Polymorphism and Polytypism of α-LiNH4SO4 Crystals. Monte Carlo

Sep 24, 2014 - Diffraction effects were calculated using the ZODS program, which calculates the Fourier transform according to the standard formula fo...
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Polymorphism and Polytypism of α‑LiNH4SO4 Crystals. Monte Carlo Modeling Based on X‑ray Diffuse Scattering Dorota Komornicka, Marek Wołcyrz,* and Adam Pietraszko Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 1410, 50-950 Wrocław 2, Poland S Supporting Information *

ABSTRACT: The crystal structure and phase transformations of α-LiNH4SO4 were investigated using powder and single-crystal samples over the temperature range of 120−713 K. An α → β reconstructive phase transition was observed between 431 and 457 K. The average crystal structure of α-LiNH4SO4 was refined at 120, 300, and 383 K. X-ray diffuse scattering, associated with structural disorder existing in the planar structure of α-LiNH4SO4 was registered and modeled by the Monte Carlo technique using two spinlike Ising models of disorder: one- and two-dimensional (1D and 2D, respectively). The nature of polytypism in αLiNH4SO4 was determined. The 1D model allowed for the simulation of a continuum of disordered polytypic structures reported previously: from a disordered 2O structure, different types of 4O polytypes with duplicate c axis, to 6O disordered structures with a triplicate c axis. The local structure of the measured crystal was described as a disordered 2O polytype with the very small addition of a 4O polytypic order. It was proven that there was no general structure of α-LiNH4SO4. The crystals, due to their metastability, can crystallize with different layer sequences, which can be easily changed during growth. Each crystal can be a specific realization of a polytypic structure ordered to a varying degree. networks. At least four structural forms of β-LAS have been observed: a high-temperature disordered paraelectric phase above 460 K (Pcmn space group), ferroelectric (Pc21n, 285− 460 K), and ferroelastic (P21/c, 28−285 K) phases, and a lowtemperature phase (Cc, below 28 K). The crystal structure of α-LAS is distinct from that of the β form. The crystal structure was solved for the first time by Pietraszko and Łukaszewicz10 as an analogue of the selenium isomorph: LiNH4SeO4 (LASe).11 It consists of isolated layers of corner-sharing SO4 and LiO4 tetrahedra perpendicular to the c axis (Figure 1a). The adjacent layers are bound to nitrogen atoms of the ammonium groups through hydrogen bonds and form a layered structure with the orthorhombic space group Pca21 and the lattice parameters at room temperature: a = 10.196 Å, b = 4.991 Å, and c = 17.010 Å (Z = 8)10. There is no ferroelectricity in α-LAS, and the only phase transition reported is an irreversible, reconstructive α → β

1. INTRODUCTION Lithium ammonium sulfate, LiNH4SO4 (LAS), is representative of the Me(I)Me(II)BX4 compound family (where Me(I) is a monovalent cation, usually alkali metal ion, and Me(II) is a cation of the same size or smaller, BX4-tetrahedron is a complex anion composed of a central atom of higher valency and oxygen, halogen, or phosphorus). LAS was synthesized from aqueous solutions of Li2SO4 and (NH4)2SO4 for the first time approximately 150 years ago1,2 and is known to crystallize in two polymorphic forms: stable β and metastable α.3 It has been shown that crystallization at a temperature near 282 K results in formation of the α modification alone, and crystallization at 303 K brings about the formation of crystals only in the β modification.3 β-LAS has been extensively studied for decades due to its ferroelectric and ferroelastic properties, and a series of structural phase transitions as a function of temperature has been observed.4−9 The structure of β-LAS is a variant of a general tridymite structure with three SO4 and three LiO4 tetrahedra forming a pseudohexagonal ring perpendicular to the c axis. Ammonium ions occupy the cavities of rings between the layered © 2014 American Chemical Society

Received: July 11, 2014 Revised: September 5, 2014 Published: September 24, 2014 5784

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Figure 1. (a) Crystal structure of α-LAS. A and B, layers forming the structure. (b) Experimental 1kl section of the reciprocal space of α-LAS showing modulated streaks of X-ray diffuse scattering running through columns of Bragg reflections. Longer frame, including reflections from 1 1 0 to 1 1 16, defines one-dimensional (1D) section which intensity profiles are presented in Figure 4. Smaller frame, including 1 1 2, 1 1 3, and 1 1 4 reflections, defines a fragment of reciprocal space used for illustrating patterns produced by six polytypes in Figure 12.

the appearance of polytypism in α-LAS leading to a specific local structure.

transformation observed at 340−350 K12,13 or alternatively at 440−450 K.3 Melnikova et al.3 explain this discrepancy by differentiating between two groups of α-LAS crystals. In accordance with the authors, the crystals of group A contain a certain amount of water adsorbed in interlayer cavities during crystal growth. The evaporation of water molecules initiates the α → β phase transition over the temperature range 340−350 K. Crystals of group B do not contain water owing to the slow process of growth or long-term storage in a closed vessel, enabling diffusional relaxation of impurities along layers toward crystal edges. In this group of crystals, the α → β transformation is observed at 440−450 K. X-ray diffraction studies of α-LAS single crystals10,14 reveal the existence of correlated structural disorder. Several extra diffraction effects can be observed: uniform streaks of diffuse scattering (see Figure 1b), streaks’ modulation and sharp local intensity enhancement, and relatively sharp additional superstructure reflections. Tomaszewski14 reported at least three types of polytypes of α-LAS with lattice parameters a, b, c (polytype 2O), a, b, 2c (polytypes 4O or 4M), and a, b, 3c (polytypes 6O), where a ≈ 10 Å, b ≈ 5 Å, and c ≈ 17 Å. Taking the above-mentioned findings into account and in anticipation of the results gathered in this study, it can be concluded that there is no general structure of α-LAS. The crystals, due to their metastability, can crystallize with different layer sequences, which can easily be altered during growth. Each crystal can be a specific realization of a polytypic structure ordered to a varying degree. One type of crystal from this wide variety was investigated in our study. The aims of the present paper were as follows: (i) to characterize, possibly completely, the α-LAS crystals of group B in our possession, including a further investigation of the average crystal structure over the temperature range of 120−383 K and the observation of phase transformations between room temperature and the decomposition temperature, 713 K; (ii) to explain, by modeling X-ray diffuse scattering data and using a Monte Carlo method, the nature of disordering phenomena and

2. EXPERIMENTAL SECTION A batch of α-LAS single crystals was grown by slow evaporation at 283 K from an aqueous solution of LiNH4SO4 obtained from equimolar quantities of Li2SO4 and (NH4)2SO4. The crystals were synthesized over 20 years ago then used in the X-ray studies described in ref 10 and have since been stored in a closed vial. Powder X-ray diffraction measurements were performed on a PANalytical X’Pert diffractometer equipped with a PIXcel solid state linear detector. X-ray diffraction diagrams were registered using Cu Kα radiation (λ = 1.5418 Å), generated at 45 kV and 35 mA, in transmission mode, with a focusing mirror forming the convergent X-ray beam irradiating samples placed in rotated capillaries. An Anton Paar HTK 1200N high-temperature chamber was used to heat samples from room temperature to above the decomposition point (713 K) and to cool them down. The temperature stability was better than ±0.1 °C. The heating procedure, consisting of 30−40 individual measurements, was repeated on two samples under continuous scanning using the θ/2θ mode over the 2θ range of 5−45° with a detector step size of 0.013°. The measurement time of an individual diagram was 35 min; the total time required to complete the entire set of measurements was 22 h. Data reduction was carried out, and crystallographic calculations were performed with PANalytical X’Pert HighScore Plus. Crystal of suitable quality and with dimensions of 0.35 × 0.31 × 0.24 mm was chosen for single crystal X-ray diffraction experiments. Data collection for both conventional crystal structure analysis and X-ray diffuse scattering modeling was performed on an Oxford Diffraction X’Calibur four-circle single crystal diffractometer equipped with a CCD camera using graphite-monochromatized Mo Kα radiation (λ = 0.71073 Å) generated at 50 kV and 25 mA. For high-temperature measurements at 300 and 383 ± 0.5 K, an Oxford Diffraction high-temperature blowing system outputting hot air was applied. Low-temperature measurements at 120 ± 0.2 K were performed with an Oxford Cryosystem “Cryostream 700” cooler using vaporized nitrogen as the cooling medium. Three-dimensional sets of X-ray diffraction data were collected over the 2θ range of 3−92°, with ω integration width 1.2° and the recording time of a single frame equals to 40 s. The lattice parameters were 5785

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Table 1. Essential Details Regarding α-LAS Structure Determination at 300 Ka

calculated from the positions of all measured reflections. The intensities of the Bragg reflections for average structure refinement were corrected for Lorentz and polarization factors. Analytical absorption correction was applied. The total numbers of collected reflections were 9972, 10393, and 9077 for measurements at 120, 300, and 383 K, respectively. Conventional X-ray crystal structure analysis was performed using SHELX-2013.15 Eighteen reciprocal planes were extracted for the analysis of X-ray diffuse scattering data from the sets of registered frames at 120, 300, and 383 K via CrysAlis.16 The final plane images were averaged over 9 adjacent slices, with the thickness covering 8% of the lattice constant (e.g., the 1kl plane was prepared as an average of 0.96kl to 1.04kl reciprocal planes). Neither absorption nor polarization corrections were applied to the data used for diffuse scattering analysis. Background was not subtracted. mm2 averaging was applied for each plane image. The set of images for 120 K is included in Supporting Information. Metropolis Monte Carlo simulations for generating test crystals were performed using ZODS.17 To calculate the interaction energy, corresponding formulas were used [they are presented later in the paper, see formulas (1) and (2)] and periodic boundary conditions were applied. T in the formula for the probability of the new system configuration acceptance [p = exp(−ΔE/kT)] was set to 1/k. Diffraction effects were calculated using the ZODS program, which calculates the Fourier transform according to the standard formula for kinematic scattering. An additional algorithm simulating reflection broadening as an effect of apparatus contribution was applied. An opensource data analysis and visualization application ParaView was employed to create a graphical representation of the diffraction effects.

formula sum formula weight crystal system space group cell parameters cell volume Z calc. density temperature h,k,l max R (reflections) wR2 (reflections)

Li N H4 S O4 121.04 g/mol orthorhombic Pca21 (29) a = 10.193(2) Å, b = 4.9967(10) Å, c = 17.127(3) Å 872.3(3) Å3 8 1.84321 g/cm3 300 K 13, 6, 23 0.0339 (2030) 0.0784 (2211)

a

Complete data for crystal structure determinations at 120, 300, and 383 K are presented in CIF file attached as Supporting Information.

phase transforms into the ferroelectric phase II.9 Further heating above 483 K does not cause any diagram changes up to 693 K, at which the β form starts to disappear. At 713 K, sample decomposition is complete. β-LAS decomposes into the Li2SO4 majority phase (approximately 90%) and an unknown minority phase. All of the above-presented results confirm that the investigated crystals of α-LAS belong to the group B described by Melnikova et al.3 The conventional crystal structure analysis performed on single crystal at three temperatures, 120, 300, and 383 K, did not reveal any structural anomalies as a function of temperature and confirmed the known average structure of α-LAS reported in ref 10. Essential details of the crystal structure at RT and refinements details are collected in Table 1 (complete data for crystal structure determinations are presented in the attached CIF file). Despite the relatively large contribution of diffuse scattering to the intensities of the Bragg reflections, the refinements results are quite good (see R factors).

3. PRELIMINARY CRYSTAL CHARACTERIZATION High-temperature X-ray powder diffraction diagrams of α-LAS obtained at a heating/cooling speed of 0.6 K/min are shown in Figure 2.

4. MODELING OF X-RAY DIFFUSE SCATTERING X-ray diffuse scattering is observed in α-LAS in the form of c*elongated modulated streaks through rows of reflections specified by odd values of h and arbitrary k values (Figures 1b and 11). This shape of diffuse scattering indicates the specific disorder between atomic layers perpendicular to the c axis and strong correlations occurring in ab planes. Comparison of diffuse scattering patterns at three measured temperatures showed no essential changes. Therefore, we decided to use 120 K data for comparing with pattern generated during modeling. Preliminary X-ray diffraction studies showed that several measured crystals from the batch demonstrated the same character and level of diffuse scattering. Streaks could be roughly treated as uniform in areas between Bragg reflections, which indicates the disordered 2O polytype (i.e., orthorhombic structure with the average unit cell containing 2 disordered layers). More careful streak examination revealed a very weak intensity enhancement in the midpoint between Bragg spots, indicating a very small contribution of the 4O polytype (i.e., an average superstructure with the unit cell containing four layers). According to these observations, we decided to model the local structure of α-LAS crystals from our batch primarily applying a model of the 2O polytype and then enhancing it slightly by adding a contribution of the 4O polytype.

Figure 2. Set of X-ray powder diffraction diagrams evidencing irreversible α→β transformation in α-LAS measured while heating to 483 K and subsequently cooling to room temperature.

It can be observed that the α → β transformation starts to occur upon heating at approximately 443 K, and at approximately 463 K, the entire sample undergoes a transition to the β form. More detailed measurements performed at a heating rate of 0.07 K/min over the temperature range of 420−470 K show the coexistence of both phases between 431 and 457 K. The β phase formed at 431−457 K exists above 457 K in its disordered, paraelectric phase I,6 and when cooling to room temperature, the 5786

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Figure 5. Pair correlation function, P(n), domain size distribution, S(d) (insets), and correlation lengths, L, for models 1−3. Figure 3. (a) Layered structure of α-LAS. Shifts in certain layers are shown in accordance with the scheme drawn on the right. (b) Scheme of 1D ANNNI model of interactions between layers constituting the basis for Monte Carlo procedure. Three interactions between first-, secondand fourth-nearest neighbors along c are shown. Broken lines denote symbolic F-type domain boundaries.

A disordered α-LAS crystal was modeled using pseudospin Ising-like models. This approach, successfully applied to describe polytypic behavior,18−24 involves a phenomenological correspondence between two states in which stacking of layers and two states of a pseudospin variable (up and down or + and −) can occur. In our case, a 1D axial next-nearest-neighbor Ising (ANNNI) model with additional fourth- (in case of 4O polytypes) or sixth-neighbor (in the case of 6O polytypes) interactions and the classical two-dimensional (2D) Ising model served as the basis for Monte Carlo simulations. It is worth emphasizing that the simulations were not aimed at studies of phase relations in the system but rather provided a tool for generating various crystal configurations or, in other words, a method for incorporating correlations into the structure, which could then be verified by experimental diffraction data. 4.1. 1D Model. The concept of 1D disorder in α-LAS is presented in Figure 3. Each plane of the A and B type can be built into the crystal during the crystallization process in its initial position or shifted along the a axis by vector 1/2a. There are no crystallochemical restrictions for such a structure; the appearance of stacking faults does not require the breaking of bonds. Figure 3b shows the resulting scheme of the 1D ANNNI model with additional fourth-neighbor interactions used to generate crystal configurations by the Monte Carlo technique. The model consists of an array of objects representing rightshifted layers, A+/B+ (arrow right), or left-shifted ones, A−/B− (arrow left). Symbolically, the shift in each layer is represented by an Ising spin variable, si = → or ←. System energy is minimized according to formula:

Table 2. Interaction Parameters Ji of 1D and 2D Models Used in Monte Carlo Simulations, Which Best Describe the Experimental Distribution of Diffuse Scattering of our α-LAS Crystala 1D J1 J2 J4 L ⟨d⟩

2D

model 1 (AF)

model 2 (F)

model 3

2 0 0 9 9.5

−2 0 0 9 9.4

−2 0.06 −0.03 9 9.0

model 4 Ja1, Jc1

2, −0.22

La , Lc ⟨da⟩, ⟨dc⟩

22, 6 11.9, 4.1

Derived crystal characteristics: L, correlation length; ⟨d⟩, average domain size. a

H = − ∑ [J1δ(si , si + 1) + J2 δ(si , si + 2) + J4 δ(si , si + 4)] i = 1..n

(1)

where the sum is over all pairs of nearest neighbors (interaction parameter J1), second neighbors (J2), and fourth neighbors (J4). δ(si, sj) is the Kronecker delta, which equals 1 whenever si = sj and 0 otherwise. n (= 240000 in our calculations) is the total number of objects in the system. Considering the simplest Ising model resulting from formula 1 for J2 and J4 = 0, one can obtain two opposing solutions: ferro order (F) for J1 < 0 and antiferro order (AF) for J1 > 0, whereas for J1 ≈ 0 random disorder occurs. These two ordered states, F and AF, can potentially be realized in growing crystals leading to

Figure 4. Comparison of the calculated intensity profiles running through 1 1 l section of the reciprocal space (models 1−4) with the experimental data.

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Figure 6. Comparison of layers of (a) LASe and (b) α-LAS structures. The layers are composed of two types of stripes, S+ and S−, also denoted by leftand right-oriented arrows.

Figure 7. Scheme of 2D Ising model of interacting stripes constituting the basis for Monte Carlo procedure. Interactions between nearest neighbors along a and c are shown. Broken lines denote domain boundaries in ab planes. Figure 8. Comparison of four calculated h2l sections of reciprocal space for 2D models with different combinations of Ja1 and Jc1 parameters. Note that quarters complement each other to form a full section.

two ordered polytypes, 2O1 (AF, with unit cell containing A+B− layers) and 2O2 (F, with unit cell A+B+), with the same Pca21 space group but slightly different atomic arrangements. However, we did not have specimens that were fully ordered 2O polytypes; our crystals were both disordered and displayed well-developed long-range periodicity. Therefore, we sought solutions with a certain level of disorder on both sides of the J1 axis: one for J1 < 0 and one for J1 > 0. Two resulting 1-parameter solutions can be compared, and a better or more justified version of these solutions can serve as a starting point for further modifications through the incorporation of additional interactions. We carefully explored a space of J parameters, searching for crystal configurations that produce the diffraction pattern fitting the experimental intensity distribution the best. Table 2 compares parameters for two simple 1-parameter models, AF and F (models 1 and 2, respectively), and 3-parameter model 3,

in which the second- and fourth-neighbor interactions are switched on. While checking the influence of interactions parameters on the intensities of diffuse scattering, we generated different 2D sections of reciprocal space (see Supporting Information). Then we compared them qualitatively with experimental data. Figure 4 shows (on chosen 1 1 l streak) that all three models have a similar fit, but in model 3, the adjustment of the diffuse intensities between Bragg reflections, which appears to be enhanced there, is slightly better. We assign this effect to the small contribution of order characteristic of the 4O polytype. However, careful inspection of several obtained sections of reciprocal space leads us to the conclusion that the existing differences are rather negligible, and as in many other problems of correlated disorder 5788

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Figure 9. Resulting crystal for 2D model 4: (a) 32 × 32 stripes fragment of the model. Individual orientations of objects (arrows) are shown. Different colors denote separate domains. (b) A complete view of 300 × 300 stripes model. Red and blue areas differentiate between individual domains elongated along a. Black rectangle marks the area presented in detail in (a).

domains are extended by an interlayer distance of ∼9.5 for models AF and F and 9 for model 3. While estimating the size of domains, we must also take into consideration the quality of experimental data to which the results of calculations have been compared. Taking into account divergence of primary beam, crystal size and quality, and detector capabilities, we have to be aware of the fact that the width of diffuse scattering profiles can be thinner. Though real crystals contain probably longer c-extended domains than those which result from modeling. 4.2. 2D Model. The structure of a single layer of α-LAS can be interpreted as being formed of two symmetrically dependent, with respect to the a-glide plane, types of stripes. The stripes, S+ and S− in Figure 6b, are infinite along b and 1/2a wide. The same kind of stripes, but of only one orientation, form the structure of LiNH4SeO4 (LASe)11 (Figure 6a). Thus, the question arises: is it possible that in the disordered structure of α-LAS, the natural stripe sequence S+ S− S+ S− can be broken and S+ S+ or S−S− faults appear? Furthermore, is it possible that any combination of the two stripes can exist to form any, wide or narrow, b− elongated domains? In particular, is it possible that in the ab layers of α-LAS LASe-like domains exist? It appears that there is no qualitative, crystallochemical reasons against all proposed phenomena. Thus, to model a disorder of this type, one can build a 2D Ising model of α-LAS composed of layers (A/B) divided into stripes (S +/S−) (Figure 7). Interacting objects are individual stripes, which interact with nearest neighbors in both the a and c directions. The orientation of each stripe is represented by an Ising spin variable, si = → or ←. System energy in this model is minimized according to formula:

Figure 10. Pair correlation functions Pa(n) and Pc(n) of 2D model 4. Insets show the domain size distributions along the a and c directions.

studied by analysis of diffuse scattering, the selection of one specific model is impossible. Careful examination of the 1 1 l profile reveals weak additional diffuse effects in between Bragg reflections, which may by partially overlapped with the background. Apart of small intensity enhancement in the midpoint between Bragg reflections, a specific shape of intensity profile (flatness or other irregularities) indicating on the contribution of not only 4O polytype but also same polytypes of longer periodicities, can be observed. The configurations of disordered crystals generated by the Monte Carlo procedure can be described by three derived characteristics: (i) pair correlation function, P(n), which describes the probability of finding layers of the same shift at a distance n between them; (ii) correlation length, L, defined as the minimum number of layers nmin such that 0.45 ≤ P(n) ≤ 0.55 for all n > nmin; and (iii) domain size distribution S(d), where d is a domain size. The numerical data are collected in Table 2, and graphical representations of all characteristics are presented in Figure 5. It can be observed that the correlation lengths for models 1 (AF), 2 (F), and 3 are the same (L = 9). The average

H=−

∑ i = 1..m

J1a δ(si , si + 1) +

∑ j = 1..n

J1c δ(sj , sj + 1) (2)

where the sum is over all pairs of nearest neighbors in the a (interaction parameter Ja1) and c direction (interaction parameter Jc1). δ(si, sj) is the Kronecker delta. m (= 300) and n (= 300 in our calculations) denote numbers of objects along a and c, respectively. 5789

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Figure 11. Comparison of chosen (a) experimental and (b) calculated (model 4) sections of reciprocal space for α-LAS.

along a, AF order is appropriate for ordered α-LAS (Ja1 > 0). F order yields additional streaks along the c* direction, at even values of h and arbitrary k values (see bottom-right quarter of Figure 8). Upper-left quarter of Figure 8 shows a section for a model having the best qualitative agreement with the experimental data (model 4, see Table 2).

The way which we treated data for 2D model was the same as for the 1D case: generation of 2D sections and qualitative comparison with experimental data. Along c, we studied only the model with F order (i.e., with Jc1 < 0). As previously mentioned, it is impossible to distinguish this model from AF, model 1, and both are equivalent. In contrast, 5790

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Table 3. Interaction Parameters Ji used in Monte Carlo Procedure for Generation of Six Disordered Polytypes of αLAS Reported by Tomaszewski.14

The calculated diffuse line profile for model 4 running through the 1 1 l section of the reciprocal space is presented in Figure 4. The agreement with the experimental profile for this model is comparable with that of 1D models. The resulting 2D distribution of objects, S+ (left arrows) and S− (right arrows) stripes, in the crystal of model 4, is shown in Figure 9. Both in the complete view in Figure 9b and in the small magnified area in Figure 9a, we can observe two types of order, which are appropriate for perfectly ordered α-LAS: F-type order along c and the AF-type one along a. The combination of these two orders leads to the appearance of three-dimensional (3D) belongated domains marked in separate colors in Figure 9. Domains have a narrow size distribution along c (mean size = 4) and a clearly wider distribution along a (mean size =12) (see Figure 10). Pair correlation functions, P(n), calculated for both the a and c directions (Pa and Pc, respectively) are shown in Figure 10. It can be observed that the correlation length along c (Lc = 6) is distinctly shorter than that along a (La = 22). The presented probability distributions are highly indicative of a system order−disorder state. In a continuum of possible states, the distributions provided information about the actual arrangement of objects in a system. Figure 11 compares the results obtained for model 4 with experimental data; analogous patterns obtained for models 1 to 3 visibly do not differ from those for model 4. As stated previously, X-ray diffraction effects for all presented models fully overlap, making the selection of any one model as superior practically impossible. However, the agreement between the calculated diffraction pattern and the experimental one is quite good for the chosen parameters, contrary to many other patterns checked by us (see Figure 8 and Supporting Information). In conclusion, our calculations performed using the 2D Ising model reveal that the diffracion experiment cannot exclude the existence of domainized ab layers. The domains, formed of bextended stripes, are mainly of the AF type along the a-direction, but a limited amount of small F-type domains, analogous to those in LASe crystals, can also exist, which do not affect the diffraction patterns. Clear experimental evidence of the existence of AF domains in the a direction would be streaks broadening along a* (in comparison with b*). However, we do not observe it. This fact can be explained in the following way: 2D modeling with different parameters shows that in the domain size range, which we propose for our sample, a relatively small change of streak width along a* can be observed. This change is probably hidden in the convolution of intrinsic streak shape and apparatus contribution (X-ray beam, crystal shape and quality, and detector capability). Therefore, we cannot directly confirm the existence of AF-domains. Hopefully, measurement of better quality could supply convincing data.

polytype J1 J2 J4 J6

2O

4O1

4O2

4O3

4O4

6O

−2 0 0 0

−1.2 0.4 −1.2 0

−1 0.45 −1 0

−2 1 −2 0

−2 2 −2 0

−2 2 0 −2

Figure 12. Scheme of intensity distribution between 1 1 2 and 1 1 4 reflections for six chosen polytypes of α-LAS after Tomaszewski14 (upper scheme) compared with intensity distributions calculated for interaction parameters collected in Table 3 and experimental data for the measured crystal.

previously in this paper can be applied. However, to take into account the whole richness of the polytypic forms, it is necessary to extend the model given by formula 1 by incorporating an additional term describing the interaction with the sixth-nearest neighbors (J6). Table 3 collects the parameters applied for six models describing the diffuse scattering distribution observed by Tomaszewski14 for six polytypes of α-LAS. Although, we are aware of the fact that polytypes of the kind proposed by Tomaszewski can be constructed in different ways, we consider the method proposed here as the most intuitive, which is why we applied the method to build test crystals. For constructing the 4O polytype, a correlation to the fourth-nearest neighbor has to be added. Consequently, for constructing the 6O polytype, a correlation to the sixth-nearest neighbor has been applied. To model different types of 4O polytypes, we have change the ratio between the J4 and J2 parameters. Looking at values in Table 3, it can be observed that the |J4|/|J2| ratio becomes smaller from polytype 4O1 to 4O4, as the sharp intensities at the midpoints of the Bragg reflections are more visible. Figure 12 compares the observed and calculated intensities. It can be observed that the 1D model of layer stacking disorder describes different types of polytypes occurring in α-LAS well qualitatively. By selecting the corresponding parameters appropriately, it is possible to cover a

5. POLYTYPISM OF α-LAS α-LAS is polytypic, as reported by Tomaszewski.14 Crystals occur in several structurally modified forms with duplicate (polytypes 4O) and triplicate (polytypes 6O) c axis and diffuse scattering present. These polytypes are collected in Figure 1 of Tomaszewski’s paper. The observed distributions of diffraction intensity are reproduced here with a slight graphical modification (Figure 12, upper scheme). As our studies demonstrate, the structure of α-LAS can be regarded as being constructed by stacking layers of identical structure, and the modifications differ only in their stacking sequence. Therefore, for the description of the different polytype structures, the 1D model presented 5791

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of domainized layers containing relatively small domains. Domains, formed of b-extended stripes, are mainly of the AF type in the a direction, but a limited amount of small F-type domains can also exist. (v) The 1D ANNNI model with fourthand sixth-neighbor interactions quite properly describes the nature of polytypism of α-LAS qualitatively and explains well all intensity distributions reported by Tomaszewski.14 (vi) The immanent feature of the applied method is a certain ambiguity in the interpretation of the X-ray diffuse scattering resulting from the generating crystal configurations. The method often allows for the rejection of erroneous crystal models but sometimes does not allow for discernment between certain configurations, which may be attributed to the equivalence of the AF and F models of αLAS.

■

APPENDIX: AVERAGE STRUCTURE OF α-LAS IN VIEW OF ITS POLYTYPISM The single crystal of α-LAS measured by us appeared to be a disordered 2O polytype with very slight contributions of the 4Otype order. We proved that for the 2O polytype, two ordered forms can exist: 2O1 (AF) and 2O2 (F), which have the same Pca21 space group but slightly different atomic arrangements. Therefore, we decided to determine whether in the course of conventional X-ray structure analysis both forms can be identified in the average structure of our α-LAS crystal. Apart from the analyses described in Preliminary Crystal Characterization, which in fact involved the use of the 2O2 (F)ordered polytype as a model of the average structure, we performed additional refinement using the 2O1 (AF)-ordered polytype as a model of the average structure based on the same diffraction data obtained at 300 K. Details of the refinement are presented in the attached CIF file (see Supporting Information). The results of the refinements show that the average structure of our α-LAS crystal can be described by the 2O2 (F)-ordered polytype as well as by the 2O1 (AF) one. Comparison of the Rfactor values obtained for both refinements (0.0381 and 0.0375, respectively) does not allow for one model to be selected over the other. Moreover, for the two refinements, certain distinct residual Fourier peaks are visible: those belonging to the AF model in the F-model refinement and those belonging to Fmodel in the AF-model refinement. Therefore, the average structure of α-LAS appears to be a composition of the F- and AFordered 2O polytype. This conclusion fully validates our model of disordered α-LAS crystal described in 1D Model.

Figure 13. Pair correlation functions P(n) for six polytypes from Table 3. Correlation length, L, is given for each polytype.

wide continuum of disordered states producing different diffraction patterns. Figure 13 is a supplement to Figure 12 and presents plots of correlation probabilities for all shown polytypes together with the corresponding correlation lengths. For polytypes of the 4O family, a certain regularity can be observed: the more visible the additional intensities become (see Figure 12), the more layers shifted in the opposite direction appear as next-nearest neighbors. This dependence is the most noticeable change in the plots presented in Figure 13.

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ASSOCIATED CONTENT

S Supporting Information *

6. CONCLUSIONS We have modeled the layer structure of α-LAS crystal by applying the Monte Carlo method for generating stacking faults appearing in both the c direction (leading to the formation of polytypes) and a direction (resulting in the formation of domains in layers themselves). Our studies have shown the following. (i) The 1D Ising or ANNNI model with additional interactions well describes the local structure of α-LAS, yielding an X-ray diffuse scattering distribution that well fits the experimental patterns. (ii) The local structure of our α-LAS specimen can be described as a disordered 2O polytype with AF or F stacking of layers, with a very small addition of order characteristic of the 4O polytype. (iii) The average structure of the α-LAS specimen can be determined to be a mixture of the 2O2 (F) polytype and the 2O1 (AF) one. (iv) The 2D Ising model calculations do not exclude the occurrence

CIF file with average structure refinement data for the F-model at 120, 300, and 383 K and for AF-model at 300 K; 18 experimental and 18 calculated 2D sections of the reciprocal space with diffuse scattering; a set of 14 calculated sections in the 1D model; and a set of 15 calculated sections in the 2D model with clearly deviating interaction parameters. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]. Tel: +48 713954146. Fax: +48 713441029. Notes

The authors declare no competing financial interest. 5792

dx.doi.org/10.1021/cg501044n | Cryst. Growth Des. 2014, 14, 5784−5793

Crystal Growth & Design

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Article

(22) Bak, P. Commensurate phases, incommensurate phases and the devil’s staircase. Rep. Prog. Phys. 1982, 45, 587−629. (23) Binder, K.; Landau, D. P. Phase diagrams and critical behavior in Ising square lattices with nearest- and next-nearest-neighbor interactions. Phys. Rev. B 1980, 21, 1941−1962. (24) Chan, E. J.; Welberry, T. R.; Heerdegen, A. P.; Goossens, D. J. Diffuse scattering study of aspirin forms (I) and (II). Acta Crystallogr., Sect. B 2010, 866, 696−707.

ACKNOWLEDGMENTS The authors are grateful to Hans-Beat Bürgi (University of Zurich), Thomas Weber (ETH Zurich), and Michał Chodkiewicz (University of Warsaw) for permission to use the unpublished ZODS program (Zurich-Oak Ridge Disorder Simulation). This general purpose tool for modeling diffuse scattering from disordered crystals has been developed at the University of Zurich, Switzerland. D.K. gratefully acknowledges the opportunity to test and learn about ZODS during a SciexNMSch fellowship at the University of Zurich (project code 11.2012).

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REFERENCES

(1) Scacchi, A. Delle combinazioni della litina con l’acido solforico. Atti d. R. Accad. d. Sci Fis. Mat. Napoli 1868, 3 (27), 1−85. (2) Wyrouboff, G. Sur quelques sulfates alcalins doubles. Bull. Soc. Min. France 1882, 5, 35−43. (3) Melnikova, S. V.; Kartashev, A. V.; Grankina, V. A.; Flerov, I. N. Investigation of the reconstructive phase transition between metastable (α) and stable (β) modifications of the LiNH4SO4 crystal. Sov. Phys. Solid State 2003, 45, 1572−1578. (4) Dollase, W. A. NH4LiSO4: A variant of the general tridymite structure. Acta Crystallogr., Sect. B 1969, 25, 2298−2302. (5) Kruglik, A. I.; Simonov, M. A.; Aleksandrov, K. S. Crystal structure of the low-temperature III phase of lithium ammonium sulfate. Sov. Phys. Crystallogr. 1978, 23, 274−276. (6) Itoh, K.; Ishikura, H.; Nakamura, E. Disordered structure of ferroelectric ammonium lithium sulphate in the high-temperature phase. Acta Crystallogr., Sect. B 1981, 37, 664−666. (7) Mashiyama, H.; Kasano, H. J. Refined crystal structure of LiNH4SO4 including hydrogen atoms in phases II and III. J. Phys. Soc. Jpn. 1993, 62, 155−162. (8) Hasebe, K.; Asahi, T. X-ray study of high pressure phases in (NH4)Li(SO4). Ferroelectrics 1994, 159, 61−66. (9) Smirnov, L. S.; Loose, A.; Dolbinina, V. V.; Yakovleva, L. M.; Grebenev, V. V. Refinement of hydrogen positions in phase II of betaLi(NH4)(SO4). J. Surf. Invest.: X-Ray, Synchrotron Neutron Tech. 2008, 2, 23−27. (10) Pietraszko, A.; Łukaszewicz, K. Crystal structure of α-LiNH4SO4 in the basic polytypic modification. Pol. J. Chem. 1992, 66, 2057−2061. (11) Waśkowska, A.; Allmann, R. Ammonium lithium selenate, NH4LiSeO4. Cryst. Struct. Commun. 1982, 11, 2029−2034. (12) Sosnowska, I.; Hilczer, B.; Piskunowicz, P. Polymorphism of LiNH4SO4 single crystals. Solid State Commun. 1990, 74, 1249−1251. (13) Połomska, M.; Hilczer, B.; Baran, J. FIR studies of α and β polymorphs of LiNH4SO4 single crystals. J. Mol. Struct. 1994, 325, 105− 110. (14) Tomaszewski, P. E. Polytypism of alpha-LiNH4SO4 crystals. Solid State Commun. 1992, 81, 333−335. (15) Sheldrik, G. M. Acta Crystallogr., Sect. A 2008, 64, 112−122. (16) CrysAlis software ver. 170.30.6, Oxford Diffraction Ltd.: Abingdon, U.K., 2007. (17) Chodkiewicz, M.; Weber, T.; Ahrenberg, L.; Bürgi, H.-B. ZODS: Zurich-Oak Ridge Disorder Simulation; University of Zurich: Switzerland, 2013. (18) Price, G. D.; Yeomans, J. The application of the ANNNI model to polytype behaviour. Acta Crystallogr., Sect. B 1984, 40, 448−454. (19) Price, G. D.; Parker, S. C.; Yeomans, J. The energetics of polytypic structures: A computer simulation of magnesium silicate spinelloids. Acta Crystallogr., Sect. B 1985, 41, 231−239. (20) Yeomans, J. What can spin models tell us about the behaviour of minerals. In Physical Properties and Thermodynamic Behaviour of Minerals; Salje, E. K. H., Reidel, D. Eds.; D. Reidel Publishing Company: Dordrecht, 1988, 119−142. (21) Plumer, M. L.; Hood, K.; Caillé, A. An Ising-like model of stacking-sequence polytypism in ABX3 compounds. J. Phys. Chem. C 1988, 21, 4189−4206. 5793

dx.doi.org/10.1021/cg501044n | Cryst. Growth Des. 2014, 14, 5784−5793