# Monoclinic mixed crystals of halogenomethanes
CBr4-nCln (n=0,...,4)

Philippe Négrier, Josep Ll Tamarit, María Barrio, Luis C. Pardo, Denise

Mondieig

► To cite this version:

Philippe Négrier, Josep Ll Tamarit, María Barrio, Luis C. Pardo, Denise Mondieig. Monoclinic mixed crystals of halogenomethanes CBr4-nCln (n=0,...,4). Chemical Physics, 2007, 336 (2-3), pp.150-156. 10.1016/j.chemphys.2007.06.008 . hal-01553017

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Submitted on 24 Jul 2017

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution - ShareAlike 4.0 International License
Philippe Negrier<sup>a</sup>, Josep Ll. Tamarit<sup>b,*</sup>, María Barrio<sup>b</sup>,
Luis C. Pardo<sup>b</sup>, Denise Mondieig<sup>a</sup>

Abstract

On the basis of the isostructural relationship between the low temperature monoclinic (C2/c, Z = 32) phases of the halogenomethane CBr4–nCln (n = 0, ..., 4), a set of mixed crystals has been analysed by means of high resolution X ray powder diffraction. It is shown that the monoclinic structure of pure and mixed crystals does not depend of the particularities of the dipolar (or dipole induced) interactions of the pure compound, neither on the composition of the mixed crystal, but on the relative content of the halogen atoms which controls the size of the molecule or the average molecule for the case of mixed crystals.

Keywords: Mixed crystals; Disorder; Vegard law; Halogenomethanes

**1. Introduction**

Simple organic compounds with molecules of almost
globular shape possess molecular reorientational freedom
within a certain range of temperatures [1]. The phase show-
ing orientational disorder is called the plastic phase and
such compounds are known as plastic crystals or orienta-
tionally disordered crystals. Among them, halogenome-
thane CBr₄ nClₙ (n 0, ..., 4) have been the focus of
many studies [2 24]. Several reasons making this series of
compounds of great interest can be claimed. They exhibit
a series of thermally induced solid solid phase transitions
before melting that are attributed to the ability to gain rota-
tional degrees of freedom in the crystalline state (orienta-
tionally disordered phases) due to the little hindrance for
reorientational processes produced by the more or less glob-
ular shape of the molecules. Moreover, minor changes in the

molecular size or in the molecular symmetry can be induced
by the change of one of the halogen atoms, giving rise to
changes in the intermolecular interactions and thus, in the
phase sequence, i.e., in the polymorphic behavior [9,14].

One of the reasons to delve into the study of the halog-
enomethane CBr₄ nClₙ (n 0, ..., 4) materials is the strong
geometrical and electrostatics similarities. As a conse-
quence it has been shown that the low-temperature
"ordered" phase of all the compounds of this family
(except for CBr₃Cl, which is unknown) are isostructural
with a complex crystal structure having 32 molecules per
monoclinic unit cell [2 6,14].

For the halogenomethanes with T_d molecular symmetry,
a high number of experimental and molecular dynamics sim-
ulation studies have been published [9 12,19,20,24] while for
the compounds with lower symmetry (C<sub>3V</sub> or C<sub>2V</sub>) only a few
preliminary experiments have been done [6,7,23].

Cohen et al. [2] were able to growth a single crystal of
CCl₄ for the low-temperature ordered phase, at 195 K.
They concluded that the asymmetric unit cell (C2/c) con-
tains four independent molecules (with Z 32 molecules

<footnote><sup>a</sup> Centre de Physique Moléculaire, Optique et Hertzienne, UMR 5798 au CNRS Université Bordeaux I, 351, Cours de la Libération, 33405 Talence Cedex, France</footnote>

<footnote><sup>b</sup> Departament de Física i Enginyeria Nuclear, E.T.S.E.I.B., Universitat Politècnica de Catalunya, Diagonal, 647, 08028 Barcelona, Catalonia, Spain</footnote>

<footnote>* Corresponding author. Tel.: +34 93 401 65 64; fax: +34 93 401 18 39.
*E mail address:* jose.luis.tamarit@upc.edu (J.Ll. Tamarit).</footnote>
per unit cell) and, within the experimental error, all of them
being regular tetrahedra. This result confirmed previous
refinement procedure of the structure performed by Scho-
maker and Trueblood [5] who considered each of the
involved molecules in the asymmetric unit as a rigid body.
As far as the low-temperature of the CBr₄ compound, Pow-
ers [25] redetermined the structure of the isostructural low-
temperature phase of CBr₄, initially solved by More et al.
[4], confirming that also for such a compound the tetrahe-
dral character of the molecule is fulfilled.

The crystal structures of the low-temperature phases of CBrCl₃ and CBr₂Cl₂ were attempted by Binbrek et al. [6] from neutron powder diffraction experiments. They described the structure of these phases on the basis of the hypothesis of the isostructural character of the low-temperature ordered phases for all four members of the CBr₄ nClₙ (n 0, ..., 4) family in terms of the C2/c space group with Z 32. For this low-temperature phase of CBrCl₃ and CBr₂Cl₂ compounds the molecules were assumed to be disordered so that sites have fractional occupancies of 0.75 and 0.25 for CBrCl₃ and 0.5 for CBr₂Cl₂ for each of Cl and Br atoms, respectively. Such an "occupational disorder" was confirmed by adiabatic calorimetry [7], which clearly revealed the existence of a glass transition at around 90 K related to the frozen of the exchange of positions between Cl and Br atoms, and by the appearance of an anomaly in the aspherism coefficient, which accounts for the anisotropy of the intermolecular interactions [14].

The isostructural relationship between low-temperature
phases of CBrCl₃ and CCl₄ has been recently confirmed
from the continuous variation of the lattice parameters as
a function of concentration (at 223.2 K) for a series of sub-
stitutional monoclinic mixed crystals of the binary phase
diagram between both compounds [23]. Data correspond-
ing to mixed crystals between CBr₂Cl₂ and CCl₄ prove
the same conclusion and make then evident the isostruc-
tural character of all the compounds of this family. These
data will be published soon [26]. It should be stressed that
substitutional mixed crystals correspond to a disordered
structure, from a compositional distribution point of view,
in which molecules of the two components are distributed
regularly over the sites of the lattice.

In terms of fundamentals, the study of this family of
substances offers the opportunity to define behavioral rules
inside the family and correlate their properties in order to
establish predictive tools without the requirement of strong
hypothesis of a physical model. Then, the principal aim of
this study is to examine to what extent the low-temperature
structures, for pure compounds as well as for their mixed
crystals will allow a uniform description of the ensemble.

**2. Experimental section**

2.1. *Samples*

The chemicals CCl₄ and CBrCl₃ were obtained from
Across with purity of 99+% and 99%, respectively, and

used as received. CBr₄ was purchased from Flucka (99%)
and CBr₂Cl₂ was obtained from Aldrich with of 98% and
fractionally distilled.

2.2. X-ray diffraction

X-ray high-resolution diffraction patterns were recorded by means of a horizontally mounted INEL cylindrical position-sensitive detector (CPS 120) [27] equipped with a liquid nitrogen 600 series Cryostream Cooler from Oxford Cryosystems. The detector, used in Debye Scherrer geometry (transmission mode), consisted of 4096 channels and enables a simultaneous recording of the profile over a 2θ range between 4° and 120° (angular step of 0.029° in 2θ). Monochromatic CuKα₁ (λ 1.54059 Å) radiation was selected by means of an asymmetric focusing incident-beam curved quartz monochromator. The generator power was set to 1.0 kW (40 kV and 25 mA). PEAKOC application from DIFFRACTINEL software was used for the calibration and for the peak position determinations after pseudo-Voigt fittings [28].

The samples were introduced into 0.3- or 0.1-mm-diam-
eter Lindemann glass capillaries in the liquid state at room
temperature and were rotated perpendicularly to the X-ray
beam during data collection to improve averaging of the
crystallites.

Patterns were obtained at 220.2 ± 1.0 K (except for CCl₄ + CBrCl₃ mixed crystals, which were measured previously at 223.2 ± 1.0 K [23]) with acquisition times of at least 120 min and a stabilization time of at least 10 min before the data acquisition.

**3. Results**

3.1. X-ray high-resolution powder diffraction measurements

X-ray powder diffraction measurements were under-
taken at 220.2 ± 1.0 K for the pure compounds CBr₄ nClₙ
for n 0, 1, 2 and 4, as well as for substitutional mixed
crystals with mole fraction values distributed along the
whole range of concentrations. The reason for the missing
compound n 3 (CBr₃Cl) is the non-commercial
availability.

As far as the CCl₄ + CBrCl₃ mixed crystals are con-
cerned values for the lattice parameters were taken from
a previous work, in spite that temperature of the measure-
ments was slightly higher (223.2 ± 1.0 K) [23]. Results for
the isobaric thermal expansion tensor of the monoclinic
phase of CBrCl₃ show that for the softer principal direction
of the tensor, on the (0k0) crystallographic plane, principal
coefficient is around 1.5 × 10⁻⁴ K⁻¹ [14]. It means that, in
the worst case, a 3 K temperature difference would imply
that lattice parameter changes less than 5 × 10⁻³ Å. Thus,
it can be concluded that such a temperature difference does
not hinder the use of published values for the aforemen-
tioned two-component system to be compared with results
of the present work.
As for the two-component systems (CCl₄) + (CBr₂Cl₂), (CBrCl₃) + (CBr₂Cl₂) and (CBr₂Cl₂) + (CBr₄), three mixed crystals with compositions along the full range were measured.

## 3.2. Structural analysis

Experimental X-ray diffraction patterns were profile refined by using the Rietveld method. The process was checked first for pure compounds CCl₄ and CBr₄, for which single-crystal structural analyses have been published [2,4]. The values obtained compare well with those published for single crystals, within the experimental error.

Next, the procedure was applied to CBrCl₃ and CBr₂Cl₂ pure compounds, following the model used by Binbrek et al. [6] which is briefly detailed. Structural parameters of CBr₄ were used as the starting point for the atomic coordinates of the four independent molecules in the asymmetric unit cell. To model the mixed crystals, the molecules were assumed to be uniformly disordered, i.e., the fractional occupancies of the halogen sites were set according to the compound analyzed: 0.25 and 0.75 for Br and Cl atoms for CBrCl₃ and 0.5 for each atom for CBr₂Cl₂. A rigid-body of a pseudo-molecule with superimposed angular tetrahedral-symmetry in which Br and Cl atoms were placed according to standard distances from the central

<figure><img src="image_2.png" /><figcaption>Fig. 1. Experimental (○, red circles) and calculated ( , black line) diffraction patterns along with the difference profile of monoclinic phase for pure compounds CBrCl₃ (at 223.2 K) and CBr₂Cl₂ (at 220.2 K). Inset correspond to a five times expanded vertical scale for the data from 2θ > 40°. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<table><caption>Table 1
Monoclinic lattice parameters (at 220.2 K) for halogenomethane compounds (CBr₄, CBr₂Cl₂) and their mixed crystals for several molar compositions (X) as a function of the fractional occupational of chlorine halogen (XCl), together with the reliability indices (Rₚ and Rᵣ) of the Rietveld refinement rᵣₗ₉₉ stands for the Associated van der Waals radii (see text)</caption><thead><tr><th rowspan="2">Pure compound or mixed crystal</th><th rowspan="2">X</th><th rowspan="2">X<sub>Cl</sub></th><th colspan="4">Lattice parameters</th><th rowspan="2">R<sub>p</sub> (%)</th><th rowspan="2">R<sub>wp</sub> (%)</th><th rowspan="2">r<sub>vdw</sub> (Å)</th></tr><tr><th>a (Å)</th><th>b (Å)</th><th>c (Å)</th><th>β (in °)</th></tr></thead><tbody><tr><td>CCl<sub>4</sub></td><td></td><td>1</td><td>20.303(4)</td><td>11.441(3)</td><td>19.903(4)</td><td>111.28(1)</td><td></td><td></td><td>2.756</td></tr><tr><td>CBrCl<sub>3</sub></td><td></td><td>0.75</td><td>20.631(3)</td><td>11.619(2)</td><td>20.201(3)</td><td>111.19(1)</td><td>4.9</td><td>6.6</td><td>2.824</td></tr><tr><td>CBr<sub>2</sub>Cl<sub>2</sub></td><td></td><td>0.50</td><td>20.872(4)</td><td>11.747(2)</td><td>20.434(4)</td><td>111.19(1)</td><td>7.1</td><td>9.9</td><td>2.888</td></tr><tr><td>CBr<sub>3</sub>Cl<sup>a</sup></td><td></td><td>0.25</td><td>21.082(4)</td><td>11.886(2)</td><td>20.659(5)</td><td>111.18(1)</td><td></td><td></td><td>2.949</td></tr><tr><td>CBr<sub>4</sub></td><td></td><td>0</td><td>21.2396(8)</td><td>12.0005(4)</td><td>20.8373(7)</td><td>111.136(2)</td><td></td><td></td><td>3.008</td></tr><tr><td rowspan="3">(CCl<sub>4</sub>)<sub>1</sub> x(CBrCl<sub>3</sub>)<sup>a</sup> (α = 0.976)</td><td>0.19</td><td>0.9525</td><td>20.393(4)</td><td>11.483(2)</td><td>19.978(3)</td><td>111.26(1)</td><td>5.7</td><td>7.5</td><td>2.843</td></tr><tr><td>0.50</td><td>0.875</td><td>20.485(10)</td><td>11.531(5)</td><td>20.051(10)</td><td>111.27(3)</td><td>4.7</td><td>6.8</td><td>2.853</td></tr><tr><td>0.80</td><td>0.800</td><td>20.580(3)</td><td>11.588(2)</td><td>20.150(3)</td><td>111.23(1)</td><td>7.4</td><td>9.6</td><td>2.869</td></tr><tr><td rowspan="3">(CCl<sub>4</sub>)<sub>1</sub> x(CBr<sub>2</sub>Cl<sub>2</sub>)<sub>1</sub> (α = 0.954)</td><td>0.30</td><td>0.850</td><td>20.505(3)</td><td>11.542(2)</td><td>20.084(3)</td><td>111.24(1)</td><td>5.2</td><td>7.3</td><td>2.797</td></tr><tr><td>0.49</td><td>0.755</td><td>20.625(3)</td><td>11.611(2)</td><td>20.200(3)</td><td>111.28(1)</td><td>5.3</td><td>7.4</td><td>2.822</td></tr><tr><td>0.73</td><td>0.635</td><td>20.750(4)</td><td>11.681(2)</td><td>20.322(4)</td><td>111.24(1)</td><td>5.1</td><td>6.8</td><td>2.853</td></tr><tr><td rowspan="3">(CBrCl<sub>3</sub>)<sub>1</sub> x(CBr<sub>2</sub>Cl<sub>2</sub>)<sub>1</sub> (α = 0.978)</td><td>0.30</td><td>0.675</td><td>20.698(5)</td><td>11.654(2)</td><td>20.262(4)</td><td>111.23(1)</td><td>5.5</td><td>7.3</td><td>2.843</td></tr><tr><td>0.45</td><td>0.6375</td><td>20.752(3)</td><td>11.687(2)</td><td>20.319(3)</td><td>111.21(1)</td><td>5.3</td><td>7.2</td><td>2.853</td></tr><tr><td>0.70</td><td>0.575</td><td>20.804(3)</td><td>11.722(2)</td><td>20.378(4)</td><td>111.21(1)</td><td>5.8</td><td>8.6</td><td>2.869</td></tr><tr><td rowspan="3">(CBr<sub>2</sub>Cl<sub>2</sub>)<sub>1</sub> x(CBr<sub>4</sub>)<sub>1</sub> (α = 0.960)</td><td>0.25</td><td>0.375</td><td>20.974(3)</td><td>11.823(2)</td><td>20.557(2)</td><td>111.19(1)</td><td>4.6</td><td>5.9</td><td>2.919</td></tr><tr><td>0.50</td><td>0.250</td><td>21.073(3)</td><td>11.886(1)</td><td>20.661(2)</td><td>111.19(1)</td><td>5.3</td><td>6.9</td><td>2.949</td></tr><tr><td>0.74</td><td>0.125</td><td>21.174(5)</td><td>11.941(2)</td><td>20.755(2)</td><td>111.18(1)</td><td>6.5</td><td>8.8</td><td>2.979</td></tr></tbody></table>

The parameter α stands for the van der Waals ratio r₁/r₂, r₂ > r₁ between the pure compounds involved in the binary mixed crystals.

<footnote>a Extrapolated values from the correlation (Table 2).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 2. Experimental (○, red circles) and calculated (■, black line) diffraction patterns along with the difference profile of monoclinic phase at 223.2 K for (a) (CCl₄)₀.₂₀(CBrCl₃)₀.₈₀ (X<sub>Cl</sub> = 0.800), (b) (CCl₄)₂.₇(CBr₂Cl₂)₀.₇₃ (X<sub>Cl</sub> = 0.635), (c) (CBrCl₃)₀.₃₀(CBr₂Cl₂)₀.₇₀ (X<sub>Cl</sub> = 0.575) and (d) (CBr₂Cl₂)₀.₂₆(CBr₄)₀.₇₄ (X<sub>Cl</sub> = 0.125). Each inset corresponds to a five times expanded vertical scale for the data from 2θ > 40°. (For interpretation of the references in color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

carbon atom (1.939 and 1.772 Å, respectively) was used. It should be noticed that the structural model imposes a dynamic disorder involving the exchange of Cl and Br atoms on fixed sites (due to the site symmetry requirements) making it impossible to define the orientation of the molecular dipole and, thus, a more detailed description of the structure. But, we must bear in mind that such a condition agrees with the results of specific heat [17], which clearly state that above the glass transition temperature (90 K) there is a dynamic disorder involving the exchange of Br and Cl atoms between fixed sites. In fact, individual molecules in the present case do not have perfect tetrahedral symmetry because of the size and bonding differences between Cl and Br atoms, but this symmetry is a reasonable constraint to impose upon the pseudo-molecule, which represents an average over all the disordered molecules.

Fig. 1 shows examples of the experimental and calculated profiles together with the difference between them for CBrCl₃ and CBr₂Cl₂ compounds (the range cutoff is due to the absence of diffraction lines at lower angles and poor signal at higher angles).

A similar process was applied to a set of mixed crystals. In these cases the fractional occupancies of the halogen atoms were set according to the composition. It means that the fractional occupancy of Chlorine halogen (X<sub>Cl</sub>) is calcu-

lated, for a given mixed crystal (CBr₄ nClₙ)₁X (CBr₄ mClₘ)X of molar composition X, as X<sub>Cl</sub> 1/4[X·m+(1-X)·n].

Continuous variation of the lattice parameters vs mole fraction X for the monoclinic mixed crystals was found whatever the analyzed case which proves the formation of monoclinic substitutional mixed crystals in the whole composition range. Table 1 summarizes lattice parameters and the R factors for the refinement procedure. Examples for some mixed crystals are given in Fig. 2.

## 4. Discussion

Vegard's law is an old empirical rule which holds a linear relation, at constant temperature, between the crystal lattice parameters of a mixed crystal and the concentration of the constituents, elements or molecules. Original law was established between the crystal lattice parameter and concentration by Vegard when analyzing ionic salt alloys [29]. It is well-known that several factors are responsible for the crystal structures of the mixed crystals: the relative atomic size of the constituents, the relative volume per valence electron in crystals of the constituents, Brillouin-zone effects or electrochemical differences between constituents. For a simple binary mixture of hard spheres, it has
been demonstrated that such a rule is based on the relative
atomic or molecular sizes constituents [30].

When analyzing the lattice parameter variation as a
function of mole fraction for each particular two-compo-
nent system studied in this work, small but noticeable devi-
ations from the Vegard's law are evidenced. Thus, whether
the structural properties of pure halogenomethane com-
pounds and their molecular mixed crystals are governed
by the relative common size or by some other particular
parameter can be disclosed by analyzing lattice parameters
as a function of the fractional occupancy of one of the hal-
ogen atoms (chlorine in our case). Fig. 3 shows the lattice
parameters for pure halogenomethanes as well as for their
mixed crystals as a function of the X<sub>Cl</sub>. Surprisingly, what-
ever the pure compound and whatever the mixed crystal is
considered, all of them are well located on the same (almost
straight) curve. It means then that packing of the low-tem-
perature phase of halogenomethanes, pure or mixed crys-
tals, are purely controlled by the relative content of the
halogen atoms. Table 2 gathers the coefficients of the poly-
nomial equations for the lattice parameters as a function of

<figure><img src="image_4.png" /><figcaption>Fig. 3. Lattice parameters as a function of the fractional occupancy of Chlorine atom for pure compounds CBr<sub>4</sub> nCl<sub>n</sub> for n = 0, 1, 2, and 4 (vertical dashed lines) as well as for molecular mixed crystals of the systems: CCl<sub>4</sub> + CBr<sub>2</sub>Cl<sub>2</sub> (Δ), CBr<sub>2</sub> Cl<sub>2</sub> + CBr<sub>4</sub> (☆) and CBr<sub>2</sub>Cl<sub>2</sub> + CBrCl<sub>3</sub> (○) at 220.2 K and CCl<sub>4</sub> + CBrCl<sub>3</sub> (●) at 223.2 K. The upper X axis corresponds to the associated van der Waals molecular radius. It should be noticed that value for X<sub>Cl</sub> = 0.25 which should correspond to the unavailable compound CBr<sub>3</sub>Cl has been obtained from (CBr<sub>2</sub>Cl<sub>2</sub>)<sub>0.5</sub> (CBr<sub>4</sub>)<sub>0.5</sub> mixed crystal. Crosses are for values from literature at different temperatures: CBr<sub>4</sub> at 296 K and CCl<sub>4</sub> at 195 K from Ref. [3], CBrCl<sub>3</sub> at 225 K and CBr<sub>2</sub>Cl<sub>2</sub> at 250 K from Ref. [6].</figcaption></figure>

$X_{\text{Cl}}$. The agreement between the calculated and experimen-
tal values has been estimated with the help of the reliability
factor, defined as $R = \sum (p_{\text{exp}} - p_c)^2/p_c^2$, where $p_{\text{exp}}$ and $p_c$
are the measured and calculated lattice parameters, respec-
tively, assuming second-order ($R_2$) or linear ($R_1$) variations.

For a fine-tuning of the Vegard's law (as a function of $X_{\text{Cl}}$) deviation, it is necessary to go deeper onto the structural details and, in particular, in the anharmonicity effects involved. Although such effects are unknown for the mixed crystals due to the lack of information about thermal expansivity or Grüneisen parameter, some reasons can be argued in a tentative way. As can be seen from Fig. 3 and from the reliability factors quoted in Table 2 for the second-order and linear fitted polynomials, the lattice parameter with the lowest deviation from a “perfect” Vegard’s line, is *b*. This result matches up with the well-known spatial distribution of intermolecular interactions for some of the pure compounds [14 20], for which the shortest distances (and then the strongest interactions) lie in the [010] plane while soft intermolecular interactions are present between (010) direction. It means then that deviation from hard-sphere behavior due to low-frequency rotation-vibrational motions associated to a certain degree of anharmonicity will be stronger in the [010] planes.

To estimate the influence of the relative size of the mol-
ecule (or the *average* molecule in the case of mixed crys-
tals), a van der Waals molecular radius has been
calculated for each entity. Molecular volumes of pure
halogenomethanes have been calculated using the values
of the van der Waals radii and interatomic chemical bonds
following Kitaigorodsky's method [31]. Owing to the fact
that we are dealing with nearly spherical or pseudo-globu-
lar molecules, a van der Waals radius was associated to
each pure halogenomethane or mixed crystal, *r*<sub>vdW</sub>. Values
of such a mean radii are consigned in Table 1, and corre-
spond to the upper *x*-axis of Fig. 3, because a linear depen-
dence, due to the detailed hypothesis, exists between the
fractional occupancy of halogen atoms and the associated
van der Waals radius.

Using the calculated molecular volumes one can calcu-
late the packing coefficient, defined as η V<sub>m</sub>/(V/Z), where
V<sub>m</sub> is the molecular volume and V/Z the volume occupied
by a molecule in the monoclinic lattice, can be calculated.
Fig. 4 depicts the variation of the packing coefficient as a
function of the fractional occupancy of Chlorine halogen.
The strong correlation evidences, in fact, that such a pack-
ing corresponds to equal-sphere closest-packing principle
as was early revealed by Powers and Rudman [3] when
comparing CCl<sub>4</sub> and CBr<sub>4</sub> monoclinic structures.

Finally, we would like to show the power of the correla-
tion between lattice parameters and the fractional occu-
pancy. In addition to the mixed crystals studied in this
work, several other can be formed, which, on the basis of
the isostructural character on the low-temperature phase
of the pure compounds, should display the monoclinic
C2/c symmetry. This would be a crystallographic require-
ment, although it is well-known that the existence of a
<caption>Table 2</caption>

Polynomial equations p $p_0 + p_1 X_{Cl} + p_2 X_{Cl}^2$ ($p$ in Å or in degree for $\beta$ parameter) to which the lattice parameters were fitted as a function of fractional occupancy of chlorine

<table><thead><tr><th>Lattice parameter</th><th>p<sub>0</sub></th><th>p<sub>1</sub></th><th>p<sub>2</sub></th><th>R<sub>2</sub> &times; 10<sup>6</sup></th><th>R<sub>1</sub> &times; 10<sup>6</sup></th></tr></thead><tbody><tr><td>a</td><td>21.233(0.009)</td><td>0.500(0.032)</td><td>0.423(0.029)</td><td>4.8</td><td>64.0</td></tr><tr><td>b</td><td>11.996(0.005)</td><td>0.403(0.021)</td><td>0.150(0.018)</td><td>6.1</td><td>29.6</td></tr><tr><td>c</td><td>20.837(0.007)</td><td>0.643(0.025)</td><td>0.286(0.022)</td><td>2.9</td><td>31.0</td></tr><tr><td>&beta;</td><td>111.141(0.011)</td><td>0.1380(0.016)</td><td></td><td></td><td>0.6</td></tr></tbody></table>

$R$ is the reliability factor (see text) for second order ($R_2$) and first order polynomial ($R_1$).

<figure><img src="image_5.png" /><figcaption>Fig. 4. Volume occupied by a molecule and packing coefficient of the monoclinic pure halogenomethane compounds and mixed crystals as a function of the fractional occupancy of the Chlorine halogen. The upper x axis corresponds to the associated van der Waals molecular radius.</figcaption></figure>

unique Gibbs function for the whole mole fraction range is also required. Nevertheless, it has been shown that the ratio between hard-sphere diameter, $\alpha \sigma_1/\sigma_2$, $\sigma_2 > \sigma_1$, univocally determines the shape of the phase diagram for a binary system of hard spheres. In such a way, for $\alpha > 0.87$ solid state displays complete miscibility, while for $\alpha < 0.87$ a eutectic, below which two solid solutions coexist (demixing region), appears [30]. For the halogenomethane molecules, assuming them as hard-spheres with an associated van der Waals radius, all the studied systems would have $\alpha$ values much higher than the limiting value for miscibility (see Table 1). Even for the binary combination between the chloro-bromo-methane compounds with the lowest $\alpha$ value, that relating $CCl_4$ and $CBr_4$, $\alpha$ 0.916, would display complete miscibility and then a continuous series of mixed crystals, if, of course, the hard-sphere hypothesis is retained.

Also from the correlation found, the lattice parameters of a possible monoclinic $C2/c$ phase of $CBr_3Cl$ can be obtained, the results being: a 21.082(4) Å, b 11.886(2) Å, c 20.659(5) Å, $\beta$ 111.18(1) at 220.2 K. It should be stressed that although the authors did not find any structural report on this compound, some works published about 50 years ago [32] showed by means of dielectric permittivity studies the close behavior, as far as phase transitions and dipole disorder are concerned, of several

halogenomethane compounds, among them $CBr_3Cl$ compound.

## 5. Conclusion

This work shows that the low-temperature phases of the halogenomethane compounds $CBr_4 nCl_n$, $n$ 0,...,4 are isostructural and can form a continuous series of mixed crystals. It has been, moreover, identified that the fractional occupancy of the halogen atoms fully controls the lattice dimensions in spite of the dipole dipole or dipole induced dipole interactions that can appear as a function of the halogen composition. Lattice parameters as a function of the fractional occupancy of chlorine atom slightly deviate from the Vegard's law, the $b$ parameter displaying the most linear behavior as a consequence of the soft van der Waals interactions in such a crystallographic direction.

## Acknowledgements

This work was financially supported by the Spanish Ministry of Science (FIS2005-00975) and by the Catalan government (SGR2005-00535). We thank Dr. S. Coco at the Chemical Physics Department, University of Valladolid, for providing us with the pure $CBr_2Cl_2$ used in this work.

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