Resolution: maximum limit of diffraction (asymmetric) crystal Y X-ray source 2θ X direct beam tan 2θ = Y X d = resolution 2d sinθ = λ detector 1
Unit Cell: two vectors in plane of image c* Observe: b* Two perpendicular 2-folds (222) Every odd spot is missing along the two major axes (2 1 screw axis) P2 1 2 1 2 1 Vector = λ = spacing between spots 2 sin tan -1 ( /F) F = detector to crystal distance 2 λ = wavelength 2
Unit Cell: third vector out of plane Observe: concentric rings cones of diffraction from a major axes Vector = nλ 1- cos ((tan -1 (r/f)) r = radius of the n th circle F = detector to crystal distance λ = wavelength Circles need to be ~concentric about direct beam 3
Assume λ = 1.246 Å and a crystal to detector distance of 15 cm. Using the above experimental details determine as much information as possible about the unit cell parameters? Direct beam 4
Data processing: obtaining F h,k,l Given crystals unit cell calculated crystal orientation and predict diffraction. If prediction fits observed data, the data is said to be INDEXED Hence the lattice plane from which each reflection originates can be identified and the intensity of each reflection measured Interpolation Background Peak Volume = Intensity F h,k,l = Intensity ½ Volume 5
Profile fitting: 1) Correct for background 2) Check peak profile (profile fit) 12 16 9 8 10 7 16 25 76 62 27 9 17 82 99 + 96 57 12 14 32 74 56 21 7 12 10 7 17 8 5 Concerns 1) Grid size 2) Counting statistics 3) Assignment of signal/background 4) Instability of detector Signal to noise assigned σ? Cutoff value of σ 6
Mosaicity of crystal: Measure of the order within the crystal Low mosaicity: highly ordered diffraction spot sharp Mosaicity: broadened diffraction (lunes) As if the diffraction data was collected with a larger oscillation angle than was used (twinning) 7
Full/partial reflection: Did the reciprocal lattice point pass completely through the sphere of diffraction Reciprocal lattice point treated as sphere Full reflection: passes completely through sphere of diffraction (fully satisfying Bragg s law) Partial reflection: only a certain volume has passed through P calc = 3q 2-2q 3 q=radius 8
Data quality: R sym = Σ I h -I h Σ I h I h = measured intensity of a reflection h I h = mean intensity of all reflections h (including symmetry related) Can be given as a % (x100) 9
Space group assignment: Relationships between Bragg reflections Friedel symmetry I h,k,l = I -h,-k,-l Orthorhombic I h,k,l = I h,-k,l = I -h,k,l I h,k,l = I h,-k,l = I -h,k,l Monoclinic 10
P2 1 2 1 2 1 I222 Then look for Systematic absences 11
A simple wave (such as X-rays) can be described by a periodic function: f(x) = F cos 2 π ( hx+ α) or f(x) = F sin 2 π ( hx+ α) (in one dimension) f(x) is the vertical height of the wave at any horizontal position x variable x is in fractions of wavelengths F is the amplitude h is the frequency the constant α specifies the phase of the wave, the position of the wave with respect to the origin of the coordinate system. 12
Fourier series: Fourier showed that even the most complex periodic functions can be described as a sum of simple sine and cosine functions whose wavelengths are integral fractions of the wavelength of the complicated function. f(x) = a 0 +a 1 cos 2π(x) + a 2 cos 2π(2x)+ + a h cos 2π(hx) +b 1 sin 2π(x) + b 2 sin 2π(2x)+ + b h sin 2π(hx) General one-dimensional Fourier h f(x)= a 0 + (a h cos 2π(hx) + b h sin 2π(hx)) 1 where h is an integer, a h and b h are constants, and x is a fraction of a period 13
General one-dimensional Fourier h f(x)= a 0 + (a h cos 2π(hx) + b h sin 2π(hx)) 1 In complex numbers: General form a +ib where i = -1 cos θ + i sin θ = e iθ θ = 2πhx h f(x)= F o + (F h e 2πi(hx) 1 14
Geometric ways of depicting phase angles: Vector F (hkl) Phase angle α (hkl) Amplitude F (hkl) Two components A (hkl) and B (hkl) Phase angle = tan α hkl = B (hkl) A (hkl) Intensity = A (hkl) 2 + B (hkl) 2 = F (hkl) 2 15
Atomic scattering factors: Since X-rays are scattered by electrons, the amplitude of the wave scattered by an atom is proportional to how many electrons there are around the atom. That is the atomic number Z. Atomic scattering factors f i are expressed as the ratio of scattering of an atom to the scattering by a single electron under the same condition. 16 Atomic scattering factor 14 12 10 8 6 4 2 S C H 0.0 0.5 1.0 1.5 Sinθ/λ 16
The structure factor is a mathematical function describing the amplitude and phase of a wave diffracted from crystal lattice planes characterised by Miller indices h,k,l. The structure factor may be expressed as where the sum is over all atoms in the unit cell, x j,y j,z j are the positional coordinates of the jth atom, f j is the scattering factor of the jth atom, and α hkl is the phase of the diffracted beam. 17
Calculation of an electron-density map (complex numbers) 1 ρ(x,y,z) = F (hkl) e -iα V hkl ib F (amplitude α (phase) a where α =2π (hx+ky+lz) and F (hkl) = F (hkl) e iα F = a + ib 1 ρ(x,y,z) = F (hkl) V hkl e -2 πi(hx+ky+lz-α ) where α = 2πα 18