Module Nuc - Phys I (Phys382) New1
Module Nuc - Phys I (Phys382) New1
Module Nuc - Phys I (Phys382) New1
1.1 Composition, Charge; Size; Mass and Angular momentum of the nucleus
An atomic nucleus is the small, heavy and central part of an atom consisting of
nucleons:
Z protons ( the atomic number) and N neutrons
A = the mass number
The basic properties of the atomic constituents(compositions)
Charge Q Mass(u) Spin( Magnetic moment(J.T-1)
Proton e 1.007276 1/2 1.411x 10-26
Neutron 0 1.008665 1/2 -9.66x 10-24
Electron -e 0.000549 1/2 9.28x 10-24
Charge: protons have a positive charge of magnitude e=1.6022x10-19 c equal and opposite to
that of the electron.
Mass: Nuclear and atomic mass are expressed in atomic mass units (u).
1 amu (u) is defined as the mass of an atom of which is the most abundant
Where L is orbital angular momentum of the nuclide and S is spin angular momentum
of the nuclide.
Size of the nucleus
Nuclei have a well- defined size. Experimentally, it is found that the volume of a nucleus
is proportional to the number of nucleons (neutrons and protons) it contains. Since the
mass number A is proportional to volume and volume is proportional to the R3, where R is
the nuclear radius, it follows that R is proportional to A1/3. We usually write
⁄
Activity
1. What are isotopes?
2. What are isotones?
3. What are isobars?
4. What is nuclide?
Atomic masses
The most abundant type of carbon atom is defined to have a mass of exactly 12 u, where u is one
atomic mass unit: 1 u = 1.6604x10-27kg = 931.48 MeV.
Atomic masses always refer to neutral atoms. In other words, atomic masses include the masses
of all of the electrons in the neutral atom.
Isotopes
Not all atoms of an element have the same mass. Isotopes are atoms of the same element having
different masses.
Nuclides
A nuclide is simply any particular nuclear species. Hydrogen and deuterium are isotopes. They
are also nuclides. Carbon-12 is a nuclide, but it is not an isotope of hydrogen. Since nuclei
contain neutrons, let's consider the neutron for a while. Neutrons were observed in 1930 and
"hypothesized" in 1932.
Chadwick in 1932 proposed that the unknown radiation could be neutral particles having about
the mass of protons.
Charge neutrality is necessary for the radiation to easily penetrate matter. Because a collision
between particles of equal mass can transfer all of the kinetic energy from the projectile to the
target, the neutrons needed to have only 5.7 MeV of energy, which was a much more reasonable
value.
Now we can fully describe the nucleus. The number of protons in a nucleus (and electrons in the
atom, if the atom is not ionized) is represented by Z.
Neutrons and protons are called nucleons, so A is the number of nucleons in a nucleus.
We identify nuclides by writing . For example, the most abundant isotope of iron has 26
protons and electrons, and a mass number of 56, so we write .
Because most physical and chemical properties are determined by the number and arrangement
of atomic electrons, isotopes of an element are very similar.
Rutherford’s nuclear theory based on the scattering of alpha particles suggested the nucleus to be
of compact structure.
Natural radioactivity suggested that the nucleus is compact; it is capable of emitting several
particles. e.g
All nuclei are positively charged and the magnitude of the electric charge is an integral
multiple (Z) of the proton charge e,
More than 99.9% of the mass of an atom is concentrated inside the tiny volume of the
nucleus,
Nuclei are spherical or nearly spherical in shape R given by
R=Ro A1/3
The striking correlation between R and A suggests that there is a universal density for
nuclear matter
⁄ ⁄
The nucleus is a tightly bound system of the nucleons with a large potential energy.
However, we want to express this quantity in terms of experimentally accessible quantities. Thus
we write the nuclear mass in terms of the atomic mass, that we can measure, ( )
[ ] , where is the atomic mass of the nucleus. We further neglect
the electronic binding energy by setting [ ]
We finally obtain the expression for the nuclear binding energy:
[ ]
Quantities of interest are also the neutron and proton separation energies:
Which are the analogous of the ionization energies in atomic physics, reflecting the energies of
the valence nucleons. We will see that these energies show signatures of the shell structure of
nuclei.
Where
Nuclear mass is less than the sum of the masses of constituent neutrons and protons. e.g.
Deuteron(bound neutron and proton):
md= 2.01355 u <mn + mp = 1.00867 + 1.00728 = 2.01595 u
Difference in mass Δm= md – (mn + mp).
We usually quote the average binding energy per nucleon (B/A). This figure shows the variation
of B/A with A:
Figure 1.1 the variation of B/A with A:
1. At low A, B/A increases with A to a broad maximum near A = 60 of about 8.6 MeV per
nucleon.
2. Beyond A = 60, there is a gradual decrease to about 7.6 MeV per nucleon for the heaviest
nuclei.
3. Nuclei with A greater than 238 are not found in significant quantities in the earth's crust.
4. Several sharp peaks below A = 30 correspond to nuclei 4He, 8Be, 12
C, 16
O, 20
Ne and
24 4
Mg. The He nucleus (α particle) is particularly stable and the A and Z of the other
nuclei is multiples of α particle. Their extra stability is taken as evidence that their
structure resembles that of a collection of α particle.
5. The approximate constancy of B/An over most of the range is indicative of the saturation
property of the nuclear force.
Exercises
1. Through an appropriate symbol, indicate the number of protons, neutrons, and electrons in an
atom of barium-135.
2. Determine the binding energy per nucleon in the 4He nucleus using the mass of the nucleus
and the binding energy per nucleon (in MeV/c2) in the 4He nucleus using the atomic mass of
4
He.(Use neutron mass = mn= 1.008665 u, atomic mass of 1H = m1H = 1.007825 u, atomic
mass of 4He = m4He = 4.002602 u).
3. How many moles of molecules are there in 12 g of carbon-12?
4. Calculate the binding energy per nucleon of nucleus (its mass being 34.9800 a.m.u.)
Given massof =1.008665 a.m.u., mass of =1.007825 a.m.u.
CHAPTER TWO
Nuclear reactions
2.1 Nuclear Reactions In General
Consider a reaction in which a target nucleus X is bombarded by a particle a, resulting in a
nucleus Y and a particle b:
As the total kinetic energy released in the reaction, Q, is equal to the difference between the
kinetic energy of the final particles and that of the initial particle, we find
( )
If Q is positive, the reaction is exothermic reaction (exoergic rxn) and if Q is negative,
endothermic reaction (supply of energy) (endoergic rxn).
For an endothermic reaction to procced the incident particle must have a minimum kinetic
energy called the threshold energy, Kth, the threshold energy is given by
The absorption cross-section can be further subdivided into a cross-section for capture and a
cross-section for fission .
The scattering cross-section can also be subdivided into elastic and inelastic scattering cross-
section.
It is important to note that these cross-sections are dependent upon neutron energy. Generally, as
neutron kinetic energy increases, probability of a reaction decreases.
d. Photo disintegration Very energetic Gamma- rays interact with the nucleus of an atom
and may be absorbed
However, because some of the neutrons are lost during slowing down and diffusion, the effective
multiplication factor, is equal to k times p, where p is the probability that the neutrons will
not be lost. If is greater than unity, the reaction will be supercritical and will progress at an
increasing rate, as in the case of the A-bomb or the case of reactor start-up. On the other hand, if
is equal to unity, the reaction is critical and fission occurs at a constant rate, as in a reactor
operating at constant power. If is less than unity, the reaction cannot continue, and the
condition is termed subcritical. The value of k depends upon the fissionable material used,
whereas the value of P depends chiefly upon the size and shape of the fuel assembly.
It is also possible to define the effective multiplication factor, , as the ratio of the production
rate of neutrons, P, to the combined rate of absorption, A, and the leakage rate, L
----------------------2
Where A represents any type of absorption, resulting in fission or just parasitic capture with
emission of gamma rays.
If the average number of neutrons emitted per fission is n while F is the fission process rate, then
---------------------------3
and Eq.2 can be rewritten
[ ⁄
] ------------------4
The amount of fissionable and non-fissionable material and the cross sections for fission and
neutron capture determine the ratio F/A, while the ability of the fissionable mass to contain and
absorb neutrons determines the ratio L/A. The quantity L/A increases without limit when the size
of the fissionable mass decreases, thus increasing neutron leakage and decreasing neutron,
absorption. The limit of is therefore decreases to zero. The quantity L/A approaches zero as
the fissionable mass increases and increases towards the limiting value nF/A.
Thus, when the composition of the fissionable material gives nF/A > 1, there exists some size for
which . The fissionable material is critical at this size so that this size is known as the
critical size and the mass at this point is the critical mass.
If a fission chain reaction is not controlled, it progresses at an increasing rate with an almost
instantaneous release of vast quantities of radiation and energy. When the reaction is controlled
and fission is made to occur at a slower and constant rate, the slower rate of liberation of energy
and nuclear radiation allows these to be harnessed for industrial use.
Chain reaction is reaction without supply of neutrons its fission reaction sustained by itself.
As we have already mentioned, at the fission of U-235 on average 2.5 neutrons are released, but
not all of these cause fission. (Neutron generation is the time that elapses between the birth of a
neutron and the birth of neutrons from the subsequent fission which is caused by the given
neutron. It’s typical value is about 10-9s [i.e. 1 nanosecond] for pure U-235.)
If the value of the multiplication factor is 1, we talk about a critical reactor. In this case the
number of neutrons in the system is constant, i.e. they cause the same number of fissions in every
second. If k<1, the reactor is subcritical and the number of neutrons is going down. On the other
hand, if k>1, the system is supercritical.
Very often the notion of reactivity is used instead of the multiplication factor. The definition of
reactivity is (k-1)/k.
Correspondingly, for a subcritical reactor , for a supercritical , and for a critical .
Obviously, the number of neutrons present in the reactor must be regulated or controlled since
this determines the rate of fission and thus the energy released per second. In order to control the
chain reaction one should use materials which tend to capture neutrons at a high probability. The
most widespread neutron absorbers are cadmium (Cd) and boron (B).
The so-called control rods are important tools of control and can be found in every reactor.
These are made of neutron absorbing materials and can be moved between the fuel assemblies.
For example, if one wishes to decrease the power of the reactor it is sufficient to push in a
control rod a little further. The control rods are particularly useful for short-term control and
stopping the reactor. For long-term regulation usually boric acid dissolved in the coolant is used.
The majority of the emitted neutrons leave the place of fission within an extremely short time.
However, about 0.64% of them (in the case of U-235) only leave the fission product significantly
later. These are the so-called delayed neutrons, which play a vital role in the controllability of
reactors.
Fig.2.1 Schematic visualization of the chain reaction in a sample of following the fission of
one nucleus by a neutron
Fig.2.2 Basic design of the most common type of Nuclear Fission Reactor
Breeder reactor is a device used to produces more fissile material than it consumes.
E.g.
̅
̅
(Pu= Plutonium)
Breeder reactors are harder to control and carry some risks e.g. through the liquid sodium.
Thermal reactor is the reactor in which energy is produced by fission of by slow neutrons.
Result:
The carbon cycle requires higher To because of the higher coulomb barrier of relative to
.
P-P cycle
is exothermic reactions-release of energy
The two positrons that are produced during the first step of the proton-proton chain collide with
two electrons; mutual annihilation of the four particles takes place, and their rest energy is
converted into 4(0.511 MeV) = 2.044 MeV of gamma radiation. Thus the total energy release is
(24.69 + 2.044) MeV = 26.73 MeV. The proton-proton chain takes place in the interior of the sun
and other stars.
The Sun is emitting at the rate of about 4x1026 Joule/sec. Because of this, the solar mass is
reducing at the rate of about 4x109 Kg/sec.
The total mass of the sun will be emitting energy at the present rate for the next one thousand
Crore (1011) years [157.8x1011year], Mass of sun=1.99x1030 kg and 1 year =3.1536 x 107 sec
About 90% of the solar mass is composed of hydrogen and Helium, and the rest 10% contains
other elements, mainly the lighter ones.
Example
Two deuterons fuse to form a triton (a nucleus of tritium, or 3H) and a proton. How much energy
is liberated?
Solution
Q = [2(2.0l4l02u) - 3.0l6049u - 1.007825u] x (93l.5MeV/u)
= 4.03 MeV
For example: hydrogen (fusion) bomb, exploded in 1952 –uncontrolled thermonuclear fusion
reaction.
2.4.2.1 A Fusion Reactor
Fusion offers several advantages over fission. One advantage is that the reserves of fusionable
isotopes are much larger than those of fissionable isotopes; in fact, they are essentially unlimited.
Another advantage is that the products of fusion reactions are less radioactive then the products
of fission reactions. Among the products of the fusion reactions listed above, only tritium and the
neutrons are radioactive. The last advantage of fusion lies in its inherent safety. There would be
very little fissionable material at any given time in the reactor and the likelihood of a runaway
reaction would thus be very small.
The basic challenges of fusion are the following:
(a) heating of the reacting mixture to a very high temperature, to overcome the repulsive
forces of positively charged nuclei;
(b) compressing the mixture to a high density so that the probability of collision (and thus
reaction) among the nuclei can be high; and
(c) keeping the reacting mixture together long enough for the fusion reaction to produce
energy at a rate that is greater than the rate of energy input (as heat and compression).
The first challenge is that of providing a huge amount of energy to the reactants. This is why
fusion is called a thermonuclear reaction. Table3.3 shows the mind-boggling temperature
thresholds (―ignition temperatures‖) needed to accomplish some of the fusion reactions shown
above.
Summary
Consider a reaction in which a target nucleus X is bombarded by a particle a, resulting in a
nucleus Y and a particle b:
The total kinetic energy released (or absorbed) in the reaction, which is called the reaction
energy, Q.
As the total kinetic energy released in the reaction, Q, is equal to the difference between the
kinetic energy of the final particles and that of the initial particle, we find
( )
For an endothermic reaction to proceed the incident particle must have a minimum kinetic
energy called the threshold energy, Kth, the threshold energy is given by
The method used to determine the probability of a reaction occurring between a neutron and
target nucleus is to represent the target as an "effective" cross-sectional area to the neutron.
This effective cross-sectional area is termed the "microscopic cross section", is
pronounced ―sigma‖.
The unit that is used for the measurement of cross section is the barn (b):
1 barn = 10-24 cm2
In terms of the mechanism of interaction nuclear reactions can be categorized as follows:
1. Elastic Scattering 7. Heavy Ion Reactions
2. Inelastic Scattering. 8. Spontaneous Decay
3. Disintegration: 9. Spallation(Fragmentation)
4. Photo disintegration Reactions
5. Radiative Capture 10. High Energy Reactions
6. Direct Reactions
The fission process is defined as the splitting of a heavy nucleus (A ~240) into two
approximately equal parts, accompanied by the release of a large amount of energy and one
or more neutrons.
Two types of fission reactions are possible:
1. Spontaneous fission
2. Induced fission
Chain reaction is reaction without supply of neutrons its fission reaction sustained by itself.
If the value of the multiplication factor is 1, we talk about a critical reactor. In this case the
number of neutrons in the system is constant, i.e. they cause the same number of fissions in
every second. If k<1, the reactor is subcritical and the number of neutrons is going down. On
the other hand, if k>1, the system is supercritical.
A nuclear fission reactor is a system designed to maintain what is called a self-sustained
chain reaction.
Nuclear fusion is a process in which two light nuclei ( combine to form a heavy
nucleus, with the release of energy.
Exercises
227 233
1. Calculate the binding energy of thermal neutrons added to the following nuclei: Th, U,
235
U, 239Pu, 238U, 242Pu. Which of these nuclei are fissionable under thermal neutrons?
2. What is the product in the reaction 59Co (p, n)?
3. (a) When lithium (7Li) is bombarded by a proton, two alpha particles (4He) are produced.
Find the reaction energy. (b) Calculate the reaction energy for the nuclear reaction.
4. Consider the nuclear reaction
Where X is a nuclide. (a) What are the values of Z and A for the nuclide X? (b) How much
energy is liberated? (c) Estimate the threshold energy for this reaction.
5. Calculate the energy released in the fission reaction
You can ignore the initial kinetic energy of the absorbed neutron. The atomic masses
are ; ; and .
6. One of the reactions is the proton cycle, . Calculate the energy released
in this reaction.
Multiple choices
1. Which statement best describes the structure of an atom?
A. A positive core surrounded by electrons packed tightly around it.
B. A particle comprised of a mixture of protons, electrons and neutrons.
C. A tiny nucleus of protons and neutrons with electrons orbiting around it.
D. A large core of protons and electrons surrounded by neutrons.
2. The nucleus radius is of the order of
A. 10−14 m B. 10−5 m C. 10−6 m D. 10−10 m
3. Which of the following statements is true for nuclear forces
A. They are equal in strength to the electromagnetic forces
B. They are short range forces
C. They obey the inverse third power law of distance
D. They obey the inverse square law of distance
4. From the three basic forces gravitational, electrostatic and nuclear which two are able to
provide an attractive force between two neutrons
A. gravitation and electrostatic D. some other forces like van der
B. electrostatic and nuclear Waals
C. gravitational and nuclear
5. What is the binding energy of ? (Given mass of proton = 1.00078 a.m.u., mass of
neutron = 1.0087 a.m.u. =931 MeV)
A. 9.2 MeV B. 92 MeV C. 920 MeV D. 0.92 Mev
6. Most suitable element for nuclear fission is the element with atomic number near
A. 92 B. 52 C. 21 D. 11
7. In order to carry out the nuclear reaction,
A. Very high temperature and low pressure would be necessary
B. Vary high temperature and relatively high pressure would be necessary
C. Moderates temperature and very high pressure will be necessary
D. Very high temperature will only be necessary
8. Moderators are used in the nuclear reactors to
A. accelerate the neutrons C. absorb neutrons
B. slow down the neutrons D. produce neutrons
9. Cadmium rods are used in a nuclear reactor to
A. generate neutrons C. slow down neutrons
B. absorb neutrons D. produce neutrons
10. Which of the following particle is used to cause fission in an atomic reactor?
A. Proton B. α –particle C. β –particle D. neutron
11. Which of the following is the best nuclear fuel?
A. Neptunium 293 B. plutonium 239 C. Uranium 236 D. Thorium 236
12. The element not occurring in nature is
A. B. C. D.
13. The phenomenon of nuclear fission is used in the construction of
A. an atom bomb C. an ordinary bomb
B. hydrogen bomb D. none of the above
14. If the binding energy of the deuterium is 2.23 Mev, the mass defect given in amu is ( Hint
1 a.m.u =931 MeV)
A. -0.0024 B. -0.0012 C. 0.0012 D. 0.0024
40 40 40
15. K , Ar and Ca are
A. Isotopes B. isobars C. isotones D. isogonal
16. In a graph between binding energy per nucleon and mass numbers small peaks indicate
that the corresponding elements are
A. Radioactive B. less stable C. comparably more stable D. more abundant
17. The order of magnitude of the density of nuclear matter is
A. 104 kg/m2 B. 1017 kg/m3
C. 10−15 kg/m3 D. 1034 kg/m3
18. The material used for absorbing the extra neutrons in a nuclear reactor is
A. zinc C. radium
B. uranium D. Cadmium
19. Which one of the following is the unit of nuclear cross section?
A. Fermi B. curie C. barn D. gray