Theory and Practice of X-ray Crystal Structure Determination

Theory and Practice of X-ray Crystal Structure Determination
Basic Crystallography Part 2 Theory and Practice of X-ray Crystal Structure Determination Charles Campana, Ph.D. Senior Applications Scientist Bruker AXS

Course Overview Introduction – Crystals and Crystallography
Basic Crystallography – Part 1 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 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 This two-hour series gives a concise introduction to modern crystal structure determination, emphasizing both its theoretical background (Part 1) and the way crystal structures are actually carried out in practice(Part 2) Our emphasis will be to develop a conceptual understanding of the concepts without using a detailed mathematical approach. The most important data collection techniques and methods of data reduction, structure solution and refinement will be discussed from a practical point of view.

Review of Part 1 Important Concepts
Let’s begin by summarizing and reviewing the most important concepts from Part 1. These concepts form the foundation for the material we will present in Part 2

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 face-centered (F) lattices. When all possible three-dimensional rotational and translational symmetry elements are combined, they form a set of 230 crystallographic space groups. 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 face-centered (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. 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 Kα and Kβ 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. 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 Unit Cell (a, b, c, , , ) Electron Density, (x, y, z) Atomic Coordinates – x, y, z Thermal Parameters – Bij Bond Lengths (A) Bond Angles (º) Crystal Faces Reciprocal Space Diffraction Pattern Reflections Integrated Intensities – I(h,k,l) Structure Factors – F(h,k,l) Phase – (h,k,l)

Introduction to X-ray Crystal Structure Determination
Now we would like to describe how a crystal structure is actually carried out.

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 The technique of single-crystal X-ray crystallography has three basic steps. The first—and often most difficult—step is to obtain an adequate crystal of the material under study. The crystal should be sufficiently large (typically larger than 0.1 mm in all dimensions), pure in composition and regular in structure, with no significant internal imperfections such as cracks or twinning. In the second step, the crystal is placed in an intense beam of X-rays, usually of a single wavelength (monochromatic X-rays), producing the regular pattern of reflections. As the crystal is gradually rotated, previous reflections disappear and new ones appear; the intensity of every spot is recorded at every orientation of the crystal. In the third step, these data are combined computationally with complementary chemical information to produce and refine a model of the arrangement of atoms within the crystal. The final, refined model of the atomic arrangement—now called a crystal structure—is usually stored in a public database. 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 Let’s discuss what type of sample we require.

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 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 The first step in a crystal structure analysis is concerned with obtaining suitable X-ray quality crystals of the material to be analyzed. The techniques required to obtain such crystals vary considerably depending upon the types of compounds to be analyzed. Stable crystals of typical organic, organometallic or coordination complexes can usually be grown by slow re-crystallization from common solvents. Other types of compounds may require the use of sublimation, zone refinement, solvent diffusion, low-temperature and/ or inert-atmosphere techniques in order to isolate suitable specimens. Once suitable crystals have been grown, a suitable specimen must be selected for analysis—ideally, a single crystal of 0.1 mm to 0.5 mm size, not cracked and not twinned.

Mounting of Samples Use micro-tools or acupuncture needles and oil to separate selected sample Mount specimen for analysis 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 The first step in a crystal structure analysis is concerned with obtaining suitable X-ray quality crystals of the material to be analyzed. The techniques required to obtain such crystals vary considerably depending upon the types of compounds to be analyzed. Stable crystals of typical organic, organometallic or coordination complexes can usually be grown by slow re-crystallization from common solvents. Other types of compounds may require the use of sublimation, zone refinement, solvent diffusion, low-temperature and/ or inert-atmosphere techniques in order to isolate suitable specimens. Once suitable crystals have been grown, a suitable specimen must be selected for analysis—ideally, a single crystal of 0.1 mm to 0.5 mm size, not cracked and not twinned.

Mounting of Sample The crystal must be mounted for measurements so that it may be held in the center of the X-ray beam and rotated. There are several methods of mounting. A modern approach is to attach the crystal to tiny loop, made of nylon and attached to a solid rod, that is then flash-frozen with liquid nitrogen. This freezing reduces the radiation damage due to the X-rays, as well as the noise in the Bragg peaks due to thermal motion of the atoms in the sample (the Debye-Waller effect). This slide shows a crystal of thaumatin mounted in a cryo-loop.

Goniometer Head Huber model 1004 goniometer head
Once a suitable specimen has been selected, it is glued or otherwise securely attached to a goniometer head (sample holder) in an arbitrary orientation. The goniometer head is then placed on the base of the goniometer assembly and the crystal is optically aligned in the center of the incident X-ray beam using a video camera or a microscope. The orthogonal X, Y, and Z translations on the goniometer head are adjusted until the specimen is centered on the cross hairs for all crystal orientations Huber model 1004 goniometer head

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. 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 χ 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 Kappa_goniometer_animation.ogg Animation showing the five motions possible with a four-circle kappa goniometer. The rotations about each of the four angles φ, κ, ω and 2θ leave the crystal within the X-ray beam, but change the crystal orientation. The detector (red box) can be moved closer or further away from the crystal, allowing higher resolution data to be taken (if closer) or better separation of the Bragg peaks (if further away).

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 X-ray 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. 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 X-ray 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. Anode Ka1 Comments Cu Å Best for organic compounds (absolute structures), small specimens, large unit cells (e.g., proteins). Mo Å Best for minerals, inorganic and solid-state compounds with strongly absorbing elements, charge density; Preferred source for routine structures. The most common metals used are copper and molybdenum. The simplest and cheapest variety of sealed x-ray tubes have stationary anodes and produce ~2 kW of x-ray radiation. For very small crystals, micro-focus sources or rotating-anode sources are often used.

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. The X-rays are usually passed through graphite-crystal monochromators or X-ray mirrors to eliminate Kβ radiation and to produce a single wavelength (Kα radiation only) and collimated to a single, intense small beam before they are allowed to strike the crystal. Collimation is done either with a collimator or with an monocapillary optics with pinholes to adjust the size and shape of the X-ray beam striking the specimen. Beam sizes may range from 120 to 500 microns. For best results, the crystal should be fully bathed in the X-ray beam.

Other Modern Laboratory X-ray Sources
Incoatec IμS microfocus source: sealed tube 30 W, air cooled. Achieve a very small focal spot. Source spot size is less than 50 microns. Rotating Anode spins the anode so you continuously have fresh (cooler) target metal at the focal spot, so it is effectively distributed along a much larger area. Allows much larger tube currents while providing sufficient cooling. ImS TXS Rotating Anode

Measurement of Intensity Data
The intensities of these reflections may be recorded with a charge-coupled device (CCD) detector. The most important and most expensive component of any modern single-crystal diffractometer system is its detector system. Most new instruments purchased since 1994 use a detector system based upon CCD (charge-coupled device) technology. The detector shown here includes a square CCD-microprocessor chip with physical dimensions of ~62 mm (4K chip) on each side. Each of these CCD chip contains 4096×4096 (4K chip) independent pixels. For protein applications, the CCD chip is bonded to a fiber-optics taper to increase the effective size to 90 mm on a side. The front end of the fiber-optics taper is attached to a phosphor, optimized for either Mo or Cu radiation (or both) to convert X-ray photons to visible wavelength photons which can be transmitted to the CCD chip. In order to reduce background noise and improve counting statistics, the CCD chip must be cooled to about –60° C using Peltier cooling methods. Typical CCD detector systems have counting efficiencies ranging from 40 to 180 electrons per X-ray photon.

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 ,g 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. 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 ,g 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 = ×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). 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 = ×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 = non-hydrogen atoms in the unit cell. C11H10O2S has 14 non-hydrogen atoms; 55.22/14 = ≈ 4.0 = Z 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 ,g 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.

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

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: 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: 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
Goniometer: 3–circle (c fixed at 54.74°) Radiation choice: Mo ( = 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

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. The recorded series of two-dimensional diffraction images, each corresponding to a different crystal orientation, must be converted into a three-dimensional array of indexed reflections, each of which has an associated intensity, I, and standard deviation, (I). This process is called data reduction. The first part of the data reduction process is called integration. This complicated procedure uses the orientation matrix and applies many corrections (e.g., background, spatial distortions, Lorentz and polarization factors, crystal decay, sample movement or misalignment, etc.) 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. 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

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. 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 combination of all these symmetry operations results in a total of 230 unique space groups describing all possible crystal symmetries. Space groups let us take advantage of the symmetry of the crystal lattice, so that we only have to determine the structure of the asymmetric unit – not the entire unit cell!

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: 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. 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: 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

Orthorhombic Primitive Unit Cell Confirmed

Space Group is determined to be P (No. 19)

hk0 layer 0kl layer

Symmetry and Space Groups
This a section of a table of space groups from the International Tables of Crystallography for only triclinic, monoclinic, and orthorhombic systems. The Hermann-Mauguin notation describes the lattice and some generators for the group. It has a shortened form called the international short symbol, which is the one most commonly used in crystallography, and usually consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, A, B, C, I, R or F). The next three describe the most prominent symmetry operation visible when projected along one of the high symmetry directions of the crystal. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, described above.

Small Molecule Example – YLID Unit Cell Contents and Z Value
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 Georg-August-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. Once the structure 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

Refinement of Structures

Refinement of Structures
When a structure is “solved”, atom types are assigned to some of the electron density peaks from the three-dimensional “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

Isotropic Refinement R1 = 7.68%

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 = with esd of WGHT

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 public-domain 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 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-1,3-indanedione (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 W. Clegg, Crystal Structure Determination, Oxford Univ. Press, 1998, ISBN W. Massa, Crystal Structure Determination, 3. Auflage 2002, Teubner, ISBN ; 2nd Edition, 2004, Springer, ISBN

Single Crystal XRD Diffracted Incident Beam Beam
Single crystal sample run on a powder diffractometer gives only one peak (maybe 2 or 3 orders) and very limited information. 72

Single Crystal XRD Each point represents Braggs law being satisfied for a different set of diffracting planes. Some of the planes belong to the same family, so their d-spacing is the same, but the orientation is different.

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 Diffracted Beam Incident Beam

Diffracted Intensity of a Powder Sample

Powder XRD in Three Dimensions

Powder XRD Tens of thousands of single crystals that are oriented randomly with respect to one another. At any incident angle, there will be a sufficient number of crystallites with every possible set of diffracting planes oriented to Bragg diffract.

Theory and Practice of X-ray Crystal Structure Determination

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