Beyond the Kubo-Toyabe and stretched exponential functions ... · Beyond the Kubo-Toyabe and stretched exponential functions: how SR can reveal spatial magnetic correlations P. Dalmas

Beyond the Kubo-Toyabe and stretched exponentialfunctions: how µSR can reveal spatial magnetic

correlations

P. Dalmas de Reotier, A. Yaouanc, and A. Maisuradze1

Institut Nanosciences et CryogenieUniversite Grenoble Alpes & CEA Grenoble, France

1Department of Physics, Tbilissi State University, Georgia

Muon Spectroscopy User Meeting: Future Developments and Site CalculationsThe Cosener’s House, Abingdon, UK

16–17 July 2018

Outline

IntroductionExperimental ZF-µSR spectraPhenomenological polarization functions

Evidencing spatial correlationsExtension of the KT modelModel-free analysisExamples

Summary and Conclusions

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ZF-µSR spectra with unconventional shapeFrom Kubo-Toyabe-like shape

Non-dilute highly disordered intermetallicalloy.

Noakes and Kalvius, PRB 56, 2352 (1997).

Yb2Ti2O7: frustrated magnet on thepyrochlore lattice, quantum spin-ice candidate;

splayed FM order below Tc = 0.24 K.Hodges et al., PRL 88, 077204 (2002).

Spectral shape reminiscent of the Kubo-Toyabefunction with Gaussian decay at short times, butweaker dip than predicted by Kubo-Toyabe model.

Pyrochlore lattice

ZF-µSR spectra with unconventional shape (2)From Kubo-Toyabe-like shape

SrCr8Ga4O19: frustrated magnet on Kagomelattice; Tg = 3.5 K.

(Undecouplable Gaussian.)Uemura et al., PRL 73, 3306 (1994)

Kagome lattice

La2Ca2MnO7: frustrated magnet on triangularlattice;

√3×√

3 AFM order below TN =2.8 K.

Dalmas de Reotier et al., SPIN 5, 1540001 (2015).

Triangular lattice

ZF-µSR spectra with unconventional shape (3)Towards exponential-like shape

Nd2Sn2O7: frustrated magnet on thepyrochlore lattice; all-in–all-out AFM order

below Tc = 0.91 K.PZ (t) = exp[−(λZ t)β ]. SrCr8Ga4O19

Uemura et al., PRL 73, 3306 (1994)

PZ (t) = exp[−(t/T1)β ],

with 1/T1 ≡ λZ .

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The Kubo-Toyabe polarization functionPolarization function from Larmor equation solution:

PstatZ (t) =

∫ [cos2 θ + sin2 θ cos(ωµt)

]Dv(B)d3B.

AssumeDv(B) = Dc(BX )Dc(BY )Dc(BZ ),

with

Dc(BX ) = Dc(BY ) = Dc(BZ ) ∝ exp

[−(BZ )2

2∆2

],

PKTZ (∆, t) =

1

3+

2

3

(1− γ2

µ∆2t2)

exp

(−γ2µ∆2t2

2

).

Dynamics is accounted for by the so-called strong collisionmodel:

PZ (t) = PKTZ (t) exp(−νc t)+νc

∫ t

0PZ (t−t′)PKT

Z (t′) exp(−νc t′)dt′.

When νc/γµ∆� 1, PZ (t)→ exp(−λZ t) with λZ ≡ 2γ2µ∆2/νc .

PZ (t).

Hayano et al., PRB 20, 850 (1979).

Phenomenological fit functions (1)The exponential-power polarization function

PZ (t) = exp[−(λZ t)β ]

I 0 < β < 1: stretched-exponential function (or Kohlrauschfunction, 1854).

A distribution of exponential relaxation functions:

exp[−(λZ t)β ] =

∫ ∞0

exp(−sλZ t)P(s, β)ds.

Physical ground for distribution P(s, β)?

I β > 1: compressed-exponential function.

Rarely appearing in physics except for β = 2.No physical backing in ZF-µSR, even for β = 2.

I β = 1/2: singular case.

PZ (t) = exp(−√λZ t

),

P(s, β) versus s.Johnston, PRB 74, 184430 (2006).

is derived for diluted magnetic systems in the extreme motional narrowing limit.Experimental confirmation by Tse and Hartmann, PRL 21, 511 (1968), Uemura et al., PRB 31, 546 (1985). . .

A complete set of high statistics data can reveal the fate ofstretched-exponential spectra:

Simultaneous fit of data with dynamical Kubo-Toyabe model:slow spin tunnelling in the paramagnetic phase of Nd2Sn2O7.Dalmas de Reotier et al., PRB 95, 134420 (2017).

Phenomenological fit functions (2)The Gaussian-broadened Gaussian polarization function. [Noakes and Kalvius, PRB 56, 2352 (1997)]

Average of Kubo-Toyabe polarization functions with Gaussian-distributed field widths:

PGbGZ (t) =

1√

2π∆GbG

∫ ∞−∞

PKTZ (∆, t) exp

(−

(∆−∆0)2

2∆2GbG

)d∆.

PGbGZ (t) as a function of R ≡ ∆GbG/∆0,

with ∆2eff ≡ ∆2

0 + ∆2GbG.

Full line at 0.200 K: PGbGZ (t).

Hodges et al., PRL 88, 077204 (2002).

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Beyond the phenomenological fit functions

Polarization function from Larmor equation solution:

PstatZ (t) =


]Dv(B)d3B

AssumeDv(B) = Dc(BX )Dc(BY )Dc(BZ )

andDc(BX ) = Dc(BY ) = Dc(BZ ).

No linear relation between field distribution and asymmetry spectrum!

Direct search for Dc(BZ ) from the data not possible.

Two routes for Dc(BZ ) determination:

I Dc(BZ ) ∝ exp

(−B2

Z2∆2

)−→ Dc(BZ ) ∝ exp

[−g(

BZδ

)]with

g(x) = 12x2 + 1

3(η3x)3 + 1

4(η4x)4.

I Direct search for distribution consistent with the data.Use of Maximum Entropy supplemented Reverse Monte Carlo (ME-RMC) algorithm.

Application to the triangular magnet La2Ca2MnO7Magnetically ordered phase

Model function simultaneously fitted to 14spectra recorded in the ordered phase.

A single value for η3 = 0.74 (2) andη4 = 0.47 (2).

Tails in the distribution.

Dalmas de Reotier et al., SPIN 5, 1540001 (2015).

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Field distribution in transverse field experimentsI Traditional method: Fourier transform of the asymmetry

spectrum

Caveats:I noise in asymmetry data (finite µ+ lifetime) is not

accounted for:apodization −→ broadening of distribution

I no error bars on the resulting distribution

I A better approach: inverse problem

PstatX =

∫cos(ωµt)Dc(BZ )dBZ

I find the distributions which provide the best fit to the dataI among the solutions, choose that with maximum entropy (ME) S

S = −∑i

Dc(BZ ,i )δBZ log

[Dc(BZ ,i )

di

](information theory)

δBZ : step in Dc (BZ ,i ); di prior estimate

Rainford and Daniell, Hyperfine Interact. 87 1129 (1994)

Riseman and Forgan, Physica B 289-290 718 (2000)

Case of zero-field asymmetry spectra

PstatZ (t) =


]Dc(BX )Dc(BY )Dc(BZ )dBXdBY dBZ

I direct search for distribution which best fits the data

I minimization ofF = χ2 − λS

where

χ2 =

Ni∑i

[Ai − a0PZ (ti )]2

σ2i

,

and λ is a Lagrange parameter.Minimization with a Reverse Monte Carlo (RMC) algorithm.

Reverse Monte Carlo algorithm

with 0.003 ∼< ε ∼< 0.03, and 0.003 ∼< p ∼< 0.03.Convergence is typically reached after ≈100 loops per degree offreedom.

Estimate of error bars

δF (r) '∑i

∂F

∂riδri +

1

2

∑i ,j

∂2F

∂ri∂rjδriδrj

' 1

2

∑i ,j

∂2F

∂ri∂rjδriδrj ≡

∑i ,j

1

2Hi ,jδriδrj .

where r is the vector formed by the free parameters [Dc(Bi ), a0, νc ,abg, . . . ] of the fit and H is the so-called Hessian matrix.The error bars σri are given by:

σ2ri

=[H−1

]ii.

Outline




Comparison of analytical model and ME-RMC analysis

I Full line: fit to the asymmetry data with

Dc(BZ ) ∝ exp[−g(

BZδ

)]and

g(x) = 12x2 + 1

3(η3x)3 + 1

4(η4x)4.

η3 = 0.73 (2), η4 = 0.46 (2).

I Red circles: ME-RMC fit to theasymmetry data.

Comparison in time domain.

Application of the ME-RMC algorithmEr2Ti2O7: a XY pyrochlore antiferromagnet with TN = 1.25 K

Dalmas de Reotier et al,PRB 86, 104424 (2012)

To be published To be published

I thanks to the ME-RMC algorithm, and the availability of a highstatistic spectrum, evidence for a weak contribution centered at12 mT

I allows for a reliable fit in time domain of the asymmetry spectrum

Evidence for short-range correlations

I Non-Gaussian distribution −→ short-range magnetic correlationsContrapositive statement of the Central Limit Theorem

I Coexistence of short-range correlations with long-range order forLa2Ca2MnO7, Yb2Ti2O7, Yb2Sn2O7, and Er2Ti2O7

Possible relation with magnetic moment fragmentationI What next?

I Quantitative information about the correlation length (Monte Carlosimulations)

I Quantitative information in terms of physical parameters entering aHamiltonianSee, e.g. Bramwell et al., PRE 63, 041106 (2001), who calculated themagnetic moment distribution for the classical XY Hamiltonian onthe 2D square lattice (BKT transition).Required:

I extension to Bloc at the muonI extension to other Hamiltonians.

Outline





I Framework for the interpretation of magnetic materials spectrawith unconventional shape

I An analytical modelI A model-free analysis using the ME-RMC algorithmI Both focussed for ZF data and isotropic distributions

Generalization to LF data straightforwardGeneralization to anisotropic distributions possible

I µSR is primarily sensitive to time correlations, but a detailedanalysis of large statistics data can unravel spatial correlations

I Despite the muons are a local probe, they can evidence spatialcorrelations

References:Yaouanc et al , Phys. Rev. B 84, 172408 (2013)Dalmas de Reotier et al , J. Phys.: Conference Series 551 012005 (2014)Maisuradze et al , Phys. Rev. B 92, 094424 (2015)Dalmas de Reotier et al , J. Phys. Soc. Jpn. 85 091010 (2016)Dalmas de Reotier et al., Phys. Rev. B 95, 134420 (2017)

Documents

Beyond the Kubo-Toyabe and stretched exponential functions ... · Beyond the Kubo-Toyabe and stretched exponential functions: how SR can reveal spatial magnetic correlations P. Dalmas