The remarkable possibilities of the ASAXS technique are based on the energy dependence of the atomic scattering factors giving selective access to the specific SAXS contributions of nano-scaled phases, which are built up by different chemical constituents in composites like for instance alloys, chemical solutions or porous multicomponent systems. In general the atomic scattering factors are energy dependent complex quantities:

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, 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 non-resonant and the resonant scattering atoms,

calculated from the electron density, , and the atomic (molecular) volumes and, respectively, where is the electron density of the entire sample. The volume represents the atomic (molecular) volume of the non-resonant scattering atoms or building groups. corresponds to the atomic volumes of the resonant scattering atoms. The functions are the number densities of the non-resonant and the resonant scattering units, respectively and represent their spatial distribution in the sample. The atomic scattering factor, , 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 . Calculating the scattering intensity by means of Equations (2) and (3) and averaging over all orientations yields the sum of three scattering contributions, , with the convolution integrals [17,18]:

Equation (4) gives the so-called non-resonant scattering, , the cross-term or mixed-resonant scattering, , originating from the superposition of the scattering amplitudes of the non-resonant and the resonant scattering atoms and finally, , which contains only the scattering contributions of the resonant scattering atom species. These basic scattering functions are based on the pair correlation functions, , and in the literature are sometimes referred to as Stuhrmann functions.

The measurement of scattering curves at three energies in the vicinity of the absorption edge of the atoms with atomic number Z constitutes the following vector equation:

where the summation is running over the index j of the matrix and vector components.  

with. The summation is running over the index j i.e. summing over the vector components A_j and the columns of the matrix M_ij in the row with index i. The transformation of the vector Equation (5) by elementary operations changes the matrix M into the triangular matrix M':

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.

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:

Iⁿ 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) the polynomial is deduced:

(10)

giving the three eigenvalues:

From the eigenvalues the eigenvectors, Eⁱ, 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 Eⁱ:

(14)

Thus the vector composed from the pair correlation functions can be represented by a linear combination of the eigenvectors Eⁱ with the linear coefficients x_i. From the vector Equation (14) the linear coefficients x_i can be calculated:

In the next step Equation (14) is inserted into Equation (6):

or, because the Eⁱ are eigenvectors of M':

From the vector Equation (16b) the three pair correlation functions can be calculated by inserting the x_i from Equation (15) into Equation (16b):

Figure 1 summarizes the results from Equations (1) to (17). The three pair correlation functions, , are plotted along the three rectangular axis of the 3-dimensional coordinate system. The eigenvectors E¹, E²(ε), E³(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 E²(ε) and E³(ε) following curvatures depending on the energy via their vector components. Thus the pair correlation functions of Equation (14) depicted by the black vector are represented by the parallelepiped

which shows the vector of pair correlation functions in eigenvector representation with the coordinates x_i/l_i in the oblique angled basic system. The l_i are the lengths of the basic vectors Eⁱ. 

From Equation (17) the so-called basic scattering functions (Stuhrmann functions) can be deduced (see Equation (4)):

or in explicit form:

(19)

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: 

1)    Calculation of the triangular matrix by elementary operations upon the vector equation established by an ASAXS measurement at in minimum three different energies.

2)    Calculation of the eigenvalues from the characteristic polynomial of the new vector equation obtained from the elementary operations.

3)    Calculation of the eigenvectors.

4)    Representation of both, the special solution vector A_i and the vector B_i by linear combination of the eigenvectors.

5)    Calculation of the pair correlation functions, , , from these representations.

6)    Calculation of the basic scattering functions (Stuhrmann functions) from the basic pair correlation functions.

With the definition of S_0R in Equation (4) the three basic scattering functions fulfill the Cauchy-Schwarz inequality because the integrals in S₀, S_0R and S_R of Equations (4) and (20) define a positive definite metric in the vector space of functions [5,14]:

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]:

where is the inverse matrix. A detailed description of the impact of the Turing number for ASAXS measurements is given in [14,20].

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 4- dimensional 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 (22) by elementary operations changes the matrix M into the triangular matrix M' of the form: 

As a consequence the fourth component of the vector on the right side must be zero, b₄ = 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!

Because b₄ is not exactly zero, the vector is located in the 4-dimensional space but outside the embedded 3- dimensional 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 i.e., ASAXS measurements at many (n > 3) energies cannot improve

the accuracy of the ASAXS sequence but require improved (!) accuracies [14,19].

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 2.2 can be used individually for each vector subspace (X-ray absorption edge). The eigenvectors in the 6-dimensional vector space are:

The physical meaning of the 3 × 3-submatrix in Equation (26) filled with zero in the top right and bottom left is, that the atomic scattering factors of component B do not change, when tuning the energy at the X-ray absorption edge of component A and vice versa. The latter is a good approximation in the case, that the X-ray absorption edges lie far apart of each other. But of course there is a small change with energy and additionally, as in the forgoing chapter, there are measurement errors which enter the right hand vector. So a realistic formula of the vector equation is: 

If the matrix elements in the top right and bottom left matrix are small enough the eigenvector problem can be solved in good approximation separately for each absorption edge as outlined in chapter 2.2. In the case of contiguous X-ray absorption edges a more sophisticated equation must be employed for the solution of the eigenvector problem as is shown in Equation (28) for the example of a binary system:

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-dimensional 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 Ni₆₈Nb₁₆Y₁₆ [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 A₀², Re(A₀A_R) and A_R² (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.