Basic Crystallography
Part 1
Theory and Practice of X-ray
Crystal Structure Determination
Charles Campana, Ph.D.
Senior Applications Scientist
Bruker AXS
Course Overview
Basic Crystallography – Part 1
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Introduction: Crystals and Crystallography
Crystal Lattices and Unit Cells
Generation and Properties of X-rays
Bragg's Law and Reciprocal Space
X-ray Diffraction Patterns from Crystals
Basic Crystallography – Part 2
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Review of Part 1
Selection and Mounting of Samples
Unit Cell Determination
Intensity Data Collection
Data Reduction
Structure Solution and Refinement
Analysis and Interpretation of Results
Introduction to
Crystallography
What are Crystals?
4
Examples of Crystals
Examples of Protein Crystals
Growing Crystals
Kirsten Böttcher and Thomas Pape
Crystal Systems and
Crystal Lattices
What are Crystals?
 A crystal or crystalline solid is a solid material
whose constituent atoms, molecules, or ions
are arranged in an orderly, repeating pattern
extending in all three spatial dimensions.
9
Foundations of Crystallography
 Crystallography is the study of crystals.
 Scientists who specialize in the study of crystals are called
crystallographers.
 Early studies of crystals were carried out by mineralogists
who studied the symmetries and shapes (morphology) of
naturally-occurring mineral specimens.
 This led to the correct idea that crystals are regular threedimensional arrays (Bravais lattices) of atoms and
molecules; a single unit cell is repeated indefinitely along
three principal directions that are not necessarily
perpendicular.
The Unit Cell Concept
Ralph Krätzner
Unit Cell Description in terms of
Lattice Parameters
 a ,b, and c define the
edge lengths and are
referred to as the
crystallographic axes.
c
a



b
 , , and  give the
angles between these
axes.
 Lattice parameters 
dimensions of the
unit cell.
Choice of the Unit Cell
Choice of the Unit Cell
A
B
A
B
C
D
No symmetry - many possible
unit cells. A primitive cell with
angles close to 90º (C or D) is
preferable.
C
The conventional C-centered
cell (C) has 90º angles, but one
of the primitive cells (B) has
two equal sides.
7 Crystal Systems - Metric Constraints
 Triclinic - none
 Monoclinic -  =  = 90,   90
 Orthorhombic -  =  =  = 90
 Tetragonal -  =  =  = 90, a = b
 Cubic -  =  =  = 90, a = b = c
 Trigonal -  =  = 90,  = 120, a = b
(hexagonal setting) or
 =  =  , a = b = c (rhombohedral setting)
 Hexagonal -  =  = 90,  = 120, a = b
Bravais Lattices
 Within each crystal system, different types of
centering produce a total of 14 different lattices.
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P – Simple
I – Body-centered
F – Face-centered
B – Base-centered (A, B, or C-centered)
 All crystalline materials can have their crystal
structure described by one of these Bravais lattices.
Bravais Lattices
Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, 3rd Ed., Addison-Wesley
Bravais Lattices
Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, 3rd Ed., Addison-Wesley
Bravais Lattices
Bravais Lattices
Crystal Families, Crystal Systems,
and Lattice Systems
Generation of and
Properties of X-rays
A New Kind of Rays
 Wilhelm Conrad Röntgen
 German physicist who
produced and detected
Röntgen rays, or X-rays,
in 1895.
 He determined that these
rays were invisible,
traveled in a straight line,
and affected photographic
film like visible light, but
they were much more
penetrating.
Properties of X-Rays
 Electromagnetic radiation
(l = 0.01 nm – 10 nm)
 Wavelengths typical for
XRD applications:
0.05 nm to 0.25 nm or
0.5 to 2.5 Å
1 nm = 10-9 meters = 10 Å
 E = ħc / l
Generation of Bremsstrahlung
Radiation
Electron
(slowed down and
changed direction)
nucleus
Fast incident
electron
electrons
Atom of the anode material
X-ray
 “Braking” radiation.
 Electron deceleration releases radiation across a spectrum of wavelengths.
Generation of Characteristic Radiation
Photoelectron
M
Emission
K
L
K
Electron
L
K
 Incoming electron
knocks out an
electron from the
inner shell of an
atom.
 Designation K,L,M
correspond to shells
with a different
principal quantum
number.
Generation of Characteristic Radiation
 Not every electron in
each of these shells
has the same energy.
The shells must be
further divided.
 K-shell vacancy can
be filled by electrons
from 2 orbitals in the
L shell, for example.
Bohr`s model
 The electron
transmission and the
characteristic
radiation emitted is
given a further
numerical subscript.
Generation of Characteristic Radiation
Energy levels (schematic) of the electrons
M
Intensity ratios
K  K  K =     
L
K
K
K
K
K
Emission Spectrum of an X-Ray Tube
Emission Spectrum of an X-Ray Tube:
Close-up of K
Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, 3rd Ed., Addison-Wesley
Sealed X-ray Tube Cross Section
Cullity, B.D. and Stock, S.R., 2001, Elements of X-Ray Diffraction, 3rd Ed., Addison-Wesley
 Sealed tube
 Cathode / Anode
 Beryllium windows
 Water cooled
Characteristic Radiation for Common
X-ray Tube Anodes
Anode
K1 (100%)
K2 (50%)
K (20%)
Cu
1.54060 Å
1.54439 Å
1.39222 Å
Mo
0.70930 Å
0.71359 Å
0.63229 Å
Modern Sealed X-ray Tube
 Tube made from
ceramic
 Beryllium window is
visible.
 Anode type and focus
type are labeled.
Sealed X-ray Tube Focus Types:
Line and Point
Target
Take-off angle
 The X-ray beam’s cross section
at a small take-off angle can be a
line shape or a spot, depending
on the tube’s orientation.
Filament
Spot
Target
Line
 The take-off angle is the targetto-beam angle, and the best
choice in terms of shape and
intensity is usually ~6°.
 A focal spot size of 0.4 × 12 mm:
0.04 × 12 mm (line)
0.4 × 1.2 mm (spot)
Interaction of X-rays
with Matter
Interactions with Matter
d
incoherent scattering
lCo (Compton-Scattering)
wavelength
lPr
intensity Io
coherent scattering
lPr (Bragg-scattering)
absorption
Beer´s law I = I0*e-µd
fluorescence
l> lPr
photoelectrons
Coherent Scattering
 Incoming X-rays are
electromagnetic waves
that exert a force on
atomic electrons.
e-
 The electrons will begin
to oscillate at the same
frequency and emit
radiation in all directions.
Constructive and Deconstructive
Interference
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Coherent Scattering by an Atom
 Coherent scattering by an
atom is the sum of this
scattering by all of the
electrons.
2q
 Electrons are at different
positions in space, so
coherent scattering from
each generally has different
phase relationships.
 At higher scattering angles,
the sum of the coherent
scattering is less.
Atomic Scattering Factor
 Scattering factor is used as an
indication of the strength of
scattering of an atom in
particular direction.
 Scattering is a maximum in the
forward scattering direction and
decreases with scattering angle.
f=
Amplitude of wave scattered by atom
Amplitude of wave scattered by one electron
X-ray Diffraction by
Crystals
Diffraction of X-rays by Crystals
 The science of X-ray
crystallography originated in
1912 with the discovery by
Max von Laue that crystals
diffract X-rays.
 Von Laue was a German
physicist who won the Nobel
Prize in Physics in 1914 for
his discovery of the diffraction
of X-rays by crystals.
Max Theodor Felix von Laue
(1879 – 1960)
X-ray Diffraction Pattern from a
Single-crystal Sample
Rotation Photograph
Diffraction of X-rays by Crystals
After Von Laue's pioneering
research, the field developed
rapidly, most notably by
physicists William Lawrence
Bragg and his father William
Henry Bragg.
In 1912-1913, the younger
Bragg developed Bragg's law,
which connects the observed
scattering with reflections from
evenly-spaced planes within
the crystal.
William Henry Bragg
William Lawrence Bragg
Bragg’s Law
 X-rays scattering coherently
from 2 of the parallel planes
separated by a distance d.
 Incident angle and reflected
(diffracted angle) are given by q.
Bragg’s Law
l
2
= d  sin q
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The condition for constructive
interference is that the path
difference leads to an integer
number of wavelengths.
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Bragg condition  concerted
constructive interference from
periodically-arranged scatterers.
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This occurs ONLY for a very
specific geometric condition.
nl = 2d sin q
Bragg’s Law
nl = 2d sin(q)
q
q
d
We can think of diffraction as reflection at sets of planes running
through the crystal. Only at certain angles 2θ are the waves
diffracted from different planes a whole number of wavelengths
apart (i.e., in phase). At other angles, the waves reflected from
different planes are out of phase and cancel one another out.
Reflection Indices
z
 These planes must intersect the
cell edges rationally, otherwise
the diffraction from the different
unit cells would interfere
destructively.
y
x
 We can index them by the
number of times h, k and l that
they cut each edge.
 The same h, k and l values are
used to index the X-ray
reflections from the planes.
Planes 3 -1 2 (or -3 1 -2)
Examples of Diffracting Planes and
their Miller Indices
 Method for identifying
diffracting planes in a crystal
system.
c
 A plane is identified by indices
(hkl) called Miller indices, that
are the reciprocals of the
fractional intercepts that the
plane makes with the
crystallographic axes (abc).
b
a
Diffraction Patterns
Two successive CCD detector images with a crystal
rotation of one degree per image:
For each X-ray reflection (black dot), indices h,k,l can be
assigned and an intensity I = F 2 measured
Reciprocal Space
 The immediate result of the X-ray diffraction
experiment is a list of X-ray reflections hkl and
their intensities I.
 We can arrange the reflections on a 3D grid based
on their h, k and l values. The smallest repeat
unit of this reciprocal lattice is known as the
reciprocal unit cell; the lengths of the edges of
this cell are inversely related to the dimensions of
the real-space unit cell.
 This concept is known as reciprocal space; it
emphasizes the inverse relationship between the
diffracted intensities and real space.
The Crystallographic Phase Problem
The Crystallographic Phase Problem
 In order to calculate an electron density map,
we require both the intensities I = F 2 and the
phases  of the reflections hkl.
 The information content of the phases is
appreciably greater than that of the intensities.
 Unfortunately, it is almost impossible to
measure the phases experimentally!
This is known as the crystallographic phase
problem and would appear to be unsolvable!
Crystal Structure Solution by
Direct Methods
 Early crystal structures were
limited to small, centrosymmetric structures with
‘heavy’ atoms. These were
solved by a vector (Patterson)
method.
 The development of ‘direct
methods’ of phase
determination made it
possible to solve non-centrosymmetric structures on ‘light
atom’ compounds
Rapid Growth in Number of Structures
in Cambridge Structural Database
The Structure Factor F and
Electron Density 
Fhkl = ∫ V xyz exp[+2i(hx+ky+lz)]dV
xyz = (1/V) hkl Fhkl exp[-2i(hx+ky+lz)]
F and  are inversely related by these Fourier transformations.
Note that  is real and positive, but F is a complex number:
in order to calculate the electron density from the diffracted
intensities I = F2, we need the PHASE () of F.
Unfortunately, it is almost impossible to measure  directly!
Real Space and Reciprocal Space
Real Space
Reciprocal Space
 Unit Cell (a, b, c, , , )
 Electron Density, (x, y, z)
 Atomic Coordinates –
x, y, z
 Thermal Parameters – Bij
 Bond Lengths (A)
 Bond Angles (°)
 Crystal Faces
 Diffraction Pattern
 Reflections
 Integrated Intensities –
I(h,k,l)
 Structure Factors –
F(h,k,l)
 Phase – (h,k,l)
X-Ray Diffraction
X-ray beam
l  1Å
(0.1 nm)
~ (0.2mm)3 crystal
~1013 unit cells,
each ~ (100Å)3
Diffraction pattern on a
CCD detector
Summary of Part 1
 Introduction to crystals and crystallography
 Definitions of unit cells, crystal systems, Bravais lattices
 Introduction to concepts of reciprocal space, Fourier
transforms, d-spacing, Miller indices
 Bragg’s Law – geometric conditions for relating the
concerted coherent scattering of monochromatic X-rays
by diffracting planes of a crystal to its d-spacing.
 Part 2 – We will use these concepts and apply them
in a practical way to demonstrate how an X-ray
crystal structure is carried out.