Lecture 4 Notes – Compton Effect, Photo-Electric Effect & Stern–Gerlach

Class Opening & Administrative Details

  • Lecture #4 (Physics, upper-level modern/quantum focus)

  • ~20 students present; instructor performed a verbal roll-call.

  • Reminder: sessions are recorded; homework #1 (on Black-Body Radiation, Compton Effect, etc.) will be posted later in the week.

  • Exams will emphasize USING formulas, but some basic derivations are still required.

Warm-Up Anecdote – A Curious Exponent Problem

  • Instructor (age 12, 1989, Russia) noticed: 24=422^4 = 4^2.
    • Question posed: Why?
    • Generalized challenge: find all distinct positive numbers $(a,b)$ such that ab=ba,  a≠b.a^b = b^a,\ \ a\neq b.
    • Requires “well-known” but non-trivial solution; offered as a brain-teaser for math-lovers.

Review of Compton Scattering

Physical Setup

  • Photon of momentum p⃗<em>i\vec p<em>i strikes a quasi-free electron (in metal) → scatters at angle θ\theta with momentum p⃗</em>f\vec p</em>f.

  • Conservation of linear momentum (2 orthogonal directions) + total energy leads to Compton relation.

Momentum–Energy Form (before wavelength substitution)

  • Derived in class (sign convention flipped from previous lecture):
    1p<em>f−1p</em>i=1−cos⁡θmec\boxed{\frac1{p<em>f}-\frac1{p</em>i}=\frac{1-\cos\theta}{m_e c}}

  • Symbols:
    • $pi,pf$ – initial & final photon momenta
    • $m_e$ – rest mass of electron
    • $c$ – speed of light
    • θ\theta – photon scattering angle (0∘≤θ≤180∘)(0^\circ\le\theta\le 180^\circ)

de Broglie Substitution

  • de Broglie POSTULATE (cannot be derived):
    λ=hp⟺p=hλ\boxed{\lambda = \frac{h}{p}}\quad\Longleftrightarrow\quad p = \frac{h}{\lambda}
    where $h$ is Planck’s constant.

  • Photon energy (Planck–Einstein): E=hν=hcλE = h\nu = \frac{h c}{\lambda}.

Final Compton Wavelength Formula

  • Insert p=h/λp = h/\lambda → obtain
    λ<em>f−λ</em>i=λ<em>C (1−cos⁡θ)\boxed{\lambda<em>f-\lambda</em>i = \lambda<em>C\,(1-\cos\theta)} with the Compton wavelength of the scatterer λ</em>C=hm<em>0c\boxed{\lambda</em>C = \frac{h}{m<em>0 c}} ($m0$ = rest mass of the particle; formula valid for electrons, protons, neutrons, etc.).

  • Interpretation: Δλ\Delta\lambda (shift) becomes observable only when the incident λ<em>i\lambda<em>i is comparable to λ</em>C\lambda</em>C.

Numerical Estimates (done in class)

Particle

$m_0$ (kg)

λC\lambda_C (m)

ν<em>C=c/λ</em>C\nu<em>C = c/\lambda</em>C (Hz)

Electron

9.11×10−319.11\times10^{-31}

2.43×10−122.43\times10^{-12}

1.24×10201.24\times10^{20}

Proton

1.67×10−271.67\times10^{-27}

1.32×10−151.32\times10^{-15}

2.27×10232.27\times10^{23}

  • Visible blue light (λ≈4×10−7 m\lambda\approx4\times10^{-7}\,\text m) ≫ λC(e)\lambda_C^{(e)} ⇒ shift negligible → classical Thomson scattering dominates.

  • Therefore Compton effect is relevant for X-ray / γ-ray photons.

Problem-Solving Advice

  • Typical exam/homework tasks:
    • Given λ<em>i\lambda<em>i & θ\theta, find λ</em>f\lambda</em>f.
    • Find scattered electron momentum, kinetic energy, or its scattering angle using full conservation equations.

  • Non-trivial versions require returning to momentum components (longitudinal & transverse) along with energy conservation.

Photo-Electric Effect

Experimental Arrangement

  • Vacuum diode: metal cathode illuminated by light; anode kept at $\,+V$ to collect emitted electrons.

  • Observation: current flows only if light frequency exceeds a threshold νc\nu_c; intensity affects current magnitude, not electron energy.

Einstein’s Energy Balance (1905 – Nobel Prize)

E<em>k,max=hν−E</em>b\boxed{E<em>{k,\text{max}} = h\nu - E</em>b}

  • Ek,maxE_{k,\text{max}} – maximum kinetic energy of emitted electrons.

  • $E_b$ – binding (work-function) energy of the metal (few eV for conduction electrons).

  • Thus, condition for emission:
    h\nu > Eb \quad \Longrightarrow \quad \nu > \nuc = \frac{E_b}{h}

  • Classical wave theory cannot account for existence of νc\nu_c.

Practical Notes

  • Typical work functions: Eb≈2–5 eVE_b \approx 2\text{–}5\,\text{eV} (alkali metals), larger for noble/transition metals.

  • IV-curve shape: for \nu<\nuc → zero current regardless of intensity; for \nu>\nuc → current rises, saturates as all photoelectrons are collected.

Stern–Gerlach Experiment (1921)

Purpose & Historical Significance

  • Demonstrated space quantization of atomic magnetic moment; led to concept of quantum spin 12\tfrac12.

  • Provided phenomenon that cannot be reproduced by classical continuous magnetic moments.

Experimental Setup

  • Neutral atomic beam (e.g.
    Ag, H, O) passes through an inhomogeneous, asymmetric magnetic field produced by specially shaped magnets (e.g.
    sharp north pole vs. broad south pole).

  • Screen collects arriving atoms → intensity pattern recorded.

Observations

  • No field: single Gaussian spot.

  • With gradient field: beam splits into two discrete spots (up & down deflection) – evidence of two allowed magnetic-moment projections.

Theoretical Outline

  1. Magnetic potential energy of an atom:
    V=−μ⃗⋅B⃗V = -\vec \mu \cdot \vec B

  2. Translational force:
    F⃗=−∇V=−∇(μ⃗⋅B⃗)\vec F = -\nabla V = -\nabla(\vec \mu \cdot \vec B)

  3. Practical field choice (dominant $x$-component varying along $y$):
    B⃗≃B<em>x(y) i^\vec B \simeq B<em>x(y)\,\hat i ⇒ only F</em>y=−μ<em>x ∂B</em>x∂yF</em>y = -\mu<em>x\,\frac{\partial B</em>x}{\partial y} significant.

  4. Quantum postulate: magnetic moment proportional to spin component
    μ<em>x=γS</em>x,Sx=±ℏ2\mu<em>x = \gamma S</em>x,\qquad S_x = \pm \frac{\hbar}{2}
    (two eigenvalues for spin-½ particles).

  5. Therefore takes two discrete values (opposite sign) → spatial splitting into two beams.

  • Classical expectation of continuous distribution fails; quantum discreteness confirmed.

Key Take-aways

  • Necessity of non-uniform ( nablaB⃗≠0\ nabla\vec B\neq 0) and asymmetric field; uniform field only exerts torque, not net translation.

  • Spin quantization (±ℏ/2\pm \hbar/2) is intrinsic; experiment cannot be explained by orbital angular momentum alone.

Miscellaneous & Pedagogical Points

  • Instructor stresses the importance of Planck’s constant and reduced throughout quantum topics.

  • Classical vs.
    Quantum boundaries are illustrated by every discussed effect:
    • Black-body spectrum (previous lecture),
    • Compton scattering,
    • Photoelectric effect,
    • Stern–Gerlach splitting.

  • Mathematical constants & values repeatedly referenced (students advised to memorize or have them on formula sheet):
    • c=3.00×108 m/sc = 3.00\times10^{8}\,\text{m/s}
    • h=6.626×10−34 J⋅sh = 6.626\times10^{-34}\,\text{J·s}
    • m<em>e=9.11×10−31 kgm<em>e = 9.11\times10^{-31}\,\text{kg}, m</em>p=1.67×10−27 kgm</em>p = 1.67\times10^{-27}\,\text{kg}.

  • Ethical/practical note: Compton & photo-electric effects underpin X-ray technology, γ-spectroscopy, solar cells, and photomultipliers.

Study & Exam Tips

  • Understand when classical formulas suffice and when quantum corrections become essential.

  • Practice deriving core relations (e.g.
    Compton shift) from basic conservation laws.

  • When given numerical tasks:

    1. List knowns (λ, ν, θ, E_b, etc.).

    2. Select the governing equation.

    3. Convert units early (eV↔J, nm↔m).

    4. Check limiting cases (θ→0, λ≫λ_C, etc.).