FAD1022 L39-L42 — Atomic & Nuclear Physics — Formula Sheet
Comprehensive formula sheet for Atomic & Nuclear Physics (Lectures 39–42). Every equation from the lecture is included below, organized by topic.
1. Bohr's Model of the Hydrogen Atom
1.1 Angular Momentum Quantization
$$L = mvr = n\frac{h}{2\pi} = n\hbar$$
| Variable | Description | Units |
|---|---|---|
| $L$ | Angular momentum | $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-1}$ |
| $m$ | Mass of electron | $9.1 \times 10^{-31}\ \text{kg}$ |
| $v$ | Orbital speed of electron | $\text{m/s}$ |
| $r$ | Orbital radius | $\text{m}$ |
| $n$ | Principal quantum number ($n = 1, 2, 3, \ldots$) | dimensionless |
| $h$ | Planck's constant | $6.63 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
| $\hbar$ | Reduced Planck constant | $1.06 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
1.2 Reduced Planck Constant
$$\hbar = \frac{h}{2\pi} \approx 1.06 \times 10^{-34}\ \text{J}\cdot\text{s}$$
1.3 Coulomb Force as Centripetal Force
From Newton's second law for circular motion, the electrostatic Coulomb force provides the centripetal acceleration:
$$\frac{e^2}{4\pi\varepsilon_0 r^2} = \frac{mv^2}{r}$$
| Variable | Description | Units |
|---|---|---|
| $e$ | Elementary charge | $1.602 \times 10^{-19}\ \text{C}$ |
| $\varepsilon_0$ | Permittivity of free space | $8.85 \times 10^{-12}\ \text{C}^2\cdot\text{N}^{-1}\cdot\text{m}^{-2}$ |
| $r$ | Orbital radius | $\text{m}$ |
| $m$ | Electron mass | $9.1 \times 10^{-31}\ \text{kg}$ |
| $v$ | Electron orbital speed | $\text{m/s}$ |
1.4 Bohr Orbital Radius
$$r_n = \left(\frac{\varepsilon_0 h^2}{\pi m e^2}\right) n^2 \quad n = 1, 2, 3, \ldots$$
Practical form (Bohr radius):
$$\boxed{r_n = (5.29 \times 10^{-11}\ \text{m}), n^2 = a_0 n^2}$$
| Variable | Description | Units |
|---|---|---|
| $r_n$ | Radius of $n$-th orbit | $\text{m}$ |
| $a_0$ | Bohr radius ($5.29 \times 10^{-11}\ \text{m}$) | $\text{m}$ |
| $n$ | Principal quantum number | dimensionless |
1.5 Total Energy of the Hydrogen Atom
$$E = K + U = \frac{1}{2}mv^2 - \frac{ke^2}{r}$$
Using force balance ($K = \frac{ke^2}{2r}$), this simplifies to:
$$E = -\frac{ke^2}{2r}$$
1.6 Quantized Energy Levels
$$E_n = -\left(\frac{2\pi^2 m k^2 e^4}{h^2}\right)\frac{1}{n^2}$$
Practical form (hydrogen energy levels):
$$\boxed{E_n = -(13.6\ \text{eV})\frac{1}{n^2} \quad n = 1, 2, 3, \ldots}$$
| Variable | Description | Units |
|---|---|---|
| $E_n$ | Energy of $n$-th level | $\text{eV}$ (or J) |
| $n$ | Principal quantum number | dimensionless |
Key energy values:
| $n$ | State | $E_n$ |
|---|---|---|
| 1 | Ground state | $-13.6\ \text{eV}$ |
| 2 | 1st excited state | $-3.4\ \text{eV}$ |
| 3 | 2nd excited state | $-1.51\ \text{eV}$ |
| 4 | 3rd excited state | $-0.85\ \text{eV}$ |
1.7 Ionization Energy
Minimum energy to remove an electron from ground state to infinity ($n=1 \rightarrow n=\infty$):
$$\Delta E = E_\infty - E_1 = 0 - (-13.6\ \text{eV}) = 13.6\ \text{eV}$$
1.8 Atomic Transitions and Photon Energy
When an electron transitions between levels, the photon energy (emitted or absorbed) is:
$$\Delta E = hf = E_i - E_f$$
Expressed in terms of initial and final quantum numbers:
$$E_{\text{light}} = -13.6\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)\ \text{eV}$$
| Variable | Description | Units |
|---|---|---|
| $\Delta E$ | Energy difference / photon energy | $\text{eV}$ or J |
| $h$ | Planck's constant | $6.63 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
| $f$ | Photon frequency | $\text{Hz}$ |
| $E_i$ | Initial energy level | $\text{eV}$ |
| $E_f$ | Final energy level | $\text{eV}$ |
| $n_i$ | Initial quantum number | dimensionless |
| $n_f$ | Final quantum number | dimensionless |
1.9 Spectral Series Summary
| Series | Transitions | Region |
|---|---|---|
| Lyman | $n \geq 2 \rightarrow n = 1$ | Ultraviolet (UV) |
| Balmer | $n \geq 3 \rightarrow n = 2$ | Visible light |
| Paschen | $n \geq 4 \rightarrow n = 3$ | Infrared (IR) |
2. Nuclear Structure
2.1 Nuclide Notation
$$^A_Z X \quad \text{where} \quad N = A - Z$$
| Symbol | Description |
|---|---|
| $A$ | Mass number (protons + neutrons) |
| $Z$ | Atomic number (protons) |
| $N$ | Neutron number |
| $X$ | Chemical symbol |
2.2 Nuclear Radius
$$R = R_0 A^{1/3}$$
where $R_0 = 1.2 \times 10^{-15}\ \text{m} = 1.2\ \text{fm}$
| Variable | Description | Units |
|---|---|---|
| $R$ | Nuclear radius | $\text{m}$ (or fm) |
| $R_0$ | Empirical constant | $1.2 \times 10^{-15}\ \text{m}$ |
| $A$ | Mass number | dimensionless |
2.3 Nuclear Volume
$$V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi A R_0^3$$
| Variable | Description | Units |
|---|---|---|
| $V$ | Nuclear volume | $\text{m}^3$ |
| $R$ | Nuclear radius | $\text{m}$ |
| $A$ | Mass number | dimensionless |
| $R_0$ | Empirical constant | $1.2 \times 10^{-15}\ \text{m}$ |
2.4 Nuclear Density
$$\rho \approx 2.3 \times 10^{17}\ \text{kg/m}^3$$
(Nuclear density is approximately constant for all nuclei.)
2.5 Subatomic Particle Masses
| Particle | Mass (kg) | Charge (C) | Atomic mass (u) |
|---|---|---|---|
| Proton ($m_p$) | $1.67262 \times 10^{-27}$ | $+e = +1.602 \times 10^{-19}$ | $1.00728$ |
| Neutron ($m_n$) | $1.67492 \times 10^{-27}$ | $0$ | $1.00867$ |
| Electron ($m_e$) | $9.10938 \times 10^{-31}$ | $-e = -1.602 \times 10^{-19}$ | $0.000549$ |
3. Mass-Energy Equivalence & Atomic Mass Unit
3.1 Einstein's Mass-Energy Relation
$$E = mc^2$$
| Variable | Description | Units |
|---|---|---|
| $E$ | Energy | J (or MeV) |
| $m$ | Mass | kg (or u) |
| $c$ | Speed of light | $3.00 \times 10^8\ \text{m/s}$ |
3.2 Atomic Mass Unit
$$1\ \text{u} = 1.6606 \times 10^{-27}\ \text{kg} \approx 1.66 \times 10^{-27}\ \text{kg}$$
(Defined as $\frac{1}{12}$ the mass of a neutral carbon-12 atom.)
3.3 Energy Equivalent of 1 u
In joules:
$$E = (1.66 \times 10^{-27})(3.00 \times 10^8)^2 = 1.49 \times 10^{-10}\ \text{J}$$
In mega-electronvolts:
$$E = \frac{1.49 \times 10^{-10}}{1.60 \times 10^{-19}} = 931.5\ \text{MeV}$$
Key conversion:
$$\boxed{1\ \text{u} = 931.5\ \text{MeV}/c^2}$$
or equivalently:
$$\boxed{c^2 = 931.5\ \text{MeV/u}}$$
3.4 Energy Unit Conversions
$$1\ \text{eV} = 1.602 \times 10^{-19}\ \text{J}$$
$$1\ \text{MeV} = 10^6\ \text{eV} = 1.602 \times 10^{-13}\ \text{J}$$
4. Mass Defect and Binding Energy
4.1 Mass Defect
The mass difference between constituent nucleons and the actual nucleus mass:
$$\boxed{\Delta m = Zm_p + Nm_n - m_N}$$
| Variable | Description | Units |
|---|---|---|
| $\Delta m$ | Mass defect | kg (or u) |
| $Z$ | Atomic number (protons) | dimensionless |
| $m_p$ | Proton mass | kg (or u) |
| $N$ | Neutron number | dimensionless |
| $m_n$ | Neutron mass | kg (or u) |
| $m_N$ | Actual mass of nucleus | kg (or u) |
4.2 Binding Energy
Energy required to break a nucleus into its constituent nucleons (or energy released when nucleons combine):
$$\boxed{E_B = (\Delta m)c^2 = [Zm_p + Nm_n - m_N]c^2}$$
In MeV (using $c^2 = 931.5\ \text{MeV/u}$):
$$\boxed{E_B = \Delta m \times 931.5\ \text{MeV/u}}$$
| Variable | Description | Units |
|---|---|---|
| $E_B$ | Binding energy | J (or MeV) |
| $\Delta m$ | Mass defect | kg (or u) |
| $c$ | Speed of light | $3.00 \times 10^8\ \text{m/s}$ |
4.3 Binding Energy per Nucleon
$$\boxed{\frac{E_B}{A}}$$
- Higher binding energy per nucleon $\rightarrow$ more stable nucleus.
- Maximum stability at Fe-56: $E_B/A \approx 8.8\ \text{MeV/nucleon}$.
| Variable | Description | Units |
|---|---|---|
| $E_B$ | Total binding energy | MeV |
| $A$ | Mass number (total nucleons) | dimensionless |
5. Radioactive Decay
5.1 Types of Radioactive Decay
| Decay Mode | Emitted Particle | Change in $Z$ | Change in $A$ | Condition |
|---|---|---|---|---|
| Alpha ($\alpha$) | $^4_2\text{He}$ | $-2$ | $-4$ | Nucleus too heavy |
| Beta minus ($\beta^-$) | $^0_{-1}e$ | $+1$ | $0$ | Too many neutrons |
| Positron ($\beta^+$) | $^0_{+1}e$ | $-1$ | $0$ | Too many protons |
| Gamma ($\gamma$) | Photon | $0$ | $0$ | Nucleus in excited state |
5.2 Decay Law (Differential Form)
The rate of decay is proportional to the number of radioactive nuclei:
$$\boxed{\frac{dN}{dt} = -\lambda N}$$
| Variable | Description | Units |
|---|---|---|
| $N$ | Number of radioactive nuclei | dimensionless |
| $t$ | Time | s (or any consistent unit) |
| $\lambda$ | Decay constant | $\text{s}^{-1}$ (or consistent inverse time) |
5.3 Activity
The rate of decay (activity):
$$\boxed{A = \lambda N = -\frac{dN}{dt}}$$
Initial activity:
$$A_0 = \lambda N_0$$
Activity decay over time:
$$\boxed{A = A_0 e^{-\lambda t}}$$
| Variable | Description | Units |
|---|---|---|
| $A$ | Activity | Bq (or Ci) |
| $\lambda$ | Decay constant | $\text{s}^{-1}$ |
| $N$ | Number of nuclei | dimensionless |
| $t$ | Time | s |
| $A_0$ | Initial activity | Bq |
| $N_0$ | Initial number of nuclei | dimensionless |
5.4 Activity Units
$$1\ \text{Bq} = 1\ \text{decay s}^{-1}$$
$$1\ \text{Ci} = 3.70 \times 10^{10}\ \text{Bq} = 3.70 \times 10^{10}\ \text{decays s}^{-1}$$
5.5 Exponential Decay Equation
Integrating the decay law:
$$\boxed{N(t) = N_0 e^{-\lambda t}}$$
| Variable | Description | Units |
|---|---|---|
| $N(t)$ | Number of nuclei at time $t$ | dimensionless |
| $N_0$ | Initial number of nuclei | dimensionless |
| $\lambda$ | Decay constant | $\text{s}^{-1}$ (or consistent inverse time) |
| $t$ | Time | s (or consistent unit) |
5.6 Half-Life
Time required for half the nuclei to decay:
$$\boxed{T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}}$$
Relationship to decay constant:
$$\lambda = \frac{0.693}{T_{1/2}}$$
| Variable | Description | Units |
|---|---|---|
| $T_{1/2}$ | Half-life | s, yr, hr, etc. |
| $\lambda$ | Decay constant | $\text{s}^{-1}$, $\text{yr}^{-1}$, etc. |
5.7 Fraction Remaining After Time $t$
From $N(t) = N_0 e^{-\lambda t}$ and $\lambda = \frac{\ln 2}{T_{1/2}}$:
$$\frac{N(t)}{N_0} = e^{-\lambda t} = \left(\frac{1}{2}\right)^{t/T_{1/2}}$$
6. Nuclear Reactions and Q-Value
6.1 Conservation Laws
In any nuclear reaction:
- Conservation of charge ($Z$): Total atomic number is conserved.
- Conservation of mass number ($A$): Total nucleon number is conserved.
- Conservation of energy: Total energy is conserved.
6.2 Q-Value (Reaction Energy)
Mass difference:
$$\boxed{\Delta m = \sum m_{\text{before}} - \sum m_{\text{after}}}$$
Reaction energy:
$$\boxed{Q = (\Delta m)c^2}$$
| Condition | Meaning |
|---|---|
| $\Delta m > 0$ or $Q > 0$ | Exothermic (exoergic) — energy released as kinetic energy of products |
| $\Delta m < 0$ or $Q < 0$ | Endothermic (endoergic) — energy must be absorbed for reaction to occur |
| Variable | Description | Units |
|---|---|---|
| $\Delta m$ | Mass difference | kg (or u) |
| $Q$ | Reaction energy | J (or MeV) |
| $c$ | Speed of light | $3.00 \times 10^8\ \text{m/s}$ |
Shortcut using atomic mass units:
$$Q = \Delta m \times 931.5\ \text{MeV/u}$$
6.3 Alpha Decay Example
$$^{226}{88}\text{Ra} \rightarrow ^{222}{86}\text{Rn} + ^{4}_{2}\text{He} + Q$$
$$\Delta m = m_{\text{Ra}} - (m_{\text{Rn}} + m_{\alpha})$$
$$Q = \Delta m \times 931.5\ \text{MeV}$$
6.4 Beta Decay Example
$$^{234}{90}\text{Th} \rightarrow ^{234}{91}\text{X} + ^{0}_{-1}e + Q$$
$$\Delta m = m_{\text{Th}} - (m_{\text{X}} + m_e)$$
$$Q = \Delta m \times 931.5\ \text{MeV}$$
7. Nuclear Fusion
7.1 Fusion Definition
Process where small nuclei combine to form larger nuclei, releasing energy.
Examples:
$$^{2}{1}\text{H} + {}^{2}{1}\text{H} \rightarrow {}^{3}{2}\text{He} + {}^{1}{0}\text{n} + \text{Energy}$$
$$^{2}{1}\text{H} + {}^{3}{1}\text{H} \rightarrow {}^{4}{2}\text{He} + {}^{1}{0}\text{n} + \text{Energy}$$
$$^{1}{1}\text{H} + {}^{2}{1}\text{H} \rightarrow {}^{3}_{2}\text{He} + \gamma$$
7.2 Fusion Energy Release
$$\text{Energy released} = (\text{Total } E_B \text{ of product}) - (\text{Total } E_B \text{ of reactants})$$
Or via mass defect:
$$\Delta m = (\text{total mass of reactants}) - (\text{total mass of products})$$
$$E = \Delta m , c^2 = \Delta m \times 931.5\ \text{MeV}$$
Worked example (two deuterons fusing):
$$^{2}{1}\text{H} + {}^{2}{1}\text{H} \rightarrow {}^{3}{1}\text{H} + {}^{1}{1}\text{H}$$
$$\Delta m = (2.0141 + 2.0141) - (3.0160 + 1.0078) = 0.0044\ \text{u}$$
$$E = 0.0044 \times 931.5 = 4.1\ \text{MeV}$$
8. Nuclear Fission
8.1 Fission Definition
A heavy nucleus splits into two lighter nuclei, releasing energy because fission products have greater average binding energy per nucleon than the parent.
Uranium-235 example:
$$^{1}{0}\text{n} + {}^{235}{92}\text{U} \rightarrow {}^{236}{92}\text{U}^* \rightarrow {}^{91}{36}\text{Kr} + {}^{142}{56}\text{Ba} + 3{}^{1}{0}\text{n} + Q$$
8.2 Fission Energy Calculation
$$\Delta m = (m_{\text{U}} + m_n) - (m_{\text{product 1}} + m_{\text{product 2}} + 3m_n)$$
$$Q = \Delta m , c^2 = \Delta m \times 931.5\ \text{MeV}$$
Worked example:
$$^{235}{92}\text{U} + {}^{1}{0}\text{n} \rightarrow {}^{85}{35}\text{Br} + {}^{148}{57}\text{La} + 3{}^{1}_{0}\text{n} + Q$$
Given masses:
- $m_n = 1.00867\ \text{u}$
- $m_{^{85}\text{Br}} = 84.91179\ \text{u}$
- $m_{^{148}\text{La}} = 147.91718\ \text{u}$
- $m_{^{235}\text{U}} = 235.04392\ \text{u}$
$$\Delta m = (235.04392 + 1.00867) - (84.91179 + 147.91718 + 3 \times 1.00867)$$
$$\Delta m = 236.05259 - 235.85498 = 0.19761\ \text{u}$$
$$Q = 0.19761 \times 931.5 = 184.074\ \text{MeV}$$
9. Carbon Dating
9.1 Carbon-14 Decay Constant
$$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{5730\ \text{yr}} = 1.21 \times 10^{-4}\ \text{yr}^{-1}$$
9.2 Age from Remaining Activity
Using $N(t) = N_0 e^{-\lambda t}$, solving for $t$:
$$\frac{N(t)}{N_0} = e^{-\lambda t}$$
$$\ln\left(\frac{N(t)}{N_0}\right) = -\lambda t$$
$$\boxed{t = -\frac{1}{\lambda}\ln\left(\frac{N(t)}{N_0}\right) = \frac{1}{\lambda}\ln\left(\frac{N_0}{N(t)}\right)}$$
Alternatively, using half-life:
$$\boxed{t = \frac{T_{1/2} \cdot \ln(N_0/N)}{\ln 2}}$$
| Variable | Description | Units |
|---|---|---|
| $t$ | Age of sample | yr |
| $\lambda$ | Decay constant | $\text{yr}^{-1}$ |
| $T_{1/2}$ | Half-life ($5730$ yr for C-14) | yr |
| $N_0$ | Initial number of nuclei / initial activity | — |
| $N(t)$ | Remaining number / remaining activity | — |
9.3 Number of Nuclei in a Sample
$$N_0 = \frac{N_A}{M} \times m_{\text{sample}}$$
For carbon in 1.0 kg:
$$N_0 = \frac{6.022 \times 10^{23}}{14} \times 1000 = 4.30 \times 10^{25}\ \text{atoms}$$
| Variable | Description | Units |
|---|---|---|
| $N_A$ | Avogadro's number | $6.022 \times 10^{23}\ \text{mol}^{-1}$ |
| $M$ | Molar mass | g/mol |
| $m_{\text{sample}}$ | Mass of sample | g |
10. Key Constants Summary
| Constant | Symbol | Value |
|---|---|---|
| Speed of light | $c$ | $3.00 \times 10^8\ \text{m/s}$ |
| Planck's constant | $h$ | $6.63 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
| Reduced Planck constant | $\hbar$ | $1.06 \times 10^{-34}\ \text{J}\cdot\text{s}$ |
| Elementary charge | $e$ | $1.602 \times 10^{-19}\ \text{C}$ |
| Electron mass | $m_e$ | $9.11 \times 10^{-31}\ \text{kg}$ |
| Proton mass | $m_p$ | $1.673 \times 10^{-27}\ \text{kg}$ |
| Neutron mass | $m_n$ | $1.675 \times 10^{-27}\ \text{kg}$ |
| Permittivity of free space | $\varepsilon_0$ | $8.85 \times 10^{-12}\ \text{C}^2\cdot\text{N}^{-1}\cdot\text{m}^{-2}$ |
| Coulomb constant | $k = \frac{1}{4\pi\varepsilon_0}$ | $8.99 \times 10^9\ \text{N}\cdot\text{m}^2\cdot\text{C}^{-2}$ |
| Atomic mass unit | $1\ \text{u}$ | $1.661 \times 10^{-27}\ \text{kg}$ |
| Energy equivalent of 1 u | $c^2$ | $931.5\ \text{MeV/u}$ |
| 1 eV in joules | — | $1.602 \times 10^{-19}\ \text{J}$ |
| Bohr radius | $a_0$ | $5.29 \times 10^{-11}\ \text{m}$ |
| Ionization energy of H | $E_\infty - E_1$ | $13.6\ \text{eV}$ |
| Avogadro's number | $N_A$ | $6.022 \times 10^{23}\ \text{mol}^{-1}$ |
| Nuclear radius constant | $R_0$ | $1.2 \times 10^{-15}\ \text{m} = 1.2\ \text{fm}$ |
Formula sheet extracted from FAD1022 L39-L42 — Atomic & Nuclear Physics — comprehensive coverage of all equations for finals revision.