Travelling Gaussians

Kocbach, L. and Liska, R., (1995) Matrix elements of travelling gaussians for applications in the theory of atomic collisions, private communication. 

P IkiPlki Pjk) of nucleus k. In this product, the traveling Gaussians have a common width parameter w. The second factor on the right-hand side in equation (1) corresponds to the electronic wave function (for N electrons)... 

The atomic spin orbitals are represented in terms of traveling Gaussian basis functions of the form... 

From a well-known result of calculus, the definite integral on the right-hand side is s/n so M is just equal to the quantity of diffusing substance. The present solution is therefore applicable to the case where M grams (or moles) per unit surface is deposited on the plane x=x at t=0. In terms of concentration, the initial distribution is an impulse function (point source) centered at x=x which evolves with time towards a gaussian distribution with standard deviation JlQit (Figure 8. 13). Since the standard deviation is the square-root of the second moment, it is often stated that the mean squared distance traveled by the diffusion species is 22t. 

Based on the manner of derivation of the Gaussian equations in Section III, we see that the dispersion parameters a-y and are originally defined for an instantaneous release and are functions of travel time from release. Since the puff equations depend on the travel time of individual puffs or releases, the dispersion coefficients depend on this time, i.e., these coefficients describe the growth of each puff about its own center. This is basically a Lagrangian formulation. 

The basic Gaussian plume dispersion parameters are ay and a. The essential theoretical result concerning the dependence of these parameters on travel time is for stationary, homogeneous turbulence (Taylor, 1921). Consider marked particles that are released from the origin in a stationary, homogeneous turbulent flow with a mean flow in the x direction. The y component, y, of the position of a fluid particle satisfies the equation... 

The diSuse scatter arises because dislocations are defects which rotate the lattice locally in either direction. This gives rise to scatter, from near-core regions, which is not travelling in quite the same direction as the diffraction from the bulk of the crystal. This adds kinematically (i.e. in intensity not amplitude) and gives a broad, shallow peak that mnst be centred on the Bragg peak of the dislocated layer or substrate since all the local rotations are centred on the lattice itself. We can model the diffuse scatter quite well by a Gaussian or a Lorentzian function of the form ... 

If solute begins its journey through a column in an infinitely sharp layer with m moles per unit cross-sectional area of the column and spreads by diffusion as it travels, then the Gaussian profile of the band is described by... 

The theory of kinematic waves, initiated by Lighthill Whitham, is taken up for the case when the concentration k and flow q are related by a series of linear equations. If the initial disturbance is hump-like it is shown that the resulting kinematic wave can be usually described by the growth of its mean and variance, the former moving with the kinematic wave velocity and the latter increasing proportionally to the distance travelled. Conditions for these moments to be calculated from the Laplace transform of the solution, without the need of inversion, are obtained and it is shown that for a large class of waves, the ultimate wave form is Gaussian. The power of the method is shown in the analysis of a kinematic temperature wave, where the Laplace transform of the solution cannot be inverted. 

Molecules of solute travel as a zone in the chromatographic system. Recording of molecules eluting from the column yields a chromatogram (Fig. 1), whose characteristics are peaks. When peaks are symmetrical (Gaussian shape), retention times are taken at peak height. Since k is dimensionless, one can record retention distances or retention volumes on the chromatogram and... 

Intermolecular collisions do not cause large deviations from the ideal gas law at STP for molecules such as N2 or He, which are well above their boiling points, but they do dramatically decrease the average distance molecules travel to a number which is far less than would be predicted from the average molecular speed. Collisions randomize the velocity vector many times in the nominal round trip time, leading to diffusional effects as discussed in Chapter 4. If all of the molecules start at time t = 0 at the position x = 0, the concentration distribution C(x,t) at later times is a Gaussian ... 

As was discussed in Chapter 1 resolution, R, is a measure of the distance between two adjacent peaks in terms of the number of average peak widths than can fit between the band (zone) centers. Assuming symmetrical (Gaussian) peaks, when R = 1, peak separation is nearly complete with only about 2% overlap. This case was shown in Chapter 1, Figure 1-4. Resolution results from the physical and chemical interactions that occur as the sample travels through the column. It should, therefore, be no surprise that resolution may also be expressed in terms of the contribution of the individual column characteristics separation factor (selectivity, a), efficiency (narrowness of peak, N), and capacity factor (residence time, k ) of the first component. The equation that describes this interrelationship is... 

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