1.

INTRODUCTION

For many years, X-ray beamlines at synchrotrons and free electron lasers have used silicon crystal optics because of the combination of highly favorable thermal, mechanical, and material properties that such crystals offer. Because silicon is a critical material for the manufacture of semiconductors, large ingots up to 125 mm in diameter and 1 m long with extremely low impurity and defect concentrations are regularly grown by industrial suppliers using the float-zone method. Silicon’s high thermal conductivity¹ allows it to dissipate high thermal loads, and because its linear thermal expansion coefficient is zero² at 120 K, silicon crystals suffer relatively little distortion under such loads when cooled by liquid nitrogen. Silicon crystals are hard enough to take a high polish, yet they can still be cut and shaped by diamond saws, and after fabrication, surface damage can be removed by etching with aqueous mixtures of nitric and hydrofluoric acid. Finally, silicon crystal optics remain undamaged through years of operation. Therefore, silicon crystals are the preferred material for X-ray monochromators, especially those exposed to beams of high power.

Germanium and diamond crystals are also sometimes used as X-ray diffracting optics, though each has some disadvantages compared with silicon. Germanium absorbs X-rays more strongly, has a higher linear thermal expansion coefficient², and has lower thermal conductivity¹ than silicon. Nonetheless, germanium crystals are frequently used along with silicon crystals as high-resolution energy analyzers for measurements of X-ray fluorescence and inelastic scattering, which do not impose high thermal loads. Large and defect-free crystals of diamond are difficult to produce and also difficult to machine and etch. They are therefore unsuitable for making crystal energy analyzers. However, their low thermal expansion³ and high thermal conductivity⁴ make them attractive as filters for removing X-rays of energy < 5 keV, which at some beamlines are not used and would impose intolerable thermal loads on optics downstream. Moreover, their low absorption makes them the material of choice for phase retarders, which alter the polarization state of an X-ray beam and thus are useful for measurements of circular dichroism, as well as for multiplexing monochromators.

All three materials, silicon, germanium and diamond, form crystal structures whose symmetry operations are those of the face-centered cubic space group . The set of symmetry operations is remarkable for its large number of rotations, mirror planes, and glide planes, as well as for including inversion (that is, these materials are centrosymmetric)⁵. As a result, the number of atomic planes in these crystals that are related by symmetry and thus have the same spacing d is much larger than in crystals whose structures have fewer symmetry operations. This particularly affects the design of crystal energy analyzers, which achieve their best resolution if they diffract X-rays close to backscattering^6,7. Since the energy of the X-rays to be analyzed is often fixed, for example at an emission line or absorption edge of some element, the likelihood that silicon and germanium can diffract those X-rays near backscattering is low. Furthermore, diffraction in backscattering from these materials is accompanied by parasitic reflections that remove the desired flux and affect the energy resolution. For instance, when the (12 4 0) atomic planes of silicon and germanium diffract in backscattering, 22 additional reflections are also excited⁸. For these reasons, non-cubic crystals that have fewer symmetry operations are being considered as possible materials for crystal energy analyzers of X-rays. Sapphire, lithium niobate, and α (or “low”) quartz have attracted the most interest⁹; they are described in Table 1. All these form trigonal crystal systems. Note that although the space groups of sapphire and lithium niobate appear similar, they are quite different. The crystal structure of sapphire is centrosymmetric, whereas that of lithium niobate is polar. The latter is therefore ferroelectric while the former is not. It is also notable that α quartz is piezoelectric; as is possible because it is also non-centrosymmetric.

Table 1:

Crystals with trigonal systems that are being considered for use as high-resolution X-ray energy analyzers. Names of space groups are as listed in the International Tables for Crystallography5. The two space groups in which the crystal structure of α quartz may exist are both chiral and are enantiomorphs (mirror images) of each other.

α quartz crystals synthetically grown by the hydrothermal process¹⁰ are commercially available. Their low defect concentration has been demonstrated^11-14. They have already been put to work as spherically diced analyzers for the Ni Kα emission line and the Ir L3 absorption edge^15,16 and as a flat crystal analyzer accompanied by a collimating Montel optic after the sample for the Ir L3 absorption edge¹⁷. In the last example, the energy resolution of the analyzer was estimated to be ~4 meV. Sapphire and lithium niobate crystals have also been tested, and although the samples currently available do not yet attain such a low concentration of defects⁹, future specimens may benefit from improved fabrication methods. Therefore, the development of methods for determining the performance of these materials as crystal energy analyzers of X-rays will be well worth the effort. In the future other crystalline materials may find applications as X-ray optics, and these methods will have to be extended to include them. Finally, for ease of use, these methods should be readily integrated with currently existing software packages for X-ray optics simulations, such as OASYS¹⁸ and SRW¹⁹.

However, to achieve this goal, crystals more complex than silicon, germanium, and diamond will have to be treated. In general, their thermal expansion will be anisotropic, as will the thermally induced vibration of their atoms. In addition, whereas the relative positions of the atoms in silicon, germanium, and diamond are independent of temperature, the same is not true for all crystals. Because crystal X-ray energy analyzers are often tuned by deliberately varying their temperature, all these factors need to be considered if the analyzer’s performance is to be predicted accurately.

A difficulty well known to crystallographers, but less so to many designers of X-ray optics, is that many crystal structures can be defined by multiple choices of origin and lattice vectors. A simple example is the choice between two possible coordinate origins for silicon, germanium, diamond, and other crystal structures belonging to the face-centered cubic space group . All these structures have a basis of two atoms separated by one-fourth of the body diagonal of the cubic unit cell. The origin may be placed either on the first atom of the basis or at the inversion center between the two atoms of the basis, and there is no universal agreement on which choice is better. For more complex crystal structures, the number of possible descriptions can be even greater, and though proposals for standardizing crystallographic data have been published^20,21, none has been completely adopted. α quartz is an extreme example that has been discussed at length^22-24.

The structure factors of a crystal’s Bragg reflections are a critical parameter for the design of diffracting X-ray optics, and they depend on both the temperature and the atomic positions. A software package PyCSFex (“Python crystal structure factor extensible”) for calculating X-ray structure factors in any crystal while leaving users free to choose their preferred description of the crystal structure has recently been published²⁵. It is available at https://github.com/DiamondLightSource/PyCSFex. Each crystal is described by a material file written by a user and processed by a set of core modules. Material files have been provided for silicon, germanium, diamond, and α quartz. The package is written entirely in Python 3 for easy integration with many already released software modules. Some less commonly discussed aspects of the theory on which PyCSFex is based are discussed below, with α quartz being used as an example.

2.

THEORY

2.1

Definition of structure factors of crystals

Crystals are indispensable for studies of all types of materials because their long-range periodicity on scales comparable to X-ray wavelengths enables them to diffract X-rays with high efficiency provided Bragg’s Law is fulfilled:

where n is a positive integer giving the order of the diffraction, λ is the wavelength of the X-rays, d is the spacing of a particular set of parallel atomic planes in the crystal, and θ is the “Bragg angle” between the atomic planes and the X-ray beam. For every crystal one may define a set of lattice vectors a, b, and c from which reciprocal lattice vectors a*, b*, and c* can be determined:

Each set of atomic planes is labelled with Miller indices (hkl). The vector H = ha* + kb* + lc* is perpendicular to all the planes in this set, and d = 1/|H|. The crystal’s lattice vectors define a unit cell whose volume V = |a · b × c|. If the unit cell is hexagonal, the Miller-Bravais indices (hkil) where i = −(h + k) are often used to make the symmetry of different sets of planes more obvious.

The characteristics of the Bragg reflection from a set of atomic planes (hkl) are given by the structure factor, which is the complex-valued scattering amplitude of one unit cell of the crystal in units of scattering from a single free electron. The structure factor depends on the momentum q transferred to the photon by the scattering process. Bragg reflection can occur if q is one of the crystal’s reciprocal lattice vectors. If the phase term of a plane wave of wave vector k is exp(−2πik · r), the structure factor at momentum transfer q is

where j denotes a particular atom within the unit cell, f_j(q) is the total atomic scattering factor of the jth atom, D_j(q) is the Debye-Waller factor of the jth atom, and r_j = X_ja + Y_jb + Z_jc is the position of the jth atom.

At a Bragg reflection from the atomic planes (hkl), q = ha* + kb* + lc*:

Some authors use exp(+2πik · r) as the phase term of a plane wave; if this is chosen, the value of F(hkl) is the complex conjugate of that defined above.

The importance of accurate calculations of structure factors in the design of diffracting crystal optics, particularly when used as monochromators, energy analyzers, and phase retarders, should be clear. The angular acceptance and peak reflectivity of a Bragg reflection depend critically on its structure factor, as is shown by many standard texts^26-28. Phase retarders, which depend on the birefringence of a crystal oriented to an angle close to a Bragg reflection, also require the structure factor of the Bragg reflection to be known for the alignment’s precision to be predicted accurately.²⁹

2.2

General thermal vibration of the atoms in a crystal

The thermal vibration of the atoms in a crystal enters the calculation of X-ray structure factors through the Debye-Waller factor D_j. In silicon and germanium, the amplitude of this vibration for all atoms is generally assumed to be isotropic and its temperature dependence is usually characterized by respective Debye temperatures of 543 ± 8 K and 290 ± 5 K³⁰. For a quartz, however, this simple assumption fails. Both Le Page et al³¹ and Kihara³² agree that the thermal vibrations of both the Si and the O atoms in α quartz are strongly anisotropic. The correct calculation of the Debye-Waller factor for each atom of α quartz therefore requires the use of the thermal displacement ellipsoids, which are explained by Downs³³. An ellipsoid is characterized by a symmetric 3 × 3 matrix β. The Debye-Waller factor D of an atom with this ellipsoid for a Bragg reflection from the (hkl) atomic planes of the crystal is given by

Often reference works give the thermal ellipsoid matrix β for only one atom of each species. If S is the 3 × 3 matrix of the symmetry operation that transforms the lattice coordinates of this atom into those of another atom in the crystal, the thermal ellipsoid of the second atom is described by the symmetric 3 × 3 matrix

where the superscript t denotes the matrix transpose.

2.3

Nonlinear temperature dependence of crystal structure

For sufficiently small temperature ranges, the temperature dependence of the lattice parameters can be taken as linear, while the temperature dependence of the atomic positions and the Debye-Waller factors can be neglected. However, this is no longer possible even for silicon, germanium, and diamond if the structure factors are to be calculated from the boiling point of liquid nitrogen at atmospheric pressure (77 K) to room temperature, which is the usual operating range of diffracting crystal optics for X-rays. In α quartz, the lattice parameters, atomic positions, and thermal displacement ellipsoids vary highly non-linearly over the temperature range from 13 K to 838 K, which is just below the transition temperature T_c = 846 K at which trigonal α (low) quartz transforms into hexagonal β (high) quartz. References are provided in Table 9 of Sutter et al²⁵. It was found there that the temperature dependence of these quantities followed the general form

where the values of the fitting parameters f₀, P, Q, and n are given in Table 10 of Sutter et al²⁵ for a right-handed coordinate system in dextro α quartz in the z(+) setting, which will be explained below. Conversions to other settings are given in Table 8 of that paper. Figure 1 shows how the theoretical fit in Equation (6) matches the experimental data on the lattice parameters a and c of α quartz.

Figure 1:

Temperature dependence of the lattice parameters a (left) and c (right) of α quartz over the range 13-838 K. The theoretical fit was obtained by fitting Equation (6) to the experimental data. See text for details.

3.

COMPUTATION AND CODING

Evidently the accurate and rapid computation of X-ray structure factors in crystals, which is vital for the design of diffractive X-ray optics, needs to account for the diversity of temperature dependencies and descriptions of the many crystal structures that may be encountered. Moreover, even when a widely respected authority speaks in favor of a particular description for a given crystal structure, there may be valid reasons to use a different one. Once again, α quartz provides an example. As mentioned before, if one is guided by the International Tables for Crystallography⁵, then in a right-handed coordinate system, the origin of the α quartz structure should be placed so that the first silicon atom is at coordinates in the dextro structure (space group P3₂21) and in the laevo structure (space group P3₁21). However, at 846 K, the dextro α quartz structure transitions to left-screw β quartz (space group P6₂22), which according to the International Tables for Crystallography should have its first silicon atom at . Likewise, the laevo α quartz structure transitions to right-screw β quartz (space group P6₄22). Again, the International Tables for Crystallography would lead one to place the first silicon atom at . This leads to a discontinuity in the c coordinate of the first silicon atom across the transition from α quartz to β quartz. Donnay and Le Page’s choice of (u, 0,0) as the coordinates of the first silicon atom in both dextro and laevo α quartz avoids this problem and was therefore adopted by Glazer²⁴ despite the high standing of the International Tables of Crystallography.

The software package PyCSFex does not attempt to take sides in these longstanding debates among crystallographers, except to require that a right-handed coordinate system be used in all cases. Other than that, it allows individual users to describe their crystals in the way that is most comfortable for them. It aims for the easiest possible application to new crystals and for the fastest possible calculation of large numbers of structure factors. Several strategies were employed to reach these goals. The first was the choice of the Python 3 programming language for writing the code. Python 3 is widely used and understood, and it makes PyCSFex compatible with OASYS¹⁸ and SRW¹⁹ as well as allowing it to function independently. The second was the modular architecture of the code. The modules of PyCSFex are separated into three clearly defined categories:

The third strategy was to avoid redundant calculations. For example, temperature-dependent parameters such as lattice parameters, atomic positions, and thermal vibrations are recalculated only when the temperature is changed. Also, each class has a method to update its attributes when required, thus avoiding the need to create a new object of that class with the new attributes.

4.

RESULTS

In the following, dextro α quartz in the z(+) setting is assumed. If laevo α quartz in the z(−) setting is assumed, the indices (hkil) are replaced by . A right-handed coordinate system is always chosen.

4.1

Comparison of isotropic and anisotropic models of thermal vibration in α quartz

The anisotropic thermal vibration of the atoms in α quartz has already been mentioned and the method for calculating the Debye-Waller factors from its characteristic thermal ellipsoids has been explained. However, the mathematical complexity of thermal ellipsoids makes it worthwhile to see how much error would be introduced to the structure factors if the thermal vibration were treated as isotropic. One simple approximation is an isotropic Debye model with a Debye temperature of 470 K as given by Gray⁴⁴. The four Bragg reflections , and are interesting because they cover a wide range of strengths from very intense to very weak. Furthermore, at the transition from α quartz to β quartz, the first pair of these reflections becomes symmetrically equivalent, so that their structure factors become equal. The same is true for the second pair of these reflections. A set of calculations was therefore performed using two separate material files, one for the isotropic approximation and the other for the full anisotropic treatment.

Figure 2 shows the results of calculations with PyCSFex with 10 keV X-rays. For the and Bragg reflections, the isotropic model of thermal vibration consistently gives higher estimates of the reflection’s strength. The discrepancy remains below 1% up to 530 K for and up to 412 K for , but as the temperature increases further, the isotropic model deviates steadily more from the anisotropic model. At the highest measured temperature of 838 K, the discrepancy has increased to 5.3%. This is still small, but for the very weak reflection , the disagreement between the two thermal models is very large, especially above 600 K. It is noteworthy that in both thermal models the structure factor of passes through a minimum, though in the isotropic model the minimum is at 562 K, while in the anisotropic model the minimum is at 536 K. The stronger reflection is much less affected by the difference in the thermal models, the largest discrepancy being only 4.3% at 643 K. Here, however, the isotropic model provides a lower estimate of the reflection’s strength than the anisotropic model.

Figure 2:

Temperature dependence from 20 K to 838 K of the squared magnitude of the structure factors of the , , and Bragg reflections of dextro α quartz in the z(+) setting with X-rays of 10 keV energy.

5.

CONCLUSIONS

Optics designers wishing to build diffractive X-ray optics such as monochromators, phase retarders, and energy analyzers must accurately calculate the X-ray structure factors of a wide variety of crystals. Even though the cubic crystals of silicon, germanium, and diamond remain by far the most popular materials for such optics because of their favorable thermal and mechanical properties, several trigonal crystals have also attracted notice for specific applications such as high-resolution energy analyzers. Sapphire, lithium niobate, and especially α quartz are being evaluated for this purpose. Of these, the material in which synthetic crystals of the best quality are available is α quartz, which therefore has already been used in experimental condensed matter studies. The use of still other crystals for X-ray energy analysis, such as topaz (orthorhombic), potassium acid phthalate (orthorhombic), and muscovite (monoclinic), has been recorded in the literature as well. However, neither the modeling of the temperature dependence nor the choice of convention used to describe the crystal structure is consistently made clear in current structure factor calculations that are aimed at X-ray optics designers. As the discussion of the obverse and reverse settings of α quartz proves, such lack of clarity can lead to gross errors. A new Python 3 package named PyCSFex has been published so that users can choose their own preferred thermal models and conventions, because even though much work has gone into the standardization of descriptions of crystal structures, some users may still have sound reasons to select a non-standard description. PyCSFex consists of a set of stable core files that are not changed, a set of material files written by the user to describe each crystal structure, and a set of scripts written by the user that apply the core and material files to carry out the desired calculations. Different thermal models and descriptions of the same crystal structure can be written in separate material files, thus minimizing the possibility of confusion. Examples of structure factor calculations on α quartz, which has an especially complex and challenging crystal structure, have been provided to demonstrate how PyCSFex can help solve practical problems in X-ray diffractive optics design.