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The explosion in high-tech medical imaging

The explosion in high-tech medical imaging. & nuclear medicine. (including particle beam cancer treatments). The constraints of limited/vanishing fossils fuels in the face of an exploding population. The constraints of limited/ vanishing fossils fuels.

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The explosion in high-tech medical imaging

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  1. The explosion in high-tech medical imaging &nuclear medicine (including particle beam cancer treatments)

  2. The constraints of limited/vanishing fossils fuels in the face of an exploding population

  3. The constraints of limited/ vanishing fossils fuels …together with undeveloped or under-developed new technologies

  4. will renew interest in nuclear power Nuclear

  5. Fission power generators will be part of the political landscape again as well as the Holy Grail of FUSION.

  6. …exciting developments in theoreticalastrophysics The evolution of stars is well-understood in terms of stellar models incorporating known nuclear processes. The observed expansion of the universe (Hubble’s Law) lead Gamow to postulate a Big Bang which predicted the Cosmic Microwave Background Radiation as well as made very specific predictions of the relative abundance of the elements (on a galactic or universal scale).

  7. 1896 1899 a, b g 1912

  8. Henri Becquerel(1852-1908) received the 1903 Nobel Prize in Physics for the discovery of natural radioactivity. Wrapped photographic plate showed clear silhouettes, when developed, of the uranium salt samples stored atop it. • 1896 While studying the photographic images of various fluorescent & phosphorescent • materials, Becquerel finds potassium-uranyl sulfate spontaneously emits radiation • capable of penetrating thick opaque black paper • aluminum plates • copper plates • Exhibited by all known compounds of uranium (phosphorescent or not) • and metallic uranium itself.

  9. 1898Marie Curie discovers thorium (90Th) Together Pierre and Marie Curie discover polonium (84Po) and radium (88Ra) 1899Ernest Rutherfordidentifies 2 distinct kinds of rays emitted by uranium  - highly ionizing, but completely absorbed by 0.006 cmaluminum foil or a few cm of air  - less ionizing, but penetrate many meters of air or up to a cm of aluminum. 1900P. Villard finds in addition to  rays, radium emits  - the least ionizing, but capable of penetrating many cm of lead, several feet of concrete

  10. a g B-field points into page b 1900-01 Studying the deflection of these rays in magnetic fields, Becquerel and the Curies establish  rays to be charged particles

  11. 1900-01 Using the procedure developed by J.J. Thomson in 1887 Becquerel determined the ratio of charge q to mass m for : q/m = 1.76×1011 coulombs/kilogram identical to the electron! : q/m = 4.8×107 coulombs/kilogram 4000 times smaller!

  12. Noting helium gas often found trapped in samples of radioactive minerals, Rutherford speculated that  particles might be doubly ionized Helium atoms (He++) 1906-1909Rutherford and T.D.Royds develop their “alpha mousetrap” to collect alpha particles and show this yields a gas with the spectral emission lines of helium! Discharge Tube Thin-walled (0.01 mm) glass tube to vacuum pump & Mercury supply Radium or Radon gas

  13. Status of particle physics early 20th century Electron J.J.Thomson 1898 nucleus ( proton) Ernest Rutherford 1908-09 a Henri Becquerel 1896 Ernest Rutherford 1899 b g P. Villard 1900 X-rays Wilhelm Roentgen 1895

  14. Periodic Table of the Elements 26 27 28 Fe Co Ni 55.86 58.93 58.71 Atomic mass values averaged over all isotopes in the proportion they naturally occur.

  15. C 6 Through lead, ~1/4 of the elements come in “single species” Isotopes are chemically identical (not separable by any chemical means) but are physically different (mass) The “last” 11 naturally occurring elements (Lead  Uranium) recur in 3 principal “radioactive series.” Z=82 92

  16.   92U23890Th234 91Pa234 92U234      92U23490Th230 88Ra226 86Rn222 84Po218 82Pb214    82Pb21483Bi214 84Po214 82Pb210    82Pb21083Bi210 84Po210 82Pb206 “Uranium I” 4.5109 years U238 “Uranium II” 2.5105 years U234 “Radium B” radioactive Pb214 “Radium G” stable Pb206

  17. Chemically separating the lead from various minerals (which suggested their origin) and comparing their masses: Thorite (thorium with traces if uranium and lead) 208 amu Pitchblende (containing uranium mineral and lead) 206 amu “ordinary” lead deposits are chiefly207 amu

  18. Masses are given in atomic mass units (amu) based on 6C12 = 12.000000

  19. Mass of bare hydrogen nucleus: 1.00727 amu Mass of electron: 0.000549 amu

  20. number of protons number of neutrons

  21. V R C A

  22. Number surviving Radioactive atoms What does  stand for?

  23. Number surviving Radioactive atoms logN time

  24. for x measured in radians (not degrees!)

  25. Let’s complete the table below (using a calculator) to check the “small angle approximation” (for angles not much bigger than ~15-20o) which ignores more than the 1st term of the series Note: the x or(in radians) = (/180o)(in degrees) Angle (degrees)Angle (radians)sin  0 0 0.000000000 1 0.017453293 0.017452406 2 0.034906585 3 0.052359878 4 0.069813170 6 8 10 15 20 25 0.034899497 0.052335956 0.069756473 0.104719755 0.104528463 0.139173101 0.173648204 0.258819045 0.342020143 0.422618262 0.139626340 0.174532952 0.261799388 0.349065850 0.436332313 25o 97% accurate!

  26. y = x y = x - x3/6 + x5/120 y = x3/6 y = x5/120 y = sinx y = x - x3/6

  27. Any power of e can be expanded as an infinite series Let’s compute some powers of e using just the above 5 terms of the series 0 0 0 1 e0 = 1 + 0 + + + = e1 = 1 + 1 + 0.500000 + 0.166667 + 0.041667 2.708334 e2 = 1 + 2 + 2.000000 + 1.333333 + 0.666667 7.000000 e2 = 7.3890560989…

  28. violin Piano, Concert C Clarinet, Concert C Miles Davis’ trumpet

  29. Similarly A Fourier series can be defined for any function over the interval 0  x 2L where Often easiest to treat n=0 cases separately

  30. Compute the Fourier series of the SQUARE WAVE function f given by p 2p Note:f(x)is an odd function ( i.e.f(-x) = -f(x)) so f(x)cosnxwill be as well, whilef(x)sinnxwill be even.

  31. change of variables: x  x'= x- periodicity: cos(X-n) = (-1)ncosX for n = 1, 3, 5,…

  32. for n = 2, 4, 6,… for n = 1, 3, 5,… change of variables: x  x'= nx

  33. periodicity: cos(X-n) = (-1)ncosX for n = 1, 3, 5,…

  34. for n = 2, 4, 6,… for n = 1, 3, 5,… change of variables: x  x'= nx for odd n for n = 1, 3, 5,…

  35. y 1 2 x

  36. Leads you through a qualitative argument in building a square wave http://mathforum.org/key/nucalc/fourier.html Add terms one by one (or as many as you want) to build fourier series approximation to a selection of periodic functions http://www.jhu.edu/~signals/fourier2/ Build Fourier series approximation to assorted periodic functions and listen to an audio playing the wave forms http://www.falstad.com/fourier/ Customize your own sound synthesizer http://www.phy.ntnu.edu.tw/java/sound/sound.html

  37. NOTE: The spatial distribution depends on the particular frequencies involved x 2  1 k k = x  Two waves of slightly different wavelength and frequency produce beats.

  38. Fourier Transforms Generalization of ordinary “Fourier expansion” or “Fourier series” Note how this pairs canonically conjugate variables  and t.

  39. Fourier transforms do allow an explicit “closed” analytic form for the Dirac delta function

  40. Let’s assume a wave packet tailored to be something like a Gaussian (or “Normal”) distribution Area within 1 68.26% 1.28 80.00% 1.64 90.00% 1.96 95.00% 2 95.44% 2.58 99.00% 3 99.46% 4 99.99%  +2 -2 -1 +1

  41. For well-behaved (continuous) functions (bounded at infiinity) likef(x)=e-x2/22 Starting from: i k g'(x) g(x)= e+kx f(x) f(x) is bounded oscillates in the complex plane over-all amplitude is damped at ±

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