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Lecture 2

Lecture 2. Cross section Conservation laws Detector principles Spectrum Signal to noise resolution. Cross section. The concept of a cross section originates in a geometrical consideration, which here is illustrated using an example from Nuclear physics:

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Lecture 2

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  1. Lecture 2 • Cross section • Conservation laws • Detector principles • Spectrum • Signal to noise • resolution

  2. Cross section The concept of a cross section originates in a geometrical consideration, which here is illustrated using an example from Nuclear physics: Assume that two 12C nuclei collide, and consider the process in a coordinate frame in which one of the nuclei is at rest. Then the geometric cross section corresponds to the area around the nucleus at rest which, if hit by the other nucleus, defines that a collision takes place. The distance between the two nuclear centres is called the impact parameter b. In the drawing the impact parameter is 2 times the radius of 12C, and at this impact parameter a peripheral collision would take place. A fully head-on collision has b=0. at rest

  3. Cross section - definition Assume that one particle/quantum a passes through a material which contains one particle A per surface area Y.Let P be the probability that this causes a particular reaction. Then the cross section s for that particular reaction is defined via: The standard unit for cross section is barn (b), (1 mb), 1 barn = 1 b = 10-28 m2 = 10-24 cm2 i.e. the dimension of an area

  4. n d Y Cross section A thin foil (thickness d, area Y) of particle A is irradiated by n particles/time. What is the reaction rate if the cross section is s?

  5. n d Y Cross section Let N be number of A-particles/volume n the number of incident a-particles volume = Yd, Total no of A-particles = NYd Probability for reaction with one a-particle and total no of reactions/unit time

  6. Detector front area defines ΔΩ Φ,azimuthal angle Θ, scattering angle Np NT Differential cross section Polar coordinates, r,Θ,φ ND 0 ° NP; number of projectile particles per second NT ; number of target particles per cm2 ND ; number of detected particles per second The result, ND, depends on NP,NT and ΔΩ. Not good for reproducibility Cross section (σ) normalizes away experiment specific parameters so you get the absolute probability for a given result σ = ND/ (NP ·NT) which has dimension area i.e unit cm2 or the more useful unit for nuclear dimensions, barn(1barn=10-24cm2)

  7. Differential cross section σ still depends on the solid angle of the detector. This is normalized away in the differential cross section: dσ/dΩ = ND/ΔΩ·NP·NT (unit: barn/steradian) or if differentiated both in angle and energy the doubly differential cross section d2σ/dΩdE = ND/ΔΩ·ΔE·NP·NT (unit: barn/steradian/eV). A result is often an energy distribution measured in a given angle. It should normally be expressed by this doubly differential cross section

  8. Differential cross section • NP can be obtained from the beam current (if picoamperes or larger). • If too low (<106 particles per sec), the particles can be counted directly with a detector in the beam. A monitor reaction with known cross section can be used to determine the product NP·NT. Elastic scattering is often used as monitor. • NT can be determined by measuring the thickness of the sample and using the density to calculate the number of nuclei per cm2. More convenient is to measure the area of the target foil and measure its weight. The unit gram/cm2 is a commonly used unit for thickness.

  9. Different types of partial cross sections • Elastic scattering: Kinetic energy conserved cross section: ss,el, example: d + 39K –> 39K + d • Inelastic scattering: Kinetic energy not conserved(excitation energy) cross section: ss,inel, example: d + 39K –> 39K* + d • Absorption reaction: cross section: sa, example: d + 39K –> B + b d ≠ b • Reaction cross section: sr = sa + ss,inel

  10. Material: carbon Total photon cross sections scoh: total photon cross section t: atomic photo-effect scoh: coherent scattering (Rayleigh) sincoh: incoherent scattering (Compton) Kn: pair production, nuclear field Ke: pair production, electron field sph: photonuclear absorption From: Thompson and Vaughan (Eds.), X-ray Data Booklet, 2nd edition, Lawrence Berkely National Laboratory 2001 Available from http://xdb.lbl.gov

  11. Photon processes Rayleigh scattering Photoelectric effect Compton scattering From: Moroi, Phys. Rev. 123, 167 (1961) From: Bjorken and Drell, Relativstic Quantum Mechanics, Mc Graw-Hill, New York 1964 From: Sakurai, Advanved Quantum Mechanics, Addison- Wesley, Reading 1967 Ejection of an atomic electron by the absorption of a photon Elastic scattering of photons by atoms Scattering of photons by free (or quasi-free) electrons

  12. Photon processes Pair production (in nuclear field) Pair production (in electron field) Photonuclear absorption From: Bjorken and Drell, Relativstic Quantum Mechanics, Mc Graw-Hill, New York 1964 Production of electron/positron pair on the field of a nucleus or an electron Absorption of a photon by a nucleus

  13. Attenuation A beam of particles that passes through a thick target is attenuated (intensity is degraded). The strength of attenuation depends on all processes possible for the beam, i,e, the sum of all different cross sections. stotal = sreaction + selastic

  14. Attenuation The change Dn of the number of particles in the beam in a segment Dx will be: Let Dx -> dx : With the solution:

  15. Attenuation length • sN has dimension 1/length • sNx = x/l l is attenuation length

  16. OUT e- IN hn(mono-energetic) Sample Photoemission Principle Schematic experiment From Energy Conservation (Esample is the total energy of the sample before and after the electron is emitted) hn + Esample(before) = Esample(after) + Ekin(e-) i.e. a Binding Energy EB (or if you like, BE) can be defined EB = hn - Ekin(e-) = Esample(after) - Esample(before) To beam line For the outgoing electrons we measure the number of electrons versus their kinetic energy. In addition the direction of the electrons may be detected (and in some cases their spin). NOTE the direction of the Binding Energy (BE) scale 

  17. Why do we see a clear signal from the surface layer in photoemission ? Bulk signal Attenuation length of soft X-ray photons in solids is of the order of 1000 Å. Is it reasonable that we see a clear signal from the surface atoms when the attenuation length of the exciting radiation is much larger than the distances between layers? Surface signal (The first atomic layer) Photon Energy

  18. Conservation laws: • Energy • Momentum • Angular momentum • Charge • Other quantum numbers

  19. Coulomb scattering • Coulomb interaction - electromagnetic force between projectile and target. • Normally the interaction is elastic, but both Coulomb excitation and disintegration can happen. In elastic Coulomb scatteringthe particle trajectories are bent in the Coulomb field.

  20. Elastic Coulomb scattering • Rutherfords formula • From classical conservation laws Rutherfords famous formula • dependence • dependence • dependence

  21. Rutherfords formula

  22. Shadowing cone

  23. The ideal detector • Sensitivity for radiation • Cross section • Size (mass) • Transparency • Response • Energy-signal • Linearity • Response function for radiation • Time • Pile-up, dead time • Resolution • Fwhm

  24. Resolution of spectral features Separation < FWHM From: Gerthsen & Vogel, Physik, 17th edition, Springer-Verlag, Berlin and Heidelberg 1993

  25. Improving the resolution is better! (but not always possible!) Modelling & curve fitting can ”increase” the resolving power to a certain extent: From: Beutler et al., Surf. Sci. 396 (1998) 117 Smedh et al., Surf. Sci. 491 (2001) 99

  26. From: Gerthsen & Vogel, Physik, 17th edition, Springer-Verlag, Berlin and Heidelberg 1993 Physical limits to the resolving power of an instrument Lens Optical microscope Minimum distance that can be resolved: dmin = 1.22 l / (2n sin a) ≈ 200 nm for optical microscopy l: wavelength of light (min. 450 nm) n: refractive index of light (often 1.56) a: collecting angle q ≈ sin q = 1.22 l/D Rayleigh criterion

  27. IS IS SN = Is / In = = ( IS + Ib )1/2 1 + 1/SB ( ) 1/2 Signal-to-noise and Signal-to-background ratios SB = Is / Ib From: J. Stöhr, NEXAFS Spectroscopy, Springer-Verlag, Berlin and Heidelberg 1992 N . From statistics: for large N the noise scales like

  28. What to optimise - SB or SN? Also depends on what you can optimise! Noise: statistical phenomenon Background: physical phenomenon! Choice of sample Choice of geometry When all external (systematic) noise has been removed the only way left is to increase the number of counts! Choice of method Counting time Choice of method

  29. From: J. Stöhr, NEXAFS Spectroscopy, Springer-Verlag, Berlin and Heidelberg 1992 Example: X-ray absorption measured using different detection methods Low count rate Good SB Intermediate count rate Bad SB High count rate Bad SB Method of choice!

  30. Unproblematic backgrounds ...

  31. ... and problematic backgrounds: background = background(x)!

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