Physics 3313 - Lecture 9
Monday February 23, 2009
Dr. Andrew Brandt
1. The Electron
2. Rutherford Scattering
3. Bohr Model of Atom
2/23/2009
3313 Andrew Brandt
1
Evolution of Atomic Models
• 1803: Dalton’s billiard ball
model
• 1897: J.J. Thompson
Discovered electrons
Cathode ray tube
– Used cathode ray tubes
– Called corpuscles
– Made a bold claim that these
make up atoms
– Measured charge/mass ratio
• 1904: J.J. Thompson
Proposed a “plum pudding”
model of atoms
– Negatively charged electrons
embedded in a uniformly
distributed positive charge
Personally I prefer chocolate chip
cookie model
Wednesday, Aug. 28, 2008
PHYS 3446, Fall 2008
Andrew Brandt
2
Millikan’s Oil Drop Experiment
• Millikan (and Fletcher) in 1909 measured charge of electron, showed that
free electric charge is in multiples of the basic charge of an electron
• By varying electric field to balance gravitational field, could determine
charge of electron
2/23/2009
3313 Andrew Brandt
3
Rutherford Experiment
• 1911: Geiger and Marsden with Rutherford performed a scattering
experiment firing alpha particles at a thin gold foil
Wednesday, Aug. 28, 2008
PHYS 3446, Fall 2008
Andrew Brandt
4
Rutherford Scattering
K
N ( ) 
sin 4
• The actual result was very different—
although most events had small angle
scattering, many wide angle scatters were
observed
• “It was almost as incredible as if you fired a 15 inch
shell at a piece of tissue paper and it came back at
you”
• Implied the existence of the nucleus.
• We perform similar experiments at
Fermilab and CERN to look for fundamental
structure

2
KE 2
Rutherford Example
• On blackboard demonstrate size of radius from distance of closest
approach
2/23/2009
3313 Andrew Brandt
6
Ruherford Atom
• 1912: Rutherford’s planetary
model, an atomic model with a
positively charged heavy core
surrounded by circling electrons
• But many questions:
a) Z=A/2, Z=atomic number
(number of electrons or protons)
what is the other half of the
atomic weight ?
b)what holds the nucleus
together?
c)how do electrons move around
the nucleus and does their
motion explain observed atomic
properties?
Wednesday, Aug. 28, 2008
PHYS 3446, Fall 2008
Andrew Brandt
7
Electron Orbit
• Electrons must be in motion or would get sucked into nucleus by Coulomb
Force
• “Assume a spherical orbit” : this implies that the centripetal force must be
balanced by the Coulomb force
2
2
e
• mv  1 e
so v 
r
4 0 r 2
4 0 mr
1 2
e2
E  mv 
2
4 0 r
• Energy of electron is kinetic energy plus potential energy
(where potential energy is defined to be 0 at infinity and negative at closer
radius since you have to input work to keep electron and proton apart)
E
e2
8 0 r

e2
4 0 r

e2
8 0 r
• Can thus determine radius of Hydrogen atom given Binding Energy (-13.6 eV)
N m2
(1.6 10 C ) 9 10
2
e 2
C
R

 5.3 1011 m
19
8 0 E 2(13.6eV 1.6 10 J / eV )
19
2/23/2009
2
9
3313 Andrew Brandt
This is known
as Bohr Radius
8
Quantum Effects
•
•
•
•
•
Classically an accelerating charge revolving with a
frequency  would radiate at the same frequency.
As it radiates, it loses energy, and radius decreases
and frequency increases (death spiral)
Law of physics in macro-world do not always apply
in micro-world
Quantum phenomena enter the picture
Evidence for quantum nature of atoms: discrete
line spectra emitted by low pressure gas when
excited (by electric current)—only certain
wavelengths emitted
A gas absorbs light at some wavelengths of
emission spectra, with the number intensity and
wavelength of absorption lines depending on
temperature, pressure, and motion of the source.
This can be used to determine elements of a star
and relative motion
2/23/2009
3313 Andrew Brandt
9
Spectral Lines
• For Hydrogen Atom (experimental observation):
1
1
1
 R( 2  2 )

nf
ni
• where nf and ni are final and initial quantum states
• R=Rydberg Constant 1.097 107 m1  0.01097 nm 1
• Balmer Series nf = 2 and ni=3,4,5 visible wavelengths
in Hydrogen spectrum 656.3, 486.3,…364.6 (limit as
n)
2/23/2009
3313 Andrew Brandt
10
Spectral Lines
2/23/2009
3313 Andrew Brandt
11
Bohr Atom
•
1)
2)
3)
Assumptions
The electron moves in circular orbits
under influence of Coulomb force
Only certain stable orbits at which
electron does not radiate
Radiates when “jumps” from a more
energetic initial state to a lower energy
final state
E  E  hv
i
•
•
f
Introduced quantum number of orbit,
can describe using de Broglie language
(he didn’t, since it didn’t exist yet)
Allowed orbits are integer number of
de Broglie wavelengths
n  2 rn
2/23/2009
non-integer
number of
wavelengths
is discontinuous,
so not physical
3313 Andrew Brandt
12
Bohr Atom Derivation
• Consider n=1, the circular orbit case: for this to be selfconsistent n  2 r implies that   2 r
n
1
e
h
v

• 
with
yields   h
4 0 mr
e
mv
•
5.3  1011 m
6.63 1034 J sec

2
19
N
m
9
31
1.6 10 C
9 10
9.1

10
kg
2
C
4 0 r
m
so finally
  33 10  11 m  2 r
• Generally
so rn  n 2 r1
2/23/2009
2 2
n
h 0
2 2
2 2
r

n   4 rn so
n
 me2
and r1  a0  5.3 1011 m
3313 Andrew Brandt
with
h 2 0
r1 
 me2
13
Bohr Atom Derivation
• Consider n=1, the circular orbit case: for this to be selfconsistent n  2 r implies that   2 r
n
1
e
h
v

• 
with
yields   h
4 0 mr
e
mv
•
5.3  1011 m
6.63 1034 J sec

2
19
N
m
9
31
1.6 10 C
9 10
9.1

10
kg
2
C
4 0 r
m
so finally
  33 1011 m  2 r
• Generally
so rn  n 2 r1
2/23/2009
2 2
n
h 0
2 2
2 2
r

n   4 rn so
n
 me2
and r1  a0  5.3 1011 m
3313 Andrew Brandt
with
h 2 0
r1 
 me2
14