Noncollinearity

Wilhelm T, Piel J and Riedle E 1997 Sub-20 fs pulses tunable across the visible from a blue-pumped single-pass noncollinear parametric converter Opt. Lett. 22 1494-6... 

Fig. 2. Stmctures of acousto-optic tunable filters (a) collinear and (b) noncollinear.

Other researchers have substantially advanced the state of the art of fracture mechanics applied to composite materials. Tetelman [6-15] and Corten [6-16] discuss fracture mechanics from the point of view of micromechanics. Sih and Chen [6-17] treat the mixed-mode fracture problem for noncollinear crack propagation. Waddoups, Eisenmann, and Kaminski [6-18] and Konish, Swedlow, and Cruse [6-19] extend the concepts of fracture mechanics to laminates. Impact resistance of unidirectional composites is discussed by Chamis, Hanson, and Serafini [6-20]. They use strain energy and fracture strength concepts along with micromechanics to assess impact resistance in longitudinal, transverse, and shear modes. 

A quadrilateral is a four-sided polygon determined by four coplanar points (three of which are noncollinear), if the line segments thus formed intersect each other only at their end points, forming four angles. 

Anton, J., Ericke, B. and Engel, E. (2004) Noncollinear and collinear relativistic density-functional program for electric and magnetic properties of molecules. Physical Review A, 69, 012505-1-012505-10. 

Figure 6.2 A fourth-order coherent Raman spectrometer constructed with a Ti sapphire regenerative amplifier (Ti sapphire) and noncollinear optical parametric amplifier (NOPA).

Defects in ferrimagnetic structures often lead to noncollinear (canted) spin structures. For example, a diamagnetic substitution or a cation vacancy can result in magnetic frustration which leads to spin-canting such that a spin may form an angle 6c with the collinear spins in the sample [80, 81]. Similarly, the reduced number of neighbor ions at the surface can also lead to spin-canting [80-83]. 

It is worth mentioning that parameter p is insensitive, to first order approximation, to modulation of the residue-specific 15N chemical shift anisotropy tensor and/or dipolar interaction, as the (d2 + c2) term in the R) / R ratio is canceled out. The noncollinearity of the CSA and dipolar tensors will require corrections to Eqs. (10) and (12) for high degrees of rotational anisotropy (D /D > 1.5), as described in detail in Ref. [22]. 

Although there are no slits associated with an AOTF device, there are some necessary compromises regarding throughput. First, there are physical restrictions on the size of available crystals, and in the case of the normal noncollinear arrangement where the inpnt beam direction is traversed by the acoustic wave front, then the open beam apertnre needs to be limited so that the transit time of the aconstic wave front across the crystal is limited, in order that a narrow optical band pass can be maintained. Typically this is of the order of a few millimeters. 

If we refrain from such a restriction and consider a spin-operator-dependent Hamiltonian, such as the 4-component KS Hamiltonian or the Dirac-Coulomb Hamiltonian, the Hamiltonian does not commute with the square of the spin operator. The square of the spin operator and the Hamiltonian then do not share the same set of eigenfunctions, and hence spin is no longer a good quantum number. In this noncollinear framework we must therefore find a different solution and may define a spin density equal to the magnetization vector (32). 

Vectors Ar, ,-, form an orthogonal trihedron. Henceforth, for a real vector Aj (k) noncollinear to k, the solution of the set of equations written above is... 

As reported by Blanco et al. (1999), neutron diffraction patterns of powder and bulk polycrystalline samples of GdCu were obtained for both structures in the cubic CsCl type of structure, which orders antiferromagnetically at 7n 150 K, a propagation vector of ( j 0) has been found with the moments probably parallel to the c-axis (note that other noncollinear magnetic structures might give rise to the same neutron-diffraction pattern). In the orthorhombic low temperature phase (7n 45 K) the available diffraction patterns... 

We performed transient absorption measurements on BP(OH>2 with a spectrometer based on two noncollinearly phase matched optical parametric amplifiers (NOPAs) pumped by an homebuilt regenerative Ti Sapphire laser system or a CPA 2001 (Clark-MXR) [1,7]. The tunable UV pump pulses are generated by frequency doubling the output of one of the NOPAs. The other NOPA provides the visible probe pulses. The cross correlation between pump and probe pulses has a typical width (FWHM) of 40 fs. The sample is a cyclohexane solution of BP(OH)2 pumped through a flow cell with a 120 pm thick channel. 

Transient spectra of solvated indole are measured in a 120 Jim liquid jet with a crosscorrelation of 80 fs by means of a white light continuum (450 - 740 nm) generated in a sapphire disc. The molecules are excited at 270 nm with pump pulses generated by frequency doubling the output of a noncollinearly phase matched optical parametric amplifier [2], Due to the short pump pulses there is a small yet finite probability for two-photon ionization in pure solvents. This allows us to study the spectral properties of the generated solvated electrons by measurements in pure solvents. The transient spectra of the indole solution are corrected for these solvent contributions. 

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