\(n^{ 2 }ln(\cfrac { 2\pi f }{ nc } )=A\cfrac { nc }{ 2\pi f } -n_{ o }^{ 2 }ln(a_{ \psi })-Aa_{ \psi }\)
since \(\lambda f=c\)
\(n^{ 2 }ln(\cfrac { 2\pi }{ n\lambda } )=A\cfrac { n\lambda }{ 2\pi } -n_{ o }^{ 2 }ln(a_{ \psi })-Aa_{ \psi }\)
which might be more convenient for data presented on the web. The following data are obtained from NIST at http://physics.nist.gov/cgi-bin/ASD/lines1.pl.
n |
\(\lambda\) (nm) | \(\cfrac { n\lambda }{ 2\pi }\) | \(n^{ 2 }ln(\cfrac { 2\pi }{ n\lambda \times10^{-9} } )\) |
|---|---|---|---|
| 1 | 121.5670 | 19.349 | 17.761 |
| 2 | 102.5728 | 32.651 | 68.950 |
| 3 | 97.2541 | 46.437 | 151.967 |
| 4 | 94.9742 | 60.464 | 265.940 |
| 5 | 93.7801 | 74.630 | 410.268 |
| 6 | 93.0751 | 88.883 | 584.494 |
| 7 | 92.6229 | 103.193 | 788.247 |
| 8 | 92.3151 | 117.543 | 1021.214 |
| 9 | 92.0970 | 131.923 | 1283.125 |
| 10 | 91.9351 | 146.326 | 1573.744 |
| 11 | 91.9349 | 160.956 | 1892.699 |
| 12 | 91.8125 | 175.354 | 2240.130 |
The lowest \(n\) is assigned to the highest \(\lambda\), arbitrarily for the time being. A preliminary plot of \(n\) vs \(\lambda\) and \(n\) vs \(f\) are given below.
Assuming that \(f\) increases with \(n\), the assignment of \(n\) to each \(\lambda\) seem to be appropriate, none of the values were folded out of place. A tabulated value for \(n^{ 2 }ln(\cfrac { 2\pi }{ n\lambda })\) and \(\cfrac { n\lambda }{ 2\pi }\) is also obtained.
A regression plot are made below,
The expression for \(a_\psi\) is, assuming that \(n^2_o=1\).
\(-ln(a_{ \psi })-Aa_{ \psi }=C\)
where \(C\) is the y-intercept,
\(ln(a_{ \psi })=-Aa_{ \psi }-C\)
As such \(a_\psi\) is given by the intersection of the line \(y=ln(a_\psi)\) and \(y=-Aa_{ \psi }-C\)
An estimate for \(a_\psi\) is obtained,
\(a_\psi=36\) nm
The problem with this value is that it is higher that the first two calculated values of \(x=\cfrac{n\lambda}{2\pi}\) in the table above. This suggests that the first two vales of \(n\) might have been mis-assigned, that both values might have been of higher values of \(n\).
More importantly, these data are presented to fit contemporary theories, which is not the same here. The temperature at which the data was obtained, for example, was not given any significance. The values of \(f\) given \(n\) are effected by temperature (cf. "Discreetly, Discrete \(\lambda\) And Discrete Frequency, \(f\)"). The first two values of the data do not fit the pattern of \(f\) presented in the post "Discreetly, Discrete \(\lambda\) And Discrete Frequency, \(f\)". In particular, the change in \(f\) theoretically, decreases between consecutive values of \(n=1\) and \(n=2\).
It could be that, as shown in the last section of the same post, that \(f\) corresponding to \(n=1\) is too high to be captured.
