WORKSHOP

X-ray crystallographyDubravka Matković-Čalogović

Laboratory of General and Inorganic ChemistryDepartment of ChemistryFaculty of ScienceUniversity of Zagreb, Croatia

Crystal

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.

Symmetry is one of the mainproperties of crystals.

Angles between the faces areconstant.

Miller indices are used to describe crystal faces.

Miller indicesThe integers hkl that characterize crystal faces (andplanes in the unit cell) are called Miller indices.

The plane shown cutsthe x axis at a/2,y axis at b and thez axis at c/3.

Intersections are at:1/2, 1, 1/3

Reciprocals of these fractions, 213, representthe Miller indices.

(213) are the Miller indices of this plane and all others parallel to it.

a) (101) b) (010)

c) (200) d) (h00)

e) hexagonal unit cell

f) (hkil), (-12-10)

i = -(h + k)

a

b

c

Triclinic a b c

90o

Monoclinic a b c

= = 90o 90o

Orthorhombic a b c

= = = 90o

Tetragonal a = b c

= = = 90o

Trigonal a = b = c

= = 90o

Hexagonal a = b c

= = 90o = 120o

Cubic a = b = c

= = = 90o

7 crystal systems

Lattice centering (includes translation)

P (primitive) x, y, z

C (centered on the (001) faces) x, y, z x+1/2, y+1/2, z

I (body centered) x, y, z x+1/2, y+1/2, z+1/2

F (face centered) x, y, z x+1/2, y+1/2, zx+1/2, y, z+1/2 x, y+1/2, z+1/2

R (rhombohedral) x, y, z x+1/3, y+2/3, z+2/3x+2/3, y+1/3, z+1/3

Triclinic Monoclinic Orthorhombic

Hexagonal P & Trigonal P, RTetragonal

Cubic

14 Bravais lattices

• Rotation axes• Inversion axes

32 Point groups

Symmetry in crystals

2 ( m) converts an_

Mirror plane symmetryobject (a molecule) into its mirror image.

z

x, y, z x, -y, z

When translation is added new symmetryoperations are obtained:

screw axes (rotation axes combined with translation)21 31 32 41 42 43 61 62 63 64 65

glide planes (reflection bymirror planes combined with translation)a b c n d

x, y, z x+1/2, -y, -zPicture shows 21 parallel to a; themolecule is translated by a/2

230 space groups

are obtained by combining:

32 point groups14 Bravais latticesscrew axesglide planes

Escher’s drawing

Unit cell is the smallest repeating unit which shows the full symmetry of the crystal structure

X-ray diffraction

X-rays diffraction on a crystal diffraction data

Wilhelm Conrad Röntgen’slaboratory

Würzburg, 1895.

• 22. 12. 1895.

New York 1897, See the bones of your hand for a nickel!!

Walter Friedrich, Paul Knipping & Max von Laue – first diffractogram 1912.

Electromagnetic radiation:

= 100 − 0,01 nm (1000−0,1 Å)

= 3·1015 − 3·1019 Hz

E = 12 eV − 120 keV

X-rays

Usefull wavelengths for X-ray diffraction: = 0,5 – 2,0 Å

X-ray tube

Coolidge side-window tube (scheme) K: filament (-) (thermionic effect)A: anode (+) Win and Wout: water inlet and outlet of the cooling device (C)

continuous (white)

characteristic

Continuous – deceleration of elecrons causes emission of X-rays– “Bremsstrahlung”.

At the edge X-ray photons acquire all E from the electrons hitting the anode.

Ephoton = hν = h(c/λ)

Eelectron = e × V

e – electron charge

V – voltage

anode – usually Mo orCu

fast electrons

fast elektronX-rays

(Cu K) = 1.5418 Å

(Mo K) = 0.7107 Å

The smallest voltage for characteristic radiation depends on the atomic number:

Mo (Z=42) V > 20 kV W (Z=74) V > 70 kV

Bremsstrahlung

Deceleration of the electron near the nucleus.

Discovered by Nikola Tesla while doing research of high frequencies.

One electron can emit more photons of smaller E.

continuous characteristic

Cu anode:

Kα1 = 1.54051 Å, Kα2 = 1.54433 Å

mean value 1.54178 Å (intensity of K α1 is twice as large as Kα2)

Kβ = 1.39217. Å

Filters

Monochromatic X-rays

absorption edge of the filter

MonochromaticBragg’s law:

on a single crystal (graphite, diamond, silicon, germanium)

sin2 hkld

SynchrotronsEuropean Synchrotron Radiation Facility (ESRF)Grenoble, France

(FOCUSING MAGNETS)

(BENDING MAGNETS)

UNDULATOR

STORAGE RING

storage ring

BEAMLINE

ESRF

X-ray diffraction

X-rays diffraction on a crystal diffraction data

The fall off in intensityis greater in large atoms.

The X-rays are scattered by electrons of an atom.The amplitude of the wavescattered by an atom, f, isproportional to the number of electrons of that atom -proportional to its atomicnumber Z.

Diffraction on a crystal- interference of waves

In phase -constructive interference

Out of phase -destructive interference

Partially out of phase

Beam 2 has to travel the extra distance BD in comparisonto beam 1. If this path difference is equal to n (n = 1, 2, 3…)constructive interference occurs.

BC/d = sinCD/d = sin

BC = CDBD = BC+CDBD = 2BCBD = n

n = 2dsin

Bragg’s law: n = 2dsin

1

2

d

d - spacing is the perpendicular distancebetween pairs of adjacent planes

Several types of reflecting planes

Reciprocal lattice

direct reciprocal

In the reciprocal lattice the point hklis drawn at a distance1/dhkl from the origin000 in direction of thenormal between a set

of planes(hkl).

X-rays

detector

Systematically absent reflectionsSystematically absent reflections due to lattice type

Lattice type Rule for reflection hkl to be observed

P noneC h + k = 2nI h + k + l = 2nF h + k = 2n, h + l = 2n, k + l = 2nR -h + k + l = 3n

Example: reflection 100 will be present only in primitive space groups (and R); 110 in P, C, I;111 in P, C, F, R

screw axes21 || a h00 h = 2n21 || b 0k0 k = 2n21 || c, 42 || c, 63 || c 00l l = 2n31 || c, 32 || c, 62 || c 00l l = 3n41 || c, 43 || c 00l l = 4n61 || c, 65 || c 00l l = 6n

glide planes || (010)a (translac. a/2) h0l h = 2nc (translac. c/2) h0l l = 2n n (a/2+c/2) h0l h+l = 2n d (a/4+c/4) h0l h+l= 4n+1,2,3

Space group determinationfrom systematic absences

P212121reflection conditions:h00: h = 2n0k0: k = 2n00l: l = 2n

Pnmareflection conditions:0kl: k+l = 2nhk0: h = 2nh00: h = 2n0k0: k = 2n00l: l = 2n

Example: NaCl

lattice is F

reflections100, 110are absent;200, 111 are present

Crystal structure analysis

A single crystalis placed on agoniometerhead on a 4-circlediffractometer.

Detector is a photographicplate, counter, imagingplate or CCD.

A crystallographercomputes a 3-D electrondensity map byFourier synthesis

A singlecrystal experiment

Diffraction image ona CCD or imaging plate

Information obtained:

unit cell dimensions

reflectionshkl and their intensity(from systematicabsences we find thespace group)

Pattersonand directmethods,Fouriersynthesis

From the electron density map we get fractional coordinates of atoms in the unit cell.

Structure factor

Re

Im

A

α

Complex plane (Argand (Fresnel) diagram)

]iexp[)sini(cos AA A

Wave function:

amplitude

phase

)(iexp)()( hklhklFhkl F

strukture factor

structure factor amplitude(structure amplitude)

relative phase of the structure

factor

Summation of two waves:

The two components can be shown as a complex number (multiplication by i is equivalent to rotation of the vector by 90° counterclockwise).

)(iexp)()( hklhklFhkl F

V

zyxlzkyhxxyzVhkl dd d )](i2exp[)()( F

electron density in the position (x, y, z)

Frakctional coordinates:

cZzbYyaXx

volume of the unit cell

Diffraction image is a Fourier transformation of theelectron density

For each reflection we can calculate the structure factor (Fc – calculated structure factor) – if we know the amplitude and phase.

The diffraction image i a Fourier transformation of one unit cell (they are all same in the crystal).

The formula shows a continuous function ρ(xyz). However addition of finite numbers is easier so the electron density is expressed by atomic scattering factors.

f3f1

f2

F(hkl)

φ(hkl)

N

jjjjj lzkyhxhklfhkl

1)](i2exp[)()(F

no. of atoms in the unit cell

atomic scattering factor of atom j

cZzbYyaXx

jj

jj

jj

Fractional coordinates of the atom:

Electron density

h k l

lzkyhxhklV

xyz )](i2exp[)(1)( F

Fourierova transformation (FT-1) of structure factor:

h k l

hkllzkyhxhklFV

xyz )](i)(i2exp[)(1)(

h k l

lzkyhxhklV

xyz )](i2exp[)(1)( F

2)(hklFI PROBLEM of

PHASES

N

jjjjj lzkyhxhklfhkl

1)](i2exp[)()(F

FT-1FT

)(iexp)()( hklhklFhkl F

reciprocal space

real space

Ramachandran i Srinivasan, Nature 190 (1961) 159.

Calculated a electron density map using phases of one structure ( ) and amplitudes of another (x).

Intenzity of diffracted ray2

2

3

int )()( hklFLPTEIkhklI oVΩ

k –constant – wavelength – volume of the crystalV – volume of the unit cellIo – intensity of X-raysL – Lorentz factor (depends on the diffraction technique)P – polarisation factor (depends on the diffraction technique)T – transmission factor (depends on the absorption coefficient of the crystal)E – extinction coefficient (depends on the mosaicity of the crystal)|F(hkl)| – amplitude of the strukture factora

2int )()( hklFhklI

Data reduction• integration of diffraction maxima

22

3

int )()( hklFLPTEIkhklI oVΩ

• Lorentz, polarisation and absorption corrections

...

...

h k l I(hkl) [I(hkl)]

determination of the space groupspace group extinctions

Solving the structure• solving the phase problem

Patterson methodaDirect methodsCharge flippingFourier recyclingMetods in direct spaceSingle isomorphous replacement (SIR)Multiple isomorphous replacement (MIR)Anomalous diffraction with single wavelength (SAD)Anomalous diffraction with multiple wavelength (MAD)Molecular replacement... • variations of the Patterson method

• mostly for biological macromolecules

Patterson method

h k l

lwkvhuhklFV

uvwP )(πi2exp)(1)( 2

Patterson function

Direct methodsCharacteristics of electron density define possible initial phases: diskrete atoms ρ(r) ≥ 0

Structure invariantskhkhhk

khkhhk EEENA )2( 2/1

)cosexp()(2)( 10 hkhkhkhk AAIP

k khkkhk

k khkkhkh )cos(

)sin()(tg

EEEE

Tangens formula

Shake-and-Bake – phase refinement in reciprocal space alternates with special procedures in direct space that include constraints – improves the phase

Fourier recycling (iteration)difference Fourier map of electron density

h k llz ky(hx hklhklFhklF

V

xyzxyzxyz

)2)(iexp)()(1)()()(

cco

co

0)( xyz

0)( xyz

in the position (x, y, z) in the model there is not enough electron density

in the position (x, y, z) in the model there is too much electron density

modeling of the missing parts

missing atom

Difference (Fo− Fc) map

Fo- map (calculated from observed structure amplitudes and calculated phases)

missing atom

Refinement of the structure

2co )( YYwMinimization of the function:

Y is usually |F|2w weight parametar (different fof diferent hkl)

• least-squares method

• refinement of coordinates (3 per atom), thermal displacement parameters (1 per atom for isotropic; 6 per atom for anisotropic), global scale factor of observed and calculated intensities

2/1

2o

2co )(

wYYYw

wR

o

co

FFF

R

2/12co )(

PN

YYwS

Evalution of the structure

• does it make chemical sense?

Weighted R-factor:

no. of data no. of refined parameters

S ≈ 1

Interpretation of the structure

From the electron density map we get fractional coordinates of atoms in the unit cell.

figures courtesy of Z. Dauter

Model building at low and high resolutionModel Model buildingbuilding atat lowlow andand high high resolutionresolution