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SN2 Reaction: Transition State02:26

SN2 Reaction: Transition State

11.7K
An SN2 reaction of an alkyl halide is a single-step process in which bond formation between the nucleophile and the substrate and bond breaking between the substrate and the halide occurs simultaneously through a transition state without forming an intermediate.
When the nucleophile approaches the electrophilic carbon with its lone pairs, the halide acts as a leaving group and moves away with the electron-pair bonded to the carbon. Dotted partial bonds represent the bonds being formed or broken...
11.7K
Predicting Products: SN1 vs. SN202:27

Predicting Products: SN1 vs. SN2

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Nucleophilic substitution reactions of alkyl halides can proceed via an SN1 or an SN2 mechanism. While in SN2 reactions, the nucleophile attacks the substrate simultaneously as the leaving group departs, in SN1 reactions, the substrate first dissociates to give the carbocation intermediate. Various factors such as the structure of the substrate, the strength of the nucleophile, and the nature of the solvent promote one mechanism over the other.
With increased substitution on the alkyl halide,...
15.9K
Properties of Transition Metals02:58

Properties of Transition Metals

29.6K
Transition metals are defined as those elements that have partially filled d orbitals. As shown in Figure 1, the d-block elements in groups 3–12 are transition elements. The f-block elements, also called inner transition metals (the lanthanides and actinides), also meet this criterion because the d orbital is partially occupied before the f orbitals.
29.6K
Phase Transitions02:31

Phase Transitions

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Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to...
22.7K
Cooperative Allosteric Transitions01:58

Cooperative Allosteric Transitions

8.6K
Cooperative allosteric transitions can occur in multimeric proteins, where each subunit of the protein has its own ligand-binding site. When a ligand binds to any of these subunits, it triggers a conformational change that affects the binding sites in the other subunits; this can change the affinity of the other sites for their respective ligands. The ability of the protein to change the shape of its binding site is attributed to the presence of a mix of flexible and stable segments in the...
8.6K
Gauss's Law: Planar Symmetry01:27

Gauss's Law: Planar Symmetry

9.4K
A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
9.4K

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Two-Dimensional J_{1}-J_{2} Clock Model: Enhanced Symmetries, Emergent Orders, and Landau-Incompatible Transitions.

Pulloor Kuttanikkad Vishnu1,2, Abhishodh Prakash3, Rajesh Narayanan1,2

  • 1Indian Institute of Technology Madras, Department of Physics, Chennai 600036, India.

Physical Review Letters
|January 20, 2026
PubMed
Summary

We studied the frustrated q-state clock model, finding new emergent phases and transitions due to competing interactions. Frustration leads to exotic stripe orders with discrete Zq spin degrees of freedom.

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Area of Science:

  • Condensed Matter Physics
  • Statistical Mechanics
  • Phase Transitions

Background:

  • The J1-J2 classical q-state clock model exhibits complex behavior driven by competing interactions.
  • Understanding emergent phenomena in frustrated magnetic systems is crucial for materials science.

Purpose of the Study:

  • To comprehensively study the frustrated J1-J2 classical q-state clock model with even q>4 on a 2D square lattice.
  • To map the complete phase diagram and identify novel emergent phases and transitions.

Main Methods:

  • Large-scale corner transfer matrix renormalization group (CTMRG) calculations.
  • Classical Monte Carlo (CMC) simulations.
  • Development of an effective field-theoretic framework.

Main Results:

  • In the unfrustrated regime (J1>2J2), standard clock model phases (ferromagnet, XY-like critical phase, paramagnet) were observed.
  • Frustration (J1<2J2) stabilized five distinct regimes: paramagnet, stripe-ordered phase, two nematic phases, and an exotic stripe phase.
  • An exotic stripe phase with emergent discrete Zq spin degrees of freedom, not present in the Hamiltonian, was discovered.
  • Various transitions including Berezinskii-Kosterlitz-Thouless, Ising, first-order, and Landau-incompatible transitions were identified.

Conclusions:

  • Frustration in the J1-J2 clock model leads to a rich variety of emergent phases and unconventional transitions.
  • The emergence of Zq order via a relevant operator highlights a nonstandard route to complex phenomena.
  • The proposed field-theoretic framework unifies these emergent orders and their transitions.