Momentum longitudinal

In this section we consider the final-state longitudinal momentum distributions of the ejected electron, recoil-ion, and projectile momentum transfer in the single... 

Normally these conditions are satisfied in fast highly charged ion-atom collisions. From Eq. (66) we can derive the equations for the singly differential cross sections with respect to the components of the longitudinal momentum distributions for the electron, recoil-ion, and projectile. The longitudinal electron momentum distribution da/dpe for a particular value of p, may be derived by integrating over the doubly differential cross section with respect to the electron energy Ek ... 

The longitudinal momentum projectile transfer da / dp P obtained by consideration of Eq. (66) is expressed as a function of the singly differential cross section of the emitted electron from the projectile ion by... 

Figure 1. Calculated longitudinal momentum distributions for the ejected electron in single ionization of He by 3.6-MeV/amu Ni24+ ions. Experimental data are from Moshammer et al. [2], Theoretical results CDW results [20], CDW-EIS1 results [20], CDW-EIS2 results [48].

Figure 2. Same as Fig. 1 but for the longitudinal momentum distribution of the recoil ion. 

Figure 7. Longitudinal momentum distributions of the emitted electron, the recoiling target ion, and the projectile after the single ionization of helium by 3.6-MeV/amu Au53+ ions. Experimental data are from Schmitt et al. [50]. Also shown are CDW-EIS calculations [50] and CTMC calculations [50].

Figure 8. Longitudinal momentum distribution for single ionization of helium by 945-keV antiproton (data points) in comparison with proton collision (full curve), (a) Electron momentum data [26] (b) recoil-ion data [26], The theoretical calculations represent antiproton collisions dotted curve, CDW results [26] broken curve, CTMC result [26], Here pze and pzr are equivalent to the notation of pey and pRy of Figs. 1 and 2, respectively.

It was also found that the CDW-EIS theory was successful in describing the experimental results for longitudinal momentum distributions of fast highly charged projectile ions by neutral target atoms produced with reaction microscopy technique. ... 

We assume that the opening is infinitely narrow and the outflow velocity of the fluid is so large that the longitudinal momentum of the jet remains finite,... 

Dv is called the longitudinal kinematic viscosity, and Dt is called the thermal diffusivity. These quantities are diffusion coefficients for the diffusion of longitudinal momentum and heat, respectively. 

Because of reflection symmetry, the set A separates into three uncoupled subsets 0(q), g z(q)), gx(q), and gy(< ). gz represents the particle flux along q. This flux is called the longitudinal momentum. gx(q) and gy < ) represent the particle flux perpendicular or transverse to q. These fluxes are called the transverse momentum. The longitudinal and transverse fluxes in an isotropic system of optically inactive molecules are therefore uncoupled. Here we have derived a result which is well known in hydrodynamics. 

We see that in addition to the convective, pressure, and viscous terms, we have an additional term, which is the gradient of the nonlinear term pu v, which represents the average transverse transport of longitudinal momentum due to the turbulent fluctuations. It appears as a pseudo-stress along with the viscous stress pdUldy, and is called the Reynolds stress. This term is usually large in most turbulent shear flows (Lieber and Giddens, 1988). 

According to the arguments presented in Section 4 it follows that the relaxation equations for these four variables consist of two coupled equations for n (g) and g (q) and two totally uncoupled equations for (q) and gy (q). This breakup is a consequence of our choice of coordinate axes such that the wave vector q defines the z axis, g (q) is the momentum density parallel to q, namely the longitudinal momentum, and g iq) and gy(q) are the momentum densities perpendicular to q, namely the transverse components of the momentum density. By symmetry, gz(q),gx(q), and gy(q) are independent, and furthermore gxiq) and gy q) should be dynamically equivalent. Let us therefore only treat g ,(q). 

Thus the longitudinal momentum propagates with propagation frequency (t>s(q) and damping constant r j(q)/2. Solution of the phenomenological Navier-Stokes equations gives the same behavior with... 

Experimentally the cross-section is usually given into a range oiq =Tn and a range of longitudinal momentum fractions denoted by ... 

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