A generalized Anderson model for a magnetic ion in a harmonic potential is formulated. The model is investigated by the numerical renormalization group (NRG) method. In addition to the conventional swave screening, the model exhibits phonon assisted p-wave Kondo effect as well as Yu and Anderson type Kondo effect. It is shown that the s-wave Kondo and the Yu-Anderson Kondo belong to the same fixed point. At the boundary between the s-wave and p-wave Kondo regions line of fixed points of the two channel Kondo effect is identified. Filled skutterudite compounds RT 4 X 12 (R = rare earth or alkaline earth element; T = Fe, Ru, Pt, or Os; X = P, As, Ge, or Sb) are characterized by their specific structures which involve a network of cages filled by guest ions. When the radius of the guest ion is smaller than the diameter of the cage, the guest ion vibrates with larger amplitude and smaller frequency than conventional localized modes. Such anharmonic local vibrations are referred to as rattling modes and may lead to novel phenomena of the coupled electron-phonon systems. Recent experiments on SmOs 4 Sb 12 show a large electronic specific heat coefficient linear in T, which is robust against magnetic field. 1) One possible scenario of the unusual behavior is the effect of strong electron-phonon coupling.When a magnetic ion couples with conduction electrons we expect Kondo effect due to the localized moment. On the other hand, it has been shown by Yu and Anderson (YA) that when conduction electrons couple strongly with ionic vibrations a different type of Kondo effect is expected.2) In this letter we will study the interplay between the conventional Kondo effect and the YA type one. Concerning the effects of coupling between a magnetic ion and ionic vibrations, Hotta has studied effects of anharmonicity in the Holstein-Anderson model.3) A lattice version of the Holstein-Anderson model is also studied. 4) In this letter, we will discuss effects of transverse vibrations on the Kondo effect rather than the breathing type vibrations.Suppose electron orbitals of a magnetic ion are described by the '-spherical-wave functions. Then the hybridization with the conduction electrons is given by overlap integrals between the '-wave localized orbitals and plane waves. To include the effect of vibrations the origin of the localized orbitals is shifted by Q which is the coordinate of the ion position. We can expand the overlap integral with respect to Q. In the zeroth order the localized '-spherical-waves hybridize with the '-partial-waves of conduction electrons. In the first order of Q, they hybridize with the ' þ 1 and ' À 1 partial waves. In the present letter we will concentrate on the simplest case of s-wave localized orbital. Then the total Hamiltonian up to the order of Q is given by the sum of

To discuss Kondo effects of a magnetic ion vibrating in the sea of conduction electrons, a generalized Anderson model is derived. The model includes a new channel of hybridization associated with phonon emission or absorption. In the simplest case of the localized electron orbital with the s-wave symmetry, hybridization with p-waves becomes possible. An interesting interplay among the conventional s-and p-wave Kondo effects and the Yu-Anderson-type Kondo effect is found, and the ground state phase diagram is determined by using the numerical renormalization group method. Two different types of stable fixed points are identified and the two-channel Kondo fixed points are generically realized at the boundary.

Effect of anharmonicity of a cage potential for a magnetic ion vibrating in a metal is investigated by the numerical renormalization group method. The cage potential is assumed to be one-dimensional and of the double-well type. In the absence of the Coulomb interaction, we find continuous crossover among the three limiting cases: Yu-Anderson-type Kondo regime, the double-well-type Kondo one, and the renormalized Fermi chain one. In the entire parameter space of the double-well potential, the ground state is described by a local Fermi liquid. In the Yu-Anderson-type Kondo regime, a quantum phase transition to the ground state with odd parity takes place passing through the two-channel Kondo fixed point when the Coulomb interaction increases. Therefore, the vibration of a magnetic ion in an oversized cage structure is a promising route to the two-channel Kondo effect.

By analyzing aerial images, we characterize the lowest order coma aberration measurements for the projection optics of a microlithography exposure apparatus based on scalar diffraction theory. Our developed method for measuring the coma aberration exploits the intensity difference between the sidelobe peaks appearing near the boundaries of the bright field ("negative") single-line or plural-line patterns. Our method further demonstrates linearity between the intensity difference of the sidelobe peaks and the amount of residual lowest order coma aberration. We analyze the coma aberration sensitivity formula and determine the duty ratio of the line-and-space pattern that realizes the highest aberration sensitivity.

We present a methodology for obtaining the analytical solution of the Gamo entropy, defined by the intensity matrix proposed by Gamo [J. Opt. Soc. Am.47, 976 (1957)JOSAAH0030-394110.1364/JOSA.47.000976]. The matrix, which consists of numerous image amplitudes at all the sampling points of the entire imaging plane, is generally infinite-dimensional. The essence of our theory is that the computational difficulties arising because of the infinite-dimensionality are avoided by introducing the inner products of two image amplitudes. The integral in continuous space plays the role of a buffer against the infinite-dimensionality. The validity of the approach is confirmed by comparing our analytical solution and Yamazoe's numerical simulations [J. Opt. Soc. Am. A28, 448 (2011)JOAOD60740-323210.1364/JOSAA.28.000448].