Particles and radiation, Particles problem-solving question

A researcher is investigating the properties of a specific isotope of Lithium, ${}^{7}_{3}\text{Li}$. (a) Calculate the specific charge of the nucleus of a ${}^{7}_{3}\text{Li}$ atom. Give your answer in $\text{C kg}^{-1}$ to an appropriate number of significant figures. (3 marks) (b) In a high-energy collision experiment, a proton strikes a lithium nucleus, resulting in the production of a neutral kaon ($K^0$) and other particles. The $K^0$ meson later decays into two pions: a $\pi^+$ and a $\pi^-$. State the quark composition of the $K^0$ meson and the $\pi^+$ meson. (2 marks) (c) The strong nuclear force is responsible for holding the ${}^{7}_{3}\text{Li}$ nucleus together. Describe how the magnitude and nature of the strong nuclear force change as the separation between two nucleons in the lithium nucleus increases from $0.2\text{ fm}$ to $4.0\text{ fm}$. (3 marks) Data for this question: Mass of proton $\approx 1.67 \times 10^{-27}\text{ kg}$ Mass of neutron $\approx 1.67 \times 10^{-27}\text{ kg}$ Elementary charge $e = 1.60 \times 10^{-19}\text{ C}$ $1\text{ fm} = 10^{-15}\text{ m}$

(a) Specific charge is defined as the ratio of charge to mass ($\frac{Q}{m}$). For the nucleus of ${}^{7}_{3}\text{Li}$: Number of protons $Z = 3$ Number of nucleons $A = 7$ Total charge $Q = Z \times e = 3 \times 1.60 \times 10^{-19}\text{ C} = 4.80 \times 10^{-19}\text{ C}$ Total mass $m \approx A \times m_{\text{nucleon}} = 7 \times 1.67 \times 10^{-27}\text{ kg} = 1.169 \times 10^{-26}\text{ kg}$ Specific charge $= \frac{4.80 \times 10^{-19}}{1.169 \times 10^{-26}} = 4.106 \times 10^7\text{ C kg}^{-1}$ To 3 significant figures: $4.11 \times 10^7\text{ C kg}^{-1}$. (b) Mesons consist of a quark-antiquark pair. The $K^0$ meson has a strangeness of $+1$ (or $-1$ for the anti-particle, but standard $K^0$ is $d\bar{s}$). Quark composition of $K^0$: $d\bar{s}$ (or $s\bar{d}$ for $\bar{K^0}$). The $\pi^+$ meson has a charge of $+1$ and strangeness $0$. Quark composition of $\pi^+$: $u\bar{d}$. (c) The strong nuclear force (SNF) behavior: 1. At $0.2\text{ fm}$ (below $0.5\text{ fm}$): The force is highly repulsive. (B1) 2. Between $0.5\text{ fm}$ and $3.0\text{ fm}$: The force becomes attractive, reaching a peak attraction at roughly $1.0\text{ fm}$. (B1) 3. Beyond $3.0\text{ fm}$: The force magnitude drops rapidly and becomes negligible/zero by $4.0\text{ fm}$. (B1)

(a) 3 marks - (M1) Correct substitution for charge: $3 \times 1.60 \times 10^{-19}$ (or $4.8 \times 10^{-19}$). - (M1) Correct substitution for mass: $7 \times 1.67 \times 10^{-27}$ (or $1.169 \times 10^{-26}$). - (A1) Final answer $4.11 \times 10^7\text{ C kg}^{-1}$ (accept $4.1 \times 10^7$ to $4.12 \times 10^7$). Must have correct units. (b) 2 marks - (B1) $K^0$: $d\bar{s}$ (accept $s\bar{d}$). - (B1) $\pi^+$: $u\bar{d}$. (c) 3 marks - (B1) States force is repulsive below $0.5\text{ fm}$ (relevant to the $0.2\text{ fm}$ start point). - (B1) States force is attractive between $0.5\text{ fm}$ and $3.0\text{ fm}$. - (B1) States force becomes negligible/zero beyond $3.0\text{ fm}$ (relevant to the $4.0\text{ fm}$ end point).

Part (a) Hint 1: Specific charge is the ratio of the total charge of the particle to its total mass. Hint 2: For a nucleus, the charge comes only from protons, but the mass comes from both protons and neutrons. Hint 3: Use the nucleon number (7) for the mass calculation and the proton number (3) for the charge. Part (b) Hint 1: Kaons and pions are mesons, meaning they must contain exactly one quark and one antiquark. Hint 2: Kaons possess the "strangeness" property; pions do not. Hint 3: Check the charges: $u = +2/3$, $d = -1/3$, $s = -1/3$. The $\pi^+$ must sum to $+1$. Part (c) Hint 1: The strong nuclear force is very short-range. Think about the specific distances where it switches from...