100:
particles, all small-angle scattering can be understood as arising from surfaces or interfaces. Normally, SAS is measured in order to detect correlations between different interfaces, and in particular, between remote surface segments of one and the same particle. This allows conclusions about the
486:
108:
is relatively large on the usual scale of SAS. In this regime, correlations between remote surface segments and inter-particle correlations are so random that they average out. Therefore one sees only the local interface
96:. In this range, the sample must not be described at an atomistic level; one rather uses a continuum description in terms of an electron density or a neutron scattering length density. In a system composed of distinct
295:
553:
94:
172:
351:
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For a specific model system, e.g. a dispersion of uncorrelated spherical particles, one can derive Porod's law by computing the scattering function
211:(the yardstick by which it is measured). In the case of a fractally rough surface area with a dimensionality d between 2-3 Porod's law becomes:
219:
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model in which the whole system can be described to be self-similar mathematically although not usually in reality in the nature.
676:
203:
mathematics it has become clear that Porod's law requires adaptation for rough interfaces because the value of the surface
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501:
58:
681:
366:
17:
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Thus if plotted logarithmically the slope of ln(I) versus ln(q) would vary between -4 and -3 for such a
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is the surface area of the particles, which can in this way be experimentally determined. The power law
97:
129:
628:
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330:
306:. Slopes less negative than -3 are also possible in fractal theory and they are described using a
652:
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188:
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52:
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25:
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481:{\displaystyle S({\vec {q}})={\frac {4\pi ^{2}}{q_{z}^{2}}}\delta (q_{x})\delta (q_{y}).}
632:
616:
670:
619:; Stanley, H. B. (1988-08-01). "X-ray and neutron scattering from rough surfaces".
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exactly, averaging over slightly different particle radii, and taking the limit
33:
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If the interface is flat, then Porod's law predicts the scattering intensity
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200:
491:
Taking the spherical average over possible directions of the vector
290:{\displaystyle \lim _{q\rightarrow \infty }I(q)\propto S'q^{-(6-d)}}
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size and shape of the particles, and their correlations.
28:, describes the asymptote of the scattering intensity
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369:. For a flat surface in the xy-plane, one obtains
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627:(4). American Physical Society (APS): 2297–2311.
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51:that are small compared to the scale of usual
8:
548:{\displaystyle S(q)={\frac {2\pi }{q^{4}}}.}
89:{\displaystyle q\lesssim 1{\text{ nm}}^{-1}}
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47:Porod's law is concerned with wave numbers
571:This has been pointed out by Sinha et al.
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187:corresponds to the factor 1/sinθ in
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60:
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564:
495:, one obtains Porod's law in the form
7:
365:as a double surface integral, using
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14:
357:by considering just an interface
167:{\displaystyle I(q)\sim Sq^{-4}}
361:Alternatively, one can express
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615:Sinha, S. K.; Sirota, E. B.;
346:{\displaystyle q\to \infty }
703:
580:Sinha et al., Eq. (2.12).
641:10.1103/physrevb.38.2297
319:as Form factor asymptote
589:Sinha et al., Eq. (3.3)
677:Small-angle scattering
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367:Ostrogradsky's theorem
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18:small-angle scattering
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207:may be a function of
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32:for large scattering
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199:Since the advent of
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16:In X-ray or neutron
633:1988PhRvB..38.2297S
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687:Neutron scattering
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621:Physical Review B
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189:Fresnel equations
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53:Bragg diffraction
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682:X-ray scattering
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195:Generalized form
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24:, discovered by
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304:surface fractal
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191:of reflection.
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308:volume fractal
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117:Standard form
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26:Günther Porod
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55:; typically
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29:
21:
15:
34:wavenumbers
22:Porod's law
671:Categories
617:Garoff, S.
597:References
314:Derivation
98:mesoscopic
649:0163-1829
527:π
457:δ
438:δ
411:π
392:→
341:∞
338:→
277:−
268:−
252:∝
235:∞
232:→
157:−
146:∼
111:roughness
79:−
66:≲
259:′
104:Porod's
74: nm
657:9946532
629:Bibcode
201:fractal
43:Context
20:(SAS),
655:
647:
179:where
559:Notes
653:PMID
645:ISSN
363:S(q)
325:S(q)
30:I(q)
637:doi
225:lim
673::
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625:38
623:.
605:^
353:.
113:.
39:.
659:.
639::
631::
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536:4
532:q
524:2
518:=
515:)
512:q
509:(
506:S
493:q
476:.
473:)
468:y
464:q
460:(
454:)
449:x
445:q
441:(
431:2
426:z
422:q
415:2
407:4
401:=
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389:q
383:(
380:S
335:q
283:)
280:d
274:6
271:(
264:q
256:S
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246:q
243:(
240:I
229:q
209:q
205:S
185:q
181:S
160:4
153:q
149:S
143:)
140:q
137:(
134:I
106:q
82:1
69:1
63:q
49:q
37:q
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