Full-Tensor Magic Angle Spectroscopy

1. Introduction

It is well-known in various types of optical spectroscopy [1,2,3,4,5], that, for oriented samples, additional information (relative to unpolarized beams) can be provided by measuring spectral dichroism with a linearly polarized photon beam. This is clear from fundamental quantum theory [8,9,10] of the interaction of light with matter and has been used productively in thousands of papers in many contexts, particularly in the UV-visible region (in biochemistry and biophysics), and in the x-ray region (in materials physics, geochemistry, condensed matter, and biophysics) e.g., [4,5,11,12,13].

The theory of the angle dependence of x-ray absorption [5,6] has been comprehensively worked out by Brouder [14], with many references to prior work. In our current paper, although we will be specifically addressing X-ray Absorption Fine Structure measurements, our same approach is expected to be applicable to other photon energies e.g., UV-Vis, IR, microwave, or THz spectroscopy.

Unless otherwise noted, we will make use of the dipole approximation. It is known that quadrupole effects can be important in the pre-edge region of XAFS spectra, particularly for transition metals, so our approach here is most applicable to such systems on the rising part of the absorption edge and above it, or for the majority of other systems that do not have significant quadrupole effects. In such cases the absorption probability transforms under rotations as a second rank cartesian tensor. Below we will make a rough estimate of an upper limit on quadrupole transitions relative to dipole transitions.

For a low-symmetry sample, in principle, up to six times as much information can be obtained compared to an isotropically averaged sample (e.g., most solutions, samples consisting of isotropically oriented microcrystals) or samples in which the absorbing sites have cubic point group symmetry (e.g., exact octahedral or tetrahedral symmetry). This is a consequence of there being six distinct values that make up a symmetric 3D cartesian tensor. Each of these six unique tensor elements is a function of energy.

X-ray linear dichroism, also known as “Polarized XAFS”, has been very productively used in systems in which there is a preferred axis such as the normal to a planar surface to which the molecules are aligned, but may be cylindrically randomly oriented, or have a 3-fold or greater axis of symmetry about the surface normal. In such a case there is a clear separation between the absorption along the direction normal to the surface, and the in-plane absorption.

Despite the extra information linear dichroism (LD) offers, it is less-often used than it might be, in part because of the additional experimental complexity, and additional data acquisition time required to do the measurements, if done in conventional ways. These involve 4-circle goniometry, and practical considerations such sample morphology and other instrumental effects come into play, because different profiles are presented to the beam, and sample thickness along the direction of beam propagation generally will vary with orientation, and needs to be taken into account. It is understandable why most experimentalists simply don’t bother with doing such experiments.

Even so, linear dichroism isn’t something that can just be ignored. If the x-ray absorbing sites (whose intrinsic absorbances have their own angular dependence) have, in addition, a higher-level statistical orientational organization within the sample (i.e., “texture”, “preferred orientation”), then systematic errors can occur if it is ignored. Fortunately, in many cases, one already has, or can prepare, a flat or film sample, in which case the systematic errors usually can be eliminated by Magic Angle Spinning [18,19].

In this paper we will use $\mu$ to refer to the spectra, and M as the tensor giving rise to them. The x-ray absorption coefficient $\mu \left(\mathcal{E}\right)$ is given by time-dependent perturbation theory [9,10,15,16,17] as

$$\mu \propto {\left|\sum _f\langle f|(\widehat{ϵ}·\overrightarrow{r})e^{i\overrightarrow{k}·\overrightarrow{r}}|i\rangle \right|}^2\approx {\left|\sum _f\langle f|(\widehat{ϵ}·\overrightarrow{r})(1+i\overrightarrow{k}·\overrightarrow{r}\dots )|i\rangle \right|}^2$$

(1)

where $|i\rangle$ is the initial state and $|f\rangle$ are the final states, and $\overrightarrow{k}$ is the k-vector of the incoming plane-polarized photon beam and $\overrightarrow{r}$ is the coordinate to be integrated over. The first term in the expansion is the dipole term, and the second gives the quadrupole term which we will treat as negligible here unless otherwise noted.

1.1. Estimate of the Relative Importance of Quadrupole vs Dipole Transitions

We can make a rough estimate of the ratio of (allowed) quadrupole transitions to (allowed) dipole transitions. Either can be zero for some states, because of symmetry; for our estimate here we presume that is not the case. The quadrupole transitions are sometimes observable in XAFS in spectral regions and geometries where otherwise-dominant dipole transitions may be forbidden. For a K-edge (or $L_1$, in the pre-edge region, if there is structural inversion symmetry of the sites, the states below the edge will contain no p-symmetry character, and dipole transitions will be forbidden in that region. If there are other states of appropriate symmetry such that quadrupole transitions are allowed, they may be observed, unmasked by dipole transitions. On the edge and above dipole absorption predominate, because there are always allowed symmetry states in the continuum. Similar considerations apply to other absorption edges.

In this subsection for convenience we will use atomic units in which ℏ, electron mass m and charge e ($e^2\equiv q^2/(4\pi {ϵ}_0$ in SI units) are all equal to 1. In atomic units (au), the length unit is Bohr radius $a_0$ and the energy unit is Hartree $=2$ Ryd $\approx 27.2$ eV. The dimensionless fine structure constant $e^2/(\hslash c)\approx 1/137.036$, which implies that in atomic units the speed of light $c\approx 137$.

The relative magnitudes of the dipole and quadrupole matrix elements above evidently are 1 and $ka$, where a is a measure of the size of the initial state ($1S$, for K-edges). This is clear if one thinks of the matrix element as an integral: the size of the initial state wave function limits the range of the integral. $kc$ is the energy of the photon, which is approximately the absorption edge energy. The initial state is, to a good approximation, that of a hydrogenic atom of atomic number Z. In atomic units we then have $a\approx 1/Z$, and the energy $kc\approx Z^2/\left(2n^2\right)$, so $ka\approx Z/(2·137)(1/n^2)$ where n is the principal quantum number of the initial state ($n=1$ for K-edges, $n=2$ for L-edges). For Fe ($Z=26$) and K-edge ($n=1$) this gives $\approx 10\%$. The absorption probability scales as the square of the matrix elements, so a rough estimate of the ratio of the allowed quadrupole transition to allowed dipole is about $1\%$. This comports with the observed experimental magnitudes where the allowed quadrupole (in the pre-edge region) of first row transition metal complexes are roughly $1\%$ of the allowed dipole (the edge step). This primitive estimate indicates the relative importance of quadrupole transitions should grow as $Z^2$: the ratio for Ru $K-edge$ would be roughly three times greater.

1.2. Dipole Absorption Tensor

Having estimated the relative magnitude of quadrupole terms, and taking just the dipole term, we can write Equation (1) as $\mu \propto \widehat{ϵ}·M·\widehat{ϵ}$ where M is a second rank cartesian tensor. Writing in spherical coordinates $\widehat{ϵ}=(sin\theta cos\varphi ,sin\theta sin\varphi ,cos\theta )$ we have

$$\begin{array}{cc}\hfill & M_{11}{sin}^2\left(\theta \right){cos}^2\left(\varphi \right)+M_{22}{sin}^2\left(\theta \right){sin}^2\left(\varphi \right)+M_{33}{cos}^2\left(\theta \right)+\hfill \\ \hfill & M_{12}{sin}^2\left(\theta \right)sin\left(2\varphi \right)+M_{23}sin\left(2\theta \right)sin\left(\varphi \right)+M_{31}sin\left(2\theta \right)cos\left(\varphi \right).\hfill \end{array}$$

(2)

This is the basic angular dependence vs polarization vector orientation of a single absorbing site, typically a molecular chromophore (e.g., Heme group in a protein), or an oriented crystallite within a material.

1.3. Averaging over the Sample

The absorption of an “elementary chromophore” or crystallite has the form given in Equation (2), but needs to be averaged over the various sites contained within the sample. The orientations of each of these may vary because of inequivalently-oriented sites within a single crystal, or different crystallites or domains with varying orientations within the sample. We will regard these as reoriented copies of the elementary M tensor that are to be summed over.

A second-rank tensor transforms as the outer product of two vectors; rotating a second rank tensor can be accomplished by sandwiching the tensor matrix between a rotation matrix $R({\omega }_f,{\omega }_i)$ and its transpose $R^T({\omega }_f,{\omega }_i)=R({\omega }_i,{\omega }_f)$, where ${\omega }_i$ and ${\omega }_f$ are the initial and final orientations. The result of averaging a second rank tensor is still a second rank tensor, but with possibly reoriented principal axes and altered principal values. Summing up such rotated tensors, weighted according to a probability distribution $P({\omega }_f,{\omega }_i)$ characteristic of the sample, can be written $\overline{M}=\int P({\omega }_f,{\omega }_i)R({\omega }_i,{\omega }_f)·M·R({\omega }_f,{\omega }_i)d{\omega }_f$, where “·” indicates matrix multiplication, and the initial orientation ${\omega }_i$ being fixed. The result $\overline{M}$ also transforms under rotations as a second rank tensor, but not generally with the values of the original prototype M. In component notation we can write the average as ${\overline{M}}_{jk}=\left(\int P({\omega }_f,{\omega }_i){\mathcal{R}}_{jklm}({\omega }_f,{\omega }_i)d{\omega }_f\right)M_{lm}$ where ${\mathcal{R}}_{jklm}$ is a fourth-rank tensor relating the two second rank tensors, and we implicitly sum over repeated indices.

We will call $\alpha$ the angle of rotation of the sample about the rotation axis. In our development here, we will specify that the sample be homogeneous, as in a uniform film.

The only $\alpha$-dependence of the measured absorption is the underlying elementary tensorial angle dependence as in Equation (2), owing to the molecular or crystalline structure. However if the sample consists of many crystallites that may themselves be oriented in various nonrandom ways, it is possible for a sample to acquire an additional $\alpha$-dependence. If this is present, it would need to be characterized and corrected for.

In this paper we consider samples in which the only angle dependence is due to the “elementary chromophore” or crystallite, and no additional $\alpha$-dependent structure in the sample is present. This will be the case if the sample is angularly ($\alpha$)-homogeneous as schematically shown in the left schematic diagram in Figure 1. The arrows indicate the orientations of elementary chromophores or crystallites. If this sample were rotated, the underlying chromophore/crystallites would rotate in the way assumed here as in Equation (2). In contrast, in the sample represented by the middle diagram of Figure 1, under rotation by an angle $\alpha$, the orientation of the measured elementary chromophores/crystallites would not change upon rotation as expected. The orientation of the crystallite/chromophore within the beam spot after the rotation by $\alpha$ would be the same as it was before the rotation. The rightmost graphic in Figure 1 shows a more complicated distribution.

This criterion may appear to be more restrictive than it actually is. Any such angular structure that is independent of $\alpha$ does not cause any difficulties, and importantly, such effects can be averaged out by magic angle spinning [18,19].

1.4. Magic Angle Spinning to Average out Anisotropies

In the most basic XAFS measurements, such as uniform polycrystalline or amorphous powders, or alloys, or solutions, it is often presumed that the local structure is isotropically averaged over the sample. The material may consist of small oriented grains, but the common assumption is the orientations of these average out isotropically. This, of course, is not necessarily the case. If particles in a sample have a preferred orientation there will be residual dichroic effects that result in systematic errors. Fortunately there is a simple way to eliminate this problem: magic angle spinning [18,19].

Consider a homogeneous sample with a flat surface that can be rotated around the surface normal. In this subsection we choose our spherical coordinate system so that its polar axis ($\theta =0$) is aligned with the rotation axis $\overrightarrow{\Omega }$. If we then average the signal, which varies according to equation 2, over a full $\varphi$ rotation, we obtain an average that is proportional to

$$\left(M_{11}+M_{22}\right){sin}^2\theta +2M_{33}{cos}^2\theta$$

(3)

Selecting $\theta$, which here is the angle between the spin axis and the electric polarization vector $\overrightarrow{ϵ}$, to be the “magic angle” ${\theta }_m=arccos(1/\sqrt{3})=arctan\sqrt{2}\approx 54.7356deg$ gives a result $\propto (M_{11}+M_{22}+M_{33})$. This is also proportional to the isotropic average

$${\int }_0^{2\pi }{\int }_0^{\pi }\widehat{ϵ}·M·\widehat{ϵ}sin\theta d\theta d\varphi ,$$

which is that normally sought in XAFS experiments.

We conclude that averaging the signal while rotating about an axis that is at an angle ${\theta }_m$ with respect to the electric polarization vector gives a result that is proportional to what would be obtained by averaging over a sphere. This special geometrical property is the reason ${\theta }_m$ is called the “magic” angle, a term originating in NMR spectroscopy.

It is important to realize that this is true regardless of how the elementary absorbing chromophore/crystallite is oriented relative to the sample normal. The magic-angle average above is proportional to the trace (sum of diagonal elements) of M. Since the transpose of a rotation matrix is also its inverse matrix, the rotation transformation of the tensor also is a similarity transformation, and the trace of a matrix is invariant under similarity transformations. This implies that if the elementary chromophore were pre-rotated to some different orientation, its trace, and therefore also the magic-angle average, would be unchanged. The chromophore orientation relative to the surface normal doesn’t affect the magic angle average. This point is sometimes misunderstood, which unfortunately may have limited the productive use of magic angle spinning.

Magic angle spinning eliminates any spurious dichroism there might be owing to residual preferred orientation within the sample. We note parenthetically that only three or more equally spaced angles in $\alpha$ are needed to accomplish this averaging effect, but in this paper we find that we can do much more with it than just calculate the average.

1.5. Full Tensor Magic Angle Spectroscopy

For simplicity and concreteness we take $\widehat{ϵ}=(1,0,0)$, i.e., the polarization along the x direction, and the rotation axis in the $x,z$ plane, so $\widehat{\Omega }=(\sqrt{{\displaystyle \frac{1}{3}}},0,\sqrt{{\displaystyle \frac{2}{3}}})$. We make use of Mathematica’s [20] RotationMatrix function to generate the transformation matrix for a rotation by angle $\alpha$ around the $\widehat{\Omega }$ vector, which can be written as

$$\left(\begin{array}{ccc}{\displaystyle \frac{1}{3}}(2cos\left(\alpha \right)+1)& -\sqrt{{\displaystyle \frac{2}{3}}}sin\left(\alpha \right)& -{\displaystyle \frac{1}{3}}\sqrt{2}(cos\left(\alpha \right)-1)\\ \sqrt{{\displaystyle \frac{2}{3}}}sin\left(\alpha \right)& cos\left(\alpha \right)& -{\displaystyle \frac{sin\left(\alpha \right)}{\sqrt{3}}}\\ -{\displaystyle \frac{1}{3}}\sqrt{2}(cos\left(\alpha \right)-1)& {\displaystyle \frac{sin\left(\alpha \right)}{\sqrt{3}}}& {\displaystyle \frac{1}{3}}(cos\left(\alpha \right)+2)\end{array}\right)$$

(4)

Figure 2. Schematic experimental geometry for ${\theta }_m$ and ${\theta }_c$. $\widehat{k}$ is the beam direction, $\widehat{ϵ}$ is x-ray electric polarization vector. All vectors lie in the $x-z$ plane. The blue line represents the edge of the sample disk with normals respectively along ${\widehat{\Omega }}_m$ and ${\widehat{\Omega }}_c$. These symmetrically straddle the ${45}^{\circ }$ orientation.

Figure 2. Schematic experimental geometry for ${\theta }_m$ and ${\theta }_c$. $\widehat{k}$ is the beam direction, $\widehat{ϵ}$ is x-ray electric polarization vector. All vectors lie in the $x-z$ plane. The blue line represents the edge of the sample disk with normals respectively along ${\widehat{\Omega }}_m$ and ${\widehat{\Omega }}_c$. These symmetrically straddle the ${45}^{\circ }$ orientation.

We then can rotate the M tensor by sandwiching it between this rotation matrix and its transpose to obtain an expression that contains only multiples of the five basis functions $1,cos\alpha ,sin\alpha ,cos2\alpha ,sin2\alpha$. The (Fourier) coefficients of these basis functions themselves depend on specific linear combinations of the M tensor elements. These are given in Table 1 for the ${\theta }_m$ magic angle orientation. We also will find it useful to compute the corresponding coefficients for the rotations around ${\theta }_c$, the complement to the magic angle: ${\theta }_c\equiv {\displaystyle \frac{\pi }{2}}-{\theta }_m$, given in Table 2.

In an experiment one can directly determine the numerical coefficients of each of these basis functions at each energy. Once found, these form a linear system of equations by which the values of the tensor elements of M can be determined. In the general case of no symmetry, six values are needed to fully determine the tensor at each energy, but these equations only provide five. But if an independent measurement is done at another angle (say ${\theta }_c$), then ten numbers are available which is more than enough to determine the 6 unique elements of the M tensor in a least-squares sense. The “extra” parameters can be used to manage experimental uncertainties such as relative starting phase angle for ${\theta }_m$ and ${\theta }_c$ scans, relative normalization of same, background shifts, and similar real-world complications. We will not get into such details here as we aim to demonstrate the potential promise of this technique.

Determining these Fourier coefficients experimentally can be done in many ways. The constant term (the coefficient of the basis function 1) is just the average value, which is what we seek for an isotropically averaged signal, as in ordinary magic angle spinning. They could be determined by analog filtering, sampling followed by simple dot product (as we do here, below), by digital bandpass filtering, or lock-in methods. Most of these can be done relatively inexpensively using modern electronics. Even discrete stepping in angle instead of continuous spinning is a feasible approach. Or the data could be simply recorded and processed afterward, as we illustrate in the section Methods and Materials.

The key point is that only rotation of the sample about the spin axis is needed to determine the full absorption tensor, and potentially can be done in only a little more time than a few normal energy scans. In the next section of this paper we will demonstrate this by doing a series of simulated experiments, by computing the spectra that would be obtained while rotating the sample in this way, treating the result as virtual experimental data; reconstructing the full $M\left(\mathcal{E}\right)$ tensor as a function of energy from it; and to test it, comparing the projections of that predicted tensor along arbitrarily chosen polarization directions with explicit calculations at that polarization, using the reference theoretical code FEFF [21]. We find excellent agreement in all cases.

An example of the 6 unique tensor elements vs energy are shown in Figure 23.

2. Materials and Methods

2.1. Computations Using FEFF

FEFF [21], a very well-established theoretical XAFS code was used to compute simulated XAFS spectra as a function of energy for a series of illustrative structures and appropriately selected polarization vectors. The FEFF8.4 version was used, which is written in Fortran 77. It was compiled in gfortran (GCC15) at optimization level 1 for Mac OS 26.5 on M2 Max MacBook Pro hardware. This is not the most sophisticated of the FEFF series of codes but it is more than sufficient for our purposes here.

The program was run in XANES Full Multiple Scattering mode, which calculates the near-edge region of the XAFS spectrum. XANES often shows interesting variations in structure, but similar calculations can be done with EXAFS, both with multiple coordination shells. Self-consistent potentials were not used here because it is irrelevant for our purpose (it is commented out in the example). The dipole approximation however is made in the FEFF calculations, as we have discussed previously. An example feff.inp file is shown below. A template string of this sort was used to automatically generate a series of jobs in which the positions of all the atoms were rotated by the variable angle $\alpha$ around the spin axis $\widehat{\Omega }$.

For each run, first a prototype local atomic structure was built, with the option to randomly perturb atomic positions so as to break all local point group symmetries, or to preserve various symmetries. Here we use a central Mn atom surrounded by 6 (or 5) oxygen atoms at various locations. No special point group symmetry or orientation is assumed, except as indicated for certain tests shown below.

Here is a sample FEFF input file:

Mn central atom, Oxygen neighbors, distorted octahedral coordination,

polarization vector (1,-2,1). This form is used to validate the computed

spectrum, synthesized from the reconstructed $M$ tensor, along that

direction.   For generating the simulated experimental  data, polarization

is fixed at (1,0,0) for all runs while the atom coordinates are rotated.

-------------------------------------------------------------------------

TITLE Illustration of Magic Angle Spinning

EDGE K 1.0

CONTROL 1 1 1 1 1 1

*SCF 3.0 1

FMS 3.0 1

XANES 8.0 0.05

Polarization  1 -2 1

POTENTIALS

*ipot z label

0 25 Mn

1 16 O

ATOMS

*x y z ipot atom

0. 0. 0. 0 Mn

-1.550676656607681 0.5279844219748255 -0.3677425161328711 1 O

1.7222347900324142 -0.12287594786724987 -0.45252829065425026 1 O

-0.05559267001253487 -1.3949939627430388 -0.07222041064169282 1 O

-0.11900081923005112 2.1574028343591256 -0.11666143758592029 1 O

-0.5190776083591475 0.28615541317596893 -2.5209783909703383 1 O

-0.7590753880744365 0.031062973781723624 2.225967883900838 1 O

END

To simulate the experiments, a series of FEFF [21] 8.4 calculations was run to calculate the spectra as a function of rotation angle $\alpha$ around the spin axis $\overrightarrow{\Omega }$. The same effect presumably could be had more easily by changing the polarization vector specified in the feff.inp file, but this approach is perhaps more intuitive, albeit more cumbersome: it corresponds to physically rotating the sample and the atoms within it. The coordinates of all the atoms were rotated using appropriate rotation matrix for the given $\alpha$ about $\widehat{\Omega }$.

Operationally this was orchestrated within Mathematica [20], with our code that generates a FEFF input file (derived from a template string in the code); then writing out the file; executing FEFF by Mathematica’s RunProcess; reading in the resulting FEFF-generated xmu.dat file; and repeating over $\alpha$ values. This results in a Mathematica data structure consisting of spectra as a function of both energy and rotation angle $\alpha$ about axis $\overrightarrow{\Omega }$. In the results shown here, 60 equally spaced values of $\alpha$ were used; the whole process takes about 18 seconds (run as a serial process) on a MacBook Pro M2 Max laptop computer.

2.2. Determining the Absorption Tensor from the Simulated Experimental Spectra

For a fixed energy value, the data are simply a function that is uniformly sampled in angle $\alpha$ 60 times between 0 and $2\pi$. Such spectra are similar to those in Figure 3; each structure generates a pair of such spectral bundles. The choice of 60 values shown here is representative, but is not critical as long as it is three or more. The basis functions are evaluated at each angle to produce a vector (1D array) and the dot products between these vectors and the sampled data directly give the numerical Fourier coefficients (once normalized by $2/60$ for the trig functions, and $1/60$ for the unit basis function). Other frequency components that might be present in experimental data that are multiples of this base frequency are orthogonal to these, and produce zero. Oversampling in $\alpha$ has the benefit of reducing aliasing effects from angular “noise”.

A set of these numerical coefficients is determined at each energy. They are equated to expressions as given in tables 1 (for ${\theta }_m$) and table 2 (for ${\theta }_c$) and numerically solved for the unknown tensor elements at that energy. The system is overdetermined so the sum of squares of the differences between corresponding tensor elements of the two estimates of the M matrix is numerically minimized, and the two tensor solutions are averaged to produce the final estimate. Alternatively this could be accomplished using a Moore-Penrose pseudo-inverse. This is done at each energy value, so the final result is a tensor-valued function of energy, in the case shown here, on a grid of 100 energy points. The choice of 100 here is not important; for EXAFS it would be several times larger. The computations at each energy are independent of each other. If certain regions of energy are questionable (for example quadrupole effects are present), those can simply be ignored/censored – errors in them don’t propagate to other energy points. The total execution time to construct the estimated tensor for all energies using the precomputed simulated experimental spectra is about 1.5 seconds on a M2 Max MacBook Pro laptop computer.

2.3. Predicting Absorption for Arbitrary Polarization Direction

This final estimate of the dipole absorption tensor $M\left(\mathcal{E}\right)$ is the key thing we are after. It can be analyzed in various ways, compared with DFT or other theory, and can be used to predict the absorption for a hypothetical experiment if the polarization vector were oriented in any chosen direction $\widehat{ϵ}$, simply by evaluating $\widehat{ϵ}·M\left(\mathcal{E}\right)·\widehat{ϵ}$. This is true even for orientations that are experimentally inaccessible, such as direct absorption measurements parallel to the plane of the sample, i.e., perpendicular to $\overrightarrow{\Omega }$. Fluorescence excitation experiments are quite feasible under such circumstances, and FTMAS would afford an opportunity to compare the two modalities. In many cases the structure and angle dependence of the fluorescence spectra are the same as in absorbance, but subtleties can arise depending on the time scales of elementary processes [7,8]. Once computed, calculating the absorbance for arbitrary polarization is virtually instantaneous. We find it is very useful for analysis, simulation, and interactive exploration using Manipulate in Mathematica.

3. Results

Here we present some examples of this procedure on a sequence of simple structures of practical relevance: Manganese-Oxygen coordination complexes, computed using full multiple scattering. The same approach applies equally well for multiple-coordination-shell complexes in the dipole approximation, and for EXAFS.

3.1. Basic Tests

In Figure 4, the averages over all $\alpha$ values of the spectra in Figure 3 are plotted as solid lines. Also plotted (with symbol “x”) is 1/3 of the trace of the reconstructed tensor as a function of energy. The latter agrees precisely with the magic angle average ${\theta }_m$, which is also the isotropic average that is normally sought in XAFS experiments. In contrast, the complementary angle ${\theta }_c$ average is quite different, as expected. The Magic Angle ${\theta }_m$ is indeed special.

All of the following examples show a spectrum that is computed for a specified electric polarization vector $\overrightarrow{ϵ}$, which corresponds to physically orienting the sample so the polarization is in that direction. In other words, they are simulated experimental spectra generated from the precomputed tensors that were separately computed from (simulated) experimental spectra, as described above.

To test these results, we also ran FEFF calculations in which the polarization vector was oriented in that same direction. This was automated similarly to that described above, but the coordinates were left unrotated, and the polarization direction was changed. We use an unnormalized polarization vector for convenience when specifying the polarization direction – it is normalized internally by our code, and also FEFF. The normalization is immaterial.

We find excellent agreement in all test cases between the spectra generated by projecting our reconstructed estimated tensor, and the independent direct calculations using FEFF. In the following plots, the + symbols are the FEFF-calculated benchmark values, and the open symbols are the spectra from our FTMAS-reconstructed tensor. Once the tensor is calculated, calculating the spectra is virtually instantaneous, and very useful for interactive exploration.

Below we present a series of plots, most corresponding to orientations of $\overrightarrow{ϵ}=(1,0,0),(0,1,0),$$(0,0,1),(-\sqrt{2},0,1)$. The choice is $(-\sqrt{2},0,1)$ corresponds to a polarization that is parallel to the planar sample surface, and which is therefore inaccessible in practice in a direct absorption experiment. A key benefit of our approach is that the spectra for any orientation of $\overrightarrow{ϵ}$ can be instantly generated, once the tensor is constructed from the experimental measurements of real data.

We start with our simplest structure, symmetric octahedral coordination. The coordinates (in Å) are given in Table 3, and results shown in Figure 5 and 6. This is a cubic point group symmetry, and the absorption is observed to be independent of orientation. Here only two orientations are shown; all others are identical.

We then break the symmetry by an orthorhombic distortion, where distances in the x, y, z directions are different, but the angles are unchanged, and inversion symmetry is preserved. Coordinates are given in Table 4, and results are shown in Figure 7, Figure 8, Figure 9 and Figure 10.

Figure 6. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\overrightarrow{ϵ}=(-\sqrt{2},0,1)$, octahedral coordination. This polarization is oriented parallel to the sample plane and normally experimentally inaccessible.

Figure 6. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\overrightarrow{ϵ}=(-\sqrt{2},0,1)$, octahedral coordination. This polarization is oriented parallel to the sample plane and normally experimentally inaccessible.

Figure 7. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\widehat{ϵ}=(1,0,0)$, orthorhombic coordination.

Figure 7. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\widehat{ϵ}=(1,0,0)$, orthorhombic coordination.

Figure 8. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\widehat{ϵ}=(0,1,0)$, orthorhombic coordination.

Figure 8. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\widehat{ϵ}=(0,1,0)$, orthorhombic coordination.

Figure 9. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\widehat{ϵ}=(0,0,1)$, orthorhombic coordination.

Figure 9. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\widehat{ϵ}=(0,0,1)$, orthorhombic coordination.

Figure 10. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\overrightarrow{ϵ}=(-\sqrt{2},0,1)$, orthorhombic coordination. This polarization is oriented parallel to the sample plane and normally experimentally inaccessible.

Figure 10. Spectrum projected from synthesized absorption tensor (circles) vs direct FEFF XANES theoretical calculation (+) for electric polarization vector $\overrightarrow{ϵ}=(-\sqrt{2},0,1)$, orthorhombic coordination. This polarization is oriented parallel to the sample plane and normally experimentally inaccessible.

We then remove one of the short bonds to leave 5-coordinate structure. This breaks inversion symmetry in the x-direction. Coordinates are given in Table 5, and results are shown in Figure 11, Figure 12, Figure 13 and Figure 14.

We then restore the 6-coordinate structure, but perturb the positions of each atom pseudo-randomly in x, y, z to break all point group symmetries. For reproducibility a random seed (1538) is used; two examples are shown. Coordinates are given in Table 6, and results are shown in Figure 15, Figure 16, Figure 17 and Figure 18.

A repeat randomly generated structure with random seed 7237 is also shown, with coordinates in Table 7, and results are shown in Figure 19, Figure 20, Figure 21 and Figure 22.

It can be seen that the reconstructed spectra in all cases are in excellent agreement with the benchmark FEFF calculations, illustrating accuracy of the reconstructed tensor.

Figure 23. The six unique elements of the reconstructed tensor vs energy for distorted octahedral structure, seed 7237.

Figure 23. The six unique elements of the reconstructed tensor vs energy for distorted octahedral structure, seed 7237.

3.2. Noisy Spectra

To test the robustness of this procedure with real-world noisy spectra, we added synthetic randomly generated noise (uniformly distributed between $\pm 0.05$) to each of the spectral points for all angles in the last example, and processed it identically to the other data sets. This $\pm 5\%$ noise level (edge steps are $\approx 1$) is considerably larger than what should be expected from experimental data, but even in such a case the tensor reconstruction is robust, as shown in Figure 24. The visible noise level shown simply reflects the added spectral noise, and no systematic shift from the FEFF-calculated reference value is seen.

Another example is given in Figure 25. This is a different projection than the previous ones, because it shows other structure (exposed using Manipulate to explore spectrum vs orientation). The reconstructed projected tensor again agrees well with the FEFF calculation at that orientation, modulo the added noise.

This indicates the Fourier decomposition and oversampling in angle worked well. This makes sense since the reconstruction fundamentally relies on sufficiently-accurate determination of the Fourier coefficients of the angular behavior. Each energy value is independent of the others. Sampling at 60 different angular values spreads much of the added noise spectrum outside the frequency range that is used for the reconstruction process, so that it does not alias to the lower angular frequencies that are used in the Fourier Decomposition.

4. Discussion

Our results show that Full Tensor Magic Angle Spectroscopy (FTMAS) is a promising new potential experimental modality to determine full dipole optical absorption tensors in real-time. An example of the tensor elements is shown vs energy in Figure 23.

Here we apply the method to the case of X-ray Absorption Fine Structure Spectra, but the dipole tensor character renders it broadly applicable to other wavelengths of light such as UV-Vis, IR, microwave. FTMAS does require a flat sample but many samples already have this character. Given the importance of thin films in industry, for many purposes e.g., semiconductors and perovskite photovoltaics, it may be useful for material characterization and process control.

FTMAS can provide information that is otherwise experimentally inaccessible – such as absorption spectra measured along any direction parallel to the sample surface – and by doing measurements that are relatively fast and easy. FTMAS has the potential to provide up to six times the information than is normally available in isotropically averaged experiments, and can make full-tensor measurements far more easily than laborious angle measurements using four-circle goniometry, which also has its own systematic errors.

We have found that even rather noisy data are handled robustly by this approach: it mostly relies on accurate Fourier decomposition of the angular dependence, for relatively low frequencies. This can be ensured by oversampling in angle to reject spurious noise components, or by analog filtering before sampling. There are other various experimental details that will need to be worked out in practice. The data acquisition seems relatively straightforward, but modulation (spin) frequency should be chosen to avoid noise sources, and be compatible with time constants in the data acquisition system; surface normal of sample must be precisely aligned with the spin axis); sample itself must be very uniform; shifts of sample position when changing between ${\theta }_c$ and ${\theta }_m$ measurements must be controlled and minimized; rotation rate must be stable; and various other experimental details must be addressed. But they can be. Some non-idealities such as shift in starting phase angle and normalization (owing to the different projected sample thickness presented to the beam) between the ${\theta }_m$ and ${\theta }_c$ measurements can calculated or absorbed using the extra parameters available: 10 quantities are available and 6 are needed. Systematic errors such as sample thickness variations as a function of $\alpha$ (which could be caused by linear thickness gradient across the sample) also must be anticipated and prevented, or corrected for.

The description we present here presumes that it is direct absorption measurements that are to be made. Fluorescence excitation spectra are also commonly used to measure XAFS, and these have similar dipolar character, to the extent the initial absorption is de-correlated with the subsequent emission. In many cases this approach will work as well with fluorescence excitation spectra, but there are nuances in this comparison that need to be considered [7,8], and this will need to be tested. The more general photon-in/photon-out aspects of Resonant Inelastic X-ray Scattering are beyond the scope of the current paper.

5. Conclusions

We have proposed Full Tensor Magic Angle Spectroscopy (FTMAS) as a promising new potential experimental modality for measuring X-ray Absorption Spectra, and other optical (UV-Vis, IR, microwave, THz), spectra in the dipole approximation. Our conclusions in this paper are based on a combination of theory and numerical simulations. Experimental R&D and testing will be needed to develop it into a practical technique. We hope others will be interested in pursuing it.

FTMAS works by measuring the time (or simply, angle) dependence of the optical (e.g., x-ray) absorption signal as a homogeneous planar film sample is rotated at a constant rate with its axis set at the magic angle $arctan\left(\sqrt{2}\right)=arccos\left({\displaystyle \frac{1}{\sqrt{3}}}\right)$ (and its complement) relative to the electric polarization vector of the linearly polarized photon beam. The Magic Angle is special – if the spin axis makes this angle with respect to the x-ray polarization vector, uniformly averaging over the angle of rotation $\alpha$ about the spin axis gives a signal proportional to the 3D isotropic average spectrum, which is 1/3 of the invariant trace of the dipole absorption tensor.

For low-symmetry samples, the full dipole tensor signal offers up to six times more information (i.e., linearly independent spectra) than the usual isotropically averaged signal. In contrast, for samples with cubic symmetry, no additional information is provided: the signal is isotropic. Deviations from cubic symmetry should be readily apparent, and often may be of interest, though.

The examples given here are to X-ray Absorption Fine Structure (XAFS) spectroscopy using the linearly polarized beams from synchrotron radiation sources. The procedure is robust against noise in the data. From the DC and audio-frequency time dependence of the signal five Fourier components can be determined: a constant value (at each energy), corresponding to the isotropic signal that is normally sought and analyzed in XAFS measurements, plus coefficients of the sine and cosine components at the spin frequency, and twice that frequency. From the time (or angle) dependence, these components are straightforward to directly determine from experimental data as a function of energy during a scan. In our initial implementation, the angles between the spin axis and the x-ray polarization vector are chosen to be the magic angle ${\theta }_m=arccos\left({\displaystyle \frac{1}{\sqrt{3}}}\right)=arctan\left(\sqrt{2}\right)$, and its complement ${\theta }_c={\displaystyle \frac{\pi }{2}}-{\theta }_m=arcsin\left({\displaystyle \frac{1}{\sqrt{3}}}\right)=arctan\left({\displaystyle \frac{1}{\sqrt{2}}}\right)$.

In the geometry considered here, with the spin axes in the plane of the beam direction $\overrightarrow{k}$and the polarization vector $\overrightarrow{ϵ}$, these two angles symmetrically straddle the ${45}^{\circ }$ degree orientation that is commonly used in fluorescence measurements. Simple equations are given to relate these numerical values to linear combinations of the tensor elements, resulting in a linear system that can be solved for the tensor elements at each energy. For the general case, with no assumed sample symmetry, six real numbers are required to specify all components of a symmetric cartesian tensor in 3D. The sixth value, plus four others can be obtained by measuring with a second spin axis, providing ten numbers per energy produced in two scans. The information from the two angle measurements easily can be merged using a least-squares criterion or Moore-Penrose pseudo-inverse. The available four extra numbers per energy point can be used to compensate for real-world experimental uncertainties such as incomplete polarization of the beam, misalignment with respect to the polarization vector, differences in normalization or precise starting phase of rotation between the two angle measurements, or other experimental issues related to sample thickness gradients, mounting misalignments, crystal glitches, etc. If necessary, a third discrete angle in addition to the magic angle and its complement easily could be added.

This paper introduces the concept to what we believe is a promising potential new technique. We hope the FTMAS method, when fully developed, will be of lasting utility in broad areas of science and technology.