Comprehensive Study Notes on Atomic and Nuclear Physics

Atomic Physics Models

• An atom is defined as the smallest indivisible part of matter.

• Four major atomic models have been proposed:

  • Plum-pudding model
  • Nuclear/planetary model
  • Bohr's model
  • Wave mechanic/electron cloud model

• Plum-Pudding Model:

  • Proposed by J.J. Thomson, who is credited with the discovery of electrons.
  • Asserts that the atom consists of a positively charged sphere with negative electrons embedded inside it.
  • The negatively charged electrons cancel out the positive charges, making the atom electrically neutral.
  • Limitation: It fails to explain the results of Rutherford's alpha-particle scattering experiment. Thomson assumed electrons were evenly distributed throughout the atom, whereas Rutherford's experiment demonstrated that electrons move in distinct orbits.

• Nuclear / Planetary Model:

  • Proposed by Ernest Rutherford, who is credited with the discovery of protons.
  • Rutherford Alpha-Particle Scattering Experiment: Bombarded a thin gold foil with fast-moving positively charged alpha particles.
  • Observations:
    • Most alpha particles passed through the thin gold foil undeflected.
    • Some alpha particles were deflected by small angles.
    • A few alpha particles were deflected straight back.
  • Conclusions:
    • Most of the space inside an atom is empty.
    • Positive charges are not uniformly distributed, but rather concentrated in specific regions.
    • The volume occupied by the positive charges is extremely small compared to the total volume of the atom.
  • Model Postulates:
  • The atom consists of a positively charged nucleus, with almost all its mass concentrated in the nucleus.
  • Electrons revolve around the nucleus in circular paths called orbits.
  • The size of the nucleus is very small compared to the size of the whole atom.
  • Limitations:
  • Fails to explain the stability of the atom. According to classical physics, an electron revolving in a circular orbit undergoes acceleration and must continuously radiate energy. As a result, it would lose energy and eventually collapse into the nucleus, causing the atom to collapse—which does not occur in reality.
  • Fails to explain the arrangement of electrons within an atom.

• Bohr's Model of the Hydrogen Atom:

  • Proposed as a modification of Rutherford's atomic model.
  • Postulates:
  • Electrons revolve around the nucleus in fixed orbits that possess fixed values of energy, known as energy levels. Starting from the nucleus, these levels are designated by integers $n = 1, 2, 3, 4 \dots$, called quantum numbers. An electron in the lowest energy level ($n = 1$) is in its ground state. An electron in a higher energy level is in an excited state (e.g., $n = 2$ corresponds to the first excited state).
  • The energy of an electron is quantized, meaning an electron cannot take arbitrary energy values, but only specific allowed energy values. As long as an electron remains in a particular energy level, its energy remains constant; energy changes only during a quantum jump between levels.
  • The electron's angular momentum is quantized and remains constant while revolving in a particular orbit: $m v r = \text{constant}$.
  • When an electron undergoes a quantum jump, it emits or absorbs a photon. The photon energy is given by $E = E_i - E_f$, where $E_i$ is the initial state energy and $E_f$ is the final state energy.
  • Key Notes on Quantum Jumps:
  • When jumping from a lower to a higher energy level, energy is absorbed (photon energy is positive).
  • When jumping from a higher to a lower energy level, energy is emitted (photon energy is negative).
  • Since frequency and wavelength cannot be negative, calculations use $E_i > E_f$ by subtracting the lower energy state from the higher energy state.
  • Limitations:
  • Works only for hydrogen and hydrogen-like single-electron atoms.
  • It is fundamentally incorrect because electrons do not move in fixed circular orbits.

• Wave Mechanic / Electron Cloud Model:

  • Electrons are not restricted to specific fixed orbits in the atom.
  • Electrons can be located at any point within an electron cloud, and it is impossible to determine the exact position of an electron at any given time.

Photons, De-Broglie Wavelength, and Quantum Mechanics

• Photons:

  • A photon is defined as a packet or bundle of light.
  • It is a massless particle traveling at the speed of light ($c$).
  • Planck's Equation: $E = h f$
  • $E$ = energy of the photon
  • $h$ = Planck's constant = $6.626 \times 10^{-34}\,\text{J\,s}$
  • $f$ = frequency of light
  • Substituting $v = f \lambda \implies c = f \lambda \implies f = \frac{c}{\lambda}$ into Planck's equation yields:     $E = \frac{h c}{\lambda}$

• Electron Volts:

  • An electron volt (eV) is the amount of kinetic energy gained by a single electron accelerated through a potential difference of one volt.
  • $e V = K = \frac{1}{2} m v^2$

• De-Broglie's Wavelength:

  • Demonstrates that moving particles (such as electrons) exhibit wave properties.
  • Derived by equating Einstein's energy equation $E = m c^2$ and Planck's equation $E = \frac{h c}{\lambda}$:     $m c^2 = \frac{h c}{\lambda} \implies \lambda = \frac{h}{m c}$
  • For a particle of mass $m$ moving at speed $v$ with momentum $p = m v$:     $\lambda = \frac{h}{m v} = \frac{h}{p}$
  • In terms of Kinetic Energy ($K$):     $K = \frac{1}{2} m v^2 = \frac{m^2 v^2}{2 m} = \frac{p^2}{2 m} \implies p = \sqrt{2 m K}$$\lambda = \frac{h}{\sqrt{2 m K}}$
  • In terms of Accelerating Potential ($V$):     Since $K = e V \implies p = \sqrt{2 m e V}$:     $\lambda = \frac{h}{\sqrt{2 m e V}}$

• Proof of Angular Momentum Quantization:

  • An electron in orbit forms a standing wave. For a complete circular orbit of radius $r$ and quantum number $n$:     $2 \pi r = n \lambda$
  • Substituting De-Broglie's wavelength $\lambda = \frac{h}{m v}$:     $2 \pi r = n \left(\frac{h}{m v}\right) \implies m v r = \frac{n h}{2 \pi}$
  • Defining $\hbar = \frac{h}{2 \pi}$, the angular momentum becomes:     $m v r = n \hbar$
  • Because $n$ and $\hbar$ are constant for a given orbit, the angular momentum of an electron in a specific orbit remains constant.

Mathematical Derivations of Bohr's Atomic Parameters

• Orbital Energy Derivation:

  • For an electron of mass $m$ and charge $e$ orbiting a nucleus of charge $e$ at radius $r$ with velocity $v$, centripetal force is provided by the electrostatic force:     $F_c = F_e \implies \frac{m v^2}{r} = \frac{k e^2}{r^2} \implies m v^2 = \frac{k e^2}{r} \quad \text{--- (1)}$
  • Potential energy ($U$) of the electron:     $U = F \cdot d = \frac{k(e)(-e)}{r^2} \times r = -\frac{k e^2}{r} \implies U = -m v^2$
  • Total mechanical energy ($E_n$):     $E_n = K + U = \frac{1}{2} m v^2 - m v^2 = -\frac{1}{2} m v^2 \quad \text{--- (2)}$
  • From angular momentum quantization $m v r = n \hbar \implies r = \frac{n \hbar}{m v}$ (3), substitute (3) into (1):     $m v^2 = \frac{k e^2}{\frac{n \hbar}{m v}} \implies v = \frac{k e^2}{n \hbar} \quad \text{--- (4)}$
  • Substitute (4) into (2):     $E_n = -\frac{1}{2} m \left(\frac{k e^2}{n \hbar}\right)^2 = -\frac{m k^2 e^4}{2 n^2 \hbar^2}$
  • Substituting $k = \frac{1}{4 \pi \varepsilon_0}$ and $\hbar = \frac{h}{2 \pi}$:     $E_n = -\frac{m e^4}{8 \varepsilon_0^2 n^2 h^2}$
  • For the ground state ($n = 1$):     $E_1 = -\frac{m e^4}{8 \varepsilon_0^2 h^2}$
  • Substituting constants:
  • $m = 9.11 \times 10^{-31}\,\text{kg}$
  • $e = 1.6 \times 10^{-19}\,\text{C}$
  • $\varepsilon_0 = 8.854 \times 10^{-12}\,\text{C}^2\,\text{N}^{-1}\,\text{m}^{-2}$
  • $h = 6.625 \times 10^{-34}\,\text{J\,s}$$E_1 = -2.169 \times 10^{-18}\,\text{J}$
  • Converting to electron volts ($1\,\text{eV} = 1.6 \times 10^{-19}\,\text{J}$):     $E_1 = \frac{-2.169 \times 10^{-18}}{1.6 \times 10^{-19}} = -13.6\,\text{eV}$
  • General expression for energy at level $n$:     $E_n = \frac{E_1}{n^2} = -\frac{13.6}{n^2}\,\text{eV}$
  • For hydrogen-like single-electron species (e.g., $\text{He}^+$, $\text{Li}^{2+}$, $\text{Be}^{3+}$):     $E_n = \frac{Z^2 E_1}{n^2}\,\text{eV}$     where $Z$ is the atomic number.
  • Note: Total energy $E_n$ increases (becomes less negative) with increasing distance from the nucleus because nuclear attraction decreases with distance.

• Orbital Radius Derivation:

  • Rearranging equation (1) for radius $r_n$:     $r_n = \frac{k e^2}{m v^2} \quad \text{--- (5)}$
  • Substitute velocity equation (4) into (5):     $r_n = \frac{n^2 \hbar^2 \varepsilon_0}{m k e^2} = \frac{n^2 h^2 \varepsilon_0}{\pi m e^2}$
  • For $n = 1$:     $r_1 = \frac{h^2 \varepsilon_0}{\pi m e^2} = 0.053\,\text{nm} = 0.53\,\text{\AA}$     where $1\,\text{\AA} = 10^{-10}\,\text{m}$ (Angstrom).
  • General expression for orbital radius:     $r_n = n^2 r_1 = 0.53 n^2\,\text{\AA}$
  • For hydrogen-like species:     $r_n = \frac{n^2 r_1}{Z}\,\text{\AA}$

• Relationship Between $E_n$ and $r_n$:   $E_n = -\frac{e^2}{8 \pi \varepsilon_0 r_n}$

• Orbital Velocity Derivation:

  • From equation (4), substituting $k = \frac{1}{4 \pi \varepsilon_0}$ and $\hbar = \frac{h}{2 \pi}$:     $v_n = \frac{e^2}{2 \varepsilon_0 n h}$
  • For $n = 1$:     $v_1 = \frac{e^2}{2 \varepsilon_0 h} = 2.2 \times 10^6\,\text{m\,s}^{-1} = 2.2\,\text{M\,m\,s}^{-1}$
  • General expression for orbital velocity:     $v_n = \frac{v_1}{n} = \frac{2.2\,\text{M}}{n}\,\text{m\,s}^{-1}$
  • For hydrogen-like species:     $v_n = \frac{Z v_1}{n}\,\text{m\,s}^{-1}$
  • Key Velocity and Energy Relationships:
  • Electron velocity decreases with increasing distance from the nucleus due to reduced nuclear attraction.
  • Kinetic energy decreases as orbital distance increases.
  • Potential energy increases with distance to maintain overall energy conservation.

Spectral Series and Atomic Transitions

• Rydberg Formula for Wavelength:

  • For an electron transitioning from higher level $n_i$ to lower level $n_f$:     $E = E_i - E_f \implies \frac{h c}{\lambda} = -E_1 \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$\frac{1}{\lambda} = R_H \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$     where $R_H = -\frac{E_1}{h c} = 1.097 \times 10^7\,\text{m}^{-1}$ is the Rydberg constant, and $\frac{1}{\lambda}$ is the wave number.
  • Expressed in terms of energy:     $E = R_e \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$     where $R_e = -E_1 = 13.6\,\text{eV}$.

• Spectral Series Classifications:

  • Lyman Series: Transitions to ground state ($n_f = 1$).
  • Balmer Series: Transitions to first excited state ($n_f = 2$).
  • Paschen Series: Transitions to second excited state ($n_f = 3$).
  • Brackett Series: Transitions to third excited state ($n_f = 4$).
  • Pfund Series: Transitions to fourth excited state ($n_f = 5$).
  • Humphrey's Series: Transitions to fifth excited state ($n_f = 6$).
  • Wavelength Trends:
  • Wavelength decreases as spectral line energy increases.
  • Longest wavelength occurs for transitions starting from the adjacent energy level ($n_i = n_f + 1$).
  • Shortest wavelength occurs for transitions starting from infinity ($n_i = \infty$).
  • Example: For the Balmer series, shortest $\lambda$ occurs at $n_i = \infty \to n_f = 2$; longest $\lambda$ occurs at $n_i = 3 \to n_f = 2$

• Ionization Energy:

  • The minimum energy required to completely remove an electron from its ground state to infinity ($I.E. = E_{\infty} - E_1$).
  • First Ionization Energy: Minimum energy required to remove the first electron from a neutral atom (e.g., $\text{He} \rightarrow \text{He}^+ \rightarrow \text{He}^{2+}$).

• Worked Example: Calculate photon energy and wavelength emitted when an electron in a hydrogen atom jumps from $n = 3$ to $n = 1$:

  • Energy method:     $E_1 = -13.6\,\text{eV}, \quad E_3 = -\frac{13.6}{3^2} = -1.51\,\text{eV}$$E = E_3 - E_1 = -1.51 - (-13.6) = 12.09\,\text{eV}$
  • Wavelength calculation:     $12.09 \times 1.6 \times 10^{-19} = \frac{(6.625 \times 10^{-34})(3 \times 10^8)}{\lambda} \implies \lambda = 1.03 \times 10^{-7}\,\text{m} = 103\,\text{nm}$
  • Alternatively, using Rydberg's formula:     $\frac{1}{\lambda} = 1.097 \times 10^7 \left(\frac{1}{1^2} - \frac{1}{3^2}\right) \implies \lambda = 1.03 \times 10^{-7}\,\text{m} = 103\,\text{nm}$

Photoelectric Effect

• Definition: The ejection of electrons (photoelectrons) from a metal surface when light falls upon it. Materials displaying this property are photoemissive.

• Key Threshold Definitions:

  • Threshold Frequency ($f_0$): The minimum frequency of electromagnetic radiation required to eject electrons from a metal surface. No electrons are emitted below $f_0$, regardless of light intensity.
  • Work Function / Threshold Energy ($\phi$): The minimum photon energy required to free an electron from the surface of a metal.     $\phi = h f_0 = \frac{h c}{\lambda_0}$
  • Threshold Wavelength ($\lambda_0$): The maximum wavelength of incident radiation capable of ejecting electrons from a metal surface.

• Einstein's Photoelectric Equation:

  • If incident photon energy equals work function ($E = \phi$), electrons are liberated but have zero kinetic energy.
  • If incident photon energy exceeds work function ($E > \phi$), excess energy becomes kinetic energy ($K$) of the photoelectron:     $K = E - \phi$$K = h f - h f_0 = h c \left(\frac{1}{\lambda} - \frac{1}{\lambda_0}\right)$

• Stopping Potential ($V_0$):

  • Setup with plates $A$ (emitter) and $B$ (collector):
  • If light with $E > \phi_A$ hits uncharged plate $A$, electrons travel toward plate $B$ at constant speed.
  • If plate $A$ is charged negative and plate $B$ positive, electrons accelerate toward $B$, giving rise to photoelectric current.
  • If plate $A$ is charged positive and plate $B$ negative, emitted electrons experience electric repulsion and slow down.
  • Stopping Potential ($V_0$): The minimum retarding voltage required to stop all photoelectrons from reaching the collector plate.
  • Electric potential energy $U = e V_0$. By energy conservation:     $K = e V_0$$e V_0 = h f - h f_0$

• Essential Principles of Photoelectric Emission:

  • Electron ejection depends exclusively on frequency and photon energy, not light intensity.
  • Intensity represents the number of photons. Higher intensity contains more photons, emitting more electrons and producing a higher photocurrent.
  • Photocurrent increases linearly with light intensity.
  • Increasing incident radiation frequency increases the kinetic energy of emitted electrons and increases the stopping potential.
  • Stopping potential stops current across plates, not the emission process itself; deflected photoelectrons are re-absorbed by the emitter plate.

• Worked Example: Caesium metal has a work function of $1.9\,\text{eV}$. Light of wavelength $4.5 \times 10^{-7}\,\text{m}$ illuminates the metal.

  • Threshold Wavelength ($\lambda_0$):     $\phi = \frac{h c}{\lambda_0} \implies 1.9 \times 1.6 \times 10^{-19} = \frac{(6.6 \times 10^{-34})(3 \times 10^8)}{\lambda_0} \implies \lambda_0 = 6.5 \times 10^{-7}\,\text{m}$
  • Maximum Kinetic Energy ($K$):     $K = \frac{h c}{\lambda} - \phi = \frac{(6.6 \times 10^{-34})(3 \times 10^8)}{4.5 \times 10^{-7}} - (1.9 \times 1.6 \times 10^{-19}) = 1.36 \times 10^{-19}\,\text{J} = 0.85\,\text{eV}$
  • Stopping Potential ($V_0$):     $K = e V_0 \implies 1.36 \times 10^{-19} = 1.6 \times 10^{-19} V_0 \implies V_0 = 0.85\,\text{V}$

Thermionic Emission and X-Ray Tubes

• Thermionic Emission: The process of ejecting electrons from a metal surface through the application of heat.

• X-Rays: Electromagnetic radiation generated when high-speed electrons collide with a heavy metal target.

• X-Ray Tube Operation:

  • A tungsten filament is heated by an electric current, releasing electrons via thermionic emission.
  • Electrons are accelerated across a high potential difference toward a positively charged anode target inside a evacuated glass envelope.
  • Upon impact at the anode, electron kinetic energy converts into ~99% heat (dissipate via oil conduction) and <1% X-radiation.

• Two Mechanisms of X-Ray Generation:

  1. Characteristic Radiation: Incident high-speed electrons knock out inner ground-state electrons of target atoms, creating vacancies. Higher-energy outer electrons fall down to fill these holes, emitting photons of characteristic X-ray wavelengths.
  2. Bremsstrahlung (Braking Radiation): Fast electrons passing near a target nucleus experience strong deceleration and deflection, emitting continuous X-ray radiation in the process.

• Energy Equations:

  • Kinetic energy of accelerating electrons: $K = e V$
  • Maximum X-ray photon energy (assuming 100% kinetic energy conversion to a single photon): $E_p = K = e V$

• Properties of X-Rays:

  • Invisible to human eyes.
  • Have zero rest mass.
  • Travel at the speed of light in a vacuum.
  • Possess extremely short wavelengths.
  • Unaffected by electric or magnetic fields.
  • Cannot be refracted.
  • Cause strong ionization of gases.

• Applications of X-Rays:

  • Radiography in medicine to image fractured bones and dental cavities.
  • Radiation therapy to destroy cancerous cells.
  • Industrial non-destructive inspection to locate cracks in metal pipes.
  • Airport security screening to detect concealed metal, weapons, and contrabands.

• Worked Example: An X-ray tube operates at a potential difference of $100\,\text{kV}$.

  • Kinetic Energy upon Impact:     $E = e V = (1.6 \times 10^{-19})(100 \times 10^3) = 1.6 \times 10^{-14}\,\text{J}$
  • Minimum Wavelength Produced:     $E_e = E_p \implies 1.6 \times 10^{-14} = \frac{(6.625 \times 10^{-34})(3 \times 10^8)}{\lambda} \implies \lambda = 1.24 \times 10^{-11}\,\text{m}$

Nuclear Physics Fundamentals

• Atomic Terminology:

  • Atomic Number ($Z$): Total number of protons in an atomic nucleus.
  • Mass Number ($A$): Total number of protons and neutrons in a nucleus ($A = Z + n$).
  • Nuclide: An atomic species characterized by specific atomic and mass numbers (${}_Z^A X$).
  • Isotopes: Atoms with identical atomic numbers ($Z$) but different mass numbers ($A$) due to differing neutron counts (e.g., ${}_{17}^{35}\text{Cl}$ and ${}_{17}^{37}\text{Cl}$).
  • Isobars: Nuclei sharing the exact same mass number ($A$).
  • Nucleons: Constituent particles of a nucleus (protons and neutrons).

• Nuclear Forces:

  • Strong Nuclear Force: Holds protons and neutrons together inside the nucleus. It is short-ranged operating only up to $\sim 1\,\text{fm}$ ($10^{-15}\,\text{m}$). At larger distances, electrostatic repulsion dominates, leading to nuclear instability.
  • Weak Nuclear Force: Operates within individual nucleons over an even shorter range; governs beta decay conversions between protons and neutrons.

• Mass Defect ($\Delta m$):

  • The difference between the sum of individual free nucleon masses and the actual bound mass of the assembled nucleus.
  • Formula: $\Delta m = Z m_p + n m_n - m_N$
  • Helium (${}_2^4\text{He}$) Example:
  • Bound atom mass = $4.0026\,\text{u}$.
  • Sum of constituent masses ($2 m_e + 2 m_p + 2 m_n$) = $2(0.00054858) + 2(1.007825) + 2(1.008665) = 4.03408\,\text{u}$.
  • Mass defect $\Delta m = 4.03408 - 4.0026 = 0.03148\,\text{u}$.

• Binding Energy ($\Delta E$):

  • Mass is converted to energy according to Einstein's relation $\Delta E = \Delta m c^2$.
  • Binding energy represents the total energy required to disintegrate a nucleus into its constituent free nucleons.
  • Binding Energy Per Nucleon: Average energy needed to extract a single nucleon from a nucleus ($\frac{\Delta E}{A}$).

• Mass-Energy Units Conversion:

  • $1\,\text{u} = 1.66 \times 10^{-27}\,\text{kg}$
  • $E = m c^2 = (1.66 \times 10^{-27})(2.998 \times 10^8)^2 = 1.492 \times 10^{-10}\,\text{J} = 931.5\,\text{MeV}$
  • Thus, $1\,\text{u} = 931.5\,\text{MeV}/c^2$

Radioactivity and Nuclear Decay

• Nuclear Stability:

  • Stable Nuclei: Strong force exceeds electrostatic repulsion due to a balanced neutron-to-proton ratio (e.g., ${}_6^{12}\text{C}$).
  • Unstable Nuclei: Electrostatic repulsion overcomes strong binding force due to excessive nucleus size or unbalanced neutron-proton ratio. Unstable nuclei spontaneously disintegrate by emitting radiation (radioactivity).
  • Examples of radioactive elements: ${}_6^{14}\text{C}$ and ${}_{92}^{238}\text{U}$.

• Types of Radioactive Decay:

  • Alpha ($\alpha$) Decay:
  • Emission of a helium nucleus (${}_2^4\text{He}$).
  • Reduces atomic number ($Z$) by 2 and mass number ($A$) by 4.
  • Example (Uranium to Thorium):       ${}_{92}^{238}\text{U} \longrightarrow {}_{90}^{234}\text{Th} + {}_2^4\text{He}$
  • Beta ($\beta$) Decay:
  • $\beta^-$ Emission: Neutron transforms into a proton, emitting an electron (${}_{-1}^0 e^-$) and an antineutrino ($\bar{\nu}$) with negligible mass and zero charge.       Example (Tritium to Helium):       ${}_1^3\text{H} \longrightarrow {}_2^3\text{He} + {}_{-1}^0 e^- + \bar{\nu}$
  • $\beta^+$ Emission: Proton transforms into a neutron, emitting a positron (${}_{+1}^0 e^+$) and a neutrino ($\nu$).       Example (Carbon-10 to Boron-10):       ${}_6^{10}\text{C} \longrightarrow {}_5^{10}\text{B} + {}_{+1}^0 e^+ + \nu$
  • Gamma ($\gamma$) Decay:
  • Excited nucleus drops to a lower energy state by emitting a high-frequency photon ($\gamma$-ray).
  • Neither atomic number nor mass number changes; no material particles are ejected.

• Radioactive Decay Law:

  • The rate of decay of a radioactive sample is directly proportional to the number of undecayed nuclei present:     $\frac{d N}{d t} = -k N$     where $k$ is the decay constant.
  • Integration derivation:     $\int_{N_0}^{N} \frac{d N}{N} = -k \int_{0}^{t} d t \implies \ln\left(\frac{N}{N_0}\right) = -k t$$N = N_0 e^{-k t}$
  • Rearranging for decay constant:     $k = \frac{\ln\left(\frac{N_0}{N}\right)}{t}$
  • Original Mass Balance: $N_0 = N + N_D$ (where $N$ is remaining quantity and $N_D$ is decayed quantity).

• Half-Life ($t_{1/2}$):

  • Time required for half of the initial radioactive atoms to decay ($N = \frac{1}{2} N_0$).     $k = \frac{\ln(2)}{t_{1/2}} \implies t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}$
  • Remaining amount after $n$ half-lives:     $N = \frac{N_0}{2^n}$
  • Total elapsed time $t$:     $t = n t_{1/2}$

• Mean Life ($\tau$):

  • Average lifetime of all nuclei in a radioactive sample:     $\tau = \frac{\int_0^{\infty} t\,d N}{N_0} = \frac{1}{k}$
  • Relationship to half-life:     $\tau = \frac{t_{1/2}}{0.693}$

• Activity ($A$):

  • Number of disintegrations occurring per unit time ($A = \left|-\frac{d N}{d t}\right| = k N$).     $A = A_0 e^{-k t}$$k = \frac{\ln\left(\frac{A_0}{A}\right)}{t}$
  • Units of Activity:
  • $1\,\text{decay/second} = 1\,\text{Becquerel\,(Bq)}$
  • $1\,\text{Curie\,(Ci)} = 3.70 \times 10^{10}\,\text{Bq}$