Optimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction data

Khouchen, M.; Klar, P.B.; Chintakindi, H.; Suresh, A.; Palatinus, L.

doi:10.1107/S2053273323005053

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ISSN: 2053-2733

Volume 79| Part 5| September 2023| Pages 427-439

https://doi.org/10.1107/S2053273323005053

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Optimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction data

Malak Khouchen,^a Paul Benjamin Klar,^b Hrushikesh Chintakindi,^a Ashwin Suresh ^a and Lukas Palatinus ^a ^*

^aInstitute of Physics of the Czech Academy of Sciences, Prague, Czech Republic, and ^bUniversity of Bremen, Bremen, Germany
^*Correspondence e-mail: palat@fzu.cz

Edited by P. M. Dominiak, University of Warsaw, Poland (Received 9 February 2023; accepted 7 June 2023; online 14 August 2023)

Estimating the error in the merged reflection intensities requires a full understanding of all the possible sources of error arising from the measurements. Most diffraction-spot integration methods focus mainly on errors arising from counting statistics for the estimation of uncertainties associated with the reflection intensities. This treatment may be incomplete and partly inadequate. In an attempt to fully understand and identify all the contributions to these errors, three methods are examined for the correction of estimated errors of reflection intensities in electron diffraction data. For a direct comparison, the three methods are applied to a set of organic and inorganic test cases. It is demonstrated that applying the corrections of a specific model that include terms dependent on the original uncertainty and the largest intensity of the symmetry-related reflections improves the overall structure quality of the given data set and improves the final R_all factor. This error model is implemented in the data reduction software PETS2.

Keywords: error modelling; error analysis; data reduction; electron diffraction.

CCDC references: 2268164; 2268165; 2268166; 2268167; 2268168; 2268169; 2268170; 2268171; 2268172; 2268173; 2268174; 2268175; 2268176; 2268177; 2268178

1. Introduction

The use of electron diffraction (ED) for crystal structure determination has grown rapidly over the past decade, particularly thanks to the introduction of 3D methods for the systematic acquisition and analysis of diffracted intensities. 3D ED techniques have been shown to be powerful for structure determination of crystals that are too small for single-crystal X-ray diffraction analysis. These techniques benefit from the strong Coulomb interaction between electrons and matter. This allows 3D single-crystal ED to be obtained from nanocrystals that are about eight orders of magnitude smaller in volume than those needed for single-crystal X-ray diffraction. Several 3D ED techniques, which share the common concept of tilting a crystal around the goniometer axis and acquiring a series of ED patterns, have been presented and developed over the years. These include automated diffraction tomography (ADT) (Kolb et al., 2007 ), rotation electron diffraction (RED) (Zhang et al., 2010 ) and precession electron diffraction tomography (PEDT) (Mugnaioli et al., 2009 ). Since the rise of sensitive detectors with negligible readout time, continuous-rotation 3D ED has become the most popular protocol for data acquisition (Nederlof et al., 2013 ; Nannenga et al., 2014 ; Wang et al., 2017 ). Several software suites are available nowadays for 3D ED data reduction, thus allowing a 3D visualization of the data set, the determination of cell parameters, reconstruction of the 3D reciprocal lattice and extraction of reflection intensities with their estimated standard uncertainties (e.s.u.'s). Determining the e.s.u.'s of the reflection intensities is challenging. Error estimates of reflection intensities from electron-counting statistics alone may underestimate the real uncertainty associated with the measurements. The accurate estimation of e.s.u.'s is important throughout the process of crystallographic structure analysis. The result of structure refinement and the accuracy of the refined parameters depend on the correct estimation of the e.s.u.'s.

The determination of reflection e.s.u.'s is an important part of the data reduction process. It has turned out that pure counting-statistics-based estimates are not optimal, and other effects must be included in the determination of e.s.u.'s. The model describing the adjustment of e.s.u.'s is called the error model. The practice of refining the error model and correcting for various effects has a long history (Diamond, 1969 ; Abrahams & Keve, 1971 ; Rossmann et al., 1979 ; Schwarzenbach et al., 1989 ; Howell & Smith, 1992 ; Leslie, 1999 ; Evans, 2006 , 2011 ).

This paper aims to analyse methods of treating the error estimates of the integrated intensities from 3D ED data for the purpose of kinematical refinement and indicate the best error model. All the structures studied in this paper are processed using PETS2 (Palatinus et al., 2019 ) and refined using JANA2020 (Petříček et al., 2023 ), but the general ideas also apply to other implementations of 3D ED data reduction software.

2. Experimental, data processing and refinement setup

Throughout the text we use three data sets to assess and validate the presented methods. The materials are the natural zeolite natrolite, (S)-(+)-ibuprofen and the amino acid L-alanine.

2.1. Natrolite

Continuous-rotation 3D ED experiments were performed at different spots of the selected crystal which showed signs of mosaicity of about 0.15°. The crystal diffracted up to a resolution d* of about 1.6 Å⁻¹ at a temperature of 293 K (see Tables 1 and 2).

Table 1
Measurement conditions of natrolite

Microscope	FEI Tecnai G2 20
Detector (type)	Olympus SIS Veleta (CCD)
3D ED data sets	One continuous rotation
λ (Å)	0.02508
T (K)	293
$[\alpha _{\min}]$ , $[\alpha _{\max}]$ , Δα (°)	−50.0, 50.0, 0.6

Table 2
Sample overview of natrolite crystal

Empirical formula	Na₂(Al₂Si₃O₁₀)(H₂O)₂
Z	8
Crystal system	Orthorhombic
Space group	Fdd2
a, b, c (Å)	18.2872 (11), 18.6661 (14), 6.6222 (3)
α, β, γ (°)	90, 90, 90
V (Å³)	2260.5 (2)
d*max (Å⁻¹)	1.6
R_int(all)	19.80%
Mosaicity (°)	0.15
Completeness	99%

A centre of symmetry was added in the averaging process in JANA2020. The default weighting scheme with weights was $[w = 1/[\sigma (I_{\rm obs} )^2 + (0.01I_{\rm obs} )^2]]$ and an extinction correction was applied in the refinement. The model was refined against F² and against all reflections. The geometry of the water molecule was restrained to distances of 0.9584 Å between H atoms and their relative O atom and to an angle of 104.45°. All non-H atoms were refined with anisotropic displacement parameters (ADPs). A riding model was used for the ADPs of H atoms, with an extension factor of 1.2. Reference covalent bond lengths of non-H atoms for the calculation of the root-mean-square deviation (RMSD) were taken from a single-crystal X-ray diffraction (XRD) study on natrolite (Capitelli & Derebe, 2007 ).

2.2. (S)-(+)-ibuprofen

Continuous-rotation 3D ED experiments were performed at different spots of two selected crystals which showed signs of mosaicity of 0.135°. Two data sets were collected and then merged. The crystals diffracted up to a resolution of about 1 Å⁻¹ at a temperature of T = 95.15 K (see Tables 3 and 4).

Table 3
Measurement conditions of (S)-(+)-ibuprofen

Microscope	FEI Tecnai G2 20
Detector (type)	ASI Cheetah
3D ED data sets	Two continuous rotations
λ (Å)	0.02508
T (K)	95.15
$[\alpha _{\min}]$ , $[\alpha _{\max}]$ , Δα (°) #1	−55.0, 30.0, 0.25
$[\alpha _{\min}]$ , $[\alpha _{\max}]$ , Δα (°) #2	−50.0, 40.0, 0.25

Table 4
Sample overview of (S)-(+)-ibuprofen crystal

Empirical formula	C₁₃H₁₈O₂
Z	4
Crystal system	Monoclinic
Space group	P2₁
a, b, c (Å)	12.368 (4), 8.021 (3), 13.536 (5)
α, β, γ (°)	90, 112.24 (3), 90
V (Å³)	1242.9 (8)
d*max (Å⁻¹)	1.0
R_int(all)	16.59%
Mosaicity (°)	0.135
Completeness	85%

XRD-based reference atomic distances for the calculation of RMSD were taken from King et al. (2011 ).

2.3. L-alanine

Continuous-rotation 3D ED experiments were performed at different spots of a selected crystal which showed signs of mosaicity of 0.3°. One data set was collected and used. The crystal diffracted up to a resolution of about 2.0 Å⁻¹ at a temperature of T = 100 K (see Tables 5 and 6).

Table 5
Measurement conditions of L-alanine

Microscope	FEI Tecnai G2 20
Detector (type)	ASI Cheetah
3D ED data sets	One continuous rotation
λ (Å)	0.02508
T (K)	100
$[\alpha _{\min}]$ , $[\alpha _{\max}]$ , Δα (°)	−40.0, 40.0, 0.5

Table 6
Sample overview of L-alanine crystal

Empirical formula	C₃H₇NO₂
Z	4
Crystal system	Orthorhombic
Space group	P2₁2₁2₁
a, b, c (Å)	5.7733 (11), 5.9524 (12), 12.247 (2)
α, β, γ (°)	90, 90, 90
V (Å³)	420.85 (14)
d*max (Å⁻¹)	2.0
R_int(all)	11.11%
Mosaicity (°)	0.3
Completeness	56%

A centre of symmetry was added in JANA2020 in the averaging process. The default weighting scheme with weights was $[w = 1/[\sigma (I_{\rm obs})^2 + (0.01I_{\rm obs})^2]]$ and an extinction correction was applied in the refinement. The model was refined against F² and against all reflections. All non-H atoms were refined with ADPs. A riding model was used for the ADPs of H atoms, with an extension factor of 1.2.

XRD-based reference atomic distances for the calculation of RMSDs were taken from Parsons et al. (2013 ).

3. Methods of adjustment of error estimates

In this section, we examine three approaches for the correction of error estimates. In PETS2, the integral intensity of a reflection on a single diffraction pattern is calculated as

$[I = \textstyle \sum \limits_S {p_i} - \sum \limits_S {b_i},\eqno(1)]$

where p_i is the detector count in pixel i in the peak area S and b_i is the estimated background value for the same pixels. The summation runs over a region S. The background values are estimated from the detector count in the rim around the peak region S: $[\langle b\rangle = [1 /( n_{\rm rim})] \sum _{j = 1}^{n_{\rm rim}} {p_j}]$ , where n_rim is the number of pixels in the rim. The total estimated background in the region S is given by $[{n_S}\langle b\rangle]$ and the background-corrected integrated intensity can then be expressed as

$[I = \sum_{i = 1}^{{n_S}} {p_i} - {{{n_S}} \over {{n_{\rm rim}}}} \sum _{j = 1}^{{n_{\rm rim}}} {p_j}.\eqno(2)]$

PETS2 employs the following formula to calculate the e.s.u. of each pixel (Waterman & Evans, 2010 ):

$[\sigma ^2\left({{p_i}} \right) = G\gamma {p_i} + \psi, \eqno(3)]$

where G, γ and $[\psi]$ are the noise parameters characterizing each detector. γ is a `cascade factor', accounting for the intensity-dependent increase of variance above the Poisson-statistics value, and $[\psi]$ is a `pixel factor' which corresponds to the variance of pixel values of a dark image. G is the gain factor of the detector used that converts the number of incident electrons to the number of counts in the digitized diffraction image (detector-readout values).

Then, using the propagation-of-errors method, the variance of the integrated intensity can be expressed as

$[\eqalignno{\sigma ^2(I )& = G\gamma \left\{ \sum_{i = 1}^{{n_S}} {p_i} + \left({{n_S} \over {n_{\rm rim}}} \right)^2 \,\sum_ {j = 1}^{n_{\rm rim}} p_j \right\}&\cr &\quad + n_S\psi \left(1 + {{n_S} \over {n_{\rm rim}}}\right).&(4)}]$

This represents the variance in the integrated intensity, taking into account the Poisson noise and detector-related increase of the variance. It is crucial to have reasonable values of G, γ and $[\psi]$ to obtain accurate error estimates (Waterman & Evans, 2010).

However, these Poisson-based standard deviations underestimate the true e.s.u.'s, and additional adjustment needs to be made to these e.s.u.'s to address any additional uncertainty introduced by other sources of errors than the Poisson noise and detector, such as instrumental instability. One way to deal with these errors is to inflate the error estimates by adding extra terms that account for the additional uncertainty.

3.1. Methodology

The starting idea is based on the analysis of intensity distribution of multiply measured and symmetry-equivalent reflections. In an ideal data set, these reflections should have identical intensities. In practice, this is not the case due to, for example, statistical noise and non-kinematical scattering. The e.s.u.'s of individual reflections should thus properly characterize any deviation from the expected equality of multiply measured reflections. Based on this concept, various methods can be devised that produce better standard uncertainty estimates. Using such a method has become a de facto standard in modern data-processing software. However, to our knowledge, its validity for 3D ED data has not yet been investigated. There are at least two reasons why it cannot be automatically assumed that the method developed for X-ray diffraction data is also valid for 3D ED data. First, the nature of errors in the 3D ED data is different. The most significant deviations from the ideal kinematical intensities do not arise from random errors and instrumental effects, but from the dynamical diffraction effects. These effects are unavoidable in 3D ED data. Strictly speaking, these effects are not sources of errors in the data, and they should be modelled as part of the calculation of calculated intensities in the refinement process (Palatinus, Petříček & Corrêa, 2015 ; Palatinus, Correa et al., 2015 ). However, when kinematical approximation is used in the refinement, these effects effectively turn into errors in reflection intensities which, although not random, may be reflected in the values of e.s.u.'s. The second reason for special consideration is that, compared with X-ray diffraction data, the spread of the equivalent intensities around the mean value is typically much larger than in X-ray data. This is again caused mainly by the dynamical diffraction, but also often by radiation damage of the crystal, and it is reflected in increased values of R_int, which frequently reach 20% or more (Bruhn et al., 2021 ). This large spread may lead to larger corrections to e.s.u.'s and may induce effects that are not significant in X-ray diffraction data. The analysis in this paper shows this is indeed the case.

In the following, we analyse and compare three models for adjusting the e.s.u.'s of three different 3D ED experimental data sets. The efficiency of each method is assessed in a number of ways. We first evaluate the quality of each error model by checking the normality of the obtained distribution of residuals. A kinematical refinement is then performed and the refinement figures of merit are compared. The assessment is also based on comparing the RMSD of refined covalent bond lengths and the RMSD of atomic shifts for all non-H atoms with reference structures.

3.2. Model 0: no adjustment

For comparison purposes, the model with no adjustment to e.s.u.'s is also included in the analysis and it is denoted model 0. In this model, the e.s.u.'s are calculated using equation (4) without further modifications.

3.3. Model 1: using equivalent errors

In the first adjustment model, the reflection e.s.u.'s are calculated from the variation of symmetry-related intensities around their mean. The e.s.u. is calculated as a sample standard deviation from the n measurements of the symmetry-equivalent reflections. Given a reflection index h with n measurements of the intensity of h or its symmetry equivalent, we define the lth measurement of h as $[{I_{{\bf h}l}}]$ . All $[I_{{\bf h}l}]$ are associated with a common e.s.u. $[\sigma(I_{{\bf h}})]$ defined as

$[\sigma (I_{\bf h} ) = \sqrt {{{ \sum_{l = 1}^n (I_{{\bf h}l} - \langle{I_{\bf h}}\rangle)^2} \over {n - 1}}}.\eqno(5)]$

The division by instead of n is known as Bessel's correction, and it corrects for the bias in the estimation of the population standard deviation from only the sample of the population. The mean of all the symmetry-related measurements is used to estimate the reflection intensity $[\langle{I_{\bf h}}\rangle]$ :

$[\langle{I_{\bf h}}\rangle = {{\sum_{l = 1}^n I_{{\bf h}l}} \over n}.\eqno(6)]$

This model does not use the original Poisson counting error estimates and assumes that a large enough sample of equivalent reflections is available to reliably estimate the uncertainty. This assumption is often not well fulfilled because 3D ED data sets from low-symmetry crystals, in particular, exhibit a limited completeness and redundancy. In extreme cases, a number of reflections may be measured only once (n = 1), and their e.s.u.'s cannot be calculated using the above relation. To include these reflections in the structure refinement, we construct a lookup table which allows estimation of the e.s.u.'s of those reflections from the e.s.u.'s of other reflections with similar resolution and intensity. Firstly, we sort the reflections in order of increasing intensity and then we divide them into ten intensity bins with N / 10 reflections in each bin (N is the total number of reflections). The second binning divides the reflections into resolution bins with a bin width of 0.2 Å⁻¹. Then the average of e.s.u.'s of all reflections in the same bin is calculated and assigned as the e.s.u. of all individually measured reflections in the same bin. The residuals $[(I-\langle I \rangle)/\sigma(I)]$ in the original data (model 0) clearly deviate from a normal distribution as shown in Fig. 1(a) for natrolite, Fig. 2(a) for (S)-(+)-ibuprofen and Fig. 3(a) for L-alanine. Figs. 1(b), 2(b), 3(b) show the analogous figures corresponding to the adjusted data according to model 1. The straight horizontal segments (constant steps) in the normal probability plot of this model at the sample quantile of −0.707 and 0.707, and the peaks in the histograms of normalized deviations at ±0.707 are due to reflections measured twice (n = 2), as follows from equations (5), (6) and (8) for the special case of n = 2. This feature and the general mismatch between the distribution of the residuals and the expected normal distribution show that model 1 is not optimal for low-multiplicity data sets. The analysis of the structure refinements shows (Section 4.1) that this model leads to a worse (ibuprofen and L-alanine) or only marginally better (natrolite) structure model than model 0.

Figure 1
Distribution of the normalized deviations of each model (blue histogram) versus the normalized Gaussian distribution (in red) for natrolite. The blue curve is the best-fit Gaussian distribution to the histogram. The normal probability plot (blue dotted line) represents the normalized deviations versus the theoretical quantiles of natrolite for all the data with N = 5362 reflections (not including eight individual single reflections having zero delta). The straight line is the best-fit line to the normal probability plot. (a) Original model normalized deviations and normal probability plot of original uncorrected error estimates data (model 0). (b) Adjusted normalized deviations and normal probability plot of model 1. (c) Adjusted normalized deviations and normal probability plot of model 2. (d) Adjusted normalized deviations and normal probability plot of model 3.

Figure 2
Distribution of the normalized deviations of each model (blue histogram) versus the normalized Gaussian distribution (in red) for (S)-(+)-ibuprofen. The blue curve is the best-fit Gaussian distribution to the histogram. The normal probability plot (blue dotted line) represents the normalized deviations versus the theoretical quantiles of (S)-(+)-ibuprofen for all the data with N = 3855 reflections (not including 37 single reflections having zero delta). The straight line is the best-fit line to the normal probability plot. (a) Original model normalized deviations and normal probability plot of original uncorrected error estimates data (model 0). (b) Adjusted normalized deviations and normal probability plot of model 1. (c) Adjusted normalized deviations and normal probability plot of model 2. (d) Adjusted normalized deviations and normal probability plot of model 3.

Figure 3
Distribution of the normalized deviations of each model (blue histogram) versus the normalized Gaussian distribution (in red) for L-alanine. The blue curve is the best-fit Gaussian distribution to the histogram. The normal probability plot (blue dotted line) represents the normalized deviations versus the theoretical quantiles of L-alanine for all the data with N = 3255 reflections (not including 197 single reflections having zero delta). The straight line is the best-fit line to the normal probability plot. (a) Original model normalized deviations and normal probability plot of original uncorrected error estimates data (model 0). (b) Adjusted normalized deviations and normal probability plot of model 1. (c) Adjusted normalized deviations and normal probability plot of model 2. (d) Adjusted normalized deviations and normal probability plot of model 3.

3.4. Model 2: three-parameter model using average intensities of symmetry-related reflections

This model was first introduced by Evans (2006). It uses the normal probability plot (Abrahams & Keve, 1971) to adjust the error estimates of the integrated intensities. The model introduces a small number (two or three) of correction parameters that are used to modify the e.s.u.'s. The correction factors are optimized to make the normal probability plot as linear as possible. In this paper, we follow the notations of the three correction parameters from Brewster et al. (2019 ). The corrected error estimates are given by

$[\sigma '({I_{{\bf h}l}} ) = {s_{\rm fac}}\sqrt {\sigma ^2({I_{{\bf h}l}} ) + s_B^2\langle I_{\bf h}\rangle + s_{\rm add}^2(\langle I_{\bf h}\rangle )^2},\eqno(7)]$

with $[s_{\rm fac}]$ , s_B and $[s_{\rm add}]$ being the correction parameters. Two of the three parameters have a physical interpretation. According to Evans (2011), $[s_{\rm fac}]$ is understood as a correction factor for unknown errors independent of the intensity value including uncertainty in the detector gain used to estimate Poissonian errors. It acts like a scaling factor. $[s_{\rm add}]$ is a parameter that accounts for any errors that are proportional to the intensity such as instrument instabilities. s_B has no direct physical meaning and it is excluded in some programs [such as XDS (Kabsch, 2010 )], but it is an obvious addition to the parameter set, and in this work we include it to provide maximum flexibility of the fitting.

3.4.1. Normalized deviation

Estimates of error such as $[\sigma ({I_{{\bf h}l}} )]$ represent the statistically expected deviation of the measurement's intensity $[I_{{\bf h}l}]$ from the unknown population mean value. Assuming a Gaussian distribution of the intensity errors, the normalized deviations of the measurements $[I_{{\bf h}l}]$ from the mean value of the symmetry-related reflections $[\langle{I_{\bf h}}\rangle]$ are expected to be distributed according to a standard normal distribution. The normalized deviations $[\delta _{{\bf h}l}]$ for $[I_{{\bf h}l}]$ are given by

$[\delta _{{\bf h}l} = {{I_{{\bf h}l} - \langle I_{\bf h}\rangle} \over {\sigma ({I_{{\bf h}l}} )}}.\eqno(8)]$

According to Evans (2006), $[\langle I_{\bf h}\rangle]$ is the mean of the measurements of h excluding the lth reflection $[I_{{\bf h}l}]$ . In this work, we consider $[\langle I_{\bf h}\rangle]$ as the average intensity over all observations of reflection h including the lth reflection:

$[\langle I_{\bf h}\rangle = {{\sum_{l = 1}^n {I_{{\bf h}l}}} \over n}.\eqno(9)]$

This means that for reflections measured only once, where $[{I_{{\bf h}l}} = \langle I_{\bf h}\rangle]$ , the corresponding normalized deviation will be 0. This choice affects mainly the refinement of the $[{s_{\rm fac}}]$ parameter. In the approach used by Evans (2006), excluding a strong reflection from the calculation of $[\langle I_{\bf h}\rangle]$ , the resulting average intensity is reduced. The normalized deviation will consequently get larger, leading to larger error estimates, higher $[{s_{\rm fac}}]$ and consequently fewer observed reflections (see Section 5.1 for discussion of the meaning of observed reflections in 3D ED data). When all reflections are included in the calculation of the average, the $[{s_{\rm fac}}]$ parameter is reduced, leading to a larger number of observed reflections. In the work of Evans (2006), the average intensity $[\langle I_{\bf h}\rangle]$ is also a weighted average, where the weights are given by inverse variance estimates of the individual observations. This approach biases the average intensity towards the weaker reflections, which have, in general, lower e.s.u.'s and hence higher weights, leading to the same effects as excluding strong reflections from the average, discussed above. We also tested this approach and concluded that the non-weighted average $[\langle I_{\bf h}\rangle]$ gives better results.

The aim of the method is adjustment of the parameters $[{s_{\rm fac}}]$ , $[{s_{\rm add}}]$ and s_B to make the distribution of the normalized deviations as close as possible to the standard normal distribution, i.e. a Gaussian distribution centred on zero with a standard deviation of 1.

3.4.2. Normal probability plot technique

The values of the correction parameters $[{s_{\rm fac}}]$ , s_B and $[{s_{\rm add}}]$ can be conveniently determined by optimizing the normal probability plot (Abrahams & Keve, 1971). The normal probability plot is a plot of the sorted normalized deviations $[{\delta _i}]$ versus the perfectly distributed quantiles x_i expected for a normal distribution. The values of x_i are known as the theoretical quantiles and they correspond to a normal distribution of zero mean and standard deviation of 1. A normal probability plot of a variable with perfect standard normal distribution is a line with intercept 0 and slope 1. To find the matching normal distribution quantiles, we first calculate the cumulative distribution function (CDF) of the standard normal distribution. It is usually denoted by $[\Phi]$ and has the general form

$[\Phi (x ) = {1 \over {\sqrt {2\pi } }} \int \limits_{ - \infty }^x \exp( - {t^2}/2)\,{\rm d}t. \eqno(10)]$

The ith element of a sorted sample with standard normal distribution represents the value with $[\Phi (x ) = i /(N + 1)]$ . Conversely, the inverse of the CDF $[{x_i} = \Phi ^{ - 1}[i / (N + 1)] ]$ gives the expected value of the ith element of a sorted sample, i.e. the so-called theoretical quantile of the standard normal distribution.

The normal probability plot thus contains N points with coordinates $[\{\Phi ^{ - 1}[i /(N + 1)],{\delta _i}\}, i = 1,2, \ldots, N]$ , where $[{\delta _i}]$ is the ith normalized deviation in the list sorted from the smallest to the largest normalized deviation. The individual reflections having $[{\delta _i} = 0]$ are not included in the list. Figs. 1, 2 and 3 show normal probability plots for different error models for all three experimental data sets. The values of $[{s_{\rm fac}}]$ , s_B and $[{s_{\rm add}}]$ can be adjusted to make the normal probability plot as close to the ideal straight line with slope 1 as possible.

3.4.3. Initial parameters

Evans (2006) proposed obtaining the initial value of $[{s_{\rm fac}}]$ by fitting the slope of the central part of the normal probability plot in the theoretical quantiles range between −0.5 and 0.5. However, we observed that the least-squares fitting procedure is so robust that good convergence is obtained even if the fit starts from the default values $[{s_{\rm fac}} = 1]$ , s_B = 0 and $[{s_{\rm add}} = 0]$ .

3.4.4. Corrected error estimates

Using equation (7) for $[\sigma '({I_{{\bf h}l}} )]$ , we calculate the adjusted error estimates using the current set of correction parameters. Then, the new normalized deviations $[\delta '_{{\bf h}l}]$ are computed using

$[\delta '_{{\bf h}l} = {{I_{{\bf h}l} - \langle I_{\bf h}\rangle} \over {\sigma '(I_{{\bf h}l})}}.\eqno(11)]$

These normalized deviations are then sorted, and a new normal plot is calculated. To obtain optimal values of the correction parameters, we minimize the quantity $[\sum _{i = 1}^N (\delta '_i - {x_i})^2]$ , which designates the sum of the squared difference between the adjusted normalized deviations and the theoretical quantiles. Note that, after one minimization, the normalized deviations must be resorted, as their order may change.

After sorting, a new minimization must be performed, and the supercycle repeated until convergence. As the number of data points is large and the number of fitted parameters is small, the convergence is usually rapid and robust. Figs. 1(a), 2(a) and 3(a) show the original and Figs. 1(c), 2(c) and 3(c) the optimized normal probability plots and the histograms of the adjusted normalized deviations and their comparison with the standard normal distribution of natrolite, (S)-(+)-ibuprofen and L-alanine, respectively.

3.5. Model 3: three-parameter model using the largest intensities of symmetry-related reflections

Model 2 provides a clear improvement in comparison with model 1 or with unmodified e.s.u.'s. However, upon closer inspection, this model suffers from certain inadequacies. As shown in Figs. 1(c), 2(c) and 3(c), the normal probability plots representing the adjusted normalized deviations versus the theoretical quantiles in the case of model 2 do not match the line of slope 1, especially at the tails. By investigating the individual cases, we realized that model 2 provides poor results, especially for reflections that exhibit a considerable variation among the intensities of the symmetry-related reflections, i.e. if the variation is large compared with the intensity value itself. Upon optimization of the three error-model parameters, this model tries to compensate for this variation by assigning very large error estimates to very strong reflections. This leads to, among other effects, a notable reduction in the number of observed reflections. This problem is not very severe for typical X-ray diffraction data, where the symmetry-related reflections tend to have very similar intensities, but it is significant in 3D ED data, where the intensity variation can be very large. To correct this problem, we propose a new model (model 3), which is very similar to model 2 except that in this correction method we use the largest intensity $[I_{\bf h}^{\rm largest}]$ of the symmetry-related reflections for the calculations of the corrected error estimates rather than the average intensity $[\langle I_{\bf h}\rangle]$ . The corrected error estimates are thus given by

$[\sigma '({I_{{\bf h}l}} ) = s_{\rm fac}\sqrt {\sigma ^2(I_{{\bf h}l}) + s_B^2I_{\bf h}^{\rm largest} + s_{\rm add}^2 (I_{\bf h}^{\rm largest})^2}.\eqno(12)]$

Here $[I_{\bf h}^{\rm largest}]$ is the largest intensity of all the symmetry-related reflections of reflection h. The procedure for the optimization of the model parameters is the same as in model 2. The effect of this change is the decrease of the fitted values s_fac, s_B and s_add. For reflection groups with little variation around $[\langle I_{\bf h}\rangle]$ this means a smaller $[\sigma '({I_{{\bf h}l}} )]$ , because the error-model parameters are smaller and $[\langle I_{\bf h}\rangle \sim I_{\bf h}^{\rm largest}]$ . However, for reflection groups with large variation, the smaller values of the error-model parameters are compensated by replacing the (smaller) $[\langle I_{\bf h}\rangle]$ by the larger $[I_{\bf h}^{\rm largest}]$ . The net result is then a relative increase of the $[\sigma '({I_{{\bf h}l}} )]$ of reflection groups with a large variation compared with the reflection groups with a small variation.

Figs. 1(d), 2(d) and 3(d) show the final normal probability plots for this model. These can be compared with the other models. There is a pronounced improvement in the fitting of the normalized deviations of the error estimates. The improvement is also visible in the distribution of the normalized deviations of this model which shows that the adjusted normalized deviations are more accurately normally distributed in model 3 than in any other model including model 2.

3.5.1. Outlier rejection

There is usually a group of errors that do not match the statistical distribution. They are most frequently caused by unpredictable experimental effects that cannot be corrected for. Measurements affected by such errors are known as outliers (Blessing, 1997 ). It is advantageous to exclude such outliers from the final data set, as they most likely indicate an erroneous measurement that cannot be properly fitted by the structure model.

We tested a number of outlier rejection algorithms, including the algorithm used in SCALA (or AIMLESS) (Evans, 2006). Finally, we converged to the use of Tukey's simple but very robust rule of thumb that is based on the quartiles of the given data set (Tukey, 1977 ): firstly, calculate the first quartile Q₁ of $[\delta '_{{\bf h}l}]$ (25% of the normalized residuals are less than or equal to this value) and the third quartile Q₃ (25% of the $[\delta '_{{\bf h}l}]$ values are greater than or equal to this value). Outliers are then defined as all values that fall outside the range

$[\left [{Q_1} - k({Q_3} - {Q_1}),{Q_3} + k({Q_3} - {Q_1}) \right]. \eqno(13)]$

Tukey proposed using k = 1.5 to identify outliers.

As the exclusion of outliers has an impact on the normal probability plot and hence on the refined error model, an iterative procedure needs to be adopted. Firstly, consider the original data set before applying any corrections to the e.s.u.'s and apply Tukey's outlier rejection procedure as described above. In the case of only two equivalent reflections, if one is marked as an outlier, the other is marked too. Then, apply the error-model refinement to calculate the values of the correction parameters s_fac, s_B and s_add, excluding the outliers from the calculation. Use the obtained values to correct the e.s.u.'s of all the reflections including the outliers. Apply Tukey's outlier rejection again to the new data set with adjusted e.s.u.'s. Iterate the error-model refinement and outlier rejection until the values of the correction parameters s_fac, s_B and s_add converge and the number of identified outliers does not change. In the tests presented in this paper, the value of k = 1.5 proved to be an appropriate value. However, in specific cases, this parameter may be adjusted to increase or reduce the number of outliers, if needed.

Kinematical refinement was carried out on each of the three data sets with outliers rejected to demonstrate the potential impact of the above-described outlier rejection approach, using model 3. The algorithm rejected 35, 66 and 93 outliers in the data of natrolite, ibuprofen and L-alanine, respectively, representing 0.65%, 1.7% and 2.69% of all measured reflections. The kinematical refinement results after applying the above outlier rejection algorithm are shown in Table 10.

4. Results

To assess the efficiency of each of the models presented above, we apply the above error correction models to data from three materials, as described in Section 2. Each data set was processed using the software PETS2. All structures could be solved from the data by the charge-flipping algorithm as implemented in Superflip (Palatinus & Chapuis, 2007 ) and refinements within the kinematical approximation were performed in JANA2020. Input data from models 0, 1, 2 and 3 were subject to refinement using the same data processing procedure, the same starting structure model, the same refinement parameters etc. The only difference was in the e.s.u.'s assigned to the intensities based on individual error models. The residual factors $[R1_{\rm obs}]$ , $[R1_{\rm all}]$ and $[wR2_{\rm all}]$ were calculated by JANA2020 based on the common definitions:

$[R1 = {{\sum |\sqrt {I_{\rm obs}} - \sqrt {I_{\rm calc}}| } \over {\sum \sqrt {I_{\rm obs}} }}]$

$[wR2 = {{\sum w({I_{\rm obs}} - {I_{\rm calc}})^2} \over {\sum w{I_{\rm obs}}}}]$

$[w = {1 \over {\sigma ({I_{\rm obs}})^2 + (u{I_{\rm obs}})^2}},]$

where u is the instability factor and the sum runs over all reflections in the case of $[R_{\rm all}]$ and $[wR_{\rm all}]$ , and only over observed reflections with $[I_{\rm obs} \,\gt\, 3\sigma ({I_{\rm obs}} )]$ for the calculation of $[{R_{\rm obs}}]$ . $[{N_{\rm obs}}]$ is the number of reflections with $[I_{\rm obs} \,\gt\, 3\sigma ({I_{\rm obs}} )]$ , $[{N_{\rm all}}]$ is the total number of reflections used in the refinement, $[R1_{\rm obs}]$ is the conventional R factor (R1) based on $[{N_{\rm obs}}]$ observed reflections, $[wR2_{\rm all}]$ is the weighted R factor based on all reflections.

The application of the above weighting scheme is equivalent to changing the value of the coefficient $[s_{\rm add}^2]$ to $[s_{\rm add}^2 + {u^2}/s_{\rm fac}^2]$ . With the default value of u used in JANA2020 (u = 0.01, see Section 2) the change is negligible. We therefore decided to keep the default settings used in JANA2020.

The accuracy of the refined model is characterized by the RMSD of the covalent bond lengths for all non-H atoms from the respective reference values. Another assessment metric is based on the RMSD of atomic shifts of all non-H atoms from atom positions in the reference structures.

4.1. Refinement results

The kinematical refinement results are shown in Table 7 for natrolite, Table 8 for (S)-(+)-ibuprofen and Table 9 for L-alanine. These tables also contain the RMSD values of the refined covalent bond lengths for each model and of the distances of all non-H atoms from the positions in the reference structure. An important remark here is that the conventional $[{R_{\rm obs}}]$ is not a particularly good measure of the refinement quality, as different error models result in a different number of observed reflections (see Section 5 for more discussion on observed reflections). $[{R_{\rm obs}}]$ tends to increase with increasing $[{N_{\rm obs}}]$ . A more robust way of assessing the different models and the quality of data in 3D ED is to compare the factors $[{R_{\rm all}}]$ instead, which are directly comparable, as the complete set of intensities is the same for all models. Tables 7, 8 and 9 show that model 3 is the best error correction model for all tested data sets and across all comparison metrics, with a single exception of the RMSD of bond lengths for L-alanine, which is better for model 2 than model 3, but the difference is marginal.

Table 7
Kinematical refinement results of natrolite for the different error estimate models (N_par is the number of refinement parameters)

Natrolite	$[{N_{\rm all}}]$	$[{N_{\rm obs}}]$	$[{N_{\rm par}}]$	$[R{1_{\rm obs}}]$ (%)	$[R{1_{\rm all}}]$ (%)	$[wR{2_{\rm all}}]$ (%)	GOF(all)	RMSD of bond lengths (Å)	RMSD of atomic shifts (Å)
Original	1289	839	93	14.58	17.20	30.52	6.26	0.0211	0.0310
Model 1	1289	876	93	13.37	16.88	26.42	2.93	0.0194	0.0247
Model 2	1289	801	93	14.90	16.48	40.60	1.48	0.0120	0.0174
Model 3	1289	1007	93	14.67	16.09	36.34	1.78	0.0092	0.0151

Table 8
Kinematical refinement results of (S)-(+)-ibuprofen for the different error estimate models

(S)-(+)-ibuprofen	$[{N_{\rm all}}]$	$[{N_{\rm obs}}]$	$[{N_{\rm par}}]$	$[R{1_{\rm obs}}]$ (%)	$[R{1_{\rm all}}]$ (%)	$[wR{2_{\rm all}}]$ (%)	GOF(all)	RMSD of bond lengths (Å)	RMSD of atomic shifts (Å)
Original	1212	825	127	17.94	21.93	31.91	4.41	0.0711	0.1113
Model 1	1212	856	127	19.11	23.36	40.71	5.46	0.0786	0.1121
Model 2	1212	883	127	18.82	21.45	42.60	2.03	0.0547	0.0702
Model 3	1212	957	127	19.01	21.20	40.97	2.36	0.0498	0.0665

Table 9
Kinematical refinement results of L-alanine for the different error estimate models

L-alanine	$[{N_{\rm all}}]$	$[{N_{\rm obs}}]$	$[{N_{\rm par}}]$	$[R{1_{\rm obs}}]$ (%)	$[R{1_{\rm all}}]$ (%)	$[wR{2_{\rm all}}]$ (%)	GOF(all)	RMSD of bond lengths (Å)	RMSD of atomic shifts (Å)
Original	1127	1036	56	14.28	14.82	34.80	8.76	0.0143	0.0244
Model 1	1127	922	56	13.92	15.71	33.76	5.35	0.0194	0.0290
Model 2	1127	845	56	13.95	15.31	34.47	2.03	0.0070	0.0210
Model 3	1127	1072	56	14.44	14.73	35.51	2.57	0.0072	0.0195

4.1.1. Natrolite

In the case of natrolite, model 1 using the standard deviation method for the error estimates introduces an evident improvement in the R factors and the goodness-of-fit parameter compared with the original model (Table 7). It further gives the best $[R1_{\rm obs}]$ factor among all other models. However, as stated earlier, a good measure of comparison for the different techniques is $[R1_{\rm all}]$ rather than $[R1_{\rm obs}]$ . The RMSDs of bond lengths and of the atomic shifts are better than those of the original model. Fig. 1(b) shows the relative improvement in the normal probability plot as well as the distribution of the adjusted normalized deviations compared with the model 0 [Fig. 1(a)].

As for model 2, in the case of natrolite, the values of the three correction parameters are: $[{s_{\rm fac}} = 0.4063]$ , s_B = 0.14643 and $[{s_{\rm add}} = 1.24655]$ . From Table 7, it is obvious that there is a drop in $[{N_{\rm obs}}]$ and a dramatic increase in $[wR2_{\rm all}]$ . This is the main drawback of this model. The refined structure model of natrolite is considerably improved upon applying the corrected errors of model 2 as can be seen from the RMSD values.

The values of the three correction parameters in the case of natrolite, model 3, are: $[{s_{\rm fac}} = 0.2994]$ , s_B = 0.0464 and $[{s_{\rm add}} = 0.8604]$ . Model 3 introduces an overall improvement of all the parameters. Firstly, the number of observed reflections is the highest of all models. As compared with model 2, this illustrates the benefit of using the largest intensity of the symmetry-related reflections for adjusting the error estimates, rather than the average intensity. This is also clear from the normal plots of natrolite (Fig. 1) where the normal plot of model 3 [Fig. 1(d)] is the most favourable. The distribution of the normalized deviations of natrolite in Fig. 1(d) also provides the best fit to a standard normal distribution. The data in this case have more positive deviations than negative deviations and this explains the slight shift of the histogram to the left [Fig. 1(d)]. The $[R1_{\rm all}]$ value is the best as well and it is 1.11 percentage points less than that of the original model. More importantly, the refined structure of natrolite corresponding to this model is the most accurate when compared with the reference model (Capitelli & Derebe, 2007), showing the best RMSD values of the bond lengths and the atomic positions of all non-H atoms. The kinematical refinement of model 3 of natrolite after the outlier rejection introduces a slight improvement in the $[R1_{\rm all}]$ factor. The RMSD values for atomic shifts and bond lengths as shown in Table 10 are almost the same.

Table 10
Kinematical refinement results of the three data sets after the outlier rejection corresponding to model 3

	$[{N_{\rm all}}]$	$[{N_{\rm obs}}]$	$[{N_{\rm par}}]$	$[R{1_{\rm obs}}]$ (%)	$[R{1_{\rm all}}]$ (%)	$[wR{2_{\rm all}}]$ (%)	GOF(all)	RMSD of bond lengths (Å)	RMSD of atomic shifts (Å)
Natrolite	1289	1008	93	14.69	16.06	36.60	1.78	0.0093	0.0151
(S)-(+)-ibuprofen	1209	972	127	19.02	21.10	41.39	2.35	0.0489	0.0656
L-alanine	1123	927	56	13.06	14.16	30.49	2.36	0.0067	0.0193

4.1.2. Ibuprofen

In the case of (S)-(+)-ibuprofen, the situation is different regarding model 1. The latter does not present any improvement from the original model. On the contrary, the structure is slightly deformed. The refinement of the H atoms of hydroxyl groups was not stable. The aromatic carbon ring is also significantly distorted, leading to worse RMSD values as compared with the reference (King et al., 2011). This is obvious from the distribution of data [Fig. 2(b)]. The two peaks at −0.707 and 0.707 indicate a high number of symmetry-related reflections (19%) that are measured only twice (n = 2).

Regarding the results of model 2 in the case of (S)-(+)-ibuprofen, the three correction parameters are: $[{s_{\rm fac}}]$ = 0.712995, s_B = 0.15164 and $[{s_{\rm add}} = 0.42056]$ . $[{N_{\rm obs}}]$ is reasonable for this data set and it is not reduced as compared with model 0. The normal plot in Fig. 2(c) shows a noticeable improvement from the original normal plot [Fig. 2(a)], but it is still not a perfectly fitting line. A positive and a negative tail are present, in addition to some other slight deviations from the theoretical line of slope 1. $[R1_{\rm all}]$ and the goodness-of-fit factors are enhanced, while $[wR2_{\rm all}]$ still records the highest value in this model. The structure model is better than the original one based on the RMSD values.

Finally, for model 3 in the case of (S)-(+)-ibuprofen, the three correction parameters are: $[{s_{\rm fac}} = 0.5498]$ , s_B = 0.0051 and $[{s_{\rm add}} = 0.3667]$ . Again, this model has the largest number of observed reflections. The $[R1_{\rm all}]$ factor is reduced by 0.73 from that of the original model. The normal plot of the normalized deviations based on this model as shown in Fig. 2(d) is the nearest to normally distributed data. Fig. 2(d) also reveals the significance of this model in adjusting the sample data to better fit a standard Gaussian distribution. Additionally, the refined structure from model 3 is the most accurate among the others as indicated by the RMSD (Table 8). The kinematical refinement of model 3 after the outlier rejection in this case improves slightly the R factors (Table 10). The RMSDs for covalent bond lengths and atomic positions from the respective reference values improve as well after discarding the outliers.

4.1.3. L-alanine

In the case of L-alanine, model 1 does not introduce any improvement of model 0. On the contrary, the value of the $[R1_{\rm all}]$ factor is increased, and the refined structure is deformed rather than enhanced by comparing the RMSD of bond lengths and atomic shifts of non-H atoms (Table 9).

As for model 2 in the case of L-alanine, the three correction parameters are: $[{s_{\rm fac}} = 1.9409]$ , s_B = 0.0668 and $[{s_{\rm add}} = 0.11835]$ . The structure of this model has indeed the best value of RMSD of the bond lengths and the goodness-of-fit factor is enhanced, but the number of observed reflections is the lowest among the other models and the $[R1_{\rm all}]$ factor is also larger than that of model 0. This case again confirms the main drawbacks of this model. The normal plot in Fig. 3(c) shows a noticeable improvement from the original normal plot [Fig. 3(a)], but it is still not a perfectly fitting line. A recognizable positive tail is present in addition to some other slight deviations from the theoretical line of slope 1.

For model 3, this last data set again confirms that this is the best error correction model among the others. The $[R1_{\rm all}]$ factor and the number of observed reflections are the best and, above all, the structure has the lowest RMSD of atomic shifts as compared with the reference (Parsons et al., 2013). Fig. 3(d) shows a noticeable improvement in the normal probability plot and the distribution of the adjusted normalized deviations. The three correction parameters are now: $[{s_{\rm fac}} = 0.6094]$ , s_B = 0.0042 and $[{s_{\rm add}} = 0.2940]$ . The kinematical refinement results in the absence of outliers have indeed been improved as shown in Table 10. The $[R1_{\rm all}]$ factor decreases from 14.73 to 14.16. The RMSDs for bond lengths and atomic shifts have been improved slightly after rejecting the 93 outliers.

5. Discussion

5.1. Observed reflections

It is customary to present refinement characteristics, typically the unweighted R value R1, calculated only on sufficiently strong reflections. A typical criterion is I > 3σ(I). The rationale behind this tradition is that weak reflections contain essentially only noise, and they do not provide useful information on the quality of the fit. The term used for reflections stronger than the selected criterion is `observed reflections'. The term `observed' refers to the experiment, and to the fact that a reflection with I > 3σ(I) is usually visible in the diffraction pattern as a distinct intensity maximum, i.e. can be observed in the pattern. This is, however, meaningful only if σ(I) is calculated from counting statistics only. As soon as σ(I) is modified to account for other errors, the term `observed' loses its original meaning. Specifically, when σ(I) is significantly increased due to the error-model correction, a reflection, which is clearly visible in the diffraction pattern, becomes formally `unobserved', i.e. has I < 3σ(I). This is not a very big problem for typical X-ray diffraction data, where the corrections to the counting statistic σ(I) are generally small and affect mostly the strong reflections. However, it becomes a problem for data with dominant systematic errors, like those caused by the dynamical effects in 3D ED data. As an example, Fig. 4 shows an image of a reflection, which, after the error-model correction, has I = 1.85σ(I). Although the reflection is nominally unobserved, it is actually quite strong and clearly visible in the experimental data. The problem becomes even more serious when the `obs' values are used for comparison between refinements. Different error models lead to different corrections to σ(I), hence to a different number of reflections with I > 3σ(I) and, as a consequence, to incomparable values of R(obs) and other obs-related statistics. As an example, error correction according to model 1 for natrolite yields 876 observed reflections with I > 3σ(I) and R1(obs) of 13.37%, while model 3 gives 1007 observed reflections (out of 1279) and R1(obs) of 14.67%. Thus, superficially, model 1 may appear to give a significantly better result, but it is just an artefact of the number of observed reflections. R1(all) as well as other statistics clearly show that the result from model 3 is superior.

Figure 4
An image of a reflection (−1, −5, −9) from the diffraction pattern of the L-alanine data set of model 3 that is clearly visible, although it has I < 3σ(I) in the correction model 3.

One could thus conclude that the use of the term `observed reflection' and related quantities is not meaningful for 3D ED data and should be discontinued. Other methods of estimating the amount of information present, e.g. the correlation-based techniques commonly used in macromolecular crystallography, may be more suitable. Until then, the scientists working with these data should be aware of the caveat just described, and use the term `observed reflections' with caution and with awareness of its limitations.

5.2. Features of the error correction models

A thorough comparison of the refinements reveals that in all cases the set of adjustments propagated in error model 3 gives significantly improved results.

Model 1 did not present any improvement in the cases of (S)-(+)-ibuprofen and L-alanine. In the case of natrolite, the structure is slightly enhanced. Model 1 is expected to be more successful with data with a high multiplicity of symmetry-related reflections since the only information it is based on is the observed variation among these reflections. Stated differently, estimates of the individual errors of this model, derived only from the standard error of the mean of the reflections, become less adequate in the case of data with a low redundancy of symmetry-related reflections. This in turn may result in serious inaccuracy of the refined structure model. (S)-(+)-ibuprofen for instance has a monoclinic crystal system, and thus many reflections have a low redundancy of 2. This explains the significant improvement in the case of natrolite, which has an orthorhombic crystal system of higher redundancy. In the case of L-alanine, although it has an orthorhombic crystal system, many reflections of this data set (25%) have a multiplicity of n = 2 due to the low completeness of the data set. This explains the inefficiency of model 1 in this case.

Model 2 clearly provides an improvement, but it still has some deficiencies that need to be worked through. In some cases (natrolite and L-alanine) an obvious drop in $[{N_{\rm obs}}]$ was evident. By looking at the correction parameters, we notice that in both cases one of these parameters is larger than 1 ( $[{s_{\rm add}} = 1.24655]$ for natrolite and $[{s_{\rm fac}} = 1.9409]$ for L-alanine). This signifies a substantial inflation of error estimates. This inflation makes the model less reliable in many situations. In the case of ibuprofen, all the correction parameters are less than 1 and the inflation of the e.s.u.'s is not so dramatic. This is due to the fact that ibuprofen is monoclinic. The variations among the intensities of the symmetry-related reflections are less prominent. This means the values of the correction parameters are not so large and thus $[{N_{\rm obs}}]$ remains reasonable for this data set and does not need to be reduced.

In fact, the correction parameters $[{s_{\rm fac}}]$ and $[{s_{\rm add}}]$ of a well processed data set should have their values close to 1 and 0, respectively. s_B is usually negligible with a value close to 0. The exact values will certainly depend on the Laue class of the corresponding data set, the redundancy of the symmetry-related reflections and the variations among their intensities. It is worth noting that the three refined parameters of model 3 in general have lower values than their analogues of model 2 for all three samples. In model 3, less compensation is needed to correct for the gain detector uncertainty and other types of errors, owing to the use of the largest intensity of symmetry-related observations instead. Model 3 adjusts the error estimates of the strongest reflections without unnecessary exaggeration of the e.s.u.'s. It provides a good compromise between adjusting the error estimates and maintaining a decent number of observed reflections.

6. Conclusion

Various models for adjustment of estimated standard uncertainties of reflection intensities in 3D ED data were investigated with the aim of verifying if the models commonly used for single-crystal X-ray diffraction data are also suitable for 3D ED data. The tests on three experimental data sets showed that the best model is model 3, which differs from the commonly used approach by employing the maximum of the symmetry-equivalent intensities in the calculation of normalized residuals rather than the average value.

It is not surprising that accurate estimates of the e.s.u.'s are useful, but it is notable how much improvement model 3 brings to the kinematical refinement compared with the case with no error-model adjustment, but also with the other tested models. The benefits of using the model include an overall enhanced accuracy of atomic positions, covalent bond lengths and improved R factors. The benefits of model 3 are expected to be most pronounced in data with low redundancy and large variation in the intensities of symmetry-related reflections. It may be expected that with data obtained by averaging a large number of individual data sets, an approach becoming more and more popular in contemporary 3D ED studies, the differences between the models would become smaller.

The procedure according to model 3 is implemented in the software package PETS2 (Palatinus et al., 2019), available at https://pets.fzu.cz/.

Supporting information

CCDC references: 2268164; 2268165; 2268166; 2268167; 2268168; 2268169; 2268170; 2268171; 2268172; 2268173; 2268174; 2268175; 2268176; 2268177; 2268178

Crystal structure: contains datablocks global, ibuprofen_model0, ibuprofen_model1, ibuprofen_model2, ibuprofen_model3, ibuprofen_model3_outliers, L-alanine_model0, L-alanine_model1, L-alanine_model2, L-alanine_model3, L-alanine_model3_outliers, natrolite_model0, natrolite_model1, natrolite_model2, natrolite_model3, natrolite_model3_outliers. DOI: https://doi.org/10.1107/S2053273323005053/pl5027sup1.cif

Structure factors: contains datablock ibuprofen_model0. DOI: https://doi.org/10.1107/S2053273323005053/pl5027ibuprofen_model0sup2.hkl

Structure factors: contains datablock ibuprofen_model1. DOI: https://doi.org/10.1107/S2053273323005053/pl5027ibuprofen_model1sup3.hkl

Structure factors: contains datablock ibuprofen_model2. DOI: https://doi.org/10.1107/S2053273323005053/pl5027ibuprofen_model2sup4.hkl

Structure factors: contains datablock ibuprofen_model3. DOI: https://doi.org/10.1107/S2053273323005053/pl5027ibuprofen_model3sup5.hkl

Structure factors: contains datablock L-alanine_model0. DOI: https://doi.org/10.1107/S2053273323005053/pl5027L-alanine_model0sup6.hkl

Structure factors: contains datablock L-alanine_model1. DOI: https://doi.org/10.1107/S2053273323005053/pl5027L-alanine_model1sup7.hkl

Structure factors: contains datablock L-alanine_model2. DOI: https://doi.org/10.1107/S2053273323005053/pl5027L-alanine_model2sup8.hkl

Structure factors: contains datablock L-alanine_model3. DOI: https://doi.org/10.1107/S2053273323005053/pl5027L-alanine_model3sup9.hkl

Structure factors: contains datablock natrolite_model0. DOI: https://doi.org/10.1107/S2053273323005053/pl5027natrolite_model0sup10.hkl

Structure factors: contains datablock natrolite_model1. DOI: https://doi.org/10.1107/S2053273323005053/pl5027natrolite_model1sup11.hkl

Structure factors: contains datablock natrolite_model2. DOI: https://doi.org/10.1107/S2053273323005053/pl5027natrolite_model2sup12.hkl

Structure factors: contains datablock natrolite_model3. DOI: https://doi.org/10.1107/S2053273323005053/pl5027natrolite_model3sup13.hkl

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S2053273323005053/pl5027ibuprofen_model3_outlierssup14.hkl

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S2053273323005053/pl5027L-alanine_model3_outlierssup15.hkl

Structure factors: contains datablock I. DOI: https://doi.org/10.1107/S2053273323005053/pl5027natrolite_model3_outlierssup16.hkl

Computing details top

For all structures, data collection: self-written scripts, OLYMPUS iTEM. Cell refinement: PETS2(ver 2.2.20220612.1425) for ibuprofen_model0, ibuprofen_model1, ibuprofen_model2, ibuprofen_model3, ibuprofen_model3_outliers; PETS2 for L-alanine_model0, L-alanine_model1, L-alanine_model2, L-alanine_model3, L-alanine_model3_outliers, natrolite_model0, natrolite_model1, natrolite_model2, natrolite_model3, natrolite_model3_outliers. Data reduction: PETS2(ver 2.2.20220612.1425) for ibuprofen_model0, ibuprofen_model1, ibuprofen_model2, ibuprofen_model3, ibuprofen_model3_outliers; PETS2 for L-alanine_model0, L-alanine_model1, L-alanine_model2, L-alanine_model3, L-alanine_model3_outliers, natrolite_model0, natrolite_model1, natrolite_model2, natrolite_model3, natrolite_model3_outliers. Program(s) used to solve structure: superflip for ibuprofen_model0, ibuprofen_model1, ibuprofen_model2, ibuprofen_model3, ibuprofen_model3_outliers, L-alanine_model0, L-alanine_model1, L-alanine_model2, L-alanine_model3, L-alanine_model3_outliers. Program(s) used to refine structure: JANA2020 for ibuprofen_model0, ibuprofen_model1, ibuprofen_model2, ibuprofen_model3, ibuprofen_model3_outliers; Jana2020 for L-alanine_model0, L-alanine_model1, L-alanine_model2, L-alanine_model3, L-alanine_model3_outliers, natrolite_model0, natrolite_model1, natrolite_model2, natrolite_model3, natrolite_model3_outliers. For all structures, molecular graphics: VESTA 3.

(S)-(+)-Ibuprofen (ibuprofen_model0) top

Crystal data top

C₁₃H₁₈O₂	Z = 4
M_r = 206.3	F(000) = 184.42
Monoclinic, P2₁	D_x = 1.102 Mg m⁻³
Hall symbol: P 2yb	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 12.368 (4) Å	Cell parameters from 3884 reflections
b = 8.021 (3) Å	θ = 0.1–0.7°
c = 13.536 (5) Å	T = 95 K
β = 112.24 (3)°	Block
V = 1242.9 (8) Å³

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.166
Radiation source: Lab6 cathode	θ_max = 0.7°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −12→12
3884 measured reflections	k = −8→8
1212 independent reflections	l = −13→13
825 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H atoms treated by a mixture of independent and constrained refinement
R[F > 3σ(F)] = 0.179	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F) = 0.319	(Δ/σ)_max = 0.047
S = 4.41	Δρ_max = 0.25 e Å⁻³
1212 reflections	Δρ_min = −0.24 e Å⁻³
127 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 2.4 (3)
139 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1a	0.344 (3)	0.514 (4)	1.164 (2)	0.059 (8)*
O2a	0.240 (3)	0.677 (3)	1.048 (2)	0.060 (8)*
H1a	0.317 (6)	0.730 (15)	1.080 (11)	0.12*
C1a	0.247 (3)	0.531 (4)	1.091 (3)	0.051 (8)*
C2a	0.149 (2)	0.397 (3)	1.066 (2)	0.040 (7)*
H2a	0.150229	0.30286	1.121346	0.0793*
C3a	0.164 (2)	0.325 (3)	0.9677 (19)	0.027 (6)*
C4a	0.239 (2)	0.171 (3)	0.987 (2)	0.037 (7)*
H4a	0.284447	0.121734	1.064232	0.0745*
C5a	0.245 (3)	0.095 (4)	0.891 (2)	0.043 (7)*
H5a	0.311599	0.005793	0.899672	0.0864*
C6a	0.169 (2)	0.1332	0.7920 (17)	0.010 (5)*
C7a	0.106 (3)	0.271 (3)	0.780 (2)	0.045 (8)*
H7a	0.070872	0.319979	0.701662	0.0902*
C8a	0.079 (3)	0.360 (4)	0.8546 (19)	0.030 (6)*
H8a	0.007645	0.443457	0.835449	0.0602*
C9a	0.167 (2)	0.034 (3)	0.6914 (16)	0.021 (6)*
H9a1	0.165206	0.118365	0.630676	0.0416*
H9a2	0.2464	−0.033998	0.710937	0.0416*
C10a	0.063 (2)	−0.087 (3)	0.6476 (18)	0.038 (7)*
H10a	−0.015368	−0.022613	0.637023	0.0757*
C11a	0.071 (3)	−0.156 (4)	0.547 (2)	0.061 (8)*
H11a	0.113587	−0.068615	0.515919	0.1216*
H11b	0.119272	−0.26894	0.565218	0.1216*
H11c	−0.014171	−0.179554	0.490623	0.1216*
C12a	0.067 (3)	−0.234 (3)	0.722 (2)	0.042 (7)*
H12a	0.005104	−0.325295	0.679731	0.0837*
H12b	0.152067	−0.287625	0.750667	0.0837*
H12c	0.048121	−0.190607	0.787662	0.0837*
C13a	0.038 (3)	0.489 (4)	1.052 (3)	0.078 (10)*
H13a	−0.032147	0.403167	1.029827	0.1556*
H13b	0.046829	0.547217	1.124998	0.1556*
H13c	0.022156	0.580348	0.991843	0.1556*
O2b	0.534 (3)	−0.294 (3)	0.224 (2)	0.040 (6)*
O1b	0.412 (2)	−0.133 (3)	0.100 (2)	0.041 (6)*
H1b	0.462 (5)	−0.359 (12)	0.206 (10)	0.0806*
C1b	0.508 (3)	−0.158 (4)	0.173 (3)	0.051 (8)*
C2b	0.605 (3)	−0.036 (4)	0.190 (2)	0.071 (10)*
H2b	0.677977	−0.110467	0.195386	0.1417*
C3b	0.609 (3)	0.072 (4)	0.294 (2)	0.045 (8)*
C4b	0.686 (3)	−0.001 (4)	0.397 (2)	0.042 (7)*
H4b	0.735548	−0.111027	0.401697	0.0833*
C5b	0.689 (2)	0.082 (3)	0.4872 (19)	0.025 (6)*
H5b	0.748524	0.041844	0.562622	0.0506*
C6b	0.612 (3)	0.225 (3)	0.483 (2)	0.048 (8)*
C7b	0.552 (3)	0.301 (4)	0.388 (2)	0.041 (7)*
H7b	0.513453	0.420727	0.379418	0.0827*
C8b	0.548 (3)	0.190 (3)	0.294 (2)	0.038 (7)*
H8b	0.482951	0.220726	0.218468	0.0767*
C9b	0.638 (2)	0.311 (3)	0.5903 (16)	0.026 (6)*
H9b1	0.652055	0.220411	0.650716	0.0511*
H9b2	0.562099	0.371268	0.59037	0.0511*
C10b	0.741 (3)	0.436 (3)	0.6260 (19)	0.042 (8)*
H10b	0.815415	0.366829	0.629367	0.0835*
C11b	0.752 (3)	0.518 (4)	0.740 (2)	0.066 (9)*
H11d	0.728972	0.427322	0.785553	0.1311*
H11e	0.695289	0.621451	0.725844	0.1311*
H11f	0.839378	0.55695	0.782082	0.1311*
C12b	0.726 (3)	0.579 (4)	0.5518 (19)	0.049 (7)*
H12d	0.796882	0.662814	0.584632	0.0972*
H12e	0.723823	0.534224	0.477342	0.0972*
H12f	0.646719	0.641323	0.540533	0.0972*
C13b	0.603 (2)	0.081 (3)	0.1053 (19)	0.034 (6)*
H13d	0.517116	0.126244	0.064935	0.0672*
H13e	0.659822	0.182806	0.140009	0.0672*
H13f	0.631702	0.019313	0.050272	0.0672*

Geometric parameters (Å, º) top

O1a—C1a	1.24 (4)	O2b—H1b	0.98 (8)
O2a—H1a	0.98 (9)	O2b—C1b	1.27 (4)
O2a—C1a	1.30 (4)	O1b—C1b	1.24 (4)
C1a—C2a	1.56 (4)	C1b—C2b	1.50 (5)
C2a—H2a	1.06	C2b—H2b	1.06
C2a—C3a	1.53 (4)	C2b—C3b	1.64 (5)
C2a—C13a	1.50 (5)	C2b—C13b	1.47 (4)
C3a—C4a	1.51 (4)	C3b—C4b	1.49 (4)
C3a—C8a	1.52 (3)	C3b—C8b	1.21 (4)
C4a—H4a	1.06	C4b—H4b	1.06
C4a—C5a	1.46 (4)	C4b—C5b	1.38 (4)
C5a—H5a	1.06	C5b—H5b	1.06
C5a—C6a	1.35 (3)	C5b—C6b	1.48 (4)
C6a—C7a	1.33 (3)	C6b—C7b	1.37 (4)
C6a—C9a	1.57 (3)	C6b—C9b	1.53 (4)
C7a—H7a	1.06	C7b—H7b	1.06
C7a—C8a	1.37 (4)	C7b—C8b	1.54 (4)
C8a—H8a	1.06	C8b—H8b	1.06
C9a—H9a1	1.06	C9b—H9b1	1.06
C9a—H9a2	1.06	C9b—H9b2	1.06
C9a—C10a	1.54 (4)	C9b—C10b	1.54 (4)
C10a—H10a	1.06	C10b—H10b	1.06
C10a—C11a	1.50 (5)	C10b—C11b	1.63 (4)
C10a—C12a	1.54 (4)	C10b—C12b	1.49 (4)
C11a—H11a	1.06	C11b—H11d	1.06
C11a—H11b	1.06	C11b—H11e	1.06
C11a—H11c	1.06	C11b—H11f	1.06
C12a—H12a	1.06	C12b—H12d	1.06
C12a—H12b	1.06	C12b—H12e	1.06
C12a—H12c	1.06	C12b—H12f	1.06
C13a—H13a	1.06	C13b—H13d	1.06
C13a—H13b	1.06	C13b—H13e	1.06
C13a—H13c	1.06	C13b—H13f	1.06

H1a—O2a—C1a	108 (7)	H1b—O2b—C1b	108 (6)
O1a—C1a—O2a	110 (3)	O2b—C1b—O1b	123 (3)
O1a—C1a—C2a	122 (3)	O2b—C1b—C2b	117 (3)
O2a—C1a—C2a	127 (3)	O1b—C1b—C2b	119 (3)
C1a—C2a—H2a	122.57	C1b—C2b—H2b	104.91
C1a—C2a—C3a	96 (3)	C1b—C2b—C3b	104 (3)
C1a—C2a—C13a	107 (2)	C1b—C2b—C13b	122 (2)
H2a—C2a—C3a	111.55	H2b—C2b—C3b	119.03
H2a—C2a—C13a	101.9	H2b—C2b—C13b	100.27
C3a—C2a—C13a	119 (2)	C3b—C2b—C13b	108 (2)
C2a—C3a—C4a	116 (2)	C2b—C3b—C4b	114 (3)
C2a—C3a—C8a	123 (2)	C2b—C3b—C8b	127 (3)
C4a—C3a—C8a	118 (2)	C4b—C3b—C8b	119 (3)
C3a—C4a—H4a	122.41	C3b—C4b—H4b	122.13
C3a—C4a—C5a	115 (2)	C3b—C4b—C5b	116 (3)
H4a—C4a—C5a	122.41	H4b—C4b—C5b	122.13
C4a—C5a—H5a	118.78	C4b—C5b—H5b	118.51
C4a—C5a—C6a	122 (3)	C4b—C5b—C6b	123 (2)
H5a—C5a—C6a	118.78	H5b—C5b—C6b	118.51
C5a—C6a—C7a	118 (2)	C5b—C6b—C7b	119 (3)
C5a—C6a—C9a	121 (2)	C5b—C6b—C9b	114 (2)
C7a—C6a—C9a	120 (2)	C7b—C6b—C9b	123 (2)
C6a—C7a—H7a	115.13	C6b—C7b—H7b	124.43
C6a—C7a—C8a	130 (3)	C6b—C7b—C8b	111 (3)
H7a—C7a—C8a	115.13	H7b—C7b—C8b	124.43
C3a—C8a—C7a	112 (2)	C3b—C8b—C7b	129 (2)
C3a—C8a—H8a	123.97	C3b—C8b—H8b	115.58
C7a—C8a—H8a	123.97	C7b—C8b—H8b	115.58
C6a—C9a—H9a1	109.47	C6b—C9b—H9b1	109.47
C6a—C9a—H9a2	109.47	C6b—C9b—H9b2	109.47
C6a—C9a—C10a	113 (2)	C6b—C9b—C10b	117 (3)
H9a1—C9a—H9a2	105.73	H9b1—C9b—H9b2	101.14
H9a1—C9a—C10a	109.47	H9b1—C9b—C10b	109.47
H9a2—C9a—C10a	109.47	H9b2—C9b—C10b	109.47
C9a—C10a—H10a	109.43	C9b—C10b—H10b	105.71
C9a—C10a—C11a	104 (2)	C9b—C10b—C11b	109 (3)
C9a—C10a—C12a	114.3 (18)	C9b—C10b—C12b	114 (2)
H10a—C10a—C11a	115.33	H10b—C10b—C11b	114.07
H10a—C10a—C12a	105.52	H10b—C10b—C12b	108.51
C11a—C10a—C12a	108 (2)	C11b—C10b—C12b	106 (2)
C10a—C11a—H11a	109.47	C10b—C11b—H11d	109.47
C10a—C11a—H11b	109.47	C10b—C11b—H11e	109.47
C10a—C11a—H11c	109.47	C10b—C11b—H11f	109.47
H11a—C11a—H11b	109.47	H11d—C11b—H11e	109.47
H11a—C11a—H11c	109.47	H11d—C11b—H11f	109.47
H11b—C11a—H11c	109.47	H11e—C11b—H11f	109.47
C10a—C12a—H12a	109.47	C10b—C12b—H12d	109.47
C10a—C12a—H12b	109.47	C10b—C12b—H12e	109.47
C10a—C12a—H12c	109.47	C10b—C12b—H12f	109.47
H12a—C12a—H12b	109.47	H12d—C12b—H12e	109.47
H12a—C12a—H12c	109.47	H12d—C12b—H12f	109.47
H12b—C12a—H12c	109.47	H12e—C12b—H12f	109.47
C2a—C13a—H13a	109.47	C2b—C13b—H13d	109.47
C2a—C13a—H13b	109.47	C2b—C13b—H13e	109.47
C2a—C13a—H13c	109.47	C2b—C13b—H13f	109.47
H13a—C13a—H13b	109.47	H13d—C13b—H13e	109.47
H13a—C13a—H13c	109.47	H13d—C13b—H13f	109.47
H13b—C13a—H13c	109.47	H13e—C13b—H13f	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O2a—H1a···O1bⁱ	0.98 (9)	1.56 (11)	2.49 (4)	158 (11)
O2a—H1a···C1bⁱ	0.98 (9)	2.39 (9)	3.37 (4)	174 (12)
O2b—H1b···O1aⁱⁱ	0.98 (8)	1.69 (8)	2.67 (4)	173 (12)

Symmetry codes: (i) x, y+1, z+1; (ii) x, y−1, z−1.

(S)-(+)-Ibuprofen (ibuprofen_model1) top

Crystal data top

C₁₃H₁₈O₂	Z = 4
M_r = 206.3	F(000) = 184.42
Monoclinic, P2₁	D_x = 1.102 Mg m⁻³
Hall symbol: P 2yb	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 12.368 (4) Å	Cell parameters from 3884 reflections
b = 8.021 (3) Å	θ = 0.1–0.7°
c = 13.536 (5) Å	T = 95 K
β = 112.24 (3)°	Block
V = 1242.9 (8) Å³

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.166
Radiation source: Lab6 cathode	θ_max = 0.7°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −12→12
3884 measured reflections	k = −8→8
1212 independent reflections	l = −13→13
856 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H atoms treated by a mixture of independent and constrained refinement
R[F > 3σ(F)] = 0.191	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F) = 0.407	(Δ/σ)_max = 1.619
S = 5.46	Δρ_max = 0.23 e Å⁻³
1212 reflections	Δρ_min = −0.27 e Å⁻³
127 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 2.0 (4)
139 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1a	0.354 (2)	0.509 (5)	1.170 (2)	0.065 (7)*
O2a	0.221 (3)	0.684 (5)	1.040 (2)	0.072 (8)*
H1a	0.304 (6)	0.72 (2)	1.067 (17)	0.1444*
C1a	0.248 (2)	0.538 (5)	1.093 (2)	0.063 (9)*
C2a	0.155 (3)	0.407 (5)	1.063 (2)	0.058 (8)*
H2a	0.156828	0.309315	1.115988	0.1168*
C3a	0.1707 (17)	0.324 (4)	0.9688 (17)	0.026 (6)*
C4a	0.2283 (19)	0.175 (4)	0.9856 (19)	0.033 (6)*
H4a	0.260951	0.11825	1.061946	0.0661*
C5a	0.244 (2)	0.094 (4)	0.8891 (18)	0.038 (7)*
H5a	0.311882	0.0068	0.900557	0.0759*
C6a	0.170 (2)	0.1332	0.7895 (17)	0.030 (7)*
C7a	0.098 (2)	0.272 (4)	0.779 (2)	0.027 (6)*
H7a	0.045285	0.313079	0.701064	0.0534*
C8a	0.0932 (19)	0.352 (4)	0.8602 (16)	0.022 (5)*
H8a	0.027634	0.444316	0.845671	0.0446*
C9a	0.1823 (17)	0.032 (4)	0.6965 (16)	0.028 (5)*
H9a1	0.188093	0.114757	0.637819	0.0563*
H9a2	0.256648	−0.046642	0.72702	0.0563*
C10a	0.060 (2)	−0.088 (4)	0.6425 (19)	0.050 (8)*
H10a	−0.021005	−0.027668	0.628838	0.1006*
C11a	0.057 (2)	−0.146 (4)	0.5391 (18)	0.042 (6)*
H11a	0.130565	−0.223385	0.551067	0.0848*
H11b	−0.020527	−0.215562	0.50007	0.0848*
H11c	0.059106	−0.042561	0.491425	0.0848*
C12a	0.080 (4)	−0.228 (6)	0.723 (3)	0.091 (12)*
H12a	0.059194	−0.343704	0.682488	0.1812*
H12b	0.168675	−0.228227	0.775992	0.1812*
H12c	0.025846	−0.209569	0.767083	0.1812*
C13a	0.037 (3)	0.494 (5)	1.054 (3)	0.077 (10)*
H13a	−0.032258	0.405684	1.028417	0.1542*
H13b	0.045257	0.540972	1.130241	0.1542*
H13c	0.01746	0.593616	0.998867	0.1542*
O2b	0.541 (3)	−0.302 (5)	0.226 (2)	0.067 (8)*
O1b	0.409 (3)	−0.137 (5)	0.103 (2)	0.069 (8)*
H1b	0.469 (9)	−0.35 (2)	0.174 (11)	0.1341*
C1b	0.519 (2)	−0.154 (5)	0.173 (2)	0.059 (8)*
C2b	0.617 (2)	−0.025 (4)	0.1949 (17)	0.039 (7)*
H2b	0.699346	−0.080259	0.207597	0.0783*
C3b	0.6056 (19)	0.069 (4)	0.2888 (18)	0.027 (6)*
C4b	0.687 (2)	0.009 (5)	0.3948 (18)	0.056 (8)*
H4b	0.746353	−0.088303	0.397637	0.112*
C5b	0.690 (2)	0.079 (5)	0.495 (2)	0.047 (7)*
H5b	0.736405	0.022522	0.569586	0.0936*
C6b	0.624 (2)	0.229 (4)	0.4854 (19)	0.038 (7)*
C7b	0.555 (3)	0.297 (6)	0.381 (2)	0.069 (9)*
H7b	0.509753	0.412387	0.369388	0.1384*
C8b	0.554 (2)	0.192 (4)	0.2965 (19)	0.030 (6)*
H8b	0.491584	0.231141	0.221735	0.0606*
C9b	0.653 (3)	0.315 (5)	0.589 (2)	0.061 (8)*
H9b1	0.666291	0.225759	0.649833	0.1213*
H9b2	0.574246	0.346463	0.599035	0.1213*
C10b	0.734 (2)	0.446 (4)	0.6275 (16)	0.038 (7)*
H10b	0.806282	0.376586	0.626647	0.0754*
C11b	0.770 (3)	0.499 (5)	0.744 (2)	0.059 (8)*
H11d	0.695148	0.531621	0.759656	0.1173*
H11e	0.826823	0.603924	0.759193	0.1173*
H11f	0.814478	0.399515	0.794602	0.1173*
C12b	0.7240 (18)	0.591 (4)	0.5555 (18)	0.033 (6)*
H12d	0.807942	0.642956	0.572354	0.0653*
H12e	0.68843	0.550636	0.474968	0.0653*
H12f	0.668594	0.682622	0.567885	0.0653*
C13b	0.5950 (19)	0.096 (4)	0.1044 (17)	0.038 (6)*
H13d	0.515208	0.15916	0.089567	0.0768*
H13e	0.66423	0.183427	0.124938	0.0768*
H13f	0.590027	0.030006	0.034862	0.0768*

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
?	?	?	?	?	?	?

Geometric parameters (Å, º) top

O1a—C1a	1.35 (3)	O2b—C1b	1.36 (5)
O2a—H1a	0.99 (9)	O1b—C1b	1.34 (4)
O2a—C1a	1.35 (5)	C1b—C2b	1.53 (5)
C1a—C2a	1.50 (5)	C2b—H2b	1.06
C2a—H2a	1.06	C2b—C3b	1.53 (4)
C2a—C3a	1.51 (4)	C2b—C13b	1.50 (4)
C2a—C13a	1.58 (5)	C3b—C4b	1.49 (3)
C3a—C4a	1.37 (4)	C3b—C8b	1.20 (4)
C3a—C8a	1.44 (3)	C4b—H4b	1.06
C4a—H4a	1.06	C4b—C5b	1.46 (4)
C4a—C5a	1.54 (4)	C5b—H5b	1.06
C5a—H5a	1.06	C5b—C6b	1.43 (5)
C5a—C6a	1.35 (3)	C6b—C7b	1.45 (4)
C6a—C7a	1.39 (3)	C6b—C9b	1.48 (4)
C6a—C9a	1.56 (3)	C7b—H7b	1.06
C7a—H7a	1.06	C7b—C8b	1.42 (5)
C7a—C8a	1.30 (4)	C8b—H8b	1.06
C8a—H8a	1.06	C9b—H9b1	1.06
C9a—H9a1	1.06	C9b—H9b2	1.06
C9a—H9a2	1.06	C9b—C10b	1.41 (4)
C10a—H10a	1.06	C10b—H10b	1.06
C10a—C11a	1.46 (4)	C10b—C11b	1.53 (4)
C10a—C12a	1.52 (5)	C10b—C12b	1.49 (4)
C11a—H11a	1.06	C11b—H11d	1.06
C11a—H11b	1.06	C11b—H11e	1.06
C11a—H11c	1.06	C11b—H11f	1.06
C12a—H12a	1.06	C12b—H12d	1.06
C12a—H12b	1.06	C12b—H12e	1.06
C12a—H12c	1.06	C12b—H12f	1.06
C13a—H13a	1.06	C13b—H13d	1.06
C13a—H13b	1.06	C13b—H13e	1.06
C13a—H13c	1.06	C13b—H13f	1.06
O2b—H1b	0.99 (12)

H1a—O2a—C1a	93 (12)	O1b—C1b—C2b	126 (3)
O1a—C1a—O2a	122 (3)	C1b—C2b—H2b	112.73
O1a—C1a—C2a	121 (3)	C1b—C2b—C3b	100 (2)
O2a—C1a—C2a	117 (2)	C1b—C2b—C13b	112.9 (19)
C1a—C2a—H2a	121.17	H2b—C2b—C3b	117.15
C1a—C2a—C3a	102 (3)	H2b—C2b—C13b	105.66
C1a—C2a—C13a	108 (3)	C3b—C2b—C13b	108 (3)
H2a—C2a—C3a	105.45	C2b—C3b—C4b	113 (2)
H2a—C2a—C13a	99.16	C2b—C3b—C8b	134 (2)
C3a—C2a—C13a	123 (2)	C4b—C3b—C8b	112 (3)
C2a—C3a—C4a	118 (2)	C3b—C4b—H4b	118.76
C2a—C3a—C8a	123 (2)	C3b—C4b—C5b	122 (3)
C4a—C3a—C8a	114 (2)	H4b—C4b—C5b	118.76
C3a—C4a—H4a	121.47	C4b—C5b—H5b	122.22
C3a—C4a—C5a	117 (2)	C4b—C5b—C6b	116 (2)
H4a—C4a—C5a	121.47	H5b—C5b—C6b	122.22
C4a—C5a—H5a	120.19	C5b—C6b—C7b	121 (3)
C4a—C5a—C6a	120 (3)	C5b—C6b—C9b	113 (2)
H5a—C5a—C6a	120.19	C7b—C6b—C9b	126 (3)
C5a—C6a—C7a	117 (2)	C6b—C7b—H7b	123.69
C5a—C6a—C9a	117 (2)	C6b—C7b—C8b	113 (3)
C7a—C6a—C9a	126.0 (19)	H7b—C7b—C8b	123.69
C6a—C7a—H7a	118.68	C3b—C8b—C7b	136 (2)
C6a—C7a—C8a	123 (2)	C3b—C8b—H8b	112.18
H7a—C7a—C8a	118.68	C7b—C8b—H8b	112.18
C3a—C8a—C7a	124 (3)	C6b—C9b—H9b1	109.47
C3a—C8a—H8a	117.81	C6b—C9b—H9b2	109.47
C7a—C8a—H8a	117.81	C6b—C9b—C10b	125 (3)
C6a—C9a—H9a1	109.47	H9b1—C9b—H9b2	86.59
C6a—C9a—H9a2	109.47	H9b1—C9b—C10b	109.47
H9a1—C9a—H9a2	111.69	H9b2—C9b—C10b	109.47
H10a—C10a—C11a	107.46	C9b—C10b—H10b	95.07
H10a—C10a—C12a	110.49	C9b—C10b—C11b	119 (3)
C11a—C10a—C12a	113 (3)	C9b—C10b—C12b	117.8 (19)
C10a—C11a—H11a	109.47	H10b—C10b—C11b	103.74
C10a—C11a—H11b	109.47	H10b—C10b—C12b	105.34
C10a—C11a—H11c	109.47	C11b—C10b—C12b	112 (3)
H11a—C11a—H11b	109.47	C10b—C11b—H11d	109.47
H11a—C11a—H11c	109.47	C10b—C11b—H11e	109.47
H11b—C11a—H11c	109.47	C10b—C11b—H11f	109.47
C10a—C12a—H12a	109.47	H11d—C11b—H11e	109.47
C10a—C12a—H12b	109.47	H11d—C11b—H11f	109.47
C10a—C12a—H12c	109.47	H11e—C11b—H11f	109.47
H12a—C12a—H12b	109.47	C10b—C12b—H12d	109.47
H12a—C12a—H12c	109.47	C10b—C12b—H12e	109.47
H12b—C12a—H12c	109.47	C10b—C12b—H12f	109.47
C2a—C13a—H13a	109.47	H12d—C12b—H12e	109.47
C2a—C13a—H13b	109.47	H12d—C12b—H12f	109.47
C2a—C13a—H13c	109.47	H12e—C12b—H12f	109.47
H13a—C13a—H13b	109.47	C2b—C13b—H13d	109.47
H13a—C13a—H13c	109.47	C2b—C13b—H13e	109.47
H13b—C13a—H13c	109.47	C2b—C13b—H13f	109.47
H1b—O2b—C1b	93 (10)	H13d—C13b—H13e	109.47
O2b—C1b—O1b	114 (3)	H13d—C13b—H13f	109.47
O2b—C1b—C2b	120 (2)	H13e—C13b—H13f	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O2a—H1a···O1bⁱ	0.99 (9)	1.67 (15)	2.58 (5)	153 (16)
O2b—H1b···O1aⁱⁱ	0.99 (12)	1.79 (15)	2.63 (5)	141 (14)

Symmetry codes: (i) x, y+1, z+1; (ii) x, y−1, z−1.

(S)-(+)-Ibuprofen (ibuprofen_model2) top

Crystal data top

C₁₃H₁₈O₂	Z = 4
M_r = 206.3	F(000) = 184.42
Monoclinic, P2₁	D_x = 1.102 Mg m⁻³
Hall symbol: P 2yb	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 12.368 (4) Å	Cell parameters from 3884 reflections
b = 8.021 (3) Å	θ = 0.1–0.7°
c = 13.536 (5) Å	T = 95 K
β = 112.24 (3)°	Block
V = 1242.9 (8) Å³

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.166
Radiation source: Lab6 cathode	θ_max = 0.7°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −12→12
3884 measured reflections	k = −8→8
1212 independent reflections	l = −13→13
883 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H atoms treated by a mixture of independent and constrained refinement
R[F > 3σ(F)] = 0.188	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F) = 0.426	(Δ/σ)_max = 0.046
S = 2.03	Δρ_max = 0.27 e Å⁻³
1212 reflections	Δρ_min = −0.26 e Å⁻³
127 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 3.7 (9)
139 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1a	0.3483 (19)	0.520 (3)	1.1738 (15)	0.047 (5)*
O2a	0.230 (2)	0.681 (3)	1.0468 (16)	0.049 (5)*
H1a	0.291 (9)	0.763 (13)	1.080 (9)	0.0977*
C1a	0.254 (2)	0.544 (3)	1.1011 (17)	0.037 (5)*
C2a	0.155 (2)	0.408 (3)	1.0677 (17)	0.043 (6)*
H2a	0.161004	0.313979	1.124761	0.0866*
C3a	0.1671 (16)	0.331 (2)	0.9703 (13)	0.020 (4)*
C4a	0.2346 (19)	0.185 (3)	0.9830 (16)	0.035 (5)*
H4a	0.279552	0.136015	1.060603	0.0709*
C5a	0.2434 (16)	0.101 (2)	0.8919 (12)	0.022 (4)*
H5a	0.309796	0.010112	0.904912	0.0435*
C6a	0.1708 (15)	0.1332	0.7919 (12)	0.009 (4)*
C7a	0.0949 (19)	0.276 (3)	0.7758 (17)	0.032 (5)*
H7a	0.044494	0.314088	0.696613	0.0631*
C8a	0.0853 (18)	0.365 (3)	0.8575 (13)	0.031 (5)*
H8a	0.020032	0.457851	0.841836	0.0622*
C9a	0.1710 (16)	0.036 (2)	0.6962 (12)	0.020 (4)*
H9a1	0.172883	0.120675	0.636497	0.0407*
H9a2	0.247725	−0.037318	0.718552	0.0407*
C10a	0.0560 (17)	−0.084 (3)	0.6472 (13)	0.029 (5)*
H10a	−0.024033	−0.02508	0.637934	0.0576*
C11a	0.062 (2)	−0.139 (3)	0.5431 (16)	0.042 (5)*
H11a	0.082053	−0.035243	0.504681	0.0844*
H11b	0.127539	−0.231109	0.557875	0.0844*
H11c	−0.019944	−0.189306	0.493457	0.0844*
C12a	0.067 (2)	−0.229 (3)	0.7203 (17)	0.042 (5)*
H12a	0.003048	−0.319715	0.680316	0.0846*
H12b	0.151229	−0.281542	0.74315	0.0846*
H12c	0.053418	−0.187212	0.789239	0.0846*
C13a	0.037 (3)	0.487 (4)	1.053 (2)	0.067 (7)*
H13a	−0.02582	0.391412	1.041545	0.1349*
H13b	0.045482	0.557643	1.121102	0.1349*
H13c	0.010264	0.565243	0.984649	0.1349*
O2b	0.541 (2)	−0.289 (3)	0.2291 (17)	0.055 (5)*
O1b	0.413 (2)	−0.124 (3)	0.1015 (16)	0.051 (5)*
H1b	0.475 (8)	−0.366 (14)	0.205 (10)	0.111*
C1b	0.512 (2)	−0.146 (3)	0.1738 (16)	0.035 (5)*
C2b	0.611 (2)	−0.023 (3)	0.1959 (16)	0.046 (6)*
H2b	0.691943	−0.083435	0.211316	0.0914*
C3b	0.611 (2)	0.073 (3)	0.2907 (16)	0.039 (5)*
C4b	0.6879 (18)	0.008 (3)	0.3981 (14)	0.036 (5)*
H4b	0.744922	−0.094793	0.406557	0.0713*
C5b	0.6804 (19)	0.088 (3)	0.4849 (17)	0.039 (5)*
H5b	0.726041	0.031082	0.559793	0.0784*
C6b	0.6217 (19)	0.233 (3)	0.4882 (15)	0.030 (5)*
C7b	0.550 (2)	0.298 (3)	0.3855 (16)	0.045 (6)*
H7b	0.500624	0.409417	0.377044	0.0904*
C8b	0.5479 (18)	0.202 (2)	0.2937 (15)	0.027 (5)*
H8b	0.486875	0.242141	0.218764	0.0543*
C9b	0.6377 (19)	0.321 (3)	0.5875 (15)	0.043 (5)*
H9b1	0.646674	0.232646	0.648523	0.0864*
H9b2	0.5577	0.376672	0.581319	0.0864*
C10b	0.731 (2)	0.443 (3)	0.6263 (15)	0.043 (6)*
H10b	0.801633	0.365674	0.628986	0.0867*
C11b	0.756 (2)	0.519 (3)	0.7400 (18)	0.057 (6)*
H11d	0.813729	0.439941	0.799233	0.1131*
H11e	0.676206	0.529633	0.752029	0.1131*
H11f	0.794042	0.638607	0.745324	0.1131*
C12b	0.7173 (19)	0.601 (3)	0.5498 (14)	0.039 (5)*
H12d	0.79347	0.675931	0.580418	0.0777*
H12e	0.705614	0.559546	0.472097	0.0777*
H12f	0.643753	0.672268	0.546386	0.0777*
C13b	0.598 (2)	0.086 (3)	0.1012 (15)	0.041 (5)*
H13d	0.50821	0.110426	0.057355	0.0812*
H13e	0.642607	0.200281	0.128271	0.0812*
H13f	0.634012	0.024541	0.05143	0.0812*

Geometric parameters (Å, º) top

O1a—C1a	1.22 (3)	O2b—H1b	0.98 (10)
O2a—H1a	0.98 (10)	O2b—C1b	1.34 (3)
O2a—C1a	1.29 (3)	O1b—C1b	1.26 (3)
C1a—C2a	1.58 (3)	C1b—C2b	1.51 (3)
C2a—H2a	1.06	C2b—H2b	1.06
C2a—C3a	1.52 (3)	C2b—C3b	1.50 (3)
C2a—C13a	1.53 (4)	C2b—C13b	1.51 (3)
C3a—C4a	1.41 (3)	C3b—C4b	1.50 (3)
C3a—C8a	1.50 (2)	C3b—C8b	1.30 (3)
C4a—H4a	1.06	C4b—H4b	1.06
C4a—C5a	1.45 (3)	C4b—C5b	1.37 (3)
C5a—H5a	1.06	C5b—H5b	1.06
C5a—C6a	1.34 (2)	C5b—C6b	1.38 (3)
C6a—C7a	1.45 (2)	C6b—C7b	1.44 (3)
C6a—C9a	1.51 (2)	C6b—C9b	1.46 (3)
C7a—H7a	1.06	C7b—H7b	1.06
C7a—C8a	1.36 (3)	C7b—C8b	1.46 (3)
C8a—H8a	1.06	C8b—H8b	1.06
C9a—H9a1	1.06	C9b—H9b1	1.06
C9a—H9a2	1.06	C9b—H9b2	1.06
C9a—C10a	1.64 (3)	C9b—C10b	1.45 (3)
C10a—H10a	1.06	C10b—H10b	1.06
C10a—C11a	1.50 (3)	C10b—C11b	1.57 (3)
C10a—C12a	1.50 (3)	C10b—C12b	1.61 (3)
C11a—H11a	1.06	C11b—H11d	1.06
C11a—H11b	1.06	C11b—H11e	1.06
C11a—H11c	1.06	C11b—H11f	1.06
C12a—H12a	1.06	C12b—H12d	1.06
C12a—H12b	1.06	C12b—H12e	1.06
C12a—H12c	1.06	C12b—H12f	1.06
C13a—H13a	1.06	C13b—H13d	1.06
C13a—H13b	1.06	C13b—H13e	1.06
C13a—H13c	1.06	C13b—H13f	1.06

H1a—O2a—C1a	110 (6)	H1b—O2b—C1b	110 (6)
O1a—C1a—O2a	123 (2)	O2b—C1b—O1b	122 (2)
O1a—C1a—C2a	121 (2)	O2b—C1b—C2b	115.0 (19)
O2a—C1a—C2a	115.9 (18)	O1b—C1b—C2b	123 (2)
C1a—C2a—H2a	116.64	C1b—C2b—H2b	111.99
C1a—C2a—C3a	102 (2)	C1b—C2b—C3b	104 (2)
C1a—C2a—C13a	111 (2)	C1b—C2b—C13b	113.2 (16)
H2a—C2a—C3a	109.9	H2b—C2b—C3b	112
H2a—C2a—C13a	101.28	H2b—C2b—C13b	102.72
C3a—C2a—C13a	117.2 (17)	C3b—C2b—C13b	113.2 (19)
C2a—C3a—C4a	118.8 (16)	C2b—C3b—C4b	116.2 (19)
C2a—C3a—C8a	124.0 (18)	C2b—C3b—C8b	129.0 (18)
C4a—C3a—C8a	114.5 (17)	C4b—C3b—C8b	115 (2)
C3a—C4a—H4a	119.48	C3b—C4b—H4b	122
C3a—C4a—C5a	121.0 (17)	C3b—C4b—C5b	116 (2)
H4a—C4a—C5a	119.48	H4b—C4b—C5b	122
C4a—C5a—H5a	118.84	C4b—C5b—H5b	115.45
C4a—C5a—C6a	122.3 (18)	C4b—C5b—C6b	129.1 (19)
H5a—C5a—C6a	118.84	H5b—C5b—C6b	115.45
C5a—C6a—C7a	117.1 (16)	C5b—C6b—C7b	115 (2)
C5a—C6a—C9a	123.4 (14)	C5b—C6b—C9b	123.0 (17)
C7a—C6a—C9a	119.4 (14)	C7b—C6b—C9b	122 (2)
C6a—C7a—H7a	118.49	C6b—C7b—H7b	122.07
C6a—C7a—C8a	123.0 (17)	C6b—C7b—C8b	116 (2)
H7a—C7a—C8a	118.49	H7b—C7b—C8b	122.07
C3a—C8a—C7a	120.0 (19)	C3b—C8b—C7b	129.1 (18)
C3a—C8a—H8a	119.99	C3b—C8b—H8b	115.45
C7a—C8a—H8a	119.99	C7b—C8b—H8b	115.45
C6a—C9a—H9a1	109.47	C6b—C9b—H9b1	109.47
C6a—C9a—H9a2	109.47	C6b—C9b—H9b2	109.47
C6a—C9a—C10a	111.5 (16)	C6b—C9b—C10b	118 (2)
H9a1—C9a—H9a2	107.37	H9b1—C9b—H9b2	99.43
H9a1—C9a—C10a	109.47	H9b1—C9b—C10b	109.47
H9a2—C9a—C10a	109.47	H9b2—C9b—C10b	109.47
C9a—C10a—H10a	114.54	C9b—C10b—H10b	98.56
C9a—C10a—C11a	102.1 (17)	C9b—C10b—C11b	117 (2)
C9a—C10a—C12a	110.1 (13)	C9b—C10b—C12b	115.1 (15)
H10a—C10a—C11a	113.14	H10b—C10b—C11b	109.93
H10a—C10a—C12a	105.58	H10b—C10b—C12b	111.66
C11a—C10a—C12a	111.6 (18)	C11b—C10b—C12b	105.0 (17)
C10a—C11a—H11a	109.47	C10b—C11b—H11d	109.47
C10a—C11a—H11b	109.47	C10b—C11b—H11e	109.47
C10a—C11a—H11c	109.47	C10b—C11b—H11f	109.47
H11a—C11a—H11b	109.47	H11d—C11b—H11e	109.47
H11a—C11a—H11c	109.47	H11d—C11b—H11f	109.47
H11b—C11a—H11c	109.47	H11e—C11b—H11f	109.47
C10a—C12a—H12a	109.47	C10b—C12b—H12d	109.47
C10a—C12a—H12b	109.47	C10b—C12b—H12e	109.47
C10a—C12a—H12c	109.47	C10b—C12b—H12f	109.47
H12a—C12a—H12b	109.47	H12d—C12b—H12e	109.47
H12a—C12a—H12c	109.47	H12d—C12b—H12f	109.47
H12b—C12a—H12c	109.47	H12e—C12b—H12f	109.47
C2a—C13a—H13a	109.47	C2b—C13b—H13d	109.47
C2a—C13a—H13b	109.47	C2b—C13b—H13e	109.47
C2a—C13a—H13c	109.47	C2b—C13b—H13f	109.47
H13a—C13a—H13b	109.47	H13d—C13b—H13e	109.47
H13a—C13a—H13c	109.47	H13d—C13b—H13f	109.47
H13b—C13a—H13c	109.47	H13e—C13b—H13f	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O2a—H1a···O1bⁱ	0.98 (10)	1.68 (11)	2.62 (3)	158 (12)
O2b—H1b···O1aⁱⁱ	0.98 (10)	1.72 (10)	2.69 (3)	170 (12)

Symmetry codes: (i) x, y+1, z+1; (ii) x, y−1, z−1.

(S)-(+)-Ibuprofen (ibuprofen_model3) top

Crystal data top

C₁₃H₁₈O₂	Z = 4
M_r = 206.3	F(000) = 184.42
Monoclinic, P2₁	D_x = 1.102 Mg m⁻³
Hall symbol: P 2yb	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 12.368 (4) Å	Cell parameters from 3884 reflections
b = 8.021 (3) Å	θ = 0.1–0.7°
c = 13.536 (5) Å	T = 95 K
β = 112.24 (3)°	Block
V = 1242.9 (8) Å³

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.166
Radiation source: Lab6 cathode	θ_max = 0.7°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −12→12
3884 measured reflections	k = −8→8
1212 independent reflections	l = −13→13
957 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H atoms treated by a mixture of independent and constrained refinement
R[F > 3σ(F)] = 0.190	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F) = 0.410	(Δ/σ)_max = 0.031
S = 2.36	Δρ_max = 0.26 e Å⁻³
1212 reflections	Δρ_min = −0.27 e Å⁻³
127 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 3.3 (7)
139 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1a	0.3481 (17)	0.518 (3)	1.1740 (14)	0.044 (5)*
O2a	0.227 (2)	0.678 (3)	1.0457 (16)	0.050 (5)*
H1a	0.294 (8)	0.754 (13)	1.077 (9)	0.1007*
C1a	0.2554 (19)	0.543 (3)	1.1008 (16)	0.037 (5)*
C2a	0.154 (2)	0.410 (3)	1.0672 (17)	0.045 (6)*
H2a	0.158665	0.316531	1.124207	0.0901*
C3a	0.1665 (15)	0.331 (2)	0.9692 (13)	0.020 (4)*
C4a	0.2362 (19)	0.186 (3)	0.9848 (16)	0.037 (5)*
H4a	0.282338	0.139272	1.06284	0.0748*
C5a	0.2439 (16)	0.100 (3)	0.8917 (12)	0.025 (4)*
H5a	0.309796	0.008596	0.903857	0.0494*
C6a	0.1697 (15)	0.1332	0.7914 (12)	0.013 (4)*
C7a	0.0948 (18)	0.275 (3)	0.7764 (16)	0.032 (5)*
H7a	0.043562	0.312385	0.697338	0.0647*
C8a	0.0859 (17)	0.366 (3)	0.8590 (13)	0.030 (5)*
H8a	0.021344	0.459591	0.843702	0.0598*
C9a	0.1714 (15)	0.039 (2)	0.6967 (12)	0.022 (4)*
H9a1	0.173159	0.123895	0.637419	0.043*
H9a2	0.247897	−0.035565	0.719715	0.043*
C10a	0.0550 (17)	−0.082 (3)	0.6470 (13)	0.035 (5)*
H10a	−0.025434	−0.0236	0.63671	0.0693*
C11a	0.062 (2)	−0.140 (3)	0.5425 (16)	0.044 (5)*
H11a	0.083217	−0.037445	0.503793	0.087*
H11b	0.127389	−0.232889	0.558385	0.087*
H11c	−0.019738	−0.190086	0.492498	0.087*
C12a	0.070 (2)	−0.228 (3)	0.7222 (18)	0.048 (6)*
H12a	0.00552	−0.31964	0.684604	0.0967*
H12b	0.154178	−0.281271	0.741319	0.0967*
H12c	0.061129	−0.186837	0.792984	0.0967*
C13a	0.038 (2)	0.487 (3)	1.054 (2)	0.065 (7)*
H13a	−0.02511	0.391528	1.041055	0.1291*
H13b	0.046876	0.553957	1.124109	0.1291*
H13c	0.011494	0.569012	0.987871	0.1291*
O2b	0.540 (2)	−0.290 (3)	0.2286 (16)	0.055 (5)*
O1b	0.414 (2)	−0.124 (3)	0.1026 (16)	0.054 (5)*
H1b	0.471 (7)	−0.361 (14)	0.198 (9)	0.1096*
C1b	0.513 (2)	−0.147 (3)	0.1736 (16)	0.039 (5)*
C2b	0.610 (2)	−0.024 (3)	0.1965 (16)	0.048 (6)*
H2b	0.689643	−0.08761	0.211948	0.0964*
C3b	0.6114 (18)	0.074 (3)	0.2909 (15)	0.034 (5)*
C4b	0.6883 (19)	0.007 (3)	0.3980 (15)	0.044 (5)*
H4b	0.745023	−0.095095	0.405713	0.0871*
C5b	0.6814 (17)	0.087 (3)	0.4866 (16)	0.037 (5)*
H5b	0.726183	0.029854	0.561585	0.0745*
C6b	0.6228 (18)	0.232 (3)	0.4872 (15)	0.030 (5)*
C7b	0.550 (2)	0.298 (3)	0.3841 (16)	0.046 (6)*
H7b	0.500169	0.408902	0.375764	0.0926*
C8b	0.5492 (18)	0.205 (3)	0.2945 (14)	0.026 (5)*
H8b	0.489581	0.246785	0.219415	0.0527*
C9b	0.6386 (19)	0.323 (3)	0.5889 (15)	0.045 (5)*
H9b1	0.649244	0.23528	0.650526	0.0902*
H9b2	0.558154	0.37732	0.582554	0.0902*
C10b	0.7300 (18)	0.446 (3)	0.6259 (14)	0.039 (5)*
H10b	0.800282	0.37035	0.626245	0.0781*
C11b	0.759 (2)	0.516 (3)	0.7412 (16)	0.054 (6)*
H11d	0.68542	0.581528	0.743684	0.1082*
H11e	0.831143	0.598531	0.761488	0.1082*
H11f	0.779622	0.41617	0.796386	0.1082*
C12b	0.7180 (18)	0.598 (3)	0.5514 (14)	0.039 (5)*
H12d	0.795402	0.6702	0.580974	0.0777*
H12e	0.70461	0.554867	0.473517	0.0777*
H12f	0.645939	0.671744	0.548596	0.0777*
C13b	0.599 (2)	0.090 (3)	0.1021 (15)	0.044 (5)*
H13d	0.509821	0.123587	0.06172	0.0889*
H13e	0.649616	0.198266	0.13044	0.0889*
H13f	0.628367	0.025194	0.048842	0.0889*

Geometric parameters (Å, º) top

O1a—C1a	1.22 (2)	O2b—H1b	0.98 (10)
O2a—H1a	0.98 (9)	O2b—C1b	1.34 (3)
O2a—C1a	1.29 (3)	O1b—C1b	1.26 (3)
C1a—C2a	1.58 (3)	C1b—C2b	1.49 (3)
C2a—H2a	1.06	C2b—H2b	1.06
C2a—C3a	1.53 (3)	C2b—C3b	1.49 (3)
C2a—C13a	1.51 (4)	C2b—C13b	1.53 (3)
C3a—C4a	1.42 (3)	C3b—C4b	1.50 (3)
C3a—C8a	1.47 (2)	C3b—C8b	1.31 (3)
C4a—H4a	1.06	C4b—H4b	1.06
C4a—C5a	1.47 (3)	C4b—C5b	1.39 (3)
C5a—H5a	1.06	C5b—H5b	1.06
C5a—C6a	1.35 (2)	C5b—C6b	1.37 (3)
C6a—C7a	1.43 (2)	C6b—C7b	1.45 (3)
C6a—C9a	1.50 (2)	C6b—C9b	1.50 (3)
C7a—H7a	1.06	C7b—H7b	1.06
C7a—C8a	1.37 (3)	C7b—C8b	1.42 (3)
C8a—H8a	1.06	C8b—H8b	1.06
C9a—H9a1	1.06	C9b—H9b1	1.06
C9a—H9a2	1.06	C9b—H9b2	1.06
C9a—C10a	1.65 (3)	C9b—C10b	1.44 (3)
C10a—H10a	1.06	C10b—H10b	1.06
C10a—C11a	1.52 (3)	C10b—C11b	1.57 (3)
C10a—C12a	1.52 (3)	C10b—C12b	1.55 (3)
C11a—H11a	1.06	C11b—H11d	1.06
C11a—H11b	1.06	C11b—H11e	1.06
C11a—H11c	1.06	C11b—H11f	1.06
C12a—H12a	1.06	C12b—H12d	1.06
C12a—H12b	1.06	C12b—H12e	1.06
C12a—H12c	1.06	C12b—H12f	1.06
C13a—H13a	1.06	C13b—H13d	1.06
C13a—H13b	1.06	C13b—H13e	1.06
C13a—H13c	1.06	C13b—H13f	1.06

H1a—O2a—C1a	106 (6)	H1b—O2b—C1b	106 (6)
O1a—C1a—O2a	125 (2)	O2b—C1b—O1b	121 (2)
O1a—C1a—C2a	122 (2)	O2b—C1b—C2b	116.2 (18)
O2a—C1a—C2a	113.2 (17)	O1b—C1b—C2b	123 (2)
C1a—C2a—H2a	116.63	C1b—C2b—H2b	109.61
C1a—C2a—C3a	101 (2)	C1b—C2b—C3b	106 (2)
C1a—C2a—C13a	112 (2)	C1b—C2b—C13b	114.0 (16)
H2a—C2a—C3a	110.17	H2b—C2b—C3b	111.99
H2a—C2a—C13a	99.4	H2b—C2b—C13b	103.62
C3a—C2a—C13a	118.1 (17)	C3b—C2b—C13b	111.7 (19)
C2a—C3a—C4a	117.8 (16)	C2b—C3b—C4b	116.1 (19)
C2a—C3a—C8a	123.3 (18)	C2b—C3b—C8b	129.2 (17)
C4a—C3a—C8a	116.5 (17)	C4b—C3b—C8b	114.5 (19)
C3a—C4a—H4a	120.3	C3b—C4b—H4b	121.73
C3a—C4a—C5a	119.4 (16)	C3b—C4b—C5b	117 (2)
H4a—C4a—C5a	120.3	H4b—C4b—C5b	121.72
C4a—C5a—H5a	119.01	C4b—C5b—H5b	116.55
C4a—C5a—C6a	122.0 (17)	C4b—C5b—C6b	126.9 (19)
H5a—C5a—C6a	119.01	H5b—C5b—C6b	116.55
C5a—C6a—C7a	117.3 (16)	C5b—C6b—C7b	116 (2)
C5a—C6a—C9a	122.5 (14)	C5b—C6b—C9b	122.1 (17)
C7a—C6a—C9a	120.0 (14)	C7b—C6b—C9b	121.4 (19)
C6a—C7a—H7a	118.25	C6b—C7b—H7b	122.36
C6a—C7a—C8a	123.5 (17)	C6b—C7b—C8b	115 (2)
H7a—C7a—C8a	118.25	H7b—C7b—C8b	122.36
C3a—C8a—C7a	119.6 (19)	C3b—C8b—C7b	129.7 (17)
C3a—C8a—H8a	120.21	C3b—C8b—H8b	115.15
C7a—C8a—H8a	120.21	C7b—C8b—H8b	115.15
C6a—C9a—H9a1	109.47	C6b—C9b—H9b1	109.47
C6a—C9a—H9a2	109.47	C6b—C9b—H9b2	109.47
C6a—C9a—C10a	110.6 (16)	C6b—C9b—C10b	118 (2)
H9a1—C9a—H9a2	108.27	H9b1—C9b—H9b2	99.79
H9a1—C9a—C10a	109.47	H9b1—C9b—C10b	109.47
H9a2—C9a—C10a	109.47	H9b2—C9b—C10b	109.47
C9a—C10a—H10a	115.17	C9b—C10b—H10b	98.11
C9a—C10a—C11a	102.1 (17)	C9b—C10b—C11b	116 (2)
C9a—C10a—C12a	108.3 (14)	C9b—C10b—C12b	116.3 (15)
H10a—C10a—C11a	113.15	H10b—C10b—C11b	108.92
H10a—C10a—C12a	107.51	H10b—C10b—C12b	109.18
C11a—C10a—C12a	110.5 (19)	C11b—C10b—C12b	107.3 (18)
C10a—C11a—H11a	109.47	C10b—C11b—H11d	109.47
C10a—C11a—H11b	109.47	C10b—C11b—H11e	109.47
C10a—C11a—H11c	109.47	C10b—C11b—H11f	109.47
H11a—C11a—H11b	109.47	H11d—C11b—H11e	109.47
H11a—C11a—H11c	109.47	H11d—C11b—H11f	109.47
H11b—C11a—H11c	109.47	H11e—C11b—H11f	109.47
C10a—C12a—H12a	109.47	C10b—C12b—H12d	109.47
C10a—C12a—H12b	109.47	C10b—C12b—H12e	109.47
C10a—C12a—H12c	109.47	C10b—C12b—H12f	109.47
H12a—C12a—H12b	109.47	H12d—C12b—H12e	109.47
H12a—C12a—H12c	109.47	H12d—C12b—H12f	109.47
H12b—C12a—H12c	109.47	H12e—C12b—H12f	109.47
C2a—C13a—H13a	109.47	C2b—C13b—H13d	109.47
C2a—C13a—H13b	109.47	C2b—C13b—H13e	109.47
C2a—C13a—H13c	109.47	C2b—C13b—H13f	109.47
H13a—C13a—H13b	109.47	H13d—C13b—H13e	109.47
H13a—C13a—H13c	109.47	H13d—C13b—H13f	109.47
H13b—C13a—H13c	109.47	H13e—C13b—H13f	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O2a—H1a···O1bⁱ	0.98 (9)	1.70 (10)	2.66 (3)	165 (12)
O2b—H1b···O1aⁱⁱ	0.98 (10)	1.73 (10)	2.69 (3)	167 (12)

Symmetry codes: (i) x, y+1, z+1; (ii) x, y−1, z−1.

(S)-(+)-Ibuprofen (ibuprofen_model3_outliers) top

Crystal data top

C₁₃H₁₈O₂	Z = 4
M_r = 206.3	F(000) = 184.42
Monoclinic, P2₁	D_x = 1.102 Mg m⁻³
Hall symbol: P 2yb	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 12.368 (4) Å	Cell parameters from 3818 reflections
b = 8.021 (3) Å	θ = 0.1–0.7°
c = 13.536 (5) Å	T = 95 K
β = 112.24 (3)°	Block
V = 1242.9 (8) Å³

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.158
Radiation source: Lab6 cathode	θ_max = 0.7°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −12→12
3818 measured reflections	k = −8→8
1209 independent reflections	l = −13→13
972 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H atoms treated by a mixture of independent and constrained refinement
R[F > 3σ(F)] = 0.190	Weighting scheme based on measured s.u.'s w = 1/[σ²(F_o²) + (0.02P)²] where P = (F_o² + 2F_c²)/3
wR(F) = 0.414	(Δ/σ)_max = 0.047
S = 2.35	Δρ_max = 0.26 e Å⁻³
1209 reflections	Δρ_min = −0.27 e Å⁻³
127 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 3.4 (7)
139 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1a	0.3488 (17)	0.518 (3)	1.1743 (14)	0.043 (4)*
O2a	0.227 (2)	0.677 (3)	1.0449 (16)	0.050 (5)*
H1a	0.292 (8)	0.754 (13)	1.077 (9)	0.1*
C1a	0.2559 (18)	0.542 (3)	1.1008 (16)	0.036 (5)*
C2a	0.155 (2)	0.411 (3)	1.0677 (16)	0.044 (6)*
H2a	0.160723	0.319241	1.125872	0.0883*
C3a	0.1663 (15)	0.331 (2)	0.9689 (13)	0.020 (4)*
C4a	0.2369 (19)	0.186 (3)	0.9851 (16)	0.037 (5)*
H4a	0.283943	0.14016	1.063165	0.0742*
C5a	0.2434 (16)	0.100 (3)	0.8912 (12)	0.025 (4)*
H5a	0.309033	0.008741	0.903009	0.0508*
C6a	0.1692 (15)	0.1332	0.7912 (12)	0.014 (4)*
C7a	0.0940 (18)	0.275 (3)	0.7759 (16)	0.032 (5)*
H7a	0.042016	0.311025	0.696825	0.0633*
C8a	0.0859 (17)	0.366 (3)	0.8596 (12)	0.029 (5)*
H8a	0.021781	0.460545	0.84455	0.0576*
C9a	0.1720 (15)	0.039 (2)	0.6970 (12)	0.022 (4)*
H9a1	0.174095	0.124362	0.637894	0.0442*
H9a2	0.247946	−0.036348	0.720715	0.0442*
C10a	0.0544 (18)	−0.081 (3)	0.6469 (14)	0.036 (5)*
H10a	−0.025973	−0.021774	0.636337	0.0719*
C11a	0.061 (2)	−0.140 (3)	0.5414 (16)	0.044 (5)*
H11a	0.080687	−0.037489	0.501712	0.0876*
H11b	0.12719	−0.23164	0.557436	0.0876*
H11c	−0.020494	−0.191954	0.492283	0.0876*
C12a	0.071 (2)	−0.227 (3)	0.7229 (18)	0.049 (6)*
H12a	0.006633	−0.319163	0.685963	0.0975*
H12b	0.155168	−0.279441	0.741458	0.0975*
H12c	0.062609	−0.18564	0.793927	0.0975*
C13a	0.038 (2)	0.487 (3)	1.054 (2)	0.065 (7)*
H13a	−0.025309	0.390626	1.040159	0.1297*
H13b	0.045419	0.552031	1.124846	0.1297*
H13c	0.010954	0.570358	0.988895	0.1297*
O2b	0.539 (2)	−0.291 (3)	0.2281 (16)	0.056 (5)*
O1b	0.414 (2)	−0.124 (3)	0.1029 (16)	0.055 (5)*
H1b	0.471 (8)	−0.363 (14)	0.196 (9)	0.1114*
C1b	0.5141 (19)	−0.147 (3)	0.1733 (15)	0.037 (5)*
C2b	0.611 (2)	−0.024 (3)	0.1974 (16)	0.048 (6)*
H2b	0.690731	−0.087699	0.214152	0.0951*
C3b	0.6114 (18)	0.074 (3)	0.2908 (14)	0.031 (5)*
C4b	0.6876 (19)	0.007 (3)	0.3974 (15)	0.046 (5)*
H4b	0.743501	−0.096507	0.404539	0.0917*
C5b	0.6820 (17)	0.086 (3)	0.4875 (16)	0.036 (5)*
H5b	0.7273	0.029411	0.562563	0.0726*
C6b	0.6233 (18)	0.231 (3)	0.4872 (15)	0.029 (5)*
C7b	0.549 (2)	0.299 (3)	0.3833 (16)	0.047 (6)*
H7b	0.498861	0.408531	0.375024	0.0934*
C8b	0.5499 (17)	0.206 (3)	0.2947 (14)	0.025 (4)*
H8b	0.491693	0.249657	0.219454	0.0494*
C9b	0.6383 (18)	0.323 (3)	0.5890 (15)	0.044 (5)*
H9b1	0.649084	0.23635	0.650935	0.0884*
H9b2	0.557602	0.377566	0.582105	0.0884*
C10b	0.7293 (17)	0.448 (3)	0.6258 (14)	0.036 (5)*
H10b	0.798804	0.371298	0.624541	0.0727*
C11b	0.760 (2)	0.515 (3)	0.7419 (16)	0.053 (6)*
H11d	0.68935	0.585373	0.745158	0.1063*
H11e	0.835563	0.591862	0.763369	0.1063*
H11f	0.777478	0.413302	0.795682	0.1063*
C12b	0.7187 (18)	0.599 (3)	0.5521 (14)	0.038 (5)*
H12d	0.796782	0.669915	0.581852	0.0759*
H12e	0.704626	0.556633	0.474015	0.0759*
H12f	0.64741	0.674397	0.549845	0.0759*
C13b	0.598 (2)	0.090 (3)	0.1023 (15)	0.046 (5)*
H13d	0.510227	0.128717	0.064474	0.091*
H13e	0.652406	0.196731	0.130031	0.091*
H13f	0.624257	0.024596	0.04693	0.091*

Geometric parameters (Å, º) top

O1a—C1a	1.22 (2)	O2b—H1b	0.98 (10)
O2a—H1a	0.98 (10)	O2b—C1b	1.34 (3)
O2a—C1a	1.29 (3)	O1b—C1b	1.26 (3)
C1a—C2a	1.56 (3)	C1b—C2b	1.49 (3)
C2a—H2a	1.06	C2b—H2b	1.06
C2a—C3a	1.54 (3)	C2b—C3b	1.49 (3)
C2a—C13a	1.52 (4)	C2b—C13b	1.54 (3)
C3a—C4a	1.42 (3)	C3b—C4b	1.49 (3)
C3a—C8a	1.46 (2)	C3b—C8b	1.32 (3)
C4a—H4a	1.06	C4b—H4b	1.06
C4a—C5a	1.47 (3)	C4b—C5b	1.40 (3)
C5a—H5a	1.06	C5b—H5b	1.06
C5a—C6a	1.34 (2)	C5b—C6b	1.37 (3)
C6a—C7a	1.43 (2)	C6b—C7b	1.46 (3)
C6a—C9a	1.49 (2)	C6b—C9b	1.51 (3)
C7a—H7a	1.06	C7b—H7b	1.06
C7a—C8a	1.38 (3)	C7b—C8b	1.41 (3)
C8a—H8a	1.06	C8b—H8b	1.06
C9a—H9a1	1.06	C9b—H9b1	1.06
C9a—H9a2	1.06	C9b—H9b2	1.06
C9a—C10a	1.66 (3)	C9b—C10b	1.44 (3)
C10a—H10a	1.06	C10b—H10b	1.06
C10a—C11a	1.54 (3)	C10b—C11b	1.57 (3)
C10a—C12a	1.53 (3)	C10b—C12b	1.55 (3)
C11a—H11a	1.06	C11b—H11d	1.06
C11a—H11b	1.06	C11b—H11e	1.06
C11a—H11c	1.06	C11b—H11f	1.06
C12a—H12a	1.06	C12b—H12d	1.06
C12a—H12b	1.06	C12b—H12e	1.06
C12a—H12c	1.06	C12b—H12f	1.06
C13a—H13a	1.06	C13b—H13d	1.06
C13a—H13b	1.06	C13b—H13e	1.06
C13a—H13c	1.06	C13b—H13f	1.06

H1a—O2a—C1a	106 (6)	H1b—O2b—C1b	106 (6)
O1a—C1a—O2a	126 (2)	O2b—C1b—O1b	120 (2)
O1a—C1a—C2a	122 (2)	O2b—C1b—C2b	116.7 (18)
O2a—C1a—C2a	112.3 (17)	O1b—C1b—C2b	123 (2)
C1a—C2a—H2a	115.58	C1b—C2b—H2b	109.55
C1a—C2a—C3a	102 (2)	C1b—C2b—C3b	107 (2)
C1a—C2a—C13a	113 (2)	C1b—C2b—C13b	113.3 (16)
H2a—C2a—C3a	110.79	H2b—C2b—C3b	111.67
H2a—C2a—C13a	99.16	H2b—C2b—C13b	104.68
C3a—C2a—C13a	117.5 (17)	C3b—C2b—C13b	111.2 (19)
C2a—C3a—C4a	117.3 (16)	C2b—C3b—C4b	115.4 (19)
C2a—C3a—C8a	123.4 (17)	C2b—C3b—C8b	130.0 (17)
C4a—C3a—C8a	117.0 (17)	C4b—C3b—C8b	114.5 (19)
C3a—C4a—H4a	120.68	C3b—C4b—H4b	121.44
C3a—C4a—C5a	118.6 (16)	C3b—C4b—C5b	117 (2)
H4a—C4a—C5a	120.68	H4b—C4b—C5b	121.44
C4a—C5a—H5a	118.74	C4b—C5b—H5b	117.13
C4a—C5a—C6a	122.5 (17)	C4b—C5b—C6b	125.7 (19)
H5a—C5a—C6a	118.74	H5b—C5b—C6b	117.13
C5a—C6a—C7a	117.5 (16)	C5b—C6b—C7b	117.2 (19)
C5a—C6a—C9a	122.2 (14)	C5b—C6b—C9b	122.0 (16)
C7a—C6a—C9a	120.0 (14)	C7b—C6b—C9b	120.7 (19)
C6a—C7a—H7a	118.48	C6b—C7b—H7b	122.65
C6a—C7a—C8a	123.0 (16)	C6b—C7b—C8b	115 (2)
H7a—C7a—C8a	118.48	H7b—C7b—C8b	122.65
C3a—C8a—C7a	119.7 (19)	C3b—C8b—C7b	130.3 (17)
C3a—C8a—H8a	120.17	C3b—C8b—H8b	114.86
C7a—C8a—H8a	120.17	C7b—C8b—H8b	114.86
C6a—C9a—H9a1	109.47	C6b—C9b—H9b1	109.47
C6a—C9a—H9a2	109.47	C6b—C9b—H9b2	109.47
C6a—C9a—C10a	110.0 (16)	C6b—C9b—C10b	118 (2)
H9a1—C9a—H9a2	108.97	H9b1—C9b—H9b2	99.89
H9a1—C9a—C10a	109.47	H9b1—C9b—C10b	109.47
H9a2—C9a—C10a	109.47	H9b2—C9b—C10b	109.47
C9a—C10a—H10a	115.52	C9b—C10b—H10b	97.31
C9a—C10a—C11a	102.3 (17)	C9b—C10b—C11b	117 (2)
C9a—C10a—C12a	107.4 (14)	C9b—C10b—C12b	117.0 (14)
H10a—C10a—C11a	112.9	H10b—C10b—C11b	108.83
H10a—C10a—C12a	108.3	H10b—C10b—C12b	108.33
C11a—C10a—C12a	110.2 (19)	C11b—C10b—C12b	107.8 (18)
C10a—C11a—H11a	109.47	C10b—C11b—H11d	109.47
C10a—C11a—H11b	109.47	C10b—C11b—H11e	109.47
C10a—C11a—H11c	109.47	C10b—C11b—H11f	109.47
H11a—C11a—H11b	109.47	H11d—C11b—H11e	109.47
H11a—C11a—H11c	109.47	H11d—C11b—H11f	109.47
H11b—C11a—H11c	109.47	H11e—C11b—H11f	109.47
C10a—C12a—H12a	109.47	C10b—C12b—H12d	109.47
C10a—C12a—H12b	109.47	C10b—C12b—H12e	109.47
C10a—C12a—H12c	109.47	C10b—C12b—H12f	109.47
H12a—C12a—H12b	109.47	H12d—C12b—H12e	109.47
H12a—C12a—H12c	109.47	H12d—C12b—H12f	109.47
H12b—C12a—H12c	109.47	H12e—C12b—H12f	109.47
C2a—C13a—H13a	109.47	C2b—C13b—H13d	109.47
C2a—C13a—H13b	109.47	C2b—C13b—H13e	109.47
C2a—C13a—H13c	109.47	C2b—C13b—H13f	109.47
H13a—C13a—H13b	109.47	H13d—C13b—H13e	109.47
H13a—C13a—H13c	109.47	H13d—C13b—H13f	109.47
H13b—C13a—H13c	109.47	H13e—C13b—H13f	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O2a—H1a···O1bⁱ	0.98 (10)	1.72 (10)	2.68 (3)	164 (12)
O2b—H1b···O1aⁱⁱ	0.98 (10)	1.72 (10)	2.67 (3)	163 (12)

Symmetry codes: (i) x, y+1, z+1; (ii) x, y−1, z−1.

DL-alanine (L-alanine_model0) top

Crystal data top

C₃H₇NO₂	Z = 4
M_r = 89.1	F(000) = 69.628
Orthorhombic, P2₁2₁2₁	D_x = 1.406 Mg m⁻³
Hall symbol: P 2xab;2ybc;2zac	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 5.7733 (11) Å	Cell parameters from 3422 reflections
b = 5.9524 (12) Å	θ = 0.1–1.4°
c = 12.2465 (2) Å	T = 100 K
V = 420.85 (14) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.111
Radiation source: Lab6 cathode	θ_max = 1.4°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −11→11
3422 measured reflections	k = −9→9
1127 independent reflections	l = −24→22
1036 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H-atom parameters constrained
R[F² > 2σ(F²)] = 0.143	Weighting scheme based on measured s.u.'s w = 1/[σ²(F_o²) + (0.02P)²] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.348	(Δ/σ)_max = 0.004
S = 8.76	Δρ_max = 0.15 e Å⁻³
1127 reflections	Δρ_min = −0.17 e Å⁻³
56 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 32 (5)
63 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1	−0.1255 (7)	0.7271 (11)	0.4154 (4)	0.0194 (16)
O2	−0.2617 (6)	0.4455 (10)	0.3150 (3)	0.0152 (13)
C1	−0.1013 (7)	0.5560 (13)	0.3592 (4)	0.0135 (16)
C2	0.1919 (8)	0.2676 (13)	0.4089 (4)	0.0198 (18)
N1	0.3149 (7)	0.6537 (10)	0.3625 (4)	0.0160 (15)
C3	0.1452 (7)	0.4687 (12)	0.3388 (4)	0.0147 (16)
H1c3	0.163048	0.41892	0.256099	0.0176
H1c2	0.364038	0.211791	0.395918	0.0237
H2c2	0.074678	0.137288	0.38825	0.0237
H3c2	0.169956	0.311443	0.492151	0.0237
H1n1	0.287175	0.782907	0.310606	0.0192
H2n1	0.293629	0.707121	0.440112	0.0192
H3n1	0.477885	0.595588	0.352624	0.0192

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
O1	0.0124 (14)	0.024 (4)	0.022 (2)	0.000 (2)	0.0009 (15)	−0.0048 (19)
O2	0.0063 (11)	0.015 (3)	0.025 (2)	−0.0004 (17)	0.0005 (13)	−0.0057 (17)
C1	0.0075 (16)	0.018 (4)	0.016 (2)	−0.002 (2)	−0.0013 (17)	0.001 (2)
C2	0.0148 (18)	0.023 (5)	0.022 (2)	0.000 (2)	−0.0012 (19)	−0.006 (2)
N1	0.0110 (14)	0.016 (4)	0.021 (2)	−0.0004 (18)	0.0007 (17)	0.0042 (17)
C3	0.0081 (15)	0.021 (4)	0.015 (2)	0.002 (2)	−0.0008 (16)	−0.005 (2)
H1c3	0.009755	0.025538	0.017559	0.002616	−0.000991	−0.006428
H1c2	0.017793	0.027562	0.025892	−0.000333	−0.001488	−0.007715
H2c2	0.017793	0.027562	0.025892	−0.000333	−0.001488	−0.007715
H3c2	0.017793	0.027562	0.025892	−0.000333	−0.001488	−0.007715
H1n1	0.013253	0.018834	0.02561	−0.000436	0.000876	0.005039
H2n1	0.013253	0.018834	0.02561	−0.000436	0.000876	0.005039
H3n1	0.013253	0.018834	0.02561	−0.000436	0.000876	0.005039

Geometric parameters (Å, º) top

O1—C1	1.237 (9)	C2—H3c2	1.06
O2—C1	1.258 (7)	N1—C3	1.502 (8)
C1—C3	1.535 (6)	N1—H1n1	1.01
C2—C3	1.498 (9)	N1—H2n1	1.01
C2—H1c2	1.06	N1—H3n1	1.01
C2—H2c2	1.06	C3—H1c3	1.06

O1—C1—O2	125.9 (5)	C3—N1—H3n1	109.47
O1—C1—C3	118.3 (5)	H1n1—N1—H2n1	109.47
O2—C1—C3	115.8 (6)	H1n1—N1—H3n1	109.47
C3—C2—H1c2	109.47	H2n1—N1—H3n1	109.47
C3—C2—H2c2	109.47	C1—C3—C2	110.1 (4)
C3—C2—H3c2	109.47	C1—C3—N1	108.9 (5)
H1c2—C2—H2c2	109.47	C1—C3—H1c3	109.92
H1c2—C2—H3c2	109.47	C2—C3—N1	111.0 (4)
H2c2—C2—H3c2	109.47	C2—C3—H1c3	107.85
C3—N1—H1n1	109.47	N1—C3—H1c3	109.04
C3—N1—H2n1	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
C3—H1c3···O1ⁱ	1.06	2.40	3.431 (7)	163.75
N1—H1n1···O2ⁱⁱ	1.01	1.82	2.799 (7)	161.34
N1—H2n1···O1ⁱⁱⁱ	1.01	1.87	2.832 (6)	157.77
N1—H3n1···O2^iv	1.01	1.81	2.802 (6)	167.00
N1—H3n1···C1^iv	1.01	2.44	3.421 (6)	162.95

Symmetry codes: (i) −x, y−1/2, −z+1/2; (ii) −x, y+1/2, −z+1/2; (iii) x+1/2, −y+3/2, −z+1; (iv) x+1, y, z.

DL-alanine (L-alanine_model1) top

Crystal data top

C₃H₇NO₂	Z = 4
M_r = 89.1	F(000) = 69.628
Orthorhombic, P2₁2₁2₁	D_x = 1.406 Mg m⁻³
Hall symbol: P 2xab;2ybc;2zac	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 5.7733 (11) Å	Cell parameters from 3422 reflections
b = 5.9524 (12) Å	θ = 0.1–1.4°
c = 12.2465 (2) Å	T = 100 K
V = 420.85 (14) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.111
Radiation source: Lab6 cathode	θ_max = 1.4°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −11→11
3422 measured reflections	k = −9→9
1127 independent reflections	l = −24→22
922 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H-atom parameters constrained
R[F² > 2σ(F²)] = 0.139	Weighting scheme based on measured s.u.'s w = 1/[σ²(F_o²) + (0.02P)²] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.338	(Δ/σ)_max = 0.031
S = 5.35	Δρ_max = 0.17 e Å⁻³
1127 reflections	Δρ_min = −0.21 e Å⁻³
56 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 15 (2)
63 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1	−0.1255 (7)	0.7261 (9)	0.4150 (3)	0.0162 (10)
O2	−0.2631 (6)	0.4407 (9)	0.3153 (3)	0.0157 (9)
C1	−0.1014 (8)	0.5568 (10)	0.3600 (3)	0.0108 (10)
C2	0.1960 (9)	0.2671 (12)	0.4091 (4)	0.0186 (15)
N1	0.3148 (7)	0.6573 (8)	0.3611 (3)	0.0136 (12)
C3	0.1445 (8)	0.4699 (11)	0.3390 (3)	0.0156 (13)
H1c3	0.159651	0.418564	0.256373	0.0187
H1c2	0.366103	0.209284	0.39265	0.0223
H2c2	0.075479	0.137857	0.391252	0.0223
H3c2	0.182477	0.311882	0.492639	0.0223
H1n1	0.299026	0.708295	0.439396	0.0163
H2n1	0.28167	0.78754	0.310604	0.0163
H3n1	0.477652	0.601256	0.348072	0.0163

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
O1	0.0103 (13)	0.022 (2)	0.0167 (11)	−0.0059 (16)	0.0005 (14)	−0.0059 (12)
O2	0.0047 (11)	0.025 (2)	0.0170 (11)	0.0024 (15)	0.0027 (10)	−0.0097 (11)
C1	0.0067 (15)	0.015 (2)	0.0104 (11)	0.006 (2)	0.0009 (14)	−0.0019 (13)
C2	0.016 (2)	0.017 (3)	0.0227 (18)	0.003 (2)	−0.0078 (19)	−0.0084 (15)
N1	0.0105 (15)	0.013 (3)	0.0177 (13)	0.0074 (17)	−0.0001 (13)	−0.0055 (11)
C3	0.0069 (15)	0.027 (3)	0.0124 (12)	0.0013 (18)	−0.0017 (14)	−0.0013 (15)
H1c3	0.008277	0.032903	0.01487	0.001522	−0.00202	−0.001569
H1c2	0.019111	0.020468	0.027206	0.004181	−0.009388	−0.010068
H2c2	0.019111	0.020468	0.027206	0.004181	−0.009388	−0.010068
H3c2	0.019111	0.020468	0.027206	0.004181	−0.009388	−0.010068
H1n1	0.012596	0.015156	0.021192	0.008933	−0.00006	−0.006614
H2n1	0.012596	0.015156	0.021192	0.008933	−0.00006	−0.006614
H3n1	0.012596	0.015156	0.021192	0.008933	−0.00006	−0.006614

Geometric parameters (Å, º) top

O1—C1	1.221 (7)	C2—H3c2	1.06
O2—C1	1.284 (6)	N1—C3	1.512 (7)
C1—C3	1.533 (7)	N1—H1n1	1.01
C2—C3	1.511 (9)	N1—H2n1	1.01
C2—H1c2	1.06	N1—H3n1	1.01
C2—H2c2	1.06	C3—H1c3	1.06

O1—C1—O2	126.6 (5)	C3—N1—H3n1	109.47
O1—C1—C3	118.5 (4)	H1n1—N1—H2n1	109.47
O2—C1—C3	114.9 (5)	H1n1—N1—H3n1	109.47
C3—C2—H1c2	109.47	H2n1—N1—H3n1	109.47
C3—C2—H2c2	109.47	C1—C3—C2	110.9 (4)
C3—C2—H3c2	109.47	C1—C3—N1	108.9 (5)
H1c2—C2—H2c2	109.47	C1—C3—H1c3	109.51
H1c2—C2—H3c2	109.47	C2—C3—N1	111.1 (4)
H2c2—C2—H3c2	109.47	C2—C3—H1c3	107.23
C3—N1—H1n1	109.47	N1—C3—H1c3	109.28
C3—N1—H2n1	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
C3—H1c3···O1ⁱ	1.06	2.40	3.434 (6)	164.89
N1—H1n1···O1ⁱⁱ	1.01	1.88	2.849 (5)	160.68
N1—H2n1···O2ⁱⁱⁱ	1.01	1.79	2.757 (6)	158.01
N1—H3n1···O2^iv	1.01	1.82	2.813 (6)	166.71
N1—H3n1···C1^iv	1.01	2.45	3.423 (6)	161.77

Symmetry codes: (i) −x, y−1/2, −z+1/2; (ii) x+1/2, −y+3/2, −z+1; (iii) −x, y+1/2, −z+1/2; (iv) x+1, y, z.

DL-alanine (L-alanine_model2) top

Crystal data top

C₃H₇NO₂	Z = 4
M_r = 89.1	F(000) = 69.628
Orthorhombic, P2₁2₁2₁	D_x = 1.406 Mg m⁻³
Hall symbol: P 2xab;2ybc;2zac	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 5.7733 (11) Å	Cell parameters from 3422 reflections
b = 5.9524 (12) Å	θ = 0.1–1.4°
c = 12.2465 (2) Å	T = 100 K
V = 420.85 (14) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.111
Radiation source: Lab6 cathode	θ_max = 1.4°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −11→11
3422 measured reflections	k = −9→9
1127 independent reflections	l = −24→22
845 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H-atom parameters constrained
R[F² > 2σ(F²)] = 0.140	Weighting scheme based on measured s.u.'s w = 1/[σ²(F_o²) + (0.02P)²] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.345	(Δ/σ)_max = 0.029
S = 2.03	Δρ_max = 0.29 e Å⁻³
1127 reflections	Δρ_min = −0.17 e Å⁻³
56 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 51 (9)
63 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1	−0.1250 (5)	0.7273 (10)	0.4159 (3)	0.0189 (12)
O2	−0.2612 (5)	0.4452 (8)	0.3155 (3)	0.0149 (10)
C1	−0.1014 (6)	0.5553 (10)	0.3595 (3)	0.0115 (11)
C2	0.1928 (7)	0.2661 (12)	0.4095 (4)	0.0202 (14)
N1	0.3142 (5)	0.6531 (8)	0.3624 (3)	0.0143 (10)
C3	0.1454 (5)	0.4701 (10)	0.3391 (3)	0.0132 (12)
H1c3	0.163455	0.422061	0.25614	0.0158
H1c2	0.365811	0.212	0.39729	0.0242
H2c2	0.077385	0.135045	0.387752	0.0242
H3c2	0.168404	0.308475	0.492782	0.0242
H1n1	0.287995	0.781509	0.309918	0.0172
H2n1	0.291942	0.707962	0.43975	0.0172
H3n1	0.477069	0.594049	0.353404	0.0172

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
O1	0.0126 (10)	0.024 (3)	0.0203 (15)	0.0004 (14)	0.0005 (11)	−0.0059 (14)
O2	0.0062 (8)	0.016 (2)	0.0228 (15)	−0.0003 (12)	0.0001 (10)	−0.0058 (12)
C1	0.0075 (11)	0.012 (3)	0.0150 (16)	−0.0012 (15)	−0.0023 (12)	−0.0003 (14)
C2	0.0159 (14)	0.022 (4)	0.0223 (18)	0.000 (2)	−0.0030 (14)	−0.0054 (17)
N1	0.0085 (10)	0.015 (3)	0.0197 (15)	−0.0017 (13)	0.0011 (12)	0.0040 (12)
C3	0.0075 (11)	0.018 (3)	0.0138 (15)	0.0025 (15)	−0.0011 (12)	−0.0020 (14)
H1c3	0.009026	0.021987	0.016535	0.003056	−0.001353	−0.002415
H1c2	0.019108	0.026869	0.026742	−0.000334	−0.003591	−0.006475
H2c2	0.019108	0.026869	0.026742	−0.000334	−0.003591	−0.006475
H3c2	0.019108	0.026869	0.026742	−0.000334	−0.003591	−0.006475
H1n1	0.010244	0.017696	0.023592	−0.002059	0.001328	0.004762
H2n1	0.010244	0.017696	0.023592	−0.002059	0.001328	0.004762
H3n1	0.010244	0.017696	0.023592	−0.002059	0.001328	0.004762

Geometric parameters (Å, º) top

O1—C1	1.243 (8)	C2—H3c2	1.06
O2—C1	1.254 (6)	N1—C3	1.489 (6)
C1—C3	1.533 (5)	N1—H1n1	1.01
C2—C3	1.514 (8)	N1—H2n1	1.01
C2—H1c2	1.06	N1—H3n1	1.01
C2—H2c2	1.06	C3—H1c3	1.06

O1—C1—O2	126.1 (4)	C3—N1—H3n1	109.47
O1—C1—C3	117.7 (4)	H1n1—N1—H2n1	109.47
O2—C1—C3	116.2 (5)	H1n1—N1—H3n1	109.47
C3—C2—H1c2	109.47	H2n1—N1—H3n1	109.47
C3—C2—H2c2	109.47	C1—C3—C2	109.9 (3)
C3—C2—H3c2	109.47	C1—C3—N1	109.6 (4)
H1c2—C2—H2c2	109.47	C1—C3—H1c3	109.69
H1c2—C2—H3c2	109.47	C2—C3—N1	111.0 (3)
H2c2—C2—H3c2	109.47	C2—C3—H1c3	108.17
C3—N1—H1n1	109.47	N1—C3—H1c3	108.49
C3—N1—H2n1	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
C3—H1c3···O1ⁱ	1.06	2.42	3.443 (6)	162.97
N1—H1n1···O2ⁱⁱ	1.01	1.83	2.804 (6)	162.19
N1—H2n1···O1ⁱⁱⁱ	1.01	1.87	2.828 (5)	156.87
N1—H3n1···O2^iv	1.01	1.81	2.805 (5)	167.05
N1—H3n1···C1^iv	1.01	2.45	3.424 (4)	162.94

Symmetry codes: (i) −x, y−1/2, −z+1/2; (ii) −x, y+1/2, −z+1/2; (iii) x+1/2, −y+3/2, −z+1; (iv) x+1, y, z.

DL-alanine (L-alanine_model3) top

Crystal data top

C₃H₇NO₂	Z = 4
M_r = 89.1	F(000) = 69.628
Orthorhombic, P2₁2₁2₁	D_x = 1.406 Mg m⁻³
Hall symbol: P 2xab;2ybc;2zac	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 5.7733 (11) Å	Cell parameters from 3422 reflections
b = 5.9524 (12) Å	θ = 0.1–1.4°
c = 12.2465 (2) Å	T = 100 K
V = 420.85 (14) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.111
Radiation source: Lab6 cathode	θ_max = 1.4°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −11→11
3422 measured reflections	k = −9→9
1127 independent reflections	l = −24→22
1072 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H-atom parameters constrained
R[F² > 2σ(F²)] = 0.144	Weighting scheme based on measured s.u.'s w = 1/[σ²(F_o²) + (0.02P)²] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.355	(Δ/σ)_max = 0.017
S = 2.57	Δρ_max = 0.23 e Å⁻³
1127 reflections	Δρ_min = −0.18 e Å⁻³
56 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 30 (6)
63 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1	−0.1250 (5)	0.7269 (9)	0.4160 (3)	0.0175 (10)
O2	−0.2613 (5)	0.4444 (8)	0.3156 (2)	0.0148 (8)
C1	−0.1023 (5)	0.5551 (9)	0.3594 (3)	0.0099 (9)
C2	0.1936 (7)	0.2658 (11)	0.4095 (3)	0.0180 (12)
N1	0.3146 (5)	0.6533 (8)	0.3623 (3)	0.0132 (9)
C3	0.1455 (5)	0.4702 (9)	0.3389 (3)	0.0114 (10)
H1c3	0.162985	0.422487	0.255901	0.0137
H1c2	0.366536	0.21167	0.396917	0.0216
H2c2	0.078036	0.134659	0.388004	0.0216
H3c2	0.170013	0.308233	0.492792	0.0216
H1n1	0.288804	0.781524	0.309708	0.0159
H2n1	0.292059	0.708341	0.439593	0.0159
H3n1	0.477399	0.593918	0.353508	0.0159

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
O1	0.0119 (10)	0.023 (2)	0.0174 (12)	−0.0003 (13)	0.0011 (10)	−0.0065 (12)
O2	0.0057 (8)	0.018 (2)	0.0203 (11)	−0.0016 (11)	0.0007 (9)	−0.0076 (11)
C1	0.0063 (10)	0.009 (2)	0.0141 (13)	−0.0001 (14)	−0.0017 (11)	−0.0010 (11)
C2	0.0136 (13)	0.019 (3)	0.0211 (15)	−0.0004 (18)	−0.0046 (13)	−0.0049 (14)
N1	0.0084 (9)	0.013 (2)	0.0178 (13)	−0.0018 (12)	0.0019 (11)	0.0040 (11)
C3	0.0061 (10)	0.016 (2)	0.0123 (12)	0.0029 (13)	−0.0011 (11)	−0.0003 (12)
H1c3	0.007367	0.018937	0.014709	0.00346	−0.001367	−0.000356
H1c2	0.016332	0.023241	0.025288	−0.000504	−0.005517	−0.005924
H2c2	0.016332	0.023241	0.025288	−0.000504	−0.005517	−0.005924
H3c2	0.016332	0.023241	0.025288	−0.000504	−0.005517	−0.005924
H1n1	0.010054	0.016175	0.021337	−0.002188	0.002271	0.00478
H2n1	0.010054	0.016175	0.021337	−0.002188	0.002271	0.00478
H3n1	0.010054	0.016175	0.021337	−0.002188	0.002271	0.00478

Geometric parameters (Å, º) top

O1—C1	1.242 (7)	C2—H3c2	1.06
O2—C1	1.251 (5)	N1—C3	1.491 (6)
C1—C3	1.538 (5)	N1—H1n1	1.01
C2—C3	1.518 (7)	N1—H2n1	1.01
C2—H1c2	1.06	N1—H3n1	1.01
C2—H2c2	1.06	C3—H1c3	1.06

O1—C1—O2	126.6 (4)	C3—N1—H3n1	109.47
O1—C1—C3	117.4 (3)	H1n1—N1—H2n1	109.47
O2—C1—C3	116.1 (4)	H1n1—N1—H3n1	109.47
C3—C2—H1c2	109.47	H2n1—N1—H3n1	109.47
C3—C2—H2c2	109.47	C1—C3—C2	109.9 (3)
C3—C2—H3c2	109.47	C1—C3—N1	109.7 (4)
H1c2—C2—H2c2	109.47	C1—C3—H1c3	109.47
H1c2—C2—H3c2	109.47	C2—C3—N1	110.9 (3)
H2c2—C2—H3c2	109.47	C2—C3—H1c3	108.29
C3—N1—H1n1	109.47	N1—C3—H1c3	108.52
C3—N1—H2n1	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
C3—H1c3···O1ⁱ	1.06	2.42	3.443 (5)	162.92
N1—H1n1···O2ⁱⁱ	1.01	1.82	2.801 (5)	162.24
N1—H2n1···O1ⁱⁱⁱ	1.01	1.87	2.829 (5)	156.78
N1—H3n1···O2^iv	1.01	1.81	2.805 (5)	166.90
N1—H3n1···C1^iv	1.01	2.44	3.417 (4)	162.97

Symmetry codes: (i) −x, y−1/2, −z+1/2; (ii) −x, y+1/2, −z+1/2; (iii) x+1/2, −y+3/2, −z+1; (iv) x+1, y, z.

DL-alanine (L-alanine_model3_outliers) top

Crystal data top

C₃H₇NO₂	Z = 4
M_r = 89.1	F(000) = 69.628
Orthorhombic, P2₁2₁2₁	D_x = 1.406 Mg m⁻³
Hall symbol: P 2xab;2ybc;2zac	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 5.7733 (11) Å	Cell parameters from 3329 reflections
b = 5.9524 (12) Å	θ = 0.1–1.4°
c = 12.2465 (2) Å	T = 100 K
V = 420.85 (14) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.099
Radiation source: Lab6 cathode	θ_max = 1.4°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −11→11
3329 measured reflections	k = −9→9
1123 independent reflections	l = −24→22
927 reflections with I > 3σ(I)

Refinement top

Refinement on F²	H-atom parameters constrained
R[F² > 2σ(F²)] = 0.131	Weighting scheme based on measured s.u.'s w = 1/[σ²(F_o²) + (0.02P)²] where P = (F_o² + 2F_c²)/3
wR(F²) = 0.305	(Δ/σ)_max = 0.038
S = 2.36	Δρ_max = 0.20 e Å⁻³
1123 reflections	Δρ_min = −0.18 e Å⁻³
56 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
0 restraints	Extinction coefficient: 27 (4)
63 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
O1	−0.1249 (5)	0.7273 (9)	0.4160 (3)	0.0176 (10)
O2	−0.2613 (4)	0.4449 (8)	0.3155 (2)	0.0149 (9)
C1	−0.1018 (5)	0.5558 (9)	0.3600 (3)	0.0109 (10)
C2	0.1935 (6)	0.2656 (11)	0.4097 (3)	0.0188 (13)
N1	0.3149 (5)	0.6527 (8)	0.3623 (3)	0.0135 (9)
C3	0.1454 (5)	0.4701 (9)	0.3390 (3)	0.0119 (10)
H1c3	0.162343	0.422281	0.256023	0.0143
H1c2	0.365583	0.210003	0.396279	0.0226
H2c2	0.07621	0.135376	0.389019	0.0226
H3c2	0.172249	0.308802	0.493053	0.0226
H1n1	0.289925	0.780649	0.309432	0.0162
H2n1	0.292019	0.708556	0.439467	0.0162
H3n1	0.47758	0.592824	0.353903	0.0162

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
O1	0.0117 (9)	0.022 (3)	0.0187 (13)	0.0001 (13)	0.0011 (10)	−0.0055 (12)
O2	0.0058 (8)	0.017 (2)	0.0214 (13)	−0.0008 (11)	0.0003 (9)	−0.0065 (12)
C1	0.0065 (10)	0.012 (2)	0.0145 (14)	−0.0004 (15)	−0.0018 (11)	−0.0015 (13)
C2	0.0136 (13)	0.022 (3)	0.0210 (16)	0.0016 (18)	−0.0035 (13)	−0.0040 (16)
N1	0.0086 (9)	0.013 (2)	0.0186 (14)	−0.0008 (12)	0.0010 (11)	0.0041 (11)
C3	0.0067 (10)	0.016 (3)	0.0127 (13)	0.0026 (14)	−0.0009 (11)	−0.0010 (13)
H1c3	0.00802	0.019564	0.01525	0.003178	−0.001132	−0.001234
H1c2	0.016282	0.0262	0.025254	0.001921	−0.004206	−0.004766
H2c2	0.016282	0.0262	0.025254	0.001921	−0.004206	−0.004766
H3c2	0.016282	0.0262	0.025254	0.001921	−0.004206	−0.004766
H1n1	0.010286	0.016108	0.022305	−0.001	0.001198	0.004899
H2n1	0.010286	0.016108	0.022305	−0.001	0.001198	0.004899
H3n1	0.010286	0.016108	0.022305	−0.001	0.001198	0.004899

Geometric parameters (Å, º) top

O1—C1	1.237 (7)	C2—H3c2	1.06
O2—C1	1.257 (5)	N1—C3	1.490 (6)
C1—C3	1.537 (5)	N1—H1n1	1.01
C2—C3	1.519 (7)	N1—H2n1	1.01
C2—H1c2	1.06	N1—H3n1	1.01
C2—H2c2	1.06	C3—H1c3	1.06

O1—C1—O2	126.5 (4)	C3—N1—H3n1	109.47
O1—C1—C3	117.8 (3)	H1n1—N1—H2n1	109.47
O2—C1—C3	115.7 (4)	H1n1—N1—H3n1	109.47
C3—C2—H1c2	109.47	H2n1—N1—H3n1	109.47
C3—C2—H2c2	109.47	C1—C3—C2	109.9 (3)
C3—C2—H3c2	109.47	C1—C3—N1	109.6 (4)
H1c2—C2—H2c2	109.47	C1—C3—H1c3	109.55
H1c2—C2—H3c2	109.47	C2—C3—N1	110.8 (3)
H2c2—C2—H3c2	109.47	C2—C3—H1c3	108.32
C3—N1—H1n1	109.47	N1—C3—H1c3	108.61
C3—N1—H2n1	109.47

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
C3—H1c3···O1ⁱ	1.06	2.41	3.443 (5)	163.19
N1—H1n1···O2ⁱⁱ	1.01	1.82	2.804 (5)	162.82
N1—H2n1···O1ⁱⁱⁱ	1.01	1.87	2.829 (5)	156.55
N1—H3n1···O2^iv	1.01	1.81	2.801 (4)	166.89
N1—H3n1···C1^iv	1.01	2.44	3.417 (4)	162.66

Symmetry codes: (i) −x, y−1/2, −z+1/2; (ii) −x, y+1/2, −z+1/2; (iii) x+1/2, −y+3/2, −z+1; (iv) x+1, y, z.

(natrolite_model0) top

Crystal data top

Al₂H₄Na₂O₁₂Si₃	Z = 8
M_r = 380.2	F(000) = 517.84
Orthorhombic, Fdd2	D_x = 2.235 Mg m⁻³
Hall symbol: F -2xuvw;-2yuvw;2z	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 18.2872 (11) Å	Cell parameters from 5213 reflections
b = 18.6660 (14) Å	θ = 0.1–1.2°
c = 6.6222 (3) Å	T = 293 K
V = 2260.5 (2) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.198
Radiation source: Lab6 cathode	θ_max = 1.2°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −29→29
5213 measured reflections	k = −29→29
1289 independent reflections	l = −10→10
839 reflections with I > 3σ(I)

Refinement top

Refinement on F²	All H-atom parameters refined
R[F² > 2σ(F²)] = 0.146	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F²) = 0.305	(Δ/σ)_max = 0.033
S = 6.26	Δρ_max = 0.25 e Å⁻³
1289 reflections	Δρ_min = −0.26 e Å⁻³
93 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 0.42 (5)
13 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Si1	0	0	0	0.023 (2)
Si2	0.0966 (2)	0.0386 (3)	−0.3722 (11)	0.0237 (17)
Al1	−0.0371 (2)	0.0936 (4)	0.3836 (12)	0.0256 (19)
Na1	−0.2186 (4)	0.0325 (5)	0.3831 (15)	0.043 (3)
O1	0.0677 (4)	0.0215 (5)	−0.1404 (14)	0.034 (3)
O2	−0.0224 (4)	0.0680 (6)	0.1347 (16)	0.040 (3)
O3	−0.0690 (4)	0.1829 (5)	0.3914 (15)	0.029 (3)
O4	−0.0987 (4)	0.0364 (6)	0.5043 (16)	0.039 (3)
O5	0.0445 (3)	0.0962 (6)	0.5312 (14)	0.030 (3)
O6	−0.3057 (6)	0.0588 (7)	0.6399 (19)	0.052 (4)
H1	−0.3520 (16)	0.070 (3)	0.579 (7)	0.0621
H2	−0.297 (3)	0.097 (2)	0.732 (8)	0.0621

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Si1	0.012 (3)	0.049 (6)	0.008 (2)	0.001 (3)	0	0
Si2	0.014 (2)	0.040 (4)	0.0172 (19)	−0.003 (2)	−0.005 (2)	0.001 (3)
Al1	0.010 (2)	0.047 (5)	0.019 (2)	0.003 (2)	0.007 (2)	0.002 (3)
Na1	0.029 (3)	0.062 (7)	0.039 (4)	−0.004 (3)	0.007 (4)	0.008 (5)
O1	0.024 (3)	0.067 (7)	0.012 (3)	0.005 (4)	0.001 (3)	−0.009 (4)
O2	0.021 (3)	0.075 (8)	0.026 (3)	−0.001 (4)	0.004 (4)	−0.020 (5)
O3	0.018 (3)	0.050 (7)	0.020 (3)	−0.009 (3)	0.006 (3)	0.009 (5)
O4	0.009 (3)	0.070 (8)	0.037 (4)	−0.003 (4)	−0.003 (3)	−0.002 (5)
O5	0.003 (3)	0.057 (7)	0.029 (3)	0.002 (3)	−0.007 (3)	0.001 (4)
O6	0.059 (6)	0.068 (9)	0.029 (4)	0.017 (5)	0.008 (5)	−0.013 (6)
H1	0.070305	0.081352	0.034582	0.019866	0.009901	−0.015483
H2	0.070305	0.081352	0.034582	0.019866	0.009901	−0.015483

Geometric parameters (Å, º) top

Si1—O1	1.599 (8)	Al1—O2	1.738 (13)
Si1—O1ⁱ	1.599 (8)	Al1—O3	1.766 (11)
Si1—O2	1.604 (11)	Al1—O4	1.745 (11)
Si1—O2ⁱ	1.604 (11)	Al1—O5	1.786 (9)
Si2—Al1ⁱⁱ	3.107 (8)	Na1—Na1^vii	3.709 (14)
Si2—Al1ⁱⁱⁱ	3.058 (8)	Na1—Na1^viii	3.709 (14)
Si2—Na1^iv	3.060 (10)	Na1—O3^ix	2.521 (12)
Si2—Na1ⁱⁱⁱ	3.545 (11)	Na1—O3^x	2.619 (13)
Si2—O1	1.655 (11)	Na1—O4	2.336 (11)
Si2—O3ⁱⁱⁱ	1.634 (8)	Na1—O5^ix	2.374 (14)
Si2—O4^iv	1.622 (12)	Na1—O6	2.381 (15)
Si2—O5ⁱⁱ	1.573 (10)	Na1—O6^vii	2.387 (16)
Al1—Na1	3.508 (9)	Na1—H2^vii	2.64 (4)
Al1—Na1^v	3.106 (12)	O6—H1	0.96 (4)
Al1—Na1^vi	3.899 (12)	O6—H2	0.96 (5)

O1—Si1—O1ⁱ	108.9 (4)	Si2^xii—Na1—O5^ix	20.8 (2)
O1—Si1—O2	108.8 (5)	Si2^xii—Na1—O6	92.2 (4)
O1—Si1—O2ⁱ	108.9 (4)	Si2^xii—Na1—O6^vii	127.1 (4)
O1ⁱ—Si1—O2	108.9 (4)	Si2^xii—Na1—H2^vii	145.5 (11)
O1ⁱ—Si1—O2ⁱ	108.8 (5)	Al1—Na1—Al1^ix	98.1 (3)
O2—Si1—O2ⁱ	112.4 (5)	Al1—Na1—Al1^x	105.6 (2)
Al1ⁱⁱ—Si2—Al1ⁱⁱⁱ	108.4 (2)	Al1—Na1—Na1^vii	113.6 (3)
Al1ⁱⁱ—Si2—Na1^iv	116.2 (3)	Al1—Na1—Na1^viii	113.5 (3)
Al1ⁱⁱ—Si2—Na1ⁱⁱⁱ	55.2 (2)	Al1—Na1—O3^ix	127.3 (4)
Al1ⁱⁱ—Si2—O1	107.1 (4)	Al1—Na1—O3^x	89.0 (3)
Al1ⁱⁱ—Si2—O3ⁱⁱⁱ	131.6 (5)	Al1—Na1—O4	26.1 (3)
Al1ⁱⁱ—Si2—O4^iv	92.3 (4)	Al1—Na1—O5^ix	66.1 (3)
Al1ⁱⁱ—Si2—O5ⁱⁱ	23.9 (4)	Al1—Na1—O6	124.4 (5)
Al1ⁱⁱⁱ—Si2—Na1^iv	79.2 (2)	Al1—Na1—O6^vii	93.3 (4)
Al1ⁱⁱⁱ—Si2—Na1ⁱⁱⁱ	63.68 (19)	Al1—Na1—H2^vii	101.2 (11)
Al1ⁱⁱⁱ—Si2—O1	110.5 (4)	Al1^ix—Na1—Al1^x	155.6 (3)
Al1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	27.0 (4)	Al1^ix—Na1—Na1^vii	69.1 (3)
Al1ⁱⁱⁱ—Si2—O4^iv	127.7 (4)	Al1^ix—Na1—Na1^viii	126.4 (3)
Al1ⁱⁱⁱ—Si2—O5ⁱⁱ	86.2 (4)	Al1^ix—Na1—O3^ix	34.6 (3)
Na1^iv—Si2—Na1ⁱⁱⁱ	131.2 (3)	Al1^ix—Na1—O3^x	169.1 (4)
Na1^iv—Si2—O1	129.8 (5)	Al1^ix—Na1—O4	122.6 (4)
Na1^iv—Si2—O3ⁱⁱⁱ	58.8 (4)	Al1^ix—Na1—O5^ix	34.9 (2)
Na1^iv—Si2—O4^iv	48.8 (3)	Al1^ix—Na1—O6	87.5 (4)
Na1^iv—Si2—O5ⁱⁱ	121.5 (5)	Al1^ix—Na1—O6^vii	104.3 (5)
Na1ⁱⁱⁱ—Si2—O1	93.7 (4)	Al1^ix—Na1—H2^vii	121.9 (11)
Na1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	90.6 (4)	Al1^x—Na1—Na1^vii	95.3 (3)
Na1ⁱⁱⁱ—Si2—O4^iv	145.7 (5)	Al1^x—Na1—Na1^viii	48.1 (2)
Na1ⁱⁱⁱ—Si2—O5ⁱⁱ	32.5 (4)	Al1^x—Na1—O3^ix	121.4 (4)
O1—Si2—O3ⁱⁱⁱ	108.3 (6)	Al1^x—Na1—O3^x	22.0 (2)
O1—Si2—O4^iv	108.0 (6)	Al1^x—Na1—O4	81.7 (4)
O1—Si2—O5ⁱⁱ	108.4 (5)	Al1^x—Na1—O5^ix	168.7 (4)
O3ⁱⁱⁱ—Si2—O4^iv	106.7 (5)	Al1^x—Na1—O6	84.3 (4)
O3ⁱⁱⁱ—Si2—O5ⁱⁱ	111.8 (6)	Al1^x—Na1—O6^vii	68.7 (4)
O4^iv—Si2—O5ⁱⁱ	113.5 (6)	Al1^x—Na1—H2^vii	47.7 (11)
Si2^xi—Al1—Si2^xii	142.3 (3)	Na1^vii—Na1—Na1^viii	126.4 (3)
Si2^xi—Al1—Na1	129.6 (3)	Na1^vii—Na1—O3^ix	44.9 (3)
Si2^xi—Al1—Na1^v	69.6 (2)	Na1^vii—Na1—O3^x	115.6 (4)
Si2^xi—Al1—Na1^vi	119.9 (2)	Na1^vii—Na1—O4	127.4 (4)
Si2^xi—Al1—O2	106.3 (4)	Na1^vii—Na1—O5^ix	95.1 (4)
Si2^xi—Al1—O3	123.8 (4)	Na1^vii—Na1—O6	119.8 (4)
Si2^xi—Al1—O4	93.9 (4)	Na1^vii—Na1—O6^vii	38.9 (3)
Si2^xi—Al1—O5	20.9 (4)	Na1^vii—Na1—H2^vii	52.8 (11)
Si2^xii—Al1—Na1	64.9 (2)	Na1^viii—Na1—O3^ix	115.1 (3)
Si2^xii—Al1—Na1^v	75.6 (2)	Na1^viii—Na1—O3^x	42.8 (3)
Si2^xii—Al1—Na1^vi	50.43 (18)	Na1^viii—Na1—O4	89.7 (4)
Si2^xii—Al1—O2	107.1 (4)	Na1^viii—Na1—O5^ix	126.7 (4)
Si2^xii—Al1—O3	24.9 (3)	Na1^viii—Na1—O6	39.0 (4)
Si2^xii—Al1—O4	89.9 (4)	Na1^viii—Na1—O6^vii	115.2 (5)
Si2^xii—Al1—O5	124.6 (5)	Na1^viii—Na1—H2^vii	94.1 (11)
Na1—Al1—Na1^v	128.6 (3)	O3^ix—Na1—O3^x	142.0 (4)
Na1—Al1—Na1^vi	108.6 (2)	O3^ix—Na1—O4	153.4 (5)
Na1—Al1—O2	93.2 (4)	O3^ix—Na1—O5^ix	69.4 (4)
Na1—Al1—O3	89.7 (3)	O3^ix—Na1—O6	85.0 (4)
Na1—Al1—O4	36.1 (3)	O3^ix—Na1—O6^vii	83.7 (5)
Na1—Al1—O5	143.2 (5)	O3^ix—Na1—H2^vii	94.4 (11)
Na1^v—Al1—Na1^vi	62.7 (3)	O3^x—Na1—O4	63.3 (4)
Na1^v—Al1—O2	130.3 (5)	O3^x—Na1—O5^ix	146.8 (5)
Na1^v—Al1—O3	54.2 (4)	O3^x—Na1—O6	81.7 (5)
Na1^v—Al1—O4	118.2 (5)	O3^x—Na1—O6^vii	83.4 (5)
Na1^v—Al1—O5	49.5 (4)	O3^x—Na1—H2^vii	64.2 (11)
Na1^vi—Al1—O2	80.9 (5)	O4—Na1—O5^ix	88.5 (4)
Na1^vi—Al1—O3	33.7 (4)	O4—Na1—O6	112.1 (5)
Na1^vi—Al1—O4	140.0 (4)	O4—Na1—O6^vii	94.6 (5)
Na1^vi—Al1—O5	100.7 (4)	O4—Na1—H2^vii	93.1 (11)
O2—Al1—O3	109.8 (6)	O5^ix—Na1—O6	94.3 (5)
O2—Al1—O4	111.5 (6)	O5^ix—Na1—O6^vii	118.0 (6)
O2—Al1—O5	113.4 (5)	O5^ix—Na1—H2^vii	139.2 (12)
O3—Al1—O4	110.6 (5)	O6—Na1—O6^vii	138.9 (5)
O3—Al1—O5	103.5 (5)	O6—Na1—H2^vii	122.2 (11)
O4—Al1—O5	107.8 (6)	O6^vii—Na1—H2^vii	21.2 (11)
Si2^xiii—Na1—Si2^xii	99.7 (3)	Si1—O1—Si2	146.7 (6)
Si2^xiii—Na1—Al1	56.72 (19)	Si1—O2—Al1	141.7 (7)
Si2^xiii—Na1—Al1^ix	154.0 (3)	Si2^xii—O3—Al1	128.1 (6)
Si2^xiii—Na1—Al1^x	50.40 (19)	Si2^xii—O3—Na1^v	129.7 (6)
Si2^xiii—Na1—Na1^vii	123.8 (3)	Si2^xii—O3—Na1^vi	88.9 (5)
Si2^xiii—Na1—Na1^viii	67.1 (2)	Al1—O3—Na1^v	91.1 (4)
Si2^xiii—Na1—O3^ix	168.1 (5)	Al1—O3—Na1^vi	124.4 (5)
Si2^xiii—Na1—O3^x	32.3 (2)	Na1^v—O3—Na1^vi	92.4 (4)
Si2^xiii—Na1—O4	31.5 (3)	Si2^xiii—O4—Al1	138.1 (5)
Si2^xiii—Na1—O5^ix	119.4 (3)	Si2^xiii—O4—Na1	99.7 (5)
Si2^xiii—Na1—O6	101.5 (5)	Al1—O4—Na1	117.8 (6)
Si2^xiii—Na1—O6^vii	85.0 (4)	Si2^xi—O5—Al1	135.3 (7)
Si2^xiii—Na1—H2^vii	73.8 (11)	Si2^xi—O5—Na1^v	126.7 (6)
Si2^xii—Na1—Al1	51.39 (17)	Al1—O5—Na1^v	95.6 (5)
Si2^xii—Na1—Al1^ix	55.2 (2)	Na1—O6—Na1^viii	102.2 (5)
Si2^xii—Na1—Al1^x	147.9 (3)	Na1—O6—H1	110 (3)
Si2^xii—Na1—Na1^vii	113.8 (3)	Na1—O6—H2	120 (3)
Si2^xii—Na1—Na1^viii	114.9 (3)	Na1^viii—O6—H1	127 (3)
Si2^xii—Na1—O3^ix	89.9 (3)	Na1^viii—O6—H2	94 (3)
Si2^xii—Na1—O3^x	126.0 (4)	H1—O6—H2	104 (4)
Si2^xii—Na1—O4	70.0 (3)	Na1^viii—H2—O6	64 (2)

Symmetry codes: (i) −x, −y, z; (ii) x, y, z−1; (iii) x+1/4, −y+1/4, z−3/4; (iv) −x, −y, z−1; (v) x+1/4, −y+1/4, z+1/4; (vi) −x−1/4, y+1/4, z−1/4; (vii) −x−1/2, −y, z−1/2; (viii) −x−1/2, −y, z+1/2; (ix) x−1/4, −y+1/4, z−1/4; (x) −x−1/4, y−1/4, z+1/4; (xi) x, y, z+1; (xii) x−1/4, −y+1/4, z+3/4; (xiii) −x, −y, z+1.

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O6—H1···O1^xiv	0.96 (4)	2.25 (4)	3.049 (14)	140 (4)
O6—H2···O2^xii	0.96 (5)	1.93 (5)	2.879 (16)	171 (4)

Symmetry codes: (xii) x−1/4, −y+1/4, z+3/4; (xiv) x−1/2, y, z+1/2.

(natrolite_model1) top

Crystal data top

Al₂H₄Na₂O₁₂Si₃	Z = 8
M_r = 380.2	F(000) = 517.84
Orthorhombic, Fdd2	D_x = 2.235 Mg m⁻³
Hall symbol: F -2xuvw;-2yuvw;2z	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 18.2872 (11) Å	Cell parameters from 5213 reflections
b = 18.6660 (14) Å	θ = 0.1–1.2°
c = 6.6222 (3) Å	T = 293 K
V = 2260.5 (2) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.198
Radiation source: Lab6 cathode	θ_max = 1.2°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −29→29
5213 measured reflections	k = −29→29
1289 independent reflections	l = −10→10
876 reflections with I > 3σ(I)

Refinement top

Refinement on F²	All H-atom parameters refined
R[F² > 2σ(F²)] = 0.134	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F²) = 0.264	(Δ/σ)_max = 2.981
S = 2.93	Δρ_max = 0.29 e Å⁻³
1289 reflections	Δρ_min = −0.29 e Å⁻³
93 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 0.099 (17)
13 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Si1	0	0	0	0.0218 (13)
Si2	0.09670 (16)	0.03891 (18)	−0.3721 (11)	0.0197 (8)
Al1	−0.03799 (17)	0.09429 (19)	0.3857 (12)	0.0212 (10)
Na1	−0.2205 (3)	0.0310 (3)	0.3799 (16)	0.0331 (15)
O1	0.0683 (3)	0.0226 (3)	−0.1437 (14)	0.0262 (15)
O2	−0.0222 (3)	0.0681 (3)	0.1344 (17)	0.0310 (17)
O3	−0.0691 (3)	0.1817 (2)	0.3883 (15)	0.0214 (14)
O4	−0.0993 (2)	0.0363 (3)	0.5011 (16)	0.0252 (15)
O5	0.0450 (2)	0.0958 (3)	0.5257 (15)	0.0280 (16)
O6	−0.3065 (4)	0.0589 (5)	0.636 (2)	0.045 (2)
H1	−0.293 (2)	0.042 (3)	0.504 (4)	0.0544
H2	−0.2680 (19)	0.091 (2)	0.673 (7)	0.0544

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Si1	0.0088 (16)	0.046 (3)	0.011 (2)	−0.0005 (14)	0	0
Si2	0.0108 (12)	0.0323 (16)	0.0161 (14)	−0.0026 (9)	−0.0005 (14)	0.0000 (16)
Al1	0.0096 (11)	0.0316 (18)	0.022 (2)	0.0038 (10)	0.0042 (18)	0.0044 (18)
Na1	0.0217 (18)	0.052 (3)	0.025 (3)	0.0005 (14)	−0.005 (3)	−0.001 (3)
O1	0.022 (2)	0.045 (3)	0.012 (3)	−0.0035 (18)	0.006 (2)	0.002 (2)
O2	0.022 (2)	0.053 (3)	0.018 (3)	−0.001 (2)	0.003 (3)	−0.007 (3)
O3	0.0155 (19)	0.023 (2)	0.026 (3)	0.0022 (13)	−0.004 (3)	0.002 (2)
O4	0.0046 (16)	0.043 (3)	0.028 (3)	−0.0021 (14)	−0.001 (2)	0.007 (3)
O5	0.0087 (19)	0.057 (3)	0.018 (3)	0.0030 (19)	−0.002 (2)	0.002 (3)
O6	0.033 (3)	0.077 (5)	0.026 (4)	0.008 (3)	0.001 (4)	0.002 (4)
H1	0.03969	0.092366	0.031226	0.009857	0.001785	0.002332
H2	0.03969	0.092366	0.031226	0.009857	0.001785	0.002332

Geometric parameters (Å, º) top

Si1—O1	1.626 (7)	Al1—O3	1.728 (6)
Si1—O1ⁱ	1.626 (7)	Al1—O4	1.736 (8)
Si1—O2	1.604 (8)	Al1—O5	1.778 (8)
Si1—O2ⁱ	1.604 (8)	Na1—Na1^vii	3.670 (14)
Si2—Al1ⁱⁱ	3.116 (7)	Na1—Na1^viii	3.670 (14)
Si2—Al1ⁱⁱⁱ	3.034 (5)	Na1—O3^ix	2.510 (11)
Si2—Na1^iv	3.086 (9)	Na1—O3^x	2.624 (11)
Si2—Na1ⁱⁱⁱ	3.580 (6)	Na1—O4	2.359 (8)
Si2—O1	1.628 (11)	Na1—O5^ix	2.418 (9)
Si2—O3ⁱⁱⁱ	1.636 (6)	Na1—O6	2.371 (14)
Si2—O4^iv	1.637 (8)	Na1—O6^vii	2.381 (14)
Si2—O5ⁱⁱ	1.575 (8)	Na1—H1	1.58 (4)
Al1—Na1	3.540 (6)	Na1—H2	2.41 (4)
Al1—Na1^v	3.092 (9)	Na1—H2^vii	2.67 (4)
Al1—Na1^vi	3.878 (8)	O6—H1	0.96 (3)
Al1—O2	1.758 (13)	O6—H2	0.96 (4)

O1—Si1—O1ⁱ	108.4 (4)	Al1—Na1—O6	122.9 (4)
O1—Si1—O2	108.3 (3)	Al1—Na1—O6^vii	92.7 (3)
O1—Si1—O2ⁱ	109.6 (3)	Al1—Na1—H1	138.1 (15)
O1ⁱ—Si1—O2	109.6 (3)	Al1—Na1—H2	100.1 (9)
O1ⁱ—Si1—O2ⁱ	108.3 (3)	Al1—Na1—H2^vii	111.4 (8)
O2—Si1—O2ⁱ	112.6 (5)	Al1^ix—Na1—Al1^x	157.1 (2)
Al1ⁱⁱ—Si2—Al1ⁱⁱⁱ	108.65 (18)	Al1^ix—Na1—Na1^vii	69.4 (2)
Al1ⁱⁱ—Si2—Na1^iv	116.5 (3)	Al1^ix—Na1—Na1^viii	126.3 (2)
Al1ⁱⁱ—Si2—Na1ⁱⁱⁱ	54.48 (16)	Al1^ix—Na1—O3^ix	33.96 (16)
Al1ⁱⁱ—Si2—O1	106.8 (3)	Al1^ix—Na1—O3^x	169.4 (4)
Al1ⁱⁱ—Si2—O3ⁱⁱⁱ	130.9 (4)	Al1^ix—Na1—O4	121.4 (3)
Al1ⁱⁱ—Si2—O4^iv	92.5 (3)	Al1^ix—Na1—O5^ix	35.03 (19)
Al1ⁱⁱ—Si2—O5ⁱⁱ	23.1 (2)	Al1^ix—Na1—O6	86.7 (3)
Al1ⁱⁱⁱ—Si2—Na1^iv	78.64 (17)	Al1^ix—Na1—O6^vii	105.0 (5)
Al1ⁱⁱⁱ—Si2—Na1ⁱⁱⁱ	64.11 (12)	Al1^ix—Na1—H1	80.5 (15)
Al1ⁱⁱⁱ—Si2—O1	109.9 (4)	Al1^ix—Na1—H2	85.6 (10)
Al1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	26.37 (19)	Al1^ix—Na1—H2^vii	110.1 (10)
Al1ⁱⁱⁱ—Si2—O4^iv	127.2 (3)	Al1^x—Na1—Na1^vii	97.1 (2)
Al1ⁱⁱⁱ—Si2—O5ⁱⁱ	86.6 (3)	Al1^x—Na1—Na1^viii	48.27 (17)
Na1^iv—Si2—Na1ⁱⁱⁱ	130.6 (2)	Al1^x—Na1—O3^ix	123.8 (2)
Na1^iv—Si2—O1	130.5 (3)	Al1^x—Na1—O3^x	21.49 (17)
Na1^iv—Si2—O3ⁱⁱⁱ	58.3 (3)	Al1^x—Na1—O4	81.5 (2)
Na1^iv—Si2—O4^iv	48.9 (2)	Al1^x—Na1—O5^ix	167.1 (4)
Na1^iv—Si2—O5ⁱⁱ	119.8 (5)	Al1^x—Na1—O6	84.9 (3)
Na1ⁱⁱⁱ—Si2—O1	93.6 (3)	Al1^x—Na1—O6^vii	69.8 (3)
Na1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	90.4 (2)	Al1^x—Na1—H1	85.8 (17)
Na1ⁱⁱⁱ—Si2—O4^iv	144.8 (4)	Al1^x—Na1—H2	94.8 (10)
Na1ⁱⁱⁱ—Si2—O5ⁱⁱ	33.1 (3)	Al1^x—Na1—H2^vii	57.3 (9)
O1—Si2—O3ⁱⁱⁱ	108.9 (6)	Na1^vii—Na1—Na1^viii	128.9 (2)
O1—Si2—O4^iv	109.0 (4)	Na1^vii—Na1—O3^ix	45.6 (2)
O1—Si2—O5ⁱⁱ	109.5 (4)	Na1^vii—Na1—O3^x	116.6 (3)
O3ⁱⁱⁱ—Si2—O4^iv	106.4 (4)	Na1^vii—Na1—O4	126.7 (4)
O3ⁱⁱⁱ—Si2—O5ⁱⁱ	110.9 (4)	Na1^vii—Na1—O5^ix	94.4 (4)
O4^iv—Si2—O5ⁱⁱ	112.1 (6)	Na1^vii—Na1—O6	121.3 (3)
Si2^xi—Al1—Si2^xii	142.8 (3)	Na1^vii—Na1—O6^vii	39.3 (3)
Si2^xi—Al1—Na1	129.8 (2)	Na1^vii—Na1—H1	105.3 (12)
Si2^xi—Al1—Na1^v	70.43 (18)	Na1^vii—Na1—H2	140.2 (9)
Si2^xi—Al1—Na1^vi	119.44 (15)	Na1^vii—Na1—H2^vii	40.9 (9)
Si2^xi—Al1—O2	105.4 (3)	Na1^viii—Na1—O3^ix	116.8 (3)
Si2^xi—Al1—O3	124.6 (4)	Na1^viii—Na1—O3^x	43.1 (2)
Si2^xi—Al1—O4	94.4 (3)	Na1^viii—Na1—O4	89.0 (4)
Si2^xi—Al1—O5	20.4 (2)	Na1^viii—Na1—O5^ix	126.3 (4)
Si2^xii—Al1—Na1	65.45 (12)	Na1^viii—Na1—O6	39.5 (3)
Si2^xii—Al1—Na1^v	75.24 (17)	Na1^viii—Na1—O6^vii	116.8 (4)
Si2^xii—Al1—Na1^vi	51.28 (14)	Na1^viii—Na1—H1	47.5 (13)
Si2^xii—Al1—O2	107.3 (4)	Na1^viii—Na1—H2	46.7 (10)
Si2^xii—Al1—O3	24.86 (19)	Na1^viii—Na1—H2^vii	99.9 (9)
Si2^xii—Al1—O4	90.4 (2)	O3^ix—Na1—O3^x	144.1 (3)
Si2^xii—Al1—O5	126.2 (3)	O3^ix—Na1—O4	151.7 (4)
Na1—Al1—Na1^v	129.3 (2)	O3^ix—Na1—O5^ix	68.8 (3)
Na1—Al1—Na1^vi	109.44 (17)	O3^ix—Na1—O6	85.3 (4)
Na1—Al1—O2	92.9 (3)	O3^ix—Na1—O6^vii	84.9 (4)
Na1—Al1—O3	90.3 (2)	O3^ix—Na1—H1	71.9 (12)
Na1—Al1—O4	35.7 (3)	O3^ix—Na1—H2	97.1 (10)
Na1—Al1—O5	144.6 (4)	O3^ix—Na1—H2^vii	81.6 (9)
Na1^v—Al1—Na1^vi	62.3 (2)	O3^x—Na1—O4	63.2 (3)
Na1^v—Al1—O2	129.7 (3)	O3^x—Na1—O5^ix	145.6 (4)
Na1^v—Al1—O3	54.2 (3)	O3^x—Na1—O6	82.6 (4)
Na1^v—Al1—O4	119.8 (5)	O3^x—Na1—O6^vii	83.6 (4)
Na1^v—Al1—O5	51.3 (3)	O3^x—Na1—H1	89.3 (15)
Na1^vi—Al1—O2	80.2 (3)	O3^x—Na1—H2	84.6 (10)
Na1^vi—Al1—O3	33.8 (3)	O3^x—Na1—H2^vii	75.7 (9)
Na1^vi—Al1—O4	141.1 (2)	O4—Na1—O5^ix	87.0 (3)
Na1^vi—Al1—O5	100.3 (3)	O4—Na1—O6	111.7 (5)
O2—Al1—O3	109.0 (5)	O4—Na1—O6^vii	93.8 (3)
O2—Al1—O4	110.4 (4)	O4—Na1—H1	127.5 (13)
O2—Al1—O5	111.0 (4)	O4—Na1—H2	92.6 (10)
O3—Al1—O4	111.8 (4)	O4—Na1—H2^vii	106.6 (8)
O3—Al1—O5	105.1 (4)	O5^ix—Na1—O6	94.1 (4)
O4—Al1—O5	109.4 (5)	O5^ix—Na1—O6^vii	116.9 (5)
Si2^xiii—Na1—Si2^xii	98.47 (19)	O5^ix—Na1—H1	96.9 (17)
Si2^xiii—Na1—Al1	56.22 (15)	O5^ix—Na1—H2	80.0 (10)
Si2^xiii—Na1—Al1^ix	152.8 (2)	O5^ix—Na1—H2^vii	132.5 (10)
Si2^xiii—Na1—Al1^x	50.09 (13)	O6—Na1—O6^vii	141.0 (4)
Si2^xiii—Na1—Na1^vii	124.3 (2)	O6—Na1—H1	16.0 (12)
Si2^xiii—Na1—Na1^viii	66.6 (2)	O6—Na1—H2	23.2 (9)
Si2^xiii—Na1—O3^ix	169.7 (4)	O6—Na1—H2^vii	120.2 (9)
Si2^xiii—Na1—O3^x	32.02 (16)	O6^vii—Na1—H1	128.5 (16)
Si2^xiii—Na1—O4	31.5 (2)	O6^vii—Na1—H2	162.2 (10)
Si2^xiii—Na1—O5^ix	117.9 (3)	O6^vii—Na1—H2^vii	20.9 (9)
Si2^xiii—Na1—O6	101.5 (4)	H1—Na1—H2	38.1 (16)
Si2^xiii—Na1—O6^vii	84.9 (3)	H1—Na1—H2^vii	108.4 (18)
Si2^xiii—Na1—H1	113.4 (13)	H2—Na1—H2^vii	141.8 (13)
Si2^xiii—Na1—H2	92.0 (10)	Si1—O1—Si2	146.8 (5)
Si2^xiii—Na1—H2^vii	88.3 (9)	Si1—O2—Al1	141.9 (5)
Si2^xii—Na1—Al1	50.43 (10)	Si2^xii—O3—Al1	128.8 (3)
Si2^xii—Na1—Al1^ix	55.10 (15)	Si2^xii—O3—Na1^v	127.6 (5)
Si2^xii—Na1—Al1^x	146.3 (3)	Si2^xii—O3—Na1^vi	89.7 (4)
Si2^xii—Na1—Na1^vii	113.2 (3)	Al1—O3—Na1^v	91.8 (4)
Si2^xii—Na1—Na1^viii	113.6 (3)	Al1—O3—Na1^vi	124.7 (5)
Si2^xii—Na1—O3^ix	89.1 (2)	Na1^v—O3—Na1^vi	91.2 (3)
Si2^xii—Na1—O3^x	124.8 (3)	Si2^xiii—O4—Al1	137.9 (3)
Si2^xii—Na1—O4	68.85 (19)	Si2^xiii—O4—Na1	99.5 (3)
Si2^xii—Na1—O5^ix	20.8 (2)	Al1—O4—Na1	118.9 (5)
Si2^xii—Na1—O6	91.3 (3)	Si2^xi—O5—Al1	136.5 (4)
Si2^xii—Na1—O6^vii	126.1 (4)	Si2^xi—O5—Na1^v	126.1 (5)
Si2^xii—Na1—H1	99.5 (17)	Al1—O5—Na1^v	93.7 (3)
Si2^xii—Na1—H2	71.7 (10)	Na1—O6—Na1^viii	101.1 (4)
Si2^xii—Na1—H2^vii	145.9 (10)	Na1—O6—H1	27 (2)
Al1—Na1—Al1^ix	97.51 (19)	Na1—O6—H2	80 (3)
Al1—Na1—Al1^x	104.93 (18)	Na1^viii—O6—H1	110 (3)
Al1—Na1—Na1^vii	113.1 (3)	Na1^viii—O6—H2	97 (3)
Al1—Na1—Na1^viii	111.9 (3)	H1—O6—H2	104 (4)
Al1—Na1—O3^ix	126.3 (3)	Na1—H1—O6	137 (3)
Al1—Na1—O3^x	88.2 (2)	Na1—H2—Na1^viii	92.4 (13)
Al1—Na1—O4	25.4 (2)	Na1—H2—O6	76 (3)
Al1—Na1—O5^ix	64.87 (17)	Na1^viii—H2—O6	62 (2)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O6—H1···Na1	0.96 (3)	1.58 (4)	2.371 (14)	137 (3)
O6—H2···O2^xii	0.96 (4)	2.20 (4)	2.892 (13)	128 (3)

Symmetry code: (xii) x−1/4, −y+1/4, z+3/4.

(natrolite_model2) top

Crystal data top

Al₂H₄Na₂O₁₂Si₃	Z = 8
M_r = 380.2	F(000) = 517.84
Orthorhombic, Fdd2	D_x = 2.235 Mg m⁻³
Hall symbol: F -2xuvw;-2yuvw;2z	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 18.2872 (11) Å	Cell parameters from 5213 reflections
b = 18.6660 (14) Å	θ = 0.1–1.2°
c = 6.6222 (3) Å	T = 293 K
V = 2260.5 (2) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.198
Radiation source: Lab6 cathode	θ_max = 1.2°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −29→29
5213 measured reflections	k = −29→29
1289 independent reflections	l = −10→10
801 reflections with I > 3σ(I)

Refinement top

Refinement on F²	All H-atom parameters refined
R[F² > 2σ(F²)] = 0.149	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F²) = 0.406	(Δ/σ)_max = 0.048
S = 1.48	Δρ_max = 0.30 e Å⁻³
1289 reflections	Δρ_min = −0.24 e Å⁻³
93 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 0.64 (16)
13 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Si1	0	0	0	0.0243 (13)
Si2	0.09715 (17)	0.0386 (2)	−0.3724 (8)	0.0240 (9)
Al1	−0.03703 (19)	0.0944 (2)	0.3853 (9)	0.0248 (10)
Na1	−0.2201 (3)	0.0310 (4)	0.3836 (11)	0.0367 (15)
O1	0.0696 (3)	0.0216 (4)	−0.1412 (10)	0.0324 (17)
O2	−0.0226 (3)	0.0675 (4)	0.1345 (13)	0.0321 (16)
O3	−0.0700 (3)	0.1818 (3)	0.3919 (12)	0.0266 (14)
O4	−0.0993 (4)	0.0358 (4)	0.5042 (14)	0.0350 (18)
O5	0.0434 (3)	0.0968 (4)	0.5272 (11)	0.0316 (16)
O6	−0.3070 (4)	0.0599 (6)	0.6385 (15)	0.045 (2)
H1	−0.3566 (10)	0.057 (3)	0.592 (8)	0.0544
H2	−0.307 (3)	0.099 (2)	0.734 (7)	0.0544

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Si1	0.0168 (18)	0.040 (3)	0.0165 (15)	−0.0007 (16)	0	0
Si2	0.0114 (13)	0.044 (2)	0.0169 (11)	−0.0017 (10)	−0.0015 (10)	−0.0037 (14)
Al1	0.0127 (13)	0.040 (2)	0.0216 (14)	0.0012 (11)	0.0000 (12)	0.0009 (15)
Na1	0.022 (2)	0.060 (4)	0.0282 (19)	0.0025 (17)	0.0018 (17)	−0.004 (2)
O1	0.016 (2)	0.065 (4)	0.0156 (18)	0.003 (2)	0.0066 (17)	−0.001 (2)
O2	0.024 (3)	0.048 (4)	0.024 (2)	0.002 (2)	0.001 (2)	−0.004 (3)
O3	0.014 (2)	0.036 (3)	0.030 (2)	0.0003 (15)	−0.003 (2)	−0.001 (2)
O4	0.017 (2)	0.049 (4)	0.039 (3)	−0.004 (2)	0.004 (2)	0.006 (3)
O5	0.013 (2)	0.060 (4)	0.0213 (19)	0.003 (2)	−0.0060 (18)	0.000 (2)
O6	0.029 (3)	0.074 (5)	0.033 (3)	0.001 (3)	−0.001 (3)	0.004 (3)
H1	0.034346	0.089062	0.039731	0.000906	−0.001612	0.004477
H2	0.034346	0.089062	0.039731	0.000906	−0.001612	0.004477

Geometric parameters (Å, º) top

Si1—O1	1.630 (6)	Al1—O2	1.755 (10)
Si1—O1ⁱ	1.630 (6)	Al1—O3	1.739 (8)
Si1—O2	1.597 (8)	Al1—O4	1.765 (9)
Si1—O2ⁱ	1.597 (8)	Al1—O5	1.746 (7)
Si2—Al1ⁱⁱ	3.112 (6)	Na1—Na1^vii	3.674 (10)
Si2—Al1ⁱⁱⁱ	3.043 (5)	Na1—Na1^viii	3.674 (10)
Si2—Na1^iv	3.058 (8)	Na1—O3^ix	2.526 (9)
Si2—Na1ⁱⁱⁱ	3.586 (8)	Na1—O3^x	2.624 (10)
Si2—O1	1.643 (8)	Na1—O4	2.350 (9)
Si2—O3ⁱⁱⁱ	1.616 (6)	Na1—O5^ix	2.401 (11)
Si2—O4^iv	1.612 (9)	Na1—O6	2.381 (11)
Si2—O5ⁱⁱ	1.609 (8)	Na1—O6^vii	2.400 (13)
Al1—Na1	3.551 (7)	Na1—H2^vii	2.66 (4)
Al1—Na1^v	3.100 (9)	O6—H1	0.96 (3)
Al1—Na1^vi	3.862 (9)	O6—H2	0.96 (5)

O1—Si1—O1ⁱ	110.0 (3)	Si2^xii—Na1—O5^ix	21.37 (19)
O1—Si1—O2	109.0 (4)	Si2^xii—Na1—O6	91.4 (3)
O1—Si1—O2ⁱ	108.3 (4)	Si2^xii—Na1—O6^vii	125.8 (3)
O1ⁱ—Si1—O2	108.3 (4)	Si2^xii—Na1—H2^vii	141.7 (11)
O1ⁱ—Si1—O2ⁱ	109.0 (4)	Al1—Na1—Al1^ix	97.1 (2)
O2—Si1—O2ⁱ	112.2 (4)	Al1—Na1—Al1^x	105.52 (18)
Al1ⁱⁱ—Si2—Al1ⁱⁱⁱ	108.49 (18)	Al1—Na1—Na1^vii	112.9 (2)
Al1ⁱⁱ—Si2—Na1^iv	116.7 (2)	Al1—Na1—Na1^viii	112.5 (2)
Al1ⁱⁱ—Si2—Na1ⁱⁱⁱ	54.59 (15)	Al1—Na1—O3^ix	126.3 (3)
Al1ⁱⁱ—Si2—O1	107.6 (3)	Al1—Na1—O3^x	88.3 (2)
Al1ⁱⁱ—Si2—O3ⁱⁱⁱ	130.9 (3)	Al1—Na1—O4	25.9 (2)
Al1ⁱⁱ—Si2—O4^iv	92.7 (3)	Al1—Na1—O5^ix	65.6 (2)
Al1ⁱⁱ—Si2—O5ⁱⁱ	22.9 (3)	Al1—Na1—O6	123.5 (4)
Al1ⁱⁱⁱ—Si2—Na1^iv	78.56 (17)	Al1—Na1—O6^vii	92.5 (3)
Al1ⁱⁱⁱ—Si2—Na1ⁱⁱⁱ	64.18 (14)	Al1—Na1—H2^vii	97.3 (11)
Al1ⁱⁱⁱ—Si2—O1	109.6 (3)	Al1^ix—Na1—Al1^x	156.7 (2)
Al1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	25.9 (2)	Al1^ix—Na1—Na1^vii	68.9 (2)
Al1ⁱⁱⁱ—Si2—O4^iv	127.6 (3)	Al1^ix—Na1—Na1^viii	126.6 (2)
Al1ⁱⁱⁱ—Si2—O5ⁱⁱ	87.0 (3)	Al1^ix—Na1—O3^ix	34.11 (18)
Na1^iv—Si2—Na1ⁱⁱⁱ	131.1 (2)	Al1^ix—Na1—O3^x	170.1 (3)
Na1^iv—Si2—O1	129.5 (3)	Al1^ix—Na1—O4	121.5 (3)
Na1^iv—Si2—O3ⁱⁱⁱ	59.1 (3)	Al1^ix—Na1—O5^ix	34.09 (19)
Na1^iv—Si2—O4^iv	49.3 (3)	Al1^ix—Na1—O6	86.7 (3)
Na1^iv—Si2—O5ⁱⁱ	121.2 (4)	Al1^ix—Na1—O6^vii	104.7 (4)
Na1ⁱⁱⁱ—Si2—O1	93.7 (3)	Al1^ix—Na1—H2^vii	123.9 (11)
Na1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	90.0 (3)	Al1^x—Na1—Na1^vii	96.6 (2)
Na1ⁱⁱⁱ—Si2—O4^iv	145.4 (4)	Al1^x—Na1—Na1^viii	48.50 (16)
Na1ⁱⁱⁱ—Si2—O5ⁱⁱ	32.9 (3)	Al1^x—Na1—O3^ix	123.1 (3)
O1—Si2—O3ⁱⁱⁱ	107.4 (5)	Al1^x—Na1—O3^x	22.11 (17)
O1—Si2—O4^iv	108.3 (5)	Al1^x—Na1—O4	81.7 (3)
O1—Si2—O5ⁱⁱ	109.1 (4)	Al1^x—Na1—O5^ix	168.5 (3)
O3ⁱⁱⁱ—Si2—O4^iv	107.5 (4)	Al1^x—Na1—O6	85.5 (3)
O3ⁱⁱⁱ—Si2—O5ⁱⁱ	111.5 (4)	Al1^x—Na1—O6^vii	69.4 (3)
O4^iv—Si2—O5ⁱⁱ	112.8 (5)	Al1^x—Na1—H2^vii	48.4 (10)
Si2^xi—Al1—Si2^xii	143.0 (2)	Na1^vii—Na1—Na1^viii	128.7 (2)
Si2^xi—Al1—Na1	129.3 (2)	Na1^vii—Na1—O3^ix	45.6 (2)
Si2^xi—Al1—Na1^v	70.52 (18)	Na1^vii—Na1—O3^x	116.6 (3)
Si2^xi—Al1—Na1^vi	119.86 (16)	Na1^vii—Na1—O4	126.7 (3)
Si2^xi—Al1—O2	105.9 (3)	Na1^vii—Na1—O5^ix	93.7 (3)
Si2^xi—Al1—O3	125.1 (3)	Na1^vii—Na1—O6	120.8 (3)
Si2^xi—Al1—O4	94.1 (3)	Na1^vii—Na1—O6^vii	39.6 (3)
Si2^xi—Al1—O5	21.0 (3)	Na1^vii—Na1—H2^vii	55.5 (11)
Si2^xii—Al1—Na1	65.36 (15)	Na1^viii—Na1—O3^ix	116.3 (3)
Si2^xii—Al1—Na1^v	75.25 (18)	Na1^viii—Na1—O3^x	43.4 (2)
Si2^xii—Al1—Na1^vi	50.89 (14)	Na1^viii—Na1—O4	89.2 (3)
Si2^xii—Al1—O2	107.1 (3)	Na1^viii—Na1—O5^ix	126.5 (3)
Si2^xii—Al1—O3	24.0 (2)	Na1^viii—Na1—O6	40.0 (3)
Si2^xii—Al1—O4	90.2 (3)	Na1^viii—Na1—O6^vii	116.7 (4)
Si2^xii—Al1—O5	125.3 (4)	Na1^viii—Na1—H2^vii	96.1 (11)
Na1—Al1—Na1^v	128.6 (2)	O3^ix—Na1—O3^x	144.0 (3)
Na1—Al1—Na1^vi	109.34 (18)	O3^ix—Na1—O4	152.1 (4)
Na1—Al1—O2	92.5 (3)	O3^ix—Na1—O5^ix	68.1 (3)
Na1—Al1—O3	89.2 (2)	O3^ix—Na1—O6	84.5 (3)
Na1—Al1—O4	35.5 (3)	O3^ix—Na1—O6^vii	85.1 (4)
Na1—Al1—O5	143.5 (4)	O3^ix—Na1—H2^vii	98.6 (11)
Na1^v—Al1—Na1^vi	62.57 (19)	O3^x—Na1—O4	62.8 (3)
Na1^v—Al1—O2	131.1 (3)	O3^x—Na1—O5^ix	146.4 (4)
Na1^v—Al1—O3	54.5 (3)	O3^x—Na1—O6	83.4 (4)
Na1^v—Al1—O4	118.9 (4)	O3^x—Na1—O6^vii	83.4 (4)
Na1^v—Al1—O5	50.4 (3)	O3^x—Na1—H2^vii	63.3 (11)
Na1^vi—Al1—O2	81.1 (3)	O4—Na1—O5^ix	88.1 (3)
Na1^vi—Al1—O3	34.6 (3)	O4—Na1—O6	112.2 (4)
Na1^vi—Al1—O4	140.6 (3)	O4—Na1—O6^vii	93.6 (4)
Na1^vi—Al1—O5	100.4 (3)	O4—Na1—H2^vii	89.2 (11)
O2—Al1—O3	110.1 (5)	O5^ix—Na1—O6	93.5 (4)
O2—Al1—O4	110.0 (4)	O5^ix—Na1—O6^vii	116.9 (4)
O2—Al1—O5	112.9 (4)	O5^ix—Na1—H2^vii	137.3 (11)
O3—Al1—O4	110.3 (4)	O6—Na1—O6^vii	141.1 (4)
O3—Al1—O5	104.8 (4)	O6—Na1—H2^vii	126.6 (11)
O4—Al1—O5	108.6 (5)	O6^vii—Na1—H2^vii	21.0 (11)
Si2^xiii—Na1—Si2^xii	98.78 (19)	Si1—O1—Si2	145.3 (4)
Si2^xiii—Na1—Al1	56.41 (15)	Si1—O2—Al1	142.4 (5)
Si2^xiii—Na1—Al1^ix	152.7 (3)	Si2^xii—O3—Al1	130.1 (4)
Si2^xiii—Na1—Al1^x	50.54 (14)	Si2^xii—O3—Na1^v	128.4 (4)
Si2^xiii—Na1—Na1^vii	124.1 (2)	Si2^xii—O3—Na1^vi	89.0 (4)
Si2^xiii—Na1—Na1^viii	66.99 (19)	Al1—O3—Na1^v	91.3 (3)
Si2^xiii—Na1—O3^ix	169.5 (3)	Al1—O3—Na1^vi	123.3 (4)
Si2^xiii—Na1—O3^x	31.91 (16)	Na1^v—O3—Na1^vi	91.0 (3)
Si2^xiii—Na1—O4	31.3 (2)	Si2^xiii—O4—Al1	138.1 (5)
Si2^xiii—Na1—O5^ix	118.9 (3)	Si2^xiii—O4—Na1	99.3 (4)
Si2^xiii—Na1—O6	102.3 (4)	Al1—O4—Na1	118.6 (5)
Si2^xiii—Na1—O6^vii	84.6 (3)	Si2^xi—O5—Al1	136.1 (5)
Si2^xiii—Na1—H2^vii	70.9 (11)	Si2^xi—O5—Na1^v	125.7 (4)
Si2^xii—Na1—Al1	50.47 (12)	Al1—O5—Na1^v	95.5 (4)
Si2^xii—Na1—Al1^ix	54.89 (15)	Na1—O6—Na1^viii	100.4 (4)
Si2^xii—Na1—Al1^x	147.1 (2)	Na1—O6—H1	113 (3)
Si2^xii—Na1—Na1^vii	112.8 (2)	Na1—O6—H2	130 (3)
Si2^xii—Na1—Na1^viii	114.1 (2)	Na1^viii—O6—H1	112 (3)
Si2^xii—Na1—O3^ix	89.0 (2)	Na1^viii—O6—H2	95 (3)
Si2^xii—Na1—O3^x	125.0 (3)	H1—O6—H2	104 (4)
Si2^xii—Na1—O4	69.2 (3)	Na1^viii—H2—O6	64 (2)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O6—H1···O1^xiv	0.96 (3)	2.16 (4)	3.005 (11)	147 (4)
O6—H2···O2^xii	0.96 (5)	1.96 (5)	2.878 (13)	159 (4)

Symmetry codes: (xii) x−1/4, −y+1/4, z+3/4; (xiv) x−1/2, y, z+1/2.

(natrolite_model3) top

Crystal data top

Al₂H₄Na₂O₁₂Si₃	Z = 8
M_r = 380.2	F(000) = 517.84
Orthorhombic, Fdd2	D_x = 2.235 Mg m⁻³
Hall symbol: F -2xuvw;-2yuvw;2z	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 18.2872 (11) Å	Cell parameters from 5213 reflections
b = 18.6660 (14) Å	θ = 0.1–1.2°
c = 6.6222 (3) Å	T = 293 K
V = 2260.5 (2) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.198
Radiation source: Lab6 cathode	θ_max = 1.2°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −29→29
5213 measured reflections	k = −29→29
1289 independent reflections	l = −10→10
1007 reflections with I > 3σ(I)

Refinement top

Refinement on F²	All H-atom parameters refined
R[F² > 2σ(F²)] = 0.147	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F²) = 0.363	(Δ/σ)_max = 0.035
S = 1.78	Δρ_max = 0.28 e Å⁻³
1289 reflections	Δρ_min = −0.24 e Å⁻³
93 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 0.40 (9)
13 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Si1	0	0	0	0.0233 (12)
Si2	0.09699 (16)	0.03848 (19)	−0.3724 (9)	0.0228 (8)
Al1	−0.03714 (18)	0.0943 (2)	0.3857 (10)	0.0235 (9)
Na1	−0.2203 (3)	0.0309 (3)	0.3829 (12)	0.0356 (14)
O1	0.0695 (3)	0.0216 (4)	−0.1412 (11)	0.0306 (15)
O2	−0.0224 (3)	0.0676 (4)	0.1353 (13)	0.0306 (15)
O3	−0.0697 (3)	0.1818 (3)	0.3919 (12)	0.0251 (13)
O4	−0.0992 (3)	0.0358 (4)	0.5026 (14)	0.0327 (17)
O5	0.0437 (3)	0.0969 (4)	0.5268 (12)	0.0303 (15)
O6	−0.3065 (4)	0.0598 (5)	0.6375 (16)	0.044 (2)
H1	−0.3560 (10)	0.059 (3)	0.590 (7)	0.0526
H2	−0.305 (2)	0.099 (2)	0.730 (7)	0.0526

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Si1	0.0148 (16)	0.039 (3)	0.0162 (17)	−0.0014 (14)	0	0
Si2	0.0107 (12)	0.0411 (19)	0.0166 (12)	−0.0015 (9)	−0.0012 (10)	−0.0023 (14)
Al1	0.0125 (12)	0.038 (2)	0.0203 (14)	0.0019 (10)	0.0006 (12)	0.0004 (14)
Na1	0.0218 (19)	0.056 (3)	0.029 (2)	0.0017 (15)	−0.0014 (18)	−0.003 (2)
O1	0.0156 (19)	0.060 (4)	0.016 (2)	0.0029 (18)	0.0049 (18)	−0.001 (2)
O2	0.022 (2)	0.049 (3)	0.021 (2)	0.0015 (18)	0.001 (2)	−0.004 (2)
O3	0.0137 (18)	0.035 (3)	0.026 (2)	0.0004 (14)	−0.0029 (19)	−0.001 (2)
O4	0.013 (2)	0.048 (3)	0.037 (3)	−0.0039 (16)	0.001 (2)	0.008 (3)
O5	0.0121 (19)	0.057 (4)	0.022 (2)	0.0028 (18)	−0.0060 (18)	0.001 (2)
O6	0.029 (3)	0.071 (5)	0.031 (3)	0.001 (3)	−0.002 (3)	0.001 (3)
H1	0.035279	0.084949	0.037716	0.000837	−0.002722	0.001305
H2	0.035279	0.084949	0.037716	0.000837	−0.002722	0.001305

Geometric parameters (Å, º) top

Si1—O1	1.628 (6)	Al1—O2	1.752 (10)
Si1—O1ⁱ	1.628 (6)	Al1—O3	1.740 (7)
Si1—O2	1.601 (7)	Al1—O4	1.755 (8)
Si1—O2ⁱ	1.601 (7)	Al1—O5	1.750 (7)
Si2—Al1ⁱⁱ	3.110 (6)	Na1—Na1^vii	3.671 (11)
Si2—Al1ⁱⁱⁱ	3.046 (5)	Na1—Na1^viii	3.671 (11)
Si2—Na1^iv	3.064 (8)	Na1—O3^ix	2.517 (9)
Si2—Na1ⁱⁱⁱ	3.589 (7)	Na1—O3^x	2.627 (10)
Si2—O1	1.642 (9)	Na1—O4	2.353 (8)
Si2—O3ⁱⁱⁱ	1.623 (6)	Na1—O5^ix	2.401 (10)
Si2—O4^iv	1.615 (9)	Na1—O6	2.370 (12)
Si2—O5ⁱⁱ	1.607 (8)	Na1—O6^vii	2.398 (12)
Al1—Na1	3.552 (6)	Na1—H2^vii	2.67 (4)
Al1—Na1^v	3.098 (8)	O6—H1	0.96 (2)
Al1—Na1^vi	3.867 (8)	O6—H2	0.96 (4)

O1—Si1—O1ⁱ	109.9 (3)	Si2^xii—Na1—O5^ix	21.24 (19)
O1—Si1—O2	109.0 (3)	Si2^xii—Na1—O6	91.2 (3)
O1—Si1—O2ⁱ	108.5 (3)	Si2^xii—Na1—O6^vii	125.9 (3)
O1ⁱ—Si1—O2	108.5 (3)	Si2^xii—Na1—H2^vii	142.3 (10)
O1ⁱ—Si1—O2ⁱ	109.0 (3)	Al1—Na1—Al1^ix	97.15 (19)
O2—Si1—O2ⁱ	111.9 (4)	Al1—Na1—Al1^x	105.34 (17)
Al1ⁱⁱ—Si2—Al1ⁱⁱⁱ	108.48 (17)	Al1—Na1—Na1^vii	112.9 (2)
Al1ⁱⁱ—Si2—Na1^iv	116.7 (3)	Al1—Na1—Na1^viii	112.3 (2)
Al1ⁱⁱ—Si2—Na1ⁱⁱⁱ	54.54 (14)	Al1—Na1—O3^ix	126.3 (3)
Al1ⁱⁱ—Si2—O1	107.6 (3)	Al1—Na1—O3^x	88.2 (2)
Al1ⁱⁱ—Si2—O3ⁱⁱⁱ	131.0 (3)	Al1—Na1—O4	25.6 (2)
Al1ⁱⁱ—Si2—O4^iv	92.5 (3)	Al1—Na1—O5^ix	65.44 (18)
Al1ⁱⁱ—Si2—O5ⁱⁱ	23.2 (3)	Al1—Na1—O6	123.2 (3)
Al1ⁱⁱⁱ—Si2—Na1^iv	78.53 (16)	Al1—Na1—O6^vii	92.6 (3)
Al1ⁱⁱⁱ—Si2—Na1ⁱⁱⁱ	64.13 (13)	Al1—Na1—H2^vii	98.2 (9)
Al1ⁱⁱⁱ—Si2—O1	109.5 (3)	Al1^ix—Na1—Al1^x	156.9 (2)
Al1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	26.0 (2)	Al1^ix—Na1—Na1^vii	69.1 (2)
Al1ⁱⁱⁱ—Si2—O4^iv	127.4 (3)	Al1^ix—Na1—Na1^viii	126.6 (2)
Al1ⁱⁱⁱ—Si2—O5ⁱⁱ	86.7 (3)	Al1^ix—Na1—O3^ix	34.15 (17)
Na1^iv—Si2—Na1ⁱⁱⁱ	130.9 (2)	Al1^ix—Na1—O3^x	169.9 (3)
Na1^iv—Si2—O1	129.6 (3)	Al1^ix—Na1—O4	121.3 (3)
Na1^iv—Si2—O3ⁱⁱⁱ	59.0 (3)	Al1^ix—Na1—O5^ix	34.23 (18)
Na1^iv—Si2—O4^iv	49.3 (3)	Al1^ix—Na1—O6	86.7 (3)
Na1^iv—Si2—O5ⁱⁱ	120.9 (4)	Al1^ix—Na1—O6^vii	104.7 (4)
Na1ⁱⁱⁱ—Si2—O1	93.8 (3)	Al1^ix—Na1—H2^vii	123.5 (10)
Na1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	90.0 (3)	Al1^x—Na1—Na1^vii	96.7 (2)
Na1ⁱⁱⁱ—Si2—O4^iv	145.1 (4)	Al1^x—Na1—Na1^viii	48.45 (16)
Na1ⁱⁱⁱ—Si2—O5ⁱⁱ	32.8 (3)	Al1^x—Na1—O3^ix	123.3 (2)
O1—Si2—O3ⁱⁱⁱ	107.4 (5)	Al1^x—Na1—O3^x	22.06 (16)
O1—Si2—O4^iv	108.8 (5)	Al1^x—Na1—O4	81.7 (3)
O1—Si2—O5ⁱⁱ	109.4 (4)	Al1^x—Na1—O5^ix	168.2 (3)
O3ⁱⁱⁱ—Si2—O4^iv	107.4 (4)	Al1^x—Na1—O6	85.4 (3)
O3ⁱⁱⁱ—Si2—O5ⁱⁱ	111.2 (4)	Al1^x—Na1—O6^vii	69.5 (3)
O4^iv—Si2—O5ⁱⁱ	112.6 (5)	Al1^x—Na1—H2^vii	48.6 (10)
Si2^xi—Al1—Si2^xii	143.0 (2)	Na1^vii—Na1—Na1^viii	128.8 (2)
Si2^xi—Al1—Na1	129.4 (2)	Na1^vii—Na1—O3^ix	45.7 (2)
Si2^xi—Al1—Na1^v	70.63 (17)	Na1^vii—Na1—O3^x	116.7 (3)
Si2^xi—Al1—Na1^vi	119.81 (15)	Na1^vii—Na1—O4	126.5 (3)
Si2^xi—Al1—O2	105.7 (3)	Na1^vii—Na1—O5^ix	93.8 (3)
Si2^xi—Al1—O3	124.9 (3)	Na1^vii—Na1—O6	121.1 (3)
Si2^xi—Al1—O4	94.3 (3)	Na1^vii—Na1—O6^vii	39.4 (3)
Si2^xi—Al1—O5	21.2 (3)	Na1^vii—Na1—H2^vii	54.8 (10)
Si2^xii—Al1—Na1	65.37 (13)	Na1^viii—Na1—O3^ix	116.4 (3)
Si2^xii—Al1—Na1^v	75.15 (16)	Na1^viii—Na1—O3^x	43.3 (2)
Si2^xii—Al1—Na1^vi	50.94 (13)	Na1^viii—Na1—O4	89.2 (3)
Si2^xii—Al1—O2	107.1 (3)	Na1^viii—Na1—O5^ix	126.5 (3)
Si2^xii—Al1—O3	24.16 (18)	Na1^viii—Na1—O6	39.9 (3)
Si2^xii—Al1—O4	90.3 (3)	Na1^viii—Na1—O6^vii	116.7 (3)
Si2^xii—Al1—O5	125.3 (3)	Na1^viii—Na1—H2^vii	96.1 (10)
Na1—Al1—Na1^v	128.7 (2)	O3^ix—Na1—O3^x	144.1 (3)
Na1—Al1—Na1^vi	109.32 (17)	O3^ix—Na1—O4	151.9 (4)
Na1—Al1—O2	92.6 (3)	O3^ix—Na1—O5^ix	68.2 (3)
Na1—Al1—O3	89.4 (2)	O3^ix—Na1—O6	84.7 (3)
Na1—Al1—O4	35.4 (3)	O3^ix—Na1—O6^vii	85.0 (4)
Na1—Al1—O5	144.0 (4)	O3^ix—Na1—H2^vii	97.8 (10)
Na1^v—Al1—Na1^vi	62.46 (19)	O3^x—Na1—O4	62.9 (3)
Na1^v—Al1—O2	130.8 (3)	O3^x—Na1—O5^ix	146.2 (4)
Na1^v—Al1—O3	54.3 (3)	O3^x—Na1—O6	83.2 (4)
Na1^v—Al1—O4	119.3 (4)	O3^x—Na1—O6^vii	83.6 (3)
Na1^v—Al1—O5	50.5 (3)	O3^x—Na1—H2^vii	63.8 (10)
Na1^vi—Al1—O2	80.9 (3)	O4—Na1—O5^ix	87.8 (3)
Na1^vi—Al1—O3	34.5 (3)	O4—Na1—O6	112.2 (4)
Na1^vi—Al1—O4	140.7 (3)	O4—Na1—O6^vii	93.6 (3)
Na1^vi—Al1—O5	100.2 (3)	O4—Na1—H2^vii	90.1 (10)
O2—Al1—O3	110.0 (5)	O5^ix—Na1—O6	93.6 (4)
O2—Al1—O4	109.9 (4)	O5^ix—Na1—O6^vii	116.8 (4)
O2—Al1—O5	112.5 (4)	O5^ix—Na1—H2^vii	137.3 (10)
O3—Al1—O4	110.6 (4)	O6—Na1—O6^vii	141.1 (4)
O3—Al1—O5	104.5 (4)	O6—Na1—H2^vii	126.2 (10)
O4—Al1—O5	109.2 (5)	O6^vii—Na1—H2^vii	20.9 (10)
Si2^xiii—Na1—Si2^xii	98.63 (18)	Si1—O1—Si2	145.3 (4)
Si2^xiii—Na1—Al1	56.23 (14)	Si1—O2—Al1	142.5 (5)
Si2^xiii—Na1—Al1^ix	152.6 (2)	Si2^xii—O3—Al1	129.8 (4)
Si2^xiii—Na1—Al1^x	50.53 (13)	Si2^xii—O3—Na1^v	128.3 (4)
Si2^xiii—Na1—Na1^vii	124.2 (2)	Si2^xii—O3—Na1^vi	89.0 (3)
Si2^xiii—Na1—Na1^viii	66.91 (19)	Al1—O3—Na1^v	91.5 (3)
Si2^xiii—Na1—O3^ix	169.6 (3)	Al1—O3—Na1^vi	123.4 (4)
Si2^xiii—Na1—O3^x	31.99 (15)	Na1^v—O3—Na1^vi	91.0 (3)
Si2^xiii—Na1—O4	31.3 (2)	Si2^xiii—O4—Al1	138.0 (4)
Si2^xiii—Na1—O5^ix	118.5 (3)	Si2^xiii—O4—Na1	99.4 (3)
Si2^xiii—Na1—O6	102.1 (4)	Al1—O4—Na1	119.0 (5)
Si2^xiii—Na1—O6^vii	84.8 (3)	Si2^xi—O5—Al1	135.7 (5)
Si2^xiii—Na1—H2^vii	71.9 (10)	Si2^xi—O5—Na1^v	126.0 (4)
Si2^xii—Na1—Al1	50.50 (11)	Al1—O5—Na1^v	95.3 (3)
Si2^xii—Na1—Al1^ix	54.83 (14)	Na1—O6—Na1^viii	100.7 (4)
Si2^xii—Na1—Al1^x	147.0 (2)	Na1—O6—H1	113 (3)
Si2^xii—Na1—Na1^vii	112.8 (2)	Na1—O6—H2	127 (3)
Si2^xii—Na1—Na1^viii	113.9 (2)	Na1^viii—O6—H1	114 (3)
Si2^xii—Na1—O3^ix	89.0 (2)	Na1^viii—O6—H2	96 (3)
Si2^xii—Na1—O3^x	124.9 (3)	H1—O6—H2	104 (4)
Si2^xii—Na1—O4	69.2 (2)	Na1^viii—H2—O6	63 (2)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O6—H1···O1^xiv	0.96 (2)	2.17 (4)	3.009 (11)	146 (4)
O6—H2···O2^xii	0.96 (4)	1.95 (4)	2.884 (12)	163 (4)

Symmetry codes: (xii) x−1/4, −y+1/4, z+3/4; (xiv) x−1/2, y, z+1/2.

(natrolite_model3_outliers) top

Crystal data top

Al₂H₄Na₂O₁₂Si₃	Z = 8
M_r = 380.2	F(000) = 517.84
Orthorhombic, Fdd2	D_x = 2.235 Mg m⁻³
Hall symbol: F -2xuvw;-2yuvw;2z	Electrons 200 KeV radiation, λ = 0.0251 Å
a = 18.2872 (11) Å	Cell parameters from 5179 reflections
b = 18.6660 (14) Å	θ = 0.1–1.2°
c = 6.6222 (3) Å	T = 293 K
V = 2260.5 (2) Å³	Irregular shape

Data collection top

TEM FEI Tecnai G2 20 diffractometer	R_int = 0.195
Radiation source: Lab6 cathode	θ_max = 1.2°, θ_min = 0.1°
continuous–rotation 3D ED scans	h = −29→29
5179 measured reflections	k = −29→29
1289 independent reflections	l = −10→10
1008 reflections with I > 3σ(I)

Refinement top

Refinement on F²	All H-atom parameters refined
R[F² > 2σ(F²)] = 0.147	Weighting scheme based on measured s.u.'s w = 1/(σ²(I) + 0.0004I²)
wR(F²) = 0.366	(Δ/σ)_max = 0.017
S = 1.78	Δρ_max = 0.28 e Å⁻³
1289 reflections	Δρ_min = −0.24 e Å⁻³
93 parameters	Extinction correction: SHELXL-2017/1 (Sheldrick, 2015), Fc^*=kFc[1+0.001xFc²λ³/sin(2θ)]^-1/4
3 restraints	Extinction coefficient: 0.40 (9)
13 constraints

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å²) top

	x	y	z	U_iso*/U_eq
Si1	0	0	0	0.0234 (12)
Si2	0.09700 (16)	0.03849 (19)	−0.3720 (9)	0.0228 (8)
Al1	−0.03715 (18)	0.0943 (2)	0.3857 (10)	0.0235 (9)
Na1	−0.2203 (3)	0.0309 (3)	0.3831 (12)	0.0355 (14)
O1	0.0695 (3)	0.0217 (4)	−0.1408 (11)	0.0306 (15)
O2	−0.0224 (3)	0.0676 (4)	0.1355 (13)	0.0306 (15)
O3	−0.0697 (3)	0.1818 (3)	0.3921 (12)	0.0253 (13)
O4	−0.0993 (3)	0.0358 (4)	0.5029 (14)	0.0328 (17)
O5	0.0437 (3)	0.0968 (4)	0.5270 (12)	0.0303 (15)
O6	−0.3065 (4)	0.0599 (5)	0.6376 (16)	0.044 (2)
H1	−0.3559 (10)	0.059 (3)	0.590 (7)	0.0525
H2	−0.305 (2)	0.099 (2)	0.730 (7)	0.0525

Atomic displacement parameters (Å²) top

	U¹¹	U²²	U³³	U¹²	U¹³	U²³
Si1	0.0149 (16)	0.038 (3)	0.0168 (17)	−0.0011 (14)	−0	−0
Si2	0.0105 (12)	0.0412 (19)	0.0166 (12)	−0.0014 (9)	−0.0013 (10)	−0.0024 (14)
Al1	0.0125 (12)	0.038 (2)	0.0205 (14)	0.0019 (10)	0.0004 (12)	0.0007 (14)
Na1	0.0216 (19)	0.056 (3)	0.029 (2)	0.0015 (15)	−0.0015 (18)	−0.004 (2)
O1	0.0155 (19)	0.060 (4)	0.016 (2)	0.0031 (18)	0.0049 (18)	−0.001 (2)
O2	0.022 (2)	0.049 (3)	0.021 (2)	0.0013 (18)	0.001 (2)	−0.003 (2)
O3	0.0139 (18)	0.035 (3)	0.027 (2)	0.0003 (14)	−0.0032 (19)	−0.001 (2)
O4	0.013 (2)	0.047 (3)	0.038 (3)	−0.0042 (17)	0.001 (2)	0.008 (3)
O5	0.0122 (19)	0.057 (4)	0.022 (2)	0.0029 (18)	−0.0058 (18)	0.001 (2)
O6	0.029 (3)	0.071 (5)	0.031 (3)	0.000 (3)	−0.002 (3)	0.001 (3)
H1	0.035211	0.084898	0.037356	0.000318	−0.00225	0.001636
H2	0.035211	0.084898	0.037356	0.000318	−0.00225	0.001636

Geometric parameters (Å, º) top

Si1—O1	1.627 (6)	Al1—O2	1.751 (10)
Si1—O1ⁱ	1.627 (6)	Al1—O3	1.740 (7)
Si1—O2	1.601 (7)	Al1—O4	1.757 (8)
Si1—O2ⁱ	1.601 (7)	Al1—O5	1.751 (7)
Si2—Al1ⁱⁱ	3.111 (6)	Na1—Na1^vii	3.671 (11)
Si2—Al1ⁱⁱⁱ	3.046 (5)	Na1—Na1^viii	3.671 (11)
Si2—Na1^iv	3.065 (8)	Na1—O3^ix	2.517 (9)
Si2—Na1ⁱⁱⁱ	3.589 (7)	Na1—O3^x	2.626 (10)
Si2—O1	1.642 (9)	Na1—O4	2.353 (8)
Si2—O3ⁱⁱⁱ	1.623 (6)	Na1—O5^ix	2.401 (10)
Si2—O4^iv	1.616 (9)	Na1—O6	2.370 (12)
Si2—O5ⁱⁱ	1.607 (8)	Na1—O6^vii	2.399 (12)
Al1—Na1	3.553 (6)	Na1—H2^vii	2.67 (4)
Al1—Na1^v	3.099 (8)	O6—H1	0.96 (2)
Al1—Na1^vi	3.867 (8)	O6—H2	0.96 (4)

O1—Si1—O1ⁱ	110.0 (3)	Si2^xii—Na1—O5^ix	21.26 (19)
O1—Si1—O2	109.0 (3)	Si2^xii—Na1—O6	91.2 (3)
O1—Si1—O2ⁱ	108.5 (3)	Si2^xii—Na1—O6^vii	125.9 (3)
O1ⁱ—Si1—O2	108.5 (3)	Si2^xii—Na1—H2^vii	142.3 (10)
O1ⁱ—Si1—O2ⁱ	109.0 (3)	Al1—Na1—Al1^ix	97.13 (19)
O2—Si1—O2ⁱ	111.8 (4)	Al1—Na1—Al1^x	105.34 (17)
Al1ⁱⁱ—Si2—Al1ⁱⁱⁱ	108.46 (17)	Al1—Na1—Na1^vii	112.8 (2)
Al1ⁱⁱ—Si2—Na1^iv	116.7 (3)	Al1—Na1—Na1^viii	112.3 (2)
Al1ⁱⁱ—Si2—Na1ⁱⁱⁱ	54.54 (15)	Al1—Na1—O3^ix	126.3 (3)
Al1ⁱⁱ—Si2—O1	107.6 (3)	Al1—Na1—O3^x	88.2 (2)
Al1ⁱⁱ—Si2—O3ⁱⁱⁱ	130.9 (3)	Al1—Na1—O4	25.6 (2)
Al1ⁱⁱ—Si2—O4^iv	92.5 (3)	Al1—Na1—O5^ix	65.42 (18)
Al1ⁱⁱ—Si2—O5ⁱⁱ	23.1 (3)	Al1—Na1—O6	123.2 (3)
Al1ⁱⁱⁱ—Si2—Na1^iv	78.51 (16)	Al1—Na1—O6^vii	92.6 (3)
Al1ⁱⁱⁱ—Si2—Na1ⁱⁱⁱ	64.14 (13)	Al1—Na1—H2^vii	98.2 (10)
Al1ⁱⁱⁱ—Si2—O1	109.6 (3)	Al1^ix—Na1—Al1^x	156.9 (2)
Al1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	26.0 (2)	Al1^ix—Na1—Na1^vii	69.1 (2)
Al1ⁱⁱⁱ—Si2—O4^iv	127.4 (3)	Al1^ix—Na1—Na1^viii	126.6 (2)
Al1ⁱⁱⁱ—Si2—O5ⁱⁱ	86.7 (3)	Al1^ix—Na1—O3^ix	34.15 (17)
Na1^iv—Si2—Na1ⁱⁱⁱ	130.9 (2)	Al1^ix—Na1—O3^x	169.9 (3)
Na1^iv—Si2—O1	129.7 (3)	Al1^ix—Na1—O4	121.3 (3)
Na1^iv—Si2—O3ⁱⁱⁱ	58.9 (3)	Al1^ix—Na1—O5^ix	34.23 (18)
Na1^iv—Si2—O4^iv	49.2 (3)	Al1^ix—Na1—O6	86.7 (3)
Na1^iv—Si2—O5ⁱⁱ	120.8 (4)	Al1^ix—Na1—O6^vii	104.7 (4)
Na1ⁱⁱⁱ—Si2—O1	93.8 (3)	Al1^ix—Na1—H2^vii	123.4 (10)
Na1ⁱⁱⁱ—Si2—O3ⁱⁱⁱ	90.1 (3)	Al1^x—Na1—Na1^vii	96.7 (2)
Na1ⁱⁱⁱ—Si2—O4^iv	145.1 (4)	Al1^x—Na1—Na1^viii	48.47 (16)
Na1ⁱⁱⁱ—Si2—O5ⁱⁱ	32.8 (3)	Al1^x—Na1—O3^ix	123.3 (2)
O1—Si2—O3ⁱⁱⁱ	107.4 (5)	Al1^x—Na1—O3^x	22.08 (16)
O1—Si2—O4^iv	108.8 (5)	Al1^x—Na1—O4	81.7 (3)
O1—Si2—O5ⁱⁱ	109.4 (4)	Al1^x—Na1—O5^ix	168.2 (3)
O3ⁱⁱⁱ—Si2—O4^iv	107.3 (4)	Al1^x—Na1—O6	85.5 (3)
O3ⁱⁱⁱ—Si2—O5ⁱⁱ	111.2 (4)	Al1^x—Na1—O6^vii	69.5 (3)
O4^iv—Si2—O5ⁱⁱ	112.6 (5)	Al1^x—Na1—H2^vii	48.6 (10)
Si2^xi—Al1—Si2^xii	142.9 (2)	Na1^vii—Na1—Na1^viii	128.9 (2)
Si2^xi—Al1—Na1	129.4 (2)	Na1^vii—Na1—O3^ix	45.7 (2)
Si2^xi—Al1—Na1^v	70.61 (17)	Na1^vii—Na1—O3^x	116.7 (3)
Si2^xi—Al1—Na1^vi	119.79 (15)	Na1^vii—Na1—O4	126.5 (3)
Si2^xi—Al1—O2	105.7 (3)	Na1^vii—Na1—O5^ix	93.8 (3)
Si2^xi—Al1—O3	124.9 (3)	Na1^vii—Na1—O6	121.1 (3)
Si2^xi—Al1—O4	94.3 (3)	Na1^vii—Na1—O6^vii	39.4 (3)
Si2^xi—Al1—O5	21.1 (3)	Na1^vii—Na1—H2^vii	54.8 (10)
Si2^xii—Al1—Na1	65.37 (13)	Na1^viii—Na1—O3^ix	116.4 (3)
Si2^xii—Al1—Na1^v	75.12 (16)	Na1^viii—Na1—O3^x	43.3 (2)
Si2^xii—Al1—Na1^vi	50.96 (13)	Na1^viii—Na1—O4	89.2 (3)
Si2^xii—Al1—O2	107.2 (3)	Na1^viii—Na1—O5^ix	126.5 (3)
Si2^xii—Al1—O3	24.15 (18)	Na1^viii—Na1—O6	40.0 (3)
Si2^xii—Al1—O4	90.2 (3)	Na1^viii—Na1—O6^vii	116.7 (3)
Si2^xii—Al1—O5	125.2 (3)	Na1^viii—Na1—H2^vii	96.1 (10)
Na1—Al1—Na1^v	128.7 (2)	O3^ix—Na1—O3^x	144.1 (3)
Na1—Al1—Na1^vi	109.34 (17)	O3^ix—Na1—O4	151.9 (4)
Na1—Al1—O2	92.6 (3)	O3^ix—Na1—O5^ix	68.2 (3)
Na1—Al1—O3	89.4 (2)	O3^ix—Na1—O6	84.7 (3)
Na1—Al1—O4	35.4 (3)	O3^ix—Na1—O6^vii	85.0 (4)
Na1—Al1—O5	143.9 (4)	O3^ix—Na1—H2^vii	97.8 (10)
Na1^v—Al1—Na1^vi	62.45 (19)	O3^x—Na1—O4	62.9 (3)
Na1^v—Al1—O2	130.8 (3)	O3^x—Na1—O5^ix	146.1 (4)
Na1^v—Al1—O3	54.3 (3)	O3^x—Na1—O6	83.2 (4)
Na1^v—Al1—O4	119.3 (4)	O3^x—Na1—O6^vii	83.6 (3)
Na1^v—Al1—O5	50.5 (3)	O3^x—Na1—H2^vii	63.8 (10)
Na1^vi—Al1—O2	80.9 (3)	O4—Na1—O5^ix	87.8 (3)
Na1^vi—Al1—O3	34.6 (3)	O4—Na1—O6	112.2 (4)
Na1^vi—Al1—O4	140.7 (3)	O4—Na1—O6^vii	93.6 (3)
Na1^vi—Al1—O5	100.2 (3)	O4—Na1—H2^vii	90.1 (10)
O2—Al1—O3	110.1 (5)	O5^ix—Na1—O6	93.6 (4)
O2—Al1—O4	109.9 (4)	O5^ix—Na1—O6^vii	116.8 (4)
O2—Al1—O5	112.5 (4)	O5^ix—Na1—H2^vii	137.3 (10)
O3—Al1—O4	110.6 (4)	O6—Na1—O6^vii	141.2 (4)
O3—Al1—O5	104.5 (4)	O6—Na1—H2^vii	126.3 (10)
O4—Al1—O5	109.1 (5)	O6^vii—Na1—H2^vii	20.9 (10)
Si2^xiii—Na1—Si2^xii	98.60 (18)	Si1—O1—Si2	145.2 (4)
Si2^xiii—Na1—Al1	56.25 (14)	Si1—O2—Al1	142.6 (5)
Si2^xiii—Na1—Al1^ix	152.6 (2)	Si2^xii—O3—Al1	129.8 (4)
Si2^xiii—Na1—Al1^x	50.53 (13)	Si2^xii—O3—Na1^v	128.3 (4)
Si2^xiii—Na1—Na1^vii	124.2 (2)	Si2^xii—O3—Na1^vi	89.1 (3)
Si2^xiii—Na1—Na1^viii	66.9 (2)	Al1—O3—Na1^v	91.6 (3)
Si2^xiii—Na1—O3^ix	169.6 (3)	Al1—O3—Na1^vi	123.4 (4)
Si2^xiii—Na1—O3^x	31.96 (15)	Na1^v—O3—Na1^vi	91.0 (3)
Si2^xiii—Na1—O4	31.3 (2)	Si2^xiii—O4—Al1	138.0 (4)
Si2^xiii—Na1—O5^ix	118.5 (3)	Si2^xiii—O4—Na1	99.4 (3)
Si2^xiii—Na1—O6	102.1 (4)	Al1—O4—Na1	118.9 (5)
Si2^xiii—Na1—O6^vii	84.8 (3)	Si2^xi—O5—Al1	135.7 (5)
Si2^xiii—Na1—H2^vii	71.9 (10)	Si2^xi—O5—Na1^v	125.9 (4)
Si2^xii—Na1—Al1	50.49 (11)	Al1—O5—Na1^v	95.3 (3)
Si2^xii—Na1—Al1^ix	54.85 (14)	Na1—O6—Na1^viii	100.6 (4)
Si2^xii—Na1—Al1^x	146.9 (2)	Na1—O6—H1	113 (3)
Si2^xii—Na1—Na1^vii	112.8 (2)	Na1—O6—H2	127 (3)
Si2^xii—Na1—Na1^viii	113.9 (2)	Na1^viii—O6—H1	114 (3)
Si2^xii—Na1—O3^ix	89.0 (2)	Na1^viii—O6—H2	96 (3)
Si2^xii—Na1—O3^x	124.9 (3)	H1—O6—H2	104 (4)
Si2^xii—Na1—O4	69.2 (2)	Na1^viii—H2—O6	63 (2)

Hydrogen-bond geometry (Å, º) top

D—H···A	D—H	H···A	D···A	D—H···A
O6—H1···O1^xiv	0.96 (2)	2.16 (4)	3.009 (11)	146 (4)
O6—H2···O2^xii	0.96 (4)	1.95 (4)	2.884 (12)	163 (4)

Symmetry codes: (xii) x−1/4, −y+1/4, z+3/4; (xiv) x−1/2, y, z+1/2.

Funding information

This research was supported by the Czech Science Foundation, project No. 21-05926X. AS and HC acknowledge funding by the H2020 ITN project NanED, grant agreement No. 956099. CzechNanoLab project LM2023051 funded by MEYS CR is acknowledged for financial support of the measurements at LNSM Research Infrastructure.

References

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Volume 79| Part 5| September 2023| Pages 427-439

https://doi.org/10.1107/S2053273323005053

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research papers\(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

Optimal estimated standard uncertainties of reflection intensities for kinematical refinement from 3D electron diffraction data

1. Introduction

2. Experimental, data processing and refinement setup

2.1. Natrolite

2.2. (S)-(+)-ibuprofen

2.3. L-alanine

3. Methods of adjustment of error estimates

3.1. Methodology

3.2. Model 0: no adjustment

3.3. Model 1: using equivalent errors

3.4. Model 2: three-parameter model using average intensities of symmetry-related reflections

3.4.1. Normalized deviation

3.4.2. Normal probability plot technique

3.4.3. Initial parameters

3.4.4. Corrected error estimates

3.5. Model 3: three-parameter model using the largest intensities of symmetry-related reflections

3.5.1. Outlier rejection

4. Results

4.1. Refinement results

4.1.1. Natrolite

4.1.2. Ibuprofen

4.1.3. L-alanine

5. Discussion

5.1. Observed reflections

5.2. Features of the error correction models

6. Conclusion

Supporting information

Funding information

References

research papers