Numerical calculation

Normally one might expect that if the transition probability vanishes on resonance it also vanishes off resonance. However, such is not the case. When the transition probability is calculated off resonance, by numerically solving Eqs. (14.16) using a Taylor expansion method, it is nonzero for both v E and v 1E.14,16 In Fig. 14.6 we show the transition probabilities obtained using two different approximations for v E, and vlE for the 17s (0,0) collisional resonance.16 To allow direct comparison to the analytic form of Eq. (14.21) we show the transition probabilities calculated with EAA = VBB = 0. For these calculations the parameters ju2l = pLz, = 156.4 ea0, b = 104ao, and v = 1.6 x 10-4 au have been used. The resulting transition probability curves are shown by the broken lines of Fig. 14.6. As shown by Fig. 14.6 these curves are symmetric about the resonance position. The vlE curve of Fig. 14.6(b) has an approximately Lorentzian form, but the v E curve of Fig. 14.6(a), while it vanishes on resonance as predicted by Eq. (14.24), has an unusual double peaked structure. 

Since the transition probabilities of Fig. 14.6 are obtained numerically, it is not appreciably more difficult to include the diagonal matrix elements AA and l BB arising from the permanent dipole moments of the atomic states in the field. The 

To convert the transition probabilities to cross sections we must integrate over impact parameter using 

It is useful to consider the case of exact resonance, which may be worked out analytically. At resonance, using either Eq. (14.18) or Eq. (14.21) the transition probability is given by 

Recalling that D we see that Eq. (14.27) resembles Eq. (14.5) which was derived in a very simple way. 

The stress is proportional to the shear gradient. A similar stress would be exerted by an elastic body of stiffness y/ro- 

Central to the view outlined the previous section is the notion that slipping layers will in most practical situations be laterally heterogeneous. For a start, we treat the slipping layer as an assembly of nanobubbles and analyze this geometry by means of a finite element analysis. 

A/ and AT can be predicted even for heterogeneous systems making use of SLA. The latter states that the complex frequency shift Af = A/ + iAT is proportional to the area-averaged tangential stress at the resonator surface (Eq. 8.5). 

18 further elaborates on the meaning of the load impedance and the electromechanical analogy. SLA is applicable to a wide range of samples. That certainly includes the Sauerbrey film. For the Sauerbrey film, the stress is mf —a/uo), where njf is the mass per unit are, uo is the oscillation amplitude, and -a uo is acceleration. The speed, u, is given as i uo. Inserting these relations, one finds the Sauerbrey equation (A/ = recovered. 

Since Af depends linearly on the load, SLA can be applied in an average sense. One may replace the stress, a, by an area-averaged stress, (cr). Assume that the sample is a complex material, such as a 

The symbol for multiplication is often omitted so that the formula would be written V = hwl. If two symbols are written side by side, it is understood that the quantities represented by the symbols are to be multiplied together. Another example is 

There are several useful rules of thumb that allow you to determine the proper number of significant digits in the result of a calculation. For multiplication of 

EXAMPLE 1.3 What is the voliune of a rectangular object whose length is given as 7.78 m, whose width is given as 3.486m, and whose height is 1.367m  

SOLUTION We denote the volume by V and obtain the volume by multiplication, using a calculator. 

The calculator delivered 10 digits, but we round the volume to 37.1 m, since the factor with the fewest significant digits has three significant digits.  

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This factor is the intermediate parameter employed in numerous calculational methods. For petroleum cuts obtained by distillation from the same crude oil, the Watson factor is generally constant when the boiling points are above 200°C. 

At sufficiently high frequency, the electromagnetic skin depth is several times smaller than a typical defect and induced currents flow in a thin skin at the conductor surface and the crack faces. It is profitable to develop a theoretical model dedicated to this regime. Making certain assumptions, a boundary value problem can be defined and solved relatively simply leading to rapid numerical calculation of eddy-current probe impedance changes due to a variety of surface cracks. 

A more subtle point concerns scaling of -xwith density. Among tlie various possibilities that exist, either employing local densities obtained from numerically calculated radial distribution fiinctions [38]... 

Predicting the solvent or density dependence of rate constants by equation (A3.6.29) or equation (A3.6.31) requires the same ingredients as the calculation of TST rate constants plus an estimate of and a suitable model for the friction coefficient y and its density dependence. While in the framework of molecular dynamics simulations it may be worthwhile to numerically calculate friction coefficients from the average of the relevant time correlation fiinctions, for practical purposes in the analysis of kinetic data it is much more convenient and instructive to use experimentally detemiined macroscopic solvent parameters. 

Often in numerical calculations we detennine solutions g (R) that solve the Scln-odinger equations but do not satisfy the asymptotic boundary condition in (A3.11.65). To solve for S, we rewrite equation (A3.11.65) and its derivative with respect to R in the more general fomi ... 

The numerical calculations have been done on a two-coordinate system with q being a radial coordinate and <() the polar coordinate. We consider a 3 x 3 non-adiabatic (vector) mabix t in which and T4, aie two components. If we assume = 0, takes the following form,... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Acids and bases are a big part of organic chemistry but the emphasis is much different from what you may be familiar with from your general chemistry course Most of the atten tion m general chemistry is given to numerical calculations pH percent loniza tion buffer problems and so on Some of this returns m organic chemistry but mostly we are concerned with the roles that acids and bases play as reactants products and catalysts m chemical reactions We 11 start by reviewing some general ideas about acids and bases... 

Before doing this, however, it is informative to compare the sensitivity of the four colligative properties in the determination of molecular weight. In the following example this is done by making the appropriate numerical calculations. 

The effect of the charge as well as that of the indifferent electrolyte, then, is contained in the term in brackets. A numerical calculation is probably the easiest way to examine this effect. This is illustrated in the following example. 

The norm is useful when doing numerical calculations. If the computer s floating-point precision is 10" , then K = 10 indicates an ill-conditioned matrix. If the floating-point precision is I0" (double precision), then a matrix with K = I0 may be ill-conditioned. Two other measures are useful and are more easily calculated ... 

Numerical techniques therefore do not yield exact results in the sense of the mathematician. Since most numerical calculations are inexact, the concept of error is an important feature. The error associated with an approximate value is defined as... 

The local equilibrium curve is in approximate agreement with the numerically calculated profiles except at very low concentrations when the isotherm becomes linear and near the peak apex. This occurs because band-spreading, in this case, is dominated by adsorption equilibrium, even if the number of transfer units is not very high. A similar treatment based on local eqnihbrinm for a two-component mixture is given by Golshau-Shirazi and Gniochou [J. Phys. Chem., 93, 4143(1989)]. 

In the numerical calculations, an elastic-perfectly-plastic ductile rod stretching at a uniform strain rate of e = lO s was treated. A flow stress of 100 MPa and a density of 2700 kg/m were assumed. A one-millimeter square cross section and a fracture energy of = 0.02 J were used. These properties are consistent with the measured behavior of soft aluminim in experimental expanding ring studies of Grady and Benson (1983). Incipient fractures were introduced into the rod randomly in both position and time. Fractures grow... 

Exploration of the region 0 < T < requires numerical calculations using eqs. (2.5)-(2.7). Since the change in /cq is small compared to that in the leading exponential term [cf. (2.14) and (2.18)], the Arrhenius plot k(P) is often drawn simply by setting ko = coo/ln (fig. 5). Typical behavior of the prefactor k and activation energy E versus temperature is presented in fig. 6. The narrow intermediate region between the Arrhenius behavior and the low-temperature limit has width... 

From the very simple WKB considerations it is clear that the tunneling rate is proportional to the Gamov factor exp —2j[2(F(s(0) — )] ds, where s Q) is a path in two dimensions Q= 61)62 ) connecting the initial and flnal states. The most probable tunneling path , or instanton, which renders the Gamov factor maximum, represents a compromise of two competing factors, the barrier height and its width. That is, one has to optimise the instanton path not only in time, as has been done in the previous section, but also in space. This complicates the problem so that numerical calculations are usually needed. 

Fig. 26. Arrhenius plot [ln(fc/a>o) against a>o /2it] for the PES (4.28) with Q = 0.1, C = 0.0357, = 1, F /a>o = 3. Solid line shows instanton result separate points, numerical calculation data from Hontscha et al. [1990] and dashed line, low-temperature limit using (3.32) for fc,D.

An example of a numerically calculated trajectory in a symmetric double well is presented in fig. 29 for the Hamiltonian... 

We have data from two independent measurements and two parameters to be fitted with these data. The more data we have, the more reliable will be the parameter fitting. Changing the values and and repeating the numerical calculation of the pressure loss by Eq. (14.126), we found that the best coincidence with the empirical data presented was obtained by... 

It should be noted that doublons, in the range of their existence (Eq. (87)), grow faster than dendrites at the same parameters A and e. This statement is confirmed by numerical calculations [94]. 

The initial development of a cellular structure from an originally flat interface has been at least partially understood [130]. Let us look only at the large-wavelength A limit (for more details see [122]). In the numerical calculations it was found [123] that for fixed cell-spacing A at increasing velocity a tail instability occurs. A side branch in the groove between two... 

T. Jung, G. Mueller. Amplitudes of doping striations comparison of numerical calculations and analytical approaches. J Cryst Growth 171 313, 1997. 

The difference between this equation and the equation for ex,wo that the internal energy of the air that is displaced by the expanded gases is taken into account. Note that, when y, is equal to yo, Eq. (6.3.13) is equivalent to Eq. (6.3.2). Aslanov and Golinskii advocate the use of the energy E bx.ag the energy of the explosion. They claim that this gives a better correlation with numerical calculations and with experiments. 

The procedure, in analyzing kinetic data by numerical integration, is to postulate a reasonable kinetic scheme, write the differential rate equations, assume estimates for the rate constants, and then to carry out the integration for comparison of the calculated concentration-time curves with the experimental results. The parameters (rate constants) are adjusted to achieve an acceptable fit to the data. Carpen-(ej-48. pp. 76-81 some numerical calculations. Farrow and Edelson and Porter... 

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