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