Space diagram

Fig. 5. Langevin trajectories for a harmonic oscillator of angular frequency u = 1 and unit mass simulated by a Verlet-like method (extended to Langevin dynamics) at a timestep of 0.1 (about 1/60 the period) for various 7. Shown for each 7 are plots for position versus time and phase-space diagrams.

Figure 4.23 Synthesis space diagram for a ternary system composed of tetraethylorthosilicate (TEOS), cetyltrimethylammonium bromide (CTAB), and sodium hydroxide (H, hexagonal phase [MCM-41] C, cubic phase [MCM-48] L, lamellar phase [MCM-50] H20/Si02 = 100, reaction temperature 100°C, reaction time 10 days). (Reprinted from Science, Vol. 267, A. Firouzi, D. Kumar, L.M. Bull, T. Besier, R Sieger, Q. Huo, S.A. Walker, J.A. Zasadzinski, C. Glinka, J. Nicol, D.l. Margolese, G.D. Stucky, B.F. Chmelka, Cooperative Organization of Inorganic-Surfactant and Biomimetic Assemblies, pp. 1138-1143. Copyright 1995. With permission of AAAS.)...

Figure 2.16 Temperature-composition space diagram of a ternary isomorphous system. Reprinted, by permission, from F. N. Rhines, Phase Diagrams in Metallurgy. Copyright 1956 by McGraw-Hill Book Co.

Figure 1- Liouville space diagram corresponding to the only term that contributes to the spontaneous light emission from a two-level system within the rotating-wave approximation [Eq. (2.7)]. Here ]g) and e) denote the ground and the excited states, respectively.

Figure 4. TW° ways to describe the time-evolution of the dipole operators single (a) and double (b) Liouville space diagrams.

A simplified parameter space diagram obtained numerically [168] is shown in Fig. 13. The dashed lines bound the region in which both the linear and nonlinear responses of period 1 coexist. The upper line marks the boundary of the linear response, and the lower line marks that for the nonlinear responses. The boundaries of hysteresis for the period 1 resonance are shown by solid lines. The region in which linear response coexists with one or two nonlinear responses of period 2 is bounded by dotted lines. This region is similar to the one bounded by dashed lines. The region of coexistence of the two resonances of period 2 is bounded by the dashed-dotted line. Chaotic states are indicated by small dots. The chaotic state appears as the result of period-doubling bifurcations, and thus corresponds to a nonhyperbolic attractor [167]. Its boundary of attraction Sfl is nonfractal and is formed by the unstable manifold of the saddle cycle of period 1 (SI). 

The mass analyzer works primarily as a velocity filter because the slope in the time-space diagram corresponds to the ion velocity. An ion moving on a line within the light gray and white areas will not be deflected and pass the analyzer, ions with other velocities will be subjected to an orthogonal electrical field and be deflected. Ions of different mass but equal energy are selected according to ... 

Since the dynamic behavior of a reaction-diffusion system may be more apparent with state-space diagrams, the temperature and concentration profiles are replaced with the spatial integral averages obtained from... 

Fig. 23. Space diagram of the copolymerization surface. Copolymer composition as a function of the monomer mixture composition and conversion (S-AN).

X m, m = 0,..., 32, f = 1,2 with the help of a fourth order Runge-Kutta scheme (see, e.g., Milne (1970)). Every single one of the 62 resulting trajectories was followed over 200 cycles of the microwave field, and the values of I and 6 after every completion of a full cycle of the microwave field were plotted as dots in a 6,1) phase-space diagram. The result is shown in Fig. 6.5. Regular and chaotic regions are clearly visible. 

Fig. 16- (a) Plane-spacing diagram, (b) and (c) Orientations of x-ray beams relative to specimen. = normal to specimen surface, Np = normal to reflecting planes. 

Figure 37. Crystal truncation rods (CTRs) for W(IOO) in ultrahigh vacuum. Open circles, experimental data. Solid and dashed lines are, respectively, fits with and without consideration of surface roughness. Inset Reciprocal space diagram. (From Robinson, I. K., Phys. Rev. B. 33, 3830, 1966, with permission.)...

Although the phase space trajectories appear as simple curves on the two-dimensional Iz,ip phase space diagram (the 0 coordinate is suppressed) most trajectories are actually quasiperiodic. The actual trajectories he on the 2-dimensional surface of a 3-dimensional invariant torus in 4-dimensional phase space. Fig. 9.14 shows such a torus. Any point on the surface of the torus is specified by two angles, 0 and. The 0 and circuits about the torus are shown, respectively, as large and small diameter circles. The diameter of the 0... 

The fixed points on the phase space diagrams or phase spheres in Fig. 9.13 are labeled A, B, Ca, and C. Each corresponds to a periodic orbit that is said to organize the surrounding region of phase space that is filled with topologically similar quasiperiodic trajectories. 

Figure 2. X-ray standing wave field formed in a crystal and above its surface by the interference of incident and Bragg-diffracted monochromatic X-ray plane waves. The inset shows the reciprocal space diagram for the Laue condition described by Equation (4).

The analysis can be performed quantitatively by means of methods explained in the section 8.3. We consider individual PFR and CSTR with the pure A as feed of Cao = 1 kmol/m The concentration of all species can be obtained as function of residence time. With this information we can plot a state-space diagram as shown in Figure 8.25. It may be seen that a PFR gives a somewhat higher concentration of B at Ca around 0.15 (conversion 0.85), while CSTR gives an optimum concentration of B at Ca around 0.3 (conversion 0.7). Thus, taking into account the performance of ideal reactors it comes out that a PFR is preferred. Now the question is if we can find a better reaction system. 

Figure 8.11 Correspondence between (a) an energy diagram and (b) a phase space diagram for the pendulum Hamiltonian (Lichtenberg and Lieberman, 1992).

Figure 2.52 Energy versus space diagram for n-type Si in flatband situation (left) and in the accumulation situation as determined by photoelectron spectroscopy (right).

If the three solubility parameter components for dispersion forces 5, dipole forces p, and hydrogen bonds plotted on a three-dimensional space diagram (Fig. 2), a system is obtained in which a vector S is defined for each solvent. The vector describes the solvent s solubility and miscibility behavior [14.28], [14.29]. Solvents that lie close to one another in this space diagram (i.e., whose vector difference is small) have similar solution properties and often a similar chemical structure. Solvents that are far apart on the diagram differ greatly in their chemical and physical characteristics they are generally immiscible [14.30]-[14.32], The solubility parameters as well as their components are shown for some solvents in Table 15. 

The limiting concentrations in a ternary system can be represented not only by a space diagram, but also by a three-coordinate system in the form of an equilateral triangle (Fig. 3). 

Particle state outside the model space are given by railed lines. Diagram (a) is irreducible, valence linked and connected, while (b) is reducible since the intermediate particle states belong to the model space. Diagram (c) is irreducible, valence linked and disconnected. 

Fig. 9.67 Recoil space diagram of the atoms through the interferometer showing the separation (exaggerated) of the atomic wavepackets. The area enclosed by the two paths is proportional to the mean value of acceleration over the path, g. [1275]...

---