Applications: Magnetic systems · 2017. 5. 15. · 23/04/2008 R. De Renzi - ISIS Muon Training...
Transcript of Applications: Magnetic systems · 2017. 5. 15. · 23/04/2008 R. De Renzi - ISIS Muon Training...
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Applications: Magnetic systems
Outline:
● Why muons in magnetic materials?● Internal fields● Different types of magnets● Magnetic relaxations: dynamic effects● Muon diffusion, site assignment● The paramagnetic state● Timescale considerations
Roberto De Renzi
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Why muons?
New material: is it magnetically ordered?● Sensitive to small moments, ~ 0.01 μB
● Sensitive in different magnetic structures:
● Zero field
● Also on powders, smaller samples than neutrons
● Low energy magnetic excitations
μ
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Internal fields
O
µr1µ
B=∑i04
mi−3mi⋅r i r i r i
3 Bhf
Dipolar (distant dipoles) Hyperfine (transferred)
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Internal fields
O
µr1µ
B=∑i04
m i−3mi⋅r i r i r i
3 Bhf
Dipolar (distant dipoles) Hyperfine (transferred)
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Internal fields
Bdip=∑i04
mi−3mi⋅r i r i r i
3 =∑i D i⋅m i
Dipolar (distant dipoles)
O
r1µµ
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Dipolar field
O
µrµ
Bdip≈0 m
4 r 13 ≈
m [B ]d 3[A3]
T
typically a few thousand Gauss
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Dipolar field
O
µrµ
typically a few thousand Gauss
Calculate precisely by dipolar sums:● electrostatic energy (for site selection)● magnetic field around that site
Bdip≈0 m
4 r 13 ≈
m [B ]d 3[A3]
T
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Hyperfine field
For 1H 1s (muonium):
Bµ=32.8 T
B∝∣r ∣2S
From Fermi interaction
(probability density forthe electron S at muon)
O
µ
typically smaller than Bdipbut ...
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LaMnO3: M(T) from Bµ(T)
µH
µC
Absolute conversionrequires
site assignment
(T)
M. Cestelli et al. Phys Rev. B 64 (2001) 064414
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LaMnO3
µH
µC
Exploit symmetry in a single crystal
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LaMnO3
µH
µC
µH
µC
Dipolar sums withconstraints:
µ-O distance ~ 1.1Å µH
µC
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LaMnO3
µH
µC
µH
µC
µH
µC
Dipolar sums withconstraints:
µ-O distance ~ 1.1Å
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M(T)Temperature dependence of order parameter
a prototype Ising antiferromagnet, CoF2:
Low T:
(spin wave excitations)
Close to TN: critical power law behaviour
R. De Renzi et al. Phys Rev. B 30 (1984) 197
1−BT B0=B∝M s T
BT ∝1−TT N
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M(T)Temperature dependence of order parameter
another Ising antiferromagnet, LaMnO3:
Close to TN: critical power law behaviour
M. Cestelli et al. Phys Rev. B 64 (2001) 064414
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Single crystal
I
Orient the crystal as in a transverse field experiment:● full precession
Orient the crystal as in a longitudinal field experiment:● non precessing relaxation
I
Bµ
Bµ
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Single crystal: relaxations
At =A0e−t /T 2cos Bt
spin-spin, carries information on:● inhomogeneous distribution of static fields● dynamics (secular part)
●spin-lattice, carries information on:● dynamics (non-secular part)
At =A0e−t /T 1
I
Bµ
Bµ
I
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Single crystal
A0cos2
A0sin2
At =A0sin2e−t /T 2cos Bt cos2e−t /T 1
Generic orientation:Bµ
Iθ
For more details:PmWikiFisica
http://www.fis.unipr.it/~derenzi/dispense/pmwiki.php?n=MuSR.MagneticSingleCrystal
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Polycrystal average
I
At =A032e−t /T 2cosBte−t /T 1
I
Simple case: no texture
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TF relaxation
HH
B(t)
At =A0e−t /T 2cosHt
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TF relaxation
HH
B(t)
At =A0e−t /T 2cosHt
Larmor frequency
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TF relaxation
1T 2=−
〈dI xdt
〉
〈I x 〉=
2∫0∞〈B ⊥ 0B ⊥ t 〉e
i H t〈Bz 0Bz t 〉dt
HH
B(t)
At =A0e−t /T 2cosHt
T2 relaxationIn the rotating frame:
Note
file:///C:/Users/docente/De Renzi/Fisica AB/note.snt
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Relaxation: summary
=1
T
J = 12∫0
∞〈B 0B t 〉ei t dt
∫−∞∞
J e−i t dt= 12
〈B 0B t 〉
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τμlog τ
Relaxation: summary
1T 1,2
=2 J B f
=1
T
Rate is maximum when Larmor ω=1/τ
1/2πγμBf
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Relaxation
B t =∑i D i⋅miAhf S
T > Tc
e S i t
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Relaxation
B t =∑i D i⋅miAhf S
T < Tc
e S i t
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ExamplesLaMnO3: T→TN
TN
Spin fluctuations slow down approaching the transition
YBa2Cu3O6+x
Tf Freezing of hole spins
Other intrinsic spin dynamics ...
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Paramagnetic state: shifts
〈B 〉=∑i D i 〈m i 〉Ahf 〈S 〉
T>Tc H > 0
e 〈S i 〉=H
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Paramagnetic state: shiftssingle crystal CoF2
T>Tc H = 0.3T
〈B 〉=∑i D i 〈m i 〉Ahf 〈S 〉
e 〈S i 〉=H
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Ferro and Ferrimagnets
B=BdipBhf 0H
∑Ls D i⋅m i
dipoles inside the Lorentz sphere
Lorentz sphere
continuum, outside the
Lorentz sphere
Lorentz field, poles on the inner surface
(the Lorentz countersphere)
demagnetization,poles on the outer surface
(depends on shape and Mmacro)
Bdem03
Mmicro
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Fe3O4: a ferrimagnetVerwey transition:
TTv metallic, delocalised Fe2+ and Fe3+ at B sites
Fe3+O
Fe2+/3+
AB
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Fe3O4: a ferrimagnet
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Fe3O4: a ferrimagnet
One spin sublattice at octahedral B sites
TV=123 K
Tetrahedral A sites: opposite spin sublattice
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Fe3O4
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Fe3O4
Muon-O bond:minima of Ue on a
sphere of radius 1.1 Åaround O
S along above TR=130 K
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Fe3O4
B=03
MmicroBhf∑Ls D i⋅〈mi 〉
0.4T ∝Mmicro∝2mB−mA
160 K < T < 240 K
two distinct field values
1x 3x
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Fe3O4
T > 240 K:muon diffusion in acubic environment
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Fe3O4
B=03
MmicroBhf∑Ls D i⋅〈mi 〉
0.4T ∝2mB−mA
T>240 K:muon diffusion in acubic environment,a single field value
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Fe3O4
B=03
MmicroBhf∑Ls D i⋅〈mi 〉
0.4T ∝2mB−mA
T
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Cuprates: YBa2Cu3O6+x
Mapping dependence of TN on doping
CuO2
CuOy
Y3+1-xCa2+x
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Cuprates: YBa2Cu3O6+x
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Cuprates: YBa2Cu3O6+xAt =
A032e−t /T 2cosBte−t /T 1
A0
A03
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Cuprates: YBa2Cu3O6+x
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Summary
Dipolar and hyperfine field.
Lorentz field.
Dipolar sums and electrostatic energy for site assignment.
Single crystal vs. polycrystals.
From static internal fields: order parameter (paramagnetic shift).
Dynamic internal fields: relaxations.
Examples of dynamics: freezing, critical, quantum.
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