60 years ago…

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).

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.

1896

1899

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.

1898 Marie Curie discovers thorium (90Th) Together Pierre and Marie Curie discover polonium (84Po) and radium (88Ra)

1899 Ernest Rutherford identifies 2 distinct kinds of rays emitted by uranium - highly ionizing, but completely absorbed by 0.006 cm aluminum foil or a few cm of air

- less ionizing, but penetrate many meters of air or up to a cm of aluminum.

1900 P. Villard finds in addition to rays, radium emits - the least ionizing, but capable of penetrating many cm of lead, several ft of concrete

B-fieldpoints

into page

1900-01 Studying the deflection of these rays in magnetic fields, Becquerel and the Curies establish rays to be charged particles

F

R

mvF

2

190sin o

R

mvqvB

2

mvqBR

BR

v

m

q

R

mvqvB

2

sin

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!

RCteQtQ /

0)( RCteVtV /

0)(

/

0)( xeNxN

/0

)( teAtA

teNtN 0)(

RCteQtQ /

0)( RCteVtV /

0)( /

0)( xeNxN

/0

)( teAtA N

um

be

r su

rviv

ing

Ra

dio

act

ive

ato

ms

What does stand for?

teNtN 0)(N

um

ber

su

rviv

ing

Rad

ioac

tive

ato

ms

time

tNN 0logloglogN

!4!3!2

1432 xxx

xex

!7!5!3

sin753 xxx

xx

for x measured in radians (not degrees!)

!6!4!2

1cos642 xxx

x

32

!3

)2)(1(

!2

)1(1)1ln( x

pppx

pppxx p

What if

x was a measurementthat carried

units?

)2sin()( ftAty

!7

)2(

!5

)2(

!3

)2(22sin

753 ftftftftft

Let’s complete the table below (using a calculator) to check the “small angle approximation” (for angles not much bigger than ~1520o)

xx sinwhich ignores more than the 1st term of the series

Note: the x or (in radians) = (/180o) (in degrees)

Angle (degrees) Angle (radians) sin

25o

0 0 0.0000000001 0.017453293 0.0174524062 0.034906585 3 0.052359878 4 0.069813170 6810152025

0.1047197550.1396263400.1745329520.2617993880.3490658500.436332313

0.0348994970.0523359560.0697564730.1045284630.1391731010.1736482040.2588190450.3420201430.42261826297% accurate!

y = sin x

y = xy = x3/6

y = x - x3/6

y = x5/120

y = x - x3/6 + x5/120

...718281828.2eAny power of e can be expanded as an infinite series

!4!3!2

1432 xxx

xex

Let’s compute some powers of e using just the above 5 terms of the series

e0 = 1 + 0 + + + =

e1 = 1 + 1 +

e2 = 1 + 2 +

0 0 0 1

0.500000 + 0.166667 + 0.041667

2.708334

2.000000 + 1.333333 + 0.666667

7.000000

e2 = 7.3890560989…

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

1

0 sincos2

)(n

nn L

xnb

L

xna

axf

where dxL

xnxf

La

L

n

2

0cos)(

1

dxL

xnxf

Lb

L

n

2

0sin)(

1

Ofteneasiestto treat

n=0 casesseparately

Compute the Fourier series of the SQUARE WAVE function f given by

)(xf

2,1

0,1

x

x

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.

dxL

xnxf

La

L

n

2

0cos)(

1)(xf

2,1

0,1

x

x

dxxfa 0cos)(1 2

00

dxdx 0cos)1(0cos11 2

0

0

dxnxdxnxan

2

0cos)1(cos1

1

dxnnxdxnx ( )coscos1

00

dxnxdxnx

00coscos

1

change of variables: x x' = x-

periodicity: cos(X+n) = (-1)ncosX

for n = 1, 3, 5,…

dxL

xnxf

La

L

n

2

0cos)(

1)(xf

2,1

0,1

x

x

00 a

dxnxan

0cos

2for n = 1, 3, 5,…

0na for n = 2, 4, 6,…

change of variables: x x' = nx

dxxn

an

n

0cos

2 0

IF f(x) is odd, all an vanish!

dxL

xnxf

Lb

L

n

2

0sin)(

1)(xf

2,1

0,1

x

x

00sin)(1 2

00 dxxfb

dxnxdxnxbn

2

0sinsin

1

dxnnxdxnx ( )sinsin1

00

periodicity: cos(X±n) = (-1)ncosX

dxnxdxnx

00sinsin

1

for n = 1, 3, 5,…

)(xf

2,1

0,1

x

x

00 b

dxnxbn

0sin

2for n = 1, 3, 5,…

0nb for n = 2, 4, 6,…

change of variables: x x' = nx

dxxn

n

0sin

2

dxL

xnxf

Lb

L

n

2

0sin)(

1

dxxn

0sin

1

for odd n

nxn

40cos

2 for n = 1, 3, 5,…

)5

5sin

3

3sin

1

sin(

4)( xxx

xf

1

2x

y

http://www.jhu.edu/~signals/fourier2/

http://www.phy.ntnu.edu.tw/java/sound/sound.html

http://mathforum.org/key/nucalc/fourier.html

http://www.falstad.com/fourier/

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 functionsand listen to an audio playing the wave forms

Customize your own sound synthesizer