Scattering from vibrating molecules

We shall express the Scattering Law in terms of the internal and external dynamics of the molecules in crystals that form the backbone of the work discussed in this book. If the molecules consist of Watom atoms each and altogether there are N ox molecules in a crystal, then the total number of atoms in the system is A = Aatom x We will consider the dynamics of the molecule and the crystal separately. 

If r t) is the time dependent atomic position vector, taken with respect to the origin of the crystallographic cell, it can be expressed as the sum of two terms, the molecular centre of mass, this is a time dependent vector, (0ext that allows a description of the vibrations of the crystal, the phonons. The second term is the position vector of the atom, H(0int, given by a Cartesian coordinate system with its origin at the molecular centre of mass. 

This convolution will have significant consequences for the predicted spectral shapes but, for the present, it will be ignored as we develop the response predicted for the internal molecular vibrations alone. (This allows us to assume that the molecule is alone in a gas-like phase but is not free to recoil.) From the definition of the scattering function given above the internal dynamics are represented by  

It is convenient develop the total internal atomic displacements into their individual vibrations in Cartesian components. The time-displacement of a particular atom / is then the sum of the displacements of the atom in each of the internal modes, labelled v = 1,2. 3iVatom - 6. 

We shall now focus on the dynamics of a single atom and momentarily suppress the subscript /. The dynamics of the molecule will eventually be the sum over the dynamics of all of its atoms taken individually. This approach is chosen because, in general, molecules are of low symmetry and contain many different types of atom. In condensed matter physics texts the dynamics is developed in terms of a simple monatomic lattice and the subscript I is removed by summation, which leads to an extra factor N outside the integral [1]. 

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The theory of neutron scattering from vibrating molecules was developed by Zemach and Glauber. Recently, the method was applied to complex biological substances (bovine pancreatic trypsin inhibitor). ... 

A small fraction of the molecules are in vibrationally excited states. Raman scattering from vibrationally excited molecules leaves the molecule in the ground state. The scattered photon appears at higher energy, as shown in Figure lb. This anti-Stokes-shifted Raman spectrum is always weaker than the Stokes-shifted spectrum, but at room temperature it is strong enough to be useful for vibrational frequencies less than about 1500 cm 1. The Stokes and anti-Stokes spectra contain the same frequency information. 

Abstract Now an incisive probe of biomolecular structure, Raman optical activity (ROA) measures a small difference in Raman scattering from chiral molecules in right- and left-circularly polarized light. As ROA spectra measure vibrational optical activity, they contain highly informative band structures sensitive to the secondary and tertiary structures of proteins, nucleic acids, viruses and carbohydrates as well as the absolute configurations of small molecules. In this review we present a survey of recent studies on biomolecular structure and dynamics using ROA and also a discussion of future applications of this powerful new technique in biomedical research. 

Light that is scattered from a molecule is primarily elastically scattered that is, the incident and the scattered photons have the same energy. A small probability exists, however, that a photon is scattered inelastically, resulting in either a net gain or loss of energy of the scattered photon. This inelastic scattering, discovered by Raman and Krishna,1 allows fundamental molecular vibrational transitions to be measured at any excitation wavelength. 

Figure 7 A displays interference between fine structure due to vibrational coupling and a broad structure which is almost identical for the three types of monolayers. The question that arises is why such vastly different size molecules show similar broad transmission of electrons. Studies of electron scattering from these molecules in the gas phase indicated that the two first electronic resonances for benzene and naphthalene are at 1.12, 4.8 eV and...

Evidence for resonances in the cross sections for electron scattering from polyatomic molecules. Including hydrocarbons, can be found In the literature as far back as the late 1920 s (1,2). The authors of these papers, however, were unaware that the pronounced low energy peaks in the cross sections of molecules such as ethylene and acetylene were due to temporary negative Ion formation. Haas (3), In 1957, was apparently the first to observe that strong vibrational excitation accompanied such a peak, and to Invoke an unstable negative Ion complex as the means through which the excitation takes place. 

The scattering from a molecule will be more complicated than for a single atom because the other molecular motions of rotation and vibration come into play. If there are no inelastic features in the measured energy transfer range studied, the vibrational term will only affect the measured intensities in the QENS domain through a Debye-Waller factor. On the other hand, the influence of the rotation on the observed profiles has to be treated in more detail. Sears has derived analytical expressions for the total differential cross-section of a molecular system, where the rotational motion is isotropic [12]. From his work, a simplified expression (Eq. 22) for the double-differential cross-section can be obtained it is spht into three terms ... 

Spiro T. G. and Stein R, Resonance effects in vibrational scattering from complex molecules, Anna. Rev. Phys. Chem., 28, 501-521, 1977. 

In Raman spectroscopy, light is scattered from the molecule in different directions and is shifted to both higher and lower frequencies. The shift in magnitude is equal to the characteristic vibration frequencies of the molecule, resulting in a unique spectrum for each molecule. Optical fibers are used as light guides for Raman spectroscopy because the optimum wavelengths for the... 

The second excitation mechanism, impact scattering, involves a short range interaction between the electron and the molecule (put simply, a collision) which scatters the electrons over a wide range of angles. The usefiil feature of impact scattering is that all vibrations may be excited and not only the dipole active ones. As in Raman spectroscopy, the electron may also take an amount of energy hv away from excited molecules and leave the surface with an energy equal to Eq + hv. 

Here, AT is a constant, f is the incoming intensity, R is the distance of the scattered wave from the molecule (in practical terms, it is the distance between the scattering center and the point of observation), i and j are the labels of atoms in the jV-atomic molecule, g contains the electron scattering amplitudes and phases of atoms, 5 is a simple function of the scattering angle and the electron wavelength, I is the mean vibrational amplitude of a pair of nuclei, r is the intemuclear distance r is the equilibrium intemuclear distance and is an effective intemuclear distance), and k is an asymmetry parameter related to anharmonicity of the vibration of a pair of nuclei. 

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