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MEASUREMENT OF ATOMIC MOMENTUM DISTRIBUTIONS BY HIGH ENERGY NEUTRON SCATTERING J Mayers (ISIS)

MEASUREMENT OF ATOMIC MOMENTUM DISTRIBUTIONS BY HIGH ENERGY NEUTRON SCATTERING J Mayers (ISIS) Lectures 1 and 2. How n(p) is measured The Impulse Approximation . Why high energy neutron scattering measures the momentum distribution n(p) of atoms. The VESUVIO instrument.

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MEASUREMENT OF ATOMIC MOMENTUM DISTRIBUTIONS BY HIGH ENERGY NEUTRON SCATTERING J Mayers (ISIS)

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  1. MEASUREMENT OF ATOMIC MOMENTUM DISTRIBUTIONS BY HIGH ENERGY NEUTRON SCATTERING J Mayers (ISIS) Lectures 1 and 2. How n(p) is measured The Impulse Approximation. Why high energy neutron scattering measures the momentum distribution n(p) of atoms. The VESUVIO instrument. Time of flight measurements. Differencing methods to determine neutron energy and momentum transfers Data correction; background, multiple scattering Fitting data to obtain sample composition, atomic kinetic energies and momentum distributions. Lectures 1 and 2. Why n(p) is measured What we can we learn from measurements of n(p) (i) Lecture 3 n(p) in the presence of Bose-Einstein condensation. (ii)Lecture 4 Examples of measurements on protons.

  2. Initial Kinetic Energy Final Kinetic Energy Momentum transfer Energy transfer • The “Impulse Approximation” states that at sufficiently high incident • neutron energy. • The neutron scatters from single atoms. • Kinetic energy and momentum are conserved in the collision. Gives momentum component along q

  3. The Impulse Approximation Kinetic energy and momentum are conserved

  4. . . . . . . ±Δr Why is scattering from a single atom? If q >> 1/Δr interference effects between different atoms average to zero. Incoherent approximation is good for q such that; Liquids S(q) ~1 q >~10Å-1 Crystalline solids – q such that Debye Waller factor ~0.

  5. Why is the incoherent S(q,ω) related to n(p)? Single particle In a potential Ef = Final energy of particle q = wave vector transfer E = Initial energy of particle ω= energy transfer

  6. IA assumes final state of the struck atoms is a plane wave.

  7. momentum distribution

  8. p in Å-1 throughout - multiply by ħ to get momentum

  9. Final state is plane wave Impulse Approximation q→∞ gives identical expressions Difference due to “Initial State Effects” Neglect of potential energy in initial state. Neglect of quantum nature of initial state.

  10. Infinite Square well

  11. T=0 ER=q2/(2MED)

  12. T=TD ER=q2/(2MED)

  13. All deviations from IA are known as Final State Effects in the literature. Can be shown that (V. F. Sears Phys. Rev. B. 30, 44 (1984). Thus FSE give further information on binding potential (but difficult to measure)

  14. Increasing q,ω Density of States

  15. FSE in Pyrolytic Graphite A L Fielding,, J Mayers and D N Timms Europhys Lett 44 255 (1998) Mean width of n(p)

  16. FSE in ZrH2 q=40.8 Å-1 q=91.2 Å-1

  17. Measurements of momentum distributions of atoms Need q >> rms p For protons rms value of p is 3-5 Å-1 q > 50 Å-1, ω > ~20 eV required Only possible at pulsed sources such as ISIS UK, SNS USA Short pulses ~1μsec at eV energies allow accurate measurement of energy and momentum transfers at eV energies.

  18. Lecture 2 • How measurements are performed

  19. Time of flight measurements Sample L0 θ v0 Source L1 v1 Detector

  20. Fixed v0 (incident v) (Direct Geometry) Fixed v1 (final v) (Inverse Geometry) Time of flight neutron measurements Wave vector transfer Energy transfer

  21. The VESUVIO Inverse Geometry Instrument

  22. VESUVIO INSTRUMENT

  23. Foil cycling method Foil out Foil in Difference E M Schoonveld, J. Mayers et al Rev. Sci. Inst. 77 95103 (2006)

  24. Cout=I0 A Foil out Cin=I0 (1-A)A Foil in C=Cout-Cin= I0 [ 1-A2]

  25. 6Li detector gold foil “in” Filter Difference Method Cts = Foil out – foil in

  26. Blue = intrinsic width of lead peak Black = measurement using Filter difference method Red = foil cycling method

  27. Foil cycling Filter difference

  28. YAP detectors give Smaller resolution width Better resolution peak shape 100 times less counts on filter in and filter out measurements Thus less detector saturation at short times Similar count rates in the differenced spectra Larger differences between foil in and foil out measurements therefore more stability over time.

  29. Comparison of chopper and resonance filter spectrometers at eV energies C Stock, R A Cowley, J W Taylor and S. M. Bennington Phys Rev B81, 024303 (2010)

  30. Θ =~160º Θ =62.5º

  31. Secondary gold foil "“out” YAP detector PrimaryGold foil Secondary gold foil “in” Gamma background Pb old Pb new

  32. Secondary gold foil "“out” YAP detector PrimaryGold foil Secondary gold foil “in” ZrH2 old ZrH2 new

  33. Need detectors on rings Rotate secondary foils keeping the foil scattering angle constant Should almost eliminate gamma background effects

  34. corrections for gamma Background Pb

  35. corrections for gamma Background ZrH2

  36. ZrH2 p2n(p)without a background correction with a background correction

  37. Multiple Scattering J. Mayers, A.L. Fielding and R. Senesi, Nucl. Inst. Methods A481, 454 (2002) Total scattering Multiple scattering

  38. Multiple Scattering Back scattering ZrH2 A=0.048. A=0.092, A=0.179,A=0.256. Forward scattering ZrH2

  39. Forward scattering Back scattering

  40. Correction for Gamma Background and Multiple Scattering Automated procedure. Requires; Sample+can transmission Atomic Masses in sample + container Correction determined by measured data 30 second input from user Correction procedure runs in ~10 minutes

  41. Uncorrected Corrected

  42. Data Analysis Impulse Approximation implies kinetic energy and momentum are conserved in the collision between a neutron and a single atom. Initial Kinetic Energy Final Kinetic Energy Momentum transfer Energy transfer Momentum along q

  43. Y scaling In the IA q and ω are no longer independent variables Any scan in q,ω space which crosses the line ω=q2/(2M) gives the same information in isotropic samples Detectors at all angles give the same information for isotropic samples

  44. Data Analysis

  45. Strictly valid only if • Atom is bound by harmonic forces • Local potential is isotropic Spectroscopy shows that both assumptions are well satisfied in ZrH2 Spectroscopy implies that wH is 4.16 ± 0.02 Å-1 VESUVIO measurements give

  46. wtd mean width= 4.141140 +- 7.7802450E-03 mean width = 4.134780 st dev= 9.9052470E-03

  47. ZrH2 Calibrations Expected ratio for ZrH1.98 is 1.98 x 81.67/6.56 =24.65 Mean value measured is 21.5 ± 0.2 Intensity shortfall in H peak of 12.7 ± 0.8%

  48. Momentum Distribution of proton

  49. Sum of 48 detectors at forward angles y in Å-1

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