1758:, until the wave packet and its phase maxima move together near the speed of light, whereas the wavelength continues to decrease without bound. Both transverse and longitudinal coherence widths (packet sizes) of such high energy electrons in the lab may be orders of magnitude larger than the ones shown here.
1493:
2171:
is the string's mass per unit length. As for the case of electromagnetic waves in vacuum, ideal strings are thus a non-dispersive medium, i.e. the phase and group velocities are equal and independent (to first order) of vibration frequency.
1380:
1712:
917:
2316:
is also non-trivial and important, being directly related to the acoustic and thermal properties of a material. For most systems, the phonons can be categorized into two main types: those whose bands become zero at the center of the
630:
1144:
2372:
studied refraction in prisms but failed to recognize the material dependence of the dispersion relation, dismissing the work of another researcher whose measurement of a prism's dispersion did not match Newton's own.
1209:
2002:
2084:
2245:
796:
1841:
1390:
2158:
1610:
288:
2101:
1943:
is the acceleration due to gravity. Deep water, in this respect, is commonly denoted as the case where the water depth is larger than half the wavelength. In this case the phase velocity is
1027:
1278:
451:
1615:
820:
82:
of each sinusoidal component of a wave in the medium, as a function of frequency. In addition to the geometry-dependent and material-dependent dispersion relations, the overarching
1934:
974:
395:
355:
225:
1051:
721:
320:
1523:
184:
2268:
937:
2282:
are possible for a given momentum and that some energies might not be available at any momentum. The collection of all possible energies and momenta is known as the
1269:
1543:
164:
2514:
P. M. Jones, G. M. Rackham and J. W. Steeds (1977). "Higher order Laue zone effects in electron diffraction and their use in lattice parameter determination".
503:
2665:
1750:. The top electron has twice the momentum, while the bottom electron has half. Note that as the momentum increases, the phase velocity decreases down to
2361:
has found application in the precise measurement of lattice parameters, beam energy, and more recently for the electronics industry: lattice strain.
1806:. In this case, the waveform will spread over time, such that a narrow pulse will become an extended pulse, i.e., be dispersed. In these materials,
2278:
In the study of solids, the study of the dispersion relation of electrons is of paramount importance. The periodicity of crystals means that many
2100:
1160:
1798:. It is possible to make the effective speed of light dependent on wavelength by making light pass through a material which has a non-constant
2497:
31:
2659:
1949:
976:) in the non-relativistic approximation. The variation has two parts: a constant part due to the de Broglie frequency of the rest mass (
2013:
2181:
1742:
This animation portrays the de
Broglie phase and group velocities (in slow motion) of three free electrons traveling over a field 0.4
736:
112:
In the presence of dispersion, a wave does not propagate with an unchanging waveform, giving rise to the distinct frequency-dependent
676:
Plane waves in vacuum are the simplest case of wave propagation: no geometric constraint, no interaction with a transmitting medium.
2570:
2472:
2447:
1736:
1870:
2338:
1746:
in width. The momentum per unit mass (proper velocity) of the middle electron is lightspeed, so that its group velocity is 0.707
2358:
2312:
Phonons are to sound waves in a solid what photons are to light: they are the quanta that carry it. The dispersion relation of
1809:
1881:
green dots propagate with the group velocity. In this deep-water case, the phase velocity is twice the group velocity. The
2680:
2384:
2122:
1548:
1045:
83:
1772:
As mentioned above, when the focus in a medium is on refraction rather than absorption—that is, on the real part of the
322:
expresses the dispersion relation of the given medium. Dispersion relations are more commonly expressed in terms of the
240:
979:
142:
Dispersion occurs when sinusoidal waves of different wavelengths have different propagation velocities, so that a
1860:
133:
2387:(1926–27) became apparent with subsequent papers on the dispersion relation's connection to causality in the
1488:{\displaystyle \omega ={\frac {m_{0}c^{2}}{\hbar }}{\sqrt {1+\left({\frac {k\hbar }{m_{0}c}}\right)^{2}}}\,.}
403:
1715:
1903:
51:
2287:
109:, have a nontrivial dispersion relation, even in the absence of geometric constraints and other media.
1714:
This gives the non-relativistic approximation discussed above. If we start with the non-relativistic
942:
363:
2630:
2595:
2519:
2377:
2346:
2295:
1147:
685:
2175:
For a nonideal string, where stiffness is taken into account, the dispersion relation is written as
1847:
and corresponds to the speed at which the peak of the pulse propagates, a value different from the
328:
192:
1799:
1789:
137:
129:
102:
35:
2535:
1894:
1041:
694:
296:
1375:{\displaystyle \omega (k)={\sqrt {k^{2}c^{2}+\left({\frac {m_{0}c^{2}}{\hbar }}\right)^{2}}}\,.}
2342:
1707:{\displaystyle \omega (k)\approx {\frac {m_{0}c^{2}}{\hbar }}+{\frac {\hbar k^{2}}{2m_{0}}}\,.}
912:{\displaystyle \omega (k)\approx {\frac {m_{0}c^{2}}{\hbar }}+{\frac {\hbar k^{2}}{2m_{0}}}\,.}
2566:
2562:
2555:
2493:
2468:
2443:
2439:
2388:
1776:—it is common to refer to the functional dependence of angular frequency on wavenumber as the
1500:
1221:
323:
169:
2253:
1387:
at non-relativistic velocity. To approximate, we pull out the rest-mass dependent frequency:
922:
2638:
2603:
2527:
2410:
2405:
2325:, since they correspond to classical sound in the limit of long wavelengths. The others are
2095:
1864:
1773:
1254:
87:
47:
2607:
2104:
Two-frequency beats of a non-dispersive transverse wave. Since the wave is non-dispersive,
730:
dispersion relation. In this case, the phase velocity and the group velocity are the same:
2322:
2634:
2599:
2523:
2354:
2326:
2318:
2283:
1848:
1844:
1743:
1528:
802:
625:{\displaystyle A(x,t)=A_{0}e^{2\pi i{\frac {x-vt}{\lambda }}}=A_{0}e^{i(kx-\omega t)},}
484:
473:
149:
117:
113:
79:
75:
2674:
2432:
2291:
2279:
1780:. For particles, this translates to a knowledge of energy as a function of momentum.
2539:
1139:{\displaystyle E^{2}=(p{\textrm {c}})^{2}+\left(m_{0}{\textrm {c}}^{2}\right)^{2}\,}
30:
17:
2621:
John S. Toll (1956). "Causality and the dispersion relation: Logical foundations".
2400:
2369:
2492:. Advanced Series on Ocean Engineering. Vol. 2. World Scientific, Singapore.
2286:
of a material. Properties of the band structure define whether the material is an
1384:
1248:
1151:
1038:
814:
143:
106:
91:
1231:
358:
146:
of mixed wavelengths tends to spread out in space. The speed of a plane wave,
101:, shallow water) or by interaction of the waves with the transmitting medium.
67:
63:
39:
2642:
1803:
98:
71:
2531:
472:) to describe the dispersion relation has become standard because both the
74:. Given the dispersion relation, one can calculate the frequency-dependent
1204:{\displaystyle E=\hbar \omega \,,\quad \mathbf {p} =\hbar \mathbf {k} \,,}
62:
on the properties of waves in a medium. A dispersion relation relates the
2662:
to help visualize dispersion surfaces, by Andrey
Chuvilin and Ute Kaiser
1869:
1735:
1037:
While applications of matter waves occur at non-relativistic velocity,
2313:
2307:
1795:
27:
Relation of wavelength/wavenumber as a function of a wave's frequency
688:
in vacuum, the angular frequency is proportional to the wavenumber:
42:
at different angles, splitting white light into a rainbow of colors.
2099:
1997:{\displaystyle v_{p}={\frac {\omega }{k}}={\sqrt {\frac {g}{k}}},}
1868:
1734:
97:
Dispersion may be caused either by geometric boundary conditions (
29:
2337:
With high-energy (e.g., 200 keV, 32 fJ) electrons in a
2079:{\displaystyle v_{g}={\frac {d\omega }{dk}}={\frac {1}{2}}v_{p}.}
1873:
Frequency dispersion of surface gravity waves on deep water. The
2240:{\displaystyle \omega ^{2}={\frac {T}{\mu }}k^{2}+\alpha k^{4},}
791:{\displaystyle v={\frac {\omega }{k}}={\frac {d\omega }{dk}}=c,}
2561:(illustrated, revised ed.). Cambridge University. p.
2116:
For an ideal string, the dispersion relation can be written as
2586:
A. D. D. Craik (2004). "The origins of water wave theory".
2329:, since they can be excited by electromagnetic radiation.
1885:
red square traverses the figure in the time it takes the
397:. Rewriting the relation above in these variables gives
661:
is a position along the wave's direction of travel, and
1836:{\displaystyle {\frac {\partial \omega }{\partial k}}}
1802:, or by using light in a non-uniform medium such as a
1722:
2256:
2184:
2125:
2016:
1952:
1906:
1812:
1794:
The name "dispersion relation" originally comes from
1618:
1551:
1531:
1503:
1393:
1281:
1257:
1163:
1054:
1044:
to derive his waves. Starting from the relativistic
982:
945:
925:
823:
739:
697:
506:
497:
The plane waves being considered can be described by
406:
366:
331:
299:
243:
195:
172:
152:
2153:{\displaystyle \omega =k{\sqrt {\frac {T}{\mu }}},}
1718:we will end up without the first, rest mass, term.
1605:{\displaystyle {\sqrt {1+x^{2}}}\approx 1+x^{2}/2,}
494:have convenient representations via this function.
2554:
2463:R. A. Serway, C. J. Moses and C. A. Moyer (1989).
2431:
2262:
2239:
2152:
2078:
1996:
1928:
1877:red square moves with the phase velocity, and the
1835:
1706:
1604:
1537:
1517:
1487:
1374:
1263:
1203:
1138:
1021:
968:
931:
911:
790:
715:
624:
445:
389:
349:
314:
283:{\displaystyle v(\lambda )=\lambda \ f(\lambda ).}
282:
219:
178:
158:
2490:Water wave mechanics for engineers and scientists
817:the frequency dispersion relation is non-linear:
2353:cross-sections of a crystal's three-dimensional
1029:) and a quadratic part due to kinetic energy.
1754:, whereas the group velocity increases up to
230:The wave's speed, wavelength, and frequency,
8:
2376:Dispersion of waves on water was studied by
1022:{\displaystyle \hbar \omega _{0}=m_{0}c^{2}}
919:The equation says the matter wave frequency
805:in vacuum, which is frequency-independent.
667:is the time at which the wave is described.
2557:Never at Rest: A Biography of Isaac Newton
2349:(CBED) patterns allows one, in effect, to
2270:is a constant that depends on the string.
1273:relativistic frequency dispersion relation
2255:
2228:
2212:
2198:
2189:
2183:
2135:
2124:
2067:
2053:
2030:
2021:
2015:
1979:
1966:
1957:
1951:
1913:
1905:
1813:
1811:
1700:
1691:
1676:
1666:
1651:
1641:
1634:
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1400:
1392:
1368:
1360:
1344:
1334:
1327:
1313:
1303:
1297:
1280:
1271:and take the square root. This gives the
1256:
1197:
1192:
1181:
1176:
1162:
1135:
1129:
1118:
1112:
1111:
1104:
1085:
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1059:
1053:
1013:
1003:
990:
981:
958:
944:
924:
905:
896:
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871:
856:
846:
839:
822:
759:
746:
738:
696:
592:
582:
552:
542:
532:
505:
405:
379:
365:
330:
298:
242:
194:
171:
166:, is a function of the wave's wavelength
151:
2341:, the energy dependence of higher-order
2167:is the tension force in the string, and
2488:R. G. Dean and R. A. Dalrymple (1991).
2467:. Philadelphia: Saunders. p. 118.
2422:
1669:
1658:
1504:
1448:
1424:
1385:Practical work with matter waves occurs
1351:
1258:
1189:
1170:
983:
874:
863:
446:{\displaystyle \omega (k)=v(k)\cdot k.}
2608:10.1146/annurev.fluid.36.050802.122118
2430:F. A. Jenkins and H. E. White (1957).
1729:phase and group velocity of electrons
2391:of all types of waves and particles.
1929:{\displaystyle \omega ={\sqrt {gk}},}
86:describe the frequency-dependence of
7:
1824:
1816:
939:in vacuum varies with wavenumber (
25:
2438:. New York: McGraw-Hill. p.
2308:Phonon § Dispersion relation
2109:
1893:The dispersion relation for deep
1886:
1878:
2588:Annual Review of Fluid Mechanics
2516:Proceedings of the Royal Society
2345:(HOLZ) lines in convergent beam
2339:transmission electron microscope
1193:
1182:
969:{\displaystyle k=2\pi /\lambda }
390:{\displaystyle k=2\pi /\lambda }
1180:
809:De Broglie dispersion relations
801:and thus both are equal to the
680:Electromagnetic waves in vacuum
59:
1628:
1622:
1291:
1285:
1082:
1068:
833:
827:
614:
596:
522:
510:
431:
425:
416:
410:
350:{\displaystyle \omega =2\pi f}
309:
303:
274:
268:
253:
247:
234:, are related by the identity
220:{\displaystyle v=v(\lambda ).}
211:
205:
1:
2553:Westfall, Richard S. (1983).
642:is the amplitude of the wave,
2666:Angular frequency calculator
1525:factor is very small so for
1150:for energy and momentum for
2112:group velocities are equal.
2105:
1889:green dot to traverse half.
1882:
1874:
1768:Frequency versus wavenumber
716:{\displaystyle \omega =ck.}
315:{\displaystyle f(\lambda )}
38:causes different colors to
2697:
2660:Poster on CBED simulations
2305:
2093:
2007:and the group velocity is
1858:
1787:
127:
1545:not too large, we expand
2643:10.1103/PhysRev.104.1760
2385:Kramers–Kronig relations
2383:The universality of the
1861:Dispersion (water waves)
1518:{\displaystyle \hbar /c}
1046:energy–momentum relation
179:{\displaystyle \lambda }
134:Dispersion (water waves)
84:Kramers–Kronig relations
2274:Electron band structure
2263:{\displaystyle \alpha }
932:{\displaystyle \omega }
815:de Broglie matter waves
58:describe the effect of
2532:10.1098/rspa.1977.0064
2434:Fundamentals of optics
2264:
2241:
2154:
2113:
2080:
1998:
1930:
1890:
1837:
1739:
1708:
1606:
1539:
1519:
1489:
1376:
1265:
1264:{\displaystyle \hbar }
1205:
1140:
1023:
970:
933:
913:
792:
717:
626:
447:
391:
351:
316:
284:
221:
180:
160:
52:electrical engineering
43:
2518:. A 354 (1677): 197.
2306:Further information:
2265:
2242:
2155:
2103:
2094:Further information:
2081:
1999:
1931:
1897:is often written as
1872:
1859:Further information:
1838:
1788:Further information:
1738:
1709:
1607:
1540:
1520:
1497:Then we see that the
1490:
1377:
1266:
1206:
1141:
1024:
971:
934:
914:
793:
718:
686:electromagnetic waves
672:Plane waves in vacuum
627:
448:
392:
352:
317:
285:
222:
181:
161:
33:
2681:Equations of physics
2378:Pierre-Simon Laplace
2347:electron diffraction
2254:
2182:
2123:
2014:
1950:
1904:
1810:
1716:Schrödinger equation
1616:
1549:
1529:
1501:
1391:
1279:
1255:
1161:
1148:de Broglie relations
1052:
980:
943:
923:
821:
737:
695:
504:
404:
364:
329:
297:
241:
193:
170:
150:
103:Elementary particles
56:dispersion relations
18:Dispersion relations
2635:1956PhRv..104.1760T
2600:2004AnRFM..36....1C
2524:1977RSPSA.354..197J
1800:index of refraction
1790:Dispersion (optics)
1778:dispersion relation
138:Acoustic dispersion
130:Dispersion (optics)
2355:dispersion surface
2260:
2237:
2150:
2114:
2076:
1994:
1926:
1891:
1833:
1740:
1704:
1602:
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1515:
1485:
1372:
1261:
1201:
1136:
1042:special relativity
1039:de Broglie applied
1019:
966:
929:
909:
788:
713:
622:
456:where we now view
443:
387:
347:
312:
280:
217:
176:
156:
44:
2499:978-981-02-0420-4
2389:scattering theory
2206:
2145:
2144:
2090:Waves on a string
2061:
2048:
1989:
1988:
1974:
1921:
1831:
1763:
1762:
1698:
1661:
1570:
1538:{\displaystyle k}
1479:
1467:
1427:
1366:
1354:
1222:angular frequency
1115:
1078:
903:
866:
777:
754:
571:
460:as a function of
324:angular frequency
264:
159:{\displaystyle v}
70:of a wave to its
48:physical sciences
16:(Redirected from
2688:
2647:
2646:
2629:(6): 1760–1770.
2618:
2612:
2611:
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2577:
2576:
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2511:
2505:
2503:
2485:
2479:
2478:
2460:
2454:
2453:
2437:
2427:
2411:Waves in plasmas
2406:Ultrashort pulse
2359:dynamical effect
2323:acoustic phonons
2280:levels of energy
2269:
2267:
2266:
2261:
2246:
2244:
2243:
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2233:
2232:
2217:
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2107:
2096:Vibrating string
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1995:
1990:
1981:
1980:
1975:
1967:
1962:
1961:
1935:
1933:
1932:
1927:
1922:
1914:
1888:
1884:
1880:
1876:
1865:Airy wave theory
1855:Deep water waves
1843:is known as the
1842:
1840:
1839:
1834:
1832:
1830:
1822:
1814:
1784:Waves and optics
1774:refractive index
1723:
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262:
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183:
182:
177:
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105:, considered as
88:wave propagation
21:
2696:
2695:
2691:
2690:
2689:
2687:
2686:
2685:
2671:
2670:
2656:
2651:
2650:
2620:
2619:
2615:
2585:
2584:
2580:
2573:
2552:
2551:
2547:
2513:
2512:
2508:
2504:See page 64–66.
2500:
2487:
2486:
2482:
2475:
2462:
2461:
2457:
2450:
2429:
2428:
2424:
2419:
2397:
2367:
2335:
2333:Electron optics
2327:optical phonons
2310:
2304:
2276:
2252:
2251:
2224:
2208:
2185:
2180:
2179:
2121:
2120:
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2017:
2012:
2011:
1953:
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1947:
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1807:
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1668:
1647:
1637:
1636:
1614:
1613:
1581:
1560:
1547:
1546:
1527:
1526:
1499:
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1453:
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1444:
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1389:
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1340:
1330:
1329:
1323:
1322:
1309:
1299:
1277:
1276:
1253:
1252:
1247:, equal to the
1237:
1235:
1234:with magnitude
1225:
1215:
1159:
1158:
1110:
1100:
1099:
1095:
1094:
1081:
1055:
1050:
1049:
1035:
1009:
999:
986:
978:
977:
941:
940:
921:
920:
892:
888:
877:
873:
852:
842:
841:
819:
818:
811:
769:
761:
735:
734:
693:
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682:
674:
650:
588:
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554:
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502:
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402:
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362:
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327:
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295:
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239:
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191:
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168:
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148:
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128:Main articles:
126:
28:
23:
22:
15:
12:
11:
5:
2694:
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2655:
2654:External links
2652:
2649:
2648:
2613:
2578:
2571:
2545:
2506:
2498:
2480:
2473:
2465:Modern Physics
2455:
2448:
2421:
2420:
2418:
2415:
2414:
2413:
2408:
2403:
2396:
2393:
2366:
2363:
2351:directly image
2334:
2331:
2319:Brillouin zone
2303:
2300:
2284:band structure
2275:
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2259:
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2005:
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1965:
1960:
1956:
1937:
1936:
1925:
1920:
1917:
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1909:
1856:
1853:
1849:phase velocity
1845:group velocity
1829:
1826:
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1785:
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1612:and multiply:
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989:
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891:
884:
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870:
865:
859:
855:
849:
845:
838:
835:
832:
829:
826:
810:
807:
803:speed of light
799:
798:
787:
784:
781:
775:
772:
767:
764:
758:
753:
750:
745:
742:
724:
723:
712:
709:
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700:
681:
678:
673:
670:
669:
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656:
648:
643:
633:
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621:
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585:
581:
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541:
535:
531:
527:
524:
521:
518:
515:
512:
509:
485:group velocity
474:phase velocity
454:
453:
442:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
386:
382:
378:
375:
372:
369:
346:
343:
340:
337:
334:
311:
308:
305:
302:
291:
290:
279:
276:
273:
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258:
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118:group velocity
114:phase velocity
80:group velocity
76:phase velocity
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
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2597:
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2579:
2574:
2572:9780521274357
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2474:0-534-49340-8
2470:
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2449:0-07-032330-5
2445:
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2292:semiconductor
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2018:
2010:
2009:
2008:
1991:
1985:
1982:
1976:
1971:
1968:
1963:
1958:
1954:
1946:
1945:
1944:
1942:
1923:
1918:
1915:
1910:
1907:
1900:
1899:
1898:
1896:
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1854:
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1314:
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1300:
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1282:
1274:
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1245:
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1233:
1228:
1223:
1218:
1198:
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1177:
1173:
1167:
1164:
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1155:
1153:
1149:
1130:
1125:
1119:
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1101:
1096:
1091:
1086:
1071:
1065:
1060:
1056:
1047:
1043:
1040:
1032:
1030:
1014:
1010:
1004:
1000:
996:
991:
987:
963:
959:
955:
952:
949:
946:
926:
906:
897:
893:
889:
882:
878:
868:
857:
853:
847:
843:
836:
830:
824:
816:
808:
806:
804:
785:
782:
779:
773:
770:
765:
762:
756:
751:
748:
743:
740:
733:
732:
731:
729:
710:
707:
704:
701:
698:
691:
690:
689:
687:
679:
677:
671:
666:
663:
660:
657:
654:
647:
644:
641:
638:
637:
636:
619:
611:
608:
605:
602:
599:
593:
589:
583:
579:
575:
568:
564:
561:
558:
555:
549:
546:
543:
539:
533:
529:
525:
519:
516:
513:
507:
500:
499:
498:
495:
493:
489:
486:
482:
478:
475:
471:
467:
464:. The use of
463:
459:
440:
437:
434:
428:
422:
419:
413:
407:
400:
399:
398:
384:
380:
376:
373:
370:
367:
360:
344:
341:
338:
335:
332:
325:
306:
300:
293:The function
277:
271:
265:
259:
256:
250:
244:
237:
236:
235:
233:
214:
208:
202:
199:
196:
189:
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187:
173:
153:
145:
139:
135:
131:
123:
121:
119:
115:
110:
108:
104:
100:
95:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
41:
37:
32:
19:
2626:
2622:
2616:
2591:
2587:
2581:
2556:
2548:
2515:
2509:
2489:
2483:
2464:
2458:
2433:
2425:
2401:Ellipsometry
2382:
2375:
2370:Isaac Newton
2368:
2350:
2336:
2311:
2277:
2249:
2174:
2168:
2164:
2162:
2115:
2006:
1940:
1938:
1892:
1793:
1777:
1771:
1755:
1751:
1747:
1741:
1726:
1496:
1383:
1272:
1251:. Divide by
1243:
1238:
1226:
1216:
1213:
1152:matter waves
1036:
812:
800:
727:
725:
683:
675:
664:
658:
652:
645:
639:
634:
496:
491:
487:
480:
476:
469:
465:
461:
457:
455:
292:
231:
229:
141:
111:
107:matter waves
96:
55:
45:
34:In a prism,
2321:are called
1895:water waves
1249:wave number
144:wave packet
92:attenuation
2417:References
2108:phase and
1727:Animation:
1232:wavevector
1033:Derivation
726:This is a
359:wavenumber
124:Dispersion
99:waveguides
68:wavenumber
64:wavelength
60:dispersion
36:dispersion
2623:Phys. Rev
2380:in 1776.
2343:Laue zone
2296:conductor
2288:insulator
2258:α
2222:α
2204:μ
2187:ω
2142:μ
2127:ω
2037:ω
1969:ω
1908:ω
1825:∂
1820:ω
1817:∂
1804:waveguide
1744:ångströms
1670:ℏ
1659:ℏ
1632:≈
1620:ω
1573:≈
1505:ℏ
1449:ℏ
1425:ℏ
1395:ω
1352:ℏ
1283:ω
1259:ℏ
1242:| =
1190:ℏ
1174:ω
1171:ℏ
988:ω
984:ℏ
964:λ
956:π
927:ω
875:ℏ
864:ℏ
837:≈
825:ω
766:ω
749:ω
699:ω
609:ω
606:−
569:λ
559:−
547:π
435:⋅
408:ω
385:λ
377:π
342:π
333:ω
307:λ
272:λ
260:λ
251:λ
209:λ
174:λ
72:frequency
2675:Category
2594:: 1–28.
2540:98158162
2395:See also
1146:use the
483:and the
2631:Bibcode
2596:Bibcode
2520:Bibcode
2365:History
2357:. This
2314:phonons
2302:Phonons
1230:is the
1220:is the
655:(0, 0),
635:where
46:In the
40:refract
2569:
2538:
2496:
2471:
2446:
2250:where
2163:where
1939:where
1796:optics
1236:|
1214:where
728:linear
263:
136:, and
2536:S2CID
2567:ISBN
2494:ISBN
2469:ISBN
2444:ISBN
1863:and
1224:and
813:For
684:For
357:and
116:and
90:and
78:and
50:and
2639:doi
2627:104
2604:doi
2563:276
2528:doi
2440:223
2294:or
66:or
2677::
2637:.
2625:.
2602:.
2592:36
2590:.
2565:.
2534:.
2526:.
2442:.
2298:.
2290:,
1851:.
1275::
1154:,
1048::
651:=
492:dk
488:dω
186::
132:,
120:.
94:.
54:,
2645:.
2641::
2633::
2610:.
2606::
2598::
2575:.
2542:.
2530::
2522::
2502:.
2477:.
2452:.
2235:,
2230:4
2226:k
2219:+
2214:2
2210:k
2201:T
2196:=
2191:2
2169:ÎĽ
2165:T
2148:,
2139:T
2133:k
2130:=
2110:â—Ź
2106:â—Ź
2074:.
2069:p
2065:v
2059:2
2056:1
2051:=
2045:k
2042:d
2034:d
2028:=
2023:g
2019:v
1992:,
1986:k
1983:g
1977:=
1972:k
1964:=
1959:p
1955:v
1941:g
1924:,
1919:k
1916:g
1911:=
1887:â—Ź
1883:â–
1879:â—Ź
1875:â–
1828:k
1756:c
1752:c
1748:c
1702:.
1693:0
1689:m
1685:2
1678:2
1674:k
1664:+
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1649:c
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1626:k
1623:(
1600:,
1597:2
1593:/
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1579:+
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1509:/
1483:.
1475:2
1470:)
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1459:0
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1398:=
1370:.
1362:2
1357:)
1346:2
1342:c
1336:0
1332:m
1325:(
1320:+
1315:2
1311:c
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1301:k
1295:=
1292:)
1289:k
1286:(
1244:k
1239:k
1227:k
1217:ω
1199:,
1194:k
1187:=
1183:p
1178:,
1168:=
1165:E
1131:2
1126:)
1120:2
1114:c
1106:0
1102:m
1097:(
1092:+
1087:2
1083:)
1077:c
1072:p
1069:(
1066:=
1061:2
1057:E
1015:2
1011:c
1005:0
1001:m
997:=
992:0
960:/
953:2
950:=
947:k
907:.
898:0
894:m
890:2
883:2
879:k
869:+
858:2
854:c
848:0
844:m
834:)
831:k
828:(
786:,
783:c
780:=
774:k
771:d
763:d
757:=
752:k
744:=
741:v
711:.
708:k
705:c
702:=
665:t
659:x
653:A
649:0
646:A
640:A
620:,
615:)
612:t
603:x
600:k
597:(
594:i
590:e
584:0
580:A
576:=
565:t
562:v
556:x
550:i
544:2
540:e
534:0
530:A
526:=
523:)
520:t
517:,
514:x
511:(
508:A
490:/
481:k
479:/
477:ω
470:k
468:(
466:ω
462:k
458:f
441:.
438:k
432:)
429:k
426:(
423:v
420:=
417:)
414:k
411:(
381:/
374:2
371:=
368:k
345:f
339:2
336:=
310:)
304:(
301:f
278:.
275:)
269:(
266:f
257:=
254:)
248:(
245:v
232:f
215:.
212:)
206:(
203:v
200:=
197:v
154:v
20:)
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