Non-particulate 2-phase systems.

Transcription

1 (See Roe Sects 5.3, 1.6) Remember: I(q) = Γ ρ (r) exp(-iqr) dr In this form, integral does not converge. So deviation from the mean <n> = η(r) = ρ(r) - <n>

2 (See Roe Sects 5.3, 1.6) Remember: I(q) = Γ ρ (r) exp(-iqr) dr In this form, integral does not converge. So deviation from the mean <ρ> = η(r) = ρ(r) - <ρ> Then Γ η (r) = η(u) η(u + r) du Γ ρ (r) = (η(u) + <ρ>) (η(u + r) + <ρ>) du

3 Then deviation from the mean <ρ> = η(r) = ρ(r) - <ρ> Γ η (r) = η(u) η(u + r) du Γ ρ (r) = (η(u) + <ρ>) (η(u + r) + <ρ>) du Γ ρ (r) = η(u) η(u + r) du + <ρ> 2 du + <ρ> η(u) du + <ρ> η(u + r) du

4 Then deviation from the mean <ρ> = η(r) = ρ(r) - <ρ> Γ η (r) = η(u) η(u + r) du Γ ρ (r) = (η(u) + <ρ>) (η(u + r) + <ρ>) du Γ ρ (r) = η(u) η(u + r) du + <ρ> 2 du + <ρ> η(u) du + <ρ> η(u + r) du 0 0

5 Then deviation from the mean <ρ> = η(r) = ρ(r) - <ρ> Γ η (r) = η(u) η(u + r) du Γ ρ (r) = (η(u) + <ρ>) (η(u + r) + <ρ>) du Γ ρ (r) = η(u) η(u + r) du + <ρ> 2 du + <ρ> η(u) du + <ρ> η(u + r) du 0 Γ ρ (r) = Γ η (r) + <ρ> 2 V 0 volume of specimen

6 Then deviation from the mean <ρ> = η(r) = ρ(r) - <ρ> Γ η (r) = η(u) η(u + r) du Γ ρ (r) = (η(u) + <ρ>) (η(u + r) + <ρ>) du Γ ρ (r) = η(u) η(u + r) du + <ρ> 2 du + <ρ> η(u) du + <ρ> η(u + r) du 0 0 Γ ρ (r) = Γ η (r) + <ρ> 2 V leads to null scattering

7 Finally Γ ρ (r) = Γ η (r) I(q) = Γ η (r) exp(-iqr) dr Normalization of Γ η (r): where γ(r) = Γ η (r)/ Γ η (0) Then: Γ η (0) = η(u) η(u + 0) du = <η 2 >V I(q) = <η 2 >V γ(r) exp(-iqr) dr

8 I(q) = <η 2 >V γ(r) exp(-iqr) dr Remember the invariant (total scattering power of specimen): Q = I(S) ds = (1/(2π) 3 ) I(q) dq = total scattering over the whole of reciprocal space For isotropic material: Q = 4π S 2 I(S) ds = (1/(2π 2 )) q 2 I(q) dq o o (S = (2 sin θ)/λ = q/2π)

9 For isotropic material: Q = 4π S 2 I(S) ds = (1/(2π 2 )) q 2 I(q) dq o Q = Γ η (r) [(1/(2π 2 )) exp(-iqr) dq ] dr = Γ η (0) = <η 2 >V o

10 Now for the model - Ideal 2-phase system: only 2 regions or phases each phase has constant scattering length density sharp phase boundary randomly mixed isotropic

11 Now for the model - Ideal 2-phase system: only 2 regions or phases each phase has constant scattering length density sharp phase boundary randomly mixed isotropic Can get: volume fractions φ i of each phase & interface area

12 Ideal 2-phase system: <ρ> = φ 1 ρ 1 + φ 2 ρ 2 η 1 = ρ 1 - <ρ> = Δρ φ 2 η 2 = ρ 2 - <ρ> = Δρ φ 1 Δρ = ρ 1 - ρ 2 = η 1 + η 2 ρ 1 <ρ> ρ 2 η 1 η 2

13 Ideal 2-phase system: <ρ> = φ 1 ρ 1 + φ 2 ρ 2 η 1 = ρ 1 - <ρ> = Δρ φ 2 η 2 = ρ 2 - <ρ> = Δρ φ 1 Δρ = ρ 1 - ρ 2 = η 1 + η 2 ρ 1 <ρ> ρ 2 η 1 η 2 Q = <η 2 >V = V (Δρ) 2 φ 1 φ 2 if ρ 1, ρ 2 known, Q --> φ 1, φ 2 if φ 1, φ 2 known, Q --> ρ 1 - ρ 2

14 Ideal 2-phase system: Most important result - Porod law: I(q) --> (2π (Δρ) 2 S)/q 4 at large q S = total boundary area betwn 2 phases log I(q) = const 4 x log q

15 Ideal 2-phase system: Most important result - Porod law: I(q) --> (2π (Δρ) 2 S)/q 4 at large q S = total boundary area betwn 2 phases log I(q) = const 4 x log q

16 Ideal 2-phase system: Most important result - Porod law: I(q) --> (2π (Δρ) 2 S)/q 4 at large q S = total boundary area betwn 2 phases log I(q) = const 4 x log q Need absolute Is If relative Is, can get invariant Q --> S/V I(q)/Q = (2π S)/(φ 1 φ 2 Vq 4 )

17 Deviations from ideal 2-phase system Ideal 2-phase system: only 2 regions or phases each phase has constant scattering length density sharp phase boundary randomly mixed isotropic

18 Deviations from ideal 2-phase system Ideal 2-phase system: only 2 regions or phases each phase has constant scattering length density sharp phase boundary randomly mixed isotropic Density fluctuations (from thermal motion): I(q) = I o (q) + I 1 (q) + I 2 (q) + I 12 (q) uniform density density fluctuations interaction betwn phases

19 Deviations from ideal 2-phase system I(q) = I o (q) + I 1 (q) + I 2 (q) + I 12 (q) uniform density density fluctuations interaction betwn phases = intensity from pure phases short range correlations - weighted by vol %s small, neglected

20 Deviations from ideal 2-phase system I(q) = I o (q) + I 1 (q) + I 2 (q) + I 12 (q) uniform density density fluctuations interaction betwn phases = intensity from pure phases short range correlations - weighted by vol %s small, neglected To correct, then, measure scattering from the 2 pure phases

21 Deviations from ideal 2-phase system Al otro lado: can make empirical correction for I 1 (q) + I 2 (q) intensity high angle region q

22 Deviations from ideal 2-phase system Diffuse boundary betwn phase regions: let ρ(r) = scattering length density in 2-phase matl w/ diffuse boundaries ρ id (r) = scattering length density in the same 2-phase matl w/ sharp boundaries (hypothetical)

23 Deviations from ideal 2-phase system Diffuse boundary betwn phase regions: Then let ρ(r) = scattering length density in 2-phase matl w/ diffuse boundaries ρ id (r) = scattering length density in the same 2-phase matl w/ sharp boundaries (hypothetical) ρ(r) = ρ id (r) * g(r) g(r) = fcn which characterizes diffuse boundaries

24 Deviations from ideal 2-phase system Diffuse boundary betwn phase regions: Then let ρ(r) = scattering length density in 2-phase matl w/ diffuse boundaries ρ id (r) = scattering length density in the same 2-phase matl w/ sharp boundaries (hypothetical) ρ(r) = ρ id (r) * g(r) g(r) = fcn which characterizes diffuse boundaries I(q) = I id (q) G 2 (q) G(q) = Fourier transform of g(r) (Fourier transform of convolution of 2 fcns = product of their transforms)

25 Deviations from ideal 2-phase system Diffuse boundary betwn phase regions: ρ(r) = ρ id (r) * g(r) g(r) = fcn which characterizes diffuse boundaries I(q) = I id (q) G 2 (q) G(q) = Fourier transform of g(r) (Fourier transform of convolution of 2 fcns = product of their transforms) Common: G(q) = exp( σ 2 /2)q 2 & for small bdy width (σ): I(q) = I id (q) (1 - σ 2 q 2 )

26 Deviations from ideal 2-phase system Diffuse boundary betwn phase regions: I(q) = I id (q) (1 - σ 2 q 2 ) q 4 I(q) = (2π (Δρ) 2 S) - (2π (Δρ) 2 S) (σ 2 q 2 ) q 4 I(q) q 2

27 Deviations from ideal 2-phase system Diffuse boundary betwn phase regions: