Phase shift radial

Hysteretic whirl. This type of whirl occurs in flexible rotors and results from shrink fits. When a radial deflection is imposed on a shaft, a neutralstrain axis is induced normal to the direction of flexure. From first-order considerations, the neutral-stress axis is coincident with the neutral-strain axis, and a restoring force is developed perpendicular to the neutral-stress axis. The restoring force is then parallel to and opposing the induced force. In actuality, internal friction exists in the shaft, which causes a phase shift in the stress. The result is that the neutral-strain axis and neutral-stress axis are displaced so that the resultant force is not parallel to the deflection. The... 

The Fourier transform of the EXAFS of Figure 5 is shown in Figure 6 as the solid curve It has two large peaks at 2.38 and 2.78 A as well as two small ones at 4.04 and 4.77 A. In this example, each peak is due to Mo—Mo backscattering. The peak positions are in excellent correspondence with the crystallographically determined radial distribution for molybdenum metal foil (bcc)— with Mo—Mo interatomic distances of2.725, 3.147, 4.450, and 5.218 A, respectively. The Fourier transform peaks are phase shifted by -0.39 A from the true distances. 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

The essence of analyzing an EXAFS spectrum is to recognize all sine contributions in x(k)- The obvious mathematical tool with which to achieve this is Fourier analysis. The argument of each sine contribution in Eq. (8) depends on k (which is known), on r (to be determined), and on the phase shift

characteristic property of the scattering atom in a certain environment, and is best derived from the EXAFS spectrum of a reference compound for which all distances are known. The EXAFS information becomes accessible, if we convert it into a radial distribution function, 0 (r), by means of Fourier transformation ... 

A straightforward Fourier transform of the EXAFS signal does not yield the true radial distribution function. First, the phase shift causes each coordination shell to peak at the incorrect distance second, due to the element-specific backscattering amplitude, the intensity may not be correct. The appropriate corrections can be made, however, when phase shift and amplitude functions are derived from reference samples or from theoretical calculations. The phase- and amplitude-corrected Fourier transform becomes ... 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

Z neighbors. Phase correction of the Fourier transform by the backscattering phase shift of one of the absorber—neighbor pairs is also extensively used. This has the effect of correcting the distances observed in the radial structure function as well as emphasizing the contributions from the chosen ab-... 

The complex phase shift can be obtained from exact numerical solution of the radial Schrodinger equation.2 The following quantities can immediately be given in terms of 8r The differential elastic cross section in the center-of-mass system... 

If the energy is raised above the second limit there are two open channels. In Fig. 20.2 at an energy WB for r > rB the wavefunction is composed of a linear combination of 0, and 02. If we put a radial box of radius rB around the ionic core we can again ask, What are the normal modes for electron scattering from the contents of the box In other words, what linear combinations of incoming coulomb wavefunctions will suffer at most a phase shift when scattering from the contents of the box There are two wavefunctions, labelled by p = 1,2. They are linear combinations of 0, and 02, given by... 

We begin our discussion with the simple case of a spinless particle of mass m and kinetic energy E = h2k2/2m in a spherical, time-independent potential V(r), so that the Schrodinger equation can be decomposed into uncoupled partial waves l. For a particular l, the scattering matrix or the S matrix is defined as S(k) = exp[2/5(A )] in terms of the phase shift 8(k). Here and in the following, the subscript l on the S matrix and the phase shift is suppressed. The asymptotic form of the time-dependent radial wavefunction is expressible as... 

For a real potential, the space part of the radial wavefunction for a real energy may be chosen to be real and the phase shift is real. Therefore, the absolute value of the S matrix is unity. This guarantees the flux conservation since, then, the incoming wave and the outgoing wave in Eq. (12) have fluxes of the same magnitude in opposite directions. 

Here, Ai(k) are phase shifts of the continuous electron radial wavefunctions Pki(r) in the field of the ionic core. 

Figure 7.4 Definition of the phase shift A as introduced by a potential. The solution of the radial function RKAr) of a wave with energy e = k2/2 (in atomic units) and with ( = 0 is shown for two situations under the influence of a repulsive potential V(r) as indicated by the shaded region (top), and for vanishing potential (bottom). In the first case one has RK((r) = FK0(r), and in the second case the radial function is equal to the spherical Bessel function, i.e., RKAr) = j0(fcr). Asymptotically, both solutions, FK0(r) and j0(Kr), differ only by a constant distance A in the r coordinate which is related to the phase shift A( as indicated. From The picture book of quantum mechanics, S. Brandt and H. D. Dahmen, 1st edition, 1985, John Wiley Sons Inc., NY. 1985 John Wiley Sons Inc.

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