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.