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1AMQ-Atoms, Molecules and Quanta
Spring Semester, 2010
-Lecturer: Zsolt Podolyák [Office06BC04]
e-mail: [email protected]
•Course provides an introduction to Modern Physics
•It provides the basis for many advanced courses including
-Quantum Mechanics, Solid State, Semiconductor
and Nuclear Physics
•It begins by introducing phenomena which led to the need
for a new mechanics, followed by the introduction of
Quantum Mechanics and how it can be used to describe
the properties of atoms.
Books:- Krane, Modern Physics, 2nd Edition, Wiley
Halliday, Resnick and Walker, Fundamentals of
Physics, 4th Edition,Wiley(Chapters 40,41,43-45)
Eisberg and Resnick, Quantum Physics,Wiley
Lectures/tutorials: slides plus notes on board. All pictures
used in class, as well as the solutions to the class tutorials
will be posted on the physics intranet
Assessment-Week 7-multiple choice test(Worth % of
total, exam = %).
Outline of Course 1AMQ
1. Introduction; Microscopic World-Sizes and Units
2. Quanta and Electromagnetic waves
3. Blackbody Radiation
4. Photoelectric Effect
5. Compton Effect
Quantum Mechanics
6. Wave-particle duality; Uncertainty principle
7. Schrodinger’s wave equation
8. Simple cases: free electrons, electrons in a box, quantum nos.
The Simplest Atom: Hydrogen
9. Spectral series for Hydrogen
10. Bohr’s Theory for the Hydrogen atom
11. Hydrogen atom in Quantum Mechanics
12. Spatial quantisation and electron spin
13. Fine Structure and Zeeman splitting
Multi-electron Atoms
14. Spectroscopic notation, Pauli principle, level ordering
15. Electron screening, shell and sub-shell structure
16. Characteristic X-rays and selection rules
17. Optical spectra of atoms and selection rules
18. Adding angular momenta for two electrons [He atom]
19. Hydrogen molecular ion
20. Hydrogen molecule and covalent bonding
• Classical Physics vs. Modern Physics
( approx. before and after ~ 1900 )
● End of 19th Century-physicists thought they had a good
grasp of the physical world with Newton’s Laws and
Maxwell’s equations for electromagnetism.
●Suddenly all this was changed by a series of discoveries:
radioactivity, X-rays, discovery of and measurements on
electrons. They also found it impossible to explain the spectra
from blackbodies.
•Reason for new paradigms: ability to make better
New phenomena observed, hence new theories
•Main Result:- The exploration of 3 extremes of Nature
- Very fast: special relativity replaces Newtonian mechanics
- Very small: quantum mechanics replaces Newtonian mechanics
- Very large: general relativity replaces Newtonian mechanics
•These new theories of modern physics are refinements of the
old ideas but are quite radical in conception.
•The old theories work perfectly well at everyday velocities
and scales.
New Experiments  New Theories  New Concepts
Measurements of
speed of light
New concepts of
Space & Time
Spectrum of Light
New ideas about
measurement and
a) from hot glowing objects
b) from electrical breakdown
in gases
Key experiments: -to do with light (very fast, c =3 x 108 ms)
-to do with atoms (very small: 10-10 m)
Here we are concerned with atoms and the theories needed
to describe their properties.We will not be concerned with
Sizes, Orders-of- magnitude and Units
•Often important to know roughly how large
something is!
-Chemical engineers need thermal constants
to 5-fig. Accuracy to predict manufacture
-Mech.engs-factor of two may make bridge fall
-Physicists-estimate whether steel is stronger
than butter from atomic constitution.
•Mass of proton= 1.66x10-27 kg
=No. x Order-of-magn. X unit
•Units-Self-consistent and comprehensible
All systems are former[SI/CGS/British]
not all are latter[see above]
● To give a precise value for any quantity we need at least
three things
No.x order-of-magnitude x units (=/- error)
1022 m
10-15 m
10-14 m
10-10 m
1019 m
10-9 m
1012 m
107 m
10-6 m
10-5 m
The Microscopic World
•ATOMS--10-10 m
•NUCLEI--10-14 m
•NUCLEONS--10-15 m
Our World View
As we have seen our World view is of a Universe with a
series of layers: each layer containing objects on a
particular length scale.
Galaxy clusters----6x1022 m
Galaxies-------------1019-1020 m
Solar system--------6x1012 m
Earth-----------------12.7x106 m
Crystals/humans---10-2 – 10 m
Atoms----------------10-10 m
Nuclei----------------10-14 m
Nucleons------------ 10-15 m
Is there a significance to this picture?
At each scale there is a dominant force. Thus gravity dictates
the motion of the planets in the solar system and the
nuclear force dictates the size of nuclei.
How do we know these sizes?
What units are appropriate?
Sizes of Atoms-Dalton’s Atomic Theory
1803-Knowledge of Chemistry was good enough for John
Dalton to propose an atomic theory. Basic idea: chemical
elements are composed of tiny, indivisible fundamental
particles, all identical, they dictate properties of the element
Dalton’s Atomic Theory
1. Chemical elements composed of extremely small particles
which retain their identities in chemical processes. Atom is
smallest mass of an element which can take part in chemical
2. Each atom has a definite weight.
3. Each element consists of a particular type of atom which differs
in weight from atoms of every other element
4. Atoms combine in simple numerical ratios.
Note: Smallest particle of a chemical compound is a molecule
e.g. HCl-Dalton’s compound atom.
Later Modification:-Discovery of ISOTOPES means that
a. An element may have atoms differing in weight.
b. It is not ATOMIC WEIGHT which characterises the element
c. It is ATOMIC NUMBER= +ve charge on the atomic nucleus
-all isotopes of an element have the same atomic number.
Avogadro’s Number
• Avogadro’s hypothesis: Equal volumes of all gases, under the
same conditions of temperature and pressure, contain identical
numbers of molecules.
• We define 1 Atomic Mass Unit (amu) as being one-twelfth
of the mass of the 12 C isotope. [Note: C chemical element,
12 is the number of protons plus neutrons in the nucleus and
6 is the number of protons.
• Now consider Avogadro’s hypothesis in terms of mass.
Define: Kilogram molecule [kmole] = amount of substance
with mass in kg equal numerically with its molecular weight
i.e. 1kmole N2 = 28.014 kg N2
• For 1kmole of any substance which has a mass M kg and
contains N0 molecules of mass m kg
M = N 0m
N0 =M/m
Since by definition M  m then N0 is the same for all
• So N0 is a universal constant = No. of molecules in 1 kmole
From experiment
N 0 = 6.022 x 1026 kmole -1 (Avogadro’s Number)
Note: 1 kmole atom contains N0 atoms
Atomic Masses and Sizes
• For 12 C: mass m = M/N0 = 12/6.022 x 10 26 = 1.99 x 10 -26 kg
By definition: 1 amu = mass(12 C)/12 = 1/ N0 = 1.66 x 10-27 kg
We find a range of atomic masses up to 4 x 10-25 kg
In general masses are well defined.
Sizes are less well defined
•Let us make an estimate: consider a solid in which the atoms are
packed closely together.
If an individual atom has a diameter of 2r,where r is the
atomic radius, then in 1m we can lay 1/2r atoms side by side.
--------------------------------In a cube of 1m side we then have (1/2r)3 atoms
Each atom occupies a volume of V=(2r)3
Now in 1kmole we have 6 x 1026 atoms and it occupies
N0 x (2r)3 m 3 and has a mass of N0 x (2r)3 x =A
where A=atomic weight and ρ =density.
r = 1/2x(A/N0 )1/3
e.g. Be: A = 9.01,  = 1.84 x 10 3 kg/m 3
 rBe = 1.0 x 10 -10 m
Now r  (A/ )1/3 which varies only slowly with A so
we expect all atoms to have radii which are a few times 10-10m
What about molecules?
Assume that they are spherical.
V = m /  = M/(N0. ) = (4/3)  r3
where m = mass of a molecule
 r = [3M/4N0 ] 1/3
= [3 x 18.015/4 x 6 x 1026 x 10 3] m for water
=1.92 x 10 -10 m
All determinations of r for small molecules give values of
r = 10 -9 - 10 -10 m
Our simple estimates tell us that atoms are approx.
10 -10 m in radius and small molecules about
r = 10 -9 - 10 -10 m
What about Nuclei?
•The idea of the nuclear atom comes from experiments in
Manchester[1911].Ernest Rutherford suggested that Geiger and
Marsden look at large angle scattering of alpha particles(4He
nuclei) from metal foils of Au.They observed that a small
number were scattered backwards.This is consistent with the
current picture of an atom as consisting of a massive,+vely
charged,central nucleus surrounded by a cloud of electrons.
Overall the atom is electrically neutral.
•The same measurement gives an idea of nuclear size.The
scattering of charged particles from the nucleus by the Coulomb
force alone is called Rutherford Scattering. As we increase the
energy of the alpha particles there comes a point where what we
observe is not consistent with RS.This is because the particle is
close enough to feel the STRONG or Nuclear force.This distance
we can define as the nuclear radius.
•If we consider scattering at 1800 then
Z1 Z2 e2/40r
 r = (Z1 Z2 e2/40) x (2/mv2)
For He(Z = 2) on Cu(Z = 29) at 5 MeV energy we get
(ε0=8.85x10-12 F/m)
r  10-14 m
Nuclear Radii
•One thing we ignored so far is the size of the electron!
Interestingly all the evidence we have so far is that it is a
genuine particle: an object with a mass and no size, which
interacts via the EM force alone.
•As a result it is the ideal tool for probing the sizes of nuclei.
Thus many experiments have been done in which high energy
electrons have been scattered from nuclei and the nuclear
radius deduced
from the results.
The picture summarises the results.The plot shows the Mean
square radius plotted against A1/3,where A is the number of
nucleons [neutrons plus protons] in the nucleus.
This can be summed up as
R = R0 A1/3,
where R0 is a constant equal to 1.2 x 10-15 m
•Note:-This is the charge radius.However other expts. tell us the
matter radius is essentially the same.
Density = Mass/volume =1.66 x 10-27/1.33x3.1412x(1.2x10-15)3
 2.3 x 10 17 kg/m3
•Systems of units must be SELF-CONSISTENT and must
be comprehensible. All systems [SI/cgs/Britsh] satisfy the first
criterion but many fail the second.
•For example: R0 = 1.2 x 10-15 m. Nuclei clearly do not belong on
this scale. So we introduce the Fermi (F) = 10-15 m (femto metre)
Similarly for atoms the Angstrom (Å)= 10-10 m
•The charge on an electron is 1.6 x 10-19 Coulombs.
The energy it acquires when it is accelerated through a potential
of 1 volt is
1/2 mv 2 = eV
So we introduce a unit of energy the electron volt = 1 eV
It turns out to be of the right size for energies in atomic systems
•Now from Special Relativity
E2 = p2c2 + m20c4
For p = 0
E = m0c2
= 511 keV for an electron
= 930 GeV for a proton
Thus it is natural in Particle Physics to talk about GeV i.e.10 9 eV
and in Nuclear Physics it is equally natural to use MeV= 10 6 eV
• Atomic Physics
Nuclear Physics
Particle Physics
1 Fermi (1F) =10 -15 m =1 fm
1 Angstrom (1A) = 10 -10 m
1eV = 1.6 x 10 -19 Joules
1 keV = 1.6 x 10 -16 Joules
1 MeV = 1.6 x 10 -13 Joules
1 GeV = 1.6 x 10 -10 Joules
1 amu = 1.66 x 10 -27 kg
Atomic sizes = 10 -10 m
Nuclear sizes = 10 -14 m
Charge on electron(proton) = 1.6 x 10 -19 C
Quanta of Light
We will now look at a series of experiments related to
electromagnetic radiation.
a) Diffraction and interference of light.
b) Photoelectric effect
c) Blackbody radiation
d) Compton Effect.
In 19th century it seemed that expts. on diffraction and
interference of light had settled question of nature of light.
Maxwell predicted that light has speed = c and was
described as a transverse wave.
1887-Hertz produced and detected such waves.
We will find that the question was not really settled and
that we now have a different view.
The electromagnetic spectrum
Electromagnetic Waves
•If charges are accelerated an electromagnetic wave is created.
The E and B fields vary with both t and r.
• Point source - spherical waves - wave fronts are spherical.
Picture shows a plane wave travelling in the +ve Z-direction
E = E0 sin (kz- t +  )
B = B0 sin (kz- t +  ) ,
wave number k = 2/, and ang. freq.  = 2 . B0=E0/c
Now c =  so we can write c = /k and the angle  is an
arbitrary phase angle.
• Note:-The wave shown is plane polarised and the energy flux
S = E x B/μ0 in the forward direction.
(μ0 =4π x 10-7 Tm/A -permeability of free space)
•S is called the Poynting vector and has units of energy/time/area
I.e. Wm -2
•Intensity  E0 2 (general property of waves)
•Intensity fluctuates with time - 2 = 2( / 2)
Normally fluctuation is too fast for us to see.For visible light
 = 10 15 oscillations per sec.
•Principle of Superposition - net effect is sum of individual
effects i. e. two waves cause disturbance at a point which is
result of combined disturbance and they emerge from the
point with all of their properties unchanged.
•This leads to Interference and Diffraction
- - - - - - - - - - - - - - - - - - - - - - - - - -- - - - -
Young’s Double Slit experiment
The observation of both interference and diffraction was
seen as a triumph for the wave theory of light. One excellent
example is Young’s double slit experiment. Here light from
a single source falls on two slits. The two slits act as coherent
sources and we observe interference on the screen behind.
=> minima
Single Slit Diffraction-A Major Success for the
wave Theory
•If the size of the slit is comparable to  then we
see a diffraction pattern not a sharp image.
•We see a central maximum.
•At the first minimum we have
a sin/2 = /2,i.e.
a sin =  (At first minimum)
Diffracton at a single slit
• Figure shows diffraction at a single slit
with a width b. Assuming a wavefront
arrives at the aperture any ray passing
through can be associated with a ray
leaving the aperture a distance b/2
away.If they are /2 out of phase then
destructive interference occurs.Then
b/2.sin1 = /2 or sin1 = /b
• If we divide the aperture into 4 parts then
b/4.sin1 = /2 or sin1 = 2/b
More generally sin1 = m/b where m = 1,2,3,4,-----Thus we get darkness on the screen at these points
and we get the diffraction pattern shown in the figure.
Interference due to the structure of the feathers