Fourier transform from r to k: Ã(k) =  A(r) e  i k r d 3 r Inverse FT from k to r: A(k) = (2  )  3  Ã(k) e +i k r d 3 k X-rays scatter off the charge density  (r), neutrons scatter off the spin density. Coherent scattering (diffraction) creates the Fourier transform of  (r) from real to reciprocal space: Ã(k) =   (r) e  i (k  k 0 ) r d 3 r à = |Ã|  e i  I = |Ã| 2 = k  k 0 = scattering vector = G hkl for periodic structures General theory of diffraction measured intensity  (r) I(k) Real space Reciprocal space k0k0 k

Structure determination by diffraction The diffraction pattern is determined by three considerations: 1)The Bragg condition (= energy and momentum conservation) determines the position of the diffraction spots in k-space. It originates from the whole crystal lattice. 2)The structure factor describes the intensity modulation of the diffraction spots by the atoms inside the unit cell. 3)The atomic scattering factor describes the diffraction at an individual atom. This is in tune with the mantra that large objects in real space correspond to small objects in k-space. 1) represents the largest possible object in real space (the infinite lattice), and it becomes the smallest possible object in k-space (a point). 2) represents a medium-sized object (the unit cell), and 3) the smallest object (an atom). They affect increasingly larger portions of k-space.

Structure factor The structure factor S hkl is given by: S hkl =   f  exp[-i G hkl r  ] =   f  exp[-i 2  (h u  +k v  +l w  )] where r  is the position of atom number  inside the unit cell and f  its atomic scattering factor. r  can be expressed in terms of the real space lattice vectors by the indices u ,v ,w  just like G is expressed by the Miller indices h,k,l. The structure factor leads to the extinction of certain spots, i.e. those for which there is destructive interference between two equal atoms in the unit cell. For example, the spot for G 100 vanishes for the fcc lattice, since the (100) planes through the corner atoms interfere destructively with those through the face-centered atoms. G 200 is the first Bragg spot on the x-axis.

Atomic scattering factor The atomic scattering factor f  is given by: f  =    (r) · exp[-i G hkl r] d 3 r where   is the charge density of a single atom inside the unit cell. The integral over the charge density of an atom is proportional to the number of electrons, i.e. the atomic number Z. Furthermore, the diffraction intensity is given by the square of the structure factor. That leads to a strong increase of the diffraction intensity for heavy atoms with high Z.

Experimental methods for structure analysis Energy and momentum conservation impose four constraints in three-dimensional diffraction. They cannot all be fulfilled by adjusting the three k components of the diffracted wave (for an arbitrary incident wave). Something else has to give. Either the energy or the direction of the incident wave needs to be flexible. This can be accomplished in several ways: 1)Incident x-rays with a continuous energy spectrum (Laue). 2)Rotate the crystal (popular with protein crystallography). 3)Use polycrystalline samples (powder diffraction, Debye-Scherrer).

Laue diffraction pattern Laue diffraction pattern of NaCl taken with neutrons. See a projection of k-space.

Powder diffraction pattern Observe rings around the incoming and outgoing beam. (Cylindrical film unfolded.) Extra diffraction rings visible for the ordered Cu 3 Au alloy. Horizontal scan across the rings for Si powder. The (100), (200) reflections are forbidden in the diamond structure, since their structure factor vanishes.

The phase problem Mathematically, the structure ( = charge density) in real space can be obtained from the amplitude of the diffracted wave in k-space by an inverse Fourier transform from k to r. However, the amplitude is a complex number of the form A = |A| · e i , which contains the phase . Only the intensity I = |A| 2 is measured, not the phase . Crystallographers have developed many tricks to retrieve the phase. For example, sulfur can be replaced by selenium in proteins, which is chemically similar. But selenium diffracts X-rays much more when the X-ray energy is tuned to be in resonance with an inner shell excitation (“anomalous scattering”). The difference between diffraction patterns on- and off-resonance provides the phase information. Simple crystal structures can be solved by calculating the diffraction pattern for trial structures containing adjustable parameters. Those are obtained by a least square fit to the diffraction intensities.

Reconstruction of a single nano-object (ptychography) With the recent advent of laser-like X-ray sources there has been great interest in Fourier-transforming the diffraction pattern of a single object, such as a protein molecule or a virus (see next slide). There is a theorem that allows the reconstruction of the phase if the object is located in a well-defined finite aperture, with no diffracting objects outside. The idea is as follows: 1) Start with arbitrary phase in k-space and perform an inverse Fourier transform from k to r. A phase error will produce a finite amplitude outside the aperture. 2) Correct the error by setting the amplitude outside the aperture to zero. 3) Perform a Fourier transform from r to k. If the resulting diffraction intensity disagrees with the data, adjust the amplitude |A| in k-space and go back to 1). This loop needs to be iterated many times, but it converges eventually to the correct amplitude and phase in both r- and k-space. Such a method allows (in principle) lensless imaging with atomic resolution, which is limited only by the wavelength of the X-rays.

Diffraction from a single object X-ray diffraction pattern of a single Mimivirus particle imaged at the LCLS at Stanford, which produces laser-like X-rays. The X-ray pulse stripped most of the electrons from the atoms, leading to a Coulomb explosion. But it was so short (< 50 femtoseconds) that the atoms did not have time to move until after this image was obtained (“diffract and destroy”). Combining thousands of such images with various orientations of the virus (tomography) provides a three-dimensional image.