W. Ubachs – Lectures MNW-1

Physics of the Atom

The Nobel Prize in Physics 1922"for his services in the investigation

of the structure of atoms and of the radiation emanating from them"

Niels Bohr

The Nobel Prize in Chemistry 1908"for his investigations into the disintegration of the elements,

and the chemistry of radioactive substances"

ErnestRutherford

W. Ubachs – Lectures MNW-1

atoms : electrically neutralthey can become chargedpositive and negative charges are aroundand some can be removed.

popular atomic model “plum-pudding” model:

Early models of the atom

W. Ubachs – Lectures MNW-1

Result fromRutherfordscattering

Applet for doing the experiment:http://www.physics.upenn.edu/courses/gladney/phys351/classes/Scattering/Rutherford_Scattering.html

( )2/sin1

441

4

22

0 θπεσ

⎟⎟⎠

⎞⎜⎜⎝

⎛=

Ω KZze

dd

Rutherford did an experiment that showed that the positively charged nucleus must be extremely small compared to the rest of the atom.

Rutherford scattering

W. Ubachs – Lectures MNW-1

the radius of the nucleus is 1/10,000 that of the atom.

the atom is mostly empty space

Rutherford scatteringthe smallness of the nucleus

Rutherford’s atomic model

W. Ubachs – Lectures MNW-1

A very thin gas heated in a discharge tube emits light only at characteristic frequencies.

Atomic Spectra: Key to the Structure of the Atom

W. Ubachs – Lectures MNW-1

Line spectra: absorption and emission

Atomic Spectra: Key to the Structure of the Atom

W. Ubachs – Lectures MNW-1

The wavelengths of electrons emitted from hydrogen have a regular pattern:

The Balmer series in atomic hydrogen

Johann Jakob Balmer

W. Ubachs – Lectures MNW-1

the Lyman series:

the Paschen series:

Lyman, Paschen and Rydberg series

Janne Rydberg

W. Ubachs – Lectures MNW-1

A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by classical atomic theory.

The Spectrum of the hydrogen Atom

W. Ubachs – Lectures MNW-1

The Bohr Model

Solution to radiative instability of the atom:1. atom exists in a discrete set of stationarystatesno radiation when atom is in such state

2. radiative transitions quantum jumpsbetween levels

3. angular momentum is quantized

fi EEhch −==λ

ν

These are ad hoc hypotheses by Bohr, against intuitions of classical physics

.

W. Ubachs – Lectures MNW-1

An electron is held in orbit by the Coulomb force: (equals centripetal force)

The Bohr Model: derivation

hnmvrL n ==

Coulombcentr FF =

20

22

4 nn rZe

rmv

πε=

Solve this equations for r:

Impose the quantisation condition:

20

2

4 mvZern πε

=

W. Ubachs – Lectures MNW-1

The Bohr Model: derivation

Solve: the size of the orbit is quantized(and we know the size of an atom !)

.

Size of an atom: 0.5 Å

W. Ubachs – Lectures MNW-1

The Bohr Model: derivation

2

2

0

2

2

2

0

2

24421

h

menZ

rZeE

nn ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=

πεπε

nnpotkinn r

Zer

ZemvEEE0

2

0

22

41

21

421

πεπε⎟⎠⎞

⎜⎝⎛ −=−=+=

2

2

0

2

24 h

emeR ⎟⎟⎠

⎞⎜⎜⎝

⎛=∞ πε

The energy of a particle in orbit

∞⎟⎟⎠

⎞⎜⎜⎝

⎛−= R

nZEn 2

2

So

Define

With the Rydberg constant

W. Ubachs – Lectures MNW-1

Reduced mass in the old Bohr model the one particle problem

Relative coordinates:

21 rrr rrr −=

Centre of Mass

021 =+ rMrm rr

Position vectors:

rMm

Mr rr

+=1

rMm

mr rr

+−=2

Velocity vectors:

vMm

Mv rr

+=1

vMm

Mv rr

+−=2

Relative velocity

dtrdvr

r =

EXTRA

W. Ubachs – Lectures MNW-1

Kinetic energy

2222

211 2

121

21 vvmvmK μ=+=

Reduced mass

MmmM+

=μ

Angular momentum

vrrvmrvmL μ=+= 222111

Centripetal force

rv

rvm

rvmF

2

2

222

1

211 μ

===

EXTRA

Quantisation of angular momentum:

hnhnvrL ===π

μ2

These are the equations for asingle (reduced) particle

W. Ubachs – Lectures MNW-1

EXTRA

Quantisation of radius in orbit:

0

2

2

20

2 4 amZn

eZnr e

n μμπε

==h

Result for the one-particle problem

Energy levels in the Bohr model:

∞−= Rmn

ZEe

nμ

2

2

Rewrite:

222

2

21 cm

mnZE e

en αμ

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

dimensionless energy

1371

≈α

Isotope effectem≈μFor all atoms:

Calculate the isotope shift on Lyman-α:

MmmM+

=μ

W. Ubachs – Lectures MNW-1

The lowest energy level is called the ground state; the others are excited states.

Optical transitions in The Bohr Model

Calculate wavelengthsand frequencies

12 EEhc

fc

−==λ

W. Ubachs – Lectures MNW-1

Quantum states and kinetics

eVJTkK B 04.0102.623 21 =×== −

kTEn

neTP /)( −=

Hydrogen at 20°C.

Estimate the average kinetic energy of whole hydrogen atoms (not just the electrons) at room temperature, and use the result to explain why nearly all H atoms are in the ground state at room temperature, and hence emit no light.

Gas at elevated temperature – probability exited state is populated:

Energy in gas:

∑

∑−

−

=

n

kTE

kTE

nn

n

n

e

eEE /

/

W. Ubachs – Lectures MNW-1

de Broglie’s Hypothesis Applied to Atoms

λπ nrn =2Electron of mass m has a wave nature Electron in orbit: a standing wave

Substitution gives the quantum condition

π2nhmvrL n ==

Another way of looking at the quantisation in the atom

W. Ubachs – Lectures MNW-1

These are circular standing waves for n = 2, 3, and 5.

de Broglie’s Hypothesis Applied to Atoms

Standing waves do not radiate;Interpretation: electron does not move(no acceleration)