Delta-function approximation

A. L. Crosbie and G. W. Davidson, Dirac-Delta Function Approximations to the Scattering Phase Function, Journal of Quantitative Spectroscopy and Radiative Transfer, 33, p. 391,1985. 

Again the delta-function approximation and the plane-wave approximation can be used. For example, using the delta-function approximation, we obtain... 

Notice that the rotational energy is BJ I + 1) and the rotational partition function is kTIB. In the delta-function approximation, it has been shown in Section IV that Eq. (167) can be expressed as... 

Magnetic-Quenching Rate Coefficients Calculated Using the Delta-Function Approximation... 

In this section, we have demonstrated how to calculate the intramolecular magnetic quenching rate of molecules. For this purpose, we have chosen 12 and used the delta-function approximation for the dissociating state wave-function. The delta-function approximation is commonly used in photodissociation of molecules. Dunn (1968) has shown that for H2 and D2, the use of the delta-function approximation gives good agreement for photodissociation from the low vibrational levels compared with results using more accurate wavefunctions. 

The tiny difference between hard pulses and their delta-function approximation can be exploited to control coherence. Variants on the magic echo that work despite a large spread in resonance offsets are demonstrated using the zeroth- and first-order average Hamiltonian terms, for C-13 NMR in Cgo- The Si-29 NMR linewidth of silicon has been reduced by a factor of about 70 000 using this approach, which also has potential applications in MR microscopy and imaging of solids. 

As demonstrated in Chapter 1, Equation (1.4.63), using the alternating lattice approach and the delta function approximation for M F), the covalent energy Ecov can be expressed in the following way ... 

We assume that the sequential errors are not correlated in time, we can write the probability of sampling a sequence of errors as the product of the individual probabilities. We further use the finite time approximation for the delta function and have ... 

The application in [24] is to celestial mechanics, in which the reduced problem for consists of the Keplerian motion of planets around the sun and in which the impulses account for interplanetary interactions. Application to MD is explored in [14]. It is not easy to find a reduced problem that can be integrated analytically however. The choice /f = 0 is always possible and this yields the simple but effective leapfrog/Stormer/Verlet method, whose use according to [22] dates back to at least 1793 [5]. This connection should allay fears concerning the quality of an approximation using Dirac delta functions. 

In the opposite case of slow flip limit, cojp co, the exponential kernel can be approximated by the delta function, exp( —cUj t ) ii 2S(r)/coj, thus renormalizing the kinetic energy and, consequently, multiplying the particle s effective mass by the factor M = 1 + X The rate constant equals the tunneling probability in the adiabatic barrier I d(Q) with the renormalized mass M, ... 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

In order to improve upon the mean-field approximation given in equation 7.112, we must somehow account for possible site-site correlations. Let us go back to the deterministic version of the basic Life rule (equation 7.110). We could take a formal expectation of this equation but we first need a way to compute expectation values of Kronecker delta functions. Schulman and Seiden [schul78] provide a simple means to do precisely that. We state their result without proof... 

Curve fitting using a delta function for the pulse input for a TAP reactor should be limited to the latter % part of the response curve for curves of FWHM < 3 times pulse width, while for curves with FWHM > 4 times pulse width, it is a fair approximation fijr most of the curve. The assumption of a zero concentration at the reactor outlet is not good evrai for a pumping speed of 1,500 Is and broad response curves with FWHM > 1000 ms. 

Using a perturbative analysis of the time-dependent signal, and focusing on the interference term between the one- and two-photon processes in Fig. 14, we consider first the limit of ultrashort pulses (in practice, short with respect to all time scales of the system). Approximating the laser pulse as a delta function of time, we have... 

The partition function, Z(4>y), cannot be calculated exactly. It could be rewritten using the integral representation of the functional Dirac delta function and evaluated within the saddle place approximation. The calculations lead to the following expression [36,126,128] ... 

The basis idea behind multi-environment models is that the mixture fraction at any location in the reactor can be approximated by a distribution function in the form of a sum of delta functions as follows ... 

With two (or more) internal coordinates, numerical approaches for the PBE using sectional methods become intractable in the context of CFD. A practical alternative is to use a finite number of samples to approximate the NDF in terms of delta functions as follows ... 

For the moment estimates, we have seen that the composition PDF, /, (delta functions (i.e., the empirical PDF in (6.210)). However, it should be intuitively apparent that this representation is unsatisfactory for understanding the behavior of fyiir) as a function of fj. In practice, the delta-function representation is replaced by a histogram using finite-sized bins in composition space (see Fig. 6.5). The histogram h, (k) for the /ctli cell in composition space is defined by... 

This implies that the exponents and y defined above are 0 = y = 2( = d) for a first-order transition. Since the symmetry around if = 0 is preserved for finite L, there is no shift of the transition. This feature is different, however, if we consider temperature-driven first-order transitions , since there is no symmetry between the disordered high-temperature phase and the ordered low-temperature phase. In order to understand the rounding of the delta-function singularity of the specific heat, which measures the latent heat for L- oo, it now is useful to consider the energy distribution, for which again a double Gaussian approximation applies ... 

A real sampler, as shown in Fig. 18.1, is closed for a finite period of time. This time of closure is usually small compared with the sampling period. Therefore the real sampler can be closely approximated by an impulse sampler. An impulse sampler is a device that converts a continuous input signal into a sequence of impulses or delta functions. Remember, these are impulses, not pulses. The height of each of these impulses is infinite. The width of each is zero. The area of the impulse or the strength of the impulse is equal to the magnitude of the input function at the sampling instant. 

Here s is the Laplac variable and c. (s) the Laplace transform of the inpu. When c. (t) can be approximated by a Dirac delta function, c (s) = 1 and the right hand side of Equation 8 is the Laplace transform of the solute concentration at any z. 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

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