EXPERIMENTAL METHODS FOR X-RAY STRUCTURE DETERMINATION

 Various experimental methods are used for analysing the structure of crystals.

1. The Powder Method

Debye and Scherrer (1916) and Hull (1917) devised a method of X-ray crystal analysis which permits the use of a substance in powder form. In this method, the wavelength is fixed by using monochromatic X-rays, with respect to the incident beam. A fine crystalline powder is taken in a thin walled capillary tube. Each particle in the powder acts as a tiny crystal oriented randomly and hence the angle & varies. Due to random orientation of particles in a crystal, the X-rays are scattered from all sets of planes. The scattered rays are further detected by using an X-ray sensitive film.

X-rays are generated in a tube and these are made to fall on a monochromatic crystal to get monochromatic beam. The beam is now passed through a slit and made incident on a sample powdered film in a thin walled glass capillary tube. Due to reflection from the various lattice planes, diffraction maxima arise. The diffraction maxima are photographed on a film (Fig. 4.10). Since the powder sample consists of microcrystal in all possible orientations, the rotation of crystal sample is not necessary.

 The diffracted X-rays form concentric cones originate from the powder. On a narrow strip of film, ares appear on either side of the bright spot at the centre. Each pair of arcs which is equidistant from the central bright spot gives the position of a reflection of a definite order from a particular plane and the value of glancing angle is calculated by taking into account the distance of the are (say S meter) from the central spot and its distance from the capillary i.e., the radius of the are. From this data, the d-spacings between the planes can be calculated.

 The glancing angle, 20 =S/r or 0 = S/2r

 0sa obtained will be in radians which can be converted into degrees.

 0 in radiation = 0× 180/pi degrees.

The photographic plate is then developed. Spots of different sizes and intensities are obtained which is characteristic of a crystal. The spots. (except

Applying Bragg's equation, nd 2d sin d can be calculated.

Laue's Method 

In this method. Dis fixed and the wavelength of X-rays in varied 2 rays from the X-ray tube are collimated into a beam by a lead shield having a hole in it moved on the circular crystal in mounted in the path of the X-rays beam. When the X-rays are incident on the pasily jonizable gos alu crystal, each atom in a plane acatters some X-rays which in turn give an impression on thisnization troduced i photographic plates The photographic plate is then developed. Spots of different sizes and intensities are obtained which is characteristic of a crystal. The spots. (Except) represent the scattering part of the original beam through various characteristic angles Different arrangements of spots.

 

 

The Bragg Method of Crystal Analysis (The Bragg Equation)

 In 1913, William Bragg and his son Lawrence Bragg worked out a mathematical relation to determine inter atomic distances from X-ray diffraction patterns. They showed:- 

1. The X-rays diffracted from atoms in crystal planes obey the laws of reflection.

2. The two rays reflected by successive planes will be in phase if the extra distance travelled by the second ray is an integral number of wavelengths.

The Braggs treated the crystal as a reflection grating rather than a diffraction grating for monochromatic X-rays. When X-rays are incident on a crystal face they penetrate into the crystal and strike the atoms in different planes. From each of these planes, X-rays are reflected. If the reflected waves, from successive layers, are in phase; constructive interference will occur and a diffraction spot will be detected from these planes. On the other hand, if the diffracted waves from successive layers are out of phase, destructive interference of the two waves will occur and no diffraction will be detected from these planes.

In order to understand the theory of this method, consider the parallel, equidistant planes AA, BB, CC, etc which constitute atomic planes of a crystal, with inter planer distance d as shown in Fig.4.9. If a beam of X-rays of wave length strikes these planes at angle 9, some of the rays are reflected from the top-most plane AA', while the rest penetrates the crystal and suffers reflection from the second, third plane etc (i.e., from the succe planes). For constructive interference, it is necessary that the waves reflected from sucenssive planes be in phase. This however would be possible only if the path difference of the two waves is an integral multiple of wavelength

The wave P'Q'R' reflected from the plane BB travelled a greater distance than the wave PQR reflected from the plane AA', since the plane BB' is below the plane AA'. However, the parts of the reflected wave QR and MR must be in phase to get constructive interference. This means that extra distance travelled by the wave P'Q'R' must be equal to an integral multiple of the wavelength of the incident X-rays. The extra distance travelled by the wave P'Q'R' can be obtained by drawing perpendiculars QL and QM on to the wave P'Q'R' from the point Q. From the figure, it is clear that PQ = P' * L and QR = M * R' The extra distance travelled by the wave P'Q'R' is then simply L * Q' + QM

For constructive interference this extra distance, ie., the path difference must be an integral multiple of the wave length, në Mathematically it follows:

L * Q' + QM = n*lambda

(1) where a is an integer value 1,2,3, and is known as the order of reflection

reflection. If follows from triangle LQQ that (L * Q')/(Q * Q') =sin theta^

or Similarly, for the triangle MQQ', we can (3) L * Q' = Q * Q' * sin theta = d * sin theta Q' * M = d * sin 0

Substituting Eqs. (2) into Eq.(1), we get d in*theta + d * sin theta = n * sin n*lambda = 2 * d * sin theta

Equation (4) is known as Bragg's equation which relates the wavelength, order of reflection a, interplanar distance d and the angle of maximum reflection 0. Thus, knowing the wavelength of X-rays and the angle of incidence 0, the interplanar distance d of a crystal can be calculated.

 

The Bragg equation states that for a given wavelength, the constructive interference would be possible only at definite 0 values and these & values depend upon the interplanas spacing d. The interplanar spacing d is independent upon the unit cell dimensions (a,b,c) and the Miller indices of the faces or planes (hkl). This relationship is expressed in different equations for the different crystal systems. For a cubic system, the interplanar distance d is related to the unit dimensiona and the Miller indices hal by the relation.

NEUTRON DIFFRACTION

Neutron diffraction supplements X-ray diffraction and is particularly helpful in locating hydrogen atoms.

The average de Broglie wavelength of thermal neutron is 149 pm at room temperature. A beam of thermal electrons, therefore, has a wavelength suitable for the diffraction studies of studies of crystals. An essentially monochromatic beam may be obtained by diffraction from a crystal monochromator that selects a small band of wavelengths from the beam of thermal electrons from a nuclear reactor. This monochromatic beam of neutrons may be scattered from a crystal in such the same way as X-rays. Although the principles of neutron diffraction are similar to those of X-ray diffraction, several complementary technique to that the x-rays diffraction, Whereas X-rays are scattered by fundamental differences between them result in neutron diffraction being a electrons, neutrons are scattered primarily by the nuclei in a crystal. Hence, the atomic scattering factors as do the no dependence on the Bragg scattering angle). This means that neutron diffraction, in scattering factors for X-rays, but, instead, have roughly the same scattering factors (with contrast to X-rays diffraction, is especially useful for accurately locating hydrogen atoms in a crystal structure. For example, in a compound such as uranium hydride, X-ray diffraction can be utilized to determine the uranium coordinates and neutron diffraction the hydrogen coordinates.

Since neutrons possess a magnetic moment by virtue of having a spin of, there is an additional scattering if the compound contains paramagnetic atoms or ions with unpaired electrons. Thus, neutron diffraction has been widely utilized to investigate structures of magnetic materials, such as MnO and Fe3O, in order to determine the arrangements of the atomic magnetic moments in ferromagnetic and antiferromagnetic crystals. Neutron diffraction is thus a specialized adjunct to X-ray diffraction.

ELECTRON DIFFRACTION

Just as the crystals can act as diffraction grating for X-rays and produce diffraction effects, the diffraction of a beam of electrons can also be used to study the internal structure of a crystal might be obtained in a manner analogous to X-rays diffraction by crystals. It was indeed In 1925. W. Elasser first suggested that the evidence of the wave nature of electrons Davisson and L. H. Germer, and by the diffraction of electrons by a thin film, about 10-6 demonstrated by the diffraction of beams of electrons by a crystal of nickel in 1927 by CJ cm thickness of metal by G. P. Thomson (1927).

When a high energy electron strikes the material, the scattering is caused primarily by interaction with atomic nuclei, ut the seattering of low energy electrons, such as electron beam with a wavelength of 1 A is caused by the interaction with outer electrons of the atoms as well as with the nuclei The electron beam with a wavelength of less than 1A 40,000 V) is used in electron diffraction studies. Since the low energy electrons can be easily absorbed by matter, samples used are usually thin films er gases and vapours.

When electron beam is passed through a gas or vapour it produces a series of concentric rings on a photographic plate due to the diffraction by the atoms within the molecule, i.e., the diffraction depends on the distances between atoms in the molecule. The dark rings on the photographic plate represent the position of maximum scattering and the lighter portions in between correspond to angles for which the electron scattering is minimum. Because of the appreciable amount of the background scattering of the electron beam, the diffraction of the techniques in electron diffraction measurements, the resolution of the bonds has been considerably improved. The scattering of electrons of two types incoherent or inelastic, due to change in equivalent wavelength of the electrons, and the coherent or elastic scattering.

Whereas the inelasting scattering is not important for the present purpose, the elastic scattering is responsible for the appearance of concentric rings on a photographic plate.

The diffraction of electrons, as in X-rays depends on spacings between targets.

EXPERIMENTAL METHOD

The instrument used for electron diffraction studies consists of

(1) an electron gun (a hot anode or cathode),

(ii) lenses to focus the electron beam on to the desired area of volume of the specimen and

photographic screen Photographie means of viewing the pattern, usually recording of the image is usually carried out by direct impart of the electron bean upon the photographic film.

In addition, the instrument is also provided with related electric power supply, vacuum pumps and controls. Most commercial electron microscopes have been used far electron diffraction studies with slight modifications. However, commercial electron microscopes are available which are exclusively used to obtain an electron diffraction pattern without moving the specimen from its normal position. The complete instrumem is large, complex and expensive.

A fine stream of accelerated electrons using a potential of about 50,000 volts is allowed to fall on a very thin film about 10 cm thickness of metal or to meet at right angles a stream of the gas or vapour at a low pressure, about 10-3 mm. Since the electrons interact very strongly with the molecules of the sample and greatly affect the photographic pate, the emerging electron beam is allowed to fall on a photographic plate for a short time, e.g. 0.5 to 5 seconds. On development, series of concentric rings is observed on the photographic plate. The plate is then held to a strong light and position of the apparent maxima and minima of scattering intensity are marked. Photometric methods have also been used.

The accuracy in estimated bond lengths and the bond angles obtained from diffraction studies for simple molecules is comparable to that obtained by X-ray diffraction studies. The accuracy is usually not better than 0.01A in cases of estimated bond lengtha for O-H. NH bonds but under favourable conditions these can be used as good as ± 0.024.

Electron diffraction studies have a number of important applications. The most important application involves the study of the diffraction of elements by substances as vapours at low pressures with a view to evaluate the bond lengths and bond angles in relatively simple molecules and to determine the molecular configurations. Electron diffraction studies on a large number of compounds, simple and complex, have been carried out which have proved to be very important to understand the structure and geometry of these compounds.

 

X-RAY DIFFRACTION

X-Ray Diffraction. Diffraction is the bending or spreading of waves that encounterin ohjeet ta barrier or an opening) in their path. When light waves pass through a narrow they scattered in such a way that the wave seems to spread out. This physica phenomenon is called diffraction. For diffraction to occur, the size of the object must be if the order of the wavelength of the incident waves, when the wavelength is much small than the size of the object, diffraction is ordinarily not observed and the object casts a shars shadow.

When light passes through many evenly spaced narrow slits fa diffraction grating), the scattered waves interact to form a series of light and dark bands known as diffraction pattern. By studying these patterns, we can learn about the diffracting obje

X-rays are generated as a result of bombardment of cathode rays on anticathos target. X-rays are electromagnetic radiation with wavelengths of the order of 0.1mm, distance which is of the same order of magnitude as the molecular diameters in gases an the roughly estimated interatomic distances in a solid. Thus, a crystal can serve as an affective diffraction grating for X-rays. X-ray diffraction results from the scattering of X rays by a regular arrangement of atoms, molecules or ions.

The atoms in a crystal form reflecting planes for X-rays. When X-rays are reflected from these planes, they show a diffraction pattern, which can be recorded on photographic plate as a series of spots. By analyzing the diffraction pattern, we can determine the positions of each atom in the unit call of the crystal. X-ray diffraction is as important method for the study of the structure of solids. The instruments used to measure X-ray diffraction, known as X-ray diffractiometers, are now computer-controlled, making the collection of diffraction data highly automated. The diffraction pattern of a crystal can be determined very accurately and quickly. Computer programs are then used to analyze the diffraction data and determine the arrangement and structure of the molecule in a crystal.

 X-rays are shorter wavelength, high energy electromagnetic radiations, produced by bombarding a metal with high energy electrons. The high energy electrons interact with the inner electrons of the inner shells of the atoms of the metal. The collision knocks an electron out of an inner shell and an electron in a higher energy shell drops into the vacancy, emitting the excess of energy as a photon of higher energy.

 The internal structure of a crystal can be studied with the help of X-rays, since the wavelength of X-rays is of the same order (10 m) as the inter atomic distances in a crystal. Hence a crystal might be used as a diffraction grating for X-rays. Actually, thi diffraction of X-rays is due to the electrons of atoms present in a crystal. Max Von laur (1912) predicted that if a beam of X-rays is allowed to fall on a crystal, it should be transmitted through the crystal as a number of diffracted beams. If a photographic plates placed behind the crystal, the image obtained on the photographic plate would show a number of spots (laue spots). From a consideration of the position of these spots, one can reach at some conclusion about the position of atoms in the crystal and thus from these Laue patterns, the crystal structure can be constructed. The study of crystal structure with the help of X-rays is called X-ray crystallography

 

The characteristics of an X-ray diffraction pattern are related to the structure of the crystal; the wavelength of the X-rays and the angle at which the radiation strikes the crystal.

 

 

 

The SOLID STATE

INTRODUCTION 

Solids are characterised by their rigidity, hardness, incompressibility and characteristic geometry. They have a definite volume and shape

Types of solid:

 Solid substances are classified into two forms

(a) crystalline (alen called true solidat,

(b) Amorphous.

A crystalline solid exista as small crystals, each crystal possesses a characteristic geometrical shape. In a crystal, the atoms, molecules or ions are arranged in regular, repeating three-dimensional pattern called the Crystal Inttice. They possess sharp melting points. Sugar and common salt are crystalline solids. An amorphous solid (Gr amorphous no form) has atoms, molecules or ions arranged at random and lacks the ordered crystalline lattice. Unlike crystalline solids, they do not exhibit sharp melting points.

Examples are rubber, plasties, glass and amorphous sulphur.

CRYSTAL LATTICE AND UNIT CELL

All crystals consist of regularly, repeating three dimensional orderly arrangement of constituent particles (atoms, ions or molecules). The positions of particles in a crystal, relative to one another in space, are designated by points signifying only the position of the centres of the particles but not their actual sizes. The over all arrangement of particles in a crystal is called the crystal lattice, space lattice or simply lattice. The positions occupied by the particles in the crystal lattice are called lattice sites or lattice points. Conceptually, a space lattice is defined as a regular three dimensional array of similar points (representing an atom, ion or a molecule) in space, arranged in such a way that a straight line passing through any two points will puss, at equal interval, through a series of similar points. The most important fundamental characteristic of a space lattice is that each point (in space lattice) has an identical surroundings throughout.

 The smallest geometrical portion of the crystal which can be used as repetitive unit to build up the whole crystal is called a unit cell. In other words, "the smallest volume of a crystal showing all the characteristics of its lattice is called a unit cell.

and c with a (alpha) as shown in Fig.4.1. A unit cell has a definite shape, has constant values for three unit cell lengths and for the angles between them. The unit cell lengths are the distances parallel to the three major axes. It is customary to represent the unit cell length parallel to x-axis with the letter 'a', parallel to y-axis with b, and parallel to z-axis with 'c'. The angle between a and b is denoted by y (gamma), between a and e with ẞ (beta) and between b and c with a.

 

 

THE CRYSTAL SYSTEMS AND FOURTEEN BRAVAIS LATTICES

The total number of possible crystal forms are On the basis of their symmetry these crystal forms are grouped into 32 classes of symmetry called point groups. The crystallographers have been able to divide 32 point groups and 14 Bravais lattices into seven crystal systems. These systems differ from one another in having different magnitudes of lengths of the various axes and the angles between them for their unit cell In other words, these systems differ from each other in their external shapes due to the arrangement of unit cells in their dimensions: There are seven such unit cells which are necessary to build the external shapes of all known crystals and according to these seven unit cells, all the crystals are classified under seven groups. These groups are called crystallo-graphic systems or simply crystal systems. The individual crystal belonging to the same system are identified from the magnitudes of the lengths of the unit cell only because the corresponding angles have the same values.

it one should know the magnitudes of lengths parallel to the three mutually perpendicular A unit cell, as already mentioned, has a definite geometric shape and to reproduce reference axes as well as the angles between them. It is customary to represent the unit cell length parallel to X-axis with the letter a, parallel to y-axis with & and parallel to Z-axis with c. The angle between a and b is denoted with y, between a and with B and between b and e with a.

 The seven crystal systems are obtained when we use a simple (primitive) unit cell The a unit cell (symbol P). The geometric characteristics of these crystal systems are listed in Table.

Besides these primitive unit cells or lattices we have other types of unit cells or lattices called as (i) Body-centred unit cell (symbol I), having one point at the centre in addition to at the corners, (ii) Face-centred unit cell (symbol F) having lattice points in the centre of each face in addition to at the corners, (iii) End centred unit cell (symbol C) having points at the corners and at the intersections of the diagonals of a pair of opposite faces. The unit cells corresponding to these space lattices are shown in Fig.

the same system are identified from the magnitudes of the lengths of the unit cell only because the corresponding angles have the same values.

it one should know the magnitudes of lengths parallel to the three mutually perpendicular A unit cell, as already mentioned, has a definite geometric shape and to reproduce reference axes as well as the angles between them. It is customary to represent the unit cell length parallel to X-axis with the letter a, parallel to y-axis with & and parallel to Z-axis with c. The angle between a and b is denoted with y, between a and with B and between b and e with a.

The seven crystal systems are obtained when we use a simple (primitive) unit cell The a unit cell (symbol P). The geometric characteristics of these crystal systems are listed in Table 3.1.

Besides these primitive unit cells or lattices we have other types of unit cells or lattices called as (i) Body-centred unit cell (symbol I), having one point at the centre in addition to at the corners, (ii) Face-centred unit cell (symbol F) having lattice points in the centre of each face in addition to at the corners, (iii) End centred unit cell (symbol C) having points at the corners and at the intersections of the diagonals of a pair of opposite faces. The unit cells corresponding to these space lattices are shown in Fig.

 

 

In 1848, Bravais showed that there are fourteen different basic arranger possible for arranging similar points in a regular continuous in three dimensional These arrangements are called Bravais lattices (spaces) and in terms of these the in terms of these are internal structure can be described. The Bravais lattices associated with the seven crystal systema are listed in Table and the unit cells of the fourteen Bravais lattices are shown in Fig.