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## 60 years ago…

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**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 …together with undeveloped or under-developed new technologies**Fission power**generators will be part of the political landscape again as well as the Holy Grail of FUSION.**…exciting developments in theoreticalastrophysics**The evolution of stars is well-understood in terms of stellar models incorporating known nuclear processes. Applying well established nuclear physics to the epoch of nuclear formation - ~3 -15 minutes after the big bang - allows the abundances of deuterium, helium, lithium and other light elements to be predicted. 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).**1896**1899 a, b g 1912**Henri Becquerel(1852-1908)**1903 Nobel Prize discovery of natural radioactivity Wrapped photographic plate showed distinct silhouettes of uranium salt samples stored atop it. 1896 While studying fluorescent & phosphorescent materials, Becquerel finds potassium-uranyl sulfate spontaneously emits radiation that can penetrate thick opaque black paper aluminum plates copper plates Exhibited by all known compounds of uranium (phosphorescent or not) and metallic uranium itself.**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 ft of concrete**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**m**R v F or**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!**V**R C A****Number surviving Radioactive atoms What does stand for?**Number surviving**Radioactive atoms logN time**for x measured in**radians (not degrees!) What if x was a measurement that carried units?**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!**y = x**y = x - x3/6 + x5/120 y = x3/6 y = x5/120 y = sinx y = x - x3/6**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…**violin**Piano, Concert C Clarinet, Concert C Miles Davis’ trumpet**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**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.**change of variables: x x'= x-**periodicity: cos(X+n) = (-1)ncosX for n = 1, 3, 5,…**for n = 2, 4, 6,…**for n = 1, 3, 5,… change of variables: x x'= nx IFf(x) is odd, all an vanish!**periodicity: cos(X±n) = (-1)ncosX**for n = 1, 3, 5,…**for n = 2, 4, 6,…**for n = 1, 3, 5,… change of variables: x x'= nx for odd n for n = 1, 3, 5,…**y**1 2 x**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