Mechanical Resolution Of The Phasmatron Spectroscope

Version 1.0 of 6/3/2004-8:10 a.m.

The mechanical resolution is determined by the minimum angle
ΔN by which the viewing telescope can turn. For PHASMATRON's
angle measuring devices, this is 10^{-3} degrees (a millionth
of a degree). Converting to radians, ΔN=1.74532*10^{-5}
rad. Next we need dE/dλ so we can approximate Δλ.

We know that dE/dn=2/cos{sin^{-1}(n_{D}/2)}. This
for n_{D}=1.72803 gives:
dE/dn=3.9724624 rad. (In the area of sodium D), and
dn/dλ=1.2702*10^{-5}/A. (again in the area of sodium D)
=> dE/dλ=(dE/dn)(dn/dλ)=5.04582*10^{-5} rad/A.
Therefore we can approximate and use ΔE/Δλ=5.04582*10^{-5}
rad/A, and since ΔE almost eqs ΔN,
Δλ=ΔN/5.04582*10^{-5} rad/A. For
ΔN=1.74532*10^{-5} rad, this gives:

Δλ_{mechanical}=0.3458942A. (In the area
of the D lines)
Compare this with Δλ_{optical}=0.3866148A.
(In the area of the sodium D lines).
(The TRUE mechanical resolution can be calculated using the program in
section Measuring Wavelengths. If you
input N=60°, M=59.097° and on another run input
N=60° and M=59.098°, you can subtract the two
wavelength values found and get thus the Δλ. This way you
will get that the true mechanical resolution in the area of sodium
lines is 0.3564453A, which is very close to the value of Δλ_{mechanical}
found above.)

Suppose you wanted to calculate instead the mechanical resolution in
the area of the blue mercury line. (4358.35A) Then:

dn/dλ=(1.76197-1.74805)/|4358.35-4799.9107|=3.1524545*10^{-5}/A. dE/dn=2/cos{sin^{-1}(n_{4358.35}/2)}=4.22704 rad. dE/dλ=4.22704 rad*3.1524545*10^{-5}/A=1.3325551*10^{-4}
rad/A. Δλ_{mechanical}=1.74532*10^{-5}
rad/1.3325551*10^{-4} rad/A=0.131A.
Compare this with Δλ_{optical}=0.1152104A from
Lord Rayleigh's formula Δλ=λ/{B(dn/dλ)},
for dn/dλ=3.1524545*10^{-5}/A, B=12*10^{-8}A,
and λ=4358.35A.
Observe that the mechanical resolution varies according to area,
roughly in the same way that the optical resolution does.