scholarly journals A Survey on the Lace Expansion for the Nearest-neighbor Models on the BCC Lattice

2020 ◽  
Vol 24 (3) ◽  
pp. 723-784
Author(s):  
Satoshi Handa ◽  
Yoshinori Kamijima ◽  
Akira Sakai
Keyword(s):  
Nearest Neighbor ◽  
Lace Expansion ◽  
Bcc Lattice

Kinetics of B2, DO3, and B32 ordering: Results from pair approximation calculations and Monte Carlo simulations

1994 ◽  
Vol 9 (2) ◽  
pp. 348-356 ◽  
Author(s):  
L. Anthony ◽  
B. Fultz
Keyword(s):  
Monte Carlo ◽  
Kinetic Theory ◽  
Monte Carlo Simulations ◽  
Nearest Neighbor ◽  
Pair Approximation ◽  
Probability Method ◽  
Critical Temperatures ◽  
Point Pair ◽  
Bcc Lattice ◽  
Kinetics Of

A kinetic theory of ordering based on the path probability method was implemented in the pair (Bethe) approximation and used to study the kinetics of short- and long-range ordering in alloys with equilibrium states of B2, DO3, or B32 order. The theory was developed in a superposition approximation for a vacancy mechanism on a bcc lattice with first- (1nn) and second-nearest neighbor (2nn) pair interactions. Chained 1nn conditional probabilities were used to account for the entropy of 2nn pair configurations. Monte Carlo simulations of ordering were also performed and their results compared to predictions of the pair approximation. Comparisons are also made with predictions from an earlier kinetic theory implemented in the point (Bragg-Williams) approximation. For all three calculations (point, pair, and Monte Carlo), critical temperatures for B2 and DO3 ordering are reported for different 1nn and 2nn interaction strengths. The influence of annealing temperature on the kinetic paths through the space of B2, DO3, and B32 order parameters was found to be strong when the thermodynamic preferences for the ordered states were of similar strengths. Transient states of intermediate order were also studied. A transient formation of B32 order in an AB3 alloy was found when 2nn interactions were strong, even when B32 order was neither a Richards-Allen-Cahn ground state nor a stable equilibrium state at that temperature. The formation of this transient B32 order can be argued consistently from a thermodynamic perspective. However, a second example of transient B2 order in an AB alloy with equilibrium B32 order cannot be explained by the same thermodynamic argument, and we believe that its origin is primarily kinetic.

Kinetic paths of B2 and DO3 order parameters: Theory

1989 ◽  
Vol 4 (5) ◽  
pp. 1132-1139 ◽  
Author(s):  
L. Anthony ◽  
B. Fultz
Keyword(s):  
Time Dependence ◽  
Nearest Neighbor ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Interatomic Interaction ◽  
Order Parameters ◽  
Rate Theory ◽  
Bcc Lattice ◽  
Activated State ◽  
State Rate

It is shown that a binary alloy with an AB3 stoichiometry on a bcc lattice may develop various combinations of B2 and DO3 order along its kinetic path toward equilibrium. The temporal evolution of these two order parameters is analyzed with an activated-state rate theory. Individual vacancy jumps are treated in a master equation formalism that involves first-nearest-neighbor (1nn) and second-nearest-neighbor (2nn) interactions. In our formulation, a set of coupled differential equations is obtained describing the time-dependence of six order parameters. These equations were integrated numerically for a variety of interatomic interactions and initial conditions. It was found that the relative rates of B2 and DO3 ordering, and hence the path of the alloy through the space spanned by the B2 and DO3 order parameters, depend on the relative strengths of the interatomic interaction potentials and on the temperature. For very strong 2nn interactions, a transient B32 structure was observed to develop at early times, although this phase disappeared as equilibrium was approached.

Generation of low tehmperature series for Ising models with nearest and next nearest neighbor interactions

1976 ◽  
Vol 54 (16) ◽  
pp. 1646-1650 ◽  
Author(s):  
M. Plischke ◽  
C. F. S. Chan
Keyword(s):  
Low Temperature ◽  
Nearest Neighbor ◽  
Tricritical Point ◽  
Ising Models ◽  
Temperature Data ◽  
Temperature Series ◽  
Bcc Lattice ◽  
Special Cases ◽  
Ising Antiferromagnet ◽  
Ferromagnetic Interactions

We have generalized the code method of Sykes et al. and applied it to the Ising model with nearest and next nearest neighbor interactions. On the bcc lattice, we have obtained the first seven low temperature polynomials for arbitrary sign of the interactions. Special cases of this model are the Ising ferromagnet and the Ising antiferromagnet with next nearest neighbor ferromagnetic interactions. The latter system exhibits a tricritical point which we plan to study using our low temperature data and high temperature series to be obtained in the future.

SPATIAL CONFIGURATION OF ATOMS WITH HIGH-ENERGY ATOMIC DISPLACEMENT CASCADE IN α-Fe

2012 ◽  
Vol 26 (29) ◽  
pp. 1250192
Author(s):  
WEI-LU WANG ◽  
XUEBANG WU ◽  
WEI LIU ◽  
Q. F. FANG ◽  
C. S. LIU ◽  
...  
Keyword(s):  
Nearest Neighbor ◽  
Spatial Configuration ◽  
Single Point ◽  
Atomic Displacement ◽  
High Energy ◽  
The Body ◽  
Characteristic Index ◽  
Bcc Lattice ◽  
Index Pair ◽  
Cascade Energy

Molecular dynamics simulations were performed to study the primary damage formation in α- Fe through collision cascades with a cascade energy of up to 100 keV. The pair analysis technique was introduced to characterize the spatial local structure distributions of atoms. The damaged microstructural unit characteristics of the body-centered cubic (bcc) crystal structure, as well as the number of point defects, followed a similar trend. Furthermore, the damaged atoms exist mostly in the microstructural characteristics of icosahedral and short-range ordering in amorphous states during and at the end of cascades. Most local spatial structures of the damaged atoms can be divided into two groups based on their corresponding non-characteristic index-pair change trends with time. The curves of the first group coincided with the vacancy (V) that exhibited one peak, whereas the curves of the second group exhibiting two peaks corresponded to the self-interstitial atoms (SIA). The maximum distance at which defects could interact with each other in space was the fifth nearest-neighbor distance of the atoms of perfect lattices in the bcc lattice. The number of local structural units of the damaged atoms that were connected with a single point defect (either V or SIA) continued to increase with increasing cascade energy by the end of the simulation. By contrast, the number of units that were connected with both V and SIA decreased. These results may help us understand the spatial configuration of atoms in the course of collision cascades.

scholarly journals NoBLE for Lattice Trees and Lattice Animals

2021 ◽  
Vol 185 (2) ◽  
Author(s):  
Robert Fitzner ◽  
Remco van der Hofstad
Keyword(s):  
Random Walk ◽  
Nearest Neighbor ◽  
Mean Field ◽  
Critical Dimension ◽  
Simple Random Walk ◽  
Computer Assisted ◽  
High Dimensions ◽  
Lace Expansion ◽  
Upper Critical Dimension ◽  
Lattice Trees

AbstractWe study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice $${\mathbb {Z}}^d$$ Z d in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension $$16$$ 16 and $$17$$ 17 , respectively. Such results have previously been obtained by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than around simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to $$d_c=8$$ d c = 8 for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier. The main ingredients in this paper are (a) a derivation of a non-backtracking lace expansion for the LT and LA two-point functions; (b) bounds on the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.

Tricritical phenomena in an Ising model with nearest- and next-nearest-neighbor interactions

1977 ◽  
Vol 55 (13) ◽  
pp. 1125-1133 ◽  
Author(s):  
M. Plischke ◽  
D. Zobin
Keyword(s):  
Field Theory ◽  
High Temperature ◽  
Ising Model ◽  
Nearest Neighbor ◽  
Tricritical Point ◽  
Mean Field Theory ◽  
Mean Field ◽  
Temperature Series ◽  
Bcc Lattice ◽  
Ferromagnetic Interactions

We report on the analysis of low and high temperature series for the Ising model with nearest-neighbor antiferromagnetic and next-nearest-neighbor ferromagnetic interactions on the bcc lattice. The high temperature series are complete to β7, the low temperature series to u37. We determine the phase diagram, locate the tricritical point, and estimate the tricritical exponents. The tricritical exponents are only in fair agreement with the predictions of tricritical mean field theory.

scholarly journals Bond-Order Potentials with Analytic Environment-Dependent Tight-Binding Integrals: Application to BCC Molybdenum

2000 ◽  
Vol 653 ◽  
Author(s):  
Matous Mrovec ◽  
Duc Nguyen-Manh ◽  
David G. Pettifor ◽  
Vaclav Vitek
Keyword(s):  
Bond Order ◽  
Phonon Mode ◽  
Nearest Neighbor ◽  
Tight Binding ◽  
The Core ◽  
Bcc Lattice ◽  
Binding Models ◽  
Bond Order Potential ◽  
Preliminary Study ◽  
Environmental Dependence

AbstractWe present a new Screened Bond-Order Potential (SBOP) for molybdenum in which the environmental dependence of two-center tight-binding bond integrals has been implemented via a recently developed analytic expression. These bond integrals reproduce very well the numerical ab-intio values of screened LMTO bond integrals. In particular, they display the large discontinuity in ddπ between the first and second nearest neighbor of the bcc lattice whereas they do not show any discontinuity in ddσ. This dependence can be traced directly to the angular character of the analytic screening function and is shown to be critical for the behavior of the second nearest neighbor force constants. The new BOP eliminates the problem of the very soft T2 phonon mode at the N point that is found in most two-center tight-binding models. Preliminary study of the core structure of 1/2<111> screw dislocations performed using SBOP indicates that the core is narrower and less asymmetric than structures found in previous studies, in agreement with recent ab-initio calculations.

Observation of Superlattice Dislocations on Cube Planes in Ni3Al

1973 ◽  
Vol 31 ◽  
pp. 174-175 ◽  
Author(s):  
J. M. Oblak ◽  
W. H. Rand
Keyword(s):  
Nearest Neighbor ◽  
Antiphase Boundary ◽  
Nearest Neighbors ◽  
High Energy ◽  
Cross Slip ◽  
Octahedral Plane ◽  
Trapping Mechanism ◽  
Slip Traces

The energy of an a/2 <110> shear antiphase. boundary in the Ll2 expected to be at a minimum on {100} cube planes because here strue ture is there is no violation of nearest-neighbor order. The latter however does involve the disruption of second nearest neighbors. It has been suggested that cross slip of paired a/2 <110> dislocations from octahedral onto cube planes is an important dislocation trapping mechanism in Ni3Al; furthermore, slip traces consistent with cube slip are observed above 920°K.Due to the high energy of the {111} antiphase boundary (> 200 mJ/m2), paired a/2 <110> dislocations are tightly constricted on the octahedral plane and cannot be individually resolved.

Application of Lattice Imaging Techniques to Amorphous Films

1975 ◽  
Vol 33 ◽  
pp. 8-9
Author(s):  
S. R. Herd ◽  
P. Chaudhari
Keyword(s):  
Nearest Neighbor ◽  
Random Network ◽  
Imaging Techniques ◽  
Network Models ◽  
Accurate Method ◽  
Direct Transmission ◽  
Lattice Imaging ◽  
Transmission Electron ◽  
Lattice Planes ◽  
Nearest Neighbor Distances

Electron diffraction and direct transmission have been used extensively to study the local atomic arrangement in amorphous solids and in particular Ge. Nearest neighbor distances had been calculated from E.D. profiles and the results have been interpreted in terms of the microcrystalline or the random network models. Direct transmission electron microscopy appears the most direct and accurate method to resolve this issue since the spacial resolution of the better instruments are of the order of 3Å. In particular the tilted beam interference method is used regularly to show fringes corresponding to 1.5 to 3Å lattice planes in crystals as resolution tests.

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