Ian W. Hamley

Small-Angle Scattering

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upper R comma upper Delta rho right-parenthesis equals left-bracket upper V upper Delta rho upper F Subscript s Baseline left-parenthesis q comma upper R right-parenthesis right-bracket squared"/> upper F Subscript s Baseline left-parenthesis q comma upper R right-parenthesis equals StartFraction 3 left-bracket sine left-parenthesis italic q upper R right-parenthesis minus italic qR cosine left-parenthesis italic q upper R right-parenthesis right-bracket Over left-parenthesis italic q upper R right-parenthesis cubed EndFraction Radius, R (upper V equals four thirds pi upper R cubed), Scattering contrast, Δρ Spherical shell P(q) = [Ks(q, R1, Δρ) − Ks(q, R2, Δρ(1 − μ))]2 Ks(q, R, Δρ) as for a homogeneous sphere as above (other parameterizations are possible) Outer radius R1 with shell scattering contrast Δρ Inner radius R2 with core scattering contrast Δρ(1−μ) Gaussian polymer chain –Debye function (see Section 1.8 for derivation) upper P left-parenthesis q right-parenthesis equals upper I 0 StartFraction 2 left-parenthesis e Superscript negative upper X Baseline plus upper X minus 1 right-parenthesis Over upper X squared EndFraction equals upper I 0 upper P Subscript upper G Baseline left-parenthesis q comma upper R Subscript g Baseline right-parenthesis upper X equals left-parenthesis q squared b squared upper N right-parenthesis slash 6 equals q squared upper R Subscript g Superscript 2 I0, forward scattering intensity Rg, radius of gyration (b, statistical segment length) Sphere with attached Gaussian chains (used for block copolymer micelles) [52, 53] upper P left-parenthesis q right-parenthesis equals upper N Subscript c Superscript 2 Baseline left-parenthesis upper Delta rho Subscript s Baseline right-parenthesis squared upper F Subscript s Superscript 2 Baseline left-parenthesis q comma upper R right-parenthesis plus upper N Subscript c Baseline left-parenthesis upper Delta rho Subscript s Baseline right-parenthesis squared upper P Subscript upper G Baseline left-parenthesis q comma upper R Subscript g Baseline right-parenthesis plus upper N Subscript c Baseline left-parenthesis upper N Subscript c Baseline minus 1 right-parenthesis left-parenthesis upper Delta rho Subscript c Baseline right-parenthesis squared upper S Subscript italic c c Baseline left-parenthesis q right-parenthesis plus 2 upper N Subscript c Superscript 2 Baseline upper Delta rho Subscript s Baseline upper Delta rho Subscript c Baseline upper S Subscript italic s c Baseline left-parenthesis q right-parenthesis where upper S Subscript italic c c Baseline left-parenthesis q right-parenthesis equals psi squared left-parenthesis q comma upper R Subscript g Baseline right-parenthesis left-bracket StartFraction sine left-bracket q left-parenthesis upper R plus d upper R Subscript g Baseline right-parenthesis right-bracket Over q left-parenthesis upper R plus d upper R Subscript g Baseline right-parenthesis EndFraction right-bracket squared and upper S Subscript italic s c Baseline left-parenthesis q right-parenthesis equals upper F Subscript s Baseline left-parenthesis q comma upper R right-parenthesis psi left-parenthesis q comma upper R Subscript g Baseline right-parenthesis StartFraction sine left-bracket q left-parenthesis upper R plus d upper R Subscript g Baseline right-parenthesis right-bracket Over q left-parenthesis upper R plus d upper R Subscript g Baseline right-parenthesis EndFraction with psi left-parenthesis q comma upper R Subscript g Baseline right-parenthesis equals StartFraction 1 minus normal e Superscript negative upper X Baseline Over upper X EndFraction upper X equals left-parenthesis q squared b squared upper N right-parenthesis slash 6 equals q squared upper R Subscript g Superscript 2 Nc, aggregation number R, core radius Rg, radius of gyration of attached chains d, displacement of chains (for non‐penetration into core, d ≈ 1) Δρs, scattering contrast of spherical core Δρc, scattering contrast of attached chains Ellipsoid of revolution (spheroid) upper P left-parenthesis q right-parenthesis equals integral left-bracket upper K Subscript s Baseline left-parenthesis q comma StartRoot upper R Subscript p Superscript 2 Baseline y squared plus nu squared upper R Subscript p Superscript 2 Baseline left-parenthesis 1 plus y right-parenthesis squared EndRoot comma upper Delta rho right-parenthesis right-bracket squared italic d y Ks(q, R, Δρ) as for a homogeneous sphere as above (other parameterizations are possible) Radius in polar direction Rp Equatorial radius Re = νRp Scattering contrast, Δρ Tri‐axial ellipsoid P(q) = ∫ [Ks(q, R(a, b, c, α, β, Δρ)]2 sin αdαdβ R(a, b, c, α, β) =[(a2sin2β + b2cos2β)sin2α + c2cos2α]1/2 Ks(q, R, Δρ) as for a homogeneous sphere as above (other parameterizations are possible) Semi axis lengths a, b, c Scattering contrast, Δρ Rectangular parallelepipedons (including cubes) upper P left-parenthesis q right-parenthesis equals upper V squared left-parenthesis upper Delta rho right-parenthesis squared integral Subscript 0 Superscript pi slash 2 Baseline integral Subscript 0 Superscript pi slash 2 Baseline left-bracket j 0 left-parenthesis q a Superscript prime Baseline right-parenthesis j 0 left-parenthesis q b Superscript prime Baseline right-parenthesis j 0 left-parenthesis italic q c Superscript prime Baseline right-parenthesis right-bracket squared sine italic alpha d alpha d beta Where j 0 left-parenthesis q a prime right-parenthesis equals StartFraction sine left-parenthesis italic q a sine alpha cosine beta right-parenthesis Over italic q a sine alpha cosine beta EndFraction j 0 left-parenthesis italic q b prime right-parenthesis equals StartFraction sine left-parenthesis italic q b sine alpha sine beta right-parenthesis Over italic q b sine alpha sine beta EndFraction j 0 left-parenthesis italic q c prime right-parenthesis equals StartFraction sine left-parenthesis italic q c cosine alpha right-parenthesis Over italic q c cosine alpha EndFraction Edge lengths a, b, c Scattering contrast, Δρ Circular cylinder upper P left-parenthesis q right-parenthesis equals upper V squared left-parenthesis upper Delta rho right-parenthesis squared integral Subscript 0 Superscript pi slash 2 Baseline left-bracket StartFraction sine left-parenthesis one half italic </p> </div><hr> <div class= Скачать книгу