text:
# Operative temperature

Operative temperature (${\displaystyle t_{o}}$), also known as Dry resultant temperature, or Resultant temperature, is defined as a uniform temperature of an imaginary black enclosure in which an occupant would exchange the same amount of heat by radiation plus convection as in the actual nonuniform environment.[1][2][3][4] Some references also use the terms 'equivalent temperature" or 'effective temperature' to describe combined effects of convective and radiant heat transfer.[5] In design, operative temperature can be defined as the average of the mean radiant and ambient air temperatures, weighted by their respective heat transfer coefficients.[6] The instrument used for assessing environmental thermal comfort in terms of operative temperature is called a eupatheoscope and was invented by A. F. Dufton in 1929.[7] Mathematically, operative temperature can be shown as;

${\displaystyle t_{o}={\frac {(h_{r}t_{mr}+h_{c}t_{a})}{h_{r}+h_{c}}}}$

where,

${\displaystyle h_{c}}$ = convective heat transfer coefficient
${\displaystyle h_{r}}$ = linear radiative heat transfer coefficient
${\displaystyle t_{a}}$ = air temperature
${\displaystyle t_{mr}}$ = mean radiant temperature

Or

${\displaystyle t_{o}={\frac {(t_{mr}+(t_{a}\times {\sqrt {10v}}))}{1+{\sqrt {10v}}}}}$[8]

where,

${\displaystyle v}$ = air velocity
${\displaystyle t_{a}}$ and ${\displaystyle t_{mr}}$ have the same meaning as above.

It is also acceptable to approximate this relationship for occupants engaged in near sedentary physical activity (with metabolic rates between 1.0 met and 1.3 met), not in direct sunlight, and not exposed to air velocities greater than 0.10 m/s (20 fpm). [9]

${\displaystyle t_{o}={\frac {(t_{a}+t_{mr})}{2}}}$

where ${\displaystyle t_{a}}$ and ${\displaystyle t_{mr}}$ have the same meaning as above.

## Application

Operative temperature is used in heat transfer and thermal comfort analysis in transportation and buildings.[10] Most psychrometric charts used in HVAC design only show the dry bulb temperature on the x-axis(abscissa), however, it is the operative temperature which is specified on the x-axis of the psychrometric chart illustrated in ANSI/ASHRAE Standard 55 – Thermal Environmental Conditions for Human occupancy.


text:
source:branches/2520_PersistenceReintegration/HeuristicLab.Persistence/3.3/Plugin.cs.frame@16485

Last change on this file since 16485 was 16485, checked in by gkronber, 15 months ago

#2520 replaced static type registration in HL.Persistence plugin with the KnownStorableTypesMap which is discovered and used by HEAL.Fossil.
As a consequence we do not need to explicitly call type registration for unit tests.
This change has been necessary because on the first access of StaticCache all StoraleTypes are registered. However, when the HL.Persistence plugin is loaded we have not yet loaded all plugins and as a consequence we would miss StorableTypes.

File size: 1.2 KB
Line
2/* HeuristicLab
3 * Copyright (C) 2002-2019 Heuristic and Evolutionary Algorithms Laboratory (HEAL)
4 *
5 * This file is part of HeuristicLab.
6 *
7 * HeuristicLab is free software: you can redistribute it and/or modify
9 * the Free Software Foundation, either version 3 of the License, or
10 * (at your option) any later version.
11 *
12 * HeuristicLab is distributed in the hope that it will be useful,
13 * but WITHOUT ANY WARRANTY; without even the implied warranty of
14 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
15 * GNU General Public License for more details.
16 *
17 * You should have received a copy of the GNU General Public License
18 * along with HeuristicLab. If not, see <http://www.gnu.org/licenses/>.
19 */
20#endregion
21
22using HeuristicLab.PluginInfrastructure;
23
24namespace HeuristicLab.Persistence {
25
26  /// <summary>
27  /// The plugin for HeuriticLab.Persistence
28  /// </summary>
29  [Plugin("HeuristicLab.Persistence", "3.3.15.$WCREV$")]
30  [PluginFile("HeuristicLab.Persistence-3.3.dll", PluginFileType.Assembly)]
31  [PluginDependency("HeuristicLab.Tracing", "3.3")]
32  public class HeuristicLabPersistencePlugin : PluginBase { }
33}
Note: See TracBrowser for help on using the repository browser.

format: plain


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# GaAs1-xBix growth on Ge: anti-phase domains, ordering, and exciton localization

## Abstract

The dilute bismide alloy GaAs1-xBix has drawn significant attention from researchers interested in its fundamental properties and the potential for infrared optoelectronics applications. To extend the study of bismides, molecular-beam heteroepitaxy of nominally 1.0 eV bandgap bismide on Ge substrates is comprehensively investigated. Analysis of atomic-resolution anti-phase domain (APD) images in the direct-epitaxy revealed a high-density of Ga vacancies and a reduced Bi content at their boundaries. This likely played a key role in the preferential dissolution of Bi atoms from the APD interiors and Bi spiking in Ge during thermal annealing. Introduction of GaAs buffer on offcut Ge largely suppressed the formation of APDs, producing high-quality bismide with single-variant CuPtB-type ordered domains as large as 200 nm. Atomic-resolution X-ray imaging showed that 2-dimensional Bi-rich (111) planes contain up to x = 9% Bi. The anomalously early onset of localization found in the temperature-dependent photoluminescence suggests enhanced interactions among Bi states, as compared to non-ordered samples. Growth of large-domain single-variant ordered GaAs1-xBix films provides new prospects for detailed analysis of the structural modulation effects and may allow to further tailor properties of this alloy for optoelectronic applications.

## Introduction

The development of dilute Bi-containing alloy GaAs1-xBix, referred to here as bismide, in recent years has been driven by its potential to extend III-V semiconductor devices into the infrared range. Incorporation of Bi into GaAs matrix leads to a large narrowing of the bandgap, up to ~90 meV/%, as compared to, e.g., by alloying with In (~10 meV/%)1,2. The bismide also has strong spin-orbit coupling effects on the electronic band structure, which is unique among III-V alloys. GaAs1-xBix with concentrations x > 10% produces a separation of the split-off valence band energy that exceeds the bandgap, leading to suppression of the Auger recombination channel involving this band3,4. The bismide can be grown on GaAs substrates and is a promising candidate for several applications, including infrared lasers, heterojunction bipolar transistors, spintronics, and multi-junction solar cells1,5,6.

GaAs1-xBix shares many similarities with GaAs1-xNx, which also produces a large bowing of the GaAs bandgap. Both bismide and the nitride show dual behavior characteristics – that of an alloy as well as of a highly-doped isoelectronic impurity material7,8,9. The bandgap reduction occurs due to the interaction of resonant Bi and N states in the proximity of GaAs valence and conduction band edges (VBE, CBE), respectively. While individual Bi and N atoms produce bandgap bowing and potential fluctuations, stochastically incorporated pairs (Bi-Bi, N-N) or clusters are thought to be a source of the local density of states that lead to the pronounced charge carrier localization and scattering in these alloys10,11,12,13. The bismide has presently emerged as a more promising candidate for photovoltaics (PV) since the trap states are formed near its VBE, whereas they occur near CBE in the nitride. High electron mobilities and lifetimes can thus be preserved in the former14,15,16,17. A lattice-matched 1.0 eV bandgap absorber material is highly desirable for III-V multi-junction (M-J) PV applications. The bandgap combination of the industry-standard 3-J device GaInP(1.9)/GaAs(1.4)/Ge(0.7) is non-ideal for space or terrestrial concentrated-sun conditions. Nearly 5% efficiency points could be added by replacing the Ge bottom cell with a 1.0 eV material, and as much as 10% by incorporating it into 4-J cells atop Ge18,19. The high absorption coefficient of 1.0 eV GaAsBi and less than 0.05% lattice-mismatch to GaAs has granted this material a thorough study within the PV community20,21.

Incorporation of Bi into III-V compounds requires low growth temperatures, typically below 400 °C, and a near-stoichiometric flux ratio of III/V elements due to the alloy miscibility gap1,22,23. There is a relatively small window of growth parameters that allow for high-quality bismides free of Bi and Ga droplets on the surface. Deviations from the growth conditions also lead to phase-separated domains and spontaneous nanostructure formations24,25,26. Furthermore, many recent studies challenge the assumption that Bi is incorporated randomly. Instead, CuPt-type ordering of group-V elements is often observed near the optimal GaAsBi growth conditions, an effect that is driven by surface reconstruction dynamics27,28,29,30. Spontaneous CuPt-type ordering in III-V semiconductor alloys typically occurs along two of the four <111> body diagonals, primarily along two <111> B, which is termed CuPtB31. Besides, the use of vicinal substrates favors the formation of a single CuPtB subvariant. The ordering reduces zincblende cubic F$$\overline{43}$$m symmetry to trigonal R3m and can lead to several effects, including bandgap narrowing, valence band splitting, the polarization of light, and anisotropic strain31,32,33,34. No such ordering-induced effects have yet been reported in GaAsBi alloys.

An attendant challenge for the bismide-based PV development is its growth on Ge substrates, as one is faced with the epitaxy of a polar material on a non-polar substrate. Mixed nucleation of group-III and V elements and random surface steps on the substrate can produce anti-phase domains (APD), which degrade device performance35,36,37. As of this writing, the attempt to grow GaAs1-xBix on Ge substrates, to the best of our knowledge, was made only once38. The authors synthesized bismides with up to 5% Bi on offcut (001) Ge substrates using Ga initiated migration-enhanced molecular-beam epitaxy (MEE-MBE). Employing a low-temperature MEE-MBE buffer in the first stage of growth, combined with a Ge substrate offcut, is a common practice to eliminate APDs in the GaAs-Ge heteroepitaxy35,39,40. Numerous triangular APDs in the first-few tens of nanometers were nevertheless found in the bismide, but an otherwise APD-pit-free surface, as well as a suppressed Ge diffusion into the film.

To extend the study of this heteroepitaxy, bulk bismide samples with nominally 1.0 eV bandgap are grown by MBE on vicinal Ge substrates and investigated using aberration-corrected scanning transmission electron microscopy (STEM) and atom-counting-based numerical techniques, X-ray diffraction and reciprocal-space mappings (XRD-RSM), temperature-dependent photoluminescence (PL), and secondary ion mass spectroscopy (SIMS). Rapid-thermal annealing (RTA) studies are also performed.

## Results and Discussion

### Direct epitaxy of GaAsBi on offcut Ge substrates

The first sample studied here is an epitaxial bismide film synthesized directly on an offcut Ge (001) substrate (as-grown sample S1, see Sample synthesis in Methods). Structural characterization using STEM high-angle annular dark-field (HAADF) imaging was performed to analyze the sample, which was prepared such that the offcut [1$$\bar{1}$$0] direction is orthogonal to the HAADF imaging axis [110], allowing for the interface to be viewed edge-on. The scanning direction in STEM is aligned to set the [001] direction parallel to the vertical direction of the page so that the interface is seen tilted with respect to the horizontal in the amount of the offcut angle. Low-magnification HAADF image in Fig. 1(a) shows typical lattice features, indicating a high-density of APDs in this sample. A more systematic study is needed for the direct epitaxy in regards to the elimination of APD defects in GaAsBi. We note that the direct epitaxy could be bypassed in the M-J photovoltaic applications by first growing a highly-doped GaAs layer, or other III-V materials with established deposition recipes, which would act as the front-surface-field before a tunnel-junction interconnect. The analysis of a bismide sample grown over a GaAs buffer layer (sample S2) will be presented in the following section. We now turn to investigate the structure of APDs generated in the direct epitaxy on Ge.

The HAADF contrast in Fig. 1(a) is dominated by large APDs with a distinctive asymmetric triangular shape. The color-coded inset shows a geometrical phase analysis (GPA) phase map of the STEM image employing [002]* reciprocal lattice vector41. It highlights the APDs as the phase changes by π crossing into their interior. The asymmetric APDs that have boundaries with outward normal vectors [1$$\bar{1}$$0]//[$$\bar{1}$$11] are common in this bismide. The darker strips in HAADF images at the APD boundaries suggests a reduced Bi content. The boundaries of these domains do not seem to reach the interface, but rather intersect smaller APDs near the interface. Figure 1(b) shows an atomic-resolution HAADF image of an APD at the interface. The (1$$\bar{1}$$0) boundary here seems to have kinks, and shifts to a (1 $$\bar{1}\,$$1) plane near the apex before intersecting the ($$\bar{1}$$11) APD boundary. Small 2–5 nm domains of single subvariant CuPtB-type ordered bismide on (1$$\bar{1}$$1) planes can be seen outside the APD. A Fourier transform of the image is shown in the inset with the marked superlattice 1/2[1$$\bar{1}$$1]* Bragg spot next to [1$$\bar{1}$$1]*, indicating the presence of ordered domains.

### STEM image analysis of GaAsBi anti-phase domains

Atomic-resolution HAADF image of a representative asymmetric triangular APD close to its vertex in sample S1 is shown in Fig. 2. GaAs crystal viewed along a <110> -type direction appears as a collection of closely-spaced Ga and As atomic-columns, so-called “dumbbells”. Pure Ga and As columns cannot be easily told apart since the HAADF contrast scales with an atomic-number as ~Z2, (Ga Z = 31, As-33, Bi-83). A large inner-collection angle of the HAADF detector was chosen here to enhance the contrast of Bi-containing columns42,43. Since Bi mainly replaces As atoms, this allows for distinguishing of group-V columns. The APD position and its boundaries in Fig. 2(a) can be clearly seen by noting the position of Bi atoms within [001] oriented dumbbells. In particular, Bi rich columns are in the lower half of each dumbbell within the APD, and this polarity switches outside the APD. Note that (1$$\bar{1}$$0) APD boundaries contain both, group III-III (Ga-Ga) and V-V (such as As-As, As-Bi, Bi-Bi) element wrong-bonds in equal numbers, whereas ($$\bar{1}$$11) boundaries have only Ga-Ga wrong bonds. The polar variant of the {111} boundary having group V-V wrong bonds was not observed in this sample, which is probably related to the dominant bismide (001) growth surface polarity. Theoretical studies of GaAs APDs suggest that {110}-type boundaries have the lowest formation energy due to their stoichiometry and charge transfer among Ga-Ga and As-As wrong bonds44. However, inclined planes, such as {111}, {112}, or {113}, can facilitate the annihilation of anti-phase boundaries and may lower the total systems’ energy. More studies are needed to elucidate whether the offcut or other growth factors favour the (1$$\bar{1}$$0)//($$\bar{1}$$11) boundaries in this bismide. Point defects can also alter the anti-phase boundary formation energies, and their distributions are investigated next.

Numerical quantification of the entire HAADF image in Fig. 2(a) was performed to gain further insights. The StatSTEM algorithm works by finding individual atomic-columns and fits each column with a Gaussian45. A new model image is thereby produced composed of a superposition of Gaussian peaks. The dynamical scattering cross-section (SCS) of an atomic-column -i is defined by integrating the volume under the ith Gaussian peak. The key factors determining the magnitude of a SCS are the type of atoms and their numbers in the column (see Methods for more details). The distribution of scattering cross-sections in Fig. 2(b) shows the count of atomic-columns in a given SCS range. Dashed vertical lines with arrows refer to various SCS regions, as mapped in Fig. 3 and discussed shortly. The histogram is tentatively fitted with three Gaussians, whose meaning can be interpreted as follows. Firstly, pure Ga and As columns, due to their similar Z-number, produce a strongly overlapping distribution of SCS, which gives rise to the tallest single Gaussian (red) centered at ~4.4 eÅ2. Plotting the spatial distribution of the lower end of the Gaussian predominantly picks Ga atomic-columns, as can be inspected from the dumbbell polarity. A fraction of As atoms are replaced by Bi, and so the SCS distribution of otherwise pure As columns is now extended with the extra Gaussian shoulders since Bi atoms scatter electrons much stronger to the high-angle annular detector. Atomic-column configurations with a different number of Bi atoms in them produce broadening of the SCS, which are tentatively represented here by two Gaussians (blue, yellow). We note that to establish SCS values of atomic-column configurations with various Bi distributions, numerical image simulations are needed to build a library for fitting to normalized experimental HAADF images. This, however, is beyond the scope of the present study.

Figure 3 shows the spatial distribution of scattering cross-sections in Fig. 2(a), plotted on the model structure by outlining the atomic-columns with colored squares. In Fig. 3(a), the distribution of atomic-columns with SCS > 5.4 eÅ2 is plotted, which corresponds to the position in Fig. 2(b) marked with a dashed vertical line where the Gaussian associated to columns with Bi atoms in them overtakes in frequency the Gaussian associated with pure Ga and As columns. This distribution indicates that (1$$\bar{1}$$0) APD boundary plane contains Bi atoms, but its concentration is diminished immediately on either side of it, forming thin 1–3 nm Bi-deficient regions. The ($$\bar{1}$$11) boundary region also shows reduced Bi content. The APD exterior exhibits small patches of CuPt-ordered domains, whereas no ordering is found in the APD interior in this projection. The outlined squares on group-III atomic-columns in Fig. 3(a) distribution indicate there a likelihood of hetero-antisite defects BiGa. Figure 3(b) shows SCS > 6.8 eÅ2, and is indicative of columns that have the largest Bi content in them. Figure 3(c) shows spatial distributions of SCS < 4.7 eÅ2, which mainly picks all group-III columns, and also group-V columns with lower SCS due to, e.g., vacancies or GaAs antisites in them. An increased likelihood of such point defects on group-V columns appears to be concentrated near the APD boundaries, which is signified by the numerous outlined group-V columns. A similar distribution mapping is performed on the group-III columns by plotting SCS < 3.9 eÅ2 in Fig. 3(d), which targets the weakest scattering Ga-columns. The likelihood of Ga vacancies is concentrated at the APD apex and near its boundaries, clearly pronouncing the APD region. We note that no significant in-plane lattice strain was found at the APD (based on GPA and StatSTEM, not shown here), which could otherwise impact the image intensities and quantification of the SCS.

### Thermal annealing effects in the direct epitaxy

The effects of GaAs1-xBix thermal annealing grown on GaAs substrates were investigated by several groups46,47. Even though the general trends were identified, the results are still ambivalent due to a spread in growth conditions. Firstly, the annealing is thought to reduce point defects generated in a low-temperature-grown GaAs lattice. The GaAsBi alloy stability at temperatures above ~600 °C and the observed onset of Bi segregation is dependent on the Bi fraction in samples, with higher Bi% typically seen as less stable, and also on the lattice quality before the annealing.

In this section, STEM-EDS is employed to examine the crystal structure and chemical changes of the bismide sample S1 annealed at 600 °C in a rapid-thermal annealing (RTA) oven (see Methods). This temperature treatment is presented here since noticeable photoluminescence changes already occur at 600 °C, as compared to the as-grown S1 sample. A low-magnification HAADF image of the bismide is shown in Fig. 4(a). Numerous asymmetric triangular APDs can be seen (indicated with arrows) with a similar density as in the as-grown S1 sample, suggesting that the APD density did not change significantly. However, the APD interiors appear darker in the Z-contrast HAADF image, indicating a lower Bi concentration. Furthermore, Bi precipitates are found near all these Bi deficient domains. Bi particles were typically reported to assume a rhombohedral crystal structure within the GaAsBi matrix, and we indeed found numerous coherently aligned Bi crystallites, as illustrated in Fig. 4(b)48. Quite a few laterally elongated phase-separated domains were found in the annealed sample. Phase-separation can occur during growth, but can also be initiated and enhanced through a dissolution of Bi during a thermal treatment. The latter appears to be the case here since phase-separated domains are accompanied by Bi precipitates, which are virtually absent in the as-grown film. Interestingly, the APD interiors are found to consistently reject Bi, which then preferentially accumulates in droplets near the interface with Ge substrate. Some of the Bi particles clearly penetrate the Ge substrate. X-ray energy dispersive chemical mapping of the sample in Fig. 4(c–g) shows Bi precipitates and confirms that large portions of the APD interior, mainly upper parts towards their vertices, are Bi deficient. Ge out-diffusion from the substrate is also noticeable in the EDS images, accompanied by reduced Ga and As concentrations. Due to large size of Bi atoms, vacancy-mediated hopping of Bi atoms has been suggested as the primary means of Bi diffusion and clustering16. The higher density of Ga and As vacancies at the anti-phase domains, as presented in the previous section, could explain the preferential Bi dissolution from the APD interiors during high-temperature annealing. On the other hand, Ge diffusion along vacancies in GaAsBi APDs and a subsequent filling of Ge vacancies with Bi could be one way to interpret the spiking of Bi within the substrate.

### Single-variant CuPt-type ordering of bismide grown with GaAs buffer layer

The as-grown sample S2 is analyzed next, which is composed of 230 nm thick nominally 1.0 eV GaAsBi deposited over 300 nm thick GaAs buffer layer on offcut Ge. The GaAs buffer was grown in three steps to reduce the density of APDs in GaAs and to minimize elemental interdiffusion across Ge-GaAs interface (see Methods)39,40. Secondary ion mass spectroscopy showed no significant diffusion of Ge into the GaAs buffer (Supplementary Fig. S1). Most of the APDs that were nevertheless generated in GaAs were buried within the buffer, leaving the bismide film virtually free of APDs. Figure 5(a) shows a representative HAADF image of the film with a GaAs buffer layer. STEM imaging direction in sample S2 is [$$\bar{1}\bar{1}0$$], and [001] is aligned vertically so that the GaAs-GaAsBi interface is inclined with respect to the horizontal. The preservation of the offcut angle and elimination of APDs had promoted the bismide growth that is uniformly single-variant CuPtB-ordered. The ordered domain sizes are at least ~230 nm (thickness of the film) with few small randomly interspersed areas showing a lower degree of ordering. As of this writing, this constitutes the largest CuPt-type domains stabilized in GaAsBi.

Atomic-column resolved EDS mapping of the bismide is presented in Fig. 5(b–f). The scanning direction in STEM was set to align the ordered (1$$\bar{1}$$1) planes horizontally. The ordering of Bi atoms is clear from Bi-M X-ray emissions in Fig. 5(c), and is also illustrated in Fig. 5(b), which is composed by overlaying EDS of all elements on the simultaneously acquired HAADF image. Line profiles across an ordered plane, shown in Fig. 5(f), indicate that Bi positions overlap As, confirming BiAs substitution. By quantifying EDS signals along the ordered Bi-rich planes, we find that Bi concentration in them reaches x′ = 9% (+/−0.5%). The XRD-RSM measurements, presented in the following section, indicate Bi concentration in the lattice is x = 5.8% Bi, while the EDS results show x = 5.5%. In general, an ordered bismide GaAs1-xBix is composed of alternating Bi-rich GaAs1-(x+η/2)Bix+η/2 and As-rich GaAs1-(x-η/2)Bix-η/2 monolayers along <111>, where η is the long-range order parameter31. Keeping that Bi concentration in the sample is x = 5.5%, while in the ordered Bi rich planes x′ = 9%, the order parameter is found η = 0.07. Note that the ηmax in a fully ordered GaAs0.945Bi0.055 would be η = 0.11, while the maximal η = 1 is achieved in a fully ordered alloy at x = 50%.

### X-ray diffraction rocking-curve and reciprocal-space mapping

Figure 6(a) shows X-ray diffraction (XRD) rocking-curve measurements of the bismide samples S1 and S2. Bi concentration in the as-grown samples S1 and S2 are x = 5.5% and x = 5.8%, respectively. The concentrations were computed using GaAs and fictitious GaBi lattice constants (see Methods), taking into account GaAsBi lattice relaxations obtained from the reciprocal-space maps (discussed below). Dashed lines in Fig. 6(a) also show XRD of the samples after annealing in RTA oven at 600 °C, indicating a decrease of Bi to 5.2% and 5.5% in S1 and S2, respectively. A decrease of Bi content in GaAsBi lattice is commonly observed in bismides after thermal annealing and can indicate Bi-segregation46,48. To investigate CuPt-type ordering, measurements of superlattice reflections were performed in skew-symmetric configurations to access all four distinct sets of {111} planes49. Only one set of planes show the superlattice reflections ½(111) in sample S2 in Fig. 6(b), which agrees with the STEM observations. This diffraction peak is almost absent in sample S1 grown directly on Ge (Fig. 6b, upper graph), suggesting that CuPt-ordered domains are small and scarce. The superlattice XRD peak intensity in sample S2 slightly decreases after thermal annealing (SI Fig. S2). Interestingly, a bismide sample grown directly on Ge substrate without an offcut showed weak 1/2{111} superlattice reflections on all four distinct sets (SI Fig. S2). This is likely due to mixed-polarity domains occurring in similar proportions.

RSM of (004) reflections of the as-grown and annealed samples S1 and S2 are presented in Fig. 6(c–f). Figure 6 (c,e) show that the as-grown samples S1 and S2 incurred a bismide layer tilt of 0.064° and 0.074°, respectively, with respect to Ge, indicating an additional leaning towards the offcut direction. This effect has been explained by Nagai, and arises here from the lattice-mismatch in the (001) direction between the epitaxial layer and Ge or GaAs, due to the offcut-introduced surface steps34,50. The epilayer tilt can be accommodated by a lattice strain without nucleation of dislocations if the lattice mismatch is not too large. Here, we find the layer tilt angles, γ, follow fairly closely the Nagai equation, $$tan\gamma /tan\beta =({a}_{L}-{a}_{S})/{a}_{S}$$, where aL and aS are the layer and substrate lattice constants (az), respectively, and β is the substrate offcut angle50,51. Following the Nagai equation, the GaAs buffer layer is indeed tilted in the opposite direction by 0.001° due to its smaller lattice constant than Ge. After annealing at 600 °C, the tilt angles decrease, with final values 0.026° in S1, and 0.007° in S2. This indicates that dislocations having a Burgers vector with a tilt component opposite to the initial tilt direction nucleated during the lattice relaxation. Such decrease in tilt angles is observed in other zincblende materials, provided there are no sources for significant asymmetry in regards to nucleation and glide of dislocations on different sets of {111} planes34. RSM (004) reflections in Fig. 6(c,e) and also (115) (in SI Fig. S2), indicate that as-grown bismide layers are in tensile strain in the out-of-plane direction with lattice parameters (az) 5.716 Å (sample S1) and 5.718 Å (sample S2). The in-plane lattice parameters (ax) are 5.657 Å (S1) and 5.660 Å (S2) (SI Fig. S2), which shows that sample S1 is fully compressively strained to the substrate, while ~9% of the S2 lattice is relaxed, probably due to their different thicknesses (100 nm S1, 230 nm S2). Annealing of both samples at 600 °C leads to further relaxation, with final lattice parameters (ax, az) in S1 (5.669, 5.704) Å, and (5.666, 5.710) Å in S2, indicating the final relaxation is R = 37% in S1, and R = 25% in S252.

### Photoluminescence properties

The photoluminescence properties of GaAsBi have been a subject of numerous studies11,12,53,54,55,56. Notable PL features of the bismide include a broad PL peak at room temperature, a red-blue-red shift of the temperature-dependent PL peak-energy position, integrated PL intensity plateau at intermediate temperatures, and the saturation of the Stokes-shift at elevated excitation powers. Present understanding asserts that two independent Gaussian-shaped local density-of-states (LDOS) components are required to explain the radiative properties. The first Gaussian distribution extends from the VBE and is attributed to the short-range disorder and fluctuations of Bi content. The second is offset by some 100 meV  from the VBE into the bandgap and is likely due to Bi-related complexes. Empirical rate-equations show that competition between excitons hopping among the LDOS and their thermalization to the mobility-edge with increasing temperature, whereby non-radiative channels become accessible, leads to the observed PL features.

The temperature-dependent PL spectra of the as-grown sample S1 is shown in Fig. 7(a). It exhibits a very broad band-edge emission peak at 1.1 eV, likely due to the non-homogenous Bi content and a highly-defective lattice, as observed in STEM images. PL from the Ge substrate is also visible below ~0.8 eV. The temperature-dependent PL first shows a blue-shift due to the lattice contraction with decreasing temperature and is then followed by the characteristic red-blue shift. The onset of this effect is accompanied by the second peak, which emerges ~150 meV lower in energy, signifying trapping of excitons in deep states. As the temperature decreases still further, the excitons are unable to reach the deepest states and remain trapped at the shallower LDOS.

The temperature-dependent PL spectra of sample S2 in Fig. 7(b) shows narrower PL peaks and reflects a more uniform GaAsBi crystal, with band-edge emission at ~1.0 eV. Nevertheless, the emission from deep states seems strongly enhanced in this sample. Deep states start to dominate the PL intensity from temperatures 220–240 K, which is in contrast to sample S1 in Fig. 7(a), whereby the S-shift occurs at 140–150 K and is more typical for bismides. The anomalous onset of below-band-edge luminescence at relatively high-temperatures in S2 may reflect the pronounced CuPt-ordering. The ordering renders 2-dimensional Bi concentrations up to x = 9%, which is expected to produce stronger interactions among localized Bi states.

The room-temperature PL spectra of samples S1 and S2 annealed at 500 °C and 600 °C in an RTA oven are shown in Fig. 7(c,d). A somewhat different PL peak position of the as-grown sample S1, as compared to the S1 temperature-dependent PL peak at 300 K, could be due to Bi inhomogeneities across the sample. After annealing at 600 °C, sample S1 shows a small decrease in the band-edge emission peak intensity as well as the peak narrowing, and a red-shift to 1.07 eV. This emission energy adheres better to the expected bismide bandgap at x = 5.5% Bi concentration, as shown using XRD in the previous section, and suggests a beneficial rearrangement of the lattice produced by the RTA. Sample S2 grown over the GaAs buffer layer does not show any significant improvement in the PL after the annealing treatments. A decrease in the emission intensity could result from the partial lattice relaxation. The blue-shift visible after 600 °C RTA is likely due to the dissolution of Bi atoms from the lattice and is corroborated by the XRD-RSM measurements. This distinct response to annealing of samples S1 and S2 suggests that optimal annealing parameters of bismides need to be tailored based on their composition and as-grown lattice quality, such as defect content and Bi distribution modes.

### Summary and conclusions

Bulk ternary bismides GaAs1-xBix with x = 5.5% and 5.8% were grown by MBE on offcut Ge substrates and analyzed in this study. The direct epitaxy on Ge lead to a high density of anti-phase domains with a characteristic asymmetric triangular shape, bounded by {111} and {110}-type crystal planes. Atomic-resolution analysis of APDs revealed that the dark contrast at their boundaries arises due to a reduced Bi content. The distribution of Ga vacancies was spatially mapped on STEM images employing computed scattering-cross sections, indicating a significant increase of these point defects at the APDs. This likely played a key role in the dissolution of Bi atoms from the interior of APDs and Bi spiking in Ge substrate during thermal annealing. Incorporation of a GaAs buffer layer was found to eliminate APDs in the bismide layer and a high-quality film was obtained. Moreover, single-variant CuPt-type ordered domains as large as 200 nm  were stabilized in the bismide. The anomalously early onset of strong localization in the temperature-dependent PL of the ordered sample suggests an underlying rearrangements of localized density of states and enhanced exciton coupling. These findings show that high-quality bismide can be grown on Ge substrates. However, a growth of subsequent layers atop the bismide can be complicated if it results in the bismides thermal annealing. Both sets of samples showed Bi seggregattion after annealing at 600 °C. The relationship of GaAs1-xBix alloy stability to its initial lattice quality needs to be better understood. On the other hand, large-domain CuPt-type ordering encourages further studies and may provide another route to tailor bismides properties for optoelectronic device applications.

## Methods

### Sample synthesis

Bismide structures were grown by solid-source molecular-beam epitaxy (MBE) on p-type epi-ready (001) Ge substrates, supplied by the Axt, Inc., with 6° offcut towards an in-plane <110> -type direction. Two different samples were produced for this study: S1 consisting of a direct bismide epitaxy on Ge, while in S2, a GaAs buffer layer was grown first on Ge, and followed by a bismide film. The samples were each divided into three pieces, so that S1 and S2 sample sets each contains the as-grown, annealed at 500 °C, and annealed at 600 °C samples. MBE growth was performed in an SVT-A reactor equipped with metallic Ga and Bi sources, and a two-zone valved-cracker source using As2 dimers. The deoxidation and sample growth temperature was monitored by a thermocouple-based controller. To remove native oxides from the substrates and prepare the Ge surface for GaAs deposition, the wafers were heated at 640 °C for 15 min under arsenic flux. In samples with GaAs buffer layers, migration-enhanced epitaxy (MEE) was used at 350 °C to deposit 20 periods consisting of 20 gallium and 20 arsenic atomic layers and was followed by 50 nm of GaAs grown at 500 °C. The remaining ~250 nm of GaAs buffer was grown at 600 °C. This three-level scheme of GaAs growth was used to reduce elemental interdiffusion across the Ge/GaAs interface, and to minimize APD density in the buffer layer35,39,40. GaAsBi films were grown at 350 °C. Before the growth was initiated, the shutter was opened to give a bismuth cover over GaAs buffer or Ge wafer surfaces. The growth rate used in both samples was 400 nm/h. The BEP ratio of As2 to Ga was fixed in the range from 1.063 to 1.1, and Bi/Ga ratio was 0.35–0.37. Post-growth ex-situ annealing of the samples was done in a rapid-thermal annealing (RTA) oven UniTemp1300 under a nitrogen atmosphere. The sample surfaces were covered by a GaAs substrate to minimize arsenic loss. Annealing duration at the set temperature (500 °C or 600 °C) was kept to 180 seconds in all samples.

### STEM

Cross-sectional TEM samples were prepared by FEI Helios Nanolab 650 dual-beam microscope equipped with an Omniprobe manipulator using the lift-out technique. The samples were thinned and polished to ~30 nm thickness and argon-oxygen plasma-cleaned before loading into the microscope. STEM high-angle annular dark-field (HAADF) images were acquired using cold-field-emission double aberration-corrected JEOL JEM-ARM200CF operated at 200 kV. Inner-collection semi-angle of the HAADF detector was set to 90 mrad, the probe convergence semi-angle was 22 mrad. X-ray energy dispersive spectroscopy (EDS) mappings were acquired using 0.98 steradian solid-angle windowless silicon drift-detector JED-2300 (JEOL, Japan) mounted in the STEM instrument. For the atomic-resolution EDS, the probe current was set to 200 pA, 0.1–0.2 msec pixel dwell time. Wiener filtering and sample drift-correction were applied, 5–6 sweeps were accumulated for the atomic-resolution EDS images.

### Image analysis with StatSTEM

HAADF image analysis was carried out using StatSTEM v3.0 software suite for Matlab45. To reduce the number of search parameters and ensure algorithm stability, single-width 2D Gaussian functions were fitted to the atomic-column positions. In contrast to a commonly employed Voronoi cell segmentation, the current approach allows for more accurate quantification of HAADF images that contain closely-spaced atomic-columns, e.g. <110> GaAs (~1.4 Å). The values of SCS primarily depend on the number and size of atoms in the column. Columnar distortions and the related broadening of SCS distributions are expected due to large Bi atoms, although this effect is expected to be small compared to e.g., influence of point defects. The positions of Bi atoms along the column will also influence the HAADF intensity and the SCS, since the electron probe intensity decreases with increasing specimen depth. We assume that the sample thickness does not vary significantly (<1 unit cell) over the field-of-view. Furthermore, experimental instabilities, such as scanning “jitter”, sample drift, and STEM electron-source fluctuations add to the SCS broadening. The magnitude of these factors is not significant, as can be inspected in the experimental HAADF images, and they are of secondary importance in the presented semi-quantitative analysis.

### XRD-RSM

X-ray diffraction (XRD) rocking curves (2theta/omega scan) were measured by a Rigaku Smartlab (Japan) diffractometer with 9 kW rotating Cu anode X-ray generator. Ge(400) × 2 monochromator was employed to select Cu Kα1 line for the diffraction experiments. Diffraction patterns were registered with scintillation detector SC-70 (for 2D measurements, a linear Dtex/Ultra detector was employed). Epitaxial layer tilt with respect to the substrate was calculated from (004) RSM and accounted for in the (115) RSM analysis. Symmetric (004) and asymmetric (115) plane RSM were used to evaluate layer relaxations, lattice parameters, and Bi concentrations. For the analysis, the following lattice constants were used: GaAs = 5.653 Å, GaBi = 6.324 Å. Measurements of ½{111} superlattice reflections were performed in skew-symmetric configurations49.

### SIMS

The secondary ion mass spectrometry (SIMS) sample characterization was performed using CAMECA IMS6F system. Cesium (Cs+) primary beam was used at 14.5 keV energy and current kept at 55 nA. The size of the raster was about 150 × 150 micrometers, secondary ions were collected from the central region 60 micrometers in diameter. Secondary ions Ge, Bi, Ga, and As as reference signals, were collected. Ge and Bi concentrations were calculated based on RSF’s for GaAs, as measured with Cs beam. Bi concentration 1e21 at/cm3 correspond to 2,3% at. content of element (i.e. x = 4.6%).

### PL

Photoluminescence (PL) measurements were performed using a monochromator (Andor SR-500i, 500 mm focal length), and a thermoelectrically cooled InGaAs photodetector. For the excitation, diode-pumped solid-state laser emitting at 532 nm wavelength and intensity of ~100 W/cm2 were used. Samples were mounted in a closed-cycle helium cryostat with a temperature controller. Spectra were normalized at each temperature to the maximum of GaAsBi PL peak and shifted vertically for clarity.

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## Acknowledgements

This work was supported by the European Regional Development Fund (project No: 01.2.2-LMT-K-718-02-0020) under a grant agreement with the Research Council of Lithuania (LMTLT). This research was also supported by the Operational Program Research and Innovation: “Independent research and development of new superhard coatings and their characterization by advanced experimental techniques” (ITMS2014+ code: 313010T598), co-financed by European Regional Development Fund.

## Author information

Authors

### Contributions

T.P. acquired STEM data, analyzed STEM and PL data. V.P. synthesized the samples. S.S. acquired and analyzed XRD-RSM data. E.D. and B.C. acquired and analyzed PL data. M.S. and A.N. prepared TEM samples analyzed STEM data. V.S. prepared samples and analyzed sample surfaces. M.C. and V.V. acquired and analyzed STEM data. R.J. acquired and analyzed SIMS data. T.P., A.G., V.P. and A.K. conceived the idea and designed the experiments. All authors contributed to writing of the manuscript.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Paulauskas, T., Pačebutas, V., Geižutis, A. et al. GaAs1-xBix growth on Ge: anti-phase domains, ordering, and exciton localization. Sci Rep 10, 2002 (2020). https://doi.org/10.1038/s41598-020-58812-y

• Accepted:

• Published:

• ### Atomic-Resolution EDX, HAADF, and EELS Study of GaAs1-xBix Alloys

• , Vaidas Pačebutas
• , Renata Butkutė
• , Bronislovas Čechavičius
• , Arnas Naujokaitis
• , Mindaugas Kamarauskas
• , Martynas Skapas
• , Jan Devenson
• , Mária Čaplovičová
• , Viliam Vretenár
• , Xiaoyan Li
• , Mathieu Kociak
•  & Arūnas Krotkus

Nanoscale Research Letters (2020)


text:
# Makeindex - Styling the \printindex?

I have two questions:

1. Is it possible to change the titel of the printindex command? It is "Index" by default.
2. How can I format the index as a multi column, alphabetic sorted list?

-

the answers depend on what document class you're using. however,

1. both the basic latex classes (book et al.) and and ams-latex classes (amsbook et al.) provide the command \indexname with a default value "Index". so

\renewcommand{\indexname}{New name}

will change the title. (the babel package does this already for a number of languages.)

2. both book.cls and amsbook.cls provide a two-column index style. if you're creating your own class, you might consider "borrowing" the code from one of those classes. documented code is given in the .dtx files classes for basic latex and amsclass for the ams-latex classes. "human readable" versions of the documented code are given in .pdf files of the same names as the .dtx files. all are on ctan, and they are probably also on your tex installation; try typing texdoc classes or texdoc amsclass at the command line.

to get the content sorted, you need to use makeindex; that is well documented in several places, including lamport's latex manual and the "companionn", as well as with texdoc makeindex. be sure to pay attention to the use of the "sort field" for math entries, entries with accented letters, and other special cases that may cause entries to diverge from the expected alphabetical order.

-
1. Redefine \indexname. For example:

\renewcommand\indexname{The Index}

2. There are imakeidx and idxlayout packages to change the style of the index. Columns and headings, and other settings.

3. You can specify a .ist file when using makeindex program to change the output of makeindex. See the document of makeindex.

A sample .ist file:

% sample.ist
% Usage
%     makeindex -s sample.ist foo.aux
preamble "
\\begin{theindex}
\\providecommand*\\indexgroup[1]{\\indexspace
\\item \\textbf{#1}\\nopagebreak}
"

postamble "\n\n\\end{theindex}\n"

group_skip "  %\n  \\indexspace\n  %\n"

-

I'm just expanding Leo Liu's item #2 a little. imakeidx can easily help you on setting a custom index title and column numbers through (from documentation):

• title: is the title that is typeset at the beginning of the specific index; if not specified, the \indexname value is used.

• columns: accepts an integer representing the number of columns in the index; this is silently ignored if the original or the twocolumn options are set; the number can even be 1.

\usepackage{imakeidx}
\makeindex[title=My custom index title,columns=2]

will do the job. If you have, say, a custom mystyle.ist style, you can also use the options option as well:

• options: is the list of options to be passed to the sorting and formatting program; this list is a balanced text of program options, separated with the syntax required by the sorting and formatting program. For example, in order to use a different makeindex sorting and formatting style mystyle.ist and avoiding any message in the screen output write options=-s mystyle.

Regarding a custom makeidx style, unfortunately I have no experience with it. I strongly recommend chapter 11 from "The LaTeX Companion" book, called Index Generation. :)

-
1. Redefine \indexname like this : \renewcommand{\indexname}{My index}.

2. You define a .ist file that stores your preferences for the style of the index. For the example you want (alphabetic list starting by the letters and dots before page numbers), you add the following lines to your .ist file :

>     heading_prefix "{\\bfseries\\hfil "

and also

delim_0 "\\dotfill"
delim_1 "\\dotfill"
delim_2 "\\dotfill"

If it based on a book derived class, the index will be on two columns anyway.

-

I am using amsmidx. I had to use

\backmatter

Don't add the index name to \printindex or you will get two indexes with each name, use \renewcommand{\indexname}{My Index Name} instead. Thanks to two commenters below, but adding a \clearpage just before the second \addcontentsline made my table of contents disappear.
The \clearpage after \backmatter is superfluous because \backmatter will issue \clearpage resp. \cleardoublepage (depending on the state of @openright). On the other hand, a \clearpage before the second \addcontentsline is missing, and this will cause a wrong page number entry in the ToC. –  lockstep Jan 1 '12 at 21:40
amsmidx is meant to be used with amsbook, and is presumed to be incompatible with any non-ams document class. when used with amsbook, the proper access is \Printindex{author}{Author Index} \Printindex{subject}{Subject Index} (note the uppercase \Printindex); the toc entries are added automatically. amsmidx was created long before imakeidx; now that imakeidx exists, it's a much better choice. –  barbara beeton Nov 8 '14 at 13:57


text:
# nLab Burnside ring is equivariant stable cohomotopy of the point

Contents

### Context

#### Representation theory

representation theory

geometric representation theory

cohomology

# Contents

## Idea

In analogy to how the representation ring of a finite group is equivalently the equivariant K-theory of the point, so the equivariant stable cohomotopy of the point is the Burnside ring.

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation ring
$KU_G(\ast) \simeq R_{\mathbb{C}}(G)$
Atiyah-Segal completion theorem
$R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant)
complex cobordism cohomology
MU$MU_G(\ast)$completion theorem for complex cobordism cohomology
$MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant)
algebraic K-theory
$K \mathbb{F}_p$representation ring
$(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$
Rector completion theorem
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

## Statement

###### Proposition

(Burnside ring is equivariant stable cohomotopy of the point)

Let $G$ be a compact Lie group (for instance a finite group). Its Burnside ring $A(G)$ is isomorphic to the equivariant stable cohomotopy cohomology ring $\mathbb{S}_G(\ast)$ of the point in degree 0, via the Lefschwetz-Dold index:

$A(G) \underoverset{\simeq}{LD}{\longrightarrow} \mathbb{S}_G(\ast) \,.$

More in detail, for $G$ a finite group, this isomorphism identifies the Burnside character on the left with the fixed locus-degrees on the right, hence for all subgroups $H \subset G$ it identifies

1. the $H$-Burnside marks $\left\vert S^H \right\vert \in \mathbb{Z}$ of virtual finite G-sets $S$

(which, as $H \subset G$ ranges, completely characterize the G-set, by this Prop.)

2. the degrees $deg\left( \left(LD(S)\right)^H\right) \in \mathbb{Z}$ at $H$-fixed points of representative equivariant Cohomotopy cocycles $LD(S) \colon S^V \to S^V$

(which completely characterize the equivariant Cohomotopy-class by the equivariant Hopf degree theorem, this Prop.)

(1)$\array{ A(G) &\underoverset{\simeq}{LD}{\longrightarrow}& \underset{\longrightarrow_{\mathrlap{V}}}{\lim} \;\; \left( \pi_0 \mathrm{Maps}^{\{0\}/} \left( S^V, S^V \right)^G \right) &=& \mathbb{S}_G(\ast) \\ S &\mapsto& LD(S) \\ \underset{ \mathclap{ \text{Burnside character} } }{ \underbrace{ \left( H \mapsto \left\vert S^H \right\vert \right) } } &=& \underset{ \mathclap{ \text{degrees on fixed strata} } }{ \underbrace{ \left( H \;\mapsto\; deg \left( S^{ dim\left( V^H\right) } \overset{\big(LD(S)\big)^H}{\longrightarrow} S^{ dim\left( V^H\right) } \right) \right) } } }$

For $G$ a compact Lie group this correspondence remains intact, with the relevant conditions on subgroups $H$ (closed subgroups such that the Weyl group $W_G(H) \coloneqq N_G(H)/H$ is a finite group) and generalizing the Burnside marks to the Euler characteristic of fixed loci.

The statement is due to Segal 71, a detailed proof making manifest the correspondence (1) is given by tom Dieck 79, theorem 8.5.1. See also tom Dieck-Petrie 78, Lück 05, theorem 1.13.

From a broader perspective of equivariant stable homotopy theory, this statement is a special case of tom Dieck splitting of equivariant suspension spectra (e.g. Schwede 15, theorem 6.14), see there.

## References

Last revised on February 20, 2019 at 09:50:03. See the history of this page for a list of all contributions to it.


text:
My watch list
my.chemeurope.com

# Fused quartz

Fused quartz and fused silica are types of glass containing primarily silica in amorphous (non-crystalline) form. They are manufactured using several different processes.

Fused quartz is made by melting high-purity naturally occurring quartz crystal at around 2000°C using either an electrically heated furnace (electrically fused) or a gas/oxygen-fuelled furnace (flame fused). Fused quartz is normally transparent.

Fused quartz can also form naturally. The naturally occurring form of fused quartz is usually referrred to as Metaquartzite and is formed under metamorphic conditions. Due to increased heat the crystals within the quartz become fused together.

Fused silica is produced using high purity silica sand as the feedstock, and is normally melted using an electric furnace, resulting in a material that is translucent or opaque. (This opacity is caused by very small air bubbles trapped within the material.)

Synthetic fused silica is made from a silicon-rich chemical precursor usually using a continuous flame hydrolysis process which involves chemical gasification of silicon, oxidation of this gas to silicon dioxide, and thermal fusion of the resulting dust (although there are alternative processes). This results in a transparent glass with an ultra-high purity and improved optical transmission in the deep ultraviolet. One common method involves adding silicon tetrachloride to a hydrogen-oxygen flame, however use of this precursor results in environmentally unfriendly by-products including chlorine and hydrochloric acid. To eliminate these by-products, new processes have been developed using an alternative feedstock, which has also resulted in a higher purity fused silica with further improved deep ultraviolet transmission.

Fumed silica is manufactured by a similar flame hydrolysis process to synthetic fused silica, however it is in the form of a fine powder/dust and is typically used in applications such as fillers for rubbers and plastics, coatings, adhesives, cements, sealants, cosmetics, pharmaceuticals, inks and abrasives.

The optical and thermal properties are superior to those of other types of glass due to its purity (or rather, its lack of impurities). For these reasons, it finds use in situations such as semiconductor fabrication and laboratory equipment. It has better ultraviolet transmission than most other glasses, and so is used to make lenses and other optics for the ultraviolet spectrum. Its low coefficient of thermal expansion also makes it a useful material for precision mirror substrates.

## Chemistry

Fused quartz is a noncrystalline form of silicon dioxide (SiO2), which is also called silica. (The crystalline form of this material is quartz).

## Applications

Specially prepared fused silica is also the key starting material used to make optical fiber for telecommunications.

Because of its strength and high melting point (compared to ordinary glass), fused silica is used as the envelope of halogen lamps, which must operate at a high envelope temperature to achieve their combination of high brightness and long life.

The combination of strength, thermal stability, and UV transparency makes it an excellent substrate for projection masks for photolithography.

Due to the thermal stability and composition it is used in the semiconductor fabrication furnaces.

Fused quartz has nearly ideal properties for fabricating first surface mirrors such as those used in telescopes. The material behaves in a predictable way and allows the optical fabricator to put a very smooth polish onto the surface and produce the desired figure with fewer testing iterations.

Fused silica as an industrial raw material is used to make various refractory shapes such as crucibles, trays, shrouds, and rollers for many high temperature thermal processes including steel making, foundries, and glass manufacture. Refractory shapes made from fused silica have excellent thermal shock resistance and are chemically inert to most elements and compounds including virtually all acids, regardless of concentration. Translucent fused silica tubes are commonly used to sheathe electric elements in room heaters, industrial furnaces and other similar applications.

## Physical properties

The extremely low coefficient of thermal expansion accounts for its remarkable ability to undergo large, rapid temperature changes without cracking (see thermal shock).

"UV grade" synthetic fused silica (sold under various tradenames including "HPFS", "Spectrosil" and "Suprasil") has a very low metallic impurity content making it transparent deeper into the ultraviolet. An optic with a thickness of 1cm will have a transmittance of about 50% at a wavelength of 170 nm, which drops to only a few percent at 160 nm. However, its infrared transmission is limited by strong water absorptions at 2.2 μm and 2.7 μm.

"IR grade" fused quartz (tradenames "Infrasil", "Vitreosil IR" and others) which is electrically fused, has a greater presence of metallic impurities, limiting its UV transmittance wavelength to around 250 nm, but a much lower water content, leading to excellent infrared transmission up to 3.6 μm wavelength. All grades of transparent fused quartz/fused silica have near-identical physical properties.

The water content (and therefore infrared transmission of fused quartz and fused silica) is determined by the manufacturing process. Flame fused material always has a higher water content due to the combination of the hydrocarbons and oxygen fuelling the furnace forming hydroxyl [OH] within the material. An IR grade material typically has an [OH] content of <10 parts per million.

## Optical properties

Dispersion of fused silica can be approximated by the following Sellmeier equation (Malitson 1965):

$\varepsilon=1+\frac{a_1\lambda^2}{\lambda^2-l_1^2}+\frac{a_2\lambda^2}{\lambda^2-l_2^2}+\frac{a_3\lambda^2}{\lambda^2-l_3^2},$

where

$a_1=0.69616630, \quad l_1=0.068404300,$
$a_2=0.40794260, \quad l_2=0.11624140,$
$a_3=0.89747940, \quad l_3=9.8961610,$

and wavelength is measured in micrometers.


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# Surface area of a solid of revolution, general approach

To calculate the surface area of a solid of revolution I have the following formulae for "4 cases" I've observed

## 1-When the curve rotates around the X-axis

i) When the limit of integration is in the X-axis

$$S = \int 2 \pi f(x) \sqrt{1+(f'(x))^2}dx\$$ |Where $$f(x)=function$$

ii) When the limit of integration is in the Y-axis

$$S = \int 2 \pi y \sqrt{1+(f'(y))^2}dy$$ |Where $$y = a\ number\ on\ the\ Y-axis$$

## 2-When the curve rotates around the Y-axis

i) When the limit of integration is in the X-axis

$$S = \int 2 \pi x \sqrt{1+(f'(x))^2}dx$$ |Where $$x = a\ number\ on\ the\ X-axis$$

ii) When the limit of integration is in the Y-axis

$$S = \int 2 \pi f(y) \sqrt{1+(f'(y))^2}dy$$ |Where $$f(y)=function$$

I was wondering if my assessment of the math was correct since I still get confused when working on exercises.

• Everything looks good to go. Just know how to translate these formulas if necessary to other axes of rotation parallel to either the $x$ or $y$ axes. – Ninad Munshi Nov 6 '19 at 6:18
• We are only working with the X and Y axes of rotation! – Max Nov 6 '19 at 6:54

The general approach you seek expressed quite succinctly in Pappus's $$(1^{st})$$ Centroid Theorem.
This theorem states that the surface area $$A$$ of a surface of revolution generated by rotating a plane curve $$C$$ about an axis external to $$C$$ and on the same plane is equal to the product of the arc length $$s$$ of $$C$$ and the distance $$d$$ traveled by its geometric centroid. Simply put, $$S=2\pi RL$$, where $$R$$ is the normal distance of the centroid to the axis of revolution and $$L$$ is curve length. The centroid of a curve is given by
$$\mathbf{R}=\frac{\int \mathbf{r}ds}{\int ds}=\frac{1}{L} \int \mathbf{r}ds$$


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# Lévy's C-curve

Lévy's C-curve
The Lévy's C-curve is a self-similar fractal.

# Basic Description

Like the Harter-Heighway Dragon Curve, the Lévy's C-curve is constructed by starting with a single straight line.

We then use this line as the hypotenuse for an isosceles right triangle (45-45-90). Then, the original line is removed, leaving the two legs of the newly created triangle. This process is illustrated below, with black being the initial lines and blue being the constructed lines.

We continue this process of using lines as bases for constructing isosceles right triangles and then removing the original lines. Again, we use black for the lines constructed before the current step and blue for the lines constructed during this step.

The first eight iterations of this process are shown below.

The twelfth iteration of the curve already has the same general shape as the main image on this page.

The curve itself is created by iterating this process infinitely. The main image is created by using a computer to generate thousands of iterations and then adjusting lighting effects.

# A More Mathematical Explanation

## Number of Sides

For the first iteration, we see that we end up with two lines, so the number of s [...]

## Number of Sides

For the first iteration, we see that we end up with two lines, so the number of sides is two.

After the second iteration, each of those two lines as turned into two more, so there are four sides.

This pattern of doubling continues, and so the number of sides of the Lévy's C-curve for any degree of iteration (k) is given by $N_k = 2^k\,$.

## Length of Sides

Looking at the first few iterations, we can see how the length of the sides changes. Assuming that the original base has length one, we can see that the two legs of the triangle must have length $\ell_1= \frac{1}{\sqrt{2}}=\frac{1}{2^{1/2}}$, since we must have that $1^2=\ell_1^2+\ell_1^2=2\ell_1^2$

At the second iteration, we have $\ell_1$ as the hypotenuse of our right triangle, so:

$\left(\frac{1}{\sqrt{2}}\right)^2= \ell_2^2+\ell_2^2=2\ell_2^2$

$\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)=\ell_2^2$

$\ell_2=\frac{1}{2}=\frac{1}{2^{2/2}}$.

For the third iteration, we now have $\ell_2$ as the hypotenuse of our right triangle and we now have

$\left(\frac{1}{2}\right)^2= \ell_3^2+\ell_3^2=2\ell_3^2$

$\left(\frac{1}{4}\right)\left(\frac{1}{2}\right)=\ell_3^2$

$\ell_2=\frac{1}{\sqrt{8}}=\frac{1}{2^{3/2}}$.

We can now see that the length of a side for each iteration is given by $\ell_k=\frac{1}{2^{k/2}}$

## Length of the Curve

The length of the curve at any iteration is simply the number of sides for that iteration times the length of each side. Therefore the length for any iteration k is $(N_k)(\ell_k) (2^k)\left(\frac{1}{2^{k/2}}\right)=2^{k/2}$.

As the curve is iterated infinitely, we see that the total length approaches infinity since $\lim_{k\rightarrow \infty} 2^{k/2}=\infty$

## Fractal Dimension

The fractal dimension of Lévy's C-curve can be calculated as follows.

We have already found that $N_k = 2^k$, so the N in the formula for fractal dimension is 2.

Since the length of the sides is given by $\ell_k=\frac{1}{2^{k/2}}$, we see that each side is scaled down by a factor of $2^{1/2}$ from the previous iteration. So we have that e in the formula for fractal dimension is $2^{1/2}$.

$D = \frac{\log N}{\log\left({e}\right)} = \frac{\log 2}{\log 2^{1/2} }=\frac{1}{1/2}=2.$

## Angle

The Lévy's C-curve iterates with a 90 degree angle but variations of the curve can be created by changing this angle.

# Teaching Materials

-A good animation of a Lévy C curve being drawn. -A picture of a Lévy C curve variation with an angle of 60 degrees.


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Browse Questions

# Refractive index for a piece of transparent material is greatest for

(a) red light

(b) violet light

(c) green light

(d) yellow light

Can you answer this question?

For the visible range wavelength for red light is greater and violet light is least
we have $\mu \alpha \large\frac{1}{\lambda}$
$\lambda _r > \lambda_v$
$\therefore \mu_v > \mu _r$
Hence b is the correct answer.

answered Jan 16, 2014 by
edited Jul 14, 2014


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# Acceptance

There’s a certain type of work I do for which I write a short proposal in single-page letter format and send it to the prospective client as a PDF attached to an email. When I started doing this type of work a few years ago, my thinking was that the client would simply email me back saying they accept the proposal and that I could start the work. But many clients wanted something more formal, or at least more analog.

They wanted something they could sign and send back to me. So I made a small PDF image I could use as a sort of stamp.

After writing the proposal in LaTeX and generating a PDF, I’d open it in PDFpen and add the stamp. At first I’d just open the stamp file, copy it out of that document, and paste it into the proposal. Then I got more clever and added the stamp to my PDFpen Library, so there was no need to open a second file.

The problem with adding the stamp this way was twofold: I had to remember to do it, and then I had to do some manual work in PDFpen. Given that the stamp would always be in the same spot—lower right corner, with a ¾″ margin from the page edges—it seemed like I could get rid of the manual part.

I probably could have written an AppleScript to control PDFpen, but when I finally go around to automating it, I used PDFtk and the stamp option I wrote about last year. What I built was a short shell script that was basically the same as the overlay script I described in that post, but with one of the PDF files being set to the acceptance stamp file, which I had to “grow out” to be a full page.

This PDFtk automation didn’t last long. It solved the manual placement problem, but I still had to remember to run the script, and it turned out that that was the more important problem to solve. I began searching for a simple way to add the stamp directly in LaTeX.

There are undoubtedly ways to add the stamp using TikZ or some similar drawing package, but I didn’t want to learn a graphics macro language just to remake a drawing I already had. What I ended up using is the pdfoverlay package, which let me put my acceptance stamp in the background and layer the proposal letter over it.1

Pdfoverlay is included in recent TeX Live and MacTeX distributions. I don’t remember how old my installation was, but I had to update it to get pdfoverlay installed and working. Once I did, though, these two lines were all I needed to add to the preamble to my proposal letters:

\usepackage{pdfoverlay}
\pdfoverlaySetPDF{/path/to/acceptance.pdf}

I edited the TextExpander snippet I use for my proposal template to include these two lines, and now I don’t have to remember anything. Which is one of the main goals of automation, right?

1. So it’s more like the inverse of a stamp.


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Is it bad that my first thought when I hear "government shutdown" is "aha! natural experiment!" ?

A Mastodon instance for maths people. The kind of people who make $\pi z^2 \times a$ jokes.

Use $ and $ for inline LaTeX, and $ and $ for display mode.


text:
### Always Two

Find all the triples of numbers a, b, c such that each one of them plus the product of the other two is always 2.

### Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

### Surds

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

# Relative Angles

##### Age 14 to 16 Short Challenge Level:

Answer: $60^\circ$

Finding angle B in terms of angle A and angle C
$B=\frac{3}{4}C$ and $B =\frac{3}{2}A$, so \begin{align} \tfrac{3}{4}C&=\tfrac{3}{2}A \\ \Rightarrow \tfrac{1}{4}C&=\tfrac{1}{2}A \\ \Rightarrow C&=2A\end{align}.

Since $A$, $B$ and $C$ are the angles of a triangle, $A+B+C=180$.

So, substituting in $B$ and $C$ in terms of $A$, \begin{align}A+\tfrac32A+2A&=180\\ \Rightarrow \tfrac{9}{2}A&=180\\ \Rightarrow \tfrac{1}{2}A&=20\\ \Rightarrow A&=40\end{align} So $B=\frac{3}{2}\times40=60$.

Finding angles A and C in terms of angle B
$B=\frac{3}{4}C$ so $C=\frac{4}{3}B$.

And $B=\frac{3}{2}A$ so $A=\frac{2}{3}B$.

Since A, B and C are the angles of a triangle, $A+B+C=180$, so \begin{align}\tfrac{2}{3}B+B+\tfrac{4}{3}B&=180\\ \Rightarrow 3B&=180\\ \Rightarrow B&=60\end{align}
You can find more short problems, arranged by curriculum topic, in our short problems collection.


text:
Simple pure python functions for simulating tennis matches

# tennisim

Simulate tennis points, games, sets and matches.

Small pure python package (no dependencies outside of standard lib) to simulate tennis using points-based modelling i.e. given a probability of a server winning a given point, simulate the outcome of:

• points
• games
• sets
• tiebreaks
• matches

with the ability to alter various parameters to gain some intuition behind how the tennis scoring system impacts matches. Using this we can answer various questions like:

• what effect does removing the second serve have on match duration?
• if the probability of winning a point on serve is reflective of skill, then how do rule alterations affect the likelihood that the more skillful player will actually end up winning the match (not just a given point)?

# Installing

pip install tennisim

# Points-based Modelling

Points-based modelling is a popular model for modelling tennis matches where predictions for games, sets and matches depends on modelling every constituent point. This can lead to a wealth of data that can be used to generate in-play match odds as we can output distributions e.g. for a given starting point, if we simulate 1000 outcomes how many show that player 1 wins the next set?

# Example

Suppose we want to simulate a game of tennis. We define the probability that the server will win a given point:

from tennisim.sim import sim_game

p = 0.7
sim_game(p)

This will simulate 1 game of tennis where the probability that the server will win any given point is 0.7. It will return a tuple containing:

• boolean result for if the server won the game
• list of tuples for the score progression of the game

We can then take this further and simulate 1,000 groups of 100 games to generate a distribution of results. This can be interesting when looking at how changes in the probability p or the length of a game impacts the servers win probability for the game.

import numpy as np
from tennisim.sim import sim_game

# set params for simulation
games = 100
sims = 1000
probabs = np.linspace(0.5, 0.8, 4)
results = {}

# for each serve win probability
for p in probabs:
# we now need to generate sims
means = []
game_lengths = []
for i in range(0, sims):
g_results = [sim_game(p) for x in range(games)]
# get mean result
mean_res = np.mean([x[0] for x in g_results])
# get mean game length
mean_length = np.mean([len(x[1]) for x in g_results])
# join to results holders
means.append(mean_res)
game_lengths.append(mean_length)
# add data to probab dict
results[p] = (means, game_lengths)

# Help

tennisim is free and open source software, distributed under the terms of the MIT license.

## Project details

Uploaded source
Uploaded py3


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May 29

## MS Office: “Word could not create the work file. Check the temp environment variable”

The error message “Word could not create the work file. Check the temp environment variable” is  caused by incorrect registry values in the key:

HKEY_CURRENT_USER\Software\Microsoft\Windows\Current Version\Explorer\User Shell Folders

Run regedit and go to the above path.

On the right hand pane, check the value data for Cache.

It has to be:

%USERPROFILE%\AppData\Local\Microsoft\Windows\Temporary Internet Files

If it is different, double click on Cache and type:

%USERPROFILE%\AppData\Local\Microsoft\Windows\Temporary Internet Files

under value data and click ok.

Close the registry (and restart the computer) and check if it works.

written by MG \\ tags: , ,

Apr 02

## PDF: Printing Multiple Pages per Sheet

To print multiple pages per sheet of paper:

2. Choose “Multiple pages per sheet” from the Page Scaling menu
3. Choose the number of pages per sheet you would like to print, and preview the layout in the Preview window

written by MG \\ tags: ,

Mar 29

## How to display Romanian characters with php/html

`First you need to specify character set for HTML output.. in your page header add:`

`<meta http-equiv="Content-Type" content="text/html; charset=utf-8">`

`in PHP you can use (this line must be first in your code): `

`<?php header("Content-type: text/html; charset=utf-8");?>`

`then you need to write a .htaccess in your root dir of your website:`

`# Set httpd charset to utf-8 `
`# Set php charset to utf-8 and setup mbstring (you may need to install mbstring module)`
php_value default_charset utf-8
php_value mbstring.internal_encoding utf-8

http://www.webdesignbox.ro/utf-8-in-paginile-web-diacritice-caractere-speciale-suport-multilingv

written by MG \\ tags: , , ,

Nov 21

## Use query_posts() and pagination

These two lines of code basically use the WordPress query_posts()
function to exclude posts in a certain category. The first line of code
fixes the pagination issue of all posts being displayed on all pages.
Again, let’s say that we want to exclude all posts from the category
with an ID of 8 from our index page. Simply place the following code in
your index.php file before this line <?php if (have_posts()) : while (have_posts()) : the_post(); ?>

 `1` `
 `2` ` ``query_posts(``"cat=-8&paged=\$paged"``); ?>`

### Issues with this method:

This method will remove posts that are ONLY in the
excluded category. If a post is in the excluded category as well as

written by MG \\ tags: ,

Nov 21

## BSD: How to recover from the Registry cannot load the hive

Stop: C0000218 {Registry File Failure} Registry cannot load the hive (file): \SystemRoot\System32\Config\SOFTWARE

Nothing is more frustrating than a Windows Blue Screen of Death. The Blue Screen of Death will keep your PC from booting and the messages are usually pretty cryptic. This message is no different. What it means is that the SOFTWARE registry key is corrupt.

Fortunately, the fix for this error is not as painful as the message would lead you to believe. The following steps must be taken to fix this error:

-Boot your PC from the Windows 2000 or Windows XP CD and go into the recovery  console by choosing “R”

-Type the following at the command prompt to replace your current SOFTWARE registry key with a repair copy.
```ren c:\windows\system32\config\software software.old copy c:\windows\repair\software c:\windows\system32\config```

If your computer is just part of a workgroup, you should be back in business. However, if you computer is part of a Domain, you may have to reinstall some applications and run:
`gpupdate /force`

This will refresh group policy. In any case the steps in outlined above should let you log back in to your PC.

written by MG \\ tags:

Nov 17

## arcconf: How to add a hotspare drive to a logical drive

If you have a IBM ServeRAID 8k controller and you have changed a defunct drive with new one,

Usage: SETSTATE <Controller#> DEVICE <Channel# ID#> <State> [LOGICALDRIVE <LD#> [LD#] … ] [noprompt]

if you are in this particular situation :

DEVICE is 1

disk is 0,5

and logical drive is 1

./arcconf setstate 1 device 0 5 hsp logicaldrive 1

written by MG \\ tags: , ,

Aug 31

## Linux: Starting the Installation in Text Mode

If you do not wish to perform a GUI installation, you can start a text  mode installation using the following boot command:

 `boot: text`

If you are having trouble booting into the graphical installation program,  you can try to boot using the no framebuffer (nofb) boot option.

At the boot command, enter the following:

 `boot: nofb`

This option allows you to use the graphical installation program without using a framebuffer.

written by MG \\ tags: , ,


text:
# Thinking about starting a new service

#### Daxton Brumfield

##### Member
I am thinking about starting a new service where you would ship a non magnetic cube to me, and I would magnetize it, set it up, and ship it back. Would anybody be interested in this?

#### KingCanyon

##### Member
Yes, I may be interested in that. It would just depend on price.

#### Daxton Brumfield

It probably does somewhere, but I don't know of anywhere that does it except like the cubicle and speedcubeshop, but you can't send in cubes that you already have there as far as I know. They also charge a ton. I would probably only charge like $5 for magnetization and set up, and I would have a big variety of magnets you could choose from to magnetize your cube, and I have all of the SCS and cubicle lubes. IDK if I will do anything for sure, just curious if anybody would actually want that before I go out and buy a ton of magnets or anything. #### Daxton Brumfield ##### Member Assuming the customer pays for shipping, do you think$5 will cover the cost of magnets, glue, lube and time spent working?

There may also be an issue with you being paid to work with hazardous materials (cyanoacrylate) while being a minor. You'd need to research the specific laws for that though, as well as whether you would be able to operate a business from your home and what taxes you would pay on your earnings.
IDK about the laws, but I am positive that $5 will make me at least a few dollars profit for each cube. Magnets are only like 50 cents for 48 off them, and glue is super cheap. I have a ton of lube that I can use as well so I am not that worried about any of that. I would have the customer pay shipping though. If I do it as a small scale business in my room I don't think I will have to pay taxes on it. This is all just theoretical right now though. #### ImmolatedMarmoset ##### Member Assuming the customer pays for shipping, do you think$5 will cover the cost of magnets, glue, lube and time spent working?

There may also be an issue with you being paid to work with hazardous materials (cyanoacrylate) while being a minor. You'd need to research the specific laws for that though, as well as whether you would be able to operate a business from your home and what taxes you would pay on your earnings.
IDK about the laws, but I am positive that \$5 will make me at least a few dollars profit for each cube. Magnets are only like 50 cents for 48 off them, and glue is super cheap. I have a ton of lube that I can use as well so I am not that worried about any of that. I would have the customer pay shipping though. If I do it as a small scale business in my room I don't think I will have to pay taxes on it. This is all just theoretical right now though.

#### Daxton Brumfield

##### Member
ok buddy. Looks like I am not gona do this buisness lol

#### ImmolatedMarmoset

##### Member
ok buddy. Looks like I am not gona do this buisness lol
why not?
the document clearly states that unless you make 12000 dollars a year, you’re fine!

Last edited:

#### ProStar

##### Member
why not?
the document clearly states that unless you make 1200 dollars a year, you’re fine!
So all you gotta do is not be successful! XD

#### Daxton Brumfield

##### Member
So all you gotta do is not be successful! XD
Ok good advice. I'm gonna start this business and have it only be kinda successful

#### ProStar

##### Member
why not?
the document clearly states that unless you make 1200 dollars a year, you’re fine!
by "make" are you(or it) talking about gross or net?

#### ImmolatedMarmoset

##### Member
by "make" are you(or it) talking about gross or net?
1. I meant to say 12000
2. not sure, believe it is gross

Ok good advice. I'm gonna start this business and have it only be kinda successful
I meant to say 12000 dollars! So you can be quite successful!

#### Etotheipi

##### Member
I meant to say 12000 dollars! So you can be quite successful!
So you can service a maximum of 2400 cubes per year.

#### ProStar

##### Member
So you can service a maximum of 2400 cubes per year.
If it refers to gross sales, then yes. If it refers to net, then it would depend on how much he'd make on each cube.

(@Daxton Brumfield I'm not asking how much you'd make, there's no reason to tell us)

#### Daxton Brumfield

##### Member
I probably would not make that much. I would mostly do it for fun because I really like to set up cubes and magnetize cubes while watching netflix or doing other stuff that does not require that much of my attention.

#### ProStar

##### Member
I probably would not make that much. I would mostly do it for fun because I really like to set up cubes and magnetize cubes while watching netflix or doing other stuff that does not require that much of my attention.
Makes since to me. You like doing it, you pass the time, and you get some money. I say go for it!

#### RyanP12

##### Member
You need to differentiate. Start looking into boron treatment and polymer coating. If you become financially stable, start an LLC. Depending on age, it’s looks pretty good in college applications, “started a legal corporate entity”.


text:
# Tag Info

0

Often, it is desirable to apply Simplify or FullSimplify to parts of an expression and then recombine the parts. A particularly effective function for this purpose is Collect. For instance, Collect[expr, a^2 - b^2, Simplify] (* (4 a^2 - 3 b^2)/(a^2 - b^2)^2 - 8/(b^2 - c^2)^2 *) Although there is no need to use FullSimplify in this case, it runs much ...

4

I think the following code will do the trick: Collect[expr, Derivative[_, _][g][_, _]] If you want Mathematica to try to simplify each "coefficient", you can use this version instead: Collect[expr, Derivative[_, _][g][_, _], Simplify] I can't claim full credit for this code — there's an example in the "Scope" section of the documentation for Collect ...

5

Just to summarize my understanding: First of all, the documentation states that Simplify assumes that variables are real when they occur algebraically in inequalities. Clearly, there are no inequalities in the logical expression y == 0 && x^2 == -1, and therefore x and y should be assumed to be general complex numbers. If they were real, then the ...

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Use ComplexExpand expr1 = E^(-2*t*g)*Sqrt[E^(4*t*(g + 2*I*l))]; expr2 = expr1 // ComplexExpand // FullSimplify[#, {l > 0, g > 0, t > 0}] & Sqrt[E^(8*Ilt)] expr1 == expr2 // FullSimplify[#, {l > 0, g > 0, t > 0}] & True

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It's a badly written example. What it is trying to show is something like the following behavior, where the local Assumptions option overrides the global $Assumptions.$Assumptions = x == -1; FullSimplify[E^(LogGamma[x] + LogGamma[y])] FullSimplify::infd : "Expression LogGamma[x] + LogGamma[y] simplified to ∞. ∞ FullSimplify[E^(LogGamma[x] + ...

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The following works on the two "Simple Examples" logSimplify = FullSimplify[#, (# > 0) & /@ Union@Cases[Cases[#, Log[z_] -> z, Infinity], z_Symbol -> z, Infinity]] & logSimplify[x Log[a] + x Log[b]] (* x Log[a b] *) logSimplify[x Log[a/b] + y Log[b/a]] (* (x - y) Log[a/b] *) and also on some variants logSimplify[x Log[a^2/b^3] + y ...

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Maybe you could extract the arguments of your Logs, then turn them into a list of assumptions to feed to e.g. FullSimplify: arguments = Cases[ {(3 Log[whatever/argument + you have] + 5)/2 + some other crud / and + many Exp[4^more] Log[expressions]}, Log[a_] -> a, Infinity ]; arguments > 0 (* Out: {expressions > 0, ...

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Will Assumptions work? Log[z^2] // FullSimplify (* Log[z^2] *) Log[z^2] // FullSimplify[#, z > 0] & (* 2 Log[z] *) Edit: Naive implementation of rules. LSimpl[f_] := f //. Log[x_*y_] -> Log[x] + Log[y] //. Log[x_^n_Integer] -> n*Log[x]; LSimpl[x Log[a/b] + y Log[b/a]] (* x (Log[a] - Log[b]) + y (-Log[a] + Log[b]) *)

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I feel I've worked on a problem like this before. Maybe there is a similar question on the site, but I can't find it. It reminds me a little of Why does Mathematica simplify $x/x\to1$?, in which the symbolic result is generically true. Analysis The method used in the sum is Method -> "IteratedSummation" so we can examine the major steps: Sum[ ...

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EDIT The CoefficientRules method given below won't work if one wants to collect by patterns like Log[_] or _Log (although it will work if the stuff inside the Log is explicitly given). In this case, my instinct is that Collect is then the way to go, and the method supplied by the OP will work pretty well as it is. CoefficientRules method Variables ...

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I can see your surprise, but Mathematica seems to be behaving as you would expect according to the documentation. In particular, it may seem careless to you, but it is clearly spelled out in the documentation that any variable involved in an inequality is implicitly considered a real number. As a reference, see: the Assumptions and Domains page of the ...

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As pointed out in the comments by "Guess who it is" and "BlacKow", the solution is to use the function ComplexExpand[], or to add ComplexExpand under Re[].

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This function lives in the system as Simplify`SimplifyCount.

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Based on Vladimirs solution I wanted to post a faster alternative to SimplifyCount which produces the same results as SimplifyCount, but is a factor 3 faster. This can be very significant in case of complicated functions, it is however still significantly slower then Automatic. myNumberComplexity[x_Integer] := If[Positive[x], IntegerLength[x] - 1, ...

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Ok so I decided to change my initial code with something less hacky. I found the relevant idea on a page. I am not sure I am allowed to post the link here, so I will put the reference in a comment. The first step is now the following function. getCommonSubexpressions[expr_, OptionsPattern[]] := Block[ {vnum, modExpr, subExprs, subExpr, mapping, ...

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text:
# What's the significance of a Killing horizon?

A Killing horizon is defined as a null hypersurface generated by a Killing vector, which is then null at that surface. Some often cited examples come from the Kerr spacetime, where the Killing vector $$\partial_t$$ becomes null at the ergosphere; one can also take a linear combination of that with $$\partial_\phi$$, which is null at the event horizon.

From this it would seem that Killing horizons are related to "special" surfaces of the spacetime, although not always: Minkowski spacetime has Killing horizons at any point, due to its high degree of symmetry. Can we make some general statements about Killing horizons besides what the definition says? Are they related to event horizons in any predictable way? If I have a metric and I find a Killing horizon, what can I say about it?

• I am not sure that Minkowski has a Killing horizon at any point, only at the lightcone May 20 '20 at 21:08
• @spiridon_the_sun_rotator but you can translate that to any point, can't you? May 20 '20 at 21:18
• You're right, observer can be located at any point of the spacetime May 20 '20 at 21:22
• It is worth pointing out that the ergosphere is not a Killing horizon. Although $\partial_t$ is null there, the ergosphere is not a null hypersurface. Only the event horizon is a Killing horizon in this case (ignoring the interior) with Killing vector $\partial_t + \Omega_H \partial_\phi$ May 21 '20 at 16:50

The "special" surfaces of spacetime defined by Killing horizons are null hypersurfaces. A null hypersurface which is a hypersurface whose normal vector at every point is a null vector (with respect to the local metric tensor).

The "boring" and trivial example is a light cone, as already mentioned. EDIT: from the comments, it is true that this statement is not true for Minkowski spacetime. Then I must say I am not sure when this applies.

In terms of other applications, I can think of two, even though they are quite interconnected: black holes event horizons, and surface gravity $$\kappa$$. A nice set of slides with useful discussions about this can be found here:

### Black holes

Taken verbatim from here:

[...] by Hawking’s rigidity theorem the event horizon of a stationary, asymptotically flat black hole spacetime (supplemented by certain additional assumptions, see [19] for a review), is a Killing horizon. In fact, one often uses the notion of a Killing horizon to provide a quasi-local definition of an equilibrium black hole.

I just wanted to show the above as a "quantitative" connection to the event horizon, as you asked. In this case, then, you can see that a physical meaning (albeit asympotically) of a Killing horizon is that it corresponds to the event horizon.

But black holes are also entering the picture through surface gravity. See below.

### Surface gravity

Surface gravity has a meaning in Newtonian/classical gravity, which is not the same in GR. Maybe the same name was used, historically, because one wanted to define the same object. But the two things have differences nowadays. Especially in black holes.

The physical meaning of the GR surface gravity $$\kappa$$ of a static Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if $$k^{a}$$ is a suitably normalised Killing vector, then the surface gravity is defined by

$$k^a ∇_a k^b = \kappa k^b ,$$

where the equation is evaluated at the horizon. Specific solutions for black hole metrics are listed here.

Surface gravity is "physically" interesting because it is related to the temperature of Hawking radiation $$T_{\mathrm{H}}$$:

$$T_{\mathrm{H}} = \frac{\hbar c\kappa}{2 \pi k_{\mathrm{B}}}.$$

• OK, this sounds like asking too much but is there a bumper sticker for Killing horizons the way there is one for event horizons? Can I say something short like I can say the following about event horizons: "things cannot return to their previous causal patch of Penrose diagram after crossing this surface"? Sure, the definition of Killing horizon itself is pretty short but something in terms of how things behave around it rather than how mathematical objects behave around it if that makes sense. May 21 '20 at 7:29
• Don’t know about that sorry May 21 '20 at 16:45
• For Minkowski space, a light cone is not a Killing horizon because there is no Killing vector tangent to it. Instead, people usually talk about Rindler space which is a wedge of Minkowski space, and its horizon is the point where a boost Killing vector becomes null. May 21 '20 at 16:52
• First small comment: the light cone is not a Killing horizon but the surface $x^2-t^2=0$ is, right? Anyway, thanks for the answer! I was mostly interested in the connection to event horizons, and the Hawking theorem seems like a good place to start. May 21 '20 at 21:32
• Yes. It turns out the derivation for the Minkowski case is on the Wikipedia page for Killing horizons May 21 '20 at 21:35