the explosion in high-tech medical imaging & nuclear medicine (including particle beam cancer...
TRANSCRIPT
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).
1896
1899
1912
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 - 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 feet 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
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!
Discharge Tube
Thin-walled(0.01 mm)glass tube
to vacuumpump &Mercurysupply
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++)
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!
Status of particle physics early 20th century
Electron J.J.Thomson 1898
nucleus ( proton) Ernest Rutherford 1908-09
Henri Becquerel 1896 Ernest Rutherford 1899
P. Villard 1900
X-rays Wilhelm Roentgen 1895
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.
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
92U238 90Th234 91Pa234 92U234
92U234 90Th230 88Ra226 86Rn222 84Po218 82Pb214
82Pb214 83Bi214 84Po214 82Pb210
82Pb210 83Bi210 84Po210 82Pb206
“Uranium I” 4.5109 years U238
“Uranium II” 2.5105 years U234
“Radium B” radioactive Pb214
“Radium G” stable Pb206
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 amuMass of electron: 0.000549 amu
number of neutrons
number of
protons
RCteQtQ /
0)( RCteVtV /
0)(
/0)( xeNxN
/0)( teAxA
teNtN 0)(
RCteQtQ /
0)( RCteVtV /
0)( /0)( xeNxN
/0)( teAxA
Nu
mb
er
surv
ivin
gR
ad
ioa
ctiv
e a
tom
s
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
)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
)(xf2,1
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
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,…
)(xf2,1
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
Two waves of slightly different wavelength and frequency produce beats.
x
x
1k
k = 2
NOTE: The spatial distribution depends on the particular frequencies involved
Fourier Transforms Generalization of ordinary “Fourier expansion” or “Fourier series”
de)(g2
1)t(f ti
de)t(f2
1)(g ti
Note how this pairs canonically conjugate variables and t.
Fourier transforms do allow an explicit “closed” analytic form for
the Dirac delta function
de2
1)t( )t(i
Area within1 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
2
2
2
)x(
e2
1x
Let’s assume a wave packet tailored to be something like aGaussian (or “Normal”) distribution
For well-behaved (continuous) functions (bounded at infiinity)
like f(x)=e-x2/22
dxexfkF ikx)(2
1)(
Starting from:
f(x) g'(x) g(x)= e+kxik
dxxgx'fxgxf )()()()(
2
1
dxek
ix'fe
k
xif ikxikx )()(
2
1
f(x) is
boundedoscillates in thecomplex plane
over-all amplitude is damped at ±
dxex'fk
ikF ikx)(
2
1)(
)()(2
1kikFdxex'f ikx
Similarly, starting from:
dkekFxf ikx)(2
1)(
)()(2
1xixfdkek'F ikx
And so, specifically for a normal distribution: f(x)=e-x2/22
differentiating: )()(2
xfx
xfdx
d
from the relation just derived: kdekF
ixf
dx
d xki ~)
~(
2
1)(
~
2'
Let’s Fourier transform THIS statement
i.e., apply: dxe ikx
21
on both sides!
dxei
kikF ikx 2
1)(
2
1 2 F'(k)e-ikxdk
~ ~~
kdkFi ~
)~
(2
' e-i(k-k)xdx
~ 1 2
(k – k)~
kdkFi
kikF~
)~
()(2
' e-i(k-k)xdx
~ 1 2
(k – k)~
)()(2
kFi
kikF ' selecting out k=k
~
rewriting as: 2
)(
/)( kkF
dkkdF
0
k
0
k
dk''
''dk'
22
2
1)0(ln)(ln kFkF
2221
)0(
)( ke
F
kF 22
21
)0()(k
eFkF
2221
)0()(k
eFkF
22 2/)( xexf Fourier transforms
of one anotherGaussian distribution
about the origin
dxexfkF ikx)(2
1)(
Now, since:
dxxfF )(2
1)0(
we expect:
10 xie
22
1)0(
22 2/
dxeF x
2221
2)(k
ekF
22 2/)( xexf Both are of the form
of a Gaussian!
x k 1/
x k 1
orgiving physical interpretation to the new variable
x px h