S-wave contributions

The s-wave contribution to the photo ionization from the 3a3p level is plotted in figure 3 and shows a quite satisfactory gauge invariance. Its peak value is in excellent agreement with that yielded by our previous STOCOS ealeulations, 346 Mb (3). 

The reason for the large difference between the values of A for positrons and electrons at an energy of 2 eV is that for positrons the s-wave phase shift passes through zero at the Ramsauer minimum and the dominant contribution to the cross section therefore comes from the p-wave, which is quite strongly peaked in the forward and backward directions. In contrast, there is no Ramsauer minimum in electron-helium scattering, and the isotropic s-wave contribution to aT is dominant at this energy. 

The s-wave contribution to partial-wave contributions are zero at the threshold. 

The elastic scattering cross section must fall as the positron energy is increased above the threshold, and it will either rise or fall as the threshold is approached from below, depending on the value of l and on the phase shift at the threshold. Furthermore, because the s-wave contribution to crPs, considered as a function of the positron energy, has an infinite slope at the threshold energy EPs, equation (3.99), so too does energy dependence has the shape of either a cusp or a downward rounded step. All other partial-wave contributions to aei, however, continue through the threshold with no discontinuity of slope. 

The most accurate values of the s-wave positronium formation cross section calculated by Humberston (1982) and Humberston et al. (1997) are shown in Figure 4.1. (The latter results are more accurate but there is no difference between the two sets of results on the scale of this figure.) This cross section is much smaller than the s-wave elastic scattering cross section and also, as we shall see, much smaller than other contributions to <7ps of low orbital angular momentum. It has recently been shown by Ward, Macek and Ovchinnikov (1998), using hidden crossing theory, that the small magnitude of the s-wave contribution to [Pg.156]

The total positronium formation cross section in the Ore gap, constructed from the addition of accurate variational results for the first three partial waves and the values given by the Born approximation for all partial waves with l > 2, is plotted in Figure 4.4. On the scale of the ordinate, the s-wave contribution is too small to be visible. A very small s-wave contribution is found to be a feature of the positronium formation cross section for several other atoms. 

The Li resonances were observed by Johnson et the peaks in both total cross section and reaction cross section are shown in Fig. 15. A similar peak is found in the total cross section of Li" (Stelson and Preston ). The inelastic scattering of neutrons by Li" has been studied by Freeman et al., who detected radiation from the 478 keV level. The shape of the excitation curve just above threshold depends on the orbital momenta of the ingoing and outgoing neutron waves and in this case indicates a strong s-wave contribution to the reaction. This type of reaction would be expected to show resonances of the compound nucleus as for other inelastic processes, and evidence for this is given by Kiehn and Goodman for scattering from Al . ... 

Since this resonance has an s-wave contribution, it is natural that the resonance enhancement of the numerator is compensated by the corresponding enhancement of the denominator. 

While it is not beyond the realm of possibility to evaluate the g s, h s, and Q for appropriate wave functions, some further simplification is necessary for our present arguments. We assume that the s are orthogonal to all y s. This is not necessarily true although the //s can be selected to make it true. Even without specific selection of /A s it is probably a good approximation for those s which contribute substantially to the polarizability. Then... 

Table 5.2. Contributions of various tip electronic states to tunneling current, in percent. Calculated for five different tip atoms. The corrugations for graphite, for five different tip atoms, are shown in the rightmost column. For the s-wave-tip theory, the corrugation is infinity. After Tersoff and Lang (1990).

Similar to the. y-wave model, the Na-atom-tip model predicts a poor resolution. The agreement of the Na-atom-tip model with the y-wave-tip model does not mean that the s-wave-tip model describes the actual experimental condition in STM. According to the analysis of Tersoff and Lang (1990), real tips are neither Na or Ca, but rather transition metals, probably contaminated with atoms from the surface (for example. Si and C are common sample materials). For a Si-atom tip, the p state dominates the Fermi-level LDOS of the tip. For a Mo-atom tip, while the p contribution is reduced, this is more than compensated by the large contribution from states of d like symmetry. The STM images from a Si, C, or Mo tip, as predicted by Tersoff and... 

There is no doubt that this field, like few others, owes very much to its founder, Ronald Gurney, because of the fast start he gave it by applying quantum mechanics to interfacial electron transfers shortly after the publication of Schrodinger s wave equation (1926). The early seminal contributions (to which must be added that of J. A. V. Butler in the same period)22 founded quantum electrochemistry and led to its broader development by Gcrischer (1960), in particular the idea of the absolute scale of potentials and the equation... 

Fig. 57. Specific heat contribution y(H) of the vortex core electrons in the mixed state (normalized by the Sommerfeld parameter /n) of the Yj[Lu jrNi2B2C samples from fig. 56 as function of the applied magnetic field (normalized by //c2(0)). The straight line y(H)

It should be noted here that the conclusion about s-wave nature of the SC order parameter is consistent with conclusion about s-wave symmetry of the SC order parameter in the bulk and d-wave symmetry at the surface of the sample of the cuprates [17]. It was noted in [17] that most conclusions about d-wave symmetry was obtained in experiments (e.g. ARPES ones) on the cuprates in which mainly surface phenomena have been used. In this sense, the resistive measurements on the cuprates (see, e.g. [4]) are essentially bulk in the nature. In addition, the electron scattering (in resistivity measurements) is sensitive to the spin disorder in the system (magnetic contribution in the electrical resistivity appears, see Sec.l). Moreover, the electron scattering permits probe not only static magnetic order but dynamical (short-lived) ones because of short characteristic times as compared e.g. with usual neutron scattering. 

Figure 4.20 The S-wave annihilation function P[p) defined by Eq. (126), p being the hyperradius, for e+ + H(1s) scattering at an energy of 10 6 a.u. above the positronium formation threshold. The total P[p) is decomposed into the contributions from the direct channels e+ + H, the positronium formation channels p + Ps, and the interference between them. Results of hyperspherical close-coupling calculations including the absorption potential —iVabs in the Hamiltonian. Figure from Ref. [16].

Figure 13 Decomposition of a plane wave into spherical waves (a) real part. The plane wave result is shown in the upper right-hand comer, the individual partial wave contributions with a given ( value in the left-hand column, the sum of partial wave contributions up to the value ( = N in the right-hand column. From The picture book of quantum mechanics, S. Brandt and H. D. Dahmen, 1st edition 1985, John Wiley Sons, Inc, 1985 John Wiley and Sons Inc.

The spin density distribution in the 2A2 excited state requires the derivation of all the contributing determinants as done for allyl radical. A full treatment is given in Exercise 8.5, while here we provide an approximate description. Already at the outset one can recall that the coefficient of the QC determinant in the excited state s wave function is zero, and we therefore expect very different spin density distribution than in the ground state. To proceed, we first express the resonance structures as products of the bonds and the odd electron. Thus... 

The realization that both matter and radiation interact as waves led Werner Heisenberg to the conclusion in 1927 that the act of observation and measurement requires the interaction of one wave with another, resulting in an inherent uncertainty in the location and momentum of particles. This inability to measure phenomena at the subatomic level is known as the Heisenberg uncertainty principle, and it applies to the location and momentum of electrons in an atom. A discussion of the principle and Heisenberg s other contributions to quantum theory is located here http //www.aip.org/historv/heisenberg/. 

Abstract. We review our recent results on higher order corrections in positronium physics. We discuss a calculation of the recoil 0(ma6) corrections to the hyperfine splitting [1] and energy levels of a positronium atom [2], 0(ma7 In2 a) contributions to the positronium S-wave energy levels [3] and Ola2) radiative corrections to the parapositronium decay rate [4],... 

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