Spectrum distribution line shape function

This is the well-known formula that states that the resolution of a diffraction grating increases (and Jv decreases) with increasing order n of diffraction and number N of lines in the grating >. For the case of three lines, or for any other spectrum, the intensity is measured with a grating spectrometer as fimction of yd (for several orders of diffraction) (see Fig. 5). The data obtained in this way are then easily converted to / (A) or I ( ) and the problem of determining the spectral distribution I ( ) is solved. It should be noted that the linewidth obtained (see Fig. 5) is influenced by the limited resolution of the instrument and that the line-widths of the three laser lines are assumed to be actually much smaller. In other words, we have been discussing the properties of the instrument line-shape function of a diffraction grating. 

The intensity, I(Hr) is proportional to the number of radicals contributing to the spectrum at the field Hr. This number was given by a distribution function P(a), for instance assumed to be a Gaussian distribution, with the distribution function, exp(asin a). A computer simulation was carried out for the orientation function, fc and the angles Xi, X2, and X3. The ESR spectrum, 1(H) can be calculated from the convolution of P(a) and the line shape function G as follows. 

Methyl radicals formed on a silica gel surface are apparently less mobile and less stable than on porous glass (56, 57). The spectral intensity is noticeably reduced if the samples are heated to —130° for 5 min. The line shape is not symmetric, and the linewidth is a function of the nuclear spin quantum number. Hence, the amplitude of the derivative spectrum does not follow the binomial distribution 1 3 3 1 which would be expected for a rapidly tumbling molecule. A quantitative comparison of the spectrum with that predicted by relaxation theory has indicated a tumbling frequency of 2 X 107 and 1.3 X 107 sec-1 for CHr and CD3-, respectively (57). 

If the spectral lines are broadened but have the same profile, the moments of the spectrum can be expressed in terms of the moments of the non-broadened spectrum and the moments of a separate line. When the moments of the separate line do not exist (for natural line shape), the global characteristics of the spectrum can be found simulating its structure by Monte-Carlo or distribution function methods and then convoluting each calculated line. 

For poly crystalline samples, the parentheses in Eq. (11) are replaced by a distribution function. The line shape of the powder spectrum has been discussed for spin / = 3/217 and / = 114 nuclei with various values of rj. In general, for / = 1 it shows singularities at (yH0(2ir) 3e2qQ( — q)fSh, shoulders or distribution edges at (yH0/2v) ... 

The comparison shows that a part of the discrepancy between experiment and theory comes from the data analysis. The left wing of the simulated distfibution (filled bars) is not completely described by the LT9.0 analysis of the computer-generated spectrum (dashed line, mean of 0.282 nm). This may be a consequence of the shape of asiX), assumed in the LT analysis to be a log normal function, the interference of the o-Ps with the e" " lifetime distribution, and the low total o-Ps intensity of 9.0% for this material. 

Fig. 6. Radial distribution function of counterions and exchange broadening in CW ESR spectra, (a) The experimental line shape of 0.5 mM FS / 8 mM PDADMAC in water-giyc-erol (solid line) cannot be fitted without considering exchange broadening (dotted line), (b) Fit residuals for the central line in the spectrum of 0.5 mM FS /7.5 mM PDADMAC in water-ethanol assuming different radial distribution functions c(r). (c) Spectrum (top trace) and fit residual (bottom trace) for c(r) r for 0.5 mM FS / 7.5 mM PDADMAC in...

---