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Chapter 2

X-ray Crystallography

Crystallography is the science of crystals which is devoted to the studyof their development and growth, their external form, internal structureand physical properties.1 Crystal structure analysis is usually based ondiffraction phenomena caused by the interaction of matter with X-rays,electrons, or neutrons.2 X-ray crystallography has application in nearlyevery branch of chemistry, physics, metallurgy and biology.

2.1 Brief history

Prior to the discovery of X-rays, crystallographers had deduced that crys-tals are made up of an orderly arrangement of atoms and could infer some-thing about this orderly arrangement from measurements of the anglesbetween crystal faces. The first contribution was from Johannes Kepler inthe year 1611. The major discovery about the true nature of crystals wasmade by Stenson. In 1669, he postulated that the angles between faces ofcrystals are constant, regardless of the size or shape of the faces. In 1839,Miller introduced the indexing of the planes of the crystals and classifica-tion of crystals was made by Hessel and Gadolin into 32 classes. Newmanshowed that the laws of symmetry hold good even for the properties exhib-ited by the crystals. Based on the atomic theory of Seeber, Bravais proved

8 X-ray Crystallography

the existence of 14 lattices which are currently named after him. Between1880 to 1891, the 230 crystallographic space groups were independentlydiscovered by the Fedorov, Schoenflies and William Barlow. The X-rayswere discovered in 1895 by Rontgen. Later Sommerfeld measured the X-raywavelength.

The discovery of X-rays gave crystallographers a powerful tool that couldsee inside of crystals and allow for a detailed determination of crystal struc-ture. The fine structure of crystals was first demonstrated by Friedrich andKnipping in 1912. This was the starting point of modern X-ray crystallo-graphy which saw a rapid development of theoretical ideas and techniquesto unravel the structures of a wide range of crystalline substances, bothnatural and synthetic.

In 1912, experiments to demonstrate diffraction of X-rays were carried outby von Laue group. Later W.L. Bragg demonstrated that the measurementof the diffraction pattern gives information on the distribution of electrondensity in the unit cell.3

In 1921, W.H. Bragg published the real work on organic structures of largenumber of compounds which included napthalene and anthracene. Com-plete structure could not be obtained but it gave the evidence that themolecules, postulated by the organic chemists, do have a real physical ex-istence in the crystal. In 1923, Dickinison and Raymond determined thecrystal structure of hexamethylene tetramine. They observed that the crys-tal is cubic in nature, the C–N bond distance is 1.44Å and showed how themolecules were arranged in the crystal. It proved the chemical formula forthe first time. Until this we knew nothing about bond-lengths and stereo-chemical aspects of the conformation. Complete X-ray analysis gives theprecise location of every atom in space.

2.2 X-rays

X-rays are a form of electromagnetic radiation, which lie between ultra-violet light and gamma radiation of electromagnetic spectrum. The ap-proximate range of wavelength of X-rays is 0.01 to 100 Å, corresponding to

2.3 Symmetry in crystals 9

frequencies in the range 3×1016 to 3×1019 Hz and energies in the range of120eV to 120keV. X-rays have varied properties, so they are used in variousapplications in science and industry.

The generation of X-rays has been performed for a long time exclusivelyin sealed high-vacuum tubes since Conrad Wilhelm Rontgen’s pioneeringwork in 1895. A focused beam of electrons, generated by an applied voltageof 20−60 kV, hits a metal target (usually pure Mo, Cu, Fe or Cr) which actsas the anode opposite to the electron emitting cathode. Because the tubeis evacuated and the cathode is at a high negative potential with respect tothe metal anode, the electrons get accelerated and reach the anode at highspeed. The target needs to be cooled with water because 99% of the electronenergy is converted to heat. The fluorescence effect of induced electrontransfer at near core levels of the target atoms produces sharp characteristiclines together with a continuous energy spectrum. The characteristic linescorrespond to transitions between discrete levels, following the selectionrules and energy differences of the shell’s fine structure. The continuousspectrum arises from the energy loss of electrons getting decelerated oninteraction with the target material. By replacing a fixed anode by a rapidlyrotating anode, considerably higher X-ray intensities can be obtained.

2.3 Symmetry in crystals

Symmetry is a property by means of which an object is brought into self-coincidence. Symmetry element is a point or a line or a plane with respectto which symmetry operation may be performed. An object may containmore than one symmetry element.2 The symmetry elements are given below

• Axes of rotational symmetry.

• Roto-translation or screw axes.

• Axes of inversion.

• Axes of roto-reflection.

• Glide plane.


A collection of interacting symmetry elements is called a point group. It isdefined as the set of symmetry elements, the operation of which leaves atleast one point unmoved. This point is called the origin, and all symmetryelements in a group pass through this point. There are 32 three dimensionalcrystallographic point groups. The set of crystals having the same pointgroup is called crystal class. Each crystal class will have crystal faces thatuniquely define the symmetry of the class.

Symmetry plays an important role in crystallography. A crystal can beconceptually regarded as

crystal structure = lattice ∗ motif

where, the motif is the structural unit which is repeated regularly in spaceand lattice is a conceptual array of points in space which serves to definethe geometrical relationship between the motif in a structure. The symbol∗ implies convolution of the lattice with motif. The two dimensional planelattices and three-dimensional space lattices are defined in terms of crys-tallographic unit vectors and crystallographic unit cells. The conventionalcrystallographic unit cell is a parallelopiped defined by the space latticeof a crystal which serves to display the symmetry of the lattice in a con-venient manner. The conventional crystallographic unit vectors are a set ofthree non-coplanar lattice vectors which define the unit cell. The directionsdefined by such crystallographic unit vectors specify the crystallographicaxes.

2.4 Scattering of X-rays

The most powerful technique for investigating both the geometrical natureof the three dimensional array of molecules within a crystal and also thenature of the molecules themselves, is X-ray diffraction.4 This sort of dif-fraction by the crystal can be related to X-ray scattering by electrons.Intensity of each diffraction maximum can be derived in terms of overallelectronic distribution within the unit cell. During the twentieth century,different types of diffraction have been found. Neutrons and electrons,which would be considered particles according to classical mechanics, canbe diffracted.5 These results complement those from X-ray diffraction stud-

2.4 Scattering of X-rays 11

ies, because mechanisms for the scattering of X-rays, neutrons and electronby atoms are different. X-rays are scattered by electrons, while neutronsare scattered by atomic nuclei. Neutron diffraction is important for thedetermination of the locations of light atoms like hydrogen atoms, whichbecause of their low electron count are poor X-ray scatterers. The in-teraction of X-rays with electrons essentially occurs either by Thomsonscattering or Compton scattering.

2.4.1 Thomson scattering

Thomson scattering is the classical explanation where we consider the in-teraction of a wave with a particle. When X-rays hit an atom, the rapidlyoscillating electric field of the radiation sets the electrons of the atom intooscillation about their nuclei. This oscillation has the same frequency asthat of the incident radiation. The result is that the electron acts as anoscillating dipole that serves as a source of secondary radiation with thefrequency of the incident beam. J.J.Thomson showed that when radiationis scattered by an electron, there is a phase shift of 180◦. This phasechange is same for scattering by all atoms in the crystal, in the absence ofunusual effects. The effect of an incident wave on an assembly of electronsensures that the phase relationship between waves scattered from differentelectrons is well defined. Hence this phenomenon is referred to as coherentscattering. Thomson scattering is often referred as coherent scattering.

2.4.2 Compton scattering

The Compton effect requires a quantum mechanical explanation unlike theThomson scattering which was dealt with as a wave particle interaction.The model used to describe Compton scattering considers the collisionbetween an electron and a quantum of electromagnetic radiation called aphoton. This may be considered as a particle-particle collision. Using thismodel, the scattered photons may be characterized by two wavelengthsjust as observed experimentally. The Compton calculation uses a particle-particle collision model, therefore rejects the concept of phase. This impliesthat when we consider the combined effect of Compton scattering on an


assembly of electrons, there are no phase relationships between the scat-tering events. Consequently, the Principle of Superposition cannot be usedto assess the overall scattering effect of an assembly of electrons. Comptonscattering is also referred to as incoherent scattering.

2.5 X-ray diffraction

Diffraction occurs as waves interact with a regular structure whose repeatdistance is about the same as the wavelength. X-rays have wavelengths ofthe order of a few angstroms, the same as typical interatomic distances incrystalline solids. The scattered X-rays from a crystalline solid can con-structively interfere, producing a diffracted beam. The molecules whichact as scattering centres within the crystalline solid are arranged in a well-defined, three-dimensional ordered array. Hence there exists a specific dis-tance between two molecules which lead to a well-defined phase relationshipbetween the wave scattered from any pair of scattering centres. The wavescattered from the complete set of molecules within the crystal will becombined together in a definite manner giving rise to a clear and precisediffraction pattern.

2.5.1 Diffraction pattern and Fourier transform

The atoms are distributed in a certain manner in unit cell of a crystal,and there exists some electron density function ρ(r) which describes thelocation of the electrons within the atoms of the unit cell. The electrondensity function will have a large value at positions corresponding to atomicnuclei, and a relatively low value between nuclei. If we can find the electrondensity function, then we may infer the positions of atoms from the electrondensity maxima and derive data such as that pertaining to bond lengths andbond angles. All the fundamental structural information of the contents ofthe unit cell is contained within the electron density function, and it is thisfunction which we wish to determine from X-ray analysis.

Let f(r) represent the effect on an incident beam from a scattering centre

2.5 X-ray diffraction 13

of volume dr. Effect of all the scattering centres over volume V is

F (k) =

∫

Vf(r) eik.rdr

The above equation is the Fourier Transform of f(r). It represents theamplitude of scattered X-rays from all centres. It is denoted by F (k) wherek is a wave vector. F (k) and f(r) are called Fourier transform mates.

F (k) ⇔ f(r)

F (k) is a complex quantity of the form

F (k) = |F (k)|eiφ

where eiφ represents the phase part. |F (k)|2 is intensity of diffracted X-rays.

2.5.2 Bragg’s law

Bragg’s law relates the wavelength of X-rays, angle of incidence and spacingbetween the crystal lattice planes of atoms. The law allows each observeddiffracted beam to be uniquely labelled with three indices (hkl) called Millerindices. Bragg6 showed that every diffracted beam that can be produced byan appropriate orientation of a crystal in an X-ray beam can be regardedas if it were a reflection from sets of planes passing through lattice points,analogous to the reflection of light from a mirror. The reflections fromadjacent planes in the set give rise to interference effects which is governedby the equation

2dhkl sin θ = λ

The above equation is called the Bragg equation. Rearrangement of theBragg equation gives

sin θ =

(

λ

2

)(

1

dhkl

)


2.5.3 Reciprocal lattice and Ewald sphere

The concept of reciprocal lattice is very helpful in explaining the X-raydiffraction phenomenon. For each direct lattice, a corresponding reciprocallattice may be postulated. When a crystal is rotated in the path of X-ray beam, it can be assumed that the reciprocal lattice associated with itis also rotated. It has same symmetry as the direct lattice, and may bederived from it. The reciprocal lattice points form a true lattice with arepresentative unit cell characterised by six parameters namely three sidesa∗, b∗, c∗ and three angles α∗, β∗, γ∗.3 The volume of the reciprocal cell isgiven by

V ∗ =K3

Vc

where Vc is the unit-cell volume and K is taken as the wavelength of theX-ray used.7 The reciprocal lattice of an infinite crystal is an infinite arrayin which reciprocal lattice sites are ideal mathematical points. The finitesize of a real lattice causes blurring or smearing out of the reciprocal latticepoints and they are no longer ideal mathematical points.

A geometrical description of diffraction which comprises the Bragg’s lawand the Laue equations was originally proposed by P.P. Ewald. Since thewavelength of the radiation remains unchanged during diffraction, the locusof possible diffracted wave vectors will sweep out a sphere with the positionof the crystal on its centre. This is called the Ewald sphere or the Reflect-ing sphere. The Ewald construction provides a geometrical relationshipbetween the orientation of the crystal and the direction of the X-ray beamsdiffracted by it.

2.5.4 Inverse Fourier transform or Fourier synthesis

The information on the crystal structure is contained in the manner inwhich the crystal perturbs the incident waves into scattered (diffracted)waves. This information is contained in the spatial distribution of diffractedwaves. If the diffraction pattern F (k) is known, the structural information

2.6 Data reduction 15

f(r) can be determined.

f(r) =

∫

all k

f(r) e−ik.r dr

2.5.5 Friedel’s law

Normally, X-ray diffraction pattern from a crystal is centrosymmetric what-ever the crystal class.4 A diffraction pattern may be thought of as a recip-rocal lattice with each point weighted by the corresponding value of I(hkl).Friedel’s law states the centrosymmetric property of the diffraction patternas

I(hkl) = I(hkl) = |F (hkl)|2

The breakdown of Friedel’s law in non-centrosymmetric crystal is of ex-treme importance in the solution of the phase problem.

Symmetry of an X-ray diffraction pattern is characterised by the symmetryof the reflection intensities and by the systematic absences. Diffractionmaxima which turn out to have zero intensity as a result of considera-tion of the symmetry of the structure of the unit cell are called systematicabsences.4 The systematic absences corresponding to lattice type unam-biguously determine the Bravais lattice. Information on screw axes andglide planes is obtained from the systematic absences. Friedel’s law helpsin the determination of presence of centre of symmetry.

2.6 Data reduction

The procedure followed to extract relative structure factor amplitudes fromraw integrated intensities is called data reduction. The primary goal of acrystallographer is to know the positions of the atoms within a unit cell.To obtain this, we measure the intensities of the reflections from a givencrystal. The measure of the total number of photons which are diffractedin the proper direction by reciprocal lattice point is termed as integrated


intensity. From these intensities, the electron density distribution in thecrystal can be analysed.

The structure factor Fhkl, is affected by certain physical and geometricalfactors. One needs to account for these influences by means of correctionsto be applied.8 They are polarization factor, Lorentz factor, absorptioncoefficient, extinction coefficient and temperature factor.

2.6.1 The Polarization factor

The incident X-ray beam is unpolarized, which means that the electricvector is in a random direction. The diffracted beam will be partially orcompletely polarized. Intensity of the diffraction maxima depends on thescattering angle. The correction for the polarization turns out to be

P =1 + cos2 2θ

2

The correction to be applied to the measured intensity is 1/P . Since graph-ite crystal monochromator is used in a diffractometer, the incident beamon the crystal itself will be partially polarized. The polarization factor is

P = (Pf )(cos2 2θm + cos2 2θhkl)

(1 + cos2 2θm)+ (1− Pf )

(cos 2θm + cos2 2θhkl)

(1 + cos 2θm)

where θm is the Bragg angle for the monochromatizing crystal and Pf isthe perfection factor for monochromatizing crystal.

2.6.2 Lorentz factor

Lorentz factor is used to correct diffraction intensities by taking into con-sideration the varying time which different Bragg reflections take to passthrough the sphere of reflection. As the point passes through the Ewaldsphere, the area of intersection will vary according to the shape of the re-ciprocal lattice point and precise geometry of interaction. When the crystal


rotates, the three-dimensional reciprocal lattice also rotates. The reciprocallattice point near the rotation axis will cut across the Ewald sphere moreslowly than the one away from the rotation axis. Further, opportunity fordiffraction of reciprocal lattice points close to uper levels is different fromthose close to equitorial plane. The Lorentz factor Lhkl is given by

Lhkl =1

λ

ω

vn=

1

sin 2θ

where 2θ is the scattering angle appropriate to the diffraction maximumindexed as hkl.

2.6.3 Absorption

According to Beer-Lambert’s law, absorption reduces the intensity of an X-ray beam traveling through a given material by an amount which dependson the material and the length of the path traveled. The path lengths aredependent on the location of the point scattering the X-rays and on theincident and scattering angle, that is, on the reflection considered. WhenX-ray beam passes through the crystal, the electric field causes electronicexcitations. Due to this the thermal energy of the electrons increase andhence the temperature of the crystal also increases. For this, energy of theincident X-ray beam is used. This attenuates the incident X-ray beam andknown as absorption.

When a crystal diffracts X-rays, the primary beam is absorbed as it entersthe crystal and the diffracted beam is absorbed as it leaves the crystal. IfIo is the intensity of incident beam, after traversing the thickness t of thecrystal, the intensity becomes

I = Ioe−µt

where µ is the linear absorption coefficient.

If the compound has very large absorption coefficient for the radiation used,the raw intensities require absorption correction. Among the methods usedfor absorption correction the numerical method14 and ψ-scan15 method are


very promising. This is a scan of diffraction data measured as the crystalrotates about the diffraction vector. The intensities can be corrected forabsorption as a function of the orientation of the crystal when the reflectionunder consideration was measured.

2.6.4 Extinction correction

Diffraction that involves multiple scattering events is called extinction. Itis due to the mosaic structure of the crystal where internal periodicity ofplanes is not perfect. There are two kinds viz. primary extinction andsecondary extinction. The lattice planes first encounterd by the primarybeam will reflect a significant fraction of the primary intensity and thedeeper planes receive less primary radiation. This causes a weakening ofthe diffracted intensity. Secondary extinction is more pronounced at lowsinθ/λ. This correction has been suggested by Darwin.16 The absorptioncoefficient is modified by adding a term g to account for secondary extinc-tion. Correction for the secondary extinction is generally affected at theend of the refinement and can be accounted for by an approximate equation

|Fc| = k|Frel|(1 + gIc)

where k is the scale constant and g is secondary extinction coefficient andis characteristic of the crystal for a given radiation.

2.6.5 Anomalous scattering

Anamalous scattering is the resonance phenomena between interaction ofX-rays with the lowest energy levels of the crystal. As the wavelength ofX-rays increases, its absorption by atom also increases slowly. At certainwavelenghts the absorption drops suddenly and then starts to rise again.These discontinuties are called absorption edges.

Due to perturbation of electron distribution, spherical symmetry and centro-symmetry are destroyed. Then the atom is said to be an anomalous scat-terer. At wavelengths just below the absorption edge, the atomic scattering


factor fj is no longer real and takes the form

fj = (fj)c +∆f ′ + i∆f ′′

where (fj)c is the form factor calculated on the assumption of sphericalsymmetry and is corrected by a real part ∆f ′ and an imaginary part ∆f ′′

both of which may be calculated theoretically. When fj is real,

|Fhkl| = |Fhkl|

which is the well known Friedel’s law. Since we measure intensities, Friedel’slaw implies

Ihkl = Ihkl

When an anomalously scattering atom is present, Friedel’s law breaks downand

Ihkl 6= Ihkl

2.6.6 Radiation damage

The exposure of a crystal to an X-ray beam is often harmful and leads toradiation damage of the crystal. Radiation damage causes Bragg reflec-tions to change intensity as a function of time. Radiation damage involvesmovemented of atoms so that each Bragg reflection is changed in a spe-cific way, depending on the crystal structure. Radiation damage may bedetected by monitoring a set of reference reflections that are measured atregular intervals throughout the data collection.

2.6.7 Thermal effects

The assumption that the nuclei are fixed at single points is valid only attemperatures near to absolute zero. This assumption is not true near roomtemperatures. Approximate corrections need to be applied. As the temper-ature increases, the ambient thermal energy causes the atoms to oscillateabout their equilibrium positions and so we cannot strictly assign point co-ordinates (xj, yj , zj) to the nucleus of any atom. Rather, (xj , yj , zj) should


be taken as the average position of the nucleus of the jth atom. Since thethermal motions are very rapid, we may represent their effect by associat-ing the nucleus with a position characterised by (xj, yj , zj) but which inreality is somewhat blurred and not an ideal mathematical point. As thenucleus oscillates about its average position, the electrons associated withthe atom also move, so the interaction of the electrons with the incidentX-rays is changed.

The effect of thermal vibrations is to reduce the effective scattering powerof an atom. If the atom is oscillating, then the expression for fj mustbe modified to take account of this. The temperature corrected atomicscattering factor (fj)T is defined as:

(fj)T = fje−Bj(sin

2 θ/λ2)

in which θ is the Bragg angle and λ is the wavelength of the incidentradiation. Bj is called temperature factor, and its value depends on theatom under study and the temperature at which the diffraction event takesplace. An implicit assumption in the form of the above equation is that thethermal vibrations of the atom j are spherically symmetric. The equationaccounts for the thermal motion by introducing a negative exponential.The term sin2 θ/λ2 is necessarily positive, and Bj is always positive. SinceBj(sin

2 θ/λ2) must be dimensionless, the dimensions of Bj are [length]2.The value of Bj is given by:

Bj =8π2

3〈u2〉

where 〈u2〉 represents the mean square displacement of the jth atom. Thisequation shows that Bj is necessarily positive.

Hence, (fj)T is always less than fj. When θ is zero, (fj)T equals fj. Also,the value of the exponential decreases quite rapidly as θ increases causing(fj)T to fall off sharply as compared to fj.

2.7 Structure solution 21

2.7 Structure solution

In order to determine the crystal structure we need both the magnitudesand phases of the structure factors. To find the structure, we have to obtainatomic positions in the unit cell from the diffraction data which is given by

ρ(x, y, z) =1

V

∑

h

∑

k

∑

l

Fhkle−2πi(hx+ky+lz)

But it is not possible in a unique and automatic way, because from the ex-perimental data only the magnitudes of the structure factors are obtained,not the phases. Therefore, in order to compute the electron density, wemust somehow derive the missing information, phase. The derivation ofthe correct values of the structure factors Fhkl, both magnitude and phase,from the relative intensities |Fhkl|2 constitutes the Phase Problem. Thephase problem can be solved in different ways, the two major methodsemployed are the Patterson method and the Direct methods.

In direct method, the approach is a more mathematical one in which therelative intensity |Fhkl|2 data are treated statistically and analytically inorder to search for certain relationships such as inequalities, which havebeen theoretically predicted. This in turn is used to determine the phases.The direct method is the most common method of determination of phasesfor small molecules.

Two important properties of the electron density function that are helpfulin obtaining the phase information are:

• it is everywhere positive, i.e., ρ(r) ≥ 0 (positivity).

• it is composed of discrete atoms (atomicity).

Possible phase angles are constrained by these two conditions, so that relat-ive phase determination hinges on the mathematical expressions for Fourierseries. In the total Fourier synthesis involving Fhkl with the correct valueof αhkl, these two conditions should apply.


2.7.1 Development of phase information

Direct methods were first put forward by Harker and Kasper in 1948 mak-ing use of Cauchy-Schwarz inequality9 to derive relationships between thestructure factors known as Harker-Kasper inequalities.2,4 In order to gen-eralize the mathematics, Unitary structure factor is defined as,

Uhkl =Fhkl∑

jfj

Further, unitary atomic scattering factor, nj is defined as,

nj =fj

∑

jfj

ThereforeUhkl =

∑

j

fnje2πi(hxj+kyj+lzj)

which is analogous to the expression for Fhkl.

i.e. Fhkl =∑

j

fje2πi(hxj+kyj+lzj)

2.7.2 Normalized structure factors

In direct methods, the measured structure factors are modified so thatthe maximum information on atomic position can be extracted by usingnormalised structure factor Ehkl, which is defined as,

Ehkl =Uhkl

〈|Uhkl|2〉1/2

where〈|Uhkl|2〉 =

1

N

∑

hkl

|Uhkl|2

2.7 Structure solution 23

in which the summation is over the N values of |Uhkl|2 corresponding to allreciprocal lattice sites, hkl. 〈|Uhkl|2〉 therefore represents the average valueof |Uhkl|2. The direct methods may be classified according to the means bywhich structure factor magnitudes are compared as follows:

• Inequalities

• Equalities

• Probabilities

2.7.3 Inequalities

The first attempt in determining the phases of the structure factors wasmade by Harker-Kasper. For this study, they used unitary structurefactors whose absolute magnitude ranges from zero to one with the effectsof atomic size removed. When magnitude of Uhkl is unity, then all atomsscatter in phase. These are based on the symmetry of the crystal. In acentrosymmetric crystal, the phase problem reduces to the allocation of theappropriate sign to each structure factor. By making use of the Cauchyinequality, a relation between the unitary structure factors is obtained asfollows:

U2h 2k 2l ≥ 2U2hkl − 1

This equation relates information concerning the reciprocal lattice site h k lto that for the site 2k 2h 2l, and if the intensity at hkl is strong enough,then the sign of U2h 2k 2l may be unambiguously deduced. In general, theygive unambiguous information only if there are a relatively large number ofintensity maxima associated with unitary structure factors of about one halfor greater. Another limitation is that the use of inequalities is restrictedto centrosymmetric crystals.

2.7.4 Equalities

The equality relationships are applicable only to the centrosymmetric crys-tals. Since only the signs of the structure factors are to be determined, we


may represent the sign associated with the structure factor Fhkl by s(hkl).By considering a structure of identical, resolved (i.e, non-overlapping) atoms,Sayre derived the following relationships between the signs of three reflec-tions indexed as hkl, h′k′l′ and h+ h′ k + k′ l + l′:

s(hkl) = s(h′k′l′) · s(h+ h′ k + k′ l + l′)

s(h+ h′ k + k′ l + l′) = s(hkl) · (h′k′l′)

Thus, knowledge of the signs s(hkl) and s(h′k′l′) permit the unambiguousdetermination of the sign s(h + h′ k + k′ l + l′). These relationships aretrue only for the hypothetical case of a structure composed of identicalatoms. For organic crystals, the predominant components carbon, oxygenand nitrogen have about the same atomic weight and hence above equationshave been shown to hold if the associated structure factors are large.

2.7.5 Probabilities

The use of probabilities as opposed to equalities is a more general approachalthough subject to errors. But if a set of signs can be identified, the errorsin the electron density may be removed by computing the Fourier synthesis.For a general centrosymmetric crystal the equation is as follows:

s(hkl) · s(h′k′l′) · s(h+ h′ k + k′ l + l′) ≈ 1

One of the problems, however, associated with the use of probabilities isthat they can give the signs of a certain structure factor only if thoseassociated with two others are already known.

The problem is more difficult for the non-centrosymmetric crystals. How-ever, the tangent formula is applicable to these cases and has been widelyused. If the phase angle associated with the structure factor Fhkl is repres-ented for concise as αh, then

tanαh ≈

∑

h′

|Eh′ ||Eh−h′ | sin(αh′ + αh−h′)

∑

h′

|Eh′ ||Eh−h′ | cos(αh′ + αh−h′)

2.8 Structure refinement 25

The estimated phase αh is thus the average taken over all the triplet re-flections involving h′.

2.7.6 Combined figure of merit (CFOM)

Direct method requires an initial set of phases to determine the structure.This can be achieved by introducing a large number of permutable phaseswhich are used to generate different phase sets. Figures of merit are usuallyused to screen the set of solutions, prior to computing Fourier transforms(E map). Since one or more figures of merit are available for each typeof structure, a combined figure of merit is effective in finding the correctsolution.

2.8 Structure refinement

The structure models obtained various methods are often incomplete and inmost cases they represent a crude first approximation of the real structure.It means the observed structure amplitudes and their relative phase angles,needed to calculate the electron density map, contain errors and these maycause misinterpretation of computed electron density map. The processof obtaining atomic parameters that are more precise than those obtainedfrom an initial model is referred to as refinement of the crystal structure.

Preliminary structure after Fourier transform is set of Fc(hkl)s where Fc

(hkl)s are calculated structure factors, one corresponding to each observedstructure factor Fo(hkl). The calculation provides values for the amplitudesand phases for Fc(hkl) i.e. |Fc(hkl)| and φc(hkl) but only Fo(hkl) areavailable through experiment not the phases φo(hkl).

The two sets of values Fo and Fc can be assessed using residual factor orR factor. It is defined as

R =

∑ ||Fo| − |Fc||∑ |Fo|

The F 2 can be used instead of F . The residual factor based on F 2 which


is widely used is

wR2 =

√

∑

w(F 2o − F 2

c )∑

w(F 2o )

2

Each reflection has its own weight w. From statistical view point it is moremeaningful than the basic R factor10.

Combination of the experimentally observed amplitudes |Fo| (which carryinformation about the true structure) and with the calculated phases φc(which are not completely correct, but are the best approximation to theunavailable φo values) produce a new model structure. This procedure isrepeated. The electron density map obtained by carrying out |Fo − Fc| iscalled difference electron density map and this process is called DifferenceFourier Synthesis. RE value describes how well the resulting electron dens-ity fits to the experiment. Generally if the RE value is less than 30% thesturcture can be expected to be solved.

The Rint value may be used as a test of the Laue group provided thatappropriate equivalent reflections have been measured. It should be below0.1 for the correct assignment. Rσ with value above 0.1 indicate data arevery weak or they have been incorrectly processed.

Rint =

∑ |F 2o − F 2

o(mean)|∑

F 2o

Rσ =

∑

σ(F 2o )

∑

F 2o

2.8.1 Precision and accuracy

Precision is a measure of how closely a series of measurements of the samequantity agree with each other. Accuracy is a measure of how close ameasurement is to the true value. Practically the measured values, whichcontain experimental errors, should lie in a random manner on either side

2.9 Molecular geometry 27

of this most probable value as expressed by the normal or Gaussian distri-bution. This distribution is a bell shaped curve that represents the numberof measurements N that have a specific value x which deviates from themean or most probable value xo by an amount x − xo, which representsthe error. Obviously, the smaller the value of x−xo, higher the probabilitythat the quantity being measured lies near most probable value xo which isat the top of the peak corresponding xo. Plot of N vs x is called a Gaussiandistribution or error curve. Mathematically it is expressed as

N =h√πe−h2(x−xo)2

The plot is symmetrical about xo. Its variance, another estimate of theerror, σ2 = 1/2h2 describes the spread for a given Gaussian distribution.σ is called standard deviation or root mean square error. Smaller the σ,the sharper the peak in the Gaussian distribution.

Another estimate of error is estimated standard deviation (e.s.d). It differsfrom the standard deviation in the fact that, mean value is used in calcu-lating the difference rather than the unknown true value xo. Greater theprecession in a measurement smaller the e.s.d or larger the weight or con-fidence that should be assigned to it. Weight can be taken as the reciprocalof the square of its e.s.d. Weighting scheme is modified further by includinga constant k which represent systematic errors such as instrumental errors.It is given by

w(hkl) =1

σ2[Fo(hkl)]or

1

σ2[Fo(hkl)] + kF 2o (hkl)

2.9 Molecular geometry

The molecular geometry such as bond lengths, bond angles, torsion anglesand the mean planes of the rings can be calculated from the atomic co-ordinates x, y, z.


2.9.1 Derived quantities

X-ray crystallography produces a detailed description of the spatial ar-rangement of atoms in molecules.

After the atomic coordinates are calculated, the interatomic distances arecalculated to check the connectivity of atoms. This shows the correctnessof a chemical formula.

Molecular constitution indicates the way in which atoms are interconnec-ted, making due distinction between single and multiple bonds.

The term structure includes the constitutional, configurational and con-formational aspects of the molecule.

2.9.2 Bond lengths and angles

A bond length is the distance between two atomic nuclei that are joinedtogether by some type of electronic bonding. Because atoms undergo vi-brational and other motions, reported bond lengths are the average orequilibrium bond lengths.

If two atoms at x1, y1, z1 and x2, y2, z2, lie in a unit cell that has edges withlengths a, b and c and interaxial angles of α, β and γ, the square of thedistance r between these two atoms is

r2 = [(x1 − x2)a]2 + [(y1 − y2)b]

2 + [(z1 − z2)c]2

+[2ab cos γ(x1 − x2)(y1 − y2)]

+[2ac cos β(x1 − x2)(z1 − z2)]

+[2bc cosα(y1 − y2)(z1 − z2)]

The internuclear bond angle, 6 ABC, is the angle between the bonds A-Band B-C formed by three atoms A,B and C, connected in that order. If thelength of A-B = l1, B-C = l2 and A−C = l3, then the angle A−B−C = δ


may be calculated from the formula

cos δ =l21 + l22 − l23

2l1l2

2.9.3 Estimated standard deviations

One way to assess the precision of a structure determination is to use theestimated standard deviation (e.s.d.) of the geometric quantity of interest.This is obtained from the least squares refinement. The more precisely ameasurement is made, the smaller is the e.s.d. of that measurement.

Estimated standard deviations of bond lengths

For two atoms, A(xA, yA, zA) and B (xB, yB, zB), σxA is the e.s.d. of xfor atom A and ∆x = xB −xA. The required e.s.d. values are obtained byleast squares refinement. The equation of estimated standard deviation ofthe bond distance between atoms A and B is:

σ2r =1

r2[[(σ2xA + σ2xB)(∆x+∆y cos γ +∆z cos β)2]

+[(σ2yA + σ2yB)(∆y +∆x cos γ +∆z cosα)2]

+[(σ2zA + σ2zB)(∆z +∆x cos β +∆y cosα)2]]

Estimated standard deviations of bond angles

The e.s.d. of the angle between the bonds A-B and B-C is calculated bythe equation

σθ =

[

σ2A(AB)2

+σ2B(AC)2

(AB)2(BC)2+

σ2C(BC)2

]1/2

where σA, σB and σC are the e.s.ds of the positions of atoms A, B and C.


2.9.4 Hydrogen bonds

The hydrogen bonding geometry is characterised by hydrogen bond-lengthsand angles jointly. The energy range of hydrogen bond is intermediatebetween Vander waals’ interaction and covalent bonds. This energy rangeis one that permits hydrogen bonds to both associate and dissociate quicklyat ambient temperature. This twin ability is the necessary condition forbiological reactions that must take place at room temperature. A hydrogenbond is denoted by X–H· · ·A, where X is donor and A is acceptor. Locationof hydrogen atom is an essential step in determining bond length. In X-ray structure determination, the distances of the hydrogen atom bondedto heavy atoms (C-H, N-H, O-H) are on average 0.1 Å–0.2 Å shorter thanthe inter nuclear distances.11 This happens because X-rays are scattered byelectrons and atomic position derived for an H atom from an X-ray analysisapproximate the centroid of electron density. But the centroid of electrondensity is not centered around hydrogen nucleus but is displaced towardsthe atom X.

2.9.5 Torsion angles

The torsion angle describes steric relationships across single bonds and thering geometry. A torsion is the extent of the twist of constituents about abond. It will indicate if the bonds to the substituent groups are eclipsed,staggered or intermediate between these. For a sequence of four atomsA, B, C, D, the torsion angle ω(ABCD) is defined as the angle betweenthe normals to the planes ABC and BCD[Fig. 2.1]. By convention, ωis positive if the sense of rotation from BA to CD, viewed down BC, isclockwise, otherwise it is negative (Fig. 2.1). The sign of a torsion angledoes not change by rotation or translation, and is reversed by reflection orinversion. According to definition,

cosω =(~a× ~b) · (~b× ~c)ab2c sinα sin γ

,b

bsinω =

(~a× ~b)× (~b× ~c)ab2c sinα sin γ


a

A

bB

c

C

D

γ α

A D

B, C

ω

Figure 2.1: Definition of the torsion angle ω

which become

cosω =cosα cos γ − sinβ

sinα sinβ, sinω =

V b

ab2c sinα sin γ

Least squares plane

A least squares plane is defined as the plane which minimizes∑

md2m,where

dm are the perpendicular distances of the m atoms from the plane

2.9.6 Ring conformation

Chemists use conformational analysis to understand the behaviour of themolecules in chemical reactions. Biochemists and molecular biologists alsouse conformational analysis to study the ways the molecules interact witheach other in living systems.

Normally, different conformations are obtained by rotation about the bonds.At a certain temperature, one or more of the possible conformations canbe observed, depending on their relative energies and the energy barriersthat have to be overcome. The potential energy and the energy barriersare influenced by intra and intermolecular forces.12

The six membered rings assume varieties of conformations. The chairconformation is the most stable of the many conformations that a six-membered ring can achieve. Each carbon in the ring has two substituents.


One of these will be in an axial position and the other will be in an equat-orial position. A substituent in an equatorial position leads to a lowerenergy conformation than one with the same substituent in axial position.For completely planar molecules, all the torsion angles will be zero. Hence,the deviation of the observed torsion angles from this ideal value gives theamount of non-planarity in the structure. These factors can be effectivelyused to describe and compare the conformations of the molecules occurringin a quantitative way.

A N -numbered ring has N −3 possible conformers, which is the number ofindependent puckering conformers.13 So for three, four, five, six and sevenmembered rings have, respectively, zero, one, two, three, and four puckeringconformers.

For example, in cyclopentane there are 5− 3 = 2 conformers E and T. viz.envelope and twisted conformation. Similarly for cyclohexane there are 6-3 =3 conformers B, T and C. viz. boat, twist and chair conformations.The analytical treatment of the conformations of a six-membered ring isgiven by the equation.

B2g : φc(i) = Zi = 6−1/2 q3 cos

[

3π(i− 1)

6

]

= 6−1/2q3(−1)i−1

where 6−1/2 and 3−1/2 are normalization factors and q3 and q2 are the meanpuckering amplitudes.12 and

E2u : φB,T (i) = Zi = 3−1/2q2 cos

(

4π(i − 1)

6+ φ2

)

where φ2 is a phase angle having values 0◦ 60◦ 120◦ 180◦ 240◦ or 300◦ forpure B conformations and 30◦ 90◦ 150◦ 210◦ or 330◦ for the pure T con-formations. The above equations describes the conformations of a six-membered ring in terms of the three parameters q3, q2 and φ2. The N–3possible conformations characterized as the three C, B and T conform-ers. The characteristic parameters (q3, q2, φ2) are generally identifies asthe N–3 generalized puckering coordinates. q3 and q2 are called puckeringamplitudes and φ2 the phase angle. In the three-dimensional space, thesix-membered ring corresponds to (q3, φ2, q2).

2.10 Instrumentation 33

2.10 Instrumentation

The methods for acquiring the X-ray diffraction data have not changedsince the first diffraction study by von Laue. The data collection of crys-tal sample started with using photographic films. But improvements havemade them much more efficient and user friendly so that crystal structuresof much larger and more complicated molecules can now be determinedroutinely. The first structural studies were carried out on crystals of simplesalts that had only a few atoms in each unit cell. Therefore, a relativelysmall number of reflections were available for measurement. But now, onecan study complicated macromolecules and determine the atomic paramet-ers for thousands of atoms in the asymmetric unit.

2.10.1 DIPLabo image plate diffractometer

The DIPLabo image plate diffractometer system as a whole and the gonio-stat and the X-ray tube are shown in the figures 2.2 and 2.3 respectively.

Figure 2.2: DIPLabo Image Plate System.


Figure 2.3: Goniostat and X-ray tube of DIPLabo IP.

The mechanical assembly in this case comprises of a kappa goniostat. Inthis type of geometry the χ circle does not exist and the goniometer headcarrying the crystal is mounted on an arm that can rotate about the κaxis, which forms an angle of 50◦ with the main instrument axis as shownin figure 2.4. The advantage of this system is that it is less cumbersomethan the Eulerian cradle and thus rotations that would cause collisionsor would produce diffracted beams that would be blocked by the χ circlein the conventional four-circle diffractometer are still measurable. Thediffractometer equipped with a normal focus, 3 kW sealed X-ray source(graphite monochromated MoKα).

Figure 2.4: Schematic diagram showing the κ geometry goniostat


Image plate

Figure 2.5: Composite structure of the Image Plate

The Image Plate (IP) is a flexible plastic plate with a uniform coating ofclusters of very small crystals (grain size: about 5 µm) of a photostim-ulable phosphor of barium fluoro-iodide containing a trace amount of bi-valent europium as a luminescence center, formulated as BaF I:Eu2+. Thecomposite structure of the IP is shown in Fig.2.5. The photostimulablephosphor is capable of storing a fraction of the absorbed X-ray energy.When later stimulated by visible light, it emits photostimulated lumines-cence (PSL), the intensity of which is proportional to the absorbed X-rayintensity.14

Figure 2.6: DIPLabo Image Plate reader.

The mechanism of PSL is as follows, when X-rays get absorbed by the im-age plate, some of the electronics in the valence band takes transition toconduction band of the phosphor crystals. These electronics are trapped


into colour centres (F-centers), which were intensionally introduced in thephosphor crystal during manufacturing process. On absorption of the vis-ible light, the trapped electronics return to the valence band of the crystal.The wavelength of the PSL (3900 Å) is reasonably separated from thatof the stimulating light (6330 Å), allowing it to be collected by a conven-tional high quantum efficiency photomultiplier tube (PMT). The process ofreading X-ray image by using phosphor reader is as shown figure 2.6. Theoutput of the PMT is amplified and converted to a digital image, which canbe processed by a computer. The residual image on the IP can be erasedcompletely by irradiation with visible light, to allow repeated use. The IPis easy to handle, because it is flexible, like a film, and can be kept underlight before exposure to X-rays.15

2.10.2 Data collection and data reduction

In the traditional approach, one collects the data first and then starts ana-lyzing the results. This strategy has a risk that there may be a grossinefficiency in the setup of the experiment, for example: the data set maybe incomplete, the reflections may overlap, the zones may overlap, a largepercent of the reflections may be overloaded, etc. Data collection is there-fore performed as a highly interactive process. Immediate data processingprovides fast feedback during data collection.

A single crystal suitable for X-ray diffraction studies was chosen carefullyand glued to a thin glass fiber with the aid of CCD camera, crystal wasoptically centered. The crystal to detector distance was fixed at 120mmwith a detector area of 441×240mm2. Thirty six frames of data werecollected at room temperature by the oscillation method. Each exposureof the image plate was set to a period of 400 s. Successive frames werescanned in step of 5◦ per minute with an oscillation range of 5◦.

The analysis and reduction of single crystal diffraction data consists ofseven major steps. These are:

1. Visualization and preliminary analysis of the original, unprocessed, diffr-action pattern.


2. Indexing of the diffraction pattern.

3. Refinement of the crystal and detector parameters.

4. Integration of the diffraction maxima.

5. Finding the relative scale factors between measurements.

6. Precise refinement of crystal parameters using the entire data set.

7. Merging and statistical analysis of the measurements related by space gr-oup symmetry.

2.10.3 Software

To analyze single crystal diffraction data collected with 2-d detectors sev-eral computer programs have been developed. The software used are,

1. Denzo16 is a program which takes raw X-ray diffraction data from ascanned image plate and reduces it to a file containing the h, k, l indicesand background and LP corrected intensity of the spot on the imagealong with an estimate of error. This process has two steps.

a. The crystal and detector orientation parameter is determined usinga single image in the collected series (frames). Crystal orientationalparameters include lattice type and unit cell dimensions. The crystalorientational parameters describe the orientation of reciprocal latticewith respect to spindle and the beam. The detector and X-ray para-meters include the wavelength, the crystal to detector distance, theprecise coordinates of the direct beam, internal scanner alignmentparameters such as non-orthogonality of the scanner head motionetc.

b. Denzo helps in controlling and selecting the parameter to be refinedin every cycle (steps 1 to 4 and 6 of data analysis).

2. Scalepack is a program which is used to reduce and merge the diffractiondata (steps 5 and 7). XdisplayF allows the crystallographer to visualise


the data in their original form and as well as to observe the progress ofdata reduction.

3. SHELXS17 is a structure solution program which includes direct meth-ods and uses the Patterson vector superposition method.

4. SHELXL18 is a program for the refinement of crystal structures fromdiffraction data. Basically, it is used for single crystal X-ray data of smallmoiety structures. It can also be used for refinement of macromoleculesat moderate resolution of about 2.5 Å and is valid for all space groups.

5. PLATON19 contains a variety of computational and graphical proced-ures for the analysis of the results of a crystal structure determination.It is designed for the automated generation of a variety of geometricalentities such as bond distances, bond angles, torsion angles, least squareplanes of rings, ring puckering parameters etc.

2.11 References 39

2.11 References

[1] Flint, Y., Essentials of crystallography, Mir publisher, Moscow, 1975.

[2] Giaccovazzo, C., Monaco, H.L., Viterbo, D., Scordari, F., Gilli, G.,Zanotti, G. and Catti, M., Fundamentals of Crystallography, IUCrtexts on Crystallography-2, Oxford University Press, Oxford, 1992.

[3] Sridhar, M.A., Structural investigations of some charge transfer com-plexes, Ph.D thesis, University of Mysore, Mysore, 1993.

[4] Sherwood, D., Crystals, X-rays and Protiens, Longman Group lim-ited, London, 1976.

[5] Glusker, J.P., Lewis, M., and Rossi, M., Crystal structure analysisfor chemists and biologists, VCH Publishers, Inc., U.S.A., 1994.

[6] Bragg, W.L.,Proc. Roy. Soc., 89, No. 610, 248–277, London, 1913.

[7] Ladd, M.F.C. and Palmer, R.A., Structure determination by X-raycrystallography, Plenum Publisher, NewYork.

[8] Stout, G.H. and Jensen, L.H., X-ray structure determination, Mac-millan Publishing Co., Inc., New York.

[9] Wallwork, S.C., Introduction to the calculation of structure factors,IUCr, University College Cardiff Press, Cardiff, Wales, 2001.

[10] Muller, P., Herbst-Irmer, R., Spek, A.L., Schneider, T.R. and Sawaya,M.R., Crystal Structure Refinement, IUCr texts on crystallography-8,Oxford University Press, Oxford.

[11] Desiraju, G.R., Thomas, S., The weak hydrogen bonds in struc turalchemistry and biology, Oxford Science Publications, 1999.

[12] Schweizer W.B., Structure Correlation, 1, 369, Eds. H.B. Burgi, J.D.Dunitz. 1994.

[13] Cremer, D. and Pople, J.A., J. Am. Chem. Soc., 97, 1354–1358, 1975.

[14] Sonoda, M., Takano, M., Miyahara, J. and Kato, H., Radiology, 148,833, 1983.


[15] Iwabuchi, Y., Mori, N., Takahashi, K., Matsuda, T. and Shionoya, S.Jpn. J. Appl. Phys., 33, 178, 1994.

[16] Otwinowski, Z., Minor, W., in Methods in Enzymology, Macromolec-ular Crystallography, Carter, C. W., Jr. and Sweet, R.M., (Eds.),Academic Press, 307, 276, 1997.

[17] Sheldrick, G.M., SHELXS-97, University of Gottingen, Germany, 1997.

[18] Sheldrick, G.M., SHELXL-97, University of Gottingen, Germany, 1997.

[19] Spek, A.L., J. Appl. Cryst., 36, 7–13, 2003.

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