PHYSICAL REVIEW E 89, 022501 (2014)
Resonant x-ray diffraction spectrum for possible structures of the smectic liquid crystal phase with a six-layer periodicity LiDong Pan,1 R. Pindak,2 and C. C. Huang3 1
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA 2 Photon Sciences Directorate, Brookhaven National Laboratory, Upton, New York 11973, USA 3 School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA (Received 12 November 2013; published 5 February 2014)
∗ ∗ (SmCd6 ) phase showing six-layer periodicity [S. Wang et al., Phys. Rev. With the discovery of the smectic-Cd6 Lett. 104, 027801 (2010)] and a recent report of the observation of a possible alternative structure, the need for a reliable and accurate method for distinguishing different possible structures is more urgent than ever. Through simulations using the tensorial structure factor method, we present the resonant x-ray diffraction (RXRD) spectra for different possible structures as proposed in several theoretical studies. Subtle distinctions between models are shown. The ability and limitations of RXRD as a technique for determining the structure of this particular phase is discussed.
DOI: 10.1103/PhysRevE.89.022501
PACS number(s): 61.30.Eb, 61.05.cc, 61.30.Gd, 64.70.mf
I. INTRODUCTION
Antiferroelectric liquid crystals (AFLCs) are interesting materials that show a rich array of intermediate liquid crystal phases between the high-temperature smectic-A (SmA) phase and the low-temperature crystalline phase [1]. Frustrations due to the competing interlayer interactions are believed to be the driving forces that stabilize those different smectic structures, often referred to as the smectic-C ∗ (SmC ∗ ) variant phases [2–6]. Different SmC ∗ variant phases are characterized by different arrangements of the azimuthal tilt direction along the layer normal, with uniform tilt amplitude across the bulk of the sample. For example, the SmC ∗ phase is a ferroelectric phase with a very long pitched (∼ μm) helical superstructure; ∗ while the SmCd4 phase is antiferroelectric, with a distorted clock structure having four-layer unit cell. A recent advancement of the field, the discovery of a SmC ∗ ∗ variant phase with six-layer periodicity (the SmCd6 phase) [7], attracted considerable attention to the study of AFLCs. In the original paper reporting the discovery of this phase, an antiferroelectric distorted clock model was proposed as the ∗ phase [illustrated in Fig. 1(a)]. On the structure for the SmCd6 other hand, recently there has been a study reporting a possible ferrielectric six-layer SmC ∗ variant phase [8]. Being the only technique that can measure directly the periodicity in the azimuthal arrangements of the tilt directions, resonant x-ray diffraction (RXRD) is regarded as the smoking gun test for the identification of SmC ∗ variant phases [9]. However, the observation of a resonant peak at the correct position in Q space alone (value of QZ /Q0 , here QZ is the x-ray photon momentum transfer along the layer normal direction, while Q0 = 2π/d is the first Bragg peak position) is not enough for a complete structure determination. For ∗ the SmCd6 phase, as an example, a total of five internal degrees of freedom remain within the unit cell (the five angles between tilt directions of neighboring layers). As a result, other complimentary experimental techniques, combined with careful RXRD studies, are needed for ultimately resolving the structures within the unit cell for SmC ∗ variant phases. In this paper we present simulation results of the polar∗ ization resolved RXRD spectra of different possible SmCd6 1539-3755/2014/89(2)/022501(5)
structures, from what have been proposed in several theoretical studies [10–12]. Based on the model of tensorial structure factors by Levelut and Pansu [13], the simulations reveal subtle differences in the RXRD spectra for different structures. We will show that, with the many internal degrees of freedom, RXRD spectrum over the entire Q space will need to be ∗ structure found examined for the identification of the SmCd6 in a particular sample. The results reported in this paper will provide useful references for future studies on this new and intriguing SmC ∗ variant phase [7,14].
II. SIMULATION RESULTS
Unlike a conventional x-ray diffraction experiment in which only the electron density modulation due to the formation of layer structures is probed, in a RXRD experiment, a heavy element atom in the molecule will produce scattering that depends on the relative orientation of the molecule with respect to the direction and polarization of the incident x-ray photons. As a result, off-diagonal terms in the x-ray structure factor become important. Thus, in a RXRD experiment, the orientational periodicity of SmC ∗ variant phases can be studied directly. In the experiments, in addition to the Bragg peaks due to the layer structures at integer multiples of Q0 = 2π/d; resonant peaks near QZ /Q0 = l + m/6 is expected for the ∗ SmCd6 phase, here l is an integer while m is the order of the resonant peaks and can take the values m = ±1, ± 2, ± 3. Since the successful application of RXRD requires the ability of tuning the incident x-ray photon energy to match the proper (e.g., K) absorption edge of the resonant element, a synchrotron light source is usually required for such experiments [15]. X-rays at synchrotron light sources are usually linearly polarized, with σ incident polarization being the most common. However, with the advancement of technology and instrumentation, full polarization control can now be achieved [16]. Such polarization resolved RXRD experiment has been shown to provide additional key information in distinguishing degenerate structures with the same periodicity [17,18]. In the remainder of this paper we present polarization resolved simulation results for both σ and π
022501-1
©2014 American Physical Society
LIDONG PAN, R. PINDAK, AND C. C. HUANG 3 2 1
6
Intensity (arb. unit)
PHYSICAL REVIEW E 89, 022501 (2014)
10
Iσσ Iσπ Iπσ Iππ
5
10
4
10
4 5 6
3
10
2
10
1
10
0
10
-1
10
(a)
1.0
1.1
1.2
6
Intensity (arb. unit)
1.3
1.4
1.5
Qz/Q0
10
4
3 21
Iσσ Iσπ Iπσ Iππ
5
10
4
10
5 6
3
10
2
10
1
10
0
10
(b)
1.0
1.1
1.2
1.3
1.4
4 3 2 5 1
6
Intensity (arb. unit)
Qz/Q0
10
5
10
Iσσ Iσπ Iπσ Iππ
4
10
6
3
10
1.5
2
10
1
10
0
10
1.0
(c)
1.1
1.2
Intensity (arb. unit)
6
10
65
5
1.4
4 32
1.5
Iσσ Iσπ Iπσ Iππ
1
10
4
10
3
10
2
10
1
10
0
10
(d)
1.0
1.1
1.2
Qz/Q0
1.3
1.4
3 2 1
6
Intensity (arb. unit)
1.3
Qz/Q0
10
Iσσ Iσπ Iπσ Iππ
5
10
4
10
4 5 6
3
10
1.5
2
10
1
10
0
10
(e)
1.0
1.1
1.2
1.3
1.4
1.5
Qz/Q0
FIG. 1. (Color online) Simulated RXRD spectra. For model used in (a) φ12 = φ23 = φ45 = φ56 = 30◦ ; (b) φ12 = φ23 = φ34 = φ56 = 30◦ ; (c) φ12 = φ23 = φ34 = φ45 = 30◦ ; (d) φ12 = φ23 = φ34 = φ45 = φ56 = 30◦ ; (e) a SmCα∗ structure is used.
polarized incident beam. The results are labeled as Iπσ for the π polarized intensity scattering from σ incident light.
Plotted in Fig. 1 are the simulated RXRD spectra for four ∗ phase [Figs. 1(a)–1(d)], as possible structures of the SmCd6 ∗ well as the SmCα phase with six-layer pitch for comparison [Fig. 1(e)]. To produce the experimentally observed split resonant peaks, a helical superstructure with pitch of 210 layers is added to all the structures. Since in the high-resolution RXRD experiments [7], no splitting of the layer Bragg peaks was reported, tilt angles are taken as constant across the model structure. In Fig. 1, both σ and π polarized scattering intensity are shown for incident beams with both polarization from QZ /Q0 = 0.95 to 1.55. The spectra are symmetric with respect to QZ /Q0 = 1.5 (as well as QZ /Q0 = 1 and 2). The schematic illustrations of the structural models used for the simulation are also shown, with the numbers representing the index of a layer in the unit cell, while the arrows represent the direction of tilt for this layer. The same layout will be used for presenting all the simulation results shown in this paper. From the simulation results, it is immediately obvious that (a) Iπσ almost overlap exactly with Iσ π over the entire Q space [19], in the following discussion these two will be referred to as the cross polarized intensities; and (b) Iππ is almost featureless except for the layer Bragg peak, for all the structures considered. Thus, we conclude that the currently used experimental geometry (σ incident light) is the most suitable for the study of AFLCs. Several other features are observed in Fig. 1. First, in addition to the Bragg scattering with preserved polarization at QZ /Q0 = 1, clear split resonant peaks with crossed polarization are observed near QZ /Q0 = 1.17 (m = 1); while higher-orders resonant peaks (m = 2, 3) are much weaker, and nearly invisible for Fig. 1(a), m = 2, and Fig. 1(b), m = 3. The direction of the splitting of the two peaks with m = 1 depends on the relative sign of the twisting direction within the unit cell as compared to the helical superstructure; when these two twist toward the same direction, the more intense m = 1 peak appears at higher QZ /Q0 value. Second, the first-order resonant peak intensity (m = 1) decreases as the structure changes from antiferroelectric toward ferroelectric. Third, cross polarized satellite resonant peaks near the Bragg peak (m = 0) also start to develop as the structure approaches ferroelectric. As a comparison, the spectrum for the SmCα∗ phase with six-layer pitch shows only first- and second-order resonant peaks (m = 1, 2) that are not split. The structures shown in Fig. 1 are the most straightforward ∗ ones one would expect for a SmCd6 phase. For those models, φ12 + φ23 + φ34 + φ45 + φ56 + φ61 = 2π , thus they would be referred to as primary structures in the following text. For those primary structures, the antiferroelectric one [Fig. 1(a)] has the highest symmetry, since the layer normal direction (Z) is also a 21 screw axis, with the unit cell size equal to six layers. The ferrielectric [Figs. 1(b) and 1(c)] and ferroelectric [Fig. 2(d)] structures can be expected to appear during field-induced transitions from the antiferroelectric state, ∗ as has been shown for the SmCd3 phase; the symmetry is correspondingly lowered with the application of external field [20]. In Fig. 2 we present the simulated RXRD spectra for structures with φ12 + φ23 + φ34 + φ45 + φ56 + φ61 > 2π ; they are thus referred to as secondary structures. All those secondary
022501-2
RESONANT X-RAY DIFFRACTION SPECTRUM FOR . . .
5 4
10
6 2 4
3
10
2
10
1
10
0
10
1.0
1.1
1.2
1.3
1.4
Iσσ Iσπ Iπσ Iππ
5
10
4
10
2 4 6
3
10
1
10
0
10
1.1
1.2
1.3
1.4
1.5
6
2
4
1
5 4
10
5 3 6
3
10
(b)
Iσσ Iσπ Iπσ Iππ
10
2 1
10
0
10
1.0
1.1
1.2
1.3
1.4
50
δ
6
30 20
400
10 0
200 0 1.15
40
1.16
1.17
1.18
Qz/Q0
0.6 0.4 0.2
0
20
δ (degrees)
40
60
∗ FIG. 3. (Color online) Simulated RXRD spectra for the SmCd6 phase with antiferroelectric primary structure (a) m = 1 split resonant peak profile with different distortion angle values. (b) Intensity ratio of the m = 1 split resonant peaks as a function of distortion angle δ; δ = 60◦ correspond to the SmCα∗ structure while δ = 0 correspond to the Ising structure.
10
(c)
5
60
1
0.0
Qz/Q0
10
4
2
0.8
2
1.0
3
1.0
10
(b)
600
(a)
5 3 1
10
800
1.5
Qz/Q0
6
Intensity (arb. unit)
3
1000x10
Iσσ Iσπ Iπσ Iππ
1
10
(a)
Intensity (arb. unit)
5
Intensity (arb. unit)
3
Intensity Ratio
Intensity (arb. unit)
6
10
PHYSICAL REVIEW E 89, 022501 (2014)
1.5
Qz/Q0
FIG. 2. (Color online) Simulated RXRD spectra for secondary structures. For model used in (a) φ12 = φ23 = φ45 = φ56 = 150◦ ; in (b) φ23 = φ45 = 150◦ ; in (c) φ12 = φ56 = 60◦ , φ23 = φ45 = 150◦ .
structures are antiferroelectric, while the most intense resonant peaks are not the first-order ones (m = 1). Instead, for the structures shown in Figs. 2(a) and 2(b), the most intense resonant peaks are located near QZ /Q0 = 1.5 (m = 3); while for the structure in Fig. 2(c), the most intense peak is the second-order one (m = 2). A more careful inspection at the proposed structures of the secondary models reveals that they can be viewed as constructed with SmCA∗ as subunit cell ∗ structures [for Figs. 2(a) and 2(b)]; and with SmCd3 structure [for Fig. 2(c)]. This explains the position of the most intense resonant scattering in Q space. While the structure in Fig. 2(a) has inversion symmetry with respect to the center of the unit cell, the two other secondary structures do not possess such symmetries. ∗ Although several different structural models of the SmCd6 phase have been proposed in theoretical studies, experimental studies using RXRD revealed the existence of the split first-order resonant peaks as well as the fact that this (m = 1) is the most intense resonant peak for this phase. Those experimental observations, together with the closeness of this ∗ phase to SmCd4 in the phase diagram, all suggest that the antiferroelectric primary structure as illustrated in Fig. 1(a) is the most likely. In the following text we inspect the features of
the simulated spectrum of this particular structural model in details. Simulations from QZ /Q0 = 1.15 to 1.19 were done for the antiferroelectric primary structure with different values of the distortion angle. The Ising structure is approached as δ approached 0; while the SmCα∗ structure appears when δ = 60◦ . The results are shown in Fig. 3. As clearly shown in the figures, the intensity ratio of the split resonant peaks strongly depend on the distortion angle and can be used to determine the value of δ in experiments, provided enough resolution can be achieved to resolve the splitting resonant peaks. As discussed, circular polarized x-ray light is now available for experiments [16]. This is incredibly useful, especially for studying chiral structures [21]. For the antiferroelectric ∗ , two structural chiralities are primary structure of SmCd6 involved. On the shorter scale, the biaxial structure within the unit cell is chiral. One can distinguish two structures with opposite chiralities from different sense of rotation between neighboring layers. On the longer scale, the helical superstructure that splits the resonant peaks and produces the satellites near the Bragg peaks is also chiral. Thus, a total of four different situations arise from the combination of those two chiralities. With the use of circular polarized x-ray, those two structural chiralities can be determined. In Figs. 4(a) and 4(b) we show the scattering intensity from right (left) circular polarized x-ray [labeled as I+ (I− )] with the same unit cell structure, but opposite sign of superstructure chiralities. In Fig. 4(c) we present the intensity difference between the I+ and I− for the two situations. From the results, it is clear that nonzero x-ray circular
022501-3
Intensity (arb. unit)
LIDONG PAN, R. PINDAK, AND C. C. HUANG
PHYSICAL REVIEW E 89, 022501 (2014)
6
III. DISCUSSION AND CONCLUSION
I+ I-
10
dichroism arises due to the helical superstructure. Maximum intensity contrast between the two circular channels appears at the Bragg peak satellite as well as the m = 2 resonant peak position [22]. From the behavior of intensity contrast as a function of QZ /Q0 , the chirality of the helical superstructure can be determined. On the other hand, as discussed earlier, the direction of splitting at m = 1 is determined by the sign of the product of the unit cell chirality and the superstructure chirality. With the latter determined, the former can easily be ascertained.
Finally, we would like to comment on the recently proposed ∗ observation of a ferrielectric structure for the SmCd6 phase. The authors of Ref. [8] presented two major points in their ∗ recent publication: (a) the SmCd6 phase observed in their experiments has a ferrielectric structure, similar to the one ∗ illustrated in Fig. 1(b); and (b) the SmCd6 phase observed in these experiments is intrinsically different from the ones observed in Ref. [7], with the reasoning that the latter ones appeared above a SmC ∗ phase, while the former ones are ∗ found below. Here we point out the fact that the SmCd6 phase observed in Ref. [7] is intimately related to the reversed phase ∗ sequence found in 10OHF [23] (a SmCd4 phase appearing ∗ above the SmC phase in cooling). As previously shown, the ∗ SmCd4 phase in 10OHF is connected in the phase diagram ∗ to the SmCd4 phase in compounds with a normal phase ∗ sequence [24,25]. Thus, the SmCd4 phase in 10OHF should be considered a normal SmC ∗ variant phase. As a result, there ∗ is no reason to expect the SmCd6 phase found in related mixtures would be any different, although it appears at higher temperatures than the SmC ∗ phase. On the other hand, a noisy region with multiple resonant ∗ peaks spanning a wide Q space is found in between the SmCd6 ∗ and the SmCd4 phase [7]. The noisy region appears over a ∗ phase in the two wider temperature window than the SmCd6 mixtures studied. Later, it was identified as a phase coexistence ∗ ∗ region between the SmCd6 and SmCd4 with sub-μm scale local orders [14]. The large interface area between the two types of local order in this model can in principle produce ferrielectric response. Thus, it is not immediately clear, without a more careful characterization with complimentary techniques, that the ferrielectric behavior found in Ref. [8] comes from a ∗ ferrielectric SmCd6 structure. In summary, we present in this paper detailed simulation of RXRD spectra of possible structures of smectic liquid crystal phase with six-layer periodicity. Spectra over a large Q phase were shown, and subtle differences between different structures were discussed. We showed that to distinguish the structures between different proposed models, the whole Q space must be inspected with great care. We also pointed out the trend of the spectra with the potential field-induced transitions from an antiferroelectric ground state. The possibility of determining the chiralities related to the structure with the use of circular polarized x-ray is also discussed. The results presented in this paper will serve as useful references for future experiments on the SmC ∗ variant phases, particularly ∗ the SmCd6 phase.
ˇ c, Rev. Mod. Phys. 82, 897 [1] H. Takezoe, E. Gorecka, and M. Cepiˇ (2010). [2] D. A. Olson, X. F. Han, A. Cady, and C. C. Huang, Phys. Rev. E 66, 021702 (2002). [3] M. B. Hamaneh and P. L. Taylor, Phys. Rev. Lett. 93, 167801 (2004). [4] J. P. F. Lagerwall, G. Heppke, and F. Giesselmann, Eur. Phys. J. E 18, 113 (2005).
[5] P. V. Dolganov, V. M. Zhilin, V. K. Dolganov, and E. I. Kats, Phys. Rev. E 82, 040701 (2010). [6] L. D. Pan, C. S. Hsu, and C. C. Huang, Phys. Rev. Lett. 108, 027801 (2012). [7] S. Wang, L. D. Pan, R. Pindak, Z. Q. Liu, H. T. Nguyen, and C. C. Huang, Phys. Rev. Lett. 104, 027801 (2010). [8] Y. Takanishi, I. Nishiyama, J. Yamamoto, Y. Ohtsuka, and A. Iida, Phys. Rev. E 87, 050503 (2013).
5
10
4
10
3
10
2
10
1
10
0
10
Intensity (arb. unit)
(a)
1.0
1.1
1.2
1.3
1.4
1.5
Qz/Q0
6
I+ I-
10
5
10
4
10
3
10
2
10
1
10
0
10
Intensity (arb. unit)
(b)
1.0
1.1
1.2
1.3
1.4
1.5
Qz/Q0
4000 2000 0
I+-I-(P = 210) I+-I-(P = -210)
-2000 -4000
(c)
1.0
1.1
1.2
1.3
1.4
1.5
Qz/Q0
FIG. 4. (Color online) Scattering from the right (I+ ) and left (I− ) circular polarized incident x-ray for the antiferroelectric primary structure with positive unit cell chirality and (a) superstructure with pitch P = 210 layer; (b) P = −210 layer; (c) intensity difference I+ -I− for the two cases.
022501-4
RESONANT X-RAY DIFFRACTION SPECTRUM FOR . . .
PHYSICAL REVIEW E 89, 022501 (2014)
[9] P. Mach, R. Pindak, A.-M. Levelut, P. Barois, H. T. Nguyen, C. C. Huang, and L. Furenlid, Phys. Rev. Lett. 81, 1015 (1998). [10] M. A. Osipov and M. V. Gorkunov, Liq. Cryst. 33, 1133 (2006). [11] M. Rjili, J. P. Marcerou, A. Gharbi, and T. Othman, Physica B 410, 162 (2013). ˇ c, AIP Conf. Proc. 1528, 337 [12] T. Y. Tan, L. H. Ong, and M. Cepiˇ (2013). [13] A.-M. Levelut and B. Pansu, Phys. Rev. E 60, 6803 (1999). [14] L. D. Pan, P. Barois, R. Pindak, Z. Q. Liu, B. K. McCoy, and C. C. Huang, Phys. Rev. Lett. 108, 037801 (2012). [15] A. Cady, J. A. Pitney, R. Pindak, L. S. Matkin, S. J. Watson, H. F. Gleeson, P. Cluzeau, P. Barois, A.-M. Levelut, W. Caliebe, J. W. Goodby, M. Hird, and C. C. Huang, Phys. Rev. E 64, 050702 (2001). [16] For example, see M. Sacchi, N. Jaouen, H. Popescu, R. Gaudemer, J. M. Tonnerre, S. G. Chiuzbaian, C. F. Hague, A. Delmotte, J. M. Dubuisson, G. Cauchon, B. Lagarde, and F. Polack, J. Phys. Conf. Ser. 425, 072018 (2013). [17] P. Fernandes, P. Barois, S. T. Wang, Z. Q. Liu, B. K. McCoy, C. C. Huang, R. Pindak, W. Caliebe, and H. T. Nguyen, Phys. Rev. Lett. 99, 227801 (2007).
[18] V. Ponsinet, P. Barois, L. D. Pan, S. Wang, C. C. Huang, S. T. Wang, R. Pindak, U. Baumeister, and W. Weissflog, Phys. Rev. E 84, 011706 (2011). [19] V. E. Dmitrienko, Acta Cryst. A39, 29 (1983). [20] S. Jaradat, P. D. Brimicombe, M. A. Osipov, R. Pindak, and H. F. Gleeson, Appl. Phys. Lett. 98, 043501 (2011). [21] Y. Tanaka and S. W. Lovesey, Eur. Phys. J. Special Topics 208, 67 (2012). [22] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevE.89.022501 for details of the calculation process and additional simulation results. [23] S. Wang, L. D. Pan, B. K. McCoy, S. T. Wang, R. Pindak, H. T. Nguyen, and C. C. Huang, Phys. Rev. E 79, 021706 (2009). [24] S. T. Wang, Z. Q. Liu, B. K. McCoy, R. Pindak, W. Caliebe, H. T. Nguyen, and C. C. Huang, Phys. Rev. Lett. 96, 097801 (2006). [25] B. K. McCoy, Z. Q. Liu, S. T. Wang, L. D. Pan, S. Wang, H. T. Nguyen, R. Pindak, and C. C. Huang, Phys. Rev. E 77, 061704 (2008).
022501-5
Resonant x-ray diffraction spectrum for possible structures of the smectic liquid crystal phase with a six-layer periodicity LiDong Pan,1 R. Pindak,2 and C. C. Huang3 1
Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA 2 Photon Sciences Directorate, Brookhaven National Laboratory, Upton, New York 11973, USA 3 School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA (Received 12 November 2013; published 5 February 2014)
∗ ∗ (SmCd6 ) phase showing six-layer periodicity [S. Wang et al., Phys. Rev. With the discovery of the smectic-Cd6 Lett. 104, 027801 (2010)] and a recent report of the observation of a possible alternative structure, the need for a reliable and accurate method for distinguishing different possible structures is more urgent than ever. Through simulations using the tensorial structure factor method, we present the resonant x-ray diffraction (RXRD) spectra for different possible structures as proposed in several theoretical studies. Subtle distinctions between models are shown. The ability and limitations of RXRD as a technique for determining the structure of this particular phase is discussed.
DOI: 10.1103/PhysRevE.89.022501
PACS number(s): 61.30.Eb, 61.05.cc, 61.30.Gd, 64.70.mf
I. INTRODUCTION
Antiferroelectric liquid crystals (AFLCs) are interesting materials that show a rich array of intermediate liquid crystal phases between the high-temperature smectic-A (SmA) phase and the low-temperature crystalline phase [1]. Frustrations due to the competing interlayer interactions are believed to be the driving forces that stabilize those different smectic structures, often referred to as the smectic-C ∗ (SmC ∗ ) variant phases [2–6]. Different SmC ∗ variant phases are characterized by different arrangements of the azimuthal tilt direction along the layer normal, with uniform tilt amplitude across the bulk of the sample. For example, the SmC ∗ phase is a ferroelectric phase with a very long pitched (∼ μm) helical superstructure; ∗ while the SmCd4 phase is antiferroelectric, with a distorted clock structure having four-layer unit cell. A recent advancement of the field, the discovery of a SmC ∗ ∗ variant phase with six-layer periodicity (the SmCd6 phase) [7], attracted considerable attention to the study of AFLCs. In the original paper reporting the discovery of this phase, an antiferroelectric distorted clock model was proposed as the ∗ phase [illustrated in Fig. 1(a)]. On the structure for the SmCd6 other hand, recently there has been a study reporting a possible ferrielectric six-layer SmC ∗ variant phase [8]. Being the only technique that can measure directly the periodicity in the azimuthal arrangements of the tilt directions, resonant x-ray diffraction (RXRD) is regarded as the smoking gun test for the identification of SmC ∗ variant phases [9]. However, the observation of a resonant peak at the correct position in Q space alone (value of QZ /Q0 , here QZ is the x-ray photon momentum transfer along the layer normal direction, while Q0 = 2π/d is the first Bragg peak position) is not enough for a complete structure determination. For ∗ the SmCd6 phase, as an example, a total of five internal degrees of freedom remain within the unit cell (the five angles between tilt directions of neighboring layers). As a result, other complimentary experimental techniques, combined with careful RXRD studies, are needed for ultimately resolving the structures within the unit cell for SmC ∗ variant phases. In this paper we present simulation results of the polar∗ ization resolved RXRD spectra of different possible SmCd6 1539-3755/2014/89(2)/022501(5)
structures, from what have been proposed in several theoretical studies [10–12]. Based on the model of tensorial structure factors by Levelut and Pansu [13], the simulations reveal subtle differences in the RXRD spectra for different structures. We will show that, with the many internal degrees of freedom, RXRD spectrum over the entire Q space will need to be ∗ structure found examined for the identification of the SmCd6 in a particular sample. The results reported in this paper will provide useful references for future studies on this new and intriguing SmC ∗ variant phase [7,14].
II. SIMULATION RESULTS
Unlike a conventional x-ray diffraction experiment in which only the electron density modulation due to the formation of layer structures is probed, in a RXRD experiment, a heavy element atom in the molecule will produce scattering that depends on the relative orientation of the molecule with respect to the direction and polarization of the incident x-ray photons. As a result, off-diagonal terms in the x-ray structure factor become important. Thus, in a RXRD experiment, the orientational periodicity of SmC ∗ variant phases can be studied directly. In the experiments, in addition to the Bragg peaks due to the layer structures at integer multiples of Q0 = 2π/d; resonant peaks near QZ /Q0 = l + m/6 is expected for the ∗ SmCd6 phase, here l is an integer while m is the order of the resonant peaks and can take the values m = ±1, ± 2, ± 3. Since the successful application of RXRD requires the ability of tuning the incident x-ray photon energy to match the proper (e.g., K) absorption edge of the resonant element, a synchrotron light source is usually required for such experiments [15]. X-rays at synchrotron light sources are usually linearly polarized, with σ incident polarization being the most common. However, with the advancement of technology and instrumentation, full polarization control can now be achieved [16]. Such polarization resolved RXRD experiment has been shown to provide additional key information in distinguishing degenerate structures with the same periodicity [17,18]. In the remainder of this paper we present polarization resolved simulation results for both σ and π
022501-1
©2014 American Physical Society
LIDONG PAN, R. PINDAK, AND C. C. HUANG 3 2 1
6
Intensity (arb. unit)
PHYSICAL REVIEW E 89, 022501 (2014)
10
Iσσ Iσπ Iπσ Iππ
5
10
4
10
4 5 6
3
10
2
10
1
10
0
10
-1
10
(a)
1.0
1.1
1.2
6
Intensity (arb. unit)
1.3
1.4
1.5
Qz/Q0
10
4
3 21
Iσσ Iσπ Iπσ Iππ
5
10
4
10
5 6
3
10
2
10
1
10
0
10
(b)
1.0
1.1
1.2
1.3
1.4
4 3 2 5 1
6
Intensity (arb. unit)
Qz/Q0
10
5
10
Iσσ Iσπ Iπσ Iππ
4
10
6
3
10
1.5
2
10
1
10
0
10
1.0
(c)
1.1
1.2
Intensity (arb. unit)
6
10
65
5
1.4
4 32
1.5
Iσσ Iσπ Iπσ Iππ
1
10
4
10
3
10
2
10
1
10
0
10
(d)
1.0
1.1
1.2
Qz/Q0
1.3
1.4
3 2 1
6
Intensity (arb. unit)
1.3
Qz/Q0
10
Iσσ Iσπ Iπσ Iππ
5
10
4
10
4 5 6
3
10
1.5
2
10
1
10
0
10
(e)
1.0
1.1
1.2
1.3
1.4
1.5
Qz/Q0
FIG. 1. (Color online) Simulated RXRD spectra. For model used in (a) φ12 = φ23 = φ45 = φ56 = 30◦ ; (b) φ12 = φ23 = φ34 = φ56 = 30◦ ; (c) φ12 = φ23 = φ34 = φ45 = 30◦ ; (d) φ12 = φ23 = φ34 = φ45 = φ56 = 30◦ ; (e) a SmCα∗ structure is used.
polarized incident beam. The results are labeled as Iπσ for the π polarized intensity scattering from σ incident light.
Plotted in Fig. 1 are the simulated RXRD spectra for four ∗ phase [Figs. 1(a)–1(d)], as possible structures of the SmCd6 ∗ well as the SmCα phase with six-layer pitch for comparison [Fig. 1(e)]. To produce the experimentally observed split resonant peaks, a helical superstructure with pitch of 210 layers is added to all the structures. Since in the high-resolution RXRD experiments [7], no splitting of the layer Bragg peaks was reported, tilt angles are taken as constant across the model structure. In Fig. 1, both σ and π polarized scattering intensity are shown for incident beams with both polarization from QZ /Q0 = 0.95 to 1.55. The spectra are symmetric with respect to QZ /Q0 = 1.5 (as well as QZ /Q0 = 1 and 2). The schematic illustrations of the structural models used for the simulation are also shown, with the numbers representing the index of a layer in the unit cell, while the arrows represent the direction of tilt for this layer. The same layout will be used for presenting all the simulation results shown in this paper. From the simulation results, it is immediately obvious that (a) Iπσ almost overlap exactly with Iσ π over the entire Q space [19], in the following discussion these two will be referred to as the cross polarized intensities; and (b) Iππ is almost featureless except for the layer Bragg peak, for all the structures considered. Thus, we conclude that the currently used experimental geometry (σ incident light) is the most suitable for the study of AFLCs. Several other features are observed in Fig. 1. First, in addition to the Bragg scattering with preserved polarization at QZ /Q0 = 1, clear split resonant peaks with crossed polarization are observed near QZ /Q0 = 1.17 (m = 1); while higher-orders resonant peaks (m = 2, 3) are much weaker, and nearly invisible for Fig. 1(a), m = 2, and Fig. 1(b), m = 3. The direction of the splitting of the two peaks with m = 1 depends on the relative sign of the twisting direction within the unit cell as compared to the helical superstructure; when these two twist toward the same direction, the more intense m = 1 peak appears at higher QZ /Q0 value. Second, the first-order resonant peak intensity (m = 1) decreases as the structure changes from antiferroelectric toward ferroelectric. Third, cross polarized satellite resonant peaks near the Bragg peak (m = 0) also start to develop as the structure approaches ferroelectric. As a comparison, the spectrum for the SmCα∗ phase with six-layer pitch shows only first- and second-order resonant peaks (m = 1, 2) that are not split. The structures shown in Fig. 1 are the most straightforward ∗ ones one would expect for a SmCd6 phase. For those models, φ12 + φ23 + φ34 + φ45 + φ56 + φ61 = 2π , thus they would be referred to as primary structures in the following text. For those primary structures, the antiferroelectric one [Fig. 1(a)] has the highest symmetry, since the layer normal direction (Z) is also a 21 screw axis, with the unit cell size equal to six layers. The ferrielectric [Figs. 1(b) and 1(c)] and ferroelectric [Fig. 2(d)] structures can be expected to appear during field-induced transitions from the antiferroelectric state, ∗ as has been shown for the SmCd3 phase; the symmetry is correspondingly lowered with the application of external field [20]. In Fig. 2 we present the simulated RXRD spectra for structures with φ12 + φ23 + φ34 + φ45 + φ56 + φ61 > 2π ; they are thus referred to as secondary structures. All those secondary
022501-2
RESONANT X-RAY DIFFRACTION SPECTRUM FOR . . .
5 4
10
6 2 4
3
10
2
10
1
10
0
10
1.0
1.1
1.2
1.3
1.4
Iσσ Iσπ Iπσ Iππ
5
10
4
10
2 4 6
3
10
1
10
0
10
1.1
1.2
1.3
1.4
1.5
6
2
4
1
5 4
10
5 3 6
3
10
(b)
Iσσ Iσπ Iπσ Iππ
10
2 1
10
0
10
1.0
1.1
1.2
1.3
1.4
50
δ
6
30 20
400
10 0
200 0 1.15
40
1.16
1.17
1.18
Qz/Q0
0.6 0.4 0.2
0
20
δ (degrees)
40
60
∗ FIG. 3. (Color online) Simulated RXRD spectra for the SmCd6 phase with antiferroelectric primary structure (a) m = 1 split resonant peak profile with different distortion angle values. (b) Intensity ratio of the m = 1 split resonant peaks as a function of distortion angle δ; δ = 60◦ correspond to the SmCα∗ structure while δ = 0 correspond to the Ising structure.
10
(c)
5
60
1
0.0
Qz/Q0
10
4
2
0.8
2
1.0
3
1.0
10
(b)
600
(a)
5 3 1
10
800
1.5
Qz/Q0
6
Intensity (arb. unit)
3
1000x10
Iσσ Iσπ Iπσ Iππ
1
10
(a)
Intensity (arb. unit)
5
Intensity (arb. unit)
3
Intensity Ratio
Intensity (arb. unit)
6
10
PHYSICAL REVIEW E 89, 022501 (2014)
1.5
Qz/Q0
FIG. 2. (Color online) Simulated RXRD spectra for secondary structures. For model used in (a) φ12 = φ23 = φ45 = φ56 = 150◦ ; in (b) φ23 = φ45 = 150◦ ; in (c) φ12 = φ56 = 60◦ , φ23 = φ45 = 150◦ .
structures are antiferroelectric, while the most intense resonant peaks are not the first-order ones (m = 1). Instead, for the structures shown in Figs. 2(a) and 2(b), the most intense resonant peaks are located near QZ /Q0 = 1.5 (m = 3); while for the structure in Fig. 2(c), the most intense peak is the second-order one (m = 2). A more careful inspection at the proposed structures of the secondary models reveals that they can be viewed as constructed with SmCA∗ as subunit cell ∗ structures [for Figs. 2(a) and 2(b)]; and with SmCd3 structure [for Fig. 2(c)]. This explains the position of the most intense resonant scattering in Q space. While the structure in Fig. 2(a) has inversion symmetry with respect to the center of the unit cell, the two other secondary structures do not possess such symmetries. ∗ Although several different structural models of the SmCd6 phase have been proposed in theoretical studies, experimental studies using RXRD revealed the existence of the split first-order resonant peaks as well as the fact that this (m = 1) is the most intense resonant peak for this phase. Those experimental observations, together with the closeness of this ∗ phase to SmCd4 in the phase diagram, all suggest that the antiferroelectric primary structure as illustrated in Fig. 1(a) is the most likely. In the following text we inspect the features of
the simulated spectrum of this particular structural model in details. Simulations from QZ /Q0 = 1.15 to 1.19 were done for the antiferroelectric primary structure with different values of the distortion angle. The Ising structure is approached as δ approached 0; while the SmCα∗ structure appears when δ = 60◦ . The results are shown in Fig. 3. As clearly shown in the figures, the intensity ratio of the split resonant peaks strongly depend on the distortion angle and can be used to determine the value of δ in experiments, provided enough resolution can be achieved to resolve the splitting resonant peaks. As discussed, circular polarized x-ray light is now available for experiments [16]. This is incredibly useful, especially for studying chiral structures [21]. For the antiferroelectric ∗ , two structural chiralities are primary structure of SmCd6 involved. On the shorter scale, the biaxial structure within the unit cell is chiral. One can distinguish two structures with opposite chiralities from different sense of rotation between neighboring layers. On the longer scale, the helical superstructure that splits the resonant peaks and produces the satellites near the Bragg peaks is also chiral. Thus, a total of four different situations arise from the combination of those two chiralities. With the use of circular polarized x-ray, those two structural chiralities can be determined. In Figs. 4(a) and 4(b) we show the scattering intensity from right (left) circular polarized x-ray [labeled as I+ (I− )] with the same unit cell structure, but opposite sign of superstructure chiralities. In Fig. 4(c) we present the intensity difference between the I+ and I− for the two situations. From the results, it is clear that nonzero x-ray circular
022501-3
Intensity (arb. unit)
LIDONG PAN, R. PINDAK, AND C. C. HUANG
PHYSICAL REVIEW E 89, 022501 (2014)
6
III. DISCUSSION AND CONCLUSION
I+ I-
10
dichroism arises due to the helical superstructure. Maximum intensity contrast between the two circular channels appears at the Bragg peak satellite as well as the m = 2 resonant peak position [22]. From the behavior of intensity contrast as a function of QZ /Q0 , the chirality of the helical superstructure can be determined. On the other hand, as discussed earlier, the direction of splitting at m = 1 is determined by the sign of the product of the unit cell chirality and the superstructure chirality. With the latter determined, the former can easily be ascertained.
Finally, we would like to comment on the recently proposed ∗ observation of a ferrielectric structure for the SmCd6 phase. The authors of Ref. [8] presented two major points in their ∗ recent publication: (a) the SmCd6 phase observed in their experiments has a ferrielectric structure, similar to the one ∗ illustrated in Fig. 1(b); and (b) the SmCd6 phase observed in these experiments is intrinsically different from the ones observed in Ref. [7], with the reasoning that the latter ones appeared above a SmC ∗ phase, while the former ones are ∗ found below. Here we point out the fact that the SmCd6 phase observed in Ref. [7] is intimately related to the reversed phase ∗ sequence found in 10OHF [23] (a SmCd4 phase appearing ∗ above the SmC phase in cooling). As previously shown, the ∗ SmCd4 phase in 10OHF is connected in the phase diagram ∗ to the SmCd4 phase in compounds with a normal phase ∗ sequence [24,25]. Thus, the SmCd4 phase in 10OHF should be considered a normal SmC ∗ variant phase. As a result, there ∗ is no reason to expect the SmCd6 phase found in related mixtures would be any different, although it appears at higher temperatures than the SmC ∗ phase. On the other hand, a noisy region with multiple resonant ∗ peaks spanning a wide Q space is found in between the SmCd6 ∗ and the SmCd4 phase [7]. The noisy region appears over a ∗ phase in the two wider temperature window than the SmCd6 mixtures studied. Later, it was identified as a phase coexistence ∗ ∗ region between the SmCd6 and SmCd4 with sub-μm scale local orders [14]. The large interface area between the two types of local order in this model can in principle produce ferrielectric response. Thus, it is not immediately clear, without a more careful characterization with complimentary techniques, that the ferrielectric behavior found in Ref. [8] comes from a ∗ ferrielectric SmCd6 structure. In summary, we present in this paper detailed simulation of RXRD spectra of possible structures of smectic liquid crystal phase with six-layer periodicity. Spectra over a large Q phase were shown, and subtle differences between different structures were discussed. We showed that to distinguish the structures between different proposed models, the whole Q space must be inspected with great care. We also pointed out the trend of the spectra with the potential field-induced transitions from an antiferroelectric ground state. The possibility of determining the chiralities related to the structure with the use of circular polarized x-ray is also discussed. The results presented in this paper will serve as useful references for future experiments on the SmC ∗ variant phases, particularly ∗ the SmCd6 phase.
ˇ c, Rev. Mod. Phys. 82, 897 [1] H. Takezoe, E. Gorecka, and M. Cepiˇ (2010). [2] D. A. Olson, X. F. Han, A. Cady, and C. C. Huang, Phys. Rev. E 66, 021702 (2002). [3] M. B. Hamaneh and P. L. Taylor, Phys. Rev. Lett. 93, 167801 (2004). [4] J. P. F. Lagerwall, G. Heppke, and F. Giesselmann, Eur. Phys. J. E 18, 113 (2005).
[5] P. V. Dolganov, V. M. Zhilin, V. K. Dolganov, and E. I. Kats, Phys. Rev. E 82, 040701 (2010). [6] L. D. Pan, C. S. Hsu, and C. C. Huang, Phys. Rev. Lett. 108, 027801 (2012). [7] S. Wang, L. D. Pan, R. Pindak, Z. Q. Liu, H. T. Nguyen, and C. C. Huang, Phys. Rev. Lett. 104, 027801 (2010). [8] Y. Takanishi, I. Nishiyama, J. Yamamoto, Y. Ohtsuka, and A. Iida, Phys. Rev. E 87, 050503 (2013).
5
10
4
10
3
10
2
10
1
10
0
10
Intensity (arb. unit)
(a)
1.0
1.1
1.2
1.3
1.4
1.5
Qz/Q0
6
I+ I-
10
5
10
4
10
3
10
2
10
1
10
0
10
Intensity (arb. unit)
(b)
1.0
1.1
1.2
1.3
1.4
1.5
Qz/Q0
4000 2000 0
I+-I-(P = 210) I+-I-(P = -210)
-2000 -4000
(c)
1.0
1.1
1.2
1.3
1.4
1.5
Qz/Q0
FIG. 4. (Color online) Scattering from the right (I+ ) and left (I− ) circular polarized incident x-ray for the antiferroelectric primary structure with positive unit cell chirality and (a) superstructure with pitch P = 210 layer; (b) P = −210 layer; (c) intensity difference I+ -I− for the two cases.
022501-4
RESONANT X-RAY DIFFRACTION SPECTRUM FOR . . .
PHYSICAL REVIEW E 89, 022501 (2014)
[9] P. Mach, R. Pindak, A.-M. Levelut, P. Barois, H. T. Nguyen, C. C. Huang, and L. Furenlid, Phys. Rev. Lett. 81, 1015 (1998). [10] M. A. Osipov and M. V. Gorkunov, Liq. Cryst. 33, 1133 (2006). [11] M. Rjili, J. P. Marcerou, A. Gharbi, and T. Othman, Physica B 410, 162 (2013). ˇ c, AIP Conf. Proc. 1528, 337 [12] T. Y. Tan, L. H. Ong, and M. Cepiˇ (2013). [13] A.-M. Levelut and B. Pansu, Phys. Rev. E 60, 6803 (1999). [14] L. D. Pan, P. Barois, R. Pindak, Z. Q. Liu, B. K. McCoy, and C. C. Huang, Phys. Rev. Lett. 108, 037801 (2012). [15] A. Cady, J. A. Pitney, R. Pindak, L. S. Matkin, S. J. Watson, H. F. Gleeson, P. Cluzeau, P. Barois, A.-M. Levelut, W. Caliebe, J. W. Goodby, M. Hird, and C. C. Huang, Phys. Rev. E 64, 050702 (2001). [16] For example, see M. Sacchi, N. Jaouen, H. Popescu, R. Gaudemer, J. M. Tonnerre, S. G. Chiuzbaian, C. F. Hague, A. Delmotte, J. M. Dubuisson, G. Cauchon, B. Lagarde, and F. Polack, J. Phys. Conf. Ser. 425, 072018 (2013). [17] P. Fernandes, P. Barois, S. T. Wang, Z. Q. Liu, B. K. McCoy, C. C. Huang, R. Pindak, W. Caliebe, and H. T. Nguyen, Phys. Rev. Lett. 99, 227801 (2007).
[18] V. Ponsinet, P. Barois, L. D. Pan, S. Wang, C. C. Huang, S. T. Wang, R. Pindak, U. Baumeister, and W. Weissflog, Phys. Rev. E 84, 011706 (2011). [19] V. E. Dmitrienko, Acta Cryst. A39, 29 (1983). [20] S. Jaradat, P. D. Brimicombe, M. A. Osipov, R. Pindak, and H. F. Gleeson, Appl. Phys. Lett. 98, 043501 (2011). [21] Y. Tanaka and S. W. Lovesey, Eur. Phys. J. Special Topics 208, 67 (2012). [22] See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevE.89.022501 for details of the calculation process and additional simulation results. [23] S. Wang, L. D. Pan, B. K. McCoy, S. T. Wang, R. Pindak, H. T. Nguyen, and C. C. Huang, Phys. Rev. E 79, 021706 (2009). [24] S. T. Wang, Z. Q. Liu, B. K. McCoy, R. Pindak, W. Caliebe, H. T. Nguyen, and C. C. Huang, Phys. Rev. Lett. 96, 097801 (2006). [25] B. K. McCoy, Z. Q. Liu, S. T. Wang, L. D. Pan, S. Wang, H. T. Nguyen, R. Pindak, and C. C. Huang, Phys. Rev. E 77, 061704 (2008).
022501-5
