through a few examples Diffuse scattering Isabelle Mirebeau Laboratoire Léon Brillouin CE-Saclay Gif-sur Yvette, FRANCE

Transcription

1 through a few examples Diffuse scattering Isabelle Mirebeau Laboratoire Léon Brillouin CE-Saclay Gif-sur Yvette, FRANCE

2 Outline General features Nuclear diffuse scattering: local chemical order and/or static displacements Magnetic diffuse scattering: local magnetic order

3 General features A simple example How to describe the diffuse scattering? Several lengthscales NO time scale: what does the spectrometer measure?

4 A simple example

5 A simple example 2 b Bragg 2 2 b b Diffuse scattering

6 N 2 b Bragg 2 2 b b Laue background A, B random Q N Towards ordering Q N Towards clustering 0 Q

7 How to describe the diffuse scattering? in real space: a deviation from the random/ordered state in reciprocal space a modulation between the Bragg peaks 2 options (+1) from the random state from the ordered state

8 How to describe the diffuse scattering? from the disordered state from the ordered state Short range order parameters Deviation from random distribution Local lattice distortions Pair distribution function (pdf) Local magnetic perturbations Correlation lengths Critical behaviours From local constraints No obvious «a priori» description! entropy Mapping and gauge fields choice of the model depends on the context Measure in absolute scale To decide if the model is realistic or not

9 magnetic intensity magnetic intensity (Tb 1-x La x ) 2 Mo 2 O 7 Tb/La Chemical Pressure T=1.4 K x=0 P=0 x=0.1 P=0 x=0.2 P= q (Å -1 ) LRO induced by Tb/La dilution Expanding the lattice x=0.2 P=3.7 GPa x=0.2 P=1.9 GPa x=0.2 P=1.05 GPa 0.00 A. Apetrei ( thèse 2007) A.Apetrei, I. M. et al PRL 97, (2006) Applied Pressure q (Å -1 ) LRO destroyed by pressure P Changing the band structure changes the magnetic frustration (sign of Mo-Mo interactions)

10 From the random state Look at the landscape! Tb 2 Mo 2 O 7 spin glass RANDOM state (paramagnetic/disordered) SRO parameters Statistical information on the first neighbour shells (occupation/ relative spin orientations) A.Apetrei, I. M. et al PRL 97, (2006)

11 Magnetic Intensity (arb. units) From the ordered state 5000 Tb Mo [001] 4000 a. x= 0.2 t m G G model of the ordered state (refined crystal/magnetic structure) correlation length (deg.) Information on the mesoscopic structure Diffuse scattering

12 The tricky cases Phase separation? (Tb 1-x La x ) 2 Mo 2 O x= P=1.9 GPa Quantum critical point Induced by pressure or concentration mesoscopic structure? x=0.1 P= q (Å -1 )

13 Several lengthscales Type Typical lengthscale Example local order Near neighbours A few unit cells 1-10 Å Binary alloys spin glasses Mesoscopic structures A few tenths of unit cells Å Critical phenomena Long Range Order Limited by exp. resolution > Å Bragg

14 NO Time scale! Energy integration (diffraction) Question : What does the spectrometer measure? i instantaneous correlations! S( q,) d =1.22 Å =4.74 Å

15 Local chemical order Deviation from random distribution Short range order parameters Nuclear Diffuse scattering Pair potentials Example 1: (single crystal) : TiC vacancies Example 2 : polycrystal: Fe-Cr alloys How improve the data quality? How improve the data treatment? The modern tool : pdf

16 Local chemical order Long Range Order Alloy: Ac A Bc B C A +C B =1 Short Range Order All sites are statistically equivalents Order parameter S n: ratio of A atoms «well placed» c A <n<1 0<S<1 P A =c A ; P B =c B S=0 S n c c B A Surstructure I Bragg S 2 A R n B origin Conditional Probability P AB ( R n ) 1 P AB ( R c B n ) SRO parameters n (Cowley-Warren)

17 Local Order parameters SRO parameters n (Cowley-Warren) Conditional Probability P AB B ( R n ) 1 P AB Alloy Ac A Bc B C A +C B =1 ( R c B n ) c A 0.5 A R n origin c c A B 1 P P P AB AB ( R ) c n n ( R ) c B B ( R n) 0 AB ( R n) cb ( ) 0 R n ( R n) 0 order clustering Random ( 0) 1 ( R n ) 0; n Calibration of the nuclear cross section (checks the consistency of the measurement) Fitting constraints 1 i1 N i i 0 Sum Rule (virtual) : Grand Canonical space

18 Chemical order parameters and nuclear diffuse scattering Cowley-Warren model Alloy : A B A: c A B : c B ). )cos( ( ) ( 2 n n n A B B A N R Q R b b c c d d Laue background Périodic Modulation In reciprocal space 0 2 s s Q Scattering Vector Cowley Phys.Rev. 77,669, (1950) Warren X ray diffraction (1968) n n n B A n n B B A A R iq b b N iq R b c b c N Q I ). exp(. ) ( ). exp(. ) ( ) ( Bragg: average lattice Diffuse scattering : deviation Diffuse cross section from chemical SRO

19 Order parameters and pair potentials Mean field model Clapp et Moss Phys. Rev. 142, 418, (1966), et suivants H 1 2 i, j [ V AA ij A i A j V Effective pair potential H Hamiltonien Ising 1 4 i, j V ij BB ij B i B j V i j ij V AB ij V ( A i AB ij B j A i 1 AA BB ij Vij ( V 2 B j )] ) V ij V ij 0 0 Diffuse cross section Order, AF Segregation, F C ( Q, T) 2c cbv ( Q 1 kt A ) Effective pair potentials (~mev) Stability of the phase diagrams

20 Local order of carbon vacancies Phase Diagram TiC x Ordered Phase Ti 2 C Long Range Order of carbon vacancies in a fcc lattice Propagation vector K= ½ ½ ½ Vacancy carbon Local Order in solid solution?

21 Ordre local order of carbon vacancies in TiC 0.64 B. Beuneu, R. Caudron (Onera) T. Priem (thèse 1988) Intensity measured in situ : T=900 C local order + lattice distortions Maxima at hkl positions ½ ½ ½ ~30 parameters Intensity from local order, calculated from the SRO parameters : SRO parameters Intensity calculated by the Mean field (Clapp-Moss) model 4 pair potentials (2 main) V 1 =15; V 2 =48; V 3 =-4; V 4 =8 mev 2 potentials

22 From single crystal to polycrystal cos( Q. R) d d c c ( b sin QR QR b sin QRi N N 2 A B B A i i i QRi ) ( R ) i i i th shell of radius R i Loss of the periodicity Shifts of the positions of the diffuse maxima Fit of a few s only FeCr alloys

23 Fe 1-x -Cr x alloys: why are they interesting? First and unique example of short range order inversion in a solid solution x >x c clustering x ~x c random x<x c ordering origin: anomaly of the band structure : local magnetism of Cr M. Hennion J. Phys. F (1982) Physical consequences Resistivity Bulk modulus Aging (irradiation) Formation energy Possible applications in metallurgy ~ 700 papers and 800 citations/year about FeCr Direct probe : Neutron diffuse scattering

24 FeCr alloys : Short range order inversion G. Parette, I. M. Phys. Rev. B (2010) Importance des corrections signal/background ratio increased by a factor ~20 with respect to first measurements IM, M. Hennion, G. Parette, PRL(1984)

25 Fe 1-x Cr x, SRO parameters and Monte-Carlo Sim. Simulation A.Caro et al Phys. Rev. Lett. 95, , (2005). P. Erhart et al Phys. Rev B(2008)

26 Fe 1-x Cr x, pair potentials and ab initio models Ab initio calculations: [ 8] M. Hennion J. Phys. F (1983) [19] Ruban et al Phys. Rev. B (2008)

27 How to improve the data quality? The secrets: Decrease environmental background Vacuum chamber.. Take care of all corrections Background ( Cd and empty sample holder) transmission angle dependent absorption multiple scattering Detector efficiency Calibrate the cross section in absolute scale Vanadium sample (Corrected) Think about possible artefacts find the suitable conditions diffuse Magnetic scattering? play with H and (q, H) angle, polarized neutrons.. inelastic (play with T) Thermodynamical conditions (quench, in situ..)

28 How to improve the data treatment? The pdf Pair Distribution Function ( for nuclear scattering) See P. Bordet, JDN15 Measured: from the TOTAL powder cross section S(Q), corrected and normalized by <b> 2 Calculated: from a structural model Needs High statistics and large Q max Peaks of G(r) position: interatomic distances r ij intensity : stucture factor b i b j width : r ij distribution, Debye-Waller, occupation disorder

29 local magnetic order Diffuse scattering and magnetic fluctuations : above T c spin waves : below T c critical behaviour: around T c Choose the right Q-scale Diffuse scattering in spin ices local constraints

30 Diffuse scattering and magnetic fluctuations : T>T C 2D fluctuations above T N in InMnO 3 T<T N magnetic order with K=(0 0 1/2) T N =120K J c =0 X. Fabrèges et al Phys. Rev. B (2011) T>T N Fit by a 2d Warren function Phys.Rev (1941)

31 From 2D to 3D order T>>T C T>T C T=T C Divergence of associated with 3d order (coupling between the planes) Collapse of the 2D diffuse scattering and onset of Bragg peaks

32 C Diffuse scattering and critical behaviour : T~T C How diffuse scattering collapses on Bragg peaks: the scaling laws T >> T C T~ TC T<< T C A

33 Towards a transition Towards a critical point at T C Chemical : transition to a superstructure Magnetic : Ferro or AF transition q S T << T C 0 A S r A( T ) A T~T C C T >> T C R ~ e I( q) A 2 2 ( q q0) 1 T T ~ ( C ) T C : correlation length T T C 0 I qa 0 I max ~1/ Width 2 q I q q 0 T T C

34 Diffuse scattering and spin waves T<T C Zn 2 VO(PO4) 2 : Quantum fluctuations in a S=1/2, 2d- Antiferromagnet Above T N Below T N 2d signal coexist with 3d order! S. M. Yusuf et al PRB (2010) Why?

35 Diffuse scattering and spin waves T<T C Zn 2 VO(PO4) 2 : Quantum fluctuations in a S=1/2, 2d- Antiferromagnet T=0 Calculation of the SW spectrum Powder average and projection in S(q) plane Sum over energies in the range 0-E i at constant Q modulus S. M. Yusuf et al PRB (2010) program from S. Petit

36 Chose the right Q scale Towards a ferromagnetic transition: from Diffuse scattering to SANS Example: Pd 1-x Co x alloys Giant moments/polarization cloud M~10µ B / atom Co Pd Co L c ~ 20 Å q~1 Å -1 diffuse scattering probes local inhomogeneities (above Tc) x>0.1% : Ferro transition T C increases by 40K per Co % q~10-3 Å -1 SANS probes large defects ( Bloch walls) I (q) q -4 I(T) M 2 (T) Neutron intensity vs. T at Constant q q~10-2 Å -1 SANS probes: the critical region the spin waves I (q) / ( 2 +q 2 ) peak of I(T) at T C I. M et al J. Mag. Mag. Mat 54, 997 (1986)

37 Diffuse scattering in Spin ices Describe the diffuse scattering from a locally ordered state Local constraints : ice rules Entropy conservation laws and pinch points mapping Positional parameters Ising spins Effective spins Water ice Spin ice Spin liquids..

38 The ice rules spins along <111 axes>, J ZZ ferro Spin ice Two in- two out Real ice Pyrochlore lattice with loosely connected tetrahedra Frozen, disordered state with ground state entropy (akin to real ice)

39 Conservation laws and diffuse scattering 2 conditions for the energy 2in-2out state yields a non divergent field T. Fennell Science (2009) Review J. Phys. F (2012) Free energy is determined by the entropy, maximized by states with M=0 The magnetic GS is a Coulomb Phase Correlations are dipolar and anisotropic Nearby (002) PINCH POINTS

40 Diffuse scattering in spin Ice and spin liquid Tb 2 Ti 2 O 7-50 mk Spin liquid (fluctuating and AF) no spectral weight at Q=0 ½ ½ ½ maxima : AF correlations Ho 2 Ti 2 O 7-50 mk Classical spin ice (static and Ferro) strong spectral weight at Q=0 T.Fennell & al, Science 326 (2009) S.Petit & al, PRB 86 (2012) S. Guitteny et al PRL(2013) Pinch points in both!

41 Analysis of the pinch points Strongly anisotropic correlations of algebric nature conservation law in TTO spin liquid analogous to the ice rules see also T. Fennell et al PRL (2012) S.Guitteny & al, PRL 111 (2013) ~ 80 Å (perpendicular to the pinch point) // ~ 8 Å (along <111>)

42 Other local constraints and pinch points ferroelectrics Positional Correlations between H-atoms spinels «Molecular modes» Tomiyasu, PRL (2013) Youngblood PRB (1978) Exp Calc.

43 Simulate the diffuse scattering in spin ices Ho 2 Ti 2 O 7 Spin ice: Exp. Monte-Carlo simulations of the near-neighbour model Fennell et al Science(2009) Bramwell et al PRL(2001) Tb 2 Ti 2 O 7 Spin liquid Exp. Mean field calculation complex Hamiltonian with several Energy terms P. Bonville et al PRB (2014)

44 Simulate the diffuse scattering in frustrated magnets Recent attempt in geometrically frustrated systems : an empirical solution to fit the diffuse scattering Paddison and Goodwin PRL(2012) Calculate powder data from exact model Fit this data by Reverse Monte Carlo technique using additionnal constraint (minimize local variance of spin orientations) Rebuild the 3d pattern (single crystal) Ideal spin ice RMC rebuilt

45 Message to bring back Diffuse scattering is crucial in many studies of condensed matter physics. For example: T>> T C local chemical order informs on pair potentials which governs physical properties, allows one to predict phase diagrams, with applications to material science T~Tc Studies of critical phenomena and scaling laws T=0 zero point fluctuations and GS entropy Low spin values, low dimensions, geometrical frustration Local constraints yield general features (such as pinch points) No a priori description! New physical cases stimulate new approaches