T1 mapping technique, called 3D SASHA (saturation‐recovery single‐shot acquisition), which combines a saturation recovery pulse with 1D‐navigator‐based‐respiratory motion compensation to acquire the whole volume of the heart in free breathing. The sequence was tested and validated both in a T1 phantom and in healthy subjects.To propose a 3D quantitative high‐resolution 3. For assessment of accuracy and precision the sequence was compared with the reference gold‐standard inversion‐recovery spin echo (IRSE) pulse sequence in a T1 phantom, while for the in vivo studies  3D SASHA was compared with the clinically used 2D MOLLI (3‐3‐5) and 2D SASHA protocols.The 3D SASHA method was implemented on a 1.5T scanner. A diaphragmatic navigator was used to allow free‐breathing acquisition and the images were acquired with a resolution of 1.4 × 1.4 × 8 mmT1 values measured in a T1 phantom with 3D SASHA and the reference IRSE pulse sequences . Mean and standard deviation of the myocardial T1 values in healthy subjects measured with 2D MOLLI, 2D SASHA, and 3D SASHA sequences were 881 ± 40 msec, 1181.3 ± 32 msec, and 1153.6 ± 28 msec respectively.There was good agreement between the T1‐mapping with T1 values in good agreement with the 2D SASHA and improved precision.The proposed 3D SASHA sequence allows for high‐resolution free‐breathing whole‐heart Level of Evidence: 2Technical Efficacy: Stage 1J. MAGN. RESON. IMAGING 2017;46:218–227 T1 mapping is a noninvasive tissue characterization technique that allows to differentiate between healthy and diseased tissues, based on the different environment of their water molecules.T1 mapping techniques have been widely used to visualize both local and diffuse fibrosis.T1‐mapping compares favorably to the late gadolinium enhancement (LGE) technique both in terms of quantification/staging of disease and fibrosis detection, as LGE can only detect the presence of focal fibrosis in the heart.T1 mapping sequences acquire multiple images with variable T1 weighting and uses pixel‐wise fitting of the imaging data to an exponential signal recovery model of the longitudinal relaxation to generate a T1 map. T1 mapping is often performed before and after the administration of a contrast agent to quantify the extracellular volume fraction.T1 mapping using either inversion‐recovery,T1 maps with high reproducibility; however, the quantification accuracy is affected by the perturbation of the longitudinal signal recovery due to the repeated image acquisitions, causing a general underestimation of myocardial T1.T1 weighting. Thanks to the use of a saturation pulse, which removes any spin history, the SASHA sequence is less susceptible to T1 errors caused by heart rate variationsMyocardial fibrosis is a common manifestation for several cardiomyopathies and it is one of the main predictors of future cardiac outcome.T1 mapping have already been proposed.T1 mapping of the whole heart, due to its high scan efficiency and quantification accuracy, and its robustness to heart‐rate variations which may occur over the course of a 5–10‐minute scan.Existing MOLLI and SASHA techniques are limited in spatial resolution and coverage due to the need for breath‐holding, which is required to minimize respiratory motion artifacts. Typically, only a single low‐resolution 2D slice can be acquired for each breathhold with both MOLLI and SASHA. Due to the increased scan time associated with higher‐resolution 3D image acquisition of the heart, free‐breathing imaging is necessary and respiratory motion compensation imperative to ensure diagnostic image quality. Some techniques for 3D In this work, we sought to develop and assess a multishot high‐resolution 3D SASHA sequence with improved spatial resolution and coverage that can be acquired in free‐breathing with respiratory motion compensation.T1 mapping in free breathing. To this end, the 2D SASHA sequencek‐space acquisition. The sampling scheme used for 3D SASHA is shown in Fig. k‐space segments with no magnetization preparation were acquired, which we term “infinity image” (as we assume it is acquired after an infinite saturation delay time during data fitting). This was followed by the interleaved segmented acquisition with preceding saturation pulse and increasing saturation delays. To ensure all k‐space segments of the infinity image were acquired at equilibrium magnetization, “pause” cardiac cycles were added between the acquisitions of these segments. During these pauses, all RF pulses and data acquisition were switched off to allow for full T1 recovery.All data were acquired on a 1.5T Ingenia MR system . The proposed 3D SASHA pulse sequence enables the acquisition of whole‐heart To compensate for respiratory motion a 1D hemi‐diaphragmatic navigator (NAV) was used. Typically, the navigator is performed immediately before the image acquisition to allow prospective motion correction and ensure high temporal correlation between NAV and image acquisition. However, for the shortest saturation time the navigator signal may be insufficient for reliable motion estimation. Therefore, the position between the saturation pulse and the navigator was swapped for the shortest saturation time (40 msec) , each followed by a dephasing gradient.To minimize the sensitivity to B3, repetition time / echo time (TR/TE) = 3.2/1.6 msec, image resolution = 1.4 × 1.4 mm2, slice thickness = 8 mm, flip angle (FA) = 35°, parallel imaging with SENSE factor of 2 in phase‐encoding direction, 10 start‐up echoes, Cartesian acquisition with radial k‐space shutters, slice‐selective RF pulse. Thirty segments of the k‐space were acquired per heartbeat with a low‐high k‐space ordering and an acquisition window duration of 90 msec.Data acquisition was performed with a segmented 3D steady‐state free‐precession (SSFP) technique and imaging parameters included: field of view (FOV) = 300 × 300 × 90 mmT1 of 1100 msec , T2 of 45 msec, 40 RF excitations per segment, FA of 35°, TR of 2.8 msec, and for different numbers of “pause” cardiac cycles.The proposed segmented multishot SASHA pulse sequence was simulated using the Bloch equations in MatLab R2014a . The simulations were performed to analyze the effect of the added “pause” cardiac cycles between acquisitions of the infinity images and to investigate the optimal number of pauses for a given heart rate to ensure complete longitudinal magnetization recovery. The simulations for an SSFP sequence were performed for heart‐rates between 60–100 bpm, 2 vials was used for imaging, with T1 values ranging from 250 msec to 1500 msec,T1 measurements between the gold standard inversion‐recovery spin echo (IRSE) technique, 2D MOLLI, 2D SASHA, and the proposed 3D SASHA. The IRSE pulse sequence was performed with 10 inversion times varying from 100–2000 msec. The reference T1 measurements were compared to 2D MOLLI, 2D SASHA, and the proposed 3D SASHA, in order to validate the imaging technique. Sequence parameters for the spin echo sequence were: FOV = 200 × 200 mm2, image resolution = 3.1 × 3.1 mm2, 10 mm slice thickness, TR = 8000 msec, and TE = 5.9 msec.A phantom with nine agar/NiCl2, TR/TE = 2.6/1.3, image resolution = 1.7 × 2.1 mm2, slice thickness = 10 mm, FA = 70°, 100–700 msec saturation time (TS) for a heart rate of 60 bpm, parallel imaging with SENSE factor of 2 in the phase‐encoding direction. For the 3D SASHA sequence the saturation times (TS) were 100–700 msec for a heart rate of 60 bpm. All three imaging techniques were reconstructed to the same in‐plane resolution of 1.25 × 1.25 mm2.The MOLLI sequence was acquired with the following parameters: FOV = 300 × 280 mm, TR/TE = 2.6/1.3, image resolution = 1.7 × 2.1 mm, slice thickness = 10 mm, FA = 35°, scan time of about 12 seconds. The acquisition parameters for the 2D SASHA sequence were: FOV = 300 × 280 mmT1‐weighted images along the Mz recovery curve compared to the conventional 2D SASHA sequence  in order to keep the total scan time within 5 minutes.The proposed 3D SASHA sequence was acquired with fewer All sequences were triggered with a simulated heart rate of 60 bpm and signal reception was performed with a 32‐channel cardiac coilTo validate the findings of the simulations and evaluate the optimal flip angle and number of “pause” heart cycles added between acquisitions of the fully recovered “infinity” images, the phantom was imaged using the 3D SASHA sequence with a different number of pauses and by changing the FA within the range from 25° to 85° with 10° increments. This experiment was repeated for 3D SASHA with 0, 1, 2, and 3 pause cycles for the infinity image.The study was performed in accordance with the Declaration of Helsinki (2000). All subjects involved in this study provided written informed consent with study approval from the Institutional Review Board (15/NS/0030). Ten healthy volunteers without a history of heart disease were imaged using 2D MOLLI, 2D SASHA, and 3D SASHA sequences. The same imaging parameters used for the phantom experiment were employed for the in vivo study. The acquisition of the 3D SASHA sequence was performed in free‐breathing with a nominal scan duration of 4:14 (min:sec) for a heart rate of 60 bpm and 100% scan efficiency. A 1D diaphragmatic navigator was used for respiratory motion compensation with a gating window of 5 mm and a tracking factor of 0.6.Following the validation with Bloch simulations and phantom studies, three “pause” cardiac cycles were used for all in vivo imaging.T1 maps were reconstructed offline using MatLab. The T1 values were calculated by fitting the image signal intensities to a three‐parameter exponential recovery curve using the fitting procedure proposed by Barral et al.T1 values were expressed as mean ± standard deviation.The T1 mapping.T1 maps was evaluated by comparing the mean T1 values with the reference measurements from the IRSE experiment using a Kruskal–Wallis test. A Bland–Altman analysis was performed to compare the different techniques and to evaluate the agreement between them. For in vivo studies the myocardial T1 values were compared with the values measured with the 2D SASHA and 2D MOLLI sequences.Accuracy and precision are the two criterions used to evaluate the efficiency of quantitative T1 measurements was used to quantify the precision, ie, the spatial variability of T1 across the selected ROI, of each imaging technique.The standard deviation of the T1 maps. Two blinded expert reviewers (Reviewer 1 with more than 10 years and Reviewer 2 with 5 years of experience in cardiovascular MRI) scored the T1 maps of all volunteers based on image quality, which includes both myocardial sharpness and signal homogeneity. The scale for scoring the image quality was: 1 poor, 2 average, 2 good, 3 very good, and 5 excellent image quality. The Wilcoxon rank t‐test was used to evaluate the statistical difference between 2D SASHA and 3D SASHA. The median was calculated and presented together with the 75% and 25% of the median itself.Sharpness of the myocardial borders was used as a criterion to quantify the performance of motion correction and was measured using dedicated software. The sharpness was calculated as a percentage of the steepness of the edges of the myocardium, where 100% corresponds to an ideal step edge.T1 values of the whole volume were measured for 16 AHA segments  in three slices . Each segment of the final model was calculated as the average of the same segment between all the volunteers.The American Heart Association (AHA) 17‐segmentP < 0.05 was used to define the statistical significance.For statistical analysis, GraphPad Prism v. 5 for Windows  was used and z magnetization for a tissue with T1 similar to healthy myocardium on 1.5T (T1 = 1100 msec) during the acquisition of the infinity image as a function of different numbers of “pause” cardiac cycles (from zero to three). The recovery of Mz depends both on the number of pauses added between the infinity images and on the heart rate. Mz returns to the equilibrium magnetization after two pause cardiac cycles for heart rates lower or equal than 60 bpm. However, three pauses are more robust for higher heart rates, and may ensure full recovery of Mz even in the presence of heart‐rate variability, which can occur during a 5–10‐minute scan. Thus three pauses were chosen for all in vivo studies.Figure T1 value calculated. The 3D SASHA T1 map appears more homogeneous and less noisy compared to the 2D SASHA T1 map  the precision of the 3D SASHA is higher, as measured by the lower standard deviation . The estimated T1 values using the 3D SASHA and the 2D SASHA sequences correlate extremely well (r = 0.9997) with the IRSE sequence . The T1 estimation for the vial with the highest T1 value deviated from the expected line of identity due to an insufficient recovery time between successive inversion pulses used for the IRSE sequence. This led to a wrong estimation of the highest T1 values with the IRSE sequence and consequently to a mismatch between the 3D SASHA and the IRSE T1 value for the vial with the highest T1.Table T1 are illustrated. The graph shows the difference between the reference T1 value and the measured T1 values, and the best agreement was obtained using an FA of 35° and three “pause” heart cycles, which confirms the results obtained from the simulations.Figure T1 maps of three volunteers obtained with the 2D MOLLI, 2D SASHA, and 3D SASHA sequences. The single T1‐weighted images acquired for Volunteer 2 are shown in Fig. T1 values measured with the 2D SASHA and the 3D SASHA sequences, while the T1 values obtained with the 2D MOLLI sequence were considerably lower  with a scan efficiency of 32%.Figure wer Fig. a. A trenwer Fig. b. The avT1 maps was significantly higher compared to 3D SASHA T1 maps (respectively 26% ± 2 and 18% ± 3).The average myocardial border sharpness measured on the 2D SASHA T1 maps, and 50% for the 2D SASHA and 3D SASHA T1 maps. In general, we observed better image quality for the 2D MOLLI T1 maps , with significant statistical difference with both 2D and 3D SASHA (respectively P = 0.0053 and P = 0.0047). There was no statistical difference between 2D SASHA  and 3D SASHA .The agreement in image quality between the observers was 80% for the 2D MOLLI T1 values measured across the left ventricle. The average myocardial T1 values are homogeneous across the left ventricle for both of the imaging techniques. The standard deviation for the 3D SASHA is lower compared to the 2D SASHA, which could be explained by the theoretical higher signal‐to‐noise ratio (SNR) for the 3D technique. However, a higher standard deviation was observed in the inferior wall of the basal slice, which could be explained by slice profile imperfection and motion artifacts.Figure T1 mapping of the whole left ventricle. The sequence was validated with simulations, phantom, and in vivo experiments.In this study we proposed and validated a free‐breathing 3D SASHA sequence for T1 mapping approach was based on the well‐established 2D SASHA pulse sequence design.k‐space acquisition was required to enable high‐resolution 3D acquisition. A Cartesian acquisition with radial k‐space shutter was used, which is commonly used for magnetization‐prepared heart imaging. However, no further analysis was conducted to investigate the effect of the acquisition scheme.The proposed 3D T1 values were obtained. The standard deviations of the 3D T1 measurements were lower compared to the 2D SASHA. Improved subjective image quality with 3D SASHA over 2D SASHA was observed, although this finding was not significant. This could be attributed to the fact that a 3D imaging sequence inherently benefits from higher SNR compared to a 2D method.The in vivo studies confirmed the phantom measurements in terms of accuracy between 2D SASHA and 3D SASHA sequences, as similar average myocardial T1 mapping imaging sequence improves accuracy compared to 2D MOLLI and precision compared to 2D SASHA. This is a promising result and warrants further evaluation of the 3D SASHA imaging technique in clinical practice. Also, a free‐breathing T1 mapping approach would be useful in patients who cannot hold their breath due to respiratory problems.As demonstrated in the phantom experiments, the proposed 3D T1 mapping techniques have already been proposed,T1 weighting. The proposed 3D SASHA technique uses a saturation recovery pulse, which has the advantage of completely erasing the history of the longitudinal magnetization between the acquisitions, thus reducing the heart rate dependency as well as improving scan efficiency. Further work will investigate other available T1 mapping techniques with the proposed 3D SASHA sequence. For the 3D SASHA imaging techniques, an incomplete and incorrect saturation could be an important source of errors in the final T1 map. In the proposed 3D SASHA sequence a composite saturation pulse was used, with a saturation efficiency of 0.989 ± 0.005, estimated from the exponential fitting curve.T1 map. This is likely due to residual respiratory motion artifacts. Improved respiratory motion compensation strategies, such as self‐navigationT1 maps. In general, better image quality and better visualization of the myocardial borders would be beneficial where trasmurality of scar should be assessed. In addition, high‐resolution T1 mapping may be especially beneficial for fibrosis quantification in the thin wall of the right ventricle. The AHA model showed in general a good homogeneity of the myocardial T1 values measured across the left ventricle. However, T1 values in the lateral wall of the mid ventricle were somewhat lower compared to other segments. This is consistent with previous findings showing lower T1 values in this region, which is likely due to susceptibility artifacts caused by the air–myocardium interface.Other 3D T1 mapping technique at the time of the study. Recently, the 5‐3 MOLLI scheme has become more popular, due to the shorter breath‐hold duration. Further work is required to compare the precision, accuracy, and clinical merits of more recent T1 mapping techniques with the proposed 3D SASHA.In this work a 3‐3‐5 MOLLI scheme was used, which was the clinically used T1 mapping.The proposed 3D SASHA imaging sequence has been extensively tested and validated with simulations, phantom studies, and in vivo experiments on healthy volunteers. Further work is required to evaluate this technique in patients with myocardial scar, as well as for postcontrast 3. The implementation of novel motion compensation techniques, such as image‐based navigation, correcting for both translational and nonrigid motion,Image resolution of the 3D SASHA sequence is mainly limited by the scan time, which is ∼12 minutes for the current spatial resolution of 1.4 × 1.4 × 8 mmT1 map, which could further improve the reliability of the T1 measurements.An additional step of retrospective image registrationk‐space segments of the infinity images. Reduction of the number of pauses could be achieved by introducing some subject‐specific correction in the fitting step. A reduced number of images acquired with the 3D SASHA sequence could be another possible solution to shorten the acquisition time, although the trade‐off between quantification precision and total acquisition time will have to be investigated further.Scan time is also affected by the number of pauses added between the T1 mapping technique allows acquiring high‐resolution myocardial T1 maps of the whole left ventricle completely in free breathing with good accuracy. While the accuracy is in the range of the 2D SASHA sequence, the precision is improved.In conclusion, the proposed 3D SASHA Additional supporting information may be found in the online version of this articleSupporting InformationClick here for additional data file.


Fd3m (Q227) which was verified by Small angle X-ray scattering (SAXS) and Transmission electron microscopy (TEM) and holds great promise for potential applications in optoelectronics, photovoltaics and thermoelectrics.Well-ordered and highly interconnected 3D semiconducting nanostructures of bismuth sulphide were prepared from inverse cubic lipid mesophases. This route offers significant advantages in terms of mild conditions, ease of use and electrode architecture over other routes to nanomaterials synthesis for device applications. The resulting 3D bicontinous nanowire network films exhibited a single diamond topology of symmetry  Hence there is an urgent need for developing approaches to their fabrication. Most of the methods reported to-date involve the use of hard templates such as porous alumina membranes (AAO)8 or polymeric substrates such as e.g. polycarbonate9 that act as templates for  deposition of the respective material. The main limitations associated with these approaches is that these templates require multiple and often complex fabrication steps and can only be removed in either acids, sodium hydroxide (NaOH) or organic solvents such as dichloromethane or dimethylformanide10 which raises safety and environmental concerns. These limitations can be overcome by using soft-templating approaches such as diblock copolymers11 or amphiphilic surfactants12. Diblock co-polymers for example have been used to produce gyroid networks of titania for applications in hybrid heterojunction solar cells13 and dye-sensitised solar cells5. The fabrication of diblock co-polymer templates however also involves synthetically complex steps and above all is very time-consuming, often taking several days13. This has prompted us to explore whether inverse cubic lipid mesophases could provide an alternative fabrication method for 3D semiconductor nanostructures.Over the past few years the emergence of synthetic strategies and fabrication technologies that afford control over the architecture of matter at nanometer length scales has provided a new tool kit for engineering functionality in materials. In particular the three-dimensional (3D) nanostructuring of metals and semiconductors has resulted in enhanced optical, magnetic and electronic properties yielding new devices and device concepts. 3D nanowire networks represent a good example and are of great technological importance for many applications including energy harvesting2S3) were prepared by electrodeposition through inverse lipid cubic phases of phytantriol at room temperature under mild aqueous conditions. A combination of SAXS and TEM in conjunction with Matlab simulations and Fast Fourier Transforms (FFTs) has been utilised to verify the formation of a single diamond topology bismuth sulphide films with symmetry Fd3m (Q227) from a double diamond phytantriol template of symmetry Q224.In the present study highly ordered 3D bicontinuous nanowire network films of the semiconductor compound bismuth sulphide , double diamond (D), and primitive (P) surfaces corresponding to the space groups Ia3d (Q230), Pn3m (Q224), and Im3m (Q229), respectively. For illustration, the bicontinuous double diamond phase is depicted in Fig. The phase diagram of phytantriol has previously been studied in detail by Baraukas 3)3, 100 mM Na2S2O3 and 200 mM EDTA through phytantriol modified gold (Au) DVDs (Delkin Devices). Bismuth sulphide electrolytes were prepared by mixing the bismuth and sulphide precursors along with EDTA in dissolved water  over night. The solution was then left to settle for 2 days to allow a formed black precipitate to settle. A clear electrolyte solution was then separated and bubbled through with Argon for a minimum of 20 minutes, to remove dissolved oxygen. The two plastic layers of the DVD were separated using mechanical force. The exposed gold layer disc was cut into 1.5 cm × 1.5 cm using scissors. Cu tape was used to make an electrical contact and polyimide tape was used to isolate a 10 mm × 10 mm working electrode area. The working electrode was dip coated in a solution of phytantriol and ethanol 1:2 by weight, allowed to dry for 30 minutes prior to being immersed in the electrolyte and was allowed to hydrate for 30 minutes in the electrolyte prior to electrodeposition. Films were deposited potentiostatically at a potential of −0.4 V vs. SCE for 4 h. A saturated calomel reference electrode and a Pt gauze counter electrode were used. Post deposition films were washed for a minimum of 30 minutes in ethanol.Nanowire network films of bismuth sulphide were prepared by potentiostatic electrodeposition at a potential of −0.4 V vs SCE from an electrolyte containing 100 mM Bi(NO2S3 deposits. Accelerating voltages up to 15 KV were applied and images recorded with a secondary electron detector. Samples were mounted as deposited (on Au DVDs) on an 18 mm SEM clip specimen mount (agar scientific). XRD Grazing incidence scans, where the source is held at a constant shallow angle to the sample and the detector is scanned through theta values, were conducted on the nanotemplated films. A Rigaku SmartLab X-Ray diffractometer was used to obtain diffraction patterns. The Rigaku Smartlab uses Cu Kα radiation with a wavelength of 1.5406 Å. Small angle x-ray scattering measurements were carried out at Diamond light source on beamline I22 with beam energy and size of 12.4 KeV and 320 µm × 80 µm respectively. A Pilatus 2 M detector28 was used to collect data over the q range of 0.05 Å−1–0.30 Å−1. Calibration was achieved by using a silver behenate29 sample. Phases were indexed by assigning Bragg peaks to known phases. Samples were prepared for transmission electron microscopy by scraping small sections of nanotemplated films with a scalpel over 400 mesh copper transmission electron microscopy grids (Sigma Aldrich). A FEI Tecnai T12 transmission electron microscopy was used to acquire transmission electron microscopy images.A Zeiss EVO LS25 ESEM scanning electron microscope (SEM) with an Oxford Instruments energy dispersive x-ray spectrometer attachment was used to determine surface topography and composition of the Bi2S3.The surface morphology of the phytantriol templated bismuth sulphide films was investigated by SEM and a representative image is shown in Fig. 2S3 film through phytantriol. The diffraction pattern has been indexed in accordance with the pdf card 9007375 for Bismuthinite from the crystallographic open database (COD) which has been superimposed in Fig. 2S3 has formed.Figure 3)3, 100 mM Na2S2O3 and 200 mM EDTA, and following electrodeposition after removal of phytantriol (4b).Small angle X-ray scattering (SAXS) in transmission mode and Transmission Electron Microscopy (TEM) were employed to verify and visualize the nanostructures. Figure Fd3m (Q227), with a lattice parameter of 139.6 ± 3.2 Å. This corresponds to approximately double the lattice parameter of phytantriol in the electrolyte (68.9 ± 0.9 Å) and is consistent with electrodeposition occurring in one water channel of the double diamond structure, resulting in a single diamond nanostructure30 with a doubled lattice parameter due to a reduction in symmetry as shown in Fig. Prior to electrodeposition we can distinguish Bragg peaks with relative positions for 1/d in the ratio √2: √3: √4: √6: √8 which are representative for the Q224 bicontinuous cubic lattice of phytantriol, with a lattice parameter of 68.9 ± 0.9 Å. Following electrodeposition and subsequent removal of phytantriol the 1D SAXS radial profiles reveal a sequence of three distinct and well-resolved Bragg diffraction peaks with 1/d spacings in the ratio of √3: √8: √11 which are characteristic of a single diamond topology of symmetry To provide further evidence TEM images in combination with MATlab projections are shown in Fig. From the MATlab projections as well as the TEM images the following lattice parameters can be estimated: a) 125 Å, b) 109 Å and c) 124 Å. The lattice parameters based on the simulations therefore matched with a lattice parameter of a similar order of magnitude as those determined by SAXS (139.6 ± 3.2 Å), however, these were smaller by 25%. This may be due to actual shrinkage of the bismuth sulphide films after detachment from the substrates. Nevertheless it can be concluded that within experimental error as evidenced by SAXS and TEM data the formation of a single diamond morphology in nanostructured bismuth sulphide has occurred.We have demonstrated for the first time that inverse lipid cubic mesophases can act as suitable templates for the room temperature production of highly ordered 3D semiconductor compound nanostructures (bismuth sulphide) as evidenced by SAXS and TEM. Fabrication via this route can be carried out in one step without the need for elaborate and complex synthetic steps resulting in well-ordered materials that are attached to an electrode surface. Hence this therefore represents an important step forward towards the fabrication of nanostructured 3D semiconductors for device applications.


The aim of this study was to evaluate experimentally the accuracy of the dose calculation algorithm AcurosXB in small field highly modulated Volumetric Modulated Arc Therapy (VMAT).n = 28) were recalculated on the phantom CT data set in the Eclipse TPS. All measurements were done on a Varian TrueBeamSTx, which can provide the jaw tracking technique (JTT). The effect of disabling the JTT, thereby fixing the jaws at static field size of 3 × 3 cm2 and applying the MLC to shape the smallest apertures, was investigated for static fields between 0.5 × 0.5−3 × 3 cm2 and for seven VMAT patients with small brain metastases. The dose calculation accuracy has been evaluated by comparing the measured and calculated dose outputs and dose distributions. The dosimetric agreement has been presented by a local gamma evaluation criterion of 2%/2 mm.The 1000SRS detector array inserted in the rotational Octavius 4D phantom (PTW) was used for 3D dose verification of VMAT treatments characterized by small to very small targets. Clinical treatment plans , is on average respectively, 0.2%, 0.8%, 6.8%, 5.7% (6 MV) and 0.1%, 1.3%, 11.7%, 21.6% (10 MV). For the seven brain mets patients was found that the smaller the target volumes, the higher the improvement in agreement between measured and calculated doses after disabling the JTT.Regarding the clinical plans, the mean ± SD of the volumetric gamma evaluation scores considering the dose levels for evaluation of 10%, 50%, 80% and 95% are (96.0 ± 6.9)%, (95.2 ± 6.8)%, (86.7 ± 14.8)% and (56.3 ± 42.3)% respectively. For the smallest field VMAT treatments, discrepancies between calculated and measured doses up to 16% are obtained. The difference between the 1000SRS central chamber measurements compared to the calculated dose outputs for static fields 3 × 3 cm2 and using the MLC with high positional accuracy to shape the smallest apertures in contrast to the JTT is currently found to be the most accurate treatment technique.Fixing the jaws at 3  In stereotactic treatments, spatial and dosimetric accuracy are inextricably linked which makes the evaluation of the dosimetric accuracy of small field delivery equally important. The IPEM report no 103VMAT plans are typically highly modulated and a considerable fraction of the total control points used for a plan consists of small subfields.3 planning target volume about 10% overdosing was detected  and more modulation in the plan was measured than calculated.Several planning and measurement studies on clinical stereotactic VMAT plans for small target volumes have been published. For instance, Lagerwaard et al.Since the investigation of Fog et al.Furthermore, the jaw tracking technique (JTT) provided by the latest types of linacs keeps jaws during dose delivery as close as possible to the MLC aperture, thereby minimizing leakage and transmission through the MLC leaves resulting in optimized organs‐at‐risk (OAR) sparing and potentially improving the dose falloff towards the surrounding critical structures.2To evaluate experimentally the accuracy of the dose calculation algorithm in the TPS in small field highly modulated VMAT treatments, 28 clinical VMAT treatment plans characterized by small target volumes and fraction doses between 2.75–30 Gy were delivered on a commercial high resolution 2D detector array in a rotational phantom for the independent 3D validation of the calculated dose distribution. Also, the influence of disabling the JTT on the dose calculation accuracy to the target is investigated for a range of small static fields as well as for a number of stereotactic radiosurgery treatments with very small target sizes. To validate the 2D detector array measurements, the small static fields were also delivered to radiochromic film  for data collection and analysis (PTW‐Freiburg). Controlled by an inclinometer, the Octavius 4D phantom rotates synchronously with the gantry, taking time‐ and gantry angle‐resolved dose measurements.2 field with no gantry rotation . Moreover, the same measurement set‐up was used to test the detector's measurement reproducibility, dose linearity and dose rate dependence  curves, independently measured with an ionization chamber in water at source‐to‐surface‐distance of 85 cm, that are used to reconstruct dose values along the ray lines that connect the relevant detectors and the focus of the beam.ence see ppendix A2.BTreatment planning was carried out with the Eclipse TPS  version 11, using the AcurosXB photon dose calculation algorithm version 10.0.28. A calculation grid spacing of 2.5 mm is the default setting and is generally applied for clinical calculations as smaller grid spacing may require impractically large computation times. In addition, the minimum available grid spacing of 1 mm has been used for very small targets.2 to 40 × 40 cm2  while the more concise set recommended by Varian contains field sizes defined by jaws between 3 × 3 cm2 and 40 × 40 cm2. As described in the Varian white paper,2 can be imported in the configuration workspace but are not needed and smaller than 2 × 2 cm2 even not used in the configuration program, while the OF below 3 × 3 cm2 are used. How these OF for fields < 3 × 3 cm2 are applied in the TPS is discussed in section 2 and 3 × 3 cm2 with a range of detectors  with effective measuring point centered on the beam axis, by scanning the 50% dose which defines the field size, using the water tank software  assuring a positioning accuracy < 0.7 mm. Multiple measurements are performed for each field size to determine the reproducibility of the measurements . The small field OF from 3 × 3 cm2 down to 1 × 1 cm2 led to comparable measurement results for the investigated dosimeters (within 0.6% and 1.3% agreement for respectively 6 MV and 10 MV photons). In the Varian Beam Configuration Reference Guide is recommended that you do not need to measure all OF, just type in some values and interpolate the rest. However, we applied our own interpolation procedure using a selfmade matlab script leading to a less coarse result: for 1 × 1 cm2 the OF for 6 MV and 10 MV photon beams is 0.624 and 0.706 respectively, compared to the average measurement value of 0.650 and 0.712, which gives a deviation of 5% and 0.9%. From these findings, we estimate an inaccuracy on the OF measurements for 1 × 1 cm2 of maximum 5%.The Eclipse TPS was originally designed for 3D conventional broad photon beam radiotherapy treatments. Details of the most important incremental improvements from the point of view of accurate dose calculation for small field sizes are presented in Torsti et al.The AcurosXB algorithm uses the multiple source modeln = 28) were chosen with various lesion sites: lung, brain, prostate and spine. All were characterized by rather small to very small target volumes (PTV between 96.5 and 0.54 cm3) and fraction doses between 2.75–30 Gy. The clinical treatment plans were projected on the artificial homogeneous phantom CT data set provided with the Octavius 4D system and recalculated in the TPS. In the TPS, the electron density of the Octavius 4D phantom relative to water has been set to 1.016 according to the manufacturer's recommendation.Clinical patients (2.C® (RA) , which is based on an inverse planning method to deliver an intensity modulated dose distribution using a MLC of which the movement is modulated to the target volume while the beam is on and the gantry moves around the patient. This MLC movement can be very complex, containing many small field segments.The treatment technique used was VMAT with RapidArcA TrueBeamSTx linac equipped with a High Definition (HD) 120‐MLC with inner leaf width of 2.5 mm  was used. This type of linac utilizes the JTT which keeps the collimator jaws during dose delivery by VMAT as close as possible to the MLC aperture, minimizing leakage and transmission through the MLC leaves.2.D2 and applying the MLC to shape the smallest apertures was investigated for static open squared fields of 3 × 3, 2 × 2, 1 × 1, 0.5 × 0.5 cm2 delivering the same amount of MU from the cross‐calibration field in an isocentric set‐up of Octavius 4D . To investigate in a static fields set‐up the differences in output for fields either created by the collimator jaws only (FSjaws) and by the MLC with jaws at a fixed 3 × 3cm2 position (FSMLC), a comparison between the 1000SRS central chambers measured dose output (Gy) and calculated by the TPS is made for both photon beam energies and for these two set‐up situations. Furthermore, the impact of JTT has been investigated for seven stereotactic patients with small brain metastases by comparing the measured and calculated dose distributions with JTT on the one hand and with collimator jaws fixed at 3 × 3 cm2 while the MLC is used to block the smaller parts on the other hand.All data were gathered in 6 MV and 10 MV photon beams. By placing the 1000SRS array's active layer in the isocenter, the distances between the ionization chambers in the inner part of the array are thus equal to the projected leaf widths of the HD 120‐MLC at the isocenter. The effect of disabling jaw tracking, thereby fixing the collimator jaws at 3 × 3 cmPrior to the measurements, the jaw position calibration has been verified using radiochromic EBT3 film  and the reproducibility of jaw position calibration with 0.1 mm can be confirmed by the regular quality control procedures following NCS report eight guidelines.2.Emax−TPS Dmax)/TPS Dmax (%) give a comparison between the maximum dose values in the 3D distribution measured by the Octavius 4D system (measured Dmax) and calculated by the TPS (TPS Dmax).To evaluate the dosimetric agreement between the 1000SRS measured and calculated dose by Eclipse, a gamma evaluation method implemented in the Verisoft 6.2 software (PTW) was used2.FSee Appendix 33.An = 28), respectively. An average pass rate of 96.0% with quite a large spread (6.9%) is found for the 10% cut‐off dose value. This may be caused by varying jaw settings in the different treatment plans. These jaw settings are the maximum jaw positions of the fields coming from the Arc Geometry tool in Eclipse, which suggests the geometry for the VMAT fields. Indeed, a trend can be observed for relative differences in the maximum dose as a function of the field size area of VMAT treatments going from very small to medium sized fields /TPS Dmax (%) for the field size areas < 9 cm2 have deviations larger than 4.5% and even up to 16.1% for the smallest fields. For the dose calculations in this figure, a default calculation grid size of 2.5 mm has been used for field size areas > 9 cm2 and a calculation grid size of 1.0 mm for the field size areas ≤ 9 cm2.The errors in the device set‐up reproducibility were minimized by making cone beam computed tomography (CBCT) images of the set‐up before each set of measurements and using the online matching procedure for shifting the corresponding CBCT with the reference planning CT scan. Regarding the VMAT treatment fields, mean ± SD of the volumetric gamma evaluation scores for the dose difference and distance to agreement criteria of 2% and 2 mm with 10%, 50%, 80% and 95% cut‐off dose values were 96.0% ± 6.9%, 95.2% ± 6.8%, 86.7% ± 14.8% and 56.3% ± 42.3% (max−TPS Dmax)/TPS Dmax (%) show the effect of using a smaller grid size: 6.5% to 15% (patient A) and 16.1% to 21.2% (patient B) for respectively grid sizes 1 mm to 2.5 mm.The impact of using a calculation grid size of 1 mm instead of the default setting of 2.5 mm in treatment plans with very small fields was demonstrated for two of the 28 patients, for which large differences between measurements and calculations occur /DTPS MLC) (%) and Δjaws = ((DOct4D jaws−DTPS jaws)/DTPS jaws)(%) describe the deviations between the measured and calculated doses for respectively static MLC defined beams  and the beams collimated by the jaws. The data belonging to FSMLC in contrast to FSjaws jaws−ΔMLC|(%) . 62% of the VMAT plans in this study have 10% or more of their fields smaller than 2 cmA possible explanation to the trend observed in Fig. norm for open or wedged fields [formula Furthermore, if we indulge in how the OF are handled in Eclipse, we have to consider the collimator backscatter factor (CBSF) table Fig. , calcula formula :23(1)CBref is the output for the reference conditions  and is equal to 1. OF is the output factor for the collimator settings in X and Y direction. D′ is the dose at the reference point as calculated by the dose calculation algorithm for collimator setting X,Y. D′ref is the dose at the reference point for the reference conditions mentioned above.With CBSF being the CBSF for the collimator settings in X and Y direction. OFcalib and Dcalib are the number of monitor units and dose under calibration conditions .MUref is the dose for the reference conditions at the calibration depth (dmax).Dnorm is the dose calculated by the TPS at the chosen normalization point for collimator setting X and Y.DWCF is a wedge correction factor for hard field wedges and not applicable in our case (WCF = 1).2, but this “jump” in the curve cannot be due to inaccuracies in the measurement of output factor at 1 × 1 cm2 alone. The differences between the physical back scatter factors and the configured values reveal information about the remaining limitations of the source model and are due to the fact that the CBSF in the source model beam data is a residual correction factor taking into account all phenomena of phantom scatter and head scatter that are not otherwise accounted for by the source model or the dose deposition engine.Obviously, the observed CBSF table shape Fig.  looks quFinally, a shortcoming in the Eclipse TPS version 11 is the fact that TrueBeam cannot be selected as Machine Type in the Parameter View of the Beam Configuration program. Varian advises the customers to use a C‐series instead although some difference in the backscatter factors between C‐Series and TrueBeam machines can be expected.2 up to 40 × 40 cm2. The dosimetric data for the smaller fields can be imported but are not used by the system for beam modeling. Currently, no asymmetrical field sizes nor MLC shaped fields are used in Eclipse when configuring the beam data. Output factors for fields smaller than 3 × 3 cm2, on the other hand, should be measured very accurately with a reliable detector, since these are applied in the dose calculation. Up to now, Varian does not recommend the use of output factors for very small fields in the configuration process.2 as minimum field size,MLC(%) for 6 MV photons in Table 2 jaws−ΔMLC|(%) in Table jaws−ΔMLC|(%) in Table 2 for the field sizes smaller than 3 × 3 cm2 and letting the MLC shape the smallest apertures. However, the 1000SRS measurements for the 0.5 × 0.5 cm2 fields (Fig. To validate the measurements done with the Octavius 4D system concerning the improvement of fixing the jaws, we compared the 1000SRS results for static fields with EBT3 film (see Appendix 52. For very small highly modulated VMAT fields larger discrepancies are obtained demonstrating the need for improvement in dose calculation for small fields. Fixing the jaws at the minimum required field size for beam data input in the TPS, i.e., 3 × 3 cm2, and using the MLC with high positional accuracy to shape the smallest apertures in contrast to jaw tracking is currently found to be the most accurate treatment technique.Doses calculated for stereotactic VMAT plans show an acceptable agreement against measurements with the 1000SRS in the Octavius 4D system for field sizes > 2 × 2 cmA calculation grid size of 1 mm is found to be superior to 2.5 mm.An upper MU constraint should be assigned during optimization to avoid plans with high modulation and very high number of MUs increasing treatment time.2 should be used for small target volumes where the JTT leads to jaw defined field sizes smaller than 3 × 3 cm2.Fixed collimator jaws at 3 × 3 cmTo conclude, our recommendations to minimize the effects of calculation uncertainties in RA plans for very small target sizes:The authors declare no conflict of interest.


SR)‐derived Ca2+ release is required to induce Ca2+ oscillations for contraction and to control cardiac development through Ca2+‐activated pathways. Here, our aim was to study how SR Ca2+ release regulates cytosolic and nuclear Ca2+ distribution and the subsequent effects on the Ca2+‐dependent localization of class IIa histone deacetylases (HDAC) and cardiac‐specific gene expression in embryonic cardiomyocytes. Confocal microscopy was used to study changes in Ca2+‐distribution and localization of immunolabeled HDAC4 and HDAC5 upon changes in SR Ca2+ release in mouse embryonic cardiomyocytes. Dynamics of translocation were also observed with a confocal microscope, using HDAC5‐green fluorescent protein transfected myocytes. Expression of class IIa HDACs in differentiating myocytes and changes in cardiac‐specific gene expression were studied using real‐time quantitative PCR. Inhibition of SR Ca2+ release caused a significant decrease in intranuclear Ca2+ concentration, a rapid nuclear import of HDAC5 and subnuclear redistribution of HDAC4. Endogenous localization of HDAC5 and HDAC4 was mostly cytosolic and at the nuclear periphery, respectively. Downregulated expression of cardiac‐specific genes was also observed upon SR Ca2+ release inhibition. Electrical stimulation of sarcolemmal Ca2+ influx was not sufficient to rescue either the HDAC localization or the gene expression changes. SR Ca2+ release controls subcellular Ca2+ distribution and regulates localization of HDAC4 and HDAC5 in embryonic cardiomyocytes. Changes in SR Ca2+ release also caused changes in expression of the developmental phase‐specific genes, which may be due to the changes in HDAC‐localization.In embryonic cardiomyocytes, sarcoplasmic reticulum ( Secondly, we aimed to study how changes in Ca2+ distribution affect the Ca2+‐dependent nuclear signaling that can lead to changes in class IIa HDAC localization and cardiac‐specific gene expression in embryonic cardiomyocytes.This study aimed to determine if SR Ca2+ mobilization and immunolabeling experiments performed at the University of Eastern Finland, using the C57BL/6JOlaHsd strain. In these experiments, where two different mouse strains were used, the results were comparable between the mouse lines. All experimental designs were approved by the Animal Use and Care Committee of the University of Oulu and the National Animal Experiment Board in Finland.The method for cell isolation and culturing of the E9‐11 embryonic cardiomyocytes has been described previously  on an imaging plane where all regions of interest  were clearly visible. The cells were perfused with culture medium containing a pharmacological inhibitor and simultaneously stimulated electrically  and line‐scanned periodically to observe changes in subcellular Ca2+ signals. Pharmacological interventions used were thapsigargin , ryanodine  and Ca2+ ionophore, A23187 . As we were interested on the acute effect of these pharmacological interventions, the cells were imaged with the treatment until Ca2+ oscillations stopped or changes in Ca2+ distribution could be clearly observed (≤20 min). The line scan images were analyzed with the FluoView 1000  and Origin 2016  programs. Fluo‐4 fluorescence intensity is expressed as an F/F0 ‐ratio, where F is the background subtracted fluorescence intensity and F0 is background subtracted minimum fluorescence value measured from each region at rest at control conditions. Fluorescent intensity in the three different intracellular locations was normalized to its own F0 value. Changes in Ca2+ signals were compared within each subcellular localization (control vs. treatment). No comparison between the localizations was conducted as this would have required calibration of the fluorescent dye at each subcellular localization.Caμmol/L A23187, 50 μmol/L ryanodine or 10 μmol/L thapsigargin) and then immediately fixed and labeled, using a method described previously  with or without pharmacological intervention . Here, after background subtraction, fluorescence measured in representative cytosolic and nuclear regions was used to calculate the n/c ratio. HDAC4 on the other hand was mainly nuclear, therefore the fluorescent signal was measured from the nuclear periphery and the center of the nucleus. These values were then normalized to the background subtracted cytosolic fluorescence.μL Opti‐MEM (Invitrogen) containing 0.2 μg HDAC5‐GFP‐plasmid and 2 μL Lipofectamine 2000 (Invitrogen). After a 5‐h incubation at 37°C, the medium was changed to Dulbecco's Modified Eagle Medium (DMEM) containing 10% fetal bovine serum  and 1% penicillin‐streptomycin  and the cells were cultured for additional 18–24 h to allow expression of the chimeric protein.After culturing for 18–24 h, E9‐11 cardiomyocytes were transfected with a plasmid containing HDAC5‐GFP , using Lipofectamine 2000 (Invitrogen) according to the manufacturer's instructions. Briefly, cells cultured on glass bottom culturing dishes were transfected with 100 2/5% CO2). Electrical stimulation (0.5 Hz) was maintained throughout the experiments with two platinum electrodes placed on opposing sides of the plate. The change in cytosolic versus nuclear fluorescence with pharmacological interventions (50 μmol/L ryanodine or 10 μmol/L thapsigargin) was followed over time and results expressed similarly to the immunolabeled HDAC5, as a n/c – ratio.A confocal inverted microscope  was used to take images of the HDAC5‐GFP with excitation at 488 nm and emission collected at 500–600 nm through a 60x objective lens. A steady temperature of 34°C was maintained by continuous superfusion with preheated culturing medium . When comparing the effect of SR Ca2+ release inhibition to cardiac‐specific gene expression, the results were further normalized to the control in each group. The primers and fluorescent probes are listed in Table Tissue samples (ventricles) from mouse embryos at different developmental stages  were used in these experiments. RNA extraction, cDNA synthesis and RT‐qPCR were performed as previously described  caused the SR‐derived Ca2+ oscillations to stop, similarly as has been shown by previous studies  from the SR. Here, stimulation was possible for a short amount of time and Ca2+ oscillations stopped as soon as the SR was depleted.In spontaneously active embryonic cardiomyocytes strong Casigargin 0 μmol/L 2+ levels both in the cytosolic region near the SR and within the nucleus  or thapsigargin (10 μmol/L), resulted in strong HDAC5 nuclear import and subnuclear redistribution of HDAC4 toward the nuclear periphery , to enable strong Ca2+ influx. This caused a rapid increase in [Ca2+] throughout the cell, with the most pronounced increase observed within the nucleus  and depletion of the SR Ca2+ stores  both caused a strong and relatively rapid nuclear accumulation of HDAC5‐GFP  and Calr were observed. Casq expression was downregulated, whereas Calr was upregulated upon SR Ca2+ release inhibition. Due to the developmental differences in Calr and Casq expression, where the latter takes over the role as the major Ca2+ buffering protein as the muscle matures . Significant changes in HDAC4 and HDAC5 intracellular distribution and cardiac‐specific gene expression were observed when SR Ca2+ release was inhibited.In this study, we show that in embryonic cardiomyocytes SR Ca2+ signaling has an essential role in the development of the early cardiac tissue and the first asynchronous Ca2+ oscillations can be observed already at the early stages of cardiac crescent development. At this stage, no sarcomeric banding or myocyte contraction can be observed , we observed strong changes in Ca2+ distribution with the diminishing Ca2+ oscillations.In embryonic cardiomyocytes, Ca2+ release‐mediated changes in [Ca2+]nuc can regulate expression of the cardiac‐specific TFs Nkx2.5, GATA4 and MEF2C  showed that HDAC5 can facilitate stabilization and nuclear accumulation of the hypoxia‐inducible factor 1α (HIF‐1α). This action was accomplished by cytosolic deacetylation of the heat shock protein 70 (Hsp70) and subsequent stimulation of a pathway that led to nuclear accumulation of HIF‐1α. This is a very interesting finding because hypoxia has been found to stimulate multiple different developmental pathways in the developing embryo (Chen et al. During the early developmental stages, strict temporal regulation of developmental‐specific gene expression is necessary for the tissue development to ensue in a correct manner. Therefore, a dynamic regulation of class IIa HDAC localization is to be expected, as the genes need to be repressed fast if, for example, activity of the cell changes. Our results do indeed show this, as uniform intracellular distribution of HDAC5 can be changed to mostly nuclear rapidly, due to the ability of differentiating myocytes to rapidly modify their [Ca2+ release can initiate spontaneous cardiomyocyte contraction and create strong Ca2+ gradients across the cell (Korhonen et al. 2+ homeostasis can have a significant effect on [Ca2+]nuc, and thereby on various Ca2+‐activated pathways, some of which can control the subcellular localization of HDAC4 and HDAC5. Furthermore, SR Ca2+ release was shown to have a significant role in regulation of cardiac‐specific gene expression. These results provide a potential link between SR Ca2+ release and cardiac development, thereby also shedding light on the embryonic lethal knock‐out models that disrupt SR Ca2+ release.In the embryonic cardiomyocytes, SR CaNone declared.


We analyze the definition of the Gibbs free energy of a nanoparticle in a reactive fluid environment, and propose an approach for predicting the size of carbon nanoparticles produced by the detonation of carbon-rich explosives that regards their condensation as a nucleation process and takes into account absolute entropy effects of the cluster population. The results are consistent with experimental observations and indicate that such entropy considerations are important for determining chemical equilibrium states in energetic materials that contain an excess of carbon. The analysis may be useful for other applications that deal with the nucleation of nanoparticles under reactive conditions. Their successful comparison of calculations based on chemical equilibrium modeling25The appearance of the condensed carbon phase in the detonation products of explosives poses a challenge for the canonical theory of the plane wave steady detonation process9n atoms, which we denote by μn); we will assume in the following that these clusters can be approximated as spherical.  Viecelli et al.μn) that is derived using standard assumptions on the applicability of the Laplace law pn) in the mother phase. Thus μn)( contains the standard surface energy contribution in the capillary approximation (where the planar surface tension is used for the properties of the cluster), along with the ideal mixing entropy39404038n immersed in a fluid matrixwhere Nn the number of clusters of size n, V the system volume, T the temperature and m the carbon atomic mass. In the case of detonation the fluid matrix is a reacting mixture, which at the C-J point should reach chemical equilibrium both within itself, and with the condensed carbon phase. If we denote by μC the chemical potential of free carbon atoms in the fluid, chemical equilibrium between the mixture and the carbon clusters requireswith C + CO2 = 2CO, CO + H2O = CO2 + H2, etc., with corresponding chemical equilibrium equations n carbon atoms. We now assume that individual carbon clusters are also in (unstable) chemical equilibrium with respect to evaporation and condensation of single carbon atoms, i.e. that they are critical nuclei. The applicable classical Gibbs condition35Chemistry in the fluid also involves reactions such as 25ρ is the system mass density, fc is the mass fraction of condensed carbon and nwhere C7H5N3O6) and 60% RDX (cyclotrimethylenetrinitramine – C3H6N6O6) that is the explosive of choice for producing nanodiamonds4C14H6N6O12), TATB (triaminotrinitrobenzene – C6H6N6O6), and BTF (benzotrifuroxan – C6N6O6). The calculations require the density ρ and temperature T of the C-J state of these materials, along with the mass fraction of the condensed carbon, fc. In principle these can be approximately determined using chemical equilibrium modeling of the detonation products25ρ and T, including calculations and experiments145464748495051fc by half of the excess carbon over the oxygen balanced stoichiometry, e.g., C7H5N3O6(TNT) → 1.5N2 + 2.5H2O + 1.75CO2 + 5.25C, fc ≃ 0.14. This is in fact roughly the amount of condensed carbon recovered in enclosed detonations853ρ and fc, due to their logarithmic contribution to 46474849505152K–4000 K for COMP-B, 2800 K–3800 K for TNT, 3100 K–4000 K for HNS, 1900 K–3000 K for TATB, and 4100 K–5700 K for BTF. We also estimate that changes of order 20% in the value of the surface energy coefficient α result in cluster size variations of approximately 10%. Independent calculations (using nm)  see  for the d, g, l} stands for diamond, graphite and liquid, μini).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Realizing controlled growth of Li2O2 is a prerequisite for high performance of Li–O2 batteries. In this work, a sandwich‐structured catalytic cathode is designed: graphene/Au‐nanoparticles/Au‐nanosheets (G/Au‐NP/Au‐NS) that enables controlled growth of Li2O2 spatially and structurally. It is found that thin‐layer Li2O2 (below 10 nm) can grow conformally on the surface of Au NPs confined in between graphene and Au NSs. This unique crystalline behavior of Li2O2 effectively relieves or defers the electrode deactivation with Li2O2 accumulation and largely reduces the contact of Li2O2 with graphene and electrolyte. As a result, Li–O2 batteries with the G/Au‐NP/Au‐NS cathode exhibit superior electrochemical performance. A stable cycling of battery can last 300 times at 400 mA g−1 when the capacity is limited at 500 mAh g−1. This work provides a practical design of catalytic cathodes capable of controlling Li2O2 growth.The working of nonaqueous Li–O Although vehicles have now been powered by Li‐ion batteries (LIBs), the driving range of electric vehicles, however, is limited by low energy density of current LIBs.2 batteries could be obviously reduced by using a bifunctional Pt/Au catalyst, where Pt and Au catalyze OER and ORR, respectively.2 battery can retain 95% of its capacity after 100 cycles with a low‐charge overpotential by using a nanoporous gold cathode.2 batteries with noble metal catalysts, such as Pd,2O2 or LiO2,2O2 or LiO2.It is widely accepted that noble metals have the best electrocatalytic activity for ORR and OER in organic systems. The work by Lu et al. showed that both charge and discharge overpotentials of Li–O2O2 or LiO2. One of the effective methods to alleviate the side reactions is to prepare carbon and/or binder‐free electrodes.x/TiN nanotube arrays cathode, which exhibited an excellent cycling stability over 300 cycles.2O2 or LiO2 over carbon has proven to be an effective measure to lessen carbon‐induced negative effects.Many strategies have been proposed to overcome the above problems related to the reactive Li2O2 only on the surface of Au particles; (3) Au NSs fix the Au particles and encapsulate Li2O2‐loaded Au NPs between the sandwich frame. This unique electrode design effectively alleviates or defers the deactivation of the electrode and reduces the contact of Li2O2 (or LiO2) with graphene and electrolyte. As a result, Li–O2 batteries with the G/Au‐NP/Au‐NS cathode can sustain 300 cycles under 400 mA g−1 at a limited capacity of 500 mAh g−1. The Li–O2 battery can also sustain 100 cycles at a full charge/discharge mode in the cutoff voltage of 2–4.5 V. This work sheds light on the design of efficient catalytic cathodes enabling controlled Li2O2 growth aiming at high‐performance Li–O2 batteries.In this work, we provide a unique design of binder‐free catalytic cathode which was prepared in ice bath. In this cathode, Au nanoparticles (Au‐NP) are sandwiched between few‐layer graphene (G) and thin Au nanosheets (Au‐NS), forming a G/Au‐NP/Au‐NS sandwich frame. The merits of the electrode design include: (1) graphene provides the electronically conducting channels for ORR and OER; (2) Au NPs catalyze the confined/conformal growth of Li2Figure2O2 can realize the confined/conformal growth within the electrode, namely, only on the surface of Au NPs.FigureFigure2 batteries with the G/Au‐NP/Au‐NS cathode at a cutoff voltage of 2–4.5 V under 400 mA g−1 (0.24 mA cm−2). A high discharge capacity of 3347 mAh g−1 is obtained with a flat discharge plateau. By contrast, Li–O2 battery with bare graphene delivers a discharge capacity of only 89 mAh g−1 at a smaller current density of 200 mA g−1 , indicating that graphene itself contributes minor to the catalytic activity of G/Au‐NP/Au‐NS, and thus can only be considered as the conducting support for ORR/OER. Therefore, the current density and specific capacity of the batteries were calculated based on the weight of Au. Figure 2 battery at a limited capacity of 500 mAh g−1. The capacity of 500 mAh g−1 can be maintained over 300 cycles at 400 mA g−1. In addition, in most of these cycles, the terminal discharge voltage is over 2.5 V, indicative of a low electrode polarization with Li2O2 deposition. The OER potentials are somewhat higher than expected which may be due to the relatively high current density used and sluggish transport rate of Li ions and oxygen in the presence of the Au film. The batteries were also tested at lower current densities of 100 and 200 mA g−1 . As expected, the OER potentials can be obviously reduced at lower current densities. A stable cycling can last 170 times when the capacity was limited at 1000 mAh g−1. A high discharge voltage of over 2.5 V is also observed for the first 120 cycles at 1000 mAh g−1, suggesting that an increased Li2O2 deposition does not cause increased electrode polarization due to the unique electrode design. By contrast, the stable cycling of Li–O2 battery with the G/Au‐NP cathode could last only 28 times .Figure2 battery with the G/Au‐NP/Au‐NS cathode in a cutoff voltage of 2–4.5 V under 800 mA g−1 (0.48 mA cm−2). Note that in this rigorous cycling mode, the Li–O2 battery can still exhibit a stable cycling. After 100 cycles, a capacity over 500 mAh g−1 is retained. The coulombic efficiency is close to 100% during cycling which suggests the reversible growth/decomposition of Li2O2. This means that the side reactions related to electrolyteRe denotes ohm resistance of the battery components, Rf and Q1 correspond to solid‐state‐electrolyte resistance and relax capacitance, Rct and Q2 represent charge transfer resistance and double layer capacitance, and Zw is related to the bulk diffusion of Li ions.Rct is not significant after discharge, implying that Li2O2 deposition does not cause obvious electrode deactivation, agreeing well with the electrochemical tests. The reduction of Rct after recharge indicates the sufficient decomposition of Li2O2.Figure2O2 particles formed. Besides, the Au NSs are visible after discharge, indicating the structural integrity of the G/Au‐NP/Au‐NS electrode. The transparent nature of the Au NSs implies that both interior and exterior surfaces of the Au NSs are Li2O2 free. In contrast, for the G/Au‐NP electrode, large Li2O2 particles or particles aggregations form after discharge . TEM images in Figure 2O2 particles grew on the surface of Au NSs. HRTEM images in Figure 2O2 layer with a thickness below 10 nm. Li 1s and O 1s XPS  indicate that the dominant discharge product is Li2O2 although a small amount of Li2CO3 also forms. The formation of Li2CO3 can be attributed to the decomposition of electrolyte. Note that repeated cycling does not lead to the accumulation of Li2CO3 obviously as seen in Figure S8 (Supporting Information), which can explain the good cycling performance of the battery. As seen in Figure S9 (Supporting Information), the battery with G/Au‐NP/Au‐NS electrode can sustain 245 cycles at 100 mA g−1, corresponding to a long working period of up to 102 d, indicating that increasing working time does not lead to obvious Li2CO3 accumulation and that the electrolyte is relatively stable during cycling. It should be noted that although Li2O2 and Li2CO3 can be qualitatively detected by XPS, the quantitative information regarding the efficiency of Li2O2 formation needs more efficient analysis tool such as differential electrochemical mass spectrometry system.2O2 can grow along the surface profiles of the Au NPs, which clearly indicates that Au NPs do catalyze the confined and conformal growth of Li2O2. This growth behavior of Li2O2 is favorable considering the facts that the contact between Li2O2 (or LiO2) with graphene can be minimized , and that the volume expansion of electrode with Li2O2 accumulation can be largely avoided. In addition, Li2O2 with a thin‐layer structure can be easily decomposed upon recharge.To further clarify the superior catalytic performance of the G/Au‐NP/Au‐NS electrode, SEM and TEM observations were conducted on the discharged electrodes. As shown 2O2 growth may be due to the insufficient electron transfer since the Au NPs that bridge graphene with Au NSs are covered with insulating Li2O2. In this case, the Au NSs may act as the separator to reduce the contact of Li2O2 (or LiO2) with electrolyte, and as the fixer to stabilize the G/Au‐NP electrode. This can explain the considerably improved cycling stability of the G/Au‐NP/Au‐NS‐catalyzed Li–O2 battery compared with the G/Au‐NP‐catalyzed battery. SEM images in Figure2O2 upon recharge can be confirmed by Li 1s and O 1s XPS . In contrast, Li2CO3 is remained after charge, suggesting that the decomposition of Li2CO3 is difficult at the applied voltage . It was also noticed that the ether peak at around 286. 3 eV appears after the first discharge, and its intensity increases with cycling, suggesting increased amount of decomposition products although the tetraethylene glycol dimethyl ether (TEGDME) electrolyte is relatively stable. Similar result was found in other work using TEGDME as electrolyte.2O2. The results indicate that controlled growth of Li2O2 can be achieved by optimizing catalytic electrode, and that the architecture and component are of equal importance in order to realize high performance of Li–O2 batteries.In the G/Au‐NP/Au‐NS electrode, graphene participates in the catalytic reactions by providing the conducting channels,32O2 realizes the conformal growth and crystallizes into thin‐layer form on the surface of Au NPs, graphene supplies the conducting channels for ORR/OER, and Au NSs stabilizes the G/Au‐NP structure and reduces the contact of Li2O2 (or LiO2) with electrolyte. The unique architecture of the G/Au‐NP/Au‐NS electrode enables controlled Li2O2 growth, easy Li2O2 decomposition, and reduced side reactions, leading to excellent electrochemical properties of Li–O2 batteries. At a limited capacity of 500 mAh g−1, the battery can sustain a stable cycling over 300 times at 400 mA g−1. The Li–O2 battery can maintain a capacity of 500 mAh g−1 after 100 cycles at 800 mA g−1 when it was tested in a full charge/discharge mode in the cutoff voltage of 2–4.5 V. The results show that the controlled growth of Li2O2 can be realized by optimizing the electrode design and that both architecture and component of the electrodes are important to achieve high performance of Li–O2 batteries.In summary, a highly efficient catalytic cathode of G/Au‐NP/Au‐NS was prepared by a facile impregnation approach in ice bath. In this electrode, Li4Electrodes Preparation and Characterization: Graphene was deposited on Ni foam by a CVD method as described previously.−2. The Ni‐supported G/Au‐NP/Au‐NS electrodes were prepared by a solution impregnation method. Briefly, the graphene‐loaded Ni foam pieces were immersed in a beaker containing HAuCl4·3H2O aqueous solution (0.16 mg mL−1) and the beaker was placed in an ice bath for 3 h. The Ni‐supported G/Au‐NP/Au‐NS electrodes were then rinsed with distilled water, dried at 60 °C for 10 h in vacuum and heated at 300 °C for 2 h in argon. For comparison, the Ni‐supported G/Au‐NP electrodes were also prepared by a similar route with the impregnation step performed at room temperature. The loading of Αu on graphene is 0.5–0.6 mg cm−2. The materials deposited on Ni foam were checked by XRD on a Rigaku D/Max‐2550pc diffractometer with Cu Kα radiation (λ = 1.541 Å). The structure of graphene on Ni was analyzed by Raman spectrum on a Jobin‐Yvon Labor Raman HR‐800 system with 514.5 nm Ar‐ion laser. The morphologies of the pristine, discharged and recharged electrodes were observed by field‐emission SEM on an S‐4800 microscope. The microstructure of electrode components exfoliated from the electrodes was characterized by TEM on a JEM 2100F microscope. The chemical states of the elements in the electrodes were examined by XPS on a Kratos Axis Ultra‐DLD spectrometer with Al Kα radiation (hν = 1486.6 eV). The electrodes after cycling were carefully handled before the XPS measurements. First, the batteries after charge or discharge were dissembled in the argon‐filled glove box. The electrodes were then washed by 1,2‐dimethoxyethane (DME) three times to remove TEGDME and LiClO4 sufficiently on the electrode surface. The DME was then removed by resting the electrodes in the small chamber of the glove box under evacuation for 15 min. After that, the electrodes were fixed on the XPS holder and sealed in a plastics box under Ar atmosphere in the glove box. For XPS tests, the electrodes were taken out from the box and transferred to the XPS chamber as quickly as possible (within 2 min). The XPS chamber was then quickly evacuated.2 Batteries Assembly and Electrochemical MeasurementsLi–O: Coin‐type Li–O2 batteries were assembled in an argon‐filled glove box using Li foils as anodes, Ni‐supported G/Au‐NP/Au‐NS  as cathodes (0.36 cm2), Celgard C480 porous films as separators, and 1 m LiClO4 in TEGDME  as electrolyte. The cathodes were dried at 80 °C under vacuum for 10 h prior to assembly of the batteries. The assembled batteries were then purged with O2 for 10 min and stayed at open voltage circuit for 5 h to achieve an equilibrium. Galvanostatic cycling was conducted on a Neware battery cycler  over a voltage range of 2.0–4.5 V (vs Li/Li+). The specific capacity (mAh g−1) and current density (mA g−1) of the G/Au‐NP/Au‐NS and G/Au‐NP electrodes were calculated based on the weight of Au. The discharge and charge are referred to lithiation and delithiation, respectively. EIS was recorded on a VersaSTAT3 electrochemistry workstation (Princeton Applied Research) by applying an ac voltage of 10 mV amplitude in a frequency range from 10−2 to 105 Hz. The electrochemical tests were all carried out at room temperature.As a service to our authors and readers, this journal provides supporting information supplied by the authors. Such materials are peer reviewed and may be re‐organized for online delivery, but are not copy‐edited or typeset. Technical support issues arising from supporting information (other than missing files) should be addressed to the authors.SupplementaryClick here for additional data file.


In 1, the large 1D channel in frameworks and window size in the nanocages can enrich methylene blue and capture CO2, exhibiting the promising applications in environmental protection. More importantly, the explorations on the cycloaddition reaction of CO2 and aziridines with various substituents suggest that 1 can serve as an efficient heterogeneous catalyst for CO2 conversion with aziridines in a solvent‐free system, which can be reused at least ten times without any obvious loss in catalytic activity. This is the first example of metal–organic framework (MOF)‐based catalysts in converting CO2 into high‐value oxazolidinones through activating aziridines and CO2, further extending the applications of MOFs materials in catalysis.Based on a novel ligand 5‐pyridin‐4‐yl)isophthalic acid (H Carbon dioxide CO, as the 2 adsorption and excellent recyclable catalytic performance.2 conversion therefore has received tremendous attention, and several MOFs have already shown high catalytic activity in CO2 conversion with the terminal alkyne activation, the hydroboration and cycloaddition of epoxides.2 conversion with aziridines still have not been reported so far.On the other hand, porous metal–organic frameworks (MOFs) can be considered as a promising candidate for the catalytic reaction as they have designable structures and modifiable functional groups, well‐defined pores and high surface area, significant capability for CO4BCP, Scheme S1, Supporting Information). Then, a unique 3D framework {[Cu2(BCP)(H2O)2]·3DMF}n (1) assembled by [Cu30] nanocages was obtained and structurally characterized. Importantly, in the cycloaddition reaction of CO2 and aziridines, 1 can act as a highly effective and recoverable catalyst, which can be reused at least ten times without any obvious loss. To our knowledge, it represents the first example of MOFs‐based catalyst for CO2 conversion with aziridines.In this contribution, we designed and synthesized a novel low symmetry organic ligand 5‐pyridin‐4‐yl)isophthalic acid . Based on H4BCP ligand, the green crystals of 1 were obtained by solvothermal methods, and the structural analyses revealed that compound 1 belongs to the trigonal system with space group R3m . The 3D framework of 1 is assembled by the unique nano‐sized [Cu30] cages 4 paddlewheel units and BCP linkers. The shape of [Cu30] cage looks like a distorted censer , which is distinctive to the reported polyhedral/spherical cages in classic MOFs, for instance, eight [Zn4O] units as vertexes construct a cubic cage in MOF‐5, 24 Zn atoms and 48 methylimidazole linkers form a spherical cage in ZIF‐8, etc.30] cage can be divided into two subcages, big cage A and small cage B 4 paddlewheel units and six BCP4− anions, displaying a distorted hexagonal bipyramid. Both top and bottom in cage A are occupied by three Cu2(O2CR)4 paddlewheel units with the window of 8.1 and 16 Å , respectively. The middle part of cage A contains six Cu2(O2CR)4 paddlewheel units with apertures of ≈21 Å . The cage B shares with three Cu2(O2CR)4 paddlewheel units from the bottom of cage A, and six BCP linkers coordinate to another three Cu2(O2CR)4 units into a hexagonal bipyramid with a pore size of 21 Å . Along the b‐axis and c‐axis directions, compound 1 exhibits irregular 1D channels, and the total potential pore volume is as high as 72.1% calculated by the PLATON program.Large organic ligand H2(O2CR)4, linking with four BCP4−, can be considered as a 4‐connected node; and each BCP4− anion bridges four Cu2(O2CR)4 unit, thus it also can be regarded as a 4‐connected node. On the basis of this simplification, the resulting network of compound 1 can be described as a ‐connected framework with the nbo‐type topology.From the topological point of view, each paddlewheel unit of Cu30] cage, the exploration on the sorption of methylene blue (MB) was carried out. First, 20 mg compound 1 was immersed into the MB solution, and the concentration of MB was monitored by UV–vis spectrophotometer. The results revealed that the concentration of MB gradually decreased as the increased time, and MB in the solution was almost completely absorbed after 7 h with the color change from dark blue to colorless , being indicative of that porous framework keeps intact until about 270 °C and possesses the capacity of capturing CO2. Therefore, porous material 1 with open Cu2+ as Lewis acidic sites displays the promising application for the coupling reaction of CO2 with aziridines.Considering the large window size and cavity in  cages in 1 can enrich or capture CO2 and substrate molecules, i.e., aziridines, and the increased concentration of reactants can accelerate the reaction; (b) the confinement effect of nano‐sized [Cu30] cages can enhance reactivity with aziridines and CO2; (c) in the stable Cu2(O2CR)4 paddlewheel unit, the steric hindrance around Cu2+ and electric neutrality of the unit effectively impede the coordination between Cu2+ and Br−, resulting in more nucleophilic attack of Br− on aziridines. Namely, in effectively catalyzing the cycloaddition reaction with aziridines and CO2, both the unique coordination environments of Cu2+ and cage‐like motif in [Cu30] nanocages play a vital role.Next step, to investigate the reaction mechanism, the different copper sources were employed in the CO2 with aziridines is proposed. First, the substrate and CO2 are enriched in the nano‐sized [Cu30] cages. Then, the coupling reaction is initiated by coordination of copper site in compound 1 with the nitrogen atom of the aziridine. Second, the nucleophilic attack of Br− can lead to the ring opening of the aziridine through two different pathways as represented by path (a) or (b) . The Cu2(O2CR)4 paddlewheel units in 1 can stabilize the obtained carbamate salt, which is converted into the corresponding oxazolidinone through intramolecular ring‐closure. In terms of selectivity, the main product 2a originates from ring‐opening of the aziridine at the most substituted carbon, which produced the more stable carbamate salt intermediate. So in most cases, the regioselectivity  exceeds 90:10. Thus, the corresponding oxazolidinone is obtained with high yield and selectivity under 1 as the catalyst.Therefore, a possible mechanism for the 1/TBAB‐catalyzed cycloaddition of CO31 with 30‐nuclear copper nanocages was synthesized and structurally characterized, possessing the capacity of capturing MB and CO2, which shows the promising applications in environmental protection field. Importantly, compound 1 was demonstrated as a new type of heterogeneous catalyst for CO2 fixation, acting as an efficient, recyclable, and environmentally friendly material to catalyze the cyclization reaction with aziridines and CO2 in a solvent‐free system. It should be noticed that robust catalyst 1 can be reused at least ten times without any obvious loss in catalytic activity. To our knowledge, it is the first example of MOF‐based catalysts for converting CO2 into oxazolidinones through activating aziridines, which expands the potential applications of MOFs in areas of catalysis and the reduction of CO2.In conclusion, a unique porous framework 42(BCP)(H2O)2]·3DMF}n (Synthesis of {[CCDC 1442640  should be addressed to the authors.SupplementaryClick here for additional data file.


This article contains the experimental data for the characterization of ∑3 boundaries in a cold deformed and annealed pure iron. Mentioned data are relevant to the research article “ The inter-connections of ∑3 boundaries in pure iron”  [1]. The characterization of ∑3 boundaries was performed through an integrated method including electron backscatter diffraction, stereology based five parameter analysis and crystallographic analysis. This method as formulated can determine the inter-connections of any type grain boundaries with fixed misorientation. Specifications TableValue of the data•The integrated method for the characterization of ∑3 boundaries provides the information of grain boundary inter-connections (GBIC) in terms of statistical significance.•Compared to the conventionally used parameters such as grain to grain misorientation and GBPO, GBIC is a more appropriate approach, describing more accurately the grain boundary character in grain boundary engineering (GBE) research.•Further investigations into GBIC for varied types of grain boundaries will be very significant to the GBE research and applications for body-centered cubic (BCC) metals and high stacking fault energy (SFE) face-centered cubic (FCC) metals.•GBIC and its distributions are structural necessities for the microstructure evolution, and GBIC research will provide some information on grain growth and texture development.1The data reported include information about the inter-connections of ∑3 boundaries in a pure iron (99.9 wt.%) annealed at 620 °C for 15 min in vacuum after a multi-directional forging with a true strain of 4. The inter-connections include (−3 2 1)/(−2 −1 3), (−3 1 2)/(−1 −2 3), (−2 −1 3)/(1 −3 2) and (−1 −2 3)/(2 −3 1) a.The dat2In order to make a comparison with that of a high purity iron 2.1A FEI Nova nano 450 field emission scanning electron microscope (FE-SEM), coupled with an Oxford Aztec electron backscatter diffraction (EBSD) analyzer were used for raw data collection. The EBSD parameters were set to an acceleration voltage of 20 kV, spot size of 4.5, working distance of 9 mm, step size of 2–4 μm and frame size of 1000 μm×800 μm for each mapping. After data collection and data processing, the two-dimensional maps of the orientation imaging microscopy (OIM) were obtained for each of the EBSD mapping and the information of each grain boundary segment was extracted accordingly.2.2A statistical analysis of GBPO concerning the ∑3 boundaries was performed by using the method of texture-corrected five parameter analysis (FPA) 2.31 k1 l1} and {h2 k2 l2} are the Miller indexes of the inter-connected two planes.Based on the GBPO of ∑3 boundaries with the misorientation of 60°/ [5]


Such lattices acquire periodicity only for rotation angles cos θ = a/c, sin θ = b/c, set by Pythagorean triples of natural numbers . While linear eigenmodes supported by lattices associated with Pythagorean triples are always extended, composite patterns generated for intermediate rotation angles allow observation of the localization-delocalization transition of eigenmodes upon modification of the relative strength of two sublattices forming the composite pattern. Sharp delocalization of supported modes for certain θ values can be used for visualization of Pythagorean triples. The effects predicted here are general and also take place in composite structures generated by two rotated hexagonal lattices.We introduce a composite optical lattice created by two mutually rotated square patterns and allowing observation of continuous transformation between incommensurate and completely periodic structures upon variation of the rotation angle  The formation and evolution of localized excitations in inhomogeneous systems, governed by the Schrödinger equation231011131415161718The fundamental relevance of aperiodic structures featuring long-range order became obvious after discovery of quasi-crystals2125The phenomenon of LDT was predicted upon analysis of the tight-binding model of incommensurable potentials14151617182811123123333435θ. By changing θ the potential can be continuously transformed from periodic to aperiodic geometries and vice versa, without any change to its rotational point symmetry. The restoration of periodicity may occur at an infinite set of the rotation angles given by Pythagorean triples, at which the linear modes turn into Bloch waves. For hexagonal sublattices, we uncover new triples leading to periodicity restoration. Using such potentials we establish the existence of previously elusive 2D LDT. Furthermore, in our case LDT occurs not only upon variation of the relative depth of two sublattices, but also upon variation of the rotation angle. We show how the restoration of periodicity affects the thresholds for formation of self-sustained solitary waves previously studied only in quasi-crystals with predetermined symmetry23373839404142434546474823505152In this paper we introduce aperiodic potentials built as a superposition pattern of two identical periodic sublattices  mutually-rotated by an angle a, b, c) satisfying the condition a2 + b2 = c2 and setting lengths of catheti and hypotenuses of a right (Pythagorean) triangle. There are 16 primitive Pythagorean triples with c < 100, including , , , etc. These are directly connected with the transition between the fully periodic and aperiodic geometries introduced here. For example, consider a potential V(r) [hereafter r =  is a two-dimensional (2D) transverse position vector with η and ζ being spatial coordinates] created by two basic square lattices with equal periods that are mutually rotated by an angle of θ:Incommensurability is at the heart of construction of aperiodic potentials. It is also one of the most important objects in number theory, which has been in use since the time of the ancient GreeksV1(r) = p1[cos(2η) + cos(2ζ)] is one of the sub-lattices, and p1 and p2 are the sublattice depths. The potential (1) is aperiodic for all values of θ except when Pythagorean triangles are formed, i.e. when cos θ = a/c and sin θ = b/c, where  is a Pythagorean triple. In this last case the periodicity of potential is restoredV(r) exhibits an aperiodic structure with long-range order  particularly convenient to study the occurrence of LDT phenomena.where N plane waves, with N being odd and selected such that the pattern in principle cannot be periodic. In our case we use two sets of four plane waves that are mutually-rotated. Their superposition can give rise to a periodic distribution for certain rotation angles and to aperiodic distributions for all other angles  is governed by the dimensionless Schrödinger equationLet us now consider a Pythagorean lattice as an optical potential for light propagation. In a paraxial approximation, a light beam with amplitude σ = −1 (σ = +1), and becomes linear at σ = 0. Here ∇ ≡  and ξ is the propagation distance.which accounts for focusing (defocusing) Kerr nonlinearity at σ = 0) are searched for in the form q = weiβξ, where the function w describes the mode profile and β is its propagation constant. The degree of transverse localization of the modal field can be characterized by the integral form-factor U = ∫|q|2dr is the energy flow. The form-factor is inversely proportional to the mode width. Hence, higher χ means stronger localization.The eigenmodes of the linear system  and on the depth p2 of the second lattice.One of our main results is illustrated in p1 a relatively sharp LDT occurs when p2 exceeds certain critical value [see typical dependence χ(p2) in θ = 3−1/2 corresponding to an aperiodic potential]. Since gradual transition from delocalized to strongly localized modes occurs within a narrow interval of p2 values, we define the LDT threshold as a p2 value at which form-factor χ exceeds 0.1. The LDT threshold depends on depths of both sublattices p1, p2 and not on their ratio, as is obvious from For a fixed p2 as they represent conventional Bloch states . Note that the α = 1 band is remarkably flat. Since the effective diffraction strength is determined by the band curvature, the flatness causes an anomalously slow broadening of the beam that excites modes from this flat band. Such effect, which was observed earlier in other lattice types5657In βk of the system  when the deviation of the rotation angle from a Pythagorean angle increases. The gap between the first and second groups of eigenvalues (former α = 1 and α = 2 bands) does not disappear abruptly and it only closes completely for deviations in θ of the order of 0.5 degrees. Therefore, phenomena associated with the presence of forbidden gaps can occur even in slightly aperiodic lattices. Thus, in a nonlinear system the persistence of a finite gap for the slight detuning of the rotation angle from a Pythagorean implies that the energy flow threshold for gap soliton existence does not disappear abruptly upon detuning.Returning to new possibilities afforded by the smooth one-parametric transition between periodic and aperiodic geometries, one may wonder what happens to the linear spectrum when the transition takes place. The answer is given in U is required to create a 2D soliton. This is applicable to model (2) with a Pythagorean lattice V(r) and with σ = −1. On the other hand, if a system supports localized linear modes, one can expect that solitons may bifurcate from such modes with an increase of the peak amplitude. Thus, for periodic and aperiodic potentials one expects qualitatively different behavior of the U(β) curves eiβξ). This is confirmed by σ = −1. Indeed, for tan θ = 3/4 corresponding to periodic potential (black curves 1 and 2), a minimal energy flow is required for soliton formation irrespectively of p2 value. However, for tan θ = 3−1/2 the U(β) curves are qualitatively different below (red curve 1) and above (red curve 2) LDT threshold in p2: in the former case minimal energy flow is still needed to form a soliton, while in the latter case energy flow goes to zero indicating a bifurcation from linear mode. Solitons are found to be stable for the intervals where dU/dβ > 0σ = +1) solitons may form in finite gaps, even for a small detuning of the rotation angle from a Pythagorean one [see U(β) curve]. Such solitons feature an energy flow threshold and are stable in the largest part of the gap, except for narrow regions close to the gap edges. Note the unusual symmetry of the soliton shapes supported by the composite lattices [The potential (1) also has an impact on the properties of nonlinear localized states. It has been provenlattices .θi = 0, 2π/3, 4π/3. Examples of such composite potentials are given in a, b, c), such that c2 = a2 + b2 + ab, which uniquely defines the rotation angle (its tangent is given by π/3-periodic χ(θ) dependencies qualitatively similar to those encountered in square rotated latticesThe above results are general in the sense that the mutual rotation of two geometrically identical structures (of any symmetry) sets the basis for the construction of one-parametric 2D potentials allowing continuous transition between periodic and aperiodic geometries and, hence, observation of LDT. To illustrate the generality of the effect we consider the potential (1) composed of two rotated hexagonal (triangular) lattices two-dimensional composite lattices created by the superposition of two mutually-rotated periodic structures. Even above the LDT threshold for given amplitudes of the sub-lattices, where all eigenmodes are localized for the majority of rotation angles, one observes sharp delocalization for rotation angles corresponding to Pythagorean triples. Thus, for specific rotation angles θ allowing periodicity restoration one always gets delocalization, while for θ values leading to aperiodic lattices localization occurs for p1, p2 values taken above the LDT threshold. Since this conclusion is based on general arguments we anticipate that the localization-delocalization transition can be observed in structures with different internal symmetries and can be experimentally realized in various systems, including optical settings and Bose-Einstein condensates. The nature of the underlying composite linear lattices also has an impact on the properties and symmetries of nonlinear self-sustained excitations, allowing, for example, thresholdless creation of two-dimensional solitons in media with Kerr nonlinearity.In summary, we have shown that LDT can occur in a new class of How to cite this article: Huang, C. et al. Localization-delocalization wavepacket transition in Pythagorean aperiodic potentials. Sci. Rep.6, 32546; doi: 10.1038/srep32546 (2016).