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Introduction |
The process of diffraction is actually a convolution of two separate and simultaneous operations, scattering and interference. The scattering function in diffraction depends only upon the interaction between the probe beam (e.g., X rays, electrons, or neutrons) and the sample. The scattered radiation beams interfere with one another because of the periodic nature of the crystalline sample. Thus, scattering studies can be carried out on matter in a variety of physical states, such as liquids, amorphous solids, as well as crystalline solids; diffraction studies can only be carried out on materials with a repeating structure such as crystalline solids. back to top |
X-ray Scattering |
X-ray photons are electromagnetic radiation. Electromagnetic radiation may be described in terms of its electric and magnetic components that oscillate in directions that are normal to the direction of propagation of the photon and each other. X Rays from conventional generators are plane polarized, that is the electric and magnetic vectors associated with a given photon have a random orientation with respect to the orientations of the electric and magnetic vectors of other photons. When the X-ray photons collide with matter, the oscillating electric component of the photons induces oscillations in charged particles of the matter. The oscillating electrons and nuclei then emit secondary "scattered" X rays of the same energy as the incident photons. back to top |
Thomson Scattering |
J. J.
Thomson (Conduction of Electricity through Gases, 2nd ed. (1906)
Cambridge University Press, p321) found that the intensity of this type
of coherent scattering could be represented by the following:
I(2q) = Io [(n e4)/(2 r2 m2 c4 ) ] [(1 + cos2 2q)/2] where n is the number of independently scattering sources (i.e., electrons), e is the charge of the particle, r is the distance from the scatterer, m is the mass of the scatterer, c is the velocity of light, and [(1 + cos22q)/2] represents the partial polarization of the scattered photon. Note that the dependence on mass greatly reduces the scattering intensity from the nuclei. Hence X rays are considered to only be scattered by the electron density of an atom. In describing this process, Thomson demonstrated that the scattered photons were shifted in phase by 180o from the incident X rays. back to top |
Atomic Scattering Factor |
The function that describes the scattering of X-rays by an atom is called the form factor or atomic scattering factor. To a first approximation, these scattering functions are determined assuming that the electron density of each atom is a discrete and spherically symmetric entity. These scattering functions are independent of the wavelength of radiation and only depend on the scattering angle and the type of atom. At zero scattering angle the value for a scattering function of a given atom has a value equal to the number of electrons in the atom. As the scattering angle is increased, the value of the scattering function is decreased. The decrease in scattering function with increasing scattering angle is reasonable because X-ray photons hitting different parts of the electron cloud of an atom are not expected to scatter in phase with one another. Also, the more diffuse the electron cloud, the more rapid will be the reduction in the scattering function with scattering angle. For example both Ca2+ and Cl- have 18 electrons. Each of these species would have scattering functions with a value of 18 at zero scattering angle. However at higher scattering angle, the Cl- species would be expected to have a smaller scattering function than Ca2+ and because it has a more diffuse electron cloud. Scattering functions in diffraction analyses are always calculated from quantum mechanical theory. The assumption of spherically symmetric scattering functions can be a poor approximation for heavy atoms with considerable amounts of d type electrons. back to top |
Displacement Factor |
The expression for the scattering function represents the scattering by an atom at rest (i.e., 0 Kelvin). Changes in temperature affect the thermal motion of atoms and this in turn affects the scattered intensities. In 1913 Peter Debye originally proposed and later Ivar Waller modified a relation describing the effect of the thermal motion of atoms on intensity (P. Debye, Verhand. Deutschen Physik. Gesell., 15, 678-689 (1913), 15, 738-752 (1913), and 17, 857-875 (1913); I. Waller, Z. Physik, 17, 398-408 (1923), Annalen der Physik, 83, 153-183 (1927)). The Debye-Waller equation assumes the form: I = Io exp[-B(sin2q/ l2)] where B = 8p2 u2 and u2 is the mean displacement of atoms. This factor only reduces the intensity of the peaks and does not change the sharpness or shape of the peaks. This displacement factor was used originally to correct calculated intensities for thermal motion of the atoms. However, this factor also takes into account a variety of other factors such as static disorder, absorption, how tightly an atom is bound in the structure, wrong scaling of measurements, and incorrect atomic scattering curves. When the displacement parameter for a given atom is expressed as a single term B, it is said to represent an isotropic model of motion. Atoms that do not vibrate the same amount in all directions may be represented with an ellipsoidal anisotropic model rather than the spherical isotropic model. Ellipsoid models require six displacement variables for each atom. back to top |
Compton Scattering |
Sometimes the incident X rays collide with matter loosing energy. These lower energy X-rays then can interact with matter as above emitting lower energy scattered X-rays. This type of incoherent scattering, called Compton scattering, always occurs whenever Thomson scattering occurs. However, the relative amount of incoherent scattering is quite low. Also, since there is no directionality in incoherent scattering, this type of scattering contributes to the general background intensity. back to top |
Diffraction of X Rays by Atoms in Crystals |
Diffraction in a crystal is a series of events that involve both interference and coherent scattering. Two mathematical descriptions of the interference effect were put forward by Max von Laue in 1912 and W. L. Bragg in 1913 (W. Friedrich, P. Knipping and M. Laue, in Structural Crystallography in Chemistry and Biology, ed. J. P. Glusker, Hutchinson & Ross:Stroudsburg, PA, pp 23-39 (1981); W. L. Bragg, Proc. Camb. Phil. Soc. 17, 43-57 (1913)). The simpler description put forward by Bragg is presented here.back to top |
Bragg's Law |
According to Bragg, X-ray diffraction can be viewed as a process that is similar to reflection from planes of atoms in the crystal. The crystal planes are illuminated at a glancing angle q and X rays are scattered with an angle of reflection also equal to q. The incident and diffracted rays are in the same plane as the normal to the crystal planes. Bragg reasoned that constructive interference would occur only when the path length difference between rays diffracting from parallel crystal planes would be an integral number of wavelengths. When the crystal planes are separated by a distance d, the path length difference would be 2d sin q. Thus, for constructive interference to occur the following relation must hold true. n l = 2 d sin q This relation is known as Bragg's Law. Thus for a given d spacing and wavelength, the first order peak will occur at a particular q value. Similarly, the q values for the second and higher order peaks can be predicted. back to top |
Reciprocal Lattice |
A concept called the reciprocal lattice helps to explain the relative positions of diffraction peaks. Consider normals to all possible direct lattice planes (hkl) to radiate from some point taken as the origin. Terminate each normal at a point a distance 1/dhkl from this origin, where dhkl is the perpendicular distance between planes of the set (hkl). The set of points so determined constitutes the reciprocal lattice. This reciprocal lattice can be represented by vectors of the form: Bhkl = ha* + kb* + lc* , |Bhkl| = K / dhkl where h, k, and l are the indices of sets of planes in the crystal, and K can assume the value of 1, l, or 2pl, depending on the user's convention. The individual lattice vectors have the following definitions: a* = K (b
x c) / (a × b
x c) a = (b*
x c*) / K (a* × b*
x c*) cosa* = (cosb cosg
- cosa)/( sinb sing) cosa = (cosb* cosg*
- cosa*)/( sinb* sing*) V = 1/V* = abc Ö (1 - cos2a - cos2b - cos2g + 2 cosa cosb cosg) V* = 1/V = a*b*c* Ö (1 - cos2a* - cos2b* - cos2g* + 2 cosa* cosb* cosg*) |
Ewald Construction |
From a knowledge of the real cell and the wavelength of the radiation, the reciprocal lattice positions can be determined. Conversely, from a knowledge of the reciprocal lattice vectors and the wavelength, the dimensions of the unit cell parameters can be determined. The reciprocal lattice is a property of the crystal. Thus a rotation of the crystal will cause a similar rotation of the reciprocal lattice. A geometrical description of diffraction that encompasses Bragg's Law and the Laue equations was originally proposed by P. Ewald (Z. Krist. 56, 129-156 (1921)). The advantage of this description, the Ewald construction, is that it allows the observer to calculate which Bragg peaks will be measurable if the orientation of the crystal with respect to the incident beam is known. As an example, consider a 2-dimensional reciprocal lattice. From Bragg's Law diffraction occurs when a set of crystal lattice planes with dhkl spacing are inclined to an angle q hkl with respect to the incident beam. A diffracted beam can be measured at an angle 2q hkl from the incident beam. The diffraction vector is perpendicular to the crystal lattice planes and has a length inversely related to the spacing between the planes, |Bhkl| = 1/dhkl = (2 sin q hkl)/l. In the Ewald construction a circle with radius 1/l is drawn centered at the crystal. In 3-dimensions the circle becomes a sphere, the Ewald sphere. The reciprocal lattice is then drawn on the same scale as the circle (or sphere) with its origin located 1/l from the center of the circle on the opposite side of the incident beam. Now, when the crystal is rotated so that a reciprocal lattice point intersects the Ewald circle (sphere), that reciprocal lattice point is in "reflecting" position. Ewald's construction and Bragg's Law tell us that for a given wavelength there is a limit to the resolution (a minimum dhkl) which can be measured. l = 2dhkl sinqhkl (dhkl)min = l/2 |Bhkl|max = 1/dhkl = 2/l A knowledge of the crystal unit cell orientation leads to a prediction of reciprocal lattice locations. These locations give no information about the locations of atoms in the unit cell. The intensity data contains the information about the atom locations. back to top |
Friedel's Law |
G. Friedel observed that the intensity distribution in diffraction patterns is centrosymmetric, Ihkl = I-h-k-l . (Comptes Rendus, Acad. Sci. (Paris) 157, 1533-1536 (1913)). Any symmetry in the intensities in the diffraction pattern other than from Friedel's Law is called Laue symmetry. The Laue symmetry displayed by a diffraction pattern is the point-group symmetry of the crystal with the addition of a center of symmetry (if not already present). For orthorhombic crystals, Ihkl = Ih-k-l = I-hk-l = I-h-kl , but for monoclinic crystals, only Ihkl = I-hk-l. If a crystal happens to have all three cell angles = 90.0o within experimental error but only Ihkl = I-hk-l then the sample has monoclinic not orthorhombic crystal system symmetry. The symmetry of the crystal system is dictated by the symmetry of the reciprocal lattice intensities not the apparent symmetry of the cell parameters. back to top |
Anomalous Scattering |
Anomalous scattering or dispersion causes a breakdown in Friedel's Law. Anomalous scattering is easily measured for crystals containing atoms that have absorption edges at or just above the wavelength of the incident radiation. These atoms strongly absorb the radiation increasing their energy state and causing a change in the scattering functions for these atoms. fanomj = fj + Dfj' + i Dfj"The values for Dfj' and Dfj" vary with wavelength and type of atom but not with scattering angle. The anomalous scattering effect is important in determining absolute chirality of a structure. This effect has also been used to help phase protein crystal structures. back to top |
Intensity Formula for Diffracted X rays |
The intensity for a given (hkl) measured by rotating the crystal with a uniform angular velocity w through a reciprocal lattice position is given by Ihkl = Io(l3 /w) (Vx L p A/V2) |Fhkl|2 where, Io = incident beam intensity; l = wavelength of radiation; w = rotation velocity of crystal; Vx = volume of the crystal; L = Lorentz factor, which depends on the relative amount of time the peak takes to pass through the Ewald sphere; p = the polarization factor; A = absorption factor ; V = volume of the unit cell; and |Fhkl| = the observed structure factor. This formula was proposed by C. G. Darwin (Phil. Mag. 27, 315-333, 675-690 (1914), Phil. Mag. 43, 800-829, (1922)). back to top |
Kinematic and Dynamic Diffraction |
Most single crystals are composed of a mosaic pattern of blocks that are each slightly misaligned relative to one another. These mosaic blocks are typically about 10-4 mm (103 Å) in diameter. A crystal with such mosaic character is considered "imperfect" because the internal periodicity is not exact. If a truly monochromatic beam of X-rays were to hit a "perfect" crystal, the Bragg condition would be satisfied only at discrete 2q values. However, because radiation is not monochromatic and most crystals exhibit a "mosaic spread," the intensity spots really occur over a small range of 2q values. Diffraction from "imperfect" crystals is said to be "kinematic" because the mosaic nature of the crystal allows all parts of the crystal to be involved in the diffraction process. Kinematic diffraction theory can only explain single scattering events and cannot explain the reduced intensities seen in crystals affected by multiple scattering events. Diffraction that involves multiple scattering events is called extinction. Extinction is very significant for perfect crystals because in these samples the intensity is proportional to the structure factor amplitude |Fhkl| rather than its square |Fhkl|2 as is usually found for crystals with mosaic character. Extinction from "perfect" crystals is called primary extinction. Primary extinction is only seen in crystals formed at high pressure over long periods of time such as certain minerals. Multiple scattering that is observed in crystals that exhibit mosaic character is called secondary extinction. Secondary extinction occurs when parts of the scattered X-ray beam are scattered a second time from parallel planes in the crystal. The multiply scattered beams reduce the intensity measured for the singly scattered spots. Secondary extinction typically occurs for large crystals and is most significant for the strong, low scattering angle peaks. For these peaks, secondary extinction is observed if the |Fc|2 are significantly greater than |Fo|2 values. Dynamic diffraction theory can sometimes be used to explain the observation of reflections that should be systematically absent. In 1937 M. Renninger (Z. Physik 106, 141-176 (1937)) observed that certain peaks that should not be observed due to systematic absences had significant intensity above background. These peaks appeared when two (or more) reciprocal lattice points are simultaneously on the Ewald sphere. In this type of multiple diffraction (n-beam diffraction) the diffracted beams interact with each other causing either an increase or decrease in the intensity over that predicted by kinematic theory. These double diffraction peaks appear in the position in reciprocal space expected for a normal peak, but their profiles are sharper in appearance than ordinary peaks. Because these double diffraction events require two reciprocal lattice points to be on the Ewald sphere at the same time, they may be eliminated by reorienting (rotating) the crystal. These events are common when crystals are oriented (aligned) for photographic work such as with Weissenberg or precession methods. This problem is uncommon for crystals that are randomly aligned on the goniometer head as is typically done in small molecule studies. back to top |