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12
Z N
X
X
m
Basic Atomic and Nuclear Physics
atom belongs to. The elements in the “Periodic table” are sorted by the number of the
protons present in their atomic nuclei. The number of neutron has nothing to do with the
chemical properties. Atoms, which have similar atomic number but different neutron number
(isotopes), are chemically identical. Although some physical properties are different, they
are essentially the same element. The number of proton in an atom is called atomic number
(represented by Z). The total number of nucleons is the mass number of the nucleus
(represented by A). The difference, A-Z is the neutron number, N. Nuclides with the same
mass number are called isobars and nuclides with the same neutron numbers are called
isotones.
There are several notations to summarize nuclear composition of an atom. The most
common is
A
where X represents the chemical symbol of the element. Chemical symbol
and atomic number carry the same information and neutron number can be calculated from
A and Z. Hence for the sake of simplicity the notation is briefed to
comprehensible. For example
137
Cs, where 137 is the mass number (A + Z), the symbol Cs
A
that is quite
represent 55th element in periodic table. The neutron number can easily be calculated
(A – Z = 82).
Table 2: Mass and charge of proton, neutron and electron
Particle Symbol Charge* Mass** Mass (kg) Energy(MeV) Relative mass
Proton p + 1 1.007276 1.6726 × 10
Neutron n 0 1.008665 1.6749 × 10
Electron e
** Unit charge = 1.6 × 10
** Mass expressed in Universal mass unit (mass of 1/12 of 12C atom )
(Data from “Particles and nuclei 1999)
–
–11
coulombs
– 1 0.000548 9.1093 × 10
-27
938.272 1836
-27
939.573 1839
-31
0.511 1
Nuclear forces
Protons in a nucleus are fairly at close distance
enormously strong repulsive force between protons. They still remain within the nucleus due
to existence of a very strong attractive force between nucleons that dominates the repulsive
force and makes the atom stable. The force must be effective in very short range and
neutrons must have an essential role in creating such force. Without neutrons protons cannot
stay in close distances.
In 1935, Yukawa proposed that the short-range strong force came about from the exchange
of particles that he called mesons. The strong nuclear force is one of the four fundamental
forces in nature that is created between nucleons by the exchange of mesons. This exchange
can be compared to constantly hitting a tennis ball back and forth between two people. As
long as this meson exchange is happening, the strong force holds the nucleons together.
1510
.This closeness results in an
Basic Atomic and Nuclear Physics
2
m.c
13
Neutrons also participate in the meson exchange and are even a bigger source of strong
force. Neutrons have no charge so they approach other nuclei without adding extra repulsive
force and meanwhile they increase the average distance between protons and help to reduce
the repulsion between them within a nucleus.
In a nucleus there is another force that is much weaker with a much shorter range and is
called week force. It is thought to be responsible for beta decay and radioactivity (discussed
in later chapters). Though our knowledge about an atom is quite clear, it is not so for a
nucleus. Table 3 summarizes the forces that are known as fundamental forces of nature. The
electromagnetic force and the weak nuclear force can be described as two different aspects
of a single electromagnetic force.
Table 3: Fundamental forces of nature
Force Relative Strength Comments
Strong (nuclear) 1 Attractive force between nucleons, holds them together
-13
-39
-2
Force between charged particles, holds atom together, responsible
Mediates beta decay
Forces between objects due to their masses, not significant at atomic
level
Electromagnetic 10
(coulomb) for chemical interactions
Weak 10
Gravitational 10
Nuclear Binding Energy and Mass Defect
Nuclear strong force is the resultant of phenomenon known as mass defect. Direct
measurements show that the mass of a nucleus is always less than the sum of the individual
masses of the constituent protons and neutrons. Using the Einstein relationship, the deficient
mass m is exactly equal to the energy required to separate the nucleons or binding energy
Eb of the nucleus.
Eb=
Where c is the speed of the light in vacuum.
The mass associated with binding energy is carried away in the form of energy that is
released during assembly of neutron and protons. Nucleons stay together in a nucleus because
they don’t have sufficient mass to be free. The nuclear mass defect, a slightly lower mass of
the nucleus compared to the sum of the masses of its constituent matter, is due to the
nuclear binding energy holding the nucleons together in the nucleus. The mass defect can be
used to calculate the nuclear binding energy, with E = mc2. The average binding energy per
nucleon is a measure of nuclear stability. The higher the average binding energy, the more
stable is the nucleus.
14
Basic Atomic and Nuclear Physics
Suggested reading
1. Physics in Nuclear Medicine, SR Cherry, JA Sorenson, ME Phelps, 3rd edition Saunders Philadelphia
USA, 2003.
2. Introductory Physics of Nuclear Medicine, R Chandra, Lea & Febiger, publisher, Philadelphia USA,
1992.
3. Fundamental Physics of Radiology, WJ Meredith and JB Massey, John Wright & Sons Limited,
Bristol, UK, 1974.
4. The physics of Radiology, HE Johns & JR Cunningham, Charles C Thomas publisher, Springfield
USA, 1969.
5. Nuclear Medicine, R Henkin (ed) Mosby, 1996.
Interaction of Radiation with Matter
G.S. Pant
Ionizing radiation transfer their energy in full or part to the medium through which they
pass by way of interactions. The significant types of interactions are excitation and ionisation
of atoms or molecules of the matter. Some of the mechanisms of energy transfer that are of
interest to us are discussed here.
Interaction of charged particles with matter
The charged particles include electron, positron, proton, alpha particles and heavy ions.
Electrons and positrons are light particles whereas the rest of them are relatively heavy. A
charged particle while passing through the matter interacts with the negatively charged
electrons and positively charged nucleus of target atom or molecule. During interaction it
loses some of its energy, which is taken up by the electrons of the target atoms falling on or
near the trajectory. The outcome of such interactions may be:
(a) Ejection of electrons from the target atoms (ionization)
(b) Excitation of electrons that take them from lower to higher energy state
(c) Molecular vibrations along the path (elastic collision) and conversion of energy
into heat
(d) Emission of electromagnetic radiation
The absorption of radiation by the target atom in the form of ionization or excitation is
of importance to us. In the energy range of 10 KeV to 10 MeV, ionization predominates over
excitation. The probability of absorption of charged particles is so high that even a small
thickness of the material can stop them completely.
The nature of interaction of all charged particles in the energy range mentioned above is
similar to each other. Light particles such as electrons deflect at larger angles than heavier
particles and there is a wide variation in their tortuous path. The path of a heavier particle is
1 5
16
Interaction of Radiation with Matter
more or less a straight line. When electrons are deflected at large angles, they transfer more
energy to the target atom and eject electrons from it. These electrons while passing through
the medium produce secondary electrons along their track (delta rays). The charged particles
undergo a large number of interactions before finally coming to rest. In each interaction
they lose a small amount of energy. These losses are called collision losses.
Energetic electrons can approach the nucleus where they get decelerated and produce
bremsstrahlung radiation (x-rays). The chance of such an interaction increases with increase
in electron energy and atomic number of the target material. Loss of electron energy in the
medium by this mode is termed as radiative loss. The energy lost per unit path length along
the track is known as the linear energy transfer (LET) and is generally expressed in keV/m.
Range of a charged particle
After traveling through a distance in the medium, the charged particle loses all its
kinetic energy and comes to rest as it has ample chance to interact with electrons or positively
charged nucleus of the atoms of the medium. The average distance travelled in a given
direction by a charged particle is known as its “range” in that medium and is influenced by
the following factors:
(a) Energy - Higher the energy of the particle more is the range
(b) Mass - Higher the mass of the charged particle smaller is the range
(c) Charge - The range is inversely proportional to square of the charge
(d) Density of the medium - Higher the density of the medium shorter is the range of
charged particle
Interaction of neutrons
As they do not experience any coulomb force, neutrons can easily enter the nucleus even
with low energy. The possible types of interactions are:
(a) Elastic collisions: In this process neutron collides with an atomic nucleus and gets
scattered after losing some of its energy to the recoil nucleus. Maximum energy is lost
during interaction with lighter nuclei such as hydrogen. The probability of such interactions
decreases rapidly up to 1 or 2 MeV beyond which it falls more slowly.
(b) Inelastic collisions: In this type of interaction neutron is first captured by a target
nucleus and then re-emitted with diminished energy in any direction. The target nucleus
finds itself in an excited state and emits a gamma photon to get back to the ground state.
(c) Nonelastic collisions: The term nonelastic is used if the particle resulting from the
interaction is not a neutron. An example of this type of interaction is 16O (n,) 13C where
projectile is neutron, target is 16O and emitted particle is alpha.
Interaction of Radiation with Matter
In biological systems, with elements such as C, N and O, inelastic and nonelastic collisions
have energy thresholds in the range of 4-12 MeV. The cross section rises sharply after the
threshold energy and becomes nearly constant beyond 10-15 MeV.
(d) Capture processes: Neutrons with energy of about 0.025 eV are called thermal
neutrons as they are in thermal equilibrium with matter. Thermal neutrons have a much
larger effective cross-section than fast neutrons, and are absorbed more easily by any atomic
nuclei that they interact with. This results in a heavier and often unstable isotope (radioisotope)
of the concerned element. The target nucleus gets one excess neutron in the process (for
example 1H (n, ) 2H). Some of the resultant nuclei become radioactive after capture process.
Most fission reactors use a neutron moderator to slow down (thermalise) the neutrons so
that they are more easily captured, causing further fission.
Both the capture process and nonelastic collisions play significant role in the production
of radionuclides. When the energy of the neutron exceeds 20 MeV, there are chances of
nuclear fragmentation.
17
Interaction of electromagnetic radiation with matter
When a beam of X or gamma rays passes through an absorbing medium, its intensity
gets attenuated. Some of the photons are completely absorbed, some are scattered and the
rest pass through the medium almost unchanged in energy and direction. The transferred
energy results in excitation and ionisation of atoms or molecules of the medium and also in
production of heat. The attenuation of the beam through a given medium may be summarised
as follows:
- Greater the thickness of the absorbing material more is the attenuation.
- Greater the atomic number of the material more is the attenuation.
- Higher the photon energy smaller is the attenuation produced by a given thickness
of material.
Linear Attenuation Coefficient
The linear attenuation coefficient () is defined as the fractional reduction in the beam
per unit thickness as determined by thin layer of the material.
The unit of is cm-1.
layer thin ain reduction Fractional
(cm) layers theof Thickness
18
Interaction of Radiation with Matter
Exponential attenuation
Exponential law can explain the attenuation of radiation beam intensity. The mathematical
derivation is given below.
Let No be the initial number of photons in the beam and N be the number recorded by
the detector placed behind the absorber with thickness x (Figure 1).
Figure 1: Attenuation of a radiation beam by an absorber. The transmitted beam is measured by
the detector ‘P’.
The number N, which gets attenuated, will be proportional to the thickness (x) of the
absorber and to the number of photons N, present in the beam.
Mathematically:
xNN
xNNor
...
Where is a constant called linear attenuation coefficient for the radiation.
The negative sign indicates that as x increases the number of photons in the beam
decreases. Equation-1 can be rearranged as:
.
N
xN
.
x
eoNN
The formal definition of attenuation coefficient (fractional reduction per unit thickness)
is derived from equation-2. Integration of equation-2 gives the following relationship.
(1)
(2)
(3)
Interaction of Radiation with Matter
19
Equation-3 can also be expressed in terms of beam intensity:
.
eoII
(4)
x
Where I and Io are the intensities of the beam as recorded by the detector with and
without absorbing material respectively. The attenuation coefficient may vary for a given
material due to non-uniformity in its thickness. This is particularly so in case the absorbing
material is malleable. It is therefore better to use the mass absorption coefficient, which is
independent of thickness of the absorbing material. Mass absorption coefficient is obtained
by dividing the linear attenuation coefficient with the density of the material. The unit of
mass attenuation coefficient is cm2/gm. The electronic and atomic attenuation coefficients
are also defined accordingly. The electronic attenuation coefficient is the fractional reduction
in x or gamma ray intensity produced by a layer of thickness 1 electron per square centimeter
whereas the atomic attenuation coefficient is the fractional reduction by a layer of thickness
1 atom per cm2. Thus the atomic attenuation coefficient will be Z times the electronic one.
Half Value Layer (HVL)
From equation-1 above, it can be seen that for a certain thickness (d
) of the absorbing
1/2
material the intensity becomes half of its original value i.e. I = Io/2. Substituting these
values, equation-1 can be rearranged as:
d
(HVL) = 0.693/ (5)
1/2
Half value layer (thickness) can be defined as the thickness of an absorbing material,
which reduces the beam intensity to half of its original value. Depending upon the energy
of radiation, various materials are used for the measurement of HVL.
Materials used for HVL measurement
Radiation upto 30 kV Cellophane
30 - 150 kV Aluminium
120 - 600 kV Copper
500 - 2 MV Lead
HVL is expressed for narrow and broad beam separately for the x-gamma radiation. Its
value for broad beam is always greater than that for the narrow beam for a given energy.
Mechanism of attenuation
There may be many ways of interaction between a photon and the matter. However,
only few of them are of practical importance in nuclear medicine and biomedical research
and are described here.
20
A
Interaction of Radiation with Matter
Photon Scattering
The scattering may or may not result in transfer of energy during interaction of photon
with the atom of the medium.
Elastic scattering
In elastic scattering or unmodified scattering the photons are scattered in different
directions without any loss of energy. The process, thus, attenuates the beam without
absorption. In this process the photon first interacts with a tightly bound electron then gets
released in any direction without transferring any energy. The contribution of this mode of
interaction is insignificant in medical applications of radiation. However, it has tremendous
application in x-ray crystallography.
Inelastic (Compton) scattering
Compton elucidated the mechanism of this type of scattering. In this process, photon
interacts with loosely bound (free) electrons. Part of the energy of the photon is used in
ejecting out the electron and rest is scattered in different direction (Figure 2).
Figure 2: Process of Compton scattering. The incoming photon ejects the electron from outer
orbit and gets scattered with reduced energy in a different direction.
In a so-called ‘head on’ collision the photon turns back along its original track (scattered
through 180 degree) and maximum energy is transferred to the recoil electron. The change
in wavelength
Where is the angle of scattering of gamma photon and
of the photon is given by:
0
)cos1{024.0 A
(6)
0
is angstrom unit for wave
length. The energy of scattered photon is expressed as:
2
001
cmEEE
e
}]cos1{/1/[
(7)
Interaction of Radiation with Matter
21
Where
E
is the energy of incident photon and
0
is that of the scattered photon,
E
1
m
e
is the mass of the electron and c is the velocity of light in vacuum/space. The Compton
scattering involves interaction between photons and electrons. The probability therefore,
depends upon the number of electrons present and independent of the atomic number. With
the exception of hydrogen, all elements contain nearly same number of electrons per gram
(practically the same electron density). The Compton scattering is, therefore, independent of
atomic number. This is the choice of interaction for radiation therapy, where the delivered
dose is aimed to be homogeneous in spite of the tissue inhomogeneity within the body. The
electron densities of some of the important elements are listed below:
Elements electron density
O
2
3.01 x 10
Ca 3.00 x 10
Pb 2.38 x 10
H
2
5.997 x 10
23
23
23
23
The total probability () for Compton process is given by
as
Where s and a are the probabilities for scattering and absorption respectively.
Summary of Compton Scattering
- Involves interaction between a photon and a free electron and is independent of Z
(depends on electron density)
- Probability of occurrence decreases with increase in energy
- More the energy of incident photon more is the energy transferred to the electron as
kinetic energy
- Most common type of interaction in soft tissue in the energy range from 100 keV to
10 MeV.
Photoelectric effect (PEE)
In this process, the photon disappears when it interacts with the bound electron. For this
type of interaction to take place, the photon energy has to be more then the binding energy
of the electron.
EnergyKineticBEhv
(8)
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