Reciprocal Space Mapping Measurements Research Articles

Homoepitaxial Regrowth of AlGaN on AlGaN Templates Prepared via Chemical Mechanical Polishing and Its Application to UV‐B Laser Diodes

In this article, the homoepitaxial regrowth of AlGaN on AlGaN templates fabricated via chemical mechanical polishing (CMP) on periodically aligned AlN nanopillars is reported on. These templates are lattice relaxed according to X‐Ray diffraction reciprocal space mapping measurements, with a low dislocation density of ≈3 × 108 cm−2. However, the AlGaN surface has hillocks exceeding 104 cm−2. Contrastingly, no hillocks are observed on the homoepitaxially regrown AlGaN on the CMP–AlGaN template and after the UV‐B laser diode (LD) layer structure is crystallized and grown on the homoepitaxially regrown AlGaN. Furthermore, when this wafer is processed for LD fabrication, room‐temperature pulsed oscillation is confirmed in the UV‐B region at a wavelength of 296 nm. The turn‐on voltage and operating voltage also reduce by ≈2 V.

Varied domain structures in 0.7Pb(Mg1/3Nb2/3)O3-0.3PbTiO3 single crystals

Single crystals of 0.7Pb(Mg1/3Nb2/3)O3-0.3PbTiO3 (PMN-30PT) with a composition close to the morphotropic phase boundary have demonstrated remarkable electromechanical properties, stimulating intensive research interest in the field of piezoelectrics. Domain structures have long been associated with their overall piezoelectric properties, thereby garnering particular research interest for the enhancement of piezoelectric properties. Here, we report on three distinct domain structures in this material. Through a combination of X-ray diffraction reciprocal space mapping and piezoresponse force microscopy measurements, an M A crystal structure has been confirmed. The three-dimensional polar vectors of the three domain structures have been reconstructed with two 4M A and one 2M A domain structures. Correlations between different domain structures and their local electrical switching properties have been revealed. Our study of the PMN-30PT single crystal examines diverse domain structures at the mesoscale in detail, which provides valuable insight into the relationship between structures and properties of the material.

Structural characterization of epitaxial γ-Ta2N thin films and AlN/γ-Ta2N heterostructures on SiC substrates

Tantalum nitride (γ-Ta2N) thin films were grown by radiofrequency plasma molecular beam epitaxy (MBE) on 3 in. diameter 6H- or 4H-SiC substrates. Epitaxial characteristics of these MBE grown layers were determined using high resolution x-ray diffraction (HRXRD) and transmission electron microscopy (TEM) measurements. HRXRD, TEM, and XRD reciprocal space map measurements show that γ-Ta2N on SiC grows pseudomorphically and stabilizes to a nearly pure γ-Ta2N phase. Structural properties of these layers are uniform across the 3 in. wafer diameter. Measured a and c lattice parameter values of a 43 nm thick γ-Ta2N film on 6H-SiC are 3.079 and 4.898 Å, respectively, and the film has an in-plane tensile strain of 1.03%. MBE growth of AlN/ γ-Ta2N /SiC heterostructures has also been demonstrated. Measured lattice a and c constants of AlN on γ-Ta2N /SiC are 3.120 and 4.974 Å, respectively. TEM and XRD show that SiC substrate and γ-Ta2N films have parallel epitaxial relation.

Strain relaxation in epitaxial SrTi0.5Fe0.5 O[formula omitted] thin films on LaAlO3: An X-ray reciprocal space mapping study

SrTi0.5Fe0.5O3−δ (STF) has emerged as a promising cathode material for solid oxide fuel cells due to its excellent high ionic and electronic conductivity. We investigated the strain-state in epitaxial STF thin films on LaAlO3 (001) using synchrotron x-ray reciprocal space mapping measurement. The STF layer consists of two groups of compressively strained domains mixed horizontally in the film plane with a strain difference of about 1.4 %. The stress-free relaxed unit cell of more compressively strained domains, where a large degree of Sr segregation is expected, is relatively larger indicating that Sr segregation causes unit cell expansion. The lattice strain becomes gradually relaxed as the film thickness increases and the in-plane domain size decreases. We conjecture that the segregated Sr from compressively strained domains diffuse to the surface to form a layer of SrOx islands.

A comparison study of InGaN/GaN multiple quantum wells grown on (111) silicon and (0001) sapphire substrates under identical conditions

Due to an increasing demand of developing III-nitride optoelectronics on silicon substrates, it is necessary to compare the growth and optical properties of III-nitride optoelectronics such as InGaN based light emitting diodes (LEDs) on silicon substrates and widely used sapphire substrates. GaN-on-silicon suffers from tensile strain, while GaN-on-sapphire exhibits compressive strain. This paper presents a comparative study of InGaN/GaN multiple quantum wells (MQWs) grown on a silicon substrate and a sapphire substrate under identical conditions. It has been found that GaN strain status has a significant influence on the growth and the optical properties of InGaN/GaN MQWs. Photoluminescence measurements indicate the InGaN/GaN MQWs grown on a silicon substrate exhibit significantly longer wavelength emission than those on a sapphire substrate. Detailed x-ray diffraction measurements including reciprocal space mapping measurements confirm that both indium content and growth rate in the InGaN MQWs on the silicon substrate are enhanced due to the tensile strain of the GaN underneath compared with those on the sapphire substrate. This work also presents an investigation on strain evolution during the InGaN MQWs growth on the two different kinds of substrates. A qualitative study based on in-situ curvature measurements indicates that a strain change on the silicon substrate is much more sensitive to a growth temperature change than that on the sapphire substrate. It is worth highlighting that the results provide useful guidance for optimising growth conditions for III-nitrides optoelectronics on silicon substrates.

Non-invasive and non-destructive characterization of MBE-grown CdZnTe/CdTe superlattice-based dislocation filtering layers

We report on the structural and optical properties of heteroepitaxial II–VI CdTe (211)B buffer layers with strained CdZnTe/CdTe superlattice layers, investigated by employing non-destructive methods including high-resolution x-ray diffraction, cathodoluminescence, and photoluminescence spectroscopy. X-ray diffraction reciprocal space mapping measurements revealed that the superlattice layers are coherently strained, leading to a spread in x-ray double-crystal rocking curve full-width at half-maximum values but better in-plane lattice-matching with HgCdTe. Both cross-sectional cathodoluminescence and photoluminescence measurements confirm the coherent growth of superlattice layers and their dislocation filtering effects. Both these techniques in CdTe layers are found to be well correlated with the dislocation density as determined by etch pit density measurements. The results indicate the potential of these non-destructive methods to be further developed into general-purpose techniques capable of characterizing the defect evolution in semiconductor heteroepitaxy.

Strain relaxation and spinodal decomposition in composition adjusted TiO2-VO2 films on TiO2(100) substrates

In this study, we investigate the process of strain relaxation and spinodal decomposition in TiO2-VO2 films on TiO2(100) substrates. Herein, the film composition was adjusted to Ti0.2V0.8O2 for the relaxation of the in-plane tensile c-axis strain in the films. A 115 nm thick Ti0.2V0.8O2 solid-solution film is epitaxially grown on a rutile-type TiO2(100) substrate using a pulsed laser deposition technique, and is annealed inside the spinodal region. Reciprocal space mapping measurements show that the in-plane tensile c-axis strain is almost fully relaxed in the solid-solution film. Scanning transmission electron microscopy (STEM) observation results reveal that the annealed film causes spinodal decomposition along the c-axis direction, forming vertically aligned multilayer structures. The STEM observations also suggest the presence of interfacial misfit dislocations along the c-axis, indicating the generation of plastic strain relaxation along the c-axis in the film. Composition analysis reveals that the Ti-V interdiffusion between the film and substrate is considerably suppressed. The overall results of this study show that the composition adjustment to Ti0.2V0.8O2 is effective for strain relaxation along the in-plane c-axis and occurrence of spinodal decomposition with reduced Ti-V interdiffusion in the TiO2-VO2 films on TiO2(100). Our study helps establish a method for the formation of nanostructures via spinodal decomposition in the TiO2-VO2 films.

Anisotropic in-Plane and out-of-Plane Strain Relaxation in Carbon-Doped Silicon Nanowires Evaluated by X-Ray Reciprocal Space Mapping

1. Background and purpose Since carbon (C) has a smaller lattice constant than silicon (Si), Si doped with C approximately 1% or less (Si:C: carbon-doped silicon) also has a smaller lattice constant than Si. By using Si:C as the source / drain material of the n-type metal-oxide-semiconductor field-effect transistor, the tensile strain can be applied to the channel region to improve electron mobility [1]. However, the nano-fabrication of Si:C causes strain relaxation, which hinders the improvement of electron mobility. We have reported the in-plane biaxial strain relaxation for Si:C nanowires evaluated with Raman spectroscopy, assuming that there is no out-of-plane strain relaxation [2]. In this study, we evaluated anisotropic strain relaxation, including not only in-plane but also out-of-plane direction, for Si:C nanowire by reciprocal lattice space mapping (RSM) measurement using synchrotron radiation X-rays. 2. Experimental method Si:C thin films were grown on (001) Si substrate by molecular beam epitaxy, and then nano-fabricated into nanowire by electron beam lithography and dry etching. The film thicknesses were 43, 50, and 37 nm, with C concentration of 0.60%, 0.83%, and 1.1%, respectively, confirmed by cross-sectional transmission electron microscope and secondary ion mass spectrometry measurements. In the Si:C nanowire, the length direction was parallel to the [110] direction and the width direction was parallel to the [-110] direction, respectively. The nanowire width (W) was varied as 1000, 500, 200, and 100 nm, while the nanowire length (L) was fixed at 10 μm. In order to obtain sufficient signals for RSM, 30,000 identical nanowires in the case of W = 1,000 and 500 nm, and 50,000 identical nanowires in the case of W = 200 and 100 nm were fabricated in approximately 1.5 mm × 1.5 mm areas, respectively. The X-ray energy was set to 10 keV. The RSMs of Si:C nanowires were obtained around 337 diffractions for C concentrations of 0.60% and 0.83% samples (Si:C0.83%), and around 115 diffraction for C concentration of 1.1% sample, respectively. 3. Results and Discussion Figures 1 (a) and (b) show the RSMs of the unprocessed and the nanowires with W of 500 nm of Si:C0.83% films, respectively. In Fig. 1 (a) and (b), the profiles near qx = 4.91 Å- 1, qz = 8.11 Å- 1 correspond to the hem of the 337 diffraction profiles for the Si substrate, and the peaks around qz = 8.165 Å- 1 are the diffraction profiles for the Si:C0.83% film or the Si:C0.83% nanowires, respectively. Here, qx and qz are components of scattering vector in the in-plane and out-of-plane directions, respectively. Figure 1 (a) shows that the profiles for Si substrate and Si:C0.83% film were obtained on the same qx, which indicates that the in-plane lattice constant of Si:C0.83% film before nano-fabrication was equal to that of Si substrate, and the tensile strain has been applied to Si:C0.83% film. Figure 1 (b) shows that qx increases and qz decreases for the Si:C0.83% nanowires with W of 500 nm as compared with the Si:C0.83% film, which indicates that the in-plane lattice constant of Si:C0.83% decreased and the out-of-plane lattice constant of Si:C0.83% increased due to the nano-fabrication. Thus, we clarified that the tensile strain applied to Si:C thin film was relaxed by nano-fabrication in both the in-plane and the out-of-plane directions. 4. Acknowledgment The RSM measurements for Si:C nanowires were performed at BL19B2 in SPring-8 approved by JASRI with proposal number of 2020A1748 and 2020A1849. References Tsung-Yang Liow, et al., IEEE Trans. Electron Devices 55, 2476 (2008).Yoshioka, et al., ECS Trans. 92 (4), 33-39 (2019). Figure 1

Strain states and relaxation for alpha-(Al_xGa_{1-x})_2O_3 thin films on prismatic planes of alpha-Al_2O_3 in the full composition range: Fundamental difference of a- and m-epitaxial planes in the manifestation of shear strain and lattice tilt

Pseudomorphic and relaxed alpha-(Al_xGa_{1-x})_2O_3 thin films are grown by combinatorial pulsed laser deposition in the entire composition range on prismatic a- and m-plane alpha-Al_2O_3 substrates. Pseudomorphic growth on m-plane sapphire has been achieved for x ge 0.45. A distinct difference between the a- and m-epitaxial plane is observed in reciprocal space map measurements being in agreement with continuum elasticity theory for rhombohedral heterostructures. While pseudomorphic layers on m-plane sapphire show a pronounced shear strain e'_5 along the c-axis direction, relaxed layers exhibit a global lattice tilt in the same direction. Both effects are not present on the a-epitaxial plane. Out-of-plane lattice constants as well as e'_5 are modeled as function of x employing elasticity theory, confirming theoretical values of the elastic stiffness tensor for alpha-Ga_2O_3, especially the non-zero value of the C_{14} component. Possible pyramidal slip systems for strain relaxation in c-axis direction are examined to explain and numerically model the difference in lattice tilt for the two substrate orientations.Graphic abstract

Two conductive mechanisms in LaMnO3 thin film: Adiabatic and nonadiabatic small polaronic hopping

We have presented the structural, surface morphology, magnetic and resistivity data for perovskite LaMnO3 epitaxial thin films which are fabricated on well-oriented (001) LaAlO3 substrates by pulsed laser deposition technique. X-ray diffraction [Formula: see text]–[Formula: see text] linear scans and reciprocal space mapping measurement confirm that the out-of-plane and in-plane epitaxial relationship are LMO(001)/LAO(001) and LMO(110)/LAO(110), respectively. Surface roughness determined by atomic force microscopy was no more than 0.3 nm. In the whole studied temperature range, all films only show a paramagnetic behavior instead of any magnetic phase transitions. Correspondingly, the electron transport behaviors always exhibit an insulting state as the temperature changes from high to low. However, we find that none of theoretical models can individually be used to understand their conductive mechanisms. Further studies indicated that charge carries of high and low temperature region obey adiabatic and nonadiabatic small polaronic hopping mechanisms, respectively. This finding offers new ways of exploiting the abnormal ferromagnetism in LaMnO3 multilayer thin films.

Effects of buffer doping on the strain relaxation of metamorphic InGaAs photodetector structures

The strain relaxation behaviors of metamorphic In0.83Ga0.17As photodetector structures on Si-doped, Be-doped and undoped InxAl1-xAs graded buffers are investigated. The cross-hatch period and surface roughness are increased by doping in buffer layers. X-ray diffraction reciprocal space mapping measurements reveal that the doping of Si and Be induces the increase of misfit dislocation density along [1−10] and [110] directions, respectively. The photoluminescence intensity of In0.83Ga0.17As photodetector structures is deteriorated by the doping in InxAl1-xAs buffers. The dark currents of fabricated photodetector chips are characterized and the In0.83Ga0.17As photodetector on undoped InAlAs buffer shows the lowest dark currents. The Si doping shows more detrimental effects on photoluminescence and dark current performances than Be doping due to the enhanced growth roughness. These results indicate that the doping of buffer layers is critical for the design and fabrication of metamorphic optoelectronic devices.

Tetragonal-like distortion and enhanced magnetic anisotropy of the cubic spinel Fe3O4 film with a thin ZnFe2O4 capping layer

25-nm-thick Fe3O4 films without and with 1.5-nm-thick ZnFe2O4 capping layer was deposited on (110) SrTiO3 substrate, and their structural and magnetic properties were studied. X-ray diffraction data showed that the position of the (110) peak shifted marginally to a higher angle after deposition of the capping layer in oxygen environment. Transmission electron microscopy images showed that the Fe3O4 film on the (110) SrTiO3 substrate appeared epitaxial, and the capping layer on the Fe3O4 film appeared polycrystalline. Reciprocal space mapping and pole-figure measurements revealed that a structural distortion of the Fe3O4 layer from cubic to tetragonal-like was presumably induced by a slight change in oxygen content during the deposition of the capping layer in oxygen environment. Compared with the cubic Fe3O4 film, magnetic hysteresis loops of the tetragonal-like Fe3O4 showed increased coercivity and enhanced uniaxial magnetic anisotropy, which were presumably attributed to an increase of crystalline anisotropy level in the tetragonal-like structure.

Growth, structural and optical properties of coherent κ-(AlxGa1−x)2O3/κ-Ga2O3 quantum well superlattice heterostructures

High quality heteroepitaxial (001)-oriented κ-(AlxGa1−x)2O3/κ-Ga2O3 quantum well superlattice heterostructures were deposited by tin-assisted pulsed laser deposition on c-sapphire substrates. Sharp superlattice fringes up to the ninth order in XRD patterns for Al-contents up to about 50 at. % confirm excellent structural quality and smooth interfaces in the multilayers on par with reports on homoepitaxial superlattices in the monoclinic modification. By employing elliptically segmented targets, the Al-content in the barrier layers of the superlattices was systematically varied in a range of 0.1 ≤ x ≤ 0.5 in a controlled and quasi-continuous manner. An in-depth investigation employing XRD 2θ-ω scans and reciprocal space map measurements on superlattices with different periods as well as single quantum well samples suggests coherent growth of the superlattices for application-relevant quantum well widths. The critical thickness for coherent growth of κ-Ga2O3 on κ-(AlxGa1−x)2O3 was further estimated to be at least 50 nm and 3 nm for x = 0.2 and x = 0.3, respectively. We determined absorption energies in optical transmission spectra for superlattices with x = 0.3 well below the bandgap of the barrier layers that decrease with increasing quantum well width suggesting transitions between localized states in the quantum wells as their origin. These results render superlattices in the metastable orthorhombic phase of Ga2O3 as a promising active layer for quantum well infrared photodetector applications.

Role of Ni substitution on structural, magnetic and electronic properties of epitaxial CoCr2O4 spinel thin films

Cubic spinel CoCr2O4 has recently attained attention due to its multiferroic properties. However, the Co site substitution effect on the structural and magnetic properties has rarely been studied in thin film form. In this work, the structural and magnetic properties of Co1–xNixCr2O4 (x= 0, 0.5) epitaxial thin films deposited on MgAl2O4 (100) and MgO (100) substrates to manipulate the nature of strain in the films using pulsed laser deposition (PLD) technique are presented. The epitaxial nature of the films was manifested through x-ray diffraction (XRD), reciprocal space mapping (RSM) and Rutherford backscattering spectrometry (RBS) measurements. Raman measurements revealed a disappearance of characteristic A1g and F2g modes of the CoCr2O4 with increase in the Ni content. Atomic force microscopy (AFM) and field emission scanning electron microscopy (FE-SEM) studies show a modification of the surface morphology upon Ni substitution. Magnetic measurements disclose that the ferrimagnetic Curie temperature (TC) of the CoCr2O4 in thin film grown on MgAl2O4 (100) and MgO (100) substrates were found to be 100.6 ± 0.5 K and 93.8 ± 0.2 K, respectively. With Ni substitution the TC values were found to be enhanced to 104.5 ± 0.4 K for MgAl2O4 (100) and 108.5 ± 0.6 K for MgO (100) substrates. X-ray photoelectron spectroscopy (XPS) suggests Cr3+ oxidation states in the films, while Co ions are present in a mixed Co2+/Co3+ oxidation state. The substitution of Ni at Co site significantly modifies the line shape of the core level as well as the valence band. Ni ions are also found to be in a mixed 2+/3+ oxidation state. O 1s core level display asymmetry related to possible defects like oxygen vacancies in the films.

Reciprocal Space Map Measurement of Ceramics Thin Films for Unequal Lattice Change at High Temperature

Ceria and zirconia are very important for their thermal, mechanical, and chemical stability, and their thin films have attracted much attention for applications such as buffer layers for growing electric devices, thermal-shield or optical coatings, corrosion-resistant coatings, oxygen sensors and ionic conductors for fuel cells. To investigate and control the thin film orientation and phase is important to improve those performances. In this study, the reciprocal space maps of CeO2/YSZ/Si(001) were obtained at high temperature by adding a heater to the sample stage. CeO2 and YSZ thin films were epitaxially grown samples. By measuring lattice constants at high temperature, it was conducted that axes of CeO2 and YSZ thin films parallel to the substrate surface showed smaller thermal coefficients than bulk reference and axes perpendicular to the surface showed larger thermal coefficients due to the underlayer and Si substrate. The distortion rate of the lattice of each film was small around at the film deposition temperature. And it could be controled the lattice parameter at the film surface by the film thickness. Therefore, when another thin film, for example, SrTiO3 is deposited on the CeO2 layer, the lattice change of CeO2 with increasing temperature may differ from that before depositing the top layer.

MBE Growth and Characterization of Strained HgTe (111) Films on CdTe/GaAs**Supported by the National Natural Science Foundation of China (Grant Nos. 11634009, 61874069, 1177041280 and 11574336), and Shanghai Science and Technology Foundation (Grant No.18JC1420401).

Strained HgTe thin films are typical three-dimensional topological insulator materials. Most works have focused on HgTe (100) films due to the topological properties resulting from uniaxial strain. In this study, strained HgTe (111) thin films are grown on GaAs (100) substrates with CdTe (111) buffer layers using molecular beam epitaxy (MBE). The optimal growth conditions for HgTe films are determined to be a growth temperature of 160°C and an Hg/Te flux ratio of 200. The strains of HgTe films with different thicknesses are investigated by high-resolution x-ray diffraction, including reciprocal space mapping measurements. The critical thickness of HgTe (111) film on CdTe/GaAs is estimated to be approximately 284 nm by Matthews’ equations, consistent with the experimental results. Reflection high-energy electron diffraction and high-resolution transmission electron microscopy investigations indicate that high-quality HgTe films are obtained. This exploration of the MBE growth of HgTe (111) films provides valuable information for further studies of HgTe-based topological insulators.

Relaxation and tilting of single and double layer structures of AlGaSb/GaSb-LPE studied by high resolution x-ray diffraction

Epitaxial AlGaSb double-layer structures grown on GaSb (001) substrates by Liquid Phase Epitaxy (LPE) were analyzed by high-resolution x-ray diffraction (HRXRD). Four AlGaSb double-layer structures grown at 450 °C were analyzed varying the thickness of the first layer and maintaining the same thickness for a second layer growth. Symmetric reciprocal space mapping measurements around the (004) reflection and asymmetric rocking curves around the (115) reflections have revealed that a subsequent Al0.15Ga0.985Sb growth on an Al0.047Ga0.953Sb layer modifies the relaxation and lattice tilting of the first layer. This behavior is attributed to the formation of dislocations within the layers during the growth and transported between them. In this work, the study was realized ex situ and is in well agreement with in situ studies and theoretical predictions on the relaxation of epitaxial films in diverse materials. The structural analysis of lattice distortion in epitaxial layers is relevant since it could modify the electrical behavior of optoelectronic devices building with them.

Temperature dependent optical properties of LaCoO3/SrTiO3 thin film studied by spectroscopic ellipsometry

We report the spectroscopy ellipsometry study for LaCoO3 epitaxy thin films grown on different SrTiO3 substrates [LCO/STO(001) and LCO/STO(111)]. Reciprocal space mapping measurements show that the as-prepared LCO/STO films are fully strained at the interface. The optical constants (the refractive index and the extinction coefficient) of LCO films from 300 K to 800 K are obtained based on the ellipsometry measurements. The optical absorption at the near-infrared region increases with heating and sharply enhances around 500 K due to the occurrence of insulator–metal transition. The observed isosbestic point reveals the evolution of the band structure along with the insulator–metal transition. The red shift of the isosbestic point from the (001) sample to the (111) sample indicates the important role interfacial strain plays in affecting the spin states. We also provide the optical conductivity and the relative effective number of interband transition electrons for different energy regions.

Имплантация ионов кремния в сапфир: малые дозы

As a result of ion implantation of silicon ions in sapphire and subsequent high-temperature annealing the precipitates of silicon and alumosilicates are observed in the subsurface sapphire region. X-ray reciprocal space mapping measurements showed the existence of compression stress with deformation -0.12% in the normal direction and tensile stress with deformation 0.2% in R-plane in that region, which reduces the mismatch with the silicon (100) lattice and can lead to an improvement in the structural perfection of epitaxial silicon films grown on such modified sapphire substrates.

In Situ Synchrotron X-ray Diffraction Reciprocal Space Mapping Measurements in the RF-MBE Growth of GaInN on GaN and InN

In this work, in situ synchrotron X-ray diffraction reciprocal space mapping (RSM) measurements were carried out for the radio-frequency plasma-assisted molecular beam epitaxy (RF-MBE) growth of GaInN on GaN and InN layers, which were also grown by RF-MBE on commercialized GaN/c-sapphire templates. In situ XRD RSM measurements were performed using an MBE apparatus directly coupled to an X-ray diffractometer at the beamline of the synchrotron radiation facility SPring-8. It was observed in situ that both lattice relaxation and compositional pulling occurred during the initial growth stage, reducing the strain of GaInN on GaN and InN. Different initial growth behaviors of GaInN on GaN and InN were also observed from the results of the evolution of GaInN integrated peak intensities.