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APTER 2

Chapter 2

X-ray Crystallography

2.1 Single crystal X-ray diffraction

2.1.1 Historical background and modern crystallography

Many scientists throughout history have pondered the mysteries of snow crystals what

they are, where they come from and why they are shaped the way they are. In 1611

Johannes Kepler published a short treatise on the Six-Cornered snowflake, which was

the first scientific reference to snow crystals. Kepler pondered the question of why

snow crystals always exhibit a six-fold symmetry [1]. Although he doesn't refer to the

atomistic view point, Kepler does speculate that the hexagonal close-packing of

spheres may have something to do with the morphology of snow crystals. He noted

that the closest packing arrangement of equal-sized balls in two dimensions is

hexagonal. He felt this model must be related in some way to the shape of the

snowflake, but was not able to account for the dendritic (multi branched) appearance

of most snowflakes. In 1665 Robert Hooke [2] published a large volume entitled

Micrographia, containing sketches of practically everything Hooke could view with

the latest invention of the day, the microscope. Included in this volume are many

snow crystal drawings, which for the first time revealed the complexity and intricate

symmetry of snow crystal structure. Philosopher and mathematician René Descartes

was the first to pen a reasonably accurate description of snow crystal morphologies,

about as well as can be done with the naked eye. These careful notes included obser-

vations of capped columns and 12-sided snowflakes, both rather rare forms [3].

Already in 1669 Niels Stensen proved the constancy of interplanar angles in quartz

crystals. Still it took more than a century before it was recognized that this property

holds for all crystal species and not just for quartz crystals. It was Jean Baptiste Louis

de Romé de l’Isle (1736-1790) who generalized this fundamental law of

crystallography that is presently known as "Steno's law" [4]. Actually this property

was incidentally discovered by de l'Isle's assistant, Arnould Carrangeot, in the course

17 2.1 X-ray Crystallography

of making terra cotta models of the crystals in de l'Isle's collection. For this purpose

in 1780 he developed a simple instrument to measure the angles between crystal

faces. The instrument that became known under the names of "Application

Goniometer" or "Contact Goniometer" was nothing more than two limbs connected

by a joint. The limbs could be applied to two adjacent crystal faces and eventually the

angle between the limbs could be measured. The accuracy of the contact Goniometer

was about 15 at best [5].

This chapter will describe the basic theory of X-ray diffraction by crystals, the

instrument used, data reduction along with the various corrections to be applied. It

also gives information used to determine the phases of the structure factors. In

addition it also gives information regarding the procedures and software used for the

analysis of diffraction data and structure determination.

2.1.2 The development of X-ray

X-radiation is also called Röntgen radiation, which is a form of electromagnetic

radiation. It has wavelengths that range from 0.1 to 10Å which is shorter than UV

radiation, but longer than γ-rays. The major use of X-rays is for diagnostic

radiography and crystallography. X-rays were first discovered in Crookes tubes.

Cathodes rays were created from electrons through the ionization of residual air in the

tube by a high DC voltage. This voltage accelerated the electrons coming from the

cathode to a sufficient velocity that created x-rays when the electrons struck the anode

or tube wall. Many early scientists did not realize the radiation from Crookes tubes

until Wilhelm Röntgen gave the first systematical investigation in 1895 [6].

Diffraction is the phenomenon of bending light passing an obstacle through a small

slit. The diffracted waves can interfere with each other giving bright and dark fringes,

depending on the phase differences of the wave. To explain the behavior of radiation,

a Dutch physicist Christian Huygens wrote a treatise on the theory of light wave

called Huygen’s principle. Max Von Laue was the first to use a crystal to diffract

X-rays in 1912 [7]. Since the crystal with the arrangement consists of parallel rows of

atoms equivalent to the parallel lines of the diffraction grating, the d-spacing could be

successfully determined from the separations of bright fringes of the diffraction

pattern. The classical Bragg law of diffraction relates the possibility of constructive

interference to the interplanar separations.


English physicists, Sir W.H. Bragg and his son Sir W.L. Bragg in 1913, derived

Bragg’s Law [8]. Some X-rays reflect off the first plane, but the rest subsequently are

reflected by succeeding planes (Figure 2.1.1) [9].

Figure 2.1.1: Diffraction of X-rays by a crystal

In first order reflection, scattered X-rays will have the difference in a whole number

wavelength. The secondary X-rays, scattered by atoms in all planes, are completely in

phase and reinforce each other. This is described as Bragg's Law:

where

λ = wavelength of x-rays

θ = glancing angle

d = inter planar separations

n = order of diffraction

The above equation can be conveniently written as:

where n is contained in dhkl and (hkl) refers to the Miller indices of the reflecting

plane.

2.1.3 X-ray crystallography

X-ray Crystallography is the study of determining the arrangement of atoms within a

crystal. The way this is accomplished is by examining the manner in which the beams

from an X-ray source scatter from the electrons within a crystal. This method can

ultimately produce a three dimensional picture of the density of the electrons within

the crystal. This can allow for determination of mean atomic positions. Diffraction of


X-ray’s in crystalline materials is a key step. Crystals are highly ordered solids in

which a particular arrangement of atoms (unit cell) repeats indefinitely along all three

axes known as basis vectors [10]. Atoms scatter X-rays primarily through their

electrons. An X-ray striking an electron produces a secondary spherical wave

emanating from the electron. This is known as scattering. A regular array of scatters

produces a regular array of spherical waves. Most of these cancel out in most

directions through destructive interference. However, some add constructively and

their direction follows Bragg’s law.

2.1.4 X-ray sources

In the X-ray source, electrons produced at a cathode are accelerated in a potential

field and bombard the metal target (Cu, for instance). Some electrons in the orbital

near the metal nucleus are removed by the bombardment. Then the electrons in the

outer shell transit into the vacated orbital and emit a characteristic x-radiation. K

series radiation is formed by electron transition from outer shell to K shell. MoKα

radiation with wavelength of 0.71073Å was used to determine the structure of the

crystals that had been reported in this thesis.

2.1.5 Diffraction of X-ray by crystals

The diffraction of X-ray by crystals was discovered by Max van Lau in 1912 [11].

Although X-rays had been discovered in 1895 by Röntgen, their nature was not

known. During the years following their discovery, a number of determined efforts

were made to prove them particles or waves. It was not, in fact, until diffraction by

crystals was observed that their wave character was proved. Diffraction effects are

observed when electromagnetic radiation impinges on periodic structures with

geometrical variations on the length scale of the wavelength of the radiation. The

inter-atomic distances in crystals and molecules amount to 0.15–0.4 nm which

correspond in the electromagnetic spectrum with the wavelength of X-rays having

photon energies between 3 and 8 Kev. Accordingly, phenomena like constructive and

destructive interference should become observable when crystalline and molecular

structures are exposed to X-rays. If X-rays are to be diffracted by a set planes the

incident ray should fall on the set of planes at a glancing angle θ is given by:


where n is an integer expressing the order of reflection from this set of planes having

interplanar spacing d and λ is the wavelength of the incident X-rays.

The above equation can be conveniently written as equation 2-2. Consider a unit cell

with translation a, b and c along its edges. It can be shown that the phase of the

wavelet scattered by an atom P. If there are N atoms in the unit cell, the resulting

amplitude of the diffracted beam is given by:

where fj is the atomic scattering factor of the jth atom. The quantity Fhkl is called the

structure factor. It’s modulus is called the structure amplitude and is defined as the

ratio of the amplitude of the radiation scattered in the order (hkl) by the contents of

the unit cell to that scattered by single classical electron kept at the origin under the

same conditions.

2.1.6 Processing the X-ray diffraction data

After a series of diffraction patterns has been recorded, the two-dimensional image

obtained must be converted to a three-dimensional model of the density of electrons

throughout the crystal. This is accomplished by a mathematical technique known as

Fourier transforms. Each spot corresponds to a different type of variation in the

electron density. The crystallographer determines which variation corresponds to

which spot (indexing), the relative strengths of each spot (merging and scaling), and

how the variations should be combined to yield the total electron density (phasing)

[11]. In order to process the data, the crystallographer must first index the reflections

within the multiple images recorded. A byproduct of indexing is determining the

symmetry of the crystal (space group). Once symmetry is assigned, the data is then

integrated. This converts hundreds of images, containing thousands of reflections, into

one file. A full data set can contain hundreds of separate images of the crystal taken at

different orientations. The images must be merged and scaled. This lets the

crystallographer know which peaks appear in two or more images and to scale the

relative images so they have a consistent intensity scale. Data collected from


diffraction experiments is a reciprocal space representation of the crystal lattice. The

position of each spot is controlled by size and shape of the unit cell and the symmetry

within the crystal. The intensity of each spot is recorded and is proportional to square

of the structure factor amplitude. Structure factor is a complex number containing

information relating to phase and amplitude of a wave.

In order to obtain an electron density map, both amplitude and phase must be known

[8]. Once an initial phase is known, an initial model can be constructed. The initial

model is used to refine the phases. Newer models are constructed and used to refine

again. This process is repeated until the best refinement is obtained. If there are few

flaws in the crystal, this ultimately leads to a structure that is reasonable [11].

2.1.7 General scattering expression for X-rays

X-ray scattering is a phenomena caused due to the interaction between the electron

and X-ray radiation. Coherent scattering happens with no X-ray energy loss and is

also called elastic scattering. Another X-ray scattering is called inelastic scattering or

Compton scattering. This inelastic scattering was proposed by Arthur Holly Compton

in 1923 [12].

The change in momentum of the X-ray radiation is due to the change in its direction

of the scattering electron. The energy of the scattered electron is thus less than the

energy of the incident X-rays. The scattering of X-rays from the electron of atoms is

possible in all the directions.

The intensity of radiation scattered by an electron is discussed by the classical

Thomson equation and called scattering power of an electron (fe). The amplitude of

wave scattered by an atom is proportional to its atomic number (Z) [13].

Also, atom scattering power, usually expressed as the ratio of scattering of an atom

factor to the scattering by a single electron under the same conditions [14-17]. Thus,

the scattering factors is expressed .The mean position of atoms in

the unit cell is vibrating at any temperature [18].

The scattering factor of an atom decreases with their amplitude of atomic vibration.

Also, the scattering factor decreases with the increase of the diffraction angle due to

the atomic size, showing the mutual destructive interference of the X-ray scattered

from the atoms. The scattering factor of an atom is given as [19]:


where f0 : scattering factor of an atom when it is rest and at 0°.

λ : wavelength of x-ray.

θ : angle of diffraction.

B : a constant (called isotropic temperature factor, which is related to the amplitude of

atomic vibration is given as where mean of square displacement of

the atom from the mean position). The exponential term is called Debye-Waller

factor).

A crystal lattice consists of atoms which form three-dimensional arrays. The structure

factor for a particular plane hkl and F(hkl) consistent angul part gj,θ and scattering

factor from the plan (hkl) and F(hkl). The structure factor of a plane hkl is expressed

as:

where

F(hkl) : Amplitude of scattered radiation from the plane hkl.

g (j, θ): Scattering factor of the atom j at the diffraction angle θ.

In an X-ray diffraction experiment, the intensity is promotional to the square of the

amplitude of the wave.

where

Io(hkl): experimental observed intensity

I(hkl): a function of the experimental conditions, Lorentz-polarization factor,

polarization, and absorption correction

N: a scale factor associated with the amplitude of scattered radiation.


2.1.8 Reciprocal lattice and Ewald sphere

The usefulness of the concept of the reciprocal lattice is to understand the diffraction

of X-rays from a crystal. A reciprocal lattice can be associated with every real

crystalline lattice. For every plane in the direct lattice, draw a normal from the origin

of the unit cell whose length is restricted to the reciprocal of the interplanar spacing.

A point is placed at the end of each limited normal and the collection of such points

constitutes a new lattice called the reciprocal lattice.

One can write the scattering vector S as:

S=hxa*+Kyb

*+Lzc*

where S is a vector in reciprocal space with the metric a*, b*, and c* (reciprocal lattice

vector). The relationship to the direct space with metric a, b, and c is still unknown.

The vector S must obey the Laue equations:

This is fulfilled only when aa* =1, hx = h and ab* and ac* =0. Similar equations can

be derived for the other two Laue conditions. Thus, vector S is a vector of a lattice in

reciprocal space. The relationship between the direct and reciprocal lattices is given

by the following:

It follows from these that a*┴ b; c; b* ┴ a; c; c*┴ a; b; and vice versa. The metric

relationships can also be derived from these relationships. They adopt the following

form for the general case of the triclinic crystal system:


This means that the inverse lattice vectors are perpendicular to the plane, which is

spanned by the two other non-inverse lattice vectors. Bragg’s law can now be derived

by inspection of figure 2.1.2.

Figure 2.1.2: Geometric representation of diffraction geometry; 2θ glance angle; θ Bragg angle.

The wave vectors for the incident wave S0 and the scattered wave s have the same

absolute value of 1/λ. Vector S must be a vector of the reciprocal lattice, and its

absolute value is equal to d*. From figure 2.1.2 the following equations 2 -13 to 2-16

can be obtaining.

The general equation for Bragg’s law is:

where n is the order of reflection and d the interplanar distance in the direct lattice.

'The Ewald sphere is a geometrical combination of the Bragg's law and the reciprocal

lattice. The Ewald construction is contained in Figure 2.1.3. A sphere of radius 1/λ is

drawn, and the origin of the reciprocal lattice is located where the wave vector s0 ends

on the Ewald sphere. The incident x-ray beam is along its diameter. The origin of the

reciprocal lattice is positioned at the point where the incident beam emerges from the

sphere.


Figure 2.1.3: Diffraction geometry in the rotation method usually applied in macromolecular X-ray diffraction systems.

A diffracted beam is generated if a reciprocal lattice vector d*hkl with an absolute

value of 1/dhkl cuts the Ewald sphere. The beam is diffracted in the direction of the

connection of the origin of the Ewald sphere and the intersection point of the

reciprocal lattice point on the Ewald sphere. The diffraction pattern of a lattice is itself

a lattice with reciprocal lattice dimensions [20].

2.1.9 Selection of single crystal

The first step in a crystal structure analysis is concerned with the selection and

mounting of a suitable specimen. Ideally, a crystal whose structure is to be determined

must be a single crystal of 0.1 mm to 0.5 mm size, not cracked and not twinned by

examining it under a polarizing microscope. Preliminary photographic examination of

the crystal provides information regarding the crystal system, unite cell dimension

space group etc.

26 2.2 Diffraction amplitude

2.2 Diffraction amplitude

2.2.1 The structure factor

The structure factor is the central concept in structure analysis by diffraction methods.

Its modulus is called the structure amplitude. The structure amplitude is a function of

the indices of the set of scattering planes h, k and l, and is defined as the amplitude of

scattering by the contents of the crystallographic unit cell, expressed in units of

scattering. The complex form of the structure factor means that the phase of the

scattered wave is not simply related to that of the incident wave. However, the

observable, which is the scattered intensity, must be real. It is proportional to the

square of the scattering amplitude [21]. The structure factor is directly related to the

distribution of scattering matter in the unit cell which, in the X-ray case, is the

electron distribution, time-averaged over the vibrational modes of the solid. The

structure factor may be represented as a complex vector [22]:

where A(hkl) and B(hkl) are the real and imaginary components of F(hkl) (Figure 2.1.

4).

The magnitude or length of the vector |F(hkl)| may then be represented as:

Alternatively, F(hkl) may be expressed as an exponential quantity:

where |F(hkl)| is the amplitude of the scattered wave and α(hkl) is its phase angle.

From Figure 2.1.4 it may be seen that:


Figure 2.1.4: Structure factor F(hkl) plotted on a diagram. α (hkl) is the phase angle and the amplitude is represented by OF.

|F(hkl)| may be calculated directly from the measured intensity I(hkl) for a reflection,

since

where K is a constant. However, the phase angle α(hkl) cannot be measured

experimentally and must therefore be obtained indirectly through a variety of

numerical techniques. The central problem in the solution of a crystal structure is the

assignment of phase angles to each reflection in the data set. The solution of the phase

problem is considerably simplified for crystals that possess crystallographic centers of

symmetry, since, to a first approximation, the imaginary components B(hkl) are zero

for centrosymmetric space groups and the phase angles are therefore restricted to

values of 0° or 180°. A structure is considered solved when a set of phase angles has

been found that allows the atoms to be located and the experimental diffraction

pattern to be matched to the calculated diffraction pattern. Since the electron density

in a crystal varies continuously and periodically in three-dimensional space, the

electron density ρ(xyz) at a point with fractional coordinates x, y, z in a unit cell of

volume V may be expressed as a three dimensional Fourier series:

If both the amplitude |F(hkl)| and the phase α(hkl) of each reflection are known, the

electron density within the unit cell of the crystal can be calculated directly. On the

other hand, if the positions of the atoms in the unit cell are known, both the structure


factor and the phase for each reflection may be calculated from the structure factor

equation:

where fj is the atomic scattering factor for the atom j and xj, yj, zj are its fractional

coordinates. In an actual structure determination both forms of the Fourier transform

equations are utilized to arrive at a model structure from which the observed

diffraction pattern can be reproduced [23].

2.2.2 Collection of intensity data

In an actual X-ray diffraction experiment, one measures the intensities Ihkl rather than

the amplitude of the reflected beam. The measure of the total number of photons

which are diffracted in the proper direction by reciprocal lattice point is known as

intensities. It is from these intensities the electron density distribution and the

positions of the atoms in the unit cell are deduced. Two general methods, ω or ω-2θ

scan are available for measuring the intensity of a diffracted beam. Either the beam

may be detected by some sort of quantum counting device, which measures the

number of photons directly or by measuring the blackening is proportion to the beam

intensity. Intensity data for crystals reported in this thesis were collected by Bruker

SMART CCD diffractometer [24] and Oxford Diffraction Xcalibur [25] diffract-

ometer.

2.2.2.1 Bruker SMART CCD diffractometer

The Bruker SMART CCD detector system is used to collect single crystal or powder

diffraction data. The Single crystal diffractometer is a 3-axis goniometer module with

SMART APEX detector, radiation safety enclosure with interlocks and warning

lights. In addition the facility also has an Oxford Cryo-systems nitrogen stream setup,

which allows data acquisition at lower temperatures (90K) see Table 2.1.1. The

external view of diffractometer and sealed X-ray tube with goniometer are shown in

figures 2.1.5 and 2.1.6 respectively [24].


Table 2.1.1: BRUKER SMART configuration.

SMART APEX Configuration

Data collection details Combination of omega and phi scans

Data Reduction Integration with the SADABS; Sheldrick 2004 and

SADABS Bruker 2001.

X-ray Source Mo Kα

Wavelength 0.71073 Å

X-ray Generator 50kV, 40mA Max.

20kV, 5mA Min.

Monochromator Graphite crystal

Collimator 0.3mm, 0.5mm

SMART can control up to 4 diffractometer axes. These are 2-theta (2θ), omega (ω),

phi (ф) and kappa (or chi (χ)). This EPICS software will also work with diffracto-

meters with fewer axes, and we have successfully used it to collect data with a simple

system with a single rotation stage, which could be called phi or omega. The diagram

of the circles is shown in figure 2.1.7.

The four circle geometry employed in the diffractometer, is used

1. for manual sample mounting on the diffractometer.

2. to keep the crystal within the beam.

3. to keep the crystal in any orientation.

4. for bring desired reflecting plane to the correct glancing angle with the

incident beam.

5. for rotating the detector arm and positioning it at an angle equal to twice the

glancing angle with respect to incident beam.


Figure 2.1.5: Overall view of Bruker Smart Apex with CCD

Figure 2.1.6: Sealed-off X-ray tube shield and goniometer of Bruker Smart Apex diffractometer.


Figure 2.1.7: Schematic diagram of a four circle diffractometer.

2.2.2.2 Oxford Diffraction Xcalibur diffractometer

The Oxford Xcalibur is a 4-circle kappa diffractometer equipped with both point and

CCD detector. The high resolution achieved and the accuracy of the data is

fundamental for electron density studies and situations where small, weak samples are

being studied. This four circle CCD (sapphire detector) diffractometer is equipped

with two low temperature devices, namely, a nitrogen cooler, CRYOJET, allowing a

routine working temperature of 90K, and a helium cooler, HELIJET , allowing to

reach a temperature as low as 15K (see Table 2.1.2). The external view of

diffractometer and sealed X-ray tube with goniometer are shown in figures 2.1.8 and

2.1.9 respectively [25].

Table 2.1.2 Oxford diffractometer configuration.

Oxford Diffractometer Configuration

Data collection details Combination of omega and phi scans to maximize reciprocal space coverage to at Least 80° 2θ.

Data Reduction Integration with the CrysAlis PRO RED software provided by Oxford Diffraction

X-ray Source MoKα, graphite

Wavelength 0.71073 Å Scan speed 0.05deg/sec, frame width 1 degree, total frames 1328. Monochromator Graphite crystal Collimator 0.3mm, 0.5mm


Figure 2.1.8: Overall view of Oxford Diffraction Xcalibur Diffractometer

Figure 2.1.9: Sealed-off X-ray tube shield and goniometer of Oxford Diffraction Xcalibur Diffractometer

33 2.3 Data reduction

2.3 Data reduction

The measured intensity, I(hkl), of diffracted X-ray beam can by calculated using the

formula of Galton Darwin for a crystal rotating with a uniform angular velocity, ω,

through a reflecting position [26]:

This equation shows that the structure factor amplitude, |F(hkl)|, is a function of the

intensity, I(hkl), of a diffracted beam. This conversion of I(hkl) to |F(hkl)| involves

the application of corrections for X-ray background intensity (I), polarization (P),

Lorentz (L), Absorption (A), Extinction and thermal motion. This process is known

as data reduction [27].

2.3.1 The polarization factor

The radiation from normal X-ray tube is unpolarized. When a totally unpolarized

beam is diffracted by a crystal, the beam will be partially polarized and the diffracted

intensity is affected by a factor called polarization factor. So the correction for

polarization turns out to be [28]:

where 1 + cos22θ denotes the polarization factor, sin 2θ describes the change in

irradiated volume of a crystal as a function of 2θ (the single crystal Lorentz factor).

The correction to be applied to the measured intensity is 1/p. Since we use graphite

crystal monochromator in a diffractometer the incident beam on the crystal itself will

be partially polarized. Hence polarization factor is:

where θm is the Bragg angle for monochromatizing crystal and pf is the perfection

factor for monochromatizing crystal.


2.3.2 The lorentz factor

The Lorentz factor takes into account the fact that for a constant angular velocity of

rotation of the crystal, different reciprocal lattice points pass through the sphere of

reflection at different rates. When the crystal rotates, the three-dimensional reciprocal

lattice also rotates. The reciprocal lattice point near rotation axis will cut across the

Ewald sphere more slowly than one away from the rotation axis. So they have

different times of reflection. For a 4-circle diffractometer measurement, Lorentz

correction is

2.3.3 Absorption

When X-rays of intensity Io pass through a material their intensities are attenuated.

The trans-mission of the x-ray beam through the crystal is given by [29]:

where ti and td are the incident and diffracted beam path lengths and μ is the linear

absorption coefficient. If the shape of the crystal is exactly known, then it is possible

to correct for absorption by calculating

where dV is an infinitesimal volume of the crystal [30].

However if' the crystal faces are not well defined it is necessary to resort to empirical

methods which attempt to measure T experimentally [31]. For intensities measured

with the Oxford Diffraction Xcalibur and BRUKER SMART diffractometers, multi-

scan [32] and Ψ-scan correction method [33] respectively are found to be successful.

In the present work multi-scan and Ψ-scan were used.

The crystal is assumed to be totally bathed by the incident X-ray beam. Consider a

point P (Figure 2.1.10) in the diffractometer plane and on the Ewald sphere. If the

crystal is rotated about the vector r* the point P will still remain on the diffracting


position and the variation in observed intensity may be attributed to absorption

effects.

Figure 2.1.10: The bisecting geometry arrangement

In practice, the φ rotation axis is brought to a direction coincident with that of the

vector r* which require the choice of a reflection with a x value of 90o. Then the

azimuthal angle Ψ is equivalent to the diffractometer angle φ.

The relative transmission factor for a given value of is given by:

where the Imax(ф) (cp) is the maximum intensity observed as φ is varied over the

360° range.

2.3.4 Extinction

Attenuation of diffracted beams may also occur due to extinction effects: primary and

secondary [34].

v Primary extinction causes weakening of intensity by multiple reflections

suffered by the incident wave for different lattice planes. Each scattering

causes a phase lag of π/2. Thus when the unscattererd radiation is joined

by the double) scattered radiation with a phase lag of π, destructive

interference results. Primary extinction is often negligible in single crystals

whereas secondary extinction is predominant in sufficiently perfect

crystals. Correction for the secondary extinction is generally considered at

the end of the refinement and can be given by approximate equation:


where k is the scale constant and g is secondary extinction coefficient and

is characteristic of the crystal for a given radiation.

v Secondary extinction accounts for the fact that lattice planes first

encountered by the primary beam will reflect a significant fraction of the

primary intensity so that the deeper planes receive less primary radiation.

This causes a weakening of the diffracted intensity, mainly observable for

high intensity reflections at low sin θ/λ values [35].

In the crystal structure refinement package SHEIXL-97 [36], an extinction parameter

x is refined by least squares,

where K is the overall scale factor.

2.3.5 Temperature factor

The size of the electron density cloud around an atomic nucleus is independent of the

temperature, at least under normal conditions. This would suggest that X-ray

scattering by a crystal would also be independent of the temperature. However, this is

not true because the atoms vibrate around an equilibrium position. The X-rays do not

meet identical atoms on exactly the same position in successive unit cells. This is

similar to an X-ray beam meeting a smeared atom on a fixed position, the size of the

atom being larger if the thermal vibration is stronger. This diminishes the scattered X-

ray intensity, especially at high scattering angles. Therefore, the atomic scattering

factor of the atoms must be multiplied by a temperature-dependent factor (Figure

2.1.11).

Figure 2.1.11: The atomic scattering factor of a carbon atom multiplied by the

appropriate.


The vibration of an atom in a reflecting plane h k l has no effect on the intensity of the

reflection (h k l). Atoms in a plane diffract in phase and, therefore, a displacement in

that plane has no effect on the scattered intensity. The component of the vibration

perpendicular to the reflecting plane does have an effect. In the simple case in which

the components of vibration are the same in all directions, the vibration is called

isotropic. Then the component perpendicular to the reflecting plane and thus along S

is equal for each (h k l), and the correction factor for the atomic scattering factor is:

Assuming isotropic and harmonic vibration, it can be shown that the thermal

parameter B is related to the mean square displacement of the atomic vibration:

If the atomic vibration is split into three perpendicular components—one

perpendicular to the reflecting plane [vibr( )] and two in the plane, vibr(||1) and

vibr(||2)—then vibr( ) is the only one giving rise to the temperature factor with

parameter [37].

For anisotropic vibration, the temperature factor is much more complicated. In this

case , depends on the direction of S. It can be shown that the temperature factor is

given by:

with U11 the value along a , U22 along b*, and U33 along c*. In general,

along a unit vector e is given by:

38 2.4 Scaling

with e1, e2, and e3 the components of e along unit axes a*, b*, and c*. The points for

which is constant form an ellipsoid: the ellipsoid of vibration. For display

purposes, the constant can be chosen such that the vibrating atom has a chance (e.g.,

50%) of being within the ellipsoid (Figure 2.1.12).

Figure 2.1.12: The plot of an organic molecule with 50% probability of thermal ellipsoids. Waleed Fadl Ali Al-eryani et al. (2010). Acta Cryst. E66, o1742.

In summary, an appropriate treatment is to describe the thermal ellipsoid by a tensor

U having six independent components in the general case. So we have the correction

due to thermal motion by using equation 2-33.

2.4 Scaling

The incident X-ray beam intensity fluctuations and possible radiation damage to the

crystal may be monitored by measuring four standard reflections of moderate

intensities at regular intervals. The average of these intensities relative to the average

of their starting values is smoothed and used to re-scale the raw intensity data. If S is

the scale factor, then the total correction applied is

where Ir is the corrected relative intensity.

39 2.5 Structure determination

2.5 Structure determination

2.5.1 The phase problem

To be able to solve the three dimensional structure of a molecule the position of each

atom in the unit cell has to be determined. The image of the scattering matter in the

point x, y, z can be represented by a Fourier summation:

where V is the volume of the unit cell, F(hkl) is the structure factor for the particular

set of h,k,1. (h, k.1 are the Miller indices with which the crystal plane or face make

intercepts a/h. b/k, c/i with the edge of the unit cell of lengths a. b, and c), x,y,z are the

fractional coordinates which are the atomic coordinates expressed as the fractions of

the unit cell length

and

Where, |F(hkl)| is the amplitude of the structure factor, |F(hkl)|2 is proportional to the

intensity of the reflections. α is the phase of the scattered beam. We can measure the

amplitude which is proportional to the square root of the intensity of the

reflections, but the direct measurement of the relative phase is not possible. So the

trick of crystallography lies in the finding of the phase.

2.5.2 Methods of structure solution

Since the electron density in a crystal varies continuously and periodically in three-

dimensional space, the electron density ρ(xyz) at a point with fractional coordinates x,

y, z in a unit cell of volume V may be expressed as a three-dimensional Fourier series:

where the summation is from -∞ to +∞ over all the hkl values and V is the unit cell

volume.


The magnitude of structure factors can be determined from the measured intensities of

the reflections, but it is not normally practicable to measure their relative phases.

Except for the phase, all quantities on the right hand side of equation (2-38) are

known. This loss of phase information constitutes the 'phase problem' in X-ray

crystallography.

An attempt to find solutions to the phase problem started around 1934 [38]. Two

substantially different approaches are used in crystallography to solve the phase

problem. While one tries to determine the phases of the reflections directly (direct

methods) by using statistical methods, the other makes use of the so-called Patterson

function which can be calculated from the experimental intensities [39].

2.5.2.1 Patterson method

In 1934, Patterson, A. L. introduced and discussed the physical significance of Fourier

series which can be directly calculate from experimental intensity data [40]. This

method can be applied to structures in which one or a few atoms are markedly heavier

than the rest. It is possible to find the locations of the heavy atoms by use of the

Patterson function which does not require a prior knowledge of the phases. The

Patterson map, commonly designated ρ(uvw), is a Fourier synthesis that uses the

indices, h,k,l, and the square of the structure factor amplitude, , of each

diffracted beam. It is usual describe the Patterson map in vector space defined by u,v,

and w, rather than x,y,z as used in electron-density maps. The Patterson method

consists of evaluating Patterson function which can be represented with the following

equation:

If the electron density is known, the Patterson function at any point u,v,w may be

calculated by multiplying the electron density at a point x, y, z with that at x+u, y+v,

z+w, doing this for all values of x, y and z and summing the products [41]:


In other words, the Patterson function is the convolution of the electron density at all

points x, y, z in the unit cell with the electron density at points . A

peak at in the Patterson map represents a vector from the origin of the

Patterson function to the point u,v,w. This means that if any two atoms in the unit cell

are separated by a vector then there will be a peak in the Patterson map at

u,v,w. The peaks in a Patterson map are inherently broader than those in a Fourier

map for finite size of atoms. Instead of N peaks in a unit cell of a Fourier map, there is

a crowding of N(N-1) peaks in the same volume of the Patterson map enhancing the

probability of superposition. However, when the structure contains one or more heavy

atoms, the peaks due to interactions of the heavy atom stand out among other peaks of

lower magnitude making interpretation comparatively easy. The Patterson technique

is therefore best applicable to heavy atom structures. The difficulty with the Patterson

method is that a unit cell with N independent atoms will have N2 inter-atomic vectors

(peaks in the Patterson map). When N is large as for a macromolecule, the peak

overlaps make it difficult to solve the structure directly using this method [42].

2.5.2.2 Direct methods

The term 'direct methods' is applied to that class of methods which seek directly to

solve the phase problem by the use of phase relationships based on the observed

intensities [43]. Direct methods, implemented in widely used highly automated

computer programs such as SHELXS [36] provide computationally efficient solutions

for structures with fewer than about 100 independent non-H atoms. Until the last

decade most of the advances in the field of direct methods concentrated on the

development of the methods [44] themselves, i.e., the intensity data used for these

developments corresponded to ideal single crystal diffraction experiments. More

recently, however, developments like the direct methods modulus sum function have

helped to extend the applicability of such methods to less favourable situations like

the crystallography of reconstructed surfaces.

In direct methods structure factor phases are derived directly from the observed

amplitudes through purely mathematical techniques; without assuming any kind of

molecular structure. When the number of structure amplitudes measured is more than

ten times the number of the parameters to be determined, direct methods, which

utilize the sign and phase relations, can be used to solve the structure. One such


relation applicable to centrosymmetric system, where phase can only be 0o or l80o, is

that the sign of F(hkl) is probably equal to the product of sign of F(h`k

`l`) and F(h-h,k-

k`,l-l

`). Possible phases can be derived from systematically searching for sets of

reflections (with high intensities) whose indices are related in this way. An electron

density map can then be calculated. Peaks are assigned for the atoms in the structure

to get the trial structure. From the trial structure the new phases are calculated. The

structure can be solved by repeating this cycle. This method is mainly used for small

molecules with up to roughly 50 non-hydrogen atoms.

The various steps involved in the direct methods are [45]:

Step I Conversion of observed structure factor |Fhkl| to normalized

structure factor |Ehkl| which are independent of θ.

Step II Setting up of phase relations using triple phase relation

(triplets) and four phase relations (quartets).

StepIII Selection of few reflections, the phase of which are assigned apriori.

Step IV Phase propagation and refinement using tangent formula.

Step V Calculation of best phase sets and expressing the reliability

of the phases in term of Combined Figure of Merit (CFOM).

Step VI Calculation of electron density map (E-map) with |Ehkl| as

the Fourier coefficient.

The electron density at a point with position vector r can be expressed as

and the quantity F( k) is given by:

where the integration is carried over the unit cell volume. For a unit cell having n

discrete atoms this becomes,

The Phase of this quantity is given by:


This is an indirect method of estimating the values of ( k)’s.

2.5.2.3 Unitary structure factor

While searching for relations between the phases and the magnitudes of |F|, we must

remember that apart from their dependence on the position of the atoms [46], |F| s are

also affected by the finite size of the atoms and thermal vibration contributes due to

these factors are evident for our purpose and the magnitudes of |F| may

advantageously stripped off of these two effects by suitable modification, so that the

modified values correspond to the idealized situation of a structure built up by

stationary point atoms. One way of doing this is to express Fh as a fraction of its

maximum value, which is

Thus we get the unitary structure factor the values of which

evidently lie between –1 and +1.

2.5.2.4 Normalized structure factor

In direct methods, the measured structure factors are modified so that the maximum

information on atomic position can be extracted from them. Other effects, such as the

falloff of intensity at high scattering angles due to atomic size and atomic vibrations

are eliminated, to be considered when the structure has been determined. The

structure-factor data are therefore converted to those expected for a structure

composed of point atoms at rest (not vibrating). The scattering factor expression for

real atoms involves the scattering of an atom with a finite size (represented by ),

and an exponential factor representing atomic vibrations and disorder. The expression

is:


In direct methods it is usual to replace this expression by one for that of point atoms,

so that X-ray scattering is now essentially independent of sinθ/λ. This is done by

dividing F(hkl) by a function of that eliminates any fall-off of intensity as a function

of sinθ/λ (Figure 2.1.13).

Figure 2.1.13: Scattering curves for regular atoms and stationary point atoms.

The resulting normalized structure factor, |E(hkl)| is:

where was defined in Equation (42-2) and the integer ε the epsilon factor is a factor

that is generally unity, but is needed to account that certain classes of Bragg

reflections, such as those with one zero index, that have, as a group, an average

intensity higher than that for the rest of the Bragg reflections [47]. Values for ε for

each type of Bragg reflection can be found in International Tables [48].

2.5.3 Structure invariant and semi-invariants

Although the values of individual phases are known to depend on the structure and the

choice of origin, certain linear combinations exist of the phases whose values are

determined by the structure alone and are independent of the choice of origin. The

linear combinations of the phases are called the structure invariants [49].


2.5.4 Refinement of structure

The atomic positions in the first solution are not the direct result of the diffraction

experiment but an interpretation of the electron density calculated from the measured

intensities and the ‘somehow-determined’ initial phase angles. New, usually more

accurate phase angles can be calculated from the atomic positions, which allow re-

determining the electron density function with a higher precision. From the updated

electron density map, more accurate atomic positions can be derived, which lead to

even better phase angles, and so forth. New atoms can be introduced into the model,

when the most recent electron density function shows a high value at a place in the

unit cell where the model does not contain an atom yet. Sometimes, atoms need to be

removed from the model when they occupy positions in the cell corresponding to a

low value in the electron density function. When the atomic model is complete, atoms

can be described as ellipsoids rather than spheres (anisotropic refinement) and

hydrogen atom positions can be determined or calculated. Every step in this process is

undertaken to improve the accuracy of the model, and the entire procedure from the

initial atomic positions to the complete, accurate and (if achievable) anisotropic model

with hydrogen positions is called the refinement. A critical point in this process is the

evaluation of the model, as the model should only be altered if a change improves its

quality. There are several mathematical approaches to define a function which is

assumed to possess a minimum for the best possible model: in the world of small

molecules (typically less than 200 independent atoms) the least-squares approach is

by far the most common method.

A few cycles of least-square refinement can lead to best adjusted parameters. The

refinement [Refinement is the process of adjusting the model to find a closer

agreement between the calculated and observed structure factors] is continued until

the shifts in the parameters are some small fraction of the estimated standard

deviation of corresponding parameters [50].

SHELXL [51] was used to refine the structures reported in this thesis. The refinement

of the structures against F2 in this program makes possible the refinement of all the

data except those known to suffer from systematic errors. At the end of the refinement

the program computes the weighted R-factor and the Goodness of fit, GOOF (also

called the standard deviation of an observation of unit weight).


2.5.4.1 Residual factors

The quality of the model can be judged with the help of various residual factors or ‘R-

factors’. These factors should converge to a minimum during the refinement and are

to be quoted when a structure is published. The three most commonly used residual

factors are [52]:

v The weighted R-factor based on F2: ωR (or ωR2 in SHELXL), which is

most closely related to the refinement against squared structure factors.

where being the standard deviation of Fo.

v Albeit based on F values and hence mostly of historical value, the most

popular one is the unweighted residual factor based on F: R (or R1 in

SHELXL), given by

Thus R measures the relative discrepancies between Fc and Fo, so that

the lower the value of R, the better is the agreement. In an ideal case R =

zero.

v Finally, there is the goodness of fit: GooF, GoF, or simply S, given by:

where NR is the number of independent reflections and NP the number of

refined parameters.

A goodness of fit S < 1 suggests the model is better than the data.

Obviously this is suspicious and usually a sign that there are some

problems with the data and/or the refinement. Frequently, failure to


perform a proper absorption correction leads to underestimated GooF

values, but refinement in the wrong space group can also have this

effect.

2.5.5 Difference Fourier synthesis

The structure factors have been calculated for a given electron density in the unit cell.

It is also possible to calculate the electron density in the unit cell by inverse operation

for a given set of structure factor, this inverse operation can be given by the three

dimensional Fourier series by [53]:

where

is electron density at (x,y,z) in the unit cell V: is the unit cell volume

(x, y, z): are the coordinates which represent crystallographic axes.

A difference synthesis can by performed by subtracting an electron density, calculated

using Fc from the usual one containing Fo.

This can be given by

where

Note that the same phases c are used both calculations.

or

where c is the phase of Fc

From the value of maximum and minimum the missing atoms or

misplaced atoms or the atom which has been considered as a different atom can be


identified. In general (xyz) for hydrogen atoms are not calculated and are fixed at

chemical acceptable positions.

2.5.6 Weighting scheme

The weighting scheme is defined as follows:

where σi is the standard deviation. The reciprocal of the variance is indeed a measure

of its reliability, and the function minimized is:

Where σ is the standard deviation and it can be shown that

where N is number of counts. In diffractometer while measuring the X-ray reflection,

left background and right background counts are considered and the standard

deviation turns out to be

where K is a scale constant, L is the Lorentz factor, P is the polarization factor and

2.5.7 Some results derived from the refined structure

The most important results that can be derived from the refined structure are the

distances between bonded atoms and the angles between pairs of bonds. It is also

necessary to find the equation of the mean plane passing through a group of atoms

supposed to lie on a plane, to check the deviation of the atoms in such a group from

the plane and perhaps also to calculate the distance of some other atoms from it.

In our computer programs for all such calculation, the fraction coordinates of any

atom X(= x,y,z) with respect to crystallographic axes are transformed into coordinates

in Å, with respect to the orthogonal coordinates whose x-axis is


coincident with the crystallographic a-axis, y-axis lies in ab-plane and z-axis along

c*-direction. These transformed coordinates are given by:

where U is transformation matrix, given by

2.5.8 Bond distances, angles and their standard deviations

Calculation of interatomic distances and angle using the orthogonal coordinate system

is too simple to be described here. If σx,σy, σz, are the estimated deviations of the

fractional coordinates x, y, z of an atom, then the matrix Po with respect to

crystallographic axes is

The variance matrix Ps with respect to orthogonal axes, as defined above, is given by

where denotes matrix transposition. For an atom with variance matrix Ps, the

variance in the direction with direction cosine L=(1,m,n) is

Now, for a distance between atoms A and B having respective variances and

in the direction of bond, the variance is given by

In case of distance of symmetry related atoms

For the angle between atoms A, B and A, C of respective lengths rAB, rAC, the

variance [30] is given by


where and are the variance of atoms B and C in the direction at right angle to

and respectively, and is the variance of A in the direction of the center of

the circle passing through B, A and C. when A and B are related by symmetry

In our calculation of standard deviation of bond angles it is assumed that the variances

of the atomic coordinates are spherically symmetric,

2.5.9 Mean planes

The equation of a plane in orthogonal coordinate system at a distance p from the

origin is:

where l, m, n are direction cosines of the normal to the plane. The distance of a point

(Xi, Yi, Zi) from the plane being

The equation of the mean plane through a group of N atoms is obtained by

minimizing the quantity

where

The condition,

gives,


or

where

showing that the mean plane passes through the centroid of group of N

atoms. Writing now,

The condition gives three liner homogeneous equation in l,m,n which

are solved by a procedure described be Blow [54], remembering that l2+m2+n2+1=0.

Thus equation for the mean plane referred to orthogonal axes as,

The deviation of the N atoms from the plane may be calculated and we

can also calculate

where

2.5.10 Hydrogen bonding

A hydrogen bond is a bond between a functional group X—H and an atom or group of

atom A in the same or a different molecule [55]. Hydrogen bonds are formed only

when X is oxygen, nitrogen, or fluorine and when A is oxygen, nitrogen or fluorine.

The oxygen may be singly or doubly bonded and the nitrogen singly, doubly, or triply

bonded. The bonds are usually, represented by dotted lines, as shown in the following

examples (Figure 2.1.14):


Figure 2.1.14: some examples in hydrogen bond.

Hydrogen bonds can exist in the solid and liquid phases in solution [56]. Hydrogen

boning has been detected in many ways, including measurements of dipole moments,

solubility behavior, freezing-point lowering, and heats of mixing, but the most

important way is by the effect of the hydrogen bond on IR [57] and other spectra.

The bond may be described in terms of the d, D, θ and r as shown in (Figure 2.1.15)

Figure 2.1.15: Definition of the geometers d, D, θ and ф for a hydrogen bond.

Hydrogen bonds are important because of the effects they have on the properties of

compounds, among them:

v Intermolecular hydrogen bonding raises boiling points and frequently

melting points.

v If hydrogen bonding is possible between solute and solvent, this greatly

increases solubility and often results in large or even infinite solubility

where none would otherwise be expected. It is interesting to speculate what

the effect on the human race would be if ethanol had the same solubility in

water as ethane or chloroethane.

v Hydrogen bonding causes lack of ideality in gas and solution and solution

laws.

v As previously mentioned hydrogen bonding changes spectral absorption

positions.

v Hydrogen bonding, specially the intermolecular variety, changes many

chemical properties.


v Table 2.1.3 lists properties of hydrogen bonds that we classify as very

strong, strong and weak. These properties are geometrical, energetic,

thermodynamic and functional in nature [58].

Table 2.1.3 Some Properties of very strong, strong and weak hydrogen bonds.

Very strong Strong Weak

Bond energy (-Kcal/mol) Examples

15-40 [F-H···F]-

[N-H···N]+

P-OH···O=P

4-15 O-H···O = C N-H···O = C O-H···O-H

<4 C-H···O O-H···π Os-H···O

Bond length H-A≈ X-H H···A > X-H H…A >> X-H Length of X-H(Å) 0.05-0.2 0.01–0.05 ≤ 0.01 D(X…A) range (Å) 1.2–1.5 1.5–2.2 2.0–3.0 Bonds shorter than vdw 100% Almost 100% 30–80% Θ(X-H···A) range (0) 175-180 130-180 90-180 KT (at room temperature) < 25 7-25 < 7

2.5.11 Torsion angles

Torsion angles (or angle of twist) of ring system or chain of atoms are the dihedral

angles between planes formed by consecutive bonds. For a sequence of atoms

A,B,C,D the torsion angle of bond B—C is the angle between the positive normals to

the planes (B—A, B —C) and (C— B, C—D), where the positive direction of a normal

is defined to be that which forms a right-handed set with the two vectors defining the

plane. The torsion angle between groups A and D is then considered to be positive if

the bond A–B is rotated in a clockwise direction through less than 180 in order that it

may eclipse the bond C–D; a negative torsion angle requires rotation in the opposite

sense.

Stereochemical arrangements corresponding to torsion angles between 0 and

90 are called syn (s), those corresponding to torsion angles between 90 and

180 anti (a). Similarly, arrangements corresponding to torsion angles between

30 and 150 or between –30 and 150 are called clinal (c) and those between 0 and

30 or 150 and 180 are called periplanar (p). The two types of terms can be

combined so as to define four ranges of torsion angle; 0 to 30 syn-periplanar (sp);

30 to 90 and –30 to –90 syn-clinalal sc); 90 to 150 , and –90 to –150 anti-clinal

(ac); 150 to 180 anti-periplanar (ap) [59] (Figure 2.1.16).


Figure 2.1.16: Torsion angles.

2.6 Computations

Data collection: SMART (Bruker, 2001) [24]; CrysAlis PRO RED; CCD (Oxford

Diffraction, 2010) [25]; cell refinement: CrysAlis RED (Oxford Diffraction, 2004)

[25]; SAINT–Plus [24]; data reduction: CrysAlis RED [25]; SAINT –Plus [25] ;

program(s) used to solve structure: SHELXS-97 [36]; program(s) used to refine

structure: SHELXL-97 [36]; molecular graphics: MERCURY [60]; ORTEP-3 [61] and

PLATON [62]; software used to prepare material for publication: WinGX [63];

SHELXL-97[36] and PLATON [62].

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2.1 Single crystal X-ray diffraction 2.1.1 Historical ...shodhganga.inflibnet.ac.in/bitstream/10603/36564/5/chapter 2.pdf · X-ray Crystallography ... 2.1.1 Historical background