The Solution of the Eigenvector Problem in Synchrotron Radiation Based Anomalous Small-Angle X-Ray Scattering

In the last three decades Synchrotron radiation became an indispensable experimental tool for chemical and structural analysis of nano-scaled properties in solid state physics, chemistry, materials science and life science thereby rendering the explanation of the macroscopic behavior of the materials and systems under investigation. Especially the techniques known as Anomalous Small-Angle X-ray Scattering provide deep insight into the materials structural architecture ac-cording to the different chemical components on lengths scales starting just above the atomic scale ( ≈ 1 nm) up to sev-eral 100 nm. The techniques sensitivity to the different chemical components makes use of the energy dependence of the atomic scattering factors, which are different for all chemical elements, thereby disentangling the nanostructure of the different chemical components by the signature of the elemental X-ray absorption edges i.e. by employing synchrotron radiation. The paper wants to focus on the application of an algorithm from linear algebra in the field of synchrotron radiation. It provides a closer look to the algebraic prerequisites, which govern the system of linear equations established by these experimental techniques and its solution by solving the eigenvector problem. The pair correlation functions of the so-called basic scattering functions are expressed as a linear combination of eigenvectors.


Introduction
Small-Angle X-ray Scattering (SAXS) experiments average over a large sample volume and give structural and quantitative information of high statistical significance on a mesoscopic length scale between 1 and hundreds of nanometers, which can be correlated to the macroscopic physical and chemical properties of the analysed condensed matter systems in solid state physics, chemistry, life science and materials science. Detailed descriptions of the experimental and theoretical aspects of Small-Angle Scattering can be found in [1][2][3]. By use of synchrotron radiation at suitable storage rings the socalled Anomalous Small-Angle X-ray Scattering (AS-AXS) can be employed, which is an excellent tool for the chemical selective structural analysis of multi-component systems.
This publication outlines in detail the basic mathematical aspects related to the scientific results of a series of former publications [4][5][6][7][8][9][10][11][12][13][14][15][16] obtained from quantitative ASAXS measurements applied to very different phys-icochemical systems. The quantitative analysis of the nano-scaled phases is correlated with the structural parameters, via the pair correlation functions and the socalled basic scattering functions, which provide (by Fourier transform) the nano-scaled architecture with very high statistical significance because scattering experiments average over up to 10 10 structural entities. The related scientific problems cannot be addressed by a classical Small-Angle X-ray Scattering (SAXS) experiment, because the specific scattering contributions of the different chemical components need to be separated. As outlined in a former publication [12] an outstanding experimental accuracy is needed in order to separate the so-called pure-resonant (element specific) scattering contribution. Additionally a suitable mathematical algorithm is employed (the Gauss elimination algorithm), which turned out to provide the best results when inverting the vector equation introduced by ASAXS measurements. The paper wants to shed some light on the algebraic basics, which govern the analysis of numerous ASAXS publications within the last decade.
 d (1) where Z represents the atomic number. When performing ASAXS measurements on multi-component systems in the vicinity of the absorption edge of one of the sample constituents the scattering amplitude is: where q is the magnitude of the scattering vector non-resonant and the resonant scattering atoms, cons , is nearly energy independent, while the atomic scattering factor, shows strong variation with the energy in the vicinity of the absorption edge of the resonant scattering atoms due to the so-called anomalous dispersion corrections (2) and (3) and averaging over all orientations yields the sum of three scattering contributions, , with the convolution integrals [17,18]: is the scattering angle, λ the X-ray wavelength and V p is the irradiated sample volume.
are the differences of electron densities of the Equation (4) gives the so-called non-resonant scattering, , the cross-term or mixed-resonant scattering, where the meaning of is [11]: , The vector Equation (6) with the triangular matrix represents the Gauss algorithm (elimination procedure) for the solution of a system of linear equations. Thus the elimination procedure is equivalent to the elementary operations performed to the matrix of a vector equation.

The Eigenvector Problem in Anomalous Small-Angle X-Ray Scattering
Up to this point the Equations (1)- (7) have been taken from [11,14] and represent nothing new. In what is to follow the underlying basics of linear algebra and the high degree of significance obtained by a suitable matrix inversion applied to experimental data (here scattering curves) will be outlined. In detail we demonstrate the calculation of the eigenvalues of the vector Equation (6) from the characteristic polynomial thereby providing the eigenvectors. Furthermore the representation of the special solution as a linear combination of the eigenvectors is demonstrated. In the next step analytical expressions for the pair correlation functions have been deduced from the eigenvector representation. Finally in the last step the basic scattering functions (Stuhrmann functions) are provided.
We start with the vector Equation (6): . The characteristic polynomial for the determination of the eigenvalues writes: I n represents the identity matrix: We will restrict the problem to n = 3, which is the minimum number of energies to be measured in order to solve the system of linear equations with three unknown quantities. From Equations (8) and (9) giving the three eigenvalues: From the eigenvalues the eigenvectors, E i , can be constructed via Equation (12): giving: Because these eigenvectors are linear independent they define a system of basic vectors in the solution space and the vectors A and B can be expressed via a linear combination of the eigenvectors E i : Thus the vector composed from the pair correlation functions can be represented by a linear combination of the eigenvectors E i with the linear coefficients x i . From the vector Equation (14) the linear coefficients x i can be calculated: 2  23 12  13  33  22  12  1  0  2  3  22  11  33  11  33  22   23  2  0  3  33  22   2   3 Re In the next step Equation (14) is inserted into Equation (6): 23 12  13  33  22  12  2  33  11  33  22  22  11  0  1  23  0  1  2  3  33  22  2  3   1 Re From the vector Equation (16b) (1) to (17). The three pair correlation functions A q are plotted along the three rectangular axis of the 3-dimensional coordinate system. The eigenvectors E 1 , E 2 (ε), E 3 (ε) build up an obliqueangled coordinate system where two of them depend on the photon energy. The blue and the red line show the eigenvectors E 2 (ε) and E 3 (ε) following curvatures depending on the energy via their vector components. Thus the pair correlation functions of Equation (14) or in explicit form:  2  2  1  2  1  2  3  2  3  1 , , , Equation (19) provides the analytical form for the calculation of the basic scattering functions from scattering curves measured at in minimum three energies. The mathematical theory behind this formula is the solution of the eigenvalue problem in six steps: is the inverse matrix. A detailed description of the impact of the Turing number for ASAXS measurements is given in [14,20].

The Eigenvector Problem in Anomalous Small-Angle X-Ray Scattering with More than Three Energies at One X-Ray Absorption Edge
This criterion is essential but not sufficient for the reliability of the basic scattering functions obtained from the matrix inversion. If it is not fulfilled, the basic scattering functions are meaningless. The significance of the special solution obtained by matrix inversion can be numerically quantified by the Turing number [19]: In the following we will discuss the system of linear equations with more than three (4…n) energies: Equation (22) represents an ASAXS sequence with scattering curves I(q,ε i ) measured at four different energies ε i . Once again the summation is running over the index j = 1, 2, 3 i.e. summing over the vector components A j and the columns with index j of the rectangular i × j-matrix M ij in the row with index i. But now the resulting vector has four components on the right side. Equation (22) represents a linear map from a 3-dimensional space into a 4-dimensional space. The possible solutions are located in the 3-dimensional subspace, which is embedded in the 4dimensional space. Because the rank of the matrix is three in minimum two of the four linear equations must be linear dependent. The transformation of the vector Equation Re Re 0 0 Because b 4 is not exactly zero, the vector is located in the 4-dimensional space but outside the embedded 3dimensional space of solution vectors. Thus no exact solution for the system of linear equations exists. Only an approximate solution exists. It is the vector in the 3-dimensional hyper plane with the shortest distance to B in the 4-dimensional space or in other words: the foot of the perpendicular constructed between B and the hyper plane (Figure 2). The meaning of Equation (24) is: no improvement can be obtained from additional measurements of scattering curves at more than three energies, because the rank of the matrix is three or in other words in case of more than three equations in the 3-dimensional vector space in minimum two of them are linear dependent. The same argumentation holds for n energies. The significance of the solution vector obtained by the matrix inversion of the system of linear equations can be calculated by the Turing number, which was outlined in a recent publication [14]. Note that the Turing number increases with the number n of equations (energies) by n i.e., ASAXS measurements at many (n > 3) energies cannot improve As a consequence the fourth component of the vector on the right side must be zero, b 4 = 0. Of course when performing measurements the latter is not exactly fulfilled due to the error bars of the measurements. Thus the value of the fourth row is not exactly zero but nearly zero! the accuracy of the ASAXS sequence but require improved (!) accuracies [14,19].

The Eigenvector Problem in Anomalous Small-Angle X-Ray Scattering with More than Three Energies but at Different X-Ray Absorption Edges
In the following we will discuss the system of linear equations established by ASAXS measurements at different X-ray absorption edges such as binary alloys composed of components A and B, glasses or chemical solutions with different metal constituents.
The I i represent six scattering curves measured at different energies, three at energies of the 1 st X-ray absorption edge, the other three at energies of the 2 nd X-ray absorption edge. The a ij again are functions of the density contrasts and the anomalous dispersion corrections of the different chemical constituents but this time from different X-ray absorption edges. The two X-ray absorption edges define two 3-dimensional vector spaces, which are orthogonal to each other i.e. the six linear equations are linear independent. Due to the different X-ray absorption edges the rank of the matrix is now six. Once again the triangular matrix can be found by elementary operation of the 6-dimensional vector equation, but for every subspace a triangular matrix can be established: For the calculation of the eigenvalues, eigenvectors, pair correlation functions and the basic scattering functions the formula of the for-going chapter 2.
The same arguments hold for N X-ray absorption edges of condensed matter systems composed by N chemical components thereby establishing a 3N × 3N matrix in a 3N-dimensional vector space. If the N X-ray absorption edges lie far apart of each other the 3N-dimensional vector space is spanned by N 3-dimensional subspaces being orthogonal to each other in good approximation and the eigenvector problem can be solved separately in each 3-dimensional subspace via the formula of chapter 2.2. In the case of M < N contiguous X-ray edges the 3Ndimensional vector space is spanned by N − M 3-dimen-sional orthogonal subspaces and a 3M-dimensional subspace. The scheme of the vector equation is shown in Figure 3. Again the significance of the solution vectors obtained by these matrix inversions must be calculated via the Turing number [14].
As an example taken from a former publication Figure  4 depicts schematically the eigenvector representation in the 9-dimensional vector space of two ternary alloys (blue vectors) of different metallurgical states with the composition Ni 68 Nb 16 Y 16 [10,11,14]. The light blue lines construct the foot points of the projection of the 9-dimensional vectors onto the three 3-dimensional subspaces of the three alloy components Nickel, Niobium and Yttrium. Because the X-ray absorption edges of the three elements  lie far apart of each other the three subspaces are orthogonal to each other in good approximation. Each subspace is spanned by the Cartesian orthogonal system of the pair correlation functions A 0 2 , Re(A 0 A R ) and A R 2 (see Figure 1). Thus within these three subspaces the pair correlation functions can be expressed in eigenvector representation i.e. each subspace provides the solution of the eigenvector problem independently from the other subspaces via the equations of chapter 2.2.

Conclusion
The mathematical basics of the system of linear equations established by Synchrotron radiation based Anomalous Small-Angle X-ray Scattering experiments have been analyzed. The linear algebraic theory provides the AS-AXS techniques with five major improvements: (1) Analytical formula for the basic scattering functions, thereby omitting in-transparent fit or inversion procedures of unknown significance (2) a mathematical unambiguous solution of the system of linear equations by employment of the (linear) Gaussian elimination algorithm (3) an essential (but not sufficient) criterion (the Cauchy Schwarz inequality) allowing direct control, whether obtained basic scattering functions can possess significance or not and (4) a sufficient criterion represented by the Turing number, which tells directly the errors of the special solution vector facilitated by the inversion algorithm, thereby revealing its degree of significance. (5) The latter can be verified directly by ab initio error calculations, which start with the experimental errors entering the scattering curves of the ASAXS experiment (for instance the statistical errors obtained via single photon count techniques). For the calculation of the error propagation, the analytical formula is mandatory.
In summary, the eigenvector-based algorithm represents a superior mathematical description, because the problem is treated in the context of a linear matrix theory thereby offering the calculation of the significance of the solution, which cannot be provided by non-linear procedures. In other words, Linear Algebra gives deep insight (first intended by [20]) into ill-conditioned mathematical problems in condensed matter research thereby giving clear hints on how to overcome these problems by experimental and mathematical improvements. The subject underlines the crucial importance of Linear Algebra for search and understanding of mathematical algorithms in condensed matter research with Synchrotron radiation.