PHGN
422: NUCLEAR PHYSICS
PHGN 422: Nuclear Physics
Lecture 3: Nuclear Radii, Masses, and Binding Energies
Prof. Kyle Leach
August 30, 2016
Slide 1
PHGN
422: NUCLEAR PHYSICS
Last Week.....
• The atomic nucleus is a very dense, positively charged object
composed of protons and neutrons
• Nuclei are organized according to their Z and N values on the
Nuclear Chart (or Chart of the Nuclides)
• Nuclei are held together by the strong interaction, and the
nuclear force is attractive at short range, but repulsive at very
short distances (we will talk about why today)
• So...back to our electron scattering experiments!
Slide 2 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Electron Scattering on Nuclei
Source: Fig. 3.1 (pg. 46) – Introductory Nuclear Physics, Ken Krane
Slide 3 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Light Scattering on an Opaque Object
Source: Department of Physics, Brock University
Slide 4 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Can This Tell Us About the Nucleus?
Opaque Object
Slide 5 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
Nuclear Matter
PHGN
422: NUCLEAR PHYSICS
The Nuclear Charge Distribution
Source: Fig. 3.4 (pg. 49) – Introductory Nuclear Physics, Ken Krane
Slide 6 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
• The angular distributions from elastic scattering of electrons from
nuclei do not show sharp minima
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
• The angular distributions from elastic scattering of electrons from
nuclei do not show sharp minima
• These minima become even less sharp with increasing Z
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
• The angular distributions from elastic scattering of electrons from
nuclei do not show sharp minima
• These minima become even less sharp with increasing Z
2
The central nuclear charge density is nearly the same for all
nuclei
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
• The angular distributions from elastic scattering of electrons from
nuclei do not show sharp minima
• These minima become even less sharp with increasing Z
2
The central nuclear charge density is nearly the same for all
nuclei
• There is no dependence on the density of charge as a function of Z
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
• The angular distributions from elastic scattering of electrons from
nuclei do not show sharp minima
• These minima become even less sharp with increasing Z
2
The central nuclear charge density is nearly the same for all
nuclei
• There is no dependence on the density of charge as a function of Z
• Nucleons do not seem to preferentially organize based on type (ie.
protons or neutrons)
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
• The angular distributions from elastic scattering of electrons from
nuclei do not show sharp minima
• These minima become even less sharp with increasing Z
2
The central nuclear charge density is nearly the same for all
nuclei
• There is no dependence on the density of charge as a function of Z
• Nucleons do not seem to preferentially organize based on type (ie.
protons or neutrons)
3
The overall matter density of all nucleons in the nucleus must
therefore be constant as well? (number of nucleons per unit
volume)
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
• The angular distributions from elastic scattering of electrons from
nuclei do not show sharp minima
• These minima become even less sharp with increasing Z
2
The central nuclear charge density is nearly the same for all
nuclei
• There is no dependence on the density of charge as a function of Z
• Nucleons do not seem to preferentially organize based on type (ie.
protons or neutrons)
3
The overall matter density of all nucleons in the nucleus must
therefore be constant as well? (number of nucleons per unit
volume)
• If this is true, we should be able to determine what the density of
nuclear matter is
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
What Does This Tell Us About The Nucleus?
1
The boundary of the nucleus is not sharp, but displays a
probability distribution
• The angular distributions from elastic scattering of electrons from
nuclei do not show sharp minima
• These minima become even less sharp with increasing Z
2
The central nuclear charge density is nearly the same for all
nuclei
• There is no dependence on the density of charge as a function of Z
• Nucleons do not seem to preferentially organize based on type (ie.
protons or neutrons)
3
The overall matter density of all nucleons in the nucleus must
therefore be constant as well? (number of nucleons per unit
volume)
• If this is true, we should be able to determine what the density of
nuclear matter is
• Also, can we find a generic way of obtaining the matter radius of a
given nucleus?
Slide 7 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Density of Nuclear Matter
Well, to start...let’s assume that the nucleus is a perfect sphere. From
here, we can estimate the volume and perhaps the density...
Proton (π)
+
+
+
Neutron (ν) +
V
=
4 3
πR
3
Slide 8 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Nuclear Matter Radius
If the nuclear matter density is also indeed constant for all nuclei:
V=
4 3
πR ≈ constant
3
Then, we can relate the radius of a nucleus to the number of
nucleons A:
R ∝ A1/3
To determine this proportionality constant, we can relate the total
nuclear matter radius R to the matter radius of the individual nucleons
R0
Slide 9 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Nuclear Matter Radius
The nucleons can also be considered spherical:
Therefore:
4
4 3
πR = A · πR30
3
3
=⇒ R = R0 · A1/3
Experimentally we know that R0 ≈ 1.2 fm. So, the nuclear matter
radius is R = 1.2 · A1/3 ! Further detailed discussion on this topic can
be found in Chapter 3.1 of Krane.
Slide 10 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Nature of Nuclear Matter
One of the most remarkable conclusions from all of this is that
nuclear matter does not seem to change density regardless of the
size of the nucleus!! In other words, the number of nucleons per unit
of volume is roughly constant for all nuclei.
How dense is nuclear matter (comparatively speaking). Well....
• Sea Water: 1.0 × 103 kg/m3
Slide 11 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Nature of Nuclear Matter
One of the most remarkable conclusions from all of this is that
nuclear matter does not seem to change density regardless of the
size of the nucleus!! In other words, the number of nucleons per unit
of volume is roughly constant for all nuclei.
How dense is nuclear matter (comparatively speaking). Well....
• Sea Water: 1.0 × 103 kg/m3
• Tin Oxide: 1.6 × 103 kg/m3
Slide 11 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Nature of Nuclear Matter
One of the most remarkable conclusions from all of this is that
nuclear matter does not seem to change density regardless of the
size of the nucleus!! In other words, the number of nucleons per unit
of volume is roughly constant for all nuclei.
How dense is nuclear matter (comparatively speaking). Well....
• Sea Water: 1.0 × 103 kg/m3
• Tin Oxide: 1.6 × 103 kg/m3
• Steel: 1.1 × 104 kg/m3
Slide 11 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Nature of Nuclear Matter
One of the most remarkable conclusions from all of this is that
nuclear matter does not seem to change density regardless of the
size of the nucleus!! In other words, the number of nucleons per unit
of volume is roughly constant for all nuclei.
How dense is nuclear matter (comparatively speaking). Well....
• Sea Water: 1.0 × 103 kg/m3
• Tin Oxide: 1.6 × 103 kg/m3
• Steel: 1.1 × 104 kg/m3
• Lead: 2.5 × 104 kg/m3
Slide 11 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Nature of Nuclear Matter
One of the most remarkable conclusions from all of this is that
nuclear matter does not seem to change density regardless of the
size of the nucleus!! In other words, the number of nucleons per unit
of volume is roughly constant for all nuclei.
How dense is nuclear matter (comparatively speaking). Well....
• Sea Water: 1.0 × 103 kg/m3
• Tin Oxide: 1.6 × 103 kg/m3
• Steel: 1.1 × 104 kg/m3
• Lead: 2.5 × 104 kg/m3
• Core of the Sun: 1.5 × 105 kg/m3
Slide 11 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Nature of Nuclear Matter
One of the most remarkable conclusions from all of this is that
nuclear matter does not seem to change density regardless of the
size of the nucleus!! In other words, the number of nucleons per unit
of volume is roughly constant for all nuclei.
How dense is nuclear matter (comparatively speaking). Well....
• Sea Water: 1.0 × 103 kg/m3
• Tin Oxide: 1.6 × 103 kg/m3
• Steel: 1.1 × 104 kg/m3
• Lead: 2.5 × 104 kg/m3
• Core of the Sun: 1.5 × 105 kg/m3
• Nuclear Matter: 2.3 × 1017 kg/m3
Slide 11 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Question:
What if the nucleus were nearly 20 orders of magnitude larger?
Slide 12 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Question:
What if the nucleus were nearly 20 orders of magnitude larger?
Well, this is not hypothetical....these are known as neutron stars
Source: NASA.gov
Slide 12 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Question:
What if the nucleus were nearly 20 orders of magnitude larger?
Well, this is not hypothetical....these are known as neutron stars
Source: NASA.gov
Slide 12 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Bound Nuclear Systems
Limits of Nuclear Existence
Putting aside neutron stars for now, let us take a look at the limits of
what nuclei can exist, and how we define it.
Slide 13 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Atomic Mass and Nuclear Binding Energy
Slide 14 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
The Atomic Mass and Nuclear Binding Energy
As we briefly mentioned last week, the mass of a given atom is not
simply the sum of neutron, proton, and electron masses, ie:
!
Z
X
A
2
2
2
2
M(Z XN )c 6= Z · mp c + N · mn c − Z · me c −
Bi
i=1
For a nucleus to exist (ie. be a bound system), the following
constraint must be satisfied (neglecting the electrons for a moment):
M(AZ XN )c2 < Z · mp c2 + N · mn c2
For the nucleons to be bound inside of the nucleus, there needs to be
some energy difference. We call this the Binding Energy.
We’ll define what we mean on the chalkboard....
Slide 15 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Mass Excess
Since the atomic mass in MeV/c2 can become a cumbersome way of
dealing with larger nuclei (ie. m(208 Pb) = 193 733 MeV/c2 )
We can define a useful experimental mass value relative to our
definition of the atomic mass unit in Lecture 1 (1u = 931.502 MeV/c2 ).
m(AZ XN )c2
=
(A · u)c2 + ∆c2
=⇒ ∆c2
= m(AZ XN )c2 − (A · u)c2
Where ∆ is referred to as the Mass Excess or Mass Defect, and
helps us to quantify how much a specific nucleus deviates from our
approximation of the atomic mass unit.
• It can be either positive or negative, as long as we satisfy
M(AZ XN )c2 < Z · mp c2 + N · mn c2 .
Slide 16 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Example:
What is the Mass Excess (∆) for 16 O in MeV?
First we’ll start with the experimentally measured mass of 16 O in u:
Slide 17 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Example:
What is the Mass Excess (∆) for 16 O in MeV?
First we’ll start with the experimentally measured mass of 16 O in u:
• Remember, we say mass 16 for
Slide 17 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
16
O, but this is not exactly true.
PHGN
422: NUCLEAR PHYSICS
Example:
What is the Mass Excess (∆) for 16 O in MeV?
First we’ll start with the experimentally measured mass of 16 O in u:
• Remember, we say mass 16 for
m(AZ XN )c2
= 15.994915 u
Slide 17 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
16
O, but this is not exactly true.
PHGN
422: NUCLEAR PHYSICS
Example:
What is the Mass Excess (∆) for 16 O in MeV?
First we’ll start with the experimentally measured mass of 16 O in u:
• Remember, we say mass 16 for
m(AZ XN )c2
16
O, but this is not exactly true.
= 15.994915 u
Now solve for the mass excess ∆, (recall 1 u = 931.505 MeV/c2 )
∆
=
m(AZ XN ) − (A · u)
=
(15.994915 − 16) · 931.505 MeV
= −4.737 MeV
Slide 17 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
The Proton and Neutron Separation Energies (Sp and Sn )
Analogous to atomic ionization energies, these separation energies
can tell us about the binding strength for an individual nucleon. We
can define these on the chalkboard:
Slide 18 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
The Proton and Neutron Separation Energies (Sp and Sn )
Analogous to atomic ionization energies, these separation energies
can tell us about the binding strength for an individual nucleon. We
can define these on the chalkboard:
We can also look at the trends of how nuclear binding changes as a
function of the mass number A
Slide 18 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
Slide 19 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
This brings us to some other revelations about the way nuclei behave:
1
Most nuclei have almost exactly the same BE/A, which is
roughly 8 MeV/A. This means the nuclear force saturates such
that only each nucleon can interact with a few of its
neighbours.
Recall that the nuclear force is strongly attractive
ONLY at short distances (∼ 1 fm).
Slide 20 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
This brings us to some other revelations about the way nuclei behave:
1
Most nuclei have almost exactly the same BE/A, which is
roughly 8 MeV/A. This means the nuclear force saturates such
that only each nucleon can interact with a few of its
neighbours.
Recall that the nuclear force is strongly attractive
ONLY at short distances (∼ 1 fm).
2
The most bound nuclei are in the region of A ∼ 56 − 62
Slide 20 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
This brings us to some other revelations about the way nuclei behave:
1
Most nuclei have almost exactly the same BE/A, which is
roughly 8 MeV/A. This means the nuclear force saturates such
that only each nucleon can interact with a few of its
neighbours.
Recall that the nuclear force is strongly attractive
ONLY at short distances (∼ 1 fm).
2
The most bound nuclei are in the region of A ∼ 56 − 62
3
Some structure in this curve also exists (particularly for 4 He) that
results from quantum effects of the nucleus. We will discuss the
shell structure of nuclei in a couple of weeks.
Slide 20 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
This brings us to some other revelations about the way nuclei behave:
1
Most nuclei have almost exactly the same BE/A, which is
roughly 8 MeV/A. This means the nuclear force saturates such
that only each nucleon can interact with a few of its
neighbours.
Recall that the nuclear force is strongly attractive
ONLY at short distances (∼ 1 fm).
2
The most bound nuclei are in the region of A ∼ 56 − 62
3
Some structure in this curve also exists (particularly for 4 He) that
results from quantum effects of the nucleus. We will discuss the
shell structure of nuclei in a couple of weeks.
4
Nuclei on the left of the peak can release energy by joining
together (Nuclear Fusion)
Slide 20 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
This brings us to some other revelations about the way nuclei behave:
1
Most nuclei have almost exactly the same BE/A, which is
roughly 8 MeV/A. This means the nuclear force saturates such
that only each nucleon can interact with a few of its
neighbours.
Recall that the nuclear force is strongly attractive
ONLY at short distances (∼ 1 fm).
2
The most bound nuclei are in the region of A ∼ 56 − 62
3
Some structure in this curve also exists (particularly for 4 He) that
results from quantum effects of the nucleus. We will discuss the
shell structure of nuclei in a couple of weeks.
4
Nuclei on the left of the peak can release energy by joining
together (Nuclear Fusion)
5
Nuclei on the right of the peak can release energy by breaking
apart (Nuclear Fission)
Slide 20 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
Source: The Open University
Slide 21 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Nuclear Fusion in Stars
Source: A.C. Phillips, The Physics of Stars, 2nd Edition (Wiley, 1999)
Slide 22 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Nuclear Fission in Reactors
Source: Department of Physics, UC Davis
Slide 23 — Prof. Kyle Leach — PHGN 422: Nuclear Physics
PHGN
422: NUCLEAR PHYSICS
Next Class...
Reading Before Next Class
• Sections 3.2 and 3.3 (first part) in Krane
Next Class Topics
• More on binding energy
• Ways to release of energy in a nuclear decay or reaction
• The experimental determination of atomic masses...and why do
we care?
Slide 24 — Prof. Kyle Leach — PHGN 422: Nuclear Physics