- a continuous range of wavelengths (Bremsstrahlung)
- sharp peaks, characterisitic of the metal, arising from collisions with electrons in inner shells of the atoms. The collision expels an electron from the inner shell and an electron of higher energy drops into the vacancy, emitting the excess energy as an X-ray photon (Fig. 11)
Fig. 10. X-rays are produced by bombarding a cooled metal target.
Fig. 11. The X-ray emission from a metal consists of a broad featureless (Bremsstrahlung) background, with sharp peaks, characterisitic of the metal. The peaks arise from collisions with electrons in inner shells of the atoms. The collision expels an electron from the inner shell and an electron of higher energy drops into the vacancy, emitting the excess energy as an X-ray photon.
X-rays are also absorbed by atoms in a wavelength dependent manner. Discontinuities (absorption edges) are found at certain wavelengths corresponding to the energies necessary to excite or eject a bound electron. This property is employed in the design of filters, used to eliminated unwanted wavelengths of a radiation produced by a given anode (Fig. 12).
Fig. 12. Principle of the filtering of the radiation produced by a Cu anode by a Ni filter.
An alernative source of X-rays are synchrotrons which produce intense beams (over a wide range of wavelengths).
In the following discussions, only monochromatic incident radiations will be discussed, to study either powders or singl crystals. There are also techniques based on the use of polychromatic (radiation spanning a range of X-ray wavelengths) incident beams but they will not be discussed here. They derive from the original method of Laue, who used a broad-band beam of X-rays to study single crystals. There is currently a resurgence of interest in the Laue technique because synchrotron radiation produces intense beams spanning a range of wavelengths.
.1. The Bragg's Law
W. Röntgen discovered X-rays in 1896. Seventeen years later Max von Laue suggested that they might be diffracted when passing through a crystal. He had realized that their wavelengths are comparable to the separation of lattice planes. Laue's suggestion was confirmed almost immediately by W. Friedrich and P. Knipping, and has grown since then into a technique of extraordinary power.
W.L. Bragg, in the summer of 1913, showed that scattered radiation from a crystal behaves as if the diffracted beam were "reflected" from a plane passing through points of the crystal lattice in a manner that makes these crystal-lattice planes analogous to mirrors. This is illustrated in fig. 13.
Fig 13. Conditions for diffraction. Scattered radiation from a crystal behaves as if the diffracted beam were "reflected" from a plane passing through points of the crystal lattice.
The conditions leading to diffraction are given by the Bragg's law, relating the angle of incidence of the radiation (theta) to the wavelength (lambda) of the incident radiation and the spacing between the crystal lattice planes (d) :
2 d sin (theta) = n lambda (Fig. 14)
Fig. 14. Geometry of diffraction and its relationship to Bragg's law. Constructive interference occurs when AB + BC is equal to an integral number of wavelengths.
The path-length difference of the two rays in Fig is AB +BC = 2d sin (theta). For many incidence angles, the path-length difference is not an integral number of wavelengths, and the waves interfere destructively. However, when the path-length difference is an integral number of wavelengths (AB +BC = n lambda), the reflected waves are in phase and interfere constructively.
.2. The Ewald sphere
A geometrical description of diffraction was originally proposed by P.Ewald. The advantage of this description is that it allows the determination of which Bragg reflections will be observed knowing the orientation of the crystal with respect to the incident beam. The Ewald construction provides a geometrical relationship between the orientation of the crystal and the direction of the X-ray beams diffracted by it. As illustrated in Fig. 14 and 15a, the Bragg condition for diffraction occurs when a set of lattice planes, with defined dhkl spacing, are inclined with respect to the incident beam by an angle theta. The diffracted beam (Bragg reflection) occurs at 2 theta from the incident beam. The diffraction vector (defined as H = (s-s0) / lambda) is perpendicular to the lattice planes
Fig. 15. The construction of the Ewald sphere, illustrated in two dimensions (Ewald circle) : a) Bragg's law and definition of s and s0
b) construction of the sphere (radius = 1 / lambda) centered on the cristal
c) orientation of the reciprocal lattice with its origin (hkl = 000) at O. The crystal (in c) can be physically oriented so that a required reciprocal lattice point can intersect the sphere. From there, s0 (the direction of the diffracted beam) can be deduced from H= (s-s0) / lambda.
In the Ewald construction, a sphere with radius 1/lambda is drawn (Fig. 15b). The reciprocal lattice, drawn at the same scale as that of the Ewald sphere, is then placed with its origin centered at O (Fig. 15c). The crystal (in c) can be physically oriented so that a required reciprocal lattice point can intersect the sphere. From there, s0 (the direction of the diffracted beam) can be deduced from H.
As the crystal is rotated, so is its crystal lattice and thus also the reciprocal lattice. If during the rotation of the crystal a reciprocal lattice point (hkl) touches the surface of the sphere, Bragg's law is satisfied. The result is a reflection in the direction s
0, with values of h, k, l corresponding both to the values of the reciprocal lattice point and for the crystal lattice planes.
The X-ray diffraction pattern of a crystal is the sampling at the reciprocal lattice points of the X-ray diffraction pattern of the contents of a single unit cell. It is only necessary to find the atomic arrangement in one unit cell, which can be derived from the overall intensity variation in the diffraction pattern. This atomic arrangement is then repeated according to the direct lattice to give the entire crystal structure. The spatial arrangement of the diffracted beams is determined by the geometry of the crystal lattice while the intensities are determined by the ar rangement of atoms within one unit cell.
.3. Powder diffraction
Principle :When the sample is a powder (polycrystalline solid), passed in a monochromatic X-rays radiation, at least some of the crystallites will be oriented so as
to satisfy the Bragg condition for each set of planes (hkl). Thus simultaneously, different diffraction directions will be possible with respect to the incident beam (ie, different families of planes will give rise to diffracted intensities for different values of theta) (Fig. 16). At the same time, crystallites satisfying Bragg's law for a given theta angle, will lie at all possible angles around the incoming beam, so the diffracted beams lie on a cone around the incoming beam of half-angle 2 theta. In principle, each set of (hkl) planes gives rise to a diffraction cone and the final diffraction pattern is a series of concentric cones.
Fig. 16. Concentric cones diffracted by a powder sample.
Detectors :
The original Debye-Scherrer method is illustrated in Fig. 17. The sample is in a capillary tube, which is rotated to ensure that the crystallites are randomly oriented. The diffraction cones are photographed as arcs of circles as illustrated in Fig 18.
Fig. 17. Debye-Scherrer came
ra. The monochromatic X-ray beam is diffracted by a powder sample placed in a capillary tube and the diffracted cones are detected by a photographic film.
Fig.18. X-r
ay powder photographs. The diffraction cones are photographed as arcs of circles.
Powder phot ographs of KCl (a) and Na Cl (b) and the indexed reflections.
Note that the film covers one-half of the circumference of the camera, the line at the top corresponding to the largest diffraction angles (smaller hkl indices).
In modern diffractometers, the sample is spread on a flat plate and the diffraction pattern (position (2 theta) and intensities) is monitored electronically.
Results :
Powder diffraction techniques are
used to identify a sample of a solid substance by comparison of the positions of the diffraction lines and their intensities with a set of standard spectra (the Joint Commitee of Powder Diffraction Standards data bank contains information on over 30 000 substances).
Powder diffraction data are also used to de
termine phase diagrams (different solid phases result in a different diffraction pattern) and to determine the relative amount of each phase present in a mixture.
The technique is also used for the initial determination of the dimension and symmetries of unit cells.
Principle :
The preliminary investigation of a
crystal studies the diffraction power (cfr resolution for macromolecules) and serves to identify the symmetry and dimensions of the unit cell. From the dimension of the unit cell (volume) and its symmetry (multiplicity) one can deduce the density of the crystal and, eventually compare it with experimental values.
d = (Z . Mw) / (0.6 . V) in g/cm3 for the volume (V) expressed in Å3.
Z is the multiplicity
Computing techniques are now available that lead to automatic determination of the symmetry, shape, and size of the unit cell and to the indexing of the reflections (diffracted spots). A data collection (result of a systematic rotation of the single cryst
al in the X-ray beam) ends up with a (complete) set of data consisting of a list of positions (hkl indices of the reflections) and the corresponding intensities. The problem is to interpret the data in terms of the structure of the crystal.
If the unit cell contains several atoms with scattering factors fi and coordinates (xia, yib, zic), then the overall amplitude of a wave diffracted b
y the (hkl) planes can be expressed as :
Fhkl = SUM fi exp i alphahkl with alphahkl = 2 pi (hxi + kyi + lzi)
The quantity Fhkl is called the structure factor and the intensity of the (hkl) reflection is proportional to |Fhkl|2. In practice, |Fhkl| can be see
n as the amplitude (height of a crest) of the diffracted wave and the relative phase angle, alphahkl, the position of the crest of the wave relative to some fixed origin.
In particular, Friedel's law states that the |Fhkl|2 of centrosymmetrically related Bragg reflections (h,k,l and -h,-k,-l) are equal, even for noncentrosymmetric structures (ie lattice planes can been seen as mirrors from both sides).
The scattering factor f of an atom is proportional to its number of electrons. Because the atom is not punctual, there is a dependence of f on the theta angle (Fig. 19) and it is only for theta = 0° that f = Nbre e-. A direct consequence is that Fhkl (SUM fi exp i alphahkl) is also dependent on the q angle and that the intensities of the diffracted spots decrease with this angle. Depending on the temperature, the volume of occupied by the atom is also different (larger atomic vibrations/disorder at higher temperature). Based on the reciprocity principle, as atoms appear to become broader, the diffraction pattern decreases in extent and becomes narrower. In other words, the wider the atom as a result of increased vibration, the more out of phase are the rays scattered by various parts of the atom. At higher scattering angles the intensity falls off more than calculated from a point atom.
a)
b)Fig. 19. a) Dependence on theta of the scattering factor. b) Punctuel vs non punctuel atom and interference between diffracted beams.
The details of the electron distribution inside a unit cell are contained in the structure factor : Fhkl depends on the type of atom (via fi) and on their location (through hxi + kyi + lzi). The electron density, rho, in the unit cell is a Fourier synthesis (Fig. 20) where the amplitudes are the structure factors for all reflections (derived from their intensities) :
rho = 1/V S Fhkl exp -i2 pi (hx + ky + lz)
Fig. 20. Principle of the Fourier synthesis. Electron density (at the bottom) as a result of the Fourier synthesis of a series of terms from h= 0 to h =10 (k = l = 0). Note the importance of the phases (here either 0 or 180°).
From Fig. 20 it is clear that the electron density is dependent on the relative phase angle (in the equation rho is related to Fhkl, ie |Fhkl| and alphahkl). From the measured intensities, however, only the magnitude of |Fhkl| can be obtained (the intensity is proportional to |Fhkl|2) but no information is provided about alphahkl. Crystallographers call it the phase problem.
The phase problem can be overcome (solved) to some extent by a variety of methods. Some of them are considered here briefely : the Patterson synthesis and its application in isomorphous replacement and molecular replacement, direct methods, and anomalous dispersion data.
Patterson synthesis
In a Patterson synthesis, |Fhkl|2, which can be obtained without ambiguity from the intensities, are used in the following expression :
P(uvw) = 1/V SUM |Fhkl|2 exp -i2 pi (hu + kv + lw) = 1/V SUM |Fhkl|2 cos 2 pi (hu + kv + lw)
The function has the same form as the equation for electron density but as only the cosine terms of the structure factor are conserved, no phases are needed and P(uvw) can be calculated directly from the direct measurement in the X-ray diffraction experiment.
The Patterson function can also be viewed as the convolution of the electron density at all points x,y,z in the unit cell with the electron density at points x+u, y+v, z+w :
P(uvw) = INTEGRAL rho(xyz) rho(x+u, y+v, z+w) dV
The outcome of the Patterson synthesis is a map of the vector separations of the atoms (distances and directions between atoms) in the unit cell (Fig. 21) If atom A is at the coordinates (xA,yA,zA), and atom B at (xB,yB,zB), then there will be a peak at (xB-xA, yB-yA, zB-zA) in the Patterson map. There will also be a peak at the negative of these coordinates, because for any vector from A to B, there is as well a verctor from B to A. The Patterson map is thus centrosymmetric, regardless of whether the space group is centro or non-centrosymmetric.
Fig. 21. The Patterson synthesis of the pattern in (a) corresponds to the map in (b). The distance and orientation of eack spot from the origin gives the orientation and separation of an atom-atom vector in (a). Note that the Patterson map is centrosymmetric, regardless of whether the space group is centro or non-centrosymmetric.
The height of the peak in the map is proportional to the product of the atomic numbers of the two atoms, ZAZB. If some atoms are heavy, they dominate the scattering and their locations may be deduced quite readily. There exist algebric tricks to enhance peaks in a Patterson map (sharpened Patterson functions) replacing |Fhkl| by values derived from it.
If a heavy atom (in comparaison with the others) is present in the cell, it will be detected in the Patterson map. It will also dominate the relative phase angles of the Bragg reflections as shown in Fig. 22. The relative phase angle (aM) deduced for the heavy atom can be combined with the observed |F| to perform a Fourier synthesis of the full electron density in the unit cell, and hence locate the light atoms as well as the heavy atoms.
Fig. 22. Relative phase angles in the presence of a heavy atom. The heavy atom (important contribution to the total vector F) has a relative phase angle (aM) that approaches the total phase angle (aT).
In the technique known as isomorphous replacement, heavy atoms are introduced artificially into a complex molecule, without affecting its structure significantly. This strategy is common in the case of macromolecules where a diffraction set of the native crystal (P) is compared to the diffracted intensities of a crystal soaked with a "heavy-atom" (heavy derivative, PH) in order to phase the native set.
FPH = FP + FH ie FH = FPH - FP
The Patterson function used has | |FPH| - |FP| |2 as coefficients. Usually more than one heavy derivative is analysed leading to the so-called multiple isomorphous replacement (MIR) technique.
Molecular replacement
The molecular replacement method is often used for protein structure determination. It implies that a first model of the molecule is available (eg a similar protein whose structure has been determined previously). Six variables (three rotational and three translational) are needed to described the transformation from one set of coordinates (in the initial model) to the new one (corresponding to the diffracted intensities). Functions derived from the Patterson synthesis can be used to determine the rotations and translations to be applied to the molecule to bring it in the unit cell.
The basic idea of the rotation function is that when the orientation of the probe model coincides with the orientation of the molecule in the crystal, the model Patterson function (Pcalc(r)) will ressemble the Patterson function calculated from the observed structure factors (Pobs(r)). The rotational search may be carried out by looking for agreement between the Patterson functions of the search and target structure as a function of their relative orientation (C is a rotation operator that rotates the coordinate system of Pcalc(r) with respect to Pobs(r).
rotation function R(C) = Integral Pobs(r) Pcalc(C r) dV
Once the orientation of the model probe in the unit cell has been established, the translation parameters corresponding to the relative positions of the molecules in the crystal must be determined. Several methods have been proposed. Some exploit the information on intermolecular atomic distances contained in the Patterson function whereas other techniques simply translate the oriented molecule to every possible position in the cell and then calculate some measure of correctness between the observed and calculated factor amplitudes.
Direct methods
Modern structural analyses also make extensive use of direct methods. The direct methods are based on statistical procedures (mathematical relationships) and depends on the possibility of treating the atoms in a unit cell as being virtually randomly distributed, and then to use statistical techniques to compute the probabilities that the phases have a particular value. It is possible to deduce relations between some structure factors and sums of others, which have the effect of constraining the phases to particular values (with high probabilities for so long as the structure factors are large).
The reliability of direct methods decreases when the number of atoms in the cell increases. In practice, direct methods are robust enough to solve almost any small molecules structures (total number of atoms <>
Anomalous dispersion data
Atoms with absorptions close to the X-ray frequency introduce an extra phase shift in the scattered X-ray. A simple way of depicting the additional phase shift is to imagine the X-rays as exciting the atom (the X-ray is near to some natural frequencies of the atom, leading to resonance), and being delayed in the process. This effect is called anomalous scattering. In these conditions, the atomic scattering factor can be rewritten as the sum of two parts: f = f' + i f" and will thus modify the expression of Fhkl = F'hkl + i F"hkl. A practical result is the breakdown of Friedel's law : Fhkl is no longer equal to F-h-k-l . For light atoms (C, N, O), the effect is negligible but for heavier atoms, a measurable effect does occur.
Two major application to anomalous scattering data are :
1) the determination of the absolute structure of a molecule (choice between enantiomorphs)
2) phase angle information. Accurate measurement of Bijvoet pairs (Fhkl and F-h-k-l) allows the calculation of an anomalous-difference Patterson synthesis with coefficients : Delta Fano = (Fhkl - F-h-k-l)2 usefull to deduce the relative phases. The choice of wavelengths must be such that Delta Fano is optimized.
Detectors :
Like for powder diffractions, photographic techniques are available to record diffraction techniques (eg oscillation camera, Weissenberg technique, precession camera technique). However, modern techniques, usually use diffractometers (for small molecules) or area detectors (for small molecules and macromolecules). In contrast to classical diffractometers where diffracted intensities are detected point-by-point (usually using a photomultiplier) area dectors like charge-coupled detectors (CCD), Image Plates (IP), Multi-Wire Proportional Counters (MWPC) are able to collect large amounts of data and are more and more used in macromolecules crystallography.
Results :
The X-ray technique depends on the availability of a single crystal of the sample. Only small crystals are needed (of side 0.1 to 0.5 mm, even smaller if synchrotron sources are used), but they need to be of high quality. This need has led to a great deal of effort to obtain suitable single crystals, especially for large, biologically important molecules.
Once a single crystal has been obtained, the indexing and determination of the unit cell characteristics are carried out. Next, the intensities are measured and interpreted as structure factors and the relative phases must be assigned. A first electron density map can then be calculated. Mathematical refinement techniques are then used to improve the approximate atom coordinates obtained from the imperfectly phased Fourier synthesis. Among different criteria used to check the progress of a refinement, crystallographers usually use the R factor that expresses the agreement between the current model and the diffracted data :
R = [ SUM (w|Fobs| - |Fcalc|) ] / SUM |Fobs|
Changes in parameters are found that increase the agreement between the measured data and those calculated from the revised model that results from those changes.
At the end of the refinement, one gets final coordinates (x, y, z or fractional x/a, y:b, z/c) and for each atomic position a displacement (sometimes called 'thermal') parameter B. An estimation of standard deviation (esd) of the values is usually listed in parentheses after the refined parameter.
The geometry of the molecule (bong lengths, valence and torsion angles) can be derived from the atomic coordinates. Information about the intermolecular forces (van der Waals interactions, H bonds) stabilizing the different molecules can also be derived from the analysis of the crystal packing.
Atomic vibrations are displacements from equilibrium positions, with frequencies (~ 1013 sec-1) slower that the frequencies of X-rays (3 1010 cm sec-1 / 1.5 10-8 cm ~ 2 1018 sec-1). As a result, the atom may vibrate and be 'viewed' by X rays as apparently stationary, but displaced randomly from its average location from cell to cell. Because the diffraction experiment involves the average of a very large number of unit cells, minor static displacements of atoms closely simulate the effect of vibration. The magnitude of this atomic displacement B is given for each atomic position. It may be either isotropic (one parameter) or anisotropic (usually 6 parameters).
Fuente: http://perso.fundp.ac.be/~jwouters/DRX/diffraction.html#chap202
Abel Colmenares
17.810.847
CRF
Note that the film covers one-half of the circumference of the camera, the line at the top corresponding to the largest diffraction angles (smaller hkl indices).
In modern diffractometers, the sample is spread on a flat plate and the diffraction pattern (position (2 theta) and intensities) is monitored electronically.
Results :
Powder diffraction techniques are
used to identify a sample of a solid substance by comparison of the positions of the diffraction lines and their intensities with a set of standard spectra (the Joint Commitee of Powder Diffraction Standards data bank contains information on over 30 000 substances).
Powder diffraction data are also used to de
termine phase diagrams (different solid phases result in a different diffraction pattern) and to determine the relative amount of each phase present in a mixture.
The technique is also used for the initial determination of the dimension and symmetries of unit cells.
.4. Single-crystal X-ray diffraction
The methods developped by W. Bragg and his son is the foundation of almost all modern work in X-ray crystallography. They used a single crystal and a monochromatic beam of X-rays, and rotated the crystal until a reflection was detected.Principle :
The preliminary investigation of a
crystal studies the diffraction power (cfr resolution for macromolecules) and serves to identify the symmetry and dimensions of the unit cell. From the dimension of the unit cell (volume) and its symmetry (multiplicity) one can deduce the density of the crystal and, eventually compare it with experimental values.
Z is the multiplicity
Computing techniques are now available that lead to automatic determination of the symmetry, shape, and size of the unit cell and to the indexing of the reflections (diffracted spots). A data collection (result of a systematic rotation of the single cryst
al in the X-ray beam) ends up with a (complete) set of data consisting of a list of positions (hkl indices of the reflections) and the corresponding intensities. The problem is to interpret the data in terms of the structure of the crystal.
If the unit cell contains several atoms with scattering factors fi and coordinates (xia, yib, zic), then the overall amplitude of a wave diffracted b
y the (hkl) planes can be expressed as :
Fhkl = SUM fi exp i alphahkl with alphahkl = 2 pi (hxi + kyi + lzi)
The quantity Fhkl is called the structure factor and the intensity of the (hkl) reflection is proportional to |Fhkl|2. In practice, |Fhkl| can be see
n as the amplitude (height of a crest) of the diffracted wave and the relative phase angle, alphahkl, the position of the crest of the wave relative to some fixed origin.
In particular, Friedel's law states that the |Fhkl|2 of centrosymmetrically related Bragg reflections (h,k,l and -h,-k,-l) are equal, even for noncentrosymmetric structures (ie lattice planes can been seen as mirrors from both sides).
The scattering factor f of an atom is proportional to its number of electrons. Because the atom is not punctual, there is a dependence of f on the theta angle (Fig. 19) and it is only for theta = 0° that f = Nbre e-. A direct consequence is that Fhkl (SUM fi exp i alphahkl) is also dependent on the q angle and that the intensities of the diffracted spots decrease with this angle. Depending on the temperature, the volume of occupied by the atom is also different (larger atomic vibrations/disorder at higher temperature). Based on the reciprocity principle, as atoms appear to become broader, the diffraction pattern decreases in extent and becomes narrower. In other words, the wider the atom as a result of increased vibration, the more out of phase are the rays scattered by various parts of the atom. At higher scattering angles the intensity falls off more than calculated from a point atom.
a)
b)The details of the electron distribution inside a unit cell are contained in the structure factor : Fhkl depends on the type of atom (via fi) and on their location (through hxi + kyi + lzi). The electron density, rho, in the unit cell is a Fourier synthesis (Fig. 20) where the amplitudes are the structure factors for all reflections (derived from their intensities) :
rho = 1/V S Fhkl exp -i2 pi (hx + ky + lz)
Fig. 20. Principle of the Fourier synthesis. Electron density (at the bottom) as a result of the Fourier synthesis of a series of terms from h= 0 to h =10 (k = l = 0). Note the importance of the phases (here either 0 or 180°).
From Fig. 20 it is clear that the electron density is dependent on the relative phase angle (in the equation rho is related to Fhkl, ie |Fhkl| and alphahkl). From the measured intensities, however, only the magnitude of |Fhkl| can be obtained (the intensity is proportional to |Fhkl|2) but no information is provided about alphahkl. Crystallographers call it the phase problem.
The phase problem can be overcome (solved) to some extent by a variety of methods. Some of them are considered here briefely : the Patterson synthesis and its application in isomorphous replacement and molecular replacement, direct methods, and anomalous dispersion data.
Patterson synthesis
In a Patterson synthesis, |Fhkl|2, which can be obtained without ambiguity from the intensities, are used in the following expression :
P(uvw) = 1/V SUM |Fhkl|2 exp -i2 pi (hu + kv + lw) = 1/V SUM |Fhkl|2 cos 2 pi (hu + kv + lw)
The function has the same form as the equation for electron density but as only the cosine terms of the structure factor are conserved, no phases are needed and P(uvw) can be calculated directly from the direct measurement in the X-ray diffraction experiment.
The Patterson function can also be viewed as the convolution of the electron density at all points x,y,z in the unit cell with the electron density at points x+u, y+v, z+w :
P(uvw) = INTEGRAL rho(xyz) rho(x+u, y+v, z+w) dV
The outcome of the Patterson synthesis is a map of the vector separations of the atoms (distances and directions between atoms) in the unit cell (Fig. 21) If atom A is at the coordinates (xA,yA,zA), and atom B at (xB,yB,zB), then there will be a peak at (xB-xA, yB-yA, zB-zA) in the Patterson map. There will also be a peak at the negative of these coordinates, because for any vector from A to B, there is as well a verctor from B to A. The Patterson map is thus centrosymmetric, regardless of whether the space group is centro or non-centrosymmetric.
Fig. 21. The Patterson synthesis of the pattern in (a) corresponds to the map in (b). The distance and orientation of eack spot from the origin gives the orientation and separation of an atom-atom vector in (a). Note that the Patterson map is centrosymmetric, regardless of whether the space group is centro or non-centrosymmetric.
The height of the peak in the map is proportional to the product of the atomic numbers of the two atoms, ZAZB. If some atoms are heavy, they dominate the scattering and their locations may be deduced quite readily. There exist algebric tricks to enhance peaks in a Patterson map (sharpened Patterson functions) replacing |Fhkl| by values derived from it.
If a heavy atom (in comparaison with the others) is present in the cell, it will be detected in the Patterson map. It will also dominate the relative phase angles of the Bragg reflections as shown in Fig. 22. The relative phase angle (aM) deduced for the heavy atom can be combined with the observed |F| to perform a Fourier synthesis of the full electron density in the unit cell, and hence locate the light atoms as well as the heavy atoms.
Fig. 22. Relative phase angles in the presence of a heavy atom. The heavy atom (important contribution to the total vector F) has a relative phase angle (aM) that approaches the total phase angle (aT).
In the technique known as isomorphous replacement, heavy atoms are introduced artificially into a complex molecule, without affecting its structure significantly. This strategy is common in the case of macromolecules where a diffraction set of the native crystal (P) is compared to the diffracted intensities of a crystal soaked with a "heavy-atom" (heavy derivative, PH) in order to phase the native set.
FPH = FP + FH ie FH = FPH - FP
The Patterson function used has | |FPH| - |FP| |2 as coefficients. Usually more than one heavy derivative is analysed leading to the so-called multiple isomorphous replacement (MIR) technique.
Molecular replacement
The molecular replacement method is often used for protein structure determination. It implies that a first model of the molecule is available (eg a similar protein whose structure has been determined previously). Six variables (three rotational and three translational) are needed to described the transformation from one set of coordinates (in the initial model) to the new one (corresponding to the diffracted intensities). Functions derived from the Patterson synthesis can be used to determine the rotations and translations to be applied to the molecule to bring it in the unit cell.
The basic idea of the rotation function is that when the orientation of the probe model coincides with the orientation of the molecule in the crystal, the model Patterson function (Pcalc(r)) will ressemble the Patterson function calculated from the observed structure factors (Pobs(r)). The rotational search may be carried out by looking for agreement between the Patterson functions of the search and target structure as a function of their relative orientation (C is a rotation operator that rotates the coordinate system of Pcalc(r) with respect to Pobs(r).
rotation function R(C) = Integral Pobs(r) Pcalc(C r) dV
Once the orientation of the model probe in the unit cell has been established, the translation parameters corresponding to the relative positions of the molecules in the crystal must be determined. Several methods have been proposed. Some exploit the information on intermolecular atomic distances contained in the Patterson function whereas other techniques simply translate the oriented molecule to every possible position in the cell and then calculate some measure of correctness between the observed and calculated factor amplitudes.
Direct methods
Modern structural analyses also make extensive use of direct methods. The direct methods are based on statistical procedures (mathematical relationships) and depends on the possibility of treating the atoms in a unit cell as being virtually randomly distributed, and then to use statistical techniques to compute the probabilities that the phases have a particular value. It is possible to deduce relations between some structure factors and sums of others, which have the effect of constraining the phases to particular values (with high probabilities for so long as the structure factors are large).
The reliability of direct methods decreases when the number of atoms in the cell increases. In practice, direct methods are robust enough to solve almost any small molecules structures (total number of atoms <>
Anomalous dispersion data
Atoms with absorptions close to the X-ray frequency introduce an extra phase shift in the scattered X-ray. A simple way of depicting the additional phase shift is to imagine the X-rays as exciting the atom (the X-ray is near to some natural frequencies of the atom, leading to resonance), and being delayed in the process. This effect is called anomalous scattering. In these conditions, the atomic scattering factor can be rewritten as the sum of two parts: f = f' + i f" and will thus modify the expression of Fhkl = F'hkl + i F"hkl. A practical result is the breakdown of Friedel's law : Fhkl is no longer equal to F-h-k-l . For light atoms (C, N, O), the effect is negligible but for heavier atoms, a measurable effect does occur.
Two major application to anomalous scattering data are :
1) the determination of the absolute structure of a molecule (choice between enantiomorphs)
2) phase angle information. Accurate measurement of Bijvoet pairs (Fhkl and F-h-k-l) allows the calculation of an anomalous-difference Patterson synthesis with coefficients : Delta Fano = (Fhkl - F-h-k-l)2 usefull to deduce the relative phases. The choice of wavelengths must be such that Delta Fano is optimized.
Detectors :
Like for powder diffractions, photographic techniques are available to record diffraction techniques (eg oscillation camera, Weissenberg technique, precession camera technique). However, modern techniques, usually use diffractometers (for small molecules) or area detectors (for small molecules and macromolecules). In contrast to classical diffractometers where diffracted intensities are detected point-by-point (usually using a photomultiplier) area dectors like charge-coupled detectors (CCD), Image Plates (IP), Multi-Wire Proportional Counters (MWPC) are able to collect large amounts of data and are more and more used in macromolecules crystallography.
Results :
The X-ray technique depends on the availability of a single crystal of the sample. Only small crystals are needed (of side 0.1 to 0.5 mm, even smaller if synchrotron sources are used), but they need to be of high quality. This need has led to a great deal of effort to obtain suitable single crystals, especially for large, biologically important molecules.
Once a single crystal has been obtained, the indexing and determination of the unit cell characteristics are carried out. Next, the intensities are measured and interpreted as structure factors and the relative phases must be assigned. A first electron density map can then be calculated. Mathematical refinement techniques are then used to improve the approximate atom coordinates obtained from the imperfectly phased Fourier synthesis. Among different criteria used to check the progress of a refinement, crystallographers usually use the R factor that expresses the agreement between the current model and the diffracted data :
R = [ SUM (w|Fobs| - |Fcalc|) ] / SUM |Fobs|
Changes in parameters are found that increase the agreement between the measured data and those calculated from the revised model that results from those changes.
At the end of the refinement, one gets final coordinates (x, y, z or fractional x/a, y:b, z/c) and for each atomic position a displacement (sometimes called 'thermal') parameter B. An estimation of standard deviation (esd) of the values is usually listed in parentheses after the refined parameter.
The geometry of the molecule (bong lengths, valence and torsion angles) can be derived from the atomic coordinates. Information about the intermolecular forces (van der Waals interactions, H bonds) stabilizing the different molecules can also be derived from the analysis of the crystal packing.
Atomic vibrations are displacements from equilibrium positions, with frequencies (~ 1013 sec-1) slower that the frequencies of X-rays (3 1010 cm sec-1 / 1.5 10-8 cm ~ 2 1018 sec-1). As a result, the atom may vibrate and be 'viewed' by X rays as apparently stationary, but displaced randomly from its average location from cell to cell. Because the diffraction experiment involves the average of a very large number of unit cells, minor static displacements of atoms closely simulate the effect of vibration. The magnitude of this atomic displacement B is given for each atomic position. It may be either isotropic (one parameter) or anisotropic (usually 6 parameters).
Fuente: http://perso.fundp.ac.be/~jwouters/DRX/diffraction.html#chap202
Abel Colmenares
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