9 Application of X-ray diffraction in the study of crystaldefects-I

Prof. P. N. Kotru

Learning Objectives

• To learn about application of X-ray diffraction in the declination of various types of imperfections in crystals.
• Grain boundaries are the defects that can be detected and characterized by the X-ray laue method using a fine-focus.
• Dislocations are detected and characterized by employing X-ray topographic techniques.
• Scanning of crystals by X-ray topographic methods detect defects like impurity aggregates, inclusions, process induced defects, other defects that originates from growth faults viz; striations, growth faces, growth body. It is explained that X-ray topographic techniques is a bulk method of directly observing defects in a crystal.
• Basic principles of X-ray diffraction Topography are expalined.
• Contrast produced in X-ray diffraction by imperfect regions of the crystals, for both Bragg as well as Laue case, leading to extinction contrast is explained.

CONTENTS:

9.1 Applicatio n of X-ray diffractio n in the study of crystal defects

9.2 X-ray Topo graphic Technique.

9.3 Basic Princ iple of x- ray Topograp hy.

9.4 Summary

9.1 Application of X-ray diffraction in the study of crys tal defects

We have already studied the application and effectiveness of x-ray diffraction as applied to exploration of the internal structure of crystals. The diffraction techniques help us to determine the lattice parameters of crystals. In order to make complete crystallographic characterization, one has to know about the internal structure which includes the exact positions of the atoms, ions or molecules in the unit cell. To be able to do that, one is required to have accurate measurement of intensities of a very large number of Bragg reflections. Analysis of crystal structure is quite a specialized topic to be dealt with and those interested may consult references given at the relevant places. The techniques as applied to this aspect of x-ray crystallography is undergoing very rapid changes both in terms of accuracy of measurements of intensities, automation and automatic computation. Crystallographic measurements are no longer dependent on a computer-controlled diffractometer. The analysis of diffraction data, no matter how long and elaborate is done through computer controlled automation.

It may, however, be noted that real crystals are not perfect and one is required to assess the crystal perfection. So far as determination of lattice parameters of crystals is concerned, it is not influenced much by the imperfections present in the sample. It, however, does depend on the type of imperfection. There are imperfections like vacancies which, if present, in limited concentration, may not have any significant effect on measurement of lattice parameters. However, it is a fact that detection, observation, assessment, extent and type of defect in a given crystal is extremely important. Automated diffractometers, software programmes and control over radiation are being continuously modified and advanced to bring about accuracy and rapidity in the solution and refinement of data recording. X-ray diffraction does provide a powerful probe for investigating defects in crystals and this is what we shall describe in this unit.

Most of the properties of crystals are structure sensitive .Imperfections, if any, can affect their properties. Evaluation of imperfections in crystals is, therefore, important. X-ray diffraction is one of the probes which can be used effectively for assessment of perfection in crystals. X-diffraction techniques will be discussed here as applied to direct observation, detection and assessment of defects in crystals.

9.2 X-ray Topog ra phic Technique.

Imperfections in crystals include (i) point defects like vacancies and impurities, (ii) one dimensional defects like dislocations ,(iii) two-dimensional grain boundaries, low angle tilt boundaries or subgrain boundaries and twin boundaries, ( iv) stacking faults and (v) three-dimensional defects like bubbles, skeltel crystals and so on. Dislocations which are classified as one-dimensional defects affect the mechanical properties of crystals. Grain boundaries are the boundaries which separate the two grains that are crystallographically misoriented with respect to each other by a very small angle of not exceeding 10°. The lattices of the two separate grains are slightly tilted with respect to each other. . This type of defect can be detected and characterized by the x-ray Laue method using a fine- focus X-ray beams which have a divergence of nearly 10^–3radians. In fact, the dislocation model of low angle tilt boundary as suggested by Vogel et al. received confirmation when the angle of tilt between the two grains of germanium as measured from etch method and x-ray diffraction matched. The low angle tilt boundaries are delineated and characterized by using x-ray Laue method where a fine-focus x-ray source and collimated with very small slits , allowing x-ray beams of divergence of nearly 10^─3 radians. Dislocations are not easily observable by x-ray techniques. However, xray diffraction topographic techniques can detect them and it is possible to characterize them also. The other type of defects like point defects and their aggregates cannot be made directly observable by using x-ray topographic techniques. Such type of defects, however, can be used by measuring diffuse Xray scattering from regions of reciprocal space close to a reciprocal lattice point and one can obtain information concerning point defects and their aggregates. Scanning of crystals by x-ray topographic methods can effectively detect defects like impurity aggregates, inclusions, process induced defects and other defects which originate from growth faults like growth facets, striations, growth bands, skeltel crystals ,air bubbles, microcrystals within the body of bulk crystal etc. X-ray topographic techniques is a bulk method of directly observing defects in crystals.

9.3. Basic Principle of x-ray Topography.

X-ray topographic technique is based on the fact that intensity of x-ray beam which is diffracted by a perfect region within a crystal is different from the one diffracted by an imperfect region in the crystal. Let us consider diffraction of a divergent beam of white x-rays from a perfect and an imperfect region of a crystal and compare the two processes. To explain this, we consider the following two different cases:

(a) Diffraction of divergent beam of white x-rays from perfect regions of the crystal (figure 9.1)

(b)Diffraction of divergent beam of white x-rays from deformed (Imperfect) regions of the crystal (figure 9.2).

Figure 9.1: Divergent bea m of white x- rays getting diffracted from a perfect area of the crystal

Figure 9.2: Divergent beam of white x-rays getting diffracted from an imperfect region of the crystal

In figures 9.1 and 9.2 we have a fairly divergent beam of white x-rays which is used to explore the crystal. When the crystal is set so as to have its perfect region exposed to x-radiation, the perfect region of the crystal will choose those rays out of the divergent x-ray beam which satisfy the Bragg law. As such, only a small fraction of the total incident beam of x-rays will get diffracted as can be ascertained from figure 9.1. However, in the defective area the diffracting planes are distorted and the normal drawn at different points in this disturbed area cannot be parallel to each other, but instead would be pointing in different directions in space (figure 9.2). Diffraction of x-rays has to follow laws of reflection .Under these conditions, the number of rays which satisfy Braggs law will be much larger as compared to those in the former case (i.e., the perfect region). It would simply mean that the intensity of diffracted x-ray beam from the disturbed or deformed areas would be much more as compared to diffraction of x-ray beam from the perfect areas /regions of the crystal.It is this variation of the intensity of diffracted x-ray beam which can be a probe for differentiating perfect regions within a crystal from the imperfect regions. So, out of the total beam that is incident and striking a perfect region of the crystal, only a small fraction of it is diffracted. If , however, a divergent white beam of x-rays strikes the imperfect or deformed region of the crystal, the normal to the lattice points at different points within the deformed regions are not parallel to each other( unlike the one striking a perfect region) and ,as such, would diffract x-rays which lie within a large angular range. It leads to much larger intensity of diffraction x-rays as compared to the one in case of diffracted x-rays from perfect regions of the crystal. The other factor is that the interplanar spacing and so the d-values of the lattice planes in the imperfect or deformed regions of the crystal varies from point to point. The incident x-ray beam being white involves several wavelengths because of which different regions select suitable wavelengths which satisfy Bragg’s law. This reinforces the intensity of the diffracted beam from imperfect regions. It is because of these factors that deformed/imperfect regions appear dark in the lighter background when the same is recorded on a photograph called as x-ray topography.

So, one may sum up that the deformed or imperfect region in a crystal pose two problems to the incident beam of x-rays. One being that: (i) the normals drawn at every point within the deformed or imperfect region will not be parallel and (ii) the interplanar spacing (i.e., d- values) will vary if one goes from point to point.

When divergent beam of white x-rays is incident on the perfect regions of the crystal it will choose some rays out of the divergent beam that satisfies Bragg’s law. Consequently, only a small fraction of the total beam gets diffracted leading to decreased intensity of the diffracted beam as compared to the one diffracted from the imperfect regions of the crystal. In short, the intensity of the diffracted rays from the imperfect regions of the crystal is expected to be much larger than the one diffracted from perfect regions of the crystal. The second difference is on account of varying interplanar spacing at different points in the deformed or imperfect regions of the crystal Since d-value is an important parameter in the Bragg equation, any variation in it is bound to have some suitable parameters for Bragg equation to be satisfied.

Now,   suppose    the   incident    x-ray   beam    is   well   collimated and monochromatic. In this case, the perfect region of the crystal (which will have uniformly carrying the same value of d at every point) will lead to large intensity of the diffracted beam as shown in figure 9.3

Figure 9.3: Well collimated monochromatic beam getting diffracted from a perfect region of the crystal

Figure 9.4: Well collimated mo nochromatic beam getting diffracted from a deformed region of the Crystal

However, variation of d-value at every point of the deformed region will not allow many x-ray beams to get diffracted , leading to decrease in intensity of the diffracted beam because of which it will appear white on a dark background when the same is recorded on the photographic film.(see figure9.4)

What has been discussed above pertains to reflection geometry in which application of Bragg equation plays the shots. The diffracted x-ray beam gets reflected on the same side on which the probing beam is incident. Now, let us consider the situation in transmission popularly known as Laue case or observation in transmission geometry. When transmission geometry is applied for diffraction from perfect crystals, several other phenomena, like anomalous transmission and Pendellosung fringes, are called into play. In this mode there are two types of situations, which are on account of the thickness and the absorption coefficient of the crystal. Let us consider the case when absorption coefficient µ and the thickness t of the crystal are such that µ t < 1. Under this situation the diffracted and the direct beams are treated as independent of each other. In the normal topographic experiments, the probing x-ray beams have a divergence much larger than that of the width of the diffraction curve as is indicated in figure 9.1 and 9.2. It is obvious, therefore, that the deformation /defects/ imperfections appear black on a grey background of the topography.

Now suppose that we are dealing with a case when absorption coefficient µ and the thickness t of the crystal are such that µ t ≥ 10. In this situation the diffracted and the direct beam are strongly coupled to each other in the crystal. The dynamical theory suggests that the direct and diffracted beams are strong enough and may be termed as a case of anomalous transmission. To make it happen the specimen has to be so held that there is diffraction from a set of lattice planes and all of whose atoms scatter in phase. It was experimentally observed by Borrmann that it happens in a narrow range of angles. It is popularly known as Borrmann effect. The dynamical theory suggests that the direct and diffracted beams should have comparable amplitudes and so result into formation of standing waves between the lattice planes from which diffraction is under consideration. On leaving the crystal surface, the direct (i.e., forward diffracted) and the diffracted beams separate out .As a result, the residual direct, the forward diffracted beams and the diffracted beam get recorded at respective places on the photographic film. It means that there are two beams in the direction of the probing x-ray beam as shown in figure 9.5. According to dynamical theory, the nodes of the standing waves on falling on the atomic sites do not get absorbed because of photoelectric emission. As a result, the waves travel without any noticeable attenuation in the crystal and on reaching the exit surface of the crystal get separated into direct and diffracted beams, making an angle of 2θB with each other as illustrated in the schematic diagram of figure 9.5.

Figure 9.5: Formation of standing waves between the lattice planes from which the diffraction occurs

Within the crystal the standing waves travel along the lattice planes, the residual probing x-ray beam making an angle θB with the direction of propagation of the standing wave. It means that the direct beam on getting diffracted does not coincide with the direction of the probing beam. It may be parallel to it, but they are spatially displaced. In order to fully understand this phenomenon one is required to go through the dynamical theory, the further details of which may not be discussed here.

It is important to know that the standing wave formation can take place only in those regions of the crystal which are perfect. That is not so in case of deformed /imperfect areas wherein the coupling between the direct and the diffracted beams gets disrupted and breaks down. As a consequence of this, the conditions are not conducive for anomalous transmission and hence these regions appear to be white on a black background in the topograph.

What has been explained above is that if we use a divergent beam of white x-rays and allow it to strike the crystal so as to be diffracted from a perfect region of the crystal; the region will make selection for diffraction of such a group of x-rays whose angle of incidence lies within narrow angular range. Thus only a small fraction of the total incident beam will get diffracted as depicted in figure 9.1. If, however, it were diffraction of a divergent white beam of x-rays from an imperfect /deformed region of the crystal, the diffracted intensity would be much larger when compared to that of a case in figure 9.1. The reason being that since the area involved is deformed , the normal to the lattice planes at different points in that deformed region would not be parallel to each other with the result that the deformed region can diffract x-rays that fall within a large angular range as depicted in figure 9.2.

If we were to consider diffraction of a well collimated beam of monochromatic x-ray beam from a perfect region of a crystal, the intensity of the diffracted x-ray beam is a large fraction of that of the incident x-ray beam as shown in figure 9.3. The The situation would be different if we were to consider diffraction of a well collimated monochromatic x-ray beam from an imperfect region of the crystal. The diffracted x-ray beam
will cover a very small fraction of the incident x-ray beam as described in figure 9.4.

It is, therefore, clear that there is contrast produced in x-ray diffraction by imperfect regions of the crystal as explained above for both Bragg as well as Laue case and is, in reality, what may be called as extinction contrast. It is this contrast which makes imperfect or deformed regions of the crystal observable. The contrast caused by imperfect /deformed regions of the crystal to the diffracted x-ray, both in Bragg geometry as well as in Laue geometry is known as extinction contrast. Other than these two modes of causing extinction contrast, there is one more means of causing contrast which is different from what has been described above. Suppose we have a crystal in which a region is crystallographically differently oriented as a whole from the rest of the remaining volume of the crystal. While Bragg’s law may not be satisfied by the imperfect or misoriented region, it may well be satisfied by the rest of the crystal which would eventually result in to creation of contrast on the topographs. However, it is a different kind of contrast which may manifest itself in the form of orientation contrast. Crystals showing orientation contrast are the ones which are imperfect.

9.4 Summary

• In this module we have discussed application of X-ray diffraction in the declination of various types of imperfections in crystals.
• That the presence of Grain boundaries can be detected and characterized by the X-ray Laue method using a fine-focus is described.
• That X-ray diffraction topographic techniques can detect and characterize Dislocations in crystals is explained.
• Impurity aggregates, inclusions, process induced defects and imperfections arising as a result of fault in the crystal growth process viz., growth facets, striations, growth bands, skeletal crystals, air bubbles, microcrystals embedded in the bulk crystal and several other can be effectively detected by scanning of crystals by X-ray topographic techniques.
• Basic principles of X-ray topography are explained. Contrast produced in X-ray diffraction topographs by imperfect regions of the crystals, known as extinction contrast are explained.

References.

1. Verma, A.R. &Srivatava,O.N:.” Crystsallography for Solid State Physics”, Wiley Eastern Ltd.,N.Delhi,1982
2. Azaroff, L.V. “ Elements of X-ray Crystallography” , McGraw-Hill,N.Y. 1968.
3. Kittel, C . : “ Introduction to Solid State Physics”, Wiley, N.Y.,1971.
4. Lal, K. “ X-ray Diffraction Topography”, J.Phys. Edu.4, 1-5 ,1976
5. Newkirk,J.B. &Wernick,J.H.(Ed.) “ Direct Observation of Imperfections in Crystals”, Intersci ence, New York,1962.
6. Nuffield, E,W: “X-ray Diffraction”,Wiley,N.Y. 1966.

Suggested Reading for more information

1. Tanner,B.K. “ X-Ray Diffraction Topography”, Pergamon,N.Y.
2. Authier, A, “ Current Topics In Materials Science”, vol.2 (Ed.) E.Kaldis&H.J.Scheel,North Holland Publishing Co., Amsterdam,1977.
3. Lang, A.R. “ Advances in X-Ray Analysis” 10,Plenum Press,N.Y.,1967.
4. Laudise,R.A.” in Crystal Growth & Characterization” ed,R. Ueda and J.B. Mullin,North Holland Publishing Co.,Amsterdam, 1975.