Physics of Atoms

Donatella Cassettari

Email: dc43 Room 218 (office)

Tel: 3109 105 (lab)

Course Plan

•Atomic theory started around ~1900

with J.J. Thomson’s model of the atom

•Succession of atomic models:

Rutherford (1911), Bohr (1913), Bohr-Sommerfeld

•Schroedinger equation for the Hydrogen atom

(1 electron, simplest case)

•Many-electron atoms (Alkali atoms, Helium atom)

•Spin-orbit interaction, spin-spin interaction

• Atoms in external fields (electric, magnetic, e.m.)

• Novel applications:

Laser cooling of atoms, Bose-Einstein condensation (BEC)

“Old” quantum theory

or “semi-classical” theory

Why is atomic physics interesting?

Richard Feynman:

“The understanding of the structure of atoms represents the

greatest success of the theory of quantum mechanics”

•Modern physics is based on the understanding of atoms

•Applications: solid state, chemistry, biophysics, astrophysics,

lasers….

Material for this course:

• Textbook:

Haken and Wolf The Physics of Atoms and Quanta 7th ed (2005)

(only selected chapters)

• These notes!

• Tutorial problems

• Practice with past exams

Models of the atom – a short history

J J Thomson’s model ~ 1900

Known that an atom with atomic number Z contains Z electrons

“Plum-pudding model”: these electrons are embedded in a

continuous spherical distribution of positive charge

amounting in total to +Ze.

Diameter of atom

is ~0.1nm

e

e

e

e

e

e

e

+

This idea though was immediately proved invalid by

Rutherford’s expts in 1911 on the scattering of α-particles

(He nuclei) by atoms:

Measure no. of particles scattered between θ and θ +d θ.

In Thomson’s model, the spread out positive charge can’t

produce a large deflection, and the electrons are too light

to do so.

However, a significant fraction (1/10,000) of the α -

particles is found to be deflected thru angles θ greater

than 90 degrees, whereas the theory says this fraction

should be negligible.

“It was quite the most incredible event that has ever

happened to me in my life. It was almost as if you fired a

fifteen inch shell at a piece of tissue paper and it came

back and hit you.”

Rutherford Model 1911

Positive charge is concentrated in a very small nucleus. So α-

particles can sometimes approach very close to the charge Ze

in the nucleus and the Coulomb force

2

)2)((

4

1

r

eZeF

oπε=

Can be large enough to cause large angle deflections.

Nuclear model of the atom

Detailed calculations now show that there is

a non-negligible probability of large angle

scattering

Stability problem

Because the electrons are in orbit, they are continuously

being accelerated and therefore, classically, should emit

radiation. They should lose energy and spiral into the

nucleus!

+e

e

Bohr’s postulates

Bohr 1913: circumvented stability problem by

making two postulates:

1. An electron in an atom moves in a circular orbit for

which the angular momentum is an integral multiple of

2. An electron in one of these orbits is stable. But if it

discontinuously changes its orbit from one where

energy is Ei to one where energy is Ef , energy is

emitted or absorbed in photons satisfying:

h

νhEE fi =−

Photon frequency

Analysis based on Bohr postulates

for hydrogen-like atom (1 electron)

r

vm

r

ZeF

o

2

0

2

2

4

1==

πε

Force of

attraction

r

ZevmKE

o

22

04

1

2

1

2

1

πε==

r

ZePE

o

2

4

1

πε−=

r

ZeETotal

o 24

1

2

πε−=

(m = electron mass) 0

+Ze

evr

Kinetic

energy

Potential

energy

E<0 because it’s

a bound state

hnvrm =0

r

Zevm

o

22

04

1

πε=

r

Ze

rm

nm

o

2

22

0

22

04

1

πε=

h

2

0

22

4Zem

nr o

hπε=

22

42

0

2 2)4(

1

hn

eZmE

oπε−=

1st postulate eliminate

v from system

Energy is quantised

For convenience we define the “Bohr Radius” as:

nmem

a o 0529.04

2

0

2

0 ==hπε

And the energy unit 1 Rydberg (1Ry) as:

eVem

o

6.132)4( 22

4

0 =hπε

0

2

aZ

nr = Ry

n

ZE

2

2

−=and

n=3

n=2

n=1

“ground state”

“excited states”

a = radius of ground state (n=1) orbit for hydrogen (Z=1)

1 Ry=13.6eV is the ground state (n=1) binding energy for

hydrogen (agrees with experiment).

For larger Z-values, radius of orbit shrinks for given n. Binding

energies get much bigger.

0

Emission Spectra

Electron makes transition from initial quantum state ni to final

state nf . The frequency ν of the photon emitted satisfies:

−=−=

222

42

0

2

11

2)4(

1

ifo

finn

eZmEEh

hπεν

−=

223

42

0

2

11

4)4(

11

ifo nnc

eZm

hππελ

Express this in terms of wavelength

−= ∞ 22

2 111

if nnZR

λ

17

3

4

0

2100974.1

4)4(

1 −∞ ×== m

c

emR

o hππε

is the Rydberg constant

There are infinite spectral lines….

Let’s organize them in families.

Family = all transitions with the same final state

Note that for each family there is a series limit for

For the Lyman series, this is at well within the

UV range

The Balmer series lies in the near UV and visible region,

and the others are all in the infrared.

nmR 911

== ∞λ

At time of Bohr’s proposal, only Balmer-Paschen series

were known, and the remaining series were therefore

predicted in advance of the discovery (triumph for Bohr

theory).

∞→in

Absorption spectrum

General formula above also applies to the case where an electron

gains the energy of a suitable photon having energy hν exactly

equal to the difference between initial and final states.

Normally, electron will start off in ground state so only the

Lyman series is observed in absorption spectrum.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Next, we are going to consider two extensions

of the Bohr model:

• Finite nuclear mass

• Elliptical orbits (Bohr-Sommerfeld model)

Finite Nuclear Mass

In the theory above, we assumed that the nucleus is so

massive that it is effectively at rest. But the mass of the

nucleus is finite and the classical model of the atom therefore

envisages that the electron and the nucleus both revolve

around their common centre of mass.

Therefore, any theory should be modified by replacing the

mass of the electron by the “reduced mass” of the system.

massnuclear

masselectron where

1

0

0

0

0

0

=

=

+

=+

=

M

m

M

m

m

Mm

Mmµ

−=

22

2 11.

1

if nnZR

λ

c

eR

o3

4

2 4)4(

1

hπ

µ

πε=

M

mMm

M

mR

R01

1

00 +=

+==

∞

µ

∞R is the limit of R as M tends to infinity

Difference between R and is small (~ 1 part in 1800)

but shows up in precise measurements of spectral lines.∞R

17100974.1 −∞ ×= mR

17100968.1 −×= mR

For hydrogen

Cf.

Isotope shift of spectral lines:

It occurs because different isotopes of the same atomic species

have different reduced masses, hence different values of R.

Example:

Positron “atom” =system containing 1 positron (like an e- with a

positive charge) and 1 electron.

How does emission spectrum differ from hydrogen?

Set M=m , so

21 0

∞∞ =

+

=R

M

m

RR

×=

−= ∞

2

111

2

122if nn

R

λprevious formula

0

All wavelengths are doubled

Effect on energy and radii of quantum states?

We replace m in the formulae above by

2

0m=µ

Therefore, all energies are halved in magnitude, radii

doubled.

0

When viewed at high resolution, transitions split.

Transitions between any two Bohr energy states involve

several spectral lines. This is known as fine structure.

Elliptical orbits (Bohr-Sommerfeld)

Explanation: each energy level actually consists of several

distinct states with almost the same energy.

The first theory that justified this was done by Wilson and

Sommerfeld: they conjectured that electron orbits can be

elliptical, of which a circular orbit is a special case.

Each orbit is specified by 2 parameters instead of 1.

Geometrically by semi-major and semi-minor axes a,b, no just

radius r. Therefore we have now two quantum numbers.

Thus, energy levels turn out to be dependent on two quantum

numbers, but only when one takes relativistic considerations

into account.Without relativity, we get the same formula for E as before.

Relativistic correction: electrons in very eccentric orbits have

large velocities when they are near the nucleus, so v/c is NOT

negligible.

= “azimuthal” quantum numberθn

a

b

n

n=θ

a

b+Ze

e

Each energy level n is split into several sub-levels

corresponding to orbits of different eccentricity (i.e. different )

fine structure

θn

Final remarks on the early atomic models

1) Bohr’s theory works! It generates the correct formula for the

energy levels in a hydrogen atom.

This is because the Bohr postulate: inadvertently

makes use of the de Broglie wave associated with an electron.

hnvrm =0

Supposing the classical orbit is circular, let’s look at that the

associated de Broglie wave following the motion:

in order for the wave to return to its initial value (i.e. we are

requiring that the wave be single valued), we must have

λπ nncecircumferer ==2

de Broglie wavelength

λ

hp =

vm

h

p

h

0

==λ

nhvrm =02π

hnvrm =0The Bohr condition

2) Inadequacies of the Bohr Theory

•Does well to describe hydrogen, but can be extended only to

1-electron atoms, i.e. hydrogen-like, with higher Z values.

•Theory does not explain rate at which transitions occur

between states, i.e. the relative intensities of spectral lines.

•The theory is ad hoc and lacks a satisfying basis.

Superseded by Quantum Mechanics, initiated by de Broglie

(1924) and Schroedinger (1926).