PyCSFex - an extensible Python three package for calculating x-ray structure factors in complex crystals

At a Glance

Metadata	Details
Publication Date	2023-10-05
Authors	John P. Sutter, J.A. Pittard, Jacob Filik, Alfred Q. R. Baron
Institutions	SPring-8, Diamond Light Source
Analysis	Full AI Review Included

PyCSFex Analysis: MPCVD Diamond Solutions for Advanced X-ray Optics

This technical documentation analyzes the requirements set forth in the research paper “PyCSFex: An extensible Python 3 package for calculating X-ray structure factors in complex crystals” and aligns them with the advanced material capabilities of 6CCVD.

Executive Summary

The accurate calculation of X-ray structure factors is critical for designing high-performance diffractive X-ray optics, including monochromators, phase retarders, and high-resolution energy analyzers.

Core Challenge: Designing next-generation X-ray optics requires materials with superior thermal and mechanical stability, necessitating precise modeling of temperature-dependent, anisotropic crystal structure factors (e.g., Debye-Waller factors).
Material Context: While silicon and germanium are common, diamond is recognized for its superior thermal conductivity and low thermal expansion, making it the material of choice for high-power beamline filters (< 5 keV) and phase retarders.
Diamond Opportunity: The paper notes that large, defect-free diamond crystals are traditionally “difficult to produce.” 6CCVD specializes in high-quality MPCVD Single Crystal Diamond (SCD) and Polycrystalline Diamond (PCD), directly addressing this supply limitation.
Thermal Modeling Requirement: Accurate structure factor calculation (as facilitated by PyCSFex) requires precise material parameters across a wide temperature range (13 K to 838 K), demanding materials with predictable and stable properties.
6CCVD Value Proposition: We provide the high-purity, custom-dimensioned, and precisely polished diamond substrates necessary to realize the next generation of high-resolution, thermally stable X-ray optics.
Customization: 6CCVD offers custom metalization and polishing (Ra < 1nm for SCD) essential for integrating diamond optics into cryogenic or high-flux thermal management systems.

Technical Specifications

The following hard data points are extracted from the research paper, highlighting the extreme precision and material stability required for advanced X-ray optics applications.

Parameter	Value	Unit	Context
X-ray Energy (Mo K$\alpha_1$)	17.479	keV	Target energy for backscattering analysis
X-ray Energy (Cu K$\alpha_1$)	8.048	keV	Target energy for backscattering analysis
Required Energy Resolution (Quartz)	~4	meV	Demonstrated high-resolution performance
Minimum Filter Energy (Diamond Use Case)	< 5	keV	For high-power beamline filtering (low absorption)
Quartz Operating Temperature Range	13 to 838	K	Range over which non-linear thermal dependence must be modeled
Silicon Thermal Expansion Coefficient	Zero	@ 120 K	Achieved when cooled by liquid nitrogen
Temperature Sensitivity (Quartz, 17.479 keV)	-3.4 to -15.9	mK/meV	Required temperature stability for energy tuning
Maximum PCD/SCD Plate Diameter	Up to 125	mm	6CCVD capability for large-area optics
SCD Polishing Specification	< 1	nm	6CCVD capability (Ra) for optical grade surfaces

Key Methodologies

The research focuses on computational methods (PyCSFex) for modeling crystal performance, emphasizing the need for accurate, temperature-dependent material inputs to predict the structure factor F(hkl).

Structure Factor Calculation: The core methodology involves calculating the complex-valued structure factor F(hkl) using the total atomic scattering factor $f_j(q)$ and the Debye-Waller factor $D_j(q)$ for each atom $j$ in the unit cell (Equation 3b).
Anisotropic Thermal Modeling: Unlike simple cubic crystals (Si, Ge) which often assume isotropic thermal vibration, complex crystals (like $\alpha$ quartz) require the use of thermal displacement ellipsoids, characterized by a symmetric 3 x 3 matrix $\beta$, to calculate the anisotropic Debye-Waller factor (Equation 4).
Non-linear Temperature Dependence: Lattice parameters, atomic positions, and thermal ellipsoids must be modeled using non-linear fitting functions (Equation 6) across the full operating range (13 K to 838 K) to ensure accuracy, especially when tuning optics via temperature variation.
Convention Management: The software package must accommodate multiple crystallographic conventions (origin placement, handedness, axis orientation) to prevent gross errors in structure factor calculation, particularly for chiral crystals like $\alpha$ quartz.
Backscattering Search: A systematic search across numerous Bragg reflections (hkil) is performed to identify optimal backscattering conditions for high-resolution energy analyzers at specific X-ray energies (e.g., Mo K$\alpha_1$ and Cu K$\alpha_1$).

6CCVD Solutions & Capabilities

The research confirms that diamond possesses the ideal combination of thermal conductivity and low thermal expansion necessary for high-power X-ray optics and low-absorption phase retarders. 6CCVD’s MPCVD capabilities directly supply the high-quality material required to overcome the traditional difficulty in sourcing large, defect-free diamond crystals.

Applicable Materials

Application Requirement	6CCVD Material Recommendation	Key Material Properties
High-Resolution Monochromators/Analyzers: Requiring near-perfect crystal structure and ultra-low surface roughness.	Optical Grade Single Crystal Diamond (SCD)	Highest purity, Ra < 1nm polish, thickness 0.1µm - 500µm. Ideal for backscattering optics where crystal perfection is paramount.
High-Power Filters/Substrates: Requiring maximum thermal dissipation and large area coverage (e.g., for X-rays < 5 keV).	Electronic Grade Polycrystalline Diamond (PCD)	Plates up to 125mm diameter, high thermal conductivity (> 2000 W/mK), Ra < 5nm polish available for inch-size wafers.
Phase Retarders: Requiring low absorption and precise thickness control.	High-Purity SCD or Thin PCD Films	Thickness control from 0.1µm up to 500µm. Low absorption coefficient is superior to Si/Ge for polarization control.

Customization Potential

To meet the stringent requirements of high-resolution X-ray optics, 6CCVD offers comprehensive customization services:

Custom Dimensions: We supply PCD plates up to 125mm in diameter and SCD substrates up to 10mm thick, allowing for the fabrication of large-area optics necessary for high-acceptance analyzers.
Precision Polishing: We guarantee ultra-smooth surfaces, achieving Ra < 1nm on SCD and Ra < 5nm on inch-size PCD, minimizing surface damage and maximizing reflectivity.
Integrated Metalization: For applications requiring robust thermal management (critical for maintaining mK/meV stability), 6CCVD provides in-house metalization stacks (Au, Pt, Pd, Ti, W, Cu). This is essential for creating reliable thermal contacts for cryogenic cooling systems (e.g., liquid nitrogen cooling mentioned for Si at 120 K).
Custom Substrate Orientation: We offer precise crystal orientation and cutting services to match specific Bragg reflection requirements (hkl) and optimize performance for backscattering or phase retardation.

Engineering Support

The complexity highlighted by the PyCSFex paper—specifically the need for accurate thermal and structural inputs—underscores the necessity of expert material consultation.

Material Specification for Modeling: 6CCVD’s in-house PhD team provides detailed material specifications, including impurity levels, thermal conductivity data, and lattice parameters, which are essential inputs for computational tools like PyCSFex.
Thermal Management Design: We assist engineers in selecting the optimal diamond grade (SCD vs. PCD) and metalization scheme to ensure the required temperature stability (mK/meV range) is maintained under high thermal load conditions typical of synchrotron beamlines.
Replication and Extension: We offer consultation for projects aiming to replicate or extend the high-resolution X-ray energy analyzer work discussed in the paper, ensuring the chosen diamond material meets the structural perfection and thermal requirements for optimal Bragg reflection performance.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

Near-perfect diffracting crystals have many uses in x-ray optics including as monochromators, energy analyzers, and phase retarders. The usefulness of a particular Bragg reflection is often related to its angular acceptance and efficiency, as is determined by the reflection’s structure factor. Silicon crystals, which belong to the same face-centered cubic space group 𝐹𝑑3̅𝑚 as germanium and diamond, are readily available in large and highly pure ingots. Combined with their high thermal conductivity and low thermal expansion, this makes them suitable for synchrotron x-ray beamlines. However, less symmetric trigonal crystals such as sapphire, lithium niobate, and α-quartz offer a better choice of high-energy-resolution Bragg reflections near backscattering with less likelihood of parasitic Bragg reflections. Because these crystals’ atoms vibrate anisotropically and shift relative to each other with temperature, the temperature dependence of their structure factors is not a given by a simple Debye-Waller factor. Also, many crystal structures may be described by several different conventions of origin and lattice vectors. A Python three software package, PyCSFex, is presented here for the rapid calculation of large numbers of structure factors of any crystal described in any convention. It can run on its own or as part of an already existing software package. Users can extend the package to new crystals by writing their own material files. α-Quartz is chosen as an example because it has already been successfully used in backscattering x-ray energy analyzers and presents the complexities previously mentioned.