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

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

Nuclear will renew interest in nuclear power

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

…exciting developments in theoretical astrophysics 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).

  1912

Henri Becquerel ( ) 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 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.

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

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

Using the procedure developed by J.J. Thomson in 1887 Becquerel determined the ratio of charge q to mass m for  : q/m = 1.76×10 11 coulombs/kilogram identical to the electron!  : q/m = 4.8×10 7 coulombs/kilogram 4000 times smaller!

Discharge Tube Thin-walled (0.01 mm) glass tube to vacuum pump & Mercury supply Radium or Radon gas Noting helium gas often found trapped in samples of radioactive minerals, Rutherford speculated that  particles might be doubly ionized Helium atoms (He ++ ) Rutherford 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!

Status of particle physics early 20th century Electron J.J.Thomson 1898 nucleus (  proton ) Ernest Rutherford  Henri Becquerel 1896 Ernest Rutherford 1899   P. Villard 1900 X-rays Wilhelm Roentgen 1895

Periodic Table of the Elements Fe Co Ni Atomic mass values averaged over all isotopes in the proportion they naturally occur.

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

92 U 238  90 Th 234   91 Pa 234   92 U 234  92 U 234  90 Th 230   88 Ra 226   86 Rn 222   84 Po 218   82 Pb 214  82 Pb 214  83 Bi 214   84 Po 214   82 Pb 210  82 Pb 210  83 Bi 210   84 Po 210   82 Pb 206  “Uranium I” 4.5  10 9 yearsU 238 “Uranium II” 2.5  10 5 yearsU 234 “Radium B” radioactivePb 214 “Radium G” stablePb 206

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 chiefly 207 amu

Masses are given in atomic mass units (amu) based on 6 C 12 =

Mass of bare hydrogen nucleus: amu Mass of electron: amu

number of neutrons number of protons

Number surviving Radioactive atoms What does stand for?

Number surviving Radioactive atoms time  log N

for x measured in radians (not degrees!)

Let’s complete the table below (using a calculator) to check the “small angle approximation” (for angles not much bigger than ~15  20 o ) which ignores more than the 1 st term of the series Note: the x or  (in radians) = (  /180 o )   (in degrees) Angle  (degrees) Angle  (radians) sin  25 o % accurate!

y = sin x y = x y = x 3 /6 y = x - x 3 /6 y = x 5 /120 y = x - x 3 /6 + x 5 /120

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 e 0 = = e 1 = e 2 = e 2 = …

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

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  22 Note: f(x) is an odd function ( i.e. f(-x) = -f(x) ) so f(x) cos nx will be as well, while f(x) sin nx will be even.

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

for n = 2, 4, 6,… change of variables: x  x' = nx

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

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

1 22 x y

Leads you through a qualitative argument in building a square wave Add terms one by one (or as many as you want) to build fourier series approximation to a selection of periodic functions Build Fourier series approximation to assorted periodic functions and listen to an audio playing the wave forms Customize your own sound synthesizer

Two waves of slightly different wavelength and frequency produce beats.  x   x x 1k1k k = 2  NOTE: The spatial distribution depends on the particular frequencies involved

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

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

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  -1  +1  +2   Let’s assume a wave packet tailored to be something like a Gaussian (or “Normal”) distribution

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

Similarly, starting from:

And so, specifically for a normal distribution: f(x)=e -x 2 /2  2 differentiating: from the relation just derived: Let’s Fourier transform THIS statement i.e., apply:on both sides! 1  2  F'(k)e -ikx dk ~ ~ ~ e -i(k-k)x dx ~ 1 2   (k – k) ~

e -i(k-k)x dx ~ 1 2   (k – k) ~ selecting out k=k ~ rewriting as: 0 k 0 k dk' ' ' '

Fourier transforms of one another Gaussian distribution about the origin Now, since: we expect: Both are of the form of a Gaussian!  x    k  1/ 

 x  k  1 or giving physical interpretation to the new variable  x  p x  h