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|R=UxYNWt w h}YwD "CU= xOW RmQtD |OOa w |y=oW}=tR; |=yVwQ R= xO=iDU=|irDNt |=yVwQ w OQ=O |}xS} w C}ty= xDWPo |=yxQwO u}kkLt ORv uoty=v |=ysDU}U

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"Ovvm |UQQ@ =Q C} RwBt=mwv=v X=wN`@=wD VRQ= =@ |=yOQ@ Q=m R= Qo}O |m} R}v |Q=DN=Uwv=v =} |Q=DN=UR} Q |R=UR=@|xDUw}B l}v=mt R= "CU= �|m} R}i |R=UxYNWt |=wU� |}x]kvwO |oDU@tyO=wt |R=UR=@ |=Q@ X |xaW= |Q=ovwDQB |wk x=Q uOQm sy=Qi |=Q@ u=wD|t |Q=t;Q=}U@ |}x]kvwO |oDU@ty `@=wD R= xO=iDU= =@ |R=UR=@ [21�18'4'2'1]"OQ@ xQy@ uoty=vCU= X |xaW= |i=QowDQB Ovv=t 'xO=iDU=OQwt |=yVwQx}k@ R= QDxv} Rysm w QDu=U;|=Q@ |oDU@ty `@=wD R= xO=iDU= XwYN QO |D=ar=]t R}v O}OH |=yQ=m QO xm|=yTma u}= |Q=t; C=aq]= R= xO=iDU= =@ xm xOW s=Hv= xvwtvl}hrDNt |=yx}q

[22'2'1]"OwW|t |R=UR=@ |Oa@xU |xvwtv |Oa@ wO

=} =yC} RwBt=m Ovv=t |@}mQD O=wt R= |a}Uw h}] QO |}xOQDUo Qw]x@ uoty=v O=wtu} QDy@ R= |Q=t; |oDU@ty `@=wD "OvwW|t Ci=} xOvR |=yCi=@ Ovv=t |a}@] O=wt'|m}v=mt X=wN |R=UxYNWt "OvDUy uoty=v O=wt *|Q=DN=UR} Q |=yQoh}YwD|[a@ R= ,=t}kDUt u=wD|t =Q uoty=v O=wt |m} R}i |rm Qw]x@ w |m} QDmr= '|}=tQo[2w1]"OQw;CUO x@ 'OvDUy|}x]kv n|oDU@ty `@=wD R= QDxOWxDN=vW xm =yQoh}YwDC@Uv\ki xm Ov=|Q=t;|oDU@ty `@=wD |x@DQt u} QDsm|}x]kvl}|oDU@ty `@=wD`} RwD OQwtQO |D=aq]= I}y |rw Ovvm|t h}YwD Q=DN=UR} Q QO =Q =yR=i |tHL|x@DQt (TPCF) 2|}x]kvwO |oDU@ty `@=wD [2w1]"Ovvm|tv x=Q= =yR=i |UOvy|UOvy |=y|oS} w R=|WN@ Ovv=wD|t xm OvDUy|}x]kvl}|oDU@ty `@=wD R= QDq=@uoty=v O=wt |}x]kvwO |oDU@ty `@=wD "OvwW pt=W =Q =yQ=DN=Uwv=v w =yQ=DN=UR} QAwmUwQm}t =} (SAXS)3lJwmx} w=R X |xaW= |oOvm =QB C=aq]= R= u=wD|t =QCUO x@ =yQ=DN=Uwv=v w =yQ=DN=UR} Q R= (SEM/TEM) 4|r=kDv=/|W@ wQ uwQDmr=xR=Ov= `} RwD C=aq]= uOQw; CUO x@ QO SAXS VwQ 'X=N Qw] x@ [6�3]"OQw;

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DOI:10.24200/J40.2018.6419

X=wN uO}HvU |=Q@ R=}v OQwt |}=OD@= |Q=t; `@=wD '|}x]kvwO |oDU@ty `@=wD

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Q=Qk i R=i QO xm |]=kv O=OaD C@Uv OwQ|t C}=yv|@ CtU x@N xm |v=tR "OwW|t|=yQ=DN=U |=Q@ "CU= (vi) R=i u; |tHL C@Uv Qou=Wv 'N x@ (Ni) OvQ}o|t|tHL |=yC@Uv w |}x]kv l} |oDU@ty `@=wD u}@ |[=} Q \@=wQ '|R=iOvJ

[25]%OwW|t u=}@ 3 |x]@=Q j@=]tpi =

NiNjN!1 = vi (3)

%x=ov; wXPi = 1 (4)

(TPCF)6 |}x]kvwO |oDU@ty `@=D "2"3pw] =@ ~r Ovv=t |iO=YD |=yQ=OQ@ R= |O=OaD ?=NDv= =@ |}x]kvwO p=tDL= `@=wDp=tDL= uOQm XNWt TBU w Q=DN=UR} Q =} Q=DN=Uwv=v pN=O r = j~rjXNWtx@ xDU@ "OwW|t x@U=Lt XNWt R=i QO 'S2 w S1 'l} Qy |=yDv= uDiQo Q=Qk%CW=O Oy=wN OwHw p=tDL= K2 |R=i K Q=DN=U |=Q@ '\NxQ=B |=yDv= wO u=mtPij(~r) =

NijNjN!1f~r = �!r2 ��!r1 ; (S1��i) \ (S2��j)g (5)

|}=yDv= |x]kv w (�i) OvwW|t ẁQW i R=i R= xmCU=|}=yQ=OQ@ O=OaDNij u; QO xmQoV}=tv ~r2 w ~r1 u}vJty "i; j = 1; 2; : : : ;K %(�j) CU= j R=i QO =yu;"Ovvm|t pYDt =Q S2 w S1 ?}DQD x@ '`HQt |xv}tR |x}L=v xm OvDUy |}=yQ=OQ@R= l}B wQDwR}= Q=DN=U Qy TPCF 'h} QaD j@] xm CU= |QwQ[ xDmv u}= QmP

[25]"CU= pkDUt xDiQo Q=Qk Q=OQ@ CyHx]kv wO xm CU= |}=yCr=L |=Q@ xDUw}B p=tDL= `} RwD `@=D Qou=}@ 5 |x]@=Qx@ xm |v=tR OvDUy ~r x=wNrO Q=OQ@ |=yDv= wO u=wva x@ S2 w S1 &OvwW|t xDN=U`@=D sQi QO x]@=Q u}= |Q=Po|=H =@ "OvQ}o|t Q=Qk Q=DN=U QO Q=@ N |iO=YD Qw]

[6]%CW=O s}y=wN |]QW p=tDL=Pij(~r) = Pf(S1��i)j(S2��j)gP (S2��j) (6)x@ "OQ=Po|tv Q}F-=D S2 Q@ S1 ẁkw p=tDL= 'r !1 'nQR@ |r}N Q=OQ@ pw] |=Q@|}x]kv l} |oDU@ty `@=D x@ |]QW p=tDL= `@=D r ! 1 |Dkw 'Qo}O CQ=@a

%|va} &O}=@|t Vy=mlimr!1Pf(S1��i)j(S2��j)g = P (S1��i) (7)

%6 |x]@=Q |U} wvR=@ =@ 'xH}Dv QOlimr!1Pij(~r) = P (S1��i)j(S2��j) = vivj (8)

?r=]t R= [25]"CU= xOW s=Hv= |}x]kv n `@=wD |=Q@ q=@ pqODU= s}taD|R=i k Q=DN=U Qy |=yTPCF xm CiQo xH}Dv u=wD|t VN@ u}= QO xOW u=wvaR= pmWDt TPCF OOa 9 |R=ixU C} RwBt=m QO 'p=Ft |=Q@ "Ov=xDU@=w sy x@\}=QW Cra x@ "OQ=O OwHw P11; P12; P13; P21; P22; P23; P31; P32; P33

[25]%OwW|t =[Q= ,=Qwi Q} R \@=wQ Ot=aDXj=1;3

Pij(~r) = vi (9)Xi=1;3

Xj=1;3

Pij(~r) = 1 (10)

|iO=YD uoty=v O=wt "2hrDNt|=yCr=L =}hrDNt O=wt R= xmCU= O=wt R=|}xDUO|iO=YDuoty=v |xO=t|B wmUwQm}t pw] T=}kt "=yp=DU} Qm|rB =} =yC} RwBt=m Ovv=t CU= xOW p}mWD|xvwtv |xYNWt pw] R= QDlJwm Q=}U@ =t= |rwmrwt T=}kt R= QDnQR@ Q=}U@xDUw}B |B wmUwQm}t T=}kt QO u=wD|t =Q uoty=v |xO=t "CU= |B wmUwQm =txO=iDU= =@ |Q=t; |=yVwQ [1]"OQm x@U=Lt =Q VQF-wt X=wN xH}Dv QO w OQm ZQiX=wN u}tND |=Q@ OvtCQOk w |rta |=yx=Q u} QDy@ R= |m} |oDU@ty `@=wD R=`@=wD hrDNt |=yx@DQt R= xO=iDU= =@ u=wD|t =Q O=wt X=wN [1]"OvDUy uoty=v O=wt|oDU@ty `@=wD pw= x@DQt |R=iOvJ O=wt QO [24w23'1]"OQm|v}@V}B |Q=t;|oDU@tyw `} RwD |xQ=@ QO |Qo}O C=aq]= w CU= hrDNt |=yR=i |tHL C@Uv Qou=Wvp}YiD x@ |Q=t; |oDU@ty `@=wD ẁ[wt xt=O= QO [1]"OyO|tv =yR=i |SwrwiQwt

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|Q=t; |oDU@ty `@=wD "3xm Xr=N Qt}rB QO |v@ Qm |xrwrwv=v Ovv=ty |}=yQ=DN=U� |R=iwO |=yQ=DN=U QO'CU= xOQm p=eW= =Q sHL R= |}x}L=v R=i Qy xm �OvW=@ X}NWD p@=k ,qt=m R=i wO=@ w v2(!) |x}L=v =@ 2 R=i w �1 |tHL C@Uv =@ v1(!) |x}L=v =@ 1 R=iOvQo}Om} ptmt x}L=v wO u}= xm u}= x@ xHwD =@ "O} Q}o@ Q_v QO =Q �2 |tHL C@Uv

"v1(!) \v2 (!) = � w v1(!) [v2 (!) = 1 %CW=O s}y=wN|iQat =@ =Hv}= QO "CU= xOW xO=O u=Wv |R=iwO |Oa@ wO |}xvwtv 1 pmW QOj=iD= ! %OwW|t h} QaD u}vJ x 2 v |=Q@ i R=i |=Q@ xm I(i)(x;!) Qou=Wv `@=D

[25]"CU= |R=iOvJ |xvwtv l} |=yR=i R= s=OmQy QO uDiQo Q=Qk

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|v=Uwv w s}vm|t xOy=Wt =Q h=Y |]N ,qt=m �CU=ylU}O `=aW D� Oa@ x@QDnQR@ swO R=i |=yx}L=v R= |Q}otWJ p@=k O=OaD xm CU= u; Qou=}@ Qt= u}= &CU}v

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|}x]kv n |oDU@ty `@=wD "3"3x@ 'OQ=O umtt Q=Okt wO \ki (I(i)(x;!)) Qou=Wv `@=D 'C@=F |xR=Ov= =@ x |=Q@R= Qo}O |[a@ |=Q@ w QiY Qou=Wv `@=D Q=Okt ! Q}O=kt |[a@ |=Q@ p=Ft u=wva|oO=U x@ I(i)(x;!) R= |r=tDL= |i}YwD "CU= 1 Qou=Wv `@=D Q=Okt ! Q}O=kt

%O};|t CUO x@ OW=@ 1 pO=at I(i)(x;!) xm u}= |=Q@PfI(i)(x;!) = 1g (11)

%O};|t CUO x@ p=tDL= ?}DQD u}O@ wPfI(i)(x;!) = 0g = 1� PfIi(x;!) = 1g (12)F (x) sm =QDt `} RwD `@=D =@ pO=at Qw]x@ u=wD|t =Q X |xDUUo |iO=YD Q}eDt

%OQm XNWt u;F (x) = PfX � xg (13)F (�1) = xmCU= x R=CU=Q xDUw}B|a@=D 'uOw@|Wy=m Q}ep}@k R=|}=y|oS} wQy u}ov=}t =} |v}@V}B Q=Okt xm OQm QmP O}=@ R}v =Q xDmv u}= "F (+1) = 1 w 0

%OQm u=}@ u}vJ u=wD|t =Q f [I(i)(x;!)] R= |a@=Df [I(i)(x;!)] = PfI(i)(x;!) = 1gf(1)+PfI(i)(x;!) = 0gf(0) (14)f [I(i)(x;!)] = 'pta QO "CU= u}ov=}t ẁtHt |vat x@ xDUmW |=yCm =Q@ xm

%xm OQ=O|t u=}@ I(i)(x)S(i)1 (x) = I

(i)(x) = PfI(i)(x) = 1g (15)`@=D x}@W ,=k}kO u; OQmrta 'I(i)(x) Qou=Wv `@=D 1 w 0 C}y=t j@] 'u}=Q@=v@S(i)1 (x) x@ xt=O= QO '�1982� 7pDU= w wD=oQwD "CU= PfI(i)(x) = 1g p=tDL=QO i R=i uDi=} Q=m u}= xm 'OvO=O `=HQ= i R=i |=Q@ |}x]kvl} p=tDL= `@=D u=wva x@|}x]kvl} |oDU@ty `@=D u=wva x@ |y=o u}vJty w 'OR=U|t umtt =Q x u=mt

"OwW|t xO=O `=HQ= R=i Qou=Wv `@=D x@'CU= v QO =yx |xty |=Q@ I(i)(x) uDUv=O x}@W vi(!) swyit uDUv=O|=yQ}eDt |xty |=H x@ vi(!) R= |iO=YD |}xDUO R= CU= umtt u}=Q@=v@vi(!) |=Q@ p=tDL= uwv=k xH}Dv QO "OwW xO=iDU= x 2 v |=Q@ I(i)(x;!)CU= OwOLt |Oa@ |=Q=O xm I(i)(x) : x 2 v |iO=YD |=yOv}=Qi `} RwD =@R= |r=YD= `} RwD =@ vi(!) R= |r=tDL= h}YwD Qo}O CQ=@a x@ "OwW|t h}YwDCU= K}LY OOa |Q}eDt xm n � 1 |=R= x@ I(i)(x1)I(i)(x2) : : : I(i)(xn)1 =} QiY Q@=Q@ I(i)(x) Q=Okt uwJ 'Ovm|t Q}}eD v |wQ x1; x2; : : : ; xn w

%Ovm|t sy=Qi =Q Cq=tDL= uOQm XNWt 'CU=PfI(i)(x1) = j1; I(i)(x2) = j2; : : : ; I(i)(xn) = jng (16)

p=Ft |=Q@ pkDUt TPCF xU =yvD Pij = Pij 'uQ=kDt |=y TPCF |=Q@xm |Dkw |R=i k Q=DN=U |=Q@ '|rm Cr=L QO "Ov=t|t |k=@ q=@ |R=i xU Q=DN=U|Dkw xm CU= K[=wQB "Ov=t|t |k=@ pkDUt TPCF ' 12 (k2 � k) 'K � 3

Pi = vi %OwW|t pY=L |}x]kvl} |oDU@ty `@=D Pii x=ou; r ! 0=@ u=wD|t =Q |Q=t; uoty Q=DN=U |=Q@ S2(r) = S(1)2 (r) |}x]kvwO `@=D|x@U=Lt w X=N |=DU=Q w r = jrj pw] =@ \NxQ=B xa]k l} |iO=YD uDN=Ov="�2 pmW� OQm x@U=Lt 'OQ}o|t Q=Qk 1 R=i uwQO \N |=yDv= \=kv xm |D=aiO O=OaDs=Hv=QU w OUQ|t r = 0 QO �i x@ S2(r) |xv}W}@ 'l}B wQDwR}= |=yQ=DN=U QO3 pmW "OUQ|t �21 Q=Okt x@ |}=tv CQwYx@ \NxQ=B pw] V}=Ri= |xt=O= =@Q=DN=U uwQO C=QP =} =y|Yr=N=v |=y|oS} w OQwt QO (S2(r)) |D=aq]= pt=WpmW w l}B wQDwR}= |xQP sDU}U |=Q@ |oDU@ty `@=wD R= xvwtv wO p=Ft |=Q@ &CU=Q=DN=U uwQO |=ylU}O xm pw= |xvwtv QO "CU= xOW xOQw; pmW u}= QO |Q=DN=U=t= CU= |v=Uwv lJwm |=yr |=Q@ S2(r) OvDUy sy R= =OH w OvQ=Ov |v=WwBsy

xm 'x=wNrO Q=DN=U l} |iO=YD Q=DN=U |=Q@ u}}=B x@DQt `@=D l}D=tW "2 pmWx@U=Lt �nvQO}iU |x}L=v� 1 R=i |=Q@ S2(r) = S(1)2 (r) |}x]kv wO p=tDL= `@=D

[25]"OwW|t

wO |=Q@ �nvQx=}U |x}L=v� 1 R=i |=Q@ S2(r) |}x]kv wO p=tDL= `@=D "3 pmWxvwtv� u=WwBsy Q}e |=ylU}O sDU}U |=Q@ '�1 = �2 = 1=2 QOhrDNt sDU}UlU}O Q]k D =Hv}= QO �u}}=B xvwtv� xOW |v=WwB sy |=ylU}O =@ sDU}U w �q=@

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Q@|QwQtx_Lqt CyH l} \ki Qo = "OvwW|t p}mWD CyH w xR=Ov= sQD wO R= |Q=t; `@=wD

|oDN=U |B wQDwR}=Q}e CU= umtt 'OW=@ |iO=YD |W}=tR; Q=DN=U Qo = |DL 'OwW|Q=t; `@=wD QO |QDW}@ |=yCyH wD=oQwD w 9u=y}W 'pmWt u}= pL |=Q@ "OW=@|xty "OQm|t hPL =Q |oDN=U |=y|B wQDwR}= Q}e |QF-wt Qw]x@ xm OvOQm |iQat|=Q@ O}OH VwQl} wD=oQwD "O};|tCUO x@ |Oa@ wO |=yTma R= |OwQw C=aq]=xaUwD |Oa@ wO C=a]k C=aq]= R= xO=iDU= =@ |iO=YD |Oa@xU p=mW= |R=UR=@|oDU@ty `@=wD w `@=D |]N x=Q |R=Uxv}y@ |=Q@ O} Q@}y |iO=YD |R=UR=@ VwQ "O=OQ=DN=U |Oa@xU |R=Ux}@W u}vJty "Ci=} xaUwD |vRxv=wH VwQ pw] QO |Oa@ wO|Q=t; C=aq]= [30]"OW xar=]t Vv=Q=mty w 10Q}|B \UwD |Oa@ wO |=yxkQw R=x@ uwQDmr= |oOvm =QB V=QB |=yR}r=v; R= p=DU} Qm CyH `} RwD w |U=vWpmW |wQO}rwD |=Q@ |OwQw u=wva x@ |y=oW}=tR; |Oa@ wO C=aq]= u}= R= "O};|t CUO|B wQDwR}= Q}e w xv=O |=yRQt pmW xm 'OwW|t xO=iDU= p=DU} Qm|rB |Oa@xU Q=DN=U

"Ovm|t ^=Lr =Q xv=O pmWxOW xU}=kt w p=ta= R}v hrDNt |R=Uxv}y@ |=yVwQ '|R=Ux}@W Ow@y@ |=Q@|=yQ=m Q@ xwqa "CU= xDW=O |QDW}@ OQ@ Q=m w G=wQ |iO=YD VwQ =yv; u}@ QO xm\UwD |DNwU pwrU Ov; Q=DN=U |R=UR=@ Q}_v |}=yQ=m QO R}v Qo}O u}kkLt 'wD=oQwD|R=UR=@ Q=m l} "Ov=xOQm xO=iDU= VwQ u}= R= [31]13|o =U=m w 12wv=m}W 11'wRwU|O=} R |=yxvwtv QO [32]"CU= xOW s=Hv= Vv=Q=mty w 14Owwrwi \UwD Qo}O ?=PHQO QDW}@ CkO |=Q@ "OvQ}o|t Q=Qk |DUQO x@ |r}rLD `@=wD |wQ |}x]kvwO `@=wDxO=iDU= `@=wD u}= VQDUo |=Q@ `} QU |x} Qwi p=kDv= R= Owwrwi '|}x]kvwO `@=wDV}=tv

"OQm=@ Q=DN=U |R=Ux}@W R= 'Q=DN=U |R=UR=@ |xv}tR QO |r@k |=yj}kLD QO ,=twta|xv}tRQO xOW s=Hv= |=yj}kLD QDW}@ "OW|t xO=iDU= |Q=t; C=aq]= R= xO=iDU=hrDNt |=yCyH QO pt=m |oDU@ty `@=wD xm |v=tR =D Ow@ |iO=YD Q=DN=U|=DU=Q =@ xm X=N CyH =@ `@=D l} \ki =yxvwtv QDW}@ QO "OvOW|tv |m}X=wN R= |}xOQDUo h}] u}vJty "CW=O OQ@ Q=m 'OW|t xO=O V}=tv |iO=YD|R=UR=@ |v=Um} |oDU@ty `@=wD R= xm |}=yQ=DN=U |=Q@ |DL xOW |v}@V}B QF-wtOwOLt Oa@ wO x@ Q=m QDW}@ 'u}= Q@ xwqa "CU= xOW xO=O K}[wD 'OvOw@ xOWQO xOW |R=Ux}@W Q=DiQ w =yQ=DN=UR} Q "Ow@v Oa@ xU x@ VQDUo p@=k w 'OW|t|=Q@ Qo}O p}rO "CW=O Oy=wN |Q=}U@ Cw=iD OvOw@ |Oa@xU xm |}=yv; =@ Oa@ wO|R=Ux}@W |=ypmW QO =y|oS} w R= |tm O=OaD R= xOW |R=UR=@ Q=DN=U |Q=O}=B=vO=} R |=ypmW OvtR=}v |Q@H |=yVwQ R= xO=iDU= =@ Q=DiQ |R=Ux}@W "O};|t xOW

"CU= sHL QoV}=tv |=yu=tr= |O=} R O=OaD =@j}kO |Q=DN=U uOQw; CUO x@ |=Q@ uoty=v O=wt |@ QHD w |OOa |R=UR=@"OQ}o Q=Qk xO=iDU= OQwt uoty=v O=wt |R=Uxv}y@ w |R=UxYNWt |=Q@ Ov=wD|tAwmUwQm}t RmQtDt uw} wDQB =} X |xaW= |i=QowDQB Ovv=t |O=} R |W}=tR; |=yVwQ|R=UR=@ |=Q@ u; R= u=wD|t xm OvQ=O OwHw (SEM/FIB) 15|W@ wQ |vwQDmr=R= |Q=t; C=aq]= '|OOa |=y|R=UR=@ |=Q@ "OQm xO=iDU= |Oa@xU |=yQ=DN=UR} Q=Q |Oa@xU |=yQ=DN=UR} Q Ov=wD|t w O};|t CUO x@ uoty=v |xO=t Q=DN=UR} Q

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16 X |xaW= |}xv=}=Q |Q=ovwDQB "1"4|R=UR=@ |=Q@ Ov=wD|t xm CU= ?QNtQ}e VwQ l} X |xaW= |}xv=}=Q |Q=ovwDQB"OQ}o Q=Qk xO=iDU= OQwt |Rri x}=B |=yC} RwBt=m Ovv=t 'wQm}t T=}kt QO uoty=v O=wt

pY=L |v}@V}B "CU= 1 =} QiY Q@=Q@ =y jk R= s=OmQy Q=Okt u; QO xm%xt=O= QO "CU= syt u}ov=}t l} I(i)(x1)I(i)(x2) : : : I(i)(xn)

S(i)n (x1; x2; : : : ; xn) = hI(i)(x1)I(i)(x2) : : : I(i)(xn)i= PfI(i)(x1) = 1; I(i)(x2) = 1; : : : ; I(i)(xn) = 1g (17)

x@ S(i)n "OvW=@ i R=i QO x1; x2; : : : ; xn |=yu=mt QO x]kv n xmv}= p=tDL= �x@ Ov=wD|t u}vJty &OwW|t xO=O `=HQ= i R=i |=Q@ |}x]kv n p=tDL= `@=D u=wva

[25]"OwW xO=O `=HQ= sy |}x]kv n |oDU@ty `@=D u=wva

|R=UR=@ "4Q=DN=U |Q=t; V}=tv |=Q@ xOW |R=Ux}@W Q=DN=U O=H}= '|Q=DN=U |R=UR=@ hOy|=Q@ 'R}v Qo}O |=yxv}tR QO "CU= |Q=DN=U |=yQoh}YwD R= |ywQo R= xO=iDU= =@R= pY=L *|Oa@ wO |=yTma R= |}xDUO R= |ak=w |Oa@ xU Q=DN=U l} uDN=U

"RmQtDt uw} wDQB |=yV}=tR; Ovv=t 'OwW|t xO=iDU= V}=tR;|xaUwD =@ |Q}otWJQw] x@ Q}N=|=yxyO QO Q=DN=U|R=UR=@ |=yVwQC}ty==@ |}=yQ=DN=U |=Q@ |[=} Q |=ypL x=Q "CU= xDi=} V}=Ri= uwo =vwo O=wt |L=Q]Q=Qk |=yu=tr= R=|ywQo R= xO=iDU= =@ Q=DN=U R=?U=vt|W}=tv OvtR=}v u}at X=wNOvv=wD|t xm |}=yQoh}YwD |v@ Qm |xrwrwv=v C} RwBt=m QO "CU= Q=DN=U uwQO xDiQo'=yxrwrwv=v pw] '=yxrwrwv=v |tHL C@Uv pt=W OvyO V}=tv |@ wN x@ =Q Q=DN=UQ=DN=U uwQO =yxrwrwv=v `} RwD |xwLv w =yxrwrwv=v |=vLv= '=yxrwrwv=v Q]k x@ pw]C@Uv'|oDU@ty |Q=t; `@=wD |=y?}DQD xrUrU QO CU= umtt =yQoh}YwD u}= "CU=xm O=wt |=yQoh}YwD =@ Q=DN=U C}=yv QO "OvwW xQ}NP |}x]kvOvJ |Q=t; `@=wD =}

"OwW|t xO=O V}=tv OwW|t xO=iDU= xOW xDio |=y|oS} w R=|=Q@ xm CU= |}=y|R=Ux}@W |=QH= 'Q=DN=U |R=UR=@ |=Q@ Qo}O |xR}ov= l}|Q=t; `@=wD Ovv=ty |}=yQoh}YwD R= [29�26]"CU= |QwQ[ X=wN uOQw; CUO x@,=t}kDUtCU= umtt|}=tQo w|m} QDmr= '|U}]=vet '|m}v=mt X=wN '|}x]kvOvJR= xO=iDU= =@ =yQoh}YwD Q=DN=U |R=UR=@ QO Qo}O |xDmv "OwW s=Hv= |R=Ux}@Wu}= R= "OwW |R=Ux}@W CU= umtt QF-wt X=wN xm CU= u; OwOLt |=RH= VwQVkv xm OwW xO=iDU= xOW |R=Ux}@W |=yQ=DN=U O}}-=D |=Q@ CU= umtt VwQ

"OQ=O O=wt |L=Q] QO |rY=QDW}@ 'CU= xOW p=@vO |R=Ux}@W |=yVwQ R= |O=} R Q=}U@ O=OaD =@ |R=UR=@'|tHLC@Uv 'xv=O xR=Ov= Ovv=ty '|}x]kvl} `@=D |x}=B Q@ |Q=t; C=aq]= R= =y u;

"Ov=xOQm xO=iDU= Qo}O OQ=wt w |iO=YD Q=DN=U 'p;xO}= pmWx_Lqt xUOvy w pmW '=yCyH `} RwD '|B wQDwR}= Q}e |=yxYNWt C=aq]='OW=@ xOW x_Lqt |Q=t; `@=wD QO CU= umtt =y|oS} w u}= xJQo = 'CU= xOWv

"Ovm|t |iQat R}v =Q |DqmWt u=tRty =t= CU= syt |Q=t; `@=wD R= |R=Ux}@WẁQW |Q=t; `@=wD R= xO=iDU= =@ =Q =yQ=DN=U |R=UR=@ |xar=]t [25w21]8wD=mQwD|iO=YD |R=Uxv}y@ =@ u}}=Bx@DQt |Q=t; `@=wD R= |iO=YD uoty=v |=yQ=DN=U "OQmxm OW xO=O u=Wv |Oa@ wO w |Oa@l} |=yQ=DN=U R= p=Ft u}OvJ "OvOW |R=UR=@|Ov@pwtQi l} "OyO|t u=Wv =Q lJwm h}] x@DQt |=Q@ |R=UR=@ VwQ C}r@=k`@=wD xm O=O u=Wv u}vJty wD=oQwD "CU= xOW u=}@ |R=UR=@ sD} Qwor= R= j}kO|[=} Q|=y?=wH =Pr w "OvW=@ uoty=v |R=iwO xO=t Qou=Wv Ovv=wD|tv |}=yvD x@ |}x]kvwO"Ow@ Oy=wNv u=Um} 'OvW=@ u}}=B x@DQt |Q=t; `@=wD \ki x}rw= \}=QW Qo = '|R=UR=@

34

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"CU= |Q=t; |x} Q_v |x@U=Lt |=Q@ |O}rm '`} RwD QDt=Q=B Qt}rB|vwQDmr= AwmUwQm}t R= u=wD|t 'C=QP |oOvm =QB OQwtQO C=aq]= uDW=O |=Q@R}r=v; |xvwtv QO xJQo = [43]"OQm xO=iDU= x |xaW= |oOvm =QB =}(TEM) 19|Qw@aC=YNWt |xty xm OQ=O ?wr]t |xH}Dv |v=tR \ki TEM |=yTma 'TEMu=Um} |oOvm =QB 'pta QO [44]"OW=@ xOW xO=O V}=tv u; QO C=QPwv=v |oOvm =QBC=QPlD R= |]wrNt =@ Q=DN=UR} Q 'OQwt u}= QO "O};|tv CUO x@ ,qwtat C=QPwv=vxm CU= xHwD p@=k "CU= xDi=} pmW CU= xQP l} R= QDW}@ pt=W xm |}xDUO w|xR=Ov= x@ |oDU@ xm OUQ@ QDtwv=v OYOvJ x@ CU= umtt C=QPwv=v |xDUO xR=Ov='u}= Q@ xwqa "OQ=O xQPwv=v w xv}tR u}@ |}=}t}W VvmsyQ@ w Ov}=Qi VwQ 'xQPwv=v'TEM VwQ =@ xOt; CUO x@ |=yTma |=Q@� xOW x@U=Lt |oDU@ty `@=wDxOW x=Q= C=L}[wD j@=]t =Q |}x]kvwO w |}x]kvl} |oDU@ty `@=wD C=aq]=C@F TEM |=yTma |}=tvnQR@ x@ COW x@ �OQw; CUO x@ u=wD|t 2 pmW |=Q@|xO=t R= |rLt x@ |oDU@ty `@=D 'O=} R |}=tvnQR@ R= xO=iDU= "OQ=O |oDU@ xOW|=yTma =}; xm u}= p=Ft |=Q@� Ov=xDiQo |B wmUwQm}t |=yTma xm CU= uoty=vKw[w 'Qo}O CQ=@a x@ [6]"�xv =} CU= C=QPwv=v |=yxDUO pt=W |@=NDv= TEM?=NDv= Tma R= |r}N xOv}=tv �sHL =}� CL=Ut =t= OW=@ q=@ Ov=wD|t Q} wYDxOvyOV}=tv C=aq]= 'sm |}=tvnQR@ R= xO=iDU= 'Tmax@ [45]"OW=@ QDnQR@ xOWCU= O=} R =yQ=t; OQwt u}= QO "OyO|tCUO x@ C=QPwv=v |oOvm =QB |xQ=@ QO |QDW}@

[37]"X |xaW= |Q=ovwDQB pwY= "4 pmW|Oa@ wO Q} w=YD w xOQm OQwNQ@ u=NQJ |xvwtv x@ X |xaW= |=ywDQB 'VwQ u}= QOQO [37w36]"�4 pmW� OvwW|t C@F xvwtv Qo}O hQ] QO QoQ=mW; l} R= xO=iDU= =@|Oa@ wO Q} w=YD ?}mQD =@ |Oa@xU |R=UR=@ �|}=Q}t |Q=ovwDQB� l}Uqm |Q=ovwDQB

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35

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I(~h) =ZZZ

~�2(r)e�ihrdV (24)

|r=oJ u=O}t Q=DN=U =@ OQi x@ QYLvt Qw] x@ 'h QO COW `} RwD xm CU= QmP p@=k%OQm C@=F [49w48] 21|=@O '|Q=t; |B wQDwR}= |x_Lqt =@ "O};|t CUO x@

he�ihri = sin(hr)hr

(25)%R= Ow@ Oy=wN CQ=@a u}ov=}t |oOvm =QB COW 'xH}Dv QO

I(h) =Z

4�r2 ~�2(r) sin(hr)hr

dr (26)

%OQm |U} wvR=@ u}vJ u=wD|t =Q 26 |xrO=at ' '|oDU@tyOwN `@=D R= xO=iDU= =@I(h) = V n20

Z4�r2:(r) sin(hr)

hrdr (27)

%=} 'CU=yuwQDmr= u}ov=}t |r=oJ n0 u; QO xm

(r) = 12�2V n201Z

0I(h) sin(hr)

hrh2dh (28)

28 |xrO=at u=wD|t '21 |xrO=at R= xO=iDU= =@ "CU= C@=F Q=Okt l} n0 =Hu}=%OQm |U} wvR=@ =Q

(r) = P 112 (r)� �21= 12�2V n20

1Z0I(h) sin(hr)

hrh2dh (29)

C=QP |}=[i `} RwD xm OyO|t u=Wv =Q |}x]kvwO p=tDL= |oDU@ty `@=D P 112 (r) w"Ovm jOY Q} R \}=QW QO O}=@ P 112 (r) "OyO|t u=Wv =Q �1 R=i�

P 112 (r) = �1 when r = 0P 112 (r) = (�1)2 when r !1 (30)

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x@ |=Q@ X |xaW= C=W}=tR; G}=Dv R= u=wD|t 30 =D 18 CqO=at R= xO=iDU= =@=yQ=DN=Uwv=v wQm}t|R=UR=@ QO xm OQm xO=iDU=|}x]kvwO|oDU@ty`@=wD uOQw;CUO

"CU= C}ty= R=L Q=}U@

|Q=t; |=yQoh}YwD R= xO=iDU= =@ |R=UR=@ "6`@=wD� OwOLt |Q=DN=UR} Q C=aq]= R= xO=iDU= =@ uoty=v O=wt |iO=YD |R=UR=@|R=UR=@ |=yVwQ "CU= |UOvyt QO ?=PH Twmat p=Ut R= |m} �|oDU@tyQO [55w34]"CU= xDi=} xaUwDu}}=B x@DQt|oDU@ty `@=wD R= xO=iDU= u=mt= |=Q@|irDNtxaUwD 22CNB R=@ |R=Uxv}y@ VwQ =@ xm |R=UR=@ |=yVwQu} QDQwyWt R=|m} 'xt=O=u=wD|t |oDU@ty `@=wD R= |ywQo R= xO=iDU= =@ "OQ}o|t Q=Qk EL@ OQwt CU= xDi=}Ov=wD|t C=aq]= u}= 'OQw; CUO x@ =Q |iO=YD uoty=v O=wt C=aq]= R= |DtUk

|oOvm =QB Cr=L 'VwQ u}= |=Q@ |v} Ro}=H u=wva x@ "CU= u}}=B Q} wYD Kw[w |rwx |xaW= |oOvm =QB |Q}oxR=Ov= =@ Ov=wD|t Qt}rB |x}=B |=yC} RwBt=mwv=v QO C=QPwv=v`} QU w u=U; VwQ l} SAXS [46]"OwW |R=UxYNWt(SAXS) lJwmx} w=RuO=O CUO R= uwO@ �q=@ Q=t;� ?amt QDt|r}t OvJ O=Oa@= x@ |tHL x@ xm CU=`} RwD |xR=Ov= 'C=QPwv=v R= xOt; CUO x@ |oOvm =QB p=vo}U "OwW|t p=ta= Kw[w=Q �Q=DN=U QwDm =i� Qo}Om} x@ C@Uv =yu; C}akwt w �QwDm =i R=� C=QPwv=v pmW w|oDU@ty `@=wD |x@U=Lt |=Q@ ,=@k=aDt u=wD|t SAXS p=vo}U R= "OyO|t u=WvOvm|t O}rwD xO=t |SwrwiQwt R= |k}kO Q=}U@ V}=tv uwJ � q=@ CkO =@ |}x]kvwO

[50�47]"OQm |Q=OQ@xQy@

|oDU@ty `@=wD w SAXS |=yxO=O u}@ \=@DQ= "5|}x]kvwO

C=QP R= xOW xOvm =QB |r=oJ x@ |mDt lJwmx} w=R x |xaW= |oOvm =QB l}vmD1000 =D 1 u}@ ,qwtat 'C=R}yHD pmW x@ xDU@ '=yu; |xR=Ov= xm CU= uoty=v|vwQDmr=|r=oJ u}@ hqDN= x@|oDU@ xOW xOvm =QB|r=oJ [53�51]"OQ=O Q=Qk QDtwv=vu}ov=}t |r=oJ =@ Ov=wD|t xm 'OQ=OVi=Q]= w xOW xOvm =QB |=y|voty=v R= � |rLt%OQm h} QaD u}vJ u=wD|t =Q uwQDmr= |r=oJ |=Q@ � |rLt Q}}eD "OwW xO=O u=Wv �� = �� (18)xm =Q |oDU@ty `@=D '`}Uw u=OvJ xv |x@DQt =@ |Q=t; l}B wQDwR}= sDU}U ZQi =@

%OQm h} QaD u}vJ u=wD|t 'Ovm|t x_Lqt =Q |r=oJ |=yu=Uwv |xvt=O

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%CU= r C}akwt x@ xDU@=w

(r) = h�(r1)�(r2)i (20)

(r = 0) = �2 \}=QW (r) |oDU@tyOwN `@=D '=y|voty=v |iO=YD `} RwD R=sDU}U |=Q@ 1 R=i 20Tv=} Q=wwm h} QaD "Ovm|t =[Q= =Q (r ! 1) = 0 w

[1]%R= CU= CQ=@a |Q=t; uoty

(r) = h�(r1)�(r2)i = P 112 (r)� �21 (21)"CU= |}x]kvwO p=tDL= `@=D P 112 w �=yxOvvmQB� 1 R=i |tHL C@Uv �1 u; QO xmC}akwt QO dV sHL u=tr= &CU= OL=w sHL QO =yuwQDmr= O=OaD �(r) u}vJtyu=wva x@ I 'x |xaW= |oOvm =QB COW "Ow@ Oy=wN uwQDmr= �(r)� dV pt=W r

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ZZZ ZZZ�(r1)�(r2)e�ihrdV1dV2 (22)

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%OQm h} QaD u}vJ u=wD|t~�2(r) =

ZZZ�(r1)�(r2)dV1 (23)

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"CU= xOW xO=O u=WvQ=Ri=sQv R= xO=iDU= w 'VwQ u}= QO xOW xDiQo Tma CYW R= xO=iDU= =@ TBUx@ xOW O}rwD |Oa@ wO |=yTma uDUw}B sy x@ R= xm |Oa@xU |rOt VCAT

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"OvDUyEL@ OQwt |}x]kv n w |}x]kv wO '|}x]kv l} |oDU@ty `@=wD `=wv= u}vJtyR= uO}UQ |=yx=Q xt=O= QO "OW sUQ u; |OQ@ Q=m pmW s=Om Qy |=Q@ w CiQo Q=Qk

Qw]x@ "OQ}o Q=Qk xO=iDU= OQwt |iO=YD uoty=v O=wt |R=UxYNWt w |R=UR=@ |=Q@R=|YNWt O=OaD =@ Q=DN=Uwv=v wQm}tl} O}rwD x@p}=t =t |R=UR=@ VwQl} QO|rmOv=wD|t|iO=YD uoty=v O=wt R= |OOa |R=UR=@ "s}DUy|}x]kvwO|oDU@ty `@=wD"OwW xO=iDU=|iO=YD xOW O}rwD Q=DN=Uwv=v/wQm}tl} |R=Uxv}y@ pmWt pL |=Q@xm CU= |R=Uxv}y@ VwQ l} CNB R=@ VwQ R= xO=iDU= =@ 23wrQ=mCvwt |R=UR=@VwQ u}= QO [55 w 34'21]"OQ}o Q=Qk xO=iDU= OQwt uoty=v O=wt |R=UR=@ |=Q@ Ov=wD|tQ_v OQwt |xvwtv =@ u=Um} |tHL C@Uv =@ |iO=YD |=yTma pw= |xrLQt QOR=i Qy pUm}B uOQm =Hx@=H |=Q@ CNB R=@ |R=Uxv}y@ VwQ TBU &OwW|t O}rwDQ=Qk Q_v OQwt |xvwtv w xOt;CUO x@ pOt |oDU@ty `@=wD u}@ =]N |R=Uxv}tm |=Q@|iO=YD |Ov@ Qm}Bl} 'OwW hOy |xvwtv x}@W|}x]kvl} `@=D xm|v=tR =D "OQ}o|t|}xQwO |RQt\}=QW uO=O Q=Qk Q_v OQwt =@ x}rw= |=tOl}TBU "OwW|t O}rwD x}rw==@|oDU@ty `@=D G}=DvTBU w x@U=Lt |Ov@ Qm}B u}= |=Q@|oDU@ty `@=D w '?=NDv=?=NDv= |iO=YD Qw] x@hrDNt |=yR=i R= pUm}B wO "OwW|t xU}=kt hOy |xvwtvTBU "OwW^iL R=i Qy|tHLC@Uv xmu}= R= u=v}t]= =@ xD@r= 'OwW|tZwa Oa@ wx]w@ Qt|oDU@ty `@=wD =@ 24u}ov=}t `@ Qt |=]N w 'x@U=Lt u=Um}|oDU@ty `@=wD|=Q@ VQ}PB Q=}at u=wva x@ 25T}rwB wQDt sD} Qwor= VwQ u}= QO "OwW|t xU}=kt%CU= pUm}B Z} waD |=Q@ VQ}PB p=tDL= P w OwW|t ?=NDv= =ypUm}B Z} waDP (Errorold ! Errornew)8

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=yCWwv=B1. statistical N-point correlation functions2. two-point correlation functions3. small angle X-ray scattering4. transmission/scanning electron microscopy5. weak-contrast, strong contrast6. two point correlation function7. Stell8. Torquato9. Sheehan10. St-Pierre11. Suzue12. Shikaono13. Kasagi14. Fullwood15. focused ion beam/scanning electron microscopy16. X-Ray computed tomography17. scattering electron microscopy18. morphology19. transmission electron microscopy20. covariance21. debye22. annealing optimization approach23. Monte Carlo24. mean square error25. Metropolis

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40

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