NUCLEAR PHYSICS
STRUCTURE OF THE ATOMIC NUCLEUS
NUCLEUS
• Composition: The nucleus contains two types of subatomic particles, protons
and neutrons, and is summarized by the Atomic Mass Number, 𝐴 = 𝑍 + 𝑁.
• Charge: The nucleus has a positive charge due to the protons. The number of
protons determines the atomic number and the element's identity.
• Mass: The nucleus accounts for nearly all an atom's mass because protons and
neutrons are much more massive than electrons.
• Binding Energy: The protons and neutrons are held together by the strong
nuclear force.
• Isotopes: Isotopes have the same number of protons (same 𝑍 number) but
different numbers of neutrons (different 𝑍 number), affecting their mass and
nuclear stability.
NUCLIDE
• All properties of a nucleus are determined by the number of protons
and neutrons it has.
• The following notation is used to represent a particular nuclide:
𝐴
𝑍 𝑋𝑁
where 𝑍 is the atomic number (number of protons), 𝑁 is the number of
neutrons, and 𝐴 is the atomic mass number, 𝐴 = 𝑍 + 𝑁.
FORCES AND STABILITY
•Electrostatic Repulsion:
•The repulsive force between positively charged protons is counteracted by
the strong nuclear force to maintain nuclear stability.
•Strong Nuclear Force:
•The force that binds protons and neutrons together.
•Extremely strong but short-ranged, acting only over distances ~10−15 meters.
•Neutrons help maintain the strong force to fight against electrostatic repulsion
of protons, but too many neutrons (heavy isotopes) causes nuclear instability.
GRAPH OF
𝑵 𝒗𝒔. 𝒁
The patterns of stable and
unstable nuclides reveal
characteristics of the nuclear
forces. The dashed line is for
𝑁 = 𝑍. Numbers along
diagonals are mass numbers 𝐴.
NUCLEAR BINDING ENERGY
• Definition: The energy needed to disassemble a nucleus (fission) into
its constituent nucleons (protons and neutrons).
• It is also the energy released during the formation of a nucleus
(fusion) from these nucleons.
• Mass Defect: The mass of a nucleus is less than the sum of the
masses of its individual protons and neutrons. This difference is
called the mass defect, ∆𝑚.
• The difference between the mass of the nucleus and the sum of
the masses of its individual nucleons. This difference is due to the
binding energy, as described by Einstein's equation:
𝐵𝐸 = ∆𝑚𝑐 2
CALCULATING BINDING ENERGY
Mass-Energy of the nucleus:
𝑍(𝑚𝑝 𝑐 2 ) + 𝑁 𝑚𝑛 𝑐 2 = 𝑚𝑛𝑢𝑐𝑙𝑒𝑢𝑠 𝑐 2 + 𝐵𝐸
The binding energy, BE, is the energy equivalent of the mass defect:
𝐵𝐸 = ∆𝑚𝑐 2 =
𝑍𝑚𝑝 + 𝑁𝑚𝑛 − 𝑚𝑡𝑜𝑡𝑎𝑙 𝑐 2
The atomic masses can be most conveniently expressed in unified
atomic mass units, 1 𝑢 = 931.5 𝑀𝑒𝑉/𝑐 2 .
A GRAPH OF AVERAGE BINDING ENERGY
PER NUCLEON, BE/A, FOR STABLE NUCLEI.
MASS
SPECTROMETER /
ENRICHMENT
𝑚𝑣
𝑟=
𝑞𝐵
Radius depends on mass
Same charge, same velocity,
same chemistry — different
mass, different path.
Atomic Bomb Cannon – Fission Device
Hydrogen Bomb – Fusion Device
NUCLEAR DECAY
ALPHA DECAY
Process: An unstable nucleus releases an alpha particle
(two protons and two neutrons, equivalent to a helium-4
nucleus).
Result: The original atom loses two protons and two
neutrons, transforming into a new element that is two
atomic numbers lower and four mass numbers lighter.
Example: Uranium-238 decays to Thorium-234 by
emitting an alpha particle.
Penetration: Alpha particles are relatively large and
heavy, so they have low penetration power and can be
stopped by a sheet of paper or human skin.
ALPHA DECAY EXAMPLE
𝛼 decay of Plutonium-239:
239
𝑃𝑢 → 235𝑈 + 4𝐻𝑒
The mass defect of this reaction is
∆𝑚 = −0.0005631 𝑢
The energy released is
𝐸 = ∆𝑚c 2 = 6.25 𝑀𝑒𝑉
This is on the order of a million times greater than typical chemical
reaction energies!
BETA DECAY
Process: A neutron in an unstable nucleus is converted into a proton, an
electron (β- particle), and an antineutrino, OR a proton is converted into
a neutron, a positron (β+ particle), and a neutrino.
Result: The atomic number changes by one (increases for βdecay and decreases for β+ decay), but the mass number
remains unchanged.
Example: Carbon-14 decays to Nitrogen-14 via βdecay, emitting an electron and an antineutrino.
Penetration: Beta particles have moderate penetration power
and can be stopped by a few millimeters of aluminum or
plastic.
TWO TYPES OF BETA DECAY
In 𝛽−​​ decay, the parent nucleus emits an electron
and an antineutrino. The daughter nucleus has one
more proton and one less neutron than its parent.
Neutrinos interact so weakly that they are almost
never directly observed, but they play a
fundamental role in particle physics.
𝛽+​​ decay is the emission of a positron that
eventually finds an electron to annihilate,
characteristically producing gammas in
opposite directions.
PET SCAN
β⁺ decay → positron meets electron →
annihilation → two γ rays at 180° →
detectors reconstruct the source
location.
X-ray shows anatomy. PET shows metabolic
activity. Different physics, complementary
information.
A THIRD TYPE OF BETA “DECAY”
• Electron capture is the third type of beta decay.
• Here, a nucleus captures an inner-shell electron and
undergoes a nuclear reaction that has the same effect
as 𝛽+ decay.
• Electrons cannot reside in the nucleus, but this is a nuclear
reaction that consumes the electron and occurs
spontaneously only when the products have less mass than
the parent plus the electron.
GAMMA DECAY
Process: An excited nucleus releases energy in the form
of gamma rays (high-energy photons) to reach a more
stable state.
Result: The atomic number and mass number remain
unchanged, but the nucleus moves to a lower energy
state.
Example: Cobalt-60 decays to an excited state of Nickel60, which then emits gamma rays to reach its ground
state.
Penetration: Gamma rays have high penetration power
and require dense materials like lead or several
centimeters of concrete to be stopped effectively.
QUESTION: REACTION BALANCING
Radium-226 undergoes alpha decay. Identify the daughter
nucleus.
A.
222
86𝑅𝑛
B.
222
88𝑅𝑎
C.
226
89𝐴𝑐
D. 230
90𝑇ℎ
226
88𝑅𝑎
→ _____ + 42𝐻𝑒
QUESTION: REACTION BALANCING
Radium-226 undergoes alpha decay. Identify the daughter
nucleus.
A.
222
86𝑅𝑛
B.
222
88𝑅𝑎
C.
226
89𝐴𝑐
D. 230
90𝑇ℎ
226
88𝑅𝑎
→ _____ + 42𝐻𝑒
PENETRATION RANGES
PENETRATION RANGES
The penetration or range of radiation depends on its energy, the material it encounters, and the type of radiation. (a) Greater energy means
greater range. (b) Radiation has a smaller range in materials with high electron density. (c) Alphas have the smallest range, betas have a
greater range, and gammas penetrate the farthest.
MEDICAL
IMAGING
• This image of the concentration of a
radioactive tracer in a patient’s body
reveals where the most active bone cells
are, an indication of bone cancer.
• A short-lived radioactive substance that
locates itself selectively is given to the
patient, and the radiation is measured
with an external detector.
• The emitted 𝛾 radiation has a sufficient
range to leave the body—the range of 𝛼s
and 𝛽s is too small for them to be
observed outside the patient. (credit:
Kieran Maher, Wikimedia Commons)
HALF-LIFE & ACTIVITY
HALF-LIFE
•Definition: The time required for half of the radioactive nuclei in a
sample to decay.
•Significance: Indicates the stability of a radioactive isotope.
•Measurement: Denoted by 𝑡1/2 ​.
•Applications:
•Radiometric dating: Determining the age of archaeological finds (e.g., Carbon-14 dating).
•Medical: Used in radioactive tracers and cancer treatments.
•Nuclear physics: Understanding decay processes and nuclear reactions.
•Examples:
•Carbon-14: 𝑡1/2 ≈ 5730 𝑦𝑒𝑎𝑟𝑠
•Uranium-238: 𝑡1/2 ≈ 4.5 𝑏𝑖𝑙𝑙𝑖𝑜𝑛 𝑦𝑒𝑎𝑟𝑠
HALF-LIFE
Number of nuclei
present after time 𝑡:
𝑁 𝑡 = 𝑁𝑜
1
2
𝑡/𝑡1
2
LINEAR PLOT
SEMI-LOG PLOT
log 10𝑛 = 𝑛
Linear Plot shows the exponential curve.
On a log scale, each step represents a ×10
change, not a +10 change.
Exponential decay becomes a straight line.
HALF-LIFE: WORKED EXAMPLE
Problem: A fossil contains 25% of its original Carbon-14.
How old is the fossil? (t₁/₂ = 5,730 years)
1
25% = =
4
1 2
→ 2 half-lives have passed
2
𝐴𝑔𝑒 = 2 × 5,730 = 𝟏𝟏, 𝟒𝟔𝟎 years
Check: 𝑁
1
= 𝑁𝑜
2
𝑡
𝑡1
2
1
= 𝑁𝑜
2
11460
5730
1 2
= 𝑁𝑜
2
𝑁𝑜
=
4
→ 25% ✓
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