Week 13: Interference and Diffraction 437
maximum. Again borrowing results from the previous section, we cansee that they
should occur at:
θminm = sin−^1(
nλ
N d)
(1052)
for the particular values:n 1 = N± 1
n 2 = 2N± 1
...
nm = mN± 1
... (1053)where the indexnmcan (as you can see) take on two values for eachm, one for the
minimum immediately before, the other for the minimum immediately after themth
principle maximum:nm=N∗m− 1 , N∗m+ 1 m= 1, 2 , 3 ... (1054)We now no longer neednm. We can directly write these angles in terms ofmalone as
(factoring):
θminm = sin−^1(
mλ
d± λ
N d)
(1055)
for each pair of values that bracket themth maximum.
We now make the small angle approximation for both the maxima and the minima. This
may well not be justified – many diffraction gratings will produce eventhe first principle
maximum at a relatively large angle – but it suffices for us to understand what they do
and the idea of “resolving power”, and we can always take the actual inverse sines if
needed for a particular actual grating. With this approximation, weget:θmaxm ≈(
mλ
d)
(1056)
and:
θmmin≈(
mλ
d ±λ
N d)
=θmaxm ±λ
N d (1057)This is just what we need to understand what a diffraction grating does: it makes an
absolutely perfectspectrometer, allowing us to cleanly resolve the spectral lines emitted
by hot glowing atoms and molecules and thereby both identify them and make many
inferences concerning their structure!
To see how this works, imagine that there are two “spectral lines”λ 1 andλ 2 being
emitted by a given atom (such as the two emitted by the Sodium atom,with D1 at
λ 1 = 589.592 nm and D2 atλ 2 = 588.995 nm, see homework). Thefirstprinciple max
forλ 1 occurs at the (presumed small) angle:θ 1 (λ 1 ) =λ 1
d(1058)
while that forλ 2 occurs at:
θ 1 (λ 2 ) =λ 2
d(1059)
These two lines areseparatedin angle by:∆θ 12 =|θ 1 −θ 2 |=λ 1 −λ 2
d