Gaussian instrument function

We may convert this relationship to half-width form by remembering that aL = 2 AxL/3 and expressing aR in terms of the Gaussian instrument function half-width AxR ... 

This technique assumes a Gaussian spreading function and thus does not take into account skewness or kurtosis resulting from instrumental considerations. It can, however, be modified to accommodate these corrections. The particle size averages reported here have been derived usino the technique as proposed by Husain, Vlachopoulos, and Hamielec 23). 

The particle size analysis techniques outlined earlier show promise in the measurement of polydispersed particle suspensions. The asumption of Gaussian instrumental spreading function is valid except when the chromatograms of standard latices are appreciably skewed. Calc ll.ation of diameter averages indicate a fair degree of insensitivity to the value of the extinction coefficient. 

Pulse-probe transient absorption data on the rise time of prompt species such as the aqueous electron can be used to measure the instrument response of the system and deduce the electron pulse width. Figure 7 shows the rise time of aqueous electron absorbance measured with the LEAF system at 800 nm in a 5 mm pathlength cell. Differentiation of the absorbance rise results in a Gaussian response function of 7.8 ps FWHM. Correcting for pathlength, the electron pulse width is 7.0 ps in this example. 

Oxford Instruments described the search algorithm used in GammaTrac (no longer available) as a correlation method using a zero-area correlation function instead of a Gaussian search function. There are practical advantages to... 

Fig. 6. The difference Compton profile for Ti and TiH experimental and theoretical for several models. The line A shows the experimental line profile difference normalised to two electrons before applying deconvolution for instrumental broadening and smoothing on row data of Ti and TiH. The solid line B shows the difference profile of Bandstructure calculations according to APW self-consistent approximation after a convolution with a Gaussian resolution function of a=0.30 a.u. The remaining lines C-F are the theoretical difference profiles of indicated model after convolution with the previous resolution function. 

The total peak profile is the sum of these two components, with a relative weight that depends on 0 and I. The Bragg component is mathematically a 5-function, but is in reality broadened by instrumental and sample imperfections. The shape of the diffuse component depends on the form of the correlation function. Many forms are possible, but two common ones are an exponential correlation function, leading to a Lorentzian line shape, and a Gaussian correlation function, leading to a Gaussian Hne shape. One can also use correlation functions that describe a preferred distance (e.g., island-island correlations) or with an in-plane anisotropy [42]. 

The time resolution of the instrument determines the wavenumber-dependent sensitivity of the Fourier-transformed, frequency-domain spectrum. A typical response of our spectrometer is 23 fs, and a Gaussian function having a half width... 

The most intense 826-cm band is broader than the other bands. The broadened band suggests a frequency distribution in the observed portion of the surface. Indeed, the symmetric peak in the imaginary part of the spectrum is fitted with a Gaussian function rather than with a Lorenz function. The bandwidth was estimated to be 56 cm by considering the instrumental resolution, 15 cm in this particular spectrum. This number is larger than the intrinsic bandwidth of the bulk modes [50]. 

In order to properly take into account the instrumental broadening, the function describing the peak shape must be considered. In the case of Lorentzian shape it is Psize = Pexp - instr while for Gaussian shape p = Pl -Pl tr- In the case of pseudo-Voigt function, Gaussian and Lorentzian contributions must be treated separately [39]. 

Lorentzians, Gaussians, and combinations of both like pseudo-Voigt functions 38Frequently the effect of instrumental broadening is tacitly considered as already eliminated. 

The complete powder XRD profile (either for an experimental pattern or a calculated pattern) is described in terms of the following components (1) the peak positions, (2) the background intensity distribution, (3) the peak widths, (4) the peak shapes, and (5) the peak intensities. The peak shape depends on characteristics of both the instrument and the sample, and different peak shape functions are appropriate under different circumstances. The most common peak shape for powder XRD is the pseudo-Voigt function, which represents a hybrid of Gaussian and Lorentzian character, although several other types of peak shape function may be applicable in different situations. These peak shape functions and the types of function commonly used to describe the 20-dependence of the peak width are described in detail elsewhere [22]. 

All impactor and filter samples were analyzed for up to 45 elements by instrumental neutron activation analysis (INAA) as described by Heft ( ). Samples were irradiated simultaneously with standard flux monitors in the 3-MW Livermore pool reactor. The x-ray spectra of the radioactive species were taken with large-volume, high-resolution Ge(Li) spectrometer systems. The spectral data were transferred to a GDC 7600 computer and analyzed with the GAMANAL code (1 ), which incorporates a background-smoothing routine and fits the peaks with Gaussian and exponential functions. 

Another approach is the characterization of peaks with a well-defined model with limited parameters. Many models are proposed, some representative examples will be deaaib i. Wefl known is the Exponentially Modified Gaussian (EMG) peak, i.e. a Gaussian convoluted with an exponential decay function. Already a few decades ago it was recognized that an instrumental contribution such as an amplifier acting as a first-order low pass system with a time constant, will exponentially modify the... 

Let us establish the required relationships more precisely. Consider a narrow idealized rectangular absorption line AT(x) = rect(x/2 AxL) having half-width AxL and centered at x = 0. Its variance is easily found to be <7l = (2 Axl/3)2. Its area is 2 AxL. Now, let us assume that this line is being used to measure an instrument response function exp( —x2/2cr2) that has Gaussian shape and variance ... 

Because the instrument response function must have unit area, the area under the narrow line is preserved by the measurement process. Recalling that this area is 2 AxL, we may write the absorptance amplitude AL of the observation (which is Gaussian) in terms of its half-width and 2 AxL ... 

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