Crystallinematerial

Intensity I = A*A. I = ³ U(x) exp(-i q·x) d3x ³ U(y) exp(i q·y) d3y
= ³ ³ U(x) U(y) exp(-i q·(x-y)) d3x d3y Substitute z = x-y I(q) = ³ ³ U(z+y) U(y) exp(-i q·z) d3z d3y
Crystalline material
• Crystalline material: Long range order between structural units.
• Crystal structure (periodic 1-, 2- or 3-D order) • Crystal: single crystal • Crystallite: small ordered domain • Polycrystalline material: consist of crystallites (of
C(z)= ³ U(z+y) U(y) d3y • In crystal structure analysis the autocorrelation function
of the electron density is called the Patterson function.
1-dim. periodic structure
• Polycrystalline material
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Bragg law
q k1
T
Path difference 2x x/d = sin ș
k2 2x = 2d sin ș = Ȝ
T
xd
Bragg law 2dhklsinT O
scattering vector
k1
k2
dhkl = distance of lattice planes
d-2 = 4/3 (h2 + hk + k2)/a2 Values of h and k for first peaks: 01, 10 11 02
Dirk Mter et al. Surfactant Self-Assembly in Cylindrical Silica Nanopores. J. Phys. Chem. Lett., 2010, 1 (9), pp 1442–1446
cell (basis).
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Autocorrelation function for a periodic function
• If a function f(x) is periodic, C is also and its period is the same as that of f.
• Let the period of f be T. Consider C(x+T): C(x+T) = f(x+y+T)f(y) dy = f(x+y)f(y) dy = C(x).
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011 310 111 201 220 211 400
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521 PE (pref.Or)
P o w d e r C e ll 2 . 2
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•Platelike crystallites, plane (00l)
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21-1 400
• Preferred orientation of crystallites affects the intensities of the diffraction peaks, not their positions.
2325 PE
•Isotropic PE sample
1163
110
P o w d e r C e ll 2 . 2
• Laue conditions present equidistant planes which are perpendicular to the lattice vector a.
• Can be extended to three-dimensional lattices.
The width of the diffraction peaks
– Symmetrical or unsymmetrical reflection – Symmetrical or unsymmetrical transmission – Perpendicular transmission – Grazing incidence diffraction GID
varying size) • Crystallinity: weight or volume fraction of
crystalline material in the sample.
• Single crystal • Crystal habit
• Nanocrystal • nanoparticle
2T = scattering angle O = wavelength
T
T
d
Scattering vector q=k2-k1 is perpendicular to the lattice planes. The lenght of the scattering vector |q| = 4ʌȜ sinș
Diffraction peakssampledet Nhomakorabeactor
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Isotropic crystalline powder
• Diffraction pattern contains rings.
• These are called Debye rings.
Diffracting planes in different WAXS/SAXS geometries
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Amplitude (real part) of a line of 10 scatterers, distance 100Å
Matlab: m = 10; a = 100; F = exp(i*(m-1)/2*q*a).*sin(m*q*a/2)./sin(q*a/2);
Intensity of a line of 10 scatterers, distance 100Å
Polyethylene (orthorhombic)
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I (arb.units)
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2T
Example. Preferred orientation
• Patterns below computed with Powcell program (public domain)
Bragg law in terms of q: d = 2ʌ/q
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About experiments
• Scattering experiment
– Intensity as a function of the scattering vector q or its magnitude q
• Measurement modes e.g.
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111
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Determination of crystal structure
• Electron density needs to be calculated for one unit cell.
• As many diffraction peaks should be measured as possible.
CH2 CH2
CH2 CH2
CH2 CH2
CH2
2-D structures
• Surfactants in solution • block.-copolymers
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Small-angle diffraction: nanoporous silica
Two-dimensional Hexagonal structure
• The width of the diffraction peak decreases as the size of the lattice increases.
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1-D structures
• Lamellar polymers • Lipid bilayers • Block copolymers • surfactants
symmetrical reflection scattering vector
symmetrical transmission
perpendicular
ș
lattice planes
ș
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Calculated diffraction pattern of isotropic polyethylene sample
• Intensities are corrected for absorption, polarization, geometry (Lorenz factor)
Intensity of elastic scattering
Amplitude proportional to Fourier transform of electron density U(x): A = ³ U(x) exp(-i q·x) d3x
peaks at 2ʌ/a, ʌ/2a, ʌ/3a,... a=100 Å
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Intensity from 1-D lattice
• The function sin(½M q·a) /sin(½ q·a) has maxima at q·a = h2ʌ, where h is integer (Laue condition)
= ³ ³ U(z+y) U(y) d3y exp(-i q·z) d3z I(q) = ³ C(z) exp(-i q·z) d3z
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Patterson function
• Intensity is proportional to I(q) = ³ C(z) exp(-iq·z) d3z • Autocorrelation function of electron density
Two-dimensional lattice
• Coatings: Monolayers (LangmuirBlodgett), surfactant proteins, nanodots
• Bulk materials: 2-d organization of long rods, e.g. self-organized polymeric materials, biopolymers as DNA
1-d lattice
Atoms at positions na, where n is an integer. Amplitude of the lattice of M unit cells F(q) = 1+exp(i q·a) + … + exp(i q·(M-1)a) Ȉm=0M-1 exp(i q·ma) = (1-exp(i q·Ma)) / (1-exp(i q·a)) 1-exp(-iq·Ma) = -2i sin(1/2Mq·a) / exp(½iMq·a) 1-exp(-iq·a) = -2i sin(½ q·a) / exp(-½iq·a) Amplitude F(q) = exp(½ i(M-1) q·a) sin (½ M q·a) /sin (½ q·a) Intensity is proportional to [sin(½Mq·a)/sin(½ q·a)]2
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Single chain
• 3-D electron density ȡ(x) = ȡ0 į(x)į(y) nį(z-nc)
• Amplitude F(qx,qy,qz) = ȡ0 į(x) exp(iqxx)į(y) exp(iqyy)
a • Lattice vector a (base vector) defines the unit cell. • Position vectors of the lattice points; R = pa,
where p is an integer. • Crystal: lattice and positions of atoms in the unit