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

Num

ber s

urvi

ving

Rad

ioac

tive

atom

s

What does stand for?

teNtN 0)(N

umbe

r sur

vivi

ngR

adio

activ

e at

oms

time

tNN 0logloglogN

!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

)2sin()( ftAty

!7

)2(!5

)2(!3

)2(22sin753 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 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…

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 Lxnb

Lxnaaxf

where dxL

xnxfL

a L

n

2

0cos)(1

dxL

xnxfL

b 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

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.

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,…

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

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,…

)(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,…

)55sin

33sin

1sin(4)(

xxxxf

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)(g21)t(f ti

de)t(f21)(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

de21)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)(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 ±

dxex'f

kikF ikx)(

21)(

)()(21 kikFdxex'f ikx

Similarly, starting from:

dkekFxf ikx)(21)(

)()(21 xixfdkek'F ikx

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)~

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

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/

x k 1

orgiving physical interpretation to the new variable

x px h