Approximation to average

Perhaps our equations are only very close approximations to average values that... 

Dose measure that uses the absorbed dose in an acrylic plastic phantom. Weighted CTDI (CTDIw) is approximation to average dose in the x-y plane for phantom and so is an approximation of the dose delivered to a cross section of the patient s anatomy. CTDI is measured in milligrays. CTDIw does not consider exposure variation along the z-axis. 

The potential energy term U, R) does also depend on the electron coordinates r, but we use the Born-Oppenheimer approximation to average out that dependence. That is why r seems to disappear from the Hamiltonian. 

For qualitative insight based on perturbation theory, the two lowest order energy eorreetions and the first-order wavefunetion eorreetions are undoubtedly the most usetlil. The first-order energy eorresponds to averaging the eflfeets of the perturbation over the approximate wavefunetion Xq, and ean usually be evaluated without diflfieulty. The sum of aJ, Wd ds preeisely equal to tlie expeetation value of the Hamiltonian over... 

Figure A3.4.4 summarizes these steps in one scheme. Different theories of elementary reactions represent different degrees of approximations to certain averages, which are observed in experiments.

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

These methods, which probably deserve more attention than they have received to date, simultaneously optimize the positions of a number of points along the reaction path. The method of Elber and Karpins [91] was developed to find transition states. It fiimishes, however, an approximation to the reaction path. In this method, a number (typically 10-20) equidistant points are chosen along an approximate reaction path coimecting two stationary points a and b, and the average of their energies is minimized under the constraint that their spacing remains equal. This is obviously a numerical quadrature of the integral s f ( (.v)where... 

In applications, one is often interested in approximating time averages over a time interval [0, T] via associated mean values of a , k = 1. ..Tfr. For T (or r) small enough, the above backward analysis may lead to much better error estimates than the worst case estimates of forward analysis. 

In the chapter on reaction rates, it was pointed out that the perfect description of a reaction would be a statistical average of all possible paths rather than just the minimum energy path. Furthermore, femtosecond spectroscopy experiments show that molecules vibrate in many dilferent directions until an energetically accessible reaction path is found. In order to examine these ideas computationally, the entire potential energy surface (PES) or an approximation to it must be computed. A PES is either a table of data or an analytic function, which gives the energy for any location of the nuclei comprising a chemical system. 

However, any average drop size is fictitious, and none is completely satisfactory. For example, there is no way in which the high surface and transfer coefficients in small drops can be made avail le to the larger drops. Hence, a process calculation based on a given droplet size describes only what happens to that size and gives at best an approximation to the total mass. 

A useful approximation to estimate the possibility of a particular reaction which depends on internal heat generation to produce the products in the proper state for separation is to ignore the heat losses from die reactor, and assumes an average heat capacity calculated from die Neumann-Kopp law... 

In order to get this expression into a more familiar form (equation 9.7), we now consider the zeroth-order approximation to /. We assume that / is locally a Maxwell-Boltzman distribution, and treat the density p, temperature T[x,t) = < V — u p> (where k is Boltzman s constant), and average velocity u all as slowly changing variables with respect to x and t. We can then write... 

The capacity of a glass vessel varies with the temperature, and it is therefore necessary to define the temperature at which its capacity is intended to be correct in the UK a temperature of 20 °C has been adopted. A subsidiary standard temperature of 27 °C is accepted by the British Standards Institution, for use in tropical climates where the ambient temperature is consistently above 20 °C. The US Bureau of Standards, Washington, in compliance with the view held by some chemists that 25 °C more nearly approximates to the average laboratory temperature in the United States, will calibrate glass volumetric apparatus marked either 20 °C or 25 °C. 

In a first approximation the average size of pore diameter has no effect on porosity, even though a superficial view leads to other conclusions. A mental experiment may be of assistance imagine a pore and its outside wall, decrease both to identical scale, then the ratio of void to outside volume remains constant. Of course the re-... 

It has been found possible to evaluate s0 theoretically by means of the following treatment (1) Each electron shell within the atom is idealised as a uniform surface charge of electricity of amount — zte on a sphere whose radius is equal to the average value of the electron-nucleus distance of the electrons in the shell. (2) The motion of the electron under consideration is then determined by the use of the old quantum theory, the azimuthal quantum number being chosen so as to produce the closest approximation to the quantum... 

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