Sample coordinate system

Sample coordinate system sl coordinate system fixed to the sample and used to specify position on the sample surface for measurement. Note The sample coordinate system is application and sample specific. The Cartesian coordinate system shown in Fig. A1 is recommended for flat samples. The origin is at the geometric center of the sample face with the Z axis normal to the sample. A fiducial mark must be shown at the... 

Incident azimuth angle (< j) the fixed 180° angle from the XB axis to the projection of the incident direction onto the XB-YB plane. Note It is convenient to use a beam coordinate system (see Fig. A2) in which = 180° because this makes the correct angle to use directly in the familiar form of the grating equation. Conversion to a sample coordinate system is straight forward, provided the sample location and rotation are known ... 

The Z and ZB axes are always the local normal to the sample face. Locations on the sample face are measured in the sample coordinate system. The incident and scatter directions are measured in the beam coordinate system. If the sample fiducial mark is not an X axis mark, the intended value must be indicated on the sample (Fig. Al). 

The orthogonal coordinate systems Si,S2,S3 and L],L2,L3, are defined as follows. S3 is perpendicular to the sample surface and S2 and S3 are parallel to the sample surface. L3 is the normal to the scattering plane hkl and makes the angle ip with S3. The (f> angle is the angle between the projection of L3 onto the sample surface and vector Sj. The measured quantity can then be converted to the sample coordinate system by the transformation ... 

The stress (a) and strains (c) in the sample coordinate system are determined using Hooke s law ... 

We assume that in (4.38) and (4.39), all velocities are measured with respect to the same coordinate system (at rest in the laboratory) and the particle velocity is normal to the shock front. When a plane shock wave propagates from one material into another the pressure (stress) and particle velocity across the interface are continuous. Therefore, the pressure-particle velocity plane representation proves a convenient framework from which to describe the plane Impact of a gun- or explosive-accelerated flyer plate with a sample target. Also of importance (and discussed below) is the interaction of plane shock waves with a free surface or higher- or lower-impedance media. 

Figure 1 shows the detailed steps of the measurement, from the perspective of a coordinate system rotating with the applied radiofrequency giq = yA)- The sample is in the magnetic field, and is placed inside an inductor of a radiofirequency circuit... 

The point at which the sample is spotted can be regarded as the origin of a coordinate system (9). The process of development is performed in two steps the first in the direction of the x-axis to a distance L. After evaporation of the solvents used, the second development will be performed in the direction of the y-axis to a distance L,. The positions of the compounds after development in the x-direction depend on the... 

We have referred to the various interactions which can cause line broadening in the solid state. One of these, which is normally not a problem in liquid state NMR, is due to the fact that the chemical shift itself is a tensor, i.e. in a coordinate system with orthogonal axes x, y and z its values along these axes can be very different. This anisotropy of the chemical shift is proportional to the magnetic field of the spectrometer (one reason why ultra-high field spectrometers are not so useful), and can lead in solid state spectra to the presence of a series of spinning sidebands, as shown in the spectra of solid polycrystalline powdered triphenylphosphine which follows (Fig. 49). In the absence of spinning, the linewidth of this sample would be around 75 ppm ... 

Thermoplastic polymer macromolecules usually tend to become oriented (molecular chain axis aligns along the extrusion direction) upon extrusion or injection moulding. This can have implications on the mechanical and physical properties of the polymer. By orienting the sample with respect to the coordinate system of the instrument and analysing the sample with polarised Raman (or infrared) light, we are able to get information on the preferred orientation of the polymer chains (see, for example, Chapter 8). Many polymers may also exist in either an amorphous or crystalline form (degree of crystallinity usually below 50%, which is a consequence of their thermal and stress history), see, for example, Chapter 7. 

Because the orientation of the reciprocal space coordinate system is rigidly coupled to the orientation of the real-space coordinate system of the sample, the reciprocal space can be explored8 by tilting and rotating the sample in the X-ray beam (cf. Chap. 9). 

Figure 10.1. USAXS observation during straining of an SBS block copolymer. Right monitor Intensity maxima on an ellipse. Raw-data coordinate system (x,y) and radial cuts for data analysis are indicated. Middle Videotaping of sample. Left Stress-strain curve. Control booth of beamline BW4, HASYLAB, Hamburg...

The charge density of dust transported through ducts and the resultant electric fields at the duct Inner walls was monitored by a Monroe Electronics Inc., Model 171 electric fieldmeter. All the electrostatic sampling In the field was performed In circular cross-section ducts. Thus, the electrostatic field Intensity, for this geometry, can be determined from Poisson s equation using the cylindrical coordinate system. 

Figure 7. A "snapshot" of a typical cellulosic chain trajectory taken from a Monte Carlo sample of cellulosic chains, all based on die conformational energy map of Fig. 6. Filled circles representing glycosidic oxygens, linked by virtud bonds spanning the sugar residues (not shown), allow one to trace the instantaneous chain trajectory in a coordinate system that is rigidly fixed to the residue at one end of the chain. Projections of the chain into three mutually orthogonal planes assist in visualization of the trajectory in three dimensions.

Scores Pla (Sample Diagnostic) The scores are the coordinates of the samples in the new coordinate system where the axes are defined by the principal components. These new axes are used to view the relevant variation in the data see m a smaller number of dimensions. The plot reveals how the samples arc rela d to each other given the measurements that have been made. Samples that are close to each other on a given score plot are similar with respect to the original measurements provided the plot displays a sufficient amount of the total variation. This mathematical proximity translates to chemical similaritySmeaningful measurements have been made. 

Fig. 3.1. Derivation of the tunneling matrix elements. A separation surface is placed between the tip and the sample. The exact position and the shape of the separation surface is not important. The coordinates for the Cartesian coordinate system and spherical coordinate system are shown, except y and 4>. (Reproduced from Chen, 1990a, with permission.)...

In the following, we show that the coefficients a , in Eq. (3.31) are related to the derivatives of the sample wavefunction i ) with respect to X, y, and z at the nucleus of the apex atom in an extremely simple way. (To simplify the notation, we take the nucleus of the apex atom as the origin of the coordinate system, i.e., xo = 0, yo = 0, and zo - 0.) This is similar to the well-known case that the expansion coefficients for a power series are simply related to the derivatives of the function at the point of expansion, the so-called Taylor series or MacLaurin series. We will then obtain the derivative rule again, from a completely different point of view. 

To determine the image, the first step is to determine the distribution of tunneling current as a function of the position of the apex atom. We set the center of the coordinate system at the nucleus of the sample atom. The tunneling matrix element as a function of the position r of the nucleus of the apex atom can be evaluated by applying the derivative rule to the Slater wavefunctions. The tunneling conductance as a function of r, g(r), is proportional to the square of the tunneling matrix element ... 

If the calibration curve is linear and the origin of the coordinate system lies within the 95% conhdence limits of the curve, the use of one standard concentration (equal to the expected sample concentration) is allowed for the analysis. 

X A) is related to the calibration samples and is termed the matrix of scores and E (/ x J) is the part of the data that is not modelled by the A factors. As mentioned above, A is the number of latent variables that are retained in the model. The loadings in P define a new coordinate system (a rotation of the original axis of the measured variables) and the scores T are the coordinates of the samples in this new coordinate system (see Figure 3.4 for a brief description). 

Let us suppose that we place a polarizing prism between the light source and the sample. If the sample is a single crystal in which all of the molecules have the same orientation relative to the crystallographic axes, we can so orient the crystal that the direction of the electric vector of the light will correspond to the x, y, or z direction in a coordinate system for the molecule. It is then possible that some transition may occur for only one or two of these orientations but not for all three. 

Consider a lamellar mesophase, being macroscopically aligned so that the symmetry axis, referred to as the director, has the same direction throughout the sample. If the transformation from the molecular coordinate system to the laboratory system is performed via the director coordinate system (D), Equation 2 reads... 

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