Complementarity: muons and neutrons

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1 Complementarity: muons and neutrons Sue Kilcoyne Salford M5 4WT

2 What can neutrons tell us? Neutrons: have wavelengths comparable to interatomic spacings ( Å) have energies comparable to structural and magnetic excitations ( mev) are scattered with a strength that varies randomly from element to element (and isotope to isotope) have a magnetic spin are deeply penetrating (bulk samples can be studied) interact only weakly with matter (so the theory is easy!) Neutron scattering is therefore an ideal probe of atomic and magnetic structures and excitations

3 Length scales... Length scale in nm atomic and magnetic structures internal strain organic surfaces and multilayers viruses molecules inhomogeneities cracks and voids magnetic defects micelles critical phenomena pharmaceuticals proteins supermolecules polymers neutron wavelength in nm

4 Time scales... Time scale (seconds) Fe CsVCl 3 Crystal fields magnons and phonons spin relaxation single particle spin fluctuations tunneling polymer reptation excitations diffusion glassy dynamics molecular excitations libration Excitation energy (ev)

5 Infra-red Dielectric spectroscopy NMR Complementarity? r / Å Inelastic X-ray Raman scattering Chopper UT3 VUV-FEL E / mev t / ps Backscattering Spin Echo Multi-Chopper Brillouin scattering Inelastic Neutron Scattering μsr Photon correlation Q / Å -1

6 Complementarity or competition? The neutron is a uniquely sensitive microscopic probe of the magnetic properties of materials The muon is a uniquely sensitive microscopic probe of the magnetic properties of materials neutrons + muons = new insights > 2 µsr offers a time window sufficiently wide for studies of fast itinerant electron spin fluctuations through to slow distributed spin relaxation in spin glasses..and μsr is sufficiently sensitive for ultra-small magnetic moments (~10-3 μ B ) and nuclear moments to be detected Neutron scattering Mossbauer μsr ac susceptibilty remanence log (relaxation time),s

7 Complementarity: Part I Slow Dynamics

8 Glassy relaxation The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition P W Anderson Science 267 (1995) and spin glasses are excellent model systems to study the glass transition Random exchange leads to a random freezing of spin at the socalled glass temperature

9 Dilute Spin Glasses: NSE and MuSR Uemura et al, Phys.Rev.B 31 (1985) 546

10 Stretched-exponential relaxation Our first μsr measurements of a canonical spin glass (AgMn) gave a muon spin relaxation function of the form P (t) = a z o G z (t) = a o exp( ( λt) β ) with λ=2γ m2 <B i2 >τ λ diverges at the glass transition T g, whilst β decreases from 1 at 4T g to 1/3 at T g Stretched exponential relaxation in μsr is now widely accepted as an clear indicator of glassy dynamics Campbell, Cywinski, Kilcoyne et al, Phys. Rev. Lett. 72 (1994) 1291

11 Kohlrausch relaxation First observed in 1854, stretched exponential or Kohrausch relaxation is variously attributed to: A distribution of relaxation rates Hierarchical relaxation Palmer et al, PRL 53 (1984) 958 Random walk on an n-dimensional hypercube Campbell et al, Physica A 230 (1996) 554 Non-extensive entropy Pickup, Cywinski et al Physica B 397 (2007), 99 β may be either temperature dependent (approaching 1/3 at T g ) or temperature independent

12 A distribution of relaxation rates In general any statistical distribution of independent (parallel) relaxation channels will lead to a stretched exponential: G(t) = w(λ)exp( -tλ)dλ exp( ( λt) As the distribution broadens β decreases from unity to 1/3 as shown on the right for a truncated log-normal w(τ) β ) β ln(δτ/τ ο ) Warning: the sum of even just two or three discrete exponential relaxations can give an apparent stretched exponential

13 Simulations of 3d Ising Spin Glasses Ogielski s MC simulations of the ±J Ising spin glass show: q(t) = x β i i t exp( t / τo ) S (0)S (t) T>T g T<T g T=T g Ogielski, Phys.Rev. B 32 (1985) 7384

14 Simulations of 3d Ising Spin Glasses Ogielski s MC simulations of the ±J Ising spin glass show: q(t) = x β i i t exp( t / τo ) S (0)S (t) x τ o β in principle these characteristic parameters can be measured with NSE and μsr... Ogielski, Phys.Rev. B 32 (1985) 7384

15 Au 1-x Fe x x = 0.20 x = 0.18 T [K] PM x = 0.16 x = 0.14 x [Fe] FM Coles,et al. Phil. Mag. B 37, 489 (1978)

16 NSE measurements on Au 86 Fe 14 1 T<T g, S(Q,t)~t -x S(Q,t)/S(Q) 0.1 T>T g, S(Q,t)~t -x exp(-(t/τ) β ) 30.7K 40.6K 45.7K 50.8K 55.8K Fourier time, t f, ns C. Pappas, F. Mezei, G. Ehlers, P. Manuel, I.A. Campbell, PRB 68, (2003)

17 μsr measurements on AuFe The μsr spectra from Au-14%Fe are also stretched exponential, with similar βs to those obtained from NSE However at concentrations higher than the percolation threshold NSE spectra are simple exponential whilst μsr spectra remain stretched exponential above T F AuFe14% Pappas, Mezei, Hillier, Cywinski in preparation

18 AuFe: A summary 6 Disordered ferromagnetic phase exponential NSE stretched exponential μsr???? T [K] spin glass phase stretched-exponential NSE and muon spectra PM x = 0.16 x = 0.14 x = 0.20 x = 0.18 x [Fe] FM λ, MHz λ, MHz λ, MHz Temperature Temperature 10 8 AuFe20% AuFe18% AuFe16% Temperature Pappas, Hillier, Cywinski et al in preparation

19 Complementarity: Part II Fast dynamics

20 YMn 2 - a frustrated itinerant magnet C15 Laves phase (Fd3m) a o =0.76nm Cywinski et al, J Phys C 3 (1991) 6473

21 Moment localisation and collapse YMn 2 YMn %Fe 5% volume expansion at T N =100K μ Mn =2.8 μ B below T N ordered in a long wavelength helix. Pauli paramagnetic above T N The antiferromagnetic phase, the volume expansion and the Mn moment itself can be destabilised by 2.5at% Fe or 3kb Cywinski, Kilcoyne et al, J Phys C 3 (1991) 6473

22 Inelastic scattering spectra from YMn 2 +5at%Fe 60 Energy transfer (mev) K 50K 100K 200K Wavevector transfer (A -1 ) Rainford, Cywinski and Dakin JMMM (1995) 805

23 A quantitative comparison... The muon relaxation rate λ can be related directly to the parameters measured by inelastic neutron scattering: G k B T χ ( q ) λ = N Γ ( q ) q where Γ(q) is the inelastic scattering linewidth, and χ(q) the susceptibility For many itinerant antiferromagnets the q-dependence of Γ is weak: G k χ L λ = B T Γ where χ = 1 χ L (q) is the local susceptibility N λ = q The neutron linewidth Γ can be compared directly with λ obtained from μsr via G k T B χ Γ with the coupling constant G as the only free parameter L Rainford in Muon Science eds Lee, Kilcoyne and Cywinski, 1999

24 A quantitative comparison with µsr For Y(Mn 0.9 Al 0.1 ) 2 Γ is Arrhenius-like: Γ( T) = Γ o exp( Ea kb T) while χ L follows a Curie law: χ L (T) = C (T + θ) with E a /k B T = 280K and θ =93K We can therefore fit the expression ct λ(t) = (T + θ)exp( E / kt) with c as the only free parameter to the muon data a Y(MnAl) 2 Neutron and muon data are in excellent agreement but while each muon spectrum takes 20min to collect, each neutron spectrum takes 12 hours Cywinski, Physica B 350 (2004) 17

25 β-mn an elemental spin fluctuator Stewart and Cywinski PRB59 (1999) 4305 Simple exponential µsr ( λt ) P(t) = a G e o KT β Mn 0.94 Al 0.06 Spin liquid phase Transition Temperature (K) Al Concentration (at%) Stretched exponential µsr β ( λt ) P(t) = a G e o KT β Mn 0.85 Al 0.15 Spin glass phase

26 Scaling of spin relaxation rates

27 β-mn an elemental spin fluctuator Neutron scattering has shown that β-mn is the first example of an elemental non-fermi liquid Stewart and Cywinski PRB59 (1999) 4305 PRL 89 (2002) 6403 Transition Temperature (K) Al Concentration (at%)

28 Complementarity: Part III Magnetic Structures

29 Small moment systems Observation of long range ferromagnetic order in a small moment magnet eg a in fully organic ferromagnetic

30 Magnetic volume fractions inhomogeneous homogeneous inhomogeneous homogeneous Amplitude = Magnetic Volume Fraction Frequency = Size of magnetic moments Damping = Inhomogeniety within magnetic regions

31 Magnetic volume fractions; URu 2 Si 2 m 2 (µ B ) 2 Neutron diffraction shows the U moment in URu 2 Si 2 decreasing in magnitude as pressure is increased

32 Magnetic volume fractions; URu 2 Si 2 m 2 (µ B ) 2 Neutron diffraction shows the U moment in URu 2 Si 2 decreasing in magnitude as pressure is increased µsr shows that this decrease is associated with a reduced antiferromagnetic volume fraction There is phase separation into magnetic and non-magnetic regions only a combined neutron + μsr study could reveal this

33 Magnetic order in ReNiO 3 compounds μ + 1 Ni Re Neutron powder diffraction magnetic models Ni Re Ni Model 1 Model 2 μ + 2 } μ 0 } μ=0 is unable to distinguish between the two distinct Garcia-Munoz, Lacorre, Cywinski PRB51 (1995) 15197

34 Magnetic order in ReNiO 3 compounds Above the MI transition there is a single paramagnetic muon site: Gz(t) = a0exp ( λt) Below the MI transition there are two equally populated but inequivalent muon sites, one magnetically ordered, the other disordered. Model 1 must be correct Garcia-Munoz, Lacorre, Cywinski PRB51 (1995) 15197

35 A cautionary tale: Ni 3 Al Ni The general consensus from neutron and other studies is that Ni 3 Al orders ferromagnetically at at 40K, with a Ni moment of ~0.076μ B and T c =40K Al Ni 3 Al - space group Pm3m L1 2 (Cu 3 Au) structure a 0 =0.356nm Dhar et al, PRB 40 (1989) 11488

36 Ni 3 Al - high temperature muon spectra At high temperature, the zf spectra are well described by B=2mT aog z(t) = a (1 3 σ 2 t 2 )e ( σ 2 2 with σ=0.15 ms -1 t 2 + and any contribution from atomic spin fluctuations is entirely motionally narrowed a b Asymmetry Ni 3 Al T=250K Time (μs) σ is consistent with the muon sitting at 1/2,1/2,1/2 in the cell B=1mT B=0

37 Ni 3 Al - low temperature spectra The atomic spin contribution to the μsr spectra remains motionally narrowed down to 40K Below 40K there are two contributions to the spectra 1 2 aog z(t) = am + (1 3 3 a G (t) nm zkt β ( ( λt ) ( λt) )e a m and a nm refer to the magnetic and nonmagnetic components respectively, and G zkt (t) = + (1 σ t )e β β 2 2 ( σ t 2 Asymmetry Time (μs) 11K 20K 29K 47K 32K

38 Ni 3 Al: Conclusions from µsr The magnetic contribution below 40K takes the form of a Voigtian Kubo- Toyabe function Crook and Cywinski J Phys Cond Matt 9 (1997) 1149 Asymmetry, a m p g T c =40K The asymmetry, a m, muon relaxation rate λ and exponent β all increase below the reported Curie temperature of T c =40K There is no evidence of long range ferromagnetic order in Ni 3 Al, but rather of an inhomogenous and relatively static magnetic ground state. But this is because the muon sits at a site of high symmetry and is insensitive to long range ferromagnetic order!! S H Kilcoyne and R Cywinski Physica B 326 (2003) 577 λ (μs -1 ) β Temperature (K) T c =40K T c =40K

39 Summary Neutrons and muons together often give more information than either technique alone - even in crystallographic studies Muons can provide crucial insights very quickly, often providing justification for a detailed and lengthy neutron experiment Where neutron and muon results disagree, there may be new physics at play