Muon relaxation functions Stephen J. Blundell Clarendon Laboratory, - - PowerPoint PPT Presentation

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Muon relaxation functions Stephen J. Blundell Clarendon Laboratory, - - PowerPoint PPT Presentation

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Muon relaxation functions Stephen J. Blundell Clarendon Laboratory, Department of Physics, University of Oxford, UK Muon training course - 2018 (Thanks to Francis Pratt for a few of the later slides on muonium-like states.) Muon training course

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Stephen J. Blundell

Clarendon Laboratory, Department of Physics, University of Oxford, UK

Muon training course - 2018

Muon relaxation functions

(Thanks to Francis Pratt for a few of the later slides on muonium-like states.)

Muon training course - 2018

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see later (data analysis sessions) Wimda

Other packages are available

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Lecture plan

  • Static distributions - what is a Kubo-Toyabe?
  • Gaussian or Lorentzian?
  • Dynamic relaxation functions - what happens when

the muons get a bit jumpy?

  • Stretched exponentials - dangerous evil or answer

to all problems?

  • When quantum mechanics shines on the

experiment!

  • Where is your muon?
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Muon spin precession

Response to a Static Field

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Response to a Static Field

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Response to a Static Field

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Distribution of Static Fields

Assume that the Bx, By and Bz components are each distributed according to a Gaussian distribution, e.g. The overall field distribution peaks near 21/2Δ/γµ

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Distribution of Static Fields

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Ryogo Kubo (1920-1995)

Static Kubo-Toyabe Function

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Minimum at t=√3/Δ 1/3 tail

Static Kubo-Toyabe Function

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Kubo-Toyabe in Field

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Kubo-Toyabe in Longitudinal Field

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2

LF Kubo-Toyabe Function

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Gaussian or Lorentzian Field Distribution?

(Dense spins)

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Gaussian or Lorentzian Field Distribution?

Dilute spins:

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Lorentzian Kubo-Toyabe (LKT)

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The ‘Dilute’ Spin Condition

e.g. metallic random alloy spin glasses e.g. complex molecular magnets

A broad range of couplings from the muon to the nearest spin is the key here

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All these functions available in Wimda

F.L. Pratt, Physica B 289-290, 710 (2000) http://shadow.nd.rl.ac.uk/wimda/

Other packages are available

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One can interpolate between statics and dynamics using a dynamical Kubo-Toyabe function

Introduce Dynamics

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Introduce Dynamics

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Introduce Dynamics

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Introduce Dynamics

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Introduce Dynamics

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Dynamical Kubo-Toyabe (DKT)

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Dynamical Kubo-Toyabe (DKT)

Since the Gz integral depends only on Gz at earlier times and the known static function gz , it can be built up sequentially in time

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Slow Hopping

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Fast Hopping

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Static Dynamic Effect of Longitudinal Field

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Zero-field Longitudinal-field Effect of Dynamics

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The Keren Function

Perturbation expansion for Pz(t) gives an analytical result valid for ν>Δ Amit Keren ZF limit: (LF Abragam) Fast ν limit: (ν>ωL)

PRB 50,10039 (94)

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Stretched Exponential Functions

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Stretched Exponential Functions Gaussian β=2 Lorentzian β=1 Stretched β<1

Stretched exponentials generally arise from: 1) Distribution of relaxation times 2) Distribution of couplings

lineshape parameter

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Distribution of Relaxation Time

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Distribution of Relaxation Times: Diffusion

Risch-Kehr function

PRB 46, 5246 (1992)

1D:

Klaus Kehr (1934-2000)

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B-dependent Relaxation and Spectral Density Φ(t) =

Correlation function for field fluctuations: Fourier transform of Φ(t) gives the spectral density S(ω) S(ω) = ν / (ν2 + ω2) λ is proportional to S(ωL) Complex relaxation processes such as those based on diffusion typically give power laws for Φ(t) and S(ω)

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B-dependent Relaxation and Spectral Density

Fe19

Blundell et al (2003)

Polyaniline

Stretched relaxation is due to the coupling distribution Stretched relaxation is due to the 1D diffusion process

Single molecule magnet

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Muons that stop closer to magnetic ions “see” a wider local field distribution (which extends to higher fields) than muons which stop at a greater distance Y.J. Uemura et al, PRB 31, 546 (1985)

Distribution of Couplings Spin Glasses

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The correct relaxation function must therefore be an average over distribution widths Δ. This leads to a root-exponential relaxation function: G(t) = G(0) exp(-(λt)1/2) where the relaxation rate λ is inversely proportional to the fluctuation rate ν.

Distribution of Couplings

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Non-magnetic host

Magnetic Non-magnetic

Distribution of Couplings

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Spin glass

Muon stops close to magnetic ion

Magnetic Non-magnetic

Distribution of Couplings

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Spin glass

Muon stops well away from magnetic ion

Magnetic Non-magnetic

Distribution of Couplings

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Muons that stop closer to magnetic ions “see” a wider local field distribution (which extends to higher fields) than muons which stop at a greater distance

Y.J. Uemura et al, PRB 31, 546 (1985)

Distribution of Couplings Spin Glasses

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Distribution of Couplings

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……but is the dogma correct?

Distribution of Couplings

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Monte-Carlo calculation

  • f distribution of Δ
  • S. J. Blundell, T. Lancaster, F. L. Pratt, C. A. Steer, M. L. Brooks and J. F. Letard, J. Phys. IV France 114, 601 (2004)

Distribution of Couplings

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  • S. J. Blundell, T. Lancaster, F. L. Pratt, C. A. Steer, M. L. Brooks and J. F. Letard, J. Phys. IV France 114, 601 (2004)

Distribution of Couplings

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1. 2. 3.

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Hints of Quantum Coherence

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F- µ+

The F-µ-F State

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The F-µ-F State

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Pz(t) = 1 6 3 + cos √ 3ωt + ✓ 1 − 1 √ 3 ◆ cos " 3 − √ 3 2 ωt # + ✓ 1 + 1 √ 3 ◆ cos " 3 + √ 3 2 ωt #!

coherent

  • scillations arising

from the magnetic dipolar interaction

F- F-

µ+

The F-µ-F State

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Fluorine: small, high nuclear moment abundant species very sensitive to r1/r2 and α

entanglement

The F-µ-F State

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state found in many ionic fluorides, and also teflon (PTFE)

The F-µ-F State

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Cs2AgF4

  • T. Lancaster, S. J. Blundell, et al.

Tc=13.95(3) K

ωd = 2π x 0.211(1) MHz F-µ separation 1.19(1) Å.

  • Phys. Rev. B 75, R220408

(2007)

The F-µ-F State

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CuF2(H2O)2(pyz)

  • Phys. Rev. Lett. 99, 267601 (2007)
  • 1. Interaction

with a single fluorine ion

The F-µ-F State

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[CuNO3(pyz)2]PF6

  • 2. Crooked FµF

bond close to a PF6 ion

  • Phys. Rev. Lett. 99, 267601 (2007)

The F-µ-F State

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[Cu(HF2)(pyz)2]X

  • 3. Interaction

with a HF2

  • ion
  • Phys. Rev. Lett. 99, 267601 (2007)

The F-µ-F State

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Analysing Asymmetry : Magnets

Polycrystalline samples

b = B / B0

LF decoupling below Tc

  • F. L. Pratt
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Analysing Asymmetry : Muonium-like States

LF decoupling

  • r ‘repolarisation’

Hyperfine tensor (A,D1,D2)

F . L. Pratt, Phil. Mag. Lett. 75, 371 (1997)

Nuclear couplings AN

  • Z. Phys. B 22, 109 (1975)

D1 D2 A (500,20,1) MHz A AN

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Analysing Asymmetry : Muonium-like States

Avoided level-crossing resonances:

Δ1 muon flip Δ0 muon-nuclear flip-flop

  • F. L. Pratt
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All these functions available in Wimda

F.L. Pratt, Physica B 289-290, 710 (2000) http://shadow.nd.rl.ac.uk/wimda/

Other packages are available

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DFT+µ = (density functional theory +µ )

  • numerically solve (lattice) structures
  • determine muon site
  • quantify perturbations

DFT+µ began with two papers (Oxford + Parma groups) studying fluorides: J.S. Mö̈ller et al., Phys. Rev. B 87, 121108(R) (2013).

  • F. Bernadini, et al., Phys. Rev. B 87, 115148 (2013).

This work has been extended to many other systems, see e.g. S.J. Blundell et al, Phys. Rev. B 88, 064423 (2013). J.S. Mö̈ller et al., Phys. Scr. 88, 068510 (2013).

  • F. Xiao et al., Phys. Rev. B 91, 144417 (2015).
  • P. Bonfà et al., J. Phys. Chem. C 119, 4278 (2015).
  • F. Lang et al., Phys. Rev. B 94, 020407(R) (2016).
  • P. Bonfà et al., J. Phys. Soc. Jpn. 85, 091014 (2016)

DFT+µ

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DFT+µ = (density functional theory +µ )

  • numerically solve (lattice) structures
  • determine muon site
  • quantify perturbations

DFT+µ can not only assess the muon site, but also any muon- induced distortion. A worst-case scenario is where magnetism arises from a non-Kramers ground state. This leads to our study

  • f quantum spin ice.
  • F. R. Foronda et al., Phys. Rev. Lett. 114, 017602 (2015).

Challenges for pyrochlores:

  • 88 atoms per unit cell
  • 4f valence electrons
  • ~102-104 cpu hours per calculation

Results:

  • typical O-H like bond with length 1 Å
  • 4f electrons influence negligible
  • r4f ≈ 2×r5s ≈ 5×(Pr-µ distance)

DFT+µ

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…it’s now time for some relaxation! After all those relaxation functions…..