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Lesson 11.3 • Competency 11.3Complete Lesson
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Wave-Particle Duality and X-Rays

Explore de Broglie matter waves (λ = h/p = h/√(2mqV)), electron diffraction evidence, Coolidge tube generation of X-rays, continuous Bremsstrahlung spectrum with Duane-Hunt cutoff (λ_min = hc/eV), and characteristic X-ray emission lines.

Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.

Learning Objectives

  • Calculate de Broglie wavelengths for moving particles and accelerated electrons: λ = h / √(2 m_e e V).
  • Explain electron diffraction and how the Davisson-Germer experiment proved the wave nature of electrons.
  • Derive the Duane-Hunt cutoff equation λ_min = hc / (eV) and explain continuous vs characteristic X-ray spectra.

Core Theory & Concepts

In 1924, Louis de Broglie reasoned by symmetry that if radiation possesses particle attributes (photons), matter particles must possess wave characteristics. The wavelength associated with any moving mass m with velocity v is λ = h / (mv). In electron microscopes, electron waves have wavelengths 100,000 times shorter than visible light, permitting atomic-resolution imaging. In X-ray tubes, the inverse process converts fast electron kinetic energy into high-energy penetrating electromagnetic photons.

De Broglie Matter Waves

For non-relativistic particles: kinetic energy K = p² / (2m) ⟹ p = √(2mK). For an electron accelerated through potential V: p = √(2 m_e e V), yielding λ = h / √(2 m_e e V) ≈ 1.227 / √V nm.

\lambda = \frac{h}{p} = \frac{h}{\sqrt{2 m_e e V}} \approx \frac{1.227}{\sqrt{V}}\text{ nm}

Continuous X-Rays & Duane-Hunt Limit

When high-speed electrons are decelerated in the target Coulomb field (Bremsstrahlung), photon energy cannot exceed the electron’s maximum kinetic energy: E_max = hc / λ_min = eV_a ⟹ λ_min = hc / (e V_a).

\lambda_{\min} = \frac{h c}{e V_a}

Common Misconceptions

Misconception

Increasing the X-ray tube filament current increases the penetrating hardness of X-rays.

Physical Clarification:Filament current solely increases electron emission rate, increasing X-ray intensity (photons/second). Penetrating hardness and cutoff wavelength λ_min depend exclusively on the accelerating high voltage V_a.

Syllabus Topic Outline

Competency 11.3
  • 1De Broglie hypothesis: matter waves λ = h / p
  • 2Electron wavelength accelerated through potential V: λ ≈ 1.227 / √V nm
  • 3Experimental verification: Davisson-Germer and thin crystal diffraction rings
  • 4Electron microscope principles and enhanced resolving power
  • 5Coolidge tube operation and high-voltage electron bombardment
  • 6Continuous Bremsstrahlung X-rays and Duane-Hunt limit: λ_min = hc / (eV)
  • 7Characteristic X-rays (K_α, K_β transitions) and target atomic signatures