8 Experimental methods for x-ray diffraction

Prof. P. N. Kotru

TABLE OF CONTENTS

8.1 Experimental Methods For X-Ray Diffraction.

8.1.1 Experimental Methods for X-Ray Diffraction.

8.1.1.1 Laue spots method.

8.1.1.2 The Powdered Crystal method.

8.1.1.3 Measurement of Bragg Angles Ѳ and Interplanar spacings d.

8.1.1.4 Indexing of Powder photograph.

8.1.1.5 The rotating crystal method.

LEARNING OBJECTIVES

Experimental methods viz., Laue spots method, rotating crystal method and powder crystal method are described.
Applications of the Laue diagram to the determination of symmetry and structure of materials, orientating single crystals and investigating distortion or polycrystallinity of materials is explained.
The principle of powdered crystal mehod is explained.
Measurements of Bragg angle and interplanar spacings in crystal are discussed.
The proced ure of indexing powder patterns is discussed.
Rotating crystal method, its experimental set-up (both simple as well as modified) and procedures, are described.

8.1 Experimental Metho ds For X-Ray Di ffraction

X─ rays have proved to be o f g reat importance on account o f the fact that they have many and varied practical app licat ions wh ich may be su mmarily classified as fo llo ws:

• Purely scientific applicat ion, such as in crystallog raphy to analyse and determine the internal structure of

crystals and investigate the perfect ion o f crystals

• Industrial application which is also g iven a general name o f rad io-metallog raphy.

• Medical applicat ion such as rad io- diagnosis (rad iography) and t reat ment (X-ray therapy).

Here, we will take up only scientific application relevant to crystallography fo r analysis of crystal structure.

i) Laue spots method

ii) Rotating crystal method

iii) Powdered crystal method.

We shall now consider these methods slight ly in more detail as under:

8.1.1.1 Laue Spots Method

In this method, single crystal is kept fixed and a continuous or white spectrum of X-rays is used. Figure 8.1 shows the experimental set up. X-rays fro m the source are made to fall on a crystal and films are appropriately positioned to detect the transmitted as well as back reflected radiation as shown in the figure. This method is used either in transmission (generally at small angles) or in reflect ion (popu larly kno wn as Laue back-reflect ion method).

Figure 8.1 : Experimental set-up o f laue method

On exposure to radiation, one finds spot on the film in directions and for such wavelengths which correspond to the conditions for diffraction. X-rays of short wavelength and carrying high energy are usually involved for transmission patterns. X-rays of longer wavelengths and lower voltages are used in reflection.

An x-ray beam with continuous range of wavelengths can be obtained by using a bulb with anticathode and high tension o f about 65,000 vo lts. One finds a series o f spots arranged on an ellipse in a Laue photograph by reflection at the planes around crystal zone. The arrangement of spots as seen in a typical Laue photograph of a simple cubic crystal is shown in a schematic d iagram of figure 8.2

Figure 8.2 : Schematic diagram sho wing arrangement of spots in a Laue photograph of a simple cubic crystal

Laue diag rams are used for:

a) Determin ing the sy mmet ry and structure of materials,

b) Orient ing sing le crystals and

c) Investigating d istortion o r polycrystallin ity o f materials.

Thick samples can be used fo r back – reflect ion method wh ich offers the advantage o f h igher resolution as co mpared to trans mission method.

Very fast diffusion -transfer films and lu minescent screens have been used which to a very great extent shorten the exposure time required for Laue photographs. Films are being replaced by Geiger – or scint illation – counter techn iques coupled with data p rocessing equip ment.

8.1.2 The Powdered Crys tal Metho d

All the methods, be it Laue method or rotating crystal method, require a single crystal specimen whose size is greater than microscopic dimensions. However, in a very large number of crystalline solids, individual crystals of the desired size as required for the rotation photograph either do not occur or are not available. Such materials are readily examined by this method. This method was devised independently by Debye and Scherrer in Germany and by Hull in America about the same time in 1916 and called powdered crystal method or powder photograph method.

The principle of this method is as follows:

Let a monochromatic x-ray beam be allowed to fall on a small specimen of the material which is ground to a fine powder. The sample is obviously in a polycrystalline form which is an aggregate of extremely tiny crystallites randomly oriented with respect to a given direction and as such all possible orientations of all lattice planes are present in the powdered sample . There is bound to be a certain number of crystal grains which will be positioned with a given set of lattice planes making the correct angle with the incident beam for reflection to occur, while another fraction of the grains will have another set of planes in the correct position for reflection and so on. Further, reflections are possible not only from the different set of planes but also in the orders for each set.

The powdered crystal method is the only technique which is readily applicable to all crystalline solids. The diffraction data which depend on the lattice parameters are unique for a given material and, therefore, can be used fo r its ident ificat ion. To identify a hu man being we make use of h is fingerprints. In the same way, diffract ion data of po wdered crystalline samp les are un ique wh ich becomes a means of their identification.

Let us consider a monochro mat ic beam o f x- rays wh ich is incident at Bragg angle Ө on a set of latt ice p lanes with interplanar spacing d in so me part icu lar crystallite so that the Bragg condit ion 2dsinӨ = λ for the above said lattice planes is satisfied. Considering the diffracted beam fro m a large number of randomly oriented crystallites , and take into account the diffraction fro m the planes with the same interplanar spacing as the first one , the locus of the diffracted beams would lie on a cone with half – apex angle 2Ө because the angle between the incident beam and the d iffracted beam is 2Ө. The situation is shown in figure 8.3

Figure 8.3 : locus of the diffracted X-ray beams due to same Bragg angle is a cone with half-apex Schematic diagram sho wing arrangement of spots in a Laue photograph of as imple cubic crystal

The diffracted beams fro m other sets of latt ice p lanes with different interp lanar spacings , say d1 d2 , d3 , d4 and so on , would lie along d ifferent cones with different half-apex ang les , 2Ө1 , 2Ө2, 2Ө3 , 2Ө4 and so on. Since the incident beam direct ion is the same for all these lattice p lanes, the cones would be coaxial.

So, the different beams from d ifferent set of planes (hkl) wou ld all lie on a circular cone. If the same is recorded on a flat plate perpendicular to the incident beam, each diffract ion from h kl p lanes would appear to be like a ring o r halo around the central spot as shown in figure 8.4.

Different film p lacements are possible. If the film is placed normal to the incident x-ray beam beyond the powder in figure 8.3, the powder diffraction pattern gets recorded in the form of concentric circles or circu lar rings ( as shown in figu re 8.4) and the Ө values can be easily evaluated. However, th is type of film placement has a disadvantage and it is that several powder lines with 2Ө > 90⁰(kno wn as high angle lines) do not get reco rded. On account o f th is limitat ion this type o f film p lacement (flat film) is not used. To overco me this limit ation of a flat film, a cy lind rical film strip is used so that all the lines corresponding to 2Ө fro m 0⁰ to 180⁰ get reco rded on the film as shown in figure 8.5 (a). This technique is due to Debye -Scherrer and the camera is named after them as “ Debye -Scherrer Camera” On un rolling the film one finds that the pattern on it is o f the type as shown in figure 8.5(b).The diffracted rays wh ich are at s mall ang les make arcs around the central spot on the film. Those wh ich get diffracted through 90⁰ and the co rresponding t race on the film is a straight line .

Figure 8.4 : The circular rings of powered photograph on a flat photographic plate.

Figure 8.5(a) & (b) : cylinderic film and traces o f p hotograp h ic film in the po wdered crystal method

The material to be investigated is ground into a fine powder which is then stuck on a hair by means of gum. It is then suspended vertically in the axis of a cylindrical camera wh ich enables sharp lines to get recorded. The photographic film fits round the inner surface of camera covering practically the whole circu mference in order to collect beams d iffracted upto nearly 180⁰. The x-rays after falling on the powdered samp le passes out of the camera th rough a ho le cut in the film, in o rder to min imize the fogging produced by the scattering of the direct beam.

The crystal structure is deduced fro m the arrangement o f the traces and their relat ive intensities .Taking Bragg ’s equation 2dsin Ө = nλ.

Differentiat ing this equat ion leads to:

∆d.s in Ө + d cosӨ. ∆ Ө= 0; n and λ being constants

Or, ∆ Ө/ ∆d = ─ tan Ө/d

When the angle of incidence Ө approaches 90⁰, ∆Ө/∆d beco mes very great which means that s mall variations in d produce large variations in Ө.

8.1.3 Meas urement of Bragg Angles Ө and Interplanar spacings

In order to find d values, it is required to measure Ө values. To do this the film is placed flat on a viewer with a linear scale fitted thereon. The film with diffraction lines is placed on the viewer which has illu minated background. The position of the diffraction lines are noted starting fro m one end along a line passing through centres of the entry and exit holes. There are two ways in which one can make measurements. One way is to note the reading R1 and R2 of the two arcs corresponding to a diffracted cone. (R₁ – R₂) is then the linear distance between arcs corresponding to one set. This gives the linear distance 2 R fro m wh ich Ө is calculated. The second way is to measure R directly. This is done by locat ing the centre o f the d irect beam wh ich co incides with the centre of the exit ho le and so corresponds to Ө = 0⁰. Posit ion of each arc fro m th is point prov ides us the value of R. We know the radius r of camera, having determined the linear d istance R through measurement, Bragg angle Ө can be calcu lated by using the relation :

4 Ө = 2 R / r

Or,

Ө = R/ 2r radians,

= R {180/ 2πr} deg rees

The above relat ion is easily derivable fro m the d iagram shown in figure 8.6

This method is applicable to any kind of crystalline matter. Since it does not require single crystals, it is very valuable and of great use in the investigation of metals and alloys, ceramics and any material in the po lycrystalline state.

In the equation Ө = R {180/2πr}, the term 180/π =180×7/ 22 which may be taken as 57.3. So, if a camera of diameter 2r = 57.3 mm is used for recording powder pattern data, 1⁰ in angle Ө would correspond to 1mm in R. This way, the geometric conversion of R into Ө is simple and straightforward.

Figure 8.6 : Conversion of linear distance on the film into Bragg angle

It is important to bear in mind that while making measurement of linear distances and then converting them to Ө values, a distinction is required to be made between low-angle diffraction lines and the high-angle diffraction lines. Scattering of radiation from the air in the camera results into the background intensity. This background intensity is maximum near Ө = 0 which corresponds to centre of the exit hole of the camera. Because of this the film near the low angle side has more of background blackening. The second distinguishing feature of low-angle diffraction lines concerns the resolution of lines corresponding to say kα1 and Kα2 components of the kα doublet for Cukα radiat ion which is generally employed for powder diffractometry. Cukα is composed of Cu kα1 with wavelength λ = 1.54050 Å and Cukα2 with wavelength λ =1.54434 Å. As a result of this, every diffract ion line is a doublet corresponding to these components of kα1 and Kα2 wavelengths. For a camera of diameter 57.3 mm, the doublet appears as a pair of two closely spaced lines which can hardly be resolved and as such appear as one thick line. The situation is different in case of cameras with larger diameters. The two components arising from kα1 and Kα2 wavelengths get better resolved only at the high angle side where the separation between the two Bragg angles is wider and so the doublet can be easily identified.

Once the Bragg angle Ө is found out, application of equation 2d sin Ө = λ y ields d values (interplanar spacing). The Interplanar spacings are related to the lattice constants a, b, c through Miller indices h, k, l by the equations given in quadrant VI section 6.3.1 fo r various crystal systems. We shall take the examp le o f a po wder photograph of a cub ic crystal taken on a camera of 57.3 mm d iameter with Cukα radiat ion ( λ =1.54 Å ). Fro m 2dsin Ө = λ, we have sin2 Ө = λ2/4d2.. For a cub ic crystal:

The linear distances of the diffraction lines are measured and Ө values determined for every line.
From these Ө values, we determine the sin2 Ө values. Division of sin 2Ө by different values of integers gives the possible N-values. A list of possible N-values is prepared and the values of sin 2Ө /N determined which should have a common factor λ²/4a² as per the above equation. Suppose that common factor turns out to be σ . Substituting for λ = 1.54 Å(say), one is able to get the appro ximate value of lattice parameter a . This procedure is adopted for a few low angle lines and then for high angle lines.

For crystallographic characterization of any material, it is extremely important to determine lattice parameters very accurately. That is done by taking measurements on the diffraction line corresponding to Ө = 90⁰. However, Ө = 90⁰ corresponds to diffracted beams which are directed back into the incident x-ray beam, making its recording impossible. So, one tries to make measurements as close to Ө = 90⁰ as possible or extrapolate the measured d values to Ө = 90⁰. The parameters required to be determined are the lattice parameters and the indices of various lines. While it is relatively simple for crystals of higher symmetry, it is quite complicated for the crystals of other systems.

8.1.4 Indexing of powder photograph.

The procedure o f indexing of the powder pattern is based on the type of crystal classes that one is dealing with . It depends on the fo llowing :

1. Substances whose unit cell is kno wn

2. Substances whose unit cell is not kno wn.

For substances in category 1, indexing is rather simple whereas for the category 2, indexing is not that simp le. For the latter type, trial and error methods are adopted for assigning indices to the powder lines. Let us first know very briefly about the method of indexing powder lines of those substances whose unit cell is known. There are two approaches-analytical approach and the graphical approach. In analytical method for substances whose unit cell is known, the assignment of the indices is done by comparing observed Ө values with those of the calculated ones.

Fro m the po wder pattern values of sin2 Ө are found. On the other hand, sin2Ө corresponding to various (hkl) indices fro m the kno wn un it cell parameter is calcu lated. Th rough comparison of the observed values of sin2ϴ and the calculated values of sin2ϴ, the values that tally correspond to the hkl ind ices. Taking simp le cub ic crystal as an examp le, the interplanar spacing d is given by:

All possible values of N are used to calcu late set of sin²ϴ values from this expression . The possible values of N =( h² + k² + l² ) fo r the cubic lattice are prov ided in the literature . It may be o f interest to know that certain possible values of N (like 7, 15, 23, 28, 31 and so on) are forbidden. If the substance does not belong to any of the cubic systems, one has to adopt a d ifferent procedure .

For example, in case of tetragonal system, the following equation between sin²ϴand hkl indices is used:

To calculate various values of sin²ϴ_hkl for different sets of (hkl) indices , a table with two sets of values , one fo r the first term contain ing possible values of A and the other for the second term containing possible values of B. Tables giv ing possible values o f sin²ϴ are available in the literature.

A particular sin²ϴ_hkl value is obtained by suitable addition of the values fro m the two sets. The calcu lated sin²ϴ_hkl values are then compared with the experimentally determined values and those which match are taken for assign ment of indices h, k, l to the observed lines.

Similar procedure is followed for indexing the lines in the powder pattern of trigonal and orthorho mb ic systems. The equat ions to be used in case of hexagonal and rho mbohed ral systems are:

For orthorho mb ic system, the equation to be used is:

For monoclinic and triclinic systems it is convenient to use expressions in terms of the reciprocal lattice parameters a* , b* , c* and α*, β*, γ* instead of direct lattice parameters and the expressions for monoclinic and triclinic systems are respectively given as follows:

Monoclin ic system: 1/d² =h² a*² + k² b* + l² c*² + 2lh c* a* cos β*; the unit cell hav ing been defined such that b-axis is perpendicu lar to the a and c axes.

Triclin ic system: 1/d² = h² a*² + k² b*² + l² c*² + 2h ka*b*cosγ* + 2klb*c* cos α*+2lhc*a*cosβ*

The said procedure is the analytical one. However, graphical method is also followed. In this method, curves are drawn between the Interplanar spacings (d spacings) and the cell dimensions for those whose unit cell is known. The experimental d spacings are plotted and compared directly with the theoretical curves. The matching between the theoretical and experimental curves is done and indexing of the powder pattern is achieved.

8.1.5 The Rot ating Crys tal Metho d

The powdered crystal method is the most useful and feasible method obtaining information concerning a material which cannot be made available as a single crystal. However, it is not suitable for the determination of internal structure on account of difficulty in indexing. Therefore, the rotation and oscillation techniques are used, provided the material becomes available in the form of a single crystal. This method enables measurement of lattice constants and indexing of reflections quite easily. The intensities of individual reflections are conveniently measured which enables us to determine the crystal structure.

The rotating crystal method was devised by Schiebold and Polanyi. Its principle is based on the fact that if a crystal is rotated slowly about a fixed axis, a large number of planes will successively come into the reflecting positions and the diffracted radiation onto the photographic plate /film in the form of a pattern of spots, popularly called as rotation photograph.

In one type of technique, a photographic plate say ‘P’ is kept at a distance of a few centimetres from the single crystal C so that its plane is normal to the incident beam. The crystal is then rotated about its pre-determined axis (say c-axis). The beams reflected from all planes parallel to this axis lie on the surfaces of a family of cones whose axes coincide with the axis of rotation and whose vertices are at the crystal. The cones on intersecting the photographic plate positioned parallel to its axis result into a series of hyperbolas. The experimental set-up is shown in figure 8.7

Figure 8.7 : Rotating crystal method

In a slight ly mod ified techn ique (see figure 8.8 (a)) the reflected beams fro m crystal C are registered on a photographic film P wh ich is bent in the fo rm of a cy linder whose axis is along the axis of rotation of the crystal. In this type of set-up the reflections from all planes which are parallel to the axis of rotation lie in a p lane normal to the axis. This plane cuts the cylindrical film in a circle . On unrolling this film the reflections are found to get registered on a horizontal line containing the registration of the incident beam. The registration of spots is seen as a series of hyperbolas above and below the horizontal line. These lines have been named as layer lines and look something like shown in figure 8.8(b)

Production of layer lines in rotation photograph on flat film and cylindrical film is shown in figures 8.9 (a,b).Here, the crystal is rotated about c-axis as shown.Planes which are parallel to c-axis will reflect rays horizontally forming spots along a horizontal row alongwith the central spot. This is called as zero layer line. There will be other reflections which would make an angle with the horizontal row of spots. Accordingly, the lines are named as zero layer line, first layer line and so on as is shown in figure 8.9

Figure 8.9:layer lines on flat and cylinderical film

With complete rotations of the crystal large numbers of spots are recorded on the film. It is, therefore, customary to rock the crystal back and forth through an angle of only 30⁰. It limits the spots on film to those of certain indices. The angular rate of rocking is, however, kept constant.

From the distances between the layer lines the lattice spacing in a direct ion parallel to the axis of rotation is determined. Taking rotation photographs with rotation of the crystal about all the three axes a, b and c separately, is a method which is helpful in the determination of size of the un it cell.

Weissenberg mod ified the technique in wh ich the crystal is rotated th rough 180⁰and back again ncontinuously wh ile the cylind rical camera fitted with the film moves at a constant speed forwards and backwards in the direction of the axis of rotation. The camera motion is so synchronized that its position corresponds to a definite angular position of the crystal as its rotation. It enables to accurately to index the spot on the film by noting its coordinates providing both the angle of reflect ion and the position of the reflecting plane.A cylinder made of a metal with a suitable and a few millimet res wide slit is p laced in bet ween the crystal and the film in a posit ion that allo ws the spots corresponding to only one layer to pass through it. In other words, the metal with an annular opening, when suitably adjusted allows only the diffracted beam for the second desired cone to get through while blocking the diffracted beams corresponding to other cones.

The x-ray diffraction technique described above lead to determination of lattice parameters. One also needs to find out the structure i.e., the position of atoms in the unit cell. For the determination of the crystal structure, accurate measurement of the intensities of a large number of Bragg reflections is necessary. Crystallographic measurements are now-a-days done on a computer controlled diffractometer.

SUMMARY

In this module we have discussed experimental for X-ray diffraction.
The methods include Laue Spots Method, Rotating crystal method and powdered crystal method.
Experimental setup of Laue method is schematically illustrated and described.
The arrangement of spots as seen in a typical Laue photograph of a simple cubic crystal are schematically illustrated and described.
Use of Laue diagrams in the determination of symmetry and structure of materials, Orienting single crystals and investigating distortion or polycrystallinity of materials is described.
The powdered crystal method as a valuable tool particularly for materials which are not available in single crystal form is explained.
Procedures involved in the measurements of Bragg angle q and interplanar spacing in crystal are discussed.
The procedure of indexing of powder photographs both for substances whose unit cell is known as well as substances whose unit cell is not known are described.

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References

Verma,A.R. &Srivastava, O.N. “ Crystallography For Solid State Physics”,Wiley Eastern Ltd., N.Delhi 1982.
Brown, F.C. “ The Physics of Solids “, W.A. Benjamin,Inc. N.Y. 1967.
Azaroff,L.V. “ Elements of X-ray crystallography”, McGraw-Hill,N.Y. 1968.
Phillips, F.C. “ An Introduction to Crystallography”, Longmans, London.

For detailed information on the topic

Woolfson, M.M.” X-ray Crystallography” Cambridge Vikas.
Abrahams, S.C. & Cohen, J.B. “ Role of Crystallography” American Institute of Physics,N.Y.1976.
Woolfson,M.M. “ Direct Methods in Crystallography”, Oxford Univ. Press,Oxford.
Henry N.F.M., Lipson ,H & Wooster, W.A.: “ The Interpretation of X-ray Diffraction Photographs, Macmillan, London.
James,R.W.: “ X-ray Crystallography, London:Methuen.