OU Crystallography Lab

Department of Chemistry & Biochemistry
Chemical Crystallography Laboratory

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Much of the material presented below follows Chapter 3 of Fundamentals of Crystallography prepared by Carmelo Giacovazzo.

An excellent web site covering X-ray diffraction methods is available at: http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html.


When X rays were discovered in 18951, their exact nature was not known. The question posed at the time was are X rays particles or are they waves like visible light? Showing that X rays could diffract would prove that X rays have a wave-like nature.

Diffraction effects were known to be observed when the repeat distances in a material were about the same order of magnitude as the wavelength of the radiation. If X rays were waves, then their wavelengths had been estimated to be roughly 0.4-0.6 Å. Thus some material was needed that would have about this same spacing to demonstrate the diffraction effect.

At that time it was strongly believed that crystals were made up of many repeating blocks, each block with the same dimensions. These repeating blocks were believed to contain a constant number of the same types of atoms. From some simple calculations using density, chemical formula, and Avogadro's number, researchers of the time were able to show that simple crystals should have repeating units of the size needed for their proof of the wave-like nature of X rays.

Friedrich, Knipping, and von Laue shot a beam of X rays at a crystal of copper sulfate and observed diffraction spots proving that X rays definitely had a wave-like nature.2

General Scattering Theory

Electromagnetic radiation, such as X-ray photons, may be described in terms of their electric and magnetic components. These components are considered to oscillate transversely and sinusoidally in directions that are normal to the direction of propagation of the photon and normal to each other.

When X-ray photons collide with matter, the oscillating electric field of the radiation causes the charged components of the atoms to oscillate with the same frequency as the incident radiation. Each oscillating dipole returns to a less energetic state by emitting an electromagnetic photon that can, in general, travel in any outward direction. Most of the scattered photons are emitted at the same energy as the energy of the incident photons; this process is called coherent scattering.

J. J. Thomson 3 derived a formula relating the intensity of this type of coherent scattering from a particle. If the incident radiation is not polarized, then his relation takes the form:

I(2θ) = Io [(e4)/(r2 m2 c4)] [(1 + cos2 2θ)/2]

where 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 + cos22θ)/2] represents the partial polarization of the scattered photon.

A proton is about 1800 times heavier than an electron. Most nuclei are made up of many nucleons that have a total mass that is greater than the mass of a proton. Because the scattered intensity is inversely proportional to the square of the mass of the particle emitting the X-ray photon, scattering of X rays from the nucleus is considered negligible compared to the scattering from the electrons of an atoms. Thus, X-ray scattering is considered to be due to the electron density of the atoms in a sample.

Thomson's theory presumes that the electrons that are bound in an atom would oscillate with a natural frequency, ω0. It is also presumed that the frequency of the incident radiation ω is much larger than ω0, but not so large that relativity corrections become important. Under these conditions, the electrons will scatter radiation exactly out of phase of the incident radiation. Hence this theory presumes that photons scattered from free electrons experience a phase shift of π radians from the phase of the incident photons. By convention, the phase shift for coherently scattered radiation is assigned this same phase shift as for free electrons.

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 coherent scattering occurs. However, the relative amount of this incoherent scattering is generally quite low. Also, since there is no directionality in the incoherent scattering, this type of scattering contributes to the measured, general background intensity. The difference in wavelength between the incident beam and a beam that has experienced Compton scattering is given by

Δλ(Å) = 0.024 (1 - cos 2θ).

Several points should be noted from the expression above. The value for Δλ is independent of the wavelength. The maximum value for Δλ is 0.048 Å achieved at 2θ = 180 ° (the backscattering region). This represents a small but significant increase to the background for high scattering angle measurements. The value of Δλ drops to 0 Å at 2θ = 0 °.

The total scattering from the electrons in a sample is due to a combination of the coherent and incoherent scattering.


When a beam of X rays hits an object, the scattered (emitted) X-ray photons travel in all directions away from the sample. This type of scattering is observed for samples of fluids and amorphous materials.

If the object is composed of many repeating blocks in 3-dimensions, as would be found for a crystal, then the waves of the photons emitted from the different repeating blocks will interfere with one another. The waves traveling in most directions will destructively interfere with the other scattered waves. However in some selected directions the wave fronts will constructively interfere. These points of constructive interference, make up an interference or diffraction pattern. The directions and dimensions of the diffraction pattern are related to the dimensions of the repeating unit cell.

A brief illustration of constructive and destructive interference prepared by Thomas Mittiga (laser.physics.sunysb.edu/~tmit/interference.html).

Constructive waves (having the same phase) add 
   together to produce a stronger signal; Destructive waves 
   (having different phases) also add but produce a composite signal of 0.

A java applet that shows the effect of constructive and destructive interference in water waves was prepared for the physics wiki.

Bragg's Law

According to W. L. Bragg4, X-ray diffraction can be viewed as a process that is similar to reflection from planes of atoms in the crystal. In Bragg's construct, the planes in the crystal are exposed to a radiation source at a glancing angle θ and X rays are scattered with an angle of reflection also equal to θ. The incident and diffracted rays are in the same plane as the normal to the crystal planes.

Bragg's Law showing that diffraction 
   occurs only when the path difference of different waves hitting parallel 
   planes in the sample is an integer multiple of wavelengths of the radiation.

Figure 1. Bragg's Law.4

Bragg reasoned that constructive interference would occur only when the path length difference between rays scattered from parallel crystal planes would be an integer number of wavelengths of the radiation. When the crystal planes are separated by a distance d, the path length difference would be 2d sin θ. Thus, for constructive interference to occur the following relation must hold true.

n λ = 2 d sin θ

This relation is known as Bragg's Law. Thus for a given d spacing and wavelength, the first order peak (n = 1) will occur at a particular θ value. Similarly, the θ values for the second (n = 2) and higher order (n > 2) peaks can be predicted.

An interactive example of Bragg's Law is presented by at Cambridgephysics.org.

The above derivation assumes that phase differences between wavelets scattered at different points depend only on path length differences. It is assumed that there is no intrinsic phase change between the incident and scattered beams or that this phase change is constant for all scattering events. This is not always the case as will be shown below (see anomalous scattering).


Crystals are solids that often have well-defined, smooth faces and straight edges. The angles between similar faces of different samples of a crystalline substance are always constant. Also it is possible to cleave some crystals along well-defined faces. These properties led Haüy and others to suggest that crystals are made of many tiny pieces that are stacked in a 3-dimensional array related by simple translation.

In current scientific thought, periodic crystals are defined as solids with long-range, 3-dimensional, internal order. Each unique piece of the 3-dimensional array is called a unit cell.

Unit Cell

Figure 2. Unit cell showing cell parameters.

The shape of any unit cell is described by six parameters. These six parameters are three axial lengths, designated a, b, and c, and three interaxial angles, α, β, and γ. The angle α is the angle between b and c; β is the angle between a and c; and γ is the angle between a and b. In most published papers the axial lengths are expressed in terms of Å (Ångströms), and the interaxial angles are expressed in terms of ° (degrees).

By convention, initial unit cell edges are chosen to be right-handed (a × b is the direction of c), to form a primitive lattice, and to have the smallest volume. The cell is chosen so that abc, and α, β, and γ all < 90 ° or all ≥ 90 °. This type of cell is called the reduced cell. Very specific rules for obtaining the reduced cell have been established by convention.5 The final cell may have a centered lattice or may have a different arrangement of the cell lengths due to space group conventions.

Some possible 2-dimensional lattices are shown below for a 2-dimensional array of commas.

possible 2D lattices for a 
   2D array of objects

Figure 3. Various 2 dimensional lattices.4

In many lattices, both the choice of origin and the choice of lattice vectors are arbitrary. A 2-dimensional example of the choice of origin is given in the transformation from (a) to (b) above. The choice of lattice vectors is shown in the transformation of (a) to (c) above.

Sets and Families of Planes

This discussion will be based on a 3-dimensional lattice of points:

Q = p a + q b + r c

where a, b, and c are lattice vectors and p, q, and r are integers.

Consider sets of parallel planes that intersect all of the points in a lattice. Some examples of these lattice planes in a 2-dimensional lattice are shown below.

Various sets of parallel planes 
      in a lattice

Figure 4. Various sets of planes.4

The intercepts of the planes with the cell edges must be fractions of the cell edge. Thus cell intercepts can be at 1/0 (= ∞), 1/1, 1/2, 1/3 ... 1/n. The conventional way of identifying these sets of planes is by using three integers that are the denominators of the intercepts along the 3 axes of the unit cell (p q r). Thus if a set of planes had intercepts at 1/2 in a, 1/3 in b, and 1/1 in c then the planes would be referred to as the (2 3 1) set of planes. This designation may also used to describe the normal to a given set of planes.

(1 0 1) planes in a lattice

(1 0 2) planes in a lattice

(1 0 -1) planes in a lattice

Figure 5. Monoclinic unit cells showing 3 sets of planes projected down the unique (b) axis. The indices of the planes are the reciprocal of the intercepts of the planes on the cell edges. The perpendicular distance between the planes is called the d-spacing.

In some crystal systems, some sets of planes are considered to have equivalent symmetry to other sets of planes, for example, in the cubic crystal system, the (1 0 0) set of planes is equivalent to the (0 1 0) set and equivalent to the (0 0 1) set. These three non-parallel but symmetry-equivalent sets may be described as the family of planes described using the expression {1 0 0}. Note that the {1 0 0} family of planes for a cubic cell is very different from the {1 0 0} family of planes for a hexagonal or tetragonal crystal, because both hexagonal and tetragonal crystal systems consider the a and b axes to be symmetry equivalent, but not equivalent to the c axis.

Particular directions in a unit cell are described with the nomenclature [p q r] where p, q, and r are integers. The a, b, and c vectors would be described as [1 0 0], [0 1 0], and [0 0 1], respectively. These directions can be easily identified as the vector occuring between the 0 0 0 lattice point and the p q r lattice point.

Crystallographers use a special way to denote negative integers. The negative sign is placed over the integer rather than being placed to the left of the number. Thus the value for -1 is written as 1 and described verbally as either "1 bar" or "bar 1", the latter being preferred in Europe.

Reciprocal Lattice

The scattering of radiation and interference between scattered photons produces a resulting pattern called a diffraction pattern. Note that from Bragg's Law there is a reciprocal relationship between the distances between rows of scattering centers d and the scattering angle θ. Thus there is a reciprocal relationship between the unit cell of the crystal and the lattice pattern of the diffracted spots. One way to conceptually construct the diffracted lattice pattern is as follows.

Consider normals to all possible direct lattice planes (h k l) to radiate from some point taken as the origin. Terminate each normal at a point that is a distance 1/dhkl from this origin, where dhkl is the perpendicular distance between planes of the set (h k l). The set of points so determined constitutes the reciprocal lattice.

2-dimensional real and reciprocal lattices

Figure 6. Real and reciprocal lattices.4

Note that a short axis in real space (the space of the crystal) leads to a large separation between spots in reciprocal (diffracted) space and that a long real axis of the unit cell leads to a short separation between spots. Also note that obtuse angles in the real unit cell lead to acute angles in the reciprocal cell, and vice versa.

The dimensions and orientation of the crystal determine the positions of the spots in a diffraction pattern. It will be shown later that the types and relative positions of atoms in the given unit cell determine the relative intensities of the diffraction maxima.

Lattice Relationships

Let the real cell vectors a, b, and c. There exists another set of vectors a*, b*, and c* such that the following relations must hold:

a* · b = a* · c = b* · a = b* · c = c* · a = c* · b = 0

a* · a = b* · b = c* · c =1

The a*, b*, and c* vectors are called the reciprocal lattice vectors.

The first set of equations above state that a* is perpendicular to the b, c plane, that b* is perpendicular to the c, a plane, and that c* is perpendicular to the a, b plane.

The second set of equations above infer that modulus (length) and direction of the reciprocal lattice vectors.

From the first set of equations above:

a* = m (b × c)

where m is a constant. One possible value for m can be obtained if the last expression above is multiplied by a on both sides producing:

a* · a = 1 = m (b × c · a) = m V

Hence, m = 1/V.

A general reciprocal lattice can be represented by vectors of the form:

Rhkl = K(ha* + kb* + lc*),
| Rhkl | = K / dhkl

where h, k, and l are integer indices of sets of planes in the crystal or integer indices of diffraction spots in a reciprocal lattice. K typically assumes the value of 1, λ, or 2πλ, depending on the user's convention (crystallography, solid-state physics, etc). In later discussions, K will be assumed to have a value of 1. K is shown in the relations below for completeness.

Thus the individual lattice vectors have the following definitions:

a* = K (b × c) / (a · b × c)
b* = K (c × a) / (a · b × c)
c* = K (a × b) / (a · b × c)

a = (b* × c*) / K (a* · b* × c*)
b = (c* × a*) / K (a* · b* × c*)
c = (a* × b*) / K (a* · b* × c*)

cosα* = (cosβ cosγ - cosα) /( sinβ sinγ)
cosβ* = (cosα cosγ - cosβ) /( sinα sinγ)
cosγ* = (cosα cosβ - cosγ) /( sinα sinβ)

cosα = (cosβ* cosγ* - cosα*) /( sinβ* sinγ*)
cosβ = (cosα* cosγ* - cosβ*) /( sinα* sinγ*)
cosγ = (cosα* cosβ* - cosγ*) /( sinα* sinβ*)

V = a · b × c = 1/V* = abc √ (1 - cos2α - cos2β - cos2γ + 2 cosα cosβ cosγ)

V* = a* · b* × c* = 1/V = a*b*c* √ (1 - cos2α* - cos2β* - cos2γ* + 2 cosα* cosβ* cosγ*)

Ewald Sphere

From the real cell and its orientation on an instrument as well as the wavelength of the radiation, the reciprocal lattice positions for a given sample can be determined. Conversely, from 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. Therefore, 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.6 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 on the instrument is known.

As an example, consider a two-dimensional reciprocal lattice. Constructive interference occurs when a set of crystal lattice planes separated by a spacing of dhkl are inclined to an angle θhkl with respect to the incident beam. A diffracted beam can be measured at an angle 2θhkl 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.

|Rhkl| = 1/dhkl = (2 sinθhkl) / λ

In the Ewald construction a sphere with diameter 1/λ is drawn centered at the crystal. The reciprocal lattice is then drawn on the same scale as the sphere with its origin located 1/λ 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 sphere, that reciprocal lattice point is in position to be measured as a point in the diffraction pattern.

Diffraction maxima can be 
   measured when points of the reciprocal lattice intersect with the Ewald 

Ewald's construction and Bragg's Law tell us that for a given wavelength

λ = 2 dhkl sinθhkl

(dhkl)min = λ / 2

|Rhkl|max = 1/(dhkl)min = 2/λ

A web page describing many of these concepts was prepared by Gervais Chapuis and Wes Hardaker entitled An Interactive Course on Symmetry and Analysis of Crystal Structure by Diffraction.

Intensity Formula for Diffracted X Rays

The intensity for a given (hkl) measured by rotating the crystal with a uniform angular velocity ω through a reciprocal lattice position is given by

Ihkl = Io3 /ω) (Vx L p A/V2) |Fhkl|2

where, Io = incident beam intensity; λ = wavelength of radiation; ω = rotation velocity of the 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.7 In practice the measured intensities are corrected for Lorentz, polarization, and absorption effects producing measured structure factors.

Note that the scattered intensity depends on λ3. Thus, if intensity data collected using Mo radiation (λ = 0.71073 Å) leads to a weak data set, then one way to increase the intensity of the data is to recollect the data using Cu radiation (λ = 1.54178 Å). In general changing from Mo to Cu radiation with a difference of a factor of ~2 should lead to an increase in the scattered intensities of about 23 = 8.

Structure Factor

The Interactive Structure Factor Tutorial prepared by Kevin Cowtan presents an interesting demonstration of many of the topics discussed below.

Consider that an atom j is located at rj from the origin in a unit cell of a crystal. This shift in origin from the center of the individual atom means that the distance r in the equation for the scattering by an atom becomes r + rj. Thus the scattering by atom j becomes:

fj = ∫ ρ(r) exp[2πi (r + rj) · S] dv

fj = fj exp(2πi rj · S)

where fj = ∫ exp(2πi r · S) dv

This last expression is called the atomic scattering factor for atom j.

The term S is called the scattering vector. S is the bisector of so, a unit vector in the incident beam direction, and s, a unit vector in the diffracted beam direction. The angle between so and s = 2θ, is called the scattering angle.

|S| = 2 sin θ / λ = 1/dhkl.

Similar expressions may be derived for all of the other atoms in the unit cell. The total scattering power of all of the atoms is given by the sum of the individual scattering amplitudes.

F(S) = ∑ fj

F(S) = ∑ fj exp(2πi rj · S)

Bragg's Law requires that the phase difference between the waves scattered by successive unit cells must be equal to an integral multiple of 2π / λ. Since the scattered wave may be considered as coming from 1/n multiples of the cell edge vectors a, b, and c, then

(2π / λ) (a · S) = 2π h / λ

(2π / λ) (b · S) = 2π k / λ

(2π / λ) (c · S) = 2π l / λ

These equations are known as the Laue equations and are a 3-dimensional representation of Bragg's Law.

The coordinates of atoms, given as the center of the nuclei, are usually represented as fractions of the unit cell edges, and, for the jth atom are labelled xj, yj, and zj. The rj vector may be written as:

rj = axj + byj + czj

Thus the product rj · S may be written as:

rj · S = xja · S + yjb · S + zjc · S

= h xj + k yj + l zj

Finally, the total scattering power for all of the atoms in the unit cell may be written as:

F(hkl) = ∑ fj exp 2π i (h xj + k yj + l zj)

The relation above is known as the structure factor expression. This relation may be recast in terms of its amplitude, |F(hkl)|, and its phase angle, φ(hkl) or in terms of its real, A, and imaginary, B components in the following expressions.

F(hkl) = |F(hkl)| exp [2πi φ(hkl)]

F(hkl) = A + iB

Electron Density

If the structure factor expression is rewritten as a continuous summation over the volume of the unit cell then the expression becomes:

F(S) = ∫ ρ(r) exp 2π i rj · S dv

By multiplying both sides by exp -2π i rj · S and integrating over the volume of diffraction space, dvr, we get an expression for the electron density of the unit cell.

ρ(r) = ∫ F(S) exp -2π i rj · S dvr

Since F(S) is nonzero only at the lattice points, The integral may be written as discrete sums over the three indices h, k, and l:

ρ(xyz) = 1/V ∑ ∑ ∑ F(hkl) exp -2π i(h x + k y + l z)

ρ(xyz) = 1/V ∑ ∑ ∑ |F|(hkl) exp -2π i[h x + k y + l z - φ(hkl)]

where the three summations run over all values of x, y, and z.

Intensities and the Phase Problem

The interaction of the electric vector of the incident radaition with charged matter in atoms generates dipoles in these charged species. The charged species then release this additional energy by emitting X-ray photons with the same energy as the incident radiation. The intensity is found experimentally to be proportional to the square of the structure factor amplitudes. Since F(S) is complex, then its square is given by F(S) × F*(S), where F*(S) is the complex conjugate of F(S).

I(S) F(S) · F*(S) |F(S)|2

Although the structure factor amplitudes may be measured directly from the diffraction experiment, all information concerning the phases of the data are lost.

In the electron density expression, the structure factor term can be rewritten into its amplitude and phase angle giving

ρ(xyz) = 1/v ∑ ∑ ∑ F(hkl) exp 2π i φ(hkl) exp -2π i (h x + k y + l z)

If both the structure factor amplitudes and the phases were known, then the electron density could be directly calculated from the expression above. But, since the phases are lost during measurement, the electron density cannot be directly calculated. This lack of knowledge of the phases is termed the phase problem in crystallography.

Phase angles can be estimated in a variety of ways. Often maps are calculated with measured structure factor amplitudes and these estimated phases to see if other features of the map can be observed.

Calculated Structure Factors

The electron density used in the structure factor expression is related to the types and positions of the atoms in the unit cell. Thus the structure factor can also be calculated as

Fhkl = ∑ fj exp[2πi (hxj + kyj + lzj)]

where the summation runs over all atoms in the unit cell. Assuming that f, the scattering factor expression is described with an isotropic model of the electron density displacements, then f = f o exp[-B(sin2θ / λ2)]

The summation above is over the j atoms in the unit cell. This calculated structure factor can be factored into a real and imaginary component.

Fhkl = Ahkl + iBhkl = ∑|F| exp[iφhkl] = ∑ fj exp[iαj]


Ahkl = ∑ fj cos 2π (h xj + k yj + l zj)

Bhkl = ∑ fj sin 2π (h xj + k yj + l zj)

|Fhkl| = [ |Ahkl|2 + |Bhkl|2 ]1/2

Argand diagram showing how the complex 
   components of the individual atom scattering vectors contribute to produce a 
   single structure factor.

Figure 7. Argand Diagram.

Fhkl can be represented graphically in an Argand diagram, Figure 7, as the sum of vectors each characterized by a modulus of fj and an angle αj with respect to the real axis. The value of φhkl depends on the moduli and the relative orientations of the vectors fj, and is said to be the phase angle of Fhkl

φhkl = tan-1(Bhkl / Ahkl)

Scattering Factor

Let the electron density at a distance r from the center of an atom be ρ(r). Consider the wave scattered at the position r in a direction s relative to the incident beam of radiation in the direction so. The scattered intensity depends on the phase difference, δ, which is 2π / λ times the path length difference, is given by:

δ = (2π / λ)[r · (s - so)] = 2π / λ r · S

where S = (s - so) / λ. The vector S is a vector in reciprocal space. The wave scattered by the volume element dv at r will have an amplitude with a maximum of ρ(r) dv. Combining this with the phase determined previously then the amplitude of the wave at r must be ρ(r) exp(2πi r · S dv. The total scattering power of the atom is given by summing over all volume elements dv of the atom giving:

f(S) = ∫ ρ(r) exp(2πi r · S) dv

This expression represents the atomic scattering factor. In practice, these scattering factors are calculated using the quantum mechanics. Attempts to measure these functions have, to date, not produced functions that are as reliable as the calculated scattering factors.

For routine diffraction experiments, atoms are approximated by discrete spherical scattering functions. The assumption of spherically-symmetric scattering functions can be a poor approximation for heavy atoms with considerable amounts of d and f type valence electrons.

X-ray scattering factor functions

Figure 8. Selected scattering factor functions.8

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 scattering function of a given atom has a value equal to the number of electrons in the atom.

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 less likely to scatter in phase with one another as the scattering angle is increased. 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 value for its scattering function than Ca2+ because Cl- has a more diffuse electron cloud than Ca2+.

Displacement Factor

The expression for the scattering factor function represents the scattering by an atom at 0 Kelvin. Changes in temperature affect the thermal motion of atoms, and this in turn affects the scattered intensities. In 1913 Peter Debye originally proposed9 and later Ivar Waller modified10 a relation describing the effect of the thermal motion of atoms on intensity. The Debye-Waller equation assumes the form:

f = f o exp[-B(sin2θ/λ2)]

where f is the corrected scattering factor for a given atom type; f o is the scattering factor for a given atom calculated at zero Kelvin; B = 8π2 u2 and u2 is the mean square displacement of the 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 functions. 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.

Anomalous Scattering

Scattering factors are calculated assuming that the frequency of the incident radiation is different from any natural absorption frequency of the electrons in the atoms. Although this is generally the case for light atoms and the types of radiation in normal use, it is often not the case for heavier atoms. It is certainly not the case for atoms with Z values just less than that of the anode material of the radiation source. The differences in the scattering of these atoms from their normal scattering factors is called anomalous scattering, anomalous dispersion, or resonant scattering. The term anomalous only implies that correction terms must be applied to the normal scattering factors. This type of scattering is primarily a function of the atom type and the type of radiation, and is generally independent of the scattering angle. Because dispersion is not a function of the scattering angle, the effect is more noticeable as the scattering angle increases and the overall intensities of peaks decreases.

Anomalous dispersion is usually applied as small real and imaginary correction terms to the scattering factor functions of the atoms. The real correction term is usually negative; the imaginary correction term is always positive.

f = (f o + Δf ' + iΔf ") exp[-B(sin2θ/λ2)]

For samples that crystallize in noncentrosymmetric space groups and contain different types of atoms, some of which are anomalous scatterers, there is a small but measurable difference between the intensities of the (h k l) and (h k l) peaks. The differences in the intensities of these data are used to determine the correct absolute structure of a sample. Johannes M. Bijvoet11 was the first to exploit this effect of anomalous scattering to determine the absolute configuration of a crystalline sample. These differences have also been used to solve protein crystal structures by means of multiple-wavelength anomalous dispersion, MAD.

A more thorough discussion of anomalous scattering is available at a site prepared by Ethan A. Merritt.

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 2θ values. However, because radiation is not exactly monochromatic and most crystals exhibit a mosaic spread, the spots really occur over a small range of 2θ values. An interesting page discussing crystal defects was prepared by Prof. Edward Goo.

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, which 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. Renninger12 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 (narrower) 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 modern studies.


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