Neutron diffraction. 1. Thermal neutrons properties and registration

Transcription

1 Neutron diffraction 1. Thermal neutrons properties and registration Fast (hot) n: high-e n generated in the nuclear reactor Collisions w. moderator H 2 O ( C) deceleration n gas Kinetic theory of gasses: Thermal (slow) n: v n ( C) E n ev n(v) Maxwell-Boltzman law: n(v) d(v)... number of n w. the v (v, v+dv) /1V N total no. of N /1V v 0 least probable v max. n(v) T [ C] λ [Ǻ] 1,8 1.3 Ǻ v 0 [10 3 ms -1 ] E [mev] 26 48

2 Neutron flux: Max: For 23 C: λ m = 1.12 Ǻ Registration of n: n- diffractometer (difractograph) Monochromator: (LiF, CuF 2, Cu, Ge, Pb plates) Bragg diffraction monochromatic n with λ Differences between n and X-ray: beam diameter: n 3-5cm, X-rays 1 mm low n(hν) n [n(hν) n n(hν) X-Ray ]!!! Sample x detector (counter) - ω : 2ω rotation Detector GM counter (BF 3 gas + 5 B 10 isotope) ionisation (U pulse)

3 Comparison of n and X-ray diffractograms ( same λ) n-record wider peaks width as ϑ? λ n 0.05 Ǻ x λ Xray 10-5 Ǻ (as E n << E X-ray ) 2. The use of thermal n for the study of solid states Basic characteristics of n: q = 0 m n = x kg I = ½ µ n = 1.91 µ N = 1.91 m e /m p x µ B λ n a (Ǻ) 1. Neutron- nucleus interactions a) q = 0, short range of nuclear forces ( m) deep penetration of n into the specimen n scattering bulk effect x X-ray scattering (surface scattering) b) elastic n-scattering n diffraction on atomic planes (Bragg law) angle distribution of diffracted n intensity atomic structure c) inelastic n- scattering E n E phonon THz vibration frequency of atoms x E X-ray >> E phonon Ε X-ray 0 due to inel. X-ray scattering 2. Neutron-electron interactions Due to µ n 0 magnetic scattering (transitive metals noncompensated d- and f-shells) a) elastic coherent scattering arrangements of magnetic moments in the x-tal b) inelastic scattering frequencies of spin waves (magnons) (describing excitations of magnetic moments) 3. Absorption of neutrons by nuclei Strong absorbing elements B, Cd, Li, some rare earth el. shielding of n-flux

4 3. Comparison of neutron and X-ray scattering Analogical theory of elastic scattering (diffraction) due to close λ of n and X-rays (Bragg law) BUT 1. Different mechanisms of scattering on atoms X-rays: scattering on el. shells n: scattering on nucleus (exception: magnetic materials n-electron scattering)? λ vs. size of scattering centres + Fourier transf. f FT (electron/nucleus/spin density of atom) FT(δ) = const. and FT(const) = δ ρ N < ρ el < ρ spin...space distribution ( localization ) of nucleus, el. or spin density of atom ρ N δ (most localized) f n const., etc. 2. Different dependence of f on A r /Z f X-Ray Z f n complex value real part nonmonotonic function of Z imaginary part absorption n-scattering on nucleus: No theory only experiment! n-scattering on nucleus: i) potential Coulomb like deflection of n on el. shell, f Z ii) resonance n penetrates to the nucleus trapped by nucleus deexcitation depends on energy levels of newly created nucleus f nonmonotonic (positive/negative) Total scattering: i + ii nonmonotonic, significantly different for neighbouring elements, even for isotopes of the same element (see figure) f Z-low f Z-high 3. Different absorption X- strong absorption (0.01 mm penetration depth) surface scattering n weak absorption (exception Cd, graphite) bulk scattering

5 4. Magnetic scattering Neutron scattering due to a) n nucleus interaction (nuclear interaction) (all atoms) Nuclear scattering of n: spherical symmetry b independent of θ (b.. atom scattering factor) Intensity of reflection F hkl 2, where F hkl structure amplitude b) n electron interaction (dipole interaction) µ n + µ at. (only atoms w. uncompesated magnetic moment noncoupled e - in d, f shells) Magnetic interaction (n+e - ): long range nonspherical symmetry

6 Scattered wave amplitude (magnetic scattering) r e.(classical) radius of e - γ gyromagnetic ratio of n (= 1.91) S spin quantum number of the atom f (sinθ/λ) atom form-factor (angular dependence of n-scattering on the atom)! Scattering depends on vector variables Amplitude of n-wave scattered by the atom f (p, α) α ε and K (vectors) K direction of magnetic moments magnetic interaction vector q q (hkl) Structure amplitude of the magnetic scattering K (hkl) p.sin α = 0 K (hkl) max. = p Study of magnetic structures Colinear magnetic structures (fero, feri, antifero) F hkl (m) str. amplitude of nuclear scattering

7 The relation of nuclear and magnetic scattering: a) independent nonpolarized n (random spins no interference) additive contributions str. factor effective atom aplitudes b) coherent (interference) polarized n λ polarization unity vector f(θ,α, λq) anisotropic mag. sc. Resumé: Determination of magnetic structure by n-diffraction magnetic structure (=arrangement of magnetic moments) symmetry systematic extinction of some reflections { } = 0 nonprimitive magnetic cell + symmetry Determination of mutual orientation of magnetic moments Str. amplitude: Direction of M f(or. µ i w. respect to x- tal axes) nonsystematic extinction { } 0

8 5. Applications of neutron diffraction in solid state physics 1. Structure of materials comprising light elements (H, oxides, carbides) Scattering amplitude XRD: f Z XRD fails Scattering amplitude n: b determined by the nucleus inner structure determination of positions of light elements in the x-tal latice H: weak coherent n-scattering + strong incoherent scattering (spin incoherency) 2 H: stronger coherent und weaker incoherent sc. determination of positions of 2 H in x-tal Examples: hydrides alcali hydrides, UH, ZrH, carbides, oxides, Be Polycrystalline specimens positions of heavier metals from XRD, light metal positions - comparison of models with experiment 2. Structure of materials comprising neighbouring elements (Z i Z j ) b nonmonotonic function of Z Examples: Ordered structure superlattice reflections FeCo, CuZn, spinels (MgAl 2 O 4 ) + anorganic cyanides (positions of atoms of C and N) 3. Magnetic structures Atoms with noncompensated magnetic moments µ i exchange interactions between µ i T ord magnetic phase transition (Curie T fero, ferimagnetics; Neel T antiferomagnetics) T < T ord - feromagnetic state T > T ord - paramagentic state Paramagnetic state elastic coherent scattering on nuclei only Feromagentic state elastic coherent scattering both on nuclei and on ordered µ i Determination of magnetic structure - 4 cases: i) magnetic properties magnetic ordering 2i) type of magnetic ordering 3i) orientation of magnetic moments 4i) value of magnetic moments i) Magnetic properties Before n experiments a) classical magnetic measurements - χ = χ(t) T C and/or T N, x-tal phase transitions, molecular field, correlations between µ i, etc. - M=M(T, B) fero/ferimagnetic transition, M s, T C, magnetocrystal anisotropy b) nonmagnetic measurements: c P, DTA, Mössbauer spectroscopy, NMR, transport and other properties of magnetic state Ambiguous results separation of nuclear and magnetic contribution to coherent scattering other techniques

9 2i) Determination of the TYPE of magnetic ordering Coherent magnetic scattering magnetic ordering Positions of reflections type of ordering Simplification: nonpolarized n beam (nuclear and magnetic scattering independent and additive), no nonmagnetic elements (contribute to nuclear scattering only) a) Colinear feromagnetics = p. q hkl exp { } Magnetic structure amplitude p...scattered wave amplitude of m.s. q... interaction vector EC XTAL = EC MAG exp {} XTAL = exp {} MAG same laws of extinction neutron dif. pattern: magnetic reflections = nuclear reflections (no extra due to magnetic sc.) Example: Magnetic structure of Fe b) Colinear antiferomagnetics EC MAG = k. EC XTAL (k= 1,2,...) symmetry of x-tal structure Different systematic extinctions of magnetic and nuclear reflections extra reflections on neutron diffraction pattern Example: - AuMn

10 - Antiphase domains Cr x-tal structure: BCC Magnetic EC: extended x-tal EC +/- cubes phase/antiphase domains +/- orientation of µ of the basal atom (other µ i antiparallel) Satellite diffraction spots c) Noncolinear feromatgnetics (NCF) NCF = Antiferomagnetics collinear ferromagnetic components Antiferomagnetics general complicated structures Special cases: d) Noncolinear and modulated antiferomagnetics Spin configuration = FT of series expansion R nν = R n + ρ ν R n... position vector of the EC ρ ν... position vector of the atom in the EC ν...indices of atoms in EC Special case: simple spiral structure ± k 0 nonzero components satellite reflections K ± k 0 Coherent magnetic scattering for K k, K...vector of reciprocal lattice of x-tal structure k...wave vector for Q ν (k) 0 for some ν General: Each point K is surrounded by a certain no of satellites corresponding to individual waves k.

11 3i) Orientation of magnetic moments with respect to x-tallographic axes = 0 Structure amplitude of magnetic scattering i) Systematic extinction of diffraction peaks exp {} = 0 (due to nonprimitive magnetic EC + elements of symmetry) ii) Random (nonsystematic) extinction (exp {} 0) F=0 due to orientation of µ with respect to K and/or x-tallographic axes Determination of orientation of µ - detail analysis nonsystematic reflections i) µ (hkl) F=0 ii) µ (hkl) F= max.... Powder polycrystals complicated identification of orientation from neutron dif. pat. Special case: colinear magnetics configuration symmetry - µ - / +/- FCC symmetry orientation of µ cannot be determined Tetragonal, hexagonal, rhomboedric structure (= single axis symmetry) only the angle between µ and main magnetic structure axis may be determined Lower configuration symmetry (orthorombic, monoclinic, triclinic) orientation may be determined High symmetry structures: measurement in the magnetic field HoN: FCC structure (NaCl type) µ <100> i) H <100> µ rotates towards <100> I (200) ii) H <111> µ unchanged I konst. TbN: (inverse example) FCC structure µ <111> Noncolinear magnetic structures configuration symmetry cannot be introduced Modeling of magnetic structures and comparison with experiment

12 4i) Value of magnetic moments Ideal case: known positions, directions and orientation of µ easy determination of µ Practice: complicated evaluation due to restrictions of experimental techniques Reminder: Techniques allowing the separation of magnetic and nuclear contribution of elastic coherent scattering: a) neutron diffraction for T > T ord and T < T ord b) reduction of magnetic contribution by H c) polarization neutron diffraction d) separation of nuclear scattering by calculation Restrictions: a) influence of thermal expansion peak shift short ordering b) nonapplicable for fero and ferimagnetics strong magnetocrystal anisotropy nonapplicable for antiferomagnetics c) on single x-tals only due to problems with n depolarization d) detail x-tal structure must be known incl. positions of nonmagnetic atoms