Measurement of the flexibility of wet cellulose fibres using atomic force microscopy

Flexibility and modulus of elasticity data for two types of wet cellulose fibres using a direct force–displacement method by means of AFM are reported for never dried wet fibres immersed in water. The flexibilities for the bleached softwood kraft pulp (BSW) fibres are in the range of 4–38 × 10¹² N^?1 m^?2 while the flexibilities for the thermomechanical pulp (TMP) fibres are about one order of magnitude lower. For BSW the modulus of elasticity ranges from 1 to 12 MPa and for TMP between 15–190 MPa. These data are lower than most other available pulp fibre data and comparable to a soft rubber band. Reasons for the difference can be that our measurements with a direct method were performed using never dried fibres immersed in water while other groups have employed indirect methods using pulp with different treatments.     

Transferable integrals in a deformation density approach to crystal orbital calculations. IV. Evaluation of angular integrals by a vector-pairing method

A general method for performing angular integrations is presented. The method depends on the fact that the integral must be invariant under rotations of the coordinate system, and it also makes use of combinatorial analysis. In most cases the method presented is computationally much faster than alternative methods of angular integration using Condon–Shortley coefficients. Applications to charge density analysis and Fourier transforms are discussed, and a general formula for the action of angular momentum projection operators on functions of the Cartesian coordinates is derived. A general angular integration formula for an m-dimensional space is also given.     

Transferable integrals in a deformation density approach to crystal calculations. I. Crystal harmonics and their properties

The term “crystal harmonic” is introduced to denote a symmetrized plane wave in the special case where the wave vector is a reciprocal lattice vector. Crystal harmonics, thus defined, have the translational symmetry of the lattice, and they also have the transformation properties of the irreducible representations of the crystal's point group. An expansion is derived expressing crystal harmonics in terms of spherical Bessel functions and in terms of the functions ??_??,ξ (eigenfunctions of L² which are also basis functions for IRS of the crystal's point group). A sum rule for the functions ??_??,ξ is derived. Methods are given for expanding periodic functions of special symmetry in terms of crystal harmonics. Methods are also presented for calculating matrix elements of the potential in a crystal using crystal harmonics as a basis and for transforming to a STO basis. It is shown that the invariant component of the product of two crystal harmonics can be expressed as a sum of a few invariant crystal harmonics, and expressions for the coefficients in the sum are derived. Orthogonality with respect to summation over networks of points and normalization are also discussed. The properties mentioned above are illustrated in detail in the case of cubic crystals with point group O_h.