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122 April 2016 RDR- - NMR and MuSR
Magnetism I: Muons and NMR
Outline - I• Un peu d'historie (Bloch, Purcell, Garwin, Ledermann)• NMR basics
o Precession and nutationo Effective fieldo Receptivity
• The muono Productiono Parity violationo Asymmetry
• Time scales
• Relaxationso Bloch equations
- Interactionso Magnetic dipoles (the muon, the nuclei, the electrons)o Fermi contact and pseudodipolar o Electric nuclear quadrupole
Roberto De Renzi– Parma (Italy)
e Sci en ze d e l l a T e rra
e Sci en ze d e l l a T e rra
222 April 2016 RDR- - NMR and MuSR
Magnetism I: Muons and NMR
Outline - II• Magnetic phase transition (static, order parameter and dynamic, fluctuations)
o MnF2,CoF2 (19F NMR, muons)
o La1-xSrxMnO3 vs. La1-xCaxMnO3 (55Mn NMR and muons)
• Dilution of magnetic momentso La2Cu1-xMxO4 with M = Mg, Zn (139La NQR, muons)
• Frustration o Ca3Co2O6 (59Co NMR)
o Exchange in an Ising magnet
• Disordero Y1-zEuzBa2Cu3O6+x (muons)
o (Y:Ca)Ba2Cu3O6+x (muons)
• Superconductorso V (muons)o YBa2Cu3O6.9
- Molecular ringso Cr8Cd
3322 April 2016 RDR- - NMR and MuSR
Un peu d'histoire: NMR
3
I.I. Rabi et al. Phys. Rev. 53 (1938) 318Nobel Prize for the year 1944
F. Bloch et al. Phys. Rev. 69, 127 (1946)E. M. Purcell et al., Phys. Rev. 69, 37 (1946)Nobel Prize for the year 1952 with E.M. Purcell
The
Ste
rn-G
erla
ch a
ppro
ach
Rad
io freq
uen
cy resonan
ce
422 April 2016 RDR- - NMR and MuSR
Un peu d'histoire: µSR
L. Garwin et al. Phys. Rev. 105, 1415 (1957)
Nobel Prize to Lee, Yang for parity violation (1957)
5522 April 2016 RDR- - NMR and MuSR
NMR cartoon
5
A magnetic moment (spin) precesses around B
B
6622 April 2016 RDR- - NMR and MuSR
NMR basics
6
Internal field
Precessing nuclear magnetization Mproduces an e.m.f. in a coil (Free Induction Decay, echoes, etc.)
7722 April 2016 RDR- - NMR and MuSR
NMR basics
7
Internal field
However M // B at equilibrium
8822 April 2016 RDR- - NMR and MuSR
NMR basics
8
A pulse of a small rf field at the Larmor frequency
makes M nutate until M ┴ B
9922 April 2016 RDR- - NMR and MuSR
NMR basics: the rf pulse
9
Nuclear magnetic moment
Larmor precessionLarmor frequency
Rotating frame at , apparent torque apparent field
effective field
101022 April 2016 RDR- - NMR and MuSR
NMR basics: the pulse in the rotating frame
B
Ba
Brf
Mn
Nutation
switch of the rf when M is in the xy plane
111122 April 2016 RDR- - NMR and MuSR
NMR basics:resonance and the effective field
In the frame rotating at the rf ω effective field: Beff = B – ω/γ + Brf
Beff ┴ B solo per B – ω/γ ≈ 0
Ba
Brf
Beff
sin(θ)= 1
√1+(B−ω/γ)2
Brf2
121222 April 2016 RDR- - NMR and MuSR
The probes
Receptivity
Group
=B
1γ=2π⋅42.8MHz /T
131322 April 2016 RDR- - NMR and MuSR
Receptivity
Amplitude A proportional to:
isotope abundance (natural or enriched)
nuclear magnetization M in a magnetic field B (Curie law)
Faraday induction M
BM=a
γ2ℏ2
3kBTB
A∝a
A∝dMdt
=ωM=aγ3ℏ2
3kBTB2 Brf
141422 April 2016 RDR- - NMR and MuSR
µSR: the muon
Charge µ- , µ+ (antiparticle)
Mass mµ = 0.1126 m
p = 206.8 m
e
Spin Iµ = ½
Gyromagnetic ratio γµ = 2π 135.5 MHz/T
Mean lifetime τµ = 2.197 µs
Production: π+ → µ+ + νµ (anti-muon, muon-neutrino)
Decay: µ+ → e+ + νe + ν
µ
151522 April 2016 RDR- - NMR and MuSR
µSR: basic aspectsproton accelerators
Pion production:
Impinge protons e.g. at E~800 MeV on solid target
π+ → µ+ + νµ
in 26 ns
http://www.fis.unipr.it/~roberto.derenzi/dispense/pmwiki.php?n=MuSR.MuSR
161622 April 2016 RDR- - NMR and MuSR
µSR: basic aspectswhere?
ISIS Chilton UK
TRIUMFVancouverCanada
JPARK TsukubaJapan
PSI Villigen CH
171722 April 2016 RDR- - NMR and MuSR
µSR: basic aspectsparity violaton
The mirror image does not exixt in nature
E.g. only negative helicity (chirality) neutrinos exist.
At work with angular momenum conservation...
181822 April 2016 RDR- - NMR and MuSR
Spin polarized muon production
@ ISIS:
2. transmission target(1 cm thick C)pions (π+) production and decay into polarized µ+
3. magnetic transport optics
1. primary proton beam
191922 April 2016 RDR- - NMR and MuSR
µ decay
4. μ+ stops in the sample. μ decay violates parity as well
e+ preferentially emitted along Sμ
probability lobe
202022 April 2016 RDR- - NMR and MuSR
Longitudinal asymmetry
θ = 0θ = π
Decadimento nel tempo
Asymmetry
5. More counts for θ = 0 , less for quello θ = π
212122 April 2016 RDR- - NMR and MuSR
%
Spin and emission probability precess at the Larmor frequency
Statistics on million events, eachrecorded vs time from its
own implantation
c
222222 April 2016 RDR- - NMR and MuSR
Continuous vs. pulsed beams
(nearly) continuous: PSI Villigen TRIUMF Vancouver
Cockroft-Walton Proton syncro-cyclotrone590 MeV β=0.8 2mA cfr. LHC 20 pA, ESRF 200mA (e-)
~ 15 m
Injector IIring
fascio primariop+
232322 April 2016 RDR- - NMR and MuSR
Continuous beams
μ+ extracted randomly from
Each μ lifetime measured individually
- Start when μ+ stops in the sample
- Stop when e+ is emitted
One at a time, τμ = 2.2 μs, up to 5τ
μ
→ not more than 50000 ev/s
~20 ns
typical resolution 80 - 1000 ps
242422 April 2016 RDR- - NMR and MuSR
Pulsed beams
ISIS: Fascio primario (~0.3 mA):
70 ns
300 ns 20 ms
~ 103 μ+/pacchetto
100 ns
400 ns 40 ms
JPARK-MUSE (~0.3 mA)
~ 5 103 μ+/pacchetto
252522 April 2016 RDR- - NMR and MuSR
Comparison
Polarization 100% Probe implanted nucleus (interstitial) in the compoundMaximumtime 50 μs 1-100 swindow
Mininum time 2-3 ns 2-3 μswindow
Detection broad band narrow band
M=γ
2ℏ
2
3k BTB
NMRμSR
262622 April 2016 RDR- - NMR and MuSR
Time windows
10-17 10-15 10-13 10-11 10-9 10-7 10-5 10-3 10-1 101 103
μSR
INS NSE
NMR
Macroscopic techniques
time (s)
electronic excitations molecular motions, diffusion
272722 April 2016 RDR- - NMR and MuSR
Relaxations: Bloch equations
Nuclear magnetization
When the equation of motion is
With a relaxation mechanism
Decay of transverse coherence
Decay towards thermodynamicequilibrium M
z0
Precession
282822 April 2016 RDR- - NMR and MuSR
Spin interactions
Magnetic interactions, between classical dipoles
S
S
Mn3+
Mu
O
Iμr
With electrons and positive nuclei, muons, finite probability that r = 0,
leading to Fermi contact interaction
eg
t2g
eg
t2g
Mn4+
292922 April 2016 RDR- - NMR and MuSR
Spin interactions: muon case, simple 1s hydrogen-like
Distant dipole field, dominant
S
S
Mn3+
Mn4+
Mu
O
Iμr
Fermi contact interaction
isotropic, Bc || S, transferred
303022 April 2016 RDR- - NMR and MuSR
Spin interactions: nuclear casemultielectron
Distant dipole field
S
S
Mn3+
Mn4+
O
139I
La
Fermi contact interaction
isotropic, Bc || S, core polarized
plus anisotropic pseudodipolar, e.g.
55I
55I
313122 April 2016 RDR- - NMR and MuSR
Electric quadrupole interactions
Nuclei with possess an electric quadrupole moment Q
, electric field gradient
I spin along majorprincipal axis
323222 April 2016 RDR- - NMR and MuSR
2I = 5 transitions
NMR + NQR, e.g. I = 5/2
B
z
+½
-½
E+3/2
-3/2
+5/2
-5/2
powders
Ix
single crystal
333322 April 2016 RDR- - NMR and MuSR
Continuous magnetic phase transitions
33
Order-disorder transition: critical behaviour at Tc
Tc
• Static aspects
343422 April 2016 RDR- - NMR and MuSR
MnF2 rutile
F-
eg
t2g
Mn2+
(spin-only, cubic Cristal Field)S=5/2 no single-ion anisotropy:
Heisenberg 3D?
Prototype localized antiferromagnet
34
353522 April 2016 RDR- - NMR and MuSR
MnF2 by 19F NMR:
P. Heller Phys. Rev. 146 (1966) 403
19 19 19ν γ B M Ts
35
0 333 3β . ( )
363622 April 2016 RDR- - NMR and MuSR
Ising 3D: muons in CoF2
TsMμBμγμν
36
Close to TN:
critical power-law behaviour
Ising
Heisenberg
1
0 33 2 , cfr.
0 327
0 367
β
μN
th
TB T
T
β . ( )
.β
.
R. De Renzi et al. Phys Rev. B 31 (1984) 186
Low T:
(spin wave excitations)
0
10
B TΔS μS Bμ
373722 April 2016 RDR- - NMR and MuSR
Ising 3D: muons in LaMnO3
Another antiferromagnet
Close to TN:
critical power-law behaviour
M. Cestelli et al. Phys Rev. B 64 (2001) 064414
37
383822 April 2016 RDR- - NMR and MuSR
Critical exponents
Critical exponents (statics)
Models
Ising 3D 1.2372(5) 0.6301(4) 0.0364(5) 0.110(1) 0.3265(3)
Ising 2D 7/4 1 1/4 1/8
Heisenberg3D
1.392(5) 0.71(1) 0.037(2) 0.367(3)
Heisenberg2D
Orders only at T=0 (see later)
XY 3D 1.317(1) 0.671(1) 0.037(1) -0.015(1)
XY 2D Kosterlitz-Thouless (not power laws)
Mean-field 1 1/2 0 0 1/2
γ
χ∝|t|−γ
ν
ξ ∝(.−t )−ν
η
G ∝|t|−d +2−η
α
C H∝|t|−α
β
|M|∝(−t )β
38
0c
c
T Tt
T
For vanish or diverge as power laws
A. Pelissetto, E. Vicari, Physics Reports 368 (2002 ) 549-727
393922 April 2016 RDR- - NMR and MuSR
Magnetic phase transitions
39
Order-disorder transition: critical behaviour at Tc
Tc
• Dynamic aspects (critical fluctuations)
404022 April 2016 RDR- - NMR and MuSR
F
µr
1µ
Mn
Interactions
Dipolar (distant dipoles)
contact (transferred)
40
414122 April 2016 RDR- - NMR and MuSR
F
µr
1µ
Mn
Interactions
Dipolar (distant dipoles)
contact (transferred)
41
424222 April 2016 RDR- - NMR and MuSR
Relaxations below and above Tc
T>Tc T<T
c
42
Different average, same fluctuations
434322 April 2016 RDR- - NMR and MuSR
NMR, muon relaxation vs neutrons
Magnetic neutron scattering cross-section
43
Relaxation rate of local probe coupled to
at the Larmor frequency!
In both the time Fourier transform of spin-spin correlations
444422 April 2016 RDR- - NMR and MuSR
NMR, muon relaxation
Relaxation rate of local probe coupled to
44
Simplest possible correlation: a decay
454522 April 2016 RDR- - NMR and MuSR
NMR, muon relaxation vs neutrons
Relaxation rate of local probe coupled to
45
For fast electronic dynamics ( )
ħ (eV)0
T (K)
A peak when
464622 April 2016 RDR- - NMR and MuSR
Dynamic critical behaviour
LaMnO3 T→T
N MnF2
(below TN) (above T
N)
Spin fluctuations slow down approaching the transition
Spin fluctuations slow down approaching the transition
46
2nν( z d η )
TN
0 67(7)n .
3D nIsing 0.717
Heisenberg 0.329
M. Cestelli et al. Phys Rev. B 64 (2001) 064414 Brown et al. Phil. Mag. Lett. 73 (1996) 195
474722 April 2016 RDR- - NMR and MuSR
La1-x
(CaySr
1-y)
xMnO
3:
FM double exchange
La5/8
Sr3/8
MnO3
La5/8
Ca3/8
MnO3
y
0
0.5
1
x0 0.5
optimal doping x = 3/8
Interplay of Double Exchange and bandwidth
Wide band
Narrow band
484822 April 2016 RDR- - NMR and MuSR
La1-x
(CaySr
1-y)
xMnO
3:
magnetic order parameter
La5/8
Sr3/8
MnO3
optimal wide band optimal narrow band
x
0 0.5 13/8 La
5/8Ca
3/8MnO
3y
Second order transition
First order truncated
First order truncated
Truncation by polaron transition
55Mn NMR
494922 April 2016 RDR- - NMR and MuSR
La1-x
(CaySr
1-y)
xMnO
3:
double exchange
La5/8
Sr3/8
MnO3
La5/8
Ca3/8
MnO3
a
c
Meskine Phys Rev B 64 094433
y
x
0 0.5 1
optimal doping x = 3/8
x=3/8
Sr-rich Ca-rich
505022 April 2016 RDR- - NMR and MuSR
1st vs. 2nd order, why?
Tp polaron localization
Tc Curie temperature
G. Allodi et al. J. Phys. Cond. Mat., 26 266004
515122 April 2016 RDR- - NMR and MuSR
Dilution of the magnetic ions
Zn, Mg
La2Cu1-xMxO4
Carretta et al. Phys. Rev. B 83 (2011) 180411(R)
51
5% Zn T = 50 K = 0.2 TN
525222 April 2016 RDR- - NMR and MuSR
Dilution of the magnetic ions
Zn, Mg
La2Cu1-xMxO4
Carretta et al. Phys. Rev. B 83 (2011) 180411(R)
52
535322 April 2016 RDR- - NMR and MuSR
Dilution of the magnetic ions
Zn, Mg
La2Cu1-xMxO4
Carretta et al. Phys. Rev. B 83 (2011) 180411(R)
53
x=0 0.2 xc = 0.4
percolation
545422 April 2016 RDR- - NMR and MuSR
Dilution of the magnetic ions
Zn, Mg
00 0.1 0.2
La2Cu1-xMxO4
x
Carretta et al. Phys. Rev. B 83 (2011) 180411(R)
54
0.3 0.4
Chernyshev Phys. Rev. B 65, 104407 dilution on a Heisenberg AF square lattice
Zn further reduction due to frustration
555522 April 2016 RDR- - NMR and MuSR
Dilution of the magnetic ions
Zn, Mg
La2Cu1-xMxO4
55
Phys. Rev. B 83 (2011) 180411(R)
Also the order parameter M (staggered) is reduced
565622 April 2016 RDR- - NMR and MuSR
Frustration
56
Ising AF on a triangular lattice…
z
?
575722 April 2016 RDR- - NMR and MuSR
Frustration
57
Ising AF on a triangular lattice…
… a pyrochlore lattice
z
?
also Ising Ferromagnetis frustrated :
Ho2Ti2O7 SPIN ICE
585822 April 2016 RDR- - NMR and MuSR
Frustration
58
Geometric: Competing interactions AF-F Ising AF on a triangular lattice… generate frustration
… a pyrochlore lattice
z
?
595922 April 2016 RDR- - NMR and MuSR
A frustrated Ising chain systemCa
3Co
2O
6
Ferromagnetic (F) Ising chainswith AF interchain coupling
Two Co sites
Co3+
Co3+
G. Allodi et al. Phys Rev. B 89 104401G. Allodi et al. Phys Rev. B 83 104408 with Stefano Agrestini and Martin R. Lees
606022 April 2016 RDR- - NMR and MuSR
Determine J1 J
2 J
3
s=0s=2s=0s=2s=0
Ising chains(Ferromagnetic intra-chain coupling)
Ising AntiFerromagnetic inter-chain couplings
?
J2,3
< 0
geometricfrustration
J1
J2
J2
616122 April 2016 RDR- - NMR and MuSR
Aim: determine J1 J
2 J
3
s=0s=2s=0s=2s=0
Ising chains(F intra-chain coupling J
1)
Ising AF inter-chain couplings, J
2 and J
3
no spin waves, Glauber (activated) Isingdynamics
J1
J2
J2
626222 April 2016 RDR- - NMR and MuSR
Frustrated ground state(s)with M = 0
µ0H
determined by J1 J
2 J
3 from
636322 April 2016 RDR- - NMR and MuSR
Ground state with increasing µ0H
µ0H
FI
BFIM
646422 April 2016 RDR- - NMR and MuSR
Ground state with increasing µ0H
µ0H
FI
BFIM
F
BFM
656522 April 2016 RDR- - NMR and MuSR
FI regime
FI F
BFIM
BFM
Co3+
µ0H
sharp quadrupolar septets
in and
µ0H
2.8 TB
d
1.2T
59B = 1.6 T59γ = 10 MHz/T
νL = 16 MHz
666622 April 2016 RDR- - NMR and MuSR
FI majority and minority chainsFM
FI F
BFIM
BFM
Co3+
FIphase
majority
minority G. Allodi et al. Phys. Rev. B 83 104408
676722 April 2016 RDR- - NMR and MuSR
Calculate the cost of spin reversal from
G. Allodi et al. Phys Rev. B 83 104408
686822 April 2016 RDR- - NMR and MuSR
Calculate the cost of spin reversal from
Ferrimagneticmajority FI↑
Ferrimagneticminority FI↓
Ferromagnetic
G. Allodi et al. Phys Rev. B 89 104401
696922 April 2016 RDR- - NMR and MuSR
The clean cuprate:YBa
2Cu
3O
6+x
69
chains
also Y-Eu, Y-Nd isoelectronic Eu3+-Nd3+ for Y3+ (up to 8% disorder)
=0.6 µB
F. Coneri et al. PRB 81 104507
707022 April 2016 RDR- - NMR and MuSR
Y1-z
CazBa
2Cu
3O
6+x
70
Same Y site: disorder with charge (up to 8%)
dramatic widening of spin glass region with disorder
S. Sanna et al. PRB 82 100503(R)
Ca
717122 April 2016 RDR- - NMR and MuSR
YBa2Cu
3O
6+x the clean case
71
F. Coneri et al. PRB 81 104507
measures h = hp
similarly Tc(h) measures h
= hp
Remember!
727222 April 2016 RDR- - NMR and MuSR
Internal doping and disorderY
1-zEu
zBa
2Cu
3O
6.35
72
Eu3+ for Y3+ isoelectronic
why?
TcT
N
0.3
0 0.5 1 O content, y
Y
Eu
b
a
chain disorder vs chain order
doping linear in chain length
Eu Y
Y Eu
O content, y
0 0.5 1
Eu 1-z Y
182
180
178
176
cell
volu
me
(Å3 )
a series of samples at fixed O
0.35
YEu
0.5
737322 April 2016 RDR- - NMR and MuSR
Chain length ℓ
NQR spectroscopy of Cu(1)
chain interior
chain end
Y
Eu
tetra ortho
→ doping linear in z
747422 April 2016 RDR- - NMR and MuSR
Y1-z
EuzBa
2Cu
3O
6.35
phase diagram
CuO2 hole density h
p
from increasing chain length
known h
p = 0.07
known h
p = 0.028
similar to O doping, but ...T
N
Tc
0 0.2 0.4 0.6 0.8 1.0
Y content, 1-z0.0 0.5 1.0 Y content, 1-z
757522 April 2016 RDR- - NMR and MuSR
Y1-z
EuzBa
2Cu
3O
6.35
phase diagram
TN for the clean
system with
these
densities
hp
hp
predicts
these
density hp
similar to O doping, but ...
Correlates with disorder
Eu YY-Eu
767622 April 2016 RDR- - NMR and MuSR
Y1-z
EuzBa
2Cu
3O
6.35
Samuele Sanna
TN reduction with structural disorder
Y-Eu
777722 April 2016 RDR- - NMR and MuSR
Type II superconductors
-M
H
Type II:
Hc
Hc1
Hc2
Flux Lattice, incommensurate to the crystal lattice
muons are interstitials in the crystal lattice
→ dense random sampling
787822 April 2016 RDR- - NMR and MuSR
GL equations predict
b
a
B maxima
B minima
saddle point
797922 April 2016 RDR- - NMR and MuSR
Textbook case: V
p(B
) arb
. un
its
0.16 T
0.20 T
0.24 T
0.29 T
808022 April 2016 RDR- - NMR and MuSR
Quick and dirty fits
Comparison with polycrystal data Second moment
B(r)exp(-r2/2σ2)
Is it worth while?
⟨ΔB 2⟩=(
Φ0
a⋅b )2
∑k≠0
K 02(k ξ)
(1+k 2λ2⏟k λ>1
)2∝
1λ4
818122 April 2016 RDR- - NMR and MuSR
σ(T): gap fits
La1.83
Sr0.17
CuO4 crystal
s-wave ns
YBCO6.95
crystal
Field cool, then shift field down(pinned flux lattice)
d-wave: lines of nodes
PRB42 8019 (1990)
YBCO6.95
powders (!)
0.05 T
828222 April 2016 RDR- - NMR and MuSR
Single molecule rings
Closed and open rings
spin singlets J ~ 15 K in zero field
Chemist can taylor couplings and play with topology
Cr8 Cr8Cd
Cr
838322 April 2016 RDR- - NMR and MuSR
848422 April 2016 RDR- - NMR and MuSR
open ring Cr8Cd exchange
anisotropy
dipolar
Zeeman
J = 1.32 meV
T. Guidi et al. Nature Comm. 2015
2 molecules per cell
B
θ
858522 April 2016 RDR- - NMR and MuSR
53Cr NMR at T = 1.4 K
nat. ab.
Iγ/2π
(MHz/T)53Cr 0.095 3/2 2.606
4 frozen inequivalent pairs
negligible quadrupole
868622 April 2016 RDR- - NMR and MuSR
53Cr resonances at 1.4 K
Field scan
Frequency scan
878722 April 2016 RDR- - NMR and MuSR
53Cr resonances at 1.4 K
888822 April 2016 RDR- - NMR and MuSR
Summarizing…
88
Local probes (nuclei, muons) can measure
• magnetic order parameters up to the transition• relaxation rates, i.e. excitations
With this tools we have seen:
• static critical exponents dynamic critical exponents• conventional magnetism: 3D Heisenberg,Ising,lower dimensions (2D,1D)• phase diagrams• magnetic dilution and frustration• exchange couplings and hyperfine couplings• superconducting penetration depth and gaps