272:, that is, that the incident field is not greatly altered within one particle so that each volume element is considered to be illuminated by an intensity and phase determined only by its position relative to the incident wave, unaffected by scattering from other volume elements.
1372:
431:
1208:
1058:
605:
1854:
Jen, Shih-Hui; Dai, Hai-Lung; Gonella, Grazia (18 February 2010). "The Effect of
Particle Size in Second Harmonic Generation from the Surface of Spherical Colloidal Particles. II: The Nonlinear Rayleigh−Gans−Debye Model".
827:
942:
496:
affecting the scattering from each volume element are dependent only on their positions with respect to the incoming wave and the scattering direction. Integrating, the scattering amplitude function thus obtains:
714:
1889:
1455:
1214:
293:
222:
268:
of the particle. The first condition allows for a simplification in expressing the material polarizability in the derivation below. The second condition is a statement of the
1069:
865:
162:
961:
113:
494:
454:
35:
of particle is close to that of the surrounding medium. The approximation holds for particles of arbitrary shape that are relatively small but can be larger than
503:
262:
242:
184:
729:
43:
1386:
1737:
Farias, T. L.; Köylü, Ü. Ö.; Carvalho, M. G. (1996). "Range of validity of the
Rayleigh–Debye–Gans theory for optics of fractal aggregates".
1577:
873:
614:
phases contributing to the scattering direction (θ, φ), remains to be solved according to the particular geometry of the scatterer. Calling
1894:
628:
1857:
1721:
1660:
1063:
Then the scattered radiation intensity, relative to the intensity of the incident wave, for each polarization can be written as:
1488:
465:
1819:
1537:
28:
46:
in 1881 and was applied to homogeneous spheres, spherical shells, radially inhomogeneous spheres and infinite cylinders.
1532:
1593:
Turner, Leaf (1973). "Rayleigh-Gans-Born Light
Scattering by Ensembles of Randomly Oriented Anisotropic Particles".
1367:{\displaystyle I_{2}/I_{0}=\left({\frac {k^{4}V^{2}}{4\pi ^{2}r^{2}}}\right)(n-1)^{2}P(\theta ,\phi )cos^{2}\theta }
1503:
426:{\displaystyle dS_{1}(\theta ,\phi )=i{\frac {3}{4\pi }}k^{3}\left({\frac {n^{2}-1}{n^{2}+2}}\right)e^{i\delta }dV}
1899:
1398:
1782:
Chong, C.S.; Colbow, Konrad (17 June 1976). "Light scattering and turbidity measurements on lipid vesicles".
189:
1709:
1389:
461:
622:
for scattering with the electric field polarization normal to the plane of incidence (s polarization) as
618:
the entire volume of the scattering object, over which this integration is performed, one can write that
1203:{\displaystyle I_{1}/I_{0}=\left({\frac {k^{4}V^{2}}{4\pi ^{2}r^{2}}}\right)(n-1)^{2}P(\theta ,\phi )}
1817:
Koch, Arthur L. (19 August 1961). "Some calculations on the turbidity of mitochondria and bacteria".
1748:
1604:
284:
280:
1053:{\displaystyle P(\theta ,\phi )=\left({\frac {1}{V^{2}}}\right)\left|\int e^{i\delta }dV\right|^{2}}
276:
36:
835:
119:
1468:
Rayleigh–Gans approximation has been applied on the calculation of the optical cross sections of
269:
55:
76:
1836:
1799:
1764:
1717:
1656:
1620:
1573:
611:
59:
479:
439:
1866:
1828:
1791:
1756:
1689:
1678:"Evaluation of the Rayleigh–Gans approximation for microwave scattering by rimed snowflakes"
1612:
1499:
265:
32:
600:{\displaystyle S_{1}(\theta ,\phi )\approx {\frac {i}{2\pi }}k^{3}(n-1)\int e^{i\delta }dV}
460:
difference due to each individual element, and the fraction in parentheses is the electric
1507:
1480:
1382:
50:
has contributed to the theory in 1881. The theory for homogeneous sphere was rederived by
1752:
1608:
1739:
1652:
1595:
1527:
247:
1883:
1832:
1795:
457:
275:
The particle is divided into small volume elements, which are treated as independent
227:
169:
51:
1542:
1515:
1477:
47:
1511:
1473:
948:
822:{\displaystyle S_{2}={\frac {i}{2\pi }}k^{3}(n-1)VR(\theta ,\phi )cos\theta }
1484:
1840:
1768:
1624:
1803:
1760:
1616:
1492:
619:
31:
by optically soft particles. Optical softness implies that the relative
1469:
1870:
937:{\displaystyle R(\theta ,\phi )={\frac {1}{V}}\int {}{}e^{i\delta }dV}
1381:
is the distance from the scatterer to the observation point. Per the
1694:
1677:
709:{\displaystyle S_{1}={\frac {i}{2\pi }}k^{3}(n-1)VR(\theta ,\phi )}
1676:
Leinonen, Jussi; Kneifel, Stefan; Hogan, Robin J. (12 June 2017).
70:
The validity conditions for the approximation can be denoted as:
1649:
The
Scattering of Light and Other Electromagnetic Radiation
1890:
Scattering, absorption and radiative transfer (optics)
838:
610:
in which only the final integral, which describes the
230:
192:
172:
1570:
Absorption and scattering of light by small particles
1401:
1217:
1072:
964:
876:
732:
631:
506:
482:
442:
296:
250:
122:
79:
1487:
calculations on biological structures such as lipid
1682:
Advances in Remote
Sensing of Rainfall and Snowfall
287:contribution from each volume element is given as:
1784:Biochimica et Biophysica Acta (BBA) - Biomembranes
1502:Rayleigh−Gans−Debye model was used to investigate
1449:
1366:
1202:
1052:
936:
859:
821:
708:
599:
488:
448:
425:
256:
236:
216:
178:
156:
107:
244:refers to the linear dimension of the particle.
1647:Kerker, Milton (1969). Loebl, Ernest M. (ed.).
955:as the squared magnitude of the form factor:
464:as found from the refractive index using the
8:
867:denotes the "form factor" of the scatterer:
1472:aggregates. The theory was also applied to
1450:{\displaystyle C_{abs}=2kV\mathbb {Im} (n)}
723:the plane of incidence (p polarization) as
54:in 1925. The approximation is analogous to
1460:which is independent of the polarization.
1693:
1434:
1431:
1430:
1406:
1400:
1355:
1321:
1292:
1282:
1267:
1257:
1250:
1237:
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1222:
1216:
1176:
1147:
1137:
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1092:
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1071:
1044:
1024:
999:
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963:
919:
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911:
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837:
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94:
80:
78:
1642:
1640:
1638:
1636:
1634:
1563:
1561:
1559:
1557:
217:{\textstyle k={\frac {2\pi }{\lambda }}}
1568:Bohren, C. F.; Huffmann, D. R. (2010).
1553:
472:, this factor can be approximated as
7:
1858:The Journal of Physical Chemistry C
1714:Light scattering by small particles
14:
1716:. New York: John Wiley and Sons.
21:Rayleigh–Gans–Debye approximation
1572:. New York: Wiley-Interscience.
186:is the wavevector of the light (
27:, is an approximate solution to
25:Rayleigh–Gans–Born approximation
1444:
1438:
1342:
1330:
1318:
1305:
1197:
1185:
1173:
1160:
980:
968:
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703:
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517:
322:
310:
144:
130:
95:
81:
1:
1820:Biochimica et Biophysica Acta
1538:Discrete dipole approximation
860:{\textstyle R(\theta ,\phi )}
1833:10.1016/0006-3002(61)90599-6
1796:10.1016/0005-2736(76)90192-9
1533:Anomalous diffraction theory
279:. For an inbound light with
157:{\displaystyle kd|n-1|\ll 1}
1895:Radio frequency propagation
1476:spheres for nanostructured
17:Rayleigh–Gans approximation
1916:
1504:second-harmonic generation
466:Clausius–Mossotti relation
108:{\displaystyle |n-1|\ll 1}
42:The theory was derived by
266:complex refractive index
489:{\displaystyle \delta }
449:{\displaystyle \delta }
1451:
1368:
1204:
1054:
947:In order to only find
938:
861:
823:
710:
601:
490:
468:. Under the condition
450:
427:
258:
238:
218:
180:
158:
109:
1452:
1369:
1205:
1055:
939:
862:
824:
719:and for polarization
711:
602:
491:
451:
428:
259:
239:
219:
181:
159:
110:
1761:10.1364/AO.35.006560
1617:10.1364/AO.12.001085
1399:
1215:
1070:
962:
874:
836:
730:
629:
620:scattering parameter
504:
480:
440:
294:
285:scattering amplitude
248:
228:
190:
170:
120:
77:
1753:1996ApOpt..35.6560F
1710:van de Hulst, H. C.
1609:1973ApOpt..12.1085T
277:Rayleigh scatterers
37:Rayleigh scattering
1447:
1364:
1200:
1050:
934:
857:
819:
706:
597:
486:
446:
423:
270:Born approximation
254:
234:
214:
176:
154:
105:
56:Born approximation
1871:10.1021/jp910144c
1865:(10): 4302–4308.
1747:(33): 6560–6567.
1579:978-3-527-40664-7
1299:
1154:
1005:
906:
759:
658:
548:
398:
344:
257:{\displaystyle n}
212:
60:quantum mechanics
1907:
1900:X-ray scattering
1875:
1874:
1851:
1845:
1844:
1814:
1808:
1807:
1779:
1773:
1772:
1734:
1728:
1727:
1706:
1700:
1699:
1697:
1673:
1667:
1666:
1644:
1629:
1628:
1603:(5): 1085–1090.
1590:
1584:
1583:
1565:
1456:
1454:
1453:
1448:
1437:
1417:
1416:
1373:
1371:
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1365:
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1039:
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935:
927:
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828:
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820:
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598:
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558:
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493:
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487:
470:(n-1) << 1
455:
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447:
432:
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177:
163:
161:
160:
155:
147:
133:
114:
112:
111:
106:
98:
84:
33:refractive index
29:light scattering
19:, also known as
1915:
1914:
1910:
1909:
1908:
1906:
1905:
1904:
1880:
1879:
1878:
1853:
1852:
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1816:
1815:
1811:
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1724:
1708:
1707:
1703:
1695:10.1002/qj.3093
1675:
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1645:
1632:
1592:
1591:
1587:
1580:
1567:
1566:
1555:
1551:
1524:
1508:malachite green
1478:polycrystalline
1466:
1402:
1397:
1396:
1383:optical theorem
1351:
1317:
1288:
1278:
1274:
1263:
1253:
1252:
1246:
1233:
1218:
1213:
1212:
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1129:
1118:
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1073:
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1067:
1020:
1016:
1012:
1011:
995:
986:
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959:
915:
872:
871:
834:
833:
761:
751:
733:
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727:
660:
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477:
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404:
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292:
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246:
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201:
188:
187:
168:
167:
118:
117:
75:
74:
68:
12:
11:
5:
1913:
1911:
1903:
1902:
1897:
1892:
1882:
1881:
1877:
1876:
1846:
1827:(3): 429–441.
1809:
1790:(2): 260–282.
1774:
1740:Applied Optics
1729:
1722:
1701:
1668:
1661:
1653:Academic Press
1630:
1596:Applied Optics
1585:
1578:
1552:
1550:
1547:
1546:
1545:
1540:
1535:
1530:
1528:Mie scattering
1523:
1520:
1465:
1462:
1458:
1457:
1446:
1443:
1440:
1436:
1433:
1429:
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1225:
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1210:
1199:
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1190:
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1171:
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1162:
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1125:
1121:
1115:
1111:
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1095:
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1080:
1076:
1061:
1060:
1047:
1042:
1038:
1035:
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1027:
1023:
1019:
1015:
1009:
1002:
998:
994:
989:
985:
982:
979:
976:
973:
970:
967:
951:we can define
945:
944:
933:
930:
925:
922:
918:
910:
905:
902:
897:
894:
891:
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562:
557:
553:
546:
543:
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534:
531:
528:
525:
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519:
514:
510:
485:
462:polarizability
445:
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433:
422:
419:
414:
411:
407:
402:
396:
393:
388:
384:
378:
375:
370:
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353:
349:
342:
339:
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324:
321:
318:
315:
312:
307:
303:
299:
281:s polarization
253:
237:{\textstyle d}
233:
211:
207:
204:
198:
195:
179:{\textstyle k}
175:
165:
164:
153:
150:
146:
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132:
128:
125:
115:
104:
101:
97:
93:
90:
87:
83:
67:
64:
13:
10:
9:
6:
4:
3:
2:
1912:
1901:
1898:
1896:
1893:
1891:
1888:
1887:
1885:
1872:
1868:
1864:
1860:
1859:
1850:
1847:
1842:
1838:
1834:
1830:
1826:
1822:
1821:
1813:
1810:
1805:
1801:
1797:
1793:
1789:
1785:
1778:
1775:
1770:
1766:
1762:
1758:
1754:
1750:
1746:
1742:
1741:
1733:
1730:
1725:
1723:9780486139753
1719:
1715:
1711:
1705:
1702:
1696:
1691:
1687:
1683:
1679:
1672:
1669:
1664:
1662:9780124045507
1658:
1654:
1650:
1643:
1641:
1639:
1637:
1635:
1631:
1626:
1622:
1618:
1614:
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1441:
1427:
1424:
1421:
1418:
1413:
1410:
1407:
1403:
1395:
1394:
1393:
1392:is given as:
1391:
1390:cross section
1388:
1384:
1380:
1361:
1356:
1352:
1348:
1345:
1339:
1336:
1333:
1327:
1322:
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1234:
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1177:
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1123:
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1021:
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551:
544:
541:
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532:
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512:
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500:
499:
498:
483:
476:. The phases
475:
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467:
463:
459:
443:
420:
417:
412:
409:
405:
400:
394:
391:
386:
382:
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