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

Transcript of Applications: Magnetic systems · 2017. 5. 15. · 23/04/2008 R. De Renzi - ISIS Muon Training...

  • 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

  • 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

  • 23/04/2008 R. De Renzi - ISIS Muon Training Course 3

    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

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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

    I

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    Single crystal

    A0cos2

    A0sin2

    At =A0sin2e−t /T 2cos Bt cos2e−t /T 1

    Generic orientation:Bµ

    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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