμ
we detect decay positrons
The positrons are
preferentially
emitted in muon
spin direction
Monitor the positron distribution to infer the muons’ polarisation after
implantation. Learn about the muons’ local environment or the muon
behaviour itself.
Muon Implantation
Therefore there is an initial spin
polarisation of 100%
The muons interact with their
local environment and the
polarisation changes with time
The average spin polarisation of an
ensemble of muons at time t after
implantation is defined as the muon
spin relaxation function, G(t)
~1-3 mm
Implantation is rapid and occurs
without loss of muon polarisation
Muon Decay
Muons decay in 2.2 μs
μ+
e++νe+νμ
νe
pe
e+
Sμ
μ+
νμ
Muon Decay
How do we detect these
positrons?
Scintillator
Light Guide
PMT
Muon decay
Lifetime:
2.19714s
Decay asymmetry:
W() = 1+a0cos
ao~0.25
Gyromagnetic ratio:
1.355342x108 x2p s-1T-1
Garwin et al Phys. Rev. 1957
Muon Decay
τμ=2.2μs
Counts
exp(-t/)
0
2
4
6
8
10
12
Time (s)
14
16
18
20
Different sources
UK:
ISIS
Canada:
TRIUMF
Japan:
JPARC
Switzerland:
PSI
Pulsed Sources
These sources are at synchrotrons and operate at a frequency ~50Hz
A proton pulse gives a burst (500+) of muons which are implanted into the
sample
Measure all the positrons from burst
Continuous Sources
These sources are at cyclotron and are quasi-continuous (50MHz)
One muon is implanted at a time
Each positron is detected
If more than one muon enters or more than one positron is detected then
event is ignored
Pros and Cons
Pulsed
Continuous
Smaller fields can be measured
Signal size decreases with increasing field
Larger fields can be measured
Know entry time for muons
Muons enter when channeled to
beamline
Small (or zero) detector background
Large detector background
Rate unlimited
Rate limited
Complementary
Limits to the field range
Timing resolution
If you have a timing resolution of 200ps, therefore period is 400ps
Max field = 1/tperoidγμ = 18.4 T
Time window
Can measure to 30μs, therefore period is 60μs
Min field = 1/ tperoidγμ = 1.2 G
Limits to range
Maximum field will be susceptible to dynamics
The lowest field will be susceptible to amount of beamtime
How many muons are available after 30 μs?
I(t)/I(0) = exp(-t/τμ)
= 0.00012%
Dynamic Range
neutron scattering
Mossbauer
SR
NMR
ac susceptibilty
remanence
0
2
4
6
10
8
log (fluctuation rate), s
12
Spin-Precession Frequency
Moment in a field
gives precession
ω=γμ/2π * B
The highest frequencies are
associated with the lightest particles
hence
ESR – microwaves
NMR – radiowaves
μSR sits nicely in between
Muons..
Muons are fundamental, charged particles with:
spin 1/2
magnetic moment 3.2 x proton
mass 0.11 x proton
µ+
1/9
H
p
d
t
1
2
3
lifetime 2.2 µs: decay into a positron (and a couple of neutrinos)
Muonium: positive muon + electron hydrogen atom analogue
Muon and muonium behaviour can be observed via the technique
of µSR . . .
muon spin rotation, relaxation and resonance
MuSR instrument at ISIS
Relaxation
Bz or B=0

F
F(t) - B(t)
Rz (t) =
= aoGz (t)
F(t) + B(t)
B
The many faces of SR
Longitudinal zero field - SR
muons
Precessing polarisation
Precessing and relaxing
polarisation
Relaxing signal
0.4
0.4
Asymmetry
Asymmetry
Asymmetry
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
-0.1
-0.1
Detector A
Detector B
00
22
44
66
Time (s)
Time (s)
88
10
10
Rotation
Bx >0

F
Rx (t) 
F( t )  B( t )
 aoG x ( t ) cos(L t )
F( t )  B( t )
B
The many faces of SR
Transverse field SR
muons
non-relaxing
polarisation
magnetic
field
relaxing polarisation
25
20
20
15
asymmetry
Y Axis Title
10
10
5
0
0
-5
-10
-10
-15
-20
-20
-25
00
1
22
3
44
5
X Axis Title
time (s)
66
7
88
Relaxation functions
Muon Spin Precession
Pz (t) = cos2 q + sin2 q cos(g m Bt)
|B| is the modulus of the local dipolar field
Muon Spin Precession
Pz (t) = cos2 q + sin2 q cos(g m Bt)
Pz(t)
1.5
1.0
θ=0
0.5
θ=π/2
0.0
-0.5
-1.0
depends on θ
-1.5
Time
Muon Spin Precession
Pz (t) = cos2 q + sin2 q cos(g m Bt)
<cos2θ>=1/3
<sin2θ>=2/3
Pz (t) =1/3+ 2/3cos(g m Bt)
1.5
1.0
0.5
Pz(t)
Angular Averages:
0.0
-0.5
-1.0
-1.5
Time
Muon Spin Precession
Muon Spin Precession
Now assume B is distributed according to a Gaussian distribution
P(B)
P(B) α B2exp(-γ2B2/2Δ2)
B
Kubo-Toyabe functions
Recovery
Apply a longitudinal field and see the recovery
Oscillation is at the applied field
Dilute Spin system
Dilute spin -> Lorentzian distribution
P(Bi ) =
gm
a
p a2 + g m2 B 2
Dynamics
“Strong Collision” approximation
Assume:
1/ local field on muon is suddenly changed and after which is not
correlated with the previous state
2/ Collision take place at a rate ν
< B(t)B(0) >
= exp(-nt)
2
< B (0) >
Summing Up
Gz(t,ν)= muons that don’t collide
+ muons that collide once
+ muons that collide twice etc
Thanks
Additional Reading:
S. Blundell cond-matt/0207699
Introduction to muons Nagamine
Muon Science: Ed Lee, Kilcoyne, Cywinski