11/10/2020 1 Nuclear Physics Lecture 2: The Nuclear Force, Radii, Masses, and Binding Energies Last Lecture..... • The atomic nucleus is composed of nucleons (protons and neutrons) • The nuclear mass A is the sum of the number of protons and neutrons in a given nucleus (A = N + Z) • The proper notation for a nuclear system is A X, where X is the chemical symbol from the periodic table • Nuclei are organized on the nuclear chart according to the number of protons and neutrons each system has
Transcript
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Nuclear PhysicsLecture 2: The Nuclear Force, Radii, Masses, and Binding Energies
Last Lecture.....
• The atomic nucleus is composed of nucleons (protons and neutrons)
• The nuclear mass A is the sum of the number of protons and neutrons in a given nucleus (A = N + Z)
• The proper notation for a nuclear system is AX, where X is the chemical symbol from the periodic table
• Nuclei are organized on the nuclear chart according to the number of protons and neutrons each system has
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The Fundamental Forces
Force Theory Mediator Relative Strength Range (m)
Strong QCD gluon (g) 1 ∼ 10−15
1 137
α = ≈ 10 − 2Electromagnetic
Weak
Gravitational
QED
Electroweak
Gravity
photon (γ)
W ± and Z bosons
unknown
∼ 10−5
∼ 10−38
∞
∼ 10−18
∞
More on the fundamental forces can be found at: http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html
Fundamental force, also called fundamental interaction, in physics, any of the four basic forces that govern how objects or particles interact and how certain particles decay. All the known forces of nature can be traced to these fundamental forces. The fundamental forces are characterized on the basis of the following four criteria: the types of particles that experience the force, the relative strength of the force, the range over which the force is effective, and the nature of the particles that mediate the force.
The up quark has a relative chargeof two-thirds. And the down quarkhas a relative charge of negativeone-third, where, of course, relativecharges are measured relative to thecharge of a proton. In other words,an up quark has a charge that is thesame sign as the charge on aproton, it’s positive. But themagnitude or size of that charge isonly two-thirds the size of the chargeon a proton.And in the same way, the downquark has a negative charge. So thesign of the charge is opposite to thatof the proton. But the magnitude orsize of that charge is one-third thatthe magnitude of the charge on a
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Forces and the History of the Universe
Known Fundamental Forces
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The Force Between Nucleons
Electrons and Atoms
• Coulomb interaction
• Electrons in classical orbitsthat have large (relative) energy spacings
• Electron distances are large(ie. small e-e interaction probability)
Nucleon-Nucleon Interaction
• Strong interaction
• Nuclear orbits (shells) havesmall (relative) energy spacings
• Due to the small nuclear size,a given nucleon will strongly interact with allnearest-neighbour nucleons
• The atomic nucleus is a very dense, positively charged object composed of protons and neutrons
• Nuclei are held together by the strong interaction, and the nuclear force is attractive at short range, but repulsive at very short distances
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 allnuclei
• 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?
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Density of Nuclear MatterWell, to start...let’s assume that the nucleus is a perfect sphere. From here, we can estimate the volume and perhaps the density...
3V =
4πR3
The Nuclear Matter Radius
If the nuclear matter density is also indeed constant for all nuclei:
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
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The Nuclear Matter RadiusThe nucleons can also be considered spherical:
Therefore:
Experimentally we know that R0 ≈ 1.2 fm. So, the nuclear matter
radius is R = 1.2 ·A1/3
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
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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
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
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The Atomic Mass and Nuclear Binding Energy
The Atomic Mass and Nuclear Binding Energy
As the mass of a given atom is not simply the sum of neutron, proton, and electron masses, ie:
For a nucleus to exist (ie. be a bound system), the following constraint must be satisfied (neglecting the electrons for a moment):
For the nucleons to be bound inside of the nucleus, there needs to be some energy difference. We call this the Binding Energy.
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Mass Excess
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).
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
The mass defect (∆) of a nucleus represents the mass of the energy binding the nucleus, and is the difference between the mass of a nucleus and the sum of the masses of the nucleons.
Example: What is the Mass Excess (∆) for 16O in MeV?
Zm(AXN ) = 15.994915u
Now solve for the mass excess ∆ , (recall 1 u = 931.505 MeV/c2)
(masses of the nucleons)
2
2
15.994915 16 ) . 931.
.
5.
4.73
( )
(
7
AZ N
MeVu u
c u
MeV
c
m X A u
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Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
Nuclear Binding Energy
Nuclear binding energy is the energy required to split a nucleus of an atom into its component parts: protons and neutrons, or, collectively, the nucleons. The binding energy of nuclei is always a positive number, since all nuclei require net energy to separate them into individual protons and neutrons.
2BE c
Binding Energy per Nucleon (BE/A)2
BEA
c
where, c2 =931.5 MeV/u
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Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)This brings us to some other revelations about the way nucleibehave:
-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 4He) that results from quantum effects of the nucleus.
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)
Characteristics of Nuclear Binding
Binding Energy per Nucleon (BE/A)
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Characteristics of Nuclear Binding
• Curve shows that EB increases with A and peaks at A = 60.• Heavier nuclei are less stable.• For heavier nuclei, energy is released when they break up
(fission). • For lighter nuclei, energy is released when they fuse
together (fusion).
Nuclear Fusion in Stars
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Nuclear Fission in Reactors
Sol.
A = 7; Z = 3; N = ?
N = A – Z
=7 – 3= 4
Number of electrons are same as number of protons.
Example: Calculate the neutron and electron number for the nucleus of a lithium atom which has a mass number of 7 and an atomic number of 3.
��4��
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Sol.
Since 1 mol of C atoms have mass of 12g and
1 mol of atoms = 6.022 × 1023 atoms
Example: show that 1 amu is equal to 1.660 × 10−27 kg.
Sol.
Example: The most common kind of iron nucleus has a mass number of 56. Find the radius, approximate mass and approximate density of the nucleus.
The radius and mass of a nucleus depend on the mass number A and density is mass divided by volume.The mass of the nucleus in atomic mass units is approximately equalto the mass number. The radius is
15 1/3
15.
(1.2 10 )(56)
104 6
R m
R m
3
15 3 43 34.6
4
3
4(3.14)( 10 ) 4.1 10
3
V R
V m m
26
43 3
17 3
9.3 10
4.1 10
2.3 10
m
V
kg
m
kgm
Since A = 56, the mass of the nucleus is approximately 56 u, or m ≈ (56) (1.66 x 10-27 kg) = 9.3 x 10-26 kg.
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Sol.
where, c2 =931.5 MeV/u
E = m c2
E = (1.00726 u)(931.5 MeV/u)
E= 938.3 MeV
Example: What is the rest mass energy of a proton (1.007276 u)?
HW: What is the rest mass energy of a neutron (1.008665 u)?
HW: What is the rest mass energy of a electron (0.00055 u)?
Sol.
Example: Calculate the mass defect for nucleon of ����
nucleus, if mass of proton mp = 1.0078 u, mass of neutron mn = 1.0087 u, mass of C12, mC = 12.0000 u.
mass defect is difference between the mass of a nucleus and the sum of the masses of the nucleons
12.0990 12 )
0.0990
( )
(
AZ N
u u
amu
m X A
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HW: Find the mass defect for the ��� nucleus of helium-4.
4.0026( ) 03AZ N um X
Sol.
Example: Find the binding energy per nucleon for helium-4. (∆= 0.030377 u)
