Atomic Spectra
Lab
Reading Assignment:
Halliday, Resnick, and Walker: Chapters 37 (especially 37.7 and 37.8) and 40 (40.6)
Introduction:
A diffraction grating essentially consists of a number of very narrow parallel
slits placed at regular intervals. A grating has a strong dispersive power, i.e., it can
separate out light of different wavelengths. Thus, we can examine the components of a
given light source by passing it through a diffraction grating. Every element in nature,
under certain conditions, emits light with a definite set of wavelengths (its emission
spectrum). This set is unique to the element in question, and so we may identify the
element by its spectrum.
Diffraction by a single slit
In the Diffraction Lab we studied the diffraction pattern obtained from a single
slit. The intensity as a function of angle  is given by
  a sin 
 sin  
I  Io  
 a sin 



with
 
2


2
   I  sin  

o

  


Eqn. 1
a sin 

Eqn. 2
where a is the slit width,  is the wavelength of the light used and Io is the intensity at 
= 0.
Diffraction by a double slit
In the Diffraction Lab we also studied the interference pattern produced by a
double slit. The transmitted intensity is given by
  a sin 
 sin  
I  4I o  
 a sin 



2


  cos d sin 
 
   


2

 sin  
2
  4 I o 
 cos 

  
2
Eqn. 3
with

d sin 

Eqn. 4
where d is the separation between slits.
Diffraction by multiple slits
When a parallel beam of monochromatic light is incident normally on a
diffraction grating, there is a Fraunhofer diffraction pattern in the transmitted light. For
the case of N slits each of width a and separation d the intensity is given by
 sin  
I  Io 

  
2
 sin N 


 sin  
2
Eqn. 5
with  and  as defined above. For N = 2 this reduces to Eqn. 3 for the double slit. The
first factor is the single slit diffraction envelope, while the second one represents the
interference term for N slits. Thus, diffraction minima are located at the same positions as
before, but the interference maxima are now located at angles given by
d sin   m
m = 0, 1, 2, . . .
Eqn. 6
where m is the order of the maxima and all angles are measured with respect to the
normal to the grating. These maxima are more intense than those in the double slit case.
As the number of slits increases, the diffraction pattern consists of a set of very sharp
principal interference maxima. Such a multiple slit device is known as a diffraction
grating.
The dispersive properties of a diffraction grating can be used to study the
spectrum of a light source in much the same way as a prism. However, good gratings
have a much greater dispersive power than prisms. Consequently they can provide a
more detailed picture of the spectra being studied. The dispersion D of a grating is a
measure of the angular separation of adjacent wavelengths and is given by
D
d
m

d d cos
m = 0, 1, 2, . . .
Eqn. 7
The resolving power of a grating R is a measure of the ability of a grating to
resolve two wavelengths that are close together and is given by
R
av
 Nm

Eqn. 8
where N is the total number of lines (slits) in the grating, av is the average wavelength
of the two spectrum lines and Δ is the wavelength difference between them. In this
experiment, you will first use the sodium dispersion spectrum to determine the value of d
for your grating and then use this same grating to study the Balmer (visible) spectrum of
hydrogen.
The Bohr Model of the Atom
Solving the Schrödinger equation for the H atom yields the energies of the
allowed states:
En  
Me 4 1
13.6
  2 eV, n  1,2,3....
2 2
2
8 0 h n
n
Eqn. 9
where n is a quantum number and M is the mass of the electron. Note that all of these
energies are negative, implying the atom is bound together. The lowest energy state
corresponds to n = 1 and is labeled the ground state. States with n = 2, 3, 4,…. are called
the excited states. An excited state can de-excite by emission of a photon that has an
energy (E = hf = hc/) equal to the difference between the initial state energy and the
final state energy:
E  hf 
 1
1 
 13.6 eV  2  2 

n f 
 ni
hc
Eqn. 10
This emission corresponds to a transition between the two energy levels. The set of
wavelengths produced by all possible transitions in an atom is called the spectrum of the
atom.
The visible part of the sodium spectrum consists of two very bright yellow lines
that are very closely spaced (λ = 589.00 nm and 589.59 nm). However, you will probably
only be able to see one line for each value of m corresponding to an average of the two
wavelengths, av = 589.30 nm. The zeroth order (m = 0) central maximum will be of the
highest intensity.
The sodium spectrum also contains a pair of lines in the infrared, λ = 818.3 nm
and 819.5 nm, which are about a tenth of the intensity of the bright yellow lines. You will
not be able to see these infrared lines visually, but you will probably detect them while
performing your experiment.
The light emitted by a hydrogen lamp has four discrete wavelengths in the visible
range. The standard lines in the Balmer series of hydrogen correspond to the transitions
of electrons from higher excited energy levels to the lowest excited state (n = 2) in the
Bohr model of the hydrogen atom. The emitted wavelengths are 656.3 nm (red), 486.1
nm (blue-green), 434.1 nm (blue) and 410.2 nm (violet).
In this lab you will measure these four emission lines and compare the energies of
the emitted photons with those expected by transitions within the H atom based on Eqn.
10.
For additional discussion and some figures, you may wish to view the following
Websites:
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/gratcal.html
http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html#c4
Atomic Spectra
Goals:
1. To study the use of a diffraction grating with sodium light
2. To measure the visible spectrum of hydrogen
3. To determine some of the energy levels in the H atom
Equipment List:
1. Spectrometer setup:
2. Optics bench (OS8518) with cm scale
3. Collimating Slits
4. Collimating lens
5. Focusing Lens
6. Aperture Bracket (CI8534) for Light sensor
with Post
7. High Sensitivity Light
Sensor (CI-6604) with
cable
8. Rotary Motion Sensor
(CI-6538)
9. Grating Mount
10. Diffraction Grating
11. Sodium Lamp
12. Hydrogen Lamp
13. Data Studio™
14. Wood blocks
Figure 1
Figure 2
1. The sodium lamp must be switched on at least 10 minutes before you use it in order to let it
warm up properly.
2. The hydrogen lamp has a short life-time, so do not keep it on when not required.
It might be helpful to review any important points in the description of the setup here (see
Figures 1 and 2). In this experiment, we send light through (1) a collimator slit and
collimator lens, (2) a fine diffraction grating, (3) a focusing lens, (4) an aperture disk, and
finally (5) a light sensor.
1. The collimators serve to change the original beam shape to a rectangular shape
with sharp edges, i.e., collimated. The slit should be set to the smallest size, slit 1
for this experiment. (If it is not already set, you can loosen the black screw and
slide the slit plate so that light passes through slit 1.) The collimating lens helps
the beam keep its size and shape over the distance it must travel to the light
sensor.
2. The diffraction grating diffracts the beam of light. You have seen what happens
when light is diffracted in a previous lab. In particular, the light waves spread out
and interfere with each other so that if we allow the resulting light to fall on a
surface, we see there are locations where there are maxima and minima. In fact,
the angle at which the light maxima occur is given by θ in Eqn. 6 above. d is the
characteristic separation between grating lines.
3. The focusing lens serves, not surprisingly, to focus the components of the highly
diffracted beam that is spreading out over the distance from the light source to the
light sensor. We would like them to be focused on the aperture in front of the
light sensor.
4. The aperture disk is the disk with various slits located just in front of the light
sensor. We would also like this set to slit 1, the smallest size.
5. The high sensitivity light sensor detects the beam hitting its aperture and sends a
signal to Data Studio™, which then returns a measurement of the intensity of that
light. The sensor has a switch for variable gains that you will have to play with to
obtain the best plots.
Lab Activity 1: Measurement of the Diffraction Grating Spacing (d) using Sodium
Light
1. Turn on the sodium lamp; it takes a while to reach full luminosity, so continue with
the procedures below.
2. Record your Grating number. There is no grating number. Leave this blank on the
template.
3. Rotate the degree plate by holding onto the knob (or screw) under the light sensor
arm until the face of the grating plate is perpendicular to the light sensor. The grating
surface should now be parallel to the zero degree line on the degree plate it sits on.
4. Make sure slit 1 is open on the first collimator and slit 1 is selected on the aperture
disk in front of the light sensor. Set the switch on top of the light sensor to a gain of
100.
5. Place the sodium lamp in front of the collimating slits, ensuring that the main beam is
not passing through the slit at an angle and that the center of the beam is directly over
the slit.
6. Once the lamp has reached full luminosity, you should see the m = 0 (“zeroth order”)
line on the white aperture holder in front of the light sensor. Try to get the line as
sharp (rectangular, rather than fuzzy) as possible. To do this, move the collimating
lens or focusing lens forward and back. The collimating lens should generally be
about 10 cm from the collimating slit and the focusing lens at the suggested markings
on the degree plate between the grating and the light sensor.
7. Start up Data Studio™ and (if they are not already in) plug in the high sensitivity
light sensor and the rotary motion sensor.
8. Open the Graph Display and the Table Window. Set both to display Intensity (in % of
maximum) versus Angle (in degrees).
9. This setup will record the light intensity from the light sensor as a function of the
angle on the degree plate. Keep in mind that the zero radians position in the Data
Studio™ graph is not necessarily the zero on the degree plate.
10. Beginning from the 50° line on the degree plate, or further out, begin recording data
while moving the light sensor across the diffraction pattern using Data Studio™.
Slowly and steadily rotate the table all the way to the opposite 50° line or further out.
You should do it slowly and move only in one direction.
11. Display your data of Intensity vs. Angle (in degrees) with the points connected.
12. Verify that the plot exhibits the behavior you might expect. If it does not, check that
no extraneous light is hitting the light sensor and that the sodium beam is properly
lined up with the collimating slits so that the beam falls directly on the aperture in
front of the light sensor.
13. Copy the graph to the template provided. Before deleting data from Data Studio™,
record the values of the angles at which the peaks lie. These angles should be read
directly off the graph in Data Studio using the Smart Tool. For each peak, identify its
corresponding value of m = 0,1,2,…
You may also add a third column to the table to identify whether each peak is a
yellow spectral line or an infrared spectral line. This is not required, but the
information will be useful for the following steps.

m
Yellow or infrared?
Add rows to this table as needed to account for all peaks you see.
14. Using only the yellow spectral lines, for each order m = 1,2,…, find the average
angular displacement from the center and record that in the following table as θm.
Then use these values to calculate a measurement of the grating line separation, d,
and enter the appropriate values in the following table.
m
θm
sin θm
d
Add rows to this table if you are able to see higher than 2nd order maxima.
15. Report your best determination of the grating spacing, d.
Lab Activity 2: Study of the Hydrogen Spectrum
1. Replace the sodium lamp with a hydrogen lamp (and use slit 3) and repeat the
previous procedure (roughly, steps 11-15) to obtain the hydrogen dispersion
spectrum. If you do not observe any spectral lines, consult your laboratory instructor
who will check and replace the bulb if necessary.

m
Color
Add rows to this table as needed to account for all peaks you see.
2. Using your value for d, complete the following table for the m = 1 order of your
hydrogen spectrum. For each color that you observed, calculate θm and sinθm as in
the previous activity, but instead of using the given wavelength to compute d, instead
use the value of d you found to compute the wavelength. For each wavelength,
calculate the corresponding photon energy and enter it into the table. The expected
wavelengths are given so that you can compare your results to the actual values, but
you should not use these expected wavelengths in your calculations.
Line# Color
sin θm
λexpected) θm
λmeas) Eγ(meas)
Red
656.28 nm
1
Cyan
486.13 nm
2
Blue
434.05 nm
3
Violet
410.17 nm
4
3. The colors of the four visible lines of the hydrogen Balmer Series spectrum are: red,
blue-green (cyan), blue and violet. For each of these colors, enter the initial energy
level number ni and the final energy level number nf in the following table. Then use
these numbers to calculate the corresponding energy levels (use ni to calculate Ei and
nf to calculate f), and then determine the difference, which should correspond to the
photon energy. Eγ(meas) should be copied from the table above.
Fill out all 4 rows of this entire table (except for Eγ(meas)) even if you did not
observe all 4 spectral lines.
Line#
1
2
3
4
Color
Red
Cyan
Blue
Violet
ni
nf
Ei
f
Ei − Ef
Eγ(meas)
4. Compare (in the above table) the measured photon energies with the calculated
photon energies. What do you conclude?