Relativistic scattering factor

The dynamic calculations include all beams with interplanar distances dhki larger than 0.75 A at 120 kV acceleration voltage and thickness between 100 A and 300 A for the different zones. The structure factors have been calculated on the basis of the relativistic Hartree - Fock electron scattering factors [14]. The thermal difiuse scattering is calculated with the Debye temperature of a-PbO 481 K [15] at 293 K with mean-square vibrational amplitude

= 0.0013 A following the techniques of Radi [16]. The inelastic scattering due to single-electron excitation (SEE) is introduced on the base of real space SEE atomic absorption potentials [17]. All calculations are carried out in zero order Laue zone approximation (ZOLZ). 

P. A. Doyle, P. S. Turner, Relativistic Hartree-Fock X-ray and Electron Scattering Factors, Acta Cryst, A24, 390-397,1968. 

Cromer, D. T. and Lieberman, D. (1970). Relativistic calculation of anomalous scattering factors for X-rays. /. Chemical Physics 53, 1891-1898. 

Relativistic X-Ray Scattering Factors for He Through Ar from Dirac-Hartree-Fock Wave Functions. 

Cromer, D. T., J. T. Waber Scattering Factors Computed from Relativistic Dirac-Slater Wave Functions. Acta Cryst. 18, 104 (1965). 

D. T. Cromer and D. Liberman, Relativistic Calculation of Anomalous Scattering Factors for X-rays, Los Alamos Scientific Laboratory, LA 4403 TID 4500, July 1970. 

To summarize, the relativistic kinematic prescription is to replace COM system momenta by their relativistic values, to replace the reduced mass by the reduced total energy (eq. 3.29), and to use the M0ller factor, eq. (3.37). This prescription yields the correct, relativistic proton-nucleus Coulomb scattering amplitude to order a. The usual proton-nucleus Coulomb potential must also be multiplied by the relativistic correction factor rj (eq. 3.56). 

Higher accelerating voltages decrease the atomic scattering factors and make kinematic theory more accurate. However, the situation is complicated by relativistic effects which become important above 200 kV. 

Only the valence Compton profiles are needed for the reconstruction of the momentum density and the occupation number density. So one has to subtract an appropriate core Compton profile. Furthermore the contribution of the multiple scattered photons to the measured spectra has to be taken into account (for example by a Monte Carlo simulation [6]). Additionally one has to take heed of the fact that the efficiency of the spectrometer is energy dependent, so the data must be corrected for energy dependent effects which are the absorption in the sample and in the air along the beam path, the vertical acceptance of the spectrometer and the reflectivity of the analyzing crystal. The relativistic derivation of the relationship between the Compton cross section and the Compton profile leads to a further correction factor [7j. Finally a background subtraction and a normalization of the valence profiles to the number of valence... 

To summarize, the p + NR elastic scattering amplitude is conveniently calculated using the relativistic DWBA formalism, where the following dianges from the procedure desoibed in ref. [Ra 88b] are carried out (1) NR distorted waves and bound state wave functions are used for the upper components (2) the lower components are set to the NR or free partide limit [i.e., no potential terms included in (a k)/ E + m)] (3) ) corresponds to the dioice of t according to eq. (6.5) and may include density dependence (4) the pA recoil factor, EJ E + ), is included. 

In addition to scattering and diffraction methods for structure determination, important experimental probes for intrinsic properties are vibrational and rotational spectroscopy. Rotational spectra will be affected by a relativistic reduction of bond length, which will reduce the moments of inertia. This lowers the rotational constant, and we should expect a relativistic red-shift of the rotational spectrum. For vibrational spectroscopy, the situation is less clear— relativistic effects may strengthen as well as weaken bonds. Thus effects of relativity on vibrational spectroscopy depend very much on the system under consideration. A further discussion of these effects is therefore postponed to chapter 22. For the diffraction and scattering techniques, relativistic effects are absorbed into atomic scattering parameters and structure factors and are thus not a primary concern of relativistic quantum chemistry. 

Similar calculations using non-relativistic Hartree-Slater wavefunctions [61] and relativistic Hartree-Slater theory [62] have also provided data for californium. The atomic form factors, the incoherent scattering functions [63], and a total Compton profile have been tabulated for californium [64]. 



---