Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 29.2): Calculate the mass density of
nuclear matter.
Consider a nucleus with one nucleon (A = 1).
r  r0 A 3  1.2 1015 m
1
The density is  
m
4 3
r
3
1.66 10  27 kg

 2.3  1017 kg/m 3 .
3
4
15
 1.2 10 m
3


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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 29.9): Find the radius and
volume of
107
the 43
Tc nucleus.
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  mc2 .
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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 29.14): (a) Find the binding energy
of the 16O nucleus.
m  mass of 8 H atoms  mass of 8 neutrons 
 mass of neutral 16 O atom
 81.0078250  1.0086649u  15.9949146u
 0.1370046u
EB  mc  127.8 MeV
2
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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/nucleo n.
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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
P
A 4
Z 2
D 
4
2
Daughter
nucleus
Alpha
particle
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 29.28): 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
P
A
Z 1
D  e  .
0
1
0
0
The reaction for beta-plus decay is
A
Z
P
A
Z 1
D  e  .
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
p n  e  
1
0
0
1
0
0
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Example (text problem 29.29): Calculate the maximum
40
kinetic energy of the beta particle when 19 K decays via
- decay.
40
The reaction is
19
K  Ca  e  .
40
20
0
1
0
0
The maximum kinetic energy of the electron is given by the
disintegration energy.
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Example continued:
The difference between the mass of the products and the
reactant is
m  M Ca  20me   me   M K  19me 
 M Ca  M K
 0.00140750 u.
The disintegration energy and the maximum KE of the
electron is
EB  m c  1.31 MeV.
2
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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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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
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 
208
81
*
Tl   .
208
81
When a nucleus has experienced alpha or beta decay, it
is not always left in the ground state.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§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   N 0 e

1

 t /
.
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   R0 e  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

  N0  

2
t
T1 / 2
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Example (text problem 29.37): 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   R0 e  t /
Solve for t:
 Rt  

t   ln 
 R0 
T1/2  Rt    5730 years   0.242 bq/gram 
  
  270 years

ln 
 ln 
ln 2  R0  
ln 2
  0.25bq/gra m 
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§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 Sv = 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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Example (text problem 29.45): An alpha particle produced in
radioactive alpha decay has a kinetic energy of typically
about 6 MeV. When an alpha particle passes through matter,
it makes ionizing collisions with molecules, giving up some of
its kinetic energy to supply the binding energy of the electron
that is removed. If a typical ionization energy for a molecule
in the body is around 20 eV, roughly how many molecules
can the alpha particle ionize before coming to rest?
total energy of alpha particle
number of ionization s 
energy lost per ionization
6 MeV

 5 105 ionization s
20 eV
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§29.6 Induced Nuclear Reactions
An unstable nucleus decays in a spontaneous nuclear
reaction. An induced nuclear reaction only takes place
because it is caused by a collision between a nucleus and
another particle.
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For example,
n  N C  p.
14
14
The above reaction is induced by the absorption of a
neutron. This process is called neutron activation.
A spontaneous nuclear reaction will always release energy.
In an induced nuclear reaction, some of the kinetic energy
of the reactants is converted into rest mass of the products.
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An exoergic reaction releases energy.
An endoergic reaction is one that requires energy to make
it proceed.
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Example (text problem 29.49): A certain nuclide absorbs a
neutron. It then emits an electron, and then breaks up into
two alpha particles. (a) Identify the original nucleus and the
two intermediate nuclei (after absorbing the neutron and
after emitting the electron).
The reactions are
1
0
n   X 1   X 2   X 3  e
e
f
 X 3     .
a
b
c
d
4
2
e
f
0
1
4
2
Since the number of nucleons is conserved e = 4 + 4 = 8.
Charge is conserved so f = 2 + 2 = 4. Identify X3 as Be.
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Example continued:
In the second step of the first reaction c= e+0 = 8 and
d = f - 1. This gives d = 3. Identify X2 as Li.
In the first step of the reaction a + 1 = c so a = 7 and
b + 0 = d = 3. Identify X1 as Li.
The nuclei involved are
X 1  Li; X 2  Li; X 3  Be.
7
3
8
3
8
4
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Example continued:
(b) Would any (anti)neutrino(s) be emitted? Explain.
In the second reaction (the beta decay of 8Li) will also
involve the emission of one antineutrino.
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§29.7 Fission
Fission is when a large nucleus splits into two smaller,
more tightly bound nuclei. This process releases energy.
Fission can either occur spontaneously when the nucleus
is very large, or it can be induced. When fission occurs by
the capture of slow moving neutrons, a chain reaction can
occur.
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§29.8 Fusion
The process of fusion takes two small nuclei to form a
larger nucleus.
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The process of fusion is the energy production mechanism
in cores of stars. Stars on the main sequence (like the Sun)
turn hydrogen into helium in their cores. This occurs by the
proton-proton cycle and by the CNO (carbon) cycle.
Overall, this process take four protons and turn them into a
helium nucleus and energy.
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Example (text problem 29.57): Consider the fusion reaction
between a proton and a deuteron (shown below). (a)
Identify the reaction product X.
1
1
H H  X
2
1
The product must have 3 nucleons and a charge (atomic
number) of 2. The element is He.
3
2
He
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Example continued:
(b) The binding energy of the deuteron is about 1.1 MeV
per nucleon and the binding energy of “X” is about 2.6
MeV per nucleon. Approximately how much energy (in
MeV) is released in this fusion reaction?
The binding energies are: 0 MeV for 1H; 2.2 MeV for
2H; and 7.8 MeV for 3He.
Energy released = difference in binding energy between
the products and reactants = 7.8 MeV – 2.2 MeV = 5.6
MeV.
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Example continued:
(c) Why is this reaction unlikely to occur at room
temperature?
At room temperature the reactants will not have
enough kinetic energy to overcome the Coulomb
repulsion between them to get close enough to fuse.
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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
•Spontaneous vs. Induced Nuclear Reactions
•Fission vs. Fusion
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