My
Chapter 29
Lecture
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Chapter 29: Nuclear Physics
•The Nucleus
•Binding Energy
•Radioactivity
•Half-life
•Biological Effects of Radiation
•Induced Nuclear Reactions
•Fission and Fusion
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§29.1 Nuclear Structure
The atomic nucleus is composed of neutrons and protons.
These particles are called nucleons.
The atom’s atomic number (Z) gives the number of protons
in its nucleus. It is the atomic number that determines an
atom’s identity.
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The nucleon number or mass number is A = Z+N, where
N is the number of neutrons.
Masses of atoms are sometimes give in terms of atomic
mass units. 1u = 1.66053910-27 kg.
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Atoms of the same element with differing numbers of
neutrons are known as isotopes.
The mass quoted for an atom in the periodic table is a
weighted average over all of the natural isotopes of that
element. The weight factors are determined by using the
relative abundance on Earth of each isotope.
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m A
For an atomic nucleus
V  A.
This implies the density of an atomic nucleus is independent
of A.
4 3
V  r  A
3
1
rA3
As an equality
r  r0 A 3
1
where r0 = 1.210-15 m = 1.2 fm
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Example (text problem 29.2): Calculate the mass density of
nuclear matter.
Consider a nucleus with one nucleon (A = 1).
r  r0 A  1.2 1015 m
1
3
The density is  
m
4 3
r
3
1.6610 27 kg

 2.3 1017 kg/m3 .
3
4
15
 1.2 10 m
3


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Example (text problem 29.9): Find the radius and volume
of the nucleus.
107
43
Tc
The radius is


r  r0 A1/3  1.2 1015 m 107  5.701015 m.
1/3
4 3
The volume is V  r  7.7  10  43 m 3 .
3
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§29.2 Binding Energy
A nucleus is held together by the strong nuclear force.
This force only acts over distances of a few fermis.
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The binding energy (EB) of a nucleus is the energy that
must be supplied to separate it into individual protons and
neutrons.
EB = Total energy of Z protons and N neutrons – total
energy of nucleus.
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Total energy of Z protons and N neutrons = (mass of Z
protons and N neutrons)c2.
Total energy of nucleus = (mass of nucleus)c2.
These can be used to define the mass defect m =
(mass of Z protons and N neutrons)  (mass of nucleus)
so that
EB  mc .
2
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Nucleons also obey the Pauli Exclusion Principle such that
only two protons (neutrons) can occupy each proton
(neutron) energy level.
Like an atom, a nucleus can be put into an excited state if it
absorbs a photon of the correct energy. The nucleus can
then emit a photon to go to a lower energy state.
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Example (text problem 29.13): (a) Find the binding energy
of the 16O nucleus.
m  mass of 8 H atoms mass of 8 neutrons
 mass of neutral16 O atom
 81.0078250 1.0086649u  15.9949146u
 0.1370046u
EB  mc2  127.8 MeV
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Example continued:
(b) What is the average binding energy per nucleon?
EB
Binding energy per nucleon 
number of nucleons
 7.986 MeV/nucleon.
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§29.3 Radioactivity
Some nuclei are unstable and decay. These nuclei are
radioactive. A nucleus can emit an alpha ray, beta ray, or
a gamma ray during its decay.
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During nuclear reactions:
1. Charge is conserved.
2. The total number of nucleons is constant.
3. Energy is conserved.
Define: disintegration energy = binding energy of
radioactive nucleus  total binding energy of products.
This is the rest mass energy that can be converted into
other forms of energy.
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Alpha rays have been identified as helium nuclei.
The reaction for alpha decay is
A
Z
Parent
nucleus
A4
Z 2
P
D 
4
2
Daughter
nucleus
Alpha
particle
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Example (text problem 29.29): Show that the spontaneous
alpha decay of 19O is not possible.
The reaction is
19
8
O C  .
15
6
4
2
The mass of the products (including electrons) is
19.01320250u.
The mass of 19O is 19.0035787u.
The mass of the products is larger than the reactant,
so this reaction cannot occur spontaneously.
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Beta rays have been identified as either electrons (-) or
positrons (+).
The reaction for beta-minus decay is
A
Z
D e  .
P
A
Z 1
0
1
0
0
The reaction for beta-plus decay is
A
Z
P
D e  .
A
Z 1
0
1
0
0
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The neutrino and antineutrino have no charge and are
nearly massless. They do not readily interact with matter.
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During beta-minus decays, a neutron is converted into a
proton.
1
0
n p e 
1
1
0
1
0
0
During beta-plus decays, a proton is converted into a
neutron.
1
1
0
0
1
0
1
0
p n e 
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During inverse beta decay (electron capture) a proton in
a nucleus captures an electron. The reaction is
0
1
e p n   .
1
1
1
0
0
0
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Gamma rays were determined to be high energy photons.
A gamma ray will be emitted when a nucleus is an excited
state when making a transition to a lower energy level. For
example,
Tl  Tl  .
208
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*
208
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When a nucleus has experienced alpha or beta decay, it
is not always left in the ground state.
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§29.4 Radioactive Decay Rates
and Half-Lives
The half-life of a sample of unstable nuclei is the time it
takes for one-half of the sample to decay. The decay
process is quantum mechanical and is based on probability.
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Each radioactive nucleus has a probability per second that it
will decay, called the decay constant.
probabilit y of decay
decay constant   
unit time
The number of nuclei that decay in a short time interval is
N   Nt.
There are statistical fluctuations in the number of decays
that occur. These fluctuations are of order N .
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The decay rate or activity is the number of radioactive
decays that occur in a sample per unit time.
number of decays
N
R

 N
unit time
t
The unit of activity is the bequerel. 1 Bq = 1 decay/sec.
Another common unit is the curie. 1 Ci = 3.71010 Bq.
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The number of nuclei remaining in a sample having N0
nuclei at t = 0 is
N t   N0et / .

1

is the mean lifetime of a nucleus.
Note: the above expression for N(t) is a way to determine
the number of remaining nuclei only. It does not tell us
which nuclei have decayed.
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The activity at time t is
Rt   R0e
t / 
where R0 is the activity at t = 0.
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It is common to write the expressions for N(t) and R(t) in
terms of half-life (T1/2).
T1/ 2   ln 2

N t   N 0  2

t
T1 / 2
1


N

0 

2
t
T1 / 2
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Example (text problem 29.35): Some bones discovered in a
crypt in Guatemala are carbon dated. The 14C activity is
measured to be 0.242 Bq per gram of carbon. Approximately
how old are the bones?
Rt   R0et /
Solve for t:
 Rt  

t   ln
 R0 
T1/2  Rt    5730years  0.242 bq/gram 
  
  270years

ln
 ln
ln 2  R0  
ln 2
  0.25bq/gram 
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§29.5 Biological Effects of
Radiation
The absorbed dose of ionizing radiation is the amount of
radiation energy absorbed per unit mass of tissue. Ionizing
radiation is radiation with enough energy to ionize an atom
or molecule.
The SI unit of absorbed dose is the Gray. 1 Gy = 1 J/kg.
Another common unit is the rad (radiation absorbed dose).
1 rad = 0.01 Gy.
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Different radiation causes different amounts of biological
damage. The biologically equivalent dose measures the
amount of damage caused by radiation exposure.
Equivalent dose (in sieverts) = absorbed dose (in grays)* QF.
Equivalent dose (in rem) = absorbed dose (in rads)* QF.
QF is a quality factor that
is a relative measure of
biological damage (200
keV x-rays have QF = 1).
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The sievert is the SI unit of biologically equivalent dose.
1 Sy = 100 rem.
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Alpha, beta, and gamma radiation penetrates to different
depths in biological materials.
•Alpha rays are stopped by a few cm of air or about 0.02
mm of aluminum.
•Beta-minus can penetrate a few cm into biological tissue.
•Gamma ray absorption is based on probability so they
can penetrate to varying depths.
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Summary
•The Nucleus (atomic & mass numbers)
•Binding Energy
•Radioactive Nuclei
•Alpha, Beta, and Gamma Radiation
•Half-life and Activity
•Absorbed Dose
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