scholarly journals Quiver Grassmannians of Type $\widetilde {D}_{n}$, Part 2: Schubert Decompositions and F-polynomials

2021 ◽  
Author(s):  
Oliver Lorscheid ◽  
Thorsten Weist
Keyword(s):  
Cluster Algebras ◽  
Indecomposable Representation ◽  
Euler Characteristics ◽  
Affine Spaces ◽  
Quiver Grassmannians ◽  
Small Defect ◽  
Quiver Grassmannian ◽  
Vertex Set ◽  
Explicit Formulae ◽  
Fixed Tree

AbstractExtending the main result of Lorscheid and Weist (2015), in the first part of this paper we show that every quiver Grassmannian of an indecomposable representation of a quiver of type $\tilde D_{n}$ D ~ n has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. F-polynomials). Along these lines, we obtain explicit formulae for all cluster variables of cluster algebras coming from quivers of type $\tilde D_{n}$ D ~ n .

scholarly journals Geometry of quiver Grassmannians of Kronecker type and applications to cluster algebras

2011 ◽  
Vol 5 (6) ◽  
pp. 777-801 ◽  
Author(s):  
Giovanni Cerulli Irelli ◽  
Francesco Esposito
Keyword(s):  
Cluster Algebras ◽  
Quiver Grassmannians

Radiation Damage in Ceramics

1982 ◽  
Vol 40 ◽  
pp. 600-603
Author(s):  
T. E. Mitchell ◽  
M. R. Pascucci ◽  
R. A. Youngman
Keyword(s):  
Radiation Damage ◽  
Outer Space ◽  
Point Of View ◽  
Nuclear Waste Storage ◽  
Waste Storage ◽  
Storage Media ◽  
Transmission Electron ◽  
Fundamental Point ◽  
Small Defect ◽  
Present Understanding

1. Introduction. Studies of radiation damage in ceramics are of interest not only from a fundamental point of view but also because it is important to understand the behavior of ceramics in various practical radiation enyironments- fission and fusion reactors, nuclear waste storage media, ion-implantation devices, outer space, etc. A great deal of work has been done on the spectroscopy of point defects and small defect clusters in ceramics, but relatively little has been performed on defect agglomeration using transmission electron microscopy (TEM) in the same kind of detail that has been so successful in metals. This article will assess our present understanding of radiation damage in ceramics with illustrations using results obtained from the authors' work.

Tem analysis of multiple-energy arsenic implanted and Rta'd silicon

1987 ◽  
Vol 45 ◽  
pp. 246-247
Author(s):  
R.A. Herring
Keyword(s):  
Device Fabrication ◽  
High Concentration ◽  
Tem Analysis ◽  
Defect Clusters ◽  
High Fluences ◽  
Small Defect ◽  
Before And After ◽  
The Matrix ◽  
The Difference ◽  
Factor Type

Rapid thermal annealing (RTA) of ion-implanted Si is important for device fabrication. The defect structures of 2.5, 4.0, and 6.0 MeV As-implanted silicon irradiated to fluences of 2E14, 4E14, and 6E14, respectively, have been analyzed by electron diffraction both before and after RTA at 1100°C for 10 seconds. At such high fluences and energies the implanted As ions change the Si from crystalline to amorphous. Three distinct amorphous regions emerge due to the three implantation energies used (Fig. 1). The amorphous regions are separated from each other by crystalline Si (marked L1, L2, and L3 in Fig. 1) which contains a high concentration of small defect clusters. The small defect clusters were similar to what had been determined earlier as being amorphous zones since their contrast was principally of the structure-factor type that arises due to the difference in extinction distance between the matrix and damage regions.

The evolution of the microstructure of 600 Mev proton irradiated materials

1990 ◽  
Vol 48 (4) ◽  
pp. 1002-1003
Author(s):  
R. Gotthardt ◽  
A. Horsewell ◽  
F. Paschoud ◽  
S. Proennecke ◽  
M. Victoria
Keyword(s):  
Nuclear Reactions ◽  
Dislocation Loops ◽  
Defect Clusters ◽  
Weak Beam ◽  
Stacking Fault Tetrahedra ◽  
Small Defect ◽  
Sufficient Intensity ◽  
Reactor Materials ◽  
Metallic Impurities ◽  
Beam Technique

Fusion reactor materials will be damaged by an intense field of energetic neutrons. There is no neutron source of sufficient intensity at these energies available at present, so the material properties are being correlated with those obtained in irradiation with other irradiation sorces. Irradiation with 600 MeV protons produces both displacement damage and impurities due to nuclear reactions. Helium and hydrogen are produced as gaseous impurities. Other metallic impurities are also created . The main elements of the microstructure observed after irradiation in the PIREX facility, are described in the following paragraphs.A. Defect clusters at low irradiation doses: In specimens irradiated to very low doses (1021-1024 protons.m-2), so that there is no superimposition of contrast, small defect clusters have been observed by the weak beam technique. Detailed analysis of the visible contrast (>0.5 nm diameter) revealed the presence of stacking fault tetrahedra, dislocation loops and a certain number of unidentified clusters . Typical results in Cu and Au are shown in Fig. 1.

Radiation-Induced Precipitation in Ti-6A1-4V

1980 ◽  
Vol 38 ◽  
pp. 154-155
Author(s):  
J. E. O'Neal ◽  
S. M. L. Sastry ◽  
J. W. Davis
Keyword(s):  
Dark Field ◽  
Dislocation Loops ◽  
Beta Titanium Alloy ◽  
Defect Clusters ◽  
Weak Beam ◽  
Beta Titanium ◽  
Small Defect ◽  
Radiation Induced ◽  
Vector Orientation ◽  
Defect Microstructures

The radiation-induced defect structure and nonequilibrium phase precipitation were studied in T1-6A1-4V (an alpha-beta titanium alloy), irradiated at 450 ± 30°C in row VII of the EBR-II to a fluence of 3.0 × 1021 neutrons/cm2 (En > 0.1 MeV). The Irradiation-induced defect microstructures were examined using bright-field, conventional dark-field, and weak-beam dark-field techniques. The nature of dislocations and dislocation loops was determined by standard-contrast experiments under two-beam conditions, and the small defect clusters were identified using the line-of-contrast criterion and black-white vector orientation criterion.

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