NUCLEAR PHYSICS AND RADIOACTIVITY
Nuclear Structure
The nucleus of the atom consists of protons and neutrons.
Collectively they are called nucleons.
The neutron was discovered in 1932 by James Chadwick. It
carries no charge and has a mass slightly larger than that of a
proton.
The number of protons in the nucleus is expressed by the
atomic number, Z. The total number of neutrons is expressed
by the neutron number, N. The total number of protons and
neutrons is called the atomic mass number, A. A multiplied by
the mass of one nucleon is approximately equal to the total
mass of the nucleus.
A=Z+N
The shorthand notation used in equations representing nuclear
reactions shows the atomic number as a subscript, the mass
number as a superscript and the chemical symbol of the
element as the main character. An isotope of Aluminum with
13 protons and 27 total nucleons would be written as:
13Al
27
Some texts put Z and A on the same side of the chemical
symbol but Z is always the subscript.
Nuclei that contain the same number of protons but different
numbers of neutrons are called isotopes. Carbon exists in three
forms. They are carbon-12, carbon-13, and carbon-14. They
each have 6 protons, but 6, 7, and 8 neutrons respectively.
The approximate radius of a nucleus can be determined from
the number of nucleons present. The formula for the radius in
meters is:
r = (1.2 x 10-15m)A1/3
Notice that the density of all nuclei will be approximately the
same. The only difference in the nuclei will be the number of
nucleons, A. This factor will divide out when total mass is
divided by volume.
The Strong Nuclear Force and Stability of the Nucleus
In the nucleus, the positively charged protons repel each other
with an electrostatic force that is many times the force of
gravity for the same particles.
Example
Compare the forces of gravitational attraction and electrostatic
repulsion for the protons and neutrons in a Helium nucleus.
Since gravity is not strong enough to hold the particles of a
nucleus together, another force called the strong nuclear force
must exist. It is a fundamental force along with gravity and the
electroweak force. They are called fundamental forces since all
other forces in nature can be explained in terms of these three.
The strong nuclear force does not depend on electric charge.
The same force exists between any two nucleons.
The strong nuclear force acts over a very short distance. The
two nucleons must be within 10-15 m of each other.
Since the strong nuclear force only acts over a short distance, it
only exists between two nucleons that are adjacent. The
electrostatic repulsive force, however, exists between all of the
protons in the nucleus.
As the number of protons increases, the number of neutrons
increases more rapidly to provide extra strong nuclear force to
help overcome the increase in electrostatic repulsion.
This graph shows number of neutrons on the Y axis and the
number of protons in the same nuclide on the X axis. Where
the slope = 1, the numbers of neutrons and protons are equal.
For elements with more than 18 protons, the number of
neutrons increases noticeably faster than the number of
protons.
Nuclei with more than 83 protons are not stable and break
apart or change their internal structure by emitting radiation.
This process is called radioactivity and was discovered by
Becquerel.
Mass Defect of the Nucleus and Nuclear Binding Energy
The nucleons in a stable nucleus are held together by the
strong nuclear force and the fact that the stable nucleus has
less energy than the separate components. In order to separate
the protons and neutrons that compose a stable nucleus an
amount of energy called the binding energy of the nucleus must
be added.
The total mass of the individual nucleons that make up a
nucleus is greater when they are separated than when they are
together in the nucleus. This difference in mass is called the
nuclear mass defect and is the matter equivalent of the nuclear
binding energy.
In Einstein's theory of special relativity, matter and energy are
shown to be different forms of the same thing. Matter can be
converted to energy and energy can be converted to matter.
The relationship is Einstein's famous equation:
E = mc2
where E and m are energy and matter and c is the speed of
light.
The total nuclear binding energy of a nucleus can be found by
replacing m in the equation with the nuclear mass defect.
Example
Determine the nuclear binding energy of the helium-4 nucleus.
Don't forget to allow for the mass of the 2 electrons.
When comparing binding energies of different nuclei, it is
generally useful to look at the binding energy per nucleon for
each nucleus. When atomic number is small, binding energy
per nucleon increases rapidly. At about A = 60, the binding
energy per nucleon reaches a maximum of 8.7 MeV. Then it
begins to decrease.
A smaller amount of binding energy is what makes
elements with atomic numbers greater than 83 unstable.
Radioactivity
The three types of naturally occurring radiation are α
rays, β rays and γ rays. They are named in order of
their ability to penetrate matter. α rays are the least
penetrating since they are stopped by .01 mm thick
lead. β rays are next and penetrate lead to a depth of 0.1
mm. γ rays are the most penetrating and can pass
through 100 mm of lead.
Notice that α and β rays are deflected by a magnetic
field which indicates they are composed of charged
particles. Since γ rays are not deflected, they must not
have a charge.
Alpha Decay
When a nucleus produces α rays it is said to undergo α
decay. An α particle consists of two protons and two
neutrons which and can be described as a He-4 nucleus.
The symbol for an α particle in nuclear equations is
4
2He . The subscript two indicates two protons and the
superscript four indicates a total of four nucleons.
The α decay of U-238 is a good example.
The equation is:
238
92U
→ 90Th234 + 2He4
The original nucleus is called the parent nucleus and the
nucleus that remains is called the daughter nucleus.
Since the atomic number of the daughter nucleus is
different from the parent nucleus, transmutation has
occurred.
During a nuclear change such as a transmutation, the
total charge before the event must be the same as the
total charge after the event. Also, the nucleon number
before the event must equal the nucleon number after
the event.
Notice in the α decay of U-238 to Th-234 that the total
charge on the left is 92 and the total charge on the right
is also 92. The total number of nucleons on the left is
238 and the total number of nucleons on the right is also
238. Both of these quantities are conserved.
Although it was not shown in the equation, energy is
also released. The amount can be calculated by finding
the nuclear mass defect and converting it to the
corresponding energy.
Example
The atomic mass of U-238 is 238.0508 u, Th-234 is
234.0436 u, and the mass of an α particle is 4.0026 u.
Determine the energy released when a U-238 atom
undergoes α decay. Notice that the mass of the α
particle includes the two electrons lost by U-238 when it
became Th-234. Also 1 u becomes 931.5 Mev when it is
converted to energy.
The energy that is released is carried away as kinetic
energy of the daughter nucleus and the α particle. Some
energy is usually released as a gamma ray. Since
momentum must be conserved, the kinetic energy
change of the alpha particle is much greater than that of
the daughter nucleus.
Beta Decay
Beta decay occurs when a nucleus emits an electron. An
example of beta decay is:
The equation is:
234
90Th
→ 91Pa234 + -1e0
The Thorium nucleus changes into a Protactinium
nucleus by losing a negative charge. The negative
charge is an electron which is thought to come from the
decay of a neutron into a proton and an electron. The
energy the electron receives from the decay process is
enough to allow it to escape the nucleus.
A second type of β decay occurs when a positron is
emitted by a nucleus. A positron has the same mass as
an electron but has a +e charge instead of -e. The
process is thought to occur when a proton emits a
positron to become a neutron.
Gamma Decay
The nucleus has discrete energy states like the electrons
that surround it. When the nucleus changes from a
higher energy state to a lower one, it emits a photon
with considerably more energy than those emitted by
electrons. A group of these high energy photons is called
a gamma ray (γ ray).
Gamma decay does not cause a change in the identity of
the atom since there is no change in the number of
protons. Since there is no change in the number of
neutrons, the mass number also remains the same.
Gamma radiation is useful in Gamma Knife surgery
and thallium stress tests.
The Neutrino (υ)
During β decay, not all of the energy produced can be
accounted for in the kinetic energy of the β particle and
the recoiling nucleus. The emission of another particle
with no charge and practically no mass was proposed to
account for this deficit.
Its existence was experimentally verified in 1956. It
turns out that an antineutrino is emitted during β- decay
and a normal neutrino is emitted during β+ decay.
Radioactive Decay and Activity
As a sample of a radioactive nuclide decays, the number
of parent nuclei decreases in a fashion that can be
represented by a smooth curve.
Although this curve looks a lot like a hyperbola, it is
actually an exponential function of time.
In order to help us compare rates at which different
nuclides decay, we use the concept of half-life.
The half-life (T½) of a nuclide is the amount of time
required for one half of the parent nucleii to turn into
daughter nuclei. In the graph above, it is easy to see that
the half-life time corresponds to a decrease in the
number of parent nuclei equal to ½ of the number
present at the beginning of the interval.
Example
Suppose 3.0 x 107 radon atoms are trapped in a
basement and the basement is sealed so that none can
enter or leave. Since the half-life of radon is 3.83 days,
how many radon atoms are left after 30.64 days?
The activity of a radioactive sample depends on the rate
at which the sample decays and the number of
radioactive nuclei present. The equation for activity is:
ΔN/Δt = -λN
where λ is the decay constant and depends on the halflife and N is the number of nuclei present. The number
of nucleii present is calculated using the equation:
N = N0e-λt
From this equation, it can be shown that λ = (ln2)/T½.
The SI unit for activity is the Becquerel (Bq) and equals
one disintegration per second. Another widely used unit
is the Curie (Ci) which equals 3.70 x 1010 Bq. One Curie
is roughly the activity of one gram of pure radium.
Example
In our last example, 3.0 x 107 radon atoms were trapped
in a basement. Find the activity (a) initially and (b) 31
days later.(T½ = 3.83 days)
Radioactive Decay
Radioactive Dating
A very useful application of radioactivity is the
determination of the age of artifacts and rocks. If the
object to be dated contains a radioactive element, it is
often possible to determine the number of half-lives that
have passed since the object came into existence.
Carbon-14 is a radioactive isotope of carbon that is
found in all living things. An assumption is made that
the level of C-14 found in living things today has not
changed for as long as 50,000 years ago.
By determining the C-14 activity of a sample of a
formerly living object, the number of half-lives that
have passed can be determined and an approximate
time since the object stopped ingesting C-14 can be
determined using the equation:
A = A0e-λt
Example
The iceman was found in a glacier in the Italian Alps in
1991. Material found with the body had a C-14 activity
of about 0.121 Bq per gram of carbon. Determine how
long ago the Iceman died. A0 = 0.23 Bq
Radioactive Decay Series
A radioactive decay series simply lists the sequence of
daughter nuclei that are produced from a radioactive
nucleus. Any time radioactive decay produces a nucleus
that is itself radioactive, the sequence continues until a
stable nucleus is produced.
As you can see in this series, more than one decay mode
is possible for certain nuclides. The end result no matter
what the path is stable Lead-206.