the explosion in high-tech medical imaging
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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The explosion in high-tech medical imaging
& nuclear medicine
(including particle beam cancer treatments)
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The constraints of limited/vanishing fossils fuels in the face of an exploding population
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…together with undeveloped or under-developed new technologies
The constraints of limited/vanishing fossils fuels
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Nuclear
will renew interest in nuclear power
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Fission power generators
will be part of the political
landscape again
as well as the Holy Grail of FUSION.
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…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).
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1896
1899
1912
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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.
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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
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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
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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!
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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!
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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
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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.
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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
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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
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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
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Masses are given in atomic mass units (amu) based on 6C12 = 12.000000
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Mass of bare hydrogen nucleus: 1.00727 amuMass of electron: 0.000549 amu
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number of neutrons
number of
protons
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RCteQtQ /0)(
RCteVtV /0)(
/0)( xeNxN
/0)( teAxA
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teNtN 0)(
RCteQtQ /0)(
RCteVtV /0)( /
0)( xeNxN
/0)( teAxA
Num
ber s
urvi
ving
Rad
ioac
tive
atom
s
What does stand for?
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teNtN 0)(N
umbe
r sur
vivi
ngR
adio
activ
e at
oms
time
tNN 0logloglogN
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!4!3!2
1432 xxxxex
!7!5!3
sin753 xxxxx
for x measured in radians (not degrees!)
!6!4!2
1cos642 xxxx
32
!3)2)(1(
!2)1(1)1ln( xpppxpppxx p
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)2sin()( ftAty
!7
)2(!5
)2(!3
)2(22sin753 ftftftftft
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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!
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y = sin x
y = xy = x3/6
y = x - x3/6
y = x5/120
y = x - x3/6 + x5/120
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...718281828.2eAny power of e can be expanded as an infinite series
!4!3!2
1432 xxxxex
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
7.000000e2 = 7.3890560989…
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Piano, Concert C
Clarinet, Concert C
Miles Davis’ trumpet
violin
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A Fourier series can be defined for any function over the interval 0 x 2L
1
0 sincos2
)(n
nn Lxnb
Lxnaaxf
where dxL
xnxfL
a L
n
2
0cos)(1
dxL
xnxfL
b L
n
2
0sin)(1
Ofteneasiestto treat
n=0 casesseparately
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Compute the Fourier series of the SQUARE WAVE function f given by
)(xf2,1
0,1
xx
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.
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dxL
xnxfL
a L
n
2
0cos)(1
)(xf2,1
0,1
xx
dxxfa 0cos)(1 2
00
dxdx 0cos)1(0cos11 2
0
0
dxnxdxnxan
2
0cos)1(cos11
dxnnxdxnx ( )coscos100
dxnxdxnx
00coscos1
change of variables: x x' = x-
periodicity: cos(X-n) = (-1)ncosX
for n = 1, 3, 5,…
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dxL
xnxfL
a L
n
2
0cos)(1
)(xf2,1
0,1
xx
00 a
dxnxan
0cos2
for n = 1, 3, 5,…
0na for n = 2, 4, 6,…
change of variables: x x' = nx
dxxn
a n
n
0cos2 0
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dxL
xnxfL
b L
n
2
0sin)(1
)(xf2,1
0,1
xx
00sin)(1 2
00 dxxfb
dxnxdxnxbn
2
0sinsin1
dxnnxdxnx ( )sinsin100
periodicity: cos(X-n) = (-1)ncosX
dxnxdxnx
00sinsin1
for n = 1, 3, 5,…
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)(xf2,1
0,1
xx
00 b
dxnxbn
0sin2
for n = 1, 3, 5,…
0nb for n = 2, 4, 6,…
change of variables: x x' = nx
dxxn
n
0sin2
dxL
xnxfL
b L
n
2
0sin)(1
dxxn
0sin1for odd n
nxn
40cos2
for n = 1, 3, 5,…
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)55sin
33sin
1sin(4)(
xxxxf
1
2x
y
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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
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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
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Fourier Transforms Generalization of ordinary “Fourier expansion” or “Fourier series”
de)(g21)t(f ti
de)t(f21)(g ti
Note how this pairs canonically conjugate variables and t.
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Fourier transforms do allow an explicit “closed” analytic form for
the Dirac delta function
de21)t( )t(i
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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
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For well-behaved (continuous) functions (bounded at infiinity)like f(x)=e-x2/22
dxexfkF ikx)(21)(
Starting from:
f(x) g'(x) g(x)= e+kxik
dxxgx'fxgxf )()()()(
21
dxekix'fe
kxif ikxikx )()(
21
f(x) is bounded
oscillates in thecomplex plane
over-all amplitude is damped at ±
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dxex'f
kikF ikx)(
21)(
)()(21 kikFdxex'f ikx
Similarly, starting from:
dkekFxf ikx)(21)(
)()(21 xixfdkek'F ikx
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And so, specifically for a normal distribution: f(x)=e-x2/22
differentiating: )()( 2 xfxxfdxd
from the relation just derived: kdekFixf
dxd xki ~)~(
21)(
~
2'
Let’s Fourier transform THIS statement
i.e., apply: dxe ikx
21
on both sides!
dxeikikF ikx 2
1)( 2
1 2 F'(k)e-ikxdk
~ ~~
kdkFi ~)~(2 '
e-i(k-k)xdx~ 1
2
(k – k)~
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kdkFikikF ~)~()(2
' e-i(k-k)xdx
~ 1 2
(k – k)~
)()(2
kFikikF ' selecting out k=k
~
rewriting as: 2
)(/)( k
kFdkkdF
0
k
0
k
dk''
''dk'
22
21)0(ln)(ln kFkF
2221
)0()( ke
FkF 22
21
)0()(keFkF
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2221
)0()(keFkF
22 2/)( xexf Fourier transformsof one another
Gaussian distributionabout the origin
dxexfkF ikx)(21)(
Now, since:
dxxfF )(21)0(
we expect:
10 xie
221)0(
22 2/
dxeF x
2221
2)(kekF
22 2/)( xexf Both are of the form of a Gaussian!
x k 1/
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x k 1
orgiving physical interpretation to the new variable
x px h