Scattering mean square atomic

Thermal properties of overlayer atoms. Measurement of the intensity of any diffracted beam with temperature and its angular profile can be interpreted in terms of a surface-atom Debye-Waller factor and phonon scattering. Mean-square vibrational amplitudes of surfece atoms can be extracted. The measurement must be made away from the parameter space at which phase transitions occur. 

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

The standard method for normalisation of diffracted intensity data into electron units, is to compute both the mean square atomic scattering factor and the mean incoherent scatter for the particular molecular repeat over a range of high two theta values (say 40°-60°) where their total value can be considered to be equivalent to the actual diffraction from the molecular system concerned. An appropriate normalisation factor is then applied to the experimental intensity data after geometrical correction and, finally, incoherent scatter is subtracted ( 1 ). 

Figure 1. Corrected equatorial trace in electron units for a PET specimen normalized over the full two theta range (]2 mean-square atomic scattering factor C incoherent scatter t total scatter)...

The INS intensity, 5 (g,fo), as calculated from the Scattering Law, Eq. (2.41), is related to the mean square atomic displacements, weighted by the incoherent scattering cross sections. What is required to calculate this quantity is the mean square atomic displacement tensor, Bi, and this can be obtained from the crystalline equivalent of L/ " ( A2.3), the normalised atomic displacements in a single molecule Eq. (4.20). This is and was introduced above, in Eq. (4.55). We have seen how... 

In this equation, w° represents the true weight fraction of crystals in the sample. The quantity P represents the mean square atomic scattering factor of the polymer summed over all atoms i (Ni is the number of atoms of type i)... 

We next describe the state-of-the-art techniques used to study both dynamical crossovers in such macromolecules of biological interest as proteins, RNA, and DNA. For both crossovers we will consider the related physics by examining experimental findings and MD simulations. Our approach to neutron scattering in biomolecules is essentially the same—with some minor adjustments—as that used in confined water. The mean squared atomic displacement (MSD) X (T) (MSD)... 

The intensity of light scattering, 7, for an isolated atom or molecule is proportional to the mean squared amplitude... 

A dynamic transition in the internal motions of proteins is seen with increasing temperamre [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24]. 

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). 

The exponent Mk depends on the mean square displacement of the atom from its equilibrium position and hence upon temperature. It is linear with (kT/m Xsin / where k is the Boltzmann constant, T the absolute temperature, the scattering angle, the wavelength and m the atomic mass (for a monatomic material). In addition there are complicated expressions dependent upon the crystal symmetry. As an example, for silicon at room temperature the /, are reduced by approximately 6%. With this correction all the equations of dynamical theory still apply. 

In such a Gaussian case the intermediate scattering function is entirely determined by the mean squared displacement of the atom (r (t)) ... 

The mean-square displacements of each of the atoms in the crystal, which affect the the X-ray scattering amplitudes, are obtained by summation over the displacements due to all normal modes, each of which is a function of ea(j Icq), as further discussed in section 2.3. The eigenvalues of D are the frequencies of the normal modes. 

It is noted that both the probability distribution of Eq. (2.16) and the temperature factor of Eq. (2.19) are Gaussian functions, but with inversely related mean-square deviations. Analogous to the relation between direct and reciprocal space, the Fourier transform of a diffuse atom is a compact function in scattering space, and vice versa. 

The damping factors take into account 1) the mean free path k(k) of the photoelectron the exponential factor selects the contributions due to those photoelectron waves which make the round trip from the central atom to the scatterer and back without energy losses 2) the mean square value of the relative displacements of the central atom and of the scatterer. This is called Debye-Waller like term since it is not referred to the laboratory frame, but it is a relative value, and it is temperature dependent, of course It is important to remember the peculiar way of probing the matter that EXAFS does the source of the probe is the excited atom which sends off a photoelectron spherical wave, the detector of the distribution of the scattering centres in the environment is again the same central atom that receives the back-diffused photoelectron amplitude. This is a unique feature since all other crystallographic probes are totally (source and detector) or partially (source or detector) external probes , i.e. the measured quantities are referred to the laboratory reference system. 

Nj and Oj in equation (2.4) represent the number of atoms in the jth shell and root-mean-square deviation of the interatomic distances over Rj which results both from static and dynamic (thermal) disordering effects respectively. The scattering amplitude, Fj(k) is given by... 



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