Download MEASUREMENT OF THE WAVELENGTHS OF LIGHT and STUDY

Brown University
Physics Department
Physics 0040
Lab 6
MEASUREMENT OF THE WAVELENGTHS OF LIGHT
and
STUDY OF THE SPECTRUM OF HYDROGEN
Method of Measuring Wave Lengths of Light
In this experiment we will use the phenomenon of diffraction to measure the wavelength of light.
Specifically, we will measure the wavelengths of four different colors of light emitted by
hydrogen excited to incandescence.
The pattern of wavelengths of light emitted by incandescent materials is not continuous,
but consists of lines of specific wavelengths, where the pattern is characteristic of the
material.
In the early part of this century, studies of these patterns, called spectra, and the numerical
relationships among the wavelengths, led to theories of atomic structure and to the
development of quantum mechanics; they changed our views of the physical nature of the
world.}
In our experiment we will study the spectrum of hydrogen, the simplest atom, which has one
proton and one electron. We will measure the wavelengths (  ) of the four spectral lines that fall
in the visible range. We will check the relation
 1
1 
 R 2  2 ,

 n1 n2 
1
(1)
in which n1 and n2 are integers, n2  n1 , and R is a constant, called the Rydberg constant. This
relationship was originally found empirically, from sets of wavelength measurements, but
eventually was established to be a consequence of quantum mechanics, with the integer values of
n being significant parameters in atomic structure.
When plane waves of light fall at normal incidence on an opaque wall containing two narrow
parallel slits a distance d apart (Fig. 1) the light spreads out upon passing through the slits. On a
distant screen the overlapping beams (waves) from the two slits undergo interference, to produce
a pattern of bright and dark fringes, depending on whether the waves arrive at a particular point
in phase and thus add together (peaks matching peaks and troughs matching troughs), or out of
phase and cancel each other (peaks matching troughs).
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Brown University
Physics Department
Physics 0040
Lab 6
P
L
L+ dsinθ
θ
Light
θ
d
C
L0
dsinθ
P’
“Wall”
Screen
when dsinθ = mλ, with m=1,2,3…, P, P’ will be bright

At C, equidistant from the slits, all wavelengths present in the light arrive in phase and
interfere constructively to produce a central image or “zero-order interference pattern”
having the same color as the original light.

At some other point P or the symmetrical point P  which is at a distance L from one slit
and L  m from the other (  is some specific wavelength present in the light beam; m
is an integer) there is also constructive interference, and a bright fringe appears with the
color pertaining to that specific wavelength.

At intermediate points where the distances from the two slits are L and L +
( 2 m  1)( / 2) respectively, destructive interference occurs for that wavelength.

Through diffraction, the light of a particular wavelength  , besides contributing to the
central spot C, has been mapped into a number of locations (symmetric about C) of
maximum intensity, interspersed with points of zero intensity.

Light of any other wavelength present in the light beam shows similar behavior, but
because the locations of maxima and minima depend on the wavelength, these will differ
for each wavelength, and the patterns for different wavelengths will be separated on the
screen.
It can be seen from Fig. 1 that the governing equation that maps the points of maximum intensity
and separates the incident light into its wavelength components is:
m  d sin 
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Brown University
Physics Department
Physics 0040
Lab 6
where m = 0, 1,2,... is an integer, provided d is very much less than the distance ( L0 ) to the
screen. This simply states that the condition for constructive interference between two beams is
that the difference, d sin  , between their path lengths, is an integral number of wavelengths,
m . In this case, m is just a symbol for the “order” of the pattern. For white light incident on
the slits, the zero-order maximum is white light (all wavelengths) at point C. The pattern
corresponding to m = 1 is the first-order spectrum, m = 2 gives the second-order spectrum,
and so on. The intensity generally falls off with increasing m, so that higher orders become
increasingly difficult to detect.
Using these principles, the number of slits, N, can be increased to hundreds or thousands, all
parallel and spaced a constant distance d apart. The result is the diffraction grating.
Substituting a grating for the two slits of Fig. 1 (still keeping the total size of the grating much
less that L0 ) does not alter the form of Eq. (2) for the bright fringes, nor the meaning of the terms
in it. The reason is that light traversing any pair of adjacent slits obeys Eq. (2), and all such
patterns are superimposed (provided that L0 is much greater than d). The availability of a great
number of slits, N, means that each bright fringe becomes much narrower, for at an angle 
differing only very slightly from one that satisfies Eq. (2) the light arrives from the various slits
thoroughly out of phase. Therefore a grating far exceeds the two-slit arrangement, not only in
the brightness of the pattern but in resolving power. That is, the bright fringes are so narrow
that two of them corresponding to spectrum lines
of only slightly different wavelength can be distinguished, or resolved.
Equipment for Hydrogen Spectroscopy
Hydrogen gas in a glass tube is made incandescent by passing an electric current through the gas
(“discharge tube”).
TWO PRECAUTIONS ARE EXTREMELY IMPORTANT: DO NOT TOUCH THE METAL
ENDS OF THE TUBE, OR ANY WIRE OR OTHER METAL THAT MAY BE CONNECTED
TO THEM. A POTENTIAL DIFFERENCE OF 5000 VOLTS IS APPLIED ACROSS THE
TUBE.
TURN THE DISCHARGE OFF WHENEVER READINGS ARE NOT ACTUALLY BEING
MADE. THE TUBE WEARS OUT IN A SHORT TIME
The diffraction grating is mounted in a square frame about 1.5 inches on a side. The number of
lines per cm should be printed on it, or supplied by the instructor. Be alert for possible unit
conversions. Several grating values are labeled in lines per inch, some in lines per cm, some in
lines per mm.
Precision Spectrometer: The core of the experiment is the measurement of the angles at which
the bright lines of the various wavelengths appear, when light from the hydrogen discharge
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Physics Department
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Lab 6
shines on the grating from the direction perpendicular (“normal”) to it. Instead of letting the
lines appear on a flat screen, the angles at which these maxima appear in the observation space
are measured directly, with an angle-measuring device called a spectrometer.
These angles are given in terms of the wavelength by the grating equation previously cited (Eq.
(2)),in which d is the grating's line spacing, the distance between any two adjacent slits, and m
the order of the spectrum. Note that the m in Eq. (2) is not related to the n1 , or n 2 , in Eq. (1),
but all three are integers. This experiment deals with the first-order (m = 1) spectrum only. The
wavelength of each line can be determined from a measurement of the angle at which it appears
in the first-order spectrum.
The spectrometer consists of a circular metal table with two movable arms, each pivoted to rotate
about the center of the table, which is where the grating is fastened. One arm carries the optical
collimator which directs parallel light onto the grating, the other a telescope through which the
lines are viewed. The angle is given by the setting of the telescope arm and can be read from an
angle scale on the rim of the table.
Source
Slit
Grating
Collimator
Lens C
θ
Telescope
Fig. 2
Light from the hydrogen discharge tube (from the central portion where the light looks red) falls
on a slit at the outer end of the collimator tube. A lens at the other end produces nearly parallel
waves of light, which fall on the grating. The telescope has a cross-hair, which can be set on the
spectrum line. The telescope can be swung around in angle, and the angle of setting can be read
from a precisely calibrated scale.
Procedure and Data
Follow the instructor's guidance: the spectrometer is expensive, precise, and somewhat delicate.
Most of the necessary adjustments will have been done before you come into the lab. You
should check the adjustments, but consult the instructor before making changes except those
specifically mentioned below.
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Brown University
Physics Department
Physics 0040
Lab 6

Adjust the eyepiece of the telescope until the cross-hairs are sharply visible. (To see that
the telescope has been focused “for infinity”, view an object the length of the room
away.).

Without changing previous adjustments, swing the telescope around the central axis until
you can see through both it and the collimator. You should now see the collimator slit
sharply. Otherwise, ask your lab instructor for help in refocusing the collimator. Then
set the slit to a nearly closed position (you may have to open it a little, later on, to get
more light), and clamp the collimator in place.

You will find the grating positioned so that the light from the collimator strikes it at
normal incidence. Do not change its position.

Turn on the hydrogen tube. Be sure it is centered on the collimator tube, and close to the
slit.
OBSERVE CAUTIONS GIVEN IN EQUIPMENT SECTION AND POSTED IN LAB.

Measure and record the angular position of the telescope when it is set on the zero-order
line (m = 0). Then move the telescope slowly in one angular direction until you
recognize the four visible lines – deep violet, blue-violet, blue-green, and red. (These are
called the “Balmer lines” of the hydrogen spectrum.) There may be many closely spaced
faint lines in the background (due to molecular hydrogen and its ion), but the four Balmer
lines are not hard to identify in the first-order spectrum.*
Take angular readings for all four Balmer lines – on both sides of the center (first and
second order). Obtain the angles for each line relative to the center. For each  , average
the two angles on either side of the center. (What error will tend to be corrected by this
procedure)? Repeat the whole set of readings at least 2 or 3 times, and take the average for
each line.
1
Analysis
(1) Calculate your measured wavelength for each of the four Balmer lines, from your
angle measurements, using Eq. (2). (In your notebook identify each one by its
color.)
(2) Plot your values on a graph of 1 /  vs. 1 /(n2 ) 2 . According to Eq. (1), this should
give a straight line of the form y  a  bx , with slope b   R and intercept
a  R / n12 .
In some setups, the deep violet line is hard to distinguish. Try for it, but if you fail to
see it, on either side, proceed with the other observations and analysis.
*
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Physics Department
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GIVEN: For all four lines, the whole number n1 (in Eq. (1)) is the same
(unspecified) while n 2 has the values 3, 4, 5 and 6.
(3) Comment on whether your graph confirms the form of Eq. (1) – that is, comment on
how closely it appears to be a straight line. (Note that our giving the values of n 2 is
a shortcut. The values could be determined by trying, for the n 2 values,
progressions with different starting values until a straight-line graph results.)
(4) From the slope of your graph determine your experimental value for R. Compare this
with the theoretical value R  1.097  10 7 m 1 .
(5) From the intercept of your graph, determine your value for the constant integer n1 , in
Eq. (1).
(6) Predict the wavelength of the next line in the series.
(7) At what angle should it occur?
(8) Why didn't you see it?
Reference: Young and Freedman, University Physics Ninth Edition, Extended Version, Chapter
40
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