The continuum elastic and atomistic viewpoints on the formation volume and strain energy of a point defect

We discuss the roles of continuum linear elasticity and atomistic calculations in determining the formation volume and the strain energy of formation of a point defect in a crystal. Our considerations bear special relevance to defect formation under stress. The elasticity treatment is based on the Green's function solution for a center of contraction or expansion in an anisotropic solid. It makes possible the precise definition of a formation volume tensor and leads to an extension of Eshelby's [Proc. R. Soc. London Ser. A 241 (1226), 376] result for the work done by an external stress during the transformation of a continuum inclusion. Parameters necessary for a complete continuum calculation of elastic fields around a point defect are obtained by comparing with an atomistic solution in the far field. However, an elasticity result makes it possible to test the validity of the formation volume that is obtained via atomistic calculations under various boundary conditions. It also yields the correction term for formation volume calculated under these boundary conditions. Using two types of boundary conditions commonly employed in atomistic calculations, a comparison is also made of the strain energies of formation predicted by continuum elasticity and atomistic calculations. The limitations of the continuum linear elastic treatment are revealed by comparing with atomistic calculations of the formation volume and strain energies of small crystals enclosing point defects.     

Relaxation methods for mixed-integer optimal control of partial differential equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.     

It was 30 years ago today when Laurens de Haan went the multivariate way

Since the publication of his masterpiece on regular variation and its application to the weak convergence of (univariate) sample extremes in 1970, Laurens de Haan (Thesis, Mathematical Centre Tract vol. 32, University of Amsterdam, 1970) is among the leading mathematicians in the world, with a particular focus on extreme value theory (EVT). On the occasion of his 70th birthday it is a great pleasure and a privilege to follow his route through multivariate EVT, which started only seven years later in 1977, when Laurens de Haan published his first paper on multivariate EVT, jointly with Sid Resnick.      

Infrared spectra of the ammonium ion in crystals: Part IV. Motions of the NH3D+ ion in ammonium perchlorate and in other weakly hydrogen-bonded ammonium compounds

Molecular motion of the ammonium ion has been investigated by IR spectroscopy in a series of ammonium salts which have been reported to be weakly hydrogen-bonded. Infrared spectra of the Isotopically isolated NH₃D⁺ ion in NH₄ClO₄, and NH₄BF₄ have been recorded between 295 and 22 K. Broadening of the N-D stretching bands, in each case, is attributed to short residence time and indicates rapid reorientation of the NH₃D⁺ ion. The band broadening persists down to the lowest temperature. The spectra of the NH₃D⁺ ion in NH₄BPh₄, NH₄SO₃F, NH₄SO₃Me and NH₄PF₆ have been recorded between 295 and 90 K. No significant lifetime broadening has been found in the spectra of the NH₃D⁺ ion in these salts.     

Die Chloride Na3xM2–xCl6 (M = La?Sm) und NaM2Cl6 (M = Nd,Sm): Derivate des UCl3-Typs. Synthese,Kristallstruktur und Röntgenabsorptionsspektroskopie (XANES)

The Chlorides Na_3xM_2–xCl₆ (M = La? Sm) and NaM₂Cl₆ (M = Nd, Sm): Derivatives of the UCl₃-Type of Structure. Synthesis, Crystal Structure and X-Ray Absorption Spectroscopy (XANES) Single crystals of the derivatives of the UCl₃-type structure Na_3xM_2–xCl₆ (M = La/x = 0.364(4); Ce/0.349(5); Pr/0.318(8); Nd/0.305(5); Sm/0.246(4)) and NaSm₂Cl₆ were grown by different methods generally under reducing conditions. They are addition [Na(Sm₂)Cl₆] and addition/substitution variants [Na_2x(Na_xM_2–x)Cl₆] of the UCl₃ structure type [□(U₂)Cl₆]. X-Ray Absorption Spectroscopy (XANES) at the L_III edge characterizes NaSm₂Cl₆ and NaNd₂Cl₆ as mixed-valence compounds with valences of +2 and +3 in statistical distribution (approximately 1:1) for Sm and Nd, respectively.     

Syntheses,constitutions and properties of stentorin and isostentorin

Summary Stentorin and Isostentorin were synthesized from 2-isopropyl-1,3,6,8-tetrahydroxyanthrone by dimerization and chromatographic separation of the resulting regioisomers. The anthrone was prepared in four steps starting from easily available properly substituted benzene derivatives; the overall yield of the stentorins was 11%. The constitutions of stentorin and isostentorin could be unequivocally assigned from the¹H NMR spectra of their potassium salts and were found to be in agreement with those derived recently by means of a rational synthesis. The spectroscopic, dissociation, and acid-base properties in ground and excited states as well as the chiroptical properties of the human serum albumin complexes were investigated and discussed comparing them with respective data of hypericin, fringelite D, and the naturalStentor pigment.

---

Synthese, Konstitution und Eigenschaften von Stentorin und Isostentorin

Zusammenfassung Stentorin und Isostentorin wurden ausgehend von 2-Isopropyl-1,3,6,8-tetrahydroxyanthron durch Dimerisierung und anschließende Chromatographie der Regioisomeren dargestellt. Das Anthron wurde in vier Schritten aus einfach zugänglichen, entprechend funktionalisierten Benzolderivaten erhalten; die Gesamtausbeute der Stentorine betrug dabei 11%. Die Konstitutionen von Stentorin und Isostentorin wurden zweifelsfrei aus den¹H NMR Spektren der Kaliumsalze abgeleitet und waren in Übereinstimmung mit einem früher erzielten Ergebnis einer Synthesestudie. Die spektroskopischen, Dissoziations- und Säure-Basen-Eigenschaften in Grund- und angeregtem Zustand, sowie die chiroptischen Eigenschaften des Human-Serumalbuminkomplexes wurden untersucht und diskutiert und mit den entsprechenden Daten von Hypericin, Fringelit D und dem natürlichenStentor-Pigment verglichen.

     

The crystal and molecular structures of the diastereomeric (4Z)- and (4E)-3-oxo-2,3-dihydrobilatrienes-abc

The molecular and crystal structures of two diastereomeric 3-oxo-2,3-dihydrobilatrienes-abc1 and2 are determined at low (1, 2) and room temperature (2). The configurations at the exocyclic double bond in position 4 are found to be (Z) for1 and (E) for2. Tautomerism, conformation and crystal packing of1 and2 can be understood on the basis of the pattern of intra- and intermolecular hydrogen bonds. Compared to1, a more open helix conformation is found for the (E) diastereomer2. An analysis of crystallographically observed temperature factors of2 yields the result that the highest flexibility is found for the saturated lactam ring.Herrn Prof.Josef Schurz zum 60. Geburtstag gewidmet.     

Stereochemic von Metallocenen, 25. Mitt.1: Über eine Methode zur Analyse des Circulardichroismus der “Ferrocenbande” optisch aktiver Ferrocenderivate

Zusammenfassung Es wird eine Methode zur Auflösung des Circulardichroismus von optisch aktiven Ferrocenderivaten im Bereich der Ferrocenbande in drei Komponenten beschrieben. Sie bedient sich der Auswertung eines Parameternetzes nach der Methode kleinster Quadrate. Ihre Anwendung wird an Hand eines Modells und am Circulardichroismus der (+)--Methyl-ferrocencarbonsäure demonstriert.

---

A method for the analysis of the circular dichroism of the ferrocene band of optically active ferrocene derivatives

A method for resolving the circular dichroism of optically active ferrocene derivatives in the region of the ferrocene band into three components is described. This method makes use of a least square analysis of a parameter-network. Its application is demonstrated with the aid of a model and the circular dichroism of (+)--methyl-ferrocenecarboxylic acid.

---

---

Mit 5 Abbildungen

---

1 24. Mitt. (zugl. 45. Mitt. über Ferrocenderivate):H. Falk, O. Hofer undK. Schlögl, Mh. Chem.100, 624 (1969).