Topic 13.2 Nuclear
Physics
5 hours
Estimating the Radii of
Nuclei
• Consider an α-particle that is on a direct
collision course with a gold nucleus and its
subsequent path. Since the gold nucleus is
much more massive than the α-particle we
can ignore any recoil of the gold nucleus.
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Energy Considerations
• The kinetic energy of the α-particle when it is a long way
from the nucleus is Ek. As it approaches the nucleus, due to
the Coulomb force, its kinetic energy is converted into
electrostatic potential energy.
kq1q2
Ep 
r
• At the distance of closest approach all the kinetic energy will
have become potential energy and the α-particle will be
momentarily at rest. Hence we have that
kq1q2 k ( Ze)(2e) 2kZe2
E p  Ek 


r
d
d
where the nucleus with atomic number Z has a charge of Ze
and the a-particle has a charge of 2e.
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• For a gold nucleus (Z = 79) and an α-particle with
kinetic energy 4.0 MeV we have that
EK
13
d
 1.2  10 m
2
2kZe
• The distance of closest approach will of course
depend on the initial kinetic energy of the αparticle. However, as the energy is increased a
point is reached where Coulomb scattering no
longer take place. The above calculation is
therefore only an estimate. It is has been
demonstrated at separations of the order of 10-15
m, the Coulomb force is overtaken by the strong
nuclear force.
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Measuring Nuclear Masses
• The measurement of nuclear (isotope) masses
is achieved using a mass spectrometer.
Positive ions of the element
under study are produced in
a high voltage discharge
tube (not shown) and pass
through a slit (S1) in the
cathode of the discharge
tube. The beam of ions is
further collimated by
passing through slit S2
which provides an entry to
the spectrometer. In the
region X, the ions move in
crossed electric and
magnetic fields.
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Measuring Nuclear Masses
The electric field is produced
by the plates P1 and P2 and the
magnetic field by a coil
arrangement. The region X
acts as a velocity selector. If
the magnitude of the electric
field strength in this region is E
and that of the magnetic field
strength is B (and the
magnitude of the charge on an
ion is e) only those ions which
have a v velocity given the
expression Ee = Bev will pass
through the slit S3 and so
enter the main body, Y, of the
instrument.
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Measuring Nuclear Masses
A uniform magnetic field, B´,
exists in region Y and in such a
direction as to make the ions
describe circular orbits. For a
particular ion the radius r of
the orbit is given by,
m v2
 B' ev
r
or
mv
r
B' e
Since all the ions have very
nearly the same velocity, ions
of different masses will
describe orbits of different
radii, the variations in value
depending only on the mass of
the ion.
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Measuring Nuclear Masses
A number of lines will
therefore be obtained on the
photographic plate P, each line
corresponding to a different
isotopic mass of the element.
The position of a line on the
plate will enable r to be
determined and as B´, e and v
are known, m can be
determined.
IB Outcome 13.2.2
- Students should be able to
draw a schematic diagram of the
Bainbridge mass spectrometer,
but the experimental details are
not required.
- Students should appreciate
that nuclear mass values provide
evidence for the existence of
isotopes.
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Nuclear Energy Levels
• The α-particles emitted in the
radioactive decay of a
particular nuclide do not
necessarily have the same
energy. For example, the
energies of the α-particles
emitted in the decay of nuclei
of the isotope bismuth-212
(a.k.a. thorium-C) have several
distinct energies, 5.973 MeV
being the greatest value and
5.481 MeV being the smallest
value. To understand this we
introduce the idea of nuclear
energy levels.
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Nuclear Energy Levels
• For example, if a nucleus of
bismuth-212 emits an
α-particle with energy 5.973
MeV, the resultant daughter
nucleus will be in its ground
state. However, if the emitted
α-particle has energy 5.481
MeV, the daughter will be in an
excited energy state and will
reach its ground state by
emitting gamma photons of
total energy 0.492 MeV.
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Nuclear Energy Levels
• The existence of nuclear energy levels
receives complete experimental verification
from the fact that γ-rays from radioactive
decay have discrete energies consistent with
the energies of the α-particles emitted by the
parent nucleus. Not all radioactive
transformations give rise to γ-emission and in
this case the emitted α-particles all have the
same energies.
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Beta Decay
• Recall that β- decay results from the decay of a neutron into a
proton and that β+ decay results from the decay of a proton in a
nucleus into a neutron, i.e.
• It is found that the energy spectrum of the β-particles is
continuous whereas that of any γ-rays involved is discrete. This
was one of the reasons that the existence of the neutrino was
postulated otherwise there is a problem with the conservation of
energy. α-decay clearly indicates the existence of nuclear energy
levels so something in β-decay has to account for any energy
difference between the maximum β-particle energy and the sum
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of the γ-ray plus intermediate β-particle energies
Beta Decay
• We can illustrate how the neutrino accounts for this discrepancy by
referring to the figure below showing the energy levels of a fictitious
daughter nucleus and possible decay routes of the parent nucleus
undergoing β+ decay. The figure shows how the neutrino accounts for the
continuous β spectrum without sacrificing the conservation of energy.
An equivalent diagram can of course be drawn for β- decay with the
neutrino being replaced by an anti-neutrino.
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The Radioactive Decay Law
• We have seen in Topic 7 that radioactive decay is a
random process. However, we are able to say that the
activity of a sample element at a particular instant is
proportional to the number of atoms of the element in
the sample at that instant. If this number is N we can
write that
N
 N
t
or
dN
 N
dt
where λ is the constant of proportionality called the
decay constant and is defined as ‘the probability of
decay of a nucleus per unit time’ and has units of s-1.
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Derivation of the
Radioactive Decay Law
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Finding the Half-Life using
the Radioactive Decay Law
• The radioactive decay law enables us to determine a relation
between the half-life of a radioactive element and the decay
constant.
• If a sample of a radioactive element initially contains No
atoms, after an interval of one half-life the sample will
contain No/2 atoms. If the half-life of the element is T½ then
t
the decay law N  Noe becomes:
Measuring the Half-Life
• The method used to measure the half-life of
an element depends on whether the half-life
is relatively long or relatively short. If the
activity of a sample stays constant over a few
hours it is safe to conclude that it has a
relatively long half-life. On the other had if its
activity drops rapidly to zero it is clear it has a
very short half-life.
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Elements with
Long Half-Lives
• Essentially the method is to measure the
activity of a known mass of a sample of the
element. The activity can be measured by a
Geiger counter and the decay equation in its
differential form is used to find the decay
constant. An example will help understand
the method.
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Elements with
Long Half-Lives
• A sample of the isotope uranium-234 has a mass of 2.0 μg. Its
activity is measured as 3.0 × 103 Bq. Find its half-life.
NOTE: 1 Bq (Becquerel) = 1 decay / second
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Elements with
Short Half-Lives (Hours)
• For elements that have half-lives of the order of
hours, the activity can be measured by
measuring the number of decays over a short
period of time (minutes) at different time
intervals. A graph of activity against time is
plotted and the half-life read straight from the
graph.
• Better is to plot the logarithm of activity against
time to yield a straight line graph whose gradient
is equal to the negative value of the decay
constant.
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Elements with
Short Half-Lives (Hours)
• For elements that have half-lives of the order of
hours, the activity can be measured by
measuring the number of decays over a short
period of time (minutes) at different time
intervals. A graph of activity against time is
plotted and the half-life read straight from the
graph.
• Better is to plot the logarithm of activity against
time to yield a straight line graph whose gradient
is equal to the negative value of the decay
constant.
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Elements with
Short Half-Lives (Seconds)
• For elements with half-lives of the order of
seconds, the ionisation properties of the
radiations can be used. If the sample is placed
in a tube across which an electric field is
applied, the radiation from the source will
ionise the air in the tube and thereby give rise
to an ionisation current. With a suitable
arrangement, the decay of the ionisation
current can be measured.
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Spherical Chickens in a
Vacuum
• The previous slides are just
an outline of the methods
available for measuring halflives and are sufficient for the
HL course.
• Clearly in some cases the
actual measurement can be very tricky. For example,
many radioactive isotopes decay into isotopes that
themselves are radioactive and these in turn decay
into other radioactive isotopes. So, although one may
start with a sample that contains only one radioactive
isotope, some time later the sample could contain
several radioactive isotopes.
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Homework:
Tsokos, Page 412
Questsions 1 to 20
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