Problem 92 - Fundamentals of Physics

Question

High-mass radionuclides, which may be either alpha or beta emitters, belong to one of four decay chains, depending on whether their mass number \(A\) is of the form \(4 n, 4 n+1,4 n+2,\) or \(4 n+3\), where \(n\) is a positive integer. (a) Justify this statement and show that if a nuclide belongs to one of these families, all its decay products belong to the same family. Classify the following nuclides as to family: (b) \({ }^{235} \mathrm{U},\) (c) \({ }^{236} \mathrm{U},\) (d) \({ }^{238} \mathrm{U},\) (e) \({ }^{239} \mathrm{Pu},\) (f) \({ }^{240} \mathrm{Pu},(\mathrm{g}){ }^{245} \mathrm{Cm}\), (h) \({ }^{246} \mathrm{Cm},\) (i) \({ }^{249} \mathrm{Cf}\), and \((\mathrm{j}){ }^{253} \mathrm{Fm}\).

Step-by-Step Solution

Answer

Each nuclide belongs to a decay chain family based on its mass number mod 4: \(^{235}U\) and \(^{239}Pu\) are in the \(4n+3\) family, \(^{236}U\) and \(^{240}Pu\) are in the \(4n\) family, \(^{238}U\) and \(^{246}Cm\) are in the \(4n+2\) family, and \(^{245}Cm\), \(^{249}Cf\), \(^{253}Fm\) are in the \(4n+1\) family.

Step 1: Understanding the Mass Number Forms

The given statement mentions four different decay chains based on the mass number \( A \) of the nuclide. These are \( 4n, 4n+1, 4n+2, \) and \( 4n+3 \). A radionuclide belongs to one of these forms if \( A \), its mass number, is in the form specified by each chain, where \( n \) is an integer. Depending on the form, the radionuclide undergoes a series of decays along one of these chains.

Step 2: Justification of Decay Chain Consistency

Every decay, alpha or beta, changes the mass number \( A \) according to specific rules: Alpha decay reduces \( A \) by 4 (\( A-4 \)), preserving the modulo 4 property. Beta decay changes \( Z \) (the number of protons) but does not change \( A \) directly, preserving the residue mod 4. Thus, if a nuclide starts in a specific family, all its decay products remain in that family, maintaining the \( A \equiv n \mod 4 \).

Step 3: Classifying Nuclides

By calculating \( A \mod 4 \), we can classify each nuclide into one of the four families:- (b) \(^{235}U\): \( 235 \mod 4 = 3 \), so it belongs to the \( 4n+3 \) family.- (c) \(^{236}U\): \( 236 \mod 4 = 0 \), so it belongs to the \( 4n \) family.- (d) \(^{238}U\): \( 238 \mod 4 = 2 \), so it belongs to the \( 4n+2 \) family.- (e) \(^{239}Pu\): \( 239 \mod 4 = 3 \), so it belongs to the \( 4n+3 \) family.- (f) \(^{240}Pu\): \( 240 \mod 4 = 0 \), so it belongs to the \( 4n \) family.- (g) \(^{245}Cm\): \( 245 \mod 4 = 1 \), so it belongs to the \( 4n+1 \) family.- (h) \(^{246}Cm\): \( 246 \mod 4 = 2 \), so it belongs to the \( 4n+2 \) family.- (i) \(^{249}Cf\): \( 249 \mod 4 = 1 \), so it belongs to the \( 4n+1 \) family.- (j) \(^{253}Fm\): \( 253 \mod 4 = 1 \), so it belongs to the \( 4n+1 \) family.

Key Concepts

Alpha Decay

Alpha decay is a type of radioactive decay in which an unstable nucleus releases an alpha particle. An alpha particle consists of 2 protons and 2 neutrons, much like a helium nucleus. This process decreases the atomic number by 2 and the mass number by 4.
During alpha decay, the heavy nucleus reduces in mass and typically becomes more stable. This is common among heavy nuclides with atomic numbers greater than 83.

• Conservation in Alpha Decay: The total number of nucleons and charge are conserved. If a parent nucleus is of mass number \( A \), the daughter nucleus will be \( A-4 \).
• Example: If a nuclide undergoes alpha decay, like \( ^{238}U \rightarrow ^{234}Th + ^{4}He \), the residual mass number and element both reflect this transformation.

This decay chain operation maintains the characteristic of the nuclear family, keeping the modulo 4 property unchanged.

Beta Decay

Beta decay is another form of radioactive decay. It involves the conversion of a neutron into a proton with the emission of a beta particle (which can be an electron or a positron).
Beta decay increases the atomic number by 1 without altering the mass number. This is because the conversion affects only the internal composition of the nucleus, changing particles but not adding or losing nucleons.
Beta decay is significant for:

• Conservation of Mass Number: While \( A \), the mass number, remains constant during beta decay, the atomic number \( Z \) increases or decreases depending on the beta decay process.
• Example: A typical beta decay process can be represented as, \( ^{14}C \rightarrow ^{14}N + e^- + \overline{u}_e \), where a neutron transforms to a proton, and an electron is emitted.

Beta decay ensures that nuclides remain in their original decay family according to modulo 4 properties.

Mass Number Modulo

The concept of mass number modulo 4 is crucial in the classification of decay chains. A mass number \( A \) can be classified into four groups depending on its remainder when divided by 4: \( 4n, 4n+1, 4n+2, \) and \( 4n+3 \). This is the modulo operation.

• A radionuclide with \( A \equiv 0 \pmod{4} \) belongs to the Thorium series.
• Radionuclides with \( A \equiv 1 \pmod{4} \) are part of the Neptunium series.
• \( A \equiv 2 \pmod{4} \) nuclides belong to the Uranium series.
• An \( A \equiv 3 \pmod{4} \) denotes the Actinium series.

This understanding helps predict how decay processes will unfold and helps keep track of a nuclide's "family" even after multiple transformations. In both alpha and beta decay, the mass number's modulo 4 property ensures consistency within these chains, maintaining the family's characteristic.

Nuclide Classification

Nuclides can be categorized into families based on their mass number when considering radioactive decay chains. Classification is essential for understanding a nuclide's pathway through different decay stages.
Nuclides are classified based on their mass number modulo 4. By determining \( A \mod 4 \), you can identify the family that a nuclide belongs to.
For example:

• If \( ^{235}U \rightarrow 235 \mod 4 = 3 \), it is part of the Actinium family.
• For \( ^{236}U \rightarrow 236 \mod 4 = 0 \), aligning it with the Thorium series.

Significance of Nuclide Classification

This categorization is particularly helpful in predicting decay products and ensuring the integrity of analytical processes involving radioactive materials. It allows scientists and researchers to interpret and manage nuclear reactions effectively.