Physics 439 – Lab in Modern Physics
Atomic Spectra Experiment
Abstract
We measured the Balmer series in hydrogen and deuterium and the splitting of the sodium
doublet, using a monochromator and a Michelson interferometer. Our study of the Balmer series using
the monochromator allow us to calculate the Rydberg constants for hydrogen and deuterium. These
were determined to be RH = (1.09705 ± 0.00040) × 107 m-1 and RD = (1.09700 ± 0.00030) ×107 m-1,
respectively, which agree with the accepted values of 1.09678 × 107 m-1 and 1.09707×107 m-1. Our
measurements of the splitting of the sodium doublet using the monochromator yield a value of (0.558 ±
0.080) nm, in agreement with the accepted value of 0.5970 nm. Our manual measurements with the
interferometer yielded a value of 0.591 ± 0.001 nm, which is not in agreement with accepted values due
to the contribution of human error which was not accounted for, while our measurements using the
interferometer yielded a value of 0.50 ± 0.1 nm, in agreement with the accepted value. An unsuccessful
attempt was made to measure the shift between the deuterium and hydrogen Balmer series.
Jason Rogers
#110233403
Pierre Turcotte-Tremblay
#260153067
October 10, 2006
1
Introduction
The theory of quantum mechanics offers deep, predictive insight into the behavior of atoms.
In particular, quantum mechanics requires that atoms only exist in certain distinct levels of energy. An
important way of transitioning to lower energy levels involves an atom's electron emitting a photon of a
particular wavelength. This phenomenon gives rise to the familiar emission spectra of atoms; it is what
causes hydrogen and sodium lamps to produce light.
A full quantum mechanical and relativistic treatment of an atom offers predictions for its
allowable transitions, and as such for the structure of its emission spectrum. The wavelengths of wellknown spectral structures such as the Balmer series in hydrogen and deuterium and the fine-structure
splitting of the Sodium doublet are predicted by the theory. As such, careful measurement of the
emission spectra of these atoms allows a comparison to predicted values that serves as a concrete,
directly observable verification of some theoretical predictions of quantum mechanics.
Two instruments that are both incredibly useful for the observation of spectral lines and widely
used are the monochromator and the Michelson Interferometer. In studying aspects of the optical
spectra of hydrogen, deuterium and sodium we will make use of both of these devices. Our inquiry will
thus serve the dual purposes of acquainting ourselves with important optical measurement technology
and allowing us to directly observe some of the atomic behavior predicted by quantum mechanics.
2
Theory
2.1
Energy Levels of Hydrogen-like Atoms
We begin by sketching a derivation of the energy levels of hydrogen-like atoms. These are
atoms with a charge Ze on the nucleus which have only one orbiting electron; atoms such as hydrogen,
deuterium and singly-ionized helium. Our derivation follows that of Townsend [1].
We begin with the Hamiltonian for a hydrogen-like atom, expressed in cgs Gaussian units,
 2 Z e 2 (Eq. 1)
p
H =
−
2  ∣r∣
Here p is the relative momentum operator, p=
m2 p 1−m1 p 2
, while μ is the reduced mass
m1m2
m1 m2
. The mass of the nucleus is m1 and the mass of the electron is m2, p 1 and p 2 are the
m1m2
momentum operators on the nucleus and the electron respectively, and r is the relative position
operator, giving the relative position of the two electrons along the line joining them. Ze is the charge
on the nucleus, with e = 1.6022 × 10-19 Coulombs.
Substituting into the radial Schrödinger equation, and making the substitution RE,l(r) = uE,l(r)/r
for the radial wave function RE,l(r) at a given energy E and angular momentum l, we get equation 2 for
uE,l(r).
=
1
[
2
]
2
ℏ d 2 l l1ℏ Ze 2
−

−
u E , l  r =Eu E ,l r  (Eq. 2)
2
r
2  dr 2
2 r
Writing E=−∣E∣ , as we are concerned with bound solutions with negative energy, we introduce the
variable ρ which we define as follows
=

8 ∣E∣
ℏ
2
r (Eq. 3)
Expressing equation 2 in terms of ρ we obtain equation 4
2
d uE , l
d
2
−
 
 1
l l1
u E , l − u E , l =0 (Eq. 4)
2
 4

where λ is given by equation 5
=
Ze
ℏ
2


(Eq. 5)
2∣E∣
With some further substitutions and manipulations, it is possible to find a power series solution for
equation 4 [2]. The terms of our power series solution exhibit a recurrence relationship of the following
form
c k1
k l1−
=
(Eq. 6)
c k  k1 k 2l2
where ck is the k-th term. In order to prevent this from growing exponentially, we require that the series
terminate. We must therefore have =1ln r where nr = 0, 1, 2, ... determines for which value of k
the series terminates.
Rearranging equation 5 for E and substituting our result for λ gives us
2
E=−
4
Z e
2
2 (Eq. 7)
2 ℏ 1lnr 
Defining the principle quantum number n as n = l + 1 + nr, we have
2 4
E n=−
Z e
2 2 (Eq. 8)
2ℏ n
where n = 1, 2, 3, ... . Equation 8 gives us an expression for the energy levels in a hydrogen-like atom.
2
2.2
Emission and Balmer Lines
One of the most direct ways of observing the energy levels of hydrogen-like atoms is by
observing the result of a transition between these energy levels. When a hydrogen-like atom makes a
transition in energy level from a state ni to a state nf, with ni > nf, the difference in energy is emitted in
the form of a photon. The energy of this photon is given by equation 9
2
E =2 ℏ =E n −E n =
1
f
Z e
2 ℏ2
4


1 1
−
(Eq. 9)
n 2f n 2i
Here we have used the expression for the n-th energy level of a hydrogen-like atom, equation 8. From
the expression relating the wavelength of a photon to its frequency, c = λυ, we can express equation 9
in terms of the photon's inverse wavelength. This gives us equation 10.
2
4

 

1 Z e
1 1
1
1
=
− 2 =R 2 − 2 (Eq. 10)
3
2
 4 c ℏ n f ni
n f ni
Here we have defined a constant R, which is given by
R=
 Z 2 e4
4 c ℏ
3
(Eq. 11)
Transforming this from Gaussian units into S.I. units, we have
2
R=
4
2 4
Z e
Z e
= 2 3 (Eq. 12)
2
3
4  0  4 c ℏ 8  0 c h
Where ε0 is the permittivity of free space.
The constant R is called the Rydberg constant of the atom. Its value is dependent on the atom in
question. The Rydberg for hydrogen, RH, can be calculated using Z = 1 and the reduced mass μ for
hydrogen. Its theoretical value is RH = 1.09678 × 107 m-1. Similarly, for deuterium, we have RD =
1.09707x107 m-1 .
If we require that nf = 2, we have the formula for the Balmer series. Described mathematically
for the first time in 1885 by Johann Balmer, this formula gives us the wavelengths of perhaps the most
famous series of spectral lines in hydrogen. Substituting n = 3, 4, 5, and so on allows us to calculate the
values of the Hα, Hβ, Hγ and other subsequent lines; alternately, measurement of these lines allows us to
calculate a value for the Rydberg. These lines are in the visible spectrum, and the stronger lines such as
Hα and Hβ are easily observed. If we use the Rydberg for deuterium in our calculation, we can calculate
its Balmer series as well; here the higher reduced mass of the deuterium atom causes the wavelengths
to be shifted downwards somewhat.
3
2.3
The Sodium Doublet and Spin-Orbit Coupling
The visible spectrum of sodium is dominated by two very bright, closely spaced yellow lines,
known as the sodium D lines or the sodium doublet. The splitting between these lines can be explained
by a quantum effect known as spin-orbit coupling.
In the rest frame of sodium's one valence electron, the
positively charged nucleus of sodium is in motion, generating
a current which produces a magnetic field. There is an energy
associated with the interaction of the magnetic field with the
electron's intrinsic spin magnetic moment. This intrinsic spin
can have two values, and combines with the orbital angular
momentum to produce two possible states with total angular
momentum j = 1/2 and j = 3/2 for our electron in the 3p
energy level. An electron in the j = 3/2 state will have a
higher-energy interaction with the magnetic field than an
electron in the j = 1/2 state; as such, the 3p energy level is
split into two closely spaced energy levels which are denoted
as 3p1/2 and 3p3/2, where the subscripts refer to the total
angular momentum. When an electron in the higher energy
Figure 1 - Splitting in the sodium
doublet due to spin-orbit interaction
3p3/2 state makes a transition to the 3s1/2 state, the emitted
[4]
photon will be of a higher energy (and as such a lower
wavelength) than the photon emitted from the corresponding
transition from the 3p1/2 state to the 3s1/2 state. The difference
in wavelength is 0.5974 nm [3]. This phenomenon is
illustrated in Figure 1.
3
Apparatus
3.1
Monochromator Experiment
A monochromator is an optical device used to allow only a certain small band of wavelengths to
pass through it while stopping all other wavelengths. The light beam going through the entrance slit
hits a mirror which reflects the light on a diffraction grating. The beam then diffracts from the grating
to another mirror which refocuses the dispersed light on the exit slit. Since the angle of reflection from
the grating is dependent on the wavelength of light, only a particular wavelength of light will be
reflected at the angle necessary to exit the monochromator. This wavelength is determined by the angle
of the diffraction grating which is controlled by the position of the monochromator's dial. By changing
the value on the dial, a specific wavelength of light can be selected for viewing.
Out setup involves a light source connected to a power supply facing the entrance slit of the
monochromator. A photomultiplier connected to another power supply faces the exit slit of the
monochromator, converting the light signal into an electric signal. This signal is then fed into both the
Labmaster analog-to-digital converter and an oscilloscope. The computer reads and records the signal
from the Labmaster while the oscilloscope shows the signal directly on its screen. The computer is also
used to control the monochromator's grating angle through a stepper motor. A program called Spectra
interfaces with the Labmaster components, allowing for control of the stepper motor and handling the
data acquisition from the photomultiplier tube. During data acquisition, the entire apparatus was
4
covered by a black cloth cloak and all main lights to the lab were turned off. This ensured that the
photomultiplier would not be exposed to strong light and it ensured no external light sources interfered
with the data. The monochromator experimental setup used is shown in Figure 2.
The light sources used where Neon, Hydrogen, Deuterium and a Sodium lamp.
Figure 2 - The Monochromator experimental setup.
3.2
Interferometer Experiment
The Michelson interferometer is a device which produces an interference pattern by splitting a
beam of light into two paths at right angles. The two separated beams hit their respective mirrors and
bounce back to recombine at the beam splitter. The key is that the paths followed by the horizontal and
vertical traveling beams can be adjusted by use of a micrometer screw so as to be of different lengths,
causing a relative shift in the two waves. In order to observe constructive interference at the target the
two waves must be in phase with each other when they recombine. This may only be achieved if the
difference in distance travelled by the waves is an integer multiple of the wavelength of the wave when
they recombine. If this is not the case the waves will be out of phase and will interfere destructively
upon superposition; this interference is at a maximum for integer plus one-half wavelengths. By
observing the interference pattern and how it varies as the micrometer screw is turned we can make
precise measurements of the wavelength of the incident light. The observations were taken either by
observing the interference pattern on a screen, by looking into the interferometer directly, or by use of a
charge-coupled device (CCD) camera. The Michelson interferometer experimental setup is shown in
Figure 3.
5
Figure 3 - The Michelson interferometer experimental setup. The target
corresponds to the location of the screen, CCD camera, or eye of the
observer.
4
Monochromator
4.1
Calibrating the Monochromator.
Before using the monochromator to study atomic spectra, it was first necessary to calibrate the
device. Such calibration involved two aspects; first, physically adjusting the experimental setup so as to
obtain the best resolution, and second, relating the wavelength measurement on the monochromator
dial to the actual transmitted wavelength.
To obtain the best resolution possible, we began by centering the light source in front of the
monochromator's entrance slit. We then used the Spectra software to find the peak intensity of the neon
line at 585.3 nm, and set the grating at the appropriate angle to transmit this peak. The monochromator
output was observed both directly by looking through the exit slit and indirectly by reading the voltage
output of the photomultiplier on an oscilloscope We then began closing the entrance aperture,
adjusting the transmitted wavelength as necessary so as to keep the monochromator centered on the
line's peak. As the entrance aperture became progressively smaller, we manually varied the transmitted
wavelength around the peak value and observed the rate at which intensity dropped off. We adjusted
until intensity dropped to a low percentage of the peak value over as small a variation of the
wavelength as possible, in order to obtain a narrow intensity profile. When the front slit had been
adjusted so that the peak profile was sufficiently sharp and further aperture reductions led to excessive
reduction of output intensity, the back slit was closed as much as was possible while still preserving the
intensity profile. The gain on the photomultiplier's amplifier was then adjusted so as to place the peak
in the 8-9 volt range. This procedure was repeated as necessary throughout the experiment for the other
6
observed elements, and occasionally also for individual spectral lines. The output amplitude and
aperture sizes were adjusted as necessary to obtain a sharp peak profile, with an amplitude high enough
to differentiate the peak from random noise, but low enough so as not to saturate that analog-to-digital
converter input.
The next step was to find a relationship between the dial reading and the wavelength of light
coming out of the exit slit. In order to do this we used a Neon light source. We scanned the intensity
of the Neon spectrum as a function of dial readings from 570 to 675 nm. The emission lines of the
Neon Spectrum are represented by peaks in intensity as shown in Figure 4.
Figure 4 - Scan of the Neon spectrum from 570 nm to 675 nm using the monochromator.
Multiple emission lines are visible.
The regions around each peak were then measured. For these and all measurements, care was
taken to never reverse direction during a set of measurements in order to reduce the effect due to
hysteresis. If we needed to reverse direction at any point, we went to at least 5 nm before where we
were to start measuring and then moved forward 5 nm, to establish a clear direction of motion. The
peaks were then fitted to a Gaussian of the form of equation 13. [5]
y= y 0
A
e
w /2
−2
x− x c 
w
2
2
(Eq. 13)
7
The relevant parameter in this Gaussian is xc, the center of the peak. By fitting the peaks to Gaussians
of this form and extracting the values of xc, we were able to find the dial positions of the emission lines.
We then plotted the accepted values of these emission lines against the measured values, and carried
out a simple linear fit. The results of are summarized in Figure 5.
Figure 5 - Calibration data for the monochromator. Accepted spectral line
values for neon are plotted against the measured values obtained from the fit
results from equation 13. The result appears linear; the parameters of the
linear fit are given in equation 14.
The fit gives us a relationship between the actual wavelength of light going through the
monochromator and the dial reading. This relation is expressed by equation 14.
λA = (0.995 ± 0.001)λM + (4.0 ± 0.7) nm (Eq. 14)
Here λM is the measured spectral wavelength from the fit and λA the corresponding wavelength in
nanometers.
We must make a comment on the nature of the error on the constant term in equation 14. While
this error does indeed mean that a given peak measurement is uncertain to approximately ± 0.7 nm, it
does not represent a truly random error; rather it represents the contribution of both random errors in
the reading of the dial position and a larger systematic error associated with the actual relationship
between dial reading and transmitted wavelength. If we were to measure two nearby peaks, such as
those of the sodium doublet, we would not expect the errors on the measurement of each peak to be
independent; if one peak was 0.7 nm higher than the accepted value, the other peak would also be
8
approximately 0.7 nm higher. The peaks would certainly never switch position in our measurements as
a result of our error term, as might happen if the errors were truly random. As such, when dealing with
differences in measured wavelength values over small ranges of data, much of the calibration error
disappears; our monochromator is more apt at measuring the differences between the wavelengths of
spectral lines than the actual values of the spectral lines themselves. It will be important to remember
this fact when we analyze the fine structure of sodium.
4.2
Balmer Series in Deuterium and Hydrogen
4.2.1 Hydrogen
With our newly calibrated monochromator, we are now ready to go about measuring the values
of some of the elements of the Balmer series in hydrogen and deuterium. We follow the same
procedure as for Neon, with the notable exception that we first adjust the data according to equation 14
so that we are dealing with the actual measured values rather than simply the dial readings. Fitting to
Gaussians allows us to extract information on the peak positions. For each value of ni multiple
measurements were taken and the fit results were averaged; each ni is the result of an average of either
two or three fits. The calibration errors from adjusting the data far outweigh the error on the fitted peak
values. The results for transitions from n = 3, 4, 5, 6 and 7 for hydrogen are summarized in Table 1.
Table 1 – Balmer Series in Hydrogen
ni
λ (nm)
Accepted λ (nm)
3
656.2 ± 0.7
656.2785
4
486.1 ± 0.7
486.133
5
433.8 ± 0.7
434.047
6
410.3 ± 0.7
410.174
7
397.1 ± 0.7
397.007
As we can see, the measured values all agreed with the accepted values within experimental
error. Such error is large, however; while the accuracy of our values is decent, their precision is
certainly lacking.
We can use these values for the hydrogen Balmer series to estimate the value of the Rydberg
constant for hydrogen. To do this, we plot the inverse wavelength λ-1 against 1/22−1/ni 2  ; as
equation 10 shows, the slope will be the value of RH, the hydrogen Rydberg constant. Figure 6
summarizes our results.
9
Figure 6 - Plot showing the Rydberg constant for hydrogen. The Rydberg constant is the
slope of the linear fit; it is given in m-1. Here our measured value is RH = (1.09705 ± 0.00020)
× 107 m-1.
From our fit, we have our measured value of the Rydberg constant for hydrogen, RH = (1.09705
± 0.00040) × 107 m-1. The theoretical value is 1.09678 × 107 m-1. Our measured value is thus in
agreement with the theoretical predictions within experimental error.
4.2.2 Deuterium
We can repeat our analysis for deuterium. Since the nucleus of deuterium contains a proton and
a neutron, the reduced mass for deuterium will be higher than that of hydrogen. A quick calculation
shows that deuterium's reduced mass will be about 1.00027 times as large as hydrogen's. Since the
reduced mass figures in the expression for the Rydberg, equation 11, the value for the Rydberg should
be slightly higher and as such the wavelengths of the spectral lines should be slightly lower. The results
for deuterium are summarized in Table 2.
10
ni
3
4
5
6
7
Table 2 – Balmer Series in Deuterium
λ (nm)
Accepted λ (nm)
656.0 ± 0.7
656.10
485.9 ± 0.7
486.00
433.9 ± 0.7
433.93
410.4 ± 0.7
410.06
397.3 ± 0.7
396.90
The values are not consistently lower than those for hydrogen, though the values for the ni = 3
and ni = 4 lines, which were the easiest to measure, are indeed lower. In any event, the uncertainties are
large, and the values are in agreement with the accepted results. The resolution is too low to effectively
determine the difference between the hydrogen Balmer series and the deuterium Balmer series.
The plot used to determine the value of the Rydberg constant for Deuterium, RD, is shown in
Figure 7.
Figure 7 - Plot showing the Rydberg Constant for Deuterium. The Rydberg constants is the slope
of the fit; the calculated value is RD = (1.09700 ± 0.00030) ×107 m-1
From the plot we obtain RD = (1.09700 ± 0.00030) ×107 m-1. The accepted value of the Rydberg
11
for deuterium is 1.09707×107 m-1. These two values are therefore in very good experimental agreement.
However, we note that our measured value for the Rydberg for deuterium is actually lower than our
measured value for the Rydberg of hydrogen. The experimental error in our apparatus is large enough
to prevent us from truly distinguishing the difference between the two line series; a more precise and
accurate apparatus, or a more thorough calibration, is necessary for such a measurement.
It is worth noticing that a possible source of our difficulties in determining two distinct values
for the Rydberg is the quality of our hydrogen tube itself. Figure 8 compares the light coming from the
two tubes. The hydrogen tube looks significantly different from the deuterium tube; it appears to have
been contaminated in some way, and a significant white light background is present. The characteristic
red glow is barely present and the tube is primarily blue. This in particular affected measurements of
the weaker lower wavelength spectral lines; the flaws in the tube may have contributed to some error
on the values measured in this experiment.
Figure 8 - The gas tubes used in our study of the Balmer series. The deuterium tube on the left is much brighter and
appears red due to the strong deuterium alpha (ni = 3) line. We would expect the same of the hydrogen tube on the
right; instead it appears reddish only in the very center, and is mostly blueish, due to some sort of defect.
12
4.3
Sodium Doublet
A similar measurement procedure was followed to measure the splitting of the sodium doublet,
with the discharge tubes replaced by a sodium lamp. We measured the region from 587 nm to 591 nm
three times. A typical set of data is plotted in Figure 9. Note the two peaks in close proximity. These
peaks should be described by a double-Gaussian. However, all attempts to fit with a custom double
Gaussian using the Origin software program's non-linear curve fitting tool [6] resulted in very poor fits
that were highly sensitive to the initial selection of parameters; we were able to vary our measured
separation distance as we wished by modifying fit parameters slightly and running the fits again.
Furthermore, the error on some parameters was huge. These observations caused us to severely doubt
the double Gaussian fitting approach, and so we settled for fitting each peak separately to single
Gaussians of the form of equation 13. The single Gaussian fits did not exhibit this same extreme
dependence on the initial parameters.
Figure 9 - Monochromator measurement of the sodium doublet. Each
peak was individually fit to a Gaussian, and the wavelength of the peak
value extracted.
Our analysis gave us a value of λ3/2 = (588.97 ± 0.70) nm for the lower wavelength peak and
13
λ1/2 = (589.52 ± 0.70) nm for the higher wavelength peak. The accepted values are 588.9953 nm and
589.5923 nm respectively. Our values are thus in good experimental agreement with the expected
values, though as before our errors are large, and our higher wavelength value is lower than we might
like. As we discussed at the end of Section 4.1, however, much of this error should not be present in the
measurement of the separation distance of two spectral lines that are reasonably closely spaced; even if
the zero of our wavelength scale is in doubt, this should not affect relative distances. As such, to find
the separation we find the difference in the two values for each of our data files and determine the error
based on the goodness of the fits of our Gaussians, rather than based on the calibration error. Our
measured result for the splitting is then Δλ = (0.558 ± 0.080 ) nm, which puts us within one standard
error of the accepted separation of 0.5970 nm. We will attempt to improve on this measurement
through the use of the interferometer.
5
Michelson Interferometer.
5.1
Calibrating the Michelson Interferometer
The first step in calibrating the Michelson interferometer is to adjust the mirrors so that an
interference pattern will be visible. To position the mirrors properly, we place a pin provided with the
interferometer in its slot next to the beam splitter, and, looking into the exit of the interferometer, we
adjust the angle of the mirrors until the two images of the pin overlap perfectly. Shining a He-Ne laser
through the interferometer then yields the expected interference pattern.
In order to complete our calibration of the Michelson interferometer we need to find the
relationship between the distance traveled by the carriage on the interferometer and the difference in
dial reading. This is expressed by equation 15.
d1 − d 2 = ( D1 − D2 )k (Eq. 15)
Here k represents the proportionality constant, d1 and d2 represents the initial and final positions of the
carriage in meters and D1 and D2 are the initial and final dial readings in meters.
In order to find the value of k we send a monochromatic HeNe laser beam of known wavelength
(λ = 632.8nm) through a diverging lens and then into the interferometer. We adjust the interferometer
so that the characteristic 'bullseye' pattern is centered on a piece of white paper held past the device's
output.
We observe that as we move the knob of the micrometer the fringes of the interference pattern
are moving inward or outward depending on the direction of rotation of the knob. We choose to take all
measurements in a clockwise direction, so that the fringes are moving outwards. Before each
measurement we back the knob up counterclockwise by an extra couple millimeters behind our initial
position, so that we never reverse the direction of motion just before taking a measurement, in order to
reduce the effects of hysteresis. Once the micrometer is set to a convenient initial position we count the
number of fringes disappearing from the pattern as a function of the dial reading. We can then
calculate the constant of proportionality using equation 16.
k=
λ∆n
(Eq. 16)
2( D1 − D2 )
14
Here λ is the known wavelength of the HeNe laser in meters, Δn is the number of fringes that have
appeared or disappeared, and D1 and D2 are the initial and final dial readings in meters respectively.
The results for our calibration measurements are expressed in table 3.
Table 3 – Calibration of the Interferometer
Δn
50
50
50
-
Error on Δn
±1
±1
±1
-
D1 (m)
0.015
0.015
0.015
-
Error on D1 (m)
± 0.000005
± 0.000005
± 0.000005
-
D2 (m)
0.0142
0.01425
0.01419
-
Error on D1 (m)
± 0.000005
± 0.000005
± 0.000005
Average
K
0.0198
0.0199
0.0195
0.0197
Error on k
± 0.0004
± 0.0004
± 0.0004
± 0.0002
The average k value for our Michelson interferometer was found to be k = 0.0197 ± 0.0002.
5.2
Sodium intensity pattern
When the incidence light passing through a Michelson interferometer consist of two closely
separated wavelengths, each wavelength will have its own intensity pattern in the interferometer. Their
respective intensity patterns will either coincide with each other causing high contrast or cancel causing
low contrast. The synchronization of the patterns will depend on the path difference of the interfering
waves defined by the position of the carriage. This can be observed by examining the interference
pattern of a sodium light source through the Michelson interferometer. If we shift the position of the
carriage, high contrast will occur when one set of fringes will superimpose with the other set and low
contrast will occur when one set of fringes will be halfway between the other set. The CCD camera
was used to take pictures of a high contrast sodium intensity pattern and a low contrast sodium
intensity pattern. Those pictures are shown in Figures 10 and 11 respectively.
Figure 10 – A region of high contrast in
the sodium doublet's interference pattern.
Figure 11 – A region of low contrast in the
sodium doublet's interference pattern
The sodium doublet lines emitted by the sodium light source have a distinct separation distance.
Two different methods will be used in this experiment to find the separation with the Michelson
Interferometer; one method involving direct measurement by looking at the interference pattern to
determine high and low contrast regions, and one involving analysis using a CCD camera.
15
5.3
Average wavelength of the sodium doublet line.
Before we attempt to measure the doublet separation for sodium we will measure the average
wavelength of the sodium doublet lines. In order to do this we must replace the HeNe laser with a
sodium lamp and reapply the procedure used to calibrate the interferometer in section 5.1. Note that
this time the value of k is known and we need to solve for λ. We will therefore simply rearrange
equation 16 in the following way.
λ =
2k ( D1 − D2 )
(Eq. 17)
∆n
The results of four trials are shown in table 4. Every time 50 fringes were counted and the
initial position of the micrometer was set to 15.0000 mm.
Δn
50
50
50
50
-
Table 4 – Average Wavelength of the Sodium Doublet
λ
Error
Error on
Error on D1
on Δn D1 (m)
D1 (m)
D2 (m)
(m)
(nm)
±1
0.015 0.000005 0.014240
0.000005
599
±1
0.015 0.000005 0.014255
0.000005
587
±1
0.015 0.000005 0.014270
0.000005
575
±1
0.015 0.000005 0.014250
0.000005
591
Average
588
Error on
λ (nm)
± 14
± 14
± 14
± 14
±7
The average wavelength of the sodium doublet was calculated to be λ = 588 ± 7 nm. The
accepted value for the average wavelength is λ = 589.3nm which agrees within the error on the
measured wavelength.
A large component of our error is due to the uncertainty in Δn; our estimation of ± 1 is
conservative. If we were to take the measurements of Δn as exact, our error measurement on the
average would be reduced to ± 2.8 nm, a somewhat more reasonable value.
5.4
Manual doublet separation measurements
Now that we have an experimental value for the average wavelength of the sodium doublet we
may attempt to measure its separation. The first method consist of finding the distance traveled by the
carriage of the interferometer as the interference pattern shifts from a position of high contrast to the
next position of high contrast. The opposite also works; we find the distance interval between the
successive positions of low contrast. The problem is that we must decide simply by looking at the
interference pattern whether it is at high contrast or low contrast. This is difficult and requires
observation skills in order to obtain accurate data. With this distance and the value of the average
wavelength of the sodium doublet we are able to solve for the separation distance using equation 18 [7]

2
 =
(Eq. 18)
2 k  D 1 −D 2 
16
where λ is the accepted average wavelength of the sodium doublet and Δλ is the separation of the
sodium doublet. The results for this procedure are shown in Table 5.
Table 5 – Manual Measurement of the Splitting of the Sodium Doublet
Runs
ΔD (D1 – D2)
Error (m)
Δλ (nm)
Error on Δλ
(m)
(nm)
High contrast (1)
.01501
0.000005
0.585
0.002
High contrast (2)
.01485
0.000005
0.591
0.002
Low contrast (1)
.01478
0.000005
0.594
0.002
Low contrast (2)
.01482
0.000005
0.592
0.002
Average
0.591
0.001
The average measured value for the separation is Δλ = .591 ± 0.001 nm while the accepted value is Δλ
= .597nm. Our error does not fall within the accepted value of the sodium doublet separation, although
the value is more accurate than that determined by the monochromator. The gap between our value and
the accepted value are due to the fact that we must decide by eye when the interference pattern is at its
high contrast or low contrast peaks. The component of human error is much larger than the
instrumental error that we have estimated here, and makes up the rest of the disagreement.
5.5
Doublet Spacing CCD Camera Method
The second method used to calculate the spacing between the sodium doublet lines involves the
CCD camera. We used the CCD camera to take pictures of the intensity pattern created by the sodium
lamp at intervals of 0.500 mm as read by the dial of the interferometer from 0 to 23 mm. Each picture
taken was analyzed by the program Eyespy and a graph of the contrast intensity going through the
center of the interference pattern as a function of the pixel position was output. An example of such a
graph is shown in Figure 12.
17
Figure 12 – The intensity of the interference pattern produced by the sodium
doublet as a function of pixel position, for the CCD camera image. The uneven
intensity is due to an imperfectly centered sodium lamp and noise, which appears
to contribute an approximately Gaussian intensity to the expected pattern.
The next step was to analyze each of these graphs and find a corresponding contrast rating to
plot as a function of the carriage position. This was done by measuring the intensity difference
between the first high peak to the right of the central peak and the following minimum and dividing by
the average of the intensities at these two positions. This quantification of the peak height compared to
the overall intensity serves us a rating of the image's contrast. We then plotted our contrast rating of
each graph as a function of carriage position; the results are in Figure 13. As expected, the contrast
varies periodically with distance.
18
Figure 13 – Contrast rating at different carriage distances. Note periodic variation
as the path difference is changed. The sinusoidal plot allows us to quantify this
period.
We fitted the graph to a sinusoidal function because the period of this function will represent
the distance between successive region of high contrast and low contrast. The period obtained from the
fit was P = 17.5 ± 0.4mm which yields a separation distance of Δλ = 0.50 ± 0.1 nm. This result is only
barely in agreement with Δλ = 0.597 nm. There are some explanations for such a poor result. First, the
quantification of the contrast level used was by no means a widely accepted definition of contrast; it
was highly prone to error and analysis of different peaks within the same graph gave results that varied
noticeably. The error bars on Figure 13 give an idea of these large errors. Secondly by looking at
Figure 13 we see that the sinusoidal fit is poor. This may be attributable to the fact that the pictures
analyzed by the Eyespy software contained noise produced by unwanted light when the pictures were
taken, or alternately it may be due to problems with the interferometer setup. The illumination from the
sodium lamp was uneven, and in addition the image began to become off-centered near the final low
contrast region; this was difficult to correct for at such low contrast and may have led to the several
very low measurements past 20 mm in Figure 13.
5.6
Hydrogen-Deuterium Balmer Alpha Line Separation
An attempt was made to measure the separation in the hydrogen and deuterium Balmer alpha
lines using the Michelson interferometer. The apparatus was set up so that the interferometer received
equal intensity light from both the hydrogen and the deuterium sources, with a number 16 Wratten
19
filter used to block the light from other The procedure used was the same as that for the measurement
of the sodium separation. This attempt failed because we did not observe a reappearance of high
contrast or low contrast in the interference pattern as we shifted the position of the carriage. When the
interference pattern was visible at position D1 it simply decreased in contrast as the distance of the
carriage was increased but never completely disappeared or returned to a level of high contrast. The
same problem occurred when trying to observe successive low contrast regions. When low contrast
was observed increasing the distance of the carriage position caused a reappearance of the interference
pattern which never disappeared afterwards. It was difficult to tell whether the observations of low
contrast were due to actual interference between the two lines or due to errors in alignment of the
interferometer; the pattern was very faint. To improve our ability to take data we tried to observe the
interference pattern through the CCD camera. The light intensity of the pattern was insufficient to
produce an image. We were thus unable to determine the separation between the Balmer alpha lines in
hydrogen and deuterium with our experimental setup.
6
Conclusion
We were successful in measuring a number of spectral structures using both the monochromator
and the interferometer, and our measurements were found to generally be in agreement with either
theoretical or accepted values. Our measurements of the Balmer lines in hydrogen and deuterium
allowed a determination of the Rydberg for these two atoms; the measured values were RH = (1.09705
± 0.00040) × 107 m-1 and RD = (1.09700 ± 0.00030) ×107 m-1, respectively. These are in agreement with
the theoretical values of 1.09678 × 107 m-1 and 1.09707×107 m-1, respectively. We were, however,
unable to measure the shift of the Balmer series in hydrogen and deuterium with either the
monochromator or the interferometer.
We were able to measure the splitting of the sodium doublet using several techniques. By using
the monochromator and fitting to Gaussians, we obtained a value of (588.97 ± 0.70) nm for the lower
wavelength 3p3/2 line in sodium and a value of (589.52 ± 0.70) nm for the higher wavelength 3p1/2 line.
The split was determined to be (0.558 ± 0.080 ) nm, which is in agreement with the accepted value of
0.5970 nm. Using the interferometer and examining the interference pattern directly, we measured a
value of 0.591 ± 0.001 nm. This was not in agreement with the accepted value, however the reason is
likely the large, unquantified contribution of human error in this measurement. Using the
interferometer and the CCD, we obtained a value of 0.50 ± 0.1 nm, which is in agreement with the
accepted value.
In carrying out these measurements, we have realized that extreme care must be taken to obtain
values with good precision. Apparatuses must be carefully calibrated and adjusted as appropriate for
every measurement; multiple measurements must be taken; for the interferometer, fringe counts must
be known precisely and a large number of fringes must be counted in a measurement, while care must
be taken to adjust the lighting conditions for the CCD camera. Obtaining precise values is challenging,
and calls for extreme and constant diligence on the part of the experimenters.
7
Acknowledgments
We would like to acknowledge the help of Dr. Fritz Buchinger, Mark Orchard-Webb and
Saverio Biunno, all of whom were incredibly helpful in getting our experiment up and running and in
20
providing troubleshooting advice and valuable insight throughout the experiment. We would also like
to thank Dusan Simic and Michael Zamfir as well as Jesse Searle and Matt Becker for their sample lab
reports on this topic, as well as Mark Orchard-Webb, Anne-Sophie Lucier and the mysterious singlenamed Po for their sample report The Thin Red Line, all of which were very useful for getting us started
with the monochromator and interferometer.
8
References
[1] For the full derivation, see pages 277-280 of: John S. Townsend, A Modern Approach to Quantum
Mechanics. University Science Books. Sausalito, California. (2000).
[2] The key is to consider the limiting behaviour for very large ρ, and to discard exponentially
increasing solutions. Details are given in Townsend (2000), pages 279-280; we have deemed the full
solution too lengthy to reproduce here.
[3] This and all other accepted values are taken from the Handbook of Basic Atomic Spectroscopic
Data of the National Institute of Standards and Technology, which can be found online at
http://physics.nist.gov/PhysRefData/Handbook/index.html.
[4] Image from Hyperphysics, http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/imgqua/Nadoub.gif
[5] The actual intensity distribution is not precisely Gaussian due to the monochromator letting in a
finite range of wavelengths that the photomultiplier cannot then discriminate between as it measures
the intensity. However, the intensity profile is still approximately Gaussian, and the fit to this equation
is easily good enough to determine the wavelength of the peak.
[6] Origin 7.0 is made and distributed by OriginLab, http://www.originlab.com/
[7] For a derivation of this equation, see Experiment 2 in the Operation and Experiment Manual of the
M – 4 Interferometer by Atomic Laboratories, Inc. (1961)
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