A diffraction grating is a regular array of scatterers, uniformly
spaced at some small distance d. In the late 19th century the
best gratings were made by ruling fine scratches on polished metal
surfaces at precisely regular intervals (these were reflection
gratings, not transmission gratings). The machines that did this
- ruling engines - were the most precise machine tools of their
time. Nowadays cheap diffraction gratings for the visible part of
the spectrum - wavelengths hundreds of nanometers - can be
easily produced by photographic and lithographic methods, but
doing similar things at the nanometer scale is part of the
emerging field of nanotechnology. (It should be noticed that
crystals are natural diffraction gratings at the angstrom scale.)
We will use a transmission grating oriented normal to the light
source. The familiar grating equation
ml = dsinq
(1)
reminds us that for small integers m, light of wavelength
l will interfere constructively in a beam deviated by
angle q from the normal to the grating of spacing d.
Since m can take various integer values, there are actually
various q's where the constructive interference occurs,
labelled by m, which is called the order of the diffracted
beam. In particular, m=0 corresponds to q = 0, i.e., no
deviation at all - this is just the beam going straight through
the grating. But for m=1, the first order diffraction, the
angle depends on l in a very simple and definite way,
making this a method for measuring l, the wavelength of
light. It is clear from the grating equation above that to get an
appreciable deviation, d, the grating spacing, must be not too
much larger than l. That is why it is not so easy to make
a grating.
The applet below shows several orders of diffraction for each of
several wavelengths, indicated roughly by their color. You can
choose different grating spacings using the little window at the
bottom.
When there are just 500 lines/cm, the diffraction
angles are almost too small to see, but if you increase the
density of lines to 1500 lines/cm (this decreases the spacing d)
you begin to see noticeable spreading out of the spectrum. The
first order diffraction is nearest to the straight-ahead
direction, and occurs on either side of zero (corresponding to m=1
and m=-1). The spreading out is more noticeable in the higher
orders. At 2500 lines/cm, the higher order diffraction spectra
have spread out so much that they begin to overlap in a slightly
confusing way. Notice how the 4th order violet is deviated less
than the 3rd order red, for example. In the experiment we will
see the Balmer lines in as many orders as possible.
The energy levels of a hydrogenic atom in Bohr theory are given by
En=-
1
2
Z2a2mc2/n2
(2)
where Z is the charge on the nucleus (Z=1 for hydrogen), the
fine-structure constant a is the dimensionless combination
a =
e2
4pe0(h/2p) c
»
1
137
(3)
the reduced mass m is given by
m =
memN
me+mN
(4)
(here mN is the nuclear mass and me is the electron mass),
c is the speed of light in vacuum, and n is the principal
quantum number in Bohr theory. For future reference, note that
1/m = 1/me+1/mN, and, since me << mN, m » me.
When the atom makes a transition between two different energy
levels, it emits a photon with an energy equal to the energy
difference between the two levels
Ephoton=En-Em=hnnm=hc/lnm
(5)
where subscripts n and m indicate the initial and final
hydrogen atom states, c is the speed of light, and h is
Planck's constant (6.67×10-34 J-s). Various transitions are indicated schematically (not to scale) in the figure below.
Combining Eqs. (2) and (5), we have the
wavelength for light emitted in the transition as
1
lnm
=
nnm
c
= RH
æ è
1
m2
-
1
n2
ö ø
(6)
where RH is the Rydberg constant. You can express RH in
terms of more fundamental constants by checking the derivation of
Eq. (6) in detail.
As an historical matter, it was already known empirically by 1913,
when Bohr proposed his theory, that the wavelengths in the
hydrogen atom spectrum were described by Eq. (6) with
a certain constant RH. What was most impressive about Bohr's
theory, which seemed wildly ad hoc in several places, was
that it ``explained" the value of RH - that is, it gave the
right number.
Of all the wavelengths lnm emitted by hydrogen atoms,
only a few with m=2 and n not too large are in the visible
part of the spectrum. These are the Balmer lines. To see them
with the Gaertner spectrometer, put a Balmer tube at the input
aperture, and look with the moveable telescope for the diffracted
beams. You should find the same lines in more than one order, as
discussed above.
Measure the angular position of all the lines you can see on each
side of the beam. Then the deviation angle q for the first
order red line, for example, can be found as half the difference
in its angular position as found on one side and the other. This
method is equivalent, in principle, to measuring the angular
difference from the central beam position, and eliminates a
systematic error in relying on the imperfectly known beam
position. In this way determine the wavelength of the red line,
and repeat in as many orders as possible. Hence arrive at a value
for its wavelength together with an estimate of uncertainty.
Repeat for the other Balmer lines.
According to Bohr theory, these measured wavelengths should fit
the form
1
ln
= RH
æ è
1
22
-
1
n2
ö ø
(7)
for n=3,4,..., i.e. 1/ln should be a linear function
of 1/n2, with a certain slope and intercept. Make the implied
plot and see if it is well fitted by a straight line. Hence find
the Rydberg constant.
File translated from
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by
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version 3.00. On 22 Aug 2001, 11:36.
