text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Various effects of magnetite on international simple glass (ISG) dissolution: implications for the long-term durability of nuclear glasses

Abstract

Understanding the effect of near-field materials, such as iron corrosion products, on the alteration of vitreous nuclear waste is essential for modeling long-term stability of these waste forms in a geological repository. This work presents experimental results for which monoliths of International Simple Glass—a six oxide borosilicate glass–, with polished and unpolished cut sides, were aged for 70 days under oxic conditions at 90 °C in a solution initially saturated in 29SiO2 at pH 7; then magnetite was added to the leaching environment. Solution and solid analyses were performed to correlate the changes in the surface features and dissolution kinetics. It was found that magnetite primarily influences the mechanically constrained surface of the non-polished sides of the monoliths, with little to no effect on the polished surfaces. This work highlights the importance of the unique chemistry within surface cracks that invokes a drastic change in alteration of glass in environments containing iron corrosion products.

Introduction

Confinement within a borosilicate glass matrix is currently proposed for the disposal of radionuclides remaining from used nuclear fuel from power reactors.1 In France, glass containing ~18% (w/w) waste is poured into a stainless steel canister. Before disposal in a deep geological repository, the canister will be placed into a several centimeter thick carbon steel over pack.2 A better understanding of the interactions between glass and iron and the associated iron corrosion products is necessary to assess the performance of these waste forms.3

Work has been performed to understand the fundamental mechanisms controlling glass dissolution under a variety of conditions and link these mechanisms to the overall kinetics of glass behavior.4 The processes that control glass dissolution in a geological repository involve a complex set of reactions, which depend on the nature of the host rock, the near field materials, the temperature, the ground water composition along with its renewal rate, and the glass composition.5,6,7,8,9,10,11,12,13 The ultimate goal for laboratory scale parametric studies of glass alteration is to build an accurate predictive model that can account for the kinetics of long-term glass alteration under a variety of relevant conditions.14,15,16 Experimental validation of these predictive models under accelerated conditions at a laboratory time scale is not possible as glass dissolution is controlled by coupled non-linear processes, therefore validation instead relies on the study of archeological and basaltic glasses altered in well understood geochemical conditions.17,18,19,20 A more thorough understanding of all mechanisms and the associated kinetics of not only the glass itself but with the surrounding environment must be achieved to build a robust model.

Literature pertaining to borosilicate glass dissolution demonstrates that silica species play a major role in the mechanisms and associated kinetics. In dilute conditions, water begins to diffuse into the glass matrix (hydration) and alkali ions exchange with the positively charged hydrogen species (interdiffusion) (Eq. 1).21,22,23,24,25 Hydrolysis of the silicon network also begins with silica being released into solution (Eqs 2 and 3).26, 27 The corresponding kinetic regime is designated as the initial or forward rate, r 0 .

$$- {\rm{Si}} - {\rm{O}} - {{\rm{M}}^ + } + {{\rm{H}}^ + } \to {\rm{Si}} - {{\rm{O}}^ - } + {{\rm{M}}^ + } + {\rm{O}}{{\rm{H}}^ - }$$
(1)
$$- {\rm{Si}} - {{\rm{O}}^ - } + {{\rm{H}}_{\rm{2}}}{\rm{O}} \to {\rm{Si}} - {\rm{OH}} + {\rm{O}}{{\rm{H}}^ - }$$
(2)
$${\rm{Si}} - {\rm{O}} - {\rm{S}}{\left( {{\rm{OH}}} \right)_3} + {\rm{O}}{{\rm{H}}^ - } \to {{\rm{H}}_4}{\rm{Si}}{{\rm{O}}_4} + {\rm{Si}} - {{\rm{O}}^ - }$$
(3)

As the dissolved silica concentration increases, the rate decreases to a residual rate, r r . The rate decrease is attributed to a decrease of the affinity of the hydrolysis of the silicate network and the formation of an amorphous layer (i.e., gel) that becomes transport-limiting in silica saturated conditions.28 Two different mechanisms are proposed for the formation of this gel layer. One mechanism is based on the precipitation of a silica rich layer on the glass surface; the second mechanism is based on in-situ re-condensation of the silicon oxygen bonds, forming a porous gel layer on the glass surface. Previous studies have made use of isotopic tracers to differentiate between these two mechanisms, allowing a greater understanding of the exchange of silicon between the bulk solution and glass surface.29,30,31,32 These studies demonstrated that a combination of both mechanisms may occur, which is dependent on the solution pH and the reaction progress.

There is also a possibility of an alteration resumption, commonly attributed to the rapid precipitation of secondary phases.33 Under this regime, zeolites, and calcium silicate hydrates form at the expense of the passivating film. Additionally, environmental components, such as organic molecules or iron corrosion products, can have detrimental effects on the passivating layer.34, 35 Glass compositions, such as those high in aluminum, and solution pH are two key factors controlling this regime.36

Iron and its associated corrosion products (primarily siderite and magnetite) have been seen to enhance glass alteration via the sorption of silica onto the surface of the iron-rich phases and the precipitation of iron silicate minerals.37,38,39,40,41,42,43 Each of these processes shifts the equilibrium established between the glass surface layers and surrounding solution; glass alteration rates are directly dependent on the availability of iron.

In addition to sorption of silica on the iron surface and precipitation of iron silicates, magnetite influences SON68 glass behavior, the French high level waste reference glass. A study describing the effect of magnetite on SON68 proposes two other processes: 1) precipitation of SiO2 on the magnetite surface and 2) iron incorporation into the glass alteration layers with potential modifications of the transport properties of the gel itself.35 The present study provides evidences that these processes occur, although iron incorporation into the alteration layer is not seen until months after the addition of magnetite. Thus, this process would be highly dependent on iron availability in solution that is resulting from magnetite dissolution. This demonstrates that the iron—glass relationship is complex and that additional information is still needed to explain mechanisms and kinetics of glass behavior.

The formation of cracks at the glass surface also contributes to the overall durability of the glass matrix, primarily because surface cracks increase the reactive surface area. While molten glass cools within the carbon steel over pack, the temperature gradient between the perimeter and core of the glass leads to thermic shock-induced cracks. These cracks (also called defaults) provide preferential sites for alteration on a small scale, such as diffusion of alkali ions into the surrounding environment.44

Experimentally, defaults within the glass matrix are also observed during cooling and sample preparation (e.g., cutting). 45, 46 These processes are mechanically violent and generate a high number of micro cracks, or Griffith cracks, due to the metastable nature of the glass structure.47 Griffith summarized that the strength of the material was dependent on defects such as these surface cracks. These Griffith cracks can spontaneously propagate due to external factors such as mechanical stress or chemical gradients. The evolution of these micro cracks into larger cracks can be compared to dislocations in crystals due to grain boundaries. Though the defaults are caused by different methods (formation vs. processing), the investigation of the alteration of non-polished glass surfaces can give insight into the alteration of the cracks seen in large-scale waste materials.

The goal of this study is to further understand the alteration of the glass waste form in the presence of iron corrosion products, specifically magnetite (iron (II,III) oxide). This study is an extension of a similar experiment recently performed, in which experimental conditions are similar to that of the present work, except that the present work includes the addition of an iron source.30 Comparison of results of this work with a reference experiment without an iron source allows for a better understanding of the alteration layer properties. The experiment was carried out in conditions initially saturated in 29Si to follow the isotopic exchange of silicon between the solution and glass to monitor changes of the alteration layer. Additionally, five of the six sides of the glass monoliths were unpolished which provides insights into the effect of surface cracks on the alteration progress of the glass.

Results

Solution analysis

Results from inductively coupled plasma—optical emission spectroscopy (ICP-OES) for individual glass constitutes and multi-channel inductively coupled plasma—mass spectrometry (MC-ICP-MS) for silicon isotope concentrations are shown in Table 1.

Figure 1 presents the silicon concentration, normalized loss (NL) based on boron solution conditions, and pH over the experiment time period. The pH was maintained during the 150 first days by monitoring the solution and correcting to pH 7 using 0.5 M HNO3 or 0.5 M KOH. The pH was measured but not adjusted after the first 150 days; the pH then slowly increased to eight. This allowed the pH to drift to a pH at which orthosilicic acid starts dissociating into anions (H3SiO4 , H2SiO4 2−…), this seems to have an insignificant effect on the glass dissolution rate (Fig. 2), which remains essentially constant between day 126 and day 395. Over the first 2 months, the glass was allowed to react in a solution saturated with amorphous silica under the same experimental conditions as seen in Gin et al.30 In the present work, magnetite was added 70 days after the experiment was initiated and a significant decrease in the concentration of silicon was observed almost immediately, with a decrease of ~60 p.p.m. of silicon within the first day after magnetite addition. This decrease extends for 2 months after the addition of magnetite, at which point the solution only contains 22 p.p.m. of silicon. The last data points (up to nine months after the addition of magnetite) show the resumption of alteration.

Figure 2 presents the alteration rate changes over the duration of the experiment based on the boron concentrations in solution. The rate continues to decrease after the addition of magnetite at day 70, although the rate never reaches the long term rates of 0.1 nm per∙day observed by Gin et al.30 After ~100 days, the alteration rate increases. The rate reaches a maximum of ~100 nm per∙day for the rest of the experiment duration. While this is a large increase, the long-term rate never reaches the value observed at day one of this experiment (500 nm∙per day).

Figure 3 demonstrates three different regimes involving the isotopic silicon concentrations. The system is in equilibrium before the addition of magnetite, where surface layers form with little or no exchange of Si species in solution, as previously demonstrated.30 The second regime lasts between days 70–126, during which the available Si in solution is consumed, either by sorption or precipitation on the magnetite surface or iron silicate precipitation. The third regime continues beyond 126 days; Si concentration increases, though only 28Si and 30Si increase while 29Si continues to decrease. This demonstrates that later time points are a resumption of alteration of the glass itself. If the increased concentrations of Si were due to dissolution of the silicon sorbed on the magnetite surface, 29Si concentrations would have risen as well.

Time-of-Flight Secondary Ion Mass Spectrometry (TOF-SIMS) analysis of the glass monoliths

In addition to monitoring changes by solution analysis, changes of mobile elements within the alteration layer were analyzed using TOF-SIMS. Monoliths were withdrawn from the solution at day 70 (immediately before the addition of magnetite), day 84 (2 weeks after addition), day 126 (2 months after addition) and day 246 (almost 6 months after addition). Figure 4 shows the analysis of the polished face of the monolith at day 70, with the three mobile species (B, Na, and Ca) showing inverse correlation with the hydrous species (H). Sodium and calcium are known to exhibit this sigmoidal profile due to preferential dissolution of these elements, with similar behavior by boron.30 While boron may not be released by the same mechanisms as the alkali species, the low energy barrier to break the B–O–Si bonds allows for a similar profile.48

The potential inclusion of iron into the alteration layers formed on the polished face of the monoliths was also studied by TOF-SIMS (Fig. 5). Significant iron incorporation into the alteration layer is not seen until day 246. Data for monoliths from day 83 and 126 demonstrate an increase of iron within the first nanometers of the sample, but this could be contributed to magnetite grains adhering to the surface instead of being incorporated into the alteration layer. While iron-containing secondary phases are commonly observed in the literature studies mentioned in the introduction, these results suggest that iron incorporation takes place after the reactions that occur at the magnetite surface. This could also be due to the time it takes for the dissolution of magnetite to introduce iron into the solution since the magnetite and glass surfaces were not in contact. These profiles also show that there is not a significant increase in alteration layer thickness, based on the boron profile, even after the addition of magnetite and calculated resumption of alteration. Since only the polished side of the monolith was analyzed by TOF-SIMS, this suggests that the behavior of the polished and unpolished sides strongly differ.

Transmission Electron Microscopy (TEM)/Scanning Electron Microscopy (SEM) of glass and magnetite surfaces

Polished faces of the monoliths withdrawn at days 70, 84, and 126 were analyzed by SEM to monitor the growth of secondary phases and any surface features. The polished side of the monolith withdrawn at day 246 was analyzed by TEM. Figure 6 shows images of a tilted sample for day 126. No secondary phases were observed until this time point. Energy dispersive spectroscopy (EDS) shows that no iron-containing secondary phases formed on this side of the monolith. A cross section of the same monolith was analyzed by SEM to calculate an alteration layer thickness of 1.26 µm.

Figure 7 shows the TEM image of a monolith withdrawn at day 246. This demonstrates an alteration layer thickness of 1 µm with a 300 nm secondary phases. Electron diffraction analysis showed that no crystalline phases were present and that significant amount of iron was incorporated in the alteration layer. EDS analysis could not identify the secondary phases due to the presence of large quantities of the deposited metals such as gold and platinum. However, based on their morphology, iron silicate minerals are highly suspected.

SEM analysis of a monolith withdrawn from the experiment at day 358 showed that significant changes at the glass surface had occurred. Figure 8 shows the alteration layers on the non-polished surfaces of the monolith that now contain sections of alteration associated with cracks in the glass surface. These cracks prove to be a critical feature in the alteration of the glass surface. The majority of the alteration takes place primarily at these points instead of a uniform alteration across all surfaces, as there does not seem to be a change in alteration thickness on the polished surface. EDS mapping shows an additional layer containing iron on the glass surface.

TEM and SEM observations were also performed on a sample of magnetite withdrawn at day 246; results are presented with Fig. 9. TEM EDS analysis shows clusters with increased concentrations of silicon outside the grain boundary confirming the precipitation of silica or iron silicate on the surface of the magnetite. Silicon peaks are seen along with potassium and calcium which indicate that silicates are precipitating on the surface of the magnetite in addition to silicon sorption. SEM images show the entire surface of the magnetite grains is covered with a silicon rich secondary phase.

Behavior of diffusive species

Monoliths withdrawn at day 70, 84, 126, and 246 were subjected to a post tracing experiment to further understand the diffusivity changes within the alteration layers of the polished faces. The behavior of iron into the surface as well as methylene blue, a non-reactive 1 nm diameter molecule, and Li, which has been shown to diffuse into the pristine glass at a rate higher than other hydrous species, were monitored.49 Figure 10 shows a comparison of a monolith before the addition of magnetite and 6 months after addition (at which time iron was seen incorporated into the alteration layer). Since these components are not within the glass composition they cannot be normalized and thus only give a qualitative view of diffusion into the alteration layers.

At day 70, just before the addition of magnetite, only lithium diffusion is seen, suggesting the pores diameters are too small to allow ingress of large molecules. Methylene blue is seen only within the first few nanometers, implying that only molecules of the dye are adhering outside of the monolith instead of diffusing into the alteration layers. These results are consistent with those published in the literature, which confirms that a dense layer made of sub-nanoporous amorphous silica acts as a molecular sieve for aqueous species.30, 31 At day 246, two distinct layers are observed, one with increased iron concentrations and the common alteration layer seen at all time points. Lithium partitions between these two layers, with a decrease at the alteration layer pristine glass interface, while it is still incorporated into the gel layer and into the pristine glass. Methylene blue is also present in the secondary iron-containing layer up to the interface of these two alteration layers. In the reference experiment, methylene blue also diffused 250 nm into a monolith sampled at day 209.30 Since this 250 nm cutoff was observed in both the reference experiment and the present work featuring iron, it suggests that iron is not responsible for this secondary layer, but that this outer 250 nm layer contain pores with diameters large enough to incorporate larger ions.

Although there is a significant interface between an iron-bearing layer and the common alteration layer, this is most likely due to the nature of the alteration layer and pore diameters of this region instead of an iron affecting the properties or porosity of the alteration layer itself. It must be noted that although iron is seen within the alteration layers, the quantity of iron is not known as this method is only qualitative. Therefore, while the exact concentration of iron is not known, iron is most likely present at a low concentration based on the lack of iron detected by EDS as shown in Fig. 7.

Discussion

Differences in alteration layer thicknesses

Analyses of the alteration layers by multiple methods demonstrate the inconsistency between thicknesses derived from boron concentration in solution and those determined by analysis of the polished side by TOF-SIMS and TEM/SEM measurements. In the reference experiment, there was a slight increase in values determined by solution concentration determination, but values for all methods were within a 15% uncertainty of the average.30 With the addition of magnetite in the present work, a much larger discrepancy was observed between the methods as shown in Fig. 11.

The three methods show agreement up until the alteration resumption. Even after this time there is not a significant change in the alteration layer thickness of the polished side that was analyzed by TOF-SIMS and SEM/TEM; the alteration layer thicknesses remain within error of the reference experiment even after the resumption of alteration. In similar experiments, a correction factor was included to account for the different rate of alteration of the rough non-polished surfaces.30, 31 It appears that the rough or cut surfaces show preferential alteration over the polished side. These reference experiments showed that the non-polished surfaces contributed to a surface area larger by a factor of 1.7, compared to calculations based on geometric considerations only. This correction is applicable to the first regime in the absence of magnetite. But even factoring in this difference, it does not explain the difference factor of 15 for the thicknesses (up to a localized factor of 70) obtained between solid and solution analysis. Moreover, a slight decrease of silicon concentration was observed between day 126 and day 246 data, but since the alteration layer should already be depleted in boron the correction factor does not explain this difference. It was found that the non-polished side had areas of large alteration that were centered on cracks in the surface that could begin to explain the discrepancies in thickness measurements.

Mechanisms of iron interaction

Many of the mechanisms detailed by Rebiscoul et al. were derived in this work, although it is difficult to distinguish between Si sorption and SiO2 precipitation onto magnetite particles by methods used in this experiment.35 It has to be assumed that both of these mechanisms (sorption and precipitation) were observed since the consumption of Si from solution was higher than can be accounted for by sorption only. The sorption capacity for magnetite has been measured to be 19 ± 14 µmol of Si per gram of magnetite.50 In this experiment, this sorption capacity only accounts for 7–80 p.p.m. of Si (allowing for the large error within the measurement provided by Philippini et al.) of the 120 p.p.m. loss observed over the first 2 months after the addition of magnetite.50 Since the mechanisms of sorption and SiO2 precipitation cannot be separated, it can be speculated that sorption takes place within the first day after magnetite was added to the system. A decrease of 60 p.p.m. silica concentration from day 70 to day 71 falls within the range of 7–80 p.p.m. that can be accounted for by the sorption capacity. Thus, both sorption and SiO2 precipitation mechanisms that occur at the magnetite surface seem to be predominant in the initial stages of the experiment, with the formation of secondary phases or a layer of iron at the glass surface only seen at later stage of the experiment.

In an additional experiment, International Simple Glass (ISG) grains of ~5 µm diameter were altered in the same conditions as those tested here (pH 7, 90 °C, solution at equilibrium with amorphous silica) until 100% of boron had been released (complete alteration of the glass grains). The resulting grains can be considered as 100% alteration layer. The solubility of these altered grains were then measured, and after 100 days the system reached an equilibrium with a concentration of 43 p.p.m. of Si at pH 7. Data demonstrates that the alteration of the glass surface resumed when the concentration of Si in this experiment reached levels below this threshold of ~40 p.p.m. Si. This is also evident from the isotopic concentrations of Si in solution. After this threshold value of Si was reached, the glass, containing primarily 28Si, begins to alter to re-establish the equilibrium in solution; this can be observed with the increase of 28Si over 29Si at day 246.

Monitoring the diffusion of methylene blue and lithium also gave insights on whether the incorporation of exogenous elements, such as iron, changed the properties of the alteration layer. It is possible that the introduction of iron into the alteration layer could reduce the porosity of the alteration layer and thus affect the ability of ions to diffuse. The GRAAL model predicts that the durability of glass waste forms are tied to the thermodynamic stability of the transport properties of the passivating layer.14 While we know that iron is detrimental to the durability glass, a layer that would restrict iron diffusion would be considered protective over time. As shown with Fig. 10, the iron rich layer did not limit the diffusion of small ions, Li in this case, into the pristine glass layer. Thus under these conditions, iron only decreases the stability of the system without providing any protective effect.

Effects of surface cracks

There is a large difference between the calculated alteration layer thicknesses by boron released and that experimentally observed on the monolith. This was due to the propagation of alteration within the cracks initially present on the rough surfaces of the glass monolith (see Supplementary data), a phenomenon that was unexpected when this study was designed. While the alteration layer on the polished side remained almost unaffected by the magnetite, the zones of alteration around the cracks extended up to 60–70 µm as shown in Fig. 8. The effects of these cracks must be carefully considered since actual waste packages are expected to have many cracks due to cooling of the glass and any possible stress fractures from movement of the canister.51 Though cracks within glass have been studied, few studies have been directly related to nuclear glasses under repository conditions.51, 52 These cracks substantially increase the surface area of the glass block along with providing areas with unique chemical environments. The localized conditions within these cracks provide a preferential site for quick alteration.

Stress fractures on the rough surfaces of the monoliths were also seen in the reference experiment, yet alteration in presence of magnetite progressed quicker than seen previously.30 This could be due to the rapid change in solution conditions when magnetite was added to the system. Within the first 2 months after the addition of magnetite, the concentration of silicon decreased from a steady state of about 140 to around 20 p.p.m., with a decrease in 60 p.p.m. during the first day. This extreme change in solution conditions caused the resumption of alteration and it was the surface cracks that provided sites for this alteration to reestablish the equilibrium between solution and the glass matrix.

Several studies have demonstrated that unique chemical environments exist within these cracks. The literature showed an increase in diffusion constants of ions, specifically Na, within the cracks when compared to the bulk glass.53, 54 The diffusion of these ions also creates a condensate region at the surface of the glass around the cracks. The pH in these regions increases compared to that of the bulk solution, which can lead to regions of increased silicate ion formation due to hydrolysis.55 Therefore, crack formation is not specific to glasses in contact with iron, but iron can take advantage of these particular areas with unique chemistry in comparison with the bulk surface. This facilitates large regions of alteration that were not seen on the time scale of the reference experiment. Since the rate of propagation increases with an increasing crack length, the formation of these areas could be detrimental to the overall durability of the bulk glass.56 Future studies will determined how the growing gel modifies the mechanical properties of the underlying glass, allowing new cracks to form.

Methods

Experimental Set-up

ISG (SiO2 56.2 wt%, B2O3 17.3 wt%, Na2O 12.2 wt%, Al2O3 6.1 wt%, CaO 5.0 wt%, ZrO2 3.3 wt%) was prepared by MoSci Corporation (Rollo, MO, USA). The glass was initially melted into ingots using platinum-rhodium crucibles at 1300 °C for about 4 h. The ingots were then annealed for 6 h at 569 °C before cooling to room temperature at a rate of 50 °C per hour. From one ingot, 8 monoliths of 2.00 × 2.00 × 0.11 cm3 and eight monoliths of 0.50 × 0.50 × 0.11 cm3 were obtained. One of the large faces of the monoliths were then polished to a mirror finish. The eight large monoliths were put into a Teflon holder and placed in the bottom of a 250 mL Teflon vessel in a vertical orientation. The smaller monoliths were wrapped in Teflon mesh and hung into solution from the top of the vessel as shown in Fig. 12.

The initial solution was saturated in amorphous 29SiO2 (Eurisotop > 95% 29SiO2) at pH90°C 7. The isotopically tagged silica was melted with KOH (Suprapur) at 600 °C and the resulting soluble potassium silicate was then dissolved in 18 MΩ cm H2O to obtain a concentration of 141 p.p.m. Si and 6900 p.p.m. K. Due to a small initial concentration of additional cations (i.e., sodium and calcium) an aliquot of the initial solution was characterized by ICP-OES for consideration of further calculations.

The total surface area was calculated as 76 cm2 by geometric considerations. A previous study has found that the resulting reactive surface area due to only one polished face of the monolith is higher by a factor of 1.7, compared to the geometric surface area; the resulting reactive surface area was 129 cm2.31 The smaller monoliths contribute only a small fraction of the surface area and these monoliths are used for sampling throughout the experiment. The pH90 °C of the solution was maintained at 7.0 ± 0.5 using additions of 0.5 M KOH and 0.5 M HNO3, after 6 months pH was measured but not controlled.

As the leaching solution was initially saturated with amorphous silica, the glass was altered in a residual rate until day 70, at which time the system was perturbed. Approximately 9.3 g of magnetite (Sigma Aldrich 0.48 µm average particle size) was added.

Solution analysis

Aliquots of solution were sampled regularly throughout the experiment for elemental concentration quantification by ICP-OES and silicon isotope ratios quantification by ICP-MS.57 All samples and standards were purified using BioRad AG50 X-12 (200–400 mesh) cation exchange resin before silicon isotope analysis, as described in detail by Georg et al.58 Silicon isotope ratio measurements were performed on a Thermo Scientific Neptune Plus Multi-Collector Inductively Coupled Plasma Mass-Spectrometer (MC-ICP-MS) at the Institut de Physique du Globe de Paris, France. All measurements were performed in static multicollection mode with Faraday cups attached to 1011 Ω amplifier resistors. The Faraday amplifier gains were calibrated daily before the analytical session, yielding long-term reproducibility better than 10 p.p.m. The instrument was operated at medium resolution to avoid polyatomic interferences (e.g., 14N16O, 14N2, 12C16O). The isotope beams of 28Si, 29Si, and 30Si were measured using L3, C and H3 cups, respectively. A tandem cyclone-Scott type spray chamber Stable Introduction System (Thermo Scientific) with a PFA micro-flow nebulizer (ESI, USA) was used as the introduction system.

Boron solution concentrations were obtained by ICP-OES and were used to calculate the NL, the equivalent thickness (E th) and subsequent rates (r) along with the altered glass fraction (AGF). The calculation for NL is shown in Eq. 4.

$${\rm{NL}}({\rm{B}}) = \frac{{{C_{\rm{B}}} \times {V_{{\rm{solution}}}}}}{{\left( {\left( {{\rm{S}}{{\rm{A}}_{{\rm{small}}}} \times {n_{{\rm{small}}}}} \right) + \left( {{\rm{S}}{{\rm{A}}_{{\rm{large}}}} \times {n_{{\rm{large}}}}} \right)} \right) \times {f_{\rm{B}}}}}$$
(4)

where C B is the concentration of boron in solution, V is the volume of solution, ρ is the density of glass which is 2.5 g cm−3, SA is the calculated surface area of the glass, n is the number of monoliths remaining, and f B is the fraction of boron in the glass (0.0537). The surface area and volume values were changed throughout the experiment as monoliths were sampled and correcting for any evaporation in the system. The equivalent thickness calculation is shown in Eq. 5 which is the NL divided by the glass density ρ glass.

$${E_{{\rm{th}}}}({\rm{B}}) = \frac{{{\rm{NL}}({\rm{B}})}}{{{\rho _{{\rm{glass}}}}}}$$
(5)

The glass dissolution rate is calculated as shown with Eq. 6. This is determined by using a three point linear regression.

$$r = \frac{{d\left( {{E_{{\rm{th}}}}\left( {\rm{B}} \right)} \right)}}{{dt}}$$
(6)

The AGF is determined by Eq. 7.

$${\rm{AGF}} = \frac{{{C_{\rm{B}}}}}{{{f_{\rm{B}}} \times \left( {{V_{{\rm{glass}}}} \times {\rho _{{\rm{glass}}}}} \right)}}$$
(7)

where C B is the concentration of boron in solution in g, f B is the fraction of boron in the glass (0.0537), and V glass is the addition of volume from each glass monolith in cm3.

The associated uncertainties for NL is 10%, E th is 15%, while r is 30%.

The surface area is calculated based on the geometric surface area alone and does not take into account the physical surface area of the monolith. A factor of 1.7 is applied to the geometric surface area to correct for surface roughness of the unpolished surfaces for the calculation of surface area of each monolith, as previously explained.31 Additionally, the propagation of cracks on the unpolished surfaces of the monolith provided additional water accessible areas that impact the overall alteration of the glass monolith.

Solid analysis

Monoliths were sampled at multiple times throughout the experiment; day 70 (before addition of magnetite), day 83 (2 weeks after addition), day 126 (2 months after addition), day 246 (6 months after addition), and day 358 (9.5 months after addition). These monoliths were cut into four pieces for different analyses.

The polished face of the first four monoliths were analyzed by TOF-SIMS (IONTOF TOF 5) to visualize elemental profiles within the alteration layer. Two sputtering beams were used O2 + (50 × 50 μm2 area) or Cs+ (40 × 40 μm2 area). The Cs+ beam was used to analyze the H, and S ions while O2 + was used for all other ions. All ions are normalized to Zr, to minimize the matrix effects, as this element is immobile during alteration.59

For monolith analysis by TEM, a thin section was extracted from the polished face of the sample and milled up to a thickness of ~100 nm. The sample was sputtered with an Au/Pd mixture and coated with Pt for protection. Observations and analyses were carried out with a Technai G2 (FEI) TEM microscope equipped with a LaB6 source operating at 200 kV. A GATAN CCD camera, a BF-DF detector, and an EDX detector EDAX Genesis were used. The spatial resolution was 0.27 nm and the EDS spot size of 5 and 10 nm. For EDX analysis, the sample was tilted of 20° to minimize the overlaying of different phases. The counting time was 20 s to limit evaporation of mobile elements. SEM-EDS (JEOL JXA-8500 F) was also used to observe the alteration layers and surface features of the day-126 (polished face) and the day 358 (rough cut face) monoliths.

A post-experiment tracing was performed on the monoliths withdrawn at days 70, 83,126, and 246. The monoliths were placed in a saturated amorphous 28SiO2 at pH 7 at room temperature for 100 h. The solution also contained 4.4 × 10−4 mol L−1 methylene blue and 0.1 mol L−1 LiCl. TOF-SIMS was performed on the polished face of the samples to study the diffusivity of these aqueous species into the developing alteration layers. This allowed for comparison to the reference sample to further understand the influence of iron on the properties of the alteration layer.

References

1. Donald, I. Waste Immobilization in Glass and Ceramic Based Hosts: Radioactive, Toxic And Hazardous Wastes. (Wiley, 2010).

2. ANDRA (Agence Nationale pour la gestion des Déchets RAdioactifs). Dossier 2005: Andra research of the geological disposal of high-level long-lived radioactive waste. Results and Perspectives (2005).

3. Gin, S. et al. An international initiative on long-term behavior of high-level nuclear waste glass. Mater. Today 16, 243–248 (2013).

4. Vienna, J. D., Ryan, J. V., Gin, S. & Inagaki, Y. Current understanding and remaining challenges in modeling long-term degradation of borosilicate nuclear waste glasses. Int. J. Appl. Glass Sci 4, 283–294 (2013).

5. Fleury, B. et al. Development of an experimental design to investigate the effects of r7t7 glass composition on the residual rate of alteration. Procedia Mater. Sci 7, 193–201 (2014).

6. Frugier, P., Martin, C., Ribet, I., Advocat, T. & Gin, S. The effect of composition on the leaching of three nuclear waste glasses: R7T7, AVM and VRZ. J. Nucl. Mater. 346, 194–207 (2005).

7. Gin, S., Beaudoux, X., Angéli, F., Jégou, C. & Godon, N. Effect of composition on the short-term and long-term dissolution rates of ten borosilicate glasses of increasing complexity from 3 to 30 oxides. J. Non-Cryst. Solids 358, 2559–2570 (2012).

8. Jantzen, C. M., Brown, K. G. & Pickett, J. B. Durable glass for thousands of years. Int. J. Appl. Glass Sci 1, 38–62 (2010).

9. Pierce, E. M. et al. Experimental determination of the effect of the ratio of B/Al on glass dissolution along the nepheline (NaAlSiO4)–malinkoite (NaBSiO4) join. Geochim. Cosmochim. Acta 74, 2634–2654 (2010).

10. Pierce, E. M., Rodriguez, E. A., Calligan, L. J., Shaw, W. J. & Pete McGrail, B. An experimental study of the dissolution rates of simulated aluminoborosilicate waste glasses as a function of pH and temperature under dilute conditions. Appl. Geochem. 23, 2559–2573 (2008).

11. Werme, L. O., Hench, L. L., Nogues, J.-L., Odelius, H. & Lodding, A. On the pH dependence of leaching of nuclear waste glasses. J. Nucl. Mater. 116, 69–77 (1983).

12. Andriambololona, Z., Godon, N. & Vernaz, E. R7T7 nuclear glass alteration in a saline medium: in situ experiments in the WIPP project. Appl. Geochem. 7, 23–32 (1992).

13. Jollivet, P. et al. Effect of clayey groundwater on the dissolution rate of the simulated nuclear waste glass SON68. J. Nucl. Mater. 420, 508–518 (2012).

14. Frugier, P. et al. SON68 nuclear glass dissolution kinetics: current state of knowledge and basis of the new GRAAL model. J. Nucl. Mater. 380, 8–21 (2008).

15. Grambow, B. & Muller, R. First-order dissolution rate law and the role of surface layers in glass performance assessment. J. Nucl. Mater. 298, 112–124 (2001).

16. McGrail, B. P., Ebert, W. L., Bakel, A. J. & Peeler, D. K. Measurement of kinetic rate law parameters on a Na-Ca-Al borosilicate glass for low-activity waste. J. Nucl. Mater. 249, 175–189 (1997).

17. Libourel, G. et al. The use of natural and archeological analogues for understanding the long-term behavior of nuclear glasses. Comptes Rendus Geosci. 343, 237–245 (2011).

18. Parruzot, B., Jollivet, P., Rébiscoul, D. & Gin, S. Long-term alteration of basaltic glass: mechanisms and rates. Geochim. Cosmochim. Acta 154, 28–48 (2015).

19. Verney-Carron, A., Gin, S., Frugier, P. & Libourel, G. Long-term modeling of alteration-transport coupling: application to a fractured Roman glass. Geochim. Cosmochim. Acta 74, 2291–2315 (2010).

20. Poinssot, C. & Gin, S. Long-term behavior science: the cornerstone approach for reliably assessing the long-term performance of nuclear waste. J. Nucl. Mater. 420, 182–192 (2012).

21. Doremus, R. H. Diffusion-controlled reaction of water with glass. J. Non-Cryst. Solids 55, 143–147 (1983).

22. Boksay, Z., Bouquet, G. & Dobos, S. Diffusion processes in surface layers of glass. Phys. Chem. Glasses 8, 140–144 (1967).

23. Ojovan, M. I., Pankov, A. & Lee, W. E. The ion exchange phase in corrosion of nuclear waste glasses. J. Nucl. Mater. 358, 57–68 (2006).

24. Gin, S. et al. The controversial role of inter-diffusion in glass alteration. Chem. Geol. 440, 115–123 (2016).

25. Doremus, R. H. Interdiffusion of hydrogen and alkali ions in a glass surface. J. Non-Cryst. Solids 19, 137–144 (1975).

26. Hench, L. Physical chemistry of glass surfaces. J. Non Cryst. Solids 25, 343–369 (1977).

27. Techer, I., Advocat, T., Lancelot, J. & Liotard, J. M. Dissolution kinetics of basaltic glasses: control by solution chemistry and protective effect of the alteration film. Chem. Geol. 176, 235–263 (2001).

28. Van Iseghem, P. et al. in Environmental Issues and Waste Management Technologies in the Materials and Nuclear Industries XII (eds Cozzi, A. & Ohji, T.) 115–126 (Wiley, 2009).

29. Geisler, T. et al. Aqueous corrosion of borosilicate glass under acidic conditions: a new corrosion mechanism. J. Non-Cryst. Solids 356, 1458–1465 (2010).

30. Gin, S. et al. Origin and consequences of silicate glass passivation by surface layers. Nat. Commun. 6, 6360 (2015).

31. Gin, S. et al. The fate of silicon during glass corrosion under alkaline conditions: a mechanistic and kinetic study with the international simple glass. Geochim. Cosmochim. Acta 151, 68–85 (2015).

32. Valle, N. et al. Elemental and isotopic (Si-29 and O-18) tracing of glass alteration mechanisms. Geochim. Cosmochim. Acta 74, 3412–3431 (2010).

33. Fournier, M., Gin, S. & Frugier, P. Resumption of nuclear glass alteration: state of the art. J. Nucl. Mater. 448, 348–363 (2014).

34. Gin, S., Godon, N., Mestre, J. P., Vernaz, E. Y. & Beaufort, D. Experimental investigation of aqueous corrosion of R7T7 nuclear glass at 90°C in the presence of organic species. Appl. Geochem. 9, 255–269 (1994).

35. Rebiscoul, D. et al. Reactive transport processes occurring during nuclear glass alteration in presence of magnetite. Appl. Geochem. 58, 26–37 (2015).

36. Ribet, S. & Gin, S. Role of neoformed phases on the mechanisms controlling the resumption of SON68 glass alteration in alkaline media. J. Nucl. Mater. 324, 152–164 (2004).

37. Bildstein, O., Trotignon, L., Perronnet, M. & Jullien, M. Modelling iron–clay interactions in deep geological disposal conditions. Phys. Chem. Earth Parts ABC 31, 618–625 (2006).

38. Burger, E. et al. Impact of iron on nuclear glass alteration in geological repository conditions: a multiscale approach. Appl. Geochem. 31, 159–170 (2013).

39. Dillmann, P., Gin, S., Neff, D., Gentaz, L. & Rebiscoul, D. Effect of natural and synthetic iron corrosion products on silicate glass alteration processes. Geochim. Cosmochim. Acta 172, 287–305 (2016).

40. Godon, N., Gin, S. & Frugier, P. SON68 glass alteration enhanced by magnetite. Procedia Earth Planet. Sci 7, 300–303 (2013).

41. McVAY, G. L. & Buckwalter, C. Q. Effect of iron on waste-glass leaching. J. Am. Ceram. Soc. 66, 170–174 (1983).

42. Michelin, A. et al. Archeological slag from Glinet: an example of silicate glass altered in an anoxic iron-rich environment. Chem. Geol. 413, 28–43 (2015).

43. Michelin, A. et al. Silicate glass alteration enhanced by iron: origin and long-term implications. Environ. Sci. Technol. 47, 750–756 (2013).

44. Kittel, C. Introduction to Solid State Physics. (Wiley, 2005).

45. Hed, P. & Edwards, D. Optical glass fabrication technology. 2: relationship between surface roughness and subsurface damage. Appl. Opt. 26, 4677–4680 (1987).

46. Preston, F. W. The structure of abraded glass surfaces. Trans. Opt. Soc. 23, 141–164 (1922).

47. Griffith, A. A. The phenomena of rupture and flow in solids. Philos. Trans. R. Soc. Lond. Ser. Contain. Pap. Math. Phys. Character 221, 163–198 (1921).

48. Zapol, P., He, H., Kwon, K. D. & Criscenti, L. J. First-principles study of hydrolysis reaction barriers in a sodium borosilicate glass. Int. J. Appl. Glass Sci 4, 395–407 (2013).

49. Neeway, J. J. et al. Low-temperature lithium diffusion in simulated high-level boroaluminosilicate nuclear waste glasses. J. Non-Cryst. Solids 405, 83–90 (2014).

50. Philippini, V., Naveau, A., Catalette, H. & Leclercq, S. Sorption of silicon on magnetite and other corrosion products of iron. J. Nucl. Mater. 348, 60–69 (2006).

51. Mallet, C., Fortin, J., Guéguen, Y. & Bouyer, F. Evolution of the crack network in glass samples submitted to brittle creep conditions. Int. J. Fract 190, 111–124 (2014).

52. Ougier-Simonin, A., Guéguen, Y., Fortin, J., Schubnel, A. & Bouyer, F. Permeability and elastic properties of cracked glass under pressure. J. Geophys. Res. Solid Earth 116, B07203 (2011).

53. Celarie, F., Ciccotti, M. & Marlière, C. Stress-enhanced ion diffusion at the vicinity of a crack tip as evidenced by atomic force microscopy in silicate glasses. J. Non-Cryst. Solids 353, 51–68 (2007).

54. Wiederhorn, S. M., Guin, J.-P. & Fett, T. The use of atomic force microscopy to study crack tips in glass. Metall. Mater. Trans. A 42, 267–278 (2010).

55. Wiederhorn, S. M., Fett, T., Guin, J.-P. & Ciccotti, M. Griffith cracks at the nanoscale. Int. J. Appl. Glass Sci 4, 76–86 (2013).

56. Freiman, S. W., Wiederhorn, S. M., Mecholsky, J. & John, J. Environmentally enhanced fracture of glass: a historical perspective. J. Am. Ceram. Soc 92, 1371–1382 (2009).

57. Alkiviadis, G. et al. Silicon isotope ratio measurements by inductively coupled plasma tandem mass spectrometry for alteration studies of nuclear waste glasses. Anal. Chim. Acta. doi:10.1016/j.aca.2016.11.063.

58. Georg, R. B., Reynolds, B. C., Frank, M. & Halliday, A. N. New sample preparation techniques for the determination of Si isotopic compositions using MC-ICPMS. Chem. Geol. 235, 95–104 (2006).

59. Cailleteau, C. et al. Insight into silicate-glass corrosion mechanisms. Nat. Mater. 7, 978–983 (2008).

Acknowledgements

This research was performed using funding received from the U.S. DOE Office of Nuclear Energy’s Nuclear Energy University Program under Project 23–3361 in addition to funding provided by the CEA, Areva, and the Chateaubriand Fellowship from the Embassy of France in the U.S. Authors warmly thank Laurent Dupuy at Biophy Research, France for the TOF-SIMS analysis, Martiane Cabié at Aix-Marseille University, France, for assistance with TEM, and CEA technical staffs for SEM analyses and Thibaut Martin for technical assistance.

Author information

Authors

Contributions

L.N. initiated and was directly responsible for the experiment; she completed data analysis with the help of S.G. The article was also co-written by these two authors. A.G., T.D., and, T.E. performed MC-ICP-MS and TEM analysis. M.F. performed analysis of the unaltered monoliths. P.J. and N.W. acted as mentors for the experimental set up and editors throughout the writing process.

Corresponding authors

Correspondence to Stéphane Gin or Nathalie A. Wall.

Ethics declarations

Competing interests

The authors declare that they have no competing financial 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

Neill, L., Gin, S., Ducasse, T. et al. Various effects of magnetite on international simple glass (ISG) dissolution: implications for the long-term durability of nuclear glasses. npj Mater Degrad 1, 1 (2017). https://doi.org/10.1038/s41529-017-0001-6

• Revised:

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41529-017-0001-6

• Long-term interactive corrosion between International Simple Glass and stainless steel

• Chandi Mohanty
• Xiaolei Guo
• Gerald S. Frankel

• S. Gin
• C. Jegou
• F. Angeli

• Microbiologically Influenced Corrosion of Q235 Carbon Steel by Aerobic Thermoacidophilic Archaeon Metallosphaera cuprina

• Hongchang Qian
• Shangyu Liu
• Dawei Zhang

Acta Metallurgica Sinica (English Letters) (2022)

• The fate of Si and Fe while nuclear glass alters with steel and clay

• C. Carriere
• P. Dillmann
• C. Martin

• Microbiologically influenced corrosion inhibition of carbon steel via biomineralization induced by Shewanella putrefaciens

• Yuntian Lou
• Weiwei Chang
• Dawei Zhang


text:
Volume 5, Issue 3
Finite Element Approximation for TV Regularization

Int. J. Numer. Anal. Mod., 5 (2008), pp. 516-526.

Published online: 2008-05

Cited by

Export citation
• Abstract

In this paper, we will develop the convergence of the solution of TV-regularization equations with regularized parameter $\varepsilon \rightarrow 0$ in BV($\Omega$) for practical purposes. Originated from the effects of regularized parameter $\varepsilon$, the error rate of finite element approximation for TV-regularization equations will be controlled by the regularized parameter $\varepsilon^{-1}$ polynomially in the energy norm when using linearization technique and duality argument. And in the $L^p$-norm, the effect of regularized parameter $\varepsilon$ will be more extremely.

35B25, 35K57, 35Q99, 65M60, 65M12

• BibTex
• RIS
• TXT
@Article{IJNAM-5-516, author = {Yao , Changhui}, title = {Finite Element Approximation for TV Regularization}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2008}, volume = {5}, number = {3}, pages = {516--526}, abstract = {

In this paper, we will develop the convergence of the solution of TV-regularization equations with regularized parameter $\varepsilon \rightarrow 0$ in BV($\Omega$) for practical purposes. Originated from the effects of regularized parameter $\varepsilon$, the error rate of finite element approximation for TV-regularization equations will be controlled by the regularized parameter $\varepsilon^{-1}$ polynomially in the energy norm when using linearization technique and duality argument. And in the $L^p$-norm, the effect of regularized parameter $\varepsilon$ will be more extremely.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/824.html} }
TY - JOUR T1 - Finite Element Approximation for TV Regularization AU - Yao , Changhui JO - International Journal of Numerical Analysis and Modeling VL - 3 SP - 516 EP - 526 PY - 2008 DA - 2008/05 SN - 5 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/824.html KW - TV-regularization, regularized parameter, finite element method. AB -

In this paper, we will develop the convergence of the solution of TV-regularization equations with regularized parameter $\varepsilon \rightarrow 0$ in BV($\Omega$) for practical purposes. Originated from the effects of regularized parameter $\varepsilon$, the error rate of finite element approximation for TV-regularization equations will be controlled by the regularized parameter $\varepsilon^{-1}$ polynomially in the energy norm when using linearization technique and duality argument. And in the $L^p$-norm, the effect of regularized parameter $\varepsilon$ will be more extremely.

Changhui Yao. (1970). Finite Element Approximation for TV Regularization. International Journal of Numerical Analysis and Modeling. 5 (3). 516-526. doi:
Copy to clipboard
The citation has been copied to your clipboard

format: plain


text:
# Minipage like Environment in latex capturing side by side float

Need a single command to capture the multiple floats. The PDF output of the floats should be set in side by side. i have tried the below macro. i'm getting

! LaTeX Error: Not in outer par mode

is there any way to produce the single command with multiple floats?

\documentclass{book}

\def\sidebysidefig#1#2{\begin{minipage}{0.45\textwidth}{#1}{#2}\end{minipage}}

\begin{document}

\chapter{Mathematical Expression $\sin\theta\,\cos\theta$}

\sidebysidefig{\begin{figure}
%\includegraphics{fig1.eps}
\caption{This is sample caption}\end{figure}}{\begin{figure}
%\includegraphics{fig2.eps}
\caption{This is sample caption}\end{figure}}

\end{document}

• you had not to close floats into minipage. you should do opposite: in figure put your \sidebysidefig: \begin{figure}\sidebysidefig{ ..}{....}\end{figure}. – Zarko May 6 at 6:57
• i do understand but, for generating the XML will be tedious task. that is the reason where I was looking for single command. – Saravanan.O May 6 at 7:29
• why is it any more tedious than your suggested markup? you only need a single figure around the outside, the content of the figure is side by side or vertical just depending if you put the items in the same paragraph or separate paragraphs. – David Carlisle May 6 at 9:49
• We have automated the figure calling methods. We are not using the \begin{figure}...\end{figures} instead, we will be using \processfigure{Position}{caption}{figure}. based on the figure width conditions, we have automated the landscape and side caption figures. if i changed the coding structure the entire process need to be updated. That is the reason where i have requested for single command with multiple floats. Hope you understand my requirements. – Saravanan.O May 6 at 10:42
• structurally there is no difference, you want a single thing around the set of side by side floats you suggest \sidebysidefig but I suggest figure then you have an environment around each float you suggest figure but I use minipage so the structure is as you want, just using different names. – David Carlisle May 6 at 11:08

figure environments need to be at the top level so they can float, but you do not need anything special to put content side-by-side:

\documentclass{book}

\usepackage{graphicx}
\begin{document}

\chapter{Mathematical Expression $\sin\theta\,\cos\theta$}

\begin{figure}
\begin{minipage}{.45\textwidth}
\includegraphics[width=\linewidth]{example-image-a}
\caption{This is sample caption}
\end{minipage}\hfill
\begin{minipage}{.45\textwidth}
\includegraphics[width=\linewidth]{example-image-b}
\caption{This is sample caption}
\end{minipage}
\end{figure}

\end{document}

• Thanks for the update. unfortunately this could not be suit for my requirements. We have more combination of floats will be available. Figure and table, figure and listing, listing and table etc. without changing the coding need to be produce the side by side floats. that is the reason where i have requested for the single command. – Saravanan.O May 6 at 10:49
• @Saravanan.O you may need to relax your requirements. All the items that are side by side need to float together so need to be in a figure or table environment as shown, you can have a table in a figure environment by using the capt-of package and using \captionof{table}{my caption} instead of \caption{my caption} although ordering with other tables is then not guaranteed – David Carlisle May 6 at 10:55


text:
# Tag Info

Accepted

### What are the "smart algorithms" applied to solve the "curse of dimensionality"?

This is a very broad question, so I am going to give a very brief overview of typical exponentially-scaling problems. I am not an expert in most of these areas, so any suggestions or improvements will ...
• 1,878
Accepted

### How to determine, a priori, whether a compound has multireference character?

This is not going to be a full answer, but maybe a starting point, a pointer to somewhat troubled systems. The general keyword here is degeneracy. Whenever you have fewer electrons than fit in the ...
• 1,561
Accepted

### Total spin and/or multiplicity for transition metal ions?

Even for the simplest transition metal diatomic molecules, the most reliable way to know the ground-state spin configuration is often by doing experiments. I will give an example where it's easy to ...
• 30.8k

### What is the largest system that has been studied by the coupled cluster method?

ORCA The local pair-natural orbital (DLPNO) based coupled cluster method have been managed to investigate large organic molecule and small proteins (linear C150H302 (452 atoms, >8800 basis ...
• 3,701

### How to incorporate the effect of spin-orbit coupling in electronic structure calculation

The inclusion of spin-orbit coupling in electronic structure calculations is done by including the interaction between the electron-spin and the orbital angular momentum in the Hamiltonian. Such ...
• 1,335
Accepted

### When should geometry relaxation not be performed?

Geometry optimization corresponds to a system in equilibrium. It is the "average" position of a molecule vibrating in a well. However, there are many cases where the system is "non-equilibrium" ...
• 7,869

### How can quantum computing accelerate materials modeling?

Quantum computers provide the possibility of simulating systems that are so mathematically complex that classical computers cannot practically be used. The difficulty comes from the fact that the ...
• 1,443
Accepted

### How to calculate energy of a molecule in an aligned electric field?

Aligning a molecule to a particular frame of reference (e.g., with the z-axis along a particular bond) is part of Avogadro for this reason: In Avogadro 1.x, there's an align tool You click one atom ...
• 8,225
Accepted

### How to incorporate the effect of spin-orbit coupling in electronic structure calculation

Basically there are two kinds of approaches which may be found in many text books, L-S coupling and j-j coupling. L-S coupling means that scalar electronic states (e.g. atomic L-S states and linear ...
• 176

### Why does orbital-free DFT scale linearly with system size?

There is linear scaling DFT, where you use density matrix as your object throughout SCF iterations instead of the wavefunctions. The linear scaling comes from the sparsity assumption/truncation on the ...
• 1,046

### How to identify a semi-metal from the band structure?

A rigorous definition of a semimetal can for example be found in the mathematical literature (e.g. this paper: arXiv:2002.01990v1 [math-ph], pdf version, assumption 2.5). The basic idea is that the ...
• 2,334
Accepted

### How to overcome the exponential wall encountered in full configurational interaction methods?

It is hard to claim that any FCI code overcomes the exponential wall, especially for strongly correlated systems. There are many algorithms, e.g. CDFCI, HCI, FCIQMC, ACI, etc., which significantly ...
• 1,046

### How to determine, a priori, whether a compound has multireference character?

As Martin has said a practical way to decide if a molecule requires multi-reference treatment is to calculate the unrestricted Hartree Fock natural orbitals.[1] Significant fractional occupancy of a ...
• 7,869
Accepted

### When do relativistic effects need to be explicitly included?

When to include relativistic corrections or modeling of any kind in computational methods is a rather complex one. Full Dirac methods as you asked about (DHF) recapture two important factors, so ...
• 832

### How can quantum computing accelerate materials modeling?

Quantum computing has been extensively discussed as long as the first quantum computers have become a reality. As already pointed out, we are still at very early stages of such development. What we ...
• 3,148
Accepted

### How is group theory used to deduce which of these integrals are equal to 0?

The question is too broad to be answered directly so I will provide a somewhat general scheme. Basically in an integral like $$\int d\mu A B C$$ one would seek to expand each part in irreducible ...
• 256

### Methods for visualising aromaticity

Multiwfn The software Multiwfn can calculated four aromaticity indices, namely FLU, FLU-pi, PDI and information-theoretic index. Multiwfn can perform wavefunction analyses based on outputted file of ...
• 20.7k
Accepted

### Gaussian-type orbitals vs Slater-type orbitals in terms of computational efficiency

Your supervisor is correct that in almost all practical cases in quantum chemistry, the cost savings of using Gaussian-type orbitals far outweigh the disadvantages of needing more orbitals. First of ...
• 30.8k

### Do bonds between the atoms influence ab-initio energies?

The bonds you draw or see in Gaussview should have no effect on the calculation (besides some minor effects in choosing internal coordinates for a geometry optimization). As you note, a bond is a ...
• 14.5k

### How can quantum computing accelerate materials modeling?

Quantum computers potentially could provide a far more efficient way to solve the electronic structure problem: finding the ground state energy (or other properties) of electrons in a compound. The ...
• 221
Accepted

### What approaches are employed to treat amorphous materials?

Creating amorphous structures is non-trivial and always rather expensive. Generally you are using large supercells such that you can achieve mid-range disorder. That means, since the cell is still ...
• 868

### What are the "smart algorithms" applied to solve the "curse of dimensionality"?

The curse of dimensionality is indeed a huge problem in quantum chemistry, since the possible ways N electrons can occupy K orbitals is a binning problem whose computational cost grows factorially (...
• 16.4k

### What is the mathematical condition that ensure that the self-consistent field (SCF) procedure must converge?

This question is a bit ill-defined: what do you mean by "the self-consistent field procedure"? If you mean the original Roothaan procedure, then the question makes sense, but it is uninteresting: ...
• 16.4k

### Is there an energy gap correction factor that needs to be used in DFT calculations?

I would like to add a few clarifications to Jack's answer: Standard DFT calculations with fixed ionic positions are actually not even 0 K. A better way to describe them is that they are static ...
• 10.7k

### FCI without Hartree-Fock

Yes, you can surely do full CI without Hartree-Fock. However, the reason why one typically starts from Hartree-Fock is that this guarantees a good conditioning of the FCI matrix, which becomes ...
• 16.4k
Accepted

### Is Heisenberg model or in its simplier form Ising model a good approximation to study magnetic systems?

Introduction Your question reminds me of a quote by Paul Dirac, The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus ...
• 4,451

### Derivation of Slater-Koster equations

Spherical harmonics are not themselves full atomic orbitals. Consider the Hydrogen wave function, which separates into a radial part and an angular part. The latter is a spherical harmonic, but the ...
• 4,451

### Is there an energy gap correction factor that needs to be used in DFT calculations?

Is this calculated indirect bandgap at room temp. or at 0K? QE is based on the density functional theory (DFT). DFT is a ground-state (0K) theory and hence the calculated bandstructure is 0K. If it ...
• 14.6k
Accepted

### Increasing k-point grid to take DOS calculation

This depends a lot on how the DOS is calculated. I don't know the options one has for this when using VASP or SIESTA, but I am aware of different approaches. The central problem is that you don't know ...
• 1,076


text:
cirq.SparseSimulatorStep¶

class cirq.SparseSimulatorStep(state_vector, measurements, qubit_map, dtype)[source]

A StepResult that includes StateVectorMixin methods.

__init__(state_vector, measurements, qubit_map, dtype)[source]

Results of a step of the simulator.

Parameters
• qubit_map – A map from the Qubits in the Circuit to the the index of this qubit for a canonical ordering. This canonical ordering is used to define the state vector (see the state_vector() method).

• measurements – A dictionary from measurement gate key to measurement results, ordered by the qubits that the measurement operates on.


text:
# The Futurama Theorem

### Dana C. Ernst Plymouth State University

#### UNO Mathematics Colloquium

February 28, 2012

Note: Just use your arrow keys to advance the slides. Also, you can get an overview of the slides by typing "m" or go to a specific slide by typing "g".

## Futurama

• Futurama is an American animated science fiction sitcom created by Matt Groening (of The Simpsons).
• The series aired on Fox from 1999-2003, and then began airing on Comedy Central in 2008. Season 7 coming in 2012.
• The show follows the adventures of a late 20th-century New York City pizza delivery boy, Philip J. Fry, who was accidentally frozen in 1999 and then thawed out on New Year's Eve 2999.
• Some main characters: Fry, Leela (one-eyed alien), Bender (robot), Professor Farnsworth (Fry's distant nephew and mad scientist), Zoidberg (lobster-like alien), Amy (spoiled and accident prone student), and more.

## Futurama

Here is short clip with the Professor Farnsworth and the Globe Trotters.

## The Prisoner of Benda

• "The Prisoner of Benda" is episode 10 of season 6 and aired on August, 2010.
• The title is a play on "The Prisoner of Zenda", a 1894 novel by Anthony Hope Hawkins.
• In the episode, Farnsworth and Amy build a machine that allows them to switch minds. However, they learn that the machine cannot be used twice on the same pair of bodies. After several characters swap minds, the rest of the episode is focused on how to return everyone's minds back to their original owner.
• With the help of two Harlem Globetrotters (and some mathematics), the irreversible mind swap problem is solved.
• Episode written by Ken Keeler, who has a PhD in applied mathematics from Harvard.
• Keeler invented and proved a mathematical theorem specifically for the show.

## The mind swap problem

Here are the swaps that take place during the episode:

• Professor ($p$) $\leftrightarrow$ Amy ($a$)
• Amy ($a$) $\leftrightarrow$ Bender ($b$)
• Professor ($p$) $\leftrightarrow$ Leela ($l$)
• Amy ($a$) $\leftrightarrow$ Wash Bucket ($w$)
• Fry ($f$) $\leftrightarrow$ Zoidberg ($z$)
• Emperor Nikolai ($e$) $\leftrightarrow$ Wash Bucket ($w$)
• Hermes ($h$)$\leftrightarrow$ Leela ($l$)

The upshot: $a \rightarrow h \rightarrow l \rightarrow p \rightarrow b \rightarrow e \rightarrow w \rightarrow a \text{ and } f \leftrightarrow z$

## The mind swap problem (continued)

The question is:

Can the machine be used to put everyone's mind back in the proper body if we are not allowed to use the machine on the same pair of bodies more than once?

We need to use math!

## The symmetric group

We can model this problem in terms of permutations (i.e., rearrangements) of numbers. It turns out that the set of all permutations of the numbers 1 through $n$ forms a "group" under composition.

A group is a set with an associative binary operation satisfying:

• #### closure:

the "product" of any two elements from the set is an element of the set.
• #### identity:

there exists a "do nothing" element.
• #### inverses:

for every element in the set, there exists another element in the set that "undoes" the original.

The group of permutations of $\{1,2,\ldots, n\}$ is called the symmetric group and is denoted by $S_n$.

## Permutation diagrams

One way of representing an element of the symmetric group is via permutation diagrams, which we illustrate by way of example.

## Permutation diagrams (continued)

Let's try multiplying.

### Example

Let $\alpha$ and $\beta$ be as on previous slide. Then

But on the other hand

We see that products of permutations do not necessarily commute (order matters).

## Permutation diagrams (continued)

However, sometimes permutations do commute.

### Example

Let $\beta$ and $\gamma$ be as before. Then

Notice that these result in the same diagram. So, $\beta$ and $\gamma$ commute.

• Order matters, but not always.
• A diagram that swaps precisely two numbers is called a transposition.
• "Disjoint" transpositions commute.
• $S_n$ is generated by the transpositions. That is, every permutation can be written as a product of some sequence of transpositions. (In fact, we only need the adjacent transpositions: $i$ swaps with $i+1$).

## The mind swaps as a diagram

If we let $a=1$, $h=2$, $l=3$, $p=4$, $b=5$, $e=6$, and $w=7$, $f=8$, and $z=9$, then the mind swaps from Futurama can be depicted as:

## Rephrasing the problem

The problem can be restated as:

Can we multiply a permutation by a sequence of transpositions that we have not yet used to obtain the identity permutation?

It turns out that, in general, the answer is yes, but to pull this off, we need to add in two more people that have not used the machine.

## The Futurama Theorem

Here is the solution presented in the show.

Note:This image is taken from The Infosphere: The Futurama Wiki.

## Cycle notation

Before describing the solution, we need to introduce a more efficient way of encoding our permutations. One such method is called cycle notation.

## Cycle notation (continued)

Let's try multiplying using cycle notation. Note: We will multiply right to left (like function composition).

### Example

Consider $\alpha, \beta$, and $\gamma$ in $S_{5}$ as in the previous examples. Then

$\alpha \beta=(1\quad 2\quad 3\quad 4\quad 5)(2\quad 4\quad 3)=(1\quad 2\quad 5).$

and

$\gamma\beta =(2\quad 4\quad 3)(1\quad 5)=(1\quad 5)(2\quad 4\quad 3).$

We saw earlier that $\beta$ and $\gamma$ commute with each other, which is why it looks like nothing happened in the second example.

• In cycle notation, the Futurama permutation is $(8\quad 9)(1\quad 2\quad 3\quad 4\quad 5\quad 6\quad 7)$.
• The interpretation of a cycle $(\ldots\quad i\quad j\quad \ldots)$ is that the body numbered $i$ passes its mind to the body $j$, etc.
• Transpositions are of the form $(i\quad j)$, which means that $i$ and $j$ swap minds.
• "Disjoint" cycles (including transpositions) commute.
• If we can figure out how to "fix" one cycle at a time, then we can likely fix them all.

## Fixing a single 7-cycle

Consider just the 7-cycle $(1\quad 2\quad 3\quad 4\quad 5\quad 6\quad 7)$. Introduce two new bodies that have not had their minds swapped with anyone else, say $x$ and $y$. To return all minds of 1-7 back to their rightful owner, multiply the cycle (on the left) by the following sequence of transpositions:

$(x\quad 7)(y\quad 1)(y\quad 2)(y\quad 3)(y\quad 4)(y\quad 5)(y\quad 6)(y\quad 7)(x\quad 1)$

Let's verify that this actually works! First, notice that all of these transpositions are distinct and since $x$ and $y$ are new, we never used any of the transpositions to obtain our original scrambling of minds.

$(x\quad 7)(y\quad 1)(y\quad 2)(y\quad 3)(y\quad 4)(y\quad 5)(y\quad 6)(y\quad 7)(x\quad 1)(1\quad 2\quad 3\quad 4\quad 5\quad 6\quad 7)=???$

Note: $x$ and $y$ now have their minds swapped with each other, but they never sat in the machine at the same time!

## Fixing an arbitrary $k$-cycle

Consider the $k$-cycle $(1\quad 2\quad \ldots\quad k)$. How do we "fix" this cycle?

As before, introduce two new bodies that have not used the machine, say $x$ and $y$. To fix a $k$-cycle, multiply the cycle (on the left) by the following:

$(x\quad k)(y\quad 1)(y\quad 2)\cdots (y\quad k-1)(y\quad k)(x\quad 1)$

Why does this work?!

• First observation: $(y\quad k)(x\quad 1)(1\quad 2\quad \ldots\quad k-1\quad k)=(1\quad 2\quad \ldots\quad k-1\quad y\quad k\quad x)$.
• Then $(y\quad k-1)$ puts the mind of $k-1$ back where it belongs.
• And then $(y\quad k-2)$ puts the mind of $k-2$ back where it belongs.
• We continue this way and in the second to last step, $(y\quad 1)$ puts the mind of 1 back where it belongs.
• And finally, we finish by swapping $x$ and $k$.
• We now have $x$ and $y$ with their minds swapped (but they never sat in the machine at the same time).

## Fixing products of disjoint cycles

What do we do when we have multiple cycles to start with?

• Use $x$ and $y$ to fix each individual cycle.
• When doing this, $x$ and $y$ will sometimes be swapped.
• If the number of disjoint cycles is even, then we are done after we fix each individual cycle.
• If the number of disjoint cycles is odd, then after fixing each individual cycle, finish by swapping $x$ and $y$.

## Let's try it out!

• We need some volunteers! Pick two people to be $x$ and $y$.
• Write your name on a name tag and on a sheet of paper. The name tag represents your body and the sheet of paper represents your mind.
• Now, randomly shuffle your sheets of paper (i.e., minds) amongst each other, but $x$ and $y$ should not swap. Other than $x$ and $y$, each person should have exactly one mind that does not belong to them.
• Determine the disjoint cycles. Hint: If someone is holding your mind, stand next to them with that person on your right.
• Number yourselves in a way that makes sense. Put these numbers on your name tag and your sheet of paper.
• Now, see if you can apply our algorithm to return each mind to the appropriate person.

## Closing remarks

• Our algorithm/proof is a special case of the one given on the board in the TV episode.
• Applying our algorithm to the arrangement of minds in Futurama requires 13 moves.
• But since Fry and Zoidberg only swapped with each other and since there is only one other cycle, we could use Fry and Zoidberg as $x$ and $y$, respectively in our algorithm. This only requires 9 moves.

### Questions to ponder

• Can we do better (i.e., fewer moves to fix the cycles)?
• Under what circumstances do we not need to introduce two "clean" people?
• Would only using one "clean" person ever be useful?
• Could you use fewer moves if you allowed more than two "clean" people?

Okay, that's it. Thank you!

These slides were made using deck.js and MathJax.

/

#


text:
# Electric Dipole

View Notes

## What is Electric Dipole?

An electric dipole is an arrangement of electric charges. Let’s say if one charge is negative then the other needs to be positive, provided that these two charges are of equal magnitude. Also, there should be a certain distance between these two charges.

Let’s suppose + q  and - q are the two charges separated by a distance ‘2a’, now joining the centre of these charges with a line, as shown below:

The mid-point of this line is the centre of the dipole. The dipole has a certain length and also the moment.

Here, we are going to discuss electric dipole and dipole moment.

### What is Electric Dipole and Electric Dipole Moment?

We understood that an electric dipole is an arrangement of equal and opposite charges. Since the two charges are separated by a distance ‘2a’, which is called the dipole length. The distance between the charge and the centre of the dipole is ‘a’.

So, what is an electric dipole moment?

The dipole moment is a vector quantity and is denoted by a symbol $\overrightarrow{p}$. Its magnitude is equal to the magnitude of either of the two charges. Since we don’t specify the sign for the dipole moment, we multiply either of the two charges with the dipole length. It is given by:

$\overrightarrow{p} = q \overrightarrow{d}$ ….(1)

Here, d  = 2a, so, we can rewrite the equation as:

$\overrightarrow{p} = q(2a)$ …..(2)

So, equation (2) is the magnitude of the dipole moment.

So, what is the SI unit of the dipole moment?

We know that the unit of the charge is ‘C’ and that of distance is ‘m’. So, the unit of the dipole moment becomes:

 S.I. unit of $\overrightarrow{p} = C.m$

Can we see the difference in equations (1) & (2)?

Yes, there is a big difference between the two, but how, let’s understand this:

See, in equation (1), we considered the distance ‘d’ as a vector quantity, however, in equation (2), both the quantities viz: charge and distance or dipole length are the scalar quantities.

Now, when the product on the R.H.S of eq (2) are scalar, so, how can dipole moment be a vector quantity?  Yes, p can be a vector quantity only when we somehow convert the distance ‘2a’ as the displacement.

So, how can a distance be displaced, as these two quantities are different?

Now, let’s look at the following arrangement:

When we take ‘2a’ displacement of the charge - q with respect to charge + q, .i.e., from L.H.S to R.H.S or of the charge + q with respect to charge - q (from R.H.S to L.H.S). Now, this  ‘2a’ becomes a vector quantity. Hence, equation (2) becomes:

$\overrightarrow{p} = q(2 \overrightarrow{a})$…..(3)

Or,                        $|\overrightarrow{p}| = q \times (2a)$ …..(4)

So, what do you mean by electric dipole moment?

From eq (3), we define electric dipole moment as the product of charge and the displacement of + q charge with respect to - q charge.

Sometimes confusion arises in considering the direction of electric dipole moment. In Chemistry, we consider the displacement of + q charge with respect to charge - q, i.e., from right-to-left, while in Physics, we take the displacement of - q w.r.t. + q, i.e., from left-to-right.

There are two more units of the dipole moment, possessing a relationship between each. These are:

• StatC . cm

• Debye (D)

Also, 1 D = 10-18 StatC . cm, which is approximately  = 3.33564 x 10-30 C.m.

There’s something ideal included in the concept of dipole moment, let’s discuss it.

### What is an Ideal Electric Dipole?

From eq (4): If the charge ‘q’ gets larger, the distance between the two charges becomes smaller and smaller,  to keep the product of these two quantities viz: ‘q’ and ‘2a’ as constant, i.e., $\overrightarrow{p}| = q \times (2a)$ = constant, this is what we call it as an ideal dipole or point dipole.

Therefore, an ideal dipole is the smallest dipole having almost no size.

### Significance of Electric Dipoles

We can find the application of electric dipoles in atoms and molecules, but how?

Consider one atom of Hydrogen and the other two of Oxygen in the water molecule. Now, what happens here is, a covalent bond forms between H and two O-atoms, and they come close to each other.

Since there is a difference in the electronegativity of H and O, there is a shift of charges; also, the centres of these atoms don’t coincide.

O is an electronegative element and a shared pair of electrons between H and O towards O. Therefore, the centre of the negative charge shifts towards O, and the positive one to the H-atom; also, a partial positive and the negative charge develops on H and O-atom, respectively. Now, the arrangement of H and O in the water molecule behaves like a dipole.

Question 1: Two charges of magnitude ± 30 C are Separated by a Distance of 10 mm, Find the Magnitude and Direction of Dipole Moment.

Answer: Given: + q = 30 C, - q = - 30 C, 2a  = 10 mm = 0.01 m

∵ p = ± q x 2a, now putting the above values, we get:

p  = 30 x 0.01 = 0.3 or 3 x 10-1  C.m

Question 2: What is the Difference Between Electric Dipole and Electric Dipole Moment?

Answer: Electric dipole is the distance between the centre of the two charges bearing equal magnitude with opposite signs, however, electric dipole moment measures the strength of the electric dipole.

Simply speaking, the dipole is the arrangement of the two charges with different polarity and the dipole moment is the measurement of the electric polarity of the arrangement of these two charges.

Question 3: What Happens When an Electric Dipole is Placed in a Uniform Electric Field?

Answer: When the dipole is placed in a uniform electric field, the positive and the negative charges experience an equal and opposite force, i.e., the net force acting on them becomes zero.

Since the torque acts in the same direction for both the charges, that’s why the net torque acting on the dipole is non-zero.

Question 4: Does CO2 have a Dipole Moment?


text:
# Strings

echo "words words words ⚑"
echo """
<html>
<head>
</head>\n\n

<body>
</body>
</html> """

proc re(s: string): string = s

echo r".""."
echo re"\b[a-z]++\b"
$nim c -r strings.nim words words words ⚑ <html> <head> <head/>\n\n <body> <body/> <html/> .". \b[a-z]++\b There are several types of string literals: • Quoted Strings: Created by wrapping the body in triple quotes, they never interpret escape codes • Raw Strings: created by prefixing the string with an r. They do not interpret escape sequences, except for "", which is interpreted as ". This means that r"\b[a-z]\b" is interpreted as \b[a-z]\b instead of failing to compile with a syntax error. • Proc Strings: raw strings, but the method name that prefixes the string is called, so that foo"12\" -> foo(r"12\"). Strings are null-terminated, so that cstring("foo") requires zero copying. However, you should be careful that the lifetime of the cstring does not exceed the lifetime of the string it is based upon. Strings can also almost be thought of as seq[char] with respect to assignment semantics. See seqs for the possible manipulations, for example: echo "abc"[0] echo "abcdefg"[0 .. 4] echo "abcdefg"[0 .. ^2] $ nim c -r strindex.nim
a
abcde
abcdef

The strutils module provides procs for handling strings.

import strutils

var a = "hello welcome,friend"

# The split proc takes a sequence of characters and splits a string based on them
echo a.split({' ', ','})

# The contains proc determines whether a string contains a substring or character
echo a.contains("hello")
\$ nim c -r strutils.nim
@["hello", "welcome", "friend"]
true

## A note about Unicode

Unicode symbols are allowed in strings, but are not treated in any special way, so if you want count glyphs or uppercase Unicode symbols, you must use the unicode module.

Strings are generally considered to be encoded as UTF-8, so because of Unicode’s backwards compatibility, can be treated exactly as ASCII, with all values above 127 ignored.


text:
# Binary symmetry

## Recommended Posts

Take the binary numbers of a cirtain length, for example.

00

01

10

11

Sort them into how many 1's they have.

00

01, 10

11

Write them in a more familiar form.

0

1, 2

3

And you have a pattern of numbers, which is more interesting with larger examples.

0

1

0

1, 2

3

0

1, 2, 4

3, 5, 6

7

0

1, 2, 4, 8

3, 5, 6, 9, 10, 12

7, 11, 13, 14

15

The first and last rows contain one element.

The 2nd row shows powers of 2.

More interestingly in all cases all but the last element of the 2nd to last row are prime.

##### Share on other sites

The number of elements in each row is according to the binomial coefficients. The 2nd row shows powers of two due to construction: It contains all values $\sum_n a_n 2^n$ for which exactly one $a_n$ is one and the others are zero. I must admit that I do not find it particularly interesting that you see an approximate pattern in the distribution of the primes. I think you could see an approximate prime pattern in any random arrangement of the numbers from 1 to 15.

Edited by timo
##### Share on other sites

The 2nd row shows powers of two due to construction: It contains all values $\sum_n a_n 2^n$

The 2nd row is an ordered set where $s_{n}$ = $2^{n}$.

How is it the sum of something times by $2^{n}$ where n is the sum counter? Isn't that going to return a number?

for which exactly one $a_n$ is one and the others are zero. I must admit that I do not find it particularly interesting that you see an approximate pattern in the distribution of the primes. I think you could see an approximate prime pattern in any random arrangement of the numbers from 1 to 15.

Are you saying that for subsequent elements in the sequence this observation of primes wouldn't hold true?

##### Share on other sites

The 2nd row is an ordered set where $s_{n}$ = $2^{n}$.

How is it the sum of something times by $2^{n}$ where n is the sum counter? Isn't that going to return a number?

I don't understand what you are asking. Any (natural) number x can be written as $x = a_0 2^0 + a_1 2^1 + a_2 2^2 + a_3 2^3 + \dots = \sum_n a_n 2^n$. If you demand that exactly one of the $a_n$ is one and the others are zero then it is obvious that the resulting x is a power of two. Iow: notice that there is another half to the half-sentence you quoted which contains important information.

Are you saying that for subsequent elements in the sequence this observation of primes wouldn't hold true?

I can probably see a pattern in every result coming out of rolling two dice if I want to. You would probably not agree that this pattern I see is an interesting find. But if you find the pattern you see interesting (I don't think I fully understand what you are saying), brute-force your arrangement on a computer to however far you get there. If you still think you see a pattern, then things might become interesting.

Edited by timo
##### Share on other sites

(from the thread title I thought you might be talking about digitally balanced numbers, which you might find interesting)

I'll try to do some brute forcing, should that help you in your quest to find something interesting.

Merged post follows:

Consecutive posts merged
The first and last rows contain one element.
This was fairly obviously as a result of you choosing numbers of the form [imath]\sum_n 1\cdot 2^n[/imath], so as that the largest number in your list would be a series of all ones.

Anyway, brute forced results up to the very, very contrived figure of 511, (or, the sum of powers of two from 0 to 8)

[size="1"]{0}
{1, 2, 4, 8, 16, 32, 64, 128, 256}
{3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 65, 66, 68, 72, 80, 96, 129, 130, 132, 136, 144, 160, 192, 257, 258, 260, 264, 272, 288, 320, 384}
{7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 131, 133, 134, 137, 138, 140, 145, 146, 148, 152, 161, 162, 164, 168, 176, 193, 194, 196, 200, 208, 224, 259, 261, 262, 265, 266, 268, 273, 274, 276, 280, 289, 290, 292, 296, 304, 321, 322, 324, 328, 336, 352, 385, 386, 388, 392, 400, 416, 448}
{15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197, 198, 201, 202, 204, 209, 210, 212, 216, 225, 226, 228, 232, 240, 263, 267, 269, 270, 275, 277, 278, 281, 282, 284, 291, 293, 294, 297, 298, 300, 305, 306, 308, 312, 323, 325, 326, 329, 330, 332, 337, 338, 340, 344, 353, 354, 356, 360, 368, 387, 389, 390, 393, 394, 396, 401, 402, 404, 408, 417, 418, 420, 424, 432, 449, 450, 452, 456, 464, 480}
{31, 47, 55, 59, 61, 62, 79, 87, 91, 93, 94, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 143, 151, 155, 157, 158, 167, 171, 173, 174, 179, 181, 182, 185, 186, 188, 199, 203, 205, 206, 211, 213, 214, 217, 218, 220, 227, 229, 230, 233, 234, 236, 241, 242, 244, 248, 271, 279, 283, 285, 286, 295, 299, 301, 302, 307, 309, 310, 313, 314, 316, 327, 331, 333, 334, 339, 341, 342, 345, 346, 348, 355, 357, 358, 361, 362, 364, 369, 370, 372, 376, 391, 395, 397, 398, 403, 405, 406, 409, 410, 412, 419, 421, 422, 425, 426, 428, 433, 434, 436, 440, 451, 453, 454, 457, 458, 460, 465, 466, 468, 472, 481, 482, 484, 488, 496}
{63, 95, 111, 119, 123, 125, 126, 159, 175, 183, 187, 189, 190, 207, 215, 219, 221, 222, 231, 235, 237, 238, 243, 245, 246, 249, 250, 252, 287, 303, 311, 315, 317, 318, 335, 343, 347, 349, 350, 359, 363, 365, 366, 371, 373, 374, 377, 378, 380, 399, 407, 411, 413, 414, 423, 427, 429, 430, 435, 437, 438, 441, 442, 444, 455, 459, 461, 462, 467, 469, 470, 473, 474, 476, 483, 485, 486, 489, 490, 492, 497, 498, 500, 504}
{127, 191, 223, 239, 247, 251, 253, 254, 319, 351, 367, 375, 379, 381, 382, 415, 431, 439, 443, 445, 446, 463, 471, 475, 477, 478, 487, 491, 493, 494, 499, 501, 502, 505, 506, 508}
{255, 383, 447, 479, 495, 503, 507, 509, 510}
{511}[/size]

Note how there is a large frequency of primes in the second to last row, but it's not all but one.

Edited by the tree
Consecutive posts merged.
##### Share on other sites

Any (natural) number x can be written as $x = a_0 2^0 + a_1 2^1 + a_2 2^2 + a_3 2^3 + \dots = \sum_n a_n 2^n$

I see, binomial coefficients? I've never come across this before. How would the number 4 be represented in this notation?

Merged post follows:

Consecutive posts merged
(from the thread title I thought you might be talking about digitally balanced numbers, which you might find interesting)

Thanks. This means that the central row of each one shows digitally balanced numbers, with varying degrees of imbalance towards the top and bottom.

This was fairly obviously as a result of you choosing numbers of the form [imath]\sum_n 1\cdot 2^n[/imath]

I think the [imath]\cdot[/imath] operator means multiply?

Note how there is a large frequency of primes in the second to last row, but it's not all but one.

It's every other one instead. Seems I was wrong, it does appear as if there is some pattern of prime distributions though. Seems to much of a coincidence.

Wolfram alpha shows that several rows including the middle one (digitally balanced numbers) produce a fractal like graph.

Edited by alan2here
Consecutive posts merged.
##### Share on other sites

I see, binomial coefficients? I've never come across this before. How would the number 4 be represented in this notation?

Four would be 100, or $a_2=1, a_{n \neq 2}=0$.

The $a_n$ are not the binomial coefficients, the number of entries in each row are. That is because the binomial coefficients $\left( \begin{array}{c} N \\ k \end{array} \right)$ Tell you how many possible combinations there are to distribute k equal elements on N places (in your case ones on the available digits).

##### Share on other sites

Just in case the OP still isn't clear on this, the expression of any natural number in any base [imath]b[/imath] expression in the form [imath]\sum_{n=0}^{\infty} a_n b^n \;| a_n \in [0\dots b-1][/imath].

For instance, 5427 = 5x103 + 4x102 + 2x101 + 7x100.

And the binomial co-efficients [imath]\binom{n}{k}= \,^{n}C_k \,=\frac{n!}{k!\,(n-k)!}[/imath] is the count of the amount of ways that, for instance, [imath]k[/imath] out of [imath]n[/imath] boxes can be ticked.

Edited by the tree
##### Share on other sites

Four would be 100, or $a_2=1, a_{n \neq 2}=0$.

Thanks. It's a differnt way to describe binary numbers.

Merged post follows:

Consecutive posts merged
Just in case the OP still isn't clear on this, the expression of any natural number in any base [imath]b[/imath] expression in the form [imath]\sum_{n=0}^{\infty} a_n b^n \;| a_n \in [0\dots b-1][/imath].

Thats a verry confusing looking equation.

For instance, 5427 = 5x103 + 4x102 + 2x101 + 7x100.

Yes, this makes sence.

Thanks. I see now how the equation works now (b = base)

A = 542 = A2*b2 + A1*b1 + A0*b0.

542 = $\sum$Anbn

And the binomial co-efficients [imath]\binom{n}{k}= \,^{n}C_k \,=\frac{n!}{k!\,(n-k)!}[/imath] is the count of the amount of ways that, for instance, [imath]k[/imath] out of [imath]n[/imath] boxes can be ticked.

So an equation calculates how many differnt K of N combinations exist is. [imath]\frac{n!}{k!\,(n-k)!}[/imath]

edit: seems to work.

Edited by alan2here
Consecutive posts merged.
##### Share on other sites

So an equation calculates how many differnt K of N combinations exist is. [imath]\frac{n!}{k!\,(n-k)!}[/imath]

edit: seems to work.

Definitely works.

It can be proved by induction, but fairly awkwardly since you have to prove it for varying n and varying k separately.

They are called binomial coefficients because of the binomial theorem:

$(x+y)^n = \sum_{i=0}^n x^{n-i}y^{i}\binom{n}{i}$

Small ones can be calculated by hand using Pascal's triangle.

## Create an account

Register a new account


text:
# Wind speed forecasting using ARIMA model in Python3

Recently, I started working on time-series models and would mention that I am very new to python and ML as a whole.

I tried to implement a time-series model on wind speed data. Being a newbie, I followed the steps given in this article: https://kanoki.org/2020/04/30/time-series-analysis-and-forecasting-with-arima-python/ and it's a great one to start with but somehow I am unable to forecast my data (or I would say the forecasted data is constant around the 5.88 value!)

# FORECAST
n=39
forecast,err,ci = results.forecast(steps=n, alpha= 0.05)
windspeed_forecast = pd.DataFrame({'forecast':forecast},index=pd.date_range(start='22/8/2020 01:00:00', periods=n, freq='MS'))
windspeed_forecast

#plot for forecast
ax = windspeed[19:].SPEED.plot(label='observed', figsize=(20, 15))
windspeed_forecast.plot(ax=ax,label='Forecast',color='r')
ax.fill_between(windspeed_forecast.index, ci[:,0], ci[:,1], color='b', alpha=.005)
ax.set_xlabel('DATE')
ax.set_ylabel('SPEED')

plt.legend()
plt.show()

-What I think is the high AIC value of 700 might be the problem!

-Also, I am unable to figure out how can I create the column of Date-time for the forecasted values same as that of the original data(i.e. hourly based data of a specific date) [As shown in the ss number 1 and as shown in ss below - I need a column starting from 22/8/2020 with hourly gaps and so on].

Also, PFA the ss of my data in jupyter notebook (out of 191 total data, 152 used as train data and rest as test data)

Any suggestion/help regarding the same will be appreciated :)

• I don't really understand what your main specific question is here. Can you clarify that (although this is an old question)? – nbro Jan 20 at 19:57


text:
# Is there a specific branch of physics that studies waves?

Is there a branch of physics that studies waves and how they propagate through air, wires etc.?

• Actually, almost all of physics studies waves in one way, or another. Many of the basic properties of harmonic oscillators and linear waves can be found over and over again in different physical systems, which means that a central part of the material physicist have to learn concerns these fundamental physical effects of harmonic motion. So it's more the other way around: every physicist has a pretty good notion of waves, no matter what they do in detail. Sep 27, 2014 at 15:00
• You might be interested in volume 3 of the Berkeley Physics Course, entitled Waves. Sep 27, 2014 at 16:55

Waves are in nearly every part of physics: sound waves, light waves, ocean waves, all of hydrodynamics really, all of magnetohydrodynamics too, stellar pulsations, gravitational waves, alternating current, coupled oscillators of any sort$\ldots$ The list goes on.


text:
# Which equation could represent the graph below?

###### Question:

Which equation could represent the graph below?

### Write 3 2/3 as improper fraction

write 3 2/3 as improper fraction...

### WHO OR WHAT DID SOMETHING IN THE PAST ABOUT SCIENCE THAT IMPACTS OUR LIFE TODAY

WHO OR WHAT DID SOMETHING IN THE PAST ABOUT SCIENCE THAT IMPACTS OUR LIFE TODAY...

### What does conjecture mean?

What does conjecture mean?...

### HELP RN 25 POINTS ITS TIMED

HELP RN 25 POINTS ITS TIMED...

### In a video game clare score 50 more points than Tyler if c is the number of points that clear score and tease the number of points that Tyler score which equations are correct select all that apply C=1.5t C=t+0.5 C=t+0.5t C=(1+0.5)t C=t+50

in a video game clare score 50 more points than Tyler if c is the number of points that clear score and tease the number of points that Tyler score which equations are correct select all that apply C=1.5t C=t+0.5 C=t+0.5t C=(1+0.5)t C=t+50...

### Which methods can be used to run a query? Check all that apply. On the Create tab, in the Queries group, click Run. In query Design view, on the Design tab, click Run. Switch to Datasheet view before any other commands. Close the Show Table dialog box in the Datasheet view. On the Create tab, in the Queries group, click Create Query.

Which methods can be used to run a query? Check all that apply. On the Create tab, in the Queries group, click Run. In query Design view, on the Design tab, click Run. Switch to Datasheet view before any other commands. Close the Show Table dialog box in the Datasheet view. On the Create tab, in the...

### Select all the equations that are equivalent to (5x+6)/2 =3-(4x+12)

select all the equations that are equivalent to (5x+6)/2 =3-(4x+12)...

### Scientists now know how much carbon dioxide was present in the atmosphere as far back as 800,000 years ago. How did they obtain this data?

Scientists now know how much carbon dioxide was present in the atmosphere as far back as 800,000 years ago. How did they obtain this data?...

### Look at this triangle.B8 cmA А22 cmСWork out length AB.

Look at this triangle.B8 cmA А22 cmСWork out length AB....

### 1) Why might investors prefer floating rate notes over a fixed rate bond? 2) Why might Hologen prefer to issue fixed rate bonds rather than floating rate notes?

1) Why might investors prefer floating rate notes over a fixed rate bond? 2) Why might Hologen prefer to issue fixed rate bonds rather than floating rate notes?...

### Earth Science! (Astrology) ANOTHER HARD QUESTION! (That I do not remember) Description: Don't have this in my notes. Lmbo Which form of Kepler’s third law can you use to relate the period T and radius r of a planet in our solar system as long as the unit year is used for the period and astronomical unit is used for the radius? ( = means proportional to in this case) -T^2=r -T=r^2 -T^2=r^3 -T^3=r^2

Earth Science! (Astrology) ANOTHER HARD QUESTION! (That I do not remember) Description: Don't have this in my notes. Lmbo Which form of Kepler’s third law can you use to relate the period T and radius r of a planet in our solar system as long as the unit year is used for the period and astronomica...

### Who's your favorite character naruto or sasuke you can pick both.

who's your favorite character naruto or sasuke you can pick both....

### Sarah went to an electronics store, purchased 3 CDs and 5 DVDs, and spent $137. Michelle went to the same store, bought 2 CDs and 7 DVDs, and spent$161. How much does each CD sell for at the store? . . a) $13 . . b)$14 . . c)$18 . . d)$19 . .

Sarah went to an electronics store, purchased 3 CDs and 5 DVDs, and spent $137. Michelle went to the same store, bought 2 CDs and 7 DVDs, and spent$161. How much does each CD sell for at the store? . . a) $13 . . b)$14 . . c)$18 . . d)$19 . ....

### How does the mentor role differ from the facilitator role in an organization explain?

How does the mentor role differ from the facilitator role in an organization explain?...

### Why do cells need both tRNA and mRNA￼ A. Only mRNA can be used to repair mistakes in the DNA sequence B. tRNA bends into a shape that can carry specific amino acids C. The tRNA is used on the ribosomes and the mRNA is used in the nucleus D. The MRNA brings information to the nucleus for the tRNA to use

Why do cells need both tRNA and mRNA￼ A. Only mRNA can be used to repair mistakes in the DNA sequence B. tRNA bends into a shape that can carry specific amino acids C. The tRNA is used on the ribosomes and the mRNA is used in the nucleus D. The MRNA brings information to the nucleus for the tRNA t...

### Incinerating or burning trash is the best way to get rid of the majority of it. True or False

Incinerating or burning trash is the best way to get rid of the majority of it. True or False...