1. Nuclear Binding, Radioactivity
SPH4UI Physics
Nuclear Binding, Radioactivity 1

2. Modern understanding: the ``onion’’ picture
Protons and neutrons
Nucleus
Atom
Let’s see what’s inside!

3. Nice Try
We are now into the realm of Quantum Field Theory.

4. Introduction: 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 nucleus)
Beta (electrons)
Gamma (high-energy photons)
1911 Rutherford, Geiger and Marsden performed scattering experiments
Established the point mass nature of the nucleus
Nuclear force was a new type of force
1919 Rutherford and coworkers first observed nuclear reactions in which naturally occurring alpha particles bombarded nitrogen nuclei to produce oxygen
1932 Cockcroft and Walton first used artificially accelerated protons to produce nuclear reactions
1932 Chadwick discovered the neutron
1933 the Curies discovered artificial radioactivity
1938 Hahn and Strassman discovered nuclear fission
1942 Fermi achieved the first controlled nuclear fission reactor

5. Some Properties of Nuclei
All nuclei are composed of protons and neutrons
Exception is ordinary hydrogen with just a proton
The atomic number (charge), Z, equals the number of protons in the nucleus
The neutron number, N, is the number of neutrons in the nucleus
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
Notation
Example:
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
where X is the chemical symbol of the element

6. Charge and mass Charge: Mass:
The electron has a single negative charge, -e (e = x C)
The proton has a single positive charge, +e
Thus, charge of a nucleus is equal to Ze
The neutron has no charge
Makes it difficult to detect
Mass:
It is convenient to use atomic mass units, u, to express masses
1 u = x kg
Based on definition that the mass of one atom of C-12 is exactly 12 u
Mass can also be expressed in MeV/c2
From ER = m c2
1 u = MeV/c2

7. Summary of Masses Masses Particle kg u MeV/c2 Proton 1.6726 x 10-27
938.28
Neutron
x 10-27
939.57
Electron
9.101 x 10-31
5.486x10-4
0.511

8. Quick problem: protons in your body
What is the order of magnitude of the number of protons in your body? Of the number of neutrons? Of the number of electrons? Take your mass approximately equal to 70 kg.
An iron nucleus (in hemoglobin) has a few more neutrons than protons, but in a typical water molecule there are eight neutrons and ten protons. So protons and neutrons are nearly equally numerous in your body, each contributing 35 kg out of a total body mass of 70 kg.
Same amount of neutrons and electrons.

9. The Size of the Nucleus or
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
The KE of the particle must be completely converted to PE or
For gold: d = 3.2 x m, for silver: d = 2 x m
Such small lengths are often expressed in femtometers where 1 fm = m (also called a fermi)

10. Size of Nucleus Example: has radius
Since the time of Rutherford, many other experiments have concluded the following
Most nuclei are approximately spherical
Average radius is
ro = 1.2 x m A
has radius
Example: Z

11. 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

12. 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 (or strong) force (QCD)
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

13. Nuclear Stability chart
Light nuclei are most stable if N = Z
Heavy nuclei are most stable when N > Z
As the number of protons increase, the Coulomb force increases and so more nucleons are needed to keep the nucleus stable
No nuclei are stable when Z > 83

14. Isotopes
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
Example:
atomic number (charge), Z
neutron number, N
nucleon number, A,

15. Binding Energy Binding Energy per Nucleon
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 separated protons and neutrons
Binding Energy per Nucleon

16. Binding Energy Plot
Iron (Fe) is most binding energy/nucleon. Lighter have too few nucleons, heavier have too many.
Fission
Fusion
BINDING ENERGY in MeV/nucleon
Fission = Breaking large atoms into small
Fusion = Combining small atoms into large

17. Binding Energy Notes
Except for light nuclei, the binding energy is about 8 MeV per nucleon
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
The curve is slowly varying at A > 40
This suggests that the nuclear force saturates
A particular nucleon can interact with only a limited number of other nucleons

18. Question
Where does the energy released in the nuclear reactions of the sun come from?
covalent bonds between atoms
binding energy of electrons to the nucleus
(3) binding energy of nucleons

19. Question Which element has the highest binding energy/nucleon?
Neon (Z=10)
Iron (Z=26)
Iodine (Z=53)

20. Question
Which of the following is most correct for the total binding energy of an Iron atom (Z=26)?
9 MeV
234 MeV
270 MeV
504 Mev
For Fe, B.E./nucleon  9MeV
has 56 nucleons
Total B.E  56x9=504 MeV

21. Proton Mass in MeV

22. Binding Energy MDeuteron = MProton + MNeutron – |Binding Energy|
Einstein’s famous equation E = m c2
Proton: mc2 = 938.3MeV
Neutron: mc2= 939.5MeV
Adding these, get MeV
Difference is Binding energy, 2.2MeV
Deuteron: mc2 =1875.6MeV
MDeuteron = MProton + MNeutron – |Binding Energy|

23. Big Problem: binding energy
Calculate the average binding energy per nucleon of
Using atomic mass units

24. Calculate the average binding energy per nucleon of
In order to compute binding energy, let’s first find the mass difference between the total mass of all protons and neutrons in Nb and subtract mass of the Nb:
Given:
mp= u
mn= u
Find:
Eb = ?
Number of protons:
1u=931.5 MeV
Number of neutrons:
Mass difference:
Thus, binding energy is

25. Radioactivity Radioactivity is the spontaneous emission of radiation
Experiments suggested that radioactivity was the result of the decay, or disintegration, of unstable nuclei
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

26. Types of Radioactivity
B field into screen
Radioactive sources
detector
a particles: nuclei
Barely penetrate a piece of paper
b- particles: electrons
Can penetrate a few mm of aluminum
Can penetrate several cm of lead
g photons (more energetic than x-rays)

27. The Decay Processes – General Rules
When one element changes into another element, the process is called spontaneous decay or transmutation
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
Conservation of mass-energy and conservation of momentum must hold

28. 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
X is called the parent nucleus
Y is called the daughter nucleus

29. Alpha Decay -- Example Decay of 226Ra
Half life for this decay is 1600 years
Excess mass is converted into kinetic energy
Momentum of the two particles is equal and opposite

30. Beta Decay
During beta decay, the daughter nucleus has the same number of nucleons as the parent, but the atomic number is one less
In addition, an electron (positron) was observed
The emission of the electron is from the nucleus
The nucleus contains protons and neutrons
The process occurs when a neutron is transformed into a proton and an electron
Energy must be conserved

31. Beta Decay – Electron Energy
The energy released in the decay process should almost all go to kinetic energy of the electron
Experiments showed that few electrons had this amount of kinetic energy
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

32. Beta Decay Symbolically  is the symbol for the neutrino
is the symbol for the antineutrino
W is the symbol for the W-boson (weak force)
To summarize, in beta decay, the following pairs of particles are emitted
An electron and an antineutrino
A positron and a neutrino

33. Gamma Decay
Gamma rays are given off when an excited nucleus “falls” to a lower energy state
Similar to the process of electron “jumps” to lower energy states and giving off photons
The excited nuclear states result from “jumps” made by a proton or neutron
The excited nuclear states may be the result of violent collision or more likely of an alpha or beta emission
Example of a decay sequence
The first decay is a beta emission
The second step is a gamma emission
The C* indicates the Carbon nucleus is in an excited state
Gamma emission doesn’t change either A or Z

34. Decay Rules g: example Nucleon Number is conserved.
Atomic Number (charge) is conserved.
Energy and momentum are conserved.
: example
recall
238 = Nucleon number conserved
92 = Charge conserved
Comment on preflight only 30% got correct that alpha decay does NOT have charge increase by 2.
: example
Neutrino needed to conserve energy and momentum.
g: example

35. Practice A nucleus undergoes  decay. Which of the following is FALSE?
Nucleon number decreases by 4
2. Neutron number decreases by 2
3. Charge on nucleus increases by 2
 decay is the emission of
A decreases by 4
Ex.
Z decreases by 2
(charge decreases!)

36. Practice The nucleus undergoes decay. Which of the following is true?
1. The number of protons in the daughter nucleus increases by one.
2. The number of neutrons in the daughter nucleus increases by one.
decay is accompanied by the emission of an electron: creation of a charge -e.
In fact, inside the nucleus, and the electron and neutrino “escape.”

37. Decay Which of the following decays is NOT allowed? 1 2 3 4
238 =
92 =
214 =
84 =
14 = 14+0
6 <> 7+0
40 =
19 = 1 2 3 4

38. Which of the following are possible reactions?
Decay
Which of the following are possible reactions?
(a) and (b). Reactions (a) and (b) both conserve total charge and total mass number as required. Reaction (c) violates conservation of mass number with the sum of the mass numbers being 240 before reaction and being only 223 after reaction.

39. 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
λ is called the decay constant and determines the rate at which the material will decay
The decay rate or activity, R, of a sample is defined as the number of decays per second

40. Radioactivity Decays per second, or “activity”
No. of nuclei present
Decays per second, or “activity”
decay constant
Start with C atoms.
After 6000 years, there are only 8 left.
How many will be left after another 6000 years?
1) 0 2) 4 3) 8
Every 6000 years ½ of atoms decay

41. Decay Curve The decay curve follows the equation
The half-life is also a useful parameter
The half-life is defined as the time it takes for half of any given number of radioactive nuclei to decay

42. Units The unit of activity, R, is the Curie, Ci
1 Ci = 3.7 x 1010 decays/second
The SI unit of activity is the Becquerel, Bq
1 Bq = 1 decay / second
Therefore, 1 Ci = 3.7 x 1010 Bq
The most commonly used units of activity are the mCi and the µCi

43. Decay Function
time

44. Practice
The half-life for beta-decay of 14C is ~6,000 years. You test a fossil and find that only 25% of its 14C is un-decayed. How old is the fossil?
3,000 years
6,000 years
12,000 years
At 0 years: 100% remains
At 6,000 years: 50% remains
At 12,000 years: 25% remains

45. Radioactivity Quantitatively
No. of nuclei present
Decays per second, or “activity”
decay constant
Survival:
No. of nuclei present at time t
No. we started with at t=0
Instead of base e we can use base 1/2:
where
Half life
Then we can write

46. Radioactivity Example
The half-life for beta-decay of 14C is 5730 years. If you start with 1000 carbon-14 nuclei, how many will be around in years?

47. Uses of Radioactivity Carbon Dating Smoke detectors Radon pollution
Beta decay of 14C is used to date organic samples
The ratio of 14C to 12C is used
Smoke detectors
Ionization type smoke detectors use a radioactive source to ionize the air in a chamber
A voltage and current are maintained
When smoke enters the chamber, the current is decreased and the alarm sounds
Radon pollution
Radon is an inert, gaseous element associated with the decay of radium
It is present in uranium mines and in certain types of rocks, bricks, etc that may be used in home building
May also come from the ground itself

49. Binding Energy M1 = Mballs + Mspring
Which system “weighs” more?
Two balls attached by a relaxed spring.
Two balls attached by a stretched spring.
They have the same weight.
M1 = Mballs + Mspring
M2 = Mballs + Mspring + Espring/c2
M2 – M1 = Espring/c2
≈ Kg

50. Strong Nuclear Force Quantum Chromodynamics
Acts on Protons and Neutrons
Also holds Quarks together inside Protons and Neutrons
performed by residual effect of gluons
Strong enough to overcome Coulomb repulsion
Acts over very short distances
Two atoms don’t feel force (why Short distances ... gluons act on gluons)
What keeps protons together? Use link

51. Q Values (nice to know)
Energy must also be conserved in nuclear reactions
The energy required to balance a nuclear reaction is called the Q value of the 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

52. Problem: nuclear reactions
Determine the product of the reaction
What is the Q value of the reaction?

53. Determine the product of the reaction
What is the Q value of the reaction?
In order to balance the reaction, the total amount of nucleons (sum of A-numbers) must be the same on both sides. Same for the Z-number.
Given:
reaction
Find:
Q = ?
Number of nucleons (A):
Number of protons (Z):
Thus, it is B, i.e.
The Q-value is then
It is easier to use atomic mass units rather than kg.

54. Interesting Mass Facts
Mass of 3 Quarks:
Mass of Proton:
If a Proton is made on 3 Quarks, why is there such a mass difference?
The Gluons that hold Quarks together have so much MeV energy that they make up the difference
A Single Quark has infinite Mev energy (due to its gluon field) when alone (if you could get it out off the Nucleus)

55. Interesting Mass Facts II
Mass of 1 electron:
Mass of 1 proton:
Mass of Hydrogen:
Hey, Electron and Proton don’t exactly add up to Hydrogen. Why?
It take energy to separated the electron from a proton to make them individual particles.
Think about the energy of the combined positive and negative electric fields cancelling out when combined in Hydrogen.

56. Summary Nuclear Reactions Decays Nucleon number conserved
Charge conserved
Energy/Momentum conserved
a particles = nucleii
b- particles = electrons
g particles = high-energy photons
Decays
Half-Life is time for ½ of atoms to decay
Survival: