Beyond the Kubo-Toyabe and stretched exponential functions: how μSR can reveal spatial magnetic correlations P. Dalmas de R´ eotier, A. Yaouanc, and A. Maisuradze 1 Institut Nanosciences et Cryog´ enie Universit´ e Grenoble Alpes & CEA Grenoble, France 1 Department of Physics, Tbilissi State University, Georgia Muon Spectroscopy User Meeting: Future Developments and Site Calculations The Cosener’s House, Abingdon, UK 16–17 July 2018
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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
Evidencing spatial correlationsExtension of the KT modelModel-free analysisExamples
Summary and Conclusions
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:
Evidencing spatial correlationsExtension of the KT modelModel-free analysisExamples
Summary and Conclusions
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) =
∫ [cos2 θ + sin2 θ cos(ωµt)
]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:
Evidencing spatial correlationsExtension of the KT modelModel-free analysisExamples
Summary and Conclusions
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.
Evidencing spatial correlationsExtension of the KT modelModel-free analysisExamples
Summary and Conclusions
Summary and Conclusions
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)