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ATOMIC NATURE OF MATTER
• All matter is composed of atoms.
• The atom is the smallest amount of matter that retains the properties of an element.
• Atoms themselves are composed of smaller particles, but these smaller particles no longer have the same properties as the overall element.
Structure of Matter
• Early Greek philosophers speculated that the earth was made up of different combinations of basic substances, or elements.
• They considered these basic elements to be earth, air, water, and fire.
• Modern science shows that the early Greeks held the correct concept that matter consists of a combination of basic elements, but they incorrectly identified the elements.
• In 1661 the English chemist Robert Boyle published the modern criterion for an element.
• He defined an element to be a basic substance that cannot be broken down into any simpler substance after it is isolated from a compound, but can be combined with other elements to form compounds.
• To date, 105 different elements have been confirmed to exist, and researchers claim to have discovered three additional elements. Of the 105 confirmed elements, 90 exist in nature and 15 are man-made.
• The modern proof for the atomic nature of matter was first proposed by the English chemist John Dalton in 1803.
• Dalton stated that each chemical element possesses a particular kind of atom, and any quantity of the element is made up of identical atoms of this kind.
• What distinguishes one element from another element is the kind of atom of which it consists, and the basic physical difference between kinds of atoms is their weight.
Subatomic Particles• For almost 100 years after Dalton established
the atomic nature of atoms, it was considered impossible to divide the atom into even smaller parts.
• All of the results of chemical experiments during this time indicated that the atom was indivisible.
• Eventually, experimentation into electricity and radioactivity indicated that particles of matter smaller than the atom did indeed exist.
• In 1906, J. J. Thompson won the Nobel Prize in physics for establishing the existence of electrons. Electrons are negatively-charged particles that have 1/1835 the mass of the hydrogen atom.
• Soon after the discovery of electrons, protons were discovered. Protons are relatively large particles that have almost the same mass as a hydrogen atom and a positive charge equal in magnitude (but opposite in sign) to that of the electron.
• The third subatomic particle to be discovered, the neutron, was not found until 1932. The neutron has almost the same mass as the proton, but it is electrically neutral.
Bohr Model of the Atom• The British physicist Ernest Rutherford
postulated that the positive charge in an atom is concentrated in a small region called a nucleus at the center of the atom with electrons existing in orbits around it.
• Niels Bohr, coupling Rutherford's postulation with the quantum theory introduced by Max Planck, proposed that the atom consists of a dense nucleus of protons surrounded by electrons travelling in discrete orbits at fixed distances from the nucleus.
• An electron in one of these orbits or shells has a specific or discrete quantity of energy (quantum).
• When an electron moves from one allowed orbit to another allowed orbit, the energy difference between the two states is emitted or absorbed in the form of a single quantum of radiant energy called a photon.
• Figure 1 is Bohr's model of the hydrogen atom showing an electron as having just dropped from the third shell to the first shell with the emission of a photon that has an energy = hv. (h = Planck's constant = 6.63 x 10-34 J-s and v = frequency of the photon.)
• Bohr's theory was the first to successfully account for the discrete energy levels of this radiation as measured in the laboratory.
• Although Bohr's atomic model is designed specifically to explain the hydrogen atom, his theories apply generally to the structure of all atoms.
Measuring Units on the Atomic Scale
• The size and mass of atoms are so small that the use of normal measuring units, is often inconvenient.
• Units of measure have been defined for mass and energy on the atomic scale to make measurements more convenient to express.
• The unit of measure for mass is the atomic mass unit (amu). One atomic mass unit is equal to 1.66 x 10^-24 grams.
• Note from Table 1 that the mass of a neutron and a proton are both about 1 amu.
• The unit for energy is the electron volt (eV). The electron volt is the amount of energy acquired by a single electron when it falls through a potential difference of one volt.
• One electron volt is equivalent to
1.602 x 10^-19 J.
Nuclides• The total number of protons in the nucleus of an
atom is called the atomic number of the atom and is given the symbol Z.
• The number of electrons in an electrically-neutral atom is the same as the number of protons in the nucleus.
• The number of neutrons in a nucleus is known as the neutron number and is given the symbol N.
• The mass number of the nucleus is the total number
• of nucleons, that is, protons and neutrons in the nucleus. The mass number is given the symbol A and can be found by the equation Z + N = A.
• Each of the chemical elements has a unique atomic number because the atoms of different elements contain a different number of protons.
• The atomic number of an atom identifies the particular element.
• Each type of atom that contains a unique combination of protons and neutrons is called a nuclide.
• Not all combinations of numbers of protons and neutrons are possible, but about 2500 specific nuclides with unique combinations of neutrons and protons have been
• identified.
• Each nuclide is denoted by the chemical symbol of the element with the atomic number written as a subscript and the mass number written as a superscript,as shown in Figure 2.
Isotopes• Isotopes are nuclides that have the same atomic
number and are therefore the same element, but differ in the number of neutrons.
• Most elements have a few stable isotopes and several unstable, radioactive isotopes. For example, oxygen has three stable isotopes that can be found in nature (oxygen-16, oxygen-17, and oxygen-18) and eight radioactive isotopes. Another example is hydrogen, which has two stable isotopes (hydrogen-1 and hydrogen-2) and a single radioactive isotope (hydrogen-3).
• The isotopes of hydrogen are unique in that they are each commonly referred to by a unique name instead of the common chemical element name.
• Hydrogen-1 is almost always referred to as hydrogen, but the term protium is infrequently used also. Hydrogen-2 is commonly called deuterium and symbolized . Hydrogen-3 is commonly called tritium and symbolized
• This text will normally use the symbologyand for deuterium and tritium respectively.
Atomic and Nuclear Radii• The size of an atom is difficult to define exactly
due to the fact that the electron cloud, formed by the electrons moving in their various orbitals, does not have a distinct outer edge.
• A reasonable measure of atomic size is given by the average distance of the outermost electron from the nucleus.
• Except for a few of the lightest atoms, the average atomic radii are approximately the same for all atoms, about 2 x 10^-8 cm.
• Like the atom the nucleus does not have a sharp outer boundary.
• Experiments have shown that the nucleus is shaped like a sphere with a radius that depends on the atomic mass number of the atom.
• The relationship between the atomic mass number and the radius of the nucleus is show in the following equation.
•The values of the nuclear radii for some light,
intermediate, and heavy nuclides are shown in Table 2.
•From the table, it is clear that the radius of a typical
atom (e.g. 2 x 10^-8 cm) is more than 25,000times larger than the radius of the largest nucleus.
Nuclear Forces
• In the Bohr model of the atom, the nucleus consists of positively-charged protons and electrically neutral neutrons.
• Since both protons and neutrons exist in the nucleus, they are both referred to as nucleons.
• One problem that the Bohr model of the atom presented was accounting for an attractive force to overcome the repulsive force between protons.
• Two forces present in the nucleus are
(1) electrostatic forces between charged particles and
(2) gravitational forces between any two objects that have mass.
• It is possible to calculate the magnitude of the gravitational force and electrostatic force based upon principles from classical physics.
• Newton stated that the gravitational force between two bodies is directly proportional to the masses of the two bodies and inversely proportional to the square of the distance between the bodies.
• This relationship is shown in the equation below.
• The equation illustrates that the larger the masses of the objects or the smaller the distance between the objects, the greater the gravitational force.
• So even though the masses of nucleons are very small, the fact that the distance between nucleons is extremely short may make the gravitational force significant.
• It is necessary to calculate the value for the gravitational force and compare it to the value for other forces to determine the significance of the gravitational force in the nucleus.
• The gravitational force between two protons that are separated by a distance of 10^-20 meters is about
10^-24 N.
• Coulomb's Law can be used to calculate the force between two protons. The electrostatic force is directly proportional to the electrical charges of the two particles and inversely proportional to the square of the distance between the particles.
• Coulomb's Law is stated as the following equation.
• Using this equation, the electrostatic force between two protons that are separated by a distance of 10 ^-20 meters is about 10^12 N.
• Comparing this result with the calculation of the gravitational force (10^-24 N) shows that the gravitational force is so small that it can be neglected.
• If only the electrostatic and gravitational forces existed in the nucleus, then it would be impossible to have stable nuclei composed of protons and neutrons.
• The gravitational forces are much too small to hold the nucleons together compared to the electrostatic forces repelling the protons.
• Since stable atoms of neutrons and protons do exist, there must be another attractive force acting within the nucleus.
• This force is called the nuclear force.
• The nuclear force is a strong attractive force that is independent of charge.
• It acts equally only between pairs of neutrons, pairs of protons, or a neutron and a proton.
• The nuclear force has a very short range; it acts only over distances approximately equal to the diameter of the nucleus (10^-13 cm).
• The attractive nuclear force between all nucleons drops off with distance much faster than the repulsive electrostatic force between protons.
• In stable atoms, the attractive and repulsive forces in the nucleus balance.
• If the forces do not balance, the atom cannot be stable, and the nucleus will emit radiation in an attempt to achieve a more stable configuration.
Atomic Nature of Matter Summary
• Atoms consist of three basic subatomic particles. These particles are the proton, the neutron, and the electron.
• Protons are particles that have a positive charge, have about the same mass as a hydrogen atom, and exist in the nucleus of an atom.
• Neutrons are particles that have no electrical charge, have about the same mass as a hydrogen atom, and exist in the nucleus of an atom.
• Electrons are particles that have a negative charge, have a mass about eighteen hundred times smaller than the mass of a hydrogen atom, and exist in orbital shells around the nucleus of an atom.
• The Bohr model of the atom consists of a dense nucleus of protons and neutrons (nucleons) surrounded by electrons travelling in discrete orbits at fixed distances from the nucleus.
• Nuclides are atoms that contain a particular number of protons and neutrons.
• Isotopes are nuclides that have the same atomic number and are therefore the same element, but differ in the number of neutrons.
• The atomic number of an atom is the number of protons in the nucleus.
• The mass number of an atom is the total number of nucleons (protons and neutrons) in the nucleus.
• The notation is used to identify a specific nuclide. "Z" represents the atomic number, which is equal to the number of protons. "A" represents the mass number, which is equal to the number of nucleons. "X" represents the chemical symbol of the element.
• Number of protons = Z
Number of electrons = Z
Number of neutrons = A – Z
• The stability of a nucleus is determined by the different forces interacting within it. The electrostatic force is a relatively long-range, strong, repulsive force that acts between the positively charged protons. The nuclear force is a relatively short-range attractive force between all nucleons. The gravitational force the long range, relatively weak attraction between masses, is negligible compared to the other forces.
Chart of the Nuclides
• A tabulated chart called the Chart of the Nuclides lists the stable and unstable nuclides in addition to pertinent information about each one.
• Figure 3 shows a small portion of a typical chart. This chart plots a box for each individual nuclide, with the number of protons (Z) on the vertical axis and the number of neutrons
(N = A - Z) on the horizontal axis.
• The completely gray squares indicate stable isotopes.
• Those in white squares are artificially radioactive, meaning that they are produced by artificial techniques and do not occur naturally.
• By consulting a complete chart, other types of isotopes can be found, such as naturally occurring radioactive types (but none are found in the region of the chart that is illustrated in Figure 3).
Information for Stable Nuclides• For the stable isotopes, in addition to the
symbol and the atomic mass number, the number percentage of each isotope in the naturally occurring element is listed, as well as the thermal neutron activation cross section and the mass in atomic mass units (amu).
• A typical block for a stable nuclide from the Chart of the Nuclides is shown in Figure 4.
Information for Unstable Nuclides
• For unstable isotopes the additional information includes the half life, the mode of decay (for example, ), the total disintegration energy in MeV (million electron volts), and the mass in amu when available.
• A typical block for an unstable nuclide from the Chart of the Nuclides is shown in Figure 5.
Neutron - Proton Ratios• Figure 6 shows the distribution of the stable nuclides
plotted on the same axes as the Chart of the Nuclides.
• As the mass numbers become higher, the ratio of neutrons to protons in the nucleus becomes larger.
• For helium-4 (2 protons and 2 neutrons) and oxygen-16 (8 protons and 8 neutrons) this ratio is unity.
• For indium-115 (49 protons and 66 neutrons) the ratio of neutrons to protons has increased to 1.35, and for uranium-238 (92 protons and 146 neutrons) the neutron to proton ratio is 1.59.
• If a heavy nucleus were to split into two fragments, each fragment would form a nucleus that would have approximately the same neutron-to-proton ratio as the heavy nucleus.
• This high neutron-to-proton ratio places the fragments below and to the right of the stability curve displayed by Figure 6.
• The instability caused by this excess of neutrons is generally rectified by successive beta emissions, each of which converts a neutron to a proton and moves the nucleus toward a more stable neutron-to-proton ratio.
Natural Abundance of Isotopes
• The relative abundance of an isotope in nature compared to other isotopes of the same element is relatively constant.
• The atomic weight for an element is defined as the average atomic weight of the isotopes of the element.
• The atomic weight for an element can be calculated by summing the products of the isotopic abundance of the isotope with the atomic mass of the isotope.
Enriched and Depleted Uranium• Enriched uranium is defined as uranium in which
the isotope uranium-235 has a concentration greater than its natural value.
• The enrichment process will also result in the by product of depleted uranium.
• Depleted uranium is defined as uranium in which the isotope uranium-235 has a concentration less than its natural value.
• Although depleted uranium is referred to as a by-product of the enrichment process, it does have uses in the nuclear field and in commercial and defense industries.
• Plays an important part in our lives
Nuclear Fission (Source of Energy from Reactors/ Weapons)
Nuclear Fusion : Maintains (nearly) all life
• Creation of all the heavy elements – Nucleosynthesis
• Possible future source of low pollution energy
• Radioactive Decay: Widely used in dating, Smoke alarms !
• Medical Applications: Diagnostic Uses Imaging
Therapeutic uses for cancer treatment
Why Study Nuclear Physics?
An understanding of nuclear physics will enable you to make
an informed contribution to the debate on the use of nuclear
materials and science and to understand their limitations
and their benefits
• Nuclear Sizes (scales, ranges)
• Nuclear Forces (magnitude of the forces,
mechanisms)
• Nuclear Models (A very brief introduction
to the types)
• Application of Quantum Mechanics to
Nuclear Phenomena
• Early Nuclear Experiments (How we know
what we know!)
what is known about nuclear physics?
Research on nucleus:
- Atomic masses : Accurate info to determine
binding energies
- Elemental radioactivity: The study of fundamental
particles
- Nuclear transmutation: The study of nuclear
characteristics such as spin, energy levels etc
- Optical spectroscopy: The study of phenomena
such as scattering, diffraction, fluorescence etc
1 Constituents of Nucleus
2 Nuclear Stability
3 Ionising Radiation
4 Interactions of Ionising Radiation
with Matter
5 Nuclear Reactions
6 Radioactivity
7 Liquid Drop Model
8 Shell Model
Course Contents
1
Properties of nucleus
Binding energy and mass defect
RELATION
c= 3.00 x 108ms-1
2
PARTICLE PROPERTIES
Particle Symbol Mass
Proton mp =1.67 x 10-27 kg / 1.007276 u
Electron me =9.11 x 10-31 kg / 0.000548 u
Neutron mn=1.67 x 10-27 kg / 1.008664 u
Hidrogen mH=1.007284 u
UNIT CONVERSION
1 u = 1.66 x 10-27kg
1eV = 1.6 x 10-19J
2 931.5MeVc =
u
1
1p
0
1e
1
0 n
1
1H
At the end of this topic, students should be able to:
State the properties of proton and neutron
Define
◦ Proton number, Z
◦ Nucleon number, A
◦ Neutron number, N
◦ Isotopes
Use to represent a nuclide
3
4
Properties of nucleus
•An atom is consists of electron and nucleus where the
electron orbiting the nucleus. (Figure 1)
•A nucleus of an atom is made up of protons and
neutrons that is also known as nucleons. (Figure 2)
Figure 1( atom) Figure 2 (nucleus)
Proton
Particle with positive charge of the nucleus
Charge : +1.60 x 10-19 C
Mass : 1.67 x 10-27 kg / 1.007276 u
Symbol :
Electron
Particle with negative charge of the atom
Charge : - 1.60 x 10-19 C
Mass : 9.11 x 10-31 kg / 5.48 x 10-4 u
Symbol :
5
1
1p
0
-1e
6
Neutron
Particle with no charge of the nucleus
Charge : -
Mass : 1.675 x 10-27 kg / 1.008664 u
Symbol : 1
0 n
Proton number
Definition: the number of protons in the nucleus.
Also called as atomic number
Symbol : Z
Nucleon number
Definition : the total number of neutrons and protons in
the nucleus.
Also called as atomic mass number
Symbol : A
Neutron number
Definition : the number of neutrons in the nucleus
Symbol : N
7
Isotope
Definition : the atoms of the same element whose nuclei
contain the same number of protons (Z) but different
number of neutrons (N).
Example :
(Hydrogen, deuterium, tritium)
8
1 2 3
1 1 1H, H, H
1
0
The atomic nucleus can be represented as
XA
Z
where X = symbol for the element
Z = atomic number (number of protons)
A = atomic mass number
= total number of protons and neutrons
Example :
56
26 Fe Element : Iron-56
Proton no, Z = 26
Nucleon no, A = 56
Neutron = 56-26 = 30
A - Z = N
1
1
Element
nuclide
Number
of protons
Number of
neutrons
Number of
electrons
8 8 8
H1
1
N14
7
Na23
11
Co59
27
Be9
4
O16
8
S31
16
Cs133
55
U238
92
Example 26.1
Complete the table below:
DEFINE the following terms:
a. Mass defect
b. Binding energy
Given the atomic mass for a nuclide and the atomic masses of a neutron, proton, and electron, CALCULATE the mass defect and binding energy of the nuclide.
The separate laws of Conservation of Mass and Conservation of Energy are not applied strictly on the nuclear level.
It is possible to convert between mass and energy.
Instead of two separate conservation laws, a single conservation law states that the sum of mass and energy is conserved.
Mass does not magically appear and disappear at random.
A decrease in mass will be accompanied by a corresponding increase in energy and vice versa.
At the end of this topic, students should be able to:
Define and determine mass defect
Define and determine binding energy,
and binding energy per nucleon,
Describe graph of binding energy per nucleon against
nucleon number.
14
p n nucleusΔm=(Zm +Nm )-m
BE
A
Careful measurements have shown that the mass of a particular atom is always slightly less than the sum of the masses of the individual neutrons, protons, and electrons of which the atom consists.
The difference between the mass of the atom and the sum of the masses of its parts is called the mass defect (m).
In calculating the mass defect it is important to use the full accuracy of mass measurements because the difference in mass is small compared to the mass of the atom.
Rounding off the masses of atoms and particles to three or four significant digits prior to the calculation will result in a calculated mass defect of zero.
19
Definition
the difference between the sum of the masses of individual
nucleons that form an atomic nucleus and the mass of the
nucleus.
Formula Δ p n nucleusm Zm Nm m
proton a of mass: pm
mass of a nucleusnucleusm
neutron a of mass: nm
neutrons of number
protons of number
N
Z
Example
From example above, can you determine the value of
mass defect ?
(Ans : 0.040475 a.m.u / 0.040475u)
20
The loss in mass, or mass defect, is due to the conversion of mass to binding energy when the nucleus is formed.
Binding energy is the energy equivalent of the mass defect.
Since the mass defect was converted to binding energy (BE) when the nucleus was formed, it is possible to calculate the binding energy using a conversion factor derived by the mass-energy relationship from Einstein's Theory of Relativity.
21
22
23
Definition
Energy required to separate a nucleus into its individual
protons and neutrons.
@ Energy released when nucleus is formed from its
individual nucleons.
Formula
Where E : Binding energy
Δm : Mass defect
c : speed of light = 3.00 x 108ms-1
There are 2 methods to determine the value of Binding
Energy, EB
Example :
Let Δm = 1 u = 1.66 x 10-27kg
=
Note : 1eV = 1.6 x 10-19J24
EB ( in unit J )
Δm ( in unit kg )
c = 3.00 x 108ms-1
EB ( in unit MeV )
Δm ( in unit u )
2 931.5MeVc =
u
2
B
-27 8 -1 2
-10 2 -2
-10
E =Δmc
=(1.66×10 kg)(3.00×10 ms )
=1.4904×10 kgm s
=1.4904×10 J
2
BE =Δmc
931.5MeV=(1 u)
u
= 931.5MeV
25
Example
Calculate
a) mass defect and
b) binding energy of the deuterium,
(Given mass nucleus deuterium 2.013553, mp= 1.007276 &
mn= 1.008664 )
p n nucleusa) Δm= Zm +Nm -m
= 1 1.007276 +1 1.008664 -2.013553
=0.002387 u
2
Bb) E =Δmc
931.5MeV =0.002387 u
u
=2.22MeV
2
1H
Solution:
Calculate binding energy of the Helium nucleus, in SI unit.
(Given mass of helium atom = 4.002603 u, mH= 1.007824 & mn= 1.008664 )
26
Example
4
2 He
27
p n nucleusΔm= Zm +Nm -m
= 2 1.007824 +2 1.008664 - 4.002603
=0.030373u
28
Question:
Calculate the mass defect and binding energy for
uranium-235. One uranium-235 atom
has a mass of 235.043924 amu.
29
2 METHODS TO FIND EB
i) Find EB in unit MeV then convert to Joule
ii) Convert Δm to unit kg then find EB in unit Joule
30
-27 -29
2
B
-29 8 2
-12
Δm=0.030373×1.66×10 =5.0419×10 kg
E =Δmc
=(5.0419×10 )(3.00×10 )
= 4.5×10 J
2
B
6 -19 -12
B
E =Δmc
931.5MeV =0.030373 u
u
=28.29MeV
E = 28.29×10 ×1.6×10 = 4.5×10 J
31
Definition
mean (average) binding energy of a nucleus
Binding energy per nucleon is measure the stability of of the
nucleus.
The greater the binding energy per nucleon, the more
stable the nucleus is.
BE
N
)number(Nucleon
)(energy Bindingnucleonper energy Binding B
A
E
A
mc2
nucleonper energy Binding
32Mass number A
Bin
din
g e
nerg
y p
er
nu
cle
on
(M
eV
/nu
cle
on
)Greatest stability
Binding energy per nucleon as
a function of mass number,A
For light nuclei the value of EB/A rises rapidly from 1
MeV/nucleon to 8 MeV/nucleon with increasing mass
number A.
For the nuclei with A between 50 and 80, the value of
EB/A ranges between 8.0 and 8.9 Mev/nucleon. The
nuclei in these range are very stable.
The nuclide has the largest binding energy per
nucleon (8.7945 MeV/nucleon).
For nuclei with A > 62, the values of EB/A decreases
slowly, indicating that the nucleons are on average, less
tightly bound.
For heavy nuclei with A between 200 to 240, the binding
energy is between 7.5 and 8.0 MeV/nucleon. These
nuclei are unstable and radioactive.
33
From the graph:
Ni62
28
The nucleus is held together by very short-range attractive forces that exist between nucleons.
On the other hand, the nucleus is being forced apart by long range repulsive electrostatic (coulomb) forces that exist between all the protons in the nucleus.
As the atomic number and the atomic mass number increase, the repulsive electrostatic forces within the nucleus increase due to the greater number of protons in the heavy elements.
To overcome this increased repulsion, the proportion of neutrons in the nucleus must increase to maintain stability.
34
This increase in the neutron-to-proton ratio only partially compensates for the growing proton-proton repulsive force in the heavier, naturally occurring elements.
Because the repulsive forces are increasing, less energy must be supplied, on the average, to remove a nucleon from the nucleus.
The BE/A has decreased. The BE/A of a nucleus is an indication of its degree of stability.
Generally, the more stable nuclides have higher BE/A than the less stable ones.
35
The increase in the BE/A as the atomic mass number decreases from 260 to 60 is the primary reason for the energy liberation in the fission process.
In addition, the increase in the BE/A as the atomic mass number increases from 1 to 60 is the reason for the energy liberation in the fusion process, which is the opposite reaction of fission.
36
The heaviest nuclei require only a small distortion from a spherical shape (small energy addition) for the relatively large coulomb forces forcing the two halves of the nucleus apart to overcome the attractive nuclear forces holding the two halves together.
Consequently, the heaviest nuclei are easily fissionable compared to lighter nuclei.
37
38
Example
Calculate the average binding energy per nucleon of the
nucleus iron-56 .
Given
56
26 Fe
56
26
p
n
mass nucleus Fe=55.93494 u
m =1.007276 u
m =1.008664 u
p n nucleusΔm= Zm +Nm -m
= 26 1.007276 +30 1.008664 -55.93494
=0.514156 u
39
2ΔmcBinding energy per nucleon=
A
931.5MeV0.514156u
u =
56
=8.55MeV/nucleon
40
Example
has an atomic mass of 22.994127 u. Calculate its
binding energy per nucleon.
Given
23
12 Mg
H
n
m =1.007824 u
m =1.008664 u
p n nucleusΔm= Zm +Nm -m
= 12 1.007824 +11 1.008664 -22.994127
=0.195065 u
41
2ΔmcBinding energy per nucleon=
A
931.5MeV0.195095u
u =
23
= 7.90MeV/nucleon
42
Exercise1) The binding energy of the neon is160.64 MeV.
Find its nucleus mass.
Given
(Ans: 19.99u)
2) Determine the total binding energy and the binding
energy per nucleon for the nitrogen -14 nucleus
Given
(Ans:104.65 MeV,7.47 MeV/nucleon)
14
7
1 1
1 1
1
0
mass 14.003074 u
mass 1.007824 u
mass 1.008664 u
N
H p
n
Ne20
10
p
n
m =1.007276 u
m =1.008664 u
3) Calculate the binding energy of an aluminum nucleus, in MeV.
(Given mass of neutron, mn=1.008664 u ; mass of proton, mp=1.007824 u ; speed of light in vacuum, c=3.00108 m s1 and atomic mass of aluminum, MAl=26.98154 u)
(Ans: 224.93 MeV)
4) Calculate the binding energy per nucleon of a boron,nucleus in J/nucleon.
(Given mass of neutron, mn=1.008664 u ; mass of proton,mp=1.007276 u ; speed of light in vacuum, c=3.00108 ms1 and nucleus mass of boron, MB=10.01294 u)
43
(E = 1.46x10 -11 J/nucleon)
27
13Al
11
5 Bo
5) Why is the uranium-238 nucleus is less stable
than carbon-12 nucleus? Give an explanation by
referring to the binding energy per nucleon.
(Given mass of neutron, mn=1.008664 u ; mass
of proton, mp=1.007824 u ; speed of light in
vacuum, c=3.00108 m s1; atomic mass of
carbon-12, MC=12.00000 u and atomic mass of
uranium-238, MU=238.05079 u )
(Ans: U think)
44
The electrons that circle the nucleus move in fairly well-defined orbits.
Some of these electrons are more tightly bound in the atom than others.
For example, only 7.38 eV is required to remove the outermost electron from a lead atom, while 88,000 eV is required to remove the innermost electron.
The process of removing an electron from an atom is called ionization, and the energy required to remove the electron is called the ionization energy.
45
In a neutral atom (number of electrons = Z) it is possible for the electrons to be in a variety of different orbits, each with a different energy level.
The state of lowest energy is the one in which the atom is normally found and is called the ground state.
When the atom possesses more energy than its ground state energy, it is said to be in an excited state.
46
An atom cannot stay in the excited state for an indefinite period of time.
An excited atom will eventually transition to either a lower-energy excited state, or directly to its ground state, by emitting a discrete bundle of electromagnetic energy called an x-ray.
The energy of the x-ray will be equal to the difference between the energy levels of the atom and will typically range from several eVto 100,000 eV in magnitude.
47
metastable state particular excited state of an atom, nucleus, or other system that has a longer lifetime than the ordinary excited states and that generally has a shorter lifetime than the lowest, often stable, energy state, called the ground state
48
The nucleons in the nucleus of an atom, like the electrons that circle the nucleus, exist in shells that correspond to energy states.
The energy shells of the nucleus are less defined and less understood than those of the electrons.
There is a state of lowest energy (the ground state) and discrete possible excited states for a nucleus.
Where the discrete energy states for the electrons of an atom are measured in eV or keV, the energy levels of the nucleus are considerably greater and typically measured in MeV.
49
A nucleus that is in the excited state will not remain at that energy level for an indefinite period.
Like the electrons in an excited atom, the nucleons in an excited nucleus will transition towards their lowest energy configuration and in doing so emit a discrete bundle of electromagnetic radiation called a gamma ray
(-ray).
The only differences between x-rays and
-rays are their energy levels and whether they are emitted from the electron shell orfrom the nucleus.
50
The ground state and the excited states of a nucleus can be depicted in a nuclear energy-level diagram.
The nuclear energy-level diagram consists of a stack of horizontal bars, one bar for each of the excited states of the nucleus.
The vertical distance between the bar representing an excited state and the bar representing the ground state is proportional to the energy level of the excited state with respect to the ground state.
51
This difference in energy between the ground state and the excited state is called the excitation energy of the excited state.
The ground state of a nuclide has zero excitation energy.
The bars for the excited states are labelled with their respective energy levels.
52
53
54
MODES OF RADIOACTIVE DECAY
• Most atoms found in nature are stable and do not emit particles or energy that change form over time.
• Some atoms, however, do not have stable nuclei.
• These atoms emit radiation in order to achieve a more stable configuration.
Stability of Nuclei• As mass numbers become larger, the ratio of
neutrons to protons in the nucleus becomes larger for the stable nuclei.
• Non-stable nuclei may have an excess or deficiency of neutrons and undergo a transformation process known as beta () decay.
• Non-stable nuclei can also undergo a variety of other processes such as alpha () or neutron (n) decay.
• As a result of these decay processes, the final nucleus is in a more stable or more tightly bound configuration.
Natural Radioactivity
• In 1896, the French physicist Becquerel discovered that crystals of a uranium salt emitted rays that were similar to x-rays in that they were highly penetrating, could affect a photographic plate, and induced electrical conductivity in gases.
• Becquerel's discovery was followed in 1898 by the identification of two other radioactive elements, polonium and radium, by Pierre and Marie Curie.
• Heavy elements, such as uranium or thorium, and their unstable decay chain elements emit radiation in their naturally occurring state.
• Uranium and thorium, present since their creation at the beginning of geological time, have an extremely slow rate of decay.
• All naturally occurring nuclides with atomic numbers greater than 82 are radioactive.
Nuclear Decay
• Whenever a nucleus can attain a more stable (i.e., more tightly bound) configuration by emitting radiation, a spontaneous disintegration process known as radioactive decay or nuclear decay may occur.
• In practice, this "radiation" may be electromagnetic radiation, particles, or both.
• Detailed studies of radioactive decay and nuclear reaction processes have led to the formulation of useful conservation principles.
• The four principles are discussed below.
• Conservation of electric charge implies that charges are neither created nor destroyed.
• Single positive and negative charges may, however, neutralize each other.
• It is also possible for a neutral particle to produce one charge of each sign.
• Conservation of mass number does not allow a net change in the number of nucleons.
• However, the conversion of a proton to a neutron and vice versa is allowed.
• Conservation of mass and energy implies that the total of the kinetic energy and the energy equivalent of the mass in a system must be conserved in all decays and reactions.
• Mass can be converted to energy and energy can be converted to mass, but the sum of mass and energy must be constant.
• Conservation of momentum is responsible for the distribution of the available kinetic energy among product nuclei, particles, and/or radiation.
• The total amount is the same before and after the reaction even though it may be distributed differently among entirely different nuclides and/or particles.
Alpha Decay, • Alpha decay is the emission of alpha particles
(helium nuclei) which may be represented as either or .
• When an unstable nucleus ejects an alpha particle, the atomic number is reduced by 2 and the mass number decreased by 4.
• An example is uranium-234 which decays by the ejection of an alpha particle accompanied by the emission of a 0.068 MeV gamma.
• The combined kinetic energy of the daughter nucleus (Thorium-230) and the particle is designated as KE.
• The sum of the KE and the gamma energy is equal to the difference in mass between the original nucleus (Uranium-234) and the final particles (equivalent to the binding energy released, since m = BE).
• The alpha particle will carry off as much as 98% of the kinetic energy and, in most cases, can be considered to carry off all the kinetic energy.
Beta Decay, • Beta decay is the emission of electrons of nuclear
rather than orbital origin. • These particles are electrons that have been expelled
by excited nuclei and may have a charge of either sign.• If both energy and momentum are to be conserved, a
third type of particle, the neutrino, , must be involved.
• The neutrino is associated with positive electron emission, and its antiparticle, the antineutrino, , is emitted with a negative electron.
• These uncharged particles have only the weakest interaction with matter, no mass, and travel at the speed of light.
• For all practical purposes, they pass through all materials with so few interactions that the energy they possess cannot be recovered.
• The neutrinos and antineutrinos are included here only because they carry a portion of the kinetic energy that would otherwise belong to the beta particle, and therefore, must be considered for energy and momentum to be conserved.
• They are normally ignored since they are not significant in the context of nuclear reactor applications.
• Negative electron emission, represented as simply as , effectively converts a neutron -to a proton, thus increasing the atomic number by one and leaving the mass number unchanged.
• This is a common mode of decay for nuclei with an excess of neutrons, such as fission fragments below and to the right of the neutron-proton stability curve.
• An example of a typical beta minus-decay reaction is shown below.
• Positively charged electrons (beta-plus) are known as positrons.
• Except for sign, they are nearly identical to their negatively charged cousins. When a positron, represented as , or simply as , is ejected from the nucleus, the atomic number is decreased by one and the mass number remains unchanged.
• A proton has been converted to a neutron. An example of a typical positron(beta-plus) decay is shown below.
Electron Capture (EC, K-capture)• Nuclei having an excess of protons may capture
an electron from one of the inner orbits which immediately combines with a proton in the nucleus to form a neutron. This process is called electron capture (EC).
• The electron is normally captured from the innermost orbit (the K-shell), and consequently, this process is sometimes called K-capture.
• The following example depicts electron capture.
• A neutrino is formed at the same time that the neutron is formed, and energy carried off by it serves to conserve momentum.
• Any energy that is available due to the atomic mass of the product being appreciably less than that of the parent will appear as gamma radiation.
• Also, there will always be characteristic x-rays given off when an electron from one of the higher energy shells moves in to fill the vacancy in the K-shell.
• Electron capture and positron emission result in the production of the same daughter product, and they exist as competing processes.
• For positron emission to occur, however, the mass of the daughter product must be less than the mass of the parent by an amount equal to at least twice the mass of an electron.
• This mass difference between the parent and daughter is necessary to account for two items present in the parent but not in the daughter.
• One item is the positron ejected from the nucleus of the parent.
• The other item is that the daughter product has one less orbital electron than the parent.
• If this requirement is not met, then orbital electron capture takes place exclusively.
Gamma Emission, • Gamma radiation is a high-energy
electromagnetic radiation that originates in the nucleus.
• It is emitted in the form of photons, discrete bundles of energy that have both wave and particle properties.
• Often a daughter nuclide is left in an excited state after a radioactive parent nucleus undergoes a transformation by alpha decay, beta decay, or electron capture.
• The nucleus will drop to the ground state by the emission of gamma radiation.
Internal Conversion• The usual method for an excited nucleus to go from the
excited state to the ground state is by emission of gamma radiation.
• However, in some cases the gamma ray (photon) emerges from the
• nucleus only to interact with one of the innermost orbital electrons and, as a result, the energy of the photon is transferred to the electron.
• The gamma ray is then said to have undergone internal conversion. The conversion electron is ejected from the atom with kinetic energy equal to the gamma energy minus the binding energy of the orbital electron.
• An orbital electron then drops to a lower energy state to fill the vacancy, and this is accompanied by the emission of characteristic x-rays.
INTERACTION OF IONIZING
RADIATIONS WITH MATTER
Writing a element in nuclide
format (carbon-14 is used as an example)
Electromagnetic radiation can also be
subdivided into ionizing and non-ionizing
radiations.
Non-ionizing radiations have wavelengths of ≥10−7 m.
Non-ionizing radiations have energies of
<12 electron volts (eV); 12 eV is
considered to be the lowest energy that an
ionizing radiation can possess.
Types of non-ionizing electromagnetic
radiation:
• Radio waves
• Microwaves
• Infrared light
• Visible light
• Ultraviolet light
Ionizing Radiation
• Ionizing (high-energy) radiation has the
ability to remove electrons from atoms;
i.e., to ionize the atoms.
• Ionizing radiation can be electromagnetic
or particulate radiation.
• Clinical radiation oncology uses photons
(electromagnetic) and electrons or (rarely)
protons or neutrons (all three of which are
particulate) as radiation in the treatment of
malignancies and some benign conditions.
Ionizing radiations
Ionizing Electromagnetic Radiation
• The electromagnetic spectrum comprises
all types of electromagnetic radiation,
ranging from radio waves (low energy,
long wavelength, low frequency) to
ionizing radiations (high energy, short
wavelength, high frequency).
Electromagnetic spectrum
• Electrons are knocked out of their atomic and molecular orbits (a process known as ionization) when high-energy radiation interacts with matter.
• Those electrons produce secondary electrons during their passage through the material.
• A mean of energy of 33.85 eV is transferred during the ionization process, which in atomic and molecular terms is a highly significant amount of energy.
• When high-energy photons are used
clinically, the resulting secondary
electrons, which have an average energy
of 60 eV per destructive event, are
transferred to cellular molecules.
Interaction of X and -rays with matter.
In order to understand the physical
basis for radiation protection, it is
necessary to know the mechanisms
by which radiations interact with
matter.
In most instances, interactions
involve a transfer of energy from the
radiation to the matter with which it
interacts.
Matter consists of atomic nuclei and
extranuclear electrons.
Radiation may interact with either or
both of these constituents of matter.
The probability of occurrence of any
particular category of interaction
depends on the type and energy of the
radiation as well as on the nature of the
absorbing medium.
In all instances, excitation and
ionization of the absorber atoms
results from their interaction with the
radiation.
Ultimately, the energy transferred
either to tissue or to a radiation
shield is dissipated as heat.
The interaction of photons is
independent of their nature of origin.
Interaction Mechanisms
There are 12 possible processes by which
photons (X-rays and -rays) may interact with
matter. These are classified in table 3.1.
Among these large number of possible
interaction mechanisms, only four major
types play an important role in radiation
measurements:
• photoelectric absorption
• Compton Scattering
• pair production
• Rayleigh scattering
All these processes lead to partial or
complete transfer of photon energy to
electron.
Photoelectric Absorption
In the photoelectric absorption process, a
photon undergoes an interaction with an
absorber atom in which the photon
completely disappears.
In its place, an energetic photoelectron is
ejected by the atom from one of its bound
shells.
The interactions with the atom as a whole
and can not take place with free electrons.
• To define it simply, when any electromagnetic radiation reaches a surface (generally a metallic surface), it transfers its energy to the electrons of that surface, which are then scattered.
• At the atomic level, the incoming radiation knocks an electron from an inner atomic orbital, propelling it from the atom.
Photoelectric effect
This is the basic interaction in diagnostic radiology.
It is dominant at energies of less than 35 kV, and in atoms with high atomic numbers (Z).
Since the atomic number of bone is higher than that of soft tissue, bone absorbs more radiation than soft tissue. This absorption difference is the basis of diagnostic radiology.
This effect also explains why metals with high atomic numbers (e.g., lead) are used to absorb low-energy X-rays and gamma rays.
The illustration of Electron photoelectric effect
For -rays of sufficient energy, the most
probable origin of the photoelectron is
the most tightly bound or k-shell of the
atom.
The photoelectron appears with an
energy given by
The photoelectric cross section strongly
depends upon the atomic number of the
absorber and the energy of the photon.
It is approximately given by
where the exponent n varies
between 4 and 5 over the gamma-
ray energy region of interest.
where h is the energy of
incident photon and Eb is the
binding energy of the photoelectron in its original shell
Compton Scattering
As the energy of photon increases above
the k-edge, cross section for the
photoelectric absorption of photon
becomes rapidly insignificant.
The important energy loss mechanism
beyond this point and up to around
1.5 MeV photons, is Compton scattering.
Compton scattering occurs when a
photon of medium energy undergoes
an elastic collision with a free, or nearly
free electron.
In matter, of course, the electrons are
bound, however, if the photon energy is
high with respect to the binding energy,
the binding energy can be ignored and
the electrons can be essentially free.
Math associated with the Compton effect
• In the Compton effect, a photon collides with an electron in an outer orbital, and the photon and electron are scattered in different directions (where θ is the angle between the directions).
• The energy of the incoming photon is transferred to the electron in the form of kinetic energy.
• The scattered electron also interacts with the outer orbital electrons of other atoms.
• After the interaction, the photon has a lower energy than it did beforehand (refer Fig.).
This is the main mechanism for the absorption of ionizing radiation in radiotherapy.
It is the dominant effect across a wide spectrum of energies, such as 35 kV–50 MV.
It has no dependency on the atomic number (Z) of the absorbent material, but it does depend on the electron density of the material.
The absorption of incoming radiation is the same for bone and soft tissues.
Illustration of the Compton effect
The incoming gamma-ray photon is
deflected through an angle with respect to
its original direction.
The photon transfers a portion of its energy
to the electron (assumed to be initially at
rest), which is then known as a recoil
electron.
Because all angles of scattering are
possible, the energy transferred to the
electron can vary from zero to a large
fraction of the gamma-ray energy.
From simultaneous equations for
conservation of energy and momentum
the expression that relates the energy
transfer and the scattering angle for any
given interaction can be shown to be:
The probability that the photon will be
Compton scattered falls off steadily with
increasing energy of photons.
Its dependence on the atomic number of
absorber has been found to be very little.
EXAMPLE : Energy lost by photons
What fraction of their energies do 1 MeV and 0.1 MeV photons lose if they are scattered through an angle of 90
(ii) In the case of 0.1 MeV gamma
rays, the energy of the scattered
photon is 0.0835MeV, and the
fractional energy loss is only 16.5%.
Notice that the photon of higher
energy loses greater fraction of its
energy in the energy range
considered.
EXAMPLE : Maximum Energy of Compton Recoil ElectronCompute the maximum energy of the Compton Recoil electrons resulting from the absorption in Al of 2.19 MeV -rays.
The recoil electron maximum energy (Emax) corresponds to the minimum energy of scattered photon(h’ min)
Substituting h = 2.19 MeV ,we obtain
Emax = 1.961MeV
Pair Production
If the gamma ray energy exceeds twice
the rest mass energy of an electron
(1.02 MeV), the process of pair
production is energetically possible.
In pair production process the photon
disappears and its materialization into an
electron-positron pair takes place.
The energy of the photon partly appears
as the rest masses of the two particles
and partly as the kinetic energies of the
electron (Ee) and positron (Ep).
Pair production and pair Annihilation
Note:-
the available energy (h - 2mc2) can be divided in any proportion
between electron and positron, but in general it is almost divided equally.
pair production can take place in the presence of a nucleus so as to
conserve momentum by recoil of nucleus.
after production of a pair, the positron and electron are projected in a
forward direction and lose their kinetic energy by excitation, ionization and
bremsstrahlung, as with any other high-energy electron.
When the positron has expended all of its kinetic energy, it combines with
an electron to produce two quanta of 0.511 MeV each of annihilation
photons.
the cross section of the pair production process increases both as energy
of photon and atomic number Z of the absorber.
This means pair production is most important process with photons
of high energy in heavy elements.
Pair Production
• This is a relatively rare effect. In it, a photon transforms into an electron and a positron near a nucleus.
• The electron sheds all of its energy by the absorption processes explained above. On the other hand, the positron propagates through the medium ionizing atoms until its energy has dropped to such a low level that it pulls a free electron close enough to combine with it, in a process called annihilation.
• This annihilation causes the appearance of a pair of photon moving in opposite directions, and each with 0.511 MeV of energy. These annihilation photons are absorbed through either photoelectric or Compton events.
The threshold photon energy level for pair production is 1.02 MeV; below this, pair production will not occur.
The probability of pair production occurring increases as Z increases.
Pair production is more frequently observed than the Compton effect at energies of more than 10 MeV.
Relationship between photon energy and absorption
coefficient for various types of interaction with matter (in air)
Rayleigh Scattering
If the photon energy is smaller than the
binding energy of the electron, then the
electron is not removed by photon from
its shell.
It only vibrates after absorbing the
photon.
The vibrating photon acts as a dipole
and thus emit photons of same energy
in the forward direction.
Because the outgoing photon energy
remain unaltered and electron of the
medium is neither removed nor
excited the process is also called
coherent scattering.
Rayleigh scattering generally
occurs at higher Z materials.
Coherent Effect (= Rayleigh Scattering, =
Thomson Scattering)• Here, an electron is scattered when an
electromagnetic wave or photon passes close to it [21]. This type of scattering is explained by the waveform of the electromagnetic radiation.
• There are two types of coherent scattering: Thomson scattering and Rayleigh scattering.
• The wave/photon only interacts with one electron in Thomson scattering, while it interacts with all of the electrons of the atom in Rayleigh scattering.
• In Rayleigh scattering, low-energy radiation interacts with an electron, causing it to vibrate at its own frequency.
• Since the vibrating electron accelerates, the atom emits radiation and returns to its steady state.
• Thus, there is no overall transfer of energy to the atom in this event, so ionization does not occur.
• The probability of coherent scattering is high in heavy (i.e., high-Z) matter and for low-energy photons.
Rayleigh scattering
The relative importance of the three major types
of photon interaction. The lines show the values
of Z and h for which the two neighbouring
effects are just equal
Photo Nuclear Reactions
When the photon energy approaches the
binding energy of the nucleon in the
nucleus it is possible for a photon to
initiate a nuclear reaction such as (,n),
(,p), (,d) etc.
The cross section of such reactions is
generally very low compared to total
cross section for interaction with atomic
electrons.
Hence energy lost by these photo
nuclear reactions is not that important up
to photon energy of nearly 10 MeV.
Gamma Ray Attenuation
Consider a beam of photons of intensity I incident
upon a plane of absorbing material as shown
At a depth x with in the material, this
intensity is reduced to I due to
interaction along its way.
within an incremental thickness dx there
is a further reduction in I by dI.
the probability for interaction with in dx
is dI=I. Thus the probability for
interaction per unit thickness is :
In terms of µ we may write the decrease in
intensity with in dx
This simple differential equation, when
solved by using the boundary condition that
I = Io at x = 0, yields
The linear attenuation coefficient µ is related
to the total microscopic cross section t by
For this reason is sometimes referred to
as macroscopic cross section.
If we neglect minor interaction processes,
the linear attenuation coefficient is the
sum of the individual attenuation coefficients
for the individual interactions
where N is the number of targets
per unit volume.
Half Value Thickness (HVT): is the thickness
of an absorber at which the intensity of the
incident beam falls to its half value.
Mean free-path: is a quantity that describes
the average distance travelled by a photon
before absorption. The mean free
path is given by
Mass Attenuation Coefficient (µm)
Mass attenuation coefficient is the probability
of interaction per unit of path length expressed
in terms of mass/area.
It is obtained by dividing the linear
attenuation coefficient by the density () of the
material.
At a given -ray energy, the mass attenuation
coefficient is independent of the physical state
of a given absorber.
For example it is the same for water whether
present in vapour or liquid form.
The mass attenuation coefficient of a
compound or mixture of elements can be
calculated from
where the wi factors represent the weight
fraction of element i in the compound or
mixture.
In terms of the mass attenuation coefficient,
the attenuation law for gamma rays now
takes the form
A unit very often used for expressing
thickness of absorbers is the surface
density or mass thickness.
This is given by the mass density of the
material times its thickness in normal units
of length.
i.e. mass thickness units are more convenient
than normal length units because they are
more closely related to the density of
interaction centers.
They thus have the effect of normalizingmaterials of different mass densities.
Note:-
Equal mass thickness of different materials will
have roughly the same effect on the same
radiation. In terms of mass thickness and mass
attenuation coefficient, the attenuation law thus
becomes
EXAMPLE
Interaction of Charged Particle with Matter
Studies of the interactions of charged
particles with the matter through which they
pass have been very informative from the
earliest scattering experiments of Rutherford
till the sophisticated high energy experiments
of today.
An understanding of these interactions has
led to a more detailed knowledge of atomic
and nuclear structure, to a better insight into
the nature of the radiations themselves, andto their effects on living systems.
A charged particle passing through matter
loses energy as a result of electromagnetic
interactions with the atoms and molecules of
the surrounding medium.
The character of these interactions and the
mechanism of the energy loss depends on
the charge and velocity of the particle and onthe characteristics of the medium.
3.2.1 Interaction Mechanisms
Two principal features characterize the
passage of charged particles in matter.
These are loss of energy by the particle
and deflection of the particle from its
incident direction.
These are the results of the following
interactions.
1. Inelastic collisions with orbital electrons
(excitation and ionization of atoms),
2. Radiative losses in the field of nuclei
(Bremsstrahlung emission),
3. Elastic scattering with nuclei and
4. Elastic scattering with orbital electrons.
Which of these interactions actually take
place is a matter of chance. However
energetic electrons lose energy mainly by
inelastic collisions which produce
ionization and excitation, and also by
radiation.
Charged particles in general lose energy
mainly by the coulomb interactions with
the atomic electrons.
If the energy transferred to the electrons
in an atom is sufficient to raise it to higher
energy state in the atom, this process is
called excitation.
If the energy transferred is more, the
electron is ejected out of this system. This
process is called ionization.
These two processes are closely
associated and together constitute the
energy loss by inelastic collision.
The ejected electron will lose its kinetic
energy and finally attach itself to another
atom thereby making it a negative ion.
These together constitute an ion pair.
Some of the electrons ejected may have
sufficient energy to produce further
ionization. Such electrons are called delta
(d) rays.
In any case, the energy for these
processes comes from the kinetic energy
of the incident particle, which is slowed
down.
Charged particles are classified mainly
into two groups:
1) heavy charged particles of mass
comparable with the nuclear mass
(protons, alpha particles, mesons,
atomic and molecular ions)
2) electrons
A striking difference, in the absorption of
heavy charged particles, electrons and
photons, is that only heavy charged
particles (and electrons only in a limited
sense) have a range.
A heavy charged particle usually loses a
relatively small fraction of its energy in a
single collision with an atomic electron.
Consequently, a monoenergetic beam of
heavy charged particles, in traversing a
certain amount of matter, loses energy
without changing the number of particles in
the beam.
Ultimately they will all be stopped after
having crossed practically the same
thickness of absorber. This minimum
amount of absorber that stops a
particle is its range.
Electrons exhibit a more complicated
behavior.
They radiate electromagnetic energy easily
because they have a large value of e/m
and hence are subject to violent
accelerations under the action of electric
forces.
Moreover, they undergo scattering to such
an extent that they follow irregular
trajectories.
For electromagnetic radiation, on the other
hand, the absorption is exponential.
Interaction of Heavy Charged Particles
Bethe Formula of Stopping Power
The stopping power (-dE/dx) of a material for a fast moving
heavy charged particle is given by the Bethe formula:
where ze is the charge of the incident particle,
v its velocity, N the number density of atoms
(number of atoms per unit volume) of the
material having atomic number Z, m the
electron rest mass and e the electron charge.
The quantity I is a material property called the
mean excitation energy, which is a logarithmic
average of the excitation energies of the medium
weighted by the corresponding oscillator
strengths.
Except for elements with very low atomic
number Z, the mean excitation energies in eV
are approximately to 10Z.
At very low velocities, i.e., when v is
comparable with the velocity of the atomic
electrons around the heavy particles (in
the case of hydrogen, v = c/137), the
heavy ion neutralizes itself by capturing
electrons for part of the time.
This results in a rapid fall off of ionization
at the very end of the range.
On the other hand, at extremely high
energies, with v ≈ c, ionization
increases for several reasons.
The relativistic contraction of the
coulomb field of the ion is one of them.
Part of the energy is carried away as
light.
It is generally assumed that chemical and
atomic aggregation phenomena affects
stopping power to a very limited extent. This
is embodied in the Bragg rule for theevaluation of the mean excitation potential:
Bragg’s Rule
where ni is the number density of electrons
associated with element and Ii is the mean
excitation potential for that element.
The implication of this relation is that the
stopping power in a compound is the sum of
the stopping powers of the individual
elements.
It is observed that chemical binding does
affect the mean excitation potential but
the effect decreases rapidly with
increasing atomic charge.
The increased validity of Bragg’s law with
increasing atomic charge relies on the
increased dependence of I on inner-shell
electrons, which are insensitive to
chemical effects.
Deviation from Bragg’s rule should be
more apparent at low energies for which
the logarithmic term in the Bethe formulae
becomes a sensitive function of I.
Scaling Laws for Stopping Power and Range
The last two relations allow us to write the
energy loss as a function of energy for any
particle, once the energy loss as a function of
energy is known for protons. In particular,
protons, deuterons, and tritium of the same
velocity have the same stopping power.
The above mentioned equations for range are
not exact, for the neutralization phenomena
occurring at the end of the range and other
corrections are neglected; but it is sufficiently
accurate for most cases, excluding very low
energies. As an example of the application of
the last equation mentioned above, we can
verify that a deuteron of energy E has twice the
range of a proton of energy E/2.
Interaction of Light Charged Particles
Light charged particles are electrons and
positrons.
As all forms of ionizing radiation eventually
result in a distribution of low-energy
electrons, the interaction of light charged
particles are of central importance in
radiation biology.
The large difference in mass between
electrons and heavy charged particles
has important consequences for
interactions.
Light charged particles deposit energy
through two mechanisms: Collisional
losses and Radiative losses
Collision Losses
• Electrons lose energy via interactions with orbital
electrons in the medium.
• This leads to excitation of the atom or ionization.
• Energy loss via these mechanisms is called
“collisional loss".
• Maximum energy transfer occurs in a “head-on"
collision between two particles of masses m and M:
and can be expressed as
• The electron collides with a particle of identical
mass and thus large scattering angles are possible.
• This results in a track that is very tortuous instead
of the straight path of a heavy charged particle.
Radiative Losses: bremsstrahlung
•A second mechanism of energy loss is possible because
of the small mass of the light charged particle (negligible
with HCPs).
•A charged particle undergoing a change in
acceleration always emits “radiative” electromagnetic
radiation called bremsstrahlung.
•The larger the change in acceleration, the more energetic
the bremsstrahlung photon.
•For electrons, the bremsstrahlung photons have a
continuous energy distribution that ranges downward
from a maximum equal to the kinetic energy of the
incoming electron.
•The efficiency of bremsstrahlung in elements of different
atomic number Z varies nearly as Z2.
•Notice that bremsstrahlung increases with electron
kinetic energy and atomic number Z.
Interaction of Neutrons
The behaviour of neutrons in matter is quite
different from that of either charged particles or
gamma rays.
Since neutrons are uncharged, no coulomb
force comes into play and a neutron
possesses free access to the nucleus of all
atoms.
For neutrons to interact with matter, i.e. for the
nuclear force to act, they must either enter the
nucleus or come sufficiently close to it.
The type of the interaction taking place between
a neutron and the nucleus differs depending
upon the kinetic energy of the incident neutron.
Energy Classification
For the purpose of study of neutron interactions;
neutrons are classified as below:
• Thermal neutrons Energy below 0.5 eV
• Intermediate neutrons 0.5 eV – 100 keV
• Fast neutrons 100 keV – 20 MeV
• High energy neutrons above 20 MeV.
Study of the neutron interaction with matter requires
the knowledge of neutron energy spectrum.
For many applications the spectrum is poorly
known. All neutrons are fast by birth and lose energy
by colliding elastically with atoms in their environment
and then after being slowed down to thermal
energies they are captured by the nuclei of the
absorbing medium.
Neutron Sources
The most prolific source of neutrons is the nuclear
reactor.
The splitting of a uranium or a plutonium
nucleus in a nuclear reactor is accompanied by the
emission of several neutrons.
These fission neutrons have a wide range of
energies, peak at ~0.7 MeV and have a mean value
of ~ 2 MeV.
Copious neutron beams may be produced in
accelerators by many different reactions.
For example, bombardment of beryllium by high-
energy deuterons in a cyclotron produces neutrons
according to the reaction:
Interaction Mechanisms
There are a number of processes by which a
neutron can interact with matter.
1. Elastic scattering (n,n)
In elastic collision both the momentum and
kinetic energy of the system of neutron and
interacting nucleus are conserved. The
process may be regarded as essentially a
billiard ball type of collision. In each
collision with a stationary nucleus, the
neutron transfers part of its kinetic energy to
the nucleus depending on the angle through
which the neutron is scattered.
2. Inelastic scattering
In the range of energy above 0.5 MeV
inelastic scattering begins to occur [(n,n), (n,
2n), (n,) type of reactions]. In this case a part
of kinetic energy of the incident neutron is
given off in the form of one or more gammas.
This process always takes place through the
formation of compound nucleus. This type of
interaction is not possible unless the neutron
energy exceeds a certain threshold
3. Nonelastic scattering (Nuclear Reactions
Involving Emission of Charged Particles)
Nonelastic reactions occur at high neutron
energies [(n,), (n,p) type]. These are the
reactions with energy thresholds in which the
neutron causes the emission of charged particles
(protons or other heavier particles) from the
target nucleus. An example of particular
importance in biological tissue is the (n,p)
reaction of 14N with slow neutrons:
This reaction produces a proton of 0.58 MeV
energy.
4. Radiative Capture (n,)
Nuclear process in which a neutron is captured
by the target nucleus and the excess energy
emitted as radiation, is called radiative capture
process. Conditions for such reactions are
especially favourable during slow neutron (with
energy < 1 eV) interaction with medium.
Cross section for these processes usually
decreases with the inverse of the neutron velocity.
It is a very common process, for it occurs with a
wide variety of nuclides from low to high mass
numbers.
This process is extensively used for the
production of isotopes by exposing stable nuclides
to slow neutrons in a nuclear reactor.
5. Spallation reaction
In this process the target nucleus is fragmented
with the emission of several
particles often including neutrons. The process
becomes significant only at neutron energies of
about 100 MeV or greater.
6. Nuclear fission(n,f)
In certain reactions involving heavy atomic
nuclei, the capture of neutron results in the
formation of an excited state of a compound
nucleus so unstable that it splits up into two
smaller nuclei. This process is of fundamental
importance for the operation of nuclear reactors.
Attenuation of neutrons
All neutrons, at the time of their birth, are fast.
Generally, fast neutrons lose energy by colliding
elastically with atoms in their environment, and
then, after being slowed down to thermal or near
thermal energies, they are captured by nuclei of
the absorbing material.
When absorbers are placed in a collimated
beam of neutrons and the transmitted neutron
intensity is measured, as was done for gamma
rays, it is found that neutrons, too, are removed
exponentially from the beam.
Instead of using linear or mass absorption
coefficients to describe the ability of a given
absorber material to remove neutrons from the
beam, it is customary to designate only the
microscopic cross section for the absorbing
material.
The product N, where N is the number of
absorber atoms per cm3, is the macroscopic
cross section .
The removal of neutrons from the beam is thus given by:
where I = Initial intensity of incident neutrons,
I = Transmitted intensity of neutrons after passing
through t cm and t thickness of the target.
EXAMPLE :
In an experiment designed to measure the total
cross section of lead for 10 MeV neutrons, it
was found that a 1 cm thick lead absorber
attenuated the neutron flux to 84.5% of its initial
value. The atomic weight of lead is 207.21, and
its specific gravity is 11.3. Calculate the total
cross section from these data.
