Lecture 8
Nuclear Reaction and Conservation
Laws
The study of nuclear reactions is important for a number of reasons. Life on earth
would not be possible without the energy provided to us by the sun. That energy is the
energy released in the nuclear reactions that drive the sun and other stars. For better or
worse, the nuclear reactions, fission and fusion, are the basis for nuclear weapons, which
have shaped much of the geopolitical dialog for the last 50 years. Apart from the
intrinsically interesting nature of these dynamic processes, their practical importance would
be enough to justify their study.
Most nuclear reactions are studied by inducing a collision between two nuclei where one of
the reacting nuclei is at rest (the target nucleus) while the other nucleus (the projectile
nucleus) is in motion. Such nuclear reactions can be described generically as
Projectile a + target X → emitted particle b and residual nucleus Y
Conservation Laws
8.1
Where (a) is the bombarding particle (projectile), X is the target nucleus, b is the emitted
particle and Y is the residual nucleus.
8.2
For example,
8.3
1
This reaction occurred by bombarding 14N by alpha particle to generate an emitted particle,
8.4
8.5
2
Q Values
We have just examined some nuclear reactions for which mass numbers and atomic
numbers must be balanced in the equations. We will now consider the energy involved in
these reactions, because energy is another important quantity that must be conserved.
We illustrate this procedure by analyzing the following nuclear reaction:
8.6
3
The total mass on the left side of the equation is the sum of the mass of (2.014 102 u) and
the mass of (14.003 074 u), which equals 16.017 176 u. Similarly, the mass on the right side
of the equation is the sum of the mass of (12.000 000 u) plus the mass of (4.002 602 u), for
a total of 16.002 602 u. Thus, the total mass before the reaction is greater than the total
mass after the reaction. The mass difference in the reaction is equal to 16.017 176 u -16.002
602 u = 0.014 574 u. This “lost” mass is converted to the kinetic energy of the nuclei
present after the reaction. In energy units, 0.014 574 u is equivalent to 13.576 MeV of
kinetic energy carried away by the carbon and helium nuclei. The energy required to
balance the equation is called the Q value of the reaction. In the above Equation, the Q
value is 13.576 MeV. Nuclear reactions in which there is a release of energy—that is,
positive Q values—are said to be exothermic reactions.
The energy balance sheet isn’t complete, however: We must also consider the kinetic
energy of the incident particle before the collision. As an example, assume that the deuteron
in Equation (8.6) has a kinetic energy of 5 MeV. Adding this to our Q value, we find that
the carbon and helium nuclei have a total kinetic energy of 18.576 MeV following the
reaction.
Now consider the reaction
8.7
Before the reaction, the total mass is the sum of the masses of the alpha particle and the
nitrogen nucleus: 4.002 602 u +14.003 074 u = 18.005 676 u. After the reaction, the total
mass is the sum of the masses of the oxygen nucleus and the proton: 16.999 133 u + 1.007
825 u = 18.006 958 u. In this case, the total mass after the reaction is greater than the total
mass before the reaction. The mass deficit is 0.001 282 u, equivalent to an energy deficit of
1.194 MeV. This deficit is expressed by the negative Q value of the reaction, -1.194 MeV.
Reactions with negative Q values are called endothermic reactions. Such reactions won’t
take place unless the incoming particle has at least enough kinetic energy to overcome the
energy deficit. If it has energy of only 1.194 MeV, energy is conserved but careful analysis
shows that momentum isn’t. If it has energy of only 1.194 MeV, energy is conserved but
careful analysis shows that momentum isn’t. It can be shown that in order to conserve both
energy and momentum, the incoming particle must have a minimum kinetic energy given
by
4
8.8
where m is the mass of the incident particle, M is the mass of the target, and the absolute
value of the Q value is used. For the reaction given by Equation 8.7, we find that
8.9
This minimum value of the kinetic energy of the incoming particle is called the threshold
energy. The nuclear reaction shown in Equation 8.7 won’t occur if the incoming alpha
particle has a kinetic energy of less than 1.535 MeV, but can occur if its kinetic energy is
equal to or greater than 1.535 MeV.
Example:
When 818O is struck by a proton, 918F and another particle are produced. (a) What is the
other particle? (b) The reaction has a Q value of -2.453 MeV, and the atomic mass of 18O is
17.999 160 u. What is the atomic mass of 18F?
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Types of Reaction:
8.1.
NUCLEAR FISSION
Nuclear fission occurs when a heavy nucleus, such as 235U, splits, or fissions, into two
smaller nuclei. In such a reaction, the total mass of the products is less than the original
mass of the heavy nucleus. Nuclear fission was first observed in 1939 by Otto Hahn and
Fritz Strassman, following some basic studies by Fermi. After bombarding uranium (Z =
92) with neutrons, Hahn and Strassman discovered two medium-mass elements, barium and
lanthanum, among the reaction products. Measurements showed that about 200 MeV of
energy is released in each fission event, and this fact was to affect the course of human
history.
The fission of 235U by slow (low-energy) neutrons can be represented by the sequence of
events
where 235U* is an intermediate state that lasts only for about 10-12 s before splitting into
nuclei X and Y, called fission fragments. There are many combinations of X and Y that
satisfy the requirements of conservation of energy and charge.
A typical reaction of this type is:
Sequence of events in a nuclear fission process described by the liquid-drop model of
the nucleus:
1. A slow neutron approaches a 235U nucleus (slow-moving) neutron Fig. a.
2. The neutron is absorbed by the 235U nucleus, changing it to 236U*, which is a 236U nucleus
in an excited state, and the excess energy of this nucleus causes it deforms and oscillates
like a liquid drop Fig.b.
3. The 236U* nucleus becomes highly elongated, and the force of repulsion between protons
in the two halves of the dumbbell-shaped nucleus tends to increase the distortion Fig.c.
4. The nucleus splits into two fragments, emitting several neutrons in the process Fig.d.
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Problem When 235U is struck by a neutron, there are various possible fission fragments.
Determine the number of neutrons produced when the fission fragments are 140Xe and 94Sr
(isotopes of xenon and strontium).
Solution:
The energy released in a typical fission process Q can be estimated.
fission event is:
Problem (a) Calculate the total energy released if 1.00 kg of 235U undergoes fission,
taking the disintegration energy per event (fission) to be Q = 208 MeV (a more accurate
value than the estimate given previously). (b) How many kilograms of 235U would be
needed to satisfy the world’s annual energy consumption (about 4 x 1020 J)?
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1. NUCLEAR REACTORS
We have seen that neutrons are emitted when 235U undergoes fission. These neutrons can in
turn trigger other nuclei to undergo fission, with the possibility of a chain reaction (As in
Fig.). Calculations show that if the chain reaction isn’t controlled, it will proceed too
rapidly and possibly result in the sudden release of an enormous amount of energy (an
explosion), even from only 1 g of 235U. If the energy in 1 kg of 235U were released, it would
equal that released by the detonation of about 20 000 tons of TNT! An uncontrolled fission
reaction, of course, is the principle behind the first nuclear bomb.
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A nuclear reactor is a system designed to maintain what is called a self-sustained chain
reaction. This important process was first achieved in 1942 by a group led by Fermi at the
University of Chicago, with natural uranium as the fuel. Most reactors in operation today
also use uranium as fuel. Natural uranium contains only about 0.7% of the 235U isotope,
with the remaining 99.3% being the 238U isotope. This is important to the operation of a
reactor because 238U almost never undergoes fission. Instead, it tends to absorb neutrons,
producing neptunium and plutonium. For this reason, reactor fuels must be artificially
enriched so that they contain several percent of the 235U isotope.
Earlier we mentioned that an average of about 2.5 neutrons is emitted in each fission event
of 235U. In order to achieve a self-sustained chain reaction, one of these neutrons must be
captured by another 235U nucleus and cause it to undergo fission. A useful parameter for
describing the level of reactor operation is the reproduction constant K, defined as the
average number of neutrons from each fission event that will cause another event. As
we have seen, K can have a maximum value of 2.5 in the fission of uranium. In practice,
however, K is less than this because of several factors, which we soon discuss.
A self-sustained chain reaction is achieved when K = 1. Under this condition, the reactor
is said to be critical. When K is less than one, the reactor is subcritical and the reaction
dies out. When K is greater than one the reactor is said to be supercritical, and a runaway
reaction occurs. In a nuclear reactor used to furnish power to a utility company, it is
necessary to maintain a K value close to one.
The basic design of a nuclear reactor is shown in Figure 8.3. The fuel elements consist of
enriched uranium. The functions of the remaining parts of the reactor and some aspects of
its design are described next.
Fig. 8.3
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Neutron Leakage
In any reactor, a fraction of the neutrons produced in fission will leak out of the core before
inducing other fission events. If the fraction leaking out is too large, the reactor will not
operate. The percentage lost is large if the reactor is very small because leakage is a
function of the ratio of surface area to volume. Therefore, a critical requirement of
reactor design is choosing the correct surface-area-to volume ratio.
Regulating Neutron Energies
The neutrons released in fission events are highly energetic (fast neutrons), with kinetic
energies of about 2 MeV. It is found that slow neutrons are far more likely than fast
neutrons to produce fission events in 235U. Further, 238U doesn’t absorb slow neutrons. In
order for the chain reaction to continue, therefore, the neutrons must be slowed down. This
is accomplished by surrounding the fuel with a substance called a moderator.
In order to understand how neutrons are slowed down, consider a collision between a
light object and a massive one. In such an event, the light object rebounds from the collision
with most of its original kinetic energy; most modern reactors use heavy water (D2O) as the
moderator.
Neutron Capture
In the process of being slowed down, neutrons may be captured by nuclei that do not
undergo fission. The most common event of this type is neutron capture by 238U. The
slowing down of the neutrons by the moderator serves the dual purpose of making them
available for reaction with 235U and decreasing their chances of being captured by 238U.
Control of Power Level
It is possible for a reactor to reach the critical stage (K= 1). However, a method of control is
needed to adjust K to a value near one. If K were to rise above this value, the heat produced
in the runaway reaction would melt the reactor. To control the power level, control rods
are inserted into the reactor core. (See Fig. 8.3.) These rods are made of materials such as
cadmium that are highly efficient in absorbing neutrons. By adjusting the number and
position of the control rods in the reactor core, the K value can be varied and any power
level within the design range of the reactor can be achieved.
A diagram of a pressurized-water reactor is shown in Figure 8.4. This type of reactor is
commonly used in electric power plants in the United States. Fission events in the reactor
core supply heat to the water contained in the primary (closed) system, which is maintained
at high pressure to keep it from boiling. This water also serves as the moderator. The hot
water is pumped through a heat exchanger, and the heat is transferred to the water contained
in the secondary system. There the hot water is converted to steam, which drives a turbine–
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generator to create electric power. Note that the water in the secondary system is
isolated from the water in the primary system in order to prevent contamination of the
secondary water and steam by radioactive nuclei from the reactor core.
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Reactor Safety:
Commercial reactors achieve safety through careful design and rigid operating procedures.
Radiation exposure and the potential health risks associated with such exposure are
controlled by three layers of containment. (1)The fuel and radioactive fission products are
contained inside the reactor vessel. (2) The reactor building acts as a second containment
structure to prevent radioactive material from contaminating the environment.(3) Finally,
the reactor facilities must be in a remote location to protect the general public from
exposure should radiation escape the reactor building.
Another potential danger in nuclear reactor operations is the possibility that the water flow
could be interrupted. Even if the nuclear fission chain reaction were stopped immediately,
residual heat could build up in the reactor to the point of melting the fuel elements.
To prevent such an unlikely chain of events, nuclear reactors are designed with emergency
core cooling systems, requiring no power, that automatically flood the reactor with water in
the event of a loss of coolant. The emergency cooling water moderates heat build-up in the
core, which in turn prevents the melting of the reactor vessel.
A continuing concern in nuclear fission reactors is the safe disposal of radioactive material
when the reactor core is replaced. This waste material contains long lived, highly
radioactive isotopes and must be stored for long periods of time in such a way that there is
no chance of environmental contamination. At present, sealing radioactive wastes in
waterproof containers and burying them in deep salt mines seems to be the most promising
solution.
Transportation of reactor fuel and reactor wastes poses additional safety risks.
However, neither the waste nor the fuel of nuclear power reactors can be used to construct a
nuclear bomb. Accidents during transportation of nuclear fuel could expose the public to
harmful levels of radiation. The Department of Energy requires stringent crash tests on all
containers used to transport nuclear materials. Container manufacturers must demonstrate
that their containers will not rupture, even in high-speed collisions.
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8.2. NUCLEAR FUSION
When two light nuclei combine to form a heavier nucleus, the process is called nuclear
fusion. Because the mass of the final nucleus is less than the masses of the original nuclei,
there is a loss of mass, accompanied by a release of energy. Although fusion power plants
have not yet been developed, a worldwide effort is under way to harness the energy from
fusion reactions in the laboratory.
Fusion in the Sun
All stars generate their energy through fusion processes. About 90% of stars, including the
Sun, fuse hydrogen, whereas some older stars fuse helium or other heavier elements. Stars
are born in regions of space containing vast clouds of dust and gas. Two conditions must be
met before fusion reactions in the star can sustain its energy needs: (1) The temperature
must be high enough (about 107 K for hydrogen) to allow the kinetic energy of the
positively charged hydrogen nuclei to overcome their mutual Coulomb repulsion as they
collide, and (2) the density of nuclei must be high enough to ensure a high rate of collision.
The proton–proton cycle is a series of three nuclear reactions that are believed to be the
stages in the liberation of energy in the Sun and other stars rich in hydrogen. An overall
view of the proton–proton cycle is that four protons combine to form an alpha particle and
two positrons, with the release of 30MeV of energy in the process.
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The energy liberated is carried primarily by gamma rays, positrons, and neutrinos, as can be
seen from the reactions. The gamma rays are soon absorbed by the dense gas, thus raising
its temperature. The positrons combine with electrons to produce gamma rays, which in
turn are also absorbed by the gas within a few centimeters. The neutrinos, however, almost
never interact with matter; hence, they escape from the star, carrying about 2% of the
energy generated with them. These energy-liberating fusion reactions are called
thermonuclear fusion reactions. The hydrogen (fusion) bomb, first exploded in 1952, is
an example of an uncontrolled thermonuclear fusion reaction.
Fusion Reactors
The fusion reactions that appear most promising in the construction of a fusion power
reactor involve deuterium (D) and tritium (T), which are isotopes of hydrogen.
These reactions are:
where the Q values refer to the amount of energy released per reaction. As noted
earlier.
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Exercies:
15
Lecture 12
Elementray Particles
The word “atom” is from the Greek word atomos, meaning “indivisible.” Discoveries in the
early part of the 20th century revealed that the atom is not elementary, but has protons,
neutrons, and electrons as its constituents. Until 1932, physicists viewed these three
constituent particles as elementary because they are highly stable. The theory soon fell
apart, however, and beginning in 1937, many new particles were discovered in experiments
involving high-energy collisions between known particles. These new particles are
characteristically unstable and have very short half-lives, ranging between 10-23 s and 10-6 s.
In the last 30 years, physicists have made tremendous advances in our knowledge of the
structure of matter by recognizing that all particles (with the exception of electrons,
photons, and a few others) are made of smaller particles called quarks. Protons and
neutrons, for example, are not truly elementary but are systems of tightly bound quarks.
12.1 THE FUNDAMENTAL FORCES OF NATURE
The key to understanding the properties of elementary particles is to be able to describe the
forces between them. All particles in nature are subject to four fundamental forces: strong,
electromagnetic, weak, and gravitational.
• Gravitational Force: This is an attraction force between two particles and it's
proportional to their masses. This force controls the motion of planets and galaxies and
determines the law of gravity.
• Electromagnetic Force: It's the combination of electrostatic and magnetic forces. It acts
between any two electrally charged particles, such as the force between an electron and a
proton. This force is responsible for holding atoms together, and most of the phenomena we
experience in life everyday.
• Weak Nuclear Force: It mediates beta decay and is transferred by W and Z bosons.
Neutrinos interact with other matter only through this force and gravity, hence, it can
penetrate large amounts of matter without being scattered. This force along with the
electromagnetic, form what is called the Electroweak force.
• Strong Nuclear Force: This force is responsible for holding together the protons and
neutrons inside the atomic nucleus. The strong force holds quarks together to form hadrons.
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12.2 POSITRONS AND OTHER ANTIPARTICLES
The general and profound implication of Dirac’s theory is that for every particle, there is
an antiparticle with the same mass as the particle, but the opposite charge. For
example, the electron’s antiparticle, the positron, has a mass of 0.511 MeV/c 2 and a
positive charge of 1.6 x 10-19 C.
30.8 CLASSIFICATION OF PARTICLES
Hadrons
All particles other than photons can be classified into two broad categories, hadrons and
leptons, according to their interactions. Particles that interact through the strong
force are called hadrons. There are two classes of hadrons, known as mesons and
baryons, distinguished by their masses and spins. All mesons are known to decay finally
into electrons, positrons, neutrinos, and photons. The pion is the lightest of known mesons,
with a mass of about 140 MeV/c 2 and a spin of 0. Another is the K meson, with a mass of
about 500 MeV/c 2 and spin 0 also.
Baryons have masses equal to or greater than the proton mass (the name baryon
means “heavy” in Greek), and their spin is always a non-integer value (1/2 or 3/2).
Protons and neutrons are baryons, as are many other particles. With the exception of the
proton, all baryons decay in such a way that the end products include a proton. Some of the
important properties of hadrons are listed in Table 12.1.
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Table 12.1
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Leptons
Leptons (from the Greek leptos, meaning “small” or “light”) are a group of particles that
participate in the weak interaction. All leptons have a spin of 1/2.
Included in this group are electrons, muons, and neutrinos, which are less massive than
the lightest hadron. Although hadrons have size and structure, leptons appear to be truly
elementary, with no structure down to the limit of resolution of experiment (about 10-19 m).
Unlike hadrons, scientists believe there are only six leptons (each having an antiparticle):
the electron, the muon, the tau, and a neutrino associated with each: The tau lepton,
discovered in 1975, has a mass about twice that of the proton. Although neutrinos have
masses of about zero, there is strong indirect evidence that the electron neutrino has a
nonzero mass of about 3 eV/c 2, or 1/180 000 of the electron mass.
CONSERVATION LAWS
A number of conservation laws are important in the study of elementary particles. Although
the two described here have no theoretical foundation, they are supported by abundant
empirical evidence.
Baryon Number
The law of conservation of baryon number tells us that whenever a baryon is created in a
reaction or decay, an antibaryon is also created. This information can be quantified by
assigning a baryon number: B = +1 for all baryons, B = -1 for all antibaryons, and B = 0 for
all other particles. Thus, the law of conservation of baryon number states that whenever a
nuclear reaction or decay occurs, the sum of the baryon numbers before the process equals
the sum of the baryon numbers after the process.
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Lepton Number
There are three conservation laws involving lepton numbers, one for each variety of lepton.
The law of conservation of electron-lepton number states that the sum of the electronlepton numbers before a reaction or decay must equal the sum of the electron-lepton
numbers after the reaction or decay. The electron and the electron neutrino are assigned a
positive electron-lepton number Le = +1, the antileptons e+ and
are assigned the electronlepton number Le = -1, and all other particles have Le = 0. For example, consider neutron
decay:
Before the decay, the electron-lepton number is Le = 0; after the decay, it is 0 + 1 + (-1) = 0,
so the electron-lepton number is conserved. It’s important to recognize that the baryon
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number must also be conserved. This can easily be seen by noting that before the decay B =
+1, whereas after the decay B = +1 + 0 + 0 = 1.
Similarly, when a decay involves muons, the muon-lepton number Lµ is conserved. The µand the are assigned Lµ = +1, the antimuons µ+ and
are assigned Lµ= -1, and all other
particles have Lµ = 0. Finally, the tau-lepton number Lτ is conserved, and similar
assignments can be made for the τ lepton and its neutrino.
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Particles
Leptons
hadrons
participate in the weak interaction
interact through the strong force
All leptons have a spin of 1/2.
There are six leptons (electron, electron neutrino,
Moun, moun neutrino, tau and tau neutrino)
less massive are electron, moun and neutrino
tau mass is twice mass of proton
Baryons
means heavy
their spin is a non-integer value(1/2 or 3/2)
Protons and neutrons are baryons
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measons
the lightest meason is Pion
the heavy meason
is K meason
Conservation laws
Baryon number
Lepton number
B = +1 for all baryons,
The electron and the electron neutrino Le = +1,
the antileptons e+ and
B = -1 for all antibaryons,
and B = 0 for all other particles
Le = -1,
and all other particles have Le = 0
The µ-and the
are Lµ = +1,
the antimuons µ+ and
B = +1
B = -1
are Lµ= -1,
and all other particles have Lµ = 0.
Finally, the tau-lepton number Lτ is conserved, and similar assignments can be made for the
τ lepton and its neutrino.
Lepton number Le =+1
=-1
Moun number Lµ = 0
=0
Tau number Lτ = 0
=0
Le = 0, Lτ = 0
Moun number Lµ = +1
=-1
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