Nuclear Physics Cheatsheet

Cheatsheet Content

### Atomic Nucleus: Its Constituents - **Definition:** The nucleus is a small, dense region consisting of protons and neutrons at the center of an atom. - **Nuclear Size:** Order of $10^{-14}$ m. - **Atomic Diameter:** Order of $10^{-10}$ m. - Most of the atom is empty space. ### Composition of a Nucleus - **Discovery of Neutrons:** By Chadwick, leading to Heisenberg's proton-neutron hypothesis in 1932. - **Main Building Blocks:** Protons and neutrons. - **Nucleus Composition:** A nucleus of mass number $A$ and atomic number $Z$ contains $Z$ protons and $(A-Z)$ neutrons. - Protons contribute positive charge and mass. Neutrons contribute mass. - **Electrical Neutrality:** The number of extra-nuclear electrons is $Z$ to neutralize the positive charge of the nucleus. - **Proton:** - Fundamental particle, nucleus of hydrogen. - Positive charge: $1.6 \times 10^{-19}$ C. - Rest mass: $m_p = 1.6726 \times 10^{-27}$ kg. (Approx. 1836 times electron mass). - Intrinsic (spin) angular momentum: $1/2$. - **Neutron:** - Charge-less fundamental particle ($q_{net} = 0$). - Rest mass: $m_n = 1.6749 \times 10^{-27}$ kg (slightly greater than proton). - Intrinsic (spin) angular momentum: $1/2$. - $m_n \approx m_p > m_e$. ### Nucleons - **Definition:** Protons and neutrons present in the nuclei of atoms are collectively known as nucleons. - **Atomic Number ($Z$):** The number of protons in the nucleus. - **Mass Number ($A$):** The total number of protons and neutrons in a nucleus. - **Relations:** - Number of protons in an atom = $Z$ - Number of electrons in an atom = $Z$ - Number of nucleons in an atom = $A$ - Number of neutrons in an atom = $A - Z$ ### Nuclear Mass & Nuclide - **Nuclear Mass:** The total mass of the protons and neutrons present in a nucleus. - **Nuclide:** An atom with a specific nuclear composition, characterized by its atomic number $Z$ and mass number $A$. - **Symbolic Representation:** $_Z^A X$ - $X$: Chemical symbol of the element. - $Z$: Atomic number. - $A$: Mass number. ### Isotopes - **Definition:** Atoms of an element with the same atomic number ($Z$) but different mass numbers ($A$). - They have the same number of protons and electrons but different numbers of neutrons. - **Examples:** - **Hydrogen:** - Protium ($^1_1 H$): 1 proton. - Deuterium ($^2_1 H$): 1 proton, 1 neutron. - Tritium ($^3_1 H$): 1 proton, 2 neutrons. - **Lithium:** $^6_3 Li$ and $^7_3 Li$. ### Average Atomic Mass (A.A.M.) - **Definition:** The weighted average of the atomic masses of all isotopes of an element, considering their relative abundances. - **Example (Chlorine):** Normal chlorine contains 75% of $^{35}_{17}Cl$ and 25% of $^{37}_{17}Cl$. - Average atomic mass of chlorine: $(0.75 \times 35) + (0.25 \times 37) = 35.5 \text{ u}$ ### Isobars - **Definition:** Atoms having the same mass number ($A$) but different atomic numbers ($Z$). - They contain different numbers of protons and electrons, leading to different chemical properties and positions in the periodic table. - **Examples:** - $^3_1 H$ and $^3_2 He$ (both $A=3$). - $^{37}_{17}Cl$ and $^{37}_{16} S$ (both $A=37$). - $^{40}_{20} Ca$ and $^{40}_{18} Ar$ (both $A=40$). ### Isotones - **Definition:** Nuclides having the same number of neutrons ($N$). - **Number of neutrons:** $N = A - Z$. - **Examples:** - $^{37}_{17}Cl$ ($N = 37-17=20$) and $^{39}_{19} K$ ($N = 39-19=20$). - $^{198}_{80} Hg$ ($N = 198-80=118$) and $^{197}_{79} Pu$ ($N = 197-79=118$). ### Atomic Masses - **Atomic Mass Unit (amu or u):** - Defined as $1/12$th of the actual mass of a carbon-12 atom. - Mass of carbon-12 atom = $1.992678 \times 10^{-26}$ kg. - $1 \text{ amu} = \frac{1}{12} \times 1.992678 \times 10^{-26} \text{ kg} = 1.660565 \times 10^{-27} \text{ kg}$. - **Mass of fundamental particles:** - Electron mass: $m_e = 0.00055 \text{ amu} = 9.11 \times 10^{-31} \text{ kg}$. - Proton mass: $m_p = 1.0073 \text{ amu} = 1.6726 \times 10^{-27} \text{ kg}$. - Neutron mass: $m_n = 1.0086 \text{ amu} = 1.6749 \times 10^{-27} \text{ kg}$. - **Mass of Hydrogen atom:** $m_H = m_p + m_e = 1.0078 \text{ amu}$. ### Electron Volt (eV) - **Definition:** The energy acquired by an electron when it is accelerated through a potential difference of 1 volt. - $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$. ### Relation between amu and MeV - **Einstein's Mass-Energy Equivalence:** $E = mc^2$. - To find energy equivalent of 1 amu: - $m = 1 \text{ amu} = 1.66 \times 10^{-27} \text{ kg}$. - Speed of light $c = 2.998 \times 10^8 \text{ m/s}$. - $E = (1.66 \times 10^{-27}) \times (2.998 \times 10^8)^2 \text{ J}$. - Convert Joules to MeV: $1 \text{ MeV} = 1.602 \times 10^{-13} \text{ J}$. - $E = \frac{(1.66 \times 10^{-27}) \times (2.998 \times 10^8)^2}{1.602 \times 10^{-13}} \text{ MeV}$. - $1 \text{ amu} \approx 931 \text{ MeV}$. ### Nuclear Size & Density - **Size of Nucleus:** - Volume of nucleus $\propto$ Mass Number ($A$). - $V \propto A \implies \frac{4}{3}\pi R^3 \propto A \implies R^3 \propto A$. - **Nuclear Radius:** $R = R_0 A^{1/3}$, where $R_0 \approx 1.2 \text{ fm}$ (fermi). - **Density of Nucleus:** - Density $\rho = \frac{\text{Mass}}{\text{Volume}} = \frac{Am}{\frac{4}{3}\pi R^3}$. - Substituting $R = R_0 A^{1/3}$: $\rho = \frac{Am}{\frac{4}{3}\pi (R_0 A^{1/3})^3} = \frac{Am}{\frac{4}{3}\pi R_0^3 A} = \frac{m}{\frac{4}{3}\pi R_0^3}$. - Nuclear density is constant and independent of the element (approx. $2.3 \times 10^{17} \text{ kg/m}^3$). ### Mass Defect - **Definition:** The difference between the sum of the rest masses of the constituent protons and neutrons in their free state and the rest mass of the stable nucleus. - **Formula:** For a nucleus $_Z^A X$ with mass $m$, mass defect $\Delta m$ is: - $\Delta m = [Z m_p + (A-Z) m_n] - m$. - Where $m_p$ is proton mass, $m_n$ is neutron mass. ### Packing Fraction (P.F.) - **Definition:** Mass defect per nucleon. - **Formula:** $P.F. = \frac{\Delta m}{A}$. - **Significance:** - **Positive P.F.:** Nucleus is unstable (e.g., mass number $ 200$). - **Negative P.F.:** Nucleus is stable. Mass has been converted into energy, binding nucleons together (e.g., mass number between $20$ and $200$). - Directly related to the availability of nuclear energy and nuclear stability. ### Binding Energy (B.E.) - **Definition:** 1. The energy required to break up a nucleus into its constituent protons and neutrons and separate them to infinite distance. 2. The surplus energy released when nucleons combine to form a nucleus. - **Formula (from mass defect):** $B.E. = \Delta m c^2 = [Z m_p + (A-Z) m_n - m_N] c^2$. - Where $m_N$ is the nuclear mass of $_Z^A X$. - **Expression in terms of atomic mass:** - Considering atomic mass $m(_Z^A X)$ and hydrogen atom mass $m_H$: - $\Delta E_b = [Z m_H + (A-Z) m_n - m(_Z^A X)] c^2$. ### Binding Energy Per Nucleon (B.E.N.) - **Definition:** The average energy required to extract one nucleon from the nucleus. - **Formula:** $B.E.N. = \frac{\Delta E_b}{A} = \frac{[Z m_H + (A-Z) m_n - m(_Z^A X)] c^2}{A}$. - Gives a measure of the force binding nucleons together inside a nucleus. ### Binding Energy Curve - A plot of B.E.N. versus mass number ($A$). - **Features:** - **Light Nuclei ($A 120$):** Gradual decrease in B.E.N. to about $7.6 \text{ MeV/nucleon}$ for $^{238}_{92} U$. This decrease is due to Coulomb repulsion between protons, making heavier nuclei less stable. ### Importance of Binding Energy Curve - Explains nuclear fission and fusion. - **Nuclear Fission:** - Heavier nuclei have smaller B.E.N. (less stable). - When a heavy nucleus splits into lighter nuclei, B.E.N. increases (e.g., $7.6 \text{ MeV}$ to $8.4 \text{ MeV}$). - The increase in B.E.N. (greater binding energy of product nuclei) results in the liberation of energy. Basis of the atom bomb. - **Nuclear Fusion:** - Light nuclei have small B.E.N. (less stable). - When two light nuclei combine to form a heavier nucleus, B.E.N. increases. - The increase in B.E.N. (higher binding energy of the product nucleus) results in the release of energy. Basis of the hydrogen bomb. ### Nuclear Reaction - **Definition:** A reaction involving the change of a stable nucleus of one element into the nucleus of another element. - Usually caused by bombarding the reacting species with suitable high-energy particles. ### Nuclear Fission - **Definition:** Phenomenon where a heavy nucleus ($A > 230$) splits into two smaller nuclei of nearly comparable masses when excited. - **Example:** Bombardment of $^{235}_{92} U$ with slow neutrons: - $^{235}_{92} U + ^1_0 n \to ^{141}_{56} Ba + ^{92}_{36} Kr + 3 (^1_0 n) + Q \text{ (Heat)}$ - Q-value (energy released) is about $200 \text{ MeV}$. - **Characteristics:** 1. A heavy nucleus splits into two smaller, nearly comparable nuclei. 2. Conditions of high temperature and pressure are NOT necessary; can be carried out on Earth. 3. Neutrons are the link particles. 4. A quick process. 5. Energy available per nucleon is small ($0.85 \text{ MeV}$). 6. Energy from unit mass of fissionable material is smaller than fusion. 7. Produces very harmful radioactive wastes. 8. Stock of fissionable material is limited. ### Nuclear Fusion - **Definition:** Process where two light nuclei combine (at extremely high temperatures) to form a single heavier nucleus. - Mass defect (difference between sum of combining nuclei masses and product nucleus mass) is released as energy ($E = \Delta m c^2$). - **Examples:** - Two protons combine: $^1_1 H + ^1_1 H \to ^2_1 H + e^+ + v + 0.42 \text{ MeV}$. - Two deuterons combine: - $^2_1 H + ^2_1 H \to ^3_2 He + ^1_0 n + 3.27 \text{ MeV}$. - $^2_1 H + ^2_1 H \to ^3_1 H + ^1_1 H + 4.03 \text{ MeV}$. - **Characteristics:** 1. Two lighter nuclei fuse to form a heavier nucleus. 2. Requires extremely high pressure and temperature; cannot be easily carried out in a laboratory. 3. Protons are the link particles. 4. Occurs in several steps with sufficient time gap. 5. Energy available per nucleon is large ($6.75 \text{ MeV}$). 6. Energy from unit mass of fusible material is large. 7. Products of fusion are harmless. 8. Fuel required for fusion is available in plenty. ### Radioactivity - **Discovery:** H. Becquerel (1896). - **Definition:** A nuclear phenomenon where an unstable nucleus undergoes a decay. - **Types of Radioactive Decay:** 1. **Alpha-decay:** Emission of a helium nucleus ($^4_2 He^{2+}$ or $\alpha^{++}$). 2. **Beta-decay:** Emission of electrons ($e^-$) or positrons ($e^+$). 3. **Gamma-decay:** Emission of high-energy photons (hundreds of keV or more). ### Questions #### Question 1 The natural chlorine is found to be a mixture of two isotopes of masses 34.98 amu and 36.98 amu respectively. Their relative abundances are 75.4% and 24.6% respectively. Find the composite atomic mass of natural chlorine. **Solution:** Average Atomic Mass (A.A.M.) = $(75.4\% \times 34.98) + (24.6\% \times 36.98)$ A.A.M. = $(\frac{75.4}{100} \times 34.98) + (\frac{24.6}{100} \times 36.98)$ A.A.M. = $26.37492 + 9.097088$ A.A.M. = $35.472008 \text{ amu}$ Rounding to two decimal places, A.A.M. = $35.47 \text{ amu}$. #### Question 2 Express 16 mg mass into equivalent energy in eV. **Solution:** Given mass $m = 16 \text{ mg} = 16 \times 10^{-6} \text{ kg}$. Speed of light $c = 3 \times 10^8 \text{ m/s}$. Using Einstein's mass-energy equivalence $E = mc^2$: $E = (16 \times 10^{-6} \text{ kg}) \times (3 \times 10^8 \text{ m/s})^2$ $E = 16 \times 10^{-6} \times 9 \times 10^{16} \text{ J}$ $E = 144 \times 10^{10} \text{ J}$ To convert Joules to electron volts (eV), use $1 \text{ eV} = 1.6 \times 10^{-19} \text{ J}$: $E = \frac{144 \times 10^{10}}{1.6 \times 10^{-19}} \text{ eV}$ $E = \frac{144}{1.6} \times 10^{10 - (-19)} \text{ eV}$ $E = 90 \times 10^{29} \text{ eV}$ $E = 9 \times 10^{30} \text{ eV}$.