High spatial resolution studies of phase transitions within organic aperiodic crystals

The understanding of the symmetry breakings within crystals that are aperiodic by construction is actually very limited. Quasicrystals and incommensurate composite crystals may potentially allow such studies. We focus on the phase transitions of the aperiodic $n$-nonadecane/urea which recovers a translational symmetry within a four-dimensional space at room temperature. High-resolution neutron and synchrotron studies are reported as a function of the temperature on this organic crystal which presents an exceptional mosaicity. They reveal the richness of such approach, showing the appearance of very long wavelength supplementary intermodulations. This work generalizes the Landau theory to incommensurate composite crystals.

Figure 1

Thermal neutron diffraction measured at the Laboratoire Léon Brillouin on a triple-axis spectrometer (1T) of the $({\mathit{a}}^{*},{\mathit{b}}^{*})$ plane of $n$-nonadecane/urea in the ferroelastic phase. The measurement was performed at $T=100$ K on a fully deuterated crystal. These two scans correspond to two measurements of the maxima of the intensity associated to the low-symmetry-induced domains. They are measured for $h=4$ and $k=0$, as defined in the schematic figure below considering the domain 1. Top: measurement along $\mathit{b}$* in $h=4$. Middle: same measurement in $h=3.86$. Bottom: schematic representation of the diffraction image, identifying the six different domains. Red, blue, and green refer to three types of symmetry-induced domains generated by the loss of the threefold symmetry element going from the hexagonal to the orthorhombic phase (domains 1, 2, and 3, respectively). A supplementary splitting is observed with a small tilt angle apart from the mean value [24], defining six perfectly separate types of domains, respectively called $1+,1-,2+,2-,3+,3-$. Further studies presented in this paper are performed in a single type of domain, conventionally called $1+$. The asterisk marks other small supplementary misoriented domains [40].

Figure 2

Cold neutron diffraction measured at the Laboratoire Léon Brillouin on a triple-axis spectrometer (4F). The sample was a fully deuterated $n$-nonadecane/urea single crystal $(T_{c1}=149.4$ K, $T_{c2}=127.8$ K). Top: Limited $q$-range scan around common reciprocal position (3 2 0 0), along ${\mathit{c}}^{*}$ from $-0.5$ to $+0.5$ in urea $c_h^{*}$ reciprocal units. Temperature decreasing from top to bottom, respectively, $T=148$, 144, 128, and 100 K. Each curve is shifted by a value of $5\times {10}^4$ counts for clarity. Bottom: At $T=128$ K, full range scan along the same reciprocal ${\mathit{c}}^{*}$ line around both the urea (3 $2-10$) and common (3 2 0 0) reciprocal position from $-1.5$ to $+0.5$ in urea $c_h^{*}$ reciprocal units.

Figure 3

Diffraction profiles measured on the inelastic x-ray backscattering spectrometer at the ESRF (ID28). Scan along the reciprocal direction ${\mathit{c}}^{*}$ around the (3 0 3 0) host reciprocal position, above $T_{c1}=160$ K (fully hydrogenated $n$-nonadecane/urea single crystal), at $T=165$ K (red circles) and 175 K (magenta triangles). The associated red (respectively magenta) solid lines are the refined profiles as described in the text. The same scan is shown in phase II $(T=158$ K, gray solid line) to show the typical Bragg peak profile that defines the momentum resolution here.

Figure 4

Diffraction profiles measured on the inelastic x-ray backscattering spectrometer at the ESRF (ID28). Scan along the reciprocal direction ${\mathit{c}}^{*}$ around the (3 0 3 0) host reciprocal position in phase II, $T=146$ K, and phase III, $T=135$ K. Measurements were performed on fully hydrogenated $n$-nonadecane/urea $(T_{c1}=160$ K, $T_{c2}=145$ K). Measured data are shown with black triangles; refined patterns are shown with solid red lines. Here, the five-dimensional notation of the Bragg peaks is used : ${\mathit{Q}}_{hklmn}$ = h ${\mathit{a}}^{*}+$ k ${\mathit{b}}^{*}+$ l ${\mathit{c}}_h^{*}+$ m ${\mathit{c}}_g^{*}+$ n ${\mathit{c}}_m^{*}$ (see text). In phase II, the fitted Bragg peaks fulfill the condition of the presence of $h+k+\mathrm{n}$ even, where the fifth supplementary parameter is ${{\mathit{c}}_m}^{*}=(0.0048\pm 0.0005){{\mathit{c}}_h}^{*}$. In phase II, we report significant satellite Bragg peaks with $\mathrm{n}=\pm 1,\pm 13$, $\pm 19$. For phase III, individual contributions of the refined Bragg peaks are also displayed. Bragg peaks appear with $\mathrm{n}=0$ and $\mathrm{n}=\pm 18$.

Figure 5

Diffraction profiles measured on the inelastic x-ray backscattering spectrometer at the ESRF (ID28). Top: Zoom on the scan along the reciprocal direction ${\mathit{c}}^{*}$ around the (3 0 3 0) host reciprocal position. Fully hydrogenated $n$-nonadecane/urea $(T_{c1}=160$ K, $T_{c2}=145$ K) in phase II and phase III. Bottom: Integrated intensity at satellite position (3 0 3 0 19) verifying the condition of the presence of $h+k+n$ even (black) and position (3 0 3 0 0) verifying the condition $h+k+n$ odd (blue) as a function of temperature, defining the clear sequence of the three phases.