Neutron scattering investigation of the magnetic ground state of PrO2

We report on a series of neutron experiments on stoichiometric PrO₂. The material, which has the cubic CaF₂ structure, orders antiferromagnetically at 14 K with a type I magnetic structure as found in UO₂. The ordered moment is 0.6 ± 0.1 μ_B/Pr atom. No lattice distortion has been observed below T_N with a high-resolution neutron-diffraction experiment. Neutron inelastic scattering at the Intense Pulsed Neutron Source, Argonne National Laboratory, has been used to determine the crystal-field splitting of 130 ± 5 meV between the Γ₈ ground state and the Γ₇ excited state of the $\text{J =}\text{5}\text{2}$ multiplet. This represents the first direct observation of a crystal-field splitting in the rare-earth or actinide dioxides, and the largest such electronic splitting ever observed by neutron scattering. We propose that the ground state of PrO₂ is the Γ₈ quartet with the degeneracy lifted by a dynamic Jahn-Teller effect. The A₄〈r⁴〉 crystal field potential term for PrO₂ is - (57 ± 3) meV, and the significance of this determination for other oxides is discussed.     

Atomic displacements in the incommensurable charge-density wave in alpha-uranium

We present a detailed examination of the incommensurable charge-density wave (ICDW) in α-uranium below 43 K. We have measured the q vectors as a function of temperature and the structure factors of a large number of first-order satellites with neutron diffraction. From the structure factors the atomic displacements that occur at the onset of the CDW have been determined in all three independent crystallographic directions. Uniaxial stress up to 3.5 kbar has been applied along 2 directions but does not change the satellite intensities, thus leaving ambiguity whether the structure is single or multi q. We discuss the relationship between the present results and the observations of Smith et al.     

Spectral effects in cyclobutane derivatives : The bathochromic-hypochromic shifts in compounds related to verbenone

The bathochromic-hypochromic spectral shifts, observed in bicyclic enones that contain a 4- or a 3-membered ring, are discussed in molecular-mechanical and molecular-orbital terms. It is proposed that the effect, not only by the cyclopropane but also by the cyclobutane ring, is due to extension of the CCCO chromophore, and not to its mere distortion from “normal” geometry. The high-energy absorption of umbellulone (3), 220 nm in ethanol, is characterized as a second π-π^*. 2-Methylbicyclo[3.2.1]oct-2-en-4-one (7) has been synthesized, and its preparation and spectrum are described.     

Subgraph statistics in subcritical graph classes

Let H be a fixed graph and a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series‐parallel graphs. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 631–673, 2017     

A rheological investigation of solid soap

Summary The rheological behaviour of solid soaps has been investigated over several orders in time and shear by means of creep experiments, measurement of stressstrain curves at constant shear rate and dynamic experiments. The modulus at small deformations is almost constant over a frequency range of 10^–3–10³ Hz; a transition from linear to non-linear behaviour has been observed at a strain of about 10^–3. The results suggest that some kind of network structure is responsible for the rheological behaviour of soap.

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Zusammenfassung Das Theologische Verhalten von Hartseifen wurde für mehrere Größenordnungen von Zeit und Schubgeschwindigkeit untersucht mittels Kriechversuche, Messung der Spannungs-Deformationskurven bei konstanter Schubgeschwindigkeit und dynamischer Experimente. Der Modul bei geringen Deformationen ist fast konstant im Frequenzbereich von 10^–3–10³ Hz; ein Übergang von linear zu nicht-linear wurde bei einer Deformation von etwa 10^–3 beobachtet. Die Ergebnisse lassen auf eine netzartige Struktur schließen, die für das Theologische Verhalten der Seife verantwortlich ist.

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The authors are indebted to Mr.C. J. Nederveen of the Central Laboratory T. N. O., Delft for the dynamic measurements, and thank Mr.D. W. Brinkman of the Unilever Research Laboratory, Vlaardingen for experimental assistance.