Linear momentum transfer

Table 1. Linear momentum transfer in gas/surface collisions.

Solution of Newton s equations of Fj g as a force completes the description of the particle dynamics. It is clear, however, that small heavy particles cannot be moved by the interface (see electron microscopy data). The maximum radius has been estimated from the conservation of linear momentum. To capture a particle of mass (m), the interface has to transfer to it a linear momentum (mv). If we assume that the particle does not move (or it moves much more slowly than the interface, which is valid for massive particles), then the total linear momentum transferred to the particle is... 

We first examine the reiationship between particie dynamics and the scattering of radiation in the case where both the energy and momentum transferred between the sampie and the incident radiation are measured. Linear response theory aiiows dynamic structure factors to be written in terms of equiiibrium flucmations of the sampie. For neutron scattering from a system of identicai particies, this is [i,5,6]... 

The problems experienced in drying process calculations can be divided into two categories the boundary layer factors outside the material and humidity conditions, and the heat transfer problem inside the material. The latter are more difficult to solve mathematically, due mostly to the moving liquid by capillary flow. Capillary flow tends to balance the moisture differences inside the material during the drying process. The mathematical discussion of capillary flow requires consideration of the linear momentum equation for water and requires knowledge of the water pressure, its dependency on moisture content and temperature, and the flow resistance force between water and the material. Due to the complex nature of this, it is not considered here. 

In the laminar sub-layer, turbulence has died out and momentum transfer is attributable solely to viscous shear. Because the layer is thin, the velocity gradient is approximately linear and equal to Uj,/Sb where m is the velocity at the outer edge of a laminar sub-layer of thickness <5 (see Chapter ll). 

Fig. 4.15 Momentum transfer (Q)-dependence of the characteristic time r(Q) of the a-relaxation obtained from the slow decay of the incoherent intermediate scattering function of the main chain protons in PI (O) (MD-simulations). The solid lines through the points show the Q-dependencies of z(Q) indicated. The estimated error bars are shown for two Q-values. The Q-dependence of the value of the non-Gaussian parameter at r(Q) is also included (filled triangle) as well as the static structure factor S(Q) on the linear scale in arbitrary units. The horizontal shadowed area marks the range of the characteristic times t mr- The values of the structural relaxation time and are indicated by the dashed-dotted and dotted lines, respectively (see the text for the definitions of the timescales). The temperature is 363 K in all cases. (Reprinted with permission from [105]. Copyright 2002 The American Physical Society)...

We now see that for a real proton the charge radius contribution has exactly the form in (6.3), where the charge radius is defined in (6.7). The only other term linear in the momentum transfer in the photon-nucleus vertex in (6.9)... 

The transfer of heat and/or mass in turbulent flow occurs mainly by eddy activity, namely the motion of gross fluid elements that carry heat and/or mass. Transfer by heat conduction and/or molecular diffusion is much smaller compared to that by eddy activity. In contrast, heat and/or mass transfer across the laminar sublayer near a wall, in which no velocity component normal to the wall exists, occurs solely by conduction and/or molecular diffusion. A similar statement holds for momentum transfer. Figure 2.5 shows the temperature profile for the case of heat transfer from a metal wall to a fluid flowing along the wall in turbulent flow. The temperature gradient in the laminar sublayer is linear and steep, because heat transfer across the laminar sublayer is solely by conduction and the thermal conductivities of fluids are much smaller those of metals. The temperature gradient in the turbulent core is much smaller, as heat transfer occurs mainly by convection - that is, by... 

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 ... 

As co) increases, the experimental scattering lobe is no longer symmetric to Zcol but turns its symmetry axis essentially into the direction of the momentum-transfer vector K=kout—kjn as indicated by 9 in Fig. 32. In Born s approximation the lobe would be exactly symmetric to K. This reflects the fact that electronic angular momentum of the atom is transferred into linear momentum of the scattered electron. 

Summary. We demonstrate that in a wide range of temperatures Coulomb drag between two weakly coupled quantum wires is dominated by processes with a small interwire momentum transfer. Such processes, not accounted for in the conventional Luttinger liquid theory, cause drag only because the electron dispersion relation is not linear. The corresponding contribution to the drag resistance scales with temperature as T2 if the wires are identical, and as T5 if the wires are different. 

To conclude, the small momentum transfer contribution dominates Coulomb drag at almost all temperatures if the distance between the wires exceeds the Fermi wavelength, see Fig. 2. Drag by small momentum transfer is possible because electron dispersion relation is not linear, and therefore can not be accounted for in the conventional Tomonaga-Luttinger model. 

The importance of the universal velocity profile is discussed in Section 12.4. From equation 12.18, for isotropic turbulence, the eddy kinematic viscosity, EaXEuE where XE is the mixing length and uE is some measure of the linear velocity of the fluid in the eddies. The momentum transfer rate per unit area in a direction perpendicular to the surface at position y is then ... 

In the dipole approximation and for linearly polarized incident light, the natural reference axis is the electric field vector, the energy transfer is equal to the photon energy, and the momentum transfer is negligible. 

In the impulse model, the excess energy Ek is transferred to an NO molecule as the momentum p0 given only to an N atom. Here, p0 is normal to the surface and Ek = p /2m, where m is mass of the N atom. Recoil of substrate Pt atoms can be ignored, because the mass of a Pt atom is much larger than that of an N atom. After desorption the momentump0 is converted to the linear momentum of the center of mass, P, and the linear momentum of the internal coordinate, p. A relationship p0 = m dri/df is satisfied in the impulse model and it can be approximated to dr2/df = 0 at the moment of the Pt-N bond breaking, where and r2 are the position vectors of N and O atoms, respectively, in an adsorbed NO molecule. 

The H + X2 reactions give non-linear surprisal plots for vibrational and translational energy disposal and are approximately quadratic in form. By including an additional minimal-momentum transfer or Frank—Condon-like constraint [250, 251], the translational distributions are reproduced by a surprisal that is Gaussian in momentum. A corresponding vibrational form can be derived [241]. 

Here, we review tight-binding models that account for the above-mentioned ET yield asymmetries in the context of the more general phenomenon of current transfer [38 -01. Current transfer is charge transfer where the transferred charge carrier maintains at least some of its linear and/or angular momentum (phase). A recent photoemission experiment [41] demonstrated current transfer a biased linear momentum distribution on a Cu (100) surface was established based on the angular distribution of the photoemitted current. 

In linear motion, we are concerned with the momentum p = mv of an object as it heads toward a particular point the linear momentum measures the impact that the object can transfer in a collision as it arrives at the point. To extend this concept to circular motion, we define the angular momentum of an object as it revolves around a point as L = mvr. This is in effect the moment of the linear momentum over the distance r, and it is a measure of the torque felt by the object as it executes angular motion. The angular momentum of an electron around a nucleus is a crucial feature of atomic structure, which is discussed in Chapter 5. 

Cluster-assembled carbon film are very porous with a pore diameter peaked at 3-4 nm, as shown by adsorption/desorption isotherm analysis [32]. The gases used in our experiments have a molecular or atomic size much smaller than the average pore size, so it is reasonable to assume that they equally diffuse in the mesoporous film network and interact with the same amount of linear carbon structures. This explains the similar Rq value observed for the three gases. The time required to reach the asymptotic value of is affected by the momentum transferred from the molecule to the film network during collisions, hence by the mass of the gas molecule. We note that H2 does not seem to chemically interact with sp structures. 

Fig. 3. The solid line represents the RPA (linear) contribution to the stopping power of equation (15), for Zj = 1 andr = 2.07, as a function of the velocity of the probe particle. Dashed and dotted lines represent contributions from momentum transfers below q < q ) and above (q > q ) the critical momentum q where the plasmon dispersion enters the e-h pair continuum.

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