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Chapter 1
basic aspects of atoms
• The compositions of atoms;
• The mass and size of the atoms and their
experimental methods;
• Isotopes;
• The periodic system of the elements;
• The electrons;
• The light/photon;
• Rutherford scattering, Rutherford model of the atoms;
• Matter waves/the wave-particle duality;
• Wave packet, probabilistic interpretation;
• Uncertainty principle
Chapter 2
Bohr’s model
The atomic models
Plum-pudding model
by Thomson
Electron cloud model
Planet model by Rutherford
Bohr’s model
Neils Bohr (1885 - 1962)
Bohr was raised in a middle class Danish family and showed
no particular talent as a child except for sports. He played soccer
at almost a professional level and was an active skier until late in
his life. Bohr studied physics as an undergraduate at the
University of Copenhagen and, upon graduation, traveled to
Cambridge to study under J. J. Thompson. Thompson was
particularly busy at this point and was not able to work with Bohr.
So Bohr took up an invitation from Rutherford to travel to
Manchester and study radioactivity.
With Rutherford, Bohr participated in some of the studies on
alpha particle scattering by heavy metals, and began to examine
the planetary ideas of the atom that had been proposed and
rejected by Rutherford.
Bohr was very familiar with the new ideas developed by Planck
and Einstein and wrote up a preliminary proposal to describe a
quantum atom in June of 1912 which he was going to discuss with
Rutherford. On talking to a friend he mentioned this talking paper
and the friend asked whether Bohr's ideas could account for the
spectrum of hydrogen. Curiously, Bohr haven't considered this
question and the friend recommended that Bohr look at Balmer's
equation to see if the model fit the data. Bohr later related "As
soon as I saw Balmer's formula the whole thing was immediately
clear to me".
Bohr's final model of the planetary atom was published in the
Philosophical Magazine in 1913
The colors of firework
Elements in flames
Robert Bunsen's work examined the spectra of elements in flames. It
was well known that many salts cause flames to assume brilliant colors.
These salts emitted only certain lines of the spectrum and not a
continuum of colors as you'd expect from a black body.
Black body and line spectra
Emission spectra
Continuous spectrum
Line spectrum
red
violet
A simple prism spectrometer
Fabry-Perot Interferometer
This interferometer makes use of multiple
reflections between two closely spaced partially
silvered surfaces. Part of the light is transmitted
each time the light reaches the second surface,
resulting in multiple offset beams which can
interfere with each other. The large number of
interfering rays produces an interferometer with
extremely high resolution, somewhat like the
multiple slits of a diffraction grating increase its
resolution.
Solar spectrum with absorption lines
Absorption spectra
Optical spectra
Continuous, band and line spectra:
• Continuous spectra are emitted by radiant
solids or high-density gases;
• Band spectra consist of groups of large
numbers of spectral lines which are very close
to on another;
• Line spectra are typical of atoms.
Observations of optical spectra:
Emission or absorption spectra
The electromagnetic spectrum
indigo
Kirchhoff and Bunsen were the first to discover
in the mid-19th century that each element
possesses its own characteristic spectrum.
Hydrogen is the slightest element, and the
Hydrogen atom is the simplest atom, consisting
of a proton and an electron. The spectra of the
Hydrogen atom have played an important role
again and again over the last 90 years in the
development of our understanding of the laws of
atomic structure and of the structure of the matter.
Balmer series of the Hydrogen atom
Balmer series:
The series of lines
fall closer and closer
together in a regular
way and approach a
short
wavelength
limit (H)
Balmer in 1885, the wavelengths of these lines could be
extremely well reproduced by a relation of the form:
 n12 
G
   2
 n1  4 
 
1 1
 RH  2  2 

2 n 
1
n1=3,4,…, G: empirical constant
Rydberg constant RH:
RH = 109677.5810 cm-1
The optical spectra of Hydrogen atoms
By Rydberg in 1889:
1
 1

'2
2 
n
n


  RH 
With n’ < n being integers
We can conclude from observation and inductive reasoning that the frequencies (or
wavenumbers) of all the spectral lines can be represented as differences of two terms
of the form R/n2. As we shall see in the following, these are just the energy levels of
the electron in a hydrogen atom. The spectral lines of the hydrogen atom can be
graphically pictured as transitions between the energy levels (terms), leading to a
spectral energy level diagram.
Left: Term diagram,
right: Grotrian diagram
Bohr’s model
Bohr’s model (in 1913) was based on the observations of
optical spectroscopy, and following the Rutherford
model, he assumed that the electrons move around the
nucleus in circular orbits of radius r with velocity, must
as the planets move around the sun in the solar system.
The fundamental difficulties of Rutherford model with
classical physics:
• From classical physics, a charge traveling in a circular
path should lose energy by emitting electromagnetic
radiation
• If the "orbiting" electron loses energy, it should end up
spiraling into the nucleus (which it does not). Therefore,
classical physical laws either don't apply or are
inadequate to explain the inner workings of the atom.
The fundamental difficulties of Rutherford
model with classical physics
Classical electron orbit
r
E pot  
e2
4 0 r 

2
dr   
e2
4 0 r
T  E pot
The orbital frequency: /2, where
Fcentripetal 
e2
4 0 r 2
 m0 r 2
 Ze2

8 0 r
Bohr’s three postulates
—to be an extremely important step towards
quantum mechanics
To avoid the discrepancy with the laws of classical physics,
Bohr proposed three postulates to describe the deviations
from classical behavior for electron in an atom.
Bohr’s three postulates
1.
Quantised orbits. The classical equations of motion are valid for
electrons in atoms. However, only certain discrete orbits with the energy
En are allowed. These are the energy levels of the atom.
2.
The motion of the electron in these quantised orbits is radiationless. An
electron can be transferred from an orbit to another orbit. Emission:
transferring from an orbit with lower binding energy En to an orbit with
higher binding energy En’; absorption: higher to lower binding energy
levels.
3.
Corresponding principle. The orbital frequency is comparison with the
frequency of emission or absorption. For large n, one can calculate the
Rydberg constant RH from atomic quantities.
In general, the corresponding principle: with increasing orbital radius r, the
laws of quantum atomic physics become identical with those of classical
physics.
Schematic representation of Bohr’s model
orbits
Bohr’s model
For the orbit,
what is the energy level, radii,
and R?
The energy term of the Hydrogen:
Rhc
En   2
n
The frequency of the emitted radiation:
1
h
  ( En  En )  Rc (
1
1

)
2
2
n n
We consider the emission between neighboring orbits, n - n’ =  = 1,
for large n:
 2
(1  n )  1  2 n  
The frequency of the emitted radiation:
 1
1
1
1
2 2RC
  Rc( 2  2 )  Rc
 2   RC 3  3
2
n n
n
n
 (n   ) n 
According to the correspondence principle, for large n, then  =
the classical orbital frequency /2. The energy level En of
electron in Bohr’s orbit is equal to that of the total energy Etot in
classical orbit: En = Etot
Rhc
En   2
n
2
e
1
4
2 1/ 3
Etot  

(
e
m

)
0
2/3
8 0 r
2(4 0 )
 2 Rc

 3
2
n
The Rydberg constant:
m0e4
RH  2 3
8 0 h c
R = (109737.318 ± 0.012) cm-1
The energy term:
Rhc
En   2
n
n 2 2 4 0
The radius rn of the nth orbital: rn 
e 2 m0
The quantum number n is called the principal quantum number
For n = 1:
The smallest orbital radius r1 = 0.529Å = Bohr radius = a0
The lowest energy state E1 = -13.59eV. Then for H atom:
1
En  13.59 2 eV
n
rn  n 2 a0
Optical spectra of Hydrogen atom
orbits
Energy level diagram, term diagram
visible
ultraviolet
infrared
Ground state and excited state
At normal temperature only the lowest energy term, n = 1, is
occupied, and only Lyman series in hydrogen is observable in
absorption.
Ground state: the lowest energy state
Excite state:
At higher temperature, the excited state, n = 2, 3, …, can be
occupied according to the Boltzmann distribution. For example,
in the sun, with a surface temperature of 6000K, only 10-8 of the
hydrogen atoms in the solar atmosphere are in the n = 2 state.
Bohr model of the atom—emission of light
If we have a hydrogen atom with its electron in an excited state (either by light absorption or by
heating) the electron may fall down to a lower orbit by emission of light. The electron may fall
into any lower orbit, and the energy it loses will be exactly equal to the energy difference
between the two orbits. In the above example, the electron falling from the third to the first
orbit emits green light. Were it to go to the second orbit, the difference in energy would be less
and the atom only emits red light.
The Balmer equation fit the energies of electrons falling from higher orbits into the second orbit.
In real life (and not the simple example above) electrons falling into the first orbit emit
ultraviolet light.
Bohr model of the atom — absorption of light
If we assume that a hydrogen atom starts out with its electron in the lowest possible
orbit, the electron may move to other obits if and only if absorbs a photon with
exactly the right amount of energy.
In the illustration above, the transition from the first orbit to the third orbit requires
exactly the amount of energy supplied by green light. The spectrum of light leaving
the hydrogen atom would be missing a line in the green.
The features of Bohr’s model
Bohr's model allowed a simple understanding of
the absorption or emission of light by atoms that
explained line spectra. But it make no statements
about processes. The classical orbital concept is
abandoned. The electron’s behavior as a function of
time is not investigated, but only its stationary
initial and final states.
Electron transitions
The Bohr model for an electron transition in hydrogen between quantized
energy levels with different quantum numbers n yields a photon by emission
with quantum energy:
This is often expressed in terms of the inverse wavelength or wave number:
 1
1 
 RH  2  2 

 n1 n2 
1
Hydrogen level energy plot
The basic structure of the hydrogen energy levels can be calculated
from the Schrodinger equation. The energy levels agree with the earlier
Bohr model, and agree with experiment within a small fraction of an
electron volt.
Hydrogen spectral tube
a hydrogen spectral tube excited by a 5000 volt transformer. The
three prominent hydrogen lines are shown at the right of the
image through a 600 lines/mm diffraction grating.
An approximate classification of
spectral colors:
•Violet (380-435nm)
•Blue(435-500 nm)
•Cyan (500-520 nm)
•Green (520-565 nm)
•Yellow (565- 590 nm)
•Orange (590-625 nm)
•Red (625-740 nm)
Measured hydrogen spectrum
Wavelength
(nm)
383.5384
388.9049
397.0072
410.174
434.047
486.133
656.272
656.2852
Relative
Intensity
5
6
8
15
30
80
120
180
Transition
Color
9 -> 2
8 -> 2
7 -> 2
6 -> 2
5 -> 2
4 -> 2
3 -> 2
3 -> 2
Violet
Violet
Violet
Violet
Violet
Bluegreen (cyan)
Red
Red
Bohr’s explanation of line spectra
Angular momentum quantisation
From Bohr’s model, an electron has velocity Vn and orbital
frequency n in the nth orbit with the orbit radius rn ,the orbital
angular momentum is quantised:
  
l rp

2
l  m0vn rn  m0 rn n  n
With n = 1,2,3,…
Hydrogen-like atoms
Hydrogen-like atoms: atoms with an electron and a nucleus with
charge +Ze, such as He+, Li++.
-e
+
+Ze
Hydrogen-like atoms
Equilibrium between the Coulomb force and the centrifugal force:
Fcentrifugal
m0v 2
Ze2


2
4 0 rn
rn
n 2 2 4 0 n 2
rn 
 a0
2
Ze m0
Z
For the possible orbital radius:
Z 2e4 m0
Z2
En  
  2 13.59eV
2
2 2
32  0 n
n
The energy states:
The wavenumber of the spectral lines:
1
Z2  1
1 
 2 2
 
En1  En2  13.59 
hc
hc  n2
n1 


Motion of the nucleus
From spectroscopic measurement RH = 109677.5810 cm-1
From theoretical calculation R = 109737.318 ± 0.012 cm1
 = R - RH  60 cm-1
The reason for the difference is the motion of the nucleus
during the revolution of the electron, which was neglected
in the above model calculation. This calculation was made
on the basis of an infinitely massive nucleus; we must now
take the finite mass of the nucleus into account.
The motion of the nucleus
The motion of two particles, of masses m1 and m2 and at the
distance r from one another, takes place around the common
center of gravity. The reduced mass:
Gravity
center
m1
m1m2

m1  m2
m2
Replace the mass of the orbiting electron, m0 by :
R  R
1
1  m0 / M
The mass of the orbiting electron m1 = m0, the mass of the
nucleus m2 = M.
Due to the motion of the nucleus, different isotopes of the same
element have slightly different spectral lines. The mass ratio M/m0
can be determined by spectroscopic observation. This so-called
isotope displacement led to the discovery of heavy hydrogen with
the mass number A = 2. It was found that each line in the spectrum
of hydrogen was actually double, for H and D.
Mproton/melectron = 1836.15
RH  R 
RD  R 
1
 109677.584cm1
1  m0 / M H
1
1  m0 / M D
 109707.419cm1
The difference in wavelengths  for corresponding lines in the spectra:
 D
  H  D  H 1 
 H

 RH
  H 1 

 RD



Spectra of Hydrogen-like atoms
According to Bohr model, the spectra
of hydrogen-like atoms, all atoms or
ions with only one electron, are the
same except for the factor Z2 and the
Rydberg number. This has been
completely verified experimentally.
Some energy levels of the atoms
H, He+, and Li2+
For He+, astronomers found the Fowler series:
1 1
 F  4 RHe  2  2 
3 n 
The Pickering series:
1 1
 P  4 RHe  2  2 
4 n 
Preparation of the hydrogen-like heavy atoms in lab: accelerate the singlyionised atoms to high energies and pass them through a thin foil, their
electrons are stripped off on passing through the foil.
For example, in order to strip all the electrons from a uranium atom and
produce the U92+ ions to recapture one electron, one can then obtain the
hydrogen-like ion U91+. The corresponding spectral lines are emitted as the
captured electron makes transitions from orbits of high n to lower orbits. For
U91+, the Lyman series has been observed in the spectral region around
100keV and the Balmer series is in the region between 15 and 35keV.
Muonic atoms
Muonic atoms: the electron was replaced by a heavier  meson
or muon.
-e
+
muon
+Ze
m = 206.8 m0 ,
Muonic atoms are extremely small in diameter and very close
to the nucleus.
Muons behave like heavy electrons, we can simply apply the
results of the Bohr model. For the orbital radii:
2
2
4


m
m0
n

2

0
0
rn ( )  2
n  rn (e )
 a0  
Ze m
m
Z m
A numerical example: Mg11+:
Electron:
muon:
0.53 o
12
r1 (e ) 

4
.
5

10
m
A
12

r
(
e
)

1
r1 (  ) 
 2.2 1014 m
207

Muonic atoms are interesting objects of nuclear physics research.
Since the muons approach the nucleus very closely, much more
than the electrons in an electronic atom, they can be used to study
details of the nuclear charge density distribution, the distribution of
the nuclear magnetic moment within the nuclear volume and of
nuclear quadrupole deformation.
Excitation of quantum jumps by collisions
—another experimental support for Bohr’s model
Lenard investigated the ionisation of atoms as early as 1902 using
electron collisions.
plate
Experimental arrangement for detecting ionisation process in gases. Only positive ions,
which are formed by collisions with electrons, can reach the plate A. The voltage are
chosen so that the electrons cannot reach the plate; they pass through the grid and are
repelled back to it. When an electron has ionised an atom of the gas in the experimental
region, however, the ion is accelerated towards the plate A. Ionisation events are thus
detected as a current to the plate.
Franck-Hertz experiment
Franck and Hertz showed for the first time in 1913 that the
existence of discrete energy levels in atoms can be
demonstrated with the help of electron collision processes
independently of optical-spectroscopic results. Inelastic
collisions of electrons with atoms can result in the transfer of
amounts of energy to the atoms which are smaller than the
ionisation energy and serve to excite the atoms without
ionising them.
Electrons pass through the grid and are carried by their momenta across a space
filled with Hg vapour to an anode A. Between the anode and the grid is a braking
voltage of about 0.5eV. Electrons which have lost most of their kinetic energy in
inelastic collisions in the gas-filled space can no longer move against this braking
potential and fall back to the grid. The anode current is the measured as a function
of the grid voltage VG at a constant braking potential VB. At a value of VG  5V
the current I is strongly increased; it then increase again up to VG  10V , where
the oscillation is repeated.
electron
electron
collision
Hg
When the electrons have reached an energy of about 5eV, they can give
up their energy to a discrete level of the mercury atoms. They have then
lost their energy and can no longer move against the braking potential.
If their energy is 10eV, this energy transfer can occur twice, etc.
One can find an intense line in emission and absorption at E = 4.85eV in
the optical spectrum of atomic mercury, corresponding to a wavelength
of 2537Å. This line was also observed by Franck and Hertz in the
optical emission spectrum of Hg vapour after excitation by electron
collisions.
Improved experiment
The resolving power for the energy loss of the electrons may be improved by
using an indirectly heated cathode and a field-free collision region. In this
way, one obtains a better uniformity of the energies of the electrons. A
number of structures can be seen in the current-voltage curve; these
correspond to further excitations of the atoms. For example 6.73eV related to
1850Å. But not all the maxima in the current-voltage curve can be correlated
with observed spectral lines.
For the optically forbidden transitions can, in some cases, be excited by
collisions.
Electron collision and optical excitation
1) From the improved Franck-Hertz experiment, the selection rules for
collision excitation of atoms are clearly not identical with those for
optical excitations.
2) The optical excitation occurs only when the light has exactly the same
energy as the quantum energy of atom, for example, Na vapour, yellow
line E = 2.11eV. Both smaller and larger quantum energy are ineffective
in producing an excitation.
3) The yellow line is emitted whenever the energy of the electrons E >=
2.11eV. Because the kinetic energy of free electrons is not quantised.
After excitation of a discrete atomic energy level by electron collision,
the exciting electron can retain an arbitrary amount of energy.
These electron collision experiments prove the existence of
discrete excitation states in atoms and thus offer an excellent
confirmation of the basic assumptions of the Bohr theory.
Electron waves and orbits
Asking why electrons can exist only in some states and not in
others is similar to asking how your guitar string knows what
pitch to produce when you pluck it. It is a standing wave
phenomenon and has to do with resonance.
Electron wave length for different states
Sommerfeld’s extension of the Bohr model
To explain the double or multiplet structure of the spectral lines,
Sommerfeld derived an extension of the Bohr model: not only
circular orbits, but also elliptical orbits are possible.
According to the Kepler’s Law of Areas, the electron sweep out
equal areas between its orbit and the nucleus in equal time. Then,
in elliptical orbit, the electron is faster when it closer to the
nucleus, more massive. The relativistic mass change of the
electron lifted the orbital degeneracy. The Sommerfeld’s
calculation:
  2Z 2  n 3 

En,k
1  2     higher order  corrections 
n k 4


Where the fine structure constant:
Z2
  Rhc 2
n
e2
1


2 0 hc 137
Sommerfeld’s extension was on one hand of great historical
importance in introducing a second quantum number k, but, on the
other hand, been made obsolete by the later quantum mechanical
treatment.
The limitation of Bohr’s model
The model only worked for hydrogen and could not be
modified to fit anything beyond lithium. It was useless in
describing bonding, and it relied on a set of arbitrary
postulates whose justification flew in the face of known
physics. At best, Bohr's model must be seen as a transitional
theory, introducing the quantum theory to the understanding
of atoms, but not a fully modern theory.
Failures of the Bohr model
While the Bohr model was a major step toward understanding the
quantum theory of the atom, it is not in fact a correct description
of the nature of electron orbits. Some of the shortcomings of the
model are:
1. It fails to provide any understanding of why certain spectral
lines are brighter than others. There is no mechanism for the
calculation of transition probabilities.
2. The Bohr model treats the electron as if it were a miniature
planet, with definite radius and momentum. This is in direct
violation of the uncertainty principle which dictates that position
and momentum cannot be simultaneously determined.
The Bohr model gives us a basic conceptual model of electrons
orbits and energies. The precise details of spectra and charge
distribution must be left to quantum mechanical calculations, as
with the Schrödinger equation.
Rydberg atoms
Rydberg atoms: atoms in which an electron has been excited to an unusually high
energy level illustrate well the logical continuity between the world of classical
physics and quantum mechanics.
Extraordinary properties:
Gigantic: 10-2 mm in diameter;
Extremely long lifetime: 1s, typical lifetime of lower excited states of atoms are
about 10-8 s;
Strongly polarised: by relatively weak electric fields.
When the outer electron of an atom is excited into a very high energy level, it
enters a spatially extended orbit, which is far outside the orbitals of all the other
electrons and the nucleus. The core, consisting of the nucleus and the inner
electrons, has a charge +e. Rydberg atoms behave in many respects like highly
excited hydrogen atoms.
In interstellar space, n up to 350;
In the laboratory, n between 10 and 290.
Rydberg atoms
Apparatus for the detection of Rydberg atoms. An
atomic beam is crossed by several laser beams.
They cause the excitation of the atoms into Rydberg
states when the sum of the quantum energies of the
laser beams corresponds to the excitation energy of
a Ryderg state. The Rydberg atoms are ionised in
the electric field of a condenser, and the ions are
then detected.
An example of the detection of Rydberg states of
the lithium atom with n = 28 to 39.
Rydberg excitation states of barium atoms with the
principal quantum number n, observed using Dopplerfree spectroscopy.
Artificial atoms
—Positonium, muonium, and antihydrogen
It is possible to make artificial atoms in which one or both
of the atomic components of hydrogen, the proton and the
electron, are replaced by other particles. All the conclusions
of the Bohr model concerning atomic radii, energy levels,
and transition frequencies should also apply to the artificial
atoms
+
e-
e+
Positronium,
(e-, e+) an atom consisting of an electron, e-, and a positron, e+, was
discovered in 1949 by M. Deutsch. Positron can be obtained from the radioactive decay
of nuclei, 22Na. positron atoms are formed when positrons pass through a gas or impinge
on solid surfaces, where the positron can capture an electron. The lifetime of
positronium is very short (1. 4·10-7s or 1. 25·10-10s), they annihilate each other with the
emission of two -quanta.
In condensed-matter physics and in modern medicine, positronium atoms are used as
probes for structures and dysfunctions, because the emission of their annihilations
radiation is dependent on their material surroundings. In medicine, positron emission
tomography is used for example to form an image of diseased tissue in the brain.
e-
+
+
Muonium, (+, e-) an atom consisting of a muon, +, and a electron,
e-. It is formed when positive + enter into a bound state with
electrons on passing through a gas or on a solid surface. + particles
are unstable, and the lifetime of muonium is correspondingly only
2.2·10-6 s. These atoms have been studied extensively by
spectroscopic methods. They are particularly relevant to the
refinements of the Bohr model by Dirac’s relativistic quantum
mechanics.
--
e+
P
Antihydrogen atom, (p-, e+) an atom consisting of a positron bound to
a negatively-charged antiproton. According to the postulates of quantum
mechanics, antimatter should behave just like ordinary matter. An
experimental test has yet to be performed, since antimatter was not
available until very recently. In 1995, the successful preparation of
antihydrogen was reported for the first time. One goal is the spectroscopic
investigation of the antihydrogen atoms, as a test of the symmetry of the
interactions between matter and antimatter.
homework
P122,
8.5, 8.6, 8.8, 8.9, 8.14, 8.15, 8.17