Analysis of stress gradients in physical vapour deposition multilayers by X-ray diffraction at fixed depth intervals
The objective of this article is to develop and apply a model for the design and evaluation of X-ray diffraction experiments to measure phase-specific residual stress profiles in multilayer systems. Using synchrotron radiation and angle-dispersive diffraction, the stress measurements are performed on the basis of the sin2ψ method. Instead of the traditional Ω or χ mode, the experiments are carried out by a simultaneous variation of the goniometer angles χ, Ω and φG to ensure that the penetration and information depth and the measuring direction φ remain unchanged when the polar angle ψ is varied. The applicability of this measuring and evaluation strategy is demonstrated by the example of a multilayer system consisting of Ti and TiAlN layers, alternately deposited on a steel substrate by means of physical vapour deposition.
Many cutting tools are coated by means of physical vapour deposition (PVD) to enhance their lifetime by an increase in wear resistance. An important trend is to manufacture multilayer coating systems in which layers of different phases, and hence of different material characteristics, alternate. This strategy enables one to combine the advantageous mechanical properties of the phases involved within one system. The aim of the development of new multilayer systems is to find optimum parameter settings for the substrate preparation, the multilayer design (material and thickness of single layers) and the deposition.
The behaviour of multilayer systems is mainly characterized by three features: resistance against wear, hardness and residual stress. Previous investigations have shown that a high hardness is accompanied by high compressive residual stresses in the brittle ceramic layers (phases), with thermal expansion coefficients (CTE) that are low compared to the CTE of metallic substrates. Additionally, the residual stresses in these layers were found to be depth dependent.
The standard nondestructive and phase-selective method to measure residual stresses in regions close to the surface is X-ray diffraction analysis. Most publications using this technique to obtain depth-dependent stress profiles focus on homogeneous materials and single layers. To this end, suitable measuring and analysis strategies have already been developed. Basically, the residual stresses are determined indirectly from measured lattice strains, which are converted to stresses on the basis of X-ray elastic constants. Two principles are used to measure the lattice strains: angle-resolved X-ray scattering using monochromatic X-rays and energy-resolved scattering with white radiation. To obtain residual stresses and stress gradients in regions near the surface or in single-layer coatings a group of methods is applied which directly provide stresses in real space. In this case, apertures are inserted into the beam to limit the diffracting volume and, thus, to avoid the averaging over gradients of the measured quantity. In the publications of Manns and Scholtes (Manns, 2010; Manns & Scholtes, 2013), the diffracting volume is limited by a system of partially absorbing slit masks deposited on the specimen surface by a photolithographic process. In an alternative approach, Denks (2008) used a system of two apertures above the specimen surface to realize a gauge volume with a rhombic cross section, which is shifted through the sample to scan depth profiles of stresses (stress scanning method). The author applied an experimental setup at the energy dispersive beamline EDDI in Berlin (BESSY) for the separation of stresses in buried sublayers of a multilayer system deposited on tungsten carbide. Another group of methods provides averaged stresses since the diffracting volume is only limited by the penetration depth of radiation. This group includes amongst others the multi-wavelength (Eigenmann, 1990; Eigenmann et al., 1990), the scattering vector (Genzel, 1994, 1999a,b), the combined ω–χ (especially at constant penetration depth) (Kumar et al., 2006a,b; Erbacher, 2006; Erbacher et al., 2008) and the LIBAD (low incidence beam angle diffraction) methods (Van Acker et al., 1994; Mohrbacher et al., 1996). Although it is not always the case that a true Laplacian transform is applied, the averaged stresses measured by these methods are referred to as Laplace space stresses. Residual stress analysis on multilayers with alternating layer stacks of different materials was carried out for the first time by using the LIBAD method (Celis et al., 1995; Saerens et al., 1997). Recently, Klaus (2009) developed the equivalent thickness model for the deconvolution of the measured quantities in multilayer systems. This model considers the thickness, the absorption properties and the contribution to the diffraction signal of each single layer to determine the intralayer residual stresses or gradients. Applying the equivalent thickness model, Klaus and co-workers (Klaus et al., 2008; Klaus, 2009) measured residual stress profiles in multilayer systems. The current article presents an alternative model to plan and evaluate diffraction experiments in order to analyse profiles of depth-dependent residual stress in multilayer systems. First, key terms and equations from the literature are summarized as required for the derivation of new model equations.
In X-ray analysis, mainly two measures are used to characterize the degree of penetration of X-rays into a material: the penetration depth and the information depth, hereafter denominated as τ and ξ, respectively. In the case of a homogeneous material the mathematical definition of the penetration depth τ is based on Beer's law, which describes the dependency of the beam intensity I on the beam path s:
I0 denotes the beam intensity at the moment when the X-ray enters the specimen and μ stands for the linear attenuation coefficient of the material. The intensity at the moment when the X-ray leaves the specimen (hereinafter called final intensity If) can be obtained by replacing the path s in equation (1) with the sum of both the path to the site of diffraction at the depth z and the path of the diffracted partial beam to the specimen surface. Thus, one obtains the following relationship between If and the depth z (Genzel, 1999b):
Here the variable k denotes the geometry factor, which can be calculated from the incidence (α) and exit (β) angle, and from the gonimeter angles Ω, χ and the Bragg angle , according to the following equation:
This equation is valid for homogeneous specimens and specimens covered by a monolayer as well as for multilayer systems. In this work, the penetration depth τ is always defined by the following condition:
The authors are aware of the fact that this so-called exp(-1) criterion defines a penetration depth τ that cannot be used as an appropriate measure to fix a depth to which the diffraction signal can be assigned.
In this expression, τ specifies a depth interval , which contributes about 63% of the diffracted intensity received by the detector in an infinitely thick, homogeneous specimen. Hence, the validity of the definition of τ in equation (5) is also restricted to homogeneous specimens and monolayers with a thickness .
Delhez et al. (1987) stressed the need for the information depth as a second measure for X-ray penetration besides the penetration depth τ. While τ specifies the depth interval that acts as a source of diffracted intensity, the information depth ξ is the depth to which the parameters derived in diffraction experiments (like composition, stress) can be assigned. Kumar et al. (2006b) defined the information depth as the average depth obtained upon weighting each depth z with an attenuation factor that accounts for the intensity reduction due to an attenuation of the signal originating from the depth z. According to Delhez et al. (1987) and Birkholz (2006), the information depth ξ is calculated from the weighted sum of all X-rays differing in the amount of damping that they are exposed to if diffracted in the depth interval :
The variable t denotes the thickness of a homogeneous specimen or the thickness of a monolayer. Based on the linear attenuation coefficient μ and the thickness t, the weight function w (k, z ) in equation (6) is defined by the following equation (see for instance Delhez et al., 1987; Birkholz, 2006):
This means that the final intensities of the diffracted partial beams determine the weight function used for z averaging.
By this definition, the information depth ξ is free of such arbitrary assumptions as are made in the case of the penetration depth τ.
The limit of the information depth ξ as the thickness t of the specimen or monolayer tends to infinity is the penetration depth τ:
At a low geometry factor k (steep beam incidence, high penetration depth) and a low coefficient of linear attenuation μ the information depth tends to t/2 (Kumar et al., 2006b):
Birkholz (2006) argues that the averaging formalism with the exponential term as the weight function, which was used to calculate the information depth ξ [see equations (6) and (7)], may be applied for any other depth-dependent property g(z). He mentions, for instance, the concentration of a chemical phase, the amorphous fraction and the crystallite size. In much earlier publications this averaging formalism was applied to the gradients of lattice spacing (Delhez et al., 1987) as well as to the tensors of strain and stress (Dölle & Hauk, 1979; Noyan & Cohen, 1987).
It is important to note that the average value of a depth-dependent property g(z), calculated by equation (12), corresponds to the value that is measured at a geometry factor k, since the same averaging takes place during the measuring process.
X-ray diffraction measurements of stress based on the method are usually performed in the Ω or χ mode. In both cases, the tilting of the specimen by the angles Ω or χ is accompanied by a permanent change in the penetration and information depths. If significant stress gradients are present beneath the surface of the material, reliable stress values can be expected only when using an evaluation procedure that takes into account the occurrence of these stress gradients. This means that the inevitable variation of the penetration depth must be utilized to get information on the stress profiles. An alternative approach is to leave the penetration depth τ unchanged during the course of the lattice strain measurement (variation of polar angle ψ) by appropriate combinations of the χ and Ω angles. This approach was proposed by Bonarski et al. (1994) and later used in several other studies (for instance, Kumar et al., 2006a,b; Erbacher et al., 2008; Klaus et al., 2008; Klaus, 2009). In consequence, all data points of the plot for a certain hkl reflection correspond to a mixture of signals which originate from the depth interval . To implement this strategy, the authors of the above-mentioned articles had to perform three main tasks when designing X-ray diffraction experiments with a constant penetration depth τ to obtain residual stress depth profiles:
(1) A set of desired penetration depths , , has to be defined on the basis of the thickness t of a homogeneous specimen or a monolayer, as well as the linear attenuation coefficients of the concerned phases.
(2) For each of these penetration depths appropriate incidence–exit angle pairs and, finally, settings of the instrumental angles Ω and χ have to be specified in such a way that a variation of the angle ψ is realized by keeping the predetermined value constant at the same time.
(3) If a non-equiaxed plane stress state is expected, the goniometer angle has to be varied concurrently so that the measuring direction (azimuth angle φ) remains constant.
The desired goniometer angles Ω and χ (item 2) can be calculated from τ, μ, ψ and θ, which are usually given while planning the experiments. The angle Ω is obtained by numerically solving the following implicit equation:
generally holds true, for the expression on the left side
is only valid in the case of homogeneous materials.
Subsequently, the angle χ can be calculated from the angles ψ, θ and Ω using the equation
To compensate for the change of the measuring direction caused by the variation of the goniometer angles Ω, and χ, which is described by the equation,
the azimuthal angle φ has to be corrected by a change of the gonimeter angle according to (item 3; see also Dümmer, 1999). Consequently, Erbacher et al. (2008) named this kind of measurement the Ω–χ–φG method.
Kumar et al. (2006a,b) used the concept of constant penetration/information depth to analyse the depth profile of equiaxed stresses in Ni monolayers. In the work of Erbacher et al. (2008), this concept was applied to a strongly graded non-equiaxed stress state in a homogeneous material (high-density alumina, α—Al2O3); such a stress state is typical for surface layers produced by friction loading or grinding. These authors applied the method of Laplacian transformation to obtain profiles in regions close to the surface of this ceramic.
Klaus and co-workers (Klaus et al., 2008; Klaus, 2009) were the first to develop model equations for studying the depth-profile distribution of residual stress in a multilayer system. They applied these equations to a multilayer consisting of three Al2O3 layers and three TiCN layers, alternately deposited on a WC/Co substrate using chemical vapour deposition. In their model, these authors have already outlined the path that was also chosen in our work to obtain a residual stress profile in multilayer systems. However, since they started with a different expression for the final beam intensity as a function of the depth of diffraction, they derived different equations to calculate the information depth and the averaged stress profile.
The multilayer model assumed for the derivation of the equations for the final intensity consists of two materials (phases a and b) alternately deposited on a substrate (phase c; see Fig. 1a). The thicknesses of each single layer tai and tbi, , and the number n of the double layers (a layer of phase b together with the next overlying layer of phase a forms a double layer) can be chosen freely. Just as in the cases of the homogeneous specimens and specimens with monolayers, the penetration depth τ, the information depth ξ and the averaged stress are calculated in our multilayer model on the basis of the intensity with which the partial beam, diffracted at a depth z, leaves the specimen (final intensity). However, instead of only one final intensity–depth function If (z ) [see equation (2)], each diffracting layer i of the multilayer system and the substrate requires a function Ii(z ) of its own. Since we assume a multilayer system with two layer phases a and b, both of which can be the current diffracting phase, a total of two sets of intensity–depth functions Iai (z ) and Ibi (z ) is needed. The index i denotes the number of the double layer to which the concerned diffracting layer belongs.
The index f formerly used to symbolize the final intensity in homogeneous materials is omitted to shorten the names of the variables. In the case of diffraction in the substrate (phase c), the index i is set to n. The index p is the phase index. Then
This formula was derived by taking into account the fact that the beam may pass step by step through both pure absorbing and further diffracting layers prior to entering and after leaving the diffracting layer (see Appendix A). The third factor in equation (18) describes the exponential z dependency of the final intensity of the partial beam, diffracted in the concerned layer and phase, assuming that the beam only passes through phase p. Therefore, the respective phase-specific linear attenuation coefficient , p = (a,b,c), is presented in this equation. For the second term in equation (18) the following three equations apply:
The variables taj and tbj with stand for the layer thickness distribution of the phases a and b, respectively. The variables and in equations (19) specify a correction of the intensity loss in all pure absorbing layers above the concerned diffracting layer. Consequently, this correction depends on the difference of the attenuation coefficients and the thicknesses of the pure absorbing layers passed through by the X-ray beam prior to entering and after leaving the layer of diffraction.
In the case of diffraction in the substrate, the expression for calculating the correction of the intensity loss [equation (20)] incorporates the linear attenuation coefficients of all phases and the thicknesses of all n double layers. The geometry factor k in equations (18)–(20) is further calculated according to equation (3).
The criterion in equation (4) which has to be met by the penetration depth τ applies not only to homogeneous materials and monolayers but also to multilayer systems. However, instead of equation (5), equation (18) is now used to obtain the following formulae to calculate the phase-specific penetration depths , and :
(1) When planning diffraction experiments, the geometry factor k has to be calculated for a given penetration depth , specified by the phase p as well as the double layer number i of the diffracting layer. The corresponding equation must be solved for k.
(2) When evaluating diffraction experiments, the geometry factor k is known and the phase-specific equation can be straightforwardly used to calculate the actual penetration depths.
The calculation of the goniometer angles Ω when planning a new experiment with a constant penetration depth is now performed with modified implicit equations. Instead of equation (13) the relations with a phase-specific and layer-specific left side gpi and an unchanged right side have to be used:
The weight functions that are required to calculate the phase-specific information depths and the average of the stress profiles , , and , , , respectively, can be calculated in the case of multilayer systems by using the same basic equation as for homogeneous specimens and specimens with monolayers [equation (8)]. However, instead of the final intensity given in equation (2), now the phase-specific final intensities of equation (18) must be included in equation (8). Additionally, the lower limit 0 and the upper limit t of the integration have to be replaced by the phase-specific lower and upper limits uaj, ubj, uc and vaj, vbj, vc, respectively:
As pointed out above, the index j denotes the number of the double layer to which the concerned diffracting layer belongs. The upper limits van and vbn specify the maximum depth of the concerned phase. In the case of the phases p = a and p = b, the integration in the denominator of equation (8), now denoted as Cp-1, yields the following equation:
In the case of the phase p = c a simpler equation for the weight function holds true:
In comparison to equation (6), the formulae to calculate the information depth in multilayer systems have to be modified regarding the weight functions and the limits of integration. Now for all three phases p the following equation holds true:
The lower and upper limits of integration upj and vpj, and the weight functions wpj, are given in equations (28)–(30), and (35) and (36), respectively. These limits and weight functions are the same as those already applied to calculate the denominators of the weight functions [cf. equations (31) and (34)]. The integration in equation (37) provides the following formula for the information depth of the phases p = a, b:
The corresponding equation in the case of the substrate phase p = c reads
The same limits of integration and weight functions as applied to calculate the information depth are also used to average the stress profile in the multilayer systems. For the averaged stress of all phases p the following relationship holds true:
The residual stress profile in the integrand is approximated by a damped second-degree polynomial:
The variables dpj (p = a,b) are defined in equations (32) and (33). Equations (43)–(48) reveal that the geometry factor k is indeed the only independent variable in function (42) for the averaged stress profile as expressed by the denotation on the left-hand side of equation (42).
Since a nonlinear ansatz function for the residual stress profile was selected [equation (41)], the averaged stress also depends nonlinearly on the set of the four unknown parameters ai, . The best-fit parameter set is determined in our evaluation program by an iterative minimization using the Levenberg–Marquardt method. For this, the vector of derivatives , , was calculated analytically on the basis on equations (42)–(48) and inserted in the corresponding routine of the least-squares algorithm.
The three multilayer systems studied in our work consist of five metallic Ti and five ceramic TiAlN layers alternately deposited by means of PVD on substrates made of the hot working steel 1.2343 (designs 1–3 in Table 1; see also Fig. 1b). In the case of design 4, only one Ti and one TiAlN layer were deposited. For each multilayer design, the thickness of the ceramic layers and the thickness of the metallic interlayers, ta and tb, respectively, were always equal. Since the uniform layer thicknesses varied from design to design, the total coating thickness of our specimens ttotal ranges from 2510 to 3000 nm. The symbol m denotes the number of stress values measured in different depth intervals. The surface pretreatment of the substrate and the PVD process are described in detail by Selvadurai et al. (2013). The following values of attenuation length were used to plan and evaluate the experiments: TiAlN, = 55 µm; Ti, = 10.75 µm; Fe, = 4.016 µm.
The diffraction experiments were carried out at beamline G3 of the synchrotron radiation facility HASYLAB at DESY, Hamburg, Germany. The reflection 111 was chosen to analyse the residual stress profile in the TiAlN layers of the multilayer systems of designs 1–4. This reflection lies in the forward-scattering range ( = 37.4°). The measurements using an energy of E = 8.0352 keV (wavelength λ = 1.54302 Å) were performed at positive and negative values of the angle ψ. Although an equiaxed plane stress state was to be expected and although no splitting of the – curves for positive and negative ψ angles was observed in the first measurements, the goniometer angle was varied to keep the measuring direction (expressed by the azimuthal angle ) constant. While planning the experiments, the same lower limit of the range of penetration depth was defined for each of the four types of multilayer design (Table 1). The upper limits were set so that a significant portion of the beam is diffracted in the substrate. The ranges of the geometry factor k, also specified in Table 1, are calculated from the τ range according to equations (21) and (22). The average stress values were estimated for each penetration depth using the method. The estimation was based on the assumption , since the problem of the unknown unstressed lattice parameters as a function of the depth is not solved.
The residual stress profiles in the TiAlN layers of the multilayer designs 1–4 were approximated by a damped first-degree polynomial. In view of the limited number m of measured stress values (see Table 1), the reduction of the number of unknown fit parameters from four to three was considered appropriate.
4.1. Penetration depth and information depth
On the basis of equations (21) and (38) and the values of the attenuation length , p = a,b,c, of our multilayer system given in §3.1, the 1/k dependency of the penetration depth and information depth were calculated for designs 3 and 4. As shown in Fig. 2(b), the τ versus 1/k plots are composed of linear graphs, the slopes of which correspond to the attenuation length . Thus, the penetration τ increases more in the TiAlN layers than in the Ti layers and in the substrate. The intercept of each line depends on both the phase-specific value and the double layer number [see also equations (21) and (22)]. In the period of planning diffraction experiments, it makes sense to take the penetration depth τ into consideration, which provides information on whether a usable signal can be expected from a given depth interval or phase. τ–1/k relationships like those plotted in Fig. 2 can be used to estimate suitable values of the reciprocal geometry factor k-1 and, from these, to calculate sets of the goniometer angles Ω, χ and the azimuthal angle φ.
In the case of phase c, the information depth ξ obeys a linear relationship with the slope (see Fig. 3b). This linearity results directly from equation (39). The limit of as 1/k tends to zero (lower limit ) is
Since ua1 = 0 holds true, is always zero for phase a (see Fig. 3b). In the case of phase b, is only low for the multilayers of designs 1–3 ( = 500 nm; Fig. 3a). For design 4, there is only one layer of phase b and the lower integration limit is relatively high (ub1 = 2500 nm). Hence, the lower limit is close to the corresponding value of the substrate phase c (Fig. 3b).
Like the penetration and information depth, the measured average stress values are plotted against the reciprocal geometry factor 1/k (Fig. 4a). These data are fitted by a versus 1/k function which was calculated using the multilayer model described in §2.6 [see equation (42)].
In Fig. 4(b), the residual stress profiles according to the ansatz function in equation (41) are plotted for the TiAlN layers of designs 1–4 using the parameters ai, , of a damped first-degree polynomial. The four curves are interrupted in each z range of the Ti interlayers. The total z range of the plotted curves approximately corresponds to the selected range of the penetration depth. The maximum information depth that was attained for each design (see also Table 1) is marked by a vertical line in the curves. In the z range the curves are drawn as dashed lines to point at possible deviations from the actual curve progression.
With the knowledge that the maximum information depth is the weighted average of depth z over all layers of phase p (p = a,b), the reliability of the extrapolated curve progression diminishes with increasing distance .
In the present work, a model that is suitable for analysing depth profiles of residual stress in multilayer systems was developed. The model equations are derived for multilayers that consist of two materials (phases) alternately deposited on a substrate. The thicknesses of each single layer and the number n of double layers can be chosen freely.
The derivation started with three phase-specific equations for the intensity Ipi (z) of a partial beam that is diffracted at a depth z at the moment when leaving the specimen (final intensity) [equations (18)–(20)]. As a next step, equations for the penetration depths , the goniometer angle Ω, and the weight function wpj (z) [equations (21)–(26), and (35) and (36), respectively] were derived directly from the final intensity functions. In order to obtain both the phase-specific information depth [equations (38) and (39)] and the average stress [equation (42)], the weight functions were subsequently used for a weighted averaging of the depth of the diffraction z and the stress profile function . was approximated by a damped second-degree polynomial [equation (41)]. The authors are aware of the fact that using this single residual stress profile to describe several depth distributions in each individual sublayer of a phase is only the first step. Finally, the vector of , , was calculated to enable the fitting of the measured () data pairs by the model functions using the Levenberg–Marquardt method.
The sequence of the above-mentioned equations shows the importance of the expressions for the final intensity functions Ipi(z ), which were derived by taking the design of the multilayer into account. They form the basis for all further equations. The penetration depth parameter that is calculated by using these final intensity functions (like the information depth parameter) retains its physical meaning as in the case of homogeneous materials. These parameters, however, are of minor importance as they are not required for either the graphical representation of the measured stress or the calculated stress profiles . To check the quality of the fitting, the measured stress values should always be plotted together with the fit function against the reciprocal geometry factor 1/k (cf. Fig. 4a). The parameter k or 1/k proved to be the only independent variable in function (42) for the averaged stress profile. To visualize the depth profile, the calculated residual stress is plotted against the depth z (Fig. 4b).
In Fig. 4(a), a strong effect of the design on the curve of the averaged stress profile is clearly recognizable. The comparison of Fig. 4(a) with Fig. 4(b) shows that for both designs the gradients of the stress profiles are higher than the gradients of the averaged stress profiles up to the maximum information depth (in first two TiAlN layers).
The advantage of our model is that it enables us to analyse the depth profiles of residual stresses without any restriction of the selection of the penetration depth τ, whereas in the approach of Laplacian transformation, τ is subject to the condition . Thus, the application of the Laplacian approach is restricted to homogeneous specimens or monolayers with thicknesses much greater than the maximum attainable or desired penetration depth . In our model, the data of the averaged stress that is measured at can be taken into account too. Here, t denotes the maximum depth of the actual phase of the multilayer. As a consequence, our measurements of the residual stress values in the TiAlN phase at penetration depths well into the substrate could be included in the analyses of the stress profiles presented in Fig. 4(b). In addition, it is not only possible but also useful to include measurements with penetration depths when varying the reciprocal geometry factor 1/k in order to obtain a greater information depth ξ (as close as possible to the maximum value).
When planning future diffraction experiments, the distance between 1/k values should be better oriented according to the 1/k dependency of the information depth ξ. A low distance is recommended in the initial 1/k range in which the information depth increases rapidly as 1/k increases (Fig. 3a: TiAlN, Ti; Fig. 3b: TiAlN). Above this range, the distance of 1/k values can be raised. If the X-rays are diffracted far from the surface, i.e. in the substrate phase Fe (design 1–4) or in the Ti layer with a thick TiAlN layer above it (design 4, Fig. 3b), equidistant 1/k values should be chosen. The maximum value of the reciprocal geometry factor should be set as high as possible. In future work, the authors will use data of nanodiffraction experiments with a sub-micrometre resolution like those carried out by Krywka et al. (Krywka et al., 2012, 2013; Bartosik et al., 2013) to check the depth profiles of the residual stresses presented in this article.
Derivation of the final intensity If
In the following, the final intensity If given in equation (18) will be derived using phase a in the double layer i as an example (cf. Fig. 5). To this end, the path of a beam that enters the specimen with an angle α (angle of incidence), penetrates the material to the depth z of diffraction and goes back to the surface with an angle β (exit angle) is considered. In Fig. 5 the beam trajectories are presented for the special case that the surface normal vector lies in the plane given by the incident beam and the diffracted beam. However, since the derivation is not based on this assumption, the derived expressions for the final intensity If apply without limitation to this special geometric case.
The paths of both the incident and the diffracted beam, sI and sO, respectively, are divided into two segments:
Segment is the beam path from the multilayer surface to the interface between the layer of phase b, belonging to the double layer i-1, and the layer in the double layer i where the diffraction takes place (belonging to phase a). is the distance travelled by the diffracted beam from this interlayer back to the multilayer surface. For these paths the following equations hold true:
sIaj, sIbj and sOaj, sObj are the segments that are passed through by the incident and diffracted beams in the layers of the phases a and b, respectively. Along the distance , the initial intensity I0 drops to
At the site of diffraction (the beam has additionally passed through the segment ), this intensity has further diminished:
Re-arranging this expression leads to
To obtain the final intensity If, the intensity loss of the beam on its way to the surface must be taken into account:
Using the expressions
As a result, for the final intensity, the following equation holds true:
We thank Mr J. Donges for his support at beamline G3 at HASYLAB@DESY in Hamburg, Germany, and Mr A. Liehr for his help in planning the experiments. The funding of this research by the German Research Foundation (DFG) under contract Nos. FI 686/8-1 and TI 343/34-1 is gratefully acknowledged.
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