Neutron Scattering in Magnetic Systems

Transcription

1 Neutron Scattering in Magnetic Systems Jim Rhyne Lujan Neutron Scattering Center Los Alamos National Laboratory 006 Spins in Solids Summer School June 19 3, 006

2 What Can Neutrons Do? Neutrons measure the space and time-dependent correlation function of atoms and spins All the Physics! Diffraction (the momentum [direction] change of the neutron is measured) Atomic Structure via nuclear positions Magnetic Structure(neutron magnetic moment interacts with internal fields) Disordered systems - radial distribution functions Depth profile of order parameters from neutron reflectivity Macro-scale structures from Small Angle Scattering (1 nm to 100 nm) Inelastic Scattering (the momentum and energy change of the neutron is measured) Dispersive and non-dispersive phonon and magnon excitations Density of states Quasi-elastic scattering

3 What do we need to do neutron scattering? Neutron Source produces neutrons Diffractometer or Spectrometer Allows neutrons to interact with sample Sorts out discrete wavelengths by monochromator (reactor) or by time of flight (pulse source) Detectors pick up neutrons scattered from sample Analysis methods to determine material properties Brain power to interpret results

4 Sources of neutrons for scattering? Lujan Neutron Scattering Center Nuclear Reactor Weapons Neutron Research Facility Proton Storage Ring Neutrons produced from fission of 35 U Fission spectrum neutrons moderated to thermal energies (e.g. with D 0) Proton Continuous source no time structure Radiography Common neutron energies mev < E < 00 mev 800 MeV Proton Linear Accelerator Proton accelerator and heavy metal target (e.g., W or U) Neutrons produced by spallation Higher energy neutrons moderated to thermal energies Neutrons come in pulses (e.g. 0 Hz at LANSCE) Wider range of incident neutron energies Isotope Production Facility

5 There are four National User Facilities for neutron scattering in the US National User Facilities HFIR 1966 NCNR 1969 IPNS 1981 Lujan 1985 (SNS 006) Intense Pulsed Neutron Source (7 kw) NIST Center for Neutron Research Local/Regional Facilities (University Reactors) MIT Missouri Manuel Lujan Jr. Neutron Scattering Center (100 kw) High-Flux Isotope Reactor Spallation Neutron Source (first neutrons in May -- operational instruments late in 006) (1000 kw)

6 Neutron scattering machines Spectrometers or diffractometers typically live in a beam room are heavily shielded to keep background low and protect us receive neutrons from the target (or reactor) correlate data with specific neutron wavelengths by time of flight accommodate sample environments (high/low temperature, magnetic fields, pressure apparatus)

7 Neutron Scattering s Moment in the Limelight

8 General Properties of the Neutron The kinetic energy of a 1.8 Å neutron is equivalent to T = 93K (warm coffee!), so it is called a thermal neutron. The relationships between wavelength (Å) and the energy (mev), and the speed (m/s, mi/hr) of the neutron are: E = / λ and v = 3960 / λ e.g. the 1.8 Å neutron has E = 5.3 mev and v = 00 m/s = 4900 mi/hr The wavelength if of the same order as the atomic separation so interference occurs between waves scattered by neighboring atoms (diffraction). Also, the energy is of same order as that of lattice vibrations (phonons) or magnetic excitations (magnons) and thus creation of annihilation of a lattice wave produces a measurable shift in neutron energy (inelastic scattering). 8

9 COMPARATIVE PROPERTIES OF X-RAY AND NEUTRON SCATTERING Property X-Rays Neutrons Wavelength Characteristic line spectra such as Cu K α λ = 1.54 Å Continuous wavelength band, or single λ = 1.1 ± 0.05 Å separated out from Maxwell spectrum by crystal monochromator or chopper Energy for λ = 1 Å h h (same order as energy of elementary excitations) Nature of scattering by atoms Electronic Form factor dependence on [sinθ]/λ Linear increase of scattering amplitude with atomic number, calculable from known electronic configurations Nuclear, Isotropic, no angular dependent factor Irregular variation with atomic number. Dependent on nuclear structure and only determined empirically by experiment Magnetic Scattering Very weak additional scattering ( 10-5 ) Additional scattering by atoms with magnetic moments (same magnitude as nuclear scattering) Amplitude of scattering falls off with increasing [sin θ]/λ Absorption coefficient Very large, true absorption much larger than scattering µ abs increases with atomic number Absorption usually very small (exceptions Gd, Cd, B ) and less than scattering µ abs 10-1 Method of Detection Solid State Detector, Image Plate Proportional 3 He counter 9

10 Golden Rule of Neutron Scattering We don t take pictures of atoms! Atoms in fcc crystal Job preservation for neutron scatterers we live in reciprocal space Intensity 10

11 How are neutrons scattered by atoms (nuclei)? Short-range scattering potential: V ( r) = πh m bδ ( r) The quantity b (or f) is the strength of the potential and is called the scattering length depends on isotopic composition Thus b varies over N nuclei can find average b defines coherent scattering amplitude b coh = b leads to diffraction turns on only at Bragg peaks But what about deviations from average? This defines the incoherent scattering b inc = Incoherent scattering doesn t depend on Bragg diffrac. condition, thus has no angular dependence leads to background (e.g., H) ( b b ) 1/ 11

12 Scattering of neutrons by nuclei A single isolated nucleus will scatter neutrons with an intensity (isotropic) I = I 0 [4πb ] where I 0 = incident neutron intensity, b = scattering amplitude for nucleus What happens when we put nucleus (atom) in lattice? Scattering from N neuclei can add up because they are on a lattice Adding is controlled by phase relationship between waves scattered from different lattice planes Intensity is no longer isotropic Bragg law gives directional dependence Intensity I (Q, or θ) is given by a scattering cross-section or scattering function 1

13 Observed Coherent Scattering Intensity of diffracted x-ray or neutron beam produces series of peaks at discrete values of θ [or d or K (also Q)] Note: d = λ/( sinθ) or K = 4πsinθ/ λ = π/d are more fundamental since values are independent of λ and thus characteristic only of material. Benzine Pattern (partial) Note: Inversion of scales - θ f(1/d) 13

14 Scattering Cross-section The measured scattered intensity in a diffraction experiment is proportional to a scattering function S(Q), which is proportional to a scattering cross-section dσ I(θ,d,or Q) S(Q) Ω= solid angle dω In turn the cross-section A(Q) A*(Q) A = scattering anplitude In second Born approximation (kinematic limit) A Fourier transform of r r scattering length density ρ(r) = ρaj (r R j ) (α atom) with the sum over j j atoms at position R j πiq r 3 πiq R then A( Q) = ρ ( r R ) e d r = f ( Q) e j where the scattering factor aj j f j ( Q) = j ρ ( r The scattering factor f(q) is the fundamental quantity describing the scattering of radiation from the material f takes different forms depending on the type of radiation f varies in magnitude depending on the scattering atom or magnetic spin aj j ' ) e πiq r' d 3 r' 14

15 Scattering Factors f. The scattering factor The Fourier transform character of the scattering factor f means that the radial extent of the scattering center density ρ aj ( r) will dictate its Q dependence. x-rays scatter from the electron cloud of dimensions comparable to λ or d ( 1/Q) Neutrons scatter from the nucleus 10-5 the dimension of λ or d ' πiq r' 3 f ( Q) = ρ ( r ) e d r' aj 15

16 Scattering Factors f, cont d For x-rays the magntude of f is proportional to Z For neutrons nuclear factors determine f, thus no regular with Z (different isotopes can have different f s) For neutrons conventionally f = b (Scattering length - constant for an element) Shaded (negative) --> π phase change 16

17 What Controls the Scattering Amplitude? I(Q) S(Q) A(Q) A*(Q) [Measured scattered intensity, S = scat.func.] where f aj = atomic scattering factor [cm -1 ] A( Q) = j f aj ( Q) e πiq R Magnitude of A(Q) is controlled by [ A(Q) A*(Q) called Structure Factor] f values for various atoms in lattice destructive interference of waves scattering from atoms at various lattice sites (calculation of above sum over atoms in lattice reveals this) fcc lattice (000, 0½½, ½0½, ½½0) bcc lattice (000, ½½½) A(Q) = 4 f a for hkl all odd or A(Q) = f a for h+k+l even hkl all even A(Q) = 0 otherwise A(Q) = 0 otherwise 17

18 Applications of Powder Diffraction Methods Scattering instruments, line shapes, resolution Rietveld Structure Refinement Analysis Magnetic Structures and Diffraction Diffuse Scattering 18

19 Powder Diffraction Experimental View Two types of diffactometers Fixed Wavelength Diffraction (x rays, neutrons at a Reactor [HFIR, NCNR]) Variable wavelength (white beam neutrons) [beam may be pulsed (LANSCE [0 Hz], IPNS [30 Hz], SNS [60 Hz]) or continuous (PNS, Geneva)] Objective (determine d or θ ) Bragg Law λ = d sinθ d = λ/( sin θ)» pulse neutron source - vary λ, keep θ fixed» x-ray or reactor neutron source -- vary θ, keep λ fixed 19

20 COMPARISON OF POWDER DIFFRACTION INSTRUMENTS *** Bragg Scattering: λ = d sin θ [we need d ] *** Reactor Source -- Monochromator selects near mono-energetic neutrons, detector moves (θ) to collect discrete (λfixed, vary θ) diffraction data Pulse Source -- White beam of moderated neutrons used, neutron time-of-flight selects wavelength, detectors grouped in banks, no moving parts (θ fixed, vary λ) λ= f(t) so d=λ/(sinθ)= f(t)/(sinθ) short t short λ small d; long t long λ large d 0

21 Rietveld Profile Refinement Least Squares Fitting Procedure for Powder Data Input Data Powder scattering pattern data Trial structure space group and approximate lattice parameters and atomic positions Line shape function and Q-dependence of resolution Output Results Lattice Parameters Refined atomic positions and occupancies Thermal parameters (Debye Waller) for each atom site Resolution parameters Background parameters R factors of fit (and other measures of fit precision) Preferential orientation, absorption, etc. More than one phase can be separately refined 1

22 Magnetic Powder Diffraction Neutron has a magnetic moment -- will interact with any magnetic fields within a solid, e.g., exchange field Magnetic scattering amplitude for an atom (equivalent to b) e γ 1 p = fgj 0.69x10 fgj ( cm) mc = where g = Lande g factor, J = total spin angular momentum, f = magnetic electrons form factor Magnetic scattering comes from polarized spins (e.g., 3d [Fe] or 4f [RE]) not from nucleus -- Therefore scattering amplitude is Q-dependent (like for x-rays) via f at Q = 0 for Fe µ = gj =. Bohr magnetons p = 0.6 (comparable to nuclear b = 0.954) all in units of 10-1 cm Mn Refinement gives moment magnitudes on + each site and x,y,z components (if symmetry permits)

23 Form Factors Experimental Calculated More Localized Moment Less Localized Moment 3

24 Magnetic Powder Diffraction II In diffraction with unpolarized neutrons (polarized scattering is a separate topic) the nuclear and magnetic cross sections are independent and additive: d dω σ S( Q) = S nucl + q S magn = S nucl + ( 1 cos α ) S magn q is a switch reflecting fact that only the component of the magnetic moment µ scattering vector K (or Q) contributes to the scattering K µ 4

25 Basic Types of Magnetic Order and Resulting Scattering a Ferromagnet (parallel spins) Single Magnetic site (e.g., Fe, Co, Tb) -- Scattering only at Bragg peak positions (adds to nuclear), but not necessarily all (q switch) Antiferromagnet (parallel spins with alternate sites reversed in direction) equivalent to new magnetic unit cell doubled in propagation direction of AFM Purely magnetic scattering peaks at half Miller index positions (e.g., 1,1,1/) c Multi Site Ferromagnet (e.g. Y 6 Fe 3 (4 distinct Fe sites) -- no new peaks in scattering Overall net magnetic moment adds 5 to 0 [job security for neutrons!!]

26 Magnetic Structures, cont d. Other more complex antiferromagnetic cells (multisite, but on each site Σµ = 0) Ferrimagnet (multi-site cell) Spins parallel on each site Sites can reverse spin direction (111) FM Sheets FM layers along (0001) Σµ 0 on each site, but overall Σµ could = 0 bcc site antiparallel to corners AFM line along (111) A B C D Ferromagnetic type scattering pattern --No new peaks 6

27 More Magnetic Structures Paramagnetism (T > T C ) spins thermally disordered no coherent magnetic scattering Periodic Moment Structures (e.g., Er, Au Mn) Σµ = 0, but periodic spin arrangement gives rise to magnetic scattering satellites (+, -) of nuclear Bragg peaks Incoherent scattering (weak) has form factor dependence 7

28 Polarized Neutron Reflectometry Incident Polarized Neutrons Specular Reflectivity Detector Al-Coil Spin Flipper Sample Al-Coil Spin Flipper Spin Polarizing Supermirror index of refraction: sensitive to µ scattering length density: used to model reflectivity non spin-flip spin-flip n n ± ± ± m ρ ( z ) = λ 1 N π λ 1 N p cos π ( k i ( b ± p sin φ ) / 4 π )[1 n φ ] reflectivity: measured quantity r (Q ) = 4 π iq Ψ ( z ) ρ ( z ) exp( ikz )dz

29 Ga 1-x Mn x As Dilute ferromagnetic semiconductor Spintronics applications Annealing increases magnetization & Tc Interstitial Mn go to the surface! K. W. Edmonds et al., PRL, 9, 3701, (004) - Auger Depth-dependence of chemical order and magnetization determined Polarized-Beam Neutron Reflectivity Compared similar as-grown and annealed films T = 13 K, H = 1 koe (in plane) J. Blinowski et al, Phys. Rev. B 67, 1104 (003)

30 GaMn x As 1-x As-Grown & Annealed t = 110 nm, x = 0.08, T C = 50 K, 10 K Reflectivity x Q 4 (Å -4 ) As-Grown Film measured R ++ fit to R ++ measured R - - fit to R - - Annealed Film measured R ++ fit to R ++ measured R - - fit to R - - Measured reflectivities & fits Spin up & spin down splitting due to sample magnetization Spin up reflectivities are different Slope at high Q different Fits are good Spin Asymmetry As-Grown Film ρ (µm - ) As-Grown Film measured SA fit from SLD model Q (Å -1 ρ ) nuc ρ mag M avg = 17 emu cm -3 moment per Mn = 1. µ B surface Depth (Å) substrate Q (Å -1 ) M (emu cm -3 ) Spin Asymmetry Annealed Film ρ (µm - ) Annealed Film measured SA fit from SLD model Q (Å -1 ) ρ nuc M avg = 48 emu cm -3 ρ mag moment per Mn = 3.4 µ B surface Depth (Å) substrate M (emu cm -3 ) Magnetic signal: spin asymmetry SA = (up down) / (up + down) Larger amplitude for annealed film Better defined for annealed film SLD Models (mag. & chem.) As-grown M doubles near surface M increases and more uniform for annealed film Both films show magnetic depletion at surface Drastic chemical change at annealed film s surface Interstitial Mn have diffused to surface! (combined with N during annealing)

31 Inelastic Scattering Inelastic Scattering (the momentum and energy change of the neutron is measured) Dispersive and non-dispersive phonon and magnon excitations Density of states Quasi-elastic scattering

32 Triple Axis Neutron Scattering Spectrometer n E Want Thermal neutrons e.g., E=14mev, λ=.4å λ i = d m sin θ m k i = π/λ i E i 1 n0 E E / kt = π kt = h k m i n kt e λ f = d a sinθ a k f = π/λ f E I f det = h k m f n ( counts ) d d σ ΩdE 3

33 MURR Triple Axis Neutron Spectrometer (TRIAX) Analyzer Assembly Beam Stop (pivots with drum and sample) Detector Shield and Collimator Sample Table and Goniometer Monochromator Drum 33

34 Inelastic Neutron Scattering *** The measurement of the functional dependence of wave vector q and the energy E = hω of an elementary excitation (e.g., magnon or phonon dispersion) Energy and Momentum Conservation Conditions: * k i = wave vector of incident neutron * k f = wave vector of scattered neutron * Neutron energies E ni h = k m i E nf h = k m f Change of E N creates ( E N < 0) or annihilates ( E N > 0) an excitation of energy ω hω E N h = m ( k k ) Neutron - excitation system momentum conservation: r r r r r k k = πτ + q Q f i f i = hω 0 P i f Constant q scan (E inc fixed) Rotate P [k f,and k i ] about 0 34

35 Physical Phenomena Studied with Inelastic Neutron Scattering *** accessible energy and momentum transfer (Q) ranges *** Note: Cold and hot sources at Reactors and particularly Spallation Neutron Sources have greatly extended the useful range of Q and E. [From J.D. Axe, Neutrons: The Kinder, Gentler Probe of Condensed Matter, Matls. Res. Symp. Proc. 166, 3 (1990)]. 35

36 Measuring Inelastic Processes Scans are usually made with q fixed and either E i or E f varying (Constant q scans) Can alternatively fix E and vary q under special conditions (constant E scans) Constant q scan: Measure intensity scattered from analyzer (and sample) as a function of E i or E f Scattered Intensity (counts/mon.int.) Γ E (mev) q fixed = q o Peak of intensity profile becomes one point on dispersion curve Energy (mev) Dispersion Curve (FM) q 0.0 o q (A -1 ) 36

37 Magnetic Scattering Cross Section General form: The Scattering Function S(Q,ω) is the Fourier transform of the space and time correlation function G(r,t) of the spins -- All the physics!! S(Q,ω) can be written in terms of the expectation values of components of Heisenberg spin operators <J α (r,t) J β (0,0)> (including the Debye Waller factor W(Q) for thermal vibrations): where J α (Q,t) is the q-space Fourier transform of the real space spin operator J α (r,t) 37

38 The integral of the diagonal component S αα over the Brillouin zone gives the average value of the α component of the magnetic moment: The diffuse (i.e. non-bragg) scattering part of S(Q,ω) can be related to the imaginary part of a generalized susceptibility χ(q,ω) via the fluctuation dissipation theorem: The cross section then directly measures the wave vector and energy-dependent susceptibility: 38

39 The imaginary part of the generalized susceptibility χ(q,ω) can be expressed as a static susceptibility χ(q) times a Dynamic Spectral Weight Function F(q,ω) [the Fourier transform of the spin relaxation function]: d σ γe 1 = gf ( Q) dωde1 moc ω[exp( hωβ ) N exp( hωβ ) 1 ( gµ ) B k exp k { W ( Q) } αβ αβ {( δ q q )[ χ ( q) F ( q, ω) ]} αβ α β k k α, β χ(q) F(q,ω) may further be separated into a longitudinal fluctuation part and a transverse part (latter gives rise to spin waves): αβ αβ αβ αβ l αβ αβ [ χ ( q ) F ( q, ω) ] = [ χ ( q) F ( q, ω) ] [ χ ( q) F ( q, ω) ] t k k k k + k k The form of F(q,ω) can not be rigorously determined so one of two forms are generally used Double Lorentzian Damped Harmonic Oscillator 39

40 Intensity E (mev) Γ Γ Intensity E (mev) Spectral Weight Function for Spin Waves At low temperature (δ functions): creation annihilation At finite temperatures linewidth (lifetime) broadening Γ occurs (Double Lorentzian form): The transverse (spin wave) part of the cross section becomes: Where the damping coefficient Γ (linewidth) in a Heisenberg magnet varies as T and q 4 : { } ] [ ] [ 1 ), ( ω ω δ ω ω δ ω + + = q q t q F + Γ + Γ + + Γ Γ = ] [ ] [ 1 ), ( q q t q F ω ω ω ω π ω [ ] [ ] + Γ + Γ + + Γ Γ = Ω ) ( ]) exp( [1 )} ( exp{ ) ( 1 q q B o o t q r g N Q W k k Q gf c m e de d d ω ω ω ω π µ ωβ ωβ γ σ h h 4 ln ) ( = Γ q k B T q T q hω

41 Dispersion in a Heisenberg Ferromagnet At low q dispersion is quadratic (to lowest order): E(q) = ω(q) h = + D q D is the spin wave stiffness parameter (may be anisotropic in q direction is the gap energy (may be 0 -- reflects crystal field anisotropy energy) Full Brillouin zone dispersion for a single nearest neighbor exchange constant J 1 is of the form: hω(q) = + 4J 1 S (3 cos q x a o cos q y a o cos q z a o ) which reduces at small q (cos x 1 - x /) to (111): hω(q 111 ) 6J 1 Sa q Dq Full zone dispersion for a fcc lattice [qqq] 80 J 1 T c In mean field approx: T c = 4 J 1 S(S+1)/k B Energy (mev) [qq0] [q00] q (rlu) 41

42 Examples of Inelastic Scattering Data Ferromagnetic Amorphous metallic glass Fe 0.86 B 0.14 Constant q scans left panel - varying T right panel - varying q) [Rhyne, Fish, and Lynn, JAP 53, 316 (198)] Quadratic dispersion (E vs. q ) [J.A. Fernandez-Baca, J.W. Lynn, J.J. Rhyne, and G.E. Fish, Physics B 136B, 53 (1986) 4

43 At low T, D renormalizes as T 5/ (Dyson) D( T) = D(0) 1 T T c 5 / ν β T D( T) = D(0) 1 Tc As T T c, D varies as: where ν-β = 0.34 for an Heisenberg ferromagnet J.A. Fernandez-Baca, J.W. Lynn, J.J. Rhyne, and G.E. Fish, Physics B 136B, 53 (1986) The spin wave linewidth Γ (lifetime) after deconvoluting the instrument resolution width is: Γ( q) = Γ T o q 4 k BT ln hω q J.A. Fernandez-Baca, J.W. Lynn, J.J. Rhyne, and G.E. Fish, Physics B 136B, 53 (1986) 43

44 Diffuse Scattering (Life Without Bragg Peaks) Amorphous Material No periodic lattice May have short range ordering (SRO) of atoms (e.g. Dense Random Packing (DRP) of Spheres, tetrahedral atom clusters (e.g., Ge) Thus no discrete Bragg Peaks, but SRO leads to features in Intensity = S(Q) Study Diffraction with White Beam (e.g., pulse source) X-ray S(Q) vs Q [K] for amorphous Fe 80 B 0 and related metallic glasses (Wagner) 44

45 Amorphous Structure Factors Reciprocal Space Single component system -- Real Space Probability Distribution of Atoms is given by Fourier Transforms of S(Q): 0 G( R) = [ S( Q) 1] Qsin QrdQ π Multicomponent system (e.g., atoms 1 and ) S(Q) is a superposition of partial structure factors S ij (Q) [Faber-Ziman formalism] S(Q) = <b> - [c 1 b 1 S 11 (Q) + c b S (Q) + c 1 c b 1 b S 1 (Q) including the scattering factor weightings Partial structure factors can often be obtained by enhancing scattering from one atom component wrt another [e.g., by isotopic substitution (neutrons) or by combining x-ray (heavy atom sensitivity) and neutron (uniform sensitivity) data Example: Amorphous metallic glass Ni 80 P 0 [Lamparter] -- Scattering Patterns: (1) Natural Ni [b = 1.03] () Ni substituted by 6 Ni [b = -0.87] (3) Natural Ni substituted by null scattering mixture [b = 0] 45

46 Amorphous Ni 80 P 0 Isotopic Substitution Patterns [Lamparter] S(Q) with various Ni atom replacements Yields Partial Structure Factors Null Ni ampl. 46

47 Amorphous Structure Factors Real Space Real Space -- Partial pair density distribution functions ρ ij (R) = #j atoms at distance R from i atom [Note: at large R --> c j ρ o where ρ o = average density of material Define partial pair correlation function G ij (R) = 4πR[ρ ij (R)/ c j - ρ o ] where the Real Space G ij (R) corresponds to the Reciprocal Space partial structure factors S ij (Q): G ij 0 ( R) = [ Sij ( Q) 1] Qsin QrdQ π The total pair correlation function is then composed of a linear superposition of the G ij (R) functions: G(R) = <b> - [c 1 b 1 G 11 (R) + c b G (R) + c 1 c b 1 b G 1 (R) and the peaks in this function give the interatomic distances R ij The RDF (radial distribution function) or partial RDF ij is the density of the system (or ij atom pairs) from R = 0 up to R = R 1 -- RDF ij = 4πR 1 ρ ij 47

48 Back to Real Space Identification of 1st, nd nearest neighbor interatomic distances [example -- amorphous YFe [D Antonio, et al.] Real space partial pair correlation functions for Ni 80 P 0 ( ), Ni 81 P 19 (.. ) [Lamparter] Relative areas give atomic coordination R numbers Z = RDF ( R) dr i R 1 ij 48