Basic Crystallography Part 2 Theory and Practice of X-ray Crystal Structure Determination

Basic CrystallographyPart 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 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

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 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 K and K 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

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

Flowchart for X-ray Structure Determination

Adapted from William Clegg “Crystal Structure Determination” Oxford 1998.

Select, mount, and optically align a suitable crystal

Evaluate crystal quality; obtain unit cell geometryand preliminary symmetry information

Measure intensity data

Data reduction

Solve the structure

Complete and refine the structure

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• 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• 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


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 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



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.

X-ray Tube Radiation Choices

Anode K1 Comments

Cu 1.54060 ÅBest for organic compounds (absolute structures), small specimens, large unit cells (e.g., proteins).

Mo 0.70930 Å

Best for minerals, inorganic and solid-state compounds with strongly absorbing elements, charge density; Preferred source for routine structures.

The most commonly-used X-ray targets for single crystal X-ray diffraction are copper and molybdenum.

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

IS TXS Rotating Anode

Measurement of Intensity Data

The intensities of these reflections may be recorded with a charge-coupled device (CCD) detector.


Small Molecule Example

Small Molecule Example – YLIDEnter Crystal Information

Small Molecule Example – YLIDOptically 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, , , 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 – YLIDAutomatic Unit Cell Determination

Measured 3 sets of 12 images (10 sec. exposure times)

Located 84 reflections above 20(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 non-hydrogen atoms in the unit cell. C11H10O2S has 14 non-hydrogen atoms; 55.22/14 = 3.944 ≈ 4.0 = Z

Small Molecule Example – YLIDIndexed 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 – YLIDIndexed 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 or )• 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 – YLIDTypical Data Collection Goniometer: 3–circle ( fixed at 54.74°) Radiation choice: Mo ( = 0.71073 Å) Crystal-to-detector distance: 6.0 cm (60 mm) Scan width: 0.5° in 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 – YLIDOne 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, (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 – YLIDScreen from Integration

Small Molecule Example – YLIDFinal 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 – YLIDAbsorption 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.

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.

Small Molecule Example – YLIDSpace Group Determination

5780 reflections read from .hkl file

Bravais lattice is Primitive


Orthorhombic Primitive Unit Cell Confirmed


Space Group is determined to be P212121 (No. 19)


hk0 layer 0kl layer

Symmetry and Space Groups

Small Molecule Example – YLIDUnit 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 – YLIDSet 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.

Small Molecule Example – YLIDResults for Direct Methods Solution

1 S atom + 13 Q-peaks

Small Molecule Example – YLIDAssignment of Atom Types

1 S atom + 11 C atoms + 2 O atoms


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 – YLIDFirst 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


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 – YLIDFinal 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 – YLIDBond Lengths and Angles

Small Molecule Example – YLIDReports

Small Molecule Example – YLIDStructure 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 – YLIDGeneration 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 – YLIDTypical Crystal Structure Diagrams

Unit-cell diagram showing the arrangement of four

molecules within the cell

Ball-and-stick diagram of one molecule

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 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 3-519-23527-7; 2nd Edition, 2004, Springer, ISBN 3-540-20644-2.

IncidentBeam

DiffractedBeam

Single Crystal XRD

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.

IncidentBeam

DiffractedBeam

Powder XRD

Diffracted Intensity of a Powder Sample

Lin

(C

ounts)

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

20 30 40 50 60

Powder XRD in Three Dimensions

Powder XRD

Date post:	06-Jan-2016
Category:	Documents
Upload:	fawn
View:	142 times
Download:	10 times

Basic Crystallography Part 2 Theory and Practice of X-ray Crystal Structure Determination

Documents