15.2 Photoelectric Effect, Photons, Atomic Spectra, X-Rays & Nuclear Reactions

Photoelectric Effect, Photons, Atomic Spectra, X-Rays & Nuclear Reactions

In the Intermediate Physics Part II syllabus and the UET ECAT entry test, modern and nuclear physics carry immense conceptual and quantitative significance. ECAT engineering questions demand lightning-fast evaluations of photon energies, photoelectric work functions, Compton wavelength shifts, Bohr orbital energy levels, Rydberg transition frequencies, Duane-Hunt X-ray cutoff wavelengths, and radioactive decay chains without numerical calculators. Mastering these quantum-mechanical models and nuclear transformations provides candidates with a powerful competitive edge.

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1. Thermal Radiation, Planck's Quantum Hypothesis & Photons

Classical Radiation Laws and the Ultraviolet Catastrophe

All matter at absolute temperatures above $0\text{ K}$ emits electromagnetic radiation due to thermal agitation of microscopic constituents. A black body is an idealized physical radiator that absorbs all incident electromagnetic radiation regardless of frequency or angle of incidence:

• Stefan-Boltzmann Law: The total radiant energy emitted per second per unit surface area of a black body is directly proportional to the fourth power of its absolute thermodynamic temperature $T$:

$E = \sigma T^4$

where $\sigma = 5.670 \times 10^{-8}\text{ W}/(\text{m}^2\cdot\text{K}^4)$ is the Stefan-Boltzmann constant.

• Wien's Displacement Law: The wavelength $\lambda_{\max}$ corresponding to peak radiant spectral emittance is inversely proportional to absolute temperature:

$\lambda_{\max} T = b \approx 2.898 \times 10^{-3}\text{ m}\cdot\text{K}$

Classical electrodynamics (the Rayleigh-Jeans law based on energy equipartition) predicted that emitted energy intensity would approach infinity as wavelength approached zero ($\lambda \to 0$, in the ultraviolet spectrum)—a catastrophic theoretical failure termed the Ultraviolet Catastrophe.

Max Planck's Quantum Hypothesis (1900)

To resolve this dilemma, Max Planck revolutionized modern physics by hypothesizing that microscopic atomic oscillators do not emit or absorb radiation continuously, but rather in discrete packets (quanta) of energy:

$E = h f = \frac{h c}{\lambda}$

where $f$ is radiation frequency, $c = 3.0 \times 10^8\text{ m/s}$ is the speed of light in vacuum, and $h$ is Planck's universal constant:

$h = 6.626 \times 10^{-34}\text{ J}\cdot\text{s} = 4.136 \times 10^{-15}\text{ eV}\cdot\text{s}$

Physical Properties of the Photon

1. Energy: $E = hf = \frac{hc}{\lambda}$.
2. Propagation Speed: Propagates in vacuum at the speed of light $c = 3.0 \times 10^8\text{ m/s}$ in all inertial frames.
3. Rest Mass: The rest mass of a photon is identically zero ($m_0 = 0$).
4. Linear Momentum: Although massless, a photon carries finite momentum determined by relativistic mechanics ($E^2 = p^2 c^2 + m_0^2 c^4 \implies E = pc$):

$p = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda}$

1. Electrical Charge: Photons carry zero electric charge ($q = 0$) and are completely undeflected by electrostatic and magnetic fields.

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2. Einstein's Photoelectric Effect

Experimental Observations

When electromagnetic radiation of sufficiently high frequency illuminates a clean metallic surface (such as sodium, potassium, zinc, or cesium), electrons are emitted from the surface. These emitted electrons are called photoelectrons. Experiments by Heinrich Hertz, Philipp Lenard, and Robert Millikan established five distinct experimental facts that completely contradicted classical wave theory:

1. Instantaneous Emission: Photoelectrons are ejected immediately upon illumination (time lag $\Delta t < 10^{-9}\text{ s}$), even under extremely low light intensity. (Classical wave theory required hours for an electron to accumulate sufficient continuous wave energy).
2. Threshold Frequency ($f_0$): For each metal, there exists a definite minimum characteristic frequency $f_0$, called the threshold frequency (or cutoff frequency), below which zero photoelectrons are emitted, regardless of how intense or prolonged the radiation is.
3. Dependence of Kinetic Energy: The maximum kinetic energy $K_{\max}$ of emitted photoelectrons depends strictly on the frequency of incident radiation and the work function of the metal; it is completely independent of light intensity.
4. Dependence of Photo-Current: For frequencies above threshold ($f > f_0$), the magnitude of the photoelectric current (number of electrons ejected per second) is directly proportional to the intensity of incident radiation.
5. Stopping Potential ($V_0$): If the collecting anode is maintained at an opposing negative potential, the photoelectric current decreases until it drops to zero at the stopping potential $V_0$. The work done halting the most energetic electron equals its maximum kinetic energy:

$K_{\max} = e V_0$

Einstein's Photoelectric Equation (1905)

Albert Einstein explained the phenomenon by postulating that light propagates and interacts with matter as localized particles (photons). An incident photon delivers its entire energy $hf$ to a single conduction electron in a one-to-one collision. Part of this energy is consumed to overcome the metallic binding potential—termed the work function $\Phi$ (or $W_0$)—and the remainder appears as the electron's maximum kinetic energy:

$hf = \Phi + K_{\max}$

Because the minimum energy required to liberate an electron at the threshold frequency $f_0$ is $\Phi = hf_0 = \frac{hc}{\lambda_0}$ (where $\lambda_0$ is the threshold wavelength):

$hf = hf_0 + K_{\max} \implies K_{\max} = hf - hf_0 = h(f - f_0)$

In terms of stopping potential $V_0$:

$e V_0 = hf - \Phi \implies V_0 = \left(\frac{h}{e}\right)f - \frac{\Phi}{e}$

ECAT Graphical Insight: A plot of stopping potential $V_0$ on the vertical axis versus incident frequency $f$ on the horizontal axis yields a straight line with:

• Slope: $\frac{h}{e}$ (a universal physical constant, identical for all metals).
• X-intercept: Threshold frequency $f_0$.
• Y-intercept: $-\frac{\Phi}{e}$.

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3. Compton Scattering & Wave-Particle Duality

The Compton Effect (1923)

Arthur H. Compton directed monochromatic high-energy X-ray photons of wavelength $\lambda$ onto a graphite target containing loosely bound valence electrons. Compton discovered that the scattered radiation contained not only the original wavelength $\lambda$, but also a modified longer wavelength $\lambda'$ ($\lambda' > \lambda$, lower frequency).

Compton treated the interaction as an elastic, relativistic particle collision between a photon of energy $hf$ and momentum $p = \frac{h}{\lambda}$ and a stationary electron of rest mass $m_0$. Applying relativistic conservation of energy and momentum yields the famous Compton Shift Equation:

$\Delta \lambda = \lambda' - \lambda = \frac{h}{m_0 c}(1 - \cos\theta)$

where $\theta$ is the scattering angle of the photon relative to its initial direction, and the constant coefficient is the Compton wavelength of the electron $\lambda_c$:

$\lambda_c = \frac{h}{m_0 c} = \frac{6.626 \times 10^{-34}\text{ J}\cdot\text{s}}{(9.109 \times 10^{-31}\text{ kg})(3.0 \times 10^8\text{ m/s})} \approx 2.426 \times 10^{-12}\text{ m} = 0.02426\text{ \AA}$

• Forward Scattering ($\theta = 0^\circ$): $\Delta \lambda = \lambda_c(1 - 1) = 0$ (no shift).
• Right-Angle Scattering ($\theta = 90^\circ$): $\Delta \lambda = \lambda_c(1 - 0) = \lambda_c \approx 0.0243\text{ \AA}$.
• Backscattering ($\theta = 180^\circ$): $\Delta \lambda = \lambda_c(1 - (-1)) = 2\lambda_c \approx 0.0485\text{ \AA}$ (maximum shift).

ECAT Note on Visible Light: For visible light ($\lambda \approx 5000\text{ \AA}$), a shift of $\Delta \lambda \sim 0.024\text{ \AA}$ represents a fractional change of less than $0.0005\%$, making the Compton effect undetectable in the optical band. It is observable strictly with high-energy X-ray and gamma-ray photons.

De Broglie's Matter Wave Hypothesis (1924)

Louis de Broglie reasoned that nature loves symmetry: if electromagnetic waves exhibit particle-like properties (momentum and collisions), then material particles (electrons, protons, neutrons) must exhibit wave-like properties. The de Broglie wavelength $\lambda$ associated with any particle of relativistic momentum $p = mv$ is:

$\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2m K}}$

For a charged particle of charge $q$ accelerated from rest through an electrostatic potential difference $V$ ($K = qV$):

$\lambda = \frac{h}{\sqrt{2m q V}}$

For an electron ($m_e = 9.11 \times 10^{-31}\text{ kg}, q = 1.602 \times 10^{-19}\text{ C}$):

$\lambda_e = \sqrt{\frac{150}{V}}\text{ \AA} = \frac{1.227}{\sqrt{V}}\text{ nm}$

Example: An electron accelerated through $V = 100\text{ V}$ has $\lambda_e = \sqrt{\frac{150}{100}} = \sqrt{1.5} \approx 1.227\text{ \AA}$—comparable to crystal lattice interatomic spacings.

Experimental Verification & Heisenberg Uncertainty Principle

• Davisson-Germer Experiment (1927): Clinton Davisson and Lester Germer scattered an electron beam from a nickel crystal target. They observed a distinct diffraction peak at an accelerating potential of $54\text{ V}$ and scattering angle $\theta = 50^\circ$. Using Bragg's law ($d = 0.91\text{ \AA}$, $2d\sin\phi = \lambda$), the experimental diffraction wavelength was $1.65\text{ \AA}$, matching de Broglie's prediction $\lambda = \frac{1.227}{\sqrt{54}} = 1.67\text{ \AA}$ within experimental error, proving the wave nature of electrons.
• Heisenberg's Uncertainty Principle (1927): Werner Heisenberg established that it is fundamentally impossible to measure simultaneously both the precise position $\Delta x$ and linear momentum $\Delta p_x$ of a microscopic particle along the same spatial coordinate:

$\Delta x \cdot \Delta p_x \ge \frac{\hbar}{2} = \frac{h}{4\pi} \quad \left(\hbar = \frac{h}{2\pi}\right)$

Similarly, for energy and time, $\Delta E \cdot \Delta t \gtrsim \frac{\hbar}{2}$. Intermediate physics texts often quote the order-of-magnitude forms $\Delta x \cdot \Delta p \approx h$ and $\Delta E \cdot \Delta t \approx h$; check which convention the answer options use.

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4. Atomic Spectra, Bohr's Hydrogen Atom & X-Ray Emission

Atomic Emission Spectra & The Rydberg Formula

When an atomic gas at low pressure (such as hydrogen) is excited by an electric discharge, it emits light at discrete characteristic frequencies, forming a line spectrum.

In 1888, Johannes Rydberg formulated an empirical equation predicting the reciprocal wavelength (wave number $\bar{\nu} = \frac{1}{\lambda}$) of all hydrogen spectral lines:

$\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$

where $R_H = 1.09737 \times 10^7\text{ m}^{-1}$ is the Rydberg constant, and $n_1, n_2$ are integers with $n_2 > n_1$:

Niels Bohr's Postulates for the Hydrogen Atom (1913)

To explain hydrogen stability and discrete line spectra, Niels Bohr introduced three quantum postulates:

1. Stationary Orbits: An electron revolves around the positive nucleus in specific circular non-radiating orbits (stationary states). While in these orbits, the electron does not emit electromagnetic radiation.
2. Quantization of Angular Momentum: The orbital angular momentum $L = m v r$ of the revolving electron can only take discrete values that are integer multiples of $\frac{h}{2\pi}$:

$L = m v r = n \frac{h}{2\pi} = n \hbar \quad (n = 1, 2, 3, \dots)$

1. Radiative Transitions: Radiation is emitted or absorbed only when an electron jumps from one stationary state of energy $E_i$ to another of energy $E_f$. The frequency $f$ of the emitted photon is:

$hf = E_i - E_f$

Quantitative Bohr Formulas for Hydrogen

Balancing Coulomb electrostatic attraction with centripetal force ($\frac{1}{4\pi\varepsilon_0}\frac{e^2}{r^2} = \frac{m v^2}{r}$) and substituting Bohr's angular momentum condition:

1. Orbital Radii:

$r_n = n^2 \left(\frac{\varepsilon_0 h^2}{\pi m e^2}\right) = n^2 r_1$

For the ground state ($n = 1$), the Bohr radius is $r_1 = 0.529\text{ \AA} = 0.0529\text{ nm}$. Radii scale with $n^2$ ($r_1 : r_2 : r_3 = 1 : 4 : 9$).

1. Energy Levels:

$E_n = -\frac{m e^4}{8\varepsilon_0^2 h^2}\left(\frac{1}{n^2}\right) = -\frac{13.6\text{ eV}}{n^2}$

• Ground state ($n = 1$): $E_1 = -13.6\text{ eV}$.
• First excited state ($n = 2$): $E_2 = -\frac{13.6}{4} = -3.4\text{ eV}$.
• Second excited state ($n = 3$): $E_3 = -\frac{13.6}{9} = -1.51\text{ eV}$.
• Continuum limit ($n = \infty$): $E_\infty = 0\text{ eV}$.
• Ionization Potential of Hydrogen: The minimum energy required to completely remove the electron from the ground state is $E_{\text{ion}} = 0 - (-13.6\text{ eV}) = +13.6\text{ eV}$.

X-Ray Production: Bremsstrahlung vs. Characteristic Lines

Discovered by Wilhelm Conrad Röntgen in 1895, X-rays are high-frequency electromagnetic waves ($\lambda \sim 0.01$–$10\text{ nm}$) produced when energetic electrons strike a heavy metallic target (such as tungsten or molybdenum) in a vacuum Coolidge tube. The emission spectrum consists of two distinct components:

1. Continuous X-Ray Spectrum (Bremsstrahlung / Braking Radiation): As fast electrons penetrate target atoms, they are rapidly decelerated by the attractive Coulomb fields of heavy nuclei. Classical electrodynamics dictates that decelerating charges emit radiation. If an electron is halted in a single catastrophic collision, its entire kinetic energy $e V$ is converted into a single photon of maximum frequency $f_{\max}$ and minimum cutoff wavelength $\lambda_{\min}$ (Duane-Hunt Law):

$e V = h f_{\max} = \frac{hc}{\lambda_{\min}} \implies \lambda_{\min} = \frac{hc}{e V} \approx \frac{12{,}400\text{ \AA}\cdot\text{V}}{V}$

The cutoff wavelength depends exclusively on accelerating potential $V$, completely independent of the target material.

1. Characteristic X-Ray Spectrum: Energetic incident electrons knock out tightly bound inner-shell electrons (e.g., from the $K$-shell) of target atoms. An electron from a higher shell drops down to fill the vacancy, emitting a photon with energy equal to the sharp difference between atomic energy levels:
   • Transition from $L$-shell ($n=2$) to $K$-shell ($n=1$): emits $K_\alpha$ line.
   • Transition from $M$-shell ($n=3$) to $K$-shell ($n=1$): emits $K_\beta$ line. These sharp spectral peaks depend strictly on the atomic number $Z$ of the target anode (Moseley's Law: $\sqrt{f} = a(Z - b)$).

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5. Nuclear Physics, Radioactivity & Nuclear Energy

Nuclear Composition, Mass Defect & Binding Energy

An atomic nucleus is designated by $^{A}_{Z}\text{X}$, where $Z$ is the atomic number (number of protons) and $A$ is the mass number (total number of nucleons: protons $+$ neutrons $N = A - Z$).

Accurate mass spectrometry proves that the bound mass $M_{\text{nucleus}}$ of any stable nucleus is always strictly less than the combined mass of its individual free constituent nucleons. This mass difference is defined as the mass defect $\Delta m$:

$\Delta m = [Z m_p + (A - Z) m_n] - M_{\text{nucleus}}$

By Einstein's mass-energy equivalence ($E = mc^2$), this missing mass is converted into the binding energy $E_b$ that binds the nucleons together against mutual electrostatic repulsion:

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

In unified atomic mass units ($1\text{ u} = 1.6605 \times 10^{-27}\text{ kg} = 931.5\text{ MeV}$):

$E_b = \Delta m\text{ (in u)} \times 931.5\text{ MeV}$

The Binding Energy Curve

The stability of a nucleus is measured by its binding energy per nucleon $\frac{E_b}{A}$:

• Light nuclei ($A < 20$) have relatively low binding energy per nucleon.
• The curve rises steeply, reaching a broad maximum plateau around iron ($^{56}\text{Fe}$) at approximately $8.8\text{ MeV/nucleon}$ (the curve peaks near $^{56}\text{Fe}$ and $^{62}\text{Ni}$, the most tightly bound nuclei).
• For heavier nuclei ($A > 100$), the curve gradually declines to about $7.6\text{ MeV/nucleon}$ for uranium ($^{238}\text{U}$) due to cumulative electrostatic repulsion among numerous protons.

ECAT Nuclear Energetics Rule:

1. Nuclear Fission: Splitting a heavy, loosely bound nucleus ($A \sim 240$) into two medium-mass fragments ($A \sim 120$) increases binding energy per nucleon from $7.6\text{ MeV}$ to $8.5\text{ MeV}$, releasing $\sim 0.9\text{ MeV/nucleon} \approx 200\text{ MeV}$ per fission event.
2. Nuclear Fusion: Fusing very light nuclei ($A \le 4$, like deuterium and tritium) into a more tightly bound nucleus ($^{4}\text{He}$) increases binding energy per nucleon dramatically, releasing massive amounts of thermonuclear energy per unit mass.

Radioactive Decay Kinetics

Radioactivity is the spontaneous, uncontrolled nuclear transmutation of unstable radionuclides via emission of ionizing radiation. In 1902, Ernest Rutherford and Frederick Soddy established the Law of Radioactive Decay: the rate of disintegration is directly proportional to the number of radioactive parent nuclei $N$ present at that instant:

$-\frac{dN}{dt} = \lambda N$

where $\lambda$ is the decay constant (characteristic disintegration probability per unit time, in $\text{s}^{-1}$). Integrating from $t = 0$ ($N = N_0$):

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

• Half-Life ($T_{1/2}$): The time required for half the active parent nuclei to disintegrate ($N = \frac{N_0}{2}$):

$\frac{N_0}{2} = N_0 e^{-\lambda T_{1/2}} \implies e^{\lambda T_{1/2}} = 2 \implies T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$

• Fraction Remaining after $n$ Half-Lives ($t = n T_{1/2}$):

$N = N_0 \left(\frac{1}{2}\right)^n = \frac{N_0}{2^{t / T_{1/2}}}$

• Mean Life (Average Life $\tau$): The average lifespan of a radioactive nucleus:

$\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln 2} = \frac{T_{1/2}}{0.693} \approx 1.44 T_{1/2}$

• Activity ($A$): The number of disintegrations occurring per second: $A(t) = \lambda N(t) = A_0 e^{-\lambda t}$. Units: becquerel (Bq) ($1\text{ Bq} = 1\text{ decay/s}$), traditional unit curie (Ci) ($1\text{ Ci} = 3.7 \times 10^{10}\text{ Bq}$).

Alpha, Beta & Gamma Decay Modes

1. Alpha Decay ($\alpha$): Emission of a helium nucleus ($^{4}_{2}\text{He}^{2+}$):

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

Mass number decreases by $4$, atomic number decreases by $2$. Highest ionizing power, lowest penetration (stopped by a sheet of paper). 2. Beta-Minus Decay ($\beta^-$): Intranuclear conversion of a neutron into a proton, emitting an electron and an electron antineutrino:

$^{1}_{0}n \longrightarrow ^{1}_{1}p + ^{\phantom{-}0}_{-1}e + \bar{\nu}_e \implies ^{A}_{Z}\text{X} \longrightarrow ^{\phantom{+}A}_{Z+1}\text{Y} + ^{\phantom{-}0}_{-1}e + \bar{\nu}_e$

Mass number $A$ remains unchanged, atomic number $Z$ increases by $1$. Moderate ionizing power and penetration. 3. Beta-Plus Decay ($\beta^+$, Positron Emission): Intranuclear conversion of a proton into a neutron, emitting a positron and an electron neutrino:

$^{1}_{1}p \longrightarrow ^{1}_{0}n + ^{\phantom{+}0}_{+1}e + \nu_e \implies ^{A}_{Z}\text{X} \longrightarrow ^{\phantom{-}A}_{Z-1}\text{Y} + ^{\phantom{+}0}_{+1}e + \nu_e$

1. Gamma Emission ($\gamma$): De-excitation of an excited metastable nucleus emitting a high-energy photon without changing $A$ or $Z$:

$^{A}_{Z}\text{X}^* \longrightarrow ^{A}_{Z}\text{X} + \gamma$

Lowest ionizing power, highest penetration (requires thick lead shielding to attenuate).

Nuclear Fission and Fusion

• Controlled Fission in Nuclear Reactors: Discovered by Otto Hahn and Fritz Strassmann in December 1938 (published early 1939) and explained as nuclear fission by Lise Meitner and Otto Frisch. Fission of $^{235}_{92}\text{U}$ by thermal neutrons:

$^{235}_{92}\text{U} + ^{1}_{0}n \longrightarrow ^{236}_{92}\text{U}^* \longrightarrow ^{141}_{56}\text{Ba} + ^{92}_{36}\text{Kr} + 3\,^{1}_{0}n + \sim 200\text{ MeV}$

To maintain a stable sustained self-propagating chain reaction ($k = 1$):

• Moderator: Heavy water ($D_2O$) or high-purity graphite slows down fast fission neutrons ($2\text{ MeV}$) to thermal speeds ($0.025\text{ eV}$).

• Control Rods: Cadmium or boron rods absorb excess neutrons to control the reproduction factor.

• Thermonuclear Fusion in Stars: The solar proton-proton cycle fuses hydrogen into helium at core temperatures $\sim 1.5 \times 10^7\text{ K}$:

$4\,^{1}_{1}\text{H} \longrightarrow ^{4}_{2}\text{He} + 2\,^{0}_{+1}e + 2\nu_e + 26.7\text{ MeV}$

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6. Modern & Nuclear Physics Summary Table