Intermediate valence behavior in CeCo9Si4

The novel ternary compound CeCo$_9$Si$_4$ has been studied by means of specific heat, magnetisation, and transport measurements. Single crystal X-ray Rietveld refinements reveal a fully ordered distribution of Ce, Co and Si atoms with the tetragonal space group I4/mcm isostructural with other RCo9Si4. The smaller lattice constants of CeCo9Si4 in comparison with the trend established by other RCo9Si4 is indicative for intermediate valence of cerium. While RCo9Si4 with R= Pr, .. Tb, and Y show ferromagnetism and LaCo9Si4 is nearly ferromagnetic, CeCo9Si4 remains paramagnetic even in external fields as large as 40 T, though its electronic specific heat coefficient (g~190 mJ/molK^2) is of similar magnitude as that of metamagnetic LaCo9Si4 and weakly ferromagnetic YCo9Si4.Comment: 2 pages, 3 figures, submitted to SCES 0

Possible canted antiferromagnetism in UCu9_9Sn4_4

We report on the new compound UCu${}_9$Sn${}_4$ which crystallizes in the tetragonal structure \emph{I}4/\emph{mcm} with lattice parameters $a = 8.600{\rm\AA}$ and $c = 12.359{\rm\AA}$. This compound is isotyp to the ferromagnetic systems RECu${}_9$Sn${}_4$ (RE = Ce, Pr, Nd) with Curie temperatures $T{}\rm_C$ = 5.5 K, 10.5 K and 15 K, respectively. UCu${}_9$Sn${}_4$ exhibits an uncommon magnetic behavior resulting in three different electronic phase transitions. Below 105 K the sample undergoes a valence transition accompanied by an entropy change of 0.5 Rln2. At 32 K a small hump in the specific heat and a flattening out in the susceptibility curve probably indicate the onset of helical spin order. To lower temperatures a second transition to antiferromagnetic ordering occurs which develops a small ferromagnetic contribution on lowering the temperature further. These results are strongly hinting for canted antiferromagnetism in UCu${}_9$Sn${}_4$.Comment: 2 pages, 3 figures, SCES0

The algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy

We discuss the algebro-geometric initial value problem for the Ablowitz-Ladik hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Ablowitz-Ladik hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for general (non-unitary) Ablowitz-Ladik Lax operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin-type equations), this yields the construction of global algebro-geometric solutions of the time-dependent Ablowitz-Ladik hierarchy. The treatment of general (non-unitary) Lax operators associated with general coefficients for the Ablowitz-Ladik hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Ablowitz-Ladik hierarchy but applies generally to (1+1)-dimensional completely integrable soliton equations of differential-difference type.Comment: 47 page

Antiferromagnetic behavior in CeCo9_{9}Ge4_{4}

We investigate the novel intermetallic ternary compounds \emph{R}Co$_{9}$Ge$_{4}$ with \emph{R} = La and Ce by means of $X$-ray diffraction, susceptibility and specific heat measurements. CeCo$_{9}$Ge$_{4}$ crystallizes in the space group ${I}$ 4/ ${mcm}$ and is characterized by the coexistence of two different magnetic sublattices. The Ce-based sublattice, with an effective moment close to the expected value for a Ce$^{3+}$-ion, exhibits a magnetically ordered ground state with $T_{\mathrm{N}}=12.5$ K. The Co-based sublattice, however, exhibits magnetic moments due to itinerant 3$d$ electrons. The magnetic specific heat contribution of the Ce-sublattice is discussed in terms of a resonance-level model implying the interplay between an antiferromagnetic phase transition and the Kondo-effect and an underlying Schottky-anomaly indicating a crystal field level scheme splitting into three twofold degenerated micro states ($\Delta_1 = 69$ K, $\Delta_2 = 133$ K).Comment: 4 pages, 3 figures, conference SCES0

Crossover from Single-Ion to Coherent Non-Fermi Liquid Behavior in Ce1−x_{1-x}Lax_xNi9_9Ge4_4

We report specific heat and magneto-resistance studies on the compound Ce${}_{1-x}$La${}_x$Ni${}_9$Ge${}_4$ for various concentrations over the entire stoichiometric range. Our data reveal single-ion scaling with Ce-concentration between $x = 0.1$ and 0.95. Furthermore, CeNi${}_9$Ge${}_4$ turns out to have the largest ever recorded value of the electronic specific heat $\Delta c/T \approx$ 5.5 J $\rm K^{-2}mol^{-1}$ at $T=0.08$ K which was found in Cerium f-electron lattice systems. In the doped samples $\Delta c/T$ increases logarithmically in the temperature range between 3 K and 50 mK typical for non-Fermi liquid (nFl) behavior, while $\rho$ exhibits a Kondo-like minimum around 30 K, followed by a single-ion local nFl behavior. In contrast to this, CeNi${}_9$Ge${}_4$ flattens out in $\Delta c/T$ below 300 mK and displays a pronounced maximum in the resistivity curve at 1.5 K indicating a coherent heavy fermion groundstate. These properties render the compound Ce${}_{1-x}$La${}_x$Ni${}_9$Ge${}_4$ a unique system on the borderline between Fermi liquid and nFl physics.Comment: 2 pages, 3 figures, SCES0

Competing magnetic interactions in CeNi9-xCoxGe4

CeNi9Ge4 exhibits outstanding heavy fermion features with remarkable non-Fermi- liquid behavior which is mainly driven by single-ion effects. The substitution of Ni by Cu causes a reduction of both, the RKKY coupling and Kondo interaction, coming along with a dramatic change of the crystal field (CF) splitting. Thereby a quasi-quartet ground state observed in CeNi9Ge4 reduces to a two-fold degenerate one in CeNi8CuGe4. This leads to a modiffcation of the effective spin degeneracy of the Kondo lattice ground state and to the appearance of antiferromagnetic (AFM) order. To obtain a better understanding of consequences resulting from a reduction of the effective spin degeneracy, we stepwise replaced Ni by Co. Thereby an increase of the Kondo and RKKY interactions through the reduction of the effective d-electron count is expected. Accordingly, a paramagnetic Fermi liquid ground state should arise. Our experimental studies, however, reveal AFM order already for small Co concentrations, which becomes even more pronounced with increasing Co content x. Thereby the modiffcation of the effective spin degeneracy seems to play a crucial role in this system

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order $s\geq0$ on the group \Diff_c(M) of compactly supported diffeomorphisms of a manifold $M$. We show that for the important special case $M=S^1$ the geodesic distance on \Diff_c(S^1) vanishes if and only if $s\leq\frac12$. For other manifolds we obtain a partial characterization: the geodesic distance on \Diff_c(M) vanishes for $M=\R\times N, s<\frac12$ and for $M=S^1\times N, s\leq\frac12$, with $N$ being a compact Riemannian manifold. On the other hand the geodesic distance on \Diff_c(M) is positive for $\dim(M)=1, s>\frac12$ and $\dim(M)\geq2, s\geq1$. For $M=\R^n$ we discuss the geodesic equations for these metrics. For $n=1$ we obtain some well known PDEs of hydrodynamics: Burgers' equation for $s=0$, the modified Constantin-Lax-Majda equation for $s=\frac 12$ and the Camassa-Holm equation for $s=1$.Comment: 16 pages. Final versio