Quasi particles

The fourth order perturbation energy (8.21) clearly equals the first term of a more general representation of the free energy of attraction according to Eqs. (3.48), (3.49). The respective dispersion function G( ) is split into four subdeterminants G( +, ) corresponding to the order of emission and absorption of photons q and r, 

We noted that the energy terms (8.8) and (8.12) of order zero and two do not depend on the separation of particles 1 and 2. Accordingly, we calculated the free energy of interaction from the fourth order energy expression (8.16). Considering this term, it is certainly consistent to calculate the average occupation numbers of the excited states from Fermi and Bose statistics. The error is at worst of order six in the interaction parameter. 

However, since the statistic weight of the excited states is affected by the electron-photon interaction, there might well arise a free energy of attraction from the energy terms of order zero and two. We have to check whether with decreasing separation there is a redistribution of occupied states, which gives rise to an energy of interaction of order four in U kiq) as well. 

Let us now check whether the need for independent particles is more appropriately met by considering quasi-particles rather than interacting electrons and photons. We noted in Section 8.2 that the energy levels corresponding to the perturbed states (8.9) are equidistant with respect to the photon occupation numbers n,. This result holds in spite of the fact that the energy terms of order four arise from two-photon exchange interactions, suggesting that at least the quasi-photons represent a system of independent particles. 

The perturbed states (8.9) arise from the unperturbed states (8.7) by a one-to-one correspondence, i.e. they are characterized by the occupation numbers / and n, of the electron states i and of the photon states q in the same way as the unperturbed states. Accordingly, we may introduce quasi-electron creation and annihilation operators and C, and quasiphoton creation and annihilation operators and A, in a fully analogous manner to the unperturbed case. We require 

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Our results fit also with a previous investigation (9) on polyenes based on a version of the 2h-lp Cl scheme restricted to the virtual one-electron states generated by a minimal basis. In our case, however, the fragmentation of lines into satellites is much more pronounced. The reason lies in the size-consistency of the ADC[3] approach (as contrasted with the size-inconsistency of any truncated form of Cl (27d), in the full handling of the virtual space, and (10) in the inclusion of correlation corrections to the reference ground state, leading to (37) a net reduction of the quasi-particle band gap of conjugated polymers. 

One obvious drawback of the LDA-based band theory is that the self-interaction term in the Coulomb interaction is not completely canceled out by the approximate self-exchange term, particularly in the case of a tightly bound electron system. Next, the discrepancy is believed to be due to the DFT which is a ground-state theory, because we have to treat quasi-particle states in the calculation of CPs. To correct these drawbacks the so-called self-interaction correction (SIC) [6] and GW-approximation (GWA) [7] are introduced in the calculations of CPs and the full-potential linearized APW (FLAPW) method [8] is employed to find out the effects. No established formula is known to take into account the SIC. 

In the present calculation the SIC potential is introduced for each angular momentum in a way similar to the SIC one for atoms [9]. The effects of the SIC are examined on the CPs of three materials, diamond, Si and Cu compared with high resolution CP experiments except diamond [10, 11]. In order to examine the quasi-particle nature of the electron system, the occupation number densities of Li and Na are evaluated from the GWA calculation and the CPs are computed by using them [12, 13]. 

As noticed from this expression, the CP calculation has to be basically carried out on the quasi-particle picture. Formally, quasi-particle energies and wave functions have to be evaluated by solving... 

As mentioned in Section 2, the CPs of solids have to be calculated on the quasi-particle scheme. In order to calculate the quasi-particle states, non-local and energy-dependent self-energy in Equation (13) must be evaluated in a real system. In practice, the exact self-energy for real systems are impossible to compute, and we always resort to approximate forms. A more realistic but relatively simple approximation to the selfenergy is the GWA proposed by Hedin [7]. In the GW A, the self-energy operator in Equation (12) is... 

It has been suggested that quasi-particle wave functions do not deviate much from LDA wave functions [26], Furthermore, in the evaluation of momentum densities shown in Figure 9, the characteristics of the quasi-particle states dominantly reflect on the occupation number densities which should be evaluated by using the general quasi-particle Green s function. In GWA, however, the corresponding occupation number densities are... 

This quasi-particle approach for CPs has been performed on Li and Na [12, 13]. In these materials, only diagonal terms of the occupation number densities are evaluated in a reasonable justification [27]. The GWA occupation number densities (denoted as N(GWA)) thus obtained are shown for the three principal directions in Figures 7 and 8 for Na and Li, respectively. For reference, the occupation number densities obtained... 

Since there is no good physical framework in which the measured hardness versus temperature data can be discussed, descriptions of it are mostly empirical in the opinion of the present author. Partial exceptions are the elemental semiconductors (Sn, Ge, Si, SIC, and C). At temperatures above their Debye temperatures, they soften and the behavior can be described, in part, in terms of thermal activation. The reason is that the chemical bonding is atomically localized in these cases so that localized kinks form along dislocation lines. These kinks are quasi-particles and are affected by local atomic vibrations. 

It is important to realize that each of the electronic-structure methods discussed above displays certain shortcomings in reproducing the correct band structure of the host crystal and consequently the positions of defect levels. Hartree-Fock methods severely overestimate the semiconductor band gap, sometimes by several electron volts (Estreicher, 1988). In semi-empirical methods, the situation is usually even worse, and the band structure may not be reliably represented (Deak and Snyder, 1987 Besson et al., 1988). Density-functional theory, on the other hand, provides a quite accurate description of the band structure, except for an underestimation of the band gap (by up to 50%). Indeed, density-functional theory predicts conduction bands and hence conduction-band-derived energy levels to be too low. This problem has been studied in great detail, and its origins are well understood (see, e.g., Hybertsen and Louie, 1986). To solve it, however, requires techniques of many-body theory and carrying out a quasi-particle calculation. Such calculational schemes are presently prohibitively complex and too computationally demanding to apply to defect calculations. 

In the above relation, quantum states of phonons are characterized by the surface-parallel wave vector kg, whereas the rest of quantum numbers are indicated by a the latter account for the polarization of a quasi-particle and its motion in the surface-normal direction, and also implicitly reflect the arrangement of atoms in the crystal unit cell. A convenient representation like this allows us to immediately take advantage of the translational symmetry of the system in the surface-parallel direction so as to define an arbitrary Cartesian projection (onto the a axis) for the... 

When metals have Raman active phonons, optical pump-probe techniques can be applied to study their coherent dynamics. Hase and coworkers observed a periodic oscillation in the reflectivity of Zn and Cd due to the coherent E2g phonons (Fig. 2.17) [56]. The amplitude of the coherent phonons of Zn decreased with raising temperature, in accordance with the photo-induced quasi-particle density n.p, which is proportional to the difference in the electronic temperature before and after the photoexcitation (Fig. 2.17). The result indicated the resonant nature of the ISRS generation of coherent phonons. Under intense (mJ/cm2) photoexcitation, the coherent Eg phonons of Zn exhibited a transient frequency shift similar to that of Bi (Fig. 2.9), which can be understood as the Fano interference [57], A transient frequency shift was aslo observed for the coherent transverse optical (TO) phonon in polycrystalline Zr film, in spite of much weaker photoexcitation [58],... 

Earlier the velocity distribution function of quasi particles of a relativistic ideal gas for a one dimensional system, for example, fluxons in thermalized Josephson systems and electrons in a high temperature plasma was found. 

A) For quasi particles (fluxon in thermalized Josephson transmision lines (JTL)) ... 

A description of nuclear matter as an ideal mixture of protons and neutrons, possibly in (5 equilibrium with electrons and neutrinos, is not sufficient to give a realistic description of dense matter. The account of the interaction between the nucleons can be performed in different ways. For instance we have effective nucleon-nucleon interactions, which reproduce empirical two-nucleon data, e.g. the PARIS and the BONN potential. On the other hand we have effective interactions like the Skyrme interaction, which are able to reproduce nuclear data within the mean-field approximation. The most advanced description is given by the Walecka model, which is based on a relativistic Lagrangian and models the nucleon-nucleon interactions by coupling to effective meson fields. Within the relativistic mean-field approximation, quasi-particles are introduced, which can be parameterized by a self-energy shift and an effective mass. 

This way we take the quasi-particle energies which are described by an effective mass and a self-energy shift and solve the Schrodinger equation for the separable Yamaguchi potential. Separating the center of mass motion, with energy p2/2 from the relative motion, with reduced mass M M /(M +... 

The modification of the three and four-particle system due to the medium can be considered in cluster-mean field approximation. Describing the medium in quasi-particle approximation, a medium-modified Faddeev equation can be derived which was already solved for the case of three-particle bound states in [9], as well as for the case of four-particle bound states in [10]. Similar to the two-particle case, due to the Pauli blocking the bound state disappears at a given temperature and total momentum at the corresponding Mott density. 

Figure 3. The quasi-particle dispersion relations at low energies in the 2SC phase (left panel) and in the g2SC phase (right panel).

It is instructive to start with the excitation spectrum in the case of the ordinary 2SC phase when dfi = 0. With the conventional choice of the gap pointing in the anti-blue direction in color space, the blue quarks are not affected by the pairing dynamics, and the other four quasi-particle excitations are linear superpositions of ur>g and dr(J quarks and holes. The quasi-particle is nearly identical with a quark at large momenta and with a hole at small momenta. We represent the quasi-particle in the form of Q(quark, hole), then the four quasiparticles can be represented explicitly as Q(ur,dg), Q(ug, dr), Q(dr,ug) and Q(dg,ur). When S/i = 0, the four quasi-particles are degenerate, and have a common gap A. 

If there is a small mismatch (dp < A) between the Fermi surfaces of the pairing u and d quarks, the excitation spectrum will change. For example, we show the excitation spectrum of Q(ur, dg) and Q(dg,ur) in the left panel of Figure 3. We can see that 5p induces two different dispersion relations, the quasi-particle Q(dg,ur) has a smaller gap A — p, and the quasi-particle Q(ur,dg) has a larger gap A //. This is similar to the case when the mismatch is induced by the mass difference of the pairing quarks [16]. 

Note that the quasi-particle energy is independent of color and flavor in this case, since we have assumed a singlet pair in flavor and color. 

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