the distribution functions are given as follows:
With jump rate distribution you describe situations where motional correlation times are distributed. This is typical for glassy polymers where correlation times can cover several decades. Weblab offers a log-Box, a log-Gauss and a Cole-Cole distribution. On a logarithmic scale these functions are all symmetric. With the center values τ0 and Ω0 for correlation time and jump rate and
the distribution functions are given as follows:
| logBox
|  |
| logGauss
|  |
| coleCole
|  |
For logBox and logGauss distributions you have to supply values for the center jump frequency Ω0 and the width of the distribution. The center value may be entered in kHz or in units of lg(Ω/kHz) or optionally of ln(Ω/kHz). The width can be specified as FWHMlg, i.e. full width at half midth, or optionally as 2σlg, both in decades (powers of ten). The σ in the top two distributions is defined as half the distance between the inflection points of the ln(x)-distribution (x=Ω0/Ω). Therefore σ = σlg * ln(10) = 2.303 σlg.
The relationship between FWHMlg = Δlg, given in decades, and σ is as follows:
The coleCole distribution has wider wings than the logGauss distribution and is characterized by the stretched exponential parameter β with values between 0 and 1.
Weblab assumes that the distribution of correlation times or jump rates is stationary and that it is allowed to superimpose spectra with different jump rates.
When pressing View, weblab shows a sketch of the distribution function, it will use in its calculation.
It is not possible, to consider a distribution of jump rates together with a distribution of cone or jump angles.
