Chapter 44
Nuclear Structure
Milestones in the Development
of Nuclear Physics

1896: the birth of nuclear physics


Becquerel discovered radioactivity in uranium
compounds
Rutherford showed the radiation had three
types:



alpha (He nuclei)
beta (electrons)
gamma (high-energy photons)
More Milestones

1911 Rutherford, Geiger and Marsden
performed scattering experiments



Established that the nucleus could be treated as a
point mass and a point charge
Most of the atomic mass was contained in the
nucleus
Nuclear force was a new type of force
Some Properties of Nuclei

All nuclei are composed of protons and
neutrons


The atomic number Z equals the number of
protons in the nucleus


Exception is ordinary hydrogen with a single
proton
Sometimes called the charge number
The neutron number N is the number of
neutrons in the nucleus
More Properties of Nuclei

The mass number A is the number of
nucleons in the nucleus



A=Z+N
Nucleon is a generic term used to refer to either a
proton or a neutron
The mass number is not the same as the mass
Symbolism
A
Z


X
X is the chemical symbol of the element
Example:
27
13





Al
Mass number is 27
Atomic number is 13
Contains 13 protons
Contains 14 (27 – 13) neutrons
The Z may be omitted since the element can be
used to determine Z
More Properties


The nuclei of all atoms of a particular element
must contain the same number of protons
They may contain varying numbers of
neutrons



Isotopes of an element have the same Z but
differing N and A values
The natural abundance of isotopes can vary
Isotope example:
11
6
13
14
C, 12
C
,
C
,
6
6
6C
Charge



The proton has a single positive charge, e
The electron has a single negative charge,
-e
The neutron has no charge



Made it difficult to detect in early experiments
Easy to detect with modern devices
e = 1.602 177 33 x 10-19 C
Mass

It is convenient to use atomic mass units, u,
to express masses



1 u = 1.660 539 x 10-27 kg
Based on definition that the mass of one atom of
12C is exactly 12 u
Mass can also be expressed in MeV/c2


From ER = mc2
1 u = 931.494 MeV/c2

Includes conversion 1 eV = 1.602 177 x 10-19 J
Some Masses in Various Units
The Size of the Nucleus



First investigated by Rutherford in scattering experiments
He found an expression for how close an alpha particle
moving toward the nucleus can come before being turned
around by the Coulomb force
From conservation of energy, the kinetic energy of the particle
must be completely converted to potential energy
Active Figure 44.1



Use the active figure to
adjust the atomic
number of the target
nucleus
Also adjust the kinetic
energy of the alpha
particle
Observe the approach
of the alpha particle
PLAY
ACTIVE FIGURE
Size of the Nucleus, cont.

d is called the distance of closest approach


d gives an upper limit for the size of the nucleus
Rutherford determined that
d  4ke

Ze 2
mv 2
For gold, he found d = 3.2 x 10-14 m
More About Size

Rutherford concluded that the positive charge
of the atom was concentrated in a sphere
whose radius was no larger than about 10-14
m


He called this sphere the nucleus
These small lengths are often expressed in
femtometers (fm) where 1 fm = 10-15 m

Also called a fermi
Size of Nucleus, Final

Since the time of Rutherford, many other
experiments have concluded the following:


Most nuclei are approximately spherical
Average radius is
r  ro A1 3


ro = 1.2 x 10-15 m
A is the mass number
Density of Nuclei



The volume of the nucleus
(assumed to be spherical) is
directly proportional to the
total number of nucleons
This suggests that all nuclei
have nearly the same
density
Nucleons combine to form a
nucleus as though they
were tightly packed spheres
Nuclear Stability

There are very large repulsive electrostatic
forces between protons


These forces should cause the nucleus to fly apart
The nuclei are stable because of the presence
of another, short-range force, called the nuclear
force


This is an attractive force that acts between all nuclear
particles
The nuclear attractive force is stronger than the
Coulomb repulsive force at the short ranges within the
nucleus
Features of the Nuclear Force


Attractive force that acts between all nuclear
particles
Very short range


It falls to zero when the separation between
particles exceeds about several fermis
Independent of charge


The nuclear force on p-p, p-n, n-n are all the
same
Does not affect electrons
Nuclear Stability, cont.


Light nuclei are most stable
if N = Z
Heavy nuclei are most
stable when N > Z



Above about Z = 20
As the number of protons
increases, the Coulomb
force increases and so
more neutrons are needed
to keep the nucleus stable
No nuclei are stable when Z
> 83
Binding Energy

The total energy of the bound system (the
nucleus) is less than the combined energy of
the separated nucleons

This difference in energy is called the binding
energy of the nucleus

It can be thought of as the amount of energy you need
to add to the nucleus to break it apart into its
components
Binding Energy, cont.

The binding energy can be calculated from
conservation of energy and the Einstein
mass-energy equivalence principle:
Eb (MeV) = [ZM(H) + Nmn – M (AZX)] x
931.494 MeV/u




M(H) is the atomic mass of the neutral hydrogen atom
M (AZX) represents the atomic mass of an atom of the
isotope (AZX)
Mn is the mass of the neutron
The masses are expressed in atomic mass units
Binding Energy per Nucleon
Notes from the Graph

The curve peaks in the vicinity of A = 60


Nuclei with mass numbers greater than or less
than 60 are not as strongly bound as those near
the middle of the periodic table
There is a decrease in binding energy per
nucleon for A > 60

Energy is released when a heavy nucleus splits or
fissions

Energy is released since each product nucleus are
more tightly bound to one another than are the
nucleons of the original nucleus
More Notes from the Graph

The binding energy is about 8 MeV per
nucleon for nuclei with A > 50



This suggests that the nuclear force
saturates
A particular nucleon can interact with only a
limited number of other nucleons
has the largest binding energy per
nucleon
62
28
Ni
Nuclear Models


Two models of the nucleus will be discussed
Liquid-drop model


Provides good agreement with observed nuclear
binding energies
Shell model

Predicts the existence of stable nuclei
Liquid-Drop Model




Nucleons are treated like molecules in a drop
of liquid
The nucleons interact strongly with one
another
They undergo frequent collisions as they
jiggle around in the nucleus
The jiggling motion is analogous to the
thermally agitated motion of molecules in a
drop of liquid
Liquid-Drop Model – Effects
Influencing Binding Energy, 1

The volume effect



The nuclear force on a given nucleon is due only
to a few nearest neighbors and not to all the other
nucleons in the nucleus
The total binding energy is proportional to A and
therefore proportional to the nuclear volume
This contribution to the binding energy of the
entire nucleus is C1A

C1 is an adjustable constant
Liquid-Drop Model –
Binding Energy Effect 2

The surface effect




Nucleons on the surface have fewer neighbors
than those in the interior
Surface nucleons reduce the binding energy by
an amount proportional to their number
The number of nucleons is proportional to the
surface area
The surface term can be expressed as –C2A2/3

C2 is a second adjustable constant
Liquid-Drop Model –
Binding Energy Effect 3

The Coulomb repulsion effect



Each proton repels every other proton in the
nucleus
The potential energy associated with the Coulomb
force is proportional to the number of protons, Z
The reduction in the binding energy due to the
Coulomb effect is –C3Z(Z - 1)/A1/3

C3 is another adjustable constant
Liquid-Drop Model –
Binding Energy Effect 4

The symmetry effect



Any large symmetry between N and Z for light nuclei
reduces the binding energy
For larger A, the value of N for stable nuclei is larger
The effect can be described by a binding energy term in
the form –C4(N - Z)2 / A
 For small A, any large asymmetry between N and Z makes
the term large
 For large A, the A in the denominator reduces the value of
the term so that it has little effect on the overall binding
energy
 C4 is another adjustable constant
Liquid-Drop Model – Binding
Energy Effect Summary

Putting these terms together results in the
semiempirical binding-energy formula:
Eb  C1A  C2 A

2 3
 C3
Z  Z  1
 C4
13
A
N
Z
A
2
The four constants are adjusted to fit the
theoretical expression to the experimental
data

For A 15, C1 = 15.7 MeV; C2 = 17.8 MeV; C3 =
0.71 MeV; and C4 = 23.6 MeV
Liquid Drop Model, Final


The equation fits the known nuclear mass
values very well
Does not account for some of the finer details
of nuclear structure


Stability
Angular momentum
Features of Binding Energy

When binding energies are studied closely it
is found that:

Most stable nuclei have an even value of A


Only 8 stable nuclei have odd values for both A and Z
There is a difference between the binding energy
per nucleon given by the semiempirical formula
and experiments
Features of Binding Energy –
Magic Numbers



The disagreement between the semiempirical
formula and experiments is plotted
Peaks appear in the graph
These peaks are at the magic numbers of
Z or N = 2, 8, 20, 28, 52, 82
Features of Binding Energy,
cont.

Studies of nuclear radii show deviations from
the expected values


Graphs of the data show peaks at values of N
equal to the magic numbers
A group of isotones is a collection of nuclei
having the same value of N and different
values of Z

When the number of stable isotones is graphed
as a function of N, there are peaks at the magic
numbers
Features of Binding Energy,
final


Several other nuclear measurements show
anomalous behavior at the magic numbers
The peaks are reminiscent of the peaks in
graphs of ionization energy of atoms and lead
to the shell model of the nucleus
Maria Goeppert-Mayer




1906 – 1972
German scientist
Best known for her
development of the shell
model of the nucleus
Shared the Nobel Prize in
1963

Shared with Hans Jensen
who simultaneously
developed a similar model
Shell Model


The shell model is also called the independentparticle model
In this model, each nucleon is assumed to exist in a
shell



Similar to atomic shells for electrons
The nucleons exist in quantized energy states
There are few collisions between nucleons
Shell Model, cont.

Each state can contain
only two protons or two
neutrons



They must have opposite
spins
They have spins of ½, so
the exclusion principle
applies
The set of allowed states
for the protons differs from
the set of allowed states
for the neutrons
Shell Model, final


Proton energy levels are farther apart than
those for neutrons due to the superposition of
the Coulomb force and the nuclear force for
the protons
The spin-orbit effect for nucleons is due to
the nuclear force

The spin-orbit effect influences the observed
characteristics of the nucleus
Shell Model Explanation of
Experimental Results

Nuclei with even numbers of protons and
neutrons are more stable



Any particular state is filled when it contains two
protons or two neutrons
An extra proton or neutron can be added only at
the expense of increasing the nucleus’s energy
This increase in energy leads to greater instability
in the nucleus
Shell Model Explanation of
Experimental Results, cont.

Nuclei tend to have more neutrons than
protons




Proton energy levels are higher
As Z increases and higher states are filled, a proton level
for a given quantum number will be much higher in energy
than the neutron level for the same quantum number
It is more energetically favorable for the nucleus to form
with neutrons in the lower energy levels than protons in the
higher levels
So, the number of neutrons is greater than the number of
protons
Marie Curie



1867 – 1934
Polish scientist
Shared Nobel Prize in 1903
for studies in radioactive
substances



Prize in physics
Shared with Pierre Curie
and Becquerel
Won Nobel Prize in 1911 for
discovery of radium and
polonium

Prize in chemistry
Radioactivity

Radioactivity is the spontaneous emission of
radiation



Discovered by Becquerel in 1896
Many experiments were conducted by Becquerel
and the Curies
Experiments suggested that radioactivity was
the result of the decay, or disintegration, of
unstable nuclei
Radioactivity – Types

Three types of radiation can be emitted

Alpha particles


The particles are 4He nuclei
Beta particles

The particles are either electrons or positrons



A positron is the antiparticle of the electron
It is similar to the electron except its charge is +e
Gamma rays

The “rays” are high energy photons
Distinguishing Types of
Radiation



The gamma particles carry
no charge
The alpha particles are
deflected upward
The beta particles are
deflected downward

A positron would be
deflected upward, but would
follow a different trajectory
than the α due to its mass
Penetrating Ability of Particles

Alpha particles


Beta particles


Barely penetrate a piece of paper
Can penetrate a few mm of aluminum
Gamma rays

Can penetrate several cm of lead
The Decay Constant

The number of particles that decay in a given
time is proportional to the total number of
particles in a radioactive sample
dN
  λN gives N  Noe  λt
dt



λ is called the decay constant and determines the
rate at which the material will decay
N is the number of undecayed radioactive nuclei
present
No is the number of undecayed nuclei at time t = 0
Decay Curve


The decay curve follows the
equation N = Noe-λt
The half-life is also a useful
parameter

The half-life is defined as
the time interval during
which half of a given
number of radioactive
nuclei decay
T1 2 
ln 2
0.693

λ
λ
Active Figure 44.9


Use the active figure to
adjust the half-life
Observe the decay
curve
PLAY
ACTIVE FIGURE
Decay Rate

The decay rate R of a sample is defined as
the number of decays per second
R 


dN
 λN  Roe  λt
dt
Ro = Noλ is the decay rate at t = 0
The decay rate is often referred to as the activity
of the sample
Units

The unit of activity, R, is the curie (Ci)


1 Ci ≡ 3.7 x 1010 decays/s
The SI unit of activity is the becquerel (Bq)

1 Bq ≡ 1 decay/s


Therefore, 1 Ci = 3.7 x 1010 Bq
The most commonly used units of activity are
the millicurie and the microcurie
Decay Processes




The blue circles are the stable
nuclei seen before
Above the line the nuclei are
neutron rich and undergo beta
decay (red)
Just below the line are proton
rich nuclei that undergo beta
(positron) emission or electron
capture (green)
Farther below the line the nuclei
are very proton rich and
undergo alpha decay (yellow)
Active Figure 44.10


Click on any colored
dot
Study the decay modes
and decay energies
PLAY
ACTIVE FIGURE
Alpha Decay

When a nucleus emits an alpha particle it
loses two protons and two neutrons




N decreases by 2
Z decreases by 2
A decreases by 4
Symbolically


A
Z
X
A 4
Z 2
Y  42 He
X is called the parent nucleus
Y is called the daughter nucleus
Decay – General Rules




The sum of the mass numbers A must be the
same on both sides of the equation
The sum of the atomic numbers Z must be
the same on both sides of the equation
When one element changes into another
element, the process is called spontaneous
decay or transmutation
Relativistic energy and momentum of the
isolated parent nucleus must be conserved
Disintegration Energy



The disintegration energy Q of a system is
defined as
Q = (Mx – My – Mα)c2
The disintegration energy appears in the form
of kinetic energy in the daughter nucleus and
the alpha particle
It is sometimes referred to as the Q value of
the nuclear decay
Alpha Decay, Example

Decay of 226 Ra
226
88


Ra 
Rn  24He
222
86
If the parent is at rest before
the decay, the total kinetic
energy of the products is
4.87 MeV
In general, less massive
particles carry off more of
the kinetic energy
Active Figure 44.11

Use the active figure to
observe the decay of
radium-226
PLAY
ACTIVE FIGURE
Alpha Decay, Notes

Experimental observations of alpha-particle
energies show a number of discrete energies
instead of a single value



The daughter nucleus may be left in an excited
quantum state
So, not all of the energy is available as kinetic energy
A negative Q value indicates that such a
proposed decay does not occur
spontaneously
Alpha Decay, Mechanism


In alpha decay, the alpha
particle tunnels though a
barrier
For higher energy particles,
the barrier is narrower and
the probability is higher for
tunneling across

This higher probability
translates into a shorter
half-life of the parent
Beta Decay


During beta decay, the daughter nucleus has
the same number of nucleons as the parent,
but the atomic number is changed by one
Symbolically
Y  e
A
Z
X
A
Z 1
A
Z
X
A
Z 1

Y  e
Beta decay is not completely described by these
equations
Beta Decay, cont.

The emission of the electron or positron is
from the nucleus


The nucleus contains protons and neutrons
The process occurs when a neutron is
transformed into a proton or a proton changes
into a neutron


The electron or positron is created in the process of
the decay
Energy must be conserved
Beta Decay – Particle Energy

The energy released in the
decay process should
almost all go to kinetic
energy of the β particle



Since the decaying nuclei
all have the same rest
mass, the Q value should
be the same for all decays
Experiments showed a
range in the amount of
kinetic energy of the emitted
particles
Were conservation laws
violated?
Neutrino



To account for this “missing” energy, in 1930
Pauli proposed the existence of another particle
Enrico Fermi later named this particle the
neutrino
Properties of the neutrino




Zero electrical charge
Mass much smaller than the electron, probably not
zero
Spin of ½
Very weak interaction with matter and so is difficult to
detect
Beta Decay – Completed

Symbolically
X
A
Z 1
A
Z
X
A
Z 1



Y  e  ν
A
Z
Y  e  ν
 is the symbol for the neutrino
ν is the symbol for the antineutrino
To summarize, in beta decay, the following
pairs of particles are emitted


An electron and an antineutrino
A positron and a neutrino
Beta Decay – Examples
Active Figure 44.15

Use the active figure to
observe the decay of
Carbon-14
PLAY
ACTIVE FIGURE
Beta Decay, Final Notes


The fundamental process of e- decay is a
neutron changing into a proton, an electron
and an antineutrino
In e+, the proton changes into a neutron,
positron and neutrino


This can only occur within a nucleus
It cannot occur for an isolated proton since its
mass is less than the mass of the neutron
Electron Capture


Electron capture is a process that competes
with e+ decay
In this case, a parent nucleus captures one of
its own orbital electrons and emits a neutrino:
A
Z

X
e
0
1
Y ν
A
Z 1
In most cases, a K-shell electron is captured, so
this is often referred to as K capture
Electron Capture, Detection

Because the neutrino is very hard to detect,
electron capture is usually observed by the xrays given off as higher-shell electrons
cascade downward to fill the vacancy created
in the K shell
Q Values for Beta Decay


For e- decay and electron capture, the Q value is Q
= (Mx – MY)c2
For e+ decay, the Q value is
Q = (Mx – MY - 2me)c2



The extra term, -2mec2, is due to the fact that the atomic
number of the parent decreases by one when the daughter
is formed
To form a neutral atom, the daughter sheds one electron
If Q is negative, the decay will not occur
Gamma Decay


Gamma rays are given off when an excited
nucleus decays to a lower energy state
The decay occurs by emitting a high-energy
photon called gamma-ray photons
A
Z


X*  X  γ
A
Z
The X* indicates a nucleus in an excited state
Typical half-life is 10-10 s
Gamma Decay – Example

Example of a decay sequence


The first decay is a beta emission
The second step is a gamma emission
12
5
B
12
6
C* 


12
6

C*  e  ν
12
6
Cγ
Gamma emission doesn’t change Z, N, or A
The emitted photon has an energy of hƒ equal to
DE between the two nuclear energy levels
Summary of Decays
Natural Radioactivity

Classification of nuclei



Three series of natural radioactivity exist




Unstable nuclei found in nature
 Give rise to natural radioactivity
Nuclei produced in the laboratory through nuclear reactions
 Exhibit artificial radioactivity
Uranium
Actinium
Thorium
Some radioactive isotopes are not part of any decay
series
Radioactive Series, Overview
Decay Series of 232Th




Series starts with 232Th
Processes through a
series of alpha and beta
decays
The series branches at
212Bi
Ends with a stable isotope
of lead, 208Pb
Nuclear Reactions

The structure of nuclei can be changed by
bombarding them with energetic particles


The changes are called nuclear reactions
As with nuclear decays, the atomic numbers
and mass numbers must balance on both
sides of the equation
Nuclear Reactions, cont.

A target nucleus, X, is bombarded by a
particle a, resulting in a daughter nucleus Y
and an outgoing particle b


a+XY+b
The reaction energy Q is defined as the total
change in mass-energy resulting from the
reaction

Q = (Ma + MX – MY – Mb)c2
Q Values for Reactions

The Q value determines the type of reaction


An exothermic reaction
 There is a mass “loss” in the reaction
 There is a release of energy
 Q is positive
An endothermic reaction
 There is a “gain” of mass in the reaction
 Energy is needed, in the form of kinetic energy of the
incoming particles
 Q is negative
 The minimum energy necessary for the reaction to occur is
called the threshold energy
Nuclear Reactions, final

If a and b are identical, so that X and Y are
also necessarily identical, the reaction is
called a scattering event


If the kinetic energy before the event is the same
as after, it is classified as elastic scattering
If the kinetic energies before and after are not the
same, it is an inelastic scattering
Conservation Rules for
Nuclear Reactions

The following must be conserved in any
nuclear reaction




Energy
Momentum
Total charge
Total number of nucleons
Nuclear Magnetic Resonance
(NMR)



A nucleus has spin
angular momentum
Shown is a vector
model giving possible
orientations of the spin
and its projection on
the z axis
The magnitude of the
spin angular
momentum is
I ( I  1)
NMR, cont.

For a nucleus with spin
½, there are only two
allowed states


Emax and Emin
It is possible to observe
transitions between two
spin states using NMR
MRI

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An MRI (Magnetic
Resonance Imaging) is
based on NMR
Because of variations in
an external field,
hydrogen atoms in
different parts of the body
have different energy
splittings between spin
states
The resonance signal can
provide information about
the positions of the
protons