Chapter 3

Radioactivity

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• Many naturally occurring and man-made
isotopes have the property of radioactivity,
which is the spontaneous transformation
(decay) of the nucleus with the emission of a
particle
• The process takes place in minerals of the
ground, in fibers of plants, in tissues of
animals, and in the air and water, all of which
contain traces of radioactive elements.
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 3.1 Nuclear Stability
• Although the repulsive Coulombic forces of
the protons attempt to separate the nucleons,
the strong nuclear forces strive to keep the
nucleus intact
• Stable nuclei are found to have a balance
between the number of repulsive protons and
the additional neutrons providing cohesion.

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• Figure 3.1 plots the atomic number versus
number of neutrons (Z vs. N) for the known
nuclides, revealing a band of nuclear stability
• Initially N ≅ Z in the belt of stability, but for
increasing Z values a greater number of
neutrons than protons is progressively
required
• Generally, nuclei with an even number of
protons and/or neutrons tend to have a higher
degree of stability.
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 Radioactive Decay
• Spontaneous Fission
• Neutron Ejection
• Alpha Decay
• Beta Decay
– Beta- Decay
– Beta+ Decay
• Electron Capture (EC)
• Gamma Decay (Tc-99m)
• Internal Conversion (IC)
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Internal Conversion (IC) – gamma emission from an excited nucleus. This
energy is directly imparted to one of the atomic electrons, thereby ejecting it
from the atom

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• Isotopes lying off the line of stability undergo
radioactive decay in an effort to reduce their
instability
• Generally, those radioactive nuclides farthest
from the belt of stability have the shortest decay
times, commonly expressed as the half-life
– Half-life (abbreviated t1⁄2) is the time required for a quantity
to reduce to half its initial value
• Those nuclides positioned above the line are
neutron deficient, and those below the line have
a neutron excess NUCLEAR ENGINEERING
• Radioactive decay seeks to rebalance the N/Z
ratio through a variety of competing decay
mechanisms, which are summarized in Table
3.1
• Sometimes even after the decay emission, the
nucleus remains in an excited state, which is
relieved through gamma (γ) emission or
internal conversion (IC)
• The emanations from radioactive decay
constitute the radiations NUCLEAR ENGINEERING
• Figure 3.1 reveals that alpha decay is more
prevalent for the heavier nuclei, but another
transformation mode exists for heavy
radionuclides: spontaneous fission
• The graph also discloses that neutron
emission tends to occur in comparatively
lighter nuclei only
• Overall, electron capture (EC) and beta and
positron emission are the dominant decay
mechanisms. NUCLEAR ENGINEERING
 3.2 Radioactive Decay
• Many heavy elements are radioactive

ALPHA DECAY:
• An example is the decay of the most abundant
isotope of uranium, in the reaction

• The particle released is the α (alpha) particle,

which is merely the helium-4 nucleus.

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BETA- DECAY
• The new isotope of thorium is also
radioactive, according to

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• The first product is the element protactinium
(Pa)
• The second is an electron, which is called the
β (beta) particle when it arises in a nuclear
process
• The nucleus does not contain electrons; they
are produced in the reaction, as discussed next
• The third is the antineutrino, symbolized by ν
(nu bar)
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• It is a neutral particle that shares with the beta
particle the reaction’s energy release. On average,
the (anti)neutrino carries two-thirds of the energy,
the electron, one-third.
• The neutrino has zero or possibly a very small mass
and readily penetrates enormous thicknesses of
matter.
• We note that the A value decreases by 4 and the Z
value by 2 on emission of an α particle, whereas the
A remains unchanged but Z increases by 1 on
emission of a β particle.
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• These two events are the start of a long sequence or
chain of disintegrations that produce isotopes of the
elements radium, polonium, and bismuth, eventually
yielding the stable lead isotope 20682Pb.
• Other chains found in nature start with 23592U and
232 Th.
90
• Hundreds of artificial radioisotopes have been
produced by bombardment of nuclei by charged
particles or neutrons and by separation of the
products of the fission process.

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• Table 3.2 gives several examples of
radioactive materials with their emissions,
product isotopes, and half-lives.
• The β particle energies are maximum values;
on average, the emitted betas have only one-
third as much energy, that is, Eβ,avg ≅Eβ,max/3.
• Included in the table are both natural and
synthetic radioactive isotopes, also called
radioisotopes
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• We note the special case of neutron decay
according to

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• A free neutron has a half-life of 10.3 min.
• The conversion of a neutron into a proton can be
regarded as the origin of beta emission in radioactive
nuclei.
• Most of the radioisotopes in nature are heavy elements.
• One exception is potassium-40, half-life 1.25 x109 y,
with abundance 0.0117% in natural potassium.
• Others are carbon-14 and hydrogen-3 (tritium), which
are produced continuously in small amounts by natural
nuclear reactions.
• All three radioisotopes are found in plants and animals.
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 +
BETA DECAY
• In addition to the radioisotopes that decay by
beta or alpha emission, there is a large group
of artificial isotopes that decay by the
emission of a positron (β+), which has the
same mass as the electron and an equal but
positive charge.
• An example is sodium-22, which decays with
2.6 y half-life into a neon isotope as

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• Whereas the electron (sometimes called
negatron) is a normal part of any atom, the
positron is not.
• It is an example of what is called an
antiparticle, because its properties are
opposite to those of the normal particle.
• Just as particles form matter, antiparticles
form antimatter.

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• The preceding Na-22 reaction can be regarded as
involving the conversion of a proton into a neutron
with the release of a positron and a neutrino by use
of excess energy in the parent nucleus.
• This is an example of the conversion of energy into
mass. Usually, the mass appears in the form of pairs
of particles of opposite charge.
• The positron–electron pair is one example. As
discussed in Section 5.4.3, an electron and a positron
will combine, and both will be annihilated to form
two γ (gamma) rays.
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GAMMA DECAY, INTERNAL CONVERSION and
ELECTRON CAPTUE
• A nucleus can get rid of excess internal energy by the
emission of a gamma ray, but in an alternate process,
called internal conversion, the energy is imparted
directly to one of the atomic electrons, thereby ejecting
it from the atom.
• In an inverse process, called electron capture, the
nucleus spontaneously absorbs one of its own orbital
electrons.
• Each of these two processes is followed by the production of X-
rays as the inner shell vacancy is filled.
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 3.3 The Decay Law
• The rate at which a radioactive substance disintegrates
(and thus the rate of release of particles) depends on the
isotopic species, but there is a definite decay law that
governs the process.
• In a given time period, say 1 second, each nucleus of a
given isotopic species has the same chance of decay.
• If we were able to watch one nucleus, it might decay in
the next instant, or a few days later, or even hundreds of
years later. The decay constant, λ (lambda), is the
“probability” that a particular nucleus will decay per unit
time.

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• We should like to know how many nuclei of a
radioactive species remain at any time. If λ is
the chance one nucleus will decay in a
second, then the chance in a time interval dt is
λdt. For N nuclei, the change in number of
nuclei is

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• Integrating, and letting the number of nuclei
at time zero be N0, yields a general formula
describing the
• number of radioisotopes at any time

• The decay constant is unaffected by such

factors as temperature, pressure, chemical
form, and physical state (gas, liquid, or solid).

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• Such statistical behavior is also described by a constant
property of the atom called the half-life.
• This time interval, symbolized by tH or t1/2 is the time
required for half of the nuclei to decay, leaving half of them
intact.
• If we start at time zero with N0 nuclei, after a length of time
tH, there will be N0/2; by the time 2tH has elapsed, there will
be N0/4; and so on.
• A graph of the number of nuclei as a function of time is
shown in Figure 3.2. For any time t on the curve, the ratio of
the number of nuclei present to the initial number is given by

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• The relationship between the decay constant
and half-life is readily found from

• We find that

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Co-60 Decay Scheme NUCLEAR ENGINEERING
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• The number of disintegrations, or decays, per
second (dps) of a radioisotope is called the
activity, A.
• Because the decay constant λ is the chance of
decay each second of one nucleus, for N
nuclei the activity is the product

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• The unit dps is called the becquerel (Bq), honoring the
scientist, Henri Becquerel, who discovered radioactivity.
• Another older and commonly used unit of activity is the curie
(Ci), named after the French scientists Pierre and Marie Curie
who studied radium.
• The curie is 3.7 x1010 Bq, which is an early measured value
of the activity per gram of radium-226.
• Because the number of radioactive nuclei changes with time,
the activity follows the same behavior

• Where A0 is the initial activity.

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• The half-life tells us how long it takes for half of the nuclei to
decay, whereas a related quantity, the mean life, t (tau), is the
average time elapsed for decay of an individual nucleus. It
turns out that (see Exercise 3.9)

• Another measure that provides a comparison between the

strength of various radioisotopes is the specific activity,
which quantifies the activity per unit mass

• This expression demonstrates that the SA is constant.

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 3.4 Radioactive Chains
• Radionuclides arise in several processes.
• They may be produced by the bombardment of
stable nuclei by charged particles as in an accelerator
or by neutrons as in a nuclear reactor.
• Or, they may come from other radionuclides, in
which the parent nuclide decays and produces a
daughter isotope.
• Still more generally, there may be a sequence of
decays between a series of radionuclides, called a
chain, leading eventually to a stable nucleus.

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 3.4.1 Buildup and Decay
• Let us examine the method of calculating
yields of some of these processes.
• The easiest case is a generation rate that is
constant in time.
• For example, suppose that neutrons absorbed
in cobalt-59 create cobalt-60 at a rate g. The
net rate of change with time of the number of
cobalt-60 atoms is

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• which may be written in the form of a differential
equation,

• If the initial number of radionuclides is zero, the

solution is

• The function rises linearly at the start and then

flattens out.
• At long times, the exponential term goes toward
zero, leaving N ≅ g/λ.
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• Figure 3.3 shows what happens if the generation is
stopped after 6 half-lives of the radionuclide.
• Prior to ceasing the production, the radionuclide is
being created by the generation process and
simultaneously depleted by the decay process.
• Once the generation mechanism is halted, only the
decay process described by Equation (3.6) remains.

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3.4.2 Compound Decay (from notes)
• In compound decay, both the parent (P) and
daughter (D) are radioactive.
• If the granddaughter (G) is stable, the process
may be depicted as

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• First, the dynamics of the decay of a parent
radionuclide to form a daughter radionuclide are
addressed.
• Let the initial number of atoms of a parent
radioisotope be NP0.
• At any time, the number of parent atoms as the result
of decay is simply found from Equation (3.6):

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• As some radionuclides decay by multiple decay modes,
let f be the fraction of parents that decay into a particular
daughter.
• Then the generation rate for the daughter is

• Substituting this generation rate into Equation (3.16)

yields

• The solution of this differential equation for the daughter

is

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Secular Equilibrium, Transient Equilibrium & No equilibrium
• There are three distinct cases depending on the relationship
between the parent and daughter half-lives:
1. Secular equilibrium occurs when the parent is very long-lived
compared to the daughter (λP<< λD). In this case, the activity
of the daughter rises up to equal the parent activity (AD ≅ AP)
within about 7 half-lives of the daughter; see Figure 3.4.
2. Transient equilibrium exists when the parent is long-lived
compared to the daughter (λP < λD). Consequently, the parent
and daughter activities are related by

3. In the general, no equilibrium case (λP > λD), the total activity
is eventually dominated by the daughter’s activity; see Figure
3.5.
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 3.4.3 Serial Decay Chains
• Natural radioactive isotopes such as uranium-238 (tH = 4.47
x109 y) and thorium-232 (tH = 1.4x1010 y) were produced
billions of years ago but still persist because of their long
half-lives. Their products form a long chain of radionuclides,
with the emission of alpha and beta particles. Those
comprising the U-238 series are:

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U-238 Series

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U-235 Series

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Neptunium Series

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Th-232 Series

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U-238 Series
• Note that radium-226 (1600 y) is fairly far down the chain.
• The final product is stable lead-206.
• Because of the very long half-life of uranium-238, the
generation rate of its daughters and their descendants are
practically constant.
• Let us write g ≅ N238 λ238 and apply the expression for the
number of atoms at long times to the radium-226, N226 ≅ g/λ226.
• Rearranging,

• the activities are approximately equal,

• And the condition is called secular equilibrium.
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• The first seven steps in the U-238 decay chain are
illustrated in Figure 3.6. The graphic conveys the
differences in alpha and beta decay.
• In the list of Equation (3.24) is the alpha particle
emitter polonium-210, half-life 138 days.
• It was the poison that caused the death in 2006 of the
former Russian KGB agent Litvinenko and is
suspected in the demise of Palestinian leader Yasser
Arafat.

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• Bateman (1910) developed a general equation
for serial decay chains such as

• Assuming that there are no progeny atoms

initially (i.e., Ni(0)=0 for i>1), the number of
radionuclides can be determined from

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 3.5 Measurement of Half-Life
• Finding the half-life of an isotope provides part of its
identification needed for beneficial use or for
protection against radiation hazard.
• Let us look at a method for measuring the half-life
of a radioactive substance.
• As in Figure 3.7, a detector that counts the number
of particles striking it is placed near the source of
radiation.
• The counting rate is computed from the number of
counts observed in a known short time interval.

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• It is proportional to the rates of emission of
particles or rays from the sample and thus to
the activity A of the source.
• The process is repeated after an elapsed time
for decay.
• The resulting values of activity may be
plotted on a semilog graph as in Figure 3.8,
and a straight line drawn through the
observed points.
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• From any pairs of points on the line, λ and tH=0.693/λ
can be calculated (see Exercise 3.10).
• The technique may be applied to mixtures of two
radioisotopes.
• After a long time has elapsed, only the isotope of
longer half-life will contribute counts.
• By extending its graph linearly back in time, one can
find the counts to be subtracted from the total to yield
the counts from the isotope of shorter half-life, a
process known as exponential peeling.

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• Activity plots cannot be used for a substance with
long half-life (e.g., strontium-90, tH=29.1 y).
• The change in activity is almost zero over the span of
time one is willing to devote to a measurement.
• However, if one knows the number of atoms present
in the sample and measures the activity, the decay
constant can be calculated from λ=A/N, from which tH
can be found.

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• The measurement of the activity of a
radioactive substance is complicated by the
presence of background radiation, which is
due to cosmic rays from outside the Earth or
from the decay of minerals in materials of
construction or in the ground.
• It is always necessary to measure the
background counts and subtract them from
those observed in the experiment.

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 3.6 Summary
• Many elements that are found in nature or are man-
made are radioactive, emitting α particles, β particles,
and γ-rays.
• The process is governed by an exponential relation,
such that half of a sample decays in a time called the
half-life tH.
• Values of tH range from fractions of a second to
billions of years among the hundreds of radioisotopes
known.
• Measurement of the activity, as the disintegration rate
of a sample, yields half-life values of importance in
radiation use and protection.
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