Interatomic transitions

The A -shell x-ray emission rates of molecules have been calculated with the DV-Xa method. The x-ray transition probabilites are evaluated in the dipole approximation by the DV-integration method using molecular wave functions. The validity of the DV-integration method is tested. The calculated values in the relaxed-orbital approximation are compared with those of the frozen-orbital approximation and the transition-state method. The contributions from the interatomic transitions are estimated. The chemical effect on the KP/Ka ratios for 3d elements is calculated and compared with the experimental data. The excitation mode dependence on the Kp/Ka ratios for 3d elements is discussed. 

For theoretical calculations of molecular x-ray emission rates, it is usual to neglect the contributions from interatomic transitions, sometimes called crossover transitions, and to use the single-center approximation [6]. This approximation is useful and can often reproduces the experimental spectra quite well. Using a simple molecular orbital (MO) approach, Urch [7,8] showed the validity of the single-center approximation for metal K x-ray emission rates in MX4 and MXg molecules. On the other hand, Adachi and Taniguchi... 

X-ray emission rates in simple molecules have been extensively studied by Larkins and his group [10,11]. Larkins and Rowlands [12] made the MO calculations with the complete-neglect-of-differential-overlap (CNDO/2) method and pointed out that there are significant contributions of interatomic transitions to the C K x-ray emission rates in CO, HCN, and CO2 molecules, but relative intensities are less sensitive to inclusion of crossover transitions. Applying the ab initio MO method to CO, they also examined [13] various factors influencing the molecular x-ray emission rate, such as choice of basis set, choice of length and velocity forms, electronic relaxation effect, and interatomic contributions. Phillips and Larkins extended their calculations to other simple molecules [14,15]. 

Recently we have estimated [16] the interatomic contributions to molecular x-ray emission rates by the use of the discrete-variational (DV) Xa MO method [17]. For CO molecule, the C K x-ray emission rate increases significantly by taking into account the interatomic transitions. On the other hand, the interatomic transitions play a minor role for K x-ray emission process in the compounds of 3d elements. 

In the calculations of x-ray emission rates in molecules with the DV-Xq method, we have used various approximations and numerical techniques. We examine several factors which influence the theoretical calculations of molecular x-ray emission rates, i.e. the accuracy of the DV integration, the electronic relaxation effect, and the contributions from the interatomic transitions. The examination of other factors, such as the choice of dipole operators, basis-set dependence, and the vibrational eflfect, has been reported by Larkins [10,11] with the ab initio method. 

Most of old theoretical calculations for molecular x-ray emission rates have been performed in the single-center approximation and the contributions from the interatomic transitions have been neglected. On the other hand, Taniguchi and Adachi [9] and Larkins [10,11] pointed out that in some cases the contributions from the interatomic transitions are appreciable. In the present work, the... 

The x-ray emission rate with the interatomic contributions is given by Eq. (2). On the other hand, the rate without the interatomic transitions can be obtained as follows. We calculate the x-ray emission rate by Eq. (2), but the summation over i in the dipole matrix element, Eq. (3), is restricted only for the atomic orbitals that belong to the same atom as the initial A -shell vacancy. This method may be different from the conventional single-center model, in which the square of D is expressed as... 

The calculations were made for A -x-ray emission rates of C and O atoms in the CO molecule. The results with and without the interatomic transitions and the relative change in the emission rate due to interatomic transitions are hsted in Table 6. In general, the A -x-ray emission rate increase by taking into account the existence of the interatomic transitions. Only one exceptional case is the 5cr la transition for the A -shell vacancy in O atom. The decrease in this transition rate due to the two-center effect has already been pointed out by Rowlands and Larkins [13] in their CNDO/2 calculations. 

The increase in the x-ray emission rates is significant for C A x rays. In this case, the single-center approximation is inadequate to predict the C A -x-ray emission process in CO. On the other hand, the interatomic transitions play less important role for O A -x-ray emission. 

We have also studied the interatomic contributions to the A -x-ray emission rates for chemical compounds of 3d elements [16]. For the compounds with octahedral symmetry, such as CrCla and MnCl2, the interatomic transitions play a minor role. On the other hand, in the case of the compounds with tetrahedral symmetry, such as CrOa and KMn04, the interatomic transitions increase x-ray intensities of 4 2, 5t2, and 6 2 components of 3d transition metals. The 4t2 component corresponds to the Kj3" peak and 6 2 to the K02,5... 

TS, and RX methods for x-ray transition probabilities was made and it is demonstrated that relative intensity ratios and shapes of x-ray emission spectra are not so sensitive to the method used. The contributions of the interatomic transitions was estimated for CO and compounds of 3d transition metals. For C K-x-iay emission rates in CO the multi-center effect is found to be important, but in the case of the Kfij Ka ratios for the 3d transition elements the interatomic transitions are almost negligible. 

For the dipole transition, Laporte selection rule (Af = 1) for the emitting or absorbing atom is valid, when the probability of interatomic transition is very small. In such a case, the atomic orbital component (or partial density of states) obtained from Mulliken population analysis is useful for a rough estimation of... 

The chemical effects on atomic x-ray spectrum has widely been investigated by DV-Xa calculations, where the contribution of interatomic transition, in other word, the cross-over transition to x-ray emission rate can essentially be estimated by eq.(ll). The abnormality in atomic Kp/Ka emission ratio for some 3d transition metal compounds has been observed. 

Thus, in a one-electron approximation the distance between individual maxima in a spectral series is equal to the difference in one-electron energies of the corresponding atomic levels. In molecules the 3 s, 3p and 3d electrons of the sulfur atom are involved in chemical bonding to form an MO system. In this case, the SKp spectrum (S3p Sis interatomic transitions), for example, corresponds to MO ->Sls transitions, and the distances between spectral maxima correspond to energy differences of the appropriate occupied molecular levels ... 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

There are two basic physical phenomena which govern atomic collisions in the keV range. First, repulsive interatomic interactions, described by the laws of classical mechanics, control the scattering and recoiling trajectories. Second, electronic transition probabilities, described by the laws of quantum mechanics, control the ion-surface charge exchange process. 

In order to perform the calculation., of the conductivity shown here we first performed a calculation of the electronic structure of the material using first-principles techniques. The problem of many electrons interacting with each other was treated in a mean field approximation using the Local Spin Density Approximation (LSDA) which has been shown to be quite accurate for determining electronic densities and interatomic distances and forces. It is also known to reliably describe the magnetic structure of transition metal systems. 

The resonating-valence-bond theory of metals discussed in this paper differs from the older theory in making use of all nine stable outer orbitals of the transition metals, for occupancy by unshared electrons and for use in bond formation the number of valency electrons is consequently considered to be much larger for these metals than has been hitherto accepted. The metallic orbital, an extra orbital necessary for unsynchronized resonance of valence bonds, is considered to be the characteristic structural feature of a metal. It has been found possible to develop a system of metallic radii that permits a detailed discussion to be given of the observed interatomic distances of a metal in terms of its electronic structure. Some peculiar metallic structures can be understood by use of the postulate that the most simple fractional bond orders correspond to the most stable modes of resonance of bonds. The existence of Brillouin zones is compatible with the resonating-valence-bond theory, and the new metallic valencies for metals and alloys with filled-zone properties can be correlated with the electron numbers for important Brillouin polyhedra. 

Strong boron-transition metal interaction interatomic distances are shorter by 5-10%, as compared to the sum of the metal radii. 

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