Center of mass momentum

Consider again non-relativistic fermions. Their BCS spectrum (for homogeneous systems) is isotropic when the polarizing field drives apart the Fermi surfaces of spin-up and down fermions the phase space overlap is lost, the pair correlations are suppressed, and eventually disappear at the Chandrasekhar-Clogston limit. The LOFF phase allows for a finite center-of-mass momentum of Cooper pairs Q and the quasiparticle spectrum is of the form... 

Figure 8. The dependence of the free energy of the combined DFS and LOFF phases on the center-of-mass momentum of the pairs Q (in units of Fermi momentum pf) and the relative deformations Se for a fixed density asymmetry [18].

For large enough asymmetries the homogeneous state becomes unstable towards formation of either the LOFF phase - a superconducting state with nonzero center-of-mass momentum of the Cooper pairs, or the DFS phase - a superconducting state which requires a quadrapole deformation of Fermi surfaces. A combined treatment of these phases in non-relativistic systems shows that while the LOFF phase corresponds to a local minimum, the DFS phase has energy lower that the LOFF phase. These phases break either the rotational, the translational or both symmetries. 

To describe how the required superposition state [Eq. (7.5)] can be constructed in the laboratory requires some introductory remarks. Note first that Eqs. (7.2) to (7.7) and the E, q, m 0) states are understood to be in the center-of-mass coordinate system and describe the relative translational motion as well as the internal state of A and B. In typical A-B scattering, separating out the center-of-mass motion comes about in a straightforward way. That is, let rA and rB denote the laboratory position of A and B and ikA, Mr denote their laboratory momenta. The relative momentum k, relative coordinate r, center-of-mass momentum K and position Rcm are defined as... 

The first such study was one by Keizer, who considered a hetero-nuclear diatomic in a viscous continuum. He showed that if the two atomic masses differ there is a dynamic coupling between the center of mass motion and the vibrational coordinate. Applying a hydrodynamic model to the reliixation of the center of mass momentum, he gave an expression for the rate of energy transfer from the vibrational degrees of freedom. Keizer used the resulting equation to calculate relaxation rates of heteronuclear... 

However, in the END approach, the projectile scattering angle is obtained by projecting the final projectile center of mass momentum on its initial momentum, i.e.. 

Imagine a well-isolated spaceship observed in a space-fixed coordinate system. Its energy is preserved, its center of mass moves along a straight line with constant velocity (the total, or center-of-mass, momentum vector is preserved), and it preserves the total angular momentum. The same is true for a molecule or atom, but the conservation laws have to be formulated in the language of quantum mechanics. 

The LAB angular distribution of ArD+ at the lowest collision energy is shown in Fig. 8.7. All the ArD+ is scattered at angles close to the direction of the incident Ar+ beam, which is expected since the center-of-mass momentum does not differ greatly from the momentum of the heavy incident (Ar+) or scattered (ArD+) particles. By transforming these data to CM a portrait of the reaction dynamics can be constructed. 

In Eq. 1 n (t) denotes the ntmiber density of the hot atoms P and p, the scalar center-of-mass momentum and reduced mass o(j/k), the total cross section for elastic scattering and inelastic scattering to final qiiantum state k d (j), the detailed state specific total reactive cross section and the primed terms designate energy restoring collisions. The second term, which accounts for hot atom depletion due to reactions, can he integrated over P to yield a time dependent chemical rate equation, ... 

In order to construct the Hamiltonian operator for the hydrogen atom we must express the kinetic energy in terms of momentum components. The center of mass momentum components are... 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

The foregoing discussion of impulse and momentum applies only when no change in rotational motion is involved. There is an analogous set of equations for angular impulse and impulse momentum. The angular momentum about an axis through the center of mass is defined as... 

The requirement I > 2 can be understood from the symmetry considerations. The case of no restoring force, 1=1, corresponds to a domain translation. Within our picture, this mode corresponds to the tunneling transition itself. The translation of the defects center of mass violates momentum conservation and thus must be accompanied by absorbing a phonon. Such resonant processes couple linearly to the lattice strain and contribute the most to the phonon absorption at the low temperatures, dominated by one-phonon processes. On the other hand, I = 0 corresponds to a uniform dilation of the shell. This mode is formally related to the domain growth at T>Tg and is described by the theory in Xia and Wolynes [ 1 ]. It is thus possible, in principle, to interpret our formalism as a multipole expansion of the interaction of the domain with the rest of the sample. Harmonics with I > 2 correspond to pure shape modulations of the membrane. 

Multiparticle collision dynamics can be generalized to treat systems with different species. While there are many different ways to introduce multiparticle collisions that distinguish between the different species [16, 17], all such rules should conserve mass, momentum, and energy. We suppose that the A-particle system contains particles of different species a=A,B,... with masses ma. Different multiparticle collisions can be used to distinguish the interactions among the species. For this purpose we let V 1 denote the center of mass velocity of particles of species a in the cell i ,3... 

The classical kinetic energy of the system has now been separated into the effect of displacement of the center of mass of the system, with momentum P and that of the relative movement of the two particles, with momentum p. In the absence of external forces, the interaction of the two (spherical) particles is only a function of (heir separation, r. That is, the potential function appearing in Eq. (37) depends only on the internal coordinates x, y, z. 

Equation (5) expresses the conservation of linear momentum that defines the position of the center of mass of the molecule, while Eq. (6) is an approximate statement of the conservation of angular momentum of the system These conditions, which are usually attributed to Eckart lead to the relation... 

In reality, the slip velocity may not be neglected (except perhaps in a microgravity environment). A drift flux model has therefore been introduced (Zuber and Findlay, 1965) which is an improvement of the homogeneous model. In the drift flux model for one-dimensional two-phase flow, equations of continuity, momentum, and energy are written for the mixture (in three equations). In addition, another continuity equation for one phase is also written, usually for the gas phase. To allow a slip velocity to take place between the two phases, a drift velocity, uGJ, or a diffusion velocity, uGM (gas velocity relative to the velocity of center of mass), is defined as... 

In Eqs. (5.1) and (5.2), m is the reduced mass of the colliding system, V is the interaction potential at ion-molecule separation r, 6 is the angle between the direction of r and the center-of-mass velocity, and the dot indicates differentiation with respect to time. Integration of (5.1) just gives the angular momentum L, which is conserved in the collision. Substitution in (5.2) gives... 

The reaction complex that is formed has angular momentum about the center of mass, which is conserved during the entire reaction. Now the products can... 

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