historical review of atomic theory rutherford’s model of
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Historical Review of Atomic Theory
Rutherford’s model of the Atom
The Bohr Atom
Leucippus
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”).
Ancient Atomic Theory
-- Against Anassagora.
Democritus
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.
“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
1743-1794
Mrs. & Mr. Lavoisier
A. Lavoisier made the first statement of“Law of conservation of Matter”.
He also invented the first periodic table (33 elements).
Dalton
1766-1844
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.
Various atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy (1808).
AvogadroThe Modern Atomic Theory
1776-1856The number of molecules in one mole is now called Avogadro's number.
Avogadro’s Law published in 1811 :“Equal volumes of gases, at the same temperature and pressure,contain the same number of molecules.”
1844-1906
Boltzmann
In 1866, Maxwell formulated the Maxwell-Boltzmann kinetic theory of gases.
van der Waals1831-1879
1837-19231910
In 1873, van der Waals put forward an "Equation of State" embracing both the gaseous and the liquid state
(Ph.D thesis)
The composition of atoms
1791-1867
1833 The discovery of the law of electrolysis by M. Faraday.
Matter consists of molecules and that molecules consists of atoms.
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.
1897 The identification of cathode rays as electrons by J.J. Thomson.
1909 The precise measurement of the electronic charge e by R. Millikan.
1913 The establish of nuclear model of atoms by E. Rutherford.
1856-19401906
e/m experiment
1868-1953
Millikan
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 ?
1871-1937
1908
Rutherford
Rutherford’s Scattering Experiment in 1911
Probe distribution of positive charge with a suitable projectile
Rutherford’s model of the Atom
Rutherford’s scattering experiment Prove Thomson model ?
Projectile :α particle w/. charge +2e.Target : Au foil.
He2+ (Helium nucleus) 2p2n
What is distribution of scattered α particles accordingto Thomson’s model?
Probability for 90o deflection :10-3500
0
vαφ
Estimated deflection angle
∆=
∆= −−
ααα
αφvm
FtanPPtan 11 t
R
Maximum force at glazing incident
2
2
2 R2
R2F ZeeZe
==
Time spent in the vicinity of atomαv
2Rt =∆
For Z(Au)=79, KEα=5MeV, R=10-10m
φ ~ 10-3 ~ 10-4 radians
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
A single encounter of α particle with a massive charge confined to a volume much smaller than size of the atom
to explain the observed large angle scattering
R10m10 ~r 4-14 −=Nucleus
All positive charges and essentially all its mass are assumed tobe concentrated in the small region.
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).
Assuming
φθ
Rutherford’s scattering model(r,θ)
bα(m,v)
Ze+ Trajectory of α particle (r,θ)Polar coordinate w/. nucleus of atom as origin (0,0)
( )( ) r̂dtdr
dtrdmr̂
r4ze2e 2
2
2
2
−=θ
πεo
Deflection due to Coulomb interaction
mvbconstantdtdmr2 ===θLConservation of angular momentum
0Fr =×=rrr
Qτ
Solving r(t) and θ(t) Both are correlated !
change variable ( ) ( )θθu
1r ≡To get trajectory r(θ)
θθθ
θθ
θ ddu
mLu
mL
ddu
u1
dtd
ddu
dudr
dtd
ddr
dtdr 2
2 −=
−===
2
2
2
222
2
2
2
2
dud
muLu
mL
dud
mL
dtd
ddu
mL
dd
dtrd
θθθ
θθ−=
−=
−=
( )( )
−=
2
2
2
2 dtdr
dtrdm
r4ze2e θ
πεo
( )( )
−−=
22
2
2
2
222
mLu
u1
dud
muLmu
4ze2e
θπεo
( )( )22
2
Lm
4ze2eu
dud
oπεθ−=+
( )( )( )
( )( )
22
2
2b2
mv4
ze2emvb
m4
ze2e
o
o
πε
πε
−=
−=
22
2
b2Du
dud
−=+θ
D
General solution u(θ) ( )θθθr
1b2Dsincos 2 =−+= BA
Initial conditions
0b2D
2 =−A 2b2D
=A
−==
∞→ vdtdr
0 r
θ
( ) v0cos0sinmL
ddu
mL
dtdr
0
−=+−−=−==∞→
BAr θθ
b1
Lmv
==BTrajectory of α particle
( ) ( ) θθθ
sinb11cos
b2D
r1
2 +−= Hyperbolic trajectory
Evaluating the scattering angle φ : r→∞ where φ + θ* = π
( ) ∗∗ +−= θθ sinb11cos
b2D0 2
−−=
∗∗∗
2sin211
b2D
2cos
2sin2 2 θθθ
D2b
2tan =
∗θ
=
−
=2
cot2
tan φφπ
φ1φ2
(r,θ)θ
= −
D2bcot2 1φ
5353115.75.710100.60.6100100φφ((oo))b/Db/D
b2α(m,v)
b1 Ze+
( )( )
2mv
4ze2e
D 2oπε=
( )
+=→=
2sin11
2DR 0
ddr
R φθ
=D , when φ=π (b=0)
=∞ , when φ=0 (b→∞)
For any trajectory, there is a close distance to the nucleus, R.
( )( )D4ze2e
2mv2
oπε=b : impact parameter
the distance of closest approach of α particle to the nucleus (Head-on collision)
( )( )
2mv
4ze2e
2o
oπε=D# of α scattered in angle range from φ to φ+dφ implying that # of incident α with impact parameter from b to b+db
( ) t bdb2 ρπ θθ )dP(
D2b
2cot =
φ
( )
−=2sin
d21
2D
2 φ
φdb
P(b)db : probability that an α particle will pass through all nuclei with impact parameter range from b to b+db
( ) ( )
( ) ( )2sind
2sin2sin
4 tD
2sin41
2cot
2 t2
P(b)db
22
2
2
φφ
φφπρ
φφφρπ
−=
−
=
dD
( )2sind sin tD
8 42
φφφρπ
−=
ρ : concentration of nucleus [ #/m3]t : thickness of Au foil [m]
# of α particles detected by detector at scattering angle Φ
( )2/sinA/r
2/mv4/(2e)(ze)
16 tI)(N 4
22
2D Φ
=Φ oπερ ( )2/sin 4 Φ∝ −
( )( )
6o4
o4
oD
oD 102.4~
2/150sin2/5sin
)5(N)(150N −×=
( )( )
m1055.4)J/eV106.1)(eV105(
)coul106.1)(coulNtm109)(79)(2(2
mv4
ze2e
DR
14196
219229
2o
o
−−
−−
×=××
××=
=≤ πε
ND
( )2/sin4 Φ
probability for large angle scattering
estimation for size of nucleus
A very dense nucleus:
• consists of Ze+
• has almost atomic mass• concentrates in a very small volume(<10-14m).
Rutherford’s model of nuclear
atomic size ~ 10-10mZe- 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)
1891-19741935
Chadwick
The model of an atom consisting of two kind of elementary particles, protons and electrons, survived for twenty years until 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 radiationleads to the ejection of high velocity protons.
Atomic model Atomic notation
XAZ
X: element symbolZ: atomic number
Number of p (or e)A: Atomic mass numberNumber sum of p and n
Electrons move in stable orbits ?
?OR
Spectrum from white light source
1824-1887
typical “continuous spectrum” of black body
Kirchhoff
1811-1899
Bunsen
Bunsen burner1855
Gaseous Emission Spectrum
Gaseous Absorption Spectrum
Schematics of energy levels and radiated spectrum of H atom1885 Balmer
Empirical formula
−
=4m
m364.6nm 2
2
mλ
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)
−= 22
nm m1
n11 R
λ17 m100968.1 −×=R
n, m integers w/. n<m
Rydberg constant
1885-1962
BohrBohr’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 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 state to another w/. frequency 。
hf fi EE −=
4. The allowed orbit for the electron where the integer number n is known as a “quantum number” which label
and characterize each atomic state。
π2nnL h
== h
Bohr atom
Ze
e-
r
rmv
rze
41 2
2
2
=oπε
Coulomb attraction Centripetal force
mvr nL == h mrnv h
=n=1,2,3,…Orbital angular momentum
substituting
Consider an atom consists of nucleus with +Ze protons and a single electron –e at radius r
o
2
2
22
22
2
az
n
mze4n
mn
ze4r
=
==
hh oo πεπε
where ao≡Bohr radius=0.529ÅAllowed radii are discrete !
Radius of allowed orbit
correct prediction for atomic sizeFor n=1and Z=1, r=ao=0.5×10-10m
Ze
e-
−+=+=
rze
41
2mvUKEE
22
oπε rze
81 2
oπε−=
o2
2
n
2
n Enr
ze8
1E zo
−=−=πε
eV6.13ae
81E
o
2
o =≡oπε
whereEnergy is quantized !
Total Energy of the electron
Conclude
o
2
n az
nr =
o2
2
n En
E z−=
orbit quantization(1)
energy quantization(2)<0 stable bound state
n=1 : “ground state”
n=2, 3, 4, …. : “excited states”
Ze
e-
Incoming photon
Only discrete energies can be absorbed
=energy difference between states
Photon Absorption Spectra Photon Emission Spectra
Ze
e- Outgoing photon
Only discrete energies can be emitted
=energy difference between states
2i
2f
2
n1
n1z1−= ∞R
λ17o m10097.1E −
∞ ×==hc
2i
2f
o2
fi
n1
n1EzEE−=
−=
hhf λ
c=
Allowed transition
(3)
RRydberg constant
Good to describe the observed spectra of any Hydrogen-like atom.
w/. nucleus charge Ze and a single orbital e-
H, He+, Li2+, …
Bohr’s sketches of electronic orbits in the early 1900s.
Bohr’s Correspondence Principle in 1923
Guide to development of quantum rulesTheory 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 nThe greater the quantum number n, the closer quantum physics approaches classical physics.
[ ] [ ]physics classicalphysics quantum limn
=∞→
For Bohr atom: radii ~n2 approach classical sizes and energy differences ~1/ n2 become essential continuous.
Classical behavior holds !
Application to radiation for large nTransition from nearest neighboring state n+1 to state n for large n
3o
2n
n2Ez
h → ∞→
( ) 22o
2fi
n1
1n1EzEE
−+
=−
=hh
f
Radiant frequency for large n transition ( ) 32
2
222 n2
n2
n1~
n111
n1
n1
1n1
=
+−=−
+
−
33
2
o
2
32oo
222
3oo
22
1 n 2m
ε4z
n2
ε4ε8mz
n2
aε8z
hh πππππ
=
=
=→+
eheehef nn
: emits radiation at orbit frequencyOrbital frequency
33
2
o
222
2o
o
2
n 2m
ε4z
mn
zε42
nε4z
r2v
h
h
h ππππ
ππ
=
==
ee
efclSAME
Classical Radiating system
1882-1964
1925
1887-1975
1925
Hertz
FranckFranck-Hertz Experiment in 1914Direct confirmation that the internal energy states of an atom are quantized
Setup
Acceleratingvoltage
Retardingvoltage
To observe current I to collector as a function of accelerated voltage Va
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.
( ) odrop V4.9eVnV +=
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
nuclear
Orbital e-
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.
from n=1 to n=2
2nd excited state6.7eV
。。。
1st excited state
nm6.253eV9.4
1240eVnm==λ
Energy levels of outer electron of Hg atom
E=010.4eV
Confirmed by photon emission4.9eV
Ground state
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.