Static limit

If it is required that the surface of the sample remains undisturbed during analysis, SIMS must be carried out at very low surface removal rates, typically about 10 monolayer/s. The terms static and dynamic are used to divide the sputtering rate of the sample into regimes where only surface species are observed (static SIMS) or where surface and bulk species are observed (dynamic SIMS). The static limit is usually considered to be <10 ions/cm impinging on the sample surface. Under these conditions, only about 1/1000 atoms on the surface of the sample are struck by a primary ion. 

In both models the rotational shift of the line 5co is the same either in the static limit, where it is equal to zero, or in the case of extreme narrowing where it reaches its maximum value coq. A slight difference in its dependence on tE is observed in the intermediate region only. The experimentally observed density dependence of the shift shown in Fig. 3.5 is in qualitative agreement with theory. 

The coupled cluster expression for frequency-dependent second hyperpolarizability, Eq. (30), can now be expanded in a Taylor series in its frequency arguments around its static limit as ... 

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

SIMS is strictly speaking a destructive technique. In the dynamic mode, used for making concentration depth profiles, several tens of monolayers are removed per minute. In the static limit, which is used to study adsorbed molecules, the rate of removal corresponds to less than one monolayer per hour, implying that the surface... 

While the above refers mainly to the static limit, new effects come into play when a moving contact line, i.e. spreading, is considered. It has been observed experimentally that the contact angle of a moving contact line 0, the dynamic contact angle, deviates from the corresponding static value 0. As an example, for a completely wettable surface (i.e. 6(, = 0), a relationship of the form... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

In lattice statics simulations all vibrational effects are neglected2 and the internal energy of the solid U is simply equal to < >, and the entropy is zero. Such minimizations give the crystal structure and internal energy (often referred to as the lattice energy) of the low-temperature phase. In the static limit at 0 K and zero pressure3 the crystal structure is thus determined by the equation... 

Thermodynamic properties at high pressures are of great interest for instance to Earth scientists who wish to understand the behaviour of the Earth s mantle, where pressures reach 100 GPa. To carry out energy minimizations in the static limit at non-zero pressures we minimize the enthalpy H = U + pV with respect to all the variables that define the structure, where p is the applied pressure and V the volume. When p is zero we regain eq. (11.7). 

Figure 11.7 shows schematically the resulting calculated variation of H with p for the NaCl-type and the CsCl-type phases of CaO. The NaCl-type structure, which is stable at low pressures, is the rock salt structure in which the Ca and O atoms are 6-coordinate. In the CsCl structure, stable at high pressures, both cation and anion are 8-coordinate. In the static limit where the entropy is set to zero, the thermodynamically most stable phase at any pressure is that with the lowest value of H at the thermodynamic transition pressure, ptrs, the enthalpies of the two phases are equal. For CaO the particular set of potentials used in Figure 11.7 indicates a transition pressure of 75 GPa between the NaCl-type and CsCl-type structures, which compares with experimental values in the range 60-70 GPa. 

Changes in phase have important consequences for other thermodynamic properties and thus geophysical implications. For example, the bulk modulus at any pressurep in the static limit is given by the value of V(d2(//dV2) (at that pressure) for CaO this increases markedly across the phase boundary. 

Dr / r2 1 the donor and acceptor cannot significantly diffuse during the excited-state lifetime of the donor. This case is called the static limit. Distinction according to the mechanism of tranfer should be made. 

It should be emphasized that, because small molecules in usual solvents have diffusion coefficients <10 cm2 s-1, the rapid diffusion limit can be attained only for donors with lifetimes of 1 ms. This is the case for lanthanide ions for instance, the lifetime of Tb3+ chelated to dipicolinate is 2.2 ms. Stryer and coworkers (1978) showed that using Tb3+ as a donor and rhodamine B as an acceptor, the concentration of rhodamine B resulting in 50% transfer was 6.7 x 10 6 M, which is three orders of magnitude less than the concentration corresponding to 50% transfer in the static limit. 

They reduce to regular energy derivatives in the static limit [48,50]. The linear response function... 

Multiplying Eq. (3.1) with the static limit of the KS response function (2.64),... 

Experimental lattice parameters for bulk kaolinite, together with those calculated in the static limit, are listed in Table 1. A difference in the length of parameter b of 2.9% is the largest discrepancy, which is reasonable since our calculation relates to an idealised clay structure. 

Dynamic and Static Limitation in Multiscale Reaction Networks, Revisited... 

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