Imaging of spin waves in atomically designed nanomagnets

The spin dynamics of all ferromagnetic materials are governed by two types of collective phenomenon: spin waves and domain walls. The fundamental processes underlying these collective modes, such as exchange interactions and magnetic anisotropy, all originate at the atomic scale. However, conventional probing techniques based on neutron1 and photon scattering2 provide high resolution in reciprocal space, and thereby poor spatial resolution. Here we present direct imaging of standing spin waves in individual chains of ferromagnetically coupled S = 2 Fe atoms, assembled one by one on a Cu2N surface using a scanning tunnelling microscope. We are able to map the spin dynamics of these designer nanomagnets with atomic resolution in two complementary ways. First, atom-to-atom variations of the amplitude of the quantized spin-wave excitations are probed using inelastic electron tunnelling spectroscopy. Second, we observe slow stochastic switching between two opposite magnetization states3, 4, whose rate varies strongly depending on the location of the tip along the chain. Our observations, combined with model calculations, reveal that switches of the chain are initiated by a spin-wave excited state that has its antinodes at the edges of the chain, followed by a domain wall shifting through the chain from one end to the other. This approach opens the way towards atomic-scale imaging of other types of spin excitation, such as spinon pairs and fractional end-states5, 6, in engineered spin chains.

Paraconsistent Multivalued Logic and Coincidentia Oppositorum: Evaluation with Complex Numbers

Paraconsistent logic admits that the contradiction can be true. Let p be the truth values and P be a proposition. In paraconsistent logic the truth values of contradiction is . This equation has no real roots but admits complex roots . This is the result which leads to develop a multivalued logic to complex truth values. The sum of truth values being isomorphic to the vector of the plane, it is natural to relate the function V to the metric of the vector space R2. We will adopt as valuations the norms of vectors. The main objective of this paper is to establish a theory of truth-value evaluation for paraconsistent logics with the goal of using in analyzing ideological, mythical, religious and mystic belief systems.