Investigation of materials
using μSR
Adrian Hillier
ISIS Muon Group
Warwick University Mar‘11
Lecture Plan
 Introduction
 Setting the muon in its historical context
 Properties
 Production of muons
 Sources
• Techniques
• Applications of μSR to materials
MuSR instrument at ISIS
Relaxation
Bz or B=0
m
F
Rz(t) 
F(t)B(t)
aoGz (t)
F(t)B(t)
B
The many faces of mSR
Longitudinal zero field - mSR
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 (ms)
Time (ms)
88
10
10
Relaxation functions
Muon Spin Precession
Pz (t)  cos 2   sin2  cos( m Bt)
|B| is the modulus of the local dipolar field

Muon Spin Precession
Pz (t)  cos 2   sin2  cos( m Bt)
1.5
Pz(t)

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)  cos 2   sin2  cos( m Bt)
Angular Averages:
<cos2θ>=1/3
<sin2θ>=2/3
Pz (t) 1/3 2/3cos( m Bt)

1.5
1.0
Pz(t)

0.5
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
Gz (t) 

 cos   sin  cos( m Bt)P(B )P(B )P(B )d B
2
2
3
x
y
z
Recovery
Apply a longitudinal field and see the recovery
Oscillation is at the applied field
Dilute Spin system
Dilute spin -> Lorentzian distribution
P(Bi ) 

m
a
 a2   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(t)
2
 B (0) 

Summing Up
Gz(t,ν)= muons that don’t collide
+ muons that collide once
+ muons that collide twice etc
Rotation
Bx >0
m
F
Rx(t) 
F(t) B(t)
 aoGx(t)cos(Lt)
F(t) B(t)
B
The many faces of mSR
Transverse field mSR
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 (ms)
66
7
88
Applications of μSR to materials
Science with muons
Muons as probe particles can inform on superconductivity, magnetism,
molecular dynamics, charge transport . . .
Muons at ISIS
MUONS:
• Versatile probes of magnetic,
superconducting, molecular
systems
• Analogues of
protons/hydrogen in
semiconductors
• Complementary to other
techniques ISIS / PSI / J-PARC
all have neutron and muon
facilities
• Around 60 groups from 20
countries using ISIS muons
Radical
studies 11%
Molecular
dynamics
2%
Light
particle
diffusion 2%
Inorganic
magnetism
31%
Spintronics
4%
Other H
studies 4%
H in other
semiconds.
7%
H in II-VIs
and oxides
9%
Other 1%
Organic
magnetism/
supercond.
15%
Polymer
charge
transport
3%
Ion/proton
transport
2%
Inorganic
supercond.
9%
Magnetism
CeRu2Al10
Khalyavin, Hillier et al Phy. Rev. B 82 100405R (2010)
CeRu2Al10
Khalyavin, Hillier et al Phy. Rev. B 82 100405R (2010)
CeRu2Al10
Khalyavin, Hillier et al Phy. Rev. B 82 100405R (2010)
Sr3ZnRhO6
Hillier et al Phys. Rev. B 83 24414 (2011)
Sr3ZnRhO6
Hillier et al Phys. Rev. B 83 24414 (2011)
NaxCoO2
Mendel et al PRL 94 136403 (2004)
Spin dynamics in frustrated systems
First order transition in the
spin dynamics of
geometrically frustrated
Yb2Ti2O7
J Hodges et al, PRL 88 (2002)
077204
• Yb2Ti2O7 - pyrochlore exhibiting
geometrical frustration
• Neutron diffraction, Mossbauer spectroscopy and mSR used to characterise
magnetic behaviour
• First order transition at 0.24K is not to long (or short) range order, but is a
change in the Yb3+ fluctuation rate.
Superconductivity
Superconducting transition temperature (K)
Superconductivity in alloys and
oxides
160
HgBa2Ca2Cu3O9
(under pressure)
140
HgBa2Ca2Cu3O9
120
TlBaCaCuO
BiCaSrCuO
100
YBa2Cu3O7
Liquid Nitrogen
temperature (77K)
80
60
(LaBa)CuO
40
20
Hg Pb Nb
1910
NbC NbN
1930
Nb3Sn
Nb3Ge
V3Si
1950
1970
1990
The two characteristic
length scales
 - the coherence length
l - the penetration depth
The length scale over
which the superconducting
wave function  varies
The length scale over
which the flux density
varies
The mixed state in Type II
superconductors
Hc1< H <Hc2
The bulk is diamagnetic but it is
threaded with normal cores
B
The flux within each core is generated
by a vortex of supercurrent
Hc1
0
-M
Hc2
H
The flux lattice
2mm
10mm
Small angle neutron scattering
R
multidetector
64x64cm2
L
B
scattering
angle 2
sample
incident
neutron beam
Probing the flux lattice with
muons
100nm
Calculating the field
distribution
The second moment of the field distribution is given by
B
2
1
2
1/ 2
0.00371o2 


l4


In this case the transverse field relaxation, Gx(t) is also Gaussian:
2t2
Gx(t)e
75780
 2
l
with  in ms-1 and l in nm.
The Gaussian approximation thus provides a simple relationship
between  and l,
-and is often a reasonable approximation
FeTe0.5Se0.5
Biswas et al Phys. Rev. B, 81 92510 (2010)
FeTe0.5Se0.5
Lu2Fe3Si5
Biswas et al Phys. Rev. B, 83 54517 (2011)
Lu2Fe3Si5
T=6.5K
T=0.3K
Biswas et al Phys. Rev. B, 83 54517 (2011)
Lu2Fe3Si5
Biswas et al Phys. Rev. B, 83 54517 (2011)
What about type I?
LaRhSi3
Anard, Hillier et al Phys. Rev. B 83 64522 (2011)
LaRhSi3
Semiconductors
Shallow donor states in II-VI
semiconductors
• First experimental evidence
of muonium (and hence
hydrogen) forming shallow
donor states in semiconductors
• Of very real interest to
semiconductor community
• Pulsed muon source essential to allow measurements to long times - ISIS
is unique in Europe.
Shallow donor muonium states in II-VI semiconductor compounds
J Gil et al, PRB 64 (2001) 075205
Particle Diffusion
Li ion mobility in battery electrode material
• Li-based cathode materials can be used
to produce solid state batteries which are
cheap and light.
• Optimisation requires knowledge of Li
ion mobility for different compositions.
• Implanted muons can be used to follow
the Li ion diffusion behaviour, revealing
composition-dependent onset
temperatures.
Li mobility in the battery cathode material Lix[Mn1.96Li0.04]O4 studied by muon spin relaxation
C Kaiser et al, Phys Rev B 62 (2000) R9236.
Sugiyama et al, Phys. Rev. Lett. 103 147601 (2009)
Ariza et al, J. Phys. Chem 107 6003 (2003)
Proton diffusion in ReO3
• HxMO3 materials have applications as catalysts,
displays, etc. Proton sites and motion are central to
their properties.
• Observe muon behaviour to inform on protons
• Muon linewidth  can be calculated given relative
positions of m+ and Re.
Proton diffusion in ReO3
B
0.10
C
0.08
-1
 ( ms )
A - static interstitial with distortion
B - rapidly rotating O-m, no long-range
motion
C - Diffusion between O-Mu sites
D - Trapping at a defect
E - diffusion between
D
E
trapping sites
A
0.12
0.06
0.04
0.02
0.00
0
100
200
300
400
500
600
Temperature (K)
A
B
C
D
Muon study of proton sites in rhenium oxide JS Lord et al.
Summary
• Muons provide a unique probe of atomic
structure/dynamics, often providing complementary
information to that given by neutrons.
• They can be used as sensitive probe particles to
investigate molecular dynamics, magnetism,
superconductivity, charge transport . . .
• They can be used as proton analogues to provide
models of hydrogen atom behaviour . . . for example in
semiconductors, proton conductors, hydrogen storage
materials, etc.