Inelastic Scattering of Neutrons or X Rays by Phonons - Essay Example

?Inelastic Scattering of Neutrons or X rays by Phonons 519208) Introduction Phonons define the thermal and electrical properties of different materials. These are defined by the degrees of freedom that are imparted in their lattice vibrations. Inelastic scattering of x rays is used in the measure of phonons. (Sinn H et al, 1999) It has been found out that x-ray scattering (RIXS) can be effected by exciting phonons at the L edge and also combining excited electrons at the K-edge. Inelastic Scattering of neutrons The process of x ray scattering involves the double differential cross section process. The instrument used to measure this form of X ray scattering is called a TOF chopper spectrometers. Inelastic scattering essentially involves the interaction between neutron and phonons and the exchange of energy and momentum that takes place. The Inelastic Neutron Scattering can be represented by the following double differential equation ( Loong Chun, 2006) Here, the scattering function S (Q, E) is directly related to the space-time correlation functions that exist between particles or the space-time constraints of the particle itself. Coherent Inelastic Scattering This involves addition of both energy and momentum. A coherent scattering experiment provides details about the frequency and space configurations of phonons. The phonons can be quantified as the vibrations of ions in their respective positions of equilibrium. The experiment manages in measuring the static and dynamic factors of both the core electrons and the electrons located in the outer periphery. These x ray diffraction experiments provide information regarding the behaviour of quantum liquids. The speed of sound in water which is function of its temperature can be ascertained using this experiment. Further density variations of fluids with temperature changes and the corresponding arrangements of ions structurally also form part of the inelastic scattering process. (Loong Chun, 2006) Coherent x-ray scattering provides an output about the dynamic structure factor. The method of measuring phonon involves relating the momentum to the inter-atomic distance. This transferred energy due to the highly excited ions provides a measure of the energy of phonons. The phonon energy is measured in terms of momentum transferred. Further the Born-Von Karman analysis provides relationship between force constants and the dispersions. (Loong Chun, 2006) Dispersion curves thus plotted using this relation provide information regarding the phonon density. This is critical in measuring thermodynamic specifics like vibration entropy and specific heat of materials.( Loong Chun, 2006) The INS Instrument-Crystal Mono-chromators and Choppers The INS instrument is capable of performing very precise measurements and calculating S (Q, E) to its absolute units. It is very critical to calculate the total energy interaction that occurs when scattering takes place. An energy filter placed at a certain collimated solid angle is used to allow these low energy scattered neutrons to pass through it. (Veenendaal M van et al, 2010) This utilizes two methods to assess the process. One involves a direct geometry which utilizes a combination of fixed incident energy and variable scattered energies. The other uses inverse geometry combining variable incident energy and fixed scattered energy. The coherent inelastic x ray spectrometer uses a mono-chromator of high energy that is utilized as the source. The analyzer is formed by a curved Si crystal. A nested crystal that is formed by a combination of Si (4 4 0)-(15 11 3) forms the mono-chromator. A Diced Silicon crystal of bending radius 6m with a combination of (18 6 0) forms the analyzer. The energy resolution obtained in this exercise is 2meV while the momentum transferred is 5A-1. The energy of the photons that are used to impact initially is in the range of 21.657 keV. (Loong Chun, 2006) Source: Phonon dispersion measurements in Berrylium along (0 0 ?) direction for the longitudinal phonons for ? values indicated for each spectrum, 2010 The resonant inelastic x-ray scattering (RIXS) technique is experimented on the cupric oxide (CuO). The principle is as follows. (Loong Chun, 2006) The X rays manage to excite the electrons in the 1s level onto the valence band which are usually the 3d and 4p states. The creation of this impact excitation leads to an alteration of charge density which leads to a combination of electrons and phonons. A monochromatic beam of 22meV was sourced using the Si (4 4 4) mono-chromator. The measurement is conducted in ambient temperature under vacuum conditions to prevent scattering due to air. The spectrometer has an energy resolution of 38meV. The measurements are repeated a few times to rule out the possibility of any errors in the instrument. Photon energies that occupied quadruple transition from the 1s energy state to the 3d energy state and also from 1s energy state to 4p energy state were collected. The energy of incidence was between the extremes of 8981 and 8981 eV since the analyzer operates at a Bragg’s angle of 89.8?. The analyzer energy is modulated using the expansion of the crystal due to temperature rise which is kept between 100K and 500K. (S. Hosokawa et al, 2001) Inferences There is ample proof to show that the interaction occurs greatest at the zonal boundary. For the cuprates a scattering angle of 16.93? along with a momentum transfer of 13.4nm-1 defines this space. Several phonon energies that have been used in scattering have been plotted in the graph below using excitation energy in the 1s to 3d excitations. Source: Fig-2, Veenendaal M van et al, 2010 The graph depicts the inelastic x-ray scattering spectrum occurring at the edge of the copper K edge in CuO materials. These have been tabulated for four energies. The Gaussian expansion is greater for hw values of 8981 eV compared to hw values at 8997 eV. Fig b denotes the area where the spectrometer was used in the experiment and it shows a maximum intensity in terms of fluorescence at the Cu-K edge in Cuprous oxide crystals.( M. D’Astuto et al, 2003) At energy of 45meV a maximum intensity is noted with intensity decreasing on either side of this energy level. However, the energy loss is not completely symmetric about the zero loss peak value; the reason being attributed due to presence of resonance. The explanations that are provided for these inelastic features are as follows 1. One of the most critical factors for this inelastic behaviour is due to lattice vibrations. Therefore the change in spectral shape may be caused due to changes in phonon relative intensities. Energies of 24 meV, 41 meV and 70 meV are said to be the three phonon modes that are visible at this particular geometry. (S. Hosokawa et al, 2001) The interaction of electrons and phonons can be defined as where akm is phonon with mode m, energy hwkm and a wavevector k. When energy of the incident rays is absorbed a sudden excitation of phonons takes place. The graph indicates that scattering intensity is close to zero both in the initial low energy ground states and the final high energy state. Scattering amplitude can again be calculated as Where a short life span of the core hole only single phonon entities are excited. Comparing the ?/hw0=3, ?/hw0=5 with the ?/hw0=10, ?/hw0=20 we find the graph flattening out giving rise to single-photon excitations. The amplitude of these vibrations is found out using the above equation. The intensity of the phonon peaks is directly proportional to ?kmhwkm. (Forte F et al, 2007) This gives a measure of the bond between the phonons and density of charges in the base state. From the graph it can be noted that inelastic phonon crests shown degrees of resonance. Resonance or excitation in high amplitudes is generally noted in the region between 1s and 3d. Source: Fig-3, Veenendaal M van et al, 2010 The energy in this region is in the range of 8981eV. Beyond this range the intensity gradually drops and is characterised by localized excitations in the 4p band. Study into the graph provides detail knowledge about electron-phonon strengths. It has been researched and found out that ?km values for 24 meV and 70meV when compared to the 41meV values need to be provided with reduction factor of 0.7. (S. Raymond et al, 2002) The figure 3 above shows the inelastic scattering for a single phonon energy transfer of energy hw0=40meV. The coupling strength is in calculated in the range of 0.15 eV. The graph depicts the ?/ hw0 for values ranging from 3, 5, 10 and 20. Applications Inelastic scattering of neutrons find varied applications in ascertaining the lattice dynamics of different material density and also in assessing the density structure of materials at different states. It is also used in calculating energy bases of rare earths by the process of magnetic scattering. (S. Hosokawa et al, 2001) Some of the materials that are used to assess these properties using x ray scattering include spinels, hydroxyapatite which are nanostructured bone minerals and xenotime (RPO4). These materials have high melting point greater than 2000?C and are highly resistant to radiation attacks. These components can be effectively used for storing nuclear waste. Having a Mohr hardness of 5.5 these are used extensively in lasers and scintillators. (Lorenzen et al, 2002) Other properties of this material include Antiferromagnetic phase changes, magnetoelastic changes and spin lattice bonding. These properties find use in sensors of different types. The inelastic neutron scattering can therefore be applied for the calculation of magnetic properties. These poperties include susceptibility and specific heat. Conclusion The inelastic scattering of x rays using phonons throws up various new possibilities of research. The bonding between phonons and the difference in charge it creates due to this inelastic scattering contributes to the study process. This is also finding increased research in the field of high temperature superconductors where coupling strengths of different phonons provide information regarding the rate and property of superconductivity. (G. Fiquet et al, 2004) The Inelastic Neutron Scattering is a very useful device to chart the atomic movements and its electronic excitations within a closed system. These are specially used in assessing the magnetic motions of different minerals. These tests can be carried out in highly critical environments where temperature and pressure are at extremes. This technique provides a good scope for measuring these excitations in extended time and spaces. A good resolution is achieved for various ranges of Energy and Spatial vectors. Time scales may range from ps to ms. The critical part of this is however not in the operation of the Spectrometers but in interpreting the data that is collected and deriving intelligent inferences from the same. Reference List 1. Sinn H et al, 1999, Inelastic Scattering of Synchrotron Radiation from Electrons and Nuclei for Lattice Dynamic Studies, Advanced Photon Source, Argonne national Laboratory. 2. Veenendaal M van et al, 2010, Observation of phonons with resonant inelastic x-ray scattering, Journal of Physics: Condensed Matter. 3. Kirov A et al, n.d, The program Neutron: Phonon Selection Rules for neutron and x-ray scattering, University del Pais Vasco 4. Loong Chun, 2006, Inelastic Neutron Scattering and Applications, Reviews in Mineralogy and Geochemistry. 5. Forte F et al, 2007, Physics Rev B 6. G. Fiquet et al, 2004, Application of inelastic X-ray scattering to the measurements of acoustic wave velocities in geophysical materials at very high pressure – Phys. Earth and Planetary Interiors 7. M. D’Astuto et al, 2003, Electron-phonon interaction in N-doped cuprates: An inelastic X-ray scattering study – Inter. J. of Modern Phys. 8. S. Hosokawa et al, 2003, Phonon dynamics of liquid Si-inelastic X-ray scattering studies. 9. M. Krisch, 2003, Status of phonon studies at high pressure by inelastic X-ray scattering - J. Raman spectrosc. 10. Lorenzen et al, 2002, Determination of phonon dispersion curves at gigapascal pressures by inelastic x-ray scattering - High Pressure Research. 11. S. Raymond et al, 2002, Phonon anomalies at the valence transition of SmS: An inelastic X-ray-scattering study under pressure. 12. S. Hosokawa et al, 2001, Inelastic X-ray scattering study on the dynamics of liquid Ge. Read More