Chapter 4 Inelastic Neutron Scattering of Phonons Chapter 4 Inelastic Neutron Scattering of Phonons Phonon dispersion curves � Phonon dispersion curves are usually measured using inelastic neutron sca[.]
Chapter 4: Inelastic Neutron Scattering of Phonons Phonon dispersion curves • Phonon dispersion curves are usually measured using inelastic neutron scattering The energies of neutrons at root temperature are about the right value for the average phonon (~ meV) • Dispersion relations can be measured in different directions in the crystal (so we are really measuring ω(K)) • Last time, we learned that the following relationship must be held with scattering: k + G = k’ +/- K Incident neutron Reciprocal lattice vector Scattered neutron Phonon wavevector (+ for phonon created, - for phonon absorbed) (Cold, thermal, and hot refer to the temp of the moderator, which can be cooled by cryogenic liquids or heated up high T (eg ~ 600 K)) Inelastic neutron scattering Triple-axis neutron spectrometer K k Defines incoming wavelength G k’ Defines outgoing wavelength, which may be different if the neutron lost a bit of energy to make a phonon Triple Axis Spectrometer Geometry of experiments • In reciprocal space, the geometry of the experiments looks a bit different • For example: for measuring the dispersion along the (100) direction, we can this in a few different ways Here is one way: Phonon created: k + G = k’ + K So k – k’ = G + K (but you can measure this in different ways by varying G The energy transfer is from changing the k and k’ vectors) k’ k (010) K (110) (210) G (000) (100) (200) Inelastic neutron scattering • Using the conservation of energy, we can define the energy of the phonon created by: Mass of neutron h 2k h k '2 = ± hω 2 2M n 2M n Incoming neutron energy • Phonon energy (sign changes if phonon is created or destroyed) Scattered neutron energy So, you can measure the incoming neutron’s wavevector and energy, and the outgoing neutron’s wavevector and energy, and then solve for the the phonon’s energy The phonon’s wavevector is solved for using: k + G = k’ +/- K Phonon dispersion curves La La La La • • • • • Phonon dispersion relations can get very complicated! Eg La2CuO4 (material used to make superconductors) The structure is tetragonal (a = b ≠ c, all angles 90 degrees) There are two important directions: the ab-plane, and the c-direction We would expect the phonons to act differently (have different dispersion curves) in different directions La La La La La La La La La La La La La La La La La c La La La La La La La La La La b La La La La a La La La La Blue Cu octahedra La La La La La Phonon dispersion relations • • • • This is the dispersion relationship in different directions, as measured by neutron scattering Why is this important? Phonons help some materials become superconducting – the lattice distortions play a role in making the electrons stick together to form Cooper pairs (which conduct electricity with zero resistance) We have to understand how phonons work to figure out how materials superconduct ab-plane ab-plane c-direction BCS Superconductivity (Bardeen, Cooper, and Schrieffer, 1957, Nobel prize) • • • • • Normally, electrons repel one another But in a superconductor, they ‘pair up’ to form Cooper Pairs In BCS superconductivity, the mechanism for this is phonons Lattice distortions create temporary clusters of positive charge, which attract electrons and enables them to pair up Many elements (eg Ti, V, etc.) are superconducting at low temperatures (< K) The high-TC superconductors, on the other hand, have much higher TCs (up to 100 K or so) Do lattice vibrations still play a role? Cooper Pair of electrons e What else phonons tell us? • • • • If a crystal structure changes, then we often see something happening to the lattice vibrations first (the atoms want to move in a slightly different way) This is called mode softening Eg : Nb3Sn has a phase transition from a cubic to a tetragonal structure at 43 K (the c-axis shrinks, and ab-plane stretches out a bit) We see something happening to the phonon dispersion relation at temperatures much higher than this transition (from x-ray diffraction) Mode softening • • • • These are phonon measurements in the ab-plane (inelastic neutron scattering) Notice how you need less and less energy to get the atoms to move in the ab-plane as you approach the transition temp This is an example of mode softening At 46 K, the mode almost disappears at low enough energies (excitations in the ab-plane) (K is called q here) (Zone Boundary) Spin waves (magnons) • Because neutrons have spin, they can also cause spin-wave excitations in ferromagnets and antiferromagnets These are called magnons An ordered ferromagnet (aligned spins at low temperatures) A spin wave (spin excitation) (Top down view) Spin waves • This is an example of a ferromagnet, EuO, where all the Eu spins align below 69 K • Neutrons can measure the dispersion relations of spin excitations • Notice how it looks like an acoustic phonon (spins moving in phase) Inelastic Neutron Scattering This is an experiment done on CuGeO3, which has 1D chains of ordered spins Low Temp (below transition to ordered state) (same as K) High Temp Notice how it looks like a sin curve: the theory for spin waves is similar to lattice vibrations