OPTION - FROM QUANTA TO QUARKS
NOTE: The notes and worksheets for this topic
are divided into two separate pages in order to keep download times within
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the link at the left marked 9.8 Continued.
the early part of the twentieth century, many experimental and theoretical
problems remained unresolved. Attempts to explain the behaviour of matter
on the atomic level with the laws of classical physics were not successful.
Phenomena, such as black-body radiation, the photoelectric effect, the
emission of sharp spectral lines by atoms in a gas discharge tube, could
not be understood within the framework of classical physics.
Between 1900 and 1930, a revolution took place and
a new more generalized formulation called quantum mechanics was developed.
This new approach was highly successful in explaining the behaviour of
atoms, molecules and nuclei. As with relativity, quantum theory requires
a modification of ideas about the physical world.
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THE DEVELOPMENT OF ATOMIC MODELS
first atomic theory of matter was introduced by the Greek philosopher
Leucippus, born around 500 BC and his pupil Democritus, who lived from
about 460 to 370 BC. However, the great success of the opposing
“continuous matter” theory proposed by Aristotle (389-321 BC) ensured
that the atomic model of matter took a back seat until about the 17th
Century AD. At this time the work and success of people like Copernicus,
Galileo and Newton undermined the authority of Aristotle and allowed the
atomistic view to be revived.
the 19th Century, John Dalton proposed an atomic model that
allowed the first really quantitative study of the atom to be attempted.
Later work by Sir J J Thomson and P Lenard led to further advances in
our knowledge of the atom.
1910 the New Zealand born physicist Ernest Rutherford, working in England,
instructed two of his students, Hans Geiger and Ernest Marsden, to investigate
alpha particle scattering from thin metal foils. What they discovered
greatly enhanced our understanding of the atom.
particles are doubly charged helium nuclei and have a mass about 7500
times that of the electron and a velocity in this scattering experiment
of about 1.6 x 107 m/s. The existing model of the atom
at that time (Thomson’s “Plum Pudding Model”) predicted that almost all
of the alpha particles fired at a metal target would simply pass straight
through the metal undeflected. To their great surprise, Geiger &
Marsden found that a significant number of alpha particles were deflected
by angles greater than 90o. That is, the alpha particles
were being reflected by the metal foil. Some even came back almost
retracing their original path.
1911, Rutherford proposed his model of the atom, based on the results
of many such scattering experiments. He proposed that the atom
consisted mainly of empty space with a tiny, positively charged nucleus,
containing most of the mass of the atom, surrounded by negative electrons
in orbit around the nucleus like planets orbiting the sun. The
electrons could not be stationary because if this were the case they would
be attracted towards the positive nucleus and be neutralized. The
Coulomb force of attraction between the positive nucleus and the negative
electrons provided the necessary centripetal force to keep the electrons
Rutherford model was a great step forward in our understanding of atomic
structure but it still had its limitations. Since the electrons
were in circular motion, they would be experiencing centripetal acceleration
and according to Maxwell’s Theory of Electromagnetism should be emitting
electromagnetic radiation. This loss of energy would cause the
electrons to gradually spiral closer and closer to the nucleus and to
eventually crash into the nucleus. Thus, matter would be very
unstable. This was clearly not the case. Also, Rutherford’s
model could not explain the observed line spectra of elements.
As electrons spiraled towards the nucleus with increasing speed, they
should emit all frequencies of radiation not just one. Thus, the
observed spectrum of the element should be a continuous spectrum
not a line spectrum.
went to work with Rutherford in 1912. During the next two years
he studied the Rutherford model of the atom. Bohr was inspired by
the work of Max Planck on quantized energy and attempted to incorporate
this idea into the atomic model to explain the discrepancies between
the observed spectra of the elements and the spectra predicted on the
basis of Rutherford’s atomic model.
we saw in the “From Ideas To Implementation” topic, in 1900 Max Planck
investigated the relationship between the intensity and frequency of
the radiation emitted by very hot objects. Planck showed that the
radiation from a hot body was emitted only in discrete quantities or “packets”
called quanta. The energy, E,
of each quantum was shown to be proportional to the frequency, f,
of the radiation emitted:
= h f
where h = Planck’s constant = 6.63 X 10-34 Js. This
idea led directly to the belief that atoms could only absorb or emit energy
in discrete quanta. Albert Einstein’s use of Planck’s quantisation
idea to successfully explain the photoelectric effect added great support
to this belief. So, Bohr was convinced that a successful atomic
model had to incorporate this energy quantisation phenomenon.
thinking on a new atomic model was also guided by the work that had been
done on the spectrum of hydrogen. Let us briefly examine
firstly what is meant by the term spectrum and secondly the understanding
of elemental spectra that existed at the time of Bohr’s work on the
an element such as hydrogen is heated to incandescence, or when
it is ionized in a gas discharge tube, it emits visible light and other
radiation that can be broken into its component parts using a spectroscope
and a glass prism or a diffraction grating. The particular radiation
emitted is known as the emission spectrum of that element and is unique
to that element. When the emission spectrum of hydrogen is examined
using a spectroscope, it is found to consist of four lines of visible
light – a red line, a green line, a blue line and a violet line on a dark
background. It can be shown that all elements produce emission
line spectra rather than the continuous spectra predicted by
the Rutherford model of the atom.
For a Practical Exercise on observing the visible
lines in the hydrogen emission spectrum click on the following link - Hydrogen
Spectrum Practical. Use your Browser's back arrow to get back
type of elemental spectrum is produced by passing white light through
the cool gas of an element. The cool gas will absorb the
same frequencies that it would otherwise emit if heated to incandescence.
This spectrum is called the absorption spectrum of an element and consists
of a continuous band of colours (different frequencies) with black lines
present where particular frequencies have been absorbed by the cool gas.
This spectrum is also unique to each element and is used to provide information
on the elemental composition of stars.
study of emission and absorption spectra of different elements provided
much information towards the understanding of atomic structure.
From 1884 to 1886 Johann Balmer, a Swiss school teacher, suggested
a mathematical formula to fit the known wavelengths of the hydrogen emission
m is an integer with a different value for each line (m = 3, 4, 5, 6) &
b is a constant with a value of 364.56 nm. This formula produces
wavelength values for the hydrogen emission spectral lines in excellent
agreement with measured values. This series of lines has become
known as the Balmer series. Balmer predicted that there should
be other series of hydrogen spectral lines and that their wavelengths could
be found by substituting values higher than the 2 shown on the right hand
side of the denominator in his formula.
1890, Johannes Rydberg produced a generalized form of Balmer’s
formula for all wavelengths emitted from excited hydrogen gas:
= Rydberg’s constant = 1.097 x 107 m-1, nf
= an integer specific to a spectral series (eg for the Balmer series nf
= 2) and ni
= 2, 3, 4, ……
other series of hydrogen emission lines besides the Balmer were found.
The following table gives the details.
of EM Spectrum
3, 4, …..
4, 5, …..
5, 6, …..
6, 7, …..
7, 8, …..
Although Rydberg’s equation was very accurate
in its predictions of the wavelengths of hydrogen emission lines, for
a long time no-one could explain why it worked – that is, the physical
significance behind the equation. Bohr was the first to do so.
In 1913, Niels Bohr proposed his model of the atom.
He postulated that:
- An electron executes circular motion around the nucleus
under the influence of the Coulomb attraction between the electron and
nucleus and in accordance with the laws of classical physics.
- The electron can occupy only certain allowed orbits
or stationary states for which the orbital angular momentum, L,
of the electron is an integral multiple of Planck’s constant divided by
- An electron in such a stationary state does not radiate
- Energy is emitted or absorbed by an atom when an electron
moves from one stationary state to another. The difference in energy
between the initial and final states is equal to the energy of the emitted
or absorbed photon and is quantised according to the Planck relationship:
= Ef – Ei = hf
The exact number and order
of these postulates is not important. Some references will give
two, some three & some four postulates for Bohr's model of the atom.
What is important to know is the basic detail contained within the postulates.
So, we could quickly summarize Bohr's postulates as: (1) Electrons orbit
a central positive, nucleus in certain allowed, circular, orbits called
stationary states from which they do not radiate energy. (2) Electrons
only move from one state to another by absorbing or emitting exactly the
right amount of energy in the form of a photon, whose energy is equal
to the difference in energy between the initial & final states, DE
The first postulate retains the basic structure that
successfully explains the results of the Rutherford alpha particle scattering
The second postulate was
necessary to explain the observed atomic emission spectra of hydrogen.
Only the separation of allowed orbits according to the second postulate
gave the experimentally observed spectra. Clearly, Bohr’s study
of the hydrogen spectrum was instrumental in the development of his model
of the atom.
Clearly, the third postulate accounts for the observed
stability of atoms. Bohr did not know why the stationary states
existed; he simply assumed that they must because of the observed stability
The fourth postulate explains
how atoms emit and absorb specific frequencies of electromagnetic radiation.
An electron in its lowest energy state (called the ground state) can only
jump to a higher energy state within the atom when it is given exactly
the right amount of energy to do so by absorbing that energy from a photon
of EM radiation of the right energy. Once the electron has jumped
to the higher level, it will remain there only briefly. As it returns
to its original lower energy level, it emits the energy that it originally
absorbed in the form of a photon of EM radiation. The frequency
of the energy emitted will have a particular value and will therefore
be measured as a single emission line of particular frequency and therefore
of particular colour if in the visible region of the EM spectrum.
Starting with these four postulates and using a
mixture of classical and quantum physics, Bohr derived equations for:
(i) the velocity of an electron in a particular stationary state; (ii)
the energy of an electron in a particular stationary state; (iii) the
energy difference between any two stationary states; (iv) the ionization
energy of hydrogen; (v) the radii of the various stationary states; (vi)
the Rydberg constant; and (vii) the Rydberg equation for the wavelengths
of hydrogen emission spectral lines.
successfully deriving the Rydberg equation from his basic postulates,
Bohr had developed a mathematical model of the atom that successfully
explained the observed emission spectrum of hydrogen and provided a physical
basis for the accuracy of the Rydberg equation. The physical
meaning of the Rydberg equation was at last revealed. The nf
and ni in the equation represented the final and initial stationary
states respectively of the electron within the atom. The hydrogen
emission spectrum consists only of particular wavelengths of radiation
because the stationary states or energy levels within every hydrogen atom
are separated by particular set distances, as described by the second
postulate. The value of the Rydberg constant calculated by Bohr
was in excellent agreement with the experimentally measured value.
Bohr’s atomic model led to a couple of useful ways
of representing the quantum jumps of electrons involved in each of the
different series of the hydrogen emission spectrum. These are shown
The following schematic diagram shows the possible
transitions of an electron in the Bohr model of the hydrogen atom (first
6 orbits only).
The diagram below is an energy level diagram
for the hydrogen atom. Possible transitions between energy states
are shown for the first six levels. The dashed line for each series
indicates the series limit, which is a transition from the state
where the electron is completely free from the nucleus (n
= infinity). The energies shown are the ionisation
energies for electrons in each energy level. This is the energy
that must be supplied to remove the electron in a given energy level from
the atom. These energies are thus written as negative values.
As you will see when
you answer question 7 on the Bohr Worksheet, the energy of an electron
in the n-th Bohr orbit is proportional to 1/n2.
Even though it is beyond the scope of the syllabus, it is worth stating
that this implies that the gaps between successive higher energy levels
get smaller and smaller (in terms of energy values) as indicated in
the above diagram. In terms of spatial arrangement of Bohr orbits,
however, the radius of the n-th Bohr orbit is proportional
to n2. So, spatially, the distance between successive higher orbits gets
larger and larger. Taken together, these two facts make good
sense, since the further the electron is from the nucleus, the less tightly
it is held and therefore the less energy is required to move the electron
from one energy level to a higher one.
LIMITATIONS OF THE BOHR MODEL:
In reality Bohr’s model was a huge breakthrough
in our understanding of the atom. For his great contribution to
atomic theory Bohr was awarded the 1922 Nobel Prize in Physics.
As with any scientific model, however, there were limitations. The
problems with the Bohr model can be summarised as follows:
Bohr used a mixture of classical and quantum physics,
mainly the former. He assumed that some laws of classical physics
worked while others did not.
The model could not explain the relative intensities
of spectral lines. Some lines were more intense than others.
- It could not explain the hyperfine structure of spectral
lines. Some spectral lines actually consist of a series of very
fine, closely spaced lines.
- It could not satisfactorily be extended to atoms with
more than one electron in their valence shell.
- It could not explain the “Zeeman splitting” of spectral
lines under the influence of a magnetic field.
WORKSHEET ON BOHR’S MODEL
State the four postulates used by Bohr to explain the nature of the atom.
Draw a sketch of the Bohr model of the atom, clearly labelling the
electronic transitions responsible for the four visible lines in the hydrogen
emission spectrum (Ha,
& Hd ).
The red line within the Balmer series has a wavelength of 6560Å.
Identify the initial and final stationary states corresponding to this
transition within the hydrogen atom?
Calculate the wavelengths of each of the visible lines in the Balmer
series for hydrogen.
Determine the frequency of the radiation emitted when an electron in a
hydrogen atom undergoes a transition from the ni =
2 energy level to the nf = 1 level.
Calculate the energy emitted by the electron in making this transition.
An electron in a hydrogen atom drops from stationary state n
= 2 of binding energy 5.43 x 10-19J
to stationary state n = 1 of binding energy 21.56
x 10-19J. Determine
the energy emitted by the electron in making this transition.
Starting with the Rydberg equation, derive equations for the frequency
and energy of the radiation emitted when an electron in a hydrogen atom
undergoes a transition from stationary state ni
to stationary state nf.
Hence write an equation for the energy of an electron in the n-th
stationary state of the hydrogen atom.
THE de BROGLIE MODEL
In 1924, Louis de Broglie, a French physicist, suggested
that the wave-particle dualism that applies to EM radiation
also applies to particles of matter. He proposed that every kind
of particle has both wave and particle properties. Hence,
electrons can be thought of as either particles or waves.
De Broglie reasoned that just as photons of EM energy
have a momentum associated with their wavelength (p = h /
particles of matter should have a wavelength associated with their momentum:
where p = momentum of particle, m = mass of particle,
v = velocity of particle and h = Planck’s constant.
impact of de Broglie’s proposal was far reaching. Its immediate impact
was to provide a physical interpretation of the Bohr quantisation of stationary
states within an atom. Its ongoing impact was to provide a new way
of describing the nature of matter, which assisted greatly in the development
of quantum mechanics. Erwin Schrodinger in 1926 used de Broglie’s
ideas on matter waves as the basis of his wave mechanics, one of several
equivalent formulations of quantum mechanics.
us examine how de Broglie’s matter wave proposal explains the Bohr quantisation
of stationary states. Bohr’s second postulate states that:
An electron can occupy only certain allowed
orbits or stationary states for which the orbital angular momentum, L,
of the electron is an integral multiple of Planck’s constant divided by
2p. Mathematically, this can be written as:
= n h / 2p
De Broglie proposed that Bohr’s allowed orbits corresponded
to radii where electrons formed standing waves around the nucleus.
The condition for a standing wave to form would be that a whole number,
n, of de Broglie wavelengths must fit around the circumference
of an orbit of radius r.
l = 2 p
Substituting for l from the de Broglie relationship above, we have:
(h / mv) = 2 p
m v r = n h / 2p
is the correct expression for the orbital angular momentum, L,
of the electron in orbit around the nucleus, de Broglie had succeeded
in showing that Bohr’s allowed orbits (or stationary states) are those
for which the circumference of the orbit can contain exactly an integral
number of de Broglie wavelengths. Thus, as shown in the figure
below, the first stationary energy state (n = 1) corresponds to an allowed
orbit containing one complete electron wavelength; the second stationary
state corresponds to an allowed orbit containing two complete electron
wavelengths; and so on.
Broglie was then able to explain the stability of electron orbits in the
Bohr atom. When an electron is in one of the allowed orbits or stationary
states, it behaves as if it is a standing wave, not a particle experiencing
centripetal acceleration. Thus, the electron does not emit EM radiation
when it is in a stationary state within the atom.
confirmation of de Broglie’s proposal on matter waves was achieved
in 1927 by Clinton Davisson and Lester Germer in the USA and by
George Thomson in Scotland. Davisson and Germer conducted
an experiment in which electrons in an electron beam produced the same
diffraction pattern as X-rays when they were scattered by a small crystal
As you will recall, diffraction
is the name given to the phenomenon in which a wave spreads out as it passes
through a small aperture or around an obstacle. Diffraction
patterns are formed when the diffracted waves interfere with one another to
produce light and dark bands on a screen or piece of film. Diffraction
patterns are most intense when the size of the aperture or obstacle is
comparable to the size of the wavelength of the wave. The electrons
in the Davisson & Germer experiment were scattered in specific directions,
which could only be explained by treating the electrons as waves with
a wavelength related to their momentum by the de Broglie relation.
Particles would have bounced off the nickel in all directions randomly.
The following is a diagram of the apparatus used by
Davisson & Germer. Electrons from filament F are accelerated
by a variable potential difference V. After scattering from the
nickel crystal, they are collected by the detector D.
D can be moved
to measure the scatter yield at any angle.
Ever wondered what the basic difference is
between classical physics and quantum mechanics? Well, take a detour
for a minute and investigate the paradox of Schrodinger's Cat. This
link takes you to the Brainteasers page of this site & to the
description of the Schrodinger's Cat paradox that resides there.
After reading the paradox, follow the links to the discussion of the
paradox and then, if you are still interested, to a description of the
domain of quantum mechanics itself. Use the back button of your
browser to get back here. Go
to Schrodinger's Cat.
WORKSHEET ON DE BROGLIE
Determine the de Broglie wavelength of the matter wave associated with a
cricket ball of mass 0.175 kg and velocity 23.6 m/s.
Use the answer to this question to explain why we do not observe the
matter waves associated with macroscopic objects.
(l = 1.6 x 10-34
Calculate the de Broglie wavelength of an electron travelling at 106
m/s. (Mass of electron is 9.11 x 10-31
kg.) Use the answer to this
question to explain why we would expect to observe the effects of the matter
waves associated with electrons. Give
one example of these effects. (l
= 7.3 x 10-10 m)
A proton is travelling at a speed of 1.5 x 107 m/s.
Determine the de Broglie wavelength of the proton, given its mass is 1.67
x 10-27 kg. (2.6
x 10-14 m)
Calculate the momentum of a neutron if it has a de Broglie wavelength of
1.59 x 10-13 m. (4.17 x 10-21 Ns)
Determine the speed of the neutron in Q.4, given that the mass of a
neutron is 1.67 x 10-27 kg. (2.50 x 106 m/s)
An electron volt (eV) is an energy unit equivalent to the work done when
an electron is moved through a potential difference of 1 volt.
If an electron has a kinetic energy of 100 eV, what is its associated de
Broglie wavelength? (Charge on an
electron is 1.6 x 10-19 C) (1.2
x 10-10 m)
Assess the contributions made by Heisenberg
to the development of atomic theory.
Once you have done that, check out the "Beam
Me Up Scotty" section on the Brainteasers page. It's an
interesting exercise on Heisenberg's Uncertainty Principle, quantum entanglement
and quantum teleportation.
THE ELECTRON MICROSCOPE -
Note that this section on the Electron Microscope is
no longer required by the Syllabus. As it is quite interesting, it has
been left here as extension material.
The smallest object we can see with the unaided eye
is about 0.1 mm. With a good light microscope we can magnify
the image of the object up to about 1500 times but we are still
limited in the detail we can see by the resolving power of the
microscope. The resolving power of a microscope is a measure
of its ability to distinguish clearly between two points very close together.
The resolving power, a,
is measured by the angular separation of two point sources that are just
detectably separated by the instrument. The smaller this angle,
the greater the resolving power.
= 1.22 l
is the wavelength of the light used and D
is the diameter of the objective lens in metres. a is measured in radians
(1 radian = 57.3o approx.). The resolving power can easily
be translated into a distance in metres from the geometry of the situation.
The best resolving power we can achieve with a light
microscope is around 0.2 mm.
Points closer together than this cannot be distinguished clearly as separate
points. Clearly, if we could reduce the wavelength of the radiation
used to view the sample, we could increase the resolution obtainable.
De Broglie’s work on matter waves provided a theoretical means of achieving
this wavelength reduction. De Broglie suggested that the matter
waves associated with electrons accelerated through large potential differences
would have wavelengths of the order of 0.5 nm. This is 1000 times
smaller than the wavelength of green light (middle of the visible spectrum).
Max Knoll and Ernst Ruska constructed the first electron
microscope around 1930. It used the wave characteristics
of a beam of electrons rather than light to study objects too
small for conventional light microscopes.
Modern electron microscopes typically consist of:
Before looking more closely at the different types
of electron microscopes, it is worth stressing that the development of
the electron microscope has had a massive impact on knowledge and understanding
in many fields of science. Modern electron microscopes can view
detail at the atomic level, 0.1 nm resolution (1000 times better than
the light microscope) at up to a million times magnification.
For excellent diagrams &
information on both TEM’s & SEM’s see the links below, as well as
those on my Useful Links page in the From Quanta To Quarks section.
The Transmission Electron Microscope
- Not Examinable
electron microscope (TEM), electrons are transmitted through a
thinly sliced specimen and form an image on a fluorescent screen or photographic
plate. Those areas of the sample that are more dense will transmit
fewer electrons (ie will scatter more electrons) and will therefore appear
darker in the image. TEM’s can magnify up to one million times and
are used extensively in Biology and Medicine to study the structure of
viruses and the cells of animals and plants. The following diagram
shows the basic structure of a TEM.
The Scanning Electron Microscope -
electron microscope (SEM), the beam is focussed to a point and
scanned over the surface of the specimen. Detectors collect the
backscattered and secondary electrons coming from the surface and convert
them into a signal that in turn is used to produce a realistic, three-dimensional
image of the specimen. The detector receives back fewer electrons
from depressions in the surface and therefore lower areas of the surface
appear darker in the image.
the specimen to be electrically conducting, so specimens are coated with
a thin layer of metal (often gold) prior to scanning. SEM’s can
magnify up to around one hundred thousand times or more and are used extensively
in many areas (Biology, Medicine, Physics, Chemistry, Engineering) to
study the 3-D structure of surfaces from metals and ceramics to blood
cells and insect bodies. A special type of SEM called the Scanning
Tunnelling Electron Microscope (STM) uses a quantum mechanical effect
called tunnelling to provide a 3-D image of the network of atoms
or molecules on the surface of a specimen.
Magnetic Lenses In Electron Microscopes
- Not Examinable
Many different types of lenses can be used to focus
and control the electron beam in electron microscopes. Electrostatic
lenses employ electric fields only to accomplish this task.
Most lenses used, however, are magnetic in nature. Magnetic
lenses can be either electromagnetic or permanent magnets.
In either case, these lenses refract (bend) electron beams in a manner
analogous to a biconvex glass lens refracting beams of light (photons).
An electromagnetic lens usually consists of
a solenoid (coil), surrounded by a soft iron casing (shroud) to enhance
the field and can also include soft iron pole pieces that further concentrate
the field. Electrons travelling at angles to the field lines
experience magnetic forces, which cause them to spiral around the field
lines as they move through the magnetic lens. The radius of
the spiral depends on the field strength. Electrons that enter the
lens along the central axis of the solenoid will initially diverge due
to repulsion from other electrons and then be refocused further along
the lens. The current supplied to the solenoid controls the effective
“shape” of the lens and therefore the focal length of the lens.
The focal point, called the cross-over point, is where the electrons converge
to a sharp focus. The diagram below shows an electromagnetic lens.
Both electromagnetic and permanent magnetic lenses
can be designed to produce various different shaped magnetic fields as
required. The diagram below shows a triangular magnetic field
of uniform intensity. The triangular shape means that electrons
that are furthest from the axis are in the magnetic field for more time
and are consequently refracted at a greater angle than electrons near
the axis. This results in the electrons in the beam being focused
to a point on the axis beyond the magnetic lens.
If a rectangular shaped field is used, the electron
beam can be focused by making the field non-uniform as shown below.
Since the magnetic field strength increases with distance from the axis,
the electrons near the axis are refracted through small angles, while
those further away from the axis experience greater angles of refraction.
MAJOR COMPONENTS OF THE NUCLEUS
quest to discover the nature of the nucleus has occupied generations
of physicists since Rutherford discovered its existence in 1911.
Rutherford determined from his scattering experiments that the nucleus
was of the order of 10-14 m in diameter. This turned
out to be about 1/10 000 of the diameter of the atom, determined by Max
von Laue in 1912 using X-ray diffraction to be 10-10 m.
Rutherford showed that the nucleus contained all of the positive charge
of the atom and most of the atom’s mass. Henry Moseley, a graduate
student working with Rutherford, found a direct correlation between an
element’s position in the Periodic Table and its nuclear charge and also
discovered that the total charge on a nucleus was equal to the total charge
of the orbiting electrons in a neutral atom. By 1914 scientists
accepted that a hydrogen ion (a hydrogen atom which has lost its electron)
consisted of a singly charged particle. Rutherford named this the
The Proton-Electron Model
possible structure for the nucleus was then suggested by Rutherford.
It was known in 1914 that an atom such as fluorine (atomic number 9) for
example, had a mass equivalent to 19 protons but a charge of only 9 protons.
So Rutherford suggested that the nucleus contained protons and electrons
to balance the charge discrepancy. A fluorine nucleus would
therefore contain 19 protons and 10 electrons – a total charge of 9 protons
and total mass of 19 protons (electron mass being negligible compared
to the proton mass).
This model was called the Proton-Electron Model.
In general, a nucleus contained A protons and (A – Z) electrons,
where A is called the mass number and Z the atomic number
of the nucleus. This model could explain how a-particles
& b-particles (electrons)
could be emitted from some radioactive nuclei but problems arose:
Energies of emitted b-particles
could not be accurately predicted.
Quantum number anomalies arose with the spin of electrons and
protons within the nucleus.
Heisenberg’s Uncertainty Principle suggested that electrons
could not be confined within the nucleus.
Hence, the model was abandoned.
In a lecture
in 1920, Rutherford suggested that a proton and an electron within the
nucleus might combine together to produce a neutral particle. He
named this particle the neutron. Experimental difficulties
associated with the detection of a neutral particle greatly hindered the
research. In 12 years of searching, no such particle was found.
In 1930, two German physicists, Bothe & Becker,
bombarded the elements boron (B) and beryllium (Be) with a-particles.
These elements, especially the Be, emitted a very penetrating form of
radiation that was much more energetic than gamma-rays.
Frederic & Irene Joliot (Irene was the daughter
of Marie Curie) found in 1932 that although this radiation
could pass through thick sheets of lead, it was stopped by water or paraffin
wax. They found that large numbers of very energetic protons
were emitted from the paraffin when it absorbed the radiation. The
Joliots assumed that the radiation must be an extremely energetic form
of gamma radiation. In the same year, the English physicist, James
Chadwick showed theoretically that gamma rays produced by a-bombardment
of Be would not have sufficient energy to knock protons out of paraffin,
and that momentum could not be conserved in such a collision between
a gamma ray and a proton.
repeated the Joliot’s experiments many times. He measured the energy
of the radiation emitted by the Be and the energies (and therefore the
velocities) of the protons coming from the paraffin. On the basis
of its great penetrating power, Chadwick proposed that the radiation emitted
from the Be was a new type of neutral particle – the neutron, as originally
proposed by Rutherford. He then applied the conservation of energy
and momentum laws to his experimental results and showed that the particles
emitted from the Be had to be neutral particles with about the same mass
as the proton. Chadwick had indeed discovered the neutron.
Chadwick explained the process occurring in the experiment
explained that when the neutrons emitted from the Be collided with the
light hydrogen nuclei in the paraffin, the neutron came to a sudden stop
and the hydrogen nucleus (proton) moved off with the same momentum as
the neutron had before the collision.
Following Chadwick’s discovery of the neutron, a new
model of the nucleus was proposed. This model suggests that the
nucleus consists of protons and neutrons. Together these
particles are called the nucleons – particles that make up the
nucleus. Protons and neutrons have approximately the same mass,
which is about 1800 times that of the electron. Protons are positively
charged and neutrons are neutral.
The number of protons in the nucleus is called
the atomic number of the nucleus and corresponds to the position
of the nucleus in the Periodic Table of Elements. The total number
of protons and neutrons in the nucleus is called the mass number
of the nucleus. Each nucleus can be represented by using nuclide
notation. A nuclide is a nucleus written in the form:
= element symbol (eg Na, Co, U), Z = atomic number and
A = mass number. Clearly, A = N + Z, where
= number of neutrons in nucleus. An alternate notation is to write
the nucleus with its mass number after it – eg U-235 for uranium
with a mass number of 235.
Isotopes of an element are atoms of that element
varying in the number of neutrons present in their nuclei. Clearly,
isotopes of the same element have the same atomic number but different
mass numbers (and therefore slightly different masses). So, U-234,
U-235 and U-238 are all isotopes of uranium – they all have
92 protons but differ in mass number.
model of the nucleus is still the basic model used today. Many
more nuclear particles have been found, however, and we will examine
some of these a little later. For now, we turn our attention to
the physical description of natural radioactivity.
Natural Radioactivity and Transmutation
Experimental work around the turn of the 20th
Century by Henri Becquerel (1896), Ernest Rutherford, Marie & Pierre
Curie, Paul Villard and many other physicists led to the discovery
of the three kinds of natural radiations – alpha particles, beta particles
and gamma rays. These radiations were emitted naturally from
certain elements (uranium, polonium, radium, actinium). Further,
it was found that the emission of natural radiations by one element usually
led to the production of a different element. For instance, radium
was produced as a result of the radioactive decay of uranium.
This change of a parent nucleus into a different
daughter nucleus is called nuclear transmutation. One element
effectively changes into another element.
When transmutation occurs, the sum of the atomic numbers
on the left hand side of the nuclear equation equals the sum of the atomic
numbers on the right hand side. Likewise, the sum of the mass numbers
on the left hand side of the nuclear equation equals the sum of the mass
numbers on the right hand side.
A nucleus of an element X changes into a nucleus of
an element Y according to:
where the helium-4 nucleus is the emitted
alpha particle. Alpha decay occurs primarily among nuclei
with atomic numbers greater than 83.
Beta Decay (The
Early attempts to explain beta decay assumed that an
electron in the nucleus (Proton-Electron Model) was emitted in a process similar
to that by which an alpha particle was emitted from a nucleus.
One problem with this explanation, however, was that although all alpha
particles emitted from a given species of nucleus had the same energy, beta
particles emitted from a given species of nucleus seemed to have a continuous
spectrum of energies.
James Chadwick, in some experiments conducted prior to
World War I, used a Geiger Counter to study beta particles emitted from a source
and then deflected by a uniform magnetic field.
He found that the beta particles had a wide range of radii of curvature
in the field, indicating that the beta particles had different velocities and
therefore different energies. Similar
experiments by many Physicists during the 1920’s and early 1930’s clearly
indicated that during the beta decay of a particular nuclear species (eg
Bi-210) electrons were emitted with a distribution of energies rather than with
a distinct single value of energy.
The following graph shows the energy spectrum for electrons
emitted during the decay of Bi-210. The
intensity (vertical axis) shows the number of electrons emitted with each
particular kinetic energy (horizontal axis).
This graph was taken from the web link:
Graphs such as this could not be explained.
Why did the beta decay of a particular nuclear species produce many
different beta particle emission energies?
How was this possible when the decay process produced exactly the same
daughter nucleus in each case?
In 1934, Italian physicist Enrico Fermi named
Pauli’s particle the neutrino (n),
meaning “little neutral one” in Italian, and formulated a theory of b-decay
using this particle. Fermi’s
theory successfully explained all experimental observations.
For instance, the shape of the energy curve shown above for Bi-210 can be
predicted from the Fermi Theory of beta decay.
Despite several ingenious attempts, the neutrino was not experimentally
observed until 1956. In that year,
two American Physicists, Cowan and Reines successfully identified the neutrino
by detecting the products of a reaction that could only have been initiated by
the neutrino. Basic details of this
experiment are provided in “Nuclear Physics” by J Joyce & R Vogt (Brooks
b-decay is often referred to as the weak interaction because it is
1012 times weaker than the strong nuclear force
that holds the nucleus together.
There are two types of
in which a neutron decays to produce a proton, an electron and an anti-neutrino
The electron and the anti-neutrino are emitted but
the proton stays behind, thus increasing the atomic number by one.
following equation for example, describes the spontaneous decay of C-14
in the upper atmosphere, as it is produced by bombardment of nitrogen
by neutrons in cosmic rays:
b+-decay in which a positron (positive electron) is emitted after a
proton decays to produce a neutron, a positron and a neutrino.
The following equation for example, describes the b-decay
of the artificially produced N-13 nucleus:
an aside, it is worth noting that the distinguishing feature between the
neutrino & anti-neutrino is their helicity. The anti-neutrino
has its spin angular momentum parallel to its linear momentum – it has
a right hand screw helicity. The neutrino has its spin angular
momentum anti-parallel to its linear momentum – it has a left hand
This usually accompanies a
or b-decay. A nucleus de-excites
by emitting a high-energy gamma ray photon. This is not
the * represents an excited nucleus.
THE STRONG NUCLEAR FORCE
It is known that there is an electrostatic Coulomb
repulsion force between any two like charges. So, in the
case of protons in the nucleus, there must be some sort of force that
holds the protons together. At first we might be tempted to
suggest that the gravitational attraction that exists between all
bodies possessing mass is responsible for holding the protons together.
However, if we evaluate the relative contributions of the electrostatic
and gravitational forces between protons, we find that the gravitational
force is millions of times smaller than the electrostatic force.
Thus, there must be another force at work.
The force responsible for holding all nucleons together
is the strong nuclear force. The graph below shows the strong
nuclear force between nucleons as a function of the separation of the
The main properties of the strong nuclear force
COMMON UNITS USED
IN NUCLEAR PHYSICS
For convenience, nuclear physicists
usually use the atomic mass unit (u) as the unit of mass and the
electron volt (eV) or Mega electron volt (MeV) as the unit of energy.
These are defined as follows.
MASS & ATOMIC MASS UNIT
present standard atom is the atom of the commonest isotope of carbon,
C-12. By definition this isotope of carbon has a mass of 12.0000
atomic mass units (u) exactly. Thus, since the mass of one C-12
atom is 1.9924 x 10-26 kg (by mass spectrograph measurements),
1.0000 u = 1.9924 x 10-26/12.0000
= 1.6603 x 10-27 kg
THE ELECTRON VOLT
The electron volt is the amount of energy gained
by an electron as it is accelerated through a potential difference of
1 eV = 1.602 x 10-19 J
(from W = qV)
1 MeV = 1.602 x 10-13 J
Clearly, using Einstein’s equation for the equivalence
of mass and energy we have:
1 u = 931.5 MeV
MASS DEFECT AND BINDING ENERGY
The experimentally measured mass of any nucleus is less than the sum of
the masses of its constituent protons and neutrons.
The mass of a proton is 1.00728 u.
The mass of a neutron is 1.00867 u.
The mass of an electron is 0.00055 u.
For example, let us consider an atom of the
commonest isotope of chlorine, Cl-35.
The actual mass of this atom, determined
by experiment, is 34.980175 u.
The combined mass of the constituent particles
may be determined as follows:
Mass of 18 neutrons = 18
x 1.00867 = 18.15606
Mass of 17 electrons
x 0.00055 = 0.00935
Combined Mass = 35.28917
The difference in mass is called the mass defect
of the atom (or nucleus, if we are dealing with the nucleus only).
In this case, the mass defect is about 0.309 u or 5.13 x 10-28
This small mass has been converted into the binding
energy of the nucleus (the energy holding the nucleus together).
The mass defect of a nucleus can therefore be defined as the mass equivalent
of the binding energy of the nucleus. The amount of binding
energy involved in this example is:
E = mc2
E = 931.5 x 0.309
E = 5.13 x 10-28 x (3 x
108)2 = 287.8 MeV
x 10-11 J
= 288.2 MeV
By definition, the binding energy of the nucleus
is the energy needed to separate the nucleus into its constituent parts.
When the nucleons come together to form the nucleus, they release the
If we take the total
binding energy of a nucleus and divide it by the total number of nucleons
in the nucleus, we get a very good measure of how tightly each individual
nucleon is held in the nucleus. This binding energy per nucleon
figure is a very good measure of the stability of the particular
nucleus. The higher the binding energy per nucleon, the more
stable the nucleus. The diagram below shows the basic shape
of the binding energy per nucleon versus mass number graph.
This will be a useful tool for explaining nuclear fission and fusion
a little later. (Note that the vertical axis has been drawn on the
right for clarity.)
Note that the binding energy per nucleon is low for
low mass number nuclei. This is because in such nuclei each nucleon
is not uniformly surrounded and thus does not experience the full effects
of the strong nuclear force. Most nuclei have binding energy per
nucleon values between 7 and 9 MeV, with the highest value being that
for Fe-56. For very high mass number nuclei the electrostatic repulsive
forces between the protons result in a gradual decrease in binding energy
per nucleon values.
The rest of the notes and worksheets for this topic are found on the