Chapter 30
Objectives
1. Determine the number of neutrons in a nuclide of known atomic
number and mass number.
2. Explain what is meant by an isotope of an element. State how
isotopes of an element differ and state the properties they have in
common.
3. Explain what is meant by the unified atomic mass unit. Calculate
the energy equivalent in MeV of an atomic mass unit.
4. Given a table of nuclear masses, calculate the binding energy of a
nucleus and the binding energy per nucleon.
5. Identify the three kinds of radiation emitted by radioactive
substances. State which radiations are deflected by electric and
magnetic fields.
6. Give the symbol used to represent each of the following: alpha
particle, beta- particle, beta+ particle, gamma ray.
7. Write a general equation to represent each of the following possible
radiation decays: alpha decay, beta+ decay, beta- decay, gamma
decay.
8. Distinguish between the parent nucleus and daughter
nucleus in a nuclear transmutation.
9. Calculate the disintegration energy (Q) for a given alpha decay.
10. List the four conservation laws which apply to radioactive
decays.
11. Write the equation which relates the half life of a substance to its
decay constant.
12. Write the equation for the law of radioactive decay. Explain
the meaning of each symbol in the equation. Solve problems related to
the law of radioactive decay.
Outline
30-1
30-2
30-3
30-4
30-5
30-6
30-7
30-8
Structure and Properties of the Nucleus
Binding Energy and Nuclear Forces
Radioactivity
Alpha Decay
Beta Decay
Gamma Decay
Conservation of Nucleon Number and Other Conservation Laws
Half-Life and Rate of Decay
30-9 Calculations Involving Decay Rates and Half-Life
30-10Decay Series
30-11Radioactive Dating
*30-12
Stability and Tunneling
30-13Detection of Radiation
Summary
Chapter 30 focuses on the nucleus, the center of the atom. The
structure and properties of the nucleus and its decay patterns are
discussed. Conservation laws and half-life are presented. Decay series
are defined. The chapter ends with a brief description of the detection
of radiation.
Major Concepts
By the end of the chapter, students should understand each of the
following and be able to demonstrate their understanding in problem
applications as well as in conceptual situations.
 Nuclei
 Nucleons
 Atomic number
 Mass number
 Isotopes
 Binding energy
 Stability
 Nuclear forces
 Strong force
 Weak force
 Radioactivity
 Alpha decay
 Beta decay
 Gamma decay
 Conservation laws
 Activity and decay rates
 Half-life
 Radioactive decay series
 Radioactive dating
 Methods for radiation detection
Lecture Note:
The nucleus consists of protons (mp = 1.6726x10-27kg, charge +e =
+1.60x10-19C) and neutrons (mn = 1.6749x10-27kg, charge zero),
collectively called nucleons.
The different species of nuclei are known as nuclides. They are
distinguished by:
 atomic number (Z), the number of protons.
 atomic mass number (A), the number of protons and neutrons.
 neutron number (N), the number of neutrons.
A
We write X where X is the chemical symbol for the element.
Z
Note: A = Z + N
Note: The mass given in periodic tables is a weighted average,
weighted according to natural abundances of the various
isotopes of an element.
Note: For a neutral atom, the number of electrons equals the
number of protons. Isotopes are nuclei of the same Z, but
different N. For example, 98.9% of the carbon found naturally is
12
13
C and 1.1% the isotope
C.
6
6
The size of a nucleus increases with atomic mass numbers, r = (1.2 x
1
10–15 m) A 3 is the radius (in m) of a nucleus with mass number A.
Thus the volume that is proportional to r3 is proportional to A, as if
nucleons were impenetrable spheres. Thus we see all nuclei have
about the same density.
Masses are often tabulated for neutral atoms (including electrons) in
unified atomic mass units (u). 1u = 1.6605x10-27kg = 931.5 MeV/c2
Protons and neutrons both have spin 1/2. The entire nucleus has a
nuclear spin I given by the vector sum of all proton and neutron spins.
h I(I  1)
The nucleus has angular momentum
2
Binding Energy and Nuclear Forces
Consider the helium atom. If we add the mass of its parts (two
protons, two neutrons, two electrons) the result is more than the mass
of the neutral atom. The missing mass has gone to binding energy by
E = mc2 of the protons and neutrons in this nucleus. The binding
energy of electrons to the nucleus is negligible. We will later make use
of the plot of binding energy per nucleon vs A.
The strong nuclear force holds the protons and neutrons together in
nucleus. This force is one of the four fundamental forces of nature,
and is stronger than electromagnetic or gravitational forces. It is,
however, a very short-ranged force, acting only at distances less than
10-15m. This force overcomes the Coulomb repulsion of protons within
the nucleus, but not between atoms.
Note: N  Z up to A = 30 or 40 for stable nuclei. Then N > Z until A
= 82, above which there are no known stable nuclei.
Note: The fourth fundamental force, the weak nuclear force, is
revealed in unstable nuclei undergoing some forms of radioactive
decay.
Radioactivity
In 1896 Becquerel discovered uranium emitted something that
exposed film wrapped to avoid exposure to light. Marie and Pierre
Curie soon discovered polonium and radium which were much more
radioactive.
We find there are three types of radioactivity.



Alpha ( α ) decay; these particles barely penetrate paper. The α
particle is the nucleus of a helium atom (two protons and two
neutrons bound together).
Beta (β) decay; these particles penetrate 3mm of aluminum.
They are high-speed electrons or positrons.
Gamma ( γ ) decay; this form of electromagnetic radiation can
penetrate several cm of lead. γ rays are energetic photons, not
material particles.
Alpha Decay
226
222
4
Ra 
Rn  He . The initial nucleus of
88
86
2
radium is the parent, while the decay product radon is the
daughter nucleus.
A typical decay is
The disintegration energy Q = (Mparent - Mdaughter - M)c2 is shared
between the α particle and daughter nucleus as KE.
241
Am02. This produces α
95
particles that ionize N2 and 02 molecules in air, allowing a current to
flow. Smoke absorbs the radiation -- less current flows, and this sets
off the alarm.
Smoke detectors contain 0.2mg of
Beta Decay
A typical decay is
14
14
C  N  e    where we write the electron
6
7
0
e as e  . This electron is not an orbital electron, but one created in
1
the nucleus as a neutron decays to a proton plus electron plus antineutrino (  ) that carries away energy. Ferrmi predicted this ghostly
particle that rarely interacts with matter.
The above was β- decay. A typical β+ decay involves the emission of a
positron or anti-matter electron.
A typical reaction is
19
19
Ne  F  e   . - decay occurs when N > Z; β+
10
9
when Z > N.
Electron capture is illustrated by
7
7
βe  e   Li  γ . There an orbital
4
3
electron is captured.
β decay is mediated by the weak nuclear force. Neutrinos () interact
only by the weak force and so can pass through light years of lead
without interaction.
Gamma Decay
A stereotypical reaction is
A x
A
X  X  γ . The nucleus does not change
Z
Z
identity, just loses energy.
 If the nucleus is originally in a meta stable state, it is an isomer.
 Often as a result of a radioactive decay, the nucleus is left in an
excited state. It will then emit one or more gamma rays.
 Internal conversion also emits γ rays when a nucleus interacts
with an orbital electron.
Conservation of Nucleon Number
In the decay processes illustrated above, note the number of nucleons
before equals the number of nucleons after the decay. Thus, A is
constant.
Half-Life and Rate of Decay
We cannot predict exactly when a given nucleus will decay, but we can
predict what percentage of a sample will decay over a period of time.
The number of decays ΔN = - λ NΔt where λ is the decay constant in
/s, t the time in s, and N the current number of nuclei.
Solving this equation we find N = N0e–t where N0 is the number of
nuclei at t = 0 and N the number at time t.
We define activity as
N
N  N   t
 N so

is the rate at which the
e
t
t  t 
activity is changing.
ΔN
decreases by a factor of two, the time required is the
Δt
half-life. We write
0.693
T1/2 =
.

When N or
Note: If N = N 0e– λt then by definition of half-life N =
N0
when t  T1 / 2
2
N0
1
1
1
 N 0 e T1 / 2 so  e  λT1 / 2 and ln    e λT1 / 2 and ln    λT1 / 2 and -.693
2
2
2
2
= - λT1 / 2 .
.693
.693
or  
Thus T1/2 =
.

T1 / 2
14
Example:
C has T1/2 = 5730 years. If a sample contains 1022 atoms
6
14
.693
of
C, find the activity λ 
= 3.83 x 10–12/s (converting T1/2 to
6
T1 / 2
seconds).
or
N
 N  3.83 x 1010 / s . This decreases by a factor of two every 5730
t
years.
The ratio of
14
12
C to
C decreases by a factor of two every 5730 years
6
6
12
C is stable. The ratio is normally 1.3 x 10-12, so we can date
6
λN 0
1
a sample to t = ln
by calculating its original activity and
λ ΔN / Δt
measuring the current activity.
also as
Decay Chains
Often 1 radioactive decay is followed by others. For example,
238
234
4
234
234
U
Th  He and
Th 
Pa  
92
90
2
90
97