Handbook of Modern Coating Technologies

Powder diffraction measurements

The kind of information that can be extracted from an X-ray measurement depends on the diffraction geometry. Traditionally only out-of-plane measurements were possible (refer Fig. 3—26), but recent advances in instrumentation have made it possible to carry out in-plane measurements as well.
As mentioned in the beginning of this chapter, thin films have been at the forefront of advancements in materials science. Determination of physical, chemical and mechanical properties of thin films is both important and challenging—primarily because in comparison to the substrate, films generally contain a miniscule amount of material and second because these parameters often follow anisotropic distribution along thickness of the film. Despite imposition of constraints when dealing with thin films, X-ray techniques offer inherent advantages, such as nondestructive nature of the procedure, flexibility in selection of sam-pling area and depth, possibility of performing analysis under conditions similar to those in the operating environment, etc.
It is worth mentioning here that while some recent X-ray techniques have been devel¬oped especially from the perspective of thin film analysis, a major chunk of the diffraction techniques were essentially meant for characterization of bulk material and subsequently adapted for thin films. Accordingly for many procedures, a clear-cut demarcation between techniques meant for bulk or thin film analysis may not exist. Various diffraction and reflec-tion techniques are discussed in following subsections, along with notes regarding their suit-ability for thin films.
■ Incident beam
■ Specimen surface normal
■ Lattice plane normal
■ Diffracted beam
| Specimen |
FIGURE 3-26 Schematic of in-plane and out-of-plane diffraction measurements.
(a)
(b)
FIGURE 3-27 Angular limits for 2в/в scan: (A) lower limit as detector will receive glare from incident beam and (B) upper limit due to physical interference between X-ray tube and detector assemblies.
3.10.1 Symmetric reflection measurement
The most commonly used out-of-plane measurement is the symmetric reflection, also known as Bragg—Brentano scan. It involves turning the detector by 2в corresponding to rotation of the specimen surface by в (or w). Some machines achieve the same relative motion by keep¬ing the specimen stationary, while the X-ray tube and detector are turned by equal angles in opposite directions. In either case, the lower value of scan angle (2в) is limited by the glare from strong incident beam, while the upper limit is imposed by physical interference between the X-ray source and detector (refer Fig. 3—27A and B).
In 2в/в scan, the detector records reflections from those lattice planes, that are oriented parallel to the specimen's surface. In other words, the diffraction vector (the vector N, which bisects the angle between incident and scattered beams) is always normal to the specimen surface. In case of a polycrystalline specimen, as в is increased, planes with lesser d-spacing (higher Miller indices) will successively contribute toward diffraction pattern. The procedure is generally used for determining lattice structure of polycrystalline materials from recorded diffraction peaks.
3.10.1.1 Structure determination
Fig. 3—28 shows a 2в/в scan using Cu Ka radiation (A = 1.54 A) for a steel specimen, wherein fluorescence has been removed by using a secondary monochromator [84]. The lat¬tice planes corresponding to the observed peaks have been identified by using extinction rules, as summarized in Table 3—4.
It is noteworthy here that since iron is already known to have BCC lattice structure at room temperature, the sin2e ratios corresponding to just three peaks have been multiplied with 2 to yield the ratios corresponding to BCC structure given in Table 3—3. The lattice parameter (a) corresponding to the cubic system (BCC in the present case) can be calculated from any one of the three peaks by rewriting Eq. (3.11) as under:
Table 3-4 Determination of lattice structure.
/ 2в; sin в; sin2 в; Ratios ^
sin2 ві Normalized ratios Lattice planes
1 44.55 0.3791 0.1437 1 2 (110)
2 64.65 0.5347 0.2859 1.99 3.98—4 (200)
3 82.35 0.6584 0.4334 3.02 6.03—6 (211)

The lattice parameter works out to be 2.87 A in the present case, which corresponds to the lattice constant of iron reported in the literature [85]. It is important to mention here that there is not much importance associated with the standard lattice parameter of pure iron, or for that matter, of individual grades of steels, since it exhibits considerable variation, depend¬ing on composition and processing.
3.10.1.2 Application to coated specimens
Owing to relatively high angle of incidence in this procedure, the X-ray beam penetrates sev-eral tens of micrometers into the specimen. As a result, the thin films would produce a weak diffraction pattern, which would be accompanied by a strong diffraction from the underlying substrate. So, the 20/0 scan is not quite suitable for characterizing thin films. Rather, the method can be successfully used for determining lattice structure of the underlying substrate, which has been coated with a thin crystalline film (with film thickness on the order of a few nanometers) or with amorphous coatings as thick as a few micrometers.
3.10.2 Asymmetric reflection measurement
Unlike symmetric reflection, which always records reflections from planes parallel to the specimen surface, the asymmetric method records reflections from planes which are tilted with respect to the specimen surface. The procedure has traditionally been used as a nonde-structive technique for estimation of residual stress. A variant of the procedure, termed as small-angle X-ray scattering (SAXS), using very small angle of incidence is suitable for char-acterization of thin films. Both these procedures are discussed below.
3.10.2.1 Estimation of residual stress
XRD is a widely used nondestructive technique for residual stress estimation [86—91]. The procedure involves determination of strain by measuring d-spacings of some suitable set of lattice planes, which are oriented differently in relation to the specimen's macroscopic direc-tions. The information thus obtained is then used for estimating the corresponding stresses. Diffraction peaks provide information about both micro- and macrostrains. Microstrains (namely, those resulting from quenching) manifest as random variation in d-spacings (corre-sponding to any given set of hkl planes) within individual grains, thereby resulting in broad-ening of corresponding diffraction peaks. On the other hand, the alteration in d-spacing of a given set of planes under the influence of macrostrain is more systematic and manifests as a shift in corresponding diffraction peak. The peak-shift depends on the orientation of these planes in relation to the direction of macrostrain. The effect of peak-shifting is more pro-nounced for diffraction from planes with higher Miller indices, though their peaks are not as intense owing to lower atomic density. Accordingly diffractometers capable of achieving 20 values in excess of 150 degrees are desirable for residual stress estimation.
Residual stress estimation can be done by tilting the specimen either about w-axis (-0-tilt) or about у-axis (refer Fig. 3—32). The x-tilt measurements are generally done with point- focus configuration using poly or monocapillary optical module for minimizing defocusing
effects. These measurements also require a more complex design of diffractometer, namely, that shown in Fig. 3—25, using an open Eulerian cradle. The ф-tilt measurements on the other hand are carried out using line-focus configuration and can be done on any diffrac-tometer that permits independent motion of specimen (w) and detector (20) axes. Only this latter method is therefore described in this section.
The systematic variation of d-spacing with orientation is conveniently described as d^ in terms of the azimuth (ф) and tilt (ф) angles (refer Fig. 3—29A and B). This dependence causes a shift in the diffraction peak for different values of ф and ф, thereby leading to the formation of noncircular Debye—Scherrer rings in the powder (polycrystalline) pattern [92].
The estimation of strain and corresponding stress in the crystal lattice is made through application of classical continuum mechanics, based on the linear elastic distortion models of Reuss or Voigt [93,94]. The Reuss model assumes stress to be homogeneous over differ-ently oriented crystallites, thereby resulting in heterogeneous distribution of strain. The Voigt model on the other hand assumes homogeneous distribution of strain, which results in het-erogeneous distribution of stress.
For a bar made of an isotropic material and loaded with uniaxial stress ay along the longi-tudinal (y) direction, the corresponding strains in various directions are given by:
5 ^ (3.14)
£x 5 єг 5 - V£y (3.15)
Here, E is Young's modulus and v is Poisson's ratio.
For a specimen subjected to tensile stress, the spacing (along longitudinal direction) for lattice planes oriented perpendicular to the surface will be increased, while due to Poisson's ratio, it will get somewhat reduced for the planes parallel to the surface. However, it is not possible to measure ЄУ by XRD, since it requires recording of diffraction by planes, which are oriented perpendicular to the longitudinal direction, that is, ф = ± 90 degrees and X-rays can penetrate only to a very limited depth in the metallic specimens. However, it is possible to measure strain over a certain range of tilt, namely, -40 # ф # 40 degrees. These mea¬surements can then be used for calculating the uniaxial internal strain ЄУ. In principle, it is possible to estimate ЄУ by recording diffraction patterns for just two values of tilt, namely, ф = 0 and some other value, say ф = 40 degrees. The corresponding measured strains are termed as є33 and Єф. This procedure is generally used in portable residual stress equipment and makes a fair trade-off between speed and accuracy. Once єу is known, the corresponding stress ay for the free surface can be estimated using the following equation:
where Sj is the Kronecker delta and A 5 ^ + ^ _ 2v) is the Lame's parameter, which is a func¬tion of the rigidity and compressibility of the material. For details of treatment, the reader can refer to Appendix A in Ref. [95].
The preceding discussion treats stress and strain in the crystallite or laboratory frame of reference. As an aid to understanding, the relationships between crystallite or laboratory coordinate system (Li) and specimen coordinate system (Si) are depicted in Fig. 3—30.
The diffraction peaks from lattice planes hkl are obtained whenever they are so oriented, as to satisfy Bragg condition in the laboratory coordinate system Li. While the laboratory and specimen coordinate systems are inherently independent of each other, it might be of inter¬est to know the stress along some direction of interest along the specimen surface—namely, direction of rolling or drawing. If the direction of interest makes an angle ф with the direction of principal stress au then estimation of stress (av) in the direction of interest requires
FIGURE 3-30 Laboratory and specimen coordinate systems.
determination of strain (є33),ф in that direction. The prime superscript signifies that the strain in above expression is in the specimen coordinate system. The following expression is obtained for (є33) ^ through application of matrix transformations on the strain vector:
In case of a free surface, the stress in the direction normal to surface is zero (i.e., G33 = 0). For such a case of biaxial stresses, Eq. (3.17) reduces to:
Eq. (3.18) can be used for deriving the following relationship between lattice spacing d,^ and stress G,:
1 @d^ 1 1 V
d0 0(єт2ф) E ^
For detailed treatment, the reader can refer to Appendices B and C in Ref. [95]. The factor (11 v)/E appearing in Eq. (3.19) relates the strain measured along a certain crystallographic direction to the macroscopic stress, known as effective elastic parameter, the value of E/(11 v) can be estimated as per ASTM E 1426 standard [96].
If d0, E and v in Eq. (3.19) are known, G, can be estimated by determining d,ф corre-sponding to two different ф-tilts. However, as mentioned in Section 3.10.1.1, for materials containing alloying elements, there does not exist a standard lattice spacing (d0). So, for practical purposes, d0 is taken to be same as d,0 which is the d-spacing corresponding to ф = 0. With this understanding, if two measurements corresponding to d,ф and d,0 are made, then in the above equation, @d,ф = d,ф — d0 while, S(sin2^ = sin^ as sin20 = 0.
If й^ф is recorded over a range of -0-tilts, a more accurate estimate of can be obtained from the slope of й^ф versus sin20 plot. As shown in Fig. 3—31A, the plot exhibits positive slope in case of tensile stresses and negative slope when the stresses are compressive in nature. Since X-rays penetrate up to certain depth into the specimen (depending on the angle of incidence), the diffraction pattern results from cumulative contribution from both surface as well as subsurface layers. Accordingly the condition of a33 = 0, assumed while deriving Eq. (3.18) may not hold true in the presence of triaxial stress states. In such situa¬tions, lattice spacings corresponding to positive and negative values of ф would not be iden¬tical [97]. As a result, the versus sin20 plot exhibits ф-splitting [84], as shown schematically in Fig. 3—31B. In the presence of crystallographic texture, the plot takes the form shown in Fig. 3—31C [88,89]. The anisotropic character associated with texturing also compromises the accuracy with which residual stresses can be estimated, since the underly¬ing models are based on the condition of isotropy of stress and strain [94].
While dealing with residual stress estimation by й^ф versus sin2-0 method, it is important to know the difference between positive and negative ф-inclination. Unlike the schematic shown in Fig. 3—29, it is not possible for detector to physically cross-over the incident beam (i.e., the X-ray tube). Rather, the detector's scan about 26 axis remains confined within a nar-row range around the diffraction peak of interest, while the ф-tilt is achieved by rotating the specimen about u-axis (refer Fig. 3—32A and B). As a result, the angle of incidence (u = 6 ± ф) becomes quite small for negative ф-tilts (Fig. 3—32B), which can lead to

FIGURE 3-32 Specimen tilt about ш and x with 29 fixed: (A) positive ф and x 5 0, (B) negative ф and x 5 0, and (C) negative ф and x 5 45 degrees.

defocusing and consequently, lower the intensity of diffracted beam. This situation can be avoided by rotating the sample 180 degrees about ф-axis and recording negative-ф peaks under positive-ф setting.
Estimation of residual stresses in thin films calls for specialized techniques, such as grazing-incidence diffraction [97,98], which is performed using combined tilts of ф and x. These procedures are discussed in subsequent sections of this chapter.
3.10.2.2 Diffraction peak location
The accuracy with which an estimation of residual stress can be obtained through XRD tech-niques depends substantially on the precision with which diffraction peak positions are determined [99]. The task is complicated both by the source of incident beam as well as imperfections in the target material. The first consideration is concerning monochromaticity of the X-ray source, since the Ka peak consists of closely spaced wavelengths corresponding to Ka1 and Ka2. The doublet remains unresolved in broad peaks and causes further increase in their breadth. Peak-broadening due to presence of microstrains is another common phe-nomenon in most quenched and fine-grained metals [100]. Estimation of residual stress requires working with relatively smaller and broader peaks corresponding to planes with higher Miller indices. So, determination of diffraction peak position generally involves ana-lyzing data pertaining to small and broad curves.
The position-sensitive detectors with good linear-response allow cumulative count of dif-fracted intensity at various locations along the peak. The recorded intensities are then normal¬ized for various effects, such as absorption, polarization, and variation in background radiation. Most of these tasks are performed automatically by the software bundled with diffractometers.
Following refinement of recorded data, the next step is to determine the peak position. A com¬mon procedure is to determine vertex of the parabola drawn by taking three points within top 15% of the peak [101]. An improvement over this procedure is to fit the parabola by least squares regression—as it makes best use of the high positioning accuracy (~ 0.0001 degrees) of modern diffractometers. The accuracy of this method gets compromised in the event of incident beam getting defocused—namely, during negative ф-tilt, or when the Ka doublet is not fully combined [102]. In such situations, it is recommended to separately fit Pearson VII functions to the Ka1 and Ka2 peaks [103]. The bundled software generally supports this procedure, along with various other methods, namely, centroid, profile function fit, middle of width at chosen height, cross¬correlation function and manual fit. In the absence of a bundled software, the data can be exported in the form of an ASCII file and processed using some spreadsheet software [104]. It should be noted that the area-integration methods are not affected by shape of the peak, but exhibit greater sensitivity to the weight of tail regions of the diffraction peak. Hence, their accuracy would be compromised in the presence of background radiation
3.10.2.3 Depth of penetration in thin film
The thin film method of asymmetric reflection makes use of parallel incident beam [105]. The incidence angle is kept fixed at some small value, while the detector is rotated indepen¬dently to record the diffracted intensities. The refractive index pertaining to X-rays is just under unity for most materials and so, X-rays do not undergo any appreciable refraction at the interface. At such an angle, the rays traverse more distance in the thin film and their depth of penetration is reduced. As a result of this longer interaction with the thin film, the intensity of X-rays diffracted by the film gets enhanced significantly. At the same time, owing to reduction in the depth of penetration, signal from underlying substrate gets suppressed. The method is suitable for structure determination and texture analysis of thin films. To dif-ferentiate the thin film method from other asymmetric reflection methods, the incidence angle is depicted by a instead of ш.
We next calculate the depth of penetration of X-rays incident at an angle a to the surface. The beam will have to traverse a distance of t/sina for penetrating to a depth t. As shown in Fig. 3—26, for crystallite planes inclined at an angle ф to the specimen surface, the angles a and в made by incident and diffracted beams respectively with the specimen surface are related to the Bragg angle of incidence в as under:
a = в — ф
в 5 в 1 ф (3.20)
.в = 2в — a
For a given set of lattice planes, let I0 and Id be the intensities of incident and diffracted beams. The ratio of intensities, I0/Id signifies reflectivity of the lattice plane under consider-ation and essentially remains constant for a given wavelength and material. If the diffracting lattice planes are located at a depth t beneath the specimen surface, then the intensity of incident beam reaching these planes will get reduced to I0' in accordance with Eq. (3.1). For the given angle of incidence a, we get:
(3.21)
The intensity of corresponding diffracted beam is given by:
I'd 5 у 10 (3.22)
J0
The intensity that would emerge from the specimen at the angle в is given by:
Qc 5 A^/2.7 X 1011p
- 11 1 1 v I sina sine
Here, Gt is the ratio of intensity diffracted from subsurface layers (Id") located at depth t to that diffracted from the top surface layer (Id). The depth of penetration (t) corresponding to a given fraction of diffracted intensity can be determined by rearranging the terms as follows:
As an example, the depth of penetration corresponding to intensity ratio Gt 5 1/100 works out to be:
4.605
1 | 1 A
sina sin(2Q — a)J
For the special case of symmetric reflection, a 5 в 5 Q, which reduces the above expres¬sion to the following form:
Film Film Film
Substrate Substrate Substrate
FIGURE 3-33 Schematic of volume sampled as a function of incidence angle.

Taking p = 0.24/pm, the corresponding depth of penetration for (110) planes of steel (26 = 44.55 degrees, refer Table 3—4) works out to be 3.64 pm for symmetric reflection, while the depth of penetration for asymmetric reflection at an incidence angle a = 1 degrees works out to be 0.33 pm.
Once the incidence angle (a) has been decided, the asymmetric scan for thin film is car¬ried out by fixing the w-axis and scanning the detector over a range of 26 (say from 20 to 100 degrees) to record diffraction peaks produced by the film. Information about texture direc¬tion can be gathered by recording multiple scans for different azimuth angle, ф. Furthermore as shown in Fig. 3—33, the volume of film material contributing toward diffracted beam can be controlled by incident angle. Thus multiple 26 scans recorded for different angles of inci-dence can provide qualitative information about change in structural properties with depth.
3.10.3 Grazing-incidence techniques
The development in semiconductor industry has been relying on technological advances in fabrication of thin film based nanostructures through physical and chemical vapor deposi-tion [106]. To establish the sequence of processing parameters, these fabrication processes require nondestructive characterization techniques. While nondestructive characterization of surface morphology is conveniently done using some scanning probe method, X-ray techni-ques remain an indispensable tool for characterizing internal and interface structures.
The in-plane techniques use a shallow angle of incidence—usually between 1 and 1.4 times the critical angle 6c and hence, are known as grazing-incidence techniques (refer Section 3.3.4). Owing to negative refractive index of materials, the refracted beam begins to travel along the specimen surface—chiefly within the thin film in case of coated specimens. As a result, the strength of signal from the thin film is enhanced and the process becomes capable of charac¬terizing films having thickness in the range of just a few nanometers.
It is worth mentioning here that the swath of specimen surface illuminated under these conditions can be substantially large in comparison to the width (W) of incident beam. For example, a beam of width W = 0.05 mm, incident at a glancing angle (a) of 1 degrees will illuminate a swath of 0.05/sin(a) or 2.9 mm. This will significantly reduce beam's brightness per unit area, while the measurements will include an integrated contribution from the entire region thus illuminated. Accordingly the resolution requirements may call for special incident optics for providing as bright a beam as possible, with highly controlled divergence. In the early years of grazing-incidence experiments, the beam with desired characteristics could ideally be obtained only from synchrotron facilities. In due course of time, this
FIGURE 3-34 Grazing-incidence techniques.
requirement has been addressed by laboratory equipment through development of portable X-ray tubes using confocal optics and rotating anodes for better cooling, which per¬mit operation at higher accelerating voltages of up to 40 kV [98].
Close to the critical angle of incidence, the depth of penetration becomes very sensitive to the angle of incidence a and is governed by the following relationship [107]:
The beam thus propagating encounters various features of the thin film, namely, dispersed par¬ticles, pores or crystal planes inclined at various angles to the surface. Pores or amorphous parti¬cles within the thin film cause SAXS, which can be detected using a detector placed at small 26 angle. This technique is known as grazing-incidence SAXS (GI-SAXS)—a reflection geometry variant of the SAXS technique (refer Fig. 3—34), which is otherwise used in transmission geometry.
The crystallite planes on the other hand, depending on their orientation, direct the scat-tered beam along various directions—both in-plane and out-of-plane. The grazing-incidence X-ray diffraction (GI-XRD) technique is based on recording the beams thus diffracted.
3.10.3.1 Grazing-incidence X-ray diffraction
GI-XRD was initially applied by Marra et al. [108] and subsequently improved by Dosch et al. [109]. By mid-1990s, the procedure had started finding application in characterization of semiconductor multilayers [110], which led to development of five-axis diffractometers incor-porating the additional 26y axis for carrying-out scans around the specimen's periphery.
In GI-XRD, the beam traveling along surface suffers diffraction from various sets of crystal planes, which can be inclined at some angle (ф ф 90 degrees) or be perpendicular (ф = 90 degrees) to the specimen surface. In the former case, the diffracted beam will emerge out of the specimen surface—a special case of asymmetric diffraction discussed earlier. In the latter case, however, the diffracted beam will be nearly in-plane with the specimen surface. Owing to reduction in depth of penetration of X-rays, the in-plane geometry effectively suppresses
signals from the substrate and hence, can record diffraction from an extremely thin film. The procedure also permits depth-profile analysis by slightly varying the angle of incidence.
The term "in-plane” is used from the perspective of specimen surface and is somewhat misleading, since the reflection geometry in this procedure is essentially noncoplanar. In-plane diffracted radiation is recorded by scanning the detector about 26y axis around the specimen, while the 26 axis is kept fixed. An important consideration is the setting for 26 in terms of incident angle ш. In accordance with the Bragg—Brentano setting, it appears that 26 should be equal to 2ш, since this geometry has traditionally been used for recording diffrac-tion peaks from lattice planes parallel to the specimen surface—that is, those lattice planes, whose normal (vector N in Fig. 3—35A and B) is collinear with the specimen surface normal
(a)
Incident
beam
Z

FIGURE 3-36 Configuration for in-plane scanning.

(vector Z) (refer Fig. 3—24 for description of various vectors and directions). However, this requirement is meaningless in grazing-incidence procedure, since we are concerned with only those lattice planes, which are nearly perpendicular to the specimen surface. Keeping 2в = 2ш would make the plane of 26x scan inclined to the specimen surface, as is evident from angle between 26x-axis and Z-axis (or ф-axis for that matter) in Fig. 3—35A, where half of the scan circle (identified as "1”) lies above while the other half (identified as "2”) lies below the specimen surface. Accordingly the exit-angle of the diffracted beam equals ш for 26x = 0 degree, decreases to zero for 26x = 90 degrees and becomes negative for 26x > 90 degrees. On the other hand, if 26 is set equal to ш, the 26x-axis aligns with the specimen sur¬face normal (Z) and scan circle becomes coplanar with the specimen surface (Fig. 3—35B).
Diffractometers not equipped with additional 26x axis can provide in-plane scanning with 26 axis by setting ш = —90 degrees and x = 90 degrees + ai (refer Fig. 3—36). The issue of slight variation in exit-angle discussed above can be addressed by using suitable optics at the receiving end.
3.10.3.2 Grazing-incidence small-angle X-ray scattering
The SAXS works with very small 26 angles. In this geometry, the diffracted beams have little angular separation from the incident beam, which makes it difficult for the detector to avoid glare from the strong incident beam. The measurements are made possible by keeping the detector at a greater distance from the specimen and using high precision collimating optics for getting an acceptable signal-to-noise ratio. The native technique, as mostly applied for characterizing pores in bulk samples (namely, coal) or size-distribution of particles dispersed in liquids, records beams diffracted over small 26 angles, which have to be detected on the other side of the sample using transmission geometry (T-SAXS). However, T-SAXS cannot be applied to thin films in the same manner, because the underlying substrate will absorb sig-nals from the film in transmission geometry. Accordingly SAXS is applied to thin films in reflection geometry using grazing incidence (refer Fig. 3—34) and the procedure is termed as GI-SAXS [68]. This configuration not only avoids absorption of signal by the underlying substrate, but also amplifies the signal of interest as the beam traverses around 1/sin(w) times longer distance within the thin film in comparison to that in transmission mode.
Owing to small scattering angles in GI-SAXS, the ability of a diffractometer to scan close to 2d = 0 becomes important. The lower-limit to 2d is imposed by divergence and spot-size of the incident beam, which imposes the requirements of a sharp and well-collimated X-ray beam.
GI-SAXS is often used in tandem with GI-XRD for determining structural anisotropy along surface or across thickness of a deposited film. The anisotropy is assessed through GI-SAXS by recording diffraction data along surface-parallel as well as along surface-normal directions using a zero-dimensional detector. The surface-parallel scan (along direction Y in Fig. 3—35) would require scanning about 2dx axis for the given diffractometer geometry, while surface- normal scan (along direction Z in Fig. 3—35) would require scanning about 2d axis (refer Fig. 3—24). The axes corresponding to scans along Y or Z directions may change for differ¬ently configured diffractometers—namely, Rigaku SuperLab uses 2d axis for scanning along Y and 2dx axis for scanning along Z direction over horizontally oriented specimen surface [98].
Using a two-dimensional detector can greatly speed-up the process of recording GI-SAXS data along Y and Z directions [111,112]. However, the 2D detector would also record the small-angle scattering by the intervening medium (air), thereby necessitating vacuum or helium environment. On the other hand, scattering by air can be eliminated by using a nar-row slit in conjunction with a 0D detector.
3.10.3.3 X-ray reflectivity measurement
Strictly speaking, X-ray reflectivity is not a diffraction technique, but an indirect way of infer-ring certain parameters, namely, surface roughness, film thickness, film density, film/sub- strate interface roughness and so on, from measurement of reflected intensity. In reflection, when the angle of incidence approaches critical angle, that is, when d = dc, the X-ray beam gets propagated along the specimen surface. In the vicinity of dc, reflectivity falls abruptly with increase in d—the fall being sharper for surfaces with higher finish and vice versa. Likewise, the d versus reflectivity curve provides an estimate of surface roughness.
Estimation of thickness and density is based on the phenomenon of interference. While experimenting on interference of X-rays on thin layers, Kiessig [51] observed that for sub-strates coated with thin films, the rays reflected from the film's surface interfere with the rays reflected from film/substrate interface to form fringes (refer schematic in Fig. 3—37).

FIGURE 3-37 Schematic of X-ray reflectivity measurements.
The phenomenon is known as Kiessig fringes and is quite similar to the optical fringes observed in thin films illuminated by monochromatic light from sodium vapor lamp.
Scanning a section across the X-ray fringes using a diffractometer will record cyclic oscillations in intensity. Since fringe-width is inversely proportional to thickness of film, the "period” (Д6) of oscillations thus recorded provides a measure of film thickness. Furthermore as film-substrate combinations having greater difference in refractive indices (i.e., densities) produce fringes with higher contrast, the amplitude of oscillations in reflected intensity serves as an estimate for film's density. A decrease in amplitude of intensity oscilla¬tions at higher 26 angles (~ 4 degrees) implies the presence of a diffusion gradient or higher roughness at the film/substrate interface [113].
3.10.4 Texture measurement
Preferred crystallographic orientation, or texturing leads to anisotropy among various physi-cal properties. It is important to determine the direction as well as degree of texturing in thin films. As a result of texturing, the intensity of Bragg peaks corresponding to individual 26 positions varies as a function of x and ф.
While the direction (axis) of texturing can be determined through repeated 26/6 scans along different directions, the degree of texturing can be estimated by recording FWHM of the corresponding diffraction peak's rocking curve in asymmetric reflection. The rocking is done both for incident angle a (i.e., about ш) as well as azimuth angle ф.
The rocking curve procedure involves keeping the X-ray source and detector fixed for the peak of interest, while the specimen is rotated (rocked) about the ш-axis to change the Bragg angle. The resulting curve of intensity versus 6, as recorded by the detector using a wide slit is known as the ш-scan rocking curve and it provides a measure for the range of orientations.
Similar to the ш-scan rocking-curve measurement for asymmetric reflection, it is possible to make ф-scan rocking measurements of in-plane radiation. The ф-scan rocking curve measures preferred orientation among the planes normal to specimen surface and is recorded by keep¬ing the detector fixed at the 26x position corresponding to the diffraction peak of interest.
A complete analysis of preferred orientations can be done by combining the above two rocking procedures—that is, by carrying out the ф measurements corresponding to each individual setting of a. The recorded intensities can be plotted on a hemispherical surface in polar coordinates a and в where в signifies azimuth angle ф with reference to some datum of interest (namely, direction of rolling for sheet specimens). For accurate quantitative analy¬sis, the recorded plot of intensities should be normalized against the plot obtained for a completely random sample. Pole figures are constructed through stereoscopic projection of the hemispherical plot on a planar surface.
3.11 Concluding remarks
The credit for development of thorough understanding about the nature of X-rays goes to select scientists, who raised mankind's knowledge bar through their arduous work. It was their quest for truth that paved the way for development of XRD techniques and their subse-quent application in nondestructive characterization of materials. Ever since the initial dif-fraction experiments performed by Laue, XRD techniques have come a long way and got established as an indispensable set of tools at the hands of a materials scientist.
Over the course of past 1 century, the laboratory X-ray sources have progressed to become more and more powerful. As a result, many of the tasks, which could only be per¬formed through synchrotron radiation in the past, can now be done using compact labora¬tory equipment. At the same time, solid state detectors have replaced the conventional photographic films, while the goniometer control has gained a lot more precision. The col-lective outcome of all these developments is that the equipment has improved by many orders of magnitude in terms of sensitivity as well as accuracy.
The recent advances in various X-ray techniques are usually facilitated through innovation by scientists and technological refinement in the equipment provided by different manufacturers. These systematic developments generally result from progress made in various allied fields. The advancements in these techniques have greatly benefitted the field of materials science, espe¬cially that of surface engineering, which relies heavily on our ability to nondestructively charac¬terize the thin films. These characterization techniques have facilitated the materials scientists to deposit coatings, whose properties can be tailored according to their specific applications.
The importance and versatility of various X-ray techniques can be gauged from the fact that a large number of scientists, who either contributed toward the development of these techniques or used them in their research work, were awarded Nobel prizes in diverse fields, including Physics, Chemistry, and Medicine. It can be anticipated that in the times to come, we can have even more powerful and compact laboratory sources of X-rays, which would make the diffraction equipment more compact, portable as well as inexpensive. It is further anticipated that the future equipment would feature some sort of artificial intelligence, by vir-tue of which, it would be able to automatically select the most suitable scan for carrying out the analysis of interest, thereby reducing human intervention, while saving equipment time.
Acknowledgments
The authors gratefully acknowledge the support received from National Facility of Texture and OIM, established under “Intensification of Research in High Priority Areas” (IRHPA) initiative taken by Department of Science & Technology, Govt. of India, at Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology Bombay, Mumbai. The authors are grateful to Prof. Indradev Samajdar, Department of Metallurgical Engineering and Materials Science, Indian Institute of Technology Bombay, Mumbai for his support and to Mr. Jagtar Singh, Senior Technician, XRD Lab, Department of SAIF/CIL, Panjab University, Chandigarh (India) for pro¬viding valuable inputs from his vast practical experience.