The scattering vector from a single atom is known as the atomic scattering factor. It is denoted f, and has length |f| and phase . The resultant scattering vector from all the atoms in the unit cell is called the structure factor, and is denoted F, with length |F| phase . fjexp[2irj.S]= fjexp[2iai]

Diffraction is an Interference Effect Diffraction is an interference effect, i.e. it results from the summation of component waves of x-rays that are scattered from individual atoms. This summation can take the form of either constructive or destructive interference. It is generally easier to consider the addition of waves in vector notation In general, F = fj F denotes the resultant (scattering from a unit cell). f denotes a component wave (scattering from an atom).

Equazione del fattore di struttura: F(S) = fj(S)exp[2irj.S] Ogni atomo j contribuisce ad F con un’ampiezza di scattering che dipende da fj ed una fase fj f2 f f1 r fj = 2irj.S F(S) = F(S)  expif Consideriamo un reticolo monodimensionale

La condizione di diffrazione ha una utilissima visualizzazione geometrica nella costruzione della Sfera di Ewald:

Ewald Construction S S s  so O   Crystallographers use it to visualize which reciprocal lattice points will be in diffraction condition for a given orientation of the crystal. A sphere (Ewald sphere) is drawn with radius 1/. The crystal is imagined to be at C (the center of the Ewald sphere). The origin of the reciprocal lattice is at O (the point at which the undeflected x-ray beam would leave the sphere). Reciprocal lattice points in diffraction condition when they cross Ewald’s sphere. When this happens, a diffracted x-ray propagates in the direction s until it hits the detector. Diffracted x-ray rl point in diffraction condition S S s  so O  X-ray beam  rl origin Bragg’s law is equivalent to a reciprocal lattice point lying on the surface of Ewald’s sphere. sin = (|S|/2) / (1/) = |S|/2 Remember that |S| = 1/d  sin = /2d;  = 2d sin Miller planes Crystal at center of sphere Radius = |so| = |s| = 1/

Diffraction when rl points touch surface of Ewald sphere Reciprocal lattice points that touch the surface of Ewald’s sphere give rise to diffracted x-ray beams. Beam trap stops direct beam and h=0, k=0, l=0 reflection O Diffracted rays that intercept the detector will be recorded Max resolution possible = /2

Lo scattering si verifica quando il vertice P del vettore di scattering S cade sulla sfera di Ewald ed F(S) è  0. In tali condizioni P è un punto del reticolo reciproco. Il reticolo reciproco ruota così come ruota il cristallo. La direzione del raggio diffratto dalcristallo è parallela a MP P Shkl è perpendicolare al piano hkl O M

Data Collection :Recording reflections Let us revisit the Ewald construction and put a detector in front of the crystal. For a Reciprocal Lattice Point (RLP) to be recorded as a reflection or diffraction spot, a number of non-trivial conditions need to be fulfilled : 1) The RLP needs to move through the Ewald sphere (ES) at least once, which is usually accomplished by turning the crystal around the vertical goniostat axis. 2) The RLP must lie within the resolution sphere (RS) of the crystal. Most protein crystals diffract to ~2.5 to 1.5 A, a few better, some worse. 3) The diffracted x-ray beam must hit the detector

RL Point A : fulfills all 3 conditions. Will be recorded RL Point A : fulfills all 3 conditions. Will be recorded. RL Point B : Still within/on ES and RS, although borderline. Weak reflection exists, but will not be recorded (does NOT hit detector). RL Point C : Although within RS, can never transverse ES and cannot be recorded in this experiment.. RL Point D : Still on ES, but outside crystal's diffraction limit. No reflection exists.

L’intensità di riflessione (hkl) è proporzionale a |Fhkl|2, quindi dalle intensità diffratte siamo in grado di determinare i moduli del fattore di struttura sperimentalmente calcolando la radice quadrata delle intensità. Se siamo in grado di ottenere informazioni anche sulla fase, potremo calcolare la distribuzione elettronica nella cella elementare. Problema della fase