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.