Problem 75 - Physics for Scientists and Engineers with Modern Physics

Question

The nuclide \({ }_{76}^{191}\) Os decays with \(\beta^{-}\) energy of 0.14 MeV accompanied by \(\gamma\) rays of energy \(0.042 \mathrm{MeV}\) and \(0.129 \mathrm{MeV}\) (a) What is the daughter nucleus? (b) Draw an energy-level diagram showing the ground states of the parent and daughter and excited states of the daughter. (c) To which of the daughter states does \(\beta^{-}\) decay of \({ }_{76}^{191}\) Os occur?

Step-by-Step Solution

Answer

(a) The daughter nucleus is \( {}_{77}^{191} \)Ir. (b) The diagram shows \( {}_{76}^{191} \)Os to \( {}_{77}^{191} \)Ir, with \( \gamma \) emissions to ground state. (c) Decay occurs to an excited state 0.171 MeV above ground of \( {}_{77}^{191} \)Ir.

Step 1: Understanding the Decay Process

The nuclide \( {}_{76}^{191} \)Os undergoes \( \beta^- \) decay. In \( \beta^- \) decay, a neutron in the nucleus decays into a proton, an electron (\( \beta^- \) particle), and an antineutrino. This increases the atomic number by 1 while keeping the mass number the same.

Step 2: Identifying the Daughter Nucleus

The nuclide after the \( \beta^- \) decay is \( {}_{77}^{191} \)Ir. The atomic number increases by 1 (76 to 77), while the mass number remains 191.

Step 3: Energy-level Diagram Analysis

The energy-level diagram shows the transitions between different energy states. After \( \beta^- \) decay, the daughter nucleus \( {}_{77}^{191} \)Ir is likely in an excited state, which then de-excites through \( \gamma \) emission (0.042 MeV and 0.129 MeV) to reach the ground state.

Step 4: Determining the Daughter State from \( \beta^- \) Decay

The \( \beta^- \) decay likely leads to an excited state of \( {}_{77}^{191} \)Ir, which then decays by emitting \( \gamma \) rays of energies 0.042 MeV and 0.129 MeV before reaching the ground state. The sum of the gamma energies suggests that the initial excited state had an energy of 0.171 MeV above the ground state.

Key Concepts

Beta Decay

In beta decay, a nucleus emits a beta particle, which is simply an electron. This results from a neutron in the nucleus transforming into a proton. As a consequence, the atomic number of the element increases by one since there is an additional proton. However, the mass number remains the same because the number of nucleons doesn't change.

• Example: In the case of the nuclide \({ }_{76}^{191}\) Os, it undergoes beta minus (\( \beta^-\)) decay. During this process, the neutron converts to a proton, and it emits an electron and an antineutrino.
• Effect: The atomic number changes from 76 to 77, forming a new element while keeping the same mass number of 191.

This is how \({ }_{76}^{191}\) Os transforms into \({ }_{77}^{191}\) Ir, involving a change in both atomic identity and energy states.

Gamma Emission

Gamma emission often occurs following beta decay. When a nucleus emits gamma rays, it is moving from a higher energy level to a lower one, releasing excess energy. Gamma rays are highly energetic electromagnetic waves, much like X-rays but with more energy. They have no mass and do not change the identity of the nucleus, only its energy state.

• Role: Gamma emission helps a nucleus shed its extra energy after a decay process like beta decay.
• Example:** After \({ }_{77}^{191}\) Ir forms from beta decay of \({ }_{76}^{191}\) Os, it might be in an excited state. The excited \({ }_{77}^{191}\) Ir then releases energy in the form of 0.042 MeV and 0.129 MeV gamma rays to settle into a stable or ground state.

By emitting gamma rays, the nucleus becomes more stable without any change in its atomic number or mass number.

Energy Level Diagram

An energy level diagram visually represents the different energy states available to a nucleus. It shows transitions between these states, often representing beta decay and subsequent gamma emissions. Each level on the diagram corresponds to a particular energy state of the nucleus.

• Purpose: It helps us understand how a nucleus progresses through various energy states during a decay process.
• Example: For \({ }_{77}^{191}\) Ir, the energy level diagram indicates that this nucleus had an initial excited state of 0.171 MeV above its ground state. It then transitions down through intermediate states, emitting gamma rays along the way.

Such diagrams make it easier to visualize the path of decay, starting from the parent nucleus to reaching the final stable configuration of the daughter nucleus.

Daughter Nucleus

When a radioactive nuclide undergoes nuclear decay, it transforms into a different nucleus, known as the 'daughter nucleus'. The identity of the daughter nucleus depends on the type of decay undergone by the original nucleus. In beta decay, the change results from an increase in the atomic number by one.

• Beta Decay Example: \({ }_{76}^{191}\) Os becomes \({ }_{77}^{191}\) Ir once it undergoes beta decay. Here, the daughter nucleus is \({ }_{77}^{191}\) Ir.
• Characteristics:** It has the same mass number as the parent but one more proton.

Understanding the transformation process aids in predicting the characteristics and behavior of the resulting material after decay.

Radioactive Nuclides

Radioactive nuclides are unstable atomic nuclei that decay over time, releasing radiation as they transform into more stable forms. Such decays can occur naturally or be induced artificially through nuclear reactions.

• Radioactivity: The process of emitting particles or electromagnetic waves is known as radioactivity, which is a characteristic behavior of unstable nuclei.
• Example: \({ }_{76}^{191}\) Os is a radioactive nuclide. As it decays through beta decay and gamma emission, it moves towards a stable state.

These transitions often result from the nucleus aiming to achieve a lower energy and more stable configuration, ultimately forming non-radioactive nuclides.