In 1923, American physicist William Duane presented[1] a discrete momentum-exchange model of the reflection of X-ray photons by a crystal lattice.
In effect, the observed scattering patterns are reproduced by a model where the possible reactions of the crystal are quantized, and the incident photons behave as free particles, as opposed to models where the incident particle acts as a wave, and the wave then 'collapses' to one of many possible outcomes.
Duane argued that the way that crystal scattering can be explained by quantization of momentum is not explicable by models based on diffraction by classical waves, as in Bragg's Law.
Duane applied his hypothesis to derive the scattering angles of X-rays by a crystal.
Subsequently, the principles that Duane advanced were also seen to provide the correct relationships for optical scattering at gratings, and the diffraction of electrons.
[2] In the early days of diffraction fine details were not observable because the detectors were inefficient, and the sources were also of low intensities.
Hence Bragg's law was the only type of diffraction observable, and Duane's approach could model it.
In 1905, Albert Einstein presented the hypothesis that the photoelectric effect could be explained if a beam of light was composed of a stream of discrete particles (photons), each with an energy (E = hf) the energy (E) of each photon being equal to the frequency (f) multiplied by the Planck constant (h).
[7] Later, in 1916 Albert Einstein also showed that the recoil of molecules during the emission and absorption of photons was consistent with, and necessary for, a quantum description of thermal radiation processes.
Each photon acts as if it imparts a momentum impulse p equal to its energy divided by the speed of light, (p = E/c).
[8] In 1925, shortly before the development of the full mathematical description of quantum mechanics, Born drew Einstein's attention to the then-new idea of "de Broglie's waves".
He wrote "It seems to me that a connection of a completely formal kind exists between these and that other mystical explanation of reflection, diffraction and interference using 'spatial' quantisation which Compton and Duane proposed and which has been more closely studied by Epstein and Ehrenfest.
"[9][10][11] Examining the hypothesis of Duane on quantized translational momentum transfer, as it accounted for X-ray diffraction by crystals,[1] and its follow-up by Compton,[12] Epstein and Ehrenfest had written "The phenomena of Fraunhofer diffraction can be treated as well on the basis of the wave theory of light as by a combination of concept of light quanta with Bohr's principle of correspondence."
"[13] Using Duane's 1923 hypothesis, the old quantum theory and the de Broglie relation, linking wavelengths and frequencies to energy and momenta, gives an account of diffraction of material particles.
The same phenomenon, considered from a different viewpoint, is described by a beam of particles of momentum p incident at angle θ upon the same array of crystal atomic planes.
The reflection is elastic, with negligible transfer of kinetic energy, because the crystal is massive.
The initial momentum of the particle in the direction perpendicular to the reflecting planes was p sin θ.
For reflection, the change of momentum of the particle in that direction must be 2p sin θ. Consequently, This agrees with the observed Bragg condition for the diffraction pattern if θ is such that It is evident that p provides information for a particle viewpoint, while λ provides information for a wave viewpoint.
For the investigation of condensed matter, neutron, X-ray and electron diffraction are nowadays commonly studied as momentum transfer processes.
The incoming and outgoing diffracted objects may be treated severally as particles or as waves.
For diffraction, classical physics usually considers the case of an incoming and an outgoing wave, not of particle beams.
When diffraction of particle beams was discovered by experiment, it seemed fitting to many writers to continue to think in terms of classical diffractors, formally belonging to the macroscopic laboratory apparatus, and of wave character belonging to the quantum object that suffers diffraction.
According to Bacciagaluppi & Crull (2009), Heisenberg in 1927 recognized that "the electron is deflected only in the discrete directions that depend on the global properties of the grating".
It seems, rather, that he thought of the diffraction as necessarily a manifestation of wave character belonging to the electron.
[38] Thus it seems possible that in 1927, Heisenberg was not thinking in terms of Duane's hypothesis of quantal transfer of translative momentum.
It was first conceived of in 1923 by William Duane, in the days of the old quantum theory, to account for diffraction of X-rays as particles according to Einstein's new conception of them, as carriers of quanta of momentum.
This is consonant with present-day quantum mechanical thinking, in which macroscopic physical bodies are conceived as supporting collective modes,[39] manifest for example in quantized quasi-particles, such as phonons.
Formally, the diffractor belongs to the quantum system, not to the classical laboratory apparatus.