1469:
that this did not lead him to think that the collective global properties of the grating should make it a diffractor with corresponding quantal properties, such as would supply the diffracted electron with a definite trajectory. It seems, rather, that he thought of the diffraction as necessarily a manifestation of wave character belonging to the electron. It seems that he felt this was necessary to explain interference when the electron was detected far from the diffractor. Thus it seems possible that in 1927, Heisenberg was not thinking in terms of Duane's hypothesis of quantal transfer of translative momentum. By 1930, however, Heisenberg thought enough of Duane's hypothesis to expound it in his textbook.
236:." Examining the hypothesis of Duane on quantized translational momentum transfer, as it accounted for X-ray diffraction by crystals, and its follow-up by Compton, 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."
22:
1490:. The quantum of translational 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, manifest for example in quantized quasi-particles, such as
1468:
It seems that
Heisenberg in 1927 was thinking in terms of a classical diffractor. 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". Nevertheless, it seems
1464:
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
1455:
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
254:
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. More recently, two slit particle diffraction has been experimentally demonstrated with single-particle buildup of electron
101:
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
387:
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
2130:
Schmidt, L.P.H., Lower, J., Jahnke, T., SchöĂler, S., Schöffler, M.S., Menssen, A., LĂ©vĂȘque, C., Sisourat, N., TaĂŻeb, 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,
661:
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. It is presented in terms of quantum mechanics by other present day writers also.
1485:
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 translational momentum, in close analogy with transfer of angular momentum in integer multiples of the
2157:: 312â319; 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."
562:
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: use the
190:
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
964:
1385:
220:". 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
116:
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.
1456:
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.
832:
768:
1083:
345:
455:
1221:
98:
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.
1123:
545:
601:
509:
1465:
terms of classical diffractors, formally belonging to the macroscopic laboratory apparatus, and of wave character belonging to the quantum object that suffers diffraction.
1172:
105:
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
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1422:
651:
1001:
1293:
1266:
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840:
120:
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
87:
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
1819:
2177:, North-Holland, Amsterdam, p. 52, "relations between dynamical variables of the particle and characteristic quantities of the associated wave".
2089:
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66:
1864:
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A quantum diffractor has an essentially quantum character. It was first conceived of in 1923 by
William Duane, in the days of the
350:
then there is constructive interference between the reflected rays, which may be observed in the interference pattern. This is
1719:
2118:
1731:
275:
upon an array of crystal atomic planes, lying in a characteristic orientation, separated by a characteristic distance
1658:. Monographs on the physics and chemistry of materials (1. publ. in paperback ed.). Oxford: Oxford Univ. Press.
773:
709:
243:, linking wavelengths and frequencies to energy and momenta, gives an account of diffraction of material particles.
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48:
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217:
1800:
1513:
305:
225:
80:
2445:
2367:
2350:
2547:
2510:
edited by H. Mark, S. Fernbach, Interscience
Publishers, New York, reprinted at pp. 335â339 in Heisenberg, W.,
1938:
1915:
1888:
1517:
1538:
The
Present Status of the Quantum Theory of Light: Proceedings of a Symposium in Honour of Jean-Pierre Vigier
976:
In diffraction, the difference of the momenta of the scattered particle and the incident particle is called
357:
The same phenomenon, considered from a different viewpoint, is described by a beam of particles of momentum
140:
diffraction are now known which cannot be explained by his approach. Hence his approach is no longer used.
415:
2019:
1179:
255:
diffraction patterns, as may be seen in the photo in this reference and with helium atoms and molecules.
970:
1091:
2437:
1809:
1740:
1697:
1494:. Formally, the diffractor belongs to the quantum system, not to the classical laboratory apparatus.
1444:
1223:
is a better measure for the typical distance resolution of the reaction than the momenta themselves.
514:
221:
153:
137:
133:
573:
472:
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2258:
2011:
1911:
1880:
1436:
1131:
983:
Such phenomena can also be considered from a wave viewpoint, by use of the reduced Planck constant
240:
229:
208:
In 1925, shortly before the development of the full mathematical description of quantum mechanics,
125:
2486:
Bacciagaluppi, G., Crull, E. (2009). Heisenberg (and Schrödinger, and Pauli) on hidden variables,
32:
2295:
2282:
2270:
2246:
1688:
1478:
1296:
1234:
2321:
Wennerstrom, H. (2014). Scattering and diffraction described using the momentum representation,
1839:
673:
36:
1684:"Ăber einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt"
1390:
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2503:
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1989:
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129:
1973:
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1432:
1425:
986:
213:
2309:
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1244:
293:
denotes the number of planes of the separation, and is called the order of diffraction. If
2508:
Properties of Matter Under Unusual Conditions (In Honor of Edward Teller's 60th Birthday),
2455:. Translated from the second German edition by J.F. Shearer, W.M. Deans at pp. 124â129 in
1795:
1780:
1679:
1487:
1230:
959:{\displaystyle {\vec {q}}={\vec {p}}_{i1}-{\vec {p}}_{f1}={\vec {p}}_{f2}-{\vec {p}}_{i2}}
564:
187:
179:
149:
2468:
2207:
2031:
1998:, translated by C. Eckart and F.C. Hoyt, University of Chicago Press, Chicago, pp. 77â78.
1957:
124:
was the only type of diffraction observable, and Duane's approach could model it. Modern
2416:
Böni, P., Furrer, A. (1999). Introduction to neutron scattering, Chapter 1, pp. 1â27 of
1813:
1744:
1701:
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1934:
1907:
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1835:
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351:
233:
121:
106:
88:
1563:
1540:, edited by Whitney, C.K., Jeffers, S., Roy, S., Vigier, J.-P., Hunter, G., Springer,
156:
could be explained if a beam of light was composed of a stream of discrete particles (
2536:
2338:
2198:, translated by C. Eckart and F.C. Hoyt, University of Chicago Press, Chicago, p. 77.
2065:
2007:
1823:
251:
398:. For reflection, the change of momentum of the particle in that direction must be
1756:
369:
upon the same array of crystal atomic planes. It is supposed that a collective of
1238:
1031:
1004:
2506:(1969/1985) The concept of "understanding" in theoretical physics, pp. 7â10 in
2069:
2147:
2049:
1086:
703:
113:
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1683:
460:
This agrees with the observed Bragg condition for the diffraction pattern if
132:
instruments are many orders of magnitude brighter, so many find details of
1961:
1860:
209:
375:
such atomic planes reflects the particle, transferring to it a momentum
281:. Two rays of the beam are reflected from planes separated by distance
1977:
1752:
246:
2088:
Tonomura, A., Endo, J., Matsuda, T., Kawasaki, T., Ezawa, H. (1989).
1491:
606:
It follows that the characteristic quantum of translational momentum
157:
2193:
2090:
Demonstration of singleâelectron buildup of an interference pattern
1993:
1380:{\displaystyle {\vec {G}}={\vec {Q}}={\vec {k}}_{f}-{\vec {k}}_{i}}
1482:
84:
83:
presented a discrete momentum-exchange model of the reflection of
2016:
Introduction to Quantum Mechanics: with Applications to Chemistry
1720:"Einstein's proposal of the photon concept: A translation of the
2273:(2010). Linear momentum quantization in periodic structures ii,
1847:
1424:. As momentum is conserved, the transfer of momentum occurs to
1536:
Bitsakis, E.(1997). The wave-particle duality, pp. 333â348 in
1447:
are nowadays commonly studied as momentum transfer processes.
247:
Young's two-slit diffraction experiment, with Fourier analysis
239:
Using Duane's 1923 hypothesis, the old quantum theory and the
15:
2249:(1967). Linear momentum quantization in periodic structures,
2399:, third edition, Cambridge University Press, Cambridge UK,
1295:, respectively, are equal. Just the direction changes by a
1125:. Often, momentum transfer is given in wavenumber units in
2150:(1948). On the notions of causality and complementarity,
1633:. Dover books on physics and chemistry. New York: Dover.
2397:
Introduction to the Theory of Thermal Neutron Scattering
1518:
The transfer in quanta of radiation momentum to matter,
670:
One may consider a particle with translational momentum
1608:(3rd ed.). Upper Saddle River, NJ: Prentice Hall.
44:
2173:, volume 1, translated by G.M. Temmer from the French
1233:. It conserves the particle energy and thus is called
1451:
Physical accounts of wave and of particle diffraction
1393:
1305:
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1134:
1094:
1039:
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989:
843:
776:
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676:
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provides information for a particle viewpoint, while
517:
475:
418:
308:
2420:, edited by A. Furrer, World Scientific, Singapore,
228:
proposed and which has been more closely studied by
195:
equal to its energy divided by the speed of light, (
41:
and explaining the responses to the fringe theories.
2488:
Studies in History and Philosophy of Modern Physics
2300:, third edition, New Age International, New Delhi,
1939:The quantum integral and diffraction by a crystal,
1785:
Mitteilungen der Physikalischen Gesellschaft ZĂŒrich
1654:Peng, L.-M.; Dudarev, S. L.; Whelan, M. J. (2011).
1176:Momentum transfer is an important quantity because
212:
drew Einstein's attention to the then-new idea of "
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339:
2364:Twentieth-Century Philosophy of Science:a History
2070:The interference of light and the quantum theory
1889:The quantum theory of the Fraunhofer diffraction
706:of two colliding particles with initial momenta
2514:, series C, volume 3, ed. W. Blum, H.-P. DĂŒrr,
2040:, Sir Isaac Pitman and Sons, London, pp. 19â22.
1656:High energy electron diffraction and microscopy
827:{\displaystyle {\vec {p}}_{f1},{\vec {p}}_{f2}}
763:{\displaystyle {\vec {p}}_{i1},{\vec {p}}_{i2}}
612:for the crystal planes of interest is given by
174:) of each photon being equal to the frequency (
2477:, Sir Isaac Pitman and Sons, London pp. 16â18.
1604:Cullity, Bernard D.; Stock, Stuart R. (2001).
2195:The Physical Principles of the Quantum Theory
1995:The Physical Principles of the Quantum Theory
8:
2347:The Historical Development of Quantum Theory
1916:Remarks on the quantum theory of diffraction
1798:(1917). "Zur Quantentheorie der Strahlung".
1783:(1916). "Zur Quantentheorie der Strahlung".
1509:
1507:
1078:{\displaystyle {\vec {k}}={\vec {p}}/\hbar }
2349:, volume 1, part 2, Springer, pp. 555â556
2216:, Sir Isaac Pitman and Sons, London, p. 20.
1871:, translated by I. Born, Macmillan, London.
567:and recall Einstein's formula for photons:
340:{\displaystyle 2d\sin \theta =n\lambda \,,}
112:Duane applied his hypothesis to derive the
2323:Advances in Colloid and Interface Science
1709:
1406:
1392:
1387:with the relation to the lattice spacing
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67:Learn how and when to remove this message
1964:, Biem, W. (1968). 'Dialog on dualism',
2227:Quantum Mechanics: a Modern Development
1503:
1192:
1072:
990:
2384:Non-equilibrium Statistical Mechanics
2056:, Prentice Hall, New York, pp. 71â73.
1718:Arons, A. B.; Peppard, M. B. (1965).
1241:of the final and incident particles,
7:
834:. The momentum transfer is given by
450:{\displaystyle 2p\sin \theta =nP\,.}
144:Early developments in quantum theory
2459:, Blackie & Son, London (1928).
1216:{\displaystyle \Delta x=\hbar /|q|}
2457:Collected papers on Wave Mechanics
2105:Dragoman, D. Dragoman, M. (2004).
1183:
969:where the last identity expresses
152:presented the hypothesis that the
14:
2440:(1927). Ăber den Comptoneffekt,
2366:, self-published by the author,
2259:doi:10.1016/0031-8914(67)90138-3
1576:10.1016/b978-044482218-5/50008-0
1481:, to account for diffraction of
1118:{\displaystyle \lambda =2\pi /k}
20:
2418:Frontiers of Neutron Scattering
2386:, Wiley, New York, pp. 258â262.
2283:doi:10.1016/j.physa.2009.12.026
2229:, World Scientific, Singapore,
1793:and a nearly identical version
1629:Warren, Bertram Eugene (1990).
540:{\displaystyle p\lambda =Pd\,.}
2294:Thankappan, V.K. (1985/2012).
1570:, Elsevier, pp. 123â144,
1365:
1343:
1327:
1312:
1209:
1201:
1061:
1046:
941:
916:
891:
866:
850:
809:
784:
745:
720:
683:
596:{\displaystyle p\lambda =h\,.}
504:{\displaystyle p/d=P/\lambda }
1:
1606:Elements of X-ray diffraction
1167:{\displaystyle Q=k_{f}-k_{i}}
770:, resulting in final momenta
79:In 1923, American physicist
47:or discuss the issue on the
2395:Squires, G.L. (1978/2012).
2107:QuantumâClassical Analogies
1732:American Journal of Physics
1564:"Diffraction from crystals"
702:In the simplest example of
2569:
1085:, which is related to the
692:{\displaystyle {\vec {p}}}
2225:Ballentine, L.E. (1998).
2018:, McGraw-Hill, New York,
1869:The Born-Einstein Letters
1801:Physikalische Zeitschrift
1431:For the investigation of
1417:{\displaystyle G=2\pi /d}
1711:10.1002/andp.19053220607
1562:COWLEY, JOHN M. (1995),
699:, a vectorial quantity.
646:{\displaystyle P=h/d\,.}
160:), each with an energy (
2133:Physical Review Letters
2074:Proc. Natl. Acad. Sci.
1941:Proc. Natl. Acad. Sci.
1920:Proc. Natl. Acad. Sci.
1893:Proc. Natl. Acad. Sci.
1865:Letter of 15 July 1925
1830:The Old Quantum Theory
1520:Proc. Natl. Acad. Sci.
1418:
1381:
1289:
1262:
1217:
1168:
1119:
1079:
1020:
997:
996:{\displaystyle \hbar }
960:
828:
764:
693:
647:
597:
541:
505:
451:
341:
39:to the mainstream view
2362:Hickey, T.J. (2014).
1419:
1382:
1290:
1288:{\displaystyle k_{i}}
1263:
1261:{\displaystyle k_{f}}
1229:occurs on the atomic
1218:
1169:
1120:
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1021:
998:
971:momentum conservation
961:
829:
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694:
648:
598:
542:
506:
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342:
269:is incident at angle
263:A wave of wavelength
2109:, Springer, Berlin,
1460:Classical diffractor
1445:electron diffraction
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987:
841:
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619:
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515:
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306:
178:) multiplied by the
154:photoelectric effect
126:electron microscopes
2175:MĂ©canique Quantique
1814:1917PhyZ...18..121E
1745:1965AmJPh..33..367A
1702:1905AnP...322..132E
1568:Diffraction Physics
550:It is evident that
241:de Broglie relation
94:The key feature of
2442:Annalen der Physik
1722:Annalen der Physik
1689:Annalen der Physik
1479:old quantum theory
1473:Quantum diffractor
1414:
1377:
1297:reciprocal lattice
1285:
1258:
1235:elastic scattering
1213:
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1016:
993:
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689:
643:
593:
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363:incident at angle
337:
186:). Later, in 1916
96:Duane's hypothesis
37:appropriate weight
2553:Quantum mechanics
2518:, Piper, Munich,
2474:Quantum Mechanics
2306:978-81-224-3357-9
2297:Quantum Mechanics
2213:Quantum Mechanics
2171:Quantum Mechanics
2037:Quantum Mechanics
1978:10.1063/1.3035103
1753:10.1119/1.1971542
1665:978-0-19-960224-7
1640:978-0-486-66317-3
1631:X-ray diffraction
1615:978-0-201-61091-8
1546:978-94-010-6396-8
1368:
1346:
1330:
1315:
1227:Bragg diffraction
1127:reciprocal length
1064:
1049:
1019:{\displaystyle k}
978:momentum transfer
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657:Quantum mechanics
259:Bragg diffraction
130:x-ray diffraction
77:
76:
69:
35:, without giving
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2453:(2)>: 257â264
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2408:
2405:978-110-764406-9
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1818:Translated here
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1755:. Archived from
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1426:crystal momentum
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679:
652:
650:
649:
644:
635:
611:
602:
600:
599:
594:
561:
555:
546:
544:
543:
538:
510:
508:
507:
502:
497:
483:
465:
456:
454:
453:
448:
409:. Consequently,
408:
397:
386:
380:
374:
368:
362:
346:
344:
343:
338:
298:
292:
286:
280:
274:
268:
204:
169:
72:
65:
61:
58:
52:
24:
23:
16:
2568:
2567:
2563:
2562:
2561:
2559:
2558:
2557:
2548:Albert Einstein
2533:
2532:
2531:
2530:
2512:Collected Works
2502:
2498:
2485:
2481:
2467:
2463:
2438:Schrödinger, E.
2436:
2432:
2415:
2411:
2394:
2390:
2378:
2374:
2361:
2357:
2337:
2333:
2320:
2316:
2293:
2289:
2269:
2265:
2245:
2241:
2224:
2220:
2206:
2202:
2188:
2181:
2165:
2161:
2146:
2142:
2129:
2125:
2104:
2100:
2092:, Am. J. Phys.
2087:
2083:
2064:
2060:
2048:
2044:
2030:
2026:
2006:
2002:
1988:
1984:
1956:
1952:
1933:
1929:
1906:
1902:
1879:
1875:
1867:, pp. 84â85 in
1859:
1855:
1822:
1794:
1779:
1778:
1774:
1765:
1763:
1759:
1726:
1717:
1716:Translated in
1678:
1677:
1673:
1666:
1653:
1652:
1648:
1641:
1628:
1627:
1623:
1616:
1603:
1602:
1598:
1590:
1588:
1586:
1561:
1560:
1556:
1535:
1531:
1512:
1505:
1500:
1488:Planck constant
1475:
1462:
1453:
1389:
1388:
1358:
1336:
1301:
1300:
1275:
1270:
1269:
1248:
1243:
1242:
1231:crystal lattice
1178:
1177:
1154:
1141:
1130:
1129:
1090:
1089:
1035:
1034:
1008:
1007:
985:
984:
934:
909:
884:
859:
839:
838:
802:
777:
772:
771:
738:
713:
708:
707:
672:
671:
668:
659:
617:
616:
607:
572:
571:
565:Planck constant
557:
551:
513:
512:
471:
470:
461:
414:
413:
399:
389:
382:
376:
370:
364:
358:
304:
303:
294:
288:
282:
276:
270:
264:
261:
249:
196:
188:Albert Einstein
180:Planck constant
161:
150:Albert Einstein
146:
73:
62:
56:
53:
45:help improve it
42:
33:fringe theories
25:
21:
12:
11:
5:
2566:
2564:
2556:
2555:
2550:
2545:
2535:
2534:
2529:
2528:
2504:Heisenberg, W.
2496:
2479:
2461:
2430:
2409:
2388:
2372:
2355:
2343:Rechenberg, H.
2331:
2314:
2287:
2263:
2239:
2218:
2200:
2190:Heisenberg, W.
2179:
2159:
2140:
2138:: 103201, 1â5.
2123:
2098:
2081:
2058:
2054:Quantum Theory
2042:
2024:
2000:
1990:Heisenberg, W.
1982:
1950:
1927:
1900:
1873:
1853:
1836:Pergamon Press
1772:
1724:paper of 1905"
1696:(6): 132â148.
1671:
1664:
1646:
1639:
1621:
1614:
1596:
1584:
1554:
1529:
1502:
1501:
1499:
1496:
1474:
1471:
1461:
1458:
1452:
1449:
1413:
1409:
1405:
1402:
1399:
1396:
1374:
1367:
1364:
1357:
1352:
1345:
1342:
1335:
1329:
1326:
1320:
1314:
1311:
1282:
1278:
1255:
1251:
1211:
1207:
1203:
1198:
1194:
1191:
1188:
1185:
1161:
1157:
1153:
1148:
1144:
1140:
1137:
1114:
1110:
1106:
1103:
1100:
1097:
1074:
1070:
1063:
1060:
1054:
1048:
1045:
1028:absolute value
1015:
992:
967:
966:
953:
950:
943:
940:
933:
928:
925:
918:
915:
908:
903:
900:
893:
890:
883:
878:
875:
868:
865:
858:
852:
849:
821:
818:
811:
808:
801:
796:
793:
786:
783:
757:
754:
747:
744:
737:
732:
729:
722:
719:
685:
682:
667:
664:
658:
655:
654:
653:
642:
638:
634:
630:
627:
624:
604:
603:
592:
588:
585:
582:
579:
548:
547:
536:
532:
529:
526:
523:
520:
500:
496:
492:
489:
486:
482:
478:
458:
457:
446:
442:
439:
436:
433:
430:
427:
424:
421:
348:
347:
336:
332:
329:
326:
323:
320:
317:
314:
311:
260:
257:
248:
245:
170:) the energy (
145:
142:
75:
74:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
2565:
2554:
2551:
2549:
2546:
2544:
2541:
2540:
2538:
2525:
2524:3-492-02927-2
2521:
2517:
2516:H. Rechenberg
2513:
2509:
2505:
2500:
2497:
2493:
2489:
2483:
2480:
2476:
2475:
2470:
2465:
2462:
2458:
2454:
2452:
2448:
2443:
2439:
2434:
2431:
2427:
2426:981-02-4069-4
2423:
2419:
2413:
2410:
2406:
2402:
2398:
2392:
2389:
2385:
2381:
2380:Prigogine, I.
2376:
2373:
2369:
2365:
2359:
2356:
2352:
2348:
2344:
2340:
2335:
2332:
2328:
2324:
2318:
2315:
2311:
2307:
2303:
2299:
2298:
2291:
2288:
2284:
2281:: 1585â1593,
2280:
2276:
2272:
2271:Van Vliet, K.
2267:
2264:
2260:
2256:
2252:
2248:
2247:Van Vliet, K.
2243:
2240:
2236:
2235:981-02-2707-8
2232:
2228:
2222:
2219:
2215:
2214:
2209:
2204:
2201:
2197:
2196:
2191:
2186:
2184:
2180:
2176:
2172:
2168:
2163:
2160:
2156:
2153:
2149:
2144:
2141:
2137:
2134:
2127:
2124:
2120:
2116:
2115:3-540-20147-5
2112:
2108:
2102:
2099:
2096:(2): 117â120.
2095:
2091:
2085:
2082:
2078:
2075:
2071:
2067:
2062:
2059:
2055:
2051:
2046:
2043:
2039:
2038:
2033:
2028:
2025:
2021:
2017:
2013:
2009:
2008:Pauling, L.C.
2004:
2001:
1997:
1996:
1991:
1986:
1983:
1979:
1975:
1971:
1967:
1966:Physics Today
1963:
1959:
1954:
1951:
1947:
1946:(11): 360â362
1945:
1942:
1936:
1935:Compton, A.H.
1931:
1928:
1924:
1921:
1917:
1914:(1924/1927).
1913:
1912:Epstein, P.S.
1909:
1908:Ehrenfest, P.
1904:
1901:
1897:
1894:
1890:
1886:
1885:Ehrenfest, P.
1882:
1881:Epstein, P.S.
1877:
1874:
1870:
1866:
1863:(1925/1971).
1862:
1857:
1854:
1849:
1845:
1841:
1837:
1832:
1831:
1825:
1820:
1815:
1811:
1807:
1803:
1802:
1797:
1790:
1786:
1782:
1776:
1773:
1762:on 2016-03-04
1758:
1754:
1750:
1746:
1742:
1738:
1734:
1733:
1725:
1723:
1712:
1707:
1703:
1699:
1695:
1691:
1690:
1685:
1681:
1675:
1672:
1667:
1661:
1657:
1650:
1647:
1642:
1636:
1632:
1625:
1622:
1617:
1611:
1607:
1600:
1597:
1587:
1585:9780444822185
1581:
1577:
1573:
1569:
1565:
1558:
1555:
1551:
1547:
1543:
1539:
1533:
1530:
1526:
1524:
1521:
1515:
1510:
1508:
1504:
1497:
1495:
1493:
1489:
1484:
1480:
1472:
1470:
1466:
1459:
1457:
1450:
1448:
1446:
1442:
1438:
1434:
1429:
1427:
1411:
1407:
1403:
1400:
1397:
1394:
1372:
1362:
1355:
1350:
1340:
1333:
1324:
1318:
1309:
1298:
1280:
1276:
1253:
1249:
1240:
1236:
1232:
1228:
1224:
1205:
1196:
1189:
1186:
1174:
1159:
1155:
1151:
1146:
1142:
1138:
1135:
1128:
1112:
1108:
1104:
1101:
1098:
1095:
1088:
1068:
1058:
1052:
1043:
1033:
1029:
1013:
1006:
981:
979:
974:
972:
951:
948:
938:
931:
926:
923:
913:
906:
901:
898:
888:
881:
876:
873:
863:
856:
847:
837:
836:
835:
819:
816:
806:
799:
794:
791:
781:
755:
752:
742:
735:
730:
727:
717:
705:
700:
680:
665:
663:
656:
640:
636:
632:
628:
625:
622:
615:
614:
613:
610:
590:
586:
583:
580:
577:
570:
569:
568:
566:
560:
554:
534:
530:
527:
524:
521:
518:
498:
494:
490:
487:
484:
480:
476:
469:
468:
467:
466:is such that
464:
444:
440:
437:
434:
431:
428:
425:
422:
419:
412:
411:
410:
407:
403:
396:
392:
385:
379:
373:
367:
361:
355:
353:
334:
330:
327:
324:
321:
318:
315:
312:
309:
302:
301:
300:
299:is such that
297:
291:
285:
279:
273:
267:
258:
256:
253:
252:Gregory Breit
244:
242:
237:
235:
231:
227:
223:
219:
215:
211:
206:
203:
199:
194:
189:
185:
181:
177:
173:
168:
164:
159:
155:
151:
143:
141:
139:
135:
131:
127:
123:
118:
115:
110:
108:
103:
99:
97:
92:
90:
86:
82:
81:William Duane
71:
68:
60:
50:
46:
40:
38:
34:
29:This article
27:
18:
17:
2511:
2507:
2499:
2491:
2487:
2482:
2473:
2464:
2456:
2450:
2446:
2441:
2433:
2417:
2412:
2396:
2391:
2383:
2375:
2363:
2358:
2346:
2334:
2326:
2322:
2317:
2296:
2290:
2278:
2274:
2266:
2254:
2250:
2242:
2226:
2221:
2212:
2203:
2194:
2174:
2170:
2162:
2154:
2151:
2143:
2135:
2132:
2126:
2106:
2101:
2093:
2084:
2076:
2073:
2061:
2053:
2045:
2036:
2027:
2015:
2012:Wilson, E.B.
2003:
1994:
1985:
1972:(8): 55â56;
1969:
1965:
1953:
1943:
1940:
1930:
1922:
1919:
1903:
1895:
1892:
1876:
1868:
1856:
1829:
1824:ter Haar, D.
1805:
1799:
1796:Einstein, A.
1788:
1784:
1781:Einstein, A.
1775:
1764:. Retrieved
1757:the original
1736:
1730:
1721:
1693:
1687:
1680:Einstein, A.
1674:
1655:
1649:
1630:
1624:
1605:
1599:
1589:, retrieved
1567:
1557:
1537:
1532:
1525:(5): 158â164
1522:
1519:
1476:
1467:
1463:
1454:
1430:
1239:wave numbers
1225:
1175:
982:
977:
975:
968:
701:
669:
660:
608:
605:
558:
552:
549:
462:
459:
405:
401:
394:
390:
383:
377:
371:
365:
359:
356:
349:
295:
289:
283:
277:
271:
265:
262:
250:
238:
207:
201:
197:
192:
183:
175:
171:
166:
162:
147:
119:
111:
104:
100:
95:
93:
78:
63:
54:
31:may present
30:
2167:Messiah, A.
2119:pp. 170â175
1838:. pp.
1808:: 121â128.
1032:wave vector
1005:wave number
666:Diffraction
352:Bragg's law
122:Bragg's law
107:Bragg's Law
89:Bragg's Law
57:August 2023
2543:Max Planck
2537:Categories
2494:: 374â382.
2444:series 4,
2329:: 105â112.
2257:: 97â106,
2152:Dialectica
2079:: 238â243.
1925:: 400â408.
1898:: 133â139.
1887:, (1924).
1766:2014-09-14
1739:(5): 367.
1591:2023-08-13
1498:References
1087:wavelength
704:scattering
214:de Broglie
114:scattering
102:outcomes.
2526:, p. 336.
2469:Landé, A.
2339:Mehra, J.
2275:Physica A
2237:, p. 136.
2208:Landé, A.
2066:Breit, G.
2032:Landé, A.
2020:pp. 34â36
1958:Landé, A.
1514:Duane, W.
1404:π
1366:→
1356:−
1344:→
1328:→
1313:→
1193:ℏ
1184:Δ
1152:−
1105:π
1096:λ
1073:ℏ
1062:→
1047:→
991:ℏ
942:→
932:−
917:→
892:→
882:−
867:→
851:→
810:→
785:→
746:→
721:→
684:→
581:λ
522:λ
499:λ
432:θ
429:
331:λ
322:θ
319:
234:Ehrenfest
148:In 1905,
49:talk page
2471:(1951).
2382:(1962).
2345:(2001).
2210:(1951).
2192:(1930).
2169:(1961).
2148:Bohr, N.
2068:(1923).
2052:(1951).
2050:Bohm, D.
2034:(1951).
2014:(1935).
1992:(1930).
1962:Born, M.
1937:(1923).
1861:Born, M.
1848:66029628
1826:(1967).
1791:: 47â62.
1682:(1905).
1516:(1923).
381:, where
287:, where
134:electron
2310:pp. 6â7
2251:Physica
1840:167â183
1821:and in
1810:Bibcode
1741:Bibcode
1698:Bibcode
1492:phonons
1437:neutron
1299:vector
1030:of the
1026:is the
230:Epstein
222:Compton
158:photons
43:Please
2522:
2424:
2403:
2304:
2233:
2113:
1846:
1662:
1637:
1612:
1582:
1550:p. 338
1544:
1483:X-rays
1237:. The
1003:. The
1760:(PDF)
1727:(PDF)
1441:X-ray
226:Duane
218:waves
138:x-ray
85:X-ray
2520:ISBN
2449:<
2422:ISBN
2401:ISBN
2368:here
2351:here
2302:ISBN
2231:ISBN
2111:ISBN
1844:LCCN
1660:ISBN
1635:ISBN
1610:ISBN
1580:ISBN
1542:ISBN
1443:and
1268:and
404:sin
393:sin
232:and
224:and
210:Born
136:and
128:and
2451:387
2327:205
2279:389
2136:111
1974:doi
1749:doi
1706:doi
1572:doi
511:or
426:sin
316:sin
216:'s
205:).
202:E/c
2539::
2492:40
2490:,
2447:82
2341:,
2325:,
2308:,
2277:,
2255:35
2253:,
2182:^
2117:,
2094:57
2072:,
2010:,
1970:21
1968:,
1960:,
1923:13
1918:,
1910:,
1896:10
1891:,
1883:,
1842:.
1834:.
1806:18
1804:.
1789:18
1787:.
1747:.
1737:33
1735:.
1729:.
1704:.
1694:17
1692:.
1686:.
1578:,
1566:,
1548:,
1506:^
1439:,
1435:,
1428:.
980:.
973:.
378:nP
354:.
284:nd
200:=
167:hf
165:=
109:.
91:.
2428:.
2407:.
2370:.
2353:.
2312:.
2285:.
2261:.
2155:2
2121:.
2077:9
2022:.
1980:.
1976::
1948:.
1944:9
1850:.
1816:.
1812::
1769:.
1751::
1743::
1714:.
1708::
1700::
1668:.
1643:.
1618:.
1574::
1552:.
1527:.
1523:9
1412:d
1408:/
1401:2
1398:=
1395:G
1373:i
1363:k
1351:f
1341:k
1334:=
1325:Q
1319:=
1310:G
1281:i
1277:k
1254:f
1250:k
1210:|
1206:q
1202:|
1197:/
1190:=
1187:x
1160:i
1156:k
1147:f
1143:k
1139:=
1136:Q
1113:k
1109:/
1102:2
1099:=
1069:/
1059:p
1053:=
1044:k
1014:k
952:2
949:i
939:p
927:2
924:f
914:p
907:=
902:1
899:f
889:p
877:1
874:i
864:p
857:=
848:q
820:2
817:f
807:p
800:,
795:1
792:f
782:p
756:2
753:i
743:p
736:,
731:1
728:i
718:p
681:p
641:.
637:d
633:/
629:h
626:=
623:P
609:P
591:.
587:h
584:=
578:p
559:λ
553:p
535:.
531:d
528:P
525:=
519:p
495:/
491:P
488:=
485:d
481:/
477:p
463:Ξ
445:.
441:P
438:n
435:=
423:p
420:2
406:Ξ
402:p
400:2
395:Ξ
391:p
384:P
372:n
366:Ξ
360:p
335:,
328:n
325:=
313:d
310:2
296:Ξ
290:n
278:d
272:Ξ
266:λ
198:p
193:p
184:h
182:(
176:f
172:E
163:E
70:)
64:(
59:)
55:(
51:.
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