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Historical Review of Atomic Theory
Rutherford’s model of the Atom
The Bohr Atom
Ancient Atomic Theory
A Greek philosopher, in around 400BC,
“There are small particles which can not be further subdivided.”
Leucippus called these indivisible particles atoms.
(from the Greek word atomos, meaning “indivisible”).
Leucippus
-- Against Anassagora.
Leucippus's atomic theory was further developed by his disciple,
Democritus who concluded that infinite divisibility of a substance
belongs only in the imaginary world of mathematics.
Democritus
“All things are composed of minute, invisible, indestructible
particles of pure matter which move about eternally in infinite
empty.”
-- Against the ancient Greek view,
“There were four elements that all thing were made from:
Earth, Air, Fire and Water.”
The Modern Atomic Theory
Mrs. & Mr. Lavoisier
A. Lavoisier made the first statement of
“Law of conservation of Matter”.
He also invented the first periodic table (33 elements).
1743-1794
Dalton
Dalton made two assertions about atoms:
(1) Atoms of each element are all identical to one another
but different from the atoms of all other elements.
(2) Atoms of different elements can combine to form more
complex substances.
1766-1844
Various atoms and molecules as depicted in John Dalton's
A New System of Chemical Philosophy (1808).
Avogadro
The Modern Atomic Theory
Avogadro’s Law published in 1811 :
“Equal volumes of gases, at the same temperature and pressure,
contain the same number of molecules.”
The number of molecules in one mole is now called Avogadro's number. 1776-1856
Boltzmann
In 1866, Maxwell formulated
the Maxwell-Boltzmann kinetic theory of gases.
1844-1906
van der Waals
1831-1879
In 1873, van der Waals put forward an "Equation of State"
embracing both the gaseous and the liquid state
(Ph.D thesis)
1837-1923
1910
The composition of atoms
9 1833 The discovery of the law of electrolysis by M. Faraday.
Matter consists of molecules and that
molecules consists of atoms.
1791-1867
Charge is quantized.
Only integral numbers of charge are transferred at the electrodes.
The subatomic parts of atoms are positive and negative charges.
The mass and the size of the charge remained unknown.
9 1897 The identification of cathode rays as electrons by J.J. Thomson.
9 1909 The precise measurement of the electronic charge e by R. Millikan.
9 1913 The establish of nuclear model of atoms by E. Rutherford.
e/m experiment
1856-1940
1906
Millikan
1868-1953
1923
Oil drop experiment
Thomson’s model of atom (plum pudding model)
1904
Since atoms are neutral, there must be positive particles
balance out the negative particles.
Atom was a sphere of positive electricity (which was diffuse)
with negative particles imbedded throughout.
(Both particles are evenly distributed.)
Constituents of atoms (known before 1910)
There are electrons with measured charge and mass.
There are positive charge to make the atom electrical neutral.
The size of atom is known to be about 10 -10 m in radius.
How is positive charge distributed ?
Rutherford
Rutherford’s Scattering Experiment in 1911
Probe distribution of positive charge
with a suitable projectile
1871-1937
Rutherford’s model of the Atom
1908
Rutherford’s scattering experiment
Projectile :α particle w/. charge +2e.
Target : Au foil.
Prove Thomson model ？
He2+ (Helium nucleus) 2p2n
What is distribution of
scattered α particles according
to Thomson’s model ？
φ
vα
R
Estimated deflection angle
 ∆Pα
φ = tan 
 Pα
−1
For Z(Au)=79, KEα=5MeV, R=10-10m
φ ~ 10-3 ~ 10-4 radians
Probability for
90o
deflection
0
:10-3500

−1  F∆t
 = tan 
 m α vα




Maximum force at glazing incident
2eZe 2 Ze 2
F= 2 = 2
R
R
Time spent in the vicinity of atom ∆t =
2R
vα
Experimental results：(Geiger and Marsden)
99% of deflected α particles have deflection angle φ≤3o.
However, there are 0.01% of α particles have larger angle φ>90o.
much larger than the expected quantity by Thomson’s model
Rutherford’s model of the structure of the atom
to explain the observed large angle scattering
A single encounter of α particle with a massive charge
confined to a volume much smaller than size of the atom
Nucleus
r ~ 10 -14 m = 10 −4 R
All positive charges and essentially all its mass are assumed to
be concentrated in the small region.
Assuming
Heavy atom : nucleus does not recoil during the scattering process.
α particle does not penetrate the nucleus.
velocity of α particle v is less than c (v~0.05c).
Rutherford’s scattering model
(r,θ)
α(m,v)
θ
b
φ
Ze+
Trajectory of α particle (r,θ)
Polar coordinate w/. nucleus of atom as origin (0,0)
2
2

(
2e )(ze )
d r  dθ  
Deflection due to Coulomb interaction
r̂ = m  2 − r   r̂
2
4πε o r
 dt  
 dt
Conservation of angular momentum
r r
Qτ = r × F = 0
r
dθ
L = mr
= constant = mvb
dt
2
Solving r(t) and θ(t)
Both are correlated !
To get trajectory r(θ)
change variable
r (θ ) ≡
1
u (θ )
dr dr dθ dr du dθ
1 du  L 2 
L du
=
=
=− 2
=
−
u


dt dθ dt du dθ dt
u dθ  m 
m dθ
d 2 r d  L du  dθ  L d 2 u  L 2 
L2 u 2 d 2 u

u =− 2
=  −
=

−
2
2 
dt
dθ  m dθ  dt  m dθ  m 
m dθ 2
2
(2e)(ze) m
d
u
2
2


u
+
=
−
(2e)(ze) = m d r − r dθ 
2
2
d
θ
4
πε
L




o
2
4πε o r 2
dt
dt
  

(2e)(ze) m
=
−
2 2
2
2 2

4πε o (mvb)2
(2e)(ze) u 2 = m − L u d u − 1  Lu  
2
2
 m 
4πε o
m
d
θ
u
(
2e )(ze )


 
−
4πε o
=
2
mv
d 2u
D
2b 2
+u =− 2
2
2
dθ
2b
D
D
1
General solution u(θ) = A cos θ + B sin θ − 2 =
2b
r (θ )
Initial conditions
D
D
A− 2 = 0
A= 2
2b
2b
 θ = 0
r → ∞ dr
dr
L du
L
v
=
−
=−
= − (− A sin 0 + B cos 0) = − v
 dt
dt r →∞
m dθ θ =0
m
mv 1
B=
=
Trajectory of α particle
L
b
1
D
1
= 2 (cos θ − 1) + sin θ Hyperbolic trajectory
r (θ ) 2b
b
Evaluating the scattering angle φ : r→∞ where φ + θ* = π
(
)
D
1
∗
0 = 2 cos θ − 1 + sin θ ∗
2b
b
∗
 θ ∗   θ ∗  D  

  
θ
2
2 sin   cos  =
1 − 1 − 2 sin   
 2   2  2b  
 2   
 θ ∗  2b
 π −φ 
φ 
= tan
tan  =
 = cot 
 2 
2
 2 D
 2b 
φ = 2 cot  
D
−1
α(m,v)
b2
(r,θ)
θ
b1
φ1
φ2
Ze+
b : impact parameter
(2e )(ze)
4πε o
D=
mv 2
2
b/D
φ(o)
100
10
1
0.6
5.7
53
mv 2 (2e )(ze )
=
2
4πε o D
the distance of closest approach of α particle
to the nucleus (Head-on collision)
For any trajectory, there is a close distance to the nucleus, R.
 =D , when φ=π (b=0)
1
D
dr
dθ
=0 →R =
R
1 +

2  sin (φ 2) 
=∞ , when φ=0 (b→∞)
(2e )(ze)
# of α scattered in angle range from φ to φ+dφ implying that
# of incident α with impact parameter from b to b+db D = 4πε o
 φ  2b
cot  =
2 D
 1

d
φ

D 2

db = −  2
2  sin (φ 2 ) 




P(b)db : probability that an α particle will pass through all nuclei with
impact parameter range from b to b+db
2πbdb (ρ t )
P(θ )dθ
ρ : concentration of nucleus [ #/m3]
t : thickness of Au foil [m]
P(b)db

 D  φ   1
dφ

= 2π (ρ t ) cot   −
2
 2   4 sin (φ 2 ) 
2
π
πρ tD 2
sin φ
dφ
2 sin φ dφ
= − ρ tD
=−
2
2
8
sin 4 (φ 2 )
4 2 sin (φ 2 ) sin (φ 2 )
mvo2
2
# of α particles detected by detector at scattering angle Φ
N D (Φ ) =
ρ tI  (2e)(ze) / 4πε o 

16 
mv 2 / 2
2
A/r 2

4
 sin (Φ / 2)
z probability for large angle scattering
N D (150o )
sin 4 (5o / 2 )
−6
=
~
4
.
2
×
10
N D (5o )
sin 4 (150o / 2 )
∝ sin −4 (Φ / 2 )
ND
¾ estimation for size of nucleus
(2e)(ze)
R≤D=
4πε o
sin 4 (Φ / 2)
mvo2
2
(2)(79)(9 ×109 Ntm 2 coul − 2 )(1.6 ×10 −19 coul) 2
−14
=
=
4
.
55
×
10
m
6
−19
(5 ×10 eV)(1.6 ×10 J/eV)
Rutherford’s model of nuclear
A very dense nucleus：
• consists of Ze+
• has almost atomic mass
• concentrates in a very small volume（<10-14m）.
atomic size ~ 10-10m
Ze- revolve around the nucleus.
Problems with Rutherford’s model
What composes the other half of the nuclear mass ?
How to keep many protons in such a minute nucleus ?
How do electrons move around the nucleus to form a stable atom ?
In 1886, Eugen Goldstein discovered " canal rays" that had properties
similar to those of cathode rays (streams of electrons) but consisted of
positively-charged particles many times heavier than the electron.
In 1913, Rutherford did α particle scattering of gas N2 and concluded that
canal rays of this atom (hydrogen) would consist of a stream of particles,
each carrying a single unit of positive charge.
He called these particles protons.
Greek “protos” (first)
The model of an atom consisting of two kind of elementary particles,
protons and electrons, survived for twenty years until
Chadwick
the discovery of the neutron by James Chadwick in 1932.
In 1930, Bothe and Becker reported that the exposure of light
elements, like Be, to α rays leads to highly penetrating radiation.
In 1931-1932, Curie and Joliot reported that the exposure of
Hydrogen –containing materials, like paraffin, to this new radiation
leads to the ejection of high velocity protons.
1891-1974
1935
Atomic model
Atomic notation
A
Z
X
X: element symbol
Z: atomic number
Number of p (or e)
A: Atomic mass number
Number sum of p and n
Electrons move in stable orbits ?
OR
?
Spectrum from white light source
Kirchhoff
1824-1887
typical “continuous spectrum” of black body
Bunsen
1855
Bunsen burner
1811-1899
Gaseous Emission Spectrum
Gaseous Absorption Spectrum
Schematics of energy levels and radiated spectrum of H atom
1885 Balmer
Empirical formula
 m2 

λm = 364.6nm 2
m −4
m=3,4,5….
Visible light∼near UV
1906-1914 Lyman, nf=1
(UV)
1908 Paschen, nf=3
1922 Brackett, nf=4
1924 Pfund, nf=5 (IR)
1890 Rydberg and Ritz formula (n<m)
1
λnm
1 
 1
= R 2 − 2 
m 
n
R = 1.0968 ×107 m −1
n, m integers w/. n<m
Rydberg constant
Bohr
Bohr’s quantum model of the Atom in 1913
Four postulates：
1. An electron in an atom moves in a circular orbit about the
nucleus under the influence of the Coulomb attraction
1885-1962
between the electron and the nucleus。
1922
2. The allowed orbit is a stationary orbit w/. a constant energy E。
3. Electron radiates only when it makes a transition from one stationary
Ei − Ef
state to another w/. frequency f =
。
h
h
4. The allowed orbit for the electron L = nh = n
where the
2π
integer number n is known as a “quantum number” which label
and characterize each atomic state。
Bohr atom
e-
Consider an atom consists of nucleus with +Ze protons
and a single electron –e at radius r
r
ze 2 mv 2
=
2
4πε o r
r
1
Ze
Coulomb attraction
Orbital angular momentum
substituting
Centripetal force
L = nh = r mv
n=1,2,3,…
nh
v=
mr
2


πε
4πε o n 2 h 2
4
h
2
o

= n  2
r=
Radius of allowed orbit
2
ze
m
 ze m 
Allowed radii are discrete !
n2
= a o where ao≡Bohr radius=0.529Å
z
For n=1and Z=1,
r=ao=0.5×10-10m correct prediction for atomic size
e-
Total Energy of the electron
2
2


1
ze
mv
1 ze
 = −
+  −
E = KE + U =
8πε o r
2
 4πε o r 
2
Ze
1 ze 2
z2
En = −
= − 2 Eo
8πε o rn
n
1 e2
where E o ≡
= 13.6eV
8πε o a o
Energy is quantized !
Conclude
n2
a o orbit quantization
(1) rn =
z 2
z
(2) E n = − 2 E o energy quantization
n
<0 stable bound state
n=1 : “ground state”
n=2, 3, 4, …. : “excited states”
Photon Absorption Spectra
Photon Emission Spectra
e-
Incoming
photon
Outgoing
photon
eZe
Ze
Only discrete energies can be absorbed Only discrete energies can be emitted
=energy difference between states
=energy difference between states
1
1
1
c
z 2Eo 1
1
2
= z R∞ 2 − 2
(3) f =
=
− 2 =
2
λ
λ
nf ni
h
h nf ni
E
Allowed transition
Rydberg constant R = o = 1.097 × 10 7 m −1
Ei − Ef
∞
hc
Good to describe the observed spectra of any Hydrogen-like atom.
w/. nucleus charge Ze and a single orbital eH, He+, Li2+, …
Bohr’s sketches of electronic orbits in the early 1900s.
Bohr’s Correspondence Principle in 1923
Guide to development of quantum rules
Theory should agree with classical physics in limit
in which quantum effects become unimportant.
For Bohr atom: radius, velocity, angular momentum, and
energy must be of a size where classical behavior holds
Typical hold for large n
The greater the quantum number n, the closer
quantum physics approaches classical physics.
lim [quantum physics] = [classical physics]
n →∞
For Bohr atom: radii ~n2 approach classical sizes and
energy differences ~1/ n2 become essential continuous.
Classical behavior holds !
Application to radiation for large n
Transition from nearest neighboring state n+1 to state n for large n
f =
Ei − Ef
h
z 2Eo
1
1
=
− 2
2
h (n + 1) n
2
z
Eo 2
→∞
n
→
h n3
−2
1
1
1   1   1 2 2
− 2 = 2 1 − 1 +  ~ 2 = 3
2
(n + 1) n n   n   n n n
Radiant frequency for large n transition
2
 ze 
 ze  2
 zeem  2
m


 3 h = 

f n +1→n = 
h
=
2  3
 4πε  2π n 3h 3
8
ε
a
n
8
ε
4
ε
n
h
π
π
π
o o 
o
o



o 

2 2
2
2 2 2
Classical Radiating system : emits radiation at orbit frequency
Orbital frequency
v  ze 

f cl =
= 
2πr  4πε o nh 
2
2
 4πε o n h   ze 
m




=
2π  2

m   4πε o  2π n 3h 3
 ze
2
2
2
SAME
Franck-Hertz Experiment in 1914
Franck
Direct confirmation that the internal energy states of
an atom are quantized
Setup
1882-1964
1925
Hertz
Accelerating
voltage
Retarding
voltage
To observe current I to collector as a function of
accelerated voltage Va
1887-1975
1925
‹ When the tube is empty, once kinetic energy of electron
acquired by acceleration in Va, is more than retarded potential,
current I will increases with increasing Va。
‹ When the tube is filled with low pressure of mercury vapor,
there are collisions between some electrons and Hg atoms.
Will current I change？
4.9V
Current I seems increases
w/. increasing Va, however,
current I shows sudden
drops at certain Va.
Vdrop = n (4.9eV ) + Vo
Observation: Steps in current I with a period of 4.9eV.
Why does I change when the tube is filled with Hg atoms?
collisions between some electrons and Hg atoms
Current drop : Partly electrons lost KE and cannot overcome eVs.
Incoming
electron
Orbital e-
nuclear
Scattered
electron
Inelastic collision, 4.9eV of KE of
incident electron excites Hg electron.
Inelastic collision leaves electron
with less than Vs, so does not
contribute to current.
Energy levels of outer electron of Hg atom
10.4eV
。。。
6.7eV
E=0
2nd excited state
1st excited state
4.9eV
Ground state
from n=1 to n=2
1240eVnm
λ=
= 253.6nm
4.9eV
Confirmed by
photon emission
Bohr’s model is usually referred as “old” quantum mechanics.
Bohr’s theory provides a simple model that gives the correct
energy levels of Hydrogen.
Critique
the theory only tells us how to treat periodic systems.
the theory does not calculate the rate at which transitions occur.
the theory is only applicable to one-electron atom, especially for H.
Even alkali metals (Li, Na, K, Rb, Cs) be treated in approximation.
entire theory somehow lack coherence.