Charge Limit Research Articles

Overcoming the thermal limits of photon counting CT resolution using focal spot multiplexing: A feasibility study.

The spatial resolution of new, photon counting detector (PCD) CT scanners is limited by the size of the focal spot. Smaller, brighter focal spots would melt the tungsten focal track of a conventional X-ray source. To propose focal spot multiplexing (FSM), an architecture to improve the power of small focal spots and thereby enable higher resolution clinical PCD CT. In FSM, the source rapidly alternates between multiple focal spot locations. The dwell time at each focal spot location is much shorter than the readout interval of the PCD, and each location is visited many times during the readout interval so that heat can be effectively distributed over a larger surface area. The PCD accumulates recorded events on the detector (prior to slip ring transmission) into multiple spatial bins, as a second dimension to conventional energy bins. We estimated the maximum power permissible for a square, 0.2 mm focal spot assuming a maximum allowed surface temperature of 2900 K and a maximum transient temperature increase at the focal spot of 1400 K. This was performed using two separate thermal simulation codes: first, a commercial finite element method (FEM) was used to estimate bulk heating; second, a custom solver linearizing the heat equation was used to estimate track heating. FSM was simulated assuming that focal spot locations lay on parallel focal tracks, that instantaneous switching between tracks was possible, and that the space charge limit was never reached. We assumed a focal track velocity of 100 m/s and a 1 mm tungsten focal track layer with TZM alloy backing. The tube power was constant over a 4 s acquisition period. Without FSM, the 0.2 mm focal spot could attain a maximum power output of 17 kW before reaching the thermal limit. FSM required a minimum switching frequency of 0.5 MHz to produce benefit. With two focal tracks and a switching frequency of 4 MHz, the power could be increased to 28 kW, and with eight focal tracks and a switching frequency of 16 MHz, the power could be increased to 98 kW. In all cases, the 1400 K transient increase thermal limit was reached before the 2900 K maximum surface temperature constraint. The relationship between maximum power, switching frequency, and number of focal tracks is discussed in relation to the underlying physics of heat transport. Focal spot multiplexing is an architecture that exploits the digital nature of photon counting to improve the power of high-resolution CT X-ray sources. With fast switching and several parallel focal tracks, it may be possible to halve the size of the focal spot while maintaining source power. This work motivates the development of new source technologies that can achieve fast electronic switching.

Flexible Array of Rigid Porous Microneedles for Bioelectric Skin Patch

Electrodermal patches that generate transdermal currents have been widely investigated for wound healing, anti-perspiration, and drug delivery. Our laboratory has recently developed a metal-free, disposable, fully organic, flexible electrodermal patch with potential for a wide range of healthcare applications. The patch has attracted considerable attention because of its ease of use, lack of need for an external power source, improved drug penetration, and, above all, its simplicity. By selecting soft materials for the patch and electrodes, a total thickness of less than 2 mm has been achieved, and the patch remains flexible enough to be used on skin surfaces and other surfaces. However, the application is limited at the present stage due to the high resistance of the stratum corneum of the outer skin layer in addition to the lower output of the on-board enzyme batteries.Microneedle array can be an attractive solution as it can penetrate through the stratum corneum without reaching blood vessels or nerves. Especially, solid porous microneedles (PMNs), which have microscopic channels throughout the needle structure, exhibit characteristic solution permeability. This serves as a pathway for transdermal iontophoresis drug delivery. Furthermore, modification of the connecting pores with electrically charged materials can generate electroosmotic flow (EOF) inside PMNs, which can enhance drug penetration into the skin without charge or size limitations. Most of the reported microneedle-based skin devices used a hard substrate with high mechanical strength as the base. However, when devices are used on sited of high curvature and deformation, contact with the skin surface is likely to be uneven. This can lead to reduced medical effectiveness, pain, and device breakage. Here, we developed a flexible microneedle array, which provides a stable skin-application effect, by combining a needle portion made of rigid material and a substrate portion made of flexible material.The flexible PMN array was fabricated by integrating rigid PMNs onto the flexible substrate such as polydimethylsiloxane (PDMS) or polyurethane (PU) sheets. The needle part was fabricated by photopolymerizing a mixture of glycidyl methacrylate, trimethylolpropane trimethacrylate and triethylene glycol dimethacrylate, together with a mixture of polyethylene glycol (PEG) and diethylene glycol monomethyl ether (DEG) as porogen. The porogen was dissolved after photopolymerization to form submicron channels networks inside the PMNs. The PMNs were integrated with flexible substrates by solidifying together directly for mechanical fixation. We then evaluated the developed flexible PMN array as follows. First, it was confirmed that the transcutaneous resistance was reduced by applying the flexible PMN patch, both on the flat and curved pig skins. Second, in combination with a bio battery consisting of an enzyme-modified fabric electrode, it was demonstrated that a stable transcutaneous current was generated from glucose and enzyme even on the curved surfaces. In addition, ex vivo experiments showed skin penetration performance while maintaining flexibility.In summary, we developed a method to fabricate rigid PMNs directly bond onto the flexible substrates. We then confirmed that the flexible PMN array can conformably fit on and penetrate curved pig skins. Such flexible and disposable skin patches combined with rigid PMNs can serve as a platform for a wide range of medical, healthcare, and cosmetic applications. Figure 1

All-loop Heavy-Heavy-Light-Light correlators in N = 4 super Yang-Mills theory

We study Heavy-Heavy-Light-Light (HHLL) correlators HHO2O2 in N = 4 super Yang-Mills theory with SU(N) gauge group at generic N. The light operator O2 is the dimension two superconformal primary in the stress tensor multiplet and H is a general half-BPS superconformal primary operator with dimension (or R-charge) ∆H. We consider the large-charge ’t Hooft limit, where ∆H→∞ with fixed ’t Hooft-like coupling λ≔∆HgYM2. We show that the L-loop contribution to the HHLL correlators in the leading large-charge limit is universal for any choice of the heavy operator H, given as λL∑ℓ=0LΦℓΦL−ℓ with an SU(N) colour factor coefficient, where Φ(ℓ) is the ladder Feynman integral, which is known to all loops. The dependence on the explicit form of the heavy operator lies only in the colour factor coefficients. We determine such colour factors for several classes of heavy operators, and show that the large charge limit leads to minimal powers of N. For the special class of “canonical heavy operators”, one can even resum the all-loop ladder integrals and determine the correlators at finite λ. Furthermore, upon integrating over the spacetime dependence the resulting integrated HHLL correlators agree with the existing results derived from supersymmetric localisation. Finally, as an application of the all-loop analytic results, we derive exact expressions for the structure constants of two heavy operators and the Konishi operator, finding intriguing connections with the integrated HHLL correlators.

Robust resistance switching performance in a Ca3Ti2O7/La0.67Sr0.33MnO3 heterostructure induced by the interface

Ruddlesden–Popper phase oxides, such as Ca3Ti2O7, have been established as hybrid improper ferroelectrics. However, investigations into Ca3Ti2O7 have primarily concentrated on their structural and ferroelectric properties. In this study, we prepared epitaxial Ca3Ti2O7 thin films via magnetron sputtering. Conducting atomic force microscopy was employed to characterize local current variations under an applied bias voltage. Electron paramagnetic resonance measurements of the Ca3Ti2O7/La0.67Sr0.33MnO3 film were conducted to assess its defect characteristics. Interestingly, the Ca3Ti2O7/La0.67Sr0.33MnO3 stacks exhibited remarkable macroscopic resistance switching, with a resistance on/off ratio reaching 100, alongside robust retention (∼2500 s) and endurance (∼2000 cycles) features. Additionally, density functional theory calculations suggest that the resistance switching is attributable to the interface barrier of the Ca3Ti2O7/La0.67Sr0.33MnO3 interface and the efficacy of space charge limitation. This work proposes an avenue for the utilization of Ruddlesden–Popper phase Ca3Ti2O7 in various applications.

Need of voltage and capacity profile reconstruction in incremental capacity analysis for second life batteries from repurposer perspective

Incremental Capacity Analysis (ICA) is a widely used non-destructive technique to predict State of Health (SoH) of Lithium-ion Battery (LiB). Till today, the SoH is judged on ICA magnitude of peaks and valleys for defining the health indicators. This research work provides the state-of-the art in voltage and capacity profile reconstruction available for ICA through Constant Capacity (CC), Constant Voltage (CV), and hybrid sampling. Here, hybrid sampling offers flexibility for the repurposer to define voltage bands for partial charge and discharge limits according to ICA peaks. This research work aims to address the scope of reducing time and efforts on partial charging and discharging tests conducted by the repurposers for SoH and Remaining useful Life (RuL) estimation of Second Life Battery (SLB). Nissan Leaf battery 2011 model taken from EV application is tested for health indicators catering insights on SoH and RuL by reconstructing the voltage and capacity profile. The necessity of a filter can be neglected as a part of ICA post-processing. The peaks and valleys from reconstructed profile is aiding the repurposer to define voltage limits for partial charging and discharging based on SoH. This led to reduction in testing time and efforts in the range of 16% to 83% during SLB characterization. The accuracy of this technique is validated by varying the sampling frequencies ranging from 0.1 Ah, 0.01 Ah and 0.001 Ah before proceeding with reconstruction. Sandia National Laboratories open source data is considered for verifying the proof of concept. Pearson correlation coefficient is deployed to differentiate strong and weak SLB based on sampled data. The actual data obtained from battery cycler fails to do the same and hence, the sampled data aids repurposer in reducing the charging and discharging time of SLB through narrowed voltage bands.

The Orbitron: A crossed-field device for co-confinement of high energy ions and electrons

To explore the confinement of high-energy ions above the space charge limit, we have developed a hybrid magnetic and electrostatic confinement device called an Orbitron. The Orbitron is a crossed-field device combining aspects of magnetic mirrors, magnetrons, and orbital ion traps. Ions are confined in orbits around a high-voltage cathode with co-rotating electrons confined by a relatively weak magnetic field. Experimental and computational investigations focus on reaching ion densities above the space charge limit through the co-confinement of electrons. The experimental apparatus and suite of diagnostics are being developed to measure the critical parameters, such as plasma density, particle energy, and fusion rate for high-energy, non-thermal plasma conditions in the Orbitron. Initial results from experimental and computational efforts have revealed the need for cathode voltages on the order of 100–300 kV, leading to the development of a custom high voltage, ultra-high vacuum bushing rated for 300 kV.

Performance investigation of a supergravity CO2 refrigeration cycle under off-design conditions

The transcritical CO2 refrigeration cycle is a research hotspot in CO2 refrigeration technology, but it suffers from significant throttling, compression, and gas cooling losses. The recently proposed supergravity CO2 refrigeration cycle, which utilizes centrifugal potential energy-pressure energy conversion in the rotating heat exchanger, provides a new approach to address these issues. However, the flow efficiency of the serpentine rotating channels requires further validation, and the operating characteristics of the system under off-design conditions need exploration. To this end, we first conducted a study on the pressure drop characteristics of a serpentine rotating channel using experimental and numerical simulation methods. We found a strong linear relationship between the friction factor and the rotation number, with the flow efficiency approaching 1.0 beyond a certain threshold, providing critical support for the cycle. Additionally, we established a distributed parameter model for the cycle and theoretically analyzed its operational performance. The results show that the COP of this cycle reaches 6.91 under air conditioning conditions. This is primarily because the local isentropic compression/expansion efficiency of CO2 in the rotating channel exceeds 99 %. Additionally, the temperature slip during the gas cooling process is only 1.2 °C, demonstrating the feasibility of achieving nearly isothermal compression through the rotating heat exchanger. Furthermore, the results showed that changes in boundary conditions resulted in minimal changes in system pressure differential, not exceeding 2 %. However, increases in CO2 charge and rotational speed led to substantial changes in system pressure differential, increasing by 39 % and 51 %, respectively. These trends reflect the effect of the centrifugal potential energy-pressure energy conversion on the operating characteristics of the system. In addition, there exists a maximum charge limit to prevent the cycle from entering the two-phase region. There is also an optimal combination of rotating heat exchanger and compressor speeds that ensures maximum COP while guaranteeing the cooling capacity. The main novelty of this paper lies in elucidating the coupling mechanism between operational parameters and the thermodynamic parameters of the supergravity CO2 refrigeration cycle.

Local charge redistribution enables single ionic conductor for fast charge solid Li battery

Solid polymer electrolytes (SPEs) have been regarded as hopeful candidate electrolyte for solid-state lithium battery. However, the low Li+ transference number and poor interface stability pose great challenges for the high rate capability of SPE. Herein, inspired from the density functional theory (DFT) calculations that local positive charge distribution can be regulated by introducing strong electron-withdrawing groups, which can selectively anchor TFSI−, largely enhance the Li+ transference number. As predicted, the obtained CPE-NO2 deliver a remarkable Li+ transference number of 0.91, equal to a single-ion conductor for Li+, largely high than that of the SPE (0.31), according well with the molecular dynamics (MD) simulations and pyridine complexation experiments. Furthermore, the PEO||Li interfacial stability, flame retardant ability, and mass transfer of Li+ in interface can also be largely enhanced by interfacial by-product, which are fast Li+ conductors according to TOF-SIMS results. Impressively, the Li|CPE-NO2|Li cell exhibit superior cyclability for 2200 h at 0.1 mA cm−2, and the solid LFP|CPE-NO2|Li battery delivers prominent capacity of 127.4 mAh g−1 at a high rate 5 C (12 mins). This work breaks the fast charge limitation of solid lithium battery, and provides a feasible approach for the construction of advanced solid electrolyte.

Charge Transfer Kinetics in Halide Perovskites: On the Constraints of Time-Resolved Spectroscopy Measurements.

Understanding photophysical processes in lead halide perovskites is an important aspect of optimizing the performance of optoelectronic devices. The determination of exact charge carrier extraction rate constants remains elusive, as there is a large and persistent discrepancy in the reported absolute values. In this review, we concentrate on experimental procedures adopted in the literature to obtain kinetic estimates of charge transfer processes and limitations imposed by the spectroscopy technique employed. Time-resolved techniques (e.g., transient absorption-reflection and time-resolved photoluminescence spectroscopy) are commonly employed to probe charge transfer at perovskite/transport layer interfaces. The variation in sample preparation and measurement conditions can produce a wide dispersion of the measured kinetic parameters. The selected time window and the kinetic fitting model employed introduce additional uncertainty. We discuss here evaluation strategies that rely on multiexponential fitting protocols (regular or stretched) and show how the dispersion in the reported values for carrier transfer rate constants can be resolved.

The unitary Fermi gas at large charge and large N

We study the unitary Fermi gas in a harmonic trapping potential starting from a microscopic theory in the limit of large charge and large number of fermion flavors N. In this regime, we present an algorithmic procedure for extracting data from perturbation theory, order-by-order, without the need for other assumptions. We perform a gradient expansion in the interior of the particle cloud, sufficiently far from the cloud edge where the particle density drops rapidly to zero. In this latter region we present the first microscopic computation characterizing the contribution of the edge terms. The microscopic theory reproduces the predictions of the superfluid eft, including the action, the form of the gap equation, and the energy of the system in a harmonic trap (which maps, via the non-relativistic state-operator correspondence, to the scaling dimension of the lowest operator of charge Q). We additionally give the Wilsonian coefficients at leading order in N up to nnlo in the large-charge expansion.

Behind-the-horizon excitations from a single 2d CFT

In this work, we consider the atypical non-equilibrium state found in [1708.06328] which holographically represents a behind-the-horizon excitation in a black hole spacetime. The special feature of this state is that it looks like an equilibrium state when probed by a class of low-energy operators. First, we retrieve this property using the uniformization mapping in the limit of a large central charge, in the process we are able to derive rather than presume approximate thermal physics. Furthermore, in the large-c and high-energy limit, we realize these excitations as elements of the commutant algebra of a GNS-representation of the light operator algebra. Instead of analytically continuing a mixed heavy-light Euclidean correlator to a Lorentzian correlator, we identify the Euclidean correlator as a GNS-linear form and interpret the Lorentzian correlator as a vacuum expectation value of representatives of the light operator algebra on the GNS-vacuum.

Natural refrigerant mixtures in low-charge heat pumps: An analysis of the potential for performance enhancements

The study investigated the characteristics of different natural refrigerants and their mixtures in a residential heat pump with low refrigerant charge. Three main evaluation criteria were utilized for comparing different mixtures: Coefficient of Performance (COP), Volumetric Heating Capacity (VHC), and a newly proposed indicator, the heating capacity at charge limit. Propane was used as a reference refrigerant. It was found that some mixtures significantly improved both COP and the heating capacity at charge limit while maintaining similar volumetric heating capacity and operating conditions. Alternative multi-criteria decision-making techniques were adopted to rank the best refrigerant mixtures. Mixtures rich in Dimethyl Ether (DME), such as DME-CO2 [0.96-0.04] and DME-Propylene [0.75-0.25] were found consistently among the best options. Those were followed by Propylene-rich mixtures such as Propylene-CO2, Propylene-Isobutane and Propylene-Butane. The levelized cost of heat (LCOH) could be improved by up to 12 %. To accelerate the transition to natural refrigerants, without compromises on efficiency and costs, further research and certification activities on non-fluorinated refrigerants based on Dimethyl Ether (R-E170), CO2 (R-744) and Propylene (R-1270) are recommended.

The role of charge in microdroplet redox chemistry

In charged water microdroplets, which occur in nature or in the lab upon ultrasonication or in electrospray processes, the thermodynamics for reactive chemistry can be dramatically altered relative to the bulk phase. Here, we provide a theoretical basis for the observation of accelerated chemistry by simulating water droplets of increasing charge imbalance to create redox agents such as hydroxyl and hydrogen radicals and solvated electrons. We compute the hydration enthalpy of OH− and H+ that controls the electron transfer process, and the corresponding changes in vertical ionization energy and vertical electron affinity of the ions, to create OH• and H• reactive species. We find that at ~ 20 − 50% of the Rayleigh limit of droplet charge the hydration enthalpy of both OH− and H+ have decreased by >50 kcal/mol such that electron transfer becomes thermodynamically favorable, in correspondence with the more favorable vertical electron affinity of H+ and the lowered vertical ionization energy of OH−. We provide scaling arguments that show that the nanoscale calculations and conclusions extend to the experimental microdroplet length scale. The relevance of the droplet charge for chemical reactivity is illustrated for the formation of H2O2, and has clear implications for other redox reactions observed to occur with enhanced rates in microdroplets.

Pulsed Electron Lenses for Space Charge Mitigation.

To produce ultimate high-brightness hadron beams, synchrotrons need to overcome a most prominent intensity limitation, i.e., space charge. This Letter characterizes the potential of pulsed electron lenses in detailed 3D tracking simulations, key to which is a realistic machine and space charge model. The space charge limit, imparted by betatron resonances, is shown to be increased by up to 50% using a low symmetric number of electron lenses in application to the Facility for Antiproton and Ion Research SIS100 synchrotron. Conceptually, a 100% increase is demonstrated with a larger number of electron lenses, which is found to rapidly saturate near the theoretical 2D limit.

Electric and dilatonic fields of a charged massive particle at rest in the field of a charged dilaton black hole

We study linear-perturbation equations for the two-body system of a charged dilaton black hole, of which dilaton coupling constant is α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}, and a static particle with mass m, electric charge q, and dilatonic charge βm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta m$$\\end{document}. We find that a consistent condition for the coupled equations corresponds to the equilibrium condition of the test particle. The expressions of classical fields are given in closed analytical formulas in the most interesting case with β=α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta =\\alpha $$\\end{document}. We examine the electrical field around a charged dilaton black hole especially in the limit of the maximum electric charge and we find the electric Meissner effect which has been found for the Reissner–Nordström black hole in the Einstein–Maxwell system.

All orders factorization and the Coulomb problem

In the limit of large nuclear charge, Z≫1, or small lepton velocity, β≪1, Coulomb corrections to nuclear beta decay and related processes are enhanced as Zα/β and become large or even nonperturbative (with α the QED fine structure constant). We provide a constructive demonstration of factorization to all orders in perturbation theory for these processes and compute the all-orders hard and soft functions appearing in the factorization formula. We clarify the relationship between effective field theory amplitudes and historical treatments of beta decay in terms of a Fermi function. Published by the American Physical Society 2024

On Minimal Entanglement Wedge Cross Section for Holographic Entanglement Negativity

We demonstrate the equivalence of two different conjectures in the literature for the holographic entanglement negativity in AdS3/CFT2, modulo certain constants. These proposals involve certain algebraic sums of bulk geodesics homologous to specific combinations of subsystems, and the entanglement wedge cross section (EWCS) backreacted by a cosmic brane for the conical defect geometry in the bulk gravitational path integral. It is observed that the former conjectures reproduce the field theory replica technique results in the large central charge limit whereas the latter involves constants related to the Markov gap. In this context, we establish an alternative construction for the EWCS of a single interval in a CFT2 at a finite temperature to resolve an issue for the latter proposal involving thermal entropy elimination for holographic entanglement negativity. Our construction for the EWCS correctly reproduces the corresponding field theory results modulo the Markov gap constant in the large central charge limit.

Quantification of plasma enabled surface cooling by electron emission from high temperature materials

In this work, the potential for hypersonic leading edge cooling by electron emission is demonstrated. To overcome space charge limitations the experiments are carried out in an argon discharge at 1 Torr. Cooling is observed with time-resolved measurements of the electron emission current and surface temperature, taking advantage of well controlled laser heating of the emitting surface and time accurate surface pyrometry. For the ignited mode of the plasma discharge, surface cooling by the electron emission is directly observed, leading to an estimated cooling rate of 1.6±0.2 MW m−2. Higher cooling rates with self sustained plasmas for space charge mitigation are expected using cesium transpiration. A two-dimensional model of heat transfer has been developed, which reproduces well the experimentally observed cooling dynamics. Parametric tests of emitter materials with various work functions show that for effective surface cooling by electron emission, the optimal work function must be less than 3.0 eV. This result indicates that electron cooling can be a promising thermal protection method for leading edges of hypersonic vehicles in flight.

The Virasoro minimal string

We introduce a critical string theory in two dimensions and demonstrate that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral. The worldsheet theory consists of Liouville CFT with central charge c≥≥ 25 coupled to timelike Liouville CFT with central charge 26-c. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, thus motivating the name Virasoro minimal string. The duality holds for any value of the continuous parameter cc and reduces to the JT gravity/matrix integral duality in the large central charge limit. It thus provides a precise stringy realization of JT gravity. The main observables of the Virasoro minimal string are quantum analogues of the Weil-Petersson volumes, which are computed as absolutely convergent integrals of worldsheet CFT correlators over the moduli space of Riemann surfaces. By exploiting a relation of the Virasoro minimal string to three-dimensional gravity and intersection theory on the moduli space of Riemann surfaces, we are able to give a direct derivation of the duality. We provide many checks, such as explicit numerical - and in special cases, analytic - integration of string diagrams, the identification of the CFT boundary conditions with asymptotic boundaries of the two-dimensional spacetime, and the matching between the leading non-perturbative corrections of the worldsheet theory and the matrix integral. As a byproduct, we discover natural conformal boundary conditions for timelike Liouville CFT.

Global conformal blocks via shadow formalism

We study \U0001d530\U0001d5292 and \U0001d530\U0001d5293 global conformal blocks on a sphere and a torus, using the shadow formalism. These blocks arise in the context of Virasoro and \U0001d4b23 conformal field theories in the large central charge limit. In the \U0001d530\U0001d5292 case, we demonstrate that the shadow formalism yields the known expressions in terms of conformal partial waves. Then, we extend this approach to the \U0001d530\U0001d5293 case and show that it allows to build simple integral representations for \U0001d530\U0001d5293 global blocks. We demonstrate this construction on two examples: the four-point block on the sphere and the one-point torus block.