Hydrogen, amplitude factor

For the quasi-elastic region ( 2 meV) that is of concern here, the vibrational motions affect only the elastic intensity through a Debye-Waller factor, exp(-Q (M )), where (u ) is the mean square hydrogen amplitude in the vibrational modes. The scattering law for rotations can be written as a sum of an elastic peak intensity and a quasi-elastic component ... 

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

This effect can be illustrated by Fig. 14.2. The effective range of local modification of the sample states is determined by the effective lateral dimension 4ff of the tip wavefunction, which also determines the lateral resolution. In analogy with the analytic result for the hydrogen molecular ion problem, the local modification makes the amplitude of the sample wavefunction increase by a factor exp( — Vi) 1.213, which is equivalent to inducing a localized state of radius r 4tf/2 superimposed on the unperturbed state of the solid surface. The local density of that state is about (4/e — 1) 0.47 times the local electron density of the original stale in the middle of the gap. This superimposed local state cannot be formed by Bloch states with the same energy eigenvalue. Because of dispersion (that is, the finite value of dEldk and... 

The reason that neutron diffraction is so much more effective than x-ray diffraction as a means for locating hydrogen atoms can be seen in the atomic scattering amplitudes given in Table 9-II (taken from reference 94, except for the neutron diffraction scattering factor for deuterons). 

Atoms in crystals seldom have isotropic environments, and a better approximation (but still an approximation) is to describe the atomic motion in terms of an ellipsoid, with larger amplitudes of vibration in some directions than in others. Six parameters, the anisotropic vibration or displacement parameters, are introduced for each atom. Three of these parameters per atom give the orientations of the principal axes of the ellipsoid with respect to the unit cell axes. One of these principal axes is the direction of maximum displacement and the other two are perpendicular to this and also to each other. The other three parameters per atom represent the amounts of displacement along these three ellipsoidal axes. Some equations used to express anisotropic displacement parameters, which may be reported as 71, Uij, or jdjj, axe listed in Table 13.1. Most crystal structure determinations of all but the largest molecules include anisotropic temperature parameters for all atoms, except hydrogen, in the least-squares refinement. Usually, for brevity, the equivalent isotropic displacement factor Ueq, is published. This is expressed as ... 

In Figure 1.6, the atomic scattering factors f(s) for hydrogen, carbon, and oxygen are plotted against s = 2(sin 6)/X. In the forward direction (s = 0) the x-ray waves scattered from different parts of the electron cloud in an atom are all in phase, and the wave amplitudes simply add up, rendering /(0) equal to the atomic number Z. As s increases, the waves from different parts of the atom develop more phase differences, and the overall amplitude begins to decrease. The exact shape of the curve f(s) reflects the shape of the electron density distribution in the atom. The... 

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