Best Pavilion Angle  

Since no pavilion angle will reflect all incident light back to the crown, it is necessary to satisfy the next best criterion. For this, Tolkowsky chose that light returned to the table should be refracted so as to produce the optimum combination of intensity and dispersion. He says that these are related to the to the external angle as follows 5:

Intensity (I) is proportional to the cosine .......... I = cosf
Dispersion (D) is proportional to the sine .......... D = sinf

The product of these factors is I•D = sinfcosf =1/2•sin2f.
...which is extreme when 2f = ±90o ... thus when... f = ±45o

The corresponding internal angle q of this ray in diamond is ...
sinq = sinf / RI = sin(±45o) / 2.417 ... thus... q = ±17o


The average of all rays incident to the table is f = 0o, for which ... q = 0o substituting q = 0o and q = ±17o into Eq.(1c) gives ...
0o±17o = 4(45o-P) ... thus... P = 45o±4.25o = 40.75o, 49.25o

If P=40.75o
qmin = -16.32o, ..... qmax = +33.32o, ..... qmax-qmin = 49.64o
fmin = -42.8o, ..... fmax = +90o, ..... fmax-fmin = 132.8o

If P=49.25o
qmin = -24.82o, ..... qmax = +7.82o, ..... qmax-qmin = 32.64o
fmin = -90o, ..... fmax = +19.2o, ..... fmax-fmin = 109.2o

    Tolkowsky's reasons for selecting the lower value (40.75o) are not theoretically sound. There are certain advantages to the higher value (49.25o). but there are more reasons to select the lower value:

  1. The stone transmits more light (fmax-fmin = 132.8o vs. 109.2o),
  2. It can be viewed through a greater angle without seeing through it (f=42.8o vs 19.2o)

    * 5. Intensity would be proportional to the cosine if there were no partial reflection of the incident ray from the table, which starts from a minimum of 17.8% at f = 0o and is very significant at large values of f; ... and how is dispersion defined quantitatively?

    Therefore this formula, which is the basis of Tolkowsky's analysis, is only intuitive.

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