Gaussian profile, normalized

As normal peaks have a Gaussian profile, which approximates to an isosceles triangle, their area can be estimated by multiplying the height by... 

The random-walk model of diffusion can also be applied to derive the shape of the penetration profile. A plot of the final position reached for each atom (provided the number of diffusing atoms, N, is large) can be approximated by a continuous function, the Gaussian or normal distribution curve2 with a form ... 

The measurements of Rouse, Yih and Humphries (1952) [1] helped to generalize the temperature and velocity relationships for turbulent plumes from small sources, and established the Gaussian profile approximation as adequate descriptions for normalized vertical velocity (w) and temperature (7), e.g. 

A good example of dispersion is a plume of smoke being swept away by the wind. This plume will normally assume a Gaussian profile, a bell-shaped curve whose width is a function of the dispersion coefficient. If the amount of smoke emitted per time S is a constant, then the concentration of material in the smoke is given by (Seinfeld, 1985)... 

We begin, then, by writing concentration c as a Gaussian (or normal distribution) profile along coordinate y... 

Equation 5.15, as the foregoing proves, is a solution to Eq. 3.42. Furthermore, it becomes infinitely narrow at t = 0 and thus acquires the necessary 6-function form at the beginning. It can be shown that this expression normalizes to unit area (which means it applies to one mole of component, or one molecule if one chooses this unit of concentration, per unit area of cross section) when the constant in Eq. 5.15 equals (47r )) 1/2. Thus the normalized Gaussian profile is... 

A statistical analysis of the Raman intensities measured in the time series is shown in Fig. 10.8. For convenient comparison, the data are normalized to the maximum intensity. The region from 0 to the maximum intensity is divided into 20 sections and the number of events found in each section was counted and presented as a frequency. Kneipp et al. compared three different experiments (a) The analysis of the Raman data of lO methanol molecules with approximately the same (but unenhanced) intensities as the dye. The analysis yielded a Gaussian profile, (b) The analysis of 100 SERS measurements for, in... 

Rachedi et al. [22] continued Zeaton et al. s work by examining the behavior of a real jet fuel (JP-10), and compared it to Zeaton et al. s carbon dioxide data. Their results showed that carbon dioxide could be used as a surrogate fluid for JP-10, since their behaviors were very similar. Both Rachedi et al. and Zeaton et al. found that the injected fluid radial concentration profile was well described by a Gaussian profile when the radius was normalized by the jethalf-radius. The jethalf-radius was defined as the radius where the concentration is half of the maximum (centerline) value. 

Two basic types of line shape are encountered in spectroscopy, described by the general Lorentzian and Gaussian profile functions. The actual fine shape function of a transition is determined by the physical nature of particle-particle and particle-photon interactions encountered in the experiment (see Box 2.4). The general forms of the Lorentzian and Gaussian fine shapes are shown in Figure 2.7 they are shown normalized for equal half-widths, i.e. Avl = Avd = Av. 

The signals at [M IM ) were observed in the majority of cases to be much wider than those of normal fragment ions and were typically of Gaussian profile. 

Fig. 23-3 (a) The normalized attenuation coefficient yp of Eq. (23-23) as a function of A for various values of the normalized bending radius RJp when the fiber has a step profile, (b) The corresponding results for a Gaussian-profile fiber calculated from Eq. (23-25). 

Figure 1.8. Schematic frequency distributions for some independent (reaction input or control) resp. dependent (reaction output) variables to show how non-Gaussian distributions can obtain for a large population of reactions (i.e., all batches of one product in 5 years), while approximate normal distributions are found for repeat measurements on one single batch. For example, the gray areas correspond to the process parameters for a given run, while the histograms give the distribution of repeat determinations on one (several) sample(s) from this run. Because of the huge costs associated with individual production batches, the number of data points measured under closely controlled conditions, i.e., validation runs, is miniscule. Distributions must be estimated from historical data, which typically suffers from ever-changing parameter combinations, such as reagent batches, operators, impurity profiles, etc.

The normal or Gaussian distribution a bell-shaped frequency profile defined by the function... 

The energy of the second harmonic (SH) is found by integrating the SH intensity over space and time while assuming Gaussian spatial and hyperbolic secant temporal profiles. The normalized efficiency,, ... 

Thus the concentration ratio c/c0 is seen to be described at all times as a function of the single parameter z- The function P(z) defined by Equation (61) is the normal or Gaussian distribution function, Equation (C.10). Example 2.5 considers how the concentration profile of the diffusing species changes with time according to the normal distribution function. 

EXAMPLE 2.5 Unsteady State Variation of Concentration Profiles Due to Diffusion Gaussian Distribution. By consulting tables of the normal distribution function, draw curves that show the broadening of a band of material with time if the substance is initially at concentration c0 and in a plug of infinitesimal thickness at x = 0. Assume that the diffusion coefficient has the value 5 10 11 m2 s for this material. Use t = 106 and t = 3 106 s to see how the concentration profile changes with time. 

Fig. 2 Positional detection and mean-square displacement (MSD). (a) The x, y-coordinates of a particle at a certain time point are derived from its diffraction limited spot by fitting a 2D-Gaussian function to its intensity profile. In this way, a positional accuracy far below the optical resolution is obtained, (b) The figure depicts a simplified scheme for the analysis of a trajectory and the corresponding plot of the time dependency of the MSD. The average of all steps within the trajectory for each time-lag At, with At = z, At = 2z,... (where z = acquisition time interval from frame to frame) gives a point in the plot of MSD = f(t). (c) The time dependence of the MSD allows the classification of several modes of motion by evaluating the best fit of the MSD plot to one of the four formulas. A linear plot indicates normal diffusion and can be described by = ADAt (D = diffusion coefficient). A quadratic dependence of on At indicates directed motion and can be fitted by = v2At2 + ADAt (v = mean velocity). An asymptotic behavior for larger At with = [1 - exp (—AA2DAt/)] indicates confined diffusion. Anomalous diffusion is indicated by a best fit with = ADAf and a < 1 (sub-diffusive)...

Figure 2.8 Several energy-distribution functions, all normalized to the same height. The Lorentzian and Gaussian distributions are shown for equal fwhm values. The Voigt profile results from the convolution of the shown Lorentzian and Gaussian functions.

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