Multiple Valency Research Articles

Solvation of Nanoions in Aqueous Solutions.

In recent years it has been increasingly recognized that different classes of large ions with multiple valency have effects conceptually similar to weakly solvated ions in the Hofmeister series, also labeled by the term chaotropic. The term "superchaotropic effect" has been coined because these effects are much more strongly pronounced for nanometer-sized ions, whose adsorption properties often resemble typical surfactants. Despite this growing interest in these nanometer-sized ions, a simple conceptual extension of the Hofmeister series toward nanoions has not been achieved because an extrapolation of the one-dimensional surface charge density scale does not lead to the superchaotropic regime. In this work, we discuss a generic model that is broadly applicable to ions of nearly spherical shape and thus includes polyoxometalates and boron clusters. We present a qualitative classification scheme in which the ion size appears as a second dimension. Ions of different sizes but the same charge density differ in their bulk solvation free energy. As the ions grow bigger at constant surface charge density, they become more stable in solution, but the adsorption behavior is still governed by the surface charge density. A detailed molecular dynamics simulation study of large ions that is based on a shifted Lennard-Jones potential is presented that supports the presented classification scheme.

Improved optical, dielectric, impedance, and magnetic properties of (BiFeO3)0.6(CaTiO3)0.4 for multifunctional utilities

The investigation of the multifaceted properties of (BiFeO3)0.6(CaTiO3)0.4, synthesized following a solid solution reaction mechanism is reported. The temperature-dependent dielectric and tanδ loss analysis sheds light on the temperature-dependent phase transitions. The structural deformation from the rhombohedral structure for pure BFO to the orthorhombic phase is evident from the Rietveld refinement. The optical energy gap investigation carried out through UV–Visible spectroscopy points out the potential useful photovoltaic applicability of the ceramic. The XPS analysis reveals the existence of multiple valency states of Fe, Ti, and oxygen defects which in turn might explain the underlying phenomena of hopping mechanisms and magnetic exchange interactions responsible for enhancing the ferroelectric, dielectric, and magnetic properties in the sample. The impedance and modulus analysis reveal the semiconducting and capacitive properties of the ceramic useful for device applications. Enhanced dielectric properties, weak ferromagnetic ordering due to suppression of G-type antiferromagnetism and double exchange paths, and reduced value of optical band gap as compared to the previous report are the novel findings in this communication.

Vanadium-Based Materials: Next Generation Electrodes Powering the Battery Revolution?

ConspectusAs the world transitions away from fossil fuels, energy storage, especially rechargeable batteries, could have a big role to play. Though rechargeable batteries have dramatically changed the energy landscape, their performance metrics still need to be further enhanced to keep pace with the changing consumer preferences along with the increasing demands from the market. For the most part, advances in battery technology rely on the continuing development of materials science, where the development of high-performance electrode materials helps to expand the world of battery innovation by pushing the limits of performance of existing batteries. This is where vanadium-based compounds (V-compounds) with intriguing properties can fit in to fill the gap of the current battery technologies.The history of experimenting with V-compounds (i.e., vanadium oxides, vanadates, vanadium-based NASICON) in various battery systems, ranging from monovalent-ion to multivalent-ion batteries, stretches back decades. They are fascinating materials that display rich redox chemistry arising from multiple valency and coordination geometries. Over the years, researchers have made use of the inherent ability of vanadium that undergoes metamorphosis between different coordination polyhedra accompanied by transitions in the oxidation state for reversible intercalation/insertion of more than one guest ions without breaking the structure apart. Such infinitely variable properties endow them with a wide range of electronic and crystallographic structures. The former attribute varies from insulators to metallic conductors while the latter feature gives rise to layered structures or 3D open tunnel frameworks that allow facile movement of a wide range of metal cations and guest species along the gallery. Accompanied by a growing stringent requirements for energy storage applications, most V-compounds face difficulty in resolving the problems of their own lack competitiveness mostly due to their intrinsically low ionic/electronic conductivity. The key to producing vanadium-based electrodes with the desired performance characteristics is the ability to fabricate and optimize them consistently to realize certain specifications through effective engineering strategies for property modulation.In this Account, we aim to provide a comprehensive article that correlates the fundamental of charge storage mechanism to crystallographic forms and design principle for V-compounds. More importantly, the essential roles played by engineering strategies in the property modulation of V-compounds are pinpointed to further explain the rationale behind their anomalous behavior. Apart from that, we further summarize the key theoretical and experimental results of some representative examples for tuning of properties. On the other hand, advances in characterization techniques are now sufficiently mature that they can be relied upon to understand the reaction mechanism of V-compounds by tracing real-time transformation and structural changes at the atomic scale during their working state. The mechanistic insights covered in this Account could be used as a fundamental guidance for several key strategies in electrode materials design in terms of dimension, morphology, composition, and architecture that govern the rate and degree of chemical reaction.

Dye degradation over the multivalent charge- and solid solution-type n-MoS2/p-WO3 based diode catalyst under dark condition with a self-supporting charge carrier transfer mechanism

Pollution of water by synthetic dyes is of great concern because of the large production and usage of dyestuffs throughout the world. For dye elimination purpose, the p-n heterojunction Sb-doped MW-x composite catalysts prepared with different Mo:W precursor molar contents were synthesized via one-pot hydrothermal method. The selection in metal precursors is intended to design a composite catalyst with multiple valency in each phase. The structural, morphological, electrochemical property, and optical band gap of the composite catalyst were studied. The catalytic performance of Sb-MW composite catalysts was studied for the degradation of MB without light irradiation. It was observed that Sb-MW-4 showed superior catalytic performance to completely degrade 20 ppm MB dye solution with 20 mg catalyst in 14 min under dark condition. The composite catalyst was composed of n-(Mo,W)(O,S)2 oxysulfide in a disordered MoS2 framework and p-(W,Mo)(S,O)3 sulfo-oxide in a WO3 structure or was abbreviatedly named as n-MoS2/p-WO3 based catalyst. The degradation reaction for MB dye had an apparent rate constant of 3.5 × 10−1 min−1. This catalyst also showed excellent stability and reusability for degrading MB dye pollutant. The degradation mechanism under dark condition is proposed.

Antiviral Activity of a Llama-Derived Single-Domain Antibody against Enterovirus A71.

In the past few decades, enterovirus A71 (EVA71) has caused devastating outbreaks in the Asia-Pacific region, resulting in serious sequelae in infected young children. No preventive or therapeutic interventions are currently available for curing EVA71 infection, highlighting a great unmet medical need for this disease. Here, we showed that one novel single-domain antibody (sdAb), F1, isolated from an immunized llama, could alleviate EVA71 infection both in vitro and in vivo We also confirmed that the sdAb clone F1 recognizes EVA71 through a novel conformational epitope comprising the highly conserved region of VP3 capsid protein by using competitive-binding and overlapping-peptide enzyme-linked immunosorbent assays (ELISAs). Because of the virion's icosahedral structure, we reasoned that adjacent epitopes must be clustered within molecular ranges that may be simultaneously bound by an engineered antibody with multiple valency. Therefore, two single-domain binding modules (F1) were fused to generate an sdAb-in-tandem design so that the capture of viral antigens could be further increased by valency effects. We showed that the tetravalent construct F1×F1-hFc, containing two sdAb-in-tandem on a fragment crystallizable (Fc) scaffold, exhibits more potent neutralization activity against EVA71 than does the bivalent sdAb F1-hFc by at least 5.8-fold. We also demonstrated that, using a human scavenger receptor class B member 2 (hSCARB2) transgenic mouse model, a half dose of the F1×F1-hFc provided better protection against EVA71 infection than did the F1-hFc. Thus, our study furnishes important insights into multivalent sdAb engineering against viral infection and provides a novel strategic deployment approach for preparedness of emerging infectious diseases such as EVA71.

Synthesis of g-C3N4/N-doped CeO2 composite for photocatalytic degradation of an herbicide

In photocatalysis, surface engineered CeO2 could be vital due to oxygen vacancies arise from multiple valency, i.e. Ce3+ and Ce4+. This study reports photocatalytic properties of g-C3N4/CeO2 composite synthesized by a facile method in the presence of l-arginine. Physicochemical properties of g-C3N4/CeO2 material were analyzed through various characterization techniques such as XRD, UV–Vis, physisorption, etc., and correlated with its photocatalytic activity. Observed bandgap of the synthesized composite material was in the visible region, around 2.8eV which is less than that of typical ceria, but higher than bandgap of exfoliated g-C3N4. On the further side, N doping into CeO2 was confirmed through XPS analysis. It is estimated that synthesis method aided for the N doping, which further played key role in lowering the bandgap of g-C3N4/CeO2 composite. Finally, Photocatalytic activity of g-C3N4/CeO2 composite was analyzed through degradation of an herbicide i.e. diuron, and the study revealed the good performance of the catalyst.

Investigation of the metal-insulator transition in NdNiO3 films by site-selective X-ray absorption spectroscopy.

In this work, multifunctional oxide NdNiO3 (NNO) thin films grown on a SrTiO3 (STO) substrate using pulsed-laser deposition are studied. Temperature dependent resistivity measurements revealed that NNO/STO samples exhibit a sharp thickness dependent metal-insulator transition (MIT) over a range of 150-200 K. It is known that the electronic properties of correlated oxides are extremely complex and sensitive to changes in orbital occupancy. To evaluate the changes in the electronic and/or crystallographic structure responsible for the MIT, a site-selective (O, Ni and Nd) X-ray absorption near edge structure (XANES) analysis is performed above and below the transition temperature. Analysis of XANES spectra suggests that: (i) in NNO films nominally trivalent Ni ions exhibit multiple valency (bond disproportionation), (ii) intermetallic hybridization plays an important role, (iii) the presence of strong O 2p-O 2p hole correlation at low temperature results in the opening of the p-p gap and (iv) the valency of Nd ions matches well with that of Nd3+. For NNO films exhibiting a sharp MIT, Ni 3d electron localization and concurrent existence of Ni 3d8 and Ni 3d8L[combining low line]2 states are responsible for the observed transition. At temperatures below the MIT the O 2p-O 2p hole correlation is strong enough to split the O 2p band stabilizing insulating phase. Temperature and thickness dependent differences observed in the site-selective XANES data are discussed in terms of possible mechanisms for the MIT (negative charge-transfer type).

(Invited) Interface Engineering Routes for a Future CMOS Ge-Based Technology

An alternative route to continue the trend of scaling of CMOS technology is to implement novel channel materials with superior transport properties. Germanium has high intrinsic mobilities for both electrons (3900 cm2/Vs) and holes (1900 cm2/Vs), and is compatible with Si process technology, making it a suitable channel material for low power, high performance devices. The smaller band gap of Ge (0.67 eV) potentially allows for lower contact resistances compared to Si due to a reduced barrier height, and hence is more suitable for voltage scaling. A challenging task is the formation of a high-quality gate stack with low interface trap density and sub-nm equivalent oxide thickness (EOT) to maintain the intrinsically high performance of Ge. The peak electron mobility has been dramatically improved for Ge MOSFETs in recent years [1,2], largely achieved by improving the quality of GeO2/Ge interface. Although thermally grown GeO2 is the most natural choice, its inherent shortcomings still remain, such as: high water solubility, low desorption temperature (~400°C), low dielectric constant (~6); unconventional growth methods. Since the dielectric constant of GeO2 is low it cannot be used as the gate dielectric for aggressively scaled devices. The focus is on high-k dielectric (k>20) for sub-nm EOT scaling. The Liverpool group has been interested in two possible routes for Ge interface engineering: (i) using high-k materials that are intimate with Ge, such as La2O3 and Y2O3, and (ii) introducing a robust ultra-thin high-k interfacial layer (IL) barrier, such as Al2O3 and Tm2O3. Concerning the first route, high reactivity of Ge with high-k allows for germanate IL formation, which role is two-fold: to reduce the interface states, and to suppress the GeO desorption at the interface. The second route involves the use of ultra-thin barrier layers, Al2O3 and Tm2O3, as oxides highly resistant to oxygen diffusion and to reaction with Ge. The rare-earth metals (La, Y, Tm) tend to possess multiple valency, such as +2 and +3 oxidation states, that can provide effective passivation of electrically active defects. Both routes lead to achieving a GeOx-free gate stack with effective Ge surface passivation.We will present an overview study of physical and electrical properties of La2O3/Ge, Y2O3/Ge, Tm2O3/Ge, Al2O3/Ge and HfO2/Al2O3/GeO2/Ge gate stacks. The interfacial composition, valence band offset, uniformity, thickness, band gap, crystallinity, dielectric function and absorption features will be shown, as ascertained using X-ray photoelectron spectroscopy, X-ray diffraction, high resolution transmission electron microscopy, medium energy ion scattering and ultra violet variable angle spectroscopic ellipsometry. The correlation of these results with electrical characterization data, will make a case for Ge interface engineering with rare-earth inclusion as a viable route to achieve high performance Ge CMOS.Acknowledgement: the authors thank the EPSRC (UK) and the EU (OSIRIS and ESTEEM2 projects) for funding.[1] S. Takagi et al., Microelectron. Eng. 109, 389 (2013); [2] A. Toriumi et al. IEEE IEDM Tech. Dig., 646 (2011).

Comment on “Taming multiple valency with density functionals: A case study of defective ceria”

Comment on “Taming multiple valency with density functionals: A case study of defective ceria”

Reply to “Comment on ‘Taming multiple valency with density functionals: A case study of defective ceria' ”

We address the issues raised in the preceding Comment by discussing the effects of an explicit account of the nonlinear core correction (NLCC) into the Ce pseudopotential employed in our previous calculations of reduced ceria [Phys. Rev. B 71, 041102(R) (2005)]. At the plain density-functional-theory (DFT) level, such an account brings our pseudopotential results in good agreement with all-electron ones, which do not predict the insulating character of reduced ceria. At the $\mathrm{DFT}+U$ level, the inclusion of the NLCC has no effect on the previously reported electronic structure and equilibrium geometries. The dependency of the $\mathrm{DFT}+\mathrm{U}$ energy on the choice of the projector functions defining the Hubbard-U functional is discussed in further detail.

Taming multiple valency with density functionals: A case study of defective ceria

Modeling multiple-valence compounds using density-functional theory has long been considered a formidable task due to the role that strong electronic correlations play in these systems. We show that, in the case of defective ceria, the main effect of these correlations is to produce a multitude of metastable low-energy states among which the one displaying the correct valence of cerium is the most stable. This ground state may be difficult to access in practice and it has in fact so far escaped a proper identification. The introduction of a Hubbard-$U$ term in the energy functional stabilizes the physical ground state and makes it easily accessible to routine calculations. When this contribution is defined in terms of maximally localized Wannier functions, the calculated energies and structural properties are independent of the value of the parameter $U$.

Extracellular matrix signaling through growth factor receptors during wound healing.

Recently, extracellular matrix components have been shown to contain domains that can interact with and activate receptors with intrinsic tyrosine kinase activity. These receptor tyrosine kinases are strong mediators of the cell responses of proliferation, migration, differentiation, and dedifferentiation. However, an interesting question is raised as to why cells would present growth factor receptor ligands in such a manner, as the majority of growth factors are small, soluble, or only transiently tethered ligands. With the exception of the discoidin domain receptors that bind collagen, the other described domains interact with a receptor that binds ubiquitous soluble peptide growth factors, the epidermal growth factor receptor. Unlike traditional growth factors, these individual "matrikine" domains within tenascin-C, laminin, collagen, and decorin possess relatively low binding affinity (high nanomolar or micromolar) and are often presented in multiple valency. The presentation of ligands within the extracellular matrix in this fashion might allow for unique biochemical and physiological outcomes. This new class of "matrikine" ligand may be critical for wound healing, as the majority of known extracellular matrix components possessing matrikines play a strong role, or are presented uniquely, during skin repair. Tenascin-C expression, for instance, is uniquely regulated spatially and has been proposed to present pro-migratory tracks during skin repair through its epidermal growth factor-like repeats. The epidermal growth factor-like repeats of laminin-5 act as cryptic ligands revealed upon matrix metalloproteinase-2 degradation of the surrounding extracellular matrix. The deletion of the discoidin domain receptors 1 and 2 for collagen have negative consequences on the role of fibroblasts and epithelial cells for matrix metalloproteinase production, migration, proliferation, and extracellular matrix turnover. Finally, decorin can bind to, inhibit, and down-regulate epidermal growth factor receptor levels and signaling, suggesting a tonic role of the epidermal growth factor binding domain of decorin in the resolution of wound healing. We provide a model framework for further studies into this emerging class of signals.

Dimerization of Fab fragments enables ready screening of phage antibodies that affect hepatocyte growth factor/scatter factor activity on target cells.

A number of applications of antibodies in diagnosis and therapy require multivalent reagents either because of the polymeric nature of the antigens or because biological activity depends on an effect on the formation of homodimeric species. Here, we report a procedure for mass screening of phage-derived monomeric antibody fragments that depend on valency for activity. As a model system, a set of 13 phage-derived human Fab fragments were first selected against mouse and human recombinant hepatocyte growth factor/scatter factor (HGF/SF), a high molecular weight polypeptide growth factor related to the blood protease plasminogen and involved in development and cancer. These Fab fragments were subsequently screened for an effect on HGF/SF activity either as monomeric fragments or after dimerization with a monoclonal antibody (9E10) directed against a peptide tag on the fragments. Fab were identified that either inhibited or enhanced HGF/SF activity on target cell lines, but dimerization was required for this effect. The approach proposed should facilitate mass screening of phage-derived antibody fragments that depend on multiple valency for activity.

Affinity enhancement of a recombinant antibody: formation of complexes with multiple valency by a single-chain Fv fragment-core streptavidin fusion.

In antigen-antibody interactions, the high avidity of antibodies depends on the affinity and number of the individual binding sites. To develop artificial antibodies with multiple valency, we have fused the single-chain antibody Fv fragments to core streptavidin. The resulting fusion protein, termed scFv::strep, was found after expression in Escherichia coli in periplasmic inclusion bodies. After purification of the recombinant product by immobilized metal affinity chromatography, refolding and size-exclusion FPLC, tetrameric complexes resembling those of mature streptavidin were formed. The purified tetrameric scFv::strep complexes demonstrated both antigen- and biotin-binding activity, were stable over a wide range of pH and did not dissociate at high temperatures (up to 70 degrees C). Surface plasmon resonance measurements in a BIAlite system showed that the pure scFv::strep tetramers bound immobilized antigen very tightly and no dissociation was measurable. The association rate constant for scFv::strep tetramers was higher than those for scFv monomers and dimers. This was also reflected in the apparent constants, which was found to be 35 times higher for pure scFv::strep tetramers than monomeric single-chain antibodies. We could also show that most of biotin binding sites were accessible and not blocked by biotinylated E.coli proteins or free biotin from the medium. These sites should therefore facilitate the construction of bispecific multivalent antibodies by the addition of biotinylated ligands.

The effect of counterions on the chain conformation of polyelectrolytes, as assessed by extensibility in elongational flow: The influence of multiple valency

AbstractAn elongational flow technique was used to determine the effect of counterions on the chain conformation of polyelectrolyte molecules in solution, by means of the extensibility of the chains in the flow field. It is demonstrated that adding excess cations of seven low molecular weight salts, NaCl, CaCl2, BaCl2, SrCl2, MgCl2, AlCl3, and SnCl4, to a very dilute solution of fully sulphonated polystyrene (NaPSS) reduces the extensibility of the chains, that is, the facility by which a chain can be extended to varying degrees, an effect associated with chain contractions. In the case of multivalent counterions, these contractions, which with monovalent counterions are primarily due to screening of charges by excess counterions, are greatly enhanced, which we attribute to the formation of intramolecular ionic bridges. When, in the case of multivalent counterions, the polymer concentration is increased, in inversion of the effect, namely increase in chain extensibility on addition of ions, is observed. We attribute this latter effect to the ionic bridges becoming increasingly intermolecular, leading to effectively large molecules, and eventually to a gel. All these effects were accentuated with increase in valency. They could also be accompanied by precipitation which were of two kinds: one due to formation of insoluble ionic associations and a second attributable to enhanced hydrophobic interaction within the contracted chain itself. © 1994 John Wiley & Sons, Inc.

Development of an artificial antibody system with multiple valency using an Fv fragment fused to a fragment of protein A.

In antigen-antibody interactions, the high avidity of antibodies for their specific antigens can be achieved both by exploiting the high affinity of each binding site and by multivalence of antibodies. In this study, we developed artificial antibodies with multiple valency. The Fv fragments of an antibody fused with one and with two domains of the Fc-binding protein A from Staphylococcus aureus (designated Fv-P and Fv-PP, respectively) were expressed in secreted forms in Escherichia coli. Their physical characteristics were examined. The Fv portions of Fv-P and Fv-PP had virtually the same affinity for the antigen as that of the original Fv molecules, and each protein A-derived domain could be bound to IgG. When Fv-P was mixed with IgG, the complex formed was composed of two Fv-P molecules and one IgG molecule. In the case of Fv-PP, a complex composed of three Fv-PP molecules and two IgG molecules was preferentially formed. Analysis of surface plasmon resonance in a BIAcore system indicated multivalency of these complexes for antigens. There was 3.5-fold decrease in the dissociation constant of the complex between Fv-PP and the antigen upon the addition of IgG. The use of these complexes as reagents in enzyme-linked immunosorbent assay and Western blotting gave much stronger signals than Fv, Fv-P, and Fv-PP alone.

X-ray photoelectron, Moessbauer, magnetic and electrical conductivity study of SnS2{CoCp2}0.31

Intercalation of SnS/sub 2/ with CoCp/sub 2/ (Cp = /eta/-C/sub 5/H/sub 5/) under mild conditions leads to the formation of SnS/sub 2/(CoCp/sub 2/)/sub 0.31/. Its physical properties have been investigated with a variety of techniques. X-ray photoelectron spectroscopy reveals the presence of Co/sup 3+/, Co/sup 2+/, Sn/sup 4+/, and probably Sn/sup 2+/. Electron paramagnetic resonance spectra showed the presence of radicals with g values close to those observed for cobaltocene intercalated into CdPS/sub 3/. Magnetic measurements showed that the sample obeyed the Curie-Weiss expression between 4 and 300 K with /Theta/ = /minus/9.17 K, and /mu//sub eff/ = 0.12 /mu//sub B/. /sup 119/Sn Moessbauer spectroscopy revealed the presence of a resonance assignable to a reduced tin site that could be formulated as Sn/sup 2+/ and an integrated Sn/sup 2+/:Sn/sup 4+/ ratio of 0.11. Diffuse reflectance spectroscopy showed the onset of a broad absorption in the near-IR region at 0.6 eV, in addition to the band gap of SnS/sub 2/ at 2.1 eV. Electrical conductivity showed that the intercalated material is 200 times more conductive than the unreacted host but still showed semiconducting behavior with an activation energy E/sub a/ = 0.11 eV, similar to that of unintercalated SnS/sub 2/. Themore » stoichiometry of the intercalated material can best be described as (Sn/sup 4+//sub 0.9/Sn/sup 2+//sub 0.1/)S/sub 2/((Co/sup 3+/Cp/sub 2/)/sup +//sub 0.2/(Co/sup 2+/Cp/sub 2/)/sub 0.1/) which is remarkable due to the multiple mixed valency of the material. 35 refs., 10 figs., 4 tabs.« less

Hydrogen chemical ionization mass spectrometry of metalloporphyrins

AbstractThe hydrogen chemical ionization (H2 CI) mass spectra of a range of metal(II) (Ni, Cu, Co, Pt), metal (III) (Al, Mn, Ga, Fe (bearing a single axial ligand)) and metal(IV) (Si, Ge, Sn (bearing two axial ligands) and V (as VO2+)) porphyrins have been determined, The spectra are highly dependent on the coordinated metal, rather than the axial ligand(s) (where present). Ni(II), Cu(II), Mn(II or III), Ga(III), Ge(IV), Fe(III) and Sn(IV) porphyrins fragment via hydrogenation and demetallation, followed by cleavage of the resulting porphyrinogens at the meso(bridge) positions to give mono‐ and di‐pyrrolic fragments. Tripyrrolic fragments are also observed in the case of Ni(II), Cu(II) and Sn(IV). Fragmentations of this type are similar to those observed for free‐base porphyrins. In the case of Pt(II), Co(II), Al(III), Si(IV) and V(IV) (as vanadyl), the dipyrrolic fragment ions are either very weak or completely absent; hence their H2CI spectra contain limited structural information. This variable CI behaviour may be related to the relative stabilities of the metalloporphyrins together with the multiple stable valency states exhibited by several metals.

Point-curve correspondences

In this paper we consider the theory of point-curve correspondences on a single surface F, i.e. correspondences in which to each point of F corresponds an algebraic curve of F. The results previously obtained for induced and extended correspondences between two surfaces require some modification here, as we can consider the self-correspondences induced on a curve C of F, and the related theory of extended correspondences, which is now complicated by the existence of the identical correspondence on C. We also develop a theory of correspondences with (non-zero) valency, and show that for a surface whose Riemann matrix is pure and without complex multiplication (and hence for surfaces of ‘general moduli’), all correspondences are valency correspondences, this result being exactly analogous to the well-known theorem for curves. We also consider generalized valency correspondences which are an extension to surfaces of the concept of correspondences of multiple valency on a curve. The analogy between our theory and the known results for curves does break down in one important respect. It is not true that every surface possesses valency correspondences even of the generalized kind, the existence of such correspondences involving restrictions on the intersection group of the surface.

Multiple Valency in the Ionization by Alpha Rays

Multiple Valency in the Ionization by Alpha Rays