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Yoshimasa Matsuno
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1415-1418
Published: 2002
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The properties of the soliton and periodic wave solutions of a nonlocal nonlinear Schrödinger equation for envelope waves are investigated by the pole expansion method. For both solutions, the dynamics of the poles are shown to be described by the first-order systems of nonlinear ordinary differential equations (ODEs). A significant result reported here is that in the case of solitons, the system is reducible to the Calogero-Moser dynamical system whereas in the case of periodic waves, the corresponding system is found to be the Calogero-Moser-Sutherland dynamical system. We then establish a purely algebraic method for solving the first-order systems of ODEs and prove their complete integrability.
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Yukitoshi Motome, Nobuo Furukawa
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1419-1422
Published: 2002
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Spin excitation spectrum of the double-exchange model is studied in the presence of randomness. Spin wave approximation in the ground state shows that randomness significantly modifies the spectrum from the cosine-like one in the pure system to that with anomalies such as broadening, anticrossing and gap opening. The origin of anomalies is speculated to be the modulation of effective ferromagnetic coupling by the Friedel oscillation. These anomalies qualitatively reproduce the spin excitation spectrum in colossal magnetoresistance manganites whose Curie temperatures are relatively low. Our results suggest that randomness control is an important notion to understand the effects of A-site substitution which has previously been understood as the bandwidth control.
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Masahiko Isobe, Emi Ninomiya, Alexander N. Vasil'ev, Yutaka Ueda
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1423-1426
Published: 2002
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We synthesized a well-characterized powder sample of NaTiSi
2O
6, which had been missing as a spin-1/2 compound in one-dimensional (1D) magnets of the pyroxene family, and measured the magnetic susceptibility, lattice parameters and specific heat as functions of temperature. We observed a typical behavior of magnetic susceptibility in a spin-1/2 1D magnet, followed by a spin-Peierls-like transition at 210 K. The most significant feature of the transition is that it occurs at a temperature higher than that at the maximum point of the Bonner-Fisher curve. This means that the short-range magnetic correlations within the chain are not fully developed and intrinsic magnetoelastic instability of a 1D system cannot be considered as a driving force for the transition. We propose an important role of the orbital degree of freedom, that is, the orbital order, based on the structural feature that the 1D chain is made up of
skew edge-sharing Ti
3+O
6 octahedra. A similar transition was observed at 230 K in LiTiSi
2O
6.
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Akihiro Koda, Wataru Higemoto, Ryosuke Kadono, Yu Kawasaki, Kenji Ishi ...
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1427-1430
Published: 2002
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Muon spin rotation measurements under high transverse fields (HTF-μSR) have been performed on polycrystalline samples of Ce
0.99Cu
2.02Si
2. The muon hyperfine coupling constant was determined to be -3.36 kOe/μ
B from a Knight shift (K) vs susceptibility (χ) plot for 2 K < T < 200 K, which is comparable to those for Cu and Si nuclei. Below ∼1 K, the Knight shift at the precession signal, showing broadening due to the magnetic A-phase, exhibited a reduction at 0.5 T, while it remained unchanged above 1.0 T, strongly suggesting the microscopic coexistence of superconductivity with the magnetic A-phase.
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Junko Hatori, Masaru Komukae, Zbigniew Czapla, Toshio Osaka
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1431-1434
Published: 2002
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The static dielectric constant of the pyridinium periodate crystal was measured at the frequency of 50 kHz in the temperature range from room temperature to 365 K. In addition, the real and imaginary parts of the complex dielectric constant were also measured between 5×10
4 Hz and 10
9 Hz in the vicinity of the para-ferroelectric phase transition temperature (T
1). It was found that pyridinium periodate shows the dielectric critical slowing-down process near T
1. The observed dielectric dispersion is characterised by a single relaxation.
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Daniil G. Sannikov, Grigorii A. Kessenikh, Hiroyuki Mashiyama
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1435-1439
Published: 2002
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Theoretical phase diagrams for {N(CH
3)
4}
2MnC
4 are calculated by using a phenomenological approach. Expressions for thermodynamical potentials of different phases and for boundaries between these phases are given in an explicit or parametric form. The theoretical temperature-pressure phase diagrams are plotted. They show sufficiently well agreement with corresponding experimental diagrams.
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Jian-Qi Shen, Hong-Yi Zhu, Hong Mao
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1440-1444
Published: 2002
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By utilizing the property of the supersymmetric structure in the two-level multiphoton Jaynes-Cummings model, an invariant is constructed in terms of the supersymmetric generators by working in the sub-Hilbert-space corresponding to a particular eigenvalue of the conserved supersymmetric generators. We obtain the exact solutions of the time-dependent Schrödinger equation which describes the time-dependent supersymmetric two-level three-photon Jaynes-Cummings model (TLTJCM) by using the invariant-related unitary transformation formulation. The case under the adiabatic approximation is also discussed.
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Kenji Toyoda, Yoshiro Takahashi, Tsutomu Yabuzaki
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1445-1449
Published: 2002
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Bose-Einstein condensates of
87Rb atoms are generated in a double-chamber configuration where for transfer of atoms between the chambers an extremely narrow tube is used, which offers good vacuum isolation between the two chambers. By performing time-of-flight absorption imaging, the condensate atom number and the temperature are determined.
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Chien C. Chang, Bin-Shei Lin
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1450-1462
Published: 2002
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In this study, we continue with a recursive renormalization group (RG) analysis of incompressible turbulence, aiming at investigating various turbulent properties of three-dimensional magneto-hydrodynamics (MHD). In particular, we are able to locate the fixed point (i.e. the invariant effective eddy viscosity) of the RG transformation under the following conditions. (i) The mean magnetic induction is relatively weak compared to the mean flow velocity. (ii) The Alfvén effect holds, that is, the fluctuating velocity and magnetic induction are nearly parallel and approximately equal in magnitude. It is found under these conditions that re-normalization does not incur an increment of the magnetic resistivity, while the coupling effect tends to reduce the invariant effective eddy viscosity. Both the velocity and magnetic energy spectra are shown to follow the Kolmogorov k
-5/3 in the inertial subrange; this is consistent with some laboratory measurements and observations in astronomical physics. By assuming further that the velocity and magnetic induction share the same specified form of energy spectrum, we are able to determine the dependence of the (magnetic) Kolmogorov constant C
K (C
M) and the model constant C
S of the Smagorinsky model for large-eddy simulation on some characteristic wavenumbers.
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Jun-ichi Fukuda, Hiroshi Yokoyama
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1463-1470
Published: 2002
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We apply the density functional formalism developed by the present authors [J. Fukuda and H. Yokoyama: J. Chem. Phys.
115 (2001) 4930] for the calculation of the free energy of rigid rods as a functional of the compositional scalar order parameter Φ and the orientational order parameter of second-rank tensor Q
ij We model rigid rods as semiflexible wormlike chains with infinite bending elasticity and the exact expression for the entropic part of the free energy up to the second order in the gradients can be obtained. Discussion is devoted to the contribution of rotational entropy to Frank elasticity. Although none of the previous molecular theories for the Frank elasticity of liquid crystals has taken rotational entropy into account, our results indicate that rotational entropy can give a non-negligible contribution to Frank elasticity.
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Yoshihiro Ishibashi, Hiroshi Orihara, David R. Tilley
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1471-1474
Published: 2002
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The exact expressions for defining the critical point of ferroelectric thin films as a function of temperature, film thickness and some other parameters are obtained on the basis of the Tilley-Zeks model. The cases of symmetric and asymmetric extrapolation lengths at the two surfaces of films are investigated. The transition temperature of a ferroelectric thin film will increase or decrease, compared with that of the bulk, according to the sign of two extrapolation lengths. But in the mixed-sign case, it is governed by the relative effectiveness, i.e., by the absolute values of the two.
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Michiaki Matsukawa, Hiromitsu Ogasawara, Toru Sasaki, Masahito Yoshiza ...
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1475-1480
Published: 2002
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Thermal expansion and magnetostriction both in the ab-plane and along the c-axis of double-layered perovskite manganite (La
1-zPr
z)
1.2Sr
1.8Mn
2O
7 (z=0, 0.2, 0.4 and 0.6) single crystals have been measured as a function of temperature and magnetic field. The anomalous lattice distortion and the giant magnetostriction associated with the CMR were observed near the PI to FM transition temperature T
c. These findings indicate that the PI to FM transition and the CMR near T
c are not associated with the anisotropic lattice striction concerning the static John-Teller effect but with the negative volume striction related to the delocalization of polarons to itinerant carriers.
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Katsurou Hanzawa
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1481-1494
Published: 2002
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The generalized periodic Anderson model with plane-wave conduction-band states is considered as a model to describe the properties of Ce, Sm, Eu, Tm and Yb compounds showing a variety of valence fluctuation phenomena. The mixing matrix elements between localized 4ƒ-electron states and the conduction-electron states, which are anisotropic in k space, are derived, and it is shown how pseudogap or real-but-anisotropic gap is formed in this model. The structures of gap or pseudogap to be derived here are considered to bear some of the characteristics of those of rare-earth compounds in such as heavy-fermion, Kondo-insulator and intermediate-valence states.
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Tamio Oguchi
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1495-1500
Published: 2002
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Systematic trend in lattice constants and heats of formation is studied for AlB
2-type diborides from first principles. Observed trend in the lattice constants and the c/a ratio can be interpreted by anisotropic bonding of pσ-pσ and sp-pπ (d-pπ) in the sp (d) metal diborides. Calculated heats of formation are in good agreement with experimental data available and their chemical trend in the d metal diborides is governed mostly by filling of d-pπ bonding and anti-bonding bands. Obtained electronic bonding mechanism may explain several structural aspects including reasons why MgB
2 has so large c/a ratio among the diborides and why there exist only few diboride phases with AlB
2 type.
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Vladimir Bubanja
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1501-1505
Published: 2002
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We consider the 3-junction single electron pump with on-chip resistors (R-pump) driven by harmonic gates. We calculate the cotunneling error of the device as a function of the external resistance. It is shown that for suitable values of the parameters metrological accuracy can be achieved.
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Hiroaki Ueda, Koichi Kitazawa, Hidenori Takagi, Takehiko Matsumoto
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1506-1510
Published: 2002
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Temperature dependent resistivities of Magnéli phases Ti
nO
2n-1 (n=4,5,6), which contains two Ti
3+ and (n-2) Ti
4+ per unit formula, are measured under various pressures up to 3.5 GPa, to investigate the nature of the two step transitions from a metal to bipolaronic insulators. We found that the pressure dependence of the transition temperatures is critically changes with the density of Ti
3+. In Ti
4O
7, the low temperature bipolaronic phase can be suppressed very rapidly with pressure and the high temperature metallic state is stabilized down to almost T=0 limit at 3.5 GPa. Ti
5O
9 shows much reduced pressure dependence. In Ti
6O
11, eventually, the transition temperatures hardly depend on the applied pressure. We propose that this critical change of the pressure effect originates from a subtle interplay between electron-lattice interaction and electron correlations (inter-site Coulomb repulsions).
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Takashi Imamura, Miki Wadati
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1511-1517
Published: 2002
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For disordered quantum wires which belong to all ten universality classes, the universal quantities of transport properties are obtained through DMPK approach. Calculated are the universal parts of one- and two-point correlation functions for probability distribution functions of transmission eigenvalues. In this analysis, the asymptotic solution of DMPK equation is used. Transport properties for new universality classes (chiral and Bogoliubov-de Gennes classes) are discussed comparing with those for standard class.
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Toshiyuki Ozawa, Yoshiyuki Ono
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1518-1528
Published: 2002
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The collision processes between two acoustic polarons, two acoustic bipolarons, and a polaron and a bipolaron in a one-dimensional electron-lattice system are analyzed by numerical simulations based on Su, Schrieffer and Heeger's model extended to include short-ranged Coulombic electron-electron interactions. The collisions between two polarons with the same spin and between two bipolarons are essentially elastic because of a rather strong repulsion due to the Pauli principle. On the other hand, those between two polarons with opposite spins and between a polaron and a bipolaron show some inelastic behaviors depending on the relative velocity of the two objects before the collision. This knowledge will serve to understand the dynamics of the system when many polarons and/or bipolarons are created in the system
e.g. by photoexcitation.
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Nobuhisa Fujima
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1529-1534
Published: 2002
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Non-collinear magnetic moments are calculated for trigonal bipyramid clusters, V
5, Cr
5, Mn
5, Fe
5, Co
5, and Ni
5 by a simple method employing the locality of the local spin-density functional approximation and the discrete variational scheme. We discuss the magnetic properties of the transition-metal clusters with the bond lengths of 80%-100% of the crystalline interatomic distance. Parallel magnetic moments appear for the Ni, Co, and Fe clusters with almost all the bond lengths. Tilting magnetic moments also appear for the Co and Fe clusters in a certain bond length. Non-collinear magnetic arrangements appear for the Mn and Cr clusters, which change to antiparallel arrangements with decreasing bond length. Only an antiparallel alignment appears for the V cluster.
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R. M. Dipasupil, M. Oda, N. Momono, M. Ido
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1535-1540
Published: 2002
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On the basis of the tunneling spectra in Bi2212/vaccum/Bi2212 junctions fabricated using STM, we report that, in the electronic excitations, there exist two kinds of pseudogaps; one with a characteristic energy comparable to the superconducting (SC) gap and another one that is 3 to 4 times larger. The smaller energy-scale pseudogap (SPG) develops progressively below temperature T
*, which nearly scales with the SC gap amplitude Δ
0 at T « T
c, in addition to the larger energy-scale pseudogap (LPG), which already exists above T
*. The SPG smoothly develops into the SC-state gap with no tendency to close at T
c.
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Hirono Fukazawa, Kosaku Yamada
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1541-1547
Published: 2002
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We study the superconducting instabilities of singlet and triplet pairing in a two-dimensional Hubbard model on the basis of the third-order perturbation theory (TOPT). We investigate the effect of the vertex correction that is given by TOPT, comparing with the study with only the second-order effective interaction. In our results, a stable p-wave pairing state spreads from low to intermediate electron density. A d
x2-y2-wave pairing is dominant for the high density near a half-filling. It is shown that the vertex correction plays an essential role in making the p-wave pairing state dominant.
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Tomio Koyama
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1548-1552
Published: 2002
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The Josephson plasma resonances in inhomogeneous high-T
c superconductors composed of two regions with different characteristic resonant frequencies are investigated in three typical configurations for applied microwaves. The calculated results are consistent with the resonant behaviors observed in Bi-2212 thin films containing inhomogeneity introduced by the partial irradiation of heavy ions. The origin of the crossover-like behavior observed in the magnetic field dependence of the resonant frequencies is discussed on the basis of the numerical results.
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Zenji Hiroi, Touru Yamauchi, Takahiro Yamada, Masafumi Hanawa, Yasuo O ...
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1553-1557
Published: 2002
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Superconducting and structural phase transitions in a pyrochlore oxide Cd
2Re
2O
7 are studied under high pressure by x-ray diffraction and electrical resistivity measurements. A rich P-T phase diagram is obtained, which contains at least two phases with the ideal and slightly distorted pyrochlore structures. It is found that the transition between them is suppressed with increasing pressure and finally disappears at a critical pressure P
c=3.5 GPa. Remarkable enhancements in the residual resistivity as well as the coefficient A of the AT
2 term in the resistivity are found around the critical pressure. Superconductivity is detected only for the phase with the structural distortion.
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Hisashi Sugie, Nobuyuki Iwata, Kay Kohn
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1558-1564
Published: 2002
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From magnetic, dielectric and magnetoelectric measurement, it is concluded that hexagonal HoMnO
3, ErMnO
3 and YbMnO
3 undergo a magnetic phase transition at a low temperature where a magnetic long-range order of rare earth ions is established. Possible magnetic structures are estimated from the experimental results and symmetry consideration. This low-temperature phase transforms successively into possibly two different phases in magnetic field. The higher-field phase is considered to have a ferrimagnetic arrangement of the rare earth magnetic moments.
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Hideki Abe, Hiroyuki Suzuki, Hideaki Kitazawa
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1565-1569
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High-field magnetization measurements up to 28 T have been performed on single crystalline TbRh
2Si
2 specimens using a hybrid steady field magnet. It was found that the magnetic isotherm along the c-axis at low temperatures takes form of a double-step function with a half height staircase between the two transition fields of 8.66 T and 15.83 T, where the saturation moment is 9.05 μ
B/f.u. Mean-field analysis has suggested that the intermediate state between the two transition fields has an incommensurate magnetic structure.
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Yusuke Tomita, Yutaka Okabe
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1570-1575
Published: 2002
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We present a detailed description of the idea and procedure for the newly proposed Monte Carlo algorithm of tuning the critical point automatically, which is called the probability-changing cluster (PCC) algorithm [Y. Tomita and Y. Okabe: Phys. Rev. Lett.
86 (2001) 572]. Using the PCC algorithm, we investigate the three-dimensional Ising model and the bond percolation problem. We employ a refined finite-size scaling analysis to make estimates of critical point and exponents. With much less efforts, we obtain the results which are consistent with the previous calculations. We argue several directions for the application of the PCC algorithm.
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Kazuyuki Matsuhira, Yukio Hinatsu, Kenichi Tenya, Hiroshi Amitsuka, To ...
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1576-1582
Published: 2002
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We report the magnetic properties of pyrochlore rare earth oxides R
2Sn
2O
7 (R=Pr, Nd, Gd, Tb, Dy, Ho, Er, Tm and Yb). The magnetic susceptibility has been measured at temperatures down to 0.1 K. Among them, some compounds show the magnetic anomaly below 1 K. Nd
2Sn
2O
7 and Gd
2Sn
2O
7 show an antiferromagnetic ordering at 0.9 K and 1.0 K, respectively. Tb
2Sn
2O
7 shows a ferromagnetic ordering at 0.87 K. Although both Er
3+ and Yb
3+ are a Kramers ion, no magnetic anomaly is observed at least down to 0.15 K. We found that Dy
2Sn
2O
7 and Ho
2Sn
2O
7 show `spin ice' behavior. In addition, it is suggested that Pr
2Sn
2O
7 is a new spin ice compound.
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Chang Seop Hong, Eun Ok Chi, Wan Seop Kim, Nam Hwi Hur, Yong Nam Choi
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1583-1588
Published: 2002
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The magnetic and structural behaviors of a single crystal of La
0.8Ca
0.2MnO
3 were investigated in terms of magnetic and neutron diffraction measurements. The frequency-dependent ac magnetic susceptibility reveals that cluster-glassy phase evolves below the Curie temperature (T
C) in this system, suggesting the existence of short-range correlation below T
C embedded in the ferromagnetic (FM) background. A correlation of the magnetic properties in La
0.8Ca
0.2MnO
3 with the buckling of MnO
6 octahedra is discussed based on its temperature-dependent neutron and magnetization data.
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Katsuhiko Nishimura, Susumu Ohya, Yoshie Kawamura, Takashi Ohtsubo, Ta ...
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1589-1593
Published: 2002
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Brute-force nuclear magnetic resonance on oriented nuclei (BF-NMRON) was used to investigate the nuclear spin-relaxation times, T
1, of
90Nb in Nb and Cu,
96Tc in Nb,
101mRh in Cu, and
110mAg in Ag with an external field of 11.922 T at about 10 mK. The observed relaxation times with known nuclear g-factors were used to estimate the converted relaxation rates, (T
1Tg
n2)
-1, which were compared with those obtained from NMR and other methods. From the comparison, it was found that (T
1Tg
n2)
-1 of
90Nb in Nb and
110mAg in Ag coincided with those of the bulk systems. (T
1Tg
n2)
-1 of
90Nb in Cu was about four times larger than that of Nb bulk system. Both
96Tc in Nb and
101mRh in Cu produced (T
1Tg
n2)
-1 values consistent with previous studies.
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Ehtesanul Islam, Akira Sakai, Akira Onodera
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1594-1597
Published: 2002
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Optical phonons of the ferroelectric-semiconductor Zn
1-xLi
xO ceramics have been studied for several Li-concentrations as a function of temperature by using a micro-probed Raman scattering. With Li-doping, the Raman peak position of the A
1(z) mode at 380 cm
-1 shifts to a lower frequency side while the E
2 mode at 437 cm
-1 does not show the Li-dependence. The generalized force constant of A
1(z) mode is obtained as a function of Li-concentration. The Li-dependence of the Raman peak positions are well explained by the change of electronic bonding force between nearest two atoms. Because the drastic phonon anomaly was not observed, it suggests that the ferroelectric-phase transition in Zn
1-xLi
xO is caused by the electronic effect rather than by the ionic instability. In addition, it is found that the Raman peak position is proportional to the lattice constant in the wide temperature region.
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Toshiaki Omori, Tsuyoshi Horiguchi
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1598-1604
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We investigate some noise effect on a neural network model proposed by Araki and Aihara for the memory recall of dynamical patterns in the hippocampus and the entorhinal cortex; the noise effect is important since the release of transmitters at synaptic clefts, the operation of gate of ion channels and so on are known as stochastic phenomena. We consider two kinds of noise effect due to a deterministic noise and a stochastic noise. By numerical simulations, we find that reasonable values of noise give better performance on the memory recall of dynamical patterns. Furthermore we investigate the effect of the strength of external inputs on the memory recall.
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Kazuo Hatakeyama, Takejiro Kaneko, Shunya Abe, Hajime Yoshida, Yasuaki ...
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1605-1606
Published: 2002
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Hiromitsu Hori, Shoji Yamamoto
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1607-1608
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Gaku Motoyama, Takashi Nishioka, Noriaki K. Sato
Article type: General Physics
Subject area: Mathematical methods in physics.
2002 Volume 71 Issue 6 Pages
1609-1610
Published: 2002
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