Growth mechanism studies of ZnO nanowire arrays via hydrothermal method

Well-controlled ZnO nanowire arrays have been synthesized using the hydrothermal method, a low temperature and low cost synthesis method. The process consists of two steps: the ZnO buffer layer deposition on the substrate by spin-coating and the growth of the ZnO nanowire array on the seed layer. We demonstrated that the microstructure and the morphology of the ZnO nanowire arrays can be significantly influenced by the main parameters of the hydrothermal method, such as pH value of the aqueous solution, growth time, and solution temperature during the ZnO nanowire growth. Scanning electron microscopy observations showed that the well oriented and homogeneous ZnO nanowire arrays can be obtained with the optimized synthesis parameters. Both x-ray diffraction spectra and high-resolution transmission electron microscopy (HRTEM) observations revealed a preferred orientation of ZnO nanowires toward the c-axis of the hexagonal Wurtzite structure, and HRTEM images also showed an excellent monocrystallinity of the as-grown ZnO nanowires. For a deposition temperature of 90 °C, two growth stages have been identified during the growth process with the rates of 10 and 3 nm/min, respectively, at the beginning and the end of the nanowire growth. The ZnO nanowires obtained with the optimized growth parameters owning a high aspect ratio about 20. We noticed that the starting temperature of seed layer can seriously influence the nanowire growth morphology; two possible growth mechanisms have been proposed for the seed layer dipped in the solution at room temperature and at a high temperature, respectively.     

Syntheses, structures and photophysical properties of heteronuclear Zn-Ln coordination complexes

Four heteronuclear Zn-Ln coordination complexes, [Nd₂Zn₂(p-toluylate)₁₀(phen)₂] (1), [Ln₂Zn₂(p-toluylate)₁₀(phen)₂]·2(HAc)_1/2 (Ln=Tb 2, Ho 3) and [PrZn₂(p-toluylate)₅(Ac)₂(phen)₂] (4) (phen=1, 10-phenanthroline), are synthesized by the hydrothermal method and their structures are measured by single-crystal X-ray diffraction. The IR and UV-vis-NIR absorption spectra and the emission spectra in the visible and near-infrared (NIR) regions of the four complexes are determined at room temperature. In the NIR region (or in the visible region), the complexes show the characteristic emission bands of Ln³⁺ ions, which may be attributed to sensitization from the ligands (the ligand directly-coordinated to Ln³⁺ ions and d-block) to Ln³⁺ ions after forming the Zn-Ln complexes. It is reported for the first time in this paper that the Zn-Pr complex 4 can exhibit the broad emission band in the NIR region. In addition, the shift, split or broadness of the ff emission bands in the NIR region of complexes 1, 3 and 4 are discussed.     

Luminescence and crystal-field fitting of two optical isomeric complexes of europium

FT-IR and Raman vibrational spectra and electronic emission spectra have been recorded for enantiomers of europium complexes with DBM: dibenzoylmethanate 1,2, and TTFA: 2-thenoyltrifluoroacetonate 3,4, employing the chiral ligands L_SS(+)- and L_RR(-)-4,5-pinene bipyridine. Contrary to the previously published X-ray data, where geometrical differences were stated to occur for particular enantiomers, the vibrational (and the emission) spectra of the individual optical isomers of a complex are not distinguishable. Using excitation into the Eu³⁺⁵D₂ multiplet term, the emission intensity is weak from ⁵D₁, whereas a complex structure is observed for the ⁵D₀→⁷F_J transitions. Features in the vibronic sidebands exhibit similar derived vibrational energies to those observed in the Raman spectra. Fittings of 25 4f⁶ crystal-field energy levels of 2 and 4 have been attempted with some approximations concerning the local Eu³⁺ environments. The ⁵D₀ emission lifetimes are monoexponential and are 0.5 (1,2) and 0.9 ms (3,4) at room temperature.     

Photoluminescent properties of heteroleptic cyclometalated platinum(II) complexes bearing 1,3-bis(3,4-dibutoxyphenyl)propane-1,3-dionate as an ancillary ligand

A new series of heteroleptic cyclometalated platinum(II) complexes Pt-1a-f was synthesized, employing 2-arylpyridine (or 1-arylisoquinoline) (HC^∧N-1) and 1,3-bis(3,4-dibutoxyphenyl)propane-1,3-dione (HO^∧O-1) for cyclometalation and as ancillary ligands, respectively, and photoluminescent properties were investigated. Focusing on red-shifted phosphorescence, C^∧N ligands containing π-extended aromatics and electron-rich heterocycles were examined. All obtained complexes exhibited photoluminescence at ambient temperature, and the emission maxima ranged from green (λ_PL=518 nm) to far red (λ_PL=708 nm). The large Stokes shifts of more than 100 nm and sub-microsecond or microsecond emission lifetimes revealed that these complexes are phosphorescent emissive. The quantum yield of Pt-1 ranged from 0.02 to 0.59 at ambient temperature and decreased as the emission maximum was red-shifted. In comparison with the reference platinum(II) complexes, Pt-2 bearing an aliphatic ancillary ligand, such as 2,2,6,6-tetramethylheptane-3,5-dionate (O^∧O-2), the ligand O^∧O-1 did not significantly affect the photoluminescence emission maxima, indicating that the energy gap between the singlet ground state and the triplet level was predominantly dependent on the C^∧N ligand.     

Synthesis,structures and near-infrared luminescence properties of Ho and Yb coordination complexes

Four Ln³⁺ coordination complexes with the formulas [Ln(p-toluylate)₂(Ac)(H₂O)]_n (Ln=Ho 1, Yb 2) and {[Ln₂(OOCCH₂CH₂COO)₃(H₂O)₄]·6H₂O}_n (Ln=Ho 3, Yb 4) were synthesized hydrothermally. Their structures were determined by single-crystal X-ray diffraction. Complexes 1 and 2 are isomorphic and form infinite 2D network structures comprising p-toluylate and acetate (Ac^–) moieties. Complexes 3 and 4 are also isomorphic and possess infinite 2D structures in which succinate acts as bridging ligands that are connected to a 3D hydrogen bonding network by O–H…O hydrogen bonds. Solid-state IR and UV-Vis-NIR spectra, excitation and emission spectra were determined for the four complexes at room temperature. Complexes 1 and 2 exhibit characteristic NIR emission bands of Ln³⁺ ions but these are shifted and split relative to the theoretical positions. This is also evident for their UV-Vis-NIR spectra. The influence of ligands on enhancing the NIR luminescence of Ln³⁺ ions in complexes is discussed.     

Prepare high efficiency Mn-doped Zn2SiO4 green phosphors in air using nano-particles

In this study, nano-scale precursors of ZnO, SiO₂, and MnO₂ powders were used to prepare mixtures with the compositions of 2ZnO+SiO₂+X mol% MnO₂ (X=MnO₂/2ZnO, abbreviated as Zn₂SiO₄-X-MnO₂), where 2≤X≤5. The mixed Zn₂SiO₄-X-MnO₂ mixtures were calcined from 900 to 1300 °C in air in order to synthesize Zn₂SiO₄:Mn²⁺ green phosphors. The X-ray diffraction patterns of Zn₂SiO₄-X-MnO₂ particles indicated that ZnO was present in the 900 °C-calcined Zn₂SiO₄-X-MnO₂ phosphors, but not in particles calcined at temperatures of 1000 °C and higher. However, the unapparent secondary phase of ZnMnO₃ was found in the 1200 and 1300 °C-calcined Zn₂SiO₄-5-MnO₂ compositions. The luminescent characteristics of Zn₂SiO₄-X-Mn²⁺ phosphors were compared with that of a commercial product (Nichia Corp., Japan). The photoluminescence (PL) intensity of 1200 °C-calcined Zn₂SiO₄-4-MnO₂ phosphors was higher and the decay times of all synthesized Zn₂SiO₄-X-MnO₂ phosphors were longer than those of the commercial product.     

Synthesis and luminescent properties of two Schiff-base boron complexes

Schiff bases N,N′-o-phenylenebis (salicylideneimine) (H₂L¹), N,N′-p-phenylenebis (salicylideneimine) (H₂L²) and their corresponding boron complexes (BF₂)₂L¹, (BF₂)₂L² were synthesized, respectively. The two boron complexes have been characterized by ¹H NMR, mass spectrometry and elemental analysis, while the luminescent properties of them were investigated with UV-VIS spectroscopy and photoluminescence spectroscopy. Then the three-layer devices [ITO/NPB (60 nm)/(BF₂)₂L¹ (50 nm)/Alq3 (20 nm)/LiF (1 nm)/Al (200 nm)] (device I) and [ITO/NPB (60 nm)/(BF₂)₂L² (50 nm)/Alq3 (20 nm)/LiF (1 nm)/Al (200 nm)] (device II) were fabricated by vacuum deposition. These two devices both exhibited blue green emission at 500 nm, but showed different luminances and efficiencies.     

Nonlinear optical absorption and fluorescence of phosphine-substituted bithiophenes in the violet-blue spectral region

The nonlinear optical absorptions of two 5,5′-bis(diphenylphosphino)-2,2′-bithiophene derivatives, Ph₂(X)P(C₄H₂S)₂P(X)Ph₂ (X = O, 1; S, 2), have been investigated by direct transmission measurement with both picosecond and nanosecond laser pulses from 420 nm to 480 nm. Saturated dichloromethane solutions of 1 and 2 exhibit strong nonlinear optical absorptions in this violet-blue spectral region with that of 2 being stronger at all wavelengths. In the picosecond regime, at 420 nm, the transmittance rapidly falls to 50% when the incident fluence is 0.22 J/cm² for 1 and 0.11 J/cm² for 2. Two-photon absorption appears to be the primary mechanism for this nonlinear absorption. The two-photon absorption coefficients β for 1 (2.1 cm/GW) and 2 (4.4 cm/GM) were obtained by fitting the measurement of transmittance as the function of incident beam intensity at 420 nm. These β values are comparable with some of the best results obtained for organic materials in the green, red and infrared spectral region. Both compounds also show fluorescence with an emission peak at 390 nm for 1 and 400 nm for 2. The fluorescence of 1 is considerably stronger than is that of 2. The combination of the wide band gap and strong fluorescence emission of 1 makes it a promising candidate as a host material for blue organic light emitting diodes.     

Optical oxygen sensing materials based on trinuclear starburst ruthenium(II) complexes assembled in mesoporous silica

The preparation and oxygen sensing properties of optical materials based on two trinuclear starburst ruthenium(II) complexes: [Ru₃(bpy)₆(TMMB)]⁶⁺ (1) and [Ru₃(phen)₆(TMMB)]⁶⁺ (2) (bpy=2,2′-bpyridine, phen=1,10-phenathroline, TMMB=1,3,5-tris[2-(2′-pyridyl)benzimidazoyl]methylbenzene) assembled in two mesoporous silicate (MS) are described in this paper. The luminescence of Ru complexes/silicate assemble materials can be quenched by molecular oxygen with good sensitivity (I₀/I₁>5 for 2/MS and I₀/I₁>3 for 1/MS), indicating that trinuclear starburst Ru(II) complexes/MS systems are sensitive to oxygen molecules.     

Magnetic and electronic properties of Cr- and Mn-doped SnO2: ab initio calculations

The ab initio calculations, based on the Korringa–Kohn–Rostoker (KKR) approximation method combined with the coherent potential approximation (CPA), indicated as KKR–CPA, have been used to study the stability of ferromagnetic and ferrimagnetic states, for systems that are SnO₂ doped and co-doped with two transition metals, that is, chromium and manganese. Our results indicate that the ferromagnetic state is more stable than the spin-glass state for the (Sn_1−xCr_xO₂; x = 0.07, 0.09, 0.12 and 0.15)-doped system, while the spin-glass state is more stable than the ferromagnetic state for the (Sn_1−xMn_xO₂; x = 0.02 and 0.05)-doped system. However, the ferromagnetic and/or the ferrimagnetic states are stable for the (Sn_0.98−xMn_0.02Cr_xO₂; x = 0.05, 0.09 and 0.13)-doped system depending on the Cr concentration. Moreover, we estimated the Curie temperature (T_c) for the Cr-doped tin dioxide (SnO₂), and we explained the origin of magnetic behaviour through the total density of states for different doped and co-doped SnO₂ systems.     

Spectroscopic and thermodynamic features of conical bubble luminescence

The influence on luminescence from conical bubble collapse (CBL) with varying Ar gas content while perturbing the liquid 1,2-Propanediol (PD) has been investigated. The temporal, spatial, and spectral features were analysed with regards to the dynamics of collapse and liquid degradation. Sulphuric acid and sodium chloride were added to disturb the liquid. The following three cases were studied: PD/Ar, (I), (PD + H₂SO₄)/Ar, (II), and (PD + H₂SO₄ + NaCl)/Ar, (III). The intensities of those cases decrease as III > II > I. Temporally, single and multiple light emissions were found to occur. The pulse shape exhibited a large variety of profiles with a main maximum and up to two local maxima around the main maximum. These local maxima resembled those generated by laser cavitation. Spatially, no radial symmetry was detected in the light emissions. Spectrally, the Swan, CH and CN lines were observed at low volumes of gas and driving pressure. The OH radical and OH-Ar bands, as well as the Na and K lines, consistently appeared superimposed on an underlying continuum that almost disappeared in (III). The Na line was observed with two satellite diffuse bands representing Na-Ar complexes in (I) and (II), whereas in (III), only the line of sodium could be seen. Weak and diffuse emission lines from the Ar atom in the near-IR region were observed in (I) and (II).The proposed mechanism of bright CBL was based on the energy transfer from electron-excited homolytic cleavage products to the chromophore molecules generated during the collapse-rebound time line (∼8200 K and ∼1 ms of collapse time from model), which had accumulated inside the liquid and remained on the walls of cavity during the repetition of the collapse. A general mechanism for the bright CBL is broached.     

Photophysical properties of a series of high luminescent europium complexes with fluorinated ligands

A series of high luminescent europium complexes have been synthesized, such as Eu(TFNB)₃phen (1), Eu(PFNP)₃phen (2), Eu(HFNH)₃phen (3) and Eu(PFND)₃phen (4), which have β-diketone ligands containing fluorinated alkyl chains with different lengths and conjugated naphthyl groups, i.e., 4,4,4-trifluoro-1-(2-naphthyl)butane-1,3-dione (TFNB); 4,4,5,5,5-pentafluoro-1-(2-naphthyl)pentane-1,3-dione (PFNP); 4,4,5,5,6,6,6-heptafluoro-1-(2-naphthyl)hexane-1,3-dione (HFNH) and 4,4,5,5,6,6,7,7,8,8,9,9,10,10,10-pentadecafluoro-1-(2-naphthyl)decane-1,3-dione (PFND). And 10-phenanthroline (phen) is coordinated as the neutral second ligand in 1-4. The crystal structures of 1 and 2 have been studied, which are typical and similar to that of 3. The results of TGA-DTA suggest that these Eu complexes have good thermal stabilities. By means of absorption and (time resolved) emission spectroscopy including determination of luminescence quantum yields, energy transfer dynamics and so on, the following results have been obtained: first, these Eu complexes show characteristic pure red color photoluminescence emission with high quantum efficiencies from the central Eu³⁺ ions through the excitation of the ligands; secondly, photophysical properties of 1, 2, 3 and 4, especially the lifetimes of excited states ⁵D₀ of Eu³⁺ ions and quantum efficiencies are influenced by the different lengths of fluorinated alkyl chains, though the singlets (S₁) and triplets (T₁) of the fluorinated ligands are almost the same.     

Relative power loss of misalignment based on electro-optical printed circuit board

In this paper, the power loss values between 45° total internal reflecting waveguide mirror (TIRWM) and the polymer optical waveguide layer are obtained by vector finite element method. There are some misalignments during an actual fabrication of electro-optical printed circuit board (EOPCB). And, b, a, c correspond to the error value of alignment in the x-axis, y-axis and z-axis direction, respectively. Another, four effective refractive indices of the 45° TIRWM and polymer optical waveguide layer are calculated, separately. And, n_eff1 = 1.425211 is uniquely chosen. Next, these relative power errors Δ are calculated, when a, b, c separately change. Moreover, these error values are plotted into some curved surface figures. By these figures, it is easy to find the relationship between a, b, c and Δ. Furthermore, it is beneficial for us to avoid the region of larger power loss during an actual fabrication of EOPCB.     

Growth of self-assembled ZnO nanowire arrays

This paper reports on ZnO nanowires arrays synthesized using Sn as a catalyst. The Sn particles were produced from the reduction of SnO₂ powders via a vapour-solid growth process. Control of growth conditions led to the formation of ZnO nanowire arrays, radial nanowire ‘flowers’ and uniaxial fuzzy nanowires. ZnO nanowire–nanobelt junctions were also grown by changing the growth direction. As-grown nanowire arrays could be fundamental materials for investigating physical and chemical properties at nano-scale dimensions.     

Eu(III) and Tb(III) luminescent lanthanide systems derived from DTPA-bisamide and DTTA-based ligands incorporating bipyridine antenna

Four new polycarboxylate ligands H₃Lⁿ have been synthesized by the attachment of two or one 2,2′-bipyridine subunits onto a diethylenetriamine pentacarboxylic acid (DTPA-bisamide derivatives: H₃L¹, H₃L²) or a diethylenetriamine tricarboxylic acid (DTTA derivatives: H₃L³, H₃L⁴) core. The neutral Eu^III and Tb^III complexes of these chelates have been prepared and studied from their UV-vis and luminescence data. The main photophysical characteristics of these complexes, i.e. the absorption and luminescence spectra, the metal-centred lifetimes and the overall luminescence yields (Φ) were measured in buffered aqueous solutions. In addition the role played by non-radiative paths (vibrational energy transfer involving coordinated water molecules, involvement of ligand-to-metal charge-transfer excited states, or metal→ligand back-transfer) was investigated. In all complexes, we found that the bidentate bipyridine chromophore is not coordinated to the lanthanide ion, allowing one (LnL¹, LnL²) or two (LnL³, LnL⁴) water molecules to penetrate the first coordination sphere of the metal. Although the bipyridine chromophore behaves as remote (from the binding site) light-harvesting unit for the lanthanide ion in these systems, a sizeable sensitization of the Eu- and Tb-centred luminescence can be effective (LnL², LnL³, Φ=16-19% in aerated D₂O solutions). Our photophysical investigations show that overall non-radiative deactivation is not dependant of thermally activated non-radiative channels but the efficiency of the ligand→Ln intramolecular energy transfer has to be taken into account to explain the obtained results.     

The nature of anharmonicity and anomalous piezoelectric properties of ferroelectric SbSBr crystal

The contribution of soft mode at Sb atom's sites, to the temperature dependences of Sb atom's equilibrium position's difference Δz(T) has been studied theoretically, when SbSBr crystal is deformed along a(x), b(y) and c(z)-axis in paraelectric phase and is deformed along c(z)-axis in ferroelectric phase. The largest change of Δz₃₃(T) occurs in the ferroelectric phase near the phase transition temperature in the range from 16 K to 21 K. The temperature dependence of Sb atom's equilibrium position's displacements Δz₃₃ is very similar to the temperature dependence of experimental piezoelectric modulus, when SbSBr crystal is deformed in the direction of c(z)-axis in ferroelectric phase.     

Structural and luminescence studies of trivalent europium complexes with pentaethylene glycol and 18-crown-6 ligands in the presence of picrate anion

Two new isostructural complexes of europium picrate (Eu-Pic) with pentaethylene glycol (EO5) and 18-crown-6 (18C6) ligands formed complexes of molecular formula [Eu(Pic)₂(18C6)]⁺(Pic)⁻I and [Eu(Pic)₂(EO5)]⁺(Pic)⁻II have been isolated and characterised. Compound I showed 10-coordination number through six oxygen atoms from the 18C6 ligand and two bidentate picrate anions. Meanwhile, compound II exhibited 9-coordination number via six oxygen atoms from EO5 ligand, two oxygen atoms from a bidentate and one oxygen atom from monodentate picrate anions. Photoluminescence (PL) spectra of the solid-state europium complexes display sharp lines which are assigned to ⁵D₀→⁷F_0-4 and ⁵D₁→⁷F_1,2,4 transitions. No emission of polyether ligands is observed, indicating that the energy transfer from the polyether ligands to the Eu³⁺ ion is quite efficient. The PL spectra of [Eu(Pic)₂(OH₂)₆]⁺(Pic)⁻·6H₂O III, [Eu(NO₃)₃(OH₂)₃]·(18C6) IV, [Eu(NO₃)₃·6H₂O] V and Eu₂O₃VI are also observed. Compounds I-IV exhibited high Ω₂ intensity parameter values, namely 16.93, 10.23, 17.10 and 12.35 (in units of 10⁻²⁰ cm²), respectively. These relatively high values reflect the hypersensitive behaviour of the ⁵D₀→⁷F₂ transition and indicate that the Eu³⁺ ion is located in a highly polarisable chemical environment.     

Photosensitized luminescence of thermostable polynuclear Eu(III) complexes

Tetranuclear europium(III) complexes, [Eu₄(μ-O)(L₁)₁₀] (L₁=2-hydroxy-4-octyloxybenzophenone,1) and [Eu₄(μ-O)(L₂)₁₀] (L₂=2-hydroxy-4-dodecyloxybenzophenone,2) were synthesized by the reaction of lanthanide nitrates with L₁ or L₂ in the presence of triethylamine in methanol. The photosensitized emission bands of the both Eu(III) complexes in THF-d₈ were observed around 579, 590, 615, 653, and 699 nm by the excitation of the ligands at 380 nm, whereas the emission from the mononuclear complex 3 containing ethanol molecules was almost quenched. The emission efficiencies were determined to be 3.1±0.1% for 1 and 3.9±0.1% for 2, respectively. The differential scanning calorimetry (DSC) measurements demonstrated that the decomposition points of 1 and 2 were 309 °C and 320 °C, respectively, indicating high thermostability of these complexes compared to the mononuclear Eu(III) complex 3 (250 °C). New strategy for designing stable rare earth compounds giving strong emission would be emphasized by introducing polynuclear complexes. Polynuclear complexes should open a wide range of molecular design for photosensitized luminescence and thermal stability.     

Reversible photomagnetism in a cobalt layered compound coupled with photochromic diarylethene

Photomagnetism is one of the most attractive topics in recent research on molecular solids. In order to produce a photo-controllable magnet, we have synthesized a novel organic-inorganic hybrid system coupled with a photochromic diarylethene anion, 2,2′-dimethyl-3,3′-(perfluorocyclopentene-1,2-diyl)bis(benzo[b]thiophene-6-sulfonate) (1a) and cobalt LDHs (layered double hydroxides). Based on the elemental analysis, the title compound, which was synthesized by the anion exchange reaction between Co₂(OH)₃(CH₃COO)·H₂O (2) and 1a, has the chemical composition, Co₄(OH)₇(1a)_0.5·3H₂O (3). Powder X-ray diffraction analysis revealed the interlayer distance of c=27.8 Å. The magnetic susceptibility measurements elucidated the ferromagnetic intra- and inter-layer interactions and the Curie temperature of T_c=9 K. By UV irradiation of 313 nm, 3 shows the photo-isomerization of diarylethene anion from the open form to the closed one in solid state, which leads to the decreases in the coercive field and the remnant magnetization. Furthermore, the photo-excited state is returned to the initial state (open form) almost reversibly by visible irradiation of 550 nm.     

TOWARDS DEEP AND SIMPLE UNDERSTANDING OF THE TRANSCENDENTAL EIGENPROBLEM OF STRUCTURAL VIBRATIONS

When using exact methods for undamped free vibration problems the generalized linear eigenvalue problem (K−ω²M) D=0 of approximate methods, e.g., finite elements, is replaced by the transcendental eigenvalue problem K (ω) D=0. Here ω is the circular frequency; D is the displacement amplitude vector; M and K are the mass and static stiffness matrices; and K (ω) is the dynamic stiffness matrix, with coefficients which include trigonometric and hyperbolic functions involving ω and mass because elements (for example, bars or beams) are analyzed exactly by solving their governing differential equations. The natural frequencies of this transcendental eigenvalue problem are generally found by the Wittrick-Williams algorithm which gives the number of natural frequencies below ω_t, a trial value of ω, as ∑J_m+s{K (ω_t)} wheres {} denotes the readily computed sign count property of K (ω) and the summation is over the clamped-clamped natural frequencies of all elements of the structure. Understanding the alternative solution forms of the transcendental eigenvalue problem is important both to accelerate convergence to natural frequencies, e.g., by plotting ∣K (ω)∣, and to improve the mode calculations, which lack the complete reliability of natural frequencies obtained by using the Wittrick-Williams algorithm. The three solution forms are: ∣K (ω)∣=0; D=0 with ∣K (ω)∣→∞; and ∣K (ω)∣≠0 with D≠0. The literature covers the first two forms thoroughly but the third form has been almost totally ignored. Therefore, it is now examined thoroughly, principally by analytical studies of simple bar structures and also by confirmatory numerical results for a rigidly jointed plane frame. Although structures are unlikely to have exactly the properties giving this form, it needs to be understood, particularly because ill-conditioning can occur for structures approximating those for which it occurs.