research article microscale fragmentation and small-angle...

Research ArticleMicroscale Fragmentation and Small-Angle Scattering fromMass Fractals

E M Anitas12

1 Joint Institute for Nuclear Research Dubna 141980 Russia2Horia Hulubei National Institute of Physics and Nuclear Engineering 077125 Bucharest-Magurele Romania

Correspondence should be addressed to E M Anitas eanitasroyahoocom

Received 21 April 2015 Revised 29 July 2015 Accepted 30 July 2015

Academic Editor Fajun Zhang

Copyright copy 2015 E M Anitas This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Using the small-angle scattering method we calculate here the mono- and polydisperse structure factor from an idealizedfragmentationmodel based on the concept of renormalizationThe system consists of a large number of fractal microobjects whichare randomly oriented and whose positions are uncorrelated It is shown that in the fractal region the monodisperse form factoris characterized by a generalized power-law decay (ie a succession of maxima and minima superimposed on a simple power-lawdecay) and whose scattering exponent coincides with the fractal dimension of the scatterer The present analysis of the scatteringstructure factor allows us to obtain the number of fragments resulted at a given iteration The results could be used to obtainadditional structural information about systems obtained through microscale fragmentation processes

1 Introduction

The number of fragments (such as those resulted from rocksweathering and explosions or produced by earthrsquos crust) asa function of their sizes over a wide range of scales usuallycan be described by a fractal distribution [1ndash3] which isresponsible for various physical properties such as hydraulicconductivity ormoisture characteristics in soils [4] Althougha quantification of these processes using the renormalizationgroup approach has been suggested in [5 6] an importantissue concerns the distribution of fragments at microscalesobtained by various methods such us in ultrasonic fragmen-tation using optoacoustic lens [7]

Small-angle scattering (SAS neutrons X-ray light) [8 9]is one of the most important techniques for investigating themicrostructure of various types of systems which addressesthe issue of size distribution including the smallest andlargest components It yields the differential elastic crosssection per unit solid angle as a function of the momentumtransferThemain advantage consists in its ability to differen-tiate betweenmass and surface fractals [10 11] and it has beensuccessfully used in studying the property of self-similarityacross nano- and microscales [2] such as various types

of membranes [12ndash15] cements [16] semiconductors [17]magnetic structures [18 19] or biological structures [20ndash22]Thus the concept of fractal geometry coupled with SAS tech-nique can give new insights concerning the structural char-acteristics of such complex systems [10 11 23ndash30] One of themain parameters which can be obtained is the fractal dimen-sion 119863

119898[1] For a mass fractal it is given by the scattering

exponent of the power-law SAS intensity 119868(119902) prop 119902minus119863119898 where

0 lt 119863119898

lt 3 For deterministic fractals additional informa-tion can be obtained such as scaling factor (from the periodof 119868(119902)119902119863119898 in the logarithmic scale) the number of fractaliteration119898 (which equals the number of periods of function119868(119902) prop 119902

minus119863119898) and the total number119873

119898of structural units of

which the fractal is composedIn this paper we develop a theoretical model based on

renormalization group approach which could describe vari-ous microscale fragmentation processesWe calculate analyt-ically the mono- and polydisperse structure factor and showhow to obtain the main structural parameter such as thefractal dimension and the number of fragments at a given iter-ation and showhow to estimate the smallest and largest radiusof the fractal from SAS data

Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2015 Article ID 501281 5 pageshttpdxdoiorg1011552015501281

2 Advances in Condensed Matter Physics

l0 l2 = l032

l1 = l03

l3 = l033

Figure 1 A two-dimensional projection of the fragmentationmodel Note that at a given iteration all the fragments have the samesize The probability that the cube with edge length 1198970 (119898 = 0initiator) will be fragmented into 27 cubes of edge length 1198971 is 119891

2 Fragmentation ModelConstruction and Properties

The construction process of three-dimensional mass fractalsis similar to that of mass generalized Cantor and Vicsekfractals (GCF and GVF) [26] in the sense that one follows atop-down approach in which an initial structure is repeatedlydivided (by a single scaling factor) into a set of smallerstructures of the same type according to a given rule which ischanged randomly from one iteration to the next one Thuswe first consider a cube of edge length 1198970 (119898 = 0 initiator) andin the first iteration (119898 = 1) the number of cubes which arekept is related to the probability 119891 in which the initiator willbe fragmented into 27 cubes of edge length 1198971 = 11989703 Figure 1illustrates the construction of a generic model of fragmen-tation where the sizes of remaining cubes at 119898th iterationare given by

119897119898=

1198970

3119898 (1)

At119898th iteration the total number of particles will be given by

119873119898= (1minus119891) (1+ 27119891+ (27119891)2 + sdot sdot sdot + (27119891)119898) (2)

and in a good approximation (119898 ≫ 1 and 27119891 gt 1) it can beshown that the fractal dimension can be written as [5]

119863119898= 3

log (27119891)log 27

(3)

with 127 lt 119891 lt 1 and thus 0 lt 119863119898lt 3 Figure 2 shows the

variation of the fractal dimension 119863119898with probability 119891 in

which the initiator will be fragmented into 27 cubes of edgelength 1198971 = 11989703

f

00 02 04 06 08 10

3

2

1

0

Dm

Figure 2 Dependence of the fractal dimension119863119898of the probabil-

ity 119891 from (3)

3 Theory

Here we restrict ourselves to two-phase systems which arecomposed of homogeneous units of mass density 120588

119898 The

units are immersed into a solid matrix of pore density 120588119901

Then [8 9] we can consider the system as if the units werefrozen in a vacuum and had the density Δ120588 = 120588

119898minus 120588119901 The

density Δ120588 is called scattering contrast and thus the scatter-ing intensity is given by

119868 (119902) = 1198991003816100381610038161003816Δ120588

1003816100381610038161003816

2119881

2⟨|119865 (q)|2⟩ (4)

where 119899 is the fractal concentration 119881 is the volume of eachfractal and 119865(q) is the normalized form factor

119865 (q) = (1119881)int119881

119890minus119894qsdotrdr (5)

obeying 119865(0) = 1 The brackets ⟨sdot sdot sdot ⟩ stand for the ensembleaveraging over all orientations of q Once a deterministicfractal is composed of the same objects say119873 cubes of edgelength 119897 then

119865 (q) =120588q1198650 (119902119897)

119873 (6)

with 120588q = sum119895119890minus119894qsdotr119895 being the Fourier component of the

density of the cubes and r119895are the center-of-mass positions of

cubes Here the cube form factor of unit edge length is givenby [8]

1198650 (t) =sin (1199051199092)

1199051199092

sin (1199051199102)

1199051199102

sin (1199051199112)

1199051199112

(7)

Therefore the scattering intensity becomes

119868 (119902) =119868 (0) 119878 (119902) 10038161003816100381610038161198650 (119902119897)

1003816100381610038161003816

2

119873 (8)

Advances in Condensed Matter Physics 3

100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

100 101 102 103 104

ql0

I mI

m(0)

f = 0296

f = 0222

f = 0148

Figure 3 The monodisperse structure factor for the fragmentationmodel at various values of probability 119891 shows a superposition ofmaxima and minima on a simple power-law decay in the fractalregionThe curves are shifted down vertically for clarity with a factorof 10minus2 (red middle curve) and respectively with 10minus4 (green lowercurve) Iteration number is119898 = 6

with 119868(0) = 119899|Δ120588|2119881

2 and

119878 (119902) equiv

⟨120588q120588minusq⟩

119873

(9)

being the structure factor and it is intimately connectedto the pair distribution function In the fractal region theform factor tends to one and this implies that the scatteringintensity is proportional to the structure factor in the fractalregion

The monodisperse form factor at 119898th iteration can bewritten as [26]

119865119898(q) = 1198650 (119902119897119898)

119898

prod

119894=1119866119894(q) (10)

with 119898 = 1 2 and 119866(q) being the generative function ofthe fractal and it specifies the positions of the scattering cubesinside the fractal Thus the scattering intensity becomes [26]

119868119898(119902)

119868119898(0)

= ⟨1003816100381610038161003816119865119898 (q)

1003816100381610038161003816

2⟩ (11)

4 Results and Discussion

For well-known systems such as Cantor sets or Vicsek fractals119866119894(q) has known analytical expressions [26] Here we use a

random distribution of cubes at each iteration for writing thegenerative function

The monodisperse structure factor for various values of119891 is shown in Figure 3 It is characterized by the presence ofthree main regions on a double logarithmic scale at low 119902

(119902 ≲ 11198970) a plateau (Guinier region) which gives informationabout the overall size of the fractal a fractal (generalized

f = 0296

f = 0222

f = 0148

100 101 102 103

ql0

10minus1

10minus1

10minus3

10minus5

10minus7

I mI

m(0)

simqminus163

simqminus189

simqminus126

Figure 4 The polydisperse structure factor for the fragmentationmodel at various values of probability 119891 shows a simple power-lawdecay in the fractal region with the scattering exponent equal to thefractal dimension of the fractal The horizontal lines indicate theasymptotic values of the structure factor (see (12)) Iteration numberis119898 = 6 and relative variance 120590

119903= 06

power law-decay) region at 11198970 ≲ 119902 ≲ 1119897119898 and an

asymptotic region

119878 (119902) ≃1119873119898

(12)

at high 119902 (1119897119898

≲ 119902) which gives information about thenumber of fragments in the fractal (see also Figure 4)

The value of the fractal dimension can be obtained byconsidering the polydispersity in size of the fractals Here weuse a distribution function 119863

119873(119897) of sizes in such a way that

119863119873(119897)d119897 gives the probability of finding a fractal whose size

falls within (119897 119897 + d119897) We consider a log-normal distribution

119863119873(119897) =

1120590119897 (2120587)12

exp(minus

[ln (1198971198970 + 12059022)]

2

21205902 ) (13)

where 120590 = [ln(1 + 1205902119903)]12 The mean length and its relative

variance are 120583 equiv ⟨119897⟩119863and 120590

119903equiv (⟨119897⟩

2119863

minus 11989720)

12120583 where

⟨sdot sdot sdot ⟩119863

= intinfin

0 sdot sdot sdot 119863119873(119897)d119897 Thus the polydisperse structure

factor is given by [26]

119868119898(119902) = 119899

1003816100381610038161003816Δ1205881003816100381610038161003816

2int

infin

0⟨1003816100381610038161003816119865119898 (q)

1003816100381610038161003816

2⟩119881

2119898119863119873(119897) d119897 (14)

where 119881119898is the total volume at the119898th iteration

Figure 4 shows the polydisperse structure obtained fromFigure 3 together with (14) As expected the influence ofpolydispersity is to smooth the scattering curves and topreserve the scattering exponent in the fractal region At high119902 the structure factor is proportional to 1119873

119898 according to

(12) Note that the value of fractal dimension is in agreementwith (3)


The results show that in an ideal SAS experiment (whichshows the presence of Guinier fractal and Porod regions) theoverall size and the smallest size in the fractal can be obtainedfrom both the mono- and polydisperse intensity through theposition of the crossover between Guinier and fractal regionand respectively between fractal and Porod regionHowevera reliable estimation of the fractal dimension can be obtainedonly in the polydisperse case

An important difference between the model developedhere and those based on exact self-similar fractals is thatfor the latter models we have a periodic distribution for theminima of monodisperse intensity [26 29 31] which allowsus to obtain the number of fractal iteration as well as the frac-tal scaling factor 120573

119904from the log-periodicity of the quantity

119868(119902)119902119863119898 versus 119902 in the fractal region For the former model

the choice of the scattering units (here cubes) according to aprobability119891 leads to an asymmetric distributionwith respectto the origin of the initiator This difference is reflected in thepolydisperse intensity by oscillations of various amplitudes(see Figure 4) around the simple power-law decay 119868(119902) prop

119902minus119863119898 However the asymptote of the fractal structure factor

gives the number of particles 119873119898inside the fractal and this

can be used in (2) together with probability 119891 from (3) (with119863119898being known from the exponent of the power-law decay)

to determine the iteration number 119898 of the fractal Thus byknowing 119898 119873

119898 and 119863

119898and using the relation [26] 119873

119898=

(1120573119904)119898119863119898 we can then still recover the scaling factor

Depending on the type and availability of the sampleand on the experimental setup various approaches could befollowed in order to extract structural information Thus ifthe structure factor given by (12) ismultiplied by a form factorcorresponding to the basic scattering unit then one obtainsthe SAS intensity in which the asymptotic region of thestructure factor is replaced by a Porod decay (119868(119902) prop 119902

minus4) andthus additional information can be obtained For examplein [32] from the calculated small-angle X-ray scatteringintensity of a fragmentation fractal silica nanoagglomeratesthe fractal structure and aggregation number have beenobtained by using the well-known Beaucage model [33] andthe specific surface area has been obtainedwith the help of theintegral parameter 119876 and the Porod constant [9]

5 Conclusions

We have developed an idealized statistical self-similar fractalmodel for describing fragmentation processes based on 3Dfractal and have used the SAS technique to characterize themicrostructural properties of such fragments The model isbased on the concept of renormalization and the suggestedapproach has important advantages since SAS is a nonde-structive technique the obtained information is averagedover a macroscopic volume and the model features allow usto obtain additional information about the fractal fragmentssuch as the number of structural units of which the fractal iscomposed the scaling factor or the fractal iteration number

We have calculated the monodisperse and polydispersestructure factor and have shown how to obtain the smallestand largest sizes inside the fractal from the position of

the crossover between Guinier and fractal region andrespectively between fractal and Porod region We haveshown that the slope of SAS intensity is in agreement withthe fractal dimension of the fractal as given by (3)

Themodel can be used in understanding various physicalproperties such as hydraulic conductivity or moisture char-acteristics for systems obtained through fragmentation and itcan be easily extended to describe more complex structuressuch as by keeping fragments of different sizes obtained atvarious iterations (surface fractals) by taking into accountmultiple scaling factors at a given iteration (multifractals)changing the scaling factors with the iteration numberaccording to a certain rule (fat fractals) or a combination ofthem

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The author acknowledges financial support from JINR-IFIN-HH projects awarded by the Romanian PlenipotentiaryRepresentative at JINR

References

[1] B B Mandelbrot The Fractal Geometry of Nature WH Free-man 1983

[2] J-F Gouyet Physics and Fractal Structures Springer 1996[3] T Vicsek Fractal Growth PhenomenaWorld Scientific Publish-

ing Co Inc Teaneck NJ Singapore 1989[4] M Bittelli G S Campbell and M Flury ldquoCharacterization of

particle-size distribution in soils with a fragmentation modelrdquoSoil Science Society of America Journal vol 63 no 4 pp 782ndash788 1999

[5] D L Turcotte ldquoFractals and fragmentationrdquo Journal of Geophys-ical Research vol 91 no 2 pp 1921ndash1926 1986

[6] D L Turcotte Fractals and Chaos in Geology and GeophysicsCambridge University Press 2nd edition 1997

[7] H W Baac J G Ok A Maxwell et al ldquoCarbon-nanotubeoptoacoustic lens for focused ultrasound generation and high-precision targeted therapyrdquo Scientific Reports vol 2 article 9892012

[8] O Glatter and O Kratky Small-Angle X-Ray Scattering Aca-demic Press London UK 1982

[9] L A Feigin andD I Svergun Structure Analysis by Small-AngleX-Ray and Neutron Scattering Springer New York NY USA1987

[10] J E Martin and A J Hurd ldquoScattering from fractalsrdquo Journal ofApplied Crystallography vol 20 no 2 pp 61ndash78 1987

[11] P W Schmidt ldquoSmall-angle scattering studies of disorderedporous and fractal systemsrdquo Journal of Applied Crystallographyvol 24 part 5 pp 414ndash435 1991

[12] M Balasoiu E M Anitas I Bica et al ldquoSANS of interactingmagneticmicro-sized Fe particles in a Stomaflex creme polymermatrixrdquo Optoelectronics and Advanced MaterialsmdashRapid Com-munications vol 2 no 11 pp 730ndash734 2008


[13] EM Anitas M Balasoiu I Bica V A Osipov and A I KuklinldquoSmall-angle neutron scattering analysis of the microstructureof stomaflex crememdashferrofluid based elastomersrdquoOptoelectron-ics and Advanced MaterialsmdashRapid Communications vol 3 no6 pp 621ndash625 2009

[14] V Maneeratana J D Bass T Azaıs et al ldquoFractal inor-ganicminusorganic interfaces in hybrid membranes for efficientproton transportrdquo Advanced Functional Materials vol 23 no22 pp 2872ndash2880 2013

[15] H Amjad Z Khan and V V Tarabara ldquoFractal structure andpermeability of membrane cake layers effect of coagulation-flocculation and settling as pretreatment stepsrdquo Separation andPurification Technology vol 143 pp 40ndash51 2015

[16] A Das S Mazumder D Sen et al ldquoSmall-angle neutron scat-tering as a probe to decide themaximum limit of chemical wasteimmobilization in a cement matrixrdquo Journal of Applied Crystal-lography vol 47 no 1 pp 421ndash429 2014

[17] K Cho P Biswas and P Fraundorf ldquoCharacterization ofnanostructured pristine and Fe- and V-doped titania synthe-sized by atomization and bubblingrdquo Journal of Industrial andEngineering Chemistry vol 20 no 2 pp 558ndash563 2014

[18] M-L Craus A K Islamov E M Anitas N Cornei and DLuca ldquoMicrostructural magnetic and transport properties ofLa05Pr02Pb03-xSrxMnO3 manganitesrdquo Journal of Alloys andCompounds vol 592 pp 121ndash126 2014

[19] T Naito H Yamamoto K Okuda et al ldquoMagnetic orderingof spin systems having fractal dimensions experimental studyrdquoThe European Physical Journal B vol 86 article 410 2014

[20] I Yadav S Kumar V K Aswal and J Kohlbrecher ldquoSmall-angle neutron scattering study of differences in phase behaviorof silica nanoparticles in the presence of lysozyme and bovineserum albumin proteinsrdquo Physical Review E vol 89 no 3Article ID 032304 9 pages 2014

[21] RGebhardt ldquoEffect of filtration forces on the structure of caseinmicellesrdquo Journal of Applied Crystallography vol 47 no 1 pp29ndash34 2014

[22] J Maji S M Bhattacharjee F Seno and A Trovato ldquoMeltingbehavior and different bound states in three-stranded DNAmodelsrdquo Physical Review E vol 89 no 1 Article ID 012121 2014

[23] H D Bale and P W Schmidt ldquoSmall-angle X-ray-scatteringinvestigation of submicroscopic porosity with fractal proper-tiesrdquo Physical Review Letters vol 53 no 6 pp 596ndash599 1984

[24] P W Schmidt and X Dacai ldquoCalculation of the small-angle x-ray and neutron scattering from nonrandom (regular) fractalsrdquoPhysical Review A vol 33 article 560 1986

[25] J Teixeira ldquoSmall-angle scattering by fractal systemsrdquo Journalof Applied Crystallography vol 21 no 6 pp 781ndash785 1988

[26] A Y Cherny E M Anitas V A Osipov and A I KuklinldquoDeterministic fractals extracting additional information fromsmall-angle scattering datardquo Physical Review E vol 84 no 3Article ID 036203 2011

[27] D Gunther D Y Borin S Gunther and S Odenbach ldquoX-raymicro-tomographic characterization of field-structured mag-netorheological elastomersrdquo Smart Materials and Structuresvol 21 no 1 Article ID 015005 2012

[28] A Y Cherny E M Anitas V A Osipov and A I KuklinldquoSmall-angle scattering from multiphase fractalsrdquo Journal ofApplied Crystallography vol 47 no 1 pp 198ndash206 2014

[29] E M Anitas ldquoSmall-angle scattering from fat fractalsrdquo TheEuropean Physical Journal B vol 87 p 139 2014

[30] I Bica E M Anitas L M E Averis and M Bunoiu ldquoMag-netodielectric effects in composite materials based on paraffincarbonyl iron and graphenerdquo Journal of Industrial and Engineer-ing Chemistry vol 21 pp 1323ndash1327 2015

[31] E M Anitas ldquoScattering structure factor from fat fractalsrdquoRomanian Journal of Physics vol 60 no 5-6 p 647 2015

[32] R Wengeler F Wolf N Dingenouts and H Nirschl ldquoChar-acterizing dispersion and fragmentation of fractal pyrogenicsilica nanoagglomerates by small-angle X-ray scatteringrdquo Lang-muir vol 23 no 8 pp 4148ndash4154 2007

[33] G Beaucage ldquoApproximations leading to a unified exponen-tialpower-law approach to small-angle scatteringrdquo Journal ofApplied Crystallography vol 28 no 6 pp 717ndash728 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


l0 l2 = l032

l1 = l03

l3 = l033

Figure 1 A two-dimensional projection of the fragmentationmodel Note that at a given iteration all the fragments have the samesize The probability that the cube with edge length 1198970 (119898 = 0initiator) will be fragmented into 27 cubes of edge length 1198971 is 119891

2 Fragmentation ModelConstruction and Properties

The construction process of three-dimensional mass fractalsis similar to that of mass generalized Cantor and Vicsekfractals (GCF and GVF) [26] in the sense that one follows atop-down approach in which an initial structure is repeatedlydivided (by a single scaling factor) into a set of smallerstructures of the same type according to a given rule which ischanged randomly from one iteration to the next one Thuswe first consider a cube of edge length 1198970 (119898 = 0 initiator) andin the first iteration (119898 = 1) the number of cubes which arekept is related to the probability 119891 in which the initiator willbe fragmented into 27 cubes of edge length 1198971 = 11989703 Figure 1illustrates the construction of a generic model of fragmen-tation where the sizes of remaining cubes at 119898th iterationare given by

119897119898=

1198970

3119898 (1)

At119898th iteration the total number of particles will be given by

119873119898= (1minus119891) (1+ 27119891+ (27119891)2 + sdot sdot sdot + (27119891)119898) (2)

and in a good approximation (119898 ≫ 1 and 27119891 gt 1) it can beshown that the fractal dimension can be written as [5]

119863119898= 3

log (27119891)log 27

(3)

with 127 lt 119891 lt 1 and thus 0 lt 119863119898lt 3 Figure 2 shows the

variation of the fractal dimension 119863119898with probability 119891 in

which the initiator will be fragmented into 27 cubes of edgelength 1198971 = 11989703

f

00 02 04 06 08 10

3

2

1

0

Dm

Figure 2 Dependence of the fractal dimension119863119898of the probabil-

ity 119891 from (3)

3 Theory

Here we restrict ourselves to two-phase systems which arecomposed of homogeneous units of mass density 120588

119898 The

units are immersed into a solid matrix of pore density 120588119901

Then [8 9] we can consider the system as if the units werefrozen in a vacuum and had the density Δ120588 = 120588

119898minus 120588119901 The

density Δ120588 is called scattering contrast and thus the scatter-ing intensity is given by

119868 (119902) = 1198991003816100381610038161003816Δ120588

1003816100381610038161003816

2119881

2⟨|119865 (q)|2⟩ (4)

where 119899 is the fractal concentration 119881 is the volume of eachfractal and 119865(q) is the normalized form factor

119865 (q) = (1119881)int119881

119890minus119894qsdotrdr (5)

obeying 119865(0) = 1 The brackets ⟨sdot sdot sdot ⟩ stand for the ensembleaveraging over all orientations of q Once a deterministicfractal is composed of the same objects say119873 cubes of edgelength 119897 then

119865 (q) =120588q1198650 (119902119897)

119873 (6)

with 120588q = sum119895119890minus119894qsdotr119895 being the Fourier component of the

density of the cubes and r119895are the center-of-mass positions of

cubes Here the cube form factor of unit edge length is givenby [8]

1198650 (t) =sin (1199051199092)

1199051199092

sin (1199051199102)

1199051199102

sin (1199051199112)

1199051199112

(7)

Therefore the scattering intensity becomes

119868 (119902) =119868 (0) 119878 (119902) 10038161003816100381610038161198650 (119902119897)

1003816100381610038161003816

2

119873 (8)


100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

100 101 102 103 104

ql0

I mI

m(0)

f = 0296

f = 0222

f = 0148


with 119868(0) = 119899|Δ120588|2119881

2 and

119878 (119902) equiv

⟨120588q120588minusq⟩

119873

(9)



119865119898(q) = 1198650 (119902119897119898)

119898

prod

119894=1119866119894(q) (10)


119868119898(119902)

119868119898(0)

= ⟨1003816100381610038161003816119865119898 (q)

1003816100381610038161003816

2⟩ (11)






f = 0296

f = 0222

f = 0148

100 101 102 103

ql0

10minus1

10minus1

10minus3

10minus5

10minus7

I mI

m(0)

simqminus163

simqminus189

simqminus126


119903= 06


asymptotic region

119878 (119902) ≃1119873119898

(12)

at high 119902 (1119897119898






119863119873(119897) =

1120590119897 (2120587)12

exp(minus

[ln (1198971198970 + 12059022)]

2

21205902 ) (13)



119903equiv (⟨119897⟩

2119863

minus 11989720)

12120583 where


= intinfin



119868119898(119902) = 119899

1003816100381610038161003816Δ1205881003816100381610038161003816

2int

infin

0⟨1003816100381610038161003816119865119898 (q)

1003816100381610038161003816

2⟩119881

2119898119863119873(119897) d119897 (14)



119898 according to












119898 and 119863


119898=




5 Conclusions







Acknowledgment


References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




100

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

100 101 102 103 104

ql0

I mI

m(0)

f = 0296

f = 0222

f = 0148


with 119868(0) = 119899|Δ120588|2119881

2 and

119878 (119902) equiv

⟨120588q120588minusq⟩

119873

(9)



119865119898(q) = 1198650 (119902119897119898)

119898

prod

119894=1119866119894(q) (10)


119868119898(119902)

119868119898(0)

= ⟨1003816100381610038161003816119865119898 (q)

1003816100381610038161003816

2⟩ (11)






f = 0296

f = 0222

f = 0148

100 101 102 103

ql0

10minus1

10minus1

10minus3

10minus5

10minus7

I mI

m(0)

simqminus163

simqminus189

simqminus126


119903= 06


asymptotic region

119878 (119902) ≃1119873119898

(12)

at high 119902 (1119897119898






119863119873(119897) =

1120590119897 (2120587)12

exp(minus

[ln (1198971198970 + 12059022)]

2

21205902 ) (13)



119903equiv (⟨119897⟩

2119863

minus 11989720)

12120583 where


= intinfin



119868119898(119902) = 119899

1003816100381610038161003816Δ1205881003816100381610038161003816

2int

infin

0⟨1003816100381610038161003816119865119898 (q)

1003816100381610038161003816

2⟩119881

2119898119863119873(119897) d119897 (14)



119898 according to












119898 and 119863


119898=




5 Conclusions







Acknowledgment


References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics













119898 and 119863


119898=




5 Conclusions







Acknowledgment


References








































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



research article microscale fragmentation and small-angle...

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