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This is obtained from an expansion of the HNC closure [50].
Exact solutions of these closure relationships are known, for instance using the PY approximation for hard spheres [50] or the Carnahan‐Starling approximation gives the expression in Eq. (1.38).
Further information on integral equation theories for liquids is available elsewhere [51].
1.7 FORM FACTORS
1.7.1 Examples of Form Factor Expressions
Analytical expressions are available for the form factor of particles of many geometries, as well as polymers discussed in Section 1.8. Some examples of more commonly used form factors are listed in Table 1.2. Extensive compilations of form factors are available [7, 17, 18, 21, 55].
Table 1.2 Expressions for common form factors.
| Form factor | Equation | Parameters |
|---|---|---|
| Homogeneous sphere |
|
Radius, R ( |
| 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) |
|
I0, forward scattering intensity Rg, radius of gyration (b, statistical segment length) |
| Sphere with attached Gaussian chains (used for block copolymer micelles) [52, 53] |
|
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 |
| Ellipsoid of revolution (spheroid) |
|
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 |
|
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