Quantal translative momentum transfer

When a beam of light or material particles impinges on a structure with spatial periodicity, such as a crystal or a standing wave of light, there occurs quantized transfer of momentum, which is observed as diffraction or scattering of the beam.

Old quantum theory

In 1905, going beyond Planck's idea of quantum generation of light, Einstein introduced the idea that light could be interpreted as propagating in quanta of energy.[1]

In 1916 Einstein further considered the light quantum concept, and indicated that it involved not only quantum energy transfer, but also quantum momentum transfer.[2]

In 1925, shortly before the invention 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."[3][4][5] Examining the hypothesis of Duane on quantized momentum transfer, as it accounted for X-ray diffraction by crystals,[6] and its follow-up by Compton,[7] 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."

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.[8][9][10][11]

Young's two-slit diffraction experiment, with Fourier analysis

Gregory Breit in 1923 pointed out that such quantum momentum transfer, examined by Fourier analysis in the old quantum theory, accounts for diffraction even by only two slits.[12] 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 reference[13][14] and with helium atoms and molecules.[15]

Bragg diffraction

A wave of wavelength λ is incident at angle θ upon an array of crystal atomic planes, lying in a characteristic orientation, separated by a characteristic distance d. Two rays of the beam are reflected from planes separated by distance nd, where n denotes the number of planes of the separation, and is called the order of diffraction. If θ is such that

2 d\sin\theta = n\lambda \,,

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 p incident at angle θ upon the same array of crystal atomic planes. It is supposed that a collective of n such atomic planes reflects the particle, transferring to it a momentum nP, where P 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 p sin θ. For reflection, the change of momentum of the particle in that direction must be 2p sin θ. Consequently

2 p\sin\theta = nP \,.

This agrees with the observed Bragg condition for the diffraction pattern if θ is such that

p/d = P/ \lambda or p\lambda = Pd \,.

It is evident that p provides information for a particle viewpoint, while λ 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:[16][17] use Planck's constant and recall Einstein's formula for photons:

p\lambda = h \,.

It follows that the characteristic quantum of momentum P for the crystal planes of interest is given by

P = h/d\,.[18][19]

Quantum mechanics

According to Ballentine, Duane's proposal of quantum momentum transfer is no longer needed as a special hypothesis; rather, it is predicted as a theorem of quantum mechanics.[20] It is presented in terms of quantum mechanics by other present day writers also.[21][22][23][24][25][26]

Diffraction

A particle has a momentum \vec p = \hbar\vec k , a vectorial quantity.

In the simplest example of scattering of two colliding particles with initial momenta \vec{p}_{i1},\vec{p}_{i2}, resulting in final momenta \vec{p}_{f1},\vec{p}_{f2}. The momentum transfer is given by

\vec q = \vec{p}_{i1} - \vec{p}_{f1} = \vec{p}_{f2} - \vec{p}_{i2}

where the last identity expresses momentum conservation.[27]

In diffraction, the difference of the momenta of the scattered particle and the incident particle is called momentum transfer. The wave number k is the absolute value of the wave vector \vec k = \vec q / \hbar and is related to the wavelength k = 2\pi / \lambda. Often, momentum transfer is given in wavenumber units in reciprocal length Q = k_f - k_i

Momentum transfer is an important quantity because \Delta x = \hbar / |q| 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, k_f and k_i, respectively, are equal. Just the direction changes by a reciprocal lattice vector \vec G = \vec Q = \vec k_f -\vec k_i with the relation to the lattice spacing G = 2\pi / d . As momentum is conserved, the transfer of momentum occurs to crystal momentum.

Practical use

Neutron scattering or diffraction is nowadays commonly studied as a momentum transfer process.[28][29]

Momentum transfer plays an important role in the evaluation of neutron, X-ray and electron diffraction for the investigation of condensed matter.

The presentation in Q-space is generic and does not depend on the type of radiation and wavelength used but only on the sample system, which allows comparison of results obtained from many different methods. Some established communities such as powder diffraction employ the diffraction angle 2\theta as the independent variable, which worked well in the early years when only a few characteristic wavelengths such as Cu-K\alpha were available. The relationship to Q-space is

Q = \frac {4 \pi \sin \theta }{\lambda}

and basically states that larger 2\theta corresponds to larger Q.

Physical accounts of wave and of particle diffraction

The phenomena may be analysed in several appropriate ways. The incoming and outgoing diffracted objects may be treated severally as particles or as waves. The diffracting object may be treated as a macroscopic classical object free of quantum features, or it may be treated as a physical object with essentially quantum character. Several cases of these forms of analysis, of which there are eight have been considered. For example, Schrödinger proposed a purely wave account of the Compton effect.[30][31]

Classical diffractor

A classical diffractor is devoid of quantum character. 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.

Quantum diffractor

A quantum diffractor has an essentially quantum character. 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. The diffractor was imagined as exhibiting quantum transfer of momentum, in close analogy with transfer of angular momentum in integer multiples of Planck's constant. The quantum of momentum was proposed to be explained by global quantum physical properties of the diffractor arising from its spatial periodicity. This is consonant with present-day quantum mechanical thinking, in which macroscopic physical bodies are conceived as supporting collective modes[32] manifest for example in quantized quasi-particles, such as phonons. Formally, the diffractor belongs to the quantum system, not to the laboratory apparatus.

References

  1. ↑ Einstein, A. (1905). "Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt". Annalen der Physik 17 (6): 132–148. Bibcode:1905AnP...322..132E. doi:10.1002/andp.19053220607. Translated in Arons, A. B.; Peppard, M. B. (1965). "Einstein's proposal of the photon concept: A translation of the Annalen der Physik paper of 1905". American Journal of Physics 33 (5): 367. Bibcode:1965AmJPh..33..367A. doi:10.1119/1.1971542.
  2. ↑ Einstein, A. (1916). "Zur Quantentheorie der Strahlung". Mitteilungen der Physikalischen Gesellschaft Zürich 18: 47–62. and a nearly identical version Einstein, A. (1917). "Zur Quantentheorie der Strahlung". Physikalische Zeitschrift 18: 121–128. Bibcode:1917PhyZ...18..121E. Translated here and in ter Haar, D. (1967). The Old Quantum Theory. Pergamon Press. pp. 167–183. LCCN 66029628.
  3. ↑ Born, M. (1925/1971). Letter of 15 July 1925, pp. 84–85 in The Born-Einstein Letters, translated by I. Born, Macmillan, London.
  4. ↑ Epstein, P.S., Ehrenfest, P., (1924). The quantum theory of the Fraunhofer diffraction, Proc. Natl. Acad. Sci. 10: 133–139.
  5. ↑ Ehrenfest, P., Epstein, P.S. (1924/1927). Remarks on the quantum theory of diffraction, Proc. Natl. Acad. Sci. 13: 400–408.
  6. ↑ Duane, W. (1923). The transfer in quanta of radiation momentum to matter, Proc. Natl. Acad. Sci. 9(5): 158–164.
  7. ↑ Compton, A.H. (1923). The quantum integral and diffraction by a crystal, Proc. Natl. Acad. Sci. 9(11): 360–362.
  8. ↑ 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.
  9. ↑ Pauling, L.C., Wilson, E.B. (1935). Introduction to Quantum Mechanics: with Applications to Chemistry, McGraw-Hill, New York, pp. 34–36.
  10. ↑ Landé, A. (1951). Quantum Mechanics, Sir Isaac Pitman and Sons, London, pp. 19–22.
  11. ↑ Bohm, D. (1951). Quantum Theory, Prentice Hall, New York, pp. 71–73.
  12. ↑ Breit, G. (1923). The interference of light and the quantum theory, Proc. Natl. Acad. Sci. 9: 238–243.
  13. ↑ Tonomura, 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.
  14. ↑ Dragoman, D. Dragoman, M. (2004). Quantum–Classical Analogies, Springer, Berlin, ISBN 3-540-20147-5, pp. 170–175.
  15. ↑ 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.
  16. ↑ 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."
  17. ↑ 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".
  18. ↑ 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.
  19. ↑ Landé, A. (1951). Quantum Mechanics, Sir Isaac Pitman and Sons, London, p. 20.
  20. ↑ Ballentine, L.E. (1998). Quantum Mechanics: a Modern Development, World Scientific, Singapore, ISBN 981-02-2707-8, p. 136.
  21. ↑ Van Vliet, K. (1967). Linear momentum quantization in periodic structures, Physica, 35: 97–106, doi:10.1016/0031-8914(67)90138-3.
  22. ↑ Van Vliet, K. (2010). Linear momentum quantization in periodic structures ii, Physica A, 389: 1585–1593, doi:10.1016/j.physa.2009.12.026.
  23. ↑ Thankappan, V.K. (1985/2012). Quantum Mechanics, third edition, New Age International, New Delhi, ISBN 978-81-224-3357-9, pp. 6–7.
  24. ↑ Wennerstrom, H. (2014). Scattering and diffraction described using the momentum representation, Advances in Colloid and Interface Science, 205: 105–112.
  25. ↑ Mehra, J., Rechenberg, H. (2001). The Historical Development of Quantum Theory, volume 1, part 2, Springer, pp. 555–556 here.
  26. ↑ Hickey, T.J. (2014). Twentieth-Century Philosophy of Science:a History, self-published by the author, here.
  27. ↑ Prigogine, I. (1962). Non-equilibrium Statistical Mechanics, Wiley, New York, pp. 258–262.
  28. ↑ Squires, G.L. (1978/2012). Introduction to the Theory of Thermal Neutron Scattering, third edition, Cambridge University Press, Cambridge UK, ISBN 978-110-764406-9.
  29. ↑ Böni, P., Furrer, A. (1999). Introduction to neutron scattering, Chapter 1, pp. 1–27 of Frontiers of Neutron Scattering, edited by A. Furrer, World Scientific, Singapore, ISBN 981-02-4069-4.
  30. ↑ Schrödinger, E. (1927). Über den Comptoneffekt, Annalen der Physik series 4, 82<387(2)>: 257–264. Translated from the second German edition by J.F. Shearer, W.M. Deans at pp. 124–129 in Collected papers on Wave Mechanics, Blackie & Son, London (1928).
  31. ↑ Landé, A. (1951). Quantum Mechanics, Sir Isaac Pitman and Sons, London pp. 16–18.
  32. ↑ Heisenberg, W. (1969/1985). The concept of "understanding" in theoretical physics, pp. 7–10 in Properties of Matter Under Unusual Conditions (In Honor of Edward Teller's 60th Birthday), edited by H. Mark, S. Fernbach, Interscience Publishers, New York, reprinted at pp. 335–339 in Heisenberg, W., Collected Works, series C, volume 3, ed. W. Blum, H.-P. Dürr, H. Rechenberg, Piper, Munich, ISBN 3-492-02927-2, p. 336.