Quanta, Matter Waves and Microscopes

Classical equipartition applied to radiation modes predicted an unbounded rise in black-body output at short wavelengths: the ultraviolet catastrophe. Planck proposed discrete energy exchanges proportional to frequency. His model suppressed the high-frequency contribution and matched the observed finite spectrum.

Einstein explained photoelectricity using photons: one photon gives energy hf to one electron. Some energy overcomes the work function and the remainder is kinetic energy. Threshold frequency, prompt emission and the dependence of maximum kinetic energy on frequency all challenge a simple classical wave-energy explanation. Increasing intensity at fixed frequency increases photon rate, not energy per photon.

De Broglie proposed a wavelength h/p for particles. For electrons accelerated non-relativistically through V, use kinetic energy eV and momentum $\sqrt{2m_e eV}$. Diffraction from a crystalline target gives rings or spots because the electron wavelength is comparable to atomic spacings.

Increasing accelerating voltage increases momentum and reduces wavelength, so diffraction angles and ring diameters decrease in a fixed apparatus. The electrons remain individually detected, but the distribution displays wave behaviour. The wavelength formula below needs relativistic correction at sufficiently high voltages.

A transmission electron microscope accelerates electrons, uses electromagnetic lenses to focus them and forms an image from electrons transmitted through a thin specimen. Short wavelengths permit detail beyond optical diffraction limits, although preparation, lens aberrations and sample damage impose practical limits.

A scanning tunnelling microscope brings a very sharp conducting tip extremely close to a conducting surface. With a small bias voltage, quantum tunnelling produces a current that depends very strongly on separation. A feedback system scans the tip while controlling current, mapping surface detail. An STM is not a TEM that simply uses a shorter wavelength; its image mechanism is tunnelling.

Try each problem before opening the worked solution. State the physical model, show substitutions with consistent units, and explain what your answer means.