CH 30 – Nuclear Energy and Elementary Particles
In Chapter 29 we discussed the binding energy of a nucleus. The mass of a nucleus is
less than the mass of its constituent protons and neutrons and this mass difference gives
the binding energy through the relationship Eb = mc2. When we calculate the binding
energy per nucleon, then we find that the nuclei with intermediate mass have greater
binding energy per nucleon than those light or very heavy nuclei. A graph of binding
energy per nucleon (also shown in the CH 29 notes) is shown below.
By combining light nuclei to form heavier nuclei or by splitting apart heavy nuclei to
form lighter nuclei, we can take advantage of the change in binding energy to convert
mass to energy. The former process is fusion and the later process is fission.
Nuclear Fission
239
In nuclear fission a heavy nucleus such as 235
92 U or 94 Pu split into two smaller nuclei
along with the production of one or more neutrons. The fission is prompted by the
capture of a slow neutron which makes the nucleus unstable. In the fission of uranium-
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235, the capture of a neutron forms uranium-236 in an excited state which subsequently
breaks apart as shown below.
1 235
236 *
0 n 92 U  92 U  X  Y  neutrons
The product nuclei produced in nuclear fission are typically radioactive.
There are many possible fission products. Some specific examples for uranium-235 are
1 235
141
92
1
0 n 92 U  56 Ba  36 Kr 30 n
1 235
140
94
1
0 n 92 U  54 Xe 38 Sr  2 0 n
An estimate of the energy released in a typical fission event can be determined as
follows. From the above graph, we see that the binding energy per nucleon for U-238 is
about 7.5 Mev and the binding energy per nucleon for a nucleus with A = 119 (= 238/2)
is about 8.5 Mev. So, the energy released per nucleon is about 1 Mev, and the total
energy released is about 238 Mev. A more precise value is Q  208 Mev. This lower
value is in part because the binding energy per nuclei of the products is smaller than 8.5
Mev since they are unstable and not typical of the values shown on the graph.
Example:
Calculate the total energy released in the fission of 1 g of U-235.
Solution:
We calculate the total number of atoms in 1 g of U-235 and multiply this by 208 Mev.
N  nN A 
1g
6.02 x10 23 / mole  2.56 x10 21
235 g / mole
E  NQ  ( 2.56 x10 21 )( 208Mev )  ( 5.33x10 29 ev )( 1.6 x10 19 J / ev )
 8.53x1010 J
Example:
A typical US household uses 1000 W of electricity. How long would the energy released
in the fission of 1 g of U-235 power a household if all the energy could be converted in to
electricity?
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Answer:
t
E 8.53x1010 J

 8.53x107 s
P
1000 J / s
Converting to years –
t  ( 8.53x10 7 s )(
1yr
)  2.7 yr
24 x3600 x365s
Nuclear Reactors
A nuclear reactor is designed to allow the sustained fission through a chain reaction. The
fission of a U-235 nucleus produces neutrons which can, in turn, be absorbed by other
nearby U-235 nuclei. On average each fission event produces about 2.5 neutrons. If an
average of one of these neutrons prompts another fission event, then this will go on until
all the U-235 nuclei have been depleted. The chain reaction is said to be critical. If more
than one neutron from each fission event prompts another fission event, then the number
of nuclei undergoing fission will rapidly increase in time and the chain reaction is super
critical. The rate at which energy is released will rapidly increase and can lead to a
nuclear explosion of a melt-down of the reactor. On the other hand, if an average of less
than one neutron produced in a fission event leads to another fission event, then the chain
reaction is not self-sustaining and is subcritical. The figure below illustrates a super
critical chain reaction.
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The neutrons released in a fission event have a
kinetic energy of about 2 Mev. This energy is too
large to be absorbed by another U-235 nucleus
and initiate another fission event. A reactor
contains a moderator material that slows down
the neutrons. A typical moderator material is
“heavy water”, D2O, where D is deuterium. An
illustration of a reactor core is shown to the right.
Reactors are designed to operate near critical
conditions. To prevent a reactor from going
super critical control rods are inserted between
fuel elements. The control rods are made of a
material such as cadmium that is a good absorber
of neutrons. The control rods can be moved up
and down as needed to keep the reactor operating near the critical condition.
The energy released in nuclear reactors is in the form of heat. A heat exchanger is used
to generate steam which drives a turbine and produces electricity.
Nuclear Fusion
In nuclear fusion lighter nuclei combine to form heavier nuclei. For A less than 60 or so
the heavy nuclei have a greater binding energy per nuclei than the light nuclei, so fusion
results in a decrease in mass with the mass difference appearing as energy released in the
process. Nuclear fusion requires overcoming the electrostatic repulsion between the
nuclei. This requires that the nuclei just collide at very high speeds, which means very
high temperatures.
Nuclear Fusion in the Sun
Nuclear fusion is the source of the radiation emitted by the sun and the stars. Stars are
formed by the gravitational collapse of dust and gas. At the gas collapses inward the
molecules gain speed and the temperature rapidly increases. Eventually when the
temperature becomes sufficiently large fusion takes place and energy is released. The
collapse of the gas (plasma) eventually ceases when the pressure due to the thermal
motion of the particles and the radiation pressure become sufficient to balance the
pressure due to the gravitational force. For the sun and other stars that are rich in
hydrogen, protons fuse to form deuterons with the release of a positron and a neutrino:
1
1
2

1 H 1H 1 D  e

Fusion takes place in the interior of the sun, where the core temperature is about 107 K.
The surface temperature is about 5,800 K, which is too low for fusion. Most collisions
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between protons in the interior do not result in fusion, but the fusion process is aided by
the very high density in the core.
Additional fusion reactions in the sun eventually produce helium-4:
1
2
3
1 H  1 D  2 He  
1
3
4

1 H  2 He  2 He  e  
3
3
4
1
2 He  2 He  2 He  2 1H
At the earth’s surface, the neutrino flux from the sun is very large, about 1011/cm2/s.
They interact very, very weakly with matter, so almost all the neutrinos that strike the
earth (and you) pass through undetected.
Fusion Reactors
Extensive research has been done and is continuing on the design of fusion reactors for
converting mass into energy. Fusion has many potential advantages over fission for
energy production. The mass change per nucleon in fusion is much greater than in fission
and there is little long-lived radioactive waste as a byproduct to worry about. In addition,
deuterium, a good fuel for fusion, is relatively abundant in nature and can be cheaply
extracted from water. The problem is how to produce the conditions such that significant
fusion can take place in a controllable manner in which energy can be usefully extracted.
A hydrogen bomb is an uncontrolled fusion reaction.
To emulate the density and temperature at the sun’s interior would require extreme
pressure. Additionally, no material could contain the high temperature plasma without
evaporating. A tokomak is a device that makes use of magnetic fields to confine a high
temperature plasma. Since the density of the plasma is much less than the density of the
sun’s core, the temperature of the plasma must be much larger than the sun’s core, ~ 108
K. In order for a fusion reactor employing a high-temperature plasma to be viable, the
energy released in the fusion process must exceed the energy required to heat the plasma
to the necessary temperature and maintain it at this temperature for a sufficient time. The
amount of time that the plasma must be held at the reaction temperature depends on the
density of the plasma.
The international community has agreed to build a new tokomak (ITER) in France that
would generate 10 times more energy than required to maintain the temperature of the
plasma at fusion conditions (http://www.iter.org/ ). The goal is to demonstrate the
viability of a fusion reactor.
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Example:
Calculate the energy released in the following fusion reactions.
2
2
3
1
1 D  1 D  1T 1 H
2
3
4
1
1 D  1T  2 He  0 n
Solution:
For the first reaction,
 m  2md  mT  mH  2( 2.014102 u )  3.016049 u  1.007825 u
 0.00433 u
Q   mc 2  ( 0.00433u )( 931.5Mev / c 2u )c 2  4.03Mev
For the second reaction,
 m  md  mT  mHe 3  mn  2.014102 u  3.016049 u  4.002603 u  1.008665 u
 0.01888 u
Q   mc 2  ( 0.01888u )( 931.5Mev / c 2u )c 2  17.6Mev
The energy released in the second reaction is considerably more than in the first, but it
requires tritium, which is very rare. The moon, however, is thought to contain a
significant amount of tritium, and mining the moon for this fusion fuel has been
proposed.
An alternative concept for a controlled fusion reactor is inertial laser confinement fusion.
In this approach a small pellet containing deuterium and tritium is blasted simultaneously
from different angles by a large number of laser pulses. The hope is that the pulses can
generate sufficient pressure on the pellet so that the density and temperature reach fusion
conditions. Research on this approach is taking place at Lawrence Livermore National
Laboratory (https://lasers.llnl.gov ).
Elementary Particles
At one time it was thought that the most elementary particle was the atom. Then it was
assumed that the elementary particles consisted of the electron, proton, neutron, and
photon. It is now known that a large variety of short-lived sub-atomic particles can be
generated by high energy collisions of other particles and by the decay of other shortlived particles. Physicists are continually trying to determine which of these particles are
the basic building blocks of nature. That is, which are truly ‘elementary’.
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An important property of particles is the force that they can experience. There are four
basic forces in nature, given in the table below.
Force
Strong
Electromagnetic
Weak
Gravitational
Relative
Strength
1
10-2
10-6
10-43
Range
Short (~ 1 fm)
Long (~ 1/r2)
Sort (~10-3 fm)
Long (~ 1/r2)
Mediating
Field Particle
Gluon
Photon
W ±, Z
Graviton
The forces vary widely in strength. The strongest is the short range “strong force” that
acts between nucleons in the nucleus. The “electromagnetic force” is the electric and
magnetic force that is based on a particle’s charge and is long-range. The “weak” force is
a short range force that is involved in beta decay. The “gravitational” force is the
weakest of all and is long range.
It is believed that these forces are mediated by the exchange of ‘virtual’ particles listed in
the right column of the table. This would be somewhat analogous to the bonding of the
two protons in a hydrogen molecule that is caused by the ‘exchange’ of the two orbital
electrons. When two particles interact by one of these fundamental forces, a virtual field
particle is being continually emitted by one particle and absorbed by the other. It is
called virtual since it is never directly detected. These virtual particles only exist during
the short time during which they are being exchanged. The existence and non-existence
of these particles would seem to violate conservation of energy; however, the uncertainty
principle, Et  /2, allows this to happen to a degree that depends on the time during
which the energy is determined. Because of the short range of the interactions and the
speed of the virtual particles, their lifetime can be short and the energies of the virtual
particles can be significant.
Classifications of Particles
All particles other than the photon can be classified as either a lepton or a hadron.
Leptons
The leptons include the electron (e-) , muon (-) , and tau ( -) particles and their
associated neutrinos - e, ,  . These six particles also have their antiparticles. All but
the muon and tau particle are stable. The muon and tau masses are about 207 and 3490
times the electron mass. The leptons have a spin of ½.
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Hadrons
The hadrons include mesons and baryons.
Mesons
There are a large number of mesons with a wide range of masses. Some have charge and
some don’t. Examples of mesons are the pion (0,  -, +) and the kaon (K0, K-, K+).
Mesons have integer spin and all are unstable.
Baryons
There are also a large number of baryons. The proton and the neutron are baryons.
Others include the lambda (0) and sigma (0, -, +) particles. Baryons have halfinteger spin (1/2, 3/2, …), and all are unstable except the proton. A free neutron has a
lifetime of about 15 min. When bound in the nucleus it is stable.
Particles can also be classified by their spin. Half-integer spin particles (leptons and
baryons) are referred to as fermions. Integer spin particles (0, 1, 2, ..) are referred to as
bosons. The mesons and the photon (spin = 1) are bosons.
Quarks
The leptons are considered to be elementary particles. The have no internal structure.
The hadrons, however, are not considered to be elementary particles and are thought to
consist of bound states of quarks. A quark has a charge of 1/3e or 2/3e. There are six
types of quarks and their antiparticles: up (u), down (d), charmed (c), strange (s), top (t),
and bottom (b). Quarks are fermions with a spin of ½. Isolated quarks have never been
observed. The mass and charge of the quarks are listed below. The antiquarks have
opposite signs.
Quark
u
d
c
s
t
b
Rest mass
energy
360
360
1500
540
173
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Charge
(e)
+2/3
-1/3
+2/3
-1/3
+2/3
-1/3
Mesons consist of 2 quarks and baryons consist of 3 quarks. The composition of the
proton is uud . Its charge is 2/3e + 2/3e – 1/3e = e. The neutron is udd and its charge is
2/3e - 1/3e – 1/3e = 0. The + is du and its charge is 1/3e+2/3e = e.
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