2.1. Crystal growth and the diffraction experiment
Single crystals of Rb
have been grown from aqueous solution (Sawada et al.
). RbCl (2.73 g, Aldrich, 99.99%) and ZnCl
(1.54 g, Aldrich, 99.999%) were dissolved in 4.5 g of ultra pure water (from a Simplicity UV system by Millipore) at T
= 323 K. The solution was slowly cooled to T
= 313 K, and crystals were obtained by slow evaporation at this temperature.
A suitable single crystal was glued to a thin glass fibre mounted on a copper pin. X-ray diffraction experiments were performed at beamline F1 of Hasylab, DESY, Hamburg, employing the radiation of a wavelength of 0.5000 Å and a MAR-CCD area detector. The temperature of the sample was maintained at T = 196 K, employing a nitrogen-flow cryostat. A large crystal-to-detector distance of 225 mm was chosen, in order to be able to resolve closely spaced reflections.
With the aid of the four-circle kappa diffractometer at beamline F1, diffraction data were collected by
scans with a scan step of 0.3° per image. Several values were chosen for the off-set of the detector and for the orientation of the crystal, thus allowing the measurement of a nearly complete data set up to a high resolution of
= 0.86 Å−1
. With the purpose of increasing the effective dynamic range of the experiment, runs with a zero detector off-set were repeated with exposure times of 2 and 8 s, and runs at higher scattering angles were repeated with 8 and 64 s exposure. The long exposure times resulted in overexposed strong (main) reflections, while they allowed weak reflections (mostly higher-order satellite reflections) to be measured.
Integrated intensities of Bragg reflections were extracted from the measured images by the software EVAL
15 (Schreurs et al.
). Absorption correction was performed with SADABS
). A fraction of the area of the CCD detector was not properly cooled during parts of the experiment. This is a technical problem that occurred for experiments of long durations (Paulmann, 2009
). As a result several pixels of the detector always gave a large intensity, which could negatively affect data quality. Therefore, the coordinates of these pixels have been determined by inspection of the images, and they were excluded from the integration. Experimental data and crystallographic information are summarized in Table 1.1
The observed volume of the unit cell is significantly smaller than reported by Aramburu et al.
), who gave
= 844.04 Å
= 7.241 (3), b
= 12.648 (5) and c
= 9.216 (3) Å. Since lattice parameters from point-detector measurements are much more accurate than from area detectors, we have employed the lattice parameters from Aramburu et al.
) in the present refinements.
Experimental and crystallographic data
The resulting data set of intensities of Bragg reflections — including satellite reflections up to fifth order — was used for structure determination, structure refinements and maximum entropy calculations.
Aramburu et al.
) have kindly supplied the diffraction data from their publication. These data will be denoted as the Aramburu data. Various models have also been tested by calculation of the values of
indices on these data.
A peculiar property of the Aramburu data is that a selection of the reflections were measured, which included all main reflections and only the strongest satellite reflections as expected on the basis of a soliton model. Satellite reflections up to order five, except fourth order, have been measured in this way by Aramburu et al.
). The result is a data set that consists of many fewer reflections than available in the present data. On the other hand, CCD detectors have a limited dynamic range so that the lower bound on measurable intensities is relatively high, resulting in the number of high-order ‘observed’ satellite reflections being comparable in the two data sets (Table 1).
2.2. Structure refinements
Structure models of different complexity have been refined against the diffraction data. They involve the basic structure coordinates
and the harmonic atomic displacement parameters (ADPs)
for each of the six crystallographically independent atoms (Fig. 1). Depending on the complexity of the model, they may include Fourier coefficients for displacement modulation (
for the sine and cosine Fourier coefficients of the order
along the direction
); anharmonic ADPs of third (
) and fourth (
) order; Fourier coefficients for the modulation of the ADPs (
for the sine and cosine Fourier coefficients of order
) as well as
(Table 2; van Smaalen, 2007
Number of parameters for the different models
Structure refinements were performed with the computer program JANA
2006 (Petricek et al.
). The model published by Aramburu et al.
) involves displacement modulation parameters of orders 1, 2, 3 and 5. Refinement of these parameters against the Aramburu data reproduced the published model within one standard uncertainty (
) of all parameters.
Model A was created to resemble the published structure model as much as possible. It includes all Fourier coefficients up to fifth order for the displacement modulation, because the availability of fourth-order satellite reflections in the present data allows the refinement of the fourth-order Fourier coefficients of the displacement modulations. Refinements were initiated with the values of the published structure model as starting parameters. Values of the refined parameters are similar to those of the published structure model, with only 12 out of 140 parameters having differences larger than
and with a maximum difference of
of atom Rb1 (cf.
Table 3 with Table VI in Aramburu et al.
Amplitudes of the displacement modulation functions of model A (relative coordinates multiplied by 105)
Model B is an extension of model A, where the first- and second-order Fourier coefficients of the modulation of the harmonic ADPs have been incorporated. Refinements with model A as starting values for the parameters gave a smooth convergence and led to a considerable improvement of the fit to all orders of reflections (Table 4).
Quality of the fit to the diffraction data after refinements of models of increasing complexity
Model B was used to create the phased observed diffraction data from the measured intensities for the MEM calculations (see §2.3
). Analysis of the MEM-derived electron-density map suggested that the next important feature was the modulation of the third-order anharmonic ADPs, while their average structure values remained zero. Model C includes, in addition to the parameters of model B, the Fourier coefficients up to
for the modulation of the third-order anharmonic ADPs,
. This refinement suffered from large correlations between parameters. Therefore, a reduced model, model Cr
, was defined, in which those Fourier coefficients
were set to zero that had values less than
in the refinement of model C. This reduced the number of coefficients
from 244 to 132 (Table 2), while models C and Cr
fitted the data almost equally well (Table 4).
Difference-Fourier maps based on the observed structure factors and those calculated for a model indicate the improvement of the fit to the data for increasing complexity of the model (Fig. 2 and Table 4). The difference-Fourier map of model B compared with that of model Cr confirms the importance of modulated third-order anharmonic ADPs, as it has been derived based on MEM density maps. The difference-Fourier map of model Cr displays structure around the Rb2 atom which, to a first approximation, is independent of the phase of the modulation. It has the signature of unmodulated fourth-order anharmonic ADPs, as they are missing in model Cr. The inclusion of fourth-order anharmonic ADPs for all atoms led to highly nonphysical values of these parameters, that is, large negative values of the joint probability distribution function for the resulting model. Model Dr was then constructed to include fourth-order anharmonic ADPs for atoms Rb1, Rb2 and Cl3 only. The improvement, compared with model Cr, of the fit to the data, in particular to the main reflections, is apparent (Table 4). Refinements of the extinction coefficient led to a negative value for this parameter, so it was fixed to zero.
The remaining discrepancies between calculated and observed structure factors can be attributed in part to the incompleteness of the model. As indicated above, the introduction of more parameters leads to nonphysical values and high correlations between them, while these additional parameters would have been required for a full characterization of the modulation. A second reason for the rather high final
values of the higher-order satellite reflections lies in the less than optimal accuracy of the present data due to limited counting statistics. This interpretation becomes apparent when the
values are considered for model Dr
on the stronger reflections of the present data [reflections with
in Table 4]. In particular, the partial
values of the higher-order satellite reflections are considerably lower than on the full data set (compare columns D
in Table 4).
The fit of the models A, B, C, Cr
to the Aramburu data has been tested by refinement of the basic structure parameters of each model against these data, while the modulation parameters and anharmonic ADPs were kept fixed to the values determined from the present data. The fit to the main reflections and first-order satellite reflections is reasonable, but it becomes worse on the introduction of modulation parameters for the (an)harmonic ADPs (models B–Dr
; Table 5). On the other hand, the latter models lead to an improvement of the fit to the third- and fifth-order satellites of the Aramburu data, but with
values that are considerably higher than those on the present data. These discrepancies can be attributed to different qualities of the sample and especially different temperatures, which will affect the shapes of the modulation functions and the contributions of modulated and anharmonic ADPs to it.
RF values on the Aramburu data of models of increasing complexity, after refinement of the scale parameter, the extinction coefficient, the ADP parameters and the atomic coordinates
Therefore, independent refinements were performed against the Aramburu data, now varying all parameters, and resulting in models A
, which differ from the corresponding models A, B, C, Cr
in the values of the parameters. The fit to the Aramburu data is dramatically improved in this way (see supplementary material
), resulting in
values comparable to
values on the present data. Exceptions are the main reflections, which are much better fitted for the present data, indicating the higher accuracy of these data compared with the Aramburu data.
Despite convergence of the refinements against the Aramburu data and the resulting low
values, the primed models suffer from high correlations between parameters and large standard uncertainties. For example, none of the modulation parameters for ADPs in model B
, which prevents a meaningful analysis of the modulation on the basis of model B
, as has already been noted by Aramburu et al.
The standard uncertainties of modulation parameters and anharmonic ADPs are a multiple of the standard uncertainties of these parameters in the corresponding unprimed models (refinements against the present data). Therefore, we refrain from a further consideration of the primed models.
2.3. MEM calculations
Phased observed structure factors corrected for anomalous scattering and scaled to the scattering power of one unit cell were obtained from the observed data and model B according to published procedures (Bagautdinov et al.
). These data were used for the calculation of a maximum-entropy-optimized generalized electron density in
-dimensional superspace [MEM density or
] with the computer program BAYMEM
(van Smaalen et al.
). A uniform prior, the Cambridge algorithm and the weights of type F2 have been used (Li et al.
). The MEM calculation converged in 69 iterations (see Table 1 for more information on the MEM calculation).
-dimensional electron-density map has been analyzed with the computer program EDMA
(van Smaalen et al.
). Physical space sections of
have been obtained for 100 equally spaced
values within one period along the fourth axis, i.e.
. Atoms in the crystal correspond to local maxima in the physical space sections of the generalized electron density.
The position of each local maximum as a function of
then provides an estimate for the modulated position of an atom. Alternatively, the center-of-charge has been determined for the atomic basins surrounding each local maximum. The dependence on
of the positions of the center-of-charge provides an alternative measure for the atomic positions. Modulation functions have been extracted from
by taking the difference between the modulated atomic position and the basic structure position as obtained from model B (Fig. 3).
Two-dimensional sections of
have been visualized by the plotting option of the computer program JANA
2006 (Petricek et al.
). The (
) section centered on the Rb2 atom clearly shows the modulated position of this atom (Fig. 4).