Nuclear Physics
Lecture 6
1
In 1896, Becquerel accidentally discovered that uranium salt crystals
emit an invisible radiation that can darken a photographic plate even if
the plate is covered to exclude light. After several such observations
under controlled conditions, he concluded that the radiation emitted by
the crystals was of a new type, one requiring no external stimulation.
This spontaneous emission of radiation was soon called
radioactivity. Subsequent experiments by other scientists showed that
other substances were also radioactive. The most significant
investigations of this type were conducted by Marie and Pierre Curie.
After several years of careful and laborious chemical separation
processes on tons of pitchblende, a radioactive ore, the Curies reported
the discovery of two previously unknown elements, both of which were
radioactive. These were named polonium and radium. Subsequent
experiments, including Rutherford’s famous work on alpha-particle
scattering, suggested that radioactivity was the result of the decay, or
disintegration, of unstable nuclei. Three types of radiation can be emitted
by a radioactive substance: alpha (α) particles, 24He, in which the emitted
particles are nuclei; beta (β) particles, in which the emitted particles are
either electrons or positrons; and gamma (γ) rays, in which the emitted
“rays”, are high-energy photons. A positron is a particle similar to the
electron in all respects, except that it has a charge of +e. (The positron is
said to be the antiparticle of the electron.) The symbol e+ is used to
designate an electron, and e+ designates a positron.
It’s possible to distinguish these three forms of radiation by using
the scheme described in Figure 1.6. The radiation from a radioactive
sample is directed into a region with a magnetic field, and the beam splits
into three components, two bending in opposite directions and the third
2
not changing direction. From this simple observation it can be concluded
that the radiation of the undeflected beam (the gamma ray) carries no
charge, the component deflected upward contains positively charged
particles (alpha particles), and the component deflected downward
contains negatively charged particles (e-). If the beam includes a positron
(e+), it is deflected upward.
The three types of radiation have quite different penetrating powers.
Alpha particles barely penetrate a sheet of paper, beta particles can
penetrate a few millimeters of aluminum, and gamma rays can penetrate
several centimeters of lead.
Figure 6.1 The radiation from a radioactive source, such as radium, can be separated into
three components using a magnetic field to deflect the charged particles. The detector array at
the right records the events. The gamma ray isn’t deflected by the magnetic field.
3
Observation has shown that if a radioactive sample contains N radioactive
nuclei at some instant, then the number of nuclei, N, that decay in a small
time interval t is proportional to N; mathematically:
6.1
6.2
Where a constant is called the decay constant. The negative sign signifies that
N decreases with time; that is, N is negative. The value of λ for any isotope
determines the rate at which that isotope will decay. The decay rate, or
activity R, of a sample is defined as the number of decays per second.
From Equation 6.2, we see that the decay rate is
Activity
R
N
t
6.3
For a random process, the activity is proportional to N:
N
  N
t
This gives (by integration)

 is the decay constant
N  N 0e   t
Where N0 is the number of nuclei at t = 0.
4
6.4
Another parameter that is useful for characterizing radioactive decay is the
half-life T1/2. The half-life of a radioactive substance is the time it takes
for half of a given number of radioactive nuclei to decay. Using the
concept of half-life, it can be shown that Equation 6.4 can also be written as
 1 


 2
N  N0
n
Where n is number of half-lives. The number n can take any non-negative
value and need not be an integer. From the definition, it follows that n is
related to time t and the half-life T1/2 by
t
n
T12
Setting N = N0/2 and t = T1/2 in Equation 6.4 gives
N0
 N 0 e   ( T1/ 2 )
2
1
 e   ( T1 / 2 )
2
Writing this in the form (e λT1/2) = 2 and taking the
natural logarithm of both sides, we get
6.6
5
6.5
In general, the following three equations can be applied to radioactivity:
Nuclei Remaining Mass Remaining
Activity R
1
R  R0  
2
n
1
N  N0  
2
t / T1 / 2
1
m  m0  
2
n
The unit of activity R is the curie (Ci), defined as
10
10
1 Ci (curie) = 3.7 x 10 decay/s = 3.7 x 10 disntegration/second
This unit was selected as the original activity unit because it is the
approximate activity of 1 g of radium. The SI unit of activity is the Becquerel
(Bq):
1 Bq = 1 decay/s = 1 disntegraion/second
[6.7]
Therefore, 1 Ci = 3.7 x 1010 Bq (activity of 1 g radium). The most
commonly used units of activity are the millcurie (10-3 Ci) and the microcurie
(10-6 Ci).
6
Example 6.1: A sample of iodine-131 has an initial activity
of 5 mCi. The half-life of I-131 is 8 days. What is the activity
of the sample 32 days later?
Example 6.2: The half-life of the radioactive nucleus
88Ra226 is 1.6x103 yr. If a sample initially contains 3.00 x
1016 such nuclei, determine (a) the initial activity in
curies, (b) the number of radium nuclei remaining after
4.8 x 103 yr, and (c) the activity at this later time.
7
Example 6.3:
Find (a) the number of remaining radium nuclei after
3.2 x 103 yr and (b) the activity at this time.
Answer (a) 7.5 x1015 nuclei (b) 2.8 μCi
8
As stated in the previous section, radioactive nuclei decay spontaneously via
alpha, beta, and gamma decay. As we’ll see in this section, these processes
are very different from each other.
6.2.1 Alpha Decay
If a nucleus emits an alpha particle (42He), it loses two protons and two
neutrons. Therefore, the neutron number N of a single nucleus decreases by 2,
Z decreases by 2, and A decreases by 4. The decay can be written
symbolically as
A
Z
X  ZA42Y  24  energy
Where X is called the parent nucleus and Y is known as the daughter nucleus.
As examples, 238U and 226Ra are both alpha emitters and decay according
to the schemes
And
226
88
4
Ra  222
Rn

86
2  energy
In order for alpha emission to occur, the mass of the parent must be greater
than the combined mass of the daughter and the alpha particle. In the decay
process, this excess mass is converted into energy of other forms and appears
in the form of kinetic energy in the daughter nucleus and the alpha particle.
Most of the kinetic energy is carried away by the alpha particle because it is
much less massive than the daughter nucleus. This can be understood by first
9
noting that a particle’s kinetic energy and momentum p are related as follows:
Because momentum is conserved, the two particles emitted in the decay of a
nucleus at rest must have equal, but oppositely directed, momenta. As a
result, the lighter particle, with the smaller mass in the denominator, has more
kinetic energy than the more massive particle.
Energy released (KE of )
Example 6.4: Calculate the energy released when 84Be
splits into two alpha particles. Beryllium-8 has an
atomic mass of 8.005305 u and that 42He of to be
4.002602 u.
10
6.2.2 Beta Decay
When a radioactive nucleus undergoes beta decay, the daughter nucleus has
the same number of nucleons as the parent nucleus, but the atomic number is
changed by 1.
a) Beta-minus Decay
Beta-minus β- decay results when a neutron decays into a proton and an
electron. Thus, the Z-number increases by one.
A
Z
X  Z A1Y  01   energy
X is parent atom and Y is daughter atom
The energy is carried away primarily by the K.E. of the electron.
b) Beta-plus Decay
Beta-plus β+ decay results when a proton decays into a neutron and a
positron. Thus, the Z-number decreases by one.
A
Z
X  Z A1Y  01   energy
X is parent atom and Y is daughter atom
The energy is carried away primarily by the K.E. of the positron.
Energy released, as KE of electron
- decay
11
Example 6.5: Calculate the maximum energy liberated in
the beta decay of radioactive 146C to 147N (mC =
14.003242 u and mN = 14.003 074 u)
c) Neutrino, 
From Example 6.5, we see that the energy released in the beta decay of 14C is
approximately 0.16 MeV. As with alpha decay, we expect the electron to
carry away virtually all of this energy as kinetic energy because, apparently, it
is the lightest particle produced in the decay. However, only a small number
of electrons have this maximum kinetic energy, most of the electrons emitted
have kinetic energies lower than that predicted value. If the daughter nucleus
and the electron aren’t carrying away this liberated energy, then where has
the energy gone? As an additional complication, further analysis of beta
decay shows that the principles of conservation of both angular momentum
and linear momentum appear to have been violated!
In 1930 Pauli proposed that a third particle must be present to carry away the
“missing” energy and to conserve momentum. Later, Enrico Fermi developed
a complete theory of beta decay and named this particle the neutrino (“little
neutral one”) because it had to be electrically neutral and have little or no
mass. Although it eluded detection for many years, the neutrino (υ) was
12
finally detected experimentally in 1956. The neutrino has the following
properties:
• Zero electric charge
• A mass much smaller than that of the electron, but probably not zero.
(Recent experiments suggest that the neutrino definitely has mass, but the
value is uncertain—perhaps less than 1 eV/c 2.
 A spin of 1/2
• Very weak interaction with matter, making it difficult to detect
With the introduction of the neutrino, we can now represent the beta decay
process of Equation 29.13 in its correct form:
The bar in the symbol ῡ indicates an antineutrino. To explain what an
antineutrino is, we first consider the following decay:
13
6.2.3 Gamma Decay
Very often a nucleus that undergoes radioactive decay is left in an excited
energy state. The nucleus can then undergo a second decay to a lower energy
state— perhaps even to the ground state—by emitting one or more highenergy photons. The process is similar to the emission of light by an atom.
An atom emits radiation to release some extra energy when an electron
“jumps” from a state of high energy to a state of lower energy. Likewise, the
nucleus uses essentially the same method to release any extra energy it may
have following decay or some other nuclear event. In nuclear de-excitation,
the “jumps” that release energy are made by protons or neutrons in the
nucleus as they move from a higher energy level to a lower level. The
photons emitted in the process are called gamma rays, which have very high
energy relative to the energy of visible light.
A nucleus may reach an excited state as the result of a violent collision with
another particle. However, it’s more common for a nucleus to be in an excited
state as a result of alpha or beta decay. The following sequence of events
typifies the gamma decay processes:
The excited carbon nucleus then decays to the ground state by emitting a
gamma ray, as indicated by Equations. Note that gamma emission doesn’t
result in any change in either Z or A.
14
As we have indicated, upon decay, a radioactive parent nucleus
produces what is called a daughter nucleus. The daughter nucleus
can either be stable or radioactive. If it is radioactive, then it
decays into a granddaughter nucleus and so on. Thus, each
radioactive parent nucleus initiates a series of decays, with each
decay-product having its own characteristic decay constant and,
therefore, a different half-life. In general, the mean life of the
parent nucleus is much longer than that of any other member of
the decay chain, and this will be important for the observations
that follow.
Consider a radioactive sample of material where the parent
nucleus has a very long life time, and therefore the number of
parent nuclei barely changes during some small time interval. Let
us suppose that the daughter, granddaughter, etc., decay
comparatively fast. After a certain lapse in time, a situation may
develop where the number of nuclei of any member of the decay
chain stops changing. In such a case, one says that radioactive
equilibrium has set in. To see when this can occur, let us denote
by N1, N2 , N3,... the number of nuclei of species 1,2,3,... in the
series, at some specified time, and by λ1, λ2, λ3,..., respectively,
the decay constants for these members of the decay chain. The
equations governing the time-evolution of the populations N1, N2,
N3,... • can be deduced from the contributions to the change in any
species, as follows. The daughter nuclei are produced at a rate of
λ1N1 due to the decay of the parent nuclei, and they in turn decay
15
at a rate of λ2N2. The difference between the two gives the net rate
of change of the daughter nuclei. For any nucleus in the chain,
there will be a similar increase in population from the feed-down
and a decrease from decay, except for the parent nucleus, for
which there is no feed-down possible. Thus, for the change in the
number of parent, daughter, granddaughter nuclei, etc., in a time
interval Δt, we can write
16
17
Radioactive nuclei are generally classified into two groups: (1)
unstable nuclei found in nature, which give rise to what is called
natural radioactivity, and (2) nuclei produced in the laboratory
through nuclear reactions, which exhibit artificial radioactivity.
Three series of naturally occurring radioactive nuclei exist (Table
29.2). Each starts with a specific long-lived radioactive isotope
with half-life exceeding that of any of its descendants. The fourth
series in Table 29.2 begins with 237Np, a transuranic element (an
element having an atomic number greater than that of uranium)
not found in nature. This element has a half-life of “only” 2.14
x 106 yr. The two uranium series are somewhat more complex
than the 232Th series (Fig. 29.12). Also, there are several other
naturally occurring radioactive isotopes, such as 14C and 40K, that
are not part of either decay series. Natural radioactivity constantly
supplies our environment with radioactive elements that would
otherwise have disappeared long ago. For example, because the
Solar System is about 5 x 109 years old, the supply of 226Ra (with
a half-life of only 1600 yr.) would have been depleted by
radioactive decay long ago were it not for the decay series that
starts with 238U, with a half-life of 4.47 x 109 yr.
18
19
6.5.1- Radioactive dating:
a) Carbon dating
Based on the reaction:
14
C 14 N + 
T1/2 = 5730 years
So the fraction of 14C nuclei remaining after one half-life is high
enough to accurately measure changes in the sample’s activity.
1. The beta
decay of 14C is commonly used to date organic
samples. Cosmic rays (high-energy particles from outer
space) in the upper atmosphere cause nuclear reactions that
create 14C from 14N.
2. In fact, the ratio of 14C to 12C (by numbers of nuclei) in the
carbon dioxide molecules of our atmosphere has a constant
value of about 1.3 x 10-12, as determined by measuring
carbon ratios in tree rings. All living organisms have the
same ratio of 14C to 12C because they continuously exchange
carbon dioxide with their surroundings.
3. When an organism dies, however, it no longer absorbs 14C
from the atmosphere, so the ratio of 14C to 12C decreases as
the result of the beta decay of 14C. It’s therefore possible to
determine the age of a material by measuring its activity per
unit mass as a result of the decay of 14C. Through carbon
dating, samples of wood, charcoal, bone, and shell have
been identified as having lived from 1 000 to 25 000 years
20
ago. This knowledge has helped researchers reconstruct the
history of living organism—including human—during that
time span.
21
b) Dating ancient rocks
Age equation:
H.W
Problem: A 50g sample of carbon is taken from the pelvis bone of a
skeleton and is found to have a carbon-14 decay rate of 200 decays/min. It is
known that carbon from a living organism has a decay rate of 15 decays/min.
g and that 14C has a half-life of 5730 yr. Find the age of the skeleton.
22
6.5.2-Smoke Detectors:
Smoke detectors are frequently
used in homes and industry for fire protection. Most of
the common ones are the ionization-type that uses
radioactive materials. (See Fig. 29.9.) A smoke detector
consists of an ionization chamber, a sensitive current
detector, and an alarm. A weak radioactive source ionizes
the air in the chamber of the detector, which creates
charged particles. A voltage is maintained between the
plates inside the chamber, setting up a small but
detectable current in the external circuit. As long as the
current is maintained, the alarm is deactivated. However,
if smoke drifts into the chamber, the ions become
attached to the smoke particles. These heavier particles
do not drift as readily as do the lighter ion, which causes
a decrease in the detector current. The external circuit
senses this decrease in current and sets off the alarm.
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6.5.3-Radon Detection:
Radioactivity can also affect
our daily lives in harmful ways. Soon after the discovery
of radium by the Curies, it was found that the air in
contact with radium compounds becomes radioactive. It
was then shown that this radioactivity came from the
radium itself, and the product was therefore called
“radium emanation.” Rutherford and Soddy succeeded in
condensing this “emanation,” confirming that it was a
real substance: the inert, gaseous element now called
radon (Rn). Later, it was discovered that the air in
uranium mines is radioactive because of the presence of
radon gas. The mines must therefore be well ventilated to
help protect the miners. Finally, the fear of radon
pollution has moved from uranium mines into our own
homes. Because certain types of rock, soil, brick, and
concrete contain small quantities of radium, some of the
resulting radon gas finds its way into our homes and other
buildings. The most serious problems arise from leakage
of radon from the ground into the structure. One practical
remedy is to exhaust the air through a pipe just above the
underlying soil or gravel directly to the outdoors by
means of a small fan or blower.
24
6.5.4
)1(
25
(2)
(3)
(4)
26
27
(5)
(6)
(7)
(8)
28