Around 37 at. % Mo in the Fe-Mo alloy system, in the temperature range between 1473 and 1761 K, there exists the intermetallic-compound R phase, whose crystal structure involves 12-, 14-, 15-, and 16-coordination polyhedra. To determine the crystal-structure units of the R phase, the crystallographic features of the bcc-to-R structural change in the (bcc -> bcc+R) reaction of the Fe-Mo system have been investigated by transmission electron microscopy. It was found that atomic columns, which were identified as the secondary structural unit, appeared first in the initial stage of the reaction. Based on this fact, the R phase is understood to have a crystal structure consisting of atomic columns, just as in the case of the mu structure [A. Hirata , Phys. Rev. B 74, 054204 (2006)]. Although only 12-coordination polyhedra form an atomic column in the mu structure, the column in the R structure consists of two kinds of polyhedra: that is, 16- and 12-coordination ones. Of these two polyhedra, in particular, the appearance of a 16-coordination polyhedron as the primary structural unit of the R structure is associated with the formation of covalent bonds in alloys.

The crystallographic features of Sr2-xPrxMnO4 samples, prepared by a coprecipitation method, have been investigated by transmission electron microscopy to understand their electronic States for 0.10 <= x <= 0.30. The orbital-ordered (OO) state lacking charge ordering existed for 0.10 <= x <= 0.25, and accompanied C-type antiferromagnetic (CAF) ordering at lower temperatures. It was also found that the suppression of CAF ordering occurred in a variant with smaller orthorhombicity, which is one of two variants forming a banded domain structure in the OO state. (C) 2007 Elsevier B.V. All rights reserved.

The structural correspondence among the bcc, ordered-bcc B2, triclinic Al(2)Fe and approximant Al(5)Fe(2) structures have been examined on the basis of the experimentally obtained orientation relationship of (110)(bcc,B2) //(2 (2) over bar1)(Al2Fe) and [(1) over bar 11](bcc,B2) // N(241)(Al2Fe), where N(hkl) denotes the normal direction of the (hkl) plane. The formation of the triclinic Al(2)Fe structure from the bcc structure is directly associated with that of Al-Fe covalent bonds; that is, hybridization between Al and Fe atoms. The covalent bond formation results in the appearance of decagonal-like atomic arrangements in the Al(2)Fe structure, which are similar to those in the approximant Al(5)Fe(2) structure.

The influence of antiphase boundaries (APBs) on the kinetic process of the L1(2) -> D0(22) structural change in an Ni3Al0.45V0.50 alloy has been investigated by transmission electron microscopy. The structural change includes two kinetic factors, i.e., the periodic introduction of antiphase-shifted planes (APSPs) in the L1(2) structure and V diffusion in the L1(2) matrix. Our results revealed that APB migration was accompanied by V diffusion as if to sweep the V atoms out of the antiphase domains (APDs). It was also found that the APBs served as the nucleation sites of the D0(22) regions, because the combination of the migrated APBs produced the APSPs in the L1(2) structure. On this basis, it is proposed that the APBs bring together the two kinetic factors in the process of the L1(2) -> D0(22) structural change, the physical origin of which is the variation in the charge density distribution of the APDs due to APB migration. (c) 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

The microstructure evolution related to the L1(2) (supersaturated Ni3Al1-xVx) -> L1(2) (Ni3Al)+D0(22) (Ni3V) phase separation of the Ni3Al0.50V0.50 alloy, with the addition of a small amount of Si (less than 0.2 mol%), was examined by transmission electron microscopy. The data showed that the L1(2) single phase was formed via the appearance and subsequent disappearance of the D0(22) regions under an isothermally holding process, as in the case of the Ni3Al0.50V0.50 alloy without Si. A striking feature of the microstructure evolution of the present alloy was that the formation of the final L1(2) single phase was accomplished within an extremely brief period of less than 10 min. Such a brief period indicated that the microstructure evolution was not accompanied by atomic diffusion, which was quite different from the case of the Si-free alloy. On this basis, we propose that the Si addition induced a pinning effect on atomic migration (i.e., suppression of long-range atomic diffusion) in the supersaturated L1(2) matrix and thus accelerated the formation of the final L1(2) single phase.

It has been found that phase separation, i.e., D0(22) precipitation in the supersaturated Ll(2) matfix to form the Ll(2) + D0(22) equilibrium state, stagnates in the Ni3Al0.52V0.48 alloy under a certain thermodynamic condition. This stagnation originates from the suppression of the long-range vanadium diffusion in the Ll(2) matrix, so-called diffusion blocking. Because diffusion blocking is inherent to the Ll(2) structure, its occurrence depends largely on the morphological features of the Ll(2) matrix. In this study, the microstructure evolution during the phase separation of the Ni3Al0.40V0.60 alloy was examined from the viewpoint of the relation between the variation in the initial microstructure and the appearance of the effect of diffusion blocking. Our results showed that cuboidal domains of about 30 nm on a side formed in the initial Ll(2) matrix and that the D0(22), regions appeared at the domain boundaries. The microstructure evolution of the alloy was found to proceed via the rearrangement, combination, and growth of these D0(22), regions, accompanying vanadium migration of about 20 nm in the Ll(2) matrix. The requisite migration length was determined by the size and density of the initial D0(22) regions, which depend on the size of the Ll(2), cuboidal domains. A shorter migration length than that in the case of the Ni3Al0.52V0.48 alloy was presumed to be advantageous to concealing the effect of diffusion blocking. On that basis, it was concluded that the occurrence of the phase separation in the Ni3Al0.40V0.60 alloy was attributed to the reduction in the size of the cuboidal domains in the initial Ll(2) matrix.

We have investigated microstructures associated with the magnetic and dielectric properties of multiferroic mixed-perovskite compounds of the formula (1 - x)BiFeO3-xBaTiO(3) by transmission electron microscopy (TEM) combined with conventional dielectric and magnetic measurements. TEM observation revealed that there exist complicated ferroelectric (FE) domains 20-30 nm in size in the FE phase of BiFeO3. It was also found that, in an x = 0.25 compound, a weak spontaneous magnetization appears in the FE phase at 20K. From the results of the X-ray and electron diffraction experiments, it is suggested that this weak spontaneous magnetization is closely associated with the decrease in the extent of rhombohedral distortion by a partial substitution of BaTiO3 for BiFeO3.

The crystallographic features of the ferroelectric domain structures in (Sr1-xBax)Bi2Ta2O9 single crystals have been examined by transmission electron microscopy. The unique 90degrees domain structures were found in these single crystals. In SrBi2Ta2O9 with x = 0, particularly the majority domains with the +/-[110](t) polarization vectors in the tetragonal notation are divided by the minority ones with the +/-[1(1) over bar 0](t) vectors, which were observed as a curved-line contrast in dark field images.

The structural phase transition in the insulating phase of La1-xSrxFeO3 has been examined by X-ray powder diffraction. Based on the Rietveld analysis of the obtained profiles, the R $(3) over bar $c-Pbnm structural transition was confirmed to take place in this oxide system. The interesting feature of the transition is that it is characterized by the appearance of the antiferroelectric displacement of the La/Sr atom, in addition to the change in the rotational-displacement pattern, R-25(x) + R-25(y) + R-25(z) --> R-25(x) + R-25(y) + M-3(z). It is thus understood that the low-temperature Pbnm phase can be identified as an antiferroelectric phase in oxides having a simple perovskite structure. (C) 2003 Elsevier B.V. All rights reserved.

The crystallographic features of the simple perovskite oxides RTiO3 with R = La, Nd, Sm and Y, which are known to be correlated-electron insulators, have been examined mainly at room temperature by transmission electron microscopy. Electron diffraction patterns indicated that the oxides with R = La and Nd have the space group Pbnm. In electron diffraction patterns of SmTiO3 and YTiO3, on the other hand, the twin splitting of reflections due to the Pbnm structure is seen along one of the (100)(c) directions, where the subscript c denotes the cubic Brillouin zone. That is, it is understood that their crystal system is lowered from an orthorhombic one to a monoclinic one. The analysis of the patterns also suggested that the Jahn-Teller atomic displacement with q = (1/2 1/2 0). should be involved in the crystal structure for R = Sm and Y, in addition to the full symmetric Jahn-Teller lattice distortion in the structure for R = La and Nd. (C) 2003 Elsevier B.V. All rights reserved.

Crystallographic features of La1-xSrxMO3 with M = Cr and Fe in the vicinity of the R3c/Pbnm phase boundary have been examined mainly at room temperature by transmission electron microscopy. When the Sr content x increases from x = 0, the change in crystal structure from Pbnm to R (3) over barc was confirmed to occur at room temperature in La1-xSrxCrO3 around x = 0.2. In La1-xSrxFeO3, On the other hand, the incommensurate phase was found to exist between the Pbnm and the R (3) over barc phases. The analysis of the electron diffraction patterns obtained in the work indicated that the normal structure of the incommensurate structure and the direction of the modulation could, respectively, be identified as the Pbnm structure and one of the <323> directions. It was also suggested that the rotational displacement Of the oxygen octahedra involved in the Pbnm structure should be modulated in the incommensurate structure. (C) 2002 Elsevier Science B.V. All rights reserved.

The structural change between substitution-type ordered structures basically includes the competition of pairwise interactions, which can describe the stability of such structures. The purpose of this study is to. investigate the frustration effect resulting from competition related to the L1(2) --> D0(22) structural change in the Ni3Al0.52V0.48 alloy and to discuss the pattern bifurcation in the pseudobinary Ni-3(Al, V) alloy. The present data obtained by transmission electron microscopy show that the change in microstructures during the structural change is characterized by the appearance and the subsequent annihilation of D0(22) regions in the L1(2) matrix. The final product of the structural change is the L1(2) single phase including topological defects, which is different from the equilibrium state known as the L1(2) + D0(22) checkerboard pattern. This indicates that the suppression of the L1(2) --> D0(22) structural change occurs in this alloy. Such suppression can be explained as being due to an inability to accommodate the strain field in the vicinity of the L1(2)/D0(22) boundaries when D0(22) precipitation occurs. The trigger of this inability is thought to be strong frustration originating from a small concentration deviation in the initial L1(2) matrix from the stoichiometry of the L1(2) structure. It is concluded that the formation of the L1(2) single phase as the final state, i.e. spontaneous L1(2) single crystallization, results from an effort by the alloy to accommodate the strain field, the occurrence of which is related to the frustration effect appearing during the L1(2) --> D0(22) structural change.

Inorganic compounds with the AB(2)X(4) spinel structure have been studied for many years, because of their unusual physical properties. The spinel crystallographic structure, first solved by Bragg in 1915(1), has cations occupying both tetrahedral (A) and octahedral (B) sites. Interesting physics arises when the B-site cations become mixed in valence. Magnetite (Fe3O4) is a classic and still unresolved example, where the tendency to form ordered arrays of Fe2+ and Fe3+ ions competes with the topological frustration of the B-site network(2). The CuIr2S4 thiospinel is another example, well known for the presence of a metal-insulator transition at 230 K with an abrupt decrease of the electrical conductivity on cooling accompanied by the loss of localized magnetic moments(3-7). Here, we report the determination of the crystallographic structure of CuIr2S4 below the metal-insulator transition. Our results indicate that CuIr2S4 undergoes a simultaneous charge-ordering and spin-dimerization transition-a rare phenomenon in three-dimensional compounds. Remarkably, the charge-ordering pattern consists of isomorphic octamers of Ir83+S24 and Ir84+S24 (as isovalent bi-capped hexagonal rings). This extraordinary arrangement leads to an elegant description of the spinel structure, but represents an increase in complexity with respect to all the known charge-ordered structures, which are typically based on stripes, slabs or chequerboard patterns.

Atomic displacements involved in a crystal structure of LaTiO3 have been investigated by means of transmission electron microscopy mainly at room temperature. Electron diffraction patterns taken from LaTiO3 were found to be characterized by the presence of the forbidden reflections for the Pnma structure. The extinction rule of these reflections including fundamental ones indicated that a space group of the crystal structure was determined to be P21/a with a monoclinic symmetry. A characteristic feature of the P21/a structure is that there exist the Jahn-Teller atomic displacement with q = 〈1/21/2 0〉c as well as the Jahn-Teller distortion. An additional displacement with q = 〈1/2 0 0〉c should also be needed. © 2002 Elsevier Science Ltd. All rights reserved.

The R (3) over bar c-to-Pbnm structural phase transition in La1-xSrxMnO3, which is characterized by changes in the rotational displacement of the MnO6 octahedra and in the crystal system, has been investigated by transmission electron microscopy. The experimental data indicate that the former change results in the appearance of the antiphase boundary and the latter produces the {110} and {112} twin structures. In addition, the structural phase transition can be reproduced on the basis of the Landau theory, in which a normal structure and an order parameter no, respectively, assumed to be the Pm3m structure and normal coordinates of the rotational displacements.

In situ observation of the dynamic local structure in La2-xSrxCuO4 has been conducted at lower temperatures by transmission electron microscopy. The observation clearly showed that around x=0.12 dynamic fluctuation of the local Pccn/LTT-tilt region in the interior of the LTO domain occurs, even at 12 K, while Pccn/LTT regions nucleated along the twin boundary are rather static. The dynamic local Pccn/LTT region is thought to play a crucial role in the slight suppression of T-C and in the appearance of the incommensurate magnetic structure, both of which have been reported to occur around x=0.12.

The microstructure produced by a low-temperature structural phase transition in La1.5Nd0.4Sr0.1CuO4 has been examined by transmission electron microscopy with the help of imaging plates. The low-temperature transition was found to be proceeded not only by the growth of the Pccn/low-temperature-tetragonal phases nucleated along the twin boundary but also by the nucleation and growth of the phases in the interior of the low-temperature-orthorhombic domain. In addition, because the map of the octahedron tilt as an order parameter is not identical to that of the spontaneous strain accompanied by the transition, the microstructure below the transition is-understood to be a very complex mixture of the low-temperature phases.

The difference between La-deficient and Sr-substituting effects for La in LaCoO3 has been investigated by using x-ray photoelectron spectroscopy (XPS). On the basis of the relationship between the intensity of the T-2(2) line and the amount of low-spin Co3+ ions in these compounds, it is pointed out that La0.9CoO3-delta is in a highly covalent low-spin state and La0.8Sr0.2CoO3-delta is in a mixed spin state. From the Co 2p and valence-band XPS results, it is pointed out that the electronic state of La0.9CoO3-delta is quite different from that of Sr-substituting La0.8Sr0.2CoO3-delta. In order to explain the difference between La-deficiency and Sr-substitution effects in valence controlling of LaCoO3, it is considered that Sr substitution produces the Co 3d holes and La deficiency yields the O 2p holes.

In order to examine a low-temperature transition in barium sodium niobate Ba2NaNb5O15 (BSN), an in situ observation was made in the temperature range between room temperature and 25 K by a transmission electron microscope. It was revealed that satellite reflection spots characterizing the incommensurate structure, which appear around 573 K, do not disappear even around 25 K. On the other hand, it was found that microdomains elongating along the [010] direction with a width of about 10 nm appear around 200 K, together with the appearance of both new satellite spots at the (h,k,l+1/2) points with h,k,l integers and new faint diffuse streaks through the (h,k+ 1/2, l+1/2) points along the [100] direction. It is concluded that the low-temperature transition in BSN is a transition from the Iq quasicommensurate state to the 2q one in which two modulated waves arise along the [100] and [010] directions.

Features of the striped-type superstructure in gamma-brass Cu-Al alloys have been investigated mainly by transmission electron microscopy in order to understand the role of a rhombohedral distortion in its stability. From dark field images taken from alloys exhibiting the superstructure, it was found that there exist two types of ferroelastic rhombohedral domains, which are in a twin relation. In other words, the superstructure is characterized by a periodic array of the ferroelastic domains as well as inversion antiphase boundaries. Because charge density waves should be responsible for the formation of the superstructure, as suggested in our previous paper [Phys. Rev. B 40, 5378 (1989)], the rhombohedral distortion must play a crucial role in their appearance. Presumably the distortion enlarges parallel portions of the Fermi surface. On the basis of the present experimental data, the interplay between the striped-type superstructure and the rhombohedral gamma-brass structure is also discussed.

In an initial stage of a eutectoid reaction of a Ti-40 at. % Al alloy, hcp-->D0(19)+L1(0), an amorphous state was found to form around an intermediate precipitate with the B-19 structure, together with a change in crystal structure of the matrix from the hcp structure to the D0(19) one. Characteristic features of the amorphous state are that the destruction of the crystal lattice mainly takes place in the (001) plane of the D0(19) structure but is very slightly extended along the [001] direction. The shape of the amorphous region is just like a thin film. On the basis of the present data, the origin of the two-dimensional-like character of the lattice destruction is simply discussed.

Features of both the ferroelectric and incommensurate transitions in Ba2NaNb5(1-x)Ta5xO15 were investigated by means of a transmission electron microscope in order to find the relationship between these transitions. An in situ observation of the transitions revealed that the ferroelectric transition takes place around 373 K in x=0.57 while the incommensurate transition occurs around 543 K. Because the ferroelectric transition does not basically accompany a change in microstructures related to the incommensurate lattice modulation, there is no interaction between these two transitions.

Features of a ferroelastic domain structure in an incommensurate phase of barium sodium niobate [Ba2NaNb2O15 (BSN)] that appear in the cooling process were investigated by means of a transmission electron microscope. The in situ observation revealed that there exists an abrupt change in the domain structure around 503 K. The ferroelastic domain structure above 503 K basically consists of two types of Iq ferroelastic microdomains with a size of about 20 nm while below it large ferroelastic domains with flat domain boundaries are formed.

Features of an incommensurate-to-commensurate transition in barium sodium niobate, which is obtained in a cooling process from a tetragonal phase, have been theoretically examined on the basis of the Ginzburg-Landau theory. The present theory which takes into account the contribution of higher-order distortion waves produced from higher-order harmonics via the umklapp process, well reproduces an experimentally obtained change in an incommensurability delta on cooling. The theory also predicts both a discommensuration with a phase slip of 2pi/4 and the incommensurate-to-commensurate transition.

Features of both structural transitions and related fluctuations in La1.875-xSmxSr0.125CuO4 have been examined by means of electron diffraction in a temperature range between about 600 K and about 20 K. As the temperature is lowered, a structural phase transition from a high-temperature tetragonal (HTT) structure to a low-temperature orthorhombic (LTO) one takes place in 0 less-than-or-equal-to x less-than-or-equal-to 0.2 and HTT --> LTO --> Pccn --> LTT successive transitions in 0.3 less-than-or-equal-to x less-than-or-equal-to 0.4 where the Pccn and low-temperature tetragonal (LTT) structures are characterized as a rotation of an oxygen octahedron about two [110] directions. An enhancement of a structural fluctuation in the LTO phase, which indicates a soft-phonon mode in the LTO-to-Pccn transition, occurs on cooling even in 0 less-than-or-equal-to x less-than-or-equal-to 0.2 as well as in 0.3 less-than-or-equal-to x less-than-or-equal-to 0.4. Because an increase in the Sm content results in two effects, a lowering of T(c) and an enhancement of the structural fluctuation, the fluctuation is understood to suppress the superconductivity. Eventually, a dip in a composition dependence of T(c) in La-Sr-Cu-O, which was experimentally found by Oda et al. [J. Phys. Soc. Jpn. 58, 1137 (1989)], is concluded to be due to the structural fluctuation related to the LTO-to-Pccn transition.

Detailed features of the structural disorder in Nd-Ce-Sr-Cu-O with the T*-structure have been investigated by diffuse scattering in electron diffraction. There exist three types of characteristic diffuse scatterings: a cylindrical-shape, a spike-shape, and spots in addition to the fundamental spots due to the T*-structure. The cylindrical-shape and spike-shape scatterings can be concluded to result from both atomic displacements of O(2) atoms along the [110] directions, on planes through the apical positions of the CuO5 pyramids, and the ordering of Nd and Sr atoms. On the other hand, the extra spots are understood to be related to displacements of O(1) or O(3) atoms along the c axis.

Diffuse scattering, reflecting structural disorder in La1.8-xSmxSr0.2CuOz, have been investigated by means of electron diffraction. In the T*-structure, in addition to the fundamental spots due to the T*-structure there exists cylindrical shape diffuse scattering due to static displacements of the apical oxygen atoms in the CuO5 pyramids. On the other hand, both thermal diffuse scattering and superlattice spots have been found in diffraction patterns obtained from the T-structure. The thermal diffuse scattering is distributed on the {100}* planes of reciprocal space. In addition, the superlattice spots appear in a range of 0.3 < x < 0.6 and are located at a zone boundary of the first Brillouin zone along the [110]* directions. On the basis of the extinction rule for the superlattice spots, the crystal system of the T-structure for 0.3 < x < 0.6 is concluded to be orthorhombic while the system for 0 < x < 0.3 is tetragonal.

Features of the crystal structures at room temperature and structural transitions in both Ba1-xKxBiO3 and Ba1-xRbxBiO3 with 0.1 &lt
ϗ ≤ 0.5 have been examined in detail by means of electron diffraction. From the analysis of diffraction patterns, it has been confirmed that at room temperature the crystal system changes with respect to ϗ cubic system for ϗ ≥ 0.35 and orthorhombic for 0.1 &lt
ϗ &lt
0.35. Furthermore, the cubic to orthorhombic structural transition with changing temperature was found to be due to the condensation of the Rx 25 and the RY 25 modes. © 1993, Taylor &amp
Francis Group, LLC. All rights reserved.

Features of structural fluctuations related to the cubic-to-orthorhombic transition in Ba1-xMxBiO3 (M = K, Rb) have been investigated in detail by means of electron diffraction. A fluctuation due to the softening of the R25 mode, which is responsible for the transition, is detected as diffuse scattering at the R point in the reciprocal space of the cubic phase. When x is reduced from x = 0.5 to a transition composition in the cubic phase, the fluctuation is enhanced remarkably at lower temperatures. Because the fluctuation of the R25 mode results in those of spontaneous strains, the fluctuations of the A1g and E(g) strains are understood to be enhanced near the superconducting transition.

Features of temperature-dependent diffuse scattering showing a structural fluctuation in Tl2Ba2Ca2Cu3O10 have been investigated in detail in a temperature range between about 20 K and room temperature by means of electron diffraction. As the temperature is lowered, the intensity of diffuse scattering increases remarkably below about 130 K, which is taken to be the onset temperature of superconductivity. Because the scattering is obviously due to a structural fluctuation, Tl2Ba2Ca2Cu3O10 is understood to exhibit an enbancement of the fluctuation in a superconducting state. In addition, a careful analysis of the features in the scattering suggests that a dynamic displacement of an apical O atom in the CuO5 pyramid along the c axis is responsible for the structural fluctuation in the superconducting state.

The crystal structures at room temperature and the structural transitions in BaPb1-xBixO3 (Ba-Pb-Bi-O) have been investigated in detail by means of electron diffraction. From an analysis of the crystal structures on the basis of electron diffraction patterns obtained experimentally, it is shown that, as the temperature is lowered, Ba-Pb-Bi-O undergoes a cubic-to-tetragonal transition resulting from the condensation of the triply degenerate R25 mode, R25x, in the lower-Bi-content range 0 less-than-or-equal-to x < 0.35, and successive transitions from the cubic to the tetragonal and to a monoclinic structure in the intermediate-Bi-content range 0.35 < x < 0.90. In the successive transitions, the cubic-to-tetragonal transition is understood to be due to the first condensation of the R25 mode, R25x, and the tetragonal-to-monoclinic transition is due to the second condensation of the R25 mode, R25y. The soft-phonon-mode behavior of the R25 mode in the cubic and tetragonal phases was actually detected as a change in an intensity of diffuse scattering appearing at the R point in the first Brillouin zone of the cubic perovskite structure. In addition, the crystal structure at room temperature in the higher-Bi-content range 0.90 < x less-than-or-equal-to 1.00 is confirmed to involve the breathing displacements resulting from the appearance of charge-density waves as well as the R25x + R25y displacements. Moreover, a simple explanation of the physical origin of the superconductivity in Ba-Pb-Bi-O is presented on the basis of the present experimental results.

One-dimensional (1D) long-period superlattices (LPSL's) in alloys, which consist of a periodic array of antiphase boundaries in the L1(0) or L1(2) ordered structure, have been examined theoretically from the viewpoint of the appearance of charge-density waves (CDW's) in a three-dimensional system. The Fermi surface of the LPSL has flat portions along the <110> directions, and the Fermi-surface nesting results in two nesting vectors in the L1(0) structure and four vectors in the L1(2) Structure. This means that the 1D LPSL is not a single first-order CDW state but is characterized by the superposition of multiple CDW's. The response of the lattice system to the CDW is further understood to take place in two ways: One is the introduction of a periodic array of antiphase boundaries as the atomic arrangement, and the other is the periodic lattice distortion as the atomic displacement. Unlike low-dimensional materials such as 2H-TaSe2, the former way plays a particularly important role in the response for the LPSL's. As a result of the appearance of the CDW's, the change in the period with respect to the composition of an alloy, the electron-atom ratio, is determined by the size of the Fermi surface along the <110> directions and can be basically explained on the basis of the same equation as that in the Sato-Toth theory [Phys. Rev. 124, 1833 (1961)]. In addition, a Ginzburg-Landau free-energy functional is proposed in order to understand the features of an incommensurate structure, such as the temperature dependence of the period and the discommensurate structures in the LPSL's. It is assumed in the theory that the normal structure is the normal ordered structure and the order parameter is the CDW. Note that the temperature dependence of the period is closely related to the phase modulation of the first-order CDW by means of the higher-order CDW's produced from the higher-order harmonics via the umklapp process. Hence the LPSL is classified into two groups in the L1(0) structure and three groups in the L1(2) structure on the basis of the positional relation between the first- and third-order spots in diffraction patterns. The temperature dependence of the period and the phase modulation of the first-order wave are then calculated by the derived free-energy expression for each group, and are found to be in good agreement with those obtained experimentally. That is, the present theory can well reproduce overall features of the LPSL's. This means that the 1D LPSL's are concluded to be the CDW state in the three-dimensional system.