A. Magerl Crystallography and Structural Physics University of Erlangen-Nürnberg Neutrons versus X-rays Novosibirsk 10. Oct. 2011

Content What is it all about? YBa 2 Cu 3 O 7 High temperature superconductor Information on an atomic/molecular scale: From structure and dynamics to functionality Y 3+ Ba 2+ O 2- Cu

Why diffraction?Interference Let us do diffraction. We shine waves onto our sample and these are scattered from each atom, i. e. each atom becomes a little (radio-) emitter. These waves are coherent and they interfere with each other. The resulting interference pattern is representative for the arrangement of the atoms (phase relations between the waves) and for the kind of atoms present (amplitudes of the waves).

elastic inelastic quasielastic I   I     I    I   scattering processes

Why diffraction?1 Å radiation For particles with mass: E kin = ½ mv 2 p = 2E kin / v = mv For particles without mass: (photons) E = hν = ћω, p = hν/c = h/λ = ћk In the following we will concentrate on X-rays and neutrons photons neutrons

Why diffraction?Wave-particle dualismus Waves and particles are two representations of the same ‘thing’. Sometimes, ‘it’ appears quantized (photoelectric effect or counting in a detector), sometimes ‘it’ appears as a wave (interference). The two representations are connected by the de Broglie equation (de Broglie 1924), which relates the momentum p (particle property) with the wavelength λ (wave property) p λ = horp = ћ k h = Planck‘s constant = 6,63 * Js = 4.14 * eVs Note, it is important and elegant to chose the appropriate physics picture when describing specific phenomena in natural science.

Why diffraction?X-rays The visible light and X-rays are just small fractions in the very wide frequency range of the electromagnetic spectrum λ [Å] = 12,4 / E [keV] Photons of 12,4 keV are needed to make light with λ = 1 Å. For comparison, visible light has an energy of 4 – 7 eV (radiation damage).

Since 40 years the increase in brilliance is 0.3% per day The development of the brilliance of x-ray sources

Why diffraction?The neutron case

Why diffraction?Interaction with matter Absorption: The incoming radiation vanishes, the energy becomes removed from the wave field, the material is damaged or warms up. Scattering: A new wave field is created without a change of the energy contained in all wave fields (at least for elastic scattering). In general the energy in the direction of the incoming wave becomes reduced while an energy flow in new direction(s) builds up. X-rays: σ tot = σ photoabs + σ Compton + σ pair + σ scattering The dominant interaction is through σ photoabs. It depends on the photon energy and on the element (characteristic absorption edges). The penetration of 1-Å x-rays into matter is limited by absorption, samples have to be small, i. g. < 1mm. Neutrons: Absorption and scattering are similar (with exceptions); The scattering amplitude does not depend on the energy. The penetration into matter is i. g. higher than for X-rays σ tot = σ nuclearabs + σ scattering

Why diffraction?The scattering process Born approximation: An incident plane wave with amplitude A = 1 and wavenumber k 0 is partially scattered by an electron (nucleus, etc.) into direction k 1 with |k 0 | = |k 1 |. f(Ω) is called scattering length. It describes the amplitude of the scattered wave and it contains all the physics. E. g. if f(Ω) is big the interaction is strong. Note, an interaction in Fourier space is described as scattering. The scattering length f(Ω) = f(δ,φ) is the Fourier transform of the interaction potential. The amplitude factor A is needed for energy normalisation. We will set A=1 at present. with

Why diffraction?Amplitude of scattered wave X-rays: Thomson scattering: Every accelerated charge emits elmag. radiation. Electrons are (loosely) bound with a binding frequency f « x-ray frequency ω 0. The electric field of the x-ray wave induces oscillations of the electrons (the motion of the nucleus is neglected) which results in dipole radiation. This can be calculated exactly within electrodynamics: = [m] is the classical electron radius. for non-polarized light: Nota: r e « 1 Å, i. e. we have a weak scattering process (Born approximation). Scattering length Neutrons: No simple dependence on the atomic number, on mass, etc.; measure it!

Scattering cross sections Neutrons X-rays (sin θ)/λ = 0 (sin θ)/λ = 0,5 Å -1 Atomic weight

Scattering cross sections

Why diffraction?Beyond Thomson The resonance behavior of a classical pendulum: X-ray region, r e = constant Absorption region with dispersion corrections f 1 and f 2 Visible region, why is the sky blue?

from Anomalous dispersion makes specific atoms particularly visible The scattering length: f(Q) = f 0 (Q) + f 1 (E) + if 2 (E) Cu: Z=29 Ionisation energy = 8920 eV Beyond the Thomson approximation

Interference of first order: The interference pattern of the electrons within one atom is called the form factor f(Q). It is the Fourier transform of the electron density distribution in an atom ρ(r): For simplicity we assume a spherical symmetric charge distribution f(Q) describes the amplitude of the scattered wave at the wave vector transfer Q. The form factor can be given as a function of Q or of (sinθ/λ). Full interference is always obtained for Q=0. f(Q=0) equals the number of electrons of the atom, each single electron scatters with the Thomson cross section. The (atomic) form factor f(Q)

Atomic form factor for F (9 electrons), Ne (10 electrons) and Na (11 electrons), and for Na + and F - (both 10 elektrons). In addition, the thermal factor is indicated for F. Form factor f(Q) K 1+ Formfaktor f für Einzelelektronen und Gesamtformfaktor 3s 2s 1s Form factor of single electrons and total form factor Ne Na 1+ F 1- Ne Na

Ensemble of atoms - unit cell Interference of second order: adding up the waves originating from a finite particle ensemble (unit cell of a crystal): structure factor (of the unit cell) F(Q): for x-rays: In crystals, the arrangement of the J atoms is named the motive (decoration) of the unit cell, and F(Q) represents the FT of the charge density in the unit cell. with the fractional coordinates a b 2 r 1 r for x-rays In principle, this is all we want to know yjyj xjxj

Single crystal Interference of third order: adding up the waves originating from a crystalline structure, i. e. of a periodic arrangement of unit cells, with translation vectors a, b, and c. This is called the lattice sum. The phase is a b 2 r 1 r a n = 0 b 2 r 1 r a b n = 1 2 r 1 r a n = 0 b n = 2 2 r 1 r a n = 1 b n = 0 2 r 1 r a n = 1 b 2 r 1 r a b n = 2

Bragg peaks In 1 dimension: Geometrical sum with solution

The squared lattice sum has within one period: 1 main maximum & (N-2) secondary maxima & (N-1) zero points All maxima have the same height The width of the main maximum is 2/N, for secondary maxima 1/N. For larger crystals the width becomes rapidly very small (Bragg peaks). Nanomaterials may retain a finite width. The lattice sum concentrates the diffracted intensity into a few spots in reciprocal space (Bragg peaks), which can be measured with sufficient intensity. The lattice sum has finite values only if (Laue equ.): Qa=n2π & Qb=n2π & Qc=n2π Discussion lattice sum (sinNx / sinx) 2 phase x=Qr [rad]

Diffraction pattern The scattering pattern is the product of: the lattice sum (the lattice determines the position of the Bragg peaks) and the structure factor (the motive determines the intensity in the Bragg peaks) for x-rays

Form factor for atoms NeutronsX-rays Neutrons see the nucleusX-rays see the electrons The scattering lengths are independent of Q and ħω (not for magnetic scattering), but depend on the isotope (incoherent scattering) The scattering lengths vary with Q and ħω, but are independent of the isotope Neutrons see easily the light elements and are ‘naturally’ sensitive to magnetism (Very powerful developments of new sources) σ x-rays ≈ σ neutrons

Ensemble of atoms - unit cell Interference of second order: adding up the waves originating from a finite particle ensemble (unit cell of a crystal): structure factor (of the unit cell) F(Q): and for neutrons:for x-rays: In single crystals, the arrangement of the J atoms is named the motive (decoration) of the unit cell, and F(Q) represents for x-rays the FT of the charge density, for neutron the FT of the nuclear potentials with the fractional coordinates a b 2 r 1 r d for neutrons 2π/d2π/d for x-rays 2π/d2π/d This is all we want to know