1736:
2547:
397:
1215:
2299:
731:
A dilute solution of a certain polymer has a unique feature: all polymers are considered independent from each other, so that interactions between polymers may be neglected. By illuminating such a solution with a ray of considerable width, a macroscopic number of chain conformations are being sampled
39:
experiments, it involves subjecting a polymeric sample to incident particles (with defined wavelengths), and studying the characteristics of the scattered particles: angular distribution, intensity polarization and so on. This method is quite simple and straightforward, and does not require special
2571:
Examining complex interactions of polymers in the solution - between themselves, and between them and the solution. Such interactions may arise if the polymers are charged, corresponding to ionic interactions, This would have a significant impact on the particles behavior, and will result in a
906:
2522:
spot emerging in usual scattering experiments. The problem is mitigated by conducting longer experiments with more exposure time, which allows the required data to "intensify". One must take care though, as to not allow the prolonged exposure to high levels of
1731:
1073:
722:
1876:
2550:
The change in the small-angle scattering signatures when the interactions between polymers are tuned. From left to right there is an increase in the amount of
Magnesium in the solution, which alleviates the coulomb repulsion (thus reducing the
2050:, a so-called "Guinier plot", we may determine the radius of gyration from the slope of this linear curve. This measure is one of many examples of how scattering experiments of polymers can reveal basic properties of those polymer chains.
1739:
A Guinier plot made with X-ray scattering in the small-angle regime. The slopes of these linear curves correspond to the radius of gyration of the polymers in the solution, while different curves correspond to different concentrations.
1535:
2656:
Nakanishi, Ryosuke; Machida, Ginpei; Kinoshita, Masaki; Sakurai, Kazuo; Akiba, Isamu (16 March 2016). "Anomalous small-angle X-ray scattering study on the spatial distribution of hydrophobic molecules in polymer micelles".
2293:
391:
2235:
1984:
1204:
748:
1394:
2353:
2429:
2391:
1600:
2131:
2494:
941:
934:
1551:
The calculation of the structure factor for cases differing from ideal polymer chains can be quite cumbersome, and sometimes impossible to complete analytically. However, when the
455:
1120:
503:
237:
567:
1749:
285:
2496:. This makes small-angle measurements in neutrons and X-rays a bit more tedious, as very small angles are needed, and the data in those angles is often "overpowered" by the
101:
1589:
2520:
2021:
2159:
560:
161:
2458:
2087:
2048:
1423:
128:
2179:
527:
1441:
733:
739:
In such a solution, where the polymer density is low (dilute) enough, homogenous and isotropic (on average), intermolecular contributions to the
2534:
On the other hand, to resolve smaller polymers and structurals subtleties, one cannot always resort to using the long-wavelength rays, as the
2242:
290:
2186:
901:{\displaystyle S({\vec {k}})={\frac {1}{N^{2}}}\langle \sum _{i,j=1}^{N}e^{i{\vec {k}}\cdot ({\vec {R}}_{i}-{\vec {R}}_{j})}\rangle }
2603:
1892:
1218:
A comparison between the Debye function (black) and the approximated
Lorentzian (blue). One sees no major differences between them.
1125:
47:
scattering experiments, where the scatterer or "target" has very distinct order, which leads to well defined patterns (presenting
1239:
2565:
Explore the distribution of the polymers in the sample - is it truly isotropic? Or does it favor certain directions on average?
2562:
Evaluating the structural and thermo-statistical behavior of a polymer, i.e. freely-jointed chain / freely-rotating chain etc.
2555:
The main purpose of such scattering experiments involving polymers is to study unique properties of the sample of interest:
2322:
936:
representing the ensemble average. This reduces to the following for an isotropic system (which is typically the case):
2618:
175:
2398:
2360:
1726:{\displaystyle S({\vec {k}})\approx {\frac {1}{N^{2}}}\langle \sum _{i,j=1}^{N}1-{\frac {1}{6}}(kR_{ij})^{2}\rangle }
55:
nature of polymer configurations and deformations (especially in a solution), gives rise to quite different results.
2535:
2763:
1881:
1068:{\displaystyle S({\vec {k}})={\frac {1}{N^{2}}}\langle \sum _{i,j=1}^{N}{\frac {\sin kR_{ij}}{kR_{ij}}}\rangle }
2613:
2102:
2463:
2686:"Small-angle neutron scattering on polymer gels: phase behavior, inhomogeneities and deformation mechanisms"
2598:
913:
2306:) Small-angle neutron scattering (SANS) and light scattering (LS) profiles of PNIPA gels and solution and (
403:
1552:
1080:
717:{\displaystyle I({\vec {k}})=|a|^{2}\sum _{i,j=1}^{N}e^{i{\vec {k}}\cdot ({\vec {R}}_{i}-{\vec {R}}_{j})}}
1871:{\displaystyle S({\vec {k}})\approx 1-{\frac {1}{3}}(kR_{g})^{2}\approx e^{-{\frac {1}{3}}(kR_{g})^{2}}}
460:
2133:
will determine the available angular spectrum of this regime. To see this one may consider the case of
2058:
In order to reap the benefits of working in this small-angle regime, one must take into consideration:
185:
1230:
transition"), the calculation of the structure can be carried out explicitly and result in a sort of
32:
242:
70:
2758:
1540:
1433:
1227:
40:
manipulations of the samples which may alter their properties, and hence compromise exact results.
2608:
2134:
1558:
1426:
171:
52:
2499:
1991:
2734:
130:. For simplicity, it is worthwhile considering identical monomers in the chain, such that all
2144:
1432:
in many practical scenarios, the above formula is approximated by the (much more convenient)
133:
2726:
2697:
2666:
2524:
740:
536:
167:
2436:
2065:
2026:
1401:
106:
2730:
2546:
2527:
the polymers (which might be a real problem when considering biological polymer samples –
44:
2164:
512:
743:
are averaged out, and only the single-molecule/polymer structure factor is preserved:
2752:
1592:
1530:{\displaystyle S_{D}({\vec {k}})\approx {\frac {1}{1+{\frac {1}{2}}(kR_{g})^{2}}}}
736:, i.e. averages over all possible configurations and deformations of the polymer.
1231:
1223:
396:
179:
1887:
We can thus approximate the scattering intensity in the small-angle regime as:
2092:
1214:
48:
36:
2717:
Pollack, Lois (2011-01-01). "SAXS Studies of Ion–Nucleic Acid
Interactions".
20:
2738:
1735:
2702:
2685:
2670:
2593:
2576:
2528:
2288:{\displaystyle {\frac {R_{g}}{\lambda }}\sin {\frac {\theta }{2}}\ll 1}
64:
28:
24:
386:{\textstyle {\vec {k}}\equiv {\vec {k}}_{final}-{\vec {k}}_{incident}}
2568:
Identifying deformations in the polymer samples and quantifying them.
732:
simultaneously. In this situation the accessible observables are all
2230:{\displaystyle k={\frac {4\pi }{\lambda }}\sin {\frac {\theta }{2}}}
2298:
1734:
395:
1226:, in the limit of very long chains (allows performing a sort of "
2580:
1979:{\displaystyle \log I(k)=-{\frac {1}{3}}k^{2}R_{g}^{2}+const.}
1199:{\displaystyle R_{ij}\equiv |{\vec {R}}_{i}-{\vec {R}}_{j}|}
1744:
and by utilising the definition of the radius of gyration:
1389:{\displaystyle S_{D}({\vec {k}})={\frac {2}{(kR_{g})^{4}}}}
1543:
of no more than 15% compared to the exact expression.
539:
293:
245:
188:
2502:
2466:
2439:
2401:
2363:
2325:
2245:
2189:
2167:
2147:
2105:
2068:
2062:
The characteristic length scale of the polymer, e.g.
2029:
1994:
1895:
1752:
1603:
1561:
1444:
1404:
1242:
1128:
1083:
944:
916:
751:
570:
515:
463:
406:
136:
109:
73:
2559:
Determine the polymers "size" - radius of gyration.
1880:where the final transition utilises once again the
27:and other sciences to study the characteristics of
2583:) that are often suspended in an aqueous solution.
2514:
2488:
2452:
2423:
2385:
2348:{\displaystyle \lambda \sim 5000{\overset {o}{A}}}
2347:
2287:
2229:
2173:
2153:
2125:
2081:
2042:
2015:
1978:
1870:
1725:
1583:
1529:
1417:
1388:
1198:
1114:
1067:
928:
900:
716:
554:
521:
497:
449:
385:
287:. This enables us to define the scattering vector
279:
231:
155:
122:
95:
529:monomers, we get the scattering intensity from a
1224:ideal gaussian chains (or freely-jointed chains)
400:The scattering geometry: scattering of a vector
239:, and is scattered by the polymer to the vector
2424:{\displaystyle \lambda \sim 1{\overset {o}{A}}}
2386:{\displaystyle \lambda \sim 3{\overset {o}{A}}}
2310:) schematic representation of the gel network.
509:By coherently summing the contributions of all
19:are one of the main scientific methods used in
8:
1720:
1645:
1062:
986:
923:
917:
895:
793:
2701:
2684:Shibayama, Mitsuhiro (17 November 2010).
2501:
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2465:
2444:
2438:
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2362:
2335:
2324:
2269:
2252:
2246:
2244:
2217:
2196:
2188:
2166:
2146:
2126:{\displaystyle {\frac {R_{g}}{\lambda }}}
2112:
2106:
2104:
2073:
2067:
2034:
2028:
1993:
1949:
1944:
1934:
1920:
1894:
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993:
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259:
248:
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202:
191:
190:
187:
141:
135:
114:
108:
81:
75:
74:
72:
2545:
2297:
1213:
2630:
2489:{\displaystyle 10-100{\overset {o}{A}}}
2141:elastic ). If the scattering angle is
35:, gels, compounds and more. As in most
1077:where two more definitions were made:
929:{\displaystyle \langle \cdot \rangle }
2731:10.1146/annurev-biophys-042910-155349
2239:so the small-angle condition becomes
450:{\displaystyle {\vec {k}}_{incident}}
7:
2636:
2634:
1547:Small-angle scattering from polymers
1115:{\displaystyle k\equiv |{\vec {k}}|}
63:We consider a polymer as a chain of
2295:, determining the relevant angles.
1595:term can be expanded so one gets:
498:{\displaystyle {\vec {k}}_{final}}
232:{\textstyle {\vec {k}}_{incident}}
14:
2604:Biological small-angle scattering
2572:significant scattering signature.
1222:If the polymers of interest are
67:, each with its position vector
280:{\textstyle {\vec {k}}_{final}}
2643:The Theory of Polymer Dynamics
2010:
2004:
1911:
1905:
1857:
1840:
1810:
1793:
1771:
1765:
1756:
1711:
1691:
1622:
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1607:
1515:
1498:
1470:
1464:
1455:
1383:
1372:
1355:
1329:
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1309:
1297:
1280:
1268:
1262:
1253:
1192:
1179:
1157:
1146:
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1101:
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963:
957:
948:
890:
878:
856:
846:
837:
770:
764:
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709:
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675:
665:
656:
605:
596:
589:
583:
574:
546:
471:
414:
350:
316:
300:
253:
196:
96:{\displaystyle {\vec {R_{i}}}}
87:
17:Polymer scattering experiments
1:
2460:values for polymers range in
1210:Ideal chains – Debye function
2719:Annual Review of Biophysics
2619:Wide-angle X-ray scattering
1584:{\displaystyle kR_{g}\ll 1}
2780:
2095:of the scattered particles
2515:{\displaystyle \theta =0}
2016:{\displaystyle \log I(k)}
1882:small-angle approximation
103:and scattering amplitude
2614:Polymer characterization
2054:Practical considerations
2641:Doi, M.; Edwards, S.F.
2599:Static light scattering
2154:{\displaystyle \theta }
555:{\textstyle {\vec {k}}}
156:{\displaystyle a_{i}=a}
2552:
2516:
2490:
2454:
2425:
2395:- For "hard" X-rays,
2387:
2349:
2319:- For visible light,
2311:
2289:
2231:
2175:
2155:
2127:
2083:
2044:
2017:
1980:
1872:
1741:
1727:
1674:
1585:
1553:small-angle scattering
1531:
1419:
1390:
1219:
1200:
1116:
1069:
1015:
930:
902:
822:
718:
641:
556:
523:
506:
499:
451:
387:
281:
233:
157:
124:
97:
2575:Studying a myriad of
2549:
2517:
2491:
2455:
2453:{\displaystyle R_{g}}
2426:
2388:
2350:
2301:
2290:
2232:
2176:
2156:
2128:
2084:
2082:{\displaystyle R_{g}}
2045:
2043:{\displaystyle k^{2}}
2018:
1981:
1873:
1738:
1728:
1648:
1586:
1532:
1420:
1418:{\displaystyle R_{g}}
1391:
1217:
1201:
1117:
1070:
989:
931:
903:
796:
719:
615:
557:
524:
500:
452:
399:
388:
282:
234:
158:
125:
123:{\displaystyle a_{i}}
98:
2500:
2464:
2437:
2399:
2361:
2323:
2243:
2187:
2165:
2145:
2103:
2066:
2027:
1992:
1893:
1750:
1601:
1559:
1442:
1425:being the polymer's
1402:
1240:
1126:
1081:
942:
914:
749:
568:
537:
513:
461:
404:
291:
243:
186:
166:An incoming ray (of
134:
107:
71:
2703:10.1038/pj.2010.110
1954:
533:, as a function of
2671:10.1038/pj.2016.32
2609:Neutron scattering
2553:
2512:
2486:
2450:
2421:
2383:
2345:
2312:
2285:
2227:
2171:
2151:
2139:even approximately
2135:elastic scattering
2123:
2079:
2040:
2013:
1976:
1940:
1868:
1742:
1723:
1581:
1555:condition is met,
1527:
1427:radius of gyration
1415:
1386:
1220:
1196:
1112:
1065:
926:
898:
714:
552:
519:
507:
495:
447:
383:
277:
229:
153:
120:
93:
51:for example), the
2645:. pp. 21–23.
2579:substances (e.g.
2538:comes into play.
2536:diffraction limit
2484:
2419:
2381:
2357:- For neutrons,
2343:
2277:
2261:
2225:
2209:
2174:{\displaystyle k}
2161:, we may express
2121:
1928:
1838:
1791:
1768:
1689:
1643:
1619:
1525:
1496:
1467:
1307:
1265:
1182:
1160:
1104:
1060:
984:
960:
881:
859:
840:
791:
767:
734:ensemble averages
700:
678:
659:
586:
549:
522:{\displaystyle N}
474:
417:
353:
319:
303:
256:
199:
90:
2771:
2764:Particle physics
2743:
2742:
2714:
2708:
2707:
2705:
2681:
2675:
2674:
2653:
2647:
2646:
2638:
2531:, for example).
2525:radiation damage
2521:
2519:
2518:
2513:
2495:
2493:
2492:
2487:
2485:
2477:
2459:
2457:
2456:
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2430:
2428:
2427:
2422:
2420:
2412:
2392:
2390:
2389:
2384:
2382:
2374:
2354:
2352:
2351:
2346:
2344:
2336:
2294:
2292:
2291:
2286:
2278:
2270:
2262:
2257:
2256:
2247:
2236:
2234:
2233:
2228:
2226:
2218:
2210:
2205:
2197:
2180:
2178:
2177:
2172:
2160:
2158:
2157:
2152:
2132:
2130:
2129:
2124:
2122:
2117:
2116:
2107:
2088:
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2085:
2080:
2078:
2077:
2049:
2047:
2046:
2041:
2039:
2038:
2022:
2020:
2019:
2014:
1988:and by plotting
1985:
1983:
1982:
1977:
1953:
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1939:
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1929:
1921:
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1708:
1690:
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1612:
1590:
1588:
1587:
1582:
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1513:
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1305:
1304:
1295:
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1275:
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1266:
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1205:
1203:
1202:
1197:
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1190:
1189:
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1119:
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1071:
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1041:
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1009:
985:
983:
982:
970:
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961:
953:
935:
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927:
907:
905:
904:
899:
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861:
860:
852:
842:
841:
833:
821:
816:
792:
790:
789:
777:
769:
768:
760:
741:structure factor
727:Dilute solutions
723:
721:
720:
715:
713:
712:
708:
707:
702:
701:
693:
686:
685:
680:
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671:
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579:
561:
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129:
127:
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100:
99:
94:
92:
91:
86:
85:
76:
45:crystallographic
2779:
2778:
2774:
2773:
2772:
2770:
2769:
2768:
2749:
2748:
2747:
2746:
2716:
2715:
2711:
2690:Polymer Journal
2683:
2682:
2678:
2659:Polymer Journal
2655:
2654:
2650:
2640:
2639:
2632:
2627:
2590:
2588:Further reading
2544:
2498:
2497:
2462:
2461:
2440:
2435:
2434:
2397:
2396:
2359:
2358:
2321:
2320:
2317:
2248:
2241:
2240:
2198:
2185:
2184:
2163:
2162:
2143:
2142:
2108:
2101:
2100:
2069:
2064:
2063:
2056:
2030:
2025:
2024:
1990:
1989:
1930:
1891:
1890:
1856:
1846:
1822:
1809:
1799:
1748:
1747:
1710:
1697:
1633:
1599:
1598:
1565:
1557:
1556:
1549:
1514:
1504:
1481:
1445:
1440:
1439:
1405:
1400:
1399:
1371:
1361:
1347:
1328:
1318:
1296:
1286:
1279:
1243:
1238:
1237:
1212:
1172:
1150:
1129:
1124:
1123:
1079:
1078:
1046:
1042:
1028:
1018:
974:
940:
939:
912:
911:
871:
849:
823:
781:
747:
746:
729:
690:
668:
642:
603:
566:
565:
535:
534:
511:
510:
464:
459:
458:
407:
402:
401:
343:
309:
289:
288:
246:
241:
240:
189:
184:
183:
137:
132:
131:
110:
105:
104:
77:
69:
68:
61:
12:
11:
5:
2777:
2775:
2767:
2766:
2761:
2751:
2750:
2745:
2744:
2725:(1): 225–242.
2709:
2676:
2665:(7): 801–806.
2648:
2629:
2628:
2626:
2623:
2622:
2621:
2616:
2611:
2606:
2601:
2596:
2589:
2586:
2585:
2584:
2573:
2569:
2566:
2563:
2560:
2543:
2540:
2511:
2508:
2505:
2483:
2480:
2475:
2472:
2469:
2447:
2443:
2433:while typical
2418:
2415:
2410:
2407:
2404:
2380:
2377:
2372:
2369:
2366:
2342:
2339:
2334:
2331:
2328:
2316:
2313:
2284:
2281:
2276:
2273:
2268:
2265:
2260:
2255:
2251:
2224:
2221:
2216:
2213:
2208:
2204:
2201:
2195:
2192:
2170:
2150:
2120:
2115:
2111:
2097:
2096:
2089:
2076:
2072:
2055:
2052:
2037:
2033:
2012:
2009:
2006:
2003:
2000:
1997:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1952:
1947:
1943:
1937:
1933:
1927:
1924:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1863:
1859:
1853:
1849:
1845:
1842:
1837:
1834:
1829:
1825:
1821:
1816:
1812:
1806:
1802:
1798:
1795:
1790:
1787:
1782:
1779:
1776:
1773:
1767:
1764:
1758:
1755:
1722:
1717:
1713:
1707:
1704:
1700:
1696:
1693:
1688:
1685:
1680:
1677:
1672:
1667:
1664:
1661:
1658:
1655:
1651:
1647:
1640:
1636:
1632:
1627:
1624:
1618:
1615:
1609:
1606:
1580:
1577:
1572:
1568:
1564:
1548:
1545:
1541:relative error
1521:
1517:
1511:
1507:
1503:
1500:
1495:
1492:
1487:
1484:
1480:
1475:
1472:
1466:
1463:
1457:
1452:
1448:
1412:
1408:
1385:
1378:
1374:
1368:
1364:
1360:
1357:
1354:
1350:
1346:
1343:
1340:
1335:
1331:
1325:
1321:
1317:
1314:
1311:
1303:
1299:
1293:
1289:
1285:
1282:
1278:
1273:
1270:
1264:
1261:
1255:
1250:
1246:
1211:
1208:
1194:
1188:
1181:
1178:
1171:
1166:
1159:
1156:
1148:
1144:
1139:
1136:
1132:
1110:
1103:
1100:
1093:
1089:
1086:
1064:
1056:
1053:
1049:
1045:
1038:
1035:
1031:
1027:
1024:
1021:
1013:
1008:
1005:
1002:
999:
996:
992:
988:
981:
977:
973:
968:
965:
959:
956:
950:
947:
925:
922:
919:
897:
892:
887:
880:
877:
870:
865:
858:
855:
848:
845:
839:
836:
830:
826:
820:
815:
812:
809:
806:
803:
799:
795:
788:
784:
780:
775:
772:
766:
763:
757:
754:
728:
725:
711:
706:
699:
696:
689:
684:
677:
674:
667:
664:
658:
655:
649:
645:
639:
634:
631:
628:
625:
622:
618:
612:
607:
602:
598:
594:
591:
585:
582:
576:
573:
548:
545:
531:single polymer
518:
492:
489:
486:
483:
480:
473:
470:
444:
441:
438:
435:
432:
429:
426:
423:
416:
413:
380:
377:
374:
371:
368:
365:
362:
359:
352:
349:
342:
337:
334:
331:
328:
325:
318:
315:
308:
302:
299:
274:
271:
268:
265:
262:
255:
252:
226:
223:
220:
217:
214:
211:
208:
205:
198:
195:
182:(or momentum)
152:
149:
144:
140:
117:
113:
89:
84:
80:
60:
57:
43:As opposed to
13:
10:
9:
6:
4:
3:
2:
2776:
2765:
2762:
2760:
2757:
2756:
2754:
2740:
2736:
2732:
2728:
2724:
2720:
2713:
2710:
2704:
2699:
2695:
2691:
2687:
2680:
2677:
2672:
2668:
2664:
2660:
2652:
2649:
2644:
2637:
2635:
2631:
2624:
2620:
2617:
2615:
2612:
2610:
2607:
2605:
2602:
2600:
2597:
2595:
2592:
2591:
2587:
2582:
2578:
2574:
2570:
2567:
2564:
2561:
2558:
2557:
2556:
2548:
2541:
2539:
2537:
2532:
2530:
2526:
2509:
2506:
2503:
2481:
2478:
2473:
2470:
2467:
2445:
2441:
2431:
2416:
2413:
2408:
2405:
2402:
2393:
2378:
2375:
2370:
2367:
2364:
2355:
2340:
2337:
2332:
2329:
2326:
2314:
2309:
2305:
2300:
2296:
2282:
2279:
2274:
2271:
2266:
2263:
2258:
2253:
2249:
2237:
2222:
2219:
2214:
2211:
2206:
2202:
2199:
2193:
2190:
2182:
2168:
2148:
2140:
2136:
2118:
2113:
2109:
2094:
2090:
2074:
2070:
2061:
2060:
2059:
2053:
2051:
2035:
2031:
2007:
2001:
1998:
1995:
1986:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1950:
1945:
1941:
1935:
1931:
1925:
1922:
1917:
1914:
1908:
1902:
1899:
1896:
1888:
1885:
1883:
1878:
1861:
1851:
1847:
1843:
1835:
1832:
1827:
1823:
1819:
1814:
1804:
1800:
1796:
1788:
1785:
1780:
1777:
1774:
1762:
1753:
1745:
1737:
1733:
1715:
1705:
1702:
1698:
1694:
1686:
1683:
1678:
1675:
1670:
1665:
1662:
1659:
1656:
1653:
1649:
1638:
1634:
1630:
1625:
1613:
1604:
1596:
1594:
1578:
1575:
1570:
1566:
1562:
1554:
1546:
1544:
1542:
1537:
1519:
1509:
1505:
1501:
1493:
1490:
1485:
1482:
1478:
1473:
1461:
1450:
1446:
1437:
1435:
1430:
1428:
1410:
1406:
1396:
1376:
1366:
1362:
1358:
1352:
1348:
1344:
1341:
1338:
1333:
1323:
1319:
1315:
1301:
1291:
1287:
1283:
1276:
1271:
1259:
1248:
1244:
1235:
1233:
1229:
1225:
1216:
1209:
1207:
1186:
1176:
1169:
1164:
1154:
1142:
1137:
1134:
1130:
1098:
1087:
1084:
1075:
1054:
1051:
1047:
1043:
1036:
1033:
1029:
1025:
1022:
1019:
1011:
1006:
1003:
1000:
997:
994:
990:
979:
975:
971:
966:
954:
945:
937:
920:
908:
885:
875:
868:
863:
853:
843:
834:
828:
824:
818:
813:
810:
807:
804:
801:
797:
786:
782:
778:
773:
761:
752:
744:
742:
737:
735:
726:
724:
704:
694:
687:
682:
672:
662:
653:
647:
643:
637:
632:
629:
626:
623:
620:
616:
610:
600:
592:
580:
571:
563:
543:
532:
516:
490:
487:
484:
481:
478:
468:
442:
439:
436:
433:
430:
427:
424:
421:
411:
398:
394:
378:
375:
372:
369:
366:
363:
360:
357:
347:
340:
335:
332:
329:
326:
323:
313:
306:
297:
272:
269:
266:
263:
260:
250:
224:
221:
218:
215:
212:
209:
206:
203:
193:
181:
177:
173:
169:
164:
150:
147:
142:
138:
115:
111:
82:
78:
66:
58:
56:
54:
50:
46:
41:
38:
34:
30:
26:
22:
18:
2722:
2718:
2712:
2693:
2689:
2679:
2662:
2658:
2651:
2642:
2554:
2542:Applications
2533:
2432:
2394:
2356:
2318:
2307:
2303:
2238:
2183:
2138:
2098:
2057:
1987:
1889:
1886:
1879:
1746:
1743:
1597:
1550:
1539:which has a
1538:
1438:
1431:
1397:
1236:
1221:
1076:
938:
909:
745:
738:
730:
564:
530:
508:
178:etc.) has a
165:
62:
42:
16:
15:
180:wave vector
49:Bragg peaks
2759:Scattering
2753:Categories
2625:References
2577:biological
2099:The ratio
2093:wavelength
1434:Lorentzian
1234:function:
53:stochastic
37:scattering
2696:: 18–34.
2504:θ
2471:−
2406:∼
2403:λ
2368:∼
2365:λ
2330:∼
2327:λ
2280:≪
2272:θ
2267:
2259:λ
2220:θ
2215:
2207:λ
2203:π
2149:θ
2119:λ
1999:
1918:−
1900:
1828:−
1820:≈
1781:−
1775:≈
1766:→
1721:⟩
1679:−
1650:∑
1646:⟨
1626:≈
1617:→
1576:≪
1474:≈
1465:→
1353:−
1339:−
1263:→
1228:continuum
1180:→
1170:−
1158:→
1143:≡
1102:→
1088:≡
1063:⟩
1023:
991:∑
987:⟨
958:→
924:⟩
921:⋅
918:⟨
896:⟩
879:→
869:−
857:→
844:⋅
838:→
798:∑
794:⟨
765:→
698:→
688:−
676:→
663:⋅
657:→
617:∑
584:→
547:→
472:→
415:→
351:→
341:−
317:→
307:≡
301:→
254:→
197:→
88:→
59:Formalism
33:solutions
31:systems:
29:polymeric
21:chemistry
2739:21332357
2594:Polymers
2529:proteins
172:neutrons
65:monomers
2551:order).
2315:Example
25:physics
2737:
1591:, the
1398:With
1232:Debye
910:with
457:into
176:X-ray
168:light
2735:PMID
2333:5000
2181:as:
2091:The
2023:vs.
1593:sinc
1122:and
2727:doi
2698:doi
2667:doi
2581:DNA
2474:100
2264:sin
2212:sin
1996:log
1897:log
1020:sin
393:.
2755::
2733:.
2723:40
2721:.
2694:43
2692:.
2688:.
2663:48
2661:.
2633:^
2468:10
1884:.
1436::
1429:.
1206:.
562::
163:.
23:,
2741:.
2729::
2706:.
2700::
2673:.
2669::
2510:0
2507:=
2482:o
2479:A
2446:g
2442:R
2417:o
2414:A
2409:1
2379:o
2376:A
2371:3
2341:o
2338:A
2308:b
2304:a
2302:(
2283:1
2275:2
2254:g
2250:R
2223:2
2200:4
2194:=
2191:k
2169:k
2137:(
2114:g
2110:R
2075:g
2071:R
2036:2
2032:k
2011:)
2008:k
2005:(
2002:I
1974:.
1971:t
1968:s
1965:n
1962:o
1959:c
1956:+
1951:2
1946:g
1942:R
1936:2
1932:k
1926:3
1923:1
1915:=
1912:)
1909:k
1906:(
1903:I
1862:2
1858:)
1852:g
1848:R
1844:k
1841:(
1836:3
1833:1
1824:e
1815:2
1811:)
1805:g
1801:R
1797:k
1794:(
1789:3
1786:1
1778:1
1772:)
1763:k
1757:(
1754:S
1716:2
1712:)
1706:j
1703:i
1699:R
1695:k
1692:(
1687:6
1684:1
1676:1
1671:N
1666:1
1663:=
1660:j
1657:,
1654:i
1639:2
1635:N
1631:1
1623:)
1614:k
1608:(
1605:S
1579:1
1571:g
1567:R
1563:k
1520:2
1516:)
1510:g
1506:R
1502:k
1499:(
1494:2
1491:1
1486:+
1483:1
1479:1
1471:)
1462:k
1456:(
1451:D
1447:S
1411:g
1407:R
1384:]
1377:2
1373:)
1367:g
1363:R
1359:k
1356:(
1349:e
1345:+
1342:1
1334:2
1330:)
1324:g
1320:R
1316:k
1313:(
1310:[
1302:4
1298:)
1292:g
1288:R
1284:k
1281:(
1277:2
1272:=
1269:)
1260:k
1254:(
1249:D
1245:S
1193:|
1187:j
1177:R
1165:i
1155:R
1147:|
1138:j
1135:i
1131:R
1109:|
1099:k
1092:|
1085:k
1055:j
1052:i
1048:R
1044:k
1037:j
1034:i
1030:R
1026:k
1012:N
1007:1
1004:=
1001:j
998:,
995:i
980:2
976:N
972:1
967:=
964:)
955:k
949:(
946:S
891:)
886:j
876:R
864:i
854:R
847:(
835:k
829:i
825:e
819:N
814:1
811:=
808:j
805:,
802:i
787:2
783:N
779:1
774:=
771:)
762:k
756:(
753:S
710:)
705:j
695:R
683:i
673:R
666:(
654:k
648:i
644:e
638:N
633:1
630:=
627:j
624:,
621:i
611:2
606:|
601:a
597:|
593:=
590:)
581:k
575:(
572:I
544:k
517:N
505:.
491:l
488:a
485:n
482:i
479:f
469:k
443:t
440:n
437:e
434:d
431:i
428:c
425:n
422:i
412:k
379:t
376:n
373:e
370:d
367:i
364:c
361:n
358:i
348:k
336:l
333:a
330:n
327:i
324:f
314:k
298:k
273:l
270:a
267:n
264:i
261:f
251:k
225:t
222:n
219:e
216:d
213:i
210:c
207:n
204:i
194:k
174:/
170:/
151:a
148:=
143:i
139:a
116:i
112:a
83:i
79:R
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