Question

Which of the following nuclides have magic numbers of both protons and neutrons: (a) helium- 4, (b) oxygen-18, (c) calcium-40, (d) zinc-66, (e) lead-208?

Short answer

Helium-4, calcium-40, and lead-208 are the nuclides that have magic numbers of both protons and neutrons.

Step-by-step solution

(a) Helium-4

For helium-4, there are 2 protons and 2 neutrons (4 - 2 = 2). Both 2 protons and 2 neutrons belong to the magic number list (2). Therefore, helium-4 does have magic numbers for both protons and neutrons.

(b) Oxygen-18

For oxygen-18, there are 8 protons and 10 neutrons (18 - 8 = 10). 8 protons belong to the magic number list (8); however, 10 is not a magic number. Therefore, oxygen-18 doesn't have magic numbers for both protons and neutrons.

(c) Calcium-40

For calcium-40, there are 20 protons and 20 neutrons (40 - 20 = 20). Both 20 protons and 20 neutrons belong to the magic number list (20). Therefore, calcium-40 does have magic numbers for both protons and neutrons.

(d) Zinc-66

For zinc-66, there are 30 protons and 36 neutrons (66 - 30 = 36). 30 protons are not in the magic number list, but 36 is also not a magic number. Therefore, zinc-66 doesn't have magic numbers for both protons and neutrons.

(e) Lead-208

For lead-208, there are 82 protons and 126 neutrons (208 - 82 = 126). Both 82 protons and 126 neutrons belong to the magic number list (82 and 126). Therefore, lead-208 does have magic numbers for both protons and neutrons. In conclusion, helium-4, calcium-40, and lead-208 are the nuclides that have magic numbers of both protons and neutrons.

Key concepts

Stability of Nuclides

The stability of nuclides is a fundamental concept in nuclear physics that helps us understand why certain atoms are more stable than others. A nuclide is considered stable when it does not change or decay over time. The key to this stability often lies in the balance between protons and neutrons within the nucleus.
Nuclides with certain numbers of protons and neutrons are more stable due to favorable configurations. These configurations reduce the energy of the system, making it more difficult for the nucleus to decay.

• Stable nuclides generally have even numbers of protons and neutrons. This is because paired nucleons (protons and neutrons) help stabilize nuclei.
• The forces within the nucleus, such as nuclear force and electrostatic repulsion, play a crucial role in determining stability.

When a nuclide is stable, it means that it is energetically favorable and less likely to undergo radioactive decay. This makes understanding the stability of nuclides crucial for predicting the behavior of atoms and the elements they form.

Proton and Neutron Count

Protons and neutrons are the building blocks of the atomic nucleus. The combination and number of these particles define the properties of the atom.
Protons carry a positive charge and determine the atomic number, which defines the element. Neutrons have no charge and contribute to the atomic mass. Together, these particles compose the mass number of the nucleus.

• An increase in protons increases the electrostatic repulsion forces in the nucleus. However, neutrons help offset this force by adding nuclear binding energy, which holds the nucleus together.
• Having the right balance between protons and neutrons is essential for the stability of the nucleus, as seen in magic numbers.

When analyzing any given nuclide, it's essential to consider both the number of protons and the number of neutrons. This will help to determine if the nuclide falls into the category of being stable or not.

Nuclear Structure

Nuclear structure refers to the arrangement and behavior of protons and neutrons within the atomic nucleus. It defines the characteristics of various elements and isotopes and is essential for understanding the properties and stability of the atom.
At the heart of nuclear structure are the forces and interactions among nucleons (protons and neutrons). The strong nuclear force is the primary factor that holds the nucleus together, being much stronger than the repulsive electrostatic forces among protons.

• Nuclear models, such as shell models, help explain how nucleons are arranged. In these models, protons and neutrons are thought to exist in "shells" similar to electron shells in atoms.
• The concept of nuclear shells is crucial in understanding why certain numbers of protons and neutrons create particularly stable configurations. This leads to the concept of magic numbers.

The understanding of nuclear structure guides scientists in predicting the behavior of isotopes and helps in fields such as energy generation and medical applications.

Magic Numbers List

Magic numbers in nuclear physics are specific numbers of protons or neutrons that result in a complete shell within the nucleus. Such configurations confer added stability to the nuclide, making it less likely to undergo radioactive decay.
The currently accepted magic numbers for both protons and neutrons are 2, 8, 20, 28, 50, 82, and 126. A nucleus with these numbers is more stable than those without. For example:

• Helium-4 has 2 protons and 2 neutrons, both magic numbers, contributing to its stability.
• Calcium-40, with 20 protons and 20 neutrons (both magic numbers), is a classic example of a stable nuclide.
• Lead-208 has 82 protons and 126 neutrons, reinforcing its stability due to these magic numbers.

Understanding magic numbers is essential for nuclear physicists as they provide insight into the stability and structure of atomic nuclei, influencing how elements are formed in the universe.