Momentum-Angle Correlation

Vertex detection precision close to the beam direction is made difficult by the increased multiple Coulomb scattering caused by the passage of charged particles through increased amounts of material (in the beam pipe and first layer of vertex detection) at small polar angles. The best strategy is to make the first 2 measurement as close as possible to the initial scatter in the beam pipe. If this is done, studies show that the detailed arrangement of silicon vertex detector planes has little effect on z resolution (see Chapter 2). 

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There is a momentum-angle correlation, (see Figure 1.6), that results in higher momentum particles at forward (in the direction of the high-energy beam) angles. 

Inequality (6.67) is the softest criterion of perturbation theory. Its physical meaning is straightforward the reorientation angle (2.30) should be small. Otherwise, a complete circle may be accomplished during the correlation time of angular momentum and the rotation may be considered to be quasi-free. Diffusional theory should not be extended to this situation. When it was nevertheless done [268], the results turned out to be qualitatively incorrect orientational relaxation time 19,2 remained finite for xj —> 00. In reality t0j2 tends to infinity in this limit [27, 269]. 

Figure 13. Density plot of the correlation between the deflection angle, 0, and the total angular momentum, J, for 1-TS (open diamonds) and 2-TS (circles) trajectories at 2.3-eV total energy. Note, only 5000 trajectories are plotted for each type for clarity.

Reasonable correlations of combustion efficiency with fuel spray momentum and spray energy in two different combustors have been shown to hold over a range of altitude-engine idling conditions 133). As different curves were obtained with different injector nozzles, spray-cone angle was thought to be a factor. Further work showed that efficiency did correlate closely with expressions representing the spray momentum or... 

Example Problem Consider the case of 240-MeV 32S interacting with 181Ta, which fissions. What would be the laboratory correlation angle between the fragments if the full linear momentum of the projectile was transferred to the fissioning system ... 

From the covariance map, in which the covariance coefficients [17] are plotted on a two-dimensional plane, we can identify the respective explosion pathways and show how the momentum components along the TOF axis of the different ion species are correlated. The characteristic patterns appearing in the double covariance map for the ion pairs (0+, 0+), (0+, C+) and (0+, 02+) ejected from C02 at the field intensity of 1015 W/cm2 were interpreted in terms of a large amplitude bending motion [18], where the widths of the bond angle distribution were found to be constant different charge states (z = 3-6). 

This correlation can be described in terms of the geometrical structure of CS3+ just before the Coulomb explosion, which is defined as the geometrical structure that reproduces the observed momentum vectors of the three fragment ions when the internuclear potentials are assumed to be purely Coulombic. The geometrical structures of CS determined from the distribution peaks (marked by a cross) in the three Ap12 — 012 plots in Fig. 1.8b-d show that the S-C-S bond angle becomes smaller (dropping from 162° to 140°) as the two C-S bonds stretch simultaneously from 1.8 A to 3.0 A [23]. 

Catapano et al. [35] studied the mass and momentum transport in hollow fiber BOs. Water and oxygen were used as the liquid and gas phases, respectively. They investigated the effect of the shell void fraction (e) and the membrane angle with respect to the main direction of liquid flow (4>). They proposed the following correlations [35] ... 

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