tomography

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Information about tomography
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Published on October 15, 2007

Author: Janelle

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Slide1:  X. Ji, PRL91, 062001 (2003) A. Belitsky, X.Ji, F. Yuan, hep-ph/0307383 Outline:  Outline Slide3:  A Brief Story of the Proton Protons, protons, everywhere:  Protons, protons, everywhere The Proton is one of the most abundant particles around us! The sun ☼ is almost entirely made of protons... And all other stars… And all atomic nuclei… The profile: Spin 1/2, making MRI (NMR) possible Mass 938.3 MeV/c2, making up ½ of our body weight Charge +1, making a H-atom by attracting an electron What’s in A Proton? (Four Nobel Prizes) :  What’s in A Proton? (Four Nobel Prizes) It was thought as a point-like particle, like electron In 1933, O. Stern measured the magnetic moment of the proton, finding 2.8N, first evidence that the proton is not point-like (Nobel prize, 1943) In 1955, R. Hofstadter measured the charge radius of the proton, about 0.8fm. (1fm = 10-13 cm, Nobel prize, 1961) In 1964, M. Gell-Mann and G. Zweig postulated that there are three quarks in the proton: two ups and one down (Nobel prize, 1969) In 1969, Friedman, Kendall, & Taylor find quarks in the proton (Nobel prize, 1990) QCD and Strong-Interactions:  QCD and Strong-Interactions Building blocks Quarks (u,d,s…, spin-1/2, mq ~ small, 3 colors) Gluons (spin-1, massless, 32 −1 colors) Interactions In the low-energy region, it represents an extremely relativistic, strongly coupled, quantum many-body problem—one of the daunting challenges in theoretical physics Clay Math. Inst., Cambridge, MA $1M prize to solve QCD! (E. Witten) The Proton in QCD:  The Proton in QCD We know a lot and we know little 2 up quarks (e = 2/3) + 1 down quark (e = −1/3) + any number of quark-antiquark pairs + any number of gluons Fundamental questions (from quarks to cosmos…) Origin of mass? ~ 90% comes from the motion of quarks & gluons ~ l0% from Higgs interactions (Tevertron, LHC) Proton spin budget? How are Elements formed? the protons & neutrons interact to form atomic nuclei Understanding the Proton:  Understanding the Proton Solving QCD Numerically simulation, like 4D stat. mech. systems  Feynman path integral  Wick rotation  Spacetime discretization  Monte Carlo simulation Effective field theories (large Nc, chiral physics,…) Experimental probes Study the quark and gluon structure through low and high-energy scattering Require clean reaction mechanism Photon, electron & perturbative QCD Slide9:  Elastic Form Factors & Charge Distributions in Space Form Factors & Microscopic Structure:  Form Factors & Microscopic Structure In studying the microscopic structure of matter, the form factor (structure factor) F(q2) is one of the most fundamental observables The Fourier Transformation (FT) of the form factor is related to the spatial charge (matter) distributions ! Examples The charge distribution in an atom/molecule The structure of crystals … The Proton Elastic Form Factors:  The Proton Elastic Form Factors k What does F1,2 tell us about the structure of the nucleon? Sachs Interpretation of Form Factors:  Sachs Interpretation of Form Factors According to Sachs, the FT of GE=F1−τF2 and GM=F1+F2 are related to charge and magnetization distributions. This is obtained by first constructing a wave packet of the proton (a spatially-fixed proton) then measure the charge density relative to the center Sachs Interpretation (Continued):  Calculate the FT of the charge density, which now depends on the wave-packet profile Additional assumptions The wave packet has no dependence on the relative momentum q |φ(P)|2 ~ δ(P) Matrix element In the Breit frame Sachs Interpretation (Continued) Up-Quark Charge Distribution:  Up-Quark Charge Distribution fm fm Effects of Relativity:  Effects of Relativity Relativistic effects The proton cannot be localized to a distance better than 1/M because of Zitterbewegung When the momentum transfer is large, the proton recoils after scattering, generating Lorentz contraction The effects are weak if 1/(RM) « 1 (R is the radius) For the proton, it is ~ 1/4. For the hydrogen atom, it is ~ 10-5 Slide16:  Feynman Quark Distribution Momentum Distributions:  Momentum Distributions While the form factors provide the static 3D picture, but they do not yield info about the dynamical motion of the constituents. To see this, we need to know the momentum space distributions of the particles. This can be measured through single-particle knock-out experiments Well-known Examples: Nuclear system: quasi-elastic scattering Liquid helium & BEC: neutron scattering Feynman Quark Distributions:  Feynman Quark Distributions Infinite Momentum Frame (IMF):  Infinite Momentum Frame (IMF) The interpretation is the simplest when the proton travels at the speed of light (momentum P∞). The quantum configurations are frozen in time because of the Lorentz dilation. Density of quarks with longitudinal momentum xP (with transverse momentum integrated over) “Feynman momentum” x takes value from –1 to 1, Negative x corresponds to antiquark. Rest-Frame Interpretation:  Rest-Frame Interpretation Quark spectral function Probability of finding a quark in the proton with energy E=k0, 3-momentum k, defined in the rest frame of the nucleon A concept well-known in many-body physics Relation to parton distributions Feynman momentum is a linear combination of quark energy and momentum projection in the rest frame. Present status :  Present status CTEQ6: J. Pumplin et al JHEP 0207, 012 (2002) Slide22:  Quantum Phase-space (Wigner) Distribution Phase-space Distribution?:  Phase-space Distribution? Wigner function:  Wigner function Short of measuring the wave function, the Wigner function contains the most complete (one-body) info about a quantum system. Simple Harmonic Oscillator:  Simple Harmonic Oscillator Husimi distribution: positive definite! N=0 N=5 Measuring Wigner function of Quantum Light:  Measuring Wigner function of Quantum Light Measuring Wigner function of the Vibrational State in a Molecule:  Measuring Wigner function of the Vibrational State in a Molecule Quantum State Tomography of Dissociateng molecules:  Quantum State Tomography of Dissociateng molecules Skovsen et al. (Denmark) PRL91, 090604 Slide29:  Quantum Phase-Space Distribution for Quarks Quarks in the Proton:  Quarks in the Proton No known experiment can measure this! 7-dimensional distribtuion a la Saches Custom-made for high-energy processes :  Custom-made for high-energy processes f(r,x) Proton images at a fixed x:  Proton images at a fixed x t= – q2  ~ qz What is a GPD?:  What is a GPD? A proton matrix element which is a hybrid of elastic form factor and Feynman distribution Depends on x: fraction of the longitudinal momentum carried by parton t=q2: t-channel momentum transfer squared ξ: skewness parameter Charge Density and Current in Phase-space:  Charge Density and Current in Phase-space Mass distribution:  Mass distribution Spin of the Proton:  Spin of the Proton Was thought to be carried by the spin of the three valence quarks Polarized deep-inelastic scattering found that only 20-30% are in the spin of the quarks. Integrate over the x-weighted phase-space current, one gets the momentum current One can calculate the total quark (orbital + spin) contribution to the spin of the proton How to measure the GPDs?:  How to measure the GPDs? Compton Scattering Complicated in general In the Bjorken limit k k’ Single quark scattering Photon wind Non-invasive surgery Deeply virtual Compton scattering First Evidence of DVCS:  First Evidence of DVCS HERA ep Collider in DESY, Hamburg Zeus detector Present and Future Experiments:  Present and Future Experiments HERMES Coll. in DESY and CLAS Coll. in Jefferson Lab has made further measurements of DVCS and related processes. COMPASS at CERN, taking data Jefferson Lab 12 GeV upgrade DVCS and related processes & hadron spectrocopy Electron-ion collider (EIC) 2010? RHIC, JLab? Slide40:  Quantum Phase-space Tomography A GPD or Wigner Function Model:  A GPD or Wigner Function Model Slide42:  x y z Up-Quark Charge Density at x=0.4 Slide43:  Surface of constant charge denstiy Slide44:  Up-Quark Charge Denstiy at x=0.01 Slide45:  Surface of Constant Charge Density A Mini-Movie:  A Mini-Movie Up Quark Density at x=0.7:  Up Quark Density at x=0.7 Slide48:  Up-Quark Density At x=0.7 Slide49:  Surface of Constant Charge Density Slide50:  Charge Denstiy at Negative x Slide51:  Charge Denstiy in the MIT Bag Comments:  Comments If one puts the pictures at all x together, one gets a spherically round nucleon! (Wigner-Eckart theorem) If one integrates over the distribution along the z direction, one gets the 2D impact parameter space pictures of Burkardt and Soper. Conclusions:  Conclusions Form factors provide the spatial distribution, Feynman distribution provide the momentum-space density. They do not provide any info on space-momentum correlation. The quark and gluon Wigner distributions are the correlated momentum & coordinate distributions, allowing us to picture the proton at every Feynman x, and are measurable!

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