## Crystallography

#### Table of Contents

- Scattering and Diffraction
- Crystals and their Properties
- Crystal Structures
- Scattering and Diffraction in Practice

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.

### Introduction

When X rays were discovered in 1895^{1}, 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) = I^{o}
[(e^{4})/(r^{2} m^{2} c^{4})] [(1
+ cos^{2} 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 +
cos^{2}2θ)/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, ω_{0}.
It is also presumed that the frequency
of the incident radiation ω is much larger than ω_{0},
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.

### Interference

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).

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. Bragg^{4}, 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.

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

*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*.

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 *a* ≤ *b* ≤ *c*, 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.

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*,

*, and*

**b***are lattice vectors and*

**c***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.

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

*, and 1/1 in*

**b***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.*

**c**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

*axes to be symmetry equivalent, but not equivalent to the*

**b***axis.*

**c**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/d*hkl*
from this origin, where d*hkl* is the perpendicular distance
between planes of the set (*h k l*). The set of points so
determined constitutes the *reciprocal lattice*.

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*,

*, and*

**b***. There exists another set of vectors*

**c****,*

**a****, and*

**b**** such that the following relations must hold:*

**c*** a** ·

*=*

**b**** ·*

**a***=*

**c**** ·*

**b***=*

**a**** ·*

**b***=*

**c**** ·*

**c***=*

**a**** ·*

**c***= 0*

**b*** a** ·

*=*

**a**** ·*

**b***=*

**b**** ·*

**c***=1*

**c**The * a**,

**, and*

**b**** vectors are called the*

**c***reciprocal lattice vectors*.

The first set of equations above state that * a** is perpendicular to the

*,*

**b***plane, that*

**c**** is perpendicular to the*

**b***,*

**c***plane, and that*

**a**** is perpendicular to the*

**c***,*

**a***plane.*

**b**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** ·

*= 1 = m (*

**a***×*

**b***·*

**c***) = m V*

**a**Hence, m = 1/V.

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

**R**_{hkl} =
K(*h** a** +

*k*

** +*

**b***l*

**),*

**c**|

**R**

_{hkl}| = K / d

_{hkl}

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**** = K (*

**b***×*

**c***) / (*

**a***·*

**a***×*

**b***)*

**c**** = K (*

**c***×*

**a***) / (*

**b***·*

**a***×*

**b***)*

**c*** a* = (

** ×*

**b****) / K (*

**c**** ·*

**a**** ×*

**b****)*

**c***= (*

**b**** ×*

**c****) / K (*

**a**** ·*

**a**** ×*

**b****)*

**c***= (*

**c**** ×*

**a****) / K (*

**b**** ·*

**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***= 1/V* = abc √ (1 - cos*

**c**^{2}α - cos

^{2}β - cos

^{2}γ + 2 cosα cosβ cosγ)

V* = * a** ·

** ×*

**b**** = 1/V = a*b*c* √ (1 - cos*

**c**^{2}α* - cos

^{2}β* - cos

^{2}γ* + 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 d*hkl* are
inclined to an angle θ*hkl* with
respect to the incident beam. A diffracted beam can be measured at
an angle 2θ_{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.

|*R**hkl*| = 1/d*hkl*
= (2 sinθ_{hkl}) /
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.

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

λ = 2 d* _{hkl}*
sinθ

_{hkl}

(d* _{hkl}*)

_{min}= λ / 2

|R* _{hkl}*|

_{max}= 1/(d

*)*

_{hkl}_{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

*I _{hkl}* =

*I*

^{o}(λ

^{3}/ω) (V

_{x}L p A/V

^{2}) |

*F*|

_{hkl}^{2}

where, I^{o} = incident beam intensity;
λ = wavelength of radiation;
ω = rotation velocity of the crystal; V_{x} =
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 |*F _{hkl}*| = 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 λ^{3}.
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 2^{3} = 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 **r**_{j} 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** +
**r**_{j}. Thus the scattering by atom j becomes:

**f**_{j} = ∫
ρ(**r**) exp[2*πi* (**r** +
**r**_{j}) · **S**] *dv*

**f**_{j} = *f*_{j}
exp(2π*i* **r**_{j} · **S**)

where *f*_{j} = ∫ exp(2π*i*
**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 **s**_{o}, a unit vector in the incident
beam direction, and **s**, a unit vector in the diffracted beam
direction. The angle between **s**_{o} and **s**
= 2θ, is called the scattering angle.

|S| = 2 sin
θ / λ = 1/d* _{hkl}*.

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**) = ∑

**f**

_{j}

** F**(

**S**) = ∑

*f*

_{j}exp(2π

*i*

**r**

_{j}·

**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**,

**, and**

*b***, then**

*c*(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 *j*th atom are labelled *x*_{j},
*y*_{j}, and *z*_{j}. The
**r**_{j} vector may be written as:

**r**_{j} =
*a**x*_{j} +
*b**y*_{j} +
*c**z*_{j}

Thus the product **r**_{j} · **S**
may be written as:

**r**_{j} · **S** =
*x*_{j}** a** ·

**S**+

*y*

_{j}

**·**

*b***S**+

*z*

_{j}

**·**

*c***S**

= *h x*_{j} + *k y*_{j} +
*l z*_{j}

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

** F**(

*hkl*) = ∑

*f*

_{j}exp 2π

*i*(

*h x*

_{j}+

*k y*

_{j}+

*l z*

_{j})

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 [2π

*i*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*

**r**

_{j}·

**S**

*dv*

By multiplying both sides by exp -2π *i* **r**_{j}
· **S** and integrating over the volume of diffraction space,
*dv*_{r}, we get an expression for the electron density of the unit
cell.

ρ(**r**) = ∫
** F**(

**S**) exp -2π

*i*

**r**

_{j}·

**S**

*dv*

_{r}

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 *h*, *k*,
and *l*.

### 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 -2*π 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* = ∑

*f*

_{j}exp[2π

*i*(

*hx*

_{j}+

*ky*

_{j}+

*lz*

_{j})]

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(sin^{2}θ / λ^{2})]

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* =

*+*

**A**hkl*i*= ∑|

**B**hkl*| exp[*

**F***i*φ

_{hkl}] = ∑

*f*

_{j}exp[

*i*α

_{j}]

where

* Ahkl* = ∑

*f*

_{j}cos 2π (

*h x*

_{j}+

*k y*

_{j}+

*l z*

_{j})

* Bhkl* = ∑

*f*

_{j}sin 2π (

*h x*

_{j}+

*k y*

_{j}+

*l z*

_{j})

|*Fhkl*| = [ |*Ahkl*|^{2} +
|*Bhkl*|^{2} ]^{1/2}

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

*f*

_{j}and an angle α

_{j}with respect to the real axis. The value of φ

_{hkl}depends on the moduli and the relative orientations of the vectors

*f*_{j}, and is said to be the phase angle of

**F**hklφ_{hkl} =
tan^{-1}(* Bhkl* /

*)*

**A**hkl### 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 **s**_{o}. 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** -
**s**_{o})] = 2π / λ
**r** · **S**

where **S** = (**s** -
**s**_{o}) / 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.

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
Ca^{2+} 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 Ca^{2+} because Cl^{-} has a more
diffuse electron cloud than Ca^{2+}.

### 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
proposed^{9} and later Ivar Waller modified^{10} 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(sin^{2}θ/λ^{2})]

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 =
8π^{2} u^{2} and
u^{2} 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} +
Δ*f* ' + iΔ*f* ")
exp[-B(sin^{2}θ/λ^{2})]

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. Bijvoet^{11} 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
(10^{3} 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 |*F*_{c}|^{2} are
significantly greater than |*F*_{o}|^{2} values.

Dynamic diffraction theory can sometimes be used
to explain the observation of reflections that should be
systematically absent. In 1937 M. Renninger^{12} 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.

### References

- W. C. Röntgen
*Sitzungs. Würzb. Phys.-Med. Ges.*,**1895**, pp132-141. English transl. A. Stanton,*Science*,**1896**,*3*, 227-231;*Nature*,**1896**,*53*, 274-276. - W. Freidrich, P. Knipping, and M. Laue,
*Proc. Bavarian Acad. Sci.*,**1912**, 303-322; Reprinted in*Naturewissenschaften*,**1952**,*39*, 367. [Laue became von Laue after the 1912 publication.] - J. J. Thomson,
*Conduction of Electricity through Gases*, 2nd ed.**1906**, Cambridge University Press, 321. - W. L. Bragg,
*Proc. Camb. Phil. Soc.*,**1913**,*17*, 43-57. - Michael Liang,
**1997**,An Introduction to the Scope, Potential and Applications of X-ray Analysis

in International Union of Crystallographers Teaching Pamphlets available at: http://www.ch.iucr.org/iucr-top/comm/cteach/pamphlets.html. - P. M. deWolff in
International Tables for Crystallography, Vol. A

, Section 9.3, Kluwer:Boston (1996) pp.741-748. - P. Ewald
*Z. Krist.*,**1921**,*56*, 129-156. - C. G. Darwin,
*Phil. Mag.*,**1914**,*27*, 315-333, 675-690;*Phil. Mag.***1922**,*43*, 800-829. - S. C. Wallwork,
**1997**, "Introduction to the Calculation of Structure Factors" in International Union of Crystallographers Teaching Pamphlets available at: http://www.ch.iucr.org/iucr-top/comm/cteach/pamphlets.html. - P. Debye,
*Verhand. Deutschen Physik. Gesell.*,**1913**,*15*, 678-689;**1913**,*15*, 738-752; and**1913**,*17*, 857-875. - I. Waller,
*Z. Physik*,**1923**,*17*, 398-408; and*Annalen der Physik*,**1927**,*83*, 153-183. - J. M. Bijvoet, A. F. Peerdeman, and A. J. Bommel,
*Nature*,**1951**,*168*, 271-272. - M. Renninger,
*Z. Physik***1937**,*106*, 141-176.