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A formalism on multiple small-angle scattering
Physica B 174 (1991) 200-205 North-Holland
A formalism on multiple small-angle scattering S. M a z u m d e r and A. Sequeira Solid State Physics Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India
The present paper reports on a formalism of multiple small-angle scattering (SAS) with a view to examining the nature of the multiple scattering profile, the nature of the extractable information from the multiple SAS experiment, and the effect of the statistical nature of the medium on multiple scattering profile.
1. Introduction One of the major applications of multiple SAS lies in enabling the study of very large size inhomogeneities which are otherwise inaccessible to a measurement because of resolution constraints of the instrument. It may be recalled [1] that the integrated scattering cross section for an inhomogeneity is proportional to the fourth power of the linear dimension of the inhomogeneity. As a result, the scattering mean free path of the radiation inside the matrix falls sharply with the increase of the linear dimension of the inhomogeneities when all other parameters like scattering density contrast and wavelength of the radiation remain fixed. So for a sample of very large size inhomogeneities, the scattering mean free path of the radiation may become comparable to the linear dimensions of the inhomogeneities. In such situations the detailed evaluation of multiple scattering effects, taking account of the statistical nature of the medium becomes a necessity since the scattering media are quite often polydisperse in nature. In the present paper, we report on a formalism for multiple SAS from a statistical medium of the aforementioned type.
2. Theory It is assumed that the sample under study consists of a large variety of scattering particles, the individual interaction of the incident radiation producing either small-angle scattering or absorption of the projectile. Each variety of scatterer is distinct in its scattering and absorption nature, characterised by the single particle differential scattering cross section and the macroscopic absorption cross section, respectively. The volume fraction of the ith type of scatterer in the sample is denoted by Pi while V~, Ri, ff~, Pi, F~ and ~i are, respectively, its volume, linear dimension, reciprocal of average chord length, number density, differential scattering cross section and macroscopic absorption coefficient. The subscript i runs from 1 to n and n indicates the degree of polydispersity of the medium. The direction of motion of the projectile at any space point is represented by a unit vector u while u 0 represents a unit vector in the incident direction. The point of incidence of the beam on the sample is assumed to be the origin of the cartesian coordinate system whose Z-axis is assumed to be parallel to u 0. The polar angle between u and the Z-axis is denoted by O and the azimuth of u by ~b. As the scattering processes are 0921-4526/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
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assumed to be elastic in nature, the momentum transfer vector q is given by q = k(u - Uo) where k is the w a v e number of the radiation with wave length A. The problem of multiple scattering, in its generality, involves the evaluation of the spatio-angular distribution of the scattered radiation as a function of the penetration depth, Z, of the radiation in the sample. Since, the scattering mean free path of the radiation is orders of magnitude higher than the de Broglie wavelength of the radiation, the interscattering interference (as distinct from the usual interparticle interference) can be neglected and the radiation (-projectile) can be treated as a classical particle as far as the description of its passage between two scattering events is concerned. We define F~(X, Y, uIZ) as the joint probability density distribution function indicating the joint probability, /7(X, Y, u ] Z ) d X d Y d u , of finding the ith type of scatterer around the point (X, Y, Z) and the projectile in the hyper-surface element d X d Y d u around the point (X, Y , Z , u ) . The spatio-angular evolution o f / 7 can be expressed [2] in terms of system of coupled integro-differential equations of the following type: cos O(OFJ OZ ) + sin 0 cos ~o(OF~/ OX) + sin 0 sin ~o(OF~/OY) + lz~F~ : (1/V~) f [F,(X, Y, u']Z) - F,(X, V, ulZ)lFii(u - u') du' - ~,/7 + p, ~ ~'i(1 - p j ) - ' F j , j=l
j¢i
with
(1)
F~(X, Y, ulO) = pi6(X)6(Y)6(u - Uo). Adding the above system of coupled equations, we obtain: cos O( OF/ OZ ) + sin 0 cos q~(OF / OX ) + sin 0 sin ~p(OF/ OY) + ( tt) F
= ~ f [F(X, Y , u ' I Z ) - F(X, g, ulZ)]QiFi(lu-u'lldu' i=1
+ ~_, f [Q,j(X, Y, u'lZ) - Q,j(X, Y, (ij)
-- E
(6)
ulz)l[(1/Vilr,(lu
-
u'l)
- ( 1 / v j ) ~ ( l u - u'l)l du' (2)
(~d~i -- I ' t j ) Q i j ,
where F = Ei"=1 17. is the spatio-angular density distribution function of the radiation in the matrix and gives the probability F(X, Y, u l Z ) d X d Y d u of finding the projectile in the hyper-surface element d X d Y d u around the point (X, Y, Z, u), {/~) = Ei~l pil.ti. The last two summations in eq. (2) are to be carried out over all possible pairs (i, j) and each of these summations will generate n(n - 1)/2 terms. The quantity Q;j = pjF i - p f j is a measure of the effect of the statistical nature of the medium corresponding to its two-state (i, j) distribution on multiple scattering profile vis-a-vis that from a statistically averaged medium which we term as 'effective medium'. For an effective medium, the linear dimensions of the scatterers are negligible in comparison with the mean free path of the radiation and Q~/tends to zero [3] so that eq. (2) reduces to cos O(OFI OZ) + sin 0 cos q~(OF/ OX) + sin 0 sin ~o(OF/ OY) + (Ix)F
= i=1 E f[F(X, r, ,,'lz)-
F(X,
Y, ull)]o rii(lu-u'l)du'.
(3)
S. Mazumder, A. Sequeira / A formalism on multiple small-angle scattering
202
A detailed analysis [1, 3-5] of the above equation shows that the nature of the extractable information from the Guinier regime of the scattering profile is (R 3 ) / ( R 2} (={Ei~ 1 (Q~R~3 )} /{2;= n 1 (p,R/2)}) and not (R4)/(R 3) as suggested by Berk and Hardman-Rhyne [6]. An asymptotic analysis on eq. (3) to delineate the nature of the Porod regime of the multiple scattering profile reveals [2] that q---~limV(qlZ)=exp[-(i.~}Z
O~~im Fi(q)/
l-exp(-N)] i=1
p~Fidu
(4)
"i=1
where N is the average number of scattering interaction the radiation has undergone while passing through the sample of thickness Z. Equation (4) indicates that the Porod law, as far as the functional dependence of the intensity on the momentum transfer vector is concerned, remains invariant under multiple scattering from an effective medium. However, the presence of the factor [1 - e x p ( - N ) ] in the right-hand side of eq. (4) indicates that the effect of multiple scattering is to increase the scattered intensity in the Porod regime of the profile. If the scatterers are assumed to be spherical with a scattering length density contrast D then for a polydisperse effective medium eq. (4) reduces to: lim F( qIZ )
q--..~e¢
= exp[-(
~>Z][1
-
exp(-N)](4'tr/AZ)( ( RZ) / ( R4) )q -4 ,
indicating that the nature of the extractable information from the Porod regime of the multiple scattering profile is given by (R 2) /(R4). Under single scattering approximation (N ~ 0), eq. (4) can be approximated to lim
q--* w
F(qIz) = e x p [ - ( p. ) ZI(6~rD2zZ)( ( R 2) / ( R3) )q -4 ,
where ~- denotes the packing fraction of the sample. The above relation indicates that the extractable information from the single scattering profile is given by (R 2) /(R3). When the linear dimensions of the inhomogeneities are not negligible compared to the mean free path of the radiation in the sample under study, the statistical nature of the medium, as seen by the radiation, becomes manifest and cannot be ignored. In such a situation the radiation transport in the medium can be described in terms of system of coupled integro-differential eq. (1). The angular distribution of the multiply scattered radiation from a bidisperse (n = 2) statistical medium can be obtained from the XY-averaged solutions of eq. (1) and is given by
F(u[Z) = ~ Fi(u]Z ) = (2"rr)-2 i=1
fg
i=1
/~i e x p ( - i r , u) dr = (2"n')-2
Y
]~ e x p ( - i r • u) d r ,
where
ffi = f Fi exp(ir- u) d u , = C,{(W1 + Az + ~1 + ~2 +/-~1)/~2} exp(W1Z) + C2{(W2 + A~ + ~ + ~2 +/~t)/~2} exp(WEZ),
(5)
S. Mazumder, A. Sequeira / A formalism on multiple small-angle scattering
203
= f F/exp(ir. u) d u , W1
=
-{(A~ + A 2 + ~r1 + ~"2"It-/--£1"+ /z2)/2} + [{(A~
W2 = - { ( A 1 +
AE
+ ~'1 + {z +/Xa +/az)/2} - [{(A1
C1 = { ~'2P2 - (W2 + A1 + ~ra +
lZa)Pa}/(W~
- A 2 + ~ - {2 + I ~ - t ~ ) z
+4{i~2)1/2]/2,
- A 2 + ~, - ~z + lz~ - laz) z + 4 { ~ { z ) ~ / z ] / 2 ,
- Wz),
and C2 = P l - C1 • The above expression for the angular distribution of the multiply scattered profile is quite involved, unlike the case for an effective medium, and no simple expression for the nature of the extractable information from the multiple scattering profile can be obtained. However, an asymptotic analysis on F ( u [ Z ) as represented by eq. (5) reveals that the Porod law would remain valid irrespective of the statistical nature of the medium as far as the functional dependence of the scattered intensity on momentum transfer vector is concerned. When the scattering medium is purely of absorbing type (A i = 0), eq. (5) reduces to F = F 1 + F 2 = [(/3, - Y ) e x p ( - f l , Z) + ( y - 132)exp(-flzZ)l/(fl
I - f12),
(6)
where Y = ~ ' , + ~ 2 +p2/z, +p,/~2'
]3, = ½ { ( ( / z ) + Y ) + [ ( ( P - ) - Y ) 2 + 4 a ] l / Z } ,
132 = l { ( ( u , ) + v) - [ ( ( u , ) - 3') 2 + 4 a ] " 2 } ,
a =p,p2(/Zl - ~ ) 2 .
The solution is same as the one represented by eq. (118) in the paper by Levermore et al. [7]. The above solution is the general solution of the problem of radiation transport in a bidisperse 'absorptiononly' Markov medium. Under 'small fluctuation limit' (Pl =P2), eq. (6) simplifies to F = [(/3, -/7,) exp(-13,Z) + ( ~ -/32) exp(-/32Z)]/(/~, - 132),
(7)
where = ~1 + ~2 + p , lz, + p z l x 2 ,
/3, = ~ {((,u,) +/.~) + [((/x) - ~)2 + 4a1~/2},
~2 ~ 1 {((],/,) .}_/,~) _ [((~u,) - ]~)2 + 4a]l/Z},
ot = p l p 2 ( l ~ 1
- tz2) 2 .
The solution is exactly the same as that (eq. (126)) obtained by Levermore et al. [7] under the head 'small fluctuation theory'. Levermore et al. [7] took a different approach for describing radiation transport in 'absorption-only' random medium. Their approach cannot account for the scattering nature of the medium. However, the present generalised formulation for radiation transport in a random medium accounts also for the scattering nature of the medium besides correctly describing the radiation transport in an 'absorption-only' medium.
S. Mazumder, A. Sequeira / A formalism on multiple small-angle scattering
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When the scattering medium is such that (A 1 + ~1) and (A: +/z2) become comparable, eq. (5) can be approximated as /~exp
I - ½ Z ~2 (A, +/.t,) 1 .
(8)
i=1
The above approximation, we will term as small noise approximation (SNA). It can be shown that under SNA, the expression of F for a polydisperse statistical medium is given by:
~.exp[_(Z/n) ~ (Ai + lxi)] .
(9)
i=1
If we assume F~ to be the Guinier approximate scattering cross section for a spherical scatterer of radius Ri, we can write from eq. (9), F ( q I Z ) : n/{5"rr ~
(Ni/piR~)}exp[-nq2/{ 5 ~ (Ni/piR~)}] exp[-(Z/n) ~ txi],
i=1
i=l
(10)
i=l
where Ni(= Z/L~) is the average number of scattering interactions the radiation has undergone with ith type of inhomogeneity while passing through the sample. The expression of F(qlZ) corresponding to eq. (10) for an effective medium can be written as F(qIZ)-1/{5"n ~
(Ni/R~)}exp[-q2/{ 5 ~
i=1
(N~/R~)}]
exp[-Z ~
i=1
P~"~I"
(11)
i=1
It is clear from the above two expressions of F(qlZ) that the nature of the extractable information from the Guinier regime of the multiple scattering profile is n/{5 E~=1(N~/piR~)} in contrast to {5 ET=1 (Ni/R~)}-~ = 2(R 3)/15ZzA2D2(R2) for an effective medium. An asymptotic analysis on eq. (9) shows that
q~
~
[1-exp{-(l/n)
i= l
~ (Ni/P i) i=1
Qi ~im 1]i(q)/pi ) -
Qil~i d u
)// Pi
(12) •
=
From eq. (12), it is clear that the extractable information for a statistical medium from the Porod 2 n (oiR~/pi)}/{E~=l (oiRi/pi)} in contrast t o {Ei= regime of the profile is given by {E~=I 4 n 1 (QiR~)} / n 4 {Ei= 1 (oiRi)} in the case of an effective medium. A comparison between the theoretical expression of a multiple scattering profile from a statistical medium with those obtained invoking the approximations, EMA and SNA, reveals that the two approximations have opposite effects as far as their modulation effect on the scattering profile is concerned. For a bidisperse scattering medium, when ( P 2 - Pl)(A1 - A2) is positive, the theoretical profile which invokes EMA will be narrower while the one invoking SNA will be broader vis-a-vis that from a statistical medium without invoking the above approximations. For a polydisperse medium, the modulation effect of the aforementioned approximations on multiple scattering profile is decided by the
S. Mazumder, A. Sequeira / A formalism on multiple small-angle scattering
205
fact that whether the quantity E(ij) (Pi - p j ) ( A j - Ai) is positive or negative. The modulation effect of EMA and SNA can cause erroneous estimation of the extracted parameter. For example, when the quantity E,j) ( p i - pj)(Aj - Ai) is positive, then (R a) / ( R 2) extracted from the Guinier regime of the profile will be underestimated while the extracted value of [n/(E~=~ (N~/p~R~)}] -~ will be overestimated. The seriousness of this erroneous estimation of the extracted parameters depends upon the extent of deviation of the quantity E(~j)(p~- pj)(Aj - Ai) from zero.
3. Conclusion
We conclude that small-angle scattering studies need not be restricted only to thin samples. Some useful structural information about the inhomogeneities can be extracted even using thick samples. We suggest that the broadening of the small-angle scattering profile due to multiple scattering effects in thick samples can, in principle, be exploited to characterise large size inhomogeneities using even relatively low-resolution instruments and in that pursuit, the present formalism would be of significant help.
References [1] [2] [3] [4] [5] [6] [7]
S. Mazumder and A. Sequeira, Phys. Rev. B 41 (1990) 6272. S. Mazumder, Ph.D. Thesis, Bombay University (1990). S. Mazumder and A. Sequeira, Phys. Rev. B 39 (1989) 6370. S. Mazumder and A. Sequeira, Pramana-J. Phys. 30 (1988) 537. S. Mazumder and A. Sequeira, Material Science Forum 27 & 28 (1988) 407. N.F. Berk and K.A. Hardman-Rhyne, J. Appl. Cryst. 18 (1985) 467. C.D. Levermore, G.C. Pomaning, D.L. Sanzo and J. Wong, J. Math. Phys. 27 (1986) 2526.
A formalism on multiple small-angle scattering S. M a z u m d e r and A. Sequeira Solid State Physics Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India
The present paper reports on a formalism of multiple small-angle scattering (SAS) with a view to examining the nature of the multiple scattering profile, the nature of the extractable information from the multiple SAS experiment, and the effect of the statistical nature of the medium on multiple scattering profile.
1. Introduction One of the major applications of multiple SAS lies in enabling the study of very large size inhomogeneities which are otherwise inaccessible to a measurement because of resolution constraints of the instrument. It may be recalled [1] that the integrated scattering cross section for an inhomogeneity is proportional to the fourth power of the linear dimension of the inhomogeneity. As a result, the scattering mean free path of the radiation inside the matrix falls sharply with the increase of the linear dimension of the inhomogeneities when all other parameters like scattering density contrast and wavelength of the radiation remain fixed. So for a sample of very large size inhomogeneities, the scattering mean free path of the radiation may become comparable to the linear dimensions of the inhomogeneities. In such situations the detailed evaluation of multiple scattering effects, taking account of the statistical nature of the medium becomes a necessity since the scattering media are quite often polydisperse in nature. In the present paper, we report on a formalism for multiple SAS from a statistical medium of the aforementioned type.
2. Theory It is assumed that the sample under study consists of a large variety of scattering particles, the individual interaction of the incident radiation producing either small-angle scattering or absorption of the projectile. Each variety of scatterer is distinct in its scattering and absorption nature, characterised by the single particle differential scattering cross section and the macroscopic absorption cross section, respectively. The volume fraction of the ith type of scatterer in the sample is denoted by Pi while V~, Ri, ff~, Pi, F~ and ~i are, respectively, its volume, linear dimension, reciprocal of average chord length, number density, differential scattering cross section and macroscopic absorption coefficient. The subscript i runs from 1 to n and n indicates the degree of polydispersity of the medium. The direction of motion of the projectile at any space point is represented by a unit vector u while u 0 represents a unit vector in the incident direction. The point of incidence of the beam on the sample is assumed to be the origin of the cartesian coordinate system whose Z-axis is assumed to be parallel to u 0. The polar angle between u and the Z-axis is denoted by O and the azimuth of u by ~b. As the scattering processes are 0921-4526/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)
s. Mazumder, A. Sequeira / A formalism on multiple small-angle scattering
201
assumed to be elastic in nature, the momentum transfer vector q is given by q = k(u - Uo) where k is the w a v e number of the radiation with wave length A. The problem of multiple scattering, in its generality, involves the evaluation of the spatio-angular distribution of the scattered radiation as a function of the penetration depth, Z, of the radiation in the sample. Since, the scattering mean free path of the radiation is orders of magnitude higher than the de Broglie wavelength of the radiation, the interscattering interference (as distinct from the usual interparticle interference) can be neglected and the radiation (-projectile) can be treated as a classical particle as far as the description of its passage between two scattering events is concerned. We define F~(X, Y, uIZ) as the joint probability density distribution function indicating the joint probability, /7(X, Y, u ] Z ) d X d Y d u , of finding the ith type of scatterer around the point (X, Y, Z) and the projectile in the hyper-surface element d X d Y d u around the point (X, Y , Z , u ) . The spatio-angular evolution o f / 7 can be expressed [2] in terms of system of coupled integro-differential equations of the following type: cos O(OFJ OZ ) + sin 0 cos ~o(OF~/ OX) + sin 0 sin ~o(OF~/OY) + lz~F~ : (1/V~) f [F,(X, Y, u']Z) - F,(X, V, ulZ)lFii(u - u') du' - ~,/7 + p, ~ ~'i(1 - p j ) - ' F j , j=l
j¢i
with
(1)
F~(X, Y, ulO) = pi6(X)6(Y)6(u - Uo). Adding the above system of coupled equations, we obtain: cos O( OF/ OZ ) + sin 0 cos q~(OF / OX ) + sin 0 sin ~p(OF/ OY) + ( tt) F
= ~ f [F(X, Y , u ' I Z ) - F(X, g, ulZ)]QiFi(lu-u'lldu' i=1
+ ~_, f [Q,j(X, Y, u'lZ) - Q,j(X, Y, (ij)
-- E
(6)
ulz)l[(1/Vilr,(lu
-
u'l)
- ( 1 / v j ) ~ ( l u - u'l)l du' (2)
(~d~i -- I ' t j ) Q i j ,
where F = Ei"=1 17. is the spatio-angular density distribution function of the radiation in the matrix and gives the probability F(X, Y, u l Z ) d X d Y d u of finding the projectile in the hyper-surface element d X d Y d u around the point (X, Y, Z, u), {/~) = Ei~l pil.ti. The last two summations in eq. (2) are to be carried out over all possible pairs (i, j) and each of these summations will generate n(n - 1)/2 terms. The quantity Q;j = pjF i - p f j is a measure of the effect of the statistical nature of the medium corresponding to its two-state (i, j) distribution on multiple scattering profile vis-a-vis that from a statistically averaged medium which we term as 'effective medium'. For an effective medium, the linear dimensions of the scatterers are negligible in comparison with the mean free path of the radiation and Q~/tends to zero [3] so that eq. (2) reduces to cos O(OFI OZ) + sin 0 cos q~(OF/ OX) + sin 0 sin ~o(OF/ OY) + (Ix)F
= i=1 E f[F(X, r, ,,'lz)-
F(X,
Y, ull)]o rii(lu-u'l)du'.
(3)
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202
A detailed analysis [1, 3-5] of the above equation shows that the nature of the extractable information from the Guinier regime of the scattering profile is (R 3 ) / ( R 2} (={Ei~ 1 (Q~R~3 )} /{2;= n 1 (p,R/2)}) and not (R4)/(R 3) as suggested by Berk and Hardman-Rhyne [6]. An asymptotic analysis on eq. (3) to delineate the nature of the Porod regime of the multiple scattering profile reveals [2] that q---~limV(qlZ)=exp[-(i.~}Z
O~~im Fi(q)/
l-exp(-N)] i=1
p~Fidu
(4)
"i=1
where N is the average number of scattering interaction the radiation has undergone while passing through the sample of thickness Z. Equation (4) indicates that the Porod law, as far as the functional dependence of the intensity on the momentum transfer vector is concerned, remains invariant under multiple scattering from an effective medium. However, the presence of the factor [1 - e x p ( - N ) ] in the right-hand side of eq. (4) indicates that the effect of multiple scattering is to increase the scattered intensity in the Porod regime of the profile. If the scatterers are assumed to be spherical with a scattering length density contrast D then for a polydisperse effective medium eq. (4) reduces to: lim F( qIZ )
q--..~e¢
= exp[-(
~>Z][1
-
exp(-N)](4'tr/AZ)( ( RZ) / ( R4) )q -4 ,
indicating that the nature of the extractable information from the Porod regime of the multiple scattering profile is given by (R 2) /(R4). Under single scattering approximation (N ~ 0), eq. (4) can be approximated to lim
q--* w
F(qIz) = e x p [ - ( p. ) ZI(6~rD2zZ)( ( R 2) / ( R3) )q -4 ,
where ~- denotes the packing fraction of the sample. The above relation indicates that the extractable information from the single scattering profile is given by (R 2) /(R3). When the linear dimensions of the inhomogeneities are not negligible compared to the mean free path of the radiation in the sample under study, the statistical nature of the medium, as seen by the radiation, becomes manifest and cannot be ignored. In such a situation the radiation transport in the medium can be described in terms of system of coupled integro-differential eq. (1). The angular distribution of the multiply scattered radiation from a bidisperse (n = 2) statistical medium can be obtained from the XY-averaged solutions of eq. (1) and is given by
F(u[Z) = ~ Fi(u]Z ) = (2"rr)-2 i=1
fg
i=1
/~i e x p ( - i r , u) dr = (2"n')-2
Y
]~ e x p ( - i r • u) d r ,
where
ffi = f Fi exp(ir- u) d u , = C,{(W1 + Az + ~1 + ~2 +/-~1)/~2} exp(W1Z) + C2{(W2 + A~ + ~ + ~2 +/~t)/~2} exp(WEZ),
(5)
S. Mazumder, A. Sequeira / A formalism on multiple small-angle scattering
203
= f F/exp(ir. u) d u , W1
=
-{(A~ + A 2 + ~r1 + ~"2"It-/--£1"+ /z2)/2} + [{(A~
W2 = - { ( A 1 +
AE
+ ~'1 + {z +/Xa +/az)/2} - [{(A1
C1 = { ~'2P2 - (W2 + A1 + ~ra +
lZa)Pa}/(W~
- A 2 + ~ - {2 + I ~ - t ~ ) z
+4{i~2)1/2]/2,
- A 2 + ~, - ~z + lz~ - laz) z + 4 { ~ { z ) ~ / z ] / 2 ,
- Wz),
and C2 = P l - C1 • The above expression for the angular distribution of the multiply scattered profile is quite involved, unlike the case for an effective medium, and no simple expression for the nature of the extractable information from the multiple scattering profile can be obtained. However, an asymptotic analysis on F ( u [ Z ) as represented by eq. (5) reveals that the Porod law would remain valid irrespective of the statistical nature of the medium as far as the functional dependence of the scattered intensity on momentum transfer vector is concerned. When the scattering medium is purely of absorbing type (A i = 0), eq. (5) reduces to F = F 1 + F 2 = [(/3, - Y ) e x p ( - f l , Z) + ( y - 132)exp(-flzZ)l/(fl
I - f12),
(6)
where Y = ~ ' , + ~ 2 +p2/z, +p,/~2'
]3, = ½ { ( ( / z ) + Y ) + [ ( ( P - ) - Y ) 2 + 4 a ] l / Z } ,
132 = l { ( ( u , ) + v) - [ ( ( u , ) - 3') 2 + 4 a ] " 2 } ,
a =p,p2(/Zl - ~ ) 2 .
The solution is same as the one represented by eq. (118) in the paper by Levermore et al. [7]. The above solution is the general solution of the problem of radiation transport in a bidisperse 'absorptiononly' Markov medium. Under 'small fluctuation limit' (Pl =P2), eq. (6) simplifies to F = [(/3, -/7,) exp(-13,Z) + ( ~ -/32) exp(-/32Z)]/(/~, - 132),
(7)
where = ~1 + ~2 + p , lz, + p z l x 2 ,
/3, = ~ {((,u,) +/.~) + [((/x) - ~)2 + 4a1~/2},
~2 ~ 1 {((],/,) .}_/,~) _ [((~u,) - ]~)2 + 4a]l/Z},
ot = p l p 2 ( l ~ 1
- tz2) 2 .
The solution is exactly the same as that (eq. (126)) obtained by Levermore et al. [7] under the head 'small fluctuation theory'. Levermore et al. [7] took a different approach for describing radiation transport in 'absorption-only' random medium. Their approach cannot account for the scattering nature of the medium. However, the present generalised formulation for radiation transport in a random medium accounts also for the scattering nature of the medium besides correctly describing the radiation transport in an 'absorption-only' medium.
S. Mazumder, A. Sequeira / A formalism on multiple small-angle scattering
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When the scattering medium is such that (A 1 + ~1) and (A: +/z2) become comparable, eq. (5) can be approximated as /~exp
I - ½ Z ~2 (A, +/.t,) 1 .
(8)
i=1
The above approximation, we will term as small noise approximation (SNA). It can be shown that under SNA, the expression of F for a polydisperse statistical medium is given by:
~.exp[_(Z/n) ~ (Ai + lxi)] .
(9)
i=1
If we assume F~ to be the Guinier approximate scattering cross section for a spherical scatterer of radius Ri, we can write from eq. (9), F ( q I Z ) : n/{5"rr ~
(Ni/piR~)}exp[-nq2/{ 5 ~ (Ni/piR~)}] exp[-(Z/n) ~ txi],
i=1
i=l
(10)
i=l
where Ni(= Z/L~) is the average number of scattering interactions the radiation has undergone with ith type of inhomogeneity while passing through the sample. The expression of F(qlZ) corresponding to eq. (10) for an effective medium can be written as F(qIZ)-1/{5"n ~
(Ni/R~)}exp[-q2/{ 5 ~
i=1
(N~/R~)}]
exp[-Z ~
i=1
P~"~I"
(11)
i=1
It is clear from the above two expressions of F(qlZ) that the nature of the extractable information from the Guinier regime of the multiple scattering profile is n/{5 E~=1(N~/piR~)} in contrast to {5 ET=1 (Ni/R~)}-~ = 2(R 3)/15ZzA2D2(R2) for an effective medium. An asymptotic analysis on eq. (9) shows that
q~
~
[1-exp{-(l/n)
i= l
~ (Ni/P i) i=1
Qi ~im 1]i(q)/pi ) -
Qil~i d u
)// Pi
(12) •
=
From eq. (12), it is clear that the extractable information for a statistical medium from the Porod 2 n (oiR~/pi)}/{E~=l (oiRi/pi)} in contrast t o {Ei= regime of the profile is given by {E~=I 4 n 1 (QiR~)} / n 4 {Ei= 1 (oiRi)} in the case of an effective medium. A comparison between the theoretical expression of a multiple scattering profile from a statistical medium with those obtained invoking the approximations, EMA and SNA, reveals that the two approximations have opposite effects as far as their modulation effect on the scattering profile is concerned. For a bidisperse scattering medium, when ( P 2 - Pl)(A1 - A2) is positive, the theoretical profile which invokes EMA will be narrower while the one invoking SNA will be broader vis-a-vis that from a statistical medium without invoking the above approximations. For a polydisperse medium, the modulation effect of the aforementioned approximations on multiple scattering profile is decided by the
S. Mazumder, A. Sequeira / A formalism on multiple small-angle scattering
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fact that whether the quantity E(ij) (Pi - p j ) ( A j - Ai) is positive or negative. The modulation effect of EMA and SNA can cause erroneous estimation of the extracted parameter. For example, when the quantity E,j) ( p i - pj)(Aj - Ai) is positive, then (R a) / ( R 2) extracted from the Guinier regime of the profile will be underestimated while the extracted value of [n/(E~=~ (N~/p~R~)}] -~ will be overestimated. The seriousness of this erroneous estimation of the extracted parameters depends upon the extent of deviation of the quantity E(~j)(p~- pj)(Aj - Ai) from zero.
3. Conclusion
We conclude that small-angle scattering studies need not be restricted only to thin samples. Some useful structural information about the inhomogeneities can be extracted even using thick samples. We suggest that the broadening of the small-angle scattering profile due to multiple scattering effects in thick samples can, in principle, be exploited to characterise large size inhomogeneities using even relatively low-resolution instruments and in that pursuit, the present formalism would be of significant help.
References [1] [2] [3] [4] [5] [6] [7]
S. Mazumder and A. Sequeira, Phys. Rev. B 41 (1990) 6272. S. Mazumder, Ph.D. Thesis, Bombay University (1990). S. Mazumder and A. Sequeira, Phys. Rev. B 39 (1989) 6370. S. Mazumder and A. Sequeira, Pramana-J. Phys. 30 (1988) 537. S. Mazumder and A. Sequeira, Material Science Forum 27 & 28 (1988) 407. N.F. Berk and K.A. Hardman-Rhyne, J. Appl. Cryst. 18 (1985) 467. C.D. Levermore, G.C. Pomaning, D.L. Sanzo and J. Wong, J. Math. Phys. 27 (1986) 2526.