Formula Sheet — Tutorial 14: Atomic Physics

1. Nuclear Structure

Mass Number

$$A = Z + N$$

Variable Meaning Units
$A$ Mass number (total nucleons) unitless
$Z$ Atomic number (protons) unitless
$N$ Neutron number unitless

Nuclear Radius

$$R = r_0 A^{1/3}$$

Variable Meaning Units
$R$ Nuclear radius m
$r_0$ Empirical constant $1.2 \times 10^{-15} \text{ m}$
$A$ Mass number unitless

Nuclear Volume

$$V = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi r_0^3 A$$

Variable Meaning Units
$V$ Nuclear volume $\text{m}^3$

2. Mass Defect & Binding Energy

Mass Defect

$$\Delta m = Z m_p + N m_n - m_{nucleus}$$

or using atomic masses:

$$\Delta m = Z m_H + N m_n - m_{atom}$$

Variable Meaning Units
$\Delta m$ Mass defect u (atomic mass unit) or kg
$m_p$ Mass of proton $1.007276 \text{ u}$
$m_n$ Mass of neutron $1.008665 \text{ u}$
$m_H$ Mass of hydrogen atom $1.007825 \text{ u}$
$m_{nucleus}$ Mass of nucleus u
$m_{atom}$ Mass of atom u

Binding Energy

$$E_b = \Delta m \cdot c^2$$

In convenient units:

$$E_b = \Delta m \times 931.5 \text{ MeV/u}$$

Variable Meaning Units
$E_b$ Binding energy MeV or J
$\Delta m$ Mass defect u
$c$ Speed of light $3.0 \times 10^8 \text{ m/s}$

Binding Energy Per Nucleon

$$\frac{E_b}{A} = \frac{\text{Total binding energy}}{\text{Mass number}}$$

Higher binding energy per nucleon means greater nuclear stability.

3. Radioactive Decay

Decay Constant

$$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693}{t_{1/2}}$$

Variable Meaning Units
$\lambda$ Decay constant $\text{s}^{-1}$ or $\text{hr}^{-1}$
$t_{1/2}$ Half-life s, hr, yr, etc.

Exponential Decay Law

$$N(t) = N_0 e^{-\lambda t}$$

Variable Meaning Units
$N(t)$ Number of nuclei remaining at time $t$ unitless
$N_0$ Initial number of nuclei unitless
$t$ Time elapsed s, hr, yr, etc.

Activity

$$A = \lambda N$$

Variable Meaning Units
$A$ Activity Bq (becquerel) = decays/s
$\lambda$ Decay constant $\text{s}^{-1}$
$N$ Number of radioactive nuclei unitless

Activity as a Function of Time

$$A(t) = A_0 e^{-\lambda t} = \lambda N_0 e^{-\lambda t}$$

Variable Meaning Units
$A_0$ Initial activity Bq

Number of Atoms from Mass

$$N = \frac{m}{M} \times N_A$$

Variable Meaning Units
$N$ Number of atoms unitless
$m$ Mass of sample g or kg
$M$ Molar mass g/mol
$N_A$ Avogadro's number $6.022 \times 10^{23} \text{ mol}^{-1}$

Time for Activity to Fall to a Fraction

$$t = \frac{t_{1/2} \ln\left(\frac{A_0}{A}\right)}{\ln 2}$$

or

$$t = \frac{1}{\lambda} \ln\left(\frac{N_0}{N}\right)$$

4. Nuclear Reactions

Q-Value (Energy Released)

$$Q = (m_{initial} - m_{final})c^2$$

or

$$Q = \Delta m \times 931.5 \text{ MeV/u}$$

Variable Meaning Units
$Q$ Energy released MeV or J
$m_{initial}$ Total mass of reactants u
$m_{final}$ Total mass of products u

Spontaneous decay requires $Q > 0$ (positive mass defect, exothermic).

Alpha Decay

$$^{A}{Z}X \rightarrow {}^{A-4}{Z-2}Y + {}^{4}_{2}\text{He}$$

Nuclear Fission

$$^{A}{Z}X + {}^{1}{0}n \rightarrow \text{fission products} + \text{neutrons} + Q$$

Conservation Laws in Nuclear Reactions

  • Conservation of nucleon number: Total $A$ is conserved
  • Conservation of charge: Total $Z$ is conserved

5. Summary Table of Key Constants

Constant Symbol Value
Nuclear radius constant $r_0$ $1.2 \times 10^{-15} \text{ m}$
Mass of proton $m_p$ $1.007276 \text{ u}$
Mass of neutron $m_n$ $1.008665 \text{ u}$
Mass of hydrogen atom $m_H$ $1.007825 \text{ u}$
1 atomic mass unit 1 u $931.5 \text{ MeV}/c^2$
Avogadro's number $N_A$ $6.022 \times 10^{23} \text{ mol}^{-1}$
Speed of light $c$ $3.0 \times 10^8 \text{ m/s}$

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