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