Nuclear Shell Model

• The difference of the shape between the proton and the neutron potentials are due to the Coulomb interaction on the proton.
• Nuclei have a Fermi energy level which is the highest energy level filled in the nucleus.
• In the ground state of a nucleus, all the energy levels below the Fermi level are filled.

The nuclear potential felt by the neutron and the proton

Neutrons are more strongly bound due to the absence of the repulsive Coulomb force

Nuclear Models

• Energy-level diagrams for ¹²C and ¹⁶O.

• Both are stable because they are even-even.

Case 1: If we add one proton to ¹²C to make

unstable

Case 2: If we add one neutron to ¹²C to make ¹³C:

stable

Nuclear Shell Model

• Even when we add another neutron to produce ¹⁴C, we find it is barely unstable.

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• Indicating neutron energy levels to be lower in energy than the corresponding proton ones.

• In this mass region, nature prefers the number of neutrons and protons to be N ≈ Z, but it doesn’t want N < Z.

This helps explain why ¹³C is stable, but not ¹³N

Nuclear shell model with well defined orbital states

(each nucleon moves in the average field of all other nucleons)

The Nobel Prize in Physics 1963.

J. Hans D. Jensen 

Maria Goeppert Mayer

Nuclear Shell Model

Magic numbers(high stability nuclei) show shell structure Goeppert-Mayer,Jensen(1963 Nobel price)

N or Z=2,8,20,28,50,82,126

Radioactivity is characteristic of elements with large atomic numbers

. Elements with at least one stable isotope are shown in light blue. Green shows elements of which the most stable isotope has a half-life measured in millions of years. Yellow and orange are progressively less stable, with half-lives in thousands or hundreds of years, down toward one day. Red and purple show highly and extremely radioactive elements where the most stable isotopes exhibit half-lives measured on the order of one day and much less.

12.6: Radioactive Decay

• The discoverers of radioactivity were Wilhelm Röntgen, Henri Becquerel, Marie Curie and her husband Pierre.
• Marie Curie and her husband Pierre discovered polonium and radium in 1898.
• The simplest decay form is that of a gamma ray, which represents the nucleus changing from an excited state to lower energy state.
• Other modes of decay include emission of α particles, β particles, protons, neutrons, and fission.

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• The disintegrations or decays per unit time (activity):

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where dN / dt is negative because total number N decreases with time.

Radioactive Decay

• SI unit of activity is the becquerel: 1 Bq = 1 decay / s
• Recent use is the Curie (Ci) 3.7 × 10¹⁰ decays / s

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• If N(t) is the number of radioactive nuclei in a sample at time t, and λ (decay constant) is the probability per unit time that any given nucleus will decay:

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• If we let N(t = 0) ≡ N₀

----- radioactive decay law

Radioactive Decay

• The activity R is

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where R₀ is the initial activity at t = 0

• It is common to refer to the half-life t_1/2 or the mean lifetime τ rather than its decay constant.

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• The half-life is

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• The mean lifetime is

Radioactive Decay

• The number of radioactive nuclei as a function of time

Euler’s number e=2.71828..

The exponential function changes by equal amounts in equal times

12.7: Alpha, Beta, and Gamma Decay

When a nucleus decays, all the conservation laws must be

observed:

• Mass-energy
• Linear momentum
• Angular momentum
• Electric charge
• Conservation of nucleons
• The total number of nucleons (A, the mass number) must be conserved in a low-energy nuclear reaction or decay.

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Alpha, Beta, and Gamma Decay

• Let the radioactive nucleus be called the parent and have the mass

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• Two or more products can be produced in the decay.
• Let the lighter one be M_y and the mass of the heavier one (daughter) be M_D.
• The conservation of energy is

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where Q is the energy released (disintegration energy) and equal to the total kinetic energy of the reaction products(note:Q(disintegration) is the negative of B(binding)

• If B > 0, a nuclide is bound and stable;
• If Q > 0, a nuclide is unbound, unstable, and may decay
• If Q < 0, decay emitting nucleons do not occur

Binding enery refers to stable, whereas disintegration energy to unstable nuclei

Alpha Decay a collection of nucleons inside a nucleus decays���

• The nucleus ⁴He has a binding energy of 28.3 MeV.
• If the last two protons and two neutrons in a nucleus are bound by less than 28.3 MeV, then the emission of an alpha particle (alpha decay) is possible.

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• If Q > 0, alpha decay is possible

EX.

The appropriate masses are

Q= 6 MeV and alpha decay is possible

Q=( 230.004u -226.025 -4.003u )c^2(931.5 MeV/c^2 u)= 6 MeV

Beta Decay

• Unstable nuclei may move closer to the line of stability by undergoing beta decay.
• The decay of a free neutron is

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• The beta decay of ¹⁴C (unstable) to form ¹⁴N, a stable nucleus, can be written as

The electron energy spectrum from the beta decay

Figure 12.13

Beta Decay

• There was a problem in neutron decay, the spin ½ neutron cannot decay to two spin ½ particles, a proton and an electron. ¹⁴C has spin 0, ¹⁴N has spin 1, and the electron has spin ½.

we cannot combine spin ½ & 1 to obtain a spin 0.

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• Wolfgang Pauli suggested a neutrino that must be produced in beta decay. It has spin quantum number ½, charge 0, and carries away the additional energy missing in Fig. (12.13).

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β⁻ decay in an atomic nucleus (the accompanying antineutrino is omitted). The inset shows beta decay of a free neutron.

Can neutrinos penetrade the earth?They come straight through the earth at nearly the speed of light, all the time, day and night, in enormous numbers. About 100 trillion neutrinos pass through our bodies every second.

Beta Decay

• An occasional electron is detected with the kinetic energy K_max required to conserve energy, but in most cases the electron’s kinetic energy is less than K_max.

the neutrino has little or no mass, and its energy may be all kinetic

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• Neutrinos have no charge and do not interact electromagnetically.
• They are not affected by the strong force of the nucleus.
• They are the weak interaction.
• The electromagnetic and weak forces are the electroweak force.

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Radioactive decay modes conservation of nucleons

Gamma Decay

• If the decay proceeds to an excited state of energy E_x rather than to the ground state, then Q for the transition to the excited state can be determined with respect to the transition to the ground state. The disintegration energy Q to the ground state Q₀.

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• Q for a transition to the excited state E_x is

Gamma Decay

• The excitation energies tend to be much larger, many keV or even MeV.
• The possibilities for the nucleus to rid itself of this extra energy is to emit a photon (gamma ray).
• The gamma-ray energy hf is given by the difference of the higher energy state E_> and lower one E_<.

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• The decay of an excited state of ^AX* (where * is an excited state) to its ground state is

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• A transition between two nuclear excited states E_> and E_< is

12.8: Radioactive Nuclides

• The unstable nuclei found in nature exhibit natural radioactivity.

Big Bang was 13.7 billion years ago

3.154^+7 s/y

Radioactive Nuclides

• The radioactive nuclides made in the laboratory exhibit artificial radioactivity.
• Heavy radioactive nuclides can change their mass number only by alpha decay (^AX → ^A−4D) but can change their charge number Z by either alpha or beta decay.
• There are only four paths that the heavy naturally occurring radioactive nuclides may take as they decay.
• Mass numbers expressed by either:
• 4n
• 4n + 1
• 4n + 2
• 4n + 3

Radioactive Nuclides

• The sequence of one of the radioactive series ²³²Th

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• ²¹²Bi can decay by either alpha or beta decay (branching).

Radon gas in the form of ²²²Rn is a health hazard�

The average indoor radon reading in Travis County, TX is predicted to be less than 2 picocuries per liter (pCi/L), so the county has been assigned EPA Radon Zone 3.

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Northern end of Lake Travis

Radon is a naturally occurring radioactive gas.

It’s produced when uranium, thorium, and radium break down in soil, rock, and water. It’s then released into the air. Radon is odorless, tasteless, and invisible.

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Curie (Ci) 3.7 × 10¹⁰ decays / s

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Radium-226 Decay Chain�

Time Dating Using Lead Isotopes

• A plot of the abundance ratio of ²⁰⁶Pb / ²⁰⁴Pb versus ²⁰⁷Pb / ²⁰⁴Pb can be a sensitive indicator of the age of lead ores. Such techniques have been used to show that meteorites, believed to be left over from the formation of the solar system, are 4.55 billion years old.
• The growth curve for lead ores from various deposits:

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The age of the specimens can be obtained from the abundance ratio of ²⁰⁶Pb/^204Pb versus ²⁰⁷Pb/²⁰⁴Pb.

Radioactive Carbon Dating

• Radioactive ¹⁴C is produced in our atmosphere by the bombardment of ¹⁴N by neutrons produced by cosmic rays.

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• When living organisms die, their intake of ¹⁴C ceases, and the ratio of ¹⁴C / ¹²C (= R) decreases as ¹⁴C decays. The period just before 9000 years ago had a higher ¹⁴C / ¹²C ratio by factor of about 1.5 than it does today.
• Because the half-life of ¹⁴C is 5730 years, it is convenient to use the ¹⁴C / ¹²C ratio to determine the age of objects over a range up to 45,000 years ago.

Calculate the binding energies of the most loosely bound neutron in the following nuclei

What is the energy released when three alpha parti-

cles combine to form 12C?

From chapter12 quiz

Which of the following reasons explains why the neutrino must exist?

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A. The neutrino is a product of gamma ray decay.

B. The neutrino is necessary to allow for the correct spin angular momentum conservation in a nuclear disintegration.

C. The neutrino is necessary to carry away a charge in a nuclear disintegration.

D. The neutrino is the force carrier that holds together quarks within protons and neutrons.

E. The neutrino decays into electrons and protons in an unstable nucleus.

If the age of the Earth is 4.5 billion years, what should the ratio of N^206 (Pb)/(N ^238 (U)) in a uranium-bearing rock as old as the Earth?