LECTURE 25 NUCLEAR STRUCTURE AND STABILITY

Instructor: Kazumi Tolich Lecture 25

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¨ 30.1 Nuclear structure ¤ Isotopes ¤ Atomic mass

¨ 30.2 Nuclear stability ¤ Biding energy

¨ 30.3 Forces and energy in the nucleus ¤ Low-� nuclei ¤ High-� nuclei Quiz: 30.1-1

¨ How many neutrons are in Li? Quiz: 30.1-1 answer

¨ How many neutrons are in Li? ¨ 8

¨ The atomic number, �, is the number of protons, the neutron number, �, is the number of neutrons, and the mass number, �, is the number of nucleons of the nucleus.

� = � + �

¨ In general, a nucleus of element � is written as �. ¨ The element is lithium, Li.

¨ The mass number � = 11 yields the number of nucleon.

¨ The atomic number � = 3 yields the number of protons. Quiz: 30.1-2

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¨ Can two different isotopes of a same element have the same nuclear mass? A. Yes B. No Quiz: 30.1-2 answer

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¨ No

¨ Each �-value in a series of nuclei with the same �-value is called an isotope.

¨ So, two different isotopes of a same element must have different number of nucleons. ¨ The atomic mass unit is u = 1.66 × 10 kg, roughly the mass of a proton or a neutron. Isotopes

¨ Deuterium is the isotope form of hydrogen; its nucleus has a proton and a neutron.

¨ The natural abundance of an isotope is the fraction of naturally occurring nuclei represented by that isotope for each element. 99.98 % abundance 0.02% abundance Trace ¨ The majority of more than 3,000 known Stable Stable Radioactive isotopes are radioactive.

¨ Only 266 isotopes are stable and occur in nature. 30.1 Atomic mass

¨ The atomic mass unit u = 1.6605 × 10 kg is defined such that the atomic mass of isotope C is exactly 12 u.

¨ The famous equation by Einstein tells us that mass is a form of energy.

� = ��

¨ The energy equivalent of 1 u of mass is � = �� = 931.49 MeV, or 1 u = = 931.49 MeV⁄�.

¨ The mass of a hydrogen atom is roughly equal to the sum of the masses of a proton and an electron.

¨ The mass of a helium atom is less than the sum of the masses of its protons, neutrons, and electrons due to the binding energy of the nucleus. Quiz: 30.1-3

¨ Boron, with atomic number � = 5, has two stable isotopes, with atomic mass numbers � = 10 and � = 11. Boron’s chemical atomic mass is 10.8. Approximately what is the natural abundance of B in percent? Quiz: 30.1-3 answer

¨ Boron, with atomic number � = 5, has two stable isotopes, with atomic mass numbers � = 10 and � = 11. Boron’s chemical atomic mass is 10.8. Approximately what is the natural abundance of B in percent? ¨ The natural abundance of an isotope is the fraction of naturally occurring nuclei represented by that isotope for each element.

¨ The chemical atomic mass shown on the periodic table is the weighted average of the atomic masses of all naturally occurring isotopes. ¨ Let � denote the natural abundance of B. ¨ B has 11 nucleons with ≈ 11 u, while B has 10 nucleons with ≈ 10 u, so the weighted average is � 11u + 1 − � 10u = 10.8u

¨ � = 0.8 or 80 % Quiz: 30.2-1

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¨ Which of the following is/are correct regarding the stability of nuclei? Choose all that apply. A. For light nuclei, the greatest stability is achieved when � ≈ �. B. For light nuclei, the greatest stability is achieved when � > �. C. For light nuclei, the greatest stability is achieved when � < �. D. For heavy nuclei, the greatest stability is achieved when � ≈ �. E. For heavy nuclei, the greatest stability is achieved when � > �. F. For heavy nuclei, the greatest stability is achieved when � < �. Quiz: 30.2-1 answer

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¨ For light nuclei, the greatest stability is achieved when � ≈ �.

¨ For heavy nuclei, the greatest stability is achieved when � > �.

¨ The stable nuclei cluster very close to the line of stability.

¨ There are no stable nuclei with � > 83 (bismuth).

¨ Heavier elements (up to � = 92, uranium) are found in nature. Quiz: 30.2-2

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¨ If a stable nucleus breaks into its constituent nucleons, was energy added to the nucleus, or was energy released by the nucleus? A. Energy was added to the nucleus. B. Energy was released by the nucleus. Quiz: 30.2-2 answer

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¨ Energy was added to the nucleus.

¨ The mass of the stable nucleus is less than the sum of the masses of the nucleons. If a nucleus could break apart spontaneously by releasing energy, it would not be stable.

¨ The binding energy of a nucleus with � protons and � neutrons is:

� = ∆�� = �� + �� − � × 931.49 MeV⁄u

¨ You can think of binding energy as the minimum energy needed to break apart a nucleus. Quiz: 30.2-3

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¨ Which nuclei is most stable? Indicate the approximate mass number on the curve of binding energy, the binding energy per nucleon. Quiz: 30.2-3 answer

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¨ � ≈ 60 where �⁄� is the maximum where the maximum energy per nucleon is needed to break apart the nucleus.

¨ In nuclear fusion, two light nuclei join together, resulting in a final nucleus with a higher binding energy per nucleon, releasing energy. ¤ The sun’s energy is from fusing hydrogen.

¨ In an alpha decay, a heavy nucleus becomes more stable by ejecting a small group of nucleons in order to decrease its mass, releasing energy.

¨ In nuclear fission very heavy nuclei split into two lighter nuclei, releasing energy. ¤ Nuclear power plants generate energy from fission of plutonium and uranium. Quiz: 30.3-1

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¨ There are 82 protons in a lead nucleus. Why doesn’t the lead nucleus burst apart due to their mutual electrostatic repulsion? A. Electrostatic repulsive force doesn’t act inside the nucleus. B. Gravity overpowers the electrostatic repulsive force inside the nucleus. C. The negatively charged neutrons balance the positively charged protons. D. The negatively charged electrons balance the positively charged protons. E. Protons lose their positive charge inside the nucleus. F. None of the above Quiz: 30.3-1 answer

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¨ None of the above

¨ Strong nuclear force binds them together. ¤ It is an attractive force between any two nucleons. ¤ Its range is only over nuclear distances (about 10-15 m). ¤ Over the range where it acts, it is stronger than the electrostatic force that tries to push two protons apart.

¨ In large nuclei, the repulsive electrostatic forces tends to make it unstable, but neutrons exert the strong force that holds the nucleus together.

¨ In small nuclei, one neutron per proton is sufficient for stability, so small nuclei have � ≈ �. 30.3 Forces and energy in the nucleus

¨ Nucleons have quantized energy levels like electrons, but their energy levels are separated by a million times more energy than the energy separation of electron energy levels.

¨ Protons and neutrons have spins, up or down. According to quantum mechanics (Pauli exclusion principle), each energy level can hold only a certain number of spin-up particles and spin-down particles. 30.3 Low-Z nuclei

¨ Low-Z nuclei (� < 8) have few protons, so we can neglect the electrostatic potential energy due to proton-proton repulsion, and the energy levels of protons and neutrons are essentially identical. ¨ C is in its lowest possible energy state. ¨ B and N could lower their energies in a beta decay. 30.3 High-Z nuclei

¨ In a nucleus with many protons, the increasing electrostatic potential energy raises the proton energy levels but not the neutron energy levels.

¨ Protons and neutrons fill the energy levels to the same height.

¨ Because neutron energy levels start at a lower energy, more neutron states are available.