X-ray Crystallography, an OverviewFrank R. Fronczek

Department of Chemistry

Louisiana State University

Baton Rouge, LA

Feb. 5, 2014

“Long before there were people on the earth, crystals were already growing in the earth’s crust. On one dayor another, a human being first came across such asparkling morsel of regularity lying on the ground or hit one with his stone tool, and it broke off and fell athis feet, and he picked it up and regarded it in his openhand, and he was amazed.”

M. C. EscherFrom “Approaches to Infinity”

Topics

1. Crystals2. Point Symmetry (Brief Review)3. Space Group Symmetry4. Diffraction and Fourier Analysis5. Intensity Data Collection6. Structure Solution and Refinement7. Absolute Structure

René Just Haüy

1743-1822

It broke into rhombohedra

Intentional breakage of a rhombohedron producessmaller and smaller rhombohedra

Calcite, CaCO3

For Calcite

This led to concept of the “unit cell”

Unit cell, in yellow, gives directions and distances of translationallyrepeating unitUnit cell, in yellow, gives directions and distances of translationallyrepeating unit

The three axes are labelled a, b, and c, and may have different lengths

If we want to indicate only the translational regularityand not the structure itself, we can do so with an arrayof points called a lattice

Important to distinguish between the structure and the lattice,which is just an array of points which indicates the regularityof the structure.

Molecules within the unit cell are related by symmetry

The asymmetric unit here is one molecule, but may be several, or less than one.

The number of molecules in the unit cell (Z) here is 4

Fractional Coordinates

b

c

a

x

y

z

x is the fractional coordinate in the a direction

y in the b direction

z in the c direction

To Completely Describe the Structure,Must Determine:

Dimensions of the Unit Cell

Symmetry of the Unit Cell

Coordinates of all the atoms in the Asymmetric Unit

0

a

b

c1

1/2

1/3

Miller Indices

Orientations of planes in space are given by indices hklwhich are the reciprocals of the fractional intercepts

For example, this is the 321 plane

0

a

b

c

1

1/21/3

321

2

2/3

1

The hkl (321 in this case) actually refers to a set of parallel planes

0

a

b

c

hkl

The perpendicular distance between the planes is called thed spacing for the set of planes

dhkl

c

a

b

Indices can be positive or negativeNote the meaning of a zero index

Natural crystal faces tend to have low-numbered Miller indices

Miller Indices of the Cubic, Octahedral, and Dodecahedral Faces of aCrystal in the Cubic System

a

b

c

SymmetrySymmetryReview of Point

Symmetry

The motif is the smallest part of a symmetric pattern, and can be any asymmetric chiralchiral* object.

1

2

3

4

To produce a rotationally symmetric pattern, place the same motif on each spoke.

The pattern produced is called a proper rotationproper rotation because it is a real rotation which produces similarity in the pattern.

*not superimposable on its mirror image, like a right hand. The pattern is produce by a

four-fold proper rotation.

Normal crystals contain only five kinds of proper rotational symmetry:1. One foldOne fold, = 360o (IdentityIdentity)2. Two foldTwo fold, = 180o

3. Three foldThree fold, = 120o

4. Four foldFour fold, = 90o

5. Six foldSix fold, = 60o

The proper rotation axis is a line and is denoted by the symbol nn (Hermann-Maugin) or CCnn (Schoenflies). Thus, the five proper crystallographic rotation axes are called 11, 22, 33, 44, 66, or CC11, CC22, CC33, CC44, CC66.

Note: molecules have proper rotation axes of any value up to

There is another, quite different way to produce a rotationally symmetric pattern:

put motifs of the opposite hand on every other spoke.

The imaginary operation required to do this is:

RotateRotate the motif through angle

InvertInvert the motif through a point on the rotational axis - this changes the chirality of the motif.

This “roto-inversion” is called an improper rotationimproper rotation

Inversion Center

The simplest is a one-fold improper rotation, which is just inversion through a center, with symbol

11–– or i

Another common type of improper rotation is the mirror

Example of a crystal with D2h symmetry

D2h is the point group, containing C2 axes, mirrors, and inversion center.

There is an infinite number of point groups

Crystals can fall into only 32 of them,because crystals can have only 1,2,3,4and 6-order rotation axes

The 32 point groups can be further categorized into7 Crystal Systems

TriclinicMonoclinicOrthorhombicTrigonalTetragonalHexagonalCubic

Importance of Getting the Symmetry Right

A canoe should have C2v symmetry, not C2h

Space Group Symmetry

Space groups

Extend 32 crystallographic point groups by adding translational symmetry elements to form new groups.

How many 3-D space groups are there?

How many orderly ways are there to pack identicalobjects of arbitrary shape to fill 3-dimensional space?

To answer this, need to extend point symmetry to include periodic structures:

Crystals!

Translational Translational SymmetrySymmetry

a a a a

a' a' a'

Can describe the repetition by the direction and distance

The set of lattice points describing the 1-D translation is a row

Net: a 2-D array of equispaced rows on a plane.

a

b

b

b

b

a

b

Unit Cell is “building block” of this 2-D lattice, and is described by a, b, and , the angle between them.

CenteredLattice

There are five 2-D lattices. Now stack up these nets to form 3-D lattices.

Get unit cell with 3 axes and 3 angles

Get several new types of centered lattices:End-centered, body-centered, face-centered

14 3-D lattices in all, called Bravais lattices

The 14 Bravais Lattices

New kind of translational symmetry element:Glide plane

Combination of mirror and translation by 1/2 cell length

Note: 2 directions associated with glide plane:Direction of mirror and direction of translation.

2211

t

In 3 Dimensions, also have combination ofrotation and translation, called Screw Axes

Can also have 3fold,4fold and 6fold screwaxes

3-Dimensional Space Groups

Combine the 14 Bravais lattices with: •32 crystallographic point groups•Screw Axes•Glide Planes

Get 230 3-D space groups

So exactly (only?) 230 orderly ways to pack identicalobjects of arbitrary shape to fill 3-dimensional space.

Are the 230 space groups equally represented byactual crystal structures? NO!

In Cambridge Structural Database (~700,000 structurescontaining “organic” carbon) 83.1% of all structures areIn just 6 space groups:

P21/c 35.1% (monoclinic)P-1 23.0% (triclinic)C2/c 8.1% (monoclinic)P212121 7.9% (orthorhombic)P21 5.5% (monoclinic)Pbca 3.5% (orthorhombic)

No other space group with >2%

All 230 space groups represented with at least one.P4mm only one structure. 25 space groups <10 structures.

X-Ray DiffractionX-Ray Diffraction

Monochromatic X-Rays from a fixed source

The crystal remains in the incident beam during rotation

Start rotating the crystal in the xray beam.

Start rotating the crystal in the xray beam.Nothing happens until ...

2

θ = Bragg angle, 2θ = scattering angle

λ = 2dhkl sin θ (Bragg Equation)

λ = wavelength of X-rays

dhkl = “interplanar spacings” in the crystal

“reflection”

dhkl

("reflections" are produced by the diffraction of X-rays)

In most orientations, there is no reflection

In most orientations, there is no reflection

In most orientations, there is no reflection, but different reflections occur at selected orientations.

In most orientations, there is no reflection, but different reflections occur at selected orientations.

In most orientations, there is no reflection, but different reflections occur at selected orientations.

In most orientations, there is no reflection, but different reflections occur at selected orientations.

In most orientations, there is no reflection, but different reflections occur at selected orientations.

In most orientations, there is no reflection, but different reflections occur at selected orientations.

A Detector records the position and intensity of each reflected beam.

A computer records the position, intensity, and crystal orientation of each reflection

Intensity Data Collection

(Single Crystal)

Kappa Apex II Diffractometer

Diffraction Pattern Recorded on Film

Each spot has indices hkl

Image from CCD DetectorIntensity of each hkl measured

Observed structure factorFo(hkl) proportional to √Ihkl

Spacing of spots gives dimensions of unit cell

Symmetry of pattern related to symmetry of crystal

Intensities of spots related to electron density in unit cell

Systematic absences yield clues about space group of crystal

Crystallographic Application of Fourier Summation

In its application to crystallography, the Fourier coefficientsare the structure factors Fhkl (derived from measured intensities)and the continuous function is 3D electron density in the unit cell

Fhkl (xyz)

Structure Factors Electron density

real

imaginary Fhkl

hkl

Structure Factor Fhkl = Ahkl + iBhkl

where

And the phase angle is:

Fhkl has an amplitude |Fhkl|

and a phase angle hkl

Electron Density by Fourier series:

where

Fourier Map

From this, we can extract xyz values for atomsand generate a mathematical model of the crystal.

Structure Solution

Phase Problem

So IF we know coordinates, phase can be calculated:

Review of structure-Factor equation:Fhkl = Ahkl + iBhkl

Where:

And we can calculate electron density by Fourier series:

Where:

But cannot calculate densityunless we know phases

So need phases to calculate density (and locate atoms)and need atom positions to calculate phases.

A Classic Catch-22!

What to do?

J. Karle, H. Hauptman (Nobel Prize in Chemistry, 1985)

They developed statistical methods for estimating phases directly from measured |Fo| values.

Called “Direct Methods” and used in almostall small-molecule structure solutions today

Example

Monoclinic space group P21/n

From Prof. Graca Vicente’s (LSU) laboratory

4508 total reflections, 340 reflections usedin direct methods calculation

Incorrect solution (wrong phases)

Correct solution (approximately correct phases)

Final structure

A Fourier map, phased on all atoms.

Note small peaks likely corresponding to H atoms

Locate H atoms using a Difference Map, in which contribution from atoms already in model subtracted out.

Difference map showing H atoms

ORTEP drawing of model after least-squares fitting

Also get details of intermolecular contacts

Example: Hydrogen bonding in D-mannitol

Structure Refinement

Linear Least Squares

• The "best" model is the one that minimizes the sum of squares of deviations with respect to the linear coefficients of the model function.

As an example, a straight line is used as the model, 2 parameters: a & b

The final result is obtained in one calculation

y

x

y = ax +b

In crystallography, the observations are thestructure-factor amplitudes |Fo|

The model is the set of adjustable parameters:xyz for each atom, displacement (thermal) parametersscale factor, etc., from which Fc are calculated.

So, we would like to find the best match between the Fo and Fc amplitudes by minimizing the sumof the squares of their differences :

( |Fo| - |Fc| )2

w(Fo2 - Fc

2)2

In practice, most software refines on F2 rather than F, and weights individual reflections according to their precision

But the equations are not linear

So they must be approximated with linear equations

and recycling is necessary until convergence is reached.

Anisotropic Displacement Parameters

Each atom modeled with oneisotropic displacement parameter

Uiso

Each atom modeled with sixanisotropic displacement parametersU11 U22 U33 U12 U13 U23

Is the result correct?

This should be small for a correct refinementTypically 0.02 - 0.05, but may be higher fora weakly-scattering crystal.

R = |Fobs – Fcalc| / |Fobs|

Crystallographic R Factor

Other indicators:

Is the model chemically reasonable?

Are the ellipsoids reasonably shaped?

Are the H atoms visible in difference maps?

Absolute Structure Determination

Enantiomers give identical intensities

And we usually get only relative configurations of chiral centers from X-ray data

R

S

R

S

S

R

But if a heavy atom is present and the wavelength is chosen carefully, enantiomeric crystals give a slightlydifferent set of intensities, and one enantiomer fits theexperimental data better than the other.

In the last few years, new methods have beendeveloped, and oxygen in CuK radiation is“heavy” enough, with high-quality crystals.

(+) Nootkatone

The major flavorant of grapefruitInsect repellent and termiticide towards Formosan termite

Absolute configuration determined by X-ray methods

Wikipedia.org

A. M. Sauer, F. R. Fronczek, B. C. R. Zhu, W. E. Crowe, G. Henderson, R. A. Laine, Acta Cryst. C59, o254-o256 (2003).