compleme ntarit y or competition · 2017. 5. 15. · ferromagnet spin glass exp(-λt) exp ... =si...
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Muon Training Course, March 2012
Professor Bob Cywinski
DeanSchool of Applied SciencesUniversity of Huddersfield
µSR and Neutron Scattering:
Compleme ntarit y or Competition ?
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What can neutrons tell us?
Neutrons:have wavelengths comparable to interatomic spacings (0.3-15 Å)
have energies comparable to structural and magnetic excitations (1-1000 meV)
are scattered with a strength that varies randomly from element to element (and isotope to isotope)
have a magnetic spin
are deeply penetrating (bulk samples can be studied)
interact only weakly with matter (so the theory is easy!)
Neutron scattering is therefore an ideal probe of atomic and magnetic structures and excitations
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Length scales.....
atomic and magnetic
structures
internal strain
organicmolecules magnetic defects pharmaceuticals supermolecules
surfaces and multilayers viruses inhomogeneities cracks and voids
micelles critical phenomena proteins polymers
Length scale in nm0.01 0.1 0.3 1.0 3.0 10 30 100
0.1 0.3 1.0 2.0neutron wavelength in nm
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Time scales.....
Excitation energy (eV)1 10-1 10-2 10-3 10-6 10-9
Time scale (seconds)10-13 10-7
Crystal fields magnons and phonons spin relaxationsingle particle spin fluctuations tunneling polymer reptationexcitations diffusion glassy dynamics molecular excitations libration
CsVCl3Fe
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Condensed matter studies with neutrons
1950 1960 1970 1980 1990
1950
1960
1970
1980
1990
2000
2000
…..and muons
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Complementarity?
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
104
10-4 10-3 10-2 10-1 100 101 102
E /
meV
Q / Å-1
108
107
106
105
104
103
102
101
100
10-1
10-2
10-3
104 103 102 101 100 10-1
r / Å
t / p
s
Spin Echo
Backscattering
Chopper
Inelastic
X-ray
Multi-Chopper
Inelastic Neutron
Scattering
Brillouinscattering
Ramanscattering
UT3
Photoncorrelation
VUV-FEL
µSR
NMR
Infra-red
Dielectric spectroscop
y
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Complementarity or competition?Complementarity or competition?
The neutron is a uniquely sensitive microscopic probe of the magnetic
properties of materials
The muon is a uniquely sensitive microscopic probe of the magnetic
properties of materials
1 + 1 > 2neutrons + muons = new insights
0 -2 -4 -6 -8 -10 -12
µµµµSR
Mossbauer
Neutron scattering
ac susceptibilty
remanence
log (relaxation time),s
µSR offers a time window sufficiently wide for studies of fast itinerant electron spin fluctuations through to slow distributed spin relaxation in spin glasses
…..and µSR is sufficiently sensitive for ultra-small magnetic moments (~10-3µB) and nuclear moments to be detected
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University of HuddersfieldInternational Institute for Accelerator Applications
Muon T
raining Course, M
arch 2012
Com
plementarity: P
art I
Slow
Dynam
ics
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Glassy relaxation
“The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition”
P W Anderson
Science 267 (1995) 1615
....and spin glasses are excellent model systems to study the glass transition
Random exchange leads to a random freezing of spin at the so-called glass temperature
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Glassy dynamics
seconds
1e-12 1e-11 1e-10 1e-9 1e-8 1e-7 1e-6
Cor
rela
tion
0.0
0.2
0.4
0.6
0.8
1.0 Exponential relaxation
seconds
1e-12 1e-11 1e-10 1e-9 1e-8 1e-7 1e-6
Cor
rela
tion
0.0
0.2
0.4
0.6
0.8
1.0“Kohlrausch” or stretched exponentialrelaxation
Fer
rom
agne
t
Spi
n gl
ass
exp(-λt)βexp(-λt)
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Kohlrausch relaxation
First observed in 1854, stretched exponential or “Kohrausch relaxation”is variously attributed to:
β may be either temperature dependent (approaching 1/3 at Tg) or temperature independent
A distribution of relaxation rates
Hierarchical relaxationPalmer et al, PRL 53 (1984) 958
Random walk on an n-dimensional hypercubeCampbell et al, Physica A 230 (1996) 554
Non-extensive entropyPickup, Cywinski et al PRL 102 (2009), 097202
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Simulations of 3d Ising Spin Glasses
Ogielski’s MC simulations of the ±J Ising spin glass show:β
ox
ii )τ/texp(t)t(S)0(S)t(q −∝= −
T=Tg
T<Tg
T>Tg
Ogielski, Phys.Rev. B 32 (1985) 7384
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Simulations of 3d Ising Spin Glasses
Ogielski’s MC simulations of the ±J Ising spin glass show:β
ox
ii )τ/texp(t)t(S)0(S)t(q −∝= −
x τo β
in principle these characteristic parameters can be measured with NSE and µSR.......
Ogielski, Phys.Rev. B 32 (1985) 7384
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Stretched-exponential relaxation
Our first µSR measurements of a canonical spin glass (AgMn) gave a muon spin relaxation function of the form
))t(exp(a)t(Ga)t(P ozozβλ−==
with λ=2γm2<Bi
2>τ
‘Stretched exponential’ relaxation in µSR is now widely accepted as an clear indicator of glassy dynamics
Campbell, Cywinski, Kilcoyne et al , Phys. Rev. Lett. 72 (1994) 1291
λ diverges at the glass transition Tg, whilst β decreases from 1 at 4Tg to 1/3 at Tg
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NSE measurements on Au 86Fe14
Fourier time, tf, ns
0.1 1 10
S(Q
,t)/S
(Q)
0.1
1
30.7K40.6K45.7K50.8K55.8K
T<Tg, S(Q,t)~t-x
T>Tg, S(Q,t)~t-xexp(-(t/τ)β)
C. Pappas, F. Mezei, G. Ehlers, P. Manuel, I.A. Campbell, PRB 68, 054431 (2003)
T=Tg
T<Tg
T>Tg
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Universal glassy relaxation ?
Combined NSE and muon measurements have enabled us to move towards a more “universal” model of (spin) glass relaxation
The model is based upon the Tsallis concept of non-extensive entropy in strongly interacting systems, and incorporates hierarchical, as well as parallel, relaxation processes
It leads to a single relaxation function
where k, the effective interaction parameter, is related to the subextensivity parameter 1<q<2, through
Note that as k→0,
and as β→0 and k→0,
Pickup, Cywinski et al PRL 102 097202 (2009)
Φ(t) = φ0[1+k(t/τ0)β]-1/k
k = (q-1)/(2-q)
Φ(t) = φ0 exp(-(t/τ0)β)
Φ(t) = φ0 exp(-t/τ0)
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Universal glassy relaxation ?
Pickup, Cywinski et al PRL 102 097202 (2009)
sube
xten
sivi
ty p
aram
eter
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But........Au 1-x Fex
FMPM
T [K
]
x [Fe]
x = 0.16
x = 0.14
x = 0.18
x = 0.20
Coles,et al. Phil. Mag. B 37, 489 (1978)
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µSR measurements on Au Fe
The µSR spectra from Au-14%Fe are also stretched exponential, with similar βs to those obtained from NSE
AuFe14%
Pappas, Mezei, Hillier, Cywinski in preparation
However at concentrations higher than the percolation threshold NSE spectra are simple exponential whilst µSR spectra remain stretched exponential above TF
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AuFe: A summary
FMPM
T [K
]
x [Fe]
x = 0.16
x = 0.14
x = 0.18
x = 0.20
Pappas, Hillier, Cywinski et al in preparation
spin glass phasestretched-exponential
NSE and muon spectra
Disordered ferromagnetic phase
exponential NSEstretched exponential
µSR
????
Temperature
0 50 100 150 200 250 300 350
λ, M
Hz
0
1
2
3
4
5
6
Temperature
0 50 100 150 200 250 300
λ , M
Hz
0
1
2
3
4
5
6
Temperature
0 50 100 150 200 250 300
λ , M
Hz
0
2
4
6
8
10
12
AuFe20%
AuFe18%
AuFe16%
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University of HuddersfieldInternational Institute for Accelerator Applications
Muon T
raining Course, M
arch 2012
Com
plementarity: P
art II
Fast dynam
ics
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YMn2 - a frustrated itinerant magnet
C15 Laves phase (Fd3m)ao=0.76nm
Cywinski et al , J Phys C 3 (1991) 6473
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Moment localisation and collapse
Cywinski, Kilcoyne et al , J Phys C 3 (1991) 6473
YMnYMn22
5% volume expansion at TN=100K
µMn=2.8 µB below TN ordered in a long wavelength helix.
Pauli paramagnetic above TN
YMnYMn22+2.5%Fe+2.5%Fe
The antiferromagnetic phase, the volume expansion and the Mn moment itself can be destabilised by 2.5at% Fe or 3kb
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Rainford , Cywinski and DakinJMMM 140-144 (1995) 805
Inelastic scattering spectra from YMn2+5at%Fe
0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6
Wavevector transfer (A-1)
60
0
-60
Ene
rgy
tran
sfer
(m
eV)
4K 50K 100K 200K
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A quantitative comparison......
The muon relaxation rate λ can be related directly to the parameters measured by inelastic neutron scattering:
∑ Γχ′
=λq
B
)q()q(
NTkG
For many itinerant antiferromagnets the q-dependence of Γ is weak:
Γχ′
=λ LBTkG
where∑χ=χ
qL )q(
N1
is the local susceptibility
Γχ′
=λ LBTkG
The neutron linewidth Γ can be compared directly with λ obtained from µSR via
with the coupling constant G´ as the only free parameterRainford in “Muon Science” eds Lee, Kilcoyne and Cywinski, 1999
where Γ(q) is the inelastic scattering linewidth, and χ(q) the susceptibility
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A quantitative comparison with µSR
)TkEexp()T( Bao −Γ=Γ
For Y(Mn0.9Al0.1)2 Γ is Arrhenius-like:
with Ea/kBT = 280K and θ =93K
while χL follows a Curie law:
)T(C)T(L θ+=χ
We can therefore fit the expression
with c as the only free parameter to the muon data
)kT/Eexp()T(cT
)T(a−θ+
=λ
Y(MnAl)2
Neutron and muon data are in excellent agreement – but while each muon spectrum takes 20min to collect, each neutron spectrum takes 12 hours Cywinski, Physica B 350 (2004) 17
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ns
Muon Training Course, March 2012
ββββ-Mn an elemental spin fluctuator
Al Concentration (at%)
0 5 10 15 20
Tra
nsit
ion
Tem
pera
ture
(K
)
0
5
10
15
20
25
30
35
40β−Mn0.94Al0.06
Spin liquidphase
Simple exponential µSR)t(
KTo eGa)t(P λ−=
Stewart and Cywinski PRB59 (1999) 4305
β−Mn0.85Al0.15
Stretched exponential µSRβλ−= )t(
KTo eGa)t(P
Spin glassphase
![Page 28: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/28.jpg)
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Scaling of spin relaxation rates
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ββββ-Mn an elemental spin fluctuator
Al Concentration (at%)
0 5 10 15 20
Tra
nsit
ion
Tem
pera
ture
(K
)
0
5
10
15
20
25
30
35
40
Stewart and Cywinski PRB59 (1999) 4305PRL 89 (2002) 6403
Neutron scattering has shown that β-Mn is the first example of an elemental non-Fermi liquid
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raining Course, M
arch 2012
Com
plementarity: P
art III
Magnetic S
tructures
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Small moment systems
eg – a in fully organic ferromagnetic
Observation of long range ferromagnetic order in a small moment magnet
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Magnetic volume fractions
inho
mog
eneo
us
h
omog
eneo
us
inho
mog
eneo
us
h
omog
eneo
us
Amplitude = Magnetic Volume FractionFrequency = Size of magnetic momentsDamping = Inhomogeniety within magnetic regions
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Magnetic volume fractions; URu 2Si2m
2(µ
B)2
Neutron diffraction shows the U moment in URu2Si2 decreasing in magnitude as pressure is increased
![Page 34: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/34.jpg)
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Magnetic volume fractions; URu 2Si2m
2(µ
B)2
Neutron diffraction shows the U moment in URu2Si2 decreasing in magnitude as pressure is increased
µSR shows that this decrease is associated with a reduced antiferromagnetic volume fraction
There is phase separation into magnetic and non-magnetic regions– only a combined neutron + µSR study could reveal this
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Magnetic order in ReNiO3 compounds
Ni
Ni
Ni
Re
Re
Model 1 Model 2
} µµµµ=0} µµµµ≠≠≠≠0
µµµµ+1
µµµµ+2
Garcia-Munoz, Lacorre, Cywinski PRB51 (1995) 15197
Neutron powder diffraction is unable to distinguish between the two distinct magnetic models
![Page 36: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/36.jpg)
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Magnetic order in ReNiO3 compounds
( )texpa)t(G 0z λ−=
Above the MI transition there is a single paramagnetic muon site:
Below the MI transition there are two equally populated but inequivalent muon sites, one magnetically ordered, the other disordered.
Model 1 must be correct
Garcia-Munoz, Lacorre, Cywinski PRB51 (1995) 15197
![Page 37: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/37.jpg)
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A cautionary tale: Ni 3Al
Dhar et al, PRB 40 (1989) 11488
The general consensus from neutron and other studies is that Ni3Al orders ferromagnetically at at 40K, with a Ni moment of ~0.076µB and Tc=40K
Ni3Al - space group Pm3mL12 (Cu3Au) structure
a0=0.356nm
NiAl
![Page 38: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/38.jpg)
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Ni3Al - high temperature muon spectra
Time (µs)
2 4 6 8 10 12
Asy
mm
etry
0.00
0.05
0.10
0.15
0.20
0.25
0.30
T=250K
B=0
B=1mT
B=2mT
Ni3Al
At high temperature, the zf spectra are well described by
b2t(22
1zo ae)t1(32
31
a)t(Ga22
+
σ−+= σ−
with σ=0.15 ms-1
σ is consistent with the muon sitting at 1/2,1/2,1/2 in the cell
…and any contribution from atomic spin fluctuations is entirely motionally narrowed
![Page 39: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/39.jpg)
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Ni3Al - low temperature spectra
Time (µs)
0 1 2 3 4 5 6A
sym
met
ry0.125
0.150
0.175
0.200
0.225
0.250
0.27511K20K29K47K32K
The atomic spin contribution to the µSR spectra remains motionally narrowed down to 40K
Below 40K there are two contributions to the spectra
( ) ( )
)t(Ga
e)t1(32
31
a)t(Ga
zKTnm
t(mzo
−
λ−+= βλ−β β
σ−+= σ− 2t(22zKT
22
e)t1(32
31
)t(G
am and anm refer to the magnetic and nonmagnetic components respectively, and
![Page 40: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/40.jpg)
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Ni3Al: Conclusions from µSR
The magnetic contribution below 40K takes the form of a Voigtian Kubo-Toyabe function
Crook and Cywinski J Phys Cond Matt 9 (1997) 1149
parameters from ni3almag.for
Asy
mm
etry
, am
0.00
0.05
0.10
0.15
λ (µ
s-1)
0
1
2
3
Temperature (K)
0 10 20 30 40 50β
0.8
0.9
1.0
1.1
1.2
1.3
Tc=40K
Tc=40K
Tc=40K
The asymmetry, am, muon relaxation rate λ and exponent βall increase below the reported Curie temperature of Tc=40K
There is no evidence of long range ferromagnetic order in Ni3Al, but rather of an inhomogenous and relatively staticmagnetic ground state.
S H Kilcoyne and R Cywinski Physica B 326 (2003) 577
But this is because the muon sits at a site of high symmetry and is insensitive to long range ferromagnetic order!!
![Page 41: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/41.jpg)
University of HuddersfieldInternational Institute for Accelerator Applications
Muon T
raining Course, M
arch 2012
Com
plementarity: P
art IV
Superconductivity
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The vortex lattice
ξ- the coherence length The length scale over which the superconducting wave function Ψ varies
λ- the penetration depth The length scale over which the flux density varies
The two characteristic length scales:
![Page 43: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/43.jpg)
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Small angle neutron scattering
R
scattering angle 2θ
sample
multi-detector64x64cm2
B
L
...measures the long range periodic flux density
![Page 44: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/44.jpg)
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Probing the flux lattice with muons
100nm100nm
µSR measures the local variations in flux density
![Page 45: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/45.jpg)
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Flux lattice melting
42 46 500
20
B (mT)
p (
B)
42 46 500
20
40
60
80
B (mT)
p (B
)
0 20 40 60 80 100-1.50
-1.00
-0.50
0.00
0.50
1.00
1.50
T (K)
T m
α =∆B 3
13
∆B 212
T<TT<Tmm<T<Tcc TTmm<T<T<T<Tcc
Lee et al, Phys Rev B55 (1997) 5666
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Muon T
raining Course, M
arch 2012
Com
plementarity: P
art V
Surface studies
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Neutron reflectometry
Peptide absorbed on lipid membranes
surfaces, bilayers, multilayers, interfaces
refractive index:
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Specular reflection
∆ρT
Bragg peak
a
QR=2π/a∆QR=2π/T
Step Thin film Multilayer
Thin filmInterference fringes
Critical edgeR=1 for QR<QCQC=4(π∆ρ)1/2
![Page 49: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/49.jpg)
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Off-specular reflection
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Low energy muons
Muons are cryogenically moderated and energy-selected to tune localisation depth within the sample:
E(keV) R(nm) ∆R(nm)
0.010 0.5 0.3
0.100 2.1 1.3
1.0 13.1 5.4
10.0 75.0 18.0
30.0 244.0 36.0
See eg, Morenzoni in “Muon Science”eds Lee, Kilcoyne and Cywinski, 1999
Incident muon rates remain relatively low (~500µ/s)
![Page 51: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/51.jpg)
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Flux distribution at surface of YBa2Cu3O7
simulated measured
Niedermayer, Forgan et alPRL83, 3935, 1999
YBa2Cu2O7
Ag
![Page 52: Compleme ntarit y or Competition · 2017. 5. 15. · Ferromagnet Spin glass exp(-λt) exp ... =Si (0)S i (t) ∝t exp(−t/ τ) ... an clear indicator of glassy dynamics Campbell,](https://reader036.vdocuments.net/reader036/viewer/2022071404/60f84112426793662376c7ef/html5/thumbnails/52.jpg)
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....and near surface dynamics
Surface dynamics of a thin polystyrene film probed by low energymuons:
Pratt et al PRB 72, 121401R 2005
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Summary
Neutrons and muons together often give more information than either technique alone - even in crystallographic studies
Muons can provide crucial insights very quickly, often providing justification for a detailed and lengthy neutron experiment
Where neutron and muon results disagree, there may be new physics at play