In 1923, American physicist William Duane presented a discrete momentum-exchange model of the reflection of X-ray photons by a crystal lattice. Duane showed that such a model gives the same scattering angles as the ones calculated via a wave diffraction model, see Bragg's Law.
The key feature of Duane's hypothesis is that a simple quantum rule based on the lattice structure alone determines the quanta of momentum that can be exchanged between the crystal lattice and an incident particle.
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.Bitsakis, E.(1997). The wave-particle duality, pp. 333–348 in The Present Status of the Quantum Theory of Light: Proceedings of a Symposium in Honour of Jean-Pierre Vigier, edited by Whitney, C.K., Jeffers, S., Roy, S., Vigier, J.-P., Hunter, G., Springer, , p. 338.
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. Modern electron microscopes and x-ray diffraction instruments are many orders of magnitude brighter, so many fine details of electron and x-ray diffraction are now known which cannot be explained by his approach. Hence his approach is no longer used.
Alfred Landé, the German-American physicist known for his 'Landé g-factor' and explanation of the anomalous Zeeman effect in atomic physics, used Duane's hypothesis as the basis for his alternative 'New Foundations of Quantum Mechanics' in his 1965 book.Alfred Landé New Foundations of Quantum Mechanics (Cambridge University Press, 1965)
Early developments in quantum theory
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 () the energy (E) of each photon being equal to the frequency (f) multiplied by the Planck constant (h). Translated in 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, (). and a nearly identical version Translated here and in
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."Born, M. (1925/1971). Letter of 15 July 1925, pp. 84–85 in The Born-Einstein Letters, translated by I. Born, Macmillan, London.Epstein, P.S., Ehrenfest, P., (1924). The quantum theory of the Fraunhofer diffraction, Proc. Natl. Acad. Sci. 10: 133–139.Ehrenfest, P., Epstein, P.S. (1924/1927). Remarks on the quantum theory of diffraction, Proc. Natl. Acad. Sci. 13: 400–408. Examining the hypothesis of Duane on quantized translational momentum transfer, as it accounted for X-ray diffraction by crystals,Duane, W. (1923). The transfer in quanta of radiation momentum to matter, Proc. Natl. Acad. Sci. 9(5): 158–164. and its follow-up by Compton,Compton, A.H. (1923). The quantum integral and diffraction by a crystal, Proc. Natl. Acad. Sci. 9(11): 360–362. 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." Later, Born and Biem wrote: "Every physicist must accept Duane's rule."Landé, A., Born, M., Biem, W. (1968). 'Dialog on dualism', Physics Today, 21(8): 55–56; .
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.Heisenberg, W. (1930). The Physical Principles of the Quantum Theory, translated by C. Eckart and F.C. Hoyt, University of Chicago Press, Chicago, pp. 77–78.Pauling, L.C., Wilson, E.B. (1935). Introduction to Quantum Mechanics: with Applications to Chemistry, McGraw-Hill, New York, pp. 34–36.Landé, A. (1951). Quantum Mechanics, Sir Isaac Pitman and Sons, London, pp. 19–22.Bohm, D. (1951). Quantum Theory, Prentice Hall, New York, pp. 71–73.
Young's two-slit diffraction experiment, with Fourier analysis
Gregory Breit in 1923 pointed out that such quantum translational momentum transfer, examined by Fourier analysis in the old quantum theory, accounts for diffraction even by only two slits.Breit, G. (1923). The interference of light and the quantum theory, Proc. Natl. Acad. Sci. 9: 238–243. More recently, two slit particle diffraction has been experimentally demonstrated with single-particle buildup of electron diffraction patterns, as may be seen in the photo in this referenceTonomura, A., Endo, J., Matsuda, T., Kawasaki, T., Ezawa, H. (1989). Demonstration of single‐electron buildup of an interference pattern, Am. J. Phys. 57(2): 117–120.Dragoman, D. Dragoman, M. (2004). Quantum–Classical Analogies, Springer, Berlin, , pp. 170–175. and with helium atoms and molecules.Schmidt, L.P.H., Lower, J., Jahnke, T., Schößler, S., Schöffler, M.S., Menssen, A., Lévêque, C., Sisourat, N., Taïeb, R., Schmidt-Böcking, H., Dörner, R. (2013). Momentum transfer to a free floating double slit: realization of a thought experiment from the Einstein-Bohr debates, Physical Review Letters 111: 103201, 1–5.
Bragg diffraction
A wave of wavelength is incident at angle upon an array of crystal atomic planes, lying in a characteristic orientation, separated by a characteristic distance . Two rays of the beam are reflected from planes separated by distance , where denotes the number of planes of the separation, and is called the order of diffraction. If is such that
then there is constructive interference between the reflected rays, which may be observed in the interference pattern. This is Bragg's law.
The same phenomenon, considered from a different viewpoint, is described by a beam of particles of momentum incident at angle upon the same array of crystal atomic planes. It is supposed that a collective of such atomic planes reflects the particle, transferring to it a momentum , where is a momentum characteristic of the reflecting planes, in the direction perpendicular to them. 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 . For reflection, the change of momentum of the particle in that direction must be . Consequently,
This agrees with the observed Bragg condition for the diffraction pattern if is such that or
It is evident that provides information for a particle viewpoint, while provides information for a wave viewpoint. Before the discovery of quantum mechanics, de Broglie in 1923 discovered how to inter-translate the particle viewpoint information and the wave viewpoint information for material particles:Bohr, N. (1948). On the notions of causality and complementarity, Dialectica 2: 312–319; p. 313: "It is further important to realize that any determination of Planck's constant rests upon the comparison between aspects of the phenomena which can be described only by means of pictures not combinable on the basis of classical theories."Messiah, A. (1961). Quantum Mechanics, volume 1, translated by G.M. Temmer from the French Mécanique Quantique, North-Holland, Amsterdam, p. 52, "relations between dynamical variables of the particle and characteristic quantities of the associated wave". use the Planck constant and recall Einstein's formula for photons:
It follows that the characteristic quantum of translational momentum for the crystal planes of interest is given by Heisenberg, W. (1930). The Physical Principles of the Quantum Theory, translated by C. Eckart and F.C. Hoyt, University of Chicago Press, Chicago, p. 77.Landé, A. (1951). Quantum Mechanics, Sir Isaac Pitman and Sons, London, p. 20.
Quantum mechanics
According to Ballentine, Duane's proposal of quantum translational momentum transfer is no longer needed as a special hypothesis; rather, it is predicted as a theorem of quantum mechanics.Ballentine, L.E. (1998). Quantum Mechanics: a Modern Development, World Scientific, Singapore, , p. 136. It is presented in terms of quantum mechanics by other present day writers also.Van Vliet, K. (1967). Linear momentum quantization in periodic structures, Physica, 35: 97–106, doi:10.1016/0031-8914(67)90138-3.Van Vliet, K. (2010). Linear momentum quantization in periodic structures ii, Physica A, 389: 1585–1593, doi:10.1016/j.physa.2009.12.026.Thankappan, V.K. (1985/2012). Quantum Mechanics, third edition, New Age International, New Delhi, , pp. 6–7.Wennerstrom, H. (2014). Scattering and diffraction described using the momentum representation, Advances in Colloid and Interface Science, 205: 105–112.Mehra, J., Rechenberg, H. (2001). The Historical Development of Quantum Theory, volume 1, part 2, Springer, pp. 555–556 here.Hickey, T.J. (2014). Twentieth-Century Philosophy of Science:a History, self-published by the author, here.
Diffraction
One may consider a particle with translational momentum , a vectorial quantity.
In the simplest example of scattering of two colliding particles with initial momenta , resulting in final momenta . The momentum transfer is given by
where the last identity expresses momentum conservation.Prigogine, I. (1962). Non-equilibrium Statistical Mechanics, Wiley, New York, pp. 258–262.
In diffraction, the difference of the momenta of the scattered particle and the incident particle is called momentum transfer.
Such phenomena can also be considered from a wave viewpoint, by use of the reduced Planck constant . The wave number is the absolute value of the wave vector , which is related to the wavelength . Often, momentum transfer is given in wavenumber units in reciprocal length
Momentum transfer is an important quantity because is a better measure for the typical distance resolution of the reaction than the momenta themselves.
Bragg diffraction occurs on the atomic crystal lattice. It conserves the particle energy and thus is called elastic scattering. The wave numbers of the final and incident particles, and , respectively, are equal. Just the direction changes by a reciprocal lattice vector with the relation to the lattice spacing . As momentum is conserved, the transfer of momentum occurs to crystal momentum.