The concept of a cross section originates in a geometrical consideration, which here is illustrated using an example from Nuclear physics: Assume that two 12 C 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 12 C, and at this impact parameter a peripheral collision would take place. A fully head-on collision has b=0. Cross section at rest
Cross section - definition barn (b), The standard unit for cross section is barn (b), (1 mb), 1 barn = 1 b = 10 -28 m 2 = 10 -24 cm 2 i.e. the dimension of an area 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 for that particular reaction is defined via:
Cross section Y d n 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 ?
Cross section Probability for reaction with one a-particle and total no of reactions/unit time Y d n Let N be number of A-particles/volume n the number of incident a-particles volume = Yd, Total no of A-particles = NYd
Polar coordinates, r,Θ,φ 0 ° Θ, scattering angle Φ,azimuthal angle Detector front area defines ΔΩ NpNp NTNT NDND N P ; number of projectile particles per second N T ; number of target particles per cm 2 N D ; number of detected particles per second The result, N D, depends on N P,N T 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 = N D / (N P ·N T ) which has dimension area i.e unit cm 2 or the more useful unit for nuclear dimensions, barn (1barn=10 -24 cm 2 ) Differential cross section
σ still depends on the solid angle of the detector. This is normalized away in the differential cross section: dσ/dΩ = N D /ΔΩ·N P ·N T (unit: barn/steradian) or if differentiated both in angle and energy the doubly differential cross section d 2 σ/dΩdE = N D /ΔΩ·ΔE·N P ·N T (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 Differential cross section
N P can be obtained from the beam current (if picoamperes or larger). If too low (<10 6 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 N P ·N T. Elastic scattering is often used as monitor. N T can be determined by measuring the thickness of the sample and using the density to calculate the number of nuclei per cm 2. More convenient is to measure the area of the target foil and measure its weight. The unit gram/cm 2 is a commonly used unit for thickness.
Different types of partial cross sections Elastic scattering: Kinetic energy conserved cross section: s,el, example: d + 39 K –> 39 K + d Inelastic scattering: Kinetic energy not conserved(excitation energy) cross section: s,inel, example: d + 39 K –> 39 K* + d Absorption reaction: cross section: a, example: d + 39 K –> B + b d ≠ b Reaction cross section: r = a + s,inel
Total photon cross sections Material: carbon coh : total photon cross section : atomic photo-effect coh : coherent scattering (Rayleigh) incoh : incoherent scattering (Compton) n : pair production, nuclear field e : pair production, electron field ph : photonuclear absorption From: Thompson and Vaughan (Eds.), X-ray Data Booklet, 2nd edition, Lawrence Berkely National Laboratory 2001 Available from http://xdb.lbl.gov
From: Sakurai, Advanved Quantum Mechanics, Addison- Wesley, Reading 1967 Rayleigh scattering Elastic scattering of photons by atoms From: Moroi, Phys. Rev. 123, 167 (1961) Photoelectric effect Ejection of an atomic electron by the absorption of a photon From: Bjorken and Drell, Relativstic Quantum Mechanics, Mc Graw-Hill, New York 1964 Compton scattering Scattering of photons by free (or quasi-free) electrons Photon processes
Pair production (in nuclear field) 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 Pair production (in electron field) Photonuclear absorption Absorption of a photon by a nucleus Photon processes
Attenuation total = reaction + elastic 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.
Attenuation With the solution: The change n of the number of particles in the beam in a segment x will be: Let x -> dx :
Attenuation length N has dimension 1/length Nx = x/ is attenuation length
Photoemission Principle IN h (mono-energetic) OUT e - Sample Schematic experiment 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 From Energy Conservation (E sample is the total energy of the sample before and after the electron is emitted) h + E sample (before) = E sample (after) + E kin (e - ) i.e. a Binding Energy E B (or if you like, BE) can be defined E B = h E kin (e - ) = E sample (after) - E sample (before) To beam line
Why do we see a clear signal from the surface layer in photoemission ? Photon Energy Surface signal (The first atomic layer) 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?
Conservation laws: Energy Momentum Angular momentum Charge Other quantum numbers
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 scattering the particle trajectories are bent in the Coulomb field.
Elastic Coulomb scattering Rutherfords formula –From classical conservation laws Rutherfords famous formula dependence
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
Detectors: Measuring ”low” energy electrons, ions, and photons Energy measurement: Sample Energy selective element (monochromator) Electron / ion / photon counting element Momentum measurement: Energy and momentum measurement: and often variable
From: Channeltron Handbook, Burle Technologies Inc., www.burle.com Electron detection From: Sevier, Low Energy Electron Spectrometry, John Wiley & Sons, New York 1972
Photon detection From: J. Stöhr, NEXAFS Spectroscopy, Springer-Verlag, Berlin and Heidelberg 1992 Depending on wavelength, for example: Prisms Fabry-Pérot interferometers Gratings Crystals
Resolution of spectral features From: Gerthsen & Vogel, Physik, 17th edition, Springer-Verlag, Berlin and Heidelberg 1993 Separation < FWHM
Modelling & curve fitting can ”increase” the resolving power to a certain extent: Improving the resolution is better! (but not always possible!) From: Beutler et al., Surf. Sci. 396 (1998) 117 Smedh et al., Surf. Sci. 491 (2001) 99
Physical limits to the resolving power of an instrument From: Gerthsen & Vogel, Physik, 17th edition, Springer-Verlag, Berlin and Heidelberg 1993 ≈ sin = 1.22 /D Rayleigh criterion LensOptical microscope d min = 1.22 / (2n sin ) ≈ 200 nm for optical microscopy : wavelength of light (min. 450 nm) n: refractive index of light (often 1.56) : collecting angle Minimum distance that can be resolved:
From: J. Stöhr, NEXAFS Spectroscopy, Springer-Verlag, Berlin and Heidelberg 1992 Signal-to-noise and Signal-to-background ratios S B = I s / I b S N = I s / I n = ISIS ( I S + I b ) 1/2 = ( ) ISIS 1 + 1/S B 1/2 From statistics: for large N the noise scales like N.
What to optimise - S B or S N ? Also depends on what you can optimise! Noise: statistical phenomenonBackground: physical phenomenon! When all external (systematic) noise has been removed the only way left is to increase the number of counts! Counting timeChoice of method Choice of sample Choice of method Choice of geometry
Low count rate Good S B High count rate Bad S B Intermediate count rate Bad S B Method of choice! Example: X-ray absorption measured using different detection methods From: J. Stöhr, NEXAFS Spectroscopy, Springer-Verlag, Berlin and Heidelberg 1992
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