Periodic systems.

1. Periodic systems. (See Roe Sect 5.5) Repeat periods of 10-1000 Å Degree of order usually low ––> smears reciprocal lattice spots Ordered domain size frequently small ––> smears reciprocal lattice spots

2. Periodic systems. (See Roe Sect 5.5) Repeat periods of 10-1000 Å Degree of order usually low ––> smears reciprocal lattice spots Ordered domain size frequently small ––> smears reciprocal lattice spots Extracting info from saxs pattern: a. periodic character - use std high angle techniques b. disorder - analysis more complex, similar to previous saxs discussions

3. Periodic systems. Repeat periods of 10-1000 Å Degree of order usually low ––> smears reciprocal lattice spots Ordered domain size frequently small ––> smears reciprocal lattice spots Extracting info from saxs pattern: a. periodic character - use std high angle techniques b. disorder - analysis more complex, similar to previous saxs discussions

4. Periodic systems. Repeat periods of 10-1000 Å Degree of order usually low ––> smears reciprocal lattice spots Ordered domain size frequently small ––> smears reciprocal lattice spots Extracting info from saxs pattern: a. periodic character - use std high angle techniques b. disorder - analysis more complex, similar to previous saxs discussions 1st: (r)=u(r) * z(r) u(r) = scattering length density for single repeated motif z(r) = fcn which describes ordering or periodicity

5. Periodic systems. 1st: (r)=u(r) * z(r) u(r) = scattering length density for single repeated motif z(r) = fcn which describes ordering or periodicity and: I(q) =|F(q)|2|Z(q)|2 (Fourier transform of convolution of 2 fcns = product of their transforms) Z(q) describes the reciprocal lattice F(q) is the "structure factor"

6. Periodic systems. Consider lamellar morphology (membranes, folded-chain crystallites, etc.) If one stack of well-ordered lamellae: saxs ––> array of points along line in reciprocal space If many randomly oriented stacks: saxs ––> isotropic pattern (rings)

7. Periodic systems. Ideal 2-phase lamellar structure (x)=u(x) * z(x/d) z(x) = (x - n) da db a b d n=∞ n=-∞

8. Periodic systems. Ideal 2-phase lamellar structure (x)=u(x) * z(x/d) z(x) = (x - n) I(q) =|F(q)|2|Z(q)|2 ––> I(q)  |F(q)|2 z(dq/2) |F(q)|2 = 4(/q)2 sin2 (daq/2) da db a b d n=∞ n=-∞

9. Periodic systems. Ideal 2-phase lamellar structure (x)=u(x) * z(x/d) z(x) = (x - n) I(q) =|F(q)|2|Z(q)|2 ––> I(q)  |F(q)|2 z(dq/2) |F(q)|2 = 4(/q)2 sin2 (daq/2) Result: (Fourier transform)2 of z(x/d) is reciprocal lattice w/ period 2/d Sharp Bragg-type peaks occur at q = 2n/d w/ intensity ~ (n)-2sin2 (na) (can get volume fractions of phases from intensities) n=∞ n=-∞

10. Periodic systems. Ideal 2-phase lamellar structure Result: (Fourier transform)2 of z(x/d) is reciprocal lattice w/ period 2/d Sharp Bragg-type peaks occur at q = 2n/d w/ intensity ~ (n)-2sin2 (na) (can get volume fractions of phases from intensities) Peaks are -fcn sharp if structure is ideal Non-ideal structures ––> peak broadening

11. Periodic systems. 2-phase structure w/ variable thickness lamellae lamellae parallel lamellae thickness varies no correlation in thickness of neighboring lamellae

12. Periodic systems. 2-phase structure w/ variable thickness lamellae lamellae parallel lamellae thickness varies no correlation in thickness of neighboring lamellae probability of thickness of A lamellae betwn a & a + da is pa(a) da similar for B lamellae Then, as before: A(q) =  Aj(q) N j=1

13. aj aj+1 a bj b = 0 xj xj+1 Periodic systems. 2-phase structure w/ variable thickness lamellae Then, as before: A(q) =  Aj(q) Aj(q) = ∫exp (-iqx) dx = (iq) exp (-iqxj)(1 – exp (-iqaj)) N j=1 xj+aj xj

14. Periodic systems. 2-phase structure w/ variable thickness lamellae Then, as before: A(q) =  Aj(q) Aj(q) = ∫exp (-iqx) dx = (iq) exp (-iqxj)(1 – exp (-iqaj)) I(q) =  Aj Ak* I(q) =  Aj Aj* +   Aj Aj+m* +Aj+m Aj*) (Nvery large) N j=1 xj+aj xj N N j=1 k=1 N N N j=1 m=1 j=1

15. Periodic systems. 2-phase structure w/ variable thickness lamellae Then, as before: A(q) =  Aj(q) Aj(q) = ∫exp (-iqx) dx = (iq) exp (-iqxj)(1 – exp (-iqaj)) I(q) =  Aj Ak* I(q) =  Aj Aj* +   Aj Aj+m* +Aj+m Aj*) (Nvery large)  Aj Aj* = (/q)2 (2 – exp (-iqaj) – exp (iqaj))  Aj Aj* = (/q)2 N(2 – <exp (-iqaj)> – <exp (iqaj)>) N j=1 xj+aj xj N N j=1 k=1 N N N j=1 m=1 j=1

16. Periodic systems. 2-phase structure w/ variable thickness lamellae Now: <exp (-iqaj)> = ∫ exp (-iqaj) pa(a) da = Pa(q) = Fourier transform of pa(a) After further similar manipulations: I(q) = 2N (/q)2 Re [(1 –Pa) (1 –Pb)/ (1 –Pa Pb)] If pa(a) known or assumed, can calc I(q) +∞ -∞

17. Periodic systems. 2-phase structure w/ variable thickness lamellae Example Suppose pa, pbGaussian: I ~ 1/(n4 (a2 + b2))

18. a b Periodic systems. 2-phase lamellar structures - correlation functions Can get obs from Fourier transform of I(q) & compare with model(x) =∫(u) (u + x) du a = b - <> = b b = b - <> = a

19. aj aj+1 a bj b = 0 xj xj+1 Periodic systems. 2-phase lamellar structures - correlation functions Can get obs from Fourier transform of I(q) & compare with model(x) =∫(u) (u + x) du If thicknesses of lamellae vary (Gaussian): smeared

20. aj aj+1 a bj b = 0 xj xj+1 Periodic systems. 2-phase lamellar structures - correlation functions Can get obs from Fourier transform of I(q) & compare with model(x) =∫(u) (u + x) du If lamellae/lamellae transitions not sharp, self correlation peak rounds smeared