Problem 14 - An Introduction to Modern Astrophysics

Question

Complete the following reaction sequences. Be sure to include any necessary leptons. (a) \(_{14}^{27} \mathrm{Si} \rightarrow \frac{2}{13} \mathrm{Al}+e^{+}+?\) (b) \(\frac{7}{13} \mathrm{Al}+_{1}^{1} \mathrm{H} \rightarrow_{12}^{24} \mathrm{Mg}+_{2}^{4} ?\) (c) \(_{17}^{35} \mathrm{Cl}+_{1}^{1} \mathrm{H} \rightarrow_{18}^{36} \mathrm{Ar}+?\)

Step-by-Step Solution

Answer

(a) Neutrino, (b) Alpha particle \(He\), (c) Neutron.

Step 1: Understanding Reaction Type (a)

The reaction given is \( ^{27}_{14}\text{Si} \rightarrow \frac{2}{13}\text{Al} + e^+ + ? \). This is a beta-plus (\(\beta^+\)) decay reaction, where a proton is converted to a neutron, and a positron \(e^+\) is emitted along with a neutrino.

Step 2: Identify Missing Particle (a)

During \(\beta^+\) decay, a neutrino \(u_e\) is emitted to conserve lepton number. So, the complete reaction is \( ^{27}_{14}\text{Si} \rightarrow \frac{2}{13}\text{Al} + e^+ + u_e \).

Step 3: Understanding Reaction Type (b)

The given reaction is \( \frac{7}{13} Al + ^{1}_{1} H \rightarrow ^{24}_{12}\text{Mg} + ^{4}_{2} ? \). This represents a nuclear reaction where a smaller nucleus combines to form a larger nucleus and a smaller particle.

Step 4: Identify Missing Particle (b)

In order to balance both atomic and mass numbers, the missing particle must be \( ^{4}_{2} \text{He} \), also known as an alpha particle. Thus, the complete reaction is \( \frac{7}{13} Al + ^{1}_{1} H \rightarrow ^{24}_{12}\text{Mg} + ^{4}_{2} He \).

Step 5: Understanding Reaction Type (c)

The given reaction is \( ^{35}_{17}\text{Cl} + ^{1}_{1} H \rightarrow ^{36}_{18}\text{Ar} + ? \). This is a typical nuclear reaction involving nucleon exchange.

Step 6: Identify Missing Particle (c)

To balance the reaction, considering both atomic and mass numbers, the missing particle is a neutron \( n \). The completed reaction is \( ^{35}_{17}\text{Cl} + ^{1}_{1} H \rightarrow ^{36}_{18}\text{Ar} + n \).

Key Concepts

Beta-plus Decay

In beta-plus decay, a proton inside an atomic nucleus is transformed into a neutron. This process involves the emission of a positron, denoted as \( e^+ \), and a neutrino, a particle with no electrical charge and very little mass. This transformation allows the element to move one place down the periodic table, as its atomic number decreases by one, while the mass number remains the same.

This type of decay is commonly observed in proton-rich nuclei or in certain artificially made isotopes. The emitted positron is the antimatter counterpart of an electron, and the neutrino (often a type known as the electron neutrino, represented as \( u_e \)) helps balance the conservation of lepton number. This means that the number of particles like positrons and neutrinos, which fall under the "lepton" category, remains the same before and after decay.

• Proton is converted to a neutron
• Emits a positron \( e^+ \) and a neutrino \( u_e \)
• Atomic number decreases, mass number remains constant

Alpha Particle

An alpha particle is essentially a helium nucleus, composed of 2 protons and 2 neutrons, symbolized as \( ^{4}_{2} \text{He} \). In nuclear reactions, such as those involving the fusion or fission of heavier elements, alpha particles are often either added to or removed from a nucleus.

These particles play a crucial role in balancing nuclear reactions. For instance, in the provided exercise, the emission of an alpha particle balances the atomic and mass numbers during the transformation from two smaller nuclei into a heavier nucleus.

The distinct features of an alpha particle include:

• High mass relative to other forms of radioactive decay particles
• Low penetration ability, can be stopped by paper or skin
• Carries a +2 charge due to its 2 protons
• Stability in terms of nuclear reactions involving heavy elements

Neutron Emission

Neutron emission involves the release of neutrons from an atomic nucleus. Unlike charged particles, such as protons or electrons, neutrons are neutral, meaning they do not carry any electrical charge. This makes them an essential component in nuclear reactions, as they can induce other nuclear interactions without being repelled by an electrical charge.

In nuclear reactions, neutron emission often accompanies reactions where a lighter nucleus is combined with a proton, resulting in a slightly heavier product and the release of a free neutron. Neutron emission stabilizes the resulting atom's nucleus and conserves both the mass and atomic numbers during the reaction.

• Neutrons are neutral particles, thus have no charge
• Can penetrate materials more deeply than charged particles
• Plays a critical role in nuclear chain reactions

Understanding these fundamentals facilitates the prediction and understanding of nuclear reactions and the resulting elements and particles involved.