Modulating coefficients

The reduced partition functions of isotopic molecules determine the isotope separation factors in all equilibrium and many non-equilibrium processes. Power series expansion of the function in terms of even powers of the molecular vibrations has given explicit relationships between the separation factor and molecular structure and molecular forces. A significant extension to the Bernoulli expansion, developed previously, which has the restriction u = hv/kT < 2n, is developed through truncated series, derived from the hyper-geometric function. The finite expansion can be written in the Bernoulli form with determinable modulating coefficients for each term. They are convergent for all values of u and yield better approximations to the reduced partition function than the Bernoulli expansion. The utility of the present method is illustrated through calcidations on numerous molecular systems. 

The values of the modulating coefficients for the Chebyshev and best Jacobi polynomials for both fixed and sub-divided ranges of the... 

Table VIII. The Bernoulli-Modulating Coefficients for Various...

The four modulating coefficients listed for each range, order and term, m, are arranged as follows ... 

This is the first rule of the mean. It holds if the coeflScients c are independent of the isotopic substitutions, as they are for the Bernoulli series. The same condition is also satisfied for the Jacobi expansions, when a quantity common to a given molecular species, such as v max of the lightest isotopic molecule, is used for evaluating the modulating coefficients. For special combinations of isotopic pairs the rule holds to higher orders and... 

Using a quantity such as v, ax of HoO for evaluating the modulating coefficients for both isotopic pairs, the one-term Jacobi expansion predicts the quantum correction to be zero, thus satisfying the first rule of the mean. The first contribution to the quantum correction arises from n = 2 in the expansion. To describe the bending vibrations adequately, however, we need at least n = 3. In Table XVI, quantum corrections predicted by expansion formulae are compared with the exact quantum correction for the disproportionation among the isotopic water molecules. No entry is made for the Bernoulli series at 300°K. because the series does not exist at this temperature. 

Table 3.12 The modulating coefficients, reduced to their simplest ratios, for the linear combinations over the uj[here and vj [here from equation 3.21 to form the group orbitals of the F3 triangle displayed in the 3rd and 4th columns of Figure 3.6 using the vector surface harmonics of Table 3.11.

The most common frequency modulated continuous wave-form employs a linear frequency pattern (Richards, 2005), where B is the swept bandwidth, f o is the starting frequency, T is the sweep duration and //the linear frequency modulation coefficient. 

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