Titel:

NMR Studies in Hexaborides Diplomarbeit in experimenteller Festkörperphysik.

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ISBN: 3835101447 ISBN: 3835101447 ISBN: 3835101447 ISBN: 3835101447

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Next: External ProFit Modules Up: Fitting Spectra Previous: Fitting Spectra

Simulating Spectra

In simulating spectra we usually know a function

$\begin{displaymath}f: (\theta, m, \nu_{\mathrm{Q}}, ... ) \longmapsto \nu_m, \end{displaymath}$

(8.1)

where $\theta$ denotes the orientation of the crystal with respect to the z-axis and the other symbols have their usual meaning. The main problem now is that our goal is to have a function $w(\nu)$ which gives us the weight (the echo intensity) in every point of the spectrum. If we further assume the physical line to be of a Lorentzian or Gaussian shape (from now on denoted by $g(\nu)$ ) and the angle $\theta$ to be distributed as well (powdered sample) we are running into a typical problem we could treat with measure theory. But we don't want to go through it. Some intuitive arguments will do it. For more formal information see [#!david!#]. As already mentioned in equation (

) for every set of parameters we know the center of the NMR line. In our example we are dealing with a uniform spacial distribution of the powder particles but it is an easy task to modify the program for oriented powders. The spectrum then is (if we assume the spacial orientation to be the only parameter with a distribution $P(\nu)$ ):

$\begin{displaymath}w(\nu) = \int_0^{\pi} g(\nu - \nu_m(\theta)) P(\theta) d\theta, \end{displaymath}$

(8.2)

with $P(\theta )$ the probability that a nucleus is oriented with the angle $\theta$ with respect to the z-axis. In the case of a uniform spacial orientation we know that $P(\theta) = \mathrm{sin} \theta$ . The integral (

) is a simple convolution and can be numerically evaluated for example by adding up in bins. The spectra due to the single transitions can be added with a relative weight factor given by $\left\vert\left<m\left\vert I_x\right\vert m-1\right>\right\vert^2$ leading to the final spectrum stored in a memory array.

Since fitting affords the evaluation of the echo intensity at many different frequencies without changing any other parameters, the memory array can be used as a lookup table -- provided that ProFit supports this idea. The next section will give an introduction into technical details about how this support is given.

Next: External ProFit Modules Up: Fitting Spectra Previous: Fitting Spectra




Einführung in die Festkörperphysik (Broschiert) von Konrad Kopitzki, Peter Herzog

Siehe auch:



Einführung in die Festkörperphysik

Festkörperphysik

Grundkurs Theoretische Physik 3: Elektrodynamik

Quantenmechanik (QM I): Eine Einführung

Festkörperphysik: Einführung in die Grundlagen (Spr...

Quantenmechanik für Fortgeschrittene (QM II)

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