Basic Crystallography
Part 2
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
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
Review of Part 1
Important Concepts
Important Concepts - Crystals
 A crystal is made up of atoms, molecules, or ions arranged in an orderly
repeating pattern extending in all three spatial dimensions.
 The crystal is similar to a 3-dimensional ‘wallpaper pattern’ made up of
millions of identical small ‘bricks’ or unit cells. The size, shape and
dimensions of the unit cell are called lattice parameters
(a, b, c, alpha, beta, and gamma).
 The lattice parameters are chosen according to accepted conventions for
the 7 crystal systems (triclinic, monoclinic, orthorhombic, trigonal,
tetragonal, hexagonal and cubic).
 Crystal lattices are also classified into 14 Bravais lattices, which include
primitive (P), end-centered (A, B, or C), body-centered (I), and facecentered (F) lattices.
 When all possible three-dimensional rotational and translational symmetry
elements are combined, they form a set of 230 crystallographic space
groups.
Important Concepts – X-rays
 X-rays are produced by accelerating high-energy electrons
toward a metal target (anode). Collision of the electrons
with the anode generates Bremsstrahlung (white) radiation
as well as characteristic Ka and Kb radiation. The useful
range of X-ray wavelengths for XRD applications: 0.05 nm
to 0.25 nm or 0.5 Å to 2.5 Å (1 nm = 10-9 meters = 10 Å).
 When monochromatic X-rays interact with the electrons of
the atoms, they undergo coherent scattering. When the
atoms are arranged in a regular array, they produce a
diffraction pattern due to constructive and destructive
interference of electromagnetic waves. The properties of
the diffraction pattern are well described by Bragg’s Law.
Important Concepts – Diffraction
Patterns and Reciprocal Space
 When X-rays are diffracted from a parallel set of planes
with Miller indices h, k, and l, they produce a reflection
with the corresponding h, k, and l indices.
 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.
Important Concepts – Fourier
Transform Relationships
Real Space
Reciprocal Space
 Unit Cell (a, b, c, a, b, )
 Electron Density, (x, y, z)
 Atomic Coordinates –
x, y, z
 Thermal Parameters – Bij
 Bond Lengths (A)
 Bond Angles (º)
 Crystal Faces
 Diffraction Pattern
 Reflections
 Integrated Intensities –
I(h,k,l)
 Structure Factors –
F(h,k,l)
 Phase – a(h,k,l)
Introduction to X-ray
Crystal Structure
Determination
Flowchart for X-ray Structure
Determination
Select, mount, and optically align a suitable crystal
Evaluate crystal quality; obtain unit cell geometry
and preliminary symmetry information
Measure intensity data
Data reduction
Solve the structure
Complete and refine the structure
Adapted from William Clegg
“Crystal Structure Determination”
Oxford 1998.
Interpret the results
X-ray Crystal Structure
Determination
Selection and Mounting of
Sample
Sample Requirements
 Prepare and purify material to be analyzed
 Grow “X-ray quality” crystals
• Slow evaporation
• Solvent or vapor diffusion
• Sublimation
 Select specimen for analysis
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Suitable size – 0.10 to 0.50 mm in all dimensions
No obvious cracks or ‘twinning’
Natural faces, if possible
Use polarizing microscope to screen specimens
Mounting of Samples
 Use micro-tools or acupuncture needles and oil to separate
selected sample
 Mount specimen for analysis
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Glass capillary - very air-sensitive samples
Glass fiber (glue) – room temperature
Cryo-Loop (Paratone-N oil) – low temperature
MiTeGen mounts (Paratone-N oil) – low temperature
Mounting of Sample
Goniometer Head
Huber model 1004 goniometer head
X-ray Crystal Structure
Determination
Hardware and
Instrumentation
3-Circle Goniometer
 The most common type of
goniometer is the “3-circle
goniometer", which offers
two angles of rotation: the ω
angle, which rotates about
an axis perpendicular to the
beam and the φ angle about
the loop/capillary axis.
The c angle is fixed at the
“magic angle” of 54.74° with
respect to the ω axis.
 The oscillations carried out
during data collection
involve either the ω axis or
the φ axis.
Model of a Kappa Goniometer
X-ray crystallography. (2010, April 19). In Wikipedia, The Free
Encyclopedia. Retrieved 16:17, April 21, 2010, from
http://en.wikipedia.org/w/index.php?title=Xray_crystallography&oldid=357005816
Kappa_goniometer_animation.ogg
Kappa_goniometer_animation.ogg
Kappa_goniometer_animation.ogg
X-ray Sources
 The experiment involves irradiating the mounted crystal
with a beam of monochromatic X-rays. The standard X-ray
source for conventional laboratory systems is a ceramic Xray tube which operates at 1500 to 2000 W. For very small
specimens, micro-focus sources or rotating anode sources
may be required.
 In all of these systems, the sources produce both
Bremsstrahlung (white) radiation and strong characteristic
Kα and Kβ lines corresponding to the energy differences
between inner-shell electrons of the target metal.
X-ray Tube Radiation Choices
The most commonly-used X-ray targets for single crystal X-ray
diffraction are copper and molybdenum.
Ka1
Comments
Cu
1.54060 Å
Best for organic compounds (absolute
structures), small specimens, large unit cells
(e.g., proteins).
Mo
Best for minerals, inorganic and solid-state
compounds with strongly absorbing elements,
0.70930 Å charge density; Preferred source for routine
structures.
Anode
X-ray Sources
 The X-rays are usually passed through monochromators or
X-ray mirrors to eliminate white radiation and Kβ radiation
and to produce a single wavelength (Kα radiation only).
 This monochromatic beam is then collimated to a single,
intense small beam before it is allowed to strike the crystal.
 Collimation is done either with a collimator or with
monocapillary optics. Pinholes may also be used to adjust
the size and shape of the X-ray beam striking the
specimen.
Other Modern Laboratory X-ray
Sources
ImS
TXS Rotating Anode
Measurement of Intensity Data
The intensities of these reflections
may be recorded with a chargecoupled device (CCD) detector.
X-ray Crystal Structure
Determination
Small Molecule Example
Small Molecule Example – YLID
Enter Crystal Information
Small Molecule Example – YLID
Optically Align Sample
Unit Cell Determination
 The experiment generally begins with the measurement of
three small sets of images (typically 12 to 30 images per
set) with the sample oriented in approximately orthogonal
positions.
 The positions of the spots (reflections) are then indexed
using an auto-indexing routine, which assigns a set of three
unique Miller indices (h, k, l) to each of the measured
reflections. At the same time, this routine determines the
dimensions (a, b, c, a, b , and V) of the crystallographic
unit cell and calculates an orientation matrix from which the
positions of all remaining reflections may be predicted.
 A by-product of indexing is determination of the unit cell
symmetry, the crystal system and the Bravais lattice.
Density, Volume and Z Value
 The density of a crystal is given by:
 = 1024ZM / NaV
where  = density in mg·m-3, Z = number of molecules in one
unit cell, M = molecular weight in Da, Na = Avogadro’s number =
6.0226×1023 and V = volume of the unit cell in Å3.
 The Z value (number of molecules per unit cell) may be
estimated by dividing the unit cell volume by 18 to obtain the
number of non-hydrogen atoms in the unit cell (Rule of 18 –
where we assume that the volume of a non-hydrogen atom is
about 18 Å3, hydrogen atoms are ignored). The result is then
divided by the number of non-hydrogen atoms in each molecule
to estimate Z (to the nearest whole number).
Small Molecule Example – YLID
Automatic Unit Cell Determination
 Measured 3 sets of 12 images
(10 sec. exposure times)
 Located 84 reflections above
20s(I)
 Indexed 82 of 84 reflections
 Determined the unit cell to be
orthorhombic P (primitive);
note that all angles are 90°
 Volume is 994 Å3; from this
we can calculate that there
are ~994/18 = 55.222 nonhydrogen atoms in the unit
cell. C11H10O2S has 14 nonhydrogen atoms; 55.22/14 =
3.944 ≈ 4.0 = Z
Small Molecule Example – YLID
Indexed Reflections
 These two slides show that
all 82 indexes reflections
lie at the center of the grid
lines 0n all three
projections.
 These 3 projections also
illustrate the concept of
reciprocal space.
• 0kl projection (k vertical, l
horizontal)
• h0l projection (l vertical, h
horizontal)
• hk0 projection (h vertical, k
horizontal)
0kl projection
Small Molecule Example – YLID
Indexed Reflections
h0l projection
hk0 projection
X-ray Crystal Structure
Determination
Data Collection
Measurement of Intensity Data
 One image of spots is insufficient to reconstruct the
whole crystal; it represents only a small slice of the full
Fourier transform.
 To collect the complete diffraction pattern, the crystal
must be rotated, in small φ or ω steps, through many
combinations of angles, with an image recorded at every
step.
 However, if the crystal has a higher symmetry, a smaller
unique data set be sufficient to solve the structure.
Data Collection Options
 Modern instruments offer many options for selecting an optimum
data collection strategy for each sample:
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Choice of wavelength – Mo or Cu
Crystal-to-detector distance (typically 4.0 to 6.0 cm.)
Scan widths (0.3 to 1.0 degrees per step in w or f)
Exposure time per image (5 to 60 sec.)
Resolution (0.84 Å max. for Cu, 0.77 Å typical for Mo)
Whole “sphere” or minimum unique dataset
Total data collection time
Sample temperature (e.g., RT or 100 K)
 Data collection strategies may depend upon:
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Size and diffracting power of specimen
Mosaicity and rocking curve
Data collection time available
Stability of compound
Length of maximum unit cell axis
Small Molecule Example – YLID
Typical Data Collection
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Goniometer: 3–circle (c fixed at 54.74°)
Radiation choice: Mo ( = 0.71073 Å)
Crystal-to-detector distance: 6.0 cm (60 mm)
Scan width: 0.5° in w
Exposure time: 10 sec. per image
Resolution: 0.77Å (55.00° in 2θ)
Whole “sphere”: 4 runs of 366 images each (512 × 512 mode)
Total data collection time: ~6 hours
Sample temperature: 23° C (296 K)
Small Molecule Example – YLID
One Image from Data Collection
X-ray Crystal Structure
Determination
Data Reduction
Data Reduction
 The recorded series of two-dimensional diffraction images must be
converted into a three-dimensional array of indexed reflections,
each of which has an associated intensity, I, and standard
deviation, s(I). This process is called data reduction.
 The first part of the data reduction process is called integration.
This procedure uses the orientation matrix and applies many
corrections as it converts the hundreds or thousands of images—
containing many thousands of reflections—into a single file,
consisting of individual records of the Miller indices, intensity with
standard deviation, and direction cosines for each reflection.
 The second part of the data reduction uses the direction cosines to
correct for absorption of X-rays by the sample, normalizes the
sigma values, scales and sorts the data for structure determination,
and performs a complete error analysis of the data.
Small Molecule Example – YLID
Screen from Integration
Small Molecule Example – YLID
Final Unit Cell Parameters
 The final unit-cell constants are calculated from the centroids of
many thousands of reflections selected from the entire data set
and typically have relative errors of less than 3/100,000.
Small Molecule Example – YLID
Absorption Correction and Scaling
Small Molecule Example – YLID
Absorption Correction and Scaling
Small Molecule Example – YLID
Absorption Correction and Scaling
X-ray Crystal Structure
Determination
Solution of Structures
Symmetry and Space Groups
 In crystallography, the space group of a crystal is a description of the
symmetry of the crystal, and can have one of 230 types.
 The space groups in three dimensions are made from combinations of
the 32 crystallographic point groups with the 14 Bravais lattices which
belong to one of 7 lattice systems. This results in a space group being
some combination of the translational symmetry of a unit cell
including lattice centering, the point group symmetry operations of
reflection, rotation and improper rotation (also called roto-inversion),
and the screw axis and glide plane symmetry operations. The
combination of all these symmetry operations results in a total of 230
unique space groups describing all possible crystal symmetries.
Space Group Determination and
Formula
 The first step in the solution of a crystal structure is the
assignment of the space group. In practice, the space group
determination is often done automatically, following the data
reduction step.
 The choice of the space group includes the following
considerations:
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The crystal system and Bravais lattice
Analysis of “systematically absent” classes of reflections
The evaluation of |E2 -1| values
Relative frequency of occurrence in CSD
 The normalized structure factors depend upon having an
approximately correct chemical formula. The ‘rule of 18’ should be
used to calculate the Z value.
Small Molecule Example – YLID
Space Group Determination
 5780 reflections read from .hkl file
 Bravais lattice is Primitive
Small Molecule Example – YLID
Space Group Determination
 Orthorhombic Primitive Unit Cell Confirmed
Small Molecule Example – YLID
Space Group Determination
Space Group is determined to be P212121 (No. 19)
Small Molecule Example – YLID
Space Group Determination
hk0 layer
0kl layer
Symmetry and Space Groups
Small Molecule Example – YLID
Unit Cell Contents and Z Value
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Chemical formula is C11H10O2S
Z value is determined to be 4.0
Density is calculated to be 1.381
The average non-H volume is calculated to be 17.7
Small Molecule Example – YLID
Set Up File for Structure Solution
Structure Refinement with
SHELX / SHELXTL
 George M. Sheldrick,
Professor of Structural
Chemistry at the GeorgAugust-Universität Göttingen
and part-time programming
technician.
 Author of public-domain
SHELX and Bruker SHELXTL
solution and refinement
software and other programs.
 Sheldrick software is used in
ca. 70% of all crystal
structure refinements.
Structure Solution
 Once the structure factor amplitudes are known, the phase
problem must be solved to find a self-consistent set of phases
that can be combined with the structure factor amplitudes to
obtain the electron density and thereby determine the structure of
the crystal.
 A number of crystallographic techniques exist for obtaining the
phases of diffracted waves; the most widely utilized approaches to
the solution of phase problem involve the use of either vector
methods based on |F(hkl)|2 or direct or statistical methods.
Typically, the solution to the structure yields only a partial or
approximate model, which must be improved by successive
applications of Fourier-transform methods before the complete
structure has been determined.
Small Molecule Example – YLID
Results for Direct Methods Solution
1 S atom + 13 Q-peaks
Small Molecule Example – YLID
Assignment of Atom Types
1 S atom + 11 C atoms + 2 O atoms
X-ray Crystal Structure
Determination
Refinement of Structures
Refinement of Structures
 When a structure is “solved”, atom types are assigned to
some of the electron density peaks from the threedimensional “Fourier map”. The atomic scattering factors
for these atoms are then used to calculate structure
factors, F(calc), which are compared with the observed
structure factors, F(obs), for the whole dataset. The
agreement is measured by an R-factor.
 The fractional coordinates are then adjusted (refined) to
obtain better agreement and to locate and assign
additional electron density peaks.
Small Molecule Example – YLID
First Refinement Run
1 S atom + 11 C atoms + 2 O atoms
Small Molecule Example – YLID
First Refinement Run
Isotropic Refinement R1 = 7.68%
Small Molecule Example – YLID
First Refinement Run
Anisotropic Refinement R1 = 4.33%
Difference peaks assigned as H atoms
Refinement of Structures
 After the entire molecular structure has been determined,
the approximate positions of the atoms are refined by
nonlinear least-squares techniques to give the best fit
between the calculated and observed intensity data for the
specimen.
 Besides positional parameters (i.e., fractional coordinates),
additional parameters are included in the refinement to
model the thermal motion of individual atoms.
Small Molecule Example – YLID
Final Refinement Run
 Final anisotropic refinement
with H atoms
 R1 = 2.03% wR2 = 5.38%
 Shown as 50% thermal
ellipsoids
 Flack x parameter = 0.0113
with esd of 0.0642
 WGHT
0.0321
0.0906
Bond Lengths and Angles
 The refinement process yields very accurate values for
atomic positions from which bond lengths, bond angles and
other structural parameters may be calculated.
 The estimated standard deviations in the unit cell
parameters and the measured intensities are used to
estimated the standard deviations in bond length, bond
angles and other derived structural parameters.
Small Molecule Example – YLID
Bond Lengths and Angles
Small Molecule Example – YLID
Reports
Small Molecule Example – YLID
Structure Validation
 After the structural refinement
process has been completed, an
analysis of the complete structure
is usually carried out with an
independent validation program
(e.g., PLATON, PublCIF) which
checks the structure for missing
information or inconsistent data.
 Warning messages are generated
that allow the authors to address
the error prior to publication.
Small Molecule Example – YLID
Generation of CIF Files
 All of the crystallographic
journals and most of the
major chemical journals have
now adopted the CIF (Crystal
Information Format) for
depositing and publishing
crystallographic data.
 Most commercial and publicdomain structure refinement
programs now generate CIF
files for validation and
deposition.
Small Molecule Example – YLID
Typical Crystal Structure Diagrams
Ball-and-stick diagram
of one molecule
Unit-cell diagram showing the
arrangement of four
molecules within the cell
Summary of 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
We demonstrated these concepts by carrying out an X-ray
crystal structure analysis on 2-Dimethylsufuranylidene-1,3indanedione (YLID)*
*Polymorphism and History of 2-Dimethylsufuranylidene-1,3-indanedione (YLID), Ilia A. Guzei,
Galina A. Bikzhanova, Lara C. Spencer, Tatiana V. Timofeeva, Tiffany L. Kinnibrugh and Charles
F. Campana, Crystal Growth & Design, Vol. 8, No. 7, 2008
Recommended Books
J. P. Glusker and K. N. Trueblood, Crystal Structure Analysis: A Primer, Oxford
Univ. Press 2nd Edition ,1985, ISBN 019-503543-7
W. Clegg, Crystal Structure Determination, Oxford Univ. Press, 1998, ISBN
019-855901-1
W. Massa, Crystal Structure Determination, 3. Auflage 2002, Teubner, ISBN 3519-23527-7; 2nd Edition, 2004, Springer, ISBN 3-540-20644-2.
Single Crystal XRD
Incident
Beam
Diffracted
Beam
Single Crystal XRD
Definition of Powder Definition
Powder diffraction is a method of X-ray diffraction analysis in which
monochromatic X-rays are incident on a sample containing a large
number of tiny crystals having random orientation, producing a
diffraction pattern that is recorded with a point detector or an area
detector.
Powder XRD
 Make the sample simultaneously consist of every possible
orientation by grinding it into a fine powder.
 The powder will consist of tens of thousands of single-crystal
grains that are randomly oriented with respect to one another.
 Every possible orientation is well-represented, and so every set of
diffracting planes has crystallites oriented such that those planes
are parallel to the sample surface.
Powder XRD
Incident
Beam
Diffracted
Beam
Diffracted Intensity of a Powder
Sample
10000
9000
8000
Lin (Counts)
7000
6000
5000
4000
3000
2000
1000
0
20
30
C2204H3 - File: C2204H3.raw - Type: 2Th/Th locked
00-005-0628 (*) - Halite, syn - NaCl - Cubic - Face-centered
40
50
Powder XRD in Three Dimensions
Powder XRD