Introduction

This document serves as a scientific and technical guide for the powder5 web application, a tool for the analysis of powder X-ray diffraction (PXRD) data via whole-pattern fitting. This technique, also known as pattern decomposition, is a crucial method in materials science and crystallography for refining structural and microstructural parameters when a complete structural model is either unknown or unnecessary.

The application facilitates the extraction of precise lattice parameters, peak profile information, and integrated intensities of Bragg reflections. It implements two principal decomposition algorithms: the iterative Le Bail method for rapid and stable convergence, and the simultaneous Pawley method for rigorous, unbiased intensity extraction. Peak profiles are modeled using phenomenological functions, including a versatile Simple pseudo-Voigt, an Asymmetric Split pseudo-Voigt, and a physically rigorous Anisotropic model (TCH) based on the Thompson-Cox-Hastings formulation with a Stephens model for anisotropic line broadening. The background is modeled using a monotonic cubic spline interpolation between user-defined points.

Getting Started: Data Input

Analysis commences with the loading of a powder diffraction data file. The application is designed to automatically parse numerous common ASCII-based file formats from major instrument manufacturers and standard crystallographic software.

• Supported Formats: Built-in parsers are included for Bruker (.brml, .uxd), PANalytical (.xrdml), Rigaku (.ras), Philips (.udf, .rd, .sd), GSAS (.esd, .gsa, .std, .xra), and Jade (.mdi). Format detection is by content first and file extension second, so a correctly formatted file will usually be recognised even if it has been renamed.
• Scientific notation (1.5E+03, 1.2e4) is accepted in the intensity or 2θ column.
• Ordering: data stored in descending 2θ is detected and sorted ascending on load. Duplicated 2θ values and non-finite points are removed, with a note in the browser console.
• Generic Data: Standard two-column ASCII files (.xy, .csv, .txt, .dat, .asc etc.) containing $2\theta$ and intensity values are also supported. The parser accommodates space, comma, or semicolon delimiters. Comment lines prefixed with #, !, ; or // are ignored, as are non-numeric header lines.
• Metadata Parsing: For many instrument-specific formats, instrument parameters such as the X-ray wavelength for Kα1 are read from the file's metadata and used to populate the relevant fields in the user interface. It is incumbent upon the user to verify the correctness of these automatically populated values.

Files & Offline Use

The application is a set of static files with no build step and no server-side component. Every file below must sit in the same directory:

Interactive Data Visualization

The diffraction pattern is rendered in an interactive plot to facilitate detailed inspection of the experimental data and the quality of the model fit. You can hide any of the plots by clicking on its legend at the top of the chart.

All navigation is implemented directly in the application, with no third-party charting plugin, so it behaves identically whether or not you have a network connection.

• Reflection Data: Hovering the cursor near a Bragg reflection marker (tick mark) displays a tooltip containing the corresponding Miller indices ($hkl$) and $d$-spacing.
• Hiding curves: Click an entry in the legend at the top of the chart to toggle that curve.

Pattern Decomposition Methodologies

Pattern decomposition enables the fitting of a powder diffraction pattern based on a unit cell and space group, without requiring a full structural model (atomic coordinates). This is essential for the precise determination of lattice parameters and the extraction of integrated intensities, which are requisite for ab initio structure determination.

The Le Bail Method

The Le Bail method is an iterative, sequential algorithm known for its computational efficiency and robust convergence. The process is as follows:

1. Initialization: A theoretical pattern is calculated from the user-supplied lattice, profile, and background parameters (defined by the spline points). Every reflection is given the same starting height. This flat start is deliberate: the method assumes no structural model, so the first decomposition partitions purely on profile overlap. Intensities are never carried over from a previous run, which would make the result depend on history rather than on the data.
2. Seeding passes: Before any parameter is allowed to move, the decomposition alone is iterated a few times so the intensities settle. The partitioning is a fixed-point iteration and converges in a handful of passes.
3. Intensity Extraction (Height Partitioning): The observed net intensity ($y_{i,obs} - y_{i,bkg}$) at each data point is partitioned among the calculated Bragg peaks contributing to that point. The contribution of each peak is proportional to its profile function value at that point. Summing these partitioned intensities for each reflection yields a new set of "observed" integrated intensities. These are then converted back to estimated peak heights using the current profile function's area.
4. Extract, then refine: The intensities are re-extracted at the top of every optimizer iteration and then held fixed for the whole of that iteration — for the baseline residual, for every column of the Jacobian, and for the trial step. The optimizer adjusts only the lattice and profile parameters; the background stays fixed at the spline points throughout.
5. Why not re-extract more often: Re-extracting inside the objective function (i.e. at every function evaluation) flattens the $\chi^2$ landscape: the peak widths can grow substantially while the extraction quietly compensates, so the residual barely responds and the width parameters become indeterminate. Holding the intensities fixed within an iteration keeps the minimum at the true width sharp. Alternating extraction with least-squares — rather than fusing them — is the actual Le Bail method.
6. Convergence: Because the alternation happens inside the iteration loop, a single press of Run Le Bail converges self-consistently; you do not need to click repeatedly for the intensities and parameters to agree. Re-running is useful only to restart from a different algorithm (see Recommended Refinement Strategy) or after changing a fixed input such as the background points.

The Pawley Method

The Pawley method employs a simultaneous, non-iterative approach. It treats the peak height ($I_{hkl}$) of each Bragg reflection as an independent, refinable variable within a single, large-scale least-squares minimization.

This means that all parameters—lattice, profile, and all individual peak heights—are refined concurrently. The background shape is fixed by the user-defined spline points and is not refined.

• Advantages: The Pawley method is considered more rigorous as it avoids the iterative bias of the Le Bail method, particularly in cases of severe peak overlap. It can yield more accurate and statistically sound integrated intensities (reported as areas) and parameter uncertainties.
• Disadvantages: The inclusion of hundreds or thousands of intensity variables significantly increases the computational complexity and memory requirements. The refinement can be susceptible to instability and parameter correlation, especially if the initial model is poor or if stochastic optimization algorithms are used with insufficient iterations.

Minimization Algorithms

The goal of refinement is to minimize the sum-of-squares objective function, $\chi^2 = \sum w_i (y_{i,obs} - y_{i,calc})^2$, where $w_i$ is the statistical weight of each data point (typically $w_i = 1/y_{i,obs}$). This application provides several algorithms to navigate the complex parameter space and find the minimum of this function.

Levenberg-Marquardt (LM)

The LM algorithm is a standard gradient-based method for non-linear least-squares problems. It effectively interpolates between the Gauss-Newton algorithm and the method of gradient descent. By calculating the Jacobian matrix (the matrix of first partial derivatives of the calculated pattern with respect to the parameters), it determines the most efficient path toward the nearest local minimum.

• Characteristics: LM is a local minimizer, exhibiting rapid quadratic convergence when the initial parameters are close to the true minimum. It is the preferred method for final, high-precision refinement and is the only algorithm here that can calculate valid estimated standard deviations (ESDs) for the refined parameters from the covariance matrix.
• Limitations: It is susceptible to converging to a local minimum if the starting model is far from the global solution.
• Parameter bounds: several parameters are bounded — profile widths and Pawley intensities below at zero, the mixing parameter $\eta$ to $[0,1]$, cell angles to $(0^\circ, 180^\circ)$. A step that would cross a bound is clipped to it, and the gain ratio $\rho = \Delta\chi^2_{\mathrm{actual}} / \Delta\chi^2_{\mathrm{predicted}}$ is formed from the step that was actually applied. This matters in Pawley mode, where weak reflections routinely park at zero intensity: the damping parameter $\lambda$ is shared by the whole model, so measuring $\rho$ against a step that was never taken would let a handful of pinned intensities throttle the cell and profile terms as well.
• Stopping at a bound: if an entire step is blocked — the model sitting in a corner of the bound box with the gradient pointing out of it — the refinement stops and reports it, rather than spending its remaining iterations raising $\lambda$ against a wall.
• Pawley Mode: Generally the recommended algorithm for Pawley refinements due to stability and efficiency.

Parallel Tempering (Replica Exchange)

Parallel Tempering, also known as Replica Exchange MCMC, is an advanced stochastic algorithm designed to overcome the slow convergence of traditional search methods on complex landscapes. Instead of a single system, Parallel Tempering simulates multiple copies (or "replicas") of the system simultaneously, each at a different, fixed temperature in a predefined ladder ($T_1 < T_2 < ... < T_N$).

• Mechanism: Each replica evolves independently according to a standard Monte Carlo or Simulated Annealing-like algorithm at its respective temperature. The high-temperature replicas explore the parameter space broadly (high mobility, escaping local minima), while the low-temperature replicas perform a fine-grained search of local minima (high precision).
• The Swap Move: Periodically, the algorithm attempts to swap the entire set of parameters between adjacent replicas (e.g., between replica $i$ at temperature $T_i$ and replica $i+1$ at $T_{i+1}$). The swap is accepted with a Metropolis-like probability that depends on the energies (costs) and temperatures of the two replicas. This crucial step allows a good solution discovered by a high-temperature replica in a distant valley to "percolate down" to the low-temperature replicas, dramatically improving the efficiency of finding the global minimum compared to single-temperature methods.
• Advantages: Significantly more efficient at global exploration than simpler stochastic methods, making it a robust choice for complex problems or when the starting model is highly uncertain (primarily in Le Bail mode).
• Le Bail intensities under PT: Unlike the LM path, which re-extracts at the top of each iteration, the stochastic search extracts the intensities once before the run and holds them fixed throughout. A stochastic walker proposes many rejected moves, and re-extracting on each would make the cost function non-stationary and the Metropolis test meaningless.
• Pawley Mode: Can be used, but may require substantially more iterations than LM to converge reliably due to the large number of intensity parameters.

Guide to Refinable Parameters

This section provides a detailed breakdown of the parameters you can control and refine.

Crystal System & Space Group

These parameters define the crystallographic symmetry of the material.

• The System selection imposes metrical constraints on the lattice parameters (e.g., for Cubic, $a=b=c$, $\alpha=\beta=\gamma=90^\circ$).
• The Space Group selection determines the systematic reflection conditions ($hkl$ absences) used to generate the list of Bragg peaks. The underlying logic for these conditions is consistent with established crystallographic libraries and was taken from Computational Crystallography Toolbox (cctbx).
• Space groups are chosen from a searchable modal listing every setting of all 230 groups. Where a group has several settings (for example No. 62: $Pnma$, $Pmnb$, $Pbnm$, $Pcmn$, $Pmcn$, $Pnam$) each is offered separately, since the reflection conditions differ between them. Rhombohedral groups are offered on hexagonal axes only, because the $d$-spacing formula is written for that indexing.
• Monoclinic settings. All three unique-axis choices are supported. The unique axis is read from the symmetry operators of the setting you select, and the lattice panel then offers the one angle that is free: $\beta$ for unique axis $b$ ($P12_1/c1$), $\gamma$ for unique axis $c$ ($P112_1/b$), $\alpha$ for unique axis $a$ ($P2_1/b11$). The other two are fixed at $90^\circ$ and are neither shown nor refined. The reciprocal metric, the reflection orbits and the multiplicities all follow that same axis, so the three settings of one lattice give identical peak positions and identical intensities. Switching between settings carries the angle across, so the cell is not lost.
• Startup default: the application opens on $Pnma$ (No. 62, standard setting) with an orthorhombic cell of $a = 8.478$, $b = 5.397$, $c = 6.958$ Å. This is only a starting point — change it to match your material before refining.

Reflection List & Multiplicities

The Bragg peak list is built from the rotation operators of the selected space-group setting. Every candidate $hkl$ inside the resolution sphere is mapped onto its orbit under the Laue group; the orbit is kept once, labelled by a canonical member, and its size becomes the multiplicity $m$ that scales the intensity, $I \propto m \cdot LP \cdot |F|^2$. Systematic absences are then tested on that representative, which decides the whole orbit, since an absence is a property of the orbit and not of any one index triple.

Working from the operators rather than from a fixed index range per crystal system matters wherever two Laue classes share a system, because the two classes have genuinely different sets of independent reflections:

In the hexagonal and trigonal systems the special forms are set by the Bravais–Miller index $i = -(h+k)$ as much as by $h$ and $k$: a reflection lies on a mirror whenever any two of $h$, $k$, $i$ are equal in magnitude, so $(1\bar{2}l)$ and $(2\bar{4}l)$ are special even though $|h| \neq |k|$. The orbit calculation accounts for this automatically.

Operators are taken from the symops field of the space-group database (or the older rotations field, which serves equally well here — only the rotation parts affect a reflection orbit). The closed group is checked against the order_p recorded for the setting before it is used. If the database carries no operators, a built-in table of the Laue-class generators is used instead; if the Laue class itself cannot be identified, the generator falls back to the lowest Laue class of the crystal system. That last case is safe rather than wrong: orbits split into smaller ones at identical $d$, so peak positions and total intensities are unaffected, and only the number of listed entries changes. The browser console records which source was used whenever it is not the first.

Instrumental Parameters

Found under the "Sample" tab, these parameters model the diffractometer configuration.

• Radiation 1/2 (Å) & Ratio: Defines the X-ray source. For divergent-beam laboratory instruments, a Kα1/Kα2 doublet is typically used. For synchrotron radiation, the Ratio is set to 0.
• Polarisation: Selects the polarisation model used in the Lorentz–polarisation factor — Lab (default), Synchrotron, or None. A second field appears beside it: the monochromator angle $2\theta_M$ for a laboratory source, or the polarised fraction $f$ for a synchrotron. This setting does not affect the fit; it governs how an extracted intensity is converted into $|F|$. See Lorentz–Polarisation Factor.
• Zero: A refinable parameter that corrects for instrumental zero-point error in the $2\theta$ axis. It is highly correlated with lattice parameters and must be refined with caution.
• 2θ Min / Max: These sliders define the refinement range. It is standard practice to exclude regions of low signal-to-noise or known artifacts from the calculation.

Lorentz–Polarisation Factor (Lp)

The integrated intensity of a powder reflection is not $|F|^2$ alone. It carries two geometric weights — the multiplicity $m$ of the reflection and the Lorentz–polarisation factor $Lp$ — on top of an arbitrary overall scale $s$:

$Lp$ is therefore the last thing standing between an extracted Le Bail or Pawley intensity and a structure factor, and it is the reason the reflection table in the report prints $m$, $Lp$ and $|F_o|$ next to the intensity rather than the intensity on its own.

The Lorentz factor

For a conventional $\theta$–$2\theta$ powder scan, Powder 5 uses

This lumps together the rate at which a crystallite passes through the reflecting condition and the fraction of the Debye–Scherrer ring intercepted by the detector. It is often written $1/(4\sin^2\theta\cos\theta)$; the factor of 4 is a constant and is swallowed by $s$, so it makes no difference to anything reported here. $L$ has no adjustable parameters and is applied for every polarisation setting, including None.

The polarisation factor

All three models are the same expression with one constant $K$, normalised so that $P(0) = 1$ in every case:

With $2\theta_M = 0$ the Lab setting reduces to $P = \tfrac12(1 + \cos^2 2\theta)$, the classical unpolarised expression, and this is what Powder 5 uses by default. Note that a fully perpendicular-polarised synchrotron beam ($f = 1$) gives $K = 0$ and therefore exactly the same arithmetic as None; the two settings are kept distinct because they describe different geometry and are labelled differently in the report.

Choosing $2\theta_M$ for a laboratory monochromator

$2\theta_M$ is the take-off angle of the monochromator crystal at the wavelength in use, not a free parameter. Common values for Cu K$\alpha$:

Choosing $f$ for a synchrotron

Bending-magnet and undulator radiation is polarised in the horizontal plane, with $f$ typically between 0.90 and 0.98 in the plane of the orbit. What matters is the orientation of the diffractometer relative to that:

• Vertical scattering plane (the usual arrangement for high-resolution powder diffraction): the electric field is perpendicular to the scattering plane, so $f \approx 0.95$–$1.0$ and $P$ is nearly flat. The correction is almost pure Lorentz, and this is precisely why the geometry is chosen.
• Horizontal scattering plane: the field lies in the scattering plane. Enter $f \approx 0.05$–$0.1$, which gives $P \to \cos^2 2\theta$ — a severe correction that goes to zero at $2\theta = 90^\circ$. Getting this backwards is a much worse error than misjudging $f$ by a few per cent.

How large is the effect?

$Lp$ falls steeply with angle whatever the model, and the polarisation term adds a further factor of two by $2\theta = 90^\circ$ for an unpolarised source. Values for Cu K$\alpha$:

$Lp$ spans a factor of about 90 between $2\theta = 10^\circ$ and $90^\circ$. Omitting it does not perturb a structure-solution calculation slightly — it weights the first few peaks roughly two orders of magnitude too heavily, and a charge-flipping map built on uncorrected intensities is dominated by them. The choice between models is a much smaller effect: at most a factor of two, and concentrated near $2\theta = 90^\circ$. Getting the correction on at all matters more than getting the model exactly right, but the models are cheap to select correctly.

Where it is applied

All of these read one shared calculation, evaluated at the corrected $2\theta$ — the same angle printed in the reflection table, including the zero shift and any displacement or transparency term. The charge-flipping worker takes the per-reflection value computed by the refinement rather than recomputing its own, so the map, the report and the prior cannot disagree about it. The charge-flipping summary panel reports the model the worker actually used, which is the check that they have not.

Background Modeling (Spline Interpolation)

The background contribution is modeled using a monotonic cubic Hermite spline interpolation between a series of user-defined points (spline points or knots). This approach provides flexibility and ensures a smooth, physically realistic background shape without introducing refinable background parameters into the least-squares minimization. The background shape is therefore considered fixed during the refinement process based on the current spline points.

Control of the background spline is located under the "Background" tab:

• Auto-generation: The application automatically estimates background points immediately upon loading a data file. You can adjust the density of these points using the Auto-points slider (10-40 points). Adjusting the slider automatically recalculates the points based on local intensity minima within intervals distributed across the current 2θ Min/Max slider range. The points at the exact 2θ Min and Max slider positions are always included and fixed to these $2\theta$ values.
• Manual Addition: Add individual points by holding Ctrl and clicking on the chart. The closest experimental point will be added to the list, provided it's within the current slider range and not an edge point.
• Editable List: The generated and manually added points appear in a list below the controls.
  • You can directly edit the $2\theta$ and Intensity values for any point, except for the $2\theta$ values of the first (Min) and last (Max) points, which are fixed by the sliders. Edits trigger recalculation of the spline.
  • Points can be deleted using the × button, except for the first and last points.
• Chart Display: The spline points can be toggled on/off on the chart using the "Show Points on Chart" checkbox. The calculated spline curve is always shown.

Simple pVoigt

This function models the peak shape as a linear combination of a Gaussian and a Lorentzian function: $pV(x) = \eta L(x) + (1-\eta)G(x)$. The angular dependence of the Full Width at Half Maximum (FWHM) for each component is modeled empirically. $$H_G^2 = GU \tan^2\theta + GV \tan\theta + GW + GP / \cos^2\theta$$ $$H_L = LX / \cos\theta$$

• GU, GV, GW, GP: Parameters describing the Gaussian FWHM ($H_G$). The terms are associated with strain ($GU$), instrumental factors ($GV, GW$), and particle size effects ($GP$). Note that $GU$, $GW$, and $GP$ should physically be non-negative.
• LX: Describes the Lorentzian FWHM ($H_L$), primarily associated with crystallite size broadening ($LX > 0$).
• eta: A simple linear mixing parameter ($0 \le \eta \le 1$; $\eta=0$ for pure Gaussian, $\eta=1$ for pure Lorentzian).
• shft & trns: Corrects for peak position shifts due to sample displacement and transparency, respectively.
  Unit and Scaling for the shft parameter:
  The refined shft parameter is a dimensionless, scaled coefficient, not a direct physical length. Its relationship to the physical specimen displacement ($s$) and the goniometer radius ($R$) is defined as follows:

  • The physical peak shift in radians is: $\Delta(2\theta)_{\text{rad}} = -2 \frac{s}{R} \cos(\theta)$
  • The program calculates this shift (in degrees) using the formula: $\Delta(2\theta)_{\text{deg}} = -(\text{shft} / 1000) \times \cos(\theta) \times (180 / \pi)$
  • Therefore, the relationship is: $\frac{\text{shft}}{1000} = \frac{2s}{R}$
  • To find the physical displacement $s$ from the refined parameter, use: $s = R \times (\text{shft} / 2000)$.
    Example: For a typical instrument with $R=240$ mm, a refined shft value of 1.0 corresponds to a physical displacement $s$ of $240 \times (1/2000) = 0.12$ mm.

TCH (Size/Strain/Aniso)

This is a physically rigorous profile function adapted from GSAS, convoluting a Thompson-Cox-Hastings (TCH) pseudo-Voigt with a Stephens model for anisotropic strain broadening.

Isotropic Broadening (TCH Model)

The TCH formulation models the FWHM of the Gaussian ($H_G$) and Lorentzian ($H_L$) components based on physical contributions to line broadening: $$H_G^2 = U \tan^2\theta + V \tan\theta + W$$ $$H_L = X \tan\theta + Y / \cos\theta$$

The total FWHM ($H$) and pseudo-Voigt mixing parameter ($\eta$) are then derived from these components using polynomial approximations. The final shape is $pV(x) = \eta L(x, H) + (1-\eta)G(x, H)$, where both functions share the same convoluted FWHM.

• U, V, W: Gaussian broadening parameters related to strain ($U, V$) and instrumental resolution ($W$). Physically, $U$ and $W$ should be non-negative.
• X, Y: Lorentzian broadening parameters related to strain ($X$) and crystallite size ($Y$). Physically, $X$ and $Y$ should be non-negative.

Peak Asymmetry

The S/L and H/L parameters introduce an angle-dependent asymmetry, primarily correcting for axial divergence effects at low $2\theta$.

Anisotropic Broadening (Stephens Model)

Anisotropic microstrain, where broadening varies with crystallographic direction, is modeled by adding terms to the Lorentzian component ($H_L$) that are dependent on the Miller indices ($hkl$). The model is a fourth-order polynomial in the reciprocal lattice vectors.

The refinable parameters (S400, S040, etc.) are the non-zero, symmetry-unique coefficients of this polynomial. The application automatically applies symmetry constraints based on the Laue class of the selected space group (e.g., for cubic, $S400=S040=S004$).

Split pVoigt (Asymmetric)

This profile function is a modification of the Simple pseudo-Voigt designed to model asymmetric peaks (e.g., from axial divergence or stacking faults). It achieves this by defining independent sets of profile width parameters for the left side (at $2\theta$ values less than the peak center) and the right side of the peak.

The shape is still a linear combination $pV(x) = \eta L(x) + (1-\eta)G(x)$, but the $H_G$ and $H_L$ parameters used in the calculation depend on whether $x$ is to the left or right of the peak center.

Gaussian Broadening (Left & Right)

The Gaussian FWHM ($H_G$) for each side is modeled as:

$$H_{G, \text{side}}^2 = GU_{\text{side}} \tan^2\theta + GV_{\text{side}} \tan\theta + GW_{\text{side}}$$

(Note: This model does not use the $GP$ term used in the Simple pVoigt profile.)

• GU-L, GV-L, GW-L: Parameters describing the Gaussian FWHM for the left side of the peak.
• GU-R, GV-R, GW-R: Parameters describing the Gaussian FWHM for the right side of the peak.

Lorentzian Broadening (Left & Right)

The Lorentzian FWHM ($H_L$) for each side is modeled as:

$$H_{L, \text{side}} = LX_{\text{side}} / \cos\theta$$

• LX-L: Describes the Lorentzian FWHM for the left side, primarily associated with size broadening.
• LX-R: Describes the Lorentzian FWHM for the right side, primarily associated with size broadening.

Peak Shape & Position

• eta (Mixing) (param: eta_split): A simple linear mixing parameter ($0 \le \eta \le 1$; $\eta=0$ for pure Gaussian, $\eta=1$ for pure Lorentzian). This single value is used for both sides of the peak.
• shft (Displ.) (param: shft_split): Corrects for peak position shifts due to sample displacement.
• trns (Transp.) (param: trns_split): Corrects for peak position shifts due to transparency.
  Unit and Scaling for the shft_split parameter:
  This parameter is a dimensionless, scaled coefficient, identical in function to the shft parameter in the Simple pVoigt profile.

  • The physical peak shift in radians is: $\Delta(2\theta)_{\text{rad}} = -2 \frac{s}{R} \cos(\theta)$
  • The program calculates this shift (in degrees) using the formula: $\Delta(2\theta)_{\text{deg}} = -(\text{shft\_split} / 1000) \times \cos(\theta) \times (180 / \pi)$
  • Therefore, the relationship is: $\frac{\text{shft\_split}}{1000} = \frac{2s}{R}$
  • To find the physical displacement $s$ from the refined parameter, use: $s = R \times (\text{shft\_split} / 2000)$.

Probabilistic Space-Group Determination

Once a Pawley refinement has completed, a small ? button becomes active on the space-group line of the control panel. It scores every space group compatible with the current Laue class against the extracted intensities and returns a ranked list of posterior probabilities. The method follows Markvardsen, David, Johnson & Shankland (2001).

The Principle

A candidate space group $S$ makes exactly one testable statement about a powder pattern: a particular subset $e$ of the reflections is systematically absent, so their true integrated intensities are identically zero. Everything else it leaves free. The question is therefore not “does this space group fit?” but “are the reflections it forbids consistent with zero, given how well the data actually determine them?”

A Pawley refinement is what makes this answerable. Because every intensity is a free least-squares parameter, the refinement returns not only the intensities $\hat{\mathbf{I}}$ but the normal matrix behind them, and therefore their full covariance. That matters enormously: in a powder pattern reflections overlap, and overlapping intensities are strongly correlated. A reflection that should be absent can carry a large apparent intensity purely by borrowing it from a neighbour it cannot be resolved from. Judging each reflection on its own $I/\sigma$ ignores this and can be badly misleading in either direction.

The Statistical Model

The true intensities are given a prior that encodes what the space group claims. Writing $\mathbf{P}_S$ for a diagonal matrix,

A variance of exactly zero forces that intensity to vanish; $\tau_j$ is the local Wilson-like intensity scale for reflections the group allows. The refinement itself contributes $\hat{\mathbf{I}} \mid \mathbf{I} \sim N(\mathbf{I}, \mathbf{Q}^{-1})$, where $\mathbf{Q}$ is the intensity precision matrix. Marginalising over the unknown true intensities collapses these into a single multivariate normal,

so the evidence for $S$ is one density evaluation. Writing $f$ for the set of reflections the group allows and dropping every term that does not depend on $S$:

The quadratic form is the evidence carried by the would-be-absent reflections, evaluated with the full correlation structure rather than reflection by reflection. The log-determinant is an Occam factor: it rewards a group for explaining more absences, and without it the group with no extinction conditions at all would win by construction, since it is never contradicted by anything.

Posterior probabilities follow by normalising over the candidate list.

Reading the Result

Each row of the table is a distinct extinction symbol, not a single space group. Powder diffraction cannot separate groups that predict identical absences — centrosymmetric and non-centrosymmetric pairs, for instance — so those share one probability and are listed together. Clicking any symbol adopts that setting. Note that the setting matters: $Pnma$ and $Pbnm$ are the same group but predict different absences relative to the fixed axes, so they appear separately.

• P(SG | data) — the posterior for that extinction symbol.
• absent — how many of the refined reflections the group forbids. A group forbidding nothing is always in the list as a baseline.
• ⟨I/σ⟩ and worst I/σ — diagnostics over the would-be-absent set, using diagonal errors only. These are the numbers a crystallographer looks at first, but they are not what drives the ranking; the probability uses the full correlation structure.

Requirements and Limitations

Two further points are worth keeping in mind.

• The Laue class is not under test. Only the absences are. The Laue class is fixed by the diffraction symmetry you have already chosen, since it determines the reflection orbits and the multiplicities. Running the test from $Pm\bar{3}m$ can never return a group belonging to Laue class $m\bar{3}$, such as $Pa\bar{3}$. If the Laue class itself is in doubt, run the test from each candidate in turn.
• Exact reflection coincidences are handled, but weaken the diagnostics. In cubic and rhombohedral cells reflections such as $(333)/(511)$ or $(300)/(221)$ fall at identical $2\theta$. Their individual intensities are then not separately determined and no covariance matrix exists. The test is formulated in precision space precisely so that it does not need one — $\mathbf{Q}$ may be singular while $\mathbf{B}_S$ never is — so the probabilities remain valid. The $I/\sigma$ columns fall back to conditional estimates in this case and are flagged as such.

Ab Initio Structure Solution: Charge Flipping

Once a Pawley refinement has extracted a set of individual integrated intensities, Powder 5 can attempt to solve the crystal structure ab initio — with no starting model — using the dual-space charge-flipping algorithm of Oszlányi and Sütő. The result is an electron-density map and a list of atomic positions, obtained purely from the diffraction intensities and the unit cell.

The Principle

A diffraction experiment measures the amplitudes \(|F_{hkl}|\) but not the phases \(\phi_{hkl}\); recovering those phases is the phase problem. Charge flipping solves it by iterating between real space (the electron density \(\rho\)) and reciprocal space (the structure factors \(F\)), enforcing one simple constraint in each:

1. Real space. A correct electron density is non-negative almost everywhere. Every voxel whose density falls below a small positive threshold \(\delta\) has its sign flipped: \(\rho \rightarrow -\rho\). This deliberately wrong perturbation is what drives the phases toward consistency.
2. Reciprocal space. Fourier-transform the flipped density, then reimpose everything that is actually known about the structure factors: the measured amplitudes, the point-group symmetry, and the systematic absences. The calculated phases are kept. Reflections that were never measured are left free — that freedom is what lets the map extend beyond the data.

Transforming back gives an improved density, and the cycle repeats. The threshold is expressed in units of the map's own RMS density \(\sigma\), so \(\delta\) is scale-independent; values around \(0.8\sigma\)–\(1.2\sigma\) work for most problems. The agreement between calculated and observed amplitudes,

\( R = \dfrac{\sum_{c} \left| \sqrt{I^{calc}_{c}} - \sqrt{I^{obs}_{c}} \right|}{\sum_{c} \sqrt{I^{obs}_{c}}} \)

is tracked each iteration. The sum runs over measured peaks \(c\), not over individual reflections: a powder pattern cannot distinguish reflections that overlap in \(2\theta\), so scoring them separately would be scoring noise. \(I^{calc}_{c}\) is the total calculated intensity of everything under peak \(c\). A run that has found the structure typically drops below \(R \approx 0.15\) and then oscillates. Powder 5 keeps the best iterate of the run, not the last.

Space-Group Constraints

Classical charge flipping deliberately works in \(P1\): it throws the symmetry away and lets the algorithm rediscover it, which is a useful safeguard when the space group is uncertain. Powder 5 keeps that mode available, but by default it uses the selected space group inside the iteration. Four separate constraints come out of it, and they are worth distinguishing because they behave very differently.

1. Orbits: symmetry equivalents share one intensity

A Pawley fit reports one intensity per symmetry-unique reflection. Its equivalents \(hR\) must be placed on the \(P1\) grid before a transform is possible, and they are generated from the actual symmetry operators of the selected group. The distinction matters: the Laue class is not the holohedry of the crystal system. \(P4\) has Laue class \(4/m\), so a general reflection has 8 equivalents, not the 16 that \(4/mmm\) would give; the same applies to \(\bar 3\) versus \(\bar 3 m\), \(6/m\) versus \(6/mmm\) and \(m\bar 3\) versus \(m\bar 3 m\). Spreading one measured intensity over twice as many positions as really exist puts half of it onto reflections that are not equivalent at all.

Within an orbit the amplitudes are equal by symmetry: \(|F| = \sqrt{I^{obs}/m}\), where \(m\) is the orbit size. This is a hard constraint, applied every cycle. Only the phases are free.

2. Overlap: distinct reflections may share one peak

The genuinely powder-specific ambiguity is different. Reflections that are not symmetry equivalent can still fall at the same \(2\theta\), either exactly (511/333 in a cubic cell) or accidentally. The pattern gives only their sum. Powder 5 groups such reflections into a cluster and re-partitions the cluster's total intensity between them each cycle in proportion to the current calculated values — the same idea as Le Bail extraction, applied inside the flipping loop. The split is free; the total is fixed.

3. Systematic absences are held at zero

Reflections forbidden by lattice centring, screw axes or glide planes are forced to \(F = 0\) every cycle. They are found directly from the operators: \(h\) is extinct if some operator \((R,\mathbf{t})\) satisfies \(hR = h\) with \(h \cdot \mathbf{t}\) non-integral. Leaving them free lets the density break the lattice centring outright, and for an \(F\) or \(I\) lattice that is most of the reciprocal grid.

4. Symmetrisation of the phases

For a real-space operator \(\mathbf{x}' = \mathbf{x}R + \mathbf{t}\), the structure factors of a correctly positioned structure satisfy

\( F(hR) = F(h)\,\exp(-2\pi i\, h \cdot \mathbf{t}) \)

Each cycle the members of an orbit are reduced to a common representative, averaged, and pushed back out. The Symmetry in loop control sets how strongly: \(F \leftarrow (1-\lambda)F + \lambda F_{sym}\). At \(\lambda = 0\) nothing is imposed and the run is classical \(P1\) charge flipping. At \(\lambda = 1\) the projection is exact, which converges faster and has a useful side effect: the solution comes out on a standard origin instead of a random one, so the automatic origin search in the Structure block becomes a verification rather than a necessity.

Intensity scaling

The refined Pawley parameter is a peak height, because the profile functions Powder 5 uses are normalised to unit height, not unit area. What charge flipping needs is the integrated intensity, so each height is multiplied by the area of its own profile (summed over the Kα₁/Kα₂ doublet) before it is used. Skipping that step underweights the high-angle reflections by the full Caglioti broadening, which is roughly a factor of two across a typical scan.

The integrated intensity then carries \(I \propto m \cdot LP \cdot |F|^2\). The multiplicity \(m\) is absorbed by spreading the intensity across the \(m\) grid points of the orbit, but the Lorentz-polarisation factor \(LP = (1+\cos^2 2\theta)/(2\sin^2\theta\cos\theta)\) is divided out explicitly. Without it the low-angle reflections are weighted several times too heavily and the map is dominated by the first few peaks.

Workflow & Controls

The controls sit in the left panel; results appear in the Charge Flipping tab of the right panel.

Recommended Settings

The defaults are chosen to solve a typical small inorganic structure from good laboratory data. The table below is a starting point by problem type; the paragraphs after it explain what to change when a run fails.

Choosing the grid

The grid must be large enough to hold every reflection in the fit: an index \(h_{max}\) needs \(N \geq 2h_{max}+2\). If it is too small the summary reports how many reflections did not fit and what \(N\) they would need, and those reflections are simply lost. Beyond that requirement, aim for a voxel of roughly 0.25–0.35 Å (the summary prints the spacing). Finer than ~0.2 Å buys nothing when the data stop at 1 Å resolution and costs \(N^3\log N\): 64³ to 128³ is about an eight-fold increase in run time.

Choosing the threshold

\(\delta/\sigma\) is the single most sensitive parameter. Too low and almost nothing is flipped, so the iteration stalls with \(R\) flat and high; too high and the perturbation is so violent that the phases never settle, giving an \(R\) that jumps around without trend. The working range is 0.8–1.2 and 1.1 is a good default. If \(R\) plateaus above 0.3 with no oscillation, lower it in steps of 0.1; if \(R\) is erratic from the first cycles, raise it. Weak or noisy data generally want a slightly lower threshold.

Iterations and random starts

Charge flipping does not converge gradually — it stays flat and then drops abruptly when the phases lock in, often after several hundred cycles. Judge a run by whether that drop happened, not by the final \(R\). More starts is usually a better investment than more iterations: a run that has not found the solution by ~1000 cycles is unlikely to find it by 3000, whereas a fresh random start might find it immediately. For a difficult problem, 10–20 starts of 1000 cycles beats 2 starts of 10000.

Choosing λ

Use \(\lambda = 1\) when the space group is established (a clean space-group test, a known compound, an unambiguous extinction pattern): it converges in fewer cycles, needs fewer random starts, and delivers the map already on a standard origin. Use \(\lambda = 0\) when the group is in doubt, or as an independent check on a \(\lambda = 1\) solution — if the \(P1\) map symmetrises to a high correlation without having been forced to, the group is right. \(\lambda = 0.5\) is the compromise and the default: it accelerates convergence appreciably while still allowing the density to disagree with a slightly wrong group.

Choosing the overlap tolerance

Set it to roughly half the FWHM of a mid-angle peak. Too small and truly overlapped reflections are treated as independently measured, which imposes intensities the data never determined; too large and reflections that were resolved get to trade intensity freely, discarding real information. 0.05° suits sharp laboratory data; broad peaks or a high-symmetry cell with many coincidences want 0.10–0.15°. The summary reports how many reflections ended up sharing a peak — if that number is close to the total, the tolerance is almost certainly too large.

Reading the Solution

Each run appends a timestamped entry to the Charge Flipping history, so earlier attempts remain available from the run selector. A solution contains:

• Best R. The amplitude agreement of the retained map, summed over measured peaks. Below ~0.15 is promising; above ~0.30 usually means the run has not converged — add random starts, adjust the threshold by ±0.1σ, or see Recommended Settings.
• Electron-density peaks. Fractional coordinates. At \(\lambda = 0\) the map is in \(P1\) with an arbitrary origin and hand, so absolute positions are meaningless — compare interatomic distances. At \(\lambda > 0\) the origin is pinned by the symmetrisation and the coordinates are directly comparable to the space group's Wyckoff positions. Relative peak heights are a rough guide to atomic number in both cases.
• Symmetry diagnostics. The summary reports which symmetry was used (operators or Laue-table fallback), how many reflections were forced to zero as systematically absent, and how many ended up sharing a peak with another reflection. A large absence count is normal for a centred lattice; an absent-but-observed warning is not, and usually means the space group is wrong.
• Density section. A slice through the map, with selectable axis and depth. The colour bar under the image is an absolute scale taken from the whole map, not from the slice on screen, so slices are directly comparable and an empty one looks empty. The caption gives the slice maximum in units of the map's \(\sigma\); anything under about 3σ is ripple, not an atom. The Stretch each slice to its own range checkbox restores per-slice auto-contrast, which is occasionally useful for tracing weak features but makes noise look like structure — the caption says so whenever it is on.
• 3D isosurface. The same map drawn as a surface of constant density, at a level you set in units of the map's \(\sigma\) — the same units as the section caption, so the two views can be read against each other. Drag to rotate, scroll to zoom. At 1–2σ you will see mostly noise; real atoms usually appear as isolated closed blobs somewhere above 3σ.
  • As solved is the raw charge-flipping density, exactly what the algorithm produced.
  • Symmetry-averaged is the same density after the Structure step has shifted it onto the space-group origin and averaged it over the symmetry operators. It is much cleaner, because averaging \(n\) equivalent copies of the cell suppresses the noise by \(\sqrt{n}\) — but it assumes the space group is right, so judge the solution on the raw map and use this one to read it. It only becomes selectable once the structure has been built.
  • Atom positions is a list under the viewer controls, one row per unique site from the Structure block: rank, colour, peak height, multiplicity, and a show/hide toggle. Every site starts hidden, so the density can be judged on its own first; click a row to bring one in, or use show all. Ten rows are shown at a time and the list scrolls beyond that. The header counts what is visible, and the caption under the viewer says how many positions are drawn and how many sites are hidden.

    Each site is drawn expanded over the symmetry operators: a site of multiplicity 4 appears four times, once beside each of its four density lobes, and hiding its row removes the whole orbit at once. The Structure table lists only one representative per orbit, so drawing that list directly would leave most of the lobes bare. The swatch in the list is the colour of the sphere — sixteen distinct colours, enough that a solution with more than eight sites does not reuse one — and the sphere radius follows the peak height, so in PbSO₄ the four Pb lobes are obvious against the lighter atoms.

    Nothing has been least-squares refined at this stage; these are located peaks. Seeing them sit centred inside the density lobes is the quickest visual confirmation that the solution hangs together, and bringing them in one at a time is the easiest way to tell which lobe belongs to which site when the cell is crowded.

  The three cell edges meeting at the origin are coloured and labelled a / b / c, so you can tell which axis you are looking down.

The two buttons beside the section viewer write the peak list (.csv) and the full density grid (.grd); the refined sites are exported separately, from the button under the Structure block.

Building a Structure in the Space Group

This runs automatically at the end of every charge-flipping run, and again whenever a run is re-selected from the history (the result is cached, so re-selecting is instant). It takes the actual space-group symmetry operators, read from the bundled database, and:

1. finds the origin shift that makes the map obey those operators — computed in closed form from a single Fourier transform, then validated by symmetrising the map and measuring the correlation;
2. averages the density over the symmetry operators;
3. reduces the peaks to one representative per symmetry orbit, reporting the site multiplicity of each (a peak that maps onto itself sits on a special position).

The symmetry correlation reported alongside the sites is the number to trust: a high value means the density is genuinely consistent with the chosen space group; a low one means either the run has not converged or the space group is wrong. The resulting sites can be exported as a .cif file.

Result Fields Explained

The Charge Flipping tab reports several quantities. Each is defined below.

Peak and site table columns

Requirements and Limitations

• Data quality. The solution is only as good as the extracted intensities. Severe peak overlap, a poor background, or an incorrect cell will all degrade or prevent a solution.
• Resolution. Atomic-resolution data (roughly \(d \lesssim 1.2\) Å) is normally needed to resolve individual atoms; lower resolution gives a blurred map.
• The space group is an assumption. With \(\lambda > 0\) it is imposed, not tested. A wrong group can still produce a low \(R\), because the constraint makes the calculated intensities agree with the observed ones by construction. Cross-check with a \(\lambda = 0\) run and with the symmetry correlation from the Structure block.
• Non-uniqueness. Random starts can land on different origins or the inverted structure. This is expected at \(\lambda = 0\); the structure builder resolves origin and hand against the space-group symmetry.
• Light atoms near heavy ones. As with any Fourier method, light atoms sitting close to much heavier ones may be lost in truncation ripples.
• Overlap is modelled, not solved. Re-partitioning the intensity of a cluster in proportion to the calculated values is a reasonable guess, not a measurement. Structures whose solution depends on resolving a specific pair of overlapped reflections may be beyond reach from powder data alone.

Recommended Refinement Strategy

A sequential and hierarchical refinement strategy is crucial for achieving a stable and physically meaningful solution. Attempting to refine all parameters simultaneously from a poor starting model will likely lead to divergence or convergence to a false minimum.

Phase 1: Initial Model Setup

1. Define the Model: Load data, select the crystal system and space group, and define the refinement range using the $2\theta$ sliders.
2. Set Background Points: The background is automatically estimated upon loading. Adjust the Auto-points slider under the "Background" tab to optimize the density of points so that they reasonably follow the experimental background. You can also edit the points manually or use Ctrl+Click on the chart. This background shape is fixed during refinement.
3. Peak Position Refinement: Using the Le Bail and LM algorithms, refine only the Lattice Parameter(s) and, if necessary, the instrumental Zero Shift. The goal is to align the calculated Bragg positions with the observed peak maxima.

Phase 2: Peak Profile Refinement

4. Isotropic Broadening: Once positions are correct, refine the primary isotropic peak shape parameters (e.g., W, Y, U, X in TCH, or GW, LX in Simple/Split pVoigt). This will account for the dominant size and strain contributions.
5. Asymmetry and Shape: Introduce asymmetry parameters (S/L in TCH, shft/trns in Simple/Split pVoigt) if there is a clear misfit, particularly at low angles. Refine the mixing parameter (eta) if needed.
6. Anisotropic Broadening (TCH only): If systematic misfits remain (e.g., some peaks are consistently broader than the model), introduce the anisotropic Stephens parameters (e.g., S400). Refine only the symmetry-independent terms.

Phase 3: Finalization and Intensity Extraction

7. Global Optimization (Optional): If the LM algorithm converges to a poor solution during the Le Bail steps, switch to Parallel Tempering (PT) for one or more Le Bail cycles to perform a global search. Afterwards, switch back to LM for a final, precise local minimization.
8. Pawley Refinement: With a stable and well-refined model from the Le Bail method, perform a final refinement using the Pawley method. It is generally recommended to use the Levenberg-Marquardt (LM) algorithm for stability, though Parallel Tempering (PT) can also be used (potentially requiring more iterations). This will provide the most statistically robust set of integrated intensities, suitable for subsequent structure solution.

Interpretation of Results

Assessing the quality of a refinement requires both statistical analysis and critical visual inspection of the fit.

Figures of Merit

Standard crystallographic R-factors are provided to quantify the quality of the fit.

• R-pattern ($R_p$): The unweighted residual error based on net intensities, sensitive primarily to the fit of high-intensity reflections. $$R_p = \frac{\sum |(y_{i,obs} - y_{i,bkg}) - (y_{i,calc} - y_{i,bkg})|}{\sum |y_{i,obs} - y_{i,bkg}|} \times 100\%$$
• Weighted R-pattern ($R_{wp}$): The primary figure of merit, weighted by the inverse of the observed gross intensities ($w_i = 1/y_{i,obs}$), which properly accounts for the counting statistics across the entire pattern. $$R_{wp} = \left[ \frac{\sum w_i (y_{i,obs} - y_{i,calc})^2}{\sum w_i y_{i,obs}^2} \right]^{1/2} \times 100\%$$
• Reduced Chi-squared ($\chi^2$, Goodness of Fit): The most statistically rigorous indicator. For a statistically perfect fit where the model correctly describes the data and the weights are accurate, $\chi^2$ should approach 1.0. $$\chi^2 = \frac{1}{N - P} \sum w_i (y_{i,obs} - y_{i,calc})^2 = \left(\frac{R_{wp}}{R_{exp}}\right)^2$$ where $N$ is the number of data points, $P$ is the number of refined parameters, and $R_{exp}$ is the statistically expected minimum $R_{wp}$.

Calculating Observed Intensities ($I_{obs}$) for Overlapping Peaks

A simple numerical integration over a fixed angular range is insufficient for accurately determining the observed integrated intensity ($I_{obs}$) of overlapping peaks. This tool employs a more robust intensity partitioning method.

At each point in the diffraction pattern, the net observed intensity ($y_{obs} - y_{bkg}$) is distributed among all contributing Bragg reflections. This distribution is proportional to the value of each peak's calculated profile function (including both Kα1 and Kα2 components, scaled by the refined peak height) at that specific point. By integrating these partitioned "slices" of intensity for each reflection across the entire pattern (using the trapezoidal rule), the method yields a reliable $I_{obs}$ value (reported as integrated area) that correctly deconvolutes contributions from neighboring peaks.

Visual Inspection

Numerical indicators can be misleading. Visual inspection of the difference plot (observed minus calculated) is the most critical step in evaluating the fit.

• The Difference Plot: A successful refinement should yield a difference plot that consists of random, uncorrelated noise centered on zero. The plot is scaled relative to the main pattern for visibility.
• Systematic Residuals: The presence of structured, non-random features in the difference plot (e.g., "M-shaped" residuals around peaks, broad humps where the spline is inadequate, or un-indexed peaks) is a clear indication of systematic errors in the model. These may arise from an incorrect peak shape, unmodeled anisotropy or asymmetry, an inadequate background model (requiring adjustment of spline points), or the presence of an unaccounted-for impurity phase.

Williamson-Hall Size-Strain Analysis

For refinements utilizing the TCH (Size/Strain/Aniso) profile function, the application can automatically perform a Williamson-Hall analysis to extract approximate microstructural information. This method separates the contributions of crystallite size and microstrain to the total peak broadening by analyzing their different dependencies on the diffraction angle, $\theta$.

The analysis is based on the linear Williamson-Hall equation, where $\beta$ is the total physical peak breadth (FWHM) in radians derived from the refined sample-only broadening parameters (U, X, Y), excluding instrumental contributions (V, W): $$\beta \cos(\theta) = \frac{K\lambda}{L} + 4\epsilon \sin(\theta)$$

This equation describes a straight line when plotting $\beta \cos(\theta)$ vs. $4\sin(\theta)$. The software performs a linear least-squares fit on the relevant data points (within the fitted 2θ range) to determine the y-intercept (related to crystallite size, $L$) and the slope (related to microstrain, $\epsilon$).

Two independent estimates are reported. The TCH profile already separates the two effects analytically — $Y$ broadens as $1/\cos\theta$ (size) while $X$ and $U$ broaden as $\tan\theta$ (strain) — so size and strain can be read straight off the refined parameters with no linear fit at all. The report prints these as from Y directly (nm) and from X,U directly (%) alongside the fitted Williamson-Hall values. Close agreement between the two is a good sign; a large discrepancy usually means the broadening is anisotropic and neither isotropic number should be trusted.

Reported Values

• Apparent Crystallite Size (nm): An estimate of the average size of the coherently scattering domains, calculated from the y-intercept of the Williamson-Hall plot (using $K=0.9$).
• Apparent Microstrain (%): An estimate of the root-mean-square strain within the crystallites, calculated from the slope of the plot.
• Linear Fit R²: The coefficient of determination for the linear regression. A value close to 1.0 indicates that the isotropic size/strain model is a good fit for the observed peak broadening. Values significantly less than 1.0 may suggest that broadening is anisotropic or that the model is otherwise inadequate.

Note: This Williamson-Hall analysis provides an approximation based only on the isotropic TCH parameters (U, X, Y). It does not account for anisotropic broadening effects (Stephens parameters) or instrumental contributions (V, W). For rigorous quantitative analysis, dedicated size/strain analysis software should be employed.

Data Export

• Save Report: Behavior depends on whether experimental data is loaded.
  • With data loaded, after a refinement: generates a comprehensive ASCII text file containing all statistical indicators, refined parameter values with their ESDs (for LM refinements), the Williamson-Hall analysis results (if applicable), the declared Lorentz–polarisation model, the list of background spline points used, the reflection table described below, and a point-by-point list of observed, calculated, background, and difference intensities across the fitted range.
  • With no data loaded: the button remains active and instead exports a theoretical reflection list for the current space group, lattice parameters, and wavelength(s) — h k l, $d$-spacing, and $2\theta$ for Kα1 (and Kα2 if the intensity ratio is greater than 0), together with the multiplicity, $Lp$ and $m \cdot Lp$ of each reflection. With no structure factors available, $m \cdot Lp$ is the whole of the predicted relative intensity. No refinement is required for this mode; the chart itself also displays this theoretical stick pattern live as you change the space group or lattice, so you can preview expected peak positions before ever loading a scan.
• Generate PDF: Creates a summary PDF document containing a high-resolution plot and tables of final parameters and statistics, suitable for archival or reporting. This requires a completed refinement. The PDF carries the same reflection table as the text report; rows too wide for the page are set in a smaller size rather than being clipped.

The reflection table

Both the text and PDF reports list every reflection inside the fitted range with the complete chain from peak to structure factor. Le Bail and Pawley share the same columns; Pawley adds one.

A legend above the table restates the polarisation model, the formulae and the value of $K$ actually used, so every $|F_o|$ can be reproduced from the printed numbers without knowing which convention the program follows.

References & Further Reading

About This Tool

The powder5 toolkit was developed by Nita Dragoe from Université Paris-Saclay as a simple browser-based implementation of powder pattern decomposition methods. It is a long-time successor of PowderV2 (Dragoe, N. (2001). J. Appl. Cryst., 34, 535) and has been updated to include the Pawley method and modern global optimization algorithms.

All numerical work — the linear solver, the Gauss-Jordan/Cholesky routines behind the covariance matrix and the ESDs, and the least-squares machinery itself — is implemented locally. An earlier version depended on math.js for these; it was removed so that a refinement can never be blocked by a content-delivery network, and so the application works fully offline. Charting uses Chart.js; all pan, zoom and pinch behaviour is implemented directly against the chart's scales, with no plugin. PDF generation uses jsPDF and html2canvas. Space-group symmetry data is derived from the Computational Crystallography Toolbox (cctbx).

This document was updated with the assistance of an AI.