Multidimensional Rietveld refinement of high-pressure neutron diffraction data of PbNCN

High-pressure neutron powder diffraction data from PbNCN were collected on the high-pressure diffraction beamline SNAP located at the Spallation Neutron Source (SNS) of Oak Ridge National Laboratory (Tennessee, USA). The diffraction data were analyzed using the novel method of multidimensional (two dimensions for now, potentially more in the future) Rietveld refinement and, for comparison, employing the conventional Rietveld method. To achieve two-dimensional analysis, a detailed description of the SNAP instrument characteristics was created, serving as an instrument parameter file, and then yielding both cell and spatial parameters as refined under pressure for the first time for solid-state cyanamides/carbodiimides. The bulk modulus B 0 = 25.1 (15) GPa and its derivative B 0 0 = 11.1 (8) were extracted for PbNCN following the Vinet equation of state. Surprisingly, an internal transition was observed beyond 2.0 (2) GPa, resulting from switching the bond


Introduction
Multidimensional (two dimensions and beyond) Rietveld refinement is a novel method being developed for analyzing neutron powder diffraction data.Originally, the method targeted the time-of-flight diffractometer POWTEX (Conrad et al., 2008;Houben et al., 2012) presently being constructed at FRM-II in Garching near Munich (Germany).After preliminary analytic work (Jacobs et al., 2015(Jacobs et al., , 2017) ) and a successful test of the POWTEX detector (Modzel et al., 2014;Houben et al., 2023) on the POWGEN (Huq et al., 2019) beamline at the Spallation Neutron Source (SNS) of Oak Ridge National Laboratory (Tennessee, USA) (Mason et al., 2006) including some 'shocking' results, we are now extending the method to further instruments, possibly all time-of-flight neutron diffractometers worldwide, simply due to the fundamental nature of this approach.In the present case discussed here, we evaluate measured data from the SNAP beamline (Calder et al., 2018;Frost et al., 2020) at the SNS and highlight the multidimensional Rietveld performance using a typical solid-state chemistry example, PbNCN, lead cyanamide.
As solid-state cyanamides and carbodiimides define a relatively new class of compounds, their high-pressure behavior has not been investigated extensively; the literature only covers studies of HgNCN, PbNCN and BaNCN (Masubuchi et al., 2022;Liu et al., 2002;Mo ¨ller et al., 2018).For PbNCN, Mo ¨ller et al. (2018) started with powder XRD studies and further predicted a few unexpected structural changes at high pressures on the basis of density functional theory (DFT) calculations.This prompted us to choose PbNCN as our object of study and to investigate its behavior under pressure, but with a better structure-analytic probe, namely time-of-flight neutron diffraction.
Within PbNCN the linear NCN 2À complex anion exhibits, due to being bonded to the Pearson soft Pb 2+ cation, two significantly different bond lengths [1.30 (3) and 1.16 (3) A ˚] and is therefore called a cyanamide anion with an N-C single and a C N triple bond, N-C N 2À .There are also plenty of solid-state carbodiimides in which the complex anion adopts an N C N 2À shape, with two N C double bonds at about 1.22 A ˚.In all of these phases, the NCN 2À complex anion somehow serves as a 'divalent nitride' (N 2À instead of the correct N 3À ).Therefore, NCN 2À may replace the O 2À oxide anion chargewise, but with a more covalent bonding character to whatever metal cation, simply because nitrogen is less electronegative than oxygen.

Synthesis
Lead cyanamide was obtained from the reaction of lead acetate and an aqueous solution of molecular cyanamide, following the synthesis described by Liu et al. (2000).An ammonia solution was added to a mixture of lead acetate and cyanamide to obtain a pH of 10 and to start precipitation of PbNCN.It crystallizes as a yellow powder which was filtered off, washed with water and dried.To collect powder X-ray diffraction patterns, the sample was ground and fixed with grease on acetal sheets which were held between split aluminium rings.Data were collected at room temperature with Cu K� 1 radiation (� = 1.54059A ˚) using a Stoe StadiP diffractometer (Stoe & Cie GmbH, Darmstadt, Germany) equipped with a MYTHEN 1K detector.The raw data were processed using the WinXPow (Stoe & Cie, 2008) software package, revealing that lead cyanamide had been obtained as a phasepure material.

Measurement strategy
The in situ high-pressure measurements were conducted at Oak Ridge National Laboratory (Oak Ridge, Tennessee, USA) using the VX5 Paris-Edinburgh (PE) press on the SNAP high-pressure diffractometer.The powder was pressed into a spherical pellet and transferred into the PE press with a MeOH / EtOH (4 : 1) mixture as the hydrostatic medium.The pressure was indirectly controlled via an external oil pressure operating on two anvils.Because the true sample pressure was not directly accessible by the instrument setup, a second sample containing a mixture of PbNCN (72.7%, 218.5 mg) and lead (27.3%, 82.3 mg) was measured and thereby served as a calibrant in the same range of oil pressures.The high-pressure behavior of lead has been described in full detail by Stra ¨ssle et al. (2014), with a bulk modulus of B 0 = 41.20 (20) GPa and a pressure derivative of B 0 0 = 5.72 (20) at 300 K. Hence, the pressures inside the sample could be straightforwardly determined according to the Vinet equation of state (Vinet et al., 1987), given as with the definition of X ¼ ðV=V 0 Þ 1=3 , the bulk modulus at ambient pressure given as B 0 ¼ À ð@p=@ ln VÞ 0 and its pressure derivative at ambient pressure being B 0 0 ¼ ð@B 0 =@pÞ 0 .

DFT calculations
All DFT-based calculations were executed using the Vienna Ab Initio Simulation Package (VASP) (Kresse & Furthmu ¨ller, 1996a,b;Kresse & Hafner, 1993) with projector-augmented waves (Blo ¨chl, 1994) for the pseudopotential setup and the GGA-like PBEsol functional (Csonka et al., 2009) for exchange and correlation.Additionally, a D3 correction with Becke-Johnson damping (Grimme et al., 2010(Grimme et al., , 2011) ) was employed to account for weaker dispersion interactions, likely to be present here taking into account the 'layered' motif of PbNCN.The reciprocal k-point meshes were created in accordance with the Monkhorst-Pack scheme (Monkhorst & Pack, 1976) and considered k-converged at 11 � 15 � 5.The integration of the Brillouin zone was done using Blo ¨chl's tetrahedron method (Blo ¨chl et al., 1994).The essentially converged plane-wave energy cutoff was defined at 500 eV.The present approach resembles what has been done before (Mo ¨ller et al., 2018), although the exchange-correlation functional and dispersion correction used now may be slightly more accurate.
Because of a persistent DFT problem (see below), structural optimization was only applied to the unit-cell volume for achieving a reasonable ground state, and the convergence criterion was 10 À 8 eV for electronic steps and a residual force of 10 À 3 eV A ˚À 1 for the ionic steps.Since PbNCN involves different C-N bond multiplicities, even gradient-corrected functionals such as PBEsol (which all suffer from the DFTtypical delocalization error) cannot reliably serve to model correctly the shape of such anions (N C-N 2À versus N C N 2À versus N-C N 2À ) (Liu et al., 2003), so the final DFT-based chemical-bonding analyses were therefore based on the experimental internal positional parameters from neutron diffraction, i.e. without further optimization.To do so, LOBSTER (Maintz et al., 2013(Maintz et al., , 2016;;Mu ¨ller et al., 2021;Nelson et al., 2020) was used to project the plane-wave-based wavefunctions resulting from DFT onto a local orbital basis and calculate the crystal orbital bond index (COBI) (Mu ¨ller et al., 2021).

Data reduction
Both the multidimensional and the conventional data treatment were carried out with the same raw event data.In order to perform a multidimensional (two dimensions in this case) Rietveld refinement, the diffraction data were first reduced using the Mantid software (Arnold et al., 2014), following a similar approach to that described by Houben et al. (2023).Therefore, the publicly available Mantid algorithm PowderReduceP2d was adapted following the SNAP instrument-specific calibration method.This algorithm yields coordinates in the strictly orthogonal (d, d ? ) diffraction space, just like in the known .p2dfile format being read by our modified version of the GSAS-II software (Toby & Von Dreele, 2013).The modifications to PowderReduceP2d were released with Mantid Version 6.8 (Mantid Project, 2023).For technical reasons, we only used the measuring data collected with one of the two detectors; further details are specified in the supporting information.

Data refinement
The 2D refinements were accomplished with a modified inhouse version of GSAS-II as previously used by Houben et al. (2023).The modified algorithms are based on the developments introduced by Jacobs et al. (2015Jacobs et al. ( , 2017)).The conventional (1D) refinements were done with SVN revision 5136 of GSAS-II (Toby & Dreele, 2013).For all single-phase refinements, the following scheme was applied for the order of refined parameters: (i) scale factor, (ii) unit-cell parameters, (iii) spatial parameters, and (iv) unit-cell and spatial parameters together.For the two-phase refinements, the order was (i) individual scale factors, (ii) unit-cell parameters (phase 1), (iii) unit-cell parameters (phase 2), (iv) scale factors together and (v) unit-cell parameters together.In the case of the 2D approach the instrument parameter DthF as given in Table 1 was refined once for the PbNCN single-phase measurement at the lowest pressure.The resulting value was used for all following refinements as the instrument does not change during the measurements.
All observed and calculated data points used with the conventional and multidimensional refinements are provided as supporting information to this article using the CIF format.Below the structural model, the final section of each CIF contains several columns with the column heads given at the beginning and then looping over all data points.The first four columns of each line hold the position of the data point in (2�, �) and (d, d ? ) coordinates.Since the d ?coordinate is not (yet) known to the CIF format, the corresponding column is named by the keyword _pd_proc_ls_special_details.Further columns list the observed (_pd_meas_intensity_ total), calculated (_pd_calc_intensity_total) and background (_pd_proc_intensity_bkg_calc) intensities.The .cif files for the conventional data sets follow a similar format.

The SNAP instrument and multidimensional data treatment
The design philosophy of SNAP corresponds to a high-flux medium-resolution neutron diffractometer dedicated to highpressure experiments, so it allows easy sample access given all sorts of pressure cells with adjustable shielding to minimize background (Calder et al., 2018).Although the angular coverage of the two Anger camera detectors of SNAP is smaller than that with general-purpose powder diffractometers with large-area detector coverage such as POWGEN (SNS) or future instruments like POWTEX (MLZ, Germany) or DREAM (ESS, Sweden) (Schweika et al., 2016), on purpose we apply the same strategy of a fundamental resolution description to SNAP as introduced by Jacobs et al. (2017).Since the SNAP instrument configuration is very flexible and allows different pressure cells to be accommodated, i.e. different sample volumes, different aperture settings etc., the derived assumptions fit mainly to our experimental setup.The detailed explanation in the next section especially addresses (SNAP) users planning to use a multidimensional data treatment with principally similar but slightly different setups, since that is mandatory for a multidimensional GSAS-II Rietveld refinement.
We will first show that there is an advantage in applying this method more generally to other time-of-flight (TOF) diffractometers (such as SNAP) in addition to those for which it was developed.Second, we also point out that considering the instrument description as part of the experiment is highly important, as the description can strongly influence the scientific evaluation of the measured data.

Multidimensional instrumental parameter description for SNAP
The fundamental parameters for calculating the instrumental resolution function for SNAP will be derived using the same formalism as used for POWGEN by Jacobs et al. (2017), but on the basis of the current design parameters of SNAP and according to the experiment performed.These parameters were taken either from Calder et al. (2018), from the SNAP homepage at the SNS, or from privately communicated information by the instrument scientists Antonio Moreira dos Santos and Malcolm Guthrie.
The primary flight path from source to sample is set to L 1 = 15 m within the < � 10 cm wide neutron guide, while the sampleto-detector distance is constantly L 2A = 0.5 m for all measurement positions of the detector center (the flat detector is a tangent, with its center at the contact point with the 2� circle and with its surface being normal to the scattered neutron beam lying in the horizontal scattering plane).Taking this geometry and the detector dimensions (see below) into account, one can calculate the average sample-to-detector distance to be L 2Avg ' 0.6 m.The parameter L 2B -used to describe the equi-angular spiral-like detector layout of POWGEN (Huq et al., 2019) -is not applicable and is set to L 2B = 0 rad À 1 .Thus, the secondary flight path arrives at a constant L 2 = L 2A for the detector center placed at 2� = 65 � .We used a Paris-Edinburgh cell, with a spherical sample of �5 mm diameter being the main effective contribution to the uncertainty in length.A Gaussian sample width of w sample = �L = 3.6 mm (FWHM) is yielded for a sphere being projected onto the detector surface.For ambient pressures, this gives a relative length uncertainty of The time uncertainty is mainly determined by the SNS moderator and, to a first approximation, is proportional to the neutron wavelength and thus also to the total time of flight.With an estimate of �t = 10 ms and an averaged total time of flight t = 3944 ms for a 2�-averaged flight path of L ' 15.6 m and for a wavelength of � = 1.0A ˚, then independent of wavelength one obtains the constant relative time resolution �t t ¼ 10 ms 3944 ms ' 2:5 � 10 À 3 : ð3Þ As described by Frost et al. (2020), the horizontal beam divergence simulated for the chosen 'middle divergence' setup was �2� div = 17.69 mrad.The angular resolution of the Anger camera detector can be calculated from the work of Calder et al. (2018).The 3 � 3 detector array has an edge length of 0.45 m.With 256 channels per detector per edge this gives as a quotient a pixel width of 0.59 mm, with a continuous uniform distribution equivalent to a Gaussian FWHM of w detector ' 0.41 mm.As above, the Gaussian sample width at ambient pressure results in w sample = 3.6 mm (FWHM).From these, the angular uncertainties are calculated by By applying equation ( 16) of Jacobs et al. (2017), we arrive at a relative resolution of �d/d = 8.55 � 10 À 3 at 2� = 90 � , which is slightly larger than the value of �d/d = 8 � 10 À 3 given by Calder et al. (2018) for 2� = 90 � without specifying the sample size.As already alluded to above, this instrument parametrization specific to our setup serves us well enough, in particular as we mainly focus on the unit-cell and spatial parameters as a function of pressure.Instrumental parameters like difC [the linear conversion factor relating the measured TOF and the corresponding d value; see Houben et al. (2023)] and the raise/decay terms of the back-to-back exponential part of the profile function, � and �, were taken from the conventional instrument parametrization.Only non-fundamental resolution parameters, such as u 1 (now dubbed DthF) and � add as used by Jacobs et al. (2017), are refineable during Rietveld analysis and also reveal the quality of the fundamental instrumental description when deviating from their expected values (DthF = 1.0; � add = 0.0).
To emphasize the importance of a well described instrument, we compare the refinement results of the first pressure point (single-phase measurements) using the final instrumental description with the same data refined with preliminary instrumental parameters which were based on an initial and still limited instrumental description.To visualize the improvement made, the difference plot obtained with the preliminary parameters was subtracted from the difference plot of the final parameters (Fig. 2).It is immediately obvious that a slight change in instrument parameters makes a significant difference for the refinement model.The highlighted area of Fig. 2 shows as an example the much better description using the new instrument parameters concerning the peak width at high d ?values (which correspond to high 2� and � of a constant d reflection, i.e. the backscattering regime).This behavior can be traced back mainly to the amendment of the time resolution �t/t (from 0.0028 to 0.0025).Further changes to the parameters contributing to the angular uncertainty result in an improved value for the correction term DthF from 0.67 (1) to 0.854 (4).Again, the target value for DthF is 1.0 according to Jacobs et al. (2017), meaning that, ideally, all effects contributing to the multidimensional resolution function are perfectly described and the derived parameters are not refined in the Rietveld analysis.If the deviations from these expected values are too strong, the instrumental description should be reconsidered instead.In this case, the above parameters predominantly influence the peak width for the low d ?range (the influence increases in the forward scattering regime, i.e. low 2�, where the time resolution becomes less important) and the resulting peak width is already sufficiently accurate (Fig. 2).The optimized instru-ment parameter file led to a slightly improved R Bragg value of 8.67%, compared with 8.74% previously.For some cell and spatial parameters, the accuracy improves by up to 0.5%.A comparison of all relevant preliminary and final instrument parameters is provided in Table 1.
The aforementioned results not only underline the importance of a well described instrument; they further illustrate the advantage of a 2D refinement using a fundamental instrument description.That is to say, certain refinement ambiguities indicated before only became visible and could be assigned to their origin in a two-dimensional plot.We expect the same approach also to enable an improved refinement of structural, physical and chemical models.

Sample pressure determination
Once all the instrument parameters were identified, the subsequent Rietveld refinements were performed on this basis.
Fig. 3 depicts the results of the 2D refinement for the first pressure point of the two-phase sample, including both PbNCN and Pb.Incidentally, this is the first time a two-phase refinement has been presented with the novel method of multidimensional Rietveld refinement.The top row shows two-dimensional data representations of the observed and calculated data and the difference plot.The bottom left shows the 2D data displayed as a projected 1D plot for an easier

Table 1
Comparison of instrument parameters for the preliminary and final instrument files.
The changes for �2� div , �2� detector , �t/t, �L/L and �2� sample were done based on the instrument characteristics, while DthF is a refined variable.comparison with the conventional 1D refinement plotted in the bottom right.The one-dimensional representation of the 2D refinement immediately manifests the superior description of the overlapping reflections around 3 A ˚compared with the 1D refinement.This relation between 1D and 2D refinements will also become apparent in the following parts of this work.The sample pressures were each calculated from lead's unitcell volume change as refined from the corresponding twophase Rietveld refinements following equation (1).The results for all volumes and the corresponding pressures can be found in Table S1 of the supporting information.The volumes of PbNCN and the sample pressures obtained from the twophase refinements were then utilized to calculate values for the bulk modulus B 0 and its pressure derivative B 0 0 according to the model of Vinet et al. (1987).The results are B 0 = 25.1 (15) GPa and B 0 0 = 11.1 (8) for the 2D refinements and B 0 = 18.9 (13) GPa and B 0 0 = 15.5 (9) for the 1D refinements.The 2D results in particular are quite similar to the bulk modulus B 0 = 23.1 (3) GPa and its derivative B 0 0 = 7.0 (3) of the structurally related and isoelectronic phase �-PbO (Giefers & Porsch, 2007).We note that the crystal structure of PbNCN can be derived from the PbO structure by a translationengleiche transition of index 2 and a klassengleiche transition of index 2.
Overall, the 2D refinement arrives at a volume change for lead from 121.2 (1) to 102.8 (4) A ˚3, and the corresponding PbNCN volumes reach from 252.59 (4) to 212.81 (16) A ˚3.The 1D refinement yields volume changes for lead from 123.06 (24) to 103.32 (10) A ˚3 and for PbNCN from 255.7 (10) to 215.6 (4) A ˚3. Thus, PbNCN's less reliable 1D refinement result is larger than the 2D result by around 3 A ˚3 (see also below).In molar units, this corresponds to a difference of about 0.5 cm 3 mol À 1 .
As the lead volume of the 1D refinement for the lowest pressure point [123.06(24) A ˚3] is significantly larger than the literature value at ambient pressure [121.29 (8) A ˚3], we decided to use the pressure values calculated from the 2D refinements for further interpretation and all plots.The lead volumes for the lowest non-ambient pressure of the 2D refinements [121.2 (1) A ˚3] are slightly lower than the literature value at ambient pressure, as expected.
After having calculated the bulk modulus and its derivative for PbNCN using the two-phase sample, the pressure inside the single-phase sample became accessible as a function of the oil pressures, again leveraging the Vinet equation of state (Vinet et al., 1987).The results are provided in Table S2 of the supporting information.As found for the two-phase sample, the PbNCN volumes from 1D refinements are roughly 3 A ˚3 (or 0.5 cm 3 mol À 1 ) larger than their 2D counterparts.

Single-phase refinement of PbNCN
Fig. 4 shows the progression of the cell parameters and the volume, each normalized by the corresponding value of the first pressure point at 0.31 (2) GPa.The standard deviations are so small that they fall inside the markers already, and they are generally even smaller for the 2D refinements by around one order of magnitude, which is remarkable.The overall trends of the 1D and 2D refinements are the same.The numerical values of the cell parameters can be found in Table S4 of the supporting information.
Fig. 5 shows as an example the refinement result for the lowest-pressure single-phase refinement, in the same style as given before (Fig. 3).One particular advantage of the 2D refinement becomes apparent when considering the strongest reflection at 2.78 A ˚: here, the 1D plot of the 2D refinement matches the observed data significantly better than the 1Drefined pattern.A detailed comparison of the spatial positions for all pressure points is given in Table S3 in the supporting information.As alluded to above and generally shown in the following, the standard deviations for the 1D refinements are always significantly larger than the 2D standard deviations, sometimes by an entire order of magnitude, for both unit-cell and spatial parameters.As they are calculated with the covariance matrix of the least-squares algorithm and the GOF (goodness of fit), which is directly affected by the degree of freedom, the substantially larger number of points in the 2D data set should not influence the difference in standard deviations this much.The GOF in GSAS-II is calculated as with � 2 the sum of the difference between the observed and calculated intensities for each data point, N obs the number of observed points and N var the number of refined parameters.We now take a closer look at the bond lengths and angle of the NCN 2À unit, as well as the environment of the lead atom.To reduce the likelihood that the observed differences between the 1D and 2D refinements are due to being trapped in a local minimum, we used the 1D-refined structure as an initial guess for the 2D refinement and vice versa.Doing that, however, did not change the final results at all -the structural behavior is real.
In order to investigate these structural peculiarities more closely in terms of electronic structure theory, we performed  total-energy calculations without structural optimization for each pressure point based on the 2D-refined structures.The resulting DFT energy differences and electrostatic Madelung energies are shown in Fig. 7(a).As expected, the energy rises during compression, at least for the first three data points.When going from 2.0 (2) to 3.7 (4) GPa (after the third pressure point), however, a change in the NCN 2À anionic shape takes place, pictured in Figs.7(b) and 7(c), and this lowers the total energy by almost 200 kJ mol À 1 .After this jump, the total energy again rises constantly.This overall trend is also mirrored by the Madelung energy (except for the last three pressure points), so this simplified point-charge model almost gets it right, despite covalency becoming more important for the highest pressure.
To evaluate the covalent impact on the system more quantitatively, we first employ a profoundly classical crystalchemical look at the bonding situation between Pb and its surroundings.The bond-valence sum (BVS) (Brese & O'Keeffe, 1991) is an easily accessible and empirical measure for the bond capacity of a certain atom within a crystal.It is calculated as the sum over all bond valences of the j coordinating atoms with interatomic distances of r ij as given by with the standard distance r 0 corresponding to a single bond (2.22A ˚for Pb-N) and a scaling length of B = 0.37 A ˚. Fig. 8 clearly indicates that the BVS of the lead atom increases with increasing pressure (the last pressure point being an exception) due to the shortened Pb-N bond lengths.It is puzzling, however, that the BVS values are generally too large, starting at almost 3; as this is significantly larger than the expected valence for a divalent Pb II atom, the literature parameter needs improvement.Alternatively, one may quantumchemically determine the BVS from first principles by means of ICOBI as implemented in the LOBSTER package.COBI, the crystal orbital bond index, is the periodic equivalent of the Wiberg-Mayer bond index for quantifying bond order in (organic) molecules, and the COBI energy integral (dubbed ICOBI) is the bond order of a pairwise contact.If summed over the entire atomic coordination sphere, this then equals the quantum-chemical BVS.The resulting pressure-dependent ICOBI plot is also given in Fig. 8, and the ICOBI trend parallels that of the previously shown empirical BVS: upon increasing pressure, the atomic valence increases.One difference is found in the absolute numbers, so ICOBI starts at 1.85 for the lowest pressure point and only exceeds the expected Pb II valence at the highest pressure point, the value being 2.06.Most notably, there is a jump between the third and fourth points, ICOBI increasing from 1.89 to 1.96, exactly where the above-mentioned internal structural change takes place.
The reason for the classical BVS values being too large, as written above, may be that the Pb-N combination implicitly relates to an N 3À anion, not the complex NCN 2À anion which is less charged, resulting in more covalency and hence a too   POWTEX project (grant Nos.05K19PA1 and 05K22PA2) is gratefully acknowledged.

Figure 2 A
Figure 2A difference plot of the difference plots of the 2D refinements done with preliminary and final instrument parameters.The highlighted area is one of many that shows a substantial change in the intensities depending on d ? .

Figure 3
Figure 3 The top row from the left shows the observed, calculated and difference data of the 2D refinement for the lowest-pressure two-phase measurement.The bottom left plot shows the 2D refinement reduced to a projected 1D plot as a comparison with the results of the conventional 1D refinement (bottom right).Blue vertical lines indicate Pb reflection positions and green vertical lines indicate PbNCN reflection positions.

4
Figure 4Normalized cell parameters for the 1D (black) and 2D (red) refinements, compared with theoretical data (gray).Error bars if not visible are inside the markers.
second and third pressure points as regards the angle.In more detail, the short (triple) N C and long (single) C-N bonds switch positions, which is seemingly coupled with a soft 'reset' of the N-C-N angle to a value of 163 � (the initial value was 168 � ).The general trend before and after this reset is comparable for both 1D and 2D refinements but is more pronounced in two dimensions: the N-C-N angle becomes more acute and the difference in the N-C bond lengths increases.This reset of the N-C-N angle is coupled with an additional internal change, namely a reorientation of the NCN 2À complex anion such that the central carbon atom is now pointing towards the layers themselves [Fig.7(c)],whereas it was tilted towards the gaps between layers before[Fig.7(b)].

Figure 5
Figure 5As in Fig.3but for the single-phase measurement.Green vertical lines indicate PbNCN reflection positions.

Figure 6 (
Figure 6 (a) The N-C bond lengths of the cyanamide unit for the 1D (black) and 2D (red) refinements.(b) The N-C-N angle as a function of pressure.If not visible, error bars are smaller than the markers.

Figure 7 (
Figure 7 (a) The DFT total energy and Madelung energy versus pressure for the structures found by the 2D refinement.The DFT energy is given relative to the first pressure point.(b) The local Pb coordination at 2.0 (2) GPa and (c) at 3.7 (4) GPa.

Figure 8
Figure 8Bond-valence sum (dubbed empirical) of lead for the 1D (black) and 2D (light red) refinements, and the sum over the respective ICOBI (red) based on the structures from the 2D refinement.If not visible, error bars are inside the markers.