B-field
points
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
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!
to vacuum
pump &
Mercury
supply
Radium or
Radon gas
Mercury
Status of particle physics
early 20th century
Electron
J.J.Thomson
1898
nucleus ( proton) Ernest Rutherford 1908-09

Henri Becquerel 1896
Ernest Rutherford 1899

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 “weight” 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

238 
92U

234 
Th
90

234 
Pa
91

234
U


230 
Th
90

226 
Ra
88
92
234
92U
222 
Rn

86

214 
Pb
82
214
83Bi



214 
Po
84
210
82Pb

210 
Pb
82
210
83Bi



210 
Po
84
206
82Pb
“Uranium I”
“Uranium II”
“Radium B”
“Radium G”
4.5109 years
2.5105 years

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
Starting from the defining relation of a Fourier transform:
1 
ikx
F (k ) 
 f ( x)e dx
2 
we can integrate this “by parts”
f(x)
g'(x)
g(x)=
i +ikx
e
k

1 



f ( x ) g ( x )    f ' ( x ) g ( x )dx

2 




1  if ( x ) ik x
i ik x 

e
  f ' ( x ) e dx

2  k
k



f(x) is
bounded
oscillates in the
complex plane
over-all amplitude is damped at ±
i 1
F (k ) 
k 2
1
2

 f ' ( x )e


 f ' ( x )e
ikx
dx
ikx
dx  ikF (k )

Similarly, starting from:
1
f ( x) 
2

 F ( k )e
ikx
dk

1
2

 F' ( k ) e

ikx
dk  ixf ( x )
And so, specifically for a normal distribution: f(x)=ex
2/22
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
1
i.e., apply:
2

 
eikxdx
 ikF (k ) 
i


2
i

2
on both sides!
1
ikx
1
~
~
~
-ikx
F'(k)e dk e dx
2 2
~ ~
~
1 ei(k-k)x
dx F' (k )dk
2
~
 (k – k)
 ikF (k ) 
i

2
~ ~
~
1 ei(k-k)x
dx F' (k )dk
2
~
 (k – k)
 ikF (k ) 
i

F' (k )
2
~
selecting out k=k
k
rewriting as:
dF ( k' ) / dk'
dk'
F ( k')
0
k
  k dk'
2
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) 
dx  2 
e
2 
f ( x)  e
 x2 / 2 2
x  
Both are of the form
of a Gaussian!
F (k )  2e
k  1/
1 2k 2
2
x k  1
or
giving physical interpretation to the new variable
x px  h