Contraction length

Length Contraction, Time Diiation, and Proper Time 

This length of an object at rest is denoted as its proper length loin order to measure the length of this object in IS, one has to measure the positions of its beginning Xi and its end X2 at the same time t. Without loss of generality we may always choose f = 0. We thus have to consider the two events 

The coordinates of any of these two events in IS and IS are related via the Lorentz boost given by Eq. (3.67). Taking advantage of fi = f2 = 0 directly 

The derivation above was focused on a Lorentz boost in the direction of the stick, i.e., the relative motion of the two frames IS and IS, described by the velocity vector v, and the stick were parallel. Due to the Lorentz boost given by Eq. (3.67) no length contraction will occur in directions perpendicular to v. This immediately yields the general expression for the length contraction. 

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Instead, we believe the electronic structure changes are a collective effect of several distinct processes. For example, at surfaces the loss of the bulk symmetry will induce electronic states with different DOS compared to bulk. As the particle sizes are decreased, the contribution of these surface related states becomes more prominent. On the other hand, the decrease of the coordination number is expected to diminish the d-d and s-d hybridization and the crystal field splitting, therefore leading to narrowing of the valence d-band. At the same time, bond length contraction (i.e. a kind of reconstruction ), which was observed in small particles [89-92], should increase the overlap of the d-orbitals of the neighboring atoms, partially restoring the width of the d-band. 

One-center expansion was first applied to whole molecules by Desclaux Pyykko in relativistic and nonrelativistic Hartree-Fock calculations for the series CH4 to PbH4 [81] and then in the Dirac-Fock calculations of CuH, AgH and AuH [82] and other molecules [83]. A large bond length contraction due to the relativistic effects was estimated. However, the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in the entire coordinate space. It is notable that the RFCP and one-center expansion approaches were considered earlier as alternatives to each other [84, 85]. 

The above descriptions of the origin of bond length contractions at surfaces are consistent with the observations made when adsorbates are deposited on these surfaces the shortened bond lengths are systematically lengthened again (sometimes to more than their bulk values) by the presence of adsorbates, as is visible in Table 6.1 (cf. Fig. 6.2). 

The important conclusion to be drawn from the foregoing discussion is that space and time coordinates are relativistically linked together in a way that compensates for apparent length contraction and time dilation... 

More recently Ziegler, Snijders, and co-workers have suggest that the bond-length contraction is independent of the orbital contraction but rather it is a direct result of the Dirac Hamiltonian (63,64)- They observed that the contraction could be computed from the nonrelativistic molecular wave... 

The origin of relativistic bond-length contraction is still somewhat of an open question. First, the precise mechanism by which it occurs has not been determined unequivocally. In addition, the computed magnitudes often vary by as much a factor of two or more from calculation to calculation. 

To date, the best ab initio all-electron molecular calculations involving heavy elements are those of Lee and McLean, who published LCAO-MO SCF relativistic calculations on AgH and AuH (75). They reported relativistic bond-length contractions of 0.08 and 0.25 A, respectively, and increases in... 

The most important difference between particles inside the bulk and in the interfacial layer comes from the surrounding environment of the particles the particles inside the bulk are in an isotropic environment, while those in the interface are in an anisotropic environment thus, in the interlayer, the forces between the particles are unbalanced. To reduce the resulting surface pressure, some additional processes occur that must be taken into account. On clean surfaces (for example, on a solid surface in vacuum), these processes are the bond-length contraction or relaxation and reconstruction of the surface particles (Somorjai 1994). It results in significantly reduced spacing between the first and second layers compared to the bulk. The perturbation caused by this movement propagates a few layers into the bulk. The other effect is that the equilibrium position of the particles changes that is the outermost layers can have different crystal structure than the bulk. This phenomenon is the reconstruction. 

It should be pointed out that Schwarz (20),using double perturbation theory,has demonstrated that it is possible to rationalize the relativistic bond length contraction in terms of the attractive Hellmann-Feynman force due to the relativistic change in electron density.In such an approach it would be necessary to analyze and get a physical picture of the relevant density changes... 

The photomechanical behavior of the networks was studied under constant stress (load) by following at constant temperature the length contraction as a function of the period of irradiation in the dark, length recovery takes place although at a lower rate... 

It should be noted that gap reduction at surfaces is always overestimated if relaxation effects are neglected, because the latters, by inducing bond-length contractions, strengthen the surface Madelung potential. The prediction of metallicity on NiO(llO) [247], based on unrelaxed geometry, should thus be reconsidered at the light of the present remark. 

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