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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
will renew interest in nuclear power
Nuclear
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).
1896
1899
1912
a, b
g
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.
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
a - highly ionizing, but completely
absorbed by 0.006 cm aluminum
foil or a few cm of air
b - less ionizing, but penetrate many
meters of air or up to a cm of
aluminum.
1900 P. Villard finds in addition to a rays, radium emits g - the least ionizing,
but capable of penetrating many cm of lead, several feet of concrete
B-field
points
into page
1900-01 Studying the deflection of these rays in magnetic fields,
Becquerel and the Curies establish a, b rays to be
charged particles
1900-01
Using the procedure developed by J.J. Thomson in 1887
Becquerel determined the ratio of charge q to mass m for
b: q/m = 1.76×1011 coulombs/kilogram
identical to the electron!
a: q/m = 4.8×107 coulombs/kilogram
4000 times smaller!
Discharge Tube
Thin-walled
(0.01 mm)
glass tube
Noting helium gas often found trapped in samples
of radioactive minerals, Rutherford speculated that
a particles might be doubly ionized Helium atoms (He++)
1906-1909 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!
to vacuum
pump &
Mercury
supply
Radium or
Radon gas
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
X-rays
Wilhelm Roentgen 1895
1900
Periodic Table of the Elements
Fe
26
55.86
Co
27
58.93
Ni
28
58.71
Atomic mass values averaged over all isotopes in the proportion they naturally occur.
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)
6
The “last” 11 naturally occurring elements (Lead Uranium)
Z=82
recur in 3 principal “radioactive series.”
92
a
238
92U
b
234
Th
90
b
234
Pa
91
a
234
U
a
230
Th
90
a
226
Ra
88
92
234
92U
222 a
Rn
86
b
214
Pb
82
214
83Bi
b
a
214
Po
84
210
82Pb
b
210
Pb
82
210
83Bi
b
a
210
Po
84
206
82Pb
“Uranium I”
“Uranium II”
“Radium B”
“Radium G”
4.5109 years
2.5105 years
a
218
Po
84
U238
U234
radioactive Pb214
stable
Pb206
214
82Pb
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 6C12 = 12.000000
Mass of bare hydrogen nucleus: 1.00727 amu
Mass of electron:
0.000549 amu
number
of
protons
number of neutrons
Q(t ) Q0e
V (t ) V0e
t / RC
t / RC
N ( x ) N 0e
A( x) A0e
x/
t /
A( x) A0e
t /
Number surviving
Radioactive atoms
N ( x ) N 0e
x/
Q(t ) Q0e
V (t ) V0e
N (t ) N 0 e
t / RC
t / RC
t
What does
stand for?
Number surviving
Radioactive atoms
logN
N (t ) N 0 e
t
log N log N 0 t
time
3
5
7
x
x
x
sin x x
3! 5! 7!
for x measured in
radians (not degrees!)
2
4
6
x
x
x
cos x 1
2! 4! 6!
2
3
4
x
x
x
e 1 x
2! 3! 4!
x
p( p 1) 2 p( p 1)( p 2) 3
ln(1 x ) 1 px
x
x
2!
3!
p
y(t ) Asin(2ft)
(2ft ) (2ft ) (2ft )
sin 2ft 2ft
3!
5!
7!
3
5
7
Let’s complete the table below (using a calculator) to check the “small
angle approximation” (for angles not much bigger than ~1520o)
sin x x
which ignores more than the 1st term of the series
Note: the x or (in radians) = (/180o) (in degrees)
Angle (degrees) Angle (radians)
25o
0
1
2
3
4
6
8
10
15
20
25
0
0.017453293
0.034906585
0.052359878
0.069813170
0.104719755
0.139626340
0.174532952
0.261799388
0.349065850
0.436332313
sin
0.000000000
0.017452406
0.034899497
0.052335956
0.069756473
0.104528463
0.139173101
0.173648204
0.258819045
0.342020143
0.422618262
97% accurate!
y=x
y = x3/6
y = x - x3/6 + x5/120
y = x5/120
y = sin x
y = x - x3/6
e 2.718281828...
Any power of e can be expanded as an infinite series
2
3
4
x
x
x
ex 1 x
2! 3! 4!
Let’s compute some powers of e using
just the above 5 terms of the series
e0 = 1 +
0 + 0 +
e1 = 1 +
1 + 0.500000 + 0.166667 + 0.041667
0 + 0 = 1
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
A Fourier series can be defined for any function over the interval 0 x 2L
a0
nx
nx
f ( x ) an cos
bn sin
2 n1
L
L
where
1 2L
n x
an 0 f ( x ) cos
dx
L
L
1 2L
n x
bn 0 f ( x ) sin
dx
L
L
Often
easiest
to treat
n=0 cases
separately
Compute the Fourier series of the SQUARE WAVE function f given by
f (x)
1 , 0 x
1 , x 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.
1 2L
n x
an 0 f ( x ) cos
dx
L
L
a0
2
f ( x ) cos 0 dx
0
1cos 0 dx ( 1) cos 0 dx 0
1
1cos nx dx ( 1) cos nx dx
1
cos nx dx cos ( nx n ) dx
an
1
f (x)
1 , 0 x
1 , x 2
1
2
0
2
0
change of variables: x x' = x-
0
0
periodicity: cos(X-n) = (-1)ncosX
cos nx dx cos nx dx
1
0
0
for n = 1, 3, 5,…
1 2L
n x
an 0 f ( x ) cos
dx
L
L
a0 0
an 0
an
2
f (x)
1 , 0 x
1 , x 2
for n = 2, 4, 6,…
cos nx dx
0
for n = 1, 3, 5,…
change of variables: x x' = nx
2 n
an
cos x dx
0
n
0
1 2L
n x
bn 0 f ( x ) sin
dx
L
L
1
f (x)
2
b0
f ( x ) sin 0 dx 0
0
bn
sin nx dx
1 , 0 x
1 , x 2
1
2
0
sin nx dx
sin nx dx sin ( nx n ) dx
1
0
0
periodicity: cos(X-n) = (-1)ncosX
sin nx dx sin nx dx
1
0
0
for n = 1, 3, 5,…
1 2L
n x
bn 0 f ( x ) sin
dx
L
L
b0 0
bn 0
bn
2
f (x)
1 , 0 x
1 , x 2
for n = 2, 4, 6,…
sin nx dx
0
for n = 1, 3, 5,…
change of variables: x x' = nx
2 n
1
sin x dx
sin x dx
0
0
n
n for odd n
2
cos x 0 n4
n
for n = 1, 3, 5,…
4 sin x sin 3x sin 5 x
f ( x) (
)
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
NOTE: The spatial distribution depends on the particular frequencies involved
x
1
x
k
k =
2
Two waves of slightly different wavelength and frequency produce beats.
Fourier Transforms
Generalization of ordinary “Fourier expansion” or “Fourier series”
f (t)
g( )
1
it
g
(
)
e
d
2
1
it
f
(
t
)
e
d
2
Note how this pairs canonically conjugate variables and t.
Fourier transforms
do allow an explicit “closed” analytic form for
the Dirac delta function
1
(t )
2
i ( t )
e
d
Let’s assume a wave packet tailored to be something like a
Gaussian (or “Normal”) distribution
Area within
1
1.28
1.64
1.96
2
2.58
3
4
68.26%
80.00%
90.00%
95.00%
95.44%
99.00%
99.46%
99.99%
-2
-1
+1
+2
1
x
e
2
( x )2
2
2
For well-behaved (continuous) functions (bounded at infiinity)
2/22
-x
like f(x)=e
Starting from:
1
ikx
F (k )
f ( x )e dx
2
f(x)
g'(x)
g(x)=
i +kx
e
k
1
f ( x ) g ( x ) f ' ( x ) g ( x )dx
2
1
2
if ( x ) ikx
i ikx
e
f ' ( x ) e dx
k
k
f(x) is
bounded
oscillates in the
complex plane
over-all amplitude is damped at ±
i 1
ikx
F (k )
f ' ( x )e dx
k 2
1
ikx
f ' ( x )e dx ikF (k )
2
Similarly, starting from:
1
ikx
f ( x)
F (k )e dk
2
1
ikx
F' (k )e dk ixf ( x )
2
2/22
And so, specifically for a normal distribution: f(x)=e-x
d
x
f ( x) 2 f ( x)
dx
d
i 1
~ ik~x ~
f ( x) 2
F' (k )e dk
dx
2
differentiating:
from the relation
just derived:
Let’s Fourier transform THIS statement
i.e., apply:
1
2
eikxdx
on both sides!
i 1
ikF (k ) 2
2
i
2
1
ikx
~
~
~
-ikx
F'(k)e dk e dx
2
~ ~
~
1 e-i(k-k)x
dx F' ( k )dk
2
~
(k – k)
ikF ( k )
i
2
~ ~
~
1 e-i(k-k)x
dx F' ( k )dk
2
~
ikF ( k )
i
2
(k – k)
F' ( k )
~
selecting out k=k
k
rewriting as:
dF ( k' ) / dk'
dk'
F ( k')
0
k
k 2dk'
0
1 2 2
ln F ( k ) ln F (0) k
2
1 2k 2
F (k )
e 2
F (0)
F (k ) F (0)e
1 2k 2
2
f ( x) e
x2 / 2 2
Fourier transforms
of one another
F (k ) F (0)e
1 2k 2
2
Gaussian distribution
about the origin
Now, since:
we expect:
1
ikx
F (k )
f
(
x
)
e
dx
2
i0 x
e
1
1
F (0)
f ( x )dx
2
1 x2 / 2 2
F (0)
e
dx 2
2
f ( x) e
x2 / 2 2
x
Both are of the form
of a Gaussian!
F (k ) 2e
k 1/
1 2k 2
2
x k 1
or
giving physical interpretation to the new variable
x px h